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AD A119440

AFWAL-TR-82-3032

DESIGN OF HIGH-PERFORMANCE TRACKING SYSTEMS

PROFESSOR B PORTER

DEPARTMENT OF AERONAUTICAL AND MECHANICAL ENGINEERING

UNIVERSITY OF SALFORD

SALFORD M5 4WT

ENGLAND

JULY 82

TECHNICAL REPORT FOR PERIOD 15 NOVEMBER 1980 - 30 SEPTEMBER 1981

APPROVED FOR PUBLIC RELEASE:DISTRIBUTION UNLIMITED

FLIGHT DYNAMICS LABORATORYAF WRIGHT AERONAUTICAL LABORATORIESAIR FORCE SYSTEMS COMMANDWRIGHT-PATTERSON AFB, OHIO 45433

0.. EUROPEAN OFFICE OF AEROSPACE RESEARCH AND DEVELOPMENT3 DEPARTMENT OF THE AIR FORCE

223-231 MARYLEBONE ROADLONDON NW1 5TH S SEP 21 982

82 09 21 056 A

NOTICE

When t.ove.nment drawings, specifications, or other data are used for any purposeother than in connection with a definitely related Government procurement operation,the United States Government thereby incurs no responsibility nor any obligationwhatsoever; and the fact that the government may have formulated, furnished, or inany way supplied the said drawings, specifications, or other data, is not to be re-garded by implication or otherwise as in any manner licensing the holder or anyother person or corporation, or conveying any rights or permission to manufactureuse,. or sell any patented invention that may in any way be related thereto.

This report has been reviewed by the Office of Public Affairs (ASD/PA) and isreleasable to the National Technical Information Service (NTIS). At NTIS, it willbe available to the general public, Including foreign nations.

This technical report has been reviewed and is approved for publication.

DUANE P. RUBURTUS, Chief EVARD H. FLN, ChiefControl Techniques Group Control Systems Development BranchControl Systm Development Branch Flight Control Division

FOR THE COMMANDER

ERNEST F. HDEREColonel, USAFChief, Flight Control Division

"If your address has .changed, if Vou wish to be removed from our mailing list, orif the addressee is no longer employed by your organization please notfyJ"J=,W-PAFB, OH 45433 to help us maintain a current mailing list*.

Copies of this report should not be returned unless return Is required bv securityconsiderations, contractual obligations, or notice on a specific document.

!m

REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM

1. Report Number 2. Govt Accession No. 3. Recipient's Catalqg Number

4. Title (and Subtitle) 5. Type of Report & Period CoveredScientific Report

DESIGN OF HIGH-PERFOPMANCE TRACKING 15 November 1980 -SYSTEMS 30 September 1981.

6. Performing Org. Report NumberUSAE/DC/120/81

7. Author(s) 8. Contract or Grant NumberProfessor Brian Porter F49620-81-C-0026

9. Performing Organization Name and Address 10. Program Element, Project, TaskDepartment of Aeronautical & Mechanical Area & Work Unit Numbers

EngineeringUniversity of Salford 62201FSalford M5 4WT, England 2304/02

11. Controlling Office Name and Address 12. Report DateAFWAL/FIGL 1 July 1982Wright-Patterson AFBOhio 45433, USA 13. Number of Pages

_ 16714. Nlonitoring Agehcy Name and Address 15.

EOARD/LNVDepartment of t Air Force223/231 Old Marylebone Road, London NW1 5

16. & 17. Distribution Statement

Approved for public release; distribution unlimited.

18. Supplementary Notes

19. Key WordsMultivariable control systemsp tracking systems; high-gain controllers;fast-sampling controllers; singular perturbation methods.

20. Abstract-7p Conceptually and computationally simple methods for the design of high-

performance tracking systems are described. These methods are equallyapplicable to the design of both analogue and digital controllers, andare based upon the exploitation of a solitary system-theoretic resultfrom the singular perturbation analysis of transfer function matrices.Illustrative examples involving the design of tracking systemsincorporating both high-qain analogue controllers and fast-sampling digitalcontrollers are presented.

FORK 14739. -. ,,

• m U m nl i Ul In ,.. [ ,1[--

PREFACE

This research was sponsored by the Air Force Flight Dynamics

Laboratory, (AFWAL/FIGL), Wright-Patterson AFB, United States

Air Force, under Contract F49620-81-C-0026 and was performed

in the Engineering Dynamics and Control Laboratory at the

University df Salford. In this work, the significant con-

ceptual contributions of Dr A Bradshaw and the computational

assistance of Dr A Hemami are gratefully acknowledged. This

report was expertly typed and collated by Mrs D Millward.

NT I "e-D7'_' ':' ;I

7-a4_

IU L . ..

Ii.

ATABLE OF CONTENTS

Page

CHAPTER I : INTRODUCTION . . . " . ...... 1

1,1 : INTRODUCTION ... 2... ........... 2

1.2 : REGULAR AND IRREGULAR PLANTS ...... 2

CHAPTER 2 : DESIGN OF TRACKING SYSTEMS INCORPORATINGHIGH GAIN ERROR ACTUATED CONTROLLERS ... 5

2.1 : INTRODUCTION ... ... ... ... .. .. 6

2.2 : ANALYSI.S .. s .............. 8

2.3 : SYNTHESIS ... ... ... .. . . . .. 11

2.4 : ILLUSTRATIVE EXAMPLE .... ... ... 12

2.5 : CONCLUSION ... ... ... ... .. . .. 14

CHAPTER 3 DESIGN OF TRACKING SYSTEMS INCORPORATINGFAST-SAMPLING ERROR-ACTUATED CONTROLLERS 33

3.1 : INTRODUCTION 35

3.2 : ANALYSIS .................. . .. 37

3.3 SYNTHESIS 39

3,4 : ILLUSTRATIVE EXAMPLE .. ... ... .. 40

3.5 : CONCLUSION .* ... ... ... ... ... 43

CHAPTER 4 : DESIGN OF TRACKING SYSTEMS INCORPORAT idGINNER-LOOP COMPENSATORS AND HIGH-GAINERROR-ACTUATED CONTROLLERS .. .. .... 62

4.1 : INTRODUCTION --- ** o ..... .... .. 63

4.2 : ANALYSIS 0. 00. ... ... *.. ... 66

4.3 : SYNTHESIS . 0 * ..... ...... 68

4.4 : ILLUSTRATIVE EXAMPLE .... .... .. 70

4.5 : CONCLUSION . .... ... ... .. 73

!, &A

it ~-7

Page

CHAPTER 5 : DESIGN OF TRACKING SYSTEMS INCORPORATINGINNER-LOOP COMPENSATORS AND FAST-SAMPLINGERROR-ACTUATED CONTROLLERS ... ... ... 111

5.1 : INTRODUCTION ... ... ... ... ... ... 112

5.2 : ANALYSIS ... . ... *..... ... 116

563 : SYNTHESIS ... ... ...... ... ... 118

5,4 : ILLUSTRATIVE EXAMPLE .... . . .. ... 120

5.5 : CONCLUSION ... *** . . .... ... 123

CHAPTER 6 : CONCLUSIONS AND RECOMMENDATIONS ... ... 161

6.1 : CONCLUSIONS ... ... ... ... ... ... 162

6.2 : RECOMMENDATIONS ..... .. ... ... 162

REFERENCES e.. ... ... ... ... ... ... 64

_,_....."_._i__,____ - - - . _ _- _ __ _ __._, .

ABBR E V I AT I ON S AND SY MB 0 L S

A plant matrix

B input matrix

C output matrix

F measurement matrix

K0 K1 controller matrices

M transducer matrix

T sampling periodcAk-l k-th Markov parameter

G(A) transfer function matrixr(A) asymptotic transfer function matrix

e error vector

f sampling frequency

g gain parameter

u control input vectorv command input vector

w measurement vector

x state vector

y output vectorz integral of error vector

C set of complex numbers

9set of real numbers

,pxq set of real pxq matricesT contlnuous-time set O,+-)TT discrete-time set (O,T,2T,...}

z t set of transmision zeros

Zs set of 'slow' modes

z fset of 'fast' modesv11

CHA PTE R

INTRODUCTION

1.1 INTRODUCTION

In spite of the extensive effort which has been expended

during the past twenty years, most currently available

techniques (Athans and Falb 1966, Rosenbrock 1974, MacFarlane

1980, Wonham 1974, Wolovich 1974, Davison and Ferguson 1981)

for the design of tracking systems incorporating linear

multivariable plants are both conceptually and computationally

rather complicated. Furthermore, such techniques are almost

exclusively concerned with the design of analogue controllers

and are therefore inapplicable to the much more important

practical task of designing digital controllers.

The design techniques described in this report are both

conceptually and computationally simple. In addition, these

techniques are equally applicable to the design of both

analogue and digital controllers. This universality and

simplicity derives from the fact that these techniques are

based upon the systematic exploitation of a solitary system-

theoretic result from the theory of singular perturbations

(Porter and Shenton 1975). This system-theoretic result

exhibits the distinctive asymptotic structure of the transfer

function matrices of linear multivariable systems with 'slow'

and 'fast' modes in a manner which is directly applicable to

the design of tracking systems incorporating either high-

gain analogue controllers or fast-sampling digital controllers.

162 REGULAR AND IRREGULAR PLANTS

In the context of the design of high-performance tracking

2

systems, it is convenient to classify linear multivariable

plants as follows:

(i) Regular plants: first Markov parameter of maximal

rank, minimum phase;

(ii) Irregular plants: first Markov parameter of non-

maximal rank, non-minimum phase.

The design of tracking systems incorporating regular plants

is considered in Chapters 2 and 3, whilst the design of

tracking systems incorporating irregular plants is considered

in Chapters 4 and 5. In the case of regular plants, arbitrarily

fast and non-interacting tracking behaviour is achievable by

appropriate selection of gain parameters or sampling periods

simply by implementing error-actuated analogue or digital

proportional-plus-integral controllers. However, in the case

of irregular plants, good tracking behaviour is achievable

only by the simultaneous implementation of inner-loop com-

pensators to provide appropriate extra measurements for

control purposes.

This result is crucially important since it makes explicit

the intuitively obvious fact that controller and transducer

designs are inseparable, and confirms the fact that the

irrelevance of much of 'modern' control theory to practical

engineering design derives from its failure to consider in-

tegrated controller/transducer system. The importance of

the difference between regular and irregular plants obviously

requires that the set of transmission zeros of linear multi-

variable plants (Porter and D'Azzo 1977) be readily computable

3

and very powerful software for this purpose has accordingly

been developed (Porter 1979).

I

. . . ... . i - i _ i _ - _ - _ . , , 4,.

- -

CHAPTER 2

DESIGN OF TRACKING SYSTEMS INCORPORATING

HIGH-GAIN ERROR-ACTUATED CONTROLLERS

4 -.

. 5

2.1 INTRODUCTION

In this chaptersingular perturbation methods are used

to exhibit the asymptotic structure of the transfer function

matrices of continuous-time tracking systems incorporating

linear multivariable plants which are amenable to high-gain

error-actuated analogue control (Porter and Bradshaw 1979a).

Such tracking systems consist of a linear multivariable plant

governed on the continuous-time set T = [O,+-) by state and

output equations of the respective forms (Porter and

Bradshaw 1979b)

x1 (t) F 11 , A121 Fx(t) FoF 1 = , I + u(t) (2.1)

L 2 (t) [A2 1 , A2 2J Lx2 (t)- 2

and

y(t) [c1 , c2 tIJ ( (2.2)

together with a high-gain error-actuated analogue controller

governed on T by a control-law equation of the form

u(t) = g{K 0e(t)+Klz(t)} (2.3)

which is required to generate the control input vector u(t)

so as to cause the output vector y(t) to track any constant

command input vector v(t) on T in the sense that the error

vector e(t) - v(t) - y(t) assumes the steady-state value

lim e(t) - lim{v(t)-y(t)} - 0 (2.4)

6

for arbitrary initial conditions. In equations (2.1),(2.2) ,(2.3),and (2.4) ,x (t) E: e-, x (t) E: RL, u (t) R' , y (t) E RL,

AlEl R(n-L)x(nL)# A1 E R(n-)xL A RLx(n- L) , A2 2 ERxk,B2 ER xL, CIERLx(n-L), c 2 ER'xL, rank C2B2 -L, e(t)IR t ,

v(t) E: RL, z(t) = z(o) + Lt edt ER1, KoC_X , K1 CRxL,

and g E R.

It is evident from equations (2.1),(2.2),(2.3) that

such continuous-time tracking systems are governed on T by

state and output equations of the respective forms

1 (t)- 1 0 , -C 1 C2z (t)1

c1 (t) =(0 A A11 A 12 x 1 (t)

x2 (t)j [B 2K1 , A2 1 -gB 2 KoC1 , A22-gB2KoC_ Lx2 (t)j

+ 0O v(t) (2.5)

and

y (t) = [O , C 1 , C2]X 1(t) .(2.6)

Lxc2 (t)J

The transfer function matrix relating the plant output vector

to the command input vector of the closed-loop continuous-

time tracking system governed by equations (2.5) and (2.6) is

clearly

71.

Xit #, C 2

G(,X)m[OC 1 1 C 21 0 , X1 n...LA 11 -A 120

L-gB 2K1, A2 1 +gB2 KoC1 , XI -A 2 2+gB2 KoC2.] LgB2KoJ

(2 7)

and the high-gain tracking characteristics of this sytem can

accordingly be elucidated by invoking the results of Porter

and Shenton (1975) from the singular perturbation analysis

of transfer function matrices.

These results yield the asymptotic form of G(X) as the

gain parameter g - - and thus greatly facilitate the deter-

mination of controller matrices K and K1 such that the

continuous-time tracking behaviour of the closed-loop system

becomes increasingly 'tight' and non-interacting as g is

increased. The frequency-response and step-response

characteristics of a continuous-time tracking system in-

corporating an open-loop unstable chemical reactor (MacFarlane

and Kouvaritakis 1977) are presented in order to illustrate

these general results.

2.2 ANALYSIS

It is evident from equation (2.7) thatby regarding

e - 1/g as the perturbation parameter, the asymptotic form

of the transfer function matrix G(A) of the continuous-time

tracking system as g - - can be determined by invoking the

results of Porter and Shenton (1975) from the singular

perturbation analysis of transfer function matrices. Indeed,

these results indicate that as g e the transfer function

matrix G(W) assumes the asymptotic form

r(A) = r(x) + r () (2.8)

where

r(X1 , Co(x)In-A o ) - I B ° (2.9)

r() - C2 (II gA4 ) gB2K° (2.10)

-K (2.11)0 = LAl 2 C2 KoI , Al _A1 2 cClc (2.11)

B o (2.12)12C2

C- [K- 1 K , o (2.13)

and

A4 - -B 2 KoC2 ( (2.14)

It follows from equations (2 .8),(2.9)and(2.11) thatthe 'slow'

modes Z. of the tracking system correspond as g - e to the

poles Z1U Z 2 of r(X)lwhere

z- 0 Cc'IxKo+K1 Imo} (2.15)

and

2 {1 1C:IXInt -A1 i+A 2 C2Cl ' 0} , (2.16)

and from equations(2.8) ,(2.10) ,and(2.14)that the 'fast'modes Zf of the tracking system correspond as g . - to I

p9

the poles Z3 of r(x) where

z ()iC:1AI +CB 2K~0 (2.17)z3 - I gC, c2Kol-01

Furthermore, it follows from equations (2.9),(2.11),(2.12),

and (2.13)that the 'slow' transfer function matrix

r( ) - 0 (2.18)

and from equation (2.10) and(2.14) that the ' fast' transfer

function matrix

-1.r)= (It+gC2 B2ro)' gC2B2Xo . (2.19)

Hence, in view of equations(2.18) and (2.19), it is

evident from equation (2.8) that as g - the transfer

function matrix G(M) of the continuous-time tracking system

assumes the asymptotic form

rF() - (xIt+gC2B2K0)- gC2B2Ko (2.20)

in consonance with the fact that only the 'fast' modes

corresponding to the poles Z3 remain both controllable and

observable as g - -. Indeed, the 'slow' transfer function

matrix i(A) vanishes precisely because the 'slow' modes

corresponding to the poles Z1 become asymptotically un-

controllable as g . - in view of the block structure of

the matrices Ao and B. in equations (2.11) and (2.12) whilst the

'slow' modes corresponding to the poles Z2 become asymptotically

unobservable as g - in view of the block structure of the

matrices Ao and CO in equations (2.11) and (2.13)

10

3. SYNTHESIS

It is evident from equations (2.5) and (2.6) that tracking

will occur in the sense of equation (4) provided only that

Zs U Zf CC (2.21)

where C" is the open left half-plane. In view of equations

(2.15),(2.16),and(2.17),the 'slow' and 'fast' modes will

satisfy the tracking requirement (2.21) for sufficiently

large gains if the controller matrices Ko and K1 are chosen

such that both ZICC- and Z3 C C in the case of minimum-

phase plants for which (Porter and D'Azzo 1977) the set of

transmission zeros

z- A c:kz n 9-A1+A2C21Cl1.0} CC- (2.22)

since it is then immediately obvious from equation (2.16)

that Z2 C C. Moreover, in such cases, tracking will become

increasingly 'tight' as g . in view of equation (2.20).

Furthermore, if K0 is chosen such that

C2B2Ko M diag{ala 2,...,O} (2.23)

where a ER + (j-1,2,...,L), it follows from equation (2.20)

that the transfer function matrix G(X) of the continuous-

time tracking system will assume the diagonal asymptotic

form

go gV2 gor, diag , ... , (2.24)

119o2

and therefore that increasingly non-interacting tracking

behaviour will occur as g . e.

2.4 ILLUSTRATIVE EXAMPLE

These general results can be conveniently illustrated

by designing a high-gain error-actuated analogue controller

for an open-loop unstable chemical reactor governed on T

by the respective state and output equations (MacFarlane

and Kouvaritakis 1977)

" 1(t) . 1.38 , -0.2077 , 6.715 ,-5.676' X1 (t),

;2(t) -0.5814 , -4.29 , 0 , 0.675 ix2()

3(t) 1.067 , 4.273 , -6.654 , j 3(t)Lc4 (t) J I .o.048 4.273 ,1.343,-2.104J JLx4( t ) -

5.679 , 0 ul(t)(+ ut)(2.25)11.136 , -3.146 [u2 (t)JL1.136 , 0

and

rx1 (t)ly 2(tlJ L, 1 , o , oJx3(t)J 1.6

t -"64 ( t)j

from which it can be readily verified that Zt - (-1.192,

-5.0391 C C" and that the first Narkov pareter

12

C 2B2 (2o.27)22 - .5.679 ..0 •(2.2

is of full rank. In case {q ,a2 } - {1,i} and K - 2Ko , itfollows from equations 2.3),t2.23)and(2.27)that the correspond-

ing high-gain error-actuated analogue controller is governed

on by the control-law equation

,ct 0 060. 76 1 e F (t) 0 .3522 iZ lt)

Lu2 (t .3179, 0 1Le2 (t)] +[0.6358, .o L 2'(-t)

.... (2.28)

and if is evident frdm eiquationis (2.17)1,(" 2.i)81, and f2.19)"that

z - {-2,-21},Z" {l.192-4.1039}, 'and Z- i {-g,-g). : It

is algo evident from: equation "(1. 2 4) that the asyptotic

transfer fuiction matrix assufis the'diagonal form

24g) 0 (2.29)r m.and therefore that the closed-loop continuous-time tracking

-"" ,l • . . :

sytmincorporating the chemical reactor will exhibitincreasingly tight' and non-interacting tracking behaviour

as g "*"when the control input vector Eul ( t ) U z(t)]T is

generated by the high-gain analogue controller governed on

T by equation (2.28).

The actual frequency-response loci G(iw) for w E(--,+ml are

shn in Figs 2.1, 2.2, and2.3 when g - 25, 50, and 100,

respectively and it is clear that the actual frequency-response

13

loci approach the asymptotic frequency-response loci r(iw)

as the gain parameter g is increased. The corresponding

step-response characteristics are shown in Fig 2.4 and Fig 2.5

when CVl(t) , v2 (t)JT - [I , T and Cv 1(t) , v2 (t)]T -

[i , 1] T, respectively, and it is evident that increasingly

'tight' and non-interacting tracking occurs as the gain

parameter g is increased.

2.5 CONCLUSION

Singular perturbation methods have been used to exhibit

the asymptotic structure of the transfer function matrices

of continuous-time tracking systems incorporating linear

multivaxiable plants which are amenable to high-gain error-

actuated analogue control. It has been shown that these

results greatly facilitate the determination of controller

matrices which ensure that the closed-loop behaviour of such

continuous-time tracking systems becomes increasingly 'tight'

and non-interacting as the gain parameter g is increased.

These general results have been illustrated by the presenta-

tion of the frequency-response and step-response character-

istics of a continuous-time tracking system incorporating

an open-loop unstable chemical reactor.

14

1.0

0.5

0.0

-0.5

-1.0 ~..-0.5 6.0 0.5 1. 1.5

Fig 2.1 (a): G 1 1 iw:q -25

V 15

1.0

0.5

-0.5

Fig 2.1I(b)~ G 12 w); 25

16

. ... . . .. . . . . . .

0.5

0.0

I

-0.5

-1.0 I I

-0.5 0.0 0.5 ie . 1.5Fig 2.1(c): G2 .(iw)., g , 25.

17 -

.. .. .. . . . . . .. . . . . . . . .

1.0

0.5

0.0

-0.5

-1.0 . ., . . . . .-0.5 0.0 0.5 1.8 1.5

Fig 2.1(d)-. G 22 (iw); g -25

0.5e

-0.0-

0.0.5.0 1.5Fig 2 .2(a): Gj 1 j) g' 50

19

1 . . . . . . . . .

Fig 2.2(b): G 1 2 (iw); g =50

20

1 .0 I I ..

1 .0 . . . . . ... . . .. .., .... .. .. . . .. . . .

0.5

-0.5 0.01 6.5 1.a 1.5Fig 2 .2 (c): G 2 1() g1 50

21

1.0

o.5

0.0

-8.5

-6.5 6.6 6.5 1.8 1.5-Fig 2.2 (d): G 22 (±w); g So5

22

0.5

0.0

-0.5-

-0.5 0.0 8.5 1.0 1.5rig .3 (a): G 1 (i-); g;m - lo

23

1.0

0.5

0.0

-0.5 -

- 1 .0 .. . I . . .

-0.,5 08. 0.5 1.0 1.5Fig 2.3(b): G12 (iw); g - 100

24

0.5

0.0

-0.5

-- 1.8 ' * . . I t | ,I | -

--. 5 8.8 0.5 1.0 1.5Fig 2 .3(a): G2 1 (iw); g - loo

25

1 .0

0.5

0.0

-0.5

-1'.3 * . .. .... S . . * I . , a

-0.5 a.0 0.5 1.0 1.5Fig 2.3(d): G2 2 (iw); g " 100

26

.,: l .. , 11 - r _ .. ..=, , . .m

* I * I * I * I * aN

4J

4.' 4J

'.4

b)

U,

'.4

II

0*'

- S

a'.4

*1*4r34

U,a

S

N CD6 0 6 0 S a a

27

ON

C~CD

co ODco(!

28,

C44

ipi

00

tp

C)

SDoINQN

29.

,)

ii

>1*

4

c*4>1u

CD

30*

Ln

co

al

-4

U*)

31

- -- * I * I * I * I * *~~~~

44)

1

G

Orl

C% GO CO (N CD C

32

CHAPTER 3

DESIGN OF TRACKING SYSTEMS INCORPORATING

FAST-SAMPLING ERROR-ACTUATED CONTROLLERS

33

t.

3.1 INTRODUCTION

In this chapter,singular perturbation methods are used


matrices of discrete-time tracking systems incorporating

linear multivariable plants which are amenable to fast-

sampling error-actuated control (Bradshaw and Porter 198Oa).

Such tracking systems consist of a linear multivariable

plant governed on the continuous-time set T = CO,+Ga) by

state and output equations of the respective forms (Bradshaw

and Porter 1980b)

= + u(t) (3.1)

Lx2 tJ [A2 1 , A2 2 j x 2 (t)

and

y x ) = C (3.2)y~t) = [ I , 2 x(t)]

together with a fast-sampling error-actuated digital con-

troller governed on the discrete-time set TT = {O,T,2T,...}

by a control-law equation of the form

u(kT) = f{K 0e(kT)+K 1 z(kT)} (3.3)

which is required to generate the piecewise-constant control

input vector u(t) - u(kT), t F[kT,(k+l)T), kT ETT, so as

to cause the output vector y(t) to track any constant command

input vector v(t) on TT in the sense that the error vector

e(t) = v(t) - y(t) assumes the steady-state value

34

lim e(kT) - lim{v(kT)-y(kT)} - 0 (3.4)k-0 k-w

for arbitrary initial conditions where f - i/T and T is

the sampling period. In equations (3.1) , (3.2) , (3.3) , and (3.4),Xl(t) E R n-1, x(t)€ R, Lu(t) E R' , y(t) ( _R' , A '7l-R (n-U)x (n-£)

Al12 R(n-Z)xL A2 1 'Rx(n-1) , A22 , B2 FRlx,

Cl Rtxln-L), C CRLx , rank C2 B2 9 ., e(t) C R , v(t) ER' ,j-k-1 j-2X1

z(kT) - z(o) +T E e(jT) E Rt , Ko0 CRx KIR and

f ER + .

It is evident from equations( 3 .1),( 3 .2 ), and (3.3) that

such discrete-time tracking systems are governed on TT by

state and output equations of the respective forms

z {(k+l) T})

22{ (k+l) T}_

[I? -TC1 -TC2 z(kT)1

[fT T2 2 l-f2KoC I, '22- f' 2 KoC 2 J x2 (kT)J

TI

fy 1 K v(kT) (35)

and

z(kT)1

y(kT) - , C1 , C21 xI(kT) (3.6)

35

where

0 11 01" exp 1 , 37ad 21 ' 02 2J LA2 1 ' A2 2jf

and

"sj[A ,A t [B2 dt .(3.8)

2A21 ' 22J

The transfer function matrix relating the plant output

vector to the command input vector of the closed-loop

discrete-time tracking system governed by equations (3.5)

and (3.6) is clearly

G(A)

X I-I TC 1 TC 2 JlITI I= [0'CI' C21 I -flKl" In-k -0ll+f~iKoCl '- 12 +fT 1Ko0C2 fT 1f Ko0

-f f2K 1', -'21 +f'2KoCl ,XAI - 2 2 +fV2 K0 C2 Lf 2K

(3.9)

and the fast-sampling tracking characteristics of this

system can accordingly be elucidated by invoking the results

of Porter and Shenton (1975) from the singular perturbation

analysis of transfer function matrices.

These results yield the asymptotic form of G(X) as the

sampling frequency f - and thus greatly facilitate the

determination of controller matrices K0 and K1 such that the

discrete-time tracking behaviour of the closed-loop system

becomes increasingly 'tight' and non-interacting as f is

36 _________

increased. The frequency-response and step-response

characteristics of a discrete-time tracking system incorporat-

ing an open-loop unstable chemical reactor (MacFarlane and

Kouvaritakis 1977) are presented in order to illustrate these

general results.

3,2 ANALYSIS

It is evident from equation (3.9) that, by regarding

e - 1/f as the perturbation parameter, the asymptotic form

of the transfer function matrix GMA) of the discrete-time

tracking system as f - - can be determined by invoking the



since it follows from equations (3.7) and (3.8) that

lim f l1In-£' 1 12 [ " 11 , A1 21 (3.10)

L 21 ' 22-I1j LA2 1 , A22 J

and

limf - (3.11)

these results indicate that as f - the transfer function

matrix G(A) assumes the asymptotic form

r(A) - r( ) + r( ) (3.12)

where

r(m) - C0 (An-I n-TAo)-I'm (3.13)

37

[A -o-0ol

o l 1 (3.15)

B - , (3.16)2 2

Co - 1KJo 1 , o] (3.17)

and

A = -B KoC . (3.18)4 20o2

It follows from equations (3.12),(3.13), and (3.15) that the

'slow' modes Z of the tracking system correspond as f *

to the poles Z1IU Z2 of r(x) where

z - {X CC: XI -I +TKoKIf=O} (3.19)

and

z2 - (X CC:II n_-ZIn_1-TA 1 1 +TA1 2 C 21 -C1 120 (3.20)

and from equations (3.12),(3.14), and (3.18) that the 'fast'

modes Z f of the tracking system correspond as f . - to the

poles Z3 of r(m) where

z - {(A CC:IAI-+B 2 K2Xo'0 . (3.21)

Furthermore, it follows from equations (3.13), (3.15) ,(3.16),

and(3.17) that the 'slow' transfer function matrix

)- 0 (3.22)

38

and from equations (3.14) and (3.18) that the 'fast' transfer

function matrix

( - (xi-I +C2B2,o) - 2 2Ko . (3.23)

Hence, in view of equations (3.22) and (3.23) , itis evident

from equation(3.12) thatas f - o.the transfer function

matrix GMd of the discrete-time tracking system assumes

the asymptotic form

r ( 1i -L ' BIl-- C B2BK (3-24)

in consonance with the fact that only the 'fast' modes

corresponding to the poles Z remain both controllable and

observable as f - m. Indeed, the 'slow' transfer function

matrix r(d vanishes precisely because the 'slow' modes

corresponding to the poles Z1 become asymptotically un-

controllable as f - - in view of the block structure of the

matrices A and B in equations (3.15) and (3.16) whilst the

'slow' modes corresponding to the poles Z2 become asymptotically

unobservable as f - in view of the block structure of the

matrices Ao and Co in equations (3.15) and (3..7).

33 SYNTHESIS

It is evident from equations (3.5) and (3.6) that tracking

will occur in the sense of equation (3.4) provided only that

Zs U Zf CP (3.25)

where 0- is the open unit disc. In viewof equations (3.19)

39

(3.20),and(3.21),the 'slow' and 'fast' modes will satisfy

the tracking requirement (3.25) for sufficiently fast sampling

frequencies if the controller matrices K° and K1 are chosen

such that both Z1C V for sufficiently small sampling

periods and Z3 C - in the case of minimum-phase plants for

which (Porter and D'Azzo 1977) the set of transmission zeros

Zt {X eC:Ixin -A +A 1 C (3.26)

where C is the open left half-plane since it is then

immediately obvious from equation (3.20)that Z2 C V- for

sufficiently small sampling periods. Moreover, in such cases,

tracking will become increasingly 'tight' as f - in view

of equation (3.24). Fufthermore, if K is chosen such that0

C2B2Ko - dUial{ala 2 ,...,at1 (3.27)

where 1-o LRflV (j-1,2,...,L), it follows from equation

(3.24)that the transfer function matrix G(X) of the discrete-time

tracking system will assume the diagonal asymptotic form

M) - dag a la 2 a ... (3.28)

and therefore that increasingly non-interacting discrete-

time tracking behaviour will occur as f . -.

3.4 ILLUSTRATIVE EXAMPLE


by designing a fast-sampling error-actuated digital controller

for an open-loop unstable chemical reactor governed on T

40

-- . .. --. . . • • : --

by the respective state and output equations (MacFarlane

and Kouvaritakis 1977)

" It)- 1.38 , -0.2077 , 6.715 , -5.676 "xl(t)1x 2 (t) m -0.5814 , -4.29 , 0 0 0.675 x 2 (t)

X3(t) ( 1.067 , 4.273 r -6.654 , 5.893 x3 (t)

4 (t). 0.048 , 4.273 , 1.343 , -2.104j Lx 4 (t),

0 015.679 , 0 Ul(t)+ ult](3.29)1.136 , -3.146 Lu2 (t)J

1.136 , 0

and

ol o 1 1 (It)1

y.(t) - 0 [ 1 x2 (t) (3.30), , 0, olx3(t)j

LX4 (t) J!!

from which it can bo readily verified that Zt - {-1.192,

-5.039} C C and that the first Markov parameter

0 -3.146"

C2B2 .679 : (3.31)

In case {l, 2 } - {1,l},and 11 - 2Ko , it follows from

equations (3.3) ,(3.27),and(2.38)that the corresponding fast-

saupling-error-actuated digital controller is governed on

TT by the control-law equation

41

M 1(T)1 [ 0 ,0.17611re1 (kT)1 1 ,o32i (kT)1mf +

U2 (kT)j 1C-O.3179, 0 e 2 (kT)" .0.6358, 0 Lz2 (kT)JJ

(3.32)

and it is evident from equations '(3.19),(3.20), and (3.21) that

Z1 {1-2T,1-2T}, Z2 - {1-1.192T,1-5.039T}, and Z = {0,O}.

It is also evident from equation (3.24) that the asymptotic

transfer function matrix assumes the diagonal form

rm - (3.33)

and therefore that the closed-loop discrete-time tracking

system incorporating the chemical reactor will exhibit

increasingly 'tight' and non-interacting tracking behaviour

as f * when the piecewise-constant control input vector

ul(t) , u2(t) ] T - ul(kT) , u2 (kT)]T, tC~kT , (k+l)T),

kT E TT, is generated by the fast-sampling digital con-

troller governed on TTby equation (3,32).

The actual frequency-response loci G(eiwT) for wT [0 , 2w]

are shown in Figs 3.1, 3.2, and3.3whenT-0.04, 0.02,and0.0l,

respectively, and it is clear that the actual frequency-

response loci approach the asymptotic frequency-response

loci r(eiWT) as the sampling frequency f is increased. The

corresponding step-response -charateriatics are shown in

Fig3.4 and Fig 3. 5whenl(t) , v 2 (t)] T _ [1 , O]T and [vl(t)

v2 (t) ] _[O , 1] T , respectively, and it is evident that in-

creasingly 'tight' and non-interacting tracking occurs as the

sampling frequency f is increased.

42

I t I I t u -- t t n I-.--

3,5 CONCLUSION



of discrete-time tracking systems incorporating linear multi-

variable plants which are amenable to fast-sampling error-

actuated digital ctzol. It has been -shown that these

results greatly facilitate the determination of controller

matrices which ensure that the closed-loop behaviour of

such discrete-time tracking system become increasingly

'tight' and non-interacting ai the sampling frequency f is

increased. These general results have been illustrated by

the presentation of the frequency-response and step-response

characteristics of a discrete-time tracking system incorporat-

ing an open-loop unstable chemical reactor.

41 •.

1 .5

6.5

-6.5

-1.0

-1 .5-1.5 -1.0 -6.5 6.6 6.5 1.0 1.5

Fig 3.1(a): G11 08 );T T - 0.04

44

1 .5

0.5

0.0

-0.5

-1.5 - 18 -0'.5 6.6 -0.5 1 .0 1.5

Fig 3.1(b): G 1 2 (e );T T'- 0.04

45I

1.

6.5

6.0

-6.5

-15 -1.0 -0.5 0.0 0.5 1.6 1.5

Fig 3.1(c):G1( w T - 0.04

46

1.5 * , , ,

1.0

0.5

0.0

-0.5

-1.0 L

-1 .5 * , I , I , I

-1 .5 -1.0 -0.5 0.0 0.5 1.0 1.5Fig 3.1(d): G2 2 (ewT; T -0.04

47

II

1.0

0.5

0.0

-0.5

-1.5 -1.0 -0 .5 0.0 0.5 1.0 1.5

Fig 3.2 (a):G(ew T =0.02

48

1.5 .

1.0-

0.5

0.0

-0.5

-1 .0

-1 5 I I

-1.5 -1.0 -0.5 0.6 0.5 1.0 1.5Fig 3.2(b): G12(iT); T - 0.02

49

1.5

0.5

0.0

-0.5

-1.5

-15 -1.0 -0.5 0.0 0.5 1.0 1.5Fig 3. 2 (c): G2 1 (e i UT T - 0.02

50

1.5'

1.0-

0.5

0.0

-0.5

-1.0

-1 .5

-1.5 -1.0 -6.5 0.0 0.5 1.0 1.5

Fig 3.2(d): G22 T - 0.02

51

1.5

10

-0.5

-1.0

-1.5 -1.0 -0.5 0.10 0.5 1.0 1.5Fig 3.3(a): G 1 1 (.±wT) T - 0.01

52

1.5| I i I *

1.0

0.5

0.0

-0.5

-1.0

-1 .5 ....-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 3.3(b): G1 2 (e ±wT T - 0.01

53

LL

1.5'

1.0

0.5

0.0

-0.5

-1.8

- 1 .5 ,AI I .

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 3.3(c): G2 1 (e~" 'r ); T - 0.01

54

1.0

0.5

0.0

-0.5

-1.0

-1.5 -1.6 -6.5 0.0 0.5 1.0 1.5Fig 3.3 (d): G 2 2 (eiWT); T - 0.01

,1 _I ll .- * i |1 I I J _

4-)

Lo

56I(~,4

0

I"

56

___ ___ __ - -~ _-~~ -- ~~-.-

E-4

U,I

0 .44 D

-4z

00

I|

In

57

-- I -II

CN

'-44

N*>1S

00 coN (24

580

44)

44

C1

0

E.4

r3

a; .D CD CD

59

- 1

0

6EA

N I --

60 • '

Lfl')

,,,--S

S

C~4 CO (S S S SS S S

9-E-,-

60 .

CHAPTER 4

DESIGN OF TRACKING SYSTEMS

INCORPORATING INNER-LOOP COMPENSATORS

AND HIGH-GAIN ERROR-ACTUATED CONTROLLERS

62

4.1 INTRODUCTION

In this chapter, singular perturbation methods are used


matrices of continuous-time tracking systems incorporating

linear multivariable plants which are amenable to high-gain

error-actuated analogue control (Porter and Bradshaw 1979)

only if extra plant output measurements are generated by

the introduction of appropriate transducers and processed

by inner-loop compensators. Such tracking systems consist

o0 a linear multivariable plant governed on the continuous-

time set T = [o,+-) by state, output, and measurement equations

of the respective forms

= [+u W A2 2 f 2] u (t) (4.1)

y(t) - c, C2 (4.2)

and

w(t) - Fl , F [] ( t ) (LX2 W_)

together with a high-gain error-actuated analogue controller

governed on T by a control-law equation of the form

u(t) - g{Koe(t)+KlZ(t)} (4.4)

which is required to generate the control input vector u(t)

so as to cause the output vector y(t) to track any constant

63

command input vector v(t) on T in the sense that

lim{v(t)-y(t)1 = 0 (4.5)

as a consequence of the fact that the error vector e(t) =

v(t) - w(t) assumes the steady-state value

lir e(t) = lim{v(t)-w(t)} 0 (4.6)

for arbitrary initial conditions. In equations (4.1), (4.2),

(4.3) ,and(4.4),xl(t) E Rn - 1, x2 (t) E R', u(t) E R', y(t) E RZ,

w(t) E R i , All(ER ( n - U~x ( n - Z) , A12 E R ( n - t ) x t , A 21 ER 1x ( n - E) ,

A2 2 ER"xL, B2 ER XL , C I E Rx(n- ), C2 ER xL, FIERtX(n-.E)

F2 ER 1XI , rank C2B2 < L, rank P2B2 = Z, e(t) R1, v(t)ZRL,

z(t) - z(o) + ot edt ER, K E RX, KI zRX, g(ER + , and

[F1 , F2] = [Cl+MAll , C2 +MA1 2] (4.7)

where M ERLx(n -L). It is evident from equations (4.2), (4.3), and

(4.7) that the vector

w(t) - y(t) = CMA 1 l , MAI. 2 ] (4.8)

of extra measurements is such that v(t) and y(t) satisfy

the tracking condition( 4.5) for any M ERx(n-L) if S(t)

satisfies the steady-state condition (4.6) since equation (4.1)

clearly implies that

lira[All AI1211 x l ( t ) 1 0(49

64p -

in any steady state. However, the condition that rank F2 B2

requires that C2 C R 1X and A1 2 E R(n - j ) x 1 are such that

M ERLx(n-L) can be chosen so that

rank F2 = rank(C2 +MAl2) - 9 . (4.10)

It is evident from equations (4.1),(4.2),(4.3), and (4.4)

that such continuous-time tracking systems are governed on

T by state and output equations of the respective forms

0(t IL ' F1 F - 2 ]Z(t)

i 1(t = , 1 1 A12 ()x2 (t) B2X1 , A21-gB2KoF1 , A2 2 -gB 2KoF 2J -x2 (t)J

+ O vlt) (4. 11)

and

[Z (t)1

Y (t) - C , C 1 1 C2 ] [xl(t)} (4. 12)



continuous-time tracking system governed by equations (4.11)


r~ ~ I , F1 , 2 -tI

G(A)-'E' lC 2] 1 0 X )I n--A 1 A1

-7gB2K11 -A214.qD2 Xo1 1, I9 -A22+gB2K0 2 g2O

65 (4.13)

and the high-gain tracking characteristics of this system

can accordingly be elucidated by invoking the results of

Porter and Shenton (1975) from the singular perturbation


These results yield the asymptotic form of G(X) as

the gain parameter g * and thus greatly facilitate the

determination of controller and transducer matrices Ko, Ki ,

and M such that the continuous-time tracking behaviour of

the closed-loop system becomes increasingly non-interacting

as g is increased. The frequency-response and step-response

characteristics of a continuous-time flight-control system

for the longitudinal dynamics of an aircraft (Kouvaritakis,

Murray, and MacFarlane 1979) are presented in order to

illustrate these general results.

4.2 ANALYSIS

It is evident from equation(4.13)that, by regarding

e = 1/g as the perturbation parameter, the asymptotic form

of the transfer function matrix G(X) of the continuous-time

tracking system as g * = can be determined by invoking the



these results indicate that as g * - the transfer function


r(m) - r(A) + r( (4. 14)

where

66

F(A) Co (AIn-Ao) -1B (4.15)

A -() - C2 (xIt-gA4 ) -gB 2K0 (4.16)

[ -o 1K1 , 0 1(A0 - El 2~u~1 I.K , A1 _A1 2 (4.17)

Bo - [A,2F-] (4.18)2 2

Co" [ -2 1 K0 1 C 1 - 2 F] , (4. 19)

and

A4 - -B 2KoF 2 (4. 20)

It follows from equations (4.14), (4.15), and (4.17) that the

'slow' modes Zs of the tracking system correspond as g -

to the poles Z 1 ,UZ 2 of r() where

zI - (AEc:IXK o+K1 10) (4. 21)

and

Z2 - {X CCIAIn_t-A11 .+A 1 2 1FI4-} (4. 22)

and from equations (4.14), (4.16), and (4.20) that the 'fast'

modes Z of the tracking system correspond as g * to the

poles Z3 of F(A) where

z 3 - (X EC:Ixl L+gF 2B 2 K0o0} . (4. 23)

Furthermore, it follows from equations (4.15), (4.17), (4.18),

and(4.19) that the 'slow' transfer function matrix

67

r(X) = (C1 -C 2 F21 F1 ) (XInk-A1 1 +A1 2 F2 F1 ) A1 2 F2

1 (4.24)

and from equations(4 .16) and(4.20) that the 'fast' transfer

function matrix

r -x) = C2F2 1 GI F2 B2 Ko ) gF2B2Ko (4.25)

Hence, in view of equations (4.24) and (4.25), it is

evident from equation(4.14)that as g - - the transfer

function matrix G(X) of the continuous-time tracking system


( = (CI-CF 1 FI) (XI -A +A F-1 F)AIF2 1

1 2 21 n-2. 11 1 2F2 F1) 12 2

+ C2F2 1 (XI z+gF 2 B2 K0 ) 1gF2 B2 K0 (4.26)

in consonance with the fact that both the 'slow' modes

corresponding to the poles Z2 and the 'fast' modes correspond-

ing to the poles Z3 possibly remain both controlldble and

observable as g - . However, the 'slow' transfer function

matrix r(A) reduces to the form expressed by equation (4.24)

precisely because the 'slow' modes corresponding to the poles

Z1 definitely become asymptotically uncontrollable as g

in view of the block structure of the matrices A and B in0 0

equations ( 4 .1 7 ) and (4.18).

4,3 SYNTHESIS


that tracking will occur in the sense of equation (4.5)

provided only that

68

zsU Zf Cc- (4.27)

where C is the open left half-plane. In view of equations(4.21),

(4.22), and (4.23), the 'slow' and 'fast' modes will

satisfy the tracking requirement(4.27)for sufficiently large

gains if the controller and transducer matrices K0 , K1 , andm are chosen such that Z1C C-, z 2 C C-, and Z 3C C" in

the case of plants for which the transducer matrix M can be

simultaneously chosen so as to satisfy the measurement con-

dition expressed by equation (4.10). Moreover, if K and M

are chosen so that both r(X) and r(X) are diagonal transfer

function matrices by requiring that

F2B2K° W (C2+MA12)B2KO - diag{a 1,a2,...,0 } (4.28)

where ao R+ (j=,2,...,Z) in the case of r(A), it follows

from equation(4.20)that the transfer function matrix G(X)

of the continuous-time tracking system will assume the

diagonal asymptotic form F(A) and therefore that increasingly

non-interacting tracking behaviour will occur as g .

Furthermore, such tracking behaviour will exhibit high

accuracy in the face of plant-parameter variations provided

that the steady-state conditions expressed by equation (4.9)

correspond to 'kinematic' relationships which hold between

the state variables as a consequence of the fundamental

dynamical structure of the plant.

However, increasingly 'tight' tracking behaviour will

not in general occur as g -P . in view of the possible presence

in the plant output vector of 'slow' modes corresponding t(

the poles Z2 of the 'slow' transfer function matrix r().

69

ilk

But the presence of any such 'slow' modes is the inevitable

consequence of the introduction of appropriate transducers

which generate extra plant output measurements so as to

ensure that rank F B2 = and thus render plants for which

rank C2B2 < I amenable to high-gain error-actuated control.

Indeed, increasingly 'tight' tracking behaviour is achievable

as g - only in the case of plants for which rank C2B2 -2

(Porter and Bradshaw 1979) and which are accordingly amenable

to high-gain error-actuated control without the necessity

for the generation of extra plant output measurements.

4,4 ILLUSTRATIVE EXAMPLE


by designing a high-gain error-actuated analogue flight

controller for the longitudinal dynamics of an aircraft

governed on T by the respective state and output equations

(Kouvaritakis, Murray, and MacFarlane 1979)

1() 0, 1.132 , 0 , 0 , 1 x 1(W

2t) 0 0 , 0 , 1 , 0 x2(t)

113 t) - 0 , -0.1712 , -0.0538 , 0 , 0.0705 x3 (t)

4(t) 0 , 0 , 0.0485 , -0.8536 , -1.013 x4 (t)

5 J , o , -0.2909 , 1.0532 , -0.6959 .xS(t)_

0 0' 0

0 , 0 ' O u1

+ -0.0012 , 1 , O .2 (t) (4.29)

0.4419 , 0 , -1.6646 Lu3 (t)

0.1575 , 0 , -0.0732J

70

and

Mx1 (t)

LY2 t J 0 0 1 , O J x (M (4.30)Y3(t)- , 1 0 , Q 4 4(t) I

x 5 (t

from which it can be readily verified that the first Markov

parameter

C2B2 O.0012 , 1 , (4. 31)

is rank defective. In case {a1 , 2 03 } - {1,1,1, K1 - Kof

and

M L O ] , (4. 32)0 0.25

it follows from equations (4.3), (4.4), and (4.28) that the

corresponding transducers and high-gain error-actuated

analogue controller are governed on T by the respective

measurement and control-law equations

"x1 (t) 7

f1 (t) 1 [1 , 0.283 ,0 , 0 ,0. 25] .2(t)

1w2(t) 0 , 0 ,1 ,0 ,0 x 3 (t) (4.33)S3(t)J . , 1 , , 0.25 , x4 (t)

x7(t)71"

and

F -28.97 0 -1.274

U2Mtg) -. 1 , -o.oo5 e2 (t)

3u3 L-7.690 , 0 , -2.740 Le3 (t),

F 28.97 ,0 ,-1.274 lzi(t)f+ -0.0348 ,1 ,-0.0015 2 (t) (4'34

L-7.690 , 0 , -2.740 -Z3 (t)i}

and it is evident from equations (4.21), (4.22), and (4.23)

that Z1 = {-1,-1,-i}, Z2 = {-4,-4}, and Z3 = {-g,-g,-g}.

It is also evident from equation(4.26) that the asymptotic


4~A 0 ,0

_o 9 o (4. 35)r( ) - , +g ,

[a0 0 '

and therefore that the closed-loop continuous-time flight-

control system for the longitudinal dynamics of the aircraft

will exhibit increasingly non-interacting tracking behaviour

as g * - when the control input vector Cul(t) , u2 (t) , u3 (t)]T

is generated by the high-gain analogue controller governed

on T by equation (4.34). However, it is apparent from equation

(4.35)that increasingly 'tight' tracking behaviour will be

achieved only in the case of Y2 (t) in view of the presence

Ln yl(t) and Y3 (t) of .he 'slow' modes corresponding to the

poles Z2.

72

J.i

The actual frequency-response loci G(iw) for w

are shown in Figs4.1,4.2,and 4.3wheng -1P,20, and 50,

respectively, and it is clear that the actual frequency-

response loci approach the asymptotic frequency-response

loci r(iw) as the gain parameter g is increased. The

corresponding step-response characteristics are shown in

Fig 4.4,Fig4.5,andFig4.6whe[vl(t) , v2 (t) , v3 (t)]T =

[1 , o , 0]T, Cvp (t) I v2 (t) v 3 (t)IT =[O , 1 , O]T, and

[Vl(t) , v2 (t) , v3 (t)]T _ [O , O , 1]T, respectively, and

it is evident that increasingly non-interacting tracking

occurs as the gain parameter q is increased but that in-

creasingly 'tight' tracking occurs only in the case of Y2 (t).

4.5 CONcujsxoN



of continuous-time tracking systems incorporating linear

multivariable plants which are amenable to high-gain error-

actuated analogue control only if extra plant output measure-

ments are generated by the introduction of appropriate trans-

ducers and processed by inner-loop compensators. It has been

shown that these results greatly facilitate the determination

of controller and transducer matrices which ensure that the

closed-loop behaviour of such continuous-time tracking systems

becomes increasingly non-interacting as the gain parameter g

is increased. These general results have been illustrated by

the presentation of the frequency-response and step-response

73

characteristics of a continuous-time flight-control system

for the longitudinal dynamics of an aircraft.

74

0.5

0.0

-0.5

-0.5 0.0 0.5 1.0 1.5Fig 4.1(a): G 1 1 (iw); g-10

75

1 .0

0.5

0.0

-0.5

- 1 , , I I , ,0, i ,-0.5 0.0 0.5 1.0 1.5

Fig 4.1(b): G 1 2 (ii); g = 10

76

1.0 . .

0.5

0.0

-0.5

-1 .0-0.5 0.0 0.5 I .0 1 .5

Fig 4.1(c) G 13 (iw) g = 10

77

1 .0 I. I ~ I

0.5

0.0

-0.5

-1.0 p

-0.5 0.0 0.5 1.0 1 .5Fig 4.1(d): G 2 1 (iw); g =10

78

1.0

0.5

0.8

-0.5

-1 .0 . f . .

-0.5 0.0 0.5 1.8 1.5

Fig.4.1(e). G 2 2 (iw); g 10

79

1.0 .1 1 1

0.5

0.0

-0.5

-0.5 0.0 0.5 1.0 1.5Fig 4.1(f): G2 3 (iw); g = 10

80

___,___23

1 ... ...

0.5

0.0r)

-0.5

"- 1 .0 . . . . . ' .. .. . , p

-0.5 0.0 6.5 1.0 1.5Fig 4.1(g): G3 1 (iw); g 10

81

dA

1 .

-0.5

-1 .8-0.5 0.0 8.5 1.0 1.5

Fig 4 . 1(h): 3 2 ('W) g 10

82

1.5

0.5

-0.5

-o.5 0.0 0.5 1.0 1.5Fig 41(i)-. G 3 3 (iw); g-10

83

0.5

0.0

-0 .5

-o.5 0.0 0.5 1.0 1.5Fig 4.2(a): Gljjiw); g =20

84

1.60 .

0.5

0.0

-0.5

-0.5 0.6 0.5 1.0 1.5rig 4.2 (b): G 1 2 (iw) g -20

85

1 .0 . . .

8.5

0.0

-0 .5

-1

.0

k

-0.5 0.0 0. 1.0Fig 4.2(c): G1 3 (iw); g =20

86

1.0

0.5

0.0

-0.5 0.0 0.5 1.0 1.5Fig 4.2 (d): G 2 1 (iw); g -20

87

1.0 ,

0.5

0.0

-0.5

*-1 .0 p p I

-0.5 0.0 0.5 1.0 1.5

Fig 4 . 2 (e): G 2 2 (iw); g =20

88

1.0 . .

0.5

0.0

-0.5

- 1 .0I . . . . . . p I I .

-0.5 0.0 0.5 1.0 1 5

Fig 4.2(f): G2 3 (iw); g - 20

7AD-A119 4.0 SALFOflO UNIV IENGLANI DEPT OF AERONAUTICAL AND MEC-EC Pie 17/7DESIGN OFm P MISW.PgNFCE TRACKING SYSTEMS. 4W

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f90

0.5

8.0

-0.5

-0.5 8.0 0.5 1.8 1.5

Fig 4.2(h): G 3 2 (iw); g -20

_ _ _ _ _ _91

1 .0 *

0.5

0.0

-0.5

-0.5 0.0 0.5 1.0 1.5Fig 4.2 (1): G 3 3 (iw); g =20

92

1.0 . .. . . . ."

0.5

0.0

I ~ -0.5

-1 .0 . . . . . . . ,. . ,A 2 .1..

-0.5 0.0 0.5 1.0 1.5Fig 4 .3(a): Gl1(iw); g - 50

93

1 0 1 T I I

0.5

-0.5

-1 0 .

-9.5 0.0 8 .5 1.0 1.5Fig 4.3 (b): G 1 2 (iw);t 50

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134

_ _-~~ <5 __

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Il -0.5

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96

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-1.1

-o.5 0.0 3.5 1.3 1.5Fig 4.3(e): G 2 2 (iw); g 50S

97 _ _ _ _ _ _ _

1.0

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98

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100

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110

CHAPTER 5

DESIGN OF TRACKING SYSTEMS

INCORPORATING INNER-LOOP COMPENSATORS

AND FAST-SAMPLING ERROR-ACTUATED CONTROLLERS

111

5.1 INTRODUCTION

In this chapter singular perturbation methods are used


matrices of discrete-time tracking systems incorporating

linear multivariable plants which are amenable to fast-

sampling error-actuated digital control (Bradshaw and Porter

1980) only if extra plant output measurements are generated

by the introduction of appropriate transducers and processed

by inner-loop compensators. Such tracking systems consist

of a linear multivariable plant governed on the continuous-

time set T = CO,+-) by state, output, and measurement

equations of the respective forms

-I+ [B u(t) (5.1)Lx 2 ( t LA21 , 22 x2(2)J

Fx(t 1= i y(t)=[C 1 1 C] LxtJ (5.2)2 (5.2)

and

w(t) = [F1 , F2] 1 (.3)

together with a fast-sampling error-actuated digital con-

troller governed on the discrete-time set TT = {0,T,2T,...}

by a control-law equation of the form

u(kT) - f{Koe(kT)+Klz(kT)} (5.4)

which is required to generate the control input vector

112

u(t) - u(kT), t ECkT , (k+l)T), kT CTT, so as to cause the

output vector y(t) to track any constant command input

vector v(t) on TT in the sense that

lim{v(kT)-y(kT)} = 0 (5.5)

as a consequence of the fact that the error vector e(t) =

v(t) - w(t) assumes the steady-state value

lim e(kT) = lim{v(kT)-w(kT)} = 0 (5.6)k-b- k-w

for arbitrary initial conditions. In equations (5.1), (5.2),

(5.3),and(5.4) ,xz(t) CR -1 x 2 (t) CE , u(t) CR , y(t) (Rz

w(t) C RI, All R(n- i)x(n- L), AI12 R(n - t)xL, A21 uRlx(n - 9),

A2 2 CR xt , B 2 E RIXI, C 1RIx(n-1), C2 ER x t , F ECRLx(n - ),

F2 ERXI, rank C B < Z,rank F 2B2 = L, e(t)C R', v(t)( R

z(kT) - z(o) + T Z e(jT) CR , KoCRXL, KC RLX, f eR ,

J-Oand

[F 1 , F2] = [Cl+MAI1, C2+MA1 2] (5.7)

where M CRLx(nh). It is evident from equations (5.2), (5.3),

and (5.7) that the vector

w(t) - y(t) - [MAll , MA1 2 ] [xt] (5.8)X2()

of extra measurements is such that v(kT) and y(kT) satisfy

the tracking condition(5.5)for any M CRLx(n - ) if e(kT)

satisfies the steady-state condition (5.6) since equation (5.1)

clearly implies that

113

lim[Azl , AI2 0o (5.9)t ®W £x2 L =

in any steady state. However, the condition that rank

F2B2 = £ requires that C2ER x z and A1 2 ER(n-I)xk are

such that M ER x(n- k) can be chosen so that

rank F 2 = rank(C 2+MA1 2 ) = z . (5.10)


that such discrete-time tracking systems are governed on TT

by state and output equations of the respective forms

z{ (k+l)T}1x (k+1)T}

2{ (k+l) T}j

9 , -TF 1 , -TF 2 1 rz(kT) 1= T K 1 ,1 1 1 -f! 1 K0 F 1 , 12 fI*KoF 2 xl(kT)

f L 2 KI , 121-fT2 KoF1, 2 2 -fI 2 KoF 2J Lx2 (kT)J

_f~;jk (5.1K1)

and

Fz(kT)1ykT)= [O , C1 , C2] Lxl(kT) (5.12)

2C2 (kT)_

where

114

" 1 0 expi1l A1 jT (5. 13)t21 02 J LA21 ' A22JJ

and

~TfexA11 Al2]ltlFdt . (5. 14)J 1LA21 A 2 [



discrete-time tracking system governed by equations (5.11)


G(X) =

X1I-I , TF I TF2 iTI[0C1' C2 ] {- f 1EK l , IIn - '' l + f ' 1E O F I ' - (' 1 2 + f ' 1E O F 2 fT 1f Ko0

L-f2K1, -021+f'2KoF1 , AIX-0 222+fY2KoF2 f 2 K 0K

(5.15)

and the fast-samling tracking characteristics of this system

can accordingly be elucidated by invoking the results of

Porter and Shenton (1975) from the singular perturbation


These results yield the asymptotic form of G(M) as the

sampling frequency f . and thus greatly facilitate the

determination of controller and transducer matrices Ko, KI,

and M such that the discrete-time tracking behaviour of the

closed-loop system becomes increasingly non-interacting as

f is increased. The frequency-response and step-response

115

characteristics of a discrete-time flight-control system

for the longitudinal dynamics of an aircraft (Kouvaritakis,

Murray, and MacFarlane 1979) are presented in order to

illustrate these general results.

5.2 ANALYSIS

It is evident from equation (5.15) that, by regarding

E = 1/f as the perturbation parameter, the asymptotic form

of the transfer function matrix G(X) of the discrete-time

tracking system as f - can be determined by invoking the

results of Porter and Shenton (1975) from the ingular


since it follows from equations (5.13) and (5.14) that

lim f ll-In-9 , 12 1 = 12I[Al A1 2] (5.16)

f c L (21 ' 22- 1 21' A2 2]

andI (

lim f = , (5.17)

these results indicate that as f ® the transfer function


r(x) = r() + r(x) (5.18)

where

r(A) = C0 ( Bn-I-TA0o , (5.19)

116

rm c 2 (XI CI A 4) B E2 K0 ,(5.20)

[Al22KI1 AII-A1 2 F2 F] (5.21)

B = , (5.22)0 [A2F 21

= EC2 F 2 QK 1 , C -C F21 F1] (5.23)

and

A 4 = -B 2 KoF 2 (5.24)

It follows from equations (5.18), (5.19), and (5.21) that the

'slow' modes Zs of the tracking system correspond as f -

to the poles Z1 U Z2 of F(A) where

z = {X C:lI-I +K-X 1KI=0} (5.25)

and

Z= { X EC:IXInt-I nZ-TAI+TAI 2 F 21 FI =O} , (5.26)

and from equations (5.8), (5.20), and (5.24)that the 'fast'

modes Z f of the tracking system correspond as f * to

the poles Z3 of r(m) where

Z3 - {X eC:IXI I z+F 2 B2 K0 I=0} (5.27)

Furthermore, it follows from equations (5.19),(5.21),(5.22),

and(5.23)that the 'slow' transfer function matrix

= (Ct-CF21F) (XIn_-In_- TAI+TA F21 F )-TA F2112 2 1 - 1 12 2 1 12 2

(5.28)117

and from equations (5.20) and(5.24)that the 'fast' transfer

function matrix

^-1 ( I£I£FBK )-1FBKfl() = C2F 2 (XIzI z+F2B2K0) F 2 . (5.29)

Hence, in view of equations (5.28) and (5.29), it is

evident from equation(5.18)that as f - - the transfer

function matrix G(X) of the discrete-time tracking system


r(A1 -( CFl )X TA +TA F-1F' -1 TAF 1F (X) = (CI-C 2 F2 IF I ) (x~n_£-In_£Tl+A2;11 IA2;1 21 n- nk 11 12 2 F1) 12 2

+ C2F21 (XII-I +F2 B2 Ko ) -1F2B2Ko (5.30)

in consonance with the fact that both the 'slow' modes

corresponding to the poles Z2 and the 'fast' modes correspond-

ing to the poles Z3 possibly remain both controllable and

observable as f - =. However, the 'slow' transfer function

matrix r(X) reduces to the form expressed by equation (5.28)

precisely because the 'slow' modes corresponding to the

poles Z1 definitely become asymptotically uncontrollable as

f - - in view of the block structure of the matrices A and0

B in equations (5.21) and (5.22).

5,3 SYNTHESIS


that tracking will occur in the sense of equation (5.5)

provided only that

ZsU Zf CD (5.31)

118

where V- is the open unit disc. In view of equations (5.25),

(5.26)iand(5.27),the 'slow' and 'fast' modes will satisfy

the tracking requirement(5. 31)for sufficiently small sampling

periods if the controller and transducer matrices K0 , Ki,

and M are chosen such that both Z1CV _ and Z2 CD- for suf-

ficiently small sampling periods and Z3 C D- in the case of

plants for which the transducer matrix M can be simultaneously

chosen so as to satisfy the measurement condition expressed

by equation(5.10). Moreover, if Ko and M are chosen so that

both r(X) and r(X) are diagonal transfer function matrices

by requiring that

F2B2K o = (C2+MA12)B2Ko = diag{al1a2 ,...,a2, (5.32)

where l-a(:Rnv- (j=l,2,...,Z) in the case of r(X), it

follows from equation (5.24)that the transfer function matrix

G(X) of the discrete-time tracking system will assume the

diagonal asymptotic form r(A) and therefore that increasingly

non-interacting tracking behaviour will occur as f - -.

Furthermore, such tracking behaviour will exhibit high accuracy

in the face of plant-parameter variations provided that the

steady-state conditions expressed by equation(5.9)correspond

to 'kinematic' relationships which hold between the state

variables as a consequence of the fundamental dynamical

structure of the plant.

However, increasingly 'tight' tracking behaviour will

not in general occur as f w in view of the possible presence

in the plant output vector of 'slow' modes corresponding to

the poles Z2 of the 'slow' transfer function matrix r(A).

But the presence of any such 'slow' modes is the inevitable119

consequence of the introduction of appropriate transducers

which generate extra plant output measurements so as to

ensure that rank F2B2 = Z and thus render plants for which

rank C2B2 < Z amenable to fast-sampling error-actuated control.

Indeed, increasingly 'tight' tracking behaviour is achievable

as f - only in the case of plants for which rank C2B2 =

(Bradshaw and Porter 1980) and which are accordingly amenable

to fast-sampling error-actuated control without the necessity

for the generation of extra plant output measurements.

5,4 ILLUSTRATIVE EXAMPLE


by designing a fast-sampling error-actuated digital flight

controller for the longitudinal dynamics of an aircraft

governed on T by the respective state and output equations

(Kouvaritakis, Murray, and MacFarlane 1979)

" (t) 7 0 , 1.132 , 0 , 0 , 1 -x (t)

2 () 0, 0 0 0 , 1 , 0 x2 (t)

x3 (t) = 0 , -0.1712 , -O.0538 , 0 , 0.0705 x 3 (t)

4(t) , 0 , 0.0485 , -0.8536 , -1.013 x 4 (t)

x 5 (t)] 0 , 0 , -0.2909 , 1.0532 , -0.6859 x 5 (t)

0 , 0

0 10 0 Fut 1+ -0.0012 , 1 , 0 u2 (t) (5.33)

0.4419 , 0 ,-1.6646 u3 (t)J

0.1575 , 0 , -0.0732

120

and

"xl(t)]L yit 1 10 0 0,0 x2(tM

2 (t) , 0 , 1 , 0 , x3 (t) (5.34)

3(t)],0 , , 0 , x4(t)X4(t)

from which it can be readily verified that the first Markov

parameter

00

C2B2 = -0.012 , 1 , (5.35)

is rank defective. In case {a 1, 2 ,a3} - {1,1,1}, K1 = K0 ,

and

M [0 , (5.36)

0 0.25

it follows from equations (5.3), (5.4), and (5.32) that the

corresponding transducers and fast-sampling error-actuated

digital controller are governed on T and TT by the respective

measurement and control-law equations

x1 (t)

Mw1t) [1 , 0.283 , 0 ,0 ,0.251 x 2t)W2(t) 0 , 0 , 1, 0 , (3t) 5.37)

3 (t) , 1 , 0 , 0.25 , x4(t)

x5 (t).

121

and E 1 (kT)1 f[28.97 , 0 , -1.274 l e 1(kT)1U2 (kT) f -0.0348 , 1 r -0.0015 e2 (kT)

u3 (kT)" -7.690 , 0 , -2.740 j Le3 (kT)I

-28.97 , , -1.274 - zl(kT)i

S-.038 ,1 ,--01 z 2(kT) (5.38)

.690 , 0 , -2.740 ilz 3 (kT)J

and it is evident from equations (5.25),(5.26),and(5.27) that

z 1 = {l-T,1-T,1-T}, Z2 = {1-4T,l-4T,1-4T}, and 73 = {0,0,0}.

It is also evident from equation (5.30) that the asymptotic


1(x) 0 [ 0 (5.39)

4T

and therefore that the closed-loop discrete-time flight-control

system for the longitudinal dynamics of the aircraft will

exhibit increasingly non-interacting tracking behaviour as

f * when the piecewise-constant control input vector

[ul(t) , u2 (t) , u3 (t)3T = [ul(kT) , u2 (kT) , u 3 (kT)] T ,

t E[kT , (k+l)T), kT ETT, is generated by the fast-sampling

digital controller governed on TT by equation (5.38). However,

it is apparent from equation (5.35)that increasingly 'tight'

tracking behaviour will be achieved only in the case of Y2 (t)

in view of the presence in yl(t) and Y3 (t) of the 'slow'

modes corresponding to the poles Z2 .

122

The actual frequency-response loci G(ei T ) for WT C[O , 2r]

are shown in Figs 5.1, 5.2,and 5.3whenT=0.i0, 0.05, and 0.02,

respectively, and it is clear that the actual frequency-response

loci approach the asymptotic frequency-response loci r(eiWT )

as the sampling frequency f is increased. The corresponding

step-response characteristics ara shown in Fig 5.4, Fig 5.5, and

Fig5.6when[vl(t) , v2(t) , v3(t)]T = [ , 0 , 0]T, [Vljt) ,

v2(t) , v 3 (t)]T = Co , 1 o] T , and [VC1(t) , v2(t) , v 3 (t)] T =

[O , O , 1]T, respectively, and it is evident that increasingly

non-interacting tracking occurs as the sampling frequency f

is increased but that increasingly 'tight' tracking occurs

only in the case of Y2 (t)"

5.5 CONCLUSION



of discrete-time tracking systems incorporating linear

multivariable plants which are amenable to fast-sampling

error-actua!:ad digital control only if extra plant output

measurement& are generated by the introduction of appropriate

transducers and processed by inner-loop compensators. It

has been shown that these results greatly facilitate the

determination of controller and transducer matrices which

ensure that the closed-loop behaviour of such discrete-time

tracking systems becomes increasingly non-interacting as the

sampling frequency f is increased. These general results

have been illustrated by the presentation of the frequency-

123

response and step-response characteristics of a discrete-

time flight-control system for the longitudinal dynamics of

an aircraft.

124

1 .5 *

1 .0

0.5

0.0

-0.5

-1.0

-1.5 * I * t ** I *

-1.5 -1.0 -0.5 0.0 0.5 1.0. 1.5

Fig S. I(a): G (e iwT T 0.10

125

1.5 * * I '

1.0

0.5

0.0

-0.5

-1 .0

-1.5 I , I , , I I

-1 .5 -1 .0 -0.5 0.0 0.5 1.0 1.5

Fig 5.1(b): G1 2 (e ; T 0.10

126

1.5 *

1 .0

0.5

6.0

-0.5

-1.0

-1.5 I

-1.5 -1.6 -6.5 0.0 0.5 1.0 1.5

Fig 5.1(C): G 1 3 (e );T T - 0. 10

127

1.5 *

1 .0

0.5

0.0

-0.5

-1 .0

-1 .5- 1.5 -1.0 -0.5 0.0 0.5 1 .0 1 .5

Fig 5.1(d): G2 eiT T=01

128

1 .0

0.5.

0.0

-0.5

-1.0

-1.5 -1.0 -0.5 0.0 6.5 1.0 1.5

Fig 5.1(6): G 2 2 (ei); T =0. 10

129

1.5

1 .0

0.5

0.0

-0.5

-1 .0

-1.5 I

-1.5 -1.0 -0.5 0.0 0.5 I.0 1.5

Fig 5.1(f): G2 3 (e i ) T ) 0.10

130

1.5 *

1 .0

0.5

0.0

-0.5

-1.0

-1.5 I

-1.5 -1.0 -0.5 0.0 0.5 1.8 1.5Fig 5.1 (g): G 3 1 (e jtT) T = 0. 10

1.31

1.5 *

1 .0

0.5

0.0

-0.5

-1 .0

-1.5 I i _ _ _ _ _ _ _ _ _ _ _ _

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 5.1 (h) : G 32e iwT) T =0. 10

132

1.5 *

1 .6

0.5

0.0

-6.5

-1 .5 I

-1.5 -1.0 -0.5 6.0 6.5 1.0 1.5

Fig 5.1(1): G 3 3 (e );T T - 0.10

133

1.5

1 .0

0.5

0.0

-0.5

-1 .0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 5.2(a): G 1 2 (e );T T =0.05

134

1.5 * I I *

1.0-

0.5

0.0

-0.5

-1.0

- 1.5 I , I , , ! , I

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 5.2(b): G1 2 (e i T); T = 0.05

135

1 .5

1 .0

0.5

0.0

-0.5

-1 .0

-1.5 *

-1 .5 -1 .0 -0.5 0.0 0.5 1.0 1.5Fig 5.2 (c): G1 3 (eiwT T=00

136

1.5 ,

1.0

0.5

0.0

-0.5

-1 .0

-1.5 I * I * p I I

-1 .5 -1.0 -0.5 0.0 0.5 1.0 1 .5

Fig 5.2(d): G2 1 (eiT); T 0 0.05

137

1 .0

0.5

0.0

-1 .5 -1.0 -0.5 0.0 0.5 -1.0 1.5

Fig 5.2(e): G 2 2 (e );T T =0.05

138

1.5 ! * I

1.0

0.5

0.0

-0.5

-1.0

-1.5 I * I * * I * 1-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 5.2(f): G2 3(e ); T = 0.0.5

139

1.5 *

1 .0

0.5

0.0

-0.5

-1.0

-1.5 I

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 5.2(g): G 31(e iwLT) T = 0.05

140

1.5 *

1 .0

0.5

0.0

-0.5

-1 .0

-1.5 I a

-1 .5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 5.2(h): G 3 2 (e );T T - 0.05

141

1.5 *

1 .0

0.5

0.0

* 1 -0.5

-1 .0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Fig 5.2(1): G 3 3 (e ) ; T -0.05

142

1.5 *

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Fig S.3(a): G 1 e );T T = 0.02

143

1 .5

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Fig 5.3 (b): G 1 2 (e );T T 0.02

144

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pig 5.3(c): G 1 3 (e );T T =0.02

145

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Fig 5.3(d): G 2 1 (e );T T =0.02

146

1 .0

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147

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148

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1494

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150

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151

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CONCLUS IONS AND REcoMENDATIONS

1L61

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6.1 CONCLUSIONS

The conceptual and computational simplicity of singular

perturbation methods (Porter and Shenton 1975) in the design

of both analogue and digital tracking systems incorporating

error-actuated controllers has been demonstrated. This

demonstration has been effected by designing such controllers

for plants which have previously been considered by other

methods (MacFarlane and Kouvaritakis 1977, Kouvaritakis,

Murray, and MacFarlane 1979) so that detailed comparisons

between alternative design techniques can readily be made,

at least in the case of error-actuated analogue controllers.

Moreover, further demonstration of the power of the

design techniques described in this report has been provided

by the design of direct digital flight-mode controllers for

the F-16 aircraft (Porter and B' adshaw 1981). In addition,

the fact that high-gain and fast-sampling error-actuated

controllers remain highly effective in the presence of gross

actuator nonlinearities has been conclusively demonstrated

(Porter 1981a, Porter 1981b, Porter 1981c, Porter 1981d).

Thus, the same basic design techniques have been shown to be

applicable to both linear and nonlinear multivariable plants.

6.2 RECOMMENDATIONS

The controllers for the F-16 aircraft designed by

singular perturbation methods (Porter and Bradshaw 1981) are

extremely robust in the sense that the same controller

remains effective throughout a wide range of different flight

162

conditions. However, in case even higher performance

characteristics are required, the use of adaptive controllers

becomes unavoidable and the study of fast-sampling adaptive

digital controllers is therefore recommended.

Finally, although fast-sampling digital controllers for

the F-16 aircraft have been successfully implemented in the

Engineering Dynamics and Control Laboratory at the University

of Salford using microprocessor hardware (Garis 1981), the

implementation of such controllers in actual aircraft and sub-

sequent flight testing is recommended. Such practical im-

plementation would obviously entail close liaison with the

aerospace industry, based initially upon the wider dissemination

of the design methodologies described in this report.

163

116

goof

ATHANS, M. and FALB, P.L., 1966, "Optimal Control".

(McGraw-Hill, New York).

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variable discrete-time tracking systems incorporating

fast-sampling error-actuated controllers", Int. J. Systems

Sci., vol. 11, pp. 817-826.

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of linear multivariable discrete-time tracking systems

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Int. J. Systems Sci., vol. 11, pp. 1279-1293.

DAVISON, E.J. and FERGUSON, I.J., 1981, "The design of

controllers for the multivariable robust servomechanism

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Automat. Control, vol. AC-26, pp. 93-110.

GARIS, A., 1981, "Fast-sampling direct digital flight-mode

control systems", Ph.D. Thesis, University of Salford.

KOUVARITAKIS, B., MURRAY, W., and MACFARLANE, A.G.J., 1979,

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vol. 29, pp. 325-358.

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165

PORTER, B., 1981, "High-gain tracking systems incorporating

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nonlinearities", Proc. 20th IEEE Conference on Decision

and Control, San Diego, USA.

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variable continuous-time tracking systems incorporating

high-gain error-actuated controllers", Int. J. Systems

Sci., vol. 10, pp. 461-469.

PORTER, B. and BRADSHAW, A., 1979, "Asymptotic properties

of linear multivariable continuous-time tracking systems

incorporating high-gain error-actuated controllers",

Int. J. Systems Sci., vol. 10, pp. 1433-1444.

PORTER, B. and BRADSHAW, A., 1981, "Design of direct digital

flight-mode control systems for high-performance aircraft",

Proc. IEEE National Aerospace and Electronics Conference,

Dayton, USA.

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linear multivariable continuous-time systems",

Electronics Lett., vol. 13, pp. 753-755.

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multivariable linear systems", Int. J. Control, vol. 21,

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Design", (Academic Press, London).

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(Springer-Verlag, New York).

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(Springer-Verlag, Berlin).

167

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