Linear Dynamical Systems

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Linear Dynamical Systems

This is Volume 135 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by William F. Ames, Georgia Institute of Technology A list of recent titles in this series appears at the end of this volume.

Linear Dynamical Systems A revised edition of

DYNAMICAL SYSTEMS AND THEIR APPLICATIONS: LINEAR THEORY

John L. Iastl International Institute for Applied Systems Analysis Laxenburq, Austria

ACADEMIC PRESS, INC. HARCOURT BRACE JOVANOVICH, PUBLISHERS Boston Orlando San Diego New York Austin London Sydney Tokyo Toronto

Copyright © 1987, 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.

ACADEMIC PRESS, INC. Orlando, Florida 32887

Library of Congress Cataloging-in-Publication Data

Casti, J. L. Linear dynamical systems. (Mathematics in science and engineering; v: 135) Rev. ed. of: Dynamical systems and their applications. 1977. Includes bibliographies and index. 1. Linear systems. I. Casti, J. L. Dynamical systems and their applications. II. Title. III. Series. QA402.C37 1986 003 86-17363 ISBN 0-12-163451-5 (alk. paper)

87 88 89 90 9 8 7 6 5 4 3 2 1 Printed in the United States of America

To the memory 0/ ALEXANDER MIKHAILOVICH LETOV Scholar, Gentleman, and Friend

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(ontents

xi xiii

Preface to the Revised Edition Preface to the First Edition

Chapter 1

Basic Concepts, Problems, and Examples 1.1 Dynamical Systems, Inputs, and Outputs 1.2 Internal Description of ~ 1.3 Realizations 1.4 Controllability and Observability 1.5 Stability and Feedback 1.6 Optimality 1.7 Stochastic Disturbances Notes and References

Chapter 2

7 11

13 17 19

Mathematical Description of Linear Dynamical Systems 2.1 2.2 2.3 2.4 2.5 2.6

Chapter 3

1 3 6

Introduction Dynamical Systems External Description Frequency-Domain Analysis Transfer Functions Impulse-Response Function Notes and References

21 21 27 28 30 31 33

Controllability and Reachability 3.1 Introduction 3.2 Basic Definitions

35 36

vii

viii

CONTENTS 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Chapter 4

58

61 62 64

68

Introduction Basic Definitions Basic Theorems Duality Functional Analytic Approach to Observability The Problem of Moments Miscellaneous Exercises Notes and References

72 73 75 81 82

83

84 85

Structure Theorems and Canonical Forms 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Chapter 6

39 43 47 52 55 57

Observability/Constructibility 4.1 4.2 4.3 4.4 4.5 4.6

Chapter 5

Time-Dependent Linear Systems Discrete-Time Systems Constant Systems Positive Controllability Relative Controllability Conditional Controllability Structural Controllability Controllability and Transfer Functions Systems with a Delay Miscellaneous Exercises Notes and References

Introduction State Variable Transformations Control Canonical Forms Observer Canonical Forms Invariance of Transfer Functions Canonical Forms and the Bezoutiant Matrix The Feedback Group and Invariant Theory Miscellaneous Exercises Notes and References

88 90 91

97 99 101 104

111 114

Realization Theory 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Introduction Algebraic Equivalence and Minimal Realizability Construction of Realizations Minimal Realization Algorithm Examples Realization of Transfer Functions Uniqueness of Minimal Realizations Partial Realizations Reduced Order Models and Balanced Realizations Miscellaneous Exercises Notes and References

117 118

124 127 128 131 132

133 138

140 145

CONTENTS

Chapter7

IX

Stability Theory 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Chapter8

147 149 152 156 162 164 169 173 174 177 179

The Linear-Quadratic-Gaussian Problem 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15

Chapter 9

Introduction Some Examples and Basic Concepts Routh-Hurwicz Methods Lyapunov Method Frequency-Domain Techniques Feedback Control Systems and Stability Modal Control Observers Structural Stability Miscellaneous Exercises Notes and References

Motivation and Examples Open-Loop Solutions The Maximum Principle Some Computational Considerations Feedback Solutions Generalized X - Y Functions Optimality versus Stability A Low-Dimensional Alternative to the Algebraic Riccati Equation Computational Approaches for Riccati Equations Structural Stability of the Optimal Closed-Loop System Inverse Problems Linear Filtering Theory and Duality The Separation Principle and Stochastic Control Theory Discrete-Time Problems Generalized X - Y Functions Revisited Miscellaneous Exercises Notes and References

182 185 187 190 192 196 204

214 216 219 220 227 231 233 234 235 240

A Geometric-Algebraic View of Linear Systems 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12

Algebra, Geometry, and Linear Systems Mathematical Description of a Linear System The Module Structu~e of n r, and X Some System-Theoretic Consequences Transfer Functions Realization of Transfer Functions The Construction of Canonical Realizations Partial Realizations Pole-Shifting and Stability Systems over Rings Some Geometric Aspects of Linear Systems Feedback, the McMillan Degree, and Kronecker Indices

246 247 249 253 257 260 263 271 273 274 278 283

x

CONTENTS 9.13 9.14 9.15 9.16

Some Additional Ideas from Algebraic Geometry Pole Placement for Linear Regulators Multivariable Nyquist Criteria Algebraic Topology and Simplicial Complex of I: Miscellaneous Exercises Notes and References

285 288 291 292 298 310

Chapter 10 Infinite-Dimensional Systems 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Index

Finiteness as a System Property Reachability and Controllability Observability and Duality Stability Theory Realization Theory The LQG Problem Operator Riccati Equations and Generalized X - Y Functions Miscellaneous Exercises Notes and References

317 319

323 325

327 330

332 335 345

347

Preface 10 the Revised Edition

When the first edition of this book, (formerly titled Dynamical Systems and their Applications: Linear Theory) was published in 1977, it presented a reasonably thorough account of the major ideas and results of linear system theory circa mid-1970s. The past decade has witnessed an explosion of interest in mathematical system theory with major advances in the understanding of systems governed by functional differential equations, n-d systems, system identification, numerical methods, and frequency-domain techniques, not to mention the creation of an elegant and comprehensive algebraic and geometric theory of linear systems. And this is just for linear systems; much more can be said about new results in bifurcation theory, chaos, fractals, and other nonlinear phenomena, some of which is treated in my volume, Nonlinear System Theory (Academic Press, 1985), a companion to the present work. The task of doing justice to this impressive body of new work in a hundred pages or less was an imposing one, and one which ultimately required many compromises. Since a detailed account of all major developments was totally out of the question, I decided to treat a few topics in some depth and then to incorporate most of the remaining results by way of augmented references and problems in the earlier chapters. Thus, the current volume includes chapter-length expositions of the algebraic and geometric theory of linear systems (Chapter 9) and the theory of infinite-dimensional systems (Chapter 10). The other topics are interwoven into Chapters 1-8, together with the correction of a number of unfortunate typos, numerical errors and, in one or two places, just plain erroneous results that marred the first edition of this book. xi

xu

PREFACE TO THE REVISED EDITION

At this time, I would like to record my thanks to numerous colleagues, friends and students who were kind enough to show me the error of my ways in the book's first edition and who served as a source of inspiration to prepare this much more comprehensive work. Their efforts and support have shaped both the content and form of this volume. John L. Casti Vienna January 1986.

Preface to the First Edition

A spin-off of the computer revolution, affecting all of modern life, is the pseudoacademic discipline of "systems analysis." The words themselves are sufficiently vague to be transferable to almost any situation arising in human affairs, yet precise enough to suggest a scientific content of sufficient depth to convince the uninitiated of the validity of the particular methodology being promoted by the analyst. The impression is often created that performing a "systems analysis" of a given situation will remove all subjectivity and vagueness from the problem, replacing fallible human intuition by objective, rational, often mechanized, "scientific" policies for future action. Valid as the above argument is for some situations, we must object to its usual mode of proof by contradiction. Implicit in the verbiage spewing forth on the general topic of systems analysis is the assumption that underlying any such analysis is a system theory whose results support the analyst's conclusions and recommendations. Unfortunately, our observations have led to the conjecture that the majority of individuals practicing under the general title of "system analyst" have very little, if any, understanding ofthe foundational material upon which their very livelihood is based. Furthermore, when this fact is even casually brought to their attention, a typical human defense mechanism is activated to the extent that the subject is brushed off with a remark such as, "Well, system theory has not yet progressed to the point where practical problems can be treated, so what are we to do when the real world demands answers?" Unfortunately, there is a germ of truth in this standard reply; but in our opinion, such a statement has an even stronger component of prejudice seeking rationality since, as noted, a majority of analysts are in no position to speak with authority as to how far system theory actually has progressed and xiii

xiv

PREFACE TO THE FIRST EDITION

what the current results do say about their problems-hence, a partial motivation for this book. While it must be confessed that, like good politics, good systems analysis is the art of the possible, it is ofprimar importance that a practitioner have a fairly clear idea of where the boundary currently lies separating the science (read: established theory) from the art (read: ad hoc techniques). In this book we address ourselves to basic foundational and operational questions underlying systems analysis, irrespective of the context in which the problem may arise. Our basic objective is to answer the question: how can mathematics contribute to systems analysis? Regarding a system as a mechanism that transforms inputs (decisions) into outputs (observations), we shall examine such basic issues as: (i) How can one construct an explanation (model) for a given input! output sequence and, if several models are possible, how can we obtain the "simplest" model? (ii) With a given model and a fixed set of admissible inputs, how can we determine the set of possible behavioral modes of the system? (iii) With a prescribed mode of observing the behavior of a process, is it possible to uniquely determine the state of the system at any time? (iv) If a criterion of performance is superimposed upon a given process, what is the best value that this criterion can be made to assume, utilizing a given set of admissible inputs?

Clearly, the above questions are far-ranging and no complete answers are likely to be forthcoming in the near future as long as we speak in such general terms. Consequently, we lower our sights in this volume and confine our attention to those systems for which there is a linear relation between the system inputs and outputs. Not only does this provide a structural setting for which a rather comprehensive theory has been developed, it also enables us to confine our mathematical pyrotechnics to a level accessible to anyone having a nodding acquaintance with linear differential equations and elementary linear algebra. Briefly speaking, the book is divided into four basic parts: introductory, structural, modeling, and behavioral. The introductory chapters (1-2) give an overview of the topics to be covered in depth later, provide motivation and examples of fundamental system concepts, and give reasonably precise definitions upon which further results are based. The structural chapters (3-5) introduce the important concepts of controllability, observability, and canonical forms. Here we find detailed information concerning the restrictions on system behavior that are imposed by purely structural obstacles associated with the way in which the system is allowed to interact with the outside world. Furthermore, such obstructions are made evident by develop-

PREFACE TO THE FIRST EDITION

xv

ment of canonical forms explicitly devised to make such system properties apparent, almost by inspection. With a firm grasp of the structural limitations inherent in a given system, the modeling chapter (6) addresses itself to the question of actually constructing a model (realization) from given input/ output behavior. A number of algorithms for carrying out such a realization are presented, and extensive attention is given to the question of how to identify "simple" models. Finally, in the behavioral chapters (7-8) we analyze questions of system dynamics. Problems associated with the stability of system behavior under perturbations of the operating environment are treated, together with the problem of choosing admissible inputs that "optimize" a specific criterion function. A student who masters the material of this volume will be well prepared to begin serious system-theoretic work either in the applications area or in graduate research in systems-related disciplines. Saratoga, California August 1976

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CHAPTER

1

Basic (oncepls, Problems, and Examples

1.1 DYNAMICAL SYSTEMS, INPUTS, AND OUTPUTS

To create an axiomatic theory of systems, dictates of logic demand the introduction of various definitions and terminology restricting the universe of discourse to a well-defined class of objects, which may then be analyzed according to the usual methods and tools of mathematics and logic. However, to anyone who has had occasion to practice in the field of" systems analysis," such a program seems self-defeating from the very outset because an attempt to verbalize precisely what one means by the intuitively understood entity "system" seems immediately to restrict the terms to such a degree that the next" system" problem encountered no longer fits within the confines of the definition. Consequently, we shall forego definition of such a vague, but basic, concept and restrict our attention in this book to the most important subset of the class of general systems-the linear systems. Having shifted the weight from one foot to the other by introducing the qualifying adjective linear, what do we mean by a linear system? Intuitively, we think of a linear system 1: as a machine that transforms inputs to outputs in a linear way. Referring to Fig. 1.1, we envision 1: as a "machine initially in a neutral or zero state, having m input terminals, and p output terminals." At any time t, a certain signal may be presented to the input terminals, the system operates on (transforms) the given input signal, and a response is observed at the output terminals. By assuming that the inputs and outputs belong to sets in which the operations of addition and scalar multiplication

2

BASIC CONCEPTS, PROBLEMS, AND EXAMPLES 1---------, ,

f

,

1

INPUTS 2

,p

:

'":

:

L

,

~'-{)2 '1

OUTPUTS

J

L

FIG. 1.1 The system ~.

are defined, the linearity of I: means that the input/output mapping is linear, i.e., if we let n be the set of admissible inputs, r the set of outputs, and

f:

n -+ r

(1.1)

the rule for transforming inputs into outputs, i.e., the input/output map of I:, then I: is linear if and only if

for any scalars ex, f3, and inputs WI> W2 E n. (Remark: The usual interpretation of nand r is that they are sequences of vectors from the vector spaces R" and RP, respectively, with the scalar being real numbers. As will be pointed out later, however, there is no need to be so restrictive as to always choose the real number field R since the same theory applies to any field k. This added generality can be critical in some applications of the theory, e.g., coding theory and systems governed by functional-differential equations). To illustrate the basic concepts, we consider a simple example. Example:

Industrial Production

Suppose we have three industries with output rates Xl' X2, X3' respectively. Say the outputs are measured in dollars per year. The output of each industry is used by itself, by the other two industries, and by the rest of the world-the consumers. Let aij be the value of product i required as input to produce one dollar's worth ofproductj, where i,j = 1,2,3. Let Yi be the rate at which consumers absorb the output of industry i, where i = 1,2,3. On the basis of these definitions, we have the relations 3

Xi

=

L

aijxj j= 1

+ Yi'

or, in vector-matrix form, X

= Ax

i = 1,2, 3,

+ y.

We may use the above model to determine the amount of industrial production x required to meet a given consumer demand Y, provided the

1.2

INTERNAL DESCRIPTION OF

1:

3

"technological coefficient" matrix A is known. Extensions and generalizations of the above setup form the foundation of what is usually termed "input/output" analysis in the economics literature. Often the matrix A is termed a "Leontief" matrix in honor of the founder of this branch of mathematical economics. In the language of our earlier discussion, this example has 0 = R 3 = F, f = (I - A) - 1, and the physical interpretation of the "machine" is that it is an industrial complex which transforms consumer demands into industrial products. The description of 1: given above is useful in some circumstances but is still quite limited. Among the important factors that such a setup omits are dynamical changes, stochastic effects,and, most importantly, the mechanism by which 1: transforms inputs to outputs. If the map f is interpreted as arising from physical experiments, the system analyst would like to know the "wiring diagram" of 1:, indicated by the part of 1: within dashed lines in Fig. 1.1, and not just the "black-box" behavior represented by f Consequently, we turn our attention to a description of 1: that allows us to deal with these questions.

1.2 INTERNAL DESCRIPTION OF 1:

To overcome the difficulties cited above, we introduce the concept of the "state" of 1: as a mathematical entity that mediates between the inputs and outputs, i.e., the inputs from 0 act on the state, which in turn generates the outputs in r. At this point it is important to emphasize the fact that the state, in general, is not a quantity that is directly measurable; it is introduced merely as a mathematical convenience in order to inject the notions of causality and internal structure into the description of 1:. There has been much confusion in the system modeling literature on this point and the source of much of the misunderstanding may be traced to a lack of attention to this critical point. The only quantities that have physical meaning are those that we can generate or observe, namely, the inputs and outputs. Another interpretation of the state (which is somewhat more intuitively satisfying) is that it is an amount of "information" which, together with the current input, uniquely determines the state at the next moment of time. Of course, this is a circular definition but does convey the intuitive flavor of the "state" concept. As noted above, it is probably safest just to regard the state as a mathematical construct without attaching any particular physical interpretation to it.

4

BASIC CONCEPTS, PROBLEMS, AND EXAMPLES

At this point we impose an additional assumption on :E. We assume that :E is finite-dimensional, i.e., that there exists a finite-dimensional vector space X of dimension n such that the following diagram is commutative:

f

Since nand r are spaces of m- and p-dimensional vector functions, respectively,the linear transformations G and H may be identified with n x m and p x n matrices, respectively. In order to account for the change in :E over time, we must also assume a linear transformation

F: X

-+

X

describing the law of motion in X for :E if no inputs are presented. Clearly, F may be identified with an n x n matrix. Putting these notions together, the internal (or state variable) description of :E is given by the system of differential equations

=

x(t)

F(t)x

+ G(t)u,

y(t) = H(t)x(t) .

(1.2)

in continuous time, or by the difference equations x(t

+

1)

=

F(t)x

y(t) = H(t)x

+ G(t)u,

(1.3)

in discrete time, where x E X", U E urn, y E yP, F, G, H, being n x n, n x m, p x n matrices, respectively. Here we have made the identifications U = B", Y = RP, and X = R", which we shall retain throughout the book, where U and Yare the spaces of input and output values, respectively. The connection between the internal description of E given by Eqs. (1.2) [or (1.3)] and the earlier external description of Eq. (1.1) is fairly clear. The input u(t) at a given time t is presented to:E and an output y(t) is observed. The external description/maps u(t) -+ y(t). On the other hand, in the internal description, y(t) is produced from u(t) by means of the differential (difference) equations (1.2) [(1.3)]. The internal description seems to contain far more information about :E than does the external description. In addition, the

1.2 INTERNAL DESCRIPTION OF 1:

5

dynamical behavior of 1: is easily accounted for through the concept of the state x and Eqs. (1.2) or (1.3) which governs its temporal behavior. Consider the following example of an internal description of 1:.

Example:

Water Reservoir Dynamics

The system is shown in Fig. 1.2, where rt(t), r2(t) are the rainfall inputs, Xt(t), X2(t), X3(t) the states of surface storage at locations 1-3 respectively, while the state of groundwater storage (including infiltration) is X4(t). The constant k is for surface water flow, while It and 12 are for infiltration. The

--@

r,ll l

RAINFALL INPUTS

u (t)

1,1"

.

~Y211)= .-~I

4

3

II)~

"J

£7

4,.

r2111~

l x -)(

Y,

II)

STREAMFLOW OUTPUT

\,;

9n u2 (l)

FIG. 1.2 Water reservoir network.

expression 13(X4 - X3) signifies the exchange between stream and groundwater. The outputs Yt, Y2 are the streamflow output and the contribution of groundwater to the streamflow respectively, and the quantities gttUt and gnu2 denote the water release. The continuity equations for this problem are

+ rt x2 = -12 x2 + r2 Xt = -/txt

x3 = 13(X4 -

:

0(0'

Also, assume that x(O)

= O. Then

lim Yet, t) = Z(SO)elsofU • t~o::

That is, M is the special case of Z where PROOF

So

is purely imaginary.

Left as an exercise.

As a result of Theorem 2.2 we see that the transfer function Z is a more salient object for system theory so, in subsequent chapters, we shall not return to the frequency-response function, although it is well to keep in mind the origins of Z in terms of the frequency response. EXERCISES

1. (a) Compute the transfer function associated with the system

(b) What is the significance of the fact that some of the elements of F, G, and H do not appear in Z(A)? 2. Show that if we allow direct output observation, i.e., y = Hx + Ju, then the transfer function Z(A) may have entries with the degree of the numerator equaling that of the denominator. (Hint: Consider Z(A) = J + H(M - F)-IG and expand in powers of IjA.) 3. Compute the transfer function matrix for the system

0 0 1 0 0 0 F= 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 1 0

0

1

1 0 G= 0 1 1 0

H =

[~

0 0

2.6 IMPULSE-RESPONSE FUNCTION

Given the linear system

x=

F(t)x

y = H(t)x,

+ G(t)u,

x(t o) = xo,

1 0 0 0 0 0

~l

32

2

MATHEMATICAL DESCRIPTION OF LINEAR DYNAMICAL SYSTEMS

it is easily verified that x(t)

=

+

r. The control

u = - G'(t)'(r, t)y will take (r, x) to (t, 0) as can be seen by substituting u into (3.3). (Necessity) Since W is symmetric, we have the orthogonal state space decomposition x = 9t'[W(r, t)] EB %[W(r, t)]. Because of linearity and the previous theorems, we need only show that no state in .¥[W(r, t)] is controllable. . Assume that x =F 0 E %[W(r, t)] and that the event (r, x) is controllable. Then we have 0= (x, W(r, t)x)

=

f"G'(s)'(r, s)x11 2 ds.

(Here we use the notation (', .) to denote the usual inner product in R".) Since the integrand is nonnegative, we must have G'(s)'(s, r)x

=0

for

almost all s E [r, t].

Since (r, x) is assumed controllable, we must have

x = - f(r, s)G(s)u(s) ds for some u. Thus

2 f(X, -(r, s)G(s)u(s)) ds.

0< IIxl1 =

However, the right side vanishes identically which yields the desired contradiction.

Corollary ~(r) = 9t'[W(r, t 1 ) ] , where t 1 is any value of t for which W(r, r) has maximal rank. The following is the corresponding reachability theorem.

Theorem 3.5 An event (r, x) is reachable some s < r, where W(s, r)

=

if and only if x E 9l[W(s, r)]

f(r, u)G(u)G'(u)'(r, u) da.

for

3

42

CONTROLLABILITY AND REACHABILITY

REMARKS (1) If G(·) is zero on (-00, r), we cannot have reachability, and if G(.) is zero on (r, (0), we cannot have controllability. (2) For F, G constant, W(r, t) = W(2r - t, r) and the integrals defining Jv, W depend only on the difference of the limits. Thus for constant l: an event (r, x) is reachable for all r if and only if it is reachable for one r ; an event is reachable if and only if it is controllable.

Since the rank condition on W (or W) may not always be easy to verify, we now present a sufficient condition for controllability of a time-varying linear system. This condition is much easier to verify than that of Theorem 3.4 but fails to be necessary. Theorem 3.6 Let F(t) E cr>. G(t) E en-Ion n x m matrix functions Qi(t) by

[0,00].

Define the sequence of

Qo(t) = G(t), Qi+ I(t) = F(t)Qi(t) - Qi(t),

i

=

0, 1, ... , n - I.

Then the linear system (3.1) is completely controllable at time ~(t)

t

if the matrix

= [Qo(t)/QI(t)1 .. ·IQn-I{t)]

has rank nfor some time t > r. PROOF

We have x(t) = {'P(r, s)G(s)u(s) ds,

(3.5)

where 'P(r, r) = - I.

o'Pjot = - 'P(t, s)F(t),

Integrating the right side of (3.5) by parts, we obtain n-I

x(t) = i~O (_I)i+ IQi(r)

t i

it t

+ (- l ]" t'P(s)Qn(S)

(s - r)i - i - '- u(s) ds

it s

(r - s)n-I

(n _ 1)! u(r)dr ds.

(3.6)

Hence, representation (3.6) shows that if the condition of the theorem is satisfied for some t 1 > r, then the rows of'P(r, t I)G(r)arelinearly independent over [r, t d, i.e., l: is controllable. EXAMPLE

Consider Hill's equation with a forcing term z(t)

+ (a - b(t))z(t) = g(t)u(t),

3.4

43

DISCRETE-TIME SYSTEMS

where b(t) is an analytic periodic function and a constant. Putting Xt{t) =

z, X2(t) = z, we have Xt(t) = X2(t), x2(t) = - [a - b(t)]xt(t)

+ g(t)u(t).

It may be that the periodicity of b(t) influences in some interesting way the controllability properties of (*). However, computing ~(t) we find

g(t)J

-g(t) ,

which implies that (*) is controllable at any r < t such that g(t) ¥- O. Thus the periodic system (*) apparently has no controllability properties which are a consequence of its periodicity. EXERCISES

1. Show that the condition of Theorem 3.5 is also necessary if F(t), G(t) are real analytic functions of t. 2. Show by example that the pair (F(t), G(t)) may be completely controllable for each fixed t, but still not be completely controllable. 3.4 DISCRETE-TIME SYSTEMS

Now we briefly consider the discrete-time, nonstationary system (3.2) and the associated reachability and controllability questions. In connection with this investigation, we define a real n x n matrix-valued function cp by . {F(k)F(k - 1)· .. FU cp(k,}) = I,

+

I)FU),

undefined,

k ?: i. j = k + I, j>k+1.

The function cp is the discrete-time analog of the continuous-time function . We have . x(k; ko, xo, u)

=

cp(k - 1, ko)xo

+

k-t

I

cp(k - l,j

+

I)GU)uU).

j=ko

We observe that in contrast to the continuous-time situation, it is possible for the set of all solutions to (3.2) with u == 0 to lie in a proper subspace of R" (e.g., let F(k) == 0). Thus the possibility of such pointwise degeneracy means that theories developed for continuous-time systems (where such degeneracies cannot occur) may not be in 1-1 correspondence with their

44

3


discrete-time analogs. The main reachability/controllability result for discrete-time systems is given in the following theorem.

Theorem 3.7 A necessary and sufficient condition for (3.2) to be completely reachable at time i in M steps is that rank[G(i - 1)1q>(i - 1, i - I)G(i - 2)1" ·Iq>(i - 1, i = n. PROOF

+M +

I)G(i - M)]

(Sufficiency) Let

Rk(i - 1) = [G(i - 1)1q>(i - 1, i - I)G(i - 2)1" '!q>(i - 1, i - k + I)G(i - k)], and suppose rank RM(i - 1) = n. Then the solution to (3.2)at time i, starting in the zero initial state at time i - M, is

l

u(i - 1)

x(i; i - M, 0, u)

= RM(i -

1) u(i

~

2)

j

~

RM(i - I)U M(i).

(3.7)

u(i - M)

Define the n-dimensional vector VM(i) by the relation

Then

which shows that we can solve for VM(i) and obtain the appropriate control sequence needed to reach any given XM(i). (Necessity) Suppose rank RM(i - 1) < n but that (3.2) is completely reachable in M steps at time i. Then there exists a vector '7 =F 0 in R" such that '7'R M (i - 1) = O. Hence, premultiplying both sides of (3.7) with '7' gives '7'x(i; i - M, 0, u) = 0 for any u. Since (3.2)is completely reachable at time i, we choose the control sequence {u(i - M), ... , u(i - I)} such that xU; i - M, 0, u) = '7. Then '7''7 = 0, contradicting '7 =F O. REMARKS (l) The criterion of Theorem 3.7 is also a sufficient condition for complete controllability of(3.2) at time i - M. However, it is not necessary unless F(k) is invertible on the interval i - M + 1 ~ k ~ i-I. This is the pointwise nondegeneracy condition referred to above. (2) The proof of Theorem 3.7 shows that complete reachability at time i in M steps implies the ability to reach any fixed state at time i from any given state (not just the origin) at time i - M.

3.4

45

DISCRETE-TIME SYSTEMS

(3) Complete reachability in M steps at time i implies complete reachability in N steps at time i for all N ~ M. This is false if reachability is replaced by controllability unless F(k) is invertible for all k ~ i + M. Example: National Settlement Planning An area in which discrete-time reachability questions play an important role is in governmental planning for national settlement policies. Several different approaches have been proposed for dealing with this sort of problem, some of them falling into the basic framework considered in this chapter. We describe one of these "system-theoretic" approaches. The government objective is to promote a desired migratory process by differential stimulation of the employment market. The state equations for the model are x(t v(t

+ 1) = + 1) =

Kx(t) Mv(t)

+ (I - M)v(t), + u(t) + z(t),

where the vector x(t) E W represents the population distribution at time t, v(t) E R" the distribution of job vacancies at time t, u(t) E W the distribution of government stimulated job vacancies, and z(t) E R" the distribution of spontaneously occurring vacancies. The matrix K is a diagonal matrix whose elements reflect the natural population growth rates within a region, while M is a migration matrix with elements mij being the probability that a job vacancy in regionj will be filled by someone living in region i; i.] = 1, ... , n. The problem, of course, is to choose u(t) so that x(t) (and possibly v(t)) follow some desired course. The budgetary and fixed immigration constraints on the choice of u(t) are given by (i) u(t) ~ 0, (ii) (u(t), r(t» ~ b, (iii) Ilu(t)11 ~ (I, t = 1,2, ... , T. Here 11·11 denotes some appropriate vector norm (e.g., [1), with r(t) being a function giving the total resource (jobs) available to be offered regionally by the government at period t, b being the total number available. By introducing the new vectors s(t)

=

L~t)J

w(t)

X(t )]

= [ v(t)

,

y(t) =

[z~)J

it is possible to rewrite the above model in the form w(t

+

1)

=

Fw(t)

+ Gs(t) + y(t),

46

3


where

Constraints (i)-(iii) restrict the region of admissible inputs s(t). Actually, on the basis of more detailed analysis, for purposes of determining reachable sets it suffices to replace inequalities (ii) and (iii) by the corresponding equality. (Physically, this fact is fairly obvious but requires a surprising amount of analysis to prove.) However, the restriction of the admissible inputs does impose added mathematical complications in determination of the reachable set. We shall take up some of these "complications" of the basic problem in a later section. An interesting question appears if one considers the discrete system (3.2) as arising from (3.1) by discretizing the time axis. For this case, in many physical processes it is natural to assume that the input u is a piecewiseconstant function, i.e., u(k)=<J.n ,

n

s. k

c n

»

o,

n,k=1,2, ....

Such an assumption is called sampling in the engineering literature since the continuous system's input and output are "sampled" at discrete-time instants of length (J. Suppose that the continuous system is completely controllable. Since the introduction of sampling is a restriction on the controls, it cannot possibly improve controllability. The question is "can sampling destroy controllability?" The answer is given by the next theorem. Theorem 3.8 Suppose that system (3.1) is constant and completely controllable. Then a sufficient condition for the sampled system (3.2) arising from (3.1) to be completely controllable also is that Im[Aj(F) - AiF)]

=1=

2nqj(J,

q

= ±1, ±2, ....

where

Re[Aj(F) - AiF)] = 0, lfm = I, this condition is also necessary. PROOF

See the references cited at the end of the chapter.

The intuitive meaning of the above result is that the periodicity inherent in sampling must not interact with the natural frequencies of the system to be controlled if controllability is to be retained. Note also that the condition can always be satisfied by choosing the sampling frequency Ij(J sufficiently large, i.e.,making a sufficientlyclose approximation to the original continuous system.

3.5

47

CONSTANT SYSTEMS

3.5 CONSTANT SYSTEMS

Often in problems of practical concern we must assume that the system under investigation is constant, since otherwise it may be impossible to determine a model for it: in an arbitrary nonconstant system, past measurements may reveal nothing about future behavior. As one might suspect, restricting our attention to constant systems results in major simplifications in the form ofthe foregoing results and also suggests several new subproblems which we shall investigate. In addition to the practical and computational aspects associated with a thorough study of the constant case, a detailed understanding of constant systems serves to point out what features to expect from a general theory of controllability and to suggest appropriate methodological tools for such a development. Thus there are several compelling reasons for a careful analysis of constant linear systems. The simplest and most useful criterion for complete controllability for a continuous-time, constant linear system ~ is Theorem 3.9. Theorem 3.9 The constant linear system (3.1) is completely controllable and only if the n x nm matrix

if

has rank n. Corollary 1 If ~ is completely controllable, then for any r, x, and s > 0, there is an input u that transfers the event (r, x) to (r + s.D), PROOF Let rc have rank n and assume ~ is not completely controllable. Thus there exists a state XI "# 0 such that XI E %(W(O, tl))' We fix t l and show, as in Theorem 3.4, that

since

Differentiating this relation n - 1 times with respect to t, and setting t = t I yields i = 0, 1, ... , n - 1,

(3.8)

which implies that X I is orthogonal to the columns of rc. But rc has n linearly independent columns. Thus Xl = 0, a contradiction for the chosen t r- But, since t 1 was arbitrary, ~ is controllable over any interval [0, t 1]'

48

3


To prove the converse, let 1: be controllable and assume rank CC < n. Then there exists a nonzero vector Xl that satisfies (3.8). By the HamiltonCayley theorem, (3.8)is actually true for all i. Therefore,

which implies that x ' E A'TW(O, t 1)], contradicting the controllability of 1:, since W must be nonsingular (Corollary to Theorem 3.4). REMARK Corollary 1 may seem somewhat surprising to practically minded engineers since it is a well-known fact that real control systems cannot be controlled in arbitrarily short intervals of time. However, the difficulty is one of practice and not theory. Theorem 3.9 assumes that the matrices F and G are known to an arbitrary degree of accuracy, in fact, that they are known exactly and the effect of the control input u is immediately felt by the system L, i.e.,there are no uncertainties or time lags in the control loop. When we apply the theory, we must be concerned about the accuracy of our data and the relevance of the simultaneity assumption before appealing to Theorem 3.9. EXAMPLES

(1) Consider the system

F =

[~ ~J

G=

[::1

We investigate the conditions on gl, g2' that imply controllability. The matrix CC is

which is of rank 2 if and only if g2(2g l - g2) -# O. (2) A less trivial example is provided by considering the longitudinal equations of motion for a VTOL-type vehicle hovering at constant altitude. These are

d dt

Ug

-O(g

U

Xu

x

q

e

o u, o

0 Xu 1

Mu

0

o

o

0 0 0

o o u,

o

1

o -g

0

Ug

0

U

o x o q o e

wl

+

0

0 15+

0

M6

0

0

0

3.5

49

CONSTANT SYSTEMS

where ug the longitudinal component of gust velocity, u velocity perturbations along the x-axis, x position along the x-axis, q pitch rate, () pitch attitude, [) control stick input, M s control stick amplification factor, M u speed stability parameter, M 9 pitch rate damping, M q control sensitivity, Xu longitudinal drag parameter, g gravitational constant, IXg wind gust break frequency, WI longitudinal component of wind speed. Clearly, the controllability of this process is unaffected by the wind speed. Thus the relevant controllability matrix is '{;=

[~.

0 0 0 M.M.

M.

0

0

-gM.

0 M"M. M.M.

0

-gM.(X u

-

M.l

-gM.(X u

-gM. 2

-gMhMu

+ M.M/

M.M/

-

M.)(X u

-gM.(X u -gM.(X u

-

M.lM u

-

M.( -gM u

-

+ M/l

M.l

M.M.(gM" - M/l

+ M/l

Hence, we see that the aircraft system is not completely controllable for any values of the parameters IXg , Xu, M u , M q , a situation more or less obvious from a glance at the dynamics, as ug is unaffected by the other states and by the control. Some additional useful corollaries of Theorem 3.9 follow. A constant system L ~ (F, G, -) is completely reachable if and only if the smallest F-invariant subspace of X containing the columns of G is X itself.

Corollary 2

Corollary 3 A constant system L = (F, G, -) is completely reachable if and only if there is no nontrivial characteristic vector of F which is orthogonal to every column of G. Corollary 4

The set of reachable states of L is a subspace of X. More precisely, the set of reachable states is the subspace of X generated by the columns of

-«

3

50 Corollary 5


Consider the extended controllability matrix

-F

0 I

0

-F

0 0

0 0

I

't=

2

of dimension n x n(n if ~ has rank n 2 •

+m

0 0 0

0 0

I

0

G

-F

G

0

0 0 0

0

0 0

0

G

G

G

0

0 0

0 0

0 0

0 0

- 1). Then .E is completely reachable

if and only

PROOF In the matrix 16, add F times the first block row to the second, then add F times the new second block row to the third, and so on. The result is a block matrix similar (by column operations) to

[

1,._ ".

:],

On.n(n- I)

which has rank equal to rank 0,

where

9 =

G)'

Fg =

(~}

Thus only interior points of the fourth quadrant are positively controllable to the origin.

3.6

53

POSITIVE CONTROLLABILITY

(3) The system 'X I

x2

cos () + X2 sin () = - X I sin () + x 2 cos ()

= XI

a
1) to be positively completely controllable is that the set of vectors (-I)P+ J FP gv, fJ = 0, 1, ... , v = 1, ... , m, form a nonnegative basis for R", where gj denotes the ith column of G, i = I, ... , m. (Definition: The vectors bl, b 2 , •.• .b, form a nonnegative basis in the subspace that they generate if any vector 5 from the subspace may be written as

,

fj =

L fJi b;,

j~

1

fJi

~ 0.)

2. Show that every controllable system may be made positively controllable by adding an additional control um+ I with vector gm+ I = - Li~ I cxig;, CXi > O.

3.7

55

RELATIVE CONTROLLABILITY

3.7

RELATIVE CONTROLLABILITY

Occasionally the requirement of being able to transfer L from any initial state xo to the origin is too strong. What we actually desire is to transfer x o to some appropriate subspace X' of X. For example, X' may represent some set of equally desirable states, a set of terminal states, and so forth. More precisely, we have the following definition.

Definition L is called controllable relative to the subspace X' = {x: Kx = O} if for every state Xo, there exists a number i < rx) and a piecewise continuous control u(t), 0 ~ t ~ i, such that Kx(t) = O. Since every controllable state Xo has the representation m

Xo =

n-l

L1 LOIlki F igb

k=

i

(3.10)

e

the states controllable to X' at t are given by KrjJ(t)xo

=

m

n-l

L L Ilki K rjJ(t )F ig

k>

k= 1 i=O

where rjJ(t)xo is the state at time t if L begins in state Xo. Since rjJ(t) is nonsingular for all t, we immediately have Theorem 3.15. Theorem 3.15 and only if

The system L is controllable relative to the subspace X' rank[KGIKFGI" ·IKFn-1GJ

if

= rank K,

i.e.

rank KC(J = rank K. EXERCISE

1. Show that in the case of an (n - I)-dimensional subspace X': k'x = 0, the condition of Theorem 3.15 takes the simpler form i = 0, I, ... , n - I,

Example:

j = L 2.... , m.

Urban Traffic Flow

The problem of regulating the flow of urban traffic in a street or freeway network provides a good illustration of a situation in which the notion of relative controllability may playa role. Consider the rectangular network depicted in Fig. 3.3. We assume that the network is oversaturated, i.e., at one or more intersections traffic demand

3

56

FIG. 3.3


Urban traffic network.

exceeds capacity. Let Xj(t) be the number of cars waiting at intersection i, and let u;(t) denote the number of cars leaving intersection i during the green light. If we assume that the travel time between two intersections is small compared to the waiting time, then the dynamics of the process are reasonably well described by the equations x(t

+ 1) =

x(t)

+

Gu(t)

+

q(t),

where the vector q(t) has components qj(t) representing the external traffic arriving at intersection i during period t. It is clear from Fig. 3.3 that the flows u 3' U6' U9' and u 1 0 are flows out of the network. The control matrix G takes the form

-1 51

G=

0 0 0 0 0 0 '1

0

0 -I 52

0 0 0 1"2

0 0 0

0 0 -1 0 0 0 0 0 0 0

0 0 0 -1 54

0 0 0 0 1"4

0 0 0 0 -1 55

0 1"5

0 0

0 0 0 0 0 -1 0 0 0 0

0 0 0 0 1"7

0 -1 0 0 57

0 1"8

0 0 0 0 0 -1 58

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 - -1

The elements I"j and s, denote the percentage of cars turning, either right or left, and going straight ahead, respectively.

3.8

57

CONDITIONAL CONTROLLABILITY

On psychological grounds, it is reasonable to impose the control constraints

u,

~

Uj(t)

u;

~

i

=

1, ... ,10,

where M i and Vi represent the minimal and maximal number of cars that may move during a given psychologically acceptable green time. The basic problem is now quite simple: given an initial state x(O), assumed to be an oversaturated condition, is there a control policy u(t) which transfers x(t) to an undersaturated region within a prescribed time T? Thus we see that the subspace f is chosen to be the smallest subspace of X containing the set of undersaturated states of X.

3.8 CONDITIONAL CONTROLLABILlTV

We have already seen that the set of controllable states forms a subspace that is generated by the columns of ctJ. In many cases, however, we are given a particular subspace .if of X and would like to determine whether or not every state in .Ii is controllable and, moreover, we desire a simple test to answer this question. This situation is particularly important in those processes for which we have either a priori knowledge about or influence over the initial state Xo and can assert that it belongs to some distinguished subspace of X. From the representation of controllable states (3.10), the following criterion for conditional controllability is obtained. Theorem 3.16 The system L is conditionally controllable from a subspace j{{xo = My, y E X} if and only if

rank[M I G I FG I·· ·\Fn - 'G]

= rank[G I FG I·· '!Fn-1G],

i.e.,

rank[MI'1&"] EXAMPLE

Let L = (F, G, -), where 1 -I

-1 -5

o

°2 -1

= rank '1&".

3

58


Assume that the initial states lie in the plane X4 = 0, X3 space AI is given by those points x = My, y E R 4 , where

M=[~

o

= O. Then the sub-

! H]. 0 0 0

Substituting these entries into Theorem 3.16, we see that the rank condition is satisfied. Thus ~ is conditionally controllable from AI. Notice, however, that had we chosen the plane x I = X 2 = 0 as the region of interest, then the rank condition would fail. 3.9 STRUCTURAL CONTROLLABILITY

In an attempt to obtain a more realistic methodology for studying system structure, we now turn to controllability questions that are dependent only on the internal connections of the process under study, and not on the specific numerical values of the system parameters. For definiteness, let us assume that the entries in the system matrices F and G are either fixed zeros or arbitrary nonzero parameters. Such a supposition is more consistent with reality since, in practice. system parameter values are never known exactly; however, the positions of fixed zeros in the structure are often known due to choice of a particular coordinate system (e.g., time derivative of position is velocity, etc.) or because physical connections between certain parts of a system are absent. From a computational point of view, the assumption of "fuzzy" parameters and "true" zeros is also desirable since digital computers can only recognize integers with exact precision. Hence it is of interest to study basic system properties that rely on numerical decisions of only the "zero/nonzero" type. We now outline an approach to the study of controllability utilizing such ideas. We first define precisely the notions of "structure" and "structural questions." Definitions A structural matrix F is a matrix having fixed zeros in some locations and arbitrary, independent entries in the remaining locations. A structured system ~ = (F, G, -) is an ordered pair of structured matrices. The two systems ~ = (F, G, -), f = (F, G, -) are structurally equivalent if there is a one-to-one correspondence between the locations of their fixed zero and nonzero entries. A system ~ = (F, G, -) is called structurally controllable if there exists a system structurally equivalent to ~ which is controllable in the usual sense.

3.9

59

STRUCTURAL CONTROLLABILITY

The foregoing definitions show that if there are N arbitrary nonzero entries in 1:, then associated with 1: is the parameter space R N and every system structurally equivalent to 1: is represented by a datum point r ERN. The properties of interest for this discussion will turn out to be true for all data points except those lying on an algebraic hypersurface in RN . To be more precise, consider a finite set of polynomials t{Jj E R[A], A = (AI"'" AN)' Then the point set V of common zeros of t{Jj(A) in RN forms what is called an algebraic variety. If V ¥ R N , then V is called proper and if V ¥ 0, V is nontrivial. A data point r E R N is typical relative to V if r E Vo, the complement of V in R N and any property n determined by the data points of R N is called generic if the set of data points for which n fails to be true forms a proper algebraic variety in R N • Thus generic system properties can be expected to hold for almost every data point in R N , i.e., they hold for all typical data points. EXAMPLE

Consider the n x nm controllability matrix C(j

=

[GIFGI· .. IP-IG].

Let t{J(A.) be the polynomial in N = n2m indeterminates AI, ... , AN defined as the sum of the squares of all possible nth order minors of C(j. Clearly, any data point r E RN such that t{J(r) = 0 implies that rank C(j < n, i.e., that 1: is not completely controllable. Hence, to show that controllability is a generic property, we need only show that V

is a proper variety in R arbitrarily.

N

•

=

{r: t{J(r)

= O}

But, this is trivial since the entries of~

can be chosen

The above example shows that it is the inclusion of" structure" into the controllability problem that makes it possible for complete controllability to fail, independently of parameter values. We now introduce the basic technical condition for studying structural controllability. Definition An n x s matrix A (s ~ n) is said to be of form (r) for some I ~ t ~ n if, for some k in the range s - t < k ~ s, A contains a zero submatrix of order (n + s - t - k + I) x k.

For example, the matrix

o

o

0 0 0 0 0 0 0 0

x x x x x x x x x x x x x x x

60

3


is of form (4) with k = 5, while x 0 0 0 0 x 0 000 x

0

0

0

0

x x x x x x x x x x is also of form (4), but with k = 4. The importance of form (1) is then seen in the following basic result. Lemma For any t, I .::;; t < n, rank A < t for every r ERN has form (r), PROOF·

if and

only

if A

See the references at the end of the chapter.

The connection between structural controllability and form (t) is now given by Theorem 3.17. Theorem 3.17 The system 1: = (F, G, -) is structurally uncontrollable and only if the extended controllability matrix I

0

0 0

-F

G

I

0 0

0

0

-F

G

0 0

0 0

0 0

G

?6'=

is ofform (n

2

0

if

o o -F G

o

0

I

0

-F G

).

PROOF The result follows from Corollary 5 to Theorem 3.9 plus the fact that 'ti has rank n 2 (generically) if and. only if ~ is not of form (n2 ). This last fact follows from the preceding lemma.

Computationally, the importance of this result cannot be overemphasized since the determination of the form of a matrix requires only that the computer be able to distinguish between zeros and nonzeros. Thus, in contrast to the usual controllability result that depends on finding the rank of f(j ~ notoriously unstable numerical operation), determination of the form of ?if or, equivalently, finding its generic rank, may be carried out with no numerical error.

3.10

61

CONTROLLABILITY AND TRANSFER FUNCTIONS

3.10 CONTROLLABILITY AND TRANSFER FUNCTIONS

Recall that the transfer function of the linear time-invariant system ~ (=(F,

G,

-»

x=

Fx

+ Gu

is given by the polynomial matrix

assuming, that x(O) = O. An interesting and important question to ask is: How we can deduce simple conditions on Z that imply the controllability of E or, conversely, given that ~ is controllable, what structural features does this impose on Z? As Z is the Laplace transform of the matrix eFtG (since x(t) = J~ eF(t-S)Gu(s) ds), it is natural to conjecture that, in view of the controllability condition given by the rank of'b', a similar type of linear independence condition on Z will be the appropriate "frequency-domain" version of Theorem 3.9. As substantiation of this "hunch," we have the next theorem. Theorem 3.18 The system ~ is completely controllable rows of(J..I - F)-IG are linearly independent.

if and only if the

(Necessity) Let ~ be controllable, fix t, and let VI, v2 , •.. , vn be the linearly independent rows of the matrix eFtG. Let VI, V 2 , ••• , vn be their transforms. Assume these transformed rows are dependent, i.e., there exist constants CI, C2"'" c., not all zero, such that PROOF

for all J. not equal to a characteristic value of F. Thus the vectors are dependent for all t since the Laplace transform is invertible (its null space is the zero vector). This implies the vectors VI, v2 , ••• , un are also dependent contradicting the original assumption. Hence (J..I - F)-IG has linearly independent rows. . (Sufficiency) Sufficiency is demonstrated in a similar fashion by reversing the above argument. SPECIAL CASE If ~ has a single input (m = I), Theorem 3.17 simplifies to the condition that no entry of the vector (J..I - F) - 19 is reducible, i.e., if

i

= 1,2, ... ,n,

is the ith entry of (AI - F)-I g, then Pp.) and q(J.) (the minimal polynomial of F) have no common factor, i = 1, 2, ... , n.

62

3 EXAMPLE

Let 1: be given by

=

F

[~

Then (fj =

Since rank of 1: is

(fj


-2J g=Gl G-2J

-3 '

[gIFg] =

I

-1 .

= 1 < n,1: is not completely coritrollable. The transfer function

Z(2)

2(2 + 2) + 1)(2 + 2) = . ). + 2 [ (2 + 1)(2 + 2) (2

Hence, cancellations occur that again show that 1: is not controllable. 3.11 SYSTEMS WITH A DELAY

A valid criticism that is often voiced against the use of ordinary differential equations to model real control systems is that such a model assumes that the action of a given controlling input is "instantaneously" felt by the system. It is manifestly true that this situation never occurs for any real system: control takes time! Thus the validity of an ordinary differential equation model is highly dependent on the time constants of the process. Even though our objectives in this book are to deal with the pure differential (or difference) equation case, we now offer a brief excursion into the differential-delay equation world in order to exhibit some of the features of these processes. It will be seen that for several basic questions the results and methods parallel the" instantaneous" case. Consider the single-input system with time lag

x=

Fx

+ Bx(t

- r)

+ gu(t),

(3.11 )

where F, B, g are constant matrices of appropriate sizes. We further assume that B is expressible in the form B = gc'. Thus the columns of B are collinear with g. To motivate the above class of linear systems, notice that if the nth order equation

3.11

63

SYSTEMS WITH A DELAY

is written in vector form, we obtain

y(t) = Fy(t)

+ By(t

- r)

+ ilu(t),

0 0

0 0

0 0

0

0

where

F=

0 0

1 0

0 1

0

0

0

!XZ

!X3

B=

g=

P.

pz

Hence, the columns of B are collinear with g. Since a differential-delay system is defined by prescribing an initial function over the interval [- r, OJ, rather than by giving only the value x(O), we must slightly modify our definition of controllability. The new definition follows.

Definition System (3.11) is completely controllable if for every T> 0, and for every piecewise-continuous function cp defined on [ - r, OJ, there exists a piecewise-continuous control u such that x(t) vanishes on [T, T + r]. Notice that the condition B = gc' implies that for complete controllability it suffices to have x(T) = 0 since we can choose the control u defined by u(t) = -c'x(t - r)fort > T.Then the system will reduce to x = FX,x(T) = 0, for t > T.

Theorem 3.19 System (3.11) is completely controllable if and only if(F, g) is completely controllable (i.e., if the system x = Fx + gu is controllable). (Sufficiency) If _(3.11) is completely controllable by taking T < r, we see that for every initial function cp there exists a control u such that we shall have x(T) = 0 for the system PROOF

o

fl'

II

u*(t) 11 2 dt,

to

unless u(t) = u*(t) almost everywhere on [to, t l ] (11·11 denotes the inner product norm for the finite-dimensional vector space Q). (b) Show that the minimum control energy E necessary to transfer Xo to Xl (assuming such a transfer is possible) is given by

66

3


7. (a) Let I: be a linear time-varying system with 0 a compact, convex set in R". Show that if x(to) = xo, the reachable set 9l(t 1) at time t 1 ~ to is compact, convex, and continuously dependent on t 1(b) (Bang-Bang Principle) Let 0 0 be a compact subset of 0, the convex hull of which coincides with the convex hull of O. Let 9l o(t1) be the reachable set for u E 0 0 , Show that 9l o(td

= 9l(t 1).

(Thus, if 0 0 = 00, only controls in 0 0 need be examined to determine the reachable set at time t 1') 8. Let I: be a time-invariant system such that 0 is a bounded set containing u = O. Prove that the set of controllable states is open if and only if I: is completely controllable. 9. Consider the nth order time-invariant linear system

Po

i= O.

Show that this system is completely controllable. The time-invariant system I: such that

10.

x=

Fx

+ qu,

is called controllable with an arbitrarily small control iffor any e > 0, and any two states X O and x', there exists a control u(t), satisfying lu(t)1 ~ s, which transfers I: from X O to Xl in a finite interval of time. Show that I: is controllable with an arbitrarily small control if and only if (a) I: is completely controllable. (b) The characteristic roots of F are purely imaginary. 11. (a) Consider the equations of motion of a point lying in a fixed plane and moving with a given circular orbit. The system dynamics are

x=

ra",(t)~,

m

where m = mo + m 1(t ), v is the gravitational constant, r the radius vector of the point, X the generalized momentum corresponding to the polar angle t/J, and ar(t), a",(t) the projections of the velocity vector relative to the radius direction and the direction orthogonal to it, respectively. Let z l' Z2, Z3 be the deviations of the state coordinates from their values along the circular orbit, i.e., Zl = r - rO,z2 = r,z3 = X - Xo·Formulatethe linear system for the variables Zl' Z2' Z3' (Hint: A change of coordinates is useful.) (b) Is the linearized system completely controllable?

67

MISCELLANEOUS EXERCISES

12. (a) Let I: be a time-varying system such that the elements of F and G are analytic and periodic functions of t with period w. Prove that for I: to be controllable on [0, t 1] it suffices that the matrix ~(w)

=

[G(O)leFwG(O)I·· 'leFn-'WG(O)]

have rank n. (Hint: Use Floquet's theorem. See Miscellaneous Exercise 2 in Chapter 7.) (b) Show by counterexample that the analyticity assumption on F and G may not be relaxed. (Hint: Consider the scalar system

x= -

tiJ(t) [I

+ t/!(t)Rr 1Rx + qu,

where R is an n x n constant matrix with sufficiently small elements, g a constant n-vector, and t/!(t) a scalar function of period to = 1 whose graph is shown in Fig. 3.4. Assume that g, (I + R)-l g, ... , (I + R)-(n-1)g are linearly independent. Now show that this system satisfies the above rank condition on ~(w) but is still not completely controllable.)

• t

o

2

FIG. 3.4 Graph of l/J(t).

13. Consider the time-varying linear system I: such that

x = a(t) [Fx + Gu], where F, G are constant and a is continuous and bounded. Show that if rank ~ = [GIFGI·· ·IFn -

1G]

= n,

then I: is completely reachable. 14. If the time-invariant system

x = Fx + Gu is controllable, then show that there exists a matrix (feedback law) K such that

x=

(F - GK)x

+ gjv(t)

is also controllable where q, is any nonzero column of G. (Here K depends, in general, on g;.)

68

3 CONTROLLABILITY AND REACHABILITY

15. If F is an n x n constant matrix, show that the two matrices [GIFGI·· ·IFn-1G]

and

have the same range spaces. 16. Suppose that for all constant k, 0 det[g

~

k

~

1, and a fixed vector e we have

+ kelF(g + ke)I·· ·IP-l(g + ke)]

=1=

O.

Does it then follow that

will be positive definite for all t 1 > 0 and all k(t) such that 0 ~ k(t) ~ 1? 17. Consider the transfer function matrix (U - F)-lG = 2(..1). Show that a necessary and sufficient condition for 2(..1) to be invertible is that the matrix

[! have rank in

+

G FG 0 G

Fn-1G P- 2G

FnG Fn-1G

0

G

2 nG

F F2n~

1

IG

P-1G

1)m. NOTES AND REFERENCES

Section 3.1 Many additional examples and insights into the controllability question are discussed in the paper by Kalman, R., Mathematical description oflinear dynamical systems, SfAM J. Controll, 152-192 (1963).

Section 3.2 The basic definitions and their algebraic implications are extensively examined by Kalman, R., "Lectures on Controllability.and Observability." C.I.M.E., Bologna, Italy, July 1968; Edizioni Cremonese, Rome, Italy, 1969, pp. 1-149.

The economic example has been adapted from McFadden, D., On the controllability of decentralized macroeconomic systems: The assignment problem. in .. Mathematical Systems Theory and Economics" (H. Kuhn, ed.), Vol. II. Springer-Verlag, Berlin and New York, 1969.

Section 3.3

The results follow

Kalman, R., "Lectures on Controllability and Observability." C.I.M.E., Bologna, Italy, July 1968; Edizioni Cremonese, Rome, Italy, 1969, pp. 1-149.

NOTES AND REFERENCES

69

For a treatment of controllability for time-varying coefficient matrices (Theorem 3.6) see also Gabasov, R., and Kirillova, F., "The Qualitative Theory of Optimal Processes." Nauka, Moscow, 1969. (Eng\. trans\., Dekker, New York, 1976.)

Section 3.4 The controllability/reachability results for discrete-time systems are taken from Weiss, L., Controllability, realization, and stability of discrete-time systems, SIAM J. Control 10, 230-251 (1972).

A more detailed treatment of the national settlement strategy problem can be found in Mehra, R., An optimal control approach to national settlement system planning, RM-75-58, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1975.

The sampling result, Theorem 3.8, was first proved by Kalman, R., Ho, Y. C., and Narendra, K., Controllability of linear dynamical systems, Contr. Diff. Eqs. I, 189-213 (1963).

Section 3.5 An interesting survey of the historical origins of the controllability concept and the genesis of the basic results is presented by Kalman, R., "Lectures on Controllability and Observability." C.I.M.E .. Bologna, Italy. July 1968; Edizioni Cremonese, Rome, Italy, 1969, pp. 1-149.

The VTOL example is taken from Kleinman, D., Baron, S., and Levison, W., A control-theoretic approach to manned vehicle systems analysis, IEEE Trans. Automatic Control AC-I6, 824--832 (1971).

Section 3.6

The problem of positive controllability is treated by

Gabasov, R., and Kirillova, F., "The Qualitative Theory of Optimal Processes." Nauka, Moscow. 1969. (Engl. trans\.: Dekker, New York, 1976.)

Somewhat more general versions of the positive controllability question are given by Brammer, R., Controllability in linear autonomous systems with positive controllers, SIAM J. Control 10, 339-353 (\ 972).

See also Maeda, H., and Shinzo, K., Reachability, cbservability and realizability of linear systems with positive constraints, Elec. Comm Japan, 63,35-42 (\982). Schmitendorf, W. and Barmish, 8., Null controllability of linear systems with constrained controls, SIAM J. Control & Optim., 18, 327-345 (1980). Perry, T. and Gunderson, R., Controllability of nearly nonnegative linear systems using positive controls, IEEE Tran. Auto. Cont., AC-22, 491 (1977). Heymann, M. and Stern, R., Controllability of linear systems with positive controls: geometric considerations, J. Math. Anal. Applic., 52, 36-41 (1975).

3

70

CONTROLLABILITY AND REACHABIUTY

It is important to emphasize the point that the positive controllability

results above only ensure that the origin can be attained in some finite time. This is in contrast to the standard situation in which complete controllability in some finite time implies controllability in an arbitrarily short time. For results on arbitrary-interval positive controllability, see Jacobson, D., "Arbitrary Interval Null-Controllability with Positive Controls," Council for Scientific and Industrial Research, Pretoria, South Africa, 1976 (preprint).

Sections 3.7-3.8

The theory of relative and conditional controllability is

covered by Gabasov, R., and Kirillova, F., "The Qualitative Theory of Optimal Processes." Nauka, Moscow. 1969.

The urban traffic example, along with other control-theoretic aspects of traffic flow is treated in the report by Stroebel, H., Transportation, automation, and the quality of urban living, RR-75-34, International Institute for Applied Systems Analysis. Laxenburg, Austria, 1975.

The concept of structural controllability was first introduced using graph-theoretic arguments by

Section 3.9

Lin, C. T., Structural controllability, IEEE Trans. Automatic Control, AC-19. 201-208 (1974).

Our approach follows Shields, R. W.• and Pearson. J. 8., Structural controllability of multi-input linear systems, IEEE Trans. Automatic Control, AC-21, 203-212 (1976).

This paper also includes an algorithm, suitable for computer implementation, which may be used to determine form (t) for an arbitrary n x s matrix A. It is of some interest to note that the basic mathematical results used to establish the lemma on form (t), as well as the proof of Theorem 3.17, are found in much earlier papers of Frobenius and Konig. See Konig, D., Graphak es Matrixok, Mat. Lapok. 389, 110-119 (1931). Konig, D., "Theorie der endlichen und unendlichen Graphen." Leipzig, 1936. Frobenius, G .• Uber Matrizen mit nicht negativen Elernenten, Berlin Akad. 23,456-477 (1912).

A fundamental dictum of practical systems analysis is that "control takes time." Thus differential-delay equations are the real substance of applied systems analysis. Such a principle is especially apparent in problems from the social sciences where long time lags are more likely to be encountered than in engineering or physics. A good account of some of these matters is given by Section 3.11

El'sgol'ts, L., and Norkin, S., "Introduction to the Theory and Application of Differential Equations with Deviating Arguments." Academic Press, New York, 1973.


71

Another basic work is by Bellman, R., and Cooke, K., .. Differential-Difference Equations." Academic Press, New York, 1963.

The controllability result cited in the text may be found in Halanay, A., On the controllability of linear difference-differential systems, in" MathematicalSystems Theory and Economics-II" (H. Kuhn and G. Szego, eds.), Vol. 12. SpringerVerlag, Berlin and New York, 1969.

See also Mallela, P., State controllability (and identifiability) of discrete stationary linear systems with arbitrary lag, Math Modelling, 3, 59-67 (1982). Artstein, L., Linear systems with delayed controls: A reduction, IEEE Tran. Auto Cant., AC-27, 869-879 (\982). Klarnka, J., On the controllability of linear systems with delays in the control, IntI. J. Control, 25, 875-883 (1977).

CHAPTER

4

ObservabilityIConstructibility

4.1 INTRODUCTION

Most modern control processes operate on the basis of feedback control mechanisms, i.e.,the controlling inputs to the system are generated by values of the state. Consequently, they implicitly assume that all values of the internal state variables may be measured at any instant of time. In most practical situations this is not the case. As a result, in order to maintain control, regulators must include a component for state determination. The state determination mechanism has two different types of data to determine the state: (a) knowledge of the system structure, e.g., transition map, output map, dimension, etc., and (b) knowledge of actual inputs and outputs of the system. In this chapter, we shall be concerned with development of results that ensure that data of type (b) may be used to obtain good estimates of the unknown state of the system. In passing, we note that in modern engineering practice it is usually assumed that data of type (a) are given a priori. When this is not the case and the data of type (a) must somehow be inferred from input/output information, then we have an adaptive control problem. The theory of adaptive systems is much talked about, but very little has been accomplished. In the nonadaptive control problem (where data on the 72

4.2

73

BASIC DEFINITIONS

system structure are given), dynamical properties of the system are assumed to be exactly known and it remains "only" to determine the instantaneous state. This is relatively easy, for structural data represents a very large amount of information, stemming from centuries of work in physics and chemistry. A machine that could provide adaptive control for arbitrary systems could also replace human beings in scientific experimentation and model building! In this chapter, we shall distinguish two kinds of state determination problems: (i) the observation problem, where the current state x(r) is to be determined from knowledge offuture outputs {y(s), s ~ r}, and (ii) the reconstruction problem, where the current state x(r) is to be determined from knowledge of past outputs {y(s), s :$; r}. In the first case we observe future effects of the present state and try to determine the cause. In the second, we attempt to reconstruct the present state without complete knowledge of the state transitions. 4.2 BASIC DEFINITIONS

As in the case of controllability/reachability, our principal definitions will be in terms of a certain event (r, x) being observable/constructible. We begin with observability. Definition 4.1 An event (r, x) in a real, continuous-time, finite-dimensional linear system ~ = (F( '), -, H(·)) is unobservable if and only if H(s)cI>(s, r)x = 0,

for all r

:$;

s
is the transition matrix associated with F(· ). The motivation for this definition is clear: the "occurrence" of an unobservable event cannot be detected by looking at the output of the system after time r. Our second concept, constructibility, complements observability just as controllability complements reachability. The precise definition is stated next. Definition 4.2 With respect to the system (r, x) is unconstructible if and only if H(u)cI>(u, r)x = 0,

~

= (F(·), -, H(· )), the event

for all a

:$;

r
r, where M(r, t)

(b)

=

{cI>'(s, r)H'(s)H(s)(s, r) ds;

unconstructible if and only if x

E

ker M(s, r)jor all s < r, where

M(s, r) = fcI>'(a, r)H'(a)H(a)(a, r) do, PROOF (a) x E ker M(r, t) ~ H(s)(s, r) = 0 for all follows similarly.

t

~ s ~ t. Part (b)

4

76

OBSERVABILITY/CONSTRUCTIBILITY

As in the last chapter, in the case of a constant, continuous-time linear system, the two notions of observability/constructibility coincide and we have a simple algebraic criterion. Theorem 4.2 If I: = (F, -, H) is a finite-dimensional, constant, continuoustime linear dynamical system, then I: is completely observable/constructible if and only if the matrix (9 = [H'IF'H'I" ·1(F,)"-lH'] has rank n. (Remark: We say that I: is completely observable/constructible whenever 0 is the only unobservable/unconstructible state.) PROOF

Word-for-word analogy to Theorem 3.9. (l)

EXAMPLES

Let F=

Then C1l{t, r)

=

diag(e 2 (t yet) =

[~~J t ),

H

=

[~ ~

e2(t- t») and

[2(te-~)

X2

oJ

1

if xC') =

c:). X2

Thus knowledge of yet) over an interval determines X2 0; however, there is no way to determine Xl O from the values of yet) over any interval t ~ r. Thus system I: = (F, -, H) is unobservable (more precisely. states of the form x' = (Xl' 0), Xl arbitrary, are unobservable). (2) (Satellite Problem) Consider the linearized dynamics of a particle in a near circular orbit in an inverse square law force field. Assuming that the distance from the center of the force field and the angle can both be measured, we have

o1 o

0 0]

0 2w 0 1 ' -2w 0 .0

1 0 0 OJ 0 1 0

H= [ 0

with y = Hx. Here ta is the angular velocity of the satellite, Yl the radial measurement, and Y2 the angular measurement. The observability matrix (9 is i o 0 0 3w 2 o 0 2 -2w _w 00100 (9= 0 1 0 0 0 o 0

r0 0 0 1 2 w

o

0

This matrix has rank 4 so I: is observable (and constructible).

4.3

77

BASIC THEOREMS

In an attempt to minimize measurements, we might consider not measuring the angle Y2' In this case, H = (l 0 0 0) and

&

~ [~ ~ I -f'J.

which has rank 3. Thus, without angular measurements, the system is not observable. In a similar way, we see that if radial measurements Yl are eliminated, :E will still be observable. In correspondence with the controllability decomposition Theorem 3.10, we have the following.

Theorem 4.3 The state space X of a real, continuous- or discrete-time, n-dimensional, linear, constant system :E = (F, -, H) may be written as a direct sum X = Xl EB X 2 with the equations of :E being decomposed as dx-fdt =

FllXb

dX2/dt = F 21Xl + F 22 X2, y(t) = H 2x 2 (t). PROOF Begin by defining X 1 as the set of all unobservable states of :E. Then proceed as in Theorem 3.10.

In discrete time, the foregoing results are expressed by the following definitions.

Definition 4.3

The discrete-time linear system x(k

+

1) = F(k)x(k)

y(k)

= H(k)x(k)

+ G(k)u(k),

(4.1)

is completely (N-step) observable at time a: ifand only if there exists a positive integer N such that knowledge of y(a: + N - 1) and u(a:), u(a: + 1), ... , u(a: + N - 2) is sufficient to determine x(a:) uniquely.

Definition 4.4 System (4.1) is completely (N-step) constructible at time a: if and only if there exists a positive integer N such that any state at time a: can be determined from knowledge of y(a: - N + 1), y(a: - N + 2), ... , y(a:) and u(a: - N + 1),... , u(a: - 1). Note that constructibility differs from observability in that in the former case we determine the" present" state from" past" data, while in the latter case we determine a "past" state from" future" measurements.

78

4


The next theorem is the main result for discrete-time systems. Theorem 4.4 System (4.1) is completely (N-step) observable at time o: only if the matrix

[H'(rx)IZ'(rx, rx)H'(rx

+

1)1·· ·IZ'(rx

+N -

2, rx)H'(rx

if and

+ N - 1)J

has rank n, where

. {F(k)F(k - 1)··· FU Z(k,j) = I,

+

k ~j, j = k + 1, j>k+l.

I)F(j),

undefined,

PROOF Identical in form to the proof of Theorem 3.7 on controllability/ reachability.

REMARK The above condition is only sufficient for complete constructibility. It becomes necessary, as well, only if the matrix F(·) is nonsingular over [rx, rx + N - 1]. Thus pointwise degeneracy would force the "present" state to be zero regardless of "past" values of y.

Example:

Input/Output Economics

Consider the very simplified dynamic Leontief system in which the production period is measured in discrete-time units. The system dynamics are

+

x(t

1) = Ax(t) .+ Ml(t),

where the production matrix has the form

a2

0 0

0

G3

0 0 0

0

0

an

0

F=

at

0

0 0

aj

~

O.

The vector x(t) represents the various products of the economic complex, with xn(t) being the finished product and Xj(t), i = 1, ... , n - 1, being intermediate products. The matrix M is assumed to be a diagonal matrix with nonnegative elements M

=

diag(ml' m2"'" m n).

The vector l(t) is the labor input to the process.

4.3

79

BASIC THEOREMS

Assume that on the basis of knowledge of the finished product xn(t) we desire to determine the level of production of the intermediate products. Thus we desire to construct the current state xlr), t ~ n, based on output measurements of xn(t). Clearly, the measured output of the economic process is given by y(t) = xn(t) = Hx(t),

where

H = (0 0· .. 0

I).

Appealing to Theorem 4.4, we compute the observability matrix

0

0

0

n

n-.

;=2

0

0

(9=

0

0

anan- t

0 an 0

0 0

0

Thus we see that the economic process is completely constructible if and only if a, # 0, i = 2, 3, ... , n. The above result also illustrates the pointwise degeneracy situation rather well since the matrix A could be singular without destroying the constructibility property. This would happen if at = O. If, however, any a., i # 1, were zero, then the system would not be completely constructible. Example:

Economics and Air Pollution

We now generalize the last example to illustrate the inclusion of nonindustrial sectors. As noted earlier, input/output analysis is a good tool for estimating the environmental and personal amenity effects of changes in the economy. The gross urban environment may be divided into several sections: the natural, community services, the sociocultural, the economic, and so on. The general problem is to show the economy's effects on the individual's environment both directly and indirectly. The first step in such an analysis is to develop linkages between particular economic activities and the affected systems. In this example, we shall look at the relation between industrial output and particulate emissions into the atmosphere. The basis approach is to postulate a dummy "atmospheric particulate matter" sector for the industrial sector. The input/output mechanism then provides a way to exhibit both the inputs and (undesired) outputs of this dummy sector.

80

4

OBSERVABILlTY/CONSTRUCTIBILlTY

We begin by expanding the original industrial sectors i = 1,2, ... , N to include antipollution activity sectors j = N + 1, N + 2, ... ,M, one for each pollutant of concern. We define output rate of industry i at time t, i = 1, ... , N, output of anitpollution activity sector i. expressed as the rate at which pollutantj is reduced,j = N + 1,... , M, rit) rate at which pollutant j is released to the air, di(t) rate of demand for industry product i to consumers, government, and export.

Xi(t) xJ{t)

The technological coefficients are aik ail ali a'm

for

input of product i required for a unit output of product k, input of product i required for a unit reduction of pollutant I, output of pollutant 1 per unit output of product i, output of pollutant 1 resulting from a unit reduction in pollutant m, i, k = 1, ... , N, I, m = N + 1,... , M.

The input/output relations are N

M

I

I

Xi(t

+

1) =

aikXk(t) + ailx/(t) k=1 /=N+1

x,(t

+

1) =

I

N

i=1

aIiXi(t)

+

+ di(t),

i

= 1, ... ,N,

M

I

m=N+1

a'mXm(t) - rl(t),

1= N

+

1,... ,M.

A plausible question to ask in the foregoing context would be whether it is possible to identify the rate at which pollution is being reduced solely on the basis of outputs from the industrial sector, i.e., we measure industrial outputs Xi(t), i = 1,... , N, and attempt to determine the pollution sector. Mathematically, the above question is equivalent to having measurements y(t)

=

Hx(t)

with x(t) = (x 1(r), ... 'XN(t), XN + 1(r),... ,XM(t))', and H = [INIOl

The system matrix F associated with the process is F = [Aik Ali

Ail ], Aim

where A ik = [aik], Ail = [ail], etc. The question of identifying the pollutant reduction may now be answered through appeal to Theorem 4.4. It is easily seen that the solution hinges critically on the properties of the matrix Ai/, linking the two sectors.

4.4

81

DUALITY

4.4 DUALITY

The reader has undoubtedly noticed a striking similarity between the definitions of the matrix functions Wand M and the functions Wand M (Theorems 3.5 and 4.1). In other words, controllability is "naturally" related (in some way) to constructibility, while observability is the natural counterpart of reachability. The most direct procedure for making this precise is to convert the integrand of W into the integrand of M. For fixed r and arbitrary real tx, the appropriate transformations are G(r

+ tx) --+ H'it

- z),

F(r

cJ>(r, r

+ tx) --+ F'(r

+ tx) --+ cI>'(r

- tx, r),

- tx).

(4.2)

Thus we take the mirror image of the graph of each function G(.), H( '), F( .) about the point t = t on the time axis, and then transpose each matrix. For controllability and constructibility, the parameter tx ~ 0, while tx $ for reachability and observa:bility. For constant systems, transformations (4.2)simplify to

°

G

--+

H',

F --+ F'.

(4.3)

The duality relations (4.2)-(4.3) are clearly one-to-one, the inverses being H

--+

G',

F

--+ F'

for constant systems and

Hit - tx)

--+

G'it

+ tx),

F(r - tx)

--+

F'(r

+ «)

for time-varying systems. In view of these remarks, we can give criteria for observability and constructibility in terms of reachability and controllability and vice versa. For example, we have Theorem 4.5. Theorem 4.5 The pair of matrix functions F(t), H(t) define a completely observable system:t at time r if and only if the matrix/unction F*(t) = F'(2r - r), G*(t) = H'(2r - r) define a completely reachable system:t* at time r.

We shall return to the "duality principle" in Chapter 6, where it will be used to help establish the canonical decomposition theorem for linear systems. This foundational result states that any finite-dimensional, linear, dynamical system may be decomposed into the four disjoint parts: (1) completely controllable/completely observable, (2) completely controllable/unobservable, (3) uncontrollable/completely observable, and (4) uncontrollable/unobservable. In Chapter 6, we shall thoroughly examine the philosophy stimulated by this basic theorem and the techniques that have been developed to find part (1)from input/output data.

82

4


EXERCISES

1. Prove that dim ker M(r, r) equals (a) n - dim range M(r, t), (b) n - rank[H'IF'H'I·· ·1(F')"-lH'] for all t > r. 2. Show that the system I: = (F, -, H) is completely constructible if and only if the system I:* = (F', H', -) is completely controllable. 4.5 FUNCTIONAL ANALYTIC APPROACH TO OBSERVABILITY

It is possible to attack the observability question from an entirely different point of view than that taken above and, as a result, to obtain additional insight into the basic concepts involved. In this section we utilize some elementary facts from functional analysis and convex sets to prove the timevarying version of Theorem 4.2 for single-output systems. As a consequence of this line of attack, we shall also obtain some results on the classical problem of moments similar to those presented in the last chapter. We begin with a slight generalization of our earlier definition of observability.

Definition 4.5 Let q be an n-dimensional vector. Ilqll = 1. Then we say the single-output system I: = (F, -, h) is observable in the direction q at time t 1 if there exists a measurable function ~(t) such that (q, xo)

= {' hx(t; xo)~(t)

(4.4)

dt

for any Xo E X. (Here x(t; xo) is the state at time t given that the system began in state Xo and no input was applied.) Since x(t; xo) = C1l(O, t)xo, where O. If (4.7) is satisfied, then setting L =

Ilqll! min fl'I(Z, '(t, O)h') I dt, 11%1151

0

we obtain p(L) ~ 0, i.e., the system is observable. Note that in the constant case we have {t,O)

=

eFI,

which, coupled with (4.7), implies Theorem 4.2. 4.6 THE PROBLEM OF MOMENTS

If we pass to a coordinatewise description of Eq. (4.4) or (4.5), then the condition

f

ll

q =

0

'(t, O)h' W) dt

(4.8)

for solvability of the problem of observability in direction q may be treated as a problem of moments and we again arrive at (4.7). Let us sharpen this result. We integrate the right side of (4.8) n times by parts. Leaving the algebraic details to the reader, we arrive at the following result.

4

84


Theorem 4.6 The observable directions of the system ~ those, have the representation

=

(F, -, h), and only

n-l

q =

L IllF')ih',

111;1
0) is possible only for xo = o. No characteristic vector x of F satisfies the condition Hx = O. There are no nonzero vectors 9 such that the expression H(aI - F)-lg is identically zero for a not a characteristic root of F. (e) For every pair of numbers t 1 and t 2 > t 1, the matrix

is positive definite. 2. The variable x is assumed to satisfy the differential equation x(t) + x(t) = O. If the values of x(t) are known for t = n, 2n, 3n, . . . , can x(O) and X(O) be uniquely determined from this data? 3. Show that the constant system

x=

Fx

+ Gu

is completely controllable if and only if the system

x=

-r

F'x,

y= G'x is completely observable. 4. Show that the constant system ~ = (F, -, H) can be completely observable only if the number of outputs (number of rows of H) is greater than or equal to the number of nontrivial invariant factors of F'.

85


5. The direction q, system

I qI

= 1 (at the point x = 0) of the single-output

x=

Fx,

y = h'x is called T -indifferent to the observation y if for all initial conditions of the form xo = aq,

1rx.1

0,

L Ip;'(Fk)'hjl =

0,

for i =1'}, i.] = 1, ... , r. NOTES AND REFERENCES

Section 4.1

The adaptive control discussion is pursued in greater detail by

Kalman, R., Falb, P., and Arbib, M., "Topics in Mathematical System Theory." McGraw-Hili, New York, 1969.

A somewhat different, but most illuminating, discussion is found in Bellman, R., "Adaptive Control Processes: A Guided Tour." Princeton University Press, Princeton, New Jersey, 1961.

Section 4.2

The basic definitions follow from

Kalman, R., "Lectures on Controllability and Observability." C.I.M.E., Bologna, Italy, July 1968; Edizioni Cremonese, Rome, Italy, 1969, pp. 1-149.

86

4

OBSERVABILITY CONSTRUCTIBILITY

See also the work Sontag, E., On the lengths of inputs necessary in order to identify a deterministic linear system, IEEE Tran. Auto. Cont., AC-25, 120-121 (1980).

The example on pharmacokinetics is taken from Jeliffe, R., et al., A mathematical study of the metabolic conversion of digitoxin to digoxin in man. USC Rep. EE-347, Univ. of Southern California, Los Angeles, 1967.

A related paper is Vajda, S., Structural equivalence of linear systems and compartmental models, Math. Biosci., 55, 39-64 (1981).

Section 4.3 The satellite example, along with numerous other problems of engineering interest, is discussed by Brockett, R., .. Finite-Dimensional Linear Systems. " Wiley, New York. 1970.

The discrete-time results are taken from Weiss, L.. Controllability, realization. and stability of discrete-time systems. SIAM J. Control 10, 230-251 (1972).

See also Hamano, F. and Basile, G., Unknown input, present state observability of discrete-time linear systems, J. Optim. Th. Applic.; 40, 293-307 (1983). Delforge, J., A sufficient condition for identifiability of a linear system, Math. Biosci.; 61, 17-28 (1982).

A deeper discussion of the input/output economics example is found in Tintner, G .. Linear economics and the Boehm-Bawerk period of production, Quart. J. Econ. 88. 127-132 (1974).

See also the book by Helly, W., "Urban Systems Models." Academic Press, New York. 1975.

for a treatment of numerous social problems possessing system-theoretic overtones. Section 4.4 The first published statement of the concept of observability and of the duality principle is given by Kalman, R.. On the general theory of control systems, Proc. lst IFAC Congr., Moscow. 481-492 (1960).

Sections 4.5-4.6 An extensive treatment of the observability question from the functional-analytic point of view is found in Krasovskii, N. N .• "Theory of Controlled Motion." Nauka, Moscow, 1968.


87

See also the treatment by Gabasov, R.. and Kirillova, F., "Qualitative Theory of Optimal Processes." Nauka, Moscow, 1971. (Engl. transl.: Dekker, New York, 1976.)

A detailed treatment of the classical moment problems and its many ramifications in mathematics and science is given by Akhiezer, N., "The Classical Moment Problem and Some Related Questions in Analysis." Hafner, New York, 1965.

CHAPTER

5

Structure Theorems and Canonical Forms t

5.1 INTRODUCTION

One of the basic tenets of science, in general. and mathematical physics, in particular, is that fundamental system properties should be independent of the coordinate system used to describe them. In other words, the properties of a system or process which we are justified in.calling "basic" should be invariant under application of an appropriate group of transformations. For example, the invariance in form of Maxwell's equations under the Lorentz group of transformations is a central aspect of relativity theory. Similarly, the invariance of the frequency of an harmonic oscillator when viewed in either a rectangular or polar coordinate system illustrates the fundamental nature of this system variable. In mathematical system theory, one of the primary objectives is to discover properties about systems which are, in some sense, fundamental or t In previous chapters it has been our policy to motivate and illuminate the basic theoretical results with numerous applications from diverse areas of human endeavor. The current chapter, however, is intended primarily for" connoisseurs" of system theory and, as a result, is almost exclusively theoretical in character with the exception of some numerical examples. The reader whose tastes do not run toward the theoretical can safely proceed to Chapter 6 at this point without loss of continuity; however. for the sake of broadening one's scientificand mathematical culture, we do recommend that this chapter at least be skimmed before proceeding to the following material.

88

5.1

INTRODUCTION

89

basic to all systems of a given class. Of course, the decision as to what constitutes a "basic" property is to a certain degree a subjective one determined by the tastes and motivations of the analyst. However, coordinatefree properties certainly have a strong claim to being regarded as basic system properties and to a large extent our discussions in this chapter will be devoted to an examination of such system features. In order to isolate coordinate-free system properties, it is necessary to specify a particular group of transformations nand p > m. It is well known that the polynomials a(s) and f3(s) are relatively prime if and only if B[IX, 13] is of full rank and that the rank deficiency of B[a, 13] equals the degree of the greatest common factor between IX and 13. To relate the Bezoutiant to linear systems, we consider a single-input! single-output system 1: = (F, g, h) having transfer function I3(s)!IX(s). The controller canonical form of this system is given by Xc = Fcx c + gcu, Y = hex",

5.6

103

CANONICAL FORMS AND THE BEZOUTIANT MATRIX

where

0 0

1

0

0

1

0 -ao

0 -al

0 -a2

0 0

(Note: here the coefficients a, are labeled to agree with the

Fc =

labeling in the polynomial

rx(s)),

-an-I

0 0 ge =

he = [ho b l

o

· · .

b.; 0··· OJ.

1 The dual of the controller form, the observer canonical form is obtained from ~e = (Fe, ge' he) by means of the relations

It is a simple task to verify that the transfer function for both systems equals fJ(s)/rx(s). In fact, there will be an uncountable number of systems ~ with transfer function fJ(s)/rx(s). Anyone of these systems is referred to as a realization of the transfer function fJ(s)/rx(s). Let us introduce the controllability and observability matrices CC and (f) as CC(F, g) = [gIFgl" ·IFn-lg],

(f)(h, F) = CC'(F', h').

A basic role in relating the Bezoutiant to the observer and controller canonical forms is played by the following result.

Theorem 5.7

The Bezoutiant matrix B satisfies the identity

where

and -al -a2

-a2

-an-I -1

-1

-an-I

-1

T=

PROOF

o

The proof is accomplished by direct computation and verification.

104

5

STRUCTURE THEOREMS AND CANONICAL FORMS

The basic result relating e, (= rtf(Fe , ge)), (20 (=rtf'(Fo', ho')), «, (=rtf(Fo' go)) and (2e (=rtf'(Fo', he'» follows. Theorem 5.8

B(~,

(d)

B(~,

PROOF

/3(Fc ) =

B(~,

/3) satisfies the following

identities

/3) = -rtf; l(2e' /3) = -rtfo(2;;1, /3) = - (2;; l(2e, /3) = -se,«;',

B(~, B(~,

(a) (b) (c)

The Bezoutiant matrix

Part (a) follows immediately from the relations rtf; 1 = T, (2c'

Part (b) follows from (a) by using the symmetry of Bta; /3) since Bi«, /3) =

B'(a, /3) = - (2o'(rtf eT 1. This fact, together with the duality relations (20' = rtf0' rtf0' = (20 establishes (b). The duality relations plus the invertibility of T = rtfe- 1 imply (c), which in

turn implies (d).

Conclusion (c) of the theorem leads to a very simple proof of the fact that any observable realization on; = (F, g, h) can be transformed to the observer canonical form by a change of state variables as xo(t) = Mx(t),

where M is the constant nonsingular matrix M =

(2;; 1(2(F,

h).

Thus, if the controller canonical form is observable, from conclusion (c) we see that B(a, /3), the Bezoutiant, is precisely' the transformation matrix needed to form the observer canonical form. EXERCISE

1. Establish the corresponding results for multi-input/multi-output systems. 5.7 THE FEEDBACK GROUP .AND INVARIANT THEORY

We have seen that the group of linear coordinate transformations in X, the state space, enables us to reduce the apparent complexity of a given system l: significantly by reducing it to a canonical form in which the inherent structure of E is more apparent. The obvious question at this stage is whether a further simplification is possible if we augment our transformation group by allowing not only basis changes in X, but also other coordinate and structural changes in l:. For a variety of reasons, some of which we shall see below, the most interesting new transformations are changes of basis in the input space 0,

5.7

105

THE FEEDBACK GROUP AND INVARIANT THEORY

and application of special inputs of the form u(t) = - Lx(t), where L is an arbitrary n x m constant matrix. Such inputs are termed "feedback" since their effect is to generate the input not as an explicit function of time, but as a function of the state x. Thus our new group of transformations consists of: (I) coordinate changes in X; (II) coordinate changes in il; (IiI) arbitrary constant feedback laws. This set of transformations forms what is generally called the feedback group !F. Given a system 1: = (F, G, H), the action of !F on 1: is (coordinate change in X) (II) (coordinate change in il) (III) (feedback law) (I)

F

-+

TFT- 1 ,

G-+GV- 1 ,

F -+ F, F

-+

H -+ HT-l,

G -+ TG,

H-+H,

G -+G,

F - GL,

H-+H,

ITI"# 0,

IVI"# 0, L arbitrary.

A particular element of !F is determined by specifying the matrices T, V, and L. Clearly, the choice V = I, L = 0, T nonsingular reduces to the case of state coordinate changes considered above. Thus the state space basis changes form a subgroup of !F. We have seen that under coordinate changes in X, the only invariants are the coefficients {IX;} of the characteristic polynomial of F (or equivalently, the characteristic values of F) and the elements of HT- 1 (assuming rank G = m). Since the feedback group allows more flexibility in modifying the structure of 1:, the number of invariants is certainly no greater than under the subgroup of state space basis changes. Our purpose is to determine the precise invariants. We begin by considering the ordered set of vectors gl, ... ,gm;Fg 1,F9 2 , ••• ,Fgm, ... ;Fn-lgI, ... ,Fn-lgm, (5.6) where gi = ith column of G. Under the assumption that the system 1: = (F, G, -) is completely reachable, the list (5.6) contains precisely n linearly independent vectors. Consider the" Young's diagram" I

F

F2

...

Fn -

91

X

X

X

.. ,

X

92

X

k2

93

X

k3

l

kl

106

5 STRUCTURE THEOREMS AND CANONICAL FORMS

where t, is the number of crosses in column (i + 1), and k j is the number of crosses in row j. Here the rule for placing crosses in the diagram is as follows: begin with row 1 and place a cross if element (i, j) is linearly independent of all vectors previously considered. Then go to row 2 and repeat the process, etc. By the complete reachability of 1:, this procedure will result inexactly n crosses being placed in the diagram. The integers {kj } , {l;} must satisfy s-1

Lk; = L t, = n.

t> 1

i."O

The vectors picked by the above procedure, namely, kj - 1 . · { gj' .•• , P gj • ]

E

M},

where M is a subset of {I, 2, ... , m} containing exactly s elements, constitute a basis for X. Let M(i)

= {tEM: k, > i},

i

= 0, 1,2, ... , n -

1.

Then M(i) has 1; elements and M(O) = M. By linear dependence, we have pk'gi

=

L

ki-1

(Xji(kj+ 1l

P k igj

+ L

L

(Xji(d l)pTg j,

iEM,

(i)

r=O jeM(Tl

jeM(k;j

e. = L (Xj;1gj'

i = 1, 2, . :. , m,

i¢M.

(ii)

jeM

We now make the transformation iEM.

(iii)

It then follows from (iii) and the definition of M(i) that gi

= {l; -

L

iEM.

'(Xji(kj+1 l0j, jeM(k;j

Upon substituting (iv) into (i) and (ii), we obtain pk,O;

=

k,-1

L L

(Xji(T+1l

p tOj'

iEM,

r=O jeM(tl

o, = L (Xji10j, jeM

i

= 1,2, ... , m,

i¢M.

(iv)

5.7

107


A new basis in X is then defined by eit

-

-

Fki-t{j

ki- l '\'

'\'

L.

i -

L. (lji(t+

1)

Fj-t{j

ieM,

j'

t=t jEM(t)

With respect to this basis, (F, G) take the form i,j, eM,

where (F .J.

~[~

1 0 0 1

-:,J

- (liil

0

(F #)ji

= [

(F #)ji

= [

(F #)ji

e Rkjxk,.

G# =

-

- (ljikj

(ljil

0

0 - (l jil

«G#)ji)'

-(ljikiJ

jeM,

...

oJ

kj > k i ,

i = 1,2, ... , m.

o o (G#L=

o

ieM,

m 1

(G.)"

~

j # i,

ieM,

ieM,

t

= 1,2, ... , k i .

108

5


If we renumber the columns of G so that k i ~ k j + l ' then F # and G # take the forms 0 0

x

1

0

x

0 1

0 0

x

0

x

F# =

0 x

x

x

0

0

x

x

0 0

1

x

0 x

) k, 'Ow,

.. , 0 1

X

0 0

... x

x

.,.

I

k,'Ow,

~

k 1 - k2 columns

0

0 1

0

0

x

x

0 x

... 0 .. ....... ... 0 ... x

0

0

0

...

0

...

0

...

0

0 0 1

)

0 x

0

...

0

x

...

x

k 2 rows

~

m-lo columns

Theorem 5.9 Relative to the feedback group fF, the invariants of the system ~ = (F, G, H) consist of the set of integers k l' k 2 ,

••• .k, and the elements hij ofR. Conversely, given any set ofm nonnegative integers k, such that LI= 1 k, = n, and a set of np real numbers hij' a system ~ is determined by transformations from fF, i.e., {kJ and {hij} constitute a complete, independent set of invariants

for~.

5.7

109


EXAMPLE

We apply Theorem 5.9 to the system

F=

1 0 2] [001

H = (0

230,

I

0).

First we verify that I: is completely controllable. Computing the controllability matrix «6, we have

I 2 I 4]

o o

2 084

10101 which has rank 3. Following the prescriptions of the theorem, we write the sequence of vectors

{g,~G), F'g,

g,

~ G),

~ G}

r«.

Fg,

~ (D,

Fg,

~ G}

~ (D}

Clearly, the first three form a linearly independent set in R 3 . The Young's diagram is 91

g2

I

x

x

f

X

F2 k l = 2, k 2 = 1. Thus we see that the Kronecker invariants for this case are k l = 2, k 2 = 1. The set M = {I, 2}. We next compute the basis (5.7). We obtain

Ito

5


The only missing quantity is the constant dependency relations 2

p2 9 1

-391

This implies that

+ 4F91

IX I J I

= -

= -

I I

I

IXIII'

This is obtained from the

IXljk P k9 j ,

j=1 k=O

=

-{IX I

Jo9 1 + IXJllF91

+ lX J2092}'

4. Thus the new basis is

If we assume that the matrices P, G, H are originally given in terms of the standard basis in R 3, i.e., the basis

and if we express the standard basis in terms of the new basis ell' it is easily verified that the matrix of this transformation is given by

T

e J 2, e2J'

~ [! ! H

Using the basis change T, we finally obtain

kl

f

~

~

TFr'

~ [--:-:--+-: R

t.

= HT-

I

= (2

The feedback law

L=[~

-~

0

=:]

0).

eliminates the second row of F which enables us to see that a complete set of invariants for the system are the integers k I = 2, k 2 = 1, together with the elements of the matrix fJ = (2 0 0). Note that in this example it was not necessary to use any Type (II) transformation because G was of full rank.

III


REMARKS (I) The importance of the control invariants k l , ... , k. is that they give the sizes of the smallest cyclic blocks into which F may be decomposed by linear" feedback" transformations. This is the only structural obstruction to arbitrarily altering F by utilization of feedback. (2) The invariants k I' ... , k. are identical to the classical Kronecker indices associated with the matrix "pencil" [zI - FIG]. In fact, Kronecker's definition for the equivalence of two pencils is

P(z) '" Q(z) ¢ > P(z) = AQ(z)C,

for nonsingular constant A and C. It is then easy to see that [zI - FIG]

if and only if C has the form C

A-I

OJ

= [ LA-I B'

= A[zI - FIG]C L arbitrary,

det B ¥- O.

Thus Kronecker's equivalence relation applied to the pencil [zI - FIG] is identical to the equivalence relation induced by the feedback group !F on pairs (F, G). EXERCISES

1. Assume that m divides n and that each k, = nlm, i = i, ... , s. Determine the subgroup of!F which leaves the Kronecker canonical forms for F and G invariant. What happens if all k, are not equal? 2. Complete the proof of Theorem 5.9 by showing that the parameter set S = {k j , !iij} constitutes an independent and complete set of invariants for the system I: = (F, G, H), i.e., for any given set S, there is some system I: whose parameter set equals S (independence) and if two systems I:I and I:2 have the same parameter set S, there exists a transformation !/ E !F such that I:I = !/I: 2 (completeness). MISCELLANEOUS EXERCISES

1. Suppose that the pair (F, G) is not completely controllable and that G ¥- O. Show that there exists a basis change in X such that F and G are transformed into

with the pair (F ll' G I) being completely controllable. Show that the rank of F II equals the rank of rc(F, G).

112

5


2. Define the single-input/single-output system ~ = (F, g, h) to be nondegenerate if it is both completely controllable and completely observable. Show that the following statements are equivalent: (a) ~ is nondegenerate. = h(A.I - F)-lg) is irreducible and the (b) The transfer function of~(Z(A) degree of the denominator equals n, the dimension of ~. (c) If any other system t of the same dimension has the same transfer function, then there exists a nonsingular T such that the two systems are related as

9= 3.

Let the system

~

be in Lur'e-Lefschetz-Letov canonical form, i.e.,

F~ [F' Fj =

Tg,

n J"

F2

G=

~2

G q

r'

F22

F,,,J....'

Gj =

nXm

G [G" ] j2

G}S(j) ni x m'

A~ I

1

Fij =

Aj 1

[g'' 1

Aj 1

Prove that vectors

~

g2u G.. = : . u g;ii nij x m

Aj

nij x nij

is controllable if and only if the set of s(i) m-dimensional row

form a linearly independent set, i = 1, 2, ... ,q. (Note that this set is composed of all the first rows of the matrices Gij associated with a given submatrix F j .) 4. Show that a single-input system ~ = (F, g, -) is completely controllable if and only if the transfer matrix Z(A) = (AI - F)-lg is irreducible, i.e., the numerator and denominator have no common factors.

113


U sing this result, check the system

F

=

-2J g=G)

[~

-3 '

for complete controllability. 5. (a) Let F, G be fixed and denote the range of G by '§, i.e., '§ = {x E R": x = Gy for some y E R m } . Further, let (FI'§) = '§ + F'§ + '" + F"-l,§ denote the controllable subspace of X for the pair(F, G). Ifdim "§ = m, show that there exist subspaces {J!;} c X such that (i) (ii)

dim("§ n J!;) = 1, i = 1, (Fj"§) = VI

E9 V2 E9

, m;

E9 Vm •

(b) If (F, G) is a controllable pair, what is the relation between the list of integers {dim VI' dim V2 , ••• , dim Vm } and the Kronecker indices k I' k 2 , · · · , km ? 6. In the theory of representations ofthe general linear group GL(R m ) , it can be shown that each set of m nonnegative integers k, ;C 0 satisfying I~ I k, = n corresponds to a certain representation of GL(R m ) . What is the connection between this fact and the control canonical form for a multi-input system? 7. Prove the following structure theorem. Let (F, G)be completely reachable, m = rank G, and k, ;C k 2 ;C ... ;C k m the ordered control invariants of (F, G). Let t/J I, ... , t/J q be arbitrary monic polynomials such that (a) t/Jdt/Ji-I,i=l, ... .a t.s s »: (b) degt/JI;C kl,degt/JI + degt/J2;C k 1 r

+ k2,etc.

Prove that there exists a feedback control law L such that F - GL has the invariant factors t/Jl"'" t/Jq. Conversely, show that the invariant factors of F - GL always satisfy (a) and (b). 8. Let k be an algebraically closed field and assume that the system I: = (F, G, -) and an arbitrary control law L are defined over k. Define a control law J to be purelyfeedforward ifand only ifXF = XF-GJ, while a control law K is purely feedback if and only if 0(, rank (!)j = rank (!)p, i > p. (3) If ~ = (F, G, H) and! = (F, G, H) are algebraically equivalent, then = iX, 13 = P, and i

= 1,2, ....

We may now utilize the sequences ~j and (!)j to give an improved version of Theorems 3.10 and 4.3 which we may then combine to give a detailed description of the structure of ~ under algebraic equivalence. Theorem 6.3 If rank ~a to ~ = (F, G, H) where

and

!q

= (F I"

= q :::;; n, then

~

= (F, G, H) is strictly equivalent

GI , HI) is completely controllable.

PROOF Let TI be a matrix whose columns form a basis for the column space of rca and let Tz be any n x (n - q) matrix whose columns along with those of T1 form a basis in R". Then the matrix T- I = [Til Tz ] is nonsingular and defines! = (P, G, H). The first q columns of F are given by

for some K I since FTI is a submatrix of rca + I' Similarly,

since G is a submatrix of ~

a'

Let

r& a be the IX-controllability matrix of ~q

=

(F II' GI' HI)' Then the forms of F and G show that

Hence, rank ~ the proof.

a

= q so that ~q = (F 11' Gl'

-)

is controllable. This completes

122

6

REALIZATION THEORY

The observability version of the above theorem follows. Theorem 6.4 If rank (fJp to ! = (P, G, H) where

F= [~:

F~J,

Also, ~q = (FIt> G1 , PROOF

= q :=:; n, then G=

= (F, G, H) is strictly equivalent

~

[g:J

H = [HI

OJ.

Ht> is completely observable.

Dualize the proof of Theorem 6.3.

Theorems 6.3 and 6.4 together enable us to prove the following canonical decomposition theorem for linear systems. Theorem 6.5 If rank (fJp~f/. = q :=:; n, then system 1: = (P, G, H), where

and ~q

=

o.; G

1,

~

= (F, G, H)

is equivalent to a

HI) is controllable and observable.

PROOF Applying Theorem 6.3 to ~ = (F, G, H) followed by Theorem 6.4 applied to the resulting controllable subsystem shows that there exists a T such that ~ is algebraically equivalent to 1:, where 1: has the indicated structure and ~qis controllable and observable. We must show that ij = q. To demonstrate that ij = q, partition ~f/. and ~p as

These matrices are partitioned conformally with the forms of F, G, H. Then ~~1 and mp1 are the IX-controllability and fJ-observability matrices of ~q = (P l l , G1, HI), respectively, and

=

(fJp~f/.

by Property 3 of ~ j and Thus

(fJ j

mp?Cf/.

=

since lJ P2 = ~

rank mpl~f/.l

f/.3

= O.

= rank (fJp~f/.

By controllability and observability of ~q, rank ~Pl~f/.l Therefore q = ij.

(!jP1?Cf/. l ,

= q.

however, we have = ij.

6.2

123

ALGEBRAIC EQUIVALENCE AND MINIMAL REALIZABILITY

Using the foregoing results, we can now deal with the question of minimal realizations. At first glance, one may ask why it is so important to produce a minimal system realization. After all, should we not be satisfied if a model of the process under study can be produced which explains all the experimental data? The answer, of course, is no! Returning to Occam's razor, we demand the simplest possible explanation. Why do we present minimality of the state space as a plausible definition of what constitutes the "simplest" explanation? The reason is Theorem 6.6 (below) which asserts that minimality is equivalent to a completely controllable and observable model. In other words, if we present a nonminimal realization as our model, then we are including features in the model that are not justified by the experimental evidence: an uncontrollable part cannot be influenced by any external input, while inclusion of an unobservable part means that irreducible internal uncertainties about the system behavior are being built into the model. Both features are clearly undesirable. Define the matrix Ki,{t) as

Kjj{t) = (fJjeFtCC j , i,j = 1,2, .... Using Kjj{t), we establish the connection between controllability, observability, and minimality. Theorem 6.6 (F, G, H).

The following statements are equivalent for constant

~ =

(1) ~n = (F, G, H) is controllable and observable. (2) rank (fJpCC a = n. (3) ~n is minimal. PROOF (2) => (1) Obvious. (1) => (2) Obvious. (3) => (2) Follows from Theorem 6.2 since if rank (f)pCC a < n, we can find a lower-order representation having the same impulse-response matrix. (2) => (3) Suppose that (2) holds and ~n is not minimal. Then there exists ~ll' with ii < n, such that ~ii has the same impulse response matrix as ~n' Since Kij{O) = (fJjCC j , we have'

Kpa(O) = ~p7jJa' But ~p has only ii columns. Thus rank (fJpCC a = rank ~p7jJa dicting (2). Corollary 6.1

~ ii

< n, contra-

Let y and (j be the first integers such that rank (f)y CC/j = rank (fJpCC a = q

~

n.

Then the minimal realizations of~n = (F, G, H) have degree q, controllability index (j, and observability index y.

6

124

REALIZATION THEORY

EXERCISES

1. Prove Properties (1)-(3) of the sequences ~i and (!]j' 2. (a) Show that the decomposition theorem 6.5 can be given in the following form: For any linear system :E = (F, G, H) (including timevarying :E) there exists a coordinate system T(t) in X such that :E is algebraically equivalent to a system t = (ft, c, R), where

(b) What does the above structure say about the internal controllability/ observability structure of E? (Hint: Draw a diagram of the input/output structure of t.) 3. Prove that all time-invariant minimal realizations of a given impulseresponse matrix can be generated from one such realization by means of constant coordinate changes in X. Is this also true for nonminimal realizations? 4. Show that the matrix (!]/l + 1~ ~: (a) is an invariant for the class of all realizations of a particular impulseresponse matrix; (b) is the only invariant necessary to determine if two representations have the same impulse response. . 6.3 CONSTRUCTION OF REALIZATIONS

We have shown that once any realization of an impulse-response matrix is given, the problem of determining all minimal realizations is effectively solved since we need only construct the complete invariant (!] p + 1~ a; and utilize Theorem 6.5, Theorem 6.6, and its corollary to reduce a nonminimal realization to one which is minimal. But, the problem of explicitly constructing some realization remains. In this section, we shall show that the suffices to calculate a minimal realization explicitly. matrix (!]/l+1~a; Our approach to the construction of a realization will be through the Hankel matrix of an infinite sequence of matrices. Definition 6.3 Let J denote a sequence of real p x m matrices J j , i = 0, 1, .... A system :E = (F, G, H) is said to be a realization of J if and only if i = 0, I, ....

(6.4)

6.3

125

CONSTRUCTION OF REALIZATIONS

To see the connection between the sequence of matrices {J;} and the realization problem in both continuous and discrete time for a given transfer function (or equivalently, impulse response), let Z(A.) denote the transfer function of a continuous-time system ~n = (F, G, H). We then have the expansion about A. = 00: 00

Z(A.) =

LJ

i= 1

(6.5)

i - 1 A.".

where J, is given by (6.4). If W(t - s) denotes the impulse response for the same system, then W(t - s)

=

L Ji(t 00

- s)i/i!.

i=O

In a similar manner, if ~n = (F, G, H) characterizes a discrete-time system, its impulse-response matrix Hi is given by while its z-transform transfer function has form (6.5). These remarks show that any procedure suitable for realizing the sequence {J;} will serve equally well,regardless ofthe specificform in which the input/output data is presented. Clearly, the first question is to determine when a given infinite series is realizable. To establish this basic point, we introduce the block Hankel matrices for J as

:Yl'ij

=

r'

J1 J2 ·

~1

Ji-

1

Jj

J)-> J.

J'+~_'

j .

The role of the Hankel matrices in realization theory follows from the fact that (6.6) when J is generated by a system ~ = (F, G, H) having observability matrix and controllability matrix ~j' In terms of the Hankel matrices, we have the next theorem.

(!}i

only

if there

all j = 1,2,... .

(6.7)

Theorem 6.7 (a) An infinite sequence J is realizable exist nonnegative integers P, tX, and n such that

rank :Yl' P« = rank :Yl' P+ 1,«+ j = n (b)

of

J.

for

if and

If J is realizable, then n is the dimension of the minimal realization

6

126

REALIZATION THEORY

(c) If'/ is realizable and {3, ~ are the first integers for which (6.7) is true, then {3 is the observability index and ~ is the controllability index of any minimal realization of ,/. PROOF (a) (Necessity) Necessity follows from (6.6) and the properties of the matrices (!)i and t:{fj' (Sufficiency) We prove sufficiency by constructing an explicit minimal realization of ,/. Let i'ij be the (i,j)th block of the Hankel matrix Jf~.p. Clearly, from the form of Jf~,p, we have

i'i+p,j = i'i,j+m,

(6.8)

i,j = 1,2, ....

Thus (6.7) implies that rank Jfp+i,~+j

= n,

i,j

= 1,2, ....

(6.9)

since the ({3 + i)th block of rows in Jf p + i. j is contained in the ({3 + i-I )st block ofrows in Jf p + i, j + I by (6.8). Let A~ denote the submatrix formed from the first n independent rows of Jfp~ and let A~* be the submatrix of Jfp+I,~ positioned p rows below A~ (i.e., ifthe ith row of A~ is the jth row of Jf p + 1, ~, then the ith row of A~ * is the U + p)th row of Jfp+1,~). The following matrices are then uniquely defined by Jf P+I,~: A

A* Al

A2

the nonsingular n x n matrix formed from the first n independent columns of A~, the n x n matrix occupying the same column positions in A~ * as A does in A~, the p x n matrix occupying the same column positions in Jf h as A does in A~, the n x m matrix occupying the first m columns of A~.

If we define F as F

= A *A - I, then it follows from

(6.9) that

j = 1,2, ... ,

where Aj and A/ are extensions (or restrictions) of Also

(6.10) A~

and

A~*

in

Jfp+I,j'

(6.11)

since the submatrix positioned m columns to the right of a given submatrix in Jfij is the same as the submatrix positioned p rows below it by (6.8). Thus, by (6.10),

6.4

127

MINIMAL REALIZATION ALGORITHM

Next, define G that

=

A 2 • It follows by repeated application of (6.10)and (6.11) j

j ~

= 1,2, ....

Define H = A 1 A - 1, use the fact that Aj spans the row space of Yepj' IX, and employ (6.9) to see that

But, Yeij = [JoIJ 11 .. ·IJj - 1 ] . Hence, we must have J, = HFiG, i = 0, 1, .... Thus we have proved that the triple

defined from the submatrices of Yep + 1./Z' realizes the infinite sequence" if (6.7) holds. Furthermore, it follows immediately that this realization is minimal and has controllability index IX and observability index p.

6.4 MINIMAL REALIZAnON ALGORITHM

Theorem 6.7 provides the following algorithm for constructing a minimal realization of the sequence": Determine integers p, IX such that rank YeP./Z = YeP+ 1./Z = n for all IX. (2) Form the matrix A/Z from the first n independent rows of Yep, /Z' (3) Form the matrix A/Z * from Yep + 1,/Z as the matrix which is positioned p rows below A/Z. (4) Form the four matrices A, A*, A 1 , A 2 as (1)

A

A* A1 A2

nonsingular n x n matrix formed from the first n independent columns of A/Z, the n x n matrix occupying the same column positions in A/Z * as A occupies in A/Z, the p x n matrix occupying the same column positions in YelIZ as A does in A/Z, the n x m matrix occupying the first m columns of A/Z.

(5) Form the minimal realization of" as ~ =

(A*A- 1 , A 2 , A 1A- 1 ) ,

128

6

REALIZAnON THEORY

6.5 EXAMPLES

(1)

Fibonacci Sequence

Let / = (1, 1,2, 3, 5, ...). Applying the above algorithm, we see that condition (1) is first satisfied for 0( = P = 1, giving rise to the one-dimensional realization 1: = (1, 1, 1). It is easy to verify, however, that this system realizes only the first two terms of the sequence /. Thus we must include more data in our search for a realization that "explains" the entire sequence /. We form the additional terms in the Hankel sequence

which shows that condition (1) of the algorithm is also satisfied for C! = P = 2, giving a two-dimensional realization. By virtue of the fact that we "secretly" know that / is generated according to the rule

it is not hard to see that regardless of how many additional terms are taken in the Hankel sequence, the rank condition will always be satisfied with some p, C! combination having n = 2. Thus this outside knowledge of the sequence allows us to conclude that the minimal realization for / is two-dimensional. Of course, on the basis of a finite data .string it will never be possible to guarantee that the infinite sequence / has a finite-dimensional realization without knowing such a realization in advance. For the Fibonacci sequence above, however, it is reasonable to conjecture that n = 2 even without knowledge of the generating formula since it is easily verified that for all 0(, p > 2, the first two columns of Yf ap always generate the remaining columns. Following steps (2)-(4) in the realization algorithm, we form the matrices A z , A z*, A, A*, A l , Az as

=

[~

A=

[~

Az

Al

=

[1

~J. ~J. 1],

A z*

=

A* =

AZ=[~J.

G~J. [~

~J.

6.5

129

EXAMPLES

According to step (5), the minimal realization for / is

F=[~

G=eJ

~J

H=[I

OJ.

This example illustrates the critical point that the success of the foregoing algorithm hinges on the assumption: / has a finite-dimensional realization. This is equivalent to saying that there exists some integer n < co, such that the rank of the sequence of Hankel matrices is bounded above by n, irrespective of the number of elements in .Yl'lJp' Of course, if / were only a sequence of numbers obtained from some experiment, then there is no a priori justification for the finite realization assumption and, in general, the best we can hope for is to realize some finite piece of the data string ,I. A more thorough discussion of this partial-realization problem is given in a later section. (2) The Natural Numbers Let / = (1,2,3,4, ...). Upon forming the relevant Hankel matrices, we see that the rank condition will be satisfied for a. f3 n 2 and, furthermore, all the Hankel matrices .Yl'ij have rank 2 for i, j > 2 (verification ?). Thus / has a two-dimensional realization. Carrying out the remaining steps of the algorithm, we find

===

=G ~) A* =G ~) A =G)'

=G ~). A=G ~). At = 2),

A2 *

A2

(I

A-

l

2

=

(-3 2) 2

-I'

Thus the realization of,l is G=

G)'

H = (l

0).

It is interesting to observe that this is not the only possible realization of ,I. For instance, the transfer function corresponding to / is

130

6

which suggests the alternate realization

~

F=

[0 IJ

-12'

G=

G)'

REALIZA nON THEORY

fJ = (0 I).

What is unique about the realization is its dimension, in this case two. However, the fact that the two realizations are related by the nonsingular transformation

-1/2t l1

o

J '

as

G=

F=

TG,

TFT- 1

suggests that a change of basis in the state space will remove all non uniqueness in the realization. We examine this point later. (3)

Multi-input-Multioutput Realization

In order to dispel any notion that a multi-input/multioutput sequence can be minimally realized by realizing each of its components separately, we consider the sequence

/ =

{(l 1), (1 2), (l 3), (2 4), (l 5), (3 6), ...}

which is made up of the two scalar sequences /

=

{I, 1, 1,2,1,3, ...}

and

/2

=

{I, 2, 3,4,5,6, ...}.

We have already seen that a minimal realization of /2 has dimension 2. It is also easy to see that /1 has a minimal realization of dimension 3; in fact, the transfer function associated with / I is A. 2+2A.+1 Z(A.) = A. 3 + A. 2 - A. - 2 . We can, of course, obtain an upper bound for the dimension of the realization of / by simply adding up the dimensions of each of its component sequences /1 and /2 which shows that a minimal realization of / has dimension no larger than 5. This upper bound may, however, be too large since by accepting nonminimal realizations of some scalar components, we may be able to reduce the dimension of the matrix sequence. Applying the above realization algorithm, it is not too difficult to see that a minimal realization of / has dimension 4. This is confirmed by examination of the irreducible transfer function

A.3+A.2_A. A. ] Z(A.) = [ (A. _ 1)2(A. + 1)2' (A. _ 1)2 .

6.6

131

REALIZATION OF TRANSFER FUNCTIONS EXERCISES

1. Complete the construction of a realization of the sequence in Example (3). 2. Show that the sequence of prime numbers f = {2, 3, 5, 7, 11, ...} has no finite-dimensional realization. 3. In purely algebraic terms, show that a 1:: is a minimal realization of the input/output map f: n ~ r if and only if there exists a finite-dimensional space X and maps g and h such that (i) g is onto, his 1-1, and (iii) the diagram (ii)

n

f

--->

r

x is commutative.

6.6 REALIZATION OF TRANSFER FUNCTIONS

The realization procedure outlined in the previous section was based purely on input/output data, as opposed to complete knowledge of the input/output map f As a result, certain unpleasant difficulties appeared in attempting to decide operationally whether: (i) the data admits a finite-dimensional realization and (ii) if so, what is the dimension of a minimal-realization. In this section we shall explore the question of how much more information on these questions is provided by the knowledge of the map f, rather than having just a finite string of data generated by f As one might expect, since knowing f is equivalent to knowing an infinite string of data, a completely satisfactory solution to the realization problem will be obtained in this case. The more difficult question as to exactly what information a finite data string provides will be treated in Section 6.8. For the present purposes, let us assume that the transfer matrix Z(l) is a strictly proper rational matrix, i.e., the degree of the numerator of each element is strictly less than the degree of the denominator. As pointed out in Theorem 6.2, this is a necessary and sufficient condition for Z(l) to be realizable by some system 1:: = (F, G, H). Such a matrix can always be written in the form &'(l)/X(l), where &'(1) is a polynomial matrix and X(l) a monic polynomial that has no factors in common with &'(1).

6

132

REALIZATION THEORY

For our main result, we need the following easy fact.

Lemma 6.1 If Z(.A.) is a strictly proper rational matrix, then X(.A.) is the minimal polynomial associated with the minimal realization of Z(.A.), i.e., if ~ = (F, G, H) is the minimal realization of Z(.A.), then X(.A.) is the minimal polynomialofF. PROOF

The proof is left as an exercise.

From Lemma 6.1 plus the fact that rank (9 + i = rank (9 and rank ~ a + 1 = rank ~(f for all i, we see that if a is the degree of the minimal polynomial of F, then the Hankel test may be terminated after a steps. This leads to the following basic result concerning realization of transfer matrices. (f

(f

Theorem 6.8 Let Z(.A.) be a strictly properrational matrix and let a = deg X(.A.). Let Jeij be the Hankel matrix associated with the expansion of Z(.A.) as in Eq. (6.5). Then the first integers f3 and IX such that rank Je Pa. = rank Je oo satisfy the realizability condition (6.7). EXERCISES

1. Compute the minimal realizations associated with the transfer matrices (a)

2.

1

Z(.A.) = .A.2 _ 1 .A.3

[2(.A. - 1)

+

.A. - 1

.A. 2 - .A.

(b)

Z(.A.) = [ (.A. _ 1)2(.A. + 1)2

(c)

Z(.A.) =

[

~ + .A.~ ~ o

+ }2 0

.A.

IJ

~

(.A.

~

il

1)2}

Determine a minimal realization for the transfer matrix 1 .A.

+1

_.A. 2 + 1 6.7 UNIQUENESS OF MINIMAL REALIZATIONS

So far we have skirted the question as to whether the realizations produced by Theorems 6.7 or 6.8 are, in some sense, unique. It is clear by Example (2) that strict uniqueness is too much to hope for. It is also highly suggestive

6.8

133

PARTIAL REALIZATIONS

that, as pointed out in Exercise (3) of Section 6.2, the impulse-response matrix (or equivalentally, the transfer matrix) is invariant under coordinate changes in the state space X. The basic result linking these observations is our next theorem. Theorem 6.9 Let,l be a realizable sequence of matrices and let the matrices A, A*, AI' A 2 be as defined in Theorem 6.7. Then any minimal realization of ,I has the form L =

(..4*..4- 1 , ..42 , .41 ..4- 1 ),

where..4 = T A,..4* = T A*,..4 2 = T A 2 with T being an arbitrary nonsinqular n x n matrix, i.e., minimal realizations are unique up to a coordinate change in the state space. PROOF The proof follows directly from Exercise (3) of Section 6.2, plus Theorem 6.7. EXERCISE

1. Use Theorem 6.9 to characterize all possible realizations of the Fibonacci sequence. 6.8 PARTIAL REALIZATIONS

From a practical point of view, the foregoing realization results are somewhat unsatisfactory as they require total knowledge of the input/output map f or, what is equivalent, an infinite string of data for their implementation. In most situations, the data is obtained from measurements made during a finite number of time periods and the objective is to utilize the data to form a model. Dictates of time, money, and manpower usually preclude use of the idealistic realization procedures presented thus far. In this section we shall give a recursive algorithm for solution of the finitedata problem. The procedure to be presented possesses the following important features: (a) Given a finite string of data {J 1, J 2' ... , J N} = ,IN' the algorithm produces a system LN which minimally realizes ,IN' (b) If an additional piece of information J N+ 1 is added to ,IN, then the algorithm produces a new minimal realization LN + 1 for the augmented string ,IN + 1 with the property that LN appears as a subsystem of LN+ 10 i.e., the matrices in LN appear as submatrices of the corresponding ones in L N+ iProperty (b) of the algorithm is quite worthy of note since it enables us to calculate only a few new elements when the sequence ,IN is extended. As a result, the algorithm will be well suited for on-line modeling of a process for

134

6

REALIZATION THEORY

which the sequence J N is generated by means of measurements produced by, for example, the output of a machine, a chemical process, a biological experiment, and so forth. For ease of notation, we consider only the single-input/single-output case, although there is little problem in extending the results to the general situation. We need a few preliminary results. Define the data sequence Y1 = (J l' J 2' ...) and let Yi = uL- 1y 1, where Ul is the left shift operator, i.e.. Ul Y1 = (J 2, J 3, ...). If e denotes the vector (... , 0, 1,0, ...) having a "1" at the zeroth position, then Y1 = f(e) and the set {uLie} spans the input space n, while the set {y;} spans the image off in the output space r. (Here, of course, f is the input/output map to be realized.) We use the numbers in Y1 to form the Hankel matrix

J1 J2

J2 J3

Jm

Jm+ 1

I I

n

n+ 1

Jf=

J m+n-

1

and denote the submatrix consisting of the first m rows and n columns by Jf mn' (N ote: The slight change in indexing Jf from the previous section has been introduced to conform to standard results on this problem.) Since the order of the system equals the number of linearly independent elements in the set {y;}, we have the following useful fact.

Lemma6.2 If dim{Yt>Y2,"'} = n, then the elements Y1,Y2, ... ,Yn are the linearly independent vectors in {y;}. Furthermore, Jf nn is nonsingular and Jf nm has rank n for all m ~ n. PROOF (Ul K

Suppose

zero, i.e.,

°

YK + 1

+ I.f= 1 CiUL- l)Yl

=

YK+2

is linearly dependent on Y 10 ... , YK, i.e., for some {c.}, Then UL of this expression is also

+ CKYK+l + ... + CIY2

= 0,

implying YK+2 is also linearly dependent on Ylo"" YK' Thus the first n must be linearly independent. Consider the last row of Jf n+ l• n• Since Yn+l is linearly dependent on Yl,"" Yn, the rank of Jf n+ l,n is n. By symmetry, the same is true for Jfn,n+ iThus the ranks of Jf nm for m ~ n are no greater than n. But, since they can be no less than n they must all equal n. The realization algorithm is based on a factorization of the type

m

~

n,

rank

Jf nm ~

n - 1,

6.8

135


where P nn is lower triangular with Is on the diagonal, i.e., ... ----- ... -

Ji

: )i+l

.:,: J ..: ·,, .!J . ·, J ·: :

'--ji-~-l-"-i

i+ 2

: :

i+m - I : ,---------, i +m

:

j +m :J -oo ..

1.:

~ r~:. lPnl

Pn2

Pn,n-I

(6.12) The factors are not unique, so by setting certain elements in Qnm equal to zero, we shalI be able to calculate the Pijs recursively. Moreover, an addition of rows and columns to .JItnm will not change the numbers already calculated. The factorization algorithm has the folIowing steps:

l.

Set q Ii = J i for all i. If n = I, we are finished and P11 = (l). Ifn > I, assume we have at the ith step calculated all the PjkS and qjkS, j = 1, ... ,i - 1. Let s(j) be the smallest integer such that qj.o(j) =1= 0, j < n. Such an s(j) exists by virtue of the rank condition on J('nm' Set qk, oW = for k > j. Equation (6.12) then leads to a set of i - I equations, one for each column s(j), j = 1, ... , i-I. Because of the previous conditions, the unknowns Pil" .. , Pi, i - I can be recursively determined one by one from these equations. The submatrix P ii , together with (6.12), determines the remaining elements of the ith row of Q"m which completes the cycle. 2.

°

As an example, consider

J(' 45

= [:

2 We have sCI)

= l.

l ~ ~ ~] . I

323

= 0, i > 1. Then P21,I + I'q21 = I=>P21

Set qjJ

Further, q22 = 0, q23 = I, q24 = -1, q25 i > 2, the first and third columns give

+ P32q23

= J5

l.

= 2, Then, since s(2) = 3, qi3 = 0, 1 => P31 = 1, = 1 => P32 = 0.

P31Qll =

1 'q13

=

136

6

REALIZATION THEORY

Continuing the process, we finally obtain

Jff 45 =

[1

I 1 1 0

J[~

1 -1

2

1

_

[P" P~1

Next we define

*

P

n-l,n-1

P31

-

q11 ]

1

q~1

= [

qn-

-1 2 1 I .

o

Pn,n-1

Hn -

,

0

1]-

P32

Pn2

Gn -

21]

I I 0 1 1 0 0 0

1

= (l

0 .. ·0).

1, 1

The basic theorem for partial realization of a sequence"n can then be stated. Theorem 6.10 Given the data" = (J l' J 2' ...), let Jff n - 1, n - 1 be nonsingular and let m be any integer such that .J'l'nm has rank n - 1. Then the sequence "n+m-1 = (J l' J 2, ... ,1n+m-l) is minimally realized by the system

I: n where H n -

1,

Gn -

1

1

= (F n -

h

Gn -

1,

Hn-

1 ),

are as above and

Fnwhere

P n- 1,n-1 is the P:- 1,n-1 is as above.

1

= P;;':l,n-1 P:-1,n-1'

(n - 1) x (n - 1) principal submatrix of

P nn,

and

PROOF By the factoring algorithm, the last row of Qnm is zero. Hence, by writing the equality between the dashed columns of Jff nm in (6.12), we obtain

for

all i.

Consider the equations x(t

+

1) = F n - 1x(t) + Gn yet) = H n - 1x(t).

1u(t),

(6.13)

With x(O) = 0, u(O) = 1, and u(t) = 0, t > 0, the first equation in (6.13) describes the consecutive states which are just the columns of Qn-l,m in

6.8

137


(6.12), while the second gives y(t) = qlt = J, for t = 1, ... , m. We must now show that (6.13) realizes the remainder of the elements in J,,+m-l' This follows from the special form of F,,_ 1, i.e.,

0 122 1 1

111 121

o o

F"-1 = 1,,-1,1

1"-1,,,-1

Applying (6.13), for t = m + 1, ... , m + n - 1, extends Q"m to (2",,,+m-l' where the last row is extended as a zero row. Multiplying this result by P"", we obtain the extension of £"m to '*'",m+1l-1> but, due to the special form of F,,-I, £".,,+111-1 will have the elements

[J: ... Thus y(t) =

qlt

=

J, for all t

~

n

+m-

1.

Any realization of J"+1I1-1 extends it indefinitely. If such a realization has order k < n - 1, then £"-1."-1 has rank k by Lemma 6.2 which contradicts the assumptions. Hence, (6.13) is minimal. The realization algorithm then takes the following form: (1) Let k be the smallest integer such that J k of; O. Take N = 2k + 1 and form Jf'k+ 1, k+ i- It has rank ~ k. (2) Apply the factoring algorithm and find Pk + 1.k+ 1 and Qk+ l,k+ iIf the last row of Q is nonzero, the rank of Jf'k+ I, k+ 1 is k + 1. Increase N by 2, form £ k + 2. k + 2' and continue the factorization. Repeat this procedure until, say, for N = 2n - 1, the last row of Q"" is zero. By Lemma 6.2 such an n exists if the sequence J admits a finite-order realization. (3) From the formulas for 1 .,, - I' G,,_I, H,,-I, and F,,_I' calculate the partial realization E"-I' (Note that P;;!I"-1 can also be calculated recursively since this matrix is lower triangular) (4) Increase N by 1. Continue the factorization for £".,,-1' If the last row of Q"." + 1 remains zero, increase N by 1, and repeat. If the last row is zero for all m, we have found the total realization. (Of course, this cannot be decided so that the algorithm in practice will never stop. A stopping rule is introduced by setting an upper limit for m.) (5) If for some m (> n) the last element in the last row of Q"m becomes nonzero, then J" +111 _ 1 is not realized by the partial realization E,,_ i- In this case, pick a new point J,,+m and form £,,+ l,m' Continue the factorization,

P:-

138

6

REALIZAnON THEORY

pick a new point, and repeat until either Qn'''' for some smallest n' ::;; m has its last row zero or n' = m and the last row is nonzero. In the first case, go to step (3); in the latter case go to step (2). EXAMPLE Consider the sequence ,I = (I, I, 1,2,3,2,3) coming from the Hankel matrix used as an example of the factorization algorithm. In step (2), n = 2 and in step (3)we obtain I: 1 = «(I), (l), (I». In step (4)an addition of J 4 = 2 makes q23 = 1 :#: O. In step (5) we pick J 5 = 1 and form Jf' 33' Returning to step (2), we take two new points, J 6 = 3, J 7 = 2. This time the last row of Jf'44 is zero and we compute E, following step (3):

1 -I

o

~),

o

0),

-1

which realizes all the numbers given in Jf'45' EXERCISE

1. Extend the foregoing realization algorithm to the case of p outputs and m inputs. 6.9 REDUCED ORDER MODELS AND BALANCED REALIZATIONS

In practice it is often the case that the canonical state-space X associated with a behavior sequence J includes states that are either "hard" to reach or "difficult" to observe. Intuitively, we might expect that such states will tend to playa minor role in the system's behavior and that an approximate model I:*, formed using a state space that excludes these "bad" states, will still capture all of the dominant behavior of the original system. This is the basic idea underlying the notion of reduced-order models. The problem that immediately arises is that a state may be difficult to reach, but easy to observe, or conversely, and so it is not totally straightforward to decide whether to neglect such a state or not. The resolution of this dilemma leads to the concept of a balanced realization. To make the above ideas precise, we first need a measure of the difficulty involved in reaching or observing a given state. Such measures are provided by recalling the reachability and observability Gramians of Chapters 3 and 4. Let I: = (F, G, H) be a reachable and observable system, with F stable, i.e., Re AiF) < 0, i = 1,2, ... , n. The reachability Gramian is defined to be W =

LX) t!tGG'eF't dt.

6.9

REDUCED ORDER MODELS AND BALANCED REALIZATIONS

139

As shown in Exercise 6 of Chapter 3, the quantity (x, W -l~) gives the minimal energy needed to transfer the origin to the state ~; thus, if this number is "large" it means that ~ is difficult to reach relative to a state for which the number is small. In a similar fashion, the observability Gramian

serves to generate the measure (x o, Mx o) giving the observation "energy" in the state X o' Thus, ifthis measure is "large" X o is easy to observe relative to an initial state for which the measure is "small". As we have noted, a state x may have (x, W-lx) large, but (x, Mx) small, or conversely. To deal with these possibilities we introduce the idea of a balanced realization. Definition 6.4 We call a realization ~ = (F, G, H) balanced if W = M, i.e, if the reachability and observability Gramians are equal.

A central result on the existence of balanced realizations is Theorem 6.11 Let ~ = (F, G, H) be any canonical realization of dimension n < 00 with F stable. Then there exists a nonsingular matrix T such that (TFT- \ TG, HT- l) = t is balanced and moreover the Gramians for t satisfy

where

with the Ai being the positive square roots of the characteristic values of the matrix MW formed from the Gramians of any canonical realization. PROOF Let W, M be the Gramians for ~. There exists a Tl E GL(n) such that W = Tl Til' Consider T~ MTl = M i - Clearly, M 1 = Mil> O. Consequently, there exists an orthogonal matrix V such that M 1 = V' A 2 V, with A2 = diag(At, A.~, ... , A;), Ai > O. Take T = A 1/2VT 1 1. Using this T, the new Gramians it: Nt are equal and take the diagonal form asserted in the theorem. Under T, the matrix MW transforms to (T- l)' MWT', which has the same characteristic values as MW In the balanced realization above, the product MW becomes A2 , completing the proof.

Now that we have a procedure for constructing a balanced realization, let us tum to the question of eliminating "difficult" states and generating a

6

140

REALIZAnON THEORY

reduced order model. Let ~ = (F, G, H) be balanced, with W = M = A, and consider the control and observation energies, EC q.

Here In denotes the 11 x 11 identity matrix, while On1,n> denotes the zero matrix. (c) Form the matrices F, G, Has

F = En.prP[aJt'rrJMEmr,n'

(Ill

x

112)

G = En,prP[Jt'rrJEmr,n'

H = Ep.pr[Jt'rr]MEmr,n' where aJt' denotes the Hankel matrix formed by left-shifting the sequence

f by one position, i.e., the Hankel matrix obtained from the sequence af = {J 2,J 3 ,

. · .}.

2. (a) Let W(t, s) be the impulse-response matrix for a time-varying system. Define ifI(t l ,

... ,

tn) =

T I, .. ·, Tn

[W(t~' W(tn'

Td

.. ;.

" " Tn)].

T 1)

•••

W(tn'

Tn)

Show that if W is realizable, a necessary and sufficient condition for W to be minimally realizable by a completely controllable system of dimension 11 is that for some time sequence t I' ' .. , t n and an arbitrary time TO' there exists a time sequence T l, ••. , Tn' T; > TO, i = 1, ... ,11, such that

(b) Apply this result to the impulse responses W(t, r) =

e/+t,

W(t,T) =

{6: - till -

r ],

Itl~I,

ITI~I,

otherwise,

to decide whether they are minimally realizable.

6

142

REALIZA nON THEORY

3. Prove the following uniqueness result for extending a finite data string: Given a finite sequence of data f N' = (J I' J 2' ... , J N') such that rank .Jft' N' N = rank

.Jft' N' + I . N

(t)

= rank .Jft' N'. N + I

for some N, N' such that N + N' = N*, the extension of the sequence ,IN' to {J)'J2' ... 'JN.'JN.+I •••.• JN.+b ... }, I ~ k ~ 00, for which rank .Jft' m'. m

= rank .Jft' N'. N,

where m' + m = N* + k, is unique. (In other words, an extension of f N' which preserves the rank condition (t) is unique.) 4. Let f N' = (J l ' ...• J N') be a finite data string and let £N' be a partial realization of f N0' Show that the dimension of a minimal partial realization satisfies the following inequality: min dim

£N0

~

N°

L

N'

rank

.Jft'j. N0+)- j -

j=1

L

rank

.Jft'j. N0- i :

j=1

where .Jft'ij is the Hankel matrix associated with f N0' 5. Let f M = {J 1••.• , J M} be a finite data string and define the integer fI(M) = n)(M)

+ ... + nM(M),

where number oflinearly independent rows in the block row [J 1... J MJ. number of linearly independent rows in the block row [J 2 ••• J M] that are also linearly independent of the rows in the block rows [J 1···JM _ 1 ] , .. ·,and number of linearly independent rows in the matrix J M which are also linearly independent ofthe rows ofthe matrices J 1•.•.• J M - iAlso define N'(M) N(M)

the first integer such that every row ofthe block row [JN' +) ••• J MJ is linearly dependent on the rows of the Hankel matrix .Jft' N'. M - N' , the first integer such that every column of the block column

••

is linearly dependent on the columns of the matrix

.Jft'M-N.N'

Prove that (a) ii(M) is the dimension of the minimal realization of the sequence f

M'

143


(b) N(M) and N'(M) are (separately) the smallest integers such that the rank condition (R)

rank .1fN'N = rank .1fN' + 1.N = rank .1fN',N+ 1

holds for some extension of f M' (c) N(M) and N'(M) are (separately) the smallest integers such that Eq. (R) holds simultaneously for all minimal extensions of f M; (d) There is a minimal extension of f M of order M'(M) = N'(M) + N(M) for which Eq. (R) is true and whose realization can be computed by the Ho algorithm (but which is, in general, non unique). (e) Every extension that is fixed up to M'(M) is uniquely determined thereafter. 6. Let R(z) be a proper rational matrix with the expansion R(z) = J oz -

1

+ J 1Z - 2 + ....

Define the McMillan degree lJ of R as

... J.] lJ = rank .1f y _

0:

l'

J,:, J 2i

with y being the degree of the least common multiple of the denominator of R(z).

Show that: (a) R(z) has a minimal realization of dimension lJ. (b) If R(z) has the partial fraction expansion y

R(z) = :LZi(Z

i= 1

+ Ai)-1,

then lJ = :L[=1 (rank Zi)' (c) Compute the McMillan degree of z

Z2

R(z) =

[

and find a minimal realization.

+

1

+ 2z +

1

1

z+2

Z2

Z2

z+ 1

+

3~

+2

]

144

6

REALIZATION THEORY

Letf: Q -+ r be a given input/output map. Consider any factorization of f = hg through a space X such that

7.

Q

---.L- r X

If g is onto and h is 1-1, the factorization is said to be canonical with a minimal state space X.

(a) Prove that a canonical factorization always exists. (b) Show that all canonical factorizations are equivalent in the sense that hg = h'g' implies the existence of a unique isomorphism q: X -+ X' such that g' = qg and h = h'q. (c) Translate the above result into the language of matrix theory. 8. What is the connection between the partial realization problem of Section 6.7 and the classical problem of Pade' approximation. (Recall: The Pade' approximation problem is that of finding a scalar rational function whose Taylor series expansion matches that of a given analytic function up to a fixed number of terms.) 9. Let f/ = {JO,J1,J 2, ...} be a finitely realizable behavior sequence. (a) Show that J, = J; if and only if there exists a signature matrix S such that FS = SF',

sa =

H'

(Recall: a signature matrix is a diagonal matrix whose diagonal elements are ±1).

(b) Prove that f/ has a canonical realization L = (F, G, H) with F = P, G = H' if and only if Yt";; = Yt";;, i = 1, 2, 3, .... 10. (a) Prove that the sequence f/ = {J l' J 2' J 3""} admits a finite-dimensional realization of dimension n if and only if there exist scalars IXl' IX2, ••• , IXn such that i = 1, 2, ....

(b)

Show that the set

{IX;}

satisfy

for any F associated with a canonical realization of f/. (c) What is the connection between the IX; and the characteristic and minimal polynomials of F?

145

NOTES AND REFERENCES NOTES AND REFERENCES

Section 6.1 The first statements of the realization problem for linear systems given in transfer matrix form and an algorithm for its solution are found in Kalman, R., Canonical structure of linear dynamical systems, Proc. Nat. Acad. Sci. U.S.A. 48, 596--600 (1962).

Gilbert, E., Controllability and observability in multivariable systems, SIAM J. Control 1, 128-151 (1963).

Kalman, R., Mathematical structure of linear dynamical systems, SIAM J. Control I, 152-192 (1963).

An alternate treatment is by Kalman, R., Irreducible realizations and the degree of a rational matrix, SIAM J. Appl. Math. 13, 520--544 (1965).

The first effective procedure for carrying out the realization procedure in the case of input/output data given in "Markov" form is presented by Ho, B. L., and Kalman, R., Effective construction of linear state-variable models from input/ output functions, Reqelunqstechnik, Prozefi-Datenoerarbeit. 14,545-548 (1966).

Important new work on realization theory in the context of identification of econometric models has been presented in Kalman, R., System-theoretic critique of dynamic models, Int. J. Policy Anal. & Info. Syst., 4, 3-22 (1980).

Kalman, R., Dynamic econometric models: A system-theoretic critique, in "New Quantitative Techniques for Economic Analysis" (Szego, G., ed.) Academic Press, New York, 19-28, 1982.

Mehra, R., Identification in control and economics, in "Current Developments in the Interface: Economics, Econometrics, Mathematics" (Hazewinkel, M., and Rinnooy Kan, A. H. G., eds.) Reidel, Dordrecht, 261-286, 1982. Picci, G., Some connections between the theory of sufficient statistics and the identifiability problem, SIAM J. Appl. Math., 33, 383-398, (1977). Deistler, M., Multivariate time series and linear dynamical systems, Advances in Stat. Anal. & Stat. Computation, vol. 1 (to appear 1986). Deistler, M., General structure and parametrization of ARMA and state-space systems and its relation to statistical problems, in "Handbook of Statistics," vol. 5 (Hannan, E., et al., eds.) Elsevier, 257-277, 1985. .

Sections 6.2-6.4 The realization algorithm, together with the basic concepts of algebraic equivalence, minimality, etc., are discussed in the paper Silverman, L., Realization of linear dynamical systems, IEEE Trans. Automatic Control AC-I6, 554-567 (1971).

The canonical structure theorem given in Exercise 2(a), first appeared in Kalman, R., Canonical structure of linear dynamical systems, Proc. Nat. Acad. Sci. U.S.A. 48, 569--600 (1962).

6

146

Section 6.6

REALIZATION THEORY

A good reference for realization of transfer matrices is

Rubio, J., "The Theory of Linear Systems," Academic Press, New York, 1971.

Additional results are given by Rosenbrock, H. H., Computation of minimal representations of a rational transfer function matrix, Proc. IEEE 115,325-327 (1968). Mayne, D., Computational procedure for the minimal realization of transfer-function matrices. Proc.IEEE 115.1368--1383 (1968). Youla, D., The synthesis of linear dynamical systems from prescribed weighting patterns, SIAM J. Appl. Math, 14, 527-549 (1966). .

Section 6.8 A detailed mathematical (and historical) account of the partial realization problem is given by Kalman, R., On partial realizations of a linear input/output map, in "Ouillemin Anniversary Volume" (N. de Claris and R. Kalman, eds.). Holt, New York, 1968.

Similar work (carried out in collaboration with R. Kalman) is reported by Tether, A., Construction of minimal linear state variable models from finite input/output data. IEEE Trans. Automatic Control AC-15, 427--436 (1970).

and by Willems, J., Minimal Realization in State Space Form from Input/Output Data. Mathematical Inst. Rep., Univ. of Groningen, Groningen, Holland, May 1973.

The rather extensive connections and interrelationships between the partial realization problem. Kronecker indices, canonical forms, continued fractions and much more are developed in detail in Kalman, R., On partial realizations, transfer functions and canonical forms, Acta Poly. Scand., Math. and Compo Ser., No. 31,9-32,1979.

The recursive realization algorithm is due to Rissanen, J., Recursive identification oflinear systems, SIAM J. Control 9, 420--430 (197 I).

Other work along the same lines is reported by Rissanen, J., and Kailath, T., Partial realization of random systems, Automatica-J, IFAC 8, 389-396 (1972).

The actual numerical computation of a canonical realization by the recursive scheme given in the text is an unstable numerical operation. The reasons for the instability, as well as a procedure to "stabilize" the Rissanen's algorithm, are covered in de Jong, L. S., Numerical aspects of recursive realization algorithms, SIAM J. Control & Optim., 16, 646-659 (1978).

See also Pace, I. S., and Barnett, S., Efficient algorithms for linear system calculations II: Minimal realization, Int. J. Sys. Sci. 5,413-424 (1974).

CHAPTER

7

Stability Theory

7.1 INTRODUCTION

Historically, the circle of questions that have now grown into the body of knowledge known in various guises as "system theory," "cybernetics," "system analysis," etc., began with consideration of the equilibria positions of dynamical processes. In rough form, the basic question was whether or not a given equilibrium position was stable under sufficientlysmall perturbations in either the system parameters or initial condition, i.e., if a system were originally in a rest position and some outside influence caused it to depart from this state, would the system return to its original position after a sufficiently large amount of time? Clearly, questions of this type are of extreme importance not only for the type of classical mechanical systems which originally motivated them, but also for numerous economic, social, and biological problems in modern life. The type of stability just mentioned might well be termed "classical" stability since, not only did it originate with so-called classical problems, but it also makes no mention of a controlled system input. Thus classical stability is a property of the internal free dynamics of the process, clearly a limited situation from the viewpoint of modern system theory, although it remains a venerable branch in the theory of ordinary differential equations. In any case, of much greater interest to the system theorist is the idea of a controlled return to equilibrium. As before, the original system is perturbed from its equilibrium state, but now we wish to allow direct interaction with 147

148

7

STABILITY THEORY

the system by means of a control law in order to have the trajectory return to equilibrium. This new concept of controlled input to achieve stability raises several basic questions, among them: (i) Does there exist any control law returning the system to equilibrium? (ii) If there is more than one stabilizing law, can we isolate a unique law by specifyingadditional restrictions such as minimum energy, minimum time, etc.? (iii) What are the possibilities for altering the stability characteristics of the original free system by means of suitably chosen feedback control laws? In connection with controlled inputs, we are also faced with the basic dichotomy of openversus closed-loop (or feedback) control laws. Our discussion in Chapters 1-6 has been primarily devoted to controls of form u = u(t), the so-called open-loop laws. We will now see, however, that it is particularly advantageous to consider laws of form u = u(y(t), t) in which the controlling action is generated by the current output (or state) of the process. Such laws were originally introduced by Maxwell in connection with regulation of the Watt steam engine and have taken on increased importance with the development of modern techniques for information processing by analog and digital computers. The motivation for the openversus closed-loop terminology is clearly indicated in Figs. 7.1 and 7.2.

"'' --1__-. . J~

r-.---yl I)

,'01 ull)

FIG. 7.1 Open-loop control.

FIG. 7.2 Closed-loop control.

A stability concept differingfrom that presented above is structural stability. Here we are concerned with a family of systems and the type of stability behavior manifested as we pass from one member of this family to another. Roughly speaking, a given system is structurally stable if all ..nearby" members of the given family exhibit the" same" type of qualitative behavior. We shall make these notions precise in a later section. The important point to note now is that structural stability is a property that seems reasonable to demand for all mathematical models pretending to represent reality. This is due to the inherent uncertainties and simplifications present in all mathematical models so, if the model is to depict faithfully what nature presents to us, it is vitally important that the stability characteristics, which are an

7.2

149

SOME EXAMPLES AND BASIC CONCEPTS

inherent part of any viable physical system, be preserved under small perturbations of the model parameters. Naturally many mathematical techniques have evolved in response to the need to characterize the stability of given systems qualitatively and quantitatively. In this chapter we shall present three basic lines of attack on these questions: Routh-Hurwicz methods, Lyapunov techniques, and frequency-domain approaches. The first is purely algebraic in character, based on the satisfaction of certain algebraic relations which imply the asymptotic stability of the process. The remaining two approaches are much more analytic in flavor, their point of view being that the stability of the process may be inferred from the behavior of certain auxiliary functions naturally arising from the physics of the original problem, an energy function for the Lyapunov approach, a transient response function in the frequency method. 7.2 SOME EXAMPLES AND BASIC CONCEPTS

To gain a clearer understanding of the types of stability we shall encounter below, consider the motion of an oscillating system (mechanical, electrical, or ?) described by the linear differential equation

x + C1 X + C z x =

a,

C z =1=

x(O) = x(O) = O.

0,

(7.1 )

Consider the characteristic polynomial of (7.1): X(z)

It has zeros

Ctt> Ctz

=

ZZ

+ ClZ + Cz.

(7.2)

and we can easily verify that the general solution of (7.1) is

where k l and kz are constants of integrations (here we assume that Ctl # Ctz). If one of the numbers Ctl or Ctz has a positive real part, then the trajectory of (7.1) becomes unbounded as L--+ 00, regardless of the value of a. If both real parts are nonpositive, then both x and x can be made arbitrarily small by suitable choice of a. Thus the equilibrium of (7.1) is stable if and only if Re a, S 0, i = 1, 2. In physical terms, the statement" (7.1) is stable" means that a small change in the external driving force a results in only a small change in the displacement x(t). Since Cl and Cz have physical interpretations as damping and spring constants, respectively, we see immediately that a sufficient condition for Re rJ.j S 0 is Cl > 0, Cz > 0, which is the natural operating condition of the process. Later we shall see that necessary and sufficient conditions for the

150

7

STABILITY THEORY

stricter requirement Re (Xi < 0 are (I) C I > 0, (2) C I C2 > 0, which, in this case, is clearly equivalent to the natural conditions just imposed. Thus we see that an undamped spring (c I = 0) cannot possibly be stable with respect to external perturbing disturbances a =F 0, confirming our physical intuition about such processes. Now assume that the oscillating circuit is described by the equation

x+ x

=

x(o) = x(o) = 0.

a sin t,

(7.3)

The explicit solution of this equation is X(f) = ta(sin t - t cos f).

Since for arbitrarily small a =F 0, the system trajectory grows without bound as f ~ 00, this system is unstable with respect to external perturbations of the type a sin t, a =F 0. Equation (7.3) represents an undamped oscillatory circuit tuned to resonance and the instability noted corresponds to the well-known physical fact that an external perturbation whose frequency equals one of the natural vibratory modes of the system will result in unbounded oscillations, i.e., the system resonates with the disturbance. The simple harmonic oscillator also illustrates the concept of structural stability. Assume the equation of motion is X(O)

= a,

(7.4)

x(O) = 0.

Thus there is no forcing term and we investigate the influence of the parameters CI and C2 on the system trajectories. On physical grounds, we consider . only the situation CI ~ 0, C2 > 0. Considering the motion of (7.4) in the (x, x)-plane, we easily see that if CI = 0, the trajectories are concentric circles with the center at the origin and radii a(c2)1/2 (Fig. 7.3). Assume now that we introduce some damping into the system. As mentioned, this means mathematically that CI > in (7.4). If c 1 2 ~ 4c2 , the phase plane portrait of the system shows that the equilibrium point x = x = is a node (Fig. 7.4); while if Cl 2 < 4c2 , it is afocus

°

°

--+-+-1F-+H---+---x

FIG. 7.3 Trajectories of Eqs. (7.4) for c. =

o.

7.2

151

SOME EXAMPLES AND BASIC CONCEPTS

-----i'-------x

FIG. 7.4 A typical system trajectory when c 1 2 ~ 4C2 and the origin is a stable node.

CI

2

FIG. 7.5 A typical system trajectory when « 4C2 and the origin is a stable focus.

(Fig. 7.5). In either case the origin is stable with respect to perturbations in C l and C 2• This behavior is in stark contrast to the undamped case in which the origin is a center and its qualitative character may be changed by an arbitrarily small change in Cl' Thus the processes with Cl =1= 0 are examples of structurally stable systems, in that the qualitative behavior of the equilibrium point (focus, node) is preserved under small changes of the system structure. The foregoing considerations illustrate the main intuitive notions surrounding the stability concepts we are interested in pursuing. To create a mathematical framework within which these ideas may be studied, we must formalize our intuition with precise definitions. Definition 7.1 The equilibrium x = 0 of.the differential equation x = f(x), x(to) = xo, is called stable (in the sense of Lyapunov) if, for each B > 0, there exists a ~ > 0 such that

"x(t)"
0 such that

lim x{t) = 0

whenever

x=

Ilxoll < n.

f(x) is called an

7

152

STABILITY THEORY

x(t)

e

FIG. 7.6 Stability in the Sense of Lyapunov. REMARK The equilibrium x = 0 is assumed here to be isolated, i.e., the neighborhood Ilxoll < h contains no points x', other than x = 0, such that f(x') = 0 for h sufficiently small. EXERCISE

1. Show by counterexample, or other technique, that the equilibrium x = 0 may be stable but not an attractor and conversely (in other words, the two concepts are independent). The type of stability of most interest to us is when the notions" stable" and "attractor " are equivalent. This is given by the next definition. The equilibrium x = 0 of the system i = f(x) is called asymptotically stable (in the sense of Lyapunooi if it is both stable and an attractor. Definition 7.3

Definitions 7.1-7.3 refer to stability of a point which is the classical concept. We defer precise definitions of structural stability to a later section. 7.3 ROUTH-HURWICZ METHODS

We begin by considering the free (uncontrolled) constant, linear dynamical system (7.5) x(O) = xo, i = Fx, where x is an n-vector of states and F an n x n constant matrix. Since we are interested in the stability of the equilibrium state x = 0, we assume Xo ¥ 0 and

7.3

153

ROUTH-HURWICZ METHODS

seek conditions on F that ensure that the trajectories of (7.5) tend to zero as t --+ co, Imitating an argument from the theory of scalar linear differential equations, let us assume that (7.5) has a solution of the form

x(t) = e).lxo'

(7.6)

Substituting (7.6) into (7.5), we find

Ae).lxo

= Fe).lxo

A./xo

or

= Fx.;

Thus system (7.5) will have a nontrivial solution if and only if the parameter A satisfies the characteristic equation (7.7)

det(A./ - F) = 0.

The roots Ai" .. ,An of (7.7) are the characteristic roots of F and, as we see from (7.6), their position in the complex plane determines the stability behavior of (7.5) as t --+ co. It is clear that the condition i

=

1, ... , n,

(7.8)

is necessary and sufficient for stability of the origin, while the stronger condition (7.9) Re(Ai) < 0, i = 1, ... , n, is needed for asymptotic stability for all xo. EXERCISE

1. If T is a nonsingular n x n matrix, show that F and T FT - 1 have the same characteristic roots. The above considerations show that one road to the study of the stability of the origin for (7.5) lies in the direction of an algebraic criterion characterizing the position of the roots of the polynomial equation (7.7). More specifically, we desire a procedure based on the coefficients of (7.7) which ensures satisfaction of (7.9) (or (7.8)). Such a method is the well-known RouthHurwicz criterion.

Theorem 7.1 Let X(z) = aoz n + aiZ n- i + ... + an-iz + an be the characteristic polynomial of F. Then a necessary condition that the roots ofx(z) have negative real parts is ai/aO > 0, a2/aO > 0, ... ,an/ao > 0. Let z 1, •.• , z; be the zeros and, in particular, let z/ be the real roots, complex ones. Then

PROOF

zZ the

XF(Z) = ao

n (z j

z/)

n (z k

zZ),

7

154

STABILITY THEORY

and, combining the complex conjugate factors, we obtain

XF(Z) = ao

n (z -

z/)

n (Z2 -

+ Iz~12).

2 Re(z~)

k

j

If all the numbers z/ and Re(z;) are negative. we can only obtain positive coefficients for the powers of z when we multiply the two products together. EXERCISE

1. Give an example of a polynomial that satisfies Theorem 7.1, but has at least one root with a nonnegative real part. As the foregoing exercise illustrates, Theorem 7.1 is, unfortunately, far from a sufficient condition for stability of the polynomial XF(Z). It does, however, provide a very simple test to discover systems that have no chance of being asymptotically stable. Our next result, given without proof, is the Routh criterion which provides necessary and sufficient conditions for stability.

Form the number array

Theorem 7.2

C22 = a4 - r2a S . C32 = a6 - r2a7' · · · . Cl3

=

C21 -

r3Cn , C23 =

C31 -

C33 =

C41 -

r3 c 32 , r3 C42'···.

i = 1,2, . . .• j = 2,3, ... ,

Ctn =

an'

Define Cm+l,O

=

Cm+I.2

= an,

Cm+l. l C m2

= =

Cm+I.3 C m3

=

= 0

0

if

if

n = 2m n=2m-1.

Then the polynomial XF(Z) has only roots with negative real parts

if Theorem 7.1 is satisfied and CII

> 0,

Cl2

>

O,,,,,Cl n

>

O.

if and only

7.3

155

ROUTH-HURWICZ METHODS

REMARK If one of the numbers Cij = 0, the Routh scheme breaks down, in which case, it is easy to see that then XF(Z) cannot be a stability polynomial.

A close relative of the Routh scheme was discovered by Hurwicz in connection with the study of the stability of a centrifugal steam engine governor. This result involves the array of numbers

A=

a1 ao 0 0 a3 a2 a1 ao as a4 a3 a2 0 0

0 0

0 0

0 0

0 0 0

0 0 0

an-1 0

an-2 an

formed from the coefficients of XF(Z), If 0 < 2i - j ~ n, the general element in the array is aij = a2;- j,otherwiseaij = O. We form the sequence of principal subdeterminants

These quantities, the so-called Hurwicz determinants, give rise to the next theorem. Theorem 7.3

The polynomial XF(Z) has all its roots with negative real parts

if and only if Theorem 7.1 holds and H 1 > 0,

H 2 > 0, ... , H n > O.

PROOF By elementary row operations, we convert the array A to lower triangular form. The numbers appearing on the main diagonal are the quantities C 11 = a 1, C 12' .•. , C 1 n = an, which are precisely the quantities from the Routh scheme of Theorem '7.2. Since elementary row operations leave the principal subdeterminants invariant, we have

and

Thus the Hurwicz determinants are all positive if and only if the same is true of the Routh numbers {Clj}'

156

7

STABILITY THEORY

EXERCISES

1. Check the following polynomials for stability:

xAz) = (b) XF(z) = (a)

(c)

XF(Z)

=

Z6 Z4

Z5

+ 5z 5 + 3z4

+ 2z + 2

-

2z 3 +

Z2

+

7z

+

6;

I;

+ 3z4 + az 3 + tz 2 + 18z + 1.

2. Use the Routh-Hurwicz criteria to state explicitly the necessary and sufficient conditions for the asymptotic stability of second-, third-, and fourth-order systems. 3. (Michailov Criterion) Define the polynomials

V(z)

=

X(z)

+ iY(z).

Show that XF(Z) = aoz n + alZ n - 1 + ... + an is a stability polynomial if and only if the zeros of X and Yare all real and separate each other. From a computational point of view, the Routh scheme is probably preferable to the Hurwicz procedure, although the Hurwicz method is particularly valuable when the influence of changes in the coefficients on the system stability is under investigation.

7.4 LYAPUNOV METHOD

In many situations, particularly when the dimension of the system is high, it is a nontrivial task to determine the coefficients of the characteristic polynomial of F. Unfortunately, it is precisely these quantities that are required in order to apply the simple algebraic tests prescribed by the Routh-Hurwicz theorems. Thus we are motivated to seek alternate procedures that operate with only the given system data, i.e., criteria that can be directly applied to F itself. Such .a procedure is the celebrated "second" method of Lyapunov. The Lyapunov procedure is intuitively based on the simple physical notion that an equilibrium point of a system is asymptotically stable if all trajectories of the process beginning sufficiently near the point move so as to minimize a suitably defined "energy" function with the minimal energy position being at the equilibrium point itself. The trick, of course, is to discover an appropriate energy function that is both complex enough to capture the relevant stability behavior and simple enough to obtain specific mathematical expressions characterizing stability or instability.

7.4

157

LYAPUNOV METHOD

To make the foregoing notions more precise, we consider a process described by the equations

x=

f(x),

x(O) =

Xo,

(7.10)

where the origin is the equilibrium point under consideration, i.e.,- f(O) = O. For simplicity, assumefis a continuous function. Definition 7.4 A function V(x) is called positive definite if V(O) = 0 and if V(x) is positive at every other point in a neighborhood U of the origin. EXERCISES

1. (a) Show that if V(x) is positive definite, then there exist continuous, strictly monotonically increasing functions q>l(r), q>2(r), r > 0, with q>1(0) = q>2(0) = 0 such that

q>l(llxll) :5 V(x)

:5

q>2(llxll).

1

(b) Show that V- (x ) satisfies

q>i 1(llxll) :5 V- 1(x ) :5 q>1 1(llxll). 2.

If u(t), v(t) satisfy the differential equation duldt = f(u)

and the differential inequality dtfdt :5 f(v),

t > 0,

respectively, show that if u(O) = v(O), then v(t) :5 u(t),

t > O.

The above exercises enable us to prove the following basic result on stability of the equilibrium x = 0 of (7.10). Theorem 7.4 If there exists a positive definite function V(x) whose derivative, when evaluated along trajectories of (7.10), is negative definite, then the equilibrium of (7.1 0) is asymptotically stable. PROOF

By assumption, we have

for some continuous, monotonically increasing function q>4. Applying Exercise I(b), we have (7.11)

158

7

STABILITY THEORY

and, hence, for x. a continuous, increasing, positive function. By Exercise 2, we see that

for some function q and a continuous, decreasing function p. Because of (7.11)

Ilxll s

lpi1(q(VO)p(t) ~

lpi 1(q(lpl(ll xo II)))p(t),

which finally yields

Ilxli for some decreasing function as t -+ 00.

0".

~ lp(lI xolI)O"(t)

Thus all trajectories of (7.10) decrease to 0

From Theorem 7.4, we are led to define the notion of a Lyapunov function for (7.10). Definition 7.5

A function Vex) such that

(a) V is positive definite for all x E R" and (b) dV/dt < 0 along trajectories of (7.10) is called a Lyapunov function for (7.10). There are many delicate nuances associated with the use of Lyapunov functions to study stability, particuiarly in the cases when V = 0 along system trajectories. However, we shall not dwell on these situations here, as our concern is with illustrating the basic ideas unencumbered by technical details. We refer to the treatises listed at the end of the chapter for a complete discussion of all issues. The main theorem associated with the use of Lyapunov functions is Theorem 7.4, which gives only sufficient conditions for stability. In addition, it says nothing about how one goes about finding a suitable function or, for that matter, whether or not such a function even exists! However, it can be shown that the existence of an appropriate Lyapunov function is also a necessary condition for stability and, what is equally important for our purposes, in special cases we can develop a systematic procedure for obtaining such functions. Since our interest in this book is with linear systems, we seek to apply the Lyapunov procedure to the linear system

x=

Fx,

x(O)

=

Xo ( =1= 0).

7.4

159

LYAPUNOV METHOD

As a candidate Lyapunov function, we choose Vex) as the quadratic form Vex) = (x, Px),

where P is an, as yet, unknown symmetric matrix. In order that Vex) be a Lyapunov function for the system, we must check condition (b) on the derivative of Vex). We obtain Vex)

= (x, Px) = (x, F'P

+ (x, Px) < 0 + PF)x) < 0,

which implies that the equation F'P

+ PF

(7.12)

= -C

must be solvable for any C > o. Furthermore, condition (a) of Definition 7.5 implies that the solution of (7.12) must be positive-definite, i.e., P > O. Hence, we have Theorem 7.5. Theorem 7.5

The equilibrium of the system

x=

Fx

is asymptotically stable if the linear matrix equation F'P+PF= -C has a positive-definite solution P for any matrix C > O.

EXERCISES

1. Show that the existence of a quadratic Lyapunov function is also a necessary condition for the stability of the origin for the equation x = Fx. 2. Prove the following strengthened version of Theorem 7.5: The equilib-

rium of x = Fx is asymptotically stable if and only if: (1) Eq. (7.11) has a positive semidefinite solution P ~ 0 for any C ~ 0, and (2) no trajectory of x = Fx vanishes identically on {x : Vex) = O} except the trajectory x == O. EXAMPLES

(I)

the solution of F' P

Let F = [- J

+ PF

= - C

_n

C = [c i j

].

It is easily verified that

is given by

for all C>

o.

Thus we verify that the system x = Fx is asymptotically stable.

160

7

STABILITY THEORY

n

(2) Let F = [_? _ C = [Cjj]. Since trace F < 0 and det F > 0, by the Routh-Hurwicz criteria, it is easily checked that F is a stability matrix. We verify this conclusion by application of Theorem 7.5. Substituting F and C into Eq. (7.12), we obtain equations for the components of P yielding

P=

[1 if C > 0, we conclude asymptotic stability of the origin for the equation x = Fx. It is interesting and instructive to also consider the stability problem for discrete linear systems, i.e., when x(t

+

I) = Fx(t),

t

=

0, 1, ....

(7.13)

Obvious modifications of Definitions 7.4 and 7.5 lead us to consider the candidate Lyapunov function V(x) = (x, Px),

where instead of

V, we form

the difference

d V = V(x(t

+

1» - V(x(t».

Using (7.13), this yields dV

= (x(t), (F'PF

- F)x(t»

=

-(x(t), Cx(t».

The foregoing considerations lead to the following result. Theorem 7.6 The origin is an asymptotically stable equilibrium position for system (7.13) if and only if there exists a positive-definite matrix P which is the unique solution of the matrix equation F'PF - F = -C

(7.14)

for all C > O.

Thus (7.14) is the Lyapunov matrix equation for discrete linear systems. In terms of the characteristic roots of F, the stability requirement is now that all characteristic roots Ai(F) must satisfy i = 1, ... , n.

(7.15)

7.4

161

LYAPUNOV METHOD

This is easily seen from the fact that (7.13) has the solution t = 1,2, ....

For arbitrary x(O), x(t) -+ 0 if and only if the bound (7.15) is satisfied by the characteristic roots of F.

EXERCISES

1. Show that the solutions of(7.12) and (7.14)are related by the transformations

where B is the Cayley transform of F' such that B

= (1 + F')(I - F')-I.

That is, if P is the solution of (7.12), then using the transformations F -+ B, C -+ 2(1 - FT IC(1 - F)-I, we may convert (7.12) into (7.14), retaining the same solution P. 2. (a) Verify that the solution of the matrix equation

dX/dt = AX

+ XB,

X(O) = C,

is given by X(t) = eAtCe Bt. Use this fact to show that the solution of AX XB = C is given by

+

assuming the integral converges. (b) What conditions must A and B satisfy to guarantee convergence of the integral for X? (c) Specialize the above results to Eq. (7.12). 3. Show that the matrix equation PF+F'P= -C

is equivalent to the vector equation [(F'

® I) + (1 ® F')]a(P) = -a(C),

162

7

STABILITY THEORY

where ® denotes the Kronecker product oftwo matrices and a the "stacking" operator which stacks the columns of an n x m matrix into a nm x 1 vector, i.e., if A = [aij], then

7.5 FREQUENCY-DOMAIN TECHNIQUES

As might be expected, stability theory also possesses a geometric side in addition to its analytic and algebraic sides as typified by the Lyapunov and Routh-Hurwicz procedures, respectively. Although we shall not go into great detail about the frequency-domain approaches in this book, they should be considered as another tool in the systems analyst's arsenal of methods to employ for the study of particular questions. Basically, the frequency methods investigate various stability properties of a system by analyzing the transfer matrix Z(A) as a function of the complex variable A.. Recall that if 1: is given by

x=

Fx

+ Gu,

x(O) = 0,

y = Hx,

(7.16)

then

Thus the characteristic roots of F coincide with the poles of the rational matrix function Z. The frequency-domain methods study the asymptotic behavior of both the open-loop system (7.16) and the closed-loop system obtained from (7.16) by using a feedback-type input u = - Kx for some m x n matrix K. One of the basic geometric results involves determination of conditions under which the closed-loop system will be asymptotically stable if the openloop system is stable. This is the so-called Nyquist theorem which gives a simple geometric criterion for stability of the closed-loop system for the case of a single-input/single-output system. To state this result we must first discuss the notion of a response diagram for a rational function of a complex variable s. Let r(s) be a proper rational function of the complex variable s, i.e., r(s) = p(s)!q(s), where p and q are monic polynomials with deg q > deg p. Then we call the locus of points r(r)

=

{u

+ iv: u =

Re[r(iw)], v

= Im[r(iw)], -

00 ~

w ~ oo}

the response diagram of r. In other words, the response diagram is the image of the line Re s = 0 under the mapping r. The basic result of Nyquist follows

7.5

163

FREQUENCY-DOMAIN TECHNIQUES

upon setting res) = Z(s) in the open-loop case, and res) = Z(s)/O + kZ(s» in the closed-loop case, with k being the output feedback gain, i.e., u(t) = ky(t) = khx(t), where the output matrix h is 1 x n. Nyquist's Theorem The closed-loop system is asymptotically stable if the open-loop system is stable and if its response diagram, traversed in the direction of increasing t» has the" critical" point ( - 1/ k, 0) on its left. Geometrically, the situation is as in Figs. 7.7 and 7.8. The proof of Nyquist's result can be obtained as a fairly straightforward application of the principle of the argument from the theory of functions of a complex variable.

-11k

FIG.7.7

Nyquist diagram: stable case.

FIG.7.8

Nyquist diagram: unstable case.

A far-reaching generalization of Nyquist's theorem, which also gives instability criteria, has been obtained under the name the "circle theorem." The basic technique used for this result is based on a scalar representation of the equations of motion. We consider the system x(t) = Fx(t) y(t) = hx(t),

+ gu(t),

(7.17)

and the related differential equation x(t) = Fx(t) - gf(t)hx(t).

(7.18)

It is easy to verify that if (7.17) is controllable, we can reduce it to control canonical form and the differential equation (7.18) may then be written in the form P(D)x(t)

+

f(t)q(D)x(t)

=

0,

where p and q are relatively prime polynomials of degree n, D the differential operator D = dldt, and f(t) a piecewise-continuous function such that (X < f(t) < {3.

7

164

Letting .@(IX, {3) represent the disk .@(IX,

+

{3) = {u

+~

iv: [u

(~

+ ~)

J

+

2

v
0 and that no characteristic values ofF lie on the line Re A = O. Iffis a piecewise-continuous function satisfying IX < f(1) < {3, then (a) All solutions of x = [F - gf(t)h]x are bounded and go to zero as t -+ 00 provided the response diagram ofZ(A) does not intersect the circle .@(IX, /3) and encircles it exactly v times in the counterclockwise sense. (b) At least one solution of x = [F - gf(t)h]x does not remain bounded as t -+ 00 provided the response diagram of Z(A) does not intersect the circle .@(IX, {3) and encircles itfewer than v times in the counterclockwise sense. EXERCISE

1. (a) Use the circle theorem criterion to show that the equation x(t)

is stable if, for some

IX

+

2x(1) + x(t)

+

f(t)x(t) = 0

> 0, we have IX 2

< f(t)

+

1
ReA p + l ~ ... ~ ReA n • Let T be the transformation that diagonalizes F. Then T FT - 1 = diag(Al , ... , An) and the rows of T are the transposes of the characteristic

7.8

173

OBSERVERS

vectors of F. Note that since (F, g) is not assumed to be controllable, the diagonalized system ~ = F~

+ (ju

is not identical with the Lur'e-Lefschetz-Letov canonical form, as 9 will have some entries equal to zero if(F, G) is not controllable. To prove necessity, we have ~i

=

A.i~i

+ Tqu,

i

=

1, ... , n.

In particular, if i :5: p, we see that ~i is controllable by u only if (Tg)i # 0. That is, the ith mode of F is not orthogonal to 9 which, by Corollaries 3 and 4 to Theorem 3.9, implies that ~i lies in the subspace generated by g, Fg, ... , Fn-1g.

Conversely, suppose

~i

lies in the subspace generated by the vectors of

°

g, Fg, ... , Fn-1g. Since ~i is just the ith row of the diagonalizing matrix T, it is clear that (~i' g) # 0, i.e., the element (Tg)i # which implies that ~i is controllable by the single-input terminal g.

EXERCISE

I. Establish Theorem 7.8 for multi-input systems. 7.8 OBSERVERS

We have noted that under the assumption of complete reachability, the poles of a system I: may be placed at arbitrary locations by means of a suitably chosen linear feedback law. There is, however, a basic assumption implicitly made in this result, namely, that the entire state vector x(t) is accessible for measurement at all times. This is clearly an unacceptable assumption in many physical problems and seriously diminishes the practical value of the pole-shifting theorem. In order to overcome the limitation of inaccessible states, the concept of an observer has been introduced. The basic idea is to replace the true, but unmeasurable, state x(t) by an estimated state ~(t), where ~(t) is constructed from the system output y(s), s :5: t, which, by definition, is totally at our disposal. The hope, which we shall justify in a moment, is that use of the estimated state ~(t) will provide the same amount of "information" as the true state x(t) (at least as far as stability properties are concerned). The justification for the above hope requires a detailed analysis using stochastic control theory. However, the following simple result provides a basis for the approach.

174

7

Theorem 7.9

STABILITY THEORY

Consider the system

x=

Fx + Gu, y(t) = Hx, ~ = Fx + L[y(t) - Hx] u(t) = - Kx(t).

+ Gu,

Then the characteristic polynomial ofthis system XT(Z) satisfies the relationship XT(Z)

=

XF-GK(Z)XF-LH(Z),

i.e.;the dynamical behavior ofthe entire system is the direct sum ofthe dynamics of the regulator loop (the matrix F - GK) and the dynamics of the estimation loop (the matrix F - LH). PROOF

X~z)

is the characteristic polynomial of the matrix [:H

Making the change of variable x we obtain the new system

F-~~~GKJ ~

x, X ~ x - X, which leaves XT invariant,

x = (F X=

- GK)x + GKx, (F - LH)x.

Since this is a triangular system, the result follows immediately. The above result suggest the following approach to stabilization of a completely controllable and completely ·constructible system ~: Pick a matrix K yielding a stable control law, i.e., XF-GK is a stability polynomial. Similarly, pick a matrix L yielding a stable state estimator, i.e., XF-LH is a stability polynomial. Define the system to be the system formed by the original dynamics plus the state estimator equations is in Theorem 7.6. Then the overall (closed-loop) system will be stable. In fact, we can actually obtain the same result as in the pole-shifting theorem since, by the controllability/constructibility hypothesis, the matrices K and L can be selected to place the overall system poles at any desired location. These results illustrate the importance of constructibility, as well as controllability, in stability analysis and regulator design. 7.9 STRUcruRAL STABILITY

As noted in the introductory section, an important feature of any dynamical system that purports to represent a real physical process is that small perturbations in the parameters of the model leave the "essential" features of the

7.9

175

STRUCTURAL STABILITY

model unchanged. Within the context of stability, a prime candidate for such an essential feature certainly is the asymptotic stability of the process, i.e., if the model is asymptotically stable, then .,nearby" systems should also possess this property. Since the borderline between asymptotically stable systems and unstable processes are systems whose dynamical matrix F possesses at least two purely imaginary characteristic roots, it is not surprising that such systems will play an essential role in formulating our main results on the structural stability of constant linear systems. First, however, let us consider certain notions of linear system equivalence which will make the job of establishing structural stability conditions particularly simple. Consider the free linear system

x=

Fx.

If h: R" -+ R ft is a one-to-one coordinate transformation (not necessarily linear), then we have the following definitions.

Definition 7.6 Two systems (I) x = Fx, (II) Y = Ay are said to be equivalent if the mapping h takes the vector function x(t) into the vector function y(t) for all t ~ O. Under these conditions, systems (I) and (II) are said to be: (a) linearly equivalent if h: R" -+ R" is a linear automorphism, i.e., h: x-+ T(t)x(t) = y(t), where T(t) are nonsingular matrices, t ~ 0; (b) differentially equivalent if h is a diffeomorphism; (c) topologically equivalent if h is a homeomorphism, i.e., h is 1-1, onto, and continuous in both directions. EXERCISES

1. Prove that linear equivalence implies differentiable equivalence which, in turn, implies topological equivalence. 2. Show that each type of equivalence indeed does define a true equivalence relation, i.e., it is reflexive, symmetric, and transitive.

We give several results about equivalence which, although they are of great importance, are not proved here as they are only stepping stones to our main questions about structural stability. The first is Theorem 7.10.

Theorem 7.10 Let the matrices F and A have simple (distinct) characteristic values. Then systems (I) and (II) are linearly equivalent if and only if the characteristic values of F and A coincide. REMARK

systems F

Simplicity of the characteristic values is essential as the two A = [A indicate. The next theorem shows that

= [A YJ and

n

7

176

STABILITY THEORY

for linear systems there is no need to distinguish between linear and differentiable equivalence. Theorem 7.11 The two systems (I) and (II) are differentiably equivalent ifand only if they are linearly equivalent.

Our final preliminary result forms the cornerstone for the main result on structural stability of linear systems. We consider the two linear systems (I) and (II), both of whose characteristic values have nonzero real parts. Let m_(F) be the number of characteristic roots of F with a negative real part, while m+(F) is the number with a positive real part, m_(F) + m+(F) = n. The central result on topological equivalence is given next. Theorem 7.12 A necessary and sufficient condition that (I) and (II) be topologically equivalent is that and REMARKS (1) This result asserts that stable nodes and foci are equivalent to each other but are not equivalent to a saddle. Thus see Fig. 7.9.

FIG.7.9

Relationship among stable nodes, foci, and saddles.

(2) The number m_ (or, of course, m+) is the unique topological invariant of a linear system. Armed with the preceding results, we return to the question of structural stability. Our goal is to determine conditions on F such that a continuous perturbation of sufficiently small magnitude leaves the qualitative features of the system trajectory invariant. Thus we ask that F and the new matrix F + eP = A have the same phase portraits for e sufficiently small and P an arbitrary, but fixed, perturbation matrix. However, Theorem 7.12 shows that the only system invariant under continuous transformations of R" is the number m _ . Thus F and A can have equivalent phase portraits if and only if m_(F) = m_(A),

which, since the characteristic roots of a matrix are continuous functions of the elements of the matrix, implies the next theorem. Theorem 7.13 The matrix F is structurally stable with respect to continuous deformations if and only ifF has no purely imaginary characteristic roots.

177


REMARK It is important to note that the magnitude of the allowable perturbation (the size of B) depends on the root of F nearest the imaginary axis. In particular, this shows the importance of the pole-shifting theorem since, by suitable feedback, we can arrange for all characteristic roots of the closed-loop system to be far away from the imaginary axis. Such a system then exhibits a high degree of structural stability as comparatively large perturbations of the system do not alter the phase portrait.


1. If the solutions of x = Fx are bounded as t -+ 00, show that the same is true of the solutions of x = (F + P(t))x provided that SO II pet) 1/ dt < 00. (Such systems are called almost constant iflimr _ 00 pet) = P, a constant matrix.) 2. (Floquet's Theorem) Show that the solution of the matrix equation

dXjdt

= P(t)X,

where pet) is periodic with period

1" and

X(t)

=

X(O) = I, continuous for all t, has the form

Q(t)eB ' ,

where B is a constant matrix and Q(t) has period 1". + alz"- 1 + ... + all be a polynomial with only simple zeros. Show that the zeros of p(z) are all real and negative if and only if

3. (Meerov Criteria) Let p(z) = aoz"

(a) the coefficients aj 2:: 0, i = 0, I, ... , n; (b) the Hurwicz determinants for jJ(z) = p(Z)2 + Zp'(Z2) are all positive. 4. Let fez) = [p(z) + (_I)"P( -z)]j[P(z) - (_I)"P( -z)]. Then p(z) is a stability polynomial if and only if (a) fez) is irreducible, (b) Re fez) > 0 when Re z > 0, (c) Re fez) < 0 when Re z < O. 5. Prove the following generalized stability theorem: The origin is stable for the system x = Fx if andonly ifRe lj(F) S; 0, i = I, ... , n and the elementary divisors ofF corresponding to those rootsfor which Re )"(F) = 0 are linear. 6. Show that the matrix

F = P-1(S - Q), where P, Q are arbitrary positive-definite matrices and S is an arbitrary skew-symmetric matrix, is always a stability matrix. Use this result to show that if F is stable, then F + A will also be a stability matrix if A is of the form A = P-1(So - Qo)' In other words, this gives a

178

7

STABILITY THEORY

sufficient condition for the sum of two stability matrices to again be a stability matrix. 7. Let p(z) = aoz n + alZn-1 + ... + an- 1z + an and let H; denote the nth Hurwicz determinant. If {Sk} are the roots of p(z), prove Orlando's formula

H;

= (_l)n(n+l)/2 a on 2 - n n (Sj

+ sd·

i5k

8. Define the matrix B = I + 2(F - I)-I. Show that F is a stability matrix if and only if B satisfies the condition Bk .... 0, k = 1,2,3, .... 9. Show that Re AiF) < - U, i = 1, ... , n, if and only if for every positive definite matrix C there is a unique positive definite Q such that F'Q +

QF

+ 2uQ = -c.

10. Let (F, G) be controllable and define

Show that if t 1 > 0, then all solutions of the linear system

x=

(F - GG'W- 1(0, t d )x

are bounded on [0, 00]. 11. Consider the linear system

x + P(t.)x = where p(t) = - p(- t) = p(t stable if

+

0,

1) ~ 0. Show that the solution is uniformly

12. (Output Feedback Stability) Let I: = (F, G, H) be a minimal realization of the transfer matrix Z(A.). Assume Re A.j(F) < 0, i = 1, ... , n, and suppose that for all real A. I - Z'( - iA)Z(iA)

Prove that if I - K'(t)K(t) ~ el >

°

x = (F -

~

0.

for all t > 0, then all solutions of

GK(t)H)x(t)

are bounded and approach zero as t .... 00.

179

NOTES AND REFERENCES NOTES AND REFERENCES

Sections 7.1-7.4 General references on stability theory both for classical, as well as controlled dynamical systems include Hahn, W., "Stability of Motion." Springer-Verlag, Berlin and New York, 1967. LaSalle, J., and Lefschetz, S., "Stability by Liapunov's Direct Method with Applications." Academic Press, New York, 1961. Bellman, R., "Stability Theory of Differential Equations." McGraw-Hill, New York, 1953. Barnett, S., and Storey, C., "Matrix Methods in Stability Theory." Nelson, London, 1970. Cesari, L., "Asymptotic Behavior and Stability Problems in Ordinary Differential Equations." Springer-Verlag, Berlin and New York, 1963.

For an extensive survey of the uses of Routh's method for the solution of a wide range of problems in stability and control, see Barnett, S. and Siljak, D., Routh's algorithm: A centennial survey, SIAM Review, 19,472-489 (1977).

For connection with realization theory, see Fuhrmann, P., On realization of linear systems and applications to some questions of stability, Math. Syst. Th., 8,132-141 (1974).

Section 7.5

The material of this section follows

Brockett, R., "Finite-Dimensional Linear Systems," Chap. 4. Wiley. New York, 1970.

For the original development of the circle theorem, see Popov, Y., Hyperstability and optimality of automatic systems with several control functions, Rev. Roumaine Sci. Tech. Sir. Electrotech. Enqerqet, 9, 629-690 (1964).

See also Naumov, B., and Tsypkin, Ya. Z., A frequency criterion for absolute process stability in nonlinear automatic control systems, Automat. Remote Control2S, 765-778 (1964). Aizerman, M., and Gantmacher, F., "Absolute Stability of Regulator Systems." Holden-Day, San Francisco, California, 1964.

An interesting use offrequency-domain ideas for the reduction of dimensionality is given in Lucas, T. and Davidson, A., Frequency-domain reduction of linear systems using schwarz approximation, Int. J. Control, 37,1167-1178 (1983).

Section 7.6 The first proof of the pole-shifting theorem (for more than a single input) was given by Wonham, W. M., On pole assignment in multi-input controllable linear systems, IEEE Trans. Automatic Control AC-12. 660-665 (1967).

A much simpler proof is found in Heymann, M., Comments on pole assignment in multi-input controllable linear systems, IEEE Trans. Automatic Control AC-13, 748-749 (\968).

180

7

STABILITY THEORY

The pole-shifting theorem raises the additional question of how many components of the system state actually need be measured in order to move the poles to desired locations. This problem has been termed the problem of" minimal control fields" and is treated by Casti, J., and Letov, A., Minimal control fields, J. Math. Anal. Appl. 43,15-25 (1973).

for nonlinear systems and in Casti, J., Minimal control fields and pole-shifting by linear feedback, Appl. Math. Comput, 2, 19-28 (1976).

for the linear case. Various refinements, extensions and generalizations of the basic Pole-Shifting Theorem continue to be of interest. For a sample of the recent results, see Armentano, Y., Eigenvalue placement for generalized linear systems, Syst. Controls Lett., 4, 199-202 (1984). Lee, E. B. and Lu, W., Coefficient assignability for linear systems with delays, IEEE Tran. Auto. Cant., AC-29, 1048-1052 (1984). Emre, E., and Khargonekar, P., Pole placement for linear systems over bezout domains, IEEE Tran. Auto. Cont., AC-29, 90-91 (1984).

Section 7.7

A detailed treatment of system stability utilizing the "modal" point of view is provided in the book by

Porter, B., and Crossley, R., "Modal Control: Theory and Applications." Barnes & Noble, New York, 1972.

The paper by Simon, J., and Mitter, S., A theory of modal control, Information and Control 13, 316--353 (1968).

should also be consulted for additional details. Section 7.8

The concept of an observer seems to first have been introduced

in the papers Luenberger, D., Observing the state of a linear system, IEEE Trans. Military Elect. MIL-8, 74-80 (1964). Luenberger, D., Observers for multivariable systems, IEEE Trans. Automatic Control AC-ll, 190--197(1966).

The results by Casti and Letov, cited under Section 7.6, also address the question of stabilization of ~ when not all components of the state may be measured. For a recent book-length treatment of the observer problem, see O'Reilly, J., "Observers for Linear Systems," Academic Press, New York, 1983.


181

Section 7.9 An excellent introduction to the basic notions of structural stability may be found in the texts by Hirsch, M., and Smale, S., "Differential Equations, Dynamical Systems, and Linear Algebra." Academic Press, New York, 1974. Arnol'd, V., "Ordinary Differential Equations." MIT Press, Cambridge, Massachusetts, 1973.

Our approach in this section follows the latter source. More detailed results, requiring a high degree of mathematical sophistication, are given in the books by Nitecki, Z., "Differentiable Dynamics." MIT Press, Cambridge, Massachusetts, 1971. Peixoto, M. (ed.), "Dynamical Systems." Academic Press, New York, 1973.

The closely related topic ofcatastrophe theory is covered for the beginner by Zeeman, E. C., Catastrophe theory, Sci. Amer. (April 1976). Amson, J. C.; Catastrophe theory: A contribution to the study of urban systems? Environment and Planning-B2, 177-221 (1975).

For a more advanced (and philosophical) treatment, the already classic work is Thorn, R., "Structural Stability and Morphogenesis." Addison-Wesley, Reading, Massachusetts, 1975.

An application of structural stability notions directly to the stability issue for linear systems is given in Bumby, R. and Sontag, E., Stabilization ofpolynomially parametrized familiesoflinear systems: The single-input case, Syst. Controls Lett., 3, 251-254 (1983).

CHAPTER

8

The Linear-Quadratie-Gaussian Problem

8.1 MOTIVATION AND EXAMPLES

At long last we are ready to turn our attention to questions of optimal selection of inputs. To this point, our concerns have been with the selection of inputs either to reach certain positions in state space (controllability) or to make the controlled system stable in some sense. Once these issues have been settled for a given system, or class of systems, and it has been determined that the set of admissible inputs n is large enough to include more than one control which will ensure stability and/or reachability, we are then faced with choosing a rationale for preferring one such input over another. It is at this point that we finally turn to optimal control theory for resolution of the dilemma. Generally speaking, the system input is chosen in order to minimize an integral criteria of system quality. For example, if we desire to transfer the system to a terminal state in minimal time, the criterion J =

f~

ds,

(8.1)

subject to the constraints x(O) = Cl' x(t) = C2 might be used, with Cl and C2 the initial and terminal states, respectively. This is an example of a so-called free time problem since the duration ofthe process is not specified in advance. Another type of problem arises when we have a quality measure g(x(t), u(t), r) specifying the cost at time t of the system being in state x(t) when the control 182

8.1

183

MOTIVATION AND EXAMPLES

u(t) is being applied. If the process duration is specified to be oflength T - to, then an appropriate criterion would be to choose u(t) to minimize J

fT g(x(t), u(t), r) dt.

=

(8.2)

10

To ensure that such a process does not terminate in an unfavorable final state x(T), criterion (8.2) is often augmented by the addition of a terminal cost h(x(T)), measuring the desirability of the final state x(T). A stricter requirement is to demand that the system terminate in a predetermined final state x(T) = d. Such a situation is encountered in various types of navigation processes. Other variations on the basic theme include the imposition of state and/or control constraints of either a local or global nature. Examples of the former are

i

= 1, ... , m,

j

= 1, ... , n,

while the latter are exemplified by forms such as

f

Tll u(S)1I ds

:0:;

M.

10

In this book we only briefly touch on variations of the above sort, since our primary interest is in the determination of the implications of linear dynamics for problems in which the cost is measured in a Euclidean norm, i.e., the costs are quadratic functions of the state and control. Thus we will be investigating the problem of minimizing the quadratic form J(to) =

fT [(x, Q(s)x) + 2(x, S(s)u) + (u, R(s)u)] ds

(8.3)

10

over all admissible input functions u(t), to are connected by the linear system

x == F(t)x + G(t)u,

:0:; t :0:;

T. It is assumed that x and u

x(t o) = c.

(8.4)

The matrices Q and R are assumed to be symmetric with further conditions to be imposed later in order to ensure the existence of an optimal u. The foregoing formulation is the so-called linear-quadratic-Gaussian (LQG) problem which is the focus of our attention throughout this chapter. It arises in a large number of areas of engineering, aerospace, and economics, as well as in situations in which initially nonlinear dynamics (and or nonquadratic costs) are linearized (quadraticized) about a nominal control and corresponding state trajectory.

8

184

THE L1NEAR-QUADRATIC-GAUSSIAN PROBLEM

An important version ofthe LQG problem is the so-called output regulation problem in which the system dynamics are given by Eq. (8.4), while the output is y(t) = H(t)x(t). (8.5) It is desired to minimize (over u) the quadratic functional J(t o) =

iT

[(y, Cy)

+ 2(y, Vu) + (u, Ru)] ds

(8.6)

to

with C > O. Clearly, the output regulation problem is equivalent to problems (8.3)-(8.4) with the identifications

Q -+ H'CH,

S -+ H'V,

R

-+

R.

Thus we shall usually consider criterion (8.3)in this chapter.

Example:

Water Quality Control

A typical sort of problem in which LQG theory arises is in the regulation of water quality in a river. To avoid complications, we assume that the river may be decoupled into k nonoverlapping reaches in such a way that the biochemical oxygen demand (BOD) and dissolved oxygen (DO) do not change with respect to the spatial distance downstream from a given reference point, i.e., the BOD and DO dynamics for a given reach involve only the single independent variable t, the time. We further assume that the reach is defined as being a stretch of the river in which there is at most a single water treatment facility of some kind. With the above assumptions, the lumped BOD-DO dynamics are described by db(t)/dt

= - Kyb(t),

which characterizes the pollution situation in the river. Here b(t), y(t) are k-dimensional vector functions representing the BOD and DO concentrations, respectively, in each reach, while K; is the BOD removal coefficient matrix, K d the deoxygenation coefficient matrix, K. the reaeration coefficient matrix, and ds the saturation level of DO, all assumed constant. The effects of adding effluents to the river have not yet been taken into account. This is accomplished by defining control vectors Ul(t) and U2(t), where Ul(t) is a k-dimensional vector representing the control of effluents by sewage treatment plants, while U2(t) is a k-dimensional vector indicating control by artificial aeration carried out along the reaches. For example, the first control might be the operation rule for a retention reservoir located after the treatment plant, while the second could be the timing schedule for the aeration brushes.

8.2

185

OPEN-LOOP SOLUTIONS

To complete the model, we define state variables and a performance measure. Since there are certain water quality standards to be satisfied during the control periods, we assume that the controlling actions are taken to minimize BOD and DO deviation from these standards. Assume that the standards are given by the constant vectors aD and aD' Then we define the state x(t) =

[X 1(t)J = X2(t)

[b(t) -

aDJ.

y(t) - aD

Thus the complete system dynamics are x(t)

=

Fx(t)

+ Gu(t),

where x(t) is as above,

since the greater the artificial aeration, the less is the oxygen deficit, and conversely. The cost function is to minimize a weighted sum of state deviations from zero, and cost of controls. Hence, we minimize J = f[(X(t), Qx(t))

+

(u(t), Ru(t))]

+

(x(T), Mx(T)),

where Q and R represent appropriate weighting matrices reflecting the relative importance of BOD and DO control in each and the relative costs of sewage treatment and aeration. The terminal cost matrix M accounts for the relative importance ofthe water quality level at the termination ofthe process. 8.2 OPEN-LOOP SOLUTIONS

We have noted earlier the fundamental conceptual (and mathematical) difference between open-IMp and closed-loop (feedback) inputs. Therefore, it should come as no surprise that the basic results associated with the LQG problem also inherit the flavor. We shall begin with a discussion of the openloop situation. It will be seen that the basic results characterizing the optimal input require the solution of a Fredholm integral equation in order to generate the optimal control. Alternatively, utilization of the Pontryagin maximum principle yields the optimal control as a function of the solution of a two-point boundary value problem. In either case, formidable computational problems may arise serving, among other reasons, to motivate a thorough study of the feedback case.

8

186

THE LINEAR-QUADRATIC-GAUSSIAN PROBLEM

At first, we consider the problem of minimizing

f.T [(x, Q(t)x) + (u, R(t)u)] dt,

J =

(8.7)

10

where

+ G(t)u,

dxjdt = F(t)x

(8.8)

The idea of our approach is to express the cost functional entirely in terms of the control u( .), from which we may employ a standard completion-of-squares argument to find the minimum, as well as conditions on R(t) and Q(t) for which a minimum exists. We write the solution of the differential equation as x(t)

= (t, tok +

f.T (t, s)G(s)u(s) ds,

(8.9)

10

where is the fundamental solution matrix of F. Substituting expression (8.9) for x(·) into the cost functional (8.7) and performing some algebra, we arrive at the expression J

= (c, Me)

- 2

f.T(U' m(s» ds 10

+

f.Tf.T(U. [R(s) b(s 10

10

s')

+ K(s, s')]u) ds ds',

.

where

M m(s)

=

f.T '(t o, s)Q(s)(s, to) ds, '0

= -

f.: I(t -

G'(s)

s)'(t, s)Q(t)(t, tok dt,

10

K(s, s') = G'(s)

f.T I(t -

s)'(t, s)Q(t)(t, s')l(t - s') dt G(s),

10

with 1(·) being the unit step function, i.e.,

I(a) = {I, 0,

a ~ 0, a < o.

It is now easy to see that there will be a unique square-integrable u(·) minimizing J if and only if %(s, s')

=

R(s)b(s - s')

+ K(s, s') >

0,

(8.10)

8.3

187

THE MAXIMUM PRINCIPLE

i.e., res, s') is strictly positive definite on the square to ~ s, s' ~ T. Furthermore, if % > 0, then from standard results in the theory of integral equations we find that the minimizing u( .) will be the unique square-integrable solution of the Fredholm integral equation R(t)u(t)

+

iT

K(t, s)u(s) ds

=

met),

(8.11 )

10

The foregoing derivation has been rather formal, proceeding under the assumption that F, G, Q, and R possess all properties necessary to ensure the square-integrability of K(·, .) plus the positive-definiteness of %(', -], But, what are these conditions? The square integrability of K(·, .) can be ensured by assuming that the matrix functions F(t) and Q(t) are piecewise continuous on the interval to ~ t ~ T. Equation (8.11) shows immediately that the uniqueness of the optimal control will be lost (in general) if R is singular. Thus we provisionaIly assume that R is nonsingular. However, to guarantee that condition (8.10) is satisfied, we should strengthen the condition on R to be R(s) > 0, to ~ s ~ T. This ensures that the first term of (8.10) is positive definite. FinaIly, a sufficient condition for the second term to be positive semidefinite is for Q(s) ~ 0, to ~ s ~ T. Notice, however, that %(', .) may stiIl be positive definite even if Q is negative definite, or even indefinite. In summary, we have established Theorem 8.1. Theorem 8.1 Let the matrices Q(s) ~ 0, R(s) > 0, F(s), and G(s) be piecewise continuous over to ~ s ~ T. Then the functional J has a unique minimum over all square-integrable functions u(s), to ~ s ~ T, and this minimum is given by the solution to the Fredholm integral equation (8.11). EXERCISE

1.

How can the above results be modified to account for (a) a terminal cost of the form (x(T), Q/x(T», (b) a cross-coupling term in J of the form 2(x, Su). 8.3 mE MAXIMUM PRINCIPLE

FoIlowing in the footsteps of Weierstrass, Legendre, and other pioneering 19th century workers in the calculus of variations, an alternate approach to the characterization of the optimal open-loop control law for the LQG problem was developed by Pontryagin, Boltyanskii, Gamkrelidze, and Mischenko in the mid-1950s. This result, termed the "maximum principle," is a substantial generalization of the classical Weierstrass condition from the calculus of variations. In essence, the maximum principle states that the

8 THE LINEAR-QUADRATIC-GAUSSIAN PROBLEM

188

optimizing control law must provide the pointwise minimization of (the Pontryagin group worked with the negative of our function J) a certain function, the Hamiltonian of the system, which is determined solely by the given system data F, G, Q, R, S. The basic idea behind the maximum principle is to augment the integral of (8.3) by adding the system dynamics (8.4), multiplied by an unknown vector function p(r), the so-called costate, which is to be determined. Thus we form the Hamiltonian Jf of the system as ,ff(x, u, p, t)

= [(x, Qx) + 2(x, Su) + (u, Ru)] + p'(t)[Fx + Gu], (8.12)

where the prime denotes, as usual, the transpose operation. The content of the maximum principle is given next. Theorem 8.2 (Weak Minimum Principle) Assume Q(s) ~ 0, R(s) > 0, to :::; s :::; T. Let u*(t) be the input which minimizes criterion (8.3), subject to the dynamics (8.4). Then the corresponding optimal trajectories x*(t), p*(t) satisfy the equations

aYf

u", p*, r),

x*(t)

=

p*(t)

= - a: (x*, u*, p", r),

ap (x",

(8.13)

(8.14)

Furthermore, u*(t) is characterized by the condition

a:ft au (* x, u '", p *) , t = 0.

(8.15)

The boundary conditions are x*(ro)

= c,

(8.16)

p*(T) = O.

In more explicit terms, Eqs. (8.13)-(8.14) take on the form x*(t)

= F(t)x* + G(t)u*,

p*(t) = -2Q(t)x*

(8.17)

+ 2S(t)u*

- F'(t)p*,

(8.18)

with the minimizing u being characterized through Eq. (8.15) in terms of p* and x* as u*(t)

= -

R - l(t) [tG'(t)p*

+ S'(t)x*].

(8.19)

Thus we may rewrite Eq. (8.18) as p*(t) = -2(Q

+ SR-1S')x*

- (SR-1G

+ F')p*,

(8.20)

8.3

189

THE MAXIMUM PRINCIPLE

while Eq. (8.17) is

x*(t)

=

(F - GR-IS')x* - tGR-IG'p*.

(8.21)

The boundary conditions are those of Eq. (8.16). Consider the problem of minimizing

EXAMPLE

J = fT[x/(t)

+ u 2(t)] dt

to

with

xl(t) = X2(t), In this problem, we have

F=[~

-~J

G=[~J

S = 0,

The Hamiltonian is Jr(x*, u*, p*, t) = x/ + u 2 Eq. (8.18), we find the costate equations are

PI *(t)

=-

o·1fIi3xl

=-2x

I*(t),

+ PIX2 -

P2 *(t) = - a·1flox2

R = 1.

P2X2 + P2U. For

=

P2*(t) - PI *(t).

The control u* must satisfy

OJr ou = 2u*

+ P2 * =

Since

0,

i.e.,

u*(t) = -tp2*(t).

0 2JrIi3u2 = 2 > 0,

we see that u" = -tp2*(t) does indeed provide the minimizing value of.1f. In the general theory of control processes, the importance of the maximum principle is that it applies even to those cases in which the input space n of admissible controls is constrained. For example, if we had demanded

Iu(t) I ::;

1,

to ::; t

s

T,

in the above example, then, the minimizing control law would still minimize the Hamiltonian .1f pointwise, subject to the constraint. EXERCISES

1. Show that if'[uirlj s; l,t o ::; t s; T,intheaboveexample,theminimizing control law would be u*(t) =

{

- t P2*(t) -1

+1

if IP2 *(t)l s 2, if P2 *(t) > 2, if P2*(t) < 2.

8

190


2. Draw a graph of the optimal control law u*(t). 3. Can the optimal control law for the constrained problem be determined by calculating the control law for the unconstrained case and allowing it to saturate whenever the stipulated boundaries are violated? 8.4 SOME COMPUTATIONAL CONSIDERATIONS

In principle, Eqs. (8.20)-(8.21), together with the boundary conditions (8.16), provide sufficient information to compute x* and p* which in turn allow us to calculate the optimal control u* by means of the minimum condition (8.19). However, the fact that (8.20)-(8.21) constitutes a two-point boundary value problem poses a nontrivial set of computational questions. The basic problem is that conditions (8.16) prescribe a value of x* at t = to, while the value for p" is given at t = T. As a result, there is not enough information at any single point to serve as initial conditions for calculating x* and p* in a recursive manner. The linearity of Eqs. (8.20)-(8.21) may be exploited to aid the computational process in the following manner. Let X H(t), PH(t) denote the solution to the homogeneous matrix system XH(t)

=

(F - GR-1S')XH

PH(t) = -2(Q

+

iGR-1G'PH,

-

SR-1S')X H - (SR-1G

+

(8.22)

F')PH ,

(8.23)

with the initial conditions Our objective is to use system (8.22)-(8.23) to determine the unknown value (X = p*(to) which, when used in Eqs. (8.20)-(8.21), will make p*(T) = O. In addition to the homogeneous matrix system (8.22)-(8.23), we must also generate one particular solution to the vector system (8.20)-(8.21). If we let x,; P« denote this solution, then X" = (F - GR-1S')x" - iGR-1G'p",

p" =

-2(Q

+ SR-1S')x"

- (SR-1G

(8.24)

+ F')p",

(8.25)

with x,,(to) = c,

p,,(to) = O.

Using the superposition principle for linear systems, we see that the complete solution of (8.20)-(8.21) is expressed by

+ (X2 xif'(t) + (XIPU'(t) + (X2pif)(t) +

x*(t} = (XtxU)(t) p*(t) =

+ lXlIX!i'(t) + x,,(t), + (XlIPW'(t) + p,,(t),

(8.26)

(8.27)

8.4

SOME COMPUTATIONAL CONSIDERATIONS

191

where the values of (Xl' ••• ' (X" of the vector (X are to be determined and xW, ~) represent the ith column at the matrices XH, PH' respectively, i = I, ... , n. Using Eq. (8.27), we see that it may be written in vector matrix form as

(8.28) where PH(t) is the n x n matrix whose ith column is PU'(t). Since we must have p*(T) = 0, from (8.28) we obtain PH(T)(X

=

-Prr(T),

or (X = Pii l(T)Prr(T).

(8.29)

Assuming that PH(T) is invertible, the value of (X from (8.29) provides the "missing" initial value for p*{to), thereby turning (8.20)-(8.21) into an initial value problem soluble by standard methods. Reexamining the foregoing procedure, we see that the computational requirements are to integrate 2n homogeneous equations and 2n particular solutions from t = to to t = T, then solve the n x n linear algebraic system (8.28). On the surface the above approach would appear to dispose neatly of the problem of determining x* and p*. However, in practice serious difficulties may arise: (i) Ifthe interval length T - to is large, then it may be difficult to compute accurately the homogeneous and particular solutions. In fact, the situation is often far worse than just a case of numerical roundoff or truncation error. This is due to the fact that we may determine the functions X H, PH' x"' Prr by either integrating forward from t = to to t = T as described above, or by integrating backwards beginning at t = T and determining the unknown value x*(T) rather than p*(O). In either case, due to the nature of the system (8.20)-(8.21), one of the equations will be integrated in an unstable direction. This is a consequence of the fact that the equation for p* is "dual" to that for x*. Thus, if the forward direction is stable for x* (i.e., x*(t) is a linear combination of decreasing exponential functions), then it is unstable for p*, and conversely, if the forward direction is stable for p*, it is unstable for x*. In either case, if the interval length is sufficiently large, numerical inaccuracies are guaranteed to appear. (ii) Even if the homogeneous and particular solutions are produced with great accuracy, it may turn out to be difficult to solve the linear algebraic system (8.28), particularly if the dimension n of the system is large. Many times the problem is theoretically solvable in that PH(T) is invertible but, for practical purposes, the problem is out of reach due to ill-conditioning. This phenomenon may produce a value of (X generating a p*(T) far from zero.

192

8


Various procedures and techniques have been proposed to circumvent the above difficulties and are described in detail in the references cited at the end of the chapter for this section. We only mention these points as motivation for development of an alternate conceptual approach to the LQG problem. This approach, utilizing the notion of closed-loop or feedback controls, will be explored in succeeding sections. EXERCISES

1. (a) Find the solution of the system x(t) = -2x(t) - p(t), P 0). The following elementary results about Eq. (8.39)will form the basis for our study of the infinite-interval problem. The first involves global existence of the solution. Theorem 8.6 Let P(t, Po, T) be the solution at time t < T of Eq. (8.39) passing through Po at time t = T. Then, given any to, P(t, Po, T) exists and is unique on to ~ t s T for all Po.

206

8


PROOF In view of standard results on local existence and uniqueness, P(t, Po, T) exists for some t :s; T. Let cI>(t, T) be the transition matrix of F. Then

P(t, Po, T) = cI>(t, T)Po i = 1, ... , n. Theorem 8.12 Let A. i be a characteristic value of V with G:J the corresponding right characteristic vector. Then - A.i is also a characteristic value of V with [~;;J being the corresponding left characteristic vector. PROOF

The prooffollows by direct verification, making use of the identity

V'=G

-IJo '

and the fact that a left characteristic vector of V associated with A. is a transposed right characteristic vector of V' associated with A.. REMARKS (l) Since V is real, its characteristic values occur in quadruples (A., A. *, - Ie, - A. *). (2) There can exist at most one stabilizing solution P; due to the symmetry of the characteristic values of V. If P, = Y X-I is such a solution, then X* y is Hermitian.

The final two results characterize the existence and uniqueness of the stabilizing solution Ps • Theorem 8.13 The stabilizing solution r, of Eq. (8.60) exists if and only (F, G) is stabilizable and Re A. # 0 for all characteristic values of V.

if

210

8

THE L1NEAR-QUADRATIC-GAUSSIAN PROBLEM

PROOF (Necessity) Suppose P; exists. It is associated with n stable roots of V and, hence, no characteristic values have a zero real part due to their symmetrical placement relative to the imaginary axis. Moreover, the matrix L = G'Ps stabilizes F, i.e., F - GG'p. is stable, which means that (F, G) is stabilizable. (Sufficiency) Suppose that the hypothesis is true and that no stabilizing solution exists. Then either (i) we cannot choose n stable characteristic values of V, which is a contradiction, or (ii) we can do so but the matrix X is singular. If (ii) is the case, let z be any nonzero vector in %(X), the null space of X. Since X*y = y*X, we have

0= Y*Xz = X*Yz. By virtue of the definition of V and (8.62),

-H'HX - F'Y = Y f.

FX - GG'y = Xf,

(8.63)

The first equation yields

z*Y*FXz - z*Y*FF'Yz = z*Y* Xfz, and, hence, G' Y z = O. But,

0= FXz - GG'Yz = Xfz, which means that %(X) is a ,I-invariant subspace of R". Thus there exists at least one nonzero vector Z E %(X) such that fZ = p.z,

where p. coincides with one of the stable roots of U. The second equation in (8.63) postmultiplied by Z yields

-F'Yz

= Yfz.

Collecting these results give us (Y2)'F

=

-p.(Yz)',

Re( -p.) > 0,

(Y2)G = O.

Thus (F, G) is not stabilizable, contradicting the original hypothesis. At last we can prove the result linking P, with P" and Pro. Theorem 8.14 The stabilizing solution P s is the only nonnegative solution of (8.60) if and only if(H, F) is detectable.

8.7

211

OPTIMALITY VERSUS STABILITY

PROOF (Sufficiency) Assuming (H, F) detectable, we show that any solution P of (8.60) is stabilizing. Suppose the contrary, i.e., a A. exists such that

ffz = ..1.z,

ReA. ~ 0,

ff = F - GG'P.

Upon rearranging Eq. (8.60), we have

PGG'P + Pff

+ ff'P + H'H =

0,

and, hence, (A. + ..1.*)z*pz = -z*PGG'Pz - z*H'Hz. Since A. + ..1.* = 2 Re A. ~ 0, the left side of this equation is nonnegative, while the right side is nonpositive. Thus both are zero and

G'Pz = 0,

Hz = 0,

which implies

Fz = ..1.z, Hz = 0.

ReA. ~ 0,

Thus (H, F) is not detectable contradicting our hypothesis. Hence, ff is stable. However, there is at most one stabilizing solution, i.e., the solution P = Ps. (Necessity) Suppose an undetectable root . 1. 1 of the pair (H, F) exists. We shall show the existence ofat least two nonnegative solutions ofEq. (8.60). By hypothesis, one of them is the stabilizing solution P s = y X-I. To form another solution PI = YIX!l, we substitute the characteristic vector [0'] of U associated with . 1. 1 for the vector [~:] associated with -..1.1> thus obtaining Also, set

x=

[x21 .. ·Ixn]

,

Now Theorem 8.13 implies.that [Zl*

Hence,

Zl *Y

=

0*][

~~J

= 0,

i = 2,3, ... , n.

0 and

o

X* as X*y

~

O.

8

212


To prove that X t is nonsingular, suppose the contrary. Then there exists a vector v :f. 0 such that Zt = Xv and, consequently,

0= ZI*Y = v*X*Y: that is, det X* Y = O. Noting that det X* Y is a principal minor of the nonnegative matrix X* Y, it is easily seen that

[XtIX]*Yv = X*(Yv) = O. Since X is nonsingular and Yv :f. 0, this is a contradiction. TABLE 8.2 Nonnegative Solutions of Eq. (8.60) Solution

Case (F, G) stabilizable, (H, F) detectable

P, exists and is the unique nonnegative definite solution P,

(F, G) stabilizable,

P, exists and there are other nonnegative definite solutions

(H, F) not detectable (F, G) not stabilizable,

=

p.

No nonnegative definite solution exists including P, and p.

(H, Fldetectable (F, G) not stabilizable, (H, F) not detectable

P, does not exist, but other nonnegative definite solutions mayor may not exist

TABLE 8.3 Asymptotic Properties of the Matrix Riccati Equation (8.39)

Case

Tfinite

Po = 0

(F, G) stabilizable, (H, F) detectable

P oc exists,

P",

(F, G) stabilizable, (H, F) not detectable

P a: exists, P", = p.

P ex>

'" P,

= P, = p.

P", exists, = any nonnegative solution depending on

Po (F, G) not stabilizable, (H, F) detectable (F, G) not stabilizable, (H, F) not detectable

P oc, does not exist P", mayor may not exist

There is always a unique solution and it is nonnegative definite

8.7

213

OPTIMALITY VERSUS STABILITY

Thus, PI does exist and PI #- P; because it corresponds to a different n-tuple of characteristic values of U. Since the foregoing results have been long and arduous, we summarize all conclusions in Tables 8.2 and 8.3. Example: Armament Races

( continued)

We return to the elementary buildup model of Section 8.5 and examine the stability ofthe optimal control for the quadratic functional (8.38). We assume that the process has been "building up" over such a time span that it is reasonable to suppose that the two countries involved are attempting to minimize

where the system dynamics are given by Eqs. (8.36)-(8.37). For simplicity, assume the grievance and goodwill coefficients are zero. The appropriate matrics for this process are F= [

- a I

kJ

G

-b'

S = 0,

= I,

R = diag(/3t, /32),

Po =

o.

The minimizing feedback control law is u*(t) = -R-tG'P*N(t),

where P* is a nonnegative definite solution of the equation

Q + PF + F'P - PGR-IG'P

=

O.

Since G = I, there is no question that (F, G) is stabilizable (it is controllable). Thus, referring to Table 8.2, we need only check the detectability of (H, F) to ascertain the closed-loop stability of the control u*. We have

@=[:F]=

((X I )1/2

0

0

((X2)1/2

- a((XI)I/2

k((XI)I/2

1((X2)1/2

- b((X2)1/2

214

8

THE LINEAR-QUADRATlC-GAUSSIAN PROBLEM

If we assume that the choice of a, b, I, and k is such that both characteristic roots of F are unstable, then the detectability condition means that (H, F) must be observable. In this case, the conditions for stability of u* are: either (i) (ii)

(iii)

1X I1X2:1=

IX I

0, or

= 0, I :1= 0, and

1X2 =

0, k :1= O.

Thus we already see the interplay between the weights each country attaches to holding down armaments production and the defense coefficients k and I. It is interesting to observe that the fatigue coefficients a and b playa role only to the extent that they determine the possibility of weaker conditions than (i)-(iii) if F has a stable root. However, satisfaction of (i)-(iii) implies a stable control law irrespective of the fatigue felt by each party. We close this section by again noting that optimality does not imply stability. The optimizing solution P* of Eq. (8.60) minimizes the cost functional whereas the stabilizing solution makes the closed-loop system asymptotically stable. These are quite different properties as the trivial example J

=

f~<XJ

(x/

+ u2 ) dt,

shows. Here an optimal control u* exists that minimizes J, since the unstable component X2 is absent from J. However, the optimal control law u* is not stable since it cannot possibly stabilize X2' The problem here, of course, is that the system is neither stabilizable nor detectable. In cases for which the dynamics and/or cost function are not so trivial, the preceding theory will be required to determine the stability of optimal laws. 8.8 A LOW-DIMENSIONAL ALTERNATIVE TO THE ALGEBRAIC RICCATI EQUATION

The generalized X- Y functions were seen to provide an alternative approach to the determination of optimal feedback controls, one which explicitly took account of redundancies in the system description to reduce the computing burden imposed by the usual matrix Riccati equation. A careful examination of the L-N system (8.41)-(8.42) shows that, unlike the Riccati equation, the infinite interval version of the equations is not well determined by the usual trick of setting L = IiI = O. Such an approach yields only the conclusion L( (0) = 0, a result that follows immediately from the representation P(t) = L(t)L'(t). An equation for N(oo), the quantity which determines the optimal feedback control, must be obtained through other means.

8.8

215

LOW-DIMENSIONAL ALTERNATIVE TO RICCATI EQUATION

We recall the basic equation to be solved is Q

+ PF + F'P

- (PG

+ S)R- 1(PG + s)'

=

o.

Our results hinge on the following basic lemma from matrix theory. Lemma 8.3 Let P, A, Q be any three matrices for which the product PAQ is defined. Then q(PAQ) = (Q' ® P)q(A), where ® denotes the Kronecker product and a the operator that" stacks" the columns of a matrix into a column vector, i.e., if A = [aij], then PROOF The proof follows from direct component-by-component verification of the asserted relation.

Using Lemma 8.3, we may manipulate the basic equation to obtain an equation for N( (0). Theorem 8.15 Assume that the matrix F - GR- 1S' has no purely imaginary characteristic roots and no real characteristic roots symmetric relative to the origin. Then the optimal steady state N( (0) = N satisfies the algebraic equation q(N) = (I ® R - 1/2G') [(F - GR -I S')' ® I

+ I ® (F PROOF

- GR- 1S,)']-l q(N'N - Q

+ SR-IS').

Collecting terms in the basic equation, we see that

(Q - SR- 1S')

+ P(F

- GR-IS')

+ (F'

- SR- 1G')P - PGR- 1G'P = O.

Applying a to both sides of this relation and using the characteristic value hypothesis, we have q(P)

= [(F -

GR-IS')'®I + I®(F - GR- IS')']-l q(N'N - Q + SR- 1S'),

where the definition N = R -1/2G' P 00 has been used. Multiplication of both sides of the above relation' by (I ® R - 1/2G') and employment of Lemma 8.3 and the definition of N completes the proof. REMARKS (l) The optimal steady state gain function K follows immediately from the relation K = R - 1/2 N + R -1 S'. (2) The importance of Theorem 8.15 is that the algebraic relation for N represents only nm equations in the unknown components of N. This is to be compared with the n(n + 1)/2 equations in the usual algebraic Riccati equation (8.60). As before, we expect a substantial computational advantage if the number of system inputs m ~ n.

216

8


8.9 COMPUTATIONAL APPROACHES FOR RICCATI EQUATIONS

Practical implementation of LQG theory requires that we devote substantial attention to the question of efficient computation of the nonnegative definite, stabilizing solution of the algebraic Riccati equation (8.60), assuming the requisite stabilizability/detectability conditions are satisfied. In this section we briefly review various techniques for the solution of Eq. (8.60). It is very difficult to label one method as being superior to another, since a given method may prove better in one application but fail in another. As usual in mathematics, it is best to have as broad an arsenal of weapons as possible to bring to bear on a given problem. Our aim is to survey the most important methods and to indicate their range of applicability, strengths, and weaknesses.

I.

Characteristic Vector Method

Since good procedures now exist for computing the characteristic vectors and values of a matrix, this method has come into increasing prominence in recent years. The essence of the procedure is described in Theorem 8.11 and its corollary. An interesting side aspect of this method is that it is the only procedure available for computing all the solutions of Eq. (8.60), nonnegative or not, by appropriate selection of subsets of the 2n characteristic vectors of U.

II.

Iterative Method

The basis of this method is a variant oftlie Newton approximation method for finding the root of a system of nonlinear equations by successive linear approximations. We assume that Eq. (8.60) possesses a unique stabilizing solution Ps. Define P, as the unique nonnegative solution of the linear algebraic system j = 0, 1, ... ,

where j

= 1,2, ... ,

and where L o is chosen such that the matrix K o = F + GL o is a stability matrix. Then it can be shown that, in the ordering induced by nonnegative matrices, j

= 0,1, ... ,

8.9

217

COMPUTATIONAL APPROACHES FOR RICCATI EQUATIONS

and lim Pj = Ps •

i« »

This method gives monotonic and quadratic convergence to P, and is believed to be one of the best methods for finding Ps •

III.

Sign Function Method

The sign function method may be regarded as a simplification of the characteristic vector method, designed to find the stabilizing solution Ps • We define the matrix sign function Z as sign Z

=

lim Zk+I'

k -+ ex.

where Zo

= Z.

Also define sign + Z

=

1 0

for all t.

In addition, let the noise processes v and w be illdependent. The initial state x(t o) = Xo is also a normally distributed random variable with mean Xo and covariance Po, and is independent of the processes v and w. It is of utmost

importance to note that we are not talking about the same problem here as in the filtering situation. The foregoing system is assumed to be a deterministic system driven by the noise process v. Thus we are considering a stochastic control process and not a pure filtering problem as before. Due to the noise in the dynamics, as well as the observations, it is not possible to pose an optimization problem requiring minimization of the quadratic form J =

JT [(x, Qx) + (u, Ru)] dt,

Q ;?: 0,

R

> 0,

10

because the performance criterion J is, itself, a random variable depending on v, w, and Xo' To deal with the situation, we replace the deterministic problem of minimizing J by the problem of minimizing its expected value B[J] = J,

232

8

THE LlNEAR-QUADRATIC-GAUSSIAN PROBLEM

where the expectation is taken over xo, v, and w. It is understood that at time t the measurements y(t), to ~ r ~ t, are available, and that the optimal control u*(t) is to be expressed in terms of y(t), to ~ r ~ t (note that u*(t) is not required to be an instantaneous function of y(t». The solution for the above problem is carried out in two steps. 1. Estimation Compute a minimum variance estimate ~(t) of x(t) at time t, using u(t), y(T), to ~ r ~ t. This estimate satisfies the equation d~/dt

= F(t)~

~(to)

= 0,

+ G(t)u + P(t)H'(t)R -l(t)[y(t) -

H(t)~],

where dP/dt = F(t)P P(t o)

=

+ PF'(t)

- PH'(t)R-1(t)H(t)P

+ (2(t)

Po·

Note that the equation for ~ is independent of the cost matrices Q and R. Although the context is quite different, the production of ~(t) by the above prescription is virtually identical with the procedure followed in the Kalman filtering context. 2. Control Compute the optimal control law u*(t) = - K(t)x(t), which would be applied if there were no noise, if x(t) were available, and if J were the performance criterion. Then use the control law u*(t)

= -

K(t)~(t),

where ~(t) is obtained from the equation above. This law will be optimal for the noisy problem. Note that calculation of K(t) is independent of H(t) and the statistics of the noise. Summarizing, we see that the optimal control is obtained by acting as if were the true state of the system and then applying the deterministic theory presented earlier in the chapter. Hence, the name "separation principle," indicating that the two phases, estimation and control, are separate problems which can be tackled independently. Schematically, we have Fig. 8.1. ~(t)

r-------r-I Noisy Linear System t-----,r"'"""'"-

Control Law from Deterministic Problem

FIG. 8.1

~ (I)

The separation principle.

8.14

233

DISCRETE-TIME PROBLEMS

8.14 DISCRETE-TIME PROBLEMS

Many problems of optimal control and filtering involve measurements that are taken at discrete moments in time. For example, economic processes in which annual data is used, sampled-data chemical process control systems in which the output of the system is analyzed only daily, and so on. In these instances, it is more natural to formulate the control/filtering problem dynamics as a finite-difference equation, rather than a differential equation. Thus the dynamics are x(k

+

I)

=

F(k)x(k)

+ G(k)u(k),

with the quadratic costs now being expressed as the finite sum J

=

N-l

L [(x(k), Q(k)x(k») + (u(k), R(k)u(k))] + (x(N), Mx(N).

1=0

In Chapters 3 and 4, we have already observed that there is no fundamental difference between basic systems-theoretic concepts for continuous- or discrete-time problems. The algebraic statement ofthe results is more complicated in discrete time, but there are no foundational issues dependent on the structure of the time set. For this reason we have usually presented only the continuous-time result, as the mathematical formalism is more compact. As illustration of the foregoing remarks, we now give the basic discretetime results for the optimal linear filtering problem. It will be a worthwhile exercise for the reader to translate them (using the duality theorem) to the control-theoretic setting. Let {x(k)} and {z(k)} be n, p-dimensional vector stochastic processes generated by the linear model x(k

+

1) = F(k)x(k) z(k) = H(k)x(k)

+ G(k)u(k), + v(k),

x(O) = xo, k ',2: 0,

where we assume xo, {u(k)}, and {v(k)} have zero mean, are uncorrelated for k > 0, and c!{xox o'} = Po, c!{u(i)u(j)'} = Q(i) 0

for

all i.

Here ~ij is the usual Kronecker delta symbol. Our objective is to obtain the best linear estimate (in the least-squares sense) of x(n), given the observations {z(O), z(l), ... , z(n - I)} and the above model. If we denote the best estimate by ~(n), then the following result summarizes the optimal discrete-time Kalman filter.

234

8

Theorem 8.20


The optimal estimate ~(k) ~(k

+

1) = F(k)~(k)

~(O)

satisfies the system

+ K(k)[z(k) -

H(k)~(k)],

= 0,

where the optimal gain matrix K(k) is given by K(k) = F(k)P(k)H'(k) [H(k)P(k)H'(k)

+ R(k)] - 1.

The error covariance matrix P(k) = 8 ((x(k) -

~(k))(x(k)

-

~(k))'),

is computed from the discrete-time Riccati equation P(k

+

1) = F(k)P(k)[I - H'(k)(H(k)P(k)H'(k)

+ G(k)Q(k)G'(k),

+ R(k))-1 H(k)P(k)]F'(k)

P(O) = Po.

8.15 GENERALIZED X-Y FUNCTIONS REVISITED

In Section 8.6 we presented an alternate approach to the Riccati equation for the solution of the LQG control problem. The functions replacing the matrix Riccati equation, termed generalized X - Y equations, were seen to have distinct computational advantages whenever the dimensions of the system input and output spaces are substantially less than the dimension of the state space. We now wish to return to this topic and present the corresponding results for discrete-time filtering problems. For simplicity, we treat only the case when the system and covariance matrices are constant, although the more general case may be dealt with in a corresponding manner, as is indicated in the references. Considering the discrete-time filtering process outlined above, we introduce the auxiliary notation

T(k) = FP(k)H',

S(k) = R

+ HP(k)H'.

Then, in this notation, the optimal filter gain matrix K(k) has the form

,K(k) = T(k)S-I(k). Using the above quantities, the following result may be obtained.

Theorem 8.21 relations

The functions S(k), T(k) may be obtained from the recursive

T(k L(k S(k U(k

+ 1) = + 1) = + 1) = + 1) =

T(k) + FL(k)U(k)-1 L(k)'H', [F - T(k + I)S(k + 1)-1 H]L{k), S(k) + HL(k)U(k)-1 L(k)'H', U(k) - L(k)'H'S(k)-IHL(k), k

~

1.

235


The initial conditions for Sand T at k = 0 are T(O)

=

FH',

S(O)

=

R

+ HPoH',

while L(O) and V(O) arefound by factoring the matrix D = FPoF'

+ GQG' -

Po - FH'(R

+ HPoHr'HF'

as M+

D = L(O) [ 0

lr

0

M- 1'(0)',

Then L(O) is the initial value for the function L(k), while U(O)

0 = [ M0 + M-

J-'

'

PROOF The proof of this important result may be found in the references cited at the end of the chapter.

From a computational point of view, the importance of Theorem 8.21 is that the sizes of the matrix functions T and S are dependent only on the dimensionality of the measurement process, i.e., T and S are n x p, p x p matrix function, respectively. In addition, we see that the dimensions of L and V are governed by a parameter r = rank D. The matrix L has dimension n x r, while V is of size r x r. Since S = S', V = V', the total number of equations is n(p + r) + t[P(p + 1) + r(r + l)J. Thus, in the event p, r ~ n, substantial computational savings may be anticipated by use of the above discrete-time generalized X - Y functions, as opposed to the Riccati equation, for computation of the optimal filter gain function K(k). MISCELLANEOUS EXERCISES

f:

1. Consider the problem of minimizing J =,

[(x, x)

+ (v, A(t)x)J dt

over all vector functions x(t) differentiable on [0, TJ with x(O) = c. (a) Show that if A(t) is constant and A > 0, then the optimal curve satisfies the Euler equation x(t) - Ax

(b)

= 0,

x(O) = c, x(T) = O.

Introducing the matrix functions sinh X = f{ex - e- X ),

cosh X = f{e x

+ e- X ),

236

8

THE LlNEAR-QUADRATIC-GAUSSIAN PROBLEM

show that the solution to the above equation is x(t)

= (cosh A 1/2T)-I(cosh A 1/2(t -

T»c,

where A 1/2 denotes the positive-definite square root of A. (c) Show that the minimal value of J is given by Jmin(T) = (c, A 1/2(tanh A 1/2T)c). (d) In the case of time-varying A(t), show that the Euler equation remains unchanged and that its solution is given by x(t) = [X 1(t) - X 2(t)X /(T)- 1 X 1'(T)]c,

where XI (t) and X 2(t) are the principal solutions of the matrix equation

X-

A(t)X = 0,

with X 1(0) = I, X1(0) = 0, X 2(0) = 0, X2 (0) = I. (Hint: It must be shown that X /(T) is nonsingular for all T for which J is positive.) (e) Show that Jmin(T) in the time-varying case is given by Jmin(T)

= (c, X 2'(T)-1 X 1'(T)c), = (c, R(T)c),

and that R(T) satisfies a matrix Riccati equation. 2. Consider the problem of minimizing J

=

foT [(x, x) + (u, u)] dt

over all u, where x = Fx + Gu, x(O) = Cl, x(T) = C2' To avoid the problem of determining those u which ensure satisfaction of the condition x(T) = c 2, consider the modified problem of minimizing

r = foT [(x, x) + (u, u)] dt + A(x(T) -

C2, x(T) - C2)

for A ~ O. The only constraint is now x(O) = c i- Study the asymptotic behavior of J~in as A -+ 00 and obtain a sufficient condition that there exist a control u such that x(T) = C2' Compare this result with the controllability results of Chapter 3. 3. (a) Consider the matrix Riccati equation -dP/dt

= Q + PF + F'P

- PGG'P

=

Y(P)

with Q, F, G constant matrices, and let Y + = {Po: Po = Po' and Y(P o) ~ O}, Y - = {Po: Po = Po' and Y(P o) =:;; O}.

(t)

237


If P +, P _ denote the supremum and infimum of Y' +, respectively, show that P +(P_) exists if and only if Y' + is nonempty and there exists a matrix L 1 E f/ _(L z E f/ _) such that F' - L 1 GG'( - F' + L z GG') is uniformly asymptotically stable. = {Po: Po - P _ > O}, prove that the matrix Riccati equation (b) If~po (t) has a global solution and as t ~ 00, P(t) ~ P + if Po E ~~_. Conversely, P(t) has a finite escape time if Po $ ~ r-: = {Po : Po - P _ ~ O} and Po - P _ is nonsingular. 4. Consider the matrix Riccati equation dP/dt = Q

+

Show that this equation matrix W such that (a) - W ® R- 1 > 0 (b) A(J¥, F, R, Q) < (b') A(J¥, F, R, Q) ~

PF

+ F'P

- PR- 1 p ,

P(o) = Po.

has a finite escape time if there exists a symmetric

°°

and or and

u(W)'u(Po) > !u(W)'W- 1 ® R(F ® I

+ I ® F)u(W)

+ t[A(J¥, F, R, Q)]l/zu(W)'U(W)/A, where A(J¥, F, R, Q) = -

[U(W)~U(W)}U(W)'(F

® I + I ® F)'W- 1

® R(F ® I + I ® F)u(W) + 4u(W)'u(Q)], and A > 0 is the smallest characteristic value of - W ® R - 1. (Here, u( .) is the column "stacking" operator introduced in Lemma 8.3 and ® is the Kronecker product.) 5. (Generalized Bass-Roth Theorem) Show that every real equilibrium solution of the algebraic Riccati equation PF

+ F'P

+ Q= 0

- PGR- 1G'P

is a solution of [ - P ® I]A(Jr) = 0, where Jr is the Hamiltonian matrix Jr=[F -Q

-GR-F'

1G'J

and A is a real polynomial of degree d possessing roots all of which are characteristic values of Jr. 6. (a) Let (F, G) be stabilizable and let AI.... ,A p be those characteristic values of Jr that are also the undetectable characteristic values of (F, H), where Q = H'H, i.e.,

Hz; = 0,

Re A;

~

0, i

=

l, ... , p.

8

238


Further, assume the set A1 , ••• , Ap consists only of cyclic characteristic values, i.e., any two characteristic vectors associated with Ai are linearly dependent. Show that under these hypotheses the algebraic Riccati equation has exactly 2P nonnegative definite solutions. (b) Let CC = {AI>"" An} be the set of all characteristic values of:K and let CCk> k = 1,2, ... , 2n denote the subsets of CC. Write P, for the solution of the algebraic Riccati equation which is generated from the stabilizing solution by replacing all - Ai by Ai E CCk : Prove that any two nonnegative solutions P k , PI satisfy if and only if i.e., the set of all nonnegative definite solutions constitutes a distributive lattice with respect to the partial ordering ~. P* is the smallest (zero element) of the lattice, while P, is the largest (identity element) of the lattice. (Note: The physical interpretation of different nonnegative solutions is that each nonnegative solution is conditionally optimizing, the condition being a certain degree of stability. Specifically, Pk stabilizes the undetectable characteristic values of (R, F) included in CCk and no others. The notion that the more undetectable characteristic values that are stabilized, the higher the cost is made rigorous via the lattice concept.) 7. If (F, G) is stabilizable, but (F, H) is not detectable, show that the solution of the matrix Riccati equation -dP/dt = Q + PF

+ F'P

- PGR- 1G'P

can be made to approach any nonnegative solution of the algebraic Riccati equation by suitable choice of Po. (Thus, in general, P( 00) is not a continuous function of Po.) 8. Show by an example that even a completely controllable and completely observable system may have structurally unstable indefinite equilibria. (Hint: Consider a system with trivial dynamics.) 9. Show that the optimal feedback control law for the problem of minimizing (x, (T), Mx(T))

+

J:

[(x, Qx)

+ (u, u)] dt,

x=

Fx

+ Gu, T
0,

10

where 0, Po ~ 0, and the Riccati group of transformations (I): £ -+ l = (TFT- 1, T-1G, T'QT, R, T'S, T'P o T), det T # 0,

= (F, GV, V'RV, SV, Po), det V # 0, (III): £ -+ l = (F + GL, G, Q + SL + L'S' + L'RL, R, S + L'R, Po), L = arbitrary, M=M', (IV): £-+l = (F, G, Q + F'M + MF, R, S + MG, Po - M), (II): £ -+ lz

it is possible to develop canonical structures and invariants for classifying the set of all LQG problems. This idea is developed in Khargonekar, P., "Canonical Forms for Linear-Quadratic Optimal Control Problems," Ph.D. Dissertation, Dept. of Electrical Engineering, U. of Florida, Gainesville, Florida, 1981.

Section 8.11 The first results on the inverse problem of linear control (for single-input systems) are given by Kalman, R., When is a linear control system optimal? J. Basic Eng. Trans. ASME, Ser.D 86D, 51---60 (1964).

More recent results for the general case are presented by Molinari, B., The stable regulatorproblem and its inverse, IEEE Trans. Automatic Control AC-18, 454-459 (1973).

Bernhard, P., and Cohen, G., Study of a frequency function occurring in a problem of optimal control with an application to the reduction of the size of the problem, Rev. RAIRO J-2, 63-85 (1973) (French).

Results relating the LQG problem and algebraic invariant theory are reported in unpublished work by R. Kalman and in Casti, J., Invariant theory, the Riccati group, and linear control problems, RM-OO-OO, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1976.

8

244

THE LINEAR-QUADRATlC-GAUSSIAN PROBLEM

Section 8.12 The pioneering works on linear estimation using a state variable approach are Kalman, R., and Bucy, R., New results in linear prediction and filtering theory, J. Basic Eng. Trans. ASME, Ser. D 83D, 95-100 (1961). Kalman, R., A new approach to linear filtering and prediction problems, J. Basic Eng. Trans. ASME, Ser. D 82D, 35-45 (1960).

More recent results, along with numerous examples, are summarized in the books by Bucy, R., and Joseph, P., "Filtering for Stochastic Processes with Applications to Guidance." Wiley (Interscience), New York, 1968. Astrom, K. J., "Introduction to Stochastic Control Theory." Academic Press, New York, 1970.

A comprehensive survey of the entire field is provided by Kailath, T., A view of three decades of linear filtering theory, IEEE Trans. Information Theory IT-20, 146-181 (\974).

and Willems, J., Recursive filtering, Statistica Neerlandica, 32, 1-39 (1978).

An alternate approach, not requiring an a priori system model, is presented by Kailath, T., and Geesey, R., An innovation approach to least-squares estimation-IV. Recursive estimation given lumped covariance functions, IEEE Trans. Automatic Control AC-16, 720-727 (\971).

For related material, see also Kailath, T., Fredholm resolvents, Wiener-Hopf equations, and Riccati differential equations, IEEE Trans. Information Theory IT-IS, 665-672 (1970).

Section 8.13

A proof of the separation principle may be found in

Bucy, R., and Joseph, P., "Filtering for Stochastic Processes with Applications to Guidance." Wiley (Interscience), New York, 196~.

Another important reference in this area is by Wonham, W. M., On the separation theorem of stochastic control, SIAM J. Control 6, 312-326 (\968).

We have seen that the separation principle is valid for Gaussian statistics and quadratic costs. These conditions are only sufficient for its validity; however, a general set of necessary conditions seems to be unknown, at present.


245

Section 8.14 The discrete-time recursive filter was first presented in the first paper cited under Section 8.12 above. Additional results and extensions can be found in the books by Bucy, R., and Joseph, P., "Filtering for Stochastic Processes with Applications to Guidance." Wiley (lnterscience), New York, 1968. Astrom, K. J., "Introduction to Stochastic Control Theory." Academic Press, New York, 1970.

Section 8.15 A substantial amount of work has been carried out to extend the discrete-time generalized X- Y functions to nonstationary processes. For some representative samples, see Morf, M., and Kailath, T., Square root algorithms for least-squares estimation, IEEE Trans. Automatic Control AC.20, 487--497 (1975).

This paper also contains extensive results on the triangular factorization approach to the solution of time-varying filtering and estimation problems. Other works dealing with the same circle of questions are Kailath, T., Dickinson, B., Morf, M., and Sidhu, G., Some new algorithms for recursive linear estimation and related problems, Proc. 1973 IEEE Dec. Control Conf., San Diego, December 1973. Lindquist, A., On Fredholm integral equations, Toeplitz equations. and Kalman-Bucy filtering, Internat, J. Appl. Math. Optimization I, 355-373 (1975). Rissanen, J., A fast algorithm for optimum predictors, IEEE Trans. Automatic Control AC-18, 555 (1973). Rissanen, J., Algorithms for triangular decompositions of block Hankel and Toeplitz matrices with application to factorizing positive matrix polynomials, Math. Compo 27, 147-154 (1973).

CHAPTER

9

AGeometric-Algebraic View of Linear Systems

9.1 ALGEBRA, GEOMETRY, AND LINEAR SYSTEMS

The basic results surrounding linear systems-reachability, observability, stability, optimality-all have a resolutely algebraic flavor. In order to determine the basic properties of ~, we must carry out operations involving the ranks of certain matrices, the spaces spanned by a given collection of vectors and so forth. In the preceding chapters, we have stayed within the confines of elementary linear algebra and matrix theory for the presentations of these results. This was primarily for pedagogical purposes; the most compact language for linear systems is that of abstract algebra (module theory) and algebraic geometry (the theory of algebraic varieties). The purpose of this chapter is to show how our earlier results can be unified and streamlined, as well as extended in a number of important directions, by stating them in the "natural" language of systems-algebra and geometry! Before embarking upon an exposition of the algebro-geometric theory of linear systems, it is worthwhile to take a moment to justify the claim that algebra and geometry are the languages of systems. Why should we go to the extra trouble and effort to develop an abstract algebraic theory of linear systems? What advantage does such a theory possess? In short, why bother? The answer to these questions are many-fold: • algebra is the tool for constructing new mathematical objects from old in a natural (i.e, canonical) way; 246

9.2

MATHEMATICAL DESCRIPTION OF A LINEAR SYSTEM

247

• algebra is compact: many of the "gadgets" of analysis (differential equations, Laplace transforms, integral representations) are only artifacts as far as linear system theory is concerned. Algebra provides a means to bypass these artifacts, while at the same time unifying the so-called "state-space" and "frequency-domain" approaches. • algebra is computationally congenial. The tools of analysis by their very nature involve limiting operations of various sorts. Since digital computers can neither exactly represent real numbers nor engage in infinite calculations, the tools and concepts of analysis cannot be directly applied in digital computers. On the other hand, the basic notions of algebra involve finite operations upon sets of elements, emphasizing structure and transformation. Such an orientation is much closer in spirit to the digital computer; • the concepts of modern algebra (homology theory, category theory, algebraic topology, etc.) are, for the most part, of rather recent vintage. Consequently, an algebraic treatment of linear systems enables us at once to make contact with some of the most active mainstreams of modern mathematics. In the sections that follow, we shall assume that the reader is familiar with the elementary concepts of algebra as found in a typical introductory undergraduate course (ring, field, module, homomorphism, etc.). For a brief refresher on these matters, any ofthe introductory texts cited in the references can be recommended. 9.2 MATHEMATICAL DESCRIPTION OF A LINEAR SYSTEM

We adopt the following standard definition of a linear system: Definition 9.1 A discrete-time, constant, linear, m-input, p-output dynamical system ~ over a field k is a composite concept (F, G, H), where

F: X -+X, G: km-+X,

H: X -+k P are abstract k-homomorphisms, with X an abstract vector space over k. The dimension of~ is, by definition, dim X. Naturally, once we fix a basis in X, the k-homomorphisms F, G, H can be identified with their corresponding matrix representations. The dynamical interpretation of ~ is given by the equations x(t

+ 1) = Fx(t) + Gu(t), yet) = Hx(t),

with t E Z, x(·) E X, u(·)

E

k", and y(.) E k".

248

9

A GEOMETRIC-ALGEBRAIC VIEW OF LINEAR SYSTEMS

The preceding definition describes what is usually termed the "internal" model ofa system E, We now define the corresponding "external", or input/ output, model. Definition 9.2

A linear, zero-state, input/output map over k is a map f: 0

such that

°

~

r

°

0 = {all k-vector sequences w: Z ~ k", such that w(t) = for all t < t - ::; and all t > 0, where t - 1 is some finite integer} (b) r = {all k-vector sequences y: Z ~ kP such that y(t) = for all t ::; O] (c) 0 and rare k-vector spaces with f a k-homomorphism. (d) f is translation invariant in the sense that the following diagram commutes, (a)

°

0-1..- r

anj

jar

0---y- r where (In and (Jr are left-shift operators defined by (In = (0,

, w( -1), w(o); 0, ...) f--+ (0, ... , w(O), 0; 0,

(Jr = (0,

,0; y(l), y(2), ...) f--+ (0, ... ,0; y(2), y(3),

), ).

REMARKS:

(1)

We may interpret the shift operators (In and (Jr as (In = shift left and append a zero (Jr = shift left and discard first symbol

(2)

The sequences [f(e;)]j = jth component of the vector f(e;), with e, = ith unit vector, i.e. [e

a .= { 0,I, k

i

=k

i #- k,

i

= 1, 2, ... , m,

will provide the same information as the impulse-response map of a continuous-time, constant linear system. Knowledge of these sequences suffices to determine the zero-state input/output behavior of a constant linear system. In our standard notation we have e, E 0 corresponding to the sequence t=O t #- 0.

9.3

THE MODULE STRUCTURE OF

n, r AND X

249

The fundamental problem oflinear system theory is to construct (realize) a canonical linear dynamical system !:, whose input/output map II:. agrees with a given input/output map f. Regarding (F, G, H) as matrices, for a moment, Definitions 9.1 and 9.2 immediately imply that E realizes f if and only if

9.3 THE MODULE STRUCTURE OF

n, r

AND X

Our main objective now is to establish the following Fundamental Theorem of Linear System Theory The natural state set X f associated with a discrete-time, linear, constant input/output map f over k admits the structure of a finitely generated module over the ring k[z] (polynomials in the indeterminate z with coefficients in k).

To prove the theorem, we shall introduce a number of definitions and constructions, which will ultimately enable us to verify that the state set X f (as defined below) satisfies the axioms for a finitely generated module over k[z]. It is most convenient to introduce the various canonical constructions in a sequence of steps. STEP 1 Q ~ km[z], regarding km[z] as a k-vector space. The explicit form of the isomorphism is

co ~

L w(t)z-t E km[z]. tEZ

Note that by Definition 9.2(a), the above sum is always finite, and t the convention adopted in Definition 9.2(a).

~

0 by

STEP 2 Q ~ km[z], regarding km[z] as a k[z]-module. In fact, Q is a free k[z]-module with the m generators {el , e 2 , ••• , em}' where

o o o 1

o o

+- i

th

position.

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9


The above claim is easily validated by defining the action of k[z] on through scalar multiplication as -: k[z]

a

x a --+ a

(n,W)Hn·w,

with

wjEk[z],)= 1,2, ... ,m.

Here nWj is the usual product of one polynomial in k[z] by another. STEP 3 On a the action of the shift operator 0"n is represented by multiplication by z. Thus, dynamical action is transformed into the algebraic operation of multiplication. STEP 4 r is isomorphic to the k-vector subspace of kP[[Z-l]] (formal power series in z - 1) consisting of all formal power series with no constant term. The explicit isomorphism is

y

~

Ly(t)z-tEkP[[Z-l]]. teZ

In general, the sum is infinite and is to be interpreted strictly algebraically with no question of convergence. The isomorphism is completed by noting that y(O) = O. STEP 5 cation as

r

has the structure of a k[z]-module by defining scalar multipli-:k[z] x

r--+r

(n, Y)Hn·y = n(O"r)Y.

This product is equivalent to the rule: multiply y by tt in the usual way and then delete all terms containing non-negative powers of z. We have now seen that a and r admit natural k[z]-module structures. It is now necessary to connect-up these structures with the input/output map f. To this end we have Definition 9.3 Given two inputs w, w' E 0, we say that lent to w', written W == fW' if and only if f(w 0 v)

= f(w' v) 0

for all v EO.

W

is Nerode equiva-

9.3

THE MODULE STRUCTURE OF

n, r AND X

251

Here "0" denotes the operation of concatenation in 0, i.e. o:OxO--+O

(co, v) f--+ O"}}lw v v,

where Iv I = length of v and v is the join operation w v co'

= (0, ... , w( -t), ... , w( -1), co'(-t), ... ,w'( -1);0, ...)

It is easily verified that

== J defines an equivalence relation on O.

Definition 9.4 The set of equivalence classes under == J' denoted X J {(w)J: we O} is the state set of the input/output map f.

=

We now return to the problem of relating the module structure on 0 and r to the map f and its state set X J. Proposition 9.1 The N erode equivalence classes X J offare isomorphic to the k[z] quotient module Olker f. PROOF

By the relation w v= 0

zlvl w

+v

and the k-linearity of f, we have few v) = few' v) 0

0

for all veO,

if and only if f(z'· co) = f(z'· co')

for all r

~

0 in Z.

There is no intrinsic reason for selecting the input space 0 to relate to X J. By duality we could just as easily have chosen the output set r, as is indicated in the problems at the end of the chapter. The preceding development shows that the state set of the input/output map f can be given the structure of a k[z]-module. Let us now consider the corresponding question for the state set Xl; of a dynamical system given in "internal" form. Proposition 9.2 The state set Xl; of the system 1: = (F, G, -) admits a k[z]-module structure. PROOF X = kn is already a k-vector space. To make it into a k[z]-module, we define scalar multiplication as

-: k[z] x k" --+ k" (n, x)f--+n(F)x.

(Here n(F) is just the polynomial n( . ) evaluated at the matrix F).

252

9


Let us now restate some basic facts of system theory in the above moduletheoretic language. Proposition 9.3 In the system L F: XI; -,XI; is given by x~z·x.

= (F, G, H) with state module X, the map

The result follows immediately from Proposition 9.2, if X X = XJ' then PROOF

x(l)

XI;' If

= Fx(O) + Gw(O)

+ Gw(O),

= F[~]J assuming x(O) results from the input input z . ~ + w(O). Hence x(l)

=

~.

This implies that x(l) results from the

= [z· ~ + w(O)]J =

z- [~]J

=

z- [~]J

+ [w(O)]J + Gw(O),

which establishes the result. We can now re-state the usual criterion for reachability in more elegant fashion. Proposition 9.4 The system L = (F, G, -) is completely reachable columns ofG generate the k[z]-module XI;'

if the

PROOF

if and only

Assume any x E XI; can be written as m

X

=

L njgj,

j=l

By Proposition 9.3, this is the same as saying m

X

=

L1 nj(F)gj'

. j=

which is equivalent to complete reachability by the usual criterion involving the matrix

It is an easy exercise in application of the basic definitions to show that the external system with state module X J is both completely reachable and completely observable. Let us now show how to obtain a module-theoretic definition of complete observability for the internal system L = (F, -, H).

9.4

SOME SYSTEM-THEORETIC CONSEQUENCES

253

Consider the k-homomorphism H: Xl; --+ Y = k". Let us extend H to a k[z]-homomorphism fl as

u. Xl;

--+

r

x ~ (Hx,

Hiz- x), H(Z2. x), ...).

From the standard definition of an observable state, we see that no non-zero element of the quotient module XJker fl is unobservable. Thus, we have The system ~ = (F, -, H) is completely observable quotient module X Jker fl is isomorphic with Xl;'

Propositon 9.5

only

if the

if and

The above reachability/observability results suggest two important modules: (i) the submodule of Xl; generated by G, i.e. k[z]G; (ii) the quotient module X Jker tt, characterizing the observable states of~.

If we are interested in states which are both reachable and observable, the obvious thing to do is factor the unobservable states out of the submodule of reachable states. This new quotient module xg = k[z]G/ker fl is called the canonical state set for the system ~ = (F, G, H). If we have X~ ~ Xl;' then we say that Xl; is canonical relative to G, H. This terminology now allows us to address the question of modeling input/ output data by an internal system ~. The first main result is Correspondence Theorem of k[z]-homomorphisms f: basis change in Xl;'

There is a bijective correspondence between the set and the set of canonical systems ~, modulo a

n --+ r

In other words, every input/output map f has associated with it an internal model ~, which is unique up to a coordinate change in the state space Xl;' 9.4 SOME SYSTEM-THEORETIC CONSEQUENCES

The above module theory framework has exhibited a number of basic system-theoretic facts in clearer and sharper detail. However, there are a number of less obvious results which can also be obtained deriving mainly from the fact that k[z] is a principal ideal domain. Here we sketch a few of the most interesting developments, leaving others to the Exercises and Problems at the end of the chapter. Let us begin by recalling the notion of a torsion module. Definition 9.5 A module M over a commutative ring R is said to be a torsion module if for every m E M, there exists an r E R such that r . m = O. If this is not the case, then M is called a free module.

9

254

Definition 9.6


If L c M, the annihilator A L of L is the set

AL

= {rER: r-l = 0

for alII E L}.

REMARKS:

(l) (2)

A L is an ideal in R. M a torsion module does not imply A L =f O.

The preceding definitions allow us to prove the following important system-theoretic fact:

Theorem 9.1 (Recall: dim PROOF

~ ~

0 and let v be the number of poles of Z(2) in Re 2 > O. Then Jl = P

+ v,

where P is the number of times the Nyquist locus encircles the Schubert hypersurface a(ker[l m' K]) in the positive direction. PROOF A direct consequence of fact that u - v is equal to the change in argument in det(Z(2)K + I), plus the fact that this number is precisely the number of encirclements of a(ker[l m' K]) made by the Nyquist locus. Details are given in the Chapter References. Note that this result recaptures the most important aspect of the scalar Nyquist criterion of Chapter Seven, namely that it involves a fixed curve obtained from Z(2) that does not need to be changed as we vary the gain K.

9.16 ALGEBRAIC TOPOLOGY AND THE SIMPLICIAL COMPLEX OF l:

To this point we have employed several algebraic and geometric objects to study the various properties oflinear systems. Each "gadget", whether it be a k[z] - module, a vector bundle or a Grassmann variety, illuminated a different aspect of the overall category called Linear Systems. In this brief section, we introduce still one more algebraic object, a simplicial complex, in order to further amplify our prejudice that the study of Linear Systems is really applied algebra in disguise. At the most basic level of set theory, a simplicial complex arises whenever we try to geometrically characterize a binary relation defined on two finite sets. Let A and B be such finite sets with card A = n, card B = m, and let 2 c A x B be a binary relation, i.e., a, is 2 - related to b, if and only if (a i , bj ) E 2, i = 1,2, ... , n;j = 1, 2, ... ,m.

9.16

ALGEBRAIC TOPOLOGY AND THE SIMPLICIAL COMPLEX OF l:

293

We can represent such a relation by an n x m incident matrix A, whose elements are given by 1, (ai' b)EA [Alj = { 0, otherwise. Thus, we identify the rows of A with the elements of A, the columns with the elements of B. Clearly, by interchanging the roles of A and B we obtain the conjugate relation A* c B x A, and the corresponding incidence matrix A* = A'. While A is an algebraic characterization of the relation A, the idea of a simplicial complex arises when we attempt to geometrically represent the relation. Let us identify the elements of A with the vertices of a complex KA(B; A), while the elements of B will represent the simplices. Thus, if the element bkEB is A-related to the elements {al' a 2 , a 3 } , for instance, then b, would constitute the 2-simplex, bk = 0, ~ 0 or indefinite, respectively. Apply this result to the quivers of system theory. 25. Consider the open-loop transfer matrix Z(A) = H(AI - F)-lG and the feedback gain matrix K. Assume 1: = (F, G, H) is canonical, i.e., Z(A) is proper. Show that the elements p, 11, v ofthe Multivariable Nyquist Theorem 9.18 are given by

+ I)], [det(Z(A)K + I)]

11 = # zeros [det(Z(A)K

v = # poles

309


as A ranges over the imaginary axis (A = iOJ, OJ real). Thus, P = J1- - v = net change in the argument of det(Z(A)K + I)/2n. 26. Let Z(A) be a non-degenerate p x m transfer matrix of Macmillan degree n = mp. For all choices (AI"'" An), show that it is possible to find d(m, p) solutions K to the equation det(Z(Ai)K

+ I = 0,

i

= 1,2, ... , n,

where

d

) _ 1!2! ... (p - 1)'21 ... (m - 1)!(mp)! 1'2' . . .... (m+p- 1)'.

(m,p -

(This problem shows the rather astounding number of solutions possible to the output pole-placement problem). 27. Let Ky(X: A) be the simplicial complex of the single-input system ~ = (F, g, -) as defined in Section 16. Given two simplices up and a; in Ky(X; A) we say they are joined by a chain of connection if there exists a finite u~s such that sequence of simplices U~I' U~2"'" (i) (ii) (iii)

is a face of up' is a face of a., U~, and U~i+ I have a common face, say, u Pi ' U~l

u~.

i = 1, 2, ... , s - 1.

If we adopt the standard terminology that dim a, = i, then we say that such a chain is a q-connection if

(a) (b)

q = min{ lXI' PI' P2,"" Ps-l' IX s } · Verify that q-connection is an equivalence relation on Ky(X; A) for each q = 0, 1, 2, ... , dim K = N. Define the structure vector Q of K as Q

= (QN' QN-l,"" Ql' Qo),

where Qi = number of distinct i classes under the relation of q-connection, i = 0, 1, ... ,N. Show that ~ is completely reachable if and only if the structure vector of Ky(X; A) has the form Q

(c)

= (11

1 .. ·1).

Let Kx(Y; A*) be the complex conjugate to Ky(X; A). Show that ~ is completely reachable if and only if the conjugate complex has the structure vector Q* = (2n

-

1,2 n -1, ... ,2n -1, 1, 1, ... ,1),

i i i

(n -1)st

{Jth

position

position

Oth position

310

9


where

p = nil k=O

(nk -- 2). 1

(Here n = dim F) and ( . ) denotes the usual binomial coefficient. 28. Consider the m-input system L = (F, G,- ), with n x nm controllability matrix C(j =

[gl"'" gm' Fg I , · · · , Fg mF2g l, .. -, F 2gm, ... , Fn-lg h

••• ,

Fn-lg m ].

Define an abstract simplicial complex K~m as an object with mn distinct vertices (corresponding to the entries of C(j) and the rule that an ordered sequence (Xl' x 2 , ••• , x q of q distinct vertices is a simplex in K~m if and only if q ~ p, p = 1, 2, ... , n.

>

(a)

Use the Mayer-Vietoris sequence to show that the homology groups of K satisfy the relation Hp(K~~l)

~ Hp(K~~ll)EBHp_l(K~m-I),

l:S;;p:s;;n-1.

(b)

Call a system p-generic if any p vectors in the set C(j are linearly independent. Assume that L = (F, G, -) is (p + I)-generic. Use the above resuslt to prove that the homology groups of the complex of L satisfy where

1)

k = (nm p+1 ' p = 1,2, ... , n - 1. (This result generalizes the single-input case treated in Section 16, at least for generic control systems). NOTES AND REFERENCES

Section 9.1 The concepts and techniques of modem algebra and geometry take their sharpest form when employed for the analysis of linear systems. Nevertheless, their principal utility lies in the ease with which most of the ideas presented in this chapter can be extended to broad classes on nonlinear processes. A recent summary of much of the work in this currently very active area of algebraic-geometric nonlinear system theory is given in Casti, 1., "Nonlinear System Theory," Academic Press, New York, 1985.


311

Some good background references to modern algebra and geometry are Birkhoff, G., and Mac Lane, S., "A Survey of Modern Algebra," 3rd ed., Macmillan, New York, 1985. Herstein, I. N., "Topics in Algebra," Blaisdell, New York, 1964. Auslander, L., and Mackenzie, R., "Introduction to Differentiable Manifolds," McGraw-Hill, New York, 1963. Singer, I. M., and Thorpe, J., "Lecture Notes on Elementary Topology and Geometry," Scott, Foresman, Glenview, Illinois, 1976.

Some excellent general references to the use of algebra in system theory are provided by Fuhrmann, P., Algebraic system theory: An analyst's point of view, J. Franklin lnst.; 301, 520-540 (1976). "Algebraic and Geometric Methods in Linear Systems Theory" (Byrnes c., and Martin, C., eds.) Lectures in Applied Math., vol. 18, American Math. Soc., Providence, R. I., 1980.

Section 9.2-9.3 The definitive statement of the module-theoretic treatment of linear systems is presented in: Kalman, R, Falb, R, and Arbib, M., "Topics in Mathematical System Theory," McGraw-Hill, New York, 1969.

The material synthesized in this volume is based upon the earlier works of R. E. Kalman, the most important being Kalman, R, Algebraic structure of linear dynamical systems, Proc. Nat. Acad. Sci. USA, 54, 1503-1508 (1965). Kalman, R., Algebraic Aspects of the Theory of Dynamical Systems in "Differential Equations and Dynamical Systems" (Hale, J., and LaSalle, J., eds.) Academic Press, New York, 1967. Kalman, R, "Lectures on Controllability and Observability," Centro Internazionale Matematieo Estivo Summer Course 1968, Cremonese, Rome.

Section 9.4 For additional examples of how a linear system can be used as a pattern recognition device, especially in the context of brain modeling, see Kalman, R, On the mathematics of model building, in "Neutral Networks" (Caianiello, E., ed.) Springer, New York, 1968.

Section 9.5 The historical use of transfer functions to describe a linear system stems from work in electrical circuit design. To some extent this is unfortunate, since a number of issues extraneous to the mathematical relationship between transfer functions and linear systems were obscured for several decades For instance, the usual discussions of transfer functions rely upon various convergence arguments for the contour integral defining the Laplace transform, conveying the erroneous impression that only stable systems can be studied by transform means. The algebraic treatment arising from the module-theoretic view shows that no questions of convergence enter

9

312


at all. The mathematical basis for transfer functions is the existence of a k[z]module structure on I" for which the input/output map f is a homomorphism. This is a purely algebraic fact. Additionally, the module machinery allows us to treat both continuous and discrete-time processes using the same formalism. Thus, both the Laplace transform and the" z-transform" methods are unified in the algebraic treatment. For some classically-oriented views on transform methods, see Horovitz, I., and Shaked, D., The superiority of transfer function over state variable methods in linear, time-invariant feedback systems designs, IEEE Tran. Auto Control, AC-20 84-97 (1975).

Zadeh, L., and Desoer, C, "Linear Systems Theory," McGraw-Hill, New York, 1963. Rubio, J., "The Theory of Linear Systems," Academic Press, New York, 1971. Smyth, M., "Linear Engineering Systems," Pergamon, New York, 1972.

Section 9.6 Two seminal semi-classical papers which led directly to the algebraic treatment of transfer function realization are Kalman, R., Mathematical description oflinear dynamical systems, SIAMJ. Control, 1, 152-192 (1963).

Kalman, R., Irreducible realizations and the degree of a rational matrix, SIAMJ. Control, 13, 520-544 (1965).

Both of these papers contain a wealth of concrete examples of the abstract realization procedures presented in the text. The Invariant Factor Theorem can be proved in many different ways at several levels of algebraic generality. For instance, see Gantmacher, F., "Matrix Theory," VoL 1, Chelsea, New York, 1959. Albert, A., "Fundamental Concepts of Higher Algebra," D. Chicago Press, Chicago, 1965. Jacobson, N., "Lectures in Abstract Algebra, VoL 2: Linear Algebra," Van Nostrand, New York, 1953.

Section 9.7 There are numerous formally different, but equivalent, methods for realizing a given transfer matrix. For example, see Rosenbrock, H., Computations of minimal representations of a rational transfer function matrix, Proc. IEEE, 115, 325-327 (1968). Mayne, D., A Computational procedure for the minimal realization of transfer function matrices, Proc. IEEE, 115, 1368-1383 (1968). Gilbert, E., Controllability and observability in multivariable systems, SIAM Control J., 1, 120-151 (1963).

The first effective procedure for carrying out the realization procedure for input/output data given in "Markov" form is Ho, B. L., and Kalman, R., Effective construction of linear state variable models from input/ output functions, Reqelunqstechnik, 14, 545-548 (1966).


313

Other procedures improving upon aspects of Ho's method are presented in Silverman, L., Realization of linear dynamical systems, IEEE Tran. Auto-Control, AC-16, 554-567 (1971). Rissanen, J. Recursive identification of linear systems, SIAM Control J., 9; 420-430 (1971). Willems, J., Minimal realization in state space form from input/output data, Mathematics Institute Report, U. of Groningen, Groningen, Netherlands, May, 1973. Guidorzi, R., Canonical structures in the identification of multivariable systems, Automatica, 11, 361-374 (1975).

A summary of much of this material and the various realization procedures is given in Kalman, R., Realization theory of linear dynamical systems, in "Control Theory and Topics in Functional Analysis," Vol. II, Int'!. Atomic Energy Agency, Vienna, 1976.

We have not touched upon the important case of non-stationary behavior sequences, for which the coefficient matrices F, G, H may be time-varying. An algebraic tretment for the realization of such behavior is developed in Kamen, E., and Hafez, K., Algebraic theory of linear time-varying systems," SIAM J. Control Optim., 17, 500-510 (1979). Kamen, E., New results in realization theory for linear time-varying analytic systems, IEEE Tran. Auto. Control, AC-24, 866-878 (1979).

Section 9.8 The idea of partial realization is actually equivalent to the classical idea of Pade approximation, which concerns the problem of finding for a given Laurent series L a.z :", a strictly proper rational function f with denominator of minimal degree whose Laurent expansion in z - 1 agrees with the given series through the first r terms, for some specified finite r. The first systematic account of the partial realization problem is given in Kalman, R., On partial realizations of a linear input/output map, in "Guillemin Anniversary Volume," (de Claris, N., and Kalman, R., eds.) Holt, Rinehart & Winston, New York, 1968.

A definitive sharpening of these results is Kalman, R., On partial realizations, transfer functions, and canonical forms, Acta Polytechnica Scandinavia, Ma 31, 9-32 (1979).

An algebraic treatment of the partial realization problem avoiding use of the Hankel matrix and the above noted Pade data constraints, but based upon a generalization of the Berlekamp-Massey algorithm for recursive decoding of cyclic codes, is found in Sain, M. K., Minimal torsion spaces and the partial input/output problem, Info. and Control, 29, 103-124 (1975).

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Section 9.9 According to popular legend, the Pole-Shifting Theorem was first proved around 1959 by J. Bertram and later by R. Bass. The first published proof for multi-input systems seems to be Wonham, W. M., On pole assignment in multi-input controllable linear systems, IEEE Tran. Auto. Control, AC-12, 660-665 (1967).

We say that ~ = (F, G) is coefficient assignable if given any monic polynomial p(z), there exists a m x n matrix K such that det(zI - F + GK) = p(z). Over a field, coefficient assignability is equivalent to pole assignability; however, over a general ring R, coefficient assignability is a much stronger property. For an example over a general ring showing that complete reach ability is not enough to ensure coefficient assignability for multi-input systems, see Bumby, R., and Sontag, E., Reachability does not imply coefficient assignability, Notice Amer. Math. Soc., 1978.

An algebraic approach to the pole-shifting problem using the transfer matrix only is given in Seshadri, V., and Sain, M. K., An approach to pole assignment by exterior algebra, Allerton Can! on Circuits & Systems, Monticello, Illinois, Sept. 1976.

Section 9.10

Two excellent surveys on systems over rings are

Sontag, E., Linear systems over commutative rings, Rich. di Auto., 7,1-34 (1976). Naude, G. and Nolte, C., A survey of the theory of linear systems over rings, NRIMS Tech. Rpt.; TWISK 161, Pretoria, South Africa, June, 1980.

See also Sontag, E., On linear systems and non-commutative rings, Math. Sys. Th., 9, 327-344 (1976).

Section 9.11 The ideas of this section have been extensively pursued and expounded by R. Hermann in a continuing series of books and research articles Hermann, R., "Interdisciplinary Mathematics," Vols. 8, 9, 11, 13, 20, 21, Maths Sci. Press, Brookline, Mass. 1974-1980.

Probably the most complete and accessible work on algebraic geometry and its uses in linear system analyses is Tannenbaum, A., "Invariance and System Theory: Algebraic and Geometric Aspects," Lecture Notes in Mathematics, Vol. 849, Springer, Berlin, 1981.


315

See also the works Hermann, R., and Martin, c., Applications of algebraic geometry to systems theory, part I, IEEE Tran. Auto. Control, AC-22, 19-25 (1977). Brockett, R., Some geometric questions in the theory of linear systems, IEEE. Tran. Auto. Control, AC-21, 449-464 (1976). Byrnes, c., and Hurt, N., On the moduli of linear dynamical systems, Adv. in Math. Supp. Series, 4, 83-122 (1979). Byrnes, c., and Falb, P., Applications of algebraic geometry in systems theory, Amer. J. M aths., 101,337-363 (1979). Hazewinkel, M., "A Partial Survey of the Uses of Algebraic Geometry in Systems and Control Theory," Report 7913/M, Erasmus u., Rotterdam, 1979. "Algebraic and Geometric Methods in Linear Systems Theory," (Byrnes, C; and Martin, C., eds.) American Math Society Lectures in Applied Math, vol. 18, Providence, R.I., 1980.

Section 9.12 The feedback group and the Kronecker indices seems to have first been introduced into linear system theory by Brunovsky in Brunovsky, P., A classification of linear controllable systems, Kybernetiku, b, 176-188 (1970).

See also, Kalman, R., Kronecker invariants and feedback, in "Ordinary Differential Equations," (Weiss, L., ed.) Academic Press, New York, 1971.

The relationship between the Kronecker indices and the McMillan degree of a transfer matrix is spelled out in detail in Martin, C, and Hermann, R., Applications of algebraic geometry to systems theory: The McMillan degree and Kronecker indices of transfer functions as topological and holomorphic invariants, SIAM Control J., 16, 743-755 (1978).

Related results are Brockett, R., Lie algebras and rational functions: Some control-theoretic questions, in "Proc, Queens' Symposium on Lie Theory and Its Applications," (Rossmann, W., ed.) Queens University, Kingston, Ontario, 1978. Brockett, R., and Byrnes, C., Multivariable Nyquist criteria, root loci and pole placement: A geometric viewpoint, IEEE Tran. Auto. Control, AC-26, 271-284 (1981). Hazelwinkel, M., Moduli and canonical forms for linear dynamical systems-III: The algebraicgeometric case, in "Geometric Control Theory," (Martin, C., and Hermann, R., eds.) Math. Sci. Press, Brookline, Mass., 1977. Friedland, S., Classification of Linear Systems, "Proc, AMS Conf. on Linear Algebra and its Role in System Theory," (Datta, M., ed.), Providence, RI (to appear 1986). Delchamps, D., Global structure of families of multivariable linear systems with an application to identification, Math. Systems Theory, 18, 329-380 (1985).

Section 9.15 The results of this section follow the work Brockett, R., and Byrnes, c., Multivariable Nyquist criteria, root loci, and pole placement: A geometric view, IEEE Tran. Auto. Cont., AC-26 (1981).

316

9


Some interesting related work is reported in De Carlo, R., and Saeks, R., The encirclement condition: An approach using algebraic topology, Int'l., J. Control, 26, 279~287 (1977). Owens, D., On structural invariants and the root-loci of linear multivariable systems, Int'l., J. Control, 28, 187~ 196 (1978).

Section 9.16 The initial results on associating a simplical complex with a linear system are reported in Casti, J., Polyhedral dynamics and the controllability of dynamical systems, J. Math. Anal. & Applic. 68, 334-346 (1979).

The extension of these results to multi-input systems is given in Ivascu, D. and Burstein, G., An exact homology sequence approach to the controllability of systems, J. Math. Anal. & Applic. (forthcoming).

C RAPTER

10

Infinite-Dimensional Systems

10.1 FINITENESS AS A SYSTEM PROPERTY

In mathematics one of the most highly-prized properties of a given structure is some type of finiteness; we can classify finite groups, but infinite groups still remain a mystery; finitely-determined function germs form the basis for the theory of singularities of smooth functions, while infinitelydetermined germs exhibit pathological behavior; the Hilbert Basis Theorem shows that the ring of invariants of a transformation group is finitely generated. In system theory, as in the rest of mathematics, finiteness properties also enter in a basic way as we have seen in the preceding chapters. For example, the Cayley-Hamilton Theorem, a finiteness condition on the subspaces generated by the powers of a matrix, plays the central role in controllability and observability conditions. Similarly, the crucial assumption underlying all realization theory results is that the external behavior sequence jr = {J l' J 2'" .}, possesses a finite-dimensional realization. In this chapter we shall explore the consequences of dropping the finiteness assumption regarding the dimension of the system state-space. As will be seen below, relaxation of this assumption greatly complicates the study of the properties of our system 1:, both from a technical as well as conceptual point of view. A variety of non-equivalent notions of controllability, observability, realization and stability emerge depending upon the choice made for the mathematical structure of the state-space, and there is no clear-cut answer as to which notion is the "right" one. 317

10 INFINITE-DIMENSIONAL SYSTEMS

318

In view of the mathematical and conceptual complications, it is reasonable to inquire as to the utility of studying such systems. Is there a broad enough class of such objects to justify the major efforts needed to create a decent theory of infinite-dimensional systems? Unfortunately, the answer to this question is an unambiguous and emphatic, yes! In fact, it is very likely the case that virtually all real systems arising in nature have infinite-dimensional state-spaces, and it is only by looking at subsystems that we encounter the finite-dimensional versions treated in the preceding chapters. For instance, any system whose space-time behavior is governed by a diffusion equation

ax at

-=Ax+f

'

x(O)

=

g

has a natural state-space that is infinite-dimensional, consisting of an appropriate function space containing the initial function g. Similarly, any system whose dynamics involve time-delays such as dx dt

= f(x(t),

x(t)

x(t - r)

+ u,

-r>o,

= get),

also has a natural infinite-dimensional state-space determined by the nature of the initial function g. There are many other examples of such systems, as well, some of which will be encountered as we proceed. Systems involving non-local interactions, multiple time-scales and time varying coefficients all lead to infinite-dimensional systems justifying the major effort needed to understand the nature and behavior of such processes and their models. In a short introduction, it is impossible to do justice to the richness and variety of the many types of issues that arise in a thorough treatment of infinite-dimensional processes. So here we shall try only to give the spirit and flavor of the nature of the questions that have to be addressed and the type of results that can be obtained. However, even to do this much it will be necessary to depart from the cozy' mathematical confines of finite-dimensional vector spaces and matrices and invoke a higher level of mathematical sophistication involving Banach and Hilbert spaces, together with the properties of various classes of linear operators defined on these spaces. Since we have no space here to develop a "crash-course" in elementary functional analysis, it will be assumed at the outset that the reader comes prepared with the equivalent of a one-semester introductory course in such matters. If not, any of the excellent texts cited in the bibliography may be consulted to fill-in background gaps.

10.2

319

REACHABILITY AND CONTROLLABILITY

10.2 REACHABILITY AND CONTROLLABILITY

We consider the system

x=

Fx

+ Gu,

x(O) = x o,

where now x E X, a reflexive Banach space. We assume that F generates a strongly continuous semigroup T(t): X -+ X, and that the operator G: U -+ X is bounded. Further, the space of controls U is also taken to be a reflexive Banach space. It is useful to write ~ in the integrated "mild" form x(t)

= T(t)x o + {T(t - s)Gu(s) ds.

Consider the operator B: U[O, t; U] -+ X

given by Bu

==

I

T(t - s)Gu(s) ds,

U E

U[O, t; U].

If we want to reach the state x* at time t using a control u, then we must require ~(B) == Range B = X,

since T(t)xo is fixed. This is a very strong condition, so strong, in fact, that ifit is satisfied it is possible to extend the semigroup T(t) to a group. Thus, generally speaking we cannot expect to.reach x* exactly, and must be content with getting arbitrarily close. In this case we need only require 9l(B) = X, and we say that ~ is approximately reachable, while if 9l(B) = X, we speak of ~ as being exactly reachable. For finite-dimensional systems, we have the very elegant, convenient and computable criterion for reachability that ~ is reachable if and only if the matrix has full rank. Our first line of attack on the infinite-dimensional case is to see to what degree the above test can be generalized. To this end, we define the set U" =

rUE

U: Gu E

n:'=l

E0(F")}.

(Here E0(T) denotes the domain of the operator T). Then we have the following result. Theorem 10.1

The system

~

is approximately reachable n

if the

= 0,1, ...}>= X,

set

320

10

INFINITE-DIMENSIONAL SYSTEMS

i.e., if the closure of the set of elements {PGU*} generates all of the statespace X. PROOF By elementary operator identities, it is easy to see that L is approximately reachable if and only if the adjoint equation

G*T*(t)x*

= 0 = x* = 0,

for all t. Thus, if L is not approximately reachable there exists an x* #- 0 such that <x*, T(t)Gu)

=0

for all u E U*.

Differentiating this identity countably many times at t = 0 yields <x*, FnGU*) = 0,

n = 0, 1,2, ... ,

implying f7I #- X, completing the proof. REMARK The converse of this theorem is not true as is seen by consideration of the case

~(F)

=

{x: x, ~;,

~:~

EX}.

Then if the system is x = Fx + glU 1 + gzu z with u 1 , U z E LZ[O, f], gl' gz EX, with gl(t), gl(t - h) vanishing, together with its derivatives outside [-h, 1], h > 0, it turns out that f7I is then a proper subspace of X. Theorem 10.1 appears to be the right generalization of the finite-dimensional rank condition, although in practice it may be difficult to apply as the set U* is not always easily obtained. Nonetheless, there are many situations in which we can readily obtain U* and, consequently, effectively determine the approximate reach ability of L. EXAMPLE

10.1 Consider the system

x = Fx + gu, where F is the right-shift operator having the matrix representation

F=

000 100 0 1 0 001

10.2

321

REACHABILITY AND CONTROLLABILITY

in the standard basis {e;}~ Then

1

of l z. Assume that g has the form g = (1 0 0 .. .y.

and, in general, Fng = en+ r- It follows that g is a cyclic vector for F and, consequently, Bl = X, implying that ~ is approximately reachable. EXAMPLE 10.2 We consider the simple one-dimensional heat equation with pointwise control

z(O, t) = z(l, t) = 0.

The semigroup generated by F

= dZjdx Z with

these boundary conditions is

00

I

T(t)qJ = J2

qJne -n

2

,,2

sin (nnx),

t

n=I

qJn

=

J2 0. Also,

1]

(here H±(I/2+E) is a Sobolev space on [0, 1]). Thus, we can easily see that G*T*(t) = J2

I

00

qJn e -

n

2

,,2

t

n=I

so that

~

sin(nnx I ) ,

is approximately reachable if 00

J2

I

n=I

qJn e-n

2 ,, 2

t

sin(nnx I )

=

°

=>

qJn

= 0,

for all t and all n. It is elementary to verify that this will be the case only if Xl is an irrational number. The preceding results have established a basis for checkingc the approximate reachability of a given system. But, what about exact reachability? Can

10

322


we give any corresponding results for determining those (rare) situations when it is possible to steer 1: to a desired state exactly? Perhaps the best general result in this direction is Theorem 10.2 Let U and X be a reflexive Banach spaces. Then 1: is exactly reachable if and only if there exists an oc > 0 such that

oc/lG*T*(· )x*llq where lip

~

+ 11q = 1. (Here /I·ll q denotes the

/Ix*/IX*, norm in the space U[O, t; U*J).

The proof of this result can be found in the Chapter References. As an illustration of the use of Theorem 10.2, consider the controlled wave equation

z(O, t) = z(l, t) =

o.

We can rewrite this in operator form as

w= The operator F = d given by T(t)

21dx 2

Fw

+ Gu.

generates the strongly continuous semigroup T(t)

W I] [21:[<WI' qJn>H cos met + ~me <w qJn>H sin mrtJqJn ] [w = . 21:[ -nn<w l, qJn>H sin nnt + <w qJn>H cos nntJqJn 2,

2

2,

acting on the Hilbert space H = L 2[0, 1]. Here we also have G =

[~}.

It is easy to verify that T*(t) = T( - t) and G* = [0 1J, so if u E L 2[0, t I; HJ, the system will be exactly reachable if and only if there exists an a > 0 such that

a/lG*T*(·)wI1 2 ~

/Iwl H ·

After some tedious calculations, this condition reduces to 2

4( t 1

-

. 22 nnt 1) SIn 4n 2 n 2

2

n n

2

~

2

(1 - cos 2nnt l) ,

together with tl > [

sin 2nntl] 2nn

.

n = 1,2, ... ,

10.3

323

OBSERVABILITY AND DUALITY

These conditions together reduce to t1 >

[

Sin met1] , me

which is satisfied for any t 1 > 0 implying that such an a> 0 can be found enabling us to conclude that the system is exactly reachable. 10.3 OBSERVABILITY AND DUALITY

One of the foundational results of linear system theory (nonlinear, too, for that matter) is the duality between reachability and observability: the system

x = Fx + Gu,

x(O)

= 0,

(L)

is completely reachable if and only if the dual system

x'=x'F', y= G'x,

(L*)

is completely observable. Loosely speaking, L* is formed from L by transposing all matrices and vectors and setting up the vector-matrix multiplications to be consistent (and also reversing the direction of time for time-dependent systems). It's natural to conjecture that by paying attention to the details, the same principles can be applied in the infinite-dimensional setting by using dual spaces and adjoint operators. However, since we have seen that there are at least two different notions of reachability, we will have to introduce corresponding concepts of observability and match them to generalize the finite-dimensional duality results. . Consider the system i

= Fz,

z(O)

=

Zo,

y=Hz,

(L*)

where F is the infinitesimal generator of a strongly continuous semigroup T(t) on a reflexive Banach space Z with Zo E ~(F), while H E L(Z, Y), with Y also a reflexive Banach space, The mild solution of L* is y

= HT(t)zo·

Z

-+

Let us define the map £1):

Lq[O, t*; Y]

Zo 1-+ HT(t)zo.

°

Definition 10.1

f3 > such that

We call L* continuously observable on [0, t*] if there exists a for all ZEZ.

324

10


Definition 10.2 The system L* is initially observable on ker @ = {O}. Now let us make the following dual identifications

u= Y*,

G=H*, 1

F=F*,

[0, t*J if

Z=X*,

1

-+-=1. P q With the foregoing definitions and identifications, we can easily prove the following duality theorem. Theorem 10.3 (a) (b)

The system ~* is initially observable on [0, t*J approximately reachable on [0, t*]. ~* is continuously initially observable on [0, t*J exactly reachable on [0, t*].

PROOF

if and only if

is

~

if and only if ~ is

See references cited at the end of the chapter.

EXAMPLE 10.3 heat equation

Using Theorem 10.2, it can be seen that the controlled

02Z

oZ

ot = ow2 z(O, t)

+ u,

(~)

= z(l, t) = 0,

is not exactly reachable on L 2[0, IJ using controls u E L 2[0, t*; Z]. As a consequence, the dual system

x(O, t)

=

x(l, t)

= 0,

(~*)

yet, w) = x(t, w),

is not continuously initiallyobservable on [0, t*]. In Chapters 3 and 4, we have drawn the distinction between reachability and controllability, as well as between observability and constructibility. Let us focus upon controllability for the moment. In this case, we are interested in whether or not there exists a control that will drive the system

x= to the origin. We say that

Fx ~

+ Gu,

x(O)

= x o =f. 0,

is exactly controllable on [0, t*J if

range fJB

:=>

range T(t*),

(~)

lOA

and

325

STABILITY THEORY ~

is completely controllable on [0, t*] if range fJI :::J range T(t*).

Conditions for these inclusions to hold can be easily obtained from the standard fact that if V, W, Z, are reflexive Banach spaces, with FE L(V, Z), G E L(W, Z), then ker(G*)

c:

ker(F*)¢>range(G)

:::J

range(F).

In terms of constructibility, the controllability concepts above correspond to the notions of continuously final and final constructibility. We say that ~* is continuously constructible on (0, t*] if there exists a fJ > 0 such that fJll(Qxllu[o,t*;Yl ~ II T(t*)xllx,

for all x E X, and

~*

is constructible on [0, t*] if ker

(!) c:

ker T(t*).

With these definitions, we can state the following duality theorem linking controllability and constructibility. Theorem 10.4 (a) (b)

The system

~*

is

constructible on [0, t*] if and only if ~ is approximately controllable; continuously constructible if and only if ~ is exactly controllable. 10.4 STABILITY THEORY

The two most important stability results for the linear system

x=

Fx

+ Gu,

x(O) = X o

*0

(~)

are (1) (2)

Asymptotic Stability (in the sense of Lyapunov), where we have Ilx(t)1I ~ 0 if and only if Re A;(F) < 0 (assuming G = 0), and the Pole-Shifting Theorem (stabilizability), which asserts that if ~ is reachable (stabilizable), there exists a feedback matrix K such that F - GK has prescribed roots (Re Ai(F - GK)

< 0).

For infinite-dimensional systems, neither of these results extends directly without the imposition of additional structure on F and/or a strengthening of our notion of stability. Let us begin with the problem of the asymptotic stability of the free motion of ~. By a well-known counterexample due to Hille and Phillips, there exists a semi-group T(t) whose generator F has an empty spectrum, although II T(t) II = e", t ~ 0, showing that there is no possibility of basing the

10

326


asymptotic stability test on the spectrum of F without further assumptions. It turns out that a convenient way to proceed is to introduce the following stronger notion of stability. Definition 10.3 The uncontrolled system there exist constants M, W > 0 such that

I T(t) II

~

S Me-rot,

is called exponentially stable if t ~

o.

We can now establish the following result Theorem 10.5 The system ~ is exponentially stable sup Re 1(F) < 0 in any of the following situations: (a) (b) (c)

if

and only

if

F bounded, T(t) an analytic semigroup, T(t) compact for some t* > O.

Instead of the spectrum test which involves finding the location of the spectrum F, it is often convenient to be able to invoke the test that ~ is asymptotically stable if and only if there exists an Hermitian matrix P > 0 such that F*P

+ PF= -Q,

for all Q > O. To extend this result to our current setting, we must assume that X = H, a Hilbert space with inner product Co). We then have Theorem 10.6 The system ~ is exponentially stable an Hermitian operator B on H such that 2Re(BFx, x) = (BFx, x)

=

if and

only

if there

exists

+ (F*Bx, x)

-lIxIl 2 •

Now let us consider the problem of stabilizing the controlled motion of ~ by means of linear feedback. We would like to be able to have a theorem along the lines that if ~ is approximately reachable, then ~ is stabilizable, and conversely. However, consider the system X = [2,

with

U

= reals,

10.5

327

REALIZATION THEORY

For any feedback control u = 0, and all x

E

E0(F).

While regular and balanced realizations of Ware quite different objects, there are some strong interconnections as the folIowing result demonstrates. Theorem 10.9 Wet) has a balanced realization if and only if it has a regular realization. Furthermore, the infinitesimal generator F can be taken to be the same in both realizations. PROOF

See the Baras and Brockett article cited in the References.

A key concept in realization theory is the idea of a canonical, i.e., reachable and observable model 1:. In our setting, we say that 1: is reachable if G*eF*tx = 0 implies x = 0 for all t ~ 0, and observable if HeFtx = 0 implies x = 0 for all t ~ O. Let us assume that we are given a (regular) realization 1: = (F, G, H) of W. How can it be reduced to a canonical realization? Theorem 10.10 Let a (regular) realization 1: = (F, G, H) be given with state space X for the weighting pattern W. Let

M = {x

E

X: HeFtx = 0, t ~ O}\

N = {x E M: PMG*eF*tx

= 0, t ~

O}',

10.5

329

REALIZATION THEORY

where PM is the orthogonal projection onto M, P N the corresponding projection onto N. Then the realization

G=

P = PNFIN'

PNG,

H=

HPN

is a canonical (regular) realization of W with state space N.

The main importance of canonical realizations is the State-Space Isomorphism Theorem, which asserts that any two canonical realizations of a given weighting pattern W differ only by a change of basis in the state-space X. Unfortunately, the natural notions of reachability and observability introduced above do not lead to canonical models admitting such a result for infinite-dimensional systems. To extend the State-Space Isomorphism Theorem, we need a more restricted concept of reachability and observability.

Definition 10.6 The system 1: = (F, G, H) is called exactly reachable if the

lim t*--+oo

r eFtGG*eF*t dt t*

Jo

exists as a bounded and boundedly invertible operator. Similarly, 1: is exactly observable if lim t*-+

00

i

tO

eF*tH*HeFt dt

0

exists as a bounded and boundedly invertible operator. We can now establish Theorem 10.11 Let 1: = (F, G, H), ~ = (p, G, H) be two realizations of W. Then 1: and ~ are similar, i.e. there exists a bounded and boundedly invertible operator P such that PF = PP,

PG=G,

if either of the following conditions are satisfied: (i) (ii)

1: and ~ are reachable and exactly observable or, 1: and t are observable and exactly reachable.

Now we come to the practical question of how to identify those weighting patterns W that possess a regular realization. Theorem 10.12 (1)

Let Wet) be a pxm weighting matrix. Then

If W admits a regular realization, each element of W must be continuous and of exponential order and

10

330

(2)


if every element of W is locally absolutely continuous and the derivative of W is of exponential order, then W admits a regular realization.

Assuming that W admits a regular realization, our final task is to explicitly construct the operators F, G, H. To this end, we introduce the Hankel operator JIf: L 2(0, 00; U) ~ L 2(0, 00; Y)

{oo W(t + a)u(o) da.

u(t)~

Assuming that W is square-integrable, JIf is well-defined and bounded. Next, introduce the left-translation operator on a space X by eFt:X~X

+ t),

f(a)~f(a

t

~

O.

An explicit regular realization of W is then given by Theorem 10.13

A regular realization

of~

of W is given by

X = range£; F

= infinitesimal generator of the left-translation semigroup on X,

(Gu)(a) = W(a)u, Hx = x(O). Moreover, this realization is reachable and exactly observable. 10.6 THE LQG PROBLEM

In correspondence with the finite-dimensional results of Chapter 8, it is reasonable to expect that with a little care in defining spaces and operators all (or, at least, almost all) the results pertaining to optimization of quadratic cost functions subject to linear dynamical side constraints can be recaptured in our current, more general setting. In view of the quadratic nature of the cost criterion, the most natural setting for such extensions is a Hilbert statespace. Consider the linear system

x=

Fx

+ Gu,

x(O) = x o,

where x(t) eX, a Hilbert space, and G is a bounded operator from a Hilbert space U to X. Assume that F generates a semigroup T(t) on X. As cost functional, we take J(u) = (Mx(T), x(T)x

+ IT«Qx, x)x + (Ru, u)u) dt,

10.6

331

THE LQG PROBLEM

where M, Q E L(X, X) are self-adjoint, non-negative operators, R E L( U, U) is such that (Ru, u) ~ ml/ul/ 2,m > oand (·,.)ydenotes the inner product in the corresponding Hilbert space Y. After an operator version of completing-the-square, whose details can be found in the standard references cited later, the (unique) solution to the above optimization problem is given by

Theorem 10.14 The unique optimizing control for the functional leu) is given in feedback form by u*(t) = -R-1G*P(t)x(t),

a.e.,

where pet) is the (unique) solution of the inner product Riccati equation d

- dt (P(t)x, y) = (Qx, y)

+ (P(t)x, Fy)

+ (Fx, P(t)y) peT)

(P(t)GR-1G*P(t)x, y),

o :5; t :5; T,

= M,

for all x, y E P}(F). Furthermore, the optimal cost l*(u*) is given by l*(u*) = (P(O)x o, xo)x. REMARK The preceding set-up does not apply to the case when the control u is exerted only upon the boundary. In this event, G is not a bounded operator from U -+ X so additional assumptions are needed to make the conclusions of Theorem 10.14 hold. See the Problems and Exercises for an account of how this can be done.

Of considerable importance is the infinite-time (T = co) version of the foregoing regulator problem. Here we wish to minimize

leu)

=

l~(QX,

X)x

+ (Ru, U)u) dt,

and it is straightforward to show that the optimal control is given by

u*(t) = -R-1G*Px(t), where P satisfies the algebraic Riccati equation

(Qx, y)

+ (Fx, Py) + (Px, Fy) -

(PGR-1G*Px, y)

provided that (F, Ql /2) is observable and (F, G) is reachable.

= 0,

332

10

EXAMPLE


Consider the controlled heat equation oW

02W

at = ow -

OX2

ow

(0 t) = -

ox'

+ U(X, t),

(1 t) = 0

ox'

0 < x < 1, t > 0,

,

W(x,O) = wo(x),

with the quadratic cost functional J(u)

= L:O

f

[w 2 (t, x)

+ u2 (t, x)] dx dt.

Letting £P;(x) = fi cos nix, £Po = 1, it is straightforward to show that the (unique) solution of the operator algebraic Riccati equation is 00

P=

L (Jn 4/ + 1 -

n 2l)£P/·, £P),

j=l

and the optimal feedback control is 00

u*(x, t) = -

L (Jn 4/ + 1 -

n2l)wj(t)£pj'

j=O

where wj is the jth coefficient in the expansion w(t, x) =

00

L wk(t, 'x)£Pk'

k=O

The optimal cost is J*(u*)

=

00

L (Jn 4j4 + 1 j=O

n 2j2)<wo, £Pj)2.

10.7 OPERATOR RICCATI EQUATIONS AND GENERALIZED X- Y FUNCTIONS

Earlier we saw that for systems in which the number of inputs and outputs are much less than the number of states, a major computational savings could be obtained by employing the so-called "generalized" X- Y functions to calculate the optimal feedback gain K = -R-1G*P, rather than solving the Riccati equation directly. This observation holds with even greater force when dealing with infinite-dimensional problems, since here it is usually the case that the controls and/or observations affect the state at only a finite number of points, or at least over some proper subset (or subspace) of X.

10.7

333

OPERATOR RICCATI EQUATIONS

Thus, we would expect the X - Y functions to provide a means for improving the computational tractability of the LQG problem. Here we sketch the means for extending these functions to the infinite-dimensional setting, i.e., the generalization to X - Y operators. Consider the linear system

x = Fx + Gu, with F, G, x, u as above, and introduce the observation equation y=Hx.

Here y E Y, a Hilbert space. We wish to minimize the quadratic cost functional J(u) = R

IT

[(y(t), y(t)y

+ (Ru, u)uJ dt,

> O. The solution as given in the last section is u*(x) = -R-1G*P(t)x = -K(t)x,

where P(t) satisfies the operator Riccati equation of Theorem 10.14 with Ql l 2 = H, (i.e., (Qx, z) = (Hx, H*z»). The generalized X-Y operators provide a means to compute the operator K directly, without the intermediate computation of the Riccati operator P. Theorem 10.15 system

The optimal feedback gain operator K(t) is given by the dK dt =

-R-1G*L*(t)L(t),

dL

dt = L(t)F + L(t)GK(t), where K(t) E L(X, U), L(t) E L(X, Y). The initial conditions are K(T) = 0,·

L(T) = H.

Furthermore, the solution of the operator Riccati equation P(t) is given by P(t)F

+ F*P(t) =

-K*(t)K(t)

+ H*H -

L*(t)L(t).

REMARK The equations for the operators K and L should be interpreted in the same inner product sense as that given in Theorem 10.14 for P. The proof is a direct extension of that given in Chapter 8 for the finite dimensional case. Again we emphasize the point that in practice the spaces U and Yare usually finite-dimensional. An easy way to see the importance of

10

334


this fact for actual generation of the optimal control is to consider the kernels k(t, x), I(t, x) of the operators K(t) and L(t) in the case when F is a diagonal operator. In this case we have the kernel equations

Ok~;

x)

Ol~~

x) = F*I(t, x)

=_

o d~

[Ix R -lg(~)I(t,

k(T, x) = 0,

]V(t, x),

+ v« x{Ixg(~)I(t,

~) d~

1

I(T, x) = h(x).

The conditions for k, 1 on ax are given by the state equation at t = 0, i.e., x(O) = Xo' The functions g(x), h(x) are determined from the operators G and Hby [Gu(t)](x)

=

gT(X)U(t),

Hx(t)

=

Ixh(~)x(t,

o d~.

Here, if k(t, x) is of dimension m l , the number of input variables, while I(t, x) is of dimension m2 , the number of system outputs, then solution of the above system involves m l + m 2 conventional partial differential equations, rather than the doubly infinite operator Riccati equation for the kernel p(t, x, ~) of the operator P(t). EXAMPLE

Consider the one-dimensional heat equation oW

02W

at = ox2 + b(X w(O, t) = w(l, t)

I

2)U(t),.

0 ~ x ~ 1,

= 0,

where the point control u(t) is exerted at the mid-point of the rod. We have the spaces X={w(x):w'EL 2[0,I;R], O~x~I}, U~

Y=R.

Let the observation be the average temperature over the rod: y(t) =

f

w(t, x)h(x) dx,

where h(x) is the weight function determined from the operator y discussed earlier. Let the control u(t) be chosen to minimize J =

faT [y2(t) + u 2(t)] dt.

= Hx,

as

335


According to our earlier results, the optimal feedback control u*(t) is given by u*(t) = -

s:

k(t, x)w(t, x) dx,

where k(t, x) is the solution of ok(t, in x) = -l( t, 1)1( 2 t, x ) , ol(t, x) = _ 02/(t, x) ot ox2

+

l( l)k( ) t, 2 t, x ,

k(t,O) = k(t, 1) = l(t,O) = l(t, 1) = 0, k(T, x) = 0,

I(T, x) = h(x).

These equations are relatively straightforward to solve numerically, once the observation weight function h(x) is prescribed. Numerical results are given in the papers cited in the references.


1. Let the input, state and output space U, X and Y be complex vector spaces and let A E C, the complex numbers. Define Z(A) to be the space of all pairs (u, y) E U X Y such that there -exists an x E X such that for a given A

AX = Fx

+ Gu,

y=Hx, where F, G, H are appropriate operators on X, U and Y, respectively. As A varies, we obtain a map C -+ ~(U

Ei1 Y),

where ~ is the Grassmann space. Call this map the transfer function of the system ~ = (F, G, H). (a)

show that if the resolvent (AI - F) - 1 exists, then ~(U

Ei1 Y)

= {(u, y): y = H(AI - F)-lGU},

so that the transfer function can be identified with the curve

A -+ H(A.! - F)-lG in the space of linear maps.

10

336


(b)

Show that usually differential operators do not satisfy the above condition. (c) Consider the linear differential equation d 2w dw a2 dt 2 +aldi+aow=f(t).

Show that for A E C, the transfer function is given by the set of all pairs (u, y) E U X Y such that Dy - Ay = uf,

where D is the differential operator d2

d

= a2 dt 2 + a 1 dt + aoI.

D

2. Consider the nonlinear diffusion equation olp

at =

jPlp ox2

+ Alp -

2 J1,lp ,

A, J1,

c: 0, 0 s

x

S 1.

Show that if A < n 2 then lp = 0 is an asymptotically stable equilibrium in the L 2 sense. (Hint: consider the candidate Lyapunov function V

= Illp 1112[0,1)'

3. Consider a transmission line with capacitance c(x), inductance l(x) > 0, resistance r(x) and conductance g(x), x E [0, 1]. The energy of a currentvoltage distribution

G)

in the line is E

=

s:

(cv 2

+ [j2) dx.

Let

Assume the line is short-circuited at x = 0 and connect the end x = 1 to a lossless reference line having c = 1 = 1. Assume that signals are sent down the reference line toward the end x = 1 of the line under study. (a) Show that the dynamical equations for this system are

:t G) FG) + =

g=HG}

Gu,

337


where F is the unbounded operator

acting on the space

~(F)

=

{G)EH:

E

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