Techniques in Discrete and Continuous Robust Systems, Volume 74: Advances in Theory and Applications

CONTROL AND DYNAMIC SYSTEMS

Advances in Theory and Applications Volume 74

CONTROL AND DYNAMIC SYSTEMS

Advances in Theory and Applications Volume 74

CONTRIBUTORS TO THIS VOLUME DE ABRE U-GAR CIA J. CHEN M UNTHER A. DAHLEH SARIT K. DAS ANTHONY N. MICHEL AHMAD A. MOHAMMED YAS UHIKO MUTOH PETER N. NIKIFORUK R. J. PATTON MILOJE S. RADENKOVIC P. K. RAJAGOPALAN TIELONG SHEN ROBERT E. SKELTON PETROS G. VOULGARIS G UOMING G. ZHU

J. A .

CONTROL AND DYNAMIC SYSq'EMS ADVANCES IN THEORY AND APPLICATIONS

Edited by

CORNELIUS T. LEONDES School of Engineering and Applied Science University of California, Los Angeles Los Angeles, California

VOLUME 74:

TECHNIQUES IN DISCRETE AND CONTINUOUS ROBUST SYSTEMS

ACADEMIC PRESS San Diego New York Boston London Sydney Tokyo Toronto

This book is printed on acid-free paper. ( ~ Copyright 9 1996 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

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United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW1 7DX

International Standard Serial Number: 0090-5267 International Standard Book Number: 0-12-012774-1

PRINTED IN THE UNITED STATES OF AMERICA 96 97 98 99 00 01 QW 9 8 7 6 5

4

3

2

1

CONTENTS

CONTRIBUTORS .................................................................................. PREFACE ................................................................................................

vii ix

Optimal and Robust Controllers for Periodic and Multirate Systems ......................................................................................

Petros G. Voulgaris and Munther A. Dahleh ................................

59

A Two-Riccati, Feasible Algorithm for Guaranteeing Output L~ Constraints ............................................................................

97

Discrete-Time Robust Adaptive Control Systems

Miloje S. Radenkovic and Anthony N. Michel

Guoming G. Zhu and Robert E. Skelton Techniques of Analysis and Robust Control via Zero-Placement of Periodically Compensated Discrete-Time Plants ................................... 133

Sarit K. Das and P. K. Rajagopalan Robust Fault Detection and Isolation (FDI) Systems

........................... 171

R. J. Patton and J. Chen Absolute Stability of Discrete Nonlinear Feedback Systems

............... 225

Yashuhiko Mutoh, Tielong Shen, and Peter N. Nikiforuk

vi

CONTENTS

Continuous Time and Discrete Time L y a p u n o v Equations" R e v i e w and N e w Directions ................................................................................ 253

Ahmad A. Mohammed and J. A. De Abreu-Garcia

I N D E X ..................................................................................................... 305

CONTRIBUTORS

Numbers in parentheses indicatethe pages on which the authors' contributions begin. J. A. De Abreu-Garcia (253), Department of Electrical Engineering, The

University of Akron, Akron, Ohio 44325 J. Chen (171), Division of Dynamics and Control, University of Strathclyde,

Glasgow, Gl lXJ, United Kingdom Munther A. Dahleh (1), Laboratory of Information and Decision Systems,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Sarit K. Das (133), Department of Electrical Engineering, Indian Institute

of Technology, Kharagpur 721302, India Anthony N. Michel (59), Department of Electrical Engineering, University

of Notre Dame, Notre Dame, Indiana 46556 Ahmad A. M o h a m m e d (253), Department of Electrical Engineering, The

University of Akron, Akron, Ohio 44325 Yasuhiko Mutoh (225), Department of Mechanical Engineering, Sophia Uni-

versity, Tokyo, Chiyoda-ku, Japan Peter N. Nikiforuk (225), Department of Mechanical Engineering, Univer-

sity of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N OWO R. J. Patton (171), Department of Electronic Engineering, University of Hull,

Hull HU6 7RX, United Kingdom Miloje S. Radenkovic (59), Department of Electrical Engineering, Univer-

sity of Colorado at Denver, Denver, Colorado 80217 P. K. Rajagopalan (133), Department of Electrical Engineering, Indian In-

stitute of Technology, Kharagpur 721302, India vii

viii

CONTRIBUTORS

Tielong Shen (225), Department of Electrical Engineering, Sophia University, Tokyo, Chiyoda-ku, Japan Robert E. Skelton (97), Space Systems Control Laboratory, Purdue University, West Lafayette, Indiana 47907 Petros G. Voulgaris (1), Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 Guoming G. Zhu (97), Cummins Engine Company, Inc., Columbus, Indiana 47202

PREFACE Effective control concepts and applications date back over millennia. One very familiar example of this is the windmill. It was designed to derive maximum benefit from windflow, a simple but highly effective optimization technique. Harold Hazen's 1932 paper in the Journal of the Franklin Institute was one of the earlier reference points wherein an analytical framework for modem control theory was established. There were many other notable items along the way, including the MIT Radiation Laboratory series volume on servomechanisms, the Brown and Campbell book, Principles of Servomechanisms, and Bode's book, Network Analysis and Synthesis Techniques, all published shortly after mid-1945. However, it remained for Kalman's papers of the late 1950s (wherein a foundation for modem state-space techniques was established) and the tremendous evolution of digital computer technology (which was founded on the continuous giant advances in integrated electronics) to establish truly powerful control systems techniques for increasingly complex systems to be developed. Today we can look forward to a future that is rich in possibilities in many systems of major significance: manufacturing, electric power, robotics, aerospace, and others with significant economic, safety, cost, and reliability implications. Thus, this volume is devoted to the most timely theme of "Techniques in Discrete and Continuous Robust Systems." The first contribution to this volume is "Optimal and Robust Controllers for Periodic and Multirate Systems," by Petros G. Voulgaris and Munther A. Dahleh. The problems of optimal disturbance rejection and robust stability in periodic and multirate systems are treated in this contribution, and techniques for dealing with these in several major cases are presented. This contribution further notes that the problem of robust stabilization in periodic and multirate systems can be treated without introducing conservatism by considering the problem for the equivalent linear time invariant (LTI) system. Other important results and techniques are also discussed, and as such it is a most appropriate chapter to begin this volume. The second contribution is "Discrete-Time Robust Adaptive Control Systems," by Miloje S. Radenkovic and Anthony N. Michel. The robust adaptive control problem has been the topic of numerous studies, publications,

x

PREFACE

and discussions over the past decade, and it has been observed that algorithms designed for the case of perfect system models cannot provide global stability of the adaptive system in the presence of unmodeled dynamics and external disturbances. Techniques are presented for establishing the global stability of adaptive systems for unmodeled system dynamics and external disturbances which are unstructured, complex, and large at high frequencies. It is worth noting that results similar to those presented in this contribution can be established for the general delay case. The next contribution is "A Two-Riccati, Feasible Algorithm for Guaranteeing Output L~ Constraints," by Guoming G. Zhu and Robert E. Skelton. This contribution presents a rather powerful new and computationally efficient methodology for designing measurement or dynamic controllers that guarantee requisite output bounds for measures of bounded input disturbances while minimizing a weighted summation of measures of the upper bound of the outputs for each control channel. The substantial effectiveness of the results presented in this contribution is illustrated by several examples. Techniques for the design of a linear, discrete periodic controller structure that has the maximum possible degrees of freedom for its order are presented in "Techniques of Analysis and Robust Control via Zero-Placement of Periodically Compensated Discrete-Time Plants," by Sarit K. Das and R. K. Rajagopalan. Analytical methods for dealing with periodic systems and the zero placement capability of periodic controllers are also presented. The role of periodic controllers in gain margin improvement of unstable plants with nonminimum phase zeros via the zero placement approach and periodic controllers is treated. Numerous examples demonstrate the effectiveness of the techniques. "Robust Fault Detection and Isolation (FDI) Systems," by R. J. Patton and J. Chen, is an in-depth treatment of the techniques in the field of robustness for fault diagnosis that rest on the model-based residual generation methods (which are defined in this contribution). This approach to fault detection and isolation offers rather significant potential, particularly in view of the role it can and probably will play in future industrial systems, process plants, and other applications of major significance. The next contribution, "Absolute Stability of Discrete Nonlinear Feedback Systems," by Yasuhiko Mutoh, Tielong Shen, and Peter N. Nikiforuk, presents various approaches for the establishment of the stability of discretetime nonlinear feedback systems. Techniques are presented for both singleinput-single-output (SISO) systems and multiple-input-multiple-output (MIMO) systems. Examples illustrate the utility of the methods presented. The final contribution to this volume is "Continuous Time and Discrete Time Lyapunov Equations: Review and New Directions," by Ahmad A. Mohammad and J. A. De Abreu-Garcia. The fundamental reasons for robust design techniques include among other factors the maintenance of satisfactory system performance in the face of model uncertainties and, perhaps more

PREFACE

xi

importantly, the maintenance of system stability. This contribution is an indepth treatment of one of the most powerful means for confirming stability for both linear and nonlinear systems. As such this is a most suitable contribution with which to conclude this volume. The contributors to this volume are all to be highly commended for their contributions to this comprehensive treatment of techniques in discrete and continuous robust systems. They have produced a work which should provide a unique and useful reference on this broad subject internationally for years to come.

This Page Intentionally Left Blank

Optimal and Robust Controllers for Periodic and Multirate Systems Petros G. Voulgaris Coordinated Science Laboratory University of Illinois at Urb~na-Chmmpaign

Munther A. Dahleh L a b o r a t o r y of I n f o r m a t i o n ~nd D e c i s i o n S y s t e m s

Massachusetts Institute of Technology

I. I N T R O D U C T I O N The study of periodically time varying systems is a topic great practical and theoretical importance. In [13] an equivalence between m-input, p-output, linear, N-periodic, causal, discrete-time systems and a class of discrete-time linear, time invariant, causal systems was established. Namely, this class consists of ruN,input, pN-output, linear time invariant (LTI) systems with A- transforms P(A) such that P(0) is a block lower triangular matrix. This equivalence is strong in the sense that it preserves the algebraic structure (isomorphism) and the norm (isometry). This equivalence is termed "lifti n ~ and the LTI system that lifting associates with the N-periodic system is called the ~lifted" system. Hence, we can effectively use the theory of LTI systems to study periodic ones. In fact, the authors in [13] use this equivalence to prove that although the performance is not improved , periodic compensators for LTI plants'offer significant advantages in terms of robustness to parametric uncertainty. In this chapter, we define the problem of optimal disturbance rejection in periodic systems and present solutions to the following three cases: I. Optimal L~~ to t~176disturbance rejection. 2. Optimal t2 to t2 disturbance rejection. 3. Optimal rejection of stochastic disturbances (the L Q G problem). Utilizing the power of lifting and the results in [3] we can easily infere that in all three cases the optimal controller for the N-periodic system can be obtained by solving the equivalent LTI problem. This problem however, includes a constraint on the optimal LTI compensator C(A), namely CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

2

PETROS G. V O U L G A R I S A N D M U N T H E R A. D A H L E H

C(0) should be block lower triangular matrix so that C corresponds to a causal N-periodic controller. This constraint makes the optimization problem a non-standard one and its solution is the theme of sections IV and V. Furthermore in section VI we demonstrate that the problem of optimal disturbance rejection for multirate sampled systems can be treated analogously. In particular, we show how with a simple modification the same approach can be used to obtain the optimal multirate compensator. Also in this chapter, we consider the problem of robust stabilization in periodic and multirate plants. We indicate that this problem can be analyzed without introducing conservatism by considering the same problem for the equivalent LTI system.

II. M A T H E M A T I C A L

PRELIMINARIES

This section presents the notation and definitions to be used throughout the chapter. Also, some important to our development mathematical results are provided. References are given to cover all of the needed mathematical background.

1

Generic

Notation

In this section we give some generic notation that is used throughtout the thesis.

p(A)

The spectral radius of the matrix A.

~[A]

The maximum singular value of the matrix A.

Izlp

The p-norm of the finite dimensional vector z = (zl z~ . . . z , ) T given as

Ixl,, = ()":~ Ix, I) ~/', p < oo i=l

Izlp = m.,~ Ix, I, p = oo. $

Ial~

The 1-norm of the m x n matrix A = (Aij) given as

FJ

IAII= /t(A)

max

i--- 1,...,m

~'~IA,#Ij=0

The A-transform of a m x n real sequence H = {H(k)}~~ defined as: oo

9(~) = ~

k~---oo

#(k)~ ~

CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS

X

I

The dual space of the normed linear space X.

BX

The closed unit ball of X.

•

The left annihilator of S C X*.

S•

The right annihilator of S C X.

IIs

The projection operator onto the subset S of the Hilbert space X.

(Z~

T*

n~

Z*)

The value of the bounded linear functional z" at point z ~ X. The adjoint of the operator T. The kth-truncation operator acting on a m x 1

~e~tor~u~d ~equen~ {~(~)}~~ a~ Hkm 9{u(0), u(1),...}

Am

: {u(0),..., u(k), 0, 0,...}.

The right shift operator acting on a m x 1 vector valued sequence {u(k) } ~~ as

A m : {u(0), u(1),...}

2

3

Some

, {0, u(0), u(1),...}.

Basic Spaces

In this section we define certain important normed linear spaces that we very frequently refer to in the course of our development. These spaces are the following (for details look at [22,16,14,28]):

l~x" 9The Banach space of all m x n matrices H each of whose entries is a right sided, absolutely summable real sequence Hij = {Hij (k) }~~ The norm is defined as:

IIHII~,

mXs

n dell" =

max E i

oo

~ IHij(t)l.

j=l k=0

: The Banach space of real rn x 1 vectors u each of whose components is a magnitude bounded reM sequence { u , ( k ) } r : 0. The norm is defined as:

II~llt: d.j miax(s~pI~,(k)l).

4

PETROS G. VOULGARIS AND MUNTHER A. DAHLEH

~~ 9The extended t~~ space: it is the space of all real right sided m x 1 vector valued sequences.

t ~ 9 The Hilbert space of real rn • 1 vectors u each of whose components is an energy bounded real sequence {u~ (h) }~~ The norm is defined as: m

oo

i=1 k=O

x , " The real Banach space of all m x n matrices H(A) such t h a t H(A) is the A-transform of an I x , x , sequence H. The norm is defined as

This space is isometrically isomorphic to lx.,x, i.e., l ~ x . _-- . A . , x . .

.A~x n 9 The Banach space of all m x n matrices H each of whose entries is a right sided, magnitude bounded real sequence Hij = {Hij(k))~~ The n o r m is defined as: m

IIHII.A:.,,..

def

max(sup IH,i(k)l).

=

dfX

cox. zero.

9The subspace of ~ x .

J

k

consisting of all elements which converge to

O0

s " The Banach space of all m x . m a t r i x valued functions F defined on the unit circle of the complex plain with def

I]F][s = ess sup ~'[F(eJ')]< oo. ,e[0,~.]

oo

7~mx. 9 The Banach space of all rn x n matrix valued functions F analytic in the open unit disk of the complex plain with def

IIFII - =

sup

max ~ [ F ( r e J ' ) ] < oo.

rE[O,1) #El0,2-]

This space can be considered as a closed subspace of s

oo

CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS

5

s 2 1 5 " The Hilbert space of matrix valued functions F defined on the unit circle of the complex plain with

IIFII~ def = [(2x) -1 fo 2" t r a c e ( F T ( e - { ' ) F ( e ~ ' ) ) d e ]

~/~ < oo.

~/~xn " The Hilbert space of all m x n matrix valued functions F analytic in the open unit disk of the complex plane with def

sup [(2r -1 JO02r t r a c e ( F T ( r e - i ' ) F ( r e ~ ' ) ) d O ] ~/2 < oo. re[o,1)

This space can be considered as a closed subspace of s

t 2 = ~2,,1. 3

Input-Output Systems

Characterization

Moreover,

of Linear

In this section we consider the input-output characterization of systems by viewing them as linear operators. 3.1

Causality

We start with the notion of a causal operator D e f i n i t i o n 3.1 L e t T : l~ ,e

n Tu =

be an operator. T is called causal i f

, ~,e

ThOu,

vk = o,

2,...,

T is called s t r i c t l y causal i f II ~ T u = IIk - m -T- InI k - I u,

V k = O , 1,2, . . . .

The class of all causal operators T on/oo,e will be denoted by s215 Such operators can be represented by infite block lower triangular matrices (Toeplitz) of the form

To(O) o "

i

""/

6

PETROS G. V O U L G A R I S AND M U N T H E R A. D A H L E H

where Tk(i) axe m • n matrices for all i, k. This representation is another way to state that these operators are convolution operators; i.e., if y = Tu then k

~(k)

= ~

n(k

-

i)~(~)

i----0

3.2 3.2.1

Stability T h e spaces ~ " ( P ) ,

~"

(t 2)

Next, we consider the notion of/a-stability where ~ = or, 2. D e f i n i t i o n 3.2 Let T be a causa/operator in s if its induced norm over t a is bounded; i.e., i f IITII

=

sup ~,:.~,o Ilulll:.

Then T is ~ - s t a b l e

< oo.

The clase of all/a-stable systems equipped with the induced norm, will be denoted by 15~T~"(t~). This class is s Banach space and in paxticulax it is a Banaeh algebra with multiplication defined as composition. In the cane where a = oo we refer to the space B~T~n(t ~176as the space of Bounded-Input-Bounded-Output (BIBO) stable, or simply, stable systems. Moreover, the following fact can be easily checked. nite rn • n block lower triangular matrices of the form

To(O) 0 ---/ r~(X) T~(O) 9

i

where Tk (i) are rn • n matrices for all i, k such that

,up I(n(k) n(k-

x) . . . n ( 0 ) ) l ~


= As in the proof of Theorem 1.1, let

TsT~ be the operator

on t 1 associated

with the stable system Si-1 and let T~si., be its adjoint on c ~ Then by defining

x = ~ r , (D) we can show that Hence ($I ~Q, X) = 0 if and only if (Q, F) = 0 which completes the proof. The functionals Xy, j = I,...,J-5 obtained by the second method will contain finitelym a n y nonzero components. Namely, Xj(m) = 0 V m > k. This is so because Fj has all its elements but the first equal to zero and hence Dy(m) equals zero for m > k. In the first method however, the obtained functionals will not, in general, have this property. They will decay though, (since they lie in co) and the rate of decay will be dictated by the poles of Ui and ~ . Hence, it seems that the linear programming problem in the dual space [5] will be simpler when the second method is used, provided that a simple way to obtain a Smith decomposition of a transfer matrix exists.

28

1.3

PETROS G. VOULGARISAND MUNTHERA. DAHLEH

Examples

Here we show that in general (OPToo) has different solution than the unconstrained problem. Example 1.1 Consider an optimization problem in a 2-periodic singleinput single-output system with the equivalent LTI problem being as follows; inf IIS - vQII

"-A/

QEll

/ 1 0 / U(A)--/A such that Q(0) is lower triangular, where/t(,~) = 0 2 ' 0 1 We first solve the unconstrained problem following [5]" The basis for the functionals that annihilate S, = {UQ "Q E t 1} consists of the following: F~={

(10) 0 0

F 2 = { l 0o

The resulting optimal solution

0

..

01 ) , 0 ,

..

~. = H-

.}

U Q . is:

(1 0)

where @12,4)22 arbitrary in t I such that II~x211,, + 114,2211,, __< 1

and

Qu= /--~b21 2--~22 / -~21

2 - ~22

Also, I1+.11~, = 1.

Now, in the constrained case, we obtain by using the second approach the following extra functional: F 3 = { I 00 01 1 , I 00 011 ,0,...} The resulting optimal solution is: ~(~)=

I I .5~ 1 0 1.5

"

CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS

with

Q(~)=

Also,

(oo) o .5

ll x then IIAGII -- IIGll > 1 and hence there exists [23] a Ax E 7/~ with II~xll < x such that the pair (AG, AI) in feedback is unstable. But then the pair ( G , A ~ is unstable with A ~ = AxA; mo~eove~ ~in~e I1~~ = II~xll < 1 and also A ~ is strictly proper then by Fact 1.2 we conclude that if IIGpII > 1 there is a destabilizing A~ E/)p with A~ being the inverse lift of A ~ In the discussion above, Gp will depend in general on the particular controller C~ we pick for the nominal unperturbed system. Hence, if we pertain to periodic controllers (i.e., periodic

CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS

1 |

,

|

W ~

! !

, ,,,,

I

! |

:Gp !

.

P,

v

|

!

| I |

| | | | | o I

1

..

| | ! ~

.

,......................................., Figure 4: Robust Stabilization. maps Gp(C~)), then in order to maximize the tolerance to perturbations in the class D the problem we have to solve is inf

C , periodic

[[Gp(C,)[[

which is a performance problem in periodic systems that can be solved with the method indicated in the previous sections. At this point, we do not know whether a general time varying Cv can give better margins than a periodic with respect to this particular class ~Dp of periodic perturbations. However, if we enlarge the class of perturbations to include generally time varying ones as then we have the following necessary and sufficient condition: T h e o r e m 2.1 The following statements are equivalent:

(i) There exists a robustly stabilizing time varying controller with respect to perturbations in l)u. (ii) There ex/sts a robustly stabilizing periodic controller with respect to perturbations in ~)~.

OiO inf

C. periodic

IlGp(C )ll "< 1.

41

42

PETROS G. VOULGARIS AND MUNTHER A. DAHLEH

Proof The (ii)=~(i)is obvious. For (i)::~(ii)consider the lifted feedback system ( G ( C ) , A) where C is the lifted controller; note that G, C, A are not shift invariant, nevertheless, all should posess a block lower triangular feedforward term. To be more specific, consider for example the relation z - Gv; then, the operation of G on v can be represented as the convolution z ( t ) - ~~176 g(t, r)v(r). G has a block lower triangular feedforward term if g(t, 0) is block lower triangular for all t. The same interpretation is valid for A, C. Now following [25] this implies that infk [[GAk[[ _< 1 for, otherwise, a strictly proper A, and hence a legitimate A~ E l)v, exists that destabilizes the system. But then, following the same arguments as in theorem 4.1 of [24] we can check that there is a controller C1 - A-t~CA t~ for some large enough integer kl, such that ]]G(C~)[] _< 1. Note that Cx has a block lower triangular feedforward term since C does and therefore it corresponds to a causal (not necessarily periodic) C1~. But then from Fact 1.4 there is a periodic controller Cp such that [[Gp(Cp)[I < 1 and therefore Cp will robustly stabilize the system for all perturbations in ~ . The (ii)r follows from the analysis of periodic perturbations in ~)p. | Hence, if we wish to maximize the tolerance to perturbations in the class/)~ the problem we have to solve is

inf

C. periodic

[[Gp(C~)] I

which is a performance problem in periodic systems that can be solved with the method indicated in section 4. Finally, we should note that the exactly the same analysis holds when we encounter the robustness problem for multirate systems.

3

Section S u m m a r y

In this section we presented the solutions to the optimal s2 to 12 disturbance rejection problem (7/~176as well as the solution to the L Q G (7/2) problem in periodic systems using the lifting technique. Both problems involved a causality condition on the optimal LTI compensator when viewed in the lifted domain. The 7/~176 problem was solved using the Nehari's theorem whereas in the 7i2 problem the solution was obtained using the Projection theorem. In particular, the 7~~176problem was solved by modifying the standard Nehari's approach in order to account for the additional causality constraint on the compensator. This modification yielded a finite dimensional convex optimization problem over a convex set that needs to be solved before applying the standard solution to the Nehari problem. The solution to the above convex finite dimensional problem can be obtained easily using standard programming techniques. In the 7g2 case the solution was obtained from the optimal (standard) unconstrained problem by projecting only the feedforward term of the standard solution to the allowable

CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS

43

Yl

Ul

K1T

L1T

K,~T

LpT

Um v

Figure 5" General Multirate System. subspace. Finally, we indicated that the small gain condition is necessary as well as sufficient for 12-stability in periodic and multirate systems and that general time varying compensation offers no advantage over periodic as far as robust stability to time varying perturbations is concerned.

VI. M U L T I R A T E SYSTEMS" O P T I M A L D I S T U R B A N C E REJECTION AND ROBUST STABILITY In this section, we utilize the methods of solutions to the problem of optimal disturbance rejection in periodic systems that we presented in sections VI and V in order to solve the same problem in the case of multirate systems. The most general configuration of a multirate system is depicted in Figure 5. In this figure each of the input sequences {uj (k) }~~ enters the system at every Tu~ seconds which might be different from, say, Tuj+l which is the rate of the input sequence {u~ +1 (k) }~~ Also, {yi (k) }~~ leaves the system at a rate of Ty~ seconds which might be different from the rates of the rest of the output and/or input sequences. A typical example of a multirate system is the multirate sampled data system (MRSD) of Figure 6 where not all of the input and output sequences are held and sampled at the same rate. In the case where not all of the rates of the input and output sequences are rational multiples of each other i.e., the sequences are not sychronized the problem of designing the controller is hard. In the case however where the rates are sychronized (i.e., any ratio of rates is rational) then the synthesis problem becomes considerably easier. In particular, lifting techniques can be used to transform a multirate problem to a LTI problem as with the case of periodic systems. Again, this lifting induces a causality constraint on the lifted controller which can be handled exactly with the same manner as in the periodic case. In the subsequent sections we give the details of

44

PETROS G. V O U L G A R | S A N D M U N T H E R A. D A H L E H

Ul . ~ ~

Uc I

Ycl

. . . .

Yl v

L~T K~T Um

G0(,)

o

Uc,~

Yc~

-I!-iii-.......

!

w

LpT

K, n T

Figure 6: General Multirate Sampled Data System. how this is done.

Lifting and Shift Invariant Equivalence Throughout this section we will deal only with the sychronous case. This means that for the system of Figure 5 the rates of all the input and output sequences can be written as integer multiples of some common time period T. In particular we have Tu~ = KiT, j = 1 , . . . , m and Ty, = L~T, i = 1 , . . . , p where we assume that the integers (L1,..., Lp, K1,...K,~) are relatively prime. The period T is called the basic time period (BTP). We will call a multirate system causal if the value of yi(k) Vi = 1 , . . . , m does not depend on any uj(l) with I > kL~/Kj Vj = 1 , . . . , p . For the multirate system of Figure 5 let N denote the least common multiple of the relatively prime integers (Li, L 2 , . . . , Lp, K1, K 2 , . . . , Kin). Also let Pi - ( N / L , ) , i -- 1 , . . . , p , Mj -- ( N / Kj ), j - 1 , . . . , m and P ~ = 1 Pi, M - ~ j = l Mj. In the sequel we introduce s lifting isomorphism [18,21] similar to the one in the periodic case, that transforms them to causal "single rate" systems. m

1.1

Lifting in multirate

systems

Before we proceed we give the following definition: Defin[tion 1.1 Given the set of integers {My )~--I, (Pi}~'=x and a P x M matrix D partitioned as S~--"

I D11D12... D1ml .

.

.

.

.

.

.

CONTROLLERSFOR PERIODICAND MULTIRATESYSTEMS

45

where each Dij is a Pi x Mj matrix we say that D satisfies the (P i, Mj) causality conditions if (D,i)o~ = 0 when (a-l) N

P~

N < O; -(/3- 1)~-j

l 1,0 < / J < 2, with all variables having finite initial conditions.

(6)

DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS

63

Regarding relation (6) we introduce the following assumptions" (A1) V(t) 6 ~+ and V(t) < do < co, for all t > 1. (A2) "Exponentially growing rule" - the sequence n,(t) E ~+ can not grow faster than exponentially, i.e., for all t _> 1

n,(t) < Con,(t - 1) + ko + ~l(t), 1 < Co < co, 0 < ko < co

(7)

where n,(t) is defined in terms of e(t), i.e., t

.o(~)- ~ ~'-J~(j + 1) ~

(8)

j=l

for a fixed 0 < A < 1. (A3) nc~(t) E ~+ and for t > 1 (i) n#(t) > An#(t- 1), no(I) r 0, and n#(0) > 0, 0 < A < 1 (ii) no(t) < Cr ne(t)+ C#~ + ~2(t), 0 < C#1, Cr < co

(9) (10)

(A4) Sequence 7(1) E @~and depends on the sequence e(t), i.e., A'-JT(j) 2

< C~

j=l

A'-Je(j + 1) 2

+ k, + ~3(t) (11)

j=l

where 0 _< C.y,k~ < co, 0 < A < I.

(As) On the subsequence at, k = I, 2, 3,..., where S(a~ -4-I) > 0 and S(ak) 0 for all t > 1. Using the definition of S(t) (Eq. (13)) and assumption (A3)(ii), from (6) one obtains

V(t + 1)+ S(t + 1) n,(t)

S(t) < v(t) + n4,(t- 1) -/~Pl

e(t + 1) 2 nr

(16)

where pl is given by assumption (A6). After summation from $ = 1 to N, from the above relation we obtain

~(t+ 1) 2

lim /~p~

S(1)

< V ( 1 ) + ,,~(0) < oo

(17)

~--1

wherefrom it follows that lim e(t + 1) 2

*-~

nr

=0

(~8)

DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS

65

Since assumption (A3)(ii) implies

n,(t) -< g'~' A l f2 i = 1,2,3

(65)

where 7 and ,~ are the same as in (61). For frequencies close to f~, conditions (64) and (65) may be difficult to achieve, since we need a filter with short transient band. In practice this is the matter of the designer skills and the specific system that has to be controlled. In our considerations we will assume that (64) and (65) are satisfied. We will also assume that W(z) is a stable polynomial. Specifically, if we design a filter with amplitude characteristics described by Eq. (64) and (65) and some of its zeros appear to be unstable, we will use a filter where the unstable zeros are replaced by their reciprocal values. These two filters have different phase characteristics but the same amplitude characteristics, and that is what we are interested in. Thus, the low-pass filter W(z) has the following form W(z -1) = w0 + wlz -1 + - " +

(66)

w, wz -"W

and we assume that (5'4) the zeros zj of t4Z(z) satisfy Izil _< )~1/2, where )~0 + )ix < )i < 1, and 0 < Ax < < ,~0 and ~0 is defined by assumption (Sx). We will stabilize the system (57) by designing the adaptive controller so that for a given reference signal y'(t), the following functional criterion is minimized J -

+ 1) - y ; ( t +

1)] +

2,

(sT)

where y/(t)-

ITV(q-1)y(t), y } ( t ) - I ~ ( q - t ) y ' ( t ) ,

(68)

while polynomials p(q-1) and Q ( q - t ) are chosen by the designer and are defined as follows

p(q-1)

=

l + ptq -1 + . . . + p,~pq-nP

Q(q-t)

=

qo + qtq - t + " ' ' + qnoq-n~

(69)

In (68), ~'V(q-1) is the low-pass FIR filter designed so that requirements (64) and (65) are satisfied and is given by Eq. (66). Using prior information related to the transfer function Be(z)/Ae(z) at low frequencies f < f2, we can choose polynomials p(q-1) and Q(q-1) so that the following assumption is satisfied:

DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS

73

(5'5) All zeros z1 of the polynomial

D(z) - Ae(z)Q(z) + So(z)bV(z)P(z) satisfy [zj[ __ 1. We assume that all initial conditions are finite. Note that the system model (57) can be written in the form

Ao(q-1)yf(t + 1) = Be(q-1)lTVu(t) + 7(t)

(70)

where "r(t)

=

[Be(q-1)A1(q-1)W1(q-1) + Ae(q-1)A2(q-l)w~.(q-l)]~V(q-1)u(t)

+

A3(q- 1) Wa(q-1 ) 1;k(q- 1)~(t)

(71)

It is not difficult to see that from (70) we can obtain e(t + 1)

where

-

[Bo(q-X)l:V(q -1) + Q(q-1)]u(t) - q[Ao(q-:) - 1]y1(t)

+ q[p(q-l)_ l]yl(t)_P(q-i)y;(t+ I)+ 7(t)

(72)

e(t + I) - P(q-1)[yl(t + I ) - y}(t + I)] + Q(q-l)u(t)

(73)

Equation (73) can be written in the form e(t + 1) = oTr

1]y/ (t) -- P(q-1)y~ (t + 1) + 7 ( t ) (74)

where

e0r -

;b0, h i , . . . , b..]

(75)

and

6(t) T = [yl(t),..., y l ( t - nA + 1); u l ( t ) , . . . , u l ( t - nB)]

(76)

u y ( t ) - ~V(q-1)u(t)

(77)

where From Eq. (74) it is obvious that when unmodelled dynamics and external disturbances are absent, control low optimal in the sense of (67) is given by 80Tr

+ Q(q-1)u(t ) + q[p(q-1) _ 1]y/(t) - P(q-1)y}(t + 1)

(78)

Based on the available prior information related to the system model at low frequencies f < f2, we introduce the following assumption:

74

MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL

(ST) The compact convex set O ~ which contains the true parameters 60, the sign of r0 = bowo + qo and a lower bound r0,min, on the magnitude of r0 are known. Without loss of generality we assume that r0 > 0 and r0,min > 0. For the estimation of 00 we propose the following algorithm ~(t + 1) - 7~{8(t)+ n,(t)r

+ 1)}, 0 < v < 2

(79)

where e(t) is given by Eq. (73), while 79{.} projects orthogonally onto (3~ so that :P{0} E O ~ for all/~ E ~,,~+-B+I, and there exists a finite constant do so that [[ 0(t) - a0 [[2_< do < oo and ~0(t) > r0,min > 0 for all t > 0. The algorithm gain sequence n4,(t ) is given by

n,(,) = ,2o + where the vector r satisfying

1)+ II r is given by Eq.

2, 1

< no < o o

(80)

(76), and A is the fixed number

A1 + Ao < A < 1

(81)

for 0 < A1 < < 1 and A0 is defined by the assumption (S1). The role of the number no in Eq. (80) is to prevent division by zero in algorithm (79), and is chosen by the designer. Since 00 is unknown, as the adaptive control law, we will use the "certainty equivalence version" of (78), i.e.,

#(t)To(t) + Q(q-1)u(t ) + q[p(q-1) _ 1]yl(t) = p(q-1)y~( t + 1)

(82)

It is not difficult to see that the estimation scheme, actually represents the algorithm for the identification of the parameters 0o of the nominal system model. Specifically, from Eq. (73) and (82) we conclude that e(t) in the algorithm (79) is given by e(t + 1) = YY (t + 1) - 8(t) r r

(83)

and represents the prediction error, which is usually used in all parameter identification schemes. In the algorithm (79) we are using measurement vector r and e(t) with filtered input and output signals. The purpose of this filtering is to attenuate the influence of the large, high frequency unmodelled dynamics. Since at high frequencies f > f2, unmodelled dynamics are unstructured and complex, they can not be estimated. Our goal is only to estimate the low frequency (f < f2) system dynamics. The obtained

DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS

75

estimates for the low frequency system dynamics do not have to be significantly affected by the high frequency unmodelled dynamics. This implies that at high frequencies, updating of the parameter estimates should be suspended (in the ideal case) or significantly slowed down. In other words, we need a frequency selective adaptive controller. In the subsequent analysis it will be shown that the proposed adaptive algorithm provides such properties. Estimation scheme (79) actually represents the normalized gradient algorithm with projection, where we use nr instead of c+ II r c > 0, as is usual in similar situations. The motivation for selecting such nr is to make possible the usefulness of the Robust Ultimate Boundedness Theorem, formulated in the previous section. Specifically, for the construction of the corresponding Lyapunov functions in the proof of the Theorem 3.1, we need property (Aa)(i) introduced in Section 2, and n~(t) given by Eq. (80) satisfies assumption (A3)(i). Let us define the following constants:

7z -

l[ Se(z)Wz(z)I?d(z) Jilt.., 72 -11Ae(z)W2(z)I~(z)[[aH ~ Ae(z) Bo(z)W(z) II Z~(z) II~, C D B - I I D(z) [!~

CD~

--

CDp

-- II P(=) II~|

,

CD~-Ii

Q(=) II~--

c~ap

-

II Ae(z)P(z)~V(~) Be(~)~(~)2P(_-) D(z) [1~**, CDBp =[I D(z) [[~/**

Cw

-

II w.(~)~r

(84)

wh~r~ II" I[~o. is the "compressed" H ~ norm defined by Eq. (5), X is given by (81) and D(z) is defined by assumption (5'5). Since assumptions (Sz) and (5'5) hold, all the above defined norms exist. Regarding the intensity of unmodelled dynamics we introduce the following assumption: (S8)

("/1

"~" ~[2 ) C D p f2, performances of the designed low-pass filter W(z)should provide [Wi(A-Zl2eil)]~r(A-X/~eil)[ < 7, i = 1,2. The following lemma will be useful for future reference. Lemma_ 4.1" Let the assumptions (Sx) - (Sa), (Ss), ($6), (Ss) hold. Then 1) (BeWP + AeQ)u(t) - Aee(t + 1) + AePy~(t + 1 ) - PT(t) (85)

2) (Bor

+ AoQ)yj(t + Z) = Ber

+ Z) + Bor

+ 1)+ Q~(t) (S6)

76

MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL

where the unit delay operator q-1 is omitted in all operator, while 7(t) and e(t) are defined by Eqs. (71) and (73) respectively.

3) II 7(t)I1~_< c~ II ~(t)IIx +k, + ~11($) (87') where II 7(t)Iix and II e(t)I1~ are defined by Eq. (1), when x(t) is equal to 7(t) and e(t + 1), respectively, while

(7, + 72)CDA

C~ = 1 -

(88)

(71 q" 72)CDP

and k~= (

1 [(7t -b 72)CDAPm1 -[" Cw 1 - (71 + 72)CDp

kw]

} ( 1 - 1A)t/2

4) II ~(z)I1~< c~ II ~(t)I1~ +k~ + e,2(t) where !! ~(t)!1~ i~ defined by Eq. (1), when z ( t ) - u(t), while C,, - CDA + CDpC.,,k,, = (1 - mlA)I/2 CDAP -[- ]r

(89) (90)

(91)

where C~ and k7 are defined by Eqs. (88) and (89), respectively. (92) 5) II yj(t)I1~< c~ II ~(t)Ii~ +~ + ~3(t) wh~,~ il y~(t)I1~ is given by Eq. (1), when ,(t) = y(t + 1), whil~, Cy = Cz, B + C~oC-,

(93)

and ml

CDBP q" kTCD Q.

(94)

In relations (87)- (92), constants 71,72, CDA, CDp, Cw, CDAP, CDB, CDQ and DDBp axe defined by Eq. (84) and the constants k~ and ml are defined by assumptions ($2) and ($6), respectively. Proof: The proof of the lemma is given in the Appendix. Important properties of the adaptive algorithm (79)-(82) will be formulated in the following lemma. L e m m a 4.2: Let the assumptions ( $ 1 ) - (Ss) hold. Then 1) ne(t) < C, n e ( t - 1)+ k, + ~14(t)

(95)

where n~(t)is defined by Eq. (2), when z(t) = e(t + 1) and e(~) is given by Eq. (73). The constants Ce and ke do not depend on the gain # of the algorithm (79), and will be specified in the proof of the lemma. At the same time, Co and ke depend on 7i, i - 1, 2, in such a way that their values decrease as 7i, i - 1, 2 decrease, where 7i i = 1, 2 are defined by Eq. (84).

DISCRETE-TIMEROBUSTADAPTIVECONTROLSYSTEMS 2) he(t) < Cr

77

(96)

C~2 or ~15($), 0 no, from (117), we obtain

Ee(t+l) 2 0 we have

84

MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL

fast adaptation and the estimator generates correct parameter estimates. If the algorithms possesses fast adaptation over the interval [vk, ~rk), correct parameter estimates obtained in the interval [at-1, rk) can be significantly deteriorated. Specifically, over the time intervals [rk, crk), the bursting function S(t + 1) < 0 and from the analysis presented in Section III, it is obvious that then the Lyapunov function V(t) may diverge. We now propose two methods of enhancing the slow adaptation over the time intervals Irk, ak). The first method requires prior information related to the constants "11,72, Cw, CDA, COp and CDAp defined by Eq. (84), and the upper bound k~ of the noise w(t). In the next remark it is pointed out that based on the available prior information related to the physical system, the constants 7x, 72, CDA and Cop can be roughly estimated by the designer. The same holds for the constants C~, Cw and ChAp. Thus, if we know k~, we can estimate k~ defined by Eq. (89) and design the following estimation algorithm: over the intervals where ne(t) > 4 k ~ / ( 1 - ~ - Cv) 2 we are using small no in Eq. (80) and in the intervals where n,(t) < 4k.~/(1- a _ Cv)2 we can use estimation scheme (79) with no > > 4~u2k~/(1 - ~ - Cx)2, or completely stop adaptation. The second method for obtaining slow adaptation over the intervals [rk, cry) is to modify the sequence n#(t) in the algorithm (89). Specifically, instead of nr given by Eq. (80), we now define

he(t) = An~(t-- 1)+ II r

=

2

+

1-A

(133)

and we obviously require the upper bound kw of the noise w(t) to be known. Such choice of the sequence n,~(t) does not influence the results derived in Sections III and IV, except that constants Co, k0, C~2 from Lemma 4.2, are different. Since in the intervals [rk, ~rk), the bursting function S(t + 1) < 0, from Eq. (13), we obtain

e(t + 1) 2 _
1

~(t + 1) = ~(t) + m~(t) p.6(f) e(t + 1),

if So(t + 1 ) > 0

/~(t + 1) =/~(t),

if So(t + 1) _< 0

(147)

where 0(I) - 01, IIOx II_ 1 2) So(t + 1) changes its sign during the algorithm operation. In the first case the statements of the theorem follow trivially from the definition of So(t + 1) (Eq. (146)) and Eq. (147). This actually means that the initial value 0(1) is a good estimate of the real parameters. Before proceeding further with the analysis, let us show that the case S0(t + 1) > 0 for all t > 1 is impossible. From (145) we obtain ^

V(t -I- I)+ So(t -I- I) < V(t) + So(t) Ie(t )____+___~~ ~36(t) m#(~) m , ( t - I) -P~ m#(t) ~ m#(t) (151) wherefrom it follows that lim e(t + 1 ) 2 / m # ( ~ ) - O. Similarly, as in Lemma 4.2 from Eqs. assumption ($7)

(152)

(85) and (86), we can obtain by

,n,(t) _ T1, So(t + 1) < 0 by which it is proved that the case S0(t + 1) > 0 for all t _ 1 is impossible. Let us now consider the case when So(t + 1) changes its sign. Similarly, as in the proof of Theorem 3.1, we define the sequences rk and ak as follows -- T1 K0 t

Z

Atm-ie(J+ 1) 2 < k~.rl2

(169)

j--~O'k

for t E [uk, r~+l). Note that from Eqs. (143) and (146) we can obtain

So(, + 1)_< Z

j~ok

+

+

(170)

for t 6 [uk,rt+1). Using the fact that So(ok) K0. Therefore, So(t + 1) _< 0 for all t _> rKo- W e have thus proved that in the estimation algorithm (147), updating of the parameter estimates will be suspended after time t _> 7Ko. By this statement 2), the theorem is proved. Since for t >_ rKo, S0(t + 1) _< 0, statement 1) of the theorem follows simply

DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS

91

from Eq. (146). Statement 3) of the theorem is the trivial consequence of statement 1). Thus the theorem is proved. !:3 Theorem 5.1 establishes that the upper bound of the performance index is bounded and that the parameter estimates are constant after some finite time To. Thus for t > To drift of the controller parameters is eliminated, i.e., we have a fixed controller which implies that bursting can not occur for t > To. These performances are achieved without ~r-modification or projection in the estimation algorithm.

VI.

CONCLUSION

Using the concept of the bursting function, new insights into the robust adaptive control problem are obtained. For the estimation of the controller parameters, a normalized gradient algorithm with projection is proposed. Global stability of the adaptive system is established for a class of unmodelled dynamics and external disturbances which are unstructured, complex and large at high frequencies. It is shown that the small algorithm gains/~ may result in unacceptably large input and output signals. The presented results also demonstrate one possible way of incorporating a pr/or/system information into the robust adaptive control design. For the sake of simplicity, the system unit delay case is considered in this chapter. It is not difficult to see that using the proposed methodology, similar results can be established for the general delay" case.

VII.

APPENDIX

P r o o f of t h e L e m m a 4.1- The first two statements of the lemma follow by simple manipulations of Eqs. (70) and (73). Let us prove the rest of the lemma. From the definition of 7(t) (Eq. (71)), by assumption (S~), it is not difficult to see that

II 7(t) IIx< ('r, + 7.~) II u(t) II~,+ c ~ II ~(t) I1~,+69(t)

(A.1)

where [I z(t)I1~, z q. {7, u,w} is given by Eq. (1) in Section II. Constants 7i, i = 1, 2 and Cw are defined by Eq. (84). Similarly, from Eq. (86), we can obtain

II ~(t)i1~< eva [Ie(t)I1~ +CoAp [[u'(t)I[~ +cop II 7(0 I1~ +6o(0 (A.2) where II y'(t)IIx is defined by Eq. (1) when z(t) = y'(t + 1). The H r162 norms CDA, CDP and CDAP are defined by Eq. (84). Statement 3) of the

92

MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL

lemma can easily be obtained by substituting (A.2) into (A.1), where we used the fact that assumption (5"8) holds and II ,,,(t)IIx_
0 .

(2.11) E!

The physical interpretation for the E O L problem is clear if one considers the inequalities in (2.8). Note that in this case

Er,

(2.12)

Ilu,

i=l

Hence, the cost function is an upper bound of the weighted summation of supremums of control signals over all L2 disturbances which belong to ~N" The EOL| problem is similar to the O L problem, but the input disturbances belong to different sets, f2 N and fir, respectively. The next section provides the solutions of the E O L problem in the continuous time case.

III. S O L U T I O N S

OF CONTINUOUS

TIME EOL

PROBLEM

In this section, we shall discuss the first order necessary conditions for static and dynamic controllers, and also present an iterative algorithm. A.

STATIC MEASUREMENT FEEDBACK CASE

For the static measurement feedback case, we seek a constant feedback gain matrix G which is the solution of the E O L problem such that

104

GUOMING G. ZHU AND ROBERTE. SKELTON (3.1)

u =Gz.

Therefore the EOL| problem becomes (3.2)

I min traceNW -~. trace R G M X M r G r P

G.W>O

P

subject to (2.1) and

[o[C,XC,r ].traceNW-I - -


0 such that i)

G = - ' f f - ' B r K X M r ( M , X M pr)-,.,

ii)

0 = KA + A r K + M r G r ' R - t G M p + CrQCp;

iii)

DrKD

iv)

Q{block diag[ C, x c r , C2 x c j ..... C x c r ]. t r a c e N W -' - F} = 0;

v)

A X + XA r + D pWD pr = 0 ,

P

P

= W - ' N W - ' ( t r a c e R G M X M rGr + traceQCpXC r)''2" P

P

(3.4a)

where "R = R. traceNW-' , Q = Q. traceNW-' , F = diag[ 2, I,,~.212 2 ..... 1~I 2 ], and l j is an identity matrix with dimension mj.

The proof of the above theorem is obvious. By using the following augmented cost function

H

trace NW -' { trace RGM pXM r G r } + +trace K[ X ( Ap

i=! + BpGMp ) r + ( Ap +

[ C x c ,rtrace NW -' - E I , (3.5)

BpGM, ) X

+ OpWD p r ],

GUARANTEEING OUTPUT L CONSTRAINTS

105

setting the partial derivatives with respect to X, G, K, W to zero, and adding one equation for inequality constraints in (3.2) (Kuhn-Tucker conditions), one can get the necessary conditions. By setting M = I , one obtains the results for state feedback case which is a special case of measurement feedback control.

Corollary 2.2 Suppose that G is an optimal solution of the EOL problem for the state feedback case, and that the EOL| problem is regular, that is, the EOL problem is not independent of the EOL cost function in (3.2). Then there exists a block diagonal positive semidefinite matrix Q such that i)

G = -'R-'BrKX;

ii)

0 = KA,, + A r K - KB,-R-'B rK + Cr'QCp ;

iii)

D ~, rKD

iv)

Q{block diag[ C, x c r , C~XC[ ..... C x c r ]. traceNW -' - F} = 0;

v)

(Ap + BpG) X + X (Ap + BpG) r + DpWDp r

P

m

p = W-I NW-I (traceRGXG r + traceQC X C ; ) |'~"

(3.6)

"-0,

g

where R , Q and F are the same as those in Theorem 3.1. The solution of the third equation in (3.3) and (3.6) is given in the following Lemma.

Lemma 3.3 Suppose that N and K are positive definite and D has full column rank. Then equation D r KD = W-INW-1 has one symmetric positive definite solution W~ = K~'[K~NK~] ~2K~-',

(3.7)

and one negative definite solution

w, where

r; ',

(3.8)

106

G U O M I N G G. Z H U A N D ROBERT E. S K E L T O N

(3.9)

K,~ = ( D r K D ) ''2 .

The proof was presented in [6]. An algorithm used to solve those necessary conditions is presented in Subsection C. B.

DYNAMIC FEEDBACK CASE Consider the following dynamic controller with order nc xc

=

AcXc + Bcz

u

=

C x +Hz.

(3.10)

It is well-known that by setting

=[APo ;x=

(3.1 la) =

;~=

;G= o

s

Ix:] [':] [5] ;y=

~

;D=

;D

=

x

"

;I)= B,,

; A

[01

(3.1 lb)

'

with compatible dimensions, the closed loop system can be written as (3.12) y

----

[d' , ,~'c" l" X-F H v

In order to obtain a bounded output L

9

norm for an L2 input, system

(3.12) should be strictly proper, i.e.,/4 = 0 or v - 0. Here we consider the case that v = 0. Then the closed loop system is in the form (1.2) with A=A+BGtr

C

[~r

21~/rGr]r

(3.13)

Let (3.14)

GUARANTEEING OUTPUT L CONSTRAINTS

107

where R = diag[r~,r2..... r ] > 0. By solving the following static measurement feedback EOL problem

trace NW -l . trace [CGl~I X~Ir G r

min

I

G.W>O

(3.15)

P

subject to (2.1) and

---

"

^T

[o[CXC ].traceNW

-1

~

2

_ e,, i = 1,2 ..... m.

one can obtain first order necessary conditions for the dynamic controller case. Hence, when v = 0 in (1.1), the EOL problem with a fixed order dynamic controller produces the static measurement feedback EOL controller as a special case. Special Case: Full Order Dynamic Controller For this special case, we consider the following strictly proper full order dynamic controller L

=

ax+

u

=

Cx.

z

(3.16)

Let

X =

IXPx-~Xc]; Y = [ ~ I ; w=IWvl'

(3.17)

then the closed loop system can be put into the form (1.2) with A -BM

I

BM c

D=

The EOL follows:

A +BC

A +BM

p

P 0

-BM

c

C;C= n

c

P 0

-A

(3.18a)

p

P.

(3.18b)

c

problem with full order dynamic controller can be stated as

108

GUOMING G. ZHU AND ROBERT E. SKELTON

min

traceNW-'.traceRC XC r, C = [ 0 , C ]

Ac ~c .Cr .W>O

(3.19)

subject to (2.1) and

"0[ C XC r ]. traceNW-1 0 such that Q{block diag[ C, x c r , C2 XC r ..... C XC r ]. traceNW-' - F} = 0,

(3.21)

and A =A,+B

C-B

M,;

(3.22)

B = ( X t , M pr + D W~2)W2-2" "~-i

C=-R

T

B K22,

where X , , and K22 satisfy X,, A ,r + A , X ,, _ ( X I M,

,r +D Wn)W~-l (X,,

M ,r

+D W~2)r +D W~,D ,r =~,

(3.23)

K=Ap + ArK= - K=B "R-'Br K= + Cr'QC = 0 , and "R = R . traceNW-' and Q = Q . traceNW-l . In addition DrKD = W-'NW-' (traceRC XC r + t r a c e Q C XCry )~2, where K is defined in (3.20) with K~2 = K22 and K~2 satisfying

(3.24)

GUARANTEEING OUTPUT L CONSTRAINTS

109

(3.25)

r ( K , , - K2:) + C~rR'C~ =0,

(K,,-K22)(A,,-BM)+(A,-BM)

and X is defined in (3.20) with X,2 = 0 and X22 satisfying

(3.26)

X22(A +B C ) r + ( A p + B p C ) X 2 2 +BW~2Br =0.

The proof is similar to the static measurement feedback case. Consider the augmented cost function m n

_~.

(3.27)

traceNW-' . traceRC XC r,, + E ' ~ [ C XC r 9traceNW-' - E,] I, : i=l

+traceK .[ AX + XA r + DWD r ].

Then (3.27) can be rewritten into H = traceXCrO.C + traceK .[ AX + XA rDWD r ] _ traceQ. F,

where 0 = block diag['Q,'ff ] and "R = R . traceNW-' , "Q = Q . traceNW-' . By setting the partial derivatives with respect to K, X, W to zero and adding the Kuhn-Tucker condition (3.21),one can complete the proof. _...

.__

Note that (3.22) and (3.23) describe an LQG controller if Q, R, and W are given. Hence, the full order E O L controller is an LQG controller with some special choice of the weighting matrices Q, R and input noise intensity matrix W. (The forthcoming algorithm determines Q, R and W). Remark 3.5 The fact that there exists a W > 0 satisfying the first order necessary conditions requires that D rKD > O. Remark 3.6 The first order necessary conditions for the OL problem only need to satisfy (3.22) and (3.23) for fixed W, which are included in the EOL| problem. Hence the OL| problem is a special case of the EOL problem.

C.

NUMERICAL ALC~RITHM

110

GUOMINGG. ZHU AND ROBERTE. SKELTON

Subsections A and B of this section provide all the necessary conditions for the design of static and dynamic controllers. The following algorithm provides a way to design the controller by iterating on these necessary conditions. This is an extension of the O L algorithm presented in [3].

THE EOL .o ALGORITHM Step I

Given system matrices Ap,Bp,Cp,Dp. Mp, initial weighting matrices Wo , Qo , R , outer product bound N, output L bounds e,, (i = 1,2 ..... m), error tolerance e , and free parameters 0 < [~< 1 and a >0. Define ~ = R. traceNW, -! ~ = Q. trace NW -~ Let i = 0

Step 2

Controller design step: Case 1- static state feedback Compute m-!

G, = - R

7"

(3.28a)

BpK,

by solving (3.28b)

K i A p + A ; K - K B p R---' i Bp"K -t- C;'QiC p = O, and then solve for X, by (Ap+BpG,)X, + X,(Ap+BpG,) r +DpWD r =0.

(3.28c)

Let D, = Op, q = Cp, and C~ = G,. Go to Step 3. Case 2: Static measurement feedback Iterate on the following three necessary conditions to obtain the optimal measurement feedback gain G i for given Qi and W

6,

- R -, ' B prK , Xi M pr ( M P X, M rP ) -''' O= K(Ap+BpG, Mp) r + K ( A p + B p G , Mp)

(3.29a)

=

+ M;
0.

(4.5)

Let w(.) in (4.2) be any e: disturbance which belongs to

a~ -

wO subject to (2.1) and o [ C XC, ]. traceNW -~ < _ E2, . . i. = . . 1. 2,

--

r

m.

Theorem 4.2 Suppose that G is an optimal solution o f the EOL problem defined in (4.17), and that the EOL problem is regular. Then, there exists a matrix Q = block diag[Q~, Q~ ..... Q= ] > 0 such that

116

GUOMING G. ZHU AND ROBERT E. SKELTON

i)

G - - ( R-- + BrpKBp) -' BrpKAFXMFr ( M pX M r ,) -l",

ii)

K = ArKA + MrG r ('R+ B;KBp)GMp +CrQCp;

iii)

D ; K D p = W - I N W -I ( t r a c e R G M p X M p r G r + trace __ QCp X C r )~,2.,

iv)

Q{ block diag[ C, X C r , C: X C r ..... C X C r ] . trace N W -' - F} : 0;

v)

A X A r + DpWD r = X ,

(4.18)

where - - = R . traceNW-', Q = Q. traceNW-', F = diag[e . 2I I ! ,e.212 2 ..... R

e21

],

(4.19)

a n d l j denotes an m j x mj identity matrix.

The proof of Theorem 4.2 is similar to that of Theorem 3.1. By setting M = I, one obtains the necessary conditions for the state feedback case, which is a special case of measurement feedback control. Corollary 4.3 Suppose that G is an optimal solution o f the E O L case, a n d that the E O L

f o r state f e e d b a c k

p r o b l e m is regular. Then there exists a matrix

Q = block diag[Q t , Q2 ..... Q. ] > 0 such that i)

G = -('R + BrKBF) -'BrKAp ;

ii)

0 = ArKAp - ArKBF ('ff+ BrKBp) -'BrKAF +Cr'QCp;

iii)

D pr K D p = W-' N W - ' ( t r a c e R G X G r + t r a c e Q C X C ; )

iv)

e { b l o c k d i a g [ C t X C r , C 2X C r ..... C X C r ]. t r a c e N W - ' - F} = 0;

v)

( Ap + B p G ) X ( A p + B pG ) r + D pW D rp= X ,

(4.20)

u2 .

where R , Q a n d I" are the s a m e as in (4.19).

The solution of the third equations in (4.18) and (4.20) can be solved by Lemma 3.3. The algorithm used to solve those necessary conditions will be given at the end of this section. B.

DYNAMIC FEEDBACK CASE We consider the fixed order dynamic controller

GUARANTEEING OUTPUT L CONSTRAINTS

x(k+l)

=

A x(k)+Bz(k)

u(k)

=

C x ( k ) + H z(k).

117

(4.21)

Let v = 0. Then the closed loop system has the following form x(k+l)

=

_ ( n - r) t h e s y s t e m w o u l d h a v e o n e zero at infinity. C o n s i d e r s u c h a M a n d a s s u m e t h a t t h e

158

SARIT K. DAS AND R K. RAJAGOPALAN

( n - - 1 ) re lo c at able zeros h a v e b e e n p l a c e d at origin, l e a d i n g to t h e e q u i v a l e n t plant configuration n

G1 = z 1n - - 1 / I ~ ( z l - - a ~ )

w h e r e zl "= z M

k-----1

F o l l o w i n g [9] (see A p p e n d i x C), it is n o t difficult to see t h a t t h e m a x i m u m n G M a t t a i n a b l e for t h i s p l a n t is (1 + r/)2/(1 - r/) 2 w h e r e r/ -- 1/1-Ikffil a ~ . Clearly, ~/ will be s m a l l e r for l a r g e r M , m a k i n g t h e G M lower. H e n c e M > ( n - r) n e e d n o t be c o n s i d e r e d . F i n a l l y , since t i m e i n v a r i a n t c o n t r o l l e r s do n o t h a v e t h e zero p l a c e m e n t c a p a b i l i t y , it is o b v i o u s t h a t for t h e case of n - r -- 1, M -- 2 is also to be considered. U] W e are n o w in a p o s i t i o n to o b t a i n t h e b e s t G M c o m p e n s a t i o n for a given p l a n t v i a t h e zero p l a c e m e n t a p p r o a c h . T h e s t e p s to be followed for this p u r p o s e are: 1. F o l l w i n g T h e o r e m 6 i d e n t i f y t h e values of M t h a t a r e to be c o n s i d e r e d . 2. F o r each of t h e s e v a l u e s of M o b t a i n t h e e q u i v a l e n t p l a n t [as given by eqn.(37)] a s s u m i n g t h a t all t h e r e l o c a t a b l e zeros h a v e b e e n p l a c e d at origin. 3. F o r e a c h of t h e s e e q u i v a l e n t p l a n t s o b t a i n G M M ( w h i c h s t a n d s for t h e b e s t G M o b t a i n a b l e u s i n g M - p e r i o d i c c o n t r o l l e r ) u s i n g t h e app r o a c h of [9] or [10]. T h e n o b t a i n t h e m a x i m u m G M as G M m a ~ -m a x [ G M M ] , M v a r y i n g over t h e r a n g e given by s t e p 1 a b o v e . M

4. F o r M = M m a x c o r r e s p o n d i n g to G M , n a x , o b t a i n t h e c o m p e n s a t o r a n d t h e c o m p l e t e C E u s i n g t h e a p p r o a c h of [10]. 5. F i n a l l y , to o b t a i n t h e p a r a m e t e r s of t h e p e r i o d i c g a i n c o n t r o l l e r t h a t a c h i e v e s t h e a b o v e C E , first check if t h e c o n t r o l l e r o r d e r rn as o b t a i n e d in s t e p 4 a b o v e satisfies t h e lower l i m i t c o n d i t i o n given by T h e o r e m 4. If it is satisfied t h e n this rn = rnmin, t h e m i n i m u m o r d e r of t h e c o n t r o l l e r r e q u i r e d . O t h e r w i s e , rnmir, will be t h e s m a l l e s t v a l u e of rn t h a t satisfies e q n . ( 3 1 ) . N e x t e q u a t e t h e coefficients of t h e like p o w e r s of t h e K - t e r m s of e q n . ( 2 8 ) to t h o s e of t h e d e s i r e d C E to g e t a set of n o n - l i n e a r e q u a t i o n s s o l v i n g which t h e c o n t r o l l e r p a r a m e t e r s c a n be obtained.

We n o w p r e s e n t a few e x a m p l e s of o p t i m u m G M c o m p e n s a t i o n of s t r i c t l y proper plants.

TECHNIQUES OF ANALYSIS AND ROBUST CONTROL

Example

4 C o n s i d e r t h e (2, 1 ) - p l a n t ( z -

159

1.5), Ibl > 1.

b)/z(z-

i) V a l u e s of M t o b e c o n s i d e r e d a r e 1 a n d 2. ii) F o r M = 2, t h e e q u i v a l e n t p l a n t w o u l d , o n z e r o r e l o c a t i o n , b e 1 / ( z t 2.25), for w h i c h G M M zt(zt

--- 6.76 a n d t h e c o r r e s p o n d i n g

C E is

-- 2 . 2 5 ) ( z t -- 1 / 2 . 2 5 ) + a K z 2 t = 0

C l e a r l y m = r n m i n = 1. N o w , o n e s e t o f t h e p a r a m e t e r

v a l u e s o f t h e first

o r d e r c o n t r o l l e r t h a t a c h i e v e s t h i s is o b t a i n e d t o b e qo(z)

-- z + 1.2b/(b

+

1.2)

qt ( z ) - - q o ( - - z ) p o ( z ) -- z + 1 / 1 . 5

pl(~) = o and the corresponding

c~ b e c o m e s

= 2(1.5 - b)(b + o . 6 6 7 ) / ( b + 1.2) iii) F o l l o w i n g [9] it is s e e n t h a t , for M

=

1, G M M

> 6.76 if a n d o n l y if

-r < b < 1.083. T h u s for b > 1.083 2 - p e r i o d i c c o n t r o l l e r s y i e l d b e t t e r results than time invariant ones. Example

5 C o n s i d e r t h e (3, 1 ) - p l a n t ( z -

b)/z(z

- 1.5)(z-

2).

i) T h e v a l u e s o f M t o b e c o n s i d e r e d a r e 1 a n d 2. ii) F o r M "- 2, t h e e q u i v a l e n t p l a n t a f t e r t w o z e r o s a r e r e l o c a t e d t o o r i g i n would be zt/(zt - 2.25)(zt -4), for w h i c h G M M -- 1.56 a n d t h e c o r r e s p o n d i n g C E is z t ( z t -- 2 . 2 5 ) ( Z l -- 4 ) ( z t -- 1 / 2 . 2 5 ) ( Z l -- 1 / 4 ) + o ~ K z ~ ( z t H e r e rn -- rn,,+in = 2. C E for b - 1.5 is

- 1 . 6 ) ( Z l - 1 / 1 . 6 ) -- 0

Now a second order controller which achieves this

q o ( z ) = 0 . 3 3 6 z 2 + 0 . 1 6 9 z + 0.946 qt ( z ) -- q o ( z ) p o ( z ) -- z 2 + 3 . 6 8 2 z + 1.109 P l ( z ) -- - - 3 . 2 6 8 z + 1.05 and the corresponding

c~ is a = 0.799.

iii) F o r M = 1, G M M

> 1.56 if a n d o n l y i f - 1 . 2 6 5

< b < 1.25.

T h u s for

values of b outside this range the 2-periodic controller yields better results.

160

SARIT K. DAS AND E K. RAJAGOPALAN

Example

6

C o n s i d e r t h e (4, 1 ) - p l a n t ( z -

b)/z2(z-

1.5)(z-

2).

i) M = 1, 2 a n d 3 n e e d o n l y be c o n s i d e r e d . ii) F o r M ---- 2, o n l y two z e r o s c a n be r e l o c a t e d to origin. T h e n t h e e q u i v a lent p l a n t w o u l d be 1 / ( Z l - 2 . 2 5 ) ( z 1 - 4 ) , for w h i c h G M M -~ 1.08, a n d t h e c o r r e s p o n d i n g C E is z2(zl

2.25)(Zl - 4 ) ( z l - 1 / 2 . 2 5 ) ( z l - 1 / 4 ) ( z l

-

+ o, K z ~ ( z l

-

+ 0 . 1 7 4 ) ( z l + 5.734) 1 . 9 2 ) ( z l - 1 / 1 . 9 2 ) -- 0

O b v i o u s l y a f o u r t h o r d e r c o n t r o l l e r will be r e q u i r e d t o a c h i e v e t h i s c o m pensation. iii) F o r M --- 3, t h e e q u i v a l e n t p l a n t a f t e r z e r o r e l o c a t i o n w o u l d b e z l / ( z , 3 . 3 7 5 ) ( z l - 8 ) , for w h i c h G M M "- 1.16 a n d t h e c o r r e s p o n d i n g C E is

-

Z2(Zl -- 3 . 3 7 5 ) ( Z l -- 8)(z I 1 / 3 . 3 7 5 ) ( Z l -- 1 / 8 ) + o l K z ~ ( z l -- 2.46)(Zl -- 1 / 2 . 4 6 ) = 0 -

-

H e r e t h e c o n t r o l l e r o r d e r n e c e s s a r y is a p p a r e n t l y 2. T h i s value, h o w e v e r , d o e s n o t s a t i s f y e q n . ( 3 1 ) w h i c h r e q u i r e s t h e m i n i m u m c o n t r o l l e r o r d e r to be 3. So, a t h i r d o r d e r c o n t r o l l e r is to b e e m p l o y e d . B u t t h e t h i r d c o n t r o l l e r zero a n d p o l e m a y b o t h b e p l a c e d at o r i g i n so t h a t t h e a b o v e C E r e m a i n s essentially the same. iv) For M = 1, t h e b e s t G M of 1.22 is o b t a i n e d w h e n Ib[---. 1. F o r b ~ 1.5 or 2, h o w e v e r , t h e a t t a i n a b l e G M is o n l y 1. T h e n 3 - p e r i o d i c c o n t r o l l e r s will be t h e b e s t . Example

7 C o n s i d e r a g a i n a (4, 1 ) - p l a n t ( z - b ) / z ( z

+ x/-~)(z

2 - 2 z + 2).

i) M -- 1, 2 a n d 3 n e e d o n l y be c o n s i d e r e d . ii) F o r M -- 2, t h e p l a n t a f t e r zero r e l o c a t i o n w o u l d b e Z l / ( Z l for w h i c h G M M -- 1.56, t h e c o r r e s p o n d i n g C E b e i n g zl(zl

-

+ r

1.5)(Zl2 4 4 ) ,

1.5)(z~ + 4)(Zl - 1 / 1 . 5 ) ( z ~ + 1 / 4 ) ( z l + 1 . 5 ) ( z l + 1 / 1 . 5 ) - 2 . 1 4 ) ( z 1 - 1 / 2 . 1 4 ) ( z 1 + 2 . 1 3 ) ( z l + 1 / 2 . 1 3 ) -- 0

C l e a r l y a 5 t h o r d e r c o n t r o l l e r is r e q u i r e d to a c h i e v e t h i s c o m p e n s a t i o n . s u c h e v e n - i n p u t c o n t r o l l e r is

One

q o ( z ) = z ( 0 . 0 0 3 4 z 4 - 4 . 4 2 3 2 z 3 - 8 . 3 3 9 5 z 2 - 6 . 0 1 2 4 z - 1.3049) ql ( z ) "- q o ( z ) po(z)

-- z s + 1.8851z 4 + 2.4966z 3 + 2.506z 2 + 1.5455z + 0.4083

pl(~)

= 0

iii) F o r M - 3, t h e z e r o r e l o c a t e d p l a n t w o u l d be z ~ / ( z l + 1.837)(z~ + 4 z l + 8), for w h i c h G M M -- 1.313, a n d t h e c o r r e s p o n d i n g C E is

Z2(Zl + 1.837)(Zl2 + 4zl + 8 ) ( z I + 1 / 1 . 8 3 7 ) ( z ~ + 0 . 5 z l + 1 / 8 ) + c ~ K z ~ ( z 2 + 3 . 1 1 8 z l + 3 . 0 5 3 ) ( z ~ + 1.02zl + 0 . 3 2 7 ) - - 0

TECHNIQUES OF ANALYSIS AND ROBUST CONTROL

161

A f o u r t h o r d e r controller would be required to achieve this. iv) For M = 1, t h e best G M of 2.53 is o b t a i n e d when Ibl ---* 1. For b ---. - x / 1 . 5 , however, the GM will be 1, a n d t h e n t h e 2-periodic controller will be t h e best.

VII

Conclusions

and Discussions

T h e c o n t e n t s of this c h a p t e r can be s u m m a r i s e d as follows: 1. A linear, discrete, M - p e r i o d i c controller s t r u c t u r e t h a t has t h e m a x -

i m u m possible degrees of freedom for its o r d e r has been presented. 2. T w o a n a l y t i c a l m e t h o d s - the lifting technique and t h e Floquet theory approach ~ have b e e n p r e s e n t e d for dealing with periodic systems. It is seen t h a t while the f o r m e r is m o r e general in t h a t it can, besides analysing stability, yield t h e i n p u t - o u t p u t relation as well, the l a t t e r yields t h e C E in a m o r e s t r a i g h t f o r w a r d fashion. 3. Based on a s t u d y of t h e features of the closed loop C E of a L D T I p l a n t c o m p e n s a t e d by a M - p e r i o d i c controller, t h e zero p l a c e m e n t c a p a b i l i t y of periodic controllers has been investigated. It is seen t h a t while periodic controllers can relocate all the zeros of a n t h o r d e r bicausal plant, they can relocate only u p t o ( n - 1) zeros ( s o m e of which might originally have been at infinity) for n t h o r d e r strictly p r o p e r plants. 4. Finally the aspect of GM i m p r o v e m e n t of u n s t a b l e p l a n t s with N M P zeros via the zero p l a c e m e n t a p p r o a c h has been i n v e s t i g a t e d leading to t h e conclusions t h a t (a) 2-periodic controllers can provide infinite G M c o m p e n s a t i o n to bicausal plants, a n d (b) for o p t i m u m G M comp e n s a t i o n of a n-pole, r-zero strictly p r o p e r plant the p e r i o d i c i t y M of t h e controller should, in general, be a p r i m e n u m b e r not g r e a t e r t h a n m a x { 2 , (n - r)}. It m a y be n o t e d t h a t zero p l a c e m e n t is not the only way t h r o u g h which G M c o m p e n s a t i o n can be provided to a plant, in fact, in [2] p e r i o d i c controllers have been used to do t h e s a m e job via t h e factorization a p p r o a c h which, as such, is m o r e general t h a n t h e zero p l a c e m e n t a p p r o a c h . T h e p r o b l e m of o p t i m u m G M c o m p e n s a t i o n using the f a c t o r i z a t i o n a p p r o a c h , however, has not so far been solved in literature. (Some results r e g a r d i n g this are p r e s e n t e d in [11].) Besides GM i m p r o v e m e n t , t h e possibilities of using periodic controllers for d i s t u r b a n c e rejection and for s i m u l t a n e o u s stabilization have also been

162

SARIT K. DAS AND P. K. RAJAGOPALAN

investigated in literature. Regarding the former, a l t h o u g h it has been shown [2][3] t h a t periodic controllers offer no a d v a n t a g e over L D T I ones so far as u n i f o r m d i s t u r b a n c e rejection is concerned, the possibility t h a t such controllers may provide superior d i s t u r b a n c e rejection at certain r e g u l a r intervals of t i m e (say, the even instants) can not be ruled out. Further, regarding the latter, although it has been claimed in [2] t h a t generically a M - p e r i o d i c controller can not stabilize a set of M plants s i m u l t a n e o u s l y for M > 2, it has been shown in [12] t h a t M - p e r i o d i c controllers can not only stabilize but can as well place the poles of M plants s i m u l t a n e o u s l y for any M.

VIII

References

1. T. Kailath, L i n e a r S y s t e m s , Prentice Hall Inc., Englewood Cliffs, N.J. (1980). 2. P. P. K h a r g o n e k a r , K. Poolla, and A. T a n n e n b a u m , " R o b u s t control of linear time invariant plants using periodic c o m p e n s a t i o n " , I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l A C - 3 0 , pp.1088-1096 (1985). 3. J. S. S h a m m a , and M. A. Dahleh, " T i m e varying versus t i m e invariant c o m p e n s a t i o n for rejection of persistent b o u n d e d d i s t u r b a n c e s and robust stabilization", I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l A C 36, pp.838-847 (1991). 4. M. A. Dahleh, P. G. Voulgaris, and L. S. Valvani, " O p t i m a l and robust controllers for periodic and m u l t i r a t e systems", I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l AC-37', pp.1734-1745 (1992). 5. S. K. Das, and P. K. R a j a g o p a l a n , "Periodic d i s c r e t e - t i m e systems: stability analysis and robust control using zero p l a c e m e n t " , I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l A C - 3 7 , pp.374-378 (1992). 6. E. T. W h i t t a k e r , A c o u r s e in m o d e r n a n a l y s i s , C a m b r i d g e University Press, N.Y. (1962). 7. B. A. Francis, and T. T. Georgieu, "Stability theory for linear timeinvariant plants with periodic digital controllers", I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l A C - 3 3 , pp.820-832 (1988). 8. S. K. Das, and P. K. R a j a g o p a l a n , "Infinite d e t e r m i n a n t m e t h o d s for stability analysis of periodic s y s t e m s " , P r o c e e d i n g s I E E P a r t - D 131, pp.189-201 (1984).

TECHNIQUES OF ANALYSIS AND ROBUST CONTROL

163

9. A. T a n n e n b a u m , "Feedback s t a b i h z a t i o n of linear d y n a m i c a l plants with u n c e r t a i n t y in the gain factor", International Journal of Control 32, pp.1-16 (1980). 10. I. Horowitz, "Design of feedback s y s t e m s with n o n - m i n i m u m - p h a s e u n s t a b l e plants", International Journal of System Science 10, pp. 1025-1040 (1979). 11. S. K. Das, and P. K. R a j a g o p a l a n , " M - p e r i o d i c controller for maxim u m gain m a r g i n for N M P s y s t e m s " , P r e s e n t e d at S I A M Conference on Linear Algebra, San Francisco, Nov. 5-8, (1990). 12. A. K. Singh, "Periodic c o m p e n s a t i o n of discrete linear time invariant plants", Ph.D. Thesis, Indian I n s t i t u t e of Technology, K h a r a g p u r , India (1994). 13. C. A. Lin, and C. W. King, "Minimal periodic realizations of transfer m a t r i c e s " , I E E E Transactions on Automatic Control A C - 3 8 , pp.462466 (1993).

IX A

Appendices SISO

Periodic

Realization

of Transfer

Matrices

In Section III-A we have seen how the equivalent M x M transfer m a t r i x ,

G(z M), c o r r e s p o n d i n g to a M - p e r i o d i c . s y s t e m can be o b t a i n e d . Here we a t t e m p t the reverse p r o b l e m of obtaining the minimal, SISO, M - p e r i o d i c realization of a given M x M C~ satisfying the causality condition (~(oo) lower triangular. It will, however, be seen t h a t such a m i n i m a l realization does not utilise all its possible degrees of freedom. Noting t h a t for a 2 x 2 transfer m a t r i x C~(z2) the (1, 1)-element, G l l , s t a n d s for the even instant input to the even instant o u t p u t t r a n s f o r m a t i o n , and, similarly, G12, G21 and G22 stand, respectively, for the odd to even, even to odd and odd to odd t r a n s f o r m a t i o n s , G may as such be realized by realizing each GU, i , j = 1, 2, individually with the a p p r o p r i a t e input signal applied to it and its o u t p u t taken at the correct instants. Fig.4 shows this realization. (Note t h a t for causality the o u t p u t s of the blocks G12 and G21 must be delayed by one sampling period.) It is clear t h a t for a M x M G(z M) to be realized in this fashion one would require (in the worst case) M2rn M - d e l a y elements (i.e., 1/z M blocks) where rn is the order of t h e c o m m o n d e n o m i n a t o r d(z M) of the elements of (~. However, it a p p e a r s logical t h a t the s a m e should be realizable using only as m a n y delay e l e m e n t s as is the highest degree of z occurring in d (i.e., M r n ) provided, of

164

SARIT K. DAS AND R K. RAJAGOPALAN

y,(.,~)

U~(~ 2) "-~ (z~)

,

!

U

J_l] 7 z

J .

I

--1

7 ~

z - 1 r.,ro(z 2 )

7 ,G"(z

)

_/

"x.

z - ~ Y o ( z 2)

F i g u r e 4: D i r e c t r e a l i z a t i o n of a 2 • 2 t r a n s f e r m a t r i x

course, t h e g a i n s are now allowed to b e c o m e p e r i o d i c . W e s h o w n e x t t h a t such i n d e e d is t h e case. F i r s t we b r i n g o u t an i m p o r t a n t p r o p e r t y of t h e G c o r r e s p o n d i n g to a c a u s a l p e r i o d i c m a p ( b e s i d e s t h e one t h a t G ( o o ) is lower t r i a n g u l a r ) [12]. 3 Given a causal, SISO, M-periodic system, if one constructs the corresponding M x M L D T I transfer matriz G ( z M) without performing any (possible) pole-zero cancellation at origin in its elements and ezpresses the 9~m~ ~ ~ ( ~ ) = ~(zU)/d(~), ~ h ~ d i~ th~ t.~.m, olth~ a r of G and 1V is a polynomial matriz, then, apart from G(oo) bein 9 lower triangular, fit(O) would be upper triangular. Conversely, if.for a given G with

Lemma

G(o~) tow~,, t,.ia,~g,,u,,, th~ .,~(o) i, . o f ,,pp~,. t,.i,,ng,,h,,., th~n it sig,,,i/ie~ that the system has a hidden mode at origin not reflected in d. Proof: For c o n v e n i e n c e we p r o v e t h e r e s u l t for M = 2. T h e p r o o f can easily be e x t e n d e d for M > 2. It was s h o w n in S e c t i o n I I I - A t h a t t h e even a n d o d d o u t p u t s of a c a u s a l 2 - p e r i o d i c m a p are r e l a t e d to t h e even a n d o d d i n p u t s by t h e t r a n s f e r m a t r i x G as given by e q n . ( 7 ) or, a l t e r n a t i v e l y , by t h e m a t r i x r as given by e q n . ( 8 ) . N o t e , however, t h a t of t h e two m a t r i c e s r a n d r G is m o r e f u n d a m e n t a l b e c a u s e it r e l a t e s t h e even a n d o d d i n p u t a n d o u t p u t s i g n a l s in t h e c o r r e c t fashion. In o t h e r words, given a 2 - p e r i o d i c m a p , its 2 • 2 L D T I r e p r e s e n t a t i o n can p h y s i c a l l y be o b t a i n e d o n l y in t h e f o r m G, a n d , conversely, a given 2 • 2 m a t r i x of t h e form of r m u s t first be t r a n s f o r m e d to t h e form r if it is to b e p h y s i c a l l y realized as a 2 - p e r i o d i c s y s t e m . Let d'(z 2) be t h e l.c.m, of t h e d e n o m i n a t o r s of Gij(z2), i , j = 1,2. Now, d e p e n d i n g on w h e t h e r G i j , i ~: j , h a v e n u m e r a t o r f a c t o r s of z 2 or n o t , four cases m a y arise:

TECHNIQUES OF ANALYSIS AND ROBUST CONTROL

165

1. Neither G12 nor G21 has a n u m e r a t o r .factor of z 2" T h e n f r o m eqns. ~,

(7) a n d (8) t h e c o m m o n d e n o m i n a t o r forms for G a n d G b e c o m e

[ z2Nll G -z N21

(~ --

z Nil z 2 N2i

zN12 ] z 2 N~2

1

N12 ] 1 z 2 N22 Z 2d'

1V d

w h e r e Nij --- G q d ' a n d d e g ( N i j ) o

Vk ~ [kL, ku]

(33)

STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS

239

where

d2

-

- b T(P T PT PdlP2)b

dl

-

2{aT(p + P f P d l P 2 ) b - bTpTPdlpn-1} Pn,n - aT (p + pT Pdl P2)a - p nT_ l P d l p n _ l

do -

-t-2aT pT Pdlpn_ 1.

(34)

Since detPd > 0, kL and ku can be obtained as two real roots of the quadratic equation q(k) =

+

+ do = 0.

(35)

Since A is assumed to be stable, q(0) > 0, which implies that do > 0. From this and the fact that d2 < 0, eq. (35) always has two real roots. This means that there always exists a set of simultaneous Lyapunov feedback gains and it is a connected closed interval [kn, ku] determined by the two real roots of q(k), kn and ku. The simultaneous Lyapunov sector obtained here is the maximum sector for a certain choice of the Lyapunov matrix equation, or more specifically for an arbitrary positive definite matrix Q in eq. (27), which means that the thus obtained sector is not necessarily the maximum simultaneous Lyapunov sector for the given nonlinear feedback system. However, it should be emphasized that the simultaneous Lyapunov sector [kL, ku] and corresponding positive definite matrix P for the Lyapunov function xT(t)Px(t) for a given nonlinear feedback system can be obtained at the same time. The following example illustrates this. Example 2 Consider the same plant of Example 1 described by

G(z)-

1

2

9

z + 0.3z - 0.4

(36)

A corresponding realization in the observability canonical form is

A_[0 04] 1

-0.3

1

cT

=

[0,1].

(37)

Note that the Hurwitz sector for this case is [-0.3, 1.4]. In the following, three sets of the simultaneously stabilizing feedback gains are evaluated according to the choice of a symmetric positive definite matrix Q.

240

YASUHIKO MUTOH ET AL.

1) Q-

[10 o] ~_[ -0.921 284, 1

-0.921 ] 1.841

(38)

a simultaneously stabilizing gain sector :

2)

[kL, kv] = [-0.263, 1.029]

(39)

] Q_[21 31] ' ~_[ -0.651 6.635 -0.651 4.635

(40)

a simultaneously stabilizing gain sector : (41)

[kL, kU] = [-0.297, 1.225]

l tp

qg(y)= 1.4y Hurwitz Sector Simultaneous Lyapunov Gain Sector qgfy)=1.225y ~y) = -0.297y =

Fig.8 Hurwitz Sector and Simultaneous Lyapunov Sector for Example 1

-0.3y

STABILITY OF D I S C R E T E N O N L I N E A R F E E D B A C K S Y S T E M S

241

3) Q-

[5 2] 2 1 '

[6.587 2.540

P-

2.540] 1.587

(42)

a simultaneously stabilizing gain sector :

[kL, ku] = [-0.084, 0.301]

(43)

Among the above cases, since the second sector is the largest, it should be taken as a set of the simultaneously stabilizing feedback gains. Fig. 8 shows the Hurwitz sector and the simultaneously stabilizing gain sector for this plant. The corresponding positive definite matrix P is given by eq. (40) for the Lyapunov function of this nonlinear feedback system. Fig. 9 shows the contour of this Lyapunov function in the state space. It must be noted that another simultaneous Lyapunov sector may exist which contains those obtained here.

/

:

\i

eq

i i

!

O

!

//, X1

Fig.9 Contour of xT Px (Case 2)

242

YASUHIKO MUTOH ET AL.

V. ALGEBRAIC RICCATI INEQUALITY APPROACH

In this section, the absolute stability of multivariable systenls is discussed based on algebraic Riccati inequality approach. It will be shown that for the absolute stability of multivariable systems, a certain transfer matrix must satisfy some H ~ -norm condition, which can be regarded as the general circle criterion for the multivariable case. The relation between the absolute stability and quadratic stability will be shown also as will the fact that if the absolute stability is guaranteed by the Hor norm condition, the stability implies the exponential stability.

A. ABSOLUTE STABILITY OF MULTIVARIABLE SYSTEM Consider a multi-input and multi-output discrete time nonlinear feedback system ~ M described by the following equations. Linear Part: z(t+l) y(t)

= =

Ax(t)+Bu(t) Cx(t) + Du(t)

(44)

Nonlinear Part: u(t) = - ~ ( y ( t ) , t )

(45)

where u(t), y(t) E R m, A E R nxn, B E R '~xm,C E R mxn and D E R mxm. It is assumed that A is stable and the linear part is controllable and observable. The nonlinear function ~ : R m x R --~ R m is assumed to satisfy p(O, t) = O, and p(g(t), t) is said to lay in sector [kL, ku], if p(y(t), t) satisfies the following quadratic constraint {~(y(t), t) - kLg(t)} T {p(y(t), t) - f u y ( t ) } 0 and the Riccati inequality - p+ATpA+CTC+(ATpB+CTD)N-I(BTpA+DTC)

< O

(47)

holds.

!

The following result gives a sufficient condition for the absolute stability of the system NM with respect to a particular sector I-k, k](k > 0). T h e o r e m 4 System NM is absolutely stable with respect to the sector 1

[-k, k] (k > O)if [[a(z)llo, < ~.

!

(Proof) Note that since kv - --kL -- k the quadratic constraint eq. (46) becomes ~T(y(t), t)~(y(t), t) 0 satisfying the Riccati inequality eq. (47) with 7 - ~. Consider the quadratic function V ( x ( t ) ) " - xT(t)Px(t) as a candidate of Lyapunov function for system EM. Then,

AV(x(t))

= :

V(x(t + 1 ) ) - V(x(t)) {Ax(t) - Bp(y(t), t)} TP{Ax(t) - BT(y(t), t)} --xT(x)Px(t) X T (t)(A T PA - P)x(t) - 2~ T (y(t), t)B T PAx(t) +pT (y(t), t)B T PB~(y(t), t). (49)

By substituting eq. (47) into eq. (49) and completing the squares gives

AV(x(t))


0, system eq. (57) with the perturbation satisfying eq. (64) is quadratically stable if there exist a scalar ~ > 0 and a positive definite matrix P > 0 such that k 2 I - A2ETpE > 0 and the Riccati inequality 1 T G + A2AT P E { k 2 I - ~2ET RE} -1 E TPA < 0 (66) - P + A TPA + ~-~G

holds.

!

From Lemma 3, it is easy to show that the condition in Corollary 2 is equivalent to the H ~ norm constraint IIC(zI - A) - Etl


_ 1) function in z with f(t, 0) - 0, vt ~ R. D e f i n i t i o n 9 System eq. (70) is said to be exponentially stable if and only if for any to and xo - x(to), there exist constants 0 < fl < 1 and p(.~0) > 0 such that the solution of eq. (70) with z(to) - xo satisfies

IIx(t)l[ < p(~0)~ (~-'~

(71)

for all t >__to.

l

D e f i n i t i o n 10 A continuous function ff 9R + ~ function, if it is strictly increasing with ~(0) - 0.

R + is called a class-K; l

The following Lemma is a slight modification of Lemma 1 in [13]. L e m m a 4 For system eq. (70), if there exists a function Y ( t , x ( t ) ) with V(t, 0) - 0 such that

1) y(t,~(t))>_ r

vt _ to.

2) A V ( t , x ( t ) ) - V(t + 1,x(t + 1 ) ) - V ( t , x ( t ) ) < M - c r V ( t , x ( t ) ) ,

vt >_ to.

for some ff E/C, and constants M > 0 and 0 < a < 1, then t-to-1

r

< w(t, ~(t)) < (1 - ~)('-to)W(to, ~(t0)) + M Z i=0

(1 - a) i. (72)

STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS

249

(Proof) From condition 2 ,

V(t,x(t)) 0

Q

:~

-{-P

(75)

1 GT G + A2AT P E N - 1E T PA} > 0 (76) + A T P A + --~

Then, for any admissible AA(t) system eq. (57) is exponentially stable with

C(xo)

"-

/ d

Xl

Pxo

(77)

v "-

x/'l - a

(78)

where a is any constant scalar satisfying (79)

(Proof) Suppose that there exist a scalar ,k > 0 and a positive definite matrix P > 0 satisfying eqs. (75) and (76). Let

r

m;

(P)llx(t)lt 2

(80)

where I1" II denotes the Euclidean norm. It is clear that ~ E/C. Define a function Y(x(t)) xT(t)Px(t). (81) -

Then, V(x) >_ ((]Ix[]) and from the proof of Theorem 6, AV(z(t)) __ -~mi~ (Q)x T (t)x(t)

(82)

250

YASUHIKOMUTOHET AL.

From eq. (82)

AV(x(t)) It has b e e n and o n l y (8)

=

if the

the

(~8)

x0,

a control states

input u with m i n i m u m

to the o r i g i n

at a g i v e n

to. shown

that

the

solution

controllability

is n o n - s i n g u l a r

[26] .

The

is p o s s i b l e

Grammian optimal

given

control

if

by Eq. law

is

given by U. where

-- B T e A T (t-t~

=

star d e n o t e s

demonstrates Grammian The

the optimal

the n e c e s s i t y

to be invertible,

condition

(19)

(to, tf) x0, control.

(19)

for the c o n t r o l l a b i l i t y i.e of full rank as a

for c o n t r o l l a b i l i t y .

observability

considering

the

Grammian

same a u t o n o m o u s

can

be

= Cx(t).

interpreted

s y s t e m with

taken as y(t)

Equation

(20)

by

the output

262

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

Now,

with

knowledge

interval state input Eq.

[t0, tf],

x 0.

of it

Without

is loss

of

both

[to,tf],

into Eq.

the

initial

the

control

for x(t)

from

(20) gives

of

Eq.

(21)

by

eAT(t-t~

T

__ e A T (t-to)c T C e A (t-to) X0,

Eq.

both

sides

re-

eAT(z-t~ CTy (~) dZ

x0 =

solution

to

can

given by

be

the

(22)

over

the

itf 0

interval

eAT(z-to)cTce A(~-t~ d~xo,

(~) d~,

necessity

for

the

order

to

invertible also

be

represents be

Eq.

eAT(~-t~

to the p r o b l e m

should

Grammian

=

Wo I (to, tf)

indicates

Grammian

of

(22)

(22) becomes

itf 0

This

Solving

the

in

integrating

It

find

over

(21)

sides

e AT(t-to) CTy (t)

which

to

output

C e A(t-t0) x0,

multiplying sults

and

generality,

zero.

(18) and s u b s t i t u t i n g =

input

desired

u can be c o n s i d e r e d

y(t)

or

the

seen

in

being noted

a measure by

(23)

(24) observability have

a unique

considered. that of

considering

the the the

observability output output

energy. energy

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

i

h

263

(25)

tf yT (~) y (I;) d~, 0

e A T (~-t0)c T c e A (~-t0) dZx0,

=

(26)

d to

=

x~W0 (to, tf) x0.

E.

THE

DISCRETE

Consider x(k+l)

The

=

x(0)

Lyapunov

time

EQUATIONS system

[15] (27)

= x 0, function

x (k) T p x (k) .

change =

V(k+l)

AV

=

x(k)T(ATpA

means

stable,

ATpA can

as (29)

-V(k),

that

the

or if one

(28)

in V is d e f i n e d

AV

This

one

the

LYAPUNOV

discrete

= Ax(k),

and define V

the

(27)

_p)x(k)

for

change

.

system

(27)

in V m u s t

be

(30) to

be

asymptotically

non-positive

definite,

writes -

p

=

derive

tinuous

time

meaning

of

-

(31)

Q,

similar

case.

stability

The

arguments only

in b o t h

to

those

difference cases

[24].

of

the

is t h a t

con-

of the

264

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

Except and

for

theorems

equations case.

are

similar

_

Wc

ATWo A

-

W o =_

time

-- -

of the

to

equations

AWcAT

the

meaning

concerning

These

where

stability, discrete

those

of

definitions

time

the

Lyapunov

continuous

time

are

BBT,

(32)

cTc,

(33)

matrices

[A,B,C]

are

those

of

the

discrete

system.

This

concludes

retical III.

aspects

the

Lyapunov

the

OF

THE

early

equation

LYAPUNOV

1960 's,

This

importance

and

review

of

the

theo-

functions/equations.

has b e e n

investigation.

extreme

historical

of L y a p u n o v

SOLUTIONS

Since of

the

the

under

is

EQUATIONS

solution

a considerable

particularly

vast

to

due

the

amount

to

their

number

of

applications

took

on

three

in

the area of controls. The

research

in

this

area

directions.

Namely,

tion

implications,

and

tions,

its

and n u m e r i c a l

In the solution, [29],

one

finds

Barnett

Barnett [16].

in this

Section

II.

et

aspects

closed

the

solu-

form e x p l i c i t

solu-

the et

key

al

al

theoretical papers

[30] . [15],

direction the

have

by

aspects

Gantmacker that most

already

interested

of

Taussky

Other m a i n

It s h o u l d be n o t e d However

of

solutions.

direction,

and O s t r o w s k i

include sults

first

theoretical

different

been reader

et

the al

references [31],

and

of the

re-

addressed

in

is r e f e r r e d

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

to

references

above

and

references

therein

265

for

more

details. In the the

second

research

when

the

Jordan gives

is

focused

system

form

in this

direction, is

or any

direction

a summary

continuous Other [34] .

It

require

in

other

be

special

form

Power

time

are

that

transformation

of

This

is

a

solution such

[32,33].

to

as

A key p a p e r Power

for both

Lyapunov

due

noted

form.

form.

form s o l u t i o n s

discrete

solutions,

the

to H.M.

contributions

the

a

form

finding

canonical

is due

should

canonical

given

and

closed

around

of s p e c i a l

time

major

the

equations.

Peter

Lancaster

explicit the

the

solutions

system

numerically

to

a

demanding

procedure. In

the

third

finds

two

main

where

the

error

direction, approachesin

imized.

For

referred

to

al

Bartels

[35],

based This

on

details the

widely A.

work et

reducing

approach

Bartels

and

the

is

methods

this

of

K.

HIGHLIGHTS

this for

presented.

first

Zietak

A matrix most

is

one

iterative

successively

min-

the

reader

is

[11-14],

Peters

et

approach

is

The

second

to

a lower

This

here

THE

one

reliable

[36].

OF

is

solutions,

approach,

[36] .

u s e d and is g i v e n

LYAPUNOV

In

of

the

Stewart

The

solution

al

the

numerical

Schur

and

approach

is is

form.

due the

to

most

in detail.

SOLUTION

OF

THE

EQUATIONS

section

a

detailed

the

solution

The

first

of the

method,

due

discussion Lyapunov

of

three

equations

to B a r n e t t

is

and Story

266

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

[15],

is

method

based

is due

on

the

to B a r t e l s

m e t h o d is due to S.J.

1.

As

Solution via

was

Barnett

This

using

the L y a p u n o v

+ ATp

=

The

[36].

second

The

third

[3] .

Equations

Products

earlier,

to s t u d y the t h e o r e t i c a l

PA

Stewart

Lyapunov

Kronecker

Story.

Consider

and

product.

Hammarling

of

mentioned

and

Kronecker

this

method

is

aspects

method

is

normally

due

utilized

of the solution.

equation

-Q,

Kronecker

to

(34)

products,

this

equation

can be w r i t t e n

as [(AT| where

+

vec(A)

transpose

(I|

T) ] v e c ( P )

defines

of the

A|

=

the v e c t o r

rows of A,

a11B a12B

--"

-

vec(Q),

f o r m e d by

(35)

stacking

alnB ,

......

the

and

. amlB

=

(36)

amnB

a s s u m i n g A is mxn. Equation A has

no

solution vec(P)

(35)

has

eigenvalues

a unique on

the

solution imaginary

if and axis

only

[15],

if the

is g i v e n by =

-

[ (AT|

+

(I|

T) ] - i v e c ( Q ) .

(37)

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

It

should

inversion

be

noted

that

of an n2xn 2 matrix.

is not u s e d for actual 2.

This

Bartels

is

one

and

of

popular MATLAB

AP

Stewart

the

most

is why this m e t h o d

Algorithm

numerically

For e x a m p l e

the

more

=

reliable

it is u s e d

and

in the

general

case

of

the

Lyapunov

-Q,

(38)

Q is symmetric.

lution

the

lets c o n s i d e r the e q u a t i o n

+ PB

where

This

involves

s o f t w a r e package.

consider

equation,

solution

calculations 9

widely used algorithms. To

the

267

if and

only

Equation

(38) has

if any e i g e n v a l u e s

a unique

[~i of

A

so-

and

~j

of B s a t i s f y (Xi + To

~j ~

solve

0 Eq.

for (38)

g e s t e d the f o l l o w i n g i)

Reduce

the

all

=

uTAu

=

and

for

P,

steps

9

matrix

unitary transformation

A

i

to

a

j.

(39)

Bartels

lower

and

Schur

Stewart

form

sug-

using

a

U as follows

Azl

0

.--

0

A21

A22

".

.

_ Apl

Ap2

---

App

where each m a t r i x Aii is at most

,

_

2x2.

(40)

268

2)

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

In

Schur

B

same

form

as

=

where 3)

the

V

The

vTBv

is

fashion,

B

B11

BI2

.--

Blq

0

B22

.--

B2q

0

-..

0

Bqq

=

also

reduce

a unitary

transformed

Q and

to

upper

triangular

(41)

_

matrix 9 P are

given

by

N

Q11 Q

=

uTQv

......

Q1q (42)

=

_

QpI

......

Qpq

P11

......

Plq

_

and

P

=

uTpv

(43)

=

Ppl 4)

Recursively,

(k=l,2,...,p; It

should

be

solve

...... for

the

Ppq _ blocks

of P

as

k-i

L-I

j=l

i=l

follows

1= 1,2,...,q). noted

that

the

(44) solution

of

Eq.

(44)

is

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

easy to carry mension

out

of 2x2.

since One

ber of c o m p u t a t i o n s B=A T.

This

Schur

should

a m a x i m u m di-

also notice

that

is r e d u c e d by almost

the num-

one half when

is needed.

of

the

computational

decomposition

(2+4g) n 3 flops,

and

where

tions

for

The actual

the

is one

due to

addition number

form

reduction

of Eq.

to

be

average

Schur

solution

is

estimated

a flop

and g is the

required

burden

is

multiplication, verge.

have

is due to the fact that only one Schur de-

composition Most

all blocks

269

the

about

and

of

one

iterato

(44) requires

conabout

7n 3 flops.

3. S.J.

Hammarling Hammarling

Algorithm

1982,

modified

the

Bartels/Stewart

a l g o r i t h m such that the solution to the L y a p u n o v equations

is

Hammarling directly

more was

numerically

able

for the

to

upper

solve

well the

conditioned.

Lyapunov

triangular

Cholesky

equations factors

of

the Grammians.

B.

SOLUTION

OF

THE

DISCRETE

LYAPUNOV

EQUATIONS As for

in the

the

possible system. reader

continuous

discrete

time

for specific For

details

time

Lyapunov

canonical of

is r e f e r r e d to N.J.

and references

case,

this

equations

solutions are

representations type

Young

t h e r e i n such as

explicit

of

of the

solutions,

[9], K e q q i a n WU

[38].

only the [37],

270

AHMAD A. MOHAMMAD AND J. A. DE A B R E U - G A R C I A

It s h o u l d be n o t e d that discrete ment

time

The

discrete

common

time

continuous rithms

procedure

efficient

practice

Lyapunov

time

can

nature

Lyapunov equation prevented

of n u m e r i c a l l y

tion.

the n o n l i n e a r

be

to

the d e v e l o p for the solu-

in n u m e r i c a l l y

equation

one w h e r e used

algorithms

solving

is to convert

efficient obtain

of the

the

it to a

and r e l i a b l e

algo-

solution.

This

the

is o u t l i n e d next.

Consider

the

discrete

Lyapunov

equation

given

by

[32,33] ATLA

-

L

=

- Q,

(45)

u s i n g the t r a n s f o r m a t i o n A

=

(B+I) (B-I)

converts

the

continuous BLb

+

discrete

LbB

=

(A-I)

it is

conditioned. for

the

tance.

=

(B_I)T

time

Lb 2 (B-I) ,

Lyapunov

(46)

equation

to

the

(47)

(47)

- Q. can

be

or

S.J.

Bartels/Stewart of

L

time L y a p u n o v e q u a t i o n

Equation However,

-I ,

should

be

solution

of

for

out

demanding

the d e v e l o p m e n t this

Lb

using

H a m m a r l i n g 's

pointed

numerically Thus,

solved

problem

that and

either

algorithm.

the

inversion

might

be

ill

of a new t e c h n i q u e is

of

great

impor-

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

C.

A

NOTE

ON

THE

DESCRIPTOR

SOLUTION

OF

Ex

= Ax

y =

EQUATIONS

system

+ Bu

(48a)

Cx,

where

(48b) -I

(sE-A)

condition The

THE

LYAPUNOV-LIKE

Consider the regular d e s c r i p t o r

exist

as

a

sufficient

and

necessary

for regularity.

expected

Grammians

controllability

for this

and

system satisfy

observability

[39]

A W c ET

+ E W c AT =

- BB T

(49)

ATWo E

+ ETWo A

- C TC.

(50)

Under

the

=

assumption

consistent

initial

eliminate

any

that

the

conditions

system

is

(initial

impulsive

behavior),

given theorems

concerning

reachability,

and

similar

stability

However,

the

unique.

In addition,

ness

with

of the

Grammians to

the

corresponding

divided to

these 1987

about

part the

of

with

conditions

that

Lewis

regular

systems.

equations [40],

the

is

not

has shown that

in general.

existence he has

two parts, the

noncausal

system part

has

observability,

He also

for these Grammians

Briefly, into

regular

Frank

for

satisfied

solution. causal

of

definition

theorems

are

those

Bender

are not

gave a more general gether

to

solution

these equations

ing

271

and

shown one

uniquethat

the

correspond-

and of

to-

the

the

other

system.

272

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

According

to

Bender,

the

teachability

Grammians

are

given by

r (AWc~ E T + ~.wo~A ~) r r

= - CoBB Tr T

(~.wo2~. ~ - Aw~2A =) r

while the o b s e r v a b i l i t y

= r162

are e x t r e m e l y applications tions.

izations, technique solution

and

the

one

hence,

to obtain to this

(54)

the coefficients

series expansion solutions This

require

For e x a m p l e

(53)

= CT-~CTCr

difficult. that

Tcr

respectively,

s o ,s -I in the Laurent

that

are given by

= - r

CT_~(~.TWo2~. - ;JWo2A) r

It is clear

(52)

Grammians

r (;JWo~. + ~.TWo~A)r

where ~-I and ~0 are,

(51)

the

can

specific

for these

solution

not

reduced

of H(s).

is a m a j o r

can not

of

obtain utilize

equations

drawback

in

of these

equa-

balanced

real-

the

order models.

balancing However,

p r o b l e m has been p r o p o s e d

a in

[2].

This

concludes

the

review

of

the

solutions

to

the

Lyapunov equations. IV.

IMPORTANT

APPLICATIONS

LYAPUNOV

Different have

already

OF

THE

the

Lyapunov

EQUATIONS

applications been

of

introduced

However,

it is

felt

that

of these

applications

equations

in the p r e v i o u s

a more

is n e e d e d

comprehensive for the

sake

section. summary of clar-

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

ity and completeness

of this

of these applications A.

LYAPUNOV

These

Consider =

theory

represent

and

a summary

here.

+ Bu,

the

in control stable

x(0)

=

first

application

system design

[41].

system given by (55 .a)

x0,

= Cx.

(55.b)

It is desired to design uTu

Thus,

CONTROLLERS

the minimal

Ax

y(t)

is p r e s e n t e d

controllers

of Lyapunov's

chapter.

273

a control

input u such that

___ 1

such

(56)

that

the

initial

state

returns

problem

starts

to the

origin

as rapidly as possible. The

solution

Lyapunov V

xTpx

=

x T[PA that

symmetric unique PA

+ ATp]x A

is

symmetric

Substituting

The

by

assuming

a

(57)

=

control

2uTBTpx. which

(58)

implies

semi-definite

positive

definite

Q,

that

there

for

any

exists

(59)

(59) into Eq. +

input

a

P such that

-Q.

Eq.

- xTQx

+

stable

positive

+ ATp

=

this

function

=

Notice

to

(58) yields

2uTBTpx. u

should

(60) be

chosen

.such

that

V

is

274

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

negative

with

the

largest

magnitude

clear that u must be parallel two

vectors

must

have

ond term negative.

to BTpx.

opposite

signs

It

However, to make

It

is

the

sec-

Thus u should be taken as

known

(61)

important

optimal LQR

is

these

BTpx

u

the

possible.

to

notice

controller

design

that

among

gives

only

this

all the

controller

controllers. linear

is The

optimal

con-

troller. B.

LYAPUNOV

EQUATIONS

ANALYS

This

has already

beginning

of

time

descriptor

earlier. suffer

It

especially responding transfer

chapter.

was

some

the

that

to

at the

discrete

also

addressed

these

extensions

problems.

descriptor

case

into two

This

is

where

the

subsystems

and p o l y n o m i a l

parts

cor-

of the

function.

LYAPUNOV

EQUATIONS OF

Controllability, ity analysis and

the

proper

MINIMALITY

have

out

to be s e p a r a t e d

to

addressed

were

serious in

been

Extensions

systems

pointed

evident

system needs

C.

this

from

STABILITY

IS

application

and

IN

already

implications

THE

systems

addressed of

these

THE

SYSTEM

observability,

of dynamic been

DEFINE

and

hence

via Lyapunov

earlier.

concepts

The in

minimalequations

importance

control

system

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

design the

were also

considered

applications

sign,

observer

lems

have

noted

design,

been

that

of these

concepts

and

discussed

the

in details

of

controller

However,

these

systems

suffers

problems.

mainly

arise

tion

and

latter

the

two

problems

solution

cases.

of

Lyapunov

These

problems

de-

probit

concepts

crete time and d e s c r i p t o r These

Among

realization

in detail.

application

therein.

the

minimal

275

was

in dis-

from serious

in the

defini-

equations have

in

been

the

briefly

introduced earlier. D .

LYAPUNOV

EQUATIONS

IN

MODEL

ORDER

REDUCTION

B.C. the

Moore

[21]

demonstrated

controllability

'balanced was

used

der m o d e l s Lyapunov

constitute

and o b s e r v a b i l i t y in the

same paper

of d y n a m i c a l

equations

and is essential tion

observability

coordinates,

controllability fact

and

algorithm

how the eigenvalues a measure

to obtain This

introduced

later;

the This

reduced

application

the most

in the d e v e l o p m e n t

in

of

of each state.

systems.

is p r o b a b l y

Grammians,

of

important

orof one

of the continuizahence

it

is

intro-

duced here in detail. 1.

Since

The

Technique

its i n t r o d u c t i o n

technique

has

[42-53].

By

niques

Balancing

triggered now,

(BT)

by B.C. Moore, an

there

to obtain b a l a n c e d

intensive

are

several

realizations

the b a l a n c i n g

wave

of

research

numerical

tech-

[see references

276

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

above] . rather

However,

than the

they

only

concept.

differ

An o u t l i n e

in

the

of this

details technique

is g i v e n next. Consider

y

the

system

[50,52,53]

=

Ax

+

Bu

(62a)

=

Cx

+

Du,

(62b)

and the c o r r e s p o n d i n g AWc

+ Wc AT

ATw o + Next,

consider

Wc = where

WoA

Lyapunov

equations

=

-BBT

(63)

=

-cTc.

(64)

the s i n g u l a r

value decomposition

of Wc

U c ~ c U T,

Uc

is

(65)

unitary,

~c

is

diagonal

with

entries

(;i~(Yi+ 1 - Let TI and

=

_

~i/2

UcLc

apply

a

,

(66)

similarity

transformation

to

system

(62)

to get

-1/2

A1

=

Zc

CI

=

CUcLc

Wcl

=

The next

Wol

I,

,.,1/2

.T..

Uc~UcZ~c

Wol

=

T 2 as

=

-1/2

Zc

1/2 T..... .., 1/2

~c

UcWoUc~c

step is to p e r f o r m

= UoiZoiUoT1-

Choosing

, B1

uTB, (67)

.

(68)

an SVD on Wol as (69)

277

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

= Uo Zo /' and

applying

system

(67)

a

(70) similarity

gives

transformation

the b a l a n c e d

using

T 2 to

system

_i14 T

Ab

-

/~oi U o l A I U o l ~ o I14 1 ,

Bb

=

~I/4UolB~ ir 2~oi

Cb

(71a)

_-I14

CiUol2~ol

=

,

(71b)

with the new G r a m m i a n s Wcb

=

_I14 T i14 Lol U o l l U o l ~ c

Wob

=

Lol

It mal

_-I14

should

and

-iI~

uTIWoIUoIEol

be

output

tained

from

_I12 ~oI =

=

noted normal

Eq.

i/2 Eol =

=

that,

the

by

E,

well

realizations

(71)

(72)

~, (~i>_(~i+l .

known

can be

applying

the

(73)

input

nor-

readily

ob-

similarity

transformations Ti

T 2 T IE ~/2

=

for input To =

normal,

nique

and

T 2 T I ~ -I/2

for output The

(74)

normal.

final is

(75)

to

step

in this

look

for

model

a break

order in

the

reduction singular

techvalues

such that (~r>>(~r + I ' and t r u n c a t e

the b a l a n c e d

system

after the

r th state.

278

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

To

illustrate

system

the

last

step,

:[A11 A12][xl

x2

A21

A22

bl

the

balanced

u (t)

b2

x2

~: [c,c.][x.]

consider

(76a) (76b)

X2

with

the G r a m m i a n s

Wc

then

=

the

'

~2

reduced

Xr

=

AllXr

Yr

=

CllXr,

with

I (Z~)>>k (Z2), min max

r th o r d e r m o d e l

(77)

can be t a k e n

as

+ bllU

(78a) (78b)

Granunians

Wcr For

o]

Wo =

-

Wor

=

simplicity,

renamed

as

2.

-- ~'r

the

(79)

reduced

order

model

matrices

are

[A r,B r,C r] .

Properties

of

the

Technique

a)

The

entries

of

the

system.

between

~i

them

t r i x H of the

of ~

are

There

a n d the system-

Balancing

[44, 4 5 , 5 2 ]

called is

an

singular

the

second

interesting values

Specifically,

order

modes

relationship

of the

Hankel

ma-

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

279

2 k i ( H H T) = ~ib)

In general,

and the

the

eigenvalues

controllability

of the

level

of

of

the

Grammians

controllability

are

and

observability a quantization

observability

of

each state. c) In balanced coordinates, as it is observable d) The subsystems

a state is as controllable

(Wo=Wc=~).

[Aii,Bi,C i] in Eq.

(76) are balanced

with Woi = Wci = ~i-

e) T r a c e ( W o o ) = ]~ D C 2

gain

f)

bound

Wco = C r o s s There

is

an

of

the

system,

Grammian. upper

on

the

frequency

error

given by ]E ( s ) ~ C g)

The

(s I-A) -IB-c I (s I-All )-IB iI~-(~j i f

i>j.

and b a l a n c e d the

input

The

system.

and output

286

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

matrices

of the CTS Bcb,

Bcb =

Bdbl~,

This

choice

Euler's

method,

trices CTS

except

Ccb can be taken as

Ccb = is

(83)

Cdb/~.

justified

for example,

if

one

does

not c h a n g e

for the m u l t i p l i c a t i o n

dynamics

matrix

Acb

is then

notices

these ma-

factor

chosen

to

that

~.

The

satisfy

the

CTS L y a p u n o v e q u a t i o n s

so

Acb%

+

~ATb

=

-- P / T - -

ATb ~

+ ~Acb

=

-

that

the

DTS.

method

and the

and

output

while

in

bilinear

the

noticing

or Eq.

transform one

same

of

the

combination

of

t r a n s f o r m method-

the

is

are

Grammians a

taken

as

values

method.

It

is

Acb

separately,

the

solution

if

one

uniqueness

of

there are two s o l u t i o n s Then

solves the

from

for

the s o l u t i o n

Euler's taken

as

worthwhile either will

Acb

Eq.

not

from

be

Eqs.

is unique.

solution,

Acb I and Acb 2 that

it follows

in are

for

simultaneously,

(84)-(85).

the

solves

(84)-(85)

Eqs.

(85)

singular

However,

show

Qc,

bilinear

unique. To

(84)

strategy

Hankel

if

(85)

-

matrices

the

that

=

have

this

method

(84)

will

Clearly,

Euler's input

CTS

Q/T

Pc

suppose

that

satisfy both

that

Acbl%

+

ZAcTbl =

-- Pc

(86)

Acb2~

+

T ~Acb2

- Pc-

(87)

Subtracting

Eq.

=

(87) f r o m Eq.

(86) and l e t t i n g

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

Acb I - Acb 2 =

287

N

yields N]~

+

]~N T =

This e q u a t i o n

reduces

(~jnij + (Yinji Similarly, must

(88)

O.

-

f r o m the

which

+

NT~

=

observability

Lyapunov

equation,

reduces

0,

N

(90)

to

(~inij + nji(~j = Comparing

Eqs.

(~i~(~j i f

i~j,

nij

=

nji

and

is

seen

it

that

(91) , that

and

the

provided

only

solution

to

the

Acb 2.

solution

solution

of

of Eqs. these

two

(84)-(85),

[61]

9

Provided

i~j,

Acb can be w r i t t e n

(~i~(~j i f

- Pcii 2(Yi

it

equations

simple p a r a m e t e r i z a t i o n

acbii =

to

is

that Acb I =

the

that

that

-- 0,

implies

Returning

(91)

0.

(89)

these two e q u a t i o n s

seen

(89)

0.

satisfy

~N

which

to

is

is

a

as (92)

288

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

for the d i a g o n a l

elements

choice

that

the

of b a l a n c e d

Clearly, (Acb,

elements

system

Ccb,

as

This

is due

and e x p l a i n s

to the

coordinates.

Dc=Dd)

it

of Acb.

is b a l a n c e d

from L y a p u n o v

Bcb,

balanced

(93)

2 2 ((~i -- (~j)

for the off d i a g o n a l fact

and

Pcijf~j -- qcij(~i

acbij =

the

of Acb,

theory,

is

satisfies

both both

the

resulting

stable, Lyapunov

system

minimal Eqs.

and

(84)

and

(86). 3.

This

Theoretical

is best

Theorem-

Aspects

summarized

For

any

given

(Adb, Bdb, Cdb, Dd, ~, T) , modes,

there

balanced

exists

CTS model

Bob = B and

Acb

/fT,

is

a

of

the

LBCT

in the following stable,

with

minimal

distinct

unique,

theorem-

balanced second

minimal,

order

stable,

(Acb, Bcb, Ccb, Dc=Dd, ~),

DTS and

where

Ccb = C b/f{,

the

unique

solution

of

the

CT

Lyapunov

equations

AcbZ

+

~ATb

=

T - BcbBcb

=

- TBdbBTb

T Acb~

+ ~Acb

=

-

T CcbCcb

=

- TC~Cdb.

the

CTS

Moreover,

the f o l l o w i n g

approximation

properties-

of

the

DTS

model

has

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

i)

The

model 2)

CTS

model

is m i n i m a l

There

minimal

a

solution

if

the

DTS

for

Eqs.

(84)-(85),

for

a

of 2) is unique.

4) The I~2 n o r m of the DTS CTS

stable

of T, B, and C.

3) The s o l u t i o n the

and

and stable.

exists

fixed choice

is

289

(notice

that

we

error

in

is equal only

to the ~HI2 n o r m

deal

with

the

of

strictly

p r o p e r part) . 5)

The

initial

the

step

response

is

identi-

the

Hankel

cally zero.

Proof

:

(i) G u a r a n t e e d by L y a p u n o v equations. (2,3) (4)

Proved A

in s e c t i o n

direct

consequence

S i n g u l a r values (5)

First,

same value

it

p r o p e r part

4.

is

noted term

the

that

D.

is i d e n t i c a l l y

Comparison

with

matching

Euler's

both

systems

Moreover,

initial

of

share the

the

the

initial strictly

zero.

Other

Techniques

the p r o p o s e d

and

from

response

with

In order to v a l i d a t e parison

of

of the DT and CT systems.

feedthrough theorem,

2.

the

technique,

Bilinear

a com-

transform meth-

ods is p r e s e n t e d next. To model

set

the

stage,

suppose

and the c o n t i n u o u s

discretization In this

case,

technique the

first

model using

that

the

are r e l a t e d a

relation between

sampling the DTS

discrete

via Euler's period

T.

and the CTS

290

A H M A D A. M O H A M M A D AND J. A. DE AB R E U - G A R C I A

state

space m a t r i c e s

Adb =

I

+ TAcb,

Substituting (I

+

Eq.

Bdb =

(94)

TAcb)~(I

where

Wcb

will

be

anced' Thus, to

is

(95a)

BobBc b

(95b)

method,

must

satisfy

T Bcb Bcb,

-

that

Hankel

t e r m in Eqw.