using asme format - dspace at koasas:...

Journal ofDynamic Systems,

Measurement,and

ControlTechnical Briefs

i

r

tiyi

E

m

m

f

n

a

d

h

Aionlly,

e

r-

ni-

-

-

l

fors

Improved Bilinear RouthApproximation Method forDiscrete-Time Systems

Younseok ChooSchool of Electronic, Electrical and ComputerEngineering, Hong-Ik University, San34-31, Sinan-RiJochiwon-Eup, Yeongi-Gun Chungnam, Korea 339-701e-mail: [email protected]

Hwang and Shieh proposed a bilinear Routh approximatmethod for reducing the order of discrete-time systems. A redumodel derived by the method is not only stable whenever an onal model is stable, but also fits the first few time-moments oforiginal one. This paper addresses the possibility of improvingmethod by letting the impulse response energy of the origmodel also be conserved in the reduced model without destrothe stability preserving and time-moments matching propert@DOI: 10.1115/1.1343912#

Keywords: Bilinear Routh Approximation, Impulse Responseergy, Routh Approximation

I IntroductionAs noted in the literature, the simplification of complex syste

plays an important role in many engineering fields. In the pthree decades, the order reduction problem has been an ampleof research and numerous techniques have been reported. Athem, the Routh approximation~RA! method@1# for continuous-time systems has received continuous attention due to its stabpreserving and time-moments matching properties.

In @2#, a bilinear Routh approximation~BRA! method has beenpresented as an extension of the RA method to discrete-timetems. The method inherits the above-mentioned properties oRA method, and provides a recursive algorithm to computereducedz-transfer functions with various order by a single striof computations.

In recent publications@3,4#, it has been shown that the RAmethod@1# and its modified version@5# can be further improvedby letting the impulse response energy of the original systembe conserved in the reduced model. In this paper a similar atteis made for the BRA method of@2#. The reduced model obtainestill possesses the stability preserving and time-moments maing properties as in the BRA method. An additional feature is tthe full impulse response energy of the original model is matained in the approximant.

Contributed by the Dynamic Systems and Control Division of THE AMERICANSOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Dynamics Sytems and Control Division January 8, 2000. Associate Editor: E. Fahrenthold.

Journal of Dynamic Systems, Measurement, and ControlCopyright © 2

oncedigi-thehenaling

es.

n-

sastareaong

ility

sys-the

theg

lsompt

tch-at

in-

This paper is organized as follows. In Section II, the BRmethod is briefly reviewed. The method is improved in SectIII, and a numerical example is presented in Section IV. Finathe paper is concluded in Section V.

II Brief Review of BRA MethodConsider a stablenth-order discrete-time system with th

z-transfer function

G~z!5b01b1z1b2z21¯1bn22zn221bn21zn21

a01a1z1a2z21¯1an21zn211anzn (1)

where anÞ0. The BRA method of@2# is based on the bilineacanonical expansion ofG(z). To this end, define a bilineartransformed function

G~s!5~z11!G~z!uz5~11s!/~12s! (2)

SinceG(s) is strictly proper, it can be expanded into the canocal form @1# as

G~s!51

s (i 51

n

b i)j 51

i

H j~s! (3)

where for j 51,2,¯ ,n

H j~s!51

a j

s1

1

a j 11

s1

1

�

an21

s1

1

an

s

(4)

with the first term in Eq.~4! replaced by 11a1 /s for H1(s). Thebilinear canonical expansion ofG(z) is, then, obtained by applying the inverse transformations5(z21)/(z11) to Eq.~2! as

G~z!51

z11G~s!U

s5~z21!/~z11!

51

z21 (i 51

n

b i)j 51

i

H j~z! (5)

where H j (z)5H j (s)us5(z21)/(z11) . The a and b parameters inEq. ~5! can be obtained directly in thez-domain using the numerator and denominator coefficients ofG(z) @2#. In addition,G(z) isstable if and only ifa i.0 for all 1< i<n.

Let H jk(z) be the function obtained fromH j (z) by settinga i50 for all k11< i<n. Then the kth-order reduced modeGk(z)5Bk(z)/Ak(z) is given by

Gk~z!51

z21 (i 51

k

b i)j 51

i

H jk~z! (6)

and can be computed by the following recursive equations:1<m<k

-

MARCH 2001, Vol. 123 Õ 125001 by ASME

et

l

t

-nd

rder

s

er

er-

Bm~z!5am~z11!Bm21~z!1~z21!2Bm22~z!1bm~z21!m21

(7)

Am~z!5am~z11!Am21~z!1~z21!2Am22~z! (8)

whereB21(z)5B0(z)50, A21(z)51/(z21) andA0(z)51. It isnoted that@2#

Gk~s!5~z11!Gk~z!uz5~11s!/~12s! (9)

is thekth-order Routh approximant ofG(s).As shown in@2#, Gk(z) is stable wheneverG(z) is stable, and

fits the firstk time-moments ofG(z), i.e., if we expandG(z) andGk(z) in Taylor series aboutz51 as

G~z!5t01t1~z21!1t2

2!~z21!21¯ (10)

Gk~z!5 l 01 l 1~z21!1l 2

2!~z21!21¯ (11)

then, l i5t i , i 50,1,2, ,k21.

III Improved BRA MethodAs briefly reviewed in Section II, thekth-order approximant

Gk(z) derived by the BRA method possesses the stability presing and time-moments matching properties. An improvementhe method is proposed in this section so that the reduced mnot only maintains those two properties, but also conservesfull impulse response energy of the original model.

As is well known @1#, the impulse response energyI of thebilinear-transformed functionG(s) in Eq. ~2! is given by

I 51

2p i E2 i`

i`

G~s!G~2s!ds51

2 (i 51

nb i

2

a i(12)

It is easily seen from Eq.~2! that

R G~z!G~z21!1

zdz5

1

2 E2 i`

i`

G~s!G~2s!ds (13)

wherer denotes the integral along the unit circle in the compplane. Consequently, the impulse response energyI of G(z) isgiven by

I 51

2p i R G~z!G~z21!1

zdz5

1

4 (i 51

nb i

2

a i(14)

Similarly, by Eq.~9!, Gk(z) has the impulse response energyI kgiven by

I k51

4 (i 51

kb i

2

a i(15)

On the other hand, it is not difficult to see from Eqs.~7! and ~8!that the time-moments ofG(z) are

t05b i

2a1

t152b1

4a12

b1

4a12 1

1

4a1•

b2

a2

A

In general, for 0<m<n21, the (m11)th time-momenttm is ofthe form

tm5 f ~a1 ,a2 , ¯ ,am ,b1 ,b2 , ¯ ,bm!

1g~a1 ,a2 , ¯ ,am!•bm11

am11(16)

which implies t0 ,t1 , ¯ ,tm are not changed as long as the rabm11 /am11 is kept the same with (a i ,b i), i 51,2,¯ ,m fixed. Ifwe replace the last parametersak andbk in Eq. ~15! by ak andbk

126 Õ Vol. 123, MARCH 2001

rv-to

odelthe

ex

io

in such a way thatbk /ak5bk /ak and I k5I , and compute thenew modelGk(z) using (a i ,b i), i 51,2,¯ ,k21 and (ak ,bk)via Eqs. ~7! and ~8!, then Gk(z) conserves the full impulse response energy ofG(z) as well as the time-moments matching astability preservation properties ofGk(z).

IV A Numerical ExampleConsider a discrete-time system described by the fourth-o

z-transfer function

G~z!50.8645z321.9002z211.3982z20.3106

z422.6z312.66z221.296z10.288(17)

Using the procedure in@2#, the~a, b! parameters are computed a

a150.21667 a250.19578 a350.31207 a450.49978

b150.43250 b250.37019 b350.50192 b450.77043

and G(z) has the impulse response energyI 50.88915. Thesecond-order approximant derived by the BRA method of@2# isgiven by

G2~z!50.45495z20.28536

1.2382z221.9152z10.84664(18)

Now, replace (a2 ,b2) by (a2 ,b2), where (a2 ,b2) are computedfrom the following two equations:

b2 /a25b2 /a251.89083 (19)

1

4S b1

2

a11

b22

a2D 50.88915 (20)

Then, a250.75330, b251.42437 and the new second-ordmodel is given by

G2~z!51.75018z21.09857

1.91652z221.67357z10.40991(21)

Table 1 and Fig. 1, respectively, compare the approximation p

Fig. 1 Comparison of frequency responses

Table 1 Comparison of mean-squared-error „MSE…

Reduced modelMSE of impulse

responseMSE of unit-step

response

G2(z) 0.4734 0.9421G2(z) 0.0242 0.0176

Transactions of the ASME

r

r

i

t

i

mo

rem

t

r

a

ve. Bydis-

TC,ts oncityre-

e-

id-ing

thery

em

eitlytoryinu-ach,thegic

-

ain.

ndlow-

s

formance ofG2(z) andG2(z) in terms of the time responses anfrequency responses. Obviously,G2(z) is an improved model.

V ConclusionsAn improved BRA method has been presented in this pape

addition to the stability preserving and time-moments fitting proerties of the BRA method, the reduced model obtained consethe impulse response energy of the original model.

References@1# Hutton, M. F., and Friedland, B. F., 1975, ‘‘Routh approximations for reduc

order of linear, time-invariant systems,’’ IEEE Trans. Autom. Control,AC-20,pp. 329–337.

@2# Hwang, C., and Hsieh, C.-S., 1990, ‘‘Order reduction of discrete-time sysvia bilinear Routh approximation,’’ ASME J. Dyn. Syst., Meas., Control,112,pp. 292–297.

@3# Choo, Y., 1999, ‘‘An improvement to the Routh approximation method usimpulse energy criterion,’’ J. Elec. Eng. Inf. Sci.,4, pp. 594–599.

@4# Choo, Y., 1999b, ‘‘Improvement to modified Routh approximation methodElectron. Lett.,35, pp. 606–607.

@5# Hsieh, C.-S., and Hwang, C., 1989, ‘‘Model reduction of continuous-tisystems using a modified Routh approximation method,’’ IEE Proc.-D: Ctrol Theory Appl.,4, pp. 151–156.

Concurrent Design of Continuous ZeroPhase Error Tracking Controllerand Sinusoidal Trajectoryfor Improved Tracking Control

Hyung-Soon ParkGraduate Studente-mail: [email protected]

Pyung Hun ChangAssociate Professore-mail: [email protected]

Doo Yong LeeAssistant Professore-mail: [email protected]

Department of Mechanical Engineering, Korea AdvanceInstitute of Science and Technology, 373-1 Kusong-dong, Yusong-gu, Taejon, Korea

A trajectory control strategy for a nonminimum phase systemproposed. A continuous-time version of the Zero Phase ETracking Controller (ZPETC), which is a well-known discrettime feedforward controller, is considered. In the continuous-ticase, the overall transfer function consisting of the ZPETC athe closed-loop plant exhibits high-pass filter characteristics. Tintroduces serious gain errors between the desired and acoutput if the desired output is made directly as the ZPETC’s inpThis paper proposes the use of a specially designed sinusotrajectory to compensate for the gain errors. The sinusoidal tjectory imparts a synergic effect to tracking performance whcombined with the continuous ZPETC. Continuous ZPETC w

Contributed by the Dynamic Systems and Control Division of THE AMERICANSOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Dynamics Sytems and Control Division November 11, 1998. Associate Editor: E. Misawa.


d

. Inp-ves

ng

em

ng

,’’

en-

d

isror-e

ndhisualut.idala-enith

sinusoidal trajectory is evaluated successfully by applying tononminimum phase plant, single link flexible arm.@DOI: 10.1115/1.1343464#

1 IntroductionZero Phase Error Tracking Controller~ZPETC!, originally de-

veloped by Tomizuka@1#, has attracted attention as an effectiand simple remedy to the problem due to the unstable zeroseliminating phase error caused by uncancelled zeros, ZPETCplays good tracking performance.

The gain error, however, which cannot be cancelled by ZPEbecomes larger for fast tracking and causes undesirable effecthe tracking performance such as residual vibration and veloerror at via points. To resolve this problem, there have beensearch works: introducing additional zeros@2#; designing ZPETCwith optimal gain@3#; and approximating unstable zeros with sries @4#.

In this paper, a continuous-time version of ZPETC is consered. The continuous-time version may have such a shortcomas noise problem due to differentiating commands. Yet, ifcommand ispredefinedas in the most of the cases of the trajectoplanning, the shortcoming is avoidable.

Instead of reducing the gain error through modifying systcharacteristics, we attempt to achieve the same goal throughtra-jectory design. By exploiting the condition that the trajectory bpredefined, we can design it in such a fashion that it can expliccompensate the gain error. To this end, we adopt a trajecplanning method that generates trajectories consisting of ssoidal components with prescribed frequencies. In our approtherefore, the control system is designed concurrently withtrajectory to compensate the gain error providing a synereffect.

2 Continuous Zero Phase Error Tracking ControllerContinuous ZPETC~CZPETC! can be designed for the reduc

tion of the phase error in a similar way with ZPETC@1#. Let usconsider a transfer function expressed in continuous-time dom

y~s!

r ~s!5

N~s!

D~s!, (1)

whereD(s) is a denominator with ordern andN(s) is a numera-tor of orderm(n>m).

For a nonminimum phase system,N(s) can be divided into twoparts.

N~s!5Na~s!Nu~s!, (2)

whereNa(s) andNu(s) represent polynomials of stable zeros aunstable zeros, respectively. Let us design CZPETC as the foling equation,

r ~s!

yd~s!5

D~s!

Na~s!

Nu~2s!

@Nu~0!#2 . (3)

Then the overall transfer function between the input,yd(s), andthe output,y(s), is

y~s!

yd~s!5

Nu~2s!Nu~s!

@Nu~0!#2 . (4)

Substitutingj v for s,

Nu~ j v!5Re~Nu!1 j Im~Nu!,

Nu~2 j v!5Re~Nu!2 j Im~Nu!, and therefore

y~ j v!

yd~ j v!5

Re2~Nu!1Im2~Nu!

@Nu~0!#2 5uNu~ j v!u2

@Nu~0!#2 . (5)-


tp

a

n

ic

i

ri-ns.tra-

iasesve-re

thetorythisthetionnc-

thendilarfre-

ed-

bet-bef thethe

inys-zeroalthewed-di-id-free

n-as

Equation~5! shows that the overall transfer function betweendesired trajectory and the output does not have an imaginarythus the phase shift is always zero for all the frequencies.

In ~3!, the numerator has an order greater than the denominwhich implies CZPETC acts as a high pass filter. Thus CZPEmay seem to be susceptible to noise amplification. But this dnot cause a problem when the input is aknown smooth functionasin the most cases of planned trajectories. In other words, theferentiation is free from noise when the derivatives ofyd(t) aregiven analytically.

If the trajectory input can be designed to have finite frequecomponents, the gain error can be compensated by the moddesign of CZPETC i.e.,

r i~s!

ydi~s!5

D~s!Nu~2s!

Na~s!uNu~ j v i !u2 , (6)

whereydi is a component of the trajectory input that has theithfrequency,v i . The method to design such a trajectory with finfrequency components, and the corresponding feedforwardtroller is discussed in the next section.

3 Design of the Sinusoidal TrajectoryConsider the following trajectory that is composed ofn sine

functions:

yd~ t !5A01(i 51

n

Ai sin~v i~ t2t0!!, ~ t0<t<t f ! (7)

where t0 , t f and Ai( i 51,2, . . . ,n) are the initial time, the finaltime and the gain of each sinusoidal function, respectively. Tunknown variables of the above equation arev i ’s and Ai ’s. Wecan assign arbitrary values tov i ’s, which are less thanp/(t f2t0) and unduplicated. Then, then unknownAi ’s can be deter-mined from the user-defined constraints att0 and t f .

A trajectory which has the position, velocity, and acceleratconstraints att0 or t f , can be determined through the followinprocedure. Substitutingt0 into the Eq. ~7! gives yd(t0)5A0 .Hence, A0 is uniquely determined by the initial position constraint. Other constraints can be expressed in the following w

A01A1 sin~v1Dt !1A2 sin~v2Dt !1A3 sin~v3Dt !

1A4 sin~v4Dt !5yd~ t f !

A1v11A2v21A3v31A4v45 yd~ t0!

A1v1 cos~v1Dt !1A2v2 cos~v2Dt !1A3v3 cos~v3Dt !

1A4v4 cos~v4Dt !5 yd~ t f !

2A1v12 sin~v1Dt !2A2v2

2 sin~v2Dt !2A3v32 sin~v3Dt !

2A4v42 sin~v4Dt !5 yd~ t f !,

where

Dt5t f2t0 .

Fig. 1 CZPETC with sinusoidal trajectory: strategy for gainerror compensation

128 Õ Vol. 123, MARCH 2001

heart;

tor,TCoes

dif-

cyified

teon-

he

ong

-ay:

In the above equations,v i ’s ( i 51,2,3,4) have different valueswhich are arbitrarily chosen to satisfyv i,p/(t f2t0) so thatyd(t) may not have a fluctuating shape. Then the unknown vaables,Ai( i 51,2,3,4), are determined from the above equatioThis procedure can be straightforwardly extended to generatejectories with more constraints.

Equation~7! has a trivial constraint on acceleration att0 , i.e.,yd(t0)50. This implies that the acceleration constraints at vpoints must be zero for the continuity of acceleration. This cauno problem with stationary to stationary motion because thelocity and the acceleration at both the initial and the final time aalways zero. For the trajectories with nonstationary via points,sinusoidal trajectory may have a disadvantage that the trajeccannot have arbitrary acceleration at the via points. However,approach still guarantees the continuity of acceleration andsmooth trajectory. If it is necessary to have nonzero acceleraat the via points, a simple combination of sine and cosine futions could be applied.

If the v i ’s are chosen to be unduplicated and less thanp/(t f2t0), the sinusoidal trajectory has a similar shape as withpolynomial-type trajectory. Both of the sinusoidal trajectory athe polynomial trajectory satisfy the constraints and have simshapes. But the sinusoidal trajectory has a finite number ofquencies whereas the polynomial trajectory does not.

The sinusoidal trajectory has a synergic effect when combinwith CZPETC ~or ZPETC!, because the gain error can be compensated for each included frequency component, resulting inter tracking performance. As shown in Fig. 1, CZPETC’s candesigned to compensate the gain errors for each component osinusoidal trajectory so that the output has a unity gain overinput trajectory.

4 Simulation

4.1 Modeling of the Selected Plant. To verify the proposedmethod, a nonminimum phase system with unstable zeroscontinuous-time domain is considered. Dispersive flexible stems with nonpropagating energy are known to have unstabledynamics@5#. They are governed by a fourth-order differentiequation, and a flexible arm is one good example. To derivedynamic equations of the flexible arm having planar motion,consider the position of a point on the beam with virtual rigibody motion and deflection with respect to the rigid-body coornates by using a Bernoulli-Euler beam model. We use the rigbody coordinates that is attached at the base with the clamp-boundary condition as shown in Fig. 2~a! @6#.

Considering the rigid body motion and the first mode, the liearized dynamic equation of the flexible arm can be writtenfollows:

Fig. 2 The model of a single link flexible arm


o

rB

t

fwithemcelsgainn-his

nceith

-

d-

rol-ol,

k-rse

ort.,

ic.

rol,

ter-gnm-s if

for

-

-

FM11 M12

M21 M22GF u1

u2G1FF1 0

0 F2GF u1

u2G1F0 0

0 KG Fu1

u2G5F t

0G . (8)

The output to be controlled is the rotating angle of the tip.

4.2 The Results of the Simulation. In the simulation, thetrajectories,yd(t), are required to satisfy the following constraints,

yd~ t0!50 deg, yd~ t f !560 deg

yd~ t0!5 yd~ t f !50 deg/s,

yd~ t0!5 yd~ t f !50 deg/s2,

wheret050 s andt f50.5 s. In addition to the above constraintthe continuity of the CZPETC’s output is considered. Therefothe sixth order polynomial trajectory and the sinusoidal trajectof 5 frequency components are designed. In the sinusoidal tratory planning, the frequencies are arbitrarily selected as 1.p,1.1p, 1.2p, 1.3p, 1.4p~Hz!. In addition, CZPETC is combinedwith a feedback controller, Time Delay Controller@7#, so that theoverall control system has robustness.

As shown in Fig. 3, CZPETC follows the desired trajectoboth the polynomial and the sinusoidal, with no phase shift.the gain error exists for the polynomial trajectory, whereCZPETC with the sinusoidal trajectory compensates the gerror.

In the simulation, CZPETC is implemented by using a digicomputer with a fast sampling rate, 250 Hz. This may seem tomathematically incorrect because CZPETC is derivedcontinuous-time domain. However, the control scheme works wwhen the sampling frequency is sufficiently large compared tobandwidth of the plant.

Fig. 3 Simulation result: position response of CZPETC withTDC „sinusoidal trajectory versus polynomial trajectory …


-

s,re,ryjec-0

y,ut

asain

albeinellthe

5 ConclusionA feedforward controller, CZPETC for trajectory control o

nonminimum phase systems is developed. It can be usedmany types of feedback controllers to give the control systrobustness. Assuming that a trajectory is given, CZPETC canthe phase error caused by the uncancelled unstable zeros. Theerror, which is problematic in trajectory control, can be compesated by the concurrent design of the sinusoidal trajectory. Tfurther improves the performance of CZPETC. The performaof the proposed methods is confirmed through the simulation wthe flexible arm, known as a nonminimum phase system.

References@1# Tomizuka, M., 1987, ‘‘Zero Phase Error Tracking Algorithm for Digital Con

trol,’’ ASME J. Dyn. Syst., Meas., Control,109, pp. 65–68.@2# Haack, B., and Tomizuka, M., 1991, ‘‘The Effect of Adding Zeros to Fee

forward Controllers,’’ ASME J. Dyn. Syst., Meas., Control,113, pp. 6–10.@3# Funahashi, Y., and Yamada, M., 1993, ‘‘Zero Phase Error Tracking Cont

lers With Optimal Gain Characteristics,’’ ASME J. Dyn. Syst., Meas., Contr115, pp. 311–318.

@4# Gross, E., and Tomizuka, M., 1994, ‘‘Experimental Flexible Beam Tip Tracing Control with a Truncated Series Approximation to Uncancelable InveDynamics,’’ IEEE Trans. Control Syst. Technol.,2, No. 4, pp. 382–391.

@5# Miu, D. K., 1991, ‘‘Physical Interpretation of Transfer Function Zeros fSimple Control Systems With Mechanical Flexibilities,’’ ASME J. Dyn. SysMeas., Control,113, pp. 419–424.

@6# Kwon, D. S., and Book, W. J., 1994, ‘‘A Time-Domain Inverse DynamTracking Control of a Single-Link Flexible Manipulator,’’ ASME J. DynSyst., Meas., Control,116, pp. 193–200.

@7# Youcef-Toumi, K., and Ito, O., 1990, ‘‘A Time Delay Controller Design foSystems With Unknown Dynamics,’’ ASME J. Dyn. Syst., Meas., Contr112, pp. 113–142.

Quantitative Feedback Design UsingForward Path Decoupling1

Edward BojeSchool of Electrical and Electronic Engineering,University of Natal, Durban, 4041, South Africae-mail: [email protected]

Osita D. I. Nwokah2

Department of Mechanical Engineering, SouthernMethodist University, Dallas, Texas

The Perron-Frobenius root is used to measure the level of inaction in MIMO quantitative feedback design (QFT). A desiapproach to reduce interaction in uncertain plants via a precopensator is presented. A decentralized stability result appliethe interaction index can be made less than unity by design.@DOI: 10.1115/1.1343463#

1 IntroductionConsider the two-degree-of-freedomm-input, m-output feed-

back control system shown in Fig. 1. The transfer functiontracking is,

TY/R5~ I1PWG!21PWGF (1)

1An earlier version of this paper appeared inSymposium on Quantitative Feedback Theory and Other Frequency Domain Methods, University of Strathclyde,1997.

2Dr. Nwokah passed away in April 1999.Contributed by the Dynamic Systems and Control Division of THE AMERICAN

SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Dynamics Systems and Control Division February 24, 1997. Associate Editor: E. Misawa.


d

pmt

p

r

i

i

l

cs

le

n-

e

ky

-

f

We consider the QFT problem~Horowitz, @1–3#! of specifying aprecompensator matrix,W, with wii 51, a diagonal feedbackcontroller, G, and a prefilter,F, to achieve certain trackingspecifications,

Ai j ~v!<uTY/R~ j v!u i j <Bi j ~v!, i , j 51,2, . . . ,m

Ai j (v) and Bi j (v) are client ~desired! specifications, for allplants,PP$P%, a set of uncertain plants.P is assumed regular withP5P21, and pi j 51/qi j .

Our interest is to design the off-diagonal elements of thenamic precompensator~also called control authority allocation odecoupler!, W, in order to ease the subsequent design ofG andreduce over-design. We address robust decoupling over theuncertainty set since decoupling based on a nominal plantmake design more difficult, especially in ill-conditioned planApproximately decoupled designs are axiomatic imperativesgeneral engineering design theory~Suh @4#!. Within the systembandwidth~low frequency! it is often possible to have high loogain, and outside the system loop bandwidth~high frequency!feedback benefits are no longer available. The critical frequerange around the loop gain crossover frequencies may betively small and often a static decoupler will be satisfactory.sparsely populated decoupler may be desirable.

There are only two independent signals and without ovdesign, the performance that can be obtained using the structuFig. 1 is obviously not different to that obtained using the QNstructure of Nwokah, et al.@5# ~or any other fully populated two-degree-of-freedom configuration!. With the two-loop design of theQNA approach, the level of interaction in the inner loop mustmade small enough to guarantee not only decoupling beforeinner loop is closed, but sufficient decoupling that the outer lowill have desirable properties when closed. This may requirenecessary over-design.

Apart from the QNA approach that motivated this work, theare several other approaches to design of off-diagonal elemenQFT. Yaniv and Chepovetsky@6# use quadratic optimization ovean enumerated set of plants to design a series decoupler. Yan@7#has used static decoupling at high frequency but over the untainty set. This requires that the high frequency plant uncertais only gain~i.e., no dead time uncertainty and fixed order!. Chenget al. @8# and Boje @9# have designed dynamic cross-feed forrotorcraft. Chang et al.@10# have applied the Perron-Frobeniubased decoupling methods summarized by Maciejowski@11# toobtain diagonal weighting~i.e., input scaling! for a nominal plant,followed by QFT-style design. There may be off-diagonal ements in the feedback path,H, for example in power system stabilizers ~Boje et al.@12#!.

Parallel connection, havingG5GO1GD , with GD diagonalandGO having zeros on the diagonal, has also been investigaHorowitz @3# mentions the use of the off-diagonal elementsreduce the interaction by usinggi j to center the template of elementqi j at the origin. Franchek et al.@13# investigated sensitivityreduction with a parallel structure.

Section 2 discusses the required Perron-Frobenius root theSection 3 details the proposed design method, including a detralized stability result, precompensator design, and commentdiagonal controller and prefilter design. Section 4 concludespaper.

2 Preliminary MathematicsNon-negative matrix theory is used and the reader is referre

Berman and Plemmons@14#; Seneta@15#; Nwokah et al. @5#;Nwokah and Perez@16#; and Limebeer@17#.

Fig. 1 Two-degree-of-freedom feedback structure

130 Õ Vol. 123, MARCH 2001

y-r

lantay

s.in

ncyela-A

er-re ofA

betheopun-

rets inrvcer-nty

as

e--

ted.to-

ory.en-on

the

d to

Definition 1: A matrix, APRn3n with A5r I2B, r .0, B>0 isa nonsingularM-matrix if r .r(B), the spectral radius ofB.

Definition 2: The comparison matrixof ZPCn3n, M (Z), haselements,

M ~Z! i j 5H uzii u ~diagonal elements!

2uzi j u iÞ j

Definition 3: The matrix,ZPCn3n is irreducible if there doesnot exist a permutation,ZPCn3n such thatPZP215@ 0

Z11Z22

Z12#,

with square submatrices,Z11 and Z22. For control problems, re-ducibility would allow separate designs for each irreducibsubsystem.

Definition 4: An irreducible matrix,ZPCn3n is an H-matrix~Hadamard matrix! if M (Z) is anM-matrix. ~Berman and Plem-mons@14#, p. 184.!

Theorem 1, ~Perron-Frobenius!: Given APRn3n, irreducibleand non-negative, with spectral radius,r~A!. There exists an ei-genvalue,lP(A) ~the Perron-Frobenius root!, such that:

~a! r(A)5lP(A).0.~b! lP(A) can be associated with positive right and left Perro

Frobenius eigenvectors,xp.0 andypT.0 respectively that can be

scaled so thatxpTxp51 andyp

Txp51.~c! Given B, with A>B>0, thenul(B)u<lP(A).~d! lP(A) is an ~algebraically! simple eigenvalue ofA.

Proof: Seneta@15#, p. 22.Theorem 2: Given APRn3n, irreducible and non-negative, th

differential sensitivity oflP(A) with respect to elements ofA, is

]lP

]ai j5

yixj

yTx.0 (3)

Proof: lP is simple~Theorem 1d! and therefore analytic withrespect toai j . The result follows from Lancaster and Tismenets@18#, p. 396 and monoticity follows from Theorem 1b.

Definition 5: A matrix, APRn3n admits aregular splitting if itcan be written asA5D2C with D21>0 andC>0. ~Berman andPlemmons@@14#, p.182.#! In addition, if D is diagonal, the split-ting is diagonal.

Theorem 3: Given ZPCn3n and regular splitting,A5M (Z)5D2C, with D21>0 and C>0. Let M5D21C. If lP(M ),1thenA has aconvergentregular splitting,A is anM-matrix andZis anH-matrix.

Proof: Berman and Plemmons@@14#, p. 138#.Definition 6: In Theorem 3,M is the interaction matrix, and

g(Z)5lP(M ) is the interaction indexof Z.Notice that if Z is row dominant,uzii u.( j 51,j Þ1

n uzi j u for all i,g~Z!,1, so that row dominance is a special case of requiringZ tobe anH-matrix, as is column dominance.

3 Application of Interaction Index to MIMO Feed-back Design

3.1 Stability Result. A robust, decentralized, inverse Nyquist stability theorem is now developed. Split diagonal~subscriptD! and off-diagonal~subscriptO! components of Eq.~1!.

~ I1PWG!TY/R5PWGF(4)

~P1WG!TY/R5~PD1G!~ I1X!TY/R5WGF

with X5(PD1G)21(PO1(W2I )G), andWD(s)5I , G(s) andPD(s) diagonal.

Theorem 4: Given:

~a! PP$P%, square and regular, havingNPO ~open loop! polesand NZO ~open-loop transmission! zeros in the closed right halplane~RHP!, and no hidden unstable modes.

~b! (I1PWG)21P is analytic at the closed RHP poles ofP~including infinity!.


-

t

t

rte

n

hi

n

re

inott offor-

ter-

e

nal

for

el-Forr. A

lting

tof

a

m’’taticimi-tedple

sns

be

~c! F, W, andG are stable andG is minimum phase by design~d! r~X!,1 for all PP$P% and forsPdD, the Nyquist contour

covering the RHP.

Then, TY/R in Eq. ~4! is stable iff each diagonal loop,l i5giqii is designed so that, for allPP$P%, the loci of 1/l i , takentogether, encircle the (21,j 0) point ND52NZO times ~in theopposite direction! ass traverses the Nyquist contour.

Proof: Following Maciejowski @11#, Theorem 2.8, let~I1PWG! have NCL RHP zeros.P has NPO RHP poles andNZO RHP zeros. Assumption ~c!⇒Darg det~G!50. r~X!,1⇒Darg~det~I1X!!50. Now, applying the principle of argument around the Nyquist contour,

D arg det~ I1PWG!5D arg det~P!1D arg det~PD1G!

1D arg det~ I1X!

5D arg det~P!1(i 51

m

~D arg~1/l i11!!

1D arg det~G!10

22p~NCL2NPO!522p~NZO2NPO!22pND

TY/R stable⇔NCL50⇔NZO52ND .

Comments

1 It is impractical and unnecessary to requirer~X!,1 for allfrequencies. Around any closed contour wherer~PWG!,1,Darg~det~I1PWG!!50. The Nyquist path can be adjustedcover the relevant portion of the RHP.

2 G may have poles at the origin if a suitable adjustment toNyquist contour is made.

3 If W andG are designed so that the~I1X! is anH-matrix,r~X!,1 aslP(uXu)>r(X). This is discussed in Section 3.2.

4 Theorem 4 applies to an enumerated set of plants. Altetively, it is sufficient to ensure that a single plant in a connecplant set is stable and deduce stability for the remaining elemof the plant set via the zero exclusion principle.

3.2 Design for Reduction of Interaction Before QFTMIMO Design. Internal stability is a minimum requirement inormal engineering design. The Perron-Frobenius root has bused for analysis of interaction in feedback design with fixplants for many years and has been applied to QFT designplants with large structured or unstructured uncertainty by Nwkah et al.@5#. The interaction index depends on the design ofWandG. In the QFT approach, Eq.~4! is written implicitly as,

TY/R5~PD1G!21WGF2XTY/R (5)

The smaller the interaction matrix elements are, the less inheover-design will occur in MIMO QFT design ifTY/R is over-bounded by its specification on the right-hand side of Eq~5!. Thisjustifies reducing interaction at appropriate frequencies overwhole set of plants if possible, independent of the requirementdecentralized stability in Theorem 4. If the plant has no unctainty, pseudo inversion of the plant dynamics around the croover frequency can be used. Reducing interaction over the wplant set may not be possible, especially if the plant isconditioned~Gjo”sæter and Foss,@19#!.

DefineS5diag$1/(11 l i)% andT5diag$li /(11li)%, the diagonalsensitivity and complementary sensitivity. The interaction indexcomplex frequency,s, is found from Eq.~4!, via the~maximum!Perron-Frobenius eigenvalue problem,

uPOPD21S1~W2I !Tux5uNux5xlP . (6)

with nii 50 and,

Journal of Dynamic Systems, Measurement, and Control

.

o

he

na-ednts

eenedforo-

rent

theforer-ss-olell-

at

uni j u5Uqj j

qi jsj1wi j t jU

5US qj j

qi j2wi j D sj1wi jU

5Uqj j

qi j

1

l j1wi jUut j u, iÞ j (7)

The following design insights follow from the fact thatlP ismonotonic with respect to the elements ofN.

1 If u l j u!1, ~outside the loop bandwidth!, feedback is ineffec-tive anduni j u'uqj j /qi j u ~the interaction index is that of the opeloop plant!.

2 When u l j u@1, ~at frequencies where feedback benefits arequired in thejth loop!, the low sensitivity ensures thatuni j u'uwi j u. This motivates choosing small off-diagonal elementsW if possible. Interaction when the loop gain is large is often na real problem for tracking as the high gain reduces the effecuncertainty and a prefilter can be used to impose tracking permance.

3 Around the crossover frequency ofl j , u l j u'1, the designdifficulties are most severe and it is desirable to have low inaction if possible. As discussed in Nwokah et al.@5#, in a slightlydifferent context, the design ofwi j may be undertaken on thinverse Nichols chart: Specify a bound,a i j ,1, for reduction ofuni j u compared to the value forwi j 50,

uni j u5uqj j /qi j sj1wi j t j u<a i j uqj j /qi j sj u, (8)

and design,

U 1

11wi j qi j gjU>a i j

21. (9)

In Eq. ~9!, an existing design forgj ~tuning! is required or (wi j gj )must be designed with sufficient roll-off so that after the diagocontroller has been designed,wi j 5wi j gj /gj is ~strictly! proper.Good design insight can be obtained by drawing value setsdifferent a i j ’s on the inverse Nichols chart.

Alternatively, becausel j is usually a designoutcome, availablesensitivity specifications,u1/(11 l j )u<b j (v), (b j.0dB), andthe Schwartz inequality can be applied to Eq.~7!, to give,

uni j u<uqj j /qi j 2wi j ub j1uwi j u (10)

This suggests seeking a low gainwi j that makes uqj j /qi j2wi j ub j small. As above, specifya i j ,1 and designu1/(12wi j qi j /qj j )u>a i j

21 with low gain wi j if possible.

4 In many problems, reducing interaction by off-diagonalements is only necessary over a restricted frequency range.example, Boje and Nwokah@20#, use QFT templates at a numbeof frequencies near the crossover frequency simultaneouslyreal-valued static precompensator to approximate the resufixed, complexW was then obtained using Kouvaritakis’ ALIGNalgorithm ~Maciejowski @11#!.

5 The interaction index differential sensitivity with respectni j , (yixj /yTx) from Eq. ~3!, can be used to judge the impact oreducing the magnitude of particular elements ofN.

6 The minimum achievable interaction index depends incomplicated way over the plant set on the elements ofX. As aresult, we should not expect to be able to construct an ‘‘optimuprecompensator using the approach above. Especially if a sprecompensator is required, it is feasible to use nonlinear optzation to improve a manual design. The outcome of automadesign can be displayed on the inverse Nichols chart, for examto establish sensitivity to component tolerances.

3.3 Diagonal Controller Design. Once a precompensator ichosen, Eq.~5! is a standard QFT design problem. Specificatioon individual loop sensitivities for interaction reduction may

MARCH 2001, Vol. 123 Õ 131

s

s

-

ta

o

le

odf

t

nm

o

on-

on-

of

n-s,’’

-

in

nce

rsus

ckal

ingnd,

msub-arethe

itor:

required in addition to uncertainty reduction and disturbancejection specifications. It may be convenient to recalculate Eq.~4!as,

TY/R5~WP1G!21GF5~P* 1G!21GF (11)

Equation ~11! represents a loop breaking between the diagocontroller and the precompensator while Eq.~4! represents a loopbreaking point at the plant input. For poorly conditioned prolems, the interaction index found from Eq.~11! may be quitedifferent to the interaction index found from Eq.~4!. This is thesubject of further investigation.

3.4 Prefilter Design for ‘‘Basically Non-Interacting’’Tracking. Once the feedback loop design has been fixed toisfy uncertainty reduction and regulation specifications, the traing response can be shaped with prefilter,F. For basically nonin-teracting specifications, if sufficiently high gain feedbackpossible, usuallyF5diag$fii% suffices. In practical designs this ioften not possible but low closed-loop interaction mayachieved. IfT5(I1L )21LPH, it can be regarded as ‘‘almoslower triangular’’ after permutation~row and column swapping!.It may then be possible to reduce interaction~off-diagonal gain! inthe tracking behavior,TY/R5TF, by designing off-diagonal elements inF ~lower triangular by the same permutation!. This ideahas been illustrated by a design in Boje and Nwokah@20#!.

4 ConclusionsThis paper has developed the application of the Perr

Frobenius root interaction index to feedback design for uncermultivariable plants. A decoupling precompensator in the forwpath, between the diagonal controller and the plant, is designerobustly reduce the level of interaction between loops befquantitative design of a diagonal feedback controller is undtaken. If the interaction index can be made less than unity bydesign, stability of the diagonal loop designs guarantees stabof the closed-loop multivariable system. The interaction index dferential sensitivity identifies the interaction matrix elements wsignificant contribution to the interaction index. Robust controldesigns for a turbofan engine~with static precompensator, Bojand Nwokah@20#!, and for a helicopter~with sparse, dynamicprecompensator, Boje@9#!, have been undertaken based on tapproach presented in this paper.

AcknowledgmentsThorough, critical review has helped to improve this paper. T

first author acknowledges the support of the South African Ntional Research Foundation~NRF! and the University of Natal.Prof. Nwokah acknowledged support from USAF AerPropulsion Lab; Wright Labs; U.S. Army Troop CommanNASA Ames-RC; and Control Analysis Division, GE AircraEngines, Avondale, OH.

References@1# Horowitz, I., 1979, ‘‘QFT for uncertain MIMO systems,’’ Int. J. Control,30,

pp. 81–106.@2# Horowitz, I., 1982, ‘‘Improved technique for uncertain MIMO systems,’’ In

J. Control,36, pp. 977–988.@3# Horowitz, I., 1991, ‘‘Survey of Quantitative Feedback Theory~QFT!,’’ Int. J.

Control,53, No. 2, pp. 255–291.@4# Suh, N. P., 1990,Principles of Design, Oxford University Press.@5# Nwokah, O. D. I., Nordgren, R. E., and Grewal, G. S., 1995, ‘‘Inverse Nyqu

Array: A quantitative theory,’’ IEE Proc.-D: Control Theory Appl.,142, No. 1,pp. 23–30.

@6# Yaniv, O., and Chepovetsky, I., 1995, ‘‘Robust control of MIMO plants: Nodiagonal QFT design,’’Quantitative and Parametric Feedback Theory Syposium, Purdue University, Aug., pp. 69–78.

@7# Yaniv, O., 1995, ‘‘MIMO QFT using non-diagonal controllers,’’ Int. J. Control, 61, No. 1, pp. 245–253.

@8# Cheng, R. P., Tischler, M. B., and Biezad, D. J., 1995, ‘‘Rotorcraft FligControl design using Quantitative Feedback Theory and Dynamic Crfeeds,’’ Quantitative and Parametric Feedback Theory Symposium, PurdueUniversity, IN, Aug., pp. 23–39.

132 Õ Vol. 123, MARCH 2001 Copyright ©

re-

nal

b-

at-ck-

is

bet

on-ainrdd tore

er-theilityif-

ither

he

hea-

-;

t

.

ist

--

-

htss-

@9# Boje, E., 1999, ‘‘Quantitative Digital Design of Crossfeed and Feedback Ctrollers for the UH-60 Black Hawk Helicopter,’’International Symposium onQuantitative Feedback Theory and Robust Frequency Domain Methods, Uni-versity of Natal, Durban, 26 and 27 Aug.

@10# Chang, J.-C, Chang Y.-H, and Chen, L.-W, 1998, ‘‘Robust Decoupled Ctroller Design with Quantitative Feedback Theory,’’American Control Con-ference, Philadelphia, Pennsylvania, June, pp. 2481–2485.

@11# Maciejowski, J. M., 1989,Multivariable Feedback Design, Addison-Wesley,Reading, MA.

@12# Boje, E., Nwokah, O. D. I., and Jennings, G., 1999, ‘‘Quantitative DesignSMIB Power System Stabilisers using Decoupling Theory,’’1999 IFAC WorldCongress, Beijing, China, 5–9 July.

@13# Franchek, M. A., Herman, P., and Nwokah, O. D. I., 1997, ‘‘Robust Nodiagonal Controller Design for Uncertain Multivariable Regulating SystemASME J. Dyn. Syst., Meas., Control,119, pp. 80–85.

@14# Berman, A., and Plemmons, R. J., 1979,Nonnegative matrices in the Mathematical Sciences, Academic Press, New York.

@15# Seneta, E., 1981,Non-negative Matrices and Markov Chains, Springer-Verlag,New York.

@16# Nwokah, O. D. I., and Perez, R., 1991, ‘‘On Multivariable Stability in the GaSpace,’’ Automatica,27, No. 6, pp. 975–983.

@17# Limebeer, D. J. N., 1982, ‘‘The application of generalised diagonal dominato linear system stability theory,’’ Int. J. Control,36, No. 2, 185–212.

@18# Lancaster, P., and Tismenetsky, M., 1985,The Theory of Matrices~SecondEdition!, Academic Press, NY.

@19# Gjo”sæter, O. B., and Foss, B. A., 1999, ‘‘On the use of diagonal control vedecoupling for ill conditioned processes,’’ Automatica,33, No. 3, pp. 427–432.

@20# Boje, E., and Nwokah, O. D. I., 1999, ‘‘Quantitative Multivariable FeedbaDesign for a Turbofan Engine with Forward Path Decoupling’’ InternationJournal of Robust and Non-linear Control,9, No. 12, pp. 857–882.

@21# Boje, E., and Nwokah, O. D. I., 1997, ‘‘Quantitative Feedback Design UsForward Path Decoupling,’’Symposium on Quantitative Feedback Theory aOther Frequency Domain Methods, University of Strathclyde, 21–22 Aug.pp. 185-191.

Stabilization of Discrete SingularlyPerturbed Systems UnderComposite Observer-Based Control

Feng-Hsiag Hsiao1

Department of Electrical Engineering,Chang Gung University,259, Wen-Hwa 1st Road,Kwei-San, Taoyuan Shian, Taiwan 333, R.O.C.

Jiing-Dong HwangDepartment of Electronic Engineering,Jin-Wen Institute of Technology,99, An Chung Road,Hsin Tien, Taipei, Taiwan 231, R.O.C.

Shing-Tai PanDepartment of Electrical Engineering,Kao Yuan Institute of Technology,1821, Chung-Shan Road,Lu-Ghu Hsiang, Kaohsiung, Taiwan 821, R.O.C.

New stability conditions for discrete singularly perturbed systeare presented in this study. The corresponding slow and fast ssystems of the original discrete singularly perturbed systemfirst derived. The observer-based controllers for the slow and

1Author to whom all correspondence should be addressed.Contributed by the Dynamic Systems and Control Division of THE AMERICAN

SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the DynamicSystems and Control Division February 2, 1999. Associate Technical EdB. Siciliano.

2001 by ASME Transactions of the ASME

-

o

p

l

v

n

d

d

eneegtdf

i

t

s

b

e

dis-ed

t:

am-delsreac-ys-tors

etc.the

ach

s

-

tem

fast subsystems are then separately designed and a compobserver-based controller for the original system is subsequesynthesized from these observer-based controllers. Finally, aquency domaine-dependent stability criterion for the original discrete singularly perturbed system under the composite obserbased controller is proposed. If any one condition of this criteriis fulfilled, stability of the original system by establishing thatits corresponding slow and fast subsystems is thus investigaAn illustrative example is given to demonstrate that the upbound of the singular perturbation parametere can be obtainedby examining this criterion.@DOI: 10.1115/1.1285759#

1 IntroductionSingularly perturbed systems have been studied by many

searchers in recent years; see, for example,@1# and the referencestherein. This is due not only to theoretical interest but also torelevance of this topic to control engineering applications. Indethe singular perturbation approach provides us with a powetool for the order reduction and separation of time-scales@2#. Afundamental feature of such an approach is that the feedbacksign problem can be broken into two design subproblems forslow and fast subsystems. The two designs are then combinegive a design for the original systems@3#.

A key to the analysis of singularly perturbed systems thusin the construction of the slow and fast subsystems. It is notedthe approximation of the original singularly perturbed systemits corresponding slow and fast subsystems is valid only whensingular perturbation parameters of this system are sufficiesmall. Therefore, it is important to find the upper bound of singlar perturbation parameters such that stability of the original stem can be investigated by establishing that of its corresponslow and fast subsystems, provided that the singular perturbaparameters are within this bound. Numerous reports in regarthis subject~without control! have been published@2,4–8#.

The most significant advantage of singularly perturbed systis the two-time-scale property which permits separate desigfeedback controls for the slow and fast subsystems and thcomposite feedback control can be synthesized from these fback controls. This simplifies the procedure of controller desiHence, in this study, the corresponding slow and fast subsysof the original discrete singularly perturbed system are firstrived. The observer-based controllers for the slow and thesubsystems are then separately designed and a compobserver-based controller for the original system is subsequesynthesized from these observer-based controllers. A frequedomain e-dependent stability criterion is proposed to examwhether the singular perturbation parametere is small enough ornot. If e is so small that any one condition of this criterionsatisfied, then stability of the slow and fast subsystems can imthat of the original system under the composite observer-bacontroller. An illustrative example is given to demonstrate thatupper bound of the singular perturbation parametere can be ob-tained by examining this criterion. This work may be viewed ageneralization of Li and Li@2# to the design of compositeobserver-based controllers for discrete singularly pertursystems.

2 Problem FormulationConsider the following discrete singularly perturbed syst

which is referred to as the C-model@9#

x1~k11!5(i 50

n

A1ix1~k2 i !1e(i 50

n

A1ix2~k2 i !1B1u~k!

(2.1a)


ositentlyfre-

ver-n

ofted.er

re-

theed,rful

de-thed to

iesthatiathetlyu-ys-ingtion

to

msof

n aed-n.

emse-astositentlyncy

ne

isplysedhe

a

ed

m

x2~k11!5(i 50

n

A2ix1~k2 i !1e(i 50

n

A2ix2~k2 i !1B2u~k!

(2.1b)

y~k!5C1x1~k!1C2x2~k!, (2.1c)

whereA1i , A2i , A1i , A2i ~for i 50,1,2,...,n!, B1 , B2 , C1 , andC2are constant matrices with appropriate dimensions. System~2.1!can be got from the slow sampling rate model as a result ofcretization or sampled-data control of singularly perturbcontinuous-time systems@2#. The small positive scalare is a sin-gular perturbation parameter subject to the following constrain

e(i 50

n

iA2i i,1 (2.2)

and it often occurs naturally due to the presence of small pareters in various physical systems, e.g., in power system mothe singular perturbation parameters can represent machinetances or transients in voltage regulators. In industrial control stems they may represent time constants of drives and actuaand in nuclear reactor models they are due to fast neutrons,

Before proceeding to the main result, a lemma is given infollowing.

Lemma 2.1@10#: For any matrixAPRm3m, if r@A#,1,2 thenudet(I6A)u.0.

Based on Lemma 2.1 and the fact that

rF e(i 50

n

A2i G<eI(i 50

n

A2i I<e(i 50

n

iA2i i,1

it is clear the matrix

S I 2e(i 50

n

A2i Dis nonsingular. Now, according to the quasi-steady-state approas that in ~@11#, p. 130! and ~@12#, p. 276! the slow and fastsubsystems of the original system~2.1! can then be derived afollows.

2.1 The Slow Subsystem. Formally, letting x2(k11)5x2(k2 i ) for i 50,1,2,...,n in Eqs.~2.1! is equivalent to assuming that after the decay of fast modes, the system of Eqs.~2.1!reaches the quasi-steady state described by the slow subsys

xs~k11!5(i 50

n

A1ixs~k2 i !1eS (i 50

n

A1i D x2~k!1B1us~k!

(2.3a)

x2~k!5(i 50

n

A2ixs~k2 i !1eS (i 50

n

A2i D x2~k!1B2us~k!

(2.3b)

ys~k!5C1xs~k!1C2x2~k! (2.3c)

where x2 denotes the discrete ‘‘quasi-steady state’’ ofx2(k).Moreover,xs , us andys are the slow components ofx1 , u andy,respectively. From~2.3b!, we have

x2~k!5S I 2e(i 50

n

A2i D 21

(i 50

n

A2ixs~k2 i !

1S I 2e(i 50

n

A2i D 21

B2us~k!. (2.4)

Substituting~2.4! into ~2.3a! and~2.3c!, the slow subsystem of theoriginal system~2.1! can be expressed as

2The notationr@A# denotes the spectral radius of the matrixA.

MARCH 2001, Vol. 123 Õ 133

s

i

t

xs~k11!5(i 50

n

Asixs~k2 i !1Bsus~k!, (2.5a)

ys~k!5(i 50

n

Csixs~k2 i !1Dsus~k!, (2.5b)

where

Asi5A1i1eS (j 50

n

A1 j D S I 2e(j 50

n

A2 j D 21

A2i ,

Bs5B11eS (i 50

n

A1i D S I 2e(i 50

n

A2i D 21

B2 ,

Cs05C11C2S I 2e(i 50

n

A2i D 21

A20,

Csi5C2S I 2e(j 50

n

A2 j D 21

A2i , Ds5C2S I 2e(i 50

n

A2i D 21

B2 .

2.2 The Fast Subsystem. To derive the fast subsystem it iassumed that the slow variables are constant during the inshort transient period in which the fast modes are active,x1(k11)5x1(k2 i )5xs(k2 i )5xs(k11) for i 50,1,2,...,n. De-fining xf(k)5x2(k)2 x2(k), yf(k)5y(k)2ys(k), uf(k)5u(k)2us(k) andus(k)5us(k2 i ), we have~from ~2.4!!

xf~k!5x2~k!2S I 2e(i 50

n

A2i D 21S (i 50

n

A2i D xs~k!

2S I 2e(i 50

n

A2i D 21

B2us~k!. (2.6)

According to~2.1b! and~2.1c!, the fast subsystem of the originasystem~2.1! is derived as follows:

xf~k11!5e(i 50

n

A2ixf~k2 i !1B2uf~k! (2.7a)

yf~k!5C2xf~k! (2.7b)

3 Observer-Based Controller Design for the Slow andthe Fast Subsystems

The design of observer-based controller for stabilizing discrtwo-time-scale systems was discussed in@13–15#. In this section,the observer-based controllers for the slow subsystem~2.5! andfor the fast subsystem~2.7! are separately designed such that bosubsystems are stable.

3.1 Controller Design for the Slow Subsystem. By defin-ing a new slow state vectorXs(k)5@xs(k),xs(k21),...,xs(k2n)#T, the slow subsystem~2.5! can be rewritten as

Xs~k11!5AsXs~k!1Bsus~k!,(3.1)

ys~k!5CsXs~k!1Dsus~k!,

where

As5F As0 As1 ¯ As~n21! Asn

I 0 ¯ 0 0

0 I ¯ 0 0

] ] � ] ]

0 0 ¯ I 0

G , Bs5F Bs

00]

0

G ,

Cs5@Cs0 Cs1 ¯ Cs~n21! Csn#, Ds5Ds .

134 Õ Vol. 123, MARCH 2001

itial.e.,

l

ete

th

Assumption 3.1:The slow subsystem~3.1! is stabilizable anddetectable.

The observer-based controller for the slow subsystem~3.1! isgiven by

Xs~k11!5~As2BsKs2FsCs!Xs~k!1Fsus~k!2FsDsus~k!,(3.2)

ys~k!5KsXs~k!,

where the two constant matricesKs andFs are chosen such thaAs2BsKs andAs2FsCs are both Hurwitz~i.e., all eigenvalues ofAs2BsKs andAs2FsCs lie inside the unit disk!. According to thefeedback interconnection as depicted in Fig. 1 and lettinges(k)[Xs(k)2Xs(k), we have

Xs~k11!5AsXs~k!1Bs@2KsXs~k!#

5~As2BsKs!Xs~k!1BsKses~k!(3.3)

es~k11!5Xs~k11!2Xs~k11!5~As2FsCs!es~k!,

or

FXs~k11!

es~k11! G5F As2BsKs BsKs

0 As2FsCsG FXs~k!

es~k! G5MsFXs~k!

es~k! G ,(3.4)

where

Ms[F As2BsKs BsKs

0 As2FsCsG .

Since the matricesAs2BsKs andAs2FsCs are both Hurwitz, theclosed-loop system~3.4! is thus stable.

Remark 3.1:If As is Hurwitz, then bothKs and Fs can bechosen to be zero.

3.2 Controller Design for the Fast Subsystem. By defin-ing a new fast state vectorXf(k)5@xf(k),xf(k2 l ),...,xf(k2n)#T, the fast subsystem~2.7! can be rewritten as

Xf~k11!5AfXf~k!1Bfuf~k!,(3.5)

yf5CfXf~k!,

where

Fig. 1 The observer-based controller for the slow subsystem„Eqs. „3.1……


e

the

Af5F eA20 eA21 ¯ eA2~n21! eA2n

I 0 ¯ 0 0

0 I ¯ 0 0

] ] � ] ]

0 0 ¯ I 0

G ,

Bf5F B2

00]

0

G , Cf5@C2 0 0 ¯ 0#.

Assumption 3.2:The fast subsystem~3.5! is stabilizable anddetectable.

The observer-based controller for the fast subsystem~3.5! isgiven by

Xf~k11!5~Af2BfK f2F fCf !Xf~k!1F fuf~k!,(3.6)

y f~k!5K fXf~k!,

where the two constant matricesK f andF f are chosen such thaAf2BfK f and Af2F fCf are both Hurwitz. Under the feedbacconfiguration of Fig. 2 and lettingef(k)[Xf(k)2Xf(k), we have

Fxf~k11!

ef~k11!G5F Af2BfK f BfK f

0 Af2F fCfG Fxf~k!

ef~k!G5M fFxf~k!

ef~k!G ,(3.7)

where

M f[F Af2BfK f BfK f

0 Af2F fCfG .

Since the matricesAf2BfK f andAf2F fCf are both Hurwitz, theclosed-loop system~3.7! is thus stable.

In the next section, a composite observer-based controllerthe original system~2.1! is subsequently synthesized from th‘‘slow’’ controller ~3.2! and the ‘‘fast’’ controller~3.6!. It is ofinterest to explore the stability relationship of the systems~3.4!and~3.7! with respect to the original system~2.1! under the com-posite observer-based controller. The corresponding theoreconsequence is stated in the next section.

Fig. 2 The observer-based controller for the fast subsystem„Eqs. „3.5……


tk

fore

tical

4 Composite Observer-Based Controller for the Origi-nal System

By defining the new state vectorsX1(k)5@x1(k),x1(k21),...,x1(k2n)#T and X2(k)5@x2(k),x2(k21),...x2(k2n)#T, the original system~2.1! can then be transformed into thfollowing form:

X1~k11!5A1X1~k!1A2X2~k!1B1u~k!

X2~k11!5A2X1~k!1A2X2~k!1B2u~k! (4.1)

y~k!5C1X1~k!1C2X2~k!,

where

A15F A10 A11 ¯ A1~n21! A1n

I 0 ¯ 0 0

0 I ¯ 0 0

] ] � ] ]

0 0 ¯ I 0

G ,

A25F eA10 eA11 ¯ eA1~n21! eA1n

0 0 ¯ 0 0

0 0 ¯ 0 0

] ] ] ] ]

0 0 ¯ 0 0

G , B15F B1

00]

0

G ,

A35F A20 A21 ¯ A2~n21! A2n

0 0 ¯ 0 0

0 0 ¯ 0 0

] ] ] ] ]

0 0 ¯ 0 0

G ,

A45F eA20 eA21 ¯ eA2~n21! eA2n

I 0 ¯ 0 0

0 I ¯ 0 0

] ] � ] ]

0 0 ¯ I 0

G , B25F B2

00]

0

G ,

C15@C1 0 0 ¯ 0#, C25@C2 0 0 ¯ 0#.

The system~4.1! can be rewritten in a more compact form:

X~k11!5AcX~k!1Bcu~k!(4.2)

y~k!5CcX~k!,

in which

X~k!5FX1~k!

X2~k!G , Ac5F A1 A2

A3 A4G , Bc5F B1

B2G ,

Cc5@C1 C2#.

Now, a composite observer-based controller is proposed infollowing lemma to stabilize the original system~4.2! for a suffi-ciently smalle.

Lemma 4.1@13,14#: If the gains ~Ks , Fs! and ~K f , F f! aredesigned such that the systems~3.4! and~3.7! are both stable, thenthe following composite observer-based controller~4.3! can stabi-lize the original system~4.2!, or equivalently system~4.1!, pro-vided thate is sufficiently small.

X~k11!5~Ac2BcKc2FcCc!X~k!1Fcu~k!(4.3)

MARCH 2001, Vol. 123 Õ 135

I

in-

-s

ilityile

by

r-:

y~k!5KcX~k!,

where

u~k!5y~k!, u~k!52 y~k!, X~k!5F X1~k!

X2~k!G ,

(4.4)

Kc~k!5FKc1~k!

Kc2~k!G , Fc~k!5FFc1~k!

Fc2~k!Gand

Kc15@ I 1K f~ I 2A4!21B2#Ks2K f~ I 2A4!21A3 ,(4.5a)

Kc25K f

and

Fc15Fs@ I 1C2~ I 2A4!21F f #2A2~ I 2A4!21F f ,(4.5b)

Fc25F f .

According to Lemma 4.1, the closed-loop system~4.1! and~4.3! is stable only for a sufficiently smalle. The criterion tocheck whethere is small enough is hence proposed, though prto this a useful lemma is first introduced as follows.

Lemma 4.2: ~Maximum Modulus Theorem! @16#. If f (z) is ana-lytic in a bounded domainD and continuous inD ~i.e., the closureof D!, thenu f (z)u takes its maximum on the boundary ofD.

We now proceed to derive a frequency domaine-dependentstability criterion for the closed-loop system~4.1! and ~4.3!.

Theorem 4.1: Under the same assumptions given in Lemma 4the closed-loop system~4.1! and ~4.3! is stable, if the singularperturbation parametere satisfies

~ I! r@D1~e,ej u!#,1, ;uP@0,2p! (4.6a)

or

~ II ! r@D2~e,ej u!#,1, ;uP@0,2p! (4.6b)

where

D1~e,eiu!5~e2 j uI 2M s~e!!21M2~e!~d~e,ej u!2~d~e,1!!M3~e!(4.7a)

and

D2~e,eiu!5~d~e,ej u!2d~e,1!!M3~e!~e2 j uI 2M s~e!!21M2~e!(4.7b)

with

d~e,z!5~ z21I 2M4~e!!21, M25F A22B1Kc2 B1Kc2

0 A22Fc1C2G ,

M35F A32B2Kc1 B2Kc1

0 A32Fc2C1G ,

M45F A42B2Kc2 B2Kc2

0 A42Fc2C2G ,

M s[F As2BsKs BsKs

0 As2FsCsG ,

Ks5Ks1G2K f~ I 2A41B2K f !21@A31B2G#,

G5K f~ I 2A41F fC2!21~A32F fC1!.

Proof: See Appendix.Remark 4.1:In principle, both the conditions~4.6a! and~4.6b!

can be used to test the stability of the closed-loop system.

136 Õ Vol. 123, MARCH 2001

ior

.1,

t is

therefore reasonable to test the stability with any one of theequalities and then, if it fails, resort to another~see the examplepresented in Section 5!.

Remark 4.2:The upper bound ofe, callede* , can be obtainedby finding the maximum value ofe that fulfills the stability con-ditions ~4.6a! or ~4.6b!. Since there is no explicit information toindicate the conservativeness of thee-bound obtained by examining ~4.6a! or ~4.6b!, the less conservativee* should be chosen athe upper bound ofe for each case in hand.

In the next section, an example is given to examine the stabcriterion of discrete singularly perturbed systems. It is worthwhto distinguish the roles of~4.6a! and ~4.6b! in this example.

5 ExampleConsider a discrete singularly perturbed system described

the following equations:

x1~k11!5(i 50

2

A1ix1~k2 i !1e(i 50

2

A1ix2~k2 i !1B1u~k!

(5.1a)

x2~k11!5(i 50

2

A2ix1~k2 i !1e(i 50

2

A2ix2~k2 i !1B2u~k!

(5.1b)

y~k!5C1x1~k!1C2x2~k!, (5.1c)

in which

A105F 0.4 0.3

0.22 0.35G , A115F0.33 0

0.3 0.25G ,A125F0.25 0.33

0 0.32G , A105F0.03 0.02

0.03 0.01G ,A115F0.024 0

0.036 0.037G , A125F0.0175 0

0.028 0.038G ,A205F20.15 0.1

0.12 20.15G , A215F20.12 0.12

20.1 0.8G ,(5.2)

A225F20.11 0

0 0.1G , A205F0.075 0.0091

0.04 0.06 G ,A215F0.06 0.0065

0.07 0.042G , A225F0.056 0.09

0.067 0.032G ,B15F 0.1

0.25G , B25F0.0080.012G ,

[email protected] 0.23#, [email protected] 0.9#.

Suppose that the value ofe is 0.02, then the composite observebased controller for the system~5.1! can be designed as follows

(i) Controller Design in the Slow Subsystem.The systemmatrices in the slow subsystem~3.1! with e50.02 are

As530.3998 0.3001 0.3298 0.0005 0.2498 0.33

0.2199 0.3499 0.2996 0.256120.0002 0.3202

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

4 ,


,-l

e

n

ore

f

ee

lyd fast

areThethe

Bs5F 0.10810.2621

0000

G , (5.3)

[email protected] 0.11620.116 0.74820.0235 0.0903#,

Ds50.0125.

It can be seen that the eigenvalues ofAs are 1.2650, 0.645220.45236 j 0.4582, 20.12796 j 0.4560. Obviously the eigenvalue 1.2650 lies outside the unit disk which means the ssubsystem ~3.1! without control is unstable. Therefore, thobserver-based controller~3.2! is used to stabilize the slow subsystem system~3.1!. Based on the pole-assignment technique,can choose the gains

[email protected] 0.8 0.9 0.8 0.7 0.7#

and

[email protected] 0.6 0.8 0.5 0.45 0.64#T (5.4)

such that the matricesAs2BsKs and As2FsCs withl(As2BsKs)520.48276 j 0.444, 20.05646 j 0.522, 0.75946 j 0.1352 and l(As2FsCs)50.6546, 0.1706, 20.13046 j 0.5008, 20.26486 j 0.3256 are both Hurwitz. Hence, thclosed-loop subsystem~3.1! and ~3.2!, or equivalently system~3.4!, is stable.

(ii) Controller Design in the Fast Subsystem.The systemmatrices in fast subsystem~3.5! with e50.02 are

Af530.0015 0.0002 0.0012 0.0001 0.0011 0.0018

0.0008 0.0012 0.0014 0.0008 0.0013 0.0006

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

4 ,

Bf5F 0.0080.012

0000

G , [email protected] 0.9 0 0 0 0#. (5.5)

Since all the eigenvalues ofAf ~i.e., 20.06926 j 0.1127, 0.1401,0.04486 j 0.0761, 20.0886! are inside the unit disk, we cachoose the gainsK f5@0 0 0 0 0 0# and F f5@0 0 0 0 0 0#T suchthat the matricesAf2BfK f andAf2F fCf are both Hurwitz. Con-sequently, the closed-loop subsystem~3.5! and ~3.6!, or equiva-lently system~3.7!, is stable.

(iii) Controller Design for the Original System.Based on~4.5! and the matrices defined in~4.1!, the gains of the compositeobserver-based controller~4.3! are

Kc15@ I 1K f~ I 2A4!21B2#Ks2K f~ I 2A4!21A3

[email protected] 0.8 0.9 0.8 0.7 0.7# (5.6a)

Kc25K f5@0 0 0 0 0 0#

and

Fc15Fs@ I 1C2~ I 2A4!21F f #2A2~ I 2A4!21F f

[email protected] 0.6 0.8 0.5 0.45 0.64#T (5.6b)


owe-we

Fc25F f5@0 0 0 0 0 0#T.

Consequently, the curves of the functionsr@D1(e,ej u)# andr@D2(e,ej u)# ~in ~4.6!! with respect tou are shown in Fig. 3. Thisfigure reveals that the condition~4.6a! is satisfied, while the in-equality~4.6b! does not hold. Based on Theorem 4.1, we therefconclude that the original system~5.1!, under the controller~4.3!with the gains given in~5.6!, is stable. In Fig. 4, the suprema othe functionsr@D1(e,ej u)# and r@D2(e,ej u)# in the rangeuP@0,2p) are depicted with respect toe. According to Fig. 4, theupper bound of the singular perturbation parametere, e* , is 1.12.

Remark 5.1:From the computer simulation, we find that threal stability bound one is 1.58 which is larger than the value* 51.12.

6 ConclusionIn this paper, two stability conditions for discrete singular

perturbed systems are presented. The corresponding slow ansubsystems of the original discrete singularly perturbed systemobtained by means of the technique of time-scale separation.observer-based controllers are first separately designed for

Fig. 3 The functions r†D1„e,e j u…‡ and r†D2„e,e j u

…‡ in „4.6…with e¿0.02

Fig. 4 The suprema of r†D1„e,e j u…‡ and r†D2„e,e j u

…‡ in therange u«†0,2p…

MARCH 2001, Vol. 123 Õ 137

o

g

st

i

f

slow and the fast subsystems. A composite observer-basedtroller for the original system is subsequently synthesized frthese observer-based controllers.

It has been shown that stability of the original discrete sinlarly perturbed system can be investigated by establishing thaits corresponding slow and fast subsystems for a sufficiently sme. A frequency domaine-dependent stability criterion is proposeto test the stability of the original system under the compoobserver-based controller. If the singular perturbation parameeis so small that any one of the stability conditions~~4.6a! or~4.6b!! is satisfied, then stability of the slow and fast subsystecan imply that of the original system under the composobserver-based controller. An illustrative example is givendemonstrate that the upper bound of the singular perturbationrametere can be obtained by examining these conditions.

AcknowledgmentThe authors wish to express sincere gratitude to the anonym

referees for their constructive comments and helpful suggestwhich led to substantial improvements of this paper. Moreovthis research work was supported by the National Science Couof the Republic of China under contract NSC 85-2213-E-182-0

AppendixProof of Theorem 4.1:~I! Rewrite the composite observer-based controller~4.3! as

F X1~k11!

X2~k11!G5F A12B1Kc12Fc1C1 A22B1Kc22Fc1C2

A32B2Kc12Fc2C1 A42B2Kc22Fc2C2G

3F X1~k!

X2~k!G1FFc1

Fc2G u~k! (A1)

y~k!5@Kc1 Kc2#F X1~k!

X2~k!G .

Based on~4.4! and defininge1(k)[X1(k)2X1(k) and e2(k)[X2(k)2X2(k), the closed-loop system~A1! and ~4.1! becomes

F X1~k11!

e1~k11!

X2~k11!

e2~k11!

G5F A12B1Kc1 B1Kc1 A22B1Kc2 B1Kc2

0 A12Fc1C1 0 A22Fc1C2

A32B2Kc1 B2Kc1 A42B2Kc2 B2Kc2

0 A32Fc2C1 0 A42Fc2C2

G3F X1~k!

e1~k!

X2~k!

e2~k!

G5FM1 M2

M3 M4GF X1~k!

e1~k!

X2~k!

e2~k!

G , (A2)

where

M15F A12B1Kc1 B1Kc1

0 A12Fc1C1G ,

M25F A22B1Kc2 B1Kc2

0 A22Fc1C2G

138 Õ Vol. 123, MARCH 2001

con-m

u-t ofall

diteer

msitetopa-

ousonser,ncil06.

M35F A32B2Kc1 B2Kc1

0 A32Fc2C1G ,

M45F A42B2Kc2 B2Kc2

0 A42Fc2C2G .

Rewrite the system~A2! as

Xe1~k11!5M1Xe1~k!1M2Xe2~k!(A3)

Xe2~k11!5M3Xe1~k!1M4Xe2~k!

where

Xe1~k!5FX1~k!

e1~k! G , Xe2~k!5FX2~k!

e2~k! G .Applying z-transform to~A3!, yields

zXe1~z!5M1Xe1~z!1M2Xe2~z!1zXe1~0! (A4a)

zXe2~z!5M3Xe1~z!1M4Xe2~z!1zXe2~0!, (A4b)

in which Xe1(0) andXe2(0) are the bounded initial conditions othe statesXe1(k) and Xe2(k), respectively. According to~A4b),we have

Xe2~z!5~zI2M4!21M3Xe1~z!1z~zI2M4!21Xe2~0!.(A5)

Substituting~A5! into ~A4a!, Xe1(z) is obtained as

Xe1~z!5zC21~z!Xe1~0!1zC21~z!M2~zI2M4!21Xe2~0!,(A6)

where

C~z!5zI2M12M2~zI2M4!21M3

5zI2M12M2~ I 2M4!21M3

2M2@~zI2M4!212~ I 2M4!21#M3

5~zI2M s!$I 2~zI2M s!21M2@~zI2M4!21

2~ I 2M4!21#M3% (A7)

with

Ms[F As2BsKs BsKs

0 As2FsCsG5M11M2~ I 2M4!21M3 ,

Ks5Ks1G2K f~ I 2A41B2K f !21@A31B2G#,

G5K f~ I 2A41F fC2!21~A32F fC1!.

Hence,~A6! can be rewritten as

Xe1~z!5zc21~z!~zI2M s!21Xe1~0!

1zc21~z!~zI2M s!21M2~zI2M4!21Xe2~0!,

(A8a)

where

c~z![I 2~zI2M s!21M2@~zI2M4!212~ I 2M4!21#M3

5I 2D1~z! (A8b)

with

D1~z![~zI2M s!21M2@~zI2M4!212~ I 2M4!21#M3 .

(A8c)


i

f

re

-

-

ly

ed

le

ndEE

sis

s,’’

a-u-

ignn-

er-

als,’’

ty

nes.velnil-is

toti-

re-b-

tems

-

SinceM s andM4 ~i.e., M f!3 are both Hurwitz, therefore to stab

lize Xe1 ~and thusXe2 is stabilized as well!, we only need to findthe condition which guarantees that all the poles ofc21(z) arewithin the unit disk, i.e., the following inequality holds:

udetc~z!u5udet@ I 2D1~z!#u.0 ;uzu>1. (A9)

Let z5 z21, thenc(z) becomes

c~z!5c~ z21!

5I 2~ z21I 2M s!21M2@~ z21I 2M4!212~ I 2M4!21#M3

5I 2D1~ z![c~ z!, (A10a)

where

D1~ z![~ z21I 2M s!21M2@~ z21I 2M4!212~ I 2M4!21#M3 .

(A10b)

Therefore, the examination of~A9! is equivalent to investigatingthe following inequality:

udetc~ z!u5udet@ I 2D1~ z!#u.0 ;uzu<1. (A11)

Introducing the singular perturbation parametere into ~A10b!,D1( z) can then be rewritten as

D1~e,z!5~ z21I 2M s~e!!21M2~e!

3@~ z21I 2M4~e!!212~ I 2M4~e!!21#M3~e!

5~ z21I 2M s~e!!21M2~e!~d~e,z!2d~e,1!!M3~e!.

(A12)

Since the matrices (zI2M s)21 and (zI2M4)21 do not have any

pole lying inside the regionuzu>1, the termD1(z) in ~A8c! doesnot have any pole lying inside the regionuzu>1 either. Conse-quently,D1(e,z)5D1( z)5D1( z21) has no poles lying inside theregion uzu<1 and the functionl i@D1(e,z)# is hence analytic andcontinuous in the bounded domainuzu<1. Therefore, if~4.6a!holds, i.e.,

r@D1~e,z!#5maxi

ul i@D1~e,z!#u,1 ;uzu51 (A13)

then we have~according to Lemma 4.2!

r@D1~e,z!#,1 ;uzu51. (A14)

Based on~A14! and Lemma 2.1, the following inequality is obtained:

udetc~ z!u5udet@ I 2D1~ z!#u5udet@ I 2D1~e,z!#u.0 ;uzu51

and then the inequality~A11!, or equivalently~A9!, is fulfilled.This implies that the closed-loop system~4.1! and ~4.3! is stable,thus completing the proof of case~I!.

~II !: Using the matrix inversion formula (I 2SL)215I 1S(I2LS)21L, the functionc21(z) in ~A8a! can be rewritten as

c21~z!5I 1~zI2M s!21M2@ I 2D2~z!#21

3@~zI2M4!212~ I 2M4!21#M3 (A15)

where

D2~z!5@~zI2M4!212~ I 2M4!21#M3~zI2M s!21M2 .

(A16)

Since all poles of the matrices (zI2M s)21 and (zI2M4)21 lie

inside the unit disk, therefore to stabilize the system~A3!, we onlyneed to find condition which guarantees that all the poles o@ I

3The fact thatM45M f can be observed by comparing~A2! with ~3.7! and usingthe matrices defined in~3.5!, ~4.1!, and~4.5!.


-

-

2D2(z)#21 are within the unit disk. Following the same procedu

as that in case~I!, the proof of case~II ! is thereby completed.

References@1# Kokotovic, P. V., Khalil, H. K., and O’Reilly, J., 1986,Singular Perturbation

Methods in Control: Analysis and Design, Academic Press, New York.@2# Li, T. H. S., and Li, J. H., 1992, ‘‘Stabilization Bound of Discrete Two-Time

Scale Systems,’’ Syst. Control Lett.,18, pp. 479–489.@3# Litkouhi, B., and Khalil, H., 1985, ‘‘Multirate and Composite Control of Two

Time-Scale Discrete-Time Systems,’’ IEEE Trans. Autom. Control.,30, pp.645–651.

@4# Feng, W., 1988, ‘‘Characterization and Computation for The Bounde* inLinear Time-invariant Singularly Perturbed Systems,’’ Syst. Control Lett.,11,pp. 195–202.

@5# Chen, B. S., and Lin, C. L., 1990, ‘‘On The Stability Bounds of SingularPerturbed Systems,’’ IEEE Trans. Autom. Control.,35, pp. 1265–1270.

@6# Sen, S., and Datta, K. B., 1993, ‘‘Stability Bounds of Singularly PerturbSystems,’’ IEEE Trans. Autom. Control.,38, pp. 302–304.

@7# Pan, S. T., Hsiao, F. H., and Teng, C. C., 1996, ‘‘Stability Bound of MultipTime-delay Singularly Perturbed Systems,’’ Electron. Lett.,32, pp. 1327–1328.

@8# Hsiao, F. H., Pan, S. T., and Teng, C. C., 1997, ‘‘D-Stabilization BouAnalysis for Discrete Multiparameter Singularly Perturbed Systems,’’ IETrans. Circuits Syst., Part I,44, pp. 347–351.

@9# Naidu, D. S., and Rao, A. K., 1985,Singular Perturbation Analysis of DiscreteControl Systems, Springer-Verlag, Berlin.

@10# Chou, J. H., and Chen, B. S., 1990, ‘‘New Approach for The Stability Analyof Interval Matrices,’’ Control-Theory and Advanced Technology,6, pp. 725–730.

@11# Mahmoud, M. S., 1982, ‘‘Order Reduction and Control of Discrete SystemIEE Proceeding—Control Theory Appl.,129, pp. 129–135.

@12# Saksena, V. R., O’Reilly, J., and Kokotovic, P. V., 1984, ‘‘Singular Perturbtions and Time Scale Methods in Control Theory—Survey 1976–1983,’’ Atomatica,20, pp. 273–293.

@13# Oloomi, H., and Sawan, M. E., 1987, ‘‘The Observer-Based Controller Desof Discrete-Time Singularly Perturbed Systems,’’ IEEE Trans. Autom. Cotrol., 32, pp. 246–248.

@14# Li, J. H., and Li, T. H. S., 1995, ‘‘On the Composite and Reduced Observbased Control of Discrete Two-Time-Scale Systems,’’ J. Franklin Inst.,332b,pp. 47–66.

@15# Wang, M. S., Li, T. H. S., and Sun, Y. Y., 1996, ‘‘Design of Near-OptimObserver-Based Controllers for Singularly Perturbed Discrete SystemJSME International Journal: Series C,39, pp. 234–241.

@16# John, W. D., 1967,Applied Complex Variables, Macmillan, New York.

Reference Trajectory Tracking ofOverhead Cranes

Kamal A. F. MoustafaDepartment of Industrial Engineering & Systems, Faculof Engineering, Zagazig University, Zagazig, Egypte-mail: [email protected]

This paper addresses the automation problem of overhead craA feedback control strategy is proposed so that the crane traand hoisting or lowering motions are forced to track a givereference trajectory while killing the payload swing angle. Stabity analysis is carried out using Lyapunov functions and itshown that the equilibrium point of the crane system is asympcally stable. @DOI: 10.1115/1.1343462#

1 IntroductionThe control and stability problem of overhead cranes has

cently attracted much attention. It is worth noting that this prolem is not the same as that of other nonlinear mechanical sys

Contributed by the Dynamic Systems and Control Division of THE AMERICANSOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Dynamics Systems and Control Division March 4, 1998. Associate Editor: E. Misawa.


i

reo

k

il

o

r

r

dge

v

alive

sed-Onn

.

and

tem

di-

r.are

~e.g., robot manipulators!. For example, cable length variationduring crane travel could introduce negative damping effect insystem dynamics. State variables in overhead cranes are of dent nature than those of robots. In robots, they are mostly anglink positions and velocities, e.g.@1,2#, whereas in overheadcranes they are the linear position of the trolley, the length ofsuspension cable and the swing angle as well as their first detives. The problem of designing automatic controllers for ovhead cranes described by highly nonlinear models has itsrelevant and, hence, merits a certain degree of attention.

A number of mathematical models for overhead cranes wdeveloped, e.g.@3,4#. These models are highly nonlinear and ausually linearized about normal operating points. The drawbaclinearization is that only small deviations of the crane variabfrom the normal operating values are allowed. There is an obvneed to develop crane automation schemes based on the nonmodel of the crane.

Most of the control schemes proposed in the literature drivecrane to a final destination@4#. Little attention has been givenhowever, to the problem of developing an automation schemedrives the crane along a reference trajectory that reflects sdesired operating conditions. This technical brief addressesproblem of developing a control strategy based on a nonlinmodel of the crane and forcing the crane to follow a given refence trajectory.

2 Nonlinear Model DescriptionWe consider the mathematical model of an overhead crane

was derived in a previous work of the author@4#. The generalizedcoordinates of the model, which are shown in Fig. 1, arex, l, andq. The crane is assumed to be driven by the two actuating foFx and Fl . Defining the generalized coordinate vectorq5@x,l ,q#T, the equations of motion can be obtained in matform as@4#

D~q!q1C~q,q!q1G~q!5F (1)

where D(q) is a positive definite inertia matrix,C(q,q)q is avector representing the centrifugal and Coriolis effects, andG(q)is the gravity term vector. The coefficient matrices of Eq.~1! andthe actuating force vector are given by

D~q!5F ax sinq l cosq

sinq a l 0

l cosq 0 l 2G

C~q,q!5F 0 q cosq ~ l cosq2 l q sinq!

0 0 2 l q

0 l q l lG

Fig. 1 Crane model and generalized coordinates

140 Õ Vol. 123, MARCH 2001

stheffer-ular

theiva-r-wn

erereof

lesousinear

the,thatmethe

earer-

that

ces

ix

G~q!5F 02g cosqgl sinq

G , F5F Fx*

Fl*0G

where ax5Mx /M , a l5Ml /M , Mx5M1mx1I x /r x2, Ml5M

1I l /r l2, Fx* 5Fx /M , Fl* 5Fl /M andg is the gravitational accel-

eration. Here,mx is the mass of the bridge,r x is the radius of thebridge motor pinion,r l is the radius of the hoisting drum,I x is theequivalent moment of inertia of the bridge motor, andI l is that ofthe hoisting motor.

3 Stability AnalysisLet the reference trajectory be defined by the desired bri

velocity xm and positionxm , the desired hoisting or loweringvelocity l m and position~cable length! l m , and the desired swingangular velocityqm and positionqm . Define the position errorvector as

«5@«x ,« l ,«q#T5q2qm

whereqm5@xm ,l m ,qm#T. It should be noted that«q5q since itis desired thatqm50. Consider the following candidate Lyapunofunction

V51/2«TD «11/2«TK«1gl~12cosq! (2)

where K5diag@kx ,kl ,kq# is a positive definite constant diagonmatrix. The above candidate Lyapunov function is clearly positsemi-definite. The time derivative of Eq.~2! is obtained as

V5 «T@D «1C« #1 «TK«1@gl~12cosq!1glq sinq# (3)

Now introduce the control laws

Fx* 52lx«x2kx«x1D1Tqm1C1

Tqm (4)

Fl* 52l l « l2kl« l1D2Tqm1C2

Tqm2g (5)

which are subject to the following constraint

D3Tqm1C3

Tqm5lqq1kqq (6)

Herelx , l l andlq are positive design parameters;DiT andCi

T ,i 51,2,3, denote, respectively, theith row of matricesD andC.

The above control laws contain PD terms similar to those uwith the controller presented in@2#. In addition, they use continuous nonlinear functions of the model states to ensure stability.the other hand, those of@2# use discontinuous functions as aadditional control component to achieve stability.

Letting l m goes to zero ast goes to infinity and using Eqs~4!–~6! in ~3!, we can get for large values oft

V52lx«x22l l « l

22lq«q2 (7)

Therefore, the closed-loop system is asymptotically stableat steady state we have

«x5 « l5 «q50 (8)

It can be shown that the above condition implies that the sysis being stabilized at the desired equilibrium pointq5qm @5#.

Finally, substituting forDiT andCi

T , i 51,2,3, the control lawsare obtained as

Fx* 52lx«x2kx«x1axxm1~sinq! l m1~q cosq! l m (9)

Fl* 52l l « l2kl« l1~sinq!xm1a l l m2g (10)

~ l cosq!xm1~ l q ! l m5lqq1kq~q2qm! (11)

It should be noted that although the payload swing is notrectly controlled by the actuating forces, the constraint of Eq.~11!indirectly provides information about the swing to the controlleNumerical simulations based on the above described scheme


.

resiosh

in

r

o

o

,

e

n

p

era-r-lateif it

osisun-ine

Kal-d

ntly

hatro-ad-

vel-s,

rtech-in-the

heir

m-e-flasts of

y-di-is

orkg ade-ain-rs,

r of

c-

essrtionog-

ardataderdi-il-

uee. Ac-

aultgad-sticingt-

s

presented in@5# where the results therein show excellent trajectotracking with the swing angle being killed in a very short time

4 ConclusionsA nonlinear mathematical model that describes the dynam

behavior of overhead cranes is presented. An automation schis developed that drives the crane along desired trajectories fobridge travel motion, payload hoisting or lowering motion as was the swing angle. The design of the feedback control lawbased on stability analysis via Lyapunov functions. The limitatof having small deviations in the crane variables from their nmal operating values, as normally required in linearization bamethods, is no longer imposed. Computer simulations that sthe usefulness of the considered automation scheme are presin @5#.

References@1# De Wit, C. C., Fixot, N., and Astrom, K. J., 1992, ‘‘Trajectory Tracking

Robot Manipulators Via Nonlinear Estimated State Feedback,’’ IEEE TraRob. Autom.,8, pp. 138–144.

@2# De Wit, C. C., and Fixot, N., 1992, ‘‘Adaptive Control of Robot ManipulatoVia Velocity Estimated Feedback,’’ IEEE Trans. Autom. Control,37, pp.1234–1237.

@3# Moustafa, K. A. F., and Abou-El-Yazid, T. G., 1996, ‘‘Load Sway ControlOverhead Cranes with Load Hoisting Via Stability Analysis,’’ InternationJournal of the Japanese Society of Mechanical Engineers~JSME!, Series C,39, pp. 34–40.

@4# Moustafa, K. A. F., and Ebeid, A. M., 1988, ‘‘Nonlinear Modeling and Controf Overhead Crane Load Sway,’’ ASME J. Dyn. Syst., Meas., Control,110,pp. 266–271.

@5# Moustafa, K. A. F., 2000, ‘‘Overhead Crane Automation Via Lyapunov Funtions,’’ Proceedings, UKACC International Conference ‘‘CONTROL 2000University of Cambridge, UK, 4–7 Sept. 2000.

An Intelligent Sensor Validation andFault Diagnostic Technique forDiesel Engines

Ehsan MesbahiDepartment of Marine Technology, Armstrong Building,University of Newcastle upon Tyne, Newcastleupon Tyne, NE1 7RU United Kingdome-mail: [email protected]

An intelligent sensor validation and on-line fault diagnosis tecnique for a 6 cylinder turbocharged diesel engine is proposed astudied. A single auto-associative 3-layer Artificial Neural Nwork (ANN), is trained to examine the accuracy of the measudata and allocate a confidence level to each signal. The saANN is used to recover the missing or faulty data with a cloapproximation. For on-line fault detection a feed-forward ANNtrained to classify and consequently recognize faulty and heabehavior of the engine for a wide range of operating conditioThe proposed technique is also equipped with an on-line learnmechanism, which is activated when the confidence level indicted fault is poor. It is hoped that a feasible, practical, anreliable sensor reading, as well as highly accurate fault diagnosystem, would be achieved.@DOI: 10.1115/1.1343461#

Contributed by the Dynamic Systems and Control Division of THE AMERICANSOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Dynamics Sytems and Control Division June 31, 1999. Associate Editor: S. Fassois.


ry

icalemethellis

onr-edow

ented

ns.

s

fal

l

c-’’

h-ndt-

redmeseislthys.ingre-dsis

1 IntroductionAs a first and essential step to improve diesel engines’ op

tional reliability, fault diagnostic accuracy, and condition monitoing precision, it is necessary to validate the acquired data, isoany failed sensor, and recover the failed critical measurementis required by other systems such as control or fault diagntools. Extended efforts have been made to apply analytical reddancy to the sensor failure detection and isolation in the jet engand nuclear power plant failure diagnosis problem@1#. In general,these approaches utilize the engine physical model and theman filter ~for state estimation! to detect and isolate the failesensor~s! @2#. Data reconciliation technique@3#, which is based onmass and energy balance and equilibrium constraints, is receapplied to chemical process monitoring@4#.

In order to implement preventive techniques, it is essential tearly warning of developing faults be provided so that the apppriate decisions may be taken and correct actions planned invance. For this reason various monitoring techniques are deoped @5#, which put more emphasis on using expert systemfailure models, and effects analysis. In 1997, Arefzadeh@6# ap-plied a combination of Fuzzy Logic and Vibration Analysis fodetection of faults in diesel engines. Case Based Reasoningnology @7# is proposed for gas turbines designed for aircraftdustry. Model-based fault diagnosis techniques have beenmost popular since time and frequency domain analysis of tbehavior are clearly defined@8#.

Artificial Neural Networks „ANN…. ANNs are widely usedas nonlinear, nonphysical, and universal modeling tool@9#. Recur-rent or dynamic ANNs are capable of capturing systems’ dynaics and can uniquely provide inverse model of dynamic timvariant systems@10#, which are of great importance in design oadaptive control systems. Their rapid developments during thetwo decades has resulted in introduction of many different clasANNs, such as Neocognitron, RBF, recurrent, cooperative, Hbrid, Sigma-Pai etc.; each is tailored to perform best in an invidual task. The network of choice for most pattern recognitiona multi-layered feedforward network. There are several netwtypes, which are useful for pattern autoassociation, allowincomplete pattern to be reconstructed when only a partial orgraded pattern is used as input. Hopfield/Tank network and BrState-in-a-Box~BSB! are of the most common pattern associatowhich both possess a single~unified! input/output layer. Theywork well on small pattern sets, but cannot store large numbepatterns without interference.

The application of ANNs for sensor validation and fault detetion as model-free diagnosis tools is not new. In 1989, Naidu@11#used ANNs for sensor failure detection in a chemical proccontrol system. Parlos@12# in 1991 described the use of ANNs fofault detection on a nuclear power plant. Recently, a combinaof Expert Systems and ANNs is proposed for gas turbine Prnostics and Diagnostics@13#.

In our approach, a single 3-layer autoassociative feedforwANN is proposed and trained for both sensor validation and drecovery purposes using real data acquired from a 6-cylinRUSTON diesel engine at healthy and faulty operational contions. Most of ANN based microprocessors commercially avaable feature classic backpropagation~BPA! as their learning algo-rithm @14#. In the proposed approach, practicality of the techniqas well as its feasibility@15# are of our main concerns. Thereforsimple backpropagation method is used for network trainingsecond 3-layer feedforward ANN is trained for engine fault reognition. The combined sensor validation, data recovery, and fdiagnostic algorithm is implemented in LabVIEW® programminenvironment, where direct communication with real sensor reings is easily made possible. In contrast to the use of simpliphysical models of a diesel engine, real data is utilized for trainand validation of ANNs. Final algorithm is capable of implemen

-


b

eh

sordn

ninges-was

ingain-ed

y,

ing an on-line real-time validation-recovery-diagnosis processwell as learning new faulty conditions as they occur.

2 Ruston Diesel: Experimental DataThe RUSTON Engine was run under healthy and many fau

conditions@16#. A large number of data was acquired from almo50 thermocouples, pressure transducers, flow meters, speedtorque sensors with a sampling rate of 10 Hz. In order to minimthe noise effect, every measurement was averaged over 10onds ~100 samples!. Engine operating conditions were in thspeed range of 470, 600, 680, 720, and 750 rpm at varying poof 120, 240, 360, 430, and 480 kW. Finally, 640 sets of data,for each condition were selected, as listed in Table 1.

3 Sensor ValidationThe concept of validating a measurement suggests that a c

parison must be made with some known and trustworthy informtion or that some logical conclusions can be drawn from the dthemselves. It is also understood that any single value readsensor must be consistent with the other sensors’ readings.valid correlation between all measured data at different optional conditions, including faulty conditions, is established, tcould lead to a valid interpretation of the available readings.reality, this correlation is represented by the system itself.

A single auto-associative feed-forward ANN with 25 nodesinput and output layers and 20 nodes in a single hidden layeselected for sensor validation purposes. By this selection, m

Table 1 Engine health conditions

Condition Description

Healthy No fault, engine running under varying operationconditions

Fault 1 Inlet air filter blockedFault 2 Fuel injector enlarged nozzle holeFault 3 Fuel injector low opening pressureFault 4 Worn fuel pump plungerFault 5 Fuel injector blocked nozzle holeFault 6 Fuel injector timing advancedFault 7 Fuel injector timing retarded

Fig. 1 Auto-associative ANN for sensor validation

142 Õ Vol. 123, MARCH 2001

as

ltystand

izesec-ewer80

om-a-

atay aIf ara-isIn

inr isany

technical and industrial objectives such as online, timely senassessment@17#, applicability on a wide operational range, ansimplicity of implementation for realtime use have beeaddressed.

500 data sets are chosen from 640 sets and used for traipurposes with only 100 training iterations. Using a P-600 procsor, it took less than 3 seconds to train the network. Input datanormalized between 0.3 and 0.7~instead of 0 and 1!, whichproved to be immensely successful in speeding up the learnprocedure as well as its generalization capability outside the tring limits. Using such simple auto-associative ANN also providus with a single functional relationship~1! for further sensitivityanalysis and simple hard/software practical implementations.

~x1 ,x2 , . . . ,xm!5 f ~x1 ,x2 , . . . ,xm! (1)

where xm is measured data andf is the nonlinear relationshipbetween sensor readings and themselves. In this case studm525. Functionf in ~2! can be represented as:

xm5sigF(i 51

h

sigS (j 51

n

xjW i , j1 1ci D Wm,i

2 1bmG (2)

al

Fig. 2 Confidence levels for 25 sensors when one sensor isfailed

Fig. 3 ANNs outputs „user interface … for faults 1 to 7 andhealthy condition, respectively

Table 2 Comparison between recovered and correct readings

SensorFailed

Correctreading

Recovered value when actual measurement is:

0%reading

20%reading

50%reading

70%reading

130%reading

Enginecondition

102 23.2 16 16 16.3 17.2 27.63 F3106 185 156 158 160 162 169.1 F6303 366 306 307 307 311 410 F7308 412 352 367 387 399 426 F4310 463 372 388 418 439 485 F2312 339 281 283 287 294 360.6 F5408 739 466 470 496 581 713 Healthy409 3.19 2.89 2.94 3.02 3.07 3.221 F7


Journal of Dyna

Table 3 Numerical presentation of faults 1–7 and healthy condition for training

20 Nodes network output

Fault1 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 1Fault2 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 1Fault3 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0Fault4 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 0 0 0 1 0Fault5 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0Fault6 0 1 1 1 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0Fault7 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0Healthy 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1

e

sdc

,

a

-

i

psu

r

on.4

ndeis

ng

at-g--

put

entIt isly50

NNard

whereh, m, and n are hidden, output, and input nodes, resptively ~m5n in this case!, W1, W2, c andb are network weights,which are frozen after training, sig~.! is sigmoidal function pre-sented in~3!:

sig~x!51

11e2x (3)

To investigate the performance of the trained ANN for senvalidation, 8 sensors~out of 25! are chosen and their actual reaings are manually changed by 0 percent, 20 percent, 50 per70 percent, and 130 percent to account for the fact that a fasensor does not necessarily transmit a value of zero.

Definition of Confidence Level „CL …. Equation~4! is usedfor calculation of a confidence level for each reading:

CLi512uSi2Si

r u

Si

(4)

whereS is the predicted value by ANN,Sr is the sensor readingandi is the sensor number. A confidence level close to 1 indicaa highly reliable sensor and a value close to zero describes a fsensor. During the test, CL for all 25 sensors is calculated, FigResults are presented in Fig. 2.

It is clearly observed that the lowest CL belongs to the sentransmitting the faulty signal. The inconsistency between thereadings results in an easily distinguishable low confidence lefor the faulty sensor. It is also noticed that the drop in othsensors’ confidence level, although inevitable, is less than thein the failed sensor.

Data Recovery. Every individual output of an autoassociative ANN is made up of all input values, with differelevels of contributions, see~2!. This could enable us to recover thfaulty signal by using all other readings, provided that not maof them failed at the same time. The ANN recovered values offaulty sensors in the previous section are presented in Table

Although not entirely accurate, all predicted values are faclose to the correct value. This may temporarily replace the reings for monitoring and control purposes.

4 Engine Fault DiagnosticsIn order to investigate the fault diagnostic capability of a sim

feed-forward ANN for diesel engines, each input data set is aciated with a particular binary output pattern, representing fafrom 1–7~Fig. 3! and healthy pattern, represented in Table 3 aFig. 4. From 640 data sets available, 400 sets are randomlylected for network training, while the rest is kept aside for validtion purposes. It is noticed that only 500 iterations are sufficienachieve a reasonably low error criterion in less than 5 secousing a P-600 processor.

Test I: Engine Fault Diagnostics. 80 unused data sets belonging to different healthy and faulty engine conditions at vaing load and speed are presented to the trained ANN as presein Fig. 4. A confidence level, similar to~5! is tagged to every fault

mic Systems, Measurement, and Control

c-

or-ent,iled

tesiled. 1.

sor25velerfall

ntenythe2.rlyad-

leso-ltsndse-

a-t tonds

-y-nted

predicted by ANN. CL of less than 0.8 is treated as no-predictiA sample of network outputs is represented in Fig. 5. Tablegives the overall results.

It is concluded that engine fault recognition using ANNs areal engine data is correct with a high level of reliability. Thgraphical interface, representing the fault numbers in digits,found very helpful when the engine condition is gradually movitowards a particular condition.

Test II: Engine Fault Diagnosis With Faulty Sensors.The validity, consistency and availability of the measured opering data is of great importance when dealing with any fault dianostic facility. ANNs’ performance in handling faulty sensor signals is discussed here by introducing faulty readings in their indata sets. The same input data~80 sets! for pattern recognition isused here, introducing faulty readings in one to three differsensors simultaneously. The results are presented in Table 5.clearly observed that the reliability of fault diagnosis is noticeabreduced, however in the majority of the cases is still more thanpercent.

In the second scenario, a combination of auto-associative Afor sensor validation and data recovery in addition to feed-forw

Fig. 4 ANN and user interface for engine fault diagnostic

Fig. 5 ANN response to untrained set of patterns representingfaults 2,4,1,5 and healthy

Table 4 Results of engine fault diagnosis using different ANNarchitectures

CorrectPrediction Healthy

Fault1

Fault2

Fault3

Fault4

Fault5

Fault6

Fault7

ANN 100% 99% 100% 100% 100% 100% 100% 100%

MARCH 2001, Vol. 123 Õ 143

ero

144 Õ Vol. 123

Table 5 Engine fault diagnosis with failed sensors and recovered data

Run% Correct

fault diagnosis

% Correct faultdiagnosis

~with data recovery! Sensor fault details

1 75% 94% Sensor 308 failed reading zero2 60% 89% Sensors 308 and 310 failed, reading zero3 53.75% 87% Sensors 102 and 104 failed reading zero4 36.25% 85% Sensors 308, 310 and 312 failed, reading z5 53.75% 89% Sensor 308 failed reading zero

fi

o

eu

h

i

ew

n

wann

ptng

V.thehe

n,A

ticis

r-ess

nd

sel

ning,’’ibi-

llyno-

s

ve

s

onearn-

and

o-ing

iveic

gineine

ti-

ANN for fault diagnostic purposes is studied. Data sets areintroduced to the validation process; all data with CL of more th0.8 are passed for fault diagnosis. For CL of less than 80 percthe recovered data, is replacing faulty/missing data in diagnprocedure. The results of this test are presented in Table 5, wa great improvement in fault detection reliability is achieved.

On-Line Adaptive Sensor Validation and Fault DiagnosticSystem. On-line learning capability of ANNs in a short timscale may be utilized to provide an adaptive platform, whenknown conditions arrive. Figure 6 proposes such utilizationsensor validation and health monitoring of diesel engines. Wthe Fault Diagnostic Confidence Level~FDCL! is less than 0.8 inany diagnosis, system will alarm the operator of an unrecogncondition and requests for a code~similar to Table 4! and retrainsthe network.

5 ConclusionsThe use of ANNs for sensor validation, data recovery and

gine fault diagnosis by using the real engine data is studied. Itobserved that an auto-associative 3-layer ANN is successfuexamination of data consistency; each reading can be taggeda ‘‘Confidence Level’’ describing the reliability of data measureIt was also experienced that if the confidence level is lower thacertain threshold, ANN can first, alarm the operator and conquently replace the faulty data with a close approximation toother users such as control or fault diagnostic systems. A sifeedforward ANN is trained for engine fault diagnosis purposand tested against untrained data, where results of more thapercent correct fault prediction is achieved. A combination of tANNs for sensor validation, data recovery, and engine fault dinostic is presented and results proved that the proposed techis providing a highly reliable monitoring system with failed sesors. The proposed fault diagnostic hardware/software is an

Fig. 6 On-line adaptive sensor validation and fault diagnosticsystem

, MARCH 2001

rstanent,sis

here

n-inen

zed

n-as

l inwithd.n ase-thegleesn 97og-ique-on-

line adaptive system where any unrecognized fault will promthe operator for a sign allocation following a complete retrainiprocess embracing new faulty situations.

AcknowledgmentThe author would like to express his gratitude to Prof. R.

Thompson and Prof. T. Ruxton for granting permission to useengine data. The efforts of Mr. T. Y. Tan in preparation of tdata for ANN training is gratefully appreciated.

References@1# Merrill, W. C., De Laat, J. C., and Burton, W. M., 1988, ‘‘Advanced detectio

isolation and accommodation of sensor failures, real-time evaluation,’’ AIAJournal of Guidance and Control Dynamics,11, p. 6.

@2# Dinca, L., Aldemir, T., and Rizzoni, G., 1999, ‘‘A model-based probabilisApproach for Fault Detection and Identification with Application to Diagnosof Automotive Engines,’’ IEEE Trans. Autom. Control,44, pp. 2200–2205.

@3# Romagnoli, J. A., and Sanchez M. C., 1999,Data Processing and Reconcili-ation for Chemical Process Operations, Academic Press, NY.

@4# Heyen, G., 2000, ‘‘Application of Data Reconciliation to Process Monitoing,’’ Proceedings of International Symposium on Computer Aided ProcEngineering, Colombia.

@5# Woud, J. K., and Boot, P., 1993, ‘‘Diesel engine condition monitoring afault diagnosis based on process models,’’ CIMAC’93,Proceedings of 21stInternational Congress on Combustion Engines, Vol. D81, pp. 1–19.

@6# Arefzadeh, S., Burgwinkel, P., and Geropp, B., 1996, ‘‘Diagnosis of DieEngines based on Vibration Analysis and Fuzzy Logic,’’Proceedings of 4thEuropean Congress on Intelligent Techniques and Soft Computing, Vol. 2, pp.1497–1500.

@7# Hargarve, S. M., and Fleming, P. J., 1998, ‘‘The Use of Case-Based ReasoTechnology to Aid Fault Isolation In Modern gas Turbine Engine DesignProceedings of ASME International Gas and Aeroengine Congress & Exhtion, Sweden.

@8# Frank, P. M., and Ding, X., 1994, ‘‘Frequency domain Approach to OptimaRobust Residual Generation and Evaluation for Model-based Fault Diagsis,’’ Automatica,30, pp. 789–804.

@9# Mesbahi, E., and Atlar, M., 1998, ‘‘Applications of Artificial Neural networkin marine design and modelling,’’Proceedings of AIOMA’98 InternationalConference on Artificial Intelligence on Marine Applications, pp. 34–46.

@10# Mesbahi, E., and Roskilly, A. P., 1998, ‘‘A Model Reference Neural AdaptiController ~MRNAC! for time-variant systems with uncertainties,’’Proceed-ings of IASTED International Conference on Control and Application, pp.48–53.

@11# Naidu, S. R., Zafiriou, E., and McAvoy, T., 1990, ‘‘Use of Neural Networkfor Sensor Detection in an Control System,’’ IEEE Control Syst. Mag.,4, pp.49–55.

@12# Parlos, A. G., Muthusami, J., and Atiya, A. F., 1994, ‘‘Incipient fault detectiand identification in Process systems using accelerated neural network ling,’’ Nucl. Technol.,105, pp. 145–161.

@13# DePold, H. R., and Gass, F. D., 1998, ‘‘The application of expert systemsneural networks to gas turbine prognostics and diagnostics,Proceedings ofASME International Gas and Aeroengine Congress & Exhibition, Sweden.

@14# Napolitabo, M. R., Silvestri, G., Windon, D. A. II, Casanova, J. L., and Inncenti, M., 1998, ‘‘Sensor Validation Using Hardware-based On-line LearnNeural Networks,’’ IEEE Trans. Aerosp. Electron. Syst.,34, pp. 456–468.

@15# Hines, J. W., Uhring, R. E., and Wrest, D. J., 1998, ‘‘Use of AutoassociatNeural Networks for signal validation,’’ Journal of Intelligent and RobotSystems: Theory and Applications,21, pp. 143–154.

@16# Isaias, L., 1989, ‘‘Performance simulation of a medium speed diesel enunder optimum and fault conditions,’’ Ph.D. thesis, Department of MarTechnology, University of Newcastle upon Tyne, UK.

@17# Malloy, D. J., Chappel, M. A., and Biegel, C., 1997, ‘‘Real-time fault idenfication for developmental turbine engine testing,’’Proceedings of ASME In-ternational Gas Turbine and Aeroengine Congress and Exposition, Sweden.


elt

t

iot

l

so

v

iu

a

v

n

de-The

wethe

tatea

f themns

ayre-

here

thericlec-olesure-stem

s

Sensor Validation and Fusion forAutomated Vehicle ControlUsing Fuzzy Techniques

Kai F. GoebelGE Corporate Research & Development, InformationSystems Laboratory, K1-5C4A, One ResearchCircle, Niskayuna, NY 12309

Alice M. AgoginoDepartment of Mechanical Engineering, University ofCalifornia Berkeley, Berkeley, CA 94720

This brief introduces a fuzzy sensor validation and fusion meodology and applies it to automated vehicle control in IntelligeVehicle Highway Systems (IVHS). Sensor measurements arsigned confidence values through sensor-specific dynamic vation curves. The validation curves attain minima of zero atboundaries of the validation gate. These in turn are determinedthe largest physically possible change a system—in our examvehicles of the IVHS—can undergo in one time step. A fuzzyponential weighted moving average time series predictor demines the location of the maximum value of the validation curvSensor fusion is then performed using a weighted average ofsor readings and confidence values, and—if available—the futionally redundant values calculated from other sensors.@DOI: 10.1115/1.1343909#

Keywords: Sensor Fusion, Data Fusion, Sensor Validation, FuFusion, Information Fusion

1 Fuzzy Sensor Validation and FusionInputs to the proposed Fuzzy Sensor Validation and Fus

~FUSVAF! algorithm are raw sensor measurements. Outputcorrected value that can be used for the machine level contras well as for supervisory control tasks@1#. Confidence values thaare assigned to all sensor measurements depend on the spsensor characteristics, the predicted value, and the physicaltations of the system. The assignment takes place in a validagate that is bounded by the physically possible changes the sycan undergo in one time step. These maximum changes aresituations of abnormal behavior such as emergency conditionthe sensor reading shows a change beyond that limit, it cannoa correct value and, consequently, is assigned a confidenceof 0. Within the region, a maximum value of 1 will be assignedreadings that coincide with the predicted value. The validatcurve is generally nonsymmetric. It is wider around the maximvalue if the sensor is known to have little variance and narrowethe sensor exhibits noisy behavior. The curves change dyncally with the operating conditions and environmental conditioaffecting sensor behavior. A validation gate and validation curare shown in Fig. 1 wherezi are the sensor measurements,s i aresensor confidence values,x(k) is the predicted value,x(k21) isthe old value at the previous time step, andv1 andv2 are the leftand right validation gate borders, respectively.

The fusion is performed through a weighted average of codence values and measurements. The sum of the confidenceues times the measurements rewards measurements closestpredicted value the most, depending on the validation curve wh

Contributed by the Dynamic Systems and Control Division of THE AMERICANSOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Dynamics Sytems and Control Division February 10, 1998. Associate Editor: S. Fassois.


th-nt

as-ida-hebypleex-er-es.sen-nc-

zzy

ions aller

ecificimi-tiontemften

s. Ift bealuetoonmr ifmi-nses

fi-val-

to theich

expresses a trust in the operation of each sensor through thesign of its shape. Measurements further away are discounted.operative equation of the FUSVAF algorithm is

xf5

(i 51

n

zis~zi !

(i 51

n

s~zi !

(1)

wherexf is the fused value,zi are the measurements, ands is theconfidence value@2#.

To calculate the expected values for the validation curves,employed a time series predictor with adaptive parameters. Inexponential weighted moving average predictor of the form

x~k11!5a x~k!1~12a!z~k!, (2)

parametera should be large when the system is in a steady sto filter out noise. It should be small when the system is intransient state to allow swift tracking of system changes@3#.These rules are encapsulated as fuzzy rules. For the design omembership functions ‘‘small system change,’’ ‘‘medium systechange,’’ and ‘‘large system change’’ triangular shaped functiowith maximum overlap for the functions were used.

2 FUSVAF Used for Automated Vehicles in IVHSThe central concept of the IVHS is to increase highw

capacity—while maintaining high speeds and safety—throughduced spacing between the vehicles. Using the scenario wvehicles move along the highway under automatic control@4#,basic functions such as longitudinal and lateral control and omaneuvering techniques are carried out by the individual vehin coordination with other vehicles in its neighborhood. To aquire the information necessary for proper operation, a whsuite of sensors is needed. Redundant individual sensor meaments need to be fused to produce a coherent input to the sycontroller.

-

Fig. 1 Validation gate for the assignment of confidence values

Fig. 2 Fuzzy fusion for split Õjoin maneuver; open-loop valida-tion and fusion of three longitudinal sensors


o

i

n

h

s

s

n

y

E

E,

ionn-

e,

al,’’

trol,e

for’’)te-n,hisin-gu-ol,

ap-ur-tors,y in

sowticn,

nalhni-of

-

Experimental results from a simple split and merge maneuare displayed in Fig. 2 which shows the separation distancetime. The radar sensor had little variance throughout the expment except for bumps around 4.5 m and 9 m which have beenattributed to a quantization error. The sonar sensor showedsmallest variance throughout its operating region but exhiboutliers which showed up above 4 m and increased with distancbetween the follower vehicle and the lead vehicle. Above 8 m novalid readings were found. The optical sensor had the highvariance of all sensors which increased with growing distabetween follower and lead vehicle but otherwise shows noverse effects. The fused value filters out the spikes of the sosensor, the bumps of the radar sensor, and the noise of all sen

Experiments were also conducted for closed-loop systewhere the sensor input was used to control the separation distbetween vehicles. The fusion performed superior comparedProbabilistic Data Association Filter and Kalman Filter in tpresence of non-Gaussian noise~Fig. 3!.

3 Final RemarksThis paper introduced a method for validation and fusion

redundant sensor information which came from several senemploying different physical principles. Noncoinciding and cotradicting measurements were resolved taking into accountproperties of the sensors as well as the state of the system.method is shown to filter out various kinds of Gaussian and nGaussian noise. The method can accept virtual measurementsvalues that are arrived at through functional redundancy. Tfeature makes the method stable against sensor failure whenure conditions revolve around spatial or physical conditions whaffect several sensors, but not the one that is functionally reddant. The FUSVAF has been applied to other systems such aturbine power plants and aircraft engines@5#. As an extension, theFUSVAF is not restricted to fusion on the data level. It can alsosuccessfully employed at the feature level or decision level@6#.

References@1# Agogino, A., Alag, S., and Goebel, K., 1995, ‘‘A Framework for Intellige

Sensor Validation, Sensor Fusion, and Supervisory Control of Automatedhicles in IVHS,’’ Intelligent Transportation: Serving the User through Deploment, Proceedings of the 1995 Annual Meeting of ITS AMERICA, Vol. 1, pp.77–87.

@2# Goebel, K., and Agogino, A., 1996, ‘‘An Architecture for Fuzzy Sensor Vadation and Fusion for Vehicle Following in Automated Highways,’’Proceed-ings of the 29th International Symposium on Automotive Technology andtomation (ISATA), Florence, Italy, pp. 203–209.

@3# Khedkar, P., and Keshav, S., 1992, ‘‘Fuzzy Prediction of Time Series,’’Pro-ceedings of the IEEE International Conference on Fuzzy Systems, San Diego,pp. 281–288.

@4# Varaiya, P., 1993, ‘‘Smart Cars on Smart Roads: Problems of Control,’’ IETrans. Autom. Control,AC-38, No. 2, pp. 195–207.

@5# Bonissone, P., Chen, Y., Goebel, K., and Khedkar, P., 1999, ‘‘Hybrid S

Fig. 3 Error spacing of follower car for closed-loop sensorvalidation and fusion in the presence of non-Gaussian noise

146 Õ Vol. 123, MARCH 2001 Copyright ©

ververeri-

thetede

estce

ad-narsors.msance

toe

ofors

n-theThe

on-, i.e.,hisfail-ichun-gas

be

tVe--

li-

Au-

E

oft

Computing Systems: Industrial and Commercial Applications,’’ Proc. IEE87, No. 9, pp. 1641–1667.

@6# Goebel, K., Badami, V., and Perera, A., 1999, ‘‘Diagnostic Information Fusfor Manufacturing Processes,’’Proceedings of the Second International Coference on Information Fusion, Fusion ’99, Vol. 1, pp. 331–336.

Extension and Simplification ofSalukvadze’s Solution tothe Deterministic NonhomogeneousLQR Problem

R. D. HamptonAssociate Professor, Department of Mechanical andAerospace Engineering, University of Alabamain Huntsville, Huntsville, AL 35899 Mem. ASME

C. R. KnospeAssociate Professor, Department of Mechanical,Aerospace, and Nuclear Engineering, University ofVirginia, Charlottesville, VA 22903

M. A. TownsendWilson Professor, Department of Mechanical, Aerospacand Nuclear Engineering, University of Virginia,Charlottesville, VA 22903 Fellow ASME

In a previous paper (Hampton, R. D., et al., 1996, ‘‘A PracticSolution to the Deterministic Nonhomogeneous LQR ProblemASME Journal of Dynamic Systems, Measurement, and ConVol. 118, pp. 354–360.) the authors presented a solution to thnonhomogeneous linear-quadratic-regulator (LQR) problem,the case of known, deterministic, persistent (‘‘non-dwindlingdisturbances. The authors used variational calculus and statransition-matrix methods to produce an optimal matric solutiofor bounded determinist forcing terms. A restricted version of tproblem (treating dwindling disturbances) was evidently firstvestigated by Salukvadze, M. E., 1962, ‘‘Analytic Design of Relators (Constant Disturbance),’’ Automation and Remote ContrVol. 22, No. 10, Mar., pp. 1147–1155, using a differential-equations approach. The present paper uses Salukvadze’sproach to extend his work to the case of non-dwindling distbances, with cross-weightings between state- and control vecand pursues the solution to the same form reported previouslHampton et al. @DOI: 10.1115/1.1344242#

IntroductionIn 1961, M. E. Salukvadze@1# used a differential equation

approach to solve the nonhomogeneous form of what is nknown as the infinite-horizon, time-invariant, linear quadraregulator~LQR! problem. He restricted his treatment to knowdeterministic disturbances that diminish with time~‘‘dwindling’’disturbances!. More recently, Hampton et al.@2# extended the so-lution to the case of non-dwindling disturbances, using variatiocalculus and state-transition-matrix methods. The present teccal brief provides an alternate solution, using a modificationSalukvadze’s historical approach.

Contributed by the Dynamic Systems and Control Division of THE AMERICANSOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the Dynamics Systems and Control Division October 22, 1998. Associate Editor: E. Misawa.

2001 by ASME Transactions of the ASME

a

n

le

e

d

-

Problem StatementGiven a stabilizable, linear time-invariant system

xI5AxI 1BuI 1E fI , (1a)

xI ~0!5xI 0 (1b)

the task is to choose optimal controlleruI 5uI * (t) to minimizeperformance index

J5 limI→`

1

2I H E0

I

@xI T uI T#FW1 W2

W2T W3

G FxIuI GdtJ (2)

for a bounded, non-dwindling disturbance vectorfI (t). Systemand weighting matrices are all constant and real-valued,W1 ispositive semidefinite~PSD!, andW3 is positive definite~PD!.

Solution by Differential Equations ApproachFirst, the performance indexJ is augmented with the state equ

tions @Eqs.~1!#, using Lagrange multiplierslI (t) ~e.g., see Elbert,@3#!. Upon setting the first variationd J equal to zero, solvingfor uI , and substituting into the state equations one obtainsfollowing:

H xI

lIJ 5F A2BW321W2

T BW321BT

W12W2W321W2

T 2~A2BW321W2

T!TG H xIlI J 1F E

OG fI ,

(3a)

subject to

xI ~0!5xI 0 (3b)

and

lI ~`!50I , (3c)

with optimal control

uI * ~ t !5W321BTlI ~ t !2W3

21W2TxI ~ t !. (4)

Next, Eqs.~3! are solved forlI . OncelI has been expressed iterms ofxI and fI , the result can be substituted into Eq.~4! to yielda useful expression for the optimal control.

Solving the homogeneous system yields

@xI T lI T#hT5exp~At !@xI 0

T lI T~0!#T, (5)

where the Hamiltonian matrix


-

the

AªF A2BW321W2

T BW321BT

W12W2W321W2

T 2~A2BW321W2

T!TG (6)

has a spectrum that is symmetric about the imaginary axis~Zhouand Doyle@4#, p. 233!. For nonsingularA there exists a similaritytransformation

J5X21AX (7)

which convertsA into block-diagonal form

J5diag~ J–,2 J– !, (8)

where J– is a Jordan canonical form containing only the stabeigenvalues ofA ~cf. Anderson and Moore@5#, p. 354!. PartitionXandX21 as@X21

X11X22

X12# and@Y21

Y11Y22

Y12#, respectively, to correspond to th

partitioning ofJ @Eqs.~7,8!#. Then Eq.~5! becomes

H xIlI J

h

5H X11eJ– tcI11X12e

2 J– tcI2

X21eJ– tcI11X22e

2 J– tcI2J , (9)

where

H cI1

cI2J 5 HY11xI 01Y12lI ~0!

Y21xI 01Y22lI ~0!J . (10)

Following Salukvadze@1#, the variation of parameters methois now used to find the general solution of Eqs.~3!. One firstsolves the equation

H X11eJ– tcI11X12e

2 J– tcI2

X21eJ– tcI11X22e

2 J– tcI2J 5 H E fI

0I J (11)

for @ cI1TcI2

T#T and then integrates to findcI 1 and cI 2 . These stepsyield

cI 15gI 11Eg3

t

e2 J–tY11E fI ~t!dt,

(12)

cI 25gI 21Eg4

t

eJ–tY21E fI ~t!dt,

with 2n integration constantsgI 1 andgI 2 , and integration limitsg3and g4 . Using Eq.~12! and the more common form for the indefinite integrals~Courant,@6#, p. 116!, the general solution is:

H xIlI J 5H X11e

J– tS gI 11E e2 J– tY11E fI ~ t !dtD1X12e2 J– tS gI 21E eJ– tY21E fI ~ t !dtD

X21eJ– tS gI 11E e2 J– tY11E fI ~ t !dtD1X22e

2 J– tS gI 21E eJ– tY21E fI ~ t !dtD J . (13)

One next eliminates unknown constantsgI 1 and gI 2 from Eq.~13!, using the constraint thatuI * is a stabilizing controller. Forthe homogeneous case, the upper partition of Eq.~13! reduces to

xI 5X11eJ– tgI 11X12e

2 J– tgI 2 . (14)

Since J– is asymptotically stable, the final term of Eq.~14! re-quires thatgI 2[0I . Accordingly, Eq.~13! simplifies to

H xIlI J 5 HX11§I 1X12cI

X21§I 1X22cIJ , (15)

where

zI 5eJ– tS gI 11E e2 J– tY11E fIdtD (16)

and

cI 5e2 J– tE eJ– tY21E fIdt. (17)

Solving now for lI in terms of xI , and assuming thatX11 isnonsingular, one obtains

MARCH 2001, Vol. 123 Õ 147

e

o

andl-

:

he

m.

t

f

lI5~X21X1121!xI 1~X222X21X11

21X12!cI , (18)

which reduces~Ortega,@@7#, p. 29#! to

lI5~X21X1121!xI 1Y22

21e2 J– tE eJ– tY21E fIdt. (19)

Application of Eq. ~4! leads to the following equation for thoptimal control:

uI * 5~W321BTX21X11

212W321W2

T!xI

1S W321BTY22

21e2 J– tE eJ– tY21E fIdtD . (20)

Define now the matrix

Pª2X21X1121. (21)

Then Eq.~20! becomes

uI * 52W321~BTP1W2

T!xI 1W321BTY22

21e2 J– tE eJ– tY21E fIdt.

(22)

Since J– is asymptotically stable,

E eJ– tY21E fIdt52Et

`

eJ–tY21E fI ~t!dt; (23)

Eq. ~22! becomes

uI * 52W321~BTP1W2

T!xI 2W321BTE

t

`

Y2221eJ–~t2t !Y21E fI ~t!dt.

(24)

It can be shown~see Appendix! that

Y2221eJ–~t2t !Y215eAT~t2t !P, (25)

where

A5A2BW321~BTP1W2

T!. (26)

With this substitution, Eq.~24! has the final form:

uI * 52W321~BTP1W2

T!xI 2W321BTE

t

`

eAT~t2t !PE fI ~t!dt.

(27)

RemarksFor the homogeneous case Eq.~27! reduces to the form

uI * 5uF/B* ª2W321~BTP1W2

T!xI . (28)

This is the well-known solution to the homogeneous LQR prolem, whereP satisfies the matric Riccati equation~MRE!:

PA1ATP2~PB1W2!W321~PB1W2!T1W150. (29)

Substituting from Eq.~28! into Eq. ~1! yields, for the homoge-neous case,

xI5@A2BW321~BTP1W2

T!#xI 5AxI . (30)

A is seen to be the dynamic system matrix for the closed-lsystem. It can be shown~see Appendix! that

AT5Y2221J–Y22. (31)

This means thatJ– is the Jordan canonical form ofAT.

ConclusionThis paper has presented a classical differential-equations s

tion to the nonhomogeneous LQR problem with known, determistic, non-dwindling disturbances. The solution was pursued tosame form reported previously in Hampton et al.@2#.

148 Õ Vol. 123, MARCH 2001

b-

op

olu-in-the

AcknowledgmentsThe authors are grateful to NASA Glenn Research Center

the Commonwealth of Virginia’s Center for Innovative Technoogy for their funding of this work.

Appendix: Simplification of the optimal preview termEquation~24! presented the following optimal-control solution

uI * ~ t !52W321~BTP1W2

T!xI ~ t !2W321BT

3Et

`

Y2221eJ–~t2t !Y21E fI ~t!dt. (A1)

This equation reduces to Eq.~27! upon simplifying the expressionY22

21eJ(t2t)Y21, as follows.The Hamiltonian matrixA is defined by

A5F A2BW321W2

T BW321BT

W12W2W321W2

T 2~A2BW321W2

T!TG . (A2)

As noted before,A can be reduced to near-diagonal form by tsimilarity transformation

A5XJX21, (A3)

J5diag~ J–,2 J– !, (A4)

and whereJ– is an asymptotically stable Jordan canonical forPartitioningX andX21 as before; substituting forX,X21, andJin ~A3!; and expanding yields the following identities:

BW321BT5X11J–Y122X12J–Y225~BW3

21BT!T (A5)

and

~A2BW321W2

T!T5X22J–Y222X21J–Y12. (A6)

Since matrixP is symmetric~Zhou and Doyle@4#, p. 236!,

P52X21X112152X11

2TX21T . (A7)

Note also~e.g., Ortega@@7#, p. 29#! that

Y22215X222X21X11

21X12 (A8)

and

Y2152Y22X21X1121. (A9)

Using Equations~A5!–~A9! one can show straightforwardly thaY22

21J–Y215ATP:

Y2221J–Y215Y22

21J–Y22P @using ~A7,A9!# (A10)

5@Y22T J–T~X222X21X11

21X12!T#TP @using ~A8!# (A11)

5@Y22T J–T~X22

T 2X12T X11

2TX21T !#TP (A12)

5@Y22T J–TX22

T 2Y12J–T~X21T 2X11

T X112TX21

T !T2Y22T J–TX12

T X112TX21

T #TP(A13)

5@Y22T J–TX22

T 2Y12T J–T~X21

T 2X11T X21X11

21!

2Y22T J–TX12

T X21X1121#TP @using ~A7!# (A14)

5@~Y22T J–TX22

T 2Y12T J–TX21

T !2~Y12T J–TX11

T 2Y22T J–TX12

T !

~2X21X1121!#TP (A15)

5@A2BW321~BTP1W2

T!#TP5ATP @using ~A5,A6,A7,26!#(A16)

It is shown now thatY2221 is a generalized eigenvector matrix o

AT:


alJ.

Y2221J–Y225~X222X21X11

21X12! J–Y22 @using ~A8!#(A17)

5X22J–Y222~X212X21X1121X11! J–Y122X21X11

21X12J–Y22(A18)

5X22J–Y222~X212X112TX21

T X11! J–Y12

2X112TX21

T X12J–Y22 @using~A7!# (A19)

5X22J–Y222X21J–Y121~2X112TX21

T !~X12J–Y222X11J–Y12!(A20)

5X22J–Y222X21J–Y121~X12J–Y222X11J–Y12!

3~2X21X1121! @using~A5,A7!# (A21)

5AT. @using~A5,A6,26!# (A22)

ThereforeY2221 is a generalized eigenvector matrix ofAT. Conse-

quently,


Y2221eJ–~t2t !Y215Y22

21eJ–~t2t !Y22~2X21X1121!

5eY2221J–~t2t !Y22P @using~A9!# (A23)

5eAT~t2t !P (A24)

Equation~27! follows immediately from Eqs.~A1! and ~A24!.

References@1# Salukvadze, M. E., 1962, ‘‘Analytic Design of Regulators~Constant Distur-

bances!,’’ Automation and Remote Control ,22, No. 10, Mar, pp. 1147–1155;Originally published inAvtomatika i Telemakhanika, 22, No. 10, Oct. 1961,pp. 1279–1287.

@2# Hampton, R. D., Knospe, C. R., and Townsend, M. A., 1996, ‘‘A PracticSolution to the Deterministic Nonhomogeneous LQR Problem,’’ ASMEDyn. Syst., Meas., Control,118, pp. 354–360.

@3# Elbert, Theodore F., 1984,Estimation and Control of Systems, Van NostrandReinhold, NY.

@4# Zhou, K., and Doyle, J. C., 1998,Essentials of Robust Control, Prentice-Hall,Upper Saddle River, NJ.

@5# Anderson, B. D. O., and Moore, John B., 1971,Linear Optimal Control,Prentice-Hall, Englewood Cliffs, NJ.

@6# Courant, R., 1937,Differential and Integral Calculus, 2nd edition, IntersciencePublishers, NY, Vol. 1.

@7# Ortega, J. M., 1987,Matrix Theory: A Second Course, Plenum Press, NY.

MARCH 2001, Vol. 123 Õ 149

using asme format - dspace at koasas:...

Documents