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ADAPTIVE SYSTEMS IN CONTROL AND SIGNAL

PROCESSING 1983 Proceedings of the /FAG Workshop

San Francisco, USA, 20-22 June 1983

Edited by I. D. LANDAU

Laboratoire d'Automatique de Grenoble, France

M. TOMIZUKA and

D. M. AUSLANDER University of California, USA

Published for the

INTERNATIONAL FEDERATION OF AUTOMATIC CONTROL by

PERGAMON PRESS OXFORD · NEW YORK · TORONTO · SYDNEY · FRANKFURT

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AU Rights Reserved. No part of this publication may be reproduced, stored in a retn'eval system or transmiited in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photoco�71f{. recording or otherwise, without permission in writing from the copyright holders.

First edition 1984 Reprinted 1985

Library of C.Ongrm Cataloging in Publication Data

Adaptivr systrms in rnnt rol & signal procrssing 1983 . "Sponsorrd by thr lntrrnational Fro«>ration of Automatic Control (IFAC). Tt>chnkal Commilll'I' on Throry. Working Group on Adaptivr Systt>ms: rn-sponsort>d by Am«>rkan Autmnatk Control Coundl (AACC). Crntrr national dt' la rrcht>rchr scit'ntifiqul' (CNRS-Fran<'«>): o rgan izt>d hy Continuing Edul'ation in Enginl'«>ring, UnivE"nity of California, 81•rkt>l«>y" P. v. 1. Adaptivf' t'l>ntrol systl'ms Congrl'll.WA. 2. Signal prc><·l's.�ing Congrl'SSl'S. I. Landau. I. n. II. Tomizuka. M. Ill. Ausland1•r. David M. IV. lntl'rnational Ft>dt>ration of Automatic Control. Tt>l'hnkal Commilll'I' on Throry. Working Group on Adaptivl' Systl'ms. V. Anwrkan Automatk Control Council. VI. C«>ntrl' national di' la rl'chl'rchl' scil'ntifiqul' (hann•) VII. Univl'nity of California. Bl'rkl'll'y. Continuing Education in Engint>t'ring. VIII. Titl«>: Adaptivf' sys11·ms in nmtrol and signal proc-«>ssing 1983. IX. St'rit>s. TJ217.A3216 1984 629.8'36 83-25691

British Library Cataloguing in Publication Data Adaptivl' syst<·ms in nmtrol and signal proc-!'ssing. 1983. 1. Adaptivl' nmtrol systl'ms I. Landau. I. 0. II. Tomizuka. M. 111. Ausland«>r. 0. M. IV. lnt«>rnational Fl'dl'ration of Automatic Control V. S.-rirs 629.8'36 1]217 ISBN 0-08-030565-2

These proceedi11,gs were reproduced by means of the photo-offset process using tht manu­scnpts supplied by the authors of the different papers. The manwcnpts have been typed using different typewriters and typefaces. The lay-out, figures and tables of some papers did not agree completely Wlih the standard requirements; consequently the reproduction does rwt display complete uniformiJy. To ensure rapid publication this discrepancy could not be changed; nor could the English be checked completely. Therefore, the readen are asked to excwe any deficiencies of this publication which may be due to the above mentioned reasons.

The Edtiors

Printed in Great Britain by A. Wheaton II Co. l.Jd., Exeter

IFAC WORKSHOP ON ADAPTIVE SYSTEMS IN CONTROL AND SIGNAL PROCESSING 1983

Organized by: Continuing Education in Engineering, University of California, Berkeley

Sponsored by: The International Federation of Automatic Control (IFAC), Technical Committee on Theory, Working Group on Adaptive Systems

Co-sponsored by: American Automatic Control Council (AACC)

Centre National de la Recherche Scientifique (CNRS - France)

International Program Committee: I. D. Landau, France (Chairman) D. M. Auslander, U.S.A. K. J. Astrom, Sweden D. Clark, U.K. B. Friedlander, U.S.A. G. C. Goodwin, Australia E. Irving, France L. Ljung, Sweden A. S. Morse, U.S.A. E. Mosca, Italy M. M'Saad, Morocco K. S. Narendra, U.S.A. R. Ortega, Mexico G. Saridis, U.S.A. M. Tomizuka, U.S.A. H. Umbehauen, Germany

National Organizing Committee: D. M. Auslander, U.S.A. (Chairman) I. D. Landau, France C. D. Mote, U.S.A. M. Tomizuka, U.S.A. L. Reid, U.S.A.

FOREWORD

. The first IFAC workshop dedicated to the field of adaptive systems was held in San Francisco, USA on June 20-22, 1983. It was initiated by the Working Group on Adaptive Systems, which is part of the IFAC Technical Committee on Theory. The organization of this workshop was motivated by the important developments that have taken place in this field in the last few years. We should note that besides the theoretical aspects of the research (and at least in part because of the intense theoretical activity) the number of applications of adaptive control is growing, and this attracts more people from the general community to this field. On the other hand, the connections between adaptive signal processing and adaptive control have also been emphasized recently. For this reason, the workshop has hosted a number of contributions in the area of adaptive signal processing.

The workshop was organized around five main topics:

- New adaptive control algorithms - Multivariable adaptive control - Robustness of adaptive control - Adaptive signal processing - Applications of adaptive control

Ten contributions addressing topics of general interest were presented in the plenary sessions, and three round tables were organized, Summaries of the round table discussions are included in these Proceedings.

The Editors

vi

Copyright © IF AC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

PLENARY SESSION 1

ADAPTIVE CONTROL OF A CLASS OF LINEAR TIME VARYING SYSTEMS

G. C. Goodwin and Earn Khwang Teoh

Department of Electrical and Computer Engineering, University of Newcastle, NSW 2308, Australia

Abstract . The key contribution of the paper is to develop a new and explicit characterisation of the concept of persistency of excitation for time invariant systems in the presence of possibly unbounded signals . The implication of this result in the adaptive control of a class of linear time varying systems is also investigated . Simulation results are presented comparing alternative algorithms for the adaptive control of time varying systems . Keywords . Adaptive control , time varying systems , identifiability , least­squares estimation.

1. INTRODUCTION One of the prime motivations for adaptive control is to provide a mechanism for deal­ing with time varying systems . However , to­date , most of the literature deals with time invariant systems , see for example , Feuer and Morse (1978 ) , Narenda and Valavani (1978) , Goodwin , Ramadge and Caines (1980 , 1981) , Morse (1980) , Narenda and Lin (1980) , Egardt (19 80) , Landau (1981) , Goodwin and Sin (1981) , Elliott and Wolovich (1978) , Kreisselmeir (1980 , 1982 ) .

Some of the algorithms with proven conver­gence properties for the time invariant case e . g . gradient type algorithms , are suitable , in principle , for slowly timevarying systems . However , other algorithms , e . g . recursive least squares , are unsuitable for the time varying case since the algorithm gain asymp­totically approaches zero . For the latter class of algorithms various ad-hoc modificat­ions have been proposed so that parameter time variations can be accommodated . One approach (Astrom, et, al. , 1977 , Goodwin and Payne , 1977) is to use recursive least squares with exponential data weighting . Various refinements (Astrom (1981) and Wittenmark and Astrom (1982 ) ) of this approach have also been proposed to avoid burst phenomena e . g . by making the weighting factor a function of the observed prediction error ( Fortescue , Kershenbaum and Ydstie , 1981 ) .

The basic consequence of using exponential data weighting is that the gain of the least squares algorithm is prevented from going to zero . A similar end result can be achieved in other ways , for example , by resetting the covariance matrix (Goodwin et , al . , 19 83); by adding an extra term to the covariance update (Vogel and Edgar , 1982 ) ; or , by using a finite or oscillating length data window (Goodwin and Payne , 1977 ) .

Another formulation that has been suggested by several authors (Weislander and Wittenmark , 1979 ) is to model the parameter time variat­ions by a state-space model and then to use the corresponding Kalman filter for estim­ation purposes . This again corresponds to adding a term to the covariance update . It has also been suggested that some of tne algorithms can be combined (Wittenmark (1979 ) ) .

Many o f the above algorithms , tailored for the time varying case , have been analyzed in the time invariant situation . This is a reasonable first step since one would have little confidence in an algorithm that was not upwards compatible to the latter case . For example , Cordero and Mayne (1981) , have shown that the variable forgetting factor one-step­ahead algorithm of Fortescue et . al . (1981) is globally covergent in the time invariant case provided the weighting factor is set to one when the covariance exceeds some prespec­ified bound. Similar results have been estab­lished by Lozano (1982 , 1983) (who uses exponential data weighting where the weight­ing is made a function of the eigenvalues of the covariance matrix) and by Goodwin , Elliott and Teoh (1983) (who use covariance resetting) .

With robustness considerations in mind , Anderson and Johnson (19 82) and Johnstone and Anderson (1982b) have established exponential convergence , subj ect to a persistent excit­ation condition, of various adaptive control algorithms of the model reference type . These results depend explicitly on the stability properties proved elsewhere (e . g . Goodwin , Ramadge and Caines (1980) ) for these algor­ithms in the time invariant case . The additional property of exponential convergence has implications for time varying systems since it has been shown (Anderson and Johnstone (19 83) ) that exponential convergence implies tracking error and parameter error bounded­ness when the plant parameters are actually slowly time varying .

2 G . C . Goodwin and Earn Khwang Tech

For stochastic systems , Caines and Chen (1982) have presented a counterexample show­ing no stable control law exists when the parameter variations are an independent process . However , i f one restricts the class o f allow­able parameter variations , then it is poss-ible to design stable controllers for example, Caines and Dorer (1980) and Caines ( 1981) have established global convergence for a stochastic approximation adaptive control algorithm when the parameter variations are modelled as a (convergent) martingale process having bounded variance , Some very prelim­inary results have also been described (Hersh and Zarrop ( 1982) for cases when the parameters undergo jump changes at prespecified instants . In the current paper we make a distinction between j ump and drift parameters . "Jump parameters" refers to the case where the parameters undergo large variations infreq­uently whereas "drift parameters" refers to the case where the parameters undergo small variations frequently . In section 2 , we will develop a new "persis­tent excitation" condition for systems having possibly unbounded signals . An important aspect of this result is that it does not rely upon first establishing boundedness of the system variables as has been the case with previous results on persistent excitat­ion (see for example Anderson and Johnson ( 1982 ) ) , The result uses a different proof technique but was inspired by a recent proof of global stability for a direct hybrid pole assignment adaptive control algorithm (Elliott , Cristi and Das (1982) ) , In the latter work a two-time-frame estim­ation scheme is employed such that the para­meters are updated at every sample point but the control law parameters are updated only every N samples , A similar idea is explicit in Goodwin, Teoh and Mcinnis ( 1982) and implicit in Johnstone and Anderson ( 1982a) . We shall also use two-time-frame estimation here and show that this leads to a relatively simple new result on persistency of excitation with poss ibly unbounded feedback signals .

We will show in section 3 that the new persis­tent excitation condition allows one to estab­lish global exponential convergence of standard indirect adaptive pole-assignment algorithms in the time invariant case . In section 4 and 5 we discuss the qualitative interpretation of these results for j ump and drift parameters respectively . In section 6 , we present some simulation studies and give comparisons of different algorithms for time varying adaptive control .

2 . A NEW PERSISTENCY OF EXCITATION CONDITION

We shall consider a single input , single out­put system described as follows :

y (t) = -a1 (t)y(.t:-l )-a2 ( t )y ( t-2) . . . . -an (t )y (t-n) + b1 ( t )u ( t-1)+ • • • • +bn (t )u (t-n) ( 2 . 1)

Note that in the above model the parameters depend upon time . In the time invariant case , the model simplifies to the standard deter­ministic autoregressive moving average model of the form :

-1 -1 A (q )y ( t ) = B (q ) u(t) ( 2 . 2) -1 where q denoter the unit delay operator ,

and A(q-1) , B ( q- ) are polynomials of order n . The model ( 2 . 1) can also b e expressed in various equivalent forms . For example we can write

1 -1 A ( t , q- )y (t ) = B ( t , q ) u (t) (2 . 3 )

The model ( 2 . 1) can also be expressed in regression form as

T y (t) = ¢ ( t-l) 8 (t ) ( 2 . 4) where

T ¢ ( t-l) [ -y ( t-1) , • • • , -y ( t-n) , u (t-l) , • • •

u ( t-n) ] ( 2 . 5 )

8 ( t)T = [ a1 (t ) , . . . , an (t ) ,b1 (t ) , . . . ,bn (t ) ] ( 2 , 6)

For the moment , we will restrict attention to the time invariant case and state a key controllability result . We shall subsequent­ly use this controllability result to develop a persistency of excitation condition for use in adaptive control , We first note that in the time invariant case , the regression vector ¢ (t) defined in equation ( 2 . 5 ) satisfies the following state space model :

¢ ( t )

I -al '''' -an• bl ''' ,bn 1

1 -0 0

-------------+----------Q,,,,.,,,0 I Q.,.,.,Q

0 1 • • 1 0

F ¢ ( t-l) + G u ( t)

¢ (t-l)+

0 0 0 - u 1 0 : 6

( 2 . 7 ) ( 2 . 8)

If we define x ( t ) as ¢ ( t-l) , then we note that we can use the model ( 2 . 8) to construct the following non-minimal 2n dimensional state space model for y (t) :

x ( t+l) F x (t) + G u ( t) ( 2 , 9) y (t) = H x (t) ( 2 . 10)

where H = [l O , , , O ] F [-a1 , • • • -a ,b1 , • • • b ] n n-( 2:11)

Linear Time Varying Systems 3

It can be verified that the model ( 2 . 9 ), ( 2 . 10) i s not completely observable . However, the following new result shows that the model ( 2 . 9 ) i�1completely reachable

provided A(q ), B (q-1 ) are relatively prime .

Lemma 2 . 1 (Key Controllability Lemma) . The 2n dimensional state space model ( 2 . 7 ) for the vector <j> (t) is completely reachable if and only if A(q-1), B (q-1) are relative­ly prime .

Proof: For details see Goodwin and Teo���983 ) · The importance of the above lemma in the con­text of persistent excitation is that it shows that the vector {<j> (t) } is 'controllable' from u (t) and thus one might expect that {u(t) } can be chosen so that {¢ (t) } spans the whole space . This is in accordance with one ' s intuitive notion of the concept of persist-ency of excitation , Concrete results of this nature will be presented below .

When the parameters of the system are known and time invariant, then the closed loop poles can be arbitrarily assigned by deter­mining the input from (see for example (Kailath (1980), Goodwin and Sin ( 1983) ) :

L ( q-1) u (t) = -P ( q-1) y (t) + v (t) ( 2 . 12 ) where L (q-1), P (q-l) are unique polynomials of order (n-1) and {v(t) } is an arbitrary external input . The feedback control law (2 , 12 ) can equiv­alently be written in terms of the vector ¢ (t) as

u(t) - K<j> (t-1) + v (t) ( 2 . 13) where K = [p1-a1P o • ···• P n-l-an-1P o ' -anpo,i1+blpO'

• • • • ,in-1+ bn-lpO,bnpO ) < 2 • 14)

With two-time frame estimation in mind, we shall assume that the feedback law ( 2 . 13) is held constant over an interval I (t0) = [t0,t0+N-l) fnd analyze the minimum eigen­value of X(t0) X(t0) where

T x(t0+1) = [¢ (t0+1) , ¢ (t0+2 ), • • • , ¢ (t0+N) J

( 2 . 15) We now have the following new result on persistency of excitation :

Theorem 2 1 (Pers istency of Excitation) Consider the system ( 2 . 7 ) and the feedback control law ( 2 . 13 ), then provided

-1 -1 ( i ) A(q ), B (q ) are relatively prime (ii) the feedback law ( 2.13) is constant over

the interval I (t0) = [t0,t0+N-1 J (iii) the external input, v (t), is of the form :

s v (t) = l rk s in (Wkt+Ok) ( 2 . 16)

k=l where wk: (O,n) ; rk f o and wj f wk ; k=l, • • • ,

s ; J=l, • • • ,s

(iv) The length of the interval, N, and the number of s inusoids, s, satis fy ( a) N � lOn (b) s � 4n

( 2 . 17 ) ( 2 . 18 )

where n is the order of the system we have \ . [X(t0+l)X(t0+i)T J <: E1 > o ( 2 . 19 ) min where E1 is independent of t0 and the initial conditions ¢ (t0) . Proof: See Goodwin & Teoh (1983) for details . VVV

The above theorem makes precise the intuit­ive noticnof persistency of excitation introduced earlier . Note that the theorem depends upon the Key Controllability Lemma (Lemma 2 . 1) , As far as the authors are aware, this is the first general persist­ency of excitation result which does not depend upon an a-priori uniform bounded­ness condition on the system response . In the next section we show how the above result can be used in a straightforward fashion to establish convergence of an indirect pole-assignment adaptive control algorithm in the time invariant case .

3 . CONVERGENCE OF A POLE ASSIGNMENT ALGORITHM IN THE TIME INVARIANT CASE

Here we shall consider an indirect pole­assignment adaptive control law using a two­time frame estimator in the linear time invariant case .

The system will be assumed to satisfy ( 2 . 2 ) subj ect to the following assumptions:

Assumption A: -1 -1 A(q ) , B (q ) are relatively prime ,

Assumption B: The order n is known •

Let N and s be chosen as to satisfy equations ( 2 . 17 ), ( 2 . 18) and let � be a prespecified arbitrary integer . Then, the two-time frame adaptive control algorithm is:

(i) Parameter Estimation Update (Least Squares )

S (t) S (t-l) + P (t-2 ) p (t-l ) e (t) 1+¢ (t-l)TP (t-2 ) ¢ (t-l)

T" e (t) = y (t) - ¢ (t-l) 8 (t-l) ; ( 3 . 1) t = 1,2 • • • and �(O ) given.

(ii) Covariance Update with Resetting

P� (t-1) = P (t-2 ) -

t

P (t-2 ) ¢ (t-1) ¢ (t-l )TP (t-2 ) 1 + <j> (t-l )TP (t-2 ) ¢ (t-l )

(3 . 2 ) I f �N i s an integer

Then resetting occurs as follows :

P (t-1) = -1-I k o Else

P (t-1 )

( 3 . 3 )

( 3 . 4)

4 G . C . Goodwin and Earn Khwang Tech

(iii) Control Law Update (in the Second Time Frame )

(a) If !. is an integer The� evaluate A -1 A( t,q )

A -1 B ( t,q )

A -1 A -n 1 + 61 (t )q + . . . 6 ( t ) q n ( 3 , 5 ) A -1 A -n 6 +l ( t ) q + • • • +62 ( t ) q n n ( 3 . 6 )

A -1 A s�1ve the following equation for L ( t,q ), P ( t,q ), each of order (n-1) : A -1 A -1 A -1 A -1 * -1 A(t,q )L (t,q ) + B ( t,q ) P ( t,q ) =A (q )

( 3 . 7 ) * -1 where A (q ) is an arbitrary stable poly-

nomial , A -1 A -1 [ In the event that A( t,q )� B ( t ,q ) are

�ot r�tatively prime, then L ( t,q-1) and P (t,q ) can be chosen arbitrarily ] .

(b ) Else put A -1 A -1 A -1 A -1 L ( t,q ) = L ( t-1,q ) ; P ( t,q ) =P (t-1,q )

( 3 . 8) (iv) Evaluation of the input

A -1 A -1 L(t,q )u ( t ) = -P ( t,q ) y ( t )+v ( t) ( 3 . 9 ) where {v (t ) } is as in ( 2 . 16)

We now have the following covergence result . Theorem 3 , 1 Consider the algorithm ( 3 . 1) to ( 3 , 9 ) appli.?ci to the system ( 2 , 21 subject to assumption (A) and (B), then 6 (t ) approaches the true value, 60, exponentially fast and {u(t ) }, {y (t ) } remain bounded for all time , Proof: Straightforward using the results of Theorem 2 . 1 and the Small Gain Theorem (Desoer and Vidyasagar (1975 ) ) . See Goodwin & Teoh ( 1983) for full details . vvv The above algorithm uses iterative least squares with covariance resetting . Three points can be made about this procedure : (i ) If resetting is not used, then the algorithm reduces to ordinary recursive least squares . In this case and for time invar­iant problems, it can still be shown that S (t ) converges to 60 but not exponentially fast . (ii) It is essential to note that ordinary least squares can not be used in the t ime varying case since the gain of the algorithm goes to zero . However, our experience is that, even for time invariant problems, re­setting is helpful s ince it captures the rapid initial convergence of least squares without having the slow asymptotic converg­ence that is well known for ordinary least squares . (iii) In the above analysis, we have reset to a scaled value of the identity matrix . However, it can be seen that an identical result is achieved if the resetting is made

to any matrix, P, satisfying :

In particular, one could reset to E1$ [P' ( t-1) -l ] . , -1 2traceP ( t-1)

( 3 . 10)

This satisfies ( 3 . 10) and has the advantage (Lozano ( 1982) ) that the directional inform­ation built up in P (t-1)-1 is retained .

4 . JUMP PARAMETERS

In the literature (see for example Wittenmark, 1979) two types of parameter variation have been considered, namely, strongly time vary­ing (or jump parameters) and slowly time varying (or drift parameters ) . This class­ification is helpful in discussing the con­vergence properties . We shall treat the former case in this section and the latter in the next section .

For our purposes we shall define j ump parameters as follows :

Definition 4 . 1 : The parameters, 6 (t ), in the model ( 2 . 1 ) are j ump parameters (having j umps at { ti : ti>ti-l' i=O,l, • • • • ,oo} ) if

(a) 6 ( t ) = 6i for ti ,,; t < ti+l ( 4 . 1)

(b) �un I ti-ti_1 I = tmin (4 . 2 ) l.

( c ) 6i E M a bounded set . ( 4 . 3) vvv

Jump parameters are often a realistic model in practical cases especially when nonlin­ear systems are approximated by linear models at different operating points . Then an abrupt change in operating point gives a j ump change to the parameters in the linear model. This type of time varying model has been the subject of several recent papers (Wittenmark, 1979 ; Wieslander and Wittenmark, 197 1 ; Fortescue et . al,, 1981 and Vogel and Edgar, 1982) . For the purpose of adaptive control, we shall further constrain the set of possible para­meter values as follows : Assumption C : For all possible paramet�l val�Is' 6i, the corresponding pair A(q ) , B (q ) are relatively prime and the magnitude of the determinant of the associated eliminant matrix is bounded below by a constant indepen-dent of t . VVV We now discuss the qualitative performance characteristics of the adaptive control algorithm ( 3 . 1) to ( 3 , 9 ) when applied to systems having j ump parameters . Our key purpose is to indicate the kind of inform­ation necessary to ensure that the system input and outputs remain bounded.

When a j ump occurs, the system response may begin to diverge. However, there is a max­imum rate at which this can occur in view of ( 4 . 3) , Moreover, we know from section 3, that

Linear Time Vary�ng Sys tems 5

since the parameters are constant between jumps there exists a finite time N such that S ( t} will be within an E neighEourhood of Bo (t) and hence one can re-establish s tabilizing control • Now, provided a suff­iciently long period passes before the next jump occurs, then the response will be brought back to its original magnitude . ( If insuff­icient time is allowed between j umps then it is easy to construct examples such that the response builds up even though a stabil­izing controller is found between the j umps) .

It is possible to compute an expression for the minimum time between jumps, t i , in terms of the following quantitiesmsg that {u(t ) }, {y ( t ) } remain bounded (i ) The diameter of the set M. (ii) The lower bound on the elimenant matrix

in assumption c. (iii)The constants k , N, s, � in the

algorithm of se�tion 3 . _1 (iv) The precise nature of A* (q ) . The explicit expression for tmi is complic­ated (Teoh ( 1983) ) and offers �ittle extra insight .

One practical point worth noting is that it is not necessary to apply the external input for all time, instead it suf fices to add this signal for a period N after a j ump has occurred . The idea of adding an external signal for a finite period when changes in the plant are perceived has been suggested by other authors, e . g . Vogel and Edgar ( 1982) . For chemical plants, ere ., it is generally not desirable to impose additional inputs continuously during steady conditions . However, the procedure suggested here only injects the external signal when uns teady conditions arise from other sources, e . g . plant time variations .

Note that we also have assumed that the order of the system remains unchanged during j umps . This assumption is certainly restrictive but to handle more general s ituations would re­quire an on-line order determination as part of the algorithm. This would lead to additional considerations well beyond the scope of the current paper .

5 . DRIFT PARAMETERS

For our purposes we shall define drift para­meters as follows : Defintion 5 . 1 The parameters, B ( t ), in the model ( 2 . 1 ) are drift parameters i f (a) I ! B (t ) - B ( t-1) 11 < o (b) B ( t ) E M a bounded set

( 5 . 1) ( S . 2)

We shall also require the following addit­ional assumption :

-1 Assumption D: For each fixed t, A(q ,B (t ) ) , B (q-1, B ( t ) ) are relatively prime

vvv Note that assumption D is necessary to ensure that the system does not drift into a region

where the order changes . As pointed out in the previous section, the more general situation, though interesting, involves considerably more complexity. A similar assumption to D appears in other papers in this general area ( see for example Anderson and Johnstone ( 1983) ) .

We now investigate the qualitative behaviour of the algorithm of section 3 when applied to the drift parameter case . Since, we have established exponential convergence in the time invariant case then we can argue as in Anderson and Johnstone (1983) to conclude that stablity is retained in the time varying case provided o in (S . l) is smaller than some fixed number depending on the size of the initial parameter error . Note the role played by exponential conver­gence in making this claim.

6 . SIMULATION STUDIES

Extensive simulation studies of the adaptive control algorithm described above have been carried out together with comparisons with exponentially weighted least squares and gradient algorithms . In this section, we present a summary of the results obtained . ( i ) The best algorithm overall appears to

be recursive least squares with cov­ariance resetting as described in section 3 .

(ii) The algorithm of section 3 is relative­ly insensitive to the resetting period, though we have found that in the case of jump parameters it is helpful to monitor the prediction error and reset when this value exceeds some threshold . In the case of drift parameters we have found that there exists an optimal resetting interval .

( iii) Recursive least squares with exponential data weighting is highly sensitive to the choice of the weighting factor A and performs extremely poorly for all A in the case of drift parameters .

( iv) Gradient s chemes are simple but con­verge extremely slowly and are there­fore unsuitable for all but very slowly varying systems .

Typical simulation results are as shown in Figures 6 , 1 and 6 , 2 for a system having s inusoidally varying parameters and set point variation as in Fig . 6 . la . Figure 6 , 1 shows the excellent performance o f the covariance resetting s cheme . Fig . 6 . 2 shows the poor performance of the exponential weighted least squares algorithm for the same problem. (Note that for the results in Fig 6 , 2 the best value of A was chosen!)

7. CONCLUSIONS

This paper has presented results in the adaptive control of linear time invariant and time varying systems . The key result is a new persistent excitation condition for systems having non-uniformly bounded signals . The implication of this result

6 G. C . Goodwin and Earn Khwang Tech

for time varying systems having either jtllllp or drift parameters has been qualitatively investigated . The results are believed to be of practical importance s ince often the real motivation for adaptive control is to provide a mechanism for dealing with time varying plants . The paper has dealt only with the determin­istic case but the algorithms have been designed with s tochastic systems in mind as an ultimate aim, In the latter case some additional features arise, for example , a rough calculation shaws that there exists an optimal covariance resetting period for both j ump and drift parameters. These questions would seem to be worthy of further s tudy as logical extensions of the determin­istic results presented here .

REFERENCES

Anderson and Johnson, (1982). Exponential convergence of adaptive identification and concrol algorithms. Automatica, 18, 1-13.

Anderson and Johnstone, (1983). Adapt:ive systems and cime­varying planes. Inc.J.Conc., 37, 367-377.

Astrom, Horrlmrnn, I.Jung and Witcrn1n111rk (1977). Theory unit application of self tuning regulators. Aut.omacica, 13,457-476.

Astrom, (1981). Theory and application of adaptive control. Proc. IFAC 8th Congress, Kyoto.

Caines and Chen, (1982). On che adaptive control of stochastic systems with random paramters: A counterexample • !SI Workshop on Adaptive Control, Florence, Italy.

Caines and Dorer, (1980). Adaptive control of systems subjecc co a class of random parameter variations and disturbances. Tech. report, McGill University.

Caines, (1981). Stochastic adaptive control: randomly varying parameters and continually dis curbed controls. IFAC • Kyoto.

Cordero and Mayne, (1981). Deterministic convergence of a self-tuning regulator with variable forgetting fact.or. Proc. IEE, 128, 19-23,

Desoer and Vidyasagar, (1975). Feedback systems: Input.­Output properties. Academic Press.

Elliocc, Crisci and Das, (1982), Global stability of a direct hybrid adaptive pole placement algorithm • Tech Report UMASS, Amherst.

Elliott and Wolovich, (1978). Parameter adaptive identif­ication and control. IEEE Trans. AC23, 592-599.

Egardc, (1980). Stability analysis of discrete time adaptive control systems. IEEE Trans. AC-25, 710-717.

Feuer and Morse, (1978). Adaptive control of single-1nput, single-output linear systems. IEEE Trans. AC-23, 557-570.

Fortescue, Kershenbaum and Ydstie, (1981). Implementation of self-tuning regulators with variable forgetting factors. A11.tomacica, 17, 831-835.

Goodwin and Payne (1977). Dynamic system identification, Academic Press.

Goodwin, Ramadge and Caines, (1980). Discrete time mulci­variable adaptive control, IEEE Trans. AC-24, 449-456.

Goodwin, Ramadge and Caines, (1981). Discrete time stochastic adaptive control. SIAM J, Cone. & Opt.19,6,829-823,

Goodwin and Sin, (1981). Adaptive control of nonminimum phase systems. IEEE Trans. AC-26, 4 78-483.

Goodwin, Elliott and Teoh, (1983). Deterministic convergence of a self tuning regulator with covariance resetting. Proc. IEE, (D), 130, pp.6-8.

Goodwin and Teoh, (1983). Persist.ency of excitation and its implicacions on the adaptive control of time varying systems". Tech. Rept. Uni. Newcastle, Australia.

Goodwin, Teoh and Mcinnis, (1982). Globally convergent adaptive controller for linear systems having arbitrary zeros. Tech. Repc. Uni. Newcastle, Australia.

Goodwin and Sin, (1983). Adaptive filtering prediction and control. Prentice Kall.

Hersh and Zarrop, (1982). Stochastic adaptive control of time varying nonminimum phase systems. Control System Centre, report .543, UMIST.

Johnstone and Anderson, (1982a). Global adaptive pole place­ment: detailed analysis of a first order system CDC, Orlando.

Johnstone and Anderson, (1982b). ''Exponential convergence of recursive least squares with exponential forgetting faccor-adaptive control Systems and Control Letters, 2, 77-82.

Kailath, (1980). Linear systems. Prentice Hall. Kreisselmeier, (1980). Adaptive control via adaptive obser­

vation and asymptotic feedback matrix synthesis. IEEE Trans. AC-25, 717-722.

Kreisselmeier, (1982). On adaptive state regulation. IEEE Trans. AC-27, 3-16.

Landau, (1981). Combining model reference adaptive controllers and stochastic self tuning regulators. IFAC Cong. Kyoco.

Lozano, (1982). Independent tracking and regulation adaptive control with forgetting factor, Automatics, 18,4,455-459.

Lozano, (1983). Convergence analysis of recursive identif­ication algorithms with forgetting factor, Auto ... 19,1,95-97.

Morse, (1980). Global stability of parameter adaptive control systems, IEEE Trans. AC-25, 433-439.

Narendra and Valavani, (1978). Stable adaptive controller designs - Direct control. IEEE Trans. AC-23, 570-583.

Narendra and Lin, (1980). Stable discrete adaptive control, IEEE Trans. AC-25, 456-461.

Teoh, (1983). Ph.D. Thesis, University of Newcastle. Aust. Vogel and Edgar, (1982). Application of an adaptive pole­

zero placement controller to chemical processes with variable dead time. A.c.c. Washington.

Wieslander and Wittenmark, (1971). An approach to adaptive control using real time identification. Auto., 7, 211-217.

Willems, (1970). Stability theory of dynamic systems, Nelson. Wittenmark, (1979). A two-level estimator for time vacying

parameters, Automatica, 85-89. Wittenmark and Astrom, (1982). Implementation aspects of

adaptive controllers and their influence on robustness, CDC, Orlando.

5.0 OUTPUT

Noise Var. =O . 1 Reset Int .= 5

Fig. 6 . l(a) -5.0 0 200 400

TIME

10.0 PARAMETER ESTIMATES

.,

., 61

Fig. 6 . l (b ) -10.

200 �00 TIME

5.0 OUTPUT

NoiseVar.=0 .1 A.=0.95

Fig. 6 . 2 (a) -5.0

0 200 �00 TIME

10.0 PARAMETER ESTIMATES

•1

.,

Fig. 6 . 2 (b )

-10.0 200 400 TIME

Copyright © IFAC Adaptive Systems in Control and Signal Pr()lfessing, San Francisco, USA 1983

ADAPTIVE SIGNAL PROCESSING FOR ADAPTIVE CONTROL

B. Widrow* and E. Walach**

*Department of Electrical Engineering, Stanford University, Stanford, California, USA **Chaim Weitzman Postdoctoral Fellow, Department of Electrical Engineering, Stanford University, Stanford,

California, USA

Abstract. A few of the well established methods of adaptive signal processing theory are modified and extended in order to address some of the basic issues of adaptive control.

An unknown plant will track an input command signal if the plant is preceded by a controller whose transfer function approximates the inverse of the plant transfer function. An adaptive inverse modeling process can be used to obtain a stable controller, whether the plant is minimum or non-minimum phase. A model-reference version of this idea allows system dynamics to closely approximate desired reference model dynamics. No direct feedback is involved. However the system output is monitored and utilized in order to adjust the parameters of the controller.

The proposed method performs very well in computer simulations of a wide range of stable plants, and it seems to be a promising alternative approach to the design of adaptive control systems.

Keywords. Adaptive control, self-adjusting system, controllers, modeling, transfer functions .

INTRODUCTION

There is a great need for learning-control systems which can adapt to the requirements of plants whose characteristics may be unknown and/or changeable in unknown ways. Two principal factors have hampered the development of adaptive controls:

a) the difficulty of dealing with learning processes embedded in feedback loops.

b) the difficulty in controlling nonminimum­phase plants.

Considerable progress has been made (see for instance works by Powell, 196D; Tse and Athans, 1972; Nakamura and Yoshida, 1973; Astrom and Wittenmark, 1973, 197 4, 1980; Landau, 1D7 4, 1976, Martin-Sanchez, 1976). However,

7

interaction between the -feedback of the learning

process and that or the signal flow path still greatly complicates the analysis which is requisite to the design or dependable control systems.

In this paper we continue with the development of an alternative approach, which was first presented by B. Widrow and his students 1978, 1981, and B. D. 0. Anderson, 1981, which circumvents many of the difficulties that have been encountered with the previous forms of adaptive control. The basic idea is to create a good transversal filter model of the plant, then to utilize it in order to obtain an inverse (or delayed inverse) of the plant. This inverse can be used as· an open loop controller of the system. Since such a controller is realized as a transversal filter, the stability of the system is assured. Moreover it can be shown that, if one is willing to allow a delay in the response of the control system, excellent control of the plant dynamics can be achieved, even for nonminimum phase plants.

8 B . Widrow and E . Walach

In this paper, the basic principles or the proposed approach will be discussed and computer simulations will be presented in order to illustrate their potential. It should be mentioned that additional intensive research has been conducted in order to address the issues or

a) cancelling the plant noise by reeding appropriate signal at the plant input

b) facilitating the modeling process by employing dither signals when sufficient ambient signal activity is lacking

c) adaptive control or MIMO (multiple input, multiple output) systems

d) detailed quantitative perCormance and stability analysis.

Progress in all these areas indicates and verifies the validity of the general approach, although the details of this research will not be included here as they go beyond the scope of the current presentation.

ADAPTIVE FILTERING

A schematic representation of an adaptive filter is depicted in Fig. 1. The filter has an input u;, an output II;, and it requires a special training signal called the "desired response" d;. The error E; is the difference between the desired and actual output responses . The filter is assumed to be transversal and its weights w 1;, . . . , w 1; are adapted in order to minimize the expected square of the error t;. Various adaptation algorithms can be utilized for that purpose. Here we will employ the LMS steepest descent algorithm of Widrow and Hoff, 1Q60, which is well known in the literature (see for instance later works by Widrow and others 1Q75, 1Q76).

PLANT MODELING

To illustrate an application of the LMS adaptive filter and to show by example how one obtains an input and a desired response in a control environment, consider the direct modeling or an unknown plant as shown in Fig. 2. When given the same input signal as that of an unknown plant, the adaptive model self-adjusts to cause its output to be a best least squares fit to the actual plant output. The unknown plant may have both poles and zeros, but the adaptive transversal filter can only realize "zeros." (The word zeros is in quotes because the adaptive filter is time variable and does not strictly have a transfer (unction. In

a quasi-static sense, the adaptive filter can be thought to have "instantaneous zeros" corresponding to the zeros that would exist .iC the weights were frozen at their instantaneous values.) However with a sufficient number of weights, an adaptive transversal filter can achieve a close fit to an unknown plant having many poles and zeros.

PLANT INVERSE MODELING

The inverse model of the unknown plant could be formed as shown in Fig. 3. The adaptive filter

input is the plant output. The filter is adapted to cause its output to be a best least squares fit to the plant input. A close fit implies that the cascade of the unknown plant and the LMS filter have a "transfer function" of essentially unit value. Close fits have been achieved by adaptive transversal inverse filters even when the unknown plant had many poles and zeros.

INVERSE MODELING OF NONMINIMUM PHASE PLANTS

If the plant itself is stable, all of its poles lie in the left half of the s-plane. But some of its zeros could lie in the right half plane, and then the plant would be nonminimum phase. The inverse of the minimum phase plant would have all of its poles in the left half plane, and there would be no problem with stability of the inverse. The non­minimum phase plant would have zeros in the right half plane and stability of the inverse would be an important issue. However, it can be shown that stable inverses for nonminimum phase plants could always be constructed if one were permitted non causal two-sided impulse responses. Furthermore, with suitable time delays, causal approximations to delayed versions of noncausal impulse responses are realizable. Thus, by allowing a delay in the modeling process (as illustrated in Fig. 3), one can obtain approximate delayed inverse models to minimum phase and nonminimum phase plants. It is not necessary to know a prion· whether the plant is or is not minimum phase. However, some knowledge of plant characteristics would be helpful when choosing the delay .£1 and the length or the transversal filter used for inverse modeling.

ADAPTIVE INVERSE CONTROL SCHEME

Using a stable delayed inverse, control is accomplished as illustrated in Fig. 4. The controller is a copy of the inverse model. The command input i;, the desired output Cor the plant, is applied as an input to the controller.

Adaptive Signal Process ing for Adaptive Control 9

The controller output is the driving function for the plant. If the controller were an exact delayed plant inverse, the plant output, assuming no noise, would be an exact copy of the input reference command, but delayed, i.e.,

1/j = i;-t;.

A step change in the command input would cause a step change in the plant output after a delay of A seconds. In order to illustrate this idea, computer simulations were performed. A nonminimum phase plant was controlled. Its impulse response is depicted in Fig. Sa. This stable underdamped plant has a small transport delay. In order to find the inverse, the scheme of Fig. 3 was used to adapt a transversal filter having 40 weights. Since the plant is nonminimum phase, a good (low error) causal inverse cannot be obtained. Hence for A=O, the error power was close to the input power. However when the delay A was increased, the error power decreased indicating that very good plant inverses were obtained. Figure Sb shows the error power as a function of the modeling delay A. For A=26, the error power decreased to below S% of the input signal power. For this value of A, the best plant inverse had the impulse response shown in Fig. Sc. Connecting this as a controller in cascade with the plant, in the manner presented in Fig. 4, the overall impulse response was as shown in Fig. Sd. Clearly the behavior of the entire system closely approximated that of a pure delay. In Fig. 6b the step response of the control system is presented, and it Qlay be compared to the ideal step response of Fig. 6a.

MODEL REFERENCE ADAPTNE CONTROL SYSTEM

Sometimes it is desired that the plant output track not the command input itself but a delayed or smoothed version of the command input. The system designer would generally know the smoothing characteristic to be used. A smoothing model can be readily incorporated into the adaptive inverse control concept, as illustrated in Fig. 7. The smoothing model is usually designated as the "reference model." [See for instance Landau, 1974.J Thus the system of Fig. 7 is a general model-reference adaptive inverse control system. The system of Fig. 4 can be viewed as a special case, when the reference model equals the delay A.

OFF-LINE MODEL REFERENCE INVERSE CONTROL

If the plant inverse model has enough weights and if the reference model contains enough delay, the

approach of Fig. 7 allows excellent control of plant dynamics (as demonstrated by the above simulation). However plant noise causes a severe degradation in the performance or this system. Indeed, any noise present in the output of the plant will automatically enter the adaptive inverse modeling process. Therefore, as was pointed by Widrow and others, 1981; the transfer function of the plant inverse, a Wiener solution, will be biased and this bias in turn will cause an erroneous control and deterioration in system performance. To resolve this difficulty, the system of Fig. 8 can be used.

The control system of Fig. 8 works in the following manner. A model P(z) of the plant P(z) is formed using the methods mentioned above. P(z) is a Wiener solution that is not biased by plant noise. An "off-line" process can then be used to obtain a controller C(z) from P(z) and the reference model M(z). This process, which could be an adaptive one, adjusts C(z) to cause the output of the cascade of P(z) and b(z) to be a best least squares match to the output or the reference model M(z) when both the cascade and the reference model are driven simultaneously by a synthetic "modeling signal" having an appropriate spectral character. The process for finding C(z) could also be non-adaptive, since C(z) is deterministically related to P(z) and M(z) for any specified modeling signal spectrum. Now given C(z ), an exact digital copy can then be used as a controller, as shown in Fig. 8. The result is a controller and plant having a cascaded dynamic response which closely approximates the dynamic response of the reference model.

The off-line process of Fig. 8 forms a model­reference inverse of the plant model P(z). We have used the model P(z) rather than the plant P(z) because the output of the real P(z) is generally corrupted by plant noise. However, since F(z) does not perfectly match P(z) at all times, use of P(z) in determination of b(z) causes errors in C(z ). However these errors can be limited by slowing the adaptation rate and thus decreasing the error in the plant estimation P(z ).

CONCLUSION

A method for adaptive inverse control unbiased by additive plant noise has been introduced. The technique is easy to implement and exhibits robust, predictable behavior. Intensive research has been conducted in this area in order to enhance the potential capabilities of the proposed approach and to perform detailed analyses or the expected behavior. The results of this additional research are now being prepared for publication.

I O B . Widrow and E . Walach

REFERENCES

B. D. 0. Anderson and R. M. Johnstone, "Convergence results for Widrow's Adaptive controller," IF AC Conr. on System Identification, 1Q81.

K. J. Astrom and B. Wittenmark, "On Seir­tuning Regulators," Automatica, Vol. 9, No. 2, March 1Q73.

K. Astrom and B. Wittenmark, "Analysis of Seir­tuning Regulator for Nonminimum Phase Systems," IF AC Symposium on Stochastic Control, Budapest, Hl7 4.

K. Astrom and B. Wittenmark, "Self-Tuning Controllers Based on Pole-Zero Placement," Proc. IEE, Vol. 127, Pt.D. , No. 3, pp. 120-130, May 1Q80.

I. D. Landau, "A Survey of Model Reference Adaptive Techniques-Theory and Applications ," Automatica, Vol 10, pp. 353-379, Hl7 4.

I. D. Landau, "Unbiased Recursive Identification Using Model Reference Adaptive Techniques," IEEE Transactions on Automatic Control, Vol. 21, April Hl76.

J. M. Martin-Sanch�z, "A New Solution to Adaptive Control," Proc. IEEE, Vol. 64, No. 8, August Hl76.

K. Nakamura and Y. Yoshida, "Learning Dual Control Under Complete State Information ," a paper presented at NSF Workshop on Learning System Theory and its Applications, October 18-20, 1Q73, in Gainesville, Florida.

F. D. Powell, "Predictive Adaptive Control," IEEE Transactions on Automatic Control, October Hl69

E. Tse and M. Athans, "Adaptive Stochastic Control for a Class of Linear Systems," IEEE Transactions on Automatic Control, February Hl72, pp. 38-51 .

B. Widrow and M. E. Hoff, "Adaptive Switching Circuits," in moo WESCON Conv. Rec., pt. 4, pp. 96-140.

B. Widrow and others, "Adaptive Noise Cancelling: Principles and Applications," Proc. IEEE, Vol. 63, pp. 1692-1716, December 1Q75.

B. Widrow and others, "Stationary and Nonstationary Learning Characteristics of the LMS Adaptive Filter," Proc. IEEE, Vol. 64, pp. 1151-1162, August IQ76.

B. Widrow and others, "Adaptive Control by Inverse Modeling," Twelfth Asilomar Conference on Circuit, Systems and Computers, November IQ78.

B. Widrow and others, "On Adaptive Inverse Control," Fifteenth Asilomar conference on Circuits, Systems and Computers, November 1981.

INPUT

INPUT

ADAPTIVE A L GOR I T H M

+

F I G . l . AN ADAPTI VE F I LTER .

P (z )

PLANT

OUTPUT

dj DES I R E D RES PONSE

1J = Yj + nl PLA N T

OUTP U T

�J + d1 = 11 �---'---' �

ADAPTIVE PLANT M ODEL

F I G . 2 . ADAPTI VE MODEL ING .

Adaptive Signal Process ing for Adaptive Control 1 1

u . J PLANT INPUT

P ( z )

P L A N T

D E L AY

c cz l (ADAPTIV E ) ' D E L AY E D PLANT INVERSE

F IG . 3. DELAYED INVERSE MODELING .

RESPONSE AMPLITUDE

(a )

PLANT

INVERSE MODELING MEAN SQUARE ERROR

TIME, j

1 1 1 1 " l111111111;i II""�" llliuiliuillillL

INVERSE MODELING DELAY t.=26 (b)

C (z) uj (COPY)

CONTROLL E R \

COMMAND INPUT

P ( z )

PLANT PLANT OUTPUT

D E L AY

F IG . 4 . AN ADAPTI VE I NVERSE CONTROL SYSTEM.

RESPONSE AMPLITUDE

(c)

RESPONSE AMPLITUDE

( d )

OPTI MAL 40-WEIGHT DELAYED PLANT INVERSE t.=26

TIME, j

( PLANT) * ( DELAYED INVERSE)

F IG . 5 . I MPULSE RES PONSE OF PLANT AND OPT IMI ZED 40-WEI GHT CONTROLLER .

1 2 B . Widrow and E . Walach

I DEAL STEP RESPONSE

( a )

CONTROL SYSTEM STEP RESPONSE

( b )

T I M E , j

i TIME, j

11=26

F IG . 6 . STEP RESPONSE OF CONTROL SYSTEM ( PLANT CASCADED WITH 40-WE I GHT INVERSE CONTROLLER) .

M ( z )

PLA NT OUTPUT

REFERENCE M ODEL

F I G . 7 . A MODEL REFERENCE ADAPTI V E I NVERSE CONTROL SYSTEM .

COMMAND INPUT

P ( z ) PLANT

PLANT MODE L

I MODELING- - - - - - - - -- - - ,

NOISE I I P c z> C < z > I

(COPY) (ADA PTIVE) I I I I I

M ( z ) I R E F E RENCE : MODEL

L _Jlff .:..l-1Nf. AQAPT lY E... E_R_Qc.f.S.5-j

PLANT OU TPUT

FIG. 8. A MODEL REFERENCE ADAPTIVE CONTROL SYSTEM FOR A NOI SY PLANT .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROBUSTNESS ISSUES IN ADAPTIVE CONTROL

R. L. Kosut

Integrated Sy1terns, Inc., 151 University Avenue, Palo Alto, CA 943 0 1 , USA

Ab strac!. The robustne s s propert ie s of adapt ive control systems are examined from an input/output view. Thi s se tting allows for the pl ant to be l inear, nonl inear, continuous or di screte . Stabil ity theorems are pres_ented that provide conditions for both gl obal and l ocal stabil ity. It i s shown that gl obal stabil ity i s guaranteed if a certain sub system operator is str ictly pos itiv e , which, unfortuna tely , has a very limited robustne s s property . The l ocal stabil ity conditions presented are not so restricted and do not require a pos it iv i ty condition .

Keywords . Adapt iv e control ; robust control .

1 . INTRODUCTION

The lure of adaptiv e control has certainly to originate from the promi sed ab i l i ty of the control l er to be se l f-tuning in such a way that , despite the presence of uncertain phenomena , a des ired l evel of performance is maintai ne d . Thi s promi se of robustn�!! to uncertainty is not unique to adapt ive control and , in fac t , is a fundamenta l property of all feedback control--adapt iv e or otherw i se . The dist ingui shing feature of the adaptive control l er i s the potenti a l to al ter its structure or parameters in accordance with a b·uil t in. ' inte l l igence , ' such that the mai ntai ne d l evel of performance i s consistently higher than i s otherwi se obtainab l e from a fixed (non-adapt ive ) control ler of s imil ar structure . The mai n i ssue i n the des ign probl em--the subj ect of thi s paper--i s to material ize the . abstract qua l i ty of int e l l i gence into a concrete methodol ogy that deal s with quantitativ e measure s at the engineering l evel .

In order to devel op such a methodology it is ne cessary to precise ly spec i fy the nature of the uncertain phenomena in the presence of which control action is needed.

Historically , in the nonadapt ive case , re­search in robust control theory has preceded from an input/output view of system s ; e . g . , ( 1 ) - ( 5) . The predominant reason to examine robustnes s i ssues in thi s way i s that the character i st ics of unmodel e d dynami c s , such as iincertain model order , are eas ily repre­sented.' On the other hand , Lyapunoy,_.theory · is not wel l suited for thi s type of uncer­tainty. Typically , p l ant uncertainty i s characterized by assuming that the pl ant be­l onas to a wel ldefined se t ; for example , a set descript i on of an uncertain l inear-time­invariant ( LTI) p l ant i s to define a 'ba l l '

1 3

in the frequency domain. The center of the ball is the nomina l plant model and the radius define s the model error . Th i s se t model descript i on is one type of a more gen­eral set descript ion, referred to as a conic­se ctor (3) . The unc ertainly in the plant induce s an uncertainty in the input /output map of the closed-loop system, which can again be characterized by a conic se ctor . Performanc e requirements for the control sys­tem can be transl ated into statement s on the conic se ctor that bounds the cl osed l oop sys­tem s , making it pos sibl e to check whe ther a g iven design meets spe cif ications and provid­ing guidel ine s for robust control ler des ign; for exampl e [ 6) •

In contrast , research in adapt ive control theory has fol l owed a state-space view ( ( 7 1-(10) ) . A s such , it is very difficul t to cons ider the effect of unmodeled dynamics or reduc ed order model ing . In fact, unmode l ed dynamics can cause a rap id deteri oration in performance and even instabi l i ty ( 11] . Thi s problem is not resolved by increas ing the or­der or complexity of the model . S ince the model of any dynamic system , by definition , i s not the actual system, i t can therefore be argued that unmodeled dynamics are always present ' ad infinitum . '

The mai n reason for these difficulties with adapt iv e control l ers i s not ent irely due to the state-space view pre cluding a sens ib l e characterization o f unmodeled dynami c s . Rather , s ince the adapt ive control l er i s by its very nature a nonl i near device, and s ince theoretica l investigati ons on the stabil ity of adapt iv e control systems have focused al­most ent irely on develop ing conditions that guaranty gl obal stab i l i ty , it fol lows that the s e cond i t i ons wil l tend to be conservativ e .

1 4 R . L . Kosut

One of the cond itions i s that a part icular sub system operator be strictly pas s ive with finite gain or , in the case of linear-time­invariant system s , the operator is strictly pos it ive real ( SPR) . This cond i t i on resul ts from appl icat ion of the Pas s ivity Theorem [ 2] , [ 12] ; spe cif ica l ly , the adapt ive system can be reconfigured into two sub system s : a ' feedback ' sub system ( the adaptation l aw) that is pas s iv e , and a ' feedforward ' sub sys­tem . Thus , if the feedforward part is SPR, then gl oba l stabil i ty fol lows , provided that some other constraints are also sat i sfied [13] . Unfortunately , however. the SPR condi­tion is very difficul t to sati sfy for an ac­tual system, ei ther because of reduced order model ing [14] , or because of unmodeled dynam­ics [11 , 13] . In fac t , even the most benign type of unmodeled dynam ics violates the SPR condit ion ( e . g . , two unmode led stable poles at arbitrarily large frequenc ies [ 13 ] ) . Al­though violation of SPR does not guaranty in­stabil ity, it has been demonstrated [12 , 1 3 ] that instabil ity can ensue i f SPR i s violated.

It i s pos s ib l e to alleviate the problem by construc ting an SPR compensator around the actual plant and attaching the adaptive con­troller to the compensator output [13b ] . In effe c t , the SPR condition is maintaine d , and thus the re sul ting adapt ive system is guar­anteed to be global ly stab l e . Stil l , thi s solut i on remains unsati sfactory for two rea­sons . First of all , the SPR property , by definition. is the oppos ite of high perform­ance ; consequently , performance level s are l imited from an adapt ive system that rel ie s on the SPR property . In fact , i t can b e arg­ued that thi s is the very reason that gl obal stabil ity i s achieved. Secondly , high-per­formance adapt ive control l ers have been de­veloped in actual appl ications where the en­vironments are such that sat i sfying an SPR condition i s out of the que s t i on; e . g . , [15 ] . It i s , therefore , compe l l ing to abandon the requirement of global stabil i ty--a require­ment that is well beyond the ne eds of any ac­tual syst em--and develop conditions for l ocal stabil ity.

Conditions for local stabil ity of adapt ive controll ers are recognized to exi s t , e . g • • pers i stent exc itation and exponential stab il­ity [ 14] , [ 16] . Also, s ince the adapt ive system is nonl ine a r , the very same input /out­put framework (menti one d before in the con­text of robust control) can be use d to de­velop l ocal stab i l i ty and robustne s s condi­tions ( see [ 17 ] for a brief sketch) .

2 . GENERAL FRAMEWORK

Function Space

Let L be a normed space of func t i ons x ( " ) : �->Rn where � is the time se t of interest and 1 1 " 1 1 i s the norm on L Associated with L i s the extended space L , cons i st ing of func t i ons � e L , w!ere Xor( t ) denotes the truncation of x ( t ) _at some finite T e � ;_ i . e . , xT( t ) = x ( t )

for t < T , and x.,.( t ) = 0 for The norm on L i s aenoted by l l x l l T := l l x; l l •

t > T

For cgnt inuous-t ime , � = R+ : = [ o , m) and L = L , the se t of integrabl e f11J1.ctions with Korm I l x l I := ( / l x ( t ) l pdt ) 'lf P , for p e [ 1 ,m) , andp I l x l I : = SfP l x ( t ) I , where l " I i s a norm onmRn • The as•oclated

extended space i s denoted by I l x l I Tp

: = I l xTI Ip

wi th norm

For di screte time , � = N : = [ 0 , 1 , 2 , • • • , m) and L = In , the se� of summa�f e functions with norm l l x l I : = ( l x ( t ) I P > P , for p e [ 1 ,m) , an8 I Ix I : = sup l x ( t ) I • m The associated In extended space is denoted by

pe

Stabil i ty and Gain

A causal operator ( or system) G : L ->L is L-stab l e i f : ( 1 ) Gu e L wheneve� e

u e L ; and ( 2) 3 finite constants k and b such that I I Gu l I i k I l u l I + b , Vu e L The sma l l est k that sati sfies the inequal­ity is referred to as the L-gain of G • and is denoted by y (G)

Certain nonl inear systems behave s imil arly to LTI systems with an exponentially decaying impul se re sponse . Such a system G : L -> L i s said to have decaying L1�mory [ 18f , if e

there exi sts a nonne gativ e , nonincreas ing function m( " ) f L1 such that

I CGu) C t > l 2 i J m( t-�) l u < � > l 2d� Vt

> 0 , Vu e L o

A s imil ar de�init ion applies to di screte sys­tems with decaying 11 memory.

Pas s iv i t.:r

A class of systems that regularly appear in the l iterature on adaptiv e control are those that are pas s iv e . Let ( " , " >T denote an inner-produc t on L = L or L = 1 Fol l owing [ 2] , if G : L2e-> L e thei! e e

G i s pass ive if 3 const ant p such that ,

VT s � Vu. e L e

G i s str i c t ly pas s iv e i f 3 const ants p > 0 and P such t�a t ,

<u. , Gu>T 2 µ l l u. l lT2 + p , Vf e � . Vu e Le If G i s L-stabl e , then G i s pos it ive or strictly pos it iv e , b y letting T - > m resp . , abov e . An important subset of the strictly pos it iv e systems are those that are str i ct ly pos it ive rea l , i . e . ,

SPR : = { G : L -> L I (u , Gu> > 0 , Vu. e L}

Note that if G i s LTI with transfer func tion matr ix G ( s ) , whose elements are strictly proper and exponent i a l ly stab l e , then G e S PR if the sma l l est eigenvalue of G ( j111) + G ' ( -j111) i s pos ith'e for a l l finite Ill

Robustness Issues in Adaptive Control 1 5

Uncertainty

Plant uncertainty ari se s from uncertain dy­namics and uncertain unmeasurab l e di sturb­anc e s . Cons ider the plant to be control led described by

y = d + Pu

where y i s the measured output , u i s the control input , d is the disturbance , and P : L -> L represent s the actual pl ant dynamics . Aenatural characterization of d i s­turbance uncertainty i s to describe a se t that contains d Let d a Sd , where

sd : = Cd a L I l lw�1d I I i l l

with Wd a known operator . For exampl e , a band-limited, continuous-time di sturbance can be represente.d in sd by the trans fer func­tion Wd( s ) = a / ( s+bJ

A set descript ion of dynamic unc ertainty ari se s natural ly by compar ing the actual dy­namics pl ant response with the output of a model P : L -> L Thus , let P a S where m e e p

S : = {P : L -> L I l lPu - P u l l T/ l lw P u l l pi 1 • VI' � � e Vu a L } m m m

e

with Wm a known operator . An equival ent expression for S i s , p

s {P : L -> L I y (AW-1 ) i l} p e e m

where A i s an L-stable operator impl icitly define d by

P : = ( I + A)P m

If, for example , P ,Pm' and Wm are LTI , and L = L2 . • then

- -1 sup a[A(j11>)Wm( j11> ) ] i 1 .., e R

where a (A) i s defined as the maximum s ing­ul ar value of the compl ex matrix A In other words , whil e the operator A i s not prec i se ly known, we do know a bound on its effect .

3 • ADAPTIVE ERROR MODEL

In thi s section we develop a generic adapt ive error mode l tha t wil l be use d in the subse­quent analy s i s .

Cons ider , for exampl e , the model reference adapt ive control (MRAC) depicted in Figure 3 .1 , cons ist ing of the uncertain pl ant P , a reference model H , and an adapt ive con­trol ler C(�) , wliere � i s the adaptive gain vector , r i s a reference input , d is a disturbance process.A. and n i s sensor noi se . Denote by H(U) the closed-::l oop sys­tem rel at ing the externa l inputs

w = ( r ' , d ' , n ' ) ' to the output error e Also, let w a S denote the adm i s s ib l e c l ass o f input slgna l s .

C(B)

ADAPTIVE LAW'

Figure 3 . 1 . A Mode l Reference Adaptive Control l er

In summary, we cons ider the muJ tivariabl e adapt iv e system described by

e = H<e> w ( 3 . 1 )

where e ( t ) a Rm i s the error s ignal t o be control l ed , w ( t ) a Rq is the ext1rnal iniut restricted to some se t W , and O( t ) a R i s the adaptive gain. The class of adaptive control l ers cons idered here are such that the adapt iv e gains mult iply elements of internal s igna l s z ( t ) a R , referred to as compen­sator s igna l s , to produce the adapt ive £2!!,­trol signa l s ,

i B [ l ,m) ( 3 .2 ) "'

where e . and z i are k . -dimens ional sub-sets of1the elements in e and z respec­t ively . Thus k = k1 + • • • + km Def ine the adapt ive gain � .

e< t > : = �< t > - e. ( 3 . 3 )

where e. B Rk i s the tuned gain ( 3 . 4 ) . Al so, define the adapt ive control � s ig­nal s ,

v : = ei' z l. i i = 1 , • • • , m • ( 3 . 4 )

An equivalent express ion i s

v = z •i ( 3 . Sa )

where the time-varying matrix Z i s def ined by

Z = b l ock diag ( z1 , z2 , • • • , zm) ( 3 . Sb )

To describe the e , z , v , and w ne ction system by ,

re l ations among the signa l s , w e introduce the intercon­H1 : (w,v) -> ( e , z ) define d

(e) (w) (H -H ) (w) : = H : = ew ev

z I v H -H v zw zv

( 3 .6 )

I n effect, thi s structure serves t o i solate the adapt ive control error v from the rest

1 6 R . L . Kosut

of the ststem . When the adapt ive . control is tuned, e = 0 and v = 0 ; consequently , the tuned error signa l is

( 3 . 7 )

We can also define a tuned compensator s igna l .

z : = H w • Z'lf ( 3 . 8)

In general , all the sub systems in HI are dependent on the tuned gains e. . The in­terconne ction system can also be written as

( 3 . 9a)

( 3 . 9b )

with v given by ( 3 . 5 ) . To complete the er­ror model requires describing the adaptatiion l aw, i . e . , the means by which �( t ) i s gen­erated. A typical law [ 7 ] - [ 10] i s :

1J; : = Ze ( 3 . 9c )

where B = B ' > O The compl ete adapt ive error system , ( 3 . 5 ) , ( 3 . 9 ) , i s shown in Figure 3 .2 . Note that the error system is composed of two sub system s : a linear sub system described by ( 3 .9a ) , and a nonl inear sub system desc ribed by ( 3 . 5 ) , ( 3 . 9b-d) .

I_ - - 1 I l I I I I I

_ _ _ ,

Figure 3 .2 . Adapt ive Error System

4 . CONDITIONS FOR GLOBAL STABILITY

The theorem stated below g ives conditions for which the adapt ive error system (Fig . 3 .2 ) i s guaranteed t o be global ly stabl e . Proof i s g iv en i n [13b ] . Heuri stically , however , the bas i s for the proof is appl ication of the Pas s iv i ty Theorem ( [ 2] , p. 1 82 ) . It turns out that the map e -> v is pas s iv e . Thus , if H i s SPR with decay ing L

!-memory , then ev the map e* -> ( e ,v) i s L2-sta le despite

the fact that � s L • Restrict ions on e., z* , and Hzv cau':: 'tr to be bounde d .

Theorem 4 .1 : Global L2(E!ponential) Stabi l ity

For the adapt ive error system shown in Figure 3 . 4 , a s sume that :

( Al ) The system i s well posed in the sense that a l l input s - w B W produce s igna l s e , v , z , �. and

'iJ in L ( A2 ) H has

m3ec�ying L1-memory . ( A3 ) Hzv

s SPR has decay1ng L1-memory . Under the se c8�ditions , i f e• i s bounded by a decay ing exponentia l , z• B Lm , and z* ,.:, i s uniformly continuous , then e , v, e and z - z* are also bounded by a decaying ex­ponent ia l .

Di scus s ion ( 1 ) Theorem 1 provides condit ions for

g l obal stabil i ty of continuous-time adapt ive system s . Identical resul ts can be obtaine d for di screte systems by repl acing Lm with lme , L1 with 11 , e t c .

( 2 ) The restr1ctions on the tuned sig­nal s e• and z* , indirectly impose re­quirement s on H and H • These latter requirements are e!ependent z�n knowledge about w B W ; for exampl e , if w is a constant , then the as sumpt ion that e* -> 0 exponenti­a l ly require s that the tuned feedback system i s a Type-I robust servomechan i sm .

( 3 ) The primary use o f Theorem 1 i s to provide the means for handl ing model error . S ince the Theorem imposes requirements on the input / output propert i e s of the interconne c­tion system , it fol l ows that the effect of model error on the se propert i e s determine s the stabil i ty robustne s s o f the adapt ive sys­tem; for exampl e , Theorem 1 requires that H a SPR Suppose , however , that H hil the form ev

H = ( I + ii )H ev ev ev ( 4 . 3 )

where H i s the proj ection onto H of the planfvuncertainty operator A ; ::d H i s a funct i on of the tune d parametr ic modelev

P* and the tuned control l er gains e.

Robustne s s of SPR Condition Condi t i ons to insure that H a SPR despite uncerta i nty in H i s provi1ed by the fol lowing : ev

Lemma 4 .1 : Let H b e given by ( 4 . 3 ) . Then H a SPR ifefhe fol lowing condit ions hol d : ev

a SPR ( i ) ( ii ) is L-stab l e such that

r2 <ii > r2 <ii ) < inf ev ev L u 8 2 Proof : Follows directly from gain and pas s iv ity .

Comments

- 2 <u, H u>/ l lu l l 2 ev

def init ions of

( 1 ) In order to apply Lemma 4 . 1 , i t i s ne cessary t o have a deta i l ed descripti on of how the plant uncertainty A proiaaates onto the interconne ction uncertai nty H • Th� type of uncertainty propagation walvexpl ored in depth by Safonov ( 3 ] _ and more sophistics-

Robustness Issues in Adaptive Control 1 7

ted expres s i ons than ( 4 .4b ) are avail able to describe the uncertain operator H .

( 2) Unfortunately , ( ii ) of Lg�a 4 . 1 places severe limitations on y2 ( H ) that are eas ily viol ated by even the mo�I benign unmodeled dynamics [13 ] . Thus , al though a robust gl obal stabil ity theory can be formu­lated (Theorem 4 . 1 , Lemma 4 .1 ) , the l imita­tions are too str ingent to account for real­i st ic model error .

S . CONDITIONS FOR LOCAL STABILITY

Since it is virtual ly impo s s ib l e to maintai n R a SPR despite unmodeled dynamics , i t i s m81e meaningful t o develop conditions for lo­£11. stab il i ty that are independent of the SPR condit ion.

The error model ( 3 .10) , ( 3 . 14) ) can be trans­formed to a more use ful form for local sta­bil ity analysis ; i . e . ,

x = i - Gf (x ) ( S . l a ) where the quant ities above are defined below by

(D c . . . ev • x : = : = -R z '0 zv •

( I + LM ) 00 + K z,.. e

c [ 1 - z;x:< z•Rev + e •Rzv ) ] ev G : = R [ 1 - z;x:< z•Rev + e •Rzv ) ] zv

K ( z•Rev + e•Rzv

K : = ( I + LM) -lL

M : = The map K can also be described by K : w 1--> � where ,

� = B(w - M�) , � ( 0) 0 or

� = Lw - LM�

( S . lb )

( S . l c )

)· ·••l · .. ·;•) R z 'K zv • -K ( S . l e )

( s . l f)

C S . l g )

( S .2 )

The model ( S . l ) i s arrived a t by separating the nonlinear cross product terms in f ( x ) from the l i near terms i n i We shal l re­fer to i as the response of the l i ne arized system. This is almost identi cal to the l i n­ear ized system studied by Rohrs et al . [ lla ] , which was arrived at by a ' f ina l approach an­alys i s , ' This model will now be ut il ized to develop l ocal stabil i ty conditions .

Theorem S . 1 : Local L -Stabil ity For some p a [ 1 , m] ind all ( e • , z. , e • • eo > 8 s assume that :

( Al ) 3 iii < "' such that l l i l lp i m ( S . 3a )

( A2 ) 3 g < "' such that yp ( G) i g ( S .3b )

(A3 ) Ym ) o , 3 a (m) ) 0 such that

l l x l l p i m => l l f ( x ) I I < a (m) l l x l l P - ( S .38) Under the se conditions , if 3 const ant s m s (m) and iii such that

( i )

( ii )

g a (m) ( 1 and,

[1 - g a (m) ] m = m

( S .3d)

( S . 3 e )

then the adaptive system i s locally L -st abl e about the se t S moreover, p

( ii i ) l l x l l i m p Proof : See [17 ] •

Discuss i on

( s . 3f )

Essent ially Theorem 4 . 1 provides sufficient conditions for local L -stabil i ty of the adapt iv e system. Thi s fol lows if a se t S exists such that < e • • z • • e • . eo > 8 s imply conditions ( 5 . 3 ) . These conditions , in fact , provide the means to expl icitly def ine the properties of S

For exampl e , condit ions ( 5 .3 a ) and ( S .3b ) re­quire , respe ctively , that the linearized sys­tem ( 3 . 7b ) and the map G ( 3 .7 c ) are locally stabl e . Exact conditions are provided in the fol lowing Lemma for the case of L..,- stabil i ty.

Lemma 5 .1 If Rev ' R v ' and K are locally

L -stab l e abouf the se t S , then condi­tions ( 4 . l a ) and ( 4 . lb ) of Theorem 4 . 1 are sati sfied.

�: See [ 17 ] .

D iscus s i on

It i s not difficul t to insure that R and Rzv ar� l ocally stable . The main is��e i s t o insure that K i s l ocally stab l e . In [ 17 ] , local stabil i ty of K is invest igated when the tuned signal signa l s are small , slowly varying and/or per s i st ently exc iting . In all case s the re sul ting conditions do not depend on Rev a SPR.

6 . CONCLUDING REMARKS

Thi s paper has presented an input/output view of mul tiv ariable adapt ive control for uncer­tain plant s . An error system of a very gen­eral form i s developed, which i s used in ana­lyz ing the stabil i ty and robustne s s proper-

1 8 R . L . Kosut

ties of the adapt ive controller in the pres­ence of unmodel ed dynamics and unmeasurab l e di sturbance s . Th e es sence o f the results is captured in Theorems 4 .1 and 5 . 1 , which pro­vide , respectively , conditions for global and local stabil ity .

Th e global stabil ity theorem extends previous resul ts for LTI systems ( e . g . , [ 7 ] -[ 10] ) so as . to account for nonl inear as well as infin­ite dimens ional systems. The structure of the theorem requires that a particul ar sub-system operator , denoted H , i s strictly positive real ( SPR) . This �Xquirement is not unique to thi s presentation--pas s ivity re­quirements in one form or another dominate proofs of global stab il ity for practically all adaptive control systems, including re­curs iv e ident ification algorithms . Unfortunately, al though Hev B SPR is robust to model error (Lemma 4 .1 ) , the bound on the model error is too small to be of practical use .

The local stability theorem doe s not require that H B SPR Inst ead, restrictions are placed 8� the behavior of signa l s in the tune d system. Restrictions such as these are to be expe cted in an actual system, and so the requirements are not unreasonab l e .

I n conclusion, then, i t would seem that the promise of building an intell igent/adapt ive controller is more well founded by restrict­ing attent ion to local rather than gl obal issue s .

REFEirnl CES

[1] G . Zames , ' On the input-Output Stabil ity of Time-Varying Nonl ine ar Feedback System s , ' IEEE Trans . on Aut . Contr .• , Part I : Vol . AC-11, No . 2 , pp. 228-23 8, April 1 966 ; Part II: Vol AC-11 , No. 3 , pp. 465-476 , July 1 966 .

[2] C . A. Desoer and M. Vidyasager, Feedback System s : Input-Ou�t Pr.Ql!ertie_!, Academic Pre s s , New York, 1 97 5 .

[ 3 ] M.G. Safonov, Stability Robustne s s of Mul tivariable Fe£dbac!_�I�te�, MIT Pre s s , New York, 197 5 .

[ 4] J . C . Doyle and G . Stein, 'Mul tivariable Feedback Des ign : Concepts for a Modern/Cl ass ical Synthe s i s , ' IEEE Trans. Aut , Contr . , Vol AC-26 , No . 1 , pp. 4-17 , February 1 981 .

[ 5 ] G . Zames and B . A. Franc i s , 'A New Approach to Cl ass ical Frequency Methods : Feedb ack and Minimax Sensitivity, ' Pro c . 20th CDC, pp . 867-874 , December 1981 , San Diego, CA .

[ 6 ] R .L. Kosut , ' Ana ly s i s o f Performanc e Robustne s s for Uncertain Mult iv ariab l e Systems, ' Proc. 21st IEEE CDC, pp . 1289-1294 , Orl ando, FL, December 1982 .

[7 ] K . J . Astrom and B . Wittenmark, ' On Sel f-Tuning Regulator s , ' Automatica, Vol . 9 , pp. 1 85-199 , 1 973 .

[ 8) L. Ljung, ' On Pos i t iv e Real Transfer Functions and the Convergence of Some Recursive Scheme s , ' IEEE Trans. A.!!!i Contr . , Vol . AC-22 , No. 4 , pp. 53 9-551 , August 1 977 .

[ 9] K . S . Narendra and L . S . Valavani , 'Dire ct and Indirect Model Reference Adapt iv e Control , ' Automatica , Vol . 1 5 , pp . 6 53 -664, 1 97 9 .

[10) Y . D. Landau, Adapt ive Control : T)� Model Reference Approa ch, Dekker, New York, 1 97 9 .

[lla] C . Rohrs , L. Val avani , M . Athans and G . Stein, ' Analytical Verification of Undes irab l e Propert i e s of Dire ct Model Reference Adapt iv e Control Algorithms, ' Proc. 20th CDC, San Diego, CA , December 1981 .

[llb] C . Rohrs , L. Valavani , M. Athans and G . Stein, 'Robustne s s o f Adapt ive Control Algori thm in the Pre senc e of Unmode led Dynamics , ' Proc . 21st IEEE CDC , Orl ando, FL De cember 1 982 .

[ 12) V.M. Popov, Hyper stabil ity of Autop�!1£ Control�_!!!.!, Springer , NY, 1973 .

[ 13a] R.L. Kosut and B . Frie dl ander, 'Performance Robustne s s Properties of Adapt iv e Control Systems , ' Proc . 21st IEEE CDC, Orl ando, FL , December 1982 .

[ 13b ] R . L . Kosut and B . Friedl ander, 'Robust Adapt iv e Control : Conditions for Global Stabil ity, ' submitted to IEEE Trans. on Aut . Con tr .

[ 14) B . D . O . Anderson and C .R. Johnson Jr. , ' Exponential Convergence of Adapt iv e Ident ification and Control Algor ithms, ' Automatica , Vol . 18 , No. 1 , 1982 .

[ 15) B . Wittenmark and K . J . Astrom, ' Impl ementation Aspe cts of Adaptive Controllers and Their Influence on Robustne s s , ' Proc . 21st IEEE CDC, Orl ando, FL, December 1982 .

[ 16 ) B . D . O. Anderson, 'Exponential Stability of Linear Equations Ari sing in Adapt ive Identification , ' IEEE Trans. Aut. Contr . , Vol . AC-22 , No . 1 , pp. 83-88, February 1 977 .

[17) R. L. Kosut , 'Robust Adaptive Control : Conditions for Local Stab il ity , ' to appear Proc . 1 983 ACC, San Francisco, CA , June 1 983 .

[18) M . Vidyasagar, Input-Output Ana lysi s of LargeScale Interconne cted Syst em s , Springer-Verlag, 1 981 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROBUST REDESIGN OF ADAPTIVE CONTROL IN THE PRESENCE OF DISTURBANCES · AND

UNMODELED DYNAMICS

P. loannou

Univrrsity of Southern Califiirnia, Department of Electrical Engineering Sy.1tern!., Los Angeles, CA 90089-0781, USA

Abstract

The effects of unmodel ed weak ly observabl e stabl e dynami cs and bounded d i sturbances on the stab i l i ty and performance of adapti ve con­trol schemes are anal yzed . A second order exampl e i s used to i l l ustrate the non-robust behav ior of the present adapti ve control l ers when di sturbances and/or unmodel ed dynami cs are presen t . An earl i er mod i fi ed adapti ve l aw [l , 2 ] i s s hown to quarantee the exi stence of a re qi on of attraction for boundedness . A new adapt i ve control l er has been i ntroduced wh i ch quarantees bound�dness for bounded i n i t i a l condi t i ons .

I ntroduction

Recent ly several attempts [ l - 1 4 ] have been made to ana l yze the stabi l i ty properti es of several adapt i ve control schemes i n the µresence of model i ng errors and/or bounded di sturbances . I t has been shown that bounded d i sturbances [3 ,4 , 1 4 ] or unmodel ed stabl e dynami cs [ l , 2 , 7 ] , [l 0-1 3 ] can make the cl osed-1 oop sy stem unstabl e. The concl u s i on of these stud i es i s that the present adapt i ve control schemes need to be redes i gned or modi fi ed for robu stnes s .

Several mod i fi ed adapt i ve l aws have been i ntroduced to counteract the effects of d i s ­turbances . I n [ 3 , 4 , 6 ] a dead zone i s i ntro­duced i n the adaptive l aws , that i s , adaptati or. i s stopped when the output error becomes smal l er than a computed bound . Thi s mod i fi ­cat i on guarantees boundedness of a l l the s i g­nal s i n the c l osed l oop but can l ead to l arge output errors if the s i ze of the dead zon e , whi c h depends on t h e un known di sturbances and pl ant parameters , is overestimated . Al terna­t i ve approaches wh i c h reta i n the potent i a l of obta i n i nq smal l output errors in the l i m i t when the d i sturbances are smal l are taken i n [ 3 , 5] . These approac hes , however , requ i re the knowl edqe of an upper bound for the de­s i red constant control l er parameter vecto r .

A mod i fi ed adapt i ve l aw wh i ch quarantees robustness wi th res pect to unmodel ed fa st dy­nami cs and/or d i sturbances i s i ntroduced i n [ l , 2 ] . Th i s i s a l i near mod i fi cation and guarantees the exi stence of a reg i on of at­tract i on from wh i c h a l l s i gnal s converge to a smal l res i dual set . I n [ 1 4 ] a boundedness resul t i s obta i n ed for an i nd i rect adapti ve control scheme under the assumption that the d i fference between the actual pl ant and the

1 9

i deal pl ant i s l i nearl y domi nated by the i n ­formation vector . The mod i f i cati on con s i sts of a dead zone , by which the adaptation i s swi tched off whenever some s i gnal s become sma 1 -1 er than a g i ven number . Local stabi l i ty has al so been proved for a reduced-order i ndi rect adapt i ve regul ator i n [ 1 2 ] .

I n th i s paper we exami ne the effects of unmodel ed wea k ly observabl e stabl e dynami cs and d i sturbances on the stabi l i ty propert i es of a conti nuous t i me d i rect adapt i ve control scheme . We fi rst use a second order exampl e to demonstrate the i nstabi l i ty phenomena wh i ch can ari se when d i sturbances and/or un­model ed wea kly observabl e dynamics are present . We then use the cr-mod i f i cation , fi rst i ntro­duced i n [l , 2 , 1 5 , 1 6] to handl e unmodel ed fast dynami cs and unmodel ed i nterconnect ions , to obta i n suffi c i ent cond i t i ons for boundedness in the presence of d i sturbances and/or un­model ed wea kly observabl e dynamics . Th i s mod i fi ed adapt i ve l aw guarantees the exi stence of a reg i on of attract ion from wh ich a l l s i g­nal s converge to a smal l res i dual set . Fur­thermore , we i ntroduce a new adaptive contra 1 1 er wh i ch guarantees boundedness for any bounded i n i t i a l condi t i ons .

I . A Scal ar Adapti ve Control Probl em

We start wi th a s i mpl e second order pl ant w i th a u n i forml y bounded i nput di stur-bance d ( t ) .

+ d ( t ) u ( t ) + ,.. ,_,

F i g . l . Second Order Pl ant

� y ( t )

where c1 , c2 > 0 and c 1 to c3 are unknown con­

stants . He assume that t:� c1 -c2 is sma 1 1 , that i s , the stabl e mode i s weak l y observabl e . The output o f the pl ant i s requi red to track the state ym of a fi rst order model

y = -a y +r ( t ) a > 0 ( l . l ) m m m m where r ( t ) i s a reference i nput , a un i forml y bounded function of t i me . T h i s exampl e i l ­l ustrates some of the stab i l i ty probl ems ari s i nq i n adapt i ve control when d i sturbances or unmodel ed dynamics are present and serves as a moti vation for an i ntroduct ion to the general methodol ogy to be devel oped i n the l ast sect i o n .

20 P . Ioannou

The state representat ion for the second order pl ant i s

X = c3x+U+Ez+d ( 1 . 2 ) ( 1 . 3 )

y = x ( 1 . 4 ) A s impl i fi ed model for the pl ant ( 1 . 2 )

to ( 1 . 4 ) i s obta i ned by assumi ng zero d i stur­bances ( d=O ) and exact zero-pol e cancel l at i on (E:=O ) , i . e . ,

( 1 . 5 ) y = x

For the s impl i fi ed model adapt i ve control l er

( 1 . 6 ) ( 1 . 5 ) , ( 1 . 6 ) the

u = -K ( t )y+r ( t ) • /:>, K ( t ) = yey , e = y-ym ' y > O

( 1 . 7 ) ( 1 . 8 )

guarantees the fol l owing propert i es . Lemma 1 : For any bounded i n i t i al cond i t i on e(O) , K(O ) , al l the s i gnal s of the cl osed l oop system ( 1 . 5 ) to ( 1 . 8 ) are bounded and 1 im e ( t ) = 0 .

t400 The questions to be answered i n th i s

paper are the fol l owing : How wi l l the adap­t i ve control l er ( l . 7 ) , ( 1 . 8 ) desi qned for the s impl i f i ed pl ant ( 1 . 5 ) , ( 1 . 6 ) behave when ap­pl i ed to the actual pl ant (1 . 2 ) to (1 . 4 ) w it h d i sturbances and/or weakl y observab 1 e unmode 1 ed dynami cs? Wi l l the properti es of Lemma 1 be preserved for smal l d i sturbances and smal l E: ? Wh i c h mod i fi cation of the adapt i ve l aw woul d hel p to preserve some of the desi rabl e pro­pert i es?

Let us fi rst cons i der the effect of d i s­turbance d ( t ) on the adapt i ve control l er ( 1 . 7 ) , ( 1 . 8 ) when appl i ed to the pl ant ( l . 2 ) to ( 1 . 4 ) wi th E: = O . The error equat i ons for ( l . 5 ) to 0 . 8) are e = -ame- ( K- K* ) ( e+ym )+d ( 1 . 9 )

K = ye ( e+ym) ( 1 . 1 0 ) where K* = c3+am . I n the d i sturbance-free case , d = 0, we can show u s i ng the Lyapunov funct ion e2 1 2 V ( e , K ) = T + 2y ( K- K* ) ( l . l l ) that the equi l i br i um e= O , K= K* i s stabl e and 1 im e ( t ) = 0 for any un i forml y bounded

t400 reference i nput s i gnal r ( t ) . I f the d i stur­bance d ( t ) i s not zero , the deri vative of ( l . l l ) sati sfi es

V ( e , K) � - I e I ( am I e I - I d I ) ( 1 . 1 2 ) Thus e ( t ) i s bounded and there exi st pos i ti ve constants c and T such that

I e ( t) J < c sup J d ( t) J , for al l t .'.'._ T ( 1 . 1 3 ) - t

However , th i s does not guarantee that K ( t ) i s bounded . For exampl e , take

-1 ( ) _ !. [ a -b ( ) -l d ( t ) = ?..r.:-:- a+t a -l +-2- a+t 4 ,_,y 5 i-(a+tf 4 ] ( 1 . 1 4 )

where a>O and b are some constants . When t h i s bounded d i sturbance , wh ich decays to zero , i s present i n the regul ati on case (y�t ) = 0, r ( t ) = 0 ) the output is sti l l regu-l ated y ( t ) = �a+t ) -� -+ O, as t400 ( l . E ) but the adapt i ve control l er i s nonrobust be-cause i

K ( t ) = (a +t ) i; +$ -+ oo as t -+ 00 ( 1 • 1 6 ) Simi l ar i nstabi l i ty phenomena can be observed for r ( t ) � 0, Ym � 0 as wel l as for the case of output d i sturbances .

I n the presence of d i sturbances as wel l as weak ly observabl e unmodel ed dynami cs , the equati ons descri b i ng the stabi l i ty properti es of the adapti ve control l er ( 1 . 7 ) , ( 1 . 8 ) are

. * e = -a e- ( K-K ) ( e+y )+E:Z +d m m i = -c2z-K ( e+ym )+r+d

K = ye ( e+y ) m

( 1 . 1 7 ) ( 1 . 1 8 ) ( 1 . 1 9 )

when both s i gnal r and d i sturbance d are con­stant and ym = r the equ i l i br i um

e = o , K = c2 ( i +c3 )r+c2d+E: ( r+d ) ( c2+E: ) r

Z = 2c2d+2E: ( r+d ) -c2c3r of ( 1 . 1 7 ) to ( 1 . 1 9 ) i s c2 ( c2+E: ) 2

. d c2 +c2+E:c2+E-E:C3 ) unstabl e 1 f - < • For l ocal r ( E:+c2 ) asymptot i c stabi l i ty , i t i s suffi c i ent that the d i sturbance-to-s i gnal rat i o and the per­turbat ion parameter E: be smal l . Because of the d i sturbances and the unmodel ed dynami cs , the adapt i ve system ( 1 . 1 7 ) to ( 1 . 1 9 ) may not converge to or may not even possess an equ i -1 i br i um for general bounded reference i nput s i gnal s . A pract i ca l goal i s then to guaran­tee some boundedness propert i es . We fi rst s how that the a-mod i fication i ntroduced i n [l , 2 , 1 5] guarantees the exi stence of a region of attraction for boundednes s . We then i ntroduced a new adapt i ve control l er wh i ch guarantees boundedness for any bounded i ni t i al cond i tions provi ded that E: is smal l . a . The a-Modi fi cat ion

The adaptive l aw ( 1 . 8 ) i s mod i fi ed to K = -ayK+yey ( 1 • 20)

where a > 0 is a des i gn parameter . W i t h ( 1 . 2 0 ) as the adapt ive l aw , t he equations de­scr i b i ng the stabi l i ty properti es of ( 1 . 2 ) to ( 1 . 4 ) wi th the contro l l er ( 1 . 7 ) are

e = -a e- ( K-K*) ( e+y )+ez+d ( 1 . 21 ) m m � = -c2z-K (e+ym) +r+d

K = -ayK+ye ( e+y ) m

( 1 . 22 ) ( 1 . 23 )

Theorem 1 : Let the reference i nput or ( t ) , state ym( t ) a n d di sturbance d ( t ) sat i sfy

Robustnes s Issues in Adaptive Control 2 1

where r1 , r2 and d1 are f i n i te pos i t i ve con­stants . Then there exi st pos i ti ve constants t1 , e:* , o , a < � , 81 to 83 , such that for a l l ! e: ! E[O ,e:*] every sol ution of ( 1 . 2 1 ) to ( 1 . 23 ) start i n q a t t = 0 frorr the set

D = { e , K , z : ! e l <8 1 ! e: ! -a . ! K l <8 2 ! e: !

-cx ,

! z l <83 ! e: l-�-a } ( l . 25 )

enters the res i dua l set a ! e ! 2 c l e: !

D0 = { e , K , z : -;- + � I K- K* ! 2+� z2

d2 ( r +d ) 2 2--1 + ! e: I 1 1 + llo ! K* ! 2} ( l . 26 ) am c2 8

at t = t1 and rema i ns i n D0 for al l t>t1 . Proof . Choos i ng the function

_ e2 { K-K*) 2 l e: I 2 V ( e , z , K ) - y � 2Y T z ( l . 27 ) we can see that for each l e: ! , c >0 , a>O the equal i ty c

0

V ( e ,z , K ) = _Q_2 ( l . 28 ) l e: ! a

defines a cl osed surface S ( e: ,a , c0 � n R3 space . The deri vati ve of V ( e , z , K ) a l onq the sol ution of ( 1 . 21 ) to ( 1 . 23 ) i s . 2 2 V ( e , z , K ) = -ame -o K ( K- K* ) - ! e: ! c2z

+e:oz+ed- ! e: ! K ( e+ym ) z+ ! e: ! (r+d ) z ( 1 . 29 ) and can b e rewri tten as

· -am 2 a 2 I e: I c2 2 V ( e , z , K ) 2_-4-e - 4( K- K* ) - -4- z 2

2 c2 le: I Ym Jtl I � I K2 1 1 2 - l e: !z [-- -- - e: - �]+o-! K* ! 2 a am am 8 2 2

+L + I e: ! ir+d ) ( l . 30 ) am c2 I n s i de S (e: ,a , c0 ) quant i t i es I e l , I K l can grow up to O ( ! e: ! -a ) , whereas ! z ! can grow up to O ( ! e: l -�-o. ) . Therefore there exi sts constants b1 ,b2 ,b3 such that

I b b b e l 2__1 , I K l 2__2 , I z ! 2__3_ ( 1 . 31 ) ! e: l a [ e: ! a ! e: ! �+a

for a l l e , K , z i ns i de S ( e: ,a , c ) . Hence ( 1 . 30 ) becomes

0 · am 2 a 2 I E I c2 2 V ( e , z , K )2_ - 4e - 4( K-K*) --4- z 1 � 1 � 2 oc -� [-2 - l e: i (/+_Q__) - ob2 ! e: ! l -2a ] a 2 m am 2

2 2 +oll[ K* ! 2 L + J e: I ir+d ) ( 1 . 32 ) 8 am c2

for a l l e , K , z i ns i de S ( e: ,a , c0 ) .

ASCSP-B

Choo s i ng a<� , i t can be shown that there exi sts constants e: 1 >0 and y1 >o such that for o > y1 J e: I ( 1 . 33 )

and each ! e: ! E[O ,e:1 J ( l . 32 ) can be wri tten as · am 2 a 2 I e: I c2 2 V ( e ,z , K )<-4e - 4( K-K* ) --4-z

2 2 �+a 18

l ! K* [ 2+ ! e: ! _l:�d ! ( 1 . 34 ) m 2

I t i s cl ear that outs i de D0 and i n s i de S ( e: ,a , c0 ) V i s stri ctl y decreas i ng . Hence there exi sts pos i t i ve constants c0 ,8 1 to o 3 and e:2 such that for each I e: I E[O ,e:*] , where e:* = mi n [e:1 ,e:2J , D i s i ns i de S ( e: ,a , c0 ) , D0 i s encl osed by D and any so­l ut i on start ing fro.m D rema i ns i n s i de S ( e: ,a ,c0 ). Si nce i ns i de D/D0 , V<O , every sol ut i on start i ng at t=O from D/D0 wi l 1 enter D0at some fi nite t ime t=t1 and rema i n i n D0 for a l l t>t1 . S i nce D0 i s uni formly bounded , the sol ut ion e ( t ) , K ( t ) , z ( t ) i s bounded for any i n i t i a l cond i t i on i n D . Remark 1 . As e:->-0 , doma i n D becomes the who 1 e space , that i s , the adapt i ve control probl em ( l . 21 ) to ( 1 . 23 ) i s wel l posed w i th respect to the unmodel ed dynami c s . Remark 2 . From ( l . 26 ) i t i s c l ear that t h e s i z e of D0 depends o n the d i sturbance d ( t ) , reference i nput s i gna 1 r ( t ) and the des i gn parameter a . Gi ven an e: , a suff i c i ent i ncrease i n r ,d and o can no l onger guarantee the property that V<O everywhere in D/ D0 . For th i s reason , our formu l ation exc l udes h i gh ampl i tude re­ference i nput s i gna l s or d i sturbances . Remark 3 . For sma l l e: , the des i gn parameter a can be sma l l and therefore i ts contri but i on to the s i ze of D0 i s smal l . However , i n the di sturbance free case and when e: = 0, a > 0 causes an output error of O(Va') . Th i s i s a trade-off between boundedness i n the presence of d i sturbances and/or unmodel ed dynami cs and the 1 oss of exact convergence of the output er­ror to zero in the absence of uncerta i nt i es .

b . New Adapt i ve Control l er I nstead of the adapti ve contra 1 1 er ( 1 . 7 ) ,

( 1 . 8 ) , we propose the contra l l er u ( t ) = -K ( t )y-K ( t )s+r ( t )

K = yes -oK t =-as+y , s ( o ) = o

( 1 . 3 5 ) ( 1 . 36 ) ( 1 . 37 )

where a>O , o>O a re des i gn parameters to be se-1 ected . The stabi l i ty properti es of ( l . 35 ) to ( l . 37 ) when appl i ed to the actual pl ant ( 1 . 2 ) to ( 1 . 4 ) are descri bed by

e = -a e- ( p+a ) (<Ps ) +e:z+d . m z = -c2z - ( p+a ) (<Ps ) - K*e+f1 ( t ) ¢ = -oy (¢+K* )+yes

( 1 . 38) ( l . 39 ) ( 1 . 40 )

where p ( · )�ddt ( · ) i s the d i fferenti al operator ,

cf>�K-K* and f1 ( t ) �r - K*ym+d .

22 P. Ioannou

Theorem 2 . There exi sts an s* > 0 such that for each s E[O ,s*] the sol ut ion e ( t ) , z ( t ) ,¢ ( t ) of ( l . 38 to ( 1 . 40 ) i s bounded for any bounded i n i ­ti a l cond i t ion . Furthermore , the sol ut ion en-ters the set 2 ame a 2 a 2 Dy = { e , z ,¢ : -4-+4(¢s ) +t( a+am )<P

c I s l 2 ( I I ) 2 2 .J..r-z2.::_ � + d+ s fl + l s l.JJ ( l . 4 1 )

m a c2 i n fi n i te t ime . Proof : Choose the pos i ti ve defi n i te funct i on -- (a+a ) 2

V ( e ,¢ , z ) = }( e+¢s ) 2+} Ym ¢ +}l s J ( z+¢s ) 2

( l . 42 ) Then , al ong the sol ution of ( 1 . 38 ) to ( 1 . 40 ) we have . 2 2 V ( e ,¢ , z ) = -ame -a ( l + l s J ) (¢s ) -a ( a+am )¢ (¢+K*)

- l s ! c2z2- J s l K*ez+sez+ed+ J s l K*¢se

+s¢sz- l s l ( a+c2 )¢sz+¢s ( d+sf1 ) +szf1 ( 1 . 43 )

Compl eting the squares we can wri te ( 1 . 43 )as · -am 2 a 2 ( a+am ) 2 c2 J s l 2 V ( e ,¢ , z ) .::_� -4(¢U -a�<P --4- z

- l s J z2 [�2 _hl ( l + I K* J ) 2- l s l ( a+c2 )2 J am 2

+ (<Ps ) 2 [� - hl -� I K* J 2 J 4 c2 a 2m 2

d2 (d+ J s J f1 ) f1 +- + I s 1 - ( l . 44 ) am a c2

:::" :ih{:[ :�:; :: ;;:::� :.h:::, 2 i ,, 1';. 1 ' '

•;2] ( l . 45 ) such that for each J s J E[O ,s*]we have

· am 2 a 2 a ( a+am ) 2 c2 2 V ( e ,¢ , z ) _::.� -4(¢s ) - 4 ¢ -4 l s l z 2 2 a ( a+am ) 2 d2 ( d+ J s J f1 ) l s l f1

3 I K* I +a a � m 2 ( 1 . 4 6 ) C l early outi sde DY , V < 0 and therefore for any bounded i ni t ia l cond i t ion , the sol ut ion e ( t ) , ¢ ( t ) , z ( t ) i s bounded and enters DY i n fi n i te t ime . Remark 4 . The new adapt ive con trol l er guaran­tees gl obal boundedness at the expense of i n­creas i ng sl i ghtly the compl exi ty of the reduced­order cont rol l er. The val ue of a in thi s case can be arb itrari l y smal l in contrast to the case a) of thi s sect i on where a > O (s ) . Remar k 5 . 1-Je note that in the absence of d i s ­turbances and unmodel e d dynam i cs the new adap­ti ve contro 1 1 er guarantees gl obal stabi l i ty and l im e ( t ) = 0 provi ded a = 0 .

t400

I I . Adapt i ve Control wi th Unmodel ed Dynamics and Di sturbances

We now cons i der the general prob 1 em of adapt i ve control of a S I SO t ime- i nvari ant pl ant of order n+m where m is the order of the weakly observabl e stabl e dynami cs and n is the order of the pl ant to be control l ed .

The pl ant i s assumed to have the fol l owing state representation

x = Ax+b1 u+sA1 2z+D1 ( 2 . l ) ( 2 . 2 )

y = ex ( 2 . 3 ) where xERn , z ER01 , F i s a stabl e matri x , s i s a smal l pos i t i ve scal ar and D1 , D2 are bounded vector d i sturbances . Such a representati on can be obtai ned from the transfer function of the pl ant in a s i mi l ar manner as in Secti on I . I n ( 2 . 3 ) we assume that the output y does not depend on d i sturbances ex pl i ci tly . \�hen out­put d i sturbances are present , we can al ways ob­ta i n the representation ( 2 . 1 ) to ( 2 . 3 ) provi ded the deri vati ve of the output d i sturbance exi sts and i s bounded , or by fi l teri ng the output u s i ng a fi rst order fi l ter [l , 1 6] .

The transfer funct ion of the s i mpl i fi ed p 1 ant obta i ned by sett i ng D1 =D2=o and s=O

l /::,, N ( s ) c ( s r -A1 )

- b1 = !�P ( s ) = KP� ( 2 . 4 ) i s assumed to sati sfy the fol l owi ng :

( i ) DP ( s ) i s a mon i c polynom ia l of degree n . ( i i ) NP ( s ) i s a mon i c Hurw i tz polynomi al of degree m .::_ n- 1 . ( i i i ) The po 1 ynomi a1 s DP ( s ) , NP ( s ) a re re 1 a­t i ve l y prime , the degree n and n� n-m

and s i gn of KP are known . The reference model i s descri bed by

xm = A X +b r , x ERn ( 2 . 5 ) m m m m

y = CTX ( 2 . 6 ) m m m whose transfer function W ( s ) i s m

T - 1 Zm ( s ) W ( s ) = C ( s I -A ) b = K � m m m m m�m\ S J ( 2 . ? ) and r ( t ) i s a uni formly bounded reference i nput s i gnal . We cons i der the s i mpl e case where n*=l and Wm ( s ) i s chosen to be stri ctly pos i t i ve real. The s i gnal generators are descri bed by the ( n-1 )th order vector d i fferent i a 1 equati ons

\)l = f\\!1 +gu ( 2 . 8 ) T wl = c ( t )'Jl ( 2 . 9 )

\) 2 = J\\)2 +gy

w2 = d0 ( t )y+dT ( t )\)2

( 2 . 1 0 ) ( 2 . l l )

Robust Redesign of Adaptive Control 23

where ti. i s an ( n- l ) x ( n- 1 ) stabl e matri x and (!i. , g ) is a control l abl e pa i r . a . The cr-Modi fi cat ion for the General Probl em

The control i nput i s g i ven by u = 8 T ( t )w ( t ) ( 2 . 1 2 )

where 8 T ( t ) � [r ( t ) , vi ( t ) ,y P ( t ) , v� ( t ) ] and 8T ( t ) � [K0 ( t ) , CT ( t ) , d0 ( t ) , dT ( t ) ] . The para­meter vector 8 ( t ) is updated us i ng the adapti ve l aw • /1 8 = -crr8 -re1 w , e1 = y-ym ( 2 . 1 3 ) It can be shown [1 7 ] that a constant vector 8* exi sts such that for 8 = 8* the transfer function of the s imp 1 i fi ed p 1 ant ( 2 . 4 ) together with the contro l l er ( 2 . 8 ) to ( 2 . 1 2 ) matches that of the reference model g i ven by ( 2 . 7 ) .

Defi n i n<:i Y = [xT ,vi ,v�]T and u s i ng 8* , the c 1 osed-1 oop sys tern becomes

Y = A Y+b [k*r+ ( 8 -8 * ) Tw]+E:A2z+D ( 2 . 1 4 ) c c 0 c ( 2 . 1 5 )

For 8 = 8* , E: = O and Dc = O , ( 2 . 1 4 ) i s a non­m in imal representat i on of the reference model

· !1 T T T T xmc = Acxmc +bcK�r , xmc = [xm ,vml ,vm2 ] ( 2 . 1 6 ) The equations for the error e �Y-x can be ex-pressed as me

e = A e+b ( 8 -8* )Tw+E:A2z+D ( 2 . 1 7 ) c c c . T z = Fz+A3e+b2 { 8 -8* ) w+f1 ( 2 . 1 8 ) e = hTe ( 2 . 1 9 ) 1 c

where he � [1 , 0 , . . . ,0]T and f1 ( t ) �

A3x +b2 K*r+D2 . The equati ons ( 2 . 1 3 ) and ( 2 . 1 7 ) me o to ( 2 . 1 9 ) descri be the stabi l i ty oroperti es of the adapt i ve contra 1 scheme in the presence of d i sturbances and unmodel ed dynam i c s . For E:=O, D =O we can show that the sol ution e ( t ) ,8 ( t ) i s bhunded for any bounded i n i ti a l condi ti on and 1 im e ( t ) = 0. When o!O , D fO the fol l owi ng t->oo c theorem g i ves suff i c i ent cond i tions for boundedness . Theorem 3 . There exi st pos i t i ve constants t1 ,E: ,cr ,a<Y, and p1 to p3 such that for each E:E[O ,E:*] every sol ution of ( 2 . 1 3 ) , ( 2 . 1 7 ) to ( 2 . 1 9 ) wh ich starts from

- . 'I 'I -a 'I 'I -a 'I 'I -Y,-a} D - { e ,8 , z . 1 e1 < p1 E , 1 8 , <p2E , , z 1 <p3E enters the res i dual set

D - { ·�I '1 2�1 *'1 2 �I '1 2 o - e , 8 ,z . 4 i e1 41 8 -8 ' +E:41 Z1 a,2 a,2

.'.:.. _l +E:_§_ 4- 1 1 8 *11 2} Al A2 2

( 2 . 20 )

( 2 . 2 1 ) at t = t1 and remai ns i n D0 for a l 1 t�t1 . I n ( 2 . 21 ) , A l ,A2 ,a2 and a5 a re pos i t i ve f i n i te constants and cr>p4E: where p4>0 .

Proof . Choose the pos i t i ve defi n i te funct ion --V(e , 8 , z ) = Y,e T P ce+Y,(8-8* )

Tr -1 ( 8-8* ) +�2z

TMz ( 2 . 22 ) where M = MT>O and Pc =P� >O , sat i sf i es the Kal ­man-Yakubov i ch l emma due to the stri ctly po­

K s i t i ve rea l ness of hT ( s I -A r1 bc =�K

P'�M ( s ) ,i . e . , c c M P cAc +A� Pc = -qq

T -v Lc , LC = L� >0 ( 2. 23 )

p b = h c c c ( 2 . 24 ) where q i s a Furthermore , t i on

vector and v>O i s a scal a r . M sati sfi es t h e Lyapunov equa-

T T FM+MF = -Q , Q = Q >O ( 2 . 25 ) The t ime deri vati ve o f V ( e , 8 , z ) a l ong the so­l ut i on of ( 2 . 1 3 ) and ( 2 . 1 7 ) to ( 2 . 1 9 ) i s

V ( e , 8 , z ) = -¥T(qqT+vLc )e-cr ( 8 -8* )T8

T T E T T +E:e PA2z+e PDc -2z Qz+E:z MA3e T T T +E:Z Mb2 ( 8 -8* ) w+E:Z Mfl ( 2 . 26 )

Cons i der the cl osed surface S ( E: , a , d0 ) de-fi ned by d

V ( e , 8 , z ) = �a ( 2 • 2 7 ) E

and note that for al l e ,z ,8 i ns i de S ( E: ,a , d0 ) , ( 2 . 26 ) can be wri tten as

-A A V ( e ,8 , z ) .'.:.. �lell 2-�l8 -8*11 2-E:-/-JJ zll 2

2 2 2 A2 ml l - 2a a3 -dl zll [-4 ---E -ql] A l A l

2 2 2 2 a4 a2 118 *11 2 a5 -ll 8 -8*ll [2:. -E -]+-=+a +E:- ( 2 . 28 ) 4 A2 Al 2 A2

where A l = �i nA ( Lc ) ,A2 = tni nA. ( M) , a1 to a5 are pos i ti ve constants obtai ned from the norm of ma­tr ices and the bounds for d i sturbances , refer­ence i nput s i gna 1 and states of the reference model, and q1 i s a pos i t i ve constant i n the bound : l 8 :l <q1 E

-a for al l 8 i ns i de S ( E: ,a , d ) . Choos i ng 2

0 4a4 a<Y, and cr> p4E where p4= �A� we can see that

2 there exi sts an E* and constants p1 to p3 such that for eac h EE[O ,E*]

V ( e ,8 , z ) .'.:.. -�Je)J 2 -�l 8-8*Ji 2-E�Jz:J 2 4 4 2 2 a2 )Je*f a5 �(J 2 +E:"-2 ( 2 . 29 )

D0 i s encl osed by D ,D is encl osed by S ( E: ,a , d0 ) and every sol ution e ( t ) , z ( t ) ,8 ( t ) wh ich starts from D rema i ns i n s i de S (E: ,a ,d0 ) . S i nce V<O everywhere i ns i de S ( E: ,a , d0 ) except poss i bly i n

24 P . Ioannou

D0 every sol ution whi c h starts from D wi l l enter D0 i n fi n i te t ime t1�o . Once i n D0 i t cannot es­cape and remai n s there for al 1 t�t1 . b . New Adapt i ve Control l er for the General Case

The control i n put i s c hosen as

where u ( t ) = e T ( t )w ( t ) +eT ( t )t; ( t ) ( 2 . 30 )

t = -�+w , s ( O ) =O ( 2 . 31 ) and a>O i's a des i gn parameter to be chosen . The parameters are updated as

e = -crre-re1 s ( 2 . 32 ) Us i nq the same procedure a s i n ( a ) o f thi s

sect ion , we can show that the equations for the error can be expressed as

e = A e+b ( p+a )¢Ts+sA2z+D ( 2 . 33 ) c c c i = Fz+A3e+b2 ( p+a )¢

Ts+f1 ( 2 . 34 )

e1 = h�e ( 2 . 35 ) where ¢� e -e * and p i s the d i fferent i a l oper-

d ator dt .

Theorem 4 . There exi sts a n s>O such that for each sE[O ,s*] and T 2 ll bcP cAc11 <x»a> 1 6 T ( 2 . 36 )

\l bcPcbc the sol ution e ( t ) , z ( t ) ,e ( t ) of ( 2 . 32 ) t o ( 2 . 35 ) i s bounded for any bounded i ni t i al cond i tion . Furthermore , there exi sts a fi n i te t ime T > 0 such that for al l t > T the sol ution e ( t ) , zTt ) , e ( t ) i s i ns i de the set D0 g i ven by

D - { . �I '1 2� ayl II T '1 2 s 'I '1 2 acr ,I '1 2 0 - e , z ,e . 41 e1 4 ¢ S i �\2 1 z, +7f. <1>1 (y2/1 Dc/l+sy3/l f1 ll l

2 EY4 acr ' I ' 1 2 < +-\- +T ' e *, } ayl 2 ( 2 . 38 )

where y1 toy4 are pos i t i ve fi n i te constants . Proof . Choose the pos i t i ve defi n i te function

( e-bc¢Ts )TPc ( e-bc¢

Ts ) V ( e ,¢ , z ) = 2

�( z-b2¢Ts ) TM (z-bc¢

Ts )�Tr-1 ¢ ( 2 . 39 ) where Pc and M are the same matri ces as i n ( 2 . 23 ) to ( 2 . 25 ) .

Al ong the sol ut ion of ( 2 . 32 ) to ( 2 . 35 ) · T T T 2 T V ( e ,¢ , z ) =-�e ( qq +vlc ) e-a ( ¢ s ) [bcPcbc

+sb;Mb2 ] -�z T Q z-acr¢ T (¢+e* )+se T P cA2z+e T P c Dc

-b�PCACe¢Ts -t:b�PcA2z¢Ts-b�PC DC¢Ts+szTMA3e

choos i ng

*- . [ay1 " 1 A l A2 a\2Y1 l s -mm 4/l b;A3!1 '

411Pc:i ( l +UA21i )'4 ( ay6+y7 )2j ( 2 . 40 )

where y1 = b�P cbc , y 6 = !1 Mb2!1 and y7 = :l b�P cA211+

:1 b;MF: l we can wri te ( 2 . 39 ) as · \1 2 ay l I T 2 s I 1 2 �I 2 V ( e ,¢ , z )_s - 4 /leii -4 ¢ s l -4\2j z,! -41 <1>1!

(y2:1 Dc11+£Y31i f1 1i l2 EY4 acr,1 *'l 2 ay + \ +Si e , ( 2 . 41 )

1 2 where y2 = 1l b�P c :l . y 3 = :1 b;M1! and y 4 = :I M:j .

Si nee D0 i s un i forml y bounded and V<O out­s i de D0 then every sol ut ion of ( 2 . 32 ) t0:2 . 35 ) with a bounded i n i t i a 1 cond i tion wi l 1 be bounded and wi l l enter D0 i n some fi n i te time t=T�O . Once i n D0 i t cannot escape but wi 1 1 rema i n there for al l t> T . Remark 6 . Theorem 4 requ i res that the des i gn parameter a has to sati sfy ( 2 . 36 ) for bounded­nes s . An overest imated 1 arge va 1 ue of a mi ght resu l t to a sma l l er s* , i . e . , to a sma l l er set of al l owabl e unmodel ed dynami cs . Remark 7 . Gi ven a fi xed s , a s i m i l ar ana l ys i s can be used to find bounds for the parameter a for boundedness .

Conc l u s i on I n th i s paper we analyzed the stabi 1 i ty

properti es of adapt i ve control schemes wi th res­pect to bounded d i sturbances and model -pl ant mi smatch caused by unmodel ed weak ly observab 1 e dynam i cs . We s howed that the adapt i ve control ­l er with the a-mod if i cation guarantees the exi s­tence of a reg i on of attracti on for boundedness . We i ntroduced a new adapt i ve contro l l er wh ich guarantees boundedness for any bounded i n i t i a l cond i tion provi ded some des i gn parameters are chosen properly . A further i nvesti gati on of thi s new contra 1 1 er and the ex tens ion of these resul ts to more general adapti ve schemes is a topi c for future researc h .

Acknowl edgement The author wou l d 1 i ke to thank Prof . Koko­

tov i c for numerous d i scuss ions . References

[l ] P . A . Ioannou and P . V . Kokotovi c , Adapt i ve Systems wi th Reduced-Model s , Spri nger­Verl ag , 1 983 .

[2 ] P . A . Ioannou and P . V . Kokotovi c , " Robust Redes i gn of Adapti ve Control , " to appear i n I EEE Trans . on Autom . Con tr . , Feb . 1 984 .

Robust Redes ign of Adaptive Control

[3] B. Egardt , "Stabi l i ty Ana lys i s of Adaptive Control Systems wi th Di sturbances , " Proc . Joi nt Automati c Control Conference , San Fran c i sco , CA, August 1 980 .

[4] B . B . Peterson and K . S . Narendra , "Bounded Error Adapt i ve Contra 1 , " I EEE Trans . on Autom . Contr . , AC-27 , December 1 982 .

[5] G . Krei sselmeier and K . S . Narendra , " Stable Model Reference Adapt i ve Control i n the Presence of Bounded Di sturbances , " I EEE Trans . on Au tom . Con tr. , AC-27 , Dec . 1 982 .

[6] C. Samson , "Stabi l i ty Ana lys i s of Adapti ­vely Control l ed Systems Subject to Bounded Di sturbances , " Automati ca , January 1 983 .

[7 ] C . E . Rohrs , L . Val avan i , M . Athans , and G. Stei n , "Anal yt i cal Veri fi cation of Un­des i rab 1 e Propert i es of Di rect Mode 1 Re­ference Adapti ve Control Al gori thms , " Proc . 20th I EEE Conference o n Dec i s i on and Control , San Di ego , CA, December 1 981 .

[8] C . R . Johnson ,Jr . and M . J . Bal as , " Reduced­Order Adapt i ve Cont ro 1 1 er Studi es , " Proc . of Joi nt Automati c Control Conference , San Franc i sco , CA , August 1 980 .

[9] B . D . O . Anderson , " Exponenti al Convergence and Pers i stent Exe i tati on , " Proc . of the 21 st I EEE Conference on Dec i s i on and Con­trol , Orl ando , FL , December 1 982 .

[1 0] B . Wi ttenmark and K . J . Astrom, " Impl emen­tation Aspects of Adapt i ve Control l ers and thei r I nfl uence on Robustness , " Proc . of the 21 st I EEE Conference on Dec i s ion and Control , Orl ando , FL , December 1 982 .

[l l ] C . E . Rohrs , L . Va 1 avan i , M . Athans , and G. Ste i n , " Robustness of Adapt i ve Control Al gori thms i n the Presence of Unmodel ed Dynami cs , " Proc . 21 st I EEE Conf . on Dec i ­s-i o n and Contro 1 , Orl ando , FL , Dec . 1 982 .

[1 2 ] G . Kre i ssel mei er , "On Adaptive State Regu­l at i on , " I EEE Trans . on Autom . Contr . , Vol . AC-27 , February 1 982 .

[1 3 ] B . Ri edl e , B . Cyr and P . V . Kokotovi c , "Sta­b i l i zation of Adaptive Systems w ith Para­s i t i cs and Di sturbances , " submi tted to the 1 983 Conference on Dec i s i on and Control .

[1 4 ] L . Pra l y , "MIMO Stochasti c Adapti ve Con ­trol : Stabi l i ty a n d Robustnes s , " Report , CAI -Ecol e des Mi nes , 35 rue Sa i nt Honor� , 77305 Fon ta i neb 1 eau , France , 1 982 .

[1 5 ] P . A . Ioannou , " Des i gn of Decentra l i zed Adapti ve Schemes , " Chapter i n the book Advances i n Large Seal e Systems-Theory and Appl i cations, Ed . by J . B . Cruz , 1 983 .

[1 6 ] P . A . I oannou and P . V . Kokotovi c , " Decen­tral i zed Adapti ve Control i n the Presence of Mul ti parameter S i ngul ar Perturbations and Bounded Di sturbances , " Ameri can Con ­trol Conference , San Franc i sco , C A , June 1 983 .

[1 7 ] K . S . Narendra , L . S . Val avan i , "Stabl e Adapt i ve Control l er Des i gn - Di rect Con ­trol , " I EEE Tra n s . o n Autom . Contr . , Vol . AC-23 , No . 4 , August 1 978 .

25

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

MODEL REFERENCE ADAPTIVE CONTROL OF MECHANICAL MANIPULATORS

M. Tomizuka and R. Horowitz

Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA

Abstract . The dynami c equati ons of mech an i ca l man i pu l ators are h i g h l y non l i near and comp l ex . Furthermore, the i nerti a charac­teri sti cs of the man i pu l ator depend on the payl oad wh i ch is often unknown . Because of these reasons , there i s an i ncreas i ng i n terest in adapt i ve contro l of mechan i c a l man i p u l ators . The obj ecti ve of thi s paper i s to present how recent model reference adapt i ve control ( MRAC ) schemes can be used for the dynami c contro l of mechani ca l man i pu l k ators . The advantages of u s i ng the MRAC schemes are demon­strated by s imu l ati on resu l ts for a three degrees of freedom mechan­i ca l man i p u l ator and experimental res u l ts for a one d i mensi onal pos i t i on contro l prob l em . Keyword s . Adapt i ve contro l , robots , mechan i ca l man i p u l ator , posi t i on contro l , d i g i ta l contro l . INTRODUCTION

Model reference adapti ve control ( MRAC ) has been recei v i ng i ncreas i ng attent i on during the l ast few years as a va l u ab l e method for dynami c control of mechan i c a l mani pu l ators . One of the major reasons for adapti ve control l i es i n the dynamics of mechani ca l man i pu l a­tors , which are h igh ly non l i near and charac­teri zed by time varyi ng/unknown parameters . Early work s on adapt i ve control of mechan i ca l man i pu l ators uti l i zed the conti nuous time MRAC techni ques ( Dubowsky and DesForges ( 1979 ) , Horow i tz and Tomi zuka ( 1980 ) , and Takegaki and Arimoto ( 1981 ) ) . Al though experimental eva l u at i ons of the MRAC schemes have not been reported , the resu l ts of com­puter s imu l ati on stud i es suggest a number of attracti ve features . The use of microcom­puters i s an economi ca l and rel i ab l e way to obta i n h i g h performance man i pu l ator control systems wi th and wi thout adapt i ve features . Thi s mot i vated research works on the d i s­crete time adapti ve control schemes for mechani ca l man i p u l ators ( Dubowsky ( 1981 ) , Koi vo and Guo ( 1981 ) and Horowi tz and Tomi zuka ( 1982 ) ) . The di screte time schemes can be obtai ned ei ther as approxi mations of conti �uous t im� MRAC schemes or by uti l i z i ng the d 1 screte t 1me MRAC theory . Si nce the M�AC syste� i s non l i near, the stab i l i ty of d � screte t 1me . schemes obtai ned as approxima­t 1 ons of cont 1 nuou s t i me schemes must be car�fu l l y exami �ed . M icrocomputer imp l emen­tat1 on of adapt1 ve schemes i s more d i rect i f they are devel oped based o n the d i screte time MRAC theory . I n thi s paper, we wi l l d i scuss the u s e of a conti nuou� t ime MRAC scheme for non l i near i ty compensat 1 on and decoup l i ng jo i nt i nteract i on and the use of a d i screte t ime MRAC scheme

27

based on the i ndependent track i ng and regu l a­t i on a l gori thm ( Lozano and L andau ( 1981 ) ) .

MAN I PULATOR MODEL In th i s paper, we wi l l consi der a three degree of freedom mechan i c a l man i p u l ator wi th three revol ute jo ints . A schemati c drawi ng of the mani pu l ator i s shown in F i g . 1 . The equat i ons for the moti on may be wri tten as !1<�p l ·iv < t ) + �(�p · �v ) + _g_(�) = _g_( t) ( l ) where ;{, = [ xp l xp2 xp3 J i s the angu l ar rotati on vector, xvT= [ xvl xv2 xv3 J i s the angu l ar vel oci ty vector, qT = [ql q2 q3 ] i s the torque i nput vector, l'l( xp ) i s a 3x3 i nerti a matr i x , �(�p · �Y )

-is a non l i near term to represent Cori o i s and centri fuga l torques , and _g_(�p ) represents the torques due to the gravi ty. I t i s known that the i nerti a matri x M i s sym­metri c and pos i t i ve defi n i te and that �(!iJ , �V ) i s i n the fol l owi ng form ( Horowi tz and Tom1 zuka ( 1980 ) ) .

�T (�p · �v ) = [ v 1 <�p • �y ) • v2 (�, �v l • v3 (�p · �v l J

and m i j ' s and n�j ' s depend on xp2 and xp3 •

28 M. Tomizuka and R . Horowitz

APPL ICATION OF CONTI NUOUS TIME MRAC I n th i s secti on , an app l i cati on of the con­t i nuous t ime MRAC techn i que to the mechani ca l mani pu l ator is presented . The objecti ve of MRAC i s to accomp l i sh nonl i neari ty compen- · sati on and decoupl i ng . The gravi ty term .9_(� ) i n Eq . ( 1 ) i s i g nored i n the devel op­ment of the MRAC ; the grav i tati onal effect wi l l be studi ed in computer s imu l at i on . Determi n i sti c Non l i neari ty Compensati on and Decoupl i ng Control I f !i( Xp ) and ..Y.<�p · �v) are known , the use of the torque i nput determi ned by

_g( t ) = !i<�p ).!!( t ) + ..Y.(�p · �v ) i n Eq. ( 1 ) resu l ts i n

ip < t ) = �v ( t ) ' iv < t ) = .!!( t )

( 3 )

( 4 ) I t shou ld b e noted that .9_(�P. ) i n E q . ( 1 ) has been set to zero and that u ( t ) i s the new externa l contro l l i ng i nput-:-Equat i on ( 4 ) represents three decoup l ed doub l e i ntegrators . Note that the second term i n Eq. ( 3 ) i s for cancel l i ng the non­l i near term �(�P. ·�v l in the man i pu l ator equati on ( 1 ) an� that the f i rst term decoupl es the i nteracti on among the three jo i nts . Adapti ve Non l i neari ty Compensati on and Decoupl i ng Control Imp l ementati on of the control l aw ( 3 ) re­qu i res that the val ues of Ji(�Q ) and ..Y.(�p ·�v ) be ei ther computed or stored for a l l �P and xv , wh i ch i s a demand i ng task for computers and becomes not pract i ca l when the payl oad i s unknown . To avo i d th i s d i ff i c u l ty, the adap­ti ve scheme summari zed i n F i g . 2 has been devel oped by wri t i ng the torque i nput as

g_(t ) = M ( t ) u (t ) + v ( t , x ) - F [x (t ) -- - --v --p '"il xPM ( t ) ] - Fv [xv (t ) - xvM ( t ) ] ( 5 )

v ( t '--vx ) T = [--vx T N 1 ( t ) x , x T N2 ( t ) x , - - --v --v -

--v

� .fl.3 ( t ) xv] ( 6 ) where the l ast two terms i n E q . ( 5 ) are for guaranteei ng the stabi l i ty of the adapti ve scheme ( see Horowi tz and Tomi zuka ( 1980 ) for the detai l s ) . Overa l l Man i pu l ator Control System One way to comp l ete the man i pu l ator control system i s to add the state vector feedback contro l l er wi th an i ntegral act i on as dep i ct­ed in F i g 3 . The feedback control g a i n s , .!S>• K v and K r can be determi ned by pol e as s i g n­ment or-l i near quadrati c synthes i s techn i que assumi ng that the man i pu l ator wi th MRAC be­haves l i ke three decoupl ed doubl e i ntegra­tors . By doi ng so , a_ stab l e overal l _system

can be obtai ned . Computer S imu l ati on F i gu re 4 shows the responses of the man i pu l a­tor control system of F i g . 3 for a step reference i nput vector, rpT = [ l 3 2 J rad and for three di fferent payl oad s . For the detai l s of s i mu l at i on and assumed va lues , see Hurowi tz and Tomi zuka ( 1980 ) . Three response curves overl ayed are i nd i sti ngu i sh­ab l e . Moreover , these responses are es sen­t i a l l y the same as the one of the i deal system wh i ch refers to three decoup l ed doub l e i ntegrators wi th the same feedback contro l l er i n F i g . 3 . These imp ly that the MRAC non l i neari ty compensator/decoupl i ng control l er i s ach i ev i ng the objecti ve. Wi thout the MRAC , the system performance i s h i g h ly sensi t i ve to vari ati ons of payl oad and mani pu l ator conf i g urati on ( F i g . 5 ) . F i gure 6 shows that the gravi ty wh i ch was neg l ected in the desi gn of MRAC does not strong ly affect the system performance.

APPL ICAT ION OF D I SCRETE TIME MRAC The use of mi crocomputers i s an economi cal and rel i ab l e way to obtai n h i gh performance man i pu l ator control systems . The conti nuous t ime MRAC in the previ ous sect i on shows that the performance of the system can be drasti ­cal l y i mproved by a s i mp l e MRAC l oop. How­ever, i t i s not necessari ly the best scheme for mi crocomputer impl ementat i on . I n thi s sect ion , we wi l l expl ore the pos s i b i l i ty of applyi ng di screte ti me MRAC al gori thms to mani pu l ators . D i screte T ime Man ipu l ator Model To app ly the di screte ti me MRAC al gori thms , we need a di screte ti me model of the mani pu­l ator . Assumi ng that changes of !1< xp ) , ..Y.(�, �� ) and .9_( Xp ) are s l ow rel ati ve to the speed of adaptati on , we wi l l wri te a model treati ng !i• ..Y. and .9. to be time i nvari ant. Then denoti ng the samp l i ng t ime by T, a di screte � ime mode l i s �p ( k+ l ) = �p ( k ) + T�v ( k ) + 0 . 5T2!1-1 [_g( k ) + .s!_] ( 7 )

�v ( k+ l ) = �v ( k ) + T!i- l [_g( k ) + .s!_J ( 8 ) where k represents the k-th sampl i ng t ime . Note that the di sturb ance term d may repre­sent the non l i near term ..Y. and gravi ty term .9. as wel l as other d i sturb ance torques such as one due to fri cti on . In order to account for addi t i ona l one step t ime del ay due to the computat i on of the torque i nput, we wri te _g( k ) = .!!( k- 1 ) ( 9 ) and treat .!!( k ) a s the control l i ng i nput. We wi l l devel op an adapt ive control scheme for a model descri bed by Eqs . ( 7 ) - ( 9 ) . M and d are treated as unknown quant i t i e s . Since

Adaptive Control of Mechanical Manipulators 2 9

these unknown quanti ti es both appear i n the vel oci ty equ ati on ( 8 ) , we wi l l i gnore Eq. ( 7 ) for the purpose of desi gn i ng an MRAC l oop : i . e. we wi l l f i rs t devel op MRAC l oop as an ( i nner) vel oci ty control l oop and then add an ( outer) pos i t i on control l oop . The MRAC l oop for the veloci ty i s desi gned based on the i n­dependent track i ng and regu l ati on al gori thm. I ndependent Track i ng and Regu l at i on A lgori thm ( known parameters ) From Equat ions (8 ) and ( 9 ) , the pl ant i s descri bed by ( l - q- 1 ,�v ( k ) = q-2 [.[�( k ) + i' J ( 1 0 )

where .[0 = �-1T , i ' = �-lTi and q-1 denotes the backward sh i ft operator . The contro l l i ng i nput �( k ) shou l d be such that D ( q-l ) [�v ( k+2 ) - �vM( k+2 ) J = Q ( 1 1 )

where the no d i mens i ona l po lynomi a l D ( q-1 ) defi nes the reg u l at i on dynami c s and i s asymp­toti ca l ly stab l e , and XvM i s the output of the model g i ven by -

AM( q- l )�vM( k ) = q-2sM ( q- l )�( k ) ( 12 ) Equ ati on ( 12 ) �ef i nes the trac k i ng dynami c s . A l though AM( q- ) and BM( q- ) can b e matri ces , they are g i ven as scal ar polynomi a l s .

The desi red contro l l i ng i s �( k ) -s�( k- 1 ) + .[0-l [D ( q- l l�vMC k+2 )

R ( q-l ) x ( k ) - d* ] ( 1 3 ) _v -where d* = S ( q-l )i' , and S ( q- 1 ) and R ( q- 1 ) sati sfy D ( q-1 ) ( l - q-l ) S ( q- 1 ) + q-2R ( q- 1 ( 14 ) S ( q- 1 ) l + s 1q- l ( - 1 < s1 < 1 ) ( 1 5 ) R ( q-1 ) ro + rlq-1 + • . . + rnRq-nR, nR = max ( O , no - 2 ) ( 16 )

F i gure 7 shows the rel at i ons among vari ous s i gna l s . MRAC A lgori thm Equ ati on ( 13 ) i nc l udes1 two unkoown quanti ­t i es , �o and i* · D ( q- ) , S ( q- 1 ) and R ( q-1 ) may be treated as known quanti t i es s i nce they can be obtai ned wi thout reference to _[0 and i* · Under these cond i t i on s , the control l i ng i nput i s

-R ( q-l )�v ( k ) - �* ( k ) J ( 1 7 ) where io< k l and �* ( k ) are the estimates of .[a and i* • respecti vel y . The esti mates are updated so that

ASCSP-B*

l i m D ( q- l ) [�v ( k+2 ) - �vM< k+2 ) J = 0 k-><><> ( 18 )

From Equati ons ( 14 ) and ( 17 ) , �f ( k } ( = D ( q- 1 ) [�v ( k ) - �vM( k ) J ) becomes

�f ( k ) [.!'._ - f( k-2 ) JT!( k-2 ) ( 1 9 ) where

e = [.[o QciJ !T ( k-2 ) = [�* ( k-2 ) a] , �* ( k-2 ) =

S ( q- l )�( k-2 ) , i* = bd · a

( 20 )

( 2 1 )

Def i n i ng the auxi l i ary error, �a ( k ) , mented error, �* ( k ) as and aug-

�a ( k } = [f( k- 2 ) -f( k ) ]Tj'.( k -2 ) and �* ( k ) = �f ( k } +�a ( k } = [f_-f( k ) ]Tj'.( k-2 ) (22 ) the adaptati on al gori thm to assure l i m �* ( k )

= 0 i s k-><><> f( k ) = f( k-1 ) + f_( k- l l!.P.( k -2 )�* T ( k ) (23 ) f-l ( k ) = Al ( k )I_-l ( k- 1 ) + A2 ( k )!( k-2 )!T ( k-2 )

( 24 ) ( 2 5 )

I t can be shown that l i m e* ( k ) = 0 i n fact k-- -

i mp l i es l i m �f ( k ) = Q ( see Landau and Rozano k-><><>

( 1981 ) for the detai l s ) . Overa l l Manipu l ator Control System

The MRAC al gori thm presented above does not depend on AM and BM expl i c i t l y . We take thi s advantage to fi n i sh the des i g n of overal l control system. If the MRAC l oop respond s fast enough , the dynami cs from XvMC k+2 } to �v ( k ) are represented by q-2 ( see F i g . 7 ) . From Eqs . ( 7 ) and ( 8 ) we obtai n �( k ) = { [0 . 5T( l +q- l ) J/ ( l -q-l ) } x�( k ) ( 26 ) Therefore, wri ti ng 0 . 5T�vMC k+2 ) as �' ( k ) and treat i ng � ' ( k ) as the s i gna l that the posi -ti on feedback contro l l er changes , u ' ( k ) and �p( k ) under perfect adaptat i on are-rel ated by �( k ) = q-2 [Bp ( q-1 ) /Ap ( q-l ) J�' ( k )

q- 2 [ ( l+q-1 ) / ( l -q-l ) J�' ( k ) ( 27 ) App ly i ng the i ndependent track i ng and regu l a­t i on des i g n once agai n , the i nput s i gna l �' ( k ) shou l d be such that Dp ( q- l ) [�p ( k+2 ) - �pM ( k+2 ) ] = Q ( 28 )

where D p ( q- l ) i s an asymptot i ca l l y stab l e P? lynom1 a l and �M i s the output of the pos i ­t 1 on reference mode l , AMp ( q-l )�pM ( k ) = q-2Bp ( q-l ) [AMp ( l ) / 2 J rp ( k ) - ( 29 )

30 M. Tomizuka and R. Horowitz

Not i ce that BMp ( q-1 ) i s set equa l to _!lp ( q-1 ) so that the contro l l er wi l l not cancel the p l ant zero at - 1 . A sca l i ng factor [AMp( l ) /2 ] makes the model stat i c g a i n un i ty.

�' ( k ) to sati sfy Eq . ( 28 ) i s g i ven by �' ( k )

where

�*pM( k+2 ) = [ l/AMp ( q-l ) ][AMp ( l ) /2 J�p ( k ) ( 3 1 ) and Sp ( q-1 ) and Rp ( q-1 ) sati sfy Dp( q- 1 ) Sp( q-l ) Ap ( q- 1 ) + q-2Rp ( q- l ) B p ( q-l ) Sp ( q-1 ) 1 1 2 ( 32 )

+ s1q- + s2q- ( 3 3 ) ( 34 )

Noti ce that Sp ( q-1 ) pl ays the ro l e of AM ( q- 1 ) i n Eq . ( 12 ) and that the second term on the ri g ht hand s i de of Eq. ( 30 ) correspond s to the r ight hand s i de of Eq . ( 12 ) . The overa l l structure of the man i pu l ator control system i s i l l ustrated i n F i g . 8 . Computer S imu l at i on Becau se of the structure of the man i p u l ator , 1-2 ( 2- 1 ) and 1 -3 ( 3- 1 ) el ements of the man i ­pu l ator i nerti a matr i x M ( xp ) i n Eq . ( 1 ) are sma l l compared to other-elements . 1-1 e l e­ment of !'.!(�p ) i s strong l y affected by x2 and x3 . Based on these observati on s , the MRAC scheme i n the prev i ous sect i on was s i mp l i f i ed by treati ng the fi rst joi nt i nde­pendently from the other two jo i nts . Th i s decompos i t i on reduces the computati onal com­p l ex i ty of the MRAC al gori thm. I nstead of u s i ng Eq. ( 24 ) , the adaptat i on ga i n was updated by

f( k ) � ( k ) [l_ - {_!( k-2 )�T ( k-2 ) } / { o ( k ) + �T ( k-2 )�( k-2 ) } ] ( 3 5 )

where � ( k ) > 0 , o ( k ) > O and � ( k- 1 ) > o ( k ) /4 . Thi s adaptati on ga i n requ i res l es s computa­t i on than the one g i ven by Eq. ( 24 ) . Si nce i t does not make f( k ) dependent upon al l the past data, it i s su i ted for the cases i n whi ch the pl ant parameters vary al l the t ime . The contro l l er samp l i ng t ime i n the s i mu l a­t i on was 0 . 01 sec . For numer i c a l val ues assumed in s i mu l at ion , see Horowi tz and Tomi zuka ( 1982 ) . The mani pu l ator dynami cs were s i mu l ated by numeri ca l ly i ntegrati ng Eq . ( 1 ) u s i ng a 4th order Runge Kutta method . The actuator mechani sm was assumed to be composed of a fast response servo motor and a speed re­ducer . Quant i zati on of the compu ted torque corrvnand and fi n i te reso l uti ons of encorder and vel oci ty counter were taken i nto account.

The f i rs t set of s i mu l ati on resu l t ( F i g . 9 ) i s for the two i nput , two output di screte t i me MRAC l oop around the second and thi rd jo i nts . The man i pu l ator was assumed i n i -ti a l l y a t rest a� the pos i t i on [ Xpl Xp2 x 3J = [0 0 O J . The i nput to the second and t� i rd jo i nt pos i t i on reference model s were both a step functi on of magn i tude 0 . l rad . The fi rst j o i nt remai ned at zero . The i n i t i al cond i t i on for the matri x �0 ( k ) i n Eq . ( 17 ) was set to �he actual value of �9 when the payl oad _ mD � s 0 . 5 �g . and the man i p u l ator i s at the i n i t i a l pos i ti on . The i n i t i a l val ue of the vector Qd ( k ) , the esti mate of bd i n Eq . ( 20 ) , was zero . Therefore, the torque due �o grav i ty i s not compensated i n i t i a l l y , and i t acts a s a step d i sturbance to the mani pu l ator . The responses of the th i rd jo i nt for three di fferent payl oad s , mp = 0 , 0 . 5 and 1 . 364 kg , a l l converge to the response of the reference model . The i n i ­t i a l l ag ob served i n the response of the man i pu l ator under a l arge payl oad cond i t i on i s attributed to the gravi tati onal forces the coupl i ng of the second and th i rd joi nts and the saturat i on of the i nput torques . F ig u re 10 shows the respo�ses of the mani pu­l �tor when th� �a� ues of �0 ( k ) and _&i ( k ) are f� xed to the � n i t i a l val ues used in the pre­vi ous case ( F i g . 9 ) . The response wi thout adaptat i on i s h i g h l y sens i t i ve to the pay­l oad changes and may become even unstab l e . F i g ure 11 shows the response of the si ng l e­i nput , si ng l e-output MRAC l oop around the fi rst joi nt. I n these s i mu l at i on s , Xp2 and Xp3 were forced to change al ong prescri bed trajectori es . I f the contro l l er parameters are fi xed , a parameter set adj usted for one conf i gurat i on becomes unacceptab l e at other conf i gurati ons . F i g ure 12 shows the response Xpl when the control parameters tuned for Xp2 = xp3 = 0 rad are not adapted .

Prel i mi nary Experi mental Eva l u at i on An i n i t i a l experi mental study of the di screte t i me MRAC scheme was conducted for the experi mental set- up dep i cted i n F i g . 1 3 . A fast response d . c . servo motor i s f i tted wi th a f lywheel g i v�ng a to�al i nerti a l l o ad of about 0 . 39xlo- Nm- sec • The motor ( I nerti al Motors Corp . , 04-037 ) i s rated at 0 . 283 Nm nomi na l torque. Vel oc i ty i s measured wi th a tachometer and pos i t i on i s at present der ived from the vel oc i ty meas urement . The torque command has a quanti zat i on l evel of 0 . 14xlo-3 Nm and the computer sampl es the tachometer output wi th a quanti zed resol uti on of 0 . 052 rad/s . The rea l -ti me adapti ve control al gori thm was wri tten i n f l oat i ng -poi nt RT-1 1 Fortran- IV and was i nterrupt dri ven. A number of modi ­f i c ati ons to the control al gori thm were made. They i nc l uded an i ntroducti on of a dry fri c­ki on model , setti ng a l ower bound Bmi n for B ( k ) and real ti me adj ustments of � and o . The samp l i ng t i me was 30 ms . I n the experi ­ments , wrong i n i t i al esti mates were g i ven for � and a, and the trans i ent response was recorded . Sampl e resu l ts are g i ven i n F i g .

Adapt ive Control o f Mechanical Manipulators 3 1

14 . The i ni t i al val ues for � and a g i ven i n F i g . 14 make the c l osed l oop system unstab l e wi thout the MRAC .

CONCLUSIONS Appl i cati ons of conti nuous t ime and d i screte t ime MRAC techni ques to mechani ca l man i pu l a­tors were di scussed . S imu l ati on studi es i n­di c ated defi ni te advantages of the MRAC approaches . The adapti ve contro l l er can make the response of the man i p u l ator i nsens i t i ve to payl oad and confi g urat i on changes . Resu l t s of prel i mi nary experi mental eva­l uati on of the d i screte MRAC scheme are encourag i ng .

ACKNOWLEDGEMENT The graduate study of R . Horowi tz i s sup­ported by a fel l owsh i p from the CON I C I T , Venezuel a . The authors thank Mr . C . H . Tham for conducti ng the experi mental study, and C armen Marsha l l for preparati on of the manuscri pt .

REFERENCES Dubowsky, S . and Desforges , D. T. , ( 1979 ) ,

of Model Reference

Fi g . 1 Three degree of freedom mani pu l ator

]!(t )

Adapt i v e Control to Roboti c Mani pu l ators , ASME J . o f Dlnami c Systems , Meas . and C ont . , Vo l . Ol , No . 3 , 1 93-200 .

Dubowsky, S . ( 1981 ) , On the Adapt i v e Contro l of Roboti c Mani pu l ators : The D i screte T ime Case , Proc . of the 1981 JACC , Vol . 1 , TA-28 .

Horowi tz , R . and Tomi zuka , M. ( 1980 ) , An Adapti ve Control Scheme for Mechani ca l Man i p u l ators - C ompensation of Non l i near­i ty and Decoup l i ng Contro l , ASME P aper 80-WA/DSC-6 .

Horowi tz , R . and Tomi zuka , M. ( 1982 ) , Di s­crete Time Model Reference Adapti ve Con­trol of Mechan i ca l Man i p u l ator, Computers i n Engi neeri ng 1982 , Vol . 2 , Robot and Roboti c s , ASME , NY , 107-112 .

Ko ivo , A . J . and Guo , T . H . ( 1981 ) , Control of Roboti c Man i pu l ator wi th Adapt i ve Contro l l er , P roc . of the 20th I EE E Conf . on Dec i s i on and Contro l , 271-27 6 .

Landa u , I . D . and Lozan o , R . ( 1 981 ) , Un i fi ca­tion and E va l uation of D i s c rete Time Expl i c i t Model Reference Adapt i ve Desi gn , Automat i c a , Vol . 1 7 , No . 4 , 593-61 l .

Takega k i , M . and Arimo to , S . ( 1 981 ) , An Ada p­t i ve Trajectory Control o f Man i pul ators , I n t . J . of Cont . , Vo l . 34 , �10 . 2 , 21 9-230 .

Fi g . 3 Mechani ca l man i p u l a tor control system

[�] c ] -

�...., [�(t � + �(tlJ 4M

[� fv] = [op!_ 0vl] ' [fp fy] = [pp!_ Pyl] ;

PARAMETER ADAPTATION ALGORITHM [�M(t � 4M(t � L----------..i d�/dt = 4M ' d4M/dt s ]!(t ) I--.&.---..,;;..,.-�

! "' ""J. g i i

Fi g .

REFERENCE rilODEL Fi g . 2 MRAC scheme for non l i near i ty compensati on and decoupl i ng

3 2) rad 10

xp2

xp3

xpl

0.5 TIME [sec}

4 Responses wi th MRAC

Payload = 5 ko --- Payload = 10 kg D a..r:; __ .....1.. ____ ....,,_ __ __. ____ � o.o 0.5 1.0 TIME [sec]

Fi g . 5 Responses wi thout t1RAC

�· ... ! ., .....

•• ,...;, •p3(t) •p3(t) "- - --:-:...--�

-- Under no i nfl uence of gravi ty � .... I -•- Under i nfl uence of riravi ty

':-... . .. �

.... ... .

• •.2 •. 4 •.e I.I I .I 1 .2 1 . 4 TINE [sec]

Fi g . 6 E ffect o f gravi ty

MRAC

!!.' ( k)

Fi g . 7

q-2/D(q-1 ) ---� [q-2

BM(q-1 ) ] /[AM( q-1 ) ] __ __.. .....

I ndependent l i near tracki ng and regul ati on des i gn

+

O . l S.-----------------. '----------! R

p(q-1 )i-----------------'

0 . 1 5..-----------------... llj,M3

0 . 1 0 ....., ... f ..... ..,0 .05

".. ... ... . . . mP" 0 .0 Kg -5>• 0 .5 Kg ---mp• 1 . 36 Kg

-0 . 05,___._ _ _,___...__,__.i-_.__....__..__ ...... __,

.....,1 0 N I ....

'l<.Q.

o . o 0 .5 1 . 0 TIME [sec] Fi g . 9 Responses wi th MRAC

....., 1 .., "' ... ..... ..., ? ;_..

. 5 ""o..

·St::==:i::=�--L---L.,--�0 o.o 1 . 0 2 . 0 3 . 0 3 . 8 TIME [sec ]

Fi g . 11 Response wi th MRAC

..... 'J . 1 0 .., • ... ..... .... O . tl5 a. ..

.• ,., f· 1·. .� l)itt3 i-.. i � . . .. .. . � ... ·. ' : r:. t·� l\ _.::. /\ r '• . / · , , - - - ·---: ,' -- ----- - - ---' I I I I I I ...... \ ,' ' , , _,

f ... . . . mp• 0 . 0 Kg x -- m • 0 . 5 Kg p 3 • • • • �· 1 . 36 Kg

-0.05 ._ __ ........ __._ ........ _.L....__.. ___ ....__._......,.1 0 . 0 0 . 5 1 . 0

TIME [sec}

Fi g . 10 Res ponses wi thout MRAC

,.....,1 0 ':' ... .....,

1 ... . .. .. .....

. ;_, . 5 ?

-St:=:=:=:::::=:::....�"-���i.-�....J1 o . o 1 . 0 2 . 0 3 . 0

TIME [sec)

Fi g . 12 Response wi thout MRAC

Fi g . 8 O ve ra l l s tructure of mani pu l a tor control sys tem

S IGNAL CONDIT ION ING

S I GNAL CONDIT ION ING

POWE R AMPL I FI E R

TACHOMETER

Fi g . 1 3 Experimental set-up

7 . 8-----------.. 601�----------..

� ;; 5.2 0 .. v

� 2 . .. � -- REFERENCE POSITION

- IWJTOR POSITION

� E ...

0 l:J I I);

---· COll1AND VELOC l1Y - IWJTOR VELOCITY

11·01eL.---s�--:'a---=11�-�1 2 • 1 00 3 6 Tll'IE (SEC)

II Tll£ (SEcl

F i g . 14 Pos i ti on and vel oci ty responses of motor l/B ( O ) = 10/B , d ( O ) = lOd

12

::c: 0 t1 � ..... rt N

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROBUSTNESS OF ADAPTIVE CONTROL ALGORITHMS 1

DISTURBANCE CANCELLATION AND DRIFT OF ADAPTIVE GAINS*

B. Riedle and P. V. Kokotovic

Coordinated Science Laboratory, University of Illinois, 1 101 W. Springfield Ave., Urbana, IL 61801, USA

Abstract . When possible the adaptive laws may adjust the controller gains to create zeros which cancel the disturbance poles . If not , controller gains will drift to in­finity in order to create a feedback loop with infinite gain . 1 . Introduction . In adaptive schemes [ l ] controller gains tend t o drift t o infinity when disturbances are present [ 2-5] . In this note we show that this tendency is due to the requirement that the output error e be exactly regulated to zero . For all but very special disturbances z the only means to achieve the exact regulation is to create a feedback loop with infinite gain . As an illustration consider

e ( s) 1 1-F ( s)W ( s) z (s) ( 1 . 1 )

where W(s ) and F (s) are the transfer func­tions of the plant and the controller , respectively (Fig . 1 ) . Suppose first that the di.s turbance is sinusoidal ,

2 w z z (s) = -2 --2 ( 1 . 2) s + w z

and that the sinusoidal steady state of e (s) is to be regulated to zero . This can be accomplished by placing a pair of ?Oles of F (s) at ±jwz , thus making the loop gain in­finite at the frequency Wz · The disturbance is cancelled because the transfer function e (

(s)) possesses a pair of zeros at ±jw We z s z

illustrate by two examples in Section 3 that disturbances which can be cancelled by zeros do not cause the parameters of F (s) to drift to infinity . We first prove in Sec­tion 2 that for more general disturbances parameters of F (s) must drift to infinity . Thus the adaptive laws adjust the controller parameters mimicking a designer who must satisfy the exact regulation criterion, but has insufficient freedom in the structure of F (s) .

* This work was supported in part by the Joint Services Electronics Program under Contract N000 14-79-C-0424 ; in part by the U . S . Air Force under Grant AFOSR 78-3633 ; and in part by the U . S . Department of Energy, Electric Energy Systems Division , under Contract DE-AC01-8 1RA50658 , with Dynamic Systems , P .O . Box 423 , Urbana , IL 6 180 1 .

33

2 . Drift of Controller Gains . With a dis­turbance z , constant reference input r , and model output Ym = gmr , the system considered is

y . 1 v • 2 v

e

T A x + b e w , p p hTx + z

1 T Av + be w , 2 Av + by ,

1 n-1 v E: It

v2€ Rn-1

r = rT > o

( 2 . la) ( 2 . lb)

( 2 . le) ( 2 . ld) ( 2 . le)

T T T 2n T lT where e = [k1 ,k2 , c ,d ] € R , w = [ r , y , v , v2T ] and (A ,b) is a companion form with de gain gl . After the model transient .has settled the only information about it is' its de gain 8m · Let g �e the c:lc gain of the plant from its input e w to its output y. Without dis­turbance , the equilibrium m�nifold M of this system is defined by x = -Ap 1bpYm/g , v1 =

1 1 2 2 glym/g , v2 = • · • = vn-1 = O , vl = glym ' v2 = • • •

v2 = 0 and n-1 1 gl 1 - k1 + k2 + - c 1 + g1d1 - " ( 2 . 2) gm g ,.,

Controller gains , which are constrained only by ( 2 . 2) can drift when z is not zero . To see this consider the functional

- 1 p = [ gm , - 1 , 0 , 0 ] r e ( 2 . 3) whose derivative along the traj ectories of ( 2 . 1 ) is

- 1 · 2 fl = [ g , - 1 , 0 , o J r e = (y-y ) m m 2 e .

Thus any output error e will force p to increase .

( 2 . 4)

Proposition : If the system ( 2 . 1 ) achieves output regulation e (t) = O as t + oo in the presence of a persistent disturbance z (t) and the controller gains remain finite , then these gains tend to constant values such that zeros of the transfer function e (s) /z (s) cancel the poles of z (s ) . If this cancella­tion is impossible , then at least one of the controller gains must drift to infinity . Proof : Since e (t) = 0 implies e = 0 the first part of the proposition follows from the transfer function analysis . To establish the second part , we note from (2 . 3) that

" e tt > " [ g , - 1 , o , 0 1 r- 1 tt - 1p ( 2 . 5) m and since e ( t) # 0 implies p + oo it follows that 8 + 00 •

34 B . Riedle and P . V . Kokotovic

3 . Simulation . For a first order plant the transfer function e (s) / z (s) cannot have ze­ros ±jwz . Therefore a sinusoidal distur­bance must produce the drift to infinity , as shown in Fig . 2 . This is not so for a third order plant whose controller is capable of producing this pair of zeros . In this case all the gains tend to finite constant values . One of them is shown in Fig . 3 . However , if the disturbance has two frequencies , then controller gains for a 3rd order plant must drift to infinity , because the controller order is not sufficient to produce the two pairs of zeros required to cancel the dis­turbance . That this drift indeed occurs is shown in Fig . 4 for the same gain as in Fig . 3 .

The equations o f the first order plant example are x = x + k2y + k1 r , y = x + z ,

k = -Sy (y-y ) 1 m k = -Sr (y-y ) 2 m

( 3 . 1 ) with r = 2 , z = . S sin 3t , x (O) = 0 , k 1 (0) = 1 , k2 (0) = -2 . The transfer functions of the third order plant ( 2 . la ,b) and the model are

2 2 W ( s ) = 3

s +is+l 2 ' Wm (s) = 3 s ;2s+2

p s +7s -S6s-64 s +7s +24s+l8 (3 . 2)

The transfer functions representing ( 2 . lc) and ( 2 . ld) are

d2s+d 1 w2 (s) = k2 + _,,_2 __ s +2s+2 ( 3 . 3)

and their parameters c 1 , c2 , d1 , d2 ,k2 and the feedforward gain k1 are adjusted according to ( 2 . l e) with r = SI . The block diagram of the system without the adaptation laws is shown in Fig . S . The data for Fig . 3 are r = 2 , z = . S sin 3t , k1 (0) = 2 , k2 (0) = - 18 , c 1 (0) = l , c2 (0) = 2 , d 1 (0) = . 1 S , d2 (0) = -3 . 3 . Tfie numerator of the transfer function e (s) / z (s) contains the factor

2 c (s) = s + (2-c2) s + (2-c 1) ( 3 , 4) in which c 1 -+ -7 , c2 -+ 2 as t -+ 00 and hence c (s) -+ s2+9 to cancel the disturbance . The plots of c 1 (t) and c2 (t) are shown in Fig . 6 and that the regulation of y (t) to f is achieved is seen from Fig . 7 . When z (t) . S sin 3t + 0. S sin St the gains drift as exem­plified by Fig. 4 and as shown in Fig . 8 , the regulation of_ y (t) cannot be achieved .

References 1. K. S . Narendra , L . S . Valavani , "Stable Adaptive

Controller Design - Direct Control , " IEEE Trans . Auto . Control , AC-23, Aug . 1978, 570-583.

2. C . E . Rohrs, L. Valavani, M. Athans, G. Stein, "Robustness of Adaptive Control Algorithms in the Presence on Unmodeled Dynamics ," Proc. 21st IEEE Conf . Dec . & Control, Orlando , FL, Dec. 1982, 3-1 1 .

3 . B . B . Peterson, K . S . Narendra , "Bounded Error Adap­tive Control," IEEE Trans . Auto . Control, AC-27, 6, Dec . 1982, 1 161-1 168 .

4. G. Kreisselmeier, K. S . Narendra , "Stable Model Reference Adaptive Control in the Presence of Bounded Disturbances , " IEEE Trans . Auto . Control , AC-2 7, 6, Dec . 1982, 1 169-1 176.

5 . P . A. Ioannou , P . V . Kokotovic, Adaptive Systems with Reduced Models, Spri

,nger-Verlag, 1983 .

F19. 1. Fndbtck Syst11111 llltll AdJust1ble F(s)

-· FIG. 2. DRll'T 01' GAi" Ki FOR FIRST

ORDER PuNT

FIG, ft, DRIFT OF GAIN fCi FOR THIRD ORDER PLANT

-+--+--+- FtG. 6. GAINS C1 AND C2 FOR

I ..

Z • .5 SIN 3c

I - ·1 I

F10. 7. OurPur REGULATION AcH1Eveo

FOR ONE FREQUENCY

j ! i

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ON THE MODEL-PROCESS MISMATCH TOLERANCE OF VARIOUS PARAMETER ADAPTATION ALGORITHMS IN DIRECT

CONTROL SCHEMES: A SECTORICITY APPROACH

R. Ortega1 and I. D. Landau

Laboratoire d'Automatique de Grenob!P (CNRS) ENS/EC, B.P. 46, 38402 Saint Martin d'Heres, France

Abstract . Designing adaptive control lers under model-process mismatched s ituations i s conside­red . A suitable representat ion for the uncertainty is expl ici tly incorporated , what renders the s tability analys is problem suitable for a sector condition formulat ion (Safonov , 1 980) . The performance from the robus tness point of view, of various currently used parameter adap­tat ion algorithms is compared . This is done by determining regions on the complex plane where an uncertainty dependent Nyquist locus must be contained to insure boundedness of the tracking error . It is shown that uniform boundedness of all s i gnals may only be proven for a weighted l eas t squares scheme under sui table weight ing coefficients select ion knowkdge of the process s teady-s tate gain is required to insure convergence to zero of the tracking error . The pro­posed framework is shown to be amenable for the s tudy of non-s tably invertible or mismatched delay sys tems .

INTRODUCTION . The bas ic requirement of a feedback des ign is to achieve certain desired levels of per­formance in spite of plant uncertainties . A major s tep in the development of sys tematic des ign techniques tolerating reasonable un­certainty levels , was the es tablishment in recent years of the global stability of d irect adaptive controllers for linear s ta­bly invertible sys tems with unknown parame­ters . However from a pract ical point o f view the hypothes is needed to rigourous ly prove the theoretical results seem to res trictive . Among the assumptions that have to be made regarding the plant , the need to know an upper bound of its order is particularly hard to es tabl ish in practice . Moreover em­barassingly high susceptib il ity to uns tabi­lity has been proven to exist g iven violation of these condit ions (Rohrs et. al, 1 982) . In light of these observations , the convergence study of reduced order adaptive algorithms that would provide a "des ign adjus tment pro­cedure" to conduct the robus tness/performan­ce tradeoff , seems essential for the succes­ful application of adaptive control theory . The problem of provid ing such a procedure for all zero cancelling direct adaptive control­lers with partially modelled sys tems was studied in (Ortega, Landau, 1 983) . The key technical device for the theory presented in that paper is the sector s tabi l ity theorem, initially introduced by zames ( 1 966) and la� ter generalized by Safonov ( 1 980) , which s tates that a feedback interconnect ion of two relations is s table if one is s trictly I R . Ortega ' s work is supported by the Natio-nal Univers ity o f Mexico .

35

ins ide a sector and the other one inside i t complement . I n this paper we focus our at­tention on the sectorcity properties o f the parameter adaptation algori thms (PAA) . The problem is formulated in a general way by decompos ing the error model . into intercon­necting blocks , leaving the P .A .A in closed­loop with a transfer function (H2 ) whose structure is defined by the uncertainty lo­cation. Provided sec tor properties can be es tabl ished for the P .A .A . , s tab i lity is insured by res tricting the feedback transfer function H2 to the compl imentary sector ( termed in the following : allowable cone) . Assuming uncertainty represented by conic bounded transfer functions , conditions were found in (Ortega, Landau , 1 983) such that H2 belongs to the allowable cone for all pos s ib le s table unmodeled dynamics . The test .cons i sts of verifying that the module of a linear operator (n . ) ( function of the un­certainty bound ana the des ired closed-loop poles ) is upperbounded by the radius of the allowable cone . By incorporationg reasonable a priori knowledge ( the uncertainty bound ) , a powerful interpretative and manipulative design too l to carry the robus tness/perfor­mance tradeoff is therefore obtained . The results presented in thi s paper genera­lize and make rigourous the previous analy­sis . I ts specific contributions may be sum­marized as follows . I ) The s tabil ity con­dition es tablished in (Ortegan Landau , 1 983 ) was defective , s ince the sector properties of the cons idered P .A .A . do not permit us to define an allowable cone independent of the regressor module and the P .A .A . gain. Here we show that such a cone may be obtained using general ized least mean squares (LMS ) P .A . A . or by a sui table cho ice of the weigh-

36 R. Ortega and I . D . Landau

ting coefficients in the weighted least squa­res (WLS) P .A .A . Z) Sector properties for the following P .A .A . ' s are es tablished : cons­tant gain, LMS , LS and WLS . The importance

from a robus tness s tandpoint, of a judi- ' cious choice of the P .A .A . in general , and of its various parameters in particular , are henceforth highlighted 3 ) A major stumbling­block to apply the design tool mentioned above is the dependence of n . on certain pa-

l. I rameters of the tuned regulator • I t is shown here that taking a regulator structu­re that adds no zero� these unknown parame­ters may be reduced to one , hence s impli­fying the design adjustment procedure . 4) It is illus trated how other problems of adapti­ve control theory , i . e . non-stably inverti­ble and mismatched delay systems , may be formulated using the conic sector framework . The problem is formulated in section Z . The generality of its framework is il lustrated with various examples and the main assump- : tions are given. In section 3 . we es tablish the sectoricity properties of the P .A .A . mentioned above and , using the sec tor s tabi­lity theorem of appendix A, conditions on the uncertain transfer function Hz are de­rived . Problem Formulation Throughout this paper , we will deal with sca­lar, linear , time invariant discrete time processes . We consider the case where the plant cannot be perfectly characterized by the assumed parametric model but we confine its transfer function to a neighborhood of known bound . We will restrict ourselves to stable conic bounded uncertainty and to cla­r,ify the ideas a specific location is consi­dered , e . g . output multiplicative . It is clear that the results to be derived are unique to the assumed form of plant pertur­bation, however mutatis mutandi s imilar re­sults may be obtained for other uncertainty

z sources . Assume the following set membership statement ·for the process transfer function� .

- I 'V - I yt = G (q ) [ l +G(q ) ] Ut (Z . l . a)

l �(ejv) I < �G (ejv) ; : for all yE[ O , TI ] (Z . l .b)

1 This term, firs t introduced in (Kosut, Fred­lander : 1 98Z) , denotes the known parameter solution of the non perturbed system. z See (Doyle, et al . 1 98Z) for a full discus-sion on . uncertainty locations and its repre• sencation. 3 . 'Ihr.ough0ut the paper, the same symbol is used for different functions . The arguments iden­tily the function as wel l as the variable . When clear from the context they wil l be omitted .

"' - I Where G, G E R(q ) - the field of rational functions in q- I . The transfer function G represents the structured part and is suppo­sed to be contained in the set of parametric models assumed for the design. �G is a known positive scalar function and � is stable. Inequality (2 . t .b ) may also be s tated as ( see appendix) '1' /::,. 'V (Z . Z ) c; E J =coNE (O ,RG) It is desired to find a controller such that th� 1 closed loop transfer function be G ER -(q ) . The design objective may be s tafed in terms of the tracking error

A /::,. - I A (Z 3 ) et+d = Gm Y t -wt+d •

W AaS U +R y ( Z , 4 ) t+d t t t t -1 with s t ,it polynomials in q with time va-rying coefficients . The following assumption is in order . A l ) There exists R, S E R [ q- 1 ] (the field of polynomial functions in q- 1 ) verifying

( Z . 5 ) Assumption A l ) i s fundamental to obtain the process separametrization, which is easily derived from (Z . l .a) and (Z . 5 ) ar

( Z . 6 ) where

y t ' yt- 1 ' . . yt-n ] T

r ns+�+Z and e E R

(Z . 7 )

From the knowledge o f nA, � Al ) we chose n� = n5 ana nR_= regulator may 5e written as

and assumption nR, hence the

A AT wt+d = et et (Z . 8 ) and the tracking error is readily seen to verify et = -HZ�t + (Hz- 1 ) Wt where � � �T A � A= ( e A e)T � A

t t-d �t-d t-d- �t-d

G- l ( I +�) H � __ m ___ _ z

G- 1 +R� m also noting that

Yt-d = Hz (Wt-�t )

u A r-d

G- 1

G- 1+R� m We can definie the regressor as

(Z . 9 )

(Z . 1 1 . a)

(Z . 1 1 .b)

Modal-Process Mismatch Tolerance 37

¢ A = H W (W -� ) t-d 2 t t where

( 2 . 1 2 )

-n � � - I / � q S /G( l +�G) W =[ l /G ( l +G) , q G ( l +G) , • • • - I -nR] T ( 2 . 1 3 ) q , • • • ' q

' I ,

Comb ining equations (2 . 9 ) , ( 2 . 1 2 ) and in the light of the s ector stabi l ity theorem A . I . The problem of es tablishing s tabil ity condi­tions for the proposed adaptive system may be formulated as follows . 1 ) Define condi­tions for the s tability of H2 and H2W . 2 ) Propose an operator T : et + e� and a P .A .A . such that the overall relation H 1 : et+�t verifies some sector conditions . 3 ) Find interpretative and manipulative indications of the apparteinance of H2 to the compli­mentary sector . Clearly no possible answer exists to point I ) without assuming further knowledge about the proces s , hence the key substantive (ho­wever natural ! ) res triction mus t be made . A2) The tuned transfer function H2 is s table . Notice that al though , as s een from (2 . 1 3 ) , the hypotesis of sys tem s table invertibility must be made , it will be shown below that the s tability aualysis may be carried without it by proper modification of the control objective and the reparametrisation. It is important to point out that the choice of ns and � is only res tricted to the exis-tence of a solution in S and R to (2 . 5 ) , hen­ce under certain conditions on G and G (i . e . minimal realization hypothesis) we Wan always choose �=O . Note that this regulator does not add any zeros to the closed-loop transfer function. What is important to re­train in our context is that the s tudy of the H2 sectoricity properties is considerably s implified when R is a scalar . For example , it can be shown (Ortega, Landau, 1 983) that for stabll invertible sys tems with known de­lay, H2 (�) is inside the allowable cone for an ?: verifying ( 2 . 1 .b ) if the followine ine-' qi'iality holds .

Ti (v) � j L-C I + I a I ( 1 -q -dR/CR)L I 1 /�G- j R/cRI

for all v E [ O , rr ]

< R a

(2 . 1 4 ) Where C , R are the center and radius of the allowabfe c�ne and L is a des igner-selected filter to be defined later . The only uncer­tain information to carry the test is the scalar R .

The remaining part of the paper i s devoted to point 2 ) , that is the establisment of sector conditions for the P .A .A . Some exam­ples wil l be given firs t to illustrate the

generality of the proposed approach . -Example I . Stably invertible sys tem with known de lay . Let G- 1

G = q-dB/A, with d (=d) known and define d q CR. The transfer function H2 is gi-m

ven by � l +G H = --'----2 -d� l +q GR/CR

Stability of H2w requires the s table inverti­bili ty hypotesis . Example 2 . Non s tably invertible sys tem with ·known delay and no unstructured uncertainty Let the system be fully described by the pa­rametric model G, which is assumed as above . In order to avoid zero cancellation some pro­visions mus t be taxen. Firs t we restate Al ) in terms of the Bezout identity . C = AS+q-dBR R Instead of (2 . 5 ) , leading to the reparame­:trization

= !_ er ¢ yt CR t-d :Notice that the choice of the tracking err.or determines the structure of H2 , hence the robustness properties of the algorithm. Set-

-d A ting Gm = q B/CR with B an a priori est ima-te B we get H2 = B/B The prior knowledge of upper bounds on the zero positions could be in this way profi­tably used . By s imple manipulations it can be shown that stab ility of H2W does not re-quire the s table invertibility condition . Example 3 . Mismatched delay sys tem. It is readil1 seen that a mismatch �n the delay (e . g . d f d) implies � = q-d+d - 1 ,

- I � d hence defining Gm = q CR we get

-d -d C +R(q -q ) R In (Ortega, Landau ; 1 983) sec tor properties were established only for interlaced P .A .A . One way to avoid the need of interlacing is to assum£ single de lay parametric models , ::that i s d= I . The price paid is the inclusion of an additional term in H2 that clearly in-0'fluences its sectoricity properties . The glo­bal uncertainty term is of the form d-d � q ( l +G )- 1

Remark 2 . 1 . In the problem formulation we have not s tated the control objective as in-

38 R. Ortega and I . D . Landau

suring et + 0 , s ince as seen from the defi­nition of finite gain s tabi l i ty ( see appen­dix) this imposes the additional assumption of H2 ( 1 ) = I (see eq . 2 . 8) . It is clear that if the s teady s tate gain is known, e . g .

'\, G ( l ) =O , such a condition is verified . F i l te-ring the tracking error by an integrator (L= l -q- 1 ) is another alternative currently under s tudy . Remark 2 . 2 . Recently (Peterson, Narendra ; 1 98 2 ) boundedness of an adaptive s cheme in­corporating a dead-zone operator T has been proven for sys tems with bounded external dis turbances . The problem formulated there clearly contains the one treated here , whe­re to the nature of the external dis turbance we have attached a modelling error • . This "more s tructured" approach al lows to incor­porate , (seemingly resonable) prior knowle­dge into the problem, abandoning the "black box" concepts that severe ly reduce the exer­cise of the des igners intuitive j udgment and experience .

PARAMETER ADAPTATION ALGORITHMS SECTOR PROPERTIES .

The problem that we will treat in this sec­tion is the determination of an operator T : e + eA and a recurs ive relation for t t Pt such that suitable sector properties may be established for the relation H 1 : e +� . t t Cons is tent with our robus tness framework we disregard generic sector properties , i . e . cones with zero radius , and in order to provide sectors amenable to frequency-do­main verification, we cons ider only conic sectors (see Safonov, 1 980) .

The inclusion of the operator T al lows to clearly differentiate the tracking error from the adaptation error . Thi s was a fun­damental contribution for the early proofs of global s tability (in the model/process matched case) , where an a posteriori error representation was needed to prove conver­gence of the augmented error . To date , i t has not been poss ible t o make a j udicious choice of T, al lowing to es tablish sector properties for H

1, when uncertainty i s in-

corporated . We will cons ider A et

- I with L E R(q ) . Its to reflect L on H2 and

(3 . I ) linearity allows us H2 + I , we will deno-

L /';, te H2 = LH2 . Thi s adaptation error has been used for the matched case with L E R (Good­win , et al . 1 98 1 ) and with L the control weighting polynomial (Ortega , M ' Saad ; 1 983) to attain input-matching convergence of a one-step-ahead optimization s cheme .

To s tudy the sectoricity conditions of the various P .A .A . we will make use , as sugges ted in (Gawthrop , 1 980 ) , of quadratic forms . The

P .A .A • . to be considered are interlaced versions of the usual P .A .A . , howeveI as no­ted in example 3 we can always take d = I , this wil l b e the case throughout the remai­ning part of the paper .

Cons tant gain P .A .A .

Al though in order to establish sectoricity properties of CG/PAA we wil l require the as sumption of bounded ¢ , ( rendering defec­tive the s tab il i ty proof ) , we anyhow consi­der this type of P .A . A . to highlight several aspects of the analysis .

Consider the quadratic form

V � I 9' F- 1 � . F = FT t 2 t t ,

Clearly (3 . 2 ) verifies 1 > 0 (3 . 2 )

6V = (� + I 69' ) T F- J /';,� t t- 1 2 t t (3 . 3 )

th�s sugges t the P .A .A . "-' A 68 t F ¢t- J

et (3 . 4 ) ( 3 . 4 ) in (3 . 3 ) , together with ( 2 . 1 1 . a) gives /';,V = � eA + I aC<f_eA) 2 ( 3 . 5 )

t t t 2 t t

where CG6 . T

<\ = q> t- 1 F ¢ t- I

I t can be readily seen that

< '(i eA/ 2+ � [ eA > >. V f all N E Z � t t t N ,,- - I ; or ' +

(3 . 6 )

where Cf � Amax (F) sup [ ¢t [ t

(3 . 7 ) henceforth provided (2 . 34 ) i s veri fied (CG I ) H 1 + fj/2 is pass ive and closed-loop s tabil ity is insured if2 for all v E [ O , n]

(CG2) H� (ejv ) is ins ide D [ I /Cf ; 1 /f ]

Remark 3 . J . I t can be readily seen that the

A 'VT operator et + 8t ¢t- l ' that is the P .A .A . with a posteriori error output , is pass ive , Notice that the a pos teriori error verifies , from (3 . 4 ) and (2 . 1 1 . a) T "-' CG A

¢ t- 1 6 t � t+ at et Hence substituting in (3 . 5 ) gives

�T A I CG( A) 2 ( e t ¢t- 1) et - 2 at et

1 The notation 6x�xt-xt , wil l be used throug­hout .

2 D [ C ; R] denotes the disk in the Z-plane with center C and radius R .

Model-Process Mismatch Tolerance 39

Note the significant difference due to the minus sign. Remark 3 . 2 The family o f allowable circles for H� (etlv) is clearly restricted to the right-half complex pl aneL hence (CG2) illl"'" plies the pass ivity of H2 • The scalar (f re­flects the speed of convergen�e of the P .A .A and the "level of excitation" of the sys tem, the smaller if , the larger, the allowable c_i circle area; If s tability is tq_be insured for the perfectly modeled case 'f .$1 . Generalized Leas t Mean Squares P .A .A . Our generalized LMS-type P .A .A . has the form.

Et is a sequence verifying E � Et > 0 , for all t E Z+

( 3 . 8 . a)

(3 . 8 .b )

(3 . 9 ) wi th E a large number fixed a priori , and pt ' introduced to avoid division by zero in (3 . 8 .b ) is given as : { => 00

if t E{ t : ¢!- 1 ¢t- 1 ? K} otherwise

Consider the quadratic form 'V et

1 /2 Introducing the multipliers ft (Desoer, Vidyasagor ; 1 975 ; pp203) we obtain �v = 'l'M eAM + I aLMS (eAM) 2

t t t 2 t t (3 . 1 0 ) where 'l'M C,f l /2'¥ . t t t ' AM � f l / 2 I u;s �,i,T f _._ et = t e t ; a

-'I' t- 1 t"' t- 1 noting from (3 . 8 ;b) and (3 . 9 ) that - LMS 6 > at , for all t E z+ we directly prove that (LMSL) H 1 + E/ 2 is passive

(3 . 1 1 )

To s tud� the s tability of the overall sys­tem we consider the equivalent scheme given below .

Fig . I . The fol lowing lemma allows us to es tablish the s tability conditions over H2 • Lemma 3 . 1 . Consider the do tted subsy$ tem of Fig . I . If for all v E [0 , TI ] H� (eJV) is

- - L - I - - I inside D[ l /E , , l /E] , e . g . [ (H2 ) -E/2 ] is s trictly passive , and f i s an increasing pos itive sequence then. t < 'l'� I Y� >N > o I J 'l'� l � iVNez+ and o > o

Proof (Gauwthrop , 1 98 1 ) Def ine � 2 zt Y tnt - 8 nt

Zm /):._ M urM _ -" (mM) 2 t yt Tt U T t �

Taking N

the sum N N j

i: t=O

fN+ I l: 2t - l: [ (l: Z t) (f . 1 -f . ) ] t=O j =O t=O J + J from the s trict pass ivity of n t + , yt and

ft we obtain the non increasing nature of N

ZM i: > 0 for al l N E Z+ t=O t and the proof is completed . In order to apply lemma 3 . 1 . we need to choose a sequence E that grows fas ter or t T equal than the sequence ¢t¢t in that case ft is non-increasing and the s tability con­dition over H2 reduces to (LMS2 ) H� (Ejv) is inside D[ l /E , 1 /E] for all v E [ O , n ] . Least Squares P .A .A Considere the usual "-' A �et = Ft ¢t- 1 et

�F- 1 = ¢ ,i,T t t- 1 '!'t-1

LS es timator

F I t is easy to show that for a quadratic form

')..T - l 'V Vt = J t\ Ft et We get �Vt = 'l'te! + { at (e�) 2 + -± 'l'� at � "'T F ,i, '!'t- 1 t '!'t- 1

(3 . 1 2 )

taking the sum and noting that atE [O , I ] for all E Z (Gawthrop , 1 980) , i t can be readily seeri that there exists a o verifying 1 � o > at� o such that

for all t E Z+ (3 . 1 3 )

(LSI ) H 1 is outs ide CONE (- 1 ,v'i-o) Henceforth from the sector s tability theo­rem the condition over H2 becomes (LS2 ) H� (ejv ) is inside D[ l /o,�o] for all v E [ O , n ] Remark 3 . 3 . Since the value of a i s not available without further assumptions on at , guidelines and not a comprehens ive des ign procedure may be developed for LS/PAA Notice that a necessary conditiyn for s ta­bility is Re {H�-{} > 0 ("" j (H�) - I I < I ) . A similar result was obtained in (Ortega, Landau ; 1 983) for a weighted-type gain re­cursive equation . The conceptual similarity

40 R. Ortega and I . D . Landau

of � (eq . 3 . 7 ) , e (eq . 3 . 1 1 ) and cr (eq . 3 . 1 3 ) is evident . Weighted Leas t Squares P .A .A . In order to avoid the allowable cone radius to vanish and to be able to es tablish o -. d d . t in epen ent sector properties we propose the fol lowing WLS/P .A .A .

<v 11 A t:iet At Ft cpt- 1 et - I = A , F- 1 + \" "' "'T T F t t t- 1 t 'Y t- 1 'Y t- 1 ; F o = F o >

, , " Requiring At > 0 in the quadratic form

Vt � f5T F- li /"),11 t t t t

we get t:ivt = '¥2+2'¥ e +\"o (eA) Z

t t t t t t Setting :>- ' t

\" t

with

" >- I \" t t- 1

> 0

1 /E t

if t E { t otherwise

' \"-\ ' :>-" t t t- 1

\" t

it can easily be proven that (WLS I ) H 1 is outs CONE (- 1 , v' ::/ ( ! �) )

With no further assumption the fol lowing condition is sufficient to insure overall system s tability ; for all v E [ O , n ]

0

(WLS2 ) H� (ejv) is INSIDE D [ ( ! +::) ; V:= ( l +::) l Remark 3 . 2 . While the s tability condition for the LMS/P .A .A . requires E to fullfil a growing condition, in the WLS/P .A .A . it is

sufficient to verify that i t is bounded from below by a posi tive non-zero constant . This provides considerable flexibility to the des igner . The sequence ft and the cons-tant K play exactly the same role for both algorithms . Since there is no upperbound res triction to E the minimum in the . real axis of the allowable zone for H2 (eJV) tends to zero , i . e . for Et = I it equals 2-�

CONCLUSIONS The performance of several P .A .A . has been compared from the point of view of the re­gion in the complex plane w�ere the uncer­tain transfer function H2 (eJV) may be allo­wed to be, preserving closed-loop s tability Al though i t is not s traightforward to de­termine where this region should be , since

the uncertain terms in H enter in diffe­rent ways , it is reasonaEle to expect that better robustness properties will be attai-:-c ned wi th those P .A .A . allowing larger zones . Furthemore as seen from (eq . 2 . 1 4 ) H2 sec to-ri city tes ts based on verifying an inequa­lity upperbounded by the allowable cone radius are available . The stability conditions derived in the pa­per require no further assumptions for the WLS/P .A .A . For the LMS/P .A .A a sequence growing fas ter or equal than cj>� cj>t must be provided . A family of o -dependent allowa­ble cones is proven to �xist for the LS/PAA being however impossible to insure a non­vanishing radius . It has been shown that the proposed frame­work , based on sectoricity prope�ties , is general enough to permit us to treat other problems of interest to the adaptive control theory , bes ides the one of unmodelled dyna­macs . The results can be directly translated for continuous time adaptive schemes , as done for systems with relative degree smal­ler than one in (Kosut , Fredlander ; 1 982) . Further research is under way in three di­rections . I ) Es tablishment by reformulation of the des ign objecttve , of less restrictive conditions over H� (eJV) . Inspired by the succes of (Peterson, Narendra , 1 982) it is our belief that a proper choice of the ope­rator T may render this poss ible . 2) Consi­deration of other P .A .A . s tructures , perhaps been reasonable to leave the integral type adaptation in order to improve robustness (Ioannou, Kokotovic ; 1 982) . 1 .Consideration of piece-wice linear regulators , which have proven extremely robus t in practice , i . e . variable structure sys tems . 3 ) On line ad­jus tment of the desired closed loop perfor­mance , i . e . CR(t) . Two important ques tions must be solved to approach the theory developed here to po­tential applications : how can bounds be given for the uncertainty in a discrete­time environment ? . How can the phase in­formation be incorporated into the problem formulation ?

REFERENCE Rohrs , CE et . al ( 1 982) . Proc . 2 1 s t . IEEE CDC . Orlando, Fl . USA Ortega, R. Landau iI . D � l 983) . IEEE Trans . Aut . Cont . (Submitted) Zames , G. ( 1 966) IEEE Trans Aut . Cont . Vol . AC- I I n° 2 , 3 . 1 966 Safonov , M .G . ( 1 980) . MIT Press . Kosut , R .L . , Friedlander , B . ( 1 982) . Proc . 2 1 s t . IEEE CDC Orlando, Fl . USA . Doyle , J . C . et . al ( 1 982) . Proc . 2 1 s t . IEEE �· Orlando , Fl . USA . Goodwin, G . C . e t . al ( 1 98 1 ) . IEEE Trans .

Model-Process Mismatch Tolerance 4 1

Aut . Cont . Vol . AC -26 , n° 6 . Pe terson, B . Narendra K . ( 1 982) . IEEE Trans . Auto . Cont . Vol . AC-27 , n°6 . Ortega , R . j l1' Saad , M . ( 1 983) . IEEE Trans . Autw Cont . (Submitted) Gawthrop , P . J . ( 1 980) . Int . J . of Contr . Vol . 3 1 , n° 5 . Desoer, C .A . Viyasagar , M . ( 1 975 ) . Ac . Pres s , N .Y . Gawthrop , P . J . ( 1 98 1 ) . In "Self-tuning &, Adaptive Control" . Ed . Harris & Billins . Peter Perigrinus U . K . Ioannou, P . Kokotovic , P . To appear in Springer-Verlag Series : Lee t . Notes in Cont. & Inf . Sc i .

APPENDIX Notation The notation and terminology used throughout the paper is s tandard in the input-output formulation (see e . g . (Desoer , Vidyasagar , 1 9 74 ) , ( Safonov, 1 980) ) . The input and output sequences 're assumed unbedded2in the named space L or i ts extension Le . Definition A . I . Let C , RE R . An I/O relation Hiut � yt is said to be i ) inside CONE (C,R) if I I y t - Cut I J N '

R I I ut I I N ; for all N E z+ ii ) strictly inside CONE (C 1R) if it is in­side CONE (C ,R ' ) for some R < R The notions of outside and s tricly outside are defined analo�ously invers ing the ine­quali ty sipn throuehout we cal l gain of H the number Y (H 1 ) defined by Definit ion A . 2 Y(H) = inf{Y E R+ : 3 ,3 such that I J Hut I J N < Y I I ut l [ N + B for all ut E L;

and for all N E Z+ Theorem A . I (Sector s tab ility cri terion) Consider the fol lowing feedback intercon­nection ut et + H2 Yt yt H I et 2 2 2 wi th H 1 , H2 : Le � Le and et , yt E Le . Let a conic sector be defined with C, R E R and I c [ > R . Under these conditions , if a) HJ is outside CONE r�) �] le -R2 c2 R:2

. b ) H2 is stric tly inside CONE (C ,R) then there exis t a K<00 such that

I I yt [ I N < K I J ut l J N ' for all ut E L� and for all N E Z +

That is the sys tem is finite gain s table .

Proof . Consider the following equivalent sys tem (Desoer, Vidyasagar , pp50) We will use throughout the properties of the CONIC sec tors (Zames , 1 966 , App .A)

- I a) � H 1 i s outside CONE (C , R)

� (H� 1 -c ) is outside CONE (O ,R) � (H� 1-C) is inside CONE (0 , 1 /R) � y{H� 1 -C)- I } .$ 1 /R

b) � (H2-c ) is s trictly inside CONE (O ,R) � Y{ (H2-c) } <: l /R

The proof follows from the small gain theo­rem. (Desoer , Vidyasagar , pp4 1 ) . Acknowledgement . We Would like to thank Mr L . Praly of the Ecole des Mines , Paris , for bringing to our attention a flaw in the previous analysis and for many fruitfull discussions .

Fig. I .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

A FREQUENCY DOMAIN ANALYSIS OF DIRECT ADAPTIVE POLE PLACEMENT ALGORITHMS IN THE PRESENCE OF

UNMODELLED DYNAMICSt

H. Elliott, M. Das and G. Ruiz

Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA OJ003, USA

Abstract . This paper examines a general class of direct adaptive pole placement algorithms in the presence of unmodelled high frequency dynamics . The analysis is broken into three stages . The first stage addresses the robustness of the fixed control structure on which most contin­uous time adapt ive control schemes are based . The second stage looks at the effect of un­modelled dynamics on the estimation model used in equation error formulations of continuous time algorithms . The last stage looks specifically at the identification process .

I . INTRODUCTION

Recently , robustness issues have been receiv­ing considerable attention in the adaptive control literature [ l ] - [ 8 ] . In [ l ] - [ 3] a linearization approach has been used to ana­lyze model reference type controllers , in [SJ singular perturbation techniques were used to analyze a specific model reference scheme . Although the work presented below is at a much less developed stage , in that we have no formal theorems regarding stability issues , we have taken a more classical ap ­proach in order to gain some phys ical ins ight into the problem . In particular , we have tried to interpret performance of pole place ­ment type adaptive control schemes from a frequency domain point of view . The class of schemes considered are those which place poles by direct estimation of the compensator poly­nomials which appear in the diophantine equation associated with pole placement , and which do this us ing equation error based identifiers . This class includes both globally stable zero cancelling pole place ­ment schemes such as discussed in [ 9 ] , and arbitrary pole assignment schemes as discussed in [ 10] . The approach taken consists of three stages . The first involves analysis of the fixed com­pensation structure used for pole placement . The structure is generic in the sense that most all continuous t ime model matching or pole placement adaptive schemes use it . I t has its roots a s a frequency domain version of pole placement via combined state estima ­tion and feedback . Before one can analyze an adaptive pole placement scheme , one must ask if the corresponding fixed version is robust in the presense of unmodelled dynamics . In this section, we give a frequency domain interpretation of this algorithm and show t

This work was supported by the National Science

43

that a key for good performance is keeping the feedback s ignal low frequency . To this end , some suggestions for improving perfor­mance are given . Stability issues are also addressed . In the second stage of the ana ­lysis , two models which are used to generate equation errors are cons idered . It is shown that estimation of parameters from these models in the presense of unmodelled dynamics is equivalent to attempting to so� phantine equation whicft-has no-solution in the formal sense . It is shown however , that if a least squares fit is made by evaluating the equation at various frequencies , then reasonable parameter estimates are obtained when the frequencies are small relative to the break frequencies associated with the unrnodelled dynamics . In light of the stage 2 analys is in stage 3 methods are discussed for limiting the frequency content of the data used for parameter estimation .

I I . PROBLEM FORMULATION AND BACKGROUND

Consider the problem of placing the poles of a s ingle- input , s ir5le -output continuous t ime system charact�rized by either of the

differential operator models :

p (D) z (t) = u (t)

y (t) r (D) z (t)

p (D)y (t) = r (D)u (t)

(la)

(lb)

(2)

where u (t) , y (t) and z (t) are the system input output and partial state . Assume p (D) to be monic , deg (p (D) ) =n , and deg (r (D) ) <n . A general compensator structure which can place poles is

s (D)u (t) = h (D) y (t) + q (D)v (t) (3)

Foundation under Grant ECS 82 14534 .

44 H. E l l iobt , M . Das and G . Ruiz

where v (t) is a reference signal . Since the closed loop system becomes

(sp - hr) z = qv y = rz

to place the poles arbitrarily , for arbitrary f (D) one must find polynomials s (D) and h (D) which satisfy the diophantine equation

sp - hr = f (4)

If the process polynomials p and r are known , and p and r are prime , then (4) can be solved by equating powers of D on both sides and solving the resulting linear equations .

In the adaptive case when r and p are unknown, the coefficients of s and h are estimated us ing input-output data . Indirect schemes first find estimates p (t , D) and r (t ,D) of p and r and then map these into estimates s (t ,D) and n (t , D) of s and h by solving (4) in real time . Direct adaptiye control schemes generate s (t , D) and h (t ,D) directly from input-output data by forming a new equi­valent system model .

To see how this is done , multiply (4) by z (t) and use (1) . This leads to a new but equiva­lent system model

s (D)u (t) - h (D)y (t) = f (D) z (t) (5)

in terms of the compensator polynomials s and h. This paper is concerned with the problem of estimating adaptive control parameters from equations of the form (5) . Actually it is connnon in the continuous time adaptive control literature to reformulate (5 ) slightly by defining k(D) = q (D) - s (D) . In this case , control law (3) becomes

q (D)u(t) = h (D)y (t) +k(D)u (t) +q (D)v(t) (6)

The polynomial q (D) is arbitrarily chosen as a Hurwitz polynomial . In this case , to place the poles one must find polynomials h (D) and k(D) satisfying the new diophantine equation

hr + kp = qp - f (7)

Furthermore , if f (D) is chosen of the form

f (D) = q (D)p* (D)

where p* (D) is an arbitrary manic Hurwitz polynomial of degree n whose roots represent desired pole locations , then control law (6) has a direct state space interpretation as implementation of combined state feedback and estimation , where the observer poles correspond to the roots of q (D) . In this case (7) becomes

hr + kp = qp - qp* (8)

Multiplying (8) by z (t) and using (1) leads to the equivalent estimation model

h (D)y (t) +k (D)u (t) =q (D)u(t) -q (D)p* (D) z (t)

(9) The remainder of this paper deals with adap­tive control schemes based upon fixed control law (6) , diophantine design equation (8) and estimation model (9) . Two specific forms for (9) wil l be considered.

An important class of adaptive controllers are those which cancel openloop zeros with closed loop poles . Model matching schemes fall into this category. In this case

p* (D) =p ' (D) r ' (D) where r' (D) is obtained by normalizing r (D) to be manic , so that for some constant g , r (D) =gr ' (D) . Equation (9) then becomes

hy+ku=qu - k 'rz g

or using (1)

h (D)y (t) +k(D) u (t)+� (D)p' (D)y (t) =q (D)u (t)

(10) This model has the unknowns h (D) , k(D) , and 1 g, the reciprocal high frequency gain .

In order to cancel zeros , they must be stable . If this cannot be assumed, then p* (D) must remain arbitrary. In this case , to estimate the unknowns h (D) and k(D) using (9) , the unmeasurable internal state z (t) must be replaced by an estimate obtained from input -output data . To construct an observer , assume a (D) and b (D) satisfy

ar + bp = 1 (11)

�1ultiplying (11) by z (t) and using (1) yields

a (D)y (t)+b (D)u(t) = z (t) (12)

Substituting (12) into (9) yields

h(D) y(t) +k (D)u(t) +a (D)q (D)p* (D) y (t) + b (D) q (D)p* (D)u (t) = q (D)u(t) (13)

In this model h (D) and k(D) as well as the observer parameters a (D) and b (D) are un­known .

When unrnodelled dynamics are present , the problem is the following . An adaptive controller is designed assuming the model (1) but in actuallity the plant is character­ized by

d (D)p (D) z (t) = u(t) (14a)

y (t) = m(D) r (D) z (t) (14b)

The unrnodelled dynamics are characterized by the transfer function

Direct Adaptive Pole Placement Algorithms 45

Tu (s) = m (s) /d(s)

where it is assumed that

and -

IT (s} I = 1 u

(15)

(16)

(17)

for 1 s 1 "small" . Furthennore d (s) is assumed Hurwitz .

Our analysis of adaptive control systems in the presense of urunodelled dynamics involves three stages . First one must understand the conditions under which a fixed pole placement controller will perfonn adequately in the presence of urunodelled dynamics . Second, it is necessary to understand how to calculate an approximation to the pole placement con­troller using a diophantine equation with urunodelled dynamics . Finally , one must deter­mine a method for parameter estimation from input-output data corrupted by the urunodelled dynamics . These issues are discussed sequen­tially in the sections to follow.

I I I . FIXED CONTROL AND UNMODELLED DYNAMICS

In the nonadaptive case , the most natural strategy would be implementation of a control based on the modelled portion of the system, neglecting the urunodelled dynamics during the design stage . In particular , use of control law (6) and diophantine design equation (8) . As will be shown in the next section , under certain assumptions it is reasonable to expect the adaptive controller to yield para­meter estimates close to the solution of (8) as well . Two issues must be addressed re­garding this strategy . First , is the resulting closed loop system stable . Second, if stable , is perfonnance similar to that expected.

To address the first issue , if h,k satisfy (8) and urunodelled are present , the closed loop characteristic polynomial is

c (s) =q (s) d (s)p (s) -h (s)m(s) r (s) -k(s ) d (s)p (s) (18)

To understand the impact of urunodelled dyna­mics on closed loop performance it is helpful to consider the simple case of a single un­modelled pole , i . e . m (s)/d (s) =y/s+y . In this case

c (s} =q (s) (s+y}p (s) -h(s)yr (s)-k (s) (s+y)p (s)

=y (q (s)p (s) -h (s) r (s) -k (s}p (s) )

+sp (s) (q (s) -k(s) )

or using (8)

c (s) =yq (s)p* (s) +sp (s) (q (s) -k (s) ) (19}

A simple root locus arguement then implies that if the openloop system is unstable , then for ,Y small , i . e . relatively low frequency

urunodelled dynamics , the closed loop system will be unstable . The root locus as y varies from 0 to 00 gives an indication of the sta­bility robustness of different pole place­ment designs .

To understand the impact on perfonnance , observe that a solution to the diophantine equation

h (s)m(s)r (s) +k (s) d(s )p (s) =q (s )d (s) (p (s) -p* (s})

(20)

is d (s)

h (s) = � h(s)

k (s) =k (s)

(2la)

(2lb)

where h (s) and k(s) satisfy (8) . A compen­sator designed using (20) , namely

h(s) k (s) u (s) =q(SJY (s) +�(s) +v(s) (22)

would place the n modelled poles at desired locations as given by the roots of p* (s) , and leave the urunodelled poles untouched . Using (21) in (22) yields

rd(s)J h (s) k (s) u (s)= L� CilsT y(s) + CilsT u(s)+v(s) (23)

The transfer function d/m is assumed unknown and furthennore dh/mq need not be proper . However , by assumption d/m ; 1 f or s "small" . Physically, this corresponds to the feedback signal

f (t) = '/, - 1 [�t�j y (sJ (24)

being low frequency . Thus if f is low fre­quency a reasonable approximation to (23) is (6) or its Laplace transfonn

u (s) = �t�� y (s) + �t�j u (s) + v (s) . (25)

Then the key to obtaining reasonable per­fonnance of the compensator (6) is keeping the feedback signal f (t) low frequency . We have been looking at two modifications to (25) to help improve the frequency content of f (t) . The first , which corresponds to a classical approach for improving steady-state tracking and disturbance rej ection , is adding integrators or other low pass filters as precompensator to the plant . The pole place­ment design is then applied to a new higher order process consisting of the cascade of the integrators or filter and the original process .

The second corresponds to increasing the relative degree of the feedback transfer function h/q . This can be done since a sufficient set of constraints to insure existance of unique solutions to diophantine

46 H. E l l iott , M. Das and G. Ruiz

equation (8) , are r and p to be prime , and for any J � O

deg(q) = n- l+J

deg(k) = n- 2+J

deg (h) = n- 1

Thus one can increase the degrees of k and q while keeping the degree of h constant .

The effect of these modifications can be sunnnarized by observing that the closed loop transfer function relating f (s) to v (s) is

mrh Tf (s) = (q-k)dp + yqp*

Thus if (q-k) dp+yqp* is stable , increasing the degrees of q,k and p relative to r and h improves cut off characteristics of the low­pass filter Tf (s) . The tradeoff is stability, this is illustrated by considering the following example .

Ex�le 1 - Stability effects of improving cu�f characteristics of Tf (s) .

Let

r (s) s+ , S PTST - sN

(s-3) (s+l)

and choose

q (s) = (s+l)J+N+l

p* (s) = (s+2) (s+l) N+l

As a measure of the stability robustness of the closed loop design we assumed m/d=y/s+y , and detennined the j w-axis crossing gain y for the root locus associated with the closed loop characteristic equation (19) as y varied from O to 00 • For J=O and N=0 , 1 , 2 , it was Ye = 1 . 4 , 2 . 65 , 5 . While for N=O J=l , 2 it was Ye ; 2 , 3 . Thus as N and J increase the value of y for which the system becomes unstable also increases .

VI . IDENTIFICATION MODELS AND UNMODELLED DYNAMICS

Assume identification models (10) and (13) are used to estimate h, k and g or h ,k ,a , and b respectively . Defining u (t) , and y (t) by (14) for the case of unmodelled dynamics , allows (10) and (13) to be written equiva­lently as

(hmr + kdp + !. qp 'r - qdp) z (t) 0 g

(hmr+kdp+aqp*mr+bqp*dp) z (t) = 0

If these are to hold for all z (t) , then h, k, 1 g, or h, k, a , b must satisfy the polynomial equations

hmr+kdp+�'mr=qdp (26) g

hmr+kdp+aqp*mr+bqp*dp = qdp (27)

Unfortunately if deg (h) =n-1 , deg (k) =n-2+J, deg (q) =n- l+J, deg (a) =n- 1 and deg (b) =n-2, there are no exact solutions to (26) and (27) . However, if one thinks of these polynomials as poly­nomials in the frequency variable s one can generate a set of linear equations which can be solved exactly by substituting K=2n distinct values of s (which are not roots of qdp) into �6) and K=4n-2 distinct values into (27) . Each value of s produces an independent equation . Similary by using K>2n or K>4n-2 estimates of- h,k ,g and h, k, a , b can be obtained by doing a least squares fit . Furthennore , if s= +j w then as will be dis­cussed in the next section, this corresponds to sinusoidal identification data consisting of these frequencies . Thus there is a physical interpretation for calculating estimates of the unknown parameters using such a procedure .

To understand the type of estimates obtained for h, k as values of s v a r y,consider the special case of m/d=y/s+y . In this case (26) and (27) can be written as : y (hr+ [�+�kp + � qp r) = y � +� qp (28)

(hr+ [�+�kp +aqp*r+�+1J bqp*p) =y [� t1Jqp (29)

Let s be the value of s of largest magnitude used in estimating h , k . Then continuity arguements imply that the values of h, k obtained using (28) , (29) converge to the solutions of (8) as the ratio s/y converges to zero . l Thus if the unmodelled dynamics is high frequency relative to the frequency content of the data used for estimation, solutions will be close to those assumed in the analysis given in the previous section . Hence it is reasona�le to expect perfonnance of the resulting control system to be com­parable .

Remark 1 : The analysis of this and the previous section are then consistant . One way to obtain reasonable perfonnance from an adaptive control system designed using the equation error formulation is to keep the frequency content of signals in the control system low relative to the unmodelled dynamics .

1Note that it has been shown in [9] and [ 10] that estimation of g , or a and b in addition to h and k does not effect the uniqueness of solutions .

Direct Adaptive Pole Placement Algori thms 47

Remark 2 : Note that stability robustness as characterized in the previous section is only a function of y , while performance is a function of the difference between frequency content of control system signals and the break frequencies of any unmodelled dynamics .

Remark 3 : We performed some studies where h and k were estimated by using values s=+jw and solving the resulting set of linear eqiiations . The results indicated that so­lutions to (28) converged faster to solutions of (8) then did those of (29) as s/y conver­ged to zero . This is consistant with sinru­lations of adaptive control systems which indicate designs based on (10) perform better then those based on (13) for the same frequency unmodelled dynamics .

v. PARAMETER ESTIMATION AND UNMODELLED DYNAMICS

Although the analysis presented below is equally applicable to equation error iden­tification schemes based upon either (10) or (13) for simplicity of presentation atten­tion will be restricted to (10) .

Using (10) , to estimate h, k and l/g , one would define the equation error

E (t)=I h. (t)Diy (t) +I k . (t) Diii(t) 1 l. i i

A - -+g (t) q (D)p ' (D)y (t) -q(D)u (t)

where

and

A L h(t ,D)=

k (t ,D)

A

l h. (t) Di . 1

� l k . (t) Di 1

i

Est (h)

= Est (k)

g (t) Est (l/g) .

-f (D)y (t) = y (t)

f (D)u(t) = u (t)

(30)

(3la)

(3lb)

(31c)

(32a)

(32b)

The filters (32) are introduced to allow causal differention in the sense that the signals Diy , Diii represent accessible filter states . Note that (30) can also be written in the usual regression form,

-T -E (t) = � (t) e (t) - q (D)u (t) (33)

By sampl ing j E (t) , ; (t) , and q (D)� (t) sequential least squares can be used for parameter estimation . Observe that if we define *t - - s z (t) , by f (D) z (t) = z (t) , and z (t) = e then

A A E (t)= {h (t , �m (s*) r (s*) +k (t , s*) d (s*)p (s*)+

g (t) q (s*)p' (s*)m(s� ) r (s*) ­q (s*) d (s*)p (s*) ) es t

The least squares algorithm tend� to minimize E (t) for all t , thus if z (t) =eS tthe global minima would occur at solutions n (t , s) , k(t ,s) , g (t) , which tend to satisfy (28) at s=s* . In the case where z (t) consists of a continuum of frequencies , then the least squares algorithm would tend to minimize the error in (28) over the same continnuum of frequencies . For rurther interpretation of equation error identifiers in terms of sinusoidal data refer to [ 11] .

The consequence of this is that to use least squares to generate estimates which are close to those which satisfy (8) , the signal z (t) nrust be of relatively low frequency . To this extent the filter polynomial l/f(s) can be thought of as a frequency weighting function . If chosen of high enough order it can cause low frequencies to be given arbitrarily large weightings relative to high frequencies and hence bias estimates closer to solutions of (8) . Note , however , that although f (s) can be used to improve parameter estimation , results of the previous sections tell us that if signals in the con­trol loop , or in particular z (t) , are not low frequency performance will still be poor . Keeping z (t) low frequency would serve to further improve parameter estimation.

VI . CONCLUDING REMARKS

The analysis given above attempts to inter­pret continuous time , equation error based adaptive control schemes using hybrid estima­tion procedures [9] from a frequency domain point of view. The conclusions are consist­ent with those presented in [3] and [4] . In particular , if unmodelled dynamics are pre­sent , for reasonable performance signals in the control loop must remain of low frequency .

Two methods are suggested for improving the frequency distribution of signals within the control system. The tradeoff is in the loss of stability robustness . Although not pre ­sented in the written version of the report , preliminary simulation studies indicate that the modifications can improve performance of resulting adaptive controllers . Methods are also discussed for improving frequency con­tent of data used for estimation.

In addition it is shown that independent of the frequency content of data , performance is ultimately limited by the stability robust ­ness of the underlying pole placement design procedure . It is the feeling of these in­vestigators that a significant payoff in the adaptive context can be made by finding pole placement control structures with improved stability robustness .

48 H . Elliott , M. Das and G . Ruiz

REFERENCES

[1] C . Roher , L . Valavani , M . Athans , "Convergence Studies of Adaptive Control Systems , Part I : Analysis" , Proc . IEEE 19th CDC Conf . , December , 1980 .

[ 2 ] C . Rohrs , L . Valavani , M. Athans , G . Stein, "Analytical Verification of Undesirable Properties of Direct Model Reference Adaptive Control Algorithms " , Proc . 20th CDC Conf . , December , 1981 .

[3] C . Rohrs , L. Valavani , M . Athans , G . Stein, "Robustness of Adaptive Control Algorithms in the Presence of Unmodelled Dynamics" , Proc of 21st CDC Conf . , December 1982 .

[4] P. A . Ioannou and P. V . Kokotovic , "Error Bands for Model Mismatch in Identifiers and Adaptive Observers" , IEEE Trans on AC , Vol . AC - 27 , August , 198 2 .

[ 5 ] P . A . Ioannou and P . V . Kokotovic , "Singular Perturbations and Robust Redesign of Adaptive Control" Proc of 21st CDC Conf . , December 1982 .

[6] C . R. Johnson and G . C . Goodwin, "Robustness Issues in Adaptive Control" , Proc of 21st CDC Conf . , December 1982 .

[ 7 ] R . L . Koust and B . Friedlander , "Per­formance Robustness Properties of Adaptive Control Systems" , Proc of 21st CDC Conf . , December 1982 .

[8] B . Wittenmark and K . J . Astrom, "Implementation Aspects of Adaptive Controllers and Their Influence on Robustness" , Proc of 21st CDC Conf . , December 1982 .

[9] H . Elliott , "Hybrid Adaptive Control of Continuous Time Systems" , IEEE Trans on AC , Vol . AC- 27 , No . 3 , �ri l 1982 .

[ 10] H . Elliott , "Direct Adaptive Pole Placement with Application to Non­minimum Phase Systems " , IEEE Trans on AC , Vol . AC- 2 7 , No . 3 , June 198 2 .

[ 11] H . Elliott and W . A . Wolovich, "A Frequency Domain t.!odel Reduction Procedure" , Automatica , Vol . 15 , March 1980 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ANALYSIS OF ROBUSTNESS OF THE INEXACT MODEL MATCHING STRUCTURE TO REDUCED

ORDER MODELLING

K. N. Shah

EG & G Torque System1, 36 Arlington Street, Watertown, MA 02 1 72, USA

Abstract . The problem of unmodel led dynamics is very cri tical in the study of adapt i ve cont rol systems because a model is merely a finite d imensional approx imat ion of the i n f i n i te-d imensional plant . In the context of model reference adapt ive control systems, wherein only the output is aval ible for measurement , thi s problem becomes a reduced order adaptive control problem, which is tackled in this paper using the concept of inexact model matching .

The inexact model match i ng st ructure , which calls for only an approximate matching between the pl ant and the model , has been shown to be globally stable (Shah and Monopoli , 19B2b) . This structure does not require a prior knc:Mledge of relat ive degree or sign of the high frequency gain of the plant and thus has tremendous potential for it ' s application to practical systems .

I t is proven in th i s paper that the system error , defined as the d ifference between the plant and the model output , remains bounded under all operating cond i t i ons . Th i s robustness property is crucial in all practical systems , wh i c h d u e to the rea s on mentioned above , a re sub j ect to reduce order modell ing .

Keywo r d s . Inexact model match ing ; unmode l l ed dynamics ; red uced order adaptive control ; bounded error adaptive control .

I . INTRODUCTION

Although cons iderable progress has recently b e e n made on the theo r e t i c a l f ro n t i n des ign ing g lobally stable adaptive control systems for l i nea r , t ime- i nvariant plants , very l i ttle prog ress has been made towards .. appl i c a t i on of these adapt i ve schemes to pract ical systems . The primary reason for this h i atus between the theory and practice of adaptive control is the unrealistic nature of assumpt ions requ i red to be made for the design of these systems .

The design proced ure requi res that relative degree and s ign of the h i gh f requency gain of the plant be known a prio r i . There is one school of thought ( Mo r s e , 1 98 2 ) i n adapt i ve cont rol wh i ch believes that i n the f ramewo rk of convent i onal model refe rence adapt i ve c o n t r o l ( MRAC ) system s , which requ i res exact match i ng between the plant and . the model or asymptotic convergence of t h e t r u e s y s t e m e r r o r to z e r o , t h e s e a s s um p t i o n s a r e i nd i spensab l e . Eve r y practit i oner o f control systems, howerver , k n o ws that these a s s ump t i o n s a r e ve ry d ifficult to establ ish in practice .

The relative degree of the plant is rarely,

49

if ever , known exactly in practice and there i s no establ i shed procedure for determining th i s paramete r . As regards the sign of the h i gh f requency gain of the plant , i t i s a dynamic va r i a b l e that may change during a d a pt i o n . Th i s wo u l d requ i r e that the pol a r i ty change be detected and the system would have to be reset, i f global stability i s to be a s s u r e d . I n l i g h t o f th i s d i fficulty of gathering the prior information about the plant , the wi de gap between the theo ry and pract ice of adapt i ve control i s not so difficult to explain.

Mo r e ov e r , an i nh e r e nt p r o b l em w i th the conventiona l MRAC system pa ramete r i zat i on i s t h a t i t r e q u i r e s t h e l i n e a r , f i n i te-d i mens i onal model to exactly match the und e r l yi ng phys ical system , wh i ch i s essent ially i n f i n i te-d imensional in nature . Th i s fund amental and apparently unresolvable c o n f l i c t between the o b j e c t i ve and the o b v i o u s i m p r o ba b i l i ty of me e t i ng the requi rements of the system to realize that o b j e c t i v e i n p r ac t i c e e x pl a i n s why the practice of adaptive control does not reflect the progress made on the theoretical front .

These l imitations of the conventional �C

50 K . N . Shah

r (t) Dg (S) [)Ts) w

qi � 1+<P� T �i�U

k:

s n - 1 °w(S)

01 (S) Dw (sl

sytstems prompted the suggestions (Peterson a n d N a r e n d r a , 1 9 8 2 ; K r e i s s e l me i e r a n d Na rendra , 198 2 ) to reformulate the adaptive c o n t r o l p r o b l e m w i t h mo r e r e a l i s t i c a s s u m p t i o n s , wh i c h m i g h t a l l e v i a t e requ i r ement o f the s t r i ngent assumpt ions me nt i o ned above . The concept of bounded error adapt i ve cont rol was thus originated . Th i s concept called for a seemingly trivial sh i f t f r om asymptotic convergence o f the system to simpl e stabi l i ty but th i s sh i f t i n t h e o b j e c t i v e o f the system a l lows a s i g n i f i c a n t change i n the eva l uation of practical systems , wh i ch a re now exami ned in terms of ' sati sfacto r y ' or ' acceptable ' behaviour , as opposed to the asymptotic convergence requ i red of the o ther system. It should be po i nted out that in practice , conve rgence to a r eg i on near a po int i s acceptable and o f ten i nd i scernible f rom convergence to a point (Anderson and Johnson , 1982) •

I n s p i r e d by the concept of bounded e r ro r control , Shah amd Monopoli ( l982a) suggested a change in the ph i losophy of control . 'lhe bas ic idea behind the suggestion was that i f l e s s w a s demanded o f t h e a d apt i v e

c o n t ro l l e r , the obj ec t i ve o f allev i a t i ng the rest r ictive requirements mentioned above could be reali zed . 'Ibis suggestion was made in the i r wo rk on ' i nexact model matching ' .

As the name suggests , the matching between the plant and the model is only approximate and the auxi l i a r y netwo r k wh i ch acted as a catalyst only d ur ing the transient stage ,

1 1 +¢�

s n -1

D (S) w n(t)

u1 {t) D u (sl DP (S)

:m(t)

D2 (s) Dw (s)

i s now called upon to provide a part of the model match i ng function . 'Ibis enabled Shah and Mo nopo l i ( l 982b) to des ign a g loba l l y stable adapt ive control system without any knowl edge of the relative degree or the sign of the high frequency gain of the plant .

One o f the basic tenets in the design o f MRAC systems is that a n a pr i o ri knowledge of exact order o f the plant i s aval ible to the designer. In view of the fact that the Irod.el i s m e r e l y a n a p p r o x i m a t i o n o f t h e

i n f i n i te-d imens ional plant , violat ion of the s a i d tenet seems mo re l i kely to be a rule than an exception in practical systems. Henc e , the relevant question to be answered by the d e s igner in the case of violat ion of the tenet would be : How does the MRAC system , wh ich has been proved to be globally s t a b l e w h e n t h e mo d e l o r d e r i s n o t

u n d e r e s t i m a t e d , s u r v i ve i n r e a l - l i f e a p p l i c a t i o n s w h e r e th i s cond i t i o n i s unque s t i o n ab l y v i o l a t ed ? Th i s issue i s addressed i n the text of the paper .

The key feature of the approach is that the objective is defined in terms of ' acceptable ' behaviour . In other wo rds , the goal is to e n s u r e t h a t t h e s y stem e r r o r rema i ns bo und ed . The r e s u l t s d e pend ma inly on c h a r a c t e r i z a t i o n o f t h e p l a n t mod e l m i smatch . The characterization used i n the pape r i s s i m i l a r to the one adopted by Ioannou and Kokotovic ( 19 8 2 ) , wherein the

model description involed only the slow modes of the plant and the mi smatch between the

Inexact Model Matching Structure 5 1

p l a n t and the mo d e l i s in terms of the f a s t e r m o d e s o f t h e p l a n t . T h e

pa r ame t e r i zat ion developed in th i s paper enables the author to develope a structure , which is robust to any mismatch between the p l a n t and the model . It has been proved that the bound on the system e r ro r , wh ich i s p r o p o r t i o n a l t o t h e m o d e l l i n g

i n a c c u r a c i e s , w o u l d b e f i n i t e . A quant i tative measure of the bound in terms o f the m i sma tch , ho wever , i s yet to be determined .

In se c t i o n I I , the problem i s fo rmulated in mo re r igorous terms and the assumptions made i n the d e s i g n o f the p r o b l ems a re stated . Section III contains a mathamatical d e s c r i pt i on of the system struc ture and deta i l s of the parameterization which yields a s o l u t i o n to the p r oblem of unmodelled d ynam i c s . Sec t i on IV conta ins the proof of stabi l i ty . Simulation results have been appended at the appropriate j unctures .

I I . DESCRIPTION OF THE PROBLEM

Th e p l a n t to be adapt ively contro lled i s descr ibed in terms of it ' s transfer function Wp (s) , where

w (s) p

Du (s)

Dp (s) (2. 1 )

Dp ( s ) be i ng a man i c pol ynom ial of degree n+k , where n is the number of modelled poles and k the number of unmodelled poles ; Du (s) is a polynomial of degree m (< n-1 ) .

The model , wh ich conta i ns a descr iption of o n l y t h e s l o w mo d e s of the pl an t , i s desc r i bed in terms of it ' s transfer function Wn (s) , where

Dg (s) Wn (s) = (2. 2)

Un (s)

Dm ( s) be i ng a hurw i t z polynomial of degree n and Dg ( s ) a hurwi tz pol ynom ial of degree r (�n-k-1 ) .

T h e d e s i g n p r o b l em i s to synth e s i z e a parameter adaptive control system which will ensure that the tr ue system error e ( t) and all the other signals in the system remain bounded .

In a d d i t i on to the above assumpt i ons , i t i s a s s u m e d t h a t Wp ( s ) i s a m i n i m a l r e p r e s en t a t i on o f t h e plant ; Du ( s ) i s a hurwi tz pol ynom ial and that an upper bound k on the number o f unmodelled poles of the plant is known .

In the d e velopement of the algo r i thm , no assumpt ion is made about the relative degree n* o f the plant . The system is designed assuming n*=l and s ince in general n* could be g reater than 2 , the aux i l l i a ry netwo rk is always used (.Monopol i , 1974 ) .

I I I . MATHEMAT I CAL DESCR I PTION OF THE SYSTEM

Henc e f o r th , the system w i l l be desc r i bed i n t e r m s of i t ' s input-output pa i r . The p l a n t i s d e sc r i bed by the d i f f e rent i a l equation :

Dp (p) yp (t) = Du (p) ul ( t ) ( 3 . 1 ) yp ( t ) and ul ( t ) be ing the output and input o f the plant respectively : p=d/dt be ing the differential operator and

n+k n+k-1 Dp (s) = s + a s +

and

1 • • + a s + a ( 3 . 2)

n+k-1 n+k

n+k-1 Du ( s ) b s

s

n+k-2 + b s i

b s + b ( 3 . 3) n+k-2 n+k-1

+ • • • +

The model i s governed by the d i ffe rent i al equation :

Un (p) ym ( t) = Dg (p) r (t) ( 3 . 4 ) where ym ( t) and r ( t) are the output and input to the model respectivel y . The refe rence i n p u t r ( t ) i s a s s umed to be p i e c ew i s e conti nuous and un i formly bounded , Un ( s) and Dg (s) being given by

n n-1 Un (s) = s + a s +

Dg (s)

+a s m (n-1 )

r

m ( l) + a

m (n)

r-1 = s +a

r ( l) s +

( 3 . 5)

• + a r ( r )

- ( 3 . 6) The m o d e l o r d e r i s enhanced to n+k by

m ul t i p l y i ng numerator and denom inator of the transfer func t i on by a polynomial Dk (s) of degree k. i . e . ,

Wn ( s )

where

Dg (s) Dk (s)

Un (s) Dk (s)

k k-1

( 3 . 7)

Dk ( s) = s +a s + • • + a ( 3 . 8 ) k l kk

The input-output desc r ipt ion of the model would then be :

Un ' (p) ym (t) = D::J ' (p) r (t) ( 3 . 9 ) whe r e Dm ' ( s ) = Dm ( s ) D k ( s ) and Dg ' ( s ) = Dg ( s ) Dk (s) .

A stable f i l ter Dw ( s ) of o rder ( n+k - 1 ) is so chosen as to preserve the strict positive rea lness ( S . P . R . ) cond i t i on . An auxil iary network (Monopo l i , 19 7 4 ) wh ich is requi red to avo i d pure d i fferentiation i s introduced and is described by the equation .

Un ' ( p) y (t) = Dw (p) w ( t ) ( 3 . 10 )

w ( t ) a n d y ( t ) be i ng the input and output of the aux i l i a ry netwo rk respectively. '!he t r ue system error e ( t ) =ym ( t ) -yp ( t ) would then be described by the equation :

52 K.N . Shah

-1 tm ' (p) e (t) = Dw (p) [ D;J ' (p)Dw (p) r ( t) -

-1 -1 Du (p) Dw (t) ul (t) + D (p) Dw (p) yp ( t ) ]

( 3 . 11)

where D�(s) = tm (s ) -Dp (s) .

Augmented erro r , which is defined as ,( t ) =e ( t) +w ( t ) , would then be goverend by

the differenc ial equation : -1

r:m• (p) '\ (t) = Dw (p) [ D;J ' (p) Dw (p) r (t) + -1 -1

w ( t) + Du (p) Dw (p) ul (t) + D�(p)Dw (p) yp ( t) ] - ( 3 . 12)

Control input ul (t) is generated as follows : -1

ul (t) = D;J ' (p) Dw (p) r (t) - u2 (t) ( 3 . 13) or

-1 D;} ' (p) Dw (p) r (t) = ul (t) + u2 (t) ( 3 . 14)

Substi tuting for Dg ' ( p ) r ( t ) i n ( 3 . 1 1 ) , we obtain

r:m • (p) '\ (t) = Dw (p) [ ul (t) + u2 (t) + • • -1 -1

D�(p) Dw (p) yp (t) - Du (p) Dw (p) ul (t) ] - (3. 15)

The design problem now is to synthesi ze the signals u2 (t) and w ( t) so that

( i ) 'l (t) -> 0 as t -> oo ( i i ) e ( t) remains bounded for t>0

As ment i oned ea r l i e r , the auxil iary network i s c a l l ed upon to a i d the model match i ng f unc t i o n a t h i g h f r equenc i e s . Th i s i s ach i eved by feed i ng the (n-l ) th derivative of ul ( t ) pa rtially to the auxiliary network and the rest as a feedbac k signal . Th i s con s t i t u tes the basic d i f ference between the system under scrutiny and a conventional MRAC system.

In mathematical terms, syntheses of signals can be described as :

1 w ( t ) = - (--2 ) k (t) cp (t)

1+ ¢N N N (3 . 16 )

and 2 q:,N u2 (t) = - (--1 )K (t)

l+ <I:>N N T

T + k (t) <I:> (t) •

1 u

+k (t) <I:> (t) ( 3 . 17)

T T

2 y

the two

k and k a re the vectors of adaptive gains - 1 2 of order (n+k-1) and (n+k) respectively;

T T q:, (t) and ¢ (t) being the vectors of - u -y

derivative of ul ( t ) and yp ( t ) respectively and

n+k-1 -1 .:p (t) = p Dw (p) ul (t) N

ul (t) can be written as :

(3 . 18 )

1 n+k-1 p

n+k-2 ul (t) + (3 p + • • + (3

1 n+k-1

Dw (p) ( 3 . 19 )

Combining ( 3 . 15) - (3 . 18 ) , we get

rm • (Pl 11 (tJ = ow (pJ [ a (tJ <I:> (tJ + • • N N

T a ( tJ<P (tJ - 1 u

T + a (tJ<I:> (tJ

-2 y T T

(3. 20 )

where [() (t) , () (t) , () (t) ] i s the N -1 -2

parameter error vector .

F i g . 1 d e p i c t s the block d i ag ram o f the system.

IV. PROOF OF STABILITY

It is wel l -known that for a system described by equation of the same form as ( 3 .2� ) , 2

integral adaptive laws (!lot>nopol i , 1974) would yield the following results :

i ) 71 (t) -> 0 as t -> GD . and i i ) The pa ramete r error vector would be bounded .

N o w , 71 ( t ) = e ( t ) + y ( t ) and r1( t ) i s u n i f o r m l y bound e d . e ( t ) and y ( t ) c a n the refore g row wi thout bounds only a t the s a me r a te and i n oppo s i te d i r ec t i o ns . l\ot>reover , e ( t) = ym (t) - yp (t) , which impl ies that if e ( t) is growing without bounds, yp (t) must a lso be g rowing unbounded at the same rate , s i nc e ym ( t ) i s uni f o rmly bound ed . In mathematical terms ,

sup { e ( r ) } - sup{y ( r) } - sup{yp ( r ) } � T t) T t) T

- ( 4 . 1 ) -

w ( t ) i s the i nput to the auxil iary network , whose transfer funct ion is IM ( s) /I)n ' (s) and s i nc e the n e t wo r k i s S . P . R . , i f y ( t ) i s g ro wi ng w i thout bound s , so must be w ( t ) (Vidyasagar and Desoer , 1975 ) . But ,

1 w ( t) = - (-'"2) K (t) q:, (t)

1+ ¢ N N N - ( 4 . 2)

The s i g n i �icance of the pos i t ive def i n i te term 1 + ¢;.rin the dei:ominator b;comes . clear at this poih"t . k (t) i s an adaptive gain

N wh i c h i s un i fo rmly bounded . Hence , even if <Pr£tl were growing without bound s , the

1 n+k-1 n+k-2

2

IM (s) = s + (3 s + + (3

e .g . k (t) i

1 n+k-1

- A ry (t) ¢ (t) i ui

• .03

. 02

i DI

I 0

� -.DI

-.02

-.03

.06

. 05

.04

.03

.02

m

- .01

.DI

.QO<l

.ooe .007 .006

-.OD!

0 "' C(,AIN> -.0001

-.00

c.0003

-.0008

-.D009

ASCSP-C

Inexact Model Matching Structure 53

P\..tMT ' W�) • (a+&.::� •• �

......_ , w.to> • � Dw(•) • (••ZX••lt)

5 10 T'.1'£ C SECCNOS >

D1(A) • Knl + K11

6 TIME C S€<DNOS ) 10

D,(o) • K_.s1 + Kns + Ku•

5 IC ,..... CSECQolOS>

'nME <SECCN>5>

Fig ,

.o

.0

.01

I � - .DI

-.02

-.03

-.DI !i � tll

- .02

-.03

.0007

.oco6

.0

.coo.

.0003

.O<Y.Q .DODI �� o �8 ,0001

c0002 -.0003 -.rt:»<

-.0005

-.0006

.C2

.01

-.Cl - .C2 -.C3 -.04

i - .05 � -.06

z !..: -.o -.OB

-.09 ,10

-.11

-.12

-.13

2

R£F'EAl:NCE l ... PUT • SQUIUil:R WAVE AMPLlTUD& < 1 UNIT l Sl:COl\ID

6 JC TIME < SECONDS)

D,(s) . Kus + 1<11

TIME C SECONDS> 10

5 JO TIME CSECONDS)

5 lO TIME (SECONDS)

54 K.N. Shah

t e r m l + �r would ensure that w ( t ) rema i ns bound ed . 1lf w ( t ) i s bounded for all times t > 0 , so must be y (t) and hence from ( 4 . 1 ) , e ( E) and yp (t) must also remain bounded .

The parameter error vecto r has been shown to be un i fo rmly bounded . Al l the s i gnals i n the system can , therefore grow no faster than an exponent i a l , i . e . , a l l the signals must belong the space L� . Equations ( 3 . 13 ) and ( 3 . 17 ) can be combined to get an equation f o r u l ( t ) i n t e r m s o f i t ' s f i l t e r ed d e r i va t i ves . From th i s equa t i on , i t can be ve ry eas i l y shown that ul ( t ) must also belong to this space . 'Ihe proof of stabil ity presented by Narend ra , Lin and Valavani ( l980 ) and i n particular Lemma 5 enables the author to conclude that the infinity norm of ul (t) must be bo unded by the i n f i n i ty norm o f yp ( t ) , provided the plant i s minimum-phase . yp ( t ) has been shown to be bounded . Hence u l ( t ) must also be bounded . I t can then be shown that

i ) //yp (t) //00� Kp ==> // <Py ( t) //00.5_ Kp i i ) //ul (t) //00 .5. Ku =>// <Pu ( t) //00 .5. Ku

- (4 . 3)

Proof of stab i l i ty for the ' inexact ' nndel match i ng s t r uc t u r e , when the system i s subj ect to reduced o rder nndelling has thus been establ ished .

Computer simulations of a third order plant , when i t ' s order i s unde r-estimated by one r ev e a l e d that the results we re i ndeed i n c o n f o r m i t y wi th the theo ry. Fig . 2 shows c o n v e r g e n c e o f t h e system e r r o r , the augmented error and the adapt i ve gains o f the system.

V. CONCLUSIONS

The primary cont r i bution of th i s paper is development o f a globally stable ' i nexact mod el match i ng ' adaptive control algorithm, when the system be i ng controlled is subject to red uced order mode l l ing . The advantage the said a lgo r i thm has above all the others i s p r i nc i pally d ue to the el im ination of two of the ma j o r restr ictive assumpt ions­prior knowl edge of the relative degree and g a i n o f the h i gh f r equency g a i n o f the p l a n t . In a dd i t i o n , the a l g o r i thm has seve ral seconda ry benefits - an improvement in the speed of convergence and in certain cases requi rement of fewer adaptive gains . Mo r eove r , the structure a r i s i ng from the algori thm i s cons id erably simpler than that of the standard MRAC systems .

REFERENCES

Anderson , B . D. O. and Johnson , C.R.Jr . ( 1982 ) . on · red uc ed-o r d e r a d apt ive o u tput e r ro r ident i f i cation and adaptive I IR filtering . IEEE Trans . Automa t . Contr . Vol . AC-27 , No . 4 , 927-932.

Ioannou P.A. and Kokotovic , P.V. ( 1982 ) . An a s ymptot i c error analys i s of ident i f i ers and adaptive obse rve rs in the presence of

pa r a s i t i c s . I EEE Trans . Automat . Contr . , Vol . AC-27, No . 4 , 921-926.

M o n o p o l i , R . V . ( 1 9 7 4 ) . Mod e l r e f e rence a d apt i v e control with an augmented e r ro r s igna l . I EEE Trans . Automat . Contr . Vol . AC-19, 474-484 .

M o r s e , A . S . ( 1 9 8 2 ) . Recent probl ems i n pa rame t e r a d a p t i v e control . Proceedings of the o pt i m i z a t i o n days , Un ive rsity of Montreal , 6-7 .

Narendra , K. S. , Lin. Y . H. and Valav?f1i , L.S. ( 19 80 ) . Stable adapt ive controller design,

Pa r t I I : P roof of stabi l i ty . IEEE Trans . Automat. Contr . Vol . AC-25, No . 3, 440-448 .

Shah , K . N . and Monopoli , R.V. ( 1982a) . Model reference adaptive control with inexact iiiOdeI match i ng . Proceed i ngs o f the ACC, Vol . 3 , 840-847.

Shah , K. N . and Monopoli , R.V. ( 1982b) . A new scheme for MRAC with i nexact model match i ng , P a r t I I : P r o o f o f s t a b i l i ty . Proceed i ng s o f t h e Workshop o n Adapt ive Control , 363-376.

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROBUSTNESS OF INDIRECT ADAPTIVE CONTROL BASED ON POLE PLACEMENT DESIGN

L. Praly Centre d'Automatique et lnformatique, Ecole Nationale Superieure de Mines de Paris, 77305 Fontainebleau,

France

Abstract . We study the robustness of an indirect adaptive control scheme based on pole placement design with respect to unmodeled dynamics , non linearities , time variations or to ill-modeled measurement disturbances . The known results about this problem show that classical adaptation mechanisms have to b e modified. Here introducing a regularized normalized least squares algorithm with a projection, we state a boundedness property in presence of mismodeling quantified in terms of noise to signal ratio . However an extra condition about the controllability of the adapted model is required.

Keywords . Adaptive control ; closed loop systems ; control theory; dis crete time systems ; iterative methods ; parameter estimation; pole placement ; stability ; system order reduction; robustness .

INTRODUCTION

Most of the proofs of stability of adaptive control algorithms available today have been established for linear time invariant plant with known order and well modeled disturbances (bounded or movering average ) . There still remains a significant gap b etween these theo­retical methodologies and the potential appli­cations . In particular it is important to determine the robustness of adaptive schemes with respect to unmodeled dynamics , non lin� arities , time variations or to ill-modeled measurement disturbances .

Several attempts have been made to formulate and analyse such problems (Kreisselmeier, 1 982 ; Gawthrop , Lim, 1 982 ; Praly , 1 983 a, 1 983 b ; Ortega , Landau, 1 983 ) . Among them let us mentionn Ioannou and Kokotovic (1 982 ) who study a singularly perturbed continuous time MRAC scheme and using a 1yapounov formulation exhib it an upperbound of the admissible para­sitic time constants in terms of initial conditions . Kosut and Friedlander (1 982 ) study an MRAC scheme for a plant with known DC gain and relative degree less than one and apply I/O stability concepts for interconnected blocks to characterize plant uncertainty by conic sector .

However the assumptions required in these results are still too restrictive. In fact, as mentionned by Rohrs and co-workers ( 1 982 ) , one of the major difficulties is due to the existence in classical adaptation mechanisms of infinite gain operators (see Remark 2 b elow ) : the operator Ji b etween the output error and the adapted pa�ameters , the operator Ji between the output error and the estimation e¥ror. Unfortunately the inver!l'e gain of J:Je limits the admissible unmodeled effects

55

(Ortega, Landau, 1 983 ; Gawthrop, Lim, 1 982 ) . And, in the presence of unmodeled dynamics, � may produce unbounded adapted parameters . I� may follow unboundedness of the complete system as mentionned by Egardt ( 1 979 ) or it makes inaccurat� the cornerstone . assumptions used by Kreisselmeier ( 1 982 ) .

To limit the gain of Ji , Egardt ( 1 979 ) and Narendra, KreisselmeierP (1 982 ) propose to keep the estimated parameters inside a compact set using a proj ection. This solu­tion only requires an a priori bound of these parameters (necessarily introduced by computer ) and does not modify the initial control ob jective. About the gain of H , we have proposed to use a normalized l�ast squares algorithm as adaptive mechanism (Praly , 1 983 a , 1 983 b ) . As a consequence the gain of � is bounded and the unmodeled effects are c!Jracterized in term of nois e to signal ratio .

To show how vrojection and normalized least squares algorithm are sufficient to get robustness o.f adaptive schemes with respect to a very wide class of unmQdeled effects, we will here study the indirect adaptive control scheme based on pole placement design proposed by Goodwin and Sin (1 981 ) . In Praly, 1 983c ) such a s tudy is presented for a direct adaptive control scheme.

ROBUSTNESS PROB LEM STATEMENT Consider a plant with u (t ) , y (t ) as scalar input and output respectively. We define a model by choosing an integer n ani a vector e

6 = (-a1 • • • -anb 1 • • • bn )T ( 1 )

56 L. Praly

We call residuals the error w(t ) b etween the true output y(t ) and the modeled output :

w(t ) = y(t ) - eT\t> (t ) (2 ) where \t> ( t ) is the following vector

\t> ( t ) = (y (t-1 ) • • • y ( t-n )u(t- 1 ) • • • u(t-n ) )T

-1 - 1 (3 ) Let A(q ) ,B (q_ ) be polynomials defined from e as :

(4 ) (5 )

The following assumption about the plant will be used :

AP : Given an integer n, a vector e and a positive scalar p , there exists 0unkno� relatively prime p8lynomials A* (q_-1 ) ,B* (q- ) such that :

(6 ) ii ) The corresponding residuals as defined by

eq_ , (2 ) satisfy

lfil:tll � where :

(7 )

s (t ) = a s (t-1 ) + Max { \ \1t> ( t ) \ \ , s } (8 ) 0 < (f < 1 • s > 0 ( g )

Inequality (7 ) characterizes a very wide class of unmodeled effects : w ( t ) may contain nonlinearities f (y (t-i ) ,u (t- j ) ) , higher order t erms ( a +.y (t-n-i )+b .u (t-n- j ) ) n i

n+J or time variations ( (a .-a . (t ) )y ( t-i )+ +(b . -b . (t ) )u (t- j ) ) • • • i i ]. J

With this assumption the robustness problem may be formulated as follows : find an adaptive control law such that

i ) :iiri > 0 : TJ <n = u ( t ) ,y ( t ) are uniformly bounded . w

ii ) If there is no residuals , the o�tput y ( t ) tracks some reference output Y ( t ) as "well" as possible.

Note that the second part of this problem deals with a tracking property with its inherent problems of delays and non minimum phase.

ADAPTIV E CONTROL

Following Goodwin and Sin ( 1 981 ) , let b e a strictly stable polynomial

M( -1 ) M -1 M -2n+1 A q = 1 +a1 q + • • • +a2n- 1 q ( 1 0 )

For any vector e (with the primeness condi-tion ) , we define a vector <)i :

<Ji=(f • • • f 1 1 e1 • • • e 1 )T o n- n- ( 1 1 )

as solution of the following linear system (Diophantine equation) :

0 b 1 0 M a1 a1

0 <)i ( 1 2) b b1 a a1 n n

b a aM . n n 2n-1

We note symbolically

a( e )q, = e ( 1 3 ) To solve the robustness problem, we propose the following indirect adaptive control scheme :

v(t ) = y ( t ) - e ( t-1 )Tlt>(t ) ( 1 4 ) g(t ) = µs��!� + lt>(t)TP(t-1 )$(t) ( 1 5 )

e ( t- t) = e (t-1 ) + g(t )P(t-1 )lt> (t )·v(t ) ( 1 6 )

P(t-t)=P ( t- 1 )-g(t )P(t-1 )\t> (t )\t>(t )TP(t-1 ) ( 1 7 ) e (t )=e +Min l 1 , p(t ) } (e (t-.L)-e ) ( 1 8 ) o

1 1 e (t-t)-eo 1 1 2 o

p ( t ) ;a. p ( t-t) ( 1 9)

<)i ( t ) = a( e ( t ) r 1 /3 (20 ) <Ji(t )

Tlt>(t+1 ) = E(t )AM(q_-

1 )yM (t ) (21 )

E ( t )

1 if IB ( t , 1 ) I > E i31t,"1 )

if not (22 )

where i ) eq_, ( 1 9 ) means that P ( t ) is any positive symetric definite matrix greater than P (t-2.J and such that : 2

0 < A0 � \min P ( t ) � \max P(t ) � A1 (23 ) For instance we can take

P ( t ) = � P ( t-t) + ( 1 -� ) A1 I

ii ) a (t ) , p ( t ) are chosen such that

0 < a � a ( t ) � 1

p (t-1 )-� l \ e ( t-t)-e (t-1 ) \\

(24 )

(25 )

� p (t ) � p (26 )

det a( e ( t ) ) '" o n

iii ) B ( t , 1 ) = l: b • ( t ) i=1 ].

(27 ) (28)

Robus tness of Indirect . Adap tive Control 57

Remark 1 : Eq. (27 ) is always possible if det d( e ) and det a��(o ) ) are different from

0 zero . We have the following Lemma 1 : sub ject to assumption AP , we · have

(E1 ) l i e ( t )I\ ..; Me V(q,k ) ' �+k l�I

t =q +1 "S{tl° ..; Vk M +k L Tlw v v

q+k l(q,k ) , l: 1 1 e (t )-e (t-1 ) i i t=q+1

(E2)

(E3 ) where Me, M , Me are positve independantsv of T) and :

constants A w

L2 = 1 + -1-v µ (29) 2

2 2 A1 Le = (2+i; ) (µ+A1 )µ (30 ) Proof : see appendix. Remark 2 : As discussed in introduction, the pro jection (1 8 ) limits the gain of lip : �f !j - e (t ) and the presence of s (t ) in eq. (1 5 ) limits the gain of )f • �(_tl - (e(t )-S* )T<P(t ) • In particular , e · ;-ct) s (t ) )f is exterior to the c9nic sector with Q�nter - 1 and radius L (compare with

v Ortega, Landau ( 1 983 ) ) . Lemma 2 : Sub ject to assumption AP , if there exists a strictly positive constant o such that : I det a( e ( t ) ) I ;;. o

then we have : \ llJ; (t )I \ ..; Ml!; ( C1 )

Vi ..; n ' lllJ;(t )-IJ;(t-i ) JI.,;; Llj;l\e (t )-e ( t-i ) l l . (C2 )

Proof : with assumption (31 ) IJ; (t ) is a differentiable function of e (t ) .

(31 )

Remark 3 : The choice of a(t ) , p ( t ) such that ineq. (27 ) is met , does not prevent lim inf det a( e ( t ) ) frr.: 111 being null . t - oo Therefore assumption tion to be considered bo11.r_dedness study.

(31 ) is an extra condi­for the forthcoming

With these bounds Me , M , Mdi and these gains Le � Lv ' Lili, we are in po�ition to state our main result :

Theorem : Sub ject to assumption · AP { if ada:(Jtive scheme defined by eq. ( 1 4 ) to (22 ) is applied and is such that ineq. is met , then the robustness problem is solved : i ) if we have :

the eq. (31 )

( (M L M )L + II M L \., < ( 1 -$ )( 1 -a ) (32 ) I!;+ 1J; e e I!; v''w 2 y n then u(� ) , y(t ) are uniformly bourded. Here � is the spectral radius of AM(q- ) , y is a positive constant which depends on AM(q-1 ) , II is a positive constant which depends on n, a, � . ii ) Moreover if T)w is equal to zero , then we have : lim AM(q-1 )y (t ) - � b . (t )r (t-i ) = O (33 ) t - "" i=1 l.

r(t ) = E(t )AM(q-1 )yM(t ) (34 ) Proof : see appendix. Discussion : Let us study expression (32 ) . About the control part of the scheme, we have the terms �MIJ;+L�M8 ) , Ml!;. Given our assumption AP ( i . e . Me ,a ) , AM (q-1 ) (i. e. � , y ) should be chosen such that ( 1 - � ) is greater and M , L are smaller . In this stability-robustn�ss �mpromiae , not only ths amplitude of the controller parameters but also its sensitivity with respect to variations of e appear. About the adaptation part of the scheme , we have LR ' L • The less Le , L are, the more rooustvthe scheme is. �wever looking at eq. (29) , (30 ) we see that Le ' L are smaller if !:!_ is smaller i . e. if t�e adap-tation ab�lity is reduced. Therefore to the classical stability-robustness compromise an adaptation-robustness compromise is added for adaptive control schemes .

CONC LUSION We have analysed stab ility of the indirect adaptive control scheme proposed by Goodwin and Sin ( 1 981 ) , when the residual between the plant and its assumed linear model is ill-modeled. More precis ely we have shown the boundedness of the input-output signals when the residual to signal ratio meets :

hltll ..; �} T) (35 ) where w(t ) is the residual , s (t ) is the norm of the input-output signals pass ed through a first order filter and Tl is a bound which can b e computed from the scheme characte­ristics .

To get this result we have b een led to intro­duce a projection and a normalization in the adaptation algorithm. In particular we have shown that these modifications limit the gain of the infinite gain operators mentionned by Rohrs , and co-workers (1 982 ) as

58 L . Praly

leading to instability. This new algorithm APPENDIX reach� the initial control ob jective when there; is no reoidual. Proof Of Lemma 1

As an import...nt consequence of our study , we have �hown that not only the classical stabi�ity-ro�ustness compromise, but also an adaptation-r9bustness compromise has to be made in adap�ive control . Note that for our result to hold , we need an extra condition about the adaptive scheme, It concerns the controllability of the estimated model.

REFERENCES Egardt , B . ( 1 979 ) , Stability of adaptive

controllers . In A.V . Balakrishnan and M. Thoma (Ed. ) , Lectures Notes in Control and Information Sciences , Vol .20, Springer Verlag, Berlin, p .p . 1 58.

Gawthrop, PJ. , and K.W. Lim ( 1 982 ). Robustness of self tuning ·controllers . IEE Proc. Vol . 1 29, Pt.D_, No , 1 _, 21 -29.

Goodwin, G . C . , and K . 8 . Sin (1 981 ), Adaptive control of non minimum phase systems , IEEE Trans Aut . Control , AC-26 , 478-483 .

Ioannou, P.A. , and P. V . Kokotov"'iC"l'1982 ) , Singular perturbations and robust red&­sign of adaptive control . Proc. 21 st IEEE Conf, on Decision and Control. Oralndo . 24-29.

Kosut , R . L. , and B . Friedlander ( 1 982 ) . Perfor­mance robustness properties of adaptive control systems. Proc. 2 1 s t IEEE Conf. on Decision and Control. Orlando . 1 8-23 .

Kreiss elmeier, G . (1 982). On adaptive state regulation. IEEE Trans . Aut . Control, AC-27 , 3-1 7 .

Kreisselmeier, G . , and K . S . Narendra ( 1 982 ). Stable model reference adaptive control in th'.e presence of bounded disturbances . IEEE Trans Aut , Control , AC-27 , 1 1 69-1 1 75 .

Ortega, R .,, and I . D . Landau ( 1 983 ), On the design of robustly performant adaptive controllers for partially modeled systems . Laboratoire d 'Automatique de Gre�ble, ENSIEG , Grenoble, France.

Praly, L. : (1 983a ) . Commande adaptive par modeie de reference : stabilite et robus­tess�. In I . D . Landau ( Ed. ) , Outils et modeles mathematigues pour 1 1 automatigue, l 'analyse de systemes et le traitement du signal, Vol . 3 , Editions du CNRS , Paris .

Praly, L, (1 983b ) , Mimo indirect adaptive control : stability and robustness . CAI-Ecole des Mines , Fontainebleau,France.

Praly , L. ( 1 983c ) . Robustness of model refe­rence adaptive control . 3rd Yale work­shop on applications of adaptive systems theory, Yale University , New Haven, Connecticut .

Rohrs , C . E. , L. Valavani , M. Athans , and G. Stein ( 1 982 ) . Robustness of adaptive control algorithms in the presence of unmodel.ed dynamics:. Proc. 21 st IEEE Conf. on Decision and Control . Orlando . 3-1 1 .

The technique used here is by now standard and we only point out the major steps Let V ( t ) be defined as follows :

v (t ) =(9 ( t )-9* )TP ( t t1 (e (t )-9* ) (A1 ) From eq. (2 ) , eq, ( 1 4 ) to eq. (22 ) , and projec­tion property, the following relations can be derived : v ( t-t) = v ( t-1 ) 2

( ) ( w(t ) ( )2 ) + g t 1 -g(t fi>(t)TP(t-1 )�(t rv t

1 �2 ) V ( t ) .;;; V ( t-2) (A3 ) l l 9 (t )-9*l l .;;; .p (t ) + p0 (M ) l l 9 ( t )-9 (t-1 ) 1 1.;;(2+'1:)g(t ) l v ( t ) l \ IP (t- 1 �(t ) ll

(A5 ) Ineq. (A4 ) directly leads to E1 and with ineq. (23 ) yields the boundedness of V ( t ) , Then eq. (A2 ) and ineq. (A3 ) , (A5 ), (23 ) lead to :

A .!!( 1 -1-) (v (t-1 )-v (t ) ) a µ ;;. r-W-)2 (A6 ) B1D

Use Schwartz inequality to get E2 , E3 .

Proof Of Theorem Notations : Let I\ . \ 1 b e the usual euclidian norm and 1 1 1 . 1 1 1 be any other equivalent norm. We have

(AS ) Lemma A : Let cp(t ), !; ( t ) be sequences of positive real numb ers such that

cp ( t +1 ) .;;; I; � t ) cp{ t ) + M cp

1f(q,k ) , �-+k l;" (t ) .. 'Ilk � + k1t; t=q+1 . If we have o .;; ri , < 1

Then cp{ t ) is uniformly bounded.

(Ag)

(A1 0 )

(A1 1 )

Proof : From assumption (A9 ) , it follows

Robustness of Indirect Adaptive Control 59

q ( II d t) )cp(O ) t=O

+ (1 + � q II /;( t ) )M ;;. cp(q+1 )

(A1 2 ) k=1 t=q+1 -k cp

But with assumption (A1 0 ) , let ;>,, = exp - (1 _ Tli;) 2

We have

k .;; K �

(A1 3 )

q M2 II C (t ) .;; exp (L:L) = M (A1 5 ) t=q +1 -k 1 -nc

I t follows K+1 cp (q+1 ) .;; ;>,,q+1 cp (O ) + (H KM+;>,,

q-K 1 -:>.. )M 1 -:>.. cp

(A1 6 ) �· (A closedloop state space representa­tion ) : Let (-a . (t ) , b . (t ) ) (resp. (f . (t ) , e . (t ) ) b e the i components o f e ( t ) i (resp.

]. . tli ( t ) ) . From the Diophantine equation (20 ) applied to u(t ) and y(t ) , it follows : L:; e. (t ) (y (t-i )-9 (t ) <l> (t- i ) ) i=O J. =AM(q-1 )y (t )

�1 T \

+ � b . (t )q,(t Jw(t+1 -i ) (A1 7 ) i=1 ]. n-1 - L:; f . (t ) (y (t-i )-9(t )Tw(t- i ) ) i=O ].

+ � a . (t )t1i (t fw c t+1 -i ) i=O ].

=AM(q-1 )u ( t ) (A1 8 )

And with eq. ( 1 4 ) , (21 ) , (34 ) this yields : n-1 L:; e. (t ) v (t- i ) i=O ].

n . + L: e . (t ) ( e (t-i-1 )-e ( t )fw (t-i )

i=1 ]. n

+ L:; b . ( t )r ( t-i ) i=O ].

n + L:; b . (t ) ( q,(t )- tli(t-i ) )Ti:t> (t+1 -i

i=1 ].

n-1 - l: f . (t )v(t-i )

i =O ]. n- 1

(A1 9)

+ L: f . (t ) ( e (t )- e (t-i- 1 ) )Tw (t-i ) i=O i

=AM(q-1 )u( t ) n + L:; a . (t )r(t-i )

i=O ]. n

+ l: a . (t ) (q,(t )-tli(t-i ) lw(t+1-i ) (A20 ) i=1 ].

Then let X(t ) b e the following vector X(t ) = (y (t- 1 ) .y: (t-2n+1 )u ( t-1 ) .u(t-2n+1 ) )T

(A21 ) We can rewrite eq. (A1 9 ) , (A20 ) in

(A22 ) where F is a companion matrix with charac­teristic polynomial AM(q-1 )2 ;�t includes the controller 0parameters e. (t ) , f . (t ) ; et includes the estimated �arameters a . (t ) ,b . (t ) ; t.Ft incorporates the following

]. ]. differences : e . ( t ) ( e ( t-i- 1 )-e (t ) ) , b . ( t ) (q,( t )-q,(t-i ) ,

]. ]. f . ( t ) ( e ( t )- e ( t-i-1 ) ) , a . ( t ) ( q, ( t )-<\i ( t-i ) ) • ]. ].

t. ( t ) = ( v ( t ) v ( t-n +1 ) ) T (A23 ) R (t ) = (r(t ) . . . r (t-n ) )T (A24 )

With the strict stability of AM(q-1 ) , there exists a norm I 1 1 . 1 1 \ such that :

� l l l x (t ) l i l l l l x Ct+1 ) \ \ I .;;

+ll l !iFtX( t )+'l\li (t )+etR(t )l \ I (A25 ) with

!;. < 1 (A26 ) �· (some inequalities) : Using E1 , C1 of lemma 1 , 2 and the norm equivalence (AS ) , we have :

y l l l !iFtX(t ) IJ I .;; 2 I J!iFtl l • l l l x (t ) l \ I (A27 ) Y1 11 l �tli ( t ) J l I .;; Y2 M<\i l lli (t ) j j (A28 ) 1 1 1 <\R(t ) J l I .;; y2 M9 \ \R ( t ) I\ (A29 ) From the definitions of <l>(t ) , X (t ) , we have :

Introducing this inequality in the definition of s (t ) yields

s ( t ) .;; o s (t-1 ) + L l \ l x (t ) \ \ I + s (A31 ) Y1 On the other hand , from the definition of t. ( t ) and property E2 of lemma 2 , we have :

'lf(q,k ) ,

�-lk l�t l .,.. _1_ 1 n 6 ""' -=L(VkM -lkL Tl ) (A32 ) t=q+1 s t 0n-1 1 -0" v v w

In the following, let us no te x(t ) = l\ l x (t ) \ 1 1 y = Y2

Y1

(A33 ) (A34 )

60 L . Praly

.§i!.lL.2. (use of Lemma A ) : let us put together ineq. (A25 ) , (A31 ) to get the following system :

(!;t>Vllt:.F t i ! )x( t ) x(t+1 ) .;;;; lit.Ct ) t'V2 (Mq, hlJ-ll s (t )+ Me ilR(t ) l i ) (A35 ) s (t ) .;;;; a s (t-1 ) + L x(t ) + s Y1 With (Al,2 ) and Rroperty E1 , C1 of lemma 1 , 2 , i!t:.(t )J I and lll:.Ftll are bounded, Then SITT there exists Mx' M8 such that :

(!;t>Viil:.Ft ! J t>VMq, ll�f !� li )x (t ) x ( t+1 ) .;;;;

+ a y2 Mq, ll�f!�I \ s (t-1 ) + Mx (A36 )

s ( t ) .;; L x(t ) + a s (t-1 ) + M Y1 B

Let cp( t ) be defined as : <p(t ) = x ( t ) + y1 a( 1-!; )s (t-1 ) (A37 )

We get from (A36 ) :

dt ) = !;t>yllt:.Ft l \+ f:-i;Mq, ll�f � � I I + a ( 1-1; ) (A38 ) Note that from the definition of t:.�t and property E1 , C1 of lemma 1 , 2 , we hAve :

n-1 'I' i=O

M,,. 2:: j[a (t )-e (t-i-1 ) \ \ l > l it:.Ft l i

+ M0 � \ \<li(t )-<li(t-i) I ! (A39) J. =1

Then from property E3 , C2, we get s.-+k 2 �=q+1 \ \t:.Ft \ \"' n (Mq,+Lq,M0 ) (1fk:M0-+k LeTJw) (MO )

Hence to meet assum�tion (A9) of lemma A, using using ineq. (A38 ) , lA32 ) , (A40 ) we 1 et :

M = I + yn2 (Mq,+Lq,Me)'Me !; + .L 1 1 -an M M 1 -1; �1 1-a q, v a

(A41 )

1 -an Mq,Lv ] an-1 (1-t)( 1-a )

(M2 ) Then we conclude that <p(t ) is bounded if assumption (A1 0 ) is met . In this condition s ( t ) is bound.ad and if TJ is equal to zero we obtain eq. (33 ) l'rom properties E2 , E3 and eq. (A1 9 ) .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

AN ON-LINE METHOD FOR IMPROVEMENT OF THE ADAPTATION TRANSIENTS IN ADAPTIVE

CONTROL

M. de la Sen* and I. D. Landau**

*Departmento de Automatica e Informatica,Universidad del Pais Vasco, Leioa, Spain **Laboratoire d'Automatique de Grenoble, Institut National Polytechnique, Grenoble, France

Abstract . A method is proposed to improve the adaptation transients in adap­t ive systems whose philosophy consists basically of the implementation of a "a posteriori" correcting action of a local nature in at least one of the parameters associated with the used adaptation algorithm. To develop this technique, one takes advantage of the freedom in the choice of such parameters , while maintaining the stability properties , in many of the commonly used parameter-adaptive algorithms . As an intermediate step , an equivalent t ime­varying and near-linear system, which is an approximate model for the whole adaptive scheme is derived in order to apply · a bang-bang (or , eventually , linear) mode suboptimal input . Keywords . Adaptive systems ; optimization; bang-bang mode control ; l inear mode contro l .

INTRODUCTION In this paper , one presents a technique, ba­sed on optimization tools , to improve the adaptation transients in adaptive systems which are known to be of poor performances

by A (q -1 ) y (t) -d - 1 q B (q ) u (t) , d > O where

- 1 q-nA + a 1 q + . . . + a nA b b - 1 b q-nB + 1 q + . . . + o nB

( 1 )

and input supplied to b e controlled plant , The proposed method consists essentially in appropriately manipulating some of the para­meters entering the .used adaptation algorithm (subsequently cal led algorithm parameters AP ' s) by taking advantage of the freedom as­sociated with their choice by maintaining the congence properties of the whole adapti­ve scheme . Al though the method is subsequent­ly developed for adaptation algorithms in direct adaptive control , it is appl icable to certain identification schemes and indirect control situations . An "a priori" condit ion for the method applicabil ity is the existen­ce of (at least one) some (time-varying) AP making the scheme asymptotically stable when chosen within a certain domain . The method need not the use of auxiliary inputs (Landau 1 979) but only an on-l ine modification of

with {u (t) } and {y ( t ) } being the input and output sequences and q-1 and d the backward shift operator and plant delay , respect ive­ly . It is wel l known (Landau and Lozano 1 98 1 , Lozano 1 982) that asymptotic tracking and regulation obj ect ives may be independently achieved either in the case of known system parame ters or in the adaptive case (by using asymptotically convergent adaptation algori­thms) . Two adaptation algorithms used in di­rect adaptive control were studied in those papers , namely :

the AP , which can be interpreted in terms of hierarchical control as being the third con­trol level (while the ordinary feedback con­trol is the f irst level and the adaptive control is the second level) .

DERIVATION OF THE ES Consider an unknown SISO discrete l inear time-invariant minimum phase plant described

ASCSP-C*

6 1

Algorithm 1 (Lozano 1 982) 9 (t) = 9 (t- 1 ) + F (t) ¢ ( t-d) Cl < t ) < e <t- 1 ) . <t><t-d) >J c (t ) + < ¢ (t-d) , F (t) ¢ (t-d) >

9 (0) e 0

( 2 )

62 M . de la Sen and I . D . Landau

Algorithm 2 (Landau and Lozano 1 98 1 )

S (t ) = S ( t - 1 ) + F (t) cjJ ( t-d) H (q - 1 ) F A

[y (t) - < 8 ( t - 1 ) , p(t-d) >J , 8 (0) 1 + < ¢ ( t -d) , F(t) cjJ ( t-d) >

e 0 (3 )

where H ( z - 1 ) - A /2 (all real 2 > A > max [A( t ) /c (t )] ) is a s trict ly real transfer funct ion , with the recursive equation for the adaptation t ime - decreasing gain matrix being in both algorithms

F (t+ 1 ) = A (�) (F (t ) -

_F_( t_)�cjJ (_t_--'d )_> _< _F-'(_t )'--'-cjJ-'(_t-_d..:...) __ J ' F ( O) > O c (t ) + < cjJ ( t -d) , F ( t ) cjJ ( t-d ) >

( symmetric) (4) Both algori thms were proved to be asymptoti­cally convergent under rather weak assump­tions

(( see Landau and Lozano 1 98 1 , Lozano

1 982) 1 ) . In algorithms before , one uses a measurement vector and a fil tered output de­f ined respec tively as :

cjJ ( t ) = [u (t) , ¢! (t ) ] T = [u ( t ) , u ( t- 1 ) , . . .

u ( t-d-nB + 1 ) , y ( t ) , y ( t- 1 ) , . . . , y ( t -nB)f

yF ( t ) = C (q- l ) y (t) = ST cjJ (t-d ) r with Cr (q - 1 ) being an asymptokically stable polynomial of degree nc and 8 ( t ) in ( 2 ) - (3) , an updated vector esti�te of the extended parameter vector (used in the direc t adapti­ve controll er implementation) :

e = [b , e T] T = [b , b s 1 + b s 2 0 0 . 0 0 0

(5) whe:e th� coeff�c ients r ( . ) and s ( . ) are defined in a unique way (Landau and Lozano 1 98 1 and Lozano 1 982) from the R(q- 1 ) and S ( q- 1 ) polynomials according to the polyno­mial equation :

- 1 - 1 - 1 -d - 1 C (q ) = A(q ) S (q ) + q R(q } r + • • • + c n c r

-n Cr q

- 1 where the degrees of the polynomials S ( q ) are �s = d-1 and nR . = max (n�- 1 , ncr-d) , res­pectively . By applying the input :

u ( t ) = _A_l_ b ( t ) 0

MF A ry ( t + d) - < e 0 ( t ) ' cp 0 ( t ) > J

(6 )

to the plant , one obtains for the f iltered error :

> + 0 as t + oo

< e - e (t ) , ¢ C t)

( 7 )

which implies that E ( t ) (y (t ) in the regula­tion case) vanishes as time increases . From ( 7 ) and ( 2) - ( 3) , one deduces that :

EF ( t+d) - EF (t+d - 1 ) = < 8 - 8 ( t- 1 ) , cjJ ( t )

- cjJ ( t- 1 ) - a (t) ¢ ( t - d) >

where

< cjJ ( t) , F (t ) p (t-d) > c (t) + < cjJ ( t -d) , F ( t ) cjJ (t-d) > for Algorithm 1

a ( t ) = <. l < $ ( t ) , F ( t ) $ ( t - d) > 1 + < ¢ (t-d) , F ( t ) cjJ ( t -d) >

, for Algorithm 2 (H(z - 1 ) = 1 )

(8 )

(9 )

Now , in order to derive an appropriate equi­valent system (ES) for the whole adapt ive system valid for applying the (proposed be­low) opt imization technique to improve the sys tem behavior during the adaptat ion tran­sient , one def ines the "a priori" variat ion interval for the c ( t )-AP : [ o o I s r:o c 1 ( t ) , c 2 ( t) = 15 ( t ) lie ( t ) , c 0 ( t ) +

lk ( t ) ]

compatible with the convergence properties of the used adaptation algorithm (for ins­tance , A (t ) = 1 , c (t ) f: (0 ,00) for Algorithm 1 and c (t ) <S ( 1 / 2 ,oo) for Algorithm 2) . Then , for such an interval , one has in (9) for Al­gorithm 1 (general i zations to Algorithm 2 follow directly) that :

'\; a(t ) = �(t) + t.a (t ) u ( t - 1 ) ( (u (t - 1 )

[- 1 , 1 ] )

with

( 1 0 )

_ 1 [ < cjJ ( t ) , F ( t ) cjJ ( t-d) > �(t ) (t.a(t ) ) - 2 0

c 1 (t ) + < cjl (t-d) , F ( t)

--- + (-) < cjJ ( t ) , F(t) cjJ ( t -d) > ] cjJ ( t - d) > c � ( t ) + < cjJ ( t-d) , F ( (t ) cjl ( t -d) >

( 1 1 )

Defining the polynomial :

one deduces from (8) that

E ( t+d) = E ( t+d - 1 ) - C (q- l ) [ E ( t+d - 1 ) r A E ( t+d - 2)] + < e - e <t - 1 ) , ¢ (t ) - ¢ ( t-O

- a ( t ) cjJ ( t-d) >

( 1 2 )

( 1 3 )

( l ) The symbols < · · · > and · >< · denote inner product of vectors and dyad of vectors .

Improvement of the Adaptation Transients 63

which , together with ( 1 1 ) , leads to the fol­lowing "a priori" ES for the whole adaptive system for all c (t) [c1 (t) , c� ( t )] : x (t ) = A x (t- 1 ) + b (t-1 ) ci'(t-1 ) + w(t- 1 )

(ci'(t- 1 ) f. [- 1 , 1 ] ) ( 1 4 ) where [ c ; A =

I ( 1 5)

with c j 1 -c 1 ; c ! c i- 1 - c . for 2 ... < i � ]. ]. n ' and c ' c c n 1 n r c + c r r x(t) [£ (t+d) , £ (t+d- 1 ) , • • • , £ (t+d-n ) ] T

c r

b (t-1 ) < 6 (t-1 ) - e , �a (t) ¢ (t-d) > e 1

w( t- 1 ) < e - e <t- 1 ) , ¢ (t) - ¢(t-O -°'M(t) ¢ (t-d) > e 1

e 1 = ( 1 , O , • • • , O) T

for both Algorithms 1-2 . OPTIMIZATION OF THE ES AND CORRECTING ACTION THE c ( . ) -AP

( 1 6)

( 1 7 )

( 1 8 ) ( 1 9 )

One establ ishes on a f inite and time sl iding orizon , the quadratic loss function for the ES :

1 J (k) = 2 < x (k+N2) Q (k+N2)x (k+N2) > + k+Nz-1

..!_ { L: < x (t) , Q (t )x (t ) > + r (t) ci'2 (t) } 2 t=k-N 1 for k=O , 1 , • • • (x (-j ) =O , ci'(-j ) =O for al l in­teger j >O) , with Q ( . ) > O and r ( . ) > 0 . Since , due to the nature of the problem at hand there is no energetic reason to weight �( . ) , r ( . ) is selected in such a way that l �< . ) I < 1 in ( 1 0) . Such a strategy al lows to use model ing eqns . ( 1 4) through ( 1 9) , while main­taining a solution of Riccati type assoc ia­ted with the minimization of (20) , which im­plie s an analytical relation between costate and state leading to a direct solution of the problem. So , one has for the ES : Bang-Bang control mode < l �(t) I = 1 ) r (t) = {< b (t) , � ( t+ 1 ) - P(t+ 1 ) [Ax ( t) + w(t)] > J - < b (t) , P (t+1 )b (t ) > (2 1 )

P ( t ) = Q (t) + ATP (t+1 ) ( I b (t ) ><P (t+ 1 )b (t ) - r(t) + < b (t) ,

P(t+1) b (t) > ] A; P (k+Nz) = Q (k+N2) (22)

� (t) = AT � (t+ 1 ) + [Q (t) - P (t ) ] A- l

[< b ( t) , � (t+1 ) > b (t ) + w(t) ] ; r (t) � (k+N2) = 0 (23)

�(t) = sgn {< b (t) , � (t + 1 ) - P (t+1 ) [Ax (t) + w(t) ] ) > } (24)

which needs the auxil iary condi tion that in (2 1 ) , r (t) > 0 in order to maintain the co­herence of (20) . If r (t) � O , one implements the subsequent linear control mode : Linear control mode ( j �(t ) I < 1 ) r (t ) = a (t ) ( smal l positive real number com­patible with the computer division by zero

( 25 ) P (t ) and � (t) are determined as in (22)- (23) and u (t) is now given by :

�(t) = < b (t) , � ( t+ 1 ) - P (t + 1 ) (Ax ( t) + w(t)] > r (t) + < b (t) , P ( t+ 1 ) b ( t ) >

(26)

Note that while in (26) , r ( t ) is specified "a priori" , in (24) is selec ted as a norma­l izing term of the input in ( 2 1 ) . So , its usefulness in this case consists in maintai­ning the Riccati type solution ( in other words (24) is equivalent to (26) using ( 2 1 )). With this philosophy , l inear mode control is only an alternative to incoherences in the bang-bang control mode appl ication . Now the question arif' hk!i s t o correct the c ( . ) -AP , based upon thed'Opt imization results for the ES . So , by equating eqns . (9)- ( 1 0) for Algori thm 1 (extension to Algorithm 2 fol lows trivially) , one obtains easily for the "a posteriori" AP :

p c� (t) c� (t) + [c0 (t) - �c (t) �(t- 1 )] c (t) = ����������������-

c0 (t) + �c (t) �(t-1 ) + < ¢ ( t-d) , F (t) ¢ (t-d) > � (2] ) < ¢ (t-d) , F (t ) ¢ (t-d) > [c� (t) , c� (t)]

About optimization scheme ' s suboptimal ity The proposed scheme has the following cha­racteristics : ( 1 ) The horizon associated with ( 20) is cons.­tituded by a correction part· [k-N 1 ,k] and a prediction part [k , k+N21 · The second one is used for natural optimization purposes . Note that the ES-model ing ( 1 4 ) through ( 1 9) toge-

64 M. de la Sen and I . D . Landau

ther with the adaptive prob lem ( 1 ) through (4) and (6 ) imply that one must work with predict ions on (k , k+N2) (deta ils are given below) . To partly compensate the assoc iated drawbacks , one takes the correction subhori­zon in which one uses real measurements . This sub-problem has not an expl ic it optimizat ion obj ective , but more exactly to make more caut ions the global strategy . Fortunately , the dual (bang-bang + l inear within the bang­bant range modes) control for the E S contri­butes to that strategy , since , according to (27 ) , it l imits the variations of the c ( . ) ­AP . (2) The ES-modeling ( 1 4) through ( 1 9) is not really l inear because of the coupl ings bet­ween its inputs and parameters resulting from the correctings strategy (27) . However , for s impl icity in the calculat ions , and com­puting t ime and computer memory capab i l ity reasons , l inearity is assumed and those cou­pl ing effects are neglected. Furthermore , by the reasons argued before , this model is inaccurate on (k , k+N2 ) . These reasons make the opt imization scheme suboptima l . On the other hand , the l inear mo­de leads always to a certain degree of subo­ptimal ity arising from the degree of arb itra­riness in the choice of a ( . ) in ( 25) . The f inite and sliding nature of the horizon a­rises d irectly from the problem nature ( s in­ce the future ES- parameters are unknown) and the inherent suboptimality of the sche-me . Approaches to the prediction problem

For model ing the ES and in order to be ab le to apply its assoc iated mode ling technique on (k , k+N2) , three approaches have been u­sed : Approach 1 : The Heuristic prediction method (HPM) , wh ich computes the future filtered output based on Taylor series expansions u­sing a f inite difference approach to appro­ximate derivat ive s . Approach 2 : The reference model aided pre­diction method (RMAPM) , which computes the f il tered output estimates using the extended vector , in a s imilar way to (5) , of the re­ference model . Approach 3 : The predict ion method based on the current parameter estimates (PEPM) , which uses for predict ion the updated para­meter vector of the adaptation algorithm. In order to l ead to caut ious prediction stra­tegies , the init ial statements of the me­thods are made on the basis of computations relat ive to the f il tered outputs . Experimen­tal work has corroborated the goodness of such a philosophy . In all cases , the output es t imates are computed from the f iltered output es t imates and the input by applying (6 ) .

Basic design rules

( 1 ) a (t) must be chosen as small as possib le, compat ible with the used computer zero per­formances , to design the l inear mode control for the E S . ( 2 ) The " a priori" variation interval for the correcting c (t )-AP action , namely , l·· o o 1 /::; l o o c 1 (t) , c2 (t) . = c ( t ) - /::;c (t) , c (t ) + /::;c ( t) I must be chosen with a small /::;c (t) .

(3 ) The convergence conditions of the used adaptat ion algorithm must be respected . In this way , asymptotic convergence is ensured. (4) If by the "on-l ine" c ( . ) -AP modif ica­t ions the stab i l ity are violated , the c ( . ) ­AP i s restarted t o an admissib le value . Discussion of results - In general , the HPM works better than the ARMPM and PEPM. This may be motivated by the fact that while the HPM constructs output estimates from its previous outputs and es­timates , the ARMPM takes as zero the predic­ted errors and the PEPM is only valid , by its proper nature , when the parameters of the reference model and the plant are only local ly deviated from each other. However , s ince the HPM do not take into account in any way u (k-d) when predict ing y (k) (u (k-i) , i > d) they are considered , in some sense, through y (k-j ) , j > 1 ) , some cautious stra­tegy for predic tion must be taken when the input varies greatly from its prior value or when the output sign changes . For instan­ce , in such isolated points either the ARMPM or the PEPM were taken in the simulated e­xamples . - If the variation of /::;c ( . ) is very small , the advantages of the use of the optimiza­tion-correct ion method are not signif icati-ve . is very large model ing errors in the ES can translate into erroneous optimi­zat ion errors which may lead to contrary effects to the suitable ones . Then , a trade­off between these undesirable facts must be chosen , for instance , (according to the si­tuat ions) from 1 0% through 20% of the nomi­nal AP-value for the HPM and under 1 0% for the ARMPM and PEPM. A real-time modifica­t ion according to the registered scheme ' s performances can be useful . This philosophy may be also applied to a real-time choice of the adequate l ength of the optimizat ion­correction horizon . - In general , the improvement in the track­ing error is achieved at the expense of greater inputs to the plant .

CONCLUSION In this paper , an optimization technique to improve the adaptation transients in adapti­ve systems has been given . The proposed me­thod consists in model ing the whole adapti-

Improvement of the Adaptation Transients 65

ve scheme by means of a near-l inear system whose inputs are related to two pos s ible choices of the so-called algorithm parame­ters ( i . e . , those parameters entering the adaptation algorithm having a variation do­main which maintains the convergence proper­ties of the whole adaptive scheme) . Then , by opt imizing these inputs and by translating these opt imizat ion into "a posteriori" modi­ficat ions of the algorithm parameters , the scheme recomputes the updated parameter es­t imates and the adaptation gain matrices over f inite and "sl iding" optimization hori­zons before generating the inputs to the plant .

ACKNOWLEDGMENTS The authors are very grateful to Dr . Lozano and Mr . Ortega by their discussions on the subj ect and to Mr . Alvarez-Lopez and Mr . Herrero by the fac ili ties g iven in the use of the 3220 PERKIN-ELMER minicomputer of the Computer Center of the Pais Vasco Univers i­ty . This work has been partly supported by the Comision Asesora de Investigac ion Cien­tifica y Tecnica (project 1 260-6) .

REFERENCES

Anderson , B . D . O . , and C . R . Johnson ( 1 98 1 ) . Exponantial convergence of identif ication and control algorithms . Automatica 1 8 , 1 .

Anderson , B . D . O . , and J . B . Moore ( 1 97 1 ) . L i near Opt imal Control . Prentice Hall , En­glewood Cliff s .

Astrom, K . J . , and B . Wittenmark ( 1 973) . On self-tuning regulators . Automat ica 9 , 1 8 5 .

Clarke, D .W. , and P . J . Gawthrop ( 1 975) . Self­tuning Controller . Proc . IEE 1 22 , 929 .

De la Sen , M. ( 1 982) . Optimisation des tran­sitoires d ' adaptation dans les sys temes adaptatif s echantillonne s . In Developpe­ment et util isat ion d ' outils et modeles mathematiques en automatique , analyse de systemes et trai tement du signal , Vol . 3 , Bel le-Ile , France , pp . 339-347 .

De l a Sen , M . ( 1 983) . A sof tware technique to design a modified adaptat ion mechanism for improving the adaptation transient in adaptive systems . In Proceedings of the f irst IASTED Sympos ium on Appl ied Informa­tics , Lille (France) .

Goodwin , G . C . , P . J . Ramadge , and P . E . Caines ( 1 980) . D iscrete t ime multivariable adap­tive control . I . E . E . E . Trans . Aut . Control , AC-25-49 .

Landau , I . D . , and H .M. S ilveira ( 1 97 9) . A stab ility theorem with appl icat ions to a­daptive control , I . E . E . E . , Trans . Aut . Control , AC-24 , 305 .

Landau , I . D . ( 1 979) . Adapt ive Control . The Model Reference Approach , Control and Systems Theory Series , Vol . VIII . Mar.eel Dekker, New York .

Landau , I . D . ( 1 980) . An extension of a sta­b il ity theorem appl icable to adaptive control . I . E . E . E . Trans . Aut . Control , AC-25 , 8 1 4 .

Landau , I . D . , and R . Lozano ( 1 98 1 ) . Unif ica­t ion and evaluation of discrete t ime ex­plicit model reference adaptive control des igns . Automatica , 1 7 , 593 .

Lozano , R . ( 1 982) . Independent tracking and regulation adaptive control with forget­ting factor . Automatica , 1 8 , 455 .

Sage , A . P . , and C . C . White ( 1 97 7 ) . Optinrum Systems Control . Prentice Hal l , Engle­wood Cliffs .

Udink ten Cate , A . J . ( 1 9 78) . Discrete model reference adapt ive control systems . Int . J . Control , 28 , 24 1 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

POSTER SESSION

DESIGN OF MODEL-REFERENCE ADAPTIVE SYSTEMS - A COMPARISON OF THE STABILITY

AND THE SENSITIVITY APPROACH

J. van Amerongen and G. Honderd

Control Laboratory, Electrical Engineering Department, Delft University of Technology, P. 0. 50 3 1 , 2600 GA Delft, The Netherlands

ABSTRACT

Design methods for model-reference adaptive systems ( MRAS ) were originally based on a sensitivity approach . Later methods based on stability theory became more popular . When both design procedures are compared it seems that the sensitivity approach determines in a more systematic way the dynamic speed of adaptation . Stability methods pay more attention to modifications in the error signal .

1 . INTRODUCTION

At present the procedure for designing a model-reference adaptive system ( MRAS ) is well known . Although mathematically correct, this procedure is not very transparent . This paper aims to give a different view on the design of MRAS rather than to propose new algorithms . Figure 1 illustrates the problem with a second-order example .

Fig. 1 A model-reference adaptive system

A system has to be designed, by appropriate adjustment of ap and bp, such that e= ( ym-yp) + 0 for t + 00• Then process and reference model yield a similar response . Let process and reference model be equal , except for the gain bp• It is easy to see that , for a positive input u , b could be adjusted by the adaptive law.

p

Stability reasons limit the adaptive better stationary results may be with an integral adaptive law.

t b = K f e dT + b ( 0 ) p 0 p

( 1 ) gain K : expected

( 2 )

This system works wel l , but fails for negative inputs . Apparently, the sign of the input signal has to be taken into account , for instance by multiplying e by sign ( u ) .

67

A similar reasoning yields a law for adjusting �· However , this would lead to adaptive iaws almost equal for all parameters . This cannot be correct . It seems more reasonable to adjust a parameter only , when the output is sensitive to changes in that particular parameter. This "dynamic speed of adaptation" is realized for b and � by the laws

p

t b p

Kl f e U dT + b ( 0 ) ( 3 )

0 p

t a p

-K f e 2 0 x 2p dT + a ( 0 ) ( 4 )

p

Small values of K 1 and K2 give reasonable results . Large adaptive gains , however , make the system unstable . This heuristic approach indicates two problems :

1 . finding dynamically adaptation

a suitable determines

signal which the speed of

2 . finding adaptive laws such that the non-linear system is stable .

total

The sensitivity approach concentrates on the first problem and stability methods mainly on the second.

2 . THE SENSITIVITY APPROACH

The sensitivity approach is based on the following reasoning ( Van Amerongen, 1 98 1 ) .

Let , for instance , the parameter b� be adjusted in order to minimize the criterion

00 C = 1 /2 J , e 2 dt

0 This is realized by making small in bp according to

bp = - 1 /2 K ac/ab p

or db /dt = -Ke (lC / (lb

( 5 )

variations

( 6 )

( 7 )

68 J. van Amerongen and G . Honderd

Because e = y -y

bp K 0 f

t e

and in a similar t

a -K f p 0

it follows that Clyp --- dT + b ( 0 ) Clbp p

way Clyp

e --- dT + ap( O ) aaP

( 8 )

( 9 )

These algorithms are similar to eqs . ( 3 ) and ( 4 ) where u and x2P have been replaced by the sensitivity coerficients , Cl yp/<l hp and ayp1 aap. The latter can be obtained by a so-called sensitivity mode l , with the same structure as the process and placed in series with it .

3 . THE STABILITY APPROACH

Two concepts are used to design a stable MRAS : Liapunov ' s second method or hyperstability theory ( Landau , 1 979 ) . Both methods yield the same algorithms . Straightforward application of the theory leads to the algorithms

b K p 0 ft .

( p 1 2e+p2 2e ) u

t

dT + b ( 0 ) p ( 1 0 )

a = -K f ( p 1 2e+p2 2�) x 2 dT + a ( 0 ) ( 1 1 ) p 0 p p

which are again similar to eqns . ( 3 ) and ( 4 ) , Instead of only the error signal e , for an n-th order system ( n- 1 ) derivatives of e are used in the adaptive laws . The hyperstability method explains that particularly the linear precompensator P, whose elements are p1 2 and P2 2 , ensures the "hyperstability" .

4. SIMULATIONS

Both approaches have been compared simulation of the system of figure 1 .

in a Figure

( 1 0 ) 2 gives the signals of eqs . ( 8 ) , ( 9 ) , and ( 1 1 ) with adaptive gains set at zero .

'

! ' '

- U I - - - � \ Obµ

, /

Fig. 2 Results of simulation

The following phenomena are observed:

when the signal e is compared with P1 2 +P22e , Clypl <lbp with u , and Clyp/ a� with x2P , it becomes clear that the signals used in the sensitivity method are similar in shape to those used in the stability design but that t�ere is a delay between the signals. This explains why the sensitivity method leads to a less stable system. The figures also demonstrate that the sensitivity model can be seen as a kind of state variable filter , which estimates the states of the process .

5 . CONCLUSIONS

The basic form of the adaptive laws can be found from heuristic reasoning. The adaptive laws consist of two parts :

- the error signal. In the stability concept not only the error signal itself but also ( n- 1 ) derivatives of the error are used. - the "dynamical speed of adaptation" , This signal takes care of adjustment of the right parameter at the right time .

It appears that the stability concept concentrates on finding a linear compensator for the error signal , while the sensitivity concept explicitly determines the "dynamic speed of adaptation" . The sensitivity model can be seen as a kind of state variable filter , which estimates the process states . Because of the low-pass character of the sensitivity model this introduces a considerable phase lag which deteriorates the system' s stability. This explains why the algorithms derived with stability methods are indeed more stable .

6 . REFERENCES

Amerongen , J , van ( 1 98 1 ) . MRAS : Model Reference Adaptive Systems . Journal A, l.a• 4 .

Landau , I . D . ( 1 979 ) . Adaptive Control - the model reference approach. Marcel Dekker Inc . , New York and Basel ,

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE MODEL REFERENCE PARAMETER TRACKING TECHNIQUE FOR AIRCRAFT

A. A. Azab* and A. Nouh**

*AIC Electrical and Special Equipment Department, Military Technical College, Cairo, Egypt **Electrical EnginPering Department, King Saud University, Riyad, Saudi Arabia

Abstract. This paper presents a proposal for adaptive control of supersonic aircraft . A modified gradient technique for a model reference adaptive con­trol system with less order model is employed. Analog computer simulation results for the aircraft adaptation are presented and discussed.

Keywords . Adaptive control ; Aerospace simulation ; Tracking systems .

INTRODUCTION

Part of the historical evolution in flight systems is that the region in which the air­craft can operate has been increased signi­ficantly, e . g . to high altitude s , high Mach numbers , and even to major changes in the geometry of the aircraft . These significant changes in the aircraft dynamics had at least partially to be overcome by means of the control system and, in particular , the controller had to operate satisfactorily for a drastically changing aircraft . Extensive theoretical and experimental studies have been devoted to obtain a fairly accurate model of the dynamic behaviour of the air­craft for all possible flight conditions (Azab , 1974 ; McRuer, 1973) . Using this knowledge of the possible parameters varia­tions a control system should be designed in a way suitably adopted to the flight condi­tion . Recent progress in the theory of adaptive control makes it applicable and well suited to solve flight control problems in an efficient way .

The available adaptive techniques are of the following type : local parametric optimiza­tion theory (Hang, 1973) , Lyapunov functions technique (Lindorff, 1973) , and hyperstabi­lity and positive concepts (Landau, 1979 ) . Most of these techniques avoid the identifi­cation problem, but stresses that the order of the reference model must be the same as that of the nonadaptive system.

In an attempt to give a perspective for the possible application of adaptive control to aircraft systems , this paper presents , a proposal of adaptive control of a supersonic aircraft.

A modified gradient method is proposed to solve the problem of adaptive model referen­ce control system when the model reference is less order than the adjustable system.

69

In this paper the method was used to control the longitudinal motion of a supersonic air­craft when the reference model was chosen less order than the aircraft system equation. The organization of the rest of the paper is as follows : in section 2 a modified gradient method for the model reference adaptive con­trol is presented . The system equations for the longitudinal motion of a hypothetical supersonic aircraft is given in section 3 . The suggested proposal where the aircraft complete transfer function is controlled to behave according to a second order chosen reference model is given in section 4 and conclusion appear in section 5 .

PROPOSED ADAPTATION TECHNIQUE

Consider the following single input/single output model reference adaptive system represented by :

a) The adjustable system

n di v di

l: CJ. . -. c ( t) l: B . -. r {t) i=o l. dt

1 i=o l. dt1

n > v (1 ) CJ. . = gi + h . i o , 1 , . . . , n ( 2 )

l. l.

where r {t) is the input, c ( t) is the output, CJ.i and Bi are functions of the changing co­efficients within the operation environments of the system, gi are the variable para­meters , and hi are the controllable para­meters .

b) The reference model

m di l: a . -. y ( t)

i-o 1 dt1

)1 di l: b . -. r (t)

i=o 1 dt1

m .:_ )1 ( 3 )

70 A.A. Azab and A . Nouh

where y (t) is the output of the reference model , ai and bi are constant parameters of the system, and (m+l ) equal to the number of controllable parameters of the adjustable system.

c ) The generalized output error

e (t) = c ( t) - y (t) ( 4 )

The parameters gi are assumed to vary over extreme ranges within the operating environ­ment of the nonadaptive system. Assuming that there exists a values of hi for which the system will behave like the chosen reference model . The objective of the adaptation mechanism is to provide these value s . Its input information i s the gene­ralized error e (t) and its output will be the instantaneous values of hi · The problem can be formulated as : find the adaptive mechanism equations which, if no limitations are placed on the values of hi , then regard­less of what values gi take , the outputs of both systems will be approximately identical whenever a quadratic function of the general­ized error will be minimized . Considering the following error function :

f (e) m 1 di 2 l: 2 [ q . -. e ( t) ]

i=o 1 dt1 ( 5 )

where qi are constant factors , Equation ( 5 ) assumed the availability of the derivatives of the error signal and it depends indirect­ly on the differences between the parameters of the adjusting system and the reference model which can be defined as :

( 6 )

If the parameters q i v?J,ry slowly as compared to the basic time constants of the adjust­able system and the reference model and the adaptation mechanism is designed to adjust the parameters hi at a rate which is much greater than the rate of variation of gi , the gradient optimization technique can be used and leads to the basic adaptation rule :

� a dt i ( 7)

where Ki are arbitrary positive constants to be chosen by the designer and depend on the particular system considered .

If the development were to be based upon equation (7 ) , the resulting design would require explicit knowledge of ai and conse­quently gi . Now suppose that ai are treated as constant, and ai are to be adjusted so as to cause the differences C i to approach the same values then the variations in ai becomes

( 8 )

The objective is not to change ai , however , to change ai · Since from equation ( 6 ) the same changes in Ci can be obtained by sub-

stracting �ai from ai , rather than adding them to ai · The resulting instantaneous rates of adjusting ai can be written as :

a 3 a . f (e)

l. ( 9 )

When equation ( 5 ) was substituted into equa­tion ( 9 ) , and the indicated partial deriva­tive was carried out equation (9 ) becomes :

m dje m a .<!;i_. y ( t) J -K . [ l: q . -. ] [2: q . -3 -ia j=o J dtJ j=o J ai dtJ

Introducing the notation :

aY uia= � i

l. o , 1 , . . . , m

( 10)

( 11)

Again, assuming that the parameters Ci are changing with slow rate s , the order of differentiation in the right-hand side of equation ( 10) canbe interchanged , yielding to

m dje m dj -K . [ l: q . -. ] [ q . -. u . ] ia j=o J dtJ j=o J dtJ ia

( 12 ) Since i t i s assumed that the adaptation mechanism will be designed so as to adjust the controllable parameters hi in a rate greater than the changes in gi • The rates of gi can be neglected and from equation ( 12) the rates of hi can be written as:

db . da . m dj m dj l. l. [ �. ] [ " "' J -- 'V -- = K . l: q

J. t.. q

J. --. u . dt - dt ia j=o dtJ j=o dtJ ia

( 13)

The only unknown quantities in equations U3) are ui • Considering the differential equation of the reference l!Odel ( 3 ) , and taken the partial derivative of both sides with respect to the parameters ai , treating the derivatives of y with respect to time as trivial partial derivative , and inter­changing the order of differentiation and employing the notation introduced in equa­tion ( 11) , the following set of equations for uia can be written as :

m diu . l: a . ___.:),£ -

i=o 1 dti o , 1 , . . . , m ( 14 )

Equation ( 14 ) represent a set of differen­tial equations with available forcing functions and their solutions provides the values of Uia for equation ( 1 3 ) .

Similarly for the case of the parameters Bi , the following adaptive mechanism equations are required .

dB . l. dt =

m -K . 8 [ l: l. .

J=O qj

j m �] [ l: dt) j=o

i=o ,

dj qj -. u . B J dt1 l. 1 , • . . , \) ( 15)

Parameter Tracking Technique for Aircraft 7 1

j = o , l , . . . , v ( 16 )

where i o , l , . . . , v ( 17)

The four sets of equations ( 13) , ( 14 ) , ( 15 ) , and ( 16 ) represent the adaptive mechanism algorithm required for adaptation of the adjusting system to follow the reference model .

MATHEMATICAL MODEL OF AIRCRAFT SYSTEM

It is well known that the short period mode can be used to represent the longitudinal dynamics of supersonic aircraft with suffi­cient accuracy due to its simplicity (Azab , 1974) • The fundamental transfer functions that describe the dynamical motions of an aircraft in the longitudinal-vertical plane of motion are given by :

2 2 p + 2duop + wo

e n 35 (p + n22 )

T = 2 2 e p (p + 2dwop + wo )

w /n22n2 3 + n32 and 0

d n22 + n33 + n30

2w 0

where :

a is the change in angle

e is the change in pitch

( 18)

( 19 )

(20)

of attack

angle

0 is the change in elevator deflec-e ti on

n . . are the coefficients for the short l.J period mode

For the purpose of illustration , consider a hypothetical aircraft with the following basic parameters :

Mass of aircraft

Wing area

Wing span

length of mean aerodynamic chord

m 8100 kg

s = 32 . 4 m2

!l 9 . 5 2 m

b 3 . 4 m

For such aircraft, the numerical values of the coefficients for the short period mode ni� were calculated for initial horizontal flight also the damping factor d and the natural frequency w0 , the calculations were

performed for different altitudes and given in table 1 . For the supersonic aircraft described above a typical classical auto­pilot can be represented by the following equation (Blakelock , 196 5 ; Duda , 1970) :

Kl 0e = p (Tp+l) [K2 ( 6- 6r) + ( E+T)P) p6 ] ( 2 1)

where

er the input reference signal

n the accelerometer gain

E the rate gyro gain

T time constant of servomechanism

Kl the amplifier gain

K2 the vertical gyro gain

TABLE 1 Coefficients for the Short Period Mode

H (km) 5 1 0 1 5 2 0 2 5 n30 0 . 68 0 . 507 0 . 28 0 . 1175 0 . 047

n22 1 . 29 1 . 06 0 . 672 0 . 34 0 . 168

n32 4 . 85 24 . 7 25 . 3 18 . 6 13 . 3

n33 1 . 5 1 . 2 3 0 . 782 0 . 391 0 . 184

n35 -15 . 9 -17 . 0 -13. 15 -7 . 17 -4 . 6 7

d 0 . 665 0 . 274 0 . 169 0 . 098 0 . 0548

w 2 . 61 5 . 1 5 . 08 4 . 32 3 . 642 0

The block diagram of the complete aircraft system is given in Fig . 1 from which the transfer function representing the total behaviour of the system can be written as :

( 22 )

where a T 0 al l+T ( n22+n33+n30)

a2 n22+n33+n 30-n35 Kl n+T ( n32 +n22n33)

a3 n22 n33+n 32-n22 n35 Kl n-n35 Kl E

a4 -n22 n35 Kl E-n35 Kl K2 as -n22 n35 Kl K2

It is clear that the behaviour of the air­craft system will vary according to the flight altitude because of the variation of the coefficients nij · For this reason the need for adaptive control for such aircraft is required . Sensitivity analysis and ana­log computer simulation for the above air­craft shows that the aircraft system para­meters which have the main effect on the transient response are the accelerometer gain n and the rate gyro gain E . In the following section a proposal for the adaptation of the aircraft is given.

7 2 A.A. Azab and A . Nouh

ADAPTIVE PROPOSAL FOR THE AIRCRAFT SYSTEM

The suggested proposal for the adaptation of the aircraft system consists of controll­ing the feedback parameters n and E by the adaptive mechanism so that the aircraft system response behave according to a second order reference model .

According to the proposed principle of adaptation given in section 2 let the refer­ence model and the error function equations be :

d2 d a.o dt2 y (t) + al dt y (t) + a.2 y (t) = 8r (t) ( 2 3 )

f (e) =� [qo e (t) + ql � ( t) + q2 e (t) ] 2 ( 24 )

The equations representing the dynamics of the adaptation mechanism are :

a4 = - cl[qo e + ql e + q2 e] [ qo ul + ql ul + q2 ill J

a3 - c2 [qo e + ql e + q2 e ] [qo u2 + ql u2 + q2 U.2 ]

and (). iil + ().1 ul + a.2 ul - y 0 (). u.2 + 1 u2 + a.2 u2 - y 0

( 25 )

(t)

( t) ( 26)

From Equations ( 2 2) , differentiating a4 and a3 with respect to time , and considering that the time derivatives of the aircraft parameters nij are neglected, the time rates at which the accelerometer gain n and the rate gyrogain E will be

(27 )

n ( 28) a4

+ _2 __ _ n22n35Kl The rate of changing the accelerometer n in equation (27 ) depends on a4 and a3 , but to secure the stability of the adaptive proces� it will be considered that n will be con­trolled only by the changes in a3 , also the coefficient [- 1 ] will be absorbed in-n22n35Kl to the coefficients c1 and c2 •

The complete adaptive system were simulated on analog computer with the reference model equation as :

0 . 0625 i + 0 . 35 y + y = 8 r ( 29)

The numerical values of the autopilot ' s co­efficient K1 and K2 and the time constant of the servosystem were chosen , as to get the best transient response , to be: K1 = 20 and K2 = 16 and T = 0 . 05 sec . ( 30)

The best function of the adaptive mechanism, where obtained for the following values of the proportionality factors :

1

c2 = 0 . 589 (31 )

The aircraft autopilot system was excited by input pulses of amplitude 0 . 09° and with 5 seconds duration in one polarity, then the convergence of autopilot ' s coefficients n and E to an optimum value , from extreme initial value s , was checked . In Fig. 2 , the aircraft was set to the parameters of alti­tude 5 km. It was found that the parameters n and E converged nearly to the same optimum value after 5 changes of the input level , from extremely large initial value s . Figure 3 represents the same trajectories o f n and E when the aircraft parameters set to the values of altitude 25 km. The system was simulated for a step input, for altitu­des ( 5 , 10 , 15 , 20 and 25 km) , with the adaptive autopilot parameters initially are n0 = 1 . 12 and E0 = 5 . 7 . Figure 4 represents the responses of the aircraft system 8 ; rate of pitch e , the changes in the angle of attack �, and the generalized error e .

CONCLUSION

The concept of a proposed gradient method is used to seek out solution of the problem of adaptive control for aircraft system where the reference model used is less order than the aircraft system. A proposal for adapt­ation of the longitudinal flight of aircraft system was presented . The suggested techni­que proved to secure the needs of aircraft stability and control . The adaptive mecha­nism is quite reasonable from the stand­point of actual physical hardware . Analog computer simulation shows that the stability of the adaptive system depends on the choice of the adaptive mechanism parameters.

REFERENCES

Azab, A.A. ( 1974 ) . Airplane Equations of Motion II Seminare modelovani dynamic­kych systemu, Fackov Sedle , CSSR.

Blakelock , J . H . ( 1965) . Automatic control of aircraft and missiles , J. Wiley, New York.

Duda, T . ( 1970) . Automaticke Rizeni Letade� VAAZ Brno , CSSR.

Hang, C . C . and P . C . Parks ( 1973) . Compara­tive Studies of Model Reference Adapt­ive Systems , IEEE Trans. AC18 , No . 5 .

Landau, Y . D . ( 1979) . Adaptive Control : The Model Reference Approach, Macel Dekker In, New York .

Lindorff , D . P . and R . L . Caroll (1973) . Sur­vey of adaptive control using Lyapunov design, In. J . Control , 18.

McRuer , D . , I . Askenas , and D . Graham (1973). Aircraft Dynamics and Automatic Control, Princeton University Press , Princeton , New Jersey .

Parameter Tracking Technique for Aircraft 73

N N c: + 0.

"

-ll .... 0 � + 0.

.. ..,

;� ' 0. 0.

0. C"

Fig . 1 Block Diagram of the Aircra ft Autopilot System

.. "' "" I::! "' � "! -;, 0 .,; . • • " e' 0 • ""'" ... C" ...

0

:2w 0

0

0 0 0

0

..... 0 0 0 0

:!l .. 0

/> "" ;g; M � - .,;

� " 0 C"

� :::: "" -;,

.,; .,; . . .

o · ..,. ..,. ... C" ...

./ M N ·�

.

0 0 � .. c w c w

Fig . 3 Convergence of the Aircraft parameters at altitude of 25 km

0 .,;

"'

0 C" �

c:i � 0 � N "'

0 C" �

0 0 0

N .. ... ... ... "'! � ... c:i ..; . . . . 0 0 � : C" ...

�" ....

0 w

... � ... N ... � ... c:i .,; . . . . .? 0 "' "' ... C" ...

�o

� ......... � � "' ..; If II I 11

� WQ � W·

"' ...

"' � .... il! .... � c:i ... .,; . . . . .? 0 .. .... ... C" ...

�

Fig . 2 Convergence of the Aircraft parameters at alti tude of 5 km

a( t )

a< t)

a(t)

•( t)

(0)

15

23 1

4 5 23

5 4

23 45

20

Fig. 4 Aircraft system responses at different altitudes

1 • 5 km Z • 10 km 3 • 15 km 4 • ZO km 5 • 25 km

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

REDUCED CONTROL EFFORT FOR SELF-TUNING REGULATORS VIA AN INPUT WINDOW

M. M. Bayoumi and J. Ballyns

Department of Electrical Engi,neering, Queen's University, Kingston, Ontario, Canada

Abstract . Self-tuning regulators ( STRs ) have proven themselves to be well suited to many process control environments , especially in the case of unknown or slowly time varying parameters . However , for processes containing moderate to excessive noise levels the basic minimum variance type STRs may prove inadequate as the resulting control action may be excessive and costly . This paper presents a method of limiting the control action via an input window without unnecessarily exceeding output variance limitations .

Keywords . Adaptive control ; computer control ; control theory ; digital control ; energy control .

INTRODUCTION

The self-tuning regulator as introduced by Astrom and Wittenmark ( 1973 ) has been exten­sively studied and has become popular in the field of adaptive control . Its ability to control processes of unknown or slowly time varying parameters has made it quite attrac­tive to systems in which the expected return for an improved controller outweighs the cost. of a computer based control system. SuccesEr ful implementation, however , has been res­stricted primarily to applications in the pulp and paper , chemical and other resource based industries where the process dynamics are significantly slower than the STR algorithm.

Clarke and Gawthrop ( 1975 ) suggested a strat­egy to minimize the control effort required by self-tuning controllers . This was based on a control law which penalized the input via a weighting parameter Q . The parameter Q would make it possible to trade off a de­crease in control effort against an increase in output variance . This approach may prove useful to processes which essentially oper­ate in a steady state mode and one wishes only to reduce the effects of process noise on the input. However for a process with frequent setpoint or load changes the use of a Q weighting parameter may also reduce the controller ' s effectiveness in handling trans­ients . This paper will present an alterna­tive method for reducing control effort with­out excessively affecting the controller ' s ability to handle transients .

THE SELF-TUNING REGULATOR

The self-tuning controller contains 3 main elements :

75

1 . A standard feedback law in the form of a difference equation acting upon meas­ured output , feedforward signals and current setpoint .

2 . A recursive parameter estimator which monitors input and output signals and computes parameter estimates describing plant dynamics in a prescribed structur· al model .

3 . A control law which uses the parameter estimates to compute the required inpu1

Model Representation and Problem Formulatio1

Consider a system defined by the following difference equation model .

- -1 A [ z ] yt

where

-1 z

d

t

yt

ut

et

A. [ z-1]

B [ z-1 ]

1

b

-d - - 1 z B [ z J ut + et ( 1 )

unit time delay operator : -1 z yt = yt-1

known time delay of the process

number of time units

output at time t

input at time t

white Gaussian noise with zero mean

n z-j + l: a .

j=l J

m z- j + . l: l b . 0 J= J

76 M.M. Bayoumi and J. Ballyns

Parameter Estimation by Recursive Least Squares Once the model orders and pure delay have been assumed , the parameters must be estima­ted . One popular method is that of recur­sive least squares because of its speed , sim­plicity and minimal memory requirements .

With knowledge of y (k ) , u (k-d) and y ref O<: ) for all k < d , we wish to find the best parameter v;ctor P (t ) to fit the relation : y (t) = P (t ) X ( t-d) + e (t) ( 2 ) where for the AR model ,

. . . , -a l n ( 3 )

X (t-d) T [ u ( t-d) , u (t-d-1 ) , . . . , u ( t-d-m) , y ( t- 1 ) , y (t- 2 ) , . . . , y ( t-n) J ( 4 )

P (t ) is chosen so as to minimize the sum of squares of errors :

v = t-k 2 L w [ y (k ) - P (t ) X (k-d) ) k=<t

( 5 )

w i s the forgetting factor which introduces an exponentially decating weighting factor on past data . (used e . g . by Keviczky (197 7 ) , Wong ( 1980 ) ) .

P (t ) which minimizes V is given by the following recursive equations .

R ( t) = �r ( t-1 ) - R (t-1 ) � ( t-d) XT ( t-d) R (t-lq w + X ( t-d) R (t-l ) X (t-d) J

( 6 )

p ( t ) , , T P (t-l ) + [y ( t) -P ( t-d) X ( t-d) ) X ( t-d) R ( t )

( 7 )

Penalized Input Controllers Penalizing the control action as well as the output variance can have some very desirable results when applied to STRs . Equations 8 and 9 represent two such loss functions .

Ll = (yr -y' ) 2+ Q (u ) 2 t+d t+d ·t t ( 8 )

( 9 )

These lead to the control equations :

( 10 )

and,

u (b2 + 0 ) - 1 [ bo (yrt+d-PlXl ) + \"\Ut-1 J t 0 ··t

( 1 1 )

where pl and xl are subsets of p and

X given by

( 1 3 ) NOTE : Estimates for future outputs yt+i

( i=l , 2 , . . . , d-1 ) are predicted using current parameter estimates and input output data .

INPUT WINDOW CONTROL

In process control of stochastic systems , an inverse relation often exists between the amount of control action and the output var­iance from target . In many industrial pro­cesses , the control objective is to minimize costs while maximizing profits . If costs increase with the amount of control action or energy then one should design a controller that minimizes control action without exceed­ing an output variance limit. This limit can be defined by the product specifications . For such a controller the operator would de­fine two control parameters , the target out­put value and the allowable output variance . The controller would then minimize control action without exceeding the allowable vari­ance . Should the allowable variance limit not be attainable the controller will default to a minimum variance controller for mimimum phase systems or a minimum Q penalized in­put control for nonminimum phase systems .

The operator-defined iance is used by the window for which ut

limit on allowable var­controller to create a must lie between . So

long as ut lies between this window the output variance will be less than or equal to the allowable variance limit .

Output Variance Regulation Via Input Window The variance of the output can be approxima­ted by the followinq equation :

<a i 2 = 1 y N Where , y

N

This can be

= l <a > 2 y N

N

mean value of y number of samples ( the more samples the better e'stimate ) rewritten as , N 2

ih (yi - Yi + Yi - y)

Expanding we get , 1 N 1 <a ) 2

ih (y i-y i ) + - L (y-y . ) 2 y N N i

2 N ih (yi-yi ) * (y-yi ) N

( 14 )

( 15 )

( 16 )

Under steady state conditions and good con­trol the following assumptions can be made :

Self-tuning Regulations via an Input Window 7 7

1. The mean value of y equals yr . 2 . The mean value of y equals yr . 3 . The model parameters are least squares

estimates . I t can be shown using the above assumptions that the following is true ,

-2 N N

PROOF :

(y . -y . ) * (y-yl.. ) l. l. 0 . 0 ( 17 )

The quantity (yi - yi ) is the estimation

error referred to as the residual . The meth­od of least squares implies that the distance or length of residual is minimized by making it orthogonal to the estimated vector y , . l.

It then follows that, (yi - yi) and yi are statistically inde-pendent. If yr is constant , then (yi - yi ) and (yr - yi ) are also statis-

tically independent .

Inv�king assumption 1 , yr can be replaced by y and hence , (yi - yi ) and (y - yi ) are statistically independent . Hence equation 17 holds true .

In order to determine the input window for limiting the control effort, on-line estim­ates for the mean of the residual squared and an operator-defined allowable output variance will be employed . The estimate for the mean of the residual squared can be made recursive by using past output estimates and current output data . This estimate shall be denoted as � which is an estimate of � given by

� = (yi - y . ) 2 ( 18 ) l. Hence ,

(a ) 2 = (yr - y . ) 2 + '¥ ( 19 ) y l. In the minimum variance control scheme we make Yi = yr as we have no control over the residual in a stochastic process . How­ever in input window control we wish only to keep the output variance below that which is allowable .

Corresponding to the allowable output vari­ance, let y . be denoted by ylimi t ( t+d) . l.

Hence , (yr - ylimit (t+d) ) 2 = (yr - y . ) 2 l. and ( a ) 2 = (a ) 2 Allowable y y (yr - ylimit ( t+d) ) 2 = (a ) 2 - '¥ ( 20 ) y ALL Our output window is defined by the two out-put limits given by :

ylimit (t+d) = yr :;: I <a > 2 - � y ALL ( 2 1 ) NOTE : . If � is complex we must resort to our penalized input control scheme with Qt = Q:nin Substitution of yr by ylimit ( t+d) into

loss functions ut and letting

8 and 9 while solving for Qt = Qmin (the minimum

value for Q to stabilize the closed loop system for a penalized input STR) , our . window limits become :

ulimit ( t)

and ,

ulimit (t )

(b�+Qminl - 1 �0 (ylimit ( t+d) -P1x� ( 2 2 )

tb6<Qminl -1 '"0 (y limit ( t+d) -', < 11 t Qmin ut-1 J ( 2 3 )

The two limits for ylimit ( t+d) together with equations 22 and 23 determine the input window for ut . The control signal is

allowed to change only if ut-l lies outside

the window limits in which case the input u is set to the nearest limit . The input win� dow therefore serves to eliminate much of the variations in the control action without exceeding the allowable output variance limits .

SIMULATION RESULTS

The following examples are for identical un­stable minimum pha�E systems with coloured noise . Minimum variance control is used for the first 100 sample periods . Figure 1 is that of a minimum variance controller , fig­ure 2 for a penalized step input controller and figure 3 for an input window controller with an allowable output variance of 0 . 06 ( the actual variance that resulted was 0 . 034 )

CONCLUSIONS

The method of using an input window for an allowable output variance is a practical means of minimizing the number and magnitude of discrete control actions for stochastic systems without substantially increasing the systems time response to changes in setpoint or load . This is an alternative to the method of using a fixed Q type STR which performs well under steady state conditions but is more sluggish in response to target or load changes as Q increases .

The method of input windows can be applied to stable , unstable , minimum and nonminimum­phase systems given the set point and allow­able output variance .

REFERENCES

Astrom, K . J . and Wittenmark , B . ( 1973 ) , On Self-tuning Regula.tors , Automatica 9 ( 2 ) , 185-199 .

Clarke , D . W . , and Gawthrop , P .J . , ( 1975 ) , Self-Tuning Controller , Proc . IEE , 122 (9 ) , 929-934 .

Keviczky , L . and Hetthessy , J . ( 1977 ) , Self­Tuning Minimum Variance Control for

78 M .M. Bayoumi and J . Ba l lyns

MIMO Discrete Time System, Autom . Control Theory and App l . 5 (1 ) , 11-17.

Wong , K . Y . (1980) , Multi- Input Multi-Output Self-Tuning Regulators and Their Applications , M . Sc . Thesis , Queen ' s University , Kingston, Ontario , Canada .

� : �1\h1�'�1���� : l�+--+ >-�+ --1---+----.--� 'o . oo 1 0 0 . 0 0 ?oo.oo 300.00 "oo.oo soo.oo soo.oo 100.00 aoo.oo T I ME I I UN I T I S O M P L E P E A I O O I

� i ��� :L_·-+ - -- '-+-�� � o . oo 1 0 0 . 00 ? 0 0 . 0 0 Joo.oo "oo.oo soo.oo soo.oo 100.00 e o o . o o

T I M E I I UN I T I SOMPL E P E A I OO I

� :��� :C�+--+--+---+-�--+---+---+---< 'o . oo 100.00 lOo.oo 300.00 "oo.oo soo.oo 500.00 100.00 eoo.oo

T I ME I I UN I T I SOMPLE P E A I OOI

F i gure 1 M in imum variance control l er

F i gure 2 Penal i zed step i nput control l er

F i gure 3 I nput wi ndow control l er

Copyright © IFAC Adaptive Systems in Control and Signal Processing; San Francisco, USA 1983

ADAPTIVE CONTROL WITH FEEDFORW ARD COMPENSATION AND REDUCED REFERENCE

MODEL

G. E. Elic;abe and G. R. Meira

INTEC (CONICET and Universidad Nacional del Litoral), C. C. No 91, Santa Fe 3000, Argentina

Abs tract . Two modifications to the s tandard adaptive model-following con­trol (Af1FC) algorith� given in Landau ( 19 79 ) are des cribed . The first re­fers to the extension of the algorithm to consider measurable dis turbance inputs in a feedforward adaptive fashion . The s econd is related to the use of reduced reference models , which consider only the more important s tate variables . This enhances the possibility of s tability and perfect model­following (PMF) in linear sys tems , even if the number of manipulated vari­ables differs from the number of s tate variables .

Keywords . Adaptive control ; direct digital control ; feedforward ; multivari­able control systems ; Popov criterion ; reduced reference model .

INTRODUCTION

AMFC techniques employ adaptive mechanisms to modify the adjustment of linear model­following control (LMFC) systems . The design of adaptation mechanisms normally require the existence of the PUF in order to insure the s tability of the controlled system [ see Oliver , Seborg and Fisher ( 1973 ) ; Landau and Courtiol ( 19 74) and Landau ( 19 79 ) ] .

This paper considers the case where only part of the total number of s tates are of in­terest from . the control point of view; mak­ing unnecessary the use of reference models of the same order as the plant . First , the modified LMFC algorithm is described , and then the corresponding Af1FC is given. In both cases , an extension by which the measur ­able disturbances can be compensated through a feedforward scheme , is also considered .

THE PROPOSED CONTROL ALGORITill1

Consider a plant model that includes a measur­able disturbance input vector d :

� = A x + B u +D d -P P -P P -P P - ( 1 )

where the vectors �P ' �p and Q are of dimen­sions n , m and p , respectively . Suppose that only a subset el ( sxl ) of ep is of interest from the control point of view , and that the remaining s tates �2[ (n-s ) xl ] can attain any finite value . Thus , the plant model may be partitioned as follows :

�l = Al l �l + Al2 �2 + Bl �p + Dl � �2 = A2 1 �l + A22 �2 + B2 �p + D2 �

( 2 . a)

( 2 . b)

79

Choosing a reference model

x = A x + B r + D d -ID m -ID m - ID -

( 3)

(where r < rxl) is the input vector) o f order s <n , the generalized error becomes :

e = x - x - -ID - 1 (4 )

The proposed control law is :

where K4 is the feedforward control matrix .

Through Eqs . ( 2 . a) , ( 3) , (4 ) and ( S) one obtains :

� = (Am - B lKs) : + (BlKl - B lKS - Al l + Am) �l +

+(B l K2- Al2) �2 -1- (-B lK3 + Bm) : +

+ (-B K + D - D ) d 1 4 m 1 - (6 )

I n order to make � null a t all times with the initial condition e (O) = 0 , it is suf-ficient tha t :

- -

B l (Kl - Ks) = All - Am B l K2 = Al2 B l K3 = Bm Bl K4 = Dm - D l

( 7 . a) ( 7 .b ) ( 7 . c ) ( 7 . d )

In the more general case when snn, a solution to Eqs . ( 7) will exist if the conditions given by Erzberger ( 1968) are verified ; and in this case the control matrices can be ob tained through the use of Bl ( the pseud�­inverse of B 1 ) . Assuming that the P:MF can be in­sured through this last procedure , consider

80 G . E . El i�abe and G . R. Meira

now the behaviour of the remaining (n-s ) s tates . Subs tituting the solution of Eqs . ( 7 ) into E q . ( S ) and it , in turn , into (Z . b), one obtains :

The subsys tem of �z will be asymptotically s table if (Azz - BzB!A1z ) is s table .

Incorporating an adaptation mechanism with signal-synthesis adaptation into the previous ­ly described Ll1FC algorithl!l, � of Eqs . (Z ) becomes �pa as follows :

u = u + I'm -pa -P -·P L'l�p = L'IKl ( t , =)�1 + L'IKz ( t , ':) �z +

+ L'IK3 ( t , ':) : + L'IK4 ( t , :) �

( 8 . a)

( 8 . b )

Following a design criteria a s des cribed in Landau ( 1 9 79 ) , the adaptation algorithl!l can be derived . Hith Eqs . (8 ) ins tead of Eq . (S ) and assuming that the PMF is possible with the control matrices Kl , Kz , K) and K; , Eq . (6) will read : 1

� = (Am - Bl Ks) : + [Bl (Kl -KJ'.-L'IK1 ) J �1 +

+ [Bl (Kz - Kz - L'IKZ) ] �z + [ B l (K) - K3 -

- L'IK3) ] : + [ B l (K4 - K4 - L'IK4) ] � (9)

The control matrices generally adopt propor­tional plus integral forms , such as :

L'IK . ( t , v) = ft ¢ . ( t , v ; r ) dT + 1/J . ( t , v) + 1 - 1 - 1 -

0 ( i=l , Z , 3 , 4)

with : v = D e

( 10)

( 1 1 )

Applying the theorem which guarantees the asymptotic hyperstability of the sys tem defined in Eqs . (9) to ( 1 1 ) , the following results can be obtained :

¢ 1 ( t , � , T ) Fal v (T ) [Gal�l (T) ] T ( l Z . a)

1/il ( t , �) F ' T ( l Z . b) a l �( t) [ G�1�1 ( t) ]

<Pz ( t , �,T) Faz v (T ) [Gaz�z (T) ] T ( l Z . c)

1/iz ( t ,�) F ' T ( lZ . d) aZ �( t) [ G�z�z ( t) ]

¢3 ( t ,� , T ) Fb �(T ) [ Gb � ( T ) ] T (lZ . e)

1/i3 ( t , �) F ' b v ( t) [ Gb r ( t ) ] T ( lZ . f)

¢4 ( t ,� , T) F v(T ) [ G d(T) ]T ( 12 . g) c - c

..µ4 ( t , �) F ' v ( t ) [G ' d ( t) ] T ( 12 .h) c - c -

where Fal • Gal • Faz • Gaz • Fb , � . Fe and Ge must be positive definite ; and the corre-spending primed matrices are positive semi-

definite . Eqs . ( 10) to ( lZ ) -represent the soup,ht adaptation mechanism, and D mus t be calculated through : -D = Bl P with P (sxs) a positive definite matrix obtained from solv­ing :

T (A - B1Kr.) P + P (A - B1K5 ) = -Q m J m and where Q (sxs) is any arbitrary positive definite matrix.

Defining L'l� ( t ,y) = [L'IK1 ( t ,y) , L'IKz ( t , y) J and choosing Fa t = Faz = Fa and F� l = F�z = F� , one obtains :

L'IK ( t ,v) = ft F v[G x ] T dT + p - a- a -p

with G a

0

+ F' v [G ' x ]T + i'IK (O) a - a -P P

r���+-�-] L 0 : Ga2 G ' a

-����

-

�

-]

0 1 G ' I a2

Note that the adaptation mechanisl!l is of

( 1 3)

the same form as that given by Landau ( 1 9 79) , but with Ga and G� having the quasi-diagonal forms shown in Eq . ( 14) .

Additionally , the discrete version of the above control algorithm was satis factorily evaluated on a s imulated example .

CONCLUSIONS

The extension of an exis ting AMFC algorithm to include feedforward compensation and re­duced reference models seems advantageous . The use of reduced reference models increases the feasibility of the PUF and sil!lplifies the necessary calculations .

REFERENCES

Erzberger , H . ( 1968) . "Analysis and Design of Model-Following Systems by State-Space Techniques" . Proc . JACC , pp . 5 72-581 .

Landau, Y . D . ( 1979) . "Adaptive Control . The Model Reference Approach" . Marcel Dekker Inc .

Landau , Y . D . and Courtiol , B . ( 1974) . "De­sign of Multivariable Adaptive Model­Following Control Systems" . Autol!latica , 1 0 , 483 .

Oliver , H .K . , Seborg , D . E . and Fisher , D . G . ( 1973) . "Hodel Reference Adaptive Control Based on Liapunov' s Direct Hethod . Part I : Theory and Control Systems Design" . Chem . Eng. Col!lm. , .!_, l ZS .

Copyright © IFAC Adaptive Systems in C.ontrol and Signal Processing, San Francisco, USA 1983

ON A CLASS OF ADAPTIVE PID REGULATORS

j. Hetthessy*, L. Keviczky** and Cs. Banyasz**

*Department of Automation, Technical University of Budapest, Budapest, Hungary **Process Control Department, Computer and Automation Institute of the Hungarian Academy of Sciences,

Budapest, Hungary

Abst ract . Based on some rel at i ons d i scovered between the determi n i st i c servo regu l ato r and the P I D des i g n a s impl e des i gn rol e i s presented for stabl e processes to ensure 5% rel at i ve overshoot for the c l osed l oop step response . The adapt i ve sol uti on i s based on the expl i c i t i d ent i fi c at i on of a second order process model wi th t ime del ay . Extendi ng the PID regu l ato r by a n appropri ate correct i ng term the poorly damped or i nverse unstabl e pro­c ess zero i s al l owed to be a zero of the cl osed loop system , as wel l . Add i t i onal pol e pl acement i s al so ava i l ab l e . Keywords . Adapt i ve control ; P I D contro l ; process control : s amp l ed data systems .

I NTRODUCT ION

The present state of the resea rch works i n the fi el d of adapt i ve control refl ects a synthesi s of the cl ass i c al pri nci p l es and the resu l t s of the sevent i es , when the i d ea of the sel f-tun i ng cont rol became so much attract i ve and popu l a r . Aimi ng at pract i ca l appl i cabi l i ty adapt i v e P I D regul ators can be cons i de red as resu l ts of thi s synthesi s . I n the control l i terature many schemes ex i st for the tun i ng of P ID regul ato rs . The s uggested methods are ma i nl y based on opti m i ­zat i on theory , but the eval uat i on proced u re i s rather time consum i ng ( I sermann , 1977 ) . Another approach i s the comb i n at i on of recur­s i v e parameter est i mat i on methods and control a l gori thms ( I sermann , 1977 : Wi ttenmark ,' 1979 ) where the suggested des i gn step is general ly an iterat i ve scheme al l owi ng to prespec i fy pol es i n the cl osed l oop system . However , t h e i nt roduced ext ra zeros o f the c l osed l oop system make d i ffi cu l t the sel ect i on of the c l osed l oop pol es . In our paper an expl i c i t adapti ve P I D regul a­tor i s presented . The expl i c i t met hod means t hat the regul ator des i g n step is based on a p rocess ident i fi cat i on step . It i s assumed that the changes i n the set po i nt duri ng the tun i ng procedu re are suff i c i entl y exc i t i ng T Th-i s w�rk was-s�pported by the Nat i ona l Sc i ence Foundat i on and the Hungari an Academy of Sc i ences under Grant No . p-1 -45 and NSF/ I NT 8120366 .

8 1

for the ident i fi cati o n . I n the sequPl we concentrate on the regu l ator des i gn step . A S IMPLE DES I GN ALGOR ITHM FOR PID REGU LATORS Cons i der a stabl e l i near process descr i bed by a second o rder model

y ( t ) = ( bo+bl z: l ) z_-d

u ( t ) = _B ( z-_1 ) u ( t-d )

- 1 -2 1 l+a1 z +a2z A( z- )

B A u ( t -d ) . ( b0:1:0 . d > O )

where t = 0 . 1 , 2 , . . denote the d i screte time i nstants . u and y stand for the process i nput and output . respect i v e ly . Th e d i screte t ime del ay d i s as sumed to be d > O . Us i ng the most common sampl ed d ata PID regu l ator g iven by -1 -2

- Po+p l z +p2z w - - - � · - - -R l - 1 -z .

the cl osed l oop system i s shown i n Fi g . 1 .

� · r � � I Fi g . l. Bl ockscheme of the c l osed l oo p system As a very good robust des i gn base ( Banyasz and Kev i czky . 1982) a cl ass of P I D regul ators g iv en by P ( z- l )=p0A ( z- l ) wi l l be cons i dered i n the sequel . To ensure a stab l e cl osed l oop behav i o u r wi th phase-advance of 'a=60° and rel ati ve overshoot of 5% a proper p0 has

82 J . Hetthes sy , L . Keviczky and Cs . Banya�z

to be _ determ i ned . However , p0 i s a compl ex funct i o n of b0 , z 1 and d , where z1=-b1 /b0 i s the zero of the process . Banyasz and Kev i c zky ( 1 982 ) showed that for z1 =o ( b1 =0 ) k l 1 Po = b = b\2d-l} ( 1 ) 0 0 g i ves a fai r ly si mpl e desi gn rol e . I n th i s case the overal l trans fer funct i on between the set poi nt ( reference si gnal - Yr ) and the control l ed _autput (y) i s k 1z

��z-l�kz -d ' where k 1 = t / ( 2d- l ) .

( 2 )

For the general case ( zi*O) f i rst recal l some resu l ts of a s impl e servo- regu l atory des i gn method ( Hetthessy , Kev i czky and Kuma r , 1983a , 1983b ) . Us i ng the des i gn pol ynom i a l s V ( z-1 ) =V= u0+u1z-l+ • • . +un z- nv and T ( z-1 ) =T=t0+t 1z-1+ . . . +tn�z-nT a regu l ator by

w = __};{ _ _ R B ( T-Vz-d ) l eads to an overal l transfer funct ion

v -d T z

( 3 )

( 4 ) I t i s seen that t o avo i d steady-state errors V ( l ) =T ( l ) mu st hol d , thu s the regu l ator by Eq . ( 3 ) i s of i ntegrat i ng type , as z=l i s a root of ( T- vz-d ) . Compar i ng Eq . ( 2 ) and Eq . ( 4 ) it i s seen that V=k i and T= l -z- l+k 1 z-d ensure the des i red cl osed l oop performance . However , the general i zat i on has been made on ly for wel l -damped process zeros so far , as the regu l ator by Eq . ( 3 ) cancel s the process zero . Hav i ng poor ly-damped o r i nverse unstabl e pro­cess zeros ( nonm i n i mum phase process ) they are supposed to be zeros of the cl osed l oop system , as wel l , whi ch l eads to an overal l t ransfer funct i on

V B -d T BID z

To ach i eve th i s a regul ator by AV

TB ( 1 ) - BVz -d s hou l d be used . Combi n i ng V=k 1 , T= l -z- l+k 1 z-d and Eq . ( 6 ) we have

( 5 )

( 6 )

Ak 1 /B ( l ) l WR = -1� • l+fz-d = WP I D • We , ( 7 )

where f ( 8 )

and the regu l ator i nvol ves a P IO regu l ator st i l l based on the si mpl e des i g n rol e k 1 =1 / ( 2d- l ) and a correct i ng term l / ( l+fz -d ) ensu ri ng the poorl y-damped or i nverse u nstabl e process zero to appear in the cl osed l oop ( F i g . 2 ) .

Fi g . 2 Bl ockscheme of the c l osed l oop system conta i n i ng the P IO regu l ator

It can be shown that for the poorly-damped zeros and for a wide range of unstabl e zeros B/B ( l ) only s l i g ht l y mod i f i es the requ i red step response of Eq . ( 2 ) . However for non­m i n i mum phase processes wi th z1 > ' 1 i t may be des i rabl e to compensate the undershoot caused by a zero cl ose to 1 by i ntroduci ng a c l osed l oop pol e Pl · Hav i ng T= ( I -z- l+k 1z-d ) T* , where T*=t0*+t 1*z-l , t0*+t *=l and P l = -t 1*/t0* , We has the �ol l owi ng form

We T*+h-d ( 9 ) where

f = k l 1 1 (y:_-µ- - 1-Z) . ( 1 0 )

1 1

As far as the adapt i ve sol ut i o n i s concerned us i ng the on- l i ne recurs i ve l east squares

' estimates i n th� cont rol l aw by Eq . ( 7 ) , real -t i me experiments gave fa i rl y ni ce resul t s . CONCLUS ION Comb i n i ng a s i mpl e des i gn rol e for PI O regu-1 at�rs and som� '.esu l ts of servo- regu l atory des i gn an expl i c i t adapt i ve cont rol l er has been suggested i n the paper . The i nt roduced adapt i ve P I O regu l ator wi th a connect i ng term proved to be very effect i ve i n the practi ce. Sev�ra l proj ects ex i st i n Hungary thi s t ime to i mpl ement such regu l ators on micropro­cessor based d i g i ta l cont rol l ers . REFERENCES Ba'nyas z , Cr. and L . Kev i czky ( 1 982 ) : Di rect

Methods for sel f-tun i ng P I O Regu l ators . I FAC Si'_mpos i um , Washi ngton , USA. Isermann , R . ( 1 97 7 ) : Qi g i t a l e Regel systeme .

Spri nger-Verl ag , Berl i n-Hei del berg-New Yo rk .

I Hetthessy , J . and L . Kev i c zky ( 1983a ) :

Adapt i ve Regu l ators for the Determ i n i st i c Servo-Regul atory Probl em. Report of De�artm�nt of E� ectr i cal Engi neer:rng; Uni vers i ty of Mi nnesot a , U . S . A . �

I

Hetthessy , J . , L . Kev i czky and K. S . P . Kumar ( 1 983b ) : On the Determi n i st i c Adapt i ve Servo Regul ators . 1 7th Annua l Conference E.!:1__J_n_!.ormat i o n Sc i ences and Systems , Bal t imore , U . S . A .

Wi ttenmark , B . ( 1 979 ) : Se l f-tun i ng P I O control l ers based on pol e pl acement . �artment of Automat i c Control , Lu nd .!_nst i t ute of Tec hnol o-g�-Sweden.

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROBUST DESIGN OF ADAPTIVE OBSERVERS IN THE PRESENCE OF PARASITICS

Y. Kawasaki**, S. L. Shah*, Z. lwai*** and D. G. Fisher**

*Department of Mechanical Engineering, Ariake Technical College, Ohmuta, Fukuoka -836, japan **Department of Chemical Engineering, Universit_� of Alberta, Edmonton, Alberta, T6G 2G6, Canada

***Department of Mechanical Engineering, Kumamoto University, Kumamoto 860, japan

Abs t ra c t . A s i n gu l a r pe r t u rba t i on approach i s used to des i gn an adapt i ve i dent i f i e r/obse rve r w i th an e xponen t i a l ra te of conve rge nce . Conve rgence ana l ys i s o f th i s obse rve r p ro v i des asymp to t i c e r ro r boun ds fo r the e s t i ma t i on e r ro r . The pe r fo rmance o f the ada p t i ve obs e r ve r i s e va l uated on sys tems w i th wea k l y and s t rong l y obse rvab l e pa ras i t i cs a n d i t i s s hown tha t the res u l t i ng pe r fo rmance i s rob us t .

Keywo rds . Adap t i ve sys tems ; obs e rve rs ; pa rame te r e s t i ma t i on ; s ta te e s t i ma t i on ; sys tem o rde r reduc t i on ; l i nea r sys tems .

I NTRODUCT I ON

Th i s pape r i s conce rned w i th the des i gn o f an adap t i ve obse rve r to es t i ma te s t a te va r i ab l es and i de n t i fy p l an t pa ra me te rs o f an unknown , S I SO , l i nea r sys tem w i th "fas t mode s " o r "pa ras i t i c" e 1 emen ts . The obse rve r des i gn

S uch a p l an t can be rep resented by the s tanda rd s i ng u l a r pe r t u rb a t i on form ( Kokotovi c and l oanno u , 1 98 1 )

(2 . 1 )

( 2 . 2 )

( 2 . 3 )

i s based o n the reduce d p l an t mode l o rde r equa l to the o rde r of the s l ow o r dom i nant pa rt o f the unknown p l an t . The red uced ­o rde r obs e r ve r s t ruc t u re cons i de re d he re i s based on a non -mi n i ma l type rep resen tat i on w i th the adap t i ve l aw dete rmi ned by eq ua t i on s tha t g i ve an e xponen t i a l rate o f conve rgence , as rece n t l y p roposed by l wa i and othe rs ( 1 98 1 ' 1 982) .

whe re x E Rn , xf E Rm , and µ i s a s ma l l pos i t i ve s ca l a r pa rame t e r assoc i a ted w i th the p resence o f fas t o r p a ras i t i c mode s .

A s i ngu l a r pe r t u rba t i on app roach to the des i gn o f l ow-o rde r adap t i ve obse rve rs to e s t i ma te s t a te va r i ab l es and i den t i fy the va l ue of the s i n g u l a r pe r t u rba t i on or the pa ras i t i c pa ramete r , µ , was f i rs t cons i de re d b y l wa i and Kawasak i ( 1 9 77) . Mo re recen t l y Koko tov i c and l oannou ( 1 98 1 , 1 98 2 ) have a l so used the s i ng u l a r pe rt u rba t i on approach fo r the des i gn of an a dapt i ve es t i ma to r and i den t i f i e r i n the p resence o f p l a n t -mode l mi s match . Howeve r , i n the i r t reatment they have not cons i de re d the conve rgence p rope rty of the i r obse rve rs . I n th i s pape r conve r­gence ana l ys i s of the a dapt i ve obse rve r i s cons i de re d by deve l op i n g asymptot i c bounds on the obse rve r e s t i ma t i on e r ro rs when app l i ed to an unknown p l an t w i th fas t para­s i t i cs . I l l us t ra t i ve e xa mp l es a re p resen ted to eva l ua te the pe r fo rmance of the adapt i ve obse rve r .

P ROBLEM FO RMULAT I ON S uppose that an ( n+m ) th o rde r , l i nea r , S I SO , t i me - i nva r i an t p l an t has n s l ow (o r "dom i nan t " ) and m fas t (o r "pa ras i t i c" ) mode s .

83

The s ta te of the dom i n a n t pa rt o f the sys tem i s cha racte r i sed by x and s ta te xf cha rac­te r i zes the fast t ra n s i en t modes of the p l an t . Not i ce that both s l ow and fas t mode s cont r i b ute t o the output . B y de f i n i ng the fas t pa ras i t i c s ta te as

(2 . 4 )

we can ob ta i n a rep res e n ta t i on o f ( 2 . 1 ) a n d ( 2 . 2 ) i n a fo rm whe re t h e dom i nan t and pa ra­s i t i c parts appea r e xp l i c i t l y ( Kokotov i c and l oannou , 1 93 1 ) as fo l l ows :

� = A0x + b0 u + A i 211 ( 2 . 5 )

µii = Af 11 + µ b f u ( 2 . 6 ) CTX + T T b f u (2 . 7) y 0 C2 11 - c2

(2 . 8 )

84 Y . Kawasaki et a l .

As s umi ng t h a t t h e pa i r ( C0 , A0 ) i s comp l e te l y obse rvab l e eqns . (2 . 5 ).- ,2 . 7 ) can be e xp ressed i n the fo l l ow i ng canon i ca l fo rm:

* = fa ; _

_ ��J x+b u+ ( cc1A f+µA i ) cr l : o . . . oj

µcr = Af a + b f a

Y = CT x CT [ ] X1 , = 1 , 0 . . . Q T T whe re A 1=A1 2-aC , b=b0+aC 2b f and n=µcr .

( 2 . 9 )

( 2 . 1 0 )

(2 . 1 1 )

The above rep resen tat i o n (eqns . ( 2 . 9 ) - (2 . 1 1 ) i s the bas i s for the des i gn of rob us t , adapt i ve obse rve rs as cons i de re d i n the fo l l ow i ng sect i ons . The a dap t i ve obse rve r s t ruct u re cons i de red he re i s a non -m i n i ma l type and i s based on the mode l o rde r eq ua l to the o rde r of the s l ow or dom i nant pa rt of the unknown p l an t . The rob us tness i s s ue i s cons i de re d he re w i th respect to mode l - p l a n t mi s match due to fas t o r "pa ras i t i c" e l ements i n the p l a n t . The " robus tness" of the des i gn i s i nves t i gated by app l i ca t i on o f the adap­t i ve obse rve r based on a l ow orde r mode l to a p l an t w i th pa ras i t i cs .

REDUCED U KDtK P LANT MO DEL I n the p resen t pape r a s i ngu l a r pe r t u rba t i on

approach i s taken to des i gn l ow-o rde r a dap­t i ve obse rve rs w i th an e xpone n t i a l rate o f conve rgence . The conve rgence rate i s re l a te d t o a s ma l 1 numbe r of pa rame te rs tha t c a n be set a rb i t ra r i l y .

By se t t i ng µ=0 i n eqns . ( 2 . 9 ) and (2 . 1 0 ) the dom i nan t pa r t of the p l an t (of o r de r n ) can be rep resented by

[ j _ _ E�:� [ ' O . . . oJ ( 3 . 1 )

( 3 . 2 )

Th i s l ow-o rde r p l an t i s now use d fo r des i gn i ng an a dap t i ve obse rve r . The des i gn and pe r­formance of the adap t i ve obse rve r i s eva l ua ­ted fo r sys tems i n wh i ch the fas t modes a re s t rong l y obse rvab l e i n the outp u t , i . e . c2�0 . The adap t i ve obse rve r app l i ca t i on when the fas t modes a re weak l y obse rvab l e i s a s pec i a l case of th i s p rece d i ng p rob l em and i ts des i gn i s obta i ne d by s i mp l y s e t t i ng C2=0 . ( s ee Appe nd i x A ) .

ADAPT I VE OBSERVER DES I GN I n some res pects , the fo l l ow i ng adap t i ve obse rve r des i gn a p p roach i s s i m i l a r to the one cons i de re d by Kokatov i c a n d l oannou ( 1 98 1 ) . Howeve r , i n the i r t reatment Kokatov i c and co-wo rke rs have not cons i de re d the conve rgence p rope rty of the i r obse rve rs . The rate of conve rgence o f the s t a te es t i ­ma tes a n d pa ramet e r i de n t i f i ca t i on i n the i r s cheme i s re l a te d to : the ga i n o f the a dap-

t i ve l aws ; the i np u t and output f i l te rs ; and on the a mp l i t ude and f re q uency of i np u t s i g­n a l s . I n the p resent pape r the a d ap t i ve obser­ve r des i gn i s based on a non-mi n i ma l type re­p resenta t i on w i th the adapt i ve l aw det e rm i ne d b y e q ua t i ons that g i ve an e xponen t i a l rate o f conve rgence as rece n t l y p roposed b y l w a i a n d othe rs ( 1 98 1 - 1 982 ) . I n a dd i t i on to th i s cons i de ra t i on , the fo l l cw­i n g adap t i ve obse rve r i s des i gned by i nc l udi ng i n the des i gn s tage the th i rd te rm on the right s i de of eqn . ( 3 . 1 ) wh i ch i s re l ated .to de r i va­t i ve s of the i np u t s i gna l , u ( t ) . Kokotov i c aid l oannou ( 1 98 1 ) have repo rted tha t the pe r fo r ­ma nce o f a n adap t i ve obse rve r de t e r i o ra te s i f the " u ( t ) te rm" i s e xc l ude d f rom the des i gn

As s ump t i ons The fo l l ow i ng bas i c a s s ump t i ons a re made for the l i nea r p l an t :

i ) A9 and A f a re H u rw i tz ma t r i ces i i ) u \ t ) and u ( t ) a re bounde d , V t � O .

Now cons i de r the fo l l ow i ng H u rw i tz n x n ma t r i x K

K = k ' I [ I J : _ _ !:! : L (4 . l )

' O . . . . 0

and ass ume tha t - A ( A>O) denotes the maxi mum va l ue of the rea l pa rt of the e i genva l ues o f K . Comb i n i ng eqns . (2 . 9 ) - ( 2 . 1 1 ) and (4 . 1 ) g i ves

* = Kx+ ( a-k ) y+bu-CC�bfu+CC�bfu T +CC2Afcr+µA1 cr

Kx+ (a-k ) y+bu+b"'cu

+CC� (b fu+Afcr ) +µA lcr ( 4 . 2 )

µa = Afcr + bfu

y = CTx T whe re b ''' = - C2bf .

The so l ut i on of eqn . (4 . 2 ) i s g i ve n by

whe re

CT - 1 ZT i )

i P i a CTK zTK

' i = 1 , 2 CTKn- 1

z!Kn- 1 I

i i ) � 3 = Kp 3 + C u p 3 (o) = 0

(4 . 3 )

( 4 . 4 )

(4 . 5 )

(4 . 7)

(4 . 8 )

b u t i n orde r to a vo i d use o f the de r i vat i ve o f the i np u t s i gna l , u , i ns tead of ( 4 . 8) we use :

Robust Des ign of Adapt ive Observers 85

and i i i )

and i v)

0

T K z 1+Cy Z 1 (o ) =O

KTZ2+C u Z2 (o ) =O

T C P 3=P 3 1

f l = lt e K ( t - ·r ) C C� (b fu+Afcr ) d-r

t- -r A d f2 = lt K ( ) e 1cr T

(4 . 9 )

( 4 . 1 0 )

( 4 . 1 1 )

( 4 . 1 2 )

( 4 . 1 3 )

( 4 . 1 4 )

(4 . 1 5 )

( 4 . 1 6 )

Fo r the sys tem des c r i bed by e q ns . ( 4 . 5) and (4 . 6 ) the adapt i ve obse rve r equa t i on s a re :

X P 1 (� -k ) +P26+p 36*

9 z: (;-k ) +z!6+z 36*

(4 . 1 ?)

(4 . 1 8 )

whe re a , 6 and 6 '" a re es t i ma tes of the un­fami 1 i a r p l a n t pa ra meters a , b and b '" respect i ve l y .

We note that the p roposed a da p t i ve obse rve r i s des i gned by cons i de r i n g the th i rd te rm on the r i gh t hand s i de of eqns . (4 . 5 ) and ( 4 . 6 ) . Th i s means that the a d ap t i ve obse rve r des i gn i s based on the re duced o r de r p l an t model , eqns . ( 3 . 1 ) and ( 3 . 2 ) . The reason fo r th i s i s so that the so l ut i on ob ta i ne d by se t t i n g µ=O i n eqns . ( 4 . 5 ) a n d ( 4 . 6 ) co i nc i de s w i th the s o l ut i on of the reduced o rde r p l an t mode l ( 3 . 1 ) and ( 3 . 2 ) .

Now , the output e s t i ma t i on e r ro r , e 1 ( t ) , can be re-exp ressed f rom eqns . ( 4 . 6 ) a n d (4 . 1 8 ) .

e l y - y

( 4 . 1 9 )

whe re

( 4 . 20 )

The a dap t i ve obse rve r p rob l em i s t o f i n d a n adap t i ve l aw s uch tha t

1 i m t; ( t ) "'O , 1 i m e ( t ) =0 (4 . 2 1 ) t+oo (e ( t ) =X-x)

w i th an a rb i t ra r i l y s pec i f i e d ra te o f con­ve rge nce .

ASCSP-D

The res u l t i ng adap t i ve l aws fo r a dj us t i ng the p a ra me te rs a re ( l wa i a n d othe rs 1 982) :

l= - ¢1; , t; (o ) =t;0 , ¢ >0 ( 4 . 22) ( ¢ i s a s ca l a r )

I n o rde r t o make th i s adap t i ve l aw rea l i z­ab l e we i n t roduce a ux i l i a ry va r i ab l es i n the fo rm o f a ( 2n + l ) xl vec to r , E ( t ) , and a symme t r i c ( 2n+l ) x ( 2n+l ) ma t r i x, R , s uch that the f i na l adapt i ve l aw becomes

- 1 - ¢ R E ( 4 . 2 3 )

whe re E a n d R a re re l a ted b y the equa t i on s :

( 4 . 24)

(� i s a s ca l a r )

R=-�R+ZZT , R (o ) i s a pos i t i ve def i n i te ma t r i x (4 . 25 )

The conve rgence p rope rt i es of th i s adapt i ve J aw (eq n . (4 . 23 ) ) a re es tab l i sh e d by the fo l l ow i ng theorem .

Theo rem 1 . As s ume that ¢ > O ; � > O ; LR . . ( t) ] i s bounde d ; R ( t ) i s nons i ng u l a r , I J - 1 a n d I I R ( t ) I I i s bounded fo r a l 1 t >, O . The n the pa rame te r adj us tme n t l aw g i ve n by eqns . (4 . 2 3 ) - ( 4 . 25 ) ens u res that

(4 . 26 )

11 e ( t ) II � (4 . 27 )

whe re A 1 = m i n { ¢ , � , A } , and mt; , mt;o ' me a n d meo a re pos i t i ve cons tan ts .

A s pec i a l case o f Theo rem 1 a n d the res u l ­t i ng conve rge nce ana l ys i s i s when µ = O . Th i s i s s ta te d as a co ro l l a ry to Theo rem 1 . Coro l l a ry 1 . Subj ect to a s s umpt i ons i n Theo rem 1 , i f µ "' 0 , then

I I t; ( t) I I � mt; e - A 1 t

Jl e ( t ) l l � me e - A1 t ( 4 . 28 ) ( 4 . 29 )

( i . e . l i m t; ( t ) =O , J i m e ( t ) =O w i th conve r ge nce t+oo t-+00

rate of A 1 )

Rema rk 1 . Co ro l l a ry 1 i s a s pec i a l case of Theo rem 1 and i s i den t i ca l to Theo rem 1 i n l wa i a n d othe rs ( 1 982) . Th i s means that when µ = 0 the a d ap t i ve obs e rve r ( 4 . 1 7 ) and ( 4 . 1 3 ) can i den t i fy the reduced o r de r p l an t mo de l ( 3 . 1 ) and ( 3 . 2 ) e xa c t l y . Howeve r , i n pract i ce µ � 0 i n wh i ch case the con s ta n t te rms , mt;o a n d meo ' asymp to t i ca l l y bound the pa rame te r e r ro rs a s e xp re s s e d i n the s t a te ­ment o f Theorem 1 . Rema rk 2 . Theorem 1 can be p roved by us i ng a rguments s i m i l a r to the p roof of Theorem I

86 Y . Kawasaki et a Z .

i n l w a i a n d othe rs ( 1 98 2 ) . The re fo re i ts p roof i s omi t ted he re fo r the sake of b rev i ty . Rema rk 3 . R ( t ) : bounded a n d nons i ng u l a r a n d I I R ( t ) - 1

1 1 i s bounde d , JJ. t � 0 a re t rue i f R (O ) > 0 a n d u ( t ) a n d O (t ) a re bounded "I- t � 0 , a nd u ( t ) i s s uff i c i en t l y r i ch (� (2n+ I ) e l ement of f req ue nc i es ) . Rema rk 4 . Note that the n umbe r of i nteg rators req u i red to cons t ruct a n adapt i ve obs e rve r i s 2n2 + I On + 3 (compa re th i s w i th the orde r ; 2 (n+m) 2 + 7 (n+m) , req u i re d fo r the non­reduced o rde r p l an t mode l ) .

Now cons i de r a n adap t i ve obse rve r des i gn method that i s based on the reduced o rde r p l ant mode l ob ta i ne d by neg l ec t i n g the th i rd te rm ( u) on the r i gh t hand s i de of eqn . ( 3 . 1 ). I n th i s case the adap t i ve obs e rve r equa t i ons a re g i ve n by :

A x = P 1 (a-k) + P2b ( 4 . 1 7)

y T A Z l ( a -k ) + zJ6 (4 . 1 8 )

and the output es t i ma t i on e r ror i s :

(4 . 1 9 )

� (4 . 20 )

( 2 . 30 )

The s i ze of £ a n d R a r e reduced to 2n x 1 , 2n x 2n f rom (2n+l ) xl , (2n+l ) x (2n+I ) respec­t i ve l y . Howeve r we note tha t f0 i s i ndepen den t o f the pa ras i t i c pa ra me te r µ , i . e . f0 i s not reduced to ze ro eve n i f µ = 0 . I n th i s case , the fo l l ow i n g Theo rem 2 and i ts coro l l a ry a re ob ta i ne d .

Theore m 2 . Unde r the same cond i t i ons a s i n Theore m 1 ,

� (4 . 26 )

( 4 . 2 7)

A s pec i a l case of Theo rem 2 i s when µ = 0 , wh i ch i s s t a ted h e re a s co ro l l a ry 2 .

Co ro l l a ry 2 . I f µ = 0 , then s ubject to a s s umpt i ons i n Theorem I

1 1 F; ( t ) 11 � n;� e-A. 1 t + n;F;

I I e ( t ) I I � m; e - A. i t + me

Rema rk 5 . I n th i s des i gn me thod , the pos i t i ve cons tants mF; a n d me rema i n as e r ror bounds e ve n when µ-. O . �

NUME R I CAL S I MULAT I ON RESULTS

The e f fect on the i den t i f i ca t i on and obs e rva­t i on res u l ts o f s t rong l y observab l e pa ra­s i t i cs , i nput s i g na 1 amp 1 i t ude and f req uency, an d the O te rm i n the obse rve r des i gn a re i l l us t ra te d by d i g i ta l s i mu l a t i on of the a dapt i ve observe r . Two examp l es a re g i ven .

Examp l e 1 . Th i s e xamp l e descr i bes a weak l y obse rvab l e pa ras i t i c sys tem (C2 = 0 ) , and i s cons i de red he re i n o rde r to demons t r a te the rob us tne s s of the p resent obse rve r des i gn me thod ve rs us tha t of l oannou an d Kokotov i c ( 1 982 ) . Cons i de r the 3 rd o rde r p l an t ( l oannou a n d Kokotov i c , 1 982) .

R • � I � �] X+� :� x,+� : :� " (5 . 1 )

µ� = -4x - 2 u f f

y = ( 1 O ] x

(5 . 2 )

(5 . 3 )

Eq ua t i ons cor re s pond i ng t o eqns . (2 .9 ) - (2 . 1 1 ) a re ob ta i ned by

x = r-. 5 l]x+ [11 u+µ fo . 9t (5 . 4 ) t 1 0 o 2 � · 5J µa = -4cr + 0 . 5 u

y = ( 1 O ] x

(5 . 5 )

( 5 . 6 )

The adap t i ve observe r i s des i gned based on 2nd o rde r mode l wh i ch i s obta i ned by sett i ng µ = 0 i n eqns . (5 . 4 ) - (5 . 6 ) :

(5 . 7)

Y = [ l O] X (5 . 8 )

The re fore the va l ues to b e i de n t i f i ed a re T T ( T a = [a 1 , a2 ] = ( -5 , - 1 0 ] and b = b 1 , b2 ]

[ l , 2 ] T . S i mu l a t i on res u l ts f rom s ta te e s t i ma t i on/paramete r i de nt i f i ca t i on a re shown i n F i g u res 1 and 2 .

The des i gn paramete rs a n d i n i t i a l cond i t i ons i n these runs a re as fo l l ows : x ( O ) = (O , O ] T , cr (O ) = 0 , R (O ) = 14 , k 1=- 1 3 , k2 =-40 ( A.=5 . o ) , $=5 . o , w=2 . o . I n the s i mu l a t i on res u l ts p resen ted here the pe rfo rmance of the two obse rve rs i s compa red to demons t ra te the robus tness of

Robust Design of Adaptive Observers 87

our des i gn me thod wi th res pect to ( i ) i np ut s i gna l amp l i t ude ( f i gure 1 ) and ( i i ) i np u t s i gna l f requency ( f i g u re 2) . From both f i gures 1 and 2 i t i s appa ren t that the conve rgence of the output and the pa rame te rs us i ng the p roposed des i gn s cheme i s re l a t i vely fas t and the e r rors neg l i g i b l e . I n f i gure 2 we compare res u l ts of the p roposed adap t i ve obse rve r to that p roposed by l oannou and Kokotovi c ( 1 982 ) and demons t rate c l ea r l y the i nsens i t i v i ty of the convergence to h i gh f re­q uency con ten t of theA i nput s i gna l . The e rrors i n e s t i ma t i ng a 2 and 62 us i ng ou r scheme a re s i gn i f i can t (of the orde r of 70%) . Howeve r , by p rovi d i ng an i nput tha t was s uff i c i en t l y r i ch (addi ng two mo re f requencies i n the i nte rmed i ate range of the i nput s i gnal ) we were ab l e to reduce these e s t i ma t i on e rrors to 7% .

Examp l e 2 . Cons i de r the 2nd orde r p l an t :

� = -5x + xf

µ� = -4x -f f

9 = x + xf

+ u (5 .9 )

2u (5 . 1 0 )

(5 . 1 1 )

Th i s examp l e desc r i bes a s t rong l y obse rvab l e pa ras i t i c sys tem ( C2 � 0 ) , and i s i nt roduced i n o rde r to con f i rm the ma i n res u l ts of Theorem 1 , and co rol l a ry l . Eqns . (5 .9 ) · � 5 . 1 1 ) i s t ransforme d i nto the form (2 . 9) -(2 . 1 1 ) :

x = -5x-2u + (-4+µ •6 ) o

µo = -4o + 0 . 5 u y = x

(5 . 1 2)

(5 . 1 3)

(5 . 1 4 )

The adapt i ve obse rve r i s des i gned based on 1 st orde r mode l

. X = -5X - 2 u - 0 . 5 u y = x

(5 . 1 5 )

(5 . 1 6 )

S i mu l a t i on res u l ts a re dep i c te d i n F i g . 3 , whe re

x(o) =O , o (o) =O , R (o) = l 3 , k 1=- 1 0 ( A= I O) ,

<f>=7 . 5 , ip=8 . 0 , u ( t ) =5s i n t+3s i n l . 3t

+5s i n2 . 5 t

The res u l ts con f i rm the fo l l ow i ng : I ) i f µ=0 , th i s adap t i ve obs e rve r can i den t i fy the reduced orde r p l a n t mode l exact l y (c f . co ro l l a ry l ) , 2) as µ i nc reases i de n t i f i ca t i on e r ro rs i nc rease (c f . Theorem I ) . Now cons i de r the des i gn of an a dap t i ve obser­ve r for the same sys tem but w i th the u te rm exc l uded . The mode l i n th i s case i s :

X -5X - 2 u

y x

As e xpected i dent i f i ca t i on e r ro rs become much g reate r , (as s ta te d i n Theorem 2 and coro l ­l a ry 2) and rema i n even when µ=0 ( F i g . 4 ) . Th i s e xamp l e i l l us t ra tes the use fu l ness of i nc l ud i n9 the u te rm i n the adapt i ve obse rver des i gn . ( Remark 5 )

CONCLUS I ONS

Conve rgence res u l ts for reduced o rde r adap­t i ve i den t i f i e rs and obse rve rs w i th exponen­t i a l rates of conve rgence have been obta i ne d fo r the gene ra l case of a s t rong l y obse rvab l e p l ant w i th fas t o r h i gh - f requency paras i t i cs . Asymptot i c es t i ma t i on e rro r bounds a re obta i ne d e ven when the ' u ' term i s cons i de red i n the des i gn stage .

S i mu l a t i on s t ud i es demons t ra te that the red uced orde r adapt i ve i den t i f i e r/obse rve r i s robus t w i th respec t to i nput s i gn a l amp-1 i tude and f req uency i n the p resence of pa ras i t i c e l ements .

ACKNOWLEDGEMENT

Fi nanc i a l ass i s tance i n the fo rm of a vi s i t i ng resea rch fe l l owsh i p to the f i rs t a uthor f rom the Japanese Educa t i on M i n i s t ry i s g ratefu l l y acknow l edged .

REFERENCES

l oannou , P . A . and P . V . Kokotovi c ( 1 982 ) . An asymptot i c e rro r ana l ys i s of i dent i f i e rs and adap t i ve obse rve rs i n the p resen ce of pa ras i t i cs . I EE E T rans . Autom . Con t rcl , 2 7 , 9 2 1 -92 7 .

lwa i , z . , K . Mano , A . I noue a nd Y . Kawasak i ( 1 98 1 ) . An a da p t i ve observe r w i th e x­ponent i a l rate of convergence fo r s i ng l e­i nput s i ng l e -output l i nea r systems . P roc . of the 8th Cong ress of I FAC , Kyoto , -­Vol . V I I , 1 087- 1 092 .

l wa i , Z . , M . Sato , A . I noue and K . Mano ( 1 982 ) . An a dap t i ve obse rve r w i th ex­ponent i a l ra te of conve rgence . T ran s . of the Soc i et of I ns t rument and Cont ro l Eng i nee rs , !_, 3 3-3 i n Japanese

l wa i , Z . and Y . Kawasak i ( 1 977) . A des i gn me thod of adapt i ve observe rs for sys tems w i th A-va r i a t i on . Trans . of the Soc i ety of I ns t rument a n d Cont ro l Eng i nee rs , 1 3 , 1 05 - 1 1 1 . ( i n Japanese) ·

-

Kokotov i c , P . V . and P .A . l oannou ( 1 98 1 ) . Robus t redes i gn of con t i n uous - t i me adap­t i ve schemes . P roc . of 1 98 1 I EEE Con f . on Dec i s i on and Cont ro l , 522-527 .

88 Y . Kawasaki e� al.

APPEND I X A Wea k l y obse rvab l e pa ras i t i cs case P l a n t eqns . (4 . 2 ) - (4 . 4 ) a re red uced to :

x = Kx+ (a -k ) y+bu+µA1cr

µcr = Afcr + bi1

y = CTx Eqns . (4 . 5 ) and (4 . 6 ) a re reduced to :

x = P 1 (a-k ) +P 2b+µf2+f 3

y zi (a-k ) +zib+µCTf2+CTf 3

( A . l )

(A . 2 )

(A . 3 )

(A . 4 )

(A . 5 )

eq ns . ( 4 . 8 ) - ( 4 . 1 0 ) , ( 4 • l 3 ) and ( 4 . l 4 ) a re unnecess a ry .

Adapt i ve obse rve r :

x = P 1 (a-k) + P 2b

v = zI (a-k) + zIG

Output es t i ma t i on e r ro r : T T T e 1 = Y-y = Z �-µC f2 -C f 3

whe re [ J �A J Z 1 a-a z = � "

Zz b-b

(A . 6 )

( A . 7 )

(A . 8 )

( A . 9 )

The s i ze o f vecto rs/ma t r i ces £ a n d R i s red uced to 2n xl and 2nx2n from (2n+l ) x l , and ( 2 n+ l ) x ( 2n+l ) res pec t i ve l y .

Theorem 3 and Co ro l l a ry 3 co r respon d i ng to Theorem 1 a n d Co ro l l a ry l can now be s ta te d as fol l ows :

Theorem 3 . Unde r the same cond i t i ons a s i n Theorem l ,

I I � (t ) I I �m�e- ;l.. 1 t+m�0

I I e ( t ) II �m*e- :>.. 1 t+m .. e .,,o Coro l l a ry 3 .

(A . 1 0 )

( A . 1 1 )

Subject to ass ump t i ons i n Theorem l * i f µ=O , then

10

£ "' G>

0

6

(�

0 0

1.5

1.0 £

<ii 0.5

00

10

' '

' '

r ...... j

jV

i, " ' •

1 · \

•' " • '

-

10

10

20 Time (sec)

20 Time (sec)

30

30

40

40

• '

! \ ____________ )___ ___________________________ ---

10

3.1%

20 Time (sec)

0.7%

30 40

I I � ( t ) I I �m�e- :>.. i t

I I e ( t ) I I �ifl:e - :>.. 1 t

(A . 1 2 ) 00�..i;:::L.:.1..-'-1�0-'--.l-J.-1-�20-'--'-'-!.-�30_,___._..___,__J40 Time (sec)

(A . 1 3)

Rema rks . I n th i s case F i g u re l :

The con d i t i on that u ( t ) : bounded IJ t � 0 i s unneces s a ry . The n umbe r o f i n te g ra to rs req u i red fo r i mp l i -men t i ng the a dapt i ve obse rve r can be reduced even mo re to (2n2 + 7n ) .

State and pa ramete r e s ti mates fo r examp l e 1 w i th u (t ) = 5 s i ne t + 1 5 s i ne 2 . 5 t . l oannou and Kokotovi c ( 1 982) obse rve r ( so l i d l i ne ) vs th i s me thod (dashed l i ne ) .

5

I.I • 0 .02, u{t} • 5 s i ne t + 5 s i ne 25t for the loanoou and Kokotovi c obse rver (sc 1 i d 1 i ne) and 1.1 = 0 .02, u ( t ) "' 5 s i ne t + 7 s i ne 2.St + 9 s i ne 3 . S t + 5 s i n� 25t for the· ob!ierver wi th exponen t i a l rate of convergence (dashed 1 1 ne)

- oo�'--'--'--'--::

1�0

-'--'--'--'--�20---'---'---'---'---130---'---'---'---'_J40 Time (sec)

10

1.5

20 Time (sec)

30 40

e; lD l-/.-�������1---����������� ii

o.s ( t 1 0

1 .4%

20 Time (sec)

30

34%

40

--------- µ, = 0 ' • " 0.5 ' , ·-·-·-·-·-·-· µ, = 0.02 ············· ····· µ, = 0.05 i l i �

u ( t ) =S s i n t+3 s i n I . 3t+5 s i n 2 . 5t 0 te rm i s cons i de re d in the des i gn p roce d u re

0 ,_.�--,=�------------------ -------- - ----------- - - - -----

0

5

5

--------- µ, = 0

10 Time (sec)

-·-·-·-·-·-·- µ, = 0.02 ··············· · · ·µ, = 0.05

---, 1 ' ' . r: f\. .

15 20

•• . :··.. _,·.t ···.... ... :°\ ......... l ··... l ... .. f ······ ··... ·· . . f ·· .......... . . \>:�:r·----�:J-·-·-·-.. :·::�1-·-·-·-·-·-<:t·--·-·-.. :/r-·-·-·-·-·> -5 1---�----------------------------------------------

0

2

<..C 0

-1

5

--------- µ, = 0

10 Time (sec)

-·-·-·-·-·-·- µ, = 0.02 ········· ·· ·· · · · · ·µ, = 0.05

-'\:"\ 0:

15

'l i\ .... ! .. ·..... r···· ...... ...... .. ... •i'. . ... t .. . ···· .. [ ······ ··· .. [ ·· ....... . .. .

20

\�:�:j�---�:�J-·-·-·--�:�Yf··-·-·-·-·-·-.s-·-·-·-·---.::-+-·-·-·-·---�> -2 1---�---------------------------------------------

0 5 10 Time (sec)

15 20 Fi gure 2 : S ta te and parame te r es t i ma tes

fo r examp l e I F i gu re 3 : State and pa rame te r es t i ma tes fo r examp l e 2

1.0 � " f ! 0.5 ! \ ' i i i

0 i

-0.5

0

50

--------- µ, = 0 -·-·-·-·-·-·-µ, = 0.02

u ( t ) =S 0 te rm des i gn

5

l! ./\ ,r.... {\ 1 ,1 -....... °' .... :,1 tf �

s i n t+3 s i n l . 3t+5 s i n 2 . 5 t i s not cons i de red i n the p roce d u re

10 Time (sec)

1 5

--------- µ, = 0 -·-·-·-·-·-·- µ, = 0.02 f\ l \ A 'i'1 i ! \ I

20

<ca f\, fJ \\ .r\, f\\ I 1', \ :,! ',\ I • f 'i\ :r I ... ' I • j ,..,,'_\. � " I '\ \ :i \''--...... !/ ',''--

'i ' ·,. I ' • • .., I ' .... ,fi.j \ ... :·:·,',\., 1'

0 - 10

30

20

(..C 10

0

0

0

I ',�·:--�',·,. f:f ' .. --...... :,' ''--��\ ! ,, • ! ,., •

;, I '� 14 r . 'l f ' \ \ : I ', ,. I .'\ : J ti \ i�

_ _ / '/ ·,! "l} • '!

5

--------- µ, = 0 -·-·-·-·-·-·- µ, = 0.02

5

10 Time (sec)

10 Time (sec)

15

1 5

Fi gure 4 : Sta te and pa ramete r es t i matl5 .for examp l e 2

20

20

l;:j (1) "' ..... � 0 ...,

� "' 't:I rt ..... < (1) 0 <T "' (1) '1 < (1) '1 "'

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

IDENTIFICATION OF A pH PROCESS REPRESENTED BY A NONLINEAR WIENER

MODEL

G. A. Pajunen

DPfJarlmenl o( ElPClrical Engineering, Hel5inki University o( Technology, Espoo, Finland

Abs t ra c t . The a i m o f t h i s paper i s t o p resent t h e resu l t s o f i dent i f i cat i on of the pH process i n a s t i r red tank reactor rep resented by a non l i nea r W i ener type mod e l based on the meas u red samp l ed va l ues of the reagent f l ow and pH of ef f l uent . A f i r s t order , s tab l e and m i n i mum phase t ransfer funct i on mode l w i th pu re t i me dea l y equ a l to one i s used to desc r i be the dynam i cs of a cont i nuous f l ow reac to r , and a stat i c , non l i nea r and con t i nuou s t i t ra t i on funct i on i s approx i ma t ed by a p i ecew i se-po l ynom i a l funct i on . Two d i f fe rent methods a re u sed : 1 ) the parameters of the 1 i nea r and non l i nea r pa r t s a re i dent i f i ed sepa rate l y u s i ng a pa ra l l e l mode l refe rence a da p t i ve sys tem s t ructu re , 2) the pa rameters of an equ i va l en t s i ng l e- i nput mu l t i -ou t pu t l i nea r sys tem a re i dent i f i ed us i ng a ser i es mod e l refe rence s t ructu re . A s pec i a l stochas t i c generator wa s deve l oped for gene rat i ng a su i tab l e exc i ta t i on s i g na l . S i mu l a t i on and expe r i men ta l resu l t s a s we l l as a va l i dat i on of the i dent i f i e d mode l s a re p resented .

Keywords . Adapt i ve systems ; i dent i f i ca t i on ; non l i ne a r sys tems ; pH p rocess ; p i ecew i ce-po l ynom i a l a pp rox i mat i on ; W i ene r mode l .

I NTRODUCT I ON

The mode l of the pH- p rocess cons i s ts of a l i nea r m i x i ng dynam i cs of the conce n t ra t i on va r i a b l es and a h i g h l y non l i ne a r momentary equ i l i b r i um re l a t i on between the pH-va r i ab l e and the concent ra t i on va r i ab l e s . Any non l i nea r mode l wh i ch cons i s t s of a l i nea r dynam i c pa r t fol l owed b y a stat i c non l i nea r i ty i s refer red to i n the l i teratu re a s the �l i ener mode l ( Habe r , Kev i czky ( 1 976) ) . Th i s process i s d i ff i cu l t to cont ro l s i nce the non l i nea r i ty can be seve re and t i me-va r i ab l e , and the l i nea r dynam i cs of the p rocess a s we l l a s the p rocess t i me del ay change w i th the f l ow . The i dent i f i ca t i on of th i s p rocess i s performed u s i ng two d i f fe rent method s p roposed by Paj unen ( 1 982) for recu rs i ve i de n t i f i ca t i on of th e pa rameters of the non l i ne a r p rocess , wh i ch can be adequa t e l y represen ted by the W i en e r mode l , the stat i c , memo ry l ess non­l i nea r i ty be i ng approx i ma ted by a po l ynom i a l of a f i n i te orde r . I n the p resent wor k , howeve r , t h e p i ecew i se-po l ynom i a l a p p rox i ma tion of the s ta t i c non l i nea r i ty wa s found to be mo re usefu l due to the fact that i n a gene ra l case a po l ynom i a l of a ve ry h i gh o r d e r must be used in order to app rox ima te the t i t ra t i on funct i on a dequa te l y i n a l a rge range of ope ra t i on . The numbe r of i nt e rva l s , the b reakpo i nt s , the p rocess o r d e r and a su i ta b l e samp l i ng i nte rva l so tha t the l i nea r dynam i c pa r t of the mode l i s the m i n i mum phase mu s t b e chosen a p r i or i .

The pa rameter e s t i ma t i ons of the l i nea r a n d

9 1

non l i nea r pa r t s sepa rate l y ( two- s tep i dent i f i ca t i on p rocedu re) and of the equ i va l en t s i ng l e- i n put mu l t i -output 1 i nea r system ( one- step i dent i f i ca t i on p rocedu re ) a re performed us i ng t h e samp l ed va l ues o f t h e reagent f l ow and p H of eff l uen t .

PROBL EM FORMULAT I ON

The pH p rocess i n a con t i nuou s f l ow reactor i s p resented i n F i g . 1 .

PROCESS UQUID

Cl., CONTROL REAGENT

F i g . 1 . pH p rocess i n a con t i nuous f l ow reacto r .

The p rocess l i qu i d i s gen e ra l l y a m i xt u re of seve ra l s t rong o r fu l l y d i s soc i at i ng ac i ds and bases , the concent ra t ions o f wh i c h a re denoted a s CA and C s respec t i ve l y , and of seve ra l wea k o r pa r t l y d i ssoc i at i ng ac i ds and bases of concen t ra t i on s Cas · I n o rd e r t o ca l cu l ate the amoun t ( o r the concent rat ion) of reagen t ( CBc o r CAc now denoted

92 G .A . Paj unen

genera 1 1 y as C R) to be added i n orde r to obta i n the des i red hyd rogen i on concen t ra t i on i n a n ef f l uent ( H+) , a l l componen ts of the so l u t i on and the i r concen t ra t i on s mu s t be known . Then the pH va l ue , wh i ch i s def i ned a s

( 1 )

can be spec i f i ed a s a fur.ct i on of the reagent concen t ra t i on CR . S i nce , howeve r , the compos i t i on of the so l u t i on and the concentra t i ons of the componen t s a re very often unknown and/or t i me-va r i ab l e , d i rect equ i l i b r i um ca l cu l a t i ons a re not pos s i b l e and some i dent i f i ca t i on method i s needed .

P rocess Mode l

The pH p rocess i n a s t i r red tank rea c tor i s best represented by the W i ener type non l i nea r mod e l shown i n F i g . 2 .

DYNAMICS CAO ---"'! OF A STIRRED r--­TANK TITRATION FUNCTION

F i g . 2 . A pH p rocess mode l .

pH

The 1 i nea r dynam i c pa rt desc r i bes the dynam i cs of a con t i nuou s f l ow reactor and the s ta t i c non l i nea r e l ement represent s a t i t rat i on funct i on . The reagent concent ra t i on on the p rocess i npu t C R and the pH of e f f l uent a re mea s u red . The concen t ra t i on of the reagent in eff l uent C Ro i s not ava i l ab l e for mea s u re­ment .

I t was found expe r i menta l l y that the l i nea r dynam i c pa rt can be represented by a f i r st order stab l e pu l se t ransfer func t i on mod e l

- 1 -2 _ 1 b0z + b 1 z cR0 ( k )

G ( z ) = _ 1 = -- ( 2 ) 1 - a 1 z C R ( k )

w i th steady s t a t e ga i n equ a l t o one .

The non l i nea r , con t i nuou s , stat i c , memo ry l es s t i t rat i on func t i on ( N i em i , Ju t i l a ( 1 977) ) i s usua l l y not su i ted to approx i ma t i on by a po l ynom i a l w i th seve ra l te rms i n a l a rge range, but i t can be a pp rox i ma ted by a p i ecew i se­po l ynom i a l funct i on

whe re

m n . pH ( k ) = :L ljJ . {:L y : . ( C RO ( k ) ] 1 } ( 3 ) a j = 1 J i = 1 1 J

1jJ . = { J 0 otherw i se

o 1 m m i s the number of i nterva l s , C RO • C RO , . . . , CRO a re the b reakpo i n t s , n i s the po l ynom ia l s o rder and y ! . a re the unknown pa rameters .

I J The p rob l em i s to e s t i ma t e the pa rameters of

th i s p roces s rep resented by the mode l g i ven b y Eqs. ( 2 ) a n d ( 3 ) based o n the mea su red reagent concen t ra t i on C R and the pH of eff 1 uen t .

I DENT I F I CAT I ON SCHEME

Two d i f fe rent method s dev e l oped by Paj unen ( 1 982 ) for i dent i f i ca t i on of the W i ener type non l i nea r sys tem a re u sed and compa red .

l Two-Step l dent i f i ca t i on P rocedu re

The pa rameters of the l i nea r and non l i nea r pa rts of the mod e l a re e s t i ma ted separa te l y u s i ng a pa ra l l e l mode l refe rence sys tem s t ructu re .

Step I . F i s t the mod e l i s 1 i nea r i zed nea r the wo rk i ng po i n t . I f the i nput s i gna l i s s u f f i c i en t l y r i ch i n i t s f requences and the ga i n of the p roces s nea r the work i ng po i n t i s not equa l to or nea r z e ro , then the asymptot i ca l l y stab l e pa ra l l e l d i s c rete- t i me recu rs i ve i n teg ra l i dent i f i e r w i th a t i me­dec reas i ng adapta t i on ga i n p roposed by Landau ( 1 979) can be u sed to obta i n unb i ased pa rameter e s t i ma tes i n the a bsence of i nput measu remen t no i se . I f there i s an i nput mea su remen t no i se , the ope ra t i ng range for wh i ch the 1 i nea r mode l i s va l i d shou l d be l a rge enough so that the i n f l uence of th i s no i se i s neg l i g i b l e . The s tocha s t i c generator has been devel oped i n o rde r to s h i ft the opera t i on f rom the non l i nea r reg i on i n to a 1 i nea r one .

Step I I . The sys tem s hown i n F i g . 3 can be u sed to e s t i mate the pa rameters i n a s tat i c pa rt of the mode l when the pa rameter e s t i mates of a l i nea r pa r t a re known . I n F i g . 3 C R.o ( k ) i s a random s i gna l wh i ch shou l d be u n i fo rm l y d i s t r i bu ted between the m i n i mum and the max i mum va l ue of the p roce s s i nput i n t h e cu r rent i nte rva l mu l t i p l i ed by the s teady state ga i n of the l i nea r pa r t of the mode l , 6CRo ( k ) i s the no i se due to the m i sma tch of the l i nea r pa rt of the p roces s and i ts e s t i ma ted mode l , 6pH ( k ) i s the non l i nea r i ty a pprox i ma t i on e r ro r and µ { k) and v ( k ) a re the i nput and outpu t measu re­ment no i ses , respect i ve l y .

Us i ng the same i dent i f i ca t i on a l go r i thm a s i n Step I . the pa rameter est i ma tes obta i ned w i l l be b i a sed , i n a genera l case due to the non l i nea r i ty approx i ma t i on e r ror , but choos i ng a su i ta b l e numbe r of i nterva l s , b reakpo i nt s and o rde r of the po l ynom i a l s , the approx i ma t i on e r ror and the b i as a re neg l i g i b l e ( Paj u nen ( 1 982) ) .

One-Step l den t i f i ca t i on P rocedu re

The pa rameters of an equ i va l en t s i ng l e- i nput mu l t i -output 1 i nea r system a re e s t i ma ted u s i ng the se r i es mode l refe rence s t ru c t u re s hown i n F i g . 4 . The se r i es e s t i ma t i on mod e l s hown i n F i g . 4 i s an i nverse of a W i ener mod e l g i ven by Egs . (2) and ( 3 ) . 6CRo ( pH (k ) ) i s s ta t i c non l i nea r i ty approx i ma t i on e r ror

Identif icat ion of a pH Process 93

and µ ( k) and v ( k ) a re the i nput and 01 1 tput mea su rement no i ses . F i r s t the pa rameters of a comb i ned p rocess mod e l a re es t i ma ted and , based on t hem , the pa rameters of the W i ener mod e l i nverse a re ca l cu l a ted . Due to redundancy i n the pa rameter va l ues the rms e r ro r i s m i n i m i zed i n orde r to obta i n a pa rt i cu l a r set of pa rameters .

A s tochas t i c gene rator has been deve l oped so tha t the proces s output i s u n i fo rm l y d i s t r i ­buted between i t s m i n i mum and max i mum va l ues i n a c u r rent i nt e rva l . The non ! i nea r i ty approx i ma t i on e r ror and the output mea su rement no i se account fo r the b i a s of the pa rameter e s t i ma tes . The sma l l e r the po l ynom i a l order n , the b i gger the b i as due to the non l i nea r i ty approx i ma t i on e r ror , but the sma l l e r the b i as due to the output mea su rement no i se , the shorter the conve rgence t i me .

F i g . 3 . A pa ra l l e l i dent i f i ca t i on scheme used fo r pa rameter es t i ma t i on of a s t a t i c non l i nea r pa rt of the mode l .

vlltl l�1ff'ffi1MiiiONMoOEL._L_ ___ , pH'"t•)' M 1 · ... . 1} , .. ;,111111 I i--00--1-1,!, "i .�, v,••1-r,.. lkll � I

I .,,.,. •• ,., .• -I

L--- ------ - - -' C:1111

ID£NTIF'IC4TION Cl11I • ALGORITHM

F i g . 4 . A se r i es i dent i f i ca t i on scheme .

S I MULAT I ON RESULTS

The fol l ow i ng data were u sed fo r p roces s s i mu l at i on : t ank vo l ume 3 1 i t res , p rocess t i me de l ay of 1 0 seconds , ma i n f l ow t h rough the sys tem of 1 1 /m i n a nd samp l i ng t i me 30 seconds . The p roces s components a re a s fo l l ows : wa ter so l u t i on o f hyd roch l o r i c ac i d of concen t ra t i on CA=0 . 0 025 , 4- n i t ropheno l ( d i s soc i a t i on con s ta n t Ka=7 . 24 · 1 0-8 ) of concent ra t i on Ca= 0 . 0 0 02 , pyr i d i ne ( d i s soc i at ion con s ta n t Kbc l . 5 · 1 0 -9 ) of concen t ra t i on C s1 =0 . 002 and ammon i a ( d i s soc i a t i on cons t a n t Kb� l . 78 · 1 0- 5 ) of concen t ra t i on c 62=0 . 00 03 5 . A l l concen t ra t i on s a n d con s t a n t s i n mo l e/ ] i t re . A wa ter so l u t i on of sod i um hyd rox i de wa s u sed as a reagen t . The i npu t measu rement n o i se i s a s s umed to be neg l i g i b l e and the output meas u rement n o i se i s s i mu l a t ed a s a wh i te

ASCSP-D*

F i g . 5 . S i mu l a t i on resu l t s a ) The t i t ra t i on c u rve ( con t i nuous l i ne)

anu i t s approx i ma t i on ( b roken l i ne) obta i ned by t he two- s tep i dent i f i ­ca t i on p rocedu re after 500 recu rs i ons in each i n t e rva l .

b) The i nvers ion of the t i t ra t i on cu rve ( con t i nuous l i ne) and i t s approx i ­ma t i on ( b roken 1 i ne) obta i ned by the one- s tep i dent i f i ca t i on p rocedu re a f t e r 1 0 00 recu rs i ons i n each i n te rval .

process w i th the va r i ance equa l to 2 . 5 · 1 0-4 The i n i t i a l ga i n for a l l a l go r i thms i n a s i mu l a t i on part was G0= 1 0 1 0

Two- Step I dent i f i ca t i on Procedu re

Step I . Ttie p rocess mod e l i s 1 i nea r i zed and the pa rame t e r s of the 1 i nea r pa rt of the mod e l and the ga i n a t the wo rk i ng po i n t a re e s t i ma ted . The ope ra t i on reg i on i s C RoE [ 0 . 0 022 , 0 . 0027 ] (mo l e/ ! i t re) . Pa rameter e s t i mates conve rge very qu i ck l y and after 1 0 0 recu rs i ons they a re very c l ose to the i r rea l va l ues . Further i mp rovements a r e , howeve r , very s l ow d u e t o the output measu rement no i se .

Step I I . The pa rame t e r s of the s t a t i c non -1 i nea r pa rt of the mod e l a re e s t i ma ted u s i ng the pa rameter est i mates of the l i nea r pa rt obta i ned i n step I and the pa ra l l e l i dent i f i ca t i on scheme s hown i n F i g . 3 . S i x i n terva l s and th i rd orde r po l ynom i a l s a re u sed i n the mode l . The ope ra t i on range i s C RoE [O , C . O l] (mo l e/ 1 i t re) or pH E: [ 3 . 68 , 1 1 . 52 ] . The b reakpo i nt s a re 2s fo l l ows � i n mo l e/ 1 4 t re) : c �o = o . o o 1 , c RO = o . 002 , c�0 = o . 0 025 , C Ro = 0 . 003 , c�0 = 0 . 00 6 . The i dent i f i ca t i on resu l t s a f t e r 500 recu r s i on s i n each i nte rva l a re s hown i n F i g . Sa . The approx i ma t i ng cu rve ma tches the t i t ra t i on cu rve very we l l .

One-S tep I dent i f i ca t i on Procedure

Con s i de r i ng t he same ope ra t i ng range , numbe r of i n terva l s , po l ynom ia l s order and b reak­po i n t s as i n t he method des c r i bed above , the pa rameters of the comb i ned p roces s mode l a re e s t i mated for each i n te rva l sepa rate l y , u s i ng a s e r i es es t i ma t i on scheme ( Paj unen ( 1 982) ) . Then the pa rame t e r s of t he 1 i ne a r and non l i nea r pa r t s f o r each i n te rva l a re ca l cu l a ted m i n i m i z i ng the rms e r ro r . Pa rame t e r e s t i mates for the l i ne a r and non-1 i nea r pa r t s of the mod e l obta i ned by th i s method a re worse than the pa ramet e r e s t i mates obta i ned by t he two- s t ep i dent i f i ­cat i on p rocedu re u s i ng the same numbe r of mea su remen t s due to the g reater i nf l uence of

94 G .A . Pajunen

measu rement no i se , the l a rger numbe r of pa rameters and the redundancy i n the va l ues of the pa ramete r s . The i nverse of the t i t ra t i on cu rve and i ts approx i ma t i on func t i on obta i ned after 1 000 recu rs i ons i n each i nte rva l a re shown i n F i g . 5 b .

EXPER I MENTAL RESULTS

The computer cont ro l l ed f l ow resea rch system ( N i em i , Jut i l a ( 1 977) ) i mp l emen ted a t t he Con t ro l Eng i neer i ng Laboratory a t He l s i nk i Un i vers i ty of Techno l ogy wa s u sed for the expe r i menta l check i ng of the above theo ret i ca l and s i mu l a t i on resu l ts . The H P- 1 000/A lO O was used as a p rocess compute r , p rog rammab l e i n rea l - t i me For t ran l anguage . The reagent concen t ra t i on adj u s tment s we re made by adj u s t i ng the con t ro l f l ow by the compu te r . Mea su red va l ues of a n i n f l uent pH , e f f l uent pH , p rocess f l ow and the con t ro l f l ow we re sent th rough the t ransm i tters to the compute r . The same chem i ca l component s we re used a s for the s imu l at i on stud i es . There was both i nput ( reagent f l ow) and ou tput ( pH of e f f l uent) measu rement no i se . S i nce the p rocess was very s l ow , on l y one i nterva l was cons i de red in a mode l and t he ope ra t i on range. was chosen so that the non l i nea r i ty was su i tab l e for approx i ma t i on by a po l ynom i a l of a l ow order in t h i s range - C R E [ 0 . 00 1 6 , 0 . 0 023 ] (mo l e/ 1 i t re) or pH E [ 5 . 6 , l . 6 ] . The i n i t i a l ga i n for a l l a l gor i thms i n a n expe r i menta l pa r t wa s Go = 1 0 1 .

Two- Step I dent i f i ca t i on P rocedu re

Step I . The same ope ra t i ng range was used a s for s i mu l a t ion a n d the pa ramete r est i ma tes conve rgence cor respond to the s i mu l a t i on resu l ts . After 200 recu rs i on s (or 1 00 m i nu tes) �he fol l ow i ng pa rameter � s t i ma tes resu l ted : a l = 0 . 85 , b0 = 0 . 1 05 and b 1 = 0 . 041 . These est i ma tes cor respond qu i te we l l w i th the phys i ca l sys tem pa ramete r s .

�� Then the stat i c non l i nea r i ty was est i ma ted us i ng the above pa ramete r est i ma tes of the l i nea r pa rt of the p rocess . The resu l t ­i ng cu rve after 200 recu rs i ons and the t heoret i ca l cu rve a re shown i n F i g . 6a . The concen t ra t i ons of the chem i ca l s i n the p rocess and t he concent ra t i on of the reagent do not need to be exact l y the same as the theoret i ca l va l ues , so the cu rve i n a rea l p rocess can be s l i gh t l y d i ffe rent f rom the t heoret i ca l one .

Va l i da t ion . The mode l i s f rozen and connected i n pa ra l l e l w i th the p rocess . The ramp i nput s i gna 1 is used for exc i ta t i on of the mode 1 and t he p rocess . The responses a re shown in F i g . 6b .

One-Step I dent i f i ca t i on P rocedu re

The pa ramete rs of the comb i ned p rocess mod e l were e s t i ma ted . A f t e r 1 00 recu rs ions pa rameter est i ma tes no l onger change much . After 400 rec u r s i ons the pa rameters of the 1 i nea r and non l i nea r pa rts we re ca l cu l a ted f rom the

... u a ) u

. ... u ...

•.. ,� H , .. , .. m , .. , .. 1.1& ,-���.�. -u,.,,_�10� .• -· fft(111ift)

...

2.10

�-10·11"'°t�jlitrel

F i g . 6 . Two-s tep i dent i f i ca t i on p rocedu re expe r i menta l resul t s .

a )

a ) E s t i ma ted s t a t i c non l i nea r i ty ( con t i nuous 1 i ne) after 200 recu rs i on s a n d t heoret i ca l l y ca l cu l a ted t i t ra t i on cu rve ( b roken l i ne) .

b) The responses of the p roces s(cont inuous 1 i ne) and the f rozen mod e l ( broken 1 i ne) to the ramp exc i ta t i on s i gna l .

.... ·/,:; ....

....

....... �-� .. -� ... --.� .• -� ... �pH ti l6 U 1.1 Tim.l��I F i g . l . One- s tep i dent i f i ca t i on p rocedu re -

expe r i me n t a l resu l t s . a ) E s t i ma ted i nverse of the stat i c non-

1 i nea r i ty (con t i nuous l i ne ) after 400 recu rs i ons and the t heo ret i ca l i nverse of the t i t ra t i on func t i on ( b roken 1 i ne ) .

b ) P rocess i npu t ( b roken 1 i ne ) and the output of the f rozen comb i ned p rocess mod e l ( con t i nuous 1 i ne ) connected i n se r i es w i th the p rocess .

pa rameter est i ma tes of the comb i ned p rocess mod e l m i n i m i z i ng rms e r ro r . The resu l t s �e re a s fo l l ows : a 1 = 0 . 81 , bo = 0 . 082 , b 1 = 0 . 045 . The est i ma ted i nverse of the stat i c non l i nea r i ty i s shown in F i g . la . The same f i gu re shows t he t heo ret i ca l l y ca l cu l a ted i nverse of the t i t ra t i on cu rve for compa r i son , a l t hough i t does not need to ma tch the non l i nea r i ty of th i s pa rt i cu l a r p rocess exac t l y .

Va l i da t i on . F i na l l y , the comb i ned mod e l of the p rocess wa s f rozen and connected i n se r i es w i th the p roces s . The i nput to the p rocess was a ramp funct i on . The p rocess output wa s used a s an i nput to the mod e l and t h e o u t p u t of the mode l was compa red w i th the i nput to the p roces s . The resu l t i s shown i n F i g . lb . The m i sma tch i s due to the measu rement no i ses and the i ns u f f i en t est i ma t i on t i me .

CONCLU S I ONS

Two methods deve l oped for on- 1 i ne i dent i f i ­cat ion of a s i ng l e- i nput s i ng l e-ou tput non-1 i nea r dynam i c system composed of a l i nea r dynam i c pa rt , fo l l owed by a pu re , known t i me de l ay i n ser i es w i th a con t i nuou s , s t a t i c , memory l ess non l i nea r i ty ( Paj unen ( 1 982) ) were a pp l i ed to the i dent i f i ca t i on

Identification of a pH Process

of the pH p rocess in a s t i r red tank reactor .

The s i mu l at ion and the expe r i menta l resu l ts ag ree w i th theoret i ca l ana l ys i s . S i nce the t i t ra t i on cu rve i s not su i tab l e for approx i ­ma t i on by a l ow order po l ynom i a l , the p i ece­w i se-po l ynom i a l approx i ma t i on must be used , so some a p r i o r i knowl edge of the shape of the non l i nea r i ty i s necessary for choos i ng su i tab l e b reakpo i nts and po l ynom i a l orde rs .

The advantage of th i s method , i n a case of pH process i dent i f i ca t i on , i s that the COllllOnents of the process so l ut i on do not need to be known . On the othe r hand i f th i s knowl edge i s ava i l a b l e , i t i s not poss i b l e to i n­corporate i t , wh i ch i s a c l ea r d i sadvantage . Due to the fact that the process i s non l i nea r , the i npu t s i gna l for process exc i ta t i on mus t b e se l ected w i th much ca re a n d a s pec i a l s toc­has t i c gene rator must be used i n order to decrease the convergence t i me .

Th i s work w i l l be con t i nued i n order to deve l op an a l go r i thm for on- l i ne ident i f i cat i on of the process t i me de l ay together w i th the process pa rameters and app l i ca t i on of th i s method for i dent i f i ca t i on i n a c l osed l oop . F i na l l y , the regu l at i on and t rack i ng object i ves w i l l be cons i dered .

REFERENCES

Habe r , R . , Kev i czky , L. ( 1 976) . I dent i f i cat i on of non l i nea r dynam i c systems . I dent i f i ­cat i on and sys tem pa rameter est i ma t i on , P roc . of the 4th I FAC Symp . , Tb i l i s i , 62 - 1 1 2 .

N i em i , A . , Ju t i l a , P . ( 1 977) . l i nea r a l gor i thms . D i g i ta l to p rocess cont ro l , P roc . I FAC/ I F I P I nt . Confe rence , 289 - 287 .

pH cont ro I by comp . appl i c . of the 5th North Ho l l and ,

Paj unen , G .A . ( 1 982 ) . App l i ca t i on of a mode l refe rence adapt i ve techn i que to the i dent i f i ca t i on of s i mp l e W i ener type non-1 i near systems . Workshop on Adapt i ve Con t ro l , P roc . of the Workshop on Adapt i ve Con t ro l , I S I S , Un i vers i ty of F l o rence , I ta l y , 3 1 1 - 3 3 3 .

Landau , Y . D . ( 1 979) . Adapt i ve con t ro l - the mode l reference approach . Ma rce l Dekke r , I nc . , N . Y . 406 pp.

95

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

DEADBEAT ADAPTIVE CONTROL IN FEEDBACK

Y. Yamane*, P. N. Nikiforuk** and M. M. Gupta**

*Mechanical Engineering, Ashikaga Institute of Technology, Ashikaga, japan **College of Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

Abstract . This paper is concerned with presenting a new procedure for design­ing adaptive control input of a linear multivariable discrete system with having unknown order for the model reference adaptive control problem. The adaptive control scheme proposed here exhibits more advanced characteristics than the previous ones . 1 . The improvement of finite convergence time . 2 . The simpler implementation of adaptive control schemes . 3 . The absence of elaborate calculation for taking out real plant parameters from parameter estimates with redundancy . Keywords . Adaptive control ; s tate feedback ; deadbeat convergence ; unknown order ; multivariable systems ; system observer ; identification ; estimation .

PROBLEM DESCRIPTION

Consider the following linear multivariable discrete system plant which is given by

x (i+l )=A x (i)+B u (i) p - p p y (i) •CTx (i) ( 1 ) p p

where XpERnp , UpERr , YpERm and np is an un­known order but np , denoting an upper limit to np , is assumed to be known as a priori knowledge . A, B , and � are also unknown para­meters and xp (i) is not accessible . The p lant ( 1 ) can be rewritten to obey an equivalent �P -th order system

Xp (i+l )=�Xp (i)+BpUp (i) yp (i)=C�xp (i)

where XpERnP and

�= [a1 • · " ·���- J . CP = H-1 . (2)

On the other hand , the reference model system is presribed as

xm (i+l)=��(i)+Bmum(i) Ym (i)=ciXm(i) (3)

where XmERnm , UmERr and YmERm . The problem we are concerned with is to f ind the plant control such that the output error between the p lant and the reference model can be regulated to go to zero in finite time as quickly as possible.

ADAPTIVE CONTROL AND SYSTEM OBSERVER

9 7

The p lant (2 ) can be changed into a q-th order time-variant system with q=�p ( l+m+r) ,

where

y (i+l ) =W (i)y(i) Yp (i) =ch (i)

(4)

W (i)= f � 1:�1-�i� : � � ·_·_1:��:>.!:���i: : �·-· : :i:r_�:j O : I : 0

__ .. - - - - - - - - - - - - _, - - - - - - - - - - __ .. o : 0 l I n= [ot�] , c1= [ c� .o .0 1 T

[ T T T T T T q y= Xp (i) ,a1 · · · · ·�·b 1 · · · · ,br ] ER W (i) is to be generated by using only input­output current signal s of the plant . From the reason that an observer for (4) is viewed as to be simultaneously workable for both iden� tifying system parameters and estimating system states , it is called an system observer for (2 ) and one obtains

y (i+l )=W (i)y (i)+H (i) [ y (i) -y (i) ] A T p p (5) yp (i) =C1y (i)

y (i) denotes the estimate ofy (i) . Particular­ly, selecting out xp from9 , we have

�P (i+l ) =� (i)�P (i)+BP (i)� (i)+H1 (i) [yp (i) -yp (i) ] (6)

A 'J'A T T T T y (i)=Cpxp (i) , H= [H1 ,H2 ,H3] AP A x , Ap and BP are estimates of x ,Ap and BP . Fgr the simplcity to synthesize Rdaptive control , Cp�� (i) is assumed to be nonsingular on the run of observing . An algorithm for implementing adaptive control s cheme is ex­pressed to be in state feedback form,

98 Y . Yamane , P . N . Nikiforuk and M.M . Gupta

u (i)=K (i)� (i)+K (i)x (i)+D (i)llui(i) ( 7 ) p p p m m m H (i) in (5) and K (i) , Kro (i) and Dm (i) in (7) remains undetermiRed yet . The closed loop resulting from applying the adaptive control to the plant becomes

� (i+l)=A (i) � (i)+B (i)�(i) n (i)=CT� (i)=ym (i)-yp (i) (8)

holds . The integer v-1 denotes the observa­bility index of (4) . Once H(i) at each time is calculated so that (9) is satisfied , � · Km and Dm is eventually determined to be as follows ,

� (i) = [ c�BP (i) J - 1c�P,i <i) ( 10) Km(i) = [ C�BP (i) J - 1c�Am ( 1 1 )

- TA - 1 T Dm(i) - [ CPBP (i) ] CmBm ( 12 ) Then, it turns out that

n (i)=O after i=v+l ( 1 3)

NUMERICAL EXAMPLES As a example , consider a third order plant and a second order reference model to show excellent performance of the proposed appro-ach. r-0 . 2 1 o�

'A = o . 3 o 1 - 0 . 4 0 0 •

A =r-0 . 1 ll m -0 . 5 oj ,

- f.1 J -T FJ -B = l�. 2

C = l� 'xp (O)=

,

Two different initial conditions for the sys­tem observer are chosen as case 1 {np =4 and y (O)= [ l , l , · · · , l , O , O ,O ]TERl2} and case 2 {np= 7 and y (O) = ( l , l , · · · , l ,0 , · · · ,0 ]TER2l } . Fig . l and Fig . 3 show input signals o f the plant and the reference model for each case. In case 1 v- 1 takes 1 0 , n (i )=O is achieved for i � 1 2 , otherwise , in case 2 v-1 takes 1 6 , n (i) =O is shown for i � 1 8 . See Fig . 2 and Fig . 4 . From the results of the both cases, i t i s clear that the procedure for designing the adaptive control scheme is very effective because of figuring out the fine performance for dead­beat convergence . When e1:Bp (i) is not nonsin­gular . in observing, simular approach will be developed in future.

REFERENCES Nikiforuk, P .N . , M .M . Gupta and Y . Yamane ,

( 1 979) . Adaptive control for single input single output linear discrete system with deadbeat convergence . JACC Proc. p . 474-480 .

Yamane , Y , P . N . Nikiforuk and M.M. Gupta ( 1 982) . Indirect adaptive control design · with deadbeat convergence for unknown order linear systems . The 4-th Internation­al Symposium on Large Engineering Systems , p . 423-428 .

\ : · ' \ I \ : \ t t l ,t ,! lq- ,/ --�--\,' AJVM�' �� \ ; . . \1 ·w. 1 - . . i - --1- _, __ ,._.,___ ,,1\ _ -_ ., , __ : ") I 'f : 10 \/ 'r' 15 'r' "r' \

-1-- J I I I \ /

' I

'

I

\r' 0----0 Um l-- - - � Up

Fig . l u and u when n =4 , 1 2 f(O ) = ( l�l , 1 , 1 , i? , • • · ,1 , O�O , OtE: R

�ig . 2 ymand y when n =4 f( O ) = ( l , l , · · · P · ,1 , 0 , o�o ) T E R1 2

0-----<l Um l-- - -l Up

�ig . 3 u and u when n =7 , v( o ) - ( lml P P T 2 1 , - , , 1 , • • • , 1 , 0 , o , o , o , o , o ) e R

3

E;_ig . 4 y and y when � =7 , T , 2 1 J( 0 ) = ( 1 �l , • • • , i1 , 0 , 0 , 0 , ij , 0 , 0 i f R

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

INFORMATION-THEORETIC ASPECTS OF PARAMETER ESTIMATION

A. I. Yashin

International Institute for Applied Systems Analysis, Laxenburg, Austria

Abstract . This paper deals with the information distance between a model and the real system it is designed to represent . The convergence properties of some Bayesian algorithms are discussed in this context .

Keywords . Parameter estimation , models , information distance , convergence of numerical methods .

INTRODUCTION Most computer or mathematical models are not exact representations of reality : lack of knowledge, technical restrictions and partic­ular modeling goals make it necessary to ap­proximate the real system in various ways . Nevertheles s , the procedures by which the mod­el parameters are tuned from observed data are usually based on the assumption that the real system differs from the model only in the val­ues of certain parameters . These particular values are usually included in the feasible set of parameter values , and this fact usually provides the identifiability property for in­dividual algorithms (Yashin, 1 98 1 ) . However , in reality both of these assumptions are generally false . Even if the structure of the system corresponds to the structure of the model, the real parameter value does not usu­ally belong to the feasible set . Moreover , mathematicians often consciously diminish this set in order to simplify the estimation algo­rithms . For example , they approximate the bounded compact set of parameter values by a set consisting of a finite number of points , thus increasing the chances that the real pa­rameter values will be excluded . It is therefore both remarkable and surprising to find that , despite these false assumptions and approximations , the parameter estimation algorithms often still converge ! The model resulting from this tuning procedure will of course not coincide with the real system, and this raises the natural question : how far is this computer model from reality? When considering this question , it is neces­sary to have some way of measuring the "distance" between individual models . One such measure of divergence was introduced by Bhattacharayya ( 1943) ; Kullback ( 1 959) also formulated some measure of information dis­tance . However, these measures were not prop­er metrics . Baram and Sandell ( 1978) later

9 9

introduced a modified version of the Kullback measure , which has been shown to be a proper distance metric . They app l�ed this approach to linear-Gaussian systems and models ; in this paper it is generalized to a . wider class of systems .

NOTATION AND DEFINITIONS

Assume that the variety of models of the real system may be characterized by a parameter S , which takes values from a parameter s e t B . I n view o f the Bayesian formulation of the problem, we will assume S to be a random vari­able defined on some probabilistic space (n, H, P) . Let s (w) ,n > O be some random n -process (observation) adapted to some nonde­creasing family of a-algebras H = (Hn)n>O in n . We shall denote by H = (Fi ) >O ' H -;- H n n n the family of a-algebras generated by the pro­cess s , n > 0 , H = a{s ,m < n} . n - n m -In the case of a continuous ( in time) observa­tion process st , t .'.':. 0 we assume the nonde-creasing , right continuous family of a-alge­bras H = (H ) >O to be given , where H = H t t . 00 and H

0 is completed by P-zero measure sets

from H . We also introduce the family of a­algebras generated by the observable process H = (Ht) t>O ' where

If the set of parameter values is denumerable or finite , we will denote by TI . (n) (or TI . ( t ) ) J J the a posteriori probabilities of events { S = S . } ' j E B , given observations s . , i < n J i -( ss , s 2 t ) .

1 00 A. I . Yashin

SOME BAYESIAN ESTIMATION ALGORITHMS

Before deriving our main results , we will first consider some Bayesian parameter estima­tion algorithms for different observation s chemes .

(a) Assume that s ,n > 0 is given by the formula n -

where 8 satisfies the recursive stochastic n equation

Here Eln ' E2n ,n .:_ 0 are sequences of indepen­dent Gaus sian random variables with zero mean and a variance of unity, and S is an unknown parameter. Assuming that S takes its values from some finite set { S1 , S2 , • . • , Sk} ' the a posteriori probabilities are

TI . (n + 1 ) = TI . (n) + J J

k TI . (n) l _i_. e = l Di (n+l )

where the mj are Kalman estimates of 8 when n n S = S . , and the D . (n) are functions of the J J conditional variance y . (n) (Kuznetsov, Lubkov , and Yashin , 198 1 ) : J

D . (n) = (b2 + C2 + s:y� (n)) l/ 2 J J J

(b) Consider the continuous (in time) obser­vation proc�ss s given by the s tochastic differential equlition

where wt , t .:_ 0 is the H-adapted Wiener process and S is an unknown parameter. Assuming again that the number of parameter values is finite, we have for TI . ( t ) P (S = S . IH ) (Liptser and J J t Shirjaev, 1978) :

d'IT . ( t ) J

TI . (O) J where

A(s�)

pj j 1 , 2 , . . . , k

k l Tii ( t )A( Si , st )

i=l (c) Consider observations made by a continu­ous-time finite-state j umping process with

unknown transition ·intensities A�j ' O�ce again assuming a finite number of values for S , we have the following a posteriori probability TI . ( t ) (Yashin , 1 9 70 ) : J

1T . ( t ) J

where

j

TI . (O) + l TI . ( s-) _ - - 1 -(A

ss , ss J J s<t J A -ss_ • ss

t . J0 TI . ( s ) (A� s - As , s ) ds J s- ' s s- s

It turns out that the a posteriori probabil­ities converge in all of these algorithms , even though the real parameter value does not belong to the feasible set ,

In order to analyze these situations we need some auxiliary results .

AUXILIARY LEMMAS

For any A E H, we denote by Px (A) , x E B the family of probability measures

Let ?X(A) , FX (A) , x E B , n > 0 be the restric­tionsnof the Px (A) on cr-algebras H, H , respectively . Assume also that for aRy x , y E B we have px -FY . Define Zxy a s a n n n Radon-Nicodim derivative

and let axy n

that if the

dpx n dPY

n

= zxY (zxy ) - 1 • It is easy to see n n- 1 (Fi 1 )- conditional distributions n-of the sn ,n .:_ o have densities fx(z \Hn_1 ) , x E B then

fx ( Sn IHn-1 )

fy ( sn \Hn-1 )

Assume that the real sys tem corresponds to a parameter value k such that k </:. B . Introduce the function Ik (x , y) = E ln axy and define the n k. n measure of distance

d (x , y) = I Ik (x,y) I n n

I&_mma 1 . (.Baram and -Sandell , 1 978) The func­tion dn ( i ,jf--is p seudo-metric. That i s , the following equalities hold:

Information-Theoretic Aspects o f Parameter Estimation I O I

(a) dn (x,y) (b) dn (x,x) (c) dn(x,k) + d (k,y) > d (x,y) n - n

Lerruna 2 . For any x ,y E B ,n ..'.:. 0 , we have Ix(x ,y) > 0 n -

Proof. have

From the definition of the Ix (x,y) we n

E ln axy = E (E (ln axY jfi 1 ) ) x n x x n n-

= E (E (�(axy) jH 1 ) ) x y n n-where �(t) = t ln t . According to the theorem of the mean , �(axy) can be represented as follows : n

where hxy varies between axy and 1 . It is not n n difficult to see that

E (�(axY) jH 1 ) = y n n-

l E ( (axy - 1 ) 2 �1�jH ) > O 2 y n hxy n-1 n

Lerruna 3. Let d (k,x) < d (k,y) . Then n n k In (x,y) > 0

k Proof. From the definition of In (x,y) , we can write :

Ek ln fx (s jH 1 ) -n n-Ek ln fy (sn ! Hn_ 1)

-Ek [ln fk(sn !Hn_1 ) - ln f� (sn !Hn_1 ) J

+ Ek [ln fk (sn !Hn_1 ) - ln fy (sn !Hn_1 ) J

-Ek ln a� + Ek ln a�Y

From Lemma 2 , kz Ek ln an ..'.:. 0 for any z E B ,

k and thus In (x,y) C! Q * d (k,y) <: d (k,x) . n n

RESULTS

Assume that the process ln axy is ergodic , i . e . , n

1 n k 1-!fil - l ln axy = Ek ln axy = I (x,y) n n m=l m �� /

Theorem. I f d (k , x) > d (k,y) then k (P a . s . )

I f it i s known that Zij + 0 n k (P a . s . ) , then

d (k ,x) ..'.:. d (k,y) . Proof. Note that , from Lemma 3 , the inequal­ity d (k,x) > d (k,y) yields Ix(x,y) < 0 and consequently

1 n lim - l ln axy

< 0 n-+<x> n m= l m

This means that 00 l

m=l and consequently

zxY + 0 n

-00

(Pk a . s . )

thus proving the first part o f the theorem. In order to prove the second part of the theo­rem we assume that zXY + 0 but that n d (k ,x) < d (k,y) . This yields

1-!fil l Y ln aij = Ik(x,y) > 0 n n m= l m

from which 00

00

and the theorem is proved by contradiction. Exa.rrrple. Assume that the sequence sn is a finite-state ergodic Markov c!!_ain . on any of the probabilistic spaces (Q, H, P1) , i E B ,

13 -where B is a finite set . Let pl , l ,m = l , k ,m be the transition probabiliti�� for one step . According t o Yashin ( 198 1 ) , aiJ i s given by the formula n

i p sn-l ' sn pj sn- 1 ' sn

Well-known results from Markov chain theory show that the process ln a!:j is ergodic . Thus , if the Bayesian algorithm for Tij (n) converges to 1 for some particular j O it means that this j

0 is the point from B that is nearest (in the sense of information distance d (k ,x) ) to the real parameter value k .

1 02 A . I . Yashin

REFERENCES

Baram, Y . , and N . R. Sandell ( 1978) . An informa­tion theoretic approach to dynaplical system modeling and identification . IEEE Trans . Automatic Contro l, VAC-23-Nl, 6 1-66 .

Bhattacharayya , A. ( 1943) . On measure _of divergence between two statistical popula­tions defined by the probability distri­butions . Bull . Calcutta Math. Soc. , 35, 99- 104.

Kullback, S . ( 1959) . Information Theory and Statistics . Wiley , New York.

Kuznetsov, N . A. , A. V . Lubkov, and A. I . Yashin ( 1981 ) . About consistency of Bayesian parameters estimators in adaptive Kalman filtration schemas . Automatic and Remote Control, 4, 47-56 (translated from Russian) .

Liptser, R. S . , and A.N . Shirj aev ( 1978) . Statistics of Random Processes. Springer­Verlag , New York ( translated from Russian) .

Yashin , A. I . ( 1 9 70) . Filtering of j umping processes . Automatic and Remote Control, 5, 52-58 ( translated from Russian) .

Yashin , A. I . ( 198 1 ) . About the consistence of Bayesian parameters estimators . The Problems of Information Transmissions, 1, 62-70 ( translated from Russian) .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

MODEL UPDATING IMPROVES PERFORMANCE OF AN MRAC DESIGN

P. P. J. van den Bosch and P. I. Tjahjadi Delft University of Technology, Department of Electrical Engineering, Laboratory for Control Engineering,

P.O. 5031 , 2600 GA Delft, The Netherlands

Abstract . We will discuss the technique of model updating to improve the performance of a MRAC design with state feedback . At specified time intervals the state of the reference model is adjusted to actual state of the system, which can improve the convergence of the parameters , reduces control effort and yields faster response times .

Keywords : Adaptive control , Adaptive systems , Aerospace control , Lyapunov methods , Model updating

INTRODUCTION

Model-reference adaptive control ( MRAC ) is a well-established design method that has demonstrated its capabilities in many applications, for example Landau ( 1 979 ) , Van Amerongen ( 1 980 ) and Van den Bosch et al . ( 1 982 ) . Asymptotic stability can be proven for linear systems . Even for nonlinear systems stability and a good performance can be obtained. Model updating can improve the performance of an MRAC design , in particular for systems whose structure does not match the structure of their reference model. Suppose that the state y of the system differs from the state x of the reference model . Then , via an adjustment of the controller parameters , MRAC will force the system to follow the reference mode l . If the structure of the system and that of the reference model differ, there is no unique parameter set for the controller that is able to maintain y at x . Consequently, oscillations of y about x can be expected, as illustrated in figure 1 . The philosophy of model updating is to reduce these oscillations . Model updating replaces at some points of time the state of the reference model by the actual state of the system. New reference trajectories are calculated , starting in the actual state of the system. Consequently, model updating avoids unnecessary control efforts and can reduce the influence of disturbances .

In this paper we will discuss two criteria for determining an appropriate point of time in which to apply model updating and we will prove that model updating does not influence the asymptotic stability of an MRAC design.

1 03

Results of a feasibility study, concerning a three-axes slew for satellites , illustrate the improvements of model updating over a standard MRAC design.

m o d e l system

� update I I I

I'. ., - . _, m o d e l . , · . , · · ·":- :-., . .., .:..:.;_ · . . �ystem ..... . .... · .

...... ,

[sec]

Fig. 1 Model Updating

UPDATE CRITERIA

Landau ( 1 979 ) has proposed an update at each sample time of MRAC for discrete systems , which yields the parallel-series structure of MRAC . Continuous systems require a different approach . Firstly, we used a fixed value for the update interval . Each Tup seconds the state of the system was introduced into the reference mode l . The choice of this fixed interval turned out to be critical (Van den Bosch and Jongkind, 1 980 ) . Later we used a criterion based on the Liapunov function v.

1 04 P . P . J . van den Bosch and P . I . Tj ahjadi

In an MRAC design this function consists of an expression dealing with the error e ( e = x-y) and an expression dealing with the difference p between the parameters of the reference model and those of the system, so

V ( e , p ) = e ' Pe + V ( p )

If we use appropriate adaptive laws we obtain (Landau , 1 979 )

V( e ) = - e ' Qe

with both P and Q positive definite matrices . Suppose �e have an update at t=ti ' Then we define ti just before :n update and t! just after an update . At t=ti we have

V ( e , p ) e ' Pe + V ( p)

vc e > -e ' Qe

and at + have t=t.j_ we

V ( e ,p) V (p )

v c e > 0

Consequently, each update decreases V ( e , p ) by e ' Pe and increases V ( e ) by e ' Qe , as ilustrated in figure 2 .

Fig. 2 An update decreases V ( e ,p) and increases 'O'( e ) .

We want to decrease V ( e ,p ) as fast as possible, which naturally leads to a choice for an update when e ' Pe reaches its maximum. This maximum occurs if the time derivative of e ' Pe becomes zero or :

Update criterion 1 : e ' Pe = 0 A disadvantage of each update is the value zero for V ( e ) , immediately after an update . We can weigh the decrease of V ( e ,p) ( advantage ) against the increase of V ( e ) ( disadvantage) via the second update criterion, which allows an update when e ' Pe/e ' Qe has a maximum. This maximum occurs if the time derivative becomes zero. Thus we have

Update criterion 2 : e ' Pe/e ' Qe-e ' Pe/e ' Qe=O

Both criteria deal with a scalar function . A zero crossing of this function can be detected easily. It turns out that both criteria offer about comparable results ( Tj ahj adi , 1 982 ) . Consequently, we prefer criterion due to its simpler method of calculation . Henceforth , we use this update criterion.

In Van den Bosch ( 1 983 ) we prove that model updating does not influence the asymptotic stability of an MRAC design , independent of the choice of an update criterion. we introduce a second reference model without adjusting its states . The error between the states of this refrence model and the states of the system are finite and disappear as the time goes to infinity.

APPLICATION

we have designed an MRAC-based controller with model updating to realize one single three-axis slew for satellites (Van den Bosch et al. 1 982 ) . A satellite is a highly nonlinear multivariable system with much interaction among the three axes . The reference model neglects some nonlinearities and the interaction due to gyroscopic coupling. Consequently , there is a significant difference between the satellite and its reference mode l . All states can be measured accurately. From this study we can conclude that : Model updating requires less control energy than an MRAC design , typically 1 0 to 30% less than MRAC and offers faster response times. The selection of the adaptive gains is less sensitive when model updating is applied. In general , large adaptive gains offer a better performance .

REFERENCES

.Arnerongen, J , van ( 1 980 ) , Model Reference Adaptive Control , Applied to Steering of Ships . Lecture Notes in Control and Information Sciences , Vol 24 , ( 1 99-208 ) ,

Springer Verlag. .Bosch , P . P . J , van den and w. Jongkind ( 1 980 ) .

Model Reference Adaptive Satellite Attitude Control . Lecture Notes in Control and Information Sciences , Vol 2 4 , ( 209-2 1 8 ) ,

Springer Verlag • • Bosch , P . P . J . van den , w . Jongkind and A.C .M. van Swieten ( 1 982 ) . An Adaptive Attitude Control System for Large-Angle Slew Manouevres . Proceedings IFAC/ESA Symposium on Automatic Control in Space , Pergamon Press , London.

,Bosch , P . P . J . van den ( 1 983 ) . Model Updating improves MRAC Performance . Submitted for publication in IEEE Trans . on Autom. Control .

,Landau ( 1 9 79 ) . Adaptive Control . Marcel Dekker Inc . , New York.

.Tj ahjadi , P . I . ( 1 982 ) . Adaptieve Standregeling voor een drie-assige Beweging van een satelliet . M. ScThesis Laboratory for Control Engineering, Delft University of Technology .

Copyright © IF AC Adaptive , Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE CONTROL FOR A NONLINEAR FERMENTATION PROCESS

Jaime Alvarez, Joaquin Alvarez and S. Moodie

Department of Electrical Engineering, Advanced Studies and Research Center, Mexico

Abstract . Linearization via optimal control (O . C . ) of a discrete nonlinear multivar iable system is analysed and an adaptive l inear control is presented . The performances of the adaptive control applied to a s imulated fermentation process l inearized by Taylor series expansion and via O .C . are compared , showing that the second option gives a very robust control structure and a better performanc e .

Keywords . Adaptive control ; optimal control ; nonlinear systems ; fermentation process .

INTRODUCTION

A classical approach for controll ing nonlinear systems is to l inearize the system around its operating point, then to design a l inear control structure . However , when the process is highly nonlinear around the operating point and input , output or parameter disturbances arise , this approach is not satisfactory .

Some optimal control algor ithms , when applied to nonlinear systems , provide a l inear closed­loop structure if the model parameters values used in the synthe sis of the O .C . law are equal to those of the process . However , if this condition is not satisfied , the control performance may be affected . In this case , it is possible to think that the application of an adaptive l inear control algor ithm to the closed-loop system might improve the robustness of the overall system .

LINEARIZATION V IA O . C .

The systems considered in this paper are those descr ibed by the following equations :

( 1 . 1 )

( 1 . 2 )

where xE: R n, u e: R

r and y e: R

m are the state ,

control and o°iltput vect;r s , respectively ; 00 �(�) and B (�) are C vector and matrix functions and C is a constant matrix of proper dimensions .

If we consider the performance index to be minimized as the following :

1 05

N- 1

J=k�o (� Q �+ (�+1 -�) T (�+1 -� l ) ( 2 )

where Q is a diagona l (mxm) positive definite matr ix , N is the observation hor izon , and �=�-:fk is the error vector at time k between tne reference (�) and output vectors , then the optimal control is given by (Alvarez and Gallegos , 1 98 1: ) :

where "o" index indicates that design pa:r;:a­meters (not the real ones) are used and M is a (mxm) -diagonal matr ix . Note that a condi­tion for the existence of a control law is that matrix CB (�) be not singular . Furthermore , i� is assumed that the state is measurable .

When it is assumed that CB (�) is not a function of the process parameters or if these parameters are exactly known , then CB (x ) = CB (� ) and the c losed-loop system equ�tj]ns are :

If the design and process parameters values are the same , the l inear closed-loop system is described by :

( 5 )

Changes i n parameter values affect the system output dyn�mic s . If these changes are not very large , the overall system follows beinq "quasi-linear" . In fac t , the output sensiti�i­ty to parameter changes is , in many cases , very much less in the O .C . closed-loop sys­tem ( Eq . 4 ) than in the open-loop case ( Eq . 1 ) .

1 06 Jaime Alvarez, Joaquin Alvarez and S . Mandie

It will be described in the next sections how to apply an adaptive controller to a fermentation process linearized by O . C . theory .

ADAPTIVE CONTROL

For convenience we use the following poly­nomial representation to describe the process :

- 1 -d - 1 C ( q ) x_ (k ) = q D ( q )� (k ) ( 6 )

where q- 1 is the backward shift operator , d - 1 - 1 i s the plant time delay and C (q_ 1 ) and D(q ) are polynomials matrix of the q operator .

As it is known in adaptive control theory, tracking and regulation obj ectives are achiev ­ed if the following equation holds :

- 1 -d Cr (q ) (x_(k) -q �(k) h_Q_ k > O ( 7 ) - 1 where C (q ) gives the error dynamics and

� (k) i� the reference vector .

We propose a linear control such that :

- 1 - 1 -1 -r - 1 D (q ) S (q )� (k ) =Cr (q ) � ( k) -q R (q ) x_ (k)

-1 -1 where S (q ) and R (q ) will completely

( 8 )

define the control law, r= 1 if d=O (no time delay in the plant) and r=O otherwise . The achievement of the obj ectives leads to the following polynomial identity :

- 1 - 1 - 1 -d-r - 1 Cr (q ) =C (q ) S (q ) +q R (q ) ( 9 )

- 1 - 1 By solving (9 ) for S (q ) and R (q ) and by substituing their values in (8 ) the control law is obtained .

Recursive least squares identification with forgetting factor is used . Let 8 be the process parameters mi/itrix and !£. the measure -ments vector , with 8 .1>_(k- 1 ) =y(k) . Then , if § is the estimated parameter �atrix : A A ( l_ (k) -B (k-1 ).1>_(k- 1 ) ) T 8 (k) =8 (k- 1 ) +F (k ) .1>_ ( k- 1 ) T 1 +!£_ (k- 1 ) F (k ).1>_(k-1 )

where

F (k+1 ) = A (�) [F (k) -

and A ( k ) = constant (Lozano , 1 98 1 ) .

( 1 0 )

F (k ) ¢ (k- 1 ) ¢T (k- 1 ) F (k) )

1 +.1>_T�k- 1 ) F�k).1>_(k- 1 ) j =

( 1 1 )

trace/trace F 1 ( k)

FERMENTATION PROCESS MODEL

The process considered is represented by :

x (µ - D ) x

s D (sa - s ) - µx R

s µ µm K + s

where x and s are the biomass and substrate co�Tentrations (g/l ) , D is the dilution rate (h ) and S the feed substrate concentra­tion (g/l ) ; a

µ , R and )': are the characteris­tic parameter� of the process .

Case 1 : Linearization by Taylor Series Expan­sion . Taylor series expansion and explicit discretization lead to the following linear model :

G

-1 - 1 or ( I - Fq ) 6� (k) = q G 6£ (k)

where F and G are parameters matrix evaluat­ed at the operating point .

The control is obtained by apropriate subs­titutions in Eqs . (8 ) and ( 9 )

6�= G -1 (cr �k+1 - (Cr+F) 4) i f Cr (q- 1) =I+Crq- 1

A where G is always nonsingular at the opera� -ing point, F and G are obtained from Eqs . ( 1 0 ) and ( 1 1 ) considering

Case 2 : Linearization via O . C . Discretiza­tion of the model by Euler method is adequate and leads to :

where H is the integration step .

The cloosed loop system is obtained from ( 4 ) and it can be written in the polynomial form as :

[ µm µmo - 1 ] - 1 ]

1 + (m1 +Hsk (-l< - - --) ) q x (k) = ( 1 +m1 q ) z ,tJ< l +sk Ko+sk

µmo - 1 ] R (K +s ) ) ) q s (k) 0 0 k

�here m1 and m2 are the diagonal elements of M , and z 1 , z2 are the reference vector compo ­nents .

We can observe that the system is linear if there are no disturbances , and that the multi.­variable problem is transformed into two mono­variable problems . When disturbances aris e , we must identify two time varying .para:roeters .

The control algorithm is obtained by consider­ing r=1 , and using Eqs . ( 8 ) and (9 ) .

Control for a Nonlinear Fermentation Process 1 0 7

z 1 k+1 =zR1 k+1 +cr 1 zR1 k- (cr1 -§ 1 J xk-m1 z 1 k

z2k+1 =ZR2k+1 +cr2ZR2k- (cr2 - B2 ) sk-m2 z2k

the reference vector for A A law . 8 1 and 82 are obtained x

In case 2 (Fig . 2 ) , the ��itial F !face is equal to 20 . 0 , and Cr (q ) =(1 -0 . 1 q )I .

The following diagrams show the controls variables D and s . (dotted lines) and the process outputs xaand s (continuous lines ) .

where (zR1 , ZR2) is the linear control from ( 1 0 ) , (11 ). · :µ 8 = m + Hs (�-m--1 1 k K+sk

µmo Ko+sk

(g/l) 3 . 4

x

µm µmo R (i< +sk) )

0 0

EXPERIMENTAL RESULTS

3 . 2

2 . 8

. 2 3 1 ·-

20 40 . 1 2 1 3

In the following experiments , we consider an operating point (maximum productivity) su�9 that : x*=3 . 0 g/l , s*=! , o g/1 , D*=0 . 1 8 1 8 h , s:=1 1 . o g/l , µm=0 . 2 h , K=0 . 1 g/l , R=0 . 3 .

s s a (g/ll 1 . 4

:·. s . ....... A....- (q/l )

1 5 . 0

A set of parametric disturbances on µm' K and R is introduced at time 1 0 and ceased at time 40 .

The reference vector changes from the operat­ing point to x=3 . 4 g/l , s=1 . 2 g/l with a 3 hours first order dynamics at time 20 . It returns to the operating point at time 50 .

In case 1 (Fig . 1 ) the i�+tial F t!::�ce is equal to 80 . 0 , and C (q ) =I+C q where C is a diagonal matrlx of dime�sion 2 whose dlaqonal terms are -0 . 7 .

x (q/l )

3 . 4

3 . 2

3 . 0

2 . 8

s (q/l )

1 . 4

1 . 2

20

. . .. ... . : · ... .. . .. . .

. . .. . . . : . . . . .. . .

. ..

x

D

40 60 t (H)

D (h- 1 ) . 2R 1 8

. 2 3 1 8

. 1 8 1 8

. 1 3 H'

s a (g/l ) 1 5 . 0

1 3 . 0

1 . 2 1 3 . 0

1 . o�--.... ��2�0���-4�0��--::;;. ... ----.... -t 1 1 . o 60 t (H)

0 . 8 9 . 0

Fiqure 2

COMMENTS

The above described experiments were also made with· an optimal control structure , giving an offset , and with an O.C structure with integrators . Case 2 results were better than the former , and allowed parameters disturbances of 50% . Inputs disturbances o f 1 0 % gave also good re ­sults . An output sensitivity analysis has been made . It has shown that the O .C . loop greatly reduces output sensitivity to parameter variations except for output s with respect to parameter R. However the general reduction is important , and it implies a reduction of the parameter variations effect .

CONCLUSION : The experiments clearly show weakness in linearizing by Taylor series expansion and using linear control in highly nonlinear cases . On the contrary, O . C . linearization provides a very satisfactory robustness and it confirms the initial idea . This new structure apparently complicated

1 . o .._ _ _._-ol+-----4i--.L.-4._..� ... ----.+ 1 1 . o becomes very simple to implement in a digital t ( H) computer .

40 0 . 8

Fi(l'ure 1

.. .. · . . �

9 . 0

1 08 Jaime Alvarez , Joaquin Alvarez and S . Mondie

REFERENCES Alvarez , J . ; and Gallegos , J .A . ( 1 982 ) .0ptimal Control of a Class of Discrete Multivariable Nonlinear Systems . Application to a Fermenta­tion Process . Trans . of the ASME-JDSMC , Vol . 1 04 . 2 1 2- 2 1 7 .

Lozano , R , ( '1 981: ) . Adaptive Control with Forgetting Factor . Proc . of the 8th IFAC World Congress . Japan . pp. 83-88 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE MULTIVARIABLE CONTROL APPLIED TO A BINARY DISTILLATION COLUMN

L. Barcenas-Uribe* and j. Alvarez-Gallegor*

*Department of Electrical Engineering, lnstituto Tecnol6gico de Queretaro, A. Postal 124, C.P. 76000, QRO, .Mexico

**Department of Electrical Engineering, Centro de Investigaci6n del JPN, A. Postal 1 4-740, 14 D.F., Mexico

Abs tract . Thi s paper d e s cribes the re sult s o f s imu lation t e s t s made to evaluate t h e p erfomanc e o f a d i s cr et e mult ivariable adapt ive c ontrol algorithm when it i s applied to a nonl inear proc e s s . I n this approach both t o p and bottom compo s i t i o n s control o f a s imulated binary d i s t i l lation c o lumn i s performed . In order to obtain a feasible appli c at ion to the pro c e s s we have s l ightly mod i f i e d the algorithm , keeping the input s and out put s within the s pace constrained by the phy s i c a l l imit s . The initial dangerous tran s ients are avo icle� by doing first an o ff l i n e pro ce s s ident ifi cation and us ing the s e ident i f i e d parameters to init i a l i z e t h e algorithm . The r e s u l t s indicate that the contro l l er adapt s to the changing cond itions in both r egulation and tracking t a s ks i n s p i t e o f the nonlin ear and t ime varying pro c e s s charac t er and the large number o f parame t er s .

Keyword s . Adapt ive control , D i s t i l l ation proc e s s e s , Mult ivaria­ble contro l .

INTRODUCT I ON Control o f d i s t i l lation c o lumns has been a popular topic for year s . Interest in the topic has intensified in the las t s years for reasons that inc lude increased raw mat erial and energy cos t s , deve lopment s in hard ware and the emerging deve lopment s in control t e chniqu e s that u s e s s ome kinds of chemi cal pro c e s s e s a s a benchmark p l an t s to evaluate the performanc e o f the new algorithms . Thi s i s the cas e o f Adaptive Control Scheme s . D i s t i l lat ion c olumns are -suitable p lant s t o d emons trate the feasibi lity of adaptive control -t e chniqu e s . Howev er , applications o f adaptive control algorithms to d i s ­t i l lat ion c o lumn s are not very corrmon ( s e e as examp l e , LIEUSON and Coworkers ( 1 9 8 0 ) , SAS TRY and Coll eagues ( 1 9 7 7 ) , UNBENHAVEN and -S CHMID ( 1 9 7 9 ) , VOGEL and EDGARD ( 198V and W I EMER and others ( 1 9 8 3 ) ) .

Thi s paper d e s cr i b e s the r e s u l t s o f simulation t e s t s made t o evaluat e the p er formanc e of the d i s crete mul t iva­riable adaptive c ontrol algorithms · ­due to G O ODWIN , RAMADGE and CAINES -( 1 9 8 0 ) when it i s applied to a non­linear proce s s . Apparent ly , the pres ent paper i s the first application o f the G O ODW I N and C o l l eagu e s ( 1 9 8 0 )

1 09

adapt ive mu ltivariable contro l t o a -d i s t i l lation c o lumn and appears that thi s p er formance s cheme may be b e t t er than s ome previous s imi lar studies in a det ermi n i s t i c ambient .

In this approach both t o p and bottom compo s i t i o n s contro l o f a s imulat ed -binary d i s t i l lation c o lumn i s p erfor­med . The nonlin ear , t ime varying para­meter mod e l . o f one p i lot d i s t i l lat ion c o lumn due t o ES PANA , ( 1 9 7 6 ) is u s e d l i k e a " Pro c e s s " . I t h a s b e e n s howed that it act s l ike the true p i lot p lant in a very wide o p erat ion c ond i ­t ions .

The two control algorithms u s e d -mono ­variabl e or S I S O and Mu lt ivariable or MIMO- drive both output tracking and regulat io n . In order to obtain a -f e a s i b l e application to the pro c e s s -we have kept t h e input s and outpu t s within t h e s pa c e con strained by t h e phy s ical l imit s by d o i n g a s l ight ly mod i f i c at io n on the algorithm . O n -the other hand , the usual large init ial tran s i ent s are avo ided by -doing first an o ff l i n e pro c e s s iden­t i f i cation and us ing the s e ident i f i ed parameters t o ini t i a l i z e the c ontrol algorithms . Thi s o f f l i n e i d ent i f i c a ­t ion i s d e s cribed in BARCENAS , ( 1 9 8 3 )

1 1 0 L . Barcenas-Uribe and J . Alvarez-Gallegos

and BARCENAS and ALVAREZ , ( 1 9 8 3 ) . The n ext s e ction deals with the pro c e s s de s cription , then it is d e s cribed t h e -s imulation st udy done to verify the performanc e o f the algorithm .

PROCE S S DESCRI PTI ON

I n this approach both top and bottom compos it ions of a s imulated bi nary -d i s t i llat i on c o lumn i s d e s ired . A -nonlinear t ime varying parame t er model of the c o lumn is used as a " Proce s s " , and has been showed that it act s l i ke a true pi lot p l ant in a v ery wide -working s pace ( E S PANA , ( 1 9 7 6 ) ) .

A reduced order mod e l ( BARCENAS and ALVAREZ ( 1 9 8 3 ) ) s imulate the change s betwen b o t h d i s t i l lat ed and r e s idual product concentrat ions ( output s ) with respect t o both chan g e s re f lux flow and heating power ( input s ) . A diagram o f the c o n s id ered d i s t i l lat ion column i s reported in Figure 1 . ° " &llHENAS /#JO ALVA i t � {lfU). The c o lumn has nine p l at e s , the mix ­ture i s feed at the fi fth plate and consists of 5 0 % wat er , 5 0 % methanol -

with a t emperature rat ed at 7 0 ° C . The standard operat ing conditions of the -

co lumn are the f o l l owing ;

Ref lux Lo 0 . 3 3 9 Heating Power Qb 1 0 1 6 4 Feed F low Lf 1 . 8 1 5

( mo l / s eg ) ( c al / s e g ) (mol / s e g )

Top Concentration XD 0 . 8 8 9 ( % Metha­nol )

Base Concentrat i on XB 0 . 1 7 1 ( % Me­thano l )

A detailed d e s cript ion o f the pro c e s s i s founded in BARCENAS , ( 1 9 8 3 ) and -BARCENAS and ALVAREZ ( 1 9 8 3 ) d e s cr i b e s t h e pro cedure u s e d t o get a t w o - ­input s -two output s , reduced order -model of the co lumn suitable t o i n i ­tial i z e t h e prop o s e d control algorithm. Thi s requires a model in ARMA form , -by s u c c e s s i v e subst itution can be re­written in "Minimun delay pred ictor" -form , The init ial mode l i s ; r, (t+ l )l �r (q-" . � J [y1 (t)J f 2 (t�1� l 0 ' 2 ( q j Y2 ( t )

+ Fi1 (q-1 ) , �2 (q-1 )] [U1 (t)l �/q-1) , �2 (q-1 ) U2 (t� ( 1 )

where : Y(t)T = (• �{t) , .a XB (t)) and U(t )T =

(• Lo(t ) , 4 Qb(t)J

°f<q-1 ) 2 . 77 3-2 . 7 7 0 q-1 + -2 = 1 . 173q

0 . 17 7 q -3

2 <q-1 ) = 2 . 247-1 . 552 q-1 + 0 . 303 q -2

0 . 102E-3 q -3

?'!. -1 0 . 717-0 . 3 73 q -1 -2 11 <q ) = 0 . 491 q +

0 . 2 0 2 q -3

,::$' -1 1 2 <q ) = -0 . 341+0 . 2 5 6 q-1 + 0 . 184 q-2

0 . 12 0 q -3

� -1 1 .387-1 . 591 q-1 -2 21 (q ) = + 0 . 188 q +

0 . 0612 q-3

/'!, -1 -1 . 171+0 . 3 2 5 q -1 -2/q ) = + 0 . 768 q

0 . 2 57E-3 q -3

ALGORITHMS

The appli cation of the adapt ive con­tro l l er i s based upon r epres ent ing -the s y s t em by means o f an ARMA model o f the form ; -d _ 1

A ( q- 1 ) Y ( t ) =[q " �n ( q ) . : J _J -1 q """'El .. ,. ( q

- 1 - 1 where A ( q ) , B i · ( q ) denot e po lynomi a l s in t�e unit del�y tor q- 1 and the factors q- d l J t e pure t ime d e lays .

• U(t) s calar o pera­indica-

G O ODWIN , RAMADGE and CAINES , ( 1 9 8 0 ) , have analyz ed .� eneral algorithms , -the s o - called " Proj ection-Algorithms" , and have shown that under c ertain ade ­quate c ondit ions they will b e globally conv ergent and have ext ended their -own result s t o a general c l a s s o f - ­s t ab l e s y s t ems ( G O ODWI N and LONG , - ­( 1 9 8 0 ) ) . G l obally convergent means -algorithms which , for all initial - ­s y s t em and algorithm stat e s , cau s e -the output s o f a given linear s y s t em to a s imptot i ca l ly track a d e s ired -output s ecuence and thi s i s achi eved with a bounded input - s ecuenc e .

The algorithms requir e s the ARMA mo ­d e l t o b e ·l'.'eWl"itten in minimun d e lay predictor form ; for the s i ngle input ­s ingle out put ( S I S O ) this reduces t o ; y ( t + d ) = °' ( q- 1 ) y ( t ) + /< q- 1 ) ( t )

- 1 - 1 < 2 )

wj_th �( q ) and ,d ( q ) p o lynomi-nals o f appropriate - d imens ion , -and in the mul t i p l e input -multiple output -( MIMO ) this r e s ult s as the e quat ion -( 1 ) . Not e that this repr e s ent at ion con s i s t s o f a set of mult i p l e - input

Mul tivariable Control of Binary Distillation Column 1 1 1

s ingle output ( MI S O ) systems having a comn:on input vector . Proj ection Algorithm I ( S I S O ), The system ( 2 ) can be densed form as ;

T y ( t+d ) = f ( t ) Go

writen in con-

( 3 )

where 1 ( t ) is the mea-surement vector and Go is the nominal parameter vec­tor . The tracking error i s

e ( t + d ) = y ( t ) T G0 - y* ( t + d ) ( 4 ) here y * ( t + d ) i s the red sequence . By that ;

" a priori " desi­u ( t ) such -

y C t ) TG0 = y* ( t + d ) ( 5 ) it i s obv ious that the tracking errO!'.' wil l be z ero . I f G0 i s unknowed , ( 5 ) may be rep laced by the adaptive algo­rithm : Oct) = �(t-1 ) + a(t) �(t-d) �+,(t-d)T j(t-d}i

. (yCtH fCt-d) �Ct-1 ll C B ) A f(t) G(t) = y* (t+d) ( 7 )

Proj ection Algorithm I ( MIMO ), Any system like

written in condensed Yi ( t +di ) = �i ( t ) G�

( 1 ) may be re­form as ; , 1 � i � m ( 8 )

where 9i ( t ) i s a vector formed by all the parameter vectors corre spond ing -to 'Ji ( t ) .

then the tracking error may be iden­tically z ero i f we choose 1.ti ( t ) such that ;

fi ( t ) T G� = y1 ( t +di ) ( 1 0 ) The adapt ive algorithm result s ; � A T �i(t)=Gi (t-1 ) + a(t) fi Ct-di )[1+yi (t-di ) Yi(t-di )) -1 . ( Yi (t)-yi(t-di) Gi (t-1)) (11)

i O � m (12)

Figure 1 shows the u s ed contro l s che­me , ( non adapt ive r epres entation ) , -This contro l l er can be applied only to systems with · stable s zeroes '

CONTROL - IMPLEMENTATION The process input an output may be -repres ented in a two dimens i on s pace a s it is shown. in figure 2 ; ·

For practical purpos e s the obs cured zones shows the mo st int ere sting ones . The input bounds are indicated in the s ame figure . The S I S O algorithm i s evaluated kee- ­ping the Heating power ( Qb ) constant and the Di st i llat e Concentrat ion ( XD ) i s drived by the reflux ( Lo ) . Figure 3 shows what happens when the des ired s e quence is a pul s e train ( T: 4 0 S ec . ) modulated by the next trans fer func-­tion ;

G ( s ) = 0 . 0 5 1 + 4 0 0 s

in Figure 4 it i s show what happens -when the input is Qb , the output i s -Xo and Loremains constant . I n both fi­gures the res idual concentrat ion XB -behaviour is i lustrat ed . The s e expe- ­riment s demonstrate the fact that i t i s obtained a better proc e s s performance if it i s controled the d i s t i l late u s ing the reflux as input that when is u s ed the heating power for the s ame purpos e . Figure 5 shows the results obtained -when deterministic perturbat ions are applied to the proce s s : Flow incre - - ­ment , liLf = ± 2 5 % L0

f and I nput concen-trat ion methanol increment , Xf = ! 2 0 % X� , I ri Fig . 5 . b . Lo i s u s ed a s con - ­trol variable keeping Qb constant , i n Figure 5 . c . the control variable i s -Qb and Lo remains constant . The MIMO control law was applied to -the process with the initial parame-­ters given by the expres ion ( 1 ) . The reference vector is a pul s e train - - ­( T: 4 0 0 0 s ec , ) modulated by next trans­fer function :

G ( s ) = 0 . 0 5 1 + 2 0 0 s

for every one of the two input s . Figu­re 6 shows the output s in a perturba­�ion f�ee environment and in Fi gure 7 i s depict ed the output s b ehaviour - - ­with a n increment of 2 5 % in Lf . The s e figures shows a ' scattering ' ef�ec� _ around the desired output valve Thi s n;-dues to the bounded input s u s ed that can �ot s at i s fy s imultaneous ly -the equations ( 1 2 ) . Then it is proposed a modification on the algorithm . A cons ­tant gain control ler i s u s ed when the prediction error becomes l e s s than - ­c ertain ' a priori ' fixed criterion -inst ead of the adaptive loop accordina to the expres ion ; �

1"1 ( t � � [G1 1 lU2 ( t j G 2 1

1 1 2 L. Barcenas-Uribe and J . Alvarez-Gallegos

wher e ; I:' G i j K _..d'�j =

1 - � � ij K k becau s e , in this case there i s a 'ltit de lay .

Figure 8 shows the propo s ed swit ching contro l ler and Figure 9 depict s the performance o f the same .

CO NC LU SIONS

The purp o s e of this paper was to i n ­v e s t i gate t h e p er formanc e charact e ­r i s t i c s o f a l inear mult ivariable adapt ive algorithm appl i ed to a non­linear proc e s s in a det ermin i s t i c en­vironment . Speci fically we wi shed to show how the global convergence pro ­pert i e s o f this algorithm can be em­ployed in a non theoreti cal s ituat i ­on . I n order to e stablish convergen ­c e we have found n e c e s s ary to mo ­dify sl ightly the algorithm pro p o s e d b y GOODWIN , RAMADGE and CAINE S ( 1 9 8 0 ) .

As it has been showed with the resulT s , the contro l l er has been succ e s s ful­

ly emp loyed t o control the t o p and bottom compo s i t i ons of a binary d i s ­t i l lation column in s imulation stud i ­e s . The s e re sult s indicate that the controller adapt s t o the changing -cond it ion s in both .regulation and con-

trol t a s k s d e s p i t e the nonl inear and t ime varying proc e s s character and the need for adapting a relatively large number of paramet ers . However , some open problems r emain unan swered ( BAR­CENAS , ( 1 9 8 3 ) ) ; a s examp l e : exponent i ­a l converg ence and knowed bounds on input s and output s .

ACKNOWLEDGEMENT

The authors acknowl edge the . financial sup port o f ' CONACYT ' and the ' In s t itu ­tp Tecno logico d e Queretaro ' and u s e o f t h e fac i l it i e s provided b y the D e ­pt . o f Electrical Enggr . a t ' C INVES ­TAV - IPN ' .

REFERENCES

BARCENAS U . L . ( 1 9 8 3 ) . Identi f i caci6n y Control Adaptable Multivariable de una Columna de Destilaci6n Bi­

nari a . Mas t er S c i ence The s e . C en­tro de Inv e s t i gaci6n d e l I PN , M e ­xico .

BARCENAS U . L . and ALVAREZ G . J . ( 1 9 8 3 ) . Ident i f i cation o f a binary d i s t i l­lation column . IFAC Workshop on A�pl ications o f PNL to optimiza­t ion and contro l . , San Fco . , Ca . , USA . , June 2 0 - 2 2 , 1 9 8 3 .

E SPANA M . D . ( 1 9 7 6 ) . Mod e l e d i namigue

de connai sance d ' une colonne a d i s ­t i ll er binaire . Int ernal report LAG-7 6 - 1 3 . I PN Grenobl e , Franc e . G O O DWIN G . C . and LONG R . S . ( 1 9 8 0 ) .

G eneral i z at ion o f result s on mul t ivariable adapt ive contro l . Tran s . IEEE-TAC , A C - 2 5 , 3 , pp . 4 4 9 - 4 5 6 .

LIEUSON H . MO RRI S A . J . ? NASER Y . and WOOD R . K . ( 1 9 8 0 ) . Experimental evaluat ion of s e l f -tuning contro­l l ers applied to pi lot plant un ­i t s . Lectur e notes in control and information s c ienc e s , 2 4 , Springer Verlag , p p . 3 0 1 - 3 0 9 .

SASTRY V . A . , S E BO RG . D . E . and WOO D R . K . ( 1 9 7 7 ) . S e l f -Tun ing r egula ­tor app l i ed to a binary d i s t i l ­lation column . Automat ica , 1 3 , pp . 4 1 7 - 4 2 4 .

�

UNBEHAVEN M . and S CHMI D CHR . ( 1 9 7 9 ) . App l ication o f adapt ive s y s t ems in pro c e s s control . Int ernat io­nal Workshop on App l i cat ions of Adapt ive Contro l , Yale Univer s i ­ty , Connect icut , USA . , Aug . 2 3 - 2 5 1 9 7 9 .

VOGEL E . F . and EDGAR T . F . ( 1 9 8 2 ) . Ap ­pl ication o f an adapt ive po l e - z e­ro placement contro l l er to che­mical pro c e s s with variable dead t ime . 1 9 8 2 American Control Con ­f erenc e , Arl ington , Virgini a , USA June 1 4 - 1 6 , 1 9 8 2 .

WIEMER P . , HAHN V . , S CHMI D C . and UNBEHAUEN M . ( 1 9 8 3 ) . Appl i cat ion o f Multivariable mod el re ference adapt ive control to a binary di s ­t i l lat ion column . IFAC workshop on Adapt ive Syst ems in Control and S ignal Pro c e s s ing , San Fco . , Ca . , USA , June 2 0 - 2 2 , 1 9 8 3 .

11- d ') y (t)

'---0--'�!.�y�t!_ _ _ _ _ _ ..... '(.o

Fig . 1 . Prop o s e d ' Dead beat ' control S cheme .

Mul tivariable Control of Binary Dis tillation Column 1 1 3

I I I _ _ ___ _ _ _ ,. _ - - - - -

1.0

�

' I I I I I

1 '1' I I

- - - - - - - - L - -

10 - - - - - - - - - , - -- - - - - - - - - ---, · o · _ _ ..._� _ _ j (l.o,Qb)

0 5 tNTEREST AREA

0 0.5

Fig . 2 . Proc e s s input and out put spa-c e s .

• 95

.85 ---------.....J

i :7

0. 15 . 000 sec

.25

.1 5 ----------./5·##�Jac;=.i. Fig . 3 . Behaviour o f XD ' dri ved by

Lo , ( Q b i s constan� ) , ( S I S O )

.94 ,...... ,...... ,......, ,,.-\v_ � II-

.84 1 5, 0 0 0

. 40

.1 5

...

I

l r

-

... J.o.

... \ I\.

Fi g . 4 . Behaviour o f X drived by Q b ( Lo i s const ant ) � ( S I S O )

...

I

'

1 1 4 L . Barcenas-Uribe and J . Alvarez-Gal legos

· ' J _ - - - - - - - - - _ __ _ J _ _ _ l .'1 8 {Q.)

,95 s �

'i C ,b.J {!.) ,95

nc � � o� u ·1 •c. •

�fig . 5 . Behavi our in regulation , ( a ) P erturbat ion Input s ( b ) Output when control input i s Lo , ( c ) Output when control input i s Q b .

Fig . 6 . 0utputs performance in track -ing ( MIMO )

����--t����--t>--��-,..�1--.,...,........,,...,..,,..__. .F ig . ·1 . Output s p ert ormanc e in 4ir��B��ec ing and regulat ion ( M IMO )

u fk) y(k )

CA: Cilllltralodllr G claplallle --nr�ca::C9llnlladar d . . stado-ol!: iW-*>' y' CJ* CDllllllllt

Fi g . B . Switc hing contro l l er Adapt ive / Constant gain . ( M IMO )

20 , 000 sec

Fig . 9 . Output s XD and X B wnen i s app! i ed the swit ching control . ( M IMO )

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

MINIMUM VARIANCE CONTROL FOR MULTIVARIABLE SYSTEMS WITH DIFFERENT

DEADTIMES IN INDIVIDUAL LOOPS

M. H. Costin and M. R. Buchner

Department of Systems Engineering, Case Institute of Technology, Case Western Reserve University, Cleveland, OH 44106, USA

Abs tract Current minimum variance control algori thms for mul tivariable systems assumes that the dead-times between the output measurements and the

control actions are all equal . This is a necessary condi tion for the in­

vertibility of a matrix used in determining the minimum variance controller , This study examines the case where at leas t one of the dead-times is larger

than the others , resulting in a nonsingular matrix to be inverted ,

The derivation of the minimum variance controller involves correcting the controller cos t criterion to take into account the added delay in some of the loop s . This s tructure for the minimum variance controller is then applied to modify the s tandard MIMO-self- tuning regulator and the MIMO-self­tuning regulator with control penalty for the non-equal dead-time case .

Key words : Adap tive contro l , Delay s , Digital contro l , Linear Op t imal Regulator , Self-Tuning regulators ,

INTRODUCTION

In two papers , Borison { 19 7 5 , 1979} derives the minimum variance control law for square multivariable systems . He then uses this min­imum variance control law to design a multi­variable self-tuning regulator (STR} . The process is assumed to be well modelled by :

-1 -1 -1 A (q } y (t} = B (q } u (t-k-l } +C (q } e {t}

-1 -1 -N A (q ) = I + A1q + • • • + ANq

-1 -1 B (q } =BO

+ Bl q

-1 C (q }

+ • • • +

+ • • • +

-M BM'l

y {t) - n - vector of system measurements

u (t) - n - vector of system inputs (con­trol actions)

e (t) - n - vector white noise sequence

( 1 )

a l l their zeroes outside the unit disc { i . e . they are stable)

k - system deadtime

The minimum variance controller for system ( 1 ) is specified for a cost criterion where Q i s a positive semi-definite matrix .

min u (t)

Q y ( t) } ( 2 )

1 1 5

Borison then derives the minimum variance controller for this system as :

F (q-1) B {q

-l} u (t) = - G (q-l) y {t) ( 3 )

- -1 - - 1 where F (q } and G (q ) are given by

C (q-1) = A (q-1

) F (q-1) + q-k-1 G (q

-1)

- -1 -1 - -1 -1 F (q ) G (q ) = G (q ) F {q )

- -1 -1 -det F {q ) = det F {q ) and F ( O ) = I

- -1 - -1 F (q ) and G (q ) are guaranteed to exist but they are not necessarily unique .

The actual control is implemented as

- 1 - -1 u {t) = - B

0 G (q ) y (t) - 1 - -1 - 1

BO F (q ) {B (q )

( 4 )

Two remarks can b e made about controller ( 3 , 4 ) . Firstly, the control i s independent of the choice of Q and secondly, B must be

. 0 nonsingular to calculate a feasible control , Borison {1979) does give an example o f a system where B0 is singular and for which a feasible minimum variance controller can still be designed, A key point of this ex­ample not discussed by Borison is that the matrix B

0 contains a zero row. For the case

where B0

has a zero row, the deadtime between

the u {t) vector and one e lement of the y (t) vector i s greater than the deadtime between the u (t) vector and the other elements o f

1 1 6 M . H . C�stin and M.R. Buchner

y (t) . In the developmehtj that follows this case can be handled usih� the methodology proposed by Borison after the system cost function is suitably modified.

I DERIVATION OF GENERALIZED MINIMUM VARIANCE

CONTROLLER

This section derives the minimum variance controller for a model of the form of equation (1 ) . However , Ihe assumptions on the matrix polynomial B (q- ) are relaxed to allow zero rows (each row will have its own deadtime ki .'.'._ k associated with it) . This requires the definition of a new measurement vector

y (t+k+l) = [;: :::::::j) y (t+k +1) n n and a new cost cri-

teria

min E{ylt+k+l) Q y (t+k+l) } u (t)

System (1 ) can be rewritten as (6) .. -1 .. -1 .. -1 A (q ) y (t) = B (q ) u (t-k-1) + C (q ) e (t)

-k . +k V = diag' {q 1 } - deadtime matrix

(6)

A(q -l) =VA(q -l) ; ; (q -l) =VB (q -1) ; c (q -l) =VC (q -l) Model (4) row has filled the zero rows of B0 by the corresponding rows of Bk . -k "

1.

The derivation of the M.V. controller can now proceed as in Borison ( 1979) . Factor C (q-1) as follows :

where f . . ii

f . . iJ

c -k .

A F + q iG

-1 1 + f . . q + • • • + f . . q u.1 iik

-1 f . . q + • • • + iJl

i -k . i f . . q iJk . i

-1 -1 -p G (q ) =Go + Gl q + • • • + Gpq

p N - 1 N > K > k

p K - K - 1 K > N

where k , K, N are defined in (1 ) . ..

-k . i

i "/' j

Additional matrices F , G and C can be deter­mined as follows :

F G ;; c

.. F c

G F - .. F A +

.. C F

-k-1 q G

- -1 -1 -where det F (q ) =det F (q ) and F (O) I . Walker ( 1982 ) and Prager and Wellstead ( 1980) give methods for determining F and G. Multiplying (6) by F yields :

F A y (t+k+l) = F B u (t) +FC e (t+k+l ) ;; c y (t+k+l) - Gy (t) = F;u (t) + CF e (t+k+l)

C (y (t+k+l) -Fe (t+k+l) ) =FBu (t) + Gy (t) ( 7 ) I

Since C =Ve implies FVc = CVF one can rewrite (7 ) as

c (y (t+k+ll -VFe Ct+k+ll l =FBu Ct l+Gy Ctl

Therefore EtYT (t+k+l) Q Y(t+k+l) }

> E{ (VFe (t+k+l) ) T Q (VF'e (t+k+l) ) } ..

with equality holding when F Bu (t) Gy (t) .

Therefore the M.V. controller for the system is

F B u (t) = - G y (t) (8)

which is implementable if s0 is nonsingular and B (q-1) is minimum phase .

If one defines the prediction of y (t+k+l) at time t as:

y (t+k+l j t ) =y (t+k+l) - V Fe (t+k+l)

then controller (8) also minimizes

' T I I min { y ( t+k+l t) Q y (t+k+l t) } u (t)

MODIFICATION INVOLVING CONTROL PENALTY

(9)

Often M.V. control requires excessive control actions . In order to reduce the variance of the control actions , cost criteria (9) is modified to a form ( 10 ) which includes a weighting on the manipulated variable ( see for example Clarke and Hasting-James (1971 ) , MacGregor and Tidwell (1977) SISO systems , Koivo ( 1980) , Bayoumi et al ( 1981) , Walker ( 1982 ) MIMO systems) .

min u (t)

{ yT (t+k+l l t) Q yT (t+k+l j t) + UT (t) R u (t)} (10)

(R positive definite)

In addition to reducing controller variance , by suitable selecting the weighing ma!fices Q ,R one can handle the case where B (q ) is non-minimum phase •

In an unpublished report Walker (1982) de­rives the solution to ( 10 ) . For the case of a zero row in B0 , one can rederive Walker ' s algorithm by modifying cost criterion (10) to :

min' {yT (t+k+l j t) Qy (t+k+l J t) +uT (t) R u (t)} (11 ) u (t)

Minimum Variance Control for Mul t ivar i able Sys tems 1 1 7

Following Walker ' s approach let

aJ 2yT (t+k+l J t) Q ay (t+k+l l t> au (t)

= au (t)

Let

.�

+ 2uT

(t) R = 0 ay ct+k+1 J t>

au (t) B

y (t+k+1 J t> =C'"1 cF iu ct> + G y (t) >

T A -1 T �-1 -u (t ) = - (R+B QB0) B Q (C G y (t)

+ c-1 P CB - B0> u <t> > c12>

0

For the case of cost criterion (10) B = B0 • If B

0 contains zero rows then B = B0 will

hold if for all k.

> k, a . . 0 n=l , • • • , ki

-k l. l.J n

h . h . . th 1 f h ' w ere a

ij is t e i3� e ement o A

n. T is

n -1 condition holds for many systems (eg. A (q ) diagonal) . For this case controller (12) becomes :

where the positive definitiveness of R and positive semi-definitiv�nesshof Q ensure the non-singularity of R + B� Q B

0 •

For the case of B � B0 , invertibility of T A

R + B � B0

can be ensured by approxim�ting B by B0 • If one looks at the term B Q, one can find a Q such that

(13) By approximating B by B0 one is ineffect

changing the weighting Q of the loop measure­ments to Q . Since Q and R are normally tuned on-line based on system performance , this approximation will not make any practical difference .

REFERENCES

Bayoumi , M.M. , K . Y . Wong and M.A. El-Bagoury , "A Self-Tuning Regulator for Multivariable Systems " , Automatica, 17 (4) , 572-592 , (1981) . Borison, u . , "Self-Tuning Regulators - Indus­trial Applications and Multivariable Theory" , Report 7513 , Dept. o f Automatic Control , Lund Institute of Technology, Lund , Sweden (1975 ) . Borison, u . , "Self-Tuning Regulators for a Class of Multivariable Systems " , Automatica, 15, 206-216 (1979) . Clarke, D ,W. and R . Hastings-Jame s , "Design of Digital Controllers for Randomly Disturbed

ASCSP-E

systems " , Proc . IEE , 118 , 1503-1506 , ( 1971) . Koivo , H . N . , "A Multivariable Self-Tuning Controller " , Automatica, 16, 351-366 ( 1980) . MacGregor , J,F . and P . Tidwe l l , "Discrete Stochastic Control with Input Constraints " , Proc . IEE , 124, 732-734 (1977 ) . Prager , D , L . , and P . E . Wellstead, "Multi­variable Pole-Assignment Regulator" , Proc . IEE P t . D , 128 , 9-18 (1981) . Walker , B . , "Multiple Input-Multiple Output Self-Tuning Regulator with Control Penalty , " Unpublished report , Department o f Aeronautics and Astronautic s , Mass . Institute of Tech . , Cambridge , Mas s . ( 1982) .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

SOME RESULTS ON INFINITE HORIZON LQG ADAPTIVE CONTROL

G. Bartolini, G. Casalino, F. Davoli and R. Minciardi

lstituto di Elettrotecnica, Univenitil di Genova, Viale F. Causa, 13, 16145 Genova, Italy

Abstrac t : An adaptive control algor :lthm is presented which is based on an infini­te horizon LQG control criterion . The algori thm makes use of the recursive i­dentification of an impl i c i t model of the Least-Squares type . Some basic re­sults about the convergence the algori thm are given .

Keywords . Adaptive systems ; identi fication ; process control ; computer central ; optimal contro l .

INTRODUCTION AND DEVELOPMENT OF THE SYSTEM IMPLICIT MODEL

Since the c lassical s tarting point of the self­-tuning regulator by l!.strom and Wi ttenmark ( 1 973 ) , a great attention has been devoted to adaptive control algori thms based on Re­cursive Least Squares identification . One of the characteristics of the s e l f-tuning scheme is the identification of an imp l ic i t model ( that i s , a model which correctly re­presents the system only in c losed-loop con­ditions ) , and not of the " true" ( open-loop ) system model.

Wh i l e the original algori thm of Astrom and Wittenmark is based on minimum variance con­trol strategy , the attainment of the infi­nite horizon LQG control law has been con­s i dered for a long time very expensive from the computational point of v i ew , s ince i t would seem t o require , at each adaptation step , the i teration t i l l convergence of a discrete-time R iccati equation .

Thi s paper presents an adaptive control algo ­r i thm which is based on an infinite horizon LQG control objective and solves a di screte­-time R i ccati equation , requiring however only a moderate computational effort at each adaptation step . The basic idea is that of interlacing the i teration of the R iccati equation with the recursive identification procedure of an impl i c i t system mode l . In this Section thi s i mp l ic i t model i s introduc ­ed and worked out , and in the second Section it w i l l be exploi ted to define the adaptive algori thm . Due to space l im i tations , in the paper only the bas i c results will b e pre­sente d , omi tting all the proofs , which can be found in ( Bartolini and others , 1983 b).

1 1 9

Throughout the paper we shall refer , only to s i ngle-input s ingle-output systems , sinc e , actual ly , the extension of our ap­proach to multivariable systems poses some addi tional technical and formal problems ( bu t , we conj ecture , no conceptual d i fficul­ties ) .

Let the system to be control le d be

-1 -1 -1 A ( q ) y , =B ( q ) u . +C ( q ) e . ( 1 )

where l. 1 l.

-1 -1 -n A ( q ) =l+a q + . . . +a q ( 2 )

- 1 1 1 n B ( q

l) =b

l q- + . . . +b q-n ( 3 )

C ( q- ) =l + c q-1 + . . . � c q-n ( 4 ) 1 n

-1 being q the uni t delay operator , and e . i s a gauss i an stat ionary white sequence . ft i s supposed that C ( z ) has i ts zeroes outside the unit c ircle ( ac tually , this assumption is certainly not too res tri ctive ) . Thi s in­put-output model can be e as i ly put into a state-space representation . For instance an observable canonical form can be chosen of the type

xi+l =Fx

i+G u

i+K ei

yi+l

=Hxi+l

+ei + l

( 5 )

( 6 )

From ( 5 ) , ( 6 ) one can derive the expression of the optimal predictor , namely

x . = ( F-KH ) x . +Gu . +Ky , l.+l l. l. l. ( 7 )

which i s asymptotically equivalent to ( 5 ) , ( 6 ) . Suppose now the system be governed by a constant ( generally s imply proper ) feedback law

U , =MX . +gy , l. l. 1 ( 8 )

up to time instant ( t-1 ) • Then , considering

1 20 G . Barto lini et aZ.

the state equation ( 7 ) and viewing ( 8 ) as the associated output equation , the follow­ing Lemma is easily deduced .

Lemma 1 . I f system ( 1 ) is governed by con­trol law ( 8 ) at least from time instant ( t-n ) up to time instant ( t-1 ) , and i f the pair [( F-KH ) , M] is completely reconstructi­ble , the system state x

t can be expressed

as

A t-1 [' t-1 x = y + u

t t-n t-n ( 9 )

i-1 . being Y . =col ( y , , . . . , y , ) , u ;i.-1 s imilar-

1-n i-1 _i-n i-n ly defined, and A and !.' matrices of sui ta-ble dimensions . 0 As a consequence of Lemma 1 , if the above hypotheses hold , one can express x as

t+l follows

x =Zir ( 10 ) t+l t+l

be ing x =col ( yt , ut and matrix Z sui-t+� t-n t-n

tably defined . Moreover , in the same condi-tions ,

y =HZx +e t+l t+l t+l

( 1 1 )

I t i s worth remarking that in Lemma 1 the choice of u

t is left " free" . However , i f

the choice o f the control law (8 ) is renewed also for time instants i <t t , then relations like ( 10 ) and ( 11 ) can be written also for x . and y . , with i � t . Thus , it makes sen-

1+1 1+1 se to consider a dynamic system with state and output equations

x =Fx +Gu +Ke i+l i i i

y =Hx +e i+l i+l i+l

( 12 )

( 13 )

where matrices F , G, K and H are defined in a straightforward manner . Equations ( 12 ) , ( 1 3 ) represent an " impl icit model " for sys­tem ( 1 ) , which holds only in the above clos­ed-loop conditions . Two additional important results are now given .

Lemma 2 . Let system ( 1 ) be governed by a control law of the type

-1 -1 R ( q )u . =P ( q ) y , ( 14 )

l. l. generally s imply proper and of order n . Be­sides , suppose that , due to this feedback relation , the system . output can be correct­ly predicted by means of the implicit predic ­tion model ( of the Least-Squares typ e ) - -1 ,.. -1 A ( q )y . =B ( q ) u . +e . ( 1 5 ) l. l. l. of order ( n+l ) . Then the control law ( 17 ) is equivalent to a s tate-observer feedback law of type ( 7 ) , ( 8 ) . D Lemma 3 . The existence of an implicit model of type ( 12 ) , ( 13 ) is a necessary and suffi­c i ent condition for the existence of an im-

plicit model of type ( 1 5 ) . [J The control problem which is dealt with in the paper , is the infinite horizon LQG one with control weight p . Assume that the 'pair ( F , G ) i s stabil izable . Then , the optimal control law i s

u . =-L[F-KH ) x . +Ky ) l. l i' (1 6 )

T -1 T where L= ( p+G TG ) G T , T being the matrix obtained by iterating till convergence the discrete-time Riccati equation associated with the problem .

Observe that the optimal control law is of the type ( 7 ) , ( 8 ) and suppose that the recon­structibili ty hypothesis of Lemma 1 holds . Then , it is possible to show that the opti­mal control law can be expressed as

(17 )

,..r- - -1- - ,... T where L=( p+G T G ) G T being T�Z TZ . I t i s possible t o give the following result to characterize matrix T. Theorem 1 . Consider system ( 1 ) put into sta­te-space representation ( 5 ) , ( 6 ) and let L be the optimal feedback gain defirirl abo'­ve . Assume that the pair [ ( F-KH ) , -L ( F+ -KH )] is reconstructible and consider the system impl icit model of type ( 12 ) , ( 13 ) which i s associated with the optimal control law ( 16 ) . Then , matrix T , above defined, is the unique convergence point ( not depend­ing from the initial ization ) of the discre­te-time R iccati equation associated with the LQG control problem corresponding to the above system implicit model and to con­trol we ight p . O THE ADAPTIVE CONTROL ALGORITHM

The results shown in the preceding Section assure that , in optimum feedback condition , the system output can be predicted via a Least-Squares type implicit prediction mo­del , which can also be put into state-space representation . Then , one is quite naturally led to define the following algori thm , which is based on the recursive identification of a prediction model of type ( 1 3 ) . Before giving the algorithm , let us note that iden­tifiability cons iderations suggest to suppo­se b

1 =HG to be known ( hence the parameter

vector does not include HG , thus removing the l inear dependence of the various compo­nents of vector x. ) .

l. Adaptive algorithm . For each control in­stant , perform the following operations : i ) update the parameters of the prediction

model ( 13 ) ( and of ( 12 ) ) by use of a Recursive Least Squares identification

Infinite Horizon LQG Adaptive Control 1 2 1

procedure ; ii ) iterate , for a single step , a discrete­

-time Riccati equation based on matri­ces F, G, H corresponding to the updat­ed prediction model , with control weight .p, and on the matrix T . obtained at the preceding i teration . Tbis itera­tion gives T. ;

:i.+lt iii ) compute matrix . as l

,... ,..T ,.., � -1 ,,...T -L . = ( p+G T . G ) G T . l l l and apply the control law

"" [ _ __ ..., - ] u . =-L . ( F-KH ) x . + Ky . l l l l

( 1 8 )

( 1 9 ) 0

The above adaptive algori thm is characteriz­ed by an interlacing of the identification procedure with the i teration of the Riccati equation for the determination of the opti­mal feedback . · For this reason , the acronym ICOF ( Interlaced Computation of the Optimal Feedback ) has been used to designate this algorithm ( Bartol ini and o thers , 1 982 ) . Clearly , the adaptive scheme can be heuristi­cally j ustified as follows . I t is known that under optimal feedback conditions , the im­plicit model ( associated with that control law ) correctly represents the system beha­viour . Then , it makes sense to try to iden­tify such a model , s ince our goal is just the attainment of the optimal control law . The convergence analysis of this algorithm is beyond the scope of this paper and i s deferred t o forthcoming work . Nevertheles s , it can be mentioned that experimental re­sults have always given a very good perfor­mance of the algori thm , even in connection with "pathological" choices of the system to be controlled , such as , for instance , non-minimum phase systems and systems which do not satisfy the positive realness condi­tion ( Bartolini and others , 1 983a ) . As a matter of fact , a fundamental problem aris­ing when the algorithm converges to an equi­librium point , is that of recognizing i f this point corresponds t o the optimum feed­back s ituation . In this connection , we can give the following result .

Theorem 2 . Suppose system ( 1 ) ( which we con­sider unknown but in its order n , and in coefficient b

1, and assume to be stabi l iza­

ble ) to be governed via the above adaptive algorithm . Assume that the algorithm conver­ges to an equilibrium point which is charac­terized by an optimum output prediction , by means of an . implicit model of type ( 13 )1 and by a stationary input-output behavior �hat i s , u . and Y . are stationary stochastic processes).1 Then , \he s tationary control law corresponding to the equi librium point of the adaptive algori thm is the ( infinite ho-

rizon ) LQG optimal one . Cl

ACKNOWLEDGMENT

This work has been supported by the National Counc il of Research of I taly ( C . N . R . ) , under the Spec ial Program on Comp�ter Science ( P . F . I . ) .

REFERENCES

Kstrom , K . J . , and B . Wittermark ( 1973 ) . On self-tuning regulators . Automatica, � . 185-199.

Bartolini , G . , G . Casalino , F. Davoli , R . Minciardi , and R . Zoppoli ( 1982 ) . Effi­cient alg:irithms for multi variable adaptive control : the ICOF scheme . Control and Computers , 10 , 58-62 .

Bartolini , G . , G . Casalino , F . Davoli , and R . Minciardi ( 1983 ) . The ICOF approach to infinite horizon LQG adaptive con­trol . Ricerche di Automatica ( to appear ).

Bartolini , G . , G. Casalino , F . Davoli , and R . Minciardi ( 1 983 ) . Basic results on the ICOF adaptive algori thm . Int . Rep . CSR-83-1 , Istituto di El ettrotecnica , U ­niversity of Genoa .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

MODEL REFERENCE ADAPTIVE CONTROL SYSTEM OF A CATALYTIC FLUIDIZED BED

REACTOR

M. S. Koutchoukali, C. Laguerie and K. Najim

Institut du Genie Chimique, Chemin de la loge, 31078 Toulouse Cedex

Abstract . This paper presents a study on the control of a f luidized bed catalytic reactor . In this study , the model reference adaptative control algorithm with indepen­dent tracking and regulation obj ectives as presented by Landau-Lozano ( 1 98 1 ) has been investigated . The purpose of the control is to maintain the temperature of the reactor as near as possible to a desired set value (490°C) . Some of the heat produced as a result of the exothermal react ion tacking place between propylene , annnonia and oxygen is carried away by a vent ilator . The control act ion is based on a s imple mathematical lower order with t ime varying unknown parameters . This method of control results in a significant ly improved control for both servo and regulatory control .

Keywords . Adaptat ive control ; chemical industry ; tracking systems ; temperature con­trol ; digital control ; stabi lity.

INTRODUCTION The analog proportionnal integral derivat ive (P . I . D . ) regulators are frequently used in industrial processes . It is frequently ne­cessary to readj ust the regulator according to the variat ion of process requirement . However , a general technique is available for adjust ing the PID parameters on l ine . In pract ice , the adj ustement of the PID knobs is very subj ective because it depends on the experience and on the knowledge of the opera­tor on the considered plant . During the last decade different adaptative approches have been developped and appl ied to provide an automatic adjustement of the control parameters to pal liate the effect of the variation of the plant dynamic and in the disturbance , such as fluctuat ion in the rea­gents purity as wel l as other varying para­meters . Among these approaches , the self tuning regu­lator ( indirect control method) and the mo­del reference adaptative system (direct con­trol method) seem to be more attract ive than the dual control method due to the fact that it is very difficult to solve the dual con­trol problem.

This study deals with the adaptative control of an f luidized bed reactor which is non li­near and non stationnary . The react ion invol­ved is the annnoxidation of propylene to give acrylonitrile . The model reference adaptative control deve­lopped by Landau-Lozano ( 1 98 1 ) is appl ied to the control of the temperature in the reactor by adjusting the velocity of the venti lator

1 23

i . e . the control device , whose obj ective is to carry away some of the heat produced by the reaction.

PROCESS DESCRIPTION The catalytic annnoxidation of propylene to produce acrylonitrile was studied in a steel f luidized bed reactor . The reactor comprises a preheater and a reac­t ion chamber ( f igure 1 ) . The preheater consists of a bed of sand f lui­dized by air and heated by two electric coils immersed in the sand.

Water fed into the preheater is instantane­ously vaporised on contact ing the hot f luidi­zed sand . Propylene and ammonia are heated separately in two coil s innnersed in the bed of sand . Reactants are mixed in the zone just above the perforated plate distributor . Two rows of deflectors are incorporated to en­hance the mixing . The reaction compartment contains a catalyst composed of tin , iron , antimony and copper . It is very fragile at high temperature and does not resist to large and sudden tempera­ture variations . The reaction is exothermic . Besides acrylo­nitrile , other products such as acroleine , acetonitrile and carbone oxides are formed . Barbouteau ( 1 979) has found that the highest yield of acrylonitrile with this type of ca­talyst occurs in temperature range of 480-5 1 0° C . ·

1 24 M . S . Koutchoukali , C . Laguerie and K. Naj im

Maintaining a constant temperature in the reactor is very important but difficult to achieve due to the characteristics of the reactor.

Two pos sible techniques can be envisaged for the removal of the heat evolved during the reaction : - variat ion of the rotational speed of the ventilator in the case of air cooling of the reactor - cooling by increasing the rate of water va­porized.

It was observed that approximately twice the quantity of steam in the feed was necessary to remove the heat evolved , but this dis­turbs cons iderab ly the bed hydrodynamics . We then have chosen the f irst technique for the heat removal .

Based on material and energy balances , the mathematical modelisation of the fluidized bed reactor , developped by Stergiou ( 1 980) , yields a set of a system of coupled non l i­near part ial differential equat ions .

For control purpose , a s implified s ingle in­put-s ingle output model of the complex dyna­mics of the reactor was assumed :

A (q- 1 ) y ( t ) = B (q- 1 ) u ( t- k) + e ( t )

with :

where k is the process time delay in integer number of sample intervals , u ( t ) is the con­trol signal (excitation vol tage) , y ( t ) i s the output variable (reactor temperature) , e ( t) is a bounded disturbance.

MODEL REFERENCE ADAPTATIVE SCHEME

We wi ll present the broad out lines of the scheme presented by Landau-Lozano . ( 1 98 1 ) .

The main obj ective of the control system is to find a control strategy in such that in tracking, the control should be elaborated such that the p lant output sat isfies the fol­lowing equation

C 1 (q- 1 ) y ( t ) q-k D ( q- 1 ) uM( t ) ( 1 ) where :

- 1 1 - 1 n -n c 1 (q ) = 1 + c 1 q + + c� q ( 2 ) i s an asymptotically stable polynomial .

- 1 - 1 -m ( ) D ( q ) = d0 + d 1 q + • • + dmq 3 and uM(t ) is a bounded reference sequence .

In regulation (uM(t ) = 0) , the control signal should be designed such that any initial per­turbation will be eliminated with the dyna­mics defined by :

c2 , (q- 1 ) y (k+ 1 ) = o (4 )

where : - 1 1 - 1 e - e c2 (q ) = 1 + c2q + . • + c2 q

is an asymptotically stable polynomial .

An appropriate control configuration used for the case of known plant parameters to realise the above obj ect ive is riven by

(5)

c2 (q- 1 ) yM( t+k) - R(q- ) y ( t ) u ( t ) = (6)

B (q- 1 ) S ( q- 1 ) where the polynomial S ( q- 1 ) and R(q- 1 ) verify the fol lowing ident ity .

( - 1 ) ( - 1 ) ( - 1 ) q-k R(q- 1 ) c2 q = A q S q +

S (q- 1 ) 1 + s 1 q- 1 + . . + s q-ns

ns + • •

( 7 )

(8)

with : n5 = d - nR = max (nA- 1 , e- 1 ) and where yM(t ) i s the output of the refe­rence mode l .

The control law (6) can be written

where

- 1 M c2 (q ) y ( t+k) (9)

�T (t ) = _[""u ( t ) , u ( t- 1 ) , • . . , u ( t-d-nB+ 1 ) , y ( t) , . • . , y ( t-nR)_7

PT = l-bO , b0s 1 + b 1 ' ' ' ' ' bnBsd- 1 ' ro • · · · •

r 7 nir When the plant parameters are unknown the vector P of the control law (9 ) cannot be computed .

Landau and Lozano have developped an exten­s ion of this l inear control ler des ign given by equation (9 ) . TQe vector P is replaced by adjustable vector P ( t) which wi ll be updated by the fol lowing adaptation algorithm.

Therefore the control law is given by : - - 1 M P ( t ) � ( t) = c2 (q ) y ( t+k) ( 1 0 )

The control obj ect ive wi ll be achieved if the fol lowing algorithm is used :

P ( t) with

F ( t+ 1 )

P ( t- 1 ) + F ( t ) � ( t-k) y* (t ) ( 1 1 )

1 = " 1 ( t ) • ( 1 2)

•rF ( ) _ F ( t ) Ht-k) �T ( t-k)F ( t ) _/ - t A 1 ( t)

A2 ( t ) + �T ( t-k)F (t ) � ( t-k)

where 0 < J. 1 ( t ) � 1 F (O ) > 0 ( 1 3)

y* (t ) represent the adaptation error defined in our app lication for (H 1 (q- 1 ) = H2 (q- 1 ) = 1) defined in Landau-Lozano by :

C2 (q- 1 ) y ( t ) - P ( t- 1 ) � ( t-k) y* (t ) = ( 1 4)

1 + �T ( t-k) F (t) � (t-k)

A Catalytic Fluidized Bed Reactor 125

CONTROL HARDWARE AND SOFTWARE The reactor temperature ( the controled varia­ble) is measured us ing a thermocouple . The speed vent ilator d . c motor is measured by a tachometer . The rotor winding of the wound rotor is connected to a thyris tor rectifier . The d . c rectifier output supplies the d . c motor . The output signal o f the thermocouple is amplified, and that of tachometer is stepped down . The reactor was interfaced with an Apple II microcomputer as sociated to digital-analog and analog-digital convecters device s . The design of the Apple I I microcomputer is based on a 6502 microprocessor . Its addres­sing capacity is 64 K. Its memory capacity (RAM) is 48 K. The data interface consists of 16 channel mult ip lexed success ive appro­ximation 1 2 bits A/D converter , with a con­version time of 25 microsecondes , 4 channel 1 2 bits D/A converter and a c lock . The configuration also involves two floppy disk drives of 1 40 K of storage capacity each, consol terminal and teletype printer . The host programm used in the control is written in Pascal . It calls external assem­bly language routines related to real time control such that : data acqui sition routi­ne , digital-analog convecter routine , and the clock reading routine for sampling pe­riod .

REAL TIME CONTROL EXPERIMENTS Several real time remote control experiments were performed on the fluidized bed cataly­tic reactor. The reactor is manual ly brought near the desired operating point . Then the control of the reactor is switched over to the computer . To avoid the inherent problem related to the implementation of adaptat ive control algo­rithm of Landau and Lozano ( 1 98 1 ) , the sam­pling period (T) was chosen to be equal to 35 sec . The time delay and the system order have been determined off line from the tran­sient response of the reactor . The obtained values are : k = 3 ; nA = 3 ; nB 2 . The polynomial C2 (q- 1 ) i s chosen to be equal to :

- 1 sz<q ) = 1 - 0 . 6 q- 1 + 0 . 1 2 q-2 - 0 . 008 q-3 The presented algorithm is used with the following initial values

A 1 (o) = A2 (o) = 0 . 95 F (O) = 1 00 . I ( I : unit matrix)

The parameter A 1 (t ) will be adapted such that the trace of the matrix F (t ) will be stay constant ( Irving , 1 979) equal 800. Figures 2 and 3 show the evolution of the reactor temperature and the speed of rota­tion of the ventilator respectively when the

ASCSP-E*

model reference adaptative technique is used . Figure 4 depits the evolut ion of the reactor temperature when the analog PID regulator is used to regulate the f luidized bed reactor temperature . We can see that the model reference adaptati­ve system gives better results , and yields in a better steady state operation of the reac­tor .

_I � Ventilator . ' · . ,�· · .:.. .. " : . : .

. . . , . . _.;;···�· --..· ��- Thermocouple ·· ·-. · . . · · · , .· "'�· . " .

heater Propylene

--+-+-1-'��-I-

Water

Fig. 1 The reactor

o c 500

480

470

1 0 20 30 40 time (mn)

Fig . 2 Response using MRAS

1 26 M. S . Koutchoukali , C . Laguerie and K . Naj im

3000

2000

1 000

RPM

1 0 20

Fig. 3 Speed venti lor vs time

oc 500 490

480

4 70

1 0

30 40 time (mn)

time (mn)

Fig . 4 Response using PID regulator

CONCLUSION The model reference adaptative technique has been successfully employed to regulate the temperature of a catalytic fluidized bed reactor . The experiments results indicate the advantage gained in us ing such algorithm which can adapt to the non-linear behaviour of the reactor.

REFERENCES Barbouteau , G . ( 1 979) . These Docteur-Inge­

nieur INP Toulouse , Ammoxydation cataly­tique du propene en acrylonitrile

Irving , E. ( 1 979) . Improving power network stability and unit stress with adaptative generator contro l . Automatica , Vol . 1 5 , pp. 3 1 -46

Landau , I . D . and Lozano , R . ( 1 98 1 ) . Unifica­tion of di screte time explicit model re­ference adaptative control design . Autornatica , Vol . 1 7 , n°4 , pp . 563-6 1 1

Stergiou, 1 . ( 1 980) . Ammoxidation catalyti­que du propene en lit f luidise . Etude c i­net ique et rnodeli sat ion du reacteur . These Docteur-Ingenieur , INP Toulouse

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ODE METHOD VERSUS MARTINGALE CONVERGENCE THEORY

D. Matko* and J. Tasic**

*Faculty of Electrical Engineering, University Edvard Kardelj in Ljubljana, Trwska 25, 61000 Ljubljana, Yugoslavia

**Institut ]. Stefan, Jamova 39, 61000 Ljubljana, Yugoslavia

Abstract . During the last few years two methods , namely The Ordinary Diffe­rential Equation method , due to Lj ung and Martingale convergence Theory which was used by several authors have been proposed for the convergence analysis of the processes governed by stochastic di fference equations . The idea of the ODE method is to associate the set of stochastic dif ference equations governing the process dynamics with a set of ordinary differential equations whose stabi lity properties contain the necessary information about the asymptotic convergence behaviour of the algorithm. An alternati­ve analysis can be done by means of the Martingale convergence theory , what is a so called "stochastic version of the Lj apunov stability theorem" . Both methods are applicable to the stochastic convergence analysis of recursive parameter estimation methods and adaptive systems . The purpose of this work is to compare both methods with respect to the assumptions needed and to the conclusions obtained . Keywords . Discrete time systems , stochastic convergence analysis .

Introduction . All parameter estimati­on methods can be described using the following generalized algorithm

� (k ) = �(k-l ) +f(k ) !'._ (k ) )e (�J e0 (k) ( 1 ) !:-l (k) =A_(k)!'..-l (k-1) +O'"(k)[t_(k) l (kl] (2 )

where 9 (k ) is an estimate of process parameter vector e , cp (k ) , and X(k ) -0 - -are vectors of process input-output and fi ltered-input-output respectively . e0 is the apostiori error, 0�1(k ) < 2 represents the weight of actual data and O<;; A.(k ) ( 1 is the supression factor for all past data .

ODE method . Following the ODE method (Ljung) eqn s ( l ) and ( 2 ) can be associ­ate with the corresponding differential equations

a'i �(t") = g-1 (TJ E{f(k,�Je0(k,e)} ( 3 )

by introducing of notations

R(k) = � P-l (k)

A't = � k

( 5 )

( 6 )

Hereby � (k , �) , e0 (k , �) I X(k , �) repre­sent the stationary processe13 which would be obtained as ! (k ) , e (k ) and

1 2 7

X(k ) respectively for a fixed constant parameter estimate e . The asymptotic stability of ( 3 ) , (4) implies conver­gence with probability one of ( 1 ) , ( 2 ) and furthermore the stable stationary points of the di fferential equations are the only possible convergence po­ints of the corresponding algorithm . Following Lj ung five assumptions are introduced : 00

2= k = 1 t(k) = co k

f. ( r(k) ) c§<oo k = 1 k

A. (k) = 1 �(k) > 0 stable process

(Al)

(A2 )

(A3) (A4) (AS )

The first and the second assumptions ensure that the fictious time interval AT ( according to difference equations ) tend to zero , but that T can take all values up to infinity . Assumptions 3 and 4 assure the boundedness and inver­tibi li ty of R (k ) and assumption 5 the existence of-stationary processes .

MGCT app:roach. For the analysis with the !1GCT the following function is uti­lized

k V(k) = A�T (k)!'._-l (k) .ti.�(k)+2 L1(i)q (i )p (i )+c2

i=o (7)

1 28 D . Matko and J , Tasi�

with 2 c < oo; 0 < €1 « 1

where q (k ) and p (k ) are de fined by

q (k) =!T (k)A�(k) + l/H (q-1 ) [e (k) -v(k)] ( 8)

p (k) = [H (q-l) - � (l + e 1 )] q (k) (9)

If condition ( 1 3 ) is violated, no­thing can be said about convergence of the parameter error to zero . But via a modified recursion for V (k )

E [V(k) l k-1] � V(k-l ) - O(k) e1(fT (k).O.�(k)] 2 +

+2;r2 (k)fT(k)�(k)f(k)0'/ ( 14)

by strengthening condition ( 10 ) to

(15 ) In the equations e (k ) and v (k ) reure­sent the aposteriori error and the in­put noise respective ly and H (q ) the corresponding fi lter depending on the used method. In order to be ab le to the a . s . convergence of V (k ) to a non-apply !lGCT , V ( k ) should be nonnegative negative finite random variable can be which can be assured bv a nositive re- concluded using MGVT . So , i f p-l (k ) re­alness -londirion on the transfer functi-mains pos itive definite for k�oo the on H ( z ) - 2 ( 1 + € ) • Straightforward a . s . boundedness of the parameter er­application or I 1GCT1yie lds : i f ror A� follows from the definition o f

00

2= k = 1 then

� V(k-1) 2- k (k-l) < oo alrrost surely (a.s) k = 1 � 't(k) ( T J 2

K�l -k- c1 f (k) A �(k) <oo a.s .

( 10)

( 11 )

( 12 )

and V (k ) /k converges a . s . to a finite nonnegative random variable . Due to the a . s . boundedne�s of the sum ( 1 1 ) V (k- l ) /k- 1 ) --0 a . s . and consequently V (k ) /k--0 a . s . The a . s . convergence of the parameter error 49-0 for k--oo can now be concluded from eq . ( 7 ) i f

. -1 lim (� (k ) /k ) > 0 (positive k-oo definite )

( 1 3 )

± . e . if the ODE-assumption 4 as fulfil­led . In case that the process is stable (ODE-assumption 5 ) , condition ( 1 3 ) (ODE­assumption 4 ) is fulfil led , if A (k ) =l ( ODE-assumption 3 ) and i f 0(k ) does not to fast decay to zero (ODE-�ssumption 1 ) , under these conditions � (k ) P (k ) � (k ) tends to zero as l /k . and condition- ( 1 0 ) becomes equivalent to the ODE-assumnti­on 2 for o=2 . These results corresp�nd perfectly to those obtained with the ODE-method . I f in addition e1>o , see ( 8 )­then the a . s . �oundedness of the sum ( 1 2 ) implies cp (k )A9 (k ) -o a . s . for k--oo and as the positive realness of H ( z-1 ) , im?lies s tabi lity of this fi l­ter , e (k ) --V (k ) a . s . for k--oo can be concluded from ( 8 ) . In case that the �rocess is not stable , it is s til l pos­sible to fulfill condition ( 1 0 ) bv a speci fic selection of O'(k ) which must now decay to zero in order to guaran­tee the boundedness of the sum in ( 1 0 ) , in this case , however , in order to guarantee the convergence of the pa­rameter error A9 to zero, the nrocess input signal must now be increased such that condition ( 1 3 ) sti ll holds .

V (k ) in ( 7 ) , even i f condition ( 1 3 ) is violated . The condition ( 1 5 ) is stron­ger than condition ( 1 0 ) in the sense that r(k ) must now always decay to ze­ro in order to guarantee the bounded­ness of V (k ) . Conclusion . The ODE method allows the analysis of stationary 9oints , local and global convergence , but the assum­�tion of process stability is needed in order to introduce stationary pro­cesses and only the convergence of the parameter error to zero can be exami­ned , i f a certain positivity conditi-on is fulfi lled . By means of HCCT the global convergence can be analyzed and under the same assumptions as for the ODE method analoauos conver0ence re­sults are obtain�d ; in addition , by me­ans of the MGCT method the converaence of the a-posteriori-error to whit� no­ise can be shown under a s lightly stron­ger positivity condition . Using MGCT the assumption of process stabi lity can be avoided by an appropriate selec­tion of weighting coefficients 7 ( k ) and even i f the condition for convergence of the parameter error to zero is vio­lated, the boundedness of the parameter estimates can be guaranteed by an ap­propriate se lection of 7(k) . So , using the MGCT , the global convergence results as obtained with the ODE approach , can be extended to a certain degree .

Re ferences . Landau , I . D . , Uear supermartingales for

convergence analysis of recursive iden­ti fication and adaptive control sche­mes . Int . J . Control . Vol . 35 , No . 2 , pp . 1 9 7-2 2 6 ( 1 9 8 2 )

Lj ung, L . ( 1 9 77a ) . On positive real trans­fer functions and the converaence of some recursive schemes . IEEE JTrans . Autom . Control , Vol .AC-2 2 , No . 4 , p:.i . 5 39-5 5 1

•

Ljung , L . ( 1 9 7 7b ) . Analysis of recursive stochastic algorithms . IEEE Tr .Au . C . Vol .AC- 2 2 , No . 4 , pp . 55 1-5 7 5

Solo , V . ( 1 9 79 ) . The convergence of AHL . IEEE Trans . Autom. Control , Vol . AC- 2 4 , No . 6 p� . 9 5 8-9 6 2

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE CONTROL OF NON-LINEAR BACTERIAL GROWTH SYSTEMS

G. Bastin, D. Dochain and M. lnstalle

Laboratoire d'Automatique et d'Analyse des Systemes, Universite de Louvain, Batiment Maxwell, B-1348 Louvain-la-Neuve, Belgium

Abstract . This paper suggests how nonlinear adaptive control of non l inear bacterial growth sys tems could be performed . The process is described by a t ime varying model ( linear in the parameters) obtained us ing usual mat erial balance equations . It does not require any specific analytic description of the bacterial growth rate. The parameters (which have a clear physical meaning) are identified in real t ime us ing a standard RLS algorithm. This parameter estimation algorithm is then combined with a Clarke-Gawthrop controller to obtain an adaptive controller . The special case of biomethanizat ion ( involving anaerobic waste water treatment) is analyzed . Three different control problems are cons idered : depollution control , methane gas product ion control and wash-out contro l . For each of these cases an adapt ive control algorithm is proposed and its effect iveness is shown by stimulation experiments .

Keywords . Model ing o f biochemical process , non linear systems , self tuning adapt ive control .

0 . INTRODUCTION A commonly used approach for the adapt ive control of non linear systems is to consider them as t ime varying l inear systems and to use black-box linear approximate models to implement the control law . This approach has been used by the authors in previous works on the control of fermentat ion proces ses [ 1 , 4] . But , s ince the underlying process is non linear , improved control can be expected by exploiting the non l inear structure of the model . Such an idea is pursued in the present paper : we suggest how non linear adapt ive control can be performed for the control of non linear bacterial growth systems . A similar idea has been recently used for the adaptive dissolved oxygen control in waste water treatment [ 1 2] , but under a somewhat different form than in the present paper. The lay out of the paper is as follows . In section I , the process is described by a non linear state-space t ime varying model obtained from usual mass balance equations . This model does not require any specific analyt ical descript ion of the bacterial growth-rate . In section 2 , the (physical ) parameters are identified in real t ime with a standard RLS algorithm. Then we specialize to the analysis of biomethanization plants ; the parameter est imation algorithm is combined with Clarke-Gawthrop controllers to obtain adapt ive controllers in three different cases : depollution control (section 3 ) ·, methane gas production control (section 4) and wash-out control (s ection S) .

1 29

I . DESCRIPTION OF THE MODEL

We consider the usual state-space representation of bacterial growth systems by mass balance equations

X (t ) (µ ( t ) -U (t ) ) X (t ) S (t ) - k1 µ (t ) X ( t ) + U (t ) (V ( t ) -S ( t ) ) ( l )

Y (t) k2µ (t ) X (t ) where X is the bacterial concentrat ion

S and V are the inner and input substrates

U is the dilution rate µ is the growth rate k 1 and k2 are yield coefficients Y is the product of the react ion .

In order to fac il itate the physical interpretat ion of the later discussions , we shall specializento the biomethanizat ion anaerobic waste treatment proces s[ I] , [ 4] where:

V ( t ) is the input organic load ( i . e . input pollution level ) .

S (t ) is the output pol lution level Y (t ) is a methane gas f low rate .

but , obviously , the discussion can also apply to other biochemical processes with the same structure .

1 30 G. Bastin, D . Dochain and M. Instal le

We could think of adopting an analytical expression for the bacterial growth rate µ (t) - The most popular expression is certainly the Monod law:

0 s Monod: µ (t) = � m

but many other expressions have been suggested like

Blackman: µ (t)

Contois : µ (t)

Haldane : µ (t)

µ (t )=O Sis� 0 s K X + S c

The choice of an appropriate model for µ (t) is far from being an easy task and is the matter of continuing research (e . g . [ 6] : Spriet[ 5] l ists nine different models for µ (t) which have been proposed in the literature , without even mentioning those which involve inhibitions like the Haldane model . Furthermore , i t is wel l known that important identifiabi lity difficulties occur when est imating the parameters (0and Km or Kb or K • • • ) from real life data(e . g . [ � , [ 3l , [ 4] ) . c Therefore we prefer to "short-circuit" the problem of this choice and to identify the parameter µ (t) in real time . Throughout the paper , we shall as sume that :

the dilution rate U (t) is the control input the organic load V(t) is an external measurable disturbance input the output pol lution level S (t) is measurable the output methane gas flowrate y (t) is measurable.

2 . ON LINE PARAMETER ESTIMATION Us ing a first-order Euler approximation for . X (t) and S (t) , with a sampling period T , we have :

(2)

We make the following approximation : (3)

Then, substituting for Xt and Xt+ l from (3) into (2) , we have :

y = t+ l at yt - T Ut y + I (4) t (\+ I 2

st+ I = kt Yt + st + T Ut (Vt-St)+e:t+ l (5) with at I + Tµt

\ k l T -k2

A time varying parameter kt is considered in order to al low for parameter variations "due to unobservable physiological or genetic events" [ 2] •

1 2 e:t+ l and e:t+ l represent errors due to noise , discretization and approximation(3) . Equations (4) and (5) constitute the basic model for the parameter estimation and the adapt ive control . S ince the basic model is l inear in the parameters a and k , recurs ive least-squares estimates cat be re�dily obtained (n 1 J ) :

Rt Yt] (y- l ) (y+Y� pt- l )+Y�Pt- 1 Pt= pt- I [ I -at---------- ]

y (y+at y� Pt-I )

O<y:; l P o > > 0

(6)

(7)

y is the forge�ting factor to allow the tracking of time varying parameters and at (equal to 0 or I ) is a switching coefficient to hold the parameter estimates constant whenever the prediction errors !t-Ytor St-St become smaller than a prespecified bound (secl ! ! ] for the detai ls) . Notice that the estimation of both parameters is decoupled but with a common gain Pt .

3 . DEPOLLUTION CONTROL The aim of the control is to regulate the output pol lution levet S at a prescribed (usually low) level S de�pite the disturbance input Vt , by acting on the dilution rate Ut . Notice that it is nit interesting to reach levels lower than S since they would necessarily correspond to unusefully low treatment rates . A dicrete-time C larke-Gawthrop control leE1 [ 9] , with a dynamic control weight "- C l -z ) in the performance index [ 1 0] , is considered . At each sampl ing time , the control input Ut is computed by minimizing the criterion :

(8)

Non Linear Bacterial Growth Systems 1 3 1

where S is a prediction of the output t+ l pollution level .

A is a free parameter From th� basic model (S) , it is natural to define st+ ! as follows :

st+ ! = kt Yt+ st + TUt (Vt-St )

The non l inear £Ontrol law is readily obtained since st+ l is l inear in ut :

(9)

A block diagram of the closed loop system is presented in fig . I .

4 . METHANE GAS PRODUCTION CONTROL . The biomethanization process can be viewed as an energy conversion process . An amount of " organic!' energy is available in the inf luent under the form of the input organic load V(t) . This energy is converted into methane gas Y (t) by the reactor . Obvious ly, the output energy Y (t) cannot , in the mean , be larger than the available input energy . When the aim of the plant is not depollution but energy production (as in industrial farms) the control obj ective is to continuous ly . adapt the output production Y (t ) to the available input load V ( ti · Therefore , the des ired gas production Y (t) is defined as fol lows :

Y* (t+ l ) = S V (t) - S S > O S > o 0 0

The coefficients S and S have to be selected careful ly by the user si&ce if , by lack of knowledge, S is chosen too large or S too smal l ( i . e . if we require from the fe�mentor more methane gas than it can actual ly provide) then the process can be driven by the controller to a wash-out steady state . [ 7] , i . e . to a state where the bacterial l ife has completely disappeared and where the reactor is definitely stopped .* The control obj ective is ihus to bring Y (t ) to follow the set-point Y (t )=S V (t )-S : it is a kind of load tracking . 0

As in the previous section , a Clarke-Gawthrop control ler is used . Ut is chosen to minimize the criterion :

( I I )

By equation (4) , we have :

and then

A block diagram of the closed-loop is shown in fig . 2 .

5 . PREVENTING A WASH-OUT

An imperative requirement , for any control scheme of continuous biomethanization processes , is to prevent a wash-out of the plant • Wash-out steady-states occur when , for a fixed dilution rate U , the input organic load drops below a critical level and becomes insufficient to maintain the bacterial l ife . In such a case , the only efficient action is to quickly decrease the dilution rate U • A wash-out steady state is characterizea by the following steady-state values :

x 0 y = 0 s = v ( 1 3 )

These express ions suggest that prevention from wash-out can be performed by monitoring (Vt-St ) : if (Vt-St ) becomes smaller than a prespecified bound :

Vt-St C -v-- � t

O<C< l

then the Clarke-Gawthrop control ler is disconnected and replaced by :

O<cx< l

( 1 4 )

( I S )

which ensures a quick decrease of dilution rate ut . Obviously, the Clarke-Gawthrop controll er i s reconnected whenever Vt- St>C Vt .

6 . CONCLUSIONS Simple adaptive control lers for a class of bacterial growth systems have been proposed . Then effectiveness has been demonstrated by s imulat ion experiments which wil l be shown during the presentation at the workshop . An advantage of the non l inear control approach of this paper is that the identified parameters correspond clearly to physical parameters (namely growth rate and yield coefficient) : tnerefore they can provide useful information , in real t ime , on the state of the biomass . Although the model ( ! ) i s well suited for industrial appl icat ions l ike most treatment in sugar industrie� where the organic load VL is acetic acid , in many other appl ications , tfie model ( I ) is only the last stage of a complex multi-stage reaction : a typical s ituation is a f ive-state twelve-parameter

(*) . model (e . g . [ 1 , 4] describing a sequence of Further detail s on wash-out steady-states together with a steady-state analysis and a three reactions ( solubilizat ion, stabi lity analysis of the biomethanization process can be found in ref . [ 7] and [ Bl .

1 32 G . Bas tin , D . Dochain and M. Installe

acidification , methanization) . This is a further reason to explore the possibility of simple control schemes for the different stages of such high-order highly non-linear systems .

8 . REFERENCES [ 1 ] BASTIN ,G . , DOCHAIN D . , HAEST M. ,

INSTALLE M. , OPDENACKER P . ( 1 982) Model ling and adaptive control of a continuous anaerobic f ermentation process . Proc . IFAC Workshop on Modell ing and Control of Biotechnical Proces ses , Helsinki , Finland, August I F l 9 , 1 982 .

[ 2] HOLMBERG A. and RANTA J . ( 1 982) Procedures for Parameter and State Est imat ion of Microbial Growth Process Models . Automatica, Vol . 1 3 , N° 2 , PP • 1 8 1 - 1 9 3 , 1 982 .

( 3] HOLMBERG A. ( 1 982) On the accuracy of estimat ing the parameters of models containing Michaelis-Menten type nonlinearit ies . Proc . IFIP Working Conference on Modell ing and Data Analysis in Biotechnology and Medical Engineering . Univers ity of Ghent , Belgium,August3 1 -September 2 , 1 982 .

( 4] BASTIN G. , DOCHAIN D . , HAEST M . , INSTALLE M. , OPDENACKER P . ( 1 982) Identificat ion and Adapt ive Control of a Biomethanization process . Proc . IFIP Working Conference on Modell ing and Data Analys is in Biotechnology and Medical Engineering -Univers i ty of Ghent Belgium, August 3 1 -September 2 , 1 982 .

( 5] SPRIET J .A . ( 1 982) Modelling of the growth of micro­organisms : a critical appraisal . in "Environmental Systems Analysis and Management" . Rinaldi Ed . , North-Holland Publ . Cy . , 1 982 .

( 6] ROQUES H . , YVE S . , SAIPANICH S . , CAPDEVILLE B . ( 1 982) Is Monod ' s approach adequate for the modelisat ion of purification processes using biological treatment? Water Resources , Vol . 1 6 , pp . 839-847 , 1 982 .

( 7] ANTUNES S . and INSTALLE M. ( 1 982) The use of phase-plane analys is in the modell ing and the control of a biomethanization proces s . Proc . VIIIth IFAC World Congres s , Kyoto , Japan , Vol . XXII , pp . 1 65-1 70 , August 1 98 1 .

[ 8] VAN DEN HEUVEL J . C . and ZOETMEYER R . J . ( 1 982) Stability of the Methane Reactor : A simple model including substrate inhibition and cel l recycle . Process Biochemistry , May-June 1 982 , pp 1 4- 1 9 .

( 9] CLARKE D . W. and GAWTHROP P . J . ( 1 979) Self-Tuning Control . Proc . IEE , 1 26 , N°6 ,pp . 633-640 , June 1 97 9 .

[ 1 0] BELANGER P . R . ( 1 983) On type I Systems and the Clarke-Gawthrop Regulator . Automatica, vol . 1 9 , N° 1 , pp . 9 1 -94 , 1 983 .

[ 1 1 ] GOODWIN G . C . and SIN K . S . ( 1 983) Adaptive Filtering , Prediction and Control . Department of Electrical and Computer Engineering - Newcastle- Austral ia- 1 980 . To be published : Prentice Hal l , 1 983 .

[ 1 2] KO . K . Y . , Mc INNIS B . C . and GOODWIN G . C . ( 1 982) Adaptive control and Identification of the Dissolved Oxygen Process . Automatica, vol . 1 8 . N ° 6 , pp . 727-730 , 1 982 .

( 1 3] GOODWIN G . C . , long R. S . and Mc INNIS B . C . Adaptive Control of bilinear Systems . Technical Report EE80 1 7 , August 1 980 , Univers ity of Newcastle , Australia.

[ 1 4] BASTIN G. Adaptive Control of Bacterial growth Systems . Laboratoroires d ' Automat ique , Universite de Louvain, Belgium, April 1 983 .

APPENDIX CONVERGENCE OF THE DEPOLLUTION CONTROL ALGORITHM.

In case of - minimum variance control ( i . e . A=O)

- constant input pollut ion level (V (t)=V)

- RLS estimation with forgetting factor y= I

The convergence of the depollution control algorithm can be proved . We give only the main steps of the proof . A detailed demonstrat ion can be found in a workpaper{I 4 ] •

STEP ! . BIBO stability of the proces s . As sume that I ) 0 � U (t ) (Umax t'�O

2) The growth-rate µ is a function of S with : µ (s)=O if S=O 0(µ (S) �0 for all S� 0

dµ O<dS <C2<oo for all s� O d 0<C3< E<"' for S=O .

(Notice that the Blackman , Monad and Haldane models presented in section I a1 1 fulfill these conditions) .

Then if O�S (O)�V and O�Y (O) �Ymax S (t) and Y (t ) , t�O , are b9ynded as fol ows :

STEP 2 .

Non Linear Bacterial Growth Systems 1 33

Convergence of the parameter estimation algorithm Assume I ) the parameter

algorithm for with y= l .

�stimation kt (section2)

2) E�+ l in (S) represents the discietization error and : J e: t+ 1 J � ti ; t � 0

otherwise Then lim sup j st-St j � ( l+Y2 P ) ti max o

t + "'

STEP 3 . Convergence o f the control algorithm Assume I ) the depollution cont,rol

algorithm of section 3 with A.= 0

2) V-S (t )> 0 t � 0 3) The contr�l law: A

S - S - ktYt u ( t) = ___ t __ _:;__ 0 T (Vt- St )

u u (t) if o�u ( t ) �u t= o o max Ut=O if U0 (t) < 0

Ut=Umax if Uo (t) >Umax Then : lim sup j st- s* j � ( l +Y2 P ) ti . max o

The proof of Step 2 is similar to that used by Goodwin and al . for the control of bilinear systems [ 1 3] • The proof of Steps I and 3 is established by exploiting the particular structure of the system.

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

SEQUENTIAL DETECTION OF ABRUPT CHANGES IN ARMA MODELS

C. Doncarli and D. Canon

Department of Automatic Control, ENSM 1 , rue de la Noe, 44072 Nantes Cedex, France

Abstrac t . The sequential detection of abrupt changes in ARMA models is an important question in multi-sensor I multi-target tracking or similar pro­blems ( failure detection , E .E . G . analysis . • • ) The authors propose an ori­ginal method based upon a self tuning fil ter which provides a decis ion variable . This variable is then proces sed by a level changes detector .

Keywords . Signal detection ; parameter estimation ; al arm sys tems ; s tochastic sys tems ; Kalman filters .

INTRODUCTION

Segmentation of s ignals into different parts with some stationnary statistic characteris­tics is pos sib le using tests on the order of the stochastic model of these signal s . But the theoretical sequential character is ba­lanced with complex computing for each step . The problem is the same when one try to de­tect the instant and the value of a j ump using Mehra and Peeschon ' s algori thm ( 1 9 7 1 ) , Wil l sky ' s one ( 1 976 , 1 980) or all methods based upon the General ized Likel ikoud Ratio theory .

Reducing the problem to the detection of the ins tant of j ump , one can use a "fil tered de­rivatives" detector or a cumulative-sum test as Shiryaev ( 1 96 1 , 1 963 , 1 965) or Hinkley ( 1 9 7 1 ) . More details can be found in Basse­ville ( 1 98 1 a , 1 98 1 b , 1 982a , 1 982b) .

We propose a sequential algori thm both tes­ting the model and estimating its parameters simul taneous ly . This method is based upon a constant order hypothesi s with abrupt changes within the parameters of an ARMA model . Thi s realistic hypothesis i s a consequence of the experimental capability of a low order ARMA model to represent correctly any Markovian proces s .

THE EXTENDED KALMAN FILTER

The basis of the method is the estimation of the parameters of the model with an Extended Kalman Filter . For thi s , we suppose we are between two success ive j umps . Then the signal can be considered as stationnary .

The process {yk} is then modeli sed with the Eq . ( 1 ) :

1 35

n n yk = E ai yk_; + Ek + E ci Ek_; , ( 1 )

i= I � . i= I �

with {ai , c i , i= I ,n} cons tant and {Ek} an independant gauss ian zero mean sequence with variance R .

Using a state space model , a new represen ta­tion wi 1 1 be :

�+ ! = F . � + G . Ek linear equation , ( 2)

H (�) + Ek non l inear equation , ( I )

[ a j " an I c1"cn l Ek.: l"Ek-J , the Ext en-ded State Vector ,

----�,, _2n n

and

F = [�· � ' o o

o ] · o o l o

We recall that the basic idea of the Exten­ded Kalman Filter is the l inearization of the Eq . ( 1 ) . I f we defin e

�=[yk-] 0 Yk-n l Ek..=1 • Ek-n l 2 i "2n] • (3 )

2 . • Ek . J_ -i (4)

with E and 2 denoting the current estimations of E and c , the l inearized form of the Eq . ( 1 ) i s

(5)

Then a factorized form (Potter) of Kalman' s equations i s applied to ( I ) and (5 ) with a current estimation of R by an exponential receeding hori zon fil ter, given by :

(6 )

1 36 C . Doncarli and D . Canon

DETECTION

Description of the Detector . Using {Ek} Estimation of the sequence {Ekl •�we define a new stochastic process Wk = EkEk- I ' its mean value T (k) with an exponential recee­ding horizon filter ( see Eq . (6 ) ) .

When the filter is assumed to be optima l , {Ek } is an independant sequence and T (k) is quasi o . But , when a change appears , the hypo­thesis of whiteness for {Ek } becomes untrue and T (k) becomes different from zero ; then , the detector is as follows : I . The test is inhibited during N steps to have convergence of the estimation of the pa­rameters . 2 . We compute I T (k) I . I f I T (k) I > A , we suppose a j ump has been detected and we update the Kalman Fi lter before going - on to I . 3 . I f not , the signal {yk} is filtered and IT (k+ I ) I is computed before going on to 2 . (here , we use a very simple detector of change in mean but any other method can be used) .

Updating the Algorithm. When a j ump is detec­ted , it is necessary to inhibite the detector for N steps to as sume a good estimation of the new parameters and to prevent redundant false alarms . A way to update the estimation algori thm is to keep the state vector which seems the best a-priori , but we correct the variance-covariance matrix.

Choice of the Decis ion Parameters : N and A. Those parameters have to satisfy contradictory properties : - N must be small not to forget j ump , but high enough to as sume the convergence of the esti­mation. - A must not be too high to minimize the non detection rate , wi thout increasing the false alarm rate .

The choice of N and A wil l be rather based upon the false alarm rate which is quite easier to reach than the non detection rate . If we suppose { Ek} is gaussian , independan t , zero mean and oI variance I , the distribution of Wk is a symmetric function derived from a Bes-sel ' s one , and one can compute the probabili­ty of false alarm

P F A = I - Prob (-A/a < Wk < A/a)

CONCLUSION

The proposed algorithm allows a sequential detection of j umps for ARMA models parameters , with a volume of operations always comp1tible with a sequential aspect .

S imulations give good results . Parameters are wel l estimed and the delay of detection is very acceptable ( 3 or 5 steps) . Without any false alarm. One can refers to fig . I and fig. 2 for which change occurs for k = 35 1 •

o, �---1e;o---··-uc,---30o-q.60 soo 60' 0 rdo Fig . I . The signal y (k)

l,.____i -• i

Fig . 2 .

--+----------.:..

3Ci 0

The detector J T (k) J alarm for k = 353 .

For references , p lease contact the authors .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

PLENARY SESSION 2

LQG SELF-TUNERS

K. j. Astrom

Department of Automatic Control, Lund Institute of Technology, Box 725, S-220 07 Lund 7, Sweden

Abstract. This paper surveys sel £ - tuning regu lators based on l i near quadrat i c gaussian < LQG ) control design and recurs ive parameter est imation. Only single- input single- output systems are considered. The l inear quadrat i c design method is reviewed. Theoretical issues like convergence, persistent excitat ion and closed loop ident i £ i a b i l i t y are d i scussed. Speci a l attention is given to the robustness issues. An application i s also given

Keywords : Adaptive control ; computer control ; cont rol systpm analysis ; i dent i £ ication ; microprocessors.

1. I NTRODUCTION

Sel£- tuning regulators are based on a very simple heur i s t ic idea. A design problem is £ i rst solved under the assumpt ion that a model 0£ the system and its envi ronment i s known. When the parameters are not known they are replaced by est imates obtained £rom a recursive parameter estimator. Sel£- tuning regu lators where the under lying design scheme i s based on li near quadratic gaussian < LQG ) control t heory are d iscussed in this paper. One advantage by this £ormulation is that the per£ormance 0£ t he control system can be characterized by a £ew parameters. In the sing l e - i nput single- output case there is in £act only two parameters : the sampl ing period and the weight i ng £actor between pen a l t y on the control signal and the output error . Another advantage is that the LQG theory is not rest ricted to any particular class 0£ systems. I t can thus eas i l y be applied to non - m i n i mum phase systems as wel l as to systems w i t h variable t i me delays.

A sel £ - t uner based on LQG was £ i rst proposed by Peterka and AstrOm ( 1 973 > . The solution was based on an interact ive solution 0£ the steady state Riccat i equation. This idea was £urther elaborated by AstrOm ( 1 974 ) , AstrOm and W i t t enmark < 1 975 > , Gustaveson C 1 980 J , Samson < 1 980 J , Belanger < 1 981 > , and Zhao - Ying and AstrOm < 1 981 > . C losely related approaches have been proposed by 'osca and Zappa < 1980 J , Menga and

1 37

Mosca < 1 980 > , ( 1 982 > .

Trul sson and Lj ung

There are comparatively £ew t heoret ical results on LQG sel£-tuners. The c losed loop ident i£ iabi l it y problems are d i scussed by Kumar < 1 981 > . A convergence proo£ is due to Moore < 1 983 > . Applicat ions 0£ LQG sel £ ­tuners have been desc r i bed in AstrOm < 1 980c > and AstrOm and Zhao - Y ing ( 1 982 ) .

This paper is organ i zed as £ollows. The LQG design £or system with known parameters i s reviewed i n Section 2. A polynomi a l approach is convenient because the treat ment is l imited to s i ngle - i nput single-out put systems. The polynomi a l approach is a l so use£ul because it is based on a model struct ure which i s suitable £or parameter estimation. Recursive estimat ion is d iscussed in Sect ion 3. The resu l t s of Sections 2 and 3 are combi ned i n Section 4 which deal s w i t h L Q G sel £ - tun ing algorithms. Theoretical issues are d iscussed in Section 5 . Robustness i s d iscussed i n Sect ion 6, where new devices to ensure robustness are proposed. An application to concen�r a t i on control i s g i ven i n Section 7 . This i l lustrates some 0£ t he advantages 0£ the LQG approac h .

2 . LQG DES I GN

The LQG design £or systems w i t h known parameters is reviewed i n t h i s

1 38 K. J . Astrom

Section. This material is wel l - known i n text Qooks. See e . g . Astrom ( 1 970 ) or Astrom and Wittenmark < 1 984 > . The problem can be formu lated either in terms of state models or input - output models. The input - output formulation i s convenient for our purposes. Con ­sider a singl e - i nput single-output system described by the model

A < q > y < t > = B < q > u < t > + C < q > e < t > ( 2. 1 )

where q is the forward shift operator.

Let the criterion be to m in i mize

1 t 2 2 E Hlll t: k� 1 c Y < k > + e u < k > l < 2 . 2 >

Not ice that the samp l i ng per iod h i s a hidden parameter in this formu­lation. A better formulation woul d be to use the cont inuous t i me criterion.

1 tf 2 • 2 E lim - C y < s > + e u < s > l ds < 2 . 2 ' > t:-+"' t 0

which is independent of the sampl i ng period. For a sampled system the criterion < 2 . 2 ' > can be t ransformed to < 2 . 2 > . The parameter e w i l l , however, depend o n t h e sampl ing period h. There w i l l also be a cross term y < k > u < k > in < 2 . 2 > . See Astrom ( 1 970 ) . We w i l l however use the simpli fied formulati on < 2 . 2 > .

The optimal feedback law which mini mizes < 2 . 2 > for ( 2 . 1 > i s gi ven by the fol lowing theorem

THEOREM 1

Consider the system < 2 . 1 > . Let the pol ynomials A < z > and C < z > have degrees n . Assume that C < z > h a s a l l its zeros i n s i d e t h e u n i t disc and assume that there is no polynomial which di vides A < z > , B < z > and C < z > . Let A2 < z > be the greatest common divisor of A < z > and B < z > and let A2 < z > of degree m, be the factor of A2 < z > which has a l l its zeros outside tne unit disc or on t he unit circle. The admissible control law which minim izes < 2 . 2 > with e >O i s then gi ven by

R < q > u < t > = - S < q > y < t > + T < q > u < t > c ( 2. 3 )

where the polynomia l s R and S sat isfy the Diophantine equat ion

A < z > R < z > + B < z > S < z > = P < z > C < z > ( 2. 4 )

with the addit ional constraints

( i ) deg R < z > = deg S < z > n + m

( i i ) A2 < z > d iv ides R < z >

* ( i i i > deg S < z > < n

The polynomial P < z > is given by

where

and

A 1 < z >

B 1 < z >

A < z > I A2 < z >

B < z > I A2 < z >

The polynomi a l T < z > is g i ven by

T < z > = t 0

where

m z C < z >

( 2 . 5 )

( 2 . 6 )

< 2. 7 )

c

A proof of Theorem 1 is g i ven i n Astrom a n d W i t t enmark < 1 984 > .

Algorithms

To solve the design prob lem it is necessary to solve t he spect ral factorizat ion problem < 2. 6 > and to solve the D iophant i ve equation < 2. 4 > . The spect ral factor izat ion problem can be solved by the algorithm due to W i lson < 1 969 > which has been retired by Vestry < 1 975 > . This a lgorithm i s i terat ive a n d h a s the advantage that the polynom i a l P, obtained at each step is guaranteed to be stable. The Diophantine equation can be solved by the Eucl idean algor ithm. This algorithm has t he advantage that it also detects common factors i n the polynomi a l s A and B. A neat implement ation at the Euc l i dean algorithm i s given by Blankinsh i p < 1963 > . Add it ional material on a lgorithms for solving t he spec t r a l factorizat ion problem and the D iophantine equat ion are g iven i n Kucera < 1 979 > .

Al ternat ive approaches

An a l terna t i v e solution to t he design problem is to use the state space formu l at io n . The control law is then obtained i n t erms of solutions of R iccati equat ions for the feed back g a i n and the K a l man f i ler. This

LQG Self-Tuners 1 39

approach is described in det a i l i n t h e t e x t books by Anderson a n d Moore < 1 971 > and AstrOm < 1970 > .

Relat ions to pole placement

The solution to the LQG problem gi ven by Theorem 1 has close relat ions to the pole p lacement design problems. The solution to the spectral factorization problem g ives the c losed loop poles. The second step in the algorithm can be i nterpreted as a pole placement problem.

It is clear from t he description of the design algor ithm that a pole placement self- tuner i s obta ined as a by - product, simply by specifying the polynomial P instead of determ i n i ng it from spectral factorization.

3 . PARAMETER EST I M A T I O N

When t h e parameters of the model C 2 . 1 > are not known they can be

estimated recurs ively by the extended least squares < ELS > method or by recursive maximum l ikel ihood C RML > . A detailed descr ipt ion of these methods are gi ven in Lj ung and SOderstrOm C l 983 > . For simplicity we w i l l here describe the ELS algorithm. I nt roduce

T e = C a1 . . . an b 1 . . . bm c 1 . . . c .l l

� C t > = c - y c t - 1 > - y c t - n >

and

e: C t + l )

u C t - 1 >

e: C t - 1 >

u < t - m >

e: < t - .t > J T

y c t + l > - �T c t + l > e c t > .

The extended least squares a lgor ithm Panuska < 1969 ) is given by

0 < t + l ) 0 C t > + K C t + l > e: < t + l )

P C t > cp < t + l > K C t + l >

1 + �T C t + l > P < t > � ( t + l )

P C t + l > = ! [P < t > - P C t > � C t + l > cp < t + l > TP < t > )

X X + � C t + l > TP C t > � < t + l ) ( 3. 1 )

I n actual implementation a square root al gorithm based on the U - D algorithm is preferable. See Bierman C 1 977 l . The number X i n C 3 . 1 > i s a forgetting factor introduced to d i s ­count past data. A better way o f d iscounting old d a t a i s gi ven b y H�gglund C l983 > .

4 . SELF -TUN I N G CONTROL

A block-diagram of a general self­tuner i s shown in F i g . 1. It can be v i ewed as an on - l ine automated design.

Design Estimator

Fig. 1 . - Block d i agram o f a sel f ­tuner .

In the LQG sel f - t uner the design is a solution to the LQG design problem and the parameter est imator i s a recursive est i mator l ike the one d i scussed in Section 3 .

Solut ion of t h e R i ccat i equat ion o r t h e spectral factorization is the major computation i n an LQG s e l f - tuner . This calculation can be made in several different ways. The Riccat i equat ion can be solved by the e i genvalue method due to Potter C l 966 l or by some i t erative method,

ordinary time iterations or by the method proposed by K l e inman < 1 968 > . The i terat ive methods w i l l in general l ead to shorter code. It i s however difficult to cope w i t h iterations in an o n - l i ne algori t h m . To guarantee t hat the calculat ions can be performed in a prescribed samp l i ng period it is necessary to truncate the iterations. It is then a necessity to guarantee that a sensible iterate is obt a i ned when the i teration is truncated. The part icular method based on spectral factorization using W i lson ' s a lgor i thm which i s d iscussed in Sect ion 2 has the advantage that the polynomial P obtained at each iter­at ion i s stable.

Programming and coding

Several d i fferent versions of the LQG sel f - t uner have been coded i n Pascal for the DEC LSI 1 1 /03. An imp lemen tat i on of the algorithms descri bed i n Sections 2 and 3 are described by AstrOm and Zhao- ying C 1 982 > . The source code i s about 1 400 l ines of Pasc a l . This includes comments and declarations. The total s i ze of the compiled code i s about 40 kbytes. In the coding f lexibi l i t y and readabi l i t y has been emphasized rather than compactness and computational speed.

Another implementation based on the solution of R i ccati equat i on was g i ven i n AstrOm < 1 974 > . I n t h i s code there was no operator communication.

1 40 K. J . Astrom

The pure foreground code compi led to about 8 kbytes on a DEC PDP - 1 5 . Another program for the DEC LS I 1 1 / 03 written in Fortran by GW1tavsson ( 1 980 > has a source code of about 1 400 l ines. Half of t hem are comments. The compiled code required about 40 kbytes. O f these about 8 kbytes was requi red for the pure foreground.

It thus appears that implementat i ons based on Riccati equat ions and polynomial man i pulat ions requi re about the same amount of code. The minimum s i ze of a dedicated implement ation with no operator communicat ion is about 8 kbytes.

5 . THEORETICAL ISSUES

The key theoretical problems for adapt ive control are stab i l i t y , con­vergence and performance. These quest ions are reasonably well under­stood for the s imple self - t uners based on least squares estimation and min imum variance cont rol, where cond i t i ons for g lobal and l ocal convergence are known. Much less is known about the algorithm d iscussed in this paper. Some ava i l able resul t s w i l l b e d iscussed in t h i s Section.

Parameter Est i mation

Est imation of the parameters i n the model ( 2. 1 > i s comp l icated even i n the off - l ine case, because i t i s non - l inear in the parameters. See AstrOm and Boh l i n < 1 965 > . When there are no i nput s ignals i t i s shown in AstrOm and SOderstrOm ( 1974 > that the asymptotic likel ihood funct i on ob­ta ined for large data sets i s unimoda l . The l i ke l i hood funct ion may, however, have local minima when there are inputs even if they are persistent l y exciting.

The difficult ies associated w i t h local m inima of the l ikel ihood funct ion are also inheri t ed by the recursive algorithms ELS and RML. It fol lows from the general resul t s on convergence of recursive algorithms that they may have the local minima as equ i l ibrium points. See Solo < 1979 > and Lj ung and SOderstrOm < 1 983 ) . The new recursive a lgorithm proposed by AstrOm and Mayne < 1982 > may perhaps avoid t h i s d i f f i c u l t y .

Persistent Exci ta t i on

To ensure a u n ique m i nimum i t i s necessary t ha t the i n p u t s i g n a l i s persistent l y exci t ing. S e e AstrOm a n d B o h l i n < 1 965 > . This i s d i f f i cu l t t o

ensure when t h e input signal i s generated by feedback unless extra perturbations are introduced or parameters are updated only �hen there i s proper excitation. These issues are d iscussed further in Section 6 when i t i s mentioned that excitat ion should be obtained by signals i n a certa i n frequency range. Compare a l so the notion of dominant excitation i n I oannou and Kokotovic ( 1 983 > .

Ident ification in c losed loop

I dent i f i a b i l i t y may a l so be lost because identification i s made in closed loop. See AstrOm and Eykhoff < 1 971 > and L j ung et a l < 1974 > . Kumar < 1981 > has shown that serious difficult ies may arise at least in the case when the parameters belong to a finite set . For a f irst order system it i s shown t hat t here i s an equ i l ibrium set for the parameters, for the LQG self- tuner, where only one point correspond to the optimal solution. This i s d i f ferent from the minimum variance sel f - tuner where all points i n the equ i l ibrium set give the optimal feedback. I t i s not clear what happens i n the general case. My conj ecture i s that the parameters may converge to the l im i t set at t he rate of 1 / t , and t hen move towards the correct solut ion a rate of 1 / ( log t > . This i s hard to analyse because t he phenomena can not be captured by t he ordinary d ifferent ial equat ions g iven by Lj ung ( 1 977 > .

Kumar has suggested a mod i f i cation of the least squares criterion, which g i ves a feedback law that converges to the true LQG solution even when the parameters are i n a d i screte set.

Results for parameters ( 1 983 ) .

the case of d i screte are a l so g iven by H i j ab

There are a l so other significant d i fferences between the case of finite and cont inuous parameter sets.

Loss of ident i f i a b i l i t y due to feedback can a lso be reduced by int roduci ng an add i t ional delay i n t h e regulator o r by introducing perturbat i on signals. Moore < 1 983 > has an algorithm which is c l a i med to converge g loba l l y .

D i rect algo r i thms

The a lgorithm given in Sect ion 4 is indirect because i t i s based on estimat ion of parameters of a process mod e l . For s i mp l er adapt i v e schemes there d irect algori t hms where the

LQG Self-Tuners 1 4 1

regul ator param•ters are updated d�rec t l y . These algorithms can _ be obtained from a reparameterizat ion of the process mode l . See e . g . Astr6m C 1980a l . Several attempts have been made to derive d irect algorithms for LQG sel f - t uners. Trulsson and L j ung < 1 982 > have suggested to use the gradient approach suggested by Tsypki n < 1 971 > . The algorithms obtained i n this way have the same complexity as the algorithms in Sect ion 4.

Another approach cal led MUSMAR i s suggested by Menga a n d Mosca ( 1 980 > and Mosca and Zappa C 1 980 l . Their d irect algori t hm reduces t o two least squares calculations. A further s i mpl i ficat ion of this scheme is proposed by Bar t o l i n i et a l < 1982 > .

6 . ROBUSTNESS

A control system should be insen­s i t i ve to measurement errors, di sturbances and model ing errors. Al though these issues are important for all control systems t hey have only lately been considered in connect ion with adapt ive systems. See Rohrs et al < 1982 > and I oannu and Koktovi c ( 1 983 > .

I A discussion of the central problems i for the adapt ive LQG regulator are g iven in this Section. The robustness of the underl ying design problem, the recursive est i mator and the combined problem are d iscussed.

Robust Control Design

Robustness propert ies are convenient ­ly d i scussed in terms of the loop gain. See the Bbode plot of a t ypical loop gain i n F i g . 2 . T h e loop g a i n i s u n i t y at the cross- over freq4ency � • A common engineering practi ce whi8h is now wel l supported by theory Horowi t z < 1963 > , Doyle and Stein < 1981 > and Lehtomahi et a l < 1981 > boils down to the fol lowing : Make the l oop gain h i gh below the cross- over frequency and make sure that the loop

Fig. 2. - Bode d iagram of the loop gain.

gain falls off rapidly above t he cross-over frequency . A high loop gain for l ow frequencies i s obtained by i nt roduci ng integral action or some resonant system which gives a h igh gain for special frequency bounds as i s indicated i n Fig. 2. The rapid rol l - of f for high frequencies i s necessary to ensure that unmodeled high frequency dynamics w i l l not cause difficulties. Computer control led systems should a lways be provided with antial iasing f i l ters to el iminate signal t ransmission above the Nyquist frequency . A wel l designed digital regulator w i l l thus not have any · signal t ransmission above the Nyquist frequency. The high frequency roll �off for a digital regulator is thus significant l y influenced b y t h e sampl i ng period.

A quantitative statement of the above d i scussion for the LQG design can be obtained as fol lows. Assume that a LQG regulator based on the model < 2. l l i s designed for a system with the t rue pulse transfer func t i on G . The following result then holds. 0

THEOREM 2

Consider a system with the pulse transfer function G • Let a regulator < 2 . 3 > be designe8 based on the approxi mate model < 2. l l . Assume that G and G = B / A have the same number of uRstable poles. The c losed loop system obt a ined i s the stable if

I Go -G I < I � I · I § I < 6• 1 l m

on the unit circle and at infinity c

The theorem is proven in Astr6m < 1980d l . The left - hand s ide i s the relative error i n the pulse transfer function. The r ight - hand s i de contains quant i t ies which can be computed when the design calculation have been performed. Notice that G i s t he open loop p u l s e t ransfer func t i on of the plant model and t hat Gm is the pulse t ransfer funct ion from the command signal to the output .

The detailed character of the i nequal ity ( 6 . l l i s highly problem dependent . Some general character istics can, however, be found by inspec t i o n . The r ight hand side of < 6. 1 > i s small when G i s less than G i . e . when the open loop gain i s lesm than the model gain. This i s the c a s e f o r frequencies around t h e cross- over frequency. Theorem 2 thus i ndicates that i t i s necessary to have a model which g i ves an accurate descript ion of the process around the cross- over frequency .

1 4 2 K. J . Astrom

Robust est i mation

When a parameter estimator i s used i n an adap t i ve scheme like the o n e shown i n Fig. 1 . it is important to make sure that good est imates are obtained. Bad data should not gene­rate poor estimates.

A special £eature 0£ t he adapt i ve control applicat ion is that a low order model is £ it ted £or a complex p lant . The mode l obta ined in such a case w i l l critically depend on the £requency content 0£ the input signa l . It was e . g . shown by Manner£elt < 1 981 > that with pure s i nusoidal exci tation the t rans£er £unct ion w i l l agree exac t l y with the p lant trans£er £unction at the excitat ion £requency . To guarantee a stable operation it £ollows £rom Theorem l that a certain precision 0£ the model i s needed around t he cross-over £requency . To ensure t h i s it is there£ore necessary t h a t t h e input signal h a s a su££ icient energy content in that £requency band. This can be monitored using the system shown in Fig. 3. The condi t ions £or persistent excitat ion, AstrOm and Bohlin < 1 965 > can be monitured instead 0£ the signal energies as shown in F i g . 3.

I £ the use£ul signal to noise ratio i s to low there are t wo opt ions : Excitat ion signals may be introduced or the parameter est imat ion may be switched 0££. Guided by the results 0£ Egardt < 1 979 > and Narendra and Peterson ( 1 980 > it is also reasonable to est i mate ouly when the absolute level 0£ the use£ul input energy is above a certain leve l . These sa£e - guards can be regarded as an i mp lement a t i on 0£ t he common sense rule : Do not estimate unless the data is good.

There are other sa£e- guards 0£ a similar nature to make sure that the data used £or est i mation i s a l ways good by excitation or that the parameter est i mation i s only made when the data is reasonable. The di££iculties not i ced by Rohrs and others ( 1982 > w i l l not arise i£ those parameters are t aken.

u

Band pass fil ter

Complementary filter

Rectifier

Rectifier

Fig. 3. - C i rc u i t £or monitoring the s ignal t o noise ratio £or est i mat ing a reduced order model

Robust adapt ive control

To obta i n a robust adapt ive control algorithm i t i s necessary to use both robust control and robust est imat ion. I n the adapt i ve problem there are a l so some new t rade-o££s to be made. Consider £or example the robustness propert ies obtained by having a high open loop gain at low £requencies. This may be obtained by having i ntegral action in the cont rol loop. I t can a lso be obtained v i a adaptat ion. An adapt ive regulator w i t h enough parameters w i l l automa­t ic a l l y introduce a h igh gain at those £requencies where there are low £requency d isturbances.

I have o£ten £ound it bene£ icial to use a design method which g ives a high gain at low £requencies and use adaption only to £ind the characteristics around the cross- over £requency. This has the addit ional advantage t hat £ewer parameters are needed . It speeds up the est i mat ion, and the degrees 0£ the polynomi a l s a r e k e p t low w h i c h improves t h e inherent numerical problems w i t h polynomial representation. One poss i b i l i t y i s to est imate a model 0£ the t ype

A < q > Vy < t > = B < q > Vu < t - 1 > + C < q > e < t > ( 6 . 2 )

where V = q - 1 is a d i££erence operator. Provided that C < l > + l t h i s mode l impl i es t hat t here a r e dri£t i ng d isturbances. A consequence 0£ t h i s i s t h a t the regulator designed £rom ( 6. 2 ) w i l l a lways have a h igh gain at low £requencies. See AstrOm < 1 982 > .

7 . AN APPLICATION

Some pract ical aspects on the LQG - tuner w i l l be given i n t h i s sect ion. S ince the LQG sel £ - tuner is more complicated t han the s i mple sel£ - t uners based on least squares and m i n i mum variance control it is l eg i t imate to quest ion t he bene£i t s 0£ the increased complexity.

Trading input and output variances

A l t hough the output variance i s o£ten 0£ maj or i nterest i n process control app l i cat i ons it i s a l so important to make sure that t he variance 0£ the control variable i s reasonable. For the simple sel £ - tuner the t rade -0££ between input and output variances i s governed by t he select ion 0£ t h e samp l i ng period h a n d the predict ion horizon d. It i s thus necessary to

LQG Self-Tuners 1 43

use t w o design parameters. I t has been demonstrated by Toivonen < 1 981 > that there are l imitat ions in t h i s approach.

Clarke and Gawthrop ( 1975, 1 979 > have proposed another way to deal with the problem. They use a c r it er ion of the type < 2. 2 > where the sum has one t erm only. This captures part of the problem. Because of the short t ime horizon there is however no guarantee that a stable System i s obtained. See Moden and SOderstrOm < 1982 ) .

The LQG self- tuner does not suffer from any of the. drawbacks d iscussed above because it i s based on an infinite hori zon solution to the LQG problem. The LQG sel f - t uner is of course a l so wel l suited for those rare problems where t here i s a natural LQG formulation. See AstrOm < 1 980c > .

Non minimum -phase plants

The simple self- tuner can not be applied to a plant where the sampled model i s not m i n i mu m - phase, because the design i s based on cancel lat ion of the process zeros. This may seem restr i c t i ve at first. It is, however, shown in AstrOm et al < 1 983 > that any stable plant which i s sampled with a sufficient ly l ong period w i l l result i n a sampled model which i s m in i mum phase . The s i mple sel f - t uner can thus always be used for stable systems if the samp l i ng period is long enough. The LQG self- tuner has no difficult ies w i t h non - m in i mum phase systems because the underlying design method i s not based on cancellat ion of process zeros.

Time delays

For the simple sel f - t uner it is ne ­cessary to know an upper bound of the process dead - t i me . This i s not needed for the LQG sel f - t uner. The problem i s c ircumvented simply by having a large number of b - parameters in the mode l . I t does not matter in the design i f the leading b - parameters are zero. The LQG sel f - t uner thus has significant advantages i f there are significant var i a t i ons in the t ime-delay.

An Example

The proper ties of the LQG sel f - tuner are i l lustrated by an application to concentrat ion contr o l .

T a p water f lows t hrough a chamber where it i s m i xed

m ixing with a

·concent rated salt solut ion. The flow rate of the salt solution is control led by a pump. The water then f l ows through a long tube and a stirred tank. The concentration at the outlet is measured with a conduct ivity ce l l . The out let flow may also be recirculated to the i nput . The amount of recirculat ion can be adj usted . The control variable is the speed of the pump. The controlled variable is the concentration at the out let. The dynamics varies with the f low rate because the t i me-delay and the t ime- constants are inversely propor ­t ional to the f l ow rate. The process gain is directly proport ional to the concent ration of the salt solut ion and i nversely proportional to the f l ow.

The i mpulse responses of the system at di fferent flow rates are shown in Fig. 4 . The figure shows clearly that there is a substant ial variat ion of t he dynamics with the flow rate. Notice in particular the response obtai ned with rec irculation.

s o 100 �-J-L>.�L_'o��-4 4�x�10�-6-m_1_s........-O SO 100 50 100

Fig. 4 . - I mpulse responses o f the process for di fferent flow rates.

The process approxi mat ively model

dynamics descri bed

can by

be the

y < t > + ay < t - l > = bdu < t - d l + bd + l u < t - d - 1 >

where the samp l i ng period i s chosen as the t ime unit and the integer d is such that the t i me - delay i s between dh and d h + h .

Si nce the number d i s n o t known a priori the fol lowing model i s used

y C t l + a1 y c t - l > =b 1 u < t - l l + . • . + br u C t - r >

where r > d is determined from an est i mate of the largest t ime -delay. Experience showed that the uncertainty in t he parameter est i ­mates increase with r . The actual numbers depend c r i t i c a l l y on the character of the input �igna l .

F i g . 5 shows t h a t a regulator constant parameters will not

w i t h work

1 44 K. J . Astrom

well if there are large flow change!6 T�e f low is f irst set to 1 4x 1 0 m I s . The process then h a s a t i me constant of 13 s and a t ime delay of 1 7 s. The curves labeled fi xed gain

show resul t s when the sel f - t uner is run for about 30 sampl ing periods. The regulator parameters are then f i xed and t he flow i s changed. I t is seen from the Fig. 5 that the regulator behaves wel l -ihe� the f low is increased to 22x10 m / s� 6 W�en the flow i s decreased to 1 0 x 1 0 m / s t h e damping decreases. The control loop becomes unstable when �ge �low i s further decreased to 8x10 m / s . The resu lts are natural because the time- delay and the t ime constants increase with decreasing flow. When the f low is sufficient ly sma l l the t i me delay is so large that the system becomes unstable.

Results from experiments with an LQG self- tuner are shown i n the curves

i :�I i ]�I i'.EMo�ptive �=:=------,--------1 "' Adaptive j � 8 o+---,-------��

30

� �� - �----�----0 100 200 Time

Fig. 5. - Results of experiments w i t h varying flow. Resu l t s w i t h a fi xed gain regulator is shown in the curves labeled fixed gain. Results w i t h an LQG self- tuner i s shown i n the curves labeled adapt ive.

labeled adapt ive i n Fig. 5 . The f igure shows clearly that the self - t urner can eas i l y cope with the parameter variat ions. The parameters used i n the self- tuner are � = 5 and A = 0 . 98. The sampl ing period i s 15s.

Fig. 5 shows that the sel f - tuner has considerably better performance than a constant gain regulator. I t i s of course possible to make such a

sel f - t uner unstable by decreasing the f low rate further.

8 . REFERENCES

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Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE CONTROLLERS FOR DISCRETE-TIME SYSTEMS WITH ARBITRARY ZEROS. A SURVEY

I. D. Landau, M. M'S"aad* and R. Ortega**

Laboratoire d'Automatique de Grenoble (CNRS) ENS/EC, B.P. 46, 38402 Saint Martin d'Heres, France

Abstract . The paper presents a comprehensive review of the problem of des igning discre te­time adaptive controllers for sys tems with arbitrary zeros (minimum or non-minimum phase systems being part icular cases ) . The authors have Rttempted to give an unified presentation of the various exis ting schemes by s tripping away the technical details and highlighting their c lose interrelation through the use of common framework . Both algori thm_ic and theoretical aspects are dis cussed .

·

Keywords Adaptive control lers ; Contro l algorithms ; Pole placement ; self-tuning control­lers ; S tabi li ty .

INTRODUCTION

The des ign of mos t dis crete-time adaptive control lers reported in the literature is based on a sys tem representation obtained by sampling the output of the process , e . g its pulse transfer function. I t is wel l known (K . J . As trom, P . Hagander , J . S ternby 1 980) that for such a representation it is more a rule than an except ion to have uns­table zeros , independently of the process nature . Many adaptive des ign techniques rely on the cancel lation of the sys tem ze­ros, and therefore these techniques can be applied only for sys tems with s table zeros (s tably invertible sys tems ) . The theory for this type of adaptive schemes is wel l deve­lopped . Adaptive schemes avoiding zeros cancellation, which are app l icab le for the general case , lead to a bilinear-in-the­parameters equation for the adaptation error (K . J . As trom, B . Wit tenmark , 1 980) hampering the application of s tandard of stab il i ty and est imation techniques .

An important research effort has been made in the las t few years in order to develop adaptive control schemes for sys tems having arbitrary zeros and to provide s tability proof for these schemes under reasonable assumptions . The purpose of this paper is to provide a review of the s tate of the art in this important domain of adaptive con­trol theory . One of our main objec tives was to es tabl ish the interrelation ship be tween the various schemes . We have s tated diffe-*

**

The work of M . M' SAAD is partly supported by Eco le Mohammadia d ' Ingenieurs - Rabat (}1aroc) R . ORTEGA ' S s tay at the LAG is sponsored

by the National University of Mexico .

1 4 7

rent contro l objec tives under a unifying framework and inspired by the original ideas of the various authors we have solved the posed problems : The approaches so related are not the only available one s , however their generic qualities seem to spain the spectrum of all jus ti fiab le s trategies .

We s tart wi th the presentation of the known parameter case in section 2 . The control objectives are properly grouped and a general contro ller s tructure , allowing us to achieve all the reported ones is introduced . The adaptive solutions for the case of unknown parameters are classi fied into direct and indirect s chemes and the corresponding for­mulations are given in sec tion 3 . Particu­lar emphasis is placed on the aspects cons­ti tuting the stumbling-bloc to rigorously solve the problem . In section 4 , the avai­lable s tabil ity resul ts are reviewed . Con� c lus ions are drawn in section 5.

KNOWN PARAMETER CASE

Given that all the adaptive regulator struc­tures that we wi l l consider in this survey are motivated by an underlying known para­meter counterpart we present firs t the pro­blem formulation and i ts solution for this case . In fac t this is the starting assump­tion for developping adaptive s cheme using the ad-hoc certainty equivalence principle .

Problem Formulation

We wil l deal with linear sys tem characteri­zed by the pul se transfer function repre­sentation

- I -d - I A(q ) y (k ) = q B (q ) u (k ) ; d > 0 *

( 2 . I )

When clear from the context the argument wi l l be omit ted

*

1 48 I . D . Landau, M. M ' Saad and R . Ortega - l where A , B E R [ q ] (The f ield o f polynomial

functions in the backward shif t operator q- l ) ; A is monic and A and B are coprime . It is wel l known (K . J . As trom, B. Wittenmark , 1 980 ) that i t is not possible with a linear controller to arb itrari ly p lace the zeros without open-loop zeros cancella­tion hence limiting the app lication of mo­del reference/minimum variance regulators to stably invertible systems ( I . D . Landau , 1 98 1 ) . Therefore the design obj ectives to treat sys tems with arbitrary zeros mus t be modified . They wil l be grouped into three broad classes termed : f i l tered sequence tracking (FST) , pole placement (PP ) and expl icit criterion minimization (E . C .M) FST The FST objectiw• is defined by; (CR B + AQ) y (k) - CRB yM(k) = 0 2 . 2 . a)

- I M Where CR, Q E R [ q ] and {y (k) } is the output of reference model described by :

M -d ...-- M � y (k) = q BM u (k) (2 . 2 .b ) The name FST becomes evident from (2 . 2 .a) and ( 2 . 2 . b) . The scheme attaining such an obj ec tive may be interpreted in two ways . l) as a one-s tep-ahead optimal control ler mi­nimizing with respect to u (k) the cri ter ion

J (u (k) ) 1 M 2 = z { [ CR (y (k) - y (k) ) ] + (Q ' u (k-d ) ) 2 } ( 2 . 3 )

where Q ' is equal up to a scalar fac tor to Q '2) as an al l zero cancel ling dedbeat control­ler for the sys tem augmented with seri.es

-d (CR) and feedforward compensator {q Q) . See (R. Kumar , J . B . Moore , 1 982) , (R. Ortega M.M' Saad , 1 983 ) and (dotted part of f ig . 2 ) PP . Here we seek only to arbi trarily ass ign the closed-loop poles leaving the open loop zeros unal tered , hence our objective beco­mes

M CR (y (k) - B y (k) ) = 0 (2 . 4 . ) I n this case Q i s unsubs tantial and there­fore we take it equal to zero .

E . C .M. The problem of zero cancel lation intrinsec to the ful l order linear controller (G . C . Goodwin, P . J Ramadge, 1 980) may b e avni.ded by fb;ing .a prio.ri the controller structure and prusuing an obj ective of s ingle s tage minimization in the contro l ler parameter space . (L . Ljung, E . Trulsson, 1 978 ) , (L . Ljung , E . Trulsson, 1 9 82 ) , e . g .

Min J (8c ) ec

(2 . 5 )

Where ec is : the regulator parameter vector . Des igti Procedure In order to s imul taneously s tudy all the objectives described above we wil l consider the following general controller s tructure (see f ig . I )

M ( .+Q) u (k) =-R y (k) + CR y (k+d) ( 2 . 6 ) with S , R E R (q- 1 ) , whose degree choice will be made according to the context .

Fig . I . In closed loop we have

CRB YM(k) y (k) -d ( 2 . 7 )

A(S+Q ) + q BR

u (k) CRA

yM(k) (2 . 8 ) A (S+Q) + q- I BR

FST Design

The objective ( 2 . 2 . a) is reli zed iff CRB = AS = q-d BR (2 . 9 )

holds . From ( 2 . 9 ) the following assumption is in order. A l ) CRB + AQ is s table Also i t is clear that the choice Q=O is ex­cluded for the case of arbi trary zeros Notice that the s tatic error may be cancel­led (even for sys tem type l e s s than one) by incorporating an integrator in Q .

I n fig . 2 . the augmented sys tem reoresenta­t ion is depicted . Notice that ( 2 . 2 ) is satis­fied i f the control ler polynomials verify . B = AS + q-d B R (2 . I 0) where B is the zero polynomial of the aug­mented sys tem defined as B = CR B + AQ ( 2 . I I )

hence under assumption A l ) no uns table zero cance llation occurs .

fig . 2 PP Des ign

In this case one has to satisfy CR = AS + q -dB R (2 . 8 ) becomes : u (k) = A yM(k+d )

(2 . 1 2 )

and a bounded control signal i s insured Notice that to avoid a s tatic erro� yM (k) has to be normalized.

Adapt ive Control lers for Discrete-time Sys tems 1 49

ECM Design : Although in (L . Lj ung, E . Trulsson, 1 978 ) the problem is formulated in a stochastic context with arbitrary contro l ler s truc ture for sake of homogeneity we will res trict our selves to the control l er structure ( 2 . 6 ) in a deterministic framework . The minimization of (2 . 5 ) is solved using a recursive proce­dure to find the zeros of i ts gradient , e . g . VJ [ (8c ) * ] = O Since the gradi�nt is no t known during the recurs ions* of ec (k), it is substituted by i ts certainty equivalent estimate , which i t s e l f is recursively evaluated In the following table the relationship between various contro ller struc tures repor­ted in the literature and (2 . 6 ) is given Table I : Correspondance of notations .

( 2 . 6 ) a - b

CR T

R s

s R

Q

c - d e

T H

* T-R F

* -s G

f - g

CR

RCR

- -d S+Q (q R- 1

Q a - K . J . As trom, B . Wi ttenmark , 1 980 .

b - L . Lj ung , E . Trul sson, 1 97 8 c - H . E l liott , 1 982 d - M . Shahrokhi , M. Morari , 1 982 e - P . E . We l lstead , S . P . Sanoff , 1 98 1 f - R . Kumar , J . B . Moore , 1 9 82 g - R. Or tega , M .M ' Saad , 1 983

ADAPTIVE CASE When A and B are unknown the fo llowing adaptive version of ( 2 . 6 ) is proposed

M (Sk + Q ) u (k ) =-� y ( k) + CR y �k+d2 (3 . l a ) where the po lynomi al sequences Sk , R_ E R

- 1 - - �K [ q ] and the coefficients of Sk , � are time function and mus t be computed such that the design obj ectives are asymptoti­cally at tained . Equation (3 . 1 . a ) may be compactly rewrit ten as: · [ Sc (k) ] T¢ (k)

wi th

*

M CR y (k+d ) - Qu (k) (3 . 1 . b )

Throughout the paper x (k) denotes the esti-mate at time k of x .

¢ (k) [ y (k ) . . . . . y (k-�) u (k) . . . . •

u (k-ns ) ] T (J. 3 )

Wher: nR and nS are the degrees o f R , S and 8c (k) is the corresponding estimated con­troller parameter vec tor Whether the primary focus of the adaptation

· mechanism is on plant or control l er parame­ter de termination, adap tive control lers are c lass i fied as indirect (expl icit) or direct ( implicit) respective ly . Several approaches reported in the l i terature for both cases will be derived below , leaving the discussion of the parameter adaptation algorithms P .A . A) to section 4 .

Indirect Schemes Indirect procedures imply the estimation of the process mode l parameters and further calcul ation of controller using a design method . In this context only PP and ECM objectives are pursued since as wil l become evident later , the FST obj ec tive was intro­duced in order to ob tain a direct scheme . PP Objec tive

The des ign method is provided by the solu­tion of the Bezout po lynomial equati on (2 . 1 2 ) . This can be done either directly , by a mat rix inversion as in (G . C . Goodwin, K . S . Sin, 1 98 1 ) , (P .E .We llstead , S .M . Sanoff , 1 98 1 ) or recurs ive ly as in (I . D . Landau , R . Lo zano , 1 982 ) , (M. Shahrokhi , M . Morari , 1 982) . In the former a recursive m1n1m1za­tion in leas t square sens is considered whi­le in the latter, the Bezout polynomi al e­quation is tranformed in a recurs ive equa­tion which asympto tical ly converge toward the exact so lution under suitable as sump­tion . E . C . M . Obj ective As mentioned above the underlying known pa­rameter des ign method leads to a recursive evaluation of the criterion gradient which is i llustrated be low for the quadratic cri­terion

1 J = 2 (y (k) Setting Q = 0 in the control ler structure (3 . 1 ) it may be readily shown that the esti­mated gradient verifies .

'l'i (k) � ()y (k) ae} (k)

hence V'J(k) = - [ '!' (k) . . . . . '.P ik) ] (y (k ) -l (k)) (3 . 4 ) 0 � ns Notice that two PAA ' s are required in this

1 50 I . D . Landau, M . M ' Saad and R . Ortega

approach , one to e s t imate the process and the other to update the controller parameters toward the gradient cancel lation .

Direct Schemes The fundamental idea to avoid the necess i ty of two s tages in the controller determina­tion is to rewri te the process model in terms of the control ler parameters . This reparame­trization is possib le provided the fol lowing prior knowledge is avai lable . A2 ) d A3 ) nA > max {deg (A) , deg (B) } Since inherent to the ECM formulation is the knowledge of the process model i t can only be approached indirectly , hence we will con­sider here only FST and PP obj ectives . FST Design From ( 2 . 9 ) and ( 3 . 2 ) we ob tain the process reparametrization .

(3 . 5 ) Where ¢c contains the control ler parameters verifying (2 . 9 ) . Defining from (2 . 2 . a) the fol lowing error .

( M � (k) = cR: k)+ Qu(k-d) - CRy (k) which form A2 ) , A3 ) and using (3 . 1 .b ) , (3 . 5 ) may b e writ ten as (see R . Ortega, M .M ' Saad , 1 983) t..(k) = - (�c (k-d) ] T ¢ (k-d ) (3 . 6 ) Where �c (k ) = Sc (k) - 8c . This equation er­ror has been thoroughly s tudied in the l iterature (I .D . Landau , 1 98 1 ) , (G .C . Good­win, P . J . Ramadge , P . E . Gaines , 1 980 ) , (Y . H . Lin, K . S . Narendra, 1 9 80 ) , Remark verifies

It can be easily shown that e (k) * e (k) = Q (u (k-d) - u (k-d ) )

where u* (k) is the one-s tep-ahead optimal input minimizing (2 . 3 ) , which satisfies

M * CR (y (k) - y (k) ) + Q u (k-d ) = 0

This error is a generalization of the input matching error discussed in (G . C . Goodwin , C . R . Johnson, J . R, R . S . Sin, 1 98 1 ) The interes t of the FST obj ective/augmented sys tem interpretation presented in section 2 . is readily seen at the light of assumption A l ) , which insures a s tably in­vertible augmented sys tem. Henceforth a class ical deadbeat adaptive design appl ied to the later wil l realize the FST objective (R. Kumar , J . B . Moore, 1 982 ) .

PP Objective Proceeding analogously to FST we get from (2 . 1 2 ) and (3 . 2 ) the reparametri zation

(3 . 7 )

Notice that this model , unlike (3 . 5 ) , i s bi­linear in the parameters hence from an i;:·s ti­mation point of view the evaluation of 8c (k) is not trivial . Two pos s ibilities have been explored to deal with this problem, ei ther seek to l inearize the model as in (K . J . As­trom, 1 980) , (H . El liot t , 1 982) or propose a recursive estimator for the bil inear model as in the former reference . An ingenuous al ternative was recently propo­sed to convert the bilinear model into a linear-in-the-parameters equation (H . E l l iott 1 982) . The key idea is to insert the follo­wing second Bezout identity . 1 = AS ' + q-d BR' (3 . 8 ) into (3 . 7 ) leading to

-d CR(AS ' + q BR' ) y (k+d) = B (Su (k )+Ry (k) ) wich allows the cancellation of B , we have therefore :

R ' (CRy (k) ) +S ' (CRu (k ) = Su (k ) + Ry (k) Noting that the leading coefficients of both S and S ' are known to be equal to c ( the leading coefficient of CR) and I , we can then write . ( �-CR) u (k) = (S ' - 1 ) (CRu (k ) + R ' (CRy (k)­

(S-c ) u (k ) - Ry (k) 0 and the following proces s reparametrizat ion is ob tained ( co-CR) u (k) = (8EC )� ¢ EC (k) (3 . 9 ) where

¢ EC (k) � [ CRu (k- 1 ) . • . CRu (k-nS , ) CRy (k) . . • • • CRy (k-�, ) u (k- 1 ) . � . u (k-ns ) y (k) . . . . .

T y (k-nR) ] and 8EC contains bes ides the controller parameters the coefficients of S ' and R' . Substituting 8EC by i ts estimate in (3 . 9 ) and subs tractring both sides we define a linear-in-the-parameters adaptation error . The model (3 . 7 ) may also be linearized un­der the certainty equivalence hypothes is of an on line identi fied model as proposed in (K . J . As trom, 1 9 80) . There the regressor vector is fil tered by the es timate of B and the controller parameters are updated dire­ctly The problem of estimating the bilinear mo­del (3 . 7 ) has been approached in (K . J . As trom, 1 980 ) as the least squares minimi­zation of the following error (See equations tJ . 1 . b ) and (3 . 7 ) ) . A Ac T CR y (k) - Bk ( 8 (k) ) · ¢ (k) in the parameter space [ Sk � Bk] with re­

M gres sor [ Bk u (k ) 1 Bk y (k) 1 CR y (k+d ) ]

Adaptive Controllers for Discrete-time Sys tems 1 5 1

STABILITY RESULTS To compiete the specification of the adap­t ive controllers d iscussed above a PAA must be derived . A fairly general PAA is given by

� (k) = �(k-1) + A3 (k) F (k) � (k-d) ep (k) (3 . 1 0 .a )

with F- I (k) A 1 (k) F- l (k-1 ) + A2 (k) � (k-d) x

�T (k-d) ; F (o ) > 0 (3 . J O .b ) Where 1 i s an integer which may be choosen equal to I (weigthed least squares ) as in (I . D . Landau , R . Lozano , 1 982) or d (inter� ced weigthed leas t squares ) as in (G . C . Goodwin, C . R . Johnson. JR, K . S . Sin, 1 98 1 ) , ep(k) , is the adaptation error and A 1 (t ) , A2 (t ) , A3 ( t ) : z +.

-+ R such the following condi tions holds : O < A/k) .;$. 1 ; 0 .,S. A2 (k) < 2 ; A3 (k) = 1

(3 . 1 0 .t) in the case of weighted least square and A 1 (k) � A2 (k) / A2 (k-d) ; A2 (k)=A3 (k) > 0

(3 . 1 0 .d ) in the case of interlaced weigthed leas t squares . When non sta ted otherwise we will assume tc be dealing with this type of P .A .A . Some exist ing stability results , both local and global will be presented below Direct Schemes To the knowledge of the authors stabili ty proofs of d irect adapt ive schemes are given only for FST obj ective In (G . C . Goodwin, C . R . Johnson. JR, K . S . Sin, 1 98 1 ) a global s tabi li ty proof of the FST scheme is siven if A l ) - 1 3 ) hold and the following prior information is known A4) bo > bo ; sign (bo ) al through the proof is given for s ingle de­lay sys tems and a scalar gain PAA , i t is claimed that it may be eas i ly extended to d > I and weigthed leas t squares PAA . The proof is done us ing the technical device of (G . C . Goodwin, P . J . Ramadge , P . E . Gaines , 1 980 ) . Recently , assumption A4) has been relaxed for the interlaced weigthed leas t squares PAA (3 . 1 0). Conic sector s tabil i ty theory was applied to obtain the convergence proof (R. Ortega ; M. M ' Saad , 1 983 ) . As clear from fig . 2 stability of B (see :quatio� 2 . 1 ) ) suffices to s tably adapt Sk and R,_ . In (R. Kumar, J . B . Moore , 1 982) a second a�aptation of Q and CR is proposed to overcome assumption A l . This however implies the adaptive stabilization of a non-s tably invertible system and furthermore i t is not clear what effect a time varying regressor

has on the s tabi l i ty proof , The most promis ing approach for the PP case is the one proposed in (H . Elliott , 1 982 ) , for which i t can be readily seen that : [S£C (k) ] T � EC (k)-----+ o

This allows to attain the obj ective (3 . 7 ) proyided that

A s ' + q-d B s '����� k k It has been conj ectured in (L . Praly , 1 983) that no such resul t may be obtained given the dependance of the information contained in the extended regression �CE (k) ,

Indirect Schemes

It has long been recognized that the key problem to prove s tabi l i ty of indirect adap­tive controllers is to show that the sequen­ce of es timated models does not have a limit point corresponding to a non-stabili zable model , e . g with unstable pole-zero cancel­lations . It is clear that under the as sump­tion of minimum realization and exact know­ledge of deg (A) , this problem could be sol­ved if a sufficient excitation condi tion, insuring convergence of the es timated model , could be imposed . This condi tion remains the main stumbling-block to solve the question of global s tability . Global convergence results have recently been es tablished (G . C . Goodwin, E .K . Teoch , B .C . Mcinnis , 1 983 ) , (B . D . O . Anderson, R .M . Johns tone , 1 983) translating the sufficient exci tat ion condition to the reference se­quence . However , in the latter an addi tional knowledge of a underbound on the sylves ter resul tant of B and A is imposed and the pro­of given in the former relies on an off-li­ne open-loop identification of questionable practical interest . An interes ting decomposition o f the two steps (Identification and Bezout identity solution)

:.involved in indirect schemes has been repor­ted in (M. Shahrokhi , M. Morari , 1 982) . It consists of rewri ting ( 2 . 1 2 ) into its matrix form in terms of E - the eleminant matrix of B and A, that is

(3 . 1 . ) where C E Rns+nR+2 and contains the coeef i­cients of the polynomial CR and n +n_+2-n S K CR addi tional zeros . Hence the following adapta­t ion error may be defined as

See (M. Shahrokni , M. Morari , 1 982 , equation 56) I t can be readi ly seen that i f the process parameter estimates converge to a value such that "' (8c)T E (k) � 0

1 52 I . D . Landau, M. M ' Saad and R. Ortega

then the adaptive law �c (k+ l ) = �c (k) + 'f(k) � (k) G

insures ' (k) 0 Hence the controller parameter tend towards a solution of the tity .

G

es timates Bezout iden-

On the ECM context a global s tability proof in terms of the ODE approach is given in (L . Ljung , E . Trulsson, 19 ) us ing a recur­sive ins trumental variable method to es ti­mate the process parameters . In order to dispose of ins truments Z (k) decorrelated to the process observation vector � (k) an auxiliary es t imator (output-equation like) is added . The input to this estimator, which is assumed decorrelated with these enters in the invertibility assumption of :

k t l: Z (i )(f ( i ) i= l

(See equation 6 . 7 in the above reference) The additional condition of :

-d A

'\ Sk + q Bk � having s table roots for all k is also impo­sed to insure boundedness of the gradient es timates (see equation 3 . 3 ) . The control ler parameters are updated using an iteration procedure of the form (3 . 1 0 . a) with ¢ (k) ep (k) � � (k) (See equat ion 3 . 3 ) and F (k) is a posit ive definite matrix that changes the es timate gradient direction to an other descent direc­tion (e . g . the Gauss-Newton one) . Assuming that the sequence of es timated mo­dels has a limi t point corresponding to a s tabilizable model , the wel l-know "self-tu­ning proper y" has been proven for the sche­me proposed in (P . E . Wells tead , S .P . Sanoff , 1 98 1 ) . Under the same hypothes is but res tric­ting to open-loop s tab le sys tems with s table models , conditions for c losed-loop stability are given for an FST scheme (R. Ortega, 1 983) in terms of the CR polynomial and the a priori knowledge of a transfer function-gain upper­bound . This provides the designer with a tool to establish us ing prior information, the ro­bus tness-performance tradeQ,ff .

CONCLUSIONS

In the previous sections algori thms whose global s tabili ty has been proven under mild assumptions , e . g . s tab ility of the known pa­rameter des ign (see Assumption A l ) ) , as well as algorithms for which succesful s imulation studies encourage us to believe that global s tab ility may be es tablished , have been discussed . (F . Alix et . al . , 1 982) Both direct and indirect schemes have been proposed to so lve the problem of adaptiveJ.y

controlling non-stably invert ible sys tems . In spi te of the ,considerable research effort and the variety o f the design objectives , " parameter adaptation algorithms and the technical devices employed to carry the ana� lysis of the problem remains , in the author ' s opinion s ti l l unsolved . We believe that in the context under which the problem is for­mulated , nei ther one of the following as-

sumptions : s tability of a process-depen­dent polynomial (see assumption Al ) ) , avai­lability of open-loop prior identification, a prior knowledge of a measure of pole-zero separation (see s tabil i ty resul ts - Indirect schemes ) nor artificial inj ection of probing s ignals on the control or reference signal s , seems reasonables . Al though i t has been c lai­med that on-l ine adaptation of the weighting polynomials is a feasible solution (R. Kumar and J .B . Moore, 1 982 ) , i t is by no means c lear how can a time-varying error model stability s tudy be carried with the usual technical tools . I t has even been conjectu­red (Morse , 1 982) that these do not exist algorithms capable to s tabilize the consi­dered type of svs tenis t.d th no further quali­fication.

Utilisation of new technical devices seem compulsory ( i . e . differential geometry to .cope with the bilinear estimation approach . No solution, besides requiring further prior knowledge, seems at hand for the linearized or the f il tered sequence tracking formula­tion . Al though the s tudy . of persis tent exci­tation has attracted a great deal of atten­tion in the las t couple of years , no defini­tive answer has been given to the key ques­tion of s tabil izab il i ty of the estimated mo­dels . On the other hand , from the point of view of engineers a reformulation of the objec tive , e . g . in terms of tracking error boundedness , may lead to schemes satisfving i t s require-ments . REFERENCES F . Alix, J .M. Dion, L . Dugard and I .D . Lan­dau ( 1 982) Florence - ITALY , pp . 445-464 , october 1 982 B . D . O . Anderson and R.M. Johns tone ( 1 982) . Private correspondance K . J . As trom and B . Wittenmark ( 1 980 ) . IEE. Proc . , .GI_, pp- 1 20- 1 30 K . J . As trom, P . Hagander and J . S ternby ( 1 9-80) , IEEE Proc . , CDC , pp . 1 -5 . K . J . As trom ( 1 980) . IEEE Proc , CDC , pp . 6 1 1 -6 1 5 D .W . Clarke and P . J . Gawthrop ( 1 979 ) , Proc . IEE , 1 26 , pp . 633-640 . H . E l l iot ( 1 980 ) , IEEE . TAC , vol . AC-27 pp . 7 20-722 , June 1 982 . G . C . Goodwin, P . J . Ramadge and P . E . Gaines ( 1 980) , IEEE . TAC , Vol . AC-25 , pp . 449-456 , June 1 980 G . C . Goodwin and P . J . Ramadge ( 1 980 ) , IEEE . TAC , Vol . AC-25 , December 1 980 .

Adaptive Control lers for Di screte-t ime Systems

G . C . Goodwin , . C .R . Johnson JR and R . S . Sin ( 1 98 1 ) , IEEE . TAC , Vol . 26 , pp . 865-872 , De­cember 1 98 1 G . C . Goodwin, E .K . Teoh and B . C . Mcinnis ( 1 982) , Private correspondance . R. Kumar and J . B . Moore ( 1 982 ) , Automatica to be published I . D . Landau ( 1 98 1 ) , Trans . ASME J , of Dyn . Sys . Means & Contro l , December 1 98 1 I . D . Landau and R . Lazano , ( 1 98 1 ) , IFAC/ IFIP sympos ium on SOCOCO - Madrid , . October 1 98 1 Y . H . Lin and K . S . Narendera ( 1 980) , IEEE , TAC , vol . AC-2 3 , n° 3 , June 1 980 . L . L jung and E . Trulsson ( 1 978) , Report LITH ISY-I-040 1 L . Ljung and E . Trulsson ( 1 982) , 6th IFAC Sympos ium on Ident ification and System Para­meter Estimation, Washington, June 1 982 . S . Morse ( 1 982) . Colloque National C .N . R . S . , Belle-Ile , France Septembre 1 982 . R. Ortega and M .M ' Saad ( 1 983) , Submitted to IEEE . TAC . R. Ortega ( 1 983) , Ricerche di Automatica, Special Issue on Adaptive Control . L . Praly ( 1 983) CAI Report , March 1 983 M. Shahrokhi and M . Morari ( 1 982) , Int . J , Control , Vol . 3 2 , n° 4 , pp . 695-7 1 0 P . E . Wells tead and S . P . Sanoff ( 1 98 1 ) ; Int . J , Control , Vol . 3 4 , n° 3 , pp . 433-455 , (For the de tailed reference l i s t , please con­tact the autors ) .

Fig. l . Linear controller scheme

- r--- - - - - -- - - · --·"-_,�, ., y\kll r<kl :

I

I I I

I I L. - - - - - - -- - - - - - - - • . • ..J ---- --------

Dead b .. c Pole plac•aat i a M) • a c1 i Q .s . ,-dii

Fig. 2 . The Augmented system representation

1 53

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

PARAMETRIZATIONS FOR MULTIV ARIABLE ADAPTIVE SYSTEMS

J. M. Dion* and L. Dugard**

*Laboratoire d'Automatique dt' Grnwlilf, B.P. 46, 38402 Saint Martin d'Hh1's, France **On {rave at the Department of Electrical and Computt'I" Engineering, Tiu' University of Newcastle,

NSW 2308, A ustralia

Abstract . This paper describes algori thms for model reference adaptive control of linear multivariable systems . A survey of recent work on MIMO direct adap­tive control is presented , focus ing on the general case of processes which are not decouplable by s tatic s tate feedback . The principal aim of this paper is to examine parametrizat ions of linear sys tem models for use in adaptive control . More precisely , comparison is made of the linear control schemes as wel l as the necessary a priori knowledge on the process in order to achieve some specified obj ectives .

INTRODUCTION The aim of this paper is to give a general idea of what has been done in direct adapt ive control of MIMO linear t ime invariant plants . The general case of plants which are not de­couplab le by static s tate feedback is cons i­dered as we ll as the non minimum phase cas e . To date the principal obj ect ive in MIMO di­rect adapt ive control is to obtain schemes requiring less a priori knowledge than the Hermite form of the transfer matrix while in­suring convergence and stability of the adap­tive algorithm . The objective of this paper is twofold . First to compare the various l i­near control schemes and second to compare the necessary prior knowledge on the process in order to achieve s table adaptive contro l . The discrete time case i s emphasized because of the increas ing use of the digital compu­ters . But there are few differences between continuous and discrete t ime case . ( i . e . add some filters , some conditions in order to prove stability ) . This paper is organized as follows . After introducing some notations and basic definitions in Section 1 , the Section 2 is devoted to the analysis o f some factoriza­t ions of the transfer matrix. A good under­standing of these factorizat ions is important in order to handle the multivariable case . Section 3 outlines the differences between the different direct multivariable control scheme s . We will successively consider the schemes developped by Dion and Dugard ( 1 982) , Goodwin and Long ( 1 980) , Elliott and Wolowich ( 1 982) , El liott Wolowich and Das ( 1 982) , Singh and Narendra ( 1 982) , Johansson ( 1 982) and Goodwin , Mclnnis and Wang ( 1 982) .

1 55

1 . NOTATIONS AND PRELIMINARIES

In this section some notations and basic re­sults which will be useful in the sequel are presented . We will start with some notations and definitions . Let R be the field of reals . Let R [ z ] be the ring of polynomials with real coefficients and R(z ) be the field of rational funct ions . Rp (s ) . denote the ring of proper rational functions .

Let Rpxm be the vector space of pxm matrices with real coefficients . Rpxm [ z ) (resp . RPxm(z ) ) represents the set of (pxm) matrices with elements in R [ z ] (resp . R (z ) ) . �xm(z) represents the set of (pxm) matrices with elements in Rp (z) . The elements T ( z ) of R�xm(z) are called proper rational matrices and are characterized by lim T ( z ) = T with T E RPxm . When T = O , T ( zfi's then called s trict ly proper . The units (invertible elements) in the ring Rmxm[ z ] are called unimodular matrices and are characterized by a non zero constant de­terminant . The units in the ring R�xm( z ) are called bicausal (or bi-proper) matrices and are characterized by the fol lowing : B(z) is a bicausal matrix i f and only if det <lim B ( z ) ) # 0 . Consider now the discrete t ime linear system (l:) : x(k+ l ) Y (k) whose

A x(k) + B u (k) x (k) 6 Rn, u (k) e. Rm

C x(k) + D u(k) y (k) e, RP trans fer matrix is T ( z ) = C ( zl-A) - 1 B+D .

By feedback group we mean s tate and input , changes of coordinate and s tate feedback .

1 56 J .M . Dion and L . Dugard

Consider the effect of the state feedback de­fined by U (k) = F x(k) + G v (k) with F Ei Rmxm and G E-Rmxm non s ingular . The c losed loop transfer function is then : TFG ( z ) = ( C + DF) (ZI - A - BF)- ! BG + DG

T (z ) ( I - F ( z I-A)- I B) - I G = T ( z ) B ( z ) where B (z ) i s eas i ly seen to be a bicausal precompensator .

Conversely Hautus anu Heymann ( 1 978) gave a complete characterization of the dynamic pre­compensators that can be imp lemented by ac­tion of the feedback group . In the seque l we wi l l use the fol lowing vers ion of this resul t .

Theorem I : Let T ( z ) be a (pxm) proper rational matrix

and (A,B,C,D) a realisation of T ( z ) . Let C ( z ) be a (mxm) proper rational precompensa­tion matrix, there exist a constant matrix F and a constant non s ingular matrix G such that T ( z ) C ( z ) = ( C+DF) ( z I-A-BF) - I BG+DG i f and only i f : C ( z ) i s bi causal and for every ! polynomial vector u(z ) such that ( zI-A) - I B u (z ) is polynomial, the vector c- l ( z ) u ( z )

lis polynomial as wel l .

Let us consider now the transmiss ion zeros of (L ) . Def init ion I : The finite transmission zeros of (Z ) are de­fined to be (Rosenbrock ( 1 970), Davison ( 1 983) ) the set of z which has the proper ty that

Rank (A�zI �) < n + min (m,p) If (Z ) has finite transmiss ion zeros which are all contained in the open unit disc then (Z) is said to be minimum phase .

If it is as sumed that ( Z) is control lab le and observab le, the finite transmiss ion zeros are the roots of the non trivial numerators of the Smith-McMi llan form of the trans fer ma­trix T ( z ) .

As it i s pointed out in Davison ( 1 983) , for square systems (m=p) the f inite transmis sion zeros are equal to the finite transmission zeros of the control lab le and observab le part of (Z ) together with the non control lab le or non observable eigenvalues of A. Then i f (Z ) is supposed to be minimum phase, this implies that ( C,A,B) i s stabi l izable and detectab l e .

Now consider the infinite zero structure of T ( z ) (or infinite structure, a proper ratio­nal matrix having no poles at infinity) i . e . the number and the degrees o f the infinite zeros . A clas s ical definit ion of the infinite structure of T ( z ) is the following, Rosenbrod< ( 1 9 70) .

Definit ion 2 : [The (pxm) rat ional matrix T ( z ) possesses an infinite zero of order k when G ( l /w) has a l finite zero of prec isely that order at w = 0 . The inf inite zeros are invariant under the transformation group of a l l input , output

and s tate changes of coordinate , state feed­back and output injection . The infinite struc­ture plays a keyrole in solving problems such as decoup ling (Descusse and Dion ( 1 982) ) , block decoupling . (Dion ( 1 983) ) , mode l fol­lowing (Malabre ( 1 982) ) .

In the next sect ion we wi l l present some use­ful factorizations of the system transfer matrix.

2 . SOME FACTORIZATIONS OF T (z)

Fraction representations of T ( z ) We will cons ider first fraction representa­tions of T ( z) introduced by Wolowich ( 1 9 74) . Recall ing the following bas ic theorem .

Theorem 2 r Let T (z) • Rpxm( z ), T ( z ) can be factorized in T ( z ) = R(z)p- 1 ( z ) where R(z ) eRpxm [ z ] and P ( z ) 6 Rmxm [ z ] . Such a factorization is cal led a right factorization of T ( z ) . More­over if R(z ) and P (z ) are re lat ive ly right coprime then R(z) and P ( z ) are unique modu­lo right multip l ication by unimodular (mxm) matrices . In this case the transmiss ion zeros of T ( z ) are the roots of the invariant factors of R(z ) and the poles are the roots of det ( P ( z) ) . Furthermore, P (z) is column proper and the maximum polynomial degrees in the columns of P (z) are equal to the control labil ity indices ni of the system .

The same result holds for left coprime facto­rizat ions .

Theorem 3 : r Let T ( z ) � R�xmiz), T ( z ) c�n be factorized in T ( z ) = p- l ( z)R( z ) where R(z ) E: RPXID[ z ] l and P (z ) E: RPXP [ z ] . Such a factorization is cal led a left factorization of T ( z ) . More-over if R ( z ) and P ( z) are relatively left coprime then R(z) and P ( z ) are unique modulo left multiplication by unimodular (pxp) ma­trices . In this case the transmiss ion zeros of T ( z ) are the roots of the invariant fac­tors of R (z ) and the poles are the roots of det (P ( z ) ) . Furthermore P ( z ) is row proper and the maximum polynomial degree.s in the rows of P ( z) are equal to the observabil ity indices Pi of the system .

foys id�ring such a prime factorization P- ( z )R ( z ) for a strictly proper system T ( z ) p I z I 0 1 p ( z ) = l ''' PPJ p + p

I ( z )

0 ' z 0

where P is non s ingular and the maximal po­lynomia� degree of the ith row of P 1 ( z ) and of R ( z ) is less than Pi •

The output i s related to the input by P (q) y ( t) = R(q) u ( t) where q- I is the ward oper:��r ·r ��?�iply�ng]on the left

p 0 '' -p 0 • q p

back­by

Parametrizations for Mul tivariable Adaptive Sys tems 1 5 7

we obtain the following ARMA model

where

- I - I A(q ) y (t) = B ( q ) u ( t ) I - I A(q- ) I + A 1 q + . . .

- I - I B (q ) B 1 q + . . . . . •

+ � q + Bk q

( I ) -k -k

k is the largest observabil ity index. In ge­neral B 1 i s not invertible even i f T ( z ) i s square and full rank .

Following the same way , from right coprime factorizations we obtain the following model :

D ( q- I ) z ( t ) = u (t ) y (t ) = N ( q- l ) z ( t) (2 )

where - I I + D 1 q + . • . + = No q- 1 + . . . +

JI, is the largest controllab il ity index.

Hermite form of T (z ) Consider now the Hermite form of T ( z ) over the ring of proper rat ional functions as de­fined by Morse ( 1 975) .

Theorem 4 : [ Let TI = z+a be a fixed polynomial of de­gree one . Let T ( z ) E: Rpxm(z) , T (z) can be factorized in T ( z ) = H�z) B ( z ) where B ( z ) is a bicausal matrix and H (z ) = [ H(z ) , O] , H( z ) e R�xr (z ) , r = rank T ( z) , the (rxr) matrix H (z ) cons ist ing of the first r li-nearly independant rows of H(z) of the form

H* (z ) [ I /TI:�, 0mr

'

hij ' ! / TI ID • •

h . . = y/TI iJ , m . . < m. , mi· ,m . . pos itive iJ iJ i iJ integers , y € R[ z ] . lH(z) is cal led the TI-Hermite form of T (z ) over the ring of proper rational funct ions and is uniquely determined by T ( z ) and TI .

A bicausal matrix has neither poles nor zeros at infinity , then the infinite s tructure of T ( z) is included in H(z ) . For TI = z the TI-Hermite form wi l l be called s imply the Hermite form. For the case of a square non s ingular trans­fer matrix, the Hermite form of T (z ) : H (z ) = H* (z ) provides the same information as the interactor � ( z ) E Rmxm [ z ] defined by Wolowich and FalB ( 1 9 76) . More precisely H(z ) = �T l ( z ) . One has the following result Theorem 5 :

Let T (z ) be a (mxm) full rank proper ra­tional transfer matrix and (A, B , C , D) a rea­lisat ion of T (z ) . There exist a constant matrix F and a constant non s ingular matrix G such that :

- I TFG(z ) = (C+DF) (z I-A-BF) BG + DG = H (z ) = - I

= i:;T (z )

ASCSP-F*

Proof : By Theorem 4 T (z ) = H (z ) B (z) . Cons ider now the precompensator B- l ( z ) , one has B- 1 ( z ) bi­causal . For every u (z) polynomial such that ( z I-A) - IB u (z ) is polynomial one has [B- l (z ) ] - I u ( z ) = H- 1 (z ) T ( z ) u (z )

= H- l (z ) [ C ( zI-A)- I B + D ] u (z ) which i s polynomial . S ince H- 1 (z ) = �T (z ) i s polynomial . Then by theorem I , proof is com­plete . We have not here dealt with s tability since H (z ) has no transmiss ion zeros , then the pro­cess transmiss ion zeros have been made non­observable by F . The c lass of reference models which can be followed by the system T ( z ) is characterized by {H (z ) T ( z ) , T ( z ) E Rpxm(z) } where H ( z ) i s a TI-Hermite form of T ( z ) , TI is chosen to be s table .

For the case where T (z ) is non minimum phas e , one can consider Hermite forms over the ring of proper rational s table functions .

In this paper we will consider that the pro­cess is non decouplable by s tatic s tate feed­back. If this is not the case the Hermite form of the transfer matrix is then diagonal , more precisely one has : Descusse and Dion ( 1 982) . Theorem 6 : l T(z ) E":R�xm (z) non s ingular is decouplable [ by static state feedback on a realization of T ( z) if and only if the orders of the in­finite zeros of T ( z ) are respectively equal to the orders of the infinite zeros of the rows T . (z ) of T ( z ) . i [ -m i l . z 0 In this case • ,, 1

H ( z ) = 0 ', z-mm

For a discrete t ime transfer function -d B (z- I ) z � , Ao = I , Bo f 0 the order of A(z )

the inf inite zero corresponds to the de lay d . In the discrete time case the m. ' s correspond to the minimum delay occuring ifi the ith row of T ( z) . In the sequel we wi l l cons ider non-diagonal Hermite forms . Due to the following reasons: i ) in order to be able to cons ider the gene­ral cas e . i i ) when us ing a state space approach H ( z ) i s generically diagonal , but i t seems that this result is not valid us ing t ransfer matrix arguments . S ingh and Narendra ( 1 982) cons i­der (2x2) transfer matrices in this last con­text . Much work remain to be done in this area in l inear system theory : i . e . : what is the appropriate topology one has to use on transfer matrices in order to derive in­teresting result s ? iii ) for robustness properties , as pointed out by Chan and Goodwin ( 1 982) .

158 J .M. Dion and L. Dugard

Smith McMillan factorizations at infinity of T(z) Let us consider now a factorization introdu­ced by Verghese ( 1 978) . Theorem :

Let T (z ) E �xm(z) , T (z ) can be factorized in T ( z) = B 1 (s) A(z )B2 (z ) , where B 1 (z) and B2 (z ) are bicausal matrices and [L'l(z ) 0 ]

A(z) = O O z , 0 [ -k 1 l

where L'l (z ) = ', -k , r = rank T (z ) . 0 ' z r

The k . ' s are the infinite zero orders of T (z ) . 1Such a factorization is called a Smith McMillan factorization at infinity of T (z ) .

Dion and Commault ( 1 982) characterized the non uniqueness of these factorizations via a multiplicative group of bicausal matrices . A(z ) turns out to be the Smith form of T (z ) over the ring of proper rational functions . When considering the Smith form over the ring of proper rational stable functions B 1 (z) and B2 (z ) are then bicausal and bistable , so A(z ) contains also the unstable poles and zeros of T (z ) .

An example Consider the system (E) = (A , B , C) r · I 0 O l

· · - rm l -3 -4 � J A = -3 4 0 0 ' c = - 1 -2 0 0 0 1

L o o -8 6

T (z ) [ I I l

,� 1 ,�, j is full rank with finite zeros .

z-3 z-4

z z- 1 z-2

no

H(z)R(z) 0 j [_z _z 1

z+2 1 6 z 2 8 z 2 L zr � <z- 1 > <z-3) <z-2) (z-4>

- I [z-3 z-41 [( z- 1 ) ( z-3) 0 1 -l R(z)P (z ) =

z- 1 z-2 0 (z-2) (z-4)

-- 1 - i"<z- l ) (z-2 ) 0 1 -1 rz-2 z- 11 P (z ) R(z) = l J l I 0 (z-3 ) (z-4) z-4 z-� [ :

+2 01 [z- 1 �

3]· -z- lj 0 z

. r �6 z 2 L (z- l ) (z-3)

z�2 l 8 z 2 j (z-2) (z-4)

[ z 0 1 - I I i;T (z) "' H (z) = z 3 +2z2 z 3 _

On has A (q-1 ) y (t ) = B (q- 1 ) u(t) - I r l-3z- 1+2z-2 0 l

where A(q ) = l -7z- 1 + 1 2z-2 ' [ - I

B ( - 1 ) = I - 2z q - I I - 4z _ z- 1 l

I - 3z- l

The controllability indices and the observa­bility indices are equal to n 1=n2=p 1=p2 = 2 . The infinite zeros orders are k 1 = I , K2 = 3 . The infinite zero orders of the rows o f T (z ) are I and 1 , then the system i s not decoupla­ble by static state feedback . In this example the infinite structure is given by the diagonal of the Hermite form. It is not always true as shown by the following

::��t:: :r l:�: ,�3i - I l [ - I 0 l [ -21 [ ' : j ' O , -4 J : : I _

3 . COMPARISON OF MULTIVARIABLE ADAPTIVE CONTROL SCHEMES

We will focus our attention mainly on the s tructural aspects of the linear control schemes , and on the asumptions made on some a priori knowledge regarding the process in order to prove stability . We will consider successively some direct schemes developped by Dion and Dugard ( 1 982) , Goodwin and Long ( 1 980) , Elliott and Wolowich ( 1 982) , Elliott , Wolowich and Das ( 1 982) , Singh and Narendra ( 1 982) , Johansson ( 1 982) , Goodwin , Mcinnis and Wang ( 1 982) . Except for the last scheme it is supposed that the number of inputs is equal to the number of outputs . It is also assumed , s ince we will require output func­tion controllability, that the transfer ma­trix is square and full rank .

Scheme developped by Dion and Dugard ( 1 982) This scheme is a generalization of the s cheme presented by Goodwin and Long ( 1 980) . The pro­posed extension lies in the fact that the ob­server poles can be choosen freely . Consider a rnultivariable linear time inva­riant system having a square , full rank , strictly proper transfer mal'Cix T (z ) . The ob­j ective of the control law is to ensure that the plant model error asymptotically goes to zero while the system inputs and outputs re­main bounded . As seen previous ly , T (z ) H(z) B(z) where

Parametrizations for Multivariable Adaptive Systems 159

B (z) is a bicausal isomorphism, consider y ( t) = H (q) y ( t) = H(q) B (q) u(t) . Using the backward operator q- 1 one has the following factorization of B (q)

A(q- 1 ) y ( t) = B (q- 1 ) u(t) - -1 - - I - -s A(q ) I + A1 q + . • • + As q - - 1 - - - 1 - -R, B(q ) = Bo + B 1q + • . • + BR, q

det (Bo ) + 0 . Now as proved in Dion and Dugard ( 1 982) the largest inf inite zero order d of T(z) will play a keyrole . In order to compute a causal control law we will use a d-step ahead output predictor . To derive this predictor , as in Goodwin and Long ( 1 980) , explicit expressions are found for two polynomial matrices S (q- 1 ) and R(q- 1 ) such that : C (q- 1 ) S (q- 1 ) A(q- 1 ) + q-d R(q- 1 ) ( 3)

- I _ - 1 -j where C(q ) - I + C 1 q + . . . + C . q - 1 - I J -d+ l S (q ) = I + s 1 q + . . . + Sd- l q

and d is the largest infinite zero order of T (z) . The tracking obj ective is to follow a desired sequence y* (t ) given by the following model * A - 1 - I - I - 1 * y (t) = H(q ) A (q ) B (q ) u ( t )

A - I m m where H (q ) = H (q) is the Hermite form of the transfer matrix! but expres sed as a poly­nomial matrix in q- . This is achieved by ta­king :

- I - - 1 - 1 u(t) = [ S (q ) B ( q ) ] - I - 1 - 1 - 1 * [C (q )� (q · · ) Bm(q ) u ( t) -

- I A- 1 1 R(q ) q-d H (q- ) y (t) ] as shown in figure 1 .

( 4)

It can be shown very eas ily that the precee­ding scheme is equivalent to the one repre­sented in figure 2 . Define

- 1 - 1 - - 1 - - 1 Nu(q ) C ( q ) B0 - S ( q ) B(q ) N (q- 1 ) = - q-d R(q- 1 ) H- l (q- 1 ) � l - 1 - I C (q ) N (q ) i s strictly causal and

c- l (q- 1 ) Nu(q- 1 ) is causal . y Consider now a right coprime factorization N(q- 1 ) D- l (q- 1 ) of the system transfer ma­trix N (q- l )D- l (q- 1 ) "' H( q- 1 ) j;- l (q- 1 ) B(q- 1 ) Multiplying on the right the diophantine equation ( 3 ) by H- l (q- 1 ) N (q- 1 ) = X- 1 (q-) B(q- ) D (q- I ) yields C(q- l )H- l (q- l )N(q- 1 ) = S (q- l )B(q- l )D (q- 1 ) +

q-dR(q- l )H- l (q- l )N (q- 1 ) = -N (q- l )D (q- 1 ) + u_ l - - I C(q )'8oD (q )

N (q- l )N (q- 1 ) • y One obtains the diophantine equation :

N (q- l )D (q- 1 ) + N (q- l ) N (q- 1 ) u - 1 - - I y A- I - I - I = C (q ) (BoD(q ) - H (q )N (q ) ) (5) As discussed in Wolowich ( 1 974) or Kailath ( 1 98 1 ) the control law ( 4) can be interpre­ted as follows . First use a linear state variable feedback via asymptotic s tate estimation where the zeros of det C (z) represent arbitrary obser­ver poles . With N ( q- 1 ) and N (q- 1 ) computed by (5) the c loseduloop transf�r function be­comes : N ( q- l ) [B: j �- l (q- l )N (q- 1 ) )- 1 = �(q- 1 ) Bo , secondly use the precompensator B; l �l (q- l ) BM(q- 1 ) in order to get the de­sired input-output behaviour .

- 1 In Goodwin and Long ( 1 980) C ( q ) a dead beat observer is used .

I , hence

In order to develop the adaptive control, the following asumptions (as in Goodwin and Long ( 1 980) have to be made : * H (z) is known * T (z) is minimum phase . * An upper bound is known for the orders of the polynomials A(q- 1 ) and B(q- 1 ) . The objective is to des ign an adaptive con­trol law such that y ( t ) and u ( t) remain boun-ded and_ 1 -d A- l _ 1 * lim[C(q ) q H (q ) (y (t ) - y ( t) ) ] = O . t-><x> Multiplying on the right (3) by �- 1 (q- 1 ) y(t ) yields C (q- I ) q-d�- I (q- l ) y ( t+d) = S (q- l )B (q- l ) u( t) +

R(q- l ) q-d it l (q- 1 ) y (t ) . ( 6 ) = 8 0 ijJ (t)

where 8 0 is a matrix containing the coeffi­cients of S (q- l )B (q- 1 ) and R(q- 1 ) and T T T -d A - 1 - 1 T 1jJ ( t ) = [u ( t ) , u ( t- 1 ) , . . . , [q ti (q ) y (t) ] ,

-dA- 1 - I T [ q H (q ) y ( t- 1 ) ] , . . • ] The control law is obtained from C(q- 1 ) q-d li- 1 (q- 1 ) y* ( t+d) = S (t) iji(t )

Starting from this point , s everal different adaptive algorithms can be used (see figure 3) . For instance the projection algorithm I �sed in

AGoodwin and Long ( 1 980) given by

6 (t ) = 8 ( t- l ) + a ( t)• -1 '-dA- 1 - I A T [C (q )q H (q )y ( t ) - 8 ( t- l )iji ( t-d) ]1jJ ( t-d) • r 1 + 1jJ ( t-d) iji ( t-d)

0 < E < a(t ) < 2 - E < 2 , With this algorithm. and subject to the asump­tions made previously and taking into account the fact that C(q- 1 ) is stable , it can be shown that ' lim C ( q - I ) q -d H- I ( q - l ) ( y ( t ) - y * ( t ) ) = 0 . t-><x> Furthermore y (t ) and u(t) are bounded for all t ime .

1 60 J . M . Dion and L . Dugard

Scheme developped by E l liot t , Wolowich and Das ( 1 982)

Elliott and Wolowich ( 1 982) have first deri­ved for continuous time systems , a control strategy for the case where the Hermite form is not diagonal . This l inear scheme is simi­lar to the control s tudied previous ly ( figu­re 2 ) , except that the observer poles are not choosen free ly . The Hermite form ( or inter­actor) is supposed to be known . Some filtered inputs and outputs are defined , but it is no more necessary in the discrete time case . The diophantine equation is equat ion ( 5 ) . The global stability analys is is performed by us ing a mult ivariable vers ion of a random sampling scheme studied by El liott ( 1 982) .

Let us focus our attention now on the direct scheme presented by Elliot t , Wolowich and Das ( 1 982) . This scheme is very general , al lowing to handle non minimum phase proces ses . But the most interesting feature of this algo­rithm is that the implementation of the sche­me requires only knowledge of the system con­trollabil ity indices , and an upper bound on the observability index . Then the knowledge of the Hermite form is replaced by the know­ledge of m integers .

The linear control scheme is s imilar to the one described previous ly ( figure 2) . In this paper both cont inuous time and di screte time case are pr.esented . Here we wi l l cons ider uniquely the discrete time case which is s im­pler (s ome filters are unnecessary) .

The authors cons idered a right factorization of the transfer matrix T ( z- 1 ) = N ( z- l ) n- l ( z- 1 ) where D (o) is triangular non s ingular . D (o) plays the role of B; I in previous section .

- 1 - 1 By choos ing N (q ) and N (q ) to satisfy the DiophantiRe equat ion . Y

- 1 - 1 - 1 - 1 N (q ) D ( q ) + N (q ) N (q ) = u - 1 y_ l - 1 - 1 C (q ) (D t o )D (q ) - Dc (q ) ( 7 )

the c losed loop transfer function of the con­tro l scheme represented in figure 4 becomes

T (z- 1 ) N ( z- 1 ) D- 1 ( z- 1 ) D- 1 (o ) c c Notice that equ. 7 is s imilar to equ . 5 and that the schemes represented by fig . 2 and fig . 4 are quite the same . The differences are mainly that in fig . 4 only bicausal com­pensators are cons idered and that this bicau­sal compensator is imp lemented uniquely by s tatic state feedback . In fig . 3 the bicausal compensator implemented by feedback is used in order to exhibit the Hermite form (by doing so the transmiss ion zeros are cance l­led) .

Consider now the adapt ive s cheme of El l iot t , Wolowich and Das ( 1 982) . A direct est imation of the control parameters as we l l as an equi­valent parametrization of the unknown system is performed . More precise ly the problem of est imating the right matrix fraction descrip­t ion N (q- 1 ) D- l (q- 1 ) i s replaced by estima-

ting the matrices A(q- 1 ) and B ( q- 1 ) of A(q- l ) N ( q- I ) + B (q- l ) D (q- l ) = U(q- l ) (8 ) h ( - 1 ) . . d 1 . ( " - 1 ) w ere U q is a unimo u ar matrix in q •

The matrices A(q- 1 ) and B ( q- 1 ) can be thought as an equivalent parametrization of the sys­tem. The estimat ion of A(q- 1 ) and B ( q- 1 ) in equation ( 8) imp lies some indetermination in the closed loop transfer matrix which beco­mes : T (z- 1 ) = N ( z- 1 ) u- 1 (z- 1 ) D- 1 ( z- : ) D- 1 (o) (9 ) c - I

c where U (z ) is given by eq . 8 .

If the contro llability indices are all equa l , then U (z- 1 ) = I and the c losed loop structu­re is unique .

This approach allows to deal with non mini­mum phase sys tems . An important contribution is that the required priori information for implementation is fairly weak : only the con­trollability indices and an upper bound on the observability index . On the other hand a unique c losed loop structure cannot be guaranteed besides only local stability re­sults have been obtained .

Scheme deve lopped by Singh and Narendra ( 1 982)

The adaptive control scheme presented by Singh and Narenda ( 1 982) is basical ly for continuous time minimum phase systems . As previous ly we present the dis crete time ver­s ion of the linear control scheme .

The linear scheme developped by Singh and Narendra ( 1 982) turns out to be a particula­rization of the s cheme presented in fir . 2 . Firs t the precompensator AMl (q- l ) BM(q- ) is restricted to be a bicausal one . Second the observer polynomial C (q- 1 ) is equal to c ( q- l ) I where c ( q- 1 ) is a monic stab le po ly­nomial of degree (v- 1 ) , v being the observa­bility index . Then C (q- 1 ) commutes with any matrix . So Nu(q- 1 ) is r=placed

_by B; l Ny (f

- 1 ) , Ny ( q- 1 ) is replaced by B; l Ny (q 1 ) and B; disapears from the feedback , as shown on figure 5 . As in Morse ( 1 98 1 ) s tab le adaptive control is pos sib le provided - The Hermite form of the process transfer matrix T ( s ) is known . - The process is minimum phase , square and ful l rank . - An upper bound is known for the observabi­lity index of T ( s ) . - The sign definiteness of the high frequen­cy gain matrix Bo is known . (existence of L = LT > 0 : L BoT + Bo L = Q > 0) . The last assumption appears solely in the continuous t ime cas e .

I n order t o e liminate the a priori knowledge of the Hermite form, when the Hermite form is not diagonal , Singh and Narendra ( 1 982)

Parametrizations for Mul tivariable Adaptive Sys tems 1 6 1

proposed t o add a diagonal precompensator in order to diagonalize the Hermite form of the controlled process transfer�matrix. The two­input , two-output case is considered in some detail . The knowledge of the Hermite form is then replaced by the knowledge of the infini­te zero order of each element of T ( s ) . Notice that even for ( 2x2) systems such a diagonal precompensator not always exists . Much work remains to be done in this area but the idea is interesting. The adapt ive control scheme is a multivariable extens ion of Narendra , Lin and Valavani ( 1 980) . When the Hermite form is not positive real auxiliary signals have to be generated and added to the model output ( this in order t o satisfy the positi­ve real condition without adding differentia­tors) .

Scheme developped by Johansson ( 1 982) The scheme developped by Johansson ( 1 982) al­lows to deal with non-minimum phase multiva­riable sys tems . The required a priori know­ledge on the process T ( z- 1 ) = B ( z- l )A- l ( z- 1 ) is the following - The Smith form of the p lant transfer matrix over the ring of proper rational stable func­tions is known . This means that the infinite zeros and the unstable finite transmiss ion zeros are known exactly . - The polynomial degree o f A(z- 1 ) and B (z- 1 ) are known . � - A diagonalization of H (z) B ( z- 1 ) by a cau­�al stable precompensator is known . (where B ( z- 1 ) is bicausal) . In the minimum phase case , the infinite zero orders have to be known . Not ice that this knowledge is weaker than the knowledge of the Hermite form and that the sum of the control­labi lity indices is greater than or equal to the sum of the infinite zero orders . (If the process is made maximally non controllable by output inj ection and maximally non obser­vable by state feedback in an appropriate way , the controllability indices of the re­duced system are the infinite zero orders of the original process) . The knowledge of a diagonalization of T ( z ) is not restrictive . If d is the largest infinite zero order if is proved in Dion , Dugard ( 1 982) that z-d

-¥- (z) is proper , so is z-d 'i\'- l (z ) H- l (z) then : H(z) �( z- 1 ) z-d � 1 ( z- 1 ) H- 1 ( z ) = z-d I . In this way the input-output delays are equal to d . If one does not want to add as much de­lay , then the knowledge of a diagonalization of H(z ) is necessary . The linear scheme is represented in figure 6 . Comparing with figure I i t follow' that the observer poles are given by T 1 (q- )AM( q- 1 ) and then not indypendant of the chosen model . (If T 1 ( q- l )�1( q- ) = I we get the dead beat observer) .

- I - I The matrices S ( q ) , R ( q ) the precompensa-

tor and the Hermite form are estimated di­rectly from input-output data .

In Johansson ( 1 982) , the formulation deve­lopped by Pernebo ( 1 98 1 ) is used extensively . Here we have used the same factorization as in the previous sect ions .

The global stability of this algorithm is not studied in Johansson ( 1 982) .

Scheme developped by Goodwin, Mcinnis and Wang ( 1 982) The scheme developped by Goodwin , Mcinnis and Wang ( 1 982) treats the case when the number of inputs is greater or smaller than the number of outputs . The Hermite form is supposed to be known . This scheme is a gene­ralization of the scheme developped by Goodwin and Long ( 1 980) . The interest of this scheme is that : * When p>m output function controllability is impossib le in general, a cost function to minimize is then introduced . * When m>p , the number of inputs is greater than the number of outputs the control law is non unique . So it is required that perfect tracking is achieved while minimizing a cost funct ion ( the input energy) .

Global convergence is proved provided the knowledge of a suitable closed , bounded , con­vex region in Rmp is avai lable (bes ides the usual a priori knowledge : Hermite form • • . ) . The parameter estimation procedure is parti­cular in the sense that it forces the es ti­mate of a gain matrix to stay in the above defined region .

CONCLUDING REMARKS

In this paper different direct MIMO adaptive control schemes have been investigated . All the linear control schemes presented before have shown to be quite s imi lar , the diffe­rences being mainly in the considered adap­tive laws . Except in Elliot t , Wolowich and Das ( 1 982) where the controller structure is non unique , all the direct schemes develop­ped without the a priori knowledge of the Hermite form, include a precompens ator in or­der to diagonalize the control led process transfer matrix . In the discrete t ime case this precompensation adds some delays . The knowledge of the Hermite form is then repla­ced by the knowledge of the infinite zero orders of the p lant transfer matrix Johansson ( 1 982) or by the infinite zero orders of the plant transfer matrix e lements . Two papers deal with non minimum phase MIMO adaptive control El liott , Wolowich and Das ( 1 982) and Johansson ( 1 982) both yielding to a linear estimation scheme by over parametri­zation . The scheme developped by El liott , Wolowich and Das ( 1 982) is more general since the unstable transmiss ion zeros must not be known exactly .

1 62 J . M. Dion and L . Dugard

Some problems are still open in part icular global stability analys is of certain schemes and extens ion of known schemes have to be cons idered .

REFERENCES Chan , S . W . , G . C . Goodwin ( 1 982 ) . "On the role

of the interactor matrix in MIMO adaptive control " . IEEE , T . A . C . , AC-2 7 , No 3 , pp . 7 1 3- 7 1 4 .

u* (t)

Davison, E . J . ( 1 983) . "Some properties of mi­nimum phase sys tems and squared-down sys­tems" . IEEE, T . A . C . , Vo l . AC-28, No 2 , H • -I -

I H(q ) B. pp . 2 2 1 -2 2 2 .

Descus s e , J . , J . M . Dion ( 1 982) . " On the struc­ture at infinity of linear square decou­pled systems " . IEEE, T .A . C . , Vol . AC-2 7 , N o 4 , pp . 97 1 - 9 7 4 .

D i o n , J . M . ( 1 983) . "Feedback decoupling and infinite s t ructure of l inear systems " . Int . J . on Control .

Dion, J . M. , C. Commaul t ( 1 982) . 11Smith­McMi l l an factorizations at infinity of rational matrix funct i ons " . Systems and Control letters, No I , pp . 3 1 2 .

Dion, J . M . , L . Dugard ( 1 982 ) . "Direct adapti­ve control of discrete time multivariable systems " . IEEE, C . D . C . , Orlando, Decem­ber .

E l l i ot t , H . , W . A . Wolowich ( 1 982) . "A para­meter adaptive control structure for li­near multi variable systems0• IEEE, T . A . G . Vol . AC- 2 7 , N o 2 , p p . 340-35 2 .

E l l iott , H . , II . A . llolowich, M . Das ( 1 982) . "Arbitrary pole p l acement for linear mul­tivariable systems " . Technical report , University of Massachus e t t s , UMASS-ECE­SE82-2 , September .

E l l iott , H. ( 1 982) . "Hybrid adaptive control of continuous time systems " . IEEE T . A . C . Vol . AC-2 7 , N o 3 , Apri 1 .

Goodwin, G . C . , R . S . Long ( 1 980) . "Generaliza­tions of results on mul tivariab le adapti­ve contro l " . IEEE , T . A . C . , Vol . AC-27 , N o 3 , Apr i l .

Goodwin, G . C . , B . C . Mclnnis , J . C . Wang ( 1 982). "Mod e l reference adaptive control for systems having non square transfer func­tions " . IEEE, C . D . C . Orland o , December .

Hautus , M . L . J . , M. Heymann ( 1 97 8 ) . "Linear feedbac k , an algebraic approach" . SIAM J . on Control , 1 6 , pp . 8 3- 1 0 5 .

Johansson, R . ( J 982) . "Parame tric models of linear multivariable systems for adaptive contro l " . IEEE, C . D . C. Orlando, December.

Kailath, T . ( 1 980) . "Linear systems " . Prenti­ce Hal l , Englewood Cliffs , N . J . , USA.

Malabre , M. ( 1 982) . "Sur la struc.tute a l ' in­fini des triplets invariants " . Int . Con f . INRIA, Versai 1 l e s , France , Decemb e r .

Morse , A . S . ( 1 975) . "System invariants under feedback and cascade control" . Proc. Int . Symp . Udine , Springer Verlag.

Morse , A . S . ( 1 9 8 1 ) . "Parame trizations for multivariable adaptive contro l " . IEEE, C . D . C . San Diego, December .

Narendra, K. S . , Y . H . Lin, L . S . Valavani ( 1 980) "Stable adaptive controller design, part I I , proof of stab i l ity" . IEEE, T . A . C . , Vol . AC-2 5 , No 3, June .

Perneb o , L. ( 1 9 8 1 ) . -An algebraic theory for the design of contro l lers for l inear mul­tivariable systems " . IEEE, T . A . C . , Vo l . AC- 2 6 , pp . 1 7 1 - 1 8 2 .

Rosenbrock , H . H . ( 1 970) . "State space and mul t i variable theory " . Nelson, London .

Singh, R. P . , K . S . Narendra ( 1 982) . "A global­ly stable adaptive controller for multi­variable systems " . Report No 8 1 1 1 , Elect . Eng . , Yale University, Apri l .

Wolowich, W. A . , P . L . Falb ( 1 976) . "Invariants and canonical forms under dynamic compen­sation" . SIAM J. Control , 1 4 , p p . 996-1008 .

Wolowich, W . A . ( 1 974) . "Linear multivariable systems " . Springer .

. u

I I

Nu(q - 1 ) • C(q- l)B. - S(q -I ) B{q- 1 ) Ny(q- 1 ) • - q-d R(q- 1 ) ij-l q- 1 )

u* t) -I - I - 1 ,.,. (q ) BM(q )

u t PLANT

'( I ...-��--, 1 ...-�����

TR(q-1 ) I Tl (q- l)"i!{q-1 ) +

I � (t)

PLANT

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Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

LATTICE STRUCTURES FOR FACTORIZATION OF SAMPLE COVARIANCE MATRICES

B. Friedlander

Systrms Control Teclmulugy, Inc., 1 801 Pagr Mill Road, Palo Alto, CA 94304, USA

Abstract Many prob l ems i n s i gn al process i n g and l i near l east-squares est imat ion involve the factor i zat i on of a s ample covar i ance matr i x or i ts i nverse . L att i ce structures prov i de an eff i c i ent computat ional tech n i que for perform i ng th i s factor i zat io n . An overv i ew of l att i ce al gor ithms for off- 1 i ne ( b atch ) and on- 1 i ne ( adapt i ve ) matr i x factor i zat ion is presented .

1 . I NTRODUCTION I n recent years there h as been a growing i nterest in l at t i ce structures and the i r appl i c at ions to estimat i on , s i gnal process i ng , system i dent i f i c at i o n and rel ated probl ems . An extens i ve l i terature exi sts on the theory and pract i ce of l att i ce f i l ters [ 1 ] - [8] . I n th i s paper we present a sel ect ive rev iew of l at t i ce structures i n l i near l east-squares estimat i o n . The central theme of our d i scuss ion wi l l be the ro l e of l att i ce structures in factor i zat ion of samp l e covar i ance matr i ces . Due to space l imi tat i ons we wi l l defer most of the deta i l s to the reference s . Many adapt i ve s i gn al proces s i ng techn i ques are based on the so l ut i on of the fo l l ow ing prototype l i near pred i ct i on prob l em : l et Yt be a d i screte-t ime st at i onary zero-mean process . We are i nterested in pred i ct i ng the current val ue of th i s process from past measurements . A l i near pred i ctor of order N wi l l have the form :

A N Yt l t- 1 = - . I AN i Yt- i ( 1 )

A l =l ' where Yt l t- l is the pred i cted val ue of Yt based on data up to t ime t-1 , and {AN i , i= l , . . . , N} are the pred i ctor

' coeff i c i ents . The d i fference between the actual val ue of the process and i s pred i cted val ue wi l l be cal l ed the pred i ct i on error ( of order N )

N £N , t=yt-Yt i t- 1 =yt+ iilAN , i Yt- i ( 2 )

The l east-squares pred i ctor i s des i gned to m i n im i ze the sum of squared pred i ct i on erfors over some t ime i nterval

t £N t£N t . The i ndex T wi l l be added t=O ' '

when necessary to spec i fy the t ime i nterva l over wh i ch the errors are m i n im i zed ( e . g . , £N , t ( T ) , AN , i ( T ) ) . Us i ng th i s notat ion we rewr ite ( 2 ) as

[£N ,O ( T ) , . . • , £N , T(T ) ]

=[ I AN , l ( T ) , . . . , AN ,N (T ) ] YN+l ,T+l ( 3 )

where

'• . r" [o . ·�· : . · . : ,� : : : : r::: J ( 4 )

As we wi l l see 1 ater , the pred i ct i o n error £ N T (T ) can be computed recurs i ve ly i n t ime anti ' i n order , us i ng a s impl e l att i ce structure [4 ] , [ 7] . The vectors of pred i ct i on errors of d i fferent orders { p=0 , 1 , . . . , N } can be comb i ned i n the

fol l ow i ng matr i x form £N , O ( T ) . . . . . .

0 . £ 0( T-N+p) p ,

I AN l { T ) . • • • • • . '

I Ap , l (T-N+p) 0 .

£ p , T -p ( T -N+p)

YN+l ,T+l

or E�+l , T+l = � ,T YN+l , T+l ( 5 )

Thi s work was supported by the Offi ce o f Naval Research under Contract No. N0001 4-81 -C-0300 ,

1 63

1 64 B . Friedlander

A bas i c property of l e ast-squares pred ict i on errors i s that they are uncorrel ated w i th past dat a . I n other words [O, . . . , O , ep , O ( T-N+p ) , . . . , ep , T-N+p ( T-N+p ) ] [ 0 , . . , 0 Yo· · · · ·YT- i ] ' = 0 for i > N-p ( 6 )

U s i ng th i s property i t i s strai ghtforward to check that E�+l , T+l YN+l ,T+l i s a l ower tr i angul ar matr i x . Post-mu l t i p l yi ng ( 5 ) by y N+ 1 , T + 1 we get A ' e uN ,TYN+l , T+l yN+l ,T+l=EN+l , T+lyN+l ,T+l ( 7 )

I where RN ,T = YN+l ,T+lYN+l , T+l i s the ( pre-wi ndowed ) s amp l e covari ance matr i x . It al so fo l l ows that

-1 _ ( e , ) -1 A RN ,T - EN+l , T+l yN+l ,T+l UN ,T ( 8 )

The l ower tri angu l ar matr i x EeY c an be normal i zed to h ave un i ty al ong the d i agonal , by premu l t i p l yi ng it by oN:r , where

o� .T = d i ag { R� .r · · · · ·Ro ,T-N } ( 9 ) where R� , T -N+p i s the sum of squared pred i ct i on errors ( of order p ) . Equat i on ( 8) can now be wr i tten as -1 - ( -e e , ) -1 -e A , 1 ) RN ,T- 0N ,TEN+l , T+1YN+l , T+l 0N ,TuN ,T 0 Note that the r i ght-hand-s ide of th i s equat ion i s a product of a u n i t d i agonal l ower· tri ang u l ar matr i x , a d i agonal matr i x and a u i nt d i agonal upper tr i angu l ar matr i x . By symmetry of RN T i t fo l l ows that the two t r i angu l ar matdct:!s are s imp l y transposes o f each other , i . e . ,

RN:T = (� ,T ) ' 0N:T � .r ( n ) We conc l ude that the l e ast-squares pred i ctor coeff i c i ents can be computed by perfor� i ng a LOU decompo s i t ion of the i nverse of the s ampl e covar i ance matr i x . I nvert i ng both s i des of equat i on ( 1 1 ) g i ves an UOL decompo s it ion of the covar i ance matr i x RN ,T= (� ,T )

-lo� .T ( u� .T )-T= ( LN ,T )

' o� . TLN , T ( 12 ) aN , N (T )

0

aO , N (T )

a · . ( T ) = I l ' l Compar i son of ( 10 ) and ( 12 ) pro v i des an immed i ate i nterpretat i on for the entr i es of the matr ix LN , T

a 0-e Ee Y ' LN , T N ,T N+l , T+l N+l , T+l ( 14 )

I n other words , the entr i es o f the UOL factors of the s amp l e covari ance matr i x are s impl y normal i zed cros s-correl at ions between pred i ct i on errors and the dat a :

ap , i ( T ) =[ O , . . . , O , ep , O ( T-N+p ) , . • . ,

ep , T-N+p ( T-N+p ) ] [ O , O ,y0 , . . . ,yT- i ] ' ( 15 )

Latt i ce structures prov ide eff i c i ent computat i ona l procedures for such cross corre 1 at i ons , as wi 1 1 be d i scussed further in sect ion 3 ( see a l so [4] , [ 5] , [ 7] ) . These structures i nvol ve the backward pred i ctor i n add i t i on t o the forward pred i ctor d i scussed so far . Let us def i ne the b ac kward pred i ctor

A N Yt-N- l l t-1 = - i:l BN , N+l- iYt- i ( 16 )

and the correspond i ng b ackward pred i ct ion error

rN , t - 1 = Yt-N- 1 - Yt-N- l l t- 1 N . E1

BN N+l- iYt- i + Yt-N- 1 l = '

( 1 7 )

By fol l ow i ng t he same steps a s i n the case of the forward pred i ctor , it c an be shown that

I . .

or

0 . . . Bp :O (T ) · · · I . • .

BN , l (T) . I N+l , T+C

( 18 )

-1 _ ( B

) ' -r B ( RN ,T - LN ,T ON ,TLN , T ' 19 )

r d . Rr r ( ) ON ' T "' l ag { 0 ' T ' . . . ' RN ' T} ' 20 where Rr T is the s um of squared backward p , pred i c t i on errors . B y i nvert i ng ( 19 ) we get

RN , T= ( L� , T )- l o� . T ( L� , T)

-T= ( U� . T ) ' O� . Tu� .T ( 21 )

where 130 o ( T ) . . s0 , p ( T ) ' .

13 UN , T sp ,p �T.) 0

13 · · ( T ) l ' l

Lattice S tructures and Sample Covariance Matrices 1 65

and 6 -r r ' UN , T = DN , T EN+ 1 , T+ 1 y N+ 1 , T + 1 ( 23 )

To summar i ze : The coeff i c i ents of the backward pred i ct or c an be computed by performing a UDL decompo s i t i on of the i nverse of the s amp l e covar i ance matr i x . The entr i es o f the LDU factor o f the covar i ance matr i x i tsel f are normal i zed cross correl at ions between backward pred i ct i on errors and the d at a . More prec i se ly ,

Bp , ; ( T ) = [r p , O ( T ) , . . . , r p , T ( T ) ] ( 24 )

[ o , . . . , o Yo · · · · Yr_ ; J ' I n the next two sect ions we wi l l br i ef l y descr ibe some l att i ce structures for comput ing the factors uA , LB , La , u6

2 . OFF-LINE FACTOR I ZATI ON OF THE COVAR INACE MATR I X

The _factor i z at i on o f the s ampl e covar i ance matr i x can be performed in two d i fferent ways : off- l i ne ( b atch proces s i ng ) or on­l i ne ( t ime-recurs i ve ) . I n t h i s sect ion we cons i der the off- l i ne c ase . It i s assumed that data are co l l ected over a t ime i nterval [ 0 , T] and used to compute a set of corre l at i on coeff i c i ents . These corre l at i on coeffi c i ents are then used as an i nput to an a l gor ithm for compu t i ng the factors of

-1 RN , T or of RN ,T .

The wel 1 -k nown Lev i nson al gor i thm i s wide ly used for comput ing the forward and backward pred i ctors for stat ionary covar i ance matr i ces . Th i s a l gor ithm is c l ose ly rel ated to a l att i ce impl emetat i on of these pred i ctors [ 9 ] . Genera l i zat ions o f the Lev i nson al gor ithm h ave been deve loped for non-stat i onary covar i ance matr i ces of the type cons idered in the prev ious sect ion [8] , [ 10]-[ 12 ] . F i gure 1 dep i cts the l att i ce f i l ter rel ated to the pre-wi ndowed vers i on of the Lev i nson a l gor i thm . Th i s f i l ter impl ements the forward and the backward pred i ct ion f i l ters of a l l orders 0 .;; p .;; N . I n other words , the impu l se

response of th i s f i l ter pro v i des al l the t . f A B en r i es o uN , T ' LN , T See [ 7 ] , [ 8 ] , [ 11 ] -

[ 13 ] , for deta i l s .

The ga i n s (ref l ect ion coeff i c i ents ) of the l att i ce f i l ter in F i gure 1 c an be computed from the samp le correl at ion coeff i c i ents us i ng the so-ca l l ed Fast Chol es ky al gor i thm [ 10] , [ 14] , [ 15 ] . F i gure 2 dep i cts the l att i ce structure rel ated to t h i s al gor i thm . Th i s structure computes recurs i ve ly its own g a i n s , wh i ch c an then be i �serted in the l att i ce pred i ctor . The s i gnal s propagat i ng i n t h i s l att i ce structure dur i ng the computat ion turn out to be prec i se l y the ent r i es of Lf� . T . I n

other words , t h i s l att i ce f i l ter performs factor i zat i on of the covari ance matr i x RN , T and at the same t ime def i nes the ga i ns

for the pred i ct ion f i l ter , wh i ch in turn prov i des factor i zat ion of RN: T The l att i ce structure computes the entr i es of . u� , T , when the i nput sequence is reversed i n order . See [ 7 ] , [ 8] for more deta i l s . Genera l i zat i ons of t h i s l att i ce structure for the c l ass of a-stat i onary cov ar i ance matr i ces are d i scussed in [8 ] , [ l l ] , [ 12] .

3 . ADAPTI VE FACTORI ZATI ON OF THE COVAR IANCE MATR I X

I n many s i gna l process i ng and contro l appl i cat i on s , i t i s des i red to perform computat ions on-l i ne i n a t ime-recurs i ve manner , r ather than off- l i ne i n b atch mode . As a typ i ca l examp l e , con s i der the prob l em of est imat i ng the parameters of an auto-regress i ve (AR) process Yt • N

Yt = - E aN ·Yt · + Vt i= : , i - i ( 2 5 ) whe�e Vt i s a wh ite no i se proces s . G i ven an est imate of the parameters aN ; at t ime T-1 we want to upd ate the est imate when the dat � po i nt Yt; becomes av ai l ab l e . By compar i son to sect ion 1 it i s c l e ar that the l east­squares est imate of the AR parameters i s prec i se l y the f i rst row of

Au� . T I n other

wo�d s , we want to upd ate uN ,T- l to get UN , T

As was the case i n the off- l i ne s i tu at ion the computat ion of the pred i ct i on f i l te; proceeds i n two steps : ( i ) The l att i ce structure depi cted i n F i gure 3 i s used to compute a set of ga i n s (refl ect ion coeff i c i ents and cert a i n pred i ct ion errors ) . ( i i ) These gains are i nserted i nto the l att i ce pred i ctor in F i gure 1 , wh i c h can then be used to compute the entr i es of U� , T ' L� , T T h e s i gna l s propagat i ng i n

t h e adapt i ve l att i ce f i l ter ( F i g . 3 ) are the forward and backward pred i c t i on error s Ep , T ( T ) , rp , T- l ( T- 1 ) d i scus sed ear l i er .

For · a more deta i l ed descr i pt i on of these adapt i ve l att i ce f i l ters and the i r app l i c at i ons see [ 7 ] and the references therei n . To m'.:>t i v ate the adapt i ve factor i zat ion of RN , T , cons i der the prob l em of est imat i ng

the par ameters of a mov i ng-average (MA) process Yt • N

Yt = E cN N . Yt . + wt i =l ' - i - i (26 ) where Wt is a wh ite no i se process . It is stra i ghtforward to check that the MA �arameters are g i ven by the f i rst row of u i n the UDL factor i z at i on of the true

1 66 B . Friedlander covar i ance matr i x of the process Yt =

RN-E : -UN ON LN 6 �. T-lJ [yT- 1 ' 0 . ,yT-NJ _

YT-N

I • cN N-1 cN , O . . '

UN . I cp , p- 1 � p ,O

0

( 2 7)

( 28)

Therefore , a cons i stent est imate of the MA coeff i c i ents i s g i ven by the f irst row of ( L;, T J ' . It can be shown that l im ap' N ( T ) = cN , p=l , . . . ,N • T+oo ' , p ( 29)

In other word s , adapt ive factor i zat ion of the s amp l e covar i ance prov i des a run n i ng est imate of the MA parameters of the process Yt · See [ 14] , [ 15 ] for more deta i l s . F i gure 4 dep icts a l att i ce structure for comput i ng a N ( T ) recurs i ve ly i n t ime and p , i n order . The parameters a N ( T ) appear as p , cert a i n ga i n s i n th i s f i l ter . See [ 16 ] , [ 17 ] for more dta i l s . Thu s , thei r computat ion i s more d i rect than the computat ion of AN , i ( T ) , BN , i (T) , wh ich req u i res a separate step for recover i ng the pred i ctor coeff i c i ents from the ga i n s of the l at t i ce f i l ter . Note that an i n d i rect way for comput i ng ( L� , T ) ' i s to compute U� , T and i nvert i t by back-sust itut i on .

A dual l attice structure can be der i ved for comput i ng the ent r i es of u� , T These entr ies can be i nterpreted as the parameters of an ant i -c ausal mov i ng average model for the process Yt [ 16 ] .

4 . CONCLUS I ONS A br i ef overv i ew of l att i ce structures for covar i ance matr i x factor i zat i on was presented . These al gor i thms h ave appl i c at i ons to var i o u s s i gal process i ng prob l ems i n c l ud i n g : adapt i ve f i l ter i ng , spectral anal ys i s and s i gn a l mode l i ng . Extens i ons of these resu l ts to model i ng and estimat i on of ARMA processes i s current 1 y under i nvest i g at i o n .

REFERENCES 1 . J . Makhou l , " St ab l e and Eff i c i ent Latt i ce Methods for L i near Pred i ct ion , " IEEE Transact ions on Acoust i c s , Speech and S i--nal Process i ng , pp . 423- 28, cto er 9 2 . L . J . Gr i ff i ths , "An Adapt ive L att i ce Structure for No i se-Cancel i ng Appl i c at i on s , " Proceed i ngs of IEEE Internat i onal Conference

4

on Acoust i c s , Speech , and S i gnal Process i n g , Tu l s a , Okl ahoma, pp . 87-90, Apr i l 1978 . 3 . M . J . Shensa , " Recurs i ve Latt i ce A l gor i thms : A Approach , " IEEE Trans . Automat . AC-26, pp . 695-702 , June 1981 .

Least-Squares Geometr i c a l Contr . , vo 1 .

4 . 0 . Lee , M . Morf , and B . Fr i edl ander , " Recur s i ve Square-Root Ladder E st i at ion A l gori thms , " IEEE Trans . on Acoust i c s , Speech , and S i gnal Proces s i n g , vo l . ASSP-29 , no . 3 , pp . 627-641 , June 1981 . Al so i n I EEE Trans . C i rcu i t s and Systes , vol . CAS-28 ,--no:-6, pp . 467-481 , June 1981 . 5 . B . Porat , B . Fr i edl ander , and M . Morf, " Square-Root Covar i ance Ladder A 1 gor ithms , " I EEE Trans . o n Automat i c Control , vol . AC-27, no . 4, 813-829, August 1982 . 6 . 0 . Lee , B . Fr i edl ander , and M . Morf, " Recurs ive Ladder Forms for ARMA Mode 1 i ng , " I EEE Trans . Automat i c Control , AC-27 , no . 4 , 753-764 , August 1982 . 7 . B . Fr i ed l ander , " L att i ce F i l ters for Adapt ive Process i ng , " Proc . IEEE , vol . 70 , no . 8, pp . 829 -867 , August 1982 . 8 . B . Fr i edl ander , " Latt i ce Methods for Spectral Est imat ion , " Proc . IEEE , vol . 70 , no . 9 , pp . 990-101 7 , September 1982 . 9 . J . D . Markel and A . H . Gray, Jr . , L i near Pred i ct i on of Speech , New York : Spr i ger­Verl ag , 1976 . 10 . M . Morf, " Fast Al gor i thms for Mul t i var i ab l e Systems , " Ph . D . d i ssertat i o n , Dept . E l ec . Eng . , Stanford Un i ver s i ty, Stanford , CA , 1974 . 1 1 . J . M . Del o sme , "A l gor i thms and Impl emetat i ons for L i near Least -Squares Estimat i on , " Ph . D . d i ssertat i on , Stanford Un i vers i ty, St anford , CA , 1982 . 12 . H . Lev -Ar i and T . Kai l ath , "On General i zed Schur and Lev i nson-Szego A l gor i thms for Quas i -St at ionary Processes , " i n Proc . 20th IEEE Conf . Dec i s i o� and Contro l ( San D i ego , CA , Dec . 1981 ) , pp . 1077- 1080 . 13 . B . Fr i ed l ander , M . Morf , T . Kai l ath , and L . Lj ung , " New I nvers ion Formu l as for Matr i ces Cl as s i f i ed i n Terms of The i r D i stance from Toep l i t z Matr i ce s , " L i near A l gebra and It s Appl i c at i o n s , vo l . 27 , pp . 31-60, 1979 . 14 . B . Fr i ed l ander , "A Latt i ce A l gor i thm for Factor i ng the Spectrum of a Mov i ng Average Proces s , " i n Proc . Con f . on I nformat ion Sc i ences and Systems , Pri nceton Un i v . , Pr i nceton , N J , Mar . 1982 . Al so , i n IEEE Tran s . Automat i c Contro l , Ju ly 1983 . ·--

15,� M D Morf� C,,.1:1 . tMur .. av.ch i,�11 P1. H . . Atoa . and J-1·1 . e Tosmo:: , Fas L;ho 1 es"r' A gor i nms ana Adapt i ve Feedback F i l ters , ' i n Proc . IEEE

Lattice Structures and Sample Covariance Matrices 1 6 7

Conf . Acoust . Speech Signal Process , ( P ar i s , France , May 1982 ) , pp . 1727-173 1 . 1 6 . B . Fr i ed l ander , "Adapt ive Structures for Factor i z at i on of Covar i ance Matr i ces , " subm i tted pub l i cat ion .

Latt ice Sampl e

for

17 . C . H . Muravch i k and M . Morf , "A New Stab l e Feedback Ladder A l gor i thm for the Ident i f i cat ion of Mov i ng Average Processes , " Proc . 1983 I EEE Conf . on Acoust i c s , Speech and S i gnal Proces s i ng , pp . 683-686 , Boston , MA , Apr i l 1983 .

APPEND I X DER I VATION OF THE LATT I CE ALGOR ITHM FOR FACTOR ING THE SAMPLE COVAR IANCE MATRI X

We present here a br i ef der i v at i on o f the l att ice al gor ithm depi cted in F i gure 4 . The der i v at i on is based on the project i on approach presented i n [ 7 ] . Due to space l im i tat i ons we defer many of the deta i l s to [16 ] . The var i ous l attice var i ab l es are obta i ned as project ions of certa in vectors on the space spanned by the dat a . We def i ne Ps as the project ion operator on the space spanned by the rows of the matr i x S ,

(A l ) The fo l l owing update formul a ho l d s for arb itrary matr i ces V , W , S , X (of compat i b l e d imens ions ) : V ( I-P { S+X} ) W ' = V ( I -Ps ) W ' -V( I -Ps ) X ' ( X ( I -P5 ) X ' ) -

1X ( I -P5) W ' (A2 )

Us i ng th i s formu l a with proper choi ces for V , W , S , and X we can gener ate the des i red l att i ce recurs ions . Tab l e 1 summar i zes al l the var i ab l e s needed for these recurs ion s . Var i ous cho i ces of X wi l l g i ve d i fferent types of updates . The fo l l owing are the most usefu l :

' .. 1 .T-N+p+l' F�:l J rp . T-N+p

= t ime and order udpate

yp+l ,T-N+p ��:�-N+� , 0'de'

I -P =[-Pyp ,T-N+p-1

Yp,T-N+p+11 0 . . . . . . t ime update

(A3 )

update(A4)

r] (A5 ) . . 0

Us i ng (A3 ) - (A5 ) and the def i n i t ions i n Tab l e 1 , we can f i l l i n the entr i es o f Tab l e 2 , wh ich conta ins the var i ous quant i t i es

i nvol ved i n the update formu l a . Each l i ne of the tab l e trans l ates i nto an update equat ion . As an exampl e , the f i rst two l i nes g i ve

ep+l , T-N+p = ep , T-N+p-Ap+l ,T-N+p • R-r p , T-N+p-1 rp, T-N+p-1

(A6)

r - r A ' p+l ,T-N+p - p ,T-N+p- 1- p+l , T-N+p (Al ) • R-e p ,T-N+p ep , T-N+p

wh ich c an be rewr i tten as

ep, T-N+p = ep+l , T-N+p -r (AS)

+ Ap+ l , T-N+p Rp , T-N+p-1 rp , T-N+p- 1 rp+l ,T-N+p = [ I -Ap+ l , T-N+pR��T-N+p

R-r ] • Ap+l , T-N+p p , T-N+p- 1 rp , T-N+p- 1 A ' R-e - p+l ,T-N+p p , T-N+p ep+l ,T-N+p

= Rr R-r p+l , T-N+p p ,T-N+p-1 rp ,T-N+p-1 I R-g

(A9) -Ap+l , T-N+p p ,T-N+p ep+l , T-N+p The th i rd l i ne g i ves

Ap+l ,T-N+p = Ap+l , T-N+p-1 + ep , T-N+p -c r ' • Yp-1 , T-N+p- 1 p , T-N+p- 1

-c Ap+ l ,T-N+p-1 + gp+ l , T-N+p Yp- 1 , T-N+p-1

• r ' + A R-r p , T-N+p-1 p+l , T-N+p p , T-N+p-1 r -c r ' • p ,T-N+p-1 Yp-1 , T-N+p- 1 p ,T-N+p-1

(AlO ) or

R-r Ap+l ,T-N+p p , T-N+p- 1

( R� , T-N+p-1 -rp , T-N+p-lY�=l ,T-N+p-l rp , T-N+p- l J =A +e -c r ' p+l ,T-N+p-1 p+l ,T-N+pYp-1 ,T-N+p- 1 p , T-N+p- 1

(All ) and f i na l l y A - [A + -c p+l , T-N+p - p+l ,T-N+p-1 ep+l ,T-N+pYp- 1 , T-N+p-1

• r ' ] R-r Rr p ,T-N+p- 1 p , T-N+p-2 p , T-N+p-1 (Al2 )

The compl ete Tab l es 3 and i n i t i al i zat i on i denti t i es were tabl e :

al gor i thm i s descri bed 4 . Tab l e 3 summar i zes procedure . The fol l owi ng used in constructi ng thi s

i n the

1 68 B . Friedlander

6�+1 ,T ( N ) =yo : T ( I -PyN , T )Yo : T=R� . T (Al5 ) These ident i t i es ho l d for al l N < T Tab l e 4 summari zes the a l gor i thm dur ing norma l operation. A more deta i l ed der i vat i on i s presented i n ( 1 6 ] , Th i s a l gori thm was previ sou ly presented i n ( 1 7 ] ,

v

'°�� ta:i N-p Yo:T

'°�� N+l Yo: T

n

Yo :T

Yo :T

Yo:T

Tabl e 1 : Defi n i t ion of Var iab l es For The Unnorma l i zed Latti ce

w

n

n

N+p Yo:T N-p Yo:T N+l Yo:T

n

N-p Yo :T N+l Yo:T

n

S • YP T-N+p . V( I-P5)W' Coment

•p, T-N+p Forward prediction error

rp,T-N+p-1 Backward prediction error

6p+l,T-N+p Cross correl ation coefficient

R� ,T-N+p Forward predict ion error covari ance

Rr Backward prediction error p, T-N+p-1 covarf ance c Likel ihood variable Yp-1 ,T-N+p-l

6�+1 , T(N) cross corre 1 at ion

Ap+l,T(N) cross correl at ion

ep , T(N) predict ion error

Tab le 3 : In i t i a l i zation of the Latt i ce Algori thm

Init ia l ization : R�,O • YoYo , •o,l • Yl ' rO,O • Yo Al, 1 (o) • R0, 1 • 'oYo • Y1Yi Al , ! (0) • Al,! • Y1Yo y�l,0 • 1

For t•l , . . ,N do Al,1 (t-1 ) = YtYO + Yt+1Yi li1 ,1( t-l) . YtYo( · Ai,o(t ) ) el ,l (t ) = Y1

Do for p=O, . . , t-2 t�+2,0(t ) = A�+l,l (t-1 ) - lip+! , ! (t-l)R�:pAp+l,p+! p+2 ,1( t-l) = Ap+l ,l( t-1) - A�+l,l( t-1 ) R�:p+l Ap+l,p+l �+2 ,l ( t) = 6�+1 ,0(t ) + •p+l,l (t ) y��p+l •p+l, p+2 p+2 ,l (t ) = •p+l,l (t ) - 6�2,l (t) R;;!l,p+2 •p+! ,p+2

y�-1 , t = y�-2,t-l - ti-1, t Rt�l,t Et-1 ,t tt , t+l = •t,l (t ) At,t " At , ! ( t-1 ) 6t+l ,0( t ) = 6t,l (t-l ) - 6t ,t Rt�! ,t-1 6t,t 6t+l , l (t ) = 6t+l ,0(t ) + tt,t+l Yt�l , t ti,t+l R� ,t = R�-1 ,t-l - 6t,t Rt�l,t 6t ,t RLt+l = 6t+l , l ( t )

Tabl e 2 : Update Formu l as for the Unnorma l i zed Lattice

. x v w V( l-P1s+x1lW' V( l-P5)W' V( l-P5)X ' X( l-P5 )X ' X( l-P5)W'

r&:i Yo :T n tp+l , T Ep, T 6p+l,T R� , T-1 r p, T-1

YQ:T p+l Yo:T n r p+l, T r p, T-1 AP+l, T R�, T Ep, T

n Yo:T r&:i 6p+!, T-1 6p+l, T c r ' tp, T Yp-1,T-l p, T-1

R�, T-1 R�,T c tp, T n Yo:T Yo :T tp, T Yp-1 , T-1

yr,} �} R�, T-2 R�,T-1 r p, T-1 c r ' n Yp-l ,T-1 p, T-1

Yo: T n n c Yp,T c EP , T R�, T tp, T Yp-l,T-1

r&:i n n c c r' R� , T-1 r Yp, T-1 Yp-l ,T-1 p, T-1 p, T-1

�} Yo :T Yo :T R�+l,T R�,T 6p+l, T Rr p, T-1 6P+t , T

Yo :T 1o:i lo�} R�+l ,T R�, T-1 AP+l ,T R�,T 6p+! ,T I

S = Yp T-N+p

n Yo:T N-p Yo :T 6�+1 , T-1 (N) A�l ,T(N) •p, T(N) c tp, T-N+p Yp-1 ,T-N+p-l

'°�� Yo:T n ep+l , T(N) ep, T(N) A�l ,T(N) Rt p, T-N+p Ep, T-N+p /

N+l Yo :T N-p A�+2 ,T(N+l ) A�+! , T(N) tip+l, T(N) Rr 6P.l ,T-N+p Yo:T Yo :T p , T-N-p-1

'°�� N+l lip+2, T(Ni tip+l, T(N) A�l ,T(N) Rt 6p+l, T-N+p Yo :T Yo :T p, T-N-p i

Tabl e 4 : The Latti ce Al gori thm for UL Factori zat i on Define t = T-N For t•l ,2 , . • . do the follo�ing

I"'"" ro,t • co,t • R�,t = R�.t-1 + ro,t rO,t ' RO,t+l • RQ,t + Yt+l Yi.+1 y�l,t • l ' eO,t+N+l (N) = Yt+l

For p-O, . . . ,N-1 do: � [ -c , ] -r r 6p+l, t+p+l" 6p+l, t+p +ep+l ,t+p+l Yp-1 , t+pr p, t+p Rp, t+p-1 Rp, t+p cp, t+p+l = cp+l ,t+p+l +ti.p+l ,t+p+l R;: t+pr p, t+p R�,t+p+l = R�.t+p + Ep,t+p+l r;:l,t+p c�.t+p+l R�+ l , t+p+ 1 •R�, t+p ·ti.;,+ 1, t+p+l Rt!p+l Ap+ 1, t+p+ 1 r p+l, t+p+l =R�+l , t+p+l R�� t+pr p, t+p-AP+t, t+p+l R�: t+p+l cp+l, t+p+l 6:.1 , t+N+l (N)•6:.1 , t+N(N)+ep, t+N+l (N)y ;;:l, t+p•p, t+p+l ep+l, t+N+l (N)=ep, t+N+l (N) -6:.1,t+p+l R;: t+p+l •p, t+p+l Y�,t+p+l = Y�-1,t+p"'EP,t+p+lR�:t+p+lcp,t+p+l «p,N (t+p+l) = R��{!p+l (6�+1 ,t+N+l (N) ) '

or «p,N( t+p+l) • (6:.l,t+N+l (N) ) ' -<N, t+N+l = eN, t+N+l (N) �. t+N+l = R=. T+N + cN, t+N+l rff�t. t+N cN, t+N+l - ( ) e/2 � 0N,N t+N+l = RN,t+N+l

-1/2 Ro,o

Input

0

Fi gure 1 : The ( pre-wi ndowed ) Latti ce Pred i cti on Fi l ter

F i g ure 2: The (pre-wi ndowed ) Ga i n Computi ng Latti ce Fi l ter

;- - - - - - - - - -- - ,

"M-1 ,T :

I I I I L------- ------.J e

N

Fi gure 3 : The Unnorma l i zed Pre-Wi ndowed Least-Squares Latti ce F i l te r

§ p.. tn � "d ,..... O> (') 0 < Pl ., .... Pl ::l (') O>

� rO,t rl ,t+l ---� rz , t+2 rN-1 ,t+N-1 __ __, �

1----- rN ,t+N h0

•z ,t+l (N)

F i g ure 4 : The Latt i ce F i l ter for UL Factori zat ion of the Samp l e Covari ance Matri x

O> "'

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1 983

REAL TIME VIBRATION CONTROL OF ROTATING CIRCULAR PLATES BY TEMPERATURE CONTROL

AND SYSTEM IDENTIFICATION

C. D. Mote, Jr.* and A. Rahimi**

''Dejmrlnu'n/ of Mnlumical btgin1'1'ring, University of Califi1rnia, lirrkl'in', C A 9-li20, USA **Shugart A.1.1wiatn, Sunny11ale, CA 940H6, USA -

Abstract . A system for rea l - t ime contro l o f the transverse v i bra­t i on of a rotat i ng c i rcu l ar p l ate, based on a therma l stres s i ng techn i que and dynami c system i denti f i cat i on , i s presented i n th i s paper. I n th i s method the p l ate natural frequency spectrum i s mod i f i ed th rough the purposefu l i ntroducti on of thermal membrane stres ses . The cri t i c a l speed is max i mi zed . In effect v i b rati on i s contro l l ed through rea l -t ime contro l of the p l ate des i g n . Eva l u ati ons wi th computer s i mu l ati on and experi menta l measure­ments on a th i n ci rcu l ar p l ate ver i fy the system capab i l i ty to control transverse vi bration i n a chang i ng thermal env i ronment . Keyword s . Vi brati on Contro l , Vi brati on , System I denti f i c at i o n , C i rcu l ar P l ates , Temperature Contro l , Rea l -T ime Des i gn Contro l I NTRODUCT I ON

A techni que i s proposed to contro l trans ­verse vi brati on of rotati ng c i rcu l ar p l ates through control of the membrane stress state. The control i s one of chang i ng the system dynami c s , i ndeed the bas i c system des i g n , i n a real - t ime manner . The theore­t i c a l pred i c t i ons are eva l u ated wi th both computer s i mu l ati ons and l aboratory measure­ments . The app l i c ati on of th i s research of par­t i cu l ar i nterest to the authors i s the control of transverse v i b rati on in c i rcu l ar saw b l ades . Approxi mately 2 5% of al l pro­cessed t imber becomes cutt i ng res i dua l at some stage . More accurate cutti ng , reduced sawb l ade kerf , and control of transverse v i b rati on duri ng cutti ng are necessary for reducti on of cutti ng res i dua l .

CR I T I CAL SP EEDS The cri ti ca l speed i n stab i l i ty is a mov i ng l oad resonance determi ned from the trans­verse vi brat ion spectrum;

w .. . (�) �' . t = m1 n cr1 Vn ,m n m 0 , 1 , 2 , . . . n = 1 , 2 , 3 , . . .

( 1 )

where m and n denote the noda l c i rc l es and d i ameters of the c i rcu l ar p l ate v i brati on modes ( n ,m ) ; see L ap i n ( 1959 ) , Mote and Ni eh ( 1971 ) , and Mote and Hol #yen ( 1975 ) .

1 7 1

The natural frequenc i es of an axi symmetr i c c i rcu l ar p l ate, apparent to a ground-based observer are wmn � n� where the p l u s and mi nus components are produced by trans l at i on of the waves defi n i ng mode ( n , m ) by rotat i on � . The h i g her frequency corresponds to the wave component in the di recti on of rotat i on , t he forward wave, and the l ower frequency corresponds to the backward wave component . The mode ( n , m ) i n wh i ch the l owest ground­based frequency van i s hes at the sma l l est rotat i on speed i s the cri t i ca l speed mode, and that rotat i on speed is termed the cri ­ti ca l speed , �cri t · Measurements i n pro­ducti on exper i ments confi rmed that the proxi mi ty of the rotat i on speed to the cri ­t i c a l speed i s di rect ly rel ated to the s awb l ade transverse vi brat i on and the pro­duct d i mensi onal accuracy ( Mote, et. a l , 1975 ' 1981 ) . The vi brati on contro l system di scussed here uses i nduced thermal memb rane stresses to control the natural frequency spectrum and max i mi ze the cri t i c a l speed . A typ i c a l control cyc l e i s descr ibed a s fol l ows : p l ate axi symmetr i c temperature di stri bu­t i on is measured in real - t ime; the natural frequenc i es are computed through a spectral observer ; control a l gori thm pred i cts the current cri t i c a l speed and i n stabi l i ty mode ; a l gori thm est imates the potenti a l benef i t f rom heati ng o r coo l i ng the p l ate; heat source commanded accord i ng ly ; see F i g . 1 . The Eu l er equati on of the stati onary va l u e prob l em

1 72 G .D . Mote , Jr and A . Rahimi

� [2lrr aJb

0 ( 2 2 (�) 2- 0ee (�)2 pwmn v - 0rr ar r2 ae

a ( 1 av 2 (ar r 3'8) ) } ] ) H r dr 0 ( 2 )

i s the fi e ld equation for the ei genva l u e prob l em govern i ng the natural frequenc i es wmn of an axi symmetri c pl ate wi th membrane stresses ( Mote, 1965 ) . The natural frequen­c i es wmn are approxi mate ly determi ned through app l i cat i on of the Ri tz method to ( 2 ) wi th mode v ( r , e ) represented as a separab l e prod uct of a rad i a l di stri b ut i o n and an ang u l ar di stri buti on contai n i ng nodal di ameters , i . e . , Cos ne . The membrane stresses resu l t i ng from axi symmetr i c thermal and rotati onal effects are

r 0rr

_ aE J T r2 a

c c r dr - 3;v pr2s-t2 + -{ + ) (3 )

0ee r

aE ( -T + ;2 f T r dr ) a

where constants c1 and c2 are determi ned from boundary cond i t i ons at r = a and r

( 4 )

b . The temperature i s descri bed by Fouri er heat conducti on with convecti on from the pl ate surfaces

Wi th a change of vari ab l es gi ven in the Nomencl ature, ( 5 ) becomes

. 2 v2f - h T + gQ_ TI_ ( 6 )

o K0 a t

For axi symmetry of temperature, t he steady state conducti on-convec t i on equat i o n from ( 6 ) i s a mod i f i ed Bes sel s equat i o n . As the mod i f i ed B i ot number ho i ncreases , the tem­perature grad i ent and the maxi mum thermal stress i ncrease . Determi nati on of the temperature di str i bu­t i on from ( 6 ) , computat i on of the associ ated thermal stresses ( 3 ) - ( 4 ) and pred i ct i on of the resu l t i ng natural frequenc i es ( 2 ) , shows that a " typi cal " temperature ri se at the pl ate peri phery sh i fts the natural frequency spectrum downward i n modes ( m, n ) for nodal

d i ameters n = 2 , 3 , 4 , • . • and al l m and upward i n modes (m, n ) for n = 0 , 1 and al l m. Conversel y , an i ncrease i n temperature near the center of the pl ate sh i fts the natural frequenc i es upwards for n = 2 , 3 , 4 • . . and downwards for n = 0 , 1 for al l m ( Mote, 196 5 ; Mote and N i eh , 1971 ) . Theoreti cal analyses and experi mental tests focused on the pred i ct i on of the control heat f l ux to max i mi ze the d i fference between the rotat i on speed and the l owest frequency of a backward travel i ng wave mode, wmn - nQ. The dependence of the backward travel i ng wave frequenci es upon the contro l heat f l u x Oc i s i l l u s trated i n F i g . 2 for a part i cu l ar case . For Oc < 1 . 25 W/cm3 the cri t i cal speed i s exceeded in mode ( n ,m) = ( 3 , 0 ) and for Oc > 1 1 . 55 W/cm3 d i vergence buck l i ng occurs i n ( 0 , 0 ) . The o�t imal heat fl u x , approxi matel y 6 . 34 W/cm , was determi ned by equal i ty of the decreasi ng ( 1 , 0 ) and i ncreas i ng ( 2 , 0 ) natural frequenci es , wh i ch max i mi zes the pl ate stabi l i ty accord i ng to the cri t i c a l speed theory ( Mote and N i eh , 1971 ) . Thi s ground based frequency was approxi mately 50 Hz for thi s pl ate for al l peri pheral heat fl u x states tested .

CONTROL A method of spectrum est imati on was deve­l oped from concepts of s i g nal model i ng . I n th i s techn i que a predi ct i on model of the v i brat i on s i g nal i s " fi t" to the observed transverse d i sp l acement data through choi ce of coeff i c i ents descri b i ng the model . The model coeff i c i ents mi n i mi ze the square of the di fference between the di sp l acement data and the model predi cti on of the data . The model coeff i c i ents are conti nuous ly updated thereby g i v i ng an adapt i ve and current model of the p l ate v i b rati o n . The natural frequenc i es that domi nate the transverse vi brat i o n are esti mated conti nuou s ly from the s i gnal model . Detai l s of thi s s i gnal model i ng research are g i ven i n references . ( L andau , et . al . , 1982 ; Rah i mi , 1982 ) . The contro l l ed vari ab l es are the pl ate natural frequenc i es , the equated frequency pai r i n parti cu l ar . The set-po i nt ( reference i npu t ) is the equal i ty of natural frequency pai r max i mi z i ng the cri ti cal speed . An adapti ve mode l for the pl ate was deve­l oped wi th Oc as i nput and the natural fre­quenc i es of the backward travel i ng waves as outpu t s . The pu l se transfer fu ncti on for th i s mode 1 i s [G l '. '- l )]

GL ( z- 1 )

( 7 )

where L equal s the number of pl ate model natural frequrnc i es . Each el ement of th i s vector, G0 ( z- ) , i s a rati onal express i on i n

Control of Rotat ing Circular Plates 1 73

the sh i ft ' operator z wi th unk nown parameters a · and b i that are i de nti f i ed i n real t ime u� i ng moae l reference adapti ve systems (MRAS ) techn i ques ;

-1 -2 -q 1 b1 z + b2z + . . . . + b z

G ( z - ) = (---------.--�9-) p - 1

for p = l , . . . . , L

1 - a 1 z p ( 8 )

"q " i s the order of the model . The s truc­ture of the feedback contro l system wi th the MRAS al gori thm for i denti fi cat i on of the pl ate model i s shown i n F i g . 3 . I n state vari ab l e form, ( 8 ) i s �( k+l ) = � �( k ) + q u ( k ) y ( k ) = � �( k )

l 0

( 9 ) ( 10)

p = [; .�] �f J. � = [ l 0 . . O J 0 0 l x q 0 9q q x q q x 1

and

91 bi ( 1 1 ) g i bi + a1 9 i - 1 for i = 2 , . . . . , q

The i nput u ( k ) i s the control heat fl ux and the output y( k ) i s one natural frequency for each p i n ( 8 ) . The control sel ected for th i s adapti ve system i s a state vector feed­back wi th i ntegral act i on on the output( s ) . The control l aw i s fi n i te ti me settl i ng con­trol ( FTSC ) for d i screte-t ime systems ; the frequency set-poi nt i s theoret i cal ly obtai ned i n " q " ti me steps for the q-order model . The feedback gai n s , � = [K1 , K2 · · · · Kq J , and the i ntegral gai n , K r , are determi ned for the mode l ( 8 ) so that the control system i s f i ­n i te t ime settl i ng ( Takahash i , e� a l . , 1975 ) ;

= K = K q- 1 I ( 12 )

The contro l l er heat fl ux output i s g i ven by

k u ( k ) = K1 I [ r ( j ) - y ( j ) J -

j=O

Kl y ( k ) - I K. x . ( k ) i =2 l l

( 13 )

where r ( j ) i s the j-th frequency set-poi nt . The state vari ab l es x2 ( k ) , . . . xq ( k ) are esti ­�ated by a state observer . The performance of FTSC deter i orates when measu rement noi se i s apprec i ab l e . The pl ate model i n (8 ) i s determi ned i n rea l - t i me through i den t if i ca­t i o n , and the control l er gai ns K 1 , and K i are updated . The des i g n of a modal control for the adap­t i ve system i s obtai ned through the transfer functi on

G ( z- l ) = � p A ( z- l ) ( 14 )

for p = 1 , . . . L . Equati on ( 14 ) , expres sed as a process i n a determi n i sti c envi ronment , becomes A ( z-1 ) y ( k+l ) = B( z-1 ) u ( k+l ) where A ( z-1 ) 1 - ai z-1 - . . . - anaz-na

B ( z- 1 ) biz- l + b2z-2 + . . . + bnb z-nb

( 15 )

The roots of B ( z-1 ) = 0 here al l l i e wi th i n the un i t ci rc l e , l z I < 1 , g i v i ng the di s­crete time equ i v al ent of a mi n i mum phase system. The procedure i s to choose a modal control l aw ( po l e- pl acement ) such that the c l osed l oop pol es are the roots of ( Egardt , 1980 ) H ( z-1 ) = 1 + h1 z- l + . . . + hnaz-na = 0 ( 16 ) For the frequency set-poi nt error e ( k+l ) = r ( k+l ) - y ( k+ l ) ( 17 ) the control object i ve i s sati sf i ed i f H ( z-1 ) e ( k+ l ) = O for al l k > 0 ( 18 ) The modal control i s ach i eved by i nput

where S ( z- 1 ) = z[ H ( z- 1 ) - A( z- 1 ) ] B* ( z-1 ) = zB ( z- 1 )

( 19 )

( 20 ) ( 2 1 )

I nc l u s i on of stochas t i c di sturbances changes the model ( 15 ) to A ( z- l ) y ( k+ l ) = B ( z- l ) u ( k+l ) + D ( z-l ) v ( k+l )

( 22 )

1 74 C . D . Mo te, Jr and A . Rahimi

where v ( k ) i s assumed to be Gaus s i an wh i te noi se ( 0 , o ) , and D ( z-1 ) i s the stochas t i c d i s turbance polynomi al D ( z-1 ) = 1 + dl z-l + . . . + dnaz- na ( 23 ) The d i are determi ned by i dent i f i cati o n . I n the stochas t i c model the obj ect i v e i s to sati sfy the mi n i mum vari ance control l aw ( Egardt , 1980 ) ; E {e2 ( k i} = mi n ( 24 ) The mi n i mum vari ance control i nput i s

( 25 )

where ( 26 )

COMPUTER S IMULAT I ON The temperature di stri buti on i s determi ned through so l u ti on of the mod i f i ed heat con­ducti on equat i on vi a an exp l i c i t di fference method . The natural frequenci es and membrane stresses are determi ned from ( 2 ) - ( 4 ) . The natural frequenc i es are the control vari ab l es and the al gori thm pred i cts the control heat f l u x to max imi ze the smal l est backward wave frequency and hence the cri t i ca l speed . Numeri cal experi ments were conducted us i ng on-off contro l , propor­t i onal pl us i ntegral contro l and adapti ve contro l . The proporti onal pl us i ntegral control l er gai n s were determi ned through the extended Z i eg l er-N i co l s ru l es for d i g i tal control l ers ( Takahash i , et . al , 1970 ) . The pl ate spec i f i cati ons used i n the s imu l a­ti ons were : the steel pl ate was unstressed and of un iform materi a l propert i es and 1 . 85 mm thi ckness ; the rim and c l amp di ameters were 432 mm and 140 mm; the rotati on speed was 1200 rpm . The samp l i ng i nterva l i s tiT = 5 s . The mai ntenance of the set-poi nt for the di fferent control algori thms i s shown i n Tab l es 1 and 2 . M i n i mum vari ance control shown i n F i g . 4 mai ntai ns the set-poi nt most effect i vely, especi a l l y i n the presence of measurement noi se . I n pol e pl acement the poles are set to zero i n thi s paper .

EXPERIMENT P l ate v i b rat ion was control l ed i n rea l -t ime on the apparatus schemat i cal ly shown i n F i g . 5 . The model was a p l ane, un i form steel p l ate , 406 . 4 mm i n di ameter , and 1 . 016 mm i n th i ck ness . The c l ampi ng di ameter was 152 . 4 mm . A vari ab l e speed motor dri ves the p l ate at 300 rpm . . An IR-thermometer scanned the axi symmetr i c temperature di s tr i buti o n . The thermometer was dri ven on a traverse at a constant vel o­c i ty of 127 mm/s al ong the p l ate rad i u s .

The temperature and the radi a l pos i t i on vari ab l es were measured and recorded on a PDP 8/E mi n i computer and the cal i brated vari ab l e error was l ess than 1% FS . Tran sverse vi brati on was exci ted by an el ectromagnet dri ven by random noi se i n the bandwidth 0- 100 Hz . The tran sverse di sp l ace­ment of the p l ate was measu red by a non­contact sensor . The control heat f l u x was supp l i ed by an IR­heat l amp pos i ti oned near the c l amp i ng col l ar . The heat l amp rad i ated energy onto a spot about 6 . 35 mm i n di ameter . The total l amp power was 0 . 75 kW g i v i ng a heat f l u x at the foca l po i n t of 1 W/mm2 . The control heat f l u x was tu ned by vo l tage control for the heat l amp . The process heat f l ux was i nduced wi th another heat l amp posi ti oned at the peri phery . The PDP 8/E mi n i computer was used for fre­quency i denti f i cat i o n , for mon i tori ng the data si gna l s from measu rement devi ces , for analyses and computati ons , and for control of the heat f l u x . Natural frequency and di sp l acement data for the test p l ate u nder process heat f l ux wi thout control are shown in F i g . 6 . The process heat f l u x was adjusted wi th the peri pheral heat l amp vo l tage . The pl ate became unstab l e i n mode ( 2 , 0 ) at t = 58 s . The decreas i ng ( 2 , 0 ) natural frequency was accompani ed by an i ncreas i ng vi brati on amp l i tude . Saturat i on of the v i b rati on ampl i tude occurred at t = 51 s . The test was di scont i nued at t = 60s because of the v i o l ent v i brat i o n .

Natural frequency control of th i s p l ate was tested wi th the on-off , the proporti ona l p l us i ntegra l , and the adapt i ve control al gori thms . M i n i mum vari ance control res u l ts are presented i n some detai l i n F i g . 7 . When the natural frequenc i es for the ( 1 , 0 ) and ( 2 , 0 ) modes are equal , the contro l i s opt i ma l . Process i denti f i cati on through determi nati on of al , b1 , b2 , b3 was performed duri ng 0- 150 s by a vers i on of on -off acti on . M i n i mum vari ance control was i n effect thereafter . Compari son of the effecti veness of the control al gori thms for frequency control i s summari zed i n Tab l e 3 . The transverse v i b rat i on record under mi n i ­mum vari ance control i s shown i n F i g . 8 . The effecti veness of the control al gori thms for v i b rat i on amp l i tude control i s sum­mari zed i n Tab l e 4 . I n thi s exper i ment , mi n i mum var i ance control mai ntai ned the cri t i ca l frequency wi th i n the control band and mai n tai ned the vi brat i o n amp l i tude bounds a greater percentage of the test durati on than the other al gori thms . Computat i on t ime on a PDP 8/E mi ni computer i s separated i nto the t ime requi red for updati ng the model i denti f i cat i on and the t i me for cal c u l at i ng the natu ral frequenci es

Control of Rotating Circular Plates 1 75

and modal d i stri buti ons . The model i den­t i f i c at i on update requ i red 1 . 45 s for na = nb = 10 and 0 . 56 s for na = nb = 6. The natural frequency computati on requ i red 1 . 5 s for fre­quency poi nts i n the 0 to 200 Hz ba,nd wi th 1 H z resol uti on , and i t requ i red 3 . 0s for fre­quency poi nts i n th i s 0 to 200 Hz b and wi th 0 . 5 Hz resol ut i on . These t i mes wi l l be reduced i n the future.

CONCLUS I ONS 1. Feedback control of the i denti f i ed fre­quency spectrum through i ntroduction of thermal stress i s an effect i ove mean s for control l i ng p l ate transverse vi brat i o n i n real -t ime . 2 . The l aboratory experiments and the s i mu ­l at i on s from theoreti c al model s i nd i c ate than an opt i mal control heat fl ux and fre­quency set-poi nt exi s t for vi brat i o n control of rotati ng symmetri ca l ci rcu l ar pl ates , and they are pred i ctab l e from the cr i ti ca l speed theory. 3 . The frequency set-poi n t maxi mi zi ng the sma l l es t natural frequency assoc i ated wi th a backward travel i ng wave corresponds to the control heat f l u x where pl ate v i b rat i on i s mi n i mi zed . 4 . Mi n i mum vari ance control was the mos t effecti ve control tested . I t reduced the v i brat i on amp l i tude of the pl ate to approxi ­mately 20% of i ts uncontrol l ed l evel .

REFERENCES

Egardt , B . ( 1980 ) . Un i f i cat i on of. some di screte t i me adapt i ve control schemes . IEEE Trans . on Automat i c Contro l , AC-25 , no . 4 , 693-697 .

--L ap i n , P . I . ( 1959 ) . Determi nati on of the

al l owab l e rotati on speed of a ci rcu l ar di s k based upon i ts strength and the natural frequenci es of vi brat i o n . L�s n . � fil, 125- 135 . --

Mote , C . D . , Jr . and Hol,iyen , S . ( 1975 ) . Confi rmat i o n of the cri t i ca l speed stab i l ­i ty theory i n symmetri c ci rcu l ar saws . ASME , J . Eng . for I ndus try, 978 , no . 3 , ITI2-lll8 .

- --Mote, C . D . , Jr . ( 1965 ) . Free vi brati on of

i n i ti a l ly stres sed ci rcu l ar d i s k s . ASME , J . Eng . for I ndu stry , 878 , 258-264 .--

Mote, C . D . , Jr . , Schajer, G . S . an Hol,.0yen , S . ( 1981 ) . C i rcu l ar saw v i brat ion control by i nduct i o n of thermal stresses . ASME , J . Eng . for I ndustry , 103 , �. ar:B9.

Mote , C . D . , Jr . and N i e h , L . T . ( 1971 ) . C ontrol of ci rcu l ar d i sc stabi l i ty wi th membrane stresses . Experi mental Mechan i cs , Q, no . 11 , 490- 498 .

R ah i mi , A . ( 1982 ) . On- l i ne spectral control of rotati ng c i rcu l ar d i scs us i ng thermal membrane s tres ses . P h . D . Di s sertat i o n , Dept . of Mec h . Engr . , Un i v . o f C al i f . , B erkel ey.

Takahash i , Y . , C h an , C. S . and Aus l ander , D. M. ( 19 70 ) . P arameter tun i ng of l i near DOC a l gori thm. ASME P aper No . 70-WA/AUT- 16 , 19 70 .

Landau , I . D . , Dugard , L . and C abera, S . ( 1982 ) . Appl i cati ons of Output Error Recur­s i ve Esti mati on Al gori thms for Adapti ve S i g na l Proces s i ng . P roceed i ngs ICASSP -82 . 639-642 .

Tak ahash i , Y . , Tomi zu k a , M. and Aus l ander , D . M . ( 1975 ) . Si mpl e D i screte Control of I ndustri a l Proces ses ( F i n i te Time Settl i ng Control Al gori thm for S i ng l e­L oop D i g i tal Control l er) . ASME , J . Dyn . Sys . Meas . and Control , 97 , 354-36 1 .

a b c e ( k ) Gp c z- l ) H h ho Ko K K1 , . . Kq m n q r ( k ) T To T x ( k ) u< k l v c r , e ) y ( k ) a p v CTrr oaee S"lcr i t n Wmn

SET-P<lNT Optimum + Noturol _ Fr�

NOME NC LA TURE cl ampi ng rad i u s peri pheral radi us spec i f i c heat error , = r ( k ) - y ( k ) transfer funct i on for µth natural

frequency d i sc th i ck ness rad i a l average convect i ve heat

transfer coeff i c i ent 2hb2 mod i f i ed B i ot number ( = "Kif) di sc conduct i v i ty D state vector feedback gai n e l ements of vector K number of nodal c i rcl es number of noda l d i ameters heat f l u x frequency set-po i n t mean pl ate temperature through

thi ck ness ambi ent temperature rel ati ve temperature, T-T0 state vector i nput heat fl ux di sc transv erse di sp l acement output natural frequency coeff i c i en t of expans i on densi ty Poi sson ' s rat i o �ri nci pal rad i a l and hoop stresses cri t i ca l rotati on speed p l ate rotati on speed natural frequency correspond i ng

to mode ( n , m )

SPECTRAL OBSE

Identifies Natural FreqU8flcies of Sowblode

Fi g . l Feedback control sys tem b l ock d i agram

1 76 C . D . Mote , Jr and A . Rahimi

4"- Reson once

N J:

300

� 200 c .. 5-� "­.. � � 0 ! u 0 ID

0

0

Fi g . 2

....

80 -

.....

I Instabil ity, n • 3

I I I I

· - · -'"":'..:::.,... �....: --+- ·--. I ' l -- · \1

5 6.34 10 Control Heel Fl u x a c. w I cm3 1 5

Backward w�ve frequenci es at control heat fl ux Q8 . � = 1 200 rpm , 2b = �32 mm , 2a = 14 mm , H = 1 . 85 mm . !/p 1 s assumed 1 . 35 kw . n = n umber of noda l d i ameters of mode ( n , o ) .

Vibration Modes

.:> n • I • n = 2

4 0 � ... - ..

Fi g . 4

I 100

I 200

Time, s

I

300

S i mul ati on of m1 n 1 mum vari ance control w i th 1 0% measurement noi se and con­ti nuous parameter i de nti fi cati on . QP i s postu l a ted .

- - - - - - - - -- -- - - - - - - - - - - - - - - - - - - - -,

: CONTROL LOOP : I I I I

1 DISTURBANCE 1 I : I I

r(k)I + CONTROLLER PROCESS y(k) 1 I I I I I I L - -- - - - - - - - - - - - ­r - - - - - - - - - - - - - - -I I I I I I I

RANOOM SIGNAL

GE N E RATOR

I I I I I I I

- - - - - - - - - - - - -' -,

-----< I I I I I I I I I I I I I I I I I I I I

IOENTIFICATION LOOP 1 L - - - - - - - - - - - - - - - - - - - - - - - - - - -J

Fi g . 3 Feedback control sys tem wi th MRAS a l gori thm for i de nti fi cati on

DRIVE

-���!��--

PERIPHERAL HEAT

Voriac

Peripheral Heat Lamp

SAW �======::::;:::!=::::l==::::::========�.-�AOE

DISPLACEMENT DATA

Fi g . 5

l/O O/A

COMPUTER INTERFACE - - - - - - - - -

POP 8/E MINl·COMPUTER

MEASUREMENT _______ _ __ ....

Exper imenta l Apparatus .

Control of Rotating Circular Plates 1 77

Vibration Modes o n :s l � � 60

I e n • 2 !Ii ,.. e II � o o o o o1 -o u 40 • � Resonance ! "'.,� 2

• • 1 Instability e - • I n • 2

m IL 00'---2"'-o--40'---·'-'s.L.o--eo'----100'----'120 Time, s

o.

Fi g . 6 Experi menta l l y observed i ns tabi l i ty of a c i rcu l ar pl ate wi th peri phera l heat fl ux . 2b = 406 mm , 2a = 1 52 mm , H = 1 . 0 mm , Q = 300 RPM .

E �i E •• 4 I .. I r � -+ -.5: I I I � 0 10 "' 30 t -+- II I I . 11 1 ' •d � -+ i5 I I I I � 40 "" 60 70

� -+- -+ � I ' ' ' � 80 90 100 1 10

+- -+ ' ' ' IZO 130 140 +-

I ' ' ' 150 180 170 110

+

Fi g . 8

+ ' I I I Ito zoo ZIO zzo

Tim•,•

Experimental measurement of transverse di spl acement of a c i rc u l a r p l ate unde r mi n imum vari a nce control . 2 b = 40G mr.i , 2a = 1 5 2 mm , fl = 1 . 0 mm , Q = 300 RPM. The natural frequenci es , heat fl uxes , and r.iodel parameters are gi ven i n Fi g. 7

Vibration Modes • n z l ! � � � SO � SET- POINT"'\

0 • \. • n •2

i � 40 ·���.._ilj ....... ___ �-c;,_ __ � al IL I I I 100 200 300 '100 Tim1, 1

Fi g . 7 Exper imental eval uati on of mi n i mum vari ance control of a c i rcul ar pl ate . 2b = 406 mm , 2a = 1 52 mm, H = 1 . 0 mm , Q = 300 RPt1. P l a te ampl i tude i s g i ven i n Fi g . 8 .

Table l Control o f Frequency Set-point with Zero

Noise by Computer Simulation

i Percent of Total Tin-e within Frequency Band

Deviation from J the frequency I Set-point* I

!. 20% ! !. 10%

!. 5%

!. n

ON-OFF

93.6**

78.2

58 . 5

43.4

Proportional Plus

I ntegral

100.0

94. l

84. l

75 . 2

• Opti mum frequency set-point • 43 Hz

Fi nite Tin-e

Settl i ng Control

97 .5

93.2

90. 5

78.6

Pole-Pl acement

100.0

94 . 5

91 . 8

7 7 . 3

•• The natural frequency o f the critical rrode l ies wi thin !. 20% of the set-point for 93.6% of the test duration.

1 78 C . D . Mote, Jr and A . Rahimi

Table 2 Control of Frequency Set-point w i th 10% Measurerrent

Noise by Computer S i mulation

Percent of Total Time with i n Frequenc_v Band

Deviation from Finite

Time Pole- Mi nimum the frequency Sett l i n g Pl acement Variance Set-point* Control

! 20% 97. 3** 100. 0 1 00 . 0

! 10% 88.4 94 . 1 95 . 8

! 5% 71 . 8 81 . 3 86. 7

! 1 % 48. 5 63. 3 71 . 4

* Opti mum frequency set-point = 43 Hz

** The natural frequency of the cri t i cal mode l ies with i n :!:_ 20% of the set-point during 97 . 3% of the test duration.

Table 3 Control of Frequency Set-point on an Experimental Sawblade

Percent of Total Time within Frequency Band

Deviation from Proportional Finite

Time Pole- Minimum Without the frequency ON-OFF Plus

Settl ing Placement Vari ance Control Set-point* Integral

Control

� 20% 80. 7** 96 . 3 84.2 100.0 100.0 41 . 2

.:!:_ 10% 61 . 2 6 8 . 7 7 2 . l 76.5 81 . 2 32 . 3

5% 48.7 52.8 51 . 3 68.6 74 . 7 21 . 8

1 % 33.2 43.6 35 . 3 59 . 8 69 . 7 1 0 . 7 L

Optimum frequency set-point "' 45 Hz The natural frequency of the critical rrode l i es within + 20% of the set-point during 80. 7% of the test duration.

-

Table 4 Control of Vibration Amplitude on an Experimental Sawblade

Percent of Total Time within Ampl itude Band

Mean Proportional

Fi n i te Vibration Time Pole- Minimum No Amp l i tude ON-OFF Plus

Settl i n g Placel'lf!nt Vari ance Control Integral < (nm)

l . O

0 . 5 0 .1 0 . 1

Control

100.0• 100.0 100.0 100.0 100.0 96.1 99 . 3 99.5 100.0 100.0 63.3 79.1 77.1 81 . 5 85 . 3 10. l 11 .1 1 8 . l 14 . 4 38. 7

Mean vibration ampl i tude is less than 1 .0 nm for 100.0% of the test duration after process identification. N "' 200 data points were coll ected with samp l i ng frequency of l KHz at 5 s time i nterva l s .

ACKNOWLEDGEMENT

68. 5 5 1 . 3 8 . 6 0 . 0

The authors thank the Nat i onal Sci ence Foundat i o n , the Un i vers i ty of Cal i forn i a Forest Products Laboratory , and the i r i ndustr i al sponsors for thei r sponsors h i p of the research . The authors al so thank C arol Brodersen for her exce l l ent ass i stance wi th preparat ion of the manu scri pt .

Copyright © IFAC Adaptive Systems in Control and Signal Procc"ing, San Francisco, USA 1 983

STOCHASTIC ADAPTIVE CONTROL

ADAPTIVE CONTROL AND IDENTIFICATION FOR STOCHASTIC SYSTEMS WITH RANDOM

PARAMETERS

H. F. Chen* and P. E. Caines**

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INTRODUCTION

The parameter-adapt ive contro l prob l em con­s i s t s in stab i l i z ing and , if pos s ib l e , opt imi­z ing the behaviour of a system with unknown parameters . As a prob lem in contro l engmeer­ing this dat e s at least from 1 9 5 1 ( s e e Draper and L i , 1 95 1 ) . After that some key event s were the fol lowing (we make no pretence here at a survey ) : In 1 973 there appeared the seminal paper by ft.s tr�m and Witt enmark (19 73 ) i n which the s e l f-tuning regulator was f irst introduced . Th i s contr ibut ion to d i screte t ime parame ter stochastic cont ro l s t imulated a great amount of theoret ical and prac t i cal work in adapt ive co�tro l . By the end o f the decade a sequence o f p apers began to appear giving rigorous proo f s of the s tab i l i z ing proper t i e s of var ious adap t ive control algo­rithms for l inear , t ime invar iant , f inite d imensional systems ; the s e paper s treated the det erministic cont inuous and d iscrete t ime cases (see e . g . Feuer and Morse , 1 9 7 8 ; Narendra and Valavani , 1 9 7 8 ; Egardt , 1 980 ; Goodwin , Ramadge , Caines 1 980) and the s to­chastic d i screte t ime cases ( s e e e . g . Goodwin, Ramadge , Caines , 1 98 1 ; Chen 1 982 ; Kumar 1 98 2 ; S in and Goodwin , 1 982 ) respect­ive ly . In (Kumar , 1982) the adap t ive control o f Markov cha ins i s treated ; this type of analy s i s requires no l inearity hypothe s e s on the system . More recently , the s tocha s t i c cont inuous t ime parameter-adapt ive LQG prob­lem with a f inite parameter set has been treat ed , see (Hijab , 1 98 3 ) and the opt ima l control law over a f inite t ime interval de­r ived . Th i s is in contrast to the previous­ly ment ioned papers where asymptotically ex­act tracking is proved for the d etermin i s t i c prob l ems and asymp to t i ca l ly minimum mean square tracking proved for the s tocha s t i c problems . A prac t ical prob lem w i t h t h e soL­t ion given by Hij ab i s that a Ka lman f i lter i s needed for each candidate parameter . Mo s t recen t ly , i n (Chen , 1 983 ) asymptot i c a l ly op­t imal adapt ive contro l laws have b een der ived for the LQG prob lem for sy s t ems with a con­t inuous t ime parameter and a cont inuum o f sy s t em parameter s .

All the work above deals with the adap t ive control of determin i s t i c and s tochas t i c sys­tems with constant unknown parame ters . How­ever the principal mo t ivat ion f or adap t ive control is the stab i l i zat ion and op t im i zation of systems who s e parameters are unknown and

1 7 9

wh ich vary in t ime (determin i s t ically or sto­chas t ically ) . A f irst s t ep towards dea l ing with thi s prob lem was descr ibed in ( Caines , 1 9 8 1 ) where the e f f i cacy of adapt ive controls o f the " GRC " type presented in ( Goodwin , Ramadge , Caine s , 1 98 1 ) wa s e s t abl i shed for sy s t ems with ( converging) mar t inga le para­meter s . Further , in ( Caine s and Chen, 1 982) the imp o s s ib i l ity o f s tab i l i z ing a system with zero mean uncorrelated ga in parame ter process wa s proven .

Th i s paper presents a s e t o f results concern­ing the adapt ive , asymptot ica l ly optimal , control of l inear autoregr e s s ive system with moving average exogenous control inputs and uncorrelat ed d i s t urbance s (cal led ARX sysrems ) with random autoregres s ive (AR) parameters .

In addit ion , condi t ions are g iven f or the parameter e s t imat e s gener ated in the adap t ive control algori thm to be s trongly con s i s tent .

It wou ld be int eresting to relate the results o f this p aper to tho se given f or the non­adap t ive LQG prob lem for random parame ter sy s t ems g iven by de Koning ( 1 982 ) . Th i s lat­ter paper gives neces sary and suffic ient con­d i t ions for a tota l ly observed l inear sys tem -who se random parame ters have known means - to p o s s e s s an opt imal f e edback contro l law wh ich g ives r i s e to a f inite total expected cost , that is to say cond i t ions for

THE STOCHASTIC SYSTEM AND THE ADAPTIVE CONTROL ALGORITHM

Cons ider the MIMO system

n n y + A

ly l+ . . . + A y

n n- p n-p

( 2 . 1 )

with the init ial cond i t ions yn

= O , un

= O ,

wn

= 0 , n < 0 • Not e that this sys tem has autoregres s ive dy­namics , a moving average operat ion on the con­trol input u and only a trivial moving aver age operat ion on the s tocha s t i c d i s turbance input w • y

n, u

n and w

n are m , £ and m d imens ional

1 80 H . F . Chen and P . E . Caines

vectors for all n E 7l . In this section the stochastic behaviour o f the parameters and the disturbance process are taken to be as follows : The matrices {A1: ; l�i�p , n E ?l } are given by i

A1: = A . + 6 A1: , l�i�p , i i i ( 2 . 2a)

where A. are constant mxm matrices and the i 6 A1: are random mxm matrices such that , for i all n ,

E ( 6 A1: 1 F 1 ) = 0 , (2 . 2b ) i n-

E ( ( 6 A1: ) ' (6 A1: ) I F l ) < 00 i i n-n T n J r

E ( ( 6 A . ) 6 A . r 1 ) = 0 , i;Oj , i J n-

(2 . 2 c )

( 2 . 2d )

where {F ; n E ZI: } is a non-decreasing family n of a-algebras on the underlying probability space (� , B , P ) . Further we require that

I I 6 Ar: II < Cl i l�i�p , n � 0 ( 2 . 2e) where a is specif ied later on . The equations ( 2 . 2 ) and all subsequent equa­tions in this paper are understood to hold a. s. P .

The disturbance process w satisfies the mar­t ingale difference conditions

E (w I F 1 ) = 0 n n-

E (w'w I F 1 ) < 00 , n n n-n;;;.l , (2 . 2f )

and the disturbances in the parameters and in the input are re lated by

T n J E (w (6 A . ) F 1 ) = 0 n i n- l�i�p (2 . 2g)

In thi s sect ion and Sect ion 3 the matrices { B1 ; l�i�q } are taken to be constant . We adopt as standard notation for this paper :

T 8 = [-A1 , . . . , -Ap , B 1 , . . . , Bq] , m x (pm + iq) , ( 2 . 3 )

-the matrix of mean values of the system parameters , and

�T = [ T T T T ] o/n yn ,yn- 1 • · · · •Yn-p+l 'un-q+l ' 1 x (pmHq ) , (2 . 4 )

-the regression vector , or state , of the sys tem at the instant n . The change in the output due to the change in the parameters at the instant n is given by

En= - (6 A�yn_1+ . • . + 6 A;yn-p)

as is shown by

( 2 . 5 )

(2 . 6 )

The MLS algorithm goes as fo llows : Recursively define

r = r 1+ 114> f , �O r = 1 , n n- n ' - 1 (2 . 7a)

and set

R_1 = (mp + iq) I (2 . 7b )

Given Rn ' � - 1 , de·fine

- T - 1 T R = R 1- ( 1+¢ R 1¢ ) R 1¢ ¢ R l (2 . 7c ) n n- n n- n n- n n n-and denote its maximum and minimum eigenvalues

n n by µmax ' µmin · R and a sequence { a ,�l } are then def ined n n by the following " logic step" :

then R R - , a n n n

otherwise

( 2 . 7d )

( 2 . 7e)

( 2 . 7 f )

By the matrix invers ion lemma , if R 1 >0 then n-R - = (R-1

1+ ¢ ¢' ) -l > 0 and so , by ( 2 . 7b ) and by n n- n n use of induct ion , we see that R >O for all n � . Finally 8n ( (pm+iq) x m) is def ined by

T T en+l= 8 + a R ¢ (y +l-¢ 8 ) n n n n n n n with any deterministic e0 •

(2 . 7g)

Observe that (2 . 7d-f) ensure that , for all n An ( An . ) - l�k when An

and An . denote ' max min 1 max min the maximum and minimum eigenvalues of R . n As in (Chen , 1 98 1 ) and ( S in and Goodwin, 1 982 ) it may be shown that

( 2 . 8)

and

Stochastic Sys tems with Random Parameters 1 8 1

(y -8T¢ 1 ) , n� 0 , n n n- (2 . 9 )

Having specif ied the MLS algorithm we now p lace conditions on w in terms of rn :

There exist constants s 1 > 0 and S2> 0 and 6 1 , 6 2s [O , l ) such that

n � 0 (2 . lOa)

(2 . lOb )

and

(2 . lOc)

where o2 is a random variab le that is almost

surely f inite . (Notice that (2 . 1 0 a , b ) are certainly satis­f ied if w is a bounded process . ) Further we need the inverse stabi l ity (mini­mum phase) requirement : £ � m,B 1 is of full rank and the zeros of the determinant of the £x£ po lynomial matrix + - + c . q B1B (z ) - B1 B1+B2z + . . . + Bqz ) l ie outs ide

the c losed unit disk . We conclude our descript ion of the adaptive contro l algor ithm by the spec ification of the F measurable adaptive contro l law u as n · n the so lut ion to the equat ion

n � 0 (2 . 1 1 )

* where { yn ; n s tz } is a bounded deterministic reference sequence . It wi l l be taken as a standing assumption for Sect ion 3 that this equation is solub le for al l n w .p . l .

ASYMPTOTICALLY OPTIMAL CONTROL : RANDOM AR PARAMETERS

In this section we show that the adapt ive control algorithm descr ibed in Section 2 for the system (2 . 1 ) , subject to the hypotheses (2 . 2 ) , ( 2 . 10) , ( 2 . 1 1 ) asymptotically stab i­lizes the system and asymptotically achieves minimum mean square tracking error . The strategy of the proof is to examine the super martingale like behavious of the quan­tity

1 e1R-l e n � 0 v ti -- Tr n n-1 n n- rn- 1

when

(3 . 1 )

e 6 e - 8 n � 0 n - n (3 . 2)

Here we give a sketch of the analysis , the reader is referred to (Chen and Caines , 1 983) for all details . From (2 . 2e ) (2 . lOc) it fol lows that r -+ 00 is n a . s . as n-+«> , because otherwise II ¢ II n -+ 0 ' b II -+ O , ll u II -+ 0 and by ( 2 . 2e ) , (2 . 5 ) , n n II s II -+ 0 and hence by (2 . 6 ) ll w II -+ 0 but n n this contradicts (2 . lOc) . By (2 . 1 ) , (2 . 5 ) , (2 . 9 ) and (2 . 12 ) we have

X ( 8T"' ) ( 1 "'T R "' ) -1 yn- n�n-1 = -an-l�n-1 n-l�n-1 - T

x (8 ¢ l+w + s ) . n n- n n

By (2 . 2e ) , (2 . 5 ) and (2 . 8 ) we obtain

(3 . 3 )

It is c lear that all zeros of the po lynomial a 1 - k(z + . . . + zp ) are outs ide the closed

unit �isk i f a -1 . . . < k3p , since in this case

I -1 p I imp lies a k3 (z + . . . z ) ( < 1 .

Hence from (3 . 3 ) there exi sts p s (0 , 1 ) such that

n ll Y II � I Pn-1C-k1 11 8 : ¢ . 1 11+ -k

1 llw . 11 n i=l 3 i i- 3 i

( 3 . 4 )

where the random variable n depends linearly on yi ' i=O , . . . , p-1 only . From ( 3 . 4 ) we have

II 11 2� 6 ¥ n-ic l 11 - T 11 2 yn "' T=iJ. l p 2 8 i¢i-l 1 i=l k3

(3 . 5 )

I I Y n 112

S ince --- � 1 it is c lear by use of (2 . 2e ) , r n (2 . 5 ) that

T s s E � < oo n � 0

rn-1

and so condit ional expectat ions for this

1 82 H . F . Chen and P . E . Caines

random variable exist . By ( 2 . 2e) and (2 . 3 ) we have

T £ £ ( E C:rn n I Fn_1)..;; 2a2

1i n \1 2 (P2 n- 1 )

n-1 n- 1 j -1

+ . . . + 2 (n-p))+ 6a2 . I I P 1-p J =n-p s=O s p

where we have used (3 . 5 ) for the last inequal­ity . Set

v n

-T - 1 -tr8nRn-l 8n

rn-1 ( 3 . 7 )

Fol lowing the calculation in (Chen , 1 98 1 ) we shall derive super mart ingale like inequali­t ies for the {v ,n � 0 } sequence . n Recall from (Chen, 1 98 1 ) and (Sin and Goodwin , 1 982)

(2 . 7h)

(l + <j>TR 1 ¢ ) R <j> = R 1 ¢ ( 2 . 7k) n n- n n n n- n

which hold when (2 . 7d) is in force . In this case , us ing (2 . 7h,k) and (2 . 9 ) ,

and then substitut ing into the right hand s ide of

we obtain

- T -1 T -8 +l (R 1+ ¢ ¢ ) 6 +1 n n- n n n

consequently , using rn ,,;;; rn+l '

2 T - - T v ,,;;; v - � e ( e q, + w + £ ) n+l n rn n n+l n+l n n+l n+l

II 9T <P 11 2 + ' n;l n

( 3 . 8 ) n

Next , when ( 2 . 7e ) holds , we have

an( l-an<j>:Rn<j>n)-1= 1 and so by (2 . 7f ) , (2 . 7g) and ( 2 . 9)

- - -T T 8 1= 6 - R ¢ (6 +l<I> + w +l+ E +l) n+ n n n n n n n

r = S -( n-lJR ¢ ( BT ¢ +w +£ . ) T

n r n-1 n n+l n n+l n+l n

and again (3 . 8 ) ho lds . By the measurabi lity of Yn+l- wn+l- En+l with respect to F and by (2 . 2b-d) we have n

Hence by (3 . 8 ) , ( 3 . 9 ) , a ,,;;; 1 and (3 . 6 ) n

E (v +l l F ),,;;; v - _!_E < l leT+l <P \12 i F ) n n n r n n n n

<j>TR <j>

1 2a2 <j>TR <j>

+ 2 n n n E ( Jlw \1 2 J F )+ - n n n rn n+l n 1-p rn

n I

j =n-p+l

(3 . 10 )

We now define a set of terms y � O , o > O n such that

s m = v +_E_ + y + o n n rn-l n

-T 11 2 n-1 6 .+1 ¢ . I II ]_ ]_

i=O r i

is a non-negat ive super-mart ingale i . e . Em < oo n for all n with E (m j F 1 )..;; m 1 , n � O . The n n- n-term yn invo lves sums of squares and condi-t ional expectations of the disturbance process w

i. ' 0 ,,;;; i < 00 , and the tracking sequence y� , . ]_

0 ,,;;; i < 00 , weighted by the sequence

q,;R. ¢ . i i i , 0 ,,;;; i < 00 • There are also some addi-r .

]_ t ional constant terms, ( see (Chen and Caines 1 983 ) ) .

'

S ince {m , n martingale and so

n � O} is a non-negat ive super it is a lmo st surely convergent

< co a . s ,

S tochastic Sys tems wi th Random Parameters 1 83

and since we established earlier that r -+oo a . s . we have n

+ 0 n-+oo a . s . ( 3 . 1 1 )

I t may now be shown that there exist finite random variables k7 ,k8 such that for sufficient ly large n

n 2 k7 n 2 n I \\y . \\ � - I li e :¢ . 1 11 + k8 i=l i n i=l i i- (3 . 12 )

with similar bounds for n 2 l fl £ . II and n i=l i

1 n 2 . - l [[u . I[ with k7 ,k8 replaced by k9 ,k10 and n i=l i k1 1 ,k1 2 respectively . From this it may be deduced that

( 3 . 1 3 )

and this yields the des ired mean square stochastic stability :

lim 1 n-+oo -n I 11y . 11 2 < 00 • A-illl .!. I 11u . 11 2 < 00 i=O i n i-0 i

A-ifil .!. I k 11 2 < 00 (3 . 14 ) n i-0 i Now we can show that

n .!. \' w'. i l 6 AkYi-k+ O , l� k� p , n-+oo , (3 . 1 5 ) n i=l i n

and 1 l ¢l-1 e w . + 0 n i=l i l im n as n-+oo . Thus n-+oo - < oo a . s . r n

( 3 . 1 6 )

II ¢ n 112

From ( 3 . 14 ) we have --- + 0 as n-+oo . This n I I ¢n 11

2 yields -r-- + 0 as n-+oo .

n Thus , setting b = a ¢'R ¢ we have n n n n n

b � a [[ ¢ 11 2 An � a [[ ¢ 11 2 _!_ +O n n n max n n r

Write z = n 6'¢ b n-1 n-1 � + 1-b (wn+

n-1 n-1

and then we get 1 I l[ z . 1[ 2 + 0 n i=l i

n

*

as n-+oo . (3 . 8 )

(3 . 1 7 )

a s n-+oo where we recall that yn-yn=zn+wn+£n . ( 3 . 18 )

And so f inally we obtain THEOREM 3 . 1 Under the hypotheses given above the adapt ive control law (2 . 1 2) results in

.,.--- 1 n * 2 2 T l n 2 n� - I llY . -y . II =cr +n.!ffi - I k II o . 1 9 ) n i=l i i n i=l i and an estimate for the last term is

n 2 2 -- n 2 A-ifil n it ll E: i II �Ca p)A-ifil � it l lY i II

2 2 � 6a P

2 Ccr2 + s�p ll/ f) ( 1-p ) k3

i

which goes to zero as a + 0 .

(3 . 20)

D

EXTENSION TO THE CASE OF RANDOM MOVING AVERAGE PARAMETERS

Subj ect to addit ional hypotheses , the analy­sis of Sections 2 and 3 can be extended to the situation where the parameters of B (z ) in (2 . 1 ) are random . We shall cons ider the ARX system

+ w , n � O , (4 . 1 ) n with the initial condit ions yn= 0 , un= O , wn= O , n < 0 , where the random parameter variat ion that is now permitted is as fol lows : Given the increasing family of a-fields { F ; n� 0} the parameter variat ion is-as in n Sect ion 2-of the mean value p lus martingale difference type where A . , l� i�p , B . , 2�i�q , i i are unknown constant matrices but B1 is a known constant matrix and 6 B�=O for all n . As in Section 3 , the disturbances are mutu­ally uncorrelated . Further , II to A� [I < a , [1 6 B� ll < S i . e . the disturbances in An ( z ) and B�z ) are uniformly bounded , where a is as in Sect ion 3 and S will be specified later . Also , as in Sections 2 , 3 , w is a . martingale difference process which i s un­correlated with the changes in the parameter s . Without loss o f general ity we can assume II B1 II < S , otherwise we may rescale un by

o h1 II cons idering un !;:_ -8- un the coefficients of B ( z )

s be mult ipl ied by llB] . 1

instead of u and n will consequently

We still need ( 2 . 2e) , ( 2 . 10) , but the mini­mum phase condition of Section 2 wil l be re­placed by the following condition Q, � m, B1 is of full rank and S is sufficien­t ly smal l that the zeros of

1 84 H . F . Chen and P . E . Caines

are uniformly greater than 1 in abso lute value . The F -measurable control action u 1 is select-n n ed so that

er� + B1u • n n n ( 4 . 3 )

where er = [A1 (n) . . . A (n)B2 (n) • • • B (n) J . It n . P q is a standing as sumption for this sect ion that ( 4 . 3 ) is solvable for all n ) 0 w .p . l . Us ing a ' ' logic step" that either switches on (a function of) u ' when u ' fO or a control n n based on past values of y and u when u�= 0 , we obtain a control act ion u that sat isf ies n

II u 11 2< n E c r n 0 ' E < 1 (4 . 4 )

An analogous l ine of reasoning t o that used in Section 3 yields

1 n 2 * - I ll z . 11 -+ 0 as n-+co where yn-yn=zn+wn+lln n 1 i

where lln generalizes En to the case of random B matrices . Then we obtain the analog o f Theorem 3 . 1

(4 . 5 )

We now briefly indicate two extensions to the case where B 1 is unknown : F irst , we consider the case of unknown constant B 1 , which imp-lies in particular that LIB� = 0 . Since B 1 is unknown we use the notation of ( 2 . 3 ) , ( 2 . 4 ) and calculate e by ( 2 . 7 ) where n e '= [A1 (n) . . • A (n)B 1 (n) . . . B (n) J . We assume n P q that , in addition to all the other standing assumptions , estimate of B1 at the instant n satisfies � B1 (n) � < c 1 �B1 � for some const­ant c 1 . A minor modification of the previous analysis then yields (4 . 5 ) . Second , cons ider the case where B 1 unknown

n but llB1 t 0 . We shall impose the condition

is defined by u� in (4 . 4 ) (with B 1 rep laced by B1 (n) ) . The modification we described earl ier is discarded in this case because of the condition II u 11

2< c 1rE . n n The analysis now proceeds along the l ines of the unknown constant B1 case described above and yields the same main result given in (4 . 5 ) .

CONSISTENCY n Let llB1=0 , but let B 1 be unknown .

so that (4 . 4) holds and assume Def ine u n

w E C-+ /rn_ 1) = o ( 5 . 1 )

rn- 1 which is the condition (weaker than ( 2 . 2f ) above ) which i s emp loyed in (Chen , 19 82 ) .

Then II s! ll <c 1r�_1 , E < ll s! l l 2 / Fn_1 )<c1r�-l ( 5 . 2 )

We need the fo l lowing condition : A ( z ) is asymptotically s tab le (zeros of det A(z) are outisde the c losed unit disk and a is small enough such that all zeros of det ( I+(llA�+A1 ) z+ . . . + (Ap+llA;) zP ) ( 5 . 3 ) are uniformly greater than 1 in the absolute value . We can now use Theorem 1 from the above mentioned paper of Chen and so

vn e -+6 a . s . if � < y (or the weaker condi-n n v . min t ion ( 1 9 ) in that paper is satisfied) where n v max and vn . min are the maximum and minimum

n eigenvalues of l � . �:+ ��- I respect ively .

i=l i i mpHq

If llB�to , but llun II< c1 r� , (see case 2 at the end of Section 4 ) , then under these condi­t ions the parameter estimate e is strongly n cons is tent .

REFERENCES

K . J . AstrBm , Introduct ion to Stochast ic Con­trol ,Academic Pres s , NYC , 1 970. K . J . AstrBm, B . Wittenmark , Automatica 9 1973 P . E . Caines , IFAC Cong .Kyoto ,Japan ,Aug . 198 1 . P . E . Caines , H . F . Chen,Workshop on Adapt ive Control , F lorence , Italy , Oct . 1 982 . H . F . Chen , (a) J . of Sys . Sci . and Math . Sc i . Vol . l ,No . 1 , 1 98 1 , 34-52 ; (b) Res . Report , Dept . ElectEng ,McGil l U , November 1 982 . (c) Int . J . Contr . 1982 , 35 ,No . 6 , (d) Res , Repor tpept . Elect . Eng . McGill--U, April 1 983 . C . S .Draper ,Y . T . Li , ASME , New York, 1 95 1 . B . Egardt , S ICOPT Vol . 1 8 ,No . 5 , Sept . 1 980 . A .Feuer , S .Morse , IEEE Trans .Autom . Cont r . AC-23 , No . 4 ,August 1 9 78 ,pp . 5 70-583 . G . C . Goodwin , P . J . Ramadge , P . E . Caines , (a) IEEE Trans . Autom . Contr . AC-25 ,No . 3 , June 1 980 pp . 449-456 (b)SICOPT 1 9 ,No . 6 ,Nov . 1 98 1 , 829-53 . P .Hal l , C . C .Heyde ,Martingale Limit Theory and its Appl ications , AP ,NYC , 1 980 . G . H .Hardy , J . E .Littlewood , G . Polya , Inequalities , CUP , 1 936 . O . B . Hij ab , IEEE Tran s . Autom, Contr .Vol .AC-28 No . 2 , 1 983 , pp . 1 7 1-1 78 . W .L .de Koning ,Automatica , Vol . 1 8 ,No . 4 , July 1 982 , pp . 443-453 . P .R . Kumar , SICOPT Vol . 20 ,No . l , Jan , 1 982 , K . S .Narendra , L . Va lavani , IEEE Trans .Autom Contr .Vol . AC-23 ,No . 4 ,Aug . 1 978 , pp . 570-83 . K . S . S in ,G . C . Goodwin ,Automatica , Vol . 1 8 , No . 3 , 1 982 , pp . 3 1 5-2 1 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

NEW SYNTHESIS TECHNIQUES FOR FINITE TIME STOCHASTIC ADAPTIVE CONTROLLERS

D. S. Bayard and .M. Eslami

/)1'/Jartrnt'nt of Eln:trical Enginl'l'ring, Stall' Unirwrsity of Nnv York, Stony Brook, NY 1 1 794, USA

Abstract . A method for stochasti c adaptive control synthesis as appli cable to finite-time problems is established. In thi s te chnique , denoted as the method of uti lity costs, control policies are generated from the dynamic programming equations for the closed-loop optimal ( CLO ) policy by replacing the optimal cost to go at each stage by an approximation . Thi s approxima­tion corresponds to the cost associated with using a prespecified control policy ( denoted as the utility control sequence ) for all future control de­cisions . Control policies synthesized by this method are actively adaptive even though utility control sequences may be chosen as passive policies . A result useful for establishing theoreti cal performance bounds on the syn­thesi zed control policy is given . The method of utility costs i s applied to defining a new class of actively adaptive control policies {cID }N-1

1 . The con-a m=

trol policy C� i s generated by specifying the utility control sequences to be a certain passive policies based on the assumption that only m future measurements will be taken. The use of passive control policies to generate active control poli cies is desirable , since passive poli cies can be deter­mined by the solution of deterministic rather than stochastic optimal con­trol problems . The as sumption of only m future measurements allows a fur­ther reduction in the computati on involved. A theoretical performance bound is given for policy c�-l demonstrating its improvement over the performance of the OLF policy. A numerical example is included to study by simulation the relative performance of the CN-l policy as well as to suggest a method (). for implementati on .

Keywords . Adaptive control ; stochasti c control ; dynamic programming.

1 . INTRODUCTION

Often in engineering design , it is desired to apply adaptive control to problems which are finite time in nature . Such appli cations include problems arising in guidance , batch processing in chemical reactors , short term economic investment , electronic countermeas­ures , etc . In these applications , the bound­ednes s of system variables is essentially guaranteed by the finite time constraint . The research objectives can therefore be shifted from stability and convergence con­siderations ( prevalent in the infinite time adaptive control literature ) , to controlling the system optimally with re spect to more general measures of desired performance .

The most general treatment of the problem of optimal control in the face of uncertainty lies in the realm of nonlinear stochastic control. As applied to the adaptive control problem , an augmented model is considered which includes the unknown parameters of the original system as states . The problem then becomes that of a type tre ated using stochas-

1 85

tic dynamic programming ( Bellman , 1961 ; Bertsekas , 1976 ) . However , difficulties due to the nonlinear character of the augmented mode l , as well as the dimensionality of the underlying state spaces have limited practi­cal results obtainable by such methods . Many researchers have therefore investigated sub­optimal stochastic control policies . Such policies include the open-loop feedback ( OLF ) policy ( Bertsekas , 1976 ; originally called OLOF in Dreyfus , 1965 ) , the m-measurement feedback ( mF ) policy ( Curry , 1969 ) , the heu­ristic certainty equivalence ( HCE) policy and cautious type control policies ( c f . , Wittenmark , 1975 ) , wide-sense dual control ( Tse , Bar-Shalom and Meier III , 1973 ) , as well as various other types ( Bar-Shalom and Tse , 1976 ) . An important distinction for a suboptimal poli cy is the capability for ac­tive learning ( Bar-Shalom and Tse , 1974 ) . Control policies having this feature active­ly seek out , or "probe" for information from the system so as to be able to control the system more e ffectively overall . This con-

1 86 D . S . Bayard and M. Eslami

cept was originally di scussed by Feldbaum ( 1961 , 1962 ) , who noted the dual character of the optimal policy in controlling the state as well as regulating learning for con­trol purpose s . Adaptive control methods can generally be clas sified as passively adaptive or actively adaptive depending upon whether the learning whi ch takes place is accidental, or the result of active learning by the con­troller.

A statement of the stochastic control prob­lem is given in Section 2 . In Section 3 , a technique for synthesis of suboptimal sto­chastic control is established, whi ch is de­noted as the method of uti lity costs . This method generates new control policies from the closed-loop optimal ( CLO ) policy (Bar­Shalom and Tse , 1976 ) , by replacing the op­timal cost to go at each stage by an approxi­mation . Thi s approximation is denoted as a uti lity cost function and corresponds to the cost as sociated with using a prespecified uti lity control sequence for all future con­trol decisions . A control policy synthesiz­ed by this method is actively ada.pti ve since the value of future information is character­ized by the approximated cost to go , and af­fects the present control deci si on . A result is presented in Theorem 3 . 1 . , whi ch is useful for establishing theoreti cal bounds on the performance of the synthesized control poli­cy. The method of utility costs is used to generate a new class of actively adaptive control policies { C�}N-l , in Section 4 . The m=l utility control sequences are chosen to be of the passive type , which is desirable sir:ce these policies can be determined by the solu­tion of deterministic rather than stochastic optimal control problems . As an initial in­vestigation into the performance of the new class of control policies , a theoreti cal per­formance bound is given for policy c�-1 . demonstrating its improved performance rela­tive to the OLF policy. A numerical example is given to demonstrate the relative perfor­mance of the c�-1 and OLF control policies , as well as to suggest a method for implemen­tation . Conclusions are deferred t o Section 5 .

An earlier presentation of these results (sub­mitted for publication at the same time as this paper , but published at an earlier date ) can be found in the literature ( Bayard and Eslami , 1983 ) . Since the proofs for many of the results presented in this paper are avail­able in that reference , these proofs will not be repeated here . Instead , a more elaborate simulation and discussion of the numerical example will be presented herein.

2 . STOCHASTIC CONTROL PROBLEM

The following state and observation equations are considered ,

xk+l=fk( xk •l\:.•wk ) ' k=O , l , . . . , N-1 , ( 2 . 1 )

Yo =ho ( xo ,vo ) ' ( 2 . 2a )

( 2 . 2b )

It i s assumed that a state space Sk , a control space Qk ' an observation space � ' and dis-turbance spaces D� and D� , are defined for

w each k such that XkESk ' 1\:_E� , YkE� , WkEDk , and v�ED�. The control uk i s constrained to � which is a non-empty subset of Qk . The sequences of random disturbances {wk } and {V){} are assumed to be white , jointly independent , and characterized by given probability meas­ures , Pv ( • ) and Pw ( • ) , k=O , . . • ,N-1 , respec-k k tively. The initial state x0 is also assumed random with probability measure Px ( • ) . 0 The information available to the controller at time k is denoted by Ik and will be called the information state . Thus , let

An admissible c?ntrol�law is defined as a se­quence of functions IT={ µ0 ,µ1 , . . . ,µN_1 } , where each function µk maps the information state Ik into the space of controls � such that , µk ( Ik ) Enk , ¥Ik , k=O , l , . . . ,N-1 . For conven­ience we define a truncated control law as ITk�{ µk • · · · ,µN-1 } .

The expected cost due to a particular admiss­ible control law IT={ µ0 , µ1 , . . . , µN-l } i s defin­ed by ,

N-1 J= E

xO ,wi ,vi+l i=O , . . . ,N-1

[ gN (xN ) + l gk (xk ,µk ( Ik ) 'wk ) ] . k=O

( 2 . 4 ) Here i t i s assumed that the real-valued func­tions , gN : SN+R , gk : Skx�xD�+R are given.

The stochasti c control problem i s then to find an admissible control law IT={ µ O ,µl • · · · µN-1} that minimizes expected cost ( 2 . 4 ) sub­j ect to system equation ( 2 . 1 ) and measurement equation ( 2 . 2 ) .

3 . METHOD OF UTILITY COSTS

It is known that the optimal stochastic con­trol for the above problem can be generated by backward recursion un the following dyna­mic programming ( DP ) equations ( cf . , Bertsekas , 1976 ) '

J��� ( IN-l )= min E [ gN (xN ) �-lEQN-1 ( 3 . la )

+gN-l ( xN-l 'uN-l 'wN-l ) \ IN-l '�-1 ) ,

J�LO ( Ik )= min E [ gk (� 'l\:. 'wk ) 1\:_EQk

+J��� ( Ik+l ) \ Ik ,1\:. ] . ( 3 . lb )

Here , the superscript CLO stands for closed

Fini te Time Stochas tic Adaptive Controllers 1 87

loop optimal, utili zing the terminology of Bar-Shalom and Tse ( 1976 ) .

The implementation of ( 3 . 1 ) to find the CLO control poli cy is often difficult due to an­alytical problems in finding simple recursive solutions and/or numerical problems involving the dimensionality of the underlying spaces . Thi s is particularly true in adaptive con­trol problems where one is dealing with a nonlinear augmented state model . In view of the stated di fficultie s , it is useful to ap­proximate the control policy given in ( 3 . 1 ) in such a way as to retain the desirable properties of the CLO solution . In particu­lar one wi shes to retain the closed-loop character of the CLO solution in order to take advantage of the active learning capa­bility present when the dual effect exi sts ( Bar-Shalom and Tse , 1974 ) .

A technique for generating suboptimal control poli cies by approximation of ( 3 . 1 ) is now in­troduced. It is noted that if at time k the system i s in state Ik ' and the cost to go J�:� ( Ik+l ) is known for all Ik+l ' then

µ�LO ( Ik ) i s found as the argument of the min-CLO imization in ( 3 . 1 ) . Here , J ( Ik 1 )

i s the expected cost of the f�ture+of the

proces s assuming that the system is in state

Ik+l and that

CLO }f!TICLO . µN-1 - k+l is

CLO the control poli cy { µk+l ' ' . . ,

used for future control de ci-

sions . The analytical expression for JCLO ( ) k+l Ik+l is given by ,

In place of knowing the future CLO poli cy IT�:� . we define a sequence of admissible

. u u 6 u funct ions { µk+l ' ' ' . , µN_1 }=TI (k+l ) . The con-

trol poli cy nu(k+l ) will be called the ut?'.l­ity contro l sequence at time k+l . Most gen­erally , nu( i ) can be chosen independently of ITu( j ) for ifj . Sometimes , however , it will be useful to de fine nu( i ) , i=l , . . . , N-1 as truncations of a single specified control policy n'l1:: { µ8, . . . , µft_1 } ( i . e . , nu( i )=IT�,i=l , . . . , N-1 ) . In this case the set of utiiity control sequences { ITu( i ) }r�l will be called consistent, and the control poli cy ITu will be called the uti lity contro l sequence gen­erator. The utility control sequence ITu(k+l ) at time k+l is used to generate a utility cost function J�+1 ( Ik+l l defined by the fol­lowing expression , J�+l ( Ik++ l=E( gN ( xN )

N-1 . + l gi ( x . , µ:1 ( r . ) ,w . ) 1 Ik+l ] , i=k+l i i i i

for k=O , . . . , N-2

( 3 . 3 )

If J�+1 ( Ik+l ) i s used as an approximation to

J�:� ( Ik+l ) , then a new control function µ=(Ik )

can be defined by the expression ,

µ� ( Ik )=arg min E [ gk ( � '� 'wk ) �£Ilk

+J�+l ( Ik+l ) I Ik '� J . ( 3 . 4 )

The specification o f the set o f all utility N-1

control sequences { Tiu ( i ) } i=l therefore de-fines

*completely a new control policy

. . ' , µN-1 } '

*6 * IT ={ µ0 '

The following theorem and corollary are o�en useful for developing bounds on the perfor­mance of control poli cies synthesized via the method of utility costs . For the proof of these result s , the reader i s referred to Bayard and Eslami ( 1983 ) .

Theorem 3 . 1 . ity control tent set of N-1 { Tiu( i ) }i=l ;

ui'l u u } . Let TI ={ µ0 , . . . , µN-l be a ut:l-sequence generator for a consis­utili ty control sequences

let J�+l ( Ik+l ) be defined by ( 3 . 3 ) for all Ik+l 'k=O , . . . ,N-2 , and let the control

. * * * . poli cy IT = { µ 0 , . . . , µN-l } be defined by ( 3 . 4 ) . Then the inequality given by ,

holds uniformly in Ik ' for all k , k=O ,l , . . . , N-1 , where J� ( I� ) i s the expected cost to go at time k associated with using IT* for fuiure control decisions .

* Corollary 3 . 1 . The total expected costs J and JU corresponding to control poli cies IT* and ITu respectively , satisfy the following inequality

J*<Ju.

These results es sentially imply that any giv­en control poli cy can be used as a utility control sequence generator to synthesize a new control poli cy which has improved perfor­mance characteri stics relative to the given policy.

4 . A NEW CLASS OF ACTIVE ADAPTIVE CONTROLLERS

A new class of active adaptive control poli­N-1

{ Cm} i s now proposed. The control policy a m=l C� i s generated by the method of utility cost s , using certain passively adaptive con­trol poli cies for the utility control seq'Ll"?n­ces .

4 . 1 . Choice of utility control sequen::e s In order to define the desired utility con-

1 88 D . S . Bayard

trol sequences , it is useful to define for j N-1

the sequence of functions { µ� l j ( Ij ) }k=j by the following expression ,

µ� I . ( I . )=arg J J k

N-1

min E [ gN( xN ) + U . E: s-2 . i i

L gi (xi ,ui ,wi ) I IJ. ] . i=j

( 4 . 1 )

Thi s expression yields the optimal control sequence for the stochasti c control problem as suming that no future measurements are to be taken . The minimization in ( 4 . 1 ) there­fore requires the solution to a determinis­tic rather than stochastic optimal control problem for each choice of j , j =O , l , . . . ,N-1 . Because the control functions defined in ( 4 . 1 ) are derived under the assumption o f no fu­ture measurements , they are of the pass ive type . These functions are useful for con­structing various passive control policie s . For example the OL policy ( cf. , Bar-Shalom and Tse , 1976 ) can be written as , rrOL = { p p p } µ0 1 0 , µ1 1 0 , . . . , µN-l l o , and the OLF policy (Dreyfus , 1965 ) , can be written as ,

( 4 . 2 )

The control functions defined by ( 4 . 1 ) are consequently very convenient for defining utility control sequences in the method of uti lity cost s . A class of utility control

N-1 sequences { rru ,m(k ) }k=l ' is now defined for each m ,m=l , . . . ,N-1 , as follows ,

u ,m( ) - { p p p rr k - µk l k ' ' ' ' , llk+m l k+m 'llk+m+l l k+m ' ' ' ' '

µ�-l l k+m} ' for k+m<N-1 ,

={µi l k ' ' ' ' , µ�-l l N-1 } , for k+m:N(t : 3 )

The control sequence rru ,m (k ) can be inter­preted as a sequence of open-loop feedback inputs from time k until time k+m , after which the inputs are chosen open-loop . The corresponding utility cost functions are then given by , }k+m] ( I )= k+l k+l

k+m-1 E [ I g . ( x . , µ� I . ( I . ) 'w . ) +

i=k+l i i i i i i

N-1 l g . ( x . , µ . l i+ ( Ik+ ) ,w . )+gN (xN ) I Ik+l ] ' j =k+m J J J m m J

for k+m<N-1 N-1

=E [ l gi ( xi ,lll l i ( Ii ) 'wi )+gN( xN) I Ik+l] i=k+l

for k+m>N-1 ( 4 . 4 )

Here , the truncation operator [ · ] i s de­fined as ,

and M. Eslami [. i , for i <N-1 , [ i ] =

N-1 , for i >N-1 .

The utility cost function J[k+m] ( I ) has k+l k+l

the following interpretation . It is the ex­pected cost to go at time k+l , associated with using pas sive control policies for all future control decisions , based on the assump­tion that only the next m measurements will be taken .

The evaluation of utility cost functions J

[k+m] 1 I ) . ( 4 4) . k+l 1 k+l in . involves e ssentially

only an integration . The integration , how­ever , i s over all possible future paths , and thi s in general will be the most difficult step in the actual implementation . One way to simplify the evaluation of ( 4 . 4 ) i s to in­tegrate only over the most likely future paths . A linearization method to do this i s demon­strated in the numerical example of Section 4 . 4 . A more direct method of evaluating ( 4 . 4 ) i s t o use a Monte Carlo type integration where the expectation is evaluated by aver­aging over several sample paths . Alternative­ly , the method of utility costs can be app)ied to "wide-sense" control ( Tse , Bar-Shalom and Meier III , 1973 ; Tse and Bar-Shalom , 1976 ) , where many of the expressions ari sing from the theory can be approximated in terms of two moments . Various computational methods are presently under investigation .

A relation between the utility cost functions defined in ( 4 . 4 ) for various m is given in the following theorem , the proof of which can be found in Bayard and Eslami ( 1983 ) .

Theorem 4 . 1 . For any k such that o:k:N-2 , the following ordering holds uniformly in Ik+l '

CLO N-1 k+2 Jk+l ( Ik+l ) 2Jk+l ( Ik+l ) 2 · · · 2Jk+l ( Ik+l )

4 . 2 .

k+l :Jk+l ( Ik+l ) .

N-1 Synthesis of { Cm} a m"'l

A new control policy denoted as � i s now con­structed using the method of utility cost s , with the utility cost functions given by ( 4 . 4 ) . The control law corresponding to cN is given at time k by the following relation ,

µ� 'm( Ik ) =arg min E [ gk ( xk '� 'wk ) �

+J���m] ( Ik+l ) I Ik ] · ( 4 . 5 )

The integer m can range from m=l to m=N-1 in ( 4 . 5 ) , giving rise to the class of control

N-1 { Cm} The one-measurement ( OM ) policy of a m=l 1 Curry ( 1969 ) corresponds exactly to C • The OM policy has been shown by simulatiog to have improved performance characteri stics

Finite Time Stochas tic Adaptive Control ler s 1 89

relative to the OLF policy for the control of a known linear system having quantized meas­urements ( Curry , 1969 , 1970 ) . From Theorem 4 . 1 . , it is noted that the optimal cost to go J��� ( Ik+l ) is better approximated in ( 4 . 5 ) by using larger values of m. It i s therefore ex­pected that the performance of control poli­cy � will improve with increasing m. By the interpretation given earlier for J���m] ( Ik+l) , the integer rn will represent in some sense -the degree to which poli cy � anticipates fu­ture measurements .

4 . 3 . Theoretical performance bounds

Upon setting m=N-1 in ( 4 . 3 ) it i s seen that the set of utility control sequences

u N-1 N-1 N-1 {IT ' ( k ) }k=l which define Ca i s consis-tent , and has the following utility control sequence generator ,

u ,N-1_ { p p p } IT - µ0 ! 0 'µ1 1 1 • · · · , µN-l l N-1 .

]Ju ,N-1 . " d t " 1 t IJOLF However , i s i e n ica o as seen by ( 4 . 2 ) , and we have as a direct consequence of Corollary 3 . 1. , the performance bound,

( 4 . 6 )

where JCLO ,Ja ,N-l , and JOLF are the expected costs of the CLO , c�-1 and OLF control poli­cies , respectively .

4 . 4 . Numerical example

In order to demonstrate the performance of the c�-1 policy relative to the OLF policy , and to suggest a method on how to evaluate ( at least approximately ) the utility cost functions , we consider the problem of con­trolling the scalar system given by ,

( 4 . 7 )

( 4 . 8 )

so as t o minimize the quadratic performance measure ,

N 2 J=E [ l a . x . ] . i=O i i ( 4 . 9 )

Here , � is the state which i s observed di­rectly , uk is the control , {wk} is a sequence of Gaussian random variables with E [wk ]=O and E [w� ]=cr2 , and b is an unknown pari:ine�er . I t is assumed that b has a known a pr�or� Gaussian probability density function with mean b0 and covariance P0 .

Using the notation b . flE [b I I . ] , Pi=E [ (b-b . f lr . J, the conditional momeiits canibe generateJ re! cursively by the following equations ,

( 4 . 10 )

( 4 . 11 )

ASCSP-G*

The generator policy for C�-l is the OLF pol­icy which i s found by performing the indica­ted minimization in ( 4 . 2 ) (to find µ� I . ) and letting k=j . The resulting expressionJi s given by ,

p OLF k -1 k ( 4 ) µk l k =µk =- [ 1 , 0 , . . . , o ] [ B ] 8 � ,':fk . . 12

H Bk R ( N-k ) x ( N-k ) d 0k RN-k ere , c_k k k an µ E: • ·ponents of B and 8 are given by ,

{ Bk }=NZkT T b (2j -i-£ )

i , £ j =l i l j i l .j k �+j

{ Dk. } N-k (2 j-i")

µ = l Ti I . bk �+J· i j=l J

The corn-

( 4 . 13 )

( 4 . 14 )

Here , the truncation operator T . I . i s defined by , i J

( 4 . 15 ) i>k

and use is made of the notation b�flE [bi l I ] . It i s seen that the OLF control law ( 4 . 12� is a linear function of the state xk , with a weighting factor which i s a function of the conditional moments of b given information state Ik . The conditional moments can in turn be expressed in terms of the conditional mean and covariance bt and Pi respectively , as generated by ( 4 . 10 ; and ( 4 . 11 ) .

In order to calculate the utility cost JN-l( Ik) from the utility control sequence Ilu(k ) = { p� l k ' " . . , µ�-l l N-l } , we consider the nonlin­ear system, bn+l bn bn+l f1 (b ,b ,P ,x ,w ) n n n n n

f2 (P ,x ) n n ( 4 . 16 )

P ( - ) - N xn+l b x +w +µ I P ,b ,x ,n-k , . . . , -1 · n n n n n n n n Here , f1 ( · ) is defined by ( 4 . 10 ) and f2 ( • ) is defined by ( 4 . 11 ) . We can rewrite ( 4 . 16 ) in vector notation as follows ,

6 . zn+l=Fn ( zn ,wn ) ' n=k , . . . ,N-1 , ( 4 . 17 )

- J T where z = [b ,b ,P ,x . A nominal traject-n n n n n . . . d " . ory is now defined from the initial con ition 0 [ - - J T . zk= bk ,bk ,Pk ,� , and the relation , 0 0 z 1=F ( z , 0 ) , n=k , . . . ,N-1 . n+ n n

0 6 0 0 0 0 T Here, zn = [bn ,bn ,Pn ,xn ] .

A perturbation ozn is defined by , 0 - J o zn=zn-zn= [ obn , obn , oPn , oxn .

( 4 . 18 )

( 4 . 19 )

The propagation o f ozn using a first order expansion i s given by ,

( 4 . 20 ) Here , A and B are the appropriate partial 4eriv�t�ve mat�ices obtained from linearizing ( 4 . 17 J about the nominal trajectory. Let

1 90 D . S . Bayard and M . Eslami /', E =Cov[ oz ] . The initial condition Ek i s

g�ven by ,n

and the covariance propagates according to the relation ,

� A � AT + B BTa2 . Gn+l= nGn n n n ( 4 . 21 )

If the ( 4 , 4 ) entry of E corresponding to n44 Cov[ ox ] is denoted as E , then the utility cost fli.�ction J� is apprgximated up to the second-order as ,

u N 0 2 44 Jk(bk ,Pk ,� ) z . l ai ( ( xi ) +Ei ) .

i=k ( 4 . 22 )

The minimization of ( 4 . 5 ) must be implement­ed using a search algorithm. In evaluating the cost corresponding to a given control � in ( 4 . 5 ) at time k , it is inconvenient to first compute all required J�+f ( Ik+1 ) . The method used here is to start the nominal trajectory at time k as if one i s computing J�-l ( Ik ) using the above method , and to use uk instead of µ� l k as the first input .

The average total cost of forty Monte Carlo trials for the OLF , c�-1 , HCE and OCKP ( op­timal control with known parameters ) control policies is tabulated in Table 1 and Table 2 for various data set s . Here , the HCE control policy i s given by the expression ,

HCE � = -bk�· The OCKP control policy i s the optimal policy for the corresponding non-adaptive problem , assuming the plant parameter is known . The performance of the OCKP policy therefore represents an unattainable lower bound on the performance of any adaptive control poli cy. The OCKP control for thi s problem is given by the expression ,

In earlier results for thi s numerical exam­ple , an unintentional bias in the simulation statisti cs led to misleadingly favorable re­sults ( Bayard and Eslami , 1983 ) . The cor­rected values for the same data sets are giv­en here in Table 1 . It is seen that for Run #1 and Run #2 the OLF policy performs very closely to optimal cost ( compare to the OCKP policy ) , leaving little room to demonstrate the improvement obtained by using the c�-l policy. In order to better expose the dif­ferences between control poli cies , the data sets of Table 2 were chosen so as to inten­tionally degrade the OLF performance . Thi s was done by increasing the initial parameter uncertainty from P= . 01 to P= . 05 , as well as exaggerating the terminal cost in increasing powers of ten , from Run #3 through Run #7 . Exaggerating the terminal cost acts to en­hance the role of the active learning fea-

ture of the c�-l policy. It is seen from Ta­ble 2 that the c�-1 policy performs close to optimal for all the runs considered , even though it is generated by the OLF policy , . whose performance deteriorates with increas­ing terminal weight . The improvement in per­formance of the c�-l policy relative to the OLF policy is predicted by Corollary 3 . 1 . , and is valid within the limits of the linear­ization approximati on used here to implement the c�-l policy . The HCE policy performs close to optimal for all runs considered here . It i s suspected that thi s is due to the simplicity of the un­derlying i dentification problem for thi s ex­ample . The simplicity of the HCE control pol­icy , as well as its relative performance in the simulations , motivates the idea of using the HCE policy as a utility control sequence generator policy in the method of utility cost s . The resulting controller , by the the­ory of Section 3 . , would show improved per­formance relative to that of the HCE policy .

TABLE 1 .

ai Average Total Cost

ai=l , 0,'.:i:'.:3 c"-l.

a OLF HCE OCKP

a4=1 4 . 065 4 . 065 4 . 065 4 . 044

s.4=10 4 . 152 4 . 159 4 . 152 4 . 132

TABLE 2 .

ai Average Total Cost

ai =1 , o::i,::3 CN-1 OLF HCE OCKP a a4=1 4 . 14 4 4 . 187 4 . 144 4. 022

a4=10 4 . 188 4. 495 4 . 188 4 . 066

a4=100 4 . 622 6 . 54 4 4 . 622 4 . 508

a4=1000 8 . 968 15 . 52 8 . 967 8 . 924

a4=10000 9 . 009 24 . 977 9 . 009 8 . 909

�1=4 ,PJ= . 05 ,ii0= . 8 ,x0=2. , o2= . 005 for Run # 3 , 4 , 5 ,6

o2= . 0005 for Run ¥7

5 . CONCLUSIONS

�un # 1

2

Run # 3

4

5

6 7

A synthesis te chnique called the method of utility costs , as well as some results con­cerning theoretical bounds on the performance of control policies synthesized via this meth­od , have been established. This method i s used t o generate a new class of active adap-N-1 tive controllers { C�}m=l where the integer m i s related to the extent to which the control policy anticipates future measurements . A theoretical performance bound i s establi shed for c�-l by application of the theory devel­oped earlier , demonstrating an improvement in performance relative to the OLF control poli­cy. The most difficult step in the implemen-

Fini te Time S tochas tic Adaptive Control lers 1 9 1 tation of controllers synthesized by the method of utility cost s , is the evaluation of the utility cost functi ons , whi ch involve integrations over the future of the process . Lineari zation techniques , Monte Carlo type integration as well as wide-sense control ideas are presently under investigation for reducing implementation overhead. A numeri­cal example is given to compare the c�-l and OLF control policies , and to suggest a meth­od for implementation by linearization . The performance advantage of the c�-1 policy over the OLF policy, as e stabli shed earlier by theoretical means , is demonstrated by simulation .

6 . REFERENCES

Aoki , M. ( 1967 ) . Optimization of Stochastic Systems . Academic Press , New York.

Bar-Shalom , Y . , and E. Tse ( 1974 ) . Dual ef­fect , certainty equivalence , and separa­tion in stochastic control . IEEE Trans . Auto. Contr . , 19 , pp . 494-500 .

Bar-Shalom , Y . , and E . Tse ( 1976 ) . Concepts and methods in stochastic control . In Leondes ( Ed. ) , Control and Dynamic Systems , vol . 12 , Academic Press , New York. pp . 99-172 .

Bayard , D . S . , and M. Eslami ( 1983 ) . On sub­optimal stochastic adaptive controllers . Proc . Conf. on Info . Sciences and Systems , The John Hopkins University , March 2 3-25 , 1983 .

Bellman , R . ( 1961 ) . Adaptive Control Pro­cesse s . Princeton University Press , New Jersey.

Bertsekas , D . P . ( 1976 ) , Dynamic Programming and Stochasti c Control . Academic Press , New York .

Curry , R . E . ( 1969 ) . A new algorithm for suboptimal stochasti c control . IEEE Trans . Auto . Contr . , 14 , pp . 5 33-5 3 6 .

Curry , R .E . ( 1970 ) . Estimation and Control with Quantized Measurements . M . I . T . Press , Cambridge , MA.

Dreyfus , S. ( 1965 ) , Dynamic Programming and the Calculus of Variations . Academic Press , New York .

Feldbaum, A .A . ( 1961 ) . Dual control theory I-II . Aut . Remote Contr . , 21 , pp . 874-880 , pp . 1033-1039.

Feldbaum, A .A . ( 1962 ) . Dual control theory III-IV. Aut . Remote Contr. , 22 , pp . 1-12 , pp . 109-121 .

Tse , E . , and Y . Bar-Shalom ( 1976) . Actively adaptive control for nonlinear stochastic systems . Proc . IEEE , 64 , pp . 1172-1181 .

Tse , E . , Y . Bar-Shalom , and L . Meier III . ( 1973 ) . Wide-sense adaptive dual control of stochastic nonlinear systems . IEEE Trans . Auto . Contr . , 18 , pp . 98-10S:----

Wittenmark , B. ( 1975 ) . Stochastic adaptive control methods : a survey. Int . J . Contr. , 21 , pp . 705-730 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

SUBOPTIMAL CONTROL LAWS OF MARKOV CHAINS: A STOCHASTIC APPROXIMATION

APPROACH*

Hai Huang

Department of Systems Science and Mat/innatics, Washington University, St. Louis, MO 6 3 130, USA

Abstract : This paper is concerned with a stochastic approximation algorithm for estimating the optimal control law of an average-cost problem of a finite­stat e controlled Markov chain , while the values of transition probabili ties are unknown . Assuming that the s tate sequence of the Markov chain is complete­ly observable , we prove that the sequence of control laws generated by the al­gorithm and depending on the state sequence is convergent to a small compact neighborhood of the optimal control law with probability one by using weak convergence technique .

Keywords : Adaptive control ; s tochastic control ; Markov process ; convergence of numerical methods ; s tabi lity criteria

1. INTRODUCTION

Consider a controlled Markov chain (� , n>O) n -taking values in a fintte s tate space S = { l , 2 , • • • , M} . The collection of control laws is the r-dimensional Euclidean space :m. r and the transition probability is

p ( i , j \ x ) n i) . ( 1)

where x s:m. r is a control law used at t ime n . n The dimension r and the size M of the state space are not necessarily identical. If they are identical and

p ( i , j \ x ) = p ( i , j \ x ( i) ) , n n where xn (i) is the ith coordinate of xn ' then the control law is closed-loop .

For each xs:m. r , let lP (x) denotes the MxM transition matrix

lP (x) = [ p ( i , j \ x) ] . . _1_. M l. , J - • • • • . Suppose

TI (x) = [TI (x \ l) , TI (x \ 2) , . . . , TI (x I M) ]

( 2)

is the unique invariant distribution of lP (x) , namely , TI (x) is the only M-dimensional row vec­tor satisfying

TI ( i \ x).'."._0 , � � ( i \ x)= l , and TI (x) lP (x) =TI (x) (3 )

To each x associate the cost

*This work was supported by the National Science Foundation under NSF-ECS 81059 78 .

1 93

G (x) 2. F (j , x) TI (j j x) , j= l

(4)

where F ( • , • ) is a real-valued function on sx:m. r . One of importan t t op ics in controlled Markov chains is t o minimize the cost G (x) over :m. r . This is equivalent to minimizing

� n.!1 E k F (�,Q, ' x,Q,) ( 5 )

among all sequence of control laws , if the invariant dis tribution TI (x) exists for each x , cf . [ 3 ) . Thi s paper i s concerned with this type of control problems, while the s tate sequence {� , n>O} is completely observab le, n but the values of transition probabilities are unknown .

Assume tnac the Markov chain ( �n , n�O_) has 2r

independent and identically distributed copies and F (j , x), p (i ,j \ x) and TI (j i x) are differen­tiable with respect to x . We proceed by a s tochastic approximation approach . The algo­rithm is a mixture of Robbins-Monro type and Kiefer-Wolfowitz type (See [ 4 ] ) . Assume the existence and uniqueness of the optimal con­trol law. Then we will prove that the output of the algorithm will be convergent to a small neighborhood of the opt imal control law .

.The method which will be exploited for proving convergence results is based on the weak con­vergence techniques developed in [ 5 ] "-' [ 6 ] , but being modified for the current prob lem .

1 94 Hai Huang

2 . ASSUMPTIONS AND THE ALGORITHM Let K denote a positive constant whose value may change from usage to usage. Let ei denote the ith unit vector defined by ei [ O , • • • , O , 1, O , • • • , O ] *E:1R r (6) where 1 is at the ith coordinate and * denotes transpose . The following assumptions (Al)�(A6) will be used throughout the paper . Assumptions (Al) sup j F(j x) ! 2_ K( l+ l x l ) 112 , for all xs1Rr .

j E:S The first derivative Fx (j , x) of F exists and satisfies sup I F (j , x) I < 00, (Note that the

j E:S x . XE:]Rr

subscript x denotes derivative with respect to x . )

(A2) For each x, there is an invariant measure TI (x) satisfying ( 3) and

I * n I n sup ei l' (x)-TI (x) 5_KY , for all n>O

l<i<M X t: JR r

where Y is a constant and O<Y<l .

( 7 )

(A3) P ( i , j l x) and TI (j j x) exists and are x . x uniformly bounded with respect to (i , j , x) . Let c denote a small positive number , Define

G (x+ce1)-G (x-ce1)

G(x , c)

where G(x) is

2c

G(x+cea)-G (x-cea) 2c

. G (x+ce ) -G(x-ce ) r r

2c defined by Equation (4) .

( 8)

2 (A4) G (x , c) = Gx (x) + c H(c , x) . Gx (x) and H(c, x) are continuous in x . And H(c, x) satisfies sup

o<c<c - 0 ! H( c , x) ! .::_ K0 < 00

xE:lR r where c0 and K0 are constants .

2 c H(c , x) has a unique (AS) � = -G (x) -x

(9 )

solution on [O, 00) frcm every initial point in lR r . An r-dimensional stochastic process (s ( t) , t.<!.O) satisfies

• 2 s (t ) =-Gx(s (t ) ) -c H(c , s ( t) ) , on [0 ,00) , with probability one , (10)

if ( f (s (t ) )-f (s (O) )+ f t f* (s (s ) ) [G (s (s) )+

0 x x 2 c H (c , s (s ) ) ] ds , t.:::_O) is a martingale, for

00 every real-valued C function f with compact support . (A6) x = -Gx(x) has a unique solution on [ 0, 00) from every initial point in lR r . More­over, there is a fixed point 8 in 1R r such that i) Gx(8) = 0 , Gx (x) 1 0 , if x 1 8 ,

* * * ii) (x-8) Gx(x) .:::_ P (x-8) (x-8) , if (x-8) (x-8)2_1 , * * (x-8) G (x) > v ! x-8 1 , if (x-8) (x-8)> 1 where x - - - --1 , V , P are positive constants . Remarks (A4) can be checked by using Taylor expansion . (AS) is the ordinary differential equation version of the Martingale Problem of Stroock and Varadhan (See [ 7 ] ) , 8 in (A6) is the unique optimal control law minimizing G(x) on 1R r . Let i be a generic element of s2r denoted by ->- .+ . - .+ . - .+ . -) i = (11 , il , • . • , 1 ' 1 ' , , . , 1 , 1 • a a r r

( 11)

Let <=n • n.:::o) be a cont�olled Xarkoy chain on s2r denoted by - + - + - + -� = <s 1 . s 1 • . . . • s . s , . . . . s , s > < 12> n n, n, n ,a n ,a n,r n,r and governed by the transition probability r 1P <= +1=1 1 = =i) = rr p (i+,j+ l x +ce ) p (i- ,j - l x -ce ) n n a=l a a n a a a n a

(13) where xn is a control law used at time n . For estimating the optimal control 8 , we pro­pose the following algorithm: Algorithm: Choose x0 in 1R r arbitrarily.

+ - + -Set sO , l = sO , l = . . . = s0 = sO , r = s0 = the initial state of the controlled Markov chain (sn ' n.:::_O) .

X =X -a n+l n n

+ -F(s +l ,x +ce )-F(s +l ,x -ce ) n ,a n a n ,a n a 2c

+ -F(s +l ,x +ce ) -F(s +l ,x -ce ) n , r n r n ,r n ,. 2c

(14)

Suboptimal Control Laws of Markov Chains 1 9 5

where c is a small constant and ( a , n�O) is n a sequence of positive numbers satisfying

Lan = 00 and L a! < oo •

- + - + Note that �n+l= (sn+l, l ' sn+l , l ' • • · • sn+l , r ' s-

+l ) is determined by the state � and the n , r n control law Xn at time n via transition pro-bability defined by ( 13) . In Section 3 we give some definitions and results of weak convergence theory , In Section 4 we prove some auxiliary results concerning uniform convergence of e�lP n (x) to n (x) . In l. x x Section 5 we prove the tightness result of the sequence (Xn ' n�O) . Let Xn ( • ) denote a stoch­astic process in :JR r defined by

Xn( ) i· f t = xn+k ' !1 a . 2_ t < ! ai i=n l. i=n

( 15)

In Section 6 we prove the tightness result of the sequence of stochastic processes (Xn ( • ) , n�O) . Then we prove that Xn will be convergent to a neighborhood of 6 with radius proportional to c with probability one in Section 7 ,

3 , ;MTHEM,A,TIC.A,L BACKGROUND ON WEN< CONVERGENCE THEORY

The theory of weak convergence of a sequence of probability measures is a substantial gen­eralization of the notion of convergence in distribution of Euclidean space random variables . Only a few results and connnents will be given here. For a full treatment see [ l] , Let � be a metric space . and V the collection of Borel sets in � . A family A of 6-valued random variables is said to be tight if for every €>0 there is a compact set K such that

lP (XEK)> 1-€ , for all XEA . In :JR. r , sup E \ X \ 8<00 implies the tightness of n n {X , n�O}, where 8 is a positive number . By n weak convergence of Xn to X in 6 we mean that for every real-valued bounded and continuous function f on 6, Ef (X )+Ef (X) , as n+oo , Suppose n A = {x , aEA} is t ight and {X , n>O} C:. A . a n - -Then there is a weakly convergent subsequence of {X , n>O} . n -Dr [ 0 , 00) denotes the space of :JR r -valued func­tions on [ O , 00) which are right continuous and have left-hand limits ; it is endowed with the Skorohod topology (See [ l ] ) . Note that Dr [ O , 00) is a complete separable metric space . Suppose {Xn( • ) } is a sequence of stochastic processes with pathes in Dr [ O , 00) . For each n (F� , t�O) is an increasing family of

a-algebras such that Xn ( t ) is F�-measurable . Let E� denote the expectation operator condi-­tioning on F� . The f ollowing special cas e · of Kurtz ' s theorem [2 , ( 4 . 20) ] will be useful for proving the tightness of {xn ( • ) } in Dr [ O , 00) . Lemma 1 . The sequence {xn ( • ) } is tight in Dr [ O , 00) , if the following two conditions hold : i) For every T > 0 and n>o there is a compact set K in :JR r such that

lim inf lP (Xn( t ) EK all 02_t2_T) >l-n. ( 16) n+oo

ii) For every T>O and n there are random variables y (8 ) such that n E� yn (8 ) � E� \ Xn ( t+u) - Xn ( t ) \ 2 (17 )

for all t , u satisfying 02_t2_T , 02_u2_<Y\(T-t ) , and that lim lim sup E Y n ( 8 ) = 0 and ( 18) 8+0 n+oo lim lim sup E \ Xn(8)-Xn(O ) \ 2 = o . ( 19 ) 8+0 n+oo

4 , SOME PRELIMINARY RESULTS

Let 3a denote the partial differential opera­tor 3/ 3xa ' where xa is the ath coordinate of xElR r , Let C be the collection of complex numbers and CM the product of M C ' s . A generic element of CM is denoted by a row vector . Lemma 2 . Under (A2) , the following statements (i)rv ( iii) hold . ( i ) vc:efl and vlP (x) v , then v an (x) , for some as C-.-

(ii ) All eigenvalues of lP (x) , except 1 , are bounded by y .

(iii) M Given usC , v ( I-lP ) = u has a solution,

say v , in CM if '!,1

ui=O , wher� u= (u1 , • . . , u1) . i=

Definition : Let H (x) denote the span of left eigenvectors and generalized left eigenvectors of lP (x) , except for the eigenvectors which correspond to the eigenvalue unity , and I (x) the lef t eigenspace of lP (x) corresponding to the eigenvalue unity . Theorem 1 Assume (A2) and (A3 ) . There are constants K>O and O<y 1<1 such that

* n n I ei3alP (x)-3an (x) J � Ky 1 , for every a=l , . . . , r , every xc: :JR r , every �O . Proof . Since I-lP (x) is nonsingular on H (x) , lP (x) is a contraction on H (x) , and n (x) 3�lP(x)

* n-£-1 "" and e . lP (x) 3 lP (x) are in H (x) , where i a £=0 , . . . ,n- 1, the result f ol lows . D

1 96 Hai Huang

5 . TIGHTNESS OF { x ' n>O} n -Without loss of generality, henceforth, assume that the optimal control law 8 in (A6) is the zero vector in lR r . L t E b h e n e t e expectation operator conditioning on (X,Q, , 3£ 2 £.::_n+l) .

Theorem 2 . Assume (Al) - (A6) . Suppose c2<min {v/K0 , l'fP/K.0 , c0 } , where c0 and KO are the constants in (A4) . Then sup E j x 1 2<00, Hence �- n �� {Xn ' n�O} is tight . n

Proof . Set S (c)=c2 ITK0/ p . Then B ( c) <t . Let A (t) denote a C00-function defined by { 1 if t>t A ( t) = between 0 and 1 , if B (t).::_t.::_T

o if t.::_B ( c) and satisfying : A ( t) >O when S (c)<t<t . Define t

2 2 2 2 V (x) = (x1+ • • . +xr) A(x1+ . , .+xr) (20) *

where (x1 , . . . , xr) = xE:lR r . Then it is easy to see tnat

* if x x>B(c) * 2 { <O

V (x) (-G (x)-c H ( c , x) ) x x =O * ( 21)

if x x.::_S ( c) . The algorithm (14) can be rewritten as

2 + -X +l=X -a [ G (X )+c H ( c ,X ) ] -a Z +a Z , (22) n n n x n n n n n n where

+ + Z- = (Z- l ' n n ,

+ . . . , z- , . . . , n ,a +

Z- ) and n , r

z± :+ F ( 1;-+l , X +ce ) -G (X +ce ) n a n- a n-- a

n ,a By (Al) , j xn+l-xn l

a K < ___!!__ - c

2c

By (A2) , ( 26) is well-defined

( 23)

(24)

( 25)

co ! ti + - at-1 V-(x,n)=+ 2 zc: llaV (x)F (j .x:tcea) t=n+l a=l J =

<i-n-1) + I I [p ( �-+l j x+ce ) -n (j x+ce ) ] n ,a , - a - a (26) <t-n-1) I Note that p - (;i , j x:±_cea) is the (i , j ) ,.., £-n-1 entry of :JP (x+ce ) • Moreover , by (Al) , - a (A2) , and (20) ,

a K i v:':<x ,n) I .:::_ � Cl+ l x l ) 3 12

Define Vn by

(27)

V = V(X ) + V+(X , n) + V- (X ,n) (28) n n n n Write En Vn+l - Vn = Tl + T2 + T3

where

+ + EnV (Xn+l'n+l) - V (Xn ,n) ,

EnV- (Xn+l 'n+l) - V (Xn ' n) . Taylor expansion , (20) , (21) , ( 25) , and Theorem 1 yield 2

"' + * a K T1<-a V ( X ) Z +a V (X ) z-+_!!_2 ( H i x I ) · - n x n n n x n n n

* + c T2 =a V (X ) Z n x n n

Hence a2K

E V +l-V < _!!_2 (l+V (X ) ) , n�O , n n n - n (29) 2 c

since l x l .::_K ( l+V (x) ) . (Note that T2 and T3 cancel out the first two terms of T1 ; that is the reason for the introduction of V+ and V-) By (27) , there is a large integer N such that

a2K a2K n n EnVn+l .:::_ ( l+ -2 )Vn + -2 , n>N ( 30) c c

This inequality implies n-1 a2K k n-1 a�K aiK EV� II ( l+ n

2 ) EVN+ II (l+-2- )-2 ,n�N ,Q,=N c =N i-=�+l c c .

Hence sup EV(Xn) <co and the tightness result n

follows . D 6 , UGHTNESS OF {x

n (_• ) ' n>o}

For each n , define m ( ' ) on [ O , 00) by n m ( t ) = ma:idk>O : a + , , ,+a +k 1<t } , (31) n - n n - -Hence Xn (t) = Xm (t)+n ' t�O .

n Lemma 3 . sup E [ max l xn (t ) I J 2<00 , T<oo , n O<t<T

for every

Suboptimal Control La�s of Markov Chains 1 97

Proof . Since

the result follows . D Corolla!}'.. The condition (i) of Lennna 1 holds .

Proof . sup F ( max ! xn(t) I >i) _2 K/.Q,2' .Q,>Q ,

n O<t<T D Theorem 3 . {xn( • ) , n�O } is tight in D

r[O, 00) .

Proof . Since the condition (i) of Lennna 2 holds , tightness of {xn( • ) , n_'.:O } is equivalent to tightness in D [ O , 00) of { f (Xn ( • ) ) , n_'.:O }

00 for every real-valued C function f with com-pact support . Henceforth f denotes a real-

oo valued C function with compact support . Define f±(x, n) by

f±(x,n)=+ ! � 2: a��l Claf (x)F(j , x±cea) i=n+l � j=l

[p (£-n-l) ( �±+l a j ! x+ce ) -TI (j \ x+ce ) ] (32) n , , - a - a

By (Al) and (AZ ) , ( 32) is well-defined and satisfies

+ a K 1 /2 ! f-(x, n) ! 2 � ( 1 + I x ! ) , ( 33)

Define f (n) by - + -f (n) = f (X )+f (X ,n)+f (X ,n) ( 34) n n n Then , as in the proof of Theorem 2 , we have

- - * 2 E f (n+l)-f (n)=-a f (X ) [G (X )+c H( c , X ) ] n n x n x n n 2 2 + a R /c , n>O , ( 35 ) n n n -

where the nth remainder R n satisfies

I Rn l � K(l+ ! Xn l ) . (36) Hence

z a £ a � 2

-c H ( c , X H) ] }+ n; E R +_.!!. E S (n ,n+k) n c n n+£ c n (37 )

I I 1/2 where i s (n ,n+k) i.2.K(l+ l xn + l xn+k )

Moreover ,

E ( f (X +k) -f (X ) ) 2..:::_E �a +iK ( l+ max I x +i i ) . n n n n� n O<i<k-1 n n Define the expectation operator Et by

Then

E� [ f (Xn (t+s ) ) -f (Xn (t) ) ] 2

m ( t+s) -1

( 38)

n <En L a .Q,K (l+ max l xn (t ) i ) , - ti=m (t ) n+ O<t<T n - -

whenever O<t<t+s<T . Define the random variable yn (o ) in the-condition (ii) of Lennna 1 by

y (o ) = K(a +o ) ( l+ max l xn (t ) i ) n n O<t<T Then the tightness result of {Xn ( • ) , n>O) follows . - D Let (X(t) , t�O) be the limit of a weakly con­vergent subsequence of {Xn( • ) , n>O } in Dr [ O , oo) , Without los s of generality , let us assume that Xn( • ) converges to X ( • ) weakly in Dr [ 0 ,00) as

00 n-+oo. For every C function f with compact support , by (37 ) and ( 38 ) , we have

E�f (Xn( t+s ) ) -f (Xn(t ) ) =E� Jt

t+sf: (x

n(u) )

n 2 n [-G (X ( u) ) -c H ( c , X (u) ) ] du x mn (t+s)-1 2 'I;\' an+£ En L 2 t Rn+£ i=m ( t ) c n an n ' + -z Et S (n, t , t+s)

(39 )

where i s ' (n , t , t+s ) l.2_K( l+ l xn (t ) ! + ! xn (t+s ) i ) 1/2

Let q denote an integer , let O<t0<t1< • • • <t <t - q-and let g ( • ) be a bounded continuous function on JR rq , Then by (36) and (39)

limEg(Xn ( t . ) , i<q ) { f (Xn (t+s) ) -f (Xn (t ) ) l. -n->oo

J t+s * 2 - f (Xn ( u) ) [-G (Xn ( u) ) -c H(c , Xn( u) ) ] du}=O

t x x

Since Xn (_• ) converges to X(• ) weakly and G ( • ) x

and H ( c , • ) are continuous in x ,

(40)

where

f t * Mf ( t ) =f (X(t ) ) -f (X(O) ) - f (X(u) ) [-G (X(u) ) 0 x x

1 98 Hai Huang

2 -c H (c , X (u) ) ) du (41)

By the arbitrariness of q, ti ' and g ( • ) , Mf (t ) is a martingale . Hence (X(t ) , t>O) is a

• 2 -solution of x=-Gx(x)-c H ( c ,x) with probability one (AS ) . Thus we have proved the following theorem.

Theorem 4 . If (X(t) , �) is a weak limit of a weakly convergent subsequence of {Xn ( • ) , �O} in Dr [ O , 00) , then X(t ) is a s olution of • 2 x=-Gx (x)-c H( c , x) , with probability one .

7 . CONVERGENCE OF THE ALGORITHM

Suppose ¢ (t ; x0) is a solution of �=-G (x) -2 x

c H(c ,x) with ¢(0 ; x0) =x0• Suppose D is a

compact set in JR r , By (20) and (21) ,

lim sup t->= x0ED Lemma 4 .

V(¢( t ; x ) ) =O . 0

V (Xn)-+O in probability , as n-+oo,

Proof . Fix T>O . Define � ( t ) by

if t>T

T-� a . <t<T-ka . L, n-i- n-i i=l =

(42�

(43)

Suppose { n ' } is a subsequence of {n} such that n ' I

{XT ( • ) } and { xn ( • ) } are weakly convergent to XT( • ) and X( • ) , respectively . Then X(O) and XT (T) are identically distributed . Since {X , n>O} lJ { limits of weakly convergent sub­n -sequences of {X , n>O } } is tight and both X(O) n -and Xr (O) are in this union ,

V (XT (T) )+O in distribution as T-+oo ,

This implies V(X(O) ) =O with probability one . The result follows . [] Set Q={xEJR r 1 1 x-8 I

2<f3(c) } . Note that 8 is the optimal control law-(A6) .

Theorem 5 . Xn converges t o Q with probability one as n-+oo ,

Proof , It is sufficient to show that V(X ) · n converges to zero with probability one , Assume 8=0 . By (30) , define W by n

00 2 00 00 2 2 a K � a.R,K al w = JI (l+ n2 )V +L JI Cl+-2 ) 2 , n>N (44) n t=n c n j =n .R-=j+l c c

Then , by (27 ) ,

(45) So l w I converges to zero in probability , On n the other hand (W , n>N) is a supermartingale n -satis fying sup E I W I <�. Hence Wn converges to

n>N n zero with probability one by using the Martin­gale Convergence Theorem. From (44) and ( 27) we have

a K a K j vn l.�.l wn l and (1-_E_ )V(X ) -_E_ < v n>N c n c - n '

Thus V (Xn) converges to zero with probability one . []

8 . CONCLUSIONS

We have derived a stochastic approximation algorithm, which is a mixture of Robbins­Monro type and Kiefer-Wolfowitz type , for estimating the optimal control law of an average-cost problem on a finite-s tate Markov chain without knowing the values of transition probabilities . The algorithm and the techni­que for proving the convergence results are important , s ince the driving force of the al­forithm is discontinuous and the "noise" se­quence is state-dependent .

ACKNOWLEDGEMENTS

The author is grateful to Professor H . J . Kushner for useful discussions .

REFERENCES

l. Billingsley , P. ( 1968) Convergence of Pro­bability Measures , Wiley , New Yo:k .

2 . Kurtz , T , G, (19 75) Semigroups of condi­tioned shifts and approximation of Markov processes , Ann , Prob . , !!_ 618-642 ,

3 . Kushner , H , J , ( 1972) Introduction to Stochastic Control , Holt , Rinehart and Winston , New York.

4 . Kushner , H . J . and D . S . Clark (19 78) Stochastic Approximation for Con­s trained and Uncons trained Sys'tems , Springer , Berlin .

5 . Kushner , H . J . and H . Huang (1981) Asymptotic properties of stochastic approximations with constant coeffi­cient s , SIAM J . Contr . Optimiz . , 19 , 87-105 .

-

6 . Kushner , H . J. and H . Huang ( 1981) On the weak convergence of a sequence of general s tochastic difference equa­tions to a diffusion , SIAM J . on Appl. Math . , 40 , 528-541 .

7 . Stroock, D . W . and-S. R. S . Varadhan (19 79 ) Multidimensional Diffusion Processes , Springer , Berlin .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

AN EFFICIENT NONLINEAR FILTER WITH APPLICATION EXPERIENCES ON

MULTIVARIABLE ADAPTIVE CONTROL AND FAULT DETECTION

J. Selkainaho, A. Halme and F. Behbehani

Laboratory of Control and Systerns Engineering, University of Oulu, Oulu, Finland

Abstract . A new nonlinear f il ter based on Bayesian MAP estimat ion criterion is introduced . The f ilter is more stable and robust in use than the extended Kalman or s imilar f il ters and needs less calculation in the on-l ine algorithm . No on-l ine covariance matrix calculation i s necessary for the gain when the f ilter is run long after initialization. Appl ication experiences are cons idered when using for s imultaneous state and parameter estimation in adaptive control and sensor/actuator faul t detection.

Keywords . Nonlinear filtering , s tate and parameter estimat ion, adaptive contro l , fault detection .

INTRODUCTION Filters are not commonly used for parameter estimation in adaptive methods probably be­cause on one hand the known theory is not strictly appl icable to such problems and on the other hand experiences on nonl inear f il­ters as algorithms have not been general ly encouraging . However , f iltering is the most natural approach to the recursive estimation espec ially when the state and parameters are estimated s imultaneously. The main intention of thi s paper is to p�esent a new nonl inear f ilter which is theoretically wel l defined and owns several nice properties as an al­gorithm, l ike good stability, robustness and relatively s imple tunability. The authors have applied the f il ter with success in adapt ive control and recently also when de­tect ing faults in sensor/actuator area of an instrumentat ion system. Experiences on these applications are briefly summarized in the paper . The filter has been orig inally presented by Halme (1980) in the continuous time form. In what fol lows it i s g iven f irst t ime in the correct discrete t ime form. The f il ter is based on a MAP estimat ion criterion which makes it poss ible to convert the ori­ginal estimat ion problem to a deterministic opt imal control problem which in turn i s solved by applying the Greens function method to the equivalent TPBVP . The algo­rithm appl ied in practice is obtained by considering the case when the filtering t i­me is inf inite or suff ic ient large , which makes it poss ible to calculate off-l ine cer­tain covariance matrices needed in the gain calculat ion. In the l inear case the f il ter result s to the same algorithm as the Kalman f ilter . In the nonl inear case the algorithm resembles that of extended Kalman f il ter but d iffers from it in certain e ssent ial detail s .

1 99

THE FILTER ALGORITHM

Problem formulation The problem is to estimate the state vector x when its dynamics is modelled by the equat ion

x (k+l ) = f (x (k) , u (k) ) +w(k) , (1 ) where f is a nonl inear function , which may include al so delayed terms of the state vec­tor x and control vector u, and w is the proces s disturbance modelled by an indepen­dent zero mean Gaus sian stochastic process having the covariance

T E (w(k)w(k) ) = Q . (2 )

The measurements from x are obtained through the observation equat ion

y (k) = h (x (k) ) +v (k) , (3) where h is a nonl inear function and v an independent zero mean Gaussian stochast ic process having the covariance

E (v (k)v (k) T) = R . (4) The dimensions of x, u and y are denoted by n, m and r re spectively. The only assumpt ion which is made on f and h at this po int i s the following : We suppose that l inear parts can be separated so that

f (x ,u) h (x)

Ax + f1 (x ,u) Cx + h1 (x)

(S) (6)

where (A, C) i s a constant completely observ­able pair . Such linear parts can be separa­ted in many different ways . How this is done in practice will be discus sed later on . Cer­tain smoothnes s properties are required on

200 J . Selkainaho , A. Halme and F . Behbehani

f1 and h1 , which are also d iscussed later on.

Est imation criterion

The system is considered on the time inter­val kE [ O . • . n+l ] . The Bayesian maximum a posteriori est imate is obtained by maximi­zing the fol lowing condit ional probabil ity at the end of the interval

P (x (n+l ) l y (n) )->max . ( 7 )

This estimate can be called a f i lter esti­mate although it more preci sely is an one step predictor .

Applying Bayes rule to the above equation results in

P (x (n+l ) l y (n) ) = P (y (n) l x{n) ) P (x (n+l ) ) /P (y (n) ) ,

where we have used the relation

P (y (n) l x (n+l ) ) = P (y (n) l x (n) ) .

(8 )

(9 )

The init ial state x (o) is supposed to be a Gauss ian stochastic variable with mean x and covariance P • Because of the Gauss i�n property of the �ariables under considera­tion the conditional probabil ity can be written as follows (see e . g . Sage and Melsa, 1971 ) .

P (x (n+l ) l y (n) ) = T -1 a exp {- ( l /2) (x (O )-x ) P (x (O) -x ) -0 0 0 n

( 1/ 2) L w (k) TQ-1w (k) -k=o ( 10) n

( 1 /2) L v (k)TR-lv (k) } , k=o

where a does not depend on x and

v (k) = y (k )-h (x (k) ) . ( 11 )

Maximizing the condit ional probabil ity ( 7 ) is equivalent to minimiz ing the quadrat ic criterion

J = ( l /2) (x (o) -x ) TP-1 (x (o ) -x ) + 0 0 0 n T -1 ( 1 /2) L [ w (k) Q w (k)+ (y (k) -

k=o h (x (k) ) ) TR-1 (y (k)-h (x (k) ) ) ]

Equivalent TPBVP

(12)

The original estimation problem is next con­verted to a deterministic opt imal control problem, where the process disturbance vec­tor w (k) is the minimiz ing variable . Standard use of discrete minimum principle leads to the fol lowing TPBVP (two point boundary va­lue problem)

x (k+l) =Ax (k )+f1 (x (k) ,u (k) ) -Qp (k+l ) , (13) T -1 p (k )= [-c-ah1 (x (k) ) / ax (k) ] R [ y (k)-Cx (k) -

T h1 (x (k) ) ] + A p (k+l ) + [ af1 (x (k) , u (k) ) / dx (k) ] Tp (k+l ) , (14)

with boundary condit ions -1 p (O) = -P0 (x (O )-x0) ,

p (n+l ) = 0 .

(15)

(16)

It is . as sumed that f1 and h1 have der ivat i­ves with respect to x .

Before solving the TPBVP the l inear part of the state equation (14) have to be rewrit­ten in forward time form and this resul t is then substituted into the state equation (13) .

Let us make notat ion simpler by us ing follo­wing extended state

'x (k) 1 Y (k) = p (k)

The TPBVP can now be simply writ ten as

( 17 )

Y (k+l ) =¢ ( l )Y (k) +F (Y (k+l ) ,Y (k) ,k) (18)

with linear boundary condit ions

M Y (o)+N Y (n+l ) = c , ( 19)

where ¢ (1 ) is the t imeinvariant state tran­sition matrix of the l inear par t ,

r A+Q (AT)-lCTR-lC -Q (AT) -1

¢ ( 1 ) ( 20)

The vector components of the nonlinear part are

T -1 T -1 Fl (k) =f 1 (x (k) , u (k) ) -Q (A ) C R [ y (k) -

(21 ) h1 (x (k) ) ] -Q (AT) -1 [ ah1 (x (k) ) /

ax (k) ] TR-l (y (k)-h (x (k) ) + Q (AT) -1 [ af1 (x (k) , u (k) ) /ax (k) ] Tp (k+l)

F2 (k)= (AT) -lCTR-1 [ y (k) -h1 (x (k) ) ] +

(AT) -1 [ ah1 (x (k) ) /ax (k) ] TR-l

[ y (k) -h (x (k) ) ] - (AT) -1 [ af1 (x (k) ,u (k) ) / dx (k) ] Tp (k+l ) ( 22)

The coefficients dit ion (19 ) are I r p l

M = O i 0 0 ! 0

N : l 0

of the linear boundary con-

(23 )

(24)

An Efficient Nonlinear Fil ter 20 1

c = x 0

� o

Solving the filter equation

(25 )

The TPBVP (18) - (1 9 ) can be rewritten further in an equivalent form (Falb and De Jong , 1 969) using the Greens functions H and G .

n Y (n+l ) =H(n+l ) c+ L G (n+l , i) F ( i) , (26 )

i=o where H(n+l ) =¢ (n+l) [M+N¢ (n+l ) ] -l and (27 )

G (n+l , i ) = ( I-H(n+l ) N) ¢ (n-i ) (28)

The filter equation can be obtained now by partitioning the equation (26 ) .

(29) n

. L [Gl l (n+l , i) F1 ( i )+G12 (n+l , i )F2 ( i) ] l=O

where H11 (n+l ) =¢11 (n+l ) -H12 (n+l ) ¢21 (n+l) , (30)

c11 (n+l , i) =¢11 (n-i)-H12 (n+l ) ¢21 (n-i)

and (3 1)

c12 (n+l , i) =¢12 (n-i)-H12 (n+l) ¢22 (n-i) (32)

The term H12 has the following properties

-H12 (0) = P 0 and (33)

-H12 (k+l ) =A[-H12 (k) -l+CTR-lC ) -lAT+Q (34)

It is easily seen tha7 -H12 is equal to the covariance of the est1mat1on error of a l i­near filter which is equivalent to the Kal­man filter .

The filtering equation (29) is next trans­ferred into a recursive form. The most re­cent term is taken out from the sum in the Eq. (29 )

n-1 x (n+l ) =H11 (n+l )x0+ . L [Gl l (n+l , i ) Fl ( i )+

i=o G12 (n+l , i) F2 (i ) ] +F1 (n)-H12 (n+l ) F2 (n) .

(35) The terms H1 1 , G1 1 and G12 have a common left multipl ier Eerm whicfi can be calcula­ted from equations (20) , (30) , (31 ) , (32) and (34) . After taking this common term out the filter equation can be represented in the following recursive form

A -1 T -1 -1 x (n+l ) =A[-H!t

(n) +C R C) [-H12 (n) ] x (n )+F1 (n) -H12 (n+l ) F2 (n)

(36)

Having in mind that the upper boundary value of the co-state is zero vector (16) , the nonl inear part s (21 ) and (22) can be further

subst ituted into the filter equation

x (n+l ) = f l (x (n) , u (n) ) +

A [-Hl 2 (n) -1+cTR-1

cJ -1[ -Hl2 (n)-1

x (n) +

T -1 A ] C R (y (n) -h1 (x (n) ) ) + -1 T -1 -1 A A[-H12 (n) +C R C ] [ 3h1 (x (n) ) /

3x (n) ) TR-l (y (n)-h (x (n) ) ) (37 )

Using the matrix inversion lemma and the abbreviation

(38)

The filter equation can be f inally written in the form

x (n+l ) =f (x (n) , u (n) ) +K(x (n) ) [y (n )-h (x (n) ) ]

x (o ) =x 0 where the gain matrix is

K(x (n) ) =AP (n) CT [ CP (n ) CT+R) -1+

-1 T -1 -1 A[ P (n) +C R C ] [ 3h1 (i(n) ) /

3x (n) ) R-l

(39) (40)

(41 )

The covariance matrix P (n) is obtained from the Riccati equation

(42)

with the init ial value

P (O) = P0 (43)

As mentioned earl ier , the estimate obtained is correct ly speaking the one step predic­tor based on the measurement y (n) . The standard notations are

i (n+l ) =x (n+l l n) and P (n+l) =P (n+l l n)

Stationary case

(44) (45)

Let us consider the dynamics of the Riccati equation (42) . Because the pair (A, C ) is time invariant , the covariance P reaches a steady state value after a short starting transient . In realtime appl ications this means that P is constant nearly all the time . Also most of the computational burden comes from updating the covariance P . So , it is no use to calculate P in real time .

If we let time index n to grow without li­mit s , we obtain the following gain matrix

K(x (n) ) =K1+K2 [ ah1 (i (n) ) / ai (n) ] K3 (46)

where K1=A P_CT [ CP_CT+R) -1,

202 J . Selkainaho , A. Halme and F . Behbehani

A P+ and R-1 .

The steady state covariance P before the measurement is obtained from the algebraic Riccati equation

p (47 ) where P+ i s the steady state covariance af­ter the measurement

p + (48) The covariance P and matrices K1 , K2 and K3 can be calcul�ted off-line .

USE IN SIMULTANEOUS STATE AND PARAMETER ESTIMATION

Let us cons ider next the problem where we l ike to estimate bes ides the state x also a parameter vector a in the dynamics

x (k+l ) = f (x (k) , u (k) ; a )+w (k) , x (49) using the same measurement equat ion (3 ) as before . To obtain a proper formulation we suppose further that the parame ter vector is a real ization from a stochastic process which can be described by the equation

a(k+l ) (SO) where w is an independent zero mean Gaus­sian stSchastic process wi th covariance Q . a Interpretation as a stochastic process means that we can not expect parameter con­vergence in the same sense as e . g . in least squares estimation . What we get i s an esti­mate a (k) which itself i s a stochas tic pro­cess and which with increas ing k can have stationary behaviour and perhaps a constant mean , provided the model explains the mea­surement data properl y . The "parameter noi­se" vector w in Eq . (SO) describes the un­certainty whfch we l ike to put on the diffe­rent components of the parameter vector . In most cases there is no physical bases to choose the covariance Q , but it is just a tuning parameter for th� user . There are two different ways to obtain the f i l ter equations for the state and parameter vectors . One way is to write the f ilter for the extended system the state of which i s formed by the state and parameter vectors together . A serious disadvantage of thi s approach is that the dimensions increase because even in the case of a l inear model the extended fi l ter cannot be separated in­to two f ilters with lower d imens ions . Pre­cisely speaking , it can be shown that this can be done in the continuous t ime case but not in the discrete time case , see (Halme , 1 981 ) . Another way i s to write the f i lters separately for the state and the parameter vectors us ing the same measurement equation

(3) for both of them . When doing so there i s one key point which, according t o the authors ' experiences , improve considerably the performance of the parameter estimation . Because the measurement y (k) i s not affected by the parameter a (k) the best uti l ization of the measurement data i s obtained by a proper phas ing so that the estimates x (k+l i k) and a (k i k) are calculated at the t ime k when the new measurement y (k) i s available . The dynamic equation (SO) i s

"then indexed as follows

x (k+l ) =f (x (k) , u (k) ; a (k) ) +wx (k) , (Sl )

The filter for the state is x (k+l i k) =X (k+l l k-l )+K (y (k) -h Cx <k J k-1 ) ) , x

(S2) x (k+l l k-l ) =f (x (k i k-1 ) , u (k) ; a <k i k ) ) , (S3)

where Kx is the gain calculated from the Eq . (46 ) . The f i l ter for the parameter i s

a <k i k )=a <k l k-l ) +Ka (y(k)-h (x (k i k-1 ) ) , (s4) a <k i k-l ) =a (k-l l k-1 ) .

Determination of the gain Ka i s not so straightforward as Kx in the case of state estimation . This i s due to the fact that the measurement equation has now to be written in the form

y (k) =h (f (x (k-l l k-2) , u (k-l ) ; a (k-l l k-l ) ) . (SS)

We have now to interpret the measurement function hof as a function of the parameter vector a . To demonstrate how the gains are calculated we consider further the special case where f and h are l inear .

The case where f and h are l inear Let us suppose that the system function has the form

f (x ,u ) = Ax + Bu , (S6)

and the measurement function the form

h (x) = Cx . (57 )

The parmneter vector now contains the ele­ments of A and B. Note that A and B are time variant when they are under estimation . Def ine the matrix

F [A B ] Then

f (x ,u)

(58)

(59)

A convenient way to define the parameter vector a is to take the matrix F row by row. For further treatment the t ime variant A(k) i s divided into two parts

An Efficient Nonlinear Fi l ter 203

A (k) = A+M(k) , (60)

where A i s a chosen constant nominal value . The dynnamical equation i s then

x (k+l ) =Ax (k) +f1 (x (k) , u (k) ) +w (k ) , ( 61 )

where

f1 (x (k) , u (k) ) =ill\x (k) +B (k) u (k)

Correspondingly the measurement equat ion (57 ) can be written in the form

(62)

y (k )=C (; ,�) a (k-l ) +h1 (x (k) ; a (k-l ) ) +v (k) ,

where

and

t I I j T T 0 I I 0 � u- 1 - T Tt , - - -1 _ o � x u 1 , o ' C (x , u) = C - - - -

' : ' - - -•

I - - 1 • , - r - i'f11 O O ; �x u J

(63)

(64)

(65)

The values ; , � are choosen constant nominal values . The gains of the state and parameter filter respect ively can now be written in the following forms

and - - - T - - - - - - T -1 K =P C (x ,u ) [ C (x , u) P C (x , u) +R] + a a a

- I - - T -1

(66)

(P -Q) C (x (k-l i k-2) -x u (k-1 ) -u) R a ' (67 ) The covariances P and P are cal culated from the Riccati �quatioRs

(68)

(69)

respectively . Note that K is a constant but K varies during esti@ation . According to th� authors ' experience the last ment ion­ed fac t has an important role in good para­meter estimation .

Tuning of the filter

The filter gain depends on the disturbance covariance matrices Q and R. Because in most case s these are not known in pract ice they are used as tuning parameter s . Usually diagonal forms are chosen . Tuning is made in principle in the same way as in the case of Kalman filter . Roughly speaking the ra­tio Q/R affect s on the gain so that a low ratio gives a low gain and vice versa . Dif­ferent component s can be weighted in a dif­ferent way . When the state and parameters are estimated s imultaneously Q have two component s � and Qa . In principle they can

be choosen independently . The opt imal gains for the parameter and state f i l ters depend on the applicat ion . Nothing very specif ic can be said . According to the authors ' ex­periences tuning is , however , relatively easy and straightforward and there is usual­ly no need to change it e . g . in adaptive control applications . Other parameters to be choosen are A, ; and . � . Best results are obt ained if they are choosen to be some fixed nominal values in the operating area . According to the experiences the choice is not critical but the f i l ter is quite robust and many different values give about the sa­me result .

APPLICATIONS IN MULTIVARIABLE ADAPTIVE CONTROL

Halme et al ( 1 982 ) have described a self­tunable multivariable state form PI-type controller which uses the nonlinear fi lter pre sented . The controller structure is shortly described in the fol lowing . The system model is supposed to be the standard l inear one

x (k+l ) =Ax (k) +Bu (k) +w (k) ( 70)

The parameter matrices A and B and the sta­te vec tor x are estimated by using the filter as described above .

By denoting

e (k) = x (k) - Xs 6e (k)= e (k ) - e (k-1 ) � (k)= x (k ) - x (k-1 ) 6u (k) = u (k) - u (k-1) 6w(k)= w (k) - w (k-1)

( 7 1 )

and supposing that the setpoint x changes rarely (relatively as to the sampfe t ime) the original model (70) can be rewritten in the form

6e (k+l ) =A6e (k) +B6u (k) +6w (k) . (7 2)

The control ler is designed and tuned us ing the quadratic criterion

{ T T I=E e (k+l ) Q1e (k+l ) +6e (k+l ) Q26e (k+l ) +

6u (k) TR 6u (k) } , ( 73 ) c which is minimized . The result is a simple PI-type multivariable controller

u (k )=u (k- l ) +P0e (k) +P1e (k-l ) , (74)

The model could also include a delay in the control variable which can be compensated in the algorithm (Halme and co-workers , 1 982) . The setpoint x has to be cal culated from the s

204 J . Selkainaho , A . Halme and F . Behbehani

measurement y . Otherwise the state estimate is used in the algorithm.

We have implemented and tested the adaptive controller in two different cases . One of the cases is a pre ssurized f lowbox shown in Fig . 1 (Halme and co-workers , 1 982) . The controlled variables are the water level x1 and the outlet pres sure x2 . This kind of device is used e . g . in papermachines with hydraulic headboxes to stabilize the fiber mass flow from the headbox . Control is made by the valve which suppres ses the air outlet u1 and the speed of the water feed pump u2 . The process has quite strong interactions and different time constants in the gross transfer functions . The proce ss is nonl inear but can wel l be treated with a l inearized model . The control ler was implemented using either a two or three dimensional state . Two of the states are in both case s the level and the pressure which are measured .

Another case is a heavy hydraul ic manipula­tor shown in Fig . 2 . It is used e . g . in the log-loaders of wood harvesting machines . The control led variables are primarily the Cartesian coordinates of the tip of the arm. Control is made by hydraul ic syl inders which move two folding and one rotating j oints . The j oint angles are measured with optical devices . A chal lenging servo control problem is to track a given path with a given variab­le velocity . In different posit ions with va­riable velocity and load the dynamics of the arm varies considerably. A three state model was used the states being the three joint angles . A given path is first transformed to the j oint coordinates and the control then performed in this coordinate system. Primary results have been reported in more detail by Vaha and Halme (1983) in a conference con­cerning advanced software for robotic s .

In both appl ications the adapt ive control ler has been implemented in a microcomputer-mini­computer environment (different systems) by us ing a two level real ization . The control­ler itself and the state filter have been implemented in the microcomputer which conununicate s with the minicomputer us ing a standard serial l ink. The parameter fil ter and control ler parameter calculat ion have been implemented in the minicomputer . In practice the controller parameter updat ing can take place less frequently than every sample interval . In fact the differentia­tion of the state and parameter estimation makes thi s easy . It is important especially in the applications l ike the manipulator where the control ler sample interval has to be very short , about 40 ms , to obtain good qual ity control . Because of shortage of t ime the parameter filter and controller parame­ter calculation cannot be run so frequently but the minimum updating interval is about 2 s . Thus the parameters are updat ing only every SOth control interval . This , of course, l imits the speed of parameter adaptation and the updating time is currently being redu­ced by replacing the computing unit with a

faster one .

Experiences with both appl icat ions have been extremely positive . The fil ter has turned out rel iable to use , relatively easy and robust to tune and the number of calcu­lations needed in the algorithms are mode­rate . The minicomputers used in the test ing systems can wel l be replaced with a micro­computer . They are now used only for prog­ramming convenience .

APPLICATION IN SENSOR/ACTUATOR FAULT DETECTION

A new area in which we have recently applied the filter is fault detection . We have tried to detect different type of faults which may occur in sensor /actuator area of an instrumentation system . The faults are not such which can be called major , i . e . a wire broken or a sensor out of range , but some­thing which can be caused by a partial da­mage such as bad dirtying of a sensor or stucking of a valve etc . These faults usual­ly cannot be recognized by the system self­checking diagnostic , because they cause either a sensible but wrong measurement sig­nal or disturbe an actuator to follow pre­cisely a sensible contro l signal . Such faults can be model led by adding step , ramp or other type di sturbance signal s to the actual signal s .

The idea we have used when detecting such faults is to follow the system with the adaptive l inear filter . With a proper tu­ning the estimation error signals are sen­sitive to faul t s , which can then be detected by using various detectors . The adaptive filter ( i . e . state fi lter with parameter estimation) has turned out especially use­ful for that purpose because the estimation error signal keeps zero mean and white noise characteristic wel l in normal condit ions in spite of proce ss parameter variat ions . The abnormal condit ions are then eas ier to detect .

Figures 3 and 4 demonstrate detection of a fault simulated by adding a slow external ramp signal into the valve posit ion signal of the pres surized flowbox (Fig . 1 ) . The ramp is about 2 % of the scale during each 4 seconds . The process is run with usual P I controllers which are connected so that the pres sure is control led by the pump and the level by the valve . Fig . 3 shows the two estimation error signal s . The detection is demonstrated by using a RC-filter based detector . The co­variance signal of the estimation error is fed to RC-filter and the level of the output tested against given treshold value . It can be shown that the detector is analog to the use of F-test when testing a change in variance of the signal . Note that an abnor­mal behaviour in the estimation error signal has only quite a short duration after which adaptation of the filter parameters compen­sates the effect of the faul t .

An Efficient Nonlinear Filter 205

CONCLUSIONS The most important new result in the paper is the filter algorithm. It has been given now first time in the correct discrete time form although presented in an earlier con­text in the continuous time form. The f il­ter algorithm is simpler and more robust than the usual extended Kalman type algo­rithms because the gain can be calculated in the stationary phase without on-line co­variance matrix calculation. The authors have used the filter in various applica­tions l ike parameter adaptive control and fault detection with extremely good expe­riences.

REFERENCES Falb, P . L . and De Jong , J .L . (1969) . Some

Succes sive Approximation Methods in Control and Oscillation Theory . Aca­demic Press , New York .

Halme , A. (1980) . Some New Algorithms for Nonlinear State Estimation and Filte­ring . In O .L .R. Jacobs & al . (Eds . ) , Analysis and Optimization of Stochas­tic Systems, Academic Press , London. PP • 373-385 .

Halme , A. (1981) . Recursive Identification of Non-Linear Systems by the Direct Method and Orthogonal Expansions . In J . E . Marshall & al . (Eds . ) , Third IMA Conference on Control Theory, Academic Press , London. pp. 587-604 .

Halme , A. , Selkainaho , J . and Soininen, J . (1982) . Implementation o f a Multiva­riable Self-Tunable PI�type Controller in a Distributed Microcomputer System. Proceedings of the Symposium on Appli­cation of Multivariable Systems Theory, The Institute of Measurement and Control , London .

Sage , A . P . and Melsa, J . L . (1971) . System Identification. Academic Press , New York.

Vaha, P . and Halme , A. (1983) . Adaptive Di� gital Control for a Heavy Manipulator. Proceedings of A. I .M . Conference on Advanced Software in Robotic s , North Holland.

ai r

1------1 --- x,

water

Fig . 1 . The flowbox proces s

MC68000

comp, link

D Fig . 2 .

Fig . 3 .

" .• -:. ,.-.+ ... r_, • '!:..·

MYORMlll ( llA(H)M(

The manipulator system

I . , . ______ _.!._..-. _____ . ____ _ J

·I -·-: • . -:,; .... .._ . .. _. Filter estimation error signals . Beginning of a 2 % fault ramp is marked with the arrow. X-scale : number of samples , 4 sec interval ; Y-scale : full measurement scale=l .

' . ' ,_

i 0 0 J_ j 1

3 0 1 3 0 .2 3 0 Fig . 4 . RC-detector output and a treshold

value corresponding 99 . 9 % confiden­ce . X-scale : the same as in Fig . 3 ; Y-scale : d�mens ionles s .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

A MV ADAPTIVE CONTROLLER FOR PLANTS WITH TIME-VARYING 1/0 TRANSPORT DELAY0

E. Mosca and G. Zappa

1Ji/1artirnento di Si1tnni e lnfimnatica, Uni1wnitrl di fiirn1u, Via S. Marta 3, 50139 Firenze, Italy

Abstract . The prob l em of adapti vely regu l at i ng a p l ant wi th an unknown or ti me-vary ing I /0 transport de l ay i s cons i dered: The de l ay i s a l l owed to ta­ke on any Dos s i b l e val ue rangi ng from Q_ uo_to d steps , or to exh i b i t t ime­vari ati ons i n such a range . A bank of T � d-Q_+l rara l l e l LS+MV se l f- tuners ( STURE ) , each i nd i v i dual l y matched to a poss i b l e de l ay , i s cons i dered . The T correspondi ng STURE outputs are thus combi ned so as to produce the actual p l ant i nput . Th i s i s accomp l i shed i n accordance w i th the cri ter i on of mi n i ­mi z i ng a MV cost functi onal whose time hori zon T i nc l udes a l l the pos s i b l e p l ant I / O transport de l ays . I f su i tab ly i mp l emented , the res u l t i ng regu l ator , referred to as MV-MUSMAR , has a compl exi ty comparabl e to that of a s i ng l e STURE . Moreover , for T l arge enough , the MY-feedback i s a pos s i b l e convergen ­ce poi nt of the MV-MUSMAR , i rresaecti ve ly of the C ( q - 1 ) - polynomi a l assoc i a­ted to the p l ant d i sturbance .

Keywords . Adapti ve contro l ; d i � i ta l control ; narameter est imati on ; t ime l ag systems ; convergence of numeri cal methods .

l . I NTRODUCT I ON

Adapti ve contro l l ers based on d i rect schemes requi re0 the a pri ori knowl edge of the p l ant de l ay ( Astrom and coworkers , 1 977 ; Landau , 1 979 ) . I n fac t , the t ime de l ay i s by a l l means the most sens i ti ve rarameter i n the de­s i gn and tun i ng of a contro l l oon . A l though i n many practi ca l i nstances i t i s poss i b l e to obta i n a n accurate est i mate of the t ime­del ay by step response or i denti fi cati on te­chni ques , there are proces ses , e . g . i n the chemi cal fi e l d , w i th i l l -defi ned or t ime- va­ryi ng t ime-del ay . I n the l atter s i tuati on , di rect adapti ve contro l l ers can cause i ns ta­bi l i ty or y i e l d an unsati sfactory perfor�an­ce for the regu l ated p l an t . In pri nc i p l e , i nd i rect adanti ve contro l l ers can be used to regu l ate pl ants w i th an unknown t ime del ay . However , the adopti on o f a n i nd i rect adapti ­ve control s cheme may subs tant i a l l y i ncrease the computat i onal comp l ex i ty of the contro l ­l er . I n th i s context , work d i rected t o deve­l op adapt i ve control a l gori thms for p l ants w i th unknown t ime de l ay has been documented i n recent years ( Lam , 1 980 ; Kurz and Goedecke , 0 Thi s work was parti a l ly supported by the I ta­

l i an Mi n i s try of Pub l i c Educati on and CNR under contracts 82 . 0 1 764 . 07 and 82 . 00832 . 07 .

207

1 98 1 ; Dumont , 1 982 ; Wong and Bayoumi , 1 982 ) .

I n th i s paper , a new al gori thm for adapti ve­ly control l i na n l ants wi th unknown i /o tran­sport del ay i s deve l oned . The proposed al go­ri thm appears to be a natural exten s i on of the LS+MV se l f-tuner and does not reou i re any dec i s i on on the actua l p l ant t ime de l ay , bei ng i ntri nseca l ly tunab l e t o any oos s i b l e t ime de l ay wi th i n a preass i gned range . Thi s i s achi eved by tun i ng the contro l l er accor­d i ng to an MV cost funct i onal i nvol v i ng a mu l ti s tep hori zon . The resu l t i ng contro l l er, though of an i nd i rect type , recurs i ve ly com­putes LS est imates of parameters g i v i ng the desi red feedback-ga i n row vector i n an a lmost d i rect way . Thi s makes the comoutat i onal compl ex i ty of the overa l l a l gori thm compara­b l e to that of a standard LS+MV se l f-tuner . Admi tted l y , the resu l ti ng a l gori thm does not appear to be as amenab l e to g l oba l convergen­ce or s tab i l i ty ana lys i s as the bas i c sel f­tune r . However , i t i s shown that the MV feedback i s a poss i b l e convergence poi nt of the a l gori thm , i rrespecti vely of the actu­al p l ant i /o transport-de l ay , and , i f the control hori zon is l arge enough , of the spe­ctra l character of the p l ant d i s turbance . The paper i s organi zed as fo l l ows .

208 E . Mosca and G. Zappa

I n Sec t . 2 the adapti ve control a l gori thm i s devel oped and i ts connecti on wi th the bas i c LS+MV sel f-tuner i s underl i ned . I n Sect . 3 asymptoti c properties of the regul ator a re analysed by Ljung ' s O . D . E . method ( Ljung , 1 977 ) . I n Sec t . 4 some concl us i ve remarks are made .

2 . PROBLEM FORMULATION AND THE MV-MUSMAR SELF-TUNER

Let us cons i der a S I SO pla.rit wi th i nputs u ( k ) and outputs y ( k ) whose dynam i c behav i or can be descri bed by an ARMAX representat i on

A (q- 1 )y ( k ) = q-dB (q - 1 ) u ( k - l ) + C (q - 1 ) e ( k ) ( 1 ) where : e ( k ) i s the i nnovati ons process of y ( k ) ; A (q - 1 ) , B (q - 1 ) , C ( q- l ) are unknown poly­nomi a l s i n the un i t-del ay operator q- 1 ,

n . -1 'i' - 1 A ( q ) = 1 + l a . q i = 1 1

-1 - 1 - 1 wi th A (q ) , B ( q ) and C ( q ) rel ati ve ly

( 2 )

prime , and C ( q- 1 ) stabl e , i . e . , the zeros of C ( q ) are outs i de the uni t ci rc l e . Further , the actual pl ant i /o tJT..a.Yf).,po!tt-de.1.a.y d , though constant , i s a l l owed to equa l any i nteger va­J ue compri sed between d and d , d , dEZ+ : = {0 , 1 , 2 , . . . } , � .s._ d ,

- -( 3 )

Wi th reference to the unknown pl ant ( 1 ) - ( 3 ) , hereafter we are concerded wi th the prob l em of se lecti ng the i nputs u ( · ) so as to make the correspondi ng outputs y ( · ) as cl ose as poss i b l e to a prespeci fi ed output �e6 eJtenee 1.i-i.grta.1 y*( · ) . As usua l , the adm-i.1.i1.i-i.ble eo n ­tJtoR.. 1.itJta-tegy i s to sel ect the i nputs u ( · ) accord i ng to a non-anti c i pati ve feedback l aw

t t-1 u ( t ) E cr [y , u J ( 4 )

where yt : = {y ( t ) , y ( t-1 ) , . . . } and cr [ytJ de ­notes the set of random vari ables meas urab l e w . r . t . the a-a l gebra generated by yt .

I f the actual p l ant i /o transport-de l ay were known , the prob l em cou l d be sol ved by a lmost any standard techn i que of adapti ve control .

I n part i c u l ar , the i nput u ( t ) asymptoti ca l ly mi n imi z i ng the Mi ni mum-Vari ance (MV ) cost functi onal

E y2 ( t+d+l ) j y , u · ( 5 ) -l t t- 1] where y ( t ) : = y ( t ) - y*( t ) , can be obtai ned by an LS+MV sel f-tuner ( Astrom and coworkers , 1 977 ) . Thi s bas i ca l ly amounts to : 1 ) compu­t i ng the Least-Squares ( LS ) estimates e ( t ) and � ( t ) based on yt , ut-d- 1 , st-d- 1 of para­meters e and � i n the fol l ow i ng d-steps ahead pred i cti ve model

y { k+d+l ) = eu ( k ) + �s ( k ) + w ( k ) ( 6 )

where w ( k ) l [u ( k ) s ' ( k ) J ' , ·

- [ k k- 1 Jltt+d+l ] ' sd ( k ) . - Yk-n+l ' uk-n-d+l ' Y t+d-n+l ( 7 )

k Yk-n : = [y ( k ) . . . y ( k-n ) J ' ; and 2 ) setti ng - 1 u ( t ) = - e ( t ) � ( t ) sd ( t ) ( 8 )

I n order to deal wi th an unknown i /o tran­sport-del ay d , i n the approach of thi s paper ( 5 ) i s rep l aced by the m�tep eo1.it &unet-i.o ­nal

T E [J i s ( t )J = E [

iL y2 ( t+i +l ) , s { t )J ( 9 )

wi th d .s.. T and

[ t t-1 Jltt+d+l ]' s ( t ) : = Yt-n+l ' ut-n-d+l ' Y t+�-n+l ( l O )

A sol uti on t o the probl em o f fi nd i ng i nputs u ( · ) mi n imi z i n g genera l mu l ti step quadrati c cost functi ona l s , i n parti cu l ar ( 9 ) , can be obtai ned by the MUSMAR sel f-tuner , deve l oped and ana lysed i n ( Menga and Mosca , 1 980 ; Mo­sca and Zappa , 1 980 ; Mosca , 1 982 ) . Thi s ada­pti ve control techn i que , as app l i ed to the mu l ti step MV cost funct i onal ( 9 ) , appears to be a natural general i zati on of the LS+MV sel f­tuner . I n fact the correspondi ng al gori thm , that wi l l be referred hereafter as the MV­MUSMAR sel f-tuner , cons i sts of: l ) comput i ng the LS �stimate� e ; ( t ) and �i ( t ) based on yt , ut-1 - l , s t-1 - l of parameters S i and �i i n the fo l l owi ng m�tep p�ed-<.et-<.v e model

y ( k+i +l ) = e . u ( k ) + � . s ( k ) + w . ( k ) 1 1 1 ( 1 1 )

i = �. �+l , . . . , Ci , . . . , T , where w; ( k )l[u ( k ) s ' ( k ) J ' ; and 2 ) sett i ng

u ( t ) = F ( t ) s ( t ) + n ( t )

A MV Adapt ive Control ler for Plants 209

T -1 T J - [ l e � ( t )] [ l e . ( t ) 1/J . ( t ) s ( t ) +n ( t ) i =d l i =d l l ( 1 2 )

I n ( 1 2 ) n ( t ) i nd i cates a l ow- i ntens i ty zero­mean whi te noi se , i ndependent of the p l ant , superimposed to the feedback control compo­nent so as to make the estimati on probl em of e . and 1/J . nons i ngu l ar . Not i ce that the feed­b�ck row�vector gai n i n ( 1 2 ) can be forma l ly rewri tten as fo 1 1 0�1s

T F ( t ) = l TI . ( t ) F . ( t ) ( 1 3 )

i =d l l

F . ( t ) . - -e � l ( t ) l/J ( t ) l l T - 1

TI . ( t ) : = . r \ 8� ( t )] 8� ( t ) , w �d , ,

( 1 4 )

( 1 5 )

I t i s to be remarked that {Tii ( t ) }�=d i s a probabi l i ty d i stri bu t i on i n that Ti i ( t ) � 0 and

T l TI . ( t ) = l

i =d l ( 1 6 )

F i g . l dep i cts a real i zati on of the MV-MUSMAR se l f-tuner i n terms of T-d+l para l l e l channel s cons i st i ng of s i ng l e-step-hori zon i -s teps ahead standard sel f- tuners whose contri buti on to the overa l l feedback i s averaged by the wei gh ts TI . . Thus , one sees that i f e i ( t ) gets c l os� to zero and hence the feedback­gai n F · ( t ) of the i -th standard se l f-tuner in the1 bank becomes excess i vely h i gh , the cor ­respondi ng wei gh t Tii , be i ng proporti onal to e? ( t ) , tends to make the i nfl uence of the i � th channel on the overa l l feedback-ga i n F ( t ) neg l i geab l e . Obv i ou s l y , th i s behavi or must be met by any adapti ve control l e r capab l e of performi ng i n an acceptable way when used wi th pl ants wi th unknown or i l l -defi ned i / o transport-del ay .

I n the MV-MUSMAR sel f- tuner ( l l ) - ( 1 2 ) , n ( t ) i s i ntroduced on purpose . Noti ce , however , that i n practi ca l i ns tances n ( t ) need not be exp l i ­ci t ly added to the i nput feedback component . I n fact , roundoff errors produced by a fi n i te word l ength d i g i ta l processor may be respon­s i b l e for the i mp l i ci t presence of the noi se component n ( t ) .

3 . THE MV FEEDBACK AS A POSS I BLE CONVERGENCE PO INT

The a im of thi s secti on i s to s how how the mu l ti step hori zon of the MV-MUSMAR sel f-tu ­ner i nfl uences the poss i b i l i ty for the a l go­ri thm to convergence to the MY-feedback . To thi s end , Ljung ' s O . D . E . method ( Lj ung , 1 977 ) i s used to ana lys i ng the al gori thm ( l l ) - ( 1 2 ) , under the assumpti on of stabi l i ty . I t can be shown ( Mosca and Zappa , 1 982 b ) that the rel evant O . D . E . associ ated to the MV-MUSMAR se l f-tuner has i n a nei ghborhood of any equ i-1 i bri um poi nt F 0 the fo l l owi ng form

_ r T _ 1 - 1 T _ F ' ( -r ) = -R 1 1 l e � J L e . R ( i +l ) ( 1 7 ) s lj =� J ; i =d , ys

where : the dot denotes time deri vati ve ; Rs : = E [ s ( t ) s ' ( t ) J ;

e i : = o�z E [y ( t+i +l ) n ( t )J F=F o R ( i + l ) ys E[y ( t+i + l ) s ( t ) J

I f we set

wi th deg F=d and deg G=n- 1 , we f i nd

- 1 - 1 - 1 - 1 y ( t+d+l ) = C ( q ) [B ( q ) F ( q ) u ( t ) +

( 1 8 )

( 1 9 )

+ G ( q - l )y ( t ) - C ( q -l )y*( t+d+l ) J +

+ v ( t+d+l )

+ v ( t+d+l ) ( 21 )

I n ( 21 ) FMV denotes the MV feedback-gai n row vector

-G ( q- l )y ( t ) -

-B ( q- l ) F ( q- l ) u ( t ) + - 1 * +C ( q )y ( t+d+l ) ,

where v ( t ) : = F ( q- l )e ( t ) , and F i nd i cates the actual feedback-gai n row-vector

u ( t ) = Fs ( t ) + n ( t )

2 1 0 E . Mosca and G . Zappa

Tak i ng i nto account ( 2 1 ) , ( 1 8 ) g i ves e . =0 , 1 i =d , . . . , d-1 . Thus , ( 1 8 ) can be rewri tten as -fo l l ows

( F ' -F ' ) ( 22 ) MV where

Eq . ( 22 ) shows that F 0 =FMV0i s a n equ i l i bri um

poi n t of ( 1 7 ) . I n order t fi nd out whether FMV i s a stabl e equ i l i bri um poi nt of ( 1 7 ) , and hence , i f B ( q - 1 ) i s a stab l e polynomi a l , a poss i b l e convergence ooi nt of the MV-MUSMAR sel f-tuner , one must check that H ( q - 1 ) i s stri ctly pos i ti ve real ( s . p . r . ) , i . e .

ReH ( ejw) > 0 , w E [Q , 211 ) .

Si nce for F=FMV ( 2 1 ) becomes

from ( 1 8 ) one fi nds e . I F = b l y i -d ' where 1 MV

y . i s impl i ci tly defi ned by 1

Thus , l l T-d .

c- (q - ) l y . q l i =0 1

= c- 1 ( q - 1 ) [c- 1 ( q )J �-d ( 24 )

- 1 T-d where [C ( q ) J denotes the sum of the l eadi ng T-d+l �erms i n the expan s i on ( 23 ) . Si nce by choos i ng T-d l arge enough H ( q - 1 ) can be made s . p . r . (Mosca and Zappa , 1 982 a ) , the conc l us i on is that , by i ncreas i ng the control hori zon beyond the maximum poss i b l e i /o tra n­sport-del ay , FMV c a n be a pos s i b l e convergen­ce poi n t of the a l gori thm i rrespecti vely of the spectral character of the p l ant d i stu r­bance . Th i s i s obv i ou s ly paid by a l arger amount of computati ons at each step of the a l gori thm .

4 . F INAL REMARKS AND CONCLUSIONS

An a l gori thm for the adapt i ve MV control of p l ants w i th unknown i /o transport del ay. has been presented . I t i s based on the i denti fi­cati on of severa l pred i cti ve model s , each matched to a d i fferent de l ay , and on the ap­propri ate combi nati on of the MV control s i ­gnal for each mode l . Hence , no deci s i on pro ­cedure on the true i /o transport de l ay i s re­q u i red . Neverthe l ess the a l gori thm sti l l a l l ows at convergence an exact MV control of the p l an t , whi l e severa l a l ternati ve approaches ( Dumont , 1 982 ; Wong and Bayoumi , 1 982 ) adopt detuned control l aws , s i nce these are l ess sens i t i ve to uncertai nti es about the true de­l ay , or to de l ay vari ati ons .

The use of a mu l ti s tep predi cti ve model , bes i ­d e a l l ow i ng the control o f p l ants whose i /o transport del ay i s unknown , yi e l ds other bene­fi ts , as by-products . I n Sect . 2 i t has been shown how the pos i ti ve rea l ness of the poly­nomi a l C, whi ch i s a necessary cond i tion for convergence of t h e standard LS+MV se l f-tu ­ner , can be re l axed . By numeri cal computer ana lys i s ( Mosca , Zappa and Manfredi , 1 983 ) , i t can a l so be s hown that the stabi l i ty condi­ti on on the polynomi a l B can be removed . I ndeed , i f the i nput energy i s wei ghted i n the cost functi onal , the control l aw generated by the present a l gori thm, can be made arbi tra­ri ly cl ose to the correspondi n g Li near Quadra­t i c so l u ti on , by extendi ng the predi cti on ho­ri zon . I n th i s way , wi th a l ow cost on the i nput energy and an extens i on of the predi ­cti rin hori zon over the actual transport de­l ay of the p l an t , i t i s pos s i b l e to get the MV contro l (wi th bounded i nput ) of non mi n i ­mum-phase p l ants .

Another feature of the present a l gori thm , whi ch i s worth to be u nderl i ned , i s i ts l ow computati onal burden , despi te the h i g h n umber of estimated parameters . I n fact , the LS estimates of the parameters of each pred i cti ve mode l have the same covari ance matri x , whi ch therefore can be computed only once at each cycl e of the a l gori thm . A deta i l ed ana lys i s o f the comput i ng comp l exi ty o f the a l gori thm i s reported i n ( Manfredi , 1 983 ) .

Pre l i mi nary s imu l ati on experiments have shown that the a l gori thm i s capab l e of tun i ng the feedback gai n coeffi ci ents to the correspon­di ng MV va l ues . Further theoreti cal studi es and computer s i mu l ati ons are presently carri ed out i n order to comp l ete ly analyze the conver­gence propert i es of the a l gori thm .

A MV Adaptive Controller for Plants 2 1 1

REFERENCES

�strom , K . J . , U. Bori s son , L. Lj ung , and B . �li ttenmark ( 1 977 ) . Theory and aop l i ca­tions of sel f-tu n i n g reg u l a tors . Auto­mati ca , _!l, 457-476 .

Dumont , A . G . ( 1 982 ) . Adapti ve dead-time compensati on . Prepri nts 5th I FAC Sympo­s i um I denti fi cati on and System Parameters Estimati on , Arl i ngton , V i rgi n i a , U . S . A . , June 7-1 1 , 397-402 .

Kurz , K . , and W . Goedecke ( 1 98 1 ) . D i g i tal parameter-adapti ve control of proces s wi th unknown dead-time . Automati ca , _l2, 245-252 .

Lam , K . P . ( 1 980 ) . Imp l i c i t and expl i c i t sel f-tun i ng control l ers . D . Ph i l . Thes i s , Oxford Uni vers i ty .

Landau , I . D . ( 1 979 ) . Adapti ve contro l . The model reference aoproach . Dekker , New York .

Lj ung , L . ( 1 977 ) . Ana lys i s of recurs i ve sto­chasti c al gori thms . I EEE Trans . Autom . Contro l , ��. 55 1 -575 .

Manfred i , C . U-D i mp l ementati on of the MUSMAR a l gori thm , I S I S , TR-2/83 , Uni vers i ta di F i renze .

Menga , G . , and E . Mosca ( 1 980 ) . MUSMAR : mu l ti ­vari ab le adapti ve regu l ators based on mu l ti s tep cost functi onal s . Advances i n Contro l , ( D . G . La i n i oti s and N . S . Tzannes Eds . ) , D . Rei del Pub l i sh i ng Company , Dor­drecht , 334-341 .

I Fd ( t ) I I I I I

-:i: I I I

FT ( t )

standard LS+MV sel f-tuners

Mosca , E . ( 1 982 ) . Mul ti var i ab l e adapti ve re ­gu l ators based on mu l ti s tep cost funct i o­nal s . NATO ASI on Non l i near Stochasti c Probl ems , ( i nvi ted paper ) , Armacao de Pe­ra , Portugal .

Mosca , E . , and G . Zappa ( 1 980 ) . MUSMAR : bas i c convergence and con s i s tency properti es . Lecture Notes i n Contro l , 28 , Spri nger­Ver l ag , 1 89- 1 9�.

Mosca , E . , and G . Zappa ( 1 982a ) . Overparame­tri zati on , pos i t i ve rea l ness , and mul ti ­s tep mi n imum-vari ance adapti ve regu l ators . NATO AS I on Non l i near Stochasti c Probl ems , Armacao de Pera , Portuga l .

Mosca , E . , and G . Zappa ( 1 982b } . Remova l of a pos i ti ve rea l ness cond i t i on i n mi n imum­vari ance adapti ve regu l ators by mu l ti step hori zons . Submi tted for pub l i cati on to I EEE Trans . on Automati c Contro l .

Mosca , E . , G . Zappa , and C . Manfredi ( 1 983 ) . Progress on mu l ti s tep hori zon sel f-tuner� the MUSMAR approach . To appear i n Ri cer­che d i Automati ca .

Wong , K . Y . , and M . M . Bayoumi ( 1 982 ) . A se l f­tun i ng a l gori thm for systems wi th unknown time-de l ay . Prepri nts 5th I FAC Sympos i um Identi fi cati on and System Parameters Estimat i on , Arl i ngton , V i rg i n i a , U . S . A . , June 7- 1 1 , 1 064-1 069 .

nd ( t )

c:: w Cl Cl ex: F ( t ) s ( t )

nT ( t )

wei ghts

F i g . l A rea l i zati on of the MV-MUSMAR sel f-tuner by a bank of paral l e l s tandard sel f-tuner

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

THE PROBLEM OF FORGETTING OLD DATA IN RECURSIVE ESTIMATION

T. Hagglund

Depart111mt of A utomatic Control, L1111d Institute of Technology, Lund, Swn/1'11

f!lll!!!;.J:i!£!;.,,,. The p ro b l eo> of forgett i ng o l d data i n recu r s i ve est i onat i o n o f t i me-va r y i ng systems i s cons i d e red . O l d p roposa l s a re rev i ewed and a new 01et hod is p resented . w h i c h cont r o l s the P mat r i x to a des i red d i agona l mat r i x . The method s o l ves p rob l ems caused by nonu n i fo >'m exc i ta t i o n . T h i s note i s a summa ry of the report Hagg l u nd ( 1 983 ) .

�!!ll'.!:!!QJ:Ql!!.::. Adapt i ve cont ro l ; data ha nd l i ng ; i dent i f i cat i on ; l east squa res app rox i mat i on ; pararroete r est i ma t i on ; t i me-va r y i ng systems .

INT RODUCT I ON

Est i mat i o n of t i o1e-va ry i ng pa rameters is a key i ssue i n a d a pt i ve cont ro l . I d ea l l y , each measu rement shou l d be weig hted in i nverse p roport i o n to i ts rel i ab i l i t y . When the pa rameters a re constant • the rel i a b i l i t y depends on t he measu rement e r ro rs a nd the mod e l i ng e r ro rs on l y . When t he pa rameters a r e t i me-va ry i ng • the rel i a b i l i t y a l so depends on the rate of change of t he pa ramet e rs . This rel i a b i l i ty i s in p ract i ce never k nown , and seve r a l ad hoc rroethods have been p roposed • most l y as mod i f i cat i ons of t he t ra d i t i ona l l east squa res method .

TRAD I T I ONAL METHODS

The equa t i ons of the recu rs i ve l east squa res a l go r i thm a re

" " 9 ( t ) Q ( t- 1 ) + P ( t ) qi ( t ) E ( t ) ( 1 )

TA E ( t ) = y ( t ) - q> ( t ) Q ( t- 1 ) ( 2 )

- 1 P l t- u

- 1 qi < t > qi < t >

T P < t > = + ( 3 )

· i t h convent i ona l nota t i ons • see e . g . Ast rooY•

a nd W i ttenma rk ( 1 973 ) . S i nce the P mat r i x i s decreas i ng , the adaptat i on a b i l i t y i s g radua l l y reduced . These equ a t i ons a re t he refore not usefu l fo r t i me-va ry i ng systems . Seve ra l proposa l s to p revent t he P io>at r i x f rom becom i ng too So'o>a l l have been g i ve n . A common a pp roach is to i nt roduce a forgett i ng factor X i n Equat i on < 3 > • i . e .

( 3 . 1 )

where 0 < X !; 1 , see Ast rom and W i ttenma r k C 1 973) . I f t he second t e rm i s sca l ed b y the va r i a nce • denoted by v C t ) , of the cor respond i ng measu rements , we get

- 1 P ( t )

The i nverse P mat r i x i s t hen a measure o f t he i nformat i on content in t he est i mato r i f the p a r a oneters a re const a nt . Such a sca l i ng w i l l be used i n t he seque l .

ASCSP-H

2 1 3

F rom Equa t i on ( 3 . 2 ) i t i s seen t hat X causes a remova l o f i n forma t i on in a l l d i rect i ons . but t ha t new i nformat i o n i s added i n t he q> ( t ) d i rect i o n o n l y . When t he re i s p e r s i stent exc i t at i on t he i ncom i ng i nf o rma t i on i s usua l l y u n i fo r m l y d i st r i buted • both i n t i me a nd i n s pace . a nd such a remova l is nat u ra l . Pa rt i cu l a r l y i n the servo p ro b l em • when t h e maJ o r e x c i ta t i on comes f roo> va r i a t i ons o f t h e command s i g na l t he re m a y b e severe p r o b l ems . The i n forma t i o n content i n d i rect i ons not rece i v i ng any new i n forma t i o n may become too sma l l . l ead i ng to nume r i c a l p 1•obl ems a nd " bu rsts" i n t he s i g na l s .

Atteo'opts t o use a t i me-va r y i ng X have been made to reduce t he p r o b l eo'O . I rv i ng C 1 979 ) suggests a X that keeps the t race of t he P ona t r i x const ant . Fortesque et a l ( 1 98 1 ) suggest a X w h i c h i s l a rge when I E I i s sma l l .

A NEW METHOD

A new mod i f i c a t i o n of Equ a t i o n ( 3 ) i s suggested i n Hagg l u nd ( 1 983) . The a i m o f the p rocedu re is t he fo l l ow i ng .

D i scount past d a t a i n such a way t ha t . i f the p a rao'ilet e rs were const a nt , a constant desi red amount of i nfo rona t i on i s ret a i ned .

I n other words t t he a i m is to t ra ns f o rm t he P mat r i x to a d i agona l mat r i x a • I • where a i s t he des i red va r i a nce o f the est i ma t es . ( D i fferent va r i a nces a re ea s i l y obta i ned by sca l i ng the e l ements i n t he e a nd qi vectors . ) I t l eads to t he fo l l ow i ng upda t i ng f o rmu l a

P C t >- 1

cx ( t ) l qi C t ) q> ( t ) T

( 3 . 3 )

where cx ( t ) i s a pos i t i ve sca l a r w h i c h det e r m i nes how much i nformat i on i s removed a t t i me t . I n Equat i o n ( 3 . 3 ) , i nforma t i o n i s removed o n l y i n d i rec t i ons whe re new i nforma t i on enters . Fo r computat i ona l reasons , the P mat r i x i s usua l l y updated d i rect l y i nstead of t he i nverse . l he equ a t i o n c o r respond i ng to ( 3 . 3 ) i s

P C t > =P < t - 1 > -P < t- l ) qi ( t ) qi ( t )

TP < t- 1 >

-----:1------:1-----T---------­<v < t > -cx < t > > +qi < t > P < t- l ) qi ( t )

( 3 . 4 )

2 1 4 T . Hagglund

!;;l:!QiS!LQf __ gitl ... The va r i a b l e cx < t > is to be chosen so t hat constant i n fo rma t i on reta i nonent i s a c h i eved . F rom Equat i on ( 3 . 3 ) i t i s obv i ous t hat cx ( t ) must b e pos i t i v e . A negat i ve cx ( t ) means t hat i nforma t i o n is added a nd not subt racted . A remova l of too much i nformat i on g i ves an unst a b l e est i mato r . The fo l l ow i ng theo rems g i ve u ppe r bounds on cx < t > .

Il:!�Q!�m-1! G i ven a sequence of mat r i ces < P < t > > w h i c h sat i sfy Equa t i on ( 3 . 4 ) . I f the i n i t i a l mat r i >< P < O > i s pos i t i ve def i n i t e • then P < t > w i l l b e pos i t i ve de f i n i te f o r a l l t p rov i ded cx < t > l i es w i t h i n the bounds

0 � cx ( t ) ( v ( t )- l + 1

---------------

Theorem 2 : Assume req�ire;ents of

qi C t > T

P C t- l > rp C t >

that cx < t > fu l f i l s Theorem 1 a nd denote

a

the the

A est i ma t i on e r ro r 9 ( t ) , i . e . 9 ( t ) = 9 ( t ) -9 ( t ) .

Then the funct i o n

decreases i n t h e no i se-f ree case i f a n d on l y i f

cx ( t ) � ------------------------------

a

Theo reoll 1 and Theorem 2 1 i m i t t he c ho i ce o f cx ( t ) . T h e pa raffoete r may be chosen f ree l y w i t h i n t hese bounds .

At t i me t i t he P mat r i >< P < t- 1 ) q> ( t ) d i rect i on . I n i s shown that i f cx ( t ) i s Ray l e i g h quot i ent equa l s

i s cha nged i n the Theo reff1 3 be l ow . i t

chosen such that the the constant a 1

q> ( t )T

P < t- 1 ) P C t > P < t - 1 ) q> ( t ) = a

t hen P < t > converges to a • I as des i red .

Il:!�Q!�fil-�! Def i ne the funct i o n w e t > by

n -r .. • w e t > L

i = 1

2 C l\ C t > a )

( 4 )

where )I. C t ) , = 1 • • n 1 a re t he e i genva l u es o f

P C t > . I f the p mat r i >< is updated acco rd i ng t o Equa t i on < 3 . 4 ) w i t h ex C t > chosen a s i n Equat i o n ( 4 ) i ns i de the bounds g i ven by Theo rems 1 a nd 2 . t hen the func t i on w < t > i s decrea s i ng w i t h t .

a

Und e r ce rta i n e >< c i t a t i o n cond i t i ons a l l the e i genva l ues of P converge to a . S i nce a symmet r i c mat r i >< w i t h equa l e i g enva l ues i s d i agona l . t he mat r i >< P w i l l converge t o the d i agona l mat r i >< a · I .

CONCLUS I ONS

The d i scussed p rob l em a nd t he p roposed s o l u t i ons a re summa r i zed i n F i g . 1 . F i g . l a shows a b l ock d i ag ram fo r t he u pda t i ng of the i nverse p mat r i >< i n the o r i g i na l LS p rocedu r e 1 i . e . Equ a t i on ( 3 ) . The mat r i >< g rows w i t hout bou nds • s i nce i t i s the output of an i nteg rator w i t h a pos i t i ve i nput

s i g na l . The system has to be cont rol l ed to obta i n a bounded i nverse P mat r i >< .

F i g . l b shows the u pd a t i ng w i t h a forget t i ng fact o r acco rd i ng to Equa t i on < 3 . 2 ) . The system i s st a b i l i zed • so the i nverse P mat r i >< stays bounded . S i nce no reference va l ue i s

spec i f i ed 1 P- l

w i l l f l uctuate depend i ng on l\ 1

qi a nd v . Not i ce i n pa rt i cu l a r t hat some e i genva l ues may a pp roach z e ro l ead i ng to the p rob l ems �1ent i oned a bove .

F i g . le desc r i bes t he method proposed i n ·Hagg l u nd < 1 983 ) . I n cont rol theoret i c te rms 1 t he 1Y1ethod cons i sts of both a feedback f rooll t he amount of sto red i nforma t i on and a feed-forward f ro�> t he amount of i ncom i ng i nformat i o n .

T h e 1Y1 a J o r advantage o f t he p roposed method i s t hat i t add resses t he p ro b l ems caused b y nonu n i fo rm p rocess e><c i ta t i on ( both i n t i me a nd i n space > . A n i ce fea tu re i s a l so t hat a p rocess rel ated cho i ce of t he forgett i ng fact o r )I. is repl aced by a pe rformance related c ho i ce of the est i mate v a r i a nce a .

REFERENCES

Astrom1 K. J . and B . W i ttenma rk ( 1 973 ) . On

sel f-tu n i ng regu l ators . a�iQmeii£e_2 • 1 85

- 189 . Fortesque 1 T . R . 1 L . S . Kershenbaum a nd B . E .

Ydst i e < 1 981 ) . I m p l ementa t i o n o f Se l f­t u n i ng Regu l at o rs w i t h Va r i a b l e Forget t i ng

Facto rs . a�tgm2tise_!Z • 831 - 835 . Hagg l u nd , T . ( 1 983 ) . Recu rs i ve l east squa res

i dent i f icat i on w i t h f o rgett i ng of o l d data . Report TFRT-725 4 • Depa rtment o f Automat i c Cont rol • L u n d I nst i t u te o f Techno l og y , Lund i Swed en .

I rv i ng 1 E . < 1 979 ) : New devel opments i n i m p rov i ng power netwo rk sta b i l i t y w i t h adapt i ve cont ro l . P roc . Workshop o n Appl i ca t i ons of Adapt i ve Cont ro l • Ya l e U n i v e rs i t y , New Have n .

a .

b .

c .

p;;, ---q> I

�L- .-� - p f ��

i l •p4J' I v I l�:b-rn- p·' .

-· ___,LJ I

1 O:J

�I -----[?]-------' p·'

Eig..__! B l ock d i ag rams desc r i b i ng the updat i ng o f t h e i nverse p mat r i >< . a . The o r i g i na l LS p roced u re . b . LS w i t h forget t i ng facto r . c . The new p roposed method .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE SIGNAL PROCESSING

THE OVERDETERMINED RECURSIVE INSTRUMENTAL VARIABLE METHOD*

B. Friedlander

Systems Control T1'c/111ology, 1801 Pagt' Mill Road, Palo Alto, CA 94304, USA

Abstract . A recurs i ve i n strumental var i ab l e al gor i thm i s der i ved for the overdetermined case, i n wh i ch the number of i n struments is qreater than the number of est imated parameter s . I n some app l i cat ion s , the al gor i thm prov i des improved parameter estimat i on acc uracy, compared to the standard recurs i ve i nstrumental var i ab l e method . The performance of the al gor ithm i s i l l ustrated by s imu l at i on res u l ts .

INTRODUCTION I nstrumental var i ab l e ( I V ) methods are used frequently to estimate the parameters of dynami cal systems . An extens i ve l i terature ex i sts by now on the theory and app l i cat i on of these method s ; see [ 1 ] - [ 7 ] and the recent survey [8] . The i nstrumental var i ab l es ( or in struments ) can be chosen in many d i fferent ways . The cho i ce of i n struments i nf l uences the accuracy of the est imates somet imes qu ite drast ica l l y [8] . The ex i stence of opt imal in struments was recent l y shown in [ 9 ] , and prev ious ly d i scussed i n [ 3] , [ 4 ] , [ 10] , [ 1 1 ] . The computat ion of the opt imal i nstruments requ ires knowl edge of the no i se dynam i c s . Therefore , opt imal IV methods invo l ve est imat i on of no i se parameters as we l l as pl ant parameters [9] . The est imat ion of no i se dynam ics i s somet imes troub l esome, and shou l d be avo ided when poss i b l e . In fact , one of the attract ive features of the or i g i n al IV concept was its ab i l i ty to prov ide con s i stent est imates of p l ant parameters , without est imat i ng no i se parameters . The use of add i t ion al i n struments prov i des an al ternat i ve way of improv i ng estimat i on acc uracy, wh i l e avo i d i ng est imat i on of no i se dynam ics [8] . Increas i ng the d imen s i on of the in strument vector l ead s to an overdetermi ned set of l i near equat i ons for the parameters , wh ich can be so l ved i n a l east-squares sense [ 2 1 ] - [ 23] . In th i s note we der i ve a recurs i ve al gor i thm for so l v i ng th i s set of equat i on s . Th i s al gor ithm appears to be espec i a l l y useful for autoregres s i ve mov i ng-average ( ARMA) model i ng of t ime-ser i es [ 12 ]- [ 14] .

PROBLEM FORMULAT ION In th i s sect i on we formul ate the overdeterm i ned i nstrumental var i ab l e prob l em . For notat iona l s imp l i c i ty we con s i der on l y the case of ARMA mode l s . Exten s ion to the more general c l ass of ARMAX mode l s [ 1 5 ] i s stra i ghtforward . Let Yt be represented equat i on

N Yt = ' - . ll i =

an ARMA process of order ( N ,M) , by the fo l l owing d i fference

M Ai yt- i + L c . et .

i =O i - i ( 1 ) where e i s a zero-mean un it-var i ance wh ite no i se prtces s . The probl em i s to est imate the parameters {A . } from a f i n i te data set . {Yn • · · : · Yt } . Eq3at ion ( 1 ) can be wr i tten in matr i x form as

( 2 ) where Xt=[y0 , . . . ,yt]

T , ( txl ) x 1 vector ( 3a ) T M

V t=[vo • · · · ·v t] ,v t= i�O C i et- i ( 3b )

a=[A1 , . . . ,AN] , N x 1 vector ( 3c )

Yo 0 0

y = t -yN- 1 -Yo t+l x N matr i x

.:yt-1 -5't-N ( 3d ) The standard l east-squares so l ut i on of ( 2 ) ,

A = [ vi vtJ-l [ vtxtJ at ( 4 )

Th i s work was supported by the Off ice o f Naval Research under contract No . N00014-8l-C-0300 .

2 1 5

2 1 6 B . Friedlander l eads to correl ated therefore , i n strumental

b i ased estimates , no i se vector Vt . the use of var i a l be matr i x

due to the t ime varying parameters [ 1 5 ] .

z = t

0 0

�o

We con s ider , an ( extended )

, ( t+l ) x K matr i x

zt- 1 tt-K ( 5 ) w ith i n struments cho sen so that E {Zivt }=O , E {ZiYt }= non s ing u l ar matr i x ( 6 ) I n the standard I V prob l em , the d imen s i on K of the i n strumental var i ab l e matr i x i s chosen equal to N . Here we assume K "° N . Pre-mul t i p l ying equat i on ( 2 ) by Zt and neg l ect ing the zivt term we get the fo l l ow ing overdetermi ned set of equat i ons

( 7 ) The est imate e of the parameter vector e i s g i ven by the l east- squares sol ution to equat ion ( 7 ) ,

at = [ YiztziYt]-l viztzixt ( 8 ) Th i s est imate i s known to be asymptot i ca l l y con s i stent , a l though not necessar i l y eff i c i ent [8] . Increas i ng the d imens i on K wi 1 1 improve the accuracy of the estimates for short data record s . See [ 1 3 ] , [ 17 ] , [ 18] and the examp l es i n sect i on 4. The instruments can be chosen in many d i fferent ways as l ong as the cond it ions of equat ion ( 6 ) are sat i sf i ed . In the ARMA prob l em the in struments wi l l usual ly be the del ayed outputs : zt = Yt-M ( c . f . IV-3 and IV-4 in [8] , and [ 14 ] , [ 1 6] ) .

I n some appl i cat ions it i s des i rab l e to co.mpute the parameter estimates

e recurs i ve ly in time . In the next sett i on we der i ve the Overdeterm i ned Recur s i ve In strumental Var i ab l e ( OR I V ) a lgor ithm , wh ich prov ides a recurs ive so l ut i on of equat i on ( 8 ) .

DER I VAT ION OF THE ORIV ALGOR ITHM The fo l l owing notat i on ind i c ates how the var ious quant i t i e s in ( 8 ) are updated :

where l.�+J denotes the lxf ro� added to Yt at time t+I ( cf . ( 3d ) ) , zt+ 1 s the lxK row added to Zt ( cf . ( 5 ) ) , and *t+l i s a scal ar quant ity ( cf . ( 3a ) ) . The param�er O " >. " 1 i s an exponent i al "forgett ing factor" wh i ch enab les the al gor i thm to track

To start the der i v at i on l et pt � ( Y�ZtZ�Yt ]-1 ( 10 )

The inverse o f th i s matr ix can be updated as fo l l ows : -1 T T pt+l=Yt+lzt+lzt+lyt+l

T T T T [ >.YtZt+l.t+l!t+l ] [ >.ZtYt+!t+l.lt+l ] 2 -1 T T =>- pt +>-��'.t+1xt+1+>-xv1�t+1 T +1.+1!t+1!t+1l.t+l

where T

�t+l = YtZt!t+l Th i s can be wr i tten more compactl y as

-1 2 -1 -1 T pt+l = >. pt + �t+l \+1 �t+l where �t+1 � L�t+1 It+1 ]

At!1 � � !]. :�J OC

/�H • r�:�Hl j . I nvers ion of equat i on ( 1 3) y ie lds

( 11 )

( 12 )

( 13 )

( 14a)

( 14b )

( 14c )

( 15 ) Th i s equat i on i s very s im i l ar to the error covar i ance update ar i s ing in the mul t i ch annel recur s i ve l east-squares ( RLS) algori thm [ 1 5 ] . To der i ve the update for et we f i rst def i ne

Lt = zixt ' st = vizt wh i ch can be recur s i ve ly updated by

Lt+l=>.Lt+!t+lxt+l ' T T St+l=>.St+�t+l!t+l

From equat i on ( 8 ) we note that 8t+1=Pt+1 5t+1 Lt+l =

T = pt+l ( >.St+l.t+l!t+l ) ( >.Lt+l+�t+lxt+l ) 2 -1 A T =>. Pt+lpt 0t + Pt+1 ( >-i'..t+1l.t+1L t+1+

( 16 )

( 17 )

T +>.�t+l xt+ 1+i'..t+1!.t+ l�t+ 1xt+1 ) ( 18) From ( 11 ) we get

The Overdetermined Recurs ive Ins trumental Variable Method 2 1 7

( 19 )

Comb i n i ng ( 18 ) and ( 19 ) we get A A T T A 6t+l = 6t+Pt+l [ A.lt+l (�t+1Lt+l-�t+1 6t )

T T A + (It+l�t+l�t+l+A�t+l ) ( xt+1 -lt+l 6t ) ] ( 20 )

Th i s equat i on can b e wr itten more compact l y as

( 21 ) where

= fzL1LJ Lxt+l J ( 22 )

Equat i ons ( 12 ) , ( 14 ) , ( 1 5 ) , ( 17 ) , ( 21 ) , ( 22 ) prov i de a compl ete set of recur,..s i ons for comput i ng the parameter est imate et . To i n i t i a l i ze the al gor ithm , we can use an exact procedure i n wh i ch equat ion ( 8 ) i s so l ved off- l i ne for , say, the f i rst k d at a po i nts :

T T T -1 Sk=YkZ k ,Lk=ZkXk ,P k= ( SkSk ) , ek=PkSkL k , ( 23 )

An approx imate i n i t i a l i zat i on procedure wh i ch does not requ ire extra computat ions i s g i ven by

1 Sk =µ[ I O ] , Pk= z I ,L k=O , ek=O ( 24 ) µ

where µ i s a scal ar parameter . The compl ete ORIV al gor i thm i s summar i zed in Tab l e 1 . The computat i ona l comp l e x i ty of the OR I V a l gor i thm i s greater than that o f a comparab l e R IV al gor i thm . A rough measure of comp lex ity i s g i ven by the number of operat i on s (mu l t i p l i c at i ons and add i t i on s ) i nvol ved in one t ime update of the al gor i thm . A standard ORIV a l gor ithm requ ires 6N2+3NK+l3N+2K operat i on s . As an examp l e , us i ng the ORIV al gor i thm with a typ i cal val ue of K = 3N , wi l l requ ire about three t imes the amount of computat ion i nvol ved i n the R I V al gor i thm . Thu s , the i ncrease i n comp l ex i ty i s not too l arge , and in some appl i cat ion s , is wel l worth the effort . The update formul a for the matr i x Pt ( 15 ) does not guarantee i t s pos i t i ve ( sem i - ) def i n iteness . I n i l l -cond i t i oned s i tuat i ons th i s may l ead to numer i c al i nstab i l i ty . Th i s prob l em can be avo ided by u s i ng a sq�?�e-root al gor i thm wh i ch propagates Pt rather than Pt . As an exampl e , the

SR I F al gor ithm descr i bed in [ Sect i on V . 2 , 19] can be used to sol ve equat i on ( 7 ) , wh i ch c an be rewr i tten as

( 2 5 ) Th i s a l gor i thm invol vet process i ng of the K x ( N+l ) matr i x [Lt , St] , wh i ch i s updated by ( cf . ( 16 ) ,

T T T J [ Lt+l 'st+1 J=A[Lt ,St]+�t+l[xt+1l.t+l The detai l s are deferred to [ 19 ] .

S IMULAT ION RESULTS

( 26 )

The performance o f the ORI V al gor i thm was tested by computer s imul at i on . Due to space l im i t at i ons we present on l y three examp l e s . Example 1 : The data was generated by Yt = /2 s i n 0 . 2nt + /2 s i n 0 . 4nt + nt

( 27 ) wh�re nt i s a un it var i ance Gauss i an wh i te no i se process . F i gure 1 dep i cts the t ime h i story of one of the AR parameters for the wel l -determi ned case ( A=l , N=K=4) , i . e . , the standard R I V al gor i thm , and for the overdeterm i ned case ( A=l , N=4 , K= l2 ) . Note the smoother behav ior of the OR IV trajectory . More important l y, the R I V parameters are con s i derab ly more b i ased than the correspond i ng OR IV parameters . Exampl e 2 : Data was generated by

Yt = 1--r-sin 0 . 3nt + nt , ( 28 ) where nt h as un it var i ance . F i g ures 2 and 3 depi cts the trajectory of one of the parameters for the wel l determ i ned case ( A=l , N=K=2 ) and the overdeterm i ned c ase ( A=l ,N=2 , K=6 ) .

As d i scussed i n [13] , the d i fference between the we l l -determ i ned and overdeterm i ned cases becomes more pronounced as the roots of A( z ) approach the un it c i rc l e . The s i nuso i d al s ignal s were chosen i n the exampl e s above to emphas i ze th i s d i fference . S im i l ar resu l ts were obtai ned for narrowband ( non determ i n st i c ) AR s i gna l s [ 20] . Exampl e 3

l+0 . 754z-2 Et + n l+0 .95z-2 t ( 29 )

where Et , nt are un it var i ance wh i te no i se prqcesses . F i g ures 4 and 5 dep i ct the parameter trajector i es for the R I V ( A=l , N=2 , K=2 ) and the ORIV ( A=l , N=2 , K=6 ) al gor i thms .

CONCLUDING REMARKS We presented a recurs i ve imp l ementat i on of the overdeterm i ned i n strumental var i ab l e method . The proposed al gor i thm appears to h ave some useful propert i e s , espec i a l l y when the i nput and output processes h ave l ong correl at i on t imes . The improvement of parameter estimat i on accuracy caused by

2 1 8 B . Friedlander i ncreas i ng the d imens ion of the i nstrument vector was demonstrated i n [ 13 ] , [ 1 7 ] , [ 18] for the off- l i n e ( non-recurs i ve ) IV method . In the recurs i ve ver s i on of the al gor i thm th i s improvement man i fests itse l f i n faster convergence of the a l gor i thm and in reduced var i ance of the f i n al est imates . A more formal ana lys i s of the accuracy aspects of the ORIV al gor i thm and its numer i cal behav i or is yet to be performed .

ACKNOWLEDGEMENT The author wi shes to than k J . O . Sm ith for prov i d i n g the s imu l at i on resu lts for th i s paper .

REFERENCES 1 . K . Y . Wong and E . Po l ak , " Ident i f i c at i on

of L i n ear Di screte Time Systems Us i ng the I n strumental Var i ab l e Approach , " I EEE Tran sact ion s on Automat i c Contro l , Vol AC-12 , pp 707-718, 1967 .

2 . P . Youn g , "An I n strumental Var i ab l e Method for Real -Time Ident i f i cat i on o f a No i sy Process , " Automat i c a , Vo l 6 , pp 271-287 , 1970 .

3 . P . Young , "Some Observat ions on I nstrument al Var i ab l e Methods of Time Ser i es Ana lys i s , " I nternat iona l Journ al of Contro l , Vol 23 , pp 593-612 , 1976 .

4 . P .C . Young and A . Jakeman , "Ref i ned I n strumental Var i ab l e Methods of Recur s i ve Time-Ser i es Ana lys i s . Part I : S i n g l e I nput , S i n g l e Output Systems , " I nt . J . Contro l , vol 29 , no 1 , pp 1-30 , 1979 .

5 . T . Soderstrom , "Convergence of !dent i f i c at ion Methods Based on the I nstrumental Var i ab l e Approach , " Automat i ca , vol 10 , pp 685-688 , 1974 .

6 . P . E . Cai nes , "On the Asymptot i c Normal i ty of In strumental Var i ab l e and Least Squares Est imators , " IEEE Tran s . Aut . Contro l , Vo l AC-21 , pp 598-600 , 1976 .

7 . B . F i n i g an and I . H . Rowe , "Strong l y Cons i stent Parameter Est imates by the I ntroduct ion of Strong I n strumental Var i ab l e s , " IEEE Tran s . Aut. Contro l , AC- 1 9 , 825-831 , 1974 .

8 . T . Soderstrom and P . Sto i c a , "Compar i son of Some I n strumental Var i ab l e Methods -Con s i stency and Accuracy Aspects , " Automat i c a , Vo l 17 , p p 101- 1 1 5 , 1981 .

9 . P . Sto1c a and T . Soderstrom , "Opt imal I nstrumenta l Var i ab l e Est imat ion and Approximate Imp l ementat i on s , " IEEE Tran s . Automat i ca Contro 1 , March 1� to appear .

10 . A . Jakeman and P . Youn g , "Ref i n ed I nstrumental Var i ab l e Methods of Recurs i ve Time-Ser i es Ana l ys i s , Part I I : Mu l t i var i ab l e Systems , " I nt . J . Contro l , Vo l . 29 , No 4 , pp 621-644, 1979 .

11 . P . Young and A . Jakeman , "Ref i ned I nstrumental Var i ab l e Methods of T ime Sere i s Ana lys i s , Part I I I : Extens i on s , " I nt . J . Contro l , Vol . 31 , pp 741-764 , 180.

12 . J . A . Cadzow , "Spectral Est imat i on : An Overdetermi ned R at i on al Model Equat i on Approach , " Proc . IEEE , Vol . 70 , No . 9 , pp 907-939 September 1982 .

1 3 . B . Fr i ed l ander and B . Porat , "A Non-Iterat i ve Method for ARMA Spectral Est imat i on , " Proc . 16th As i l omar Conference on C i rcu its Systems and Computers , Pac i f i c Grove , Cal i forn i a , November 1982 , to appear .

1 4 . B . Fr i ed l ander , " I n strumental Var i ab l e Methods for ARMA Spectral Estimat i on , " I EEE Tran s . Acoust i c Speech and S ign al Processing , to appear .

1 5 . T . Soderstrom , L . Lj ung and I . Gustavsson , "A Theoret i cal Anal ys i s of Recurs i ve Ident i f i c at i on Methods , "Automat i c a , vo l 14 , pp 231-244 , 1978 .

16 . P . C . Young , "Comments on "On-L i ne Iden t i f i c at i on of L i near Dynami c Systems with Appl i c at ions to Kalman F i l ter i ng" , IEEE Tran s . Automat i c Contro l , vol AC-1 7 , pp 269-270, Apr i l 197 2 .

17 . J . A . Cadzow, " H i gh Performance Spectral Est imat ion - A New ARMA Method , " I EEE Tran s . Acoust i c s , Speech and S ignaf Process mg , vol ASSP-28 , no . 5 , pp 524-529, October 1980 .

18 . S . M . Kay , "A New ARMA Spectral Estimator , " I EEE Tran s . Acoust ics Speech and Sign al Processing , vol ASSP 28 , no . 5 , pp 585- 588 , October 1980 .

19 . G . J . Bi erman , Factor i zat ion Methods for D i screte Sequent i al Estimat ion , Academic Pres s , 1977 .

20 . B . Fr i ed l ander , "Adapt ive Est imat ion of Autoregres s i ve S i gn al P arameter s , " IEEE Tran s . Acoust ics Speech and S ignal' Process 1 n g , subm1 tted for publicat 1on .

21 . R . I sermann and U . Bauer , "Two-step Process I dent i f i c at ion with Correl at i on Ana lys i s and Least Squares Parameter Est imat ion , " Tran s . ASME ; ser . G . , 96 , pp 425-432 , 1974 . ' 22 . V . Peterka and A . Hal ouskova , "Tal l y Est imate o f Astrom Model for Stochast i c Systems , " Proc . 2nd I FAC Symp . Ident i f i cat ion and Process Parameter Estimat i on , " Prague , 1970 .

23 . P . Sto i c a and T . Soderstrom , "Comments on the Wong and Po l ak M i n imax Approach to Accuracy Opt im i zat ion of I nstrumental Var i ab l e Methods , " I EEE Transact ions on Automat i c Contro l , vol AC-27, No . 5 , pp 1138-1139 , October 1982 .

l 0

The Overdetermined Recursive Ins trumental Variable Method

Tabl e 1 : The Overdetermi ned Recursi ve I nstrumental Vari abl e Al gori thm

I n i t i a l i zation : T T T - 1 sk = vkzk , Lk = zkxk , Pk = ( Sksk) , ek = PkSkLk

or sk = µ[ 1 : 0], Lk = o, Pk = !, 1 . ek = o µ

At each t ime step do :

Kt+l

�t+l = 5t!.t+l

5t+l T = >.St + lt+ l�t+ 1

•t+l = [�t+l lt+l]

>.2'\ = +l e·:'"' :] ( 2 T 1-l = Pt+t+l >. '\+1 + +t+lPt+t+l

T 2 pt+l = [Pt - Kt+l+t+lpt+l]/>. '

' • �:•! L� t+l •t+l

Lt+ l = >.Lt + �t+ l • t.+ l 0t+l = Ot + Kt+ l ( Vt+l T • - 'i t+! \ I

, N x l

N x K

N x 2

, 2 x 2

, Nx2

N x N

• 2 x 1

� x

N x

�50 590 7S0 Tl 11E: <SFl1PLE:Sl

Fi gure 1 . The Parameter Trajectori e s , Exampl e 1

RIV - ::-:: ORIV

1000

2 1 9

220 2

- 1

2

- 1

B . Friedlander

25 Se ?5

F i gure 2 . R I V Parometer Traj ectori es , Exampl e 2 1 00

TIME

.-2 _1____,--,---,--,---,--.-,-..--.-.--.-�..----,.----,-.-,--.--.--,

A2

2

A2

25 F igure 3 .

5f} ?5 ORIV Parameter Trajectori es , Examp l e 2

1 60 TIME

- - - - - - ---- - - - - - - -- - - - - - - - - - ·- - - - - - - - - - - - -

256 500 TIME <SAMPLES>

Fi gure 4 . RIV Parameter Troj ectori es , Exampl e 3

·-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

256 500 TI ME <SAMPLE'S>

F i gure 5 . ORIV P�rameter Trajectori es , Examp l e 3

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE IDENTIFICATION OF STOCHASTIC TRANSMISSION CHANNELS

L. H. Sibul*, J. A. Tague* and E. L. Titlebaum**

*Applied Research Laboratory and Electrical Engineering Departmeut, The Pmnsvlvania State Uni1iasity,

Universitv Park, PA 1 6802, USA

**Electrical Engineering Department,-

UniFersit_v of Rochester, Rochester, NY 1 4620, USA

Abstract . In this paper , we address the problem of identifying stochastic transmi ssion channels that are characterized by bounded linear stochastic operators iw , t < ) , such that

i ( ) = L ( ) + R ( ) w , t t w

where Lt ( ) are linear , deterministic and time-varying operators and Rw( ) are linear random operators . That is , Lt ( ) produces deterministic propagation losses and Rw( ) produces zero mean random fluctuations of the propagated or scattered signals . The probing signals are completely known and their specification is a part of the design of the channel identification experiments .

We show that minimization of the mean square error between the propagation channel output and assumed reference model output identifies the deterministic part of the propagation operator . The problem can be solved by least squares identification of ARMA parameters . In the scalar case , the correlation function of the random operator Rw( ) can be calculated from the correlation function of estimation error and additive noise . Both of these functions can be measured directly.

In the vector/matrix cas e , the components of the correlation matrix of the random operator can be identified through use of specially designed probing signal s . Hence , in both cases we have a method for identification of the second order statistics of a stochastic transmission channel .

Keywords . Identification ; stochastic systems ; adaptive systems ; correlation methods ; modeling ; random processes ; signal processing ; propagation .

INTRODUCTION

Adaptive system identification methods have been used in man� co�trol sys tem applications . (Astrom and Eykhoff , 197 1 ; Eykhoff , 1974 , 1981 ; IEEE Trans . on Autom. Cont . 1974 ; Isermann , 1979 , 198 1 ; Lee , 1964 . ) Some of these methods can be extended to identify randomly fluctuating propagation channels . To design effective signals and systems for transmission of information one needs to develop models that allow identification of both the determinis tic and stochastic properties of the propagation channel . Modeling and identification of the deterministic part of the propagation operator is closely related to the widely studied system identification problems . Only limited attention has been paid to the identification of the stochastic operators .

A linear stochastic operator is a linear operator which itself is a random function . For example , if the stochastic operator can be represented by a matrix , then the elements o f the matrix are random variables . (Adomian , 1970 , 1 98 3 ; Adomian and Sibul , 1 98 1 ; Sibul , 1967 , 1968 , 1970 ; Sibul and Comstock , 1 97 7 . ) Identification of a system represented by the stochastic operator means determination of the statis tics of the stochastic operator from which the corresponding output statistics can be determined if the input to the system is known . Input signals can be either deterministic or random. By second order " identification" of a stochastic operator we mean modeling of the statistics of the s tochastic operator to the extent that we can determine the second order statistics of output process of the stochastic propagation

2 2 1 ASCSP-H"'

222 L . H . Sibul , J . A. Tague and E . L . Titlebaum

channel when it is excited with a known input signal or stochastic process .

STATEMENT OF THE PROBLEM

We address the problem of identifying stochastic transmission channels that are characterized by bounded linear stochastic operators iw , t < ) , such that

£ ( ) = Lt ( ) + R ( ) w , t w ( 1 )

where Lt ( ) are linear , deterministic and time-varying operators and Rw( ) are linear random operators . That is , Lt ( ) produces deterministic propagation losses and Rw( ) produces zero mean random fluctuations of the propagated or scattered signals . The probing s ignals are completely known and their specification is a part of the design of the channel identification experiments .

We formulate the problem in the space of finite variance complex stochastic processes which are the elements of a separable Hilbert space . We assume that iw t < ) is a bounded linear operator . It can be either an integral operator (L2 space ) or a matrix ( i2 space ) . Since complex separable Hilbert spaces are isomorphic and isometric ( Lusternic and Sobolev , 1961 ) , we can use either integral or matrix operators at our convenience . It is convenient to use matrix operators for detai led calculations , since some of the results are more transparent in matrix notation and we can use specific results from the discrete-time control system theory in our development .

We show that minimization of the mean square error between the propagat ion channel output and assumed reference model output identifies the deterministic part of the propagation operator . The correlation function of the random operator Rw( ) can be calculated from the correlation function of estimation error and addit ive nois e . Both of these funct ions can be measured directly. Hence , we have a method for identif ication of the second order stat istics of a stochastic transmiss ion channel .

MOTIVATlON AND APPLICAT I ONS

We note that the fo rmulation of the sys tern ident if i cat ion problem in terms of operator equat ions as shown by Equat ion I encompasses a fairly wide range of models . It is well known that the solution of both deterministic and stochas tic linear state-space equat ions can be expressed by integral operators as shown in Equation 1 . By the deterministic state-space equation we mean tlie equations where the state transition matrices are deterministic mat rices and by the stochas tic state-space equat ions we mean the cases where the state transition matrices contain e lements which are stochas tic processes (Adomian and Sibul , 198 1 ; Sibul , 1 967 , 1968 ,

1 970 ; Sibul and Comstock , 1977 ) . It can be shown that usual autoregressive (AR) , moving average (MA) , and autoregres sive moving average (ARMA) models are subsets of Equation 1 . Need for the identification of the operators in Equation 1 arises in the signal extraction and detect ion problem. Let the received s ignal vector v( t ) be given by

( 2 ) where £ is a matrix operator , u( t ) is the s ignal-to be detected , and n(t}i:S the interfering noise with the covariance matrix Q(t , T ) . To illustrate the basic connection between the identification problem and signal detection/extraction problem , let us assume that v( t ) is a Gaussian process . In this example we assume that signal and noise are complex , zero mean , mutually uncorrelated processes . It can be shown ( Sibul and Sohie , 1983) that the maximum likelihood detector computes the log likelihood function given by

2i = ( E {! � iH} ( E {! � iH} + gJ -l�]

H g

-1�

( 3 )

where U = u uH. The factors ( E { £ U-tH}-+-q] -1 and q-1 can be adaptively determined from the received data , however , E {i U f H} represents prior knowledge which must-b�either known or determined through the i.dentif ication procedure . Above comments provide the basic motivation for the problem addres sed in this paper .

DEF I NITIONS AND NOTATION

In this paper we wi ll consider finite variance complex stochas tic proces ses x( t , w) , y (t ,w ) , x( t ,w ) , y( t , cu ) , t E: T and 1.i E: II defined on the probabi lity space ( Sl , F , P ) . Lower case letters denote scalars , underlined lower case letters vectors , capital letters operators , and underlined capital letters mat rix operators . Late r , for notational convenience , we shall drop arguments w and t i f they are not needed for clarity. In this paper , we wi ll consider stochastic processes and operators def ined on complex � 2 or 12 Hilbe rt spaces wi th inne r products

and

<x( t , w ) , y( t , w ) >1 2

( 4 )

x( t , 1u )y*( t , w )dt ( 5 )

where superscript T denotes transpose and * denotes complex conj ugate . The norm of x( t , 1J ) denoted by ll x( t , ,.J ) ll , in the Hilbert space is by def init ion <x ( t ,w ) , x( t , w ) > l/2 and the norm of a bounded linear operator A is sup{ :t Axll ; ll x ll = l} .

Adaptive Identification o f S tochastic Transmiss ion Channels 223

IDENTIFICATION OF THE DETERMINISTIC COMPONENT OF THE OPERATOR

General

The simplest identification problem is shown in Figure 1 . This problem illustrates the basic ideas of identification of the deterministic part of operator £ . We want to identify L so that

' 2 E{ j x( t ) - x ( t ) j }

is minimized . Let

T y ( t ) = � ( t ) u ( t )

where

( 6 )

( 7 )

( 8 )

�( t ) [u( t ) , u (t - 1 ) • • • u(t - k) )

[u0 (t ) , u 1 ( t ) • • • uk( t ) ) ( 9 )

T y ( t ) = � u( t ) ( 10 )

and

y ( t ) = i_ �( t ) + E_( t ) • ( 1 1 )

To minimize the mean square error , as given by Equation (6 ) the orthogonality condition must be satisfied for all elements of vector u ( Tretter , 1976 ; Kailath , 1981 ) . This condition can be expressed as a vector matrix equation

T H E { (x ( t ) - �l_ u( t ) ) � ( s ) } = _Q E{�) (u( t ) uH( s ) ) + E{n( t ) uH(s ) }

T H = � u( t ) u ( s ) •

Let (u( t ) uH( s ))= U( t ,�, then we have

UNKNOWN STOCHASTIC PROPAGATION CHANNB.

KNOWN PROllNG SIGNAi..------� uftl

MOOD. l 1 I

ADDITATIVE

NOISE

nltl

llltl

( 12 )

( 1 3 )

( 14 )

eftl

Fig . 1 . Model-reference ident i fication of a stochastic system.

where we have used the relation E{i} = L and E{n ( t ) uH( s ) } = 0 . If the inverse of U( t , s ) exis�

( 15 )

We have shown how the deterministic part of L can be identified through a mean square error minimization . The correlation matrix of the s tochastic operator can be determined as well . This will be considered later in the paper .

Identification of the Determinis tic Part of i

We have shown that minimizing the mean square prediction error leads to an estimate of the deterministic part of £ . Next , we consider three methods of implementation . Each uses two all zero linear predictive filters to estimate the poles and zeros of £ .

It is assumed that i can be modeled by a linear , discrete time , autoregressive moving average process . Its output at time interval n may be written as

N N y ( t ) l: a . y ( t - i ) + l: b . u (t - i)

i=l l i=l l

T I. + .!?_T ( 1 6 ) a . . � ,

where y__ and � are the output and input signal vectors , and a and b are the associated deterministic-:- "slowly time variant " weighting vectors . The problem examined in this section is the estimation of a and b given noisy measurements of the channel output and a known set of probing signals u.

Figure 2 illustrates the first method . The probing signal is input to the channel under examination and linear predictor L1 . The output of i is corrupted by addit ive white gaussian noise n(t ) . The observable quant ity is called x( t ) . The difference between x( t ) and the output of 11 forms prediction error e1 ( t ) . A second prediction error e2( t ) is the difference between e1 ( t ) and the output o f predictive filter 12 . It can be shown that the mean square mi �imi zation of the prediction errors , e 1 ( t ) and e2( t ) gives the fo l lowing biased estimates of a and b.

ultl

nm

Fig . 2 . Joint process estimation technique .

224 L . H . S ibul , J . A. Tague and E . L . Titlebaum

-1 • R -uu

( 17 )

( 18 )

where �· �· � are cross correlation matrices , ana ..&iu and !xx_ are auto-correlation matrices . ( In other sections we have also used the symbol U for ..&iu· ) Clearly the major disadvantage of this method is that it gives biased estimates of � and b.

The second technique is shown in Figure 3 . I t can be shown that this technique does not identify the poles and zeroes separately .

uttl

nttl

Fig . 3 . Joint process/whitening filter method .

The third method is illustrated in Figure 4 . This method has been previously discussed by Astrom and Eykhoff ( 197 1 ) . Consider a vector of measurements x :

xT = [x ( t ) x( t-1 ) • • • x( t-N) ] • ( 19 )

x = qJ . B ( 20 ) -

where

13T [a l a2 • • • aN bl • • • bN ] ( 2 1 )

and ¢ i s a data matrix :

x (t-1 ) x( t-2 ) • • • x( t-N) u(t-1 ) • • • u(t-N)

x( t-N- 1 ) • • • x( t�2N) u(f-N- l ) • • u( f-2N)

nttl

uttl xttl

eftl

Fig . 4 . Multi-observation estimation method .

The prediction error vector � is written as :

e = x - x

We want to minimize the sum of the mean square prediction errors :

E { i e ( t ) i 2 } + E{ i e ( t-1 ) 1 2} + • • .

( 22 )

+ E{ i e ( t-N) i 2} = E{�H • e} , . ( 23 )

A necessary and sufficient condition to minimize ( 23 ) is that

�

E{<l>H • (� - ¢ • �) }

0 .

Solving for B gives :

( 24 )

( 25 )

The modeling error over the N observations is

H E {� • �} E{xH • x} -

E{xH ¢} E { (!H

• E{ <!JH • x} ( 26 )

We see this method gives the most complete identification of the three methods examined . Equation ( 25 ) shows that the estimate of unknown vector B is in the form of standard least squares estimate . Since there are a large number of recursive least-squares estimation algorithms that can be used to estimate B ( Goodwin and Payne , 19 7 7 ; Friedlander , 1982 ; Nehorai and Morf , 1983 ) , we can turn our attention to identification of the stochastic component of the linear operator . Adaptive identification of Lt by recursive least-squares estimation algorithms and j oint process predictors will be discussed in the full paper .

IDENTIFICATION OF THE STOCHASTIC OPERATOR

Contribution of the Stochastic Component

To see the effect of the stochastic component of the operator , let us assume that we have perfectly identified the deterministic component of the operator and let us compute the total mean square output error :

uH E{RH R} u + E{nH n} - - - - - - ( 2 7 )

We see from above that total mean square output error consists of two terms ; the first

Adaptive Identi f ication of S tochastic Transmiss ion Channels 225

term is the increase of the mean square output error due to the random component of the operator and the second term is due to the additive noise. We can determine the last component by measuring E{ EH E } with the input signal turned off . Hence, -

H H H H � E{! !} � = E{E_ E_} - E{.!!_ .!!_} ( 28 )

From this relation we can determine the norm of the random operator R by

ll Rll 2 2 2 supll! �II / ll u ll

H H H sup (E{E_ E_} - E{.!!_ .!!_} ) /� � . ( 29 )

Clearly , the norm of the random operator can be determined from the measured data .

Wide-Sense Stationary Scalar Case

In this case we can write

E ( t ) = z ( t ) + n( t ) ( 30 )

where

( 3 1 )

we also have assumed that E{ z ( t ) n* ( t ) } = E{ z* ( t ) n( t ) } 0 • ( 32 )

The covariance o f the error is

( 33 )

or in the terms of respective power spectra we have

or

G ( f ) + G ( f ) zz nn

E{ j HR( f ) j2} Guu( f ) + Gnn( f )

( 34 )

G ( f ) /G ( f ) zz uu G ( f ) - G ( f ) EE nn ( 35 )

or the covariance of the random operator is

G ( f ) - G ( f ) F-1 E E uu

Guu( f ) ( 36 )

where f- 1 is the inverse Fourier transform . Thus , we have , in principle , a method for calculating the mean of f and the covariance of the random part of ]• In the next section we turn our attention to the vector-matrix case .

Stochastic Operator in Vector Matrix Notation

In this section we consider the case where the channel output l.. can be expressed by an m dimensional vector

= L u + R n + .!!_ , ( 3 7 )

where 1 and R are mxm matrices . We assume that 1-has been identified previously . The covariance matrix E{X.. X..H} is given by

E{X.. X..H} = � � �H �H + E{! � �H !H} + E{n nH}

= � .!!_ �H + E{! .!!_ !H} + E {.!!_ .!!_H} , ( 38 )

where we have used the assumption that the elements of matrix R are zero mean random elements of R and the noise vector is uncorrelated:- The problem is to "identify" the elements of the matrix R, using a test signal under our control , so that when another signal u is transmitted we can estimate the new E{ R U RH} . As discussed previously , this problem arises in the detection/estimation problem. The matrix R is in the form

rl l r l 2 rlm r2 1 r22 r2m

R

rml r ( 39 ) mm

where the elements rij are zero mean random variables . To avoid extraneous complexity complete identification of the E{R U RH} can be accomplished by expanding E{R u R:li} in terms of the signal matrix . For simplicity we illustrate the essential idea for 2x2 matrices . Extension to larger matrices is s traight forward but tedious .

E{! .!!_ !H} =

E{ j r 1 1 ! 2} E{ r 1 1 * r2 1 }

ul l E { r2 1 *

r 1 1 } 2 E{ j r2 1 J }

E{ r1 1 *

rl2} E{ r l l * r22 }

+ ul 2 * * E{ r2 1 r l2} E{ r2 1 r22}

E{ r l 2 * r 1 1 } E{ r l2

* r2 1 } + u2 1 E { r22

* r 1 1 } E{ r22

* r2 1 }

2 E{ i r 12 i } E{ rl 2 *

r22} + u22 E { r22

* 2 r l2} E{ I r 22 I } ( 40 )

226 L . H . Sibul , J . A. Tague and E . L . Titlebaum

To identify all the terms in these matrices , we transmit a sequence of signal vectors that wil l have only one non-zero component . This wil l allow us to uniquely identify each of the terms in the above matrices . We note that the matrices following u1 2 and u21 terms contain elements that are conj ugate pairs . Hence , elements of only one of these matrices need to be determined . Once all of the elements E {rij r�1} have been estimated we can reconstruct the E {R U RH} matrix for any known but arbitrary sign;l� Hence we have completed the second order identification of stochastic operator. Frequently , we know from the physics of the problem that

0 ( 4 1 )

for i * k and j * £ . Then matrices mul tiplying uij i * j vanish and E {R U RH} becomes E{R R } .!!d where E{� �H} is_ a_ -diagonal matrix with typical diagonal terms E { riirii } and Ud is a diagonal matrix with typical terms Uii • In this case we can identify the terms of the random matrix by

( 42)

where E{R U RH} is determined from measured data and _!!d from the known probing signa l . O f course , inversion of the diagonal matrix .!!d is trivial .

CONCLUSIONS

Many communication and control systems can be characterized by linear operators that can be decomposed into deterministic and stochastic linear operators . For proper design of the communication and control system, one needs to identify both the deterministic and stochastic components of the operator . We have known that minimization of the mean square error between the output of the operator and the assumed reference model output identifies the deterministic part of the operator . To be more definite , we assumed that the deterministic operator can be represented by an ARMA model that can be identified by a recursive least squares algorithm. The stochastic part of the operator can only be "identified" in the statistical sense . In the scalar case , the correlation function of the random operator can be cal culated from the correlation function of the estimation error and additive noise . Both of these functions can be measured directly . Simi larly , the correlation matrix of the random operator can be determined from measured data provided that specially designed signals are used . Details of the signal design wil l be discussed in the ful l-length version of this paper .

The deterministic part of the operator can be identified by standard parameter identification techniques . In this paper we have outlined methods for second order

identification of the stochastic part of the operator . Computational detail s wil l be presented in the longer paper . The concepts developed in this paper have communication and control system applications .

REFERENCES

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�

Adomian , G . ( 1983) . Stochastic Systems . Academic Press , New York .

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Adaptive Identif ication of S tochas tic Transmission Channels

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Sibul , L. H. ( 1 970) . Sensitivity analysis of linear control systems with random plant parameters . IEEE Trans . Autom Cont . , AC-15 , PP • 459-462 .

Sibul , L . H. and C . Comstock ( 1 977 ) . Stochastic differential equations and their application to randomly time-varying systems . Int . J . Control , 25 , pp. 647-656 .

Sibul , L. H. and G . R. L. Sohie ( 1 983 ) . Application of operator theory to signal modeling and processing . Neuvieme Colloque nor le Traitement du Signal et ses Applications . GRETSI , Cagnes sur Mer .

Tret ter , S . A. ( 1 976 ) . Introduction to Discrete-Time Signal Processing . Wiley , New York , Chapter 12 , pp . 333-354 .

227

Copyright © JFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

CONVERGENCE PROPERTIES FOR A FAMILY OF BOUNDED FIXED STEP-SIZE ALGORITHMS

D. C. Farden

Def)(lrtrnenl o/ 1';/ectriml Enginl'l'ring, The University of Rochester, Rocltesler, NY 1 4627, USA

Abstrac t . Con ver g e n c e a n d t r ac k i n g p r oper t i e s a r e d i s c ussed f or· a f a m i l y of a d a p t i v e s i g n a l p r oc e s s i n g a l g or i t h ms . T h e f am i l y t r e a t e d i n c l u d e s b o u n d ed f i x ed g a i n ver s i on s of b o t h t h e RM a n d KW s t o c h a s ­t i c: a p p r ox i ma t i on p r oc e d u r es . N e w r es u l t s a r e p r esen ted t h a t a r e ap p l i c o-;b l E? 1-ih e n t h e t r· a i n i n q d a t a i s c:or- r- E� l atJ=!d << n d t h •:=! " op t i m<i. l " sol u t i on i s n ot u n i q u e a n d / o r t i me-va r v i n g . T h e non-un i q ueness i s o·f i n t e!r .. Pst i n e s t a b l i s h i n g " c:: onver q E;nc:: t;, " f or· c:: on st r· a i n 0?d a l qo­r .. :i. t i·im"' a s 1-ie l l '" s d e t t:; r .. m i n i n n • • .,,, :i n q l 1?.-.. modE! b 12!hav :i. or . . • • .

Keywor d s . A d d p t i v e .,;; y !s t em!;; ; ;;,u::l ap t:. :i. v P c: o n t r· o l ; c o v ,;;1 1·- :i. i'i. n c: e d e c:: i'1 Y -r· ;;; t e c on d i t i on s ; c on v e r g en c e .

I NTRODUCT I ON

Con s i der t h e a l g or i t h m

<:I. a )

l>J k + :i. ""f" ( W k + l ) " i :l. b l p where W k E C ( c:: omp � e x p - s p a c: e l , u 1 s a

r F; a l p u s :i. t i. v e c: o n s t ;.,1 n t. , a n d P : C P -:;...s i s a c l i p p i n g upe r a t i on wh i c h e n s u r e s t h a t C W k } i s a b u u n d e d sequen c e . T h e set S i s a c:: ump a c t subset of w :i. t h i n wh i c h the ' ' d t:!!;; i r . . ed '' p a r a m1=!ter .. W r em a i n s f ur a l l k . T h e c: or r ec: -o , k

r;P

l i on t er· m Z k ( Y k , W k l i s c P ·-vi:t l t.1.f.�d f u n c t i un uf d a t a Y k a n d t h e e s t i m a t e

W , , of W 1, t h e a l gor i t h m o b t a i n ed a l t·. o , ·. i t:. er· a t. :i. cm k . N o t e t h a t 1-i i. t h t h e

i�. 1 q o-

r i t h m C l ) r e p r esen t s f i x ed s t ep s i z e bounded ver s i on s o f b o t h t h e R M C 1 95 1 1 a n d KW C 1 9 5'.? I p r· oc ed u r es; . I n most a d ap t i ve s i g n a l p r c� es s i n q i mp l emen­t at i u n s of ( 1 ) , Z k w i l l c on t i n u e to

h a ve s t o c h a s t i c: f l u c t u a t i on s f or a l l k ; h e n c: f.� , 1 1 c: o n v f.��r· q e n c: e 11 :i. n t h e! u�;u,3 J sense i s p r ec l uded f or t h ese f i x ed step s i. Z !?. a Lg cw i t h m!;; . T h e• b e s t. b e h a v i ur t h a t o n e may h o p e f or i s t h a t. t h e " d i. s t ,,. n c: e " b e t. ween t h e d e s i r ed p a r amet er W , a n d t h e est i -o , r:: mat e Wk ob t a i n e d b y t h e a l g or i t h m

bec omes s ma l l f or l a r g e k a n d t h a t. t h i s " d i s t a n c e " c: an b e d ec:r· eased ( a t l east i n t h e s t a t i o n a r y c a se ) b y u s i n q a s ma l l er '-' • I n c a s e W ,, i s CJ , I'·. un i q u e , a mean -sq u a r e " d i. st a n c e " mea­s u r e i s a p p r o p r i a t e . When W . , i s n o t o , •< un i que or when une w i s h e s t o ex am i n e

229

t h e t r �: k i n q p Pr f o r m a n c: e of a s i n g l e

une t h en n ee d s a

and a f a m i l y of wo ,k s . T h e f o l l ow i n g

an a l y s i s p r o v i d e s t e c h n i ques f or u p -p E!r .. b o u n d :i. n q v k '""E ( I S ( 1.; k ) I 2 l ,

V k = W k -w o , k ' S i s a p x p c omp l e x m a t r i x

'".i:i . t h l l S ll :;; :I. , <:<. n d E ( • ) d P n o t. 1?.s; s t a t i s-­t i c: a l ex p e c t a t i on .

ANALYS I S

A =W - W Assume t h a t k D ,. k + :I. o , k t h i.� t l f" ( X ) --f" ( Y l l :"; I X -Y I f DF"

a l l X . Y E C P , a n d P I X l = X f or a l l X m S . . S i n c:: e Wo , k + l E S , we h a v e

I S ( V ) I "" I S ( P ( W > --- W k + l A k + l 0 , k + l ,,;; I S ( W k + l -wo k + l ) I ( 2 ) . - , . = I S < V k -µ T + µ T-µ Z k -Ak l

an where T i s ( f or n o w )

e l ement of c P , a n d Z k = Z k < Y k , W k l . Let H k , k "=S ( µ ( T - Z k I ) • a n d

Assume t h a t. t h e r e ex i st f i n i t e p o s i ­t i ve c on s t a n t s µ 1 , b .l , b � , b � s a t i s f y i n q ( �=� ( µ ) ( 1 ) • . L -

''"' ,_, l f -\ l "':=;� I S < V k l 1 :: E ( I H -S ( A l 1 ·"" l +2E I Ah S ( A l I

_k , k 2 k A k k

sb 1 u +b 2µ 6 +b 3 6 =f ( µ , 6 1

( :3 )

( 4 ) f or a l 1 k , w k , w I � s . 0�6 ( µ ( µ 1 , a , .. � . Sub s t i t u t i n g i n t o ( 2 ) we h a ve

v k + l�� v k + f ( µ , 6 , � l +2 E C ReA:Hk , k } . I t e r a t i n g 1 5 > f or k = l , 2 , • • • , n

( 5 )

230 D . C . Farden

( 6 )

A suwnat i on b y p ar t s y i e l d s ,:n n - k h n - 1 h -

_..,. 'l' A H = '!' A H t;;; 1 k k , k 1 1 , n

n - 1 +"-- n - k - 1 . h . h - ,

,...,.

'l'

f. A t-- + ·I H t + l -'l' ?\ H 1, + 1 , 11 1 . r:;; 1 •. . . , n

Not e t:. h ,;1t. 1 ri' 1 H-'(A

1 1 H ' l ::;;'( I A l l H--·H '' I ( 8 ) k + l k k + ( ( 1 ·-'( ) I A k + l l +'( l t\ : + l -P.1 k I ) I H I .

I n m a n y c h n i C: f:i! �::;

ap p l i c at :i on s , a p p 1·· op r i a t e

c a n f or T a n d a d e c o m p os i t i on of b e f ound t h a t e n ab l e Lemma 1

b e l ow t:. o be ap p l i ed to ( 6 ) - ( 8 ) . LEMMA 1 . Let:. { X k } a n d { C' \, } c e s of c P - v a l ued r a n d o m w i t h E C X k l =O . Def i n e

be !5!? q u e n -

v a r· i •� b l e s

..,.D. 5 t. , n = > :X. m :�·;; k

I f E I G k l 2 �g 2 a n d I e ( k , k +u ) I ,;;c ( 1 + I u I ) i.-· f or some O< v< l a n d a l l i n t eg e r s k , u :

n - l I I E r ··c:; · . n - k - l r, ·1,.., \ I I .. , .. 'I' ·"I '"' ! -1 j ) ' f-;;; 1 < < .. . , n

,,; ( T�'.<:,. ) 1 �:.::··r ·-2r, ( :?--� ) < l -·'l' l -'..?+v 1 2 .

Proof . See F a r d e n ( 1 98 1 ) .

EXAMPLE

A f i x ed s t e p -s i z e KW p r oc e d u r e . Let { d k } a n d { X k } b e j o i n t l v d i s t r i b u t ed s t a t i o n a r y s t o c h ast i c p r oc esses w i t h X k e c P a n d d k c c 1 . L e t � k =Wh X k . A n y

W=W0 wh i c h m i n i m i z es � ( W J =E ( l d k -d k l L l i s a s o l ut i on t o R W0= P , wher e

R=E < X k X� l a n d P=E < d: X k l . Assume t h a t R i s ( st r i c t l y ) p o s i t i ve d ef i n i t e , l et P l • ) b e a component b y c o m p o n e n t c l i p p i n g op er a t i on , a n d l et S b e t h e i d ent i t y . Assume t h a t v a l ues of � k < W > = l d k -Wh X k l 2 are a v a i l ab l e f or one v a l ue of W at eac h t i me k W e con s i d er a s c h eme s i m i l ar t o t h e s i n g l e r e c e i ver system of Can t o n i 1 1 9 80 ) . L e t C U . } b e a d e t e r m i n i st i c

J p er i od i c sequen c e w i t h p er i od K ,

U . e c P , a n d s a t i sf y i n g I U .. 1 .. l =u ,

J .

.. K t' "'·· V V ' ·.>

·. U . U T=O .. ·� .. > u . u . =u·-�:: r ,

J / u 1. =0 ,

j = l .J J j = 1 J . J;; l ..

a n d f or e a c h j c { 1 1 2 1 • • • , K } eH i s t s an :i ( j ) D { 1 1 2 1 • • • , fC· ,

t h e r e i ( j ) # j

s u c h t h at U . . . 1 = - U . . Let l. I J J

.• · y W l l L •,· ( 1.,. I =--:::;-t . . Ku'·

( k + l l f< ""·- U . ·�- . ( W+l.I . l t;;k K + l .i ·· J J

w h e r e a l l i +mK . Some H , , k "" 1..1 T - p Z 1 •.

m i ss i n g sub sc r i p t s a r e a l g e b r a ver :i f :i es t h a t

as r e q u i r ed . T h e f a c t t·. , •, t·. t h a t { W J./ i s; a bounded SE? q u e n c e

e n ab l es o n e t o a pp l y Lemma 1 t o b y u s i n g ( 8 ) a n d sp l i t t i n g H 1 m , < t h e t h r ee t er ms spec i f i ed b y 1 9 ) .

CONCLUS I ON

C7 ) :i n t o

T h e m a i n r e su l t s of t h i s paper a r e summar i z ed b y ( 5-8 ) . W i t h t h e a i d of Lemma 1 , t h e se r esu l t s c an be ap p l i ed i n m a n y p r ac t i c a l c ases t o esta b l i sh suf f i c i en t c o n d i t i on s f or t h e " d e s i n;:! d b e h av i or· " of an < Hl ap t i ve s i g n a l p r oc e ss i n g a l g or i t h m . Us i n g t h ese r e su l t s i t i s p o s s i b l e to o b ­t a i n .,,1 n u p p er b o u n d cm t. h f.? " d i st a n c e " b e t ween t h e d e s i r ed p a r ameter a n d t h e v a l ue ob t a i n ed b y t h e a l g or i t h m , a n d i l l u s t r a t e t h e t r adeof f s i n v o l ved i n c h oos i n g var i ous a l g or i t h m p arame t e r s t o ac h i eve a c c ep t ab l e a l g or i t h m p e r ­f or ma n c e < e . g . i n c r e a s i n g 1..1 t o speed " c on v er·· g E? n c e " r e su l t s i n <H l i n c: r·ease i n the " st. e a d y - s t ;a t e m i s;,� d j u!O; t m en t " l .

ACKNOWLEDGMENT

T h i s wor k was s u p p o r t ed by Syrac use Resear c h Cor p or at i on a n d B a l l i st i c: m i s s i l e D e f e n s e A d v a n c ed Tec h n o l o g y Cen t er u n d e r Con t r ac t D A S G 60-8 1 -C-00'..?6 .

REFERENCES

C <a n t:. cm i , A . < 1 980 ) . Ap p l i c a t i on of o r ­t h ogon a l p er t ur b at i on seq uen c e s t o a d a p t i ve b ea m f o r m i n g . I E E E Tran s . An t e n n a s Pro pag a t . ,

v o l . AP-28 , 1 9 1 -2 02 . F a r d en , D . C . ( 1 9 8 1 ) . T r a c k i n g

p r o p er t i es of a d ap t i v e s i g n a l p r oc ess i n g a l g or i t h ms . I E E E Tran s . R c o u s t . . Speec h , S i g n a l

P r o c . , v o l . ASSP- 2 9 , 439-4 4 6 . K i ef er· , .J . , ,,. n d Wed f ow i t z , ,J . ( 1 952 ) .

S t o c h ast i c est i m a t i on of t h e max i mum of a r e g r e ss i o n f u n c t i on . Rn n . M a t h . S t a t i s t . , 23 , 4 62-466 •

R ob b i n s , H . , a n d Mon r o , S . C 1 9 5 1 l . A s t o c h ast i c a p p r oH i ma t i on met h o d . R n n . M a t h . S t a t i s t • . 23 , 4 �)-407 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1 983

A STUDY OF ADPCM USING AN RML PARAMETER ESTIMATOR

C. R. Johnson, Jr.*, J. P. Lyons, Jr. and C. Heegard**

School of Unlriml Engirtr'fring, Cornfll Vnivtr.1ily, !thaw, NY 1 4853, USA

Abstract Thi s paper prov i des further study of a

recently s uggested al gori thm for the adapt i on of an I I R fi l ter used as a sequenti a l l y adaptive backward predi ctor wi th i n an adapti ve di fferenti a l pul se code modul at i on (ADPCM ) s cheme . 1 . I ntroducti on

The probl em of adapt i ve di fferenti a l pu l se code modul at i o n ( ADPCM) has attracted s i g n i f i cant research and devel opment , espec i a l l y i n speech process i ng [ l ] . Major studi es are current ly u nderway to establ i s h i nternati onal standards i n 3 2 kbi t/second ADP CM for tel ephonj [ 2 ] .

Di fferenti a l pu l se code modul ati on ( DPCM) schemes typi cal l y empl oy a predi ctor to remove the redundancy from the source and quant i ze the ( hopefu l l y ) l ower vari ance pred i cti on error . This l ower vari ance s i gna l can be quanti zed wi th fewer bi ts and then reconstructed wi th the same fidel i ty as strai ght PCM of the ori g i na l source s i gnal at the hi gher quanti zer word l ength . When e i ther the quanti zer step poi nts or the predi ctor parameters are ada pted th i s i s terr.;ed adapti ve DPCM o r ADPCM .

Thi s paper recons i ders a mod i f i cati o n , suggested i n [ 3] , of the recurs i ve max imum l i ke l i hood ( RML ) parameter est i mator [4] that al l ows i ts appl i cat i on as a sequenti a l l y adapti ve backward predi ctor for ADPCM use . The termi nol ogy-sequenti a l l y adaptive backward pred i ctor is borrowed from [5 ] . The l abel "sequenti a l " i mpl i es an updati ng of the predi ctor coeff i c i ents at each samp l e time , rather than batch proces s i ng the data i n mul ti p l e samp l e frames . The term " backward " impl i es that the adapt i on i s dri ven on ly by i nformati on avai l ab l e at both the transmi tter ( encoder ) and rece i ver ( decoder ) . Typ i cal l y the o n l y i nformation ass umed ava i l ab l e a t the transmi tter and the recei ver i s the transmi tted quanti zed predi cti on error . See Fi gure 1 . Thus the l abe l " backward" i s i n accord wi th the encoder structure that effecti vely feeds back the past quanti zed predi cti on errors throuah the predi ctor to *C . R . Johnson , J r . i s s upported i:>y NSF Grant ECS-81 19312 . **C . Heegard i s supported by NSF Grant ECS-8204886 .

23 1

form a current pred i cti on and thus the current pred i cti on error .

The novel ty of the adapti ve predi ctor proposed i n [3] and cons i dered here is that i t has an i nfi n i te- impul se-response ( I I R ) structure , i n whi ch both the pol es and zeros are i ndependently adapted . See F i gure 2 . Th i s I I R pred i ctor structure i s parti cul arly app l i cabl e to sources more reasonably model ed as noi se dri ven autoregres s i ve , movi ng-average ( ARMA ) model s than the more typi cal l y used stri ctly AR mode l s , to whi ch Fi gure 1 appl i es . 2 . Open I ssues and Practi cal Concerns

Due to space l i mi tati ons we must refer the reader to [3] for the compl ete al gori thm statement i l l ustrati ng the mi nor mod i fi cati on of RML needed for th i s ADPCM appl i cati on . Note that an AML [6] variant i s poss i bl e but convergence is assured on ly if [ l+C ( q- l ) J -1 is strict ly pos i ti ve real , wh i ch is a severe restri cti on for thi s s i gnal proces s i ng appl i cati o n . Our pre l i mi nary s i mu l ati on experi ence appears successful i n that wi thout e l aborate fi ne-tun i ng our al gori thm ' s performance exceeds that of al gori thms wi thout the e componen t . Experiments on s imul ated data were summari zed i n [ 3] and on real speech data , the same as used i n [ 4] , are to be summari zed i n [7 ] . However , th i s experi ence does rai se a number of open theoreti cal i ss ues and practi cal concerns . These questi ons i nc l ude the effects of quanti zation err9r , the assurance of the s tabi l i ty of [ l-A ( q- l ) J - 1 , the robustness of the proj ect i on fac i l i ty , the effects of channel errors , and the computati onal burden of RML .

Our s imu l ati on experi ence i ndi cates that quanti zati on noi se does not drasti cal l y degrade the predi ctor performance . Such perturbati ons are not common ly analyzed i n the system i denti fi cati on use of RML . The i r i nfl uence on predi ctor performance and ADPCM performance is the pri nci pal subj ect of [ 7 ] . Note that thi s error is transmi tted to the rece i ver so the adapti on at both encoder and recei ver s houl d sti l l fol l ow the same trajector i es .

An i ss ue that ari ses i n the ADPCM appl i cati on , but not i n the normal system i denti fi cat i on use of RML , i s the need for

232 C. R. Johnson, J r . e t al

assuring the stabi l i ty of [ l-A(q- l ) J- 1 . Thi s i s i n add i tion to the stabi l i ty check and associ ated projecti on of [ l+e { q- 1 ) ] - 1 requ i red by RML . The need for a stabi l i ty check on [ l -A ( q- l ) J - 1 ari ses for two reasons . The fi rst i s commonly recogni zed i n the ADPCM l i terature , e . g . [8] , as the need for stabi l i ty at the rece i ver . The rece i ver operates on E wi th [ l+C ( q- 1 ) ] / [ 1 -A ( q- 1 ) ] i n a n attempt to recover y from E ( � e + w) . Thus i f [ l-A(q- l ) J- 1 were momentari l y unstabl e , g i ven channel errors the recei ver output cou l d qui ckly exceed a l l reasonabl e l i mi ts . The second reason for cequ i ri ng the i nstantaneous stabi l i ty of [ l-A(q- l ) J - 1 ari ses when the quanti zer saturates such that e can be growi Qg but E rema i ns c l amped . At the transmi tter y can grow wi thout bound , wh i ch wi l l cause e to conti nue growi ng . Thi s undes i rabl e behavi or wi l l a l so be repl i cated at the rece i ver . Thus as an i nterim fi x we used a stabi l i t� check and proj ecti on faci l i ty on [ l -A(q- l ) J- 1 i n our s i mu l ati ons .

I n our s i mu l ati ons wi th speech data we noti ced that we needed to addAa safety margi n to QUr stabi l i ty check on [ l+C ( q- l ) J -1 ( and [l -A(q- l ) J - 1 ) . By only reacti ng once [l+C(q- l ) J - 1 had pol es outs i de the uni t c i rcl e , i ts pol es were abl e to creep right up to the stabi l i ty boundary and , if our numeri cal eval uati on of stabi l i ty was the l east bi t i naccurate , i nstabi l i ty cou l d resul t and e woul d stal l wi thoutAour detecti ng the i nstabi l i ty of [ ltl ( q- 1 ) ]- 1 . Addi ng a check for pol es of [ l+C { q- 1 ) ] - 1 outsi de a c i rc l e of rad i us r< l rai ses questi ons

.about how to choose r and about i ts effect on performance . Furthermore , no uni versal l y acceptabl e projecti on scheme i s known , once " i nstabi l i ty" i s detecte51 , that can guarantee the absence of trappi ng C i n a nearly unstabl e l i mi t cyc l e behav i or .

As noted earl i e r , channel errors can res u l t i n recei ver i nstabi l i ty , i f [ l-A ( q- l ) J- 1 i s unstabl e , even though the transmi tter cou l d remai n stabl e . Di ffi cul ti es wi th channel errors are common to al l ADPCM schemes us i ng a backward predi ctor at the transmi tter that resu l ts in an I I R rece i ver . Furthermore , these channel errors are never ful l y fl us hed even from a stabl e I I R recei ver . One suggest ion [8] i s to use l eaky i ntegrati on in the adapt i ve parameter esti mator . Unfortu­nately ADPCM performance i s rap i d l y degraded as thi s scal ar i s decreased even s l i ghtly bel ow uni ty . However , a sca l a r factor appreci ably l ess than one i s needed to rap i d l y remove channel errors . Thus channel errors represent a s i gni fi cant open i ssue .

predi ction quanti zed source error pred i cti on error

y e E quanti zer

Y ..------f A( q - 1 l __ _, E

Fi g . 1 : Current ADPCM Encoder Structure for AR Sources

The computati onal burden of RML for the pol e-zero pred i c tor of Fi gure 2 i s A s i gn i f i cant comRared to schemes where only A i s adapted and C = 0 . The s tabi l i ty checks themsel ves represent a maj or cos t . One s ug£es tion i s to use l a tti ce forms , the stabi l i ty checks on whi ch req u i re only assuri ng that the appropri ate refl ecti on coeff i c i ents are l ess than u n i ty . I t appears that thi s s impl e check only appl i es to the MA l atti ce and not as di rectly to i ts ARMA form needed i n our appl i cati on .

Though the precedi ng ( i ncompl ete ) l i st of open questi ons argues agai nst the i mmed i ate appl i cati on of our RML- based ADPCM s cheme , the pos s i bi l i ty of adapti ng both pol es and zeros i n the backward predi ctor i s s o enti c i ng that we hope others wi l l become i nteres ted i n i nvesti gati ng the ADPCM s cheme of [ 3] . The real i zati on that many of these i ss ues are gener i c to many ADPCM schemes , and not j ust ours , adds some hope that adapt i ng both pol es and zeros i n an ADPCM scheme i s not as unreasonabl e as previ ous ly ass umed .

3 . References

[ 1 ] J . D. Gi bson , "Adapti ve pred i cti on i n speech d i fferent i a l encodi ng systems , " Proc . I EEE , pp . 488-525 , Apri l 1 980 .

-­

m- X . Mai tre and T . Aoyama , "Speech cod i ng acti v i ti e s wi th i n CC I TT : Status and trends , " Proc . 1982 I EEE I CASSP , Pari s , France , May 1982 , p p . 954-959 . [3] C . R . Johnson , J r . , J . P . Lyons , J r . , and C . Heegard , "A new adapti ve parar.1eter estimati on s tructure appl i cabl e to ADPCM , " Proc . 1983 I EEE I CASSP , Boston , MA , Apri l 1983 , pp . 1-4 . [ 4 ] B . Fri edl ander , " System i denti fi cati on techni ques for adapti ve s i gnal proces s i ng , " I EEE T-ASSP , pp . 240-246 , Apri l 1982 . [5] J . D . Gi bson , " Sequenti al l y adapti ve backward pred i cti on i n ADPCM speech coders , " I EEE Trans . on Comm . , p p . 145- 150 , Jan . 1978 . [6] V . Sol o , " The convergence of AML , " I EEE T-AC , pp . 958-962 , December 1979. [7] C . Heegard , C . R . Johnson , J r . , and J . P. Lyons , J r . , "Quanti zer effects in RML­based ADPCM , " Proc . 22nd I EEE CDC , San Antoni o , TX , December 1983 . [8] D . Coi ntot , "A 32 k- bi t/sec ADPCM coder robust to channel errors , " Proc . 1982 I EEE I CASSP , Pari s , France , May 1982 , pp . 964-967 .

wh i te noi se

y

quant i zer

Fi g . 2 : Proposed ADPCM Encoder Structu re for ARMA Sources

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

PERFORMANCE OF AN ADAPTIVE ARRAY PROCESSOR SUBJECTED TO TIME-VARYING

INTERFERENCE

F. B. Tuteur

/J1'fmrt11u'nl of 1�·1ntriwl 1�·nginPering, Yafr Uniiwnity, Nnu Hmwn, CT 06520, USA

Ab s t ract . Array p rocessors for pass ive detect ion o f directional wide-band s ignals are commonly used in s onar , seismic work , and radio as tronomy . The per fo rmance of s uch p rocessors is degraded by direct ional noise source s re ferred to as inte rferences and adap tive pro ces sors to eliminat e the e f fect o f interferences have b een used for many years . The issue addressed in this paper is movement o f the interfe rence and the e f fect that this has on adap tive loop design . The processor cons idered in the pape r is an a rray o f adj ust ab le F I R filters using the Widrow LMS algori thm to adj ust the filter weights . Changes in inter ference patterns a f fect b oth the covariance mat rix of the obse rved signal and the op timum weight vec tor . For the p urposes o f this paper the opt imum weigh t ve ctor i s mode l led a s an independent-increment p ro cess . This model permi ts the rigo rous derivat ion o f the o p timum adap tive­loop gain parame ter and of wors t-case performance . In gene ral it is f ound that interfe rence movement results in two e rror component s : a gradient-noise component and a t racking component . Increases in adap tive loop gain de crease the t racking component but increase the gradient-noise component . Thus for a given set of inte r fe rence-mo t ion statistics an optimum gain can be fo und .

KeywC?_rds . Adap tive , filter, array , LMS , inter fe rence , array processing, signal processing .

1 . I NTRODUCTION

The sys tem cons ide red in this pape r is an adap tive processor designed fo r use with direct ional signals received by an array of sensors . Each of the array elements are assume d to be conne cted to a s imple t r ans­versal ( F I R) filter cons isting o f a de lay line , variable- gain ampli fiers (weigh t s ) and and s umme r to generate the fi lter output

through ampli fiers wi th a variable gain Wij k '

( See Fig. 1 ) . The filter outputs are summed , squared , and smoothed , and the smoothed out­put is compared to a thre sho ld to decide whe ther a signal is pre sent in the desired direction . In addition to the desired signal (which may or may not be p re sent ) there is also noise and interference whose precise characteris tics are unknown , and s ubject to change . The purpose o f the adapt ive p roces­sor is to maximize the signal- to-noise ratio at the output in the face of this unknown and variable noise fiel d .

The main issue addressed b y this paper i s the specific e f fe ct of time variat ion o f the noise fie ld on the performance and optimum design o f the adaptive p rocessor .

We assume that the receiving array consists o f M e lements e ach of which is connected to the input o f a t ransversal filter cons isting of an n-section t apped delay l ine with iden­tical delay e lements each p roviding a delay 6t . The n delayed t ap o utputs are passed

233

where i6t is the current t ime , j = 1 . . . M is the array-element inde x , and k = l . . . n is the delay-line index . The output of all o f these ampli fiers are s ummed t o p roduce the outp ut signal z

i.

The signal x . ( t ) from the j th array e lement J appears wi th a de lay k6t at the kth del ay­line t ap of the j th delay line . Hence all· the delay-line-tap o utput signals at t ime i6t can b e represented by the Mn dimens ional ve ctor

x. -i ( 1 )

where the s up erscript T stands for matrix t ransposit ion . Simil arly the variable gains W . ' k can be combined into an Mn dimens ional

l.J weight vector

The adap tation is b as e d on the Widrow LMS algo ri thm [ 1 , 2 ] , modified as in [ 3 , 4 ] t o t ake into account the fact that in the array­process ing app li cat ion an "e rro r signal" is not e xp lici t ly present . The p urpose o f the adap tat ion is to convert the weight ve ctor �i f rom some initial val ue to the o p t imum

234 F . B . Tuteur

value W� . -1 We start by considering an updating algo­rithm of the form:

�i+l (3 )

where µ is the adaptive loop gain and ve: -1 is the gradient of the mean square error at time instant i . I f the desired signal and the filter output are respectively given by s . and z . then 1 1

(4)

The filter output signal is given by T -z . = w . x . 1 -1-1 (5 )

and therefore the elements of the gradient vector are

( 6 )

We suppose now that the signal component of xijk is si+k . -k ; in other words we regard

J the desired signal at the input to the j th delay line to have a negative delay of k . J units . This also means that in each delay line there is a delay k = k . at which the J signal component is the desired signal si . The delay k . is precisely the delay needed in J the j th delay line to steer the array in the direction from which the desired signal is expected . 1

We take signal and noise to be statistically independent and zero mean . Then

( 7)

where Rs ( · ) is the autocorrelation function of the signal . The values of R (k-k . ) con-s J tain information about the signal temporal spectrum and direction that the filter needs to look for the desired signal . Thus Rs ( · ) is assumed to be known ; i . e . the filter is

1 since the delay lines provide only discrete delays the array can only be steered in a finite number of discrete directions . The effect of discrete steering on a continuous­ly variable direction (for both target and interference) is not considered in this pa­per ; in e ffect we assume that the delay line has enough delays to act like a continuous delay line .

designed to look for a signal with the specified Rs ( · ) . On the other hand the ex-pectation z . x . " k in Eq . (6 ) contains noise 1 1] covariance information that we assume not · to be known . This expectation cannot , there­fore , be evaluated .

Following the approach of stochastic approx­imation we therefore replace z . x . "k by 1 1] z . x . . k . It is well known [5 ] that z . x . . k 1 1] 1 1] is , in fact , an unbiased estimate of the correlation z . x . . k We finally obtain the 1 1] updating algorithm in the form

(8)

where

[Rs (k1-l )Rs (k1-2)Rs (k1-3) . . .

Rs (k1-n )Rs (k2-1 ) . . . Rs (k2-n) . . .

Rs (kM-1) . . . Rs (�-n) ] ( 9 )

is the desired-s ignal covariance vector .

Since the chief issue addressed in this pa­per is the e ffect of changes in the noise field we take the signal characteristics to be constant in t ime and space . Hence the covariance vector P . is not a function of the -1 time index i and will henceforth be written as P .

On the other hand the optimum weight vector w . 0 is a function of time because of varia­-1 tions in the noise field ; hence the subscript i . The optimum weight vector satis fies the Wiener relation

w . 0 -1 R. -lp -1 (10)

where

R. x . x . T

-1 -1-1 (11)

is the covariance matrix of the received sig­nal .

We take the optimum weight vector to be an independent-increment process ; thus

Also the variations slow relative to the processor ; transient changes of the noise ered.

in w . 0 are assumed to be -1 adaptation time of the effects caused by rapid field are not consid-

Performance of an Adaptive Array Processor 235

A second fundamental assumption used in the analysis is that the !i are statistically uncorrelated , zero-mean Gaussian vector sequences . This assumption is basic to all previous analyses of adaptive system per­formance [6 ] . The conditions on both w . 0 and

-]_ !i are restrictive and are not always satis-fied in practice . However , computer simu­lations indicate that relaxation o f the re­quirement that the data be uncorrelated re­suts in only a small change in system per­formance [ 7 ] . Equation (5 ) can be used in Eq . ( 8) to yield an update equation in the form

( 1 3) I f the system to w� then in -]_

is stable and i f W . converges the steady state tfie average

weight vector is o Wo. [ T o ] �i+l = �i = -]_ + 2 µ .!'._-!i!i�i

or T o = R .W� !i!i�i p -]_-]_

(14)

where the equality on the right follows from Eq . ( 10) . Also , by use of Eq . ( 11) we get the result :

The weight-error vector is de fined by

and there fore

V. W . -w . 0 -]_ -]_ -]_

(15 )

(16)

(17)

We see from Eq . ( 1 3) that �i depends on !i_1 . Hence , since !i is independent from !i-l by assumption

( 18)

Then , by use of Eq . (15 ) we have that in the steady state

--- -x . x . Tv . X . X . T

v . R .V . = 0 (19) -]_-]_ ]_ -]_-]_ -]_ -]_-]_

Equations (15 ) and (19) suggest that in the steady state both �io and yi behave as if they were statisti cally independent o f Xi ; also because �i is nonsingular Eq . (19) im­plies that

lim V . i + 00 -]_

0 ( 20 )

2 . STABILITY By use o f Eq . ( 14) the update equation ( 1 3) can be written in terms o f the weight error as

yi+l 0 0 -(�i+l-�i ) (21)

This equat ion has the general form analyzed in Weiss and Mitra [ 8 ] , Narendra and Peter­son [9 ] , etc . i f we define

( 22 )

N . is a random disturbance similar to the -]_ "gradient noise" mentioned in Widrow [ 10 ] . Since Wi is uncorrelated with Xi , and there-fore also with Ri ' it is easily seen that the mean value of N. is zero . The term

]_ 6W . 0 = W�+1-w . 0 is the change in optimum

--]_ -]_ -]_ weight vector resulting from�ange in the noise field. By hypothesis 6W� = O. Thus , -]_

taking the mean o f Eq . ( 21 ) results in the homogeneous equation

( 2 3)

Since !i is a real , positive de finite and symmetric matrix it can be diagonalized by means of the unitary trans formation g_i :

-1 �i = Qi �iQi c 2 4 )

where �i is the diagonal matrix of eigen­values and Qi is a matrix of eigenvectors . We assume that Q is normalized so that g_-l = QT . We de fine the t rans formed error vector by

Yi = QiYi (25 ) so that Eq . ( 17 ) is t rans formed into

( 26 )

or , i f vK(i ) is the Kth element of yi

( 2 7) It easily fol.lows that the solution of the homogeneous equation is asymptotically stable i f

or i f 0 < µ < --1- for all K and i ( 28) :\K(i )

236 F . B . Tuteur

Suppose that for all i and K

(29)

Then the s tability requirement is

1 0 < µ < >. max ( 30)

a slight extension of the well-known result of Widrow (10 ] to a t ime-varying covariance matrix B:i · By not ing that

( 31 )

one can replace the right hand side o f Eq . ( 30) by the somewhat looser bound

1 0 < µ < ( tr R. ) for all i ( 32 ) -1

For changes in the noise field that do not affect the total noise power ; e . g . changes in the direction of inter ference source s , tr R. is not a funct ion o f i and can b e re--1 placed by t r R, the total power o f the re­ceived signal-:- I f the power fluctuates , then an upper bound for µ is obtained by replacing tr B:i by (tr B:)max in Eq . ( 32) . Although Eq . (32) guarantees stability of the mean error , it does not necessarily guarantee s tability of higher-order moment s . In fact , by analyzing the asymptotic behavior o f the effor covariance , Weinste in ( 11 ] has recently shown that Eq . ( 30 ) is not suf fi cient for s tability of the second moments and that µ should not exceed between 1 /3 to 1/2 o f the value given in Eq . ( 32 ) . Weinstein ' s anal­ysis assumes a constant noise covariance mat rix R and to our knowledge no analysis for variable R has been performed . We assume in the sequel that µ is always small enough so that the system is s table .

3 . STEADY-STATE PARAMETER-ERROR COVARIANCE

By adding and subtract ing a term T o 211 (B:i-!i!i )�i , Eq . (21 ) can be rewritten in

the form: T T o o

Yi+l = (l_-2w!i!i )yi+211 CB:i-!i!i )�i-6�i ( 33)

Then the error covariance takes the form :

where : A

B

c D

T T T ( I-2µX , X . ) V . V . ( I-2µX , X . ) -i-1 -i-i - -i-i 2 T o o T 4 v CB:i-!i!i )�i�i CB:i-!i!i )

6W�6W�T ( 35 a) -1 -1

T oT T 211 <I-2v�!i )Yi.!ii CB:i-!i!i )

E ( 3Sb )

F

The s ix terms in Eq . ( 35a ,b ) are evaluated in the appendix under the assumptions o f X . X . = 0 for i � j . Under the further -1-J assumpt ion that w? is an independent­-1 increment proce s s we find product terms D, E , and F generally will not vanish

that the cross­all vanish . (They for other varia-

tions of w� . particularly -1 zero) . Then

cov Yi+l A+B+C

i f 6W� is non -1

cov yi-2µ [B:icov yi+(cov yi )B:i l 2

+4µ [B:it r B:icov yi+2B:i ( cov yi)B:i l

+4µ2 (PPT+R . PTR . -lP ] - -1- -1 -( 36 )

In the s teady s tate cov Yi+l cov V . so that -1

-2µ (B:icov yi )+2B:i ( cov yi ) Ri ] T T -1 1 2 [PP +B:if B:i f]+ 2µ cov 6Wi ( 37 )

It is again convenient to trans form this ex­pression so that � is diagonalized . Using the trans formation Q of Eq . (24) in Eq . ( 37 ) result s in

A . cov V . +cov V . A . -1 -1 -1-i

-2µ ( 6it r.!l_icov yi+2_i ( cov yi)� ]

- - T -T - T 1 -= 2 [PP +£'._if +Ai!'. AiP ]+ 211 cov 6� ( 38)

where Yi ' f, �i are the respective trans­formed vectors . The diagonalization of B:i does not necessarily diagonalize cov y1 . However , the diagonal elements o f cov Yi satis fy :

2 n 2 2 2 2o >. -2µ>. L J.KoKK-4µ>.p opp pp p p K=l n

= 2µ [P 2+A L A P 2 ]+ L ti2 p p K=l K p 2µ pp ( 39 )

where a 2 i s the pq element o f cov V . , >. pq -1 p

is the pp element o f A , ---p is the ment of P and 6 pq

P is the pth ele­p pq element o f

cov 6W . • -:-1 Also the o f f-diagonal terms o f cov vi s atisfy 2 2

a ( J. +>. ) -4µo >. A pq p q pq p q (40 )

Performance of an Adaptive Array Processor 2 3 7

We observe that there is no coupling between the off-diagonal terms and the diagonal terms ; thus Eqs . ( 39 ) and ( 40 ) can be dealt with separately . We are , in fact , inter­ested mainly in the diagonal terms as is shown by the following discussion .

If the desired signal s . is present at the filter input , the optimtm output is

( 41 )

where A0 is a gain constant and �io is the

minimum output noise. Small changes from optimal adj ustment should initially cause mainly an increase in output noise ; i . e . the steering function of the filter should re­main relatively unaffected, but the inter­ference would no longer be properly can­celled . Thus , if

(42)

is the optimum mean-square filter output , then small perturbations of the filter weights away from their optimum values should result in an increase in J . This suggests that the quantity

(43)

can be regarded as a measure of performance degradation . It is nonnegative and vanishes for optimal adj ustment . Because of the in­dependence of �i and �i it is easily shown that

6J tr (R . cov v . ) -i -i tr (J\ . cov Yi) -i n 2 L: ;\ a ( 44)

p=l p pp

Thus the performance degradation 6J depends only on the diagonal terms of cov Vi . By dividing out the common factor 2:\ from both p sides of Eq . ( 39 ) we can put this equation into the form

Ta = _g_

where

T = I-2µJ\ - . [�]

[ :\l · · · "n l - - .

]J

(45 )

(46 )

i s an n x n matrix , and where � and _g_ are vectors whose pth elements are respectively

[0 ] - p 2 CT pp (4 7 )

and

[ g ] = ]J

p (48)

Equation ( 45 ) is easily solved by use of the matrix inversion lemma

(49 )

After some algebra we find that the perform­ance degradation is given by

6J = t r(_g_ cov Y)

]J

m 1 - L:

p=l

(50)

where m = Mn is the dimensionality of the weight vector W . . By converting back to the nondiagonaltzed form of R. this can be put into the form: -i

6J

-1 T -1 1 o tr{_l-2µ_g_) [ µ!'.!'._ (l_+_g_ tr _g_)+ �ov 61!!_ ] }

-1 1 - JJ tr [ ( I-2µ_g_) B:J

(51)

where the subscript i has been omitted for simplicity .

Equations (50) and ( 5 1 ) are the main result of this paper . The result is similar to that given in Widrow et al [ 2 ] but di ffers in de tail and is derived under different conditions . We see that the performance degradation 6J has a "gradient noise" com­ponent involving P which decreases with JJ and a "tracking" component which increases as µ becomes small . Equation (51 ) can be simplified somewhat under certain commo�ly encountered conditions . For instance i f µ is very small the term (I-2µR)-l in both numerator and denominator will have negli­gible effect and can be omitted. Also , for small enough µ the entire denominator can be replaced by unity . Finally , it can be seen from Eq . (50) that for large m the coeffi­cient of ppT can be simplified to R-ltr R. With all of these simpli fications we obtain the approximate expression

6J � JJfTB:-lf tr _g_ + ZJJ t r cov 61!!_0

,

( 52 ) for µ tr B: < < 1 , n >> 1

An optimum value for µ is easily found from Eq . (52 ) by differentiation and is given by

1 tr cov 6W0 � z PTR-lP-tr R

(53)

238 F . B . Tuteur

and the resulting minimum performance degradation is

6J . �( tr cov 6�0) ·.!'._TB:-l.!'._ tr B: (S4) min

Since the processor is intended to be adaptive it may be unrealistic to assume precise knowledge o f the parameters .!'._ and �o of Eqs . ( S 3) and (S4) . The optimum gain and minimum performance degradation could then be based on worst-case values however . I f the eigenvalues of B: are approximately equal except possibly for a few that are larger , Eq . (SO) can be reduced to forms that permit easy computation of 6J as a function of µ . We specifically consider the case of a single large eigenvalue ; i . e .

/c l Mic M > 1 /c 2 !c 3 le !c , m m

Then Eq . (S3) reduces to

6J = p 2 [2+ m-1 ] 1 M M+m

µ {

1 + -4µ

+ ---l-2Mµ/c 1-2µ/c

2 611 1 --- + --l-2Mµ A 1-2µ/c

[l _ Mµlc l-2Mµ/c

(m-1) µ /c ] 1-2µ/c

m I

i=2

>> M

P . 2 } l

(SS)

This equation has been plotted as a function of µle for various values of M and m and with the further s impli fication of P12 0 ,

m 2 m 2 I P . / le = 1 . The assignment I P . / /c 1 i=2 l i=l l

is equivalent to replacing 6J by its nor­malized value 6J/J . This follows from the de finition o f J in Eq . (42 ) . By use o f Eq . ( 14) we see than an alternate expression for J is

J (S6) and by invoking the separation properties given in Eqs . (lS) to ( 19 ) this becomes

(S 7 )

where , a s in Eq . (SO) the Pi are the ele­ments of the trans formed P matrix . Although 6J clearly depends on the ratio o f the 6 ' s to the P ' s , the distribution o f 6 ' s o r o f P ' s over the various eigenfunctions of R does not appear to matter too much ; hence

the choice of P12 = 0 is not particularly

signi ficant . Also , it is clear that the case o f all equal eigenvalues is simply a special case o f Eq . (S2 ) with M = 1 . Finally , the main s impli fication in Eq . (SS) is the reduction in the number o f arbitrary parameters that have to be assigned to per­mit computation ; s imilar results could be obtained if there were more than one large eigenvalue as long as all the eigenvalues were related in some s imple way . Typical plots o f Eq . ( S 8 ) are shown in Figs . 2 and 3 . They all show the expected large performance degradation for values of µ that are too small to permit the system to t rack variations in noise environment . As µ is increased further , eventually a point is reached where the denominator of Eq . ( SS ) vanishes ; this is the stability limit . This limit can be easily calculated by solving the quadratic equation in µle that results when the denominator is set equal to zero . For large m and M an approximate solution to this equation is

µle 1 (l+ 2M(m+2 ) + . . . ) m+3M+l ( 3M+m+l) 2

1 ( l+ 2M) (S8)

m+3M m For m-1 eigenvalues of le and 1 eigenvalue o f Mic the stability limit µ t r R = 1 would be equivalent to µle = l/ (m-l+M) . We see that the s tability limit obtained here is slight­ly dif ferent , although the di fference is signi ficant only if M/m is fairly large . The optimum value of µ depends on the ratio o f

2 2 I6ii to L:Pi ; or equivalently to the ratio of tr cov 6W0 to PTR-1P . The simple square­root relati(;"nship-of Eq. ( S 3) holds for sufficiently small cov 6W0 as is shown for by straight-line portions at the le ft of the plot o f Fig . 3 . For larger values o f t r cov 6�0 the approximations of Eq . ( S 2 ) and ( S 3) become invalid and the optimum value for µ approaches a saturation value roughly equal to one half of the stability limit . Also , once this saturating value of cov 6�0 is reached the performance degradation increases very rapidly .

4 . SUMMARY AND CONCLUSIONS

We have considered the performance of a sim­ple adaptive array processor in which each array element is passed through a transversal f ilter with adj us table weights . The adjust­ment of the weights uses the Widrow LMS algorithms . It can be shown that the optimum f ilter that results from the operation of the adaptive algorithm is a noise canceller that generates nulls in directions of large noise concentrations . As these concentrations move , so must the nulls generated by the fil­ter. These filter changes are mani fested by changes in the optimum filter weight vector

Performance o f an Adaptive Array Processor 239

w0 • For the purposes of our analysis the weight vector is assumed to be an inde­pendent-increment s tochastic process ; also the elements of the received-signal vector are assumed to be independent . Under these assumptions relatively simple expressions for the performance degradation caused by movement of the noise field have been ob­tained . They show the expected dependence on adaptive loop gain ; i . e . larger than the minimum for gain that are either too small to permit proper tracking of the motion , or too large if gradient noise is the major factor . For very large changes in the noise environ­ment it was .found that the optimum loop gain approaches a saturation value of one half of the stability limi t . It is believed that the qualitative behavior of the performance degradation or the optimum loop gain would not be changed if some of the simplifying assumption made to facilitate the analysis were removed .

REFERENCES [ l ] B. Widrow, "Adaptive Filters " in.

Aspects of Network and Systems Theory , R. E . Kalman and N . De claris Eds . , Holt Rinehart and Winston , N . Y . , 19 71 , pp . 5 63-587 .

f2 ] B . Widrow , J .M. McCool , M .G . Larrimore , and C .R . Johnson , Jr . , "Stationary and Nonstationary Characteristics of the LMS Adaptive Filter" , Proc . IEEE �. 8, pp. 1151-1162 , Aug . 19 76 .

[ 3 ] J .H . Chang and F . B . Tuteur , "A New Class of Adaptive Array Processors " , J. Acoust . Soc. Am. 49 , No . 3 (Pt . 1) pp . 6 39-649 , March 19 7y;-

f4 ] L . J . Griffiths , "A Simple Adaptive Al­gorithm for Real-Time Processing of Antenna Arrays" , IEEE Proc . Vol . 5 7 , No . 10 , pp . 1696-1704 , 1969 .

[S ] B . Widrow e t al . "Adaptive Noise Can­celling, Principles and Applications" Pro c . IEEE , Vol . 6 3 , No . 12 , pp. 1692-1 716 , Dec . 1975 .

[6 ] K. Senne , "Adaptive linear discrete-time estimation" , Ph . D. dissertation , Stan­ford University , S tanford , CA. , June 196 8 .

[ 7 ] ibid. [ 8 ] A . Weiss and D. Mitra , "Digital Adaptive

Filters ; Conditions for Convergence , Rates of Convergence , Effects o f Noise and Errors Arising from the Implementa­tion" , IEEE Trans . IT-25 , No . 6 , pp . 6 37-652 , Nov. 1979 .

f9 ] B . B . Peterson and K. S . Narendra , "Bounded Error Adaptive Control" , IEEE Trans . AC-27 , No . 6 , pp . 1161-1168 , Dec . 1982 .

[10] B . Widrow , J .M . McCool , M . G . Larrimore , and C . R. Johnson , Jr . op . cit .

( 1 1 ] E . Weinstein, "Stability Analysis o f LMS Adaptive Fil ters" t o appear in IEEE Transactions for Acoustics , Speech , and Signal Processing .

APPENDIX 1 . EVALUATION OF THE PARA­METER-ERROR COVARIANCE

Al-1 : The squared terms A ,B , C

A T T T ( I-2µX. X. ) V . V . ( I-2µX . X . ) 1 1 1 1 1 1

2 T T +4 xixi vivi xixi

Term 1 : V . V . T = cov V . -1-1 -1

Term 2 :

The first equality follows from the in­dependence of Xi and Vi (Eq . 18) .

Term 3 : v . v:x . x: cov V . R. -1 -1 1 1 1 1

Term 4 : x . x .T

v . v . x . x .T

1 1 1 1 1 1

Then

This term is the product of three matrices . We drop the subscript i for convenience ; then the pq term of this matrix product is

because of the independence of �i and v . . We as sume X to be a Gaussian vec­-1 tor; then the 4th-order moment can be expanded as

-- -- --L L xp� vkvQ. xQ.xq is the pq element k Q.

of Bo( cov :DBo· Similarly and since both R and l: l: k Q.

cov V are symmetric matrices

�( cov }'_)Bo. The term L L x x �xQ. vkvQ. is k JI, p q

the pq element of Bo tr(R cov V) .

240 F . B . Tuteur

Combining these partial results and adding the coefficients results in

Term B :

2 o o T o o T T 4µ [RiWiWi Ri-RiWiWi xixi T o oT T oT T -xixiwiwi Ri+xixiwiwi xixi J

The last term o f B is

where z 0 evaluate X. and z -i 0

X. z z x: --i o o i oT Ri � is the filter output . To

the quadruple moment we assume that are zero-mean Gaussian ; this is

approximately true if the variation in W0 -i

is small , bandwidth

X. z z x: -i o o-i

random , and slow compared to the of xi . Then

-- --T T -;z 2Xizo zoXi + XiXi o T o oT T Z 2xixiwi

• wi xixi + B:i zo

2 PPT + R z2 - i 0

If we treat Z0 as the output after the fil­ter weights have settled , or if for the pur­pose of approximating this term we regard Xi and � as approximately independent , we can use the Wiener-Filter result

Combining all of these results finally yields :

B = 4µ2 [PPT + R PTR-1P J - -i- -i -o oT The squared term C = �Wi�Wi

is not �urther reducible .

Al-2 : The Cross-product terms

oT 2 T oT i5 = 2µ�� B:i - 4µ �!i.Y.iRi B:i

oT T 2 T oT T -2µ �� !i!i + 4µ ��.Y.iRi �!i

Term 1 :

because V . -]_

Term 2 :

[x. x:w . - x. x:w� ] PT -]_-]_-]_ --i-i-i -

0

But by Eq . (12) Ri is a function o f xi-1 ' X . 2 • • • ; hence by the assumed independence ].-of the !i ' !i and Ri are independent . --- -- -x . x:w . = X. X: W . = R. W� . --i-i--i -].-]_ -]. -]_-]_

Also , from Eq . (13)

term 2 vanishes . x. x:w� -]_-].-].

Then

Thus

Term 3 : V. PT = 0 -]_-

The fact that V . is independent from the -i other three factors in this term follows from Eq . (15 ) .

Term 4 :

Dropping the subscript i for convenience we find that the pq element of the six-fold vector product is

where the break in the overbar follows from the fact that v . is independent of X . and

-]_ -]_ W� . Then since for all k -;:;k = 0 , we find -]_

that term 4 vanishes . Thus D = 0 - T oT E = (.!_-2µ!i!i).Y_i�Wi = 0

Since �W� is an independent , zero mean , in­-i crement by hypothesis .

- T o oT F = 2µ (B:i-!i!i)Ri�wi = 0

s ince �W� is an independent-increment process . -].

Thus all the cross-product terms vanish in the the steady state if W� is an independent­.increment process . -i

Performance of an Adaptive Array Processor 24 1

ARRAY ELEMENTS

DELAY LINES

• • • •

Fig. 1 Block diagram of Adaptive Array Processor.

. 01 ...... ��-'-��--'���..i....�� ..... . 0001 .001 .01 0.1

np.'A

10

.01 """'":::--�""'""'::--�..&...�� ... ��....L.��....J 10- 8 10-7 10-6 10-5 10-4 10-3

tr(cov AW0)

Fig. 3 Optimum gain and corresponding perform­ance as a function o f tr (cov Wo ) . (n = 105 , M = 102 , Al = MA1 , ;i 6i1=0) •

Fig . 2 Variation of performance degradation with 5 2 adaptive loop gain (n = 10 , M = 10 , pl2 6i1 = 0 , Al = MA ) .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE TECHNIQUES FOR TIME DELAY ESTIMATION AND TRACKING

R. A. David and S. D. Stearns

Sandia National Laboratories, Organization 7125, Albuquerque, NM 87185, USA

Ab s tract . Th i s p aper d i s cu s s e s the pro b l em o f adapt i ve t ime de l ay e s t imat ion j_n the s p e c i f i c c a s e where the r e c orded input s i gnals are s inus o i d s with addi t ive no i s e . We cons i der two s imple adapt ive t e chnique s whi c h do not require t rans forms and are c ap ab l e of t ra c king a non s t a t i onary , t ime-varying delay . F�r s t a

_me �hod whi c h make s u s e o f a two-p o l e , al l-p a s s adapt ive

f i l t e r is introdu c e d . An LMS- type of adaptive algori thm i s derive d , and the as s o c iated error surface i s di s c us s e d . The performanc e o f this adapt ive s t ruc ture i s then c ompared to that o� the � adapt ive de lay e lement " r e c e n t ly introdu c e d by E t t e r . S imulat i o n r e s u l t s are c omp are d on the b a s i s o f d e l ay e s t imat e accuracy , rate o f c o nvergenc e , and the ab i l i t y t o track t ime­vary ing pha s e shift s .

Keywords . Adap t i ve s y s t ems ; Time-varying s y s tems · S i gnal pro c e s s ing ; I t e rative method s ; Corre l a t i o n method� .

INTRODUC TION

Adap t ive e s t imat ion of the t ime de l ay be tween two narrowb and s igna l s re­ce ived s imultaneous ly at s ep arate sensors in the pre s en c e o f unc orre­lated no i s e i s a c ommon prob lem in s e i smic and acou s t i c a l data p r o c e s s ­ing . Computing t h e cros s - c orre lation funct ion and s e l e c t ing the peak of the correlogram as an e s t imate of the t ime d i ffe renc e is probab ly the mo s t widely used s o lution ( I ann i e l l o , 1 9 8 2 ; B�adley , 1 9 8 2 ; Hert z , 1 9 8 2 ) , but a va­riety o f other me thods have app eared in the l i terature . A te chnique in­troduced re cently b y Youn and c ol­le ague s ( 19 8 2 ) use s the LMS algori thm in a s y s tem ide nt i f i c at ion c on fi gu­rat i on . The p rob lem o f e s t imat ing the t ime de lay func t ion is formu l at e d as o n e of identi fying a t ime-varying s y s t em who s e imp u l s e re sponse reache s i t s maximum at a t ime whi c h c orre­sponds to the de l ay e s t imate . The de lay e st imat ion proc e s s there fore require s adap t at ion of the FIR c oe f­fic ient s , interpolat i on of the im­pulse respons e , and d e t e c t ion of the pea� c orre sponding to the t ime d e l ay e s t imate . Here we are intere s te d in s imp le adap t ive t e chniques whi c h do not require trans forms and provide a dire ct e s t imate o f the t ime d e l ay at e a ch iterat i o n .

243

I n t h i s paper we c on s i de r the spe c i f­i c c a s e whe re the two input s i gnals are s inus o i da l and the noise is whi t e . The mo de l t o b e u s e d fo r t h i s d i s c u s s i on i s i l lu s t r a t e d i n F i g . 1 . The unknown pha s e s h i ft , s , may b e e i th e r f i x e d o r t ime -varying ; b u t b e c au s e t h e s ignal i s s inus o i da l , w e a r e c o n c e rned o n l y f o r t h e range where the magnitude o f 8 is l e s s than n radi an s ( i . e . ± 1 /2 c y c l e t ime d e l ay ) . The goal of the adapt ive proc e s s is t o minimi ze t h� mean­s quared output e r ro r , E [ E ] , thereb y gene rat ing an e s t imate o fkthi s delay .

To imp l ement the e s t imation p r o c e s s i n Fig . 1 , w e w i l l introduce b e l ow a new t e chnique involving the u s e o f a two-p o l e , al l-p a s s adap t ive fi l t e r . The p o le s l i e on a fixed radius within the unit c ir c l e , and a s ingle adap t ive we ight i s used t o adj u s t the ang le o f the poles t o as c e rtain the re lat ive phase b e twe e n the two s ig­nals . We w i l l d e s cribe the e rror surface as s o c i a t e d with this I I R s t ru c ture and de rive a n LMS-type o f a l go rithm f o r adap t i n g the p o le angle .

We w i l l a l s o d i s cu s s the " ad ap t ive de lay e lement " i n t ro du c e d b y E t t e r ( 1 9 8 1 ) a s a means fo r synch roni zing two h igh ly corre lated s i gnal s .

244 R.A. David and S . D. S tearns

Using thi s t e chnique , the de lay e st i ­mate i s up dated a t e ac h . t ime s t ep v i a a gradient s e arch a lgorithm . Al­though the adapt i ve de l ay e lement it­s e l f al lows on ly an i n t e gral numb e r o f de l ay unit s , the me thod h a s b e e n extende d t o p rovide a c ontinuous de­l ay e st imate .

Computer s imu lations were perfo rmed us ing e ach of the t e chniques de s c rib­e d

-above t o generate the adapt ive de­

l ay e s t ima{;e for the mode l in Fig . 1 . The re sult s o f thi s e xpe riment were c ompare d on the b as i s of de l ay e s t i ­mate accurac y , rate o f convergenc e , and ab i li t y to t rack t ime-varying phase shift s .

x s i n ( y k ) + n

1 ( k )

s in ( y k + e ) + n2

( k )

Adap t ive D e l ay

E s t imat i on

+

F i g . 1 . Bas i c s t ruc ture for adap t ive t ime de lay e s t imat i on .

ADAPTIVE DELAY ESTIMATION US ING THE ALL-PASS F I LTER

For t h i s d i s c u s s i on we wi l l u s e the general b l o c k d iagram of Fig . 1 with an APF ( a ll -pas s f i l te r ) ne twork p ro­viding the adaot ive de lay e s t imate . The t ran s fe r fun c t i on o f the s e c ond order APF is given b y

H ( z ) ( 1 )

whe re

for a l l w ( 2 )

( 3 )

In thi s app l i c a t i on , we use a c on s t ant

b 2 =r2 whe re r is the p o le rddius in

Eq . 1 . The s ingle adap t ive c o e f fi­c ient , b

1, i s adj u s t e d t o minimi ze

the mean- squared output e rror , and an e s t imate of the r e l at ive pha s e s h i f t b etwe en x 1 and x 2 i s c ompu t e d

from Eq , 3 . H e r e w e a s s ume knowledge of the i nput line frequency , y . An e s t imate of thi s frequency is eas i ly ob tained by imp l ement ing the s y s tem i l l u s trated i n Fig . 2 , where the ALE ( adapt ive l ine enhance r ) is a two­p o l e f i l t e r whi ch forms a re s onant p e ak at the inc oming s igna l fre quency ( David , 1 9 8 3 ) . The ALE provides an

� s t imate o f y and a l s o imp rove s the input S NR for the TDE ( t ime de l ay e s t imat ion ) proce s s .

F i g . 2 .

ALE

!'!( z )

s ame transfer fun c t i on

ALE

'!le z )

Adap t ive t ime de lay e s t i ­mat ion u s i ng the APF .

Performanc e Fun c t ion Analys is

Here we use the s t ructure i l l u s t rated in Fig . 1 , with the t rans fer funct ion d e fined i n Eq . 1 providing the adap­t ive de l ay e s t imat e . In the t rans ­form domain ( argument z omit t e d for notational simp li c it y ) , we c an write

where s 1 = s inyk and s 2= s in ( yk+ e ) .

For this analy s i s we w i l l a s s ume equal i nout SNRs with ¢ = ¢ s = · s l

sl s 2 2

( 4 )'

¢ S S 2

and whi t e n o i s e with ¢ = ¢ -n ln

l n 2n2 crn ' Us ing Eq . 2 and the definit ions

above , we have

( 5 )

whe re

2 -j� ¢ ( w ) = rr cr o ( w-y ) e y • ( 6 )

s 1s

2 s

2 2 De fining S =SNR=cr /cr and u s ing the s n re l at ionship

Time Delay Estimation and Tracking 245

together with Eq . 4 - 6 , we c an derive the fol lowing expre s s ion for t he nor­mal i z e d error s urface :

� - input power ,!, { 1 + ( 8 )

2 0 ain 2[i - tan-�( l:�:��::�:��� The plots in Fi g . 3 i ll u s t rat e the shape of this performan c e funct i on for different values o f b 2 ( i . e . dif-

ferent pole radi i ) with 8 = 1 and a fixed de lay o f 6 = 3 radians . Note that as the p o l e s move t oward the unit c ircle the gradient b e c omes in­creas ingly nonuniform , thus making the s ur fa c e more difficult to s earch us ing s t eepest-des cent t e chnique s .

"' <IJ :E � t .8 N "" ,-< "' E 0 "'

-2 -1 a 1

Adaptive Coefficient b 1

Fig . 3 . Norma l i z e d e rror surface o f APF with 8 = 1 , y = . 5 , 6 = 3 .

Phase Cons t raint s

As stated previous ly , we are inter­e s t e d in measuring relat ive phase shifts ranging from - TI to TI radian s . The range o f de lay s achievab l e with the two-pole APF i s , however , s ome­what more r e s t r i c t e d . To show thi s , we wil l de fine

( 9 )

and rewrit e Eq . 3 as

tan(!) ( lo )

Using the s t ab i lity const raint l b 1 I <

l+b 2 , we obt ain

ASCSP-1

r ange [- t an� =�::� [t an� , t an(;+�� ( 11 )

We note that as b 2 appro ache s zero ,

the l imi t s o f - ¢ are approximate ly w and w + TI radi ans . The p l o t s in Ffg . 4 i l lus t rate the d e l ay s achievab l e for di�ferent radii with w = y = . 5 radian s .

Adaptive Coefficient b 1

Fig . 4 . De lays achievab le with y= . 5 i n a two-p o l e APF .

To c over the p o t en t ial d e l ay range from - TI to TI radians , we imp l ement the s ys tem i l l us t rat ed in F ig . 5 . The value of 6 i s c omput ed t o provide a de l ay of approximat e l y 1 / 2 c y c l e . Note that in many c a s e s the o p t imal d e l ay wil l b e within the achievab l e range o f more than o n e APF , thus pro­vi ding redundant result s . Whi le it has not been proven analyt i c al ly , we c an show exp erimentally that if the opt imal d e l ay is not achievab l e by an APF , the adap t ive c o e f f i c i ent t ends t o lock onto a v alue near the s t ab i l­i t y l imi t . Thi s provi d e s a s impl e c ri t e rion t o e s t ab li s h t h e validlty o f the de lay e s t imat e .

Fig . 5 . APF s y s tem for t ime de l ay e s t imation .

246 R .A . David and S . D . Stearns

APF Adapt ive Algorithm

Us ing the s i gnal definit ions of Fi g . l and the APF t rans fer fun c t i on from Eq , 1 we c an wri t e the fo l l owing equat ions :

y C k ) = b 2x 2 C k ) + b 1 C k ) x 2 C k- l ) +

x 2 C k- 2 ) - b 1 C k ) y C k- l ) - b 2y C k- 2 )

( 1 2 )

d k ) x 1 C k ) - y C k ) . c 1 3 )

We wil l u s e the NRLMS algorithm C Davi d , 19 8 3 ) and expr e s s the coeffi­c ient update as

b C k 1 ) b C k ) + P E C k ) a C k ) C l 4 ) 1 + = 1 ¢ ( k )

where p i s a parame t e r c ontro l l ing the rate of c onvergence , E C k ) is de­fined b y Eq . 1 3 , a C k ) i s the p art ial derivative

a C k ) = a y C k )

� x 2 C k- l ) - y C k- 1 )

C l 5 )

- b 1 C k ) a C k- l ) - b 2 a C k- 2 ) ,

¢ C k ) i s a norma l i z at ion fac tor de­fined by

¢ C k ) = v ¢ ( k- l ) + a2 C k ) , C l 6 )

and v i s a " forge t t ing fac t o r " in the range O <v < l . S imul at ion re sul t s wi l l be u s e d b e l ow t o demons trate the per­formanc e o f this algorithm .

ADAPTIVE DELAY E LEMENT

A de t a i l e d devel opment of the a l go­rithm for the adapt ive d e l ay e l ement c an be found in E t t e r C l9 8 1 ) . Here we state the algorithm in t e rms of the notat i on in F i g . 1 where d i s now t he adapt ive d e l ay e s t imat e :

d'C k+ l ) = 8C k ) + ( 1 7 )

µ C x1 C k ) -y C k ) ) C y C k- l ) -y C k+ l ) )

where y C k ) = x 2 C k- i ) , i i s t he i nt e ­

ger p o r t i o n o f dC k ) , and µ i s a con­vergence p arame t er s imi lar to p in Eq . 1 4 .

EXPERIMENTAL RESULTS

A c omputer s imulation invo lving both s t at ionary and nonst at ionary delays was used t o evaluat e the performanc e of the two algorithms d i s c u s s e d ab ove . The results are p rovided in Fi g . 6 . The pha s e shift from the APF a l go­rithm has b e en trans lat e d into a t ime qelay to enab l e a dire c t compari son o f c onvergenc e c harac t eri s t i c s . The ac tual de lay , which range s from 3 to 5 t ime s teps C l . 5 to 2 . 5 rad i ans for y = , 5 radians ) , is p lo t t e d together with the adapt ive delay in each c ase . The c onvergence parameters were ad­j u s t e d experiment ally in an att empt to provide opt imal c onvergence char­acteris t i c s . It is apparent that b o t h algorithms converge rapidly and are ab le to t rack the nonstationary de lay . To provide an overal l measure of e s t imat e ac curacy , the MSE was com­puted by averaging C d-d) 2 over the expe riment . The MSE o f the APF algo­ri thm was app roximat e ly 1/3 that of the adapt ive d e l ay e l ement in this s imulat ion .

250 680 750 Data Sample

1 000

a ) APF algorithm with p = . 5 , v = . 9 7 5 C MSE= . 1 3 3 )

� p. w .µ "' w E .... .µ � .... � "' '""' w "'

e 268 see 760 1 eee Data Sample

b ) Adapt i ve delay e lement with µ = , 0 4 C MSE= . 3 9 8 )

Fig . 6 . Conv e rgenc e result s from experiment with input SNR= l , y = . 5 rad i ans .

Time Delay Estimation and Tracking

CONCLUSI ONS

The adap t ive algorit hms dis cus s ed in this paper are s imp le to impl ement , provide a de l ay e s t imate at e ach iteration , and are c apab l e o f tracking non s t at ionary delay s . The adap t ive de lay e lement has s ome advantage s s ince i t i s simp ler and does not re­quire a priori know ledge o f the i nput s ignal frequency . In addition , a s ingle e lement c an adapt to any un­known delay within the range of ± 1/ 2 cycl e . The APF algorithm doe s , how­ever , appear to provide a more a c c u­rat e de lay e s t imat e .

Whi l e the re s u l t s presented in this paper are not c ons i dered t o b e c on­c lus ive , they indi c a t e that both algorithms should b e use ful in app li ­c at ions where a continual e s t imate o f the t ime de lay b e twe en two narrowb and s ignals is require d ,

REFERENCES

Bradley , J , N . and R . L . Kirlin ( 1 9 8 2 ) , De l ay Est ima t i on Simu lat ions and a Norma l i z ed C omp ari son of Pub ­l i shed Resul t s . IEEE Trans . on ASSP , Vo l , ASSP- 3 0 , No . 3 , 5 0 8-5 1 1 ,

David , R . A . , S . D . S t e arn s , G . R . El liot t , and D . M . Etter ( 1 9 8 3 ) . I I R Algori thms for Adap t ive Line Enhancement . Pro c . of the Int l . Conf , on ASSP , Vo l . 1 , 1 7 -2 0 .

Ette r , D . M . , and S . D . S t e arns ( 19 8 1 ) . Adapt ive E s t imat i on o f Time De lays in Samp led Data S y s t ems , IEEE Trans . on ASSP , Vo l . ASSP-� No . 3 , 5 8 2 - 5 87 .

Hert z , D . and J , Re i s s ( 19 8 2 ) , An Exp l i c i t E s t imat e of Time De l ay Between Two Signals with an Un­known Re l at ive Phase Shift . IEEE Trans . on ASSP , Vol , ASSP- 3 0 , No , 6 , 1006-100 7 .

Iannie l l o , J . P , ( 1 9 8 2 ) . Time De l ay E s t imat i on via Cro s s -C orre lat i on in the Pre se n c e o f Large E s t ima­t ion Error s . IEEE Trans . on ASSP , Vol , ASSP- 3 0 , No , 6 , 9 9 8- 10 0 3 .

Youn , D . H , , N , Ahmed , and G . C . Cart e r ( 19 82 ) , On Us ing the LMS Algo­rithm for Time De lay E s t imat io n . IEEE Tran s . on ASSP , Vol , ASSP- 3 0 , No . 5 , 7 9 8-80 1 .

247

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE ESTIMATOR OF A FILTER AND ITS INVERSE

W. Kofman and A . . Silvent

CEl'llAG, JU'. 46, 3N402 St\ Martin D 'llrr1;.1, Frana

Abs trac t . This paper describes the L .M . S . adaptive f il te r , wh ich compensates for the effect of a l inear f i lter on a s ignal . The adaptive f i l ter e s t imates s imul taneous ly the f i l ter response and its inverse . We show the computer s imu lations of th is sys tem app l ied to the mu lti­path transmis s ion channe l .

Keyword s . Adapt ive sys tems ; f i ltering ; l inear sys tem ; impu lse re sponse ; inver se f i lter .

INTRODUCTION

The adaptive method of s ignal e s t imation by no ise subtract ion us ing the L . M. S . algorithm (Widrow and Hof f , 1960 ; Widrow e t al . , 19 7 6 ) has been extens ive ly s tud ied and appl ied t o s tud ies of adaptive antennas , the e l imination of spurious s igna l s , echoe suppres s ion , etc . (Widrow et al . , 19 75 ; Sondhi and Berk ley , 1980 ) .

We describe the adaptation of the L . M. S . al ­gor i thm to e s t imate the impu lse re sponse of a f i lter and its inver s e .

Algor ithms

The problem that we had to solve was to com­pensate a transmiss ion channe l with an adap­tive f i l ter , which has to be connected in series and in front of the channe l . This can be mode led by the b lock d i agram shown in F ig . 1 .

x

Fig. 1 Adaptive filter W and transmission channel F

The equations describing the

Y . = x:w . -J '•, =.i-J s igna l are

Zj !_T_!:j

X . = [X . , X . 1 , . . . Xj k 1 J =J -J -J - - - +

°!TJ. = [X . , X . l , • • • • • • • , X . N 1) -J - J - -J - - ( 1 )

249

The error and the gradient of the error is

Ej = Xj -Z j = Xj - !_T

�T

j�j

V'w ( E� ) = -2Ej lj with Yj = �!:.. ( 2 )

The iterat ive bui ldup of the vec tor W is

W . l= W . + 2µE . Y ' . -J + -J J- J ( 3 )

The vec tor Yi represents the output o f the f i lter F for the input X . The quantity y ' i s not ava i lable during the actual experi­ment . Th is imp l ies that we have to e s t imate the impulse response of the f i l ter and its inverse at the same t ime . We adapted the a lgor ithm for wh ich the b lock d iagram is shown in F ig . 2 . In this b lock d iagram , F

COPY OF Wl

y•

x

y W2

DELAY p "9 l'ig. 2 Block diagram of our algorithm

repre sents the unknown f i lter . The f i l ter Wl g ives the e s t imate of the channe l and f i l ter W2 the e s t imate of i ts inver se . To e s t imate Y ' we use the e s t imated impul s e response o f Wl . E quat ion (4 ) descr ibes the adaptive process (with in itia l cond i t ions W2 ,j, 0 ) .

250 W. Kofman and A. Si lvent

z : = Wl:Y . J - J-J

Wl . l Wl . - J + - J

Y x: W2 . j -J - J E2 . X . -Z . J J -p J

W2 . l = W2 . + 2 •µ2 • E2 . Y '. - J + J J-J

( 4 )

Ver y often the inverse f ilter i s a noncausal one or not entirely causal (case of multi­path channel ) . In this case , we de lay the reference input , which permits us to obtain the noncausal part of the inverse filter . This delay serves also to compensate for the channel de lay . After the convergence , if the error E2 is small , the two f i l ters W2 and F (connected in ser ies ) can be descr ibed by the re lation

W2*F = o [ t- (p-m ) ]

where m i s a propagation delay . It should be noted that if we exchange the position of F and W , we don ' t need to estimate the f ilter response to obtain the inverse fi lter , due to the fact that we have direct access to Y ' .

Computer Simulations

We performed computer s imulations of our algorithm . The multipath channel was s imu­lated by the f ilter F (Fig . 1 ) . At the in­put X of the system , white noise was applied and the output of channel (F ) was perturbed by an additive noise B , not correlated with the input s ignal . The system parameters were

µl , µ2

p

N

the L .M. S . algorithm coeffic ients

de lay of reference s ignal

dimension of the system .

We studied the behavior o f the system a s a function of its intrinsic error , which is the difference between the input of the sys­tem and output of the corrected channel after subtracting the perturbing noise . The choice of parameter p is not very critical ; p has to be only bigger than the delay introduced by the transmiss ion channel (Kofman and S ilvent , 1982 ) . In practice , we chose p in order to center the causal and noncausal parts of the input response between N points of the trans­verse filter W2 .

In F ig . 3 , the spectra of s ignals at differ­ent points of the system are shown . The spectra were obtained by averaging 10 points of the power spectra . This figure shows the effect of the channel on the s ignal and how the system compensates for this effect .

In F igs . 4 and 5 are shown the temporal be­haviors of the system for the case of a two­path channel with d ifferent s ignal-to-noise ratios . F ig . 4a shows the channel impulse response estimate and 4b its invers e . In F ig . 4c is shown the temporal evolution of instantaneous error . We can see (Fig . 4 and

0.3

0.2

1 .0

SPECTRUM OF X

SPECTRUM OF Y WITH COMPENSATION 0.3

0.2

RELATIVE FREQUENCY - Hz

SPECTRUM OF Z WITH COMPENSATION

Fig. 3 Spectra of signals at different points in the system

.. z 0 � 0

t:

"' � -1

I I W1 AFTER 15,000 ITERATIONS

� DIRECT FILTEA --W1

lal ::; 1 �-----------�

�

W2 AFTER 15,000 ITERATIONS

� 0 1--��--"'"n-'"'-rr"'-���--l ;!!

INVERSE F I LTE R - - W2 lb}

-• '---------''-------'-' 0 50 100

le} 5,000 10,000

NUMBER OF ITERATIONS

µ1 = µ2 = 10-3 N • 100 p = 50

A1 ,. 1.0 A2 = 0.5 � = +14 dB

Fig. 4 Temporal behavior of the system, + 14 dB

� -1

W1 AFTER 15,000 ITERATIONS

- - -

(•}

W2 AFTER 15,000 ITE RATIONS

lb} _, ._ _____ .__ _____ .._. 50 100

(c} 5,000 10,000

µ1 "' µ2 • 10-3 N • 100 p • 50

� : 6:� � · O dB

Fig. 5 Temporal behavior of the system, 0 d B

Adaptive E s timator of a Fil ter 25 1

5 ) that the output no ise increases when the s igna l - to-noise rat io decreases .

We s tudied the performance of our adaptive sys tem as a func t ion of s igna l - to-no ise ratio for a two-path channe l .

PE/Px represents the mean intr ins ic error normalized to the s igna l power . This error was calculated by averaging 1000 points of the E2 (Fig . 1 ) output taken when j > 4 /µ 2 . F ig . 6 shows th is error as a func tion of no ise-to-s ignal ratio and the convergence parame ters . The error is re lative l y insen­s it ive to µ l .

_µ_1 _ _ µ_2_

• 1.25 10-4 1.25 10-4

• 2.5 10-4 2.5 10-4

• 5 10-4 5 10-4

6 1 10-3 1 10-3

• 5 10-3 1 10-3

0 5 10-4 2.5 10-4

10-l 0 2.5 10-4 5 10-4

10-3 �---�-�-�--'---'---'--"'---'--'-------'

Fig. 6

0.1 0.5 PB/Px

I ntrinsic error as a function of noise to signal ratio

For the se reasons we s imu lated the s imp l i f ied algor ithm in which we rep laced El , E2 , Y , Y ' in the calculat ion of the we ight vectors Wl , W2 , only by the ir s igns . The mos t interes t ­ing results are shown i n F ig . 7 , and one can see very sma l l degradat ions of the perform­ances when El , Y , and Y ' were rep laced by the ir s igns ( Schwar tz and Malah , 1 9 79 ) .

CONCLUS IONS

In th is work , we tested an adaptive sys tem that compensates the transmi s s ion channe l . The computer s imu lat ions showed that in spite of two s imul taneous e s t imate s of the impul se response of the f i l ter and its in­verse , the sys tem converges rapidly and i s relative ly insens it ive t o the value of the convergence coe f f ic ient µ l . We f ind a power­law dependence be tween the error and no ise­to-s ignal rat ios and almost l inear depend ­ence between the error and convergence co­eff ic ient s (µ l = µ2 ) . The re sul ts obtained with the s imp l i f ied s y s tem , in which we use the " sign" function in the algorithm , show the s teps for bui ld ing the e lec tron ic devices .

E1 x Y E2 x Y'

Sign (Et) x Sign (Y) E2 x Sign (Y')

NUMBER OF ITERATIONS 1o3

E1 x Sign (Y) E2 x Sign (Y')

2 3 1o3 NUMBER OF ITERATIONS

µ1 = 3.10-3 µ2 .. 10-3 N • 100 p .. 1 3

A1 = 1 B .. 0 A2 = 0.5

Fig. 7 Results of the simplified algorithms

REFERENCES

Kofman , W . , A . S i lvent ( 1982 ) , U t i l isation de l ' algorithme L . M . S . pour identif ier s imultanement un f i l tre e t son inver s e Rapport CEPHAG n • 46 /82 .

Schwar t z , T . , and D . Malah ( 19 79 ) , Hybrid rea l i s at ion of an adaptive f i l ter for rea l - t ime no ise- canc e l l ing app l ications , E le c tronics Letters , Vo l . 15 , n ° 21 .

Sondh i , M . M . , and D . A . Berkley ( 1980 ) , S i lenc ing echoes on the telephone ne t­work , Proceed ings of the IEEE , Vol . 68 , n° 8 .

Widrow , B . , J . R . G lover , J . M . McCoo l , J . Haunitz , C . S . Wi l l iams , R . M . Mearn , ( 19 75 ) , Adapt ive noise cance l l ing pr in­ciples and app l ications , Proceed ings of the IEEE ; Vol . 63 , n° 12 .

Widrow , B . , and M . Hoff , Jr . ( 1960 ) , Ad apt­ive switch ing c ircuits , IRE WESCON , Conv . Rec . , pt 4 , pp . 96-104 .

Widrow , B . , J . M. McCoo l , M . G . Lainmore , and C . R . Johnson , J r . ( 19 76 ) , S tation­ary and non-s tat ionary l earning charac ­ter i s t ic s of the L . M . S . adap t ive f i l ter , Proceed ings of the IEEE , Vo l . 64 , n ° 8 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROBUSTNESS OF ADAPTIVE CONTROL

ALGORITHMS 2

EFFECTS OF MODEL STRUCTURE, NONZERO D.C.-VALUE AND MEASUREABLE DISTURBANCE

ON ADAPTIVE CONTROL

D. R. Yang* and W.-K. Lee**

*Department of Chemical Engineering, Korea Advanced Institute of _Science and Technology, Seoul, Korm

**Department of Chemical Engineeriug, The Ohw State Umverszty, Columbus, Ohw, USA

Abstract . This paper is concerned with the sensit ivity of d iscrete-time adaptive control algorithms to the model order and t ime delays , nonzero d . c . ­values and measurable disturbances . In the s imulat ion study the adapt ive control system was relat ively insensit ive to the model order , but the under­estimated maximum delay failed to give sat isfactory performances . The use of d . c . -value est imat ion method was found to be more effective in the MIMO system for accommodat ing nonzero d . c . -value than in the SISO system . Feed­forward adapt ive control system performed well despite the measurable d is­turbance but it required a precise knowledge of t ime delays between distur­bance and outputs . On the whole the simulation results indicate a general robustness of adapt ive control . Keywords . Adaptive control ; delays ; linear systems ; models ; sensitivity analys is .

INTRODUCTION In recent years many different adapt ive con­trollers have been proposed to control pro­cesses with unknown and constant or slowly time-varying parameters . These various schemes differ in the manner in which pro­cess ident if icat ion and adaptat ion are car­ried out . Two approaches seem to attract considerable research interest , namely , the model reference adaptive control (MRAC) and the self-tuning regulator (STR) . As has been indicated by Landau ( 1982) as well as , among others , Shah and Fisher ( 1 98 0) , and Egardt ( 1980) , there are close similarit ies between the two approaches leading to their possible unification (Landau , 1 982) . With the availability of very inexpensive but powerful microprocessors in the last few years , new possibil ities for the industrial applicat ion of adaptive control have been opened up .

However , several problems l imit the applica­t ion to real processes : a prec ise knowledge of the model structure is required a pr iori . Those two approaches have been found to work well when parameters of the model structure are known , but fail to give stable control performance when there are modell ing errors . To obtain conf idence in pract ical appl ica­t ion of adapt ive control , it is necessary to examine how sensitive adapt ive control sys­tem is to parameters of the model structure . In des ign of a discrete-t ime adapt ive con­trol system, the process is assumed to be

ASCSP-1* 253

represent ed by a l inear , f inite-d imensional discrete-time model in which an upper bound for the order of each polynomial and t ime delays should be specif ied a priori . There are c ircumstances where the model order could be set to a lower value th�n that of the pro­cess to simpl ify the resulting control law or due to imprecise knowledge of the process . In smoe cases the process order may vary de­pending on operat ing conditions . Time delays are usually known , at least in cases where the sampl ing per iod is not too small . In many cases , however , it is difficult to spec­ify prec ise values of t ime delays a priori or they are t ime-variant . It would thus be of interest to have theory which gives insight into propert ies of adapt ive controllers based on model structures that may be d ifferent from the real process . Unfortunately , there are very few result s of this type available due to the d iff icult ies associated with rig­orous convergence proofs ( Goodwin and S in , 1 9 7 9 ; Astrom, 1 980) . Therefore , to tackle the problems approaches of on-line search for these structure parameters have been reported (Kurz , 1 97 9 ; Schuman, Lachmann and Isermann , l981 ) , which may be l imited to the single­input single-output systems ( SISO) due to possibly high computat ional effort . In addit ion , the real process may have non­zero d . c . -value or measurable disturbances . In process control control input s are bounded by virtue of physical constraints . Bes ides there arises a need for set t ing hard limits for theoret ical and pract ical reasons . These

254 D . R. Yang and W . -K . Lee

are al so points of great prac t ica l importance which should be taken into cons iderat ion in impl ement at ion o f adapt ive contro l .

In t his paper a d i scret e-t ime adapt ive con­t ro l algor ithm is eva luat ed via computer s im­ulat ion in such an env ironment , i . e . , in the presence o f errors in the model error and t ime d elays , nonz ero d . c . -value, measurable d isturbances .

DI SCRETE-TIME MULTIVARIABLE ADAPTIVE CONTROL ALGOKI THMS

The t heory of d iscrete- t ime adapt ive control is de sc r ibed in detail elsewher e , but can be outl ined here in its s impl e s t form . It is furt her assumed t hat the reader i s famil iar with the d isc rete- t ime adapt ive control scheme introduced by Goodwin , Ramadge and Ca ines ( 1 980) . However , their bound ednes s cond i t ion is too str ingent for some of the real proc esses to sat isfy . To tackl e this pro blem the extended metho d us ing a long-t erm pred ic tor has been proposed (Lee and Lee, 1 98 2 ) .

T t is as sumed that the dynamic behav ior of a proc ess can be represented by a d iscrete-t ime mul t i- input mul t i-output ( MIMO) , f inite d i­mensional model

A . ( q-1

) 1 + a i -1 + . . + i where a 1 q n

A ]_

- 1 b

ij b

ij -1 B ij

( q ) + q + . . + 0 1

b ij q-nB ' b ij # o , 0 n

B -1

q is a unit delay o perat or and d . . ' s are lJ t ime d elays . In this process mod el of Eq . ( 1 ) , an upper bound for the o rder of each poly�omial and t ime delays d ij 's are as sumed to be know" a prior i . Parameters o f Eq . ( 1 ) are in i t ia l l y unknown , and the adapt ive con­trol st rat egy is to ident ify t hese on- l ine and use these e s t ima tes to cons �ruc t U ( k) suc h t hat the proc ess output Y ( k) f o l l ows a referenc e output s equenc e yr ( k) in the f o l ­lowing manner .

l im Y ( k) = Yr

( k) k + 00 ( 2 )

'!'he sol u t ion o f this adapt ive control pro b l em is well e stabl ished . Goodwin , Ramadge and Ca ines ( 1 98 0) proposed d i sc rete-t ime adapt ive control al gor ithms , which ensure an overal l stab il ity o f the f eedback system , and a

boundedness cond it ion for control inputs . The ir boundedness cond i t ion i�

z �U (z ) . . • z • B1n ( z ) [ d l -dl l -1 dn -

dl n -1 ] d -d

: d -d

: # 0 n nl - 1 n nn -1

z Bn1 ( z ) • 0 0 z Bnn ( z )

where d i

for \ z \ '.'.. I d . . # o : lJ

( 3 )

This cond it ion, however , i s too str ingent for many proc ess es to sat isf y . I f a l l o f the maximum d e l a y i n each row would appear in one column in the process model , Eq . ( 1 ) , the cond i t ion of Eq . (3 ) can never be sat is­f ied and the det erminant always becomes z ero as l z \ approaches inf inity . This corres­ponds to a proc ess that has a control input giv ing larger d elays to a l l outputs t han the others .

To tackle this pro blem, an ext ended adapt ive control method ha s been propo sed (Lee and Lee , 1982) by introduc ing a concept of long­term pred ictor . A long-t erm pred ic tor in the extended adapt ive cont rol is devised to generate the f uture control inputs which make the output error z ero at any future t ime . Cons id er a mu lt i- input singl e-output ( MISO) system from the MIMO system o f Eq . ( 1 )

- 1 A . ( q ) y . ( k) + " ' +

]_ ]_

Us ing the f o l lowing polynomial ident ity

( 4 )

-1 -1 -m · J 1 S . ( q ) A . ( q ) + q 1R . ( q - ) ( 5) ]_ ]_ ]_

and

m1. :2: min d . . lJ 1 S j Sn

a pred ic t or is construc t ed for eac h Yi ( • )

y . ( k+m . ) l ]_

m . -d · 1 - 1 q l l p

i l ( q ) u

l ( k) +

m -d + i inp ( -l ) u ( k)

q in q n

-1 + R . ( q ) y . ( k) l ]_ -1

where P ij (q )

-1 - l S , ( q ) B . . ( q )

]_ lJ

( 6)

( 7 )

( 8)

S ince m . ' s .::re chosen as in Eq . ( 6) , t here may exiilit t ime l ead in Eq . ( 7 ) . I t is thus nec essary to extract the maximum t ime l ead as soc iat ed wi t h each Uj ( ' ) f rom Eq . ( 7 ) .

By d ef ining dj = max (m .-di . ) l .s i:;;:n 1 J ( 9 )

Effects of Model Structure 255

where £ . . : m . -d : . -dj . Control1tnput§ af� obtained by equat ing Yi (k+mi) to YI ( k+mi) and sub stituting Eq . (8) into Eq . ( 10)

1 . .Q,11 -1 [u1 ( �+d )]:[q

.Q,

Bln ( q )

n ln -1 u ( k+d ) q B1 ( q ) n n

1 [�l�:l:

Sn ( q-1)

-1 Rl ( q ) yl ( k)) ] -1 Rn ( q ) yn ( k))

( 1 1 )

The adapt iv e control algor ithm b y Goodwin , Ramadge and Ca ines ( 1 980) is then seen to be a spec ial case of the extended method with mi:d i .

It can be clearly seen from E q . ( 1 1 ) that the boundedness condit ion for control inputs is changed to

for l z I .'.'. I ( 1 2 )

This cond it ion allev iat es a possible unbound­edness problem to be encountered in the con­vent ional method by properly choos ing mi ' s . In add ition, a process with d irect transmis­s ion can be dealt with in the extended method by choosing m . ' s great er than z ero while not in the convenfional method .

The implementat ion algor ithm of the extended multivar iable adapt ive control is descr ibed as follows . Def ine

I 1 n • [u . ( k+d ) , • • • , un ( k+d ) : 1 I

y . ( k) , y . ( k�l) ' · · · ] 1 1

( 1 3 )

T � T ' ' T . P . ( k) : [ a . (k) : b . ( k) J , 1:1 , • • • , n (14 ) 1 1 I 1 Using a parameter updat ing routine

P i( k-1) + Fi9 i ( k-mi) x

[ yi (k) -PiT ( k-1 ) 8i ( k-mi) ]

[ 1 +e . T ( k-m . ) F . ( k) 8 . ( k-m . ) ] ( 15)

1 1 1 1 1

F . (k) 9 . ( k-m . ) 9 . T ( k-m )F . ( k) 1 1 1 1 1 1 HS T ( k-m . ) F . ( k) 8 . ( k-m . ) i 1 1 1 1

l ( 1 6)

where o < A�l , F . (o) > O , and P i (o ) : P i , control input veftor U ( k) can be detergined by

U ( k)

A 1' l bl (k) V1 ( k)

- b T (k) V ( k) n n ,

SIMTJLA1'ION RESULTS A�m DISr,usSION

( 1 7 )

S ince discrete-t ime adaptive control algo­rithms in both convent ional and extended methods require prec ise knowledge of the mod­el o rder and t ime d elays a priori for t heir impl ementa t io n , it is nec essary to examine the sensitivity of adaptive control to these design paramet ers . I n addit ion , a real pro­c ess may have measurable disturbanc es or non­z ero d . c . -values . And also contro l inputs are bounded by virtue o f physical constra ints . A digital computer s imulation was used to investigate the effects of different des ign parameters and to study possible methods for mea surable d isturbanc es , nonzero d . c . -values and bounded control inpu t s , respect ively . A process of 2 x 2 MU�O system has beep. select­ed in which the s econd con trol input shows larger t ime delays to a l l outputs than the f irst one .

-1 -2 [( l+0 . 2q -0 . 24q ) 0 J -1 -2 0 (l-0 . 2q - 0 . 3 5 q )

[yl ( k)] y2 ( k)

[

-1 - 1 - 3 -1 ] q ( 4+q ) q ( l+O . Sq ) - -2 -1 -4 -1 q ( 2- l . 4 q )q ( l . S+0 . 6q )

[ u1 ( k)J u2 ( k)

+

q (-1 .0+0 . Sq - 0 . 0Sq_ v ( k) [

- 3 -1 - 2 )] q-4 ( 1 . 0-0 . 7q-40 . 0lq 2 )

256 D . R . Yang and W . -K . Lee

-1 -2 + ( 1 . 0-q + 0 . 25q ) [w1 ( k)]

w2 (k) ( 18 )

where v ( k) i s a PRBS with magnitude added as measurable disturbanc e and a sequence of pseudo-random numbers ro mean and c ovarianc e of 0 . 4 5 used

of 0 . 5 w i ( k) is with ze­as no ise .

This process is a stable and minimum phase system, and is then to be adj usted to track a g iven refer ence model

- 1 -2 r (4-4q +q ) yl (k)

-1 - 2 r (8-8q +2q ) y2 ( k)

-3 r - 1 r q ( 2 . �+q ) u1 ( k)

-4 -1 r q ( 2 . 5-q ) u2 (k) ( 1 9 )

using discrete-time adapt ive control scheme . The referenc e inputs , uf ( k) and uz ( k) , are var ied in the magnitude between 0 and 1 in a stepwise manner with an interval of 3 0 samp­l ing steps for u1 r ( k) and 25 sampl ing steps for u2r (k) , respec t ively . The extended dis­crete-time adapt ive control scheme is appl i­ed with an initial value of ga in matrix o f 1 00 and constant fo rget ting factors of 0 . 99 . Contro l act ions are hounded to l ie between -3 . 0 and 3 . 0 .

The results are shown in Fig . 1 which com­pares the performanc es for deterministic and stochastic cases in the absenc e of measurable d isturbanc e and nonzero d . c . -value . Perfect tracking performanc es are obta ined for both outputs with no apparent intera c t ion prob­lems , which should he taken into considera­t ion in other multivariahle control schemes .

Ef fect of Model Order

The effec t of errors in the model order was invest igated using three different sets o f upper bounds for the order of po lynomial s , nA = 4 and DB = 2 , nA = 2 and nB = 0 , and nA = 0 and nB = 0 while the upper bounds for the real proc ess model are nA = 2 and nB = 1 .

The s imulat ion resul ts are shown in Figs . 2-1 and 2-2 where the responses of Yl were not affec t ed at all , and the tracking performanc e of Y2 wa s degraded sl ightly hut still sat is­factory us ing a model of lower order . It is not eworthy that better responses of the d is­crete-t ime adapt ive control ler were obta ined in the MIMO system us ing a lower-o rder model than in the S ISO system . This ind icates that the adapt ive control scheme is less s ensitive to the ac curacy o f a model o rder in the MIMO system than in the SISO system partly because larger number of paramet ers i s used in the MIMO cas e . This o bservat ion is of substant ially prac t ical s ignif icanc e because it supports the use o f s impl e models (Astrom, 1 98 0 ; Kershenbaum and Yd stie, 1 98 1 ) .

Effect of Time Delays

In the real app l icat ions of adaptive control schemes , it is nec essary to have a prec ise

knowledge of t ime d elays in the process a prio r i . These schemes have been found to work well when the t ime d elays are constant and known . However , c lo sed-loop instabil ity can result when the t ime d elays are unknown or varying (Vogel and Edgar , 1 980) . Figures 3-1 and 3-2 show the tracking performance of the outputs for the mod.el with four d iffer ent sets of values of t ime delays which are cl if­f erent from the true values , a1 1 = 1, a1 2 = 3 , d1 2 = 2 and d22 = 4 .

In the case where there is no error in the maximum delay between inputs and each output with an overest imated delay between Y2 and u1, the responses are found to be very close to those with the prec ise knowledge as illust ra­t ed in Figs . 3-l ( a) and 3-2(a) . Figures 3-l (b) and 3-2 (b) compare a sl ightly poor trac king performance of Y2 to a perfect track­ing of Yl when the t ime delay between Yl and u2 is underest imat ed which corr esponds to having an underest imated maximum delay be­tween Yl and inputs . However , the adapt ive control system completely lost a tracking power as shown in Fig s . 3-l (c ) and 3-2 ( c ) when the maximum delay of the process is un­�erestima t�d . When the maximum d elay in y2 is overest imated (d22 = 5) , a poor response of Y2 was obtained while y1 tracked the ref­erence output perfectly with an initial poor r esponse .

It is clearly shown here that the c lo sed-loop response does not achieve the desir ed perfor­mance when the maximum delay of the process is underestimated . In any case, t ime delays should no c ircumstanc es be underest imated but may be overestimated . A prio r i prec ise know­ledge o f time delays is required in des ign of adaptive controller to achieve its sat isfac­tory performanc e . This requirement l imits its real appl icat ions . To tackle this prob­lem several t echniques for estimat ing model paramet ers inc lud ing t ime delays have been proposed . �any of the techniques were r eport­ed to be unsat isfac tory either due to poor performance or exc essive computat ional re­quirements (Vogel and Edgar , 1 980) . Part icu­larly Kur z ( 1 97 9 ) proposed an on-l ine search procedure for the t ime delay , and Schumann , Lachmann and I s ermann ( 1 981) extended this on-l ine method for both model order and t ime delay . However , its extens ion to the MIMO system may not be f ea s ible due to high compu­tat ional effort and because it still requires provid ing bounds for model order and t ime de­lays a priori .

Nonzero D . C . -Value

In parameter est imat ion algor ithms the var ia­t ions of the process input and output signals have to be used . However , there are some cases in which their d . c . -values are not known a prio r i or t ime-varying . I n these cases they have to be estimated . This can be done with inclusion of a bias term, cdc • in �q . ( 1 8 ) followed by r edef ining Eqs . ( 1 3 ) and ( 14 ) (Kurz , I sermann and Schumann , 1 980)

E ffects o f Model Structure 2 5 7

T T [U ( k) , Vi (k) , l ]

[ a . T ( k) , b . T ( k) , cT ] 1 1

( 1 3 I )

( 14 ' )

where c is a parameter for estimat ion of Cdc ·

To study the effec t o f nonzero d . c . -value , the model , Eq . (18) , is used without mea sur­able disturbance and noise t erms r -1 -2 J (l+0 . 2q - 0 . 24q ) 0

-1 - 2 0 ( l- 0 . 2q - 0 . 3 5q )

-1 -1 -3 -1 [yl ( k)] =[q ( 4+q ) q (l+0 . 5q ) ]

x -2 - 1 -4 -1 y2 ( k) q ( 2- 1 . 4q ) q ( l . 5+0 . 6q ) [u1 ( k)] + [( 0 . 2+0 . 00lk) J u2 (k) ( 0 . 1 5 + C . Olk)

(20 )

As can be seen from Eq . (2 0 ) , steady- state values o f the proc ess outputs are b iased and furthermore , t ime-vary ing . Figure 4 shows a sl ightly poor response o f the proc ess outputs when the nonzero d . c . value wa s not taken in­to considerat ion . However , F igure 5 illus­trates the improved responses with incorpo­rat ion of on-l ine estimat ion method for d . c . ­values . I t is not eworthy that inclusion of d . c . -value est imat ion in adap t ive control schemes proved to be more effec t ive in the MIMO system than in the SISO system . Al­though there may be a sl ight increas e in computational t ime , a cons iderably improved response can be obta ined by applying d . c . ­value estimat ion method when there are un­c erta int ies in the s teady-state values o f the process input and output s ignals or t ime­vary ing d . c . -values .

Measureable D isturbances

The ef fect of measurable disturbance on the performance of adap t ive control is illustra­t ed in Fig . 6 where poor t racking perfor­manc es were obta ined . Figure 7 , however , shows clearly the improved responses with addit ion of feedforward approach .

In implement ing the f eedforward adapt ive con­trol t ime delay between Yl and v correspond­ing to dvl = 3 is used and dv2 = 4 between Y2 and v . The s ensit ivity o f the f eedfor­ward adapt ive control system is illustrated in Fig . 8 suggesting that t ime d elays be­tween the pro c es s output s and mea surable d isturba nces should no c ircumstances be under or overestimated .

CONCLUSIONS

A d i screte- t ime mult ivariable adapt ive con­trol scheme has been evaluated v ia d ig ital computer s imulat ion to d etermine its s ensi­t ivity to the cho ic e of the model o rder and t ime d elays , and to tackle the eff ec t s o f

nonz ero d . c . -values and measurable d istur­bances . In the s imulation study the ad�pt ive control system performed well in bot h deter­ministic and stocha s t ic cases with a perfect knowledge of the proc ess model . And the a­dapt ive control system was abl e to achiev e sat isfactory t racking performanc es even if the model order used wa s lower t han that of the controlled proc ess . However , it failed to g ive sat isfactory responses when the maxi­mum d elay of the process was underest ima t ed . : t is clear from this result that a prec ise knowledge of t ime delays is of the mo st s ig­n if icance to achieve the desired performance .

Nonz ero o r t ime-varying d . c . -value was found to sl ightly affec t the performanc e of adap­t ive control , but incorporation of a d . c . ­value est imat ion method markedly improved the control performanc e in the �'IMO system with­out caus ing much increase in computat ional ef fort . The addition o f f eedfo rward adap t ive control proved to be ef f ec t ive at min imiz ing the eff ec t of measurable d isturbance but it required the prec ise knowl edge of t ime d elays present between measurable d isturbanc e and putputs to be eff ect ive .

. REFERENCES

Astrom, K . J . ( 1 980) . Self-tun ing control of a f ixed-bed chemical r eactor system . Int . J . Control , 3 2 , 221-256 .

Egardt, S . ( 1 980) . u;-if icat ion of some d is­crete-t ime adapt ive control schemes . IEEE Trans . Autom. Contr . , AC-2 5 , 693-69 7 .

Goodwin , G . C . and K . S . S in ( 19 79).�Effec t of model , system and contro ller order on adapt ive control . Proc . 18th IEEE Conf . on Dec i s ion and Control , Fort Lauderdale , 202-205 .

Goodwin , G . C . , P . J . Ramadge, and P . E . Ca ines ( 1 980) . D iscret e-t ime mul t ivar iable a­dapt ive control . IEEE Trans . Autom. �ont r . , AC-2 5 , 449-4 5 6 .

Kershenbaum, L . S . , and B . E . Ydstie ( 1981) . Implementat ion o f adapt ive self-tun ing controller s to large p ilot scale proc ess systems . 74th Annual AIChE meet ing , New 0rl eans .

Kurz , H . ( 1 9 7 9) . Digital paramet er-adapt ive control of proc esses with unknown con­stant or timevarying deadt ime . 5th IFAC Sympos ium on Identificat ion and System Parameter Estima t io n , Darmstadt , Pergamon Pres s , Oxf .

Kurz , H . , R . I sermann , and R . Schumann ( 1 98 0) . Experimental comparison and appl icat ion o f various paramet er-adaptive control algo­rithms . Automatica , 1.2_, 117-1 3 3 .

Landau , I . D . ( 1 982 ) . Combining model refer­enc e adapt ive controllers and s tocha s t ic self-tuning regulators . Automa t ic a , l� 7 7-84 .

Lee , K . S . , and Won-Kyoo Lee ( 1982) . Extended discrete- t ime mult ivariable adaptive con­trol u s ing long-t erm predictor . Int . J . Control ( to be publ ished) .

Schumann , R . , K-H Lachmann and R . I sermann ( 1 98 1 ) . �oward appl icabil ity o f param­et er-adapt ive control algorithms .

2S8 D . R . Yang and W . -K . Lee

Preprints of 8th I FAC Congres s , ]_, 22-28 . Shah, S . , and D . G . Fisher ( 1 98 0) . Algorthmic

and structural s imilarit ies between model reference adapt ive and self-tuning control­lers . Proc . 1 98 0 JACC .

Vogel , E . F . , and R . F . Edgar ( 1 980) . An adap­t ive dead time compensator for process control . Proc . 1 98 0 JACC .

,,Jf �Model -4 - plant -6 ' '

y,j��-�� -4�• , I �. , I �. -6'-'-'----'---'--'--'---'--'---'-.J_"---''--'--'--'---'--'--'--....____J

(a)

T ime (b )

0 00

Fig . 1 . Well-designed MIMO adapt ive control result s : (a) deterministic case ; (b) stochastic case .

J� _4 plant

- gr ( h ) Y1

-( �

-4 !ant -6 ' ' ' 6 ( c ) 4 2

so 1 00 Time

ISO 200

Fig . 2-1 . Effec t s of error in the model order on y1 : (a ) nA=4 , nB=2 ; (b) nA=2 , nB= O ; ( c ) nA= O , nB=O

6� 4 (a ) 2 0

_2 Mod el

=i� 6 (b ) 4 2 Y2 0 �-· ·

_2 \ �Model �plant

c

so 100 T ime

150 200

Fig . 2-2 . Effec t s of error in the model error on y2 : (a ) nA=4 , nB=2 ; (b ) nA= 2 , nB=O ; ( c ) nA= O , nB=O

Y1 ( c )

0 -2 M -4 p -6

6 ( d ) 4 2 0

-2 -4 -6 0 200

Time Fig . 3-1 . Effec t s of error in the t ime d elay

on y1 : (a) d . . =l , 3 , 3 , 4 ; (b) d . =1 , 2, l.] iJ

3 , 4 ; ( c ) d . . =1 , 3 , 3 , 4 ; (d ) di . =1, 3 ,3,S l.J J

Effects o f Model S tructure

J� . . -4 plant -6 .

6 ( b) 4 �-:�

Model •

Time F ig . 6 . Effec t s of measurabl e d is turbance

2 59

Fig . 3-2 . Effects o f error in the t ime delay ( a ) d . . =l , 3 , 3 , 4 ; (b) d . . = l ,

,,Jcr:vy

-4�� -���===============�================: on Y 2 : lJ lJ 2 , 3 , 4 ; ( c ) d ij =l , 3 , 3 , 3 ; (d ) d =l ij , 3 , 3 , _s

4 2 y 2 0 h-+"'-f-..,....

-2 -4 -6"--''--'__._---'-r5�0--'---'-_._�1�0�0--'-_.__..__�1rl5�0r-_._"-7;'2nr;oo

Fig . 7 . T ime

Results with f eedforward adapt ive control

6 Y1 0 4 -2 � Y2 6 -4 .. · - _jJ

l�� -2 -g :===:::::'.::::'.:::::::::::::::'.::::::::::::::=::'.::::::::::::'.::::'.:::::'.::::: � =� WLI.L...._,__....,..l,.-..__.___..__._

T,,.-:i1;-;m

,..e

--'---'-�1+5on'--

'---"---;;2"';;-o-C:o Y 2 j �' Mod el �

Fig . 4 . Effec t s of nonz ero d . c . -value -4 plant

yl 0 �� -2 �M:d -4 �

, ,]�· 1 -4 -

- 5 0 50 1 00 1 5 0 2�0 Time

Fig . 5 . Result s with on-l ine nonz ero d . c . ­value est imat ion method

-6 I I I I I I (a ) 6 �������������� .. �-.-�.

4

T ime ( b)

Fig . 8 . Effec t s o f error in the t ime d elay o f disturbanc e on f eedforward adapt ive contro l : (a) d . = 3 , 3; ( b) d .=4 , 4

V l V l

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

HOPF BIFURCATION IN AN ADAPTIVE SYSTEM WITH INMODELED DYNAMICS*

B. Cyr. B. Riedle, and P. V. Kokotovic

Corrrrlinatnl Srimff /,abomtmy, Uni1wnity of lllirwi.1, / / {) / W. Sprirt{!;(idd AvP., Urbana, IL 61801, USA

Abstract . Adapt ive schemes can exhibit a "nonlinear" instability in which the linear system with f ixed parameters is stable . This instability is a Hopf bifurcation caused by unmodeled dynamics .

1 . Introduction . A qualitative description of an instability mechanism for adaptive systems with disturbances and unmodeled dynamics was recently given by several authors [ 1-5 ] . A persistent disturbance causes a slow drift of controller gains to­ward infinity . When the gains reach some large values the stability is lost due to the effects of unmodeled dynamics ( "parasi­tics") . This is a linear instability because the system is unstable even when the adaptation is disconnected .

In this note we show that a type of nonlinear instability is also possible in which the linear system with fixed gains is exponentially stable , but the equilibrium of the nonlinear adaptive system is unstable . �fe illustrate this using a f irst order model for a second order plant , that is introducing a "parasitic" pole . When the constant ref­erence input r is as large as the square root of the parasit ic pole the equilibrium bifur­cates into a limit cycle . This Hopf bifurca­tion [ 6 ] does not necessarily imply unbound­edness because , as we show by simulation , the limit cycle can be an attractor .

In the considered example an isolated equilibrium exists because the b parameter of the plant is assumed to be known . If, instead, the de gain of the plant is known and a modified adaptation law is used , the equilibrium can be any point on a linear manifold . Then the parasitics do not cause nonlinear instabilities , but the equilibrium manifold opens the possibility for a para­meter drift and , hence , linear instability .

2 . Nonlinear Instability . In contrast to the ideal first order plant with one unknown parameter a , we let the actual plant be of second order with an addit ional pole at -fi : *

a , b > 0 m m ( 2 . 1 )

This work was supported in part by the Joint Services Electronics Program under Contract N000 14-79-C-0424 ; in part by the U . S . Air Force under Grant AFOSR 78-3633 ; and in part by the U . S . Department of Energy , Electric Energy Systems Division , under Contract DE­AC0 1-8 1RA50658 , with Dynamic Systems , P . O . Box 423 , Urbana f I L 6 1 80 1 .

26 1

ideal plant : y ay + bu ( 2 . 2) actual plant : y ay + bz ( 2 . 3)

µz -z + u ( 2 . 4) control : � -ky + r ( 2 . 5)

adaptive law : k yy (y-ym) ( 2 . 6) For brevity we let am = bm = b = 1 and , e = y-Ym· Then the equilibrium of

ae + z + (a+l) ym-r ye (e + ym)

µ z -z - k ( e + y m) + r for constant r and y: = r is

e* = O , k* = l+a , z* = -ar

(2 . 7) ( 2 . 8) ( 2 . 9)

(2 . 10) and it is exponentially stable for r < r0 and unstable for r > r 0 , where

2 r 0 1 - (l-µa) , µy µa < 1 , ( 2 . 1 1 )

_1 that is , if the signal r is O (µ �) or larger. The instability for r > r0 occurs in spite of the fact that the actual linear system with gain k* is exponentially stable . It can be shown that for r = r0 the system ( 2 . 7) , ( 2 . 8), ( 2 . 9) satisfies the Hopf Bifurcation Theorem [ 6 ] , that i s , its linearization has a pair of eigenvalues with ReA (r 0) = 0 and d

d Re). (r) t > 0 r r=r 0 ( 2 . 12 )

while the third eigenvalue is real and negative . This implies that there exists a segment [ r0 , r 1 l such that to every r E. [ r 0 , r 1 J there corresponds a unique limit cycle at a distance O ( /r 1-r0) from the equi­librium ( 2 . 10) , and of period ImA �; ) + O (r 1-r0) . We illustrate this by s iifiulation .

In Fig. la , the traj ectory for a=l , µ=0 . 1 , y=5 , r0= 1 . 34 , Ym(O) =y (O) = z (O) =k (O) =O. and r=r<r converges to the equilibrium e*=O , k*=� , while the traj ectory for r=2>r0 in Fig. lb converges to a limit cycle around the equilibrium.

3 . A !iodified Scheme . If we assume that the de gain - % of the plant (2 . 3) is known and in place of ( 2 . 5) and (2 . 6) apply the control and adaptive law

b u = (k - �)� r - ky ( 3 . 1 ) b a

k

m b

y (y-y ) (y - � r) m a m ( 3 . 2)

262 B. Cyr, B. Riedle and P. Kokotovic

then the ideal controlled plant becomes b

y = (a-bk)y - (a-bk) � r ( 3 . 3) a m and the adjustment of k affects only the pole location , while the de gain remains constant . Letting am = bm = 1 , e = y-ym , - a z = z +b r , em = ym-r ' we rewrite ( 2 . 3) , ( 2 . 4) , ( 3 . 1) , ( 3 . 2) as

.:. ]JZ

ae + bz + (a+l ) e m ye (e + em) -ke-z-kem + ]J � r .

( 3 . 4) ( 3 . 5)

( 3 . 6)

S ince the inputs are em and r rather than r itself , the system has no forced response for r = const . The equilibrium manifold of (3 . 4) , (3 . 5) , (3 . 6) for constant r is

e* = 0 , z* = 0 , ( 3 . 7) and k* is arbitrary because it does not affect the plant de gain . In contrast to the case with an isolated equilibrium, the stability of the equilibrium manifold (3 . 7) is not affected by the magnitude of r . We can prove that for all e (O) , z (O) , k (O) within the hyperboloid

- 2 2 2 2 b - 2 b - 2 V = (e+bµz) + b µ z +y- (k-k) < y (1'1-k) ( 3 . 8) where - a a . fl};--:::---1 1 k > b and l'l = \-i;- l ry µa<a- k- l ) - z-l (3 . 9) the traj ectories of ( 3 . 4) , (3 . 5) , (3 . 6) with em = r = 0 are bounded and converge to the equilibrium manifold (3 . 7) .

In Fig . 2 the traj ectory of the system ( 2 . 3) , ( 2 . 4) , (3 . 1 ) , (3 . 2) for a= l , µ=0 . 1 , y=5 , y (O) = y (O) = z (O) = k (O) = O , r=2 , ( the same v�lues as in Fig . 1 ) converges to the equi­librium manifold e* = 0 , k* > - 1 , z* = -2 . The undesirable osc illations seen in Fig . lb are eliminated . 4 . Concluding Remarks . As pointed out in [ 5 ] the existence of an exponentially stable equilibrium improves the robustness with respect to dis turbances , but we have shown that it may lead to Hopf bifurcation , a form of nonrobustness with respect to parasitic s .

References [ l ] B . B . Peterson and K. S . Narendra ,

"Bounded Error Adaptive Control , " IEEE Trans . on Automatic Control , Vol . AC-2 7 , pp . 1 16 1 - 1 168 , Dec . 1 982 .

[ 2 ] G . Kreisselmeier and K . S . Narendra , "Stable Model Reference Adaptive Control in the Presence of Bounded Disturbances , " ibid . , pp . 1 169-1 1 7 6 .

[ 3 ] C . E . Rohrs , e t al . , "Robustness of Adapt ive Control Algorithsm in the Pre­sence of Unmodeled Dynamics , " Proc . 2 1st IEEE Conf . on Decision an<fCO'Utrol, Orlando , FL , Dec . 1982 .

[ 4 ] P . Ioannou and P . V . Kokotovic , Adapt ive Systems with Reduced Models , Springer Verlag , 1 982 .

[ 5 ] B . D . 0 . Anderson and C . R . Johnson , Jr. , "Exponential Convergence of Adaptive Identification and Control Algorithms , " Automatica , Vol . 1 8 , pp . 1- 1 3 , Jan . 1982 .

[ 6 ] J . E . Marsden and M . McCracken , Hopf Bifurcation and its Applications-,-­Springer Verlag , 1976 .

-e . e -e . 4 -e . 2 e . e e . 2 e . 4 e . e e . e e

F I G , la . ASYMPTOT I C STABI LITY FOR

2 _[ __

J I I �t " I .. .. [ ' I .. ' -2 . e -1 .Ii -1 .e -e . s e . e e . s t . a e

F I G . lb . LIMIT CYCLE FOR r > r0 , s--..---�-

4-t----+-..,,,..-"""-+----i----+--�· , __ ___,,___--ii

3 _ _ _ _L __ _J 2

0-4-��.....1-��"'"4��..,..;:i:::;=.....,...,...;��� ���.-....��� -1 . 00 -lil . 75 -lil . 60 -lil.25 lil . lillil 0 .25 Iii . Se e e . ·

FIG, 2. RESPONSE O F MOD I F IED SCHEME

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

DESIGN OF ADAPTIVE TRACKING SYSTEMS FOR PLANTS OF UNKNOWN ORDER

N. Minamide*, P. N. Nikiforuk** and M. M. Gupta**

*DefHLrlrnenl of Eln:lriml l'.nginPning, Faculty of Fnginfl'ring, Nagoya University, Furo-clw, Chikwa-ku, Nagoya, .Japan

**Cylwrnl'lirs lfrsmrdi /,abomlory, Collfg!' o(Enginl'fring, Uniwrsily o(Sa.1kalclu'wan, Saskatoon, Sa.1katclu'wa11, Canada

1 . INTRODUCTION

Many successful app l ications of indirect adap­tive control schemes have been reported in the literature [ 1 -4 ] . There sti l l , however , re­mains a theoret ical difficulty in establ ish­ing a global convergence of such schemes . By an indirect approach , s ince the stab i l i z ­ing feedback compensator is designed based on the mode l ' s dynamics , the model system need be contro l lab l e , or at least stab i l i zab le asymptotical ly . What has been establ ished so far rel ies on the hypothesis that such singu­lar situations do not occur . Thi s hypothesis , however , imposes certain restrictions on the plant parameters and the c l ass of the input appl ied . In part icular , such restric­tions are essenti al when only the upper bound of the p lant order is assumed to be known . This is the case of interest here .

The present paper aims to estab l i sh a globally stable adapt ive tracking algorithm for a s in­gle- input s ingle output discrete-time l inear plant with the knowledge of the upper bound of the p lant order . In thP. modified approach proposed here , a c l ass of paral le l mode l s is first characteri zed using an adj usting param­eter vector . Extraction of the suitab le sta­bi l i zable model and determination of the feed­back gains of the pole-placing functional ob­server are then attained by solving the asso­c iated minimizat ion prob lem . Gains of a feed­forward compensator are final ly determined so that the output of the resulting closed- loop system may track a reference signal generated from a known dynamics . A global stab i l ity of the overal l control configulation is estab­l ished .

2 . PROBLEM STATEMENT

Consider a l inear time- invariant p lant having autoregress ive moving average representat ion of the form

- 1 A* (q ) yt - 1 B * (q ) ut ( 2 . 1 )

where Ut and Yt are scalar plant input and output , respectively , q-1 i s a unit de lay op­erator and A* (q- 1 ) , B* (q- 1 ) are scalar poly­nomials given by

1 - 1 -n A* ( q- ) l + a:<-q + - 1 - 1 1

+ a�q -m ( 2 . 2 )

B * (q ) = b iq + . . .

where the coefficients + b;q

{ a� } i= l , 2 , · · · , n and ]_

263

{bi} i = l , 2 , · · · , m are unknown .

The obj ectives of the control we are consid­ering are tracking and regulation : 1 ) Tracking The output of the system ( 2 . 1 ) should track a reference signal Yt * that obeys the dif­ference equation

g (q- l ) y * = 0 t where g (q- 1 ) = l + g1 q- l+ . · · +g q-s= l + g ' � ( s) is assumed to be known . Here ,s� (s ) = [q- 1 . . . q-s ] • . 2) Regulation In regulation CYt*=O) , an init ial disturbance should be el iminated with the dynamics de­fined by

- 1 D (q ) yt = 0 where D (q- l ) = l+d1q- l + . · · +drq-r=l + d ' � (r) .

In the seque l , the fol lowing assumptions on the system wi l l be used . A

1 ) The upper bound r of the p lant order r*= max (n ,m) is known .

A2 ) A* (q- 1 ) and B* (q- 1 ) are re latively prime . A3) A scalar polynomial D (q- 1 ) defining the

des ired closed loop characteristic equa­tion is stable and has at least one real root .

A4) g (q- 1 ) and B* (q- 1 ) are re lat ively prime . A5 ) g (q- 1 ) and D (q- 1 ) are re latively prime .

From assumption A1 the input Ut and the out ­put Yt satisfying ( 2 . 1 ) may be cons idered to obey the fol lowing model of order r :

A (q- 1 ) Yt B (q- l ) ut ( 2 . 3) where

A ( q- 1 ) -r + . . . + arq

B (q- 1 ) b - 1 = lq + = 1 + a ' � (r)

· · · + b q-r = b ' � (r) r Here , it i s to be noted that the pair ( a , b) satisfying ( 2 . 3) may not be determined uniquely and there may be infinite such pairs in general .

3 . A REDUCED ORDER ADAPTIVE OBSERVER

Let ¢k- l= [yk- 1 yk- 2 · · · Yk-r uk- 1 uk- 2 · · · uk-r] '

Then , we have by ( 2 . 3) yk = h ' ¢k- l (k=l , 2 , · · · , t )

where h= [ -a ' b ' ] ' and {uk} ' {yk} are data

2 64 N. Minamide , P . N . Nikiforuk and M .M . Gupta

sati sfying (2 . 1 ) . Consider the l east square adaptation scheme defined by A

pt - 1¢t- l (yt- ¢t - lht - l ) ht= ht - 1+

At- 1 + ¢�- l pt - 1¢t - l p ¢ ¢ '

p = [ I _ t - 1 t - 1 t- 1 J P p = t \ + ¢ ' P ¢ t - 1 O

t - 1 t - 1 t- 1 t - 1 where 0 < At- ls £0 .

( 3 . 1 )

Lemma 3 . 1 : The adaptation scheme ( 3 . 1 ) has the properties : 1 ) {h } and {P } are uniformly bounded conver-t t gent sequences . 2 ) For any uniformly bounded vector sequence

{ v } , 1 et t A A A A \) p

\) - h

\)' A-ht = ht + t

\)t ' yt - t �t- 1

Then , there exists {a } such that A t I Yt - Y� I s at (� + I I ¢t _ 1 1 1 ) , lim at= O

t->00 3) For any vector h E R2r satisfying p

yt= h�¢t - l (t = 1 , 2 , . . . ) there exists a uniformly bounded vector se­q��ce ivt } such that ht h + p \) = h t t t p (t=l , 2 , . . . )

With the parameter estimates generated by ( 3 . 1 ) , an implicit-type reduced order adaptive observer is now defined by

( 3 . 2 )

where A\) A\) A\) A\) A\) A\) h [ - a ' b ' ] ' a = [ a . . · a ] ' t t t ' -t 2t ' ' nt �t - 1= [yt-1 · · • Yt-r+l ut - 1 · · · ut-r+l ] '

L (� = �2 . : : an

a · O A n _1

Lemma 3 . 2 : xt obeys the di fference A\) A

xt = At- lxt- 1 + f;V + � t- 1 ut - 1 t- 1

equation ( 3 . 3)

where Av Av At ate + D , c = [ 1 O · · · O ] ' with D a shifting matrix , and �t - l sati sfies the estimate of the form

I I �t _ 1 I I s st I I ¢t_ 1 I I , 4 . ADAPTIVE CONTROL

Eq . ( 3 . 3) can be regarded as the state space representation of the paral le l model having the adj usting parameter vector Vt . Thus , if the input ut is generated by the causal state feedback control l aw

ut = - f�xt + vt (4 . 1 ) so that the characteristic polynomial of ( 3 . 3) may become

I r - q- 1 c�v - bvf ' ) I = D (q- 1 ) (4 . 2) t t t the desired pol e-pl acement wi l l be attained . Lemma 4 . 1 : There exist vectors \it and f satisfying (4 . 2) . Such vectors can be cffarac-

terized as any solution to the fol l owing min­imization prob lem :

min min X ( f ,v ; ht , Pt ) l l f l [ s M [ [ v i [ S M

where M is a constant independent of t and X ( f ,v ;ht , Pt ) is defined by A A Av X (f , v ; ht , Pt ) = [ [ H (a� , b�) f - d + at l l Here , H (a ,b ) i s an rxr symmetric matrix given by

H (a , b) = L (b ) { I + V (a) } - L (a) V (b) where lo a · · · a l 1 r- 1

V (a) = • : 0 · a

01

Theorem 4 . 1 : Cons ider the fol lowing algorithm : 1 ) Compute ht and Pt by ( 3 . 1 ) . 2 ) �et ft= ft_ 1and Vt= vt- l ' and compute \) A

ht = ht + Pt\!t A A 3) Compute X (ft ,Vt ; ht , Pt ) . I f X ( ft ,Vt ; ht , Pt)

s 8 , go to step 4) . Otherw�se , find (ft , Vt ) sati sfying X (ft , Vt ; ht , Pt ) s o /2 by applying a suitable optimi zation method .

- l A\) A\) A\) A\) 4) Compute w = H ( g , bt ) {H Cat , bt ) ft + at - g}

where zeros should be appended to make the operation l egal if necessary . Compute then kt = [w1 { -L (g_)�} ' ] ' . I f H (g , b�) is sin­gular , go to step 3) by letting o/2 ->- o ,

5) Generate the control input ut = - ftxt + k l tYt + · · · +ks /t-s o +l where s o = max (r , s ) and Xt i s given by ( 3 . 2) .

Suppose that some recursive optimi zation meth­od is avai lable for complet ing step 3) for sufficient ly smal l o . Then step 4) is feasi ­b l e . Moreover , the state �t and ¢t are uni­formly bounded , Yt approaches Yt and the co­efficient vector of the resulting c losed- loop characteristic polynomial approaches that of D ( q- 1 ) with norm error at most o, where 8 can be specified arbitrari ly smal l .

4 . CONCLUSION

Using an indirect approach , a discrete -time adaptive tracking and regulation problem has been investigated , and the global convergence of the al gorithm has been shown . Although the al gorithm requires minimization of the func­tional , the convergence analysis does not as­sume that the plant is minimum phase -type or that the p lant degree is known . The es sential assumption made is that an upper bound on the plant order is known .

REFERENCES

1 . Ph . de Larminat , "Unconditional stabi l i zer for nonminimum phase systems" , Lecture Notes 24 , pp . 54-6 3 , Springer-Verlarg , 1980 .

2 . G . C . Goodwin and K . S . Sin , "Adaptive con­tro l of nonminimum phase systems" , I EEE Trans . Automat . Contr . , vol . AC -26 , pp . 478-483 , 1981 .

3 . C . Samson , "An adaptive LQ control ler for non-minimum-phase systems" , Int . J . Contr . , vol . 35 , pp . 1 - 2 8 , 1982 .

4 . G . Kreisselmeier, "On adaptive state regu­l at ion" , I EEE trans . Automat . Contr . , 1982 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

REDUCED ORDER ADAPTIVE POLE PLACEMENT FOR MULTIVARIABLE SYSTEMS

T. Djaferis, M. Das a_nd H. Elliott*

/)1'/mrtment of l�frrtriwl and Cmn/mli'r f\'ngirtl'l'ring, Uniwnity of Mas.1athu1ett.1, A mherst, MA ()/()()3, USA

Abstract . A method i s s uggested for adapti ve ly ass i gn i n g the cl osed l oop pol es of a conti n uous t ime l i near mul t i vari ab l e system . The res u l t i ng des i gn s i g n i f i cant l y reduces the comp l e x i ty i n compari son wi th any other k nown mu l t i ­vari ab le scheme . For i mpl ementat ion one needs to know the order of the sys ­tem and an upper bound on the observab i l i ty i ndex . Keywords . Pol e Ass i gnment , Adapti ve Control *Th i s research has been s upported by AFOSR under grant AFOSR-80-01 55 , and by NSF under grant ECS-821 4534

1 . INTRODUCT ION Adapti ve pol e ass i gnment is one of the a l ter­nati ve strateg ies for adapt i ve ly contro l l i ng l i near mul t i vari abl e systems . Recent contri ­buti ons i n c l ude [ l , 2 ] . Even though g l obal s tabi l i ty has not been shown as yet for such schemes , they nonthel ess are appea l i n g due to the fact that these techn i ques a l l ow i mpl e­mentati on u s i ng poss i b l y l ess restri ct i ve apriori assumpt i ons . Th i s i s very i mportant i n appl i cati ons where a concerted effort must be made to keep the control techn i que as s i m­pl e as pos s i b l e wi thout sacri f i c i ng perfor­mance . I n many d i rect adapti ve control schemes est i ­mates of control l er parameters are fi rst ob­tai ned from i nput-output data and then used for control . I t i s therefore apparent that the order of s uch a contro l l er d i rect ly i m­pacts the s i mpl i c i ty of the control strategy . I n general any scheme that reduces the n umber of est imated vari abl es needed for some a l gor­i thm , w i thout i ntroduc i ng adverse performance characteri st i cs , i s des i rabl e . Recent ly E l l i ott , Hol ov i ch and Das [3] s ug­gested a techn i que for mu l t i var i ab le adapti ve pol e ass i gnment . Thei r a l gori thm i s based on a fi xed control strategy wh i ch for a known m- i nput , p-output , n-s tate , stri ct ly proper system w i th control l ab i l i ty i ndec i es µ ; 1 < i < m and observab i l i ty i ndeci es '' i 1 < i < p (v = max { v . } ) , req u i res a con-- - l tro 1 1 er of order m (v - 1 ) . For the adapti ve i mpl ementati on one needs to have apri ori knowl edge of the control l ab i l i ty i ndeci es µ i and an upper bound on v . However ex i st i ng l i terature [4 , 5 ] for known systems s uggest methods that res u l t i n arb i trary pol e ass i gn ­ment wi th contro l l ers of order v - 1 . A s a matter of fact i n a "generi c " sett i ng one can 265

do better than that ( [6 ] and references there i n ) . The bas i c i dea beh i nd most of these approaches evol ves a round the fact that a pxm system can be made contro l l ab l e by a s i ng l e i nput . I n essence the pxm probl em i s " reduced" to a pxl prob l em . One s ugoest ion for mul t i vari abl e adapti ve pol e ass i gnment becomes apparent : fi rst " re­duce" the pxm prob l em i nto a pxl probl em and then apply the res u l ts i n [ 3 ] . I n th i s re­port we demonstrate that s uch a procedure does l ead to an adapt i ve pol e ass i gnment strategy wh i ch requ i res on ly a 1J- l order con­trol l er . The contro l l ab i l i ty i ndeci es need not be known apri ori . What i s requ i red i s the system order n and v . Add i t i ona l s i mpl i ­fi cat i on res u l ts because r [ 3] need not be estimated s i n ce r = 1 for the pxl case . Moreover the s i ng l e i nput structure may l ead to even fewer requi red parameter est imates ( s ee Remark 2. 1 ) . Sect i on 2 s ummari zes the scheme g i ven i n [ 3] as i t appl i es to the pxl cas e . Secti on 3 s uggests a techn i que for "re­duc i ng " the pxm prob l em to a pxl prob l em . No gl oba l s tab i l i ty ana lys i s is carri ed out but s imu l at i on res u l ts are encoura g i n g .

2 . T H E px l CASE A . F i xed Control Strategy Suppose that a system mode l p ( D ) z ( t ) = u ( t ) y ( t ) = R ( D ) z ( t ) , R ( D ) ( pxl ) , p ( D ) ( l xl ) i s known where R ( D ) and p ( D ) are ri ght copri ne and u ( t ) , y ( t ) and z ( t ) are the system i nput , output and part i a l state respecti ve l y . Then a fi xed control strategy [8] , i n d i fferent i a l operator form : ( l i near state vari ab le feed­back via asymptot i c state est imat i on ) q ( D ) s ( t ) =H ( D ) y ( t )+k ( D ) u ( t ) ,u( t )=s ( t ) +v ( t ) ( 2 . 1 )

2 66 T . Dj aferis , M . Das and H . E l l io tt

res u l ts i n the cl osed l oop system equat i ons : [q ( D ) p ( D ) -H ( D ) R ( D ) - k ( D ) p ( D ) ]z ( t ) =q ( D ) v ( t )

y ( t ) = R ( D ) z ( t ) . ( 2 . 2 ) By choos i ng H ( D ) , k ( D ) to sat i s fy : H ( D ) R ( D )+k ( D ) p ( D ) =q ( D ) ( p ( D ) - pd ( D ) ) ( 2 . 3 ) the cl osed l oop transfer functi on becomes :

Tc£ ( s ) = R ( s ) pd ( s f1 .

Therefore pd ( D ) i s the c l osest l oop charac­teri st i c polynomi a l , ( q ( D ) conta i ns the ar­b i trari ly chosen observe r ool es ) . I n order for the sol ut ion to correspond to a rea l i zab l e system ( di fferenti ator free ) some degree constra i nts need be i mposed ( i . e . , the order of the control l er be h i qher then or equal to v- 1 ) . One sol ut i on i s to . l et

\)- l . H ( D ) I H . D1

i =O 1 v- l .

k ( D ) = I k . D1 i =O 1

a ( q ( D ) ) = v - l , a ( pd ( D ) ) = n B . Adapti ve Control Strategy

( 2 . 4 )

The adapt i ve i mpl ementati on o f ( 2 . 2 ) becomes : v - 1 A o A i q ( D ) s ( t ) = _ l H i ( t ) D1y ( t ) + k i ( t ) D u ( t ) . A A 1 =O If H i ( t ) , k i converqe to constant matri ces

Hi ' k i sati s fy i nq ( 2 . 3 ) , ( 2 . 4 ) th i s control l aw wi l l ass i gn the des i red cl osed l oop pol e s . A A For est imati ng appropri ate H . ( t ) , k . ( t ) defi ne the fi l tered s i ana l s : 1 1

f ( D )z \ t ) z ( t } f ( D )u( t ) u ( t ) f ( D ) Iy( t ) = y ( t )

a ( f ( D ) ) ::._ v- 1 + 2µ ( µ = max ( µ i ) ) I t can be shown that

p ( o )z( t ) = u( t ) ( E J y( t ) = R ( D )�( t ) ( s )

(A ( t ) = B ( t ) ( E ) denotes that A ( t ) - B ( t ) decreases_exponenti a l ly fast . ) Mul t i ply i ng ( 2 . 3 ) by z ( t ) g i ves

H ( D )y( t ) + k ( D ) )u( t ) + q ( D ) pd ( D )z( t ) = q ( D )u( t ) ( s ) ( 2 . 5 ) whe re z( t ) must be est i mated from y( t ) , u( t ) . S i n ce R ( D ) , p ( D ) ri ght copri me there exi st A ( D ) , b ( D ) s uch that

A ( D ) R ( D ) + b ( D ) p ( D ) = l . ( 2 . 6 ) Th i s means that an asymptot i c est imate of z( t ) i s g i ven by :

A ( D )y( t ) + b ( D)u( t ) = z( t ) ( s ) ( 2 ; 7 ) wh i ch combi ned w i th ( 2 . 5 ) i mpl i es : H ( D )y( t )+k ( D )u( t ) + A ( D ) q ( D ) pd ( D )y( t ) +

b ( D ) q ( D )�( D )u( t ) = q ( D )u( t ) ( s ) . Now i f x ( t ) i s the 2 ( p+l ) v .�ol umn y�ctor contai n i ng the s ub ve ctors Dly ( t ) , Dl u ( t ) , oi q ( D ) prl ( D )y( t ) and oi q ( D ) p0 ( D )u( t ) and e i s the 2 ( m+p ) v row vector conta i n i ng H i , k i , Ai , b i then th i s re l at i onsh i p can be expressed as :

e x ( t ) = q ( D )u( t ) ( s ) .

The entri es of e can be est imated through any one of many l i near est imat i on techn i ques by defi n i ng the error

e ( t ) = e ( t ) x ( t ) - q ( D )u( t ) . Remark 2 . l I t i s i nterest i ng to note that i f two entri es i n [� � � � ] are known to be coprime polynomi a l s one can obta i n a n est imate o f z( t ) i n ( 2 . 7 ) i n a s impl e r manner exmpoy i ng only two entri es of [A ( D ) , b ( D ) ] and sett i n g the rest equal to zero . Th i s res ul ts i n fewer re­qu i red est imates . S im i l ar sav i n g can occur i f one knows more about the ran k structure of the res ul tant of R ( D ) and p ( D ) ( one onl y needs as many para­meters i n H ( D ) , k ( D ) as the n umber of cl osed l oop pol es pl us observer po les ) .

3 . THE pxm CASE A. F i xed Contro l Strategy I f one s urveys the l i te rature on a rb i tra ry pol e ass i g nment of systems w i th known para­meters ( both by state and output feedback , stat i c or dynami c ) i t becomes apparent that many of the methods suggested share a common ph i l osophy : formul ate the probl em i n s uch a way so that c l osed l oop characte ri st i c poly­nomi a l coeff i c i ents depend l i nearly on the contro l l e r parameters . Such a formul at i on wou l d l ead to l i near equati ons , thus fac i l i ­tat i n g the sol ut i on . At the same t i me one may al so wi sh to keep the order of the con­trol l er as sma l l a� pos s i b l e . I n the case of pxl (or l xm ) systems both these requ i re­ments can be met . A natural method of approach for dea l i ng w i th pxm systems i s to somehow " reduce " that probl em to a pxl ( or l xm) prob l em . I n s o doi na one must guarantee that certa i n copr imeness cond i t i ons are mai n­tai ned ( eq u i va l en t ly , guarantee i ng that the resu l t i ng s i ng l e i n put system i s contro l l ab l e ). The fol l ow i n a two res u l ts [7 ] i nd i cate how th i s can be accompl i shed . F i rs t a defi n i ti on .

Reduc ed Order Adap t ive Po l e P lacement 267

Defi n i t i on A sq uare non-s i ng u l a r polynomi a l matri x i s s i mpl e i f i t has on ly one non- uni ty i nvari ­ant polynomi al . We note that a constant matri x A i s cyc l i c i f the polynomi a l matri x s I -A i s s i mp l e . Lemma 3 . l ( [ 7] ) I f ( D1 , N 1 ) are l eft coprime and D1 s i mp l e then ( D1 , N1 g ) are a l so l eft copri me for "a l most any " constant vector g . I n parti cul ar th i s means that i f ( N , D ) are

- 1 - 1 ri ght coprime and D1 N 1 = N D then ( N (Adj D ) g , det D ) are ri aht copri me . Lemma 3 . 2 ( [ 7] ) If Ul , D ) are rf oht coprime , then " a l most any " constant F wi l l make Dk = D + FN s i mp l e . I t i s apparent from the above that by "pre­proces s i ng " the g i ven system by fi rst con­stant output feedback F (mak i ng Dk s i mp l e ) wh i ch i s then fol l owed by mul t i pl i cat ion by an appropri ate constant vector g , " reduces " the oi ven pxm system to a pxl system . Remark 3 . l Actual ly i t can be shown [9] that " a lmost any" F and q = [ 1 , 0 , . . . O ]T is an appropri ate pai r . It shoul d be ment ioned that if one knows apri ori that the system i s control l ab l e by some i np ut ( say the fi rst one ) then a n ob­vi ous choi ce wou l d be F = 0 and ci= [ l , O , . . . o]T .

Remark 3 . 2 It i s a l s o true [ 5 ] that for a "aeneri c " tern the choi ce

F

r o o 0

0 i s an appropri ate one . B . Adapti ve Control Strategy

sys-

One method of adapti ve pol e ass i gnment i s : l ) Choose F to make Pk = P + FR s i mp l e . 2 ) Choose q to make ( NAdj Pkg ) , det Pk ri ght coprime . 3 ) Apply the Adapti ve Contro l strategy of

sect i on 2 . \·!hen the system i s known there are ( compl i ­cated ) procedures for comput i n g appropri ate F and q , From Lemmata 3 . l , 3 . 2 one does concl ude that for an arb i trary system a " ran­dom" choi ce of F and g wi l l be an appropri ate

one wi th "probab i l i ty one " . Th i s i s a l so true when the system i s not known ( even though one cannot be ass ured) . S i mu l a t i on res u l ts however seem to i nd i cate that th i s i s not a prob lem . A scheme for detect i n g i nappropri ate choi ces woul d be very des i r­ab l e and the whol e i ss ue i s presently under cons i de rat i o n . Another a l ternat i ve wou l d b e to dev i s e a n adapti ve strategy where F a n d g woul d a l so be estimated .

4 . CONCLUSI ONS If adapti ve control strateg ies are to ga i n a w i der acceptab i l i ty i n i ndustry , every effort must be made to keep them as s i mp l e as pos s i ­b l e . I n th i s report we suggest a mul t i vari ­ab l e adapti ve pol e ass i gnment scheme that re­s u l ts i n a cons i derab l e redu ct i on i n the num­ber of parameters that need to be est imated , when compared to the method reported i n [ 3 ] . Questi ons of g l obal stab i l i ty have not been addressed . S imul at ion resu l ts however are encouragi n g .

REFERENCES [ l ] H . E l l i ott , " Di rect adapti ve pol e p l ace­

ment wi th appl i cat ion to non-mi n i mum phase sys terns , " I EEE Trans . on AC , Vol . AC-26 , June 1 982 .

[ 2 ] D . L . Prager, P . E . We l l s tead , "Mu l t i ­vari ab l e pol e ass i gnment sel f-tun i ng regu l ators , " I E E Proceedi ngs , Vol . 28 , tlo . l , January 1 980 .

[ 3 ] H . E l l i ott , W . �Jol ovi ch , M . Das , "Arb i ­trary adapti ve pol e pl acement for l i near mu l t i vari abl e systems , " I EEE Tran s . on AC ( to appear ) . --

[ 4 ]

[ 5 ]

F . M . Brasch , J . B . Pearson , " Po l e pl ace­ment us i nq dynami c compensators , " I EEE Trans . on AC , Vol . AC- 1 5 , No . l , February 1 9 70 . T . E . Dj aferi s , "Generi c pol e ass i gn­ment us i ng dynami c output feedback , " I nt . J . Control , Vol . 37 , No . l , 1 27- 1 44 , 1 983 .

[ 6 ] T . E . Dj aferi s , A . Narayana , "A new suffi ci ent condi t i on for generi c pol e ass i gnment by output feedback , " Tech . Report #UMASS-ECE -DEC82 -l .

[ 7 ] T . Kai l ath , L i near Systems , Prent i ce Hal l , Engl ewood Cl i ffs , New Jersey , 1 980 .

[8] W . Wol ovi ch , L i near Mul t i vari ab l e Systems , Spri nger Ver l ag , New York .

[ 9 ] C . D i ng , F . Brasch , J . B . Pearson , "On mul t i var iab le l i near systems , " I E E E Tran s . on AC , Vol . A -C 1 5 , No . -1 ,­Februa ry l 970 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

SOME PRACTICAL SOLUTIONS FOR THE ROBUSTNESS PROBLEM OF MULTIVARIABLE

ADAPTIVE CONTROL

E. Irving, H. Dang Van Mien and M. Redjah

Elertriciti' de Fm11t1', 1 m'. du GI de G11 11/le, 92 1 4 1 , C/11111art, Frana

I . Introduction I t is presented in th i s paper two succ e s s f u l l app l i c at ions of mult i­var i able d i s c r ete t ime adapt ive control of stable and unstable non-min imum phase system . The f ir st app l i c at ion i s the mult ivar i able control of a steam gene­r ator of a f ast br eeder r eactor wh i ch is a stable system with an inver s e poss ibly unstable . The se cond app l i c at ion deals w ith the terminal vo ltage contr o l of the turbo-g ener ator unit wh ich eventually may be unstable w ith an inver s e hav i�g weakly damped osc i llatory mode s . Th i s latter char acter i s t i c produces t h e s ame problems as non-min imum phase systems . The two previous app l i c at i ons ar e the i l lustr at ions of two ways for r obus­t i fy ing control prob lems . The f i r st examp le f e atur es a very ef f i c i ent solut ion f or stable damped systems wh i ch cons ists e s s ent ially in pr ecanc ellat i on of the system poles . The s econd examp le descr ibes another s o lut ion for even­tually unstable systemq : the extended Smith pred i ctor pr inc iple wh ich has the interest of g iv ing a bas i c tool to make wor k control des igned on s imple l inea­r i zed system on h ighly comp l i cated l inear or non- l inear r eal systems . The r e is only one as sumpt ion to be s at i s f ied by the add itional parts of the systems wh i ch added to the s imp l i f i ed system make the comp l i cated one : they must be stab le and damped wh ich r epresents an acceptable hypothes i s . Th i s last idea r epr e s ents a s imple s o lution for one of the bas i c i s s ues of modern control theory . Mor e­over , each est imat ion algor ithm has been mod i f i ed so as to inc lude a pr ior i informat ions on the aver age and the standard dev i at ion of each est imated par ameter . Th i s last char acter i s t i c makes a poster ior i adapt ive contr o l to compete v i ctor ious ly w ith s cheduled gains adapt ive contr o l wh ich opens a very lar ge f ie ld of app l i c at i ons .

I I . The contro11ed system mathemat ica1 1110de1 F r om partial der ivat ive equat i ons [ l ] r epresent ing the phys i c al phenomenas of the steam gene r ator , it is poss ible to obtain step r e s pons es wh i ch allows ident i f i c at ion of a very s imple poly­nomial model w ith thr e e inputs and thr ee outputs descr ibed by the f o l lowing equat ion : yk=A1 1Yk- 1 +A1 2 Yk- 2 +B l 1uk- 1 +Bl 2 uk- 2 ( l )

269

equ i valent to the f ol lowing left poly­nom i al representat i on :

yk= A� 1 ( z- 1 ) Bl ( z- 1) uk ( 2 )

uk i s the thr ee components vector contr o l var i able wh ich ar e var i at ions of f low r ates and steam valve apertur e , Yk i s the thr ee components var i at i ons of the contro lled var i able wh ich ar e the gener ated steam f low r ate , steam p r e s ­s u r e and output sod ium temper ature [ l ) . The control led system descr ibed by the previous equa� ion is stable but is not min imum-phas e , ma inly due to s amp l ing e f f ect . Th i s is gener ally the s ituat i on when the initial t ime der ivat ives of the step r e sponses ar e z er o . I ndeed , the least squar e min imum-phase system appro­x imat ion of the contr o l led system has step r es pons es w ith no mor e than 10% r e lat ive error r e f f er ing to step r e s ­pons es o f the pr e c i s e mode l . Th i s small d i f f er ence suggests than a r obust ad apt ive contr o l s o lut i on may b e obt a i ­n e d i f t h e ma in control loop c an b e r e al i z ed on t h e min imum-phas e p a r t of the system the r est of the system be ing stable and small c an be des igned o uts ide th i s main loop .

I I I . The control scheme ideas

The f ir st step of the contr o l s c heme i s to u s e the Smith pr ed ictor pr inc i p le , it i s meant that two control loops ar e u s ed , th e f ir st internal loop manadges w ith the min imum phase part of the cont r ol led system wh i ch can g i ve a very r obust adapt ive control [ l ] , the exter nal loop i s des igned to tackle the d i sturbance problem : it i s a f e edback wh i ch uses the d i f f er ence between the output o f the r eal system and the output of the ident i f i ed system wh i ch has purpo s e ly the structure of the ent ir e mathemat i cal model of the contro lled system ident i cal to the equat ion ( 1 ) . To ut i l i z e conven i ently th i s previous f ir st ide a , the f o l lowing second idea is very ef f i c i ent : ident ify the contro lled system w ith the left form po lynom ial r epr es entat ion wh i ch i s pr e c i s e ly the structure o f equat i on ( 1 ) , but r e al i z e the contr o l loop on the r ight f orm r epr e s entat i on equ ivalent to the f ir st f orm and has the f o l lowing structur e [ 2 ] :

or :

pk-Ar 1Pk- 1+Ar 2Pk- 2 +uk ( J ) yk=Br 1Pk- 1 +Br 2Pk- 2 ( 4 )

( 5 )

270 E. Irving , H. Dang Van Mien and M. Redj ah

The algor ithm to obta in the r ight form matr i c e s f r om the left f orm i s standar d [ 3 ] and needs neces s ar i ly the contr o l­lab i l ity cond it ion wh ich i s obtained by realiz ing its equ ivalence w ith the po lynomial matr i c e left copr imeness of the po lynomial matr ices A1 and B i . The gr eatest left common f actor G of the previous polynomial matr i c e s is obtained by the pr ev ious algor ithm wh i ch g ives in the s ame t ime the r ight copr ime po ly­nom i al matr i c e s P , Q , and Ar , Br sat i s ­fy ing the f o llow ing equat ions :

A1P + B 1Q = G

AlBr - B lAr = 0

( 6 ) ( 7 )

Hav ing the r educed r ight r epres entat i on it i s , now , easy to use the Smith pr e­d i ctor ideas . The contr o ller has two contr o l loops ( f ig . 1 ) : the internal loop wh i ch master s the s impl i f i ed part of the mathemat ical model and the d i stur bance loop wh i ch e l iminates the d i stur bances on the output of the r eal system . It can be s een that the contr o l ler str ucture i s very s imple thus , r obust . The idea that it is pos s ible to br ing outs ide in ser i e s w ith the ma in control loop the comp l i cate but stable parts of the system has been extended in a par allel structure by us ing the r ight po lynomial matr i c e r ep r e s en­tat i on . Th i s has the inter est of putt ing outs ide the internal loop the non-mini­mum phase part of the system . The inter­nal loop hav i ng to master a s impl i f i ed min imum phase system i s very ef f ic i ent . Due to the prope r t i e s of the system to be contro lled , the perf ormanc es of the internal loop ar e s l ight ly d i stur bed by the outs ide par ts .

Set point Control D is t u rbance

Adjustme t

algorithm

F ig . l . Mult ivar i able adapt ive pr ed i ctor for stable systems

Smith

It has been f ound that cance llat ion of stable poles i s a very r obust idea wh ich g ives the f o l lowing contr oller :

( 9 ) A pr ior i knowledge on est imated par a­mete r s i s u s ed by the f o l lowing f orm of adj u s tment law :

ek= aek_1+.eep , k- 1 +K€k ( 10 )

ak i s the vector of est imated par ameter s and 8p k the ir a pr i or i knowledge at t ime k , a i s' a d i agonal matr ix chosen stable , .B is cho s en to obtain un ity stat i c gain between a pr ior i and a poster ior i e s t i ­mates , €k i s the ident i f ied mode l output error . For unstable systems , a predi ctor stab i­l i z ing loop i s necessary . A very s imple exemp le of s econd order system w ith cons ide r at ion of computat ional de lay and d i stur bances on the input and the output is r epresented on f igur e 2 .

F ig . 2 . Smith pr ed i ctor f or unstable systems

I V . Conclus ions

Due to lack of r oom , it is impos s ible to g ive mor e deta i ls on the perf ormances and numer ical r e sults , but t i ll now , all s e ems very prom i s ing and so lves , mor e­over , some numer ical problems r e lated to excess of complex ity of the contr oller .

v . References [ l ] E . I r v ing , H . Dang Van M i en ,

Mu lt ivar i able adapt ive contr o l of nu­c le ar p lants . I FAC Kyoto , 198 1 .

[ 2 ] T . Kai lath , Linear systems , Pr en­t i ce Hal l , 19 8 0 .

[ 3 ] v . Kucer a , D i s c r ete l inear con­tr o l , W i ley , 197 9 .

[ 4 ] I . D . Landau , Adaptive systems : the r ef er ence mode l approach , Mar c e l Dekker , 197 9 .

[ 5 ] K . J . Astr om , Theory and appl i ­cat i ons of adapt ive contr o l , I FAC , Kyoto , 1 9 8 1

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

NEW ALGORITHMS

AUTOMATIC TUNING OF SIMPLE REGULATORS FOR PHASE AND AMPLITUDE MARGINS

SPECIFICATIONS

K. J. Astrom and T. Hagglund

Department u{ Automatic Control, Lund Institute o( Technology, Lund, Swrdm.

Abs t rac t . The p a p e r desc r i bes p roced u res fo r a u t oma t i c t u n i ng of reg u l a t o r s o f th;;;-PID-type to spec i f i cat i ons o n phase a nd a1llp l i tu d e ma rg i ns . T h e key i dea i s a s i m p l e method fo r es t i ma t i ng t he c r i t i ca l g a i n a nd t he c r i t i ca l f requency . The p rocedu re w i l l a u t omat i ca l l y generate t he a p p ro p r i at e test s i g na l s . The method is non-pa ramet r i c a nd i nsens i t i ve t o mode l i ng e r ro rs a nd d i st u r b a nces . It may be used fo r a u t omat i c t u n i ng of s i mp l e reg u l a t o r s as we l l a s i n i t i a l i zat i o n o f ma re comp l i ca ted adapt i ve reg u l a t o rs .

�!lll�Q.!:Q�.!. Adapt i ve cont ro l ! Cont rol non l i nea r i t i es ! Desc r i b i ng funct i o n ! I d e nt i f i cat i o n ! L i m i t cyc l e s ! Nyqu i st c r i t e r i on ! P I D cont r o l ! R e l a y cont ro l .

1 . I NTRODUC T I ON

Ada pt i ve techni ques may be used i n ma ny d i ffe rent ways . In t he a p p l i c a t i o ns d i scussed i n t h i s pape r t he pu rpose is t o obta i n techni ques fo r a u tomat i c tu n i ng o f regu l ato rs . The word 2\!i9=ik!!Ji!l9 i s used t o empha s i ze t h i s .

Many o f t he p roposed a d a pt i ve regu l a t o rs requ i re cons i d e r a b l e a p r i o r i i n f o rma t i o n . I t i s often necessa ry t o spec i fy t he sam p l i ng pe r i od . des i red c l osed l oop ba ndw i dt h • forget t i ng facto rs a nd des i red regu l at o r co1np l e x i t y . T h i s means t h a t i t m a y b e a cons i d e ra b l e eng i nee r i ng e f fo rt to co111m i ss i o n such a d a pt i ve regu l a t o rs .

T h i s paper p roposes a s i m p l e 1llet hod fo r automa t i c tu n i ng of s i mp l e reg u l at o rs of t he P I D type . w h i c h was f i rst suggested i n Ast r6111 ( 1 98 1 ) . The ba s i c i dea i s t h e obse rva t i o n t ha t many s i m p l e regu l at o rs requ i re i n fo rmat i o n a bou t t he po i nt w h e re t he open l oop Nyqu i st cu rve o f t he p rocess t 1•a nsfe r func t i o n i nt e rsects t he neg a t i ve rea l a x i s . T h i s po i t1t i s colfnYton l y desc r i bed i n t e r1Y1s o f t he c r i t i ca l g a i n k a nd t he c r i t i c a l

c

f requency w • See Z i eg l e r a nd N i c ho l s ( 1 943 ) . c

A ve ry s i m p l e est i ma t i o n p rocedu r e w h i c h g i ves t hese pa r a1ne t e r s i s p roposed . The method has t he a d va n t age t h a t a pert u r b a t i o n s i g na l i s gene r a t ed a u tomat i ca l l y . T h i s pe rtu rbat i on s i g na l i s i n fact c l ose t o a n opt i m a l i nput s i g n a l f o r t h e pa rt i cu l a r est i ma t i o n p ro b l e m .

S i mp l e regu l at o rs c a n a l so b e tu ned u s i ng convent i ona l se l f-tu ners or mod e l refe rence a d a p t i ve cont r o l . See W i ttenma r k a n d Ast rom C 1 980 ) • Ast rem a nd Wi t t e nma r k ( 1 973 , 1 ·�80 l a n d

Landau ( 1 9 79 > . T h e 1Y1ethod p r o posed i n t h i s pape r has two a d va ntages ove r t hese a p p roaches .

Convent i ona l a d a pt i ve cont r o l based o n pa ramet e r est i mat i o n needs a p r i o r i k now l ed g e o f t he dom i na t i ng t i 1ne const a nts . T h i s i s req u i red i n o r d e r t o obta i n a guess o f t h e samp l i ng pe r i od . T he re a re tech n i ques t o a d J u st t he samp l i ng pe r i od a u toma t i ca l l y • s e e Ku r z ( 1 979) a nd Ast r601 a nd Z ha oy i ng ( 1 98 1 ) . These tec h n i ques w i l l howeve r not wo r k i f t h e 2 7 1

i n i t i a l guess i s o f f by a n o rde r o f i1'1ag n i t ud e . T he methods p roposed i n t h i s p a pe r w i l l not req u i re p r i o r k now l ed g e o f t he t i me sca l e o f t he p rocess . The refo re t hey may a l so be used to i n i t i a l i ze m o r e soph i s t i cated s e l f - t u n e rs .

Convent i ona l a p p roaches to se l f-tu n i ng P I D cont r o l resu l t i n a m i c rop roce s s o r code o f a few f( bytes . The code f o r t he p r o posed schemes is at l ea s t an o rd e r o f mag n i tu d e sma l l e r . T he p roposed methods m a y t he re f o r e conve n i e nt l y be i nc o r po rated e v e n i n v e r y s i m p l e regu l a t o rs .

The maJ o r d ra w ba c k o f t h e methods p r o posed i n t h i s pape r co111pa red t o convent i o na l a d a pt i ve t ec h n i qu es i s t ha t t hey a re l i m i ted to t u n i ng of s i m p l e cont r o l l a ws of t h e P I D type .

The pape r i s o rg a n i zed as f o l l ows : The est i m a t i o n method i s desc r i bed i n Sect i on 2 . S i m p l e a l go r i t hms f o r a u t oma t i c t u n i ng t o a mp l i tu d e ma rg i n a nd phase ma rg i n spec i f i c a t i ons a re g i ven i n Sect i ons 3 a nd 4 . Resu l t s f rom l a bo ra t o ry a nd i ndust r i a l e x pe r i ments w i t h t he a l g o r i t h1ns a re p resented in Sect i on 5 .

2 . THE BAS I C I DEA

T he Z i eg l e r- N i c ho l s ru l e fo r tu n i ng P I D regu l a t o rs was based o n t h e obse rva t i o n t h a t t he regu l at o r pa ramet e r s cou l d be d e t e rm i ned f rom k now ledge of one po l'nt on the Nyq u i s t cu rve o f t h e ope11 l o o p system . T h i s p o i nt i s t rad i t i o na l l y desc r i bed i n t e rms o f t h e

£.!:iii£2!_g2in a nd t h e £.!:iii£2!_f.!:!19\!!1!l£ll · I n t he o r i g i na l Z i eg l e r- N i c h o l s scheme • desc r i bed i n Z i eg l e r a nJ N i c ho l s ( 1 943) , t h e c r i t i c a l g a i n a nd t h e c r i t i c a l f requency a re dete r m i ned i n t h e f o l l ow i ng way . A p ropo rt i ona l regu l a t o r i s connected to t h e system . T h e g a i n i s g ra d u a l l y i nc reased u nt i l a n osc i l l at i on i s obta i ned . I t i s d i f f i cu l t t o pe r f o rm t h i s e x pe r i ment i n such a way t ha t t he a m p l i t u d e o f t he osc i l l a t i o n i s kept u nd e r cont ro l . Anot her '"ethod f o r a utomat i c dete rm i na t i o n o f s pec i f i c po i hts o n t h e Nyq u i s t cu 1•ve i s t he re fo re p roposed .

The method i s based on t h e observa t i o n t h a t a

272 K. J . Astrom and T . Hagglund

system w i t h a phase l ag of at l east n at h i g h f requenc i es may osc i l l a t e w i t h f requency w

c

u nde r r e l a y cont r o l . To d e t e r m i ne the c r i t i ca l ga i n and the c r i t i ca l f requency t he system i s connected i n a feed back l oo p w i t h a r e l a y as is shown i n F i g . 1 . The e r ro r e i s t hen a pe r i od i c s i g na l a nd t he pa ramet e rs k

c

a nd w can be det e rm i ned f roor1 t he f i rst c

h a r mo n i c component of t he osci l l a t i o n .

Let d b e the re l a y a 1'<1p l i t ude and l et a b e t h e a m p l i tude of t he f i rst ha rmo n i c o f the e r ro r s i g na l . A s i m p l e Fou r i e r s e r i es e x pa ns i o n o f t he r e l a y output t h e n shows t h a t t he re l a y 1oa y b e desc r i bed b y t he equ i va l e nt ga i n

k r

4d

na

I t fo l l ows f rom the p r i nc i p l e o f ha ro'()o n i c ba l a nce t hat t h i s g a i n i s equa l t o t h e c r i t i c a l ga i n k • I t i s poss i b l e t o mod i fy

c

t he p roced u re to d e t e rm i ne othe r po i nt s o n t he Nyq u i st cu rve . An i nteg ra t o r m a y be connected i n - t he l oop after t he r e l a y t o obta i n t he po i nt w he re the Nyqu i s t cu rve i nte rsects t he nega t i ve i mag i na ry a x i s . Othe r po i nts on the Nyq u i s t cu rve can be dete r1'<1 i 11ed b y repea t i ng t he p rocedu re w i t h l i nea r syst e1\ls i nt i·oduced i nt o t he l oo p .

An exact e x p ress i o n osc i l l a t i on is g i ven t heo re1n .

fo r by

t he the

pe r i od o f f o l l ow i ng

T heorem 1 . Cons i de r a l i nea r t i 111e- i 11va r i a 11t ;�;�;;-���e r re l a y cont ro l . Let H ( T , z > be t he pu l se t ra t1s f e r fu nct i o n of t he l i nea r system w i t h a sam p l e and ho l d . Assume t hat t he re i s a l i m i t cyc l e w i t h pe r i od t • Then t i s

g i ven by

p p

H ( t / 2 , - 1 > p

0 a

The p roof i s g i ven i n Ast rom 0 983 > .

A s i m p l e r e l a y cont r o l e x pe r i ment t hu s g i ves t he i nfo rmat i on a bout t he p rocess w h i c h i s needed i n o r d e r t o a p p l y the des i g n methods . T h i s method has t he a d va ntage t hat i t i s easy t o cont r o l t he amp l i t u d e o f t he l i m i t cyc l e by a n a pp rop r i a t e cho i ce o f t he r e l a y a m p l i tu d e . Not i ce a l so t hat t h e est i ma t i on method w i l l a u t omat i ca l l y generate an i np u t s i g na l t o t he p rocess w h i c h h a s a s i g n i f i c a n t f requency cot1tent a t w • T h i s ensu res t h a t

c

t he c r i t i ca l po i nt c a n be dete r m i ned accu rate l y . See Manne r fe l t < 1 98 1 ) .

F i g . 1 .

P I O

Process y

- 1

B l oc k d i ag ram o f t h e a u t o-tu ne r . The system ope rates as

. a re �a y

cont r o l l e r i n t he tu n i ng mode <.� ) a nd as a n o rd i na ry P I O reg u l at o r i n t he cont ro l 1l'1ode C c > .

When t he c r i t i c a l p o i nt on t he Nyqu i s t cu rve is k nown it is st ra i g ht fo rwa rd to a p p l y the c l as s i c a l Z i eg l e r-N i cho l s tu n i ng ru l es . It i s a l so poss i b l e t o d ev i se many other des i g n scheoroes based on t he know l edge o f t he c r i t i ca l po i nt . A l g o r i t hms f o r autoia t i c t u n i ng o f s i mp l e regu l a t o rs based on , t he a m p l i tu d e a nd phase ma rg i n c r i t e r i a w i l l be g i ven in Sect i ons 3 a nd 4 .

·Met hods for a u t omat i c dete r m i n a t i o n o f f requency a n d t he ampl i tu d e of osc i l l a t i on w i l l be g i ven to com p l ete desc r i pt i on o f t he est i m a t i o n met hod .

t he t he t he

The pe r i od of an osc i l l a t i on c a n eas i l y be dete r111 i ned by -.-"easu r i ng t he t i mes between z e ro-c ross i ng s . The amp l i tude · may be dete rm i ned by measu r i ng t he peak-to-peak va l u e s . These est i mat i o n methods a re easy to i m p l ement because t hey a re based on cou nt i ng a nd co1npa r i so11s o n l y .

S i nce t he desc r i b i ng f u nct i o n a na l ys i s i s based on the f i rst ha rmon i c o f the osc i l l a t i on t he s i m p l e est i mat i on t e c h n i ques requ i re t hat t he f i rst h a r ol\O ll i c do"i i nates . I f t h i s i s not t he case i t 1nay be necessa ry t o f i l t e r the s i g na l before measu r i ng . See e . g . Ast rom c 1 ·�75 ) .

The p r ec i s i o n of t he met hod can be i mp roved cons i d e r a b l y by i nt roduc i ng a s i m p l e c o r re l a t i o n w i t h t h e r e l ay amp l i t u d e . T h i s c a n b e i or1 p l e11·1et1ted s i m p l y b y u p-down count i ng .

More e l a bo rate est i m a t i o n schemes l i ke l east squa res est i 111a t i o n a nd ex tended Ka l ma n f i l te r i ng m a y a l so b e used to dete ron i ne the a m p l i tu d e a n d t he f requency o f t he l i m i t cyc l e osc i l l a t i o n . Th i s i s d i scussed i n Ast roon < 1 982 ) a nd l<a i S i ew 0 982 > .

3 . AMPL I TUDE MARG I N AUTO-TUNERS

When t he c r i t i ca l p o i nt is k nown it i s s t r a i g ht forwa rd t o f i nd a reg u l a t o r w h i c h g i ves a des i red aon p l i tude ma rg i n . The s i m p l est way is to choose a p ropo1• t i o na l regu l at o r w i t h t he g a i n

k k / A < 2 > c m

where A i s t he d e s i red amp l i tu d e ma rg i n and m

k i s t he c r i t i ca l g a i n . c

Somet i mes t h i s s o l u t i o n i s not sat i sfacto r y because i nteg ra l act i o n may be requ i red . S i nce t he f requency response of a P I O regu l a t o r c a n b e w r i t t en a s

[1 G < i w > k + R

i t f o l l ows t hat a ny g a i n g i ven by ( 2 ) a n d

w2

T c i

1 w

2T T > ) --- ( 1 - ( 3 )

i wT i d i

P I O regu l a t o r w i t h t h e

( 4 )

a l so g i ves t he d e s i red a mp l i tude ma rg i n . The

i nteg ra t i on t i oroe can t hen be c hosen a r b i t ra r i l y • e . g . i nve rse l y p ropo r t i ona l t o w , a nd t he de r i va t i on t i me is t hen g i ve n by

c

Equa t i on ( 4 ) .

Automatic Tuning of S imple Regulations 2 7 3

4 . PHASE MA RGIN AUTO-TUNERS

Cons i de r a s i tuat i o n when one po i nt on the Nyqu i s t cu rve for the open l oop system i s k nown . W i t h P I , P D o r P I O cont rol i t i s then poss i b l e to move t he g i ven po i nt on the Nyqu i st cu rve to an a rb i t ra ry pos i t i o n i n the com p l e K p l a ne . T h i s is i nd i cated i n F i g . 2 . B y chang i ng the ga i n i t i s poss i b l e t o move t he Nyqu i st cu rve i n the d i rect i on of G C i w ) . The p o i nt A may be moved i n t he o rthogona l d i rect i o n by chang i ng i nteg ra l or d e r i va t i ve ga i n . I t is t hus poss i b l e to move a spec i f i ed po i nt to an a rb i t ra ry pos i t i o n . T h i s i dea c a n b e used to obta i n des i g n methods . B y mov i ng A to a po i nt on the u n i t c i rc l e i t i s e . g . poss i b l e t o obta i n systems w i t h a p resc r i bed phase-ma rg i n . An e K a m p l e i s g i ve n be l ow .

sl!.!!!!!E!l!!Ll Cons i de r a p rocess w i t h t he t ransfer funct i o n G ( s ) . The l oop t ra nsfe r funct ion w i t h P I O cont rol i s

k C l + s T + _!_ ) G < s l d sT . l

AsSUl'ioe that t he po i nt w where t he Nyqu i st c

cu rve of G i nte rsects t he is k nown . Requ i r i ng t hat l oop t ra nsfer funct i o n

neg a t i ve rea l a K i s the a rgument o f t he

at w is + - n t he c m

fo l l ow i ng cond i t i o n is obta i ned

w T c d w T

c i

t a n + m

( 5 )

There a re matiy T a nd T w h i c h d

sa t i sfy t h i s

cond i t i o n . Dtie poss i b i l i t y i s to choose T

a nd Td

so that

T "' T d

Equa t i o n ( 5 )

eq uat i on for T d

t hen g i ves a second w h i c h has the so l u t i o n

t a n +m

+ /; + 2 t a n +

m

2 w c

.,!_ G(iw) / W

Im G

Re G iw G(iw)

( 6 )

o rde r

( 7 )

F i g . 2 . Shows t ha t a g i ven po i nt o n t he Nyqu i s t cu rve oroay be moved to a n a rb i t ra ry pos i t i on i t1 t he G- p l a ne by P I • PD o r PIO cont rel . The po i nt A may be moved i n the d i rec t i ons G < i w ) , G ( i w ) / i w a nd i wG < i w ) by chang i ng p roport i o na l , i nt eg ra l and d e r i vat i ve ga i n respect i ve l y .

S i m p l e c a l cu l at i ons show t ha t t he l oo p t ra ns fe r func t i o n ha s u n i t g a i n a t w i f the

c

regu l a t o r ga i n i s chosen as

cos +

k ------�� = k cos • . . ( 8 ) I G < i w l l c m

c

whe re k is the c r i t i c a l ga i n . The des i g n c

ru l es a re t hus g i ven by t he equat i ons ( 5 ) • ( 6 ) , ( 7 ) and ( 8 ) .

There a re many other pa raonet e r T may e . g . be

i

has a g i ven v a l ue .

poss i b i l i t i es . The chosen so t ha t w T

c

a

So fa r • i t has been assumed t ha t the nonl i tiea r i ty i nt roduced i n the feed back l oop i s a re l a y , a nd t he po i nt whe re t he Nyqu i st cu rve i nt e rsects t he neg a t i ve rea l a K i s has been i d e nt i f i ed . Other no n l i nea r i t i es ca n a l so be used . I n Hagg l u nd < 1 98 1 ) a r e l a y w i t h hyste res i s i s used t o t u ne t he system t o a des i red phase ma rg i n . The i nverse desc r i b i ng func t i on of a re l ay w i t h hyst e res i s i s

N C a l

n /a

2

4d

I 2

h nh

4d (9)

where d i s t he re l a y amp l i tude a nd h i s t he hyst e res i s w i d th . I n t he com p l e x p l a ne t h i s funct i o n may b e desc r i bed as a s t ra i g ht l i ne pa ra l l e l to t he rea l a K i s , see F i g . 3. By choos i ng t he re l a t i o n between h and d it i s t he re f o re poss i b l e t o determ i ne a po i nt on t he Nyqu i s t cu rve w i t h a spec i f i ed i mag i na ry pa rt . I n t he next examp l e • t h i s p r o pe rty i s used t o obta i n a regu l at o r w h i c h g i ves a desi red phase ma rg i n of a system .

!;;J!,;!!llE!l!L'5: Cons i de r a p rocess w i th t ra ns fe r funct i o n G < s l • cont r o l l ed b y a p ropo r t i on a l regu l ato r . The l oop t ra nsfe1· funct i o n i s thus k · G ( s ) . Assume t ha t t he des i g n goa l is t o obta i n a c l osed l oop systel'io w i t h the phase m a rg i n + Choose t he relay cha racte r i st i cs

m

so that t he neg a t i ve i nverse desc r i b i ng funct i on goes t h rou g h the po i nt p d e f i ned i n F i g . 3 . The pa raoroet e rs a re t hen

d

*

* na

4

* h = a s i n < +

m

where a is t he desi red a mp l i tude of t he

osc i l l a t i ons . The desi red phase ma rg i n i s

1 N(A )

F i g . 3 . The pos i t i on of t he i nverse desc r i b i ng funct i on and t he d e s i red loca t i o n of t he Nyq u i st cu rve .

274 K . J . As trom and T . Hagglund

obta i ned i f t he Nyq v i st cu rve goes t h rough t he po i nt p in F i g . 3. S i nce t he i nt e rsect i o n between - 1 / N ( a ) a nd k · G ( i .., ) can b e determ i ned f rom t he amp l i tude o f the osc i l l a t i o n • t h i s po i nt c a n be reached e . g . by i te ra t i ve l y c h a ng i ng the g a i n k . The Reg u l a fa l s i method g i ves t he fo rmu l a

k n+ 1

k n

( a n

* a )

k - k n n-1

a a n n- 1

( 1 0 )

w h i c h has a quad rat i c conve rgence rate nea r t he so l u t i o n . I nteg ra l a nd de r i vat i ve act i o n ca n b e i nc l uded , u s i ng t he methods p roposed i n E x a.r1 p l e 1 .

a

There a re m a ny poss i b l e va r i a t i ons of t h e g i ve n des i g n methods fo r P I O- regu l a t o rs . A l l methods a re c l os e l y r e l a ted because t hey a r e based o n i n fo rma t i o n a bout t he p rocess to be cont r o l l ed in te rms o f one po i nt on the Nyqu i st c u r ve o f the p rocess . The po i nts whe1·e t he Nyq u i st cu rve i nt e rsects t he rea l a x es or s t r a i g ht l i nes pa ra l l e l to t he rea l a xes a re s i m p l e cho i ces . The des i g n methods may be mod i f i e d . Other r e l a t i ons between T

a nd Td

t ha n t hose g i ve n by ( 6 ) may e . g . be

used . Othe r c r i t e r i a l i k e da m p i ng o r band w i d t h ma y be chosen i nstead o f t he phase­or a m p l i tude-ma rg i ns . It is a l so poss i b l e to have des i g n met hods w h i c h a re based on k now l ed g e o f mo re po i nts on t he Nyq u i s t cu rve ..

5 . E X P E R I MENTS

A nuff1ber of s i mu l a t i o ns a nd e x pe r i ments have been pe r fo rmed i n o rd e r to f i nd out i f a usefu l auto- t u ne r c a n be des i g ned based on t he i deas desc r i bed in t he p rev i ous sect i o ns . The resu l t s a re summa r i zed i n t h i s sect i o n . Some rep resent a t i v e e xa m p l es a re a l so p resented .

T h e re a re seve ra l must be s a l ved auto-tu ne r . I t

p ract i c a l p r o b l ems w h i c h i n o r d e r to i mp l ement a n

i s necessary t o account fo r 1)·1easu remet1t satu rat i ot1 a d j u stment osc i l l a t i on .

no i se , l ev e l a d j u stment , o f actuators and a u tomat i c o f t he amp l i t u d e o f the

Measu rement no i se ma y g i ve e r ro r s i n detect i o n of pea k s and z e r o c ross i ng s . A hysteres i s i n t h e r e l a y is a s i mp l e way t o reduce t he i n f l u ence of measurement no i se . F i l t e r i ng i s a no t he r poss i b i l i t y . The est i mat i on schemes based on l ea s t squa res a nd e x t ended Ka l m a n f i l t e r i ng can be made l ess sens i t i ve to no i s e .

When t he regu l a t o r i s swi tched on i t ma y happen t hat t he p rocess output i s fa r f rom t he des i red eq u i l i b r i um cond i t i o n . I t wou l d be des i r a b l e t o have a system w h i c h reaches t he eq u i l i b r i um automat i ca l l y . Fo r a p rocess w i t h f i n i te l ow-f requency g a i n t h e r e is no gua rantee t h a t t he des i red steady state w i l l be a c h i eved w i t h r e l a y cont r o l u n l ess t he re l a y amp l i t ude i s su f F i c i e nt l y l a rge . To g ua rantee t ha t t he output c a n actua l l y rea c h t he reference va l ue i t m a y b e necessa ry t o i nt roduce manua l o r automat i c reset .

I t i s des i ra b l e t ha t t he r e l ay am p l i tu d e i s adJ usted aut omat i ca l l y . A reaso nab l e a p p roa c h i s t o requ i re t hat t he osc i l l a t i o n i s a g i ve n pe rcentage o F t he a dm i ss i b l e sw i ng i n t h e o u t pu t s i g na 1 .

The consequences of u s i ng est i mat i o n schemes o f d i ffe rent comp l e K i t y have been e x p l o red by s i mu l at i on . See Ast rom ( 1 982 ) a nd Ka i S i ew ( 1 982 ) . l t1 t hese e x pe r i ments p rocesses hav i ng

d i ffe rent dyna m i cs have been reg u l ated w i t h d i ffe rent types o f auto-tu ners . The effects o f 1neasu re<oet1t no i s e a t1d l oad d i stu rbances have been i nvest i g a ted . A l t hough work s t i l l rema i ns t o be done t he e K pe r i ments have i nd i cated t hat t he s i m p l e est i ff1at i on method based on z e ro-cross i ng a nd pea k detect i on w o r k s very we l l . The e x pe r i ments a l so i nd i cate t ha t s i m p l e m i nded l eve l adJ UStment methods often a re sa t i sfact o r y .

T h e auto-t u ne rs have been i m p l emented o n seve ra l d i ffe rent compu t e rs . A DEC LSI 1 1 /03 was used i n some ea r l y e x pe r i ments . See E l fg ren ( 1 98 1 ) . The a l go r i t h1ns were coded i n Pasc a l w i t h a rea l t i me k e r ne l . Sma l l l a boratory p rocesses we re cont ro l l ed . The e x pe r i ments showed t hat t he s i m p l e a l g o r i thms we re robust a nd t hat t hey wo rked wel l .

The a l go r i thms w e re a l so coded i n Bas i c on a n App l e I I . T h i s i mp l ementa t i on was very easy to use because o f the g ra p h i cs a nd t he i nt e ract i ve man mach i ne i nte rface .

An e x pe r i ment made w i t h t he A p p l e I I i m p l ement a t i o n w i l l now be p resented . F i g . 4 shows t he resu l t when the auto-t u ne r was a p p l i ed to a tank w i t h a f ree out l et . The

i n l e t va l ve to t he t a n k was cont ro l l ed f rom ff1easu rements of t he wate r l evel i n t he t a n k . The tu n i ng p roced u re c a n be d i v i ded i nt o two phases • see F i g . 4 . The f i rst phase is a n i n i t i a l phase w h i c h moves t he p rocess t o eq u i l i b r i u m • i . e . t o t he d e s i red reference l eve l . The second phase i s t he f i na l tu n i ng phase . The two phases a re desc r i bed i n ma re deta i l be l ow .

�bA��-1£ When t he p rocess dynam i cs i s tota l l y u nk nown , i s is natu ra l to ma ke sma l l step c h a nges of t he cont ro l s i g na l to dete r m i ne t he mag n i tude of t he ga i n a nd t he domi nat i ng t i me constant of t he system . T h i s is done i n i t i a l l y i n phase 1 . Based on t h i s rou g h cha racte r i z a t i o n of t he p rocess , a

0.5 0.4 0.3 0. 2

0. 6 0.4 02

0

0

0

Phase 1 Phase 2

200 400

200

F i g . 4 . E x pe r i ments ff1ade on the t a n k p rocess .

Automatic Tunin g o f S imple Regulations 2 7 5 conse rvat i ve P I cont ro l l e r i s d e s i g ned w h i c h ramps t he system to t h e eq u i l i b r i u m w i t h a s l ope dete r m i ned f rom t h e est i ma t ed t i m e consta nt . T h i s f i rst phase c a n b e o<o i t t ed i f t he p rocess i s manua l l y moved t o t h e equ i l i b r i um .

Ebi!.§!L;6.!. When t he desi red l eve l is reached , t he est i ma t i o n p roced u re sta rts . A re l a y w i t h a sma l l hyst e res i s i s i nt roduced i n t he l oo p as shown i n F i g . 1 . T h e r e l ay amp l i tu d e i s adJ usted automat i ca l l y so t h a t a n osc i l l a t i o n w i t h d e s i red amp l i tude i s obta i ned . The amp l i tude a nd t h e f requency of the osc i l l a t i o n a re est i ma t ed by pea k detec t i o n a nd dete rm i nat i o n o f t he t i ff1es between z e r o c ross i ngs of t he cont r o l e r ro r .

The des i g n ff1ethod i n t h i s e x am p l e was based o n a comb i na t i o n of phase- a nd a mp l i tu de-ma rg i n s pec i f i ca t i o n . I t was requ i red t ha t t he Nyqu i st cu rve i nt e rsects t he c i rc l e w i t h rad i us 0 . 5 a t an a ng l e of 225 " . Two step responses of the system cont r o l l ed by t he est i mated cont ro l l e r a r e s hown i n F i g . 4 . T h e l a c k o f symmet ry d e pends o n t he no n l i nea r i t i es . The h i g h f requency d i stu rbance i ti t he cont r o l s i g na l is caused by t he l ow reso l u t i o n i n t he AD-conve rt e r , 8 b i t s o n l y .

The App l e I I i m p l ement a t i o n w a s used i n a sug a r ref i ne ry to test t he fea s i b i l i t y o f auto-tu n i ng i n a n i ndust r i a l env i ronment . The auto-tuner was a p p l i ed t o seve ra l feed b a c k l oops . I n F i g . 5 a n e x a m p l e of a tempe ratu re cont r o l l oop i s g i ve n . O n l y phase 2 was used .

The e x pe r i ments i n t he sug a r r e f i ne ry were very f ru i t fu l . They showed t ha t the auto-tut1e1· wo 1•ked very w e l l . T he est i 1r1ated P I D pa ramet e rs va r i ed o n l y ± l OX between coo1pa ra b l e e x pe r i me tit s . Accepta b l e cont ro l l e rs we re obt a i ned even i n l oops w h i c h w e re cons i d e red a s v e r y ha rd t o cont ro l .

as•

ao•

0 2 [min]

0.4

0 .2

a 2 [min]

F i g . 5 . E x p e r i ments on a tempe ratu r e cont r o l l oop i n a suga r ref i ne ry . Phase 2 fo l l owed by a step response .

6 . L I M I TAT I ONS

The p i·oposed s i m p l e sc he<oe can obv i ou s l y not w o r k fo r a l l systems . The re a re seve ra l t h i ng s t hat may go w rong . The pa ra<oet e r est i mat i o n ff1ay fa i l i f t h e r e i s n o l j. m i t cyc l e . T h i s occu rs e . g . fo r systems w h i c h a re st r i ct l y pos i t i ve r ea l . Such syste<os poses l i t t l e p ro b l em because t hey a re easy t o cont r o l a nywa y . T h e a dd i t i o n o f a n hyste res i s may a l so b r i ng S P R systems t o osci l l a t i ons . The p re c i s e cond i t i ons fo r osc i l l a t i o n a re g i ve n Theo rem 1 . The system may a l so fa i l t o g o i nt o a l i m i t cyc l e osc i l l a t i o n i f t h e r e l a y st i ck s to o n e va l u e . T h i s m a y h a p pe n fo r st rong l y u nst a b l e systems .

Phase a nd a mp l i t u d e ma rg i ns a re . a l so fa i r l y c ru d e d es i g n c r i t e r i a . I t i s we l l - k nown t h a t systems w i t h t h e sa1ne ma rg i ns may e x h i b i t d rast i ca l l y d i ffe rent beha v i ou r .

7 . CONCLUS I ONS

The re a re many poss i b i l i t i e s to t u ne reg u l a t o rs automat i ca l l y . Se l f-tu n i ng regu l at o rs based on m i n i mum va r i a nce cont ro l , po l e p l a cement o r LQG d e s i g n methods may be conf i g u red to g i ve P I D cont ro l . S u c h a p p roaches h a v e b e e n cons i d e red b y W i t t e nma r k e t a l ( 1 980) a nd W i t t enma r k a nd Ast rom < 1 980 ) . T hese regu l a t o rs have t h e

d i sa d va ntage t ha t soff1e i nfo rma t i o n a bout t h e t i me sca l e o f t he p rocess m u s t be p ro v i ded a p r i o r i to g i ve a reasona b l e est i ma t e of t he samp l i ng pe r i od i n t he regu l a t o r . The r e a re some poss i b i l i t i e s to t u ne t he s a mp l i ng pe r i od a u toma t i ca l l y . D i fferent schemes have been p r o posed by Ku r z ( 1 979) and Ast rom and Z haoy i ng ( 1 9 8 1 > . These methods w i l l • howeve r , o n l y wo r k fo r mod e r a t e c ha nges i n t he p rocess t i 1ne const a nt s .

The method p roposed i n t h i s pape r does not su ffe r f rom t h i s d i s a d v a n t a g e . It may be a pp l i e d t o p rocesses ha v i ng w i d e l y d i ffe rent t i me sca l es . The test s i g na l w h i c h i s genera ted automat i ca l l y b y t he a l g o r i t h m w i l l have a cons i de ra b l e ene rgy a t the c rossov e r f requency of t he p rocess .

Convent i o na l s e l f-tu n i ng reg u l at o rs based o n recu rs i ve est i ma t i o n of a pa ramet r i c mod e l req u i res a compu t e r code o f a few K bytes .

T he a l go r i t hms p roposed i n t h i s pa pe r wh i c h a re based a n det e rm i na t i o n of z e ro c ross i ngs a nd peak detect i o ti may be p ro g r a <..,rned in a few hund red bytes . It is t hu s poss i b l e t o use t hese met hods even i n very s i m p l e reg u l a t o r s .

The methods p roposed w i l l of cou rse i nhe r i t t he l i m i t at i ons o f t he P I D-a l g o r i t hrns . They w i l l not wo r k we l l for p ro b l ems w h e r e m o r e comp l i ca ted regu l a t o rs a re requ i red .

The e x pe r i ences repo rted a l so i nd i c a t e t h e s i m p l e ve rs i o ns of t he a l g o r i t hms w e l l a nd t ha t t hey a re robu s t . I t a ppea rs wo r t h w h i l e t o e x p l o re a l go r i t hms f u r t he r .

t h a t w o r k t hu s

t hese

T he a l g o r i t h<os d i scussed i n t h i s pape r may be used i n seve ra l d i fferent ways . They may be i nco r po ra t ed i n- s i ng l e l oo·p cont ro l l e r s to p rov i d e an opt i o n fo r a utoma t i c t u n i ng . They may a l so be used t o p ro v i d e t he a p r i o r i i nfo rma t i o n w h i c h i s req u i red b y 1no r e soph i s t i cated a d a p t i ve a l g o r i t hms . When com b i ned w i t h a bandw i d t h se l f-t u n e r l i k e the one d i scussed in Ast rom C 1 979) it i s poss i b l e t o obta i n a n a d a pt i ve regu l a t o r w h i c h may set a su i t a b l e c l osed l oop b a ndw i d t h a u toff1a t i ca l l y .

276 K . J . Astrom and T. Hagglund

8 . ACf(NOWLEDGEMENTS

T h i s w o r k was pa rt i a l l y s u p p o rted by resea rch g ra nt 82-3430 f rom the Swed i s h Boa rd of Techn i c a l deve l o p.nent ( STU) . Th i s s u p po rt i s g ra t e fu l l y a c k no w l edged . We wou l d a l so l i ke t o t h a n k Ka l l e Ast ram who p rog ra.r.med t h e A p p l e I I i m p l ementat i o n .

9 . REFERENCES

Ast ra,·,, , K. J . < 1 ''i'75 ) . Lectu res o n system i dent i f i cat i o n . Cha pte r 3. F requency response a na l ys i s . Repo rt TFRT-7504 . Dept of Aut oo>a t i c Cont ro l , Lund I nst i tute o f Techno l og y • Lu nd • Swed e n .

Ast ram . K . J . ( 1 979l . S i m p l e sel f-t u n e rs 1 . Repo rt TFRT-7 1 8 4 . Dept o f Automa t i c Cont r o l • Lund I nst i tute o f Techno l ogy , Lund . Swed en .

Ast ram • K . J . ( 1 9 8 1 ) . M o r e stu p i d . Repo rt TFRT-72 1 4 . Dept o f Autoii>a t i c Cont ro l , Lund I nst i tute of Techno l og y . Lund • Swed en .

Ast ra"" f( . J . l 1 98 2 l . Z i eg l e r-N i c ho l s Auto-Tu ne rs . Report TFRT-3 1 6 7 . Dept o f Automa t i c Cont r o l • L u nd I tis t i tu t e o f Techtiol ogy .• Lund • Swed e n .

Ast ra''" K . J . ( 1 983) . Lectu res o n Adapt i ve Cont r o l - Chapt e r 4 : Se l f-osc i l l at i ng a d a pt i v e systems . Dept of Automa t i c Cont r o l • Lund I nst i tute o f Techno l o g y , Lund , Swed e n .

Ast ram . K . J . a nd B . W i t t enma rk C 1 973l . On s e l f-tu n i ng reg u l a t o rs . BY�Qffi��!g�_2 • 1 85- 1 99 .

Ast ram . K . J . a nd B . W i t t enma r k C 1 980 l .

Sel f-tu n i ng cont ro l l e rs based on p o l e - z e r o p l acement . !��-frgg&_!aZ • 1 20- 1 30 .

Ast ram , K . J . a nd z . Z ha o y i ng ( 1 98 1 ) . Sel f-tune rs w i t h au toma t i c a d j u stment o f t he samp l i ng pe r i od fo r p rocesses w i t h t i me d e l ays . Repo rt TFRT-722''i' . Dept o f Automat i c Cont ro l • Lund I nst i tute o f Techno l ogy , Lund . Swed e n .

E l fg ren . L . -G . ( 1 98 1 ) . Tempe ratu r reg l e r i ng med SJa l v i nsta l l a nde P I O- regu l a t o r . ( Tempe ratu re cont ro l w i t h a sel f-tu n i ng P I O-cont ro l l e r ) . MS T hes i s . Report TFRT-5254 . Dept o f Automa t i c Cont ro l . Lund I nst i tute of Techno l og y , Lund • Swed e n .

Hag g l u nd • T . ( 1 981 ) . A P I O t u n e r based on phase ma rg i n spec i f i cat i ons . Repo rt TFRT-7224 . Dept of Auto1nat ic Cont ro l , Lutid I nst i tute of Techno l og y • Lund • Swed e n .

f(a i S i ew • W . ( 1 982) . Methods f o r automat i c tu n i ng o f P I O regu l a t o rs . M S Thes i s . Repo rt TFRT-528 8 . Dept o f Automa t i c Cont ro l • Lund I nst i tute o f Techno l ogy , Lund • Sweden .

Ku r z , H . ( 1 979 l . D i g i ta l pa ramet e r-ada pt i ve cont rol of p rocesses w i t h u nk nown const ant or t i meva ry i ng dead t i •Yie . P re p r i nts 5th I FAC Sympos i u m on I dent i f i c a t i on a nd Pa ramet e r Est i ma t i on . Da r••>Stadt .

Landa u . I . D . ( 1 ''179 ) . Adapt i ve cont r o l - The mod e l refe rence a p p roac h . Ma rce l Dek k e r . New Yo r k .

Matine rfe l t , C . F . < l ''i'8 1 ) . Robust cont r o l des i g n f o r s i m p l i f i ed mode l s . P h D Thes i s . Report TFRT- 1 02 1 . D e p t o f Automa t i c Cont r o l , Lu tid I nst i tu t e o f Tec h no l og y • Lund • Swed e n .

Z i eg l e r 1 J . G . a nd N . B . N i c ho l s C 1 943 J .

Opt i .num set t i ng s fo r aut o.na t i c cont ro l l e rs . Ir�D§&_B§��-�� 433-444 .

W i tte�na rk , B . 1 P . Hagande r a nd I . Gusta vsson C 1 980l . STUP I D - I m p l ementa t i on of a se l f-tu n i ng P I O-cont ro l l e r . Re po rt TFRT-72 0 1 . Dept of Automa t i c Cont r o l • Lund I nst i t ute of Techno l og y • Lund • Swed e n .

W i ttenma r k • B . a nd K . J . Ast ram ( 1 980 ) . S i m p l e se l f-tu n i ng cont ro l l e rs i n Unbehaueti • H ( ed i t o r > . Methods a nd App l i cat i ons i n Adapt ive Cont ro l • S p r i nge r , Be r l i n .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROBUSTNESS OF MULTIVARIABLE NON-LINEAR ADAPTIVE FEEDBACK STABILIZATION

A. Hmamed and L. Radouane

LEESA, Faculft! des Sciences, B.P. 1 014 , Rabat, Morocco

Abstract . Here considered i s the stabilazing property o f the adaptive non­linear dynamic feedback control proposed in (Hmamed and Radouane , 1 983) for

multivariable systems in the presence of uncertainty . A multivariable system is decomposed into a set of single-input subsystems and the second Luenberger canonical form is obtained . This representation is suitable for non-l inear adaptive control of unknnwn systems . The measure of robustness of the proposed scheme is defined in terms of bounds of allowable perturbations such that the s tability is preserved . It is shown that the system can always remain s table for a large class of perturbations .

Keywords . multivariable systems , non-linear adaptive stabil ization, robustnes s , eventual stability.

INTRODUCTION

Usually, in practice when dealing with a con­trol problem, a designer is faced with uncer­tainty which can be defined in terms of accurate model ling availability, parameter variation, l inear and non-linear perturbations . . Therefore , in the case of incomplete informa­tion on the cons idered system, i t is necessary to investigate the robustnes s of the control design by obtaining a quantitative measure of the perturbations under which the system sta­bility is preserved . The problem of des igning robust controllers for linear 111.ul tivariable sys tems has been studied by many authors (Davison, 1 9 7 6 ; Staats and Pearson, 1 974; Francis , Sebakhy and Wonham, 1 974) . Most of these works has been devoted to the Linear Quadratic Gauss ian design (Anderson and Moore , 1 97 1 ; Barnett and Storey, 1 966) , Other works are not specifically concerned with robustness in an LQSF design • Bounds

ASCSP-J 277

on the quadratic cost of optimal regulators have been determined (Langholz and Hoffman, 1 978 , 1 979) , which contain explicitly infor­mation pertaining to some parameters of the sys tem, e . g the weighting matrices of the quadratic cos t . S tructural reliability , robus­tness properties of optimal l inear quadratic mult ivariable have been studied (Wong , Stein and Athans , 1 9 7 6 ; Patel , Toda and Sridhar ,

1 977 � Bourles and Mercier , 1 982) . In the present paper , we are interested in the evaluation of robustness of the adapt ive non-linear control scheme proposed in (Hmamed and Radouane , 1 983) for multivariable systems with uncertainty . A mul tivariable system is transformed into a set of coupled single-input subsystems giving the second Luenberger canonical form . It is then assumed that the parameters are unknown and that bounds on the perturbations are available . A decentralized and a centra­lized adapt ive control are then proposed . It is shown that the system can always be s ta-

278 A . Hmamed and L. Radouane

bilized .

PROBLEM STATEMENT

Cons ider a system whose dynamics are expressed by differential equations of the form

x ( t ) =Arnx (t ) +Brnu (t )+f (x (t) , u (t) ) ( 1 )

Where x ( t ) E: Rn, u ( t ) E: Rm, A and B denote rn rn the system model parameters and f is a non-1 inear vector funct ion . A differential equa­tion of this form may be cons idered as the l inearizat ion result of a general non-l inear equation of the form

x ( t ) g (x ( t ) , u (t) ) (2 )

Arn and Brn denot ing the Jacohian of g with respect to the state vector x (t) and the control vector u (t ) respectively , and f deno­ting higher order terms . Assuming that the exact express ion of the non­l inear function f is not usually available but only some bound may be evaluated , we shall be interested in the study of the robustness of a rnultivariable non-l inear adaptive feed­back des ign for the l inear model

x (t ) = Arn x (t) + Brn u ( t ) ( 3 )

i n the presence of some non-l inear perturba­tion f (x (t ) , u ( t) ) . The pair (Arn , Bm) is as sumed to be control lable and the r col unns of Brn are linearly independent .

In this case, (3) is deco�posed into a set of coupled s ingle-input systems , spec ifi­cally (Luenberger , 1 967 ; S inha and Rozsa, 1 976) asserts that there exist a nonsingular trans­format ion matrix S of the s tate vector and a nons ingular transformation C of the input vector which reduce the sys tem ( 3 ) to the following second Luenberger canonical fomn

y (t) = Ay ( t) + Bv (t) where

A=

O· - 1,- -- 00 - -- -

' 0 - - - ' 1

- - 0

x x - - x ', x x - - - - xxx ' '

0 - - - O·o 1 0 - - - - 0 I I

I

� x xxx - - - - - x - x - - - - x

(4)

B

0 - - - - -1 0 0 0 0 k

00 -' '

' ' - - - '

0 I . 0

v(t ) = C u (t )

The x ' s reoresent possible nonzero elements .

DECENTRALIZED STABILIZATION

The system (4) can be decomposed into rn dyna­mic subsystems Si described by

rn y . (t ) =A . y . ( t )+ � A . . y . ( t )+b . v . (t ) l l l j�i lJ J l l j =l i= 1 , 2 , . . ,rn

( 5 )

Phere y . € Rni l is the state and v . E:ll is the l input of Si . The state y ( Rn and the input v ( Rm are such that

y T '!' T T <Y l ' Yz · . . ym)

T v = (v 1 ,v2 , • . ,vrn)

The (nixni) matrix A i and the ni vector bi are given in the following cornpa n ion form

A . = l

n 1 0 - - - - n I 'o I '

0 '

" ' 1

i a n . l

(6 )

and the (n . xn . ) matrix l J is such that

aij = 0 pq p= l , 2 , • . . , nf1 ( 7 ) q= l , 2 , • • • , nj

It is assumed because of system uncertainty , that all tl'e parameters are not exactly known or unknown . The decoupled subsystems are described by

y . ( t ) =A . y . ( t )+b . v . i= 1 , 2 , . • ,rn l l l l l

Let A . = [� 1'� O +b . a:-=A�+b . a! l I ' , l l l l l

I '1 O- - - - o _

T i i i where ai = (a 1 , a2 , . . . , �n . )

l To stabilize system -'.8) , we propose the fnllowing non-l inear adaptive controller

T v . ( t ) = - (k. + A k . ) y . ( t ) l l l l

(8)

(9 )

( 1 0)

Mult ivariable Non-l inear Adaptive Feedback 279

i= l , 2 , • • ,m

where

The matrix P . is the solution of the follo­i. wing Riccati equation.

with the conventional (A� , D:) observable .

l. l. T i

<Pi (t ) = (<P I ' is such that

Remark

co i 1 1 <P · I dt< oo 0 J ' ( 1 2)

We remark that the knowledge of the parame­ters ai (9) is not necessary to solve equa­tion ( 1 1 ) . Moreover , it can eas ily be solved by using the attractive Newton iterative method . The term �i ( t ) is introduced in order to apply the theorem of eventual sta­bility (Hmamed and Radouane , 1 983) . The closed loop system is then

• 0 T T T m y . ( t )= (A . +b . a . -b . b . P . -b . tiR) y . + 1: A . . y . ( t)

l. l. l. l. l. l. l. l. l l. j;!i l.J J

where

j = I +hi (y (t ) ) i= l , 2 , • • ,m

• T i'lk . = P . (y . (k . y . ) +<P . ( t ) ) l. l. l. l. l. l.

T h (y (t ) ) = (h1 (y ( ) ) , • • hm(y (t ) )

( 1 3)

h (y (t ) ) = Sf(S- ly (t) , c- 1v(t ) )

In the augmented state space, the state (yi=o , t,ki=o , i= l , 2 , • • ,m) may not be an equil ibrium state ;then in this case ., we can not speak of the stability in the sense of Lyapunov, so other kind of stabilities should be considered . The definition of eventual stability introduced by Lasalle and Rath (Lasalle and Rath, 1 9 63) treats stabi­lity of states which are not equilibrium states but neverthless act more and more l ike equil ibrium s tates as time goes on . This definition is suitable for adaptive systems .

EVENTUAL STABILITY

Definition (Lasalle and Rath, 1 963) Consider the differential equation

x (t) = F (x , t ) , t? t0 ( 1 4)

The origin of ( 1 4) is said to be eventually stable if given E>o ; there exist numbers f and T such that I I x�< cf impliesllx<1; ,ta xc1<E

for all t> t � T . x (t , t , x ) denotes the solu-o 0 cf t ion of ( 1 4) that starts at t ime t0 at x0 • Consider the system

:ic<t> y ( t )

F ( t , x, y) G ( t , x ,y)

( 1 5 )

Where x and y are m1 and m2 vectors respecti­vely • It is assumed that F ( t , x, y) is bounded For bounded x and yand a l l t � t0 .Let v (x , y)be Lyapunov function satisfying the following conditions : a) v(x,y) is positive definite and has conti-

nuous partial derivatives b) v (x ,y)-- as I ! x i 1

2+ 1 I Y I 12

--c) v (x,y)� -w(x) +h l ( t ) q (x ,y )+h2 ( t )v (x ,y ) where 1) w (x) is continuous and positive definite 2) q (x,y) is continuous 3) �00 l hi < t > I dt<00 i=1 , 2 ,

Theorem I If for the system ( 1 5 ) there exists a function v (x,y) satisfying conditions a )-c) , then the state x=o , y=o is eventually stable and corres­ponding to each r ) o there exists a T, such

2 2 2 that I l x ( t0) I I + I l y ( t0) I I <r for some t0�T , implies that y (t ) i s bounded and x ( t ) tends to zero as t tends infinity . If in addition d) for some k> o and some o <n < I

then all solutions y ( t ) are bounded and all x (t)+ o as t+ co • In the following section we shall apply this theorem to the present scheme .

Theorem 2 The system ( 1 3) is eventual ly stable (all x (t )+ o and tik remains bounded as t + oo ) if the matrix f is a (-M) matrix and if the non-l inear vector function hi (y ( t ) ) satisfies

the condition

( 1 6)

i= l , 2 , . , , ,m

280 A. Hmamed and L . Radouane

Hj

Proof Let v(y, Ak) be a luapunov function candidate for sys tem ( 1 3)

v(y, &) m

= . f d . v . i= i i where di a�e posiiive

Tscalars ior all i

Ak = ( Ak 1 , Ak! ,1

• • i

. , t.km) v . (y . , t.k . ) = - y . ( t) P . y . ( t ) i i Ti- l � i i i

1... (a . - t.k . ) P . (a . - t.k . ) 2.. i i i i i +

( I 7)

The time derivative of vl along motion ( I 3) is

, _ _, . T T , _ _ T - 1 • v . - _ (y . P . y . +y . P . y . ) (a . t.k . ) P . t.k . i z. i i l. i i i i i i i

_, T " T T o T = 2.-y . ( (A . -b . b . P) P . + P . (A . -b . b . P) ) y . i i i i i i i i i i i T T - 1 ' + (ai -t.k . ) (y . (Fb . ) y . - P . t.k) + i i i i i i

T T + y . P . 2,.A . . y .+ y . P . h . (y (t ) ) i i j � � iJ J i i i

By tacking into account equtions ( 1 1 ) and ( 1 3) , we obtain

v · = - ·} y�(Q . + P . b . b�P . ) y-t (a . -t.k . ) Tep . ( t )+ i � i i i i 1 i i i i i T T

+ y . P . L A . . y . + y . P . h (y ( t ) ) i i iJ J i i . By us ing equation ( 1 6) ,we ha�e

• 1 T T v . � -,.y .D .y . + (a . - l\k . ) q> . ( t) + i ' i 1 i i i 1 y::P . L A . . y . +_!.J. . (D . ) I y l 2

i i iJ J � min i i So we can write

2 ,;..c ,.\. ( .! n. - � (D. ) Ii. ) I y l + i nun Z.. i min i i

I a. - t.k. 114> . < t )j + I Y. 1 1 p .\ rf·A . . 1 1 Y . I i i i i i iJ J where I A ! = A (ATA ) max

! xi is the euclidean norm of vector x

By using the inequality

A . (P� ) I a. -t. k, j' d a. - t. k. t?J(a. -t. k. ):i: 2v. min i i i • i i i i i i we have

I a. -n. k lzv./1>.� ) i i ¥' .. min i

For the over system the time derivative of the as sociated Lyapunov function is

. m T .r.-: - 1 v = Ld . v .� Z D F Z + L d i .P i &. /V� (P. ) i i i t i min i

where T z = < fy1\ , IY21 , . . . . . . . , \vi ) D =diag (d 1 , dz , . . . . . . . . �)

It is clear that v satisfies the condition of theorem 1 and theorem 2 is proved .

Given that the system parameters are assumed to be unknown ,we can not guarantee that the matrix F is a (-M) matrix . In this case it i s possible t o adjust the linear gain k�of each subsystem (Hmamed and Radouane 1 983) or to modify the controller dynamic by introdu­-cing another non l inear term .

FIRST AMELIORATION

Let the control be of the form vi (t ) = - (k. (13 . )+ t.k. ) Y. ( t ) ( 1 8 ) i i 1 1

"'T'" ki (Si ) = Pi (Si ) bi , Pi (Si ) =Pi (Si ) > 0 P. ( s . ) i s the solution of the Riccati i i equation

o r o (A. +s . I. ) P. (S . ) +P. (13 . ) (A. +s . I. ) -1 i i i i 1 i i i 1

T Pi (Si )bibi Pi (Si ) + Qi =O ( 1 9)

where I • is the ( n. n . ) unit matrix and the 1. 1• l parameter S · represents the degree of stabi-

l o T -lity of the matrix (Ar -bi ki_l �i)) Theorem 3 IF the matrix F is a (-M) matrix and i f the non-linear funct ion hi (y ( t ) ) satisfies the condition

I hi <y <t> ) I 1 ----� 2 I Yi (t ) I

i=l , 2 , • • • ,m then the system ( 1 3) with the control ( 1 8) i s eventually s table where F= (F . . ) ,F . . = iJ iJ

\ - � <-21 <n. - � <n� r . > > -min i min • i

12 , (P . (!3 . ) - � P . (13 . ) ) i=j � i i i min i i

!Pi (13 i � !Ad i;'j

Multivariable Non-linear Adapt ive Feedback 28 1

T D . = Q . +P . (B . )b . b . Pi ( ,i. ) i i i i i i

The proof is similar to that of theorem 2 . llemark The condition (20) is the one obtained in (Patel , Toda and Sridhar , 19 77 ) in the case of a single system (m= l ) .

SECOND AMELIORATION Let the control be of the form

T . A T v . ( t )=- (k . (13 ,. ) ( l +y . (P . (13 . ) y . )+'-'k . ) y . (t ) i i i i i i i i i

( 2 1 ) where P . (13 . ) i s given b y ( 1 9 ) .

i i

Following the same steps of calculations as before, we can state the following theorem

Theorem 4 If the matrix F is a (-M) matrix and if the non-linear function h . (y ( t ) ) satisfies the

i

condition

� (D. ) 1 (P . (13 . ) ) min i + 13 . min i i

I P . <13 . ) I i I P . <13 . ) I i i i i

"- (D _0 )"- min (P . (13 . ) ) 2 + min i 'i i i I y . I I P . <13 . ) I i

i i

i=l , 2 , • • • ,m

+

then the system ( 1 3 ) with the control ( 2 1 ) is eventually stable .

where 1 r I- - A. -- . (-2 (D . - . (D. ) I . ) -F-F _ min i min i i - ij1-a · '- . (P . (13 . ) - "- . P . (a . ) )

µ i min i i min i µ i

I P . (a . ) 1 1 A • . I i µ i iJ I

i=j

The previous condition necessate that the matrix F ( theorem 2 , 3 and 4) be a (-M) matrix. This is a constrainte where a decentral ized control is used . However this constrainte may be removed by letting each control be a func­tion of the whole state vector .

CENTRALIZED CONTROL

Let us recal l system ( 1 3) . Assuming that each controller observes the whole state vector ,we propose the following control law

T l3 T v . ( t ) =- (k . (13 . ) ( I +y . p . ( . ) y . ) +& k. ) y . ( t ) -i i i i i i i i i

m T I: e . . y . ( t ) j= I iJ J

j ;!i

( 22)

• T t1 k . = P . (y . (k. (ll . ) y . ) -4\j> . ( t ) ) ( 23) i i i i i i i

� . . = P . (y . ( t ) (P . (s . ) b . ) Ty . ) ( 24 ) iJ i J -1 i i i

The closed loop becomes

y. ( t ) = (A�+b . a:-b . b: (P . (fl . ) -b .ll k:) y . + i i 1 l. i i i i i i i

m T I: b . (a . . -e . . ) y . ( t ) +h . (y ( t ) ) ( 25 ) • J_ . i i J i J J i J r i

j= l where a . . is defined as

i

:ij = 1-� � �-���� -- �-

= b . a: . • . . . . . i i] iJ i] l.J a l , a2 , • • . ; anj

T a . . l.J

Theorem 5

ij ij aij. ) (a l , a2 • · · · · • nJ

Consider the system ( 1 3) with the control defined by ( 22) , ( 23) and ( 24) . If the condi­t ion

I hi ( t ( t ) )I I � 2 I Yi < t )I

+ "-. (D. -Q . ) min i i

\ (D . ) A. p . ( 13_, ) min i + 13 min i l..... i p . rA . )! i I p. ( 13. ) I

i � ip ) ]_ i

"- min ( i � i I y . ( d 2

I P . (f3 j i

]_ ]_ i= l , 2 , • • • • ,m

(26)

is satis fied , then system ( 1 3) is eventually stable . Proof m Let V(Y , ll K,E) = I: d . v .

i= l i ]_

where E -= {_ '- • j } 1 T I T - I v . = -2 y . ( t ) P . y . ( t ) + -2 (a .-i> k . ) P . ( a . -tik . ) +

]_ i i i ]_ ]_ ]_ ]_ i

1 m T - I -2 I: (a . . -e . . ) P . (a . . -e . . ) • J_ • ·l.J l.J � iJ iJ

] T l.

j = l i= l , 2 , • • • ,m

The t ime derivathTe of vi along motion ( 2S) is • 1 T [ o T v . ( t ) = -2 y . P . ( 13. ) (A . + 13. I . -b . b . P . ( 13. ) ) +

i ]_ i ]_ ]_ i ]_ ]_ ]_ ]_ i

(A?+ f3. I . -b . b!P . ( 13. ) ) TP . e13. ) J y . i ]_ ]_ ]_ i ]_ ]_ i i ]_

+ (a .-ll k . ) T [y . ( (P . ( S. ) b . ) Ty . ) -P :- 1 (13. ) ti.-, + -i ]_ ]_ ]_ ]_ 1 ]_ i ]_ ]_ "' - T [ T -. r. (a . . -e . . ) y . ( t ) (P . ( 13. )b . ) y . -

J 7"l. -l.J l.J J 1 1 i 1 j = l - I ·1 T P . ( 13. ) e . . + y . P . ( 13. )h . (y ( t ) ) 1 1 1 ] _ 1 1 1 1

By us ing relations ( 23) , ( 24 ) and (26) ,we obtain

282 A. Hmamed and L . Radouane

; . {t ) � - r:>... c-21 cn . - A. (D . ) I . ) - 8. A. P . ( 8. )1

i Eiin i min i i i min i i

I 2 1 121 � 2 \hi (y {t ) )l y . ( d +v . �. ( t + IY . ( d IP . ( 8. >I · ( ) i i i i i i I Yi t I It is clear that under condition ( 26) , the Lyapunov funct ion satisfies the conditions of theorem I and theorem 5 is proved .

CONCLUSION

A non l inear adapt ive control scheme for non

linear systems is proposed . The non l inear

system is decomposed into a set of coupled

single-input subsystems . Sufficient conditions

of s tabil ity are derived us ing the notion of

eventual stability . The robustness of the

scheme is defined in terms of bounds on the

the perturbations such that the s tabil ity is

preserved . It is shown that the control dynamic

can be adjusted such that the system stability

is preserved .

REFERENCES Anderson , B . D . O . , and Moore , J . B . ( 1 97 1 ) .Linear

Optimal control . Parentice -Hall , Enp;law>O" Cliff s , N . J ,

Barnett , S . , and S tory , C . ( 1 966) . Insensitivity

of optimal linear control systems to persistent changes in parameters . Int , J , Cont , ,Vol . 4 , pp 1 79- 1 84 ,

Bourles , H , , and Mercier , O , L . ( 1 982) .Marp;es de s tability et robustesse structurelle generalisees des regulateurs l ineaires quadratiques mult ivariables , R ,A . I ,R ,O .Aut , Syst ,Analy . and Cont . ,Vol 1 6 , pp . 49-70 .

Davison, E . J ( 1 976 ) , The robust control of a servomechanism problem for l inear t ime invariant multivariable systems . IEEE Trans ,Aut , Cont , ,Vol . 2 1 , pp , 25-34 .

Franci s , B . , Sebakhy , O ;A , , and Wonham,W .M .

( 1 974) . Synthesis of mul tivariable regulators . The internal model princi­

ple ,Appl ,Math . optimi z . Vol , 1 1pp , 64-86 ,

Rmamed ,A . , and Radouane ,L . ( 1 983) . Decentralized nonlinear adaptive feedback stabiliza­tion of large scale interconnected systems . IEE Proc . ,Vol . 1 30 ,Pt .D ,N0 . 2 , pp . 57-62

Langholz , G . , and Hoffmann,A . ( 1 978) .Lower bounds on the quadratic cost of optimal regulators .Automatica ,Vol . 1 4 ,pp . 1 7 1 - 1 7 5

Langholz ,G . , and Hoffmann ,A . ( 1 979) .Lower and upper bounds for the quadratic cost of linear regulators . Int . J . Cont . ,Vol . 29 , N0 . 5 ,pp . 797-802 .

Lasal le , J . P , and Rath ,R , J . ( 1 963) .Eventual stability . Proc .of Sec . Bong .of the I .F .A . C ,pp . 556-560 .

Le.enberger , D . G . ( 1 967 ) . Canonical forms for linear multivariable systems . IEEE Trans Aut . Cont . , pp . 290-293 .

Patel ,R .V . ,Toda ,M . , and Sridhar , B . ( 1 977) . Robustness of linear quadratic s tate feedback designs in the presence of system uncertainty . IEEE Trans .Aut . Cont . Vol .AC-22 ,No . 6 , pp . 945-949 .

Staats ; P .W . , and Pearson . ( 1 974) .Robust solution of the l inear servomechanism problem .Dep .Elec .Eng . ,Rice Univ .Houston TX,Rep . 740 1 .

Sinha ,N .K . , and Rosza , P , ( 1 976) . Some canonic­al forms of linear multivariable syste­

-ms . Int . J , Cont . ,Vol . 23 ,No . 6 ,pp . 865-883 .

Wong , P . K , , Stein , G . , and Athans ,M . ( 1 978) . Structural reliability and robustnes s properties of optimal linear-quadratic multivariable regulators . Proc . 7 I . F .A . C Congress .Helsinki . pp . 1 797-1 805 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE MODEL ALGORITHMIC CONTROL 1 •2

W. E. Larimore, Shahjahan Mahmood and R. K. Mehra

Scientific S_vstnm Inc , Cambridge, MA 021 40, USA

Abstract . �bdel Algorithmic Control (MAC) is a relat ively new design methodology successfully used by indust ries for the last s everal years . The obj ect ive o f th is paper is to invest igate robustness propert ies of MAC , and evaluate the use o f adapt ive methods for real-t ime identificat ion o f the plant under close d-loop control . Some theoret ical robustness propert.ies o f MAC are g iven in terms o f clas sical qualities such as gain margin and phase margin for a wide c lass o f systems . Although MAC is an output-feedback controlle r , it has a guaranteed cont inuous-t ime equivalent phas e mar­gin of 60° , and the upward gain margin can be made arb i t rarily large by 5low in g down the referen ce t raj ectory . Some robustness p roperties o f MAC are also given b y a perturbat ion analysis of a miss-modeled p lant impuls e response . Preliminary results are discussed for on-l ine identif icat ion of the clo sed-loop plant us in g the canonical variate method . Performance o f the identificat ion o f the plant in the presence of both input and measurement noise is g iven .

Keywords . Adapt ive contro l ; ident ifi cat ion ; robustness ; canonical variate analy s is ; model algorithmic cont rol .

INTRODUCTION

The MAC methodology generates a control sequence by on-line optimization o f a cost-functional , and the algorithm is suit­able for implementat ion on microprocessors . One o f the attract ive features o f MAC is the clear and transparent relat ionship between system performance and var ious des ign para­meters embedded in the des ign procedure . MAC has been described elabo rately in the l iterature (Mehra e t . al . ( 1 9 7 7 , 1 979 , 1980) , Mereau e t . a l . ( 1 9 78) , Richalet et . al . ( 1978) , and Rouhani and Mehra ( 1 982) ) , and therefore only a brief descript ion o f MAC is g iven below. The z-transform or s-trans form of a time function is denoted by replacing the time-argument by z or s respect ively ; for example y ( z ) deno tes the z-trans form o f y (n) . For the sake o f s impl ic ity a s ingle­input s ingle-output sys tem i s considered although the extens ion to multiinput mult i­output plants is conceptually s traight­forward .

There are five basi c elements in MAC ;

( i ) An actual plant with a casual pulse response funct ion h ( t ) = {h . , i = l , . . . N } , l.

This work was supported by the Air Force Wright Aeronautical Laboratory .

283

an input u ( t ) and an output y ( t ) to be control led . The y ( t ) are related through a convolut ion operator ( *)

or ,

( ii )

with

o r ,

N y ( t ) = h ( t) * u ( t ) = /: h . u (t-i )

i= l l.

y ( z ) h (z ) u ( z ) , h ( z )

A model of the plant

output y( t) and input

N y ( t ) /: h . u ( t-i)

i= l l.

y ( z ) h ( z ) u ( z ) , h ( z)

N • /: h . z- i

i= l l.

h ( t ) = {h . , i l.

u ( t ) s o that

N /: h . z- i

i = l l.

( 1 . 1 )

= l , • • • N}

( 1 . 2 )

( iii) A smooth t raj ectory y (t) ini t iated on r

the current output y ( t ) that l eads y ( t) to a possibly t ime varying set point c . The y ( t)

r

2 Repr inted from Proc . IFAC Workshop on �daptive Systems in Control and S ignal Processing , June 20 22 , 1983 , San Francisco .

284 W . E . Larimore , S , Mahmood and R . K. Mehra

evolves as follows :

yr ( t+l) = o:yr ( t) + ( 1-o:) c ( t) , yr ( t )

- 1 - 1 y/z ) = o:z y ( z ) + ( 1-o:) z c ( z )

y ( t ) ( l . 3a)

( 1 . 3b)

where o: is a constant determining the speed of response ;

( iv) a closed loop predict ion s cheme for pre­dict ing the future output y ( t ) of the plant p according to the scheme

y ( t+ l ) = y ( t+ l ) + y ( t ) - y ( t ) p ' -1 y ( z ) = y ( z) + z ( y ( z ) - y ( z ) ) p

( 1 . 4a)

( l . 4b)

and finally

(v) a quadrat ic cost funct ional J based on the error between y ( t ) and y ( t ) over a finite P r horizon T :

J i= l

[ ( y ( t+i) -y ( t+i)) 2 w ( i ) p r

+ u2 ( t+i-l ) r ( i- l )) J ( 1 . 5 )

where w ( i) and r ( i ) a r e t ime varying weights . Usually r ( i ) is chosen to be zero .

Given ( i) - (v) , MAC finds an optimal control sequence {u* ( t+i- 1 ) , i= l , . . . T- 1 } by minimizing J over the admiss ible input sequence 1.u (t+i- l ) t..Sl ( i ) , i = 1, • . . T- 1 } . Once the opt imal cont rol sequence is computed, the first element of the sequence is applied to the actual plant and the process repeats all over again .

To invest igate the theoret ical propert ies o f MAC and t o interpret MAC from the classical control viewpoint we make the following assumpt ions :

( i) the actual plant h ( z ) is min imum phase ;

( ii) Sl( i)

there are no input constra ints , i . e . R for all i , where R is the real line ;

( iii) the op t imizat ion is carried over one future s tep ahead i . e . , ( T= l ) ; under this condition MAC is a one-step ahead predictive controller .

Under these s impl ifying assump t ions , it is sufficient to select u* ( t ) sat is fying

y ( t+l ) = Yr ( t+l ) for all t ( 1 . 6) p for a minimum o f the cost funct ion J. The assumpt ions ( i ) - ( ii ) ensure the existence o f an optimum control u * ( t ) sat is fying U .6 ) . ut' ( t ) i s then implicitly generated b y y ( z ) = y ( z ) so that

p r

( z - l ) h ( z ) u ( z ) + ( 1-o:) h ( z ) u ( z )

= ( 1-o:) c ( z ) ( 1 . 7)

By further manipulat ion ( 1 . 7) can be express­ed as

u ( z ) 1 -o: c ( z ) = ( z-l ) h ( z ) + ( 1 -o:) h ( z ) ( 1 . 8a)

� _ h ( z) ( 1 -o:) c ( z ) - (z- l ) h (z ) + ( 1-o:) h ( z ) ( 1 . Sb)

Equat ions ( 1 . 8) imply that MAC under assump­t ions ( i) - ( iii) above is equivalent to the following class ical unity feedback configura­t ion in an input-output sense .

l Compensator y(z) - --,

Plant

h(z)

Fig . 1 . MAC as a Classical Controller

This interpretation of MAC is the basis of our analys is o f MAC in terms o f class ical contro l .

PHASE AN D GAIN MARGINS

The block within the dashed l ine can be cons idered as a dynamic controller of the class ical type . The loop trans fer funct ion at poin t 1 is

L( z ) h ( z ) ( 1-o:) fi' ( z ) ( z- 1 )

and the return difference funct ion is

h ( z ) ( z - 1 ) + h ( z ) ( l -o:) l+L( z) = ���-�-���� h ( z ) ( z - 1 )

( 2 . la)

( 2 . lb)

The error e ( z ) g iven by

c ( z) - y ( z) in t racking is

e ( z ) - 1 ( l+L( z ) ) c ( z )

s o that the s teady state error due t o a step input is

e ( t ) S S L im ( l+L ( z) ) -l = ( 1 + L( l ) ) - l z-+l

0

which is a consequence of a builtin inte­grator in the compensat o r . It may be noted that u s ing the set-up o f Fig. 1 and by treat� ing ( 1 -o:) as a gain , the usual classical root-locus technique can be appl ied to analyze the behavior o f the closed-loop poles as o: changes from 0 to 1 . To make the root­locus pict ure complet e , the characteris t ic polynomial can be rearranged with a modified gain S = o:/ ( 1 -o:) s o that as o: changes from 0 to 1 , S changes from 0 to infinity .

Adaptive Model Al gorithmic Control 285

It may be noted from Fig. 1 that at poin t 2 , x ( z ) =y ( z ) when h ( z) = h ( z) , where y ( z ) is the reference signal . This shows wh� perfect tracking is possible under perfect ident ificat ion . We will , however , not pursue 'this approach here .

It is obvious f rom ( 1 . 8 ) and ( 2 . 1 ) that the closed-loop system is internally asymptotic­ally stable if the roots of the rat ional function

¢cl ( z ) = ( z - l ) h ( z ) + ( 1 -a) h ( z ) ( 2 . 2 )

are within the open un it disk ! z l < l , and these roots are also the roots of the return difference function 1 + L ( z ) . We can there­fore f ind the stability margin in terms of the gain margin ( GM) and phase margin ( PM) from the Bode plot or Nyquist plot of the loop transfer func t ion L( z ) evaluated on the Nyquist car. tour z = exp (j w) appropriately indented around the poles on this contour . Recal l that in continuous-time , the GM and PM are those values of k and ¢ respec tively such that the perturbed loop L( s ) = kexp (j ¢) L( s ) is s table , where L( s ) is the nominal loop and s is the Laplace variable . A s imilar interpretation goes for the dis­crete-t ime systems (Kuo ( 1 980) ) ; but the PM, unless it is an integral value of the sampling interval , does not have any physical significan ce . Strictly speaking the complex cons tant kexp (j ¢) in continuous time should be replaced by kz-n , n an integer, for measur­ing GM or PM of the discrete-time system.

Another way to compare with other con tinuous­time domain des ign techniques is that each element of the disc rete-time loop should be transformed into an equ ivalent continuious-t ime element using bilinear trans format ion , and PM of the fictit ious continuous- t ime loop can be taken as the PM of the discrete-time loop . In this paper the word PM is used to mean the continuous-time equ ivalent phase margin . We can now state

Theorem 1 :

Under assumptions ( i) - ( iii) , MAC has _ 1 GM = ( 0 , 2 / ( 1-a)), equivalent PM ?= Cos ( 1 -a) / 2 , and unity gain cross-over f requency

u.1 = 2 s irt - l 0-a) /2 . 0

Proo f : The proof is trivial if we recall that PM and GM are measured on a nominal loop . Here we can assume that the nominal plant h ( z ) = h ( z ) , which impl ies h . = h . and N = N because both h ( z ) and h ( z ) afe po�er series in z- 1 . The nominal loop transfer function from (2 . la) is then

1-a L ( z ) = z-l ( 2 . 3)

i . e . an integrator delayed by one-s tep . Evaluating on z = exp ( j 1JJ) , we get

ASCSP-J*

L (exp ( jw)) 1 -a 2

1 -a w j -2- co t 2 ( 2 . 4 )

and IL ( exp ( jWo ) J 1 . 0 implies the unity gain cross -over frequency at

w 0 -1 1 -a 2 s in -2- ( 2 . 5 )

The Nyquist plot of the dis crete-time loop ( 2 .4 ) is quite s imple and f rom the plot it is easy to see that the system is s table for all gain s (O , 2 / 1 -a), and a pure delay ¢ = 90° - S in - 1 ( 1-a) /2 will change the number o f enc irclement by the Nyquist con­tour , thus making the system unstable .

To get the equivalent PM we t ransform each element of the loop usin g the bilinear trans ­formation s = ( z - 1 ) / ( z+l ) -l to g e t the equivalent continuous loop

1 -a 1 L ( s ) = -2 C; - 1 ) . ( 2 . 6 )

From the Nyquist p lot of L ( s ) it is obvious that GM s ( O , 2 / ( 1 -a) ( the same as found by analyzing the discrete-t ime Nyquist plot ) and a PM = Cos- 1 ( 1-a) /2 .

Theorem 1 , although very s imple , reveals some intuitively appealing resul ts about GM and PM o f MAC . We can make the following remarks .

Remarks :

( 1 ) S ince as [ 0 , 1 ) , the guaranteed upward GM is 2 and the PM is 60° respectivel y .

( ii) W e can always t rade-o f f robustness against the speed o f response . As response speed is in crease d by decreas in g a , BW 111 = 2 s in- 1 ( 1-a) / 2 inc reases (which makes sgnse) with a consequent reduct ion o f robustness in terms o f G M and PM .

( iii) We get this remarkable PM even though MAC is an output-feedback con t roller poss ibly because the plant is inverted causally through the use o f an optimization algorithm in the sense that at each t ime the a lgorithm provides the controller with the enti re future input sequence . For the same reason , the discrete-time loop has a one pole roll­off for all frequencies - which is rather unusual .

( iv) Theorem 1 ensures that the contro ller can stabil ize the loop for all the plant s { h . � belongin g t o the set { i I - -h i h i = khi , i= l , . . . N , kE(O , 2 / ( 1 -a)) } .

PLANT ROBUSTNESS ANALYS I S

The nominal model h ( z) is usually different from the actual plant h ( z) for various reasons . Somet imes h ( z ) is del iberately made s imple to facilitate the cont ro l compu­tat ion by retain in g the modes in the act ive frequency range . On many o ccasions it is

286 W . E . Lar imore, S . Mahmood and R . K . Mehra

diff icult to model high frequency modes , and these are s imply neglected . Due to age­ing , �tc . , the modes of the a ctual plant drifts slowly thus introducing low-frequency error . Thus the model ling error e ( z ) has in almos t every cas e , a dynamic s t ructure ; and the information about e (z ) mus t be incorpor­ated in des igning a nominal loop . As a minimum amount of informat ion e ( z ) is expres­sed as an upperbound on f e (exp ( j w)) I ; and the purpose of robustness analys is is to find a requirement on the nominal loop in terms of this upperbound so that the closed loop performance and stabil ity is maintained in the face o f modelling uncertainty .

Usually the admiss ible uncertain t ies are expressed in two way s : additively or multi­plicatively . , If we take h(z ) as the nominal plant , then in an additively un certain model , we express the actual plant h ( z ) a s

h ( z ) = h ( z ) + Ah ( z ) ) a ( 3 . 1 ) and in a mult iplicatively un certain mode l , the actual plant h ( z ) is

h ( z ) h ( z ) ( 1 + Ah ( z ) ( 3 . 2 a ) m

or h(z ) = h ( z) Ah ( z ) ( 3 . 2b) m

For s ingle-loop systems the orde r of mult i­plicat ion in ( 3 . 2 ) is irrelevant , but for MIMO cases the order is important because of the non-commutat ivity o f mat rices where input channel ( left ) uncertainty and output-channel ( right) uncertainty mus t be distinguished . Both of the mult ipl icat ive forms in ( 3 . 2 ) are often used in analys is , but in this paper we sha ll he u s ing ( 3 . 2b) . Note that at nom­inal values of the plant , Ah ( z ) = Ah ( z ) = 0 a m and Ah ( z ) = 1 . Also note that the class ical m GM and PM ensures the stability o f a perturb -ed plant of the form ( 3 . 2b ) . I f the GM is k, then Ah (z) = k, and if the PM = n ( in the sense of dis crete-data system) , Ah ( z ) = z-n . These are undoubtedly a limited cl�s s o f allowable perturbat ions and we must cons ider other poss ible error-st ructures in des igning the nominal loop. The framework of ( 3 . 1 ) and ( 3 . 2 ) is more general in the sense that it can handle a constant , non-constant and even dynamic mo del mismatch ( say for example unmodelled poles , et c . ) . Let us rewr ite h ( z ) and h ( z ) as

where

and

N h ( z ) = l: h . z- l

i= l ]_

h ( z ) p

h ( z )

h ( z ) p

N 2, i= l

-N z h

N

h . z N-i 1

( z ) ' p !f- i

l: h. z i= l ]_

z-N h ( z ) p ( 3 . 3a)

a polynominal in z ,

( 3 . 3b )

Then b y straight forward manipulat ion , the closed loop characteristic polynominal is

¢ ( z) c l , p

N -z ( z - l ) h ( z) p

N + z ( 1-o:) h ( z) p ( 3 . 4 )

For closed-loop stab ilit y , ¢cl , p ( z) must have all the roots strictly ins ide the un�t disk l z.I = 1 . For perfect identification N = N , - N ( -hp{ z) = hp ( z ) , and ¢cl , ( z ) = z zNo:) hp ( z ) .

Of course the zeros of h ( z ) and z will be cancelled eventually lea�ing the only closed loop pole at z = o:. However N, the order of the t rue plant , is usually unknown . In real­world s ituat ions , ( 3 . 4 ) can not be evaluated . The actual plant h ( z ) must be considered as a perturbat ion of the nominal plant h ( z ) , and the stabil ity condit ions must be derived in terms of the nominal sequence {\11 } and the perturbat ion Aha ( z ) or Ahm( z ) . Let us assume that Aha ( z) and Ahm( z ) can be expressed as in (3 . 3) , i. e . ,

Ah ( z ) a

N a

I. i= l

- i h . z ai

-N Ah ( z ) , Ah ( z ) a polynomial ( 3 . 4a) z ap ap in z

N m -i -N {\h ( z) l: Ah . z z m h ( z ) ( 3 . 4b ) m i= l mi mp

although the fol lowing theorem can be develop­ed without such an explicit form. No te that the index in ( 3 . 4b ) must s tart from 0 to accomodate constant mult iplicat ive perturba­tion . We have the following theorem on robustness :

Theorem 2 :

( i) The system is closed-loop stable for all addit ive perturbat ions Aha ( z ) satisfying

I Ah ( z ) [ ap I 2 -2<1 Cos (<' + a2

< l il ( z ) I , p 1 -a

z = exp ( j u.V ( 3 . Sa)

( ii) The system is closed-loop stable for all mult iplicative perturbat ions Ahm ( z ) satisfying

N I Ah ( z ) -z m r mp

< 1z-Ci I 1-a ( 3 . Sb)

on the unit circle , where Ah (z) and Ah ( z ) ap mp are g iven by ( 3 . 4 ) .

Proo f : The proof is s t raightforward if we express h ( z ) using the form ( 3 . 3) - ( 3 . 4 ) , f ind the corresponding closed-loop characteris­t ic polynomial, and f inally use Rouche ' s theorem to prove ( 3 . S ) on the assumpt ion that

Adapt ive Model Algorithmic Control 287

the nominal loop is internally s table and hence ( z-a) h ( z ) has all the roots s trictly inside the uni� disk l z l = 1 . The tests of the type ( 3 . 5 ) are sufficient conditions and generally tend to be conservative . Nevertheless we can make the following remarks : ( i) Both tes ts ( 3 . 5a) and ( 3 . 5b) are useful. For example when an actual known model {h . , i= l , . . . N } is truncated to obtain c i - - l { hi ' i= l , . . . N, N 5:_ N _ , so that llhai = hi , i = N , N + 1 , . • • N and llh i = 0 , i < N} , a -stability around {h , } can be obtained from ( 3 . 5a) . i

( ii) For constant mult iplicative gain mis­match , i . e . h . = kh . for all i, {llh . = k i i mi when i=O and l\h . = 0 when i>O } , so that

N mi llh ( z ) = kz m and test ( 3 . 5b) yields that mp the system is stable for all k such that

l k - l I < � , z = exp ( j w) ( 3 . 6)

But it is easy to see that min l exp ( jw) - a l = 1 -� so that ( 3 . 6 ) becomes / k-1 < 1 which implies kc(0 , 2 ) . This clearly shows that these test are conservative . ( See remarks ( iv) of the previous sect ion) . CLOSED-LOOP IDENTIFICATION The results of identificat ion of the plant under closed-loop control us ing MAC are described in this sect ion . The major dif­ficulty in closed-loop identification is that the future plant inputs are correlated with the past outputs due to the feedback. Many ident ificat ion procedures assume the absence of such correlation , and produce biased estimates or have other difficult ies in their presence . Maximum likelihood will handle such correlation , but can be computa­t ionally expensive especially if not prov­ided with good init ial estimates of the parameters .

For identification in this s tudy , the can­onical variate analysis method was used. This approach to stochastic realization was first proposed by Akaike ( 1 975 ) . A recent generalizat ion (Larimore ( 1983) ) extends the method to input-output identificat ion in the presence of noise . The method is based upon a decomposit ion of the covariance matrix of the past p ( t ) and future f ( t ) of the plant input proces s u ( t ) and output proces s y ( t ) where

T T T T ( u ( t ) ' y ( t ) ' u ( t-1 ), y ( t-1 ) ' . . . ) T T (y ( t+l ) , y ( t+2 ) , . . . )

A canonical variate decompos it ion of the covariance of p ( t ) and f ( t ) determines the important linear combinations of p ( t ) for predict ion of f ( t ) . From this a state-space

model is constructed of the form

where w and v are white noise processed that are independent with covariance matrices Q and R respectively. These white noise processes model the covariance structure of the error in predicting y from u. Computa­tionally , a s ingular value decomposit ion of the sample covariance matrix between p ( t ) and f ( t ) is used . This decomposition i s numerically very well condit ioned and stable .

To demonstrate the identificat ion algorithm, the feedback system under MAC control il lus­trated in Fig. 2 was considered where there is input white noise added prior to observ­ing the plant input and output white noise added prior to observing the plant output with power spectral densit ies SI and S0 respect ively . The particular plant consid­ered is the very lightly damped miss i le dynamics model (Mehra et . al . ) ( 1980) (x1)

(- 1 . 4868 l . OO)(x1) ( 0 )

x2 =

-149 . 4 3 0 x2 +

28 1 . 1 1 u

y x 1

where the s tates are x1 = a the angle o f attack ( rad) , x2 = P the perturbed pitch rate (rad/s ) , input u = oa the elevator angle ( rad) , and output y = a the angle of attack (rad) . An analysis of the dynamics gives a natural frequency of 12 . 24 r/s ( 1 . 95 Hz) and a damping ratio ( �) of 0 . 06 1 .

The canonical variate method was used to identify a second-order system while operat­ing under MAC control with input and measure­ment noise . No other input nor change in the set point was present , and the system was in statist ical steady-state. The pre­sence of an input or varying set point would improve the ability of the algorithms to identify the plant . The plant was approx­imated by a dis crete time system using the exponent ial transformation at a sample rate of 10 Hz . This was used for the actual plant in the discrete time s imulation , and in the MAC control computations the discrete t ime impulse response was used out to 5 seconds and set to zero at longer times . The true and ident ified plant models are shown using sample s izes of 100 and 900 in Fig . 2 . Note that the identified plant is close to the true even for a substantial amount of measurement noise . REFERENCES Akaike , H . ( 1 975 ) . Markovian Representation

of Stochas tic Processes by Canonical Vari­ables, SIAM J. Contr . , Vol. 1 3 , pp. 162- 1 7 3 .

288 W . E . Larimore, S . Mahmood and R . K . Mehra

Input Noise

Feedback

Second

Order

Plant

MAC

C ontroller

(a) S imulat i0n Model

20

- 1 0

FREQUENCY C HZ l

Output Measurement

Noise

( c ) Magn i tude Transfer Funct ion

2

- 1

T I ME( SEC I ( b ) Pu l se Re sponse

� Cl � 0 1-������-->.."-r-��������--1 UJ Cf) a: I CL

-"Tf

FREQUENCY C HZ l ( d ) Phase Transfer Func t ion

0

Fig . 2 . I dent i f icat ion Under MAC Contro l , True Plant ( So l id ) , Ident i f ied P l ant for N- 100 , s1= 1 , S

0= 0 . 0 l (Dashed) , and for N=900 , s1= 1 , S0= 0 . 5 ( Do t t ed) .

Kuo , B . C . , ( 1 980) . Digital Cont rol Sys tems ,

Ho l t , Rinehart and Wins ton , pp . 407 - 409 . Larimore, W. E . ( 1 983) . Syst em Identif icat ion ,

Reduced-order Filte ring and Mode l ing Via Canon ical Var iate Analysis , Proc . 1 983 Ame rican Cont rol Conference San Francisco, CA , June 22-24

Mehra , R. K . , W . C . Kessel , A. Rault and J. Richalet ( 1 9 7 7 ) . Model Algor ithmic Con trol Us ing IDCOM for the F- 1 00 Jet Engine Mul t ivariable Con trol Des ign Problem. Int erna tional Forum o f Al ternat ives for Multivar iable Cont rol .

Mehra , R. K . , R. Rouhani . A. Rault and J . G . Reid ( 1 9 7 9 ) . Model Algorithmic Contro l : Theoretical Resul t s on Robus tness . Pro c . Jo int Automa t ic Cont rol Conference P P • 387-392 .

Meh ra , R. K . , J . S . Et erno , R . Rouhan i , R . B . Washburn , Jr . , D . B . S t illman and L . Praly ( 1 980) . Basic Research in Digital Stochas t ic Model Al gorithmic Cont rol , Techn ical Report AFWAL-TR-80-3 1 2 5 , Air Force Wright Aeronaut ical Laboratories , Wr ight -Pa t terson AFB , Ohio 4 5433 . DTIC Document AD-Al02 14 5 .

Mereau , P . , D . Guil lamme and R . K . Mehra ( 1 9 78) . Fl igh t Control Appl icat ion o f MAC with IDCOM ( iden t i f icat ion and com­mand) . Pro c . I EEE Conf . on Decis ion and Control , pp . 9 7 7-982 .

Richal et , J . , A. Raul t , J . L . Testud and J . Papon ( 1 9 78 ) . Model predict ive heuristic con t rol : applicat ions to indus t rial processes . Automat ica , Vol . 14 , pp . 4 1 3 .

Rouhani , R . , and R . K . Mehra ( 1 9 82) . Model Algori thmic Cont rol (MAC) ; Basic Theoret ical Properties , Automa t i ca Vol . 1 8 . pp . 4 0 1 -4 14 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

DESIGN OF DISCRETE-TIME ADAPTIVE SYSTEMS BASED ON NONLINEAR PROGRAMMING

C. -B. Feng and H. Li Nanjing Institute of Technology, Nanjing, Jiangsu, People's Republic of China

Abs tract . Design me thod of d ete rm i n i s tic d is crete - time a d a ptive system b a s e d on nonlinear prog ramming is s tu d i ed . A pos i ti ve de­fin ite quad ra tic. cos.t function i s formed by u s ing th e vP.ctor out­put e r ror betwe e n th e i d e n tifier a n d the un known s y s tem, and then d ifferen t g rad i ent me thods are applied to c a l cu l a te the pa r a mP te rs of the sys tem . In order to in cre a se the convergence spe<>d of the ide n ti f i ca t i o n proce s s an inc.re a se of the d imen s i on of the ou tput e r r o r vector is su gge sted . The pro po sed. me thod i s a l s o a p pl i ed to d e s ign of ad aptive con trol s y s tem s .

Keyword s . Ad a p tive sy s tem s ; ada p ti ve con trol ; identi fica tion ;

IN TRODU '.:TIQN D e te rm i n is tic. a d a p t ive sy s tems in cl u ­d in g ad aptive observe r:s and rnod el -r:e­fe ren ce a d a p tive con trol s y s tems are u su a l l y d e s i g n on the bas i s cf stabi ­l i ty theory . Fo r th i s pu r pose the Lya ­punov fun c tion method or Po pov ' s hy pP r ­stabi l i ty theory can be used to d efine the parameter ad a p t a ti on laws (La n d a u , 1 979 ; N a randra , 1 980 ) . Al though t h e g l o bal st.abil i ty o f s u ch s y s tems i s guaran teed , the convergence of the e s ­timate·cL parameters to thei r true value s d epend s u pon wheth er the cond i tion o f " pers i s ten t exc i ting " is p rovid ed o r n o t ( Eykh off , 1 977 ; Morg_an , 1 977 ) . The pa rameter con vergence proces s w i l l b e in te rru pted i.f th e- s y s tem is no l on g_e r exci ted . In general , the convergence s peed of the p arame ter es ti ma tion pro­ce s s. i s very l ow . Vec to r ou tpu t e rr.or i s used by K r e i s s e lmeier ( 1977 ) to i n ­cre a s e th e s pe ed of p ar amete r estima­tion for. ad a ptive observe rs , which is. an L'll porta nt imP.rovement.. Applicati on of n o n l i n e ar programm i n g i s. s u g g e s te d by Feng ( 1982 ) to d e term ine the u n known paramete rs of conti nuou s -time s y s tems. . A c:onvex function can be e a s i l y fo rmed by u s in g the vector output ernoi:. Then d i fferen t g rad ient method s of n o nl i n ear prog rammi n g can be u s ed to dete rm ine the s y s tem paramete r s . The re i s a wide choice of method s. . In this p a per tr.his method is u sed to d e term i n e t h e paramete rs of the di screte ad a pti v.e o b serve-rs an d of the con trol l ers •

The a p pl i cat i on of n o n i n e a r prog ramming to d e si g n of adaptive observers i s. f i rs t d i scus sed in t h i s paper. D e s i g n sch emes for s tate-vari a b l e mod e l and ARMA mod e l are given res pective l y . A pos.i tiv,e d e -

289

fin i te quad r a ti c cost function is form­ed by u sin g the vector output error be­tween the o b se rver and the ori gi n al sys­tem , and then d iffe ren t grad ient methods are a pplied to ca l cu l a te the pa rameters of the i d e n tified system . A comparison of the-se m ethod s i s d i scus s ed . In order to in creas e the s peed of convergPnCIP. of the i terative calcul a tion a method of improv ing the cost function is sug� gested . The propo s ed method is al so ap­pl i ed to the d e s i g n of mod e l - reference ad aptive con trol sy s tem s . Two feasible design schemes are sug g e sted .

D ES I GN Of ADAPTIV� OBS ERV� D e s i g n s fo r s t.ate-v a r i a b l e modPl a n d ARMA mod e l of s ingle-input and s ingle­-outpu t s y s tems a re given res pectively . The output error equ ation between the o b s e rv e r and identified s u s t"'m i s fi rst es t a b li s h ed . From thi s equation the ou tput error vect or is o b t a ined .

E:rr o r Equation

S t.a te-vari able mode l . As sume that the s t.ate-variable equation of a singl"'­-inpuQ and single-outpu t s y s tPm is a s fo l lows :

x ( k+ 1 ) =Ax (k ) +bu ( k ) , x (0 ) =x0 } y ( k ) =cTx (k )

( 1 ) where x ( k) E Rn i s the s tate vecto r , u ( k } and y ( k ) a re th e sca l a r fnpu t . and output. As sume that the s.y tem 1 s com ple tely con trol l ab l e and observalbe . Withou t loss of generality the matr ices. A, b and c can he expres sed in the c anon i cal form a s :

290 c . -B . Feng and H. Li

A= [a :- �g ::J]E R nxn

a= (a 1 , a2 , . . · , a�T € R n

b= [b l ' bz . . . . ' b�T E Rn

c= [ 1 ' 0 ' . . . , oT E Rn

w h e re a a n d b a re u n k n own p a r a m e t e r vi:>cto r s .

Thi:> LuPn b e r g e r o b se rve r o f th i s s y s tem h a s t h e fo l l ow i n g fo rm

x( k+ 1 ) �Fx ( kj + p 1 y ( k ) +p2u ( k ) , x ( O ) =x0 l ( 2 )

y ( k }=cTx ( k ) w h e r e x ( k ) i s a n e s tima te o f x ( k } ; p l '

p2E Rn a r e the pa rame ter v e c t o r s o f the

o b server to be d e t e rm ined ; FE R n x n i s a kn own s ta bl e m a t r i x w h i c h h a s the form

� [f:-�fl:.11 f={f1 , f2 , · · · , f�T I 0 ,

If p 1 and p2 s a t i s fy tth e m a tc h in g con ­

d i ti ons , we h ave

T F=A- p 1 c , o r p 1 =a -f , ( 3 ) P2=b .

r.he refore , w h en f i s g i ven , t h e s y s tem p a r am P. t e r s can bP. d e t e rrn i n Pd o n l y w h e n p 1 and p2 a re fou n d .

From ( 2 ) we have

k k-I k . 1 x ( k } =F x0+ � F -J - y ( j ) p 1 + � Fk-j -1 u ( j ) p2 ( 4 ) J:. O

D e f i n e two n x n - d i m Pn s i o n a l m a t r ice s s 1 ( k ) and s2 ( k ) a s fo l l ow s :

S 1 ( k )= �� Fk-j - l y ( j ) ( 5 ) S ( k ) = f: Fk - j - l u ( j ) ( 6 ) 2 j-:.O

Th e s e two m a t r i ce s s a t i s fy t h e fo l l ow­ing P.qu a t i on s respec t i v e l y :

s 1 ( k+1 ) =FS 1 ( k ) + 1 ny ( k ) , s 1 ( o ) =O }( ? )

S2 ( k+ l ) =FS2 ( k } + l nu ( k ) , S 2 ( 0 } =0

U s in g ( 5 ) and ( 6 ) , !::'q . ( 4 ) can be e x ­pr,,.s sed a s

� ( k ) =s 1 ( k ) p 1 +s2 ( k ) p2+Fkx0

w h e re

( ) " k� =S k e+F x0

S ( k )={s 1 ( k ) , s2 ( k )] E Rn x 2n

" [ T T ]if 2n 8= P 1 ' P2 E R

( 8 )

From ( 2 ) a n d ( 8 ) WP c an obtain

y ( k ) =cf} ( k ) S+cTFkx0 , ( 9 ) w h e re

<\:>( k ) =ST( k } cER2n ·

�q s . ( 8 ) and ( 9 ) d efinP the pa ramet r i z ed obse rve r f o r d i s c re te - t ime s y s tem a s g iven b y Ta k a s h i a n d co-wo rkers ( 1 980 ) .

-i< -l<T �· T JT N e w l <> t 8 ={p 1 , p2 rl enotes the f)8 r arne-t�r v � c t o r wh i c h s a t i s f i e s the matc h i n o cond i t i on ( 3 ) , t h e n the sys tem s t a te v x ( k ) and ou tpu t y ( k ) �ay b"' e x p r e s s ed r e s pec t i vely a s fo l l ows :

* k x ( k ) =S ( k ) 8 +f x0 , ( 1 0 � T * T k y ( k ) = <P ( k ) e +c F x0 • ( 1 1 )

Su b t r a c t i ng ( 1 1 ) f r om ( 9 ) we o b t a in the ou t pu t e r ro r e ( k ) :

e ( k ) =q} ( k )B+cTFkx,, , ( 1 2 ) u

w h e r e �=@-if d en o te s tho p a r amete r error v e c to r , x ( k )=x ( k ) -x ( k ) d eno tes the S t A te e rro r V P. c t o r . Be ca u s e F i s a s ta b le m a t r i x t h e e r r o r dua to th<> i n i tial s t a te orror xc w i l l f a d e a way to ze ro .

When u ( k ) and y ( k ) ar0 me a s u ra b l � ,� ( k ) is k n own . Then the identi fica tion of the s y s tP m is c h an ged to the p r o b l em of f i nd in g p 1 and p2 which fit the ra-r a m e te r ma tch i n g cond i t i c n s s o a s t o m a k e � and e ( k ) ., q u a l t o zero .

ARMA mod e l . The �; e n i:> r a l forrr o f the t, R M 4 mod e l i s

n rn y ( k ) = � a . y ( k-i ) + t: b . u ( k-j ) , ( 1 3 )

l = I 1 )= I J �i thou t l o s s of gen e r a l i ty we may l e t m=n . The u n k n own s y s tem pa rametP r vcc� to r s a re :

a= [a 1 , a 2 , . · - , a�T

b= [b 1 , b2 · · · · , bJ For A R M A mod al a n o b s e rver with the s ame s t r u c tu re as t h e o r i g in a l s y s tem is a d o p ted . I ts eq ua t ion i s

n n "' y ( k ) = L'. 13 . y ( k- i ) + � b . u ( k - j ) , ( 1 4 )

i :. j 1 J= I J w h e re a and b a re t h e e s t i ma tes of a and b r e s p e c t i ve l y .

S u b t r a c ti n g ( 1 3 ) f rom ( 1 11 ) w e have

e ( k ) =�T ( k } B ( 1 5 )

w h e r e ¢ ( k ) = [ y ( k- 1 ) , · · " , y ( k-n ) , u ( k- 1 ) ,

· . . ,

u ( k -n ) ] T

e={a1 1 · · ,an ,'b l , · · · , ti =[C a- a ) : ('b- b )TJT

(15 ) and (1 2 ) a re s i m i l a r in f o rm . They

Design of Discrete-time Adaptive Sys tems 2 9 1

are the same when the error d u e t o the initial s tate error has faded away . Re ­cause F i s s table t h e error d u e t o the i nitial s tate error w i l l s u rely fade away to zero . In practica l a p p l i ca­ti on the� adaptive obse-rve r: can be fi rst exci ted , and when this initial e rror has pi::actica l ly faded away to ze ro , then d a ta are reg i s te red for identifi­ca tion calculation . For conc i seness the term cT Fk x w i l l be omi tted i n the fol l owingaHa l y s is . Thu s the error equa­tions for these two mod e l s are the same in form and formu l ae for c a l cu l a­tion of paramete rs a re a l so the s ame .

Cost Function

Define an ou tput error vecto r a s

E ( k ) =[e ( k-2n + 1 ) ; . . , e ( k ) J TEO R 2 n • We have

E( k )=W ( k )S W ( k ) = q} ( k-2n + 1 )

q,T ( k- 2n + 2 )

ql ( .k )

( 1 6 ) ( 1 7 )

A pos i tive def in i te q uadratic cost fu nction is formed by u s ing ( 1 6 )

L ( S ) =tET ( k ) QE( k ) =tSTWT ( k ) QW ( k ) S =teTR ( k ) � ( 1 8 )

where

R ( k ) =WT ( k ) QW ( k ) E R2 nx2n , Q=QT> O . In the 2n-d imen s ional 8 vector s pace !:q . ( 1 8 ) represents a perfect hy per­e l l i p soid w i th its center at the o ri­gin .

Appl ication Of Non l i near Programmi ng

Fnom ( 1 8 ) we know that R ( k ) i s s ymme­tric pos i t ive d�fin i te�and L ( g ) has its m inimum a t 8=0 . L ( 9 ) i s a s t ric tly convex function , who s e min imum i s uni­que . �he refore if the value o f � which min imize s L( � ) i s found , then the true va l ues of the sys tem parameters are obta ined . Thus the pa rameter identi­fication i s converted to a problem of minimization of the cos t fu nc tion . This problem can be solved by u s ing d iffe rent g radient method s of nonlinear prog ramming .

The partial�de rivatives of L(S) w i th res pect to 8 are

� L ( e ) /� e�vL ( S ) =R ( k ) S=WT ( k ) QE ( k ) ( 1 9 ) a2 L( S ) /ae2� v2 L (B ) =R ( k ) ( 20 )

All the partial derivatives of o rd e r 3 a n d h i g h e r are e q u a l t o ze ro .

The formu lae of recu rs ive calcu l a t ions for d iffe rent g rad ient method s a re as fol lows ( Avriel , 1 9 76 ) . S teepe s t descent method ( S DM ) .

A />. ?( 8t=8 t- 1 -�tV L( � t- 1 ) ( 2 1 )

>..t=VLT 0\_ 1 ) v Lct\_ 1 ) /v L r c et_,) R(k)vL < �-� "' ( 2 2 )

Here et denotes the value of 8 a fter tth recu rs ive iteration . Du ring the re­curs ive calculat ion k rem a i n s unchanged .

( 2 1 ) defines the correct ion o f e in­stead of the correction of 8 . Bu t s ince et-et- 1 = Cet-e* ) - C ©t- 1 -e� ) =et-9t- 1 • Therefore the correction of § is equal to the correction of � . This i s the same for all t h e form u l a e in the fo l ­lowing .

Conjugate grad ient method ( CGM )

( 2 3 )

( 2 4 ) z t+ l =-VL(et ) + �tzt ( 2 5 )

h=VL T ( S� \IL( E\) /1J L T (et_1) Q L (-et-� ( 2 6 ' 2n where ztE R denotes the vecto r for one

d imen s i onal optimization , which satis­fies ziR ( k ) zj =O ( i¢j ) . Eq s . ( 2 3 ) - ( 2 6 ) define a complete algorithm for uncon­s bained m i n imi zation .

Theoretica l l y the conj ugate g rad ient me thod guaran tees that the t ru e p a ra­meters can be obta ined after 2n i t era­tion s , where 2 n is the number of the parameters to be es timated . Bu t in practice because of the round-off e rror of compu ter the true parameters can be obtained after 2n i terations on l y when the cond i t ion number o f R ( k ) i s not too large .

Newton method ( NM ) ,-. A - 1 c ( � ) et=et_ 1 -R k ) � L et- l ( 2 7 )

from ( 1 9 ) and (27 ) we obtain A ,-. - 1 � A (" *) � e =8 -R R =a - A -8 =8 . t t-1 -1 , - 1 't- 1 '

This �quation s hows . that the t ru � para­meters w i l l be obtained w i thout i tera­tion . But for th i s pu rpose the inverse o f R ( k ) should be ca lc�lated . There may be some troubl e i f the d imen s ion of R ( k ) i s h ig h .

Variable metric method ( VMM)

et=et _ , +A.t 2 t �t=-z�VL (�t- l ) /z�R ( k ) z t

( 2 e )

( 29 )

292 C . -B . Feng and H. Li

zt=-Htv l (�t ) ( 3 0 ) T T

!:it =H + dt..,dt ... _Ht °tt+i( Ht 1't+1) ( 3 l ) ' t+ 1 t T T v dt+I �ti Ot+I Ht 0ttl d t=e

t-et- 1 (3 2 ) �t= VL (et ) -VL (et-1 ) ( 3 3 )

]h i s method has a s pe c i f i c feature : H t begins w ith iden t i ty matrix a n d s c a l a r optim izat ion is carried out a l o n g the conju gate d i recti�� and a f t e r 2n- 1 ite­rations H2 n_ 1 = R ( k ] is obta ined . Fin a l l y the l a t e r i s i d e n t i c a l t o t h e Newton d i rection . Therefo�e the true pa rame te rs can be obta i ned afte r 2n i teration s .

S imul a t ion � e s u l t s

Simu l a t ion h a s b e e n made o n a d i g i ta l computer w i th 4 8 b i t s for a s y s tem of 2 n d orde r . Q= 1 i s taken and 1 k ind s o f i n pu t s i g n a l a re used :

u 1 ( k )=Sin ( C . 2 k ) +S in ( C . 5 K ) u 2 ( k ) =Si n ( 0 . 5 k ) + 1 0S in ( k ) u3 ( k ) =2 E [1 ( k - 1 0n ) - 1 ( k- 1 0n - 5 )]

ri:o

The i n i t i a l pa rame ters of the ad a p tive observer for the s t a te -v a r i a b l e mod e l a re tak<>n t o be §T =[-f1 ,-L, , O , O ] . S imula-o L T tion resu l ts for d ifferent. f= [f 1 , f2] are l i s ted in T a b l e 1 . Fo r A R M A mod e l the i n i ti a l pa rameters of the ad 2 ptive ob­s e rver i s ta ken to be 0 . T he s imu l ation resu l t s are l i s ted in Table 2 . In t.h�se tables fl =C ond ( R ) denotes the condi tion numb<>r of matr ix R. Cond ( R ) = Amo.x(R ) /t,,,in(Rl when• Ama.x ( R ) and Amin( R ) a ri:> th� l ar g e s t and sma l l e s t e i g enva l u e s of R . t!i d enotes the value a ft e r i th ite ration .

The fol l owing con c l u s i ons can be drawn u pon a n a l y s i sof the s imu l a t i on resu lts :

( i ) . The v a l u e of fl has a s trong influ­ence on the convergence pro perty of the ca l cu lation .

( i i ) . Con v e r g e n ce of the steepe s t des­cen t me thod i s very poor .

( i i i ) . Conj u g a te g rad ient method and variable me t r i c method po s s e s s g ood con­vergence pro pe rty . The conve rgence pro­perty of the va r i a b l e metric method seems even bette r .

( iv ) . In ou r example Newton method a p­pears to be very effective . Hence i t s hou ld b e c ons ide rred t o b e a good method for s y stems of l ow o rde r . Improvemen t of l'he Co s t Funct ion

Simu l ation shows that the cond i t ion num­ber of matr ix R stron g l y affect the c:mnv e rg e n ce of the i terat ion calculat ion . The converg ence s peed w i l l ,be increased by reduc ing th is cond i t ion number. We know R ( k ) =Wlk ) QW ( k ) , wh�rP �( k ) depends upon the input and can ' t be change a r­bi trari l y . Reducing the condi tion number by a proper choice of Q is a l s o very d i ­ffi cul t . U su al l y for s impl ic ity Q i s ta ken to be t h e id�ntity matri x . Then we have R ( k )::.1.l( k )W ( k ) . In th is case from ( 1 7 ) we have

R ( k ) = �¢ ( k-i )q;'°( k- i ) ( 34 ) I" 0

This eq uation s hows that R ( k ) i s com­po sed of 2n symmetric ROS i t i ve semide­fi n i te ma trices ¢ ( k-i }q:l( k- i )( i=O , 1 , - · · , 2n- 1 ) . Suppose that the vector � ( k- i ) ' s are linearly ind e pendent. Then a l l the s.emid efin Lte pasi tive matrices are a l l

TA3 LE 1 S imu l a tion Re s u l t s of State - Var i a bl e Model

f;= [1 . 49 , -0 . 5 5f T 11'= [o . 5 , -0 . 5 5]T , 8= (0 . 03 , -0 . 0 5 , C . 43 , -0 . 3 5] Ef= [1 . 02 , -0 . 05 , 0 . 43 , -0 . 3 5]

C GM Y2 =4 1 2 4 [u 2 ( k ) ] ;r ,.. ll=94 5 [ u z. ( k )] T @44= [o . 02 99 , -o . 0499, o . 4 2 99 , -o . 3 499} e1t{1 · 0 1 9 9 , -o . 0499 , o . 4299 , -o . 3499] ,.. Y{,= 1 5 62 4 [u 1 ( k )] ;r ,.. tl =26 1 8 U ( k ) T VMM e55= [ C . 03 08 , -0 . 0 5 0 , 0 . 43 0 , -0 . 3 5 0� 64=� . 0 1 99 , -0 . 04 99 , C . 4 3 0 , -0 . 3 5 0)

� = 1 5 6 2 4 [ u , ( k )] NM § 1 = [0 . 03 , - 0 . 0 5 , 0 . 4299 , -0 . 3 4 9 9]

TAB LE 2 Simul ation K e s u l t s of ARMA Mod e l

CGM

SDM

* [ T e = 1 . 52 , -0 . 6 , 0 . 43 , -0 . 3 5] fl=6372 [u,( k)] T §8= � . 5 1 99 , -0 .5 999 , 0 . 4299 , -0 . 349�

e*= (0 . 4 , -Q . Q3 , 0 . 89 , -0 0 65]T

}Z=2 1 90 u ( k ) T e8= [0 . 40 , -0 .030 , 0 . 8899 , -0 . 6499] _

fl. =27 fu3 ( k )] T @407= [0 . 3 3 85 , -0 . 1 088 , o . 901 7 . -0 . 593 9]

Design of Discrete-time Adaptive Systems 293

di fferent. F�om the v iew point of [eo� metry we can obtain a perfect hy per­e l l i psaid in the 2n-dimen sional para ­meter error vector. s pace from ma.tr.ix R ( k ) . Th is hypere l l ipsoid i s exanded in all 2n independent di re1:tions . from each posi tive semidefin ite matrix we obtain an im perfect hy pere l l ipsoid wh ich is not expanded in s.-ame direc­tions . Hence i ts con d i tion numb�r is oo. Since al l the cp( k-i ) ' s are independen t. therefore the di rections in wh ich each impe rfect hypere l l i psoid is no.t ex pand­ed are differen t . Thu s these 2n differ­Pnt impe rfect hy pere l l ips.oids al l toge­ther form a pe rfect hypere l l i ps.oid wh ich i s ex panded fully and it.s condi­tion number i s finite . N.ow let u s in� crease the d imen s ion of output error vector E"( k ) by one . We have

R ( k }= � cp( k-i )<PT( k-i ) t � - 1

Suppose that <l>( k+1 ) is al so independent of the other vectors . Then the u nexpan­ded d i rection s of the hy pere l l ipsoid for.med by 4>( k+1 )4:t( k + 1 ) are al so. d i ffer­ent from the unexpanded directions of a l l those limperfect hyr;>ere l l ipsoids . Thus the add ition of .P ( k+1 )cjiT( k+ 1 ) to the former R ( k ) cau ses the perfect hy­per.e l l ipsoid t:o be more "s,to.ut" and neduces its ccnd i tion number .

Simuia�ion has heJOJn made to verify the foregoing argument.. Calculation has be11>n done for the s ta,te-vari.able model with u1( k ) as in put . The resu l t. i s shown ii1 Fi g . 1 • Cu rve I shows the change of the cond ition numben of R ( k ) with the dimen s i on of f( k ) unchanged . • Cu rve lI shows thP. change of the cond ition num­ber of R ( k ) with the d imen s ion of f( k ) con tinu0u s l y increase d .

Cond ( R) ) 5000 1 0 00 0 I

n k 6 i 8 q 10 I I 12. 13 '14. 15 5000

Fig .

S imulation shows tha.t the increse of the d imen sion of E ( k ) is truely an P.'.f�ctive method for �educin g the con­d 1 t1on number of R ( k ) . Con sidering the foregoing d i scu s sion we can change the method of identification calculation as fol lows : If the sys tem parameters can ' t be obtained du ring a definit.ein­terval of time ( for example , d urin g 1 sampling period ) we can increase the d imens ion of E ( k } by one and continue the calculation as before .

DES IGN OF ADAPT I VE CONTROLLER

Tue foregoi n g method of calc.ulation for

parameter identification can equally appl ied to the des ign of adaptive con­tro l l ers . S ince we are dealing wi th determin is t i c syst1=>ms , the system para­meters under the closed loop cond ition can lue identified by the foregoing me­t.hod and then different: k inds of sel f­-t.un ing controller can be designed . Erut the foregoing method can also be d i rect­ly ap plied to the d P si gn of model-re­ference adaptive control systems (MRACS ). Two feas i bl e schemes of design a re pre­sen ted below .

MRAC.S Scheme I

The des ign scheme shown in Fig . 2 i s adopted .

fig .

'Tihe control l�d obj ect is defined by a( q-1 ) /A ( q- 1 ) which is u nk nown . As sume that

The parameters of the control ler are to be determined as fol l ows : Assume that

( - 1 ) - 1 -n L q -1 0+1 1 q + . . ·+lnq

H ( q - 1 } =h 1 q- 1 + " "+hnq -n ( 3 8 )

If A and B are kno\Oln 1 then take L=El and A+H==C , we h ave y � t ) =Y'( t ) , 'Ir/ t . In order to obtain the parameters of the control led obj ect we apply our method to i dentify the fol lowing system :

B ( q - 1 ) z ( t ) =A{ q - 1 ) y ( t } ( 3 9 )

'Tih is equ a tion can be changed to

L( q -1 ) z ( t )= (C (q - l )-H (q- l )] y ( t ) ( 40 ) or (41 ) where

q,T ( t )=[y (t- 1 ) ; · · , y (t-n) , z ( t ) ; · · , z (t-n)] (4Z )

8 T={h,-q, h 2 -c2 I . . , hn-cn ' lo ; . . , ln] ( 43 ) Parameters h . and 1 . can be determined 1 1

294 C . -B . Feng and H. Li

by ao lving (41 J , using the me thod p ro­po sed in this pape r . Thus the paname ­ters of the controller can be d e te r­m i n e"CI d i rec:tly .

MRACS Scheme-II

Another scheme of des ign i s shown i n fi g . 3 .

u(t) (t )

Fig . 3

From th is block d i ag ram we h ave

y ( t )=(o(q� / ( A (q1) D(g'(��(q;l , H (q1�} u ( t) (44 )

Let this eq u ation be i den tically equi­v a l ent to the following equat:i.orr:

then WP have

B ( q -1 )=D ( q -1 )+L( q - 1 )

A ( q -1 ) =C ( q -1 ) -H ( q - 1 )

( 4 5- }

(46 )

( 4 7 )

where D ( q-1 ) and C(q-1 ) are g iven . Now use z ( t ) a n d y ( t ) as inp u t and outpu t and s-0lve the fol low i ng e-quation :

Lll( q'"') +L( qi ] z ( t )�(q1) -lt(q1)J y ( t ) ( 4 8 )

then the o pe rato rs L(q - 1 ) and lt(q -1 ) can be de te rm ined .

CONCLUSION Appli cat ion of nonlinear progI'.amming to the d e s ign of i d e n tifier a nd. ad ap .,, tive control ler for di screte-time sys­tem i s pr:esentecl in th is pape r . The conve rgence pr�pe rti e'S of the differ•

e n t g r.ad i'en t methods are anal y zed and compare"CI with each other. Simu l a tion s hows that if the numbe r of parameters to be id en tified i s not toCl la rge and the inve rse of the matrix R ( k ) is easy to obtain , then the New ton method is. a good onP. . If the invB rs e of R ( k ) i s d i fficu l t to calculate , then the con­

j uga te g rad i en t me thod and variable metric method may be applied . They possp s s good convergence property , wh i lP the con verg Pnce property of the steepest descent method is u s u al ly very poor .

The con d i tion number of the matrix R ( k ) has strong influence on the sp�ed of

conve r g e n c o of thP i d e n t i f i ca t i on pro­CP. s s . Incn,,a se of tho d im o n s i on of the ou tpu t error VPCtor CT k ) i s i:affect i ve for reducin� thp cond i tion number of R ( k ) .

The pro po sed method has the fol lowing ac:Wen tage s :

( i) . ThP p a rametors o f tho i d o n t i fi o r o r of t h e con tro l l e r can d e te rm i ned from on ly 2 n o r a few moro i n pu t a n d o u tp u t sam pl i ng d a ta . Th e cond i t i on o f " pe rs i ste n t e x c i t i n g " i s a v o i d o d . T he l a te r i s u sual l y n e ce s s a ry f o r t h P c o n ­vergence o f recu rs i v� i d e n t i f i c a t i o n p roce s s .

( i i ) . Si nce the pa rame te rs c a n b o d o t o r­m ined b y u s i n g l i m i ted s a m p l i n g d a ta , therefore the id en t i fi ca t ion a n d c o n ­tro l o f u n s t a b l e s y s tPm bPCOmP pos s i b l o .

� i i i ) . . When t h o a d a r tivo con t rol s y s tPm is d€ si g n e d on the b a s is o f rocu rs i vo ca l cu l� ti on, . the q u o s ti on of g l o b a l s tabi � i ty s hou l d be a n sw e re d . T h i s pro­blem i s a h ard onP . In ou r d o s i qn tho whol e s y s tem is l i n e a r b o fo ro a n d a f­t e r th 0 change of the c o n t ro l l P r pa r a ­m e te rs , a n d the pa rame tPrs aro c h a n c "' d on l y o n cp . T h e roforo t h P pro b l om of

"

g l o b a l s tab i l i ty is avoidod . Th i s i s somew h a t l i ko thP hy br id a d a p t ivo c o n ­tro l recen t l y d eve l oped ( El l i o t t , 1 982 \

R CFEREN C C Avri P l , M . ( 1 97G ) . N on l i n .., a r rroc r am ­

m ing . F· r.,.n tic e - H a l l , In c . E l l io t t , H , ( 1 982 ) . Hy b r i d A d a p t ivo

Con trol of Con t i n u ou s Tim"' S y s tem s . IEEE T r . AC- 2 7 , 4 1 9-4 2 6 .

:y k h off, P . ( 1 9 7 7 � . Sys t"'m I d o n t i fi c a ­�· J o h n �:i l o y a n d S o n s . Fen g , � . - B . ( 1 982 ) . Id on t i f i o r d o s i q n

v i a T im o - V a ry in g N o n l i n P a r Pron ram re ing . R P c o n t 0Pvolopmo n ts in ccintro) T h o e r and Its A l i cat i on s . G o r d on a n d B r0 a ch , S c i o n c o r u b ] i s o r s , Inc . 3 1 1 -3 1 8 .

Kre i s s p lme ie r , G . ( 1 977 ) . Ad a p t i vo Ob­s e rv o r wi th [x ponen tial R a t o for Conve rg o nc� . IEr:E T r . AC-2 2 , N o . 1 .

Lan dau , I . D . ( 1 97 9 ) . Ad aptive C o n t ro l . D e k kP r , In c .

RarPn d ra , K . S . , Y . -H . L i n , L . S . Va l va n i ( 1 9 80) . Sta b l o Ada p t i v� Con trol l o r D� s i gn - - P a r t II : P roof of Stab i l i ty . I� Tr. AC-25 , 440-449 .

Nar1:>ndra , K . S . , Y . -H . L in ( 1 980 ) . St ab l e D is c retP Ad a p t ive Con trol . I E EE T r . AC -Z5 , 4'6 -4 6 1 .

T ak a�u zu k i , T a ku m i N a k amu ra , M a s a n b ­ri Kog a , ( 1 980 1 D i scroto Ada pt ive O b s P rvPr with Fa s t C o nve rgenc e . In t . J . Control, Vo l . 3 1 , N o . 6 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

A STABLE ADAPTIVE CONTROL FOR LINEAR PLANT WITH UNKNOWN RELATIVE DEGREE

S. Shin and T. Kitamori

Department of Mathematical Engineering and Instrumentation Physics, Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo 1 1 3, Japan

Abstract . This paper presents a method to realize a tracking type stable adaptive control without knowing the p lant ' s relative degree . A structure estimator for the unknown relative degree is introduced to the ordinary adaptive control which consists of the parameter estimator and the control input generator. The relative degree is es t imated with a multip le identi­f ication model using a criterion whether the parameter estimation is worked well or not . The global s tability will be ensured by combining the estab­lished method and the nature of the criterion . Through simulation s tudy , it is verified that the adaptive control sys tem works well even if the relative degree is changed suddenly .

Keywords . Adapt ive control ; relative degree ; delays ; structure es t imation ; stability ; tracking system; identification .

INTRODUCTION systems .

Since Monopoli ( 1974) proposed an adaptive controller with the augmented error signal, there has been considerable progress in the field of the adaptive control. The global s tability is proved in many adaptiv� control systems for the linear p lant (Egart , 1979 ; Goodwin, 1980 ; Morse , 1980 ; Narendra , 1980) . Egardt ( 1979) also constructed a basic s tructure of the adaptive control system through comparative s tudy between the Model Reference Adaptive Control (MRAC) and the Self Tuning Regulator (STR) . However , there remain many problems unsolved yet in practical application and in theoretical analysis .

This paper presents a stable deterministic adaptive control for the discrete-time linear plant with the unknown relative degree . The relative degree is one of the structures , and a structure estimator must be introduced to the ordinary adaptive controller which consists of the parameter estimator and the control input generator . The introduction should be done keeping the global stability of the total control system.

It is one of the problems that the tracking type adaptive controller requires a priori exact knowledge of the relative degree , which is the difference between the numbers of the poles and the zeros of the plant transfer function. The requirement is too strong to be satisfied in practice and contradicts itself even in the theoretical framework since the relative degree to be known is determined by the coefficients of the transfer function which is assumed to be unknown a priori .

Inspite of such importance , there have been a few literatures considering in the point . Kurz ( 1979) proposed a heuristic approach to overcome the difficulty. Fessel and Karny ( 1979) also proposed a Baysian approach in the s tochastic enviroment . They , however , have not been successful in analyzing the closed loop property of the designed control

295

Figure 1 proposed

y \ t + O l

Fig . 1 .

shows a adaptive

eslim1111t>n f( t ) error 1

thrC'SIOld 1/ 1 )

input UC\)

block diagram of control system .

I .- - · - - - - - - - -. I

output ---• Y ( t )

:- -: _ _ _ _ _ _ _ _ _ l - : L MODEL u '

Multiple identification model adaptive control system. The model number corresponds to the assumed relative degree .

the It

296 S. Shin and T. Kitamori

includes the multiple identification model composed of models each of which has one of possible structures . The relative degree , i . e . , the structure is estimated through two stages . At the t-th step , the estimator selects a subset I (t) of models from the multiple identification model as the first stage (THRESHOLD LOGIC) . A criterion for the selection is obtained in relation to stability proof of MRAC . On the second s tage , one identification model i� ·chosen from the set I (t) according to a prespecified preferece index (INDEX [ . ] ) . The control input is synthesized based on the estimated parameters and the structure of the chosen model . In the later sections , we will show detailed description and asymptotic property of the control system .

CONTROL SYSTEM

The plant is assumed to be a discrete-time linear system

- filtl -d y (t )- A (z) z u ( t ) ,

n n-1 A(z )= z + a 1 z +

B ( z ) = boz n + b i z n-1 +

. . . + ( 1-1) a n ( 1-2)

• • • + b n ( 1-3)

where bo•o , y ( t) and u(t ) are the plant output and input respectively at the t-th step and z is the forward shift operator . The unknown relative degree corresponds to the time delay d in the discrete-time system. Coefficients ak, bk and the plant order n are also unknown. However , B ( z ) is assumed to be a stable polynominal . Degrees n and d are assumed not to exceed known constants N and D respectively. Therefore , d is a member of the possible structure set Io = {O , l , . . . , D L

Equation (1) can b e rewritten a s follows with a state variable f ilter Q (z ) , a m6nic s table polynominal whose order nq is larger than (N+D) . u ( t )= bo1y(t+d) - bo1 ���� u(t )

-1 � d - bo Q ( z ) z y ( t)

where P (z ) and Q ( z) are defined as through a quotient polynominal S ( z ) , Q ( z )= A(z) S (z ) + R(z ) , 3 S ( z )= n - n , 3R ( z ) < n q P ( z )= B ( z ) S (z) - b 0Q (z ) ClP (z) < n q

( 2 ) Eq . (3 )

(3 ) where dX(z) means the order of the polynominal X(z) . The plant is parametrized as the input error formulation (Goodwin and co-workers , 1981) :

-1 u(t-d )= bo y ( t+d-D) + o ' � ( t-D)

where -1 -1 -1 o ' = [-bo P O -b o pn -l 0 · · · 0 -bo ro

-1 q -b0 rn-l O · • • 0 ) ,

N+D-1

(4)

n -1 � ' ( t )= [ �(�) u ( t) 1

Q (z) u ( t) z Q ( z ) }f (t )

1 • • • -- y ( t ) ] Q ( z ) '

and scalors pk and rk are P (z ) and R(z) respectively Eq . (1 ) .

coefficients as defined

(5) of in

The unknown structure d and the parameters bO and o are estimated multiple identification model : ii . ( t ) = 6 . ' ( t ) v . ( t-D) , 't/i E io , l. l. l.

A A

unknown by the

( 6) where ui (t) and 8i(t) are estimates of ui( t-D) and 8d respectively at the t-th step and v '. ( t ) = [ y (t+i) � · (t ) ]

l.

The plant is represented backward form in Eq . (4) , and u ( t-D) are available The estimate Si (t) is deterministic adaptive law,

(7 ) as the D step

so that vi( t-D) at the t-th step . updated by the

E: ? ( t) e . ( t+l) = e . ( t ) + ---=-""1'------- ( t D) l. l. h+ v '. (t-D)v . ( t-D) vi - '

l. l.

Vi E I0 , ( 8) where E:Oi (t) is a priori defined by

estimacion error

E:� ( t ) = u(t-D) - ui (t)

u ( t-D) - 8'. ( t )v . ( t-D) l. l.

A posteriori estimation error Ei (t) , E: , ( t ) = u ( t-D) - 8 '. ( t+l) v . ( t-D), l. l. l.

(9)

( 10) plays a key role on the convergence proof in the later section .

On constructing the multiple identification model adaptive control system, the most difficult point is how to produce the �ontrol input u ( t ) from the D+l estimates 8:1.(t ) , 'iiEIO . The control system is

designed in such a way that one identification model j ( t) is selected through a criterion and a preference index , and that the control input u (t) is generateft with the estimated parameter vector � ( t ) of the selected model j (t) , so

Stable Adaptive Control for Linear Plant 297

that the plant output will track a bounded reference input y* ( t) asymptotically . The criterion and the preference index are described below precisely .

A posteriori error can be represented as Eq . ( 11) in the true model , i . e . , in the model d ( see APPENDIX) .

where S d ( O) is an estimate and fd ( t ) by

initial value is one of fi ( t )

f ' ( t )= v '. ( t-D) 0 . ( t+l ) , i l. l.

1 0 . ( t+l ) = 0 . ( t) - 1 + v '. ( t-D)v . ( t-D) l. l. l. l.

0i (O) = E , 1h E IO ,

( 11) of the created

( 12) where E is the unit matrix . It is evident that the true mode l satisfies

( 13) where I I · I I means a vector norm and M is an appropriate constant , which is determined by an , initial parameter estimation error 8d-8d (O) and is assumed to be known for a while . The unkown case will be considered in the later section . Using Eq . ( 13) as the criterion , the available model set I ( t+l) is generated with the measurable information up to the t-th step by

I ( t+l ) = { i i I E: i ( t ) I � I I f i ( t ) l l · M ,

\fiE I ( t ) } , I (O) = I o ,

The preference index , for instance , ( 14)

index ( I (t+l ) ] = max ( i I \f:iE:I ( t+l) ] • or

( 15-1)

index [ I ( t+l) ]= min ( i l l;fiE I ( t+l) J , ( 15-2) selects one identif ication model j ( t+l) from the set I ( t+l) in order to generate the control input u ( t+l) with 8 . ( t+l) ( t+l) . Accordingly , J

j ( t+l) = index [ I ( t+l ) ] , ( 16) corresponds to an estimate of the unknown structure d .

If the plant were completely known, the plant output y ( t) would track the reference input y* ( t ) with the control input ;

-1 * u ( t )= b o y ( t+d) + o ' � ( t )

Since d , bO and o are unknown, input u ( t ) is subsitited by

(17) the control

u (t )= &j ( t ) ( t )y;' ( t+ j C t ) ) + o j ( t ) ( t ) U t ) •

( 18)

where

( 19) It is a certaintly equivalence type control law.

Then , the total control system is constructed as the multiple identification model (Eq . (6 ) ) , the adapt ive law (Eq . (8) ) , the structure estimator (Eqs . ( 14) and ( 16) ) , and the control law (Eq . ( 18) ) . The asymp totic property of the control system is shown in the next section .

PROPERTY

Since the available model set I ( t+l) is defined by Eq . ( 14) , it is non-empty set and is monotonically decreasing as

{ d } � • • • � I ( t+l) � I ( t ) � • • • s;; I (O) ( 20)

From Eq . (20) and the fact that I ( t ) is a discrete set ,

I ( t ) = 3I , ( 2 1 ) I t is evident from Eq . ( 2 1 ) that after sufficient steps have passed , the estimated structure j (t ) becomes equal to a constant j which belongs to I . The constant j may be or may not be equal to d, but it does not concern the s tability proof .

Le t ¢ i (t ) be a positive definite defined by

function

¢ i ( t ) = t r a c e ( 0 � ( t ) 0 i ( t )

The difference of ¢i ( t ) is ¢ i ( t+ l ) - ¢ i ( t )

- ( 2 + v '. ( t - D ) v . ( t - D ) ] f '. ( t ) f . ( t ) l. l. l. l.

( 22 )

< 0

Therefore , properties ;

f i ( t ) has the (23)

following

l i m t-+oo

l im t -+oo

E t = O

1 1 f . ( t ) l.

1 1 f . ( t ) l.

I I f . < t ) l.

1 1 = 0

( 24) 1 1 · I I v . ( t - D ) 1 1 = 0 '

l. (25)

I I . I I v i ( t - D ) 1 1 < 00 (26)

From Eq . ( 14) , the model which belongs to I satisfies

I E: i ( t ) I � I I f i C t ) I I · M ,

From Eqs . (24) - ( 27) , um I E: . ( t) I = o, l. t-+oo

l im 1 £ . ( t ) l · l l v . ( t-D) l l l. l. 0 ,

(27 )

(28)

(29)

298 S . Shin and T. Ki tamo ri

[ E . ( t ) [ · [ [ v . ( t-D) [ [ < oo 1. 1.

't/i E I Since the adap t ive law can be rewritten as

( 3 0 ) defined b y Eq . ( 8 )

the selected model j ( t ) satisfie s

l i m E j ( t ) ( t ) = 0 , t-+«>

and

�_: e j C t) c t ) = e *

where 8* is a b ounded cons tant vector .

( 31 )

( 3 2 )

( 3 3 )

B y subs tituting u ( t ) of Eq . ( 18 ) into Eq. ( 10) , a pos teriori estimation error of the selec ted model can be wri tten as

Ej ( t ) ( t ) = u ( t-D ) - 8 j ( t+l) ( t+l) vj ( t+l) ( t+l)

a. . ( t-D) yi' ( t-D+j ( t-D) ) J ( t-D) - o.j ( t+l) ( t+l) y ( t-D+j ( t+D ) )

+ oJ ( t-D) ( t-D) � ( t-D)

- o J ( t+l) ( t+l) � ( t-D) Then ,

l im S ( t ) E . ( t ) ( t ) t-+«> J

l im S ( t ) [ o. . ( ) ( t-D) y* ( t-D+j ( t-D) ) J t-D t-+«>

- o.j ( t+l) ( t+l) y ( t-D+j ( t- 1 ) ) ]

+ lim S ( t ) [ o j ( t-D ) ( t-D) t->-oo

- oj ( t+l) ( t+l) ] ' � ( t-D) ,

where

2

( 3 4 )

( 3 5 )

( 3 6 ) I n Eq . (35) , the l e f t hand s ide and the second term of the right hand side are equal to zero owing to Eqs . ( 3 2 ) - ( 3 3 ) and the fact that S ( t ) � ( t-D ) is always bounded. Therefore , the second t erm o f the right hand side of Eq . ( 3 5 ) should be equal to zero , that is ,

l im 8 ( t ) [ o.j ( t-D) ( t-D) y* ( t-D ) t->«>

o.j ( t+l) ( t+l) y ( t-D+j ( t+l) ]

= l im S ( t ) o. . ( t ) e ( t-D+j ) = 0 t->= J ( 3 7 )

where e ( t ) i s the output error defined by

e ( t ) = y* ( t ) - y ( t ) ( 3 8 ) From Eq . ( 3 7 ) and the lemma g iven b y Goodwin and co-workers ( 1980) , it is assured

that the designed adaptive control system is s table and the output error e ( t ) converges to zero asymptotically .

D ISCUSSION

In the former se ctions , we assumed that M in Eq . ( 1 3 ) was known . I f it ' s unknown , M can be chosen , to be an arbitrary positive cons tant M. There is no problem in the theore t ical analysis if at least one identificat ion model satisfies Eq . ( 2 7 ) wit� M in p lace of M. The inadequate choice of M will cause a troub le when I ( t ) is emp ty , since the index [ • ] can not choose any iden t if icat ion model from the empty se t .

The con trol algorithm i s s lightly modif ied in such a way that after f inite step s , the set I ( t ) becomes emp t y . , Whenever I ( t ) is emp ty, the prespecif ied M is increased by a f inite value 6M and I ( t ) is rese t to the initial value I O :

M = M + 6M

I ( t ) = IO

( 3 9 )

(40) Since M is finit e , M will exceed M after a f inite number of increments . I t means that empty I ( t ) can occure only in f inite steps period and that I ( t ) will become thereaf ter non-emp t y . Therefore , the modified control system has the same asymp totic property as the sys tem analysed in the previous sect ion af t er the suf f icient steps have passed . The modif icat ion corresp onds to int roduct ion of an estimator for the unknown constant M.

There is another way in the modif icat ion of the control sys tem. It is a r e-init ializ ing method . Instead of increasing M, 0 i ( t ) and I ( t ) are set to be the initial value 0 i ( O ) (= E ) and IO respectively whenever I ( t ) becomes emp ty :

0 . ( t ) = E 1.

I ( t ) = IO

(41)

( 42 ) �ince the parameter estimation error G d ( t ) -8d o f the new initial s tep must be smaller than that o f the former initial s tep , M defined at the new initial point can 9e smaller than the f ormer M . Consquently , M becomes greater than M after f inite re-initialization s . Therefore , I ( t ) becomes non-emp ty a f t er the f inite steps have passed . The method is effective not only in the unknown M case but also in the case when the p lant parameter d may change abrup tly, since the control system is re-initialized when I ( t ) becomes emp t y .

The condit ion , emp ty I ( t ) , is equivalent that the p lant is not correct ly controlled . It means that the p lant violates one o f the assump t ions : linear , noiseless , t ime-invariant , contro llable , observabl e , s tably inver tib l e , D known , N known and M known sys tem . Therefore , N and D can not need to b e known if the as sumed upper bounds N and D are incremented whenever I ( t )

Stable Adaptive Control for L inear Plant 299

becomes empty.

NUMERICAL EXAMPLE

In order to illustrate the control scheme , a numerical s imulation is excuted on the 2nd order linear plant :

y( t) 0 . 54z2 + 0 . 2lz + 0 . 00042 -d z u ( t) 22 - 0 . 2 7 z + 0 . 018

(43) Two cases are considered about the unknown time delay d as follows ,

d � { 1 0 � t � 30 CASE A : -2 31 � t 1 0 < t � 30

CASE B : d - { -2 31 � t

( 44-1)

( 44-2 ) The upper bounds N , D to be known are set to be 2 , 3 re spectively . Therefore , the multiple identification model set consists of 4 models , i . e . , MODEL 0 , MODEL 1 , MODEL 2 , and MODEL 3 , where the numbers of the models correspond to the assumed time delays . The control object is to track the command input :

0 0 < t � D y* (t) = {

s in ( 211 /t) D � t ( 45) In the simulation , the unknown M is treated with the re-ini� ializ ing method . The initial value ,of M is chosen as 1 , so that the specified M is too small as the required M either at the initial step or at the step when the time delay has changed . The preference index is def ined as Eq . ( 15-1) . The other parameters to be set initially are as followings Q ( z ) = z 5

e ' ( 0 ) = ( 1 0 . . . . O ] vi E IO i

(46)

(47) Figure 2 shows the s imulation result of the case A. At the 4th step , the adaptive controller selects MODEL 1 whose asumed t ime delay is equal to the unknown t ime delay 1 and the output error goes to zero . At the 33rd step , the available model set I ( t ) becomes empty . It means that the controller detects the time delay change occured at the 31st step and that the control system is re-initialized at the 33rd step. The available model set I ( t) converges to the set { 2 , 3 } and MODEL 3 is selected as the estimated structure . the output error e ( t ) converges to zero as proved in the theoretical analysi s , even if the estimated structure j (t) does not converge to the true structure 2 .

The case B shown in Fig . 3 i s the as the case A but that after the changes at the 31st step , the speed is lower than the case A . rate convergence may be expected

almost same time delay convergence The higher in the case

f 1 M E Of.LAY t S CHANGto 1 . 0 FR().1 l TO 2

. ......... -JL�···+--20 30 \; 40 so 60 steo

-1.0 t 0 : SELECTED MODEL

MODEL 0 f---1 .-----MODEL l >--+-----G-----l MOCfL 2t----i r--------MODEL 31-&1

Fig . 2 . Transitions of the output error e ( t) and the available model set I ( t ) . Time delay is changed from 1 to 2 at the 31st step .

OUTPUT ERROR

TIME DELAY IS CHANGED

1. 0 l- I FROM 2 TO I

0 f\ /· J L""\.··.,

/\ o or��-·-· .. ···· I 1 , I \, y�-... 10 20 30 40 I 90 10C ......_ l \G 120 v "'°

�l 0 i 0 SELECTED MODEL le t > MODELO 1--1 1-----t

MOO::L J 1----1 >------; >----MODEL 2 J-----t ;----f'-OOEL 3 1----f 1----€---1 l----0--i

Fig. 3. Transitions of the output error e (t ) and the available model set I ( t) . Time delay i s changed from 2 to 1 at the 31st s tep .

B with the choice of preference index as Eq . (15-2) .

CONCLUSION It is designed and analysed that the s table adaptive control for the linear p lant with unknown relative degree in the discrete time system. The control system can be rewritten in the continuous form with a slight modification . The introduction of the structure estimation will be effective not only in the tracking type adaptive control system but also in all f ield of the adaptive control systems .

REFERENCES Egardt , B . (1979) . Stability of adaEtive

controllers , Springer Verlag , Berlin . Fessel , T . ' and M. Karny (1979) . Choice of

models for self-tuning regulators . Proc. 4th IFAC SymE · on Identification and Parameter Estimation , Darmstadt ,

300 S . �hin and T . Ki tamori

1179-1186 . Goodwin , G . c. , P . J . Ramage , and P . E .

Caines (1980) . multivariable adaptive Trans . Automat . 449-456 .

Discrete-time control . IEEE

Contr . , AC-25 ,

Goodwin , G . C . , c. R. Johnson , Jr . , and K. S . Sin ( 1981) . Global convergence for adaptive one-step-ahead' optimal controllers based on input matching . IEEE Trans . Automat . Contr . , AC-26 , 1267-1273 .

Kur z , H . ( 1979 ) . Digital adaptive control on t ime varing dead t ime . Proc . 4th IFAC Symp . on Identification and _P_a_r_a_m_e_t_e_r ___ E_s_t_i_m_a_t_i_o_n , Darmstadt , 1187-1193 .

Monopoli , R. (1974) . Model reference adaptive control with an augmented error signal . IEEE Trans . Automat . Contr . , AC-19 , 474-484 .

Morse , A. S . ( 1980) . Global s tabi ity of parameter-adaptive control sys tems . IEEE Trans . Automat . Contr . , AC-25 , 433-439 .

Narendra , K. S . , Y . H. ( 1980) .

Lin, S table

and L . S . adaptive

proof of Automat .

Valavani controller design part I I ; stability . IEEE Trans . _g_�, AC-25 , 440-448 .

APPENDIX Let 8d ( t ) be the parameter on the true model ;

estimation error

-ed ( t ) = ed - ed ( t )

(A-1) From Eqs . (8 ) and ( 9 ) ,

E: d ( t ) 8d ( t ) - vd ( t-D)

l + v� ( t-D)vd ( t-D)

vd ( t-D) vd ( t-D) _ ed ( t ) - ed ( t )

l + vd ( t-D) vd ( t-D) -1 8d ( t ) - (E + vd ( t-D)vd (t-D) )

(E +

(E + · (E +

t -1 -= [ IT ( E + vd (r-D)v� ( r-D) ) ] 8d (O) r=O (A-2)

Eq . (12 ) , Applying the same recursion 8d ( t ) can be represented as

to

t -1 8 d ( t) = IT ( E + vd (r-D)v� (r-D ) ) 8d (O) r=O

t -1 = IT ( E + vd ( r-D)v� (r-D) ) r=O (A-3)

Therefore , sd (t ) can be rewritten as v� ( t-D) 8d ( t+l) fd ( t ) 8/0)

(A-4)

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

DISTRIBUTED CONTROL USING SELF-TUNING REGULATORS

M. H. Costin and M. R. Buchner

Depart111e11t o( Systems E11gi11l'l'ri11g, Case Wes/1'1"11 R1'Sl'l"l'<' Univn,ity, Cle11e/1111d, OH 44 1 06, USA

Abstrac t . A modified self-tuning regulator algorithm is proposed for the case of decentralized controllers . The modification involves using the measurements of the other subsystems in the updating algorithm of the STR, Information about the other subsystems is not required for the controller portion of the algorithm, Simulation results show improvements in the control obtained by the new control algorithm. The proposed algorithm is especially effective in reducing the variance of the manipulated variables associated with each controller ,

Key words : Adaptive control , Decentralized Control , Least Squares Estimation , Self-tuning Regulators , Suboptimal Contro l .

1 . INTRODUCTION

In the last few years , two trends have stim­ulated considerable interest in industrial process control. The first is the appearance and general acceptance of distributed digital control networks for industrial applications , The second is the increase in research and applications of adaptive control schemes , expecially the self-tuning regulator algorithm, This paper attempts to unify these concepts by presenting a modified self-tuner algorithm especially developed for dis tributed control situations .

The development of dis tributed control sys tems has been spurred by a significant increase in the reliability and capabilities of digital computers along with significant decreases in the cost of the underlying hardware . (Buchner and Lefkowitz ( 1982) , Kahne , Lefkowitz and Rose (1979) ) , Unfortunately , control theory for decentralized control of time varying or unknown systems has lagged far behind the industrial application of the hardware . Only Davidson ( 1978) has examined the problem of tuning decentralized controllers for an unknown sys tem. Davidson' s control s cheme is limited because each local controller is tuned sequentially and once the controller has been tuned , it remains fixed and is not adj usted again, These controllers are not able to adapt for a time varying system or to improve their tuning as more information is gained about the system,

This paper describes an adaptive control s cheme which has been developed explicitly for a distributed control s tructure , The proposed adaptive control scheme is a modification of the basic single input s ingle output self-

30 1

tuning regulator (STR) (Astrom and Witten­mark (1973) ) , Recently , many papers have appeared extending the algorithm and its modifications to multivariable systems , The multivariable STR algorithm is given below for later comparison to the decen­tralized algorithms .

For the design of multivariable STR' s , the multi-input/multi-output (MIMO) process to be controlled is assumed to be adequately modelled by the linear sys tem given in equation ( 1 . 1) ,

-1 ( I+A(z ) ) y ( t)

-1 -1 d J.l ( z ) J: (t-f-l )+(!+� ( z ) ) e (t ) /\/ (1 . 1 ) where i < t) n - vector , sys tem output

(measurement) i:< t) p vector , sys tem input e ( t ) n - vector , zero mean random

dis turbance -1 -1 -1 �(z ) , !l ( z ) and g ( z ) are process poly-

nomial matrices of appropriate dimensions , in the backward shift operator z-1 , which are unknown and/or time varying , (Note on notation : all underlined characters repre­sent matrices except for the vectors l • � and �) .

-1 -N �l z + • , . + �z

n -1 + B -M �O + �l z + • • • �Mz

-1 -K Qlz + • • • + QKz

302 M . H . Cos t in and M . R . Buchner

f = sys tem dead time ( f sampling units )

E { � ( t ) � ' ( t) } =e_ ; E { � ( t) � ' ( t-k) } = 0 for k#O

'\/ = 1 - z-l ( the b ackward difference oper­ator)

d order o f non-s tat ionary dis t urbances

d 1 models s tep dis turbances in the mean o f y ( t) . Taking d=l f orces the con­troller developed for this model to include integral action.

The basic S ISO-STR controller was firs t extend­ed to mul tivariable sys tems by Borison ( 19 75 , 19 79) . Borison ass umes that the sys tem has an e� u�l number o f inputs

_�md outputs , � ( z-1) is minimum phase (det B ( z ) has all i ts zeroes s trictly o uts ide the uni t circle) and B is nons ingular. Borison ' s controllers co;Q verge to the minimum variance (K V . ) con­troller where the M. V . controller is defined as the controller which minimi zes the variance of the regulated variab l e , This algori thm is limi ted in that it cannot handle non-minimum phase sys tems or sys tems with d i f fe rent amounts of deadtime in the individual loop s ; s i tuations that are very common in mul ti­variable systems ,

Koi vo ( 1980) extended Boris on 1 s MIMO-STR algorithm to include a cos t on th e control actions . Koivo ' s algori thm allows one to control sys tems which are non-minimum phase or have non-equal deadtimes . Koivo assumed that fio is non-singular , howeve r , this assumption is not necessary for the actual implementation of the algori thm. Bayoumi et al (19 81) extend this algorithm to a sys tem which has an uneq ual number o f meas urements and controls , For this case d mus t b e taken as zero ,

The MIMO-STR developed by the above mentioned authors is given by eq uations ( 1 , 2) - ( 1 . 4) b elow , Cost Criteri on :

m!n <i ' ( t+f+l l t> gi < t+f+l l t ) +vd�, ( t ) gvd�( t)

'\/ u ( t ) - ( 1 . 2 )

where y ( t+f+l J t) is the estimate o f x < t+f+l) given information up to time t ,

g and g are wei gh ting matrices selected by the des i gner . Koivo s tates that his algori thm i s minimi zing the expectation o f equation ( 1 , 2 ) , Thi s is incorrect as discussed by MacGregor and Tidwell ( 1 9 7 7 ) for the SISO cas e .

The STR algorithm is divided into two parts a l inear control equation (equation 1 . 3) ) and an adaptive mechanism ( equation ( 1 , 4) ) .

d - 1 '\/ u ( t) = - (R+B 'QB ) - - -o --o �og < � ( t) �( ( t) ) (L 3)

where § ( t ) (an • • • • a ' Rl • • • • S ) -� -m � - Q,

matrix o f controller parame ters to b e e stimated .

x ( t ) = (z� C t ) , i ' C t-1 ) • • • , l ' C t-m-l) , vdll, ( t-l ) • • ,

vdu ' ( t- £-1) )

§ ( t ) = § ( t- l ) +�( t ) (y ( t ) - � ( t-l)x� ( t-f-1) ) ( 1 . 4 )

The � ( t ) matrix i s usually generated using the recursive least squares algori thm (Borison (19 79) ) . This matrix has identical columns thus s implifying the calculations ,

2 , DISCUSSION OF DECENTRALIZED SELF-TUNING REGULATORS

The problem under consideration in this paper i s to control the sys tem given by equation ( 1 . 1) , for n = p ( the same number of control variables as measurements) , using an algo­ri thm suitable for imp l ementation in a dis tributed computer control architecture . I f the sys tem i s time varying or unknown , i t i s des irable to implement a control which would be ab le to adapt as the sys tem changed or as more information is gained about the sys tem . One possible s olution is to use n S ISO-STR contro llers . S ection 2 , 1 examines the p erformance of n S ISO-STR ' s when they are imp lemented as completely decentralized con­trollers . In Section 2 . 2 , a modification to the updating p ortion of the STR algorithm is p roposed . This modif ication enables the n­STR ' s to better coordinate their actions leading to an improvement in the overall performance o f the decentralized control network,

2 . 1 STANDARD SISO SELF-TUNING REGULATOR

The standard STR algori thm can converge to the minimum variance controller for a system only if the order o f the controller portion o f the STR algori thm matches the order o f the system' s minimum variance (M.V . ) con­tro ller . I f the regulator and the M . V . con­trol law are of different orders then the STR will no t minimize the output variance for the given regulator s tructur e . (Goodwin and Ramadge (1979) ) .

In comp le tely decentralized control of a multivariab le sys tem o f order n , the control­lers are restricted to a set of n-S ISO con­trollers . S ince ea�h controller is given only local information , the set of SISO con­trollers cannot have the same s tructure as the sys tem' s minimum variance controller (excep t if the system i s fully decoupled ( diagonal �. � and g matrice s) ) . Therefore, using S ISO-STR ' s for decentralized control will not , in general , reduce the variance of the measurements to the minimum value one can achieve by using a set of decentral­ized controllers .

Astrom and Wittenmark ( 1 9 7 3 ) prove that an

D i s t r ibu ted Control Usin g S e l f - tuning Regul ators 303

STR , cont roll ing loop i , whose p arameters have converged has the p roperty o f removing the correlations given by equation ( 2 . 1) .

f+l , • • • , f+mi+l

di E { y . ( t+T ) V u . ( t ) } l l 0 T f+l , • • • , f+£ . +l l

The mi and ,Q,i parameters are determined by the order o f the controller portion o f the STR which is the s calar vers ion o f equat ion ( 1 . 3) .

Equation ( 2 . 1) determines the convergence points of all the controller p arame ters and there fore the performance o f the converged contro ller . Thi s equat ion , however , contains no information about the correlat ions between loop i and the other s ubsys tems .

Thus the level o f subsys tem interac tions are not regula ted and there fore the de cent rali zed S ISO-STR sys tem canno t be des i gned to take into account and moderate these interactions .

2 . 2 MODIFIED STR FOR DI STRIBUTED CONTROL ARCHITECTURES

The p revious sec tion ind icates why the n- S I SO­STR s t ruc ture wi l l no t be able to take into account interac t ion b e tween the sub sys tems . Usual ly i t is des i rab le to minimize subsystem interac t ions by coordinat ing the individual local cont rollers . Th is reduces the possi­bility o f individual cont ro l lers regulating their local measurements at the expense o f causing upse ts t o o ther subsys tems .

This sec tion p roposes that the coordinat ion be done by modi fying the up dating equat ion used to calculate the controller parame ters o f the STR. The updat ing equat ion becomes :

e . ( t ) =e . ( t- l ) +K . ( t ) ( w . ( t ) - e . ( t- l ) x� ( t- f . . - 1 ) ) l l l l l l J l ( 2 . 2)

Equation ( 2 . 2 ) is formed by replac ing yi ( t ) i n equat ion ( l . 4a) b y wi ( t ) .

n w . ( t ) =y . ( t- f . . +f . . ) + i: q . k l l J l ll k=l l

kh

Bki (O) B . . ( O ) yk ( t- fj i+fki )

l l ( 2 . 3 )

where j i s defined b y fj i � fki ' k=l , • • • , n (j i s the loop containing the larges t amount of delay between y . and u . ) . By using this J l modified updat ing equation , the correlations removed by the STR control would be given by equa tion ( 2 . 4) .

E { w . ( t+T ) y . ( t ) } = 0 l l T = f . . +l , , • • , f . . +m . +l J l J l l ( 2 . 4 ) d ·

E { w . ( t+T ) V 1u . ( t ) } = O T = f . . +l , • • • , f . . H . +l l l J l J l l

The physica l s igni ficance o f equa t ion ( 2 . 3 )

is that each y . ( t ) has been rep laced by a l in­ear combinatioB o f the entire s e t o f sys tem measurements . These measurements are weigh ted by the q . k terms which are selec ted by the control §ys tem des igner . They are also weigh ted by the Bk . ( O) / B . . (0) terms which l dll weigh ts the e f fect o f V 1ui ( t ) on yk ( t ) and y . ( t ) . This particular form for w . ( t ) is de-l l rived by examining the desi red cos t cri terion for the mul t ivariab l e sys tem given by equation ( 2 c 5 ) •

2 min E { y . ( t+f . . + l ) + vct . 1 1 1

n 2 z q . . y . ( t+f . . +l) } ( 2 , 5 )

1 u . ( t ) l j =l lJ J J l j #i

where q i i = 1 . and q ij = l . / qj i

Cost cri terion ( 2 . 5 ) s t a tes that each control . di act ion V ui ( t ) is t rying to minimi ze a linear

comb ination of the variance of the local meas­urement yi and the variance of the measurements of the o ther subsys tems . Each non- local sub­sys tem variance is we igh ted by q . . , a factor p i cke d by the des igne r . l J

In the results p resented below , c o s t cri terion ( 2 . 5 ) is s tudied for a model of the form o f equation ( 1 . 1 ) . Eq uat ion ( 1 . 1 ) is general i zed s lightly to a l low for di f ferent deadt imes in each loop , deno ted by f . . • For simp l i c i ty , lJ a 2 x 2 sys tem i s assumed. The mathematics can be immediately generalized to an n x n sys tem. The subscripts on A , B and C refer to the e lemen ts o f the matrices i);, � and S · For the 2 x 2 sys tem the cost criteria ( 2 . 5 ) can be wri t ten for contro l ler 1 :

( 2 . 6 )

Equation ( 2 , 6 ) can then b e minimi zed by taking its deriva tive with re spect toV dlu 1 ( t ) at time

t . The de rivatives o f y . ( t+f . . +l) are : J J l

3y1 ( t+f1 1+1 ) = B11 ( 0 ) ( 2 . 7 )

� av u1 ( t )

Relation ( 2 . 7 ) assumes that i )

o .

0

304 M. H . Co stin and M.R . Buchner

Because �(O) =!) and ii)

Similar assumptions can be formed for the derivatives of y2 ( t) . Assump tion i) holds

-1 -1 i f f11 .::_ f21 or A12 ( z ) = O , or B 21 ( z )

dl This condition s tates that 'iJ u1 ( t ) must

o .

affect loop one at leas t a s soon as i t affects loop two . For f12 .'.:_ f11 and a de­centralized control environment , condition ii) always holds . In decentralized control , each u ( t) is calculated independently of the currenE and future u

l( t) ' s . For the case of

fl 2 < fll ' the calcu ation of u2 ( t-fl2+fll)

may not be independent of u1 ( t) due to the presence of feedback .

The remainder of this discussion assumes that conditions i ) and ii) hold . However , the actual control resul ts for the newly de­rived algorithm will be shown to be indepen­dent of these assumptions , Thus , with these assumptions , the derivative of equation ( 2 . 7 ) i s given by :

E{ 2B1 1 C0);1 C t+f11+l i �)+2q12B21 C 0) ;2 (t+f21+I i t )}

( 2 , 8)

are discussed by Astrom (1980) while the neg­ative attributes , with a counter example , are presented by Goodwin and Ramadge (1979) .

The only modification to the standard STR in­volved in the derivation presented in this section is the replacement of y . ( t) by �i ( t ) in equation ( 1 . 4) . �i ( t ) iis calculated using the measurements from all the subsys tems . In typical decentralized digital control applications this information is usually available as each loop measurement is of ten sent to the centralized operator interface . There is very li ttle difference in terms of communication cos t involved in broadcasting this information to all the controllers or j ust to the operator .

The next section applies the new algorithm in several examples .

3 . snruLATION RESULTS

In order to investigate the effect of using self tuning regulators in decentralized con­trol , equations ( 2 . 2) - ( 2 . 3) , the original and modified STR algori thms , are compared using simulation and analytical results .

Example A: For the ini tial case s tudy , the sys tem given by equation ( 3 . 1) is examined .

-1 In order to minimize cost criterion ( 2 , 5 ) one (1-bz ) y1 ( t ) =u1 ( t-l )+w0u2 ( t-l)+e1 ( t ) must find the control whi ch sets equation (2 . 8) equal to zero .

� � E{yl ( t+fll+l) +ql2B21 (0) /Bll (O )y2 ( t-f21+1) }=0 . )

( 3 . 1 )

Using the recursive leas t squares (RLS) algo- For this and all t3� following cases , it is rithm (equations 2 . 2-3) will , for the given con- desired for each 'iJ iu . ( t ) to minimize cost troller s tructure find the control which sets cri teria (2 . 5 ) where �ach q . . = 1 . (equal i] the square of the term inside the expectation brackets , less 8 . ( t-l)x . ( t- f . . -1) , as close i i J i as possible to zero ,

No te that the proposed algori thm wil l no t be minimizing the cost criterion ( 2 . 5 ) ( except in the special case where the decentrali zed controller will achieve minimum variance con­trol (diagonal A, B and C matrices ) ) but will be removing the cor�elations as given in equation ( 2 . 4) . Also the removal of the indicated correlations is independent of assumptions i) and ii) which were used to derive equation ( 2 . 7 ) .

Because the new algorithm is not minimizing cos t cri teria ( 2 , 5) as desired , there can be some ques tion as to the e ffectiveness of equations (2 . 2) - ( 2 . 3) as a controller. However , assuming that the structure of the controller used to regulate the system i s Ghosen sufficiently large s o that the sys­tem is well represented , then removing the correlations as in equation ( 2 , 4) should yield effective control results , Little research has been done on the e ffects of using a lower order controller than that needed to achieve M.V . control , Some of the positive attributes of low order controllers

weights for each subsystem measurement) .

The minimum variance controllers for w0 = 0 ( fully decoup led system) are given by u ( t ) = K . y . ( t ) where K . = -b . The same 1 i i i proportional s tructure was selected for the decentralized STR 1 s , Following equation ( 2 . 4 ) , the relationships given in equation ( 3 . 2) hold for the converged STR ' s .

E { (y . ( t) + µy . ( t ) ) y . ( t-1) } = 0 i J i i=l , 2

E { (y . ( t )+µy . ( t ) ) u . ( t-1) } = 0 i J i

( 3 . 2)

The µ parameter is equivalent to the q ik Bki (O) /Bii (O) term of equation ( 2 . 2) . For this example , q . . = 1 , Bk . (O) = w0 and B . . (O) i] i ii = 1, therefore the desired value of µ would be equal to w0 •

Because the model ( 3 0 1) is symmetrical and the same value of µ is used in each STR updating algorithm, the additional assumptions of E {yi ( t ) }=E{y� ( t ) } , E { yl (t ) y1 ( t-1) }=

D i s tr ibuted Con trol Us ing Self-tun ing Regul ators 305

F,{ (y2 ( t) y 2 ( t- l ) } etc , and K1 = K2 hold , The relations given by ( 3 , 2 ) and t he above assump tions can be used to generate a s e t o f simul taneous equations for K , var ( u ( t ) ) and var (y ( t) ) based on the value selected for µ ,

Figure 2 shows the variances calculated for u ( t ) and y ( t ) for various values o f K , for the case o f b = 0 , 5 , w0 = 0 , 8 , For µ = O ,

the standard STR algori thm, K would converge to K = -b , the s ame value as the algori thm finds for the M , V , contro l ler for w0 = O . The values found for the variances are var (y ( t ) ) = 1 , 19 and var ( u ( t ) ) = 0 , 30 , However , when µ = w0 = 0 , 8 i s selecte d , the algori thm then f inds K = -0 , 2 75 which yields var (y ( t ) ) = 1 , 09 5 and var ( u ( t ) ) = 0 . 08 3 , From Figure 1 , the minimum variance o f the output achievable for this sys tem us i ng decentrali zed p roport ional cont ro l lers i s var ( y ( t ) ) = 1 . 084 with corre sponding var ( u ( t ) ) = 0 . 0 9 5 ,

Fo r this case , using the centrali zed MIMO­STR wi l l achieve minimum variance control o f

- 1 u ( t ) = -�0 �l y ( t ) , The minimum variance for the measurement would be var { y ( t ) } = l, with a corresponding control action variance of var { u ( t ) } = 3 , 1 6 , The variance o f the contro l act ions i s almost 40 t imes l arger than that achieved by the modi fied decentrali zed STR algori thm, Thi s i s quite a large cos t to reduce the variance of the measurements an additional 8 , 4 % .

I n this examp le , the modi fied di s t ributed STR (µ = 0 , 8 ) reduced the variance of the control actions p roduced by the standard STR in a distributed environment and by the MUIO-STR con troller , In mo s t indus trial situations it i s highly des i rable to maintain the s i ze and f requency o f t he control actions as small as possib le , This is because there is generally some cos t associated with implement ing a control action , This could be a cost in energy o r o ther re­sources needed to accomplish the change in contro l , In addi tion, l arge control act ions may cause upsets in related equipment or processes , or undue wear and tear on equip­men t .

The p roposed algori thm has variances f o r y ( t ) which a r e very c l o s e to the min imum variance achievable by decentrali zed control ( less than 0 , 1% dif ference) and has a variance o f u ( t ) approxima tely 1 0 % less than t h e minimum variance decentralized controller , The decen tral i zed control ler with the minimum variance has Ki = -0 . 294 . This gain will be the convergence point for a modi fied S TR with µ = 0 , 68 , S imulat ion s tudies for µ = 0 and µ = 0 , 8 confirmed the above analytical resul t s ,

Exampl e B : The second sys tem s tudied i s :

y . ( t ) = 0 . 5 u . ( t-1 ) + WO -1 - u . ( t-1) +

l-0 . 5z- l J ]_ l-0 . 5 z ]_

e . ( t ) ]_ i 1 , 2 ----::y-1-z

I?. I

When the subsystems are no t i nterconnec ted (w0 = 0) , the minimum variance con trollers

for each loop are di yi tal PI controller s : ( l-0 , 5 z- ) Vu . ( t ) = - --0 5 --- y . ( t ) , The PI

]_ . ]_

s truct ure is chosen for each local controlle r ,

B y us ing the following relations :

E { ( yl ( t ) +µy2 ( t ) )

E { (y l ( t ) +µy 2 ( t ) )

y ( t-1 ) } 1 y l ( t- 2 ) }

0

0 ( 3 . 3 )

toge ther with co rresponding rela tions for y2 ( t ) + µy1 (t) and maki ng s imilar assump tions about symme trical sys tems as in Examp le A, one can derive the results given in Fi gure 2 , showing the variances for y ( t ) versus w0 • Figure 2 details var ( y ( t ) ) for µ = 0 the old STR algori thm and µ = w0 /0 , 5 , the new algo­ri thm as well as the analyt ical resul ts for the central i zed M . V . (MIMO-STR) con troller , For the centrali zed case , var { y ( t ) } = 1 . for all values o f w0 < 0 . 5 . Also , examp le simula t ion results for var { y 1 ( t ) } and

var { y 2 ( t ) } are included for comparison , F igure 3 p resents the corresponding results for var ( Vu ( t ) ) ,

Figure 2 indicates that as w0 , the level o f inte rac t ion , is increased , the variances associated with y ( t ) increase for both cases (µ = 0 and µ = w0/ 0 , 5 ) . In all cases , the

new algori thm ( )J = w0/ 0 , 5 ) has a smaller variance than for the standard STR (µ = 0) . Even more no tewor thy are the variances associated w i th the control ac t ions , p resented i n Figure 3 , For the s t andard STR cas e , µ = 0 , and the MIMO-STR , the variances fo the control actions increase as the interac t ion gain in­creases , However , for the new algori thm , the variances of the control actions ac tually decrease with increas ing interac t ion gains ( excep t at h i gh values of w ) , The added in­formation in the updat ing a2gori thm al lows the individual local controllers to take ad­vantage of each o ther ' s control ac t ions , ( i . e . coop erating instead o f f ighting one ano ther) , The control variances for MIMO-M . V , , control are considerably higher than e i ther of the decentralized con t rol lers ,

The s imula t ion resul ts p resented in Figures 2 and 3 do no t exactly match the analytical

306 M .H . Cos tin and M . R . Buchner

results but do follow the same general trend . The differences are due to the use o f the discounted recursive least squares algorithm used as the updating algorithm and the use of a finite number of samples to calculate the variances . Each controller is simulated for 400 samples using the same set of random dis turbances e ( t) . The variances are cal­culated using- the las t 200 points . The discounted least square algorithm uses a forgetting factor of 0 . 9 8 . which causes the past data to be forgot ten exponentially with a time constant o f 50 samples . Because o f the discounting and the finite number of samples , the simulation would have variances of y ( t) different than those obtained in the analyti­cal solution . All the other simulations re­ported in this sec tion use the same conditions of forgetting factor and number of samples . The forgetting factor is used in the simula­tions in order to study a practical application of the modified STR algorithm, All experi­mental uses of an STR algori thm have discount­ing of past data in order to track time varying parameters ; thus , tuning the control­lers based on more recent data. Figure 4 shows simulation results for a sen­sitivity analysis on the estimate of w0 • The sys tem is modelled wi th w0 = 0 . 3 and equation ( 2 . 3 ) is implemented wi th a dif ferent value �0 • The results �or w0 = 0 to 0 . 6 are shown in Figure 4 , As w0 increased , the variance �ssociated with the Vu (t ) decreased and for w0 > w0 the variance associated wi th the � ( t) increased . This trend stopped for w0 > 0 , 6 as for this case simulations became unstable . Analytically , no solution was found for the system of equations developed from equation ( 3 . 3 ) for �O > 0 . 6 .

4 . CONCLUS IONS AND FUTURE RESEARCH

The simulation results presented in Sec tion I II show that the proposed algorithm can yield signi ficant improvements over a com­pletely decentralized set of STR' s for controlling a coupled multivariable process . The quantitative improvements appear in reductions o f both the control variances and weighted sum o f the output variances of the controlled plant . Furthermore , the algo­rithm can be implemented naturally in a distributed computer control sys tem. As a result o f the structure of the algorithm and the pattern of information exchange between local controller , the communication sys tem requirements are decreased as compared to a full multivariable minimum variance controller . In addition , if the computer­to-computer communication line fails , then the resulting control s tructure will main­tain decentralized adaptive contr0l on the process through the set of isolated STR controllers .

REFERENCES As trom , K . J . , "Self-Tuning Control o f a Fixed

Bed Chemical Reactor System , " Int . J . Contr . , 32 ( 2) , 221-256 ( 19 80) .

As trom, K . J . and B . Wittenmark , "On Self­Tuning Regulators , " Automatica , 2. ( 2 ) , 185-194 (1973) .

Bayoumi , M .M . , K . Y . Wong and M.A . El-Bagoury , "A Self-Tuning Regulator for Multivariable Systems , " Automatica , 1:2_ (4 ) , 5 7 2-59 2 , ( 19 81 ) .

Borison , U . , "Self-Tuning Regulators - In­dustrial Applications and Multivariable Theory" , Report 751 3 , Dept . of Automatic Contro l , Lund Ins titute of Technology , Lund , Sweden ( 19 75 ) .

Borison , U . , "Self-Tuning Regulators for a Class o f Multivariable Sys tems" , Automatica , 12_, 206-216 ' (1979 ) .

Buchner , M .R . and I . Lefkowitz , "Distributed Computer Control for Industrial Process Systems : Characteristics , Attributes and an Experimental Facility , " Contr . ���!..• CSM- 2 , 8-14 (1982 ) . Clarke , D . W . and P . J . Gawthrop , "Self-Tuning Controller" , ,I'ro c . IEE , 122 , 929-9 34 , (1975 ) .

Davidson , E . J . , "Decentralized Robust Control o f Unknown Systems Using Tuning Regulators , " Trans . Autom. Contr . , AC-23 , 273-289 (1978) .

Kahne , s . , I . Lefkowitz and C . W. Rose , "Automatic Control by Dis tributed Intelli­gence Control" , Scientific American , June (1979) .

Koivo , H .N , , "A Multivariable Self-Tuning Controller" , Automatica , 16 , 351-366 ( 1980 ) .

MacGregor , J . F , and P . Tidwell , "Discrete Stochastic Control with Input Constraints" , Proc . IEE, 124 , 7 32-734 ( 1977 ) .

---

->.

-lo... ro

>

---

->.

-lo... ro

>

Distributed Control Using Self-tuning Regulators

1 . 3

1 . 2

1 . 1

. . I ' w0=µ�0 . s; ! F0 . 6 8

' K=-0 . 2 1 5: ' '

' ' ' ' ' ' ' ' ' : :

. ' ' : l ' ' ' ' ' ' ' ' ' '

l : ' ' ' '

0 . 3

0 . 2 < � .., -c: --

--

0 . 1

o . -0. 1 - 0 . 2 - o . 3 - o . 4 - o . 5

FIGURE l

S i m u l a t i o n s + ,).l=O.

2 . 0

1 . 5

o. 0 . 1 0 . 2

WO

FIGURE 2

K

..

0 . 3

-........ ......

->. .......,

lo... )J"-B .. ( O ) / B .. (OJ C\1 J II >

0 . 4 o . s

1 5 .

---

-::J I>

1 0 . -lo... ro

>

0 .

2 .

1 .

o . 0 . 1

S i m u l a t i o n s

x u=B .. ( 0 ) / B . . ( 0 ) / }I II

.. x

0 . 1 0 . 2

FIGURE 3

0.2 0 . 3

A WO

FIGURE 4

307

+ ..

i

0 . 3 0 . 4 0 . 5

WO

8 .

6.

< Pl ...,

........ <l

4. c: ---.......,

2 .

o.

0.4 0.5 0 . 6

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

APPLICATIONS OF ADAPTIVE CONTROL

APPLICATION OF MULTIVARIABLE MODEL REFERENCE ADAPTIVE CONTROL TO A BINARY

DISTILLATION COLUMN

P. Wiemer, V. Hahn, Chr. Schmid and H. Unbehauen

Depa r/111e11/ of Elertrirnl E11gi111'1'ri11g. R 11hr-U11ii 'l'l:lity Bor/111111, Po.\lfi1111 ]() 21 48, D-4630 Bor/111111 I , Fnln11/ Rt'/mblir of C1'n1111 n\'

Abstract . Application of a multivariable model reference adaptive control strategy to a pilot distillation column is described . The control obj ective is to maintain the top and bottom compositions constant under different dis­turbances at the inlet via manipulation of the reflux ratio and heating power . On account of the nonlinearity and initial uncertainty of the plant dynamic s , adaptive control is wel l suited in this case . Smooth control actions are achieved by augmenting the plant output with a signal deduced from filtered control input . This results in an inexact model matching problem . The con­trol scheme yields satisfactory disturbance rejection when only a little a priori knowledge about the plant dynamics is available .

Keywords . Adaptive control ; chemical industry ; multivariable control systems ; process control .

INTRODUCTION

In this paper an application of a model refe­rence adaptive control strategy to a pi lot di­stillation column at Bayer AG , Leverkusen, is described . As on one hand this process is very often found in chemical industry , and on the other hand , in most cases , only a little a priori knowledge about the process dynamic s is available, the demonstration of the feasibili­ty of adaptive control techniques here is of great interest . By using adaptive schemes the controller design procedure is simplified . Moreover , the controller adapts itself to the slowly time-variant behaviour of the plant

and the power of the heating at the bottom of the column . Thus , smooth control actions are required which can be achieved if the plant output is augmented by filtered plant input signals .

and to setpoint changes .

Dahlqvist ( 1 980 ) investigated the application of self-tuning regulators to distillation col­umns . While Dahlqvist uses single input - sin­gle output and multi input - multi output regu­lators , in the present study only multivari­able control schemes are considered due to the strong interaction among the control loops . Moreover, a model reference adaptive control scheme with a fixed gain estimation algorithm is chosen . This is far simpler than the recursive least squares method combined with the self­tuning controller . The control scheme is based on a discrete version of the algorithm pro­posed by Monopol i and Hsing ( 1 9 7 S ) . The main control obj ective is to maintain the top and bottom produc t compositions constant under varying f low rate , composition and tem­perature at the inlet . Setpoint changes are of minor interest . However , it is difficult to measure the top and bottom product composi­tions directly . As these compositions are strongly related to the top and bottom tempe­ratures under isobaric conditions , these tem­peratures are controlled via manipulation of the reflux ratio of the distil late at the top

ASCSP-K 309

In the following section , the theory of the control scheme is briefly outlined . Then the plant is described and in the last section the results are discussed .

THE CONTROLLER

Consider the plant with q inputs � and q out­puts � described by the qxq transfer function matrix

-1 I Q ( z i = I

- 1 Bi j ( z )

- 1 Ai j ( z ) - i i z I i , j = l , . . . , q

so that - 1 � ( z ) = �( z ) u ( z ) .

Q ( z- 1 ) can easily be factorized as

- 1 G ( z )

using

- 1 � ( z )

and

A . ( z ) l.

- 1 B ( z )

and

- 1 Bi j ( z )

- 1 - 1 - 1 - 1 � ( z ) �( z ) z

diag { Ai ( z ) } ; i=l , . . . , q

least common multiple {A . . ( z - l ) } ; 1.:._j.:._q l. J

{ B . . ( z- l ) } ; . . 1 l. J l , J = , . . . , q

q II - - 1 A . . ( z ) l. J - - 1 j = l B . . ( z ) =------l.J A . ( z- l

) l.

( 1 )

( 2 )

( 3 )

( 4a)

(4b)

( Sa )

( Sb )

3 1 0 P . Wiemer e t a l .

� ( z- l ) contains the least common denominator of each row i of the trans fer function matrix Q ( z- 1 ) at ( i , i ) . The matrix � ( z- 1 ) contains the extended numerator polynomials . Then , eq . ( 2 ) can be written in a more general form

- 1 - 1 - 1 � ( z ) x_ ( z ) = � ( z ) z _l:!_ ( Z ) + .::'._( Z ) , ( 6 ) where v ( z ) denotes the qxl disturbance vector . Since A ( z- 1 ) is diagonal eq . ( 6 ) can be sepa­rated into q equations

- 1 A . ( z ) y . ( z ) 1. 1. b� ( z- 1 ) u ( z ) + v . ( z ) -]. - 1. ( 7 )

for i=l , . . . , q , where bT ( z- 1) i s the i-th row

of the maxtrix � ( z- l ) �i

The plant outputs yi are augmented by the signals

wherein the filters B . ( z- 1 ) C l. - 1 Gci ( z ) = _ 1 z i=l , . . . , q A . ( z ) C l.

( 8 )

( 9 )

must b e stable . Such filters were originally introduced in order to make nonmimimum phase control possible (Hahn and Unbehauen ( 1 982 ) ) and are called "correction networks" . The augmented plant outputs

are compared with the output s ignals

- 1 y . ( z ) mi

B . ( z - l ) mi _ 1 z- 1w . ( z ) ; i= l , . . . , q A . ( z ) 1.

mi

( 10 )

( 1 1 )

of the stable reference mode l . w . ( z- 1 ) are the reference input signals . In ordef to reject the influence of deterministic disturbances which have the property

lim .::'._ ( k ) = const , k->oo

( 1 2 )

i t i s convenient to force the controller to have integral action . This can easi ly be done by multiplying eq . ( 6 ) with the factor ( l -z- 1 ) on both sides . Therefore , eq . ( 6 ) is modified to

- 1 �I ( z ) x_ ( z ) - 1 - 1 - 1 B ( z ) z u ( z ) +v ( z ) - -I -I

with

- 1 A ( z ) -I - 1 K - 1 ( 1 -z ) � ( z )

- 1 K - 1 ( 1 -z ) d iag{ Ai ( z ) } ,

- 1 - 1 K .1:!.I ( z ) ( 1 -z ) !:!. ( z l , - 1 - 1 K .::'..I ( z ) ( 1 - z ) .::'._ ( Z ) ,

K = {0 , 1 } .

( 1 3 )

( 14a )

( 1 4b )

( 1 4c )

( 1 4d )

Note that under the assumption o f eq . ( 1 2 )

l im .::'..I ( k ) k->oo

o for K=l . ( 1 5 )

The basic structure o f the control scheme i s shown in Fig . 1 .

With the notation

- 1 Ali ( z ) l +A* . ( z- 1 ) z- 1 ,· · 1 I i i= , . . . , q , ( 1 6a )

- 1 B . . ( z ) l.J

* - 1 - 1 . . b . . +B . . ( z ) z ; i. , J =l , . . . , q ( 1 6b ) l. JO l. J - 1 A . ( z ) C l. - 1 B . ( z ) C l.

* - 1 - 1 . l +Ac i ( z ) z ; i=l , . . . , q ,

- 1K * - 1 - 1 ( 1 -z J (b . +B . ( z ) z ; Cl.0 C l. i= l , . . . , q,

A . ( z- 1 ) = l +A* . ( z- 1 ) z- 1 - · 1 mi mi ' i= ' · · . , q ,

the error signals

( 1 6c )

( 1 6d)

( 1 6e)

ei ( z ) = ymi ( z ) -yai ( z )

i= l , . . . , q ;

ymi ( z ) -yi ( z ) -yci ( z )

( 1 7 ) can b e expressed by

A . ( z- 1 ) e . ( z ) = e . ( z ) ; i=l , . . . . , q mi i mi -1 - 1 emi ( z ) ¢i ( z ) z - (biiO+bciO) uii ( z ) z

q - 1 q * - 1 -2 - 1 - l b . . UI . ( Z ) Z - l B . . ( Z ) z u . ( z )

j=l l. JO J j = l

l. J I ]

j fi

( 1 8a)

* - 1 * - 1 -1 + (Al i ( z ) -Ami ( z ) ) z y ( z ) -vi i ( z ) , ( 1 8b )

wherein

¢i ( z ) =

the signals ¢i ( z ) are defined by

B . ( z- 1 ) w . ( z ) -B* . ( z- 1 ) z- 1u . ( z ) mi i ci I i * - 1 + Aci ( z ) yci ( z ) · ( 1 9 )

Note that in eq . ( 1 6d ) each of the polynomials Bci ( z- 1 ) is forced to have a zero at z=l if integral action is required . The signals ¢i ( z ) are generated only with known parame­ters and measurable plant input and output signals . For i= l , . . . , q the control signals may be computed by

q Ul i ( z ) = IP . ( z ) - l l S . b . . ] ' u . ( z ) i

j =O 1. l. J O I J j fi

with

Si = l / ( bi iO+bciO) ' and

u . ( z ) 1. ----,-1- u . ( z ) •

( 1 -z- ) K I i

( 20a )

( 20c )

( 2 1 )

The sign ,, _ ,, indicates that the true parame­ters which depend on the unknown plant para­meters are replaced by their estimates . Com­bining eq . ( 20a ) with eq . ( 1 8b ) , an estimation problem is defined , which is very well studied at least in the disturbance free and white noise case ( e . g . Lozano and Landau, 1 9 8 1 ) . Note that by the definition of ¢ . ( z ) by eq . ( 1 9 ) the number of estimated par�meters is al­ways equal to the number of unknown plant pa­rameters and does not depend on the design of the correction networks .

Binary Disti l lation Column 3 1 1

In our approa�h the algorithm proposed by Ionescu and Monopol i ( 1 977 ) is used . This i s a " c lassica l " model reference scheme with a non decrea.sing ( fixed ) gain estimation a lgo­rithm . If no disturbances are present (v . ( k ) =O) , the convergence o f the fi ltered errBfs emi ( z ) i s guaranteed . As the reference model is stable, the convergence of the original errors ei ( z ) direc�ly fol lows . The influence of step disturbances to the estimation proce­dure is at least asymptotically rej ected i f an explicit integrator ( K=l ) is used because then eq . ( 1 5 ) holds .

The convergence of the error signals ei ( z ) means that the augmented plant output signa ls Yai ( z ) follow the outputs Ymi ( z ) of the refe­rence model as t-+<><> . Under the assumption

lim ui

( k ) = canst ; i=l , . . . , q , k-+<><>

the outputs yc i ( z ) of the correction networks vanish for t-+<><> if all polynomials Bc i ( z- 1 ) have at least one zero at z=l . In thi s case , for the plant outputs asymptotic mode l match­ing for constant reference s igna ls wi ( z ) and step di sturbances vi ( z ) is achieved .

Each control signal ui i ( k ) cannot directly be generated by eq . ( 20a ) , because it depends on all other control signals u1 j ( k ) ( j +i ) lead­ing to an algebraic loop . There for e , the q e­quations ( 20a ) are •.vri tten in the vector-ma­trix form

K u ( z ) = 4l ( z ) - -I - ( 2 2 )

wherein the matrix K contains the estimate·j parameters [13 . b . . ] - . Eq. ( 2 2 ) can easily be solved provid�dltge matrix R is nonsingular . A necessary condition for this i s that the corresponding matrix � built up by the true parameters is nonsingular . This can always be achieved by choosing appropr iate coeffic ients bciO for the correction networks .

Instead of choosing a model reference scheme with explicit integral action , it is possible· to cascade the model reference adaptive con­troller with decoupled P I-control lers in or­der to rej ect deterministic disturbances . This is shown in Fig . 2 . However, the question of stability of those schemes is not completely clear at present . The gains of the P I-con­trollers are tuned on- line . At the beginning they should be chosen rather sma l l so that the PI- loops are approximately open , and may be increased a fter a successfull estimation of the plant parameter s .

THE PLANT

The plant is a binary pilot distillation col­umn of Bayer AG , Leverkusen . A schematic dia­gram of the column is shown in Fig . 3 . The in­coming produc t consis ting of two components is heated at the bottom of the column . The component with the higher boiling temperature is accumulated at the bottom , whereas the other more volatile component is condensed at the top of the column . A part of this conden­sate is fed back to the column . The reflux f low ratio p , defined as the ratio of the quantity fed back to the quantity of output

condensate , can be changed by means of an actuator valve and is chosen as one control variable . Another control variable is the heating power h , which can also be easi ly manipulated .

As already mentioned , the controlled variab­les ar e the temperature Tt at some point at the top and the temperature Tb at some point at the bottom of the column . Disturbances at the inlet are s imulated by changing the speed of the inlet pump ( feed dis turbance ) and switching the feed between two containers filled with different concentrations of the components of the fed product (concentration disturbanc e ) . It is also possible to change the temperature at the inlet, but as this disturbance has only a very sma l l effect on the controlled variable s , it has not been con­sidered in thi s study .

The control obj ective is to maintain the top and bottom temperature constant . However , only smooth control ac tions are al lowed .

The dynamic behaviour of the plant may be ex­pressed by a 2x2 transfer matr ix . Three e le­ments of thi s matrix are estimated as trans­fer func tions of 2nd order and one transfer function is assumed to be only of first order . For the scheme discus sed in the previous sec­tion , 21 parameter s have to be estimated on­l ine . For setpoint changes the two measured temperatures Tt and Tb have rise-times from 4 5 to 70 minutes . However , according to the dynamics of some fast disturbances a sampl ing time between 1 0 sec and 30 sec is chosen .

EXPERIMENTAL RESULTS

S imulations using an a pr ior i model indicated that with s imple correction networks

B . ( z - l

) C l

- 1 A . ( z )

Cl

( 2 3 )

a satis factory disturbance rej ec tion by using only smooth control actions can be rea l ized . Eq . ( 2 3 ) means that the plant outputs yi ( z ) are augmented by the increments o f the control signals ui ( z ) weighted by A i · Moreover , the simulations show that best results can be achieved at a sampl ing rate of T=30 sec using a ( fast) reference model

- 1 y

mi ( z ) = z w

i ( z ) ; i=l , 2 ,

i . e . Ami

( z ) = 1 and Bmi

( z ) = 1 ; i=l , 2 .

For a faster sampling rate , for example , T= l O sec , the control loop is difficult to stabi­lize . This may be due to the fact that the augmentation of the plant by the correction network of eq . ( 2 3 ) yields a nonminimum phase behaviour for all A i · Also the estimation of the discrete plant parameters , which are very small for small sampling times , leads to nu­merical problems .

The experimental results are shown in F igures 4 and 5 . The y-axes are sca led in relative values related to a certain setpoint . Because of the plotter used , the time bases of the

3 1 2 P . Wieme r et a l .

different y-axis are s l ightly shifted to each other .

First the dynamic behaviour of the c losed loop system under feed disturbance at the inlet is investigated using controllers wi th and with­out integral action . At the beginning the con­troller parameters are adj usted using an a priori model of the plant . The results are shown in Fig . 4a and 4b . In both cases the influence of the disturbance is completely re­j ec ted mainly by increasing the heating power . It is surpr ising that no steady state errors occur when a controller without integral ac­tion is applied . In this case steady state errors can only vanish because of the inte­gral properties of the nonlinear adaptation procedure which detunes the controller para­meters in an appropr iate way . Using a control­ler with explicit integration has the disad­vantage of increasing the variances of the control signals . In experiments with a model reference controiler with constant parameter s , i . e . when the adaptation i s switched off , steady state errors are indeed observed when a controller without integral action is ap­plied . In the present application these steady state errors are rather small , when the con­tro ller is adj usted to suit the a priori mo­del of the plant . However , if no a priori knowledge is available and if the controller is much detuned from i ts correct values , steady state offsets may become a problem . This is shown in Fig . Sa , where the influence of feed and concentration disturbances with­out a priori knowledge is studied keeping the setpoints constant. At the beginning of this experiment all controller parameters are set to zero . The controller has no integral ac­tion . Already the first disturbance step is rej ected in a satisfactory manner . Because of the adaptation of the controller parameter s , the performanc e of the control system is con­siderably improved a fter further disturbance changes . However, it can be seen that the top temperature Tt tends to oscillate whereas the bottom temperature Tb shows a dynamic beha­viour with steady state errors depending on the dis turbances . Therefor e , at the time 6 the adaptive gains in the estimation algo­r ithm are changed . For the first control loop of the top temperature the gains are decreas­ed so that the influence of the parameter estimation is reduced . As can be seen in Fig . Sa the oscillations then vanish . For the se­cond control loop of the bottom temperature Tb , the adaptive gains are increased leading to a reduction of the steady state errors .

In Fig . Sb the results of model reference a­daptive control in cascade with decoupled PI control lers ( see Fig . 2 ) are presented . The inner adaptive loop has no integrator . All controller parameter s of the adaptive part are set to zero at the beginning . For the first disturbance s tep the deviation of the top and bottom temperatures from the setpoints is smaller than in the experiment without in­tegrator as shown in Fig . S a . Because the PI controllers reduce the excitation of the adap­tive loop , the rate of convergence of the esti­mation procedure is decreased . Moreover , sta-

bility of the cascaded scheme is not guaren­teed , and the choice of the PI controller gains may be difficul t . In the present case , these gains are experimentally determined . I f no a priori information is available , these gains should be chosen rather small at the begin­ning and may be increased on-line as adapta­tion goes on.

so far only the disturbance behaviour of the adaptive control scheme has been described because this is very important in chemical industry . If the controller is implemented in a distil lation column with great uncer­tainty about its dynamic behaviour , it is con­venient to fasten the adaptation by exc iting the control system with small setpoint chang­es . As shown in Fig . S c , small setpoint chang­es about lK are suf fic ient for a fast estima­tion of the controlle� parameters . In order to avoid heavy control actions , the setpoint changes are filtered by low pass filters of first order with time constants about 270 sec for top and 1 90 sec for bottom temperature . At the time 6 the adaptive gains in the esti­mation procedure for the bottom temperature control loop are increased .

In Fig . 4c the dynamic behaviour of the con­trol loop is shown when the controller para­meters are wel l ad j usted . The influence of the dis turbances is well rejected by the con­troller . In order to keep the top and bottom temperatures constant, the temperature pro­file in the column must change due to changes at the inle t . In Fig . 4c this is indicated by two other temperatur es T 1 and T2 measured a­bove and below the inlet ( see Fig . 3 ) .

The performance of this model reference adap­tive control scheme has been compared with that of generalized minimum variance self tuning control proposed by Koivo ( 1 98 1 ) which has also been applied to the same column . The results of both schemes are similar . However , the full least squares estimation combined with self- tuning control leads to larger stor­age requirements and a computation time which is about five times greater than that for the model referenc e scheme . A satis factory cont­trol is also achieved with a linear state feedback controller using an observer . How­ever , the design of thi s controller is more difficult and its parameters have to be re­tuned from time to time .

CONCLUS ION

A multivar iable model reference adaptive con­troller has been succesfully applied to a bi­nary disti l lation column. Though model refe­rence schemes have originally been designed for servo problems , with slight modifications they can also be used for regulation . The comparison with a self tuning controller shows nearly identical results . However , the realisation of the mode l reference scheme is found to be simpler . In contrast to linear control lers , the implementation of adaptive controllers is simplified because only a lit­tle a priori knowledge about the plant dyna­mics is needed . A retuning of the controller parameter s , which has to be done with l inear

B inary D i s t i l l ation Column 3 1 3

control lers from time to time , is not neces­sary .

REFERENCES

Dahlqvist, S . A . ( 1 980) . Application of self­tuning regulators to the control of distil­lation columns . Pepr . 6th IFAC/IFIP I nt . Conf . on Digital Computer Appl . to Proc ess Control , Dusseldorf .

Hahn, V . ; Unbehauen, H . ( 1 9 82 ) . D irect adap­tive control of non-minimum phase systems . Pepr . IEEE Conf . on Applic ations of Adap­tive and Multivariable Control, Hul l , pp . 1 70- 1 7 5 .

Ionescu , T . ; Monopol i , R . V . ( 1 977 ) . Discrete model reference adaptive control with an augmented error signal . Automatica 1 3 , No . 5 , pp . 507- 5 1 7 .

-

Koivo, H . N . ( 1 980) . A multivariable self­tuning control ler . Automatica .!:..§_, No . 4 , pp . 3 5 1 - 366 .

Lozano , R . ; Landau , I . D . ( 1 98 1 ) . Redesign of adaptive control scheme s . I ntern . J .

Contr . 22_, No . 2 , pp . 247-268 . Monopol i , R . V . ; Hsing , C . C . ( 1 97 5 ) . Parameter

adaptive control of multivariable systems . I nter . J . Contr . �, No . 3 , pp . 3 1 3-327 .

ACKNOWLEDGEMENTS

This study was supported by DFG project Un25/ 2 1 . We would l ike to thank Bayer AG, Lever­kusen for making the experiments possible .

p l a n t

c o rrec t i o n Ye + netw o r k +

!:! ¥a co m pen sator

a d a pt i o n �

+

'!!. r e fe r e n ce �m m o d e l

y

Fig . 1 . Basic structure of the adaptive model referenc e scheme us ing a correction network and an explic it integrator

.---......,...,..........,., w'

d i a9 LS.J 11-z-1\ a d ap t i ve c o nt r o l

l oo p

Fig . 2 . Adaptive control scheme cascaded with decoupled PI control lers

c ,

T,

- h

Fig . 3 . Schematic diagram of the disti l lation column . c

1; c 2

: containers with different con­centrations of the feed product p . : i nlet pump h� heating p : reflux flow ratio T

t: top temperature

Tb

: bottom temperature

3 1 4

0 20 40 60 80 t/min

JiMtHfl¥ �!=118"1 fhr i i i d -+2- : . . . : ± F -2??/ K··· ! I :

� f I + 2 . b ) 0 20 40 60 80 t/min

JtK1111w1 - 5 �hl0/o , ; : : : : '

+ 5

I:' 'y r .

':

• c )

O io 40 6b 80 t/min Fig . 4 . Performance with well adjusted controller parameters a) without integrator 1 b) with integrator 1 c ) without integrator 2

P . Wiemer et a l .

-li�.! I 11IN1Y·�f '11t1it'�ff J +1 - . : t : ' ' � : .� . -.- - _f ' t . I t ' ' ' -51/ih/O/� ; ! : : ' : ; ' ; : : i: { �r ! 7 I t : : ' J "\ .s , l 'P> f 1 )- F(

I I ' E ; ' 1· : ' : :

-2 0A1K: f . : :• � :4 1 : . J : 1'±7Hi · 1 'i i �

+1 -5 +S

7� :-s ' j£ ' i t ' t o. )

O 20 4 0. 60 80 1 00 1 20 1 1. 0 1 60 t /m i n Fig . 5 . Performance of adaptive transients with zero initial controller parameters

2 2 a ) without integrator b) cascaded with PI-controllers c) without integrator , setpoint changes +/- lK ( t/+ ) � 2 5 % feed disturbance ( t/+ )

2 5 % feed disturbance and 2 0 % concentration dist . ( t/+ )

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

MODEL REFERENCE ADAPTIVE CONTROL OF AN INDUSTRIAL PHOSPHATE DRYING FURNACE

B. Dahhou, K. Nijim, M. M'Saad and B. Youlal

L11bomtoire d'Electroniqw' et d'Et11d1' des Systhnes A uto11111tiq111's, Farnltt des Sciences, B.P. 1014 , Rabat, Momffo

Abstrac t . Thi s paper presents experimental results o f a mode l reference adap­tive control algori thm wi th independent tracking and regulat ion obj ectives presented in (Landau, Lozano , 1 98 1 ) to the control of a phosphate drying pro­cess at the Beni-Idir Fac tory of the OCP (Office Cherifien des Phosphates -Maroc ) . The main control obj ective is to keep the mois ture content of dried phosphate at a prescribed value ( l , 5 %) , independently of external perturbations acting on the drying process . The plant dynamic characteristics vary under the effect of variations of the input material characteristics such as the phosphate nature and humidity that vary from one layer to another . The implementation of the adaptive algori thm was based on a reduced order plant model previous ly checked and uses a small s ize minicomputer . An energy saving c lose to 4 , 5 % and ten times reduction of the variance of the output humidity error wi th respect to the des ired one were obtained . Thi s led to the motivation of introduc ing an advanced computer control in Moroccan Phosphate Industry .

Keywords . Adapt ive Control ; Model Reference Process ing ; Drying Furnace .

Energy Saving Phosphate

INTRODUCTION

During the pas t few years different approaches to adapt ive control have been suggested, stu­died and applied . Among these approaches , the Mode l Reference Adaptive Sys tem and the Se lf­tuning Regulator seem to be the most attrac­tive ones .

This paper deals wi th the app lication of a Mode l Reference Adaptive Control Algori thm , presented in (Landau , Lozano , 1 98 1 ) , to the contro l of a Phosphate Drying Process at the Beni-Idir Factory of OCP . The phosphate , independently of its way of extraction has about 1 7% humidity . Be fore being sold , thi s high humidity has to be re­duced to around 1 , 5 % in Rotary Drying Furna­ces . The drying process is one of industrial opera­tions that requires a great consumption of energy , hence an increase in the price of the produced dried material . The obj ective of this s tudy is to keep the hu­midity of the dried phosphate close to the prescribed value ( 1 , 5%) , independent ly of raw material humidity variat ions ( 7 � 20%) ; feed f low rate variat ions ( 1 00 � 240 t/h) and other perturbations that may affect the drying pro­cess . There is , invariably , some uncertainty in the

3 1 5

characteri stics of the proces sed phosphate that can be attributed to variable moisture content and the nature of the d�mp product . The phosphate drying proces s i s therefore non­linear and non-s tationnary in its nature . The change in dynami c characteris tics wi th opera­ting conditions is such that a fixed parame­ter control ler i s inedequate to achieve satis­factory performances in the entire range over whi ch the characteristics of the process may vary . An adaptive contro l holds obvious at­tractiveness in such s i tuation because con­troller parameters are adj usted during opera­tion to maintain speci fied dynamic performan­ces . A Model Reference Adaptive Control Scheme , developed by I . D . Landau and R. Lozano and based on reduced order plant model , previous­ly checked was implemented us ing smale s i ze minicomputer . The main motivations of such control s cheme are the following : - It i s s imple : i . e . i t can be implemented

even on microcomputer . - It ensures the asymptotic convergence of

the plant output (the humidity of the dried phosphate) to the reference sequence and the boundness of the contro l appl ied to the plant .

- It al lows to solve the problem of

3 1 6 B . Dahhou et a l .

independent specif ication of tracking and regulation obj ectives .

This paper is organized as fol lows . In sec­tion I I , we provide physical description of the used drying proces s . In section I I I , a mathematical model of the dry ing furnace is formulated . In sect ion IV, the adaptive con­trol scheme used to control the phosphate drying furnace is presented whi le in sect ion V, the hardware and software faci l i ties are described and the furnace control performan­ces us ing the Model Reference Adapt ive Con­trol Algori thm are reported .

PROCESS DESCRIPTION

The phosphate drying furnace is mainly cons­tituted of the following components (fig . I ) : - Feed ing sys tem. - Combus tion chamber . - Drying tube - Dus t ing chamber - Vent ilator and chimney . These elements are described in the fol lowing . Feeding Sys tem . The main part of the feeding sys tem is a cons­tant speed moving belt that carries the raw phosphate into the furnace . A large container spreads the phosphate over the belt at regula­ted rate by contro lling the opening of the container to the belt . This wi l l al low the phosphate to be fed into the furnace at the rate needed for production. Combus t ion Chamber . The combust ion chamber produces the hot gas needed for the dry ing process . The heavy fue l is initial ly heated to 1 00° C by steam . To fa­ci li t � t e its mixing with the air , the fue l is pulveri sed by the aid of auxi liary j et of steam. The necessary oxygen for the combus­t ion is produced by the primary air inj ected under low pres sure by a vent i lator in the combus t ion chamber . The heat produced is lransfered into the drying tube by secondary air current .

Drying Tube . This is an hor izontal tube of 25m length , its rotation veloci ty is constant ; its produc­t ion capac ity is in the order of 1 5 0 ton/hr . The tube has cascades in its inner s ide ar­ranged helically , to faci l itate the thermal exchange between the hot gas and the phospha­te, and also they help in dr iving the phos­phate to the output of the tube . Contrary to cement furnaces the movement of the phosphate and the hot gas occurs in the same d irect ion in the drying furnace , from the combus tion chamber to the dus ting chamber .

Dus ting Chamber . The dus ting chamber is made up mainly of shel­ved tubes whose primary function is to s low down and recapture the phosphate f ine particu-1 es which are carried into the dus t ing cham-

ber by the hot gas . These fine part icules ma­ke up about 30% of the dried phosphate .

Vent ilator and Chimney . The main role of the vent ilator is to create a reduction in the pres sure at the head of the drying tube to induce a secondary air cur­rent and to prevent trapping of the phosphate in the drying tube . The chimney act ion wi l l serve as evacuator of the hot gas out of the furnace . The final product is received at the exi t of the dust ing chamber by the main conveyor . The exi st ing conventional control loops on the phosphate dry process are shown in figu­re 1 . The flows of primary air and steam are adj us­ted wi th respect to the fuel f low in order to ensure a complete combus t ion.

PROCESS MODEL

Several mode ls have been developed in (K . Naj im and al l , 1 97 6 , 1 97 7 , 1 978 , 1 979) to des­cribe the dynamic behaviour of the phosphate drying furnace . We have chosen a s ingle input­single output one , by letting the product feed rate to be kept cons tant (e . g . maximum production) . The fuel flow (the control va­riable) and the humid ity of the dried phos­phate (the output var iable) are the key va­riab les for suitable s ingle input - single output mode l of the furnace . A s imple repre­sentation of the simplified model can be written as :

- I A (q ) y ( t ) -d - I q B (q ) u (t) + w (t )

- I -n b +b 1 q + . . . +b q B o nB b -Fo 0

where

( 1 )

( 2 )

{q- 1 } i s the backward shift operator, {d } represents the process time de lay , {u (t ) } and {y (t ) } are the process input (the fue l f low) and output (the humidity of the dried phosphate) respectively , and w(t) is a boun­ded disturbance . Th is model is mos t adaptable to adaptive con­tro l sys tem which we have adopted . Moreover , it uses the var iables to which the operating of the furnace i s the most sens i t ive . The samp ling per iod T and the proces s time delay have been determined from an a priori characterizat ion study of the proce s s , while the process model order has been chosen to al low satisfactory performances of adapt ive contro l system . The obtained values are T = 45 s d = 2 ; nB = 1 ; and nA = 3

PRESENTATION OF ADAPTIVE CONTROL SCHEME

We wi l l use the notation of (Landau, Lozano ,

An Industrial Pho sphate Drying Furnace 3 1 7

1 98 1 ) and g ive only a b r ief out l ine of the basic theory of the contro l scheme adop t ed . The theory and des i gn of this s cheme i s wi de­ly discussed in the above reference . The main obj e c t ive of the control sys tem i s to f i nd a control l aw so that a n ini t i a l e r­ror between the p lant output (described by the equat i ons ( I ) and ( 2 ) and assume d to be a minimum phase p l an t ) and a reference sequen­ce {yM (k) } or an i ni t i a l output d i s turbance conve rgence to zero wi th the dynami c s of the CR - po lynomi a l , i . e . ,

- I M - I CR (q ) (y (k+ d ) - y (k+d) ) = S (q ) w ( t ) ( 3 ) where

- I CR (q ) ( 4 )

i s an asymptotically s t ab le po lynomi a l and the po lynomi a l S is so that

- I S (q ) w ( t ) = 0 for k � k* (5 )

The reference sequence c a n b e re a l i zed by the output of a reference model described by

- I M -d - I M CT (q ) y (k ) = q D (q ) u (k )

where - I T - I T CT (q ) = I + C l q + . . . + CnC

( 6 )

T is an asymp t o t i c a l ly s t ab le po lynomial and

- I - I - n D (q ) = d + d 1 q + . . . +d q D o nD

( 8 )

An app ropri ate control conf i gura t i on u s e d for the case of known p l ant p arame ters to rea l i s e the obj e c t i f ( 3 ) i s given by

u (k ) - 1 M - I CR (q ) y (k+d) -R (q ) y (k )

B (q- l ) S (q- 1 ) ( 9 )

where t h e po lynomials S (q- 1 ) and R (q- 1 ) veri­fy the f o l l owing iden t i ty .

- I - I - I -d - I CR (q ) = A (q ) S (q ) + q R (q ) ( I O)

- I -n l +S 1 q + . . . +s n q s s

- I -nR r +r 1 q + . . . + r q o nR

( 1 1 )

( 1 2 )

wh ich has an unique so lut ion for the po lyno­�ials S (q- 1 ) and R (q- 1 ) for a given CR (q- 1 ) if one choos es :

and n d - I s

Th e control l aw ( 9 ) can be wri t ten

where

ASCSP-K*

- 1 M CR ( q ) y (k+d )

( 1 3 )

( 1 4)

T ¢ ( t ) = [ u ( t ) , u ( t- I ) , . • , u ( t-d-�+ l ) , y ( t ) , . • • , y (t-nR) ] ( I S )

PT [b , b s 1 +b 1 , . . . , b sd- l ' r , . . . r ] o o nB o nR ( 1 6)

When the p lant parameters are unknown the p a­rame te r vector p of the control l aw ( 1 4 ) g iven b y E q ( I 6) can n o t b e computed . Landau and Lozano have deve loped an extension of the l i near contro l l er des i gn given by E q ( I 4) whi ch is ap p l i c ab le to minimum phase p lants and for

which only the t ime de l ay {d} and u� perbounds of the degrees of po lynomi a l s A (q- ) and B (q- 1 ) denoted nA and nB are known .

The parameter p i n E q ( I 4 ) i s rep l aced by adj ustab le parameter vector p (k ) wh i ch wi l l b e updated b y the adaptat ion mechani sm . Therefore the control law i s g iven by :

T - I M p (k ) ¢ (k ) = CR (q ) y (k+d ) ( 1 7 )

and the d e s i gn obj ec t i f ( 3 ) wi l l be asymp t o t i ­c a l ly ach i eved i f :

w (k ) = 0 ( 1 8 ) and i f t h e f o l l owing adap t a t i on algorithm i s used

p (k ) = p (k- 1 ) + F (k ) ¢ (k-d) y* ( k ) ( 1 9 ) wi th (2J F (k+ I ) = _I_ [F (k ) - F (k ) p (k-d ) p T (k-d ) F (k)

A (k) A (k) I � ¢T (k-d ) F (k ) ¢ (k-d ) 2

whe re O<A. 1 (k) � 1 ; 0< 2 (k ) �2 ; F ( l ) >O ( 2 1 )

and y* (k) i s the adap tation error defined as : - I H I (q ) - I H z (q )

(22)

- ! - 1 where H 1 (q ) and H 2 (q ) are asymp to t i c a l l y s t ab l e manic p o lynomi als and should b e chosen s uch that the t rans fer func t i on

i s s t r i c t ly pos i tive real func t ion with

for k < k < oo 0

( 2 3 )

( 2 4 )

and E* (k) i s the augmented error d e f i ned as :

(25 ) The adap t ive cont ro l algori thm whi ch has been adopted for the control of the phosphate dry i ng furnace i s der ived from the previous one for :

- I - I H 1 (q ) = H 2 (q ) = ( 2 6 )

The posi t iv i ty cond i t ion in E q ( 2 3 ) i s auto­mat ical ly ver i f ied and the exp res s i on for the adaptation error in Eq (22) becomes :

3 1 8 B . Dahhou e t al.

- I T CR (q ) y (k ) -p (k- 1 ) ¢ (k-d) 1 +¢T (k-d ) F (k ) ¢ (k-d)

Figure 2 shows the b lock diagram of the adap­tive control scheme .

l'RACTICAL ASPECTS OF THE CONTROL SYSTEM

Computer Hardware and Sof tware Faci lities . The DDC computer hardware used for imp lemen­ting the contro l ler algori thm was based on a D . E . C . LSI- I I microcomputer . The configura­tion invo lves a 1 6 bit microproces sor wi th the minimum hardware ari thmetic faci lit ies , i . e . a l l integer and f l oating point multip li­cat ion and divis ion performed by software , 64 K memory , dual f loppy disc mass storage , con­sole terminal and te letype printer . The experimental data interface cons is ted of a 16 channe l multiplexed succes sive approxima­tion A/D converter , 4 D/A converters a l l wi th 1 2 bit resolution and programmab le real-time c lock counter . The standard DEC real- time operating sys tem RT- I I was used to deve lop the programme and to contro l i ts execut ion , us ing the real- I I Fortran software faci l i t ies . The f lowchart of the real-t ime algori thm wi th the interface between the process and compu­ter is shown in fig . 3 .

The Choice of the CR Po lynomial . - 1 The choice of the polynomial CR (q ) results

from a compromi se be tween the track ing error and the control value . Indeed , we have ob ser­ved that when the tracking error decreases quickly after any perturbation, the control becomes more energetic . In the case of our experiment , the fol lowing po lynomial

- I -2 -3 l -0 . 85q +0 . 25q -0 . 0585q

has been chosen in order to avoid abrupt chan­ges in the plant output . "Start-up" of the Control System. The initialisation of the control system has been done as fo l lows

[O , • . . , 0 ) [ UN , . . • , UN ,HN , . . . ,HN )

where UN and HN represents the fue l f low and the humidity of the dried phosphate respecti­vely at the operating point

F ( I ) = 1 000 I

The use of such initial values lead to a con­tro l too important for the process this in­duces us to fix the contro l to its nominal va­lue UN unt i l the computed control is c lose to an interval around its nominal value UN and

this in cons tant way ( the control may remains in the prescribed interval for about ten ite­rat ions ) . This beeing done , the control sys­tem operated with the "decreas ing gain" algo­rithm (A 1 (k) = A2 (�) = 0 : 95 ) as . lo�g as the trace of the adaptive gain matrix is greater than a prescribed value . If not so , the con­tro l sys tem operated wi th "cons tant trace" al­gori thm (A 1 (k ) = A2 (k ) and A 1 (k) is such tra­ce (F (k) ) = cons tant ) .

Results . In order to compare the performances of the adap tive control scheme wi th those achieved when using conventional PID control lers , the fol lowing experiments have been carried out - the PID control lers are used to control the

phosphate drying furnace , its parameters are adjus ted by an operator in order to pro­vide acceptable performances . The microcom­puter i s used only to supervise the furnace operating and for product ion management .

- The adaptive control system presented above is used to control the phosphate drying fur­nace . The microcomputer is then used to con­trol and supervise the furnace operating and for production management .

The operating condit ions of the dryer for toth adaptive contro l system and conventional PID control l ers were the most common ones : at the input , the product feed rate was c lose to 220 t/h and its moisture content was sub­j ect to random variations . The range of these variations is between 1 0 and 1 5 % . The recorded curves of the humidity of the damp and dried phosphate and the fuel f low obtained by the two experiments are shown in figures 4 and 5 . Table I summaries statistical results that a l lows to appreciate the performances by u­sing the two control systems .

-- ------it.cord• Stu:i.stical

Couvu1cional Ch..r.ctu·itc:ics Caatralhr 4dapti� coo.crolhr tlpn�anc• 12,3% 1 2 , 70%

1)-.p pl!.asphau lalaidity Va:rimca 0 , 1 8 0 , 7 1

Driad pboaphaca !spar.ace 1 ;1a: 1 , 41%

b.-1dity Varianca 0,.57 O, l.5 !'!aan COftSUlllptiou 1 1 ,JS 10,9 ·-1 fl�

Varieca 0 , 1 6 0,039

Tab le 1 Recorder statistical characterist ics

An Indus trial Phosphate Drying Furnace 3 1 9

CONCLUSION

The control studies reported in this paper de­monstrate a successful application of model reference adaptive control ler to an industrial phosphate dryer . The results of the experimentation i llustrate the key features of the mode l reference adap­tive control ler , espec ially its potentiality to ensure sui table performances when changes of the plant dynamic characteristics occur . On the other hand , the adaptive contro l sys­tem presented above al lows , an energy saving of 4 , 5% and satisfactory quality of regula­tion which involves the material saving , be­cause of l ess thermic solici tat ions leading to a longer period between revisions . Acknowledgement . The authors grateful ly acknowledge the finan­cial and material support of the OCP of Mo­rocco .

rr� ... ry lr�ondary

fturner Combuotlon Chamher

REFERENCES

Landau , I . D . , and R. Lozano ( 1 98 1 ) . Unifica­tion of discrete time explicit model refe­rence adaptive control des igns . Automatica vol . 1 7 , n° 4 , pp . 543-6 1 1 .

Naj im, K . , M. Naj im, B . Koehret , and T . Ouazzani ( 1 976) . Modelisation and s imu­lat ion of a phosphate drying furnace . 7th Annual Pittsburgh Conf . on Model ing and Simulation. Pittsburgh - USA. April .

Naj im, K . , and D . Jouhari ( 1 97 7 ) . Identifica­tion of a mul tivariable industrial system : A phosphate drying furnace . 20th Mid West Symposium on Circui ts and Systems Lubbock, Texas , U . S . A . , August .

Naj im, K . , M . Naj im and D . Jouhari ( 1 978) . Identification of a phosphate drying fur­nace . JACC 1 8-20 Oct . Phi ladelphia, USA.

Naj im , K . ( 1 979) . Commande des systemes com­plexes par apprenti ssage stochast ique . These de Docteur-es-Sciences , Universite Paul Sabatier , Toulouse , Mai .

llOtary Drlng Tube Duat Chamber

vent l l •tor

Fig . l Drying Surface nr1�d produCt

,. lr• o:: ll i n g 11101Jc l IC e g u l a t i ou

•01ltt 1

� c�J

Fig . 2 Block diagram of the adaptive control sys tem

v(t)

__!(t),!(t-d) I "1 T (t-d)F(l 'j6 (t-d)

320

:� HOO

B . Dahhou et a i.

tai.ci.&liutiou

Oat& acquilicioG

Thr.,hold overcak.io.1?

Coacro lhr par ... ceu

Htiucion ind c:aacrol C0'9Dutatioa

troducdoa maaace•at gcf •U.Ci.atic.al t'U\ll.t•

updu.ia1

, IHulca touiac

�Coit 'itwi Ot oa-tlae di.1lllicu•

Vah for cloclr.

"'""-----! AID coa.verter

>----- 0/A COllft:l"tU

Oaca dhelay 1nd e1h. t �i>G OD u•• ••mo�

U1u· tauraccioll

Fig . 3 Flowchart of the real-t ime algorithm

(hours)

. ' (h0'1:'1)

Fig . 4 Typical conventional control recordings

1 4 l l

12 I I

An Industrial Phosphate Drying Furnace

tll2o (Dried pnesphate humidity)

2100 2600

2400

cruel !'!aw)

Fig . 5 Typical adapt ive control recordings

32 1

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1 983

ADAPTIVE CONTROL OF CHEMICAL ENGINEERING PROCESSES

L. Hallager and S. Bay Jj;jrgensen

/Jrfiarl11t1'Ul of Chrmiad J�'ngin1'ni11g, Tnlwiml Uuivrni/_\' of Dnwuirk, Buildiug 229, 2800 Ly11gby, D1'11111111ll

Abstract . A MIMO selftuning controller (MIMOSC ) is proposed for multivariable , nonlinear chemical engineering processes which may have distributed character . MIMOSC consists of a l inear time series model combined with a linear quadratic optimal control strategy modified to permit on-line solution of the Riccati equation . The model structure is selected based upon .§: priori chemical engi­neering process dynamics knowledge and the model parameters are estimated using a recursive extended least squares method . MIMOSC is investigated experimen­tally on a fixed-bed chemical reactor . The ability to handle load change and a trackin,e; problem is investigated .

Keywords . Adaptive control , process dynamics , chemical reactor control , process identification , recursive parameter estimation , quadratic optimal control .

INTHODUCTION

Chemical engineering plants contain in gener­al a number of energy and mass exchange pro­cesses combined with chemical reaction pro­cesses . In either type of process at least two streams or components are brought into close ( direct or indirect ) contact whereupon the exchange or reaction occurs . The proces­ses are most often carried out in tubular units or equipment with many stages , hence the processes tend to have distributed char­acter . In addition the processes are bilinear and often nonlinear due to the thermodynamic or reaction kinetic relationships . In indus­trial practice slow changes of ten occur in internal process characteristics such as dirt deposition on heat exchange surfaces and catalyst or biological activity changes . The control problems for chemical processes are mainly to maintain product quality within specified limits and if that fail s then to maintain production if at all possibl e . In principle these control problems are multi­variable due to the number of streams or components involved . Traditionally however they have often been solved as mult i loop­problems . This solution has been satisfactory in the vast majority of the cases . However a number of processes have remained difficult to control thus requiring a multivariable design technique . With the increased tendency towards process intensification many more con­trol problems may advantageously be solved while recognizing their multivariable nature .

During the last decade it has been demonstrat­ed experimentally th_ ct many chemical engin­eering processes often can be satisfactorily controlled using rnultivariable techniques .

323

The control designs applied generally have been based upon off line identified linear­ized mathematical models . These design methods however have not been applied signif­icantly in process control practice ; probably due to the effort required to develop a math­ematical model from basic physics and chemis­try using conservation principles . Further­more it is often difficult and costly to identify such a model , partly because esti­mates of some parameters may be difficult to obtain from experiments , and partly because the model often will be non-linear in the pa­rameters . Finally the parameters in chemical processes are usually time varying , for in­stance due to changes in internal process characteristics thus necessitating more or less regular updating of parameters . Conse­quently there are obvious opportunities for multivariable adaptive control design methods .

The method presented in this paper is based upon an indirect or explicit approach where simple linear relationships are postulated between measurements and control inputs . The relationships are based upon a uriori chemical engineering process knowledge

-c�ng bulk

flow directions , energy and mass transfers ai1d rough estimates of the time delays involv­ed . By estimating the coefficients of these relationships on line it is possible for the model to adapt to slow changes either in inter­nal process characteristics or in external loads . The uppropriate feedback gains are evaluated based upon the identified model using some specific control strategy . Rela­tively few investigators have investigated algorithms for tuning of controllers in the multivariable case . Peterka and Astr12J111 ( 1973 ) proposed a multivariable selftuning

324 L . Hal lager and S . Bay J�rgensen

regulator based on linear quadratic optimal control of processes with uncertain parameters . Borison ( 1975 , 1979) extended the basic mini­ITRIDl variance selftuning controller to the multivariable case . Keviczky and Hetthessy ( 1977 ) used a dead time transformation . Koivo ( 1980) extended Clarke and Ga<Vthrop ' s method to a MIMO selftuning controller with an equal number of inputs and outputs . Buchhol t and Klirrrnel ( 198 ) investigated a one step control method on a two stage evaporator . Bayoumi et al . ( 1982 ) presented an algorithm for model order determination and considered processes where the time delay is the same for all con­trol signals .

In chemical engineering processes it is often advantageous to use more measurements than controls . The algorithm presented herein i s based upon a n incremental multivariable time series model where the parameters are esti­mated using recursive extended least squares . The controller design is based upon linear quadratic optimal control .

EXPERIMENTAL PROCESS

A chemical reactor is used as an example of a typical chemical engineering process . The pilot plant reactor considered consists of a tubular fixed bed of' catalyst particles , de­signed to be essentially adiabatic and with negligible heat capacity of the reactor wall . It is described in detail by Hansen and J0rgensen ( 1976 ) . In this series of experi­ments a stream of hydrogen containing 0 . 25-0. 75 mole % oxygen is fed to the packed cata­lyst bed at a temperature of approximately 82 °c . T2e steady state total flow rate is 2 . 6 mg/cm /sec . The adiabatic temperature rise during water formation i s 169 °c pr . mole % reacted oxygen .

Qualitative Pynamical Description

In the gas phase fixed bed catalytic reactor , investigated here , the heat capacity of the gas is negligible compared to the heat capa­city of the catalyst pellets , thus the thermal residence time is several hundred times great­er than the gas residence time . Furthermore it i s known (Hansen , 1973 ) that the mass transport processes are sufficiently fast to make quasi stationary descriptions of the mass-balances reasonable . Finally there is almost thermal equilibrium between the cata­lyst pellets and the gas and within the pel­lets . A reasonable simplified mathematical model of the reactor therefore is a pseudo homogeneous model consisting of one dynamic non-linear partial differential equation for the temperature and one quasi stationary dif­ferential equation for the oxygen balance . Both including dispersive terms .

The two important dynamical features of the reactor system are thus

a . A fast propagation of concentration and flow rate changes . The mass balance i s a

quasi stationary functional of the inlet conditions and the temperature profile .

b . A slow propagation of temperature changes . The thermal wave passes through the reac­tor in 11-30 minutes dependent on the flow rate .

The Control Problem

The control problem considered in this paper is control of the chemical reactor ( see Fig . 1 ) . The plant has three inputs-temperature , composition and flow rate of the entering gas - in which disturbances can enter and control actions be taken . The possible outputs are 11 equidistant temperature measurements along the reactor axi s , composition measurements at the inlet and at one of 5 axially equidistant points through the reactor and a flow rate measurement . In industrial practice only a limited number of measurements are available ( 2-5 temperature measurements and possibly a concentration measurement ) . Hence the temper­ature profile is known only to a limited ex­tent . The purpose of control may be defined as :

I . Disturbance rej ection :

a . to maintain all o r part of the temper­ature profile , to reduce the thermal load on the process .

b . to maintain the exit concentration as a measure of product quality .

II . Set point tracking :

c . to track changes in the set point smoothly with reasonable speed .

THEORY

The problem of designing an adaptive control­ler for a chemical process i s divided into three parts .

i . Model structure selection

ii . Parameter estimation

iii . Control design

The selection of a model structure which i s relevant for the given process is considered to be very important in order to pose a well conditioned estimation problem with a reason­able low number of parameters . In separating the process identification ( i and i i ) from the control design ( i i i ) the separation theo­ry is used although it is not strictly valid . The certainty equivalence principle is used as an ad hoc design principle in that the control design is handled deterministically, not accounting for the parameter uncertainty .

Model Structure

A qualitative model structure may be obtained from chemical engineering knowledge of quali­tative process dynamics . The ingredients in such a dynamic process description are knowl-

Adaptive Control of Chemica� Engineering Processes 325

edge of :

i ) The sufficient number of conservation balances for momentum , mass and ener­gy necessary tu describe the process state with a for practical purposes reasonEible accuracy .

i i ) The significant capacities .

iii ) The bulk flow directions .

iv) The presence of couplings among the balances .

For many chemical engineering processes the Dynamic Process Intraactions among the ba­lances and the prucess inputs may be depicted clearly in a diagram ( a DYPID ) as shown in Fig . 1 for the example process . The heat balance having a significant capacity is shown as a box which is rectangular in order to indicate the axial coordinate of the re­actor . The total mass and oxygen flows are shown as horizontal lines .

Given the process controls and measurements it is possible to select a reasonably low rnunber of' parameters in the multi variable time series models described below provided the process is observable . When a state of a balance with predominantly convective flow is measured all inputs which affect the par­ticular balance are observable . Usin,_c; the DYPID it is thus possible to investigate whether essential process inputs are quali­tatively observable . For the example system measurements of internal oxygen concentra­tion and temperature contain information from all inputs whereas measurement of total flow rate e .g . at the reactor outlet only provides information about inlet flow rate . It is assumed that the input-output data from the chemical reactor can be described by a dis­crete-time multivariable time-series model :

*- -1 *- -1 J[ x ( t ) = A(q )x ( t-1 ) + B(q )u ( t-d)

+ C ( q-1

) e ( t )

where

x*( t ) ( dim n) i s x( t ) -x ( t-1 ) where x ( t ) i s a measurement vector

u*( t ) (dim m) is u ( t ) -u( t-1) where u( t ) i s an input vector

e ( t ) ( dim n ) is a zero means , white noise sequence

d is Lhe delay fl'om input to output

-1 -1 -1 A(q ) , B ( q ) and C ( q ) are matrix poly-nomials of order a , b and

_l in the backward shift operator q i . e .

-1 -1 a A(q ) Ao + q Al + • . • + q-Aa

I

-b + q �

The dimensions ( n , m , a , b , c ) and the delay d are assumed known and the matrices Ai , i = 0 ,

• . . , a Bi , i = O , . . . , b and Ci , i = l , . . . , c are assumed unknown but constant or slowly varying .

The size of the model is defined when n , m , a , b , c and d are fixed and it is decided which measurements and controls are to be used . If the model is not further structured it would be necessary to estimate n(n· ( a+l+c ) +m( b+l ) + l ) parameters ( including constant terms ) .

In the example process the number of parame­ters is reduced using DYPID ( Fig . 1 ) :

a . The distributed disturbances ( concentra­tion and flow rate ) influences all meas­urements with a delay of 1 sampling peri­od ( due to the sampling of the system) .

b . The thermal wave as observed by a number of temperature measurements is described essentially as a delay from the tempera­ture input to the first measurement and from one measurement to the next . This behaviour necessitates a proper selection of sampling time in relation to model order .

The simplest approach is to select equi­distant measurements , a sampling time equal to ( or slightly larger than) the delay between two neighbouring measure­ments and set a=O , to describe convective propagation .

A more elaborate approach is to select a shorter sampling time but use additional old measurement ( i . e . a>O) , thus includ­ing a description of dispersive effects . This description adds flexibility in the selection of sampling time and allows for larger flow rate disturbances .

In this way the dominant thermal dynamics of the reactor and the influence of the inputs can be modelled with a reasonable number of parameters .

Parameter Estimation

The i ' th state in equation ( 1 ) can be written :

T x . ( t ) = • . ( t ) 8 . + e . ( t ) 1 1 1 1

where :

;: ( t ) 1

and

8 . 1

T T T {x ( t-1 ) ,x ( t-2 ) , . . . ,x ( t-a-1 ) ,

T T u ( t-d) , . . . ,u ( t-d-b) }

( superscript i means the i ' th row of the matrix) .

The parameter vector 8 . is estimated using a recursive least square§ algorithm with vari­able forgetting ( Fortescue et al . 1981 , Hallager 1983 ) :

326 L. Ha llager and S. Bay J¢rgensen

@ . ( t+l ) 1 E . ( t+l ) 1 K . ( t+ l ) 1 s . ( t+l ) 1 H . ( t+l ) 1

where : P . ( t+l )

1

§ . + K . ( t+l ) E . ( t+l ) 1 1 1 x . ( t+l ) - �'. ( t+l ) @ . ( t ) 1 1 1 P . ( t ) � . ( t ) ( S . ( t+l ) +a . ( t+l ) ) -l

1 1 1 1 �'. ( t+l ) P . ( t ) � . ( t+ l ) 1 1 1

2 1-S . ( t+l ) -E . ( t+l ) /V . 0 1 1 1 1 2 Yo = -2 [ H . ( t+l ) + (H. ( t+1 ) +4S . ( t+ 1 ) ) 2 ] 1 1 1

is the normalized covariance matrix of the estimate error ( e . -13 . ( t+l ) ) for a . = 1 . 1 1 1

K . ( t+ l ) is the corresponding gain vector 1 ai ( t+l ) is the variable forgetting factor

which keeps a weighted sum of the squared a posteriori errors fixed at a target value ViO '

In this algorithm the forgetting factor stays within reasonable limits except if S . ( t+ l ) = 0 and EI ( t+l ) /Vci:l occurs simultaneously , which is not likely . 1he algorithm is initiated with :

§ . ( 0 ) = 0 and 1 1 P . ( 0 ) = - I 1 s

The target value ViO may be chosen initially based upon the desired memory length and the process noise level . With a reasonable value of ViO the value of S is of minor importance .

By extending � . ( t ) and 0 . appropriately the coefficients i� the syst§m noise polynomial matrix may be estimated using the algorithm given above . 1he convergence properties of the resulting extended least squares algorithm is however more restricted than for the simple recursive weighted least squares ( SoderstrCim et al . 1978 ) .

In general all parameters are assumed comple­tely unknown hence it is reasonable to initi­ate all covariance matrices at the same value and thus reduce computations considerably as it only will be necessary to calculate one estimator gain and one covariance matrix in each step (Borison , 1979 ) .

In the present case however extensive use is made of structural zeros , i . e . on the basis of a priori knowledge some parameters are set to zero . 1his may lead to differences between the different � . , K . and P . ' s as the corre­sponding terms k t.ernoved1from � . , and thus may necessitate the updating on n1estimator gains and n covariance matrices . However each estimator gain and covariance matrix will be of reduced dimension , compared to the ful l parameter case and therefore possibly require less computer memory and time .

Control

In order to eliminate drift which is a conman

feature of chemical processes a method for offset elimination is essential . Here inte­gral action of selected measurements is used . Consequently the identified model is reformu­lated to the following state space form :

y(t+l ) =A*y( t ) +s'\i.*( t )

where

*T *T *T y( t ) = {x ( t-1 ) , . • . ,x ( t-a-1 ) , u ( t-d-1 ) , it:T T

• . • u ( t-d-b) , z ( t-1 ) }

* y( t ) =z ( t-l ) +Ey ( t )

and A*, s* are shown in Table 1 .

Minimization of the control criterion :

gives

where

T = y ( N ) QoY(N) + N-1 T . T l [y ( k ) Q1y(k) + u* ( k ) Q2u ( k ) ] k=O u*( k ) = - L ( k ) y ( k )

*T * -1 *T * L ( k ) = ( Q2 + B · S (k+l ) ·B ) · B · S ( k+l ) 'A

and S(k) is calculated by solving the Riccati equation :

*T * * S ( k ) =Ql + A · S ( k+ l ) ' (A -8 · L ( k ) )

subj ect to the condition : S ( N ) = o0 •

In the general selftuning case this solution is not strictly usable because the system ma­trices A* and s* are not constant and their future variation is not known as the parame­ters are estimated on-line . If N is chosen as 1 , i . e . a one-step criterion is employed , the solution is applicable . In the present case , however , there is known to be a thermal time delay from inlet to outlet , which usual­ly is 5-10 times the sampling time . It seems reasonable that the control criterion should at least cover this delay in order to account for the primary convective effects of a given control input . Consequently the following strategy is attempted:

N is set to infinity indicating that the sta­tionary solution is searched for , the Riccati -equation is iterated ( the 'wrong ' way) once at each sampling instant ( starting from the last solution) and the newest value of S ( k) is used in the calculation of the feedback gains .

When the system parameters are constant the feedback gains converge to the stationary optimal gains . 1herefore , if the parameters converge the feedback gains also will con­verge to the corresponding optimal stationary values .

1he purpose of this strategy is to approach the true optimal solution asymptotically and

Adap tive Control o f Chemical Engineering Processes 327

yet only use modest amounts o f computer time . It does not seem reasonable to invest a lot of computer time in finding the true station­ary solution of the Riccati-equation as long as the parameters , which are the basis of the calculations , are only varying , uncertain approximations .

A consequence of using this strategy is a transient period during startup in which an unstable process may have to be stabilized by a simple controller if the feedback-gains are far from optimal .

EXPERIMENTAL SPECIFICATIONS

The Multi Input-Multi Output Selftuning Con­troller (MIMOSC ) presented above has been implemented on a DEC-1 103 system which con­trols the fixed bed chemical reactor . The necessary engineering input encompasses spec­ifications of model structure and control purpose . The rationale for these specifica­tions is demonstrated below for the example process .

Model Structures

The five model structures shown in Table 2 were investigated . In all models the delay d was set to 1 and a single diagonal c-matrix was used. In models I , III-V five equidis­tant temperature measurements were used . In models I and III the sampling time of 4 min nnd one A matrix ( a=O ) were used . The samp­ling time is 10-2o% larger than 1/5 of the menn thennal residence time at thi s operating point .

The A0 matrix was structured as :

x 0 0 0 0 x x o o o o x x o o o o x x o o o o x x

( 2 )

where X denotes a parameter to be estimated . In addition to the delay tenn from one meas­urement to the next ( - subdiagonal elements ) , an autoregressive tenn in each measurement is included ( - diagonal elements ) . This expan­sion was employed to add some flexibility to the model .

The B matrix for model III was structured as : x x x o x x o x x o x x o x x

( 3 )

I n model I were only temperature control is employed , the two last columns were deleted from this matrix .

In model II only the measurements at the axial positions 0 . 2 ancl 1 . 0 were used with a sampling time of 4 min . The four A-matrices were structured as :

The single tenns in A3 & A4 accounts for the delay .

In model IV and V a sampling time of 3 and 2 min and two and three A and B matrices were used . The A0 and B0 were structured as above . The following A and B matrices accounted for the delay and dispersive effects :

0 0 0 0 0 · x o o:

x o o o o o o o : O X O O O I and B . = 1 o o x o o : l

o o o x o l J

Control Purpose

0 0 0 0 0 0

The purpose of control was selected as keep­ing or reaching a desired temperature profile . The weighting matrices were diagonal . The integral states were infinite summations of T0 . 2 for model I and II and of T0 . 2 ,

_T0 . 6 and

T1 0 for models III-V. The state weights were always 0 . 001 , the weights on the integral states were 0 . 25 except for model V . 2 where 0 . 1 was used on all three integral states . The input weights are shown in Table 2 . In the control examples given below no effort was put into tuning the control parameters . The experiments were undertaken without prior simulations .

Procedure

Initially all parameters were set to zero , v . 0 to the values shown in Table 2 and 13=1 . 1

During identification independent Pseudo-Ran­dom Binary Sequences were applied simultane­ously to each input . After a few samples solution of the Riccati-equation was started . When the model parameters had become nearly constant the PRBS generation was stopped . Model III was identified in closed loop while the other models were identified in open loop . Now the loop was closed , if not before , and the controller allowed to bring the process to the desired steady state . The parameters were stored and the experimental program was undertaken .

RESULTS AND DISCUSSION

Identification

Generally the a priori specified model struc­ture proved to be very reasonable at low oxygen level as the models with the estimated parameters were able to explain 8o% or more of the observed variance . The first measure­ments were however not well modelled by mod­els I-III . In an earlier version Hallager and Jorgensen ( 1981 ) applied an absolute val­ue model instead of the incremental model applied here . They found that the first measurement was well modelled with a model similar in structure to model III , but with a

328 L. Hallager and S . Bay J�rgensen

second B-matrix.

On-line tuning of ViO was performed in sane instances in order to keep the forgetting factors from changing too often .

In agreement with the earlier analysis , where a model structure similar to model III was analysed , it is concluded that the models are well structured.

Models I-V . 1 were estimated at an inlet oxy­gen concentration around 0 . 25 mole% whereas model V . 2 was estimated at 0 . 75 mole% oxygen . At this relatively high oxygen level the model was able to explain 60% or more of the ob­served variance . The change in inlet concen­tration gives rise to an increase in the tem­perature gradient over the reactor length from - 30 °c to 140 °c . This increase in temperature level gives rise to a significant increase in process gain and due to the cou­pling between the heat and oxygen balance also to significantly increased nonminimum phase character .

Control

The behaviour of MIMOSC has been investigated with several different disturbances . Results will be presented for step changes in the up­stream temperature and described for a set point change .

Regulatory Behaviour

Fig . 2 shows the exit temperature and temper­ature control for models I , and II where the number of measurements used are five and two respectively . The responses show that there in this case is very little difference be­tween the two model structures . Due to the convective nature of the thermal disturbances , these controllers can only handle the deter­ministic low frequency part of the disturbance . All the temperature control can do is to start a thermal wave to chase but never reach the disturbance . Integral action gives in both cases a very sluggish response due to the windup occurring while the disturbance is in the system. In this case offset elimina­tion by weighting the absolute states and the incremental control action is to be preferred (Hallager and Jr;,rgensen ( 1981 ) ) . The effect of using multiple controls has also been dis­cussed in the above paper and can also be seen here through comparison of Fig. 2 with Fig . 3 . The same disturbance is used , and inlet concen­tration and flow rate are included as manipu­lated variables . Comparing the outlet temper­ature with the previous cases , the present cases are obviously better with smaller excur­sions in exit temperature and control signal s . The improvement stems from the utilization of the concentracion and flow rate control s . Con­centration and flow rate changes immediately influencing all of the reactor and is used in this case as a feed-forward-like action to late measurements .

The similar behaviour of all three models III

-V . 1 is obvious in Fig. 3 ; the differences are mainly due to lack of tuning of the con­trol weights .

Servo behaviour

The servo behaviour of MIMOSC was investi­gated through a gradual set-point change of the integral states , corresponding to an in­crease in the inlet oxygen concentration from 0 . 25 mole% - ca . 0 . 75 mole% over a time period of two hours . The controller stabi­lized the reactor efficiently at the high oxygen level using the controller weights shown in Table 2 and specified above .

CONCLUSION

Model structures containing model order , sampling time and structural zeros may be specified a priori based upon qualitative chemical engineering process knowledge . A number of this type of models with different number of measurement positions and inputs are demonstrated to describe the example chemical reactor well .

The recursive extended least squares parame­ter estimation algorithm with variable for­getting factor proved simple to tune and worked very efficiently.

The asymptotic stationary optimal control strategy with iterative solution of the Riccati-equation worked well . Offset elimi­nation using weights on incremental controls and absolute states are probably to be pre­f erred over integral control for chemical engineering processes .

The successful application of the MIMOSC approach to a typical chemical engineering distributed and nonlinear process indicates that this approach may be successfully applied to many other chemical engineering processes . The special advantage of the approach is that the model structure is spec­ified a priori on the basis of chemical en­gineering knowledge of process dynamics .

REFERENCES

Bayoumi , M .M . , K .Y . Wong and M .A . El-Bagoung ( 1981 ) . A self-tuning regulator for multi­variable systems . Automatica , 17 , 575 .

Borisson , U . ( 1975 ) . Self-tuning regulators -industrial application and multivariable theory . Report 7513 , Dept . of Automatic Control , LTH , Lund , Sweden .

Bor:Lsson , U . ( 1979 ) . Self-tuning regulators for a class of multivariable systems . Automatica , 15 , 209-15 .

Buchholt , F . and M . Kllirmel ( 1981 ) . A multi­variable selftuning regulator to control a double effect evaporator . Automatica, 17 , 737 .

Fortescue , T . R . , L . S . Kershenbaum and B . E . Ydstie ( 1981 ) . Implementation of self-tuning

regulators with variable forgetting fac-

Adaptive Control of Chemical Engineering Processes

tors . Automatica , 17, 831-835 . Hallager, L . ( 1983 ) . Multivariable self-tun­

ing control of a fixed-bed chemical reac­tor using structured, linear models . Ph . D . Thesis , under preparation .

Hallager, L . and S .B . J0rgensen ( 1981 ) . Ex­perimental investigation of self-tuning control of a gas phase fixed-bed cataly­tic reactor with multiple inputs . Paper no . 106 . 2 at 8 ' th IFAC World Congress , Kyoto , Japan .

Hansen , K . ( 1973 ) . Simulation of the transi­ent behaviour of a pilot plant fixed-bed reactor . Chem.Eng. Sci . , 28 , 723-734 .

Hansen , K . and S .B . J0rgensen ( 1976 ) . Dynam­ic modelling of a gas phase catalytic fixed-bed reactor . I-III . Chem .Eng. Sci . , 31 , 473-479 and 579-598 .

Keviczky, L . and J . Hetthessy ( 1977 ) . Self­tuning minimum variance control of mimo discrete time systems . Automatic Control Theory Appl . , �' 11-17 .

Peterka , V. and K . J • .l\strom ( 1973 ) . Control of multivariable systems with unknown but constant parameters . Preprints of 3rd IFAC Symposium on Identification and sys­tems parameter estimation, p . 535-44 . The ll<lbrue , Netherlands .

SOderstri:im, T . , L . Ljung and I . Gustavsson ( 1978 ) . A theoretical analysis of recur­sive identification methods . Automatica, 14 ,

r AO I , -

0 A* = 0

0 -

231-44 .

( a+l )n

Al A . . . a - - - - 0

I 0

- - 0

- 0

b·m i · r · '

Bl

Bb 0 '1

0 - - - 0 0 0 s* = I - - - 0 ,,

0 - - I 0

" BO 0 ( a+l )n

0 I 0 b ·m

0 0 -E 0 - -I 0 } i

' Table 1 . Matrices for control problem . i is the

number of integral states .

Model lit Measurement m(u) min n positions

I 4 5 0 . 2 ; 0 . 4 ; 0 . 6 ; 0 . 8 ; 1 .0 l ( T ) c II 4 2 0 . 2 ; 1 . 0 l ( T ) c

III 4 5 as I 3 ( T X F ) c c c IV 3 5 as I as III

V . l 2 5 as I as III

V . 2 as V . l

a b

0 0

3 0

0 0

1 1

2 2

c ViO

1 40 , 5 , 20 , 25 , 25

1 40 , 20

1 40 , 5 , 20 , 5 , 5

1 40 , 10 , 20 , 10 , 10

1 20 , 5 , 10 , 5 , 5

20 , 20 , 40 , 20 , 20

Control , Input weight

1 . 00

2 . 00

2 . 0 , 1 . 0 , 1 .0

2 .0 , 1 . 0 , 1 .0

4 . 0 , 2 .0 , 2 . 0

4 . 0 , 2 .0 , 2 .0

Table 2 . List of model structures investigated and the corresponding sampling times .

(Note that the number of matrices used are a+l , b+l and c )

329

330 L. Hal l ager and S . Bay J�rgensen

DYNAM I C PROCESS I NTRAACTION DIAGRAM: ( DYP I D ) F

Upst ream

r c '-'c 'T'c ----------­Con t r o l s

'----·----- --�--· Trmpera t u rc me<lsu r�mPnts

Fig. 1 . Dynamic Process Intraaction Diagram ( DYPID) for the fixed-bed reactor example . The state variables are shown at the ri,g,ht hand side . The total set of temperature measurement positions considered in this paper are indicated by crosses .

I . <

Fig . 2 . Reactor outlet temperature and temperature control for Models I and II in response to upstream temperature dis­turbance of +5 °c at time O min . Vertical axis unit 1 0c , horizontal axis unit 5 min. Fig. 3 . Reactor outlet temperature and

control signals flow ( Fe ) , Con­centration ( Cc ) and Temperature ( Tc ) for Models III-V in response to upstream temperature disturbance of +5 °c at time 0 min . Units as Fig. 2 plus 0 .01 mole% oxygen for concentration and 0 . 1 mg/cm2/sec for flow rate .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

GLOBAL ADAPTIVE POLE PLACEMENT FOR NON­MINIMUM PHASE PLANTS APPLICATION TO A

THERMAL PROCESS

R. Lozano and M. Bonilla

[),,j,lo, ,f,, Ing. Elhtrica, CIEA rll'I IPN, AjJ. P1J.1tal 1 4-740, 0 7000 M/xico 14 D.F., Mexico

Abstract . An adaptive pole placement for non-minimum phase plants i s presented . Global convergence is ensured by introduc ing persistent excitation on the control input . An application to a Thermal process is presented .

Keyword s . Adaptive Control , Pole Placement , Temperature Control .

Introduction . In (3) a locally convergent adaptive control algorithm for non­minimum phase systems was presented . The local nature of the proof was due to the fact the identified plant model may converge to an un ­stabilizable model . Globally convergent control algorithms can be obtained by introducing persistent excitation conditions on the observation vector so as to guarantee unbiased estimates . In the first part of this paper it is shown that persistent excitation conditions can be introduced in a plant controlled by feedback to obtain an adaptive pole placement . In the second part of the paper we present the application of the adaptive control algorithm to a thermal process .

I I . Pole Placement of a Known Plant Consider the following discrete-time plant

- 1 - 1 A (q ) yt=B (q ) ut ( 1 )

wher� 1yt and ut are the plant output and in­put , q is the backward shif t operator and

- 1 - 1 -n A (q ) = 1 +a1 q + . . . +anq

- 1 - 1 -m B (q ) = b1 q + . . . +bmq

are co-prime polynomials .

( 2 )

( 3 )

A control law achieving pol e placement i s :

( 4 )

where yr is a bounded reference sequence and S and Rtare polinomials of order (r- 1 ) , r= max (n ,m ) such that :

- 1 - 1 - 1 - 1 - 1 C (q ) =S (q ) A (q ) +B (q ) R (q ) ( 5 )

where C is a monic asymptotically stable poly­nomial of order n . Multiplying eq . (4 ) by B and combining it with eqs . ( 1 ) and ( 5 ) it readily follows that :

- 1 - 1 r C (q ) yt = B (q ) yt ( 6 ) So , the closed loop p�les �re 2+ven by the zeroes of the polynomial z C (z ) .

I I I . Parametric Identification . The plant in eqs . ( 1 ) , ( 2 ) and ( 3 ) can be rewritten as :

( 7 )

3 3 1

T r ) where G = tb 1 , . . . , bm , a 1 , . . . , an

¢�- 1 = (ut- 1 ' " " ' ut-m ' yt- 1 · · · yt-n)

It is shown in ( 1 ) that if ¢t verifies the following persistent excitation condition

� ¢ . ¢� > a I > o lf t : n+m i=t-�-� - 1 -

(8 )

(9 )

( 1 0 )

then , the identi fication algorithm with forgett-

and C < o < A < 1 -t: ; o < € < < 1 t -

( 1 1 )

( 1 2 )

( 1 3 )

provides estimates that converge exponentially fast to the true parameters in a deterministic frame work . It can be shown ( see (2) ) that if at least one of the b . is non zero , then persistent excita­tion conaitions on the inputs :

t - -T l: ui ui : a2

I.? 0 'ft > n+m

i=t-m

with ;;'7 = (u . , . . . , u . 1 ) i i i-m+

( 1 4 )

ensure persistent excitation conditions on the observation vector ¢ as in eq . ( 1 0 ) .

IV .Adaptive Pole Placement . We assu�e that : A1 ) An upper bound for r is known A2 ) A lower bound a , for l det M (A , B ) I is

known , where M 1A , B ) is the eeliminant of A and B (see (4f l . Consider the following Control law :

1 ) t;p date the estimates using_lj'qs . ( 1 1 ) - ( ! ;p . 2 ) Define the polynomials At ( � ) and B (q )

using et i f l det M (A , B ) I >a3 > o . t e !:1 t - - 1 Otherwise define At (q ) =At- l ( q ) ,

- 1 - 1 Bt (q ) =Bt-1 (q ) .

3 ) Solve the polynomial identity

C = S At + Bt R ( 1 5 )

332

4) To guarantee persistent inputs as in expression control input from :

S u =yr+!:::. - R y t t t t

R . Lozano and M. Bonilla

excitation on the ( 1 4 ) , compute the

( 1 6 )

where !::::.t is a small change in ut . Go to step 1 .

and the adaptive control law is used . The forgetting factor At was such that trace (Ft) =SO_�t . MNote that the trackinq error (yt-B(q ) yt/�( 1 ) ) converges to zero . Fig . 2 shows the tracking error with the same para­meters change but usinq the non-adaptive scheme of section I I tuned for P and ravina

. t 1 . . . h ( 1 - 1 1, -1) ( -1 an in eqra action , i . e . wit S q ) =S (q 1 -q ) Remark . Independent vectors u . . u

as required in expression ( 1 4 ) can be oBtll/in � ed by introducing small changes !::::. . in u . i=t-2m+1 , . . , t , such that the dete?minanfs of all the principal minors of the matrix (ut-m ' ' ' ut) are , in absolute value , greater than a given threshold .

in eqs . ( 4 ) and ( 5 ) . In fiq . 3 the plant para­meters were chanaed from P3 to P 1 and the adaptive controller was used . A singularity in the solution of the polynomial equation ( 1 5 ) was encountered . This problem has been solved s imply by boundina the control input . References

\1) Lozano R . ( 1 983 ) . Convergence Analysis of Recursive Identification Alaorithms with foroettinq factor . Automatica , Vol 1 9 , No . 1 pp . 95-97 .

Given that the estimates converge to the true plant parameter s , the plant + adaptive controller converges to an asymptotically stable system. Thus it can be shown (see (3) ) that

lim t -+ 00

( 1 7 ) (l2J Moore J . B . ( 1 98 3 ) . Persistence Excitation in

Extended Least Souares . IEEE-TAC , Vol 28 , No . 1 , pp . 60-68 ( 3) Goodwin G . C . , K . S . Sin ( 1 9 8 1 ) .Adaptive Control for Non Minimum Phase Systems . IEEE-TAC 26 pp . 478-48 3 .

with ut and yt bounded . V .Application to a Thermal Process

The adaptive control law was appl ied to a real process in which a heatinq element feeds hea�

Wolovich W.A . ( 1 974 ) . Linear Multivariable Systems . New York , Springer-Verlao. Landau I . D . , R. Lozano . ( 1 98 1 ) .Unification into the airstream circulated by a centrifugal (ls)

fan alonq a tube . The control input is the power in the heater and the output is the air temperature at the end of the tube . The transfer

and Evaluation of Di screte Time Explicit M . R . A . C . Desiqns . Automatica 1 7 , No . 4 , 593-6 1 1

fu�X�ion of the process can be aproximated by 10 r� Ke I ( Ts+ 1 ) . !VJ ! , J p1! !>� . ·

The discrete-time model i s : - 1 -d - 1 -d ( 1 +a 1 q ) y = (b q +b o ) u

where -T/T a = -e 1

P 1 -0 . 88 P2 -0 . 94 P3 -0 . 94

t d . d+1 - t

b = K ( 1 -e -rT/T ) d

0 . 003 0 . 0 1 4 0 . 1 7 0 . 0025 0 . 003 0 . 45 0 . 0025 0 . 00 1 o . 72

we 9tve chosen _ 1 c ( q ) = 1 -0 . 5 q

{ 1 8a)

( 1 Rb)

( 1 9 ) In order to asymp�otically track a desired output sequence y , we have defined ( see eq. (6 ) ) : t

30

Fig . 1

' ' 10 : .; ' l

•t (�) JC

Fig . 2

r - 1 M ,.., yt = C ( q ) yt I B ( 1 ) ( 20 ) we have set !::::. =O s ince it was not needed in the experiment . On the other hand since the

10 yt B {q" 1 ! II f1 ''.'-�d� -< f,.�t_-_•_<.,• :"'o""Y-•....i:...l,,""'"----...;t;,;.c 11..;�-o time spent in computing the con�rol input was 10'-�--���������t-<•�> . . .... . 30 • . , . . • , • • . • 1 • • - 1 5-0 comparable to the sampling period. we chanqed 'l.oos h <t>-a <tlb t<t> · 1 -1 • 1. 1 ot . R by q R in eqs . ( 4 ) , ( 5 ) , ( 1 5 ) and ( 1 6) . Thus cvi · : · . . the computation of can be initiated at tim3 � ut Jo i;0 t- 1 since ut is no lonaer a function of yt .

Fig. 1 shows the plant behaviour when the plant parameters are switched from P 1 to P2

:lOOS

Fig . 3

Copyright © JFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ADAPTIVE CONTROL OF A SYNCHRONOUS GENERATOR

Raymond Hanus,* J.-C. Maun** and M. Kinnaert**

*Sernia d'Automatique, C.P. 1 65, Universite Libre de Bruxelles, 50 avenue F.D. Roosevelt, 1 050 Bruxelles, Belgium

**Sernict' de Glnit' Electriqu1', C.P. 1 65, Univenite Libre de Bruxelles 50, avenue F.D. Roosevelt, 1 050 Bruxelles, Belgium

Ab s t r a c t . Theo re t i c al a s p e c t s are b a s e d o n PARK ' s e q ua t i on s wh i l e the exp e r imen t a l part i s a c h i e ve d f rom o n- l i ne mach ine f r e q uency r e s p o n s e s o n s ma l l s i gn a l s f or d i f f e r e n t a c t i ve o r r e a c t i ve powe r s . The c o n c o r d a n t r e s u l t s show t h a t a t h i r d o r d e r mu l t i -i n p u t - mu l t i -o u t p u t non m i n imum p h a s e s y s te m c h a r ac t e r i z e s c o r r e c t l y t h e dynami c b ehavi o u r o f t h e s yn c h ro n o u s machine and that t h i s s y s tem v a r i e s g re a t l y wi th the l o a d . The s y s tem c h ar a c t e r i s t i c s mi l i t a t e a d ap t i ve c o n t r o l l e r .

i n f avor o f u s i n g an

I t is the s ub j e c t o f the s e c o n d p a r t of the p ap e r . The numar i c a l c o n t r o l l e r i s comp o s e d o f two p a r t s . The f i r s t one i s c o n c e r ned w i t h the s t a t i c non- l i ne ar b e h av i o ur o f the s y s t e m . The s e c o n d o n e mu s t c o r r e c t t h e d e f e c t ivene s s o f the s t a t i c mo d e l and s t ab i l i z e the s y s t e m aro und a s t e ady s t a t e p o i n t d e f i n e d by the s ame s t a t i c mo d e l . Th i s dynam i c � o n t r o l p a r t i s g e t t i n g more comp l e x , a s the c o n t r o l l e d s y s t em i s mul t i - i n p u t - mu l t i - o u t p u t , d e p e n d s on s t e ady s t a t e p o i n t , and h a s s o me t i me s a non-min imum p h a s e b ehav i o ur . The c h o s e n c o n t r o l s t r u c t u r e i s o f p a r e l l e l type , w i t h a memo ry l e s s d i r e c t a c t i o n i n o rd e r t o c o mp e n s a t e t h e s e co n d a r y e f f e c t s o f p a r ame t r i c ad j u s t i n g s a n d t o a c c o u n t imp l i c i t e l y f o r t h e s t r o n g n o n l i n e a r i t i e s o f the c omman d . Examp l e s o f s yn c h r o n o u s mach i n e s i mu l a t i on show the g o o d b e havi o u r o f t h e c o n t r o l s y s tem i n c a s e o f min imum p h a s e .

Keywo �� · Adap t i ve c o n t r o l ; D e c o up l i n g ; E l e c t r i c var i ab l e s c o n t ro l ; Gen e r a t o r s , e l e c t r i c I d e n t i f i c a t i o n ; Mu l t i var i ab l e s s y s t ems ; Non l i ne a r s y s t ems ; P r e d i c t i ve c o n t r o l .

333

334 R. Hanus , J . -C . Maun and M. Kinnaert

I . I NTRODUCT I ON Th i s p ap e r d e a l s w i th t h e p r i ma r y con t r o l o f the g en e r a t o r s b a s e d o n l o c a l me a s u r emen t . Un s t ab i l i t y r i s k s i n c r e a s e w i th t h e un i t i mp o r t a n c e s and the d i s t an c e b e tween gen e r a t o r s a n d l o a d s . The p r e s en t c o n t ro l l e r s a r e the r e s u l t o f s u c c e s s i ve i mp roveme n t s made i n o r d e r t o s o l ve t h e s e p r o b l ems ( 3 ) . T h e r e f o r e , i t s e ems t o b e i n t e -r e s t i n g t o d e s i gn , d i r e c t l y f r o m t he p r o c e s s cha r a c t e r i s t i c s , a d i g i t a l c o n t r o l l e r w i th a s imp l e r s t ruc t ur e , s a t i s fy i n g t h e f o l l ow i n g s p e c i f i c a t i o n s ( 4 ) - c o n s e rve a s u f f i c i e n t o p t i mum c o n t r o l

f o r any o p e r a t i n g p o i n t a n d f o r e ach o u t p u t ,

- ma i n t a i n the s t eady s t a t e v a l u e s o f b u s vo l t age and l o ad a n g l e .

Coup l e d wi th a ne twork , t h e s yn c h r o n o u s ma c h i n e fo rms a non- l i n e a r s y s t e m ( I ) A f i xed c o n t r o l l er w o u l d b e op t i ma l f o r o n l y o n e l o ad . Th e r e fore , i t i s more advi s ab l e t o u s e a t i me -var y i n g c o n t r o l l e r ad ap t e d t o t h e dynami c s c o r r e s p o n d i n g t o any o p e r a t i n g p o i n t ( 8 ) .

2 . OFF-L I NE IDENT I F I CAT I ON

2 . 1 S y s t em d e s c r i p t i on

The powe r ne two rk i s d e f i n e d b y i t s THE VEN I N ' s e q u i v a l e n c e .

R • + j X · I I

I

F i gur e I .

The p ow e r uni t conne c t e d t o t h e n e twork i s a s y s t e m w i t h the f o l l o ­w i n g i np u t s

- t h e f i e l d vo l t age - t h e me c h an i c a l t o r q ue : a n d t h e f o l l ow i n g o u t p u t s - t h e ma c h i n e b u s vo l t age - the l o ad a n g l e : o

T m

B o t h p u t p u t s d e p e n d o n b o th i np ut s . Th i s s y s t e m c a n b e d e s cr i b ed by f o u r t r an s f e r f u nc t i o n s . The PARK ' s e q ua t i o n s ( I ) a r e u s e d i n o r d e r t o c h ar a c t e r i z e t h e dynami c b ehaviour o f t h e s yn c h r o n o u s mac h i n e . The r e s u l t i n g s y s t em i s non- l i n e a r ( 4 ) .

v cod v coq VT d VT q

Vco s in oco v co s o co co vT s i n o

VT c o s o

s � d -ww + ( R +R2 ) i d co coq a ww d + s w + ( R + R2 ) i co coq a q vcod-R£i d- s L 2i d+ wL £ i q vcoq -R£ i q - s L £ i q -wL £ i d

2 H s w = Tm + �T d i q - �T q i d The f l uxe s and t h e c u r r e n t s a r e r e l a t e d b y t h e op e r a t i on a l i mp e d an c e s

�T d ( s ) � T q ( s ) �cod ( S ) � ( s ) coq

Xd ( s ) i d + G f ( s ) v f x ( s ) i q q � T d ( s ) + L £ i d

F o r a g i ven l o a d and f o r s u f f i c i e n t s ma l l s i gn a l s , t h e s y n c h r o n o u s mac h i n e c a n b e c o n s i d e r e d a s a l in e a r p r o c e s s . The c l a s s i c a l l i n e a r t h e o r y c a n b e u s ed . I n r e g a r d t o i t s a c c uracy and i t s p r ac t i c ab i l i ty , t h e f r e q uency r e s p o n s e app r o a c h c o n s t i t u t e s a p r i v i l e ge d o f f ­l in e me t ho d .

Adaptive Control of a Synchronous Generator 335

2 . 2 T h e o r e t i c d l d e t e r mi n a t i on o f f r equency r e s p o n s e s

The mach i n e e q u a t i on s a r e l in e ar i z e d around each o p e r a t i n g p o i n t f o r s ma l l va r i a t i on s o f t h e f o l l ow i n g s e c ondary s i gn a l s

!'ii d !'ii d - and q- axi s c u r r en t s q tivTd ' tivT q vo l t a g e s

ti 'I' co d ' ti 'I' f l uxe s ooq M i an g l e b e tw e e n n e t w o rk 00

vo l t a g e and ma c h i ne r o t o r d e p e n d i n g o n t h e i n p u t s i gn a l s tiTm , ti v f The o u t p u t s , tivT and ti o a r e expr e s s e d i n t e rm o f s e c ondary va r i ab l e s

tivT = tivTd s i n o 0 + tivT q c o s o 0 VT ti o =tivTd c o s o 0 - tivT q s i n o 0 0 The four t r an s f e r fun c t i on s

ti T m

a r e comp l e t e l y e v a l u a t e d , w i t h o u t a n y e l s e s imp l i fy i n g a s s ump t i ons .

2 . 3 E x p e r imen t a l r e s u l t s The me a s u r eme n t h a s b e e n r e a l i z e d o n a 1 0 k V A l ab o r a t o r y ma c h i n e d r iven by a DC mo t o r ( 7 ) . The mach i ne i s coup l e d , t hro ugh a t h r e e - p h a s e l i ne mo d e l , w i th the i n f i n i t e busbar o f

• t h e U . L . B . e xp e r i me n t a l and d i d a c t i c ne twork . A vo l t a ge c o n t r o l l e d c h o p p e r f i xe s the f i e l d vo l t a g e . A c u r r e n t c o n t ro l l e d c h o p p e r s u p p l i e s t h e D C mo t o r . A d i g i t a l d a t a a c q u i s i t i o n i s p e r f o rmed i n a f r e q u e n c y band f r o m 0 . 1 to I O H z

mach i n e b u s vo l t a g e s V00R i n f i n i t e b u s vo l t age 6 ang l e b e tw e e n i n f i n i t e b u s

vo l t age a n d r o t o r p o s i t i on i n p u t s

• Unive r s i t e L i b r e d e B ruxe l l e s ( Fre n c h F r e e U n i v e r s i ty o f B r u s s e l s )

3

0

- I

B u s vo l t a g e VT and l o a d a n g l e 6 a r e t h e n c o mp u t e d . E a ch i n p u t and o u t p u t s i gn a l i s i d en t i f i e d t o a f un c t i o n o f A + B s i n ( wr t + ¢ ) t yp e , by l e a s t s q uar e s . The f o u r FOURIER ' s t r ans f e r fun c t i o n s are eva l ua t e d i n th i s way .

2 . 4 E va l u a t i o n o f the LAPLACE ' s t r ans f e r f un c t i o n s

T h e p a s s a ge f r om f r eq uency r e s p o n s e t o t ra n s f e r f un c t i o n c o n s t i t u t e s a c l a s s i c a l p r ob l em . I t c a n b e done b y a l e a s t s q ua r e i d en t i f i c a t o r p r o g ram . The p o l e s and z e r o s o f t h e s y s t em are comp u t e d by a NEWTON- B A I R S T OW ' s me t h o d . O n l y t h e domi n an t p o l e s and z e r o s o f t h e p ro c e s s a r e c o n s e r ved .

2 . 5 Theo r e t i c a l and ex p e r i men t a l r e s u l t s

The t h e o r e t i c a l and e xp r i me n t a l amp l i t u d e s o f the f o ur t r a n s f e r func­t i o n s a r e s h own i n f i gu r e s 2 t o 5 , f o r a l o a d P = p . u . and Q = O . p . u .

. 3

F i g u r e 2 :

I .

I d e n t i f i c a t i o n

( H z ) I 0

T r a n s f e r f u n c t i o n

b e t w e e n VT

and vf

( H z ) 1 0

F i g u r e 3 : T r a n s f e r f u n c t i o n b e tw e en 6 a n d v

f

336 R . Hanus , J . -C . Maun and M. Kinnaert

3

0

- I . 3

0

- I

-2

-3 . 3

Ident i f i c a t i on

J . 3 • F i gure 4 : Tr ans�er func t i on b e twe en o and T m

Iden t i f i c a t i o n

J .

( H z ) I 0

F i gure S : Tran s f e r f unc t i on b e tween VT and Tm

The four t r an s f e r f unc t i ons , b o th t h e o re t i c a l and e xp er imen t a l , e xh ib i t the s ame t h r e e p o l e s f o r a g i v e n l o ad : a r e a l and two c omp l e x conj u g a t e one s . The r e a l p o l e c a n b e c o me un s t ab l e i n func t i on o f t h e l o a d . T h i s un s t a-b i l i ty o f synchronous mac h i n e i s we l l known f o r g r e a t l o ad a n g l e . The four t r ans f e r func t i on s a r e d i f fe r en t i a t e d b y t h e i r z e ro s . They a r e one or two . r e a l or c o mp l e x conj u ga t e , i n the p o s i t i ve o r n e g a t i v e p a r t o f the s - p l an ( 4 ) .

3 . I

3 . ADAP T I VE CONTROL OF THE SYN CHRONOU S GENERATOR

I n t roduc t i on The t h e o r e t i c a l and e xp e r i me n t a l deve l opment h ave s hown t h a t t h e di f fe r e n t t r ans f e r func t i ons c o u l d b e wr i t t e n : Y ( s ) = F ( s ) . U ( s ) w i th Y T = { �VT ' � c }

UT = { �vf ' �T�

and F . . ( s ) = K . . l. J ]. J

whe r e c i 2 = 0

O r a f t e r s amp l i n g

2 l +d . . s + c . . s l. J l. J

Y ( z ) = F ( z ) . U ( z )

w i t h F ( z )

whe r e B . a r e ma t r i c e s 2 x2 and a . ]. J s c a l ar s .

A f i r s t a t t emp t t o d e t ermine an a d a p t ive c o n t r o l sy s t em was made by us i n g t h e s t a t e var i ab l e d e s c r i p t i on . A s t ud y o f t h i s a l go r i thm c a n b e f o und i n ( 8 ) . I n t h i s p ap e r , w e a r e c o n c e r n e d i n t h e d i s c r e t e t r a n s f e r func t i on d e s c r i p t i on .

3 . 2 P r e v i s i o n a l c o n t r o l w i t h l ar g e s i gna l s

I n r e gard t o l ar g e s i gn al s , t h e s y n c h r o nous gene r a t o r mod e l i s n o n ­l i n e a r .

I t can b e s t ud i e d o f f- l i ne and g i v e s a g o o d a p p r o x i ma t i o n o f l ar ge s i gn a l v a r i a t i on s , f o r a n y s t eady s t at e p o i n t . Thanks t o t h i s mod e l , i t i s p o s s i b l e t o b u i l d a non- l i ne a r p r e v i s i ona l c o n t r o l l e r b a s e d cont i ­nuou s l y on t h e f o l l ow i n g i n s t a n t aneous q u an t i t i e s s e t p o i nt , i n p u t and output me a s ureme n t s and me a s urab l e d i s t ur b an c e s . Th i s p a r t o f t h e c o n t r o l i s s ummar i ze d i n f i gu r e 6 .

w ;- 1 s

F i gure 6

Adaptive Control of a Synchronous Generator 337

w i t h f a c t u a l s t a t i c c harac t e r i s t i c s o f the p ro c e s s

� - I f a mo d e l o f the i nve r s e s t a t i c s

w charac t e r i s t i c s e t p o i n t ve c t o r

U s t eady s t a t e i n p u t ve c t o r s t eady s t a t e o u t p u t ve c t o r

I n ab s ence o f s t eady s t a t e mo de l app r ox ima t i on s o f any d i s turbanc e s , and o f dynami c e f f e c t s , t h i s o p e n­l oop c on t r o l s t ruc ture wou l d b e ab s o -l u t e l y p e r f e c t . We a r e ob l i ge d t o i n t roduce a f e e d -b ack c on t r o l i n o r d e r t o a c c oun t f o r the s e d i f f e r e n t pheno-rne na .

The con t r o l s t r u c t ure b e c ome s

w +

- C

Figure 7

w i th f a c t ua l comp l e t e ( s t a t i c and dynami c ) chara c t e r i s t i c o f the p r o c e s s

U a c t u a l i n pu t ve c t o r Y ac t u a l o u t p u t ve c t o r u i npu t ve c t o r i n var i a t i o n s y o u t p u t ve c t o r i n var i a t i on s C fe edb ack ma t r i x ( f unc t i on o f

the s e t p o i nt W )

Th i s s t ruc ture i s e q u i va l en t t o the f o l l ow ing one

+ y A +

-C

Figure 8

+

. h B Wl. t A t ran s f e r f un c t i o n o f the p r o c e s s w i t h s ma l l s i gn a l s

x und i s t u rb ed o u t p u t ve c t o r n d i s t urban c e , image o f t h e

d i f fe r e n t b e tween W and Y .

Th e f e edback ma t r i x d e t e rmi n a t i on i s t h e t o p i c o f p a r a graph 3 . 4 .

3 . 3 P a r a l l e l c on t r o l s t ruc t ure

For s ev e r a l r e a s o n s , i t i s p o s s i b l e t h a t the c on t ro l l e r o u t p u t U c anno t b e f u l l y app l i e d t o t h e p r o c e s s ( manu a l swi t c h i n g , a c t ua t o r s a t u r a t i on s , e t c . . . ) . In t h e s e c i r c on s t an c e s , t h e a c t u a l p r o c e s s i n p u t Ur i s t emp o r a ry d i f fe r e n t f r om the con t r o l o u t p u t U . Mo r e o ve r , the c o n t r o l l e r d e f i n e d by the t r an s f e r ma t r i x C is t i me -vary i n g , a s i t d e p end s o n the s e t -p o i n t var i a t i on . Th i s t i me - vary i n g s t r u c ture , i n i t s e l f , gene r a t e s endo genous p e r tu rb a t i on s . I t h a s b e e n p r oved t h a t t h e p a r a l l e l s t ruc ture i s we l l adap t e d t o t h e s e two p r ob l ems . ( 5 ) Th i s s t ru c t ur e i s i d e n t i c a l t o the s t ruc t ure o f f i gure 7 p r o v i d e d

r - I U = U and C00= l imC ( z ) z -+oo

U r - y N . L . +

+

Figure 9

338 R . Hanus , J . -C . Maun and M. Kinnaer·t

Th i s l a s t c o nd i t i on c o n c e r n i n g t h e cho i c e o f the c on s t an t ma t r i x C00 i s . n e c e s s a ry i n o rd e r t o b u i l d a r e a l i ­s ab l e c o n t r o l l o o p ( a t l e a s t one d e ad - t ime i n the l o op ) . I t i s e a s y t o p r ove the e q u i v a l e n c e o f the t w o s t ru c t u r e s . I t i s t o p o i n t o u t t h a t t h e s t a t i c e r r o r van i s h e s w i t h o u t e xp l i c i t e l y u s i ng an i n t e g r a l e f f e c t .

3 . 4 De t e rmi n a t i on o f the f e e db a ck ma t r i x C

The t r an s f e r fun c t i o n w i t h sma l l s i gn a l s i s g i ven b y

w i t h a = I 0

wh e r e B i s a p o l ynom i a l ma t r i x and A a p o l y nomi a l s c a l a r . We t ry t o f i n d a c o n t r o l l e r w i th the s ame f o rm :

w i t h r = I 0

where S i s a l s o a p o lynomi a l ma t r ix R a p o l y nomi al s c a l a r . In the s e c o nd i t i o n s , t h e t r an s f e r func t i on o f the c l o s e d - l o o p b e c ome s ( AR] + BS ) yk wh e r e ] i s the i d e n t i t y ma t r i x o f d i mens i on m . O r , in o r d e r t o p o i n t o u t the p o l e s and z e r o s o f th i s c l o s e d l o o p

yk ad j ( ARJ + E S ) AR nk d e t ( AR] + BS )

We s e e t h a t the c l o s e d l o o p i s o f the s ame f o rm t o o :

wh e r e Q = a d j ( AR] + l3 S) AR i s a p o l ynom i a l ma t r i x and P = d e t ( ARJ+B$ ) i s a p o l yn o mi a l s c a l ar .

T h e c l o s e d l o o p w i l l b e s t ab l e p ro v i ­d e d t h e p o l ynomi a l P h a s a l l i t s z e r o s o u t s i d e t h e uni t c i r c l e . I t i s p o s s ib l e t o r e d u c e t h e d e t e r m i -

� 1 n a t i on o f t h e m a t r i x S ( z ) and t h e s c a l ar R ( z - 1 ) t o a s c a l ar p r o b l em i f the f o l l ow i n g cho i ce f o r S i s mad e .

- I - I S ( z ) = S 1 ( z ) - I whe r e s 1 ( z ) i s

W i t h t h i s c ho i ce , b e co m e s

AR yk AR + S I d e t B

a d j B ( z - 1 ) a s c a l ar .

t h e c l o s e d l o o p

nk

The p rob l em i s w e l l r e d u c e d t o a s c a l ar p r ob l e m , w i th an e q u i v a l e n t p ro c e s s d e t B / A a n d an e q u i va l e n t c o n t r o l l e r s 1 / R .

I f t h e p o l ynomi a l s A and d e t B have s ome z e r o s i n s i d e the un i t c i r c l e , the s e z e r o s c anno t b e c an c e l l ed b y z e r o s o f s 1 o r R . N o o t h e r r e s t r i c t i o n mus t b e done c o n c e r n i n g t h i s s t r u c t ure , even i f s o me mo d e l app r o x i ma t i o n s are mad e .

On b a s e s o f e s t ima t e s B and A o f B and A , p o l yn o mi a l s s 1 and R are d e t er m i ne d , a s w e l l a s mat r i x S . B y s ub s t i t u t i o n , the a c t u a l c o n t r o l l o o p b e c o m e s

,. a d j ( AR J + s 1 B a d j B ) AR

d e t ( AR J + S 1 B ad j B ) '

w i th a d j ( AR ] + s 1 B a d j B ) "' ( m- 1 ) ( AR+ s 1 d e t B ) J

' b e c au s e B a d j B "' d e t B . J and d e t ( AR J + s 1 B a d j B

"'m ( AR + S I d e t B )

f o r t he s ame r e a s o n ; w i t h m the d i me n s i o n o f t h e mat r i c e s B and S .

B e c a u s e R and s 1 w e r e cho s e� i n s u c h a w a y t h a t t h e p o lynomi a l ( AR + s 1 d e t B ) h a s a l l i t s z e r o s o u t s i d e t h e un i t

Adaptive Control of a Synchronous Generator 339

c i rc l e � we may s imp l i fy b y ( m- l ) (AR + s 1 <l e t B ) , a n d o b t a i n

� AR AR+ s 1 d e t B

wh i ch i s we l l the s c a l ar r ed uc t i on .

I t i s t o p o i n t o u t t h a t the a d j o i nt o f the ma t r i x B mu s t n o t b e c a l c u­l a t e d in t h e p ar a l l e l s t r u c t u r e . Indeed , o n l y the i nve r s e ma t r i x C - I mu s t b e eva l u a t e d . And i f

S I c

R a d j B

we have c - 1 B d e t B

3 . 5 App l i c a t i on on t h e synchronous ge n e r a t o r

The s t udy o f t h e synchronous g e n e r a t o r shows t h a t t h e sys t em i s a lways o f non-m i n i mum p h a s e t y p e ( a t l e a s t f o r one t r ans f e r function o u t o f f o u r ) and some t imes un s t ab l e .

,-------·---------·--··-·-·--··-··---··----··--·-·· I PERTURBATION RESPONSE �OR ROOT [ OET B l > J '

T ""-, ..

-( 20.

-21. I 1-. ..

F i gure 1 0

The z e r o s o f ma t r i x B d e t erminant a r e o u t s i d e o r i n s i de uni t c i r c l e , d ep en d i n g o n s e t p o i n t . When z e r o s o f d e t B a r e o ut s i d e , t h e r e s u l t s ob t a i n e d i n s imu l a t i o n are e x a c t l y as e xp e c t e d , even i f t h e ma t r i x B were o f non-min i mum p h a s e a n d the s y s t e m even t ua l l y uns t ab l e . Un f o r t una t e l y , whe n the z e r o s o f d e t 3 a r e i n s i d e t h e uni t ci r c l e , c l o s e d l o o p i s a lway s un s t ab l e .

t h e

I t i s t o r e m a r k t h a t t h i s un s t ab i l i t y should not be due to an eventual cancellation of process non-minimum zeros by controller poles , because we have taken the precaution to keep these zeros in closed-loop . The e s t ab l i s h e d un s t ab i l i ty i s o f mo n o t o n o u s t y p e and a p p e a r s o n l y a f t e r a very l o n g t ime . T h e c au s e o f t h e un s t ab i l i t y h a s n o c onne c t i o n wi t h

" an a p p r o x i ma t i on o f B i n re g a r d t o B a s e xp e r imen t s s h ow t h a t t h e uns t ab l e b eh a v i o u r i s i n d ep e n d a n t f r om t h e l e v e l c f t h i s a p p r o x i ma t i o n ( s e e f i g . I 0 and I I ) .

l-38.

... ...

' ________ IFL���

F i gure I 1

340 R. Hanus , J . -C . Maun and M. Kinnaert

A f t e r a new theore t i c a l s tudy , i t appears that the instability should not be due to d i g i t a l a p p r o x i ma t i o n s o f the c o mp u t e r i t s e l f and ab s o l ut e l y n o t to a p p r o x i ma t i ons o f t h e e s t i m a t e B o r / and A , n e i th e r t o t h e c o n t r o l s t r u c t u r e u s e d . For e xamp l e , t h e c a l c u l a t e d va l ue o f d e t B i s n e c e s s ar i l y s l i gh t l y d i f f e rent , b u t y e t d i f f e r e n t , f r om the a c t u a l v a l ue of t h i s d e t e r m i n an t . And the mo re p r e c i s e the comput e r , and t h e l on g e r the un s t ab l e l a t e n � t i me . F o r the moment , we do n o t s e e how w e shall b e ab l e t o ove r come t h i s f un d amen­tal d i f f i c u l t y .

{ 1 }

{2 }

{3 }

{ 4 }

{ 5 }

BIBLIOGRAPHY

B . ADKINS , R . G . HARLEY The general theory of alternating current machines Chapman and Hall-London 1 9 75 P. BARRET et al . Modell ing and tests at Fessenheim power station of a 1 0 8 0 MVA turbo­generator and of its excitation system IEEE Trans . on P . A . S . vol PAS- 1 00 , n°8 , August 1 98 1 , pp 3993-4006

P . BARRET Regimes transi toires des machines tournantes electriques Editions Eyrolles - Paris - 1 98 1

M . DE PAEPE Reglage parametrique des groupes de production Prix SRBE 1 982 - Accepted to publish Revue E - SRBE

I . COHEN Contribution a l ' etude des reglages adaptatifs These de doctorat - ULB- 1 982

{6 } I . COHEN , R . HANUS Mise en application et extension de methodes de regulation adaptative Analysis and optimization of systems Lecture Notes in Control and Information Sciences Vol 4 4 , pp 335-352 Springer Verlag, 1 982

{ 7 } J . C . MAUN Measurement of the dynamic parameters of synchronous machines IEE conference publication n°2 1 3 , 1 982 pp 94-98

{ 8 } J . C . MAUN , R . HANUS Etude d ' un regulateur parametrique pour un groupe de production . CONUMEL Toulouse mai 1 98 3 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

MULTIVARIABLE ADAPTIVE CONTROL

MULTIVARIABLE WEIGHTED MINIMUM VARIANCE SELF-TUNING CONTROLLERS

M. J. Grimble and T. J. Moir

Industrial Control Unit, Department of Electronic and Electrical Engineering, University of Strathclyde, George Street, Glasgow, UK

Abstrac t . The we ighted minimum variance control ler i s employed in a self­tuning control sys tem for plants which can be mul t ivariable and non-square . Both explicit and imp l icit se l f-tuning strategies are discussed . Keywords . Se l f-tuning control , optimal contro l , mult ivariable control sys tems , minimum variance control , adaptive control .

INTRODUCTION

Recently a new Weighted Minimum Variance (WMV) control ler has been proposed for non­square mu l t ivariable sys tems Grimble and Moir ( 1 983 ) , derived from that for sca lar sys tems , Grimble ( 1 98 1 ) . The related General ised Minimum Variance ( GMV) control­ler developed by Has tings-James ( 1 970) and used in the succes s fu l Clarke , Gawthrop ( 1 975 ) sel f-tuning control ler is unstab le when low control we ight ings are used and the plant is nonminimum phase . Th is unfortunate operating characteristic doe s not apply to the WMV contro l ler which is stab le for such cases . If high control weight ings are used with the GMV control ler the plant can always be stabi l ised i f it is open-loop stab le but this may not be pos s ible if it is open- loop unstable . Thus , it may not be easy to find a range of control we ight ing values for the GMV des ign which leads to a stab le closed­loop system , part icularly if the plant para­meters are uncertain . The advantages of the WMV controller in this s ituation are obvious and both exp l icit and implicit sel f-tuning schemes were therefore developed , us ing this control ler and are reported here .

SYSTEM DESCRIPTION The mul t ivariable l inear time-invariant sys tem can be non-square and is as sumed to be free of unstable hidden modes . The sys tem is represented as :

- 1 - 1 - 1 A (z )y ( t ) = B ( z )�( t ) + C ( z )S,( t ) ( l ) - 1 rxr - 1 - 1 rxr - 1 The matrix A (z ) £R ( z ) , C ( z ) £R ( z )

and B ( z- l ) £Rrxm( z- l ) i s assumed to be of normal ful l rank . The output y ( t ) £Rr , control u ( t ) £Rm and the dis turbance � ( t ) £Rq ( q � r ) . The s ignal { � ( t ) } i s a sequence of norma l ly dis tributed i.ndependent random vectors with zero mean and covariance E{5_( t )5_T( t ) } = Q > 0 . The polynomial

ASCSP-L 34 1

matrices A , B , C are assumed to be of known degrees na ,nb = nb0+ k , nc , respect ive ly . The polynomial matrix C ( z- 1 ) is also assumed to have a stable inverse . Let

A( z - l )

B ( z - l )

C ( z- l )

- 1 -na Ir+A 1 z + . . . +An z ( 2 ) - 1 a -nbo -k ( B +B 1 z + . . . +B z ) z ( 3 ) o nbo - 1 -n Ir+c 1 z + . . . +en z c

c ( 4 )

where the delay k � l . prob lems , the de lay k to unity which is the realisab i l i ty .

I f , in sel f-tuning is unknown k may be set minimum for phys ical

. f . ( - 1 ) It is neces sary to actorise the B z matrix into minimum ( B i ( z- 1 )) and nonminimum phase ( B2 ( z- l ) ) spectral factors . Any m-square polynomial matrix B ( z- 1 ) of ful l rank can be wr itten as :

B = E2AbE l = E2diag{b 1 , b2 . . . , bm}E 1 ( 5 )

The polynomials bi ( z- l ) are the invariant polynomial s of B and E 1 ( z- l ) and E2 ( z- 1 ) are unimodular . The factorisation B = B2B l gives:

B l A 1 E 1 = diag{b 1 1 , b 1 2 , . . . , b 1m}E 1 ( 6 ) -k B2 E2A2z = E2diag{b2 1 , b22 . . . , b2m} ( 7 )

where bi = b2 ib l i and b i i ( z- l ) is the minimum phase term and b l i ( O ) � l for i = 1 , 2 , . . . ,m . The orders o f B 1 and B2 are denoted by n 1 and n2 , respec tive ly and n0 �n2-k; also :

- 1 - 1 -no -k B2 ( z ) = ( B20+B2 1 z + . . . +B2n z ) z ( 8 ) 0

I f the sys tem is non-square :

or

-k B = B2 . B 1 = E2A 2z [ I r

( for r < m) ( 9 )

342 M. J . Grimble and T . J . Mo ir

( for m < r ) ( 1 0 ) Define A 1 ( o ) = I => B 1 ( o ) = E 1 ( 0 ) is m-square and ful l rank .

- rxr - 1 Summaris ing , B B2TB 1 where B2£R ( z ) , TERrxm( z- 1 ) and , ( a ) B2 � E2/\2z -k r=m,

( b ) r<m, B2 � E2/\2z -k

( c ) r>m , B'2 �E2 [�2 0 r Ir-m

-k '

T LI I = m T LI [ I = r T � [:m] O ]

( 1 1 ) The input e ( t ) � r ( t )-y ( t ) , where r ( t ) represents-a zero mean stochas tic-reference s ignal that can be mode l led by a dynamical system dr iven by white noise :

- 1 - 1 - 1 E_( t ) = A( z ) E ( z )W( t ) ( 1 2 ) The closed-loop sys tem i s shown in Fig . 1 . The s ignal w( t ) represents a known set-point s ignal . The covar iance matrix for the white noise w( t ) is denoted by Q0 and w( t ) and � ( t ) are assumed to be uncorre lated . The Innovations s ignal representat ion for the plant becomes :

- 1 - 1 - 1 A( z )�( t ) = D ( z )�( t ) - B ( z );;:( t ) ( 1 3 ) where the s table spectral factor D ( z- 1 ) can be calculated us ing :

D ( z - l ) R D T ( z ) = E ( z - 1) Q ET ( z ) + C ( z - 1) QC T ( z ) £ 0 ( 1 4 )

and D( o ) � Ir . The matrix RE > 0 in ( 1 4 ) denotes the covar iance matrix for the white noise innovat ions s ignal £ ( t ) .

PERFORMANCE CRITERION

The WMV contro l law must minimise a s ingle­s tage cost funct ion of the form :

J = E {t1 ( t+k)Tt1 ( t+k ) l t } ( 1 5 ) The s ignal s :

� < t ) �E. < t ) -z < t ) < 1 6 ) i'\ ( t+k) LI P ( z- 1 ) e ( t ) +P 1 ( z- 1 )w( t ) ( 1 7 ) - = 0 - - 1 -t1 ( t+k) �t( t+k)-P2 ( z ) ;;:( t ) ( 1 8 )

The weight ing matrices P0 , P ] and P2 can be speci fied to achieve des ired response characteristics , and

P ( z- 1 ) LI B ( z- 1 )A ( z- l ) - l 0 = c c

The matrix BcE Rrx r( z- 1 ) defined as : Be = (B�) - l DcH- 1 , i s related to B2 · The matrices De and Ac affect the zero and pole s truc ture of P0 . These matr ices are assumed to have s tab le inverses and Dc ( o ) =Ac ( o )= Ir ; A 1 ( z- l ) �A( z- 1 )Ac ( z- 1 ) .

WEIGHTED MINIMUM VARIANCE CONTROL LAW

Theorem 1 : The WMV control law t o minimise ( 1 5 ) for the sys tem ( 1 3 ) , shown in Fig . 1 , can be calculated as : Case ( i ) Square Systems ( r=m)

0 - 1 ;!_ ( t ) = (B 1 +D0P2 ) ( G� ( t ) +D0P 1�( t ) ] ( 1 9 )

Case ( ii ) Non-square Systems ( r>m) u0 ( t ) = (VTD- l (TB 1 +D P2 ) ) - l (YTD- l (TGe ( t ) - 0 0 0 0 0 -c

+D0P 1�( t ) ) ] ( 20 )

Case ( i i i ) Non-Square Systems ( r<m) 0 t ;!. ( t ) = ( TB 1 +D0P2 ) [ TG� ( t ) +D0P l�·( t ) ] ( 2 1 )

The matrix D fol lows by writ ing D- l B�=B2D� l where <let D0°= <let D , D0 ( o ) = Ir and nd0 � nd and {G ,H } satisfy :

A 1 H + B2G = DDc ( 2 2 ) - 1 - 1 The s ignal �c ( t ) �H Ac �( t ) , and

V0 � TB 1 ( o ) + P2 ( 0 ) . The right inverse of the matrix ( TB lc+ D0P2 ) in ( 2 1 ) is denoted by ( TB 1 + D0P2 )' . That is , ( 2 1 ) becomes :

0 ;!_ ( t ) = W(G 1 1� ( t ) +D0P 1�( t ) ) ( 2 3 ) where

W � (TB 1 +Do p 2 ) T ( ( TB 1 +Do p 2 ) (TB 1 +Do p 2 ) T ) - 1

and in 23 , G l l � TG

Proof : Grimble and Moir ( 1 983 )

EXPLICIT SELF-TUNING CONTROL

( 24 )

( 25 ) 0 0

Assume now that the process ( 1 ) is constant but unknown and that the orders of the poly­nomial matr ices na , Ilb and nc are known . The est imat ion algori thm m�y be appl ied as fol lows .

Extended recursi ve least squares (ELS ) From the innovat ions sys tem descript ion ( 1 3 ) write

e ( t ) = -A e ( t- 1 ) -A e ( t-2 ) - • • • -A e ( t-n ) - 1- 2- n - a a

-B u( t-k)-B u ( t-k- 1 )- • • • -B u( t-k-n ) o- 1- nbo bo +D E ( t- l ) +D £ ( t-2 ) + • • • +D E ( t-n ) 1- 2- nd- d +£( t ) ( 2 6 )

and denote the i th parameter vector by :

a.�LI ith row o f [ A l , A2 , . . . ,A ! B , B 1 . . . , B -i= n� o nbo

: D l , D2 . . . ,D ] I Ild

and the data vec tor by :

( 2 7 )

Self-tuning Contro l l ers 343

6 T ( t- 1 ) = [ -e T ( t- 1 ) , -e T ( t-2 ) , . . . , -e T ( t-n ) - - - - a : T T T 1 -!:.1_ ( t-k) , -!:.1_ ( t-k- 1 ) , . . . , -!:.1_ ( t-k-nb ' T T T T o ;£ ( t- 1 ) ,£ ( t-2 ) , . . . ,£ ( t-nd) ] ( 28 )

then the ith output of the sys tem ( 1 ) takes the form:

T y . ( t ) = qi ( t- 1 )& . ( t ) +E . ( t ) i - -i i ( 2 9 )

for i= l , 2 , . . . , r . The parameter vector & . may b e e s t imated via the ELS algorithm :-i

ELS Algori thm

Set i = 1 l . Parameter update:

!; ( t ) = � . ( t- l ) +K . ( t ) � . ( t ) � -i i i

2 . Gain update: P . ( t ) q\ ( t- 1 ) i -

K . ( t ) i ( A+q\T ( t- l ) P . ( t- l ) q\ ( t- 1 ) ) i -3 . Error co variance updat e :

l Pi ( t ) = );" [ P i ( t- 1 )

P . ( t- l ) q\T ( t- l ) q\ ( t- l ) P . ( t- 1 ) i i A+q\T ( t- l ) P . ( t- l ) q\ ( t- 1 ) i

4 . Estima te of residual : /\ T A € . ( t ) = y . ( t )-6 ( t- 1 )& . ( t- l ) i i - -i

5 . i = i + l .

6 . I f i < r GOTO l .

( 30 )

( 3 1 )

( 3 2 )

( 33 )

D

The calculat ions mus t be performed in the order : P . ( t- 1 ) -+ K . ( t ) -+ � . ( t ) -+ & . ( t ) -+ P . ( t ) i i i -i i The � parameter is an exponential forgett ing fac tor ( 0 . 99 � A � l .

Solut ion o f the Diophant ine Equat ion

Cases r � m - - 1 - 1 - 1 Let D ( z ) �D ( z )Dc ( z ) and cons ider only

the cases where a unique solut ion exists to ( 22 ) when the orders nh �n2- l (n2 � k+n0 ) and ng �max(na 1 - l , nd+nd -n2 ) , respect ive ly . The diophant ine equ�t ion has the form :

- 1 -na - 1 -nh ( I +A 1 1 z + . . . +A 1 z ) ( I +H 1 z + . . . +H z ) r � r � - 1 -no + ( B2o+B2 l z + ... +B2n z )

0 -k - 1 -ng x z ( G G l z + . . . +G z ) o n g

( 34 )

This may b e writ ten as : LX = F

where the GT GT

0 l

T T T solut ion for X� [ H l H2 . . . Hnh T ]T . . d U d h Gn i s require . n er t e g

( 35 )

above as sumpt ion L is ful l rank and X may be calculated as X = L- l F , ( for r=m) or X = L+F where the left inverse L + � (LTL)- lL TF , ( for r>m) . For the case �<m write :

A 1H+B2G = D ( 36 ) and recall B2 = B2T B2 [ Ir O J . Partit ion G into

T GT I T ( 3 7 ) G = [ G l 1 1 2 then the equat ion becomes :

( 38 ) - rxr - 1 rxr - 1 where B2ER ( z ) and G 1 1 ER ( z ) . Thus ,

the diophant ine equat ion ( 38 ) has the same form as that for the square sys tem case and the solut ion may be found as previously described .

Note: The greatest common left divisor of A 1 and B2 is a unimodular matrix and this i s suffic ient to guarantee that a solut ion exists to the diophant ine equat ion ( 22 ) . However , this does not imply that a solut ion can always be found with the orders of H and G as fixed above (L may not be ful l rank) . During s e l f-tuning , as Prager and We lstead ( 1 980 ) note , the s tochas t ic e lement wi l l ensure that L remains ful l rank .

WMV Expl icit Self-Tuning Control Algori thm

Square System: ( r = m) l . Select the we ighting matrices P2 and P 1 . 2 . E s t imate the sys tem matrices uskng the

�LS algo�ithm and hxnce obtain A( z- 1 ) , B ( z- 1 ) , D ( z- 1 ) and D0( z- l ) .

3 . Compute B2 ( z- l ) and B 1 ( z- 1 ) from B ( z- 1 ) us ing a fac torisat ion algorithm (Kucera , \ 1 97 9 )/ . Alternat ive ly , identify either B 1 or B2 directly from the plant when appropriate measurements are avai lable .

4 . Compute �( z- 1 ) and a ( z- l ) from the Sylves ter form of xhe dAoph�nt ine equat ion ( 3 5 ) , as X = (L ) - l F .

5 . Us ing the pr inc iple o f certainty equivalence , compute the control at t ime t from , Ao /\ /\ - 1 u ( t ) = ( B 1 +D0P2 )

ng A A /\ o : G . e ( t-j ) +D P 1w( t ) ) j=o J-C o -

where � ( t ) � �- 1 e ( t ) -c

or � ( t ) = e ( t ) --c

nh /\ /\ E H . e ( t- i ) i-c i= l Return to Step 2 .

344 M. J. Grimble and T. J. Moir

Non-Square : ( r > m) . As abovx buX J;ePli!;cx the calculat ion in Step 4 by X = (L°h)- 1 LTF , and replace Step 5 by the opt imal control given by ( 20 ) , ( evaluated using estimated parameters ) .

Non-Square : ( r < m) . As above but so 1 ve the �iop�antAne equat ion ( 38 ) in Step 4 us ing X = (L) - 1 F , and replace Step 5 by the optimal control given by ( 2 3 ) .

Example 1 : WMV Se l f-Tuning Regulator

This process is both unstable and nonminimum phase with two inputs (m = 2 ) and one output ( r = 1 ) :

- 1 - 1 -2 A (z ) = l +a 1 z +a2z

= ( l - l . 5z- 1 ) ( 1 -0 . 7z- l )

where a 1 = -2 . 2 and

B ( z- 1 ) = z-k [ l

a2 = 1 . 05 . The matrix - 1 -2 l ) ( b0+b 1 z +b2z )

where b0 = 1 , b 1 = -2 . 6 , b2 = 1 . 2 , k = 1 and C ( z- 1 ) = 1 . The control we ight ing matrix Dc ( z- 1 ) � l -z- l +o . sz-2 . The reference s ignal Q0 = 0 , D ( z- 1 ) = 1 and Q = 0 . 0 1 .

Solution :

B( z - l ) - 1 - 1 B2 ( z ) B 1 ( z )

n g

- 1 - 1 [ - 1 ( l -2z ) z [ l O J �l -0 . 6z )

max(na- 1 , nd +nd+n2 ) c

�] [� :J

The contro ller parameters for a_known plant model can be evaluated from AH+B2G 1 1 = CDc to obtain :

u0( t )

-4 . 49 , g l 2 . 988 , h l 5 . 6923 .

Simula tion Resul ts

The control and output signals for the self­tuning system ( shown in Fig . 2 ) are almost ident ical to those for the optimal case . The computed control ler parameters are shown in Fig . 3 . The coeffic ients of B2 and B 1 were found by on-line factorisat ion .

IMPLICIT SELF-TUNING CONTROL

To derive an implicit WMV self-tuning control scheme ( Grimble ( 1 982 ) ) let e ( t) ti A- 1 e ( t )

-0 = c -

and assume that P 1 = 0 , P2 = 0 . The des ired implicit model follows from ( 1 3 ) , ( 1 7 ) and ( 2 2 ) :

e ( t ) = HD- l e: ( t ) +HD- lD- 1 B2 (Ge ( t ) -B 1 u ( t ) ) -0 c - c -c -

" - 1 Let T li H-D , e: ( t ) li D e: ( t ) and write - l e = - l e - .:J � 1- - 1 G ( z )H (z ) = H 1 ( z ) G 1 ( z ) where

( 3 9 )

det tt 1 = det H , H 1 ( o ) = Im and nh 1 = nh ' and

N 1 � H 1 B 1 . Cons ider the cases r � m and not ice that u0 ( t ) sets the final term in ( 3 9 ) to zero . After s impli ficat ion ( 3 9 ) gives :

" e ( t ) = e: ( t ) +T e: ( t ) +B2 (G l e ( t ) -N 1u( t ) ) -0 - c- -0 - (40)

The est imat ion equat ion may now be obtained as :

� C t l t- 1 ) = T e:'°C t ) +B2 ( G 1 e ( t ) -N 1 u( t ) ) -0 c- -0 -

= X( t- 1 ),'l- ( 4 1 )

where the prediction error ;0 ( t l t- l ) � e: ( t ) . The optimal control signal - --

o /\- l /\ � ( t ) = N 1 G 1�( t ) .

Example 2 : Implicit Se lf-Tuning Control for a Square unstable and Nonminimum Phase Process [ - 1 1 -0 . 9z

- 1 0 . 5z

0 . 5z - 1 l - 1 l -0 . 2z - 1 - 1 - 1 B ( z ) = z ( 1 -2z ) 12 , - 1 C( z ) = I2

The cost-funct ion D ( z- l ) - 1 -9

- 1 Ac ( z ) = 12 and P 1 ( z ) = P2 ( z ) = 0 . Q = diag {0 . 0 1 , 0 . 0 1 } and Q 0

The covariances = 0 .

Solution :

Let n = n - 1 = 1 and n = n - 1 = 0 thence - l h b2 - 1 g - l a

H( z ) = I2+H 1 z and G ( z ) = G0 • Write - 1 -T - 1 - 1 - 1 - 1 G ( z )H (z ) = H 1 ( z ) G 1 ( z ) where - 1 - 1 - 1 H 1 ( z ) = I2+H 1 1 z and G 1 ( z ) = G 1 0 .

- 1 - 1 - 1 - 1 De fine N 1 ( z ) = H 1 ( z )B 1 ( z ) H 1 ( z ) (B 1 = 12 ) . For s implic ity the

- 1 - 1 B2 ( z ) = b2 ( z ) 12 matrix is obtained here by a first stage identificat ion . estimation equat ion fol lows from

- 1 letting yb ( t ) = b2 ( z )y( t ) and - 1 �b( t ) = b2 ( z )�( t ) :

The (40) by

�( t )-y( t ) = �( t ) + (H 1�( t- l ) -G l o1.b( t )

The estimation o f the controller parameters now fo llow from two separate scalar extended least squares calculat ions us ing ub 1

C t ) -y 1 ( t ) = e: 1 ( t ) +X( t- 1 )�1 and ub2

( t ) -yz< t) = e:2 C t ) +X (t- 1 )�2 . The system

Self-tuning Control l ers 345

responses are shown in Fig . 4 and Fig 5 and the contro l lers parameter estimates in Fig . 6 and Fig . 7 .

CONCLUSIONS A multivariab le se l f-tuning controller has been described which can inc lude set point and stochastic re ference s ignals and which is based upon a cost- function inc luding a contro l penalty term . The sys tem can be both nonminimum phase and open- loop unstable , and the transport de lays can be unknown and different in different loops . Imp l icit and expl ic i t vers ions have been desc ibed and these have been demons trated on both square and non-square systems .

Set po int

�

+

Feedf orward control

- 1 c , ( z )

Cascade control

+

REFERENCES

Clarke , D . W . and Gawthrop , P . J . ( 1 9 7 5 ) . Se l f-tuning contro l ler . Proc IEE , 1 22 , 9 , 929-934 .

Gr imble , 11 . J . ( 1 98 1 ) . A control we ighted minimum-variance control ler for non­minimum phase systems . Int . J. Contr . , Vol 3 3 , 4 , 75 1 -762 .

Grimb l e , M . J-:- ( 1 982 ) . Weighted minimum­variance sel f-tuning contro l . Int . J. Contr . , Vol 36 , 4 , 597-609 .

Grimb l e , M . J . and Moir , T . J . ( 1 983 ) . We ighted minimum variance contro l ler for non-square mu ltivariable sys tems , Res . Report , Univ . of Strathc lyde , ICU/2 1 / 1 983 .

Has tings , J . R . ( 1 9 70) . Re search Report CN/ 70/ 3 , Univ of Cambridge , Dept of Engineering .

Kucera , v . ( 1 97 9 ) . Discrete linear control , John Wiley & Sons , Chichester .

Prager , D . L . and We l l stead , P . E . ( 1 980 ) . Mu ltivariable po le-ass ignment se l f­tuning regulators . Proc IEE, 1 28 , Pt D , J_ , 9- 1 7 .

P l a n t

t I . I - 1 I �D i sturbance

I T <:! '

Fig . 1 . Tracking/servomechanism prob lem .

ASCSP-M

346 M. J . Grimble and T . J . Moir

\ �� i ;·:� ,��'r.:l/lf�\Wr p, 1 .00

-2 .00

� ;::l � 0.00 g 2 . 00

-'LOO

Fig . 2 . Output and control s ignals ( expl icit ).

R �) I 6.00 , ,...-- ------ ---- - ---·-- ----- --+---- -·--• ----- -------4' (i i lj , 00 " 7.00 0.00

-2.00

·6.00 R.�l-

----------- - - - - -49-----------.. --- - - - -- -<ft ., ,

llW I · - · I - - 'iiio I .. j . -- f .... .. -j 4"lf)

1 tt1F l sr r � 1

____ ... __________ ., 'f,,

Fig . 3 . Computer control ler parameters .

Fig . 4 . Output and contro l s ignals ( impl icit l

Fig . 5 . Output and control s igna l s ( impl icit ) .

Fig . 6 . Control ler estimates for G 1 0 .

7 . 5

7 .0 r ·· ·- ' -- - - · -···· -

1 .0 r r L 5

i �/I - - · ·- ' --0. 5 : r 0.0�1:1 '

-0. 5 -i [11. I 60

I 170 - 1 .0 1 � \\ -;:;] v- ·-· _ __ _J - · - U- -------#- - -

I 180

Fig . 7 . Control ler estimates for H 1 .

i. : :

"" "

· I

i.; : "

" "

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

AN INDIRECT ADAPTIVE CONTROL SCHEME FOR MIMO SYSTEMS

J. M. Dion* and R. Lozano**

*Laboratoire d' Autornatique de Grerwble, B.P. 46, 38402 Saint Martin d'Heres, France **CIEA de[ /mtituto fiolitecnico rwciorwl, Ap. postal 1 4 . 740, Mfxico 14 D. F., Mexico

Abstract . In this short paper we consider the adaptive control of multivaria­ble discrete t ime minimum phase systems . The system under consideration is not supposed to be decouplable by static state feedback . The adaptive control design presented here is based on pole zero p lacement . The a priori knowledge of the p lant transfer matrix Hermite form is not necessary , this knowledge is replaced by that of the largest infinite zero order of the p lant transfer matrix.

1 . INTRODUCTION

In the last few years , numerous papers have dealt with MIMO adaptive contro l . Several papers have dealt with direct MIMO adaptive control for multivariable non decouplable (by static state feedback) systems : Elliott and Wolowich ( 1 982) , Goodwin and Long ( 1 980) , Dion and Dugard ( 1 982) , Goodwin Uc Innis and Wang ( 1 982 ) . Implementation of these adapti­ve control schemes requires the prior know­ledge of the plant trans fer matrix Hermite form, which seems to be restrictive . Elliott Wolowich and Das ( 1 982) have deve lopped a new parametrization requiring only the know­ledge of the controllability indices of the system. Singh and Narendra ( 1 982) adding a precompensator required the knowledge of the infinite zero order of each e lement of the p lant transfer matrix. Johansson ( 1 982) ad­ding also a precompensator , but est imating direct ly this precompensator , required the exact knowledge of the infinite zero struc­ture of the plant transfer matrix ( in the mi­nimum phase case) .

Prager and Wellstead considered an indirect approach , focus ing uniquely on the regulation case .

We wi ll consider also an indirect scheme . The prior knowledge of the Hermite form is not necessary , rather one needs to know the lar­gest infinite zero order of the system trans­fer matrix or an upper bound on this integer . The proposed adaptive s cheme allows to speci­fy independant tracking and regulat ion obj ec­tives . The paper is structured as follows . In section 2 are presented some preliminaries which will be useful in the seque l .

347

Section 3 is devoted to the study of the l i­near control ler structure (known parameters case) . This part wil l be presented in some detai ls . Some new results relating the infi­nite structure of the transfer matrix with the usual left coprime factorization are given .

The corresponding adaptive control scheme is presented in section 4 .

2 . NOTATIONS AND PRELIMINARIES

We will need some notations and prelimina­ries such as the definition of the infinite structure , of the Hermite form, of some fac­torization of the transfer matrix. In order to be short , the reader is refered to the paper by Dion and Dugard in the same procee­dings , where some direct MIMO adaptive sche­mes are studied .

3 . LINEAR CONTROL SCHEME

Cons ider the adaptive control of a discrete linear time invariant finite dimensional sys­tem having strictly proper transfer function . The obj ective of the control law is mainly to ensure that the p lant model error asymp­totically goes to zero . Now we wil l consider the l inear control scheme (known parameters) . Consider T ( z ) E. R�xm(z ) non singular , the p lant transfer matrix is assumed to be mini­mum phase .

Consider the following factorization T ( z ) = H(z ) B ( z ) , where B ( z ) is bicausal and H ( z ) is the Hermite form of T (z ) . Al l the tra­cking models are given by TM(z ) = H ( z ) P ( z ) where P ( z ) i s proper (and stable) . I t i s

348 J.M. Dion and R . Lozano

shown in Dion and Dugard ( 1 982) that z-d JH- l ( z ) is proper when d 1 is greater or equal than the largest infinite zero order d of the p lant transfer matrix. Then we can choose .

-d - 1 -d TM(z) = H(z) z H (z ) = z I The controlled process can follow any input sequence but delayed on d steps .

Cons ider now a right coprime factorizat ion N (z- l ) D- l ( z- 1 ) ( see Sect ion 2) of T ( z ) . D (z- 1 ) is bicausal then N(z- 1 ) = H ( z ) B 1 ( z ) where B 1 ( z ) is b icausal . SoAthe Her�ite form H(z ) (which can be written H (z- 1 ) , H ( z- 1 ) i s a polynomial matrix in z- 1 ) is contained in N(z- 1 ) , It fol lows that z-d N- l (z- 1 ) is pro­per .

In order to derive an indirect scheme , where a left coprime factorizat ion of T ( z ) : T (z ) = A-l (z- 1 ) B (z- 1 ) is estimated , a simi­lar result is needed for left coprime facto­rizations .

More precisely one can state the following .

Theorem 1

Let T ( z ) be a full rank strictly proper (mxm) polynomial matrix. Cons ider the left coprime factorizat ion A- l (z- l ) B ( z- 1 ) of T (z) , where

- 1 - 1 -k A(z ) = I + Al

z + . . . + � z - 1 - I -2 -k B ( z ) = B

1z + B2z + . • . +Bkz

The infinite structure of T ( z) is the in­finite structure of B ( z- 1 ) expressed as an element of Rmxm (z) . Furthermore z-d 1 B- l (z- 1 ) i� proper if d 1 is greater or equal than the largest infinite zero order (d) of T ( z ) .

Proof :

Consider A(z- 1 ) , one has det < l im (A(z- 1 ) ) .f O z->oo

then A(z- 1 ) is bicausal , so is A- l tz- 1 ) , A bicausal matrix has neither poles nor zeros at infinity so the first part of the theorem is proved .

More precisely concerning the pole and zero structure ( in z) one has : The poles of A- l (z- 1 ) are the process poles A- 1 (z- 1 ) possesses m zeros of order

P l • · · · Pm at zero , where the Pi ' s are the observab ility indices of the system. possesses m poles of order P l • · · · Pm at zero . is proper and then has no poles at infinity .

The finite zeros of B ( z- 1 ) are the finite system transmiss ion zeros .

The infinite zeros of B ( z- 1 ) are the infini­te systems zeros .

Then the zero structure of T (z ) is totally included in the zero structure of B ( z- 1 ) ,

-d 1 - 1 - 1 Suppose now that z B (z ) is proper which is equevalent to z-d 1 B- 1 (z-1 ) has no poles at infinity which is also equivalent to zd l B ( z- 1 ) has no zeros at infinity .

Let B 1 (z) A(z ) B2 (z ) a Smith McMillan facto­rization at infinity of B (z- 1 ) , Where B 1 ( z ) and Bz (z ) are b icausal and [ -m 1 z 1 o

A (z ) = '• , , -� j 0 ' z

the m . ' s are the infinite zero orders of T ( z) . i

has no zeros at infinity if d is greater or equal than the largest inflnite zero or­der d = m�x (mi ) of T (z) . Then the result

i is proved .

We will use now the yroper precompensator above defined z-d B- (z- 1 ) .

Cons ider ;(t ) d - 1 q B ( q ) u ( t)

Then - 1 - I -d - I - 1 - -d -A(q ) y (t) = B (q ) q B ( q ) u(t ) = q u (t ) .

The set "process + precompensato,Ed has an Hermite form which is equal to q I and then is more easy to control .

Us ing the d ivis ion algorithm, explicit ex­pressions are found for two polynomial ma­trices S ( q- 1 ) and R ( q- 1 ) such that

C (q- 1 ) = S (q- l )A(q- 1 ) + q-d R (q- 1 ) where

- 1 - 1 -N C (q ) = I + c 1 q + . . . + CN q - 1 - 1 -d+ l S ( q ) = I + S 1

q + • • • + Sd- lq

- 1 - 1 R ( q ) = R0 + R 1q + . . , + RT q-T

T = Max (N-d , k- 1 ) .

It fol lows that - 1 - 1 - 1 -d - 1 C ( q ) y ( t) = S (q )A (q ) y ( t ) + q R (q ) y ( t)

= q-d [ S (q- l )u ( t ) + R (q- l )y ( t ) ]

By taking - - I - 1 - 1 - 1 u (t ) = S ( q ) (C (q ) y* (t+d) - R(q ) y ( t ) )

Control Scheme for Mirna Sys tems 349

where y* (t ) i s the des ired sequer.ce to be followed . One achieves the following

C (q- I ) (y (t) - y* (t) ) = 0

Considering now an additive uni formly boun­ded disturbance w (t ) on the process output y ( t ) .

One has A(q- 1 )y (t) = B (q- 1 ) u (t ) + w (t) with the control above defined one achieves the following

- I * - I C ( q ) (y (t ) - y ( t ) ) = S ( q )w(t ) .

In regulat ion y* (t) = 0 we obtain - I - I - I y(t ) = C ( q ) S (q )w (t ) .

The regulat ion poles are then chosen as the zeros of C (q- 1 ) and correspond to the ob­server poles (see Dion and Dugard , same pro­ceedings ) .

The linear control scheme is represented in figure I .

4 . ADAPTIVE CONTROL SCHEME

The adaptive control law is described as follows : ( sketchy) .

A - I A - I * A first part i s direct , S ( q ) and R(q ) are estimated from input , output data (u(t) and y ( t) ) . More precisely one has : C (q- l )y ( t+d) = S (q- l ) u (t ) + R(q- l )y ( t ) = 8 ¢ (t) where 8 is a matrix containing the coeffi­cients of S (q- 1 ) and R(q- 1 ) and T -T -T T T ¢ ( t ) = [u ( t ) , u (t- 1 ) , . . . ,y ( t ) , y ( t- 1 ) , . . . ]

from this point the parameters may be es t i­mate by standard methods .

* A second part is indirect , B (q- 1 ) is esti­mated recurs ively from input output data (u(t ) and y ( t) ) .

At each step two estimation problems are sol­ved , the second is indirect . In order to prove stab il ity we need pers istence of exci­tation on the p lant input (see J . B . Moore ( 1 983) ) .

At this point the following remarks have to be made .

Remark I : In order to be able to prove sta­bility without persistant excitation, i t would b e very interesting to develop a d i­rect scheme on these bases . It seems possi­ble to do this by est imating directly the precompensator , but us ing a scheme where the precompensator is outer the feedback loop . This , with the same prior knowledge on the plant trans fer matrix .

Remark 2 : In the non-minimum phase case the proper precompensator

- I qn-d-k det (B�q ) ) • B- 1 (q- 1 )

- I n-d-k adj (B(q ) ) q B can b e used ,

where 6 { I i f b ( I ) = 0 b ( I ) otherwise

- I det (B(q ) ) .

6 is computed in order to assure a nul l stea­dy state tracking error .

n-k is equal to the number of process poles minus the number of process transmiss ion zeros . So n-k is equal to the sum of the in­f inite zero orders .

Remark 3 : The first part of the algorithm is direct in order to avoid the resolut ion of the Bezout equation for the plant parame­ters estimates .

5 . CONCLUDING REMARKS This paper has proposed an indirect adaptive control algorithm for MIMO square ful l rank minimum phase systems . The knowledge of the largest infinite zero order of the p lant transfer matrix is required . More work has to be done , following the same l ines , in order to obtain a direct adaptive control with the same a priori knowledge .

REFERENCES

Dion , J . M. , L. Dugard ( 1 982 ) . "Direct adap­t ive control of discrete t ime mul tiva­riable systems " . IEEE , C . D . C . Orlando , December .

Dion , J . M. , L . Dugard ( 1 983) . "Parametriza­t ions for multivariable adaptive sys­tems " . Same proceedings .

E l l iott , H . , W .A . Wolowich ( 1 982) . "A para­meter adaptive control structure for l inear multivariable systems" . IEEE T .A . C . Vol . AC-27 N . 2 , pp . 340-352 .

E lliot t , H . , W .A . Wolowich , M . Das ( 1 982 ) . "Arbitrary pole placement for l inear multivariable systems " . Technical report . Univers i ty of Massachusetts . UMASS-ECE­SE82-2 , September .

Goodwin , G . C . , R . S . Long ( 1 980) . "Generali­zations of results on mul tivariab le adaptive control" . IEEE T .A . C . Vol . AC-2 7 , No 3 , April .

Goodwin, G . C . , B . C . Mcinnis , J . C . Wang ( 1 982). "Mode l reference adaptive control for systems having non square trans fer func­t ions " . IEEE , C .D . C . , Orlando , December .

Johansson , R . ( 1 982) . "Parametric model s of l inear multivariable systems for adapti­ve control" . IEEE , C . D . C . , Orlando , De­cember .

350 J . M. Dion and R . Lozano

Moore , J . B . ( 1 983) . "Persistence of excita­tion in extended least squares" . IEEE , T .A . C . , Vol . AC-28 , No 1 , January .

Prager , D .L . , P . E . Wel lstead ( 1 98 1 ) . "Multi­variable pole assignement self tuning regulators" . IEE Proceedings , Vol . 1 28 , pp . 9- 1 8 .

S ingh , R .P . , K . S . Narendra ( 1 982) . "A global­ly stab le adaptive controller for mul ti­variable systems " . Report No 8 1 1 1 , E lect . Eng . Yale University , April .

r :� -d q I

plant

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

STOCHASTIC ADAPTIVE CONTROL WITH KNOWN AND UNKNOWN INTERACTOR

MATRICES

G. C. Goodwin* and Luc Dugard**

*Defmrtrnmt of Elfftriwl and Corn/JU!Pr EnginPrring;

University of .Newcastle, ,�s\¥.. 23?8, A ustralia

**Laboratoire d'Autornatzque de Grenoble, B . I . 46, 38402 St. Martm d lleie.1, hance.

Abs t ract . Thi s paper analyzes a s tochastic adaptive control algorithm for linear multivariable systems . The system i s ass umed to be s tably invertible so that pole-zero placement techniques can be used . It i s also assumed that the system interactor matrix is known. The latter assumpt ion . i s di�cussed in detail and a method is p resented for removing this ass umption whilst retaining global convergence of the adapt ive control law.

�ord�. Adaptive control , Stochastic control , Linear systems , Multivariable systems .

1 . INTRODUCTION It has been shown (Elliott and Wolovich ( 1982) and Goodwin and Long (1980) ) that the system interactor matrix (Wolovich and Falb (1976) ) is the appropriate multivariable generalizat­ion of the system delay in the s ingle input s ingle output case . By assuming knowledge of the system interactor matrix, several authors (e . g . Goodwin and Long 1980) , Elliott and Wolovich (1982 ) ) have established global con­vergence of multivariable adaptive controllers of the one-s tep-ahead type .

The s tochast ic generalization of the determ­inistic one-step-ahead controllers i s the class of algorithms generally known as Minim­um Variance Regulators (Astrom ( 19 70) ) . Adap­tive forms of these algorithms have been des­cribed by many authors , see for example (As trom and Wittenrnark (1973) and Astrom, Borisson, Lj ung and Wittenmark (1977 ) . In the multivariable stochastic case , the inter­actor matrix again emerges as the appropriate generalization of the delay in the s ingle in­put s ingle output cas e . Early work on s toch­astic multivariable adaptive control assumed that the interactor matrix had a s implified structure . For example , Borisson ( 1979 ) assum­ed that the interactor matrix was o f the form zdI . Thi s amounted to associating a single delay with all inputs . A natural generaliz­ation of this was given in Goodwin , Ramadge and Caines ( 19 82 ) where the interactor was d res tricted to be of the form : diag ( zd 1, • • • z � i . e . a different delay was associated with each output . In the general cas e , the interactor matrix, � (z ) , has the form:

� (z ) H ( z )D ( z ) ( 1 . 1) where H ( z ) i s a lower triangular unimodular

35 1

matrix with l ' s on the main diago�l and D ( z ) i s of the form diag . ( zd , . . • z ID) . To the author ' s knowledge , there has to date been no s imple derivat ion of a s tochastic minimum variance controller in the case of systems having a general interactor matrix. A derivation i s given in Goodwin and S in (1983) but this relies upon relatively comp­licated arguments via the Kalman Filter . One of the contributions of the current paper is to present a s imple derivation of a s tochas­tic minimum variance controller using nothing more than the division algorithm of algebra. We also show that a globally convergent adap­t ive control law can be developed based on the resulting s tochastic minimum variance controller .

A feature of the stochastic adaptive control law described above is that the interactor matrix is assumed known . This is perhaps reasonable for those systems for which the interactor has the form � ( z ) = diag ( zd 1, • • • , zdm) . However , in general , the interactor matrix contains real variables and it is difficult to know how these could be predet­ermined without full knowledge of the system model . With this as motivation , we present a poss ible method for eliminating the need to know the interactor matrix. Thi s method appears to off er promis e though further work is necessary to fully investigate the comp­lete practical implications of the algorithm.

2 . MINIMUM VARIANCE CONTROLLER FOR MULTIVARIABLE SYSTEMS

We consider the following discrete t ime linear t ime invariant proces s :

y ( t) = T (q )u ( t ) + N(q )w(t ) ( 2 . 1)

352 G . C . Goodwin and L . Dugard

described in terms of transfer function mat­rices T (q ) and N(q) . In ( 2 . 1) , {y ( t) } , {u ( t) } , {w (t ) } denote the output , input and innovations sequence respectively .

The m x m matrix T (z) is assumed to be stric­tly proper and to have full rank a . e z ; this being a necessary condition for arbitr­ary output tracking (Goodwin and Sin (1983) ) . The transfer function T (z) and N (z) are assumed to be stably invertible .

It is known (see Wolovich and Falb ( 1976) ) that there exists a matrix l; ( z ) having the s tructure given in equation (1 . 1 ) and such that lim l; (z ) T ( z ) = K ( finite and nonsingular) z->oo (2 . 2 ) A key s tep in deriving a stochastic minimum variance controller for the system (2 . 1) is to develop a predictor for � (q)y (t ) . This is described in the following Lennna .

Lemma 2 . 1 The optimal s teady s tate predict­ion, z (t ) , for the quantity z ( t) = !;: (q )y ( t ) satisfies an equation of the form

c (q-1) � ( t ) = /3 (q-1 ) u (t) + a (q-1 )y (t ) ( 2 . 3) where

- -1 - -1 - -n C ( q ) = I + c1 q + . . . C n q ( 2 . 4)

S(q-l) = Bo + Slq-1 + . . . + Bmq-m (2 . 5 ) -8 = K 0 (finite and nonsingular)

- -1 p a (q ) = a + . . . + a q-0 p

( 2 . 6 )

(2 . 7 )

Proof: Multiplying equation ( 2 . 1) on the left by l; (q) gives

!;: (q)y ( t ) = l; (q ) T (q )u ( t ) + !;: (q) N (q )w(t ) (2 . 8 ) The noise term l; (q )N (q ) can be factored into future and past noise by use of the division algorithm of algebra , i . e . we write

l; (q)N (q) = F (q) + R(q) ( 2 . 9 ) where

d F (q) = Fdq + . . • F1q and R(q) is a proper transfer function. Substituting (2 . 9 ) into (2 . 8) gives

l; (q )y (t ) = !;: (q )T (q)u ( t ) + F (q )w(t ) + R(q )w(t ) (2 . 10)

Since {w (t ) } is the innovations sequence it can be causally obtained from {y (t ) } . An appropriate expression for w (t ) is obtained by invert ing (2 . 1 ) , i . e .

w (t ) = N (q) -1 [y ( t ) - T ( q) u (t ) ] Substituting ( 2 . 11) into ( 2 . 10) gives

( 2 . 11)

or

l; (q )y ( t ) = l; (q) T (q )u ( t ) + F (q )w(t ) + R(q)N (q) -l [y (t ) -T ( q )u (t ) ] [ l; (q ) T (q ) - R (q )N (q-l) T (q) ] u (t ) + R(q )N (q) -ly ( t ) + F (q )W(t )

( 2 . 12)

[ l; (q )y (t ) -F (q)w (t ) ] = [ l; (q) -R (q )N (qf1 ] T (q )u ( t ) -1 +�fq)N(q ) y ( t ) -1 [ l; (q )N(q ) -R (q) ] N (q) T (q )u ( t )+R(q )N (q) y (t)

-1 -1 F (q )N (q) T (q )u ( t ) + R(q )N (q) y ( t ) (2 . 13 )

where F (q )w (t ) denotes the future (unpredict­able) noise .

From equation (2 . 2) it follows that F (q )N (q)1T (q) is a proper transfer function . Similarly , R (q )N (q)1 is a proper transfer function by definition of R(q) . Hence the composite matrix [ F (q )N (q)11 (q) ; R(q )N (qf1 ] can be described by a causal left matrix frac­tion description as follows : [ F ( q )N (qf1T (q ) ; R (q)N (qf1 J =

c(q-1) -1 [ /3(q-1 ) ;a (q-1) l c 2 . 14 ) - -1 - -1 - -1 where C(q ) , S ( q ) , a (q ) are polynomial

matrices in the unit delay operator q- 1 such that the zeros of C(q- 1 ) are the transmission zeros of N(q) .

Substituting ( 2 . 14) into ( 2 . 13) innnediately gives - -1 C(q ) [ l; (q )y ( t ) -F (q) w( t) ] scq-1

> u < t> - -1 + a (q ) y ( t) ( 2 . 15)

By choosing C (0) = I , equation (2 . 2 ) . This tor (2 . 3 )

we get S(o) = K from establishes the predic-

The above result leads to the following theorem. Theorem 2 . 1

vvv

The minimum variance controller for regulating l; (q )y ( t ) about l; (q) y* ( t ) is given by - -1 - -1 - -1 * S ( q ) u (;)+a (q ) y ( t ) = C(q ) l; (q )y ( t ) ( 2 • 16) where { y (t ) } is given reference traj ectory .

* The tracking error y ( t) -y ( t ) satisfies c(q-1) i; (q) [ y (t ) -y*

( t ) J =c (q-1) F (q )w ( t) (2 . 17 ) which asymptotically gives

* � -1 y ( t) - y ( t) = F ( q )w (t ) ( 2 . 18)

where F � (q-l ) is a finite moving average polynomial satisfying

Proof: F� ( q-l) = i; (q) -lF (q) (2 . 19)

Straightforward using the results of Lemma ( 2 . 1)

vvv Remark 2 . 1 The optimal steady s tate predict­or given in (2 . 3 ) can also be derived by s imple algebraic manipulation s tarting from the usual ARMAX model : A (q-l) y ( t ) =B (q-l ) u (t ) + C (q-l)w ( t) ( 2 . 20) In this cas e , we need the following 3 inequal­ities : A(q-1) -lC (q-l ) = C ( q-l )A(q-1) -1 ( 2 . 21) i; (q) C( q -l) = F (q)A( q-1 ) + G ( q -l) ( 2 . 22)

Stochastic Adaptive Control 353

( F (q )C ( q-l)� G(q-l) C (q-1) -1 ) = c (q-1) -1 ( F (q) ,G (q-1) J

3 . ADAPTIVE CONTROL SCHEMES

( 2 . 23) 'iJ'iJ'iJ

Given the structure of the predictor ( 2 . 3 ) it is a relatively s traightfoward matter to derive a globally convergent adaptive control law . For example , one can use the division algorithm as in Sin, Goodwin , Bitmead (1981) to convert ( 2 . 3) into an equivalent form and then to use the modified least squares proc­edure described in Sin and Goodwin (1982) . Global convergence can be established for a non-interlaced algorithm using interlacing of Lyapunov functions as originally developed in Moore and Kumar (1980) and described in Goodwin and Sin (1983) . In the above adaptive control law it is nec­essary to assume prior knowledge of s (q) . This is reasonable in cases where s (q) = diag (qd 1 • • • qdm) . However , in the general case s (q) contains real variables and knowledge of these variables is tantam­ount to knowing the system transfer function . This raises doubts about the utility of adap­tive control in the general case . For this reason, in the next section we describe an alternative approach aimed at avoiding know­ledge of the interactor matrix whilst retain­ing the simplicity of the minimum variance control law (which is the main reason for its widespread use) .

4 . A SIMPLE CONTROL LAW WHEN THE INTERACTOR MATRIX IS UNKNOWN

The method presented here was inspired by an idea originally sugges ted by Sargent (19 82) for the single-input single-output case .

'� For simplicity , we shall take y ( t ) = 0 (*egulation) . [ The more general case when y (t ) is a non-zero constant can easily be handled by adding s low acting integral feed­back to keep the mean value of y ( t ) at*y* and by then working wi th Liy ( t) =y ( t) - y ] Most commonly used stochastic parameter est­imation schemes (Goodwin and Sin (19 83) ) allow one to fit the Kalman Filter directly to the data . Thus , let us consider the corr­esponding s teady s tate innovations model in state space form , i . e . � ( t+l) = A� (t) + Bu ( t ) + Kw ( t) ( 4 . 1) where w(t ) y ( t) - C� ( t ) ( 4 . 2) Let d be an upper bound on the maximum for­ward shift in the interactor matrix for T (q) T . In the sequel we will assume knowledge of d but not of the interactor matrix itsel f . Note that x (t+l) is actually the best estim­ate of x (t+l) given data up to time t . We can now evaluate the best estimate , i ( t+d l t ) , of x (t+d) given data up to time t as

ASCSP-M*

A x < t+d I t ) dA A x ( t )

d-1 . + l Ad-l-J Bu ( t+j )

j =O + Ad-lKW (t ) (4 . 3 )

The corresponding d-step ahead prediction , for y (t) is y (t+d j t) , where A d d-l d 1 • d 1 y ( t+d j t ) = CA � (t ) + l CA - -J Bu ( t+j )+CA - Kw(t:)

j =O (4 . 4)

In the special case when the interactor is s (z ) = zdI , t:hen the right hand side of (4 . 4 ) will contain only u ( t: ) . More generally , however , y ( t:+d l t) will be a function of u ( t: ) , . . • , u ( t+d-1) . For the general case , t:he cont:rol law bring­ing y (t+d j t ) to zero will be non-unique . We t:herefore choose t:he sequence u (t ) , • • • , u( t+d-1) having the least energy . This is described in the following :

Theorem 4 . 1 A (a) The control law bringing y ( t+d l t ) to zero and having least energy is given by u ( t+j ) = - L . (Adi (t )+Ad-lKw(t ) ) ; j =O , . . . d-1 J (4 . 5 ) where

1_ 1 _l . d-1 L . = (CA d-1-J B) T l (CA d-1-kB) (CA d-1-kB) T C J k=O - -( 4 . 6) (b) The matrix inverse in to exist by the choise of

( 4 . 6 ) d .

i s guaranteed

Proof: (a) St:raight:forward using Lagrange mult:ipl­

ers to minimize d-1

J = I 1 1 u < t+j > I I 2 j =O ( 4 . 7 )

A subj ect to the constraint y ( t+d j t ) =O

(b) Let 6 . = CAd-l-j B J and put:

j =O , . . • d-1

M = [ So ' . . . , Sd-1 ] ( 4 . 8) Then the matrix in f4 . 6 ) which we require

t:o be nonsingular is MM . Now by definition of s (z ) we have : lim s (z ) T (z ) T K finit:e and z->oo nonsingular (4 . 9 )

Li d where s (z ) = soz + . . . sd-lz (4 . 10) Putting (4 .1) , (4 . 2 ) int:o ( 4 . 4) gives T ( z ) :

d -1 -1 d-l . -1 T ( z ) = ( z I-a ( z ) ] [ l 6 . zJ +6 � ( z ) ] ( 4 . 11) j =O J

for _polynomial a ( z-1 ) , S � ( z- 1 ) of the form: -I -1 � -1 � -1 � -2 a ( z ) = a0+a1z + . . . ; S (z ) =S1z +S2z + . . .

Substit:ut:ing (4 . 10) , (4 . 11) into (4 . 9 ) gives . d T -<l T -1 hmf s0z + . . . sd

_1z l f 60 z + . . . Sd-lz

z->oo +z -dB � ( z-1) T l [ I- a(z -1) Tz -d ) -1

= [ s0 • • • Sd-l ]MT ( 4 . 12)

354 G . C . Goodwin and L . Dugard

i . e . T [ t;0 • • • E;d_1JM = K finite and nonsingular

( 4 . 13 ) Now from Sylvesters inequality w e know that the rank of a product is less than or equal to the minimum of the ranks . Thus from (4 . 13) T Rank M = dimension o f u (t ) and hence MMT i s nons ingular a s required .

'V'V'V The system response resulting from the feed­back control law ( 4 . 5) is characterized in the following theorem. Theorem 4 . 2 (a) The response o f the system with time­varying feedback law (4 . 5 ) satisfies the following stationary (nd) dimensional Markov model : x(N) = A.x cN-1) + K.wcN-1) N=O , l , where - A T A T T x (N) = [ x ( t+l) , • . . , x ( t+d) ] ; t=Nd w (N-l )= [w(t ) T, w ( t+l )T, . . . w(t+d- l ) T ] A.=

( 4 . 14)

( 4 . 15 ) t=Nd ( 4 . 16)

0 - - - - 0 (A-BL Ad) 0

K.= d-1 (K-BL0A K) O..,;::- - - -

d-1' , .... ' (AK-ABL A K , -.. 0 d-1 ' ..... .... -BL1A K) , (K) ,

' �Ad-lK-Ad-lBLoAd-lK . . . -BLd-lAd-�) "

(4 . 17 )

� 1 : (4 . 18 )

I ' O

(K)

(b) The system is asymptotically stable provided the nd x nd state transition matrix A has all its eigenvalues inside the unit circle . (c) then t=l , d .

Subj ect t o I \ (A) I < l ; i=l , • . . nd , the process {ylk)=C� (k)+w(k) ; k=Nd+t ; • • • , d } is cyclostationary with period

Pi>oof: (a) Immediate from ( 4 . 5 ) , ( 4 . 1 ) by success­

ive substitution. (b) From part (a) . (c ) At is clear from ( 4 . 14) that the vector

x (N) is stationary . Hence since

y ( t ) = C�(t) + w ( t ) , it follows that {y (k) } is cyclostationary with period d .

Remark 4 . 1 The feedback control law describ-ed above can be readily made adaptive by using the estimated values of A, B and K obtained from an on-line algorithm ( e . g . RMLl �) . We have not checked the details , but we conj ect­ure that the resulting indirect adaptive alg­orithm can be shown using standard tools (see Goodwin and Sin (1983) ) to be globally con­vergent subj ect to the prior knowledge of d and provided condition (b) of Theorem 4 . 2 is satisfied for the true system . Thus we expect the adaptive algorithm to work under precisely the same conditions as the non­adaptive algorithm. Moreover , in practice it would seem that the performance of the system could often b e preferable to the exact minim­um variance control solution due to the limit­ation on input energy . Of course , the output variance will increase commensurately so a compromise is necessary . Remark 4 . 2 Of course , in the adaptive case one cannot check that the eigenvalues of A are inside the unit circle . However , this is no different from the conditions for the usual self tuning regulator which depends upon a minimum phase property which is equally un­checkable . The key point is that both ass­umptions give precise conditions on the true system for the adaptive algorithm to converge . For d=l , the conditions on A reduce to the minimum phase condition and for d large , the conditions on A are satisfied if the original system is stable . In this sense , d plays a similar role as regards stability as does the weighting on the input in weighted one-step-ahead control ( Clarke and Gawthrop (1975) , Goodwin , Johnson and Sin (1981) ) .

5 . CONCLUSIONS This paper has presented two results : a simple derivation of a stochastic minimum variance controller for linear multivariable systems with arbitrary interactor matrix , and a method for eliminating the need to know the interactor matrix in multivariable stochastic adaptive control . The latter algorithm app­ears quite promising from a practical point of view and probably deserves further study .

6 . REFERENCES Astrom, K . J . (1970) , Introduction to Stochas­

tic Control Theory, Academic Pres s , New York .

Astrom, K . J . & Wittenmark, B . (1973) , "On self tuning regulators" , Automatica , Vol . 9 pp . 195-199 .

Astrom, K . J . , Borisson, U . , Lj ung , L . and Wittenmark , B . "Theory and application of self tuning regulators" , Automatica , Vol . 13, No . 5 , pp . 457-4 7 6 .

Goodwin , G . C . , Ramadge , P . J . & Caines , P . E . , "Discrete Time Stochastic Adaptive Control", SIAM Jnl . Control and Optimization . Vol . 19 , No . 6 , pp . 829-85 3 .

S tochastic Adaptive Control

Sin , K . S . , Goodwin, G . C . & Bitmead , R . R . , "An adaptive d-step ahead predictor based on least squares" , IEEE Trans . Auto . Cont . Vol . C-25 , No . 6 , Dec . 1980 . Also Proc . 19th CDC Conf . Albuquerque , New Mexico, Dec . 1980 .

Moore , J . B . & R . Kumar , (1982 ) , " Converg­ence of Weighted Minimum Variance N-Step Ahead Prediction Error Schemes" , Tech . Rpt . 8009 , Dept . Elec . Eng . , Univ . of Newcastle .

Sargent , R .W . (1982) - Seminar at University of Newcastle .

Clarke , D .W . & J . P . Gawhtrop (1975) , " Self Tuning Controller" , Proc . IEE , Vol . 122 (a) pp . 929-934 .

Goodwin, G . C . , C . R. Johnson & K . S . Sin (1981) "Global convergence for adaptive one step ahead optimal controllers based on input matching" , IEEE Trans . Aut:o . Cont:rol , Vol . AC , No . 6 , pp . 1269-1273 .

Borisson , U . , "Self-tuning regulators for a class of mult:ivariable systems " , Automatica Vol . 15 , No . 2 , 1979 , pp . 209-217 .

Elliot:t , H . & W .A . Wolovich , "A parameter adaptive control struct:ure for linear multivariable systems" , IEEE Trans . Auto . Control , Vol . AC-27 , pp . 340-352 .

Goodwin, G . C . & R . S . Long , (1980 ) , "General­ization of results on multivariable adapt­ive control" , IEEE Trans . Auto . Control , Vol . AC-25 , No . 6 .

Goodwin , G . C . & K . S . Sin , Adaptive Filtering, Prediction and Cont:rol , Prentice Hall , 1983 .

Wolovich , W .A . & Falb , P . L . "Invariants and Canonical Forms Under Dynamic Compensation' SIAM J . Cont . & Optimization, Vol . 14 , No . 6 , pp . 996-1008 , 19 7 6 .

355

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

DISCRETE DIRECT MULTIVARIABLE ADAPTIVE CONTROL

I. Bar-Kana and H. Kaufman

Electrical, Computer, and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 1 2 1 80, USA

Abstract . Direct mult ivariable model reference adaptive control (DMMRAC) procedures have been successfully used in continuous-time systems . Previous discrete versions of the algorithm required apriori knowledge of a stab ilizing feedback gain matrix for the controlled plant and also an estimate of the ideal model following control in order to guarantee a bounded output tracking error . In this paper a discrete version of the DMMRAC algorithm is shown to guarantee stability of the adaptive system as well as asymptotically perfect output model following provided that the system satisfies a positive realness con­dition . No a priori information is needed for implementation . Keywords . Adaptive control ; reduced order model ; discrete-time systems ; multivariable control systems ; linear systems ; nonlinear equations ; stab ility ; invariance principle . * This material is based upon work supported by the National Science Foundation under Grant No . ECS80-16173 .

1 . INTRODUCTION equations : x (k+l) A x (k) + B u (k) p p p p p y (k) C x (k) + D u (k) p p p p p and the plant output y (k) is required to p

(1 )

(2 )

A simple direct multivariable model reference adaptive control (DMMRAC) procedure developed by Sobel (1980) and by Sobel , Kaufman and nabius (l980) , and extended by Bar-Kana and Kaufman (1982a , 1982b) was shown to guarantee stab ility of continuous-time adaptive control systems . Discrete versions of the adaptive algorithm required a priori knowledge of a stabilizing feedback gain matrix and also an estimate of the ideal model following con­troller in order to guarantee boundedness of the output tracking error .

follow the output ym(k) of the asymptotically

In this paper it is shown that a new discrete version of the DMMRAC algorithm guarantees the stability of the adaptive systems as well as asymptotically perfect output model follow­ing , provided that there exists a constant gain feedback matrix (unknown and not needed for implementation) such that the equivalent closed-loop system is (simply rather than strictly) positive real . No a priori infor­mation about the controlled system is required for implementation of the algorithm. It is also shown that some systems which do not satisfy the positive realness condition (like non-minimum phase systems) can be con­trolled by using a supplimentary direct input-output feed forward gain matrix .

2 . FORMULATION OF THE PROBLEM The controlled plant is represented by the

357

stable reference model :

y (k) m C x (k) + D u (k) m m m m where it is permissible to have

(3 ) (4 )

dim(x ) » dim(x ) (5 ) p m We represent the input commands u (k) as m outputs of a command generating system of the form vm(k+l) Avvm(k) ( 6 ) u (k) c v (k) ( 7 ) m v m where Av is not necessarily stable . The representation ( 6 ) - ( 7 ) is only needed for the subsequent analysis . The matrices A and C v v are unknown and only measurements of the input u (k) are permitted . m When the reference model (3) is supplied

358 I. Bar-Kana and H . Kaufman

with an input of the form ( 6 )- (7 ) its solu­tion can be written as x (k) = E v (k) + Ak o (8) m m m o By substituting a solution of the form (8) into Equation (3 ) it can be shown that the matrix E satisfies the equation A E - EA + B C = 0 m v m v and that

(9)

(10) For the subsequent stability analysis we de­fine some bounded "ideal curves" x* (k) of the form p * xp (k) = xllxm(k) + xl2vm (k) (11)

* such that x (k) satisfies the plant equation p (1) when the plant is forced by an "ideal input" control u* (k) of the form p * - -u (k) = K x (k) + K u (k) (12) p x m u m * At the same time it is desired that x (k) p satisfy the output tracking equation

* * * y (k) = C x (k) + D u (k) = C x (k) + P P P P p m m D u (k) = y (k) m m m (13) In general curves defined by Equation (11) cannot simultaneously satisfy the plant equation (1) and the output equation (13) . However , by comparing the difference equation obtained by substituting Equation (11) into Equation (1) with the equation obtained by direct differencing of Equation (11) it can be shown that , if the following relations hold [ (ApXll - XllAm + BpKx) E + ApX12 - Xl2Av

- x11B c + B K c ] vm (k) =O m v p u v (14)

[ (c x11+D K -c ) E+C x12+D K c -D c ] v (k) =O p p x m p p u v m v m

then the ideal curves (11) satisfy the equations

(15)

* * * - k x (k+l) =A x (k)+B u (k) - (A xll-xllA +B K ) A 0 p p p p p p m p x m o * - k y (k) =y (k) + (C x11+D K -C ) A o p m p p x m m o

(16) (17)

Equations (14) - (15) can be simplified . If we define X = x11E + x12 K = K E + K C x u v

(18) (19)

and use Equation (9) , then Equations (14)­(15) become

r XA 1 I C E+: c J m m v

(20)

A solution for Equation (20) exists , in general , if dim[v ] < dim[u ] + dim[x ] . * m - m m In that case x (k) defined by Equation (11) p can eventually satisfy Equation (1) as le+<'> , * and , also , y (k) + y (k) as le+<'> . - p m We thus require that the actual traj ectory x (k) of the plant satisfy : p x (k) p y (k) p

* + x (k) as k + oo p * y (k) + y (k) as k + oo P m

3 . THE DISCRETE ADAPTIVE ALGORITHM

The state error is defined as * e (k) = x (k) - x (k) x p p

and the output tracking error is then e (k) = y (k) - y (k) y m p

(21) (22)

(23)

(24)

The adaptive algorithm generates the follow­ing plant control : u (k) = K(k+l) r (k) p where rT (k) � [ eT (k) y K(k) = [Ke (k) K (k) x and

= KI (k+l) + Kp (k+l)

uT (k) ] m K (k) ] u

K(k+l) KI(k+l) K (k+l)

= KI (k) + ey (k) rT (k) T T -= e (k) r (k) T p y

From Equation (24) we get * * e (k) = C x (k) + D u (k) - C x (k) y p p p p p p

- k - D u (k) - (C x11+D K -C ) A o p p p p x m m o

(25)

(26) (27)

(28) (29) ( 30)

C e (k) - D (K(k+l) -K) r (k) -D K e (k) p x p p e y

- (c x11+n K. -c ) Ak o p p x m m o or e (k) y c e (k) -n (K(k+l ) -K) r (k)+EAk o (31) p x p m o where

K. � [K. e

c /:::,. ( I p

-K x K. l u

+ D K ) -l p e c p

n � p (I + D K ) -l D P e P

(32)

(33)

(34)

Discrete Direct Mult ivariable Adaptive Control 359

£ � ( r+D K. ) -1 (c x11+D K. -c ) Ak o p e p p x m m o (35)

The difference equation of the state error is then

* ex (k+l) = x (k+l) - x (k+l) p p * * - k =A x (k)+B u (k) - (A xll-xllA +B K ) A 8 p p p p p m p x m o

-A x (k) - B u (k) p p p p

e (k+l) = A e (k) - B (K (k+l) -K) r (k) x p x p - - k -B K e (k) - (A xll-XllA +B K )A 8 p e y p m p x m o (36)

Substituting e (k) from Equation (29) finally gives y

ex (k+l) = A e (k) -B (K(k+l)-K) r (k)+FAko (37) p x p m o where K � K ( I + D K ) -l

ec e p e

A � A - B K c p p p ec p

B � B ( I-K D ) p p e p

( 38)

( 39)

(40)

-L - - -F= (A xll-xllA +B K ) -B K (C xll+D K -c ) p m p x p ec p p x m (41)

4 . STABILITY ANALYSIS

The following discrete quadratic Lyapunov function is used to prove stability of the adaptive system represented by Equation (29) and Equation (37) : V(k) = eT (k) Pe (k) x x +tr [ S (KI (k) -K) T-l (KI (k) -K) TST ] LV(k) = V(k+l) - V (k) LV (k) = eT (k+l) Pe (k+l) -eT (k) Pe (k) x x x x

+tr [ S (KI (k+l) -K) T-l (KI (k+l) -K) TST ]

-tr [ S (KI (k+l) -K) T-l (KI (k) -K) TST ]

(42) (43)

(44) By substituting ex (k+l) and KI (k+l) from Equation (29) and Equation ( 37) and manipu­lating the resulting algebraic expressions , we get : LV(k) = eT (k) (ATP A -P) e (k) x p p x

-2eT (k) ATPB (K(k+l) -K) r (k) x p p +rT (k) (K(k+l) -K) TB TP B (K(k+l) -K) r (k) p p

-2eT (k+l)PF Ak 8 x m o

-rT (k) (K(k+l) -K) T [DT ( STS)+(STS)D ] (K (k+l) -K)· p p r (k) - eT (k) ( STS ) e (k) rT (k) (2T+T) r (k) y y

-2rT (k) ) K(k+l) -K) T ( STS) EAk 8 (45) m o If the following relations are satisfied

ATPA - p = -LTL < 0 p p

DT ( STS) + ( STS) D - BTPB = WTW > 0 p p p p

we finally get LV (k) =- [ eT (k) LT-rT (k) (K(k+l) -K) TWT ] . x

[Lex (k) -W(K(k+l) -K) r (k) ]

-eT (k) (STS) e (k) rT (k) (2T+T) r (k) y y

-2e (k+l) PFAk 8 x m o

(46)

(47)

(48)

(49)

Note that LV(k) is not necessarily negative definite or semidefinite , due to the last two terms in Equation (49) . However , LV (k) can still be used to prove s tability of the sys­tem . To this end , by using Equation (37) and Equation (42) it can be shown (Bar-Kana , 1983) that , if I l e (k) I I or I I K (k+l) -K I I x become large enough , we get from Equation (49)

V (k+l) < V (k)+a.V (k) 1 1 A 1 1 k for some a. > 00 (50) - m or

V(k+l) < ( l+a. I I A I l k) V(k) - m It is clear that

( Sl)

V(k) .2_ W (k) (S2) where W (k) is defined by the difference equation

W(k+l) = ( l+a.I I A I l k) W (k) ; W(O) = V(O) m ( S3) Using Equation (S3) for k=0 , 1 , 2 . . . , i t can be seen that

W (k+l) = (l+a.1 1 A 1 1 k) (l+a.1 1 A 1 1 k-l) . . • ( l+a.)W(O) m m and from (S4) (54)

( l+a. 1 1 A 1 1 n-2) . . . ( l+a. 1 1 A I J ) ( l+a.) W (O ) (SS ) m m

360 I . Bar-Kana and H. Kaufman

We can apply the ratio test to the series obtained from Eq . (55) for k + 00 , to show that the series converges and that W (k) is bounded .

From Eq . (52/ i.t can be seen that V (k) is bounded for all k ; then , the quadratic form Eq . (42) of V (k) guarantees that the gains KI (k) , the state error ex (k) , and the output error ey (k) are bounded . In that case the last two terms in Eq . (49) vanish as k + 00 • This fact permits a subsequent application of a modified La Salle ' s Invariance Principle for discrete nonlinear nonautonomous systems (LaSalle , 1977 ; Bar-Kana , 1983) to get the following theorem of stability for the dis­crete adaptive system. Theorem

Assume that there exist a positive def inie matrix P and a gain matrix Ke (not needed for implementation) such that relations Eqs . (46)­(48) are satisfied ; also , assume that Equa­tions (18) - ( 20) have a solution for the gain matrices x11 , x12 , Kx ' and Ku .

Then , all states and gains of the adaptive system are bounded and the output tracking error vanishes asymptotically .

Conditions Eqs . (46) - (48) are equivalent to requiring that the closed loop input-output transfer function

Z ( z) - - -1-

D + C ( zI - A ) B p p p p ( 56)

be (simply rather than strictly) positive real .

Special attention must be paid to systems with D =O . As seen from Eq . (48) , in that case tRe discrete positive realness condition cannot be satisfied . Therefore , we may try to use supplemetary gains D and D in order to satisfy the positivity cgnditioWs .

However , even in that case we want

C x (k) + C x (k) as k + 00 p p m m and get instead C x (k) + D u (k) + C x (k) + D u (k) p p p p m m m m

(57)

as k + oo (58)

In general , condition ( 58) does not imply condition ( 57) , because , if we want

and

c x (k) + c x (k) p p m m

D u (k) + D u (k) p p m m then Eq . ( 20) must be replaced by

( 59)

(60)

(61)

Note that the above relation has more equa­tions than variables and it does not have a solution , in general . For the particular case of asymptotically constant inputs

lim u (k) = U = constant k+oo m

we get by substituting Eqs . (14) - (15) : [A X + B p p

C X-C E p m D K-D C p m v

K-AAv l A 0 • v

u 0

(62)

c I in v

(63)

where U is a vector of constant coefficient .

Equations (66) have a solution in general if dim [ um(k) ) > 1 or if U = O. When dim [ u (k) ) = 1 an unknown value D which m m satisfies Eq . (63 exists and might be found experil!lentally .

5 . E��MI'LES

A minimum-phase and two non-minimum phase un­stable systems are used for application of the digital Dl1MRAC to regulate and control digital control systems . An implicit loop was imple­mented , in order to satisfy Eqs . ( 24) - ( 30) . The reference model is represented in all cases by

xm(k+l " [ �2S :] xm(k) + [ }m(k) (64)

(65)

Example I The first example is the minimum phase plant

x (k+l) = p l . 8824 : . ,J xp (k) + [:}, (k)

y (k) = [ -1 . p 2 . ) x (k) + 2 . u (k) p p The adaptation gain matrices defined in Eqs . ( 29) - ( 30) are

T = T = . 1 I

(66) (67)

(68)

A s inusoidal input was used to control the system. The results of the digital simula­tion are represented in Fig . 1 . ResuLts show good output model tracking .

Discrete Direct Multivariable Adapt ive Control 36 1

Example II The second example is an unstable plant which, without the direct feedf orward, would have been non-minimum phase

x (k+l) p [ O l ] xp (k) + [0] u (k) - . 625 1 . 75 1

p

y (k) = [ -2 . p l . ] x (k) - 6 . u (k) p p The adaptation gain matrices are

T = T = I .

(69) ( 70)

(71)

The results of the servo following test with unit step input are shown in Fig. 2 and with a sinusoidal input are represented in Fig . 3 .

Observations show good output model follow­ing .

Example III

The third example is a non-minimum phase unstable system represented by

x (k+l) p - l . 625

1 J x, (k) {0}, (k) 1 . 75 1

y (k) = [ 2 . p l . ] x (k) p

( 72) ( 73)

The system ( 72) - (73) does not satisfy the positive realness conditions ( 46 ) - ( 48) , therefore Eq . (73) is replaced by

y (k) = [ 2 . p l . ] x (k) + 7u (k) p p ( 76)

such that the new system satisfies the posi­tivity conditions .

For the subsequent presentation of the re­sults we define the following values

- (k) c x (k) yp p p ( 75)

- (k) c x (k) ym m m (76)

e (k) y - (k) ym - y (k) p ( 7 7)

The adaptation gain matrices are

T = T = . 1 I . (78)

The results of the servo following test with a unit step input are shown in Figs . 4-5 and with a sinusoidal input are shown in Figs . 6-7 . Observations indicate that Yp (k) + Ym(k) while e (k) is finite and bounded . S ince y dim (u (k) ) = 1 we do not expect e (k) to m y vanish , as shown by Eq . (63) and the conclu­siond following it . However since the system is stable , a gain D exists for s tep inputs such that E q . (63) �s satisfied .

Such a value for D was found experimentally to be - . 08 . m

The results of the servo following test with a step input for the adjusted model are represented in Figs . 8-9 . Observations show that both conditions (59) and ( 60) , are sat­isfied in this case .

CONCLUSIONS In this paper the feasibility of direct mul­tivariable adaptive model reference proce­dures was extended for discrete-time systems . The stability of the adaptive system is guar­anteed provided that a positive realness condition is satisfied . No a priori infor­mation is required for implementation of the algorithm . A supplementary direct feedfor­ward gain may be used to control systems like non-minimum phase which to not satisfy the positive realness conditions .

This extension , together with the s implicity of implementation and the low order of the controller make the D�'.!RAC algorithm a use­ful adaptive control method suitable expecially for large scale systems .

REFERENCES Bar-Kana , I . , H . Kaufman ( 1982) . Model Ref­

erence Adaptive Control for Time-Variable Input Corrnnands . Proc . 1982 Conf . on Inf . Sciences and Systems , Princeton , NJ .

Bar-Kana , I . , H . Kaufman ( 1982) . Multivar­iable Direct Adaptive Control for a General Class of Time-Variable Corrnnand s , Proc . 2 1 s t IEEE CDC , Fl .

Bar-Kana , I . ( 1983) . Direct Multivariable Model Reference Adaptive Control with Applications to Large Structural Systems , Ph . D . Thesis , RPI , Troy , NY .

LaSalle , J , P . (1977) . The Stability of Dynamical Systems . S IAM .

Sobel , K . (1980) . Model Reference Adaptive Control for Multi-Input Multi-Output Systems . _ Ph . D . Thesis , RPI , Troy , NY .

Sobel , K . , H . Kaufman and L . Mabius (1980) . Model Reference Output Adaptive Control Systems Without Parameters Identification . Proc . 18th IEEE CDC , Fl .

362

Fig . 1 : Example I S inusoidal input tracking . Outputs y (t) , y (t) m p

- - ... T

Fig. 4 : Example III S tep input tracking Output y (t) , y ( t) m p

Fig. 7 : Example III S inusoidal input tracking Outputs y (t) , y (t) m p

I . Bar-Kana and H. Kaufman

Fig. 2 : Example II

!] , . !f

!l I 2!\ I iod' , S ' \

�1 I

�j . ' ..

S tep input tracking Outputs y (t) , y (t) m p

... T

Fig. 5 : Example III S tep input tracking Outputs y ( t) , y (t) m p

Fig . 8 : Example III D = - . 08 m Outputs y (t) , y (t) m p

!f

Fig. 3 : Example II S inusoidal input tracking

I I !l

Outputs y (t) ,y (t) m P

�fVJ!0Jv �1 . I ,....�-,,,=---=--, ... �, ,--,,,.,,...-..,,,.,,.-�

Fig . 6 : Example III S inusoidal input tracking

!, I I !t

Outputs y (t) ,y ( t) m p

Fig . 9 : Example III D = - . 08 m Outputs y (t) ,y (t) m P

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

DISCRETE TIME MULTIVARIABLE ADAPTIVE CONTROL FOR NON-MINIMUM PHASE PLANTS

WITH UNKNOWN DEAD TIME

N. Mizuno and S. Fujii

Depart111n1t of Meclwnirnl Enginfl'ring, Nagoya Institute of' Teclmology, Nago_va, japan

Abstract . A new method for designing a discrete time multivariable adaptive control system is presented . The controlled plant is a multi-input , multi­output s table , non-minimum phase p lant with unknown dead t ime . In the proposed method an adaptive control is carried out using a controller designed by a certain decomposed representation of the unknown plant . This method requires no a priori knowledge regarding the dead t ime of the plant and the plant s tructure , and may in many cases require less compu­tation than other methods which have been proposed for non-minimum phase systems . The stability of the proposed adaptive control system is considered . Moreover , the relation between the proposed method and the conventional model reference adaptive control or adaptive pole placement method is clarified . Finally , the results of computer s imulation of adaptive control , applied to minimum phase plant without dead time and non­minimum phase plant with dead t ime , are included to illustrate the effectiveness of the proposed method . Keywords . Adaptive contro l ; multivariable system; discrete t ime system ; non-minimum phase system; dead t ime .

INTRODUCTION

In the recent pas t , several approaches for a discrete time adaptive control system have been developed , and many of them have re­solved the s tability problem under certain assumptions (Goodwin , 1980 ; Morse , 1980 ; Narendra , 19 80) . However , from the practical point of view, there are many problems to investigate . For example , in the model reference adaptive control system (MRACS) , it mus t be assumed that the plant is a minimum phase system, s ince the adaptive controller is designed based on cancellation of all the plant zeros . This assumption res tricts the application of the MRACS to plants controlled with a digital computer in discrete time , because the discrete time form of the minimum phase continuous time p lant often becomes a non-minimum phase system. In the multivariable cas e , the assumption is more restrictive . For this problem in the s calar case , the conventional MRACS has been extended to handle non-minimum phase systems by modifi­cation in terms of zero shifting (Johnstone , 1980 ; Fuj ii , 198la) . However , i t may be difficult to design the controller for an unknown plant so that the zeros of the augmented system lie inside of the unit circle of the z-plane , and so that the adaptive control system has fast convergence .

363

Especially , in the multivariable case , the procedure is complex and not clear .

On the other hand , different types of adap­tive control algorithms applicable to non­minimum phase sys tems have been proposed by several authors (Goodwin , 1981 ; Astrom, 19-80 ; Ellio t t , 1982 ; Lozano , 1982 ; Suzuki , 19-83) . These methods are based on the adap­tive pole placement . In the indirect me­thods (Goodwin, 1981 ; As trom, 1980 ) , para­meters of a p lant are recursively estimated and then parameters of a controller deter­mined by a design calculation . In the di­rect methods (Astr�m, 1980 ; Ellio t t , 1982 ) , parameters of a controller are updated di­rectly using special types of plant repre­sentation . In o ther methods (Lozano , 1982 ; Suzuki , 1983) , the parameters of p lant and controller are updated alternately by two independent algorithms . In either case , s ince these methods require more computation than that of the conventional MRACS , it is diff icult to extend these methods to handle multivariable systems . Moreover , in the conventional MRACS , it mus t be assumed that the dead t ime of the unknown plant is known . Even in the s calar case , the dead time of the p lant is not exactly known , and in the multivariable case , it is impossible to de­termine the dead t ime of the p lant unless a priori knowledge of the plant s tructure is obtained .

364 N . Mizuno and S . Fuj i i

For these problems , Fuj ii has proposed an adaptive control method using an autoregre­ssive plant model with dead t ime (198lb) , and the method has been extended to handle the multivariable case . On-line computer ex­periments were carried out using a plant tes­ting the performance of a refrigerant com­pressor (Fuj ii , 1982a) . The present authors also have proposed, in the scalar case , a new design method for a discrete t ime adaptive control system for non-minimum phase plants with unknown dead time (Fuj i i , 1982b) in which an adaptive control is carried out us­ing a controller designed by a new plant re­presentation with special structure .

In the present s tudy , the above method i s ex­tended to handle multi-input , multi-output plant s . First , a new plant representation composed of two subsystems is considered ; the one is a minimum phase system which has the same D . C . gain and the same poles as the original system and the other has the zero D . C . gain and the same poles . Then , the de­sign method of the conventional MRACS is applied to minimum phase sub system of the original system. By using this plant repre­sentation , the adaptive controller construct­ed can be applicable to a multivariable non­minimum phase plant with unknown dead time , and the parameters of the controller can be

· directly calculated from input , output measurement of the plant with small compu­tation . Next , i t is shown that the s ta­bility of the proposed method is ensured when the estimated parameters converge to their true values or in some region in the para­meter space . Moreover , from the equivalent representation of the control system, it is s.hown that the proposed method has in part the same structure as the conventional MRACS , and has the related structure to the adaptive pole placement method . Finally , in order to illustrate the effectiveness of the proposed method, computer s imulations are performed for different types of plants .

STATEMENT OF PROBLEM

Consider the multi-input ,multi-output , dis­crete time , linear , time invariant system (plant) having the following ARMA repre­sentation .

y (k)

<b

where [y (k) ] , [u (k) ] are the vector system output and input , respectively , and di . re-

- J presents pure dead t ime and Ai (q1 ) ( i=l , • • ,m) , Bij (q-1 ) ( i=l , • • ,m ; j=l , . • , m) denote scalar

polynomials in the unit delay operator q� as follows .

A. ( q-1 ) =1-a � q-1- . • . -a ��v; ( i=l , . . , m) (2) i i i - -1 - 1 2 -1 n;j -nij+l Bij (q ) -bi/bij q + . . . bij q

( i=l , . . , m; j =l , . . , m) ( 3 )

The following assumptions are made about the system.

a)

b)

c)

n=max (v . , n . . +d . . ) is known . i iJ iJ A. (q1) is a s table polynomial but A. (ql ) i i

- 1 and Bij (q ) have unknown coefficients .

The system (1) satisfies the following condition . [ Bll (1) . . . 'Elm Cl) l det . . -;o

ii 1 <1) . . . B (1) m mm (C . 1)

Note that assumption c) implies that the system can be s tatically decoupled and is less restrictive than an assumption that the system has a stable inverse .

It can be seen that Eq . (l) consists of a set of multi-input , s ingle-output systems with unknown dead time having a common input vector . Equation (1 ) can also be represent­ed as

n n y (k+l) = l: A . y (k+l-i)+ l: B . u (k+l-i) (4) i =l i i=l i

where A . and B . ( i=l , . . , n) are mxm matrixes i i with elements in the form

A . =diag (a1i , . . . , ai ) i m

i if i>vj , aj=O

i Bi= bjk

( i=l , . . ,n) (S )

i bjk=O ( i=l , . . n; j =l , . . ,m ; k=l , . . , m) (6 )

The problem here is how to design an adap­tive control system which will cause the error between the reference sequence [y* (k) ] and the plant output [y (k) ] to approach zero when the plant is a non-minimum phase system with unknown dead time . In order to solve this problem, let us introduce the new representation of the plant of the form

n n y (k+l )= .l: Aiy (k+l-i)+B . l: B '. u(k+l-i) •=1 g 1•1 i

n-1 + l: B i [ u (k+l-i )-u (k-i) ] (7 ) ial C where B and B i ( i=l , • • , n-1) are the un-g c known mxm matrixes related to the matrixes Bi ( i=l , . • , n) in Eq . (4) as follows .

Discrete Time Mul tivariable Adaptive Control 365

B . B '. + B . -B = B g 1 C1 ci-1 i B . B ' g n

(8)

where Bi ( i=l , . • , n) are mxm any constant n matrixes which are chosen

i-1 so that det [ L B ' i=1 i

z ] fO for j z j .s. l . From Eq . ( 8) , we have

B g n n -1 ( L B . ) . ( L B '. )

i= l 1 i=l 1 (9 ) and it is shown that the matrix B is non­singular from the assumption c) . g The matrix B is introduced to avoid a steady g state tracking error when the reference se­quence converge to the constant value .

CONTROL OF THE KNOWN PLANT

First , when all parameters of the plant are assumed to be known , let us design a control system for non-minimum phase plants . Define the output error as

* e (k) = y (k) -y (k) (10) * * * T where y (k) = [ y1 (k) , . . , ym(k) ] is a known

bounded mxl reference sequence . From Eqs . ( 7 ) and (10) , we obtain

* n n e (k+l) =y (k+l) - L A . y (k+l-i) -B L B '. . i=l 1 g i:1 1 n-1 u(k+l-i)- i� Bci [ u(k+l-i)-u(k-i) ]

(11) In order to ensure that the control obj ective can be achieved with a bounded control input , let us introduce an augmented error s ignal n (k) and an auxiliary signal w (k) .

n (k) = e (k)+ w(k) (12) _ n _ n-1 w(k) = L A .w (k-i)+ L B . [u (k-i )-u (k-1-i) ] i = I 1 i=1 C1

(13) Then, from Eqs . ( 11) , (12) and (13) ,

* n -n (k+l)=y (k+l)- ;�1 Ai [y (k+l-i ) - w(k+l-i) ] -B � B '. u (k+l-i) ( 14 ) g i = l 1

Next , the control input u (k) should be deter­mined so as to achieve n(k+l ) =O .

- 1 - 1 * n u (k) =B1' . B [y (k+l)- L A . {y (k+l-i) g i= 1 1 - w(k+l-i) } ] - 1 B-

1•1 . B '. u (k+l-i) ( 15) i=2 1

By using this control input , i t is ensured that the plant output [y (k) ] , and input [u (k) ] are bounded and e (k )+ 0 as

* y (k) + constant .

ADAPTIVE CONTROL OF THE UNKNOWN PLANT S ince the parameters and the dead time are unknown, the control law will be recursive­ly estimated . To estimate the parameters of° the plant , let us rewrite Eq . ( 7 ) as

(16)

where (17 )

T T T T T_ jT jT B = [b 1 , • . ,b ] , B . - [b 1 , • • , b ] (18) g g gm CJ c cm

b bj mxl vectors (J' = l , . • , n-1) gi' ci

cSTi (k) = [y . (k) , • • , yi (k+l-n) , { .1 B '. u (k+l-j ) }; 1 J=1 J

uT (k) -uT (k-l) , . . , uT (k+2-n) -uT (k+l-n) ] ( i=l , • . ,m) (19 )

The following adaptation algorithm (Landau , 1981) is used to estimate ei as ei (k) from the plant input [ u (k) ] , output [ y (k ) ] and plant representation (16) .

A A * e . (k) =e i (k-l)+F . (k-l) cS . (k-l) vi (k) ( 20 ) 1 1 1

-1 -1 T Fi (k) =Ali (k) Fi (k-l)+A2i (k) cS i (k-l) cSi (k-l)

* vi (k) = l+o: (k-l) F . (k-l) cS . (k-1) 1 1 1

(21)

(22 )

AT A l An A A l ei (k) = [ ai (k) , • • , ai (k) ,bgi (k) , bci (k) , • . ,

bn-l (k) ] ( 23) ci

��ing the estimated parameters ;I (k) , and b�i (k) , the adaptive version of the aux­iliary signal w(k) , denoted as w(k) = [wl (k) ,

T . • ,w (k) ] , will become m n A • n-1 A • wi (k+l) = .L a� (k) w . (k+l-j ) + .L bJ i (k) .

J=1 1 1 J=1 c [ u (k+l-j ) -u (k-j ) ] ( i=l , . . , m) (24 )

Next , the control input u (k) = [ u1 (k) , . • , T um(k) ] is determined in the adaptive case

considering the restriction of its amplitude (Fuj ii , 1982c) .

366 N. Mizuno and S . Fuj i i { u . (k) Cl. Risgn [uci (k) ]

l u . (k) i <R . Cl. - l.

I u . (k) I >R. Cl. l. ( i=l , . . , m) (25)

where Ri>O is the known maximum permissible value of the plant input . The signal u (k) =

T c [ u 1 (k) , . • , u (k) ] i s calculated directly c cm using es timated parameters of the plant as

,1 'T ' T 11 -u (k) =B1 . [b 1 (k) , . . ,b (k) ] . u (k) c g gm - .£ BI1B '. u (k+l-j )

T J = 2 J -;:;- (k) = [u1 (k) , . . . ,um(k) J

(26)

(27 )

u . (k) =y� (k+l) - .1 �� (k) [y . (k+l-j ) -w . (k+l-j ) ] l. l. J=I l. l. l. ( i=l , . . , m) (28)

'T 'T T where [b 1 (k) , . . , b (k) ] is assumed to be g gm non-singular for all k . Substituting Eqs . ( 25 ) - (28) into Eq . (16) and denoting the adaptive control error n . (k) as

* n . (k) =y . (k) -y . (k) +w . (k) , l. l. l. l. we obtain

l.

( 29 )

n . (k+l) = [ e . (k) -e . J To . (k) -b . (k) [ u (k) -u (k) J l. l. l. l. gl. c ( i=l , . . , m) ( 30)

In the plant representation (4) , if it is i i assumed that aj =ak for all i , j and k , then

a s imple algori thm can be adopted to es timate the parameters (Mizuno , 1983) . Using the adaptation algorithm : Eqs . ( 20) - ( 22) , and the control law : Eqs . ( 24 ) - ( 28) , the con­trol obj ective i s achieved when the refer­ence sequence [y* (k) ] is realizable by a certain plant input within the amplitude re­s triction , and e (k)+ 0 as [y* (k) ]+ cons tant .

In the des ign of the adaptive controller des­cribed in this section , the controller para­meters are directly estimated from the plant representation with special s tructure . An adaptive control system can also be designed in which the plant parameters are f irst es ti­mated by Eq . (4) and then the controller para­meters are calculated us ing Eq . ( 8) . Even in this case , it i s to be noted that the propos­ed method requires less computation than the other direct methods des igned for non-minimum phase sys tems .

In the next section , let us consider the s tability of the proposed control sys tem .

STABILITY OF CONTROL SYSTEM In the above proposed method , the control is arranged so as to cause the s ignal [y (k)­w (k) ] to track exactly the reference se­quence [y* (k) ] . Thus , if the transfer ma­trix between the plant input [u (k) ] and the s ignal [y (k) -w (k) ] has the s table in­verse as k+ oo, the boundednes s of all s ig­nals in the control sys tem i s ensured as follows . Let G ( z) denote the transfer matrix of the p plant as follows :

G ( z) = ( Izn- E A . zn-i) -1 . ( E B . zn-i) p i=l l. i=l l. = ( Izn- n n-i -1 n B ' n-i l: A . z ) . [B l: . z +

i:l l. g i= l l. n·I n-1-i ( l: B . z ) ( zI-I ) ] i:l Cl. ( 31)

And the transfer matrix G (z) between U ( z ) w and W ( z ) is given from Eq . (24) for constant es timated parameters as

n n ' n-i -1 n-1 ' n-1-i G ( z ) = ( I z - l: A . z ) . ( l: B . z ) . W i=l l. i=l Cl.

( zI-I) ( 32)

I f the es timated parameters converge to their true values , from Eqs . ( 31) and ( 32) , the transfer matrix G ' ( z) between U ( z ) and p [Y ( z ) -W ( z) ] becomes

n n n-i -1 n n-i G ' ( z ) = (Iz - l: A . z ) . (B l: B '. z ) ( 33) p i = l l. g i=l l. n i-1 In Eq . ( 33) , s ince det [ . l: B '. z ]#0 for IZI � 1

• = 1 l. and det [B ] fO , G ' (z ) has the stable inverse g p and the boundednes s of the s ignal is ensur­ed (Fuj i i , 19 8la) . And from the continuity of the G ' ( z ) for estimated parameters B . (k) , p l. there exists a region in the parameter space such that if the initial values of the e . (k) are within the region , then G ' ( z) l. p has the stable inverse for all k , and the signal is bounded . However , from the s tand­point of the control problem, it suffices that although the initial values of the e . (k) are out of this region , the estimated

l. parameters e . (k) will converge to certain l. values which assure that G ' ( z ) has the p s table inverse as k+ 00 •

STRUCTURE OF CONTROL SYSTEM In this section the relation is clarified between the proposed method and the con­ventional model reference adaptive control or adaptive pole placement method .

Di screte Time Multivar iable Adaptive Control 367

For simplicity , the matrixes Bi=I , Bi=O are used for i�2 in this section . The adaptive control system can be written as shown in Fig . 1 when the estimated parameters converge to their true values .

G ( z )• ( I zn- 1 ) -·1 ( � A lzn - l ) • i•L

G ( z )•( Izn - �A1zn- l ) '"".1 ( z l - I ) w l• L

· c n f: 1ec l

zn- 1 - l ) ,.,

Y<z>

Fig . 1 An equivalent representation of the proposed control system

In this figure , G ( z ) is the discrete trans-p fer matrix of the plant , and G� ( z ) and Gw (z ) are indicated in the figure . The transfer matrix between U ( z ) and [Y ( z ) ­W(z) J i s given by

n n n-i -1 n-1 G (z)-G ( z )= (Iz - L A . z ) . B z (34) p w � i g Regarding Eq . ( 34) , the relation between y* (z ) and Y(z) [=Y ( z )-W(z) ] becomes

Y( z ) = [ I+( Izn- 1 A . zn-i) -1 . B zn-l . B-1 . i:l i g g

(Izn-1) -1 . ( � A . zn-i) ] -1 . B zn-1 . B-l . i= l i g g * zI . Y (z )

* =Y (z) (35) Hence , the proposed method has in part the same structure as the conventional MRACS . On the other hand , the relation between Y* ( z ) and Y ( z) is given as

n n n-i -1 n n-i n -1 Y ( z ) = ( Iz - L A . z ) . ( L B . z ) . (B z ) • i:l i 1:1 i g n n n-i * (Iz - L A . z ) . zl . Y ( z )

i=l i If the following equalities hold ,

( Z B . zn-i) . ( Izn- � A . zn-i) -l i = l i i=l i

n -1 n n n-i (B z ) . ( Iz - L A . z ) = g 1=1 i n n n-i n -1 (Iz - . L A . z ) . (B z )

1=1 i g then Eq . (36) is reduced to the form

(36)

(37)

n n-i n-1 -1 * Y ( z) = ( i B . z ) . (B z ) .Y (z ) (38 ) 1=1 i g Therefore , the proposed method has the struc­ture of the adaptive pole placement method in the sense that all plant poles are can­celled and all plant zeros are preserved . Note that Eq . (37) does not always hold in the multivariable case . For example , in Eq . (17 )

if a�=aki for all i , j and k , that is , Ai= J . ail (ai : scalar) , then Eq . (38) holds . In

the scalar case , Eq . (37 ) is always satis­fied and Eq . (38) is identical to the result of Lozano and Landau (Lozano , 1982) .

SIMULATION RESULTS In order to illustrate the effectiveness of the proposed method , computer s imulations are performed for two different types of plants . The first example is the minimum phase plant without dead time in Eq . (l ) with A:1 (q1 ) =1-1 . 68q1+0 . 684q2 A:2 (q1 ) =1-l . 50q1+o . 527q2 Bll (ql ) =O . Oll+0 . 0009 7 3q1 B (q1 ) =-o . 000221-o . 000195q1 12 ]21 (q, ) =-o . oos14-o . oo658q, B22 (q1 ) =0 . 0l63+0 . 0l32q1 , d11=d12=d21=d22=1 and the reference sequence [y* (k) ] will be generated by (l- l . 72q1+0 . 756q2 ) y� (k)

= ( O . Ol82ql+o . 0166q2 ) u� (k) * (1-1 . 54qi+o . 6 70q2 ) y2 (k)

= (0 . 0694q1+0 . 0607q2 ) u; (k) * where u . are square waves of the amplitude

1 . 0 forii=l and 2 . 0 for i=2 respectively , and with a period of 100 steps . In the adaptation algorithm (20) - (22) , Ali (k) =0 . 9 5 and

8A2i (k) =l . O are used for i=l , 2 and Fi (O)

=10 I for i=l , 2 . In the plant represen­tation ( 7 ) , n=2 and the matrixes Bi=I , Bz=O , and in E q . (25) , R . =100 for i=l , 2 . The ini­tial values of th� adaptive system are set at zero except for B (O )=diag ( 0 . 0182 , 0 . 0166) . g Figure 2 shows the simulation results for the f irst example . In this figure c shows the reference s equence and + denotes the plant output .

l.O \ • r 2 6 a • �ti . - • "' . 0 0 ;

2 • • •

\ J \ "' • - I

·� ' ' ' ' ' ' ' ' 20 40 60 80 100 120 140 160 180 200

... 2.0 � • ,"- • . N ! : "' . . 0.0 � : 2 ii'

N •. ; "' -2.0 ;v . .

20 40 60 80 k 100 120 140 160 180 200

Fig . 2 S imulation results for the first example

368 N. Mizuno and S . Fuj ii

The next example is the non-minimum phase plant with dead time in Eq . (l) with X1 (q-1 ) =l-1 . 66q-1+o . 6 70cf2 A.2 (q-1 ) =l-1 . 52q-l+O . 549q2 B11 <<!1 ) =0 . 0011+0 . ooos61q1 B12 <<!1 ) =-o . oo335+0 . 0017<11 B21 (q� ) =-0 . 0115-0 . 00945q1 B22 <cf1 ) =0 . 0165+0 . 0135cf1 , d11=d12= 2 , d21=

d22=1 This plant is the discrete time form of the non-minimum phase continuous time plant . The reference sequence [y* (k) ] and the adap­tation algorithm are the same as those of the first example . Figure 3 shows the s imu­lation results for the second example .

20 40 60

20 40 60

' · · .. . .

·� ""----� 80 100 120 140 160 180 200

!'-.... __ __

� I � I

80 100 120 140 160 180 200

Fig . 3 Simulation results for the second example

CONCLUSION The discrete t ime multivariable adaptive control system presented here can be appli­cable to non-minimum phase plants with un­known dead time . The proposed method has a very simple structure and requires little computation , so the adaptive controller is easily used with a digital computer which has a l imited computing capacity like a low­cost microprocessor .

REFERENCES Goodwin , G . C . , Ramadge , P . J . , and Caines ,

P . E . (1980) . Discrete-time multivariable adaptive control . IEEE Trans . Autom. Control , AC-25 , 449-456 .

Morse , A . S . ( 1980) . Global s tability o f para­meters adaptive control . IEEE Trans . Autom . Control , AC-25 , 443-440 .

Narendra , K . S . , and Lin , Y . H . (1980 ) . Stable discrete adaptive control . IEEE Trans . Autom . Control , AC-25 , 456-461 .

Johnstone , R .M . , Shah , S .H . , and Fisher , D . J . ( 1980 ) . An extension of hyper­stable adaptive control to non-minimum phase systems . Int . J . Control , 3 1 ,

5 39-545 . Fuj ii , S ., and Mizuno , N . ( 198la) . A design

method of discrete model reference adaptive control system for non-mini­mum phase systems . Trans . So . Instrum. and Control Eng . , 17-3 , 449-451 .

Goodwin , G . C . , and Sin, K . S . (1981) . Adap­tive control of nonminimum phase sys­tems . IEEE Trans . Autom . Control , AC-26 , 478-483 . -

Astra;;:- K . J . (1980) . Direct methods for nonminimum phase systems . Proc . of 19th IEEE CDC , Albuquerque , 611-615 .

Elliott , H . (1982) . Direct adaptive pole placement with application to nonmini­mum phase systems . IEEE Trans . Autom. Control , AC-2 7 , 720-722 .

Lozano Leal , R�d Landau , I . D . ( 1982) . Quasi-direct adaptive control for non­minimum phase systems . Trans . ASME , ser . G , 104 , 311-316 .

Suzuk� and Shinnaka , S . ( 1983) . A de­sign method for adaptive pole-place­ment sys tems . Trans . So . Ins trum . and Control Eng. , 19-1 , 28-35 .

Fuj i i , S . , and Mizuno , N . (198lb) . A dis­crete model reference adaptive control using an autoregressive model with dead time of the plant . Preprints of 8th IFAC Congress , VII , 120-125 .

Fuj ii , S . , and Mizuno , N . (1982a) . Multi­variable discrete model reference adaptive control using an autoregres­sive model with dead time of the plant and its application . Trans . So . Instrum. and Control E�, 18-3 , 238-245 .

Fuj ii , S . , and Mizuno , N . (1982b ) . A design method of discrete adaptive control system for non-minimum phase plants with unknown dead time . Trans . So . Instrum. and Control Eng. , 18-12 , 1165-1172 .

Landau, I . D . , and Tomizuka , M . (1981) . Theory & practice of adaptive control sytems . Ohm Co . Ltd .

Fuj i i , S . , and Mizuno , N . (1982c) . Const­ruction of a discrete MRACS for plants with the restriction of input ampli­tude . Trans . So . Ins trum. and Control Eng . , 18-8 , 859-861 .

Mizuno , N . , and Fuj i i , S . (1983) . An improv­ed identification algorithm for linear discrete t ime multivariable systems with matrix and scalar parameters . ..§..£. Trans . Instrum . and Control Eng. , 19- 7 .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

PLENARY SESSION 3 ROUND TABLE DISCUSSION REPORTS

ROUND TABLE DISCUSSION ON ROBUSTNESS OF ADAPTIVE CONTROL

P. V. Kokotovic

Depar/111nit of Ell'ftrirnl Enginl'l'ri11g and Coordinated Scimce Laboratory, University of Illinois, 1 101 W. Springfield Avenue, Urbana, IL 61801, USA

One of the main issues of this workshop was robustness of present adaptive schemes with respect to modeling uncertainties . Two sessions and a round table discussion were devoted to this topic . In most present schemes a linear combination of adjustable parameters equals the integral of the square of the adaptation error . Hence , if this error cannot be made zero due to plant­model mismatch or some external disturbances, at least one of the adjustable parameters grows unbounded . Present research follows two related directions . First , analytical tools are being developed to characterize various types of non-robustness and 11:uarantee robustness bounds . Second , modifications of the existing algorithms are sought to prevent the parameter drift . Both d irec­tions require reformulations of the adaptation goals . The discussion at the round table centered on these reformulations . A prominent approach is to introduce a set character­ization of uncertainties and derive some robustness bounds in the form of sectoricity conditions . Another approach is to guarantee boundedness via a condition imposed on a "signal-to-modeling error" ratio . S imilar , but more specific characterization is via singular perturba­tions and weak observability . Further discussion is needed to examine similarities and differences of these approaches and their handling of the available a priori information

It appears that the goal of zeroing the error i . e . asymptotic stability is to be replaced by more realistic bounds on the residual error and parameters . It is an open question whether the theory should insist on a global character of these more modest properties or , instead , attempt to guarantee their local validity in an "operating ring . " At this workshop the adaptive control has demonstrated its validity by giving new problems whose solution will bring the

369

theory closer to the conditions of the real world .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROUND TABLE DISCUSSION ON ADAPTIVE SIGNAL PROCESSING

B. Friedlander

Syslnn1 Control Tn:ltrwlogy, Inc., 1801 Pagf M ill Road, Palo A lto, CA 94304, USA

Adapt i ve s i gnal process i ng h as many poi nts of s im i l ar i ty to adapt i ve contro l . There are , however , many important d i fferences in emphas i s lead i ng to di fferences in the al gor i thms and i n the ana lys i s . The fo l l owi ng poi nts were d i scussed at the round tab l e : 1 . In adapt i ve control we h ave measurements of both the i nput and the output of the p 1 ant . In ad apt i ve process i ng the " p l ant" i s a s i gnal model whose i nput is unknown . 2 . In contro l appl i c at i on the p l ant i s often stab l e and m i n i mum phas e . The mode l s encountered i n s i gnal process i ng h ave pol es near or on the uni t c i rc l e and are often non-m i n i mum phas e . 3 . In adapt i ve f i l ters i t i s often necessary to impose var i ous constrai nts on the f i lter parameters { l i near phase character i st i c s , equ i r i pp l e pass-band , etc . ) . These constrai nts compl i c ate the der i v at i on and analys i s of the adapt i ve al gor i thm . 4 . In s i gn al process i ng prob l em the measurements are often extremely no i sy . I n control prob l ems the measurement system i s usual ly des i gned to prov i de rel at i ve ly noi se-free i nformat i on . A number of probl ems of mutual i nterest to adapt i ve s i gn al process i ng and ad apt i ve contra 1 were brought out i n the d i scus s i on : 1 . The study of quant i z at i on effects on adapt i ve al gor i thms . Th i s i s of great prac t i c al i mport ance for hardware implementat i on i n f i xed-po i nt ar i t hmet i c . 2 . Study i ng convergence rates of adapt i ve al gor i thms . Current ly avai l ab l e resu l ts are almost al ways about the asymptot i c behav i or of the al gor i thm . 3 . Deve lopment of robust methods for on-1 i ne model order est imat i on .

37 1

There was general agreement on the benef i ts of stronger cooperat i on between the s i gnal proces s i ng and control commun i t i es wor k i ng on adapt i ve proces s i ng . I t i s not c l e ar , however , how to br i ng about such i nterac t i on .

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

ROUND TABLE DISCUSSION ON APPLICATION OF ADAPTIVE CONTROL

K. J. Astrom Department uf Automatic Control, Lund Institute of Technology, Bux 725, S - 220 07 Lund, Sweden

The d i scus s i on was wel l attended . There were parti c i pants representi ng manufacturers , users and researchers . Many i s s ues were brought up in a l i vely d i s cuss i on . The fol l owi ng is an attempt to s ummari ze the major areas .

Can adapti ve control be used i n i ndustry ?

Th i s top i c created a heated debate . There are several adapti ve regul ators on the market . Experi ences of the i r use and performance i s a l s o i ncreas i ng . Good exampl es o f super i or performance of adapt i ve control as compared to conventi onal were quoted . The wi despread use of adapti ve techn i ques i n s i gna l proces­s i ng appl i cati ons was a l s o menti oned . These practi cal emp i ri ca l facts were contrasted w i th the concerns on robustness ra i s ed by recent theoret i cal work . Th i s d i s cus s i on l ed to the next topi c .

Securi ty i ssues It was menti oned by severa l practi ti oners that securi ty is important for a l l types of regul ators . To ach i eve th i s even for constant gai n regu lators it is necessary to make many th i ngs i nto account wh i ch i s not covered by standard l i near theory . Actuator nonl i neari ­ti es l i ke sti cti on and back l ash were menti o­ned as typ i ca l examples . Many d i fferent fi xes are i ntroduced in conventi onal regu l ators to handle thi s . These are often cons idered pro­pri etory . Adapt ive control di ffers only i n the respect that there i s much l ess experi en­ce . Moni tori ng of the estimator and the regu­l ator performance was menti oned as one i mpor­tant factor .

How rel evant i s the current theoreti cal research ?

The danger of work i ng w i th too s i mpl i sti c mo­del s was menti oned . I t woul d be of i nterest to cons i der nonl i near i t i es and w i de operati ng ranges j ust to get some i ns i ght i nto what may happen even i f g l obal asymptoti c stabi l i ty can not be proven .

Educati on Adapti ve control contai ns many concepts and i deas that are novel to i ndustri a l i sts e . g .

373

m1 n 1mum vari ance contro l , pole p l acement and estimation al gori thms . It was deemed cri t i ­ca l ly important to be ab le to p resent these i deas i n such a way that potent ia l users feel comfortab l e to the extent that they can ope­rate and commi s s i on adaptive contro l l ers . There were d ivergent op i n i ons on how much of the deta i l s of the al gori thms a user need to know . Shou l d he accept a b l ack box on meri ts or shou l d he know the detai l s of i ts func­t io ns ?

Copyright © IFAC Adaptive Systems in Control and Signal Processing, San Francisco, USA 1983

Alvarez , Jaime 105 Alvarez , Joaquin 105 Alvarez-Gallegos , J . 109 A.strom, K. J . 1 3 7 , 271 , 3 7 3 Azab, A. A. 69

Ballyns , J . 75 Banyasz , cs . 81 Bar-Kana , I . 357 Barcenas-Uribe , L . 109 Bartolini , G. 119 Bastin , G . 129 Bay J¢rgensen , s. 323 Bayard , D. s . 185

75 199

3 31

Bayourni , M . M . Behbehani , F . Bonilla , M . Buchner, M . R . 115 , 301

Caines , P . E. 179 Canon , D. 135 Casalino , G . 1 1 9 Chen, H . F . 179 Costin , M. H. 115, 301 Cyr , B. 2 61

Dahhou , B . 315 Dang Van Mien , H . 269 Das , M. 43 , 265 David, R . A. 243 Davoli , F . 119 de la Sen , M . 61 Dion , J . M . 155, 347 Dj aferis , T . 265 Dochain , D . 129 Doncarl i , c . 135 Dugard, L. 155 , 351

79 Eli�abe , G . E. Elliott , H . 43 , 265 Eslami , M . 185

Farden , D. c. 229 Feng, c . -B . 289 Fisher, D. G. 83 Friedlander , B . Fuj ii, S . 363

163 , 2 1 5 , 371

AUTHOR INDEX

Goodwin , G. C . Grirnble , M . J . Gupta , M . M .

1 , 351 341

97, 263

Hagglund, T . 213 , Hahn , v . 309 Hallager, L . 323 Halme , A . 199 Hanus , R . 3 3 3 Heegard, c . 231 Hetthessy , J . 81 Hrnamed, A . 277 Honderd, G. 67 Horowitz , R. 27 Huang, H. 193

Installe , M. 129 Ioannou , P. 1 9 Irving , E . 269 Iwai , Z . 83

271

Johnson , c. R . 2 3 1

Kaufman , H. 357 Kawasaki , Y . 83 Keviczky , L . 81 Kinnaert, M. 333 Kitamori , T . 295 Kofman, w. 249 Kokotovic , P . v . 3 3 , Kosut , R . L. 1 3

261 ,

Koutchoukali , M . s . 123

Lagueri e , c . 123 Landau , I. D . Larimore , W . E . Lee , W . -K . 253 Li , H . 289

3 5 , 61 , 14 7 283

Lozano , R. Lyons , J. P .

331 , 347 2 31

37 5

369

376

M ' Saad, M. 147 , 315

Mahmood, s . 283

Matko, D . 1 27

Maun , J .-c . 333

Mehra , R. K . 283

Meira , G . R . 7 9

Minamide , N . 263

Minciardi , R . 119

Mizuno , N . 363

Moir, T. J . 3 41

Mondie , s . 1 05

Mosca , E . 207

Mote , c. D. 171

Naj im . K. 123 , 315

Nikiforuk , P. N. 97 , 263

Nouh , A. 69

Ortega , R. 35, 147

Pajunen , G. A. Praly, L . 55

91

Radouane , L. 277

Rahimi , A . 1 7 1

Redjah, M . 269

Riedle , B . 3 3 , 261

Ruiz , G . 4 3

Author Index

Schmid, Chr . 3 0 9

Selkainaho , J . 199 Shah , K. N . 49 Shah , S . L . 83

Shin, S . 295

Sibul , L . H. 221

Silvent, A . 249

Stearns , s . D . 243

Tague , J. A. 221

Tasil: , J . 127

Teoh, E . K. 1

221

103

Titlebaum, E . L. Tjahjadi , P . I . Tomizuka , M . 2 7

Tuteur, F . B . 233

Umbehauen , H . 309

van Amerongen , J . 67

van den Bosch , P. P. J .

Walach , E . Widrow, B . Wiemer, P .

7

7

309

Yamane , Y . 9 7 Yang, D . R. 253 Yashin , A. I . 99

Youlal , B . 315

Zappa , G . 207 I I

1 0 3