Three Decades of Mathematical System Theory: A Collection of Surveys at the Occasion of the 50th Birthday of Jan C. Willems (Lecture Notes in Control and Iinformation Sciences)

Lecture Notes in Control and Information Sciences Edited by M.Thomaand A.Wyner

135 H. Nijmeijer J. M. Schumacher (Eds.)

Three Decades of Mathematical System Theory A Collection of Surveys at the Occasion of the 50th Birthday of Jan C. Willems

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

Series Editors M. Thoma • A. Wyner Advisory Board L D. Davisson. A. G. J. MacFarlane - H. Kwakernaak J. L. Massey. Ya Z. Tsypkin • A. J. Viterbi

Editors Hendrik Nijmeijer Department of Applied Mathematics University of Twente P. O. Box 217 7500 AE Enschede The Netherlands Johannes M. Schumacher Centre for Mathematics and Computer Science P. O. Box 4079 1009 AB Amsterdam The Netherlands and Department of Economics Tilburg University P.O. Box 90153 5000 LE Tilburg The Netherlands

ISBN 3-540-51605-0 Spdnger-Verlag Berlin Heidelberg New York ISBN 0-387-51605-0 Spdnger-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of trenslation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its version of June 24,1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. ~) Spdnger-Vedag Bedln, Heidelberg 1989 Printed In Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Offsetprinting: Mercedes-Druck, Bedin Binding: B. Helm, Berlin 216113020-543210 Printed on acid-free paper.

Preface

The year of birth of a scientific discipline is not often clearly defined. However, in the case of mathematical system theory, the year 1959 is a strong candidate, with only 1958 and 1960 as serious opponents. One may take as evidence the testimony of George S. Axelby, the founding editor of both the IEEE Transactions on Automatic Control and the IFAC journal Automatica: The year 1959 was the prelude to drastic changes in the control field. (...) Then, the first IFAC Congress was held in Moscow, USSR, in June 1960. Three papers were presented that were to revolutionize the theory of automatic control and set the direction of research for years to come. They were the papers by Kalman, Bellman, and Pontryagin. It seemed that almost immediately after the IFAC Congress all papers were involved in modem control theory and the use of state variables with the theorem, lemma, proof format. (Quoted from a speech held in Los Angeles on December 10, 1987, at the 24th IEEE Conference on Decision and Control; IEEE Control Systems Magazine 8-2 (1988), p. 98.) The above words are about control, but, of course, system theory concerns more. Here's a description of what happened to filtering in the same period, following the early contributions of Wiener and Kolmogorov: As could be expected, there were many researchers who advanced, reexamined, reconsidered, generalized, etc., this Wiener-Kolmogorov theory and many applications of it were reported. In the opinions of some, these efforts had long reached the point of diminishing returns and the IEEE Transactions on Information Theory in fact felt it to be necessary to publish an editorial which essentially told these authors - - albeit in nicer terms - - to "get off it and get on with something else". This happened in 1958 at about the same time that Kalman and Bucy were putting together their papers which would result in one of the most rapid shifts of attention ever to be wimessed in a research field! These lines come from the paper "Recursive Filtering", Statistica Neerlandica 32 (1978), pp. 1-39 (the quote is on pp. 6-7). The author is Jan C. Willems, who himself turned twenty in 1959 and was largely innocent of the great turnover so aptly described by him. This was soon going to change. In 1968, Jan Willems received his doctorate in electrical engineering from MIT and started a career that would make him one of the great contributors to the emerging field. His early work is concerned with the stability analysis of nonlinear systems using Lyapunov techniques. Later on, he undertook a general study of dissipativity and the linear-quadratic optimal control problem, contributed to realization theory, and introduced the highly fruitful concept of 'almost invariance" in geometric control theory. In recent years, he has been systematically working out the consequences of his definition of a system as a 'behavior', a set of trajectories which may be described in various ways. A man of many ideas, Jan Willems is guided by a strong intuition and always looks for the basic issues. As a Professor at the University of Groningen, he has been responsible to a large extent for the flourishing of System Theory in the Netherlands; this has included the founding of the Dutch National Graduate School in System and Control Theory, of which he is Chairman. His leading position in the international system

IV theory community is illustrated by the fact that he is Managing Editor of two prominent journals in the field, the SlAM Journal on Control atut Optimization and Systems and Control Leuers. This year, Jan Willems will be fifty, and the field of mathematical system theory is three decades old. This volume has been compiled to let it serve as a surprise gift for Jan on his birthday, September 18, but also to look back on what system theory has achieved in thirty years, and to look ahead for the new challenges that are facing us. The contributions were written by invitation. All subjects covered in this book are related in some way to work done by Jan Willems, and all authors are related in some way to Jan Willems himself. In a volume on system theory, these are inactive constraints. The present book covers the wide area of mathematical system theory, and in particular indicates the great variety of methods that are being applied in this field. The contributors have been asked to write a survey-like paper, discussing past, present and future of a particular research field. We have also encouraged the authors, who are all responsible for leading contributions in their field of writing, to apply personal taste in selecting trends that they feel are important. Jan Willems has many friends, and many among these are prominent system theorists. We, as the editors of this volume, have tried to keep the anniversary project manageable. We have not asked all of Jan's friends to contribute, and our selection is to some extent arbitrary. We would like to thank the authors, for their enthousiastic response and for their fine papers. In particular, we are grateful to Jacques L. Willems, who happens to turn fifty on the same day as Jan, for his willingness to write a contribution with a more personal flavour than one usually finds in scientific articles. Furthermore, we thank the reviewers, who have helped the authors and us a lot to polish the contributions. Finally, our thanks go to Springer Verlag, for its willingness to publish the volume in the series Lecture Notes in Control and Information Sciences with its wide distribution.

Enschede/Amsterdam, June 1989

Henk Nijmeijer Hans Schumacher

Contents

A. C. Antoulas The Cascade Structure in System Theory R. W. Brockett Smooth Dynamical Systems which Realize Arithmetical and Logical Operations

19

C. I. Bymes Pole Assignment by Output Feedback

31

G. Conte, A. M. Perdon Zeros, Poles and Modules in Linear System Theory

79

R. F. Curtain Representations of Infinite-Dimensional Systems

101

M. Deistler Symmetric Modeling in System Identification

129

P. A. Fuhrmann Elements of Factorization Theory From a Polynomial Point of View

148

K. Glover, J. C. Doyle A State Space Approach to H.. Optimal Control

179

M. lkeda Decentralized Control of Large Scale Systems

219

H. Kimura State Space Approach to the Classical Interpolation Problem and Its Applications

243

V. Kucera Generalized State-Space Systems and Proper Stable Matrix Fractions

276

S. K. Mitter, Y. B. Avniel Scattering Theory, Unitary Dilations and Gaussian Processes

302

H. Nijmeijer On the Theory of Nonlinear Control Systems

339

G. Picci Aggregation of Linear Systems in a Completely Deterministic Framework

358

J. M. Schumacher Linear System Representations

382

VI H. J. Sussmann Optimal Control

409

A.I. van der Schaft System Theory and Mechanics

426

J'. Vandcwalle, L. Vandenberghe, M. Moonen The Impact of the Singular Value Decomposition in System Theory, Signal Processing, and Circuit Theory

453

J.H. van Schuppen Stochastic Realization Problems

480

J. L. Willems Robust Stabilization of Uncertain Dynamic Systems

524

W. M. Wonham On the Control of Discrete-Event Systems

542

The Cascade Structure in System Theory A. C. Antoulas Department of Electrical and Computer Engineering Rice University Houston, Texas 77251-1892, U. S. A. and Mathematical System Theory E. T. H. Zi3rich CH-8092 Z0rich, Switzerland

Abstract. An overview is presented of results which show the central role of the cascade structure in linear system theory. The first group of results is related to the recursive realization problem while the second group of results is related to passive network synthesis.

1. INTRODUCTION. The cascade interconnection of two-pairs (systems having two inputs and two outputs) has played a major role in network theory. The impetus was given by the landmark result of Darlington's in 1939, who showed that every positive real (or bounded real) function can be synthesized as a cascade interconnection of lossless two-pairs terminated by a resistor. A quarter of a century later, Belevitch generalized this result to matrix-valued positive-real (or bounded-real) functions, whereby the scalar two-pairs are replaced by matrix two-pairs. This topic is also closely related to scattering matrix synthesis which has recently been investigated in detail in a series of papers by Dewilde and Dym (see e. g. Dewilde and Dym [ 1989]). A few years ago Delsarte, Genin, and Kamp [1981] pointed out that the celebrated Nevanlinna-Pick recursive interpolation algorithm is closely related to the cascade structure. Furthermore, since stability tests for polynomials can be formulated equivalently as NevanlinnaPick problems, they are also related to the cascade structure. Actually, Vaidyanathan and Mitra [ 1987] showed that the classical Schur-Cohn test (for polynomials in the discrete transform variable z) as well as the Routh-Hurwitz test (for polynomials in the continuous transform variable s) can be interpreted in a unified way in terms of the cascade structure. Moreover, the parameters Support was provid~ by N.S.F. through Grant ECS - 05293.

2 which define this decomposition can be used to determine the root distribution of these polynomials with respect to the unit circle and the imaginary axis, respectively. Independently of the above developments, it has been recognized in the past decade, that the cascade structure plays a fundamental role in system theory; this is apparently unrelated with the passive network synthesis results. This came about by noticing that the cascade strucure is closely connected with the problem of recursive realization of a finite or infinite sequence of matrices or, equivalently, of a formal power series. Actually, Kalman [ 1979] noticed that the scalar version of the recursive realization problem can be represented as a ladder interconnection, which is equivalent to a continued fraction decomposition (well known in network theory). A few years later it was shown that the general reeursive realization problem for matrix formal power series can be solved using the cascade interconnection of matrix two-pairs (Antoulas [1986]). It follows that in the scalar case the ladder interconnection is a special case of the cascade interconneetion. This also shows that several results obtained in connection with the scalar recursive realization problem are actually connected with the cascade structure. Very recently (c.f. Willems and Antoulas [ 1989]) it has been recognized that the cascade structure is actually involved in the solution of the general (deterministic) modeling problem of arbitrary time series. These results are based on the powerful new framework of time series modeling introduced by Willems [ 1986-1987]. Besides the fact that the structure of the recursive realization problem is completely revealed (and the recursive Berlekamp-Massey algorithm shown to be connected to the cascade structure) there are further results of interest. Introducing the concepts of junks, the structure of an arbitrary sequence of numbers (or formal power series) is revealed. From this the remarkable result follows that the Cauehy index of a rational function can be recursively computed from its cascade decomposition. Consequently, the classical test for the stability of a polynomial (obtained by decomposing it to even and odd part) has an interpretation in terms of the cascade structure. Finally, a connection between the cascade structure and geometric control theory can also be established (see Kalman [1979] and Antoulas and Bishop [1987]). In the next section the cascade interconnection of two-pair systems is defined. In the first part of section 3, the consequences of a unimodularity assumption on the cascade interconnection are explored. This leads to the results related to rccursive realization which are summarized in section 4; in section 4.1 results in connection with ladder intercormections are displayed. In the second part of section 3, lossless two-pair systems and related concepts are introduced; they are used in section 5, to give a brief overview of the relationship between the cascade structure, the Nevanlinna-Pick algorithm and various stability tests.

2. GENERAL DEFINITIONS AND NOTATION. We will consider linear time-invariant systems, denoted by X, with two sets of inputs u, and two sets of outputs y, .~, with dimensions

u,~ ERm, y , ~ ~ R p.

u

_1 -I

Y"

I I_

--Y

3 Such systems are known in network theory as two-port or two-pair systems. A scalar two-pair system is one for which m = p = 1. These systems can be described by means of the chain parameters or the transfer parameters, which are rational matrices in the variable o. The former

ale: x(o) E ~×v(o),

L(o) ~

a'×e(o),

Y(o) ~ ReX"(o), M(o) ~ R'~Xm(o), det M ( o ) ~ : 0 ,

(2.1a)

arranged in the chain parameter matrix

v(o)

IX(o) Y(o)]

RfP +re)x(? +m)(o),

(2.1b)

I. L (o) M(o)

which satisfies the relationship

y(o) [~(o)l. .(o)] -- v(o) Ly(o)]

(2.10

The latter are:

Z(o) E RpX"(o),

OI2(o) ~ RpXV(o),

o2t(o) E R " X ' ( o ) , 022(0) ~ RmXV(o), det O21(o):~ O,

(2.2a)

arranged in the transfer parameter matrix

O(o):= [ Z(o) On(o)]

R~, +,,,) x ~ + m)(o)'

021 (O) 022(0)

(2.2b)

which satisfies the relationship

y(o) j

/

(2.20

Notice that the (1, l) clcmcnt of O(o) is singlcd out and denoted by Z instead of 0 I1 as it will bc of special significance in thc sequel. The chain and transfer parameters are related as follows:

Z(o) = Y(o)M--l(o), 012(o) = X(o) -- Y(o)M-l(o)L(o), 021(o) = M - I ( a ) ,

022(o) = -- M-I(o)L(o),

(2.3a)

and X(o) = On(a) - Z(o)(02J(o))-~O22(a), Y(o) = Z(a)(O21(a)) -~, (2.3b) L (a) ---~ -- (O21((7))- 1022 (i:0'

M (o) = (O21(o)) - 1

In the sequel the dependence of the various quantities defined above on o will be dropped to

keep the notation simple. Several two-pair systems Xk, k = i, i + 1.... , j - 1, j , as defined above, can be interconnected by letting

Yk

2

"1

-

I

Zi

[uk+i

~i+|

I~

' k=i,i+l

..... j - - l .

" .........

I. . . . . . . . . .

~j--I <

I

~j I-

I

I-I-

The overall system, denoted by Y'i,j, will be referred to as the cascade intereonnection of the subsystems 2~k, k = i, i + 1, ..., j. It readily follows that the chain parameter matrix Vi.j of Zi4 is

Vi,j = ViVi+ 1 "'" Vj_IVj" ~

(2.4)

in other words

The (I, 1) entries Z k of the Ok matrices are related in terms of linear fractional transfor-

mations, sometimes referred to as homographic traasfornuztions: Z..~, = (Y.,# + X.,BZ#+I,~,)(M.,Ij + La,#Z/~_I,v) - I = : V..#[ZB+1.~] ,

(2.5)

for all i ~ a <~ fl < y ~g j . This formula is sometimes interpreted as an extraction of the twopair described by O.,t~ from the system described by Z,.y with remainder the system described by Z p + ~,r.

3. S P E C I A L P R O P E R T I E S O F T H E CHAIN PARAMETER MATRICES. In this paper we will focus our attention on cascade interconnections where the chain parameter matrices are either polynomial unimodular or Iossless. Some general properties of the corresponding cascade interconneetions will be described next. 3.1 P O L Y N O M I A L UN1MODULAR CItAIN PARAMETER MATRICES. If the chain parameter matrix Vk of Xk is polynomial unimodular, its inverse, denoted by Wk, is also polynomial unimodular; it will be partitioned in the same way as Vk:

Wk =

-Uk

Rk

'

From dot Mk : # 0 follows det T~ :/: 0. It will be assumed that

M~. 1Lk, Uk T f I are strictly proper ratiooal.

(3. i b)

Let the degree df the jth column of Mk, i th row of T k (i. e. i tj' column of the transposed matrix Tk') be sy(k) = deg (Mkb, j E n~, ui(k) = deg (Tk')i, i E p.

(3.2)

(For a positive integer r, r : = { 1, 2, - . . , r}.) We will assume without loss of generality, that M k, Tk satisfy the following normalization conditions: ~. gy(k) = deg dot Mk, ~ i,~(k) = deg det Tk. jC-m

i~p

Such normalized matrices are called column reduced, row reduced respectively. In this case the indices sy(k), pi(k) can be interpreted as the Kronecker or reachability indices, the dualKronecker or observabili 9, indices of Zk, respectively. As a consequence of the above assumptions

M k I, T f I are pro17er rational. For details on reduced polynomial matrices and the Kronecker indices see Kailath [ 1980, ch. 6.3]. From the unimodularity of Vk it follows that the transfer parameters become:

Zk = Y k M k I = Tt-Iak,

O~2 = Tk1,

O21 = M f I,

0 22 = -- M f l L k .

(3.3)

The McMillan degree 6(Z) of a rational matrix Z is defined as the dimension of any minimal state space realization of Z. The unimodularity of Vk implies 8(Ok) = 8(Zk).

(3.4a)

Furthermore, since the polynomial factorizations of Zk given in (3.3) are coprime, we have 6(Zk) = deg det Mk = deg det Tk.

(3.4b)

(3.4a) implies that Zk uniquely determines the two-pair Y., and consequently the corresponding chain parameter matrix Vk as well. For details, and a proof of (3.4a), see Antoulas [1986]. Consider the cascade interconnection of the two two-pair systems Y-~, ~a+ l- If the quantity LaZo+ 1 M f I is strictly proper rational, i.e.

L , Z ~ + 1 M ~ I = C i o - I + C20-2 + . . .

, Ct E R n'xm,

(3.5)

it can be shown that (cf. Antoulas [1986])

8(Za.a+l) = 8(Za) + 8(Z.+ 1)-

(3.6a)

If the above rclationship holds for a = i, i + 1, - • • , j - 1, then ~(zi.j) ~ ~(zi) + ~(z~+~) + .-- + 8(zj_~) + 8(zj).

(3.6b)

Therefore, if the chain parameter matrices of a cascade interconneetion are polynomial unimodular and (3.5) is satisfied, the complexity of the overall system is equal to the sum of the complexities of each one of the subsystems.

Under the assumption (3.5) a minimal state space realization of Za.a+l can be written down in terms of minimal state space realizations of O. and Z . + I, by inspection. Let O,,(o) =

(ol -- F , , ) - I ( G a

G.)

+

(3.7a)

,

ha

)o

Za+l(a) = H ~ + l ( a I - F a + I ) - I G ~ + I ,

(3.7b)

be minimal state space realizations. Then Fa,a÷l

:=

,

a.+ift~

F~+l

Gaa+l

:=

'

(3.7c)

,

a~+lJ.

H,,,,,+I : = (H~ J . H . + I ) , is a minimal state space realization of Z.,~+l

(3.7d)

= V.IZ.÷d.

These formulae can be extended to the cascade intcrconnection of several subsystems Y.a, a = i, i + 1..... j - I, j . If these subsystems are such that the corresponding transfer parameter matrices are strictly proper rational (condition which is always satisfied for scalar two-pairs) the following triple

r,

0~i+1

ci+lh~

F,-~ I

0

Gi÷lai+2 F~+2

Fj-.2

~Tj-2Hj- 1

6-, c/uj_ j a~4 = ( G /

0

0 ...

0)',

H j j = (It,.

0

0...

©

0),

is a minimal state space realization of Zi4. The above formulae constitute the cascade canonical form, which shows explicitly how each Fa is ncvted in the overall Fi4. For details see Antoulas and Bishop [ 1987]. As a consequence of the fact that (3.7a) is a minimal realization of Z.,a+ l, the corresponding state spaces, namely X., Xa+ ~, X~,~+ 1, satisfy X~.~ l =

X~ • X~+l.

(3.8)

The isomorphism which appears in the above formula can be converted into equality by using

7 polynomial spaces as introduced by Fuhrmarm [1976]. Given a non-singular n Xn polynomial matrix A (a), the set

X a := {x ~ R " [ o ] : A - l x isstrictlyproperrational}, is a finite dimensional vector space, containing polynomial vectors as elements, and dim Xx = deg det A. With this notation, it turns out that we can define

Xo : = Xr.,

X~+~ : = Xr.~,,

x ~ . . + l : = xr~..,.

Isomorphism (3.8) then becomes

x . . . +, = r . + , x r . + xT..,.

(3.9)

The above formulae can easily be extended to the cascade interconnection of any number of subsystems (see Antoulas and Bishop [ 1987]). 3.2 L O S S L E S 8 CHAIN PARAMETER MATRICES. The concepts and results presented in this sub-sectlon are valid for discrete-time systems (i. e. in this section o = z, the discrete-time transform variable). With appropriate modifications however, they arc valid for continuous-time systems as well. In the sequel D will denote the open unit disc in the complex plane C, D its closure, Dc the complement of D, and OD the unit circle. Superscript * denotes complex conjugation of a scalar or a matrix. Given a rational matrix A (o), A Ca). : = A ' ( o - ~),

where prime denotes complex conjugation followed by transposition. A two-pair system Y is called all-pass if its transfer parameter matrix 19 satisfies 19(o).19(o) = / ,

for all a.

is bounded real (11t?)iff 0 is a bounded real matrix, i. e. all its poles are in D (is stable) and 19'(o)o(~) ~< I, o ~ B~.

Y- is Iossless bounded real (LBR) iff in addition O is unitary on 3D, i.c.

o'O)190) = l, o ~ 0D. Notice that Y is lossless bounded real iff it is stable all-pass. In section 5 we will consider elementa~ scalar two-pair systems with chain parameter matrices of the following form: v(o) =

~*

0j

~rO) '

8 where ] ¢ [ 5~ 1 and ~r(o) is the following elementary scalar all-pass function: • ,(o) =

1 - o~'*

The corresponding transfer parameter matrix is: (3.10b)

If ! -- ~t~* > 0, we can normalize the above two-pair as follows: P(o)

-

1

(1 - ~ * ) ~

0.11a)

v(a).

The resulting normalized transfer parameter matrix o(a)

=

0.11b) (1 - ~ * ) "

- ~*

o ~a)

is all-pass, i. e. o(,o.§(,r)

OA le)

= t.

If in addition, [ ~"[ < 1, the normalized two-pair is LBR. We will refer to • with chain parameter matrix defined by (3.10a) as an elementary unnormalized Iossless two-pair, and to ~1 with chain parameter matrix defined by (3.1 la) as an elementary lossless two-pair.

4. RECURSIVE REALIZATION AND RELATED TOPICS. Consider the sequence of p × m constant matrices SI.N_ 1 :

(,4 I, "42 ..... A N - I),

(4.h)

which will also be referred to as a sequence of Markovparameters. The realization problem consists in obtaining a parametrization of a/l p × m rational matrices ZLN _ ~(o) such that Z 1 . N - i ( a ) = A io -1 + A z a - 2 + . . .

+ AN_la -N+I + ... ,

(4.1b)

i. e. the coefficients of the first N - l terms of the formal power series expansion (or equivalently, the Laurent expansion in the neighborhood of infinity) of the solution match the given N - 1 Markov parameters. The complexity 8(ZI.N-1), which is defined as the McMillan degree (see sec. 3.1), is to be the parameter. Of particular interest are the minimal complexity realizations. The basic problem of existence and uniqueness of realizations was first studied by Kalman (see Kalman, Falb, and Arbib [1968]). In the sequel we will address a deeper question, namely, how Z1.N and 8(Z1.N) depend on N. In other words, we are looking for the function q, such that ZI,N = q'(ZI.N- 0.

(4.2)

This question is closcly related to the recursive realization problem: given Z u v _ 1 and the new Markov parameter A~, compute the updated solution Zi.t¢ as a function of Zt,r¢-l. As it turns out, the cascade structure provides the key to obtaining explicit answers to the above questions. Let Z Lr¢- I be a minimal realization of the sequence S L^'- I. Consider the coprime polynomial faetorizations -I = Ti-,~-1Q1,1¢-! = Zl,iv-1 = YL/c-1ML~v-!

= AI 0-1

-t-AN_.I a - N 4 1 + A N f f - N

-I- " ' "

(4.3)

"4" " ' " .

Because of copfimeness, there exist polynomial matrices Xi,t¢-i, Li,t¢-l, UI,N-I, R L t c - b such that conditions (3.1a, b) are satisfied. It will be assumed that MLN _ t, TLN- I, are column, row reduced (cf. (3.2a, b)). Thus to the original system, characterized by the transfer matrix Z tdv - 1, we attach a two-pair system 2:Uv - l whose chain parameter matrix is [XLt¢-I VI,N_ ! =

ILl,N_

YL~-t J I MI.N-I

"

From (3A) it follows that ~(O1,# - j) = 8(Zlm

- 0-

Recall the definition of a homographic (or linear fractional) transformation given in (2.5). We can state the following basic (4.4) Theorem. Given Z E Rt'Xm(o), : = V u v - I [ ~z] = ( Y l , # - I + X I , # - I Z ) ( M 1 , # - - z + L L i v - I Z ) - I ,

(4.5)

is a realization of S I,~r- 1 if and only if

(4.6)

L I.~ - t Z'Mi.~ - 1

is strictly proper rational. This result shows that given the fact that ZLN-J encodes the information contained in S m y - I , the parameter Z is in one-to-one correspondence with the continuations of S my- I,

namely I-I

~

( A # , A # ÷ I , "'" ).

(4.7)

Consequen@, if any of the Markov parameters At, t ~ N, is changed, on/y Z will be thereby affected, not 2l.t¢- I. The cascade interpretation of this formula is

Z ----~

From (3.6a) follows that

01,~-,

"

:

10

8 ( ~ ) = 8 ( z ~ , N _ 1) + ,~(Ez);

(4.8)

this implies that Z is minimal if and only if Z is minimal. The above result establishes the fundamental connection between the cascade structure and the realization problem. It shows that the function ~ in (4.2) is in essence the homographic or linear fractional transformation. The complete solution of the recursive realization problem can now be obtained, based on the above theorem. Given S1.N - i and a minimal realization ZI.N -1 thereof, our goal is obtain a parametrization of all realizations of the updated sequence SI,N = ( A I , A2, "'" , A N - I ,

AN),

and in particular, of all minimal ones. Recall (4.3) as well as the fact that M I . N - 1 and T1.N_ I are column, row reduced with column, row indices K)(I,N -- 1), j E m, u i ( l , N - 1), i E p . Let AK : = diag (o K'(1'N-

I),

. ..

A,, : = diag (O"#'l(l'N-l), . . .

,OK.(I.N

-- 1)),

,Opp(l.N-1)),

(4.9a) (4.9b)

while (4.9c)

Mhc @ R mxm, Thr E R e×p ,

are the constant matrices made out of the highest column, row coefficients of n l , N - 1, T 1 , N -. 1. According to Theorem (4.4), given A~,, there exists a rational matrix Z, denoted in this case by ZN, such that (4.10a)

Zl. N = Vl.N_l[ZN] ,

is a realization of S J,N and

8(Z~,N) = ~ ( Z ~ , # - ~ )

(4.10b)

+ 8(Zn).

Moreover, all realizations of S I.N can be obtained in this way. Thcre remains to show how Zt~ can be determined. Let E : = Thr(A~v - A#)Thc

= :

(cq).

(4.11a)

The sequence

SN : = (El,

"'" , Ep), p : = m a x { N - ~ j ( l , N - 1) - ~,(I,N - 1)} i4"

(4.1 l b )

is defined as follows: 'i,j iff t = N - Kj(I,N -- 1) -- v i ( l , N - 1) > 0, (El)J4

:=

0 i . f f t = 1,2, . . . , N -

Kj(I,N--I)--

? otherwise,

v,(I,N-I)>O,

(4.11e)

11 for j ~ m, i ~ p, where "?" stands for an element which can be chosen freely. In the scalar case p = m = 1, according to whether p = N - 2n, n : = d(Z1.N_ 0 = deg MI.N-I, is positive or not, Sr¢ = (0, " " , 0, ~) or S~ = (9.); in the latter ease the sequence is completely undetermined. We can now state the following fundamental (4.12) Theorem. (a) Z i.t¢ is a (minimaO realization o f S 1.t¢ if and only i f Z ~ is a (mh~imat) realization o f S~. (b) One minimal realization of the sequence S ~ is given by the formula Z~ : = A,(Eo-tV)A~.

(4.13)

Notice that Sjv depends on N, the Kronecker, the dual Kronecker indices, and the difference between the desired and the actual N th Markov parameters. Thus by using the cascade structure, th6 computational complexity of updating a minimal realization is the same as the computational complexity of determining the realization of a single-term-sequence. The second part of the theorem provides a closed formula expression for one minimal update. The above theorem has interesting consequences. The jt'~ Kronecker index of the updated system Zz.jv depends on the Kronecker and the dual Kronceker indices of Zl.~v-I as follows: one of the three relationships given below will occur Kj(1,JV) = K/(1,JV -- l);

(4.14a)

x j ( I , N ) = Ki(I,N -- 1) > ~j(I,N - !);

(4.14b)

Ky(I,N) = N -- v i ( l , N - 1) > g j ( I , N - 1).

(4.14c)

A similar result holds for the dual Kroneeker indices. The above relationships show that the Kronecker (and the dual Kronecker) indices of the overall minimal realization are non-decreasing functions of N. They also show how the fine structure of the complexity builds up as successive Markov parameters A t are supplied. Thus Z~.~_ l already determines all possible line structures of the updated system Zl,t~. Which one actually occurs depends on thc value of the particular A~v. The Kronecker indices of Z l,~v, Z I.Jv- 1 satisfy a further relationship, namely ~¢j(1,N) = Jcj(1,N - 1) + KI(N),

(4.15)

where Kj(N) is the jth Kronecker index of Z~. This shows that for minima/realizations, besides the McMillan degrees, the fine structure of the complexity (the Kronecker indices) is additive as well. Again, (4.15) holds also for the dual Kronecker indices. We have shown how given Sl.jv_.l, a minimal realization Z I j v _ 1 thereof, and the updated sequence S l.,v, we can construct a minimal Z,v such that Zl.. = gl,tc-dZ~],

or equivalently, V|,N = V I , N - I V N ,

represents a minimal realization of S j.N. Repeating the procedure, we can take care of further updatings A ~ + i, A^, +2, "" " , using the cascade structure. For details and proofs, see Antoulas

12 [19861.

The above recursive realization for the scalar case m = p = 1, and in particular formula (4.13), is known as the BerlekanwMassey algorithm. The matrix case presented above was solved by Antoulas [1986]; Remark (5.19i) of this reference gives a short history of the problem. We can now summarize the connection between recursive realization and the cascade structure in the following (4.16) Theorem. Consider the sequence of p × m constant matrices

S = ( A 1 , A2, " " , A u ,

""),

and the associatedformal power series Z(o) = A l o - I + A 2 o - 2 + . . . + A~vo -to + . . . . There is a bijective correspondence between S, Z and a (possibly infinite) sequence of two-pair systen~ ~i, described by the corresponding chain parameter matrices Vi(o). These systen~ are interconnected in cascade. Furthermore, for every t > 0 there exists apasitive integer it, t > O, such that

Zl. t = Yl.tMl..tI, where Via = Vii"2 "'" Vt =

I Xl.t Yi.t Ll.t MI,t

],

is a minimal realization of the following subsequence of S: SI. t = (.41, A2, "'" , Air). The Kronecker hzdices, the dual Kronecker indices and consequently the McMillan degrees of the subsystenzs add up to the corresponding quantities for any finite part of the overall system. Raaonali.ty of Z (o) is equivalent to the finiteness of the cascade structw'e. 4.1 CASCADE AND LADDER STRUCTURES. Whenever the Markov paramcters are scalar quantities, i.e. m = p = I, the cascade structure can be simplified. First we notice that the chain parameter matrix of the i th subsystem in the cascade decomposition has the form vi(o) =

i 0, ] -

1 mi(o)

where mi is a polynomial of degree 8i. Consequently, the corresponding transfer parameter matrix is 8i(a)-

1 m~(o)

[ I1 ] 1 1 "

13

It readily follows that the formal power series Z ( o ) ---- a l o -1 + a 2 o - 2 -4- • . .

can be expressed as a

c o n t i n u e d f r a c t i o n in terms of the polynomials mi:

1 z(a) =

1 m~(a) +

I

m2(o) + - -

As stated in Theorem (4.16), rationality of Z is equivalent to the finiteness of the cascade decomposition, which in turn is equivalent to the finiteness of the above continued fraction decomposition. Thus the cascade (or continued fraction) decomposition in the scalar ease becomes:

In network theory the above structure is recognized as a l a d d e r realization (see e.g. Balabanian and Bickart [1981, ch. 12]). Clearly, both the cascade and the ladder structures are related to the E u c l i d e a n algorithm. Let Z be rational: Z (o) = .V(o) / m (o), deg m (o) > deg y (o). Applying the Euclidean algorithm to the (eoprime) pair m, ),, the successive quotients turn out to be the polynomials mi: qi - ! = qimi + qi + I, i ~ k ,

where q0 : = m and ql : = Y , and q~ = 1. The equivalence of the cascade decomposition with continued fractions in the scalar case suggests a way of defining a continued fraction decomposition of rational matrices. Actually, as shown in Antoulas [1986], one can define the m a t r i x continued fraction decomposition as the cascade decomposition. Consequently, the cascade decomposition provides a means of defining a matrix Euclidean algorithm (see Antoulas [1986]). For a different way of defining matrix

14 continued fractions and a matrix Euclidean algorithm, see Fuhrmann [ 1983]. Next we will derive the consequences of relationships (4.14) for the scalar case. Let 8(1, t) denote the dimension of the minimal realizations of the sequence (a i, "'" , at). Clearly, if the sequence is rational, for large enough t we have

8(1, t ) = 8 1 + 8 2 +

"-.

The positive integer Ni will be called the ith jump point of S iff 8(1, t ) = 8 1

+ 8 2 + "'" + 8 ,

for all N i < ~ t < N i + ] ,

i.e. iff the dimension of minimal realizations of all sequences (a I, "'" , at) with t in the above interval, is equal to the sum of the first i 8's. Moreover, 8 i will be referred to as the ithjump o f S. Clearly, d(1, t) is a staircase function of t, with jumps of size 8i taking place at the points t = N i. Because of (4.14), for any sequence, there is a bijectivc correspondence between the set of j u m p sizes and the set of jump points. Let : = ( 8 1 82 - - . 8k)' , N : = ( N

1 N 2 - - - Ark)'.

It follows that N =Ad, where A is a lower triangular Trplitz matrix with first column equal to (1 2 2 • -. 2)'; A - 1 is also a lower triangular Trplitz matrix with first column equal to (1 - 2 2 - . . ( - 1 ) * 2 ) . These relationships imply in particular 8i+1 + 8i = Ni+] - Ni; This means that the larger the interval between jumps, the larger the new j u n ~ . It is interesting to notice that for a generic sequence 3i = 1, for all i ~

N i = 2 i - 1, for all L

In conclusion, to any arbitraly sequence of numbers S = (a], a2, • • • ), one can associate two sets of positive numbers 8 , Ni which satisfy the above relationship. This result for scalar sequences was first derived by Kalman [1979]. An issue of interest ifi applied mathematics, which is related to the cascade structure is that of the Cauchy index of a rational function Z(a) = p ( a ) / q ( a ) = a l o -1

+ a2 a-2

+

"..

,

which is defined as the number of jumps of Z from - oo to + oo minus the number of jumps from + oo to - oo. The Cauchy index is useful in investigating questions involving the distribution of roots of polynomials with respect to a given region. A classical result asserts that the Cauchy index of Z is equal to the signature of the Hankel matrix associated with the sequence (a I, a2, "'" ). (Recall that the signature of a symmetric matrix is equal to the number of positive minus the number of negative eigenvalues.) It was first shown by Kalman [1979] that the cascade decomposition of Z provides a way of

15 computing the Cauchy index or equivalently the signature of the associated Hankel matrix. Let mi(o ) = pio~' + lower order terms.

The distribution of the positive and the negative eigenvalues of the corresponding Hankel matrix can be recursivcly computed as a function of di and #i: (a) If ~i is even, the signature remains unchanged, i.e. the number of positive and the number of negative eigenvalues increase by 8i/2. (b) If 8i is odd, the signature changes by one. If /d > 0, the signature increases by one, i.e. the number of positive eigenvalues increases by (8~ + 1)/2, while the number of negative eigenvalues increases by (~$i - 1)/2. If p i < 0, the signature decreases by one, i.e. the number of positive eigenvalues increases by (8i - I ) / 2 while that of negative eigenvalues increases by (8~ + 1)/2. It is well known (see e.g. Gantmacher [1960]) that the above procedure applied to the rational function

m~en(a) moda(o) Z(o) = ~ or Z(o) = - moad(o) m,~,(o)

--

alO-I

q- a 2 0 . - 2 -{- . . .

,

where m~en, moad are the even, odd parts of a given polynomial m, leads to a count of the number of roots of m in the left-, right-half of the complex plane. The polynomial m is stable, i. e. has all its roots in the left-half of the complex plane if and only if all the jumps of the sequence (al, a2, "'" ) are equal to one (i.e. the sequence is generic) and the leading coefficients /d of all subsystems of the cascade (ladder) decomposition of Z are positive. By combining the above result with the general state-space representation of systems decomposed in cascade, a canonical form of (F, G, H) is derived; the elements of F are the Hurwitz determinants, i.e. stability checking is reduced to checking the positivity of the non-zero elements of F (see Kalman [ 1979]). A final application of the cascade interconnection is in geometric control (see Wonham [1979]). Given a scalar triple (F, g, h) subspaccs V of the state-space satisfying F V C V + im g,

V C ker h, dim V: nmximal, are of central importance. It is easy to show that there is a unique subspaee, dcnoted by V*, which satisfies the above conditions. Let Z(o):= h(oI-

F ) - I g = a l o -1 + a20-2 + . . . .

Consider the cascade decomposition of Z. It follows that V* is equal to the state-space of the interconnection of all but the first subsystem in this decomposition (see Kalman [1979]). This result can be generalized to the matrix case (see Antoulas and Bishop [1987]). A similar result was shown by Fuhrmann [1983].

16 5. STABILITY ISSUES. In this section we will show that the well-kown Nevanlinna-Pick rccursivc algorithm is closely related to the cascade structure. We will also show that various stability tests for polynomials have also an interpretation in terms of the cascade structure. 5.1 T H E NEVANLINNA-PICK ALGORITHM. Consider the array of pairs (oi, q'i), i ~ N, [o/[ > 1. For simplicity of exposition we will assume that oi 5/= oj, i :g:j. The Nevanlinna-Pick inte~olationproblem consists in parametrizing all rational functions ~(o) such that

q~(ai) =ep~, i E N ,

and

[q~a)[ <~M,

[a[ > 1,

where M is a given positive constant. With 9*(o) : = nl~(o)/d*(o), k = O, 1, " " , N, let 1

Vk(a) : =

M

0

)* M

where I t ( o ) =

1 - - oa/,.* O" - -

,for k = 1,2, -.- , N , and

O"k

nk-I(a) M d k - ! (o)

n*(o) ], = v,(o) Md*(o)J

with 9 ° : = ~i, i ~ N. Using these relationships we can recursively construct the constants

q'~z:=gk(o,,,),for

k = 1,2, - . . , N - - I ,

and m = k + l ,

--- ,N.

Thus the matrices Vk, k E N, are completely determined. Recall notation (2.5). It follows that for a fixed, arbitrary ~lv, the function

~ o ) : = ~°(o) = vl(o)v2(o) --. V~v(a)[¢N(o)], interpolates the given points. Furthermore

I~(~)1 <~M iff I~$-~1 < M , k

~ N , and [~v(o)] ~ M ,

for [a{ > 1.

The above considerations show that ~ a ) can be synthesized, in system theoretic terms, as a cascade intereonnection of two-pairs with chain parameter matrices V~ terminated by any bounded real system having transfer function ~N(o). A solution to the Nevanlinna-Piek interpolation problem exists if the parameters q,k - I of each one of these subsystems have magnitude less than M. If a solution exists, by normalizing the chain parameter matrices we can achieve that the

17 two-pairs involved are all-pass systems (cf. sec. 3.2). The normalized chain parameter matrices arc

Pk(o) :=

i

(1-

I , ~ - ~ / M I 2 ) '~

vk(o).

while the corresponding transfer parameter matrices are

Ok(a) =

(1-

I,~-~/MI2)"

- (~-~/M)"

~(o) '

The above considerations can be summarized in (5.1) Theorem. The Nevanlinna-Pick h;terpolation problem is solvable if and only if the interpolating

function ccm be represented as a cascade interconnection of all-pass two-pairs, terminated with some BR traasfer function. 5.2 ON T H E R O O T DISTRIBUTION OF POLYNOMIALS. The method described in the previous subsection can also be used in order to determine the distribution of the roots of a given polynomial d(o) with respect to the unit circle. Let deg d -- N. Wc define the all-pass function

~o) := ~ a ( ° - b a(o) Choose N points o i outside the unit disc, and define ~'i : = ~k(ai). It readily follows that d(o) has all its roots inside the unit circle if and only if the Nevanlinna-Pick problem corresponding to the pairs (oi, q'i), i E N, defined above, is solvable with M = I. If we make use of the unnormalized decomposition of subsection 3.2, it is actually possible to determine, iteratively, the distribution of the roots of an arbitrary polynomial d(o) with respect to the unit circle. For this we need the sequence of functions

q/(o) = nk(o)ldk(o), k ~ N, introduced in the previous sub-section. Let ~+ (p) denote the number of roots of the polynomial p(o) inside the unit circle. The following rules, applied iteratively for k = N, N -- 1, --- , 2, 1, are simple consequences of Rouche's theorem: h 4 ( d k) = ~,+(dk+l), ;~+(d ~) = 1 + h~(nk+~),

if

I~, ~l z < l, if I,/,~.-~ 12 > 1.

Note that h+ (d k) + X+ (n *) = k, i.e. there are no roots on the unit circle. Suppose that only one parameter q ~ - I has magnitude greater than unity. It follows that the number of roots of d(o) outside the unit circle is N -- k + 1; therefore if k = 1, all roots of d(o) lie outside the unit circle, while if k = N, only one root lies outside the unit disc. For a detailed exposition on this topic the reader is referred to Vaidyanathan and Mitra [1987], and Delsarte, Genin, and Kamp I1981].

18 5.3 DARLINGTON SYNTHESIS. Finally, we would like to mention that the famous Darlington synthesis for bounded real functions, which is the cornerstone of passive network synthesis, is also closely related to the cascade structure. (5.2) Theorem. Every bounded real function can be expressed as the transfer function of a cascade interconnection of Iossless bounded real fanctions which in addition are reciprocal as well, terminated by systems of wzity trat~fer function. (5.3) Remark. The three results mentioned in this section, i.e. the Nevanlinna-Piek algorithm, checking the root distribution of polynomials, and the Dadington synthesis, have been generalized to the matrix case. Just as in the scalar case, there is a very close connection of the matrix version of these results with the cascade structure. For details, see e.g. Delsarte, Genin, and, Kamp [! 979], Vaidyanathan and Mitra [1987], and the references therein.

6. REFERENCES. A.C.Antoulas [1986], On recursiveness and related topics in linear systen~, IEEE-AC 31: 1121-1135. A. C. Antoulas and IL H. Bishop [ 1987], Continued fraction decon~osition of linear systems in the state space, Systems and Control Letters, 9: 43-53. N. Balabanian and T. Bickart [ ! 981], Linear network theory, Matrix Publishers. Ph. Delsarte, Y. Genin, and Y. Kamp [1979], The Nevanlinna-Pick problem for matrix-valued functions, SIAM J. AppI. Math., 36: 47-61. Ph. Delsarte, Y. Genin, and Y. Kamp [I 981], On the role of the Nevanlinna-Pickproblem in circuit andsystem theory, Int. J. Circuit Theory Appl., 9: 177-187. P. Dewilde and H. Dym [ 1989], hzverse scattering and networks, short course given during the MTNS-89 Symposium, Amsterdam, June 19-23. P.A. Fuhrmann !1983], A matrix Euclidean algorithm and matrix continued fraction expansion, Systems & Control Letters, 3: 263-271. F. IL Gantmacher [ 1960], Matrix theory, 2 volumes, Chelsea. R.E. Kalman, P.L. Falb, and M.A.Arbib [1969], Topics in mathematical system theory, McGraw-Hill. R.E. Kalman [19791, On pat;tial realizations, transfer functions, and canonical for,~, Aeta Polytechnica Scandinavica, Ma31: 9-32. T. Kailath [ 1980], Linear systems, Prentice Hall. P.P.Vaidyanathan and S. K. Mitra [1987], A unified structural interpretation of some wellknown stability-test procedures for linear systenu, Proc. IEEE, 75: 478-497. J. C. Willems and A. C. Antoulas [1989], Recursive time series modeling, Technical Report, Department of Mathematics, E.T.H. Zurich. J. C. Willems [ 1986-1987], From time series to linear system, Automatiea; Part I: Finite dimensional linear time invariant sl~sten~, 22: 561-580; Part I1: Exact modeling, 22: 675-694; Part II1: Approximate modeling, 23:87-115.

Smooth Dynamical Systems which Realize Arithmetical and Logical Operations R. W. Brockett Division of Applied Sciences, Harvard University Cambridge, MA 02138, U. S. A.

D e d i c a t e d t o J a n W i l l e m s o n t h e O c c a s i o n o f his 5 0 t h B i r t h d a y Abstract x

Although many biological and man-made systems combine aspects of digital and analog processing, until recently there has been very little theoretical work on models this type and many basic questions remain unresolved. In this paper we describe input-output systems governed by ordinary differential equations i) whose behavior is robust in the sense that certain well-defined qualitative aspects of the output depend only on certain well-defined qualitative aspects of the input and ii) are capable of generating behavior of the type one usually associates with digital systems. It is show that rather simple differential equation models can robustly execute arithmetical and logical operations; in particular, we show that continuous-time dynamical systems can simulate arbitrary finite automata.

1This work was supported in part by the U.S. Department of the Air Force under grant AFOSR-96001971 in part by the U.S. Army Research Office under grant DAAL03-86-K-0171, and in part by tile National Science Foundation under grant CDR-85-00108.

2O 1. I n t r o d u c t i o n Over the last 25 years we have learned a great deal about the properties of the differential equation models for systems with a given input-output behavior. Having its roots in passive network synthesis, the work of Kalman and Youla put the linear realization theory on its present course. The stochastic theory for second order statistics developed rapidly in the late 1960's and brought to the subject of stochastic differential equations some of the fundamental insights of Bode and Shannon. A little later developments in the theory of integral representations for input-output maps via Volterra series, resulted in a mathematical theory for nonlinear realization having sufficient specificity so as to bring Wiener's conceptual ideas on nonlinear input-output descriptions within reach of practically-minded people. Aspects of these developments have made their way into textbooks and can now be regarded as fundamental results in the subject of mathematical modeling. On the other hand, there are numerous input-output synthesis problems arising in electronics, especially digital electronics, calling for a different formalism. The realization of analog counters and quantizers are examples of the kind of "signal-to-symbol" transduction problems that we have in mind. In this paper we develop a formalism targeted at such questions. We claim to show that it is possible to combine analog and digital computation in a theory which is both computationally powerful and aesthetically pleasing. Ordinary analog computing is completely non-robust in that the values of the computed quantities depend on the values of the input and the system parameters. Here we want to develop models which are robust in the sense that small changes in the input produce little or no change in the output, and any changes which do occur do not propagate through successive stages of the signal processing. This suggests that one might want to attempt to find systems whose output behavior is governed more by some "topological" aspect of the input (e.g. a winding number} rather than by the details of its time history. For example, if the space of admissible input values is not simply connected, then one can attempt to find systems having the property that qualitative features of their output depend only on the homotopy type of the input curve. The performance of such systems would be robust in a very strong sense and would, at the same time, provide a mathematically precise link between aspects of analog and digital computing. In order to make this somewhat more concrete, consider figure 1. We show there a curve which is parametrized by t and which lies in an annulus in (u, fi)-space. Trajectories in this space are subject to certain constraints by virtue of the fact that when

du/dt is

positive, u must be increasing, and when du/dt is negative, u must be decreasing. This is true regardless of the details about how t parametrizes the curve. The problem of counting the number of zero crossings of a function of time is, for unrestricted functions, non-robust;

21 the function m a y come a r b i t r a r i l y close to zero without crossing zero, or it m a y have a narrow spike which crosses zero for an extremely brief p e r i o d of time b u t is otherwise safely removed from zero. However, functions which are constrained to the annulus of figure 1 have welbdefined zero crossings, the exclusion of t h e inner disk means t h a t when the value of t h e function is near u = 0, the velocity is large enough so t h a t it will pass through zero in an u n a m b i g u o u s way a n d the exclusion of the outer region prohibits high frequency spikes. W i t h i n this annulus the idea of a zero crossing can justifiably be called robust. In a later section we will construct a differential equation which counts the zero crossings of a curve u p r o v i d e d it satisfies a restriction of this type. In o u r t r e a t m e n t of the realization question for a u t o m a t a we will use a similar b u t slightly more complicated input space. The counter a n d the a u t o m a t a provide examples of solving a realization p r o b l e m in which the o u t p u t of the d y n a m i c a l s y s t e m is sensitive only to qualitative aspects of the input.

u

F i g u r e 1: T h e input space for a s y s t e m which counts zero crossings. T h e differential equations we will use to realize these qualitatively defined i n p u t - o u t p u t maps will typically have a large number of stable equilibria whose locations are insensitive to the value of the input. Roughly speaking we will set up a situation in which the input forces the s t a t e from one equilibrium to another, reminiscent of finite a u t o m a t a . F r o m this point of view one could think of our construction as being a way to a d d flesh in the form of differential equations to the skeleton provided by a finite a u t o m a t o n . W h a t distinguishes this work from s t a n d a r d digital electronics are the fact t h a t we do not make a h a r d and fast distinction between the logical model and the circuit model and the fact t h a t we require no explicit sampling or synchronizing clock.

22

2. P r e l i m i n a r y R e s u l t s We collect here a few ideas about the differential equations we will be working with in the later sections. We use ][A[[ to denote the square root of the sum the squares of the entries of A. Let Sym{AI, A2,..., A, } denote the set of all real n by n symmetric matrices with eigenvalues A1, As,..., An and let [A,B] = A B - B A .

In a recent paper [1]

wc investigated the equation in the space of symmetric matrices, / : / = [H, [H, NIl

(1)

where N = N r. Our attention here is devoted to some variations on this theme. Equation (1) has a number of remarkable properties which have to do with its origins as a steepest descent equation on the orthogonal group [2]. Some of these are explored in [1]. The more recent work of Bloch [3] and Bloch et al. [4] take the study of this equation in other directions. For our present purposes we want to consider the addition of a linear term of the form [H, fl] so that we get

= [H, [H, N]] + [R,

(2)

where, again, H and N are symmetric and f / i s restricted to be is skew symmetric. Lemrna 1: The eigenvalues of the solution of equation (2) do not change as time evolves.

Considered as a differential equation on Sym{Al(H(0)), A2(H(0)),..., A,(It(O))}, if N is diagonal and has no repeated entries and if H(0) has no repeated eigenvalues, then there exists ¢ > 0 such that for []fl][ < e equation (2) has nl equilibrium points exactly one of which is asymptotically stable. Proof: Expressing f I as f I = [ H , [ H , N ] + fl] = [ H , f ( H ) ] we see that this is the

standard isospectral form. If fl = 0 we know from [I] that the equilibrium points take the form H = diag (Af(1),A={2), . . . , At(,)) with ~r being any permutation and that the linearization of [H, [H, NI] at these points has eigenvalues of the form (A, - A i ) ( n , ( , } n~ff). As discussed in [1], all n! of the equilibria are nondegenerate and exactly one of the equilibria is asymptotically stable. For [~ # 0 we see that

[H, [H, #]] + in,

= 0

is satisfied if [H, N] = fl. The symmetric solutions of this equation take the form H=D+A

where D is diagonal and A is given by

"i)

23 To first order in wlj we obtain the correct eigenvalues for D + A A~(2), . . . , A~(,,)).

by letting D = diag (A,0I,

There are n! such solutions, corresponding to the n! choices of the

permutation 7r. Because the zeros of [H, [H, N]] = 0 are nondegenerate and isolated there are no other solutions of [//[//, ]V]] + [fl,//] : 0 for IJflH sufficiently small. For reasons which will a p p e a r later, we consider in more detail a special case of equation (2). Remark

i : Let H be two by two and suppose H(0) has +1 and -1 as its eigenvalues. If

N takes the form N = diag (0,u), then for hll = cos(0), equation (2) takes the form 0 =~a+usin0 Proof: This is just a calculation. We have

hu

=

-uh~,2 +

~h12

h12

=

u ( h l l - h2a)h12 - w h l l

hzz

=

uh~2 -

wh12

B e c a u s e / ; / = [H, [H, NIl + lit, H] evolves in such a way as to keep the eigenvalues of H constant we see that h u + h2~ = 0 and h l i h 2 2 - h~2 = - 1

Thus if we use h n = -h22 = cos0 and let h12 = sin0, we get d ~ c o s O = - O s i n 0 = wsinO + usin2 0 or

= ~a + usin 0 This calculation allows us, in this case, to lift equation (2) from the circle to the real line. T h a t is to say, we can regard 0 as being real valued, not just circle valued. This will be important in the next section. We now consider a simple class of systems which, unlike the ones considered above, have a large number of (locally) asymptotically stable equilibria. By diag H we understand the matrix whose diagonal entires are the same as those o f / / a n d whose off-diagonal terms are zero. Lemma

2: Let H and L be n by n matrices. T h e solution of the coupled equations

_f/= [H, [H, diag Lll

L = [r.,[L, diag HI]

(3)

24 evolves in such a way as to leave the eigenvalues of H and L unchanged. If H and L are

symmetric, tr(diag H)(diag L) satisfies d t r ( d i a g (H)diag (L)) =

d ~ t r ( L d i a g (H))

=

d ~-~tr(Hdiag (L))

=

I[[H, diag (L)][['+ [[[L, diag (H)[[ z

If H(0) and L(0) are symmetric with eigenvalues A, < A2 < ... < A., and #t < ... < # , (unrepeated in each ease), then as an equation on Sym{A1,),2,...,A,} x Sym{~l, # z , . . . , #,}, this pair of equations has (n!) 2 nondegenerate equilibrium points

/~2 <

corresponding to each of the possible orderings of the eigenvalues along the diagonals. Exactly u! of these equilibria are asymptotically stable. The remaining stationary points occur at points where diag H and/or diag L have repeated entries; none of these equilibria are asymptotically stable. Proof: Isospectrality follows from the fact that any solution of ~i = [A,/(A)] is necessarily isospectral. The derivative computation is completely analogous to the one given in [1] and uses the fact that the Lie bracket of two symmetric matrices is necessarily skew symmetric. There are nt different diagonal matrices having eigenvalues A1, A2,..., A, and n! different diagonal matrices having eigenvalues #1, #2,...,/~,. Thus there are (n!) 2 pairs of diagonal matrices which satisfy [H, L] = 0. If diag(L) has unrepeated entries, then the only matrices which commute with it are diagonal. On the other hand, if diag(L) has repeated entries, then there are non-diagonal H's which commute with L and if diag(H) has repeated entries, there are non-diagonal L's which commute with diag(H), however the Liapunov function tr(diag Hdiag L) shows that none of these can be asymptotically stable. Remark 2:. (Compare with [1]) If (Ax, As. . . . . A,} is a set of n real numbers, we denote by C(A:, As. . . . , A,} the convex hull of the n! points in It" of the form (A~(t},A~(2},..., ~,(,)), where ~r ranges over the set of all permutations. The Shur-Horn theorem asserts that the possible diagonals of elements of Sym{Al, A2,..., A,) coincide with this set. It is not difficult to see that equation (3) evolves in such a way as to solve the following problem. Let x E C{A,(H(0)), A2(H(0)),..., A,(H(0))} and let y E C{A~(L(0)), A,(L(0)),..., A,(L(0))}. Find x and y subject to these constraints such that the euclidean inner product (x, y) is as large as possible. Of course there are at least n! optimizing values for x and y. Our result shows that if the A~(H(O)) are distinct and the Ai(L(0)) are distinct, then there are exactly n! points of local (and global) maxima.

25 3. C o u n t i n g Z e r o C r o s s i n g s As an illustration of what we mean by a dynamical realization of an arithmetical operation, we consider the question of finding a dynamical system which counts. More precisely, we will define a dynamical system which generates a running total of the number of zero crossings of a scalar function u as it evolves in time. Because the output y will be a continuous function of time, we need to explain in what sense this, possibly nonintegervalued, output can be taken to be a count of anything. In fact, what we will show is that for a suitable restriction on u there is an inequality of the form

n-

1 <_ yCt)
+c

with n being the numbcr of times u crosses zero on [0, t). It will also happen that as the time since the last change in the sign of the input goes to infinity, the output yCt) will approach a value which is, to within some assignable tolerance, the true count ~. The differential equation we will use for counting zero crossings is the real line version of the two by two form of equation C2) which we explored above, i.e. the equation -- w -i- u s i n 0 The output y(t) is just O(t)/~r. Our first result applies to the case where u(.) is piecewise constant.

Lcmma 3: Let co be a given real number co E (0,1). If u(.) is piecewise constant and if it takes on values in

[u(t)[ > 1/co with the points of discontinuity being separated by at

least ~r/co units of time, then the solution of the input-output system

O(t)=w+u(t)sinO(t)

;

0(0)=0

;

y(t)=O(t)/~r

is such that - 1) < y(t)

where

< Ct) + sin- (co

act) is the number of times u changes sign on [0,t]. tOur convention is to declare

that a jump occurs at t = 0.) Proof: (See figure 2). W e analyze the case where

u(t) is initiallypositive, the opposite case is similar. It is not difficultto show that if 0(0) -- 0 and if initiallyu(t) > l/w, then because the time between jumps exceeds ~r/w, at the firstjump time T ;> 0,

O(T) =

Z oJ + u(t)

sin 0 dt

is between ~r and ~r -{- sin-l(w~). At time T u jumps to a negative value which is less than

-1/w. Because u sin 0(T) will necessarily be positive, 0 increases over the next ~r/co units of time until it is to within sin-l(w 2) units of 2~r. Continuing in this way leads to the result claimed.

26

posilive ~0 r~ga~ve

input positive

negative ~ positive

i n p u t negative

F i g u r e 2: The function of w + u(t}sinO for u(t) = 9=1 showing stable (= s) and unstable (= u) equilibrium values. The intervals on which w+u(t) sin 0 are positive for any admissible choices of u(t) are the 0-values for which switching can

occur.

l u(t)

LJ

y(t}

y(t)

5 4 3 2

/

/

/

/

1

0

F i g u r e 3: The response of 8 = w + u sin 0 ; y = O/~r to smoothcd and piecewise constant inputs. This argument shows that at no time does 8(t)/~r exceed the number of zero crossings by more than lr-x sin-Z(w 2} and at no time does it fall more than one behind. Clearly if u is constant, the solution approaches the appropriate value monotonically.

27 By an annular region in (u, du/dt)-space we understand a space which is formed by removing from a simply connected open set a second simply connected open set which is intcrior to the original. Using the fact that the solutions of this type of differential equation defines a continuous mapping from a function space L~(O,t) where u lies to the function space L~ where 0 lies, we can assert the following theorem (whose proof is omitted).

Theorem i: There exists an annular region A in (u, d)-spaee such that for u in A the equation 0 = w + u sin 0 counts the zero crossings of u in the above sense. In figure 3 wc show the results of simulating this system for an ideal and a "generic" input.

4. Quantization w i t h o u t Sampling

Quantizers are ordinarily part of any system which uses digital computers to control analog systcms. Usually quantizers operate by sampling the signal which is to be quantized and then operating on the sampled value. Subsequent stages of processing associated with the system are often synchronized with the sampling operation. Because there may not be any obvious synchronization signal present in some applications, the qucstion arises as to tim fcasibility of quantization without sampling. The input space will be topologically "intcrcsting" as suggested by figure 4. Our realization will again use a form of equation (2). This time the input u enters as /'t = [HIH, No + until + [[1, H]

;

y = tr(N:H)

with No, NI and N2 being diagonal matrices and f~ being a small skew symmetric matrix.

F i g u r e 4: The qualitative form for the input space of a quantizer.

Lemma ~: Given n -

1 real numbers a I < a2 < ... < an-i and n real numbers

bl,b~ . . . . , b . , thore exlsts a n U(0) a=a a system of the f o r m / ~ ---

IIS, ISt, No + ~N,I] ;

y = tr(N2H) such that for each constant value of u which is not equal to any of the a's

28 therc is a unique equilibrium point H(u) o f / ; / = [ H , [ H , No + uN1]] such t h a t H(u) is a s y m p t o t i c a l l y stable, a n d for all a i - i < u ,< a~ y -- tr(N2H(u)) = bl. (For u < as ;

tr(H(u)/V2) = bl a n d for a . - 1 < u ; tr(H(~)/V3) = b..)

Proof: We c o n s t r u c t the s y s t e m in the following way. No = diag ( a l , a 2 , . . . , a , , - 1 , 0 ) , N l = diag ( 0 , 0 , . . . , 0 , 1) a n d N~ = NI. H(0) has eigenvalues (bl,b2 . . . . ,b,). T h e results then follow from t h e o r e m 4 of [1]. This l e m m a does n o t always provide a practical solution to the quantization p r o b l e m because the solution H m a y s p e n d t o o much t i m e escaping from an u n s t a b l e equilibrium point. Experience with simulations suggests t h a t it is b e t t e r to a d d a linear t e r m so t h a t a change in u a f f e c t s / ' t additively. We use

f I = [1t, IH1No + ~ 1 1 1 + In, H l

;

y -- t r ( N 3 H )

This has an i m p o r t a n t implication. T h e value of y for a c o n s t a n t value of u is no longer exclusively d e p e n d e n t on the interval (ai, ai+l) to which u belongs. T h e r e is a dependence on the precise u value in this range. This sensitivity to u can be m a d c as small as one likes b y taking 12 to b c small. This defect is, however, c o m p e n s a t e d for by an increased speed of response. A simulation of a seven level quantizer is shown in figure 5.

F i g u r e 5: T h e response of a seven level quantizer to a sin t input.

5. R e a l i z i n g A r b i t r a r y

Finite Automata

In this final section we show how to define continuous t i m e systems which can perform any c o m p u t a t i o n which a finite a u t o m a t o n can perform. This shows how one might use a continuous time (unsampled) s y s t e m to perform any c o m p u t a t i o n t h a t a digital c o m p u t e r can perform.

29 Let U and X be finite sets; U = {ux,u2 .... ,urn); X = (zt, x~ ..... xn}. Suppose that we are given ~ : X × U --~ X and wish to simulate the difference equation

x(k + 1) -- 6 ( x ( k ) , , , ( k ) ) We can think of this equation as defining a finite a u t o m a t o n with i n p u t a l p h a b e t U and state space X . We describe a s y s t e m governed by o r d i n a r y differential equations which simulates this machine. T h e differential equations we will use are a v a r i a t i o n of equation (3) and take the form

[I = [tf, [tI, O(u,L)] l + [ a t , H] HI] + Ill2, L}

], = [L, [ / , ¢ o ( u ) d i a g

with H a n d L being p by p s y m m e t r i c matrices with eigenvalues ( 1 , 2 , . . . ,p). T h e idea is this. Let p be an integer such t h a t p! > n. Let f be a function f associates w i t h each s t a t e xl of the given a u t o m a t o n a p e r m u t a t i o n , ~rl ¢ ~ri if

: X-*

H which

f(xi) = ~ri such t h a t

xl ~ x i. Associate with each ui the integer i. Let ~b(u, L) b c a function of u and

L which takes on values in t h e space of diagonal matrices. Wc t a k e ¢ ( 0 , L) = 0.

uitu~2 . . . u l , as a time function with the zero values

Code a generic i n p u t sequence

being used to m a r k the end of a letter; i.e. associate to the above word the time function i,

u(t) =

; 0_
o

; TI
i~

; e~ < t < T=

0

; T2<_t
. , .

, , ,

In this way, the i n p u t s are coded as integer valued levels with 0 interspersed to m a r k time. We will assume t h a t P~ - Ti and Ti+x -

Pi are b o t h greater t h a n some fixed T. We t a k e ¢0

to be a s m o o t h function such as e -~"2 which is one when u is zero and very small when u is larger than, say,

1/2.

We are now ready to show how to choose ¢ so as to m a k e the s y s t e m mimic the given automaton. permutation

Li be the diagonal m a t r i x with entries ( 1 , 2 , 3 . . . . . n) p e r m u t e d by the f(xi). Define [$(i,j) so t h a t

Let

~r~(i,d) = fC6Cxi, ud)) and let 0 b c a s m o o t h function such t h a t

~b(ui, Lj) = diag (~r¢(,d)(1) , ~r~{i,i)(2) , . . . ,

~rp(~,i)(n)). W i t h this definition we can see t h a t if u = f :~ 0, then 1~ is effectively zero a n d the H equation will flow in such a way as to p e r m u t e the diagonal o f / / s o

as to make

3O it match the diagonal of ¢(ui,Lj). When u switches back to zero, /'I is zero and the L equation flows to match the diagonal of L with that of H. If u remains zero for a suitable time, then H and L are both nearly diagonal and their diagonals are matchcd. When a nonzero u appears, this "shuts down" the L equation and realigns the diagonals of H; when it returns to zero, this "shuts down" the H equation and updates the L equation so that it again matches H .

Conclusions The question of finding differential equations which evolve in such a way as to robustly carry out a calculation has considerable scientific interest because of its connection with problems in biology and computing. In this paper we have considered a class of inputoutput models which have a large number of equilibria and have given a construction which shows that smooth dynamical systems of this type can simulate one general computational model, namely that of a finite automaton.

References

[11 R. W. Brockett, "Dynamical Systems that Sort Lists,Diagonalize Matrices and solve Linear Programming Problems," Proceedings o/ the 1988 IEEE Conference on Decision and Control, December 1988. [2] R. W. Brockett, "Least Squares Matching Problems," Journal of Linear Algebra and

Its Applications, to appear. [3] A. M. Bloch, "Steepest Descent, Linear Programming and Ilamiltonian Flows," submitted for publication. [4] A. M. Bloch, It. W. Brockett, and T. Ratiu, "A New Formulation of the Generalized Toda Lattice Equations and their Fixed Point Analysis via the Moment Map," submitted for publication.

Pole Assignment by Output Feedback C. I. Byrnes Department of Electrical and Computer Engineering and Department of Mathematics Arizona State University Tempe, AZ 85287, U. S. A.

1. I n t r o d u c t i o n . The problem of tuning the natural frequencies of a linear system using output measurement has long been rccognized ([1]-[3]) as central in both classical and modern automatic control. Fundamental work by Nyquist in 1932 provided graphical stability and instability criteria for output feedback, while the graphical root-locus method of Evans also gives a clean set of necessary and sufficient conditions for a self conjugate set ( s l , . . . ,8~} of frequencies to be assignable as the poles of an nth degree transfer function g(s); viz. g(si) = g(8i). For multivariable systems, this problem has been the object of study for many authors using an imprcssive variety of techniques from linear systems theory, combinatorics, complex function theory ~nd, most recently, geometry and topology. Beginning in the early 1970's, methods from algebraic geometry were applied to several open problems in linear systems, in part because more standard techniques (see chapter 2) were not able to resolve some of the fundamental nonlinear features surprisingly underlying what were phrased and regarded as problems about linear systems. Among the most striking examples of this phenomenon are the essential role played by algebraic geometry and topology in the solution of the "parametrization" problem and the related questions concerning canonical forms for realization (see e.g. *Dedicated, in homage, to my collaborator,friend and teacher, Jall Willems **Research supported in part by grants from A F O S R aim NSF.

32 [4]-[7]) and the applications of algebraic geometry to questions about output feedback stabilization and pole-assignment for multivariable linear systems, which we survey in this article. As early as 1972, the language of hypersurfaces, generic point, etc. was used (bee e.g. I81) to derive certain general position results concerning multiple roots of a closed-loop characteristic polynomial, while in [9] Anderson, Bose and Jury showed that the solvability of particular pole-assignment problems could be decided using polynomial criteria. Within just a few years, there were several quite sophisticated and novel applications of algebraic geometry to this problem. Indeed, the most striking message of the work by A. G. J. MacFarlane et al. (see e.g. [10]) is the recognition that, for scalar gains and square multivariable systems, the behaviour of multivariable root-loci--which had previously been elusive--is a problem best stated and solved using the theory of rational functions defined on Riemann surfaces, allowing for a clear understanding of precisely how multivariable root-loci evolve with changes in the output gain. Simultaneously, in a series of seminal papers (see e.g. [11]-[13]) R. IIermann and C. F. Martin introduced a variety of useful tools and concepts from algebraic geometry. Perhaps the most widely appreciated of these techniques, at the time, was the application of the "dominant morphism theorem" proving that for generic systems, in the correct dimension range, pole-assignment is achievable using complex feedback (see 4.2). In retrospect, the Hermann-Martin curve [12] turned out to be an essential piece of this problem. However, the "dominant morphism" result coupled with a positive pole assignment result proved by Kimura [14] using combinatorial and geometric methods (§3), sparked renewed interest in the problem of arbitrary pole-assignment by real feedback. In a fundamental calculation [15], J. C. Willems and W. H. Hesselink showed by example that the positive results obtained over the complex field, failed to hold for real feedback systems in a serious way (see §4.3). Their results show that for 2 × 2 multivariable systems of degree 4, in general there will be two pole assigning feedback laws which are either both real or complex conjugates "with equal probability." The quadratic nature of this problem, undcrscorlng some of the nonlinear features, also implies that linear formulae do not exist for pole-assigning output feedback laws, in contrast to the state feedba~:k situation. This discovery considerably expanded the problem of pole-assignment by static or dynamic output feedback beyond the simple question of existence of real or even complex solutions to the pole-assignment equations, including questions such as the nature of possible formulae for pole assigning gains if these should exist and the behaviour of various nonlinear computational schemes for their numerical solution.

33 About a decade ago, inspired by a profound geometric construction due to Hermann and Martin (see §5.4) and by the questions raised by Willerns and Hesselink, we developed a geometric framework for analyzing pole-assignment problems which incorporates the earlier algebraic geometric methods with the combinatorial methods pioneered by Itautus and by Kimura, making systematic use of classical enumerative geometry and its modern version, intersection theory. Using this framework, Brockett and Byrnes [16] were able to provide a multivariable Nyquist criterion for nonscalar gains and to explicitly enumerate the number of gains (possibly complex) which assign a given set of closed loop poles for the general p × ra system of degree m p = n, viz. 1!..-(p - 1)!(mp)! 1)!

dm,p = m ! . . . (m + p -

(1.1)

While dz,2 = 2, in harmony with Willems-Hesselink, d2,3 = 5 so that real pole-assigning gains exist for real data (see §5.1), a result which was unexpected at the time. One can also glean a great deal about the structure of the solution to the pole assignment equation from the analysis (see §5.2) underlying the determirtation of (1.1). For example, it is possible to compute the Galois groups for several generic problems (§6.3) thereby answering open questions about the kinds of formulae which might exist for poleassigning gains. Moreover, this analysis also makes it possible to prove convergence for certain numerical schemes, e.g. homotopy continuation methods (see §5.3), for the solution of the pole-assignment equations. Finally, these techniques have been refined to study pole-assignability and stabilizability by real feedback whert m p > n [see §5.4). While, az we shall point out, we are still far from a complete solution to this important problem through the work of many people the results obtained using geometric methods contain, to the best of my knowledge, all existing results. On the other hand it is also fair to say that the major results of the geometric theory have still not been obtained by other methods. In section 2.1, we define the pole assignment problem and develop some preliminaries. Section 2.2 contains a treatment of the rank one situation, which can still be treated by standard linear systems theory. Djaferis [17] made the observation that rank one methods could be used to study special pole-assignment problems using dynamic compensation and in section 2.3 we use this approach to give a simpler proof of the Brasch-Pearson theorem, also discussing related results due to Seraji [18], Stevens [19], Vidyasagar-Viswanadham [20] and Ghosh [21]. In chapter 3, we describe the combinatorial geometric interpretation of the problem, following Kimura [14] and Hautus [22], as well a s s o m e recent improvements due to Rosenthal [23]. After a brief introduction, Chapter 4 concentrates on applications of affine algebraic geometric techniques. Specifically, in section 4.2, following Hermann-Martin we compute the differential of the

34

pole-assignment maps both for output and for state feedback obtaining necessary conditions for pole assignment which are generically sufficient over C, using the ~dominant morphism theorem". Over It, with a little more work we obtain a new, and rather clean, proof of Heymann's Lemma. In 4.3, we derive the Willems-Hesselink results through an explicit, general elimination of the pole-assignment equation, using a multilinear algebraic approach due to Anderson, Morse and Wolovich [24]. Following Byrnes and Anderson [35] we then use elimination theoretic methods to show that this necessary condition for pole-assignment is also necessary for generic stabilizability. In 5.1 we cast the pole-assignment problem as a problem in enumerative geometry to which the classical methods of Schubert calculus formally apply. Putting the Schubert calculus on a firm foundation was, of course, Hilbert's 15th Problem, which was solved by van de Waerden, by Ehresmann, by Hodge and by Chern in its modern realization as intersection theory on Grassmannians. In section 5.2, we give an exposition of our geometric framework for pole-assignment which combines the earlier combinatorial approaches and algebraic geometric methods with the tools of the modern Schubert calculus. In particular, section 5.2 contains a sketch of the proofs of the main results announced in section 5.1. In section 5.3 we prove, using geometry, Galois theory and numerical methods that, in general, pole-assigning gains cannot be solved for in terms of rational expressions and the extraction of n-th roots. Nonetheless, the geometric analysis in section 5.2 does provide a convergcncc proof for the numerical solution of problems by the homotopy continuation method, as is illustrated by an example in 5.3. As we note in section 5.4, using Schubert calculus and the notion of Ljusternik-~nirel'mann category from the calculus of variations, it is possible to obtain some significant refinements of the results derived for m p =

n, amounting to what are currently the sharpest results

known about pole-assignment by real output feedback. It is a pleasure to thank many people for stimulating correspondence, conversations, preprints and suggestions: J. Ackermann, B. D. O. Anderson, I. Berstein, R. W. Brockett, T. Djaferis, B. K. Ghosh, J. Harris, M. L. J. Hautus, R. Hcrmann, W. Hesselink, H. Itillcr, T. Kailath, H. Kimura, C. F. Martin, S. K. Mitter, A. S. Morse, D. Mumford, J. Rosenthal, J.-P. Serre, P. K. Stevens, R. E. Stong and X.-C. Wang. Finally, I want especially to acknowledge the profound influence Jan Willems has had on my way of thinking about this problem and about systems and control theory in general. Happy birthday, maestro! 2. R a n k O n e M e t h o d s for Pole A s s i g n m e n t by S t a t i c o r D y n a m i c O u t p u t Feedback 2.1 P r e l i m i n a r i e s . We begin with a statement of the problem of pole-assignment by

35 output feedback, first stated in the more elementary, but less natural, state-space form. Suppose x E R n, u E R m, y E R p and that the n × n matrix A, the n × m matrix B and the p × n matrix correspond to the linear control system

,~ = Ax + Bu y=Cx which provided a minimal dimension realization

G(s) = C ( s I -

A)-XB

of the transfer function

tC )

C2.1)

=

The problem of pole-assignment by o u t p u t feedback formalizes the problem of arbitrarily tuning the natural frequencies of a linear system using only memoryless sensor feedback: Given a self-conjugate set

{sx. . . . . ~.}, of

complex numbers, perhaps counted with

multiplicity, find an o u t p u t feedback law, u = ky, which places these frequencies, in the sense that the roots of the characteristic polynomial of A + B K G coincide with {s 1,... , an}. Given the input-output nature of the problem of tuning natural frequencies by output feedback it is, not surprisingly, more convenient to express an n dimensional p x m linear system in terms of its transfer function, (2.1) which we shall always assume is a matrix-valued rational function, vanishing at oo. After constant o u t p u t feedback u =

-Ky+

v

the closed-loop transfer function GK(s) takes the familiar form

Gk(s) = G(s)(I + gGCs)) -I -- ( x + G(s)IC)-te(s)

(2.2)

leading to a "characterization" of the closed-loop poles as the zeros of the return difference determinant d e t ( I + gG(s)) = 0

(2.3)

Of course (2.3) is a little unsatisfactory with respect to open-loop poles, which correspond for example to the law K = 0. This can be rectified by remembering that the left-hand side of (2.3) is a rational function with poles at the open loop poles and zeroes at the closed-loop poles, some of which may however cancel. While we shall typically work with (2.3) keeping this caveat in mind, it is also easy to remedy this lacuna using the m e t h o d of coprime factorization (see for example [26]). Explicitly, there exist polynomial matrices g ( s ) , Dis ) satisfying

G(s) = g ( ~ ) D ( s ) -1

i2.4)

36 which are coprime in the sense that there cxist polynomial solutions to the equation

UCs)N(s) + VCs)D(s) = I Moreover the poles of G(s) are defined as the roots of the equation act D(s) = 0

(2.5)

From (2.2), one may derive the representation

GKCa) =- NCs)(D(,) + K N ( 8 ) ) -1

(2.6)

which is also a coprime factorization and hence we have a formula for the closed-loop poles det(D(s) + K N ( 8 ) ) = 0

(2.7)

which givcs the corrcct answer if K = 0, suggesting that the correct form for (2.3) should be det(I + KG(~)) det D(s) = 0

(2.3)'

which incorporates pole-zero cancellations. We note that, in the scalar input-scalar output case, (2.7) gives a simple characterization of those self-conjugate sets { s l , . . . ,~,} of frequencies which can be assigned b y output feedback, viz. g(~i) = gC~J),

Vi, j.

For multivariable systems the heart of the pole assignment problem lies in the fact that for fixed so E C the solution set d e t ( I + gG(so)) = 0

(2.8)

is in general very difficult to describe. In geometric tcrms, (2.8) dcfines a hypersurface but there is of course a class of hypersurfaccs which are easy to analyze; viz. hyperplanes. To put it algebraically, if for examplc G(So) has rank 1 thcn (2.8) takes the vastly more simple, affine form det(I + gCCso)) = 1 - trCgC(so)) In particular if G(s) has rank _< 1, (2.3)' takes the form (see e.g. [27]): get D(~) - tr(KadjG{~)) = 0

(2.9)

so that the closcd-loop characteristic polynomial is afflne in the gain, generalizing the scalar input-output case. In this chapter, we exploit this simple obscrvation to develop simplc proofs or improvcmcnts of several fundamental theorems.

37 2.2 P o l e - a s s i g n m e n t b y O u t p u t F e e d b a c k in t h e C a s e min(m,p) = 1. If min(m,p) = 1, then G(s) is either a row or a column vector and hence the affine formula for the return diffcrence determinants applies. We shall assume m = 1, the situation being symmetric in rn and p. In this case, the coprime faetorization (2.4) takes the form aCs) r = [n~ ( a ) / d C a ) , . . . , n~Cs)Id(~)l

(2.9)

and the dosed loop polynomial associated to the output feedback gain K

=

is simply

p

dCs) -

= dCa) - E KiniCa) = 0 (2.10) i=l In particular, to place the closed-loop poles at (al . . . . . an) is to solve the affine equation p dCs) - drCa) = ~ Kini(~) C2.11) i=1 where n dr Ca) = 1-[ Ca - ai) i=1 Since dr (a) is arbitrary, one obtains the following necessary and sufficient conditions for pole assignment.

T h e o r e m 2.1. Suppose min(m,p) = i and set r = dim span {nl ( s ) , . . . , nmax(m,p)(~)}. (a) If max(re, p) > n, a necessary and sut"l]cient condition for pole assignment is r = n.

In partlcular, if max(re, p) = n, then it is necessary and sufficient that the entries of a numerator coprime factor be linearly independent. (b) K max(m, p) < n, then arbitrary pole-assignment is not possible.

Ilowever, one

can place any self-conjugate set { s l , . . . ,st} of closed-loop poles by a real output feedback law. P r o o f . Only the last claim in (b) has not been discussed. From (2.11) we see that to place a pole at si is to define an affine hyperplane H(si) in the linear space of gains K . To say that r of the ni(a ) are linearly independent is to say that H ( a l ) , . . . , H(ar) are not parallel and in fact intersect nontrivially. 2.3 P o l e A s s i g n m e n t b y D y n a m i c C o m p e n s a t i o n Suppose (A, B, C) is a minimal triple in a state-space realization of a p × m matrix valued rational function G(8) having degree n, so that

G(a) = C ( s I -

A)-IB

38 We denote by ~;i,i = 1 , . . . , m and v i , j = 1 , . . . ,p the controllability and observability indices of (A, B) and (A, G), respectively. Using some fundamental work by Forney [42], it is possible to compute these partitions of the integer n directly from the transfer function G(8), but it is more traditional to interpret these o u t p u t feedback invariants in terms of a state-space realization. In this section we will present pole-placement criteria due independently to Seraji [18] and Stevens [19] generalizing the well-known result in terms of the controllability and observability indices, due to Brasch-Pearson [28]. First, we recall the nontrivial fact that for any G ( s ) of degree n there exist constant gain o u t p u t feedback laws K such that the closed-loop transfer function G K ( s ) has n distinct (simple) poles (see also [28]). In the scalar case, this follows directly from a rootlocus plot which shows also that this in fact holds for all but finitely many gains. In the pxm

case, supposing p > m, for almost all r n x p matrices F , the square transfer function

FG(,)

has degree n, reducing the rectangular case to the square case by "squaring

down". In the square case, we can again appeal to a root-locus argument using the beautiful multivariable root-locus theory developed by Postlethwaite and MacFarlane [10]. Our first formal result generalizes this observation, yielding a variant of Heymann's Lemma. Lemma

2.2. For a n y p x m transfer f u n c t i o n G ( s ) w i t h degree 6(Gk(*)), t h e r e e x i s t s

an o u t p u t f e e d b a c k law K and an m vector v such t h a t

:Proof. Choosing K so that GKCs) has n = 6(GKCs)) simple poles S l , . . . ,sn, we can write cK(,)

=

n

Ri

i=1

S -- s i

where Ri = / 2 i whenever si = ~j. To say 6 ( G K ( s ) ) = n is of course to say rank (Ri) = 1, i = 1 , . . . , n. For any v 6 R m we have GK(s)v

Riv

= i=1 8

-

-

8i

and thercfore 6 ( G g ( s ) v ) < n if, and only if, R/v = 0 for some i. Therefore, the choice of any v not in the proper subset i =t~1 ker (14) of 1%r~ proves the assertion. Remark

2.3. (i) Our proof and remarks actually show that there is an open dense set

of K and an open dense set of v for which the lemma holds. (ii) Denoting A - B K C

by AK we have

aK(,)v = c(,I - A

)-IBv

39 In particular, the observability indices of Gg(s)v coincide with the observability indices of G(s). (iii} The same results hold, of course, for wtGK(s) with observability indices replaced by controllability indices. We can now give an elementary proof of the following classical result. Theorem

2.4 (Brazch-Pearson [28]). Arbitrary pole az~ignmen~ for G(s) can be

achieved using dynamic compensation of order q = min(tCm~x, t/m=) - 1. P r o o f [20].

Suppose a self-conjugate set of desired poles, {s~,... ,sr,+q} is given.

Choosing K and v as in Lemma 2.2, we seek a I x p compensator K(s) placing these poles. If GK(s)v = N(s)D(s) -I and K(s) = Q(s)-lP(s) are coprime factorizations, then the return difference determinant reduces to the constraint n+q

Q(s)D(s) + P(s)N(s) = I~ (s - ,i)

(2.12)

i.= l

which is linear in the polynomial pair (Q(s),P(s)) thought of as an element of a (q + 1)(p + 1) dimensional vector space. In particular, arbitrary pole assignment by compensator v K ( s ) is possible if, and only if, the linear (Sylvester) map Sq : 1-~(q't-l)(p't-1) ~ 1~(rt-l-qq-1),

obtained by equating coefficients in (2.12), is surjective. On the other hand, Bitmead et ah [30] have given an elegant formula for the rank of Sq in terms of the observability indices of GK(s)v, which of course coincide with those of G(s): rank Sq-= (p-t-1)(q-t-1)-

~

(q+l--vi)

vi
Therefore, G(s) can be pole-assigned by a compensator of order q, where q satisfies (q+l)p-

~ (q+l-r'i) vi
>n

(2.13)

However, the left-hand side of (2.13) is a nondecreasing function of q, achieving its maximum value, viz. n I, when q = r'max - 1. Since, by duality, the same results hold mutatis mutandis for q = ~raax - 1, the assertion follows. Since min(xmax, Vmax) _< n, one very easy corollary of the Brasch-Pearson T h e o r e m is the well-known result that one can always pole-assign by a dynamic compensator of order q < n - 1 . In fact for systems having rank C = 1 and rank B = 1, tCm~x = t/max = n so that we must take q = rt - 1. These rank conditions are however quite restrictive, or

40 "nongeneric', and for generic systems one can obtain a much less conservative estimate for min(~:max,vm~x), l~ecall that a property P of points in an affine space R N or C N is called generic provided the set of points which do not enjoy P lie in a proper subset X of the form

X={x:h(x):O

,

j=l,...,e}

(2.14)

where fj(x) are polynomials. A set of the form (2.14) is called an algebraic subset (of either 1~n or Ca). For example the set of all n × m matrices B having rank less than or equal to 1 is an algebraic subset of R rtm. Also, those pairs (A, B) corresponding to a four-dimensional state space system having two inputs, which have controllability indices (4, 0) or (3, 1) arc nongcncric while those pairs which have controllability indices (2, 2) are generic. More generally, for systems having the generic choice of controllability indices it follows that q- m > n

==~

ICmax> q

(2.15)

q .p > n

==~

b'max> q

(2.15)'

By duality we also have

leading to a generic form of the Brasch-Pearson Theorem. C o r o l l a r y 2.5. ArbiLrary pole assignment holds for the generic degree n, p x m / / n e a r systems using q-th order dynamic output compensation, where q is any natural number

satisfying max(m,p)(q + 1) >_ n

(2.16)

Of course, if min(m,p) = 1 then Corollary 2.5 also implies that the generic system can be pole assigned using constant gain o u t p u t feedback, provided max(re, p) > n. More generally, by combining Corollary 2.5 with the technique used in the proof of the Brasch-Pearson result, we obtain a stronger form of (2.16). First, consider the problem of placing n + q distinct real poles: s x , . . . ,Sn+q. If p < m, the algorithm proceeds as follows (i) Place the subset { S l , . . . ,~p-1} by output feedback K , as in Corollary 2.5; (ii) If Ri is the residue of GK(~) at s = si, choose a p-vector w so that wTRi = O,

i=l,...,p--1. For generic G(a), wTG(s) has the generic set of controllability indices and McMillan degree n - p + l . According to Corollary 2.5 we can place the remaining poles sp,... , sn+q by a compensator of order q, where q satisfies

41

Recalling that p = rain(re, p) and m = max(rn, p), we can state some generic stabilization results. T h e o r e m 2.6 (Stevens). For the generic p × m s y s t e m of degree n any set of n + q distinct real poles can be ~ s i g n e d by a real dynamic compensator of order q for any q satisfying

(q + 1) max(re, p) + min(m,p) - 1 __ n

(2.16)

In particular, the generic p x m s y s t e m of degree n can be stabilized by a q-th compensator, for any q satisfying (2.16).

The only difference arising when considering a self-conjugate set of complex poles, is that if min(m,p) is even then the subset of min(m,p) - 1 poles selected in (i) must contain a real pole. This of course is not an obstruction if n + q is odd. T h e o r e m 2.7 (Stevens). The generic p × m s y s t e m of degree n can be generically pole assigned by a compensator of order q if

( q + 1) max(re, p) + m i n ( m , p ) - 1 >_ n + e

(2.17)

where e E {0, 1} with e = 0 unless both min(m,p) and r~ + q are even. This calculation also gives a proof of a related theorem due to Seraji [18 I. T h e o r e m 2.8 (Seraji). /f (2.16) holds then using a q-th order compensator closedloop poles can be generically assigned at any set of n + q frequencies containing a self-conjugate set of min(m, p) - 1 frequencies.

Remark

2.9.

In [31], Youla, Bongiorno and Lu treat the interesting problem of

stabilizing a scalar input-scalar output system using stable dynamic compensation, discovering the hard constraint that between every pair of real, nonncgative open loop zeroes there should exist an even number of real open-loop poles. This should, of course, underscore some of the differences between dynamic output compcnsation and observer based stabilization schemes. Saeks and Murray [32] discovered that this result was equivalent to a result on the existence of a single dynamic compensator simultaneously stabilizing two given plants. Because of its use as a method for designing reliable, or fault-tolerant, stabilization schemes, the simultaneous stabilization problem was formulated and researched in the general sctting of stabilizing r given p × m systems having McMillan degrees hi, i = 1 , . . . ,r, by a single (nonswitching) compensator. In particular, Vidyasagar and Viswanadham [20] showed, using the "YBJ parameterization" of all stabilizing compensators for a given plant, that when r -- 2 and

42 max(m, p) > 1 the generic pair of systems is always simultaneously stabilizable, in sharp contrast to the case max(ra, p) = 1 studied in [32]. In [33], Ghosh and Byrnes proved that max(re, p) _> r is sufficient for generic simultaneous stabilizability, generalizing the results of [20]. Moreover, in his thesis ([21]), Ghosh obtained a generalization of Theorems 2.6-2.8, yielding an explicit estimate for the order of the simultaneously stabilizing compensator

q(m~x(m,p) + 1 - ~) + max(re,p) > ~ ~

(2.17)

i=1

Finally, in the cases r < max(m, p) one can use transcendental methods, related to what we now call H°°-methods, to obtain simultaneous stabilizability criteria for fixed open sets of r-tuples of systems (see [341). 3. C o m b i n a t o r i a l G e o m e t r i c M e t h o d s . In 1975, H. Kimura published a pole-a~signment result which essentially generalizes both the rank 1 case (Corollary 2.5) and the classical results on cigenvalue assignment by state feedback [14]. Kimura's motivation was the problem of stabilizing mechanical systenxs by output feedback which proved intractible, for very good reason. Such a system typically has r~ = 2rn degrees of freedom, rn inputs and m outputs but more generally, one might hope the condition, m + p >_ n, would suffice for pole assignability, for the generic system. Actually, a fundamental computation due to Willems and Hesselink ([15], see also section 4.3) shows that this fails in the first nontrivial case, m = p = 2 and n = 4. Nonetheless, Kimura was able to show that something close to the intuition gleaned from mechanics does hold (see also Davison-Wang [35], for an independent derivation). T h e o r e m 3.1 (Kimura). If m + p - 1 > n, the generic s y s t e m can be assigned any self-conjugate set of distinct poles, different from the open-loop poles.

R e m a r k 3.2. Actually one can sharpen this result to obtain arbitrary pole-assignment for the generic system. The original proof of Theorem 3.1 given by Kimura contains an interesting combination of the geometry of linear subspaces and combinatorics involving dimension counting for subspaces with various incidence properties. There are of course some antecedents for such methods in the literature on eigenvalue assignment by state feedback, especially by M.L.J. ttautus [22]. In [22], IIautus is able to express the eigenvalue assignment as the problem of choosing n linearly independent (n + m) vectors, one each from a set of n preassigned n-dimensional subspaces of (n + m)-space. Hautus' criterion for the existence of such a selection is, quite remarkably, a linear algebra analogue of a result

43 of Rado's [36], generalizing Ph. Hall's famous solution of the "marriage problem", and is equivalent to controllability of the underlying linear system. Kimura's situation is similar, but of course more complicated since he requires output feedback. More explicitly, from the return differcnce determinant we see that to say 8i, i = 1 , . . . , n is a closed-loop pole for the output feedback gain - K is to say = 0

det

(3.1)

K Equivalently, we have col. sp.

n col. sp.

# (0)

(3.2)

when si is not a pole of G(~) and in general

LD(si)

where G(s) = N(s)D(8) - l is a coprime factorization. Regarding K and G(si) as linear maps K:C

p - ~ c "~ ,

G(si):C m ~ c

~

which define subspaces of C p+rn via

gr (K) = {(y,u):u = Ky} gr

=

y = aCs,)

)

the pole-assignment condition (3.1) may be expressed as gr ( - K ) VI gr G(si) ~ (0)

,

i = 1 , . . . ,n.

(3.3)

In this language, the pole-assignment problem m a y be stated as follows:

Given the n m-planes gr G(si) in C p+m find a p plane W , having the form W = gr (K), such that dim(WN gra(si))_~ 1

,

i=l,...,n

(3.4)

For state feedback, G(s) = (el- A)-tB and the approach taken by Hautus [22] was to construct W by choosing n linearly independent vectors, wi 6 gr G(si), setting W = sp {wl,... ,wn}. For output feedback and real poles, an alternate construction of W is given in the following result, due to Rosenthal.

44 L e m m a 3.3 [231. Given (rn + p - 1) m-planes Vi in It m+p there exists a p-plane W

intersecting each Vi nontrivially. Proof.

Suppose rn < p. If l~ is a (p - m + 1)-plane intersecting V 1 , . . . , Vp-rn+l

nontrivially, for any two planes Vi, V/one finds a vector vi so that lfi/@ sp {vi} N Vi ~t (0), W ~B sp {vi} N V j ¢ (0). By adding rn - 1 vectors, one therefore intersects (p - rn + 1) + 2(rn - 1) p l a n e s

As Roscnthal shows, it is possible to improve on this result by choosing I~ more carefully. For example [23], if V1,... ,Vs C It s are 2-planes the VI -{- V2 q- V3 cannot be a direct sum; i.e. there is a 2-plane I~z intersecting V1,V2,Va nontrivially. Because l?f + V4 + Vs is not a direct sum, there is a 3 plane intersecting each Vi nontrivially. We need a little notation in order to formalize this conclusion: For a, b E Z, b ~t 0, the Gauss bracket [a/b] denotes the greatest integer less than or equal to a/b.

the condition

is sumcient for pole placement of real, distinct poles. A rigorous proof of Theorem 3.4 requires a proof that if a p-plane W can be constructed, meeting each Vi = gr G(~i), then there exists such a p-plane having the form, W = gr (K). This "general position" argument will be discussed in sections 5.2, 5.4. 4. A f f i n e A l g e b r a i c M e t h o d s f o r P o l e A s s i g n m e n t 4.1 I n t r o d u c t i o n . When rank G(s) >_ I, equations (2.3)-(2.3)' are no longer affine. This, of course, is also the case for eigenvalue assignment by state feedback and while we will examine some analogies with this problem in section 4.3 the differences between pole assignment by output and by state feedback should become clear from a variety of viewpoints. One of the possible differences was anticipated in [9], where several questions concerning the finer structure of the problem are raised, concerning for example what kinds of formulae would exist for pole-assigning feedback gains, assuming such feedback laws would exist. For example, Anderson, Bose and J u r y [9] ask whether, as in the state feedback case, there would exist affine straight-line formulae for K i.e. universal formulae for K that are affine in the closed-loop poles with coefficients which can be "prcprocessed" rationally in terms of the plant parameters. As we have seen, this does hold in the case min(rn, p) = 1. One of the corollaries of the now famous calculation by Willcms and Hcssclink [15] is that the answer to this question is negative; indeed,

45

in the general 2 × 2 case one requires straight-line formulae using square roots (see Remark 4.10.) Furthermore, Byrncs and Stevens [37] showed in the 2 × 3 case that it is impossible to find solutions using straight-line formulae with radicals (see section 5.3). We now turn to the technical details, beginning with the fundamental analysis by tIermann and Martin of pole-assignment over G. 4.2 I n f i n i t e s i m a l A n a l y s i s . In terms of a minimal state-space realization G(s) = C ( s I - A ) - I B

(4.1)

of G(s) with A an n x n matrix, B an n x m matrix and C a p x n matrix, the pole assignment equations (2.3)-(2.3)' take the form detCM - ,4 - B K C ) = s a + c,(K)~ ~ - ' + . . . + ca(K)

(4.2)

Equating coefficients (4.2) defines a polynomial map X : k mp ~ k ~,

(4.3)

x ( K ) = (ci(K))~= 1

(4.3)'

defined for k = It. or G via

There is a nontrivial consequence of the formulation of the pole-assignment problem as a problem concerning a polynomial mapping (4.3), apparently first noticed by Willems and Hesselink [15]. Namely, the condition

mp >_

(4.4)

is necessary for arbitrary pole-assignment. While one might appeal to a "dimension count" in this case, such arguments are usually far more subtle than just counting a few numbcrs. A careful proof would even show that (4.4) is necessary for image (X) to contain an open set. We denote by dX[K the Jacobian of X at a point K . A point y E k a is then called regular if either X - l ( y ) is empty or if rank dX[K, for all K E X - l ( y ) , has the maximum possible value. Supposing that image (X) contains an open set, Sard's theorem asserts that there exists a regular value in the image of X, from which (4.4) then follows using the implicit function theorem. P r o p o s i t i o n 4.1 (Willems-Hesselink). Tile condition m p > n is necessary /'or poleassignment by output feedback. Turning to sufficient conditions, any proposition asserting that rank dXlKo = n

(4.5)

46 for some K0 will have lots of corollaries over both It and C. As we noted, over either R or C, from the implicit theorem it follows that (4.5) implies image (X) contains an open neighborhood of X(K0); i.e. that one can assign all poles sufficiently close to the roots of

Pxo(S) ----s a -}- c, (Ko)~ n-' + . . - -{- cn(Ko) Since (4.3) is a polynomial map, over C a more remarkable fact follows from what is called either "the fundamental openness theorem" [38] or the "dominant morphism theorem" [39]; viz. (4.5) implies that image (X) contains an open dense set of monic complex polynomials. In other words, over C the existence of o n e / t o satisfying (4.5) implies that one can place a generic, or almost any, set of desired poles. The generic existence of such K0 follows from a beautiful calculation given in [11]. T h e o r e m 4.2 (Hermann-Martin). For any p x ra strictly proper transfer function of degree n,

rank dx[o -= dim span { C B , C A B , . . . , CA'*-IB}

(4.6)

P r o o f . From (2.3) we compute the directional derivative of X at 0, in the direction K, ;LS

dXIo(K) -- det D ( s ) l i m

det (I + e g G ( s ) - det (I)

~0

E

= det DCs ) lim

trCeKG(s)) = _ det DCs)trCKGCs))

E--*0

E

= det D ( s ) < - K , G(s)> where ( K , M ) = t r ( K M ) is the canonical representation of the p x ra matrix K as a linear functional on the space of ra x p matrices M . Expanding G(s) in a Laurent series oo

cC ) = i=l

we find that rank dxIo is given by the dimcnsion of space of n-vectors

(<-K,HI>,..., (-K, Ha)) with K arbitrary. Since Hi = C A i - I B , (4.6) follows. Of course, if rap >_ n the property that the right hand side of (4.6) is n-dimensional is generic. C o r o l l a r y 4.3 (Willems-Hesselink). For the generic real system, the condition rap > n

is necessary and sumeient to be able to assign an open neighborhood of the open-loop characteristic polynomial via real output feedback.

47 C o r o l l a r y 4.4 (Hermann-Martin). For the generic real or complex system, the condition mp >_ n is necessary and sufficient to be able to assign an open, dense set of complex closed-loop characteristic polynomials via complex output feedback. R e m a r k 4.5. Results such as Corollary 4.4 are often described as providing generic pole-assignment for the generic system. In fact, it is known [43] that if rnp >_ n for the generic system the polynomial map X is proper, i.e. X-1 (F) is compact for compact sets F. Therefore, X maps closed sets to closed sets and, in particular, image X is also closed (cf. section 5). In summary, any result on generic pole-assignment for generic systems can be sharpened to arbitrary pole-assignment for generic systems. Moreover, in many cases the particular generic property of such systems is explicitly known, just as in the ease of state feedback and reachable systems. The geometric analysis of eigenvalue assignability via state feedback is similar but, naturally, much simpler. In fact, using infinitesimal methods and one of the techniques presented in chapter 2 one can give an elementary, clean proof of Heymann's Lemma. T h e o r e m 4.(} (Heymann). The pair (A, B) is reachable if, and only if, there exists a state feedback law K and a vector v E R m such that the pair (A + B F , By) is reachable. Since eigenvalue assignment, via state feedback, for scalar reachable systems is quite easy to prove, the principal application of Heymann's Lemma is Wonham's celebrated result asserting that reachability is equivalent to arbitrary eigenvalue assignability. To prove Hcymann's Lemma we first differentiate a rational map, as in the proof of Theorem 4.2, again following Martin and Hermann [12]. We need some notation: Gl•(It) = (real n x n matrices T : det T ¢ 0) Mm,~ : (real m × n matrices} [ s l , s2] = 51s~. - s ~ s l

,

s~ e M . , =

Consider the rational map

defined via @(T,F) = T ( A + B F ) T -1

(4.7)

A straightforward, but enjoyable, calculation shows that the directional derivative of the map A ~ T A T -1 in the direction S E Mn,n is just [S, A]. More generally we have d¢I(I,o)(S,F ) = [S,A] + B F

48 To say d(I)J(i,0) : M•,n x Mm,,~ "--* M,~,n is onto is to say (d~)* : M ~ ,

n --+

M~,.

x

M~n,,,

is one-to-one, but if L E M~, n - Ma,n we have

d ¢ [ i L o ) ( L ) ( S , F ) = ([S,A] + BF, L) = tr([S,A]L + B F L )

(4.8)

If d¢iI,0 ) (L) is zero, then (4.8) vanishes for all (S, F). In particular tr([S, A], L) = tr([A, LIS ) = 0

VS

and

tr(BFL)

tr(nBF) = 0

=

VF

In other words, we must have [A, L] = 0

and

LB = 0

so that

L(B, AB,...,

Ar~-IB) = O.

L e m m a 4.7 (Hermann-Martin). For '~ defined in (4.7)

n 2 = rank dCJ(l,0) ¢~ dim col span (B, A B , . . . , A a - I B ) = n

(4.9)

Hcrmann and Martin used (4.9) to prove an analogue of Corollary (4.4) using the dominant morphism theorem, viz.

reachability implies almost arbitrary eigenvalue

assignability via complex state feedback. Over R, for (A,B) reachable, the implicit function theorem shows that image (~) contains an open set U of n x n matrices. Since the set X of matrices with nondistinct eigenvalues is a proper, algebraic subset of M , , , , X has no interior. In particular, U - U n X -~ ¢ so that if (A, B) is reachable there exists F such that (A + B F ) has distinct eigenvalues. As in Lemma 2.2, we note that reachability of this pair is equivalent to

GF(S ) = ( s l -- A - B F ) - I B

= ~ i=1 S

Ri - -

Si

with ~i distinct, rank Ri = 1 and Ri = Rj whencver si = ~j. Therefore any v satisfying

Riv ¢ 0

i = 1 , . . . ,n

49 will also render (A + B F , Bv) reachable, proving Heymann% L e m m a and implying Wonham's Theorem. We conclude this section with a recent extension [40]-I41 ] of these kinds of results to a more general class of additive inverse eigenvalue problems.

In a state-space

representation, we consider the problem of making the spectrum of

A+L,

Le£

(4.7)

arbitrary where/~ is a subspace of the vector space Mn of complex n x n matrices. For pole-assignment by o u t p u t feedback, we have £o = { B K C : fl, C fixed, K arbitrary} In general, defining

xa:£

-' c

(4.s)

via xA(L) = det (sI - A - L) an extension of the proof of Proposition 4.1 shows that the condition rank £ > n is necessary for arbitrary eigenvalue assignment, for any fixed A. This of course is not sufficient, even for generic A, as the example

sea = {L : tr L = O} shows in a dramatic way, since dim sen = n 2 - 1. A remarkable result, derived in [40], shows that these two considerations are in fact the essence of this problem. T h e o r e m 4.8 (Wang). Necessary and sumcient conditions for image XA to contain an open dense set of polynomials for the generic A, are (i) dim Z > n (ii) £ ¢: se,,. We note that £0 c se,~ if, and only if, C B = 0.

Moreover, for generic B, C, we

compute dim £0 = rap. In [41], Wang also gives explicit description of those A for which XA is "almost onto". The method of proof for Theorem 4.8 begins similarly, with an application of the "dominant morphism theorem," from which one derives equivalent conditions for Xa to be "almost onto" for generic A: There exists an Lo 6 f such that

5O (a) rank d×A(Lo) = n; (b) The linear map ¢ : L --* C a, ¢(L) = ( t r ( L ) , t r ( A + L o ) L , . . . , t r ( A + Lo))*-IL) is onto; (c) Denoting by Q the Lie algebra of matrices generated by ( I , A + L0,... (A + L0) a-l} consider the restriction of the moment map ,~ : ~ --* Q*, O ( L ) M = t r ( L M ) , then dim Q = n and (I)(£) = Q*; (d) £ + I m ad(A+Lo) = Mn, where adA+Lo(M) = [A + Lo,M] = (A + L o ) M - M ( A + L0); (e) £J- N Ker ad(A+i,o)t = (0), where £J- = ( M E M~ : tr(M~L) = 0 V L e / ~ ) . When ~1 = { B K : B fixed, K arbitrary}, Hermann and Martin [11] proved that condition (e) is equivalent to controllability of (A,B). For C0 as defined as above, Willems and Hesselink [15] proved that (a), (b) and (e) are equivalent if A + Lo has a cyclic vector. Wang's proof [41] of the general result proceeds from (c), using the Lie theoretic interpretation of the moment map from Hamiltonian mechanics and the conjugacy of Cartan subalgebras of M)) to eliminate the dependence of criteria (a)-(e) on L0. 4.3 E l i m i n a t i o n T h e o r e t i c Techniques. In the aftermath of the introduction of infinitesimal (complex) algebraic geometric methods, some rather serious questions arose concerning pole-assignment using real output feedback. In the light of rank one methods and Kimura's theorem, the first nontrivial case to analyze is pole assignment for 2 × 2 systems having McMillan degree 4. Here, the system (4.2) of algebraic equations consists of 4 equations, of degrees 4, 3, 2 and 1 in 4 unknowns. General wisdom (gleaned from Bezout's Theorem) would then have it that generically there should be 4! = 24 solutions. In contrast, eliminating all but one of the variables, Willems and Itesselink [15] found that generically the resulting equation, or "eliminant," had degree 2, so that there are 2 < < 24 solutions. This shows that the pole-asslgnment problem has a highly nongeneric special structure, which is probably most easily derived from the return difference determinant using the Binet-Cauchy expansion of an (m + p) × (m + p) determinant as an inner product of complementary m-th order and p-th order minors:

51 In this case, there are s i x 2 × 2 minors of (K/), of which one is constant, four are linear and one quadratic. Thus, assuming sl distinct (4.9) yields, for i = 1,... ,4, four equations in K which contain the same quadratic term; i.e. we encounter four linear equations subject to a quadratic constraint. R e m a r k 4.9. In general, the pole assignment equations consist of n linear equations

< (7')> v, ma,

= 0

,

i = 1,... ,rt

(4.9)

subject to quadratic equations (i.e. Plficker relations) reflecting the fact that v is a vector consisting of the p-th order minors of a (p + m) × p matrix. This was first discovered in the general case by Brockett and Byrnes ([16], see also [43]) using both projective algebraic geometric methods (see chapter 5) and the Binet-Cauchy formula, and also independently by Morse, Anderson and Wolovich [24] using multilinear algebra. This observation has also been the starting point of a recent assault on the pole-assignment problem by Karcanias et al. [44]-[45]. In order to find an explicit form of the eliminant in the case m = p = 2, n = 4, we follow the derivation in [24}, with the obvious notation

1

[PI(s)

P2(s)] , "i(s)polynomial P4(s) J

G(s)=X~(s)tPa(s) Ps(s) = A(s)det G(s)

and Ks = det K = KIK4 - K2K3

(4.1o)

In particular, (4.2) may be expressed as 4

= IIC

5

=

+

i=1

pA,)K 2"=1

In tcrms of the expansion py = ~ PjiS i i

(4.2) reduces to

(4.11) J

52 together with the quadratic relation (4.10) defining Ks. We denote a general solution of (4.11) by Ki = K°+"/ei where (ei) is a basis element of the (generically) one-dimensional kernel of the linear map defined in (4.11). The quadratic relation (4.10) thus becomes a7 2 - fl'), + o = 0 where O~ -~- g l e 4

--

e2e 3

t~ = K°ea + K°e, - q K ° - e 4 K f a =

0

0

IQ K 4

-

0 0 K 2 K~ -

+ e5

K °

In particular, the explicit form of the solution, assuming a -~ 0 is 2a Remark

ei

(4.12)

4.10. The condition a # 0 is precisely the "nondegencracy" condition used

in [16], [43] for m p < n. This generic condition implies that image (X) will be closed; e.g. that sequences of solutions don't "go off to infinity" for convergent sequences of data (sce also Remark 4.5). Also, the generic condition a # 0 then implies that X will be 2 to 1 and that any expression for pole-assigning gains will require the use of radicals, answering the questions raised by Anderson, Bose and Jury [9] in the negative (cf. section 4.1). Since examples for which a # 0 and a # 0, exist in great profusion, we have several corollaries of the calculation in [15] which hold for the generic system. Theorem

4.11 (Willerns-ttesselink). For the generic 2 × 2 system having McMillan

degree 4, there are two output feedback gains, counted with multiplicity, which assign a given set of closed-loop poles. Moreover, for the generic real system: (i) There is an open set, of infinite Lebesgue measure, of real characteristic polynomials which cannot be assigned using real output feedback; (ii) The set of assignable, real, closed-loop characteristic polynomials is a dosed set, containing an open subset having infinite Lebesgue measure, and (iii) There is no straight-line, linear formula for pole assigning gains using just rational preprocessing of the system parameters and the desired characteristic coefficients. As one might expect, the situation becomes far more complicated for larger rn, p, n and straightforward elimination soon becomes prohibitive. Before turning to the general case, it is therefore important to understand what elimination theory can and cannot

53 imply. We will also illustrate this with some quite different applications of elimination methods to the pole-assignment problem. Put geometrically, the main problem of elimination theory can be described as follows. Suppose Z C k N × k M is an algebraic set and consider the projection map Pl : Z ---* .KN defined via Pl(~, Y) = X

The main problem is to describe the subset P1 (Z) = {x : B y such t h a t (x, y) E Z} as explicitly as possible. In other words x E p l ( Z ) if, and only if, for some y the point (x, y) satisfies the polynomial equations defining Z. Therefore, if pl (Z) were itself described by polynomial equations we would have succeeded in eliminating y from the equations defining Z. This is not always the case, since projections are not closed maps-take, for example,

z = {(~,y) : ~ v

-

1 = 0} c c x c.

There are nevertheless several results describing Pl (Z) which are useful. If k = C, then Chevalley's Theorem [39] asserts that p l ( Z ) is "constructible"; i.e. Pi (Z) can be described by polynomial equations

A(~) = 0

i= 1,...,~

(4.13)

for some j = 1 , . . . ,~

(4.14)

and polynomial inequations gj(x) ¢ 0

If Pl (Z) is closed, it is algebraic but this is not always the case. The main theorem of elimination theory, over C, is that pl (Z) will be closed if the equations h~(:~,u) = o

k = i,... ,t

defining Z are homogeneous in y (see [46]).

(4.1s)

Remarkably, for the pole-assignment

problem this turns out to be the case for "nondegenerate" systems (see section 5). If k = R, then Pl (Z) may not even be constructible, as the basic example z = { ( ~ , y ) : u = =2} c ~ x r~

54 shows. The fundamental theorem of Tarski-Seidenberg [47] asserts, however, that Pl (Z) will always be "semialgebraic'; i.e. pI(Z) is described by (4.13)-(4.14) together with polynomial inequalities pm(z,y) > 0

some m = 1,... ,t~

(4.16)

some n = 1 , . . . , v

(4.17)

or

qtt(x, y) >_0

The main theorem of elimination theory also holds over 1~, so that Pl (Z) is closed if (4.15) is homogeneous in y. In this case, a recent refinement of Tarski-Seidenberg, by Delzell [48], asserts that p,(Z) earl be described by (4.13) and (4.17), as in the second example given above. Finally, there exist somewhat more symmetric versions of these results: Over C, the image of a constructible set is constructible and, over It, the image of a semialgebraic set is semialgebraic under a projection or, slightly more generally, a polynomial map. For example, since the open left-half complex plane is semialgebraic and hence so is its n-fold product, the set of n-th degree Hurwitz polynomials is semialgebraic, being the image of this product space under the polynomial map n

where

a C, - , j ) = 1

i=O

And, the Routh-Hurwitz conditions explicitly define this open scmialgehraic set. R e m a r k 4.12. Semialgebraic sets are often called "decidable" since membership in such sets can be decided in a finite number of polynomial operations. For example, it is possible to decide for which (A, B, C) a given characteristic polynomial p(s) can be assigned via output feedback, i.e. if

Z = {((A,B,C),K):

d e t ( s I - A - B K C ) = p(s)}

and Pl ((A, B, C), K) = (A, B, C) then Pl (Z) is semialgebraic (see [9] for several refinements of such arguments). As another illustration, we consider the problem of generic stabilizability [25]. Using a standard bilinear transformation, it is not hard to show that if stabilization by output feedback can be achieved for the generic p x m continuous-time system of degree n, then it is also possible for the generic p × m discrete-time system of degree n. A similar calculation also enables one to achieve generic stabilization with a pre-assigned margin of stability; i.e., for all K E Z + it must also follow that for the generic system one

55 can assign poles somewhere in the disc about 0 of radius 1 / K . Using the fact that, generically, image (X) is closed (cf. Remark 4.5) and the Baire Category Theorem, it follows that for a dense set of systems it must be possible to place the closed-loop poles all at 0. Moreover, an application of the Tarski-Seidenberg theorem then allows one to conclude that it must also be possible, for an open dense set of systems, to place the closed-loop poles all at 0. Finally, since the dimension of the algebraic set of n x r~ nilpotent matrices is n 2 - n (see [49]), a dimension count as in the proof of Theorem 4.1 before yields the surprising result that (4.4) is in fact also necessary for generic stabilization. T h e o r e m 4.13 (Anderson-Byrnes). The condition mp >_ n is necessary for output feedback stabilization of the generic p x m system having degree n. We remark that all known examples suggest that, for given m, n and p, stabilizability may hold generically if, and only if, pole-assignability holds generically. For example, P. Molander (unpublished) has modified the Willems-Hesselink calculation to show that generic stabilizability does not hold for m = p = 2 and n = 4. This, of course, could also be seen from the formulae (4.12) and in [25] an explicit example of a (nondegenerate, i.e. a ~ 0) system which cannot be stabilized is given. Using (4.12) and the RouthHurwitz inequalities, one concludes that small perturbations of this system also cannot be stabilized. P r o p o s i t i o n 4.14 (Molander, Anderson-Byrnes). I f m = p = 2, n = 4 there is an open set of systems which cannot be stabilized by output feedback. In particular, for each such system the set of characteristic polynomials which cannot be assigned contains a set having infinite Lebesgue measure. I~emark 4.15. There is of course an open set of 2 × 2 systems having degree 4 which can be pole assigned and therefore stabilized. However, the existence of such open sets also follows from more classical methods; e.g. m × m minimum phase systems with invertible high frequency gain can always be stabilized using output feedback. This can be seen using a multivariable root-locus plot (see [10D or using geometric linear systems theory by intepreting zeroes in terms of (A, B), or almost (A, B), invariant subspaces

(see [50]). In the next chapter we will turn to the general case m p = n, obtaining the explicit formula (1.1) for the degree of the eliminant, calculated by Brockett-Byrnes without using elimination theory. Thls implies several positive results. For example, if m = 2 and p = 3, the eliminant has degree 5, a result obtained independently by Anderson, Morse and Wolovich by explicit elimination. This result implies, or course, that the

56 generic 2 × 3 system having degree < 6 can be arbitrarily pole-assigned using real o u t p u t feedback. Remark

4.16. It is not surprising, yet nontrivial to prove (see [37]), that the Galois

group of the generic 2 x 3, degree 6 problem is in fact the symmetric group on 5 letters, 5'5, which of course is not a solvable group. In particular, there do not exist formulae for pole-assigning gains which use rational preprocessing and the extraction of r-th roots, underscoring the nonlinear nature of this problem. 5. P r o j e c t i v e A l g e b r a i c G e o m e t r i c M e t h o d s 5.1 E n u m e r a t i v e

Geometry and the Schubert Calculus

Having analyzed the case m = p = 2, rt = 4 in section 4.3 using an explicit elimination argument, the next nontrivial case would correspond to the choices rain(re, p ) = 2

,

max(re,p)=3

n=mp=6

(5.1)

We know both from the general formula (1.1) and an explicit elimination argument [24] in this case that the system (4.3) of 6 equations in 6 unknowns generically will have an eliminant having degree 5. This is remarkable since, in general, a system of six equations of degrees 1, 2 , . . . , 6 will have an eliminant of degree 6! = 720, implying that for the pole assignment equation 715 unexpected cancellations of a rather complicated nature occur, suggesting that correctly carrying out an elimination argument even for modest sizes of m, n and p will be extremely difficult. On the other hand, as the earlier work by Hautus and by Kimura suggests, the degree of the eliminant might possibly be interpreted in combinatorial geometric terms and therefore be computed without resorting to an explicit elimination. In fact, using methods of enumerative geometry, Brockett and Byrnes [16] were able to compute the number (counted with multiplicities)

dm,p of possibly complex feedback laws placing a given set of poles of a (specific) generic class of p × m systems having degree rt = rap. Explicitly, the degree dm,p of the eliminant is given by the formula 1!--. (p - 1)!(rap)!

dm,p = m ! . . . (m + p - 1)!

(5.2)

We note that if min(m,p) = 1, then dm,p = 1, in harmony with the results of chapter 2. Moreover, d2,2 = 2, while d2,3 = d3,2 = 5. More generally, d2,p is the p-th Catalan number which is odd precisely when p = 2 r - 1.

For example, if min(m,p) = 2

and max(re, p) = 7 there are 429 solutions of the pole-assignment equations and, in particular, at least one real feedback law for a real system and a self-conjugate set of desired poles. Formula (5.2), and several very useful generalizations, can be derived in a formal way from the Schubert calculus of enumerative geometry, but it is known that there are

57 several technical conditions which must be verified to ensure that such calculations will be correct.

Indeed, Hilbert's 15th Problem, first solved by van der Waerden

in 1929, asked for the rigorous justification of the Schubert calculus and of general methods in enumerative geometry. In this section we outline the basic framework, and necessary technical conditions, one obtains from Schubert's calculus. In 1886 Schubert [51] addressed the problem: Given rnp m-planes Vi in C re+p, how m a n y p-planes W intersect each V/nontrivially? According to Schubert, if the planes V/are in "general position" then this number is given by (5.2). Moreover, taking V~ = gr(G(si)), we see from (3.4) that, when it applies, (5.2) gives an upper bound on the number of possibly complex feedback laws K placing the closed-loop poles at s = si. Indeed, if each such p-plane W were of the form W = gr(K)

(5.3)

then (5.2) would yield the exact number of pole-assigning gains. There are then two points which need clarification. First, in rigorous treatments of the Schubert calculus (see [52] and also the survey [53]), "general position" can be interpreted as the condition that only a finite number of p-planes W meet the planes no,trivially. The second is to guarantee a priori that every such W has the form (5.3). Fortunately, these conditions are implied by a simple, system theoretic condition which involves some interesting geometric techniques (see sections 5.2-5.3). Not surprisingly, the condition rnp < n is necessary for the n m-planes gr G(si) to be in "general position". Explicitly, "general position" will be implied by

det

- 0

K~. G(a)

~. rk

< p

(5.4)

K2

In [16], condition (5.4) is referred to as "nondegeneracy" of G(a) and it is known that nondegenerate systems are generic if m p <_ n (see [43]). Since nondegeneracy also implies image (X) is closed, we can deduce T h e o r e m 5.1 (Brockett-Byrnes). Suppose m p = n. For any n o , degenerate p x rn system having degree n and for any monic polynomial p(s) of degree r~ there are, counting with multiplicity, dm,p output feedback laws assigning p(s) as a closed-loop characteristic polynomial.

In particular, if min(rn, p) = 1 or if m i n ( m , p )

= 2 and

max(m, p) = 2 r - l , for nondegenerate real transfer functions and real p(s) there is atways a real output feedback law assigning p(s) as the closed-loop characteristic polynomial.

58 E x a m p l e 5.2. Consider the 3 x 2 system having degree 6

---i 2 3

A=

--

1

1

2

2

3

2 --3

-I 1 --2

2 --3 --1

--1

1

3

1 3 --1 --i

i' 1 1 --2

1 2

--3 1

oi ,

0

0 1 0

B=

0 0 1

0 0 0

0 0 0 0 0 07 0 0 0 0

Ii°

(5.s)

In general, nondegeneracy is equivalent [43] to the non-existence of rn linearly independent functionals ¢1,.-. , ¢ ~ such that det [¢i(gj{s))] =-- 0

where gj(s) is the j - t h column of G(s). Using this criterion, in the case rn -- 2, it is tedious but straightforward (see [37]) to check that this system is nondegcnerate. Moreover, there exists 5 feedback laws, three real and one complex conjugate pair, yielding a characteristic polynomial equal to the open-loop characteristic polynomial. Explicitly, one may compute, with maximal error +0.01, the solutions

[0001

/(i ---- L - - J 0

, K2--- [ 2.47

[27.6

-00

3.86

-12.36

K3 = [58.29 -27.6

[ -.91 + j2.52 K4,s---- [-1.76:1:jl.36

36.24J

-10.35] -18.18J

2.3 + j4.5 -2.33 + j2.94] .91-{-j2.52 -.65±j6.7 J

We remark that the error bounds show that these are simple roots; i.e. roots with multiplicity one. We shall return to this example in section 5.3. 5.2 T h e G e n e r a l P o s i t i o n L e m m a In this section, we outline a rigorous justification for the calculation (5.2) of the number drn,p of roots to the pole-assignment equations for "nondcgenerate" systems. In this program, we need to make a significant use of the concept, and properties, of a Grassmann variety (or Grassmannian) for two reasons. First, the modern formulation and justification of the Schubert calculus-in particular of formula (5.2)-is inextricably interwoven with the geometry of Grassmannians and, second, the only method we currently know to show that generically the m-planes gr G(s¢) are in general position is through the Hermann-Martin map, a geometric interpretation of matrix transfer functions as curves in Grassmannians. We use the notation Grass (m, N) = {V C c N : V a subspace of dimension m}

(5.6)

59 The special case m = 1, corresponding to the projective space of lines through 0 in C N, is typically denoted by p N - 1 . Of course, the notations Grass ( n, N) and p/V-x make sense for any field of scalars and, when we need to restrict ourselves to real d a t a and real objects we will use the notation G r a s s a ( m , N ) and ItP N-1. One very useful point of view is t h a t Grass (m, N ) is a natural "compactification" of the space MN_m,m of ( N -- m) × m matrices. To see this, choose an m-dimensional space U C C N and any complementary subspace Y C C N so t h a t C N = U @ Y. For any m-plane V there are two possibilities; either V is c o m p l e m e n t a r y to Y or V is contained in the set

a(Y)={V:

dimV=m,

dim(VNY)

>1}

(5.7)

If V is c o m p l e m e n t a r y to Y then for some linear G V=gr

(G)

when

G:U-+Y.

Hence, there is a natural correspondence Grass (m, N) - a(Y) ,

,

(5.8)

As we vary Y, (5.8) defines a cover of Grass (m, N ) by subsets in natural correspondence with M N _ m , m ~- C m(N-ra}. For example, if m = I and N = 2 one has

pt = (pt _ a(Y))Ua(Y)

(5.9)

where a(Y} is a single point, while p1 _ a(Y) is in natural correspondence with C. Varying Y gives a covering of p1 which induces the natural structure of a compact manifold on p1. On the other hand (5.9) exhibits p1 as the one-point compactification of C, i.e. p1 m a y be naturally identified with the 2-sphere S 2. correspondence, ltP 1 m a y be identified with the circle, S 1.

Under the same

In general (5.8) allows

one to regard Grass (re, N) as a compact complex manifold of dimension m ( g - m)

containing MN-m,m as an open dense submanifold. D e f i n i t i o n 5.3. For any N - m plane Y, the subset a(Y) defined in (5.7) is referred to as a Schubert hypersurface. R e m a r k 5.4. When m = 1, a Schubert hypersurface a(Y) in p N - t consists of the lines l C Y. Since dim Y = N - 1, we see t h a t

a(Y) "" pN-2 C p N - 1 In general, however, Schubert hypersurfaces are not smooth. If, for example, m = 2 and N = 4 then there are two kinds of points V E a ( Y ) for any fixed Y: those V such that dim (V N Y) = 1

60

a(Y). If Yi, Y2 are 2-planes in a(Yx) f3 a(Y2) is nonsingulax. In fact,

and V = Y, which is the unique singular point on general position in C 4, i.e. if if

V E a(YI) A a(Y2)

YI @ Yz

= C 4, then

then V = V N Y1 @ V N Y~, d i m V n l~ = 1 so t h a t

0.(yl) I~10.(y2) ..~ p1 x p1

(5.10)

In particular, over It we have a 2-torus o~[~)

n o~(YI)

(5.11)

-~ s ~ x s I _ T 2

Our interest is in Schubert hypersurfaces in Grass (p, m + p) defined by

(5.12)

aCsl) = a ( g r G(si)] = { V : dim V N g r GCsi) >_ 1}

since, according to (3.4), to say K place the closed-loop poles at ~ = ~i, for i -- 1 , . . . ,n, is to say n

gr (g) ~ i =n io(~)

(s.13)

Before analyzing the general case, it is interesting to interpret the cases m p and rain(re, p) = methods.

1 or m

- n

= p = 2, already understood in sections 2 and 4 by other

First suppose p = 1, m - n. As above any Schubert hypersurface

linear h y p e r p l a n e p ~ - I C P~; i.e. to

o(sl)

corresponds an n-plane

V(si)

a(si) is a

in C t'+l such

that

o(~) = Since

V(si) mad V(~j)

{lines ~ c

c '~+~

: e c v(~))

are either equal or have a codimension 2 intersection, we see that

imposing each (linear) condition, gr ( g ) E m o s t one; i.e.

a(si)

amounts to a loss in dimension of at

tt

dim Ct a ( s i ) _> 0 i=1 As we have seen it is generically the case t h a t dim n

a(si) = O,

i=1

occurring precisely when the numerators are linearly independent,

in which case

rt

in=lo(si)

= {V). If V were not of the form V = gr ( I f ) , this would impose another con-

dition on V, viz. V = Y, which does not hold generically. Finally, if the set {~1, . . . . *n} is self-conjugate then we have rt

~--x

tg

i

1°

) = (v}

so that v = gr (~) and hence K ~ real (compare Theorem 2.1).

61 The case m = p = 2, n = 4, is of course nonlinear. However, assuming the Schubert hypersurfaces a(si), i = 1 , . . . ,4 are in general position, this case can also be analyzed just from the definitions [23]. We consider the real situation, taking si E It. As above, the first two constraints on V

are equivalent to the fact that the 2-plane V lie on a 2-torus. Moreover, as in section 4.3 the additional constraint

V e aRCs3) n Ca~Cs~) n aRC~2)) is linear; i.e.this intersection is the graph of a linear m a p ¢3:RP I~RP

1 , gr (¢3) C R P 1 x R P I

Explicitly, n Co

(,1) n

= gr

i = 3, 4.

Thinking of the 2-torus T 2 = S 1 x S I = / L P 1 × RP 1 as a rectangle with opposite side identified we have one of two pictures

RP I × RP 1

Rp 1 × RP 1

so that gr (Ca) intersects gr (4'4) either in 2 real points or not at all (compare T h e o r e m 4.11). There are several basic facts a b o u t Schubert hypersurfaces which are quite useful analyzing the intersection (5.13). T h e o r e m 5.5. Consider n ( N - p).planes 1~ in C N. r

(1) If dim iO=la(Yi) = d, then r.+ l

dim

n aCYi) > _ d - 1 i=l

62 (2) I f n =

p(N -

p), then n o(~) i=1

dim

=

o

if, and only if, n

n ,7(~) i : 1

is finite, in which case

# in, acY~)= =

1 ! . . . (p-- 1 ) ! ( p ( N - p))[ ( N - p)! - .. (N - i)[

counted with multiplicity. Remark

5.6.

Assertion (i), that the dimension of an intersection with a Schubert

hypersurface can go down by at most one, is not a general fact about hypersurfaces. This follows from an analysis of the P1Ocker imbedding

p : Crass (p, N) -~ eC~)-I exhibiting P (Grass (p, N)) as a (projective) algebraic set defined by homogeneous quadratic (Plficker) equations. Under the Plficker imbedding any a ( Y ) can be realized as the intersection of P(Grass (p,N)) with a projective hyperplane.

It is in this sense

that one may interpret the precise meaning of dimension, as well as claim (1), see [39]. Assertion (2) follows from an interpretation of Na(Y~) in terms of any of the now standard intersection theories: homology, cohomology, algebraic intersection rings, see [53]. Now consider a p x m transfer function G(s), having degree n = rap. We shall denote C ra by U and C v by Y and consider the Schubert hypersurfaces a(si) C Grass (p, r e + p ) as defined in (5.12), for si 6 C distinct. We note that, if s = oo, the notation a(oo) still makes sense, in fact since G(oo) = 0 we have

U=grG(oo) c Y @ U In particular,~,(~o) = ,,(U) and therefore to say V e o(oo) is to s~y

dimCVNU)_> 1

,i.e.

V : f i g r (K)

G e n e r a l P o s i t i o n L e m m a 5.7. Suppose m p = n. I f G(s) is nondegenerate, then for any choice of distinct si, i = 1, . . . . n #f'lla(S,)= < o o -

-

(5.14)

63 Furthermore, the generic system is nondegenerate. P r o o f of ,5.1'/. Consider the rational equation in Ki

det

[KI C(s)] K~.

=0

(5.15)

=0

(5.15)'

I

or, equivalently, the polynomial equation

act

where G ( s ) = N(~)D(~) -1 is a coprime factorization and rank [KK~]= p. For G(s)fixed, if Kt = I, (5.15) I has roots at the closed-loop poles corresponding to K2. Therefore, if rank 1/1 = p, (5.15) t is a polynomial equation of degree n. By continuity, for any K , the degree of (5.15)' is at most n. The nondegeneracy condition (5.4) implies that (5.15)' cannot be the zero polynomial and can therefore have at most n roots for any p-plane V = col. span

[K,1

.

(5.16)

LK2J In particular, oo cannot be a root of (5.15). P u t geometrically, this asserts n

n

n o(co) = ¢

i=l

since for any V in this intersection (5.15) would have the roots si, i = 1 , . . . , n and co. By Theorem 5.5(1)

n

dim n a(si) >_0 i=l

If this dimension were positive, (5.17) would have nonnegative dimension, by Theorem 5.5(1). Therefore, n

dim N a(si) = 0 i----1

and Theorem 5.5(2) applies, implying the theorem of Brockett and Byrnes for nondegenerate systems. It remains, however, to show that nondegenerax:y is generic; i.e. for m, n and p given there exists a (real) nondegcnerate system. Before demonstrating this fact, we turn to another interesting corollary of the General Position Lemma. C o r o l l a r y 5.8 [43]. For a nondegenerate system, image (X) c C n is a closed subset. In particular, for a nondegenerate real system the set of assignable reaI monic polynomials is closed. Proof. We noted that the degree of (5.15y is at most n, the degree being less than n reflecting the fact some of the closed-loop poles have gone off to infinity in the high

64 gain limit, det K, --* 0. This can be analyzed by homogenizing (5.15); i.e. replacing s by

8/t

and multiplying by t r, r the highest power of t -1, yielding a map

[K1] ~-+~(s't}K2

(5.18)

where ¢(s, t) is homogeneous of degree n, never zero if the system is nondegenerate. As in (5.16), this induces an extension of X ~: V ~ span {ff(s,t)},

where span {~} denotes the line through • in the n + 1-dimensional space of n-th degree homogeneous polynomials; i.e. : Grass (p, m + p) --* P~ Restricting ~ to the complement of a(oo) we recover

X : Cmp -'~ Ca as before. We claim image (X) = image (2) f3 C n

(5.19)

and since image (2) is always closed, the Corollary follows. But (5.19) is implied by the assertion t h a t if a p-plane V assigns a set of n finite poles, counted with multiplicity, then V = gr (K). If V # gr (K) then V 6 a(oo) implying that (5.15) has n finite roots and one infinite root, counted with multiplicity, contradicting nondegeneracy. We conclude this section with the construction of a nondegenerate system whenever

mp = n.

For this we need the Herman Martin map : pX __~ Grass (m, m + p)

defined for s 6 C by

G(a)=col.span [N(s)] LD(8) while

0(

,=col sp

Each p-plane V defines a Schubert hypersurface

a(V)

in Grass (m, m + p) and, by

definition,

e o W)

v"

oC,)

(5.19)

65 In particular, recalling that Y = gr (0) we see [12]

~-1(~(p1) n o(y)) = {Poles of GC~)} so that, counted with multiplicity,

# ~ - 1 ( ~ ( I ,~) n o(Y)) = ,~ In other words, (~(p1) is a (rational) curve of degree n in Grass (m, re+p), the HermanMartin curve [12]. Moreover [12], every rational curve of degree n satisfying

~Coo) = v corresponds to a p × m transfer function of degree n. This correspondence allows for the application of both constructions and results from classical algebraic geometry to problems of linear systems. For example [43], nondegeneracy originally was formulated in this context. In the light of (5.19), to say G(8) is nondcgenerate is to say that (~(pi) is not contained in any Schubert hypersurface. In particular, if m = 1 and p = n, G(s) is a rational curve in pr, not contained in any hyperplane, the classical algcbraic geometric definition of nondegeneracy. The classical example of such a curve

=col sp r]:

[

(5201

arises from the transfer function

1Is n e(,~) -:~ 1/'sa-1

(5.20) t

(5.20) is often referred to as the rational normal curve [53], but for n = 3 is called the twisted cubic, being an example of a space curve which lies in no plane [46]. In general, the rational normal curve is nondegenerate; i.e. n

~-~ a i s i : 0

~

ai : O

i=0

corresponding to linear independence of the numerators ni(~ ) as in T h e o r e m 2.1. One can construct other nondegenerate curves, derived from the rational normal curve as follows. Consider the rational normal curve . pl

_.+ p p + 1

66

To each point if(a) on ~/there corresponds a tangent line l(s) C pp+l. Being a line in pp+l means of course t h a t £(~) corresponds to a 2-plane V(a) in CP+2: =

{e c c'+2:

c

Therefore, we have a derived curve ./(1) : pl __, Grass (2, 2 + p) which is also nondegenerate and has degree 2p. More generally, to the rational normal curve ,,/ : p l .._¢ p r o + p - 1

we associate the derived curve ~/(rn--*) : p1 ~ Grass (rn, rn + p) where "l(m-1)(s) is the osculating ( m - 1)-plane to " / a t ~/(~), see [53]. And, ~(m-1) is a nondegenerate curve in Grass (m, m -}- p), having degree n = rnp. This proves that the generic property, nondegeneracy, is not vacuous and hence holds generically. 5.3 S o l u t i o n a n d C o m p u t a t i o n o f P o l e - A s s i g n m e n t P r o b l e m s In this section we shall focus on the possible explicit form of solutions to the poleassignment problem in the case m p = n and on the computation of feedback gains for explicit problems. As we have seen, the calculation by Willems-Hesselink in the case m = p = 2, n = 4 shows that, in general, linear formulae for pole assigning gains will not exist. Rather, any explicit formula will require the extraction of square roots. This is in fact the only case where the extraction of square roots suffices [25]. Furthermore, as we will demonstrate here, for rain(m, p) = 2, max(m, p) = 3 and n = 6, the pole assignment equations are not even solvable by radicals [37]. Although there exist general formulae for the roots of an n-th degree polynomial in terms of theta functions, it is clear that most pole-assignment problems will need to be solved numerically. P r o p o s i t i o n 5.9 [37]. If rnp = n and G(s) is nondegenerate, the system of algebraic

equations x(K)

=

c

can be solved numerically by the homotopy continuation me~hod. P r o o f . We begin with the observation that the proof of Corollary 5.8 shows more, viz. for a nondegenerate system and any compact set F of monic polynomials, the set X-I (F)

67 is compact. As in I54] and [55], the homotopy continuation method-which allows one to deform a solution to a nominal problem, e.g. for (A0, B0, Co) one takes the solution X(0) = P0(s) = open-loop characteristic polynomial and continue it to a solution for (A1, Bl, C1) of the problem, xCK) =

along paths from (Ao, Bo, Co) to (Ah B1,6'1) and P0(s) to pl(s)-will work, without a bifurcation analysis at the branch points, provided there is a path from (A0, B0, Co) to (A,, Bl, (7,) along which X remains proper. Since over C the generic set of nondegenerate (A,B, C) is necessarily connected and since X is always proper for (A, B, C) in this set, the homotopy continuation method applies. E x a m p l e 5.2 (bis). Consider the 3 × 2 nondegenerate system (5.5), having degree 6. Consider the following path linking the open loop characteristic polynomial to the polynomial s ~ ,~6 _ 5t~s + 4t~4 + 12t~3 _ 87is + 623ts -- 246t

0 < t < I

The solutions Ki(t), i = 1,... , 5 to the pole placement problem, az computed by the homotopy continuation method, can be schematically represented az follows: Open-loop , t = 1 gl(t)

~ K2(t)

~ ga(t)

~: g4(t) ~ [f,l(t)

s6

~K2(t)

~K3(t)

cg4(t)~gs(t)

,t=OK,(t)

with a unique branch point at to = 0.603 4- 0.001. The solution at the branch point to takes the form K , ( t 0 ) = ( 1.25 3.8 0.98) -1.42

0.75

-1.33

Kz(to)=(-3.14 -5.06 41.681 3.14 5.05 36.70J K3(to)=(2.81

3.89

509

-084 /

-1.91 /

-0,gj 1.34 --1.24

~0~ 81

\--2.40 3.24 3.37 J with maximal error 4-0.01. The roots K4(/), Ks(t) are real for t < to. For general re,p, mp = n one obtains the map X : crop "-'* Cn

68 and if E = C(Cl) and F = C(k~j) are the fields of rational functions, then composition with X gives a map

x*:E~

F,

x'(f) = f o x.

Since X is surjectlve if (A, B, C) is nondegenerate by T h e o r e m 5.1, X* is injective so one can regard

E~-x*ECF as an extension of fields. For example, to say that there exists rational formulae for the entries kff in terms of the ci is to say E -- F . From (5.2) it follows that this is the case if, and only if, rain(m, p) = 1. Indeed F is a vector space over the subfield E and from

([39],

Theorems 6-7 on pp. 116-117) it follows that dimE(F) = I F : E] ----d e g c ( x ) = dra,p

(5.20}

We first illustrate the use of (5.20) in conjunction with Galois theory to prove: The conditions min(m,p) = 1 or m = p = 2 are necessary for the existence of formulae expressing pole assigning gains in terms of rational expressions and square roots [25]. For, by Galois theory, it is necessary that IF : E] is a power of 2; i.e. dm,p = 2 r for some r. We claim that if rain(re,p) > 2 and m + p > 5, then drn,p is divisible by an odd prime. Indeed, by the strong form of Bertrand's postulate [56] there is a prime q, necessarily odd, such that

r e + p - - 1 < q < 2(m + p) - 4. Clearly, q cannot divide the denominator of (5.2) but on the other hand if min(m, p) > 2 then m p > q so that q divides the numerator of dm,p. Therefore, q divides dm,p. We note, for example that if m = 2, p = 3 then q = 5 is the unique prime satisfying this inequality. We shall now use these computations to calculate the Galois group of the poleplacement equations when m = 2, p = 3 and n = 6. Consider the fixed, but generic (indeed, a nondegenerate) system (A, B, C), where A is a 6 x 6, B is a 6 x 2, and C is a 3 x 6 real matrix. We shall prove that the pole-placement equations X(A,B,c)(K)

= (c, .....

= c

cannot be solved by radicals. Thus, if m = 2, p = 3 then (5.20) reduces to [F : E] = 5, therefore the minimal polynomial over C(c/) of kij(c), where x(K(c)) = c, has degree 5 for generic c ([39], pp. 116-117). And, since X extends to a globally defined map 2 on G r a s s c ( p , m + p), the minimal polynomial has its coefficients in C[c/]. Moreover, if ci E It then the coefficients of the minimal polynomial of kff(c) are real polynomials in the ci.

69

T h e o r e m 5.10. Ifmin(m,p) = 2, max(re, p} = 3 and n -- 6, then for generic ( A , B , C ) a n d / o r generic (c~) E R e, the equation x ( K ) -- (cl)

is not solvable by radicals. P r o o f . To say that x ( K ) = (ci) is solvuble by radicals, is to say that the minimal polynomial of kq(c) is solvable by radicals. Since this is an equation of prime order defined over a subfield of R, by Galois theory [57] one has a dichotomy provided the Galois group is in fact solvable: either (i) all 5 roots klj(c) are real; or (ii) just 1 root kij(c ) is real. In terms of the extended map, which is globally defined if (A, B, C) is nondegcnerate, : GrassR(3, 5) --~ RP 6 this is the assertion: (i) ~ is 5 to 1 on an open subset, 1 to 1 on its complement, or (ii) ~ is 1 - 1 everywhere. L e m m a 5.11. Suppose min(m,p) = 2, max(re, p) = 3 and n = 6. If for an open se~

of (A,B, C) the equation x ( g ) = (ci) is not solvable by radicals for an open set of (c~) of (A,B, C), then this equation is not solvable by radicals for the generic choice of (ci)

(A, B, C). P r o o f . Denote by V C C 66 the open, dense subset of nondegenerate ( A , B , C ) and consider the map : V × Grassc(3,5) ~ V x p6 defined by ~((A,B,C),YI) = ~t(A,B,C)(II) for a 3-plane II C C s. If K1 dcnotes the field of rational functions on V x CP ~ and K2 denotes the field of rational functions on V x Grassc(3,5) then ~*K1 C K2 and it follows from the formula (5.3) that deg [Ks : x*K1] = 5. Moreover, the extcnsion K2/x*KI is solvable if, and only if, the extension F / E defined in (5.20) is solvable for generic (A, B,C), by elementary Galois theory ([58], pp. 244-249). This, in turn, is solvable if, and only if, the extension field associated to the equation x(K) = (ci) is solvable for generic (ci), again by Galois theory. From these statements, the assertion in the Lemma follows by taking contrapositives. Turning to the proof of Theorem 5.10, one can see for purely topological reasons that (ii) can never occur for a nondegenerate system. T h a t is, if ~ were 1 - 1 then since

70 is continuous and Grassa(3,5) is compact. : Gra~sR(3,5) --~ RP 6 would be a homeomorphism which is easily seen to be false by comparing higher homotopy groups. On the other hand, Example 5.1 shows t h a t (i) is false for (5.5) and, since the roots Ki are simple, by a perturbation argument it follows that (i) is false for an open set of systems. In fact, we can prove more. T h e o r e m 5.12. Let n -- 6, max(rn,p) --- 3 and mln(m,p) -- 2. For generic ( A , B , C ) and generic (ci), the Galois group of the equation =

is the full symmetric group Ss on 5 letters.

Proof.

It follows from the above argument for the generic ( A , B , C ) and a generic

choice of (ci), that the equation x ( g ) = (ci)

is not solvable by radicals. Moreover, the minimal polynomial of the entries kii of K has degree 5 so t h a t the Galois group G is a nonsolvable subgroup of Ss. It is a well known and straightforward proposition that the only such subgroups are As, the alternating subgroup, and $5. Thus, we shall have G = $6 if we can prove that G contains a simple transposition. Now, by elementary Galois theory ([58] pp. 244-249), it suffices to find a particular choice of nondegenerate (A, B, C) and cl such that G -~ S~, and for this example we return to (5.5), leading to the map : Grass(3, 5) --~ P~. By Lemma 5.11, the Gatois group of the equation

is nonsolvable for generic (ci). We prove that G contains a simple transposition by using two results due to Joe Harris [59]: L e m m a 5.13. Let H : Y -* X be a holomorphic map of degree n. I f there exists a point p 6 X such that the fiber of Y over p consists exactly o f n -

1 distinct points--i.e, n - 2

simple points ql~... ,q~-2 and one double point q~_1--and if Y is locally irreducible at q•-l, then the monodromy group M of II contains a simple transposition.

71 L e m m a 5.4. ff X, Y are irreducible algebraic varieties of the same dimension over the complex numbers e, and II : Y ~ X is a m a p of degree d > O, the m o n o d r o m y group equals the Galois group.

On the other hand, we have already shown numerically that there exists (A, B, C) and a closed-loop characteristic polynomial for which there are three distinct solutions-three real simple roots I Q , K 2 ~ K 3 - - a n d one real double solution K4 to the poleassignment equation. We have thus shown that the Galois group of the equation x ( K ) = (ei), and thus of the extension field x * E C F, is

calCFIE)

= ss.

For generic (A, B, C) and generic (ci) the Galois group G of the pole-placement equation is a subgroup

G c Ss, while for fixed nondegenerate (A, B, C) and (ci) the Galois group G' is a homomorphic image of G. In particular

G~S5 is surjective and therefore, by a counting argument, one has

G=

for generic

Ss

(A,B, C) and (ci).

5.4 T o p o l o g i c a l M e t h o d s for P o l e - A s s i g n m e n t a n d S t a b i l i z a b i l i t y There is a simple set-theoretic alternative to the interpretation of the poleassignment problem as an intersection problem on Grass (p, m + p).

We shall work

over the real field. Explicitly, given G(s) and a set of desired, distinct real poles si, i = 1 , . . . ,n, to say

tt

¢ = i~1o~(~) c CrassR(p,m + p)

(5.21)

is to say the open sets U¢ = Grassrt(p, m + p) - art(Si) cover GrazsrL(p, m + p) n

u v~ = C r a ~ ( p , m + p).

(5.2z)'

i=1

As we have seen in section 5.2, the complement of a Schubert hypersurface is an open set, diffeomorphic with Mm,p(It) -~ It mp and hence contractible. In particular, to say there is no real output feedback law placing the poles at si is to say that one can cover

72 Gra~sR(p,m + p) by n open subsets, contractible in Grassa(p,m + p). This simple observation is quite powerful in conjunction with general position arguments, such as Lemma 5.7. In their study of global methods in the calculus of variations, Ljusternik and ~nirel'mann [60] discovered an important invariant of smooth manifolds: Definition 5.15. Suppose X is a smooth n-manifold. The Ljusternik-~nirel'mann category of X, denoted by L-S-cat (X) is the minimum cardinality of an open cover of X N

UUi=X

i=1

by open sets Ui which are contractible in X. Using a generalization of the General Position Lemma to the case rnp > n, one can prove [61] a generic pole-assignment result; placing poles at generic real frequencies. In particular, one h a a generic stabilizability result. T h e o r e m 5.16 [61]. Fix m , n and p. The generic system can be stabilized by real output feedback provided km,p = L-S-cat (CrassR(p, m + p)) _> n + 1.

(5.22)

Since L - S - cat (X) _< dim X + 1, (5.22) reflects the necessary condition given in Theorem 4.13 for generic stabilizability. Indeed, one can always assert m W p - l ~_ km,p <_ mp since, according to Eilcnbcrg [621, L-S-cat (X) can be bounded from below topologically; in fact, by the maximum number of terms in a nontrivial product in any cohomology ring of X. Not surprisingly, for X = Grassrt(p, m + p), this can be estimated in terms of the Schubert calculus, although over 1l the combinatorics becomes fairly involved (see [63]-[65]). Before turning to the problem of pole-assignment, we give several examples of what can be deduced from (5.22). First, define the integer s by 2 s < m + p ~ 2s+l

(5.23)

C o r o l l a r y 5.17 [61]. I f m i n ( m , p ) = 2, then max(m, p) + 2 s -- 1 :> n implies generic stabilizabUity. C o r o l l a r y 5.18 [61]. If min(m, p) = 3, each of the following conditions imply generic stabilizability (i) r n + p = 2

s+l-2 r+l

and2 s+2-3(2 r-1)-4_~n,

73 (ii) r a + p = 2

s+l-2 r+2+t

(iii) r e + p = 2

s+l and2 s + 2 - 5 > n

whereO
r-l-2and2

~+2-3(2 r-1)-2+t>n;or

C o r o l l a r y 5.19 [61]. I f min(m, p) = 4, each o f the following conditions i m p l y genetic stabilizability

(i) r e + p = 2

s+land2

s+1+2 s-7>n;or

(ii) rn+p = 2 s + 2 r q-jq-1 where s > r > O, 0 <<_j < 2 r - 1 and 2 s+l + 2 s + s r+l + j - 7

> n.

There is a slightly more refined invariant which plays a similar role in the problem of pole assignment by real o u t p u t feedback. If f:X-~Y

is a continuous map, the category of f , denoted by cat ( f ) , is defined as the minimum cardinality of an open cover (Ui) of X such that f[tr~ is homotopic to a constant. If f = i x , the identity map on X, then of course cat (ix) = L-S cat (x) and in gcneral one

has cat (f) < L-S cat (X). An argument as before [66] shows that (5.21) implies c. ,p = eat (Pro,p) < m p

(5.24)

where Pra,p is the Pl[tckcr imbedding [m4"p~ --I

Pra,p : GrassR(p, m + p) ---* ItPt r J

Theorem 5.20 [66]. Fix m, n an d p. cra,v > n implies generic pole assignability by real output feedback.

As before one can estimate crn,p from below using Schubcrt calculus; see [63]. Explicitly, with ~ as defined in (5.23) set , era,p =

2 +' - 1 2 '+l - 2 2s+l_l 2 k+l

if rain(re, p) = 2, max(re, p) # 2' - 1 if rain(m, p) = 2, m a x ( m , p) = 2s - 1 if m i n ( m , p ) = 3, m + p = 2K + l otherwise

C o r o l l a r y 5.21 [66]. F i x m, n and p. The condition c~,p' > 2 [ n - 1 / 2 ] + 1 _ is sufficient for pole assignment for the generic p × m s y s t e m of degree n.

C o r o l l a r y 5.22 (Brockett-Byrnes). If rnp = n, then the conditions min(m,p) = 1 or min(rn, p) = 2 and max(re, p) = 2 r - 1

74

are sufficient for pole-assignment of the generic p x rn system of degree n. C o r o l l a r y 5.~3 (Kimura). / f r o - b p - 1 _> n, thon the generic p x rn system of degree n can be arbitrarily pole assigned. Bibliography [1] G. B. Airy, "On the regulator of the clock-work for effecting uniform movement of equatoreals," Memoirs of the Royal Astronomical Society 11 (1840) 249-267. [2] G. B. Airy, "Supplement to the paper 'On the regulator of the clock-work for effecting uniform movement of equatoreals~'" Memoirs of the Royal Astronomical Society 20 (1851) 115-119. [3] J. C. Maxwell, "On governors," Proc. Royal Society London 16 (1868) 270-283. [4] R. E. Kalman, "Algebraic geometric description of the class of linear systems of constant dimension," 8th Annual Princeton Conference on Information Sciences and Systems, Princeton, N J, 1974. [5] M. Hazewinkel and R. E. Kalman, "On invariants, canonical forms and moduli for linear, constant, finite-dimensional dynamical systems," in Proc. of CNR-CISM Syrup. on Alg. Sys. Th., Udine (1975), Lect. Notes in Econ. Math. Sys. Th., Vol. 131, Springer-Verlag, Heidelberg, 1976, 48-60. [6] J. M. C. Clark, "The consistent selection of local coordinates in linear system identification," Proc. JACC, Purdue U., 1976. [7] C. I. Byrnes, "On the moduli space for linear dynamical systems," in Proc. of 1976 NASA-Ames Conference on Geometric Control Theory, (C. F. Martin and R. Hermann, eds.) Math. Sci. Press, 1977, 229-276. [8] E. J. Davison and S.-H. Wang, "Properties of linear time-invariant multivariable systems subject to arbitrary output and state feedback," IEEE Trans. Aut. Control 18 (1973) 24-32. [9] B. D. O. Anderson, N. K. Bose and E. I. Jury, "Output feedback stabilization and related problems-solution via decision algebra methods," IEEE Trans. Aut. Control, AC-20 (1975) 53-66. [10] I. Postlethwaite and A. G. J. Mac Farlane, ~A complex variable approach to the analysis of linear multivariable feedback systems," Lecture Notes in Control and Inf. Sciences 12 Springer-Verlag, New York 1979. [11] R. Hermann and C. F. Martin, "Applications of algebraic geometry to system theory-Part I," IEEE Trans. Aut. Control, A C - 2 2 (1977) 19-25.

75 [12] C. F. Martin and R. Hermann, "Applications of algebraic geometry to system theory: The McMillan degree and Kronecker indices as topological and holomorphlc invariants," S I A M J. Control 16 (1978) 743-755. [13] It. Hermann and C. F. Martin, "Applications of algebraic geometry to system theoryl Part II: Feedback and pole-placement for linear Hamiltonian systems," Proc. of IEEE 65 (1977) 841-848. [14] H. Kimura, "Pole assignment by gain output feedback," IEEE Trans. Aut. Control AC-20 (1975) 509-516. [15] J. C. Willems and W. H. Hcssclink, "Generic properties of the pole-placement problem," Proc. of the 7th IFAC Congress (1978) 1725-1729. [16] R. W. Brockett and C. I. Byrnes, "Multivariable Nyquist criteria, root loci and pole placement: A geometric viewpoint," IEEE Trans. Aut. Control A.C-26 (1981) 271-284. [17] T. E. Djaferis, "Generic pole assignment using dynamic output feedback," In~. J. Control 37 (1983) 127-144.

[18l

H. Seraji, "Design of pole placement compensators for multivariable systems," Automatica 16 (1980) 335-338.

[19] P. K. Stevens, A1gebro-Geometrie Methods for Linear Multivariable Feedback Systems, Ph.D. Dissertation, Harvard University, 1982. [20] M. Vidyasagar and N. Viswanadhan, "Algebraic design techniques for reliable stabilization," IEEE Trans. Aut. Cont. A C - 2 7 (1982) 1085-1095. [21] B. K. Ghosh, "Simultaneous stabilization and pole-placement of a multimode linear dynamical system," Ph.D. dissertation, Harvard Univ., 1983. [22] M. L. J. Hautus, "Stabilization, controllability, and observability of linear autonomous systems," Proc. Kon. Nederl. Akadamie van Wetensehappen-Amsterdam, Series A, 73 (1970) 448--455. [23] J. Rosenthal, "Tuning natural frequencies by output feedback," Computation and Control (K. Bowers and J. Lund, eds.) Birkh~.user, Boston, 1989. [24] A. S. Morse, W. A. Wolovich and B. D. O. Anderson, "Generic pole assignment: Preliminary results," Proc. 20th IEEE Conf. on Decision and Control, San Diego, 1981. [25] C. I. Byrnes and B. D. O. Anderson, "Output feedback and generic stabilizability," S I A M J. on Control 22 (1984) 362-380.

76 [26] D. F. Delchamps, State Space and Input-Output Linear Systems, Sprlnger-Verlag, New York, 1988. [27] C. I. Byrnes, "On the control of certain infinite dimensional systems by algebrogeometric techniques," Amer. J. Math. 100 (1978) 1333-1381. [28] F. M. Brasch nnd J. B. Pearson, "Pole placement using dynamic compensation,~ IEEE Trans. Aut. Control AC-15 (1970) 34-43. [29] C. I. Byrnes and P. K. Stevens, "Pole placement by static and dynamic output feedback," Proe. of 21st IEEE Conf. on Dec. and Control Orlando, 1982. [30] R. 1l. Bitmead, S. Y. Kung, B. D. O. Anderson and T. Kailath, "Greatest common divisors via generalized Sylv~ter and Bezout matrices," IEEE Trans. Aut. Contr. A C - 2 3 (1978) 1043-1047. [31] D. Youla, J. Bonglorno and C. Lu, "Single loop feedback stabilization of linear multivariable dynamic plants," Automatica 10 (1974) 159--173. [32] R. Saeks and J. J. Murray, "Fractional representation, algebraic geometry, and the simultaneous stabilization problem," IEEE Trans. Aut. Control, A C - 2 7 (1982), 895-903. [33] B. K. Ghosh and C. I. Byrnes, "Simultaneous stabilization and simultaneous poleplacement by non-switching dynamic compensation," IEEE Trans. Aut. Control AC-28 (1983) 733-741. [34] B. K. Ghosh, "Transcendentaland interpolationmethods in simultaneousstabilization and simultaneouspartial pole placementproblems," SIAM J. Control and Opt. 24 (1986) 1091-1109. [35] E. J. Davison and S.-II. Wang, "On pole-assignmentin linear multivariablesystems using output feedback," IEEE Trans. Aut. Contr. AC-20 (1975) 516-518. [36] R. Rado, "A theorem on independencerelations," Quart J. Math 13 (1062) 83-80. [37] C. L Byrnes and P. K. Stevens, "Global properties of the root-locus map," in Feedback Control of Linear and Nonlinear Systems (D. Hinrichsen and A. Isidori, eds.), Springer-Verlag Lecture Notes in Control and Information Sciences 39, Berlin, 1982. [38] D. Mumford, Algebraic Geometry I: Complez Projective Varieties, Springer-Verlag, NY, 1976. [39] I. R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, NY, 1974. [40] X.-C. Wang, "Geometric inverse eigenvalue problems," Computation and Control

77 (K. Bowers and J. Lund, eds.) Birkh~.user-Boston, 1989. [41] X.-C. Wang, Ph.D. dissertation, Arizona State University, 1989. 142] G. D. Forney, "Minimal bases of rational vector spaces with applications to multivariable linear systems," SIAM J. Control 13 (1975) 493-520. [43] C. I. Byrnes, "Algebraic and geometric aspects of the analysis of feedback systems," in Geometric Methods in Linear Systems Theory (C. I. Byrnes and C. F. Martin, eds.), D. Reidel, Dordrecht 1980, 85-124.

[441

C. Giannakopoulis and N. Kareanias, "Pole assignment of strictly proper and proper linear systems by constant output feedback," Int. J. Control 42 (1985), 543-565.

[45] N. Karcanias and C. Giannakopoulis, "Grassmann matrices, decomposability of multivectors and the determinantal assignment problem," Linear Circuits, Systems and Signal Processing: Theory and Applications (C. I. Byrncs, C. F. Martin and R. Sacks, eds.), North-Holland, 1988, 307-312. [46] D. Mumford, Introduction to Algebraic Geometry, Harvard Univ., 1964. [47] N. Jacobson, Lectures in Abstract Algebra Vol. I, Van Nostrand, New York, 1953. [48] C. Delzcll, Ph.D. Disscrtation, Stanford Univ., 1980. [49] B. Kostant, "Lie group representaLions on polynomial rings," Amer. J. Math. 85 (1963), 327-404. [50] J. M. Schumadmr, "Almost stabilizability subspaces and high gain feedback," IEEE Trans. Aut. Control A C - 2 9 (1984), 620-629. [51] H. Schubert, "Anzahlbcstimmungen ffir lineare R~ume beliebiger Dimension," Acta Math 8 (1886), 97-118. [52] P. Griffiths and J. Harris, Principles of Algebraic Geometry, J. Wiley and Sons, New York, 1978. [53] S. L. Kleiman, "Problem 15. Rigorous foundation of Schubert's enumerative calculus," Proc. of Syrup. in Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, 1976. [54] F. J. Drexler, "A homotopy method for the calculation of all zeroes of zero dimensional polynomial ideals," in Continuation Methods (H. Wacker, ed.), Academic Press, NY, 1978, 69-93. [55] Continuation Methods (It. Wacker, ed.), Academic Press, NY, 1078. [56] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers 5th ed.,

78 Oxford, 1979. [57] E. Artin, Galois Theory, Univ. of Notre Dame Press, Notre Dame, 1971. [58] S. Lang, Algebra, Addison-Wesley, Reading, MA, 1971. [59] J. Harris, "Galois Groups of Enumerative Problems," Duke Math. J. 46 (1979), 685-724. [60] L. Ljusternik and L. ~nirel'mann, Mdthodes Topologiques dans les Probl~mes Vari. ationnels, Hermann, Paris, 1934. [61] C. I. Byrncs, "On the stabilizability of multivariable systems and the Ljusternik~nirel'mann category of real Grassmannians," Systems and Control Letters 3 (1983), 255-262. [62] S. Eilenberg, "Sur un th6or~me topologique de M.L. Schnirelmann," Mat. Sb I (1936), s57-s59. [63] H. I. Hiller, "On the height of the first Stiefel-Whitney class," Proc. Amer. Math. Soc. 79 (1980), 495-498. [64] R. E. Stong, "Cup products in Grassmannians," Topology and its Applications 13 (19s2), lO3-113. [65] I. Berstein, "On the Ljusternik-Schnirel'mann category of Grassmannians," Math. Proc. Camb. PhIL Soc. 79 (1976), 129-134. I66] C. I. Byrnes, "Control theory, inverse spectral problcms, and real algebraic geometry," Diff. Geom. Methods in Control Theory (R. W. Brockett, It. S. Millman and H. J. Sussmann, eds.) Birkh£user, Boston, 1982.

Zeros, Poles and Modules in Linear System Theory G. Conte Dip. Matematica, Universit& di Genova Via L. B. Alberti 4, 16132 Genova, Italy

A. M. Perdon Dip. Metodi Modelli e Malematici per le Scienze Applicate Universit& di Padova Via Belzoni 7, 35100 Padova, Italy

INTROD U C T I O N

The aim of this paper is to present an account of the module theoretic approach to the notions of pole and of zero of a linear transfer function and to show its naturalness and usefulness. The motivations for developing this approach arise from noticing that tile classical definitions of pole and zero are not "rich" enough to allow a deep, satisfactory analysis of certain design problems. On tile other hand, the notion of pole module and of zero module, which will be described in the paper, provide a powerful algebraic tool for investigating the behavior of poles and zeros in many interesting situations. The definition of zeros of a multivariable systems is originally due to H. Rosenbrock ([20]), whose fu,~damental work stimulated in the past two decades a great number of researches on such concept (a review of notable publications on this topic in the period 1970-1987 can be found in [21]).The idea of defining a module which captures tile concept of zero of a multivariable transfer function was introduced by B. Wyman and M. Sain in [27]. In that paper the zero module was employed to study the relationship between the zero structure of an invertible transfer function and the pole structure of its inverses. The result was a satisfactory description of such a relationship, which agrees with the basic intuition developed in the scalar case or in the square and invertible one. The original treatment considered only the finite zeros and poles and was extended to include poles and zeros at infinity by the authors (see [4]), by introducing the notions of infinite pole module and infinite zero module. Subsequently, various applications to transfer function equations ([5,91) and to particular control and design problems ([6,7,8,28,29]) have been developed (see also [3,13,31,32]). The paper is organized as follows. Section I contains some preliminaries and notations. The algebraic notions which will be used throughout the paper are very briefly recalled, emphasizing only the features we are interested in. In Section 2, after a review of the classical definitions of multivariable finite pole and zero of a

80 transfer function, we discuss two examples concerning the inversion problem and a more general transfer function equation. We show that a complete understanding of the relationship between the pole/zero structure of the data and the pole/zero structure of the solutions of the inversion problem or of the considered transfer function equation cannot be obtained, except in the scalar case or in the "trivial" nonsingular one, by means of the classical notions. Therefore we introduce in Section 3 two algebraic objects, uamely the finite pole module X(T) and the finite zero module Z(T), associated with a transfer function T(z), which contain more information than the classical concepts of pole and zero. The notion of pole module can be easily described by means of the state module of the minimal realization of the considered transfer function. The notion of zero module, although quite natural, is less familiar and deserves to be investigated more deeply. We describe tile way in which it is related to tile transmission blocking properties and to the controlled invariant subspaces of the minimal realization. In particular, we have that the zero module Z(T) of a strictly proper transfer function T(z), with minimal realization (X,A,B,C), is naturally isomorphic to the quotient V*/P,,*, provided with the module structure over tile ring of polynomials induced by (A+BF), where V* is the the maximum controlled invariant subspace contained in Ker C, R'is the maximum controllability subspace contained in Ker C and F is such that (A+BF)V*c V*. In Section 4, the algebraic tools we have developed are applied to tile study of the transfer function equation T(z) = lt(z)G(z), where T(z) and G(z) are two given transfer functions with the same input space and I l(z) is sought. This equation generalizes the examples discussed in Sec.2 and our aim is to clarify the rehttionship between the pole/zero structure of the involved transfer functions by employing the notions of pole module and of zero module. In particular, we want to know if there exists an "essential" pole structure which appears in any solution H(z). Moreover, in the affirmative case, we are interested in obtaining a description of it in terms of the pole/zero structure of T(z) and G(z) and in investigating the existence of solutions all of whose poles are essential. The relevance of the above questions in a design problem whose solution can be characterized as a solution H(z) of T(z) = H(z)G(z) (e.g. inversion problem, factorization through G(z), model matching) is clear, since the pole structure determines the dynamical properties of the minimal realization of a transfer function (e.g. its stability or dead-beat properties). The main result we obtaiu says that there exists a module P, described abstractly in terms ofT(z) and G(z), which is naturally contained in the pole module of any solution H(z). The invariant factors of P characterize the essential part of the pole structure of any solution H(z). More concretely, P turns out to consist, in an algebraically meaningful sense, exactly of the poles of T(z) which are not poles of G(z) and of the zeros of G(z) which do not appear as zeros of iT(z)t G(z)t] t. A simple procedure to compute tile invariant factors of P in terms of coprime fractional representations of T(z) and G(z) is described. Then we show by construction that there exist essential solutions H(z), i.e. solutions all of whose poles are essential. In Section 5, we briefly describe the extension of this approach to the case of poles and zeros at infinity and to the case of poles and zeros in a specific region of interest. This provides useful results for the design problems we have mentioned above and gives us the algebraic tools, represented by the infinite pole module and the infinite zero module, for obtaining a complete picture, including finite poles and zeros and poles and zeros at infinity, of the classical pole/zero structure of a linear transfer function. Further constructions are however needed in order to get a complete understanding of the relations between the total number of poles and the total number of zeros of a transfer function. Therefore, following

81 [33], we introduce, in Section 6, new spaces associated with T(z) whose dimensions measure the difference, called defect, between the total number of poles and the total number of zeros when T(z) fails to be invertible. Such spaces, which are finite dimensional K-vector spaces without any natural or canonical module structure, can be interpreted as spaces of generic zeros, where generic means not located at any finite point nor at infinity, related to the kernel and cokernel ofT(z). In this way, we can say that the total number of poles of any transfer function equals the total number of zeros, when everything is counted properly, that is when also the generic zeros are counted. Moreover, this result is stated in terms of existence of maps of algebraic objects and of exactness of a sequence, rather than in terms of equality of numbers. It therefore refines previous results which express the defect of T(z) in terms of Kronecker indices or Wedderbum numbers.

1. P R E L I M I N A R I E S AND NOTATIONS Let K be a field and let K[z] and K(z) denote respectively the ring of polynomials and the field of rational functions in the indeterminate z over the field K. Given a K-vector space V = K n, we denote by V(z) the K(z)-vector space V ®K K(z), whose elements are n-tuples with entries from K(z), and we denote by E2V the K[z]-module V ®K K[z], whose elements are n-tuples with entries from K[z]. f2V is, i,~ an obvious way, a K[zl-submodule of V(z) and we denote by FV the quotient K[z]-module V(z)/f2V. Any clement y(z) in FV can be uniquely represented as an n-tuple of strictly proper rational functions whose formal Laureat series expansion has the form v(z) = Ei= 1. . . . vt z-t , with v t ~ V. The K[z]-module structure is given, in this representation, by the usual product followed by the deletion of the polynomial part. We assume that th.e reader is familiar with the theory of modules over a principal ideal domain, more specifically over K[z], and with the algebra of polynomial matrices (see, e.g. [ 1,14,18 ]). In particular, we will make use of the notions of Smith form of a polynomial matrix, torsion submodule of a K[z]-module, inv~riant factor~ of a finitely generated torsion K[z]-module, and short exact seouences of K[z]-modules a~2dmorphisms, which are briefly recalled below. Smith form Given a pxm polynomial matrix M(z) (more generally a matrix with entries in a principal ideal ( d i a g { p l ( z ) ..... pr(z) } 0) domain), it is possible to factor M(z) as M(z) = Bl(z ) 0 0 B2(z) , where B l(z), B2(z) are polynomial unimodular matrices (that is : polynomial nonsingular matrices with a polynomial inverse, or, equivalently, with constant nonzero determinant), r = rank M(z), the pi(z)'s are monic polynomials such that pi(z) I Pi+l(Z) (pi(z) divides Pi+l(Z) ) for i = 1..... r. The pi(z)'s are uniquely dctennincal and the matrix blockdiag( diag (pi(z)},0 } is called the Smith foma of M(z). Torsion submodule - lnvariant faelors If we consider the morphism M(z) : ~ U ~ ~ Y induced between I2U and V2Y, for U=Km and Y=KP, by the polynomial matrix M(z) with respect to the canonical bases, we have that the factorization of M(z) related to the Smith fore1 induces a canonical isomorphism between the quotient module f2Y/MX"2U and the direct sum ~ i = l , r K[z]/Pi(z)K[zl ~) (K[z]) p'r- The submodule of OY/ME,~U isomorphic to ~i=l,r K[z]/Pi(z)K[z] is the torsion submt'xhde tK[zI(~Y/MX"2U)of E2Y/Mf2U.

82 It is characterized by the fact that any element in it is annihilated by a suitable element of K[z] (in this case a factor of the product of the pi(z)'s ). The polynomials pi(z)'s, which are the dements of the Smith form of M(z), are called the invariant factors of the module tK[z](f~Y/Mf~U). More generally, any finitely generated module M over K[z] has a unique direct sum decomposition of the form ~ i = l,q K[z]/pi(z)K[z] ~ (K[z])s, where the pi(z)'s are monic polynomials such that pi(z) I Pi+l~Z). Two modules are isomorphic iff their decompositions are equal. In particular, if M is a finitely generated torsion module, in particular if M consists of a finite dimensional K-vector space provided with a K[z]-module structure, we have s--0 and, as an abstract module, M is completely characterized by its invariant factors pi(z). Exact seauence Given a sequence of K[z]-modules and morphisms M 1 - - f---) M 2 - - g --~ M 3 we say that the sequence is exact at M 2 if Im f coincides with Ker g. In the following we will be interested in sequences of the form 0

) M 1 - - f--~ M 2 - - g --~ M 3

)0

which arc exact at M1, at M 2 and at M 3. Such a sequence is called a short exact seauence. Exactness of the sequence at M 1 is equivalent to f being injective and exactness of the sequence at M 3 is equivalent to g being surjective. Together with exactness at M2, this means that M 1 is isomorphic to Ker g and that M 3 is isomorphic to M2/f(M1). Taking into account only the underlying structure of K-vector space of the modules M1, M 2 and M 3 , the fact that the short sequence is exact implies that, as a K-vector space, M 2 is isomorphic to the direct sum M 1 • M 3 _=M 1 • M2/f(MI). We will make use of the following result: given the diagrmn of modules and module morphisms il A $ f

Pl >B

(1)

> B/A

.l,g

A'

) B'/A'

i2

P2

where il, i2 are inclusions and Pl, P2 are canonical projections, if (1) is commutative, i.e. g i I = i2 f, then there exists an induced map Ii: B/A ~ B'IA' such that h Pl = P2gTransfer function - Fractional reoresentations Given two K-vector spaces U = K m and Y = KP, by a transfer fimction we mean a K(z)-linear map T(z) : U(z) ---) Y(z) or, equivalently, tile p x m matrix of rational functions which represents T(z) with respect to the canonical K(z)-basis of U(z) and Y(z). By a left (resp. right) coorime fractional renresentatlon of T(z) we mean a representation of the form T(z) = D-l(z)N(z) (resp. T(z) = N l ( z ) D l - l ( z ) ) , where D(z), N(z) (resp. D l ( z ) , N l ( z ) ) are polynomial matrices whose common left (right) factors are unimodular and D(z) (Dl(z)) is nonsingular. Rczdization diagram We assume that the reader is familiar also with the algebraic realization theory of T(z), or with the so-called polynomial model approach, described in [12,15,30]. In particular, we will frequently

83 make reference to the i-¢alization diagram, whose construction is described in detail in [2], induced by any coprimc left fractional representation T(z) = D -1 (z)N(z):

N(z) T(z)

D(z)

U(z)

) Y(z)

jT

$~. T#

1.1

> Y(z)

f2 U

D(z) ~ 1-'Y=Y(z)/f2Y

U

)

X=Ker D

B

$~.

A

) FY

) Y C

The notations arc as follows: i andj are canonical inclusions, g is the canonical projection, p is defined by p( Zt= 1. . . . Ytz-t ) = Yl; T#: f2U --~ r'Y is defined by T ~ = ~_ T j , D : FY --~ FY is defined as the action of D(z) followed by the deletion of the polynomial part; Ker D, which consists of the strictly proper clements of Y(z) whose image by D(z) is polynomial, is a finitely gencratcxi torsion K[z]-modulc, hence it is a finite dimensional K-vector space ; : Ker D -~ 1-'Yis the inclusion and B : flU ~ Ker D is uniquely determined by the condition (2 B =

forgetting the K[z]-module structure, we denote by X the underlying K-vector space (the equality X = Ker D in 1.1 is an abuse of notation, since it concerns only the K-vector space structure); B : U ~ X is defined by B = B i , C : X --4 Y is defined by C = p ~ , A : X --~ X is defined using the K[z]structure

of Kcr D by Ax = z x.

It is well known that the dynamical system (X,A,B,C) arising from the realization diagram above, whose evolution is described by the equations

84

{

x ( t ) = A x ( t ) + B u(t)

y(t)

C x(t)

or

(continous time)

x ( t + l ) = A x(t) + B u(t) y(t) = C x(t) (discrete time)

is a minimal realization of tile strictly proper part of T(z), which coincides with C(i;I-A)-IB. Remark that the K[z]-module structure of Ker D coincides with the one induced on the K-vector space X by z x = Ax. Denoting by (X,A) the K[z]-module consisting of X provided with the K[z]-structure induced by A, we therefore have that CX,A) is K[z]-isomorphie to Ker D. More generally, we have the following result: 1.2 PROPOSITION Let T(z) = D-I (z)N(z) = N 1(z)D 1" 1(z) be eoprime fractional representations and let (X,A,B,C) be a minimal realization of T(z). Then, we have the following K[z]-isomorphisms: (X,A) _-Ker D = f2Y,/Df~Y = f~U/DI f2U. Proof. The first two isomorphisms are proved in [2]. The last one follows, for instance, by [16] 6.5. In dealing with matrices and morphisms induced by matrices, we use a superscript "t" to denote the transposition. Since no confusion can arise, we use a superscript "-I" to denote both the counterimage and, when the matrix or the morphism is invertible, the inverse.

2. SOME E X A M P L E S AND MOTIVATIONS In this section we want to show that the classical concepts of poles and zeros of a multivariable transfer function are not rich enough to allow a deep analysis of some design problems. This motivates the introduction, in the next section, of the more structured notions of pole module and zero module. Let us start by recalling the classical definition of finite poles and zeros of the linear tr,msfer function T(z) (see, for example, [ 16] ). To this aim, we assume, without loss of generality, that the coefficient field K is algebraically closed, e.g. K = ~;. If T(z) describes the input/output behaviour of a single-input/singleoutput system, it can be written as T(z) = p(z)/q(z), where p(z) and q(z) are coprime polynomials. Then, if tx~ K is a root of q(z) of multiplicity Pet, we say that T(z) has a pole at tx of order Pot and, if 13is a root of p(z) of multiplicity pl3, we say that T(z) has a zero at 13of order pp. Roughly speaking, we can say that the poles of T(z) are the roots of its denominator and that the zeros of T(z) are the roots of its numerator. Actually, "the notion of zero, or of pole, consists of a datum, expressed by an element of K, which determines the location of the zero, or of file pole, and of a multiplicity. In the muhivariable case, the definitions are given by means of the Smith-McMillan form of T(z). Recall that T(z) can be decomposed as

T(z) = B 1 (z)

( d i a g { c t ( z ) / ~ l ( Z ) ..... Cr(Z)/~r(Z)} 0

0)132(z),whereB1(z) 0

and B2(z) ar e

polynomial unimodular matrices, r = rank T(z), el(Z) and Vi(z) are coprime monic polynomials such that el(Z) I el+ 1(z) and ~i+l (z) I ~i(z) for i = 1,2..... r. The elements ei(z ) and Vi(z) of the Smith-McMillan form are uniquely determined. Then, if ct e K is a root of the Vi(z)'s of multiplicities (Pl . . . . . Pr)tx, we say

85 that T(z) has a pole at ct of total order Epi and we call the r-tuple (-Pl . . . . . -Pr)ct the pole stmct~r~ ofT(z) at o~ ( remark that Pi > Pi+l and that some of the pi's may be zero). If o~ is a root of the ei(z)'s of multiplicities (61 ..... 6r)o~ we say that T(z) has a zero at ct of total order g o i and we call the r-tuple (o I, .... Or)0~ the zero structure ofT(z) at tx. Thus, the notion of multivariable zero, or pole, consists of a datum, expressed by an element of K, which determines the location of the zero, or of the pole, and of a string of multiplicities. Remark that, in the multivariable case, T(z) may have, at the same time, a pole and a zero at a given ~ We speak in this case of numerically coincident pole and zero. In the multivariable case, a characterization in terms of numerator and denominator,in a suitable sense, of T(z) is expressed by the following 2.1 PROPOSITION Let T(z) = D-l(z) N(z) = Nl(z) Dl-l(z) be coprime fractional representations. Then, the nontrivial elements in the Smith forms of D(z) and of Dl(z) coincide with the ~i(z)'s and the non trivial elements in the Smith forms of N(z) and of Nl(z) coincide with the ¢i(z)'s. Proof. See [16] Sect. 6.5. The dynamical interpretation of poles and zeros is well known (see [16] and also [34] and the references therein). Let us recall, in particular, that, assuming that T(z) is strictly proper and that (X,A,B,C) is its minimal realization, the poles ofT(z) coincide with the eigenvalues erA. Therefore, they characterize the free dynamics of the system and, for example, its stability or dead-beat properties. In this sense, for K = C, we will speak of unstable ( i.e., in continous time, right half plane ) poles and of stable ones. On the other hand, the zeros describe the transmission blocking properties of the system (X,A,B,C). If T(z) has a zero at ct, in fact, the system matrix

0 loses rank at z = ct. Then, there exists a vector

Go)

such

O

that(tt~ A " B ' ~ ( x ° ' ~ = 0 . This means, in the discrete time model, that, taking theinput sequenceu(t) =

ojt, uo)

~tuo for t=0,1 ..... then the response corresponding to this input and to the initial state xo is y(t)=0. In

(uo)

other terms, the state xo blocks the transmission of the input u(t). An analogous result holds in the eontinous time situation. The vector

o

is called in [19] a zero direction associated to

Moreover, if we denote by V* and by R* respectively the maximum controlled invariant subspacc of Ker C and the ma×imum controllability subspaee of Ker 12 of the geometric theory (see [24]), we have that the zeros coincide with the eigenvalues of the automorphism of V'/R* induced by A + BF, where F: X ~ U is any feedback such that ( A + BE ) V* c V* (such an F is called a "friend" of V*). The relevance of poles and zeros in the design of linear systems is illustrated in the following examples. They also show, by pointing out the differences between the scalar ease and the multivariable case, the inadeguacy of the classical notions we recalled above in providing a clear description of the bchaviour of multivariable poles and zeros in various interesting situations. .2.2 EXAMPLE - System inversion Many problems in control theory reduce ultimately to the inversion of a linear system. For an account of the literature on this, the reader is referred to [27]. Briefly, assuming that T(z) is a right or left inverdble transfer function, one is interested in the solutions, respectively, of 2.3

T(z) G(z) = I

or of

86 2.4

F(z) T(z) = I.

Clearly, one would like to know from the analysis of T(z) which are the design limitation concerning G(z) or F(z), or, in other terms, which are the dynamical features of the inverses. The analysis of the case in which T(z) is square and nonsingular, in particular when it is a scalar transfer function, presents no difficulty. In such a situation, in fact, there exists a unique inverse, whose pole structure coincides with the zero structure of T(z). This is obvious in the scalar case and follows easily in the multivariable case since, except for the ordering, the Smith-McMillan form of T-I (z) is the inverse of the Smith-McMillan form of T(z). In other terms, we have that when T(z) is square and nonsingular th¢ dynamics of its inverse is completely described by the zero structure ofT(z). When T(z) is not square, we have many inverses with different pole structures. Consider, for instance, the left invertible transfer function T(z) = [ z/(z+l) z/(z+2) ] t , whose Smith-McMillan form is [z/(z+l)(z+2) 0 It. Both the transfer functions Fl(z) = [ (z+l)(z+2)/z -(z+l)(z+2)/z ] and F2(z) =

[ z+l)/(z-l) -(z+2)/z(z-1) ] are left inverses of T(z), and their Smith-McMillan forms are given respectively by [ (z+l)(z+2)/z 0 ] and [ llz(z-l) 0 ]. Hence, F l ( z ) has a pole structure consisting of a pole at 0 of order 1, which coincides with the zero structure ofT(z), while F2(z) has also an unstable pole at 1 of order 1. In this situation, it is therefore natural to consider the following questions: i)

is there an "essendar' pole structure which appears in every inverse of T(z) ?

ii)

given an affirmative answer to i), does there exist an inverse all of whose poles are essential ? An answer to these questions would provide a characterization of the dynamical properties which are

necessarily shared by all the inverses. In particular, it would allow one to investigate the existence of, e.g., stable inverses or dead-beat inverses. Clearly, what we expect is that, in accordance with the scalar case and, more generally, the multivariable square nonsingular case, an essential pole structure does exist and coincides with the zero structure ofT(z). Roughly speaking, this is to say that the zeros ofT(z) are, in some suitable sense, poles of any inverse. Unfortunately, mainly because of the presence of the unimodular matrices Bl(z ) and B2(z) which are not uniquely determined, it is difficult to obtain a satisfactory answer to i) and ii) using the classical definitions we have seen. 2.5 EXAMPLE - Model matchin~ and factorizatioa oroblem A class of problems more general than the system inversion is described by means of the equation 2.6

T(z) = H(z) G(z)

2.7

T(z) = F(z) H(z)

or, dually,

where T(z) and G(z) are two given transfer functions with the same input space (respectively, T(z) and F(z) are two given transfer functions with the same output space ) and H(z) is soughL Such equations arise, for instance, in the model matching problem and in various factorization problems (see [5,9] and ~he references therein). In the case in which the existence of solutions H(z) is assured, we are interested, in order to satisfy the design requirements, in their poles, and we can again formulate the same questions as in 2.2 i) and ii) :

87 i)

is there an "essential" pole structure which appears in any solution H(z) of T(z) = H(z) G(z) (respectively T(z) = F(z) H(z) ) ?

ii) given an affirmative answer to i), is there any solution H(z) all of whose poles are essential ? Intuitively and roughly speaking, we expect that the essential part of the pole structure of any solution H(z) must supply the poles of T(z) which do not already appear in G(z), as well as the poles needed to cancel the zeros of G(z) which do not appear in [ T(z) t G(z)t ]t (note that it is necessary to compare the zeros of G(z) with those of [ T(z) t G(z)t i t , and not only with those of T(z), since some of the first may fail to appear as zeros of T(z) for non dynamical reasons, i.e. without being canceled by a pole of H(z), if T(z) is not full column rank). In fact, similarly to 2.2, this is what happens clearly in the scalar case. If T(z) = p(z)/q(z) and G(z) = p'(z)/q'(z) are coprime representations, then the unique solution of 2.6 or 2.7 has a representation H(z) = p(z)q'(z)/q(z)p'(z), which is not necessarily reduced. However, the poles of H(z) consist of the roots of q(z), i.e. poles of T(z), which do not already appear as roots of q'(z), i.e. poles of G(z), together with the roots of p'(z), i.e. zeros of G(z), which do not appear as roots of p(z), i.e. zeros of T(z). A straightforward generalization of these observations to the multivariable case, in which often the solution is not unique, is not possible. Numerator and denominator matrices, in fact, may not be invertible and, furthermore, numerically coincident poles and zeros do not necessarily cancel. Therefore, although some result on the location in K of the poles which are necessarily present in any solution H(z), called fixed poles, can be proved (see [25]), the infommtion on the whole pole structure as well as a complete answer to i) and ii) seems difficult to obtain. To close this section, we can conclude that, in the design problems mentioned in 2.2 atad 2.5, it appears difficult, if we remain inside the framework characterized by the classical definitions, to extend beyond the scalar case the results suggested by the basic intuition about poles and zeros.

3. POLE MODULE AND ZERO MODULE

In this section we introduce the notions of pole module and of zero module. Overcoming the inadequacy of the classical definitions, they will allow us to develop, in the next section, a satisfactory algebraic treatment of the previously mentioned problems. Since tile zero module is, in some sense, a concept less familiar than the pole module, we will spend more time in describing it and in comparing it with file objects of the geometric theory. To begin with, let us remark that, given T(z) with a minimal realization of its strictly proper part (X,A,B,C), the whole information on the poles of T(z) can be given by means of the K[z]-module (X,A), consisting of the vector space X with the module slructure induced by A. This, as we have recalled in 1.2, is isomorphic to I'2Y/D(z)f2Y and to X"2U/DI(z)X'-2U, where T(z) = D-l(z) N(z) = Nl(z) D l - l ( z ) are coprime fractional representations, and hence its invariant factors are, by 2.1, exactly the non trivial ~i(z)'s of the Smith-McMillan form ofT(z). Let us clarify this point: the key fact is that the information contained in a set of polynomials, in particular in the ~gi(z)'s, can be expressed by means of an abstract module having those polynomials as invariant factors. Hence, the poles can be computed directly from (X,A). In this sense

88 we can speak of the pole module associated to T(z). Furthermore, since the problems we are dealing with are expressed at a transfer function level without employing the realizations, we choose, for the pole module, the representation given by the K[z]-module X(T) = f l u / ( T ' I (E2Y) n flU). This is justified by the existence of an obvious isomorphism between (X,A) and X(T) which can be checked directly on 1.I. Formally, we have the following ~. ! DEFINITION The ffinite~ pole module of a transfer function T(z) : U(z) --~ Y(z) is the K[z]-module flU X(T) defined by X(T) = T_l.fly_ {. ) n flU To introduce the notion of zero module we follow the original treatment of B. Wyman and M. Sain which appeared in [27]. It is helpful to start by considering the scalar case, representing T(z) as T(z) = p(z)/q(z) with p(z) and q(z) coprime polynomials. Now, if the input u(z) e U(z) has a representation u(z) = a(z)Fo(z), with a(z) and b(z) coprime polynomials, and if b(z) and p(z) have nonunit factors in common, then the "modes" of u(z) represented by these factors fail to appear in the corresponding output y(z) = T(z) u(z) = p(z)a(z)/q(z)b(z) , because of cancellation. In other words, this means that the factors of the nunaerator, i.e. the zeros, can be revealed by looking at the modes of the inputs which fail to appear in the outputs. Henceforth, let us focus on the inputs which can produce no modes whatsoever in the response. In the scalar case we are considering, they are of the form u(z) = a(z)q(z)/p(z), for a(z) in K[z], and produce responses y(z) = a(z) e K[z] having no modes. These excitations, whose modes describe the zeros of T(z), are characterized by being elements of T-1 (K[z]). Generalizing to the multivariable case, we will take into account the elements of T- 1(fly). Now, remark that if u(z) is an element of K[z], no zero effect could be observed, si,lce u(z) has no modes which can fail to appear in the output. At the same time, note that identically zero output is of little interest in the above discussion, since what is important is the failure of certain exciting modes to appear in the response. Although this last point has little significance in the scalar case, when there are no non-zero inputs which can produce zero responses, it is important in the muhivariable generalization, where Ker T is not necessarily zero. Combining the above remarks, we have that, in the definition of an abstract module which captures the notion of zeros, the elements of Ker T and of flU, which generalizes K[zJ, can be safely neglected. This can be accomplished by forming an algebraic quotient modulo Ker T + flU. Thus, we are led to the following 3.2 DEFINITION The (finite} zero module of a transfer function T(z) : U(z) ~ Y(z) is the K[z]-module T-I(E2Y) + flU Z(T) defined by Z("I') = Ker T + f l U Remark that the addend flU in the numerator of the quotient is provided for consistency, so that the denominator is contained in the numerator. Clearly we have now to show that Z(T) contains, in particular, the whole information on the numerator polynomials £i(z)'s of the Smith-McMillan form ofT(z). This is proved by the following 3.3 PROPOSITION Let T(z): U(z) --->Y(z) have the coprime fractional representation T(z) = D'l(z) N(z). Then, the zero module Z(T) is isomorphic to the torsion submodule tK[z] (fIY/Nf2U) of f2Y/NflU.

89 The fundamental result stated by 3.3 says, by 2.1, that the invariant factors of Z(T) are exactly the nontrivial ei(z)'s of the Smith-McMillan form of T(z). Therefore, as for the pole module, we have an algebraic object described directly in terms of the transfer function which, in particular, contains the whole information on the zeros. Furthermore, the isomorphism with tK[z] ( f l y / Nf~U), which is a finitely generated torsion K[z]-module, displays the structure of Z(T), showing that the latter is finitely generated over K with dimension as a K-vector space given by Y-'ideg (ci(z)). The proof of 3.3 is given in [27] Th. I and will not be repeated here. However, as an example of the algebraic techniques which will be used in the following, it is interesting to remark that the isomorphism between Z(T) and tK[z]( f l y / NflU ) can be defined by the map induced (dotted arrow) in the diagram below by the commutativity of (1) (see Sec. 1) Ker T + ~ U

) T" l(flY) + flU

) Z(T) I

N~

(1)

N flU

N$ >

flY

>~Y/NflU

~.4 REMARK Before going further, let us remark explicitly that Z(T) and X(T) contain more information than the sets { ci(z) } and { Vi(z) }, since they are defined not simply as abstract modules with a given set of invariant factors, but using a specific representation in terms of input and output spaces. However, it is important to note that these representations are not intended and are not suitable for computational purposes. In fact, it may be very difficult to compute the pole module or the zero module by means of their definitions, except in the scalar case (see 3.5 below). Thus, to determine explicitly the pole structure and the zero structure we have to use the coprime fractional representations and the isomorphism between X(T) and ~Y/DDY and between Z(1) and tK[z](flY/NDU). On the other hand,the way in which the pole module and the zero module are defined and represented is an essential point in proving the results of the next Section. .q.5 EXAMPLE Let us compute the zero module Z(T) in the scalar case T(z) = p(z)/q(z). By definition, since Kcr T = 0, we have Z(T) = (T- l(K[z]) + K[z]) / K[z]. Now, T-I(K[zl) + K[z] = { a(z)q(z)/p(z) + b(z), a(z) and b(z) ~ K[z] } = { (a(z)q(z) + b(z)p(z) )/p(z), a(z) and b(z) • K[z] } = I/p(z) K[z]. The last step follows from the fact that p(z) and q(z) are relatively prime, so that any polynomial r(z) in K[z] can be written as r(z) = a(z)q(z) + b(z)p(z) for suitable a(z) and h(z) in K[z]. Therefore we have Z(T) = (l/p(z) K[z])/K[z]. Remark that there exists a K[z]-module isomorphism between (1/p(z)K[z])/K[z] and K[z]/p(z)K[z] given by [r(z)/p(z)]mo d K[z] -'* [r(z)]mod p(z)K[z]. Thus we obtain the expected result, namely Z f I ) _= K[z]/p(z)K[z]. In addition to 3.3, which states technically the relationship between Z(T) and the classical notion of zero, there is a natural connection between the concept of zero module and the transmission blocking properties of the minimal realization of T(z), when T(z) is strictly proper. To clarify this, let us write, for any u(z) • U(z), u(z) = Upo1 + Usp, where Upo 1 ~ flU is the polynomial part of u(z) and Usp is its strictly proper part. Now, any nontrivial element in Z(T) has the fore1 [u(z)] ,where T(u(z)) is a nonzero element in flY. The fact that T(z) is strictly proper implies that y(z) = T(usp) is strictly proper and, since T(upol) = T(u(z)) - y(z), we have that UpoI is nonzero. The polynomial input UpoI sets up a state x o = B(upo 1) of the minimal realization ofT(z) (see 1.1), and the output corresponding to the initial condition x(0) = x o and to

00 the input sequence Usp is given by 0-.(Xo) + ~_T(usp) = x_T(u(z)) -- 0. In other words, the state xo blocks the transmission of the input sequence Usp and we can say that any non trivial element in Z(T) determines at least one pair (state, input sequence) with such a property. Comparing with the comments following 2.1, dfis shows that the zero module captures also the notion of zero direction described in [19]. By the previous discussion, it turns out in particular that the state xo = B(upo I) belongs to Ker C and has the property of being weakly unobservable, that is : there exists an input sequence, in this case Usp, such that the corresponding output, with initial state x(0) = xo, is zero. It is known that the set of weakly unobservable states coincides with the maximum controlled invariant subspace V* contained in Ker C of the geometric theory [24]. Henceforth, we have a slightly different interpretation of the situation described above which point out a connection between ZfO and V*. This is made precise in the following : 3.6 PROPOSITION Let T(z): U(z) ~ Y(z) be a strictly proper transfer function with minimal realization (X,A,B,C). Denote respectively by V* and R* the maximum controlled invariant subspace of Ker C and the maximum controllability subspace contained in Ker C and let F: U ~ X be such that (A + BF) V* c V*. Then, there exists a natural K[z]-isomorphism between Z(T) and V*/R* provided with the K[z]-module structure induced by A + BF. Two slightly different proofs of 3.6 are given in [31] Th.1 and in [3] Sect.4 (compare also with [23]). Itere it is important to remark that 3.6 does not simply state the existence of an isomorphism between Z(T) and (V'/R*, A + BF ) as abstract modules. This, on the other hand, is already known, since the invariant factors of the two modules are the same. The emphasis, in 3.6, is on the existence of a natural isomorphism, where "natural" means induced directly by the realization diagram 1.1. Actually, such an isomorphism is induced by the correspondence [u(z)] = [Upo1 + Usp] --¢ B(upo 1) we have already discussed. Alternatively, its inverse can be characterized as follows. For v e V*, let a(z) e ~ U be such that B(a(z)) = v and let b(z) = El= 1. . . . F(A + BF)i-l(v) z"i, where F is such that (A + BF)V*c V*. It is possible to show that (a(z) + b(z)) ¢ T-I(12Y) and that the correspondence v --~ [a(z) + b(z) ] ~ Z(T) between V* and ZfO is well defined and induces the isomorphism we are speaking of.

4. FIXED POLES OF TRANSFER FUNCTION EQUATIONS We have now the algebraic tools needed to tackle the problems stated in 2.2 and 2.5 and to give an answer to i) and it). In particul~Lrwe focus on the equation 2.6, namely T(z) =H(z) G(z) where T(z) and G(z) are given, and, when a suitable necessary and sufficient condition for the existence of solutions H(z) is satisfied, we provide a description of the essential part of the pole structure of any solution. The equation 2.7 is treated in detail in [9]. The inversion problem of 2.2 is obviously a particular case of 2.6 and 2.7 and has been investigated in [27,31]. A somewhat less satisfactory treatment, if compared with that of [9], of 2.6 and 2.7 is contained in [5]. Before going further, let us recall that the equation 2.6, where T(z) : U(z) ~ Y(z) and G(z) : U(z) ~ W(z) are given, has solutions H(z) : W(z) --~ Y(z) if and only if (A)

KerG c KerT .

91

Condition (A) simply means that, if an input produces zero response through G(z), then it produces zero response also through T(z). In the sequel we assume that the pair of transfer functions T(z) and G(z) we are dealing with, verifies the condition (A). The approach we will follow in this Section can be summarized in the following way. First, we define abstractly, in tenns of the data T(z) and G(z), a K[z]-module P which is shown to be contained in the p01e module X(t 1) of any solution H(z). Therefore, P represents the essential pole structure of the solutions of 2.6. Then, we prove that P can be described in terms of poles and zeros of T(z) and G(z) in a way which agrees with the basic intuition explained in 2.5. Given the equation 2.6 and assuming that (A) holds, let us consider the K[z]-module P defined by G-1(f2W) P = G- 1(f2W) n T- 1( O y ) " The following proposition justifies the introduction of P. 4.1 PROPOSITION For any solution H(z) of 2.6 there exists a natural inclusion j : P ~ X(H). Proof. The map j : P --o X(H) is the map induced (dotted arrow) in the diagram below by the commutativity of (1). G-I(f~W) n T-I(~Y)

.I.G

) G-I(~w)

(1)

.I.G

)P

j

t

,

4-

f2W n H-I(~Y)

~

I'2W

) X(H)

Assumi~lg that G(T-1 (X'-2Y))is contained in H-1 (~y), we have used implicitly the fact that H(z) is a solution of 2.6, i.e. T(z) = It(z) G(z). It remains now to show that j is injeetive. For this purpose, let [u(z)] be an element of P such that j([u(z)] = 0. By definition, this implies that G(u(z)) belongs to ~ W n H-I(~Y), hence T(u(z)) = HG(u(z)) belongs to ~ Y . Thus, u(z) belongs to G-I(x"2W) c~ T-I(f2Y) and, as a consequence, [u(z)] = 0. By 4.1 we have immediately that the module P provides an answer to the question i) of 2.5. The invariant factors of P describe a pole structure which appears in any solution H(z). Therefore, we have the following definition (compare with [25]): G-I(f~W) 4.2 DEFINITION The module P = G _ I ( ~ w ) n T - I ( ~ Y ) is called the ~modul¢ 9f fixed pole~ of the equation 2.6. 4.3 EXAMPLE Let T(z) = I2, the 2 x 2 identity matrix, and let

G(z) =

(z/(z+l) 2 0 z/(z+l) 2

0 ) (z2+2z)/(z+l) 3 . It is easily seen that the columns of G(z) are independent, hence (z2+2z)/(z+ 1)3

Ker G = 0 and the condition (A) is trivially satisfied by T(z) and G(z). The equation 2.6 is, in this case, I = H(z) G(z) and actually what we have is an inversion problem. In particular, T-I(I2Y) = I2Y and P = G-I(f2W) and therefore, using a canonical isomorphism and the fact that Ker G = 0, we have, in G-I(f~W) c~ ~ Y

92 accordance with [27], P = G - I ( f w ) + f~Y = Z(G). Using 3.3 and the coprime fractional representation flY G(z) = D-1 (z)N(z) =

form of N(z) is

-(z3+3z2+3z) 0

0

z(z+2)

0

0

1 0

z3+3z2+3z (z+ 1)3

0 z(z+l)

z(z+2) z(z+2)

, since the Smith

, we obtain Z(G) = K|zl / zK[zl • K[zl/z(z+2)K[zl. Therefore, any left

inverse of G(z) must have a pole at 0 of total order 2 and structure (-1,-I) and a pole at -2 of total order 1 and structure (-1,0). 4.4 (~OMPI,JTATION OF P Before giving another example, let us show how to compute practically P, or better its invariant factors, in the general case. For this purpose, assume that the data T(z) and G(z) of 2.6 have coprime fractional representations T(z) = N(z)D-l(z), G(z) = NG(z)DG-1 (z) and let M(z) = D(z)A(z) = DG(z)B(z) be the minimum common left multiple of D(z) and DG(Z). The product by M(z) induces an isomorphism, described by [u(z)] ---¢[M(uz)], between the module

(NGB)-I(nw) (NGB)-I(f2W) n (NA)-I(K2Y)

and P.

Note that the multiplication by M(z) was the technique used in [5] in order to simplify the analysis of 2.6. Now, let S(z) be the maximum (nonsingular) common fight divisor of NG(z)B(z) and N(z)A(z), in particular let NG(z)B(z) = N'(z)S(z). Then, we have (NGB)-I (~W) n (NA)-I(flY) = S-I(f~U) and hence P - (N'S)-I(f2W)/S-I(fu). As before, the multiplication by N'(z)S(z) induces an isomorphism between P and ( ~ W n Im N') / N'(~L1). At this point it is easy to see that the last module is isomorphic to the torsion submodule tK[z]( ~ W [ N ' ( f U ) ) of f W [ N ' ( f U ) and that, as a consequence, the invariant factors of P are the nontrivial elements in the Smith form of N'(z). 4.5 EXAMPLE Consider the transfer functions

1/(2z+2)

-11(2z+2)

and G(z) =

l/z 1/(z2+z)

l/(z+l) 1/(z+l)

.

It is easy to see that both T(z) and G(z) are full rank, hence the condition (A) is trivially satisfied since Ker G = 0 and Ker T = 0. We compute the invariant factors of P, the module of fixed poles of the equation T(z) = H(z)G(z), using the procedure explained in 4.4. Coprime fractional representations for T(z) and G(z) are the following:

1

,DG(z ) =

The minimum common left multiple of D(z) and DG(Z) is given by M(z)= D(z)A(z)= DG(z)B(z) =

z3+3z2+2z 2z2+2z / 0 -2(z+l) '

.

93 with A(z)= ( z2/2z 0z+l ] and B(z) = ( z + 2 0 2 0 ) . Then, since the matrices

(z2+2z)/2

z+l

and NG(Z)B(z) =

z2+3z+2 z+2

2z 0

turn out to be right coprime, their maximum common right divisor S(z) is unimodular. Hence P_= ( N G B ) - I ( ~ w ) I ~ U = tK[z] ( £2W/NGB(~U ) ) and the invariant factors of P are the nontrivial elements in the Smith fonn of NG(z)B(z), which is given by

(10) 0 0

z(z+2) 0

. This means, in particular,

that any solution II(z) of 2.6 has a pole of total order 1 at 0 whose structure is (-1,0) and a pole of total order 1 at -2, whose structure is (-1,0/. That agrees with the fact that T(z) has a pole of total order 1 at -2 which does not appear as pole of G ( z ) , and G(z) has a zero of total order 1 at 0 which is not a zero of

Let us show that the abstract module P has a more concrete description in terms of zeros and poles of the data T(z) and G(z). Since our aim is to extend the intuitive interpretation we have in the scalar case, we need first to clarify what we mean by "the poles ofT(z) which do not appear in G(zy and by "the zeros of G(z) which do not appear in [ T(z) t G(z) t ]t,,. This can be done quite naturally in the framework we have built up. First, since (A) holds, the zero module of the transfer function [ T(z) t G(z) t ]t : U(z) --->(Y@W)(z) can be represented as / T -\ I Z(~)

=

+

Ker(T)+"U

T-I(~Y) n G - I ( ~ w ) + ~ U KerT~

KerG +~U

~U = T-I(c~Y) n G - I ( ~ w ) + ~ U Ker G + ~ U

Therefore, as Z ( G ) = G'I(f2W) Ker G + +~ U~ U , we have a natural inclusion i : Z ( T )

.~ Z(G) induced by the

obvious inclusion of the numerator modules. On the other hand, we have a natural projection p : x

X

-,

(~)

x(o

,wooo

f~U OU and X(G) = induced by the obvious = T-I(F2Y) n G-I(f2W) n f2U G'I(f~W) n ~ U '

inclusion of the numerator modules. Here the term "natural" means that the existence of the involved maps follows directly from the representations of the zero and pole modules that have been chosen. 4,7 NQTATIQN$ i) We denote by Z tile K[z]-module defined, up to isomorphism, by the short exact sequence

94 O~Z

g)

1-> Z(G) ~ Z ~ 0

ii) We denote by X the K[z]-module defined, up to isomorphism, by the short exact sequence

0 ~ X --> X ( T ) __p--> X ( G ) ~ 0 . The module 7_,,cokernel of i, is identified with the quotient Z(G)/Z ( ~ ) and clearly it can be viewed as representing the zeros of G which are not zeros of [ T(z) t G(z) t ]t. In particular we have that, as a Kvector space, Z(G)is isomorphic to Z ( T ) OK Z. In other terms Z represents exactly the zeros that one expects to be cancelled by the poles of any solution H(z)of 2.6. Analogously. since X ( T ) describes the union of the poles of T(z) and of G(z), the kernel X of p can be viewed as representing the poles of T(z) which are not poles of G(z) (see also [5] 3.5). They are exactly the poles that one expects to see as poles of any solution H(z) of 2.6. More precisely, we can state the following 4.8 PRQPQ$1TION There exists a natural inclusion ~: X --->P and a natural projection ~ : P --->Z such that the following sequence O ~ X ~--~

P - - W - - > Z --> O

is exact (i.e., in particular, ~(X) = Ker V). Proof. See [9] 3.9. From the above proposition it follows, in particular, that P =_X @K Z as a K-vector space. Then we can say, in a precise algebraic sense which extends and generalizes what we have seen in the scalar case, that the module of fixed poles of the equation 2.6 consists exactly of the zeros of G(z) which are not zeros of [ T(z) t G(z) t It and of the poles ofT(z) which are not poles of G(z). It remains to give an answer to 2.5 ii). If we call essential a solution H(z) whose pole module X(H) contains only the fixed poles represented by P, i.e. for which the map j of 4.1 is an isomorphism, this amounts to investigating the existence of essential solutions. The result we have is contained in the following 4.9 PRQPQSITION There exists a solution H(z) of 2.6 such thatj : P ~ X(H) is an isomorphism. Proof. For a complete proof, the reader is referred to [9] 4.4, here we simply show a possible construction of an essential H(z). First, let us choose a K[z]-basis {m I ..... mr,Vr+ I ..... Vq} of [2W, such that {m 1..... mr} is a K[z]-basis of G(U(z)) n ~W. Such a basis exists because both G(U(z)) n f2W and ~W are free K[z]-modules (see [27] and [5] 4.1 for details) and, moreover, it is also a K(z)G(U(z)) n f~W basis of W(z). N o w , let u I (z) ..... Ur(Z) be elements in U(z) such that G(ui(z)) = m i for i=l ..... r , and define H(z) : W(z) --> Y(z) as follows It(mi) = T(u(i(z)) i = 1..... r H(vk) = 0 k = r+l ..... q. It turns out that H(z) is a solution of 2.6 such that j : P ~ X(H) is an isomorphism. 4.10 REMAR~ Different choices of the elements mi's, Vk'S, ui(z)'s in the above construction lead to different essential solutions. However, not all the essential solutions can be obtained in this way. A complete description of the set of all the essential solutions is given in [9] 4.4, 4.5.

95 Summarizing the results of this Section, we can say, by 4.1 and 4.6, that the design problem represented by the equation 2.6 has solutions, if (A) holds, whose pole structure simply consists, in a meaningful way, of the poles of T(z) which do not appear in G(z) together with the zeros of G(z) which do not appear in [ T(z) t G(z) t it. Moreover, this pole structure can be a priori computed by means of P, as we saw in 4.5, and a procedure to find these essential solutions can be given, as we mentioned in the proof of 4.9. The relevance of such results in solving the design problem, especially when solutions with particular properties as for instance, stability are sought, is obvious. 4.11 EXAMPLE Let us consider again the transfer functions of example 4.6. In this case we have, for instance , G(U(z)) n ~ W = span {(1,z+l,1)t,(0,1,0) t} and G ( ( z ( z + l ) , 0) t) = (1,z+l,1) t, G((z+l, -(z+l)/z )t) = (0,1,0)t. Adding the vector (0,0,1) t, we obtain the basis for W(z) needed to define H(z) as H((1,z;+l,l) t) = T((z(z+l),0) t) = ( z(z+l)/(z+2), z]2) t H((0,1,0) t) = T((z+l ,-(z+l)/z) t ) = ((z2-z-4)/(z(z+2)),(z2+3z+4)/(2z(z+2)))t H((0,0, I) t) = 0. With respect to the canonical basis of W(z), we have C H(z) =

(z+l)(z+4)/(z(z+2)) (_2z2.7z_4)/(2z(z+2))

(z2-z'4)/(z(z+2)) (z2+3z+4)/(2z(z+2))

0 0

) and then it is not difficult to verify

that H(z) is a solution of 2.6 whose pole module coincides with the fixed pole ,nodule P, computed as in 4.6. Hence, H(z) is an essential solution of the equation we are considering.

5. FURTIIER D E V E L O P M E N T S AND G E N E R A L I Z A T I O N S

In the previous Sections we have considered only the finite poles and zeros, i.e. those which axe located at a point tx of K. In addition one can consider also the poles and zeros at infinity (see [16] for generalities). They play, in fact, a fundamental role in various design problems and the same questions concerning the finite poles and zeros we have seen can be formulated in the case of poles and zeros at infinity. We will not develop here a complete treatement of the case of poles and zeros at infinity (see [4,6,7,9,10,26] for applications), but we will simply point out its general lines, assuming that the reader is familiar with the classical notions. As in the finite case, the classical definitions, which are given in terms of the Smith-McMillan foml at infinity, involve a string of multiplicities and, implicitly, a "location" at infinity. The same motivations we have described in 2.2 and 2.5 justify the attempt of developing a module theoretic approach. The key fact, in introducing the concept of module of poles at infinity (briefly: infinite pole module) and of module of zeros at infinity (infinite zero module), is that the ring of polynomials K[z] must be replaced, in our framework, by the ring O0o of proper rational functions, i.e. rational functions of the form p(z)/q(z) with deg p < deg q. In fact, in considering the structure at infinity, we have to take into account only the elements of K(z) which are "regular" (as meromorphic functions if, for instance, K = ~:) at infinity, that is the proper rational functions, in the same way as, in considering the structure at every finite point of K, we had to take into account the elements of K(z) which are regular at every point of K, that is the polynomials. Note that O** is, like K[z], a principal ideal domain, therefore all the results concerning the theory of modules

96 and the algebra of matrices which have been used in the previous sections hold without changes. In fact, given the K-vector space V = K n , denoting by x"/ooV the O.o-module V ®K Ooo (whose elements are ntuples with entries from O~), we have the following definitions 5.1 DEFINITION The infinite pole module of T(z) is the Oo.-module Xoo(T) defined by Xoo(T) -T" 1( ~ . . y ) ~ f2ooU .5,2 DEFINITIQI'q The infinite zero module of T(z) is the Ooo-module Zoo(T) defined by Zoo(T) -T" 1 (C2**Y) + ~ . ~ U Ker T + f2ooU The analogy between 5.1 and 3.1 and between 5.2 and 3.2 is obvious. The invariant factors of X~(T) over O** describe the structure of the poles at infinity and the invariant factors of Z**(T) describe the structure of the zeros at infinity. Moreover, if T(z) is strictly proper with minimal realization (X,A,B,C) and S* denotes the minimal conditionally invariant subspace of X containing Im B, we have that Zoo(T) is Ooo-isomorphic to S*/R* provided with the Oo.-structure given by the shift operator z-1. Considering again the equation 2.6, we can introduce the following : 5.3 DEFINITION The Oo.-module P** =

G-l(oooW) G-l(f2ooW) ~ T-I(f2ooY)

is called the module of fixed Doles at

infinity of tile equation 2.6. Then we can prove, as in section 4: 5.4 PROPOSITION i) For any solution H(z) of 2.6 there exists a natural inclusion jo. : P . . --~ X~o(H). ii) There exists a solution H(z) of 2.6 such that j~. : P.. ~ X,,.0i) is an isomorphism. The result of 5.4 can be applied when one is interested in the existence of a proper solution. A transfer function, in fact, is proper iff it has no poles at infinity, that is iff its infinite pole module is zero. Then, there exists a proper solution H(z) of 2.6 iff Poo = 0. 5.5 REMARK It is not possible to combine 4.9 (existence of essential solutions) together with 5.4 ii) in order to obtain an existence result for solutions H(z) having, at the same time, a pole module X(H) isomorphic to P and an infinite pole module Xo.(H) isomorphic to P,,.. The two propositions, in fact, concern two different frameworks, the polynomial one and the proper rational one, which, from the point of view of poles and zeros, are mutuaUy exclusive. Furthemmre, such solutions, in general, do not exist. A further generalization, which leads to useful applications, can now be easily described (see [26]). Assume that we are interested in considering the pole/zero structure at every point of a proper subset S of K t..) {,~} which does not coincide with K (finite case) nor with {~} (case at infinity). For instance, if K = and we want to investigate the existence of proper and stable solutions, the set S we have to deal with consists of all the points with positive real part together with the point 0o Noting that the subset of elements of K(z) which are regular at every point of S forms a ring O S which is a principal ideal domain, it is not

97 difficult to realize that one can develop an Os-theory following the general lines along which the K[z]theory and the O~-theory have been constructed. Therefore, using notations which are obvious, we have, in particular:. 5.6 D E F I N I T I O N The S-pole module of T(z) is the O s - m o d u l e X s ( T ) defined by

Xs(T ) =

f2sU T-I(I2sY ) c~ I2sU" G-I(K2sVO ~,7 DEFINITION The Os-module PS = G . I ( f 2 s W ) n T ' I ( f 2 s Y ) is called the modul~ of fixed p01~ in $ for 2.6 5.8 PROPOSITION i) For any solution H(z) of 2.6 there exists a natural inclusion jS : PS ~ X5(It); ii) There exists a solution H(z) of 2.6 such that JS : PS ~ Xs(H) is an isomorphism. As a consequence of 5.8 we have, for instance, that there exist proper stable solutions H(z) of 2.6 iff PS = 0 for S = { ct ~ ~: u {*~}, Re tx > 0 or 0t = 0,, }, or that there exists a dead beat solution H(z) of 2.6 iff PS = 0 for S = { cz e K u {-0}, tx ~: 0 }. The reader can easily list other examples. What is important to note is that, although the framework is different, the analysis of 2.6 with respect to a subset S of K u {o.} can be carried out, with the obvious modifications of notations, as we have seen in Section 4.

6. GENERIC ZEROS AND THE W E D D E R B U R N . F O R N E Y CONSTRUCION Forgetting tile module structures, the pole and zero modules, viewed as K-vector spaces, can be combined to form the global pol~ space X(T) = X(T) G)K Xoo(T) and the global zero sp~¢~ Z(T) = Z(T) @K Zoo(T). The dimensions of X(T) and of Z(T) are the total number of poles and the total number of zeros of T(z). In general, dim X(T) may be greater than dim Z(T) and the difference between the two is called the defect of T(z). This can be expressed in terms of Kronecker indices or Wedderburn numbers as in [11] and [161 Th. 6.5-11 and can be related to the failure ofT(z) to be surjective or to be injcctive. In the cascade composition of two transfer functions T(z)G(z), the presence of a nontrivial Kernel or Cokernel of one of the two may cause new zeros, that is zeros which are not in T(z) norin G(z), to appear. This phenomenon, investigated in [8] and [17], suggests the idea of associating to Ker T and Cok T a notion of generic zero, not located at any point of K nor at infinity, of T(z). One can think that the Krouccker indices or the Wedderburn numbers we have mentioned above describe the number of such zeros. A precise algebraic formalization of these ideas again requires the use of the module theoretic approach. Following [331, given a K-vector space V = K n, we consider the K-linear map it_ : V(z) --->I"V described in Section 1 which associates with any element of V(z) its strictly proper part, then, for any K(z)subspace S c V(z), we have the following 6.1 DEFINITION The Wedderburn-Fornev soace associated with S c V(z) is the K-vector space W(S) ~.(S) defined by W(S) = S n z-lf2ooV "

98 For any S c V(z), W(S) turns out to be a finite dimensional K-vector space. Therefore, for a given transfer function T(z), the construction of the Wedderburn-Forney spaces W(Ker T) and W(Im T) associated with Ker T and with Im T yields two objects of the same kind as X(T) and Z(T). We will think of the spaces W(Ker T) and W(lm T), which are trivial iff respectively T(z) is injeetive or surjective, as of spaces of generic zeros. Now, denoting by x+ : V(z) ~ f2V the K-linear map which associates with any element of V(z) its polynomial part, we have that the mapping u(z) ---}(x.u(z),x+u(z)) induces a K-linear map h : W(Ker T) ~ X(T). It turns out that h is injective, so that W(Ker T) can be viewed as a subspace of X(T) and we have tile following 6.2 PROPOSITION For any transfer function T(z) there exists an exact sequence of K-vector spaces X(T) 0 --} Z ( T ) - - f --~ W(Ker T) - - g where f

--~ W(Im T) --} 0 ,

X(T)

: Z(T) --} W(Ker T) is the map induced by (u(z),v(z)) ~ (~+(u(z)+v(z)),n_(u(z)+v(z))) for

X(T)

u(z) ~ T ' I ( ~ Y ) and v(z) ~ T-I(z'lO~,,Y) and g : W(KerT) --~ W(lm T) is the map induced by (u(z),v(z)) --~ x_T(z)(u(z)+v(z)) for u(z) ~ I/U and v(z) ~ z - l ~ u . Proof. See [33] Main Th. 5.1. As a direct consequence of 6.2 we have for any transfer function T(z) that dim X(T) = dim Z(T) + dim W(Ker T) + dim W(Im T). So, if we think of the dimensions over K of W(Ker T) and of W(Im T) as of the number of generic zeros of T(z), we have that the total number of poles equals the total number of zeros, including the generic ones. The above Proposition can therefore be viewed as a refinement, in structural terms, of the numerical results about the defect and the Kronecker indices or the Wedderburn numbers we mentioned at the beginning of the Section. Moreover, the above exact sequence can be viewed as a complete concise description of the relationship between zeros and poles of any transfer function.

REFERENCES [1]

M.F. Atiyah and I.G. MacDonald - Introduction to Commutative Algebra, Addison-Wesley,

[2]

G. Conte and A.M. Perdon - On polynomial matrices and finitely generated torsion K[z]-module,

Reading (1969) A.M.S. Lectures in Appl. Math., Vol.18 (1980) [3]

G. Conte and A.M. Perdon An algebraic notion of zero for system over rings, Lecture Notes in Control and lnfomaation Science, Springer-Verlag, 58 (1984)

[4]

G. Conte and A.M. Perdon - Infinite zero module and infinite pole module, Lecture Notes in Control and Infomlation Science, Springer-Verlag, 62 (1984)

[5]

G. Conte and A.M. Perdon - Zero module and factorization problems, A.M.S. Series in Contemporary Mathematics, Vol. 47 (1985)

[6]

G. Conte and A.M. Perdon - On the minimum delay problem, Systems and (1985)

Control Letters, 5

99 [7]

G. Conte and A.M. Pcrdon - On the causal factorization problem, IEEE Trans. Aut. Control, AC-30

[8]

(1985) G. Conte and A.M. Perdon - Zeros of cascade composition, in Frequency Domain and State Space Methods for Linear Systems, C.Byrnes and A.Lindquist Eds., North-Holland,(1985)

[9]

G. Conte, A.M. Perdon and B. Wyman - Fixed poles in transfer function equations, SIAM J. Control Opt., 26 (1988)

[10]

G. Conte, A.M. Perdon and B. Wyman - Zero/pole structure of linear transfer functions, Proe. 24th

IEEE CDC, Fort Lauderdale (1985) [11] D.G. Forney - Minimal bases of rational vector spaces with applications to multivariable linear systems, SIAM J. Control, 13 (1975) [12]

P. Fuhrmann - Algebraic system theory : An analyst's point of view, J. Franklin Inst., 301 (1976)

[13]

P. Fuhrmann and M. Hautus - On the zero module of rational matrix functions, Proc. 19th IEEE

[14]

CDC, New York (1980) B. Hartley and T.O. Hawkles - Rings, Modules and Linear Algebra, Chapman and Hall, London

[15]

(1970) M. Hautus and M. Heymann - Linear feedback. An algebraic approach, SIAM J. on Control and Opt., 16 (1978)

[16]

T. Kailath - Linear Systems, Prentice HaU (1980)

[17]

S. Kung and T. Kailath - Some notes on valuation theory in linear systems, Proc. 17th IEEE CDC,

[18]

San Diego (1978) S. Lang- Algebra, Addison-Wesley, Reading (1965)

[19]

A. MaeFarlane and N.Karcanias - Poles and zeros of linear multivariable systems : a survey of the

[20]

H. Rosenbrock - State space and multivariable theory, John Wiley and Sons, N.Y., (1970)

algebraic, geometric and complex-variable theory, Int J. Control, 24, (1976) [21]

C. Schrader and M. Sain - Research on system zeros : a survey, Proc. 27th IEEE Conf. on Decision and Control, Austin, (1988)

[22]

G. Verghese - Infinite-frequency behaviour in generalized dynamical systems, Ph.D. Thesis, Dept. Electrical Eng., Stanford Univ., (1978)

[23] J.C. Willems and P. Fuhmlann - A study of (A,B)-invariant subspaces via polynomial models, Int. J. Control, vol.31 (1980) [24] [25] [26]

W.M. Wonham - Linear multivariable control: A geometric approach, Springer-Verlag, (1979) W. Wolovich, P. Antsaklis and H. Elliott - On the stability of the solutions to minimal and nonminimal design problems, IEEE Trans. Autom. Control, AC-27 (1977) B. Wyman, G. Come and A.M. Perdon - Local and global linear system theory, in Frequency Domain and State Space Methods for Linear Systems, C.Byrnes and A.Lindquist Eds., NorthHolland,(1985)

[27]

B. Wyman and M. Sain - Tile zero module and essential inverse systems IEEE Trans. Circuit and Systems, CAS-28 (1981)

[28]

B. Wyman and M. Sain - Exact sequences for pole zero cancellation, Proc. M'INS 1981, Santa Monica ,(1981)

I00

[29]

B. Wyman, M. Sain et al. - The total synthesis problem of linear multivariable controi,Proc. 20th Joint Autom. Control Conf., (1981)

[30]

B. Wyman and M. Sain - Internal zeros and the system matrix, Proc. 20th Allerton Conf., Urbana

[31 ]

B. Wyman and M. Sain - The zero module of a minimal realization, Linear Algebra and Appl., 50

[32]

(1983) B. Wyman and M. Sain - A unified pole-zero module for linear transfer functions, System and

[33]

Control Letters, 5 (1984) B. Wyman, M. Sain, G. Conte and A.M. Perdon - On the zeros and poles of a transfer function,

(1982)

Linear Algebra and Its Application, to appear 134] Outils et ModUles Mathematiques pour i'Automatique, l'Analyse des Syst~mes et le Traitement du Signal, Ed. du C.N.R.S., Vol. 1 (1981)

Representations of Infinite-Dimensional Systems R. F. Curtain Mathematics Institute, University of Groningen P. O. Box 800, 9700 AV Groningen, the Netherlands

TIds article is dedicated to Marghcrita Willcms.

1. I n t r o d u c t i o n Thc

aim

coacel)t F,'db

of

and

of

an

this

is

infinite-dimensional

Arbib

coitlxolled

delay

the

decades

three

article

[15], and

and

the

1)artial

of

to

clarify

dynamical more

in

the

of

as

proposed

in

Salamou

Earlier

research

[28]-[29], tended

to

diffcrenqal

state

space

p.d.e.'s

delay

equations

or

abstract

[31], Salamon

realization

[33]-[36],

concentrate

formulations

[16], Lasiecka and Triggiani {23], Washburn

Weiss

theories

(Lions

and

[22],

in

in

the

and

literature.

either

Curtain

Curtain

of

and

Despite

sytems,

and

Weiss on

it [7].

various

well-posedness

Pritclla.rd

for

in an elegant

exclusively

questions

tim

Kahnan,

thcot'ics

infinite-dimensional

is only recently that this question has been completely resolved manner

between

well-posedness

equations area

relationships

system

recent

differential

research

tile

[3],

of

Lasiecka

[17]-[21], llo and Russell [14], [,ions and Magenes [27] to menl, ion a representative for

transfer

functions

(Itelton

sample) [12],

or

on

¥amamoto

[37], Fuhrman [11], Barns and Brockett [1]-[2]) and the twain took a long time to meet,. The importance of a unified theory embracing both state space and frequency

domain

~pccts

of

control

synthesis

and

design

for

102

infinite-dimensional

systems

has

special

emphasis

was

Mready

exists

substantial

a

laid

on

bcen

pointed

special

classes

theory,

tlere

out

in

of

Curtain

systems

we discuss a

for

much

[6],

where

which

there

larger

class of

infinite-dimensional linear systems. There

are

many

ways

of

representing

lincar

infinite-dimensional systems,

for example: (i)

directly in terms of p.d.e.'s or diffcrcntial delay equations;

(ii)

in terms of a quadruple (A,B,C,D)

of abstract operators on a Banach (or

llilbert) spacc;

(~ii) (iv)

as a frequency domain relationship between inputs and outputs; as

a

dynamical

system

in

the

think

of

sense

of

l(alman

(an

abstract

linear

system).

It

is perhaps

useful

to

(i)

as a differential representation,

of

(ii) as a semigroup representation (since A is usually rcquired to generate a C0-scmigrou p on some Banach spacc) and of (iv) as an integral representation. (Following Wciss [33]-[3G] we shall use

(iv).)

These

domain

are

all

state

representation.

pro'pose of

this

Each

article

representations

from

the

thercin)

transfer

functions

references

(i)

function

its own

survey,

differential and

the

term

"abstract

representations, much

Much research

mertis

and

less

obtaining

(see

the

or

the

scmigroup

was

not

investigated

system"

for

and all

it

is not

aspects

of

the these

done

on obtaining semigroup

(see

Salamon

semigroup

Fuhrmann

tlowever,

uses

compare,

has been

linear

(iii) is a frequency

whereas

representations

on

description

therein),

representation transfer

has

to

different representations.

refercnccs

space

[11],

question

[27]-[28]

realizations

Yamamoto whether

(ii)

representation systematically

Salamon [28]. Even more recently the link between

from

the

and

the

[37]

the

differential

a

well-defined

have

until

and

fairly

recently

in

the intergral representation

(iv} and the reprcsentations (ii) and (iii) was made in the series of papers by

Salamon

[29],

aspect

we

linear

system

linear

system,

Weiss [33]-[36]

shall

discuss

as

the

one

in

this

and

fundamental obtains

a

Curtain

article.

and

Using

representation

very

elegant

Weiss

the

[7],

concept of

theory

an

and of

it an

is this abstract

infinite-dimensional which

clarifies

the

relationships with the other three representations. Let

us

illustrate

the

above

ideas

with

the

well-known

linear system whose differential representation is given by

finite-dimensionN

103 x(t) = Ax{t) + Bu(t)

(1.1)

y(t) = Cx(t) + Du(t)

(1.2)

where x ( t ) e X = R n, tile state space, u ( t ) e U = R "~, the input space and y ( ~ ) e Y = R k, the

(A,B,C,D)

output

space

and

dimensions,

which

coincides

An important the

unique

(1.1)

for

property

of

continuously any

given

is

with the

a

the

quadruple abstract

differential

matrices

operation

representation

differentiable

solution

initial

x(0) = xoeX.

state

of

of

matching

is

x(t)

that

differential

The

(ii).

representation (i)

the

of

is

equation

frequency

domain

representation (iii) is in terms of the transfer function

G(s) = C(sl-A)-aB The

corrresponding

abstract

linear

(1.3)

system

representation

is

given

by

~ = ($,,I),L,F) defined by Tt

• tu =

(L,.x)(t)

={

ft 0

= eAt

{1.4)

eA(t-~,)Bu( a)ga

(1.5)

ceAtx

; t e [0,7"}

0

; t_>r

(1.6)

Cft eA(t-a)Bu(a)&r +Du(t) (Gu)(t)

=

; t e [0,r)

0

(1.7)

0

which

explains

relationship to

(iii)

between

are

realization

trivial

theory

relationships

why

; t>r

we

described

representations and

the

which does

between

the

it

as

an

integral

[i)

and

(ii)

and

the

implication

to

(ii)

is

implication not

abstract

of

concern linear

(iii)

us here. system

representation.

In order

representation

the to and

The

(ii)

of

subject explain the

of the

other

three we first need to define this concept. We follow Weiss [33]-[36]. An

essential

role

is

played

by

the

composition

properties

concatenation of inputs, u and v, which is defined for any r > 0 by

under

the

104

u(t)

for t e [0,v)

v(t-r)

for t ~ v

(u 0 v)(t) = r

(1.8)

loc

for any u,veL2 (0,co; U). 1 We shall also use the projection

operators P~ from

L tOC.~

z ~v,cv; U) onto Lz(0)r; U) which are defined for any r > 0 by u(t)

for t e [0,r)

0

for t_>r

(Pru)(t) =

Definition

1.1

(1.9)

)-]=(T,db,L,F) is a abstract linear system on the Hilbert spaces

U, X and Y) if

(i)

T = ( T r ) r > 0 is a strongly continuous semigroup of linear operators

on X,

i.e., TLx is continuous in t for all x~X,

O.io)

TeTr = Tt+ r

for any t,T_>0 and T 0 = I (tile identity}.

(ii)

~--('I'r)r_> e is a family of bounded linear operators from L2(0,oo;U ) to X

such that @l+r(u O v) = Tt¢I,~u + 'I'tv

(1.11)

T

for any u, veL2(0,oo; U)

(iii)

L=(Lr)rzo

and any r, t>0.

is a family of bounded linear operators

from X to L2(0,oo; Y)

such that

(1.12)

Lr+tx = Lrx<~LtTrX for any x e X and any t,v>_0, and L0=0.

(iv) F=(Fth_>0 is a family of bounded linear operators from L2(0,co;U ) to L2(0,oo; Y) such that

a (u,oo; U) denotes the space of functions f on (0,oo) every T > 0 the restriction of f to (0,T) belongs to L2(0,T; 0).

such

that

for

105

FT+t(U 0 v) =

Fru ¢' (Lt',bru+Ftv)

T

(1.13)

T

for any u, veL2(O,oo; U) and any t,r>_0 and Fo=0. U is called the input The

operators

maps

and

,I,t

Ft

axe

are

space, X the state space and Y the output

called

called

teachability

input-output

v, aps,

It

maps.

The

the

causality

are

observability

called

conditions

space.

(1.4),

(1.5)

and

reachability

and

(1.6) are called composition properties.

Notice

that

this

definition

implies

of

the

input-output maps, i.e., for any t > 0

'I'tPt = "I't; FtPt = Ft.

(t.14)

PtLr = Lt;

(1.15)

In addition there holds

P f r = Ft

for arty 0 _
lr

L2(O,eo; Y)

if

and St

Fr is

have

a

limit

exponentially

as

r-~co,

stable

or

where

the

otherwise

in

convergence

is

the

space

Frdchet

in

L~°~(0,~; Y): L~

for any x ~ X

$:~ov = limF,-v

= limL~.x;

1.16)

and any vEL2{O,oo; U). In fact, Fr and Lr are defined uniquely by

F® and I_~ respectively through (1.15), if we take r = o o there. Returning readily verified via (1.4)-(1.7)

to

our

(A,B,C,D)

that and

finite-dimensional

linear

defines an

system

abstract

conversely that ~ on the finite-

abstract

(ii)

and

(1.4)-(1.7). linear this

system is

In

other

words,

is equivalent equivalent

the

with

with

integral the

the

see

that

it

is

linear system E=('i-,~,I.,F) dimensional spaces U, X

and Y uniquely determines matrices Aef-.(X), B e £ ( U , X ) , satisfying

we

Ce£.(X,Y) and DE£(U,Y) representation

abstract

operator

differential

(iv)

of

an

representation

representation

(i).

106

Moreover, the transfer function given by (1.3) and the input-output map given by

(1.7)

are

isomm'phic using

shifted

Laplace

transforms.

motivated much of the research into representations for

infinite-dimensional

which

uses

the

linear

concept

systems,

of

an

including

abstract

This

example

has

and weU-posedness studies

the

theme

linear

system

of

this

article,

as

the

central

representation. We show that given an abstract linear system ~: on Hilbert spaces U, X and Y there

(A,Ba,Ca) of operators satisfying the equations

exists a unique triple

(1.4)-(1.6)

in

a

suitably

generalized

setting.

Conversely

we

give

sufficient

(A,B,C) of operators to generate a family of abstract In general a triple (A,B,C) will not determine a unique

conditions for a triple linear

systems.

transfer

function

input-output transfer

(or input-output map),

formula (1.7) exists, function.

representation function

in

regularity regular

general.

property

abstract

which leads

of

it

of

also

depends

most

abstract

linear

great

simplifications

allow for

Moreover

differential

of the

complicated and depends on the

generalization

but to

systems

(1.1)-(1.7).

solution

a

exists,

Surprisingly,

linear

all the formulas continuous

it is rather

Shnilarly,

(1.1)-(1.2)

and although a generalization

a

natural,

tlle

differential

on

the

systems

in the

transfer satisfy

theory.

a

These

simple generalization

of

x(t) has the interpretation as a strong equation

of

the

form

(1.1)

and

(1.2}

holds. These results and much more can be found in Salamon [29], Curtain and Weiss [7] and Weiss [33]-[36]. Salamon [29] and Curtain and Weiss [7] consider the

case

where

trajectories

X,

are

Y and

U are

locally L2 and

Hilbert

for

spaces

simplicity

and

this

is

the

input

the

case

and

output

treated

here.

In [33]-[36] Weiss extended this to the more general situation where X, Y and U are

Banach

spaces

and

the

arbitrary

pe[1,oo).

Moreover

analysis

of

observation

the

input

the

and

results

output of

operator

spaces

Weiss

and

are

contain

related

locally

Lp for

much

deeper

of

linear

a

aspects

infinite-dimensional linear systems, which we shall not go into here. Following relationship and

delay

this

between

extended the

equations

(1.11).

Since

the

p.d.e.'s

and

control

operator"

abstract

linear

usual

and

the

concept

differential here system.

introduction,

in

well-posedness existence

of

of

section concepts

a

we

prefer

for

is

to

use

section

3

we

examine

the

discuss

controlled map

usually the

which we show is compatible with In

we

rea~hability

"well-posedness"

systems

2

fit

the dual

p.d.e.'s

satisfying

associated term

the

with

"admissible

concept of an concept

of

an

"admissible observation operator" and the existence of an observability map I.~

I07

satisfying operator

(1.12). is

We

compatible

show

that

the

with

that

of

concept an

of

abstract

an

admissible

linear

system.

observation In

addition,

the key concept of the Lebesgue extension of the abstract observation operator is introduced. The representation is then treated generalization four

frequency

linear

the

representation

(1.1)-(1.7)

types

domain

and

of

that

representations of

an

class of

differential

result

is

that

representation

triples

linear

for

regular

is always possible. abstract

observation

space

abstract

system)

linear

operator

equations

plays

(1.1)-(1.2),

albeit between

operator,

are

clarified.

function and they correspond A surprising

systems

of the stone simple form as (1.3)-(1.7)

state

possible,

(A,B~C) which we call "input-output

representation

Lebesgue extension abstract

always

(differential,

abstract

Abstract linear systems define a unique transfer and a

a somewhat complicated

is

function of the system. Moreover, the relationships

different

to a certain

system in its entirety

in section 4, where it is shown that of

using the transfer the

of an abstract

a

and

most

elegant are)

a

is possible in which the

crucial

have

for almost all t with x(t) as the unique strong

(and

admissible"

a

role.

Moreover,

pointwise

the

interpretation

continuous solution of

(1.1).

The concepts are illustrated by a detailed example in section 5.

2. Admissible The control

control

motivation operator

operators for

B

stems

introducing from

the

the

concept

need

to

of

make

admissibility sense

of

the

of

the

abstract

differential equation

Jc(t) = Ax(t)+Bu(t)

(2.1)

where x(t) denotes the state of the system at time t, x(t)~X, (the state

space),

and

T={l"r}r>_0 on X, u(t) (the input space)

B~£(U,X),

then

A is

the

infinitesimal

generator

of

a Itilbert space the

Co-semigroup

is the input which takes values in the Hilbert space U

and B is a linear map on U. If B is a bounded operator, it

i s ~vell-known,

(Curtain

and Pritchard

ueL2(O,r; U) the following variation-of-parameters

[3]),

that

for

any

formula provides a suitable

interpretation as the continuous solution of (2.1) on [0,~-]

108

x(t) = Ttx(O )

(2.2)

+ f Tt.aBu(a)d~. 0

X, but into a larger VoX, and (2.2) may not be well-defined. It is customary to call such control maps unbounded, although this does not agree with the meaning of However, in many applications B does not map into

space,

unbounded

in functional

in systems

described

differential-dclay literature such

on

solution chapter and

in

by

on

some

control

p.d.e.'s

with

with

delayed

control

question

and

Unbounded

linear

systcms

the

systems

analysis.

as

to

appropriate

boundary

conditions

sense,

for

arise action

and in

There

exists

a

mathematical

under

which

example

naturally

control

action.

the appropriate

sufficient

operators

(2.2)

Curtain

formulation

of

will

a

and

define

Pritchard

8, Curtain and Salamon [4], Desch, Lasiecka and Schappacher

Russell

[14],

Lasiecka

[16],

Lasiecka

and

Triggiani

vast

[17]-[20],

[3],

[8], If0

Lions

and

Magenes [23], Russell [25], Salamon [27]-[28], Washburn [31]. The concept of a "solution"

of

an

abstract

differential

equation

(2.1)

is

an

interesting

topic

in its own right and it depends very much on the mathematical formulation one chooses.

Usually

regularity example

a

p.d.e,

properties Lasiecka

and

and

formulation so

the

Triggiani

enables

strongest

[17]-]21]). Later

be able to say more about the interpretation in an

abstract

sense.

one

In the natural

to

concept on

obtain of

the

strongest

solution

in this

(see

section

for

we shall

of (2.2) as a solution to (2.1)

formulation

of p.d.e,

systems and

delay

systems one is not presented with an (A,B) pair and Hilbert spaces X and V; a crucial

step

lies

in

formulating

(A,B) description for appropriate Hilbert

the

spaces X and Y. We shall not treat this aspect here, Salamon

[27]

and

[28] for

one

methodology of

but refer the reader to

obtaining

an

(A,B)

abstract

formulation for suitable X, V spaces for delay and p.d.e, systems. The Salamon methodology may be seen as a synthesis and extension of ~existing aimed

at

obtaining

infinite-dimensional

an

systems.

abstract In

semigroup

particular,

this

formulation

methodology

techniques for

coupled

linear with

the

Carleson measure test in Ilo and Russell [14] and Weiss [32] is very useful for establishing

the

well-posedness

of

p.d.e,

systems

generated

by

a

spectral

operator and having finite-rank inputs and outputs. We should at the stone time point

out

that

for

(A,B) or abstract relationships

between

many control semigroup

applications

formulation.

well-posed

systems it is a necessary first step.

p.d.e,

and

it

is not

However, delay

in

necessary order

systems

and

to

to

use an

clarify

abstract

the

linear

109 At

this

stage

we

are

only

unbounded control operators.

Tt

with

making

It is clearly necessary that

sense the

of

(2.1)

integral term

values in X for any input ueLz(O,r; U), and this in turn

{2.2) have that

concerned

have

a

continuous

extension

on

V.

This

motivates

the

for in

requires following

defintion of an admissible control operator. Suppose that A generates the C0-semigrou p T=(Tr)r>o on

Definition 2.1

the

llilbert space X and that BeL(U,V) where U and V are Hilbert spaces. If X is dense in V and $ has a continous extension to V, then we call B an admissible

control operator for T with respect to V if the teachability maps ( ~ r ) r > 0

are

bounded from L2(0,oo; U) to X for all finite r > 0 , where t

(2.3)

• ~u := f Tt.aBu(a)da 0

The

terminology "admissible control operator"

comes from Weiss [33] and

it is equivalent to the assumption $2 of Salamon [27], [28] and in some sense to the many other concepts of "well-posedness" which abound in the literature. Our concern is not their respective merits, lead

to

well-defined

abstract

It is readily verified that respect

to

V

assumptions

in

according Definition

linear

if B is an to

in

the

sense

of

admissible control o p e r a t o r

Definition 2.1,

1.1; we

but the type of (A,B)-pairs which

systems

shall

then say

the

it

pair

generates

(T,,I,) an

Definition 1.i. for

]" with

satisfies

abstract

the

linear

control system on X and U. Definition 2.2

An abstract linear control systems on the Hilbert spaces X and

Y is a pair (T, 0 is a C0-semigrou p on X and =(~/>r)r> 0 is a family of

bounded

operators

from L2(O,oo; U) to X such

that

the composition

property (1.11) holds. Clearly an 1.1. Not

only

a b s t r a c t linear control system is a subset is

(T,,I,)

continuous family of

causal

operators

(see and

(1.14)),

the

but

function

of ~

'I~--(~I,~)~>_0 is

of Definition a

strongly

"b(t,u):=~tu is continuous on

[0,co)×L2(0,oo; U), (Weiss, [33, Prop. 2.3]). The

question

we

are

interested

in

here

is

whether

an

abstract

linear

control system determines an admissible control o p e r a t o r B. This was shown to

110

be true in Salamon [29] for the p = 2

and Hilbert space case and in Weiss [33]

for the Banach space case and any pe[1,oo). A related representation result for p.d.e,

systems

Schappachcr

with

boundary

[8]. The

main

control

result

also

appeared

in Weiss

[33]

is

in

Desch,

that

any

Lasiecka abstract

and linear

control system has a representation (T,,I,) with ,I, given by the form (2.3) for a unique admissible control operator B a e £ ( U , X q ) , where X q is the completion of X induced by the following norm:

(2.4)

Ilxll-x = 11(/3I-A)-Xxll for a flop(A). Notice that

(2.4) defines equivalent

norms for different

fi's in p(A)

and

so X_l is independent of fi and X is automatically dense in X_1. Furthermore) I has a the

unique continuous extension

isomorphism

literature

on

(flI-A)

to Xq

from X to

well-posedness

of

which coincides with its image via

Xq.

controlled

In view of p.d.e,

and

the delay

complex,

technical

equations)

this is

a surprisingly simple and elegant result. Theorem

2.3

Let X

and U be Hilbert spaces and

suppose

that

('~,e2) is an

abstract linear control system on X and U. Then there exists a unique operator Ba~E(U~X_I) such that for any t>O~ ~I,t has the representation t

(2.5)

• tu = f L-~ B~u(a)da 0

for

any

u~L2(0,co; U) and

B e is

an

admissible

control

operator

/or

"[ with

respect to X 1. Moreover, f o r any x o e X and ueL2(0)oo; U) the function defined by

(2.6)

x(t) = Ttx o + ~d~ is

the

unique

strong

continuous

solution

of

(2.1)

with B = B ~

and

the

initial

condition x(O) = x o.

By tile concept "strong continuous solution" is meant tile following one. Definition

2.4

We say that the function x(.) is a strong continuous

solution

111 of (2.1) on [0,r] if x(.)~C(O,r; X) and for any t e [ 0 , r ] the following holds t

(2.7)

x(Q-x(O) = f [Ax(s)+Bu(s)]ds. 0

it was shown that x(t) given by (2.6) is the

In Salamon [28], Lemma2.5, unique

strong

continuous

solution

of

(2.1)

with

the

original

admissible

control o p e r a t o r B. Although B and Ba generate the same reachability map ,I,, they need

not

be

equal.

Sufficient

admissible control o p e r a t o r

conditions

for

B=B a axe that

B

for l" with respect to V as in Definition

be

an

2.1 and

A have a continuous extension to an o p e r a t o r in £(X,V). For then BE£(U,X_I) and since Ba is tmique, B=B a (Weiss I33], Remark3.14).

An imporLant special

case is when B is a bounded operator. To avoid any confusion we introduce the following definition and notation.

Definition 2.5

Given the abstract linear control system (T~,I~) oil the Hilbert

spaces X and U we call the unique o p e r a t o r Ba~£(U,X_I) of Theorem 2.3 the

abstract control operator. It is defined by Bay = l i m ~1 (¢~v).

(2.8)

2.4 it is natural to ask whether any Be£,(U,X_I)

In the light of Theorem

and C0-semigrou p ]" on X define

an abstract

This conjccture

if Be£(U,X_I)

control system all t), then always

be

is false. (],(I,)

In

(i.e. ,I,~ defined by

often

2.3

(2.6)

does is not

not

(T,~I,) via

define

an

driven

out

has

considerable

of on

an

the

arbitrarily

primarily theoretical burden

of

state

value

verifying

space

in £(L2(0,oo; U), X)

short and

that

B

X

by

some

time-interval.

for can input

Consequently

does

not

relieve

is

an

admissible

operator for 1- with respect to V in p.d.e, and delay examples.

{2.7).

abstract

(as shown in Weiss [33]) the state t r a j c c t o r y given by (2.6)

ueL2(0,v; U)nC°~([O,T), U) Theorem

fact

control system

us

of

the

control

112

3.

Admissible The

observation

motivation

operators

arises

for

operators introducing

from

the

the

need

to

concept

of

formulate

admissible

the

following

observation type

of

observability map y(t) = CTtx

; t>O

(3.1)

where T=(Tr)T>_o is a Co-semigroup on the Hilbert space X (the state space),

x e X , and C:W->Y is a linear operator from a subspace W of X to another Hilbert space Y (the output space). If C is bounded on X then (3.1) is defined for all

x ~ X and for all t>_O, but in many applications C is only defined on a proper subspace of X and so may be unbounded. Unbounded operators occur when one modcls boundary or point observations for systems described by linear p.d.e.'s or when one has delayed observations in delay equations. There is an extensive literature

dealing

example

Curtain

L a s i e c k a and Seidman function

with

systems

having

unbounded

and

Pritchard

[3],

Triggiani

[18],

Pritchard

[30],

Yamamoto

[37].

Dolecki

The

and

and

observations

operators,

Russell

Fuhrmann

Wirth

usual

[9],

[24],

Salamon

interpretation

of

(3.1)

[27], is

for [11], [28],

as

a

~oc.

from W to z,2 (0,0o; Y) and we follow the formulation used in Weiss

[34].

Definition

3.1

Suppose

T = (Tr)rz 0

is

a

C0-semigrou p

with

infinitesimal

generator A on the Hilbert space X and that IV is a dense Y-invariant subspace of X. Then C is an admissible observation operator for $ with respect to W if for

some

(and

hence

any)

r>0,

the

observability

map Lr has a continuous

extension to a bounded map from X to Lz(0,oo; Y), where LT is defined for xetV by

] CTtx;

t e [O,r)

(t,-x)(t)

(3.2) 0

Notice that

t_>T.

the reachability and observability maps # r

and Lr

are

duals

T

of

each

other

with

respect

to

the

pairing:

< u , y > = f u,v*dt. 0

Consequently B is an admissible control operator for T, with respect to X_1 if

113 and only if B* is an

admissible

observation

(X*)I, where A'_I is defined by (2.4) and

operator

for

T~with respect

to

(X*)I is' the domain of A*~ D(A*),

with the graph norm. This duality was used extensively by Salamon [27]~ [28] to deduce

results

about

control operators, slightly

admissible

observation

operators

from dual

admissible

where he used the terminolbgy "H1 and H2 hypotheses" and

different,

but

equivalent

definitions.

More

general

duality

results

can be found in Weiss [34]. FirSt we examine the relationship of the concept of

an

admissible

observation

operator

to

that

of

an

abstract

linear

observation system. Definition 3.2

Let X and Y be Hilbert spaces. An abstract linear observation

system on Y and X is a pair (L,T) where T=(T~)r_> o is a Co-semigrou p on X and

[=(Lr)r>0

is a family of bounded maps from X to L2(0,oo;Y) such that

the

composition property (1.12) holds and Lo=O.

It is easy to verify that $t with respect

if C is an admissible observation operator

to W as in Definition 3.1, then

it defines an

abstract

for

linear

observation system with Lt defined by (3.2). The

following

systems is contained

representation in Salamon

theorem

for

abstract

linear

[29] in a slightly different

form;

observation the

given

vcrsion follows Weiss [34] where he proved it for the more general situation in which X and Y are Banach spaces and the output functions are in loc Lp (0,co; Y), where p~[1,co]. Let us denote by X 1 the space D(A) with the graph norm. Theorem 3.3

Let X and Y be Hilbert spaces and suppose

abstract

observation

linear

system

on

Y

and

X.

Then

that (L~T) is an

there

exists

a

unique

C ~ £ ( X I , Y ) such that for any x e X l and any t>_O

(L~)(t) = C~Ttx.

Since

Ca

will

in

general

be

different

(3.3)

from

an

original

admissible

observation operator C, we introduce the following definition.

Definition 3.4

Given

the abstract linear observation system

(L,T) on

tlle

114 Hilbert spaces X and Y, we call the unique o p e r a t o r Ca~£(XI~Y ) of Theorem 3.3 the abstract observation operator. It is defined by for xED(A).

Cax = (Ecox)(O)

Clearly Ca is an

admissible observation

operator

for

(3.4)

]'t

with

respect

to X1

according to Definition 3.1. So linear

the

results

on

observation

systems

parallel those

abstract

far

admissible

linear control systems and

observation for

operators

admissible control

and

abstract

operators

the duality between Theorems

and

2.3 and 3.3

is obvious. The

discussion which now follows has

control

systems

systems,

which

and was

it on]y

is

a

special

recently

no counterpart

feature

discovered

of by

for

abstract Weiss

abstract

linear

[3,I].

It

linear

observation holds

more

~[OC

generally for X and Y Banach spaces and with t,p (0,oo; Y) as output trajectory space for p~[1,oo].

D e f i n i t i o n 3.5

Let X and Y be Hilbert spaces~ 1" a C0-semigrou p on X and

suppose that CE£(Xj,Y). Then the Lebesgue extension of C (with respect to l),

CL: D(CL)-->Y is defined by 1

1¢

Ckx = 1im C -~f T~xda •r-~ 0

(3.5)

0

D(CL) = {x~X [ the limit in (3.5) exists}.

Weiss showed that CL is an extension of C in the sense that

x~ c.., D(CL) ~ X. Since C is typically not

closed,

D(CL)

will not

(3.6) be complete under

norm, but it does become a Banach space under the following norm

the graph

115

Ilxllc = Ilxll + With respect

to

this

CLe£(D(CL),Y)

norm

(3.7)

,-.(o,~]sup IIC!r!~'r,,xdall. and

the

embeddings

in

(3.6)

it

makes

are

continuous. The possible

significance to

give

a

of

the

Lebesgue

pointwise

simple

extension,

CL,

interpretation

is

of

the

that

observation

(3.1) for every x in the original state space, X and almost all t_>0,

it map

whereas

Theorem 3.3 only applies to x in the smaller space D(A).

Under the same assumptions as in Theorem 3.3, if CL is the

Theorem 3.6

Lebesgue extension of Ca, then for any x e X ,

t>_O we have that TtxeD(Cz) if and

only if U_,,x has a Lebesgue point in t. Furthermore, ([~x)(t) = CLTtX

(3.8)

almost everywhere in [O,co). 2 The

above

is

a

surprisingly

simple,

elegant

representation

for

the

observability map which one could never have anticipated by examining the many examples of well-posed observation maps which abound in the literature. next

section

representation

we of

shall the

use

transfer

this

Lebesgue

function

of

extension the

to

input-output

obtain map

In the

a

simple

for

regular

abstract linear systems.

4. I n p u t - o u t p u t In

section

linear control operators

and

admissibility 2

we

systems abstract

considered and

admissible

in section

linear

control

operators

3 we considered

observation

systems.

Here

admissible

and

abstract

observation

we consider

the

loc

2We recall that y ~ L 2 (O,oo; Y) has a Lebesgue point in t if the limit of 1 t+r

-~f

y(a)da exists a~ r-*O. Almost every t_>O is a Lebesgue point for y

L

and the limit equals y(t) a.e.

full

116

abstract linear system ~ = (T,(I,,L,F) of Definition 1.1. An obvious example of an abstract linear system on the Ililbert spaces U, X

and

Y can

be

obtained

analogously

to

the

finite-dimensional

case

(cf.

(1.10)-(1.15)) by considering the following state space system:

~c(t) = Ax(t)+Bu(t)

(4.1)

y(t) = Cx(t)+Du(t)

(4.2}

where A generates the Co-semigroup l on X and BeE(U,X), Cef.(X,Y), Def.(O,Y). This defines an abstract linear system E=(T,~I,,L,F), where the reachability and observability maps, ,I, and L, are given by (2.4) and (3.2} respectively and the input-output map F is given by l

C f T~_aBu(a)da+Du(t ) (F~u)(~) =

; 0<_t
0

(4.3)

0 When one

tries

;t>v

to

obtain

a~l expression like

(4.3)

for

the

input-output

map F for unbounded, but admissible control and observation operators B and C, $

one is confronted with the problem that f Tt_aBu(a)da will not be in D(C) in 0 general. In facL having admissible control and observation operators B and C is not

sufficient

to guarantee

This was realized for

a

frequency-domain to

of a

bounded

input-output

in Salamon [28], where he imposed a time-domain

"well-posed"

relationship

the existence

system.

This

condition

the

in

concept

of

was

replaced

Curtain

[5]

an

abstract

by

a

and linear

more

Salamon system

~

map.

condition

easily

verifiable

[29],

and

the

was clarified

in

Curtain and Weiss [7], where they used the terminology "well-posedness of tile triple in

(A,B,C)

the

connection

with

terminology condition

(in

the

introduction

sense we

differential

"input-output needed

for

of

feel

linear that

equations

systems and

so

of

the

admissibility input-output

theory}".

"well-posedness" here

admissibility

As remarked is

we

triple of

already prefer

(A,B,C)". a

triple

to

earlier

used

in

use

the

The

extra

(A,B,C)

is

expressed in terms of its transfer functions. Definition 4.1

Let U~ X, Y, V and W be tlilbert spaces such that W c X c V and let

Be£(U,V), Ce£(W,Y) and T = ( T r ) r > 0 be a C0-semigroup on X. Suppose that B is an

117

a&nissible

control

operator

for

]-t

with

respect

to

V and

that

C

is

an

admissible observation o p e r a t o r for $t with respect to W. Then we define the

transfer functions of the triple (A,B,C) to be the solutions, G: p(A)-)£(U,Y) of

(4.4)

G(s ) -G(fl) = _ C ( s [ - A)-l(flI- A)-aB s-fl for s, fl~p(A), s¢fl.

We remark that since B is an admissible control o p e r a t o r with respect to

It,

(flI-A)-lB

admissible

is

an

£(U,X)-vMued

observation

operator

analytic

with

function

respect

£{X,Y)-valued analytic function. Both (flI-A)-lB some

right

always

half-plane

exist

as

Co = { s e C : Res>0}.

£(U,Y)-vaiued

to

and l't,

since

is

an

is

a

and C ( s I - A ) -1 a r e analytic on

Consequently

functions

C

C ( s I - A ) -L

which

the

are

transfer

analytic

in

functions

some

right

half-plane, Ca. They differ only by an additive constant, DeE(U,Y). The point is that they need not necessarily be bounded on any right half-plane, Co. We impose

this

as

input-output

an

extra

assumption

admissibility. Notice that

on as

the a

(A,B,C)

triple

consequence

and

call

of Theorems

this

2.3 and

3.3, C and B in (4.4) may be replaced by the a b s t r a c t observation and control operators Ca and Ba respectively. Under the same assumptions as in Definition 4.1,

Definition 4.2 the

(A,B,C)

triple

operator

for

observation

$

is

with

operator

input-output respect for

T

to with

admissible the

Hilbert

respect

to

if

B

is

space the

an

V, Hilbert

we say that

admissible

C

is

an

space

control

admissible tV and

its

transfer functions are bounded on some h a l f - p l a n e Ca . The main result in Curtain and Weiss [7] is that is input-out, put

admissible

corresponds

to

the

notion

a triple (A,B,C)

which

of

linear

an

abstract

system.

Theorem 4.3

(i) An iT~puS-output admissible Sripl¢ (A,B,C) determines a family

of absSract linear systems ~=(T,~I,,i.,F) where I is the Co-scrnigroup generated

118 by A and ,I~ and L are the teachability and observability maps defined by (2.3) and (3.2) respectively. The family of input-output maps, F, is defined by the equation t

(F~u)(t) : Co[f Tt-,)Bau((7)da- (~I-A)-'Bau(t)] + G(/3)u(t)

(4,5)

0

• 1)2 , u~wo,zociu,co; U) 3 and I=~ is defined by (1.8) with "r=co.

for

(ii) An abstract linear systgTn ~ = ( T ) , L , F )

determines the unique input-output

admissible triple (A,Ba,Ca) where Ca and Ba are the abstract observation and control operators associated with (T,~) and (k,T) respectively.

We

emphasize

that

an

input-output

admissible

triple

(A,B,C)

only

determines G and hence F0D up to an arbitrary additive constant DEL(U,Y). To a given

transfer

function

G

corresponds

a

unique

F

defined

by

(4.5)

and

conversely to a given F corresponds exactly one transfer function G. The key equation

(4.5),

was extended

was

initially

derived

~loc. ^

to all "u.EL2 {U,o0; U )

in Salamon

[29]

for

smooth

inputs

and

in Weiss [35], provided Ca is replaced by

its Lebesgue extension, CL. While Ba in (4.5) may be replaced by B, (4.5) may not hold for the original C operator

(unless the domain of C includes D(A)).

To understand the main idea of Theorem 4.3 let us consider the case where T is an exponentially

stable

C0-semigroup. Then Theorem

4.3 says essentially

that

the input-output map F¢0 is bounded from L2(0,oo; U) to L2(0,oo; Y) if and only if

its

Laplace

half-plane. that

transform,

(An analagous

T is not

stable.)

the

transfer

G,

function,

is

but more complicated statement

This seemingly obvious fact

is not

bounded holds for

on

some

the

case

so straightforward

to prove as one might think. Two different proofs are given in Salamon [29] and Curtain and Weiss [7]. So

3.

1,2

Theorem

4.3

clarifies

the

relationship

between

an

abstract

Wo, loe denotes the~oc space of absolutely continuous functions whose derivatives are in L 2 (O,oo;U), and which are zero at t=O. It is a dense T loc subspace of b 2 (O,co; U).

linear

119

~=(T,~,,L,F) on the Hilbert spaces (], X and Y and input-output

system

admissible triples the

(A,B~C) and

finite-dimensional

identified

the

corresponding give the

formulas

abstract

integral (abstract

we have (1.3)

operator

obtained to

(1.7).

appropriate

domain

sought

In

other

representation

linear system) representation

frequency

the

differential

words,

representation

we

corresponding

(ii)

(iv) and at

representation

generalization

of

have

to

the

the same time the

(iii)

via

(4.5).

(i)

and

the

It

remains

to

generalization

of

the finite- dimensional formulas (1.1) and (1.2). Theorem output

4.4

Let ~=(T,,I,,I_,F)

spaces

generator

of

U T,

and Ba

Y is

be an abstract

respectively. the

linear system with input

Suppose

abstract

control

that

A

is

the

operator,

Ca

is

observation operator and C L is its Lebesgue extension. xoeX

and

any

~lOC

u ~ L 2 (O,oo;U),

abstract

any flep(A), riOts

x: [O,c~)-~X and

the functions

infinitesimal the

Then for

and

yeL,2 I0,¢¢;Y)

defined by x(t) =

Ttxo + "I,tu

(4.6)

y = 4 , z o + F~u

(4.7)

satisfy the following equations a.e. in t>O

(4.8)

x(t) = Ax(t)+Bau(t )

y(t) = c d x ( O - (~I-A)-~a,,u(t)] Moreover,

x( t )

is

the

unique

strong

+a(~)u(t).

continuous

(4.9)

solution

of

(4.8)

under

tl~

iuitial condition x ( O ) = x o.

So

(4.8)

and

dimensional formulas differential

(4.9)

are

(1.1) and

representation

for

the (1.2) an

sought and

generalizations

we have

abstract

established

linear

system

to the on

the

finite-

appropriate the

Hilbert

spaces X, Y and U. Theorems 4.3 and 4.4 show that an input-output admissible triple satisfies

(A,B,C)

the

corresponds differential

via

F.

equations

to

the (4.8),

admissible (4.9)

terminology "well-posed" used in Curtain and Weiss [7].

and

triple this

(A,Ba,CL)

motivated

which the

120

Although between

we

the

have

four

achieved

different

types

linear systems, the equations have

wished

and

(1.2),

for.

they

Unlike depend

defined

up

to

the

fact

class of abstract

of

on

life

of

clarifying

representations

corresponding

the

for

the

relationships

infinite-

parameter

finite-dimensional fl

and

the

dimensional

additive constant to

be

borne.

(A,B,C).

by

However,

formulas

transfer

function G and the input-output

arbitrary

an

is a

of

goal

(4.5) and (4.9) are not as elegant as we might

Moreover, both the transfer situation

our

(1.7)

function

G.

map F are only In

general

in Weiss [35],

a

linear systems was identified which has the property

this

natural that

determines a unique feedthrough operator and for which (4.5) and (4.9) can be simplified considerably. Definition

4.5

Let ~ be an abstract

linear system with input space U and

output space Y. We say that /2 is regular if for any ueU, the corresponding

step response has a Lebesgue point at 0. By the step response corresponding to v~U is meant the function Yv -- Fray where

in

(4.10)

v

denotes

that

the

the

constant

(4.10) function

on

[0,co)

equal

to

v

everywhere.

We

remark

generalization

of

infinite-dimensional of

~

being

the

step

response

finite-dimensional

defined

by

concept,

(4.10) except

is

that

case Yv need not be continuous. An hnportant

regular

is

that

the

the

feedthrough operator De£(U,Y)

natural in

the

consequence is

uniquely

defined by the following limit

1

/"

Dv = "1"40 lim ~ J" yv(a)do.

(4.11)

0

Most systems arising in practice are regular and the assumption that ~ be regular

eliminates

defining

regular

proposition.

the

mathematically

abstract

linear

pathological systems

are

cases. given

Alternative in

the

ways

of

following

121

Suppose that ~=(T,~I,,L,F) is an abstract linear system with

Proposition 4.6

input space U, state space X and output space Y. I f A is the in[initesimal generator of T, Ba and Ca are the abstract control and observation operators and CL is

the Lebesgue

extension

of

Ca, then

the following

conditions

are

equivalent. (i)

~ is regular;

(ii}

CL(sI-A)-JBa is an analytic £(U,Y)-valued function of s on p(A);

(iii)

for stone sep(A) and any veU, (sI-A)-IBaveD(Ct.). We remark

admissible

that

Proposition

(A,B,C)

triple

4.6 provides

generates

regular

a

way of

abstract

testing

linear

whether

an

using

for

systems

example the results in Salamon [28]) and it motivates the following defintion. Definition Definition conditions

Let (A~B,C) be an input-output

4.7

4.2. Then (ii)

or

(iii)

of

admissible

triple according to

(A,B,C) a regular admissible triple if either

we call

Proposition

4.5

hold,

where

Ba and

Ca are

of the

abstract control and observation operators and CL is the Lebesgue extension of

ca. Regular abstract

linear systems have a nice simple representation

for the

input-output map which is independent of the transfer function. Theorem 4.8

Suppose that ~ is a regular abstract linear system with input and

output spaces U and Y respectively. I f by (4.11) and u ~

r[OC

D is its feedthrough operator defined

(0,oo; U), then for almost all t>_O t

f Tt-~Bau(a)daeD(CL) 0

and t

(F©u)(t) = CL] Tt_aBau(a)da+Du(t)

(4.12)

0

Conversely,

a

regular

admissible

triple

(A,B,C)

together

with

a

given

122

D e £ ( U , Y ) determines a regular abstract linear system ~ = ( T , I , , L , F ) where ,I~ and L are defined by (2.3) and (3.3) (or (3.8)) as before and F is now defined by (4.12).

This

is

now

a

nice

generalization

of

the

finite-dimensional

case

(1.7). So here we see the crucial role played by the Lebesgue extension Ct. of the abstract observation operator Co; with Ca we can say no more than (4.5), in which we need the transfer function. Combining all the previous results we arrive

at

an

elegant

representation

theorem

for

regular

abstract

linear

systems, which generalizes theorem for systems with bounded operators given by (4.1), (4.2). T h e o r e m 4.8 and

output

generator

spaces of

observation operator

Let ~=(T,,I,,k,F) T,

U and Bo

operator,

of

~.

is

Y

respectively.

the

CL is

Then

be a regular abstract linear systevn with input abstract

its

for

any

Suppose

control

that A is the infinitesimal

operator,

Lebesgue extension xo~X

and

Co

and

is

D is

abstract

the feedthrouyh

r ]OC

any

the

u e ~ 2 (0,co;U)

the

functions

loc

x: [0,oo)-->X and y ~ L 2 (0,oo; Y) defined by x(t) = Ttx o + fftu

(4.13)

y = t . z 0 + F.u

(4.14)

satisfy the following equations a.e. in t > 0

In

particular,

the

~c(t) = Ax(t)+Bau(t)

(4.15)

y(t) = C L x ( t ) + P u ( t ) .

(4.16)

function

x

is

the

unique

strong

continuous

solution

of

(4.15) under the initial condition x ( O ) = x o and x(t)eD(CL) a.e. in t>O. Notice that to

~,

then

the

if (A,B,C) trajectory

is an x(t)

input-output

defined

by

admissible

(4.13)

triple

also satisfies

corresponding (4.15)

almost

everywhere with Ba replaced by B (Salamon [28], Lemma 2.5). However, Ct. in {4.16)

may

not

Theorem 4.8 terms

of

the

be

replaced

is that regular

regular

by

C in general.

abstract

admissible

linear

triple

The

correct

systems have

(A,Bo,Ca)

and

a

interpretation

a representation unique

of in

feedthrough

123

operator,

D,

differential

and

their

equations

x(t)

trajectories

(4.15)

and

(4.16).

and In

y(t)

satisfy

particular,

if

the

B

abstract

and

C

are

bounded, then B = Ba and C = Ca = CL. A regular system also has a simple frequency domain representation.

With the notation of Theorem 4.8 if ueL2(0,oo; U), then,

Proposition 4.9

tile

Laplace transform of y, exists and for s e C vaith Re s sufficiently large ~(s) = C ( s l - A f l x o + G ( s ) f z ( s )

(4.17)

where the transfer functiou G is given by G(s) = CL(SI-A)-IB+D

(4.18)

lim G(A)v = Dv

(4.19)

arid moreovcr, for any y o u

A-..~

~here A is assumed to be real.

So we see representations (1.1)-(1.7).

It

that

regular

which is

a

systems have

are

reminiscent

pleasing

and

rather

differential of

the

unexpected

and

frequency

domain

finite-dimensional

ones,

result

many

which

took

years to be unravelled.

5. An example (see Curtain and Weiss [7] for details) Le~ X = g 2

U=Y=E,

and

define

tile

following

operators

for

sequences

x = ( x k ) e X and u ~ C (Ax)k = -kxk;

Q x = E xk; k~N

(Ou)k = u

(5.1)

C2x = ~ (-1)kxk

(5.2)

ken

124 Then A generates the C0-semigrou p on g2 given by

(Ttx)k

=

e'ktXk;

ken

(,5.3)

for any x - - ( x D e g 2.

X-i = {(xD: (-~)EG}; Xl={(xk): (kxk)e~2} and

B~£.(U, Xq), Cl and C2~£(X1,Y ). Using the Carleson measure criterion (see

iio and Russcl [14] and Weiss [32]) it follows that B is an admissible control operator for T with respect to X_~ and C~ and C2 are admissible observation operators

for

T

with

respect

to

X i.

For

input-output

admissibility

triples wc need to evaluate the transfer function from (4.4). For

of

the

(A,B, Cx) we

obtain Gt given by

- k~. ~

a , ( s ) = GI(O)

and

G1(s) is not

input-output

bounded

on

any

(5.4)

s

half-plane,

admissible. On the other hand,

so

that

(A,B,C1) is not

(A,B,C2) has the transfer function

G~ given by

G2(s)

= G2(O) -

E ken

and

this

generates

is bounded abstract

on

linear

Co, and systems

so

(-1)k(~- s+-V~) 1

15.5)

(A,B,C~) is input-output a&nissible and

(depending

on

D).

From

Proposition

7.2

in

Weiss [34] we have that the domain of Lebesgue extension C/, of C2 contains the space

{(xk): E

(-1)kXk converges}

ken

and for all x in this space we have

CLX for xED(CL). It follows that

=

E (-1)kx~

ken

125

(sl -A)-IBv = (-}-(--~)eD(CI,) v for any sep(A) and all v~C, so by Proposition 4.6, all systems ~] generated by

{A,B,C2) are regular and the input-output map has the representation (4.12), where

D is

the

arbitrary

feedthrough

term.

The

transfer

function

has

the

representation Gz(s ) = CL(st-A)-IB+D.

(5.6)

For another completely worked example (the one-dimensional heat equation with Dirichlet boundary control and point observation) see Curtain and Weiss [7]. Further examples can be found in Salamon [28] and Curtain [5] including some on retarded equations and hyperbolic p.d.e.'s.

Acknowledgements The author is grateful to Dietmar Salarnon, Hans Schumacher, George Weiss and Hans Zwart for their valuable comments and suggestions.

References and

and

J.S. Baraa~

2.

Realization Theory, SIAM J. Control 13, 1975~ 221-224. J.S. Baras, R.W. Brockett, and P.A. Fuhrmann: State Infinite Dimensional Systems, in "Mathematical C. Marchesini

R.W. Brockett:

H2-Functions

i.

and

S.K. Mitter,

eds.

Infinite

Dimensional

Space

Models for

Systems

pp. 204-225,

Theory",

LNEMS

131,

Springer-Verlag, New York, 1976. 3.

R.F. Curtain,

A.J.

Pritchard:

Infinite Dimensional Linear Systems Theory,

Lecture Notes in Information Sciences 8, Spriager-Verlag, 1978. 4.

D. Salamon: Finite Dimensional Compensators for Infinite Dimensional Systems with Unbounded Input Operators, SIAM J. Control &

R . F . Curtain,

Optim. 24, 1986, pp. 797-816. 5.

R.F. Curtain:

Time and

Well- Posedness of Infinite Frequency Domain, Research

Groningen, August 1988.

Dimensional Linear Systems Report TW-287, University

in of

126 6.

A Synthesis of Time and Frequency Domain Methods for the

R.F. Curtain:

Control of Infinite-Dimensional Systems: A System Theoretic Approach (to appear in the series SIAM Frontiers in Applied Mathematics). 7.

R.F. Curtain,

Sense

of

Well Posedness

G. Weiss:

Linear

Systems

Theory),

of

Triples

Proceedings

of of

Operators the

(in

the

Conference

on

Distributed Parameter Systems, Vorau, Austria, July 1988, to appear. I. Lasiecka,

W. Schappacher: Feedback Boundary Control Problems

8.

W. Desch,

9.

for Linear Semigroups, Israel J. of Math. 51, 1985, pp. 177-207. S. Dolecki, D.L. Russell: A General Theory of Observation

and

Control,

SIAM J. Control & Optim. 15, 1977, pp. 185-220.

Boundary

10. H.O. Fattorini:

Control Systems,

SIAM

J.

Control

6,

1968,

pp. 349-385. 11. P. Fuhrman:

Linear Systems and Operators in Hilbcrt Space,

McGraw-Hill,

New York. 1981.

Systems with Infinite- Dimensional State Space Approach, Proc. IEEE, 64, 1976, pp. 145-160.

12. J.W. Helton: 13. J.W. Helton:

Engineering,

Operator Theory, Regional

Analytic

Conference

Functions,

Series

in

Space:

The

IIilbert

Matrices and Electrical

Mathematics,

No. 68,

AMS,

Providence, RI, 1987.

Admissible Input Elements for Systems in Hilbert Space and a Carlcson Measure Criterion, SIAM J. Control & Optim. 21, 1983,

14. L.F. l|o,

D.L. Russell:

pp. 614-640. 15. R.E. Kahnan,

P.L. F~lb,

M.A. Arbib:

Topics in Mathematical System Theory,

McGraw-Hill, New York, 1969. 16. I. Laziecka: Unified Theory for Abstract Parabolic Boundary

S~'migroup Approach, Appl. Math. Optim. 6, 1980, pp. 287-333. R. Triggiani: A Cosine Operator Approch

17. I. Laziecka,

Problems to

A

Modelling

L2(0,T; L2(F))-Boundary Input Hyperbolic Equations, Apph Math. Optim. 7, 1981, pp. 35-93. 18. I. Lasiecka,

Stabilization

R. Triggiani:

of Neumann

Boundary

Feedback of

Parabolic Equations: The Case of Trace in the Feedback Loop, Appl. Math. Optim. 10, 1983, pp. 307-350. 19. I. Lasiecka,

Regularity of Hyperbolic Equations under Boundary Terms, Appl. Math. Optim. 10, 1983,

R. Triggiani:

L2(0,T; L2(F))-Dirichlet pp. 275-286. 20. I. Lasiecka,

R. Triggiani:

Feedback Semigroups

and

Cosine

Operators

for

Boundary Feedback Parabolic and tlyperbolic Equations, J. of Diff. Eq. 47,

127

1983, pp. 246-272. 21. I. Lasiecka,

J.L. Lions,

R. Triggiani:

Non

tlomogcneous Boundary

Value

Problems for Second Order Hyperbolic Operators, J. Math. pures et appl. 65, 1986, pp. 149-192.

Optimal Control of

22. J.L. Lions:

Equations,

Die

Systems

Grundlehren

der

Governed by

mathematischen

Partial

Differential

Wissenschaften

in

Einzeldarstellungen 170, Springer-Verlag, 1971. 23. J.L. Lions,

Non

E. Magcnes:

Applications,

Vol. I

and

Homogeneous Boundary

Value Problems and

II,

der

Die

Grundlehren

mathematischen

Wissenschaften in Einzeldarstellungen 181 and 182, Springer-Verlag, 1972. 24. A.J. Pritchard,

h. Wirth:

Unbounded Control and Observation Systems

and

their Duality, SIAM J. Control & Optim. 16, 1978, pp. 535-545. 25. D.L. Russell:

A Unified Boundary Controllability Theory for IIyperbolic and

Parabolic Partial

Differential

Equations, Studies

in

Applied

Mathematics

52, 1973, pp. 189-211. 26. D.L. Russell:

Dimensional

Some Remarks on

Transfer

Function

L i n e a r Systems,

Research

Report,

Methods for Virginia

Infinite

Polytechnic

Institute and State University, October 1988. 27. D. Salamon:

Control and Observation of Neutral Systems,

Pitman,

London,

1984. 28. D. Salamon:

Observation:

Infinite

Dimensional Systems

with

Unbounded Control and

A Functional Analytic Approach, Transactions

of

the

A.M.S.

300, 1987, pp. 383-431. 29. D. Salamon: Realization Theory in Hilbert Space, Matt1. Systems Theory 21, 1989, p. 147-164. 30. T. Seidman:

Observation and

Prediction for

One-Dimensional

Diffusion

Equations, J. Math. An. Appl. 51, 1975, pp. 165-175. 31. D. Washburn: A Bound on the Boundary Input Map for Parabolic Equations

with Applications to Time Optimal Control, SIAM J. Control & Optim. 17, 1979, pp. 652-671. 32. G. Weiss: Systenrs

Admissibility of Input Elements for Diagonal Scmigroups on d2, and

Control

Letters

10,

1988,

pp. 79-82.

33.

C. Weiss:

Admissibility of Unbounded Control Operators, SIAM J. Control & Optim. 27, 1989 (to appear). 34. G. Weiss:

Admissible Observation Operators for

Linear Semigroups,

Israel

J. Math. 65, 1989, pp. 17-43. 35. C. Weiss: The Representation of Regular Linear Systems on Hilbert Spaces,

128

Proceedings

of

the

Conference

on

Distributed

Parameter

Systems,

Vorau,

Austria, July 1988, to appear. 36. G. Weiss:

The

Representation

of

Linear

Systems

on

Banach

Spaces,

in

preparation. 37. Y. Yamamoto:

Realization

Theory

of

Infinite- Dimensimml

Linear

Systems,

Part I, Math. Systems Theory 15, 1981, pp. 55-77. Part II, Math. Systems Theory 15, 1981, pp. 169-190.

Symmetric Modeling in System Identification M. Deistler Technische Universit~t Wien Institut fLir Oekonornetrie und Operations Research Argentinierstrasze 8/119, A-1040 Wien, Austria

The paper is c o n c e r n e d with the r e a l i za t i o n p r o b l e m for linear dynamic e r r o r s - i n - v a r i a b l e s models where the component p r o c e s s e s of the n o i s e term are m u t u a l l y u n c o r r e lated. The a n a l y s i s is b a s e d on the second m o m e n t s of the observations.

I. Introduction

In identification

of linear

of noise m o d e l i n g

is to add all noise

the same for our purposes

(dynamic)

is called the e r r o r s - i n - e q u a t i o n s

is p r e d i c t i o n

the o b s e r v e d

inputs.

general and more where

However

(in principle)

may

This is called the e r r o r s - i n - v a r i a b l e s approach.

This a p p r o a c h

also

allows

symmetric way of s y s t e m m o d e l i n g , the c l a s s i f i c a t i o n

of a p p l i c a t i o n s

of n o i s e m o d e l i n g

all v a r i a b l e s

is

aim of

a more

is a p p r o p r i a t e ,

be c o n t a m i n a t e d

by noise.

(EV) or l a t e n t v a r i a b l e s

for a more g e n e r a l

where

of the v a r i a b l e s

approach

if the p r i m a r y

is

this

from their own past and from

in a n u m b e r

way

This

way

(or w h i c h

In e c o n o m e t r i c s

(EE) approach.

in p a r t i c u l a r

of the o u t p u t s

symmetric

the c o n v e n t i o n a l

to the o u t p u t s

to the equations).

widely used and is a p p r o p r i a t e modeling

systems

into

the n u m b e r inputs

a n d more

of e q u a t i o n s

and o u t p u t s

and

is ob-

130 tained from data rather than f r o m a priori knowledge.

The m a i n cases

where the EV approach is appropriate are: (i) If we are i n t e r e s t e d in the " t r u e " s y s t e m g e n e r a t i n g the data (rather than in p r e d i c t i o n or in just c o d i n g data by s y s t e m parameters)

and we cannot be sure a priori that the o b s e r v e d inputs are

not corrupted by noise. For i n s t a n c e the "true" s y s t e m may relate to a c e r t a i n "physical" t h e o r y fictions,

[of course,

but they may be g o o d ones]

"true" systems are always

and the n o i s e may c o r r e s p o n d to

m e a s u r e m e n t noise. This is the c l a s s i c a l m o t i v a t i o n for EV modeling. (ii}

If we want to explain the "essential part" of a h i g h d i m e n s i o n a l

data v e c t o r by a r e l a t i v e l y small n u m b e r of variables

(or factors).

(in general unobserved)

This is the c l a s s i c a l m o t i v a t i o n for factor

analysis or for p r i n c i p a l c o m p o n e n t analysis,

the most f a m i l i a r

a p p l i c a t i o n of which are m e n t a l tests w h e r e the data c o r r e s p o n d to the test scores and the factors to m e n t a l factors. (iii) If we have no s u f f i c i e n t a priori i n f o r m a t i o n about the number of equations in the system, about the c l a s s i f i c a t i o n of the v a r i a b l e s into inputs and outputs or about causalitydirections, we have to p e r f o r m a more symmetric system modeling, w h i c h demands a more m o d e l i n g in order to avoid "prejudices"

symmetric noise

(Kalman 1982). This g i v e s a

fairly general setting for the i d e n t i f i c a t i o n p r o b l e m case, which is the only case we c o n s i d e r in this paper]

[in the linear where in parti-

cular the number of e q u a t i o n s may be d e t e r m i n e d f r o m the data rather than from a priori knowledge. The statistical a n a l y s i s for the EV case turns out to be significantly more c o m p l i c a t e d c o m p a r e d to the EE case, m a i n l y because the structure of the relation between the

(population)

s e c o n d m o m e n t s of the ob-

servations and the system p a r a m e t e r s is more complicated. central p r o b l e m s in this c o n t e x t is a basic the sense that in general

One of the

"non-identifiability"

in

the system p a r a m e t e r s are not u n i q u e l y

d e t e r m i n e d from the second m o m e n t s of the o b s e r v a t i o n s ,

since the

separation between the system and the noise part is not unique without imposing a s s u m p t i o n s w h i c h in general are rather restrictive. The kind and amount of a priori a s s u m p t i o n s

imposed clearly varies

with the p r o b l e m setting. The two e x t r e m e p h i l o s o p h i e s are either to add as little a d d i t i o n a l a s s u m p t i o n s on the noise as p o s s i b l e to a v o i d uniqueness

"prejudice"

(Kalman 1982)

in order

and to d e s c r i b e the r e s u l t i n g non-

in the r e a l i z a t i o n p r o b l e m or, on the other hand,

to impose

131 sufficiently many a d d i t i o n a l a s s u m p t i o n s

in o r d e r to g u a r a n t e e identi-

fiability. We here will mainly take the first point of view, w h i c h has been e m p h a s i z e d in p a r t i c u l a r by Kalman

(1982,

should be said, that in a number of a p p l i c a t i o n s und M i t t a g

1986, Fuller

1987),

1983). However,

(see e.g. S c h n e e w e i B

for i n s t a n c e if r e p e a t e d m e a s u r e m e n t s

for fixed true values are possible,

to m e n t i o n one case,

identifiability

may be o b t a i n e d without i m p o s i n g p r e j u d i c i a l assumptions. approach,

it

For the first

lack of k n o w l e d g e c o n c e r n i n g the "true" system is c a u s e d by

two facts; first by s a m p l i n g e r r o r s and second by n o n - i d e n t i f i a b i l i t y . In a n o n - s t o c h a s t i c world,

the n o n - i d e n t i f i a b i l i t y m e n t i o n e d above

arises from a v a r i a t i o n of norms in d e f i n i n g a m e a s u r e of

(mis)fit

of a system to data. Another feature of i d e n t i f i c a t i o n in the EV case is that non Gaussian case)

e.g. m o m e n t s of order unequal to two

may provide additional

identifying information

(in the (and one)

(Geary 1942,

Reiers~l

1950). Whereas,

for the stable case at least,

the theory of i d e n t i f i c a t i o n

for the EE case has r e a c h e d a c e r t a i n stage of c o m p l e t e n e s s now (see e.g. Hannan and D e i s t l e r

1988), due to the p r o b l e m s m e n t i o n e d

above, in i d e n t i f i c a t i o n in the EV case there is still a great number of u n s o l v e d problems. their great appeal,

Accordingly,

in actual a p p l i c a t i o n s ,

in the p r o p e r l y dynamic case at least,

despite of such models

have been used to a limited extend only. For the static case,

the EV p r o b l e m has a long history. More than

hundred years ago, A d c o c k

(1878)

recommended,

that in fitting a straight

line to a scatter plot when the errors to the v a r i a b l e s are uncorrelated and have the same variances,

the d i s t a n c e of a point to the line

should be taken o r t h o g o n a l to the line [ r a t h e r than p a r a l l e l to the y-axis as in the typical r e g r e s s i o n case ]. In a more g e n e r a l context, where the v a r i a n c e s are unknown, Gini [1921]

this p r o b l e m has been a n a l y s e d by

and for more than two v a r i a b l e s by Frisch [1934]

and

Koopmans [1937]. A second r o o t is factor analysis, which, in the beginning, has been d e v e l o p e d and used m a i n l y in psychology,

specifically

in order to p r o v i d e m o d e l s for h u m a n ability and behavior.

The e a r l i e s t

papers here are Pearson

(1901) and Spearman

(1904). F r o m an a b s t r a c t

point of view the EV model and the factor model are equivalent.

How-

ever, in "classical" EV a n a l y s i s emphasis was given to models with one or with a few linear relations,

whereas

in factor a n a l y s i s e m p h a s i s

was given to m o d e l s w i t h one or few factors [and thus with many relations,

in general].

132 It was r e c o g n i z e d by Reiers~l

(1941) and G e a r y

(1943)

that correlation

in the true inputs may help to identify the slope p a r a m e t e r of a line. Systematic investigation of d y n a m i c EV models on the other hand is quite recent [see Geweke (1981), Maravall Novak

(1977), Sargent and Sims

(1979), S S d e r s t r 6 m

(1985), Picci and Pinzoni

(1977), B r i l l i n g e r

(1980), A n d e r s o n and D e i s t l e r

(1984)j

(1986)].

2. The Statement of the P r o b l e m

We consider a d e t e r m i n i s t i c

system of the form

^

w(ZlZ t = 0

(2.1)

where zt is the n - d i m e n s i o n a l v e c t o r of latent unobserved)

variables, where

(i.e. z(zt I t 6~) w(z)

z is used for the b a c k w a r d - s h i f t operator 0,

= (zt-11 t C ~ ) ) =

(i.e. in general

Z W zj j=_~3

and finally where ;

W.6~mxn 3

(2.2)

We will call w the relation f u n c t i o n of the exact relation

(2.1).

Without restriction of g e n e r a l i t y we will assume that m
that w contains no linear d e p e n d e n t rows. form

Clearly systems of the

(2.1) allow for a s y m m e t r i c treatment of all v a r i a b l e s in the

sense that neither a c l a s s i f i c a t i o n of the v a r i a b l e s

into inputs

and outputs nor c a u s a l i t y d i r e c t i o n s have to be specified a priori. Systems of this form are s t u d i e d in detail and Blomberg and Ylinen (1983) .

in W i l l e m s

(1979)

(1986)

The observations are of the form

zt = zt + ut

(2.3)

where u t is noise.

Here, unless the c o n t r a r y is stated e x p l i e i t e l y we will assume

(i)

All processes c o n s i d e r e d are

(wide sense)

stationary

[In addition

all limits of random v a r i a b l e s are u n d e r s t o o d in the sense of mean squares convergence]

133 ^

(ii)

Ez t = Eu t = 0 ^

(iii) Eztu ~

=

(iv)

(zt) and

(zt),

0

(ut) are AIU~A p r o c e s s e s

(i.e. their spectral

^

densities f, f and ~ r e s p e c t i v e l y are rational) matrix. Clearly Eu t = 0 and

and w(z)

is a rational

(iii) are innocent assumptions. A s s u m p t i o n

clearly is a restriction of g e n e r a l i t y are ruled out), nevertheless, assumption is quite common.

(for instance unstable systems

for the statistical analysis such an

Most of the results p r e s e n t e d in this

paper also hold without i m p o s i n g

(iv); again in the statistical ana-

lysis such a rationality a s s u m p t i o n

is frequently imposed and is

justified by the a p p r o x i m a t i o n p r o p e r t i e s of rational functions.

The

assumption E~ t = 0 can e a s i l y be d r o p p e d and is imposed for n o t a t i o n a l convenience only.

In this paper we assume that the information from the o b s e r v a t i o n s consists of their second m o m e n t s only. However it should be noted, that in this context,

in the n o n - G a u s s i a n case, e.g. higher order

moments contain additional Deistler 1986).

i n f o r m a t i o n about the system

In addition,

unless the contrary

(Akaike 1966,

is explicitely

stated, we impose the i d e a l i z i n g a s s u m p t i o n that the population second moments of the o b s e r v a t i o n s are a v a i l a b l e und thus we do not consider sampling aspects. Let us repeat that f, f and ~ d ~ o t e U~e spectral densities of (zt), (it) and (ut) res~-ective3y. Since the raticnal extensionof a raticnal f~iction

from

the unit circle to the complex plane ~ is unique we will identify these functions

(where spectral d e n s i t i e s are c o n s i d e r e d as functions

of e -il rather than of I ) with their e x t e n s i o n s to ~. Note that the ^

rational functions form a field. M a t r i c e s like f, f or w are c o n s i d e r e d as matrices over such fields.

Notions like linear independence,

rank, ^

ect. are u n d e r s t o o d in this sense. Clearly in this sense, e.g. f has rank r if and only if f(k) has rank r k-a.e.

From

(2.3) we have:

f =f

+?

In this framework our p r o b l e m can be stated as follows:

(2.4)

(i)

134 Given f,

(a) c h a r a c t e r i z e the set of all f

(and ~) c o m p a t i b l e with given f

(b) find the m a x i m u m c o r a n k of f ,m* say, among all d e c o m p o s i t i o n s (2.4) of f. Sometimes we also use the n o t a t i o n mc(f)

for m*.

(c) Under which a d d i t i o n a l a s s u m p t i o n s are f and w unique.

In the first step we a n a l y s e the s y s t e m r e p r e s e n t a t i o n in p a r t i c u l a r the d e t e r m i n i s t i c

(2.1) and

r e a l i z a t i o n p r o b l e m under the fiction

^

that z t has been observed.

This p r o b l e m has been a n a l y s e d in a more

general framework in W i l l e m s

(1986,

1987).

Clearly, from an a b s t r a c t p o i n t of view,

: (-

"

-~11,

O,

z~

~'

^

...)

is c o n t a i n e d

( L ~ ) ~, where L 2 is the usual

(2.1) means that

in a linear subspace

(Hilbert)

space of square integrable

functions defined on the u n d e r l y i n g p r o b a b i l i t y a l t e r n a t i v e concrete r e p r e s e n t a t i o n

space

(~,A,P). An

is the factor model

(see e.g.

Picci and Pinconi 1986)

zt = a(z)"ft

(2.5)

where Qo

a(z)

and where

--

X A.z 3 j=_~ ]

(ft) is the

;

A. 6JRnx(n-m) ]

(n-m) d i m e n s i o n a l

factor process.

In this way

the subspace for z is e x p r e s s e d as the range rather as the kernel of a certain matrix. Given f and no a d d i t i o n a l higher order moments of inputs and outputs,

information

[exceeding

(i)-(iv) ] e.g. on

(i t) or on an a priori c l a s s i f i c a t i o n

into

every r a t i o n a l m a t r i x w satisfying

^

w.f = 0

(2.6)

gives rise to a r e p r e s e n t a t i o n see Picci and Pinzoni

(2.1)

[In the sense of a weak realization,

(1986), van Putten and van Schuppen

(1983), i.e. a

r e a l i z a t i o n which is c o m p a t i b l e w i t h the p r o b a b i l i t y structure of

135 the observations]. possible,

Here we w a n t to explain by the system as much as

in the sense,

that the number of equations should be maximal.

In other words, given ~, we choose w in a way such that its rows form a basis for the

(left) kernel of ~. Clearly then, under our assumptions,

every such a basis is c o m p a t i b l e with given ~ and thus for given ~, w is unique up to left m u l t i p l i c a t i o n by a n o n s i n g u l a r rational matrix. Since w can be w r i t t e n as w=p

.c

where p is a least c o m m o n d e n o m i n a t o r polynomial of w, we can always find a polynomial b a s i s c for the kernel of ~. Now let us rearrange A the entries of z t is such a way that the first m columns of c (after rearrangement) form a n o n s i n g u l a r ^l ~t accordingly as (Yt' ~t)' we can write

submatrix, a say. P a r t i t i o n i n g

a(z)~t = b(z)~ t where 9t are outputs,

(2.7)

~t inputs and

(2.7)

is a usual

("ARMA")

representation of a linear s y s t e m with transfer function k = a

-I

.b.

In this way we have d e f i n e d the inputs as those c o m p o n e n t s of^~ t w h i c h correspond to a maximal

set of linearely independent rows of f and

this definition is not unique and not f is known

(at best)

unraveling f from f in

in general. However,

in general only f

and the main c o m p l i c a t i o n arises in

(2.4).

Let us first consider the static case,

i.e. the case where all p r o c e s s e s

considered are white noise and where w is constant. write

In this case we

(2.4) as

z = ~ + 7

(2.8)

where Z,~ and ~ are the c o v a r i a n c e m a t r i c e s of zt,z t and u t respectively. It is s t r a i g h t f o r w a r d to show that if Z is nensingular, rely chosen singular c o v a r i a n c e m a t r i x is a matrix

~ compatible with

solution to the problem.

{, c.~,

Z; thus any

for an arbitra-

for suitably chosen c>0

(static)

system w is a

For the special case n=2 this c o r r e s p o n d s

to the problem of fitting a least squares line into a scatter plot, where the distance of a point to the line is defined by a n o r m with a unit ball c o r r e s p o n d i n g to an a r b i t r a r y ellipse in

~2.

Here, as is

easily seen, unless all p o i n t s lie e x a c t l y on a straight line, every

136 line can be o b t a i n e d Thus,

in order

ness", point

by a s u i t a b l e

to give our p r o b l e m

additional

structure

ones

is no u n i v e r s a l l y

words

such a s s u m p t i o n s

otherwise

a certain

since,

justifiable have

prejudice

"non a r b i t r a r i F r o m the

seem to be the m o s t

at least

in the a u t h o r ' s

assumption

of this kind.

to be j u s t i f i e d

they may e x p r e s s

degree of

has to be imposed.

such a s s u m p t i o n s

in our context,

there

of a norm.

on the n o i s e

of view a p p l i c a t i o n s

delicate

choice

in the actual

rather

opinion, In other

application,

than real a p r i o r i

know-

ledge.

In this paper we will a s s u m e

(v)

i.e.

f is d i a g o n a l

all c o m p o n e n t

By this a s s u m p t i o n

processes we c o v e r

such that an a n a l y s i s

of this k i n d

pretation

of the a s s u m p t i o n

by saying

that the c o m m o n

and the individual of a s s u m p t i o n model,

(a) To assume

is m o r e

(b) f is d e f i n e d also

assumption

general

The case n=2 Anderson

where

(see e.g.

than

criterion.

may

(2.1). Two

the blocks c o r r e s p o n d Picci and P i n z o n i sense,

1986).

however

of z t into inputs a n d outputs

to f with a given

impose

"prejudice"

of a certain

variation

rank.

Clearly,

by the choice

It is one of the flavours

of our

in the a p p r o x i -

can be a n a l y s e d . understood

Deistler

now

(see A n d e r s o n

1986 a n d D e i s t l e r

the p r o b l e m

of d e t e r m i n i n g

is r e f e r r e d

to A n d e r s o n

the reader

see Kalman

(1982)

and D e l s t l e r

and A n d e r s o n

case n = 3 where

case

noise

system model

(v) in a certain

time, static

features

"symmetric"

(v) are:

as a b e s t . a p p r o x i m a t i o n

is well

1985,

symmetric

of noise,

to the s y s t e m w(z)

One of the nice

of the e n t r i e s

(v) that the e f f e c t s

criteria

the

One inter-

it gives a d e f i n i t i o n

are a t t r i b u t e d

diagonal,

this a s s u m p t i o n

of the a p p r o x i m a t i o n

mation

is that

respectively

an a p r i o r i c l a s s i f i c a t i o n is required.

in general

with

are uncorrelated.

range of a p p l i c a t i o n s

to be justified.

to the errors.

that ~ is b l o c k

This a s s u m p t i o n

seems

to a s s u m p t i o n

and outputs

process wide

it g i v e s a c o m p l e t e l y

is c o m p a t i b l e

common a l t e r n a t i v e s

to inputs

(v}

effects

effects

(v) is that

which

of the noise a reasonably

m* arises

and Deistler

and A n d e r s o n

(1984).

1989);

1984, for the

for the first (1987);

for the

137 This p a p e r

is o r g a n i z e d

properties

of

given

the

results

lower

If x is a vector,

Properties

In this

as w e l l

is w r i t t e n

where

than

below

of

letter,

fixed

Z =

Z + D

special

complete 5 some

e.g.

to d e n o t e

Solution

as in the

some

general

emphasis analysis

is for

identifiability

described.

letters

the

but

its

A, we use

its e n t r i e s ,

j-th

the c o r r e s e.g.

aij.

element.

Set

following

frequency

section ~. The

, the a n a l y s i s

basic

equation

is g i v e n (2.4)

then

(3.1)

D respectively

spectral

for

For g i v e n

in s e c t i o n

by a capital case

where

4 a rather

Finally

are

section

as

Z,~ a n d

rather

derived,

we u s e x. to d e n o t e 3

3. Some

for a r b i t r a r y

cases

denoted

indexed

set a r e

In the n e x t

= I. In s e c t i o n

is given.

special

For a n y m a t r i x ponding

m*

m* = n-1 for

follows:

solution

to the c a s e

the case

as

variing

i is

(constant) The

complex

generalization

Hermitean of the

matrices

results

straightforward.

Z, the m a t r i c e s

0 < Z-

are

densities.

~ or

D are

called

feasible

if

D < Z

(3.2)

^

where

Z - D = Z is s i n g u l a r

out we w i l l

and

D is d i a g o n a l .

For

simplicity,

through-

assume

(vi)

Z>0

and (vii)

oij

Clearly,

these

~ 0

i,j

assumptions

are

generically

= I .... ,n fulfilled.

^

Due to

(vii)

be e x p r e s s e d we c a l l

and

since

a solution

component

Z is s i n g u l a r ,

as a l i n e a r

x I equal

combination

(corresponding to one,

such

the f i r s t

r o w of

of the o t h e r

to Z) a n y v e c t o r that

there

~ can

rows.

always

Accordingly

x 6~n,^with

is a f e a s i b l e

first

Z satisfying

138 ^

x~ = 0 Clearly,

(3.3)

every solution c o r r e s p o n d s

matrix w in

to a (row of

a ) relation-

(2.1). The set of all solutions corresponding

Z is called the solution

set ~

set of all feasible matrices

; analogously

D corresponding

Our main aim now is to give a description

we define ~

to a given as the

to Z.

of the solution

set~.

Let S = Z-I and let S = ( ~ij.Slj -I) i,j = I ..... n" Thereby we in addition will assume throughout q,

(viii) Then,

S.. + 13

0

as is easily

i,j

-- 1

(0 ..... 0, s1_j

, 0 ..... 0)

sj say, of S correspond

to the case where all variables to be observed free of noise. elementary Now,

= 1,...,n

seen from %'

s ~= j the rows,

,

to the elementary

in positions

regressions,

unequal

Correspondingly

we call

s. the j-th 3

solution.

let us investigate x7 = xD;

x6~

the relation between ~ a n d

~

which is defined by

, D6~

(3.4)

Let =

(~I I '

a n 'E12

~I 21

,

, \ 12,

of

n

^

Z12'

~22/

be partitionings ~(n-1)x(n-1)

;

Z, ~ and D respectively

0

=

D22

where

Z22"

~22" D22 6

and let x = (I,x2), x26~ n-1. Note that under our

12) --

=

D

Z22

assumptions

^

i.e.

to j are assumed

X

In- I

T.22

(-X 2 '

,

I

n_

1 )

139 ^

holds a n d

thus

corresponding

~2 _

has

the same

vectors x

x are

2 ^ Z22

In p a r t i c u l a r

x is u n i q u e then

dimensional

affine

Conversely,

for

given

as

Z. T h u s

for g i v e n

D6~,

the

the

by

= - E12

has c o r a n k

m,

rank

(3.5)

if { has

corank

I; more

set of all c o r r e s p o n d i n g

generally

solutions

if

is an m-1

space.

given

x, the

corresponding

matrices

D are

given

by

x2~22 = x2D22

(3.6)

o11

(3.7)

and

Sometimes

+ x2z1~

we use

particular, unequal

= d11

the

symbol

D is u n i q u e l y

to

zero.

if one e n t r y

Let

~j

of x, xj

D x to

indicate

determined ($j)

say,

that

D corresponds

for g i v e n

denote

the

is e q u a l

j-th

to

x if e v e r y

row o f X

zero,

then

~i

to x.

entry

(~).

Conversely,

can be e x p r e s s e d

^

as a l i n e a r this c a s e

combination

we may

show t h a t

of

the rows ~ ,

put d..=0;

^

other r o w s

of

then

o.=o.

33

Z. On

$. c a n

3

the o t h e r

be m a d e

hand

linearly

i~I,

i#j.

is l i n e a r l y

dependent

from%j

For the n e x t

is c o n c e r n e d w i t h

Anderson

Theorem

which

and Deistler

independent

of t h e

' and every

d.. 33 w i t h

the

to

a certain

0-< d .33-. < d mJJ ax

c a s e m*=l,

see

(1986b).

3. I

(a) mc(E)

= I if a n d o n l y

(b) For mc(Z) subset (c) For mc

= 1,~is

of ]R2n( ~ ~n}

if t h e r e

bounded

(a)

and closed

(Z) = 7, the r e l a t i o n

has

been

is no x e ~ w i t h and w h e n

it is of d i m e n s i o n

a homeomorphism.

Proof:

in

straightforward

by s u b s t r a c t i n g

say, positive n u m3b e r , d ~3J x gives r i s e to a f e a s i b l e D.

theorem,

In p a r t i c u l a r

3

it is q u i t e

shown

above.

between

~

a zero

entry.

considered

as a

n-1. and~

defined

by

In

of x is

(3.4)

is

T40 (c) : T h a t tinuity

the r e l a t i o n

in b o t h

is b i j e c t i v e

directions

is e a s y

has

been

shown

to see f r o m

(3.5)

(b) F i r s t it is s t r a i g h t f o r w a r d to s h o w t h a t ~ of d i m e n s i o n n-1. T h e r e s t f o l l o w s f r o m (c).

It can set that

also

be

shrinks

real

case

result

Kalman

terization

(Anderson

to a s i n g l e t o n

in the

satisfactory e.g.

shown

1982). of ~

Let

us r e t u r n

the

solution

if t h e

(i.e.

than

and Deistler

when

Theorem

noise

3.1

In p a r t i c u l a r ,

1989b)

real

and

that f goes

entries)

for m * = l

for t h e

The

con-

(3.6).

is b o u n d e d

spectrum

~ has

above.

closed

the to

solution zero.

a much

Note

more

c a n be

obtained

case

a complete

real

and

(see charac-

is a v a i l a b l e .

to the

case

of

general

set we c o n s i d e r

m

. In o r d e r

to f u r t h e r

the q u e s t i o n

which

part

is c o n t a i n e d

in ~ .

(Note

of the

investigate (complex)

line

ex +

(1-e)y

connecting in e£~

~2n

two p o i n t s

G ~n

(1-~)y)E

be s a t i s f i e d

Here

the

case

we h a v e

plane).

Dj

the

= exD x

special

= diag

= D s' = d i a g J first

+

for a D e ~

simplest

D I = Dsl

Then

x,y~

is a(real)

This

that

is the same as a s k i n g

such

a line

for w h i c h

the e q u a t i o n .

(ex +

can

; ~ 6

equation

(1-~)yDy

(i.e.

case

=

D~0,

is w h e n

(sx + ( 1 - a ) y ) D

(3.8)

E-D~0).

x=s 1 a n d y=sj,

j>1.

d!~ )JJ " 0, .... 0} + j-th p o s i t i o n s y s t e m (3.8) is of the

form

In t h i s

{d ~I) I ' 0 '''" ,0] {0,..°,0,

in the

(3.9) Analogously,

equation

( 1 - ~ ) s J3d !3Jj)

=

j in

( Slj

(3.8)

÷

is of

(1-~)sjj)djj

the f o r m

141 and thus

1

• d ! J ) = d..

Sjl

o~ I+-I-~

33

(3.10)

33

sjj

-I> where s'1"sjj]

0. F r o m

i~I, i+j w h i c h

the r e m a i n i n g

equations

we can p u t d ! ~ ) = 0 ;

gives a feasible

c h o i c e of D. N o w n o t e that d ~ ) is ]i (I) _> dl I h o l d s for any d11 c o r r e s p o n d i n g n~ximal in the sense t h a t d11 (3) . This is straightforward to D 6 ~ a n d t h a t the same h o l d s for djj from d e t

(Z-D)=0.

for e v e r y

eel0,1]

Thus

(3.9)

and

(3.10)

in t h i s w a y we o b t a i n

imply e£[0,1]. a feasible

Conversely,

D, as c a n b e

checked f r o m the m i n o r s . In a c o m p l e t e l y

x

analogous

= as i + (1-a)sj,

In this c a s e

for the

w a y we c a n

eel,

investigate

the r e a l p l a n e

i,j > I

j-th e q u a t i o n

of

(3.8) we o b t a i n

I :

I +

I-~

and o n l y S

> 0

(3.11)

t h r e e cases:

, .

]i > 0 t h e n x £ ~ i f S

if

S.. 33

Thus we can d i s t i n g u i s h S

39

. .

31

(a) If

d

s.. 99

and thus x E ~ i f

]-a

33

.q . -ji

~

,

and only

if eE[0,1]

.

]3 S . .

(b) If

31 < 0 t h e n x s 6 ~ s.. 33

if a n d o n l y

if a (-~,0JU[I, ~)

s c)

If

J' is n o t r e a l then e c o r r e s p o n d s ]J

( 3.1 1 11 in

R 2.

to a c u r v e

(described by

142 4. The

Case

mc(E)

For

the c a s e

For

the p r o o f s

referred ponds next

mc(Z) of

theorem

Bekker

4.1.

(b) T h e r e

exists

(Tij)i,j

and

well

exists

(tij)i,j

tiktjl

a rather in this

(1989a).

complete

section Clearly

a one d i m e n s i o n a l

straightforward for a long

description.

the r e a d e r m*

factor

= n-1

process.

generalization time

for the

is

corresThe

of a r e s u l t

real

case

(see

1987).

The

following

a diagonal

unitary

is real

statements

matrix

with

are e q u i v a l e n t :

U such t h a t U * Z U

all e n t r i e s

= 0

;

i,j,k,l

Tji ~ 0

;

i,j,k

a diagonal

= 1,...,n

and

Tij

=

positive

- tiltjk

the c a s e

n=3,

m*=2

solution

this

for

corresponding 4.2.

(a) t h e r e

~m*

holds.

is real

= 0

of

set

unitary

(4.1)

all d i f f e r e n t matrix

with

U such

all off

;

i,j,k,l

see A n d e r s o n

the c a s e ease.

m*

(4.2) that

diagonal

By ~ m

to a Z w i t h

is a u n i q u e

T,7 m*

U*Z-Iu

=

elements

(Z - Zm,)

different

we give

we d e n o t e

corank

say,

all

and D e i s t l e r

= n-1

m

Let n_>3 a n d m c ( ~ ) = m * = n - l .

= ker

all d i f f e r e n t

satisfying

characterization

Theorem

with

known

- ~il. Tjk

negative

x£~

given

Deistler

= 1,...,n

~jk - ~ik

(c) T h e r e =

model

give

satisfying

Tik. Tjl

~ii

and

L e t n>3.

= n-1

For

the r e s u l t s

a n d de L e e u w

(a) mc(Z)

=

we can

is a r a t h e r

is a l r e a d y

Theorem

= n-1

to A n d e r s o n

to a f a c t o r

which e.g.

= n-1

such

(i.e. Then that

(1987).

(4.3)

After

a description

the

set of all

to m o u t p u t s ) . we h a v e

of

this the

solutions

143 (b) Let xj = a*s I + (1-a*)sj

;

a* = (I -

s

13 ) s23

(for all

Then all e n t r i e s

of xj n o t in p o s i t i o n s

-I (4.4)

j>l)

I a n d j are e q u a l

to zero

and n

~m*

=

j

n

Z=28jXj

;

(C) ~ # ~ m C ~ m *

'

(d) ~J1 c a n be w r i t t e n

(4.5)

ZS. = I 2 21

i < j < n-1 as ~ 1 , u L ) ~ 1 , 1

where

n

~1,u

= {SSl

n

T. I ~,Bj~[o,I], j=2 ~jxj c

* (l-e)

T~B 2 3

=

I}

(4.6)

and

~I

,

Note that,

once

the c l a s s i f i c a t i o n

input a n d the o u t p u t s system.

n n ~ S.s. [ ~8 = I, j=2JJ 2 J

l = {as1+(1-~)

This,

is e v i d e n t l y

It is e a s y to s h o w t h a t f o r n ~ 3 set of all

Z w i t h mc(D = n-1

in the set of all known, e s t i m a t i o n

Z which

of the s o l u t i o n

uniquely

determined

forward.

From

ly d e p e n d s done

m* c o r r e s p o n d s

(Anderson and D e i s t l e r of

dimension

(real)

1989a)

dimension

n 2. Once m*

= n-1

set ~ a n d in p a r t i c u l a r

system corresponding

(4.4) a n d

to a u n i q u e

not true for the c a s e n = 2.

is a m a n i f o l d

is of

~6[0,a*]}

of the e n t r i e s of zt into an

has b e e n made,

in g e n e r a l

n ~S s 6~, 2J3

to m * , a r e

rather

the 3n is

of the straight-

(4.5) we see that such a s y s t e m c o n t i n u o u s -

on ~. Thus t h e m a i n p r o b l e m

for i n s t a n c e b y a p r o c e d u r e

using

is to e s t i m a t e

m*,

which

an i n f o r m a t i o n

criterion.

can be

144 5. Some Results on I d e n t i f i a b i l i t ~

In this short section we discuss two

(rather special)

examples

where i d e n t i f i a b i l i t y is o b t a i n e d using the rational s t r u c t u r e of the spectral densities.

Thereby the idea is due to S 6 d e r s t r 6 m

(1980)

(see also A n d e r s o n and D e i s t l e r 1984).

(i)

If we k n o w that

(ut) is white noise we can d e t e r m i n e the poles

of every main d i a g o n a l element ^ fii of f since they are also the poles of f... A

If we in addition assume that for the rational

ll

f u n c t i o n s fii the r e s p e c t i v e numerator degrees are strictly smaller than the c o r r e s p o n d i n g denominator degrees,

then f is

u n i q u e l y d e t e r m i n e d from f.

(ii) If zt is known to be purely a u t o r e g r e s s i v e and u t is a m o v i n g ^ average process, poles of f...

than again from the poles of the f .. we k n o w the

If every f.. has at least one pole

ll

unit circle)

(within the

ll

then the c o r r e s p o n d i n g innovation v a r i a n c e can be

uniquely d e t e r m i n e d and we again have identifiability.

I want to thank B . D . O .

A n d e r s o n and J.H.

van S c h u p p e n for v a l u a b l e

discussions.

References

Adcock,

R.J.

(1878). A p r o b l e m in least squares. A n a l y s t

5, 53-54

Akaike,

II. (1966). On the use of n o n - G a u s s i a n process in the identi-

fication of a linear dynamic system. Annals of the I n s t i t u t e of Statistical Mathematics Anderson,

B.D.O. (1985).

18, 269-276

I d e n t i f i c a t i o n of scalar e r r o r s - i n - v a r i a b l e s

models with dynamics. A u t o m a t i c a 21, 709-716

Anderson,

B.D.O.

and M. D e i s t l e r

e r r o r s - i n - v a r i a b l e s models.

(1984). I d e n t i f i a b i l i t y

in dynamic

Journal of Time Series A n a l y s i s

5, 1-13

145 Anderson, B.D.O. and M. Deistler systems with three variables. Anderson, B.D.O. and M. Deistler

(1987). Dynamic errors-in-variables Automatica (1989a).

23, 611-616

Identification of dynamic

systems from noisy data: The case m*=1. Mimeo Anderson, B.D.O. and M. Deistler

(1989b). Identification of dynamic

systems from noisy data: Some further results. Mimeo. Anderson, T.W.

(1984). Estimating linear statistical relationships.

Annals of Statistics

12, 1-45

Bekker, P. and J. de Leeuw matrices, Blomberg,

Psychometrika

H. and R. Ylinen

(1987). The rank of reduced dispersion 52, 125-135 (1983). Algebraic theory for multivariate

linear systems. Academic Press, London Brillinger, D.R.

(1981). Time Series:

Expanded Edition. Deistler, M.

Data analysis and theory.

Holden Day, San Francisco

(1986). Linear errors-in-variables models.

In

(S.Bittanti,

ed.) Time series and linear systems, Lecture Notes in Control and Information Sciences,

Springer-Verlag,

Deistler, M. and B.D.O. Anderson variables models: of Econometrics Frisch, R.

Berlin,

37-67

(1989). Linear dynamic errors-in-

Some structure theory. To appear in: Journal

41.

(1934). Statistical confluence analysis by means of complete

regression systems.

Publication No. 5, University of Oslo, E c o -

nomic Institute Fuller, W.A.

(1987). Measurement error models. Wiley, New York

Geary, R.C.

(1942). Inherent relations between random variables.

Proceedings of the Royal Irish Academy, Geary, R.C.

Sec. A, 47, 63-76

(1943). Relations between statistics: The general and the

sampling problem when the samples are large. Proceedings of the Royal Irish Academy.

Sec. A, 49, 177-196

146 Geweke, J.

(1977). The dynamic factor analysis of econometric time

series models. In

(Aigner, D.Jand A.S., Goldberger eds.) Latent

variables in socioeconomic models. North Holland, A m s t e r d a m

Gini, C.

(1921). Sull' interpolazione di una retta quando i valori della

variabile indipendente sono affetti da errori accidentali. Metron I, 63-82 Hannan, E.J. and M. Deistler

(1988). The statistical theory of linear

systems. Wiley, New York

Kalman, R.E.

(1982). System identification from noisy data, in

(A.

Bednarek and L. Cesari, eds.) Dynamical Systems II, a University of Florida International Symposium, Academic Press, New York Kalman, R.E.

(1983). Identifiability and modeling in econometrics,

in (Krishnaiah, P.R., ed.) Developments in Statistics, Vol.4, Academic Press, New York Koopmans, T.C.

(1937). Linear regression analysis of economic time

series. Netherlands Economic Institute, Haarlem

Maravall, A.

(1979). Identification in dynamic shock-error models,

Springer-Verlag, Berlin

Nowak, E.

(1985). Global identification of the dynamic shock-error

model. Journal of E c o n o m e t r i c s 27, 211-219

Pearson, K.

(1901). On lines and planes of closest fit to systems of

points in space. Philosophical Magazine 2, 559-572

Picci, G. and S. Pinzoni stationary processes.

(1986). Dynamic factor-analysis models for IMA J. Math. Control and Information 3,

185-210 Reiers~l, O.

(1941). Confluence analysis by means of lag moments and

other methods of confluence analysis. Econometrica 9, 1-24

Reiers~l, O.

(1950). Identifiability of a linear relation between

variables which are subject to error. Econometrica 18, 375-389

147 Sargent, T.J. and C.A. Sims

(1977). Businesscycle modeling without

pretending to have too much a priori economic theory. In

(Sims, C.A.

ed.) New Methods in Business Cycle Research. Fe. Reserve Bank, Minneapolis

SchneeweiB, H. und H.J. Mittag

(1986). Lineare Modelle mit fehlerbehafte-

ten Daten. Physica Verlag, Heidelberg S~derstr6m, T.

(1980). Spectral decompositions with application to

identification. In (Archetti, F. and M. Dugiani, eds.) Numerical Techniques for Stochastic Systems, North Holland P.C., Amsterdam

Spearman, C.

(1904). General intelligence, objectively determined and

measured. American Journal of Psychology 15, 201-293 van Putten, C. and J.ll. van Schuppen

(1983). The weak and strong

Gaussian probabilistic realization problem. Journal of Multivariate Analysis

13, 118-137

Willems, J.C.

(1979). System theoretic models for the analysis of physi-

cal systems. Richerche di Automatica 10, 71-106

Willems, J.C.

(1986). From time series to linear systems

-

Part

I,

Automatica 22, 561-580

Willems, J.C.

(1987). From time series to linear systems - Part III,

Automatica 23, 87-115

Elements of Factorization Theory From a Polynomial Point of View P. A. Fuhrmann Department of Mathematics, Ben-Gurion Univsersity of the Negev Beer Sheva, Israel

This paper

is d e d i c a t e d

to Jan

C. Willerns

Abstract Tile paper outlines a coherent development of factorization theory in tile framework of polynomial model theory. Starting from tile most elementary factorizations of polynomial matrices we build up tile connections to invariant subspace theory, factorizations of transfer functions, Wiener-Hopf factorizations. We pass on to spectral factorizations of polynomial matrices and rational functions and the connection with the analysis of the algebraic Riccati equation. Finally we study inner/outer factorizations for a cla~s of transfer functions and the derivation of state space formulas. The aim throughout is to highlight the logical interconnections and the technique rather than the derivation of the most general results.

1

INTRODUCTION

F a c t o r i z a t i o n t h e o r y is a typical m a t h e m a t i c a l s u b j e c t where t h e i n t e r e s t i n g questions are of the inverse p r o b l e m type. Once a m u l t i p l i c a t i o n o p e r a t i o n is defined t h e q u e s t i o n of t h e r e p r e s e n t a t i o n of a n o b j c c t a.s a p r o d u c t becomes a n a t u r a l question. T h i s becomes evcn more i n t e r e s t i n g when we look for a c o m p l e t e factoriz, a t i o n into irreducible factors, or ~ l t c r a a t i v c t y if c e r t a i n c o n s t r a i n t s arc p u t on the factors. F a c t o r i z a t i o n p r o b l e m s are quite often i n t i m a t e l y c o n n c c t e d to s t r u c t u r M p r o p e r t i e s a n d to a variety of c~nonical forms.

149 In this paper we want to outline the factorization problems most often encountered in the area of systems theory. We will start from the most elementary factorization problem, that of nonsingular polynomial matrices, and study its relation to the analysis of invariant subspaces. We will see what symmetry constraints add to this problem. This will relate Hamiltonian, or parahermitian symmetry to special classes of invariant subspaces of symplectic spaces. Passing from the polynomial level to that of rational functions we analyze the factorization problem for causally invertible transfer functions. We will give a short description of Wiener-Ilopf factorizations. Our main emphasis will be on the circle o f i d e ~ centered around spectral factorizations and the algebraic Riccati quation. The central importance of this set of problems in systems theory needs no elaboration. It is a central tool in optimal control and filtering theory. Our approach is based on the spectral factorization of polynomial matrices. Finally we focus on inner/outer factorizations of rational matrix functions. Throughout we try to outline the connection between external data, polynomial matrices, transfer functions and faetorizations etc., and the internal data given in state space terms. Tile machinery that makes this connection clearest is the theory of polynomial models developed in Fuhrmann [1976-1988]. The history of the various factorizations is both old and rich. There is no intention of trying at a complete account or a full referencing. The following is the barest of outlines. The connection between polynomial matrices and module theory is quite old and can be found for example in MeDuffee [1956} where further references to early contributors can be found. The connection between factorization of polynomial matrices and invariant subspaces derives from the work of Livsic and Brodskii in operator theory. In the context of polynomial models it can be found in Antoulas [19791 and Fuhrmann [19791. The standard result on factorization of rational transfer functions was derived in Bart, Cohberg, Kaashoek and Van Dooren [1980], and a fuller account is given in Bart, Gohberg, and Kaashock [1979]. Our development takes a polynomial approach and is bascd on Shamir and Fuhrmann [1984]. The analysis of spectral and inner-outer factorizations owes a lot to, one might say it is actually based on, Francis [1988] and Chcn and Francis [1988]. In fact it arose out of trying to understand the intuition behind some of the state space formulas. Finally it is a pleasure to acknowlcdge the influence of Jan Willcms' fundamental and pathbreaking paper, Willems [1971]. It is here that we see most clearly the convergencc of spectral factorizations and state space methods, namely the FOccati equation, as alternative tools in optimization problems. In view of his long and many outstanding contributions it is hard to bclieve that Jan has only now reached the ripe old age of fifty. On this occasion this modest contribution is dedicatcd to him.

2

POLYNOMIAL

MODELS

Our starting point is this basic result about free modules. T h e o r e m 2.1 Let It be a principal ideal domain and M a free left R - m o d u l e with n basis elements. Thcn cvery R-submodule N of M is free and has al most n basis elements. If V is a finite dimensional vector space over a field 1; then V[z], the space of vector polynomials is a free finitely generated module over the polynomial ring F[z]. Throughout

150

we will assume a basis has been chosen and thus V will be identified with F " and similarly V[z] with f"[z}. Also we will identify F'~[z] and (F[z]) n and speak of its elements as polynomial vectors. Similarly elements of Fm×n[z] will be referred to as polynomial matrices. Bccause of the nature of the factorization results we are interested in, and for the consistency of notation, we will identify the field F with the real field R , noting that some of the results hold in greater generality. The next theorem, whose easy proof follows directly from Theorem 2.1, is the basic representation theorem for submodules. T h e o r e m 2.2 A subset M of Itn[z] is a submodule of Rn[z] if and only if M = DRn[zl for some D in l't~x'~[z]. The following is the basic theorem that relates submodule inclusion to factorization. T h e o r e m 2.3 Let M = DR"[z] and N = EI'U~[z] then M C N if and only if O = EG fo~" s o m ~ a

in

R"~"[d.

Let 7r+ and r _ denote the projections of 1Un((z-l)) the space of truncated Laurent series on lUn[z] and z-ll-tm[[z-l]], the space of formal power series vanishing at infinity, respectively. Since

rt=((z-1)) = rt'~[z] ~ : l r t ' [ [ z - l l l

(1)

7r+ and 7r_ are complementary projections. Given a nonsingular polynomial matrix D in R"Xm[z] we define two projections 7tO in R'~[z] and ~rD in z-XR'~[[z-1]] by

7rDf = D~r_D-I f 7rDf = Ir_D-'Tr+Dh

for f E R'[z] f o r h • z-lRml[z-X]]

(2) (3)

and define two linear subspaces of Rm[z] and z-lRm[[z-1]] by X o = Im~ro.

(4)

X O = lrnTr D.

(5)

and An element f of Rm[z] belongs to X n if and only if ~r+D-lf = O, i.e. if and only if D - i f is a strictly proper rational vector function. We turn XD into an R [ z ] - m o d u l c by defining v.

/

=

~Dpf

1o~

v •

hid, f • x n .

(o)

Since KcrvrD = DRm[z] it follows that XD is isomorphic to Rm[z]/DRm[z]. T h e o r e m 2.4 l¥(lh the previously defined module structure, XD is isontorphic to the quotient module R'*tz]/ DRn[z]. In X o we will focus on a special map SO, an abstraction of the classical companion matrix, which corresponds to the action of the identity polynomial z, i.e.,

S D f = 7rDzf

for

f E D.

Thus the module structure in X o is identical to the module structure induced by So through p . f = p ( S o ) f . With this definition the study of So is identical to the study

151

of the module structure of X/). In particular the invariant subsp~ccs of St) are just the submodules of X/) which we proceed to investigate. The interpretation of factorization of polynomial matrices on the level of polynomial models is described next. T h e o r e m 2.5 A subset M of XI) is a submodule, or equivalently an S o invariant subspace, if and only if M = D1XD2 for some factorization D = D1D2 with Di E IU'x'~[z]. We summarize now the connection between the geometry of invariant subspaces and the arithmetic of polynomial matrices. T h e o r e m 2.6 Let Mi, i : 1 , . . . ,s be submodules of X o , having the representations Mi : EiXF,, that correspond to the factorizations

D=EiF,. Then the following statements are true. (i) 11,[1 C 1~12 if and only if Et = E2R, i.e. if and only i.f E2 is a left factor of El. (ii) f']i=xl~fz " - has the representation E,,)t'F~ with E,, the least common right multiple (l.e.r.m.) of the Ei and 1,', the g.c.r.d, of the Ft. (iii) MI + .." + Ms has the representation EuXF~ , with E~ the greatest common left divisor (g.c.Ld.) of the El and F~, the l.c.l.m, of all the Ft. C o r o l l a r y 2.1 Let D = EiFi, for i = 1 , . . . ,s. Then (i) We have X o = E t X r , + ... + E, XF, if and only if the Ei are left coprime. (::) We have N'i'=lEiX~: = 0 i f and only if the Fi are right coprime. (it 0 The decomposition X o = ExXt'~ @ . . . @ E~Xp, is a direct sum if and only if D = EiFi for all i, the Ei are left coprime and the Fi are right eoprime. The next result summarizes tile relation betwecn factorization and the spectral decomposition of linear maps. T h e o r e m 2.7 Let D(z) E ItnXn[z] be nonsingular and let d(z) = detD(z) be its characterislic polynomial. Suppose d has a factorization d = e t ' " e s with the ei pairwise coprime. Then D admits factorizations D = DiEi with detDi = di, detEi = ei and such that

XD = DxX~, ~ . . . ~) D~Xs,

(7)

del(SDIDiXE,) = ei.

(8)

Moreover

152

Denoting by 7~ the transpose of the matrix T we define, for an element A(z) = ~ = - o 0 AjzJ of R'~×'~((z-t)), .4 by

~i(~) = ~ aj~.

(9)

In lrtr"((z-l)) × 1-tm((z-1)) we define a symmetric bilinear form [f,g] by [f,g]=

fi

/ja-j-1

(10)

where f ( z ) = ~,j=_oo f j z a n d g ( z ) = )"]1=-o0 gjz . As both f and g are truncatcd Laurent series, it is clear that the sum in (6) is well defined, containing only a finite number of nonzero terms. We denote by T* the adjoint of a map T relative to the bilinear form of (6), i.e. T* is the unique map that satisfies

IT f , g] = If, T'g]

(11)

for all f , g E II.m((z-1)). We use this global bilinear form to obtain a concrete representation of X b , the dual space of XD. T h e o r e m 2.8 The dual space of XD, to be denoted by X b , can be identified with X b under the pairing < f , g > = [ D - i f , g] (12)

for f E XD and g E XD. through

Moreover the module structures of X17 and X[~ are relalcd S b = S/).

(13)

The following result shows how the characterization of submodules of X/9 and their relation to faztorizations is reflected by duality. T h e o r e m 2.9 Let M C XD be a submodule, represented as M = E X G for some faetorizalion D = E G into nonsingular factors. Then the orthogonal subspace 1~ig" is a submodule of X D and is given by M ± = G X $ . Recall that a vector space V is symplectic if it is equipped with a nondegenerate, alternating bilinear form. A linear map H in a symplectic space V is Hamiltonian if H" = - 1 I relative to this form. A map R in V is a symplectic if it is invertible and leaves the form invariant. The canonical example of a symplectic spa~ce is r t 2n with the bilinear form induced by J=

(0/i) -

0

"

A Ilamiltonian map II in this case is given by

1f=

O

_~

with P, Q symmetric. R is symplectic if and only if [~JR = R.

153

Contrary to inner product spaces, there are self orthogonal elements and subspaces in symplectic spaces. A subspacc /:: of a symplcctic space V is called I,agrangian if it is a maximal self orthogonal subspacc. We modify now the previous approach to duality to accommodate the study of sympleetie spaces. To this end we introduce now a global alternating form on R " ~ ( ( z - l ) ) x l't'~((z-1)) in the following way. We define now a new bilinear form on l'tm((z-1)) x

Rm((=-l)) by {f,g} = [rf, g] for f,g e I:U"((z-1))

(16)

where ~-: IU'*((z-1)) --, lq.m((z-1)) is defined by

(Tf)(z) = f(--z).

(17)

L e m m a 2.1 The bilinear form defined by (16) on R"~((z-l)) i.e. { f , g } = -{g,f}.

x Rm((z-1)) is alternating, (18)

Given a rational matrix function (I) we define its llamiltonian or parahermitian conjugate, 4~. by

,.I:,.(z) = (~(--z).

(19)

This implications of this s y m m e t r y were originally studied by Drockett and Rahimi [1972]. An extensivc study of ltamiltonian s y m m e t r y and its associated rcalization theory can be found in F u h r m a n n [1984]. Given a m a p Z : R ' ~ ( ( z - 1 ) ) --* R"~((z-1)) we will denote by Z v the map, assuming it cxists, which satisfics {Z/',g} = {/, ZVg} (20) for all f , g e R m ( ( z - 1 ) ) . Since the form [, ], and hence also { , }, is nondegcnerate the map Z v , if it exists, is unique. The m a p Z V will be called the Hamillonian adjoint of Z. A~sumc now that we have two nondcgenerate bilinear forms on V x V* and V* x V which satisfy < < x,y > > - - - < < y , x > > for all 0c (5 V and y E V °. Note t h a t while the two forms arc distinct wc do not distinguish between them, as it is always clcar from the context which form wc use. We say x is orthogonal to y if < < x,y > > = 0. Given a subset M C V we define M j- as usual by M -L={yEV*

I <<x,Y>>=0

for a l t x E M } .

(21)

bull submodules of 1-U~[z] are submodules of the form DR'~[z] with D a nonsingular polynomial matrix. It is of interest to characterize their orthogonal complements. The result should be compared with Theorem 2.9. Wc can use these isomorphisms to define a pairing between XD and X(D.) by T h e o r e m 2.10 The dual space of XD, to be denoted by X ~ , can be identified with X(D.)

under the pairing << f,g : > > = { D - l f , 9'} = [JD-lf, g]

(22)

for all f E XD and g G -¥(D.). Moreover the module structures of XD and X(D,) are related through S~) = -S(o°).

(23)

154

We note at this point that if D is a nonsingular Hamiltonian symmetric polynomial matrix, i.e. if D, = D, then XD with the metric of (22) is a sympleetic space. Much use of this will be made later. The pairing of elements of XD and X o . given by (22) allows us to compute, for a subset V of XD, the annihilator V ±, i.e., the set of all g E XD. such that < < f , g > > = 0 for all f E V. Since S b = - S o , , it follows that if V C XD is a submodule, then so is V ± in X/).. However, we know that submodules are relatcd to factorizations of D into nonsingular factors. The annihilator of V can be concretely identified. T h e o r e m 2,11 Let V C XD be a submodule. Then V ± is a submodule of X(D.). Moreover if V = E X F where D = E F is a faclorization of D into nonsingular factors, then v * = F, X E . .

(24)

If we assume D ( - z ) = D(z), i.e. D, = D, then XD is a symplectic space and SD a IIamiltonian map. For details see Fuhrmann [1984]. One way to obtain Hamiltonian symmetric polynomial matrices is to study those which have a factorization of the form D(z) = F , ( - z ) E ( z ) or D = E , E . T h e o r e m 2.12 Let D = E . E . XI~.

Then E . X E is a Lagrangian Sv-invariant subspace of

D e f i n i t i o n 2.1 Let P, = P be a tlamiltonian symmetric polynomial matrix. A factorization P = N.~N (2~) with a signature matrix ~2 is called a llamiltonian symmetric factorization. It is called an unmixed factorization if a(N) n a(N,) = 0 (26) or equivalently if detN and detN. are relatively prime. The following theorem, a special case of an unmixed factorization of a Hamiltonian symmetric polynomial matrix, is central to the whole theory of spectral factorizations. It goes back to the work of Jacubovich [1970] and Coppel [1972]. A more accessible account is Gohberg, Lanc,'mter and Rodman [1982] where Coppel's proof is reproduced. T h e o r e m 2.13 Let Q E Rnxnlz] be a polynowial matrix such that Q(it) = Q(it) > 0 for all real t. Then there exists a real polynomial matrix P such that O(z) = P ( z ) P , ( z ) . Moreover, P can be chosen such that all zeroes of det P lie in the open left (right) half plane.

155

3

REALIZATION

THEORY

As usual, given a proper rational matrix G we will say a system (A, B, C, D) is a realization of G if G --~ D + C ( z I - A ) - I B . We will use the notation G = [A, B, C, D]. We will be interested in realizations associated with rational functions having the following representations

G(z) = V ( z ) T ( z ) - ' U ( z ) + W(z).

(27)

Our approach to the analysis of these systems is to associate with each representation of the form (27), a state space realization in the following way. We choose X T as the state space and define the triple (A, B, C), with A : X T ~ X T , B : I~ m ~ X T , and C : X~, , R p by A=ST

(28)

B~ = ~TU~,

Cf = (VT-lf)_~ D = G(oo). We call this the associatcd realization to the polynomial matrix P given by P=

-V

W

"

T h e o r e m 3.1 The system given by (28) is a realization of G = V T - 1 U + ~V. This realization is reachable i f and only if T and U are left coprime and observable if and only if T and V arc right coprime. The following result as well as its dual, due to Hautus and Heymann [1978], are extremely useful. T h e o r e m 3.2 Let (A, C) be an observable pair, G ( z ) = C( z l - A ) -1 be the corresponding state to output transfer function and let G(z) = T ( z ) - * U ( z ) . be a left eoprime matrix fraction representation. Then, given any polynomial matrix N , T - i N is strictly proper if and only if there exists a constant matrix K for which N ( z ) : U ( z ) K . This is cqui~,atcnt to the columns o f U being a basis for X T . T h e o r e m 3.3 Let C = N D -1 have the reachable realization ( A , B , C ) M D -~ . Then G' has a realization (A,B, C0) for some Co.

and let G I =

Let Gl = [At, B~, CL, Dll and G2 = [As, B2, C2,/92] be two transfer functions rcalized in the state spaces X1 and X2 respectively. If the number of inputs of the second system equals the nu,nbcr of outputs of tile first we can feed those outputs to the second system. This gives rise to the series coupling and the corresponding transfer function is G2G1 =

B2Cx

A2

'

B2D1

'

We will use also the notation G2G ~ = [A2, B2, C2, D2] x [A1, Bl , Cl , Dl ].

156

D e f i n i t i o n 3.1 We say that (A, B, C, D) acting in R 2n is a Hamiltonian system if, with J defined by (14), we have

fi.J JB

= =

0

D

=

b

-JA

(30)

The main result concerning Hnmiltonian realizations is the following by Brockett and Ihdfimi [1972]. T h e o r e m 3.4 (i) A transfer function G has a Hamiltonian realization if and only i f it is Hamiltonian symmetric. (it) Two minimal Hamiltonian realizations of G ave symplcctically equivalent. A modification of representation (27) leads to IIamiltonian realizations via polynomial models. T h e o r e m 3.5 Let G, assumed to be llamiltonian symmetric, strictly proper and rational, have the represenialion

+ P(z)

a(z) = X.(z)Q(z)-lX(z)

(31)

where X , P and Q are polynomial matrices, with Q nonsingular and Hamilionian symmetric. Then, in the symplectic space XQ the associated realization to (31) is Hamiltonian. It is minimal if and only if X and Q are left copvime.

T h e o r e m 3.6 Let G be a p × m strictly proper rational transfer function having the reprcscnlation

a(~) = V(z)T(~)-'V(z)

+ w(~).

(32)

Let the associated realization (A, B, C), in the slate space XT, be defined by A

Bu Cf

=

ST

= =

7rT.Uu u E R m (VT-II)-I for f EXT.

(33)

Then the Hamiltonian adjoint of the realization ( A, B, C) associated with this representation is ( - A v, C v, B v) given by A v = ST. = - S t

(34)

Bvf = (U.(T.)-II)_x

(35)

C V u = 7r(T,) V . u

(36)

and and it is the realization associated with c.(~) = U,(~)(T,(~))-'V,(~) + W,(z).

(37)

157

We pass now to the polynomial characterization of conditioned invariant subsp~ccs. We will need this in the study of the factorization problem. Given a pair (C, A), a subspace V of the statc space X is called a conditioned invariant subspace if there exists a linear transformation L such that

(A + L C ) V C V.

(38)

T h e o r e m 3.7 Let (A, B, G) be thc observable realization associated with the transfer function G(z) = T ( z ) - i U ( z ) . Then a subspacc V C X T is a conditioned invariant subspace if and only if V = EIXF1 (39)

where 7"1 = ElF1 is such that T~-IT is a bieausal isomorphism. The following results will be necded in the study of the factorization problem. T h e o r e m 3.8 (i) Let M1, AI2 be two conditioned invariant subspaces of X T with the representations A I i = L'iXF,. Then M1 C M2 if and only if E1 = E2Y for some polynomial matrix Y . (ii) Let M h M2 C X T be conditioned invariant subspaccs and lct M~ = EiXI.;, i = 1,2. Let M = M1 A M2. Then M = )(7" f3 EFP[z] (40)

where E is the l.c.r.m, of El,l~½. (iii) Let M1, M2 C XT be conditioned invariant subspaces, and let A'[i = EiXF,. Let M be the smallest conditioned invariant subspace containing both ltIt and M2. Then M = X T A E F P [ z ] where E is a g.c.l.d, of E1 andE2. (iv) Let T and S be p × p nonsingular polynomial matrices. Then X T f3 SFr[z] = {0}

(41)

if and only if all the right Wiener-IIopf faclorizalion indices at infinity of S - I T are nonpositive, see l,'uhrmann and Willems [19"[9], or equivalently if there exists a unimodular matrix U such that u S - i T is proper rational.

4

INVERSE

SYSTEMS

The following result is well known. While it has a trivial state space proof we still find it of interest to obtain a polynomial model proof of this result. This is done in order to have the polynomiM approach reprcscntcd in a unified and self contained form. T h e o r e m 4.1 Let G -= [A, B , G , I ] be a normalized bicausal isomorph;~m. Assume the realization is minimal. Then G - l ( z ) = [A ×, B , - C , t ] where A x = A - BC.

158

Proof." Let T - 1 U be a left coprime factorization of rr_G. Then a ----[ + T - I u = T - I ( T + U) = T - I D

(42)

with D defined by D=T+U.

By our assumption T - 1 D is a bicausal isomorphism, which implies that the polynomial models XD and ),'7" contain the same elements. Since (42) implies G -1 = D - i T = D - I ( D - U) = I - D-1U,

the shift realizations of G and G -~ have the same state space. Let (A, B, C) and (A1, B1,6'1) be the shift realizations of ~r_G and 7r_G -1 respectively, i.e. A-=ST,

B~

=

U~ = ~rTU~

Cf

=

(T-If)_i

AI

--- SD,

and

Clf

=

(D-af)-l.

Clearly Bl = - B and, since for f 6 X o C l f = ( D - l f ) - I = (D - I T T - l f ) - I = ( T - l f ) - I = C f,

by tile fact that the constant term of D - I T is the identity, it follows that C1 = C. Finally recall that given f 6 X7, S r f = zf - T~I

for some constant vector Q. Clearly ~i is given by Q = 7r+zT-lf = (T-Xf)_l.

Now SDf

= = = =

ST+Uf = z f - - (T + U)((T + U ) - l f ) _ l z f - (T + U ) ( ( T + U ) - I T T - l f ) _ I z f - (T-Jr V ) ( T - l f ) - i (z f - T ( T - t f ) _ l ) U(T-lf)_I

=

ST f - - U ( T - l f ) - i

or

Al = A-

BC=

A x.

Thus we have recovered the theorem modulo the trivial isomorphism given by - I .

Wc pass on to the study of onc sided inverses. Of special interest will be the characterization of the class of transfer functions for which the singularities of the inverse system are localized in some special subsets of the complex plane. We recall that H ~ is the IIardy space of all bounded analytic functions in the right half plane 1I+. By R H °° we will denote tile real subspacc of ~dl real rational functions with all poles in the open left half plane I/_. For a state space approach to the result below, see for instance Bengtsson [1974].

159

L e m m a 4.1 A s s u m e G E R H ¢¢ has minimal realization [A, B , C , D]. right inverse of D. Then the following statements are equivalent. (i) G has a right inverse in R H ~ . (it) D is surjeetive and for some F we have

Let D + be any

G + OF = 0

(43)

and

X+(A +

B F ) = {01,

(44)

(iii} D is surjective and

X+(A - BD+U) C< A - BD+C, BKerD >.

(45)

I f F is such that (it) is satisfied then a right inverse G + is given by G = [A + BF, - B D +, F, D+].

Proof; (i) ~ (iii) ~ (ii). Assume G has a right inverse in R I I °°. Then G has full row rank in II+. In particular D is surjective. Let G(z) = D + N ( z ) E ( z ) -1. Then to G is associated the polynomial system matrix

P=

~v(z)

D

and it has full row rank for z E II+. This implies the same for the polynomial system matrix =

N(z)

D

=

0

I

N(z)

D

"

Now tile fact that the first row of pi is of full rank in H+ means, by a generalization of the ttautus controllability test, that the pair (SE_D+I~ , rE_O+ N • ( I -- D+ D)) is stabilizable. On the other hand this pair is, by the Hautus Heymann [1978] polynomial characterization of state feedback, isomorphic to (A - B D + C , i3(1 - D+D)). Now the stabilizability of this pair is equivalent to X+(A-

BD+G) C< A-

BD+C, BIm(I-

D+D) > .

and since I r n ( I - D + D ) --- K e r D there exists a feedback map K such that A - B D + C + B ( I - D + D ) K is stable. Putting F = - D + C + (I - D + D ) K , then A + B F is clearly stable. Moreover C+DF

---- C - . - D D + C + ( D - D D + D ) K =

( I -- D D + ) ( C

+ DK)

= O.

(it) ~ (i) Assume D is surjcctive and there exists an F for which (431 and (44) hold. Let

D + he any righL inverse of D. Equality (431 implies D(F + D+C) = 0

160

i.e. I m ( F + D+C) C K c r D = I m ( l - D+D). Now define G+(z) by G+(z) = (I - F ( z I - A - B F ) - I B ) D +. Clearly G+(z) E R H °°. We will show that GG + = I. As GG + = D[A, B, C, I] x [A + BF, B, F, I]D+ it suffices to show that [ A , B , C , I ] x [A + BF, B , F , I ] = I. Using the series connection we have [ A , B , C , I ] x [A + BF, B , F , I ]

=

BI,"

A

'

B

,

F

D+C

,I

IJsing the similarity matrix ( t fhIe p r0I e )v i ° u s s e r i e s c ° n n e c t i ° n i s i s ° m ° r p h i c t ° -

0

A

'

0

,

0

C

,I

Ilence GG + = I.

5

TRANSFER

FUNCTION

FACTOR.IZATION

Given (Ai, Bi, C,), i = 1,2, canonical realizations of Gi with 6(Gi) = dimXi the McMillan degree of Gi, then we have a realization of G2G1 in the state space Xt @ X2. This realization may not be canonical, but it gives an upper bound on tile McMillan degree of a product of rational matrices. Specifically we have the inequality

6(02c~) s ~(c~) + ~(02). We tackle now the inverse problem. Namely given a transfer function, when can it bc faetorcd into the product of two transfer functions. We will assume throughout this scction that the transfer function is proper with costant term equal to the identity. Wc will call such a transfer function a normalized bicausal isomorphism. In particular the inversc of a normalized bicausal isomorphism is also a normalized bicausal isomorphism. In much the same way as factorizations of polynomial matrices were related to invariant subspaces and controlled or conditioned invariant subspaccs one expects some such relation in the c ~ c of factorizations of transfer functions. As wc shall see this indeed turns out to bc the case. Wc pass now to the basic theorem concerning factorization of transfcr functions due to Bart, Gohbcrg, Kaazhoek and Van Dooren [1980].

161 T h e o r e m 5.1 ( D a r t , G o h b e r g , K a a s h o e k a n d V a n D o o r e n ) Let G = [ A , B , C , I ] be a rational normalized bieausal isomorphism with the realizalion, in the state space X , assumed to be minimal. Then a necessary and su~cient condition for C to admit a factorization G ----a 2 g l

(46)

wilh Ci normalized bicausal isomorphisms is that

X = M1 (9 M2

(47)

with MI an A - i n v a r i a n t subspaec, M2 an A x - i n v a r i a n t , where A x is defined by A x =A-BC.

(48)

Proof: We base our analysis of the factorization problem on polynomial models and this yields a concrete representation of the factors in polynomial terms. A representation of the factors in state space terms was derived in Bart, Gohberg, Kaashoek and Van Dooren [1980]. Assume (46) is a minimal factorization such that both Gi are normalized bicausal isomorphisms. Let G1 and G2 have the following left coprime factorizations. a l = T~ID1

(49)

G2 = Ti-1 D2.

(50)

and Ilcnce G = 7~-l)-)lTl-ID2. Since the factorization (,16) is minimM it follows that D n T 1 are right coprlmc. Let T{'ID1 be a left coprime factorization of DIT~ -1 then G = T 2 1 T ~ l D 1 0 2 : (T1T2)-I(D1D2) : T - 1 D

(51)

a -1 = (D1D2)-'(TIT2).

(52)

and For the shift rcMization of G, based on the factorization in (51), the subspace TIXT~ of X,r is ST--invariant. Similarly DIXD2 is S/)-invariant. However since S v f = S T f -- w r D ( T - l f ) - l ,

(53)

I")IXD2 is a conditioned invariant subspace. Note that here ST and .5"o correspond to A and A x respectively. %re will .show now the direct sum representation X T : T1XT~ @ D t X t h ,

(54)

TIXT2 Q D I X D 2 -- {0}.

(55)

T1XT2n D~XD2 = XT n ErtP[-]

(56)

and as a tirst step we show that

We know, by Theorem 3.8, that

162

where E is a l.e.r.m, of T1 and D1. Thus E = T1A = D t B for a pair A, B of right coprime polynomial matrices. Since Ti-' D, = A B -~ = - g ~ l I

(57)

it follows that for some unimodular matrix U, A = D I U and B = 7;iU. Hence

E - 1 T = E-1T1T2 = A-1T2 = u-l(Dl"1T2).

(58)

Thus E - I T is the product of a unimodular matrix and a bicausM isomorphism and so all its right Wiener-Hopf factorization indices arc zero. By Theorem 3.8 this implies (55). ~-~--1 • To show equality (54) wc use a dimcnsionality argument. From the fact that T 1 D2 is a bicausal isomorphism it follows that deg det T1 -- deg det D2. Also the equality T~'IDI = DIT~ -1 of the two matrix fraction representations implies degdetT1 = degdetT1. So taking into consideration (55) we have

dim(TiXT2 + DIXD2) ---'-degdet T2 + degdet D2 = deg det 712 + deg det T1 = deg det T2 + deg det T1 = degdet T1T2 = dim XT.

(59)

}lence equality follows. Conversely assume now that the normalized bicausal isomorphism G is represented by the left coprime matrix fraction G -- T - 1 D such that T -- TIT2 and D -- D1D2. Furthermore assume X T ~-- TIXT2 (]3 Di XD2. (60) Then wc have to show that G ha~ a factorization G --- G2G1 with Gi also normalized bicausal isomorphisms. Applying Theorem 3.8 it follows from the equality

X T = TIXT2 + D1XI)2

(61)

that T1 and DI are left coprime. Let E be the 1.c.r.m. of T1 and D1. Then, as TIXT2 N D1XD2 = {0}, it follows that XT' N ErO'[z] = {0} and hence the right Wiener-Hopf factorization indices arc nonpositive. Let

E = T1D1 = D1T1.

(62)

Since E is the l.c.r.m, then D h T 1 are right coprime. Now deg det E = dcg det TI + deg dot D1 --deg dct T1 + deg det D1 = deg dct Tl + d c g det 7"2. llere the equality deg det Dl = deg dct T2 was a consequence of the direct sum representation (54). Thus necessarily E-1T1T2 has trivial right factorization indices. Thus

E-1TIT2 = g r

(63)

for some unimodular U and bicausal isomorphism F which without loss of generality we can assume normalized. Since XT, T2 = XIAD2 also E - I D 1 D 2 = U1FI as before. However

163 and

E - t Di D2 = ( Dfff l ) - l Dt D2 = T~I D2 = UiFt. The equality T1DI = D I T t implies

T~-ID1 = D1T-~ 1 and hence

G = T.TtT~-IDID2 = ( T ~ t D I ) ( T t ' I D 2 ) = F - I U - I U I F I . This in turn implies F G I ' i -t = U-1UI. established with GI = FI and G2 = F.

Thus the factorization G = G2G1 has been II

A remark is in order. A 1.c.r.m. is only defined up to a right unimodular factor. Thus w i t h o u t loss of generality wc can, by (63), assume E-1T1T2 is a normalized bicausal isomorphism. Thus XF, = XT~T2 ~nd since

E = T1D1 = D1T1 with Tt, Dt left coprime and T1, D1 right coprimc we have X E = "1"1X-~l ~ Dl X ~ . This means t h a t on X T wc can redefine the l ' t [ z ] - m o d u l e structure so t h a t both TIXT2 and D1XD~ both bccome submodulcs. T h u s the invariant subspace TtXT= and the conditioned invariant subspace DtXD2 both considered as conditioned invariant subspaces are compatible. This should come as no surprise as wc know t h a t two linearly independent controlled iuvariaat subspaces are compatible. Hence also two conditioned invariant subspaces whose sum equals the whole state space are compatible in the sense t h a t for some constant H both arc (A + H C ) - i n v a r i a n t , as was the case here. We focus now on a special case of W i e n e r - I i o p f factorizations. D e f i n i t i o n 5.1 Let G E R H °° bc such that G, G -1 have no poles on JR. A factorization l

G=G+G_ such that G_, G -1 arc stable and G+, G+ t are antistable is called a canonical factorization. Tile following theorem has been proved in Bart, Gohberg and Kaashock [19791. T h e o r e ~ n 5.2 A normalized bieausal isomorphism G with minimal realization [A, 13, C, I] has a canonical faclorization if and only if X _ ( A x) and X+(A) are complementary sub-

~paces. Proof: Assume G = G+G.: is a canonical factorization. Such a factorization is a u t o m a t i cally minimal. Let G+ = T + 1 D + and G _ = T_-ID_ be a left coprime factorization. Let D + r~-l 7 _ = T Z 1 D + with the last one a left coprime factorization. Clearly .7'- is stable and D+ antistable. Thus

G+G_ = T+IT_-t D+D_ = (T_T+ ) - t ( D+D_)

164 By T h e o r e m 5.1 we must have XT_T+ = T - X T + @ D + X D _

(64)

But T-XT+ = X + ( A ) and D+XD_ = X - ( A X ) . Conversely, assume X = X + ( A ) @ X _ ( A × ) . Thus a factorization exists. Let G = T - t D . Factor T = T_T+ and D = D+D_ such t h a t D _ , T_ are stable and T+, D+ are antistable. Such ractorizatlons exist. Also T_XT+ = X + ( A ) and D+XD_ = X _ ( A × ) . By our assumption XT_T+ = T-XT+ @D+XD_ so by T h e o r e m 5.1 this implies a factorization G = G+G_. l

6

SPECTRAL

FACTOI%IZATION

In this section we study the problem of spectral factorization and its relation to the algebraic Riccati equation. O u r main object is to obtain state space formulas for the spectral factors. D e f i n i t i o n 6.1 Given a Hamillonian symmetric normalized bicausal isomorphism 62 we define a s y m m e t r i c factorization of 62 as a factorization 62(z) = V , ( z ) V ( z )

(65)

such that V is also a normalized bicausal isomorphism. The factorization is called minimal if6(qb) = 26(V), i.e. if there are no pole-zero cancellations between V and V,. 6.1 Let G be a Ilamiltonian symmetric transfer function having no poles or zeroes on i R and such that G(oo) > O. Then G(z) admits a spectral factorization of the form G(z) = G _ ( - z ) G _ ( z ) (66) Theorem

with G _ , G "1 stable. This is called a spectral factorization and G_ is called a spectral factor. Proof: Let d be the characteristic polynomial of G, i.e. the generator in any minimal realization of G, and let d a stable and an antistable factors. Let G = M D - I be G. T h e n D admits a factorization D = D+D_ with d+ "~-I + N be a left coprime factorization of M D + 1 So

tile characteristic polynomial of = d+d_ be its factorization into a right coprihm factorization of = dctD+ and d_ = de~D_. Let

G(z) = M D + I D - l = -D+IND_:I

(67)

From the fact t h a t G is llanfiltonian we obtain

~--l -D+(z)-' N ( z ) O _ ( z ) -1 = f ) _ ( - z ) - l lfl(-z)-D+ ( - z )

(68)

It follows that, up to a unimodular factor which wc incorporate in the i)olynomial matrix N , we have D+(z) =/)_(-z) (69)

165

and hence

V(z) = D _ ( - z ) - l N D _ ( z )

-1

(70)

Again, the fact that G is ttamiltonian implies N ( - z ) = N(z). Now the a~zumption that G is invertible on iR implies that N is invertible there, i.e. there are no imaginary zeroes. Since

N(it) = b _ ( - i t ) G ( i t ) D _ ( i t )

(71)

is positive, it follows from Theorem 2.13, that N has a spectral factorization

N(z) = E_(-z)E_(z).

(72)

With G_(z) = E _ ( z ) D _ ( z ) -~, this in turn leads to the spectral factorization (66). Clearly the boundedness of G at infinity implies that of G_. We quote next the main results on the Riccati equation needed for our purposes. Assume now that we are given the the algebraic Riccati equation (ARE)

/ I X + X A - X B [ 3 X + Q = 0.

(73)

With the Riccati equation we associate the Hamiltonian matrix

ii=(

A_Q - B f 3

(74)

well as the rational matrix function qS(z) dei}ned by

¢(z) = I - [?(zI + A ) - X Q ( z I - A ) - I B .

(75)

Clearly q) is an 7t x a causally invertible rational matrix function satisfying @(oo) = I, i.e. 4, is a normalized bicausal isomorphism. Moreover it is easily established that ¢ is Itamiltonlan symmetric. L e m m a 6.1 (i) For dd defined by

¢)(z) = I - [3(zl + ,71)-lQ(zI - A)-I B. y2e h a v e

¢(z)=~-(,o OT

-Q

J))

(z, Q A zr+2 0 )'(). o

o)(.)(

-A

'

0

(ii} For (~ as above we have

i.e.

(0

,

0

h

)]

,~.

(76)

(77) (Ts)

166

Observe also that (I, dcfincd by Equation (75) has a realization in R 2n. In particular its MeMiltan degree 6((I)) is bounded by 2n. Our standard assumption throughout this paper is that 6((I,) = 2n. In particular this implies the assumption of reaehability of the pair (A, B), for otherwise ( z l - A ) - X B is not ;1 (left) coprime factorization and we could reduce common left factors. Similarly, if Q is nonnegative definite, one can write Q = C(7. In any such factorL.,~ion (A, C) is necessarily an observable pair. Under these assumptions we can a.~sume without loss of generality that the spectra of A and H arc disjoint. We will not dwell on this since all the details can be found in Molinari's paper [1973b]. Observe that the reaJizations of (I, and 4 -1 given by (77) and (79) respectively are liamiltonian realizations in the sense of Brockett and I~ahimi [1972]. In this connection see also Fuhrmann [1984]. D e f i n i t i o n 6.2 A real symmetric solution K of A R E is called an unmixed solution if

a(A - B f 3 K ) n a ( - i t + KB[3) = 0.

(81)

The concept of unmixed solution was introduced by Shayman [1983] as a generalization of extremal solutions of the ARE studied by Willems [1971]. We are ready now to state the following thcorem, for a proof sec Fuhrmann [1985]. T h e o r e m 6.2 Let the matrices A, B and Q be given with (A, B) reachable and Q symmetric. Let ¢I, be defined by

a2(z) = I - [3(zI + .71)-XQ(zI - A ) - I B .

(82)

Let H be the Hamillonian matrix H= ( and lcl d be given by J=

A_Q - B [ d

(0, ') -

0

(83)

"

Then there exists a bijeclive correspondence between the following: (i} Real symmetric solutions of the ARE. (ii) Minimal symmetric factorizations of q2(z). (iii) 11amiltonian symmetric factorizations of ( z I -

H)J.

(iv) Lagrangian H-invariant subspaces of R 2n. (71) lnvariant subspaces of A - B[3K for any unmixed real symmetric solution K of the A RE. (vi) Polynomial matrix solutions N of the Polynomial Riccati Equation ( P R E ) N , N = D , D + II, QII

(85)

for which N D -1 is an normalized bicausal isomorphism and whcT~ D, H arc right coprime polynomial matrices satisfying (zI - A ) - I B = l I ( z ) D ( z ) -1.

(86)

167

Some remarks concerning the history of this theorem are appropriate. The equivalence of (i) and (it) is essentially contained in Andersoxl [1969] and Willems [1971] and is fully presented in Molinari [1973b]. That (i) implies (iv) is due to Potter [1966] and others. That (iv) implies (i) as well az the equivalence of (iv) and (v) is due to Shayman [t983]. A result similar to the implication of (iv) by (i) appears in Lancaster and R.odman [1980], The equivalence to (vi) is from Fuhrmann [1985]. T h e o r e m 6.;I Assume ( A, B) is reachable. Then the following statements are equivalent:

(i) The algebraic Riccati equation has a real symmetric stabilizing solution f(, i.e. a(A - B[3fi) C H_. (it} The algebraic Riccati equation has a real symmetric antistabilizing solution )(, i.e. o-(A -/3//)~') C l-I+. (iii) l'br cp defined by we have (l,(io.,) > e [ l ( - i w l - f l ) - l ( i w I - A)-113

(87)

for some ¢ > O. (iv) For the llamiltonian matrix II we have a(H) N iR = 0.

Clearly the reach,ability of the pair (A, 13) is not necessary for the existence of a real symmetric stabilizing solution. The following result makes this prccise. T h e o r e m 6.4 The algebraic Riccati equation has a real symmetric stabilizing solution if and only if the following two conditions hold: O) The pair (A, 13) in slabilizable. (ii) For the Hamillonian matrix H we have a( ll) f3 iR = ~.

Next we obtain a state space representation of the spectral factors. T h e o r e m 6.5 Let G = [A, 13, C, D] be a IIamillonian symmetric transfer function having no poles or zeroes on iR and such that G(oo) > O. [,el G(z) = D + ll+(z) + H_(z)

(88)

be such that H+ and H_ are strictly proper and stable and antistable respectively. Let II = IA_, 13_, C_] with the realization assumed minimal. Then the right spectral factor of G is given by G_ : [ A _ , B _ , D - ~ ( C _ - [ ] _ X ) , D ~ ] where D½ is the positive square root of D and X is the unique stabilizing solution of the algebraic Riccali equation X(A- -13-D-~C-)+(fl-

- G - 1 ) - t f 3 - ) X + X B - D - ~ [ 3 - X + ( 7 - D - I C - =O,

(89)

168 Proof: Let G(oo) = D. Let d be the characteristic polynomial of G, and let d = d+d_ be its factorization into a stable and an antistable factors respectively. Let G(z) = D + H + ( z ) + H _ ( z ) be a partial fraction decomposition of G. So H_ and H+ are strictly proper and have their poles in the left and right half planes respectively. As G is tIamiltonian c l e a r l y / ) =- D and tI+(z) = H _ ( - z ) . Let lI_ = g _ D -1 be a right eoprimc factorization of It_. Associate with tI_ a minimal realization H_(z) = C_(zl - A_)-IB_.

(oo)

H+(z) = - 1 3 _ ( z I + A _ ) - I O _ .

(m)

Clearly Now G(z)

= = =

D + D - l f i l _ + N_D-_ 1 D - ' I { D _ D D _ + N _ D _ + D _ N _ } D -1 D-1ND__-1

(92)

Obviously N is lIamiltonian symmetric. Now tile assumption that G is invertlble on i R implies that N is invcrtiblc there too, i.e. there arc no imaginary zeroes. Since N ( i t ) = D_ ( - i t ) G ( i t ) D _ (it)

(93)

is positive, it follows from Theorem 2.13, that N has a spectral factorization N(z) = E_(-z)E_(z).

(94)

With G_(z) = E _ ( z ) D _ ( z ) - l , this in turn leads to the spcctrM factorization (66). Olcarly the boundcdness of G at infinity implies that of G_. Now G(z) is the parallel connection of H+(z) and H _ ( z ) and has constant term D, so it has a realization

On the other hand we proved G has a factorization

V(z) = V+(z)V_(z) = d _ ( - z ) e _ ( z )

06)

where G _ ( z ) = E _ ( z ) D _ ( z ) -1. Comparing this with I I _ ( z ) = N _ ( z ) D _ ( z ) -1 it follows that G_ has a realization [A_, B_, Co, D 6] where Co has to be determined. This realization implies tile reMization i - A - , C 0 , - / ) _ , D ½ ] for G _ ( - z ) . Ilence G as a series coupling of G_ and ( ~ _ ( - z ) has a realization

[(A °)( " ) ( CoCo-71_

'

doD~

'

D~Co-[3_

)] ,D

.

(97)

As both realizations (95) and (97) are canonical they are, by the state space isomorphism theorem, isomorphic. Based on the study of skew primeness, we will look for a map of the form

XI

I0 ) , with X symmetric, intertwining the two realizations. The intertwining /

condition reduces to the following two matrix equations:

(, 0)(L 0) (A 0)(, 0) X

I

~'oC0

-if_

=

0

-.4"_

X

I

'

169

as well as

(c_

-~_ )

= ( ~0

x ~

-~-)

~1

The third relation follows from the previous one by symmetry. Thus we must have

;t_X + XA_

= -CoCo

(100)

and

C- - [}_X = D½Co

(101)

!

Thus Co = D - i ( C _ - [~_X) and substituting this back into equation (100) we obtain the algebraic Riceati equation

X(A_ - B_D-IC_) + (A_ - O_D-' [I_)X + XB_D-* [3_X + C _ D - t C _ = 0, (102) i.e. X must be a solution to the ARE that corrcspond:s to the Ilamiltonian matrix

II = ( A_ - I3_D-IC_ C_D-'C_

-13_D-'[3_) -(fl_ -C_D-tfJ_)

(103)

Now, from the realization (95) it follows that G -1 is realized by

_oo,0 C-D-IC( D-'C'_

-("I- - 0-D-I/3-)

-C-D-t

'

'

(I0,I)

-D-'/3_ ),D-']

llowever G - t ( z ) = D_(f3_E_)-ID_ has a Hamiltonian realization in X~_~_ with generator S $ _ E , so tile two realizations are symplectically equivalent. Since tile last map has a Lagrangian invariant subspace, namely E - X E _ , so has the matrix H. Hence by Theorem 6.2, the corresponding Riceati equation is solvable. Since G_ = E_DY. t is a spectral factor both G and G - t are stable. Now as G_ has the realization [A_, B_, Co, D½] it follows that G "-I = [A_ - B - D - ~ C' o , B - D - ' ,!D - ~ C o , D - ½ ] . 1

The stability Of G_.E1 means that A_ - B-D-~Go is also stable. computed that

d_D-~C_

-(A_- d_o-~O_)

x

However it is easily

f

(, o)(A__._o-,co =

so that

x

;

o

-(A_

X

X _ ( H ) then necessarily U is invcrtible and X = VU -1.

) - d0o-~l)_)

V

is any basis matrix for

II

An obvious way to construct Hamiltonlan symmetric transfer functions is to study transfer functions of the form G ( - z ) G ( z ) where G is an arbitrary real transfer function. The next theorem tackles spectral factorizatioas of functions given in this form.

170 T h e o r e m 6.6 Let G E R t I ~ be an proper transfer function, and assume G(it) is injcetivc for all real t. Further assume G has a realization [A, B, C, D]. l ~ t D__= D D . Then G,G has a spectral factorization G _ , G - , with the spectral factor G - realized by

(105)

[A, B, D - } (3(7 - / ) X ) , D~]

and X is the stabilizing solution of the algebraic Rieeati equation corresponding to the Hamiltonian matrix A - BD-1DC

-(CO

-

)

-BD-I[3

~ D -1 DC) - ( A

-

(106)

~"D_D_D-1B)]

Proof: Since we assume G is left invertible it follows that O(oo) = D is injective. Clearly D is invertible. Let G = N E -1 (107) be a right eoprime factorization. As G 6 R H °° it follows that E_ is a stable polynomial matrix. Thus GG = E_ ~,-1 N - N E -t and a spectral factorization of GG is obtained from a spectral factorization of NN, i.e. /qN = N _ N _ . Such a factorization exists by Theorem 2.13 as we assume N ( i l ) is injective for all real t which implies /~'N > 0 on the imaginary axis. Thus G_(z) N_(z)E_(z) -l (108) =

and hence,comparing (107) and (108) and as

d_(oo)C_(oo)

=

O(oo)C(oo)

=

DD = D

we must have G_(oo) = 193 and, applying Theorem 3.3, G_ is realized by [A,B,Co,D3], i.e. a_(~) = _03 + c o ( z z - A)-~Z~. This implies ~ " ! G _ ( - z ) G _ ( z ) = (D~ - [3(zl + A~ ) - 1 Co)(D* + Co(zl - A ) - ' B )

(109)

A product of transfer functioas is realized by thc series coupling and so G _ ( - z ) G _ ( z ) is realized by [ ( ACoC0-/i0 ) ( '

G'0D2 B ) " (D½C°'

-B)

, D]

(110)

But on the other hand the equality G , G = G _ , G _ shows that there is an equivalent realization given by

[(a0)(-) OC

-ft

'

OO

,(De

]

-[1),DD

.

The two realizations (110) and (111) are isomorphic. The map isomorphism of the two realizations if and only if A

0

i0

(lid

[ X 0 '~ defines I i

an

171 and ( [)G

-[3 )

X

I

= (D½Go

-[3 ) .

(113)

hold. These two matrix equations reduce to A X + X A + (~0G0 - CG' = 0 and bc

- Bx

=

~Co.

(114) (i15)

Solving tile last equation for O0 yields

Co = D-½(/)C -/}X).

(116)

which upon substituting bagk into equation (114) leads to o

=

.4X+XA-OC+OoCo

=

f i X + X A - OO + ( C D - X B ) D - I ( [ g C - / 3 A ' ) .

(i17)

This is the algebraic Kiccati equation

X(A - liD-' D O ) + (ei_ODD_-'f3)X - C O + O D D - ' [ 3 O + X B D - I [ 3 X = 0. (i18) This algebraic lticcati equation is associated with the llarniltonian matrix

A - BD-1DC

-BD-lf3

-(CO- ~D_-I/)O) -(fi - ~"D_D_D-'/.)))

(119)

From an inspection of the realization (111) it is clear that H is the generator of a realization

of (O(-z)c(~))-'.

But

( G ( - z ) G ( z ) ) -t = E _ ( - z ) ( I q ( - z ) g ( z ) ) - l E _ ( z ) . So H is symplcctically equivalent to SgN. The last map clearly has an invariant Lagrangian subspace and therefore so has H. Thus, by Theorem i].2, the algebraic Riceati equation (118) has a symmetric solution. Moreover there exists a unique stabilizing solution i.e. one for which A - BD-][)C - BD-I[3X is stable.

I

The next theorem discusses a special case of Theorem 6. In this case tile algebraic Riccati equation reduces to a homogeneous one which is equiva]ent to the Liapunov equation.

T h e o r e m 6.7 Let G be a rational proper transfer function with realization [A, B,C, I]. Assume A is stable and A × = A - B C is antistable. Then G . G has a spectral factorization G_,G_, with the spectral factor G_ realized by

[A,B,C-

IIX, I]

(12(I)

and X is the unique stabilizing solution of the homogeneous algebraic Riccati equation (A - C'I])X + X ( A - BC) + X B [ 3 X = 0.

(121)

172

Proof: Let

a(z) = N+(~)E: 1(~)

(122)

then G ( z ) -~ = E _ ( z ) N + ( z ) -~. By thc assumptions of the thcorcm C_ is stablc and N+ is antistable. A spectral factorization of G . G is obtained from a spectral factorization N_N_ of N+N+. Since lg+ZN+ is just the antistable invariant subspacc of SI~+N+ then thc spectral factorization problem reduces to finding the complementary stable invariant subspace. This reduces to a Liapunov equation, which we proceed to derive. From (122) it follows that (7_ is realized by [A, B, Co, I]. Computing G ( - z ) G ( z ) = G _ ( - z ) _ G ( z ) we have two realizations of G . G given by

Cc

-~

'

C

o0c0

-a

'

Co

'-

(123)

- "

and

respectively. The map ( X/

0[ )

'-

,i

.

(12'0

providesanisomorphismofthetworealizationsifand

only if the two matrix equations

(,0)

and

x

,

= ( Co

)

hold. These two matrix cquations reduce to

Co = C - hA"

(127)

and

Ax +

X A + doC0 - d c

= o

(12s)

Substituting (1.27) into (128) we obtain the homogeneous Riccati equation (A - O [ I ) X + X ( A - B C ) + X B ] 3 X

= 0.

(129)

But, as A - B C is antistablc, there exists, by Liapunov's theorem, a positive definite solution Y to the equation Y ( A - C/f3) + ( A - B C ) Y = Bf3

Lctting X = y - Z we get a solution to the homogeneous algebraic Riccati cquation.

(130) |

Assume G E R H °°. Certainly if 7 > HGHo~then 7 2 I - G , G > 0 on JR. Thus a spectral factorization exists. Thc following result gives a statc spacc formula for the spectral factor of G.

173 Theorem 6.8 Let C E R I t °° have the realization [ A , B , C , D ] and let 7 > IIGIIoo. Let D = ( 7 2 / - ~)D)½. Then G,G has a spectral factorization G_,G_, with the spectral factor G_ realized by [Z, B,_D_-I(DC - B X ) , D_] (131)

where X is the unique stabilizing solution of the algebraic Riceali equation associated with the Hamiltonlan matrix A - BD-2DC

-BD-Zf3

- ( d e - ~__o-2bc)

(132)

- ( 2 - ~O_D-2~) /

Proof: Assume G(z) = N ( z ) E _ ( z ) - l and by our assumption E_ is stable. As C I - ~ ( - ~ ) a ( ~ ) = ,r~r - ( 9 - ~9(~I + . ~ ) - ~ d ) ( D + C(~.r - A ) - ~ B )

(133)

it follows that 721 - G ( - z ) G ( z ) is realized by [(

0 ) ( B )

A Oc

-~

'

( -DC

dD

'-

13_ ) , 7 2 I - D D ]

"

(134)

On the other hand wc have also

7 2 1 - G,G

= 7 2 I - ( E _ , ) - I N , N E -1 = (E_,)_1172E_,E - _ N , N ] E _ 1

(135)

and a~ the middle factor is positive on iR wc have a spectral factorizatioa

72E_,E_ - N . N = g _ , N _ .

(136)

721 -- G,G = [(E_, )-~ N_,][ N _ E : l] = G_,G_.

(137)

Itence From the representation G_ = N _ E - I it follows that G_ has a rcalization G_(z) = D+Co(zl A ) - I B with D = (721 - / ) D ) ½ . So -

751 - G ( - z ) G ( z ) = G_,(z)G_(z) = (D__- f3(zI + fi.)-IG'0)(D + Co(zI - A ) - I B). (138) This implies the realization

OoCo -~

'

do~

'-

(139)

-'

for 7 2 1 - G , G . The two realizations are isomorphic and we try an isomorphism of the form I X

0 "~ This map will indeed be I ) "

isomorphism provided the following equalities

(i 0)(A 0) (i0) X

and

an

I

(b

G'oCo - A

0

=

x

do

=

0)(, 0)

-.,4

X

I

(1,1)

174

hold. These two matrix equations imply

.4X + XA + CoCo - CO = 0

(142)

DC - / 3 X = DCo.

(143)

and Solving the last equation for Co yields Co = D - t ( D C

- BX).

(144)

which upon substituting b~ck into equation (142) leads to

X ( A - B D - 2 D C ) + (Jl_ - C D D - 2 ~ t ) X - C'tS'-I- C D D - 2 D C + X B D - 2 [ L X = 0, (14,5) i.e. this is thc algebraic I~.iccati equation associated with tile Hamiltonian matrix

( H =

A- BD-2Dc - ( C C - dDD_-2DC)

-BD-213 ) - ( ; t - ~OO-~[~)

(146)

Now H is symplecticMly equivalent to the map S.pE_,E__N. N = SN_.N_. Clearly N - , X N _ is a Lagrangian invariant subspace and SIv_,Iv_ I N - , X I v , which is isomorphic to SN_, is stable.. On thc other hand in the isomorphism of H and S2v_,N_ the Lagrangian subspace N_,X~v_ corresponds to the image space Ad of

X

"

dard calculation shows that H [ .A4 is isomorphic to A - B D - 2 D C - BD-213X. So the last map is stable and X is the stabilizing sotutlon. I

7

INNER-OUTER

FACTORIZATIONS

Inner-outer f~tctorizations of analytic functions belonging to vectorial IIardy spaces play an important role both in operator theory as well as its applications in the analysis of infinite dimcnsionM systems and in modern H °° control theory, more specifically in the various forms of the model matching problem. In this section we study a spccial casc of inner-outer f~torizations in the casc that the function to bc factored is in R H °°. We will deal here only with the squarc case and wc modify the definitions accordingly. Thus a function G 6 R H °° will bc called inner if G . G = I on the imaginary axis, i.e. it is unitary there. A function G E R H °° will be called outer if GHS(C ") is dense in H2(Cn). IIowever we will restrict our usage of the term outer to functions G that have an inverse in H°~(Cn). This restriction excludes the possibility of zeroes on the imaginary axis.

Under these terms the product of an inner fnnction and an outer function is invertible in the right half planc with the cxception of at most a finitc number of points. Thc ncxt theorenl focuses on factorization resuls and _-.tate space formulas for the factors. Let G 6 R H °° b c a rational matrix, which we assume to satisfy G(co) = I and let G have a realization [ A , B , C , 1]. Let G = N E Z t bc a right coprime factorization. As

175 G E RH 'x', detE_ has all its zeroes in the open left half plane, i.e. detE_ is a stable polynomial. Let N = D+N_ b c a factorization of N such that delD+ has its zeroes in thc right half plane whcrcas detN_ has them in the left half plane. Let Gi = D+D.[ l be the inner factor of G. Then wc can write G~ = ( D + D - 1 ) ( D - N _ E - I ) . Now Gi

--- D + D -1

= (D+IV_)(~'-'~: 1) =

(147)

(D+N_)('D_N_)-'.

Note that the associated realization corresponding to tile last matrix fraction representation of Gi is reachable but not ncccssari}y observable. Since Gi is a bicausM isomorphism W e h&ve

G'[ 1 = (D_N_)( D+N_) -1.

(148)

G -1 = (E_)(D+N_) -1.

(149)

Comparing (148) with and noting that G - I = [ A - BC, B, - C , I], it follows that G~-l = [ A - BC, B, Co, I]. Using again the realization of inverse system it follows that [A - B(C + Co), B, -Co, I] realizes Gi. Putting F = - ( C + Co) the inner function Gi is realized by (tl + BF, B , C + F,I). Next we procced to characterize F . Sittce Gi is an inner filnction, it zatisiies G~-I = Gi,. Now from the rcalization [A + BF, B , C + F,I] of Gi it follows that Gi. is realized by [ - ( A +~-BF), ( C ~ F ) , - B , I ] . On the other hand G~"1 is, by the characterization of inverse systems, realized by

[A + B F

-

B(C + F), B, - ( C + F ) , I] = [A - BC, B , - ( C + F), I].

We look for a possiblc isomorphism X between the two systems, i.e. an invertiblc map for which the diagram Rm

/

n

\

C+['

X

R '~ A - BC

--.,

R"

1

J,

-(fi, -I-/~'13)

(150)

X % - ( c + F)

/ R "~

-I)

is commutative. The commutativity of this diagram is equivalent to the equations

X ( A - BC) + (A + F B ) X = 0

(150

- 1)X = - ( C + F ) .

(15~)

and

176 Thus F = - C + / 3 X which substituted back into equation (151) leads to

X ( A - BC) + (7i + ( X B - C)f3)X

(153)

= X ( A - B C ) + (,4 - O f 3 ) X + . ¥ U [ 3 X

= o.

The algebraic Riccati equation (153) is associated with the Hamiltonian matrix

If=(

A-BUO

-(??cf3))"

(154)

lIowevcr

A-BC 0

155,

But if wc compute the series realization of G,C we obtain

cc

-.~

'

-o

(156)

'-

.

This implies that ( C , C ) -1 has a realization

0

o )'-(

c ~ ),

.

(1~7/

But on the other hand ( C , C ) -1 = E _ ( N , N ) - I E _ , . Thus the algebraic Riccati equation (153) is solvable if and only if N , N has a spectral factorization. Clearly such a faetorization exists if N , N > 0 on JR. This is guaranteed if we assume the injectivity of N(z) on the imaginary axis. 7.1 Let the square rational proper transfer function O E R H ~ be injective on iR have a realization [A, B, C, I]. Then C.G has a spectral factorization G_,G_, with the spectral factor C_ realized by [ A , B , C - [IX, I] (158)

Theorem

and X is the unique stabilizing solution of the homogeneous algebraic Riccati equation (A - C B ) X + X ( A - B e ) + X B f ; X = 0

(159)

associated with the Humillonian matrix o

-(

~)

"

An inner-outer faetorization exists. The outer and inner faclors arc given,by Go(z) = G _ ( z )

(161)

c~(~) := [A + ~F, n , C + F,I]

(1~2)

and

l~spcctivcly, where F = - C + f3X.

177

References [1979] A. C. Antoulas, "A polynomial matrix approach to F mod G-invariant subspaces", Doctoral Dissertation, Dept. of Mathematic% ETIt Zurich. [1979] 1I. Dart, I. Gohberg and M. A. Kaazhoek, Minimal Faetorization of Matrix and Operator Functions, Birkhauser, Basel. [1980] 1I. Bart, I. Gohberg, M. A. Kaazhoek and P. Van Doorea, "Faetorizations of transfer functions", S l A M J. Contr. Optim., 18, 675-696. [1974] (3. Bengtsson, "Minimal system inverses for linear multivariable systems", d. Math. Anal. Appl. 46, 261-274. [1972] R.W.Brockett and A.Rahimi, "Lie Algebras and Linear Differential Equations", in Ordinary Differential Equations, (L. Wc~zz, Ed.) A,'ademic Press, New York, 1972. [1988] T. Chen and B. Francis, "Spectral and inner-outer factorizations of rational matrices", to appear. [1974a] W. A. Coppel, "Matrix quadratic equations", Bull. Austr. Math. Soc., 10, 377-401. [1974b] W. A. Coppel, "Matrices of rational functions", Bull. Austral. Math. Soe. 11, 89-113.

[1980] E. Emre, "Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces", S I A M J. Contr. Optimiz. 18,288-296. [1980] E. Emre and M. L. J. llautus, "A polynomial characterization of (A,B)-invariant and re0~chability subspaces", S l A M d. Contr. Optimiz., 18, 420-436. [1988] 13. Francis, " H °° Control Theory", Springer. [1976] P. A. Fuhrmann, "Algebraic system theory: An analyst's point of view", J. Franklin Inst., 301,521-540. [1977] P. A. Fuhrmann, "On strict system equivalence and similarity", Int. d. Contr. 25,5-10. [1978] P. A. Fuhrmann, "Simulation of linear systems and factorization of matrix polynomials", Int. d. Contr., 28,689-705. [1979] P. A. Fuhrmaun, "Linear feedback via polynomial models", 1st. J. Contr. 30,363377. [198l] P. A. Fuhrmann, "Duality in polynomial models with some applications to geometric coutrol theory," Trans_ Aut. Control, AC-26,284-295. [1983] P. A. l"uhrmann, "On symmetric rational transfer functions", Linear Algebra and Appl., 50,167-250. [1983] P. A. Fuhrmann, "A matrix Euclidean algorithm and matrix continued fractions", Systems and Control Letlers, 3, 263-271. [1984] P. A. Fuhrmann, "On llamiltoaian transfer functions", Lin. Alg. Appl., 84, 1-93.

178

[1985] P. A. Fuhrmann, "The algebraic Riccati equation - a polynomial approach", Systems and Control Letters, , 369-376. [1979] P. A. Fuhrmann and J. C. Willems, "Factorization indices at infinity for rational matrix functions", Integral Equal. and Oper. Theory, 2,287-301. [1980] P. A. Fuhrmann and J. C. Willems, "A study of (A,B)-invariant subspaces via polynomiM models", Int. J. Contr. 31,467-494. [1959] F. R. Gantmacher, Matriz Theory, Chelsea, New York. [1982] I. Gohberg, P. Lancaster and L. Rodman, "Factorization ofselfadjoiat matrix polynomials with constant signature", Lin. and Multilin. Alg. , 11,209-224. [1978] M. L. J. IIautus and M. Iteymann, "Linear feedback-an algebraic approach", SIAM J. Cortlrol 16,83-105. [1989] U. tIelmke and P. A. Fuhrmaan, "Bezoutians", to appear, Lin. Alg. Appl.. [1970] V. A. Jacubovich, "Factorization of symmetric matrix polynomials", Soviet Afath. Dokl., 11, 1261-1264. [1982] P. P. Khargonekar and E. 13rare, "Further results on polynomial characterization of (F,G)-invariant and teachability subspaces', IEEE Trans. Aut. Control, 27, 352-366. [1980] P. Lancaster and L. Rodman, "Existence and uniqueness theorems for the algebraic Riccati equation", S I A M J. Cont.. [1956] C. C. MacDuffce, The Theory of Matrices , Chelsea, New York. [1971] K. Martensson, "On the matrix Riccati equation", Inform. Sci. 3,17-,t9. [1973a] B. P. Molinari, "The stabilizing solution of the algebraic Riccati cquation", SIAM J. Cont., 11,262-271. [1973b] B. P. Molinari, "Equivalence relations for the algebraic Riceati equation", SIAM J. Cont., 11,272-285. [1966] J. E. Potter, "Matrix quadratic solutions", S I A M J. Appl. Math., 14,496-501. [1985] T. Shamir attd P. A. Fuhrmann, "Minimal factorizations of rational matrix functions in terms of polynomial models", , Lin. Alg. Appl., 68, 67-91. [1983] M. A. Shayman, "Geometry of the algebraic R.iccati equation, Part I", S I A M J. Cont., 21,375-394. [1971] J. C. Willems, "Least squares stationary optimal control and the algebraic Riccati equation", Trans. Automat. Contr., 16,621-634. [1966] D. C. Youla and P. Tissi, "N-port synthesis via reactance extraction- part I", IEEE Inter. Convention Rccord, 183-205.

A State Space Approach to

OptimalControl

K. Glover Department of Engineering, University of Cambridge Trumpington Street, Cambridge CB2 1 PZ, United Kingdom

J. C. Doyle Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125, USA

Abstract

Simple state-spa.ce formulae are derived for all controllers solvi,tg a stand~u'd 'Hoo problem: for a given number 7 > 0, lind all controllers such that the 'Ho0 norm of tile closed-loop transfer function is < 7. Under these conditions, a paramctrization of all controllers solving the problem is given as a linear fractional tra.nsformation (LFT) on a contractive, stable free parameter. The state dime,tsion of tile coellicient ,natrix for tile LFT equals th;rt of tl, e plant, and has a sel)~mLtion structure ,'eminiscent of classical LQG (i.e., ~/2) theory. Indeed, thewhole devclol)ment is very remil,iscent of earlier 7/2 results, especially those of Willems (t971). This l)apcr direct, ly generalizes thc results in Doyle, Glover, Kh~trgonekitr, and Francis, 1989, and Glover and Doyle, 1988. Some ~l)ects of the optimM ease (< 7) are considered.

1 1.1

Introduction Overview

The 7-/o0 norm dcfined in tile frequency-domain for ~, stablc transfer ma.trix G(~) is

IlCll~

:= supV[a(j~o)]

( ~ := n,aximun, shlgll]ar vahle )

tl/

The problem of analysis and synthesis of control systems using this norm ariscs in a number of ways. Wc t'LssLIInetile reader eithcr is fa..niliar with the cnginccring motivation for thcsc problems, or is interested ill the results of this paper for some other reason. This paper considcrs particula.r 7-/o0 optimal control problcms that are direct generalizations of those considered in Doyle, Glovcr, Khargonekar, and Francis (1989), and Glover and Doyle (1988), herea, fl,cr referred to a.s D G K F and fAD, respectively. The basic block diagram uscd ill this paper is

180

where G is tile generalized plant and K is the controller. Only finite dimcnsim~al linear timc-invariant (LTI) systems and controllers will bc considered in this paper. Tile gctteralizcd plant G contains what is usually called tile plant in a control i)roblcm plus all weighting functions. Tim signal w contains all external inputs, iuclnding disturbances, sensor noise, and commands, the output z is an error signal, y is the measured variables, and u is the control input. Tile diagram is also referred to as a linear fractional transformation (LFT) on K and G is called the coefficient matrix for the LFT. The resulting closed loop transfer function from w to z is denoted by T~w = .T~(G, K) . The main 7-/00 output feedback results of this paper as described in tbe abstract are presented in Scctioll 4. Tim proofs of these results exploit the "separation" structure of the controller. If full information (x and w) is available, then the ccntra.l controller is simply a gain matrix Foo, obtainc(l through finding a certain st~blc inwu'iant subspace of a llamiltonian matrix. Also, the optimal output cstima.tor is an observer whose gain is obtained ill a similar way from a dual IIamiltonian matrix. These special cases arc described in Section 3. In the general output feedback case the controller can be interpreted ,as an optimal cstima.tor for . ~ x . Furthermore, tim two llamiltonians involved in this solution can bc associated with full information and output estimation problems. The proofs of these results arc constructed out of a series of lcmmas, several of which have some independent interest, particularly those involving sta.te-spacc characterizations of mixed IIankcl-Tocplitz operators. A possible contribution of this paper, beyond the new formulae and theorems, may be some of tiffs tcclmical machinery, most of which is developed in Section 2. The rcst,lt is that the proofs of both tlm" theorems al,d the lemma~ leading to I.hcnl are quite short, let,rthernu)re, the development is reasonably selfcontained, and the prima.ry backgrout~d required is a knowledge of demcutary aspects of state-space theory, /~2 spaces, and operators on g2, including projections and adjoints. More specialized knowledge a.bout tile connections between Riccati equations, sl)ectral factorization, and Hamiltonian matrices would also be useful. As mentioned, this paper is a direct generalization of DGKF, and contains a subst;mtial repetition of material. Roughly speaking, we prove those results in GD which were stated without proof, using DCKF machinery, which considered a less general p,'oblem. An alternative approa.ch in relaxlng sortie of thc assumptions in DGKF is to use loop-shifting teclmiques as in Zhou and I(ha.rgonekar (1988), GD, and more completely in Salbnov el al. (1989). We also organize this i)aper much dilferently than DCKF. 'rile results are presented in a conventional bottom-up linear order, with lemmas and theorems followed by their proofs, which in turn only use lemmas and theorems alrca.dy I)roven. Readers interested in pursuing all tile details of the proofs may find it more convenient than DG KF. This I)aper lacks the tutorial llavor of DGKF and the explicit connections with the more familiar ?'{2 problem, although the 7-(2 theory will be found lurking at every corner. We also consider some aspects of gcneralizations to the < ca.se, primarily to indicate tile problems eucou,ltered in the optimal ca.se. A detailed derivation of the necessity the generalized conditions for the Full Information problem is given. [n keeping with the style

181

of GD and DGKF, we don't present a complcte treatment of the < case, but lcave it for yet another day. Complete dcrivations of the optimal output feedback casc can bc found in Glover ctal. (1989) u~ing different techniques.

1.2

Historical perspective

This section is not intended as ~ review of the literature in 7-/00 theory, nor even all attempt to outline thc work that most closcly touches on this papcr. For a I)ibliogr,lphy and rcvicw of the early ?Coo litcraturc, the interested rcadcr might sce [Francis, 1987] and [Francis and Doyle, 1987], and an historical account of the results leading up to those in this papcr may bc found in DGKF. Instcad, wc will offer a slightly revisionist history, which lacks somc factual accuracy, but has the advantage of more clearly emphasizing statcspace methods and, more spccifically, Willems' central role in 7-/~ theory. This mildly fictionalized reconstruction tclls things as they could have been, if only we'd been morc clever, and thus contains a certain truth ~ valuable as that of a more factually accurate accounting. Besides, "historical perspectives" are often revisionist anyway, we're just admitting to it. Zamcs' (1981) original formulation of ~oo optimal control theory was in an inputoutput setting. Most sohttion techniques available at that time iHvolvcd analytic functions (Nevanlinna-Pick interpolation) or operator-theoretic methods [Sa.rason, 1967; Adam jan el al., 1978; Ball and llelton, 1983]. Indeed, 7-/00 theory scemcd to many to signal the begimfing of the end for the state-space methods which had domhmtcd control for the previous 20 ycars. Unfortunately, the standard &equency-domain apl)roachcs to 7-/~ started running into significant obstacles in dealing with multi-input-output (MIMO) systems, both mathematically aud computationally, much as the 7ffz theory of the 1950's had. Not surprisingly, the first solution to a general rational MIMO 7-/~o optinml control problem, presented in [Doylc, 1984], rclicd heavily on state-space methods, although more as a computation,-d tool than in any essential way. The steps in this solution were ,~s follows: parametrize all internally-stabilizing controllcrs via [Youla ct al., 1976]; obtain realizations of the closed-loop transfer matrix; convert t.he resulting modcl-qm.tching problem into the equivalent 2 × 2-block gcncral distance or best approximation problem involving mixed IIankcl-Tocplitz operators; reducc to the Nehari problem (Ilankel only); solve the Nchari problem by the proccdure of Glover (198,1). Both [Francis, 1987] and [Francis and Doyle, 1987] give expositions of this approach, which will be referred to as the "1984" apl)roach. In a mathematical sense, the 1984 procedure "solved" the 7-Qo optimal control problem. Uufortuuatcly, it involved a peculiar patchwork of tcchniqucs and the associated complexity of computation was substantial, involving scveral lticcati equations of iacrcasing dimension, and fornmlac for the resulting controllers tended to bc vet3, complicated and have high state dimension. Nevertheless, much of the subsequent work in 7-/oo control thcory focused on the 2 × 2-block problems, either in the model-matching or gcncral distance forms. This contimzed to provide a context for a stimulating interchange with operator tltcory, the benefits of wlrich will hopefidly contiauc to accrllc. But fiom a control perspective, the 7-/~ theory seemed once again to be headed into a cul-dc-sac, but now with a Q in the corner. The solution has turned out to involve an cvcn more radical

182

emphasis on statc-spacc theory. In addition to providing controller formulac that are simplc and exprcsscd in terms of plant data~ the methods in DGKF and this paper are a fundamenta.l depart.ure from the earlier work dcscribed above. In particular, the Youla paramctrization and the resulting 2 × 2-block model-matching problem of the 1984 solution are avoided entirely; replaced by a more purely state-space approach involving observer-based compensators, a pair of 2 × 1 block problems, and a scparation argumcnt. The opcrator thcory still plays a ccntral role (as does Redheffer's work [R.edheffer, 1960] on linear fractional transformations), but its use is more straightforward. The key to this was a return to simplc and familiar state-space tools, in the style of Willems (1971), such as completing the square, and the connection betwccn frcqucncy domain inequalities ( e.g. IIG[1¢¢ < 1), Riccati cquations, and spectral factorization. In esscncc, one only needed to think about how Willcms would do it, and the rest is simply tedmical detail. The state-space theory of 7-/¢¢ can be carried much furthcr, by generalizing timcinvariant to time-varying, infinite horizon to finite horizon, and finite dimensional to infinite dimensional. A flourish of activity has begun on these problems and the ah'eady numcrons results indicate, not surprisingly, that many of the results of this paper gcncrMize mulalis mutaadis, to thcsc cases. In fact, a cynic might express a sense of ddja vu, that despite all the rhctoric, 7/~ theory has come to look much likc LQG, circa 1970 (or cvcn more specifically, LQ differential games). A morc dlaritablc vicw might bc that currcnt 7-/oo theory, rathcr than cnding the reign of state-space, rcalfirms the power of its computational madlincry and the wisdom of its visionaries, excmplified by Jan Willcms.

1.3

Notation

The notation is fairly standard. Tile Ilardy spaces 7-/2 and H~ consist of square-integrablc functions on tile imagi,,ary axis with analytic continuation into, respectively, the right and left half-l)lane. Tl,e Hardy space 7-(oo consists of bounded functions with analytic continuation into the right half-plane. The Lcbcsguc spaces E2 = E2(-co, co), 122+ = E2[0, oo) , and £2- = £2(-0%0] consist, respectively of squarc-intcgrablc functions on ( - c o , co), [0, oa), and (-co,0], and £ ~ consists of boundcd functions on ( - c o , co). As intcrprcted in this'pa,pcr, £oo will consist of functions of frcqucncy, £2+ and E2- functions of time, and E2 will bc uscd for both. We will make liberal use of the Ililbert space isomorphism, via the Laplace transform and thc Paley-Wicner theorcm, of/22 = £2+ @£2- in thc time-domain with £2 = 7t2 ®7"/~ in the frequcncy-domain and of E~+ with 7-[2 and E2- with 7-/~. In fact, wc will normally not make any distinction betwccn a time-domain signal and its transform. Thus we may write w E E2+ and then trcat w as if w E H2. This style strcamlincs tile dcvelopnlcut, as wcll as the notation, but when any possibility of confusion could arise, we will make it clcar whcther we are working in the time- or fi'cquency- domain. All matrices and vectors will be assulncd to be complex. A transfcr matrix in terms of state-space data is denoted ) - ' H -{- D

183

For a matrix M 6 Cv×~, M' denotes its conjugate transpose, ~(M) = p ( M ' M ) 1/2 denotes its maximum singular value, p(M) denotes its spectral radius (if p = r), and 111t denotes the Moore-Pcnrose psuedoinverse of M . hn denotes image, kcr denotes kernel, and G~(s) := G ( - g ) ' . For operators, F* denotes the adjolnt of F. The prefix/3 denotes thc open unit ball and the prefix T/~ denotes complex-rational. The orthogonal projections P+ a.nd P_ map £2 to, respectively, 7-/z and "H~ (or £:+ aad f2-). For G 6 £:~, the Laurent or multiplication opcrator Ilia : £.~ --* 12.~ for frequncy-domain w 6 /22 is defined by M a w = Gw. The i,ornls on £oo alld ].~2 ill the frequency-domain were defined in Section 1.1. Note that both norms apply to matrix or vector-valued functions. The unsubscripted norm 11• II will denote the standard Euclidean norm on vectors. We will omit all vector and matrix dimensions throughout, and assume that all quantities have compatible dimensions. 1.4

Problem

statement

Consider tile system described by the block diagram

•Z~/)

Both G and K are complex-rational and proper, K is constrained 1,o provide internal stability. We will denote the transfer functions fi'om w to z ,as T,,, ill general and for a feedback connection (LFT) as above we also write T,,u = .Tt(G, K ) . This section discusses the assumptions on G that will be used. In our application we shall have state models of G and K. Then internal slability will mean that the states of G and K go to zero fl'om all initial values when w = 0. Since we will restrict our attention exclusively to proper, complex-rational controllers which arc stabilizable and detectable, these properties will be assumed throughout. Thus the term controller will be taken to mean a controller which satisfi~ these prol)ertics. Controllers that have the additional property of being intern,'dly-s~abiliziag will bc said to bc admissible. Although wc are taking everything to bc complex, in the special case where the origiaal data. is real (e.g. G is real-rational) then atl the of tl,c results (such a.s K) will also be real. The problem to be considered is to find all admissil)le l((s) such that IIT_-~II~ < 7 (-< 7). The realization of the transfer matrix G is tMicn to be of the 5,rna G(s)=

Ci O,l C~ Dzl

Ol~ 0

=

compatible with the dimensions z(l) 6 Cv', y(t) 6 Cw, w(l) 6 C ''~, u(l) 6 C'''2, and the state x(t) 6 C". The following assumptions are made: (A1) (A, B2) is stabilizablc and (C~, A) is detectable

1 B4

(m)

D12 is full column rank with [ D12 D± ] unitary imd D2i is full row rank with [D21 D± ] unitary.

(A3) [ A-j~IC1 D12B2] h a s f u l l c ° l u m n r a n k f ° r a l l w " (A4) [ A-jwIC2 D21B1]

hasfullr°wrankf°rallw"

Assumption (A1) is necessary for the existence of stabilizing controllers. The assumptions in (A2) mean that the penalty on z = Clx+Dl~u includes a nonsingular, normalized penalty on the control u, and that tim exogenous signal w includes both plant disturbance and sensor noise, and the sensor noise weighting is normalized and nonsingular. Relaxation of (A2) leads to singular control problems. Assumption (A3) relaxes the DGKF assunaptions that (Ct, A) is detectable and D{2C~= 0, and (A,t) relaxes (A, Bx) stabilizable and 131D~ = 0. Assumptions (A3) and (A,I) arc made for a technical reason: together with. (A1) it guarantees that the two tIamiltonian matrices in the corresponding 9/2 problem belong to dom(Ric). It is tempting to suggest that (A3) and (A4) can be dropped, but they are, in some sense, necessary for the methods in this paper to be applicable. A hLrther discussion of the assumptions and their possible relaxation will be discussed in Section 5.2. It can be assumed, without loss of generality, that ~ = 1 since this is acidcved by the scalings 7 - t D n , 7-l/2Hi, 7-q2C1, 7t/2B2, 7L/2C2, aaLd 7-1K. This will be done implicitly for ln&ny of tile proofs and statements of this paper.

2

Preliminaries

This scction reviews some mathematical preliminaries, ill particular tile computation of the various norms of a transfer matrix G. Consider the transfer matrix

with A stable (i.e., all eigenvalues in tile left half-plane). The norm I[G'H~oarises in a number of ways. Suppose that we apply an input w E £2 and conskler tile output z E/.:2. Then a standard result is that IIGII~ is the induced norm of the multiplication operator Ma, as well as the Toeplitz operator P+Ma : 7-{2-~ ~2. Ilalloo =

va6.13£.2

II lh =

sup

wE/3£:2+

II_P÷ l[

=

sup

wet37/2

IIP+MGol[

The rest of this section involves additional characterizations of the norms in terms of state-space descriptions. Section 2.1 collects some basic material on the Riccati eqnation and the Riccati operator which play all csscntial role in the development of both theories. Sectiou 2.3 reviews some results on Hmtkcl operators and introduces the 2 x t-block mixed Ilankcl-Tocplitz operator result that will play a key role in the 7"/0o FI problem. Section 2.4 includes two lcmma~ on characterizing inner transfer functions mid their role in ccrtain LFT's and Scction 2.5 cousidcrs the stabilizability and detectability of feedback systems.

185

2.1

The

Riccati

operator

Let A, Q, R be complex n x n matrices with Q and R tlermitian. Dcfine the 2n x 2n Ilamiltonian matrix //:=

Q

-A'

If wc bcgin by ,assuming H has no cigcnvalucs on thc imaghmry axis, thcn it must have n cigcnvalucs in Re s < 0 and n in Rc s > 0. Considcr thc two n-dimcnsional spcctral subspaccs A_(H) und X+(H): thc formcr is the invariant subspace corresponding to cigcnvalucs in Rc s < 0; thc lattcr, to cigcnvalucs in Rc s > 0. Finding a b~is for X..(H), stacking thc basis vectors up to form a matrix, and partitioning the matrix, wc gct

X-(II)=Im[

]X~ X1

(2.2)

where X1, X2 E C"x'', and

X2

=

X2

7x,

Re Ai(Tx) < 0 V i

(2.3)

If X1 is nonsingular, or equivalently, if the two subspaees

,¥_(II),

hn [ 0]I

(2.4)

are complcmeatary, we can set X := X2X~ "~. Then X is uniquely determined by 11, i.e., H ~ X is a function, which will be dcnoted Ric; thus, X = Ric(11). Wc will t~ke the domain of ltic, denoted dom.(Ric), to consist of llamiltonian nmtrices 11 with two properties, namely, H has no eigenvalues on the imaginary axis and the two subspaces in (2.,1) are complementary, l"or ease of reference, these will be called the stability property and the conq)lementarity property, respectively. The following well-known results give some properties of X ~ well as verifiable conditions under which H belongs to dom(Ric). Scc, for example, Section 7.2 i, [l~rancis, 1987], Theorem 12.2 in [Wonham, 1985], and [Kuccra, 1972]. L e m m a 2.1 Suppose 1I C dom(Ric) and X = Ric(H). Then

(a) X is Ilcrmitian (b) X satisfies the algcbmic Riccati equation A'X + X A + X1LV - Q = 0 (c) A + R X is slab&

L e m m a 2.2 Suppose H has no imaginary eigcnvalues, R is either positive semi-definite

or ucgativc semi.definite, and (A,R) is stabilizablc. Then H E dom(Ric).

186

L e m m a 2.3 Suppose H has the form [

H =

Z

-BB']

-C'C

-W

will,. (A, B) stabilizablc. The,, tl C dom(Ric), X = mc(lI) > O, a,,d ker(X) C X := stable unobscrvable subspace. By stable unobservable subspace we mean tile intersection of the stable invariant subspace of A with the unobservable subspace of (A, C). Note that if (C, - A ) is detectable, then Ric(H) > 0. Also, note that ker(X) C X C ker(C), so that tile equation X M = C' always has a solution for M, for example the le~t-squares solution given by X t C '. We may extend the domain of Ric by relaxing the stability requirement. Even if H has eigenvalues on tile imaginary axis, it must have at least n eigenvalues in Re s <_ 0. Suppose that we now choose some n-dimensional invariant subspace, again denoted by ¢'t-(//), corresponding to n eigenvalues in Re s < 0 and a corresponding basis as in (2.2), but now satisfying

This subspacc is not uniquely determined by H, but if it still satisfies the complementarity property, then we can set X := X~Xi -l as before, if this X is also llermitian. We may thus define a new map R~, whose domain dom(R-~) will be taken to consist of llamiltonian matrices H with the property that an A'_(H) exists satisfying the complementarity condition and with the resulting X := X~X~ 1 Hermitian. To show that this is actually a map, we have to verify that X is uniquely determined, which is not always the ease. In fact, the conditions under which R ~ is actually a map arc intimately connected with the conditions on existence of Ho~ optimal controllers. It turns out that for the cases of interest in the present paper, whenever H is in dom(R~), the subspace will be uniquely determined. Thus whenever Ric is needed, it will be a well-defined map, but this must be proven. Fortunately, these cases can essentially be reduced to spectral fa.ctorization problems and standard theory can be al)l)lied (e.g. Gohl)erg, Lanca.ster and l{.odmaa (1986)). We may further extend the doma.in of R ~ by relaxing the coml)lcmentarity condition. TILe minimal requircment we will place on ,Y_(H) is that (2.5) hold and t h a t X ' ¥,. 2 = X ~ X x

(2.6)

is Hermitian. Note that this condition also does not depend on the particular choice of basis taken in (2.2). It is convenient to define dom(Rie) to be the set of those H for which a subspace ,.V__(H) exists and s~tisfics (2.5) and (2.6). Once again, the map R~, from dom(R-~) to n dimensional subspaces of C2'' (this is a Grassman manifold) does not always ('xist as the subspace is not uniquely determined I)y H. The smnc remarks about Ric as a map apply hcrc to Ric. These have bccn iutroduccd in order to treat the optimal ea.sc, but their use will be limited ~s this case it not ana.lyscd in detail. Note that dom(Ric) C dom(R~) C dom(I~Tc). Also, if II E dom( Ric) then R ~ and Ric are obviously wcll-delincd maps and Ric(H) = R-~(II).

187 2.2

C o m p u t i n g 7/0o n o r m

For the transfer matrix G(s) in (2./), with A stable, define the I lamiltonian matrix

[ H

:=

A+BR-ID'C - C ' ( [ - DD')-IC

=

-C'C

-A'

+

BR-'B' ] - A + BR-1D'C ' -C'D

R - ' [ D'C

(2.7)

]

(2.8)

whcrc R = I - D ' D . The following Icmma is essentially from [Anderson, 1967], [\Villeins, 1971], and I Boyd ct al., 1989]. L e m m a 2.4 L The followin 9 coltditio~ts arc equivalent: (,,) IIGII~ < 1

(b) H ha.s no eigcnvalucs on the imaginary axis (c) H E dom( l~ic) (d) 11 E dom(Ric) aud Ric(ll) >_0 (Ric(11) > 0 if ( e , A ) is observable) IL The followiu 9 conditions arc equivalent: ('O 116'll~ < 1 a,t,t ~(O) < 1

(b) 'zz c ,to,,~(l~) (c) 11 E dom(R~c) and R~c(ll) is unique with R~(II) > 0 (R~(I1) > 0 if(G,A) is observable) P r o o f From (t-G~G)(s)

=

[z

-C'C D'C

-A' B'

it is immcdiatc that H is tile A-matrix of ( [ - G~G) -I. It is easy to check using the PBtl tcst that this rcalization has no uncontrollable or unobscrvablc modes on the imaginary ,axis. Thus H has 11o eigenvalucs on tile imaginary axis iff ( I - G ~ G ) -1 has 11o poles there, i.e., (I - G~G) -l E 7~/20o. So to prove tile equivalence of (la) and (Ib) it sulficcs to prove that IlCll~ < l ~ ( I - C~C) -1 e ~ z ; ~ If IIGII~ < 1, thc~a t - G(jw)'G(jw) > 0, Yw, and hence (I G~G) -1 E ~£o~. Convcrsely, if IlGl[oo > I, thcu a[G(jw)] = t for some w, i.e., t is au eigcnvaluc of G(jw)'G(jw), so 1 - G(jw)'G(jw)is singular. Thus (Is) and (Ib) are equiwdcnt. Tile equivalence of (Ib) and (Ic) follows fiom Lemma 2.2, and the cquiwdence of (Ic) and (Id) follows from Lemma 2.1 and standard results for solutions of Lyapunov equations. -

188

The proof of part II is more involved and is given by the established results on spectral factorization as in Gohbcrg cl a/.(1986), since I - G~G > 0 for all s = jw. • In part II it w ~ ~ s u m c d that O(D) < I so that the llamiltonian matrix could be defined. Alternatives that avoid this are to consider Linear Matrix Inequalities or the deflating subspaccs of matrix pencils. This is discussed more in Section 5.2.5. Lemma 2.4 suggests the following way to compute an 7{~ norm: select a positive number q'; test if Ila[l~ < "f by calculating the cigcnvalues of H; increase or decrease 7 accordingly; repeat. Thus 7/¢~ norm computation requires a search, over either 7 or w. Wc should not be surprised by similar characteristics of the ~¢o-optimal control problem. A somewhat analogous situation occurs for matrices with the norms IIMII~ = trace(M°-M)

and llMIIo~ = e[MI. In principle, IIMII~ Cal, be computed exactly with a finite numbe," of operations, as can the test for whether &(M) < 7 (e.g. 7 2 I - M * M > 0), but the value o f / r ( M ) cannot. To compute (r(M) we must use some type o1" iterativc algorithm.

2.3

Mixed

Hankel-Toeplitz

Operators

It will bc useful to characterize some additional induced norms of G(~) in (2.1) and its associated differential equation J: =

A x + Bw

z

Cx+Dw

=

(2.9)

with A stable. We will prove several lcmmas that will be useful in the rest of the pal)er. It is convenient to describe all the results in the frequency-domain and give all the proofs in time-domain. Consider first the l)roblem of using an input w E £~_ to maximize I[P+zll~. This is exactly the standard problem of computing the Ilankel norm of G (i.e., the induced norm of the Hankel operator IQMe : ~ ~ 7J:), and can be expressed in terms of the Gramians L~ and Lo

AL~ + L¢A' + BB' = 0

A'Lo + LoA + C'C = 0

(2.10)

Although this result is well-known, we will include a time-domain proof similar in technique to the proofs of the new results in this paper. L e m m a 2.5

sup IlP+~ll~ = sup IIP+McwI[~ =p(LoL~) wElJ£~_ wEI3"H~

P r o o f Assume ( A , B ) is controllable; otherwise, restrict attention to the controllable subsl)a.ce, rl'hen Lc is invertible and w E £2- can bc used to produce any x(0) = x0 given :r(-oo) = 0. The proof is in two steps. First, inf {llwllN I x ( o ) = x o } =x'oLy'xo

(2.11)

"1'o show this, wc can differentia.re x(t)'Lytx(t) along the solution of (2.9) for any given i,,l)ut w as follows: d , -t -~(x L¢ x) = .~'L[tx + x ' L [ ' x = x'(A'L[ 1 + L'~'A)x + w ' B ' r [ ' x + x'L[.'Bw

189 Use of (2.10) to s u b s t i t u t e for A'L'[ 1 + L'[IA and completion of the squares give d , L[ 1x) = Hwll2 - Hw - B'L'['xH 2 --~(x Integration fi'om l = - o o to t = 0 with x ( - c ~ ) = 0 and x(0) = x0 gives ! --I xon~ Xo =

]lwH~

-

[Iw -

B'L-[~wlJ~ <_ IIw[l~

Ifw(t) = B'c-'~'tL[~xo = B'L[%CA+nmLY')Lx o on ( - c a , 0], then w = B'L~.lx and equaliLy is achieved, thus proving (2.11). Second, given x(O) = xo and w = O, the n o r m of z(t) = CeAtxo can be found from

If

=

'

xoLoxo

These two results can bc combined as in Section 2 of [Glover, 198@ sup

IIP+=II~ =

~up

IIP+zll~

x(|Loxo

- m a x - - '

--

p(LoLD

,,

if Ilalloo < ~ t h e , by L c m m a s 2.1 and 2.,1, the IIamiltonian maU'ix I1 in (2.8) is in do,,~(m~), X = m ~ ( H ) > 0, A + ~ B ' X i~ stable a,,d

A ' X + X A + C'C + ( X B + C ' D ) R - ' ( B ' X + D'C) = 0

(2.12)

Similarly, if 5-(D) < 1 and Ilallo~ _< 1 then by L e m m a 2.,1, the Ilamiltonian m a t r i x H in (2.8) is in d o m ( R ~ ) , X = R ~ ( H ) > 0, A + B B ' X has eigen values in the closed left hail plane and (2.12) holds. The following l e m m a offers additional consequence of bounds on IIC,%. ~,, f~ct, this simple t i m e - d o m a i n characterization and its proof form the basis for the entire development to follow. L e m n m 2.6 1. Suppose

IIC,'llco <

1 and x(0) = x o. Thcn

~.p (11=11~ -I1.,11+) = +oX.~:o

w6£2+

and I/~csup is a c h i c v c d . H. Suppose th.l llollo~< i, a(D) < I, a,~,lx(O) = zo. The.

~.p

wE£~4

(11~-I1~-I1,,,11-~)

= ~oXxo

P r o o f : We can differentiate x(O'X.c(O as abovc, usc the Riccati equation (2.12) to substitutc for A~X + X A, and comp[cte tim squares to get d(dxx)

= -Ilzll ~ + Ilwll ~ - IllFi/'2{mo - (13'X + D'C)xIII"

If w E £~+, then x E £2+, so integrating k o m l = 0 to l = oo gives I1:11~ - Ilwll~ = xoX.~.o - II/~-'/~[nw -

(n'x + D'C)xlll~ < x'oXxo

(2.13)

190

For Part I, if we lct w = - R - I ( B ' X + D'C)x = B'Xe[A+Bn-I(B'X+D'O)I'xo, then w E £~+ bccause A + BR-I(B~X + D'C) is stable. Thus the inequality in (2.13) can bc made an cquality and thc proof is complete. Notc that the sup is achieved for a w which is a linear function of the state. For Part II, A + B R - 1 ( B ' X + D'C) may havc imaginary axis cigcnvalucs, hcncc the inequality in (2.13) is still valid, but may not give thc suprcnmm. A sequence of functions w, can howcvcr be constructcd to approach the suprcmum by considcring X, = Ric(If~) where

H.=

-c'c-A'

+

-C'D

The, for ~. = (~ + ~ 1 ) - ' ( n ' X . + D'C)x 11:1122 -

(:2 W

Ilwcll~ = = ~ X , x o +

2~

t

•

, ~ _ x o X xo

Finally taking the limit as c ~ 0 gives the result by uniqueness of X = llm,_o X,. • G~(s) ], and w is partitioned conformally. Theu IIC;~ll~ < 1 ifr

G(s) = [ G'~(s)

-C'C

flw :=

-A'

+

-C'D~

is in d o m ( R i c ) , where 1~,~ := l - D'~D~. Similarly, 8(D2) < 1 and IIC'lloo _< 1 iff n w e dom(ll.T~). In either case, dcfine W = R~(Hw), which will be unique, and let w E W :=

{[w']l 102

wl E ~L,w~ E £2

)

(2.14)

Wc arc interested in a test for sup~et~w ]]P+z]]2 < 1 ~up Ilrwll.~ < 1

toE/3W

(< 1), or cquivMcntly

(< 1)

(2.15)

where F = P+[Mc, Ma=] : W ~ "H2 is a mixed Ilankcl-Toeplitz operator:

F[w,l w~

= P+

Ol

G~

l(,v, 1 w~

wlE~,

w~E/22

Notc that F is the sum of the llankcl operator P+M~P_ with the Toeplitz operator P+Ma2P+. The following lcmm~ gcneralizcs Lemma 2.4 (B1 = 0, DI = 0) a.nd Lemma 2.5 (B2 = 0,

D2

=

0).

L c m m a 2.7 L (2.15) holds wilh < iff the following lwo condilions hold:

(i) llw E dom(Ric) (iO p ( W L J < l

IL (2.15) hohl~ with < iff the followin,9 two conditions hold: (i) 11w E dom( R~)

191

(ii) p(WL~) <_ 1 P r o o f As in Lcmma 2.5, assume (A, B) is controllable; othcrwi~c, restrict attention to the controllable subspace. By Lemma 2.4, condition (i) is necessary for (2.15) for both cases, so we will prove that given condition (i), (2.15) holds iff condition (ii) holds. By definition of 14,', if w E 141 then

liP+dig -Ilwllg = liP+dig - I I p + w d l l - l i p - w i l l Note that the last tcnn only contributes to then Lemma 2.6 and (2.11) yield

liP+rill

~[ [121

'

wfildJ

through x(0). Thus if L~ is invertible,

'

-''

(2.16)

For part 1 we will prove the equivalent statement that p(I,VL~) > 1 ifl'sup,~zL~w IIr,,~ll= > t. The suprcnmm is achicvcd in (2.16) for some w E Id) that can bc constructed front the previous lemmas. Since p(I,VL~) > I iff 3 xu ¢ 0 such that the right-lured side of (2.16) is > 0, we have, by (2.16), that p(I'VL¢) > 1 iff 3 w E W, w 7~ 0 such that II&=lll > Ilwllg. But this is true iff sup,oe~w Ilrwll= _> 1. For part II, note that (2.15) holds with < iff sup Ilr-,ll~ - Ilwll~ _< o

wEBW

which by (226) is iff p(WL~) < 1. • The Fl proof of Section 3.3 will make use of the adjoint F* : ~2 ~ IV, which is given

by p'z =

GT~

=

G7

z

(2.17)

where P_Gz := P_(Gz) = (P_Mc)z. That the cxprcssiou in (2.17) is actually the adjoint of F is easily vcrified from thc definition of thc immr product (m vcctor-valucd Z2~, cxprcsscd in the fl'equency-domain as 1 too

(2.18)

< xl,x2 > : = 2"~ f-oo x,(flo)*x.2(jw)dw

The adjoint of 1" : )'V --* 7-/~ is the operator F* :7-ta ~ 14; such that < z, Fw > = < F'z, w > for all w E Fg, z E 7-{> Directly usiz~g the definition in (2.18), we get

= =

=+ + < aTz,w.2 >

G2w2>

192

2.4

LFT~s

and

inner

matrices

A transfer function G in 7~7"/oo, is called inner if G~G = I, and hcnce G(jw)*G(jw) = I for all to. Note that G inncr implies that G has at least as many rows as columns. For G inner, and any q e Cm, w E £2, then [[G(jw)ql] = [[q[[, Vw, and [[Gw[[2 = []w[[2. Because of these norm preserving properties inner matrices will be central to several of the proofs. In this section we give a characterization of inner functions and some properties of linear fractional transformations. First, we present a state-sl)ace characterization of inner transfer functions analogous to Lcmma 2.4 that is well-known and simple to verify

(sce [Anderson, 1967], [Wonham, 1985], [Glovcr, 198,t]).

[ _N2_l

L e m m a 2.8 Suppose G = [ C ] D ] with (C, A) detectable and Lo = L" sati4cs

A'Lo + LoA + C'C = O. 2'hen (,t) L o > 0 iff A is slablc

(b) D'C + B'Lo = 0 implies G~G = D'D (c) Lo > O, (A, B) cont,'ollablc, and G~G = D'D implies D'C + B'Lo = O. The next lcmma considcrs lincar fractional transformations with imacr matrices and is based on the work of Redheffcr (1960). L e m m a 2.9 Consider the followiug feedback syslcm,

Z~ I .%D ~..]Iu 1).~_[ Pll /012I E R T / ~ r ~_]__j

v

P21 P22

Suppose lhal P ~ P = I, l~-l' E RT[~o, and Q is a proper ralional matrix. 'lhcn lhe following arc equivalent: (a) The sy.~lcm is internally stable and wcll-poscd, and IIT~lloo < 1.

P r o o f (b) ~ (a). Intcrnal stability and well-poscdncss follow fl'om P,Q E T~7too, itt~2l]~ < 1, [IQIIco < 1, and a small gain argument. To show that IIT~,l]oo < I consider thc closcd-loop systcm at any ficqucncy s = j w with thc signals fixed as complex constant vectors. Lct IlQHoo =: e < i and note that T ~ = P ~ ' ( I - P22Q) E T~7~. Also let a := [[T~[[oo. Then [[wl[ < a[[r[[, and P inner implies that [[z[[2 + [[r[[~ = [[w[[~ + [[vil2. Therefore, Ilzll =

Ilwll

+

-

l)ll,.ll

[1 - (1 -

=

193

which implies 117;41oo < 1. (a) =¢, (b). To show that IIQIloo < t s ~ p p o ~ there exist a (real or infinite) frequency w and a const~mt nonzero vector r such that at ~ = jw , IIO"ll >- I1,'11. Then setting w = 1~-1t(1 - 1½:Q)r, v = Qr gives v = Toww. But ~ above, P immr implies that II~ll~ + II,-II~ = II,,,ll ~ + Ilvll ~ ~nd hence I1~11= > I1,1,11~, which is impossible since IIT=AI~ < 1.

It follo,vs tl,at ,~(Q(jw)) < 1 for ~H ,,, i.e., IIGIIoo < 1, since Q is rational. Finally, Q has a right-coprime factorization Q = N/11 -l with N, 111 c g'Hoo. We shall show that M -~ E gT-/oo. Since T,,,P~ t = Q(1 - I~2Q) -t it has the right-coprime f~ctorization To,,l~q ~ = N(M-P.2.2N) -~ But since 2'o~,1~-i~ E 7 ~ o o , so does (M-la.2.,N) -t. This implies that the winding number of d c t ( M - P:2N), as ~ traverses the Nyquist contour, equals zero. Furthermore, since d e t ( M - ,xl~:N) ¢ 0 for all a in [0,1] and all s = jw (this uses the fact that II.n~211oo< 1 and IlOll~ < 1), we have that the winding number of (let 51 equals zero too. Therefore, Q E 7~7"{oo ~nd the proof is COml)lctc. •

2.5

LFT's and

stability

In this section, wc consider the stabilizability and detectability of feedback systems. The i)roofs in this section are very routine and usc standard tcchniqu(~% I~rin(:il)~tlly the PBll test for controllability or obscrvability, so they will only bc sketched. Recall the realization of G fi'om Section 1.4 and SUl)posc that A E C ''×'', and tlmt z, y, w and u have dimension ]h, pz, ml, and m~, respectively. Thus Ct E O '~×', Bz E C "×'~2, and so on. Now supposc wc a.pply a controllcr K with stabilizable and dctect,~ble realization to G to obtain :/'~,. I"or the following lcmma., wc do not need the •~sumptions fi'om Section 1.4 on G for the output feedback problem. L e m m a 2.10 Th.c feedback connection of the realizations for G and K is,

(a) dclcclablc ~f rank [ A -Cl Al (b) stabilizable if rank [ A -C,.~ ,,1

Bz ] = n + m2 for all ReA >0. Dv~ 13, ] = n + P'2 for all l~cA >_ O. D'et

P r o o f Form the c]oscd-loop st~tc-spacc matrices and perform a PBIt test for controllability and obscrvability. It is easily chcckcd that any unobscrvablc or uncontrollablc modcs must occur at A violating the abovc rank conditions (see Limcbcer and llalikias (1988) or Glovcr(1989) for more details), hence giving the results. •

3

Full

Information

and

Full

Control

Problems

In this section wc discuss [our problems from which thc output feedback solutions will be constructed via. ~ scparatlon argunmnt. Thcsc spccial l)roblcms arc centred to the whole approach taken ill this paper, and a~ we ~hall see, they are also iml)ortant in their own right. All pertain to tire st~,ndard block diagram,

194 4

.@u Z

•

G

I; "W

but with different structures for G. The problems are labeled FI. Full information FC. Full control DF. Disturbancc feedforward (to be considered in section 4.t) OE. Output estimation (to be considered in section 4.1) FC arm OE are natural duals of FI and DF, respectively. The DF solution can be easily obtained from the F1 solution, as shown in Section 4.1. The output feedback solutions will be constructed out of the FI and OE results. A dual derivation could use the FC and DF results. The I"I and FC problems are not, strictly speaking, special cases of the output feedback l)roblem, as they do not satisfy all of the assumptions. Eadl of the four problems inherits certain of the assumptions A1-A4 from Section 1.4 as appropriate. The terminology and assumptions will bc discussed in the subsections for each problem, hi each of the four cases, the results are necessary and sufficient conditions for the existence of a controller such that IlT~dloo < "r ,,nd the family of all controllcrs such that IIT,,olI~ < ~. h, all cases, K must bc admissible. The 7-/00 solution involves two tlamiltonian matrices, 11oo and J ~ which arc defined as follows: R

:=

/'l := H~

D 'I o D I . - [ ~/2L''' 0 00 ] '

where

D ° I D,° , -

where

-C~CI

:=

00 ' 0 [ "/21~,, ]

-A'

-BIB'l -A

Joo :=

-

Dt.:=[Dn D.I :=

D~2]

[°"1 D21

-C~DI. -BID'ol

(a.~)

,

(3.2)

If Hoo E dorn(Ric) then let XI, .¥2 be any matrices such that

X -

=

.¥2

Tx,

X[X~=X2X~ , ReAi(Tx)_<0Vi

(3.3)

2

Similarly if J~ E dom(RTc) then let )~, 1/2 be any matrices such that

1I~

[ ] ["] )Y~

=

1.

Tr,

)~'13 = )~'r, ReAi(Tv) g 0 v i

Further if in addition Hoo E dom(Ric) and/or Joo E dom(Ric) then deline,

(3.4)

195

Xo~ :=

X2X7 l,

I ~ :=

Y2YF x

(3.5)

Finally define the 'state feedback' and 'output injection' matrices as F

:=

F2

:=

L := [ L, 3.1

Problem

- R - l [DI'CI + B'X~]

]:=-[B,D', + rooC'ln-'

(3.6) (3.7)

FI: F u l l I n f o r m a t i o n

In the FI special problem G has the following form.

a(~) =

A CI

BL

311

B2 DI2

(3.s)

0 It is seen that thc controUcr is provided wiLh

Full hiformation sincc y =

X ). hi SOLIIC It)

cascs, a suboptimal controllcr may cxist which uses just the state feedback x, but this will not always be possiblc. While Llxest~tc feedback problem is more traditional, we believe that the full information problcm is more fundamental and more natural than the slate feedback problem, once one gets outside the pure 7/: setting. The assumptions rclcv,~lt to the FI problem which are inhcritcd from the output fccdback problem are (AI)

(A, B2) is stabiliz,d, le.

(A2) D,2 is full cohmm ,'auk with [ D,2 Dj. ]unitary. (A3) [ A - flQIC'

DI2B'~] has full c°h'mn raLlk f°r all w"

The rcsults for the Full Information case are as follows: T h e o r e m 3.1

Suppose G is given by (3.8) and satisfies AI-A3. 1hen

(b) lf e(n~Dn) < 1 then 31¢ such that IIT~.,[I~ < x ,~ /I~ ~ dom(l£ic), X°,.\'.2 = x;x,

> o. x , ,,,,d x~ ,,~ ,t41,,~ i,, (3.3).

(c) All admissible K(s) such thal ]VI',wH~ < 1 ave give,, by

fo,.Q c n o n e , IJQII~ < i. Note that the sullicicucy proof for part (b) is omitted. We will prove the 1"I results ~md tile FC results follow by duality.

196

3.2

Motivation

of the proofs

for P r o b l e m

FI

Wc will first motivate the proof by considering a completion of the squares assuming that Xoo > 0 and exists. Lct us fiLcLor

0 =:

=~ T2

I][Ti 0

(3.9) (3.10)

T'JT

=

7'2

=

D I' ~ D n , T;Tt = I -

! Now I - D tt t D ± D t D lt

I

DllD H DI~DH

=

DiIDl~I

]

(3.11) (3.12)

D u' D ± D ±'D n

> 0 since ,~t a = oo

[°1

7,(6', K ) ( ~ ) = O . + D , ~ K ( ~ )

1

DAId 1 > V(2](G,K)(oo)) > y ( D k D , I ). Now consider Lhc Riccltti cqua.tion for Xoo,

[el (3.13)

=~ X o o A + A ' X ~ + C~(2"z - F ' R F = 0

and observe that

([1) ([]) w

-

Fx

R

It

w

-

Fx

"{t

=

'

- w'w + x'(CID1. + X J3) ~t

w IL

+ [ w' u ' ] ( D ; . C , + B ' X ~ ) x + x ' F ' I ~ F x =

~ ~

-

w w + x XooB

+

g'X~x

+ :c'l'"l~,l"x

It

=

z z - w'w + x'XcoAx + x'A'X~x

+ z'X~13

w

u'

11

=

d

z'z - w'w + :a(z

,t

2~x)

Integrating from t = 0 to oo with :r(0) = x(oo) = 0 gives

II~-llg- II,,l[~ = tlTh~o+ , - [ 7h I ] F.,'II~ - l i T , ( , , , - F.~)II.~.

(3.14)

u,~.ce ~o obt,~in II=ll~ < II'oll~ ,,'~ require II~rhw + . - [ 7h 1 ] 1~:~I1~ < II:Z~(w lfix)ll2 .,,d we ~ce that in some sense the "worst w" is l;'~x, whereas the "best u' is - 7 ~ w + [ T2 I ] F x . Notice that in the ca.se T~ # 0 (:::> lh~ ¢ 0) the natt, ral full infornlation controller uses both w and x.

197

3.3

Proofs

for Problem

FI: Necessity

(a) 1]"there exists an admissible controller such that IITMI~ < 1, the,, H¢o c ,lo.~(ai4.

m~(~,~,¢o)>_ o

(3.15)

(b) If there exists an admissible controller such that IIT,,,IIo~ _< 1, then I&o E dom(R~),

XIX~ = X~X~ > O.

(3.16)

We will prove a slightly stronger result, but before that, we need some preliminary results. Let us first consider

-cc,z A'O]-[

O'.C,.

where 7' and d are given in (3.10). Notc that

DI.T ~1 = BT -l 1 - DI.R-ID'I.

[ Dj_D~DI,Ti-',

D,z

]

I - - DI:D,12+ D±D,zr~ ,o-t,p,-t,~, n n, £]11/1 -t 1 UIIIJ±L/± Ir --I = D±(I + D±D,,(T;T,) D,,Dz)D'z ~_.

I

•

I

= D±S-'D:

(3.~)

whel.'e

' ' - DxDHDu

S:=I llence

II~ =

[

D±>O.

']

N -C'IDj.S-ID~Ca

-B2B~ + B~B'~ -N'

where N :=

A - B:D'vzC, + BIT~ I)IID±D±Ct "

•

I--1

l

I

Next we will show that we can assume without loss of generality that tile pair (D~tCl, - N is detectable. This simplifies tile technical details of tile proof. Thus suppose that the pair (D~C1,-N) is not detectable or equivalently that (D~C1,-A + 13,~D~26'~) is not delectable. That is, ( A - 132D~2CI) has stable modes that are not observable from D~CI (note that modes of (A - 132D~2Gi) on the imaginary axis are observable from D~CI by A3). If we now change st~te coordina.tes so that

[ A] B ] " 6'1 Dto

=

Z~ A,21B,,

A~t A,2~ BI2 C11 Cl~ DH

B2,] B22 DI~

198 t with AI= - B2aD,uC,~ = O, D~C12 ~ O, (D'ICH,-AH + B:~ID t12Cll) detectable and (Au2 - B22D'12C12) stable, then the state equations for the system witll controller K =

A r ~[_: 1~ l t,

1

can be writtcn as

kl = Allxl + BHw + B=1(D'12CI2x2+ u) Z

=

Cllxl + DllW +

D12(D'12C12x2+ u)

X'2

=

A22x2 + A21Xl +

Bl2W + B221t

X t u + D12C~z2 =

¢ X C'~:+ b l z l + b2z2 + [~3w + Dl~C~

If the controller, li, is Mmissible with I1~',(0,2")11= < a (-< state equations show that the subsystem G~ =

Ct= DH

t),

then the

abo~

D~

controller, 1(1 (givcn by the final three equations above), which satisfies II~-~(c,, K,)llo~ < 1 (< 1). Furthcrmorc, suppose wc can find a suitable stable invariant subspace

Nil X~l

]

for the IIamiltonian for G~ then Xtt

0

0 I

X2~ 0 0 0 will be suitable for G since (A22 - B22D~2C=2) is stable. We will thcrcfore a.ssume that (D'zC~,-A + B,~D'vzC~) is detectablc for thc rcmaindcr of the uccessity proof. Tile proof also requires a prcliminary change of variables to v := u - Fox This changc of va6ablcs will ncithcr change internal stability nor the achievable norm since the st~tcs ca.n bc meamtrcd. The matrix Fo is the optimal state feedback matrix for a corrcsponding ~2 problem as given below. By Lemnla 2.3 the IIamiltonian matrix

Ilo :=

[ A - B2D~2C, -B2B~ ] -C~D±D'±CI - ( A - B2D',2CI )'

belongs to do,a(l{ic) since (A,/32) is stabilizable, and Xo := Ric(Ilo) > 0 since (DzC~,' - A + BaD'12Cl ) is detectable. Define TM

14 := -(D'~2C1 +/3'~Xo),

AFo := A + 13.~Fo, 6'tF0 := 6't + Dt21;'o

[ A,,~ I B, ] c'~(~)

:= t

c,,~o I D,,

Suppose D l is any matrix making [D,2 D±] an orthogonal matrix, and define

199

[u

u.l=tc,,;0

-~o

',1~',

(3.19)

o,

Then the transfer funct,ion from w and u to z becomes z=

C1~

Du

D12

v

The last result needed for the proof is the following lemma which is easily proven using Lemma 2.8 by obtaining a state-space realization, and then eliminating uncontrollable states using a little algebra involving the Riccati equation for )to. Letr,.ma 3.2 [U U±] is squarc and ira, or and a realization f o r G ~ [ U

c';[u

v.]=

o;,o.

Uj. ] is

o ,o,

This implics that U and U± are each inner, and botll U~G~ and U~G~ are in T ¢ ~ . We are now ready to state and prove the main result. P r o p o s i t i o n 3.3 I. If

II. If

sup wCaB£.2+

mln vE£2÷

sup

,,in 11=112< I

wEB/:z+ uE£~4

Ilzll=< 1

±heniio~ e do,n.(nic) antl l~i4H~ ) > O.

Ute,, Iloo E dom(RTc) and X',X.~ = X~.V, > O. Xt and

--

X2 arc dcflucd i. (3.3). P r o o f of P r o p o s i t i o n II[U u.l~=lk, and

Since [U U,] is square and inner by Lemma 3.2, [[=112 :

[U~GJo+" ] Since u E ~2, its optimal value is ~, = -P+U~Gew and the hypotheses of the proposition imply that

we~n,

U~G~BIw

112 < I

(< 1)

Mixed llankel-Tocplitz operators of this type wcre considered in Section 2.3. We can define the adjoint operator I'" : 122+ ~ 14' (14; from (2.1,1)) by

P'w =

U'ZG~w

=

U2

of the operator F : 14,' --* 7-/2 given by

q2 IIence

: l+(O~ (Uq, + U, q2)) = P+G;

U U,

q2

200 sup Ilrqll~ < i

(< l)

qEBW

This is just the condition (2.15), so from Lcmmas 2.3 and 2.7 and (3.21) we have that IlaTV±lloo < i

(< i)

and hcnce Hw E dom(Ric) ( or Hw E dom(R~)) where (substituting for the Riccafi I r l I ~l • I equation for Xo and noting that Bl.~o + DllCtF o = T~BI.~o + D II I D I D.I.CI , see (3.17),

(Hw)n

r--I

I

r Ir

I

--1

~l

I

#

= AEo + ( - X o C , D ~ ) D I D n ( T ; T , ) (T~BIXO + D~,DIDkC,) =-Xo'(A _ B~D',2C,)'Xo _ X.o- , ,ClD±D.cCl ,

-'~o C1D~D.DuT; --1 ]31.~o r--1

I

¢

•

~l

r

t --i I -.Xr--I o C1DIS D ± D n D nt D x D ±I C I

=

_X

lN,Xo

-

X. -oI C ,i D i S

-J

, D±CI

(Hw)l'2 = Xol C'l D i S -l D'i C~-V~~ I I r It1 rvl -I r (llw).~, = -(Xo[3,T, + C,D± D±D,,)(TI'J, ) - - ] (%,Bl);o + D ,I j D . D"z1 C ,1 ) = -Xo[31B1)¢o - N'Xo - XoN + .¥oB2B~Xo - G1Da_b Da_Cl "

$

•

It is now immediate that

llw = T-IHooT where T =

•

[_, x.} -Xo

0

'

I

=

r

¥1

-

rf__

1

" t

[0 xol] .Vo

Tile apl)ropriate sta.ble invariant subspace for llw will be hn

-I

1 ] ~md hence that W

for 1Ioo will be

Moreover Lemma 2.7 will give that p(l,VXo' ) < 1 (< 1) and hence Xo > H: (X0 _> W) giving that

( 1 - Xo IW)'Xo = X o - W > O ( > 0 ) or

Xoo = Xo(Xo - W)-~Xo > 0 in case (a). This completes the necessity proof for both parts (a) and (b).

201

3.4

Proofs

for Problem

FI: Sufficiency

Art admis,it, t, I¢(~) ~u& th,,t IlT.dloo < 1 a,v given by 1,-(,) =

[-@(,)

[,, o][,,, -,] 7,

F2

I

0

for Q ~ n o ~ , , II@[Ioo< 1.

Note that this contains the/,/'part of (a). Before beginning the proof, we will perform a change of va.riablcs suggested by Section 3.2. Firstly change the input variable to

with tlle corresponding controller I<,,,,(~)

I~'(s) + [

[ r~ I ] r

r

]

and state equ~tions •~ = /IFx + (B~ - B aT2)w + B2v z = CtFX+ DiD'.tDltw+ Dl2v where

AF:=(A+Ba[T2

lib');

C,r=Ct+Dta[T2

I]F

Also define the new feedback variable ,-:= T~(w - & . )

Now suppose ~,'.,,,,(.~) = @(~):z: [ -1,',

,];

that is

This gives the following feedback configuration ill which one would expect fi'om (3.1,t) that _v~." = 1 ~ i , ~ DII'~ -I1"111 = II"llg -II"ll~ and this is now proven together with the stability of A~-.

.Z~D

Ar p =

BI-B2T2

B2 ]

D±D~D|I DI2 -T:&

7~

L e m m a 3.4 P ~E~7-{oo , P ~ P = I and p,~l E 7~eTgoo.

0

J

(3.22)

202

P r o o f The observability Grmlai~m of P is Xoo since

A'FX°° + X°°AF + C'IFCIF + F' [ T~ ] [ =

0]F

(BiX~ + O'.q)

A'X~ + X ~ A + C',C, + F'

+(X~B~+CID,2)[ "1"2 I ]F I = =

F ' -

2

TIT2

T~

T~

I

00

j)

+R

+

0

F

0

0

where we have nscd tile identity -I3~Xoo - D;~Ct = [ "1"2 I ] F. Furthermorc, since Xoo > 0 and (F~, At:) is detectable (note AF + (B1 -- B2T2)l~ = A + BF is stable sincc Xoo = Ric(ll~o)) wc have that AF is stable by Lemma 2.8(a). Also

= =

0

I

0

tIence by Lelnlna 2.8(b),

[,0] =

0

I

as claimed. It is also easily shown that 1~ I 6 ~7-/~ since its poles are AI(A + BF). • The proof of sufficiency for Theorem 3.1 (a) and the class of all controllers given in Theorem 3.1(c) can now be completed. Let K be any admissible controller such that IIT,~ollo~< 1. Theu T~w E T~/~ and T~, = I~x + P127;~. Now define O = (I+'1;,oI~]IP22)-~T,,wP~ l so that Q(I-P22Q)-tP2~ = T,,,,, aad T~, = Pa1+P12Q(I-P22Q)-'P2~. Since P22 is strictly proper all the above are well-posed and Q is real-rational and proper. Hence Lemma 2.9 implies that Q E 7-¢7"/,o with IIQII~ < ~. This verifies that all transfer functions 7Lw and hence T,,~,, can be represented in this way. •

Remark Ill the optimal case of I)a,rt (b) t,he lU'OO[ of sufficie,cy is more delicate and to ilhtstr~ttc the difficulty the following example is given. Let

203

G=

then,

Iloo=

[lo] [x,] [o] -1

-1

'

X2

1

, Tx

-1,

X~X2

0>0.

An optimal controller is given by

u=Fx-w,~

k=(F+l)x,

zl=z,

z2=tt=Fx-w,

where F + 1 < 0 but F is otherwise arbitrary. Clearly for this controller a: = 0 and hence Z1 =

0~ Z 2 ----- --'//3,

If the controller for the suboptimal case with 7 -2 = l - ez is applied (see D C K F item F1.5), then, 1 + v ' T + (' Xoo

----

62

An admissible optimal controller is obtained ,as c ~ 0 iff Q(s) = - 1 , in which case - 1 ].

U(~)--, [ - ( l + , / l + e ~ ) 3.5

Problem

FC:

Full Control

The FC problem has G given by,

G(s)=

A

Bt

I

C,

D,,

0 I

C~ D21

0

0 0

and is the dual of the Full Information case: the G for the FC problem has the same form as the transpose of G for the FI problem. The term Full Control is used because the controller has full access to both the state through output injection and to the output z. The only restriction ou the controller is that it must work with the mcasurcnlcnt y. The assumptions that the FC problem inherits fi'om the output feedback problem are just the dual of those in the FI problem: (hi) (C2,A) is detectable (A2) 1)'2, is full row ra, k with

CA,l)

[A-jw[ C~

L

/)±

1

unitary.

11, ] h,m full row ra,,k for all

D~,

J

204 Necessary and stdlicicnt conditions for the FC case are given in the following corollary. The family of all controllers can be obtained from the dual of Theorem 3.1 but these will not be required in the sequel and arc hence omitted. C o r o l l a r y 3.5 Suppose G is given by (3.23) and satisfies A1, A2 and A4. Then

(a) 3K such that IlT,~lloo < 1 ~

Joo E dom(Ric), Ric(Jo~) > 0

(b) 3I( such that II"1"~lloo <_ ~ ¢, J~ E dom(R~), X~X~ = X ; X , > O. X , and X~ are de~fi,~cd i,~ (3.4).

4

M a i n Results: O u t p u t feedback

The solution to the Full hfformation problem of section 3 is used in this section to solve the output feedback problem. Firstly in Theorem 4.2 a so-called disturbance fccdforward problem is solved. In this.problem one component of the disturbance, w2, can be estimated exactly from y using an observer, and the other coml)oncnt of the disturbance, wt, doc~ not affect the state or the output. The conditions for the existence of a controller satisfying a closed-loop ~ - n o r m constraint is then identical to the Fl case. The solution to the general output feedback problem can then bc derived from the transpose of Theorem 4.1 (Corollary 4.3) by a suitable chauge of variables which is based on Xo~ and the completion of the squares argument given in Section 3.2 and the characterization of all solutions given in Section 3.4. The main result is now statc(l in terms of the matrices defined in sccl, ion 3 involving the solutions of the X ~ and ~1~ Riccati equations together with the "sta.tc feedback" and "output injection" matrices F and L. It will further bc convenient to additionally assume unitary changes of coordinates on w a , d z have been carried out to give the following partitions of D, l;'l and L].

I F;, 1,,'~1~] L2

(4.1)

0

T h e o r e m 4.1 Suppose G satisfies the assumptions A1-A4 of section 1.4.

(a) There exists an admissible conlmllc," I((s) such lhal II~-e(G, K)lk~ < 7 (/.e. IlT~lloo < 7) if and only if (i) 7 > max(a[D[,n, D, ..,, ], o[D]m, D',,=,]) 60 H~ ~ ~lo,,,U~i~) with X ~ = ~,:c(11~) > 0 6ii) J~ e ~lo,-,,(Ri~) with Y~ = mc(J~) >_0

6 # p(XooV~) < C. (b) Given that the conditions oil, art (a) are satisfied, then all rational internally stabilizing controllers l((s) satisfying ]]Yt(G', K)]I~ < 7 arc givcT~ by I," = 7t(1;.,¢,)

/o,. a,.bU,.ar,j ~) ~ no'too

~,,~h that

il~'il~ < 7

205

where

I¢,~=

8'1 .bn b12 d~ b~, 0

D H = -wl121 "" D ' m i t "t

~I

- D m l D m, l )

-1

D i n 2 - DH22,

Dr2 E C"2x'~ and 1)~ ~ Cm×w arc any matrices (e.g. Cholcsky factors) satisfying

b,.~b',~ ^

= I - D,t:t(7~I - D',mD,,rx)-' D~,~,,

^

D~D.~ = I -

'

~ _ n

D'

~-x n

and

0~ = - b . . ( c ~ + F,~), b, = -Zoo " -' L2 + .b~b?.2 b, 5, ^

~

Cl = F2 + DtID211C'~, ,A = A + B F + [3~ b~ ~C~, whe~e

zoo = (1 - 7 - : ~ x o o ) .

(Note that if Dll = 0 then the fornmlac arc considcrably simplified.) The proof of this main result is via some special l)roblems that are siml)ler special cascs of the general problem and can be derived fi'om the FI and FC problems. A sel)aration type argument can then give the solution to the general problem from these special problems. It can be ~ s u m e d , without loss of gencrality, that 7 = 1 since this is achieved by the scalings 7-1DH, 7-1/~B1, 7-1/~C1, 7J/zB2, 71/~C:, 7-aXo~, 7-1Y~-~ and 7 - 1 K . All the proofs will be given for the case 7 = 1.

4.1

Disturbance

Feedforward

Ill tile Disturbance Feedforward problem one component of tile disturbance, zot, does not alfcct the state or thc output. The other component of the disturbance, w2 ( and hence tile state x), can be estimated exactly fiom y using all observer. Tile conditions for tile existence of a controller satisfying a closed-loop 7-/~-norm constraint is thcn idcntical to the Full Information case. T h e o r e m 4.2 (Disturbance Feedforward) Thco~cm 4.1 is lruc under the additional assumptions that B,D~_ = 0;

A - B,D'2~C.~ is stable.

In lhis case,

,,v;, ]

(4.2)

206 Proof

(a) T h e necessity of the conditions is immediate from Theorem 3.1 since the existence of an output feedback controller intplies ~ the existence of a state feedback controller. Further, the additional condition - ~ ( D n D ' , ) < 1 is clearly necessary by considering s = co. Theorem 3.1 also shows that all controllers satisfying H.TI(G, I()[[~ < 1 arc given by ,,

=

Q(~)T,(,,, - F , = ) + T = ( & x - w ) + & x

,.

=

T,(w - F,x)

"O ~

QF

For any Q E R'Hoo, diagram

iIQII~< 1. Also the transfer function T,~, is obtained from the block

Z~U

r

v

as

T,~, = ( I - O P z a ) - l Q P 2 l

[-~

(4.3)

and hencc (4A)

u = ( ( I - QI½.a)-'QP2, - T.z)w + T2Fxx + F2.~

We uccd to find a Q ( s ) that c~m be written as an output feedback. T h e assumption of (4.1) will give the following realization for G,

G

W

=

A

0

Bv~

Ctl C,2 C=

Din,

D n l2

Dll21 D1122 I

Wl]

=

B~ 0

0

I

0

W2

Ilence wx affects z but neither x nor y and we must firstly find a that T,,,, is zero. Since 5I~.,o~ is zero wc laced

Tu~,, =

(I - Q&2)-'(Q(P2,

+

Q(s) in

P22T~)- T:)D', = 0

(,1.4) such

(4.5)

Using the slate space realization tllat for [ P~t P2a ] it, (3.22) gives [I½, + P22T2]D~ = TD , ±" "' ~ Q'l~ D" i = ' / ~' / ) '±

Aga.in without loss of gener,~lity we can assmne that

1 2~3 12

.T~ = wherc

0

(4.6)

207 $ *

t

t

T~It = I - DnD±DID.

and hence

T~,T,, T2Di

= =

Tn Tn

= ~I';~Ti~ =

I-

D'unDnn

DnDnDa.

= Dn21

-D'nuDl112 I - D',, n D u I , T ~ ' T [ [ ' D ' n , , D , , ,2 - D'I, , 2 D , , ,2 =:

Q~ ],

IIence (4.6)implics that for Q = [ e l Qt = D.~Tg

b'~,b~,

~

and Q Q ~ < I implies that t

=

(I + Duzl(I

:=

DI2DI2

--1

-

t

'

Dll.Dlllt

-

, D1121 D1121) -1 D .,. ~ l ) -1

where the indicated inverses exist by (a)(i). IIence

Q2 = b,2Q~ for Q~ e , ~ ,

llQ~lloo < i.

We have hence shown that all controllcrs can bc written as fccdback from w~ iuid x by substituting for Q into (4.4)

+ [ D,,2i,

DI, n ] ( F l x - w ) + F 2 x

=

b,2Q~b~,(,.2

=

b,2Q3b~,(w: - O 2 , F , x ) + (--Wll2l(l

- D2,/:'~x) + D u 2 , T ~ ' [ T , 2 w 2 - [T,,Tr~lFt;r]

t

-- D I I n Dim l l )

-1

t

DIlI1DI115

-- W l 1 2 2 ) ( W 2 -- D ' 2 l F l . z )

+ F2x

=

(b,, + b,2Q3b~i)(w: - u,2x) + F.:.

This gives the complete family of controllers in terms of x and w2. The disturbance, w~, and stale x can bc exactly estimated from the measurement, y, by means of an observer ,as follows, 1

x

=

lb2 = 7' =

Ak + Bntb2 + B~u -C2k+y Q.3q

tt follows thaL

208

(5: - ~:) = (A - B,2C2)(x - 3:)

and hence for x(0) = 5:(0) = 0, .~(t) = x(l) and tb(t) = w(t) for all t > 0. Furtl,ermore, intcrnal stability will follow from the stability of A - B12C2. Finally it is straightforward to verify that this family of controllers corresponds exactly to those of Theorem 4.1 with Y~ = 0, Z = I, and since

[0

]

and 0 = ll,ic(doo).

The transpose of Theorem 4.2 can now be stated to obtain another special case of Theorem 4.1. C o r o l l a r y 4.3 (Output Estimation) Theorem .[.1 is truc vndcr the additional assumptions that D i C t = O, A - B2D'r2C] is stablc. In this case

4.2

Converting

Output

Feedback

to Output

Estimation

Tile output fee(lback case when the disturbance, w, cannot bc estimated fi'om the ouLput is reduced to the c~tse of Corollary 4.3 by a suitable change of wu'iablcs. Since we showed ill (3.14) that

I1~11~- Ilwll,~ = Ilvll:~ -I1,'11~ wherc

r =

T~(w-Flx)

Wc will perform tile change of variM)lcs with v replacing z and r replacing w. ltcncc 5: =

(A + B , F , ) x + 13~7]-b. + B~u

v =

u + 7 ' 2 T V b ' - V2x

y

C 2 x + D217~-tr + D21Flx

=

209

:=

B,T,-'

A+BIFI -F2

T:T,-' 1 C2 + D2t lit D~tTi-1 0

] (4.7)

Similarly substituting v for u in the equation for G gives that the transfer function equivalence of the first two of the following block diagrams, with 7;r given by the third OllC.

y~u

7"~0

y

it

L e m m a 4.4 Lct G salisfy AI-A4, and assume that Xoo cxists and .V¢o >_O. Then thc following arc equivalent:

(a) 1( internally stabilizes G and [I.T'e(G,]()lloo < 1, (b) K internally stabilizes G~ur,, and [I.Te(G.u~., K)l]oo < X, (c) I( internally stabilizes Gt,,,p and II.~t(Gt,,,o,¢)11¢o < l, where G~u~u is given by (4.7) and G~,,,I, :=

-D,~F2 C2+DzIFI

D . Dl~ D~z 0

•

Proof (a) ¢* (b) Rcferring to thc abovc block diagram for P and T~r, it is scea by Lcmma 2.9 that Tz~, E RT-/~ with 117"~wl[~ < l iff T,,, E R'Ho~ with IIT ,II < s. (Recall that P ~ P = I, P E RT/oo, and 1~-~1 E RT-/o~). In order to prove intcrnM stability of both systcms wc note that this is equivalent to the realizations bclng stabillzablc and detectable. The realization of T,~ is detectable since the system zeros of (G,~,,)l~ arc the cigcnvalucs of A + BF (scc Lcnuna 2.10). Further the rcalisation of T,, is stabilizablc from r i f f the rcalisation of 7',~ is stabilizablc from w since they are related by state feedback. Finally if the realisation of T~ is internally stable with HT.~[[~o < 1 then thc above block diagram for T~, = .~e(P,.Te(G,,u~,,, K)) is intcrnally stable by a small gain argument and hence so is that for .We(G,K). (b) ¢:~ (c) Internal stability of both systems is equivalent since the closcd-loop Am~triccs arc identical. Further notc that G"P= aad recall that

[ DJ-0 Dr20 O] I T ~[D'~D, - ' I G~u~

01 [7~0 0]i

210

T~TI

=I

DIID±D~D,1.

-

Hence

DkDn

]

arid

I

-

~'e(Gt,,,p,

K)~.~e(Gt,,u,, K) = T~(I .Tt(G,v,,,, K)~.Tt(G~v,,,, K))2] -

hence giving the equivalence of (b) and (c).

The importance of the above constructions for G.~F, and Gt,,u, is that they satisfy the assumptions for the output estimation probleln (Corollary ,1.3) since A + BF is stable. IIence we are now able to prove Theorem 4.1. P r o o f of T h e o r e m 4.1 ( O u t p u t F e e d b a c k ) (a). Tile necessity of tile conditions will be first proved. Let K be a proper controller satisfying IITZ.lltHO0 < 1, then the controller h'[ C= D2, ] solve~ the full information

pr°blcmandhence(ii) h°lds" Similarly [ Ks°lvesthcfullc°ntr°ll'r°blcmandDt2 B2 ] hence (iii) holds. From Lemma 4.,1 I( stabilizes G,,,p with II.rdG,.,,,, K)lloo < 1, which satisfies the assumptions for the output estimation problem of Corollary ,1.3, since A4 implies that

rank [ A + B t I ' ] - j w I C2 + D21F1

131 ] D~l = n + mx

Hence wc rcquire dtn,p E

dom(Ric)

and )'~,,,p := R i c ( J t , , , p ) > 0

where Jtmp

A' + I ?tI BLt =

-BIB~

J.:=[

BIFI

-

t

,_xo]

0

-A

-DnF2 ] -I'2DI2 C2 + I ID21 [~-1 -BID n -BID21 D21B1 C2 + D2t Ft

--

We el,aim that

]

0

I

Joo

,,'

l

0

1

= Jtmp

211

where J~o was defincd in (3.2)

0] - [ -11~D',1 c, jR_,[ D'~BI

-BxB'~ - A

Joo :=

c]

To verify this clMm let M E

[

-D12F21

(4.8)

:= C~ + D21FI := D . I B ' ] A % + C - 3 1 := Dt.F + Ct

(4.9) Substituting for B'lXoo from (3.7) gives BIX~

=

E=

(4.ao)

F~-- DhN

D"(F~-D"'iV)+[

C'+D':F2 ]-D:,F,

[ I-- DHDh ] i~I =

_D~tD~l

and hence

- "i~-l E =

0

(4.it)

Now consider thc claim componcnt by component. Clcarly (J,.)~, = ('J~)2, = (g.,,,,)~l Secondly (Y~)~ = - A - BIB~X~ + B,D'.I[t-'(M + E) = - A + B,D'.~R-'M - B,(B;X~ + D'./V)

= (J~,,,,,h~ by

(4.t0). Finally (J-h~ - (a.,,phz

Substitute from

= X ~ A + A'X¢o + XooB~B'IX~ - (M' + l;')i~-'(M + 6) + M ' i t - ' M

(3.13):

XooA + A'X~o =

-C'~C, + I,"RF

=

-(IV'

=

_fi¢'/¢+

Equation (4.10) gives

F'D',.)(N

D~.F) + F D , . D , . I - I'tF,

, . + .ff'D,oF - FIr'1 F , DI.N

212

XooBiB~Xoo = (1~ - IV'D,,)(F~ - D~,1¢I) and (4.11) and (4.8) give

-M'k-'

E - E'k-' M - E'k-' E

=

-F~D',2N - / ¢ ' D , 2 F 2

-/:'(D.D'.

- I)5+

Adding these three expressions gives (J~)12 = (J,,,,p)~2 and the claim that J~ = gt,,,~ is verified. Since

J~

[']['] y~

--

y~

(A'+CL'),

we have

I-X~Y~

Jtmp [ -Xoo~oo

]

(A' + C ' L ' )

and li,,,v := lti,:(d,,,,,,) = ~%(~, - x o o Y + ) - '

_> 0.

It is readily verified that this implies and is implied 1>3, (iv), that p(Xo~Voo) < 1. % see this, consider Y~o = [Y°~' 0

- ' - -\'~1 > 0; conversely 0 ] ' Yo~x>0, a n d n o t e t h a t ) :¢~1

note that Xoo~¢o " : = (I -I- ~m,i,Aoo] ~+ v ~-xyt,,,,. .~-. . and hence Yt,,,p > _ 0 implies p(X¢o]%) < 1. Therefore the necessity of the condition is proven. Sufficiency also follows immediately because of the equivalence of the G and Gt,np problems. (b) C h a r a c t e r i z a t i o n of all s o l u t i o n s To characterize all controllers for G we just need to characterize all controllers for G,,,,. Y '2'1-1 whct'c using Corollary ,t.3, with ] rt,,,p = ~¢o.-~o

L..p

= =

- ( B~D'~ + Y , m / l t ' ) h - ' - Z £ ' ( B , D : , + Yoo(=XooB1Dt., + M ' ) ) f t - '

=

-Zgol(B,O :, + ~ % ( c ' - E'))R-'

--

Ft,.p

=

z:o't-z:o'Y [

[0] F~

,

Xt,.p=0;

0] Zt,.v=l

Wc call now substitute in the formulae of Theorem 4.1 with values to obtain the class of controllers.

f32 = = =

Gtmp and

the above (*)t,,,v

(132 + Z~o'L, Dr2 - ZL'l%oN'D,~)b,2 Z~o'(B, - )%XooB, + L,D,2 - ]%(I:'D'lo + Z~I(B~ + L , D 1 , ) f ) , ,

C',)D,,)D,,~

by (3.7). The expressions for Cl, G'2,/)1 a n d / i are then obtained by a direct transcription of the above exl)ressions and are hence omitted. This completes the prooL •

213

5

Generalizations

In this section we indicate how the results of section 4 can be extended to more general cases. Firstly the optimal case is considered when a variety of new phenomena arc encountered. Secondly the removal of assumptions A1-A4 is discussed. Finally some comments are included for the case when the optimal ~o~-norm is necessarily achieved at z = c¢. 5.1

The

Optimal

Case

Ill the optimal case any combination of the conditions of Theorem 4.1 (a) may bc violated. In order that tile Hamiltonian matrices II~o and Joo can be defined wc will assume that condition (a)(i) in Theorem 4.1 is satisfied and will state the result proven in Glover ct

.t. (1980). P

Firstly if II¢o, Joo

'1

E d o , n ( R ~ ) t h e n there exist [ X , ] satisfying cquatiOll (3.3) X~ d L

a,ld [Yx y: ] satisfyingequation(3.4). I n t h c o l ) t i n l a l c a . s c X l and/or YI naaybcsillgular so that X¢o := X 2 X ? 1 and Yo~ := )~]~-i may not exist, and if these inverses exist Zoo := I - 7-2Yo~Xoo may be singular. In order to avoid taking these inverses we wil] modify the definitions of the 'state-feedback' matrix, F in (3.7), and the 'output injection' matrix, L in (3.7), as follows. F °

:=

L ° :=

-R

-t /¢

[D',.C,X, +

B'X~]

I

-DaBxD.~ + ]~'C'][I-'

Furthermore as in (4.1) we assume that D, F °, and L ° havc been transformed and partitioned as follows.

1~, ' L °~ D

=

1'7./ F~°'

L~.2 Dl1~1 Dn~2 L; 0 1

I 0

The solution to the output fecdback problem in the optimal case can now bc statcd (Glovcr cl al. (1989)). T h e o r e m 5.1 Suppose G satisfics thc assumptions A I-A4 of scction 1./j and

7 > max(~r[Dm,,Dlm,l,(T[D',,,,, Dh~,l). (a) Thc,~c cxists au admis.sible cont,'ollcr l((s) such that II.rdG, l")ll~ < "r (i.c. 1lT'.-~ll~ _< 7 ) if a , . t o , @ if

(i) H ~ E do,n('R~) with X~, X2 satisfyin9 (g.g) such that X[X2 > O. (ii) do~ C dom(~) with Y~, Y~ satisfying (8.4) such that Y~'I~ > O. (iii)

-' ""'/ [ 7 -X~X, >° t l~'X,, 7 1~'%'~1 Y~ -

214

(b) Given that the conditions of part (a) are satisfied, then all rational internally stabilizing cont,vllers K(8) satisfying II~-dG, K)ll¢o <- 7 a,~ given by K = ~'t(K,,¢b) such that

for arbitrary a2 E TOil'too

IIOll~ _<7, d e t ( I - (I¢~)~(o0)¢,(oo)) 4 o.

wh, eFe

# denotes a suitable pseudo inverse, Dij are defined in Theorem 4.1, and B~ :=

(]q'B2 + Ll2)Dl~ o

^

&~ := b7 := -L~ +/~;D#bll ~ .= F; +b,,b.7,'&.~ hoh-,d~o

I3Tx + - , ' ~ 2 , :=

"~2

, , ~

= T~.E +

hoh-,&o

--2--12 ~ l

I~'X, - -r-2:l~.k'2

The descriptor form of tile equations for the controllers has been used as proposed for optimal IIankcl-norm approximation by 3afonov el al. (1987). At optimality/~ will typically be singular and the state-space equations of Theorem 4.1 are not pos.siblc. Moreover the matrix (sE-.zi °) may be singular for all s, but the transfer function I(,(s) nevcrtheless remains uniquely defined. The condition that d e t ( l - (l/~)2~(oo)d)(oo)) :fl 0 is required so that this LFT is well-posed. It is often the case that all the controllers ca.n be charactcrized by • = lt41~Ihltl2 for non-square constant matrices,/I'/~M, = I and M2M~ = I, with (I)l 6 7~7-/~ such that I1~,11~ _< 7. The optimal case may also occur when Iloo or Joo have eigen-values on the imaginary axis but 11oo, Joo E dom(R'~). In this case Theorem 5.1 can give regular sta.te-space equations with/3, X,, and Y1 all invertible. The stable invariant subspace of 11oo or Joo will only bc unique when the additional constraint that XiX2 and Yt'lq are Ilermitian is illcludcd, and this requires some special purpose algorithms (see Section 5.2.5).

5.2

Relaxing Assumptions

5.2.1

Relaxing A3 and A4

A1-A4

[00]

Suppose that, G=

0 0 1 1 1 0

which violates both A3 and A4 and corresponds to tile robust stabilization of an integrator. If the controller u = - c x , for c > 0 is used then

215

•s -t- (~

Hence the norm can bc made arbitrarily small as e --* 0, but ~ = 0 is not admissible since it is not stabilizing. This may be thought of as a case where the 7"/o0-optimum is not achieved on the set of admissible controllers. Of course, for this system, 7/~ optimal control is a silly problem, although the suboptimal case is not obviously so. If one simply drops the requirement that contt'ollcrs bc admissible and removes assumptions A3 arid A4, then the formulae in this paper will yield u = 0 for both the optimal controller and the suboptimM controller with ~ = 0. This illustrates that assumptions A3 and A4 are necessary for the techniques in this paper to be directly applicable. An alternative is to develop a theory which maintains the same notion of admissibility, but relaxes A3 and A4. Thc easiest way to do this would be to pursue the suboptimal case introducing ~ perturbations so that A3 and A4 are satisfied.

5.2.2

Relaxing A1

If assumption AI is violated, then it is obvious that no admissible controllers exist. SUl)pose A1 is relaxed to allow uustabilizable a n d / o r undetectable modes on the jw axis, and internal stability is also relaxed to also allow closed-loop jw axis poles, but A2-A,1 is still satisfied. It can bc easily shown tlmt under thcse conditions the closed-loop 7-(00 norln cannot be made finite, and in particular, that the unstabilizable and/or undetcctable modes on the jw axis must show up as poles in the closed-loop system. 5.2.3

V i o l a t i n g A1 and e i t h e r o r b o t h o f A3 a n d A4

Sensible control problems can be posed which violate A1 and either or both of A3 and A4. For example, cazes when A has modes at s = 0 which are tmstabilizable through B2 and/or undetectable through Cz arise when an intcgrator is included ill a weight on a disturbance input or an error term. In these cases, cither A3 or A,I are also viola.ted, or tile closed-loop 7"/00 norm cannot bc made finite. In many applications such 1)roblcms can bc reformulated so that the integrator occurs inside the loop (csscnti~dly using the internal model principle), and is hence detectable and st~d)ilizablc. An alternative approach to such problems which could potentially avoid the problem rcformulation would be pursue the techniques in this paper, but relax internal stability to the requircment that all closed-loop modes be in the closed left half plane. Clearly, to have finite 7-/o0 norm these closed-loop modes could not appear ,as poles in T~,o. The formulae given in this paper will often yicld controllers compatible with these assumptions. The user would then have to decide whether closed-loop poles on the imaginary axis wcrc due to weights and hencc ~cceptable or duc to the problem being poorly posed as in the above example. A third alternative is to again introduce e perturbations so that AI, A3 and A4 arc satisfied, l~.oughly speaking, this would produce sensible answers for sensible prohlcms, but the bchaviour as c ~ 0 could be problematic.

216 5.2.4

Relaxing A2

Iu the cases that either D~2 is not full column rank or D2~ is not full row ra.nk thc,l improper controllers can give boundcd ~¢~-norm for T~,o, althougll will not bc admissible as defined in section 1.4. Such singular filtering and control problems have been wellstudicd in 7~2 thcory and many of the same techniques go over to the ~oo-case (e.g. Willems(1981), Willems et a/.(1986) and tIautus and Silverman(1983)). In particular the structure Mgorithm of Silvcrman (1969) could bc used to make the terms Dl2 and Dn full rank by the introduction of suitable differcntiators in the controller. 5.2.5

B e h a v i o u r a t s = oo

It has becn assumcd in Theorem 5.1 that

3' > max(~[Dii,,,D,l,2,],

~[O',,,,, D',12,])

and a necessary condition for a solution is that this holds with >. If equality holds then onc or both of the IIamiltonian matrices cannot bc dcfincd. This corresponds to the casc

i,,f a ( ~ t ( G ' ( ~ o ) , K ( ~ ) ) ) IC(oo)

=

1

where K(c~) is just considered to be au arbitrary n, atrix. If inr ~r(.T'e(a(jw),K(jw))) < 1, for some w = wo K(j~) thcu a bilincar transformation from thc right half plane to the right half plane that ,novcs the point jWo to co will enable the IIamiltonians to be defined. One of them will however have an eigen value at the point on tim imaginary axis to which tim point at (x) h ~ bccn transformcd. A morc intricate situation arises when inf ~(:Fe(G(j~),K(j~,))) = 1, V w. lc (j~) llerc thc corrcsponding J-factorization problem (scc Grccn ct al.(1988)) or spcctral factorization problem is rank deficicnt for all w. Thc thcory of spc~:tral factorization for such cascs can bc dcrivcd via the solutions to a Linear Matrix IncquMity (Willems(1971)), or via the stable dcflating subspace of the zero pcncil (scc Van Doorcn (1981) and Clcmcnts and Glovcr(1989)). A c k n o w l e d g e m e n t The second author would like to thank the first author for his meticulous attention to the technical minutiae and thc first author would like to thank the sccond author for his carcful typing of parts of tlm manuscript. Both a.uthors gratefully acknowledge financial support fi'om AFOSR, NASA, NSF, ONR (USA), and SERC(UK).

References Adamjan, V.M., D.Z. Arov, and M.G. Krcin (1978). "Infinite block Ilankel matrices and relatcd cxtcnsion problcms," A M S Trausl., vol. 111, pp. 133-156.

217

Anderson, B.D.O. (1967). "An algebraic solution to tile spectral factorization problem," IEEE Trans. Auto. Control, vol. AC-12, pp. 410-414. Ball, J.A. and J.W. tlclton (1983). "A Beurling-Lax theorem for the Lie group U(m,n) whidl contains most classical interpolation theory," J. Op. Theory, vol. 9, pp. 107-142. Boyd, S., V. Balakrishnan, and P. Kabamba (1989). "A bisection method for computing the 7"(~ norm of a transfer matrix and rclatcd problems," Math. Control, Signals, and Syslem.~, vol. 2, no. 3, pp. 207-220. Clcments, D.J. and K. Glover (1989) "Spectral Factorization via tIermitian Pencils', to appear Linear Algebra and ils Applicalions, Linear Systcms Spccial Issue. Doyle, J.C. (1984). "Lecture notes in advances in multivariable control," ONR//IIoncywcll Workshop, Minneapolis. Doyle, J.C., K. Glovcr, P.P. Khargonckar, B.A. Francis (1988). " State-space solutions to standard ~.~ and ~oo control problems," IEEE Trans. AT,lo. Coulrol, vol. AC-34, no. 8. A preliminary vcrsion appeared in Proc. 1988 American Control Conference, Atlanta, June, 1988. Francis, B.A. (1987). A coursc in 7-lo~ conlrol theory, Lecture Notes in Control and Iuformation Scicuces, vo[. 88, Springcr-Vcrlag, Berlin. Francis, B.A. and J.C. Doyle (1987). "Linear control theory with an 7-/~o opthnality critcrion," SDIM J. Control Opl., vol. 25, pp. 815-84,1. Glovcr, K. (1984). "All optimal tIankel-norm approximations of linear multivariablc systcms and their ~oo-error bounds," [nl. J. ConlTvl, vol. 39, pp. 1115-1193. Glover, K. (1989). "Tutorial on llankcl-norm approximation," to appear in From Dala lo Model (J.C. Willcms cd.), Springcr-Verlltg, 1989. Glover, K. and J. Doyle (1988}. "State-space formulae for all stabilizing controllcrs that satisfy an ~oo norm bound and relations to risk sensitivity," Syslcms and Control LcUers, vol. 11, pp. 167-172. Glovcr, K. and D. MustMa (1989). "Derivation of the Maximum Entropy "Hco-controller and a Statc-spaco formula for its Entropy," hd. J. Conl~vl, to appear. Glover, K., D.J.N. Limcbeer, J.C. Doyle, E.M. Kasenally, and M.G. Safonov (19SS,). "A diaractcrization of all solutions to the four block gcaeral distanco problem," submitted to SIAM J. ContTvl Opt.. Cohbcrg, I., P. Lancaster and L. Rodman (1986), "On the IIcrmitian solutions of the symmetric algebraic Riccati equation," SIAM J. Control and Optim., vol. 24, no. 6, pp. 1323-1334. Green, M., K. Glovcr, D.J.N. Limebcer and J.C. Doyle (19SS), "A J-spectra| factori~ation approach to Iloo control," submitted to SIAM J. Control Opt.. llautus, M.L.J. and L.M. Silvcrmau (1983). "System structure and singular control." Lil~car Algebra Applic., vol. 50, pp 369-402. Kuccra V. (1972), "A coatribution to matrix quadratic equations," IEEE 71"ans. Auto. Conlrol, AC-17, No. 3, 34,1-347.

218

Limebeer, D.J.N. and G.D. I lalikias (1988). "A controller degree bound for 7"t,,o-optimal control problems of the second kind," SIAM J. Control Opt., vol. 26, no. 3, pp. 646-677. Mustafn, D. and K. Glover (1988). "Controllers which satisfy a closed-loop 7-/oo norm bound and maximize an entropy integral," Proc. eTth [EEE Conf. on Dcciaion and Control, Austin, Texas. Redheffer, R.M. (1960). "On a certain linear fractional transformation," J. Math. and Physics, vol. 39, pp. 269-286. Safonov, M.G., R.Y. Chiang and D.J.N. Limebeer (1987), Ilankel model reduction without balancing: a descriptor approach, Proc. 26th IEEE Conf. Dec. and Cont., Los Angeles. Safonov, M.G., D.J.N. Limebeer, and R.Y. Chiang (1989). "Simplifying the "/-too theory via loop shifting, matrix pencil and descriptor concepts," submittcd to lnt. d. ControL A preliminary vcrsion appearcd in Plvc. 27th IEEE Conf. Decision and Control, Austin, eI~ x ~s .

Sarason, D. (1967). "Generalized interpolation in "/-/co," Trans. AMS., vol. 127, pp. 179-203. Van Dooren, P. (1981). "A generalized eigenwlue ~pproach for solving R.icc,a.ti equations", SIAM d. Sci. Comput., 2, pp. 121-135. Silverman, L.M. (1969). "Invcrsion of multivariable lincar systcms," IEEE Twns. Auto. Conhvl, vol. AC-14, pp 270-276. Willems, J.C. (1971). "Least-squares stationary optimal control and the algebraic Riccati cqu~tion," IEEE Trans. Auto. Control, vol. AC-16, pp. 621-634. Willems, J.C. (1981). "Almost invariant subspaccs: an approach to high gain feedback design - Part I: almost coutrollcd invariaut subspaces." llgl~'E Trans. Auto. Control, vol. AC-26, pp235-252. Willems, J.C., A. Kitapci and L.M. Silverman(19S6). "Singular optimal control: a geometric approach." SlAM J. Control Optim., vol. 24, pp 323-337. Youla, D.C., H.A. Jabr, and J.J. Bongiorno (1976). "Modern Wiener-Hopf design of optimal controllers: part II," 1EEE Trans. Auto. Control, vol. AC-21, pp. 319-338. Wonham, W.M. (1985). Linear Multivariable Controh A Gcomctric Approach, third edition, Springer-Verlag, New York. Zames, G. (1981). "Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses," IEEE Trans. Auto. Control, vol. AC-26, pp. 301-320.

Decentralized Control of Large Scale Systems M. Ikeda Department of Systems Engineering Kobe University, Kobe 657, Japan

A b s t r a c t : This paper is conccrncd with thc decentralized stabilization problem for interconnected systcms. Following the history of rescarch, we first survcy time-domain mcthods for dcccntralized state fccdback and observcrs. "We then outlitlc ml s-domain mcthod of designing dcccntralizcd output fcedback, which is an application of the rcccnt factorization approach.

1

INTRODUCTION

In this paper, wc refer to a systcm as a large scale systcm if it is more appropriate to considcr tile systcm as an intcrconncctlon of smM1 subsystcms them dealing with it as a whole. "Large scale" does not necessarily mcan high dimcnsionality of the systcm hcrc. Thcrc arc mm~y actual systcms, for example, cncrgy systems, socioeconomic systems, transportation systems, water systcms, etc. (e.g., Siljak, 1978; Singh and Title, 1978; ffamshidi, 1983; Gcering and Mansour, 1987; Tamura mid Yoshikawa, 1989), to which in~crconncctcd modeling is suitable. When we consider such large scMe systems, we usually acknowledge the autonomy of subsystcms as dcsirable cxplicitly or implicitly. Rathcr often in practical systcms dcscribcd by intcrcommctcd modcls, autonomy of individual subsystcms is a crucial requircmcnt. If wc would like to prcscrvc thc autonomy in control systcms, the control

220

stratcgy we can apply is restrictcd to bc a decentralized one with an information structure constraint conformable to the subsystems. Each subsystem is controlled locally using its own available information. One of the most fundamcntM problems in control is stabilization. In the case of decentralized control for intcrconncctcd systems, it is commonly required that not only the overall closed-loop system is stable, but also the closed-loop subsystems are auto,mmously stable. Furthcrmorc, stability is dcsired even when perturbations occur in intcrconncctions among subsystems. This kind of robust stability is cMlcd connective stability (Siljak, 1978). A la.rge number of papers have been published concerning connective stabilization. The objective of this pal)cr is to make an overview of those results for future research. When wc require stability of subsystems as well as s~ability of the overall system, which is the casc of this paper, the well-known fixed mode condition (Wrong and Davison, 1973) for dcccntralizcd stabilization is necessary, but nog sufficient. The readers who arc less interested in stability of subsystems should consult other surveys written in the context of fixed modes (e.g., Davison, 1987; Yoshikawa, 1989). The control law cmi)loycd first for connective stabilization was state feedback in individual subsystcms (Davison, 1974), wlfich is reasonable if we look back into the history of modern control theory. The state feedback has been considered extensively until the middle of the eighties (Siljak and Vukucevic, 1977; Ikcda, Umcfuji mid Kodmna, 1978; Sczcr and Siljak, 1981; Ikcda, Silj;fl~and Yasuda, 1983; Shi and Gao, 1986). Much effort has bccu made to broaden the class of stabilizable systems. All the obtained stabilizability conditions arc sufficient ones. The decentralized state feedback has been considered under the assumption that the states of subsystems are available. If wc camlot measure the state of a subsystem directly, we nccd to estimate it from the input and output data using an observer. For autonomy of the subsystem the s[atc of which is to bc estimated, the associated observer has to compute the estimate using locM informations only. Since the subsystem is connected with other subsystems, its behavior is influenced by the behaviors of adjacent subsystems which m'e unknown to the local observer. Therefore, the state esti,nation is not an easy t~mk in the decentralized case. Disturbance SUl)prcssing observers arc adopted to construct output feedback controllers utilizing the results obtmncd by state feedback (Viswanadham and Ramakrishna, 1982; Willems and Ikeda, 1984; Ikcda and Willcms, 1987). Conditions for the existence of such observers restrict the class of stabilizablc systems. Recently, a new method of designing decentralized controllers of output feedback type has bccn proposed (Tan and Ikcda, 1987; Ikcda and Tan, 1989) using the proper

221

stable factorization approach (Vidyasagar, 1985). The most significant result of the factorization approach is the paramctrization of all ccntralizcd stabilizing controllcrs for a givcn systcm. Wc apply this result to each subsystem to dcfinc a local stabilizing controller with an unspecified paramctcr. Then, we tune the parameter to mal:c the overall closed-loop system connectively stable. This can bc done under a certain condition.

2

SYSTEM DESCRIPTION

The large scale system we deal with is a so-cMled input-output decentralized system (Siljak, 1978) dcscribcd by N

S:

Xi Yi

"~" Aixi + Biui + ~.~ A i j x j j=l = CiXi, i 1,2, .... N, f

(2.1)

.

which is COml)osed of N subsystems Si:

5"i =

Aixi + Biul

Yi =

Cixi,

(2.2)

i = 1,2 ..... N N

interacting with cach other through the static intcrconncction ~ A i j x j . In (2.1) and j=l

(2.2), xi is the statc, ul is the input, and Yl is the output of the subsystem Si, which are of appropriate dimensions. Thc matrices Ai, Bi, Ci and Aij arc constant and also of appropriate dimensions. We assunm that the pair (A~, Bi) is stabilizablc and (C~, A~) is dctcctablc. In some cases, wc describe the interconncction matrices Aij as (2.3)

Ais = G i E i j I t j ,

where the rnatriccs Gi and Hj are defined so that imGi = kerHj

=

imAii -t- imAi2 + ... + i m A i g

kerAlj n kerA2j n . . . n kerANj,

(2.4)

mid Eij is selected for (2.3) to hold. Then, we rewrite the overall system 5 as N

S :

"+i =

Aixi + 13iui + ~ G i E i j t I j x j

y~ =

C~a:i,

j=l

i = 1,2,...,N

(2.5)

222

and the subsystems Si as Si:

?ci =

A i x i + Biu~ + Givl

Yl

=

Cixl

wl

=

Hixi,

i = 1, 2,...,N

(2.o)

which arc connected via N

vi = ~

E~jwj,

i = 1 , 2 .... ,IV,

(2.7)

j----I

where vi is the interconnection input of thc subsystem S i and w j is the intcrconncction output of Sj. For coavcnicncc, wc ~dso use thc compact form S:

5J = ,j

ADX + BDU + G D E c H D x

=

(2.s)

where

AD =

diag{A~, A~, . . . , AN},

BD =

diag{B1, B2, . . . , BN}

Cj9 =

diag{C~, C2, . . . , CN},

Go

=

diag{G~, G2, . . . , GN}

Ec

=

[Eij]

Ito

=

dia g{Hl, I I 2 , . . . ,

,J

=

[ui

HN},

...

Throughout this paper, wc assume that the subsystems are not overlapping with each othcr, that is, no part of the given system is shared by a number of subsystems in common. There arc, however, actuM large scale systems for which it is reasonable to consider decompositions into overlapping subsystems. For manipulating such unorthodox decompositions of systcms, a suitable mathcmatical framework is the inclusion principle (Ikcda and Siljak, 1980a; Ikcda, Siljak and White, 1981, 1984). Using the principle wc can treat ovcrlapping subsystcms as non-ovcrla.pping ones.

3

STABILITY CONDITIONS

A typical scheme of stability analysis for large scale systems is as follows (e.g., Si/jak, 1978; Araki, 1978; Michcl and Millcr, 1977). Wc first disconnect all subsystems, and cnsurc their stability by applying common techniques developed for centralized systcms. Then, using aggrcg~ttcd inform~tions about the subsystcms and intcrconncctions, we define m~d t c s t a condition for stability of the ovcr,'dl system. The conditions obtained

223

in this way usuaUy rcquire a certain matrix to be an M-matrix (Araki, 1975), and are satisfied whcn the magnitudes of intcrconnections arc sufflcicntly small. In this sense, the conditions are of the smMl gain type (Zamcs, 1966). For the discussions of following sections, we prcscnt two stability conditions here, one in the time domain and the other in the s domain. Lct us considcr the large scale systcm N

S:

:~i = Aixi + ~_,Aijxj,

i=l,2,...,N,

(3.1)

j=l

which is the systcm of (2.1) with no subsystcm inputs nor outputs. We assume that each subsystcm 5i: ki = Aixi (3.2) is stable. Then, thcrc cxists a positive dcfinitc mah'i× Pi which satisfies the Liapunov equation AIPI + PiAi = - [ , (3.3) whcrc I is an identity matrix of a propcr dimcnsion. (We may considcr any positivc definitc matrix Qi instead of I in the right hand side of (3.3). However, I produces thc lcast rcstrictivc stability condition (Patcl and Toda, 1980; Ikeda and Silja.k, 1987).) The xnatrix Pi defines a Liapunov function

v,(x,) = ~ P , ~

(3.4)

for the subsystem S i of (3.2), and the time deriwtive is

+,(~) = - . ~ .

(3.5)

Wc consider the linear combination of vi(xi), N

,,(x) = ~ d,,,,(x,),

(3.6)

i:I

as a caaadidatc Liapunov function for the overall system S of (3.I), wherc di are positive numbcrs, which are not yet specified. The time derivative of v(x) with respect to (3.1) is computcd mid can be majorizcd as N

+(x)

=

N

- F_, d , ( x l x , - ~ j=l

2.~:P,A,s~s)

i:I

<_ - ~ ' ( W ' P D + D P W ) ~ ,

(3.7)

where W" = [wij] is an aggregated matrix defined by { 1/2AM(PI)-- HA/ill, i = j

wij =

_][Ao[[,

i # j,

(3.8)

224

and

= ff =

[llx, II I1 11 ... IIxNtl]' diag{Ar,~(P1), AM(P2), . . . , AM(Ply)}

D

diag{dl, d2, . . . , dN}.

=

(3.9)

tIere, we use the Euclidean vector norm and the corresponding induced matrix norm. The notation AM means the maximum eigenvalue of the indicated matrix. The Liapunov's stability theory (e.g., SiljM~, 1969) implies that we can conclude stability of the overall system if the matrix I'WffD + D f i W is positive definite. To state a stability theorem obtained in this way, we introduce the notion of Mmatrix, which is a kind of quasi diagonal dominant matrices. D E F I N I T I O N 3.1 A matrix W with nonTositive off-diagonal elcment~ is 8aid to be an M-matrix if its leading principal minors are all positive. The most importaalt property of an M-matrix for stability analysis of l ~ g e scale systems is given by (Tartar, 1971; Araki and Kondo, 1972; Araki, 1975): L E M M A 3.1 If W is an M.matrix, then/here exists a positive diagonal matrix D such that W t D + DI'V is positive definite, When W defined in (3.8) is an M-matrix, P W is also an M-matrix. Then, this lemma immediatcly implies the following stability condition. T H E O 1 Z E M 3,1 If the matrix W of (3.8) is an M-matrix, then the system S of (3.1) is stable. Wc note that the M-matrix property of W is preserved even when the intcrconncction matrices Aij change, provided their norms do not increase. This means that stability guaranteed by Theorem 3.1 is robust to the perturbations in couplings among subsystems if they do not become stronger. \Vc also note that Theorem 3.1 holds even if the testing matrix I'V of (3.8) is defined using any upper bounds of the norms of intcrconnection matrices instead of the norms thcmsclvcs. However, the rcsultant stability condition is more conservative. Now, we present another stability condition in the s domain. Let us consider the system description (3.1o) S : :~ = ADx + GDEGHDX,

225

which is (2.8) without input and output. V,re again assume stability of the subsystems, th,~t is, AD is a stable m,~trix. T h e characteristic polynomial of this system is c o m p u t c d [IS

dct(s/-

AD -- GDEcHD)

=

dct(sI - AD)dct(I - (~I- AD)-IGt)EotlD)

=

d e t ( ~ I - A D ) d e t ( I - H o ( s I -- A D ) - I G D E c ) ,

(3.u) which implies that if ( I - HD(SI -- A D ) - I G D E c ) is nonsingular for all s in Re s >_ 0, then this polynomial docs not havc unstablc roots and wc can concludc stability of S. To derive a condition for ( I - HD(~I -- AD)-~GDEv) to bc nonsingular in Re s _> 0, wc considcr the c o n t r ~ y situation. If it is singular for some s ~ in Re ~ > 0, thcn there exists

a

nonzero vector y = [y~ y~ . . .

y~v]~ such that

y,

T~(~')E,~

T,(s')E,~

...

T,(s')E,N

Yl

Y2

T2(~')E~x

T2(s')En

...

T2(s')E2N

Y2

:

:

:

...

TN(~')E~

i

YN

Z~.'(S')EN1 TN(~')Em

(3.1z)

YN

whcrc Ti(s) = H ~ ( s I - A;)-'G~ is thc tr.'msfer m a t r i x of the subsystem 5 i of (2.6) from the in~crconncction input v~ to the interconncction output wi. Taldng the n o r m of each component ,'rod using the n~aximum modulus principle for mmlytie functions, we obtain from this equation, the vector inequality V9 _< 0,

(3.13)

whcre V = [vii] is dcfincd by

,,j =

1 - I I T i H lIE.H,

i= j

-tiT, II liE,ill,

i ~ /

(3.14)

and [[T,-][ =

,J =

sup [IT~(jw)[[

[lly, l[ Hy~IJ ---

Ity~ll ]'-

(3.1~)

Here, we usc the L1, L2, or Loo vector n o r m for Yi and the corresponding induced m a t r i x norm for Ti(jw) and Zij. Now, wc use the following p r o p e r t y of M-matrices (Araki and Kondo, 1972). LEMMA

3.2 I f V is an M-matrix, then it has an inverse matrix whose elements arc

all nonncga~ivc.

226

This implies that when V is an M-matrix, we have 9 -< 0, which is scen by multiplying thc inequality (3.13) by V -1 fl'om the left. This contradicts nonncgativity of thc nonzcr0 vector y, and lmncc thc cquation (3.12) does not hold. Thus, we obtain the following stability condition. T H E O R E M 3.2 /f the matrix V of(3.14) is an M-matrix, then the system S of(3.10) is stable. Stability guaranteed by this theorem also is robust to the pcrturbations of intcrconncction matrices in thc same sense as thc robustness implied by Theorem 3.1. This theorem is still valid cven if in the definition (3.14) of the testing matrix V, thc norms of intcrconncction matrices are rcplaccd by their upper bounds. Stability analysis for interconnected systems has been a major subject of large-scale systcm thcory in the scvcntics. The dccomposition-aggrcga.tion schcmc adopted here has bccn ;~ common tcchniquc. Grc~t efforts have bccn made to obtain less restrictive stability conditions. Rcadcrs who arc intcrcsted in those results should rcfcr to, for example, Michel and Miller (1977), Siljak(1978), and Araki(1978).

4

DECENTRALIZED STATE FEEDBACK

We start our discussion of dcccntralized stabilization by state fecdback control under thc assumption that thc subsystems are controllable and their states are available f()r control synthesis. In this case, wc can arbitrarily assign the eigcnvalues with large negative real parts in each closed-loop subsystem. It might sccm that such stt~bilization of subsystems may always result in stability of the overall system. This is not true, however (Ikcda and Siljak, 1979). The validity depends on the interconncction patterns mnong subsystems. Many efforts have bccn nmdc to define ,'rod broaden the class of intcrconncctlon structures for which appropriate cigcnvMuc assignment in the subsystems stabilizes the ovcrM1 systcm (c.g., Sczcr and Siljak, 1981; Shi and Gao, 1986). For simplicity, wc consider the case in which the subsystems Si of (2.2) are single-input ones and each one is dcscribcd in the compmlion form

A i

0

1

0

...

0

0

0

0

1

...

0

0

:

:

._

:

:

"_

0

0

0

0 i

1 _~i

•

--C'~*I

i

--a2

...

i

--(~3

. . . .

ani-1

,

rti

Bi

=

(4.1)

227

where nl is the dimension of Si. We apply tim local state feedback (4.2)

Ui -~- I k i x i

to Si and obtain the closed-loop subsystem S~:

.:ci = (Ai + B i K i ) x , .

(4.3)

The feedback g a i n / f i is determined so that the matrix (Ai + B i K i ) has a set of distinct rcal cigcnvalucs defined by (:~, = { - a ' a ~ ,

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

(4.4)

, < a ~i < . . . < a , , , . i The positive number a is common whcrca>__ 1 , v i > 0 a n d 0 < a ~ to all subsystems, and is used to ch~'mge the subsystem eigenvalues so tlaat the overall closed-loop system N

5c:

ki=(AiTBilfl)xiT~-~Aijxj,

i = 1,2,...,N,

(4.5)

j=l

becomes stable, vi are introduced to adjust the rates of cigenvalues among subsystems when a is changed, which axe selected appropriately later, a], can be arbitrarily fixed to dctcrnfine the nominal patterns of the subsystcm-eigenvalues. Although the choice of subsystcm-eigenvalues defined by ,I,i of (4.4) may restrict the class of stabilizable large scale systems, it is appropriate in order to define the structures of intcrconncction matrices Aij for which there always exist local feedback gains Ki which stabilize the overall closed-loop system S c of (4.5). To investigate stability of the closed-loop system S c of (4.5), we transform the state s p a c e ,'m

5:i = (Fi)-'xl, to obtain

i = 1,2,...,N

(4.6)

N

~c:

xi=Ai.~i+~-~.~ij:~.i,

i = 1,2,...,N,

(4.7)

j=l

where Fi is the Vandcrmondc matrix (e.g., 1,2ailath, 1980) dcfincd by the sct ~I,i of (4.4), m~d

= ~,j~.. =

- d m g { a al, a~'a'2, .. . , a,~"a,,,)i •

,vi

F~IAijFi.

i

'

"

(4.s)

228 T h e m a t r i x Fi can be factorized (Siljak and Vukcevic, 1977) as Pi

=

II?Iq

Hi

=

diag{1, a ' , 1

1

i --Cr 1

~O'~

° -"

1

, . .

--O'n~

•

:

(--~i)n'-'

(4~)

..., a"("i-1)},

(__O.~)n/--1

i

: ...

: (__o.i ]ni--I \ rill

Wc n o w a p p l y T h e o r e m 3.1 to the closed-loop s y s t e m S¢ of (4.7). solution Pi to the Linpunov equation of (3.3) with

Ai

In this case, the

= Ai is

P~ = (ll2)di~.g{(lla~),~ -~', (11,'~)~ -~' , . . . ,

(l/cr,,)o' ' -,, }

(4.10)

aria ;~M(P~) = (1/2o~)~ '-~'. To caku1=to the te~ting matri~ W of (3.S), wo neod .1so tl,¢ Eli, which are m a j o r i z e d as

nol"ms of the interconnection m a t r i c e s

II--151l -< II'I'; -' IIII%IIIIn~-'A;JIAI-

(4.11)

We note tlmL in the right hand side, only the last norm depcnds on o, For c o m p u t i n g an u p p e r b o u n d of the n o r m IIH?'Aurbl I in (4.11) when a' is changed, we characterize the s t r u c t u r e of the original interconnection m a t r i x Aij = [a~j] by employing the integers 1 _< p~i < l,~J < . . . < p;.,¼, < ni a n d 1 < qjJ < q~i < . . . < qi,j, < nj which arc defined as follows. ij (i) I f a t qi j -¢ 0 for some q, t h e n p ~ "" = 1. Ifa~q = 0 f o r a l l q, t h c n l , ~"" is t h e l a r g c s t ;J = 0 for all p < p~ and q. integer such t h a t %q

(ii)

q~J is

ij = 0 for a l l p the sm~tllest integer such t h a t %q

ij ij < Pk and q > q k , k =

1, 2 , . . . , g i j .

(ill) Pkij is the largest integer such t h a t

"" a'~]q =

ij 0 for all 1) < Pki j a n d q > qk-~, k =

2 , 3~ ...~ K i j . tj T h e pairs (Pk, qkt 3 ), k = 1 , 2 , . . . , ~ij, define a b o u n d a r y for nonzero e l e m e n t s of A 0 as

(plj'qP)

O

229

where the shadowed portion indicates the area of nonzero elements allowed. By an easy calculation, we obtain Tti

117

IInE'A,jnJll <

I

•,

i)

I

ij

-,)l.

(4.z3)

p=l q=l

In the right hand side of (4.13), the maximum is takcn with respect to k. Using this majorization and AM(P/) = ( 1 / 2 a i ) a -~', we define the matrix W = [wii} of (3.8) for testing stability of the closed-loop system S c of (4.7) as

Wij

=

{ a[a,"' - ~ila ....... (-pk'+q~'), I-7 i-7 __(ijC¢

..... t-"*(Pk --1)4""-7(qk --1)1,

where

nt

i= j i

(4.14)

# j

n]

1 %;Jl •

¢,j -- II'I' -Xllll'X'Jll

(4.15)

p=l q=l

Theorem 3.1 implies that if this IV is an M-matrix, then the closed-loop system ge of (4.7) is stable, and hence the given open-loop system S of (2.1) is dccentrally stabilizablc. Under a certain condition on the pairs (l'~, q~i) defined for the intcrconnection nmtriccs Aij of S, we can nmke IV an M-matrix by increasing thc positive a,. To present the condition, wc use a directed graph which describes the intcrconncction pattern among thc subsystems in S. In the graph, which we denote by G, node i represents the subsystem Si and the dircctcd (orientcd) branch fl'om node j to node i means that there is a connection from Sj to S i. Wc note that the leading principal minors of W are composed of multiplications of wij Mong the directed loops in G. Therefore, the minors are all positive for a sufflcicntly large a so that W is an M-matrix if

~.. ln,~'L'v.[--l/iplkj 'b tej(q~kj -- 1)] <: 0

(4.1G)

t,2

holds with respect to all the directed loops in G. Thus, we state the following (Sczcr and Siljak, 1981): T I I E O R E M 4.1 The system S of(2.1) is dcccntraIIy stabilizablc if there exist positive numbers ui, i = 1,2, ..., N, which satisfy the inequalities (4.16) along all the directed loops

in G. It sccms that testing tile condition of this theorem is not an c ~ y task. Actually, we nccd to employ linear programming tcclmiqucs to treat general cases (Sczcr and SiljM~,

23O

1981). However, there are nontrivial classes of interconncction matrices for which testing is not needed. For example, if all All arc of the forms

1

A

,

(4.17)

whcrc the shadowed portions indicate the a rca of nonzero clements allowed and a~q = 0, p < q, then these conditions always hold for ui = 1, i = 1, 2 , . . . , N. Another example is described by lower triangular forms as

Aij =

Aij =

for wllich these conditions hold for ui = hi, seen in Sezer and Siljak (19Sl).

i = 1,2,...,N.

,

(4.18)

More general examples arc

We have considered only single-input subsystems to obtain Theorem 4.1. The theorem can be generalized for the case of multi-input subsystems as follows. Wc describe each multi-input subsystem in the controllability canonical form (Luenbcrgcr, 1967), and reduce it to a set of single-input components by employing preliminary local state feedback (Ikcda m~d Siljak, 19S0b). Then, we consider the single-input components as subsystems, and decompose interconnection matrices correspondingly. Thus, we can apl)ly the theorem. The resem'ch of decentralized stabilization by state feedback was initiated by Davison (1974). The structure of intcrconncction matrices hc considered explicitly was such that RangcAij C RmlgcBi, for all i, j

(4.19)

which means that tile intcrconncction cffccts on each subsystem can be cancelled out by the local control input. This is a special structure included in the class of (4.18). T h e class of intcrconncction matrices with which large-scale systems are deccntrally stabilizablc, haa bccn broadened by a number of reseaxehcrs. The structure of (4.17) was presented by Siljak and Vukccvie (1977). Ikcda m~d Siljak (1980b) indicated that

231

the boundaries of nonzero clcments may bc uppcr and more right in some A 0. if those of others arc lower and more left. Then, Sczcr and Siljak (1981) showed that the slopes of the boundaries nccd not bc - 1 if negative. The structure of (4.18) is a special case of their result. These rcsults arc obtained by applying state feedback of a high gMn type as we did here. The a r c ~ where nonzero elements arc allowed in intcrconncction matrices are essentially of triangular forms including the lower left comer. Shi and Gao (1986) used low gain local feedback as well by removing the restriction of positivity on ui in (4.4) when wc ~ s i g n cigenvalucs in the closed-loop subsystem. In this case, the permissible areas of nonzero elements can be represented as triangles including the lower right, upper Hght, or upper left corners. All these results have been obtained using the cigenvMuc-assignment method presented in this section. The class of dcccntralIy stabilizable large-scale systems can be broadened more (Willems and Ikcda, 1984) using the system description S of (2.5) and applying ~he Mmost disturbaJlce deeoupling technique (Willems, 1981) to the subsystems S{ of (2.6). The approach is the d u n of that tM~cn in the next section for the state estimation. Like stabilization, optimization of large-scale systems has been similarly dcalt with employing the decentralized state feedback conformable to subsystems (Ozguncr, 1975; Ikcda and Siljak, 1982). It has been shown (Ikcda, Siljak, m~d Yasuda, 1983) that under the condiLions of Theorem 4.1, there exists a decentralized control law which is optimal to a quadratic performance index. Although this result is obt,'-dncd in the context of the inverse rcguhttor problem and wc generally cannot choose the performance index arbitrarily, the rcsultm~t optimal control system has the same robustness properties as the optimal centralized control system has (Safonov and Athans, 1977).

5

DECENTRALIZED

OBSERVER

To implement tile decentralized state feedback, we need to know the states of subsystems. When the state of a subsystem is not measurable directly, wc have to estimate it fl'om the control input and the available measured output. In the c ~ c of centralized control, it is common to use an observer for such a purpose (Luenbcrger, 1971). The necessary and sufficient condition under which wc cml obtain the state of a system via an observer is detectability of the system. In tile decentralized control ca~e, to satisfy the information structurc constraints, wc consider a set of disjoint local observers associated with the individual subsystcms.

232

Then, the application of the observer theory developed for centralized control systems is not as straightforward. Since the essence of the observer is an on-line real-time simulating model, each local observer needs all the informatiori about inputs affecting the behavior of the subsystem the state of which is to be estimated. However, the subsystem has an interconnection input, which is generally not measurable, as well as a control input. Thc local obscrver docs not have enough information for estimating the state. This severely complicates the decentralized estimation problem. Somehow, however, we have to estimate the subsystem state from the local control input and measured output. In this situation, what we can do is to consider the interconnection input as an unknown disturbance and try to suppress its influence on the state estimate. Wc employ an ahnost disturbance dccoupling observer (Willems, 1982) for this purpose. Let us consider the subsystem description of (2.6), Si :

xi =

A i x i -~- J~iY, i "or GiVi

y;

Gz~

=

(5.1)

where we omit the interconnection output wi because it is not necessary for designing an observer. For this subsystem, wc consider Obi :

~i = (Ai - LiCi)£'i + B i u l + L i y i ,

(5.2)

which works as an observer when the subsystem S i is disconncctcd from other subsystems and vi = O, where xi is the estinmte of x i mad L i is a gain matrix such that (Ai - LiCi) is stable. Then, the estimation error ci = ]ci - xi is governed by b,i = (Ai - L i C i ) c l - Girl.

(5.3)

This cquation implies that the intcrconncction input vl affects el, and in general el may not decay ~ t goes on. Through vi the state estianation problem is coupled with the control problem, and the separation property between estimation and control does not hold. Although the estimation error el does not decay, we expect that if the error ei is small enough, then the estimate ~i can serve for the real state xi in the decentralized stale feedback. For this purpose, to make the transfer matrix ( s I - A i + L i C i ) - 1 G i fi'om vi to cl in (5.3) sufficiently small, we use tlm following lemma which is ilnplicd by the results of Willems (1982), and Ha utus and Silverman (1983). T h e lcmma is dual to the result of perfect regulation (Kimura, 1981). Wc can cmptoy any induced matrix norm to state the following:

233 L E M M A 5.1 For any e > 0 there ezi~t~ a gain m a t r i x Li ~uch that the matrix (At LiCi) is stable and

II(.~Z - Ai + L i e i ) - ' a i l l

(5.4)

< ~., f o r aU s in R e s >__0

ff r=,.

r ,I- A,

[

Ci

=

:.,,

.°,,k,

0 J

:or

°,,.,,..e. > O.

(5.5)

Rouglfly speaking, this lcmma mcans that unstable modcs (if present) in tim interconncction input vl can bc supprcsscd arbitrarily in the cstimatc &i if thc subsystcm Si satisfies the condition (5.5) which is of a ,ninimum phase type. Then, Ob i of (5.2) works as an observer for Si, though approximately, cven whca S i is connected with other subsystems. We collect such local observers to construct a decentralized observer for the overall system, which we describe using the notations in (2.9) as Ob :

(5.~)

& -= (AD -- LDCD).f: "J- .l)Ou "4- .LDy

wherc

Lu =

diag{Lt, L~, ..., LN}.

(5.7)

Now, we employ this dccentralizcd observer to implcmcnt the deccntra,lized state fcedback u = KDx

(5.S)

which, we assume, stabilizes the overall system $ of (2.8), where I @ = dia g{Kl, I(~ . . . . .

(5.~)

KN}

and Ki, i = 1, 2 . . . . . N, are local state fcedback gains. T h e state estimate ~ gcneratcd by 0b is uscd for tim state x in (5.8). Then we have the observer-based output feedback controllcr X" ~ (AD LDCD "]- BDI(D)3:"31-L D y

(5.1o)

U

=

KD;r,,

and the resultant overall closed-loop system is written a,s g¢:

.

x,

=

.

LDCD

AD - LDCt) + B u K o

(5.11)

}

I11 this case, stability of gc is not atttomatically implied. This is a significant difference from the centralized case.

234

To investigate stability of the overall closed-loop system ~c, we calculate the charactcristlc polynomial of the system matrix as

[ sIdet =

AD - GI)Ec, IID -Lt)Co

-Bulf~

]

s I - At) + LDCo -- BDKt)

d c t ( s I - Ao - BDKD -- G o E c H o ) d e ~ ( s I - At) + LDCD) •det[I + E c H D ( s I - AD

-- J~DI(D

--

GDEoHD)-IBDI(O

•( s l -- AO + LDCD)-IGD].

(5.12)

Since I k o is assumed to be a stabilizing state feedback gain for the overM1 system, the first determinant of tile right hand side h ~ no root in Re a :> O. We can say the same thing for the second de~erminmat because (AD -- L o C o ) is stable. It is also seen tlmt when the locM observer gain Li in Ob i is chosen to satisfy (5.4) with a positive e such that e < [ sup [IEctlD(s[ - AD -- BDI(D -- GDEcIto)-IB~)Kt)][] -~, (5.13) Res_>O

the third determinant does not become 0 in Re s >_ 0. Thus, stability of the closed-loop system gc of (5.11) is concluded. We note here that the choice of the observer gain LD is not independent of the state feedback gain I(.D. "v\re can now state: T H E O R E M 5.1 If the 3ubsy3tcms S i o]'(5.1) sati~fy ghc condition (5.5) in Lemma 5.1, then there exists a decentralized observer Ob of(5.6) which provides local siate estimates tha~ can bc used in M.abilizing decentralized stale feedback. Wc note tlm~ the condition (5.5) implies detectability of the pair (Ci, At), but the converse is not vMid. IIcncc, the condition for a decentralized observer to exist is more restrictive than in the centralized case. A way of relaxing the condition is to estimate the function Ifixi instead of the full s~atc xl. Once a stabilizing gain If9 = cling{K1, I(2, . . . , If/v} of decentralized state feedback is determined, it suffices to estimate Iflxi suppressing the influence of the interconncction input vi on this linear function. T h e existence condition for such an observer has bccn given (Ikcda and Willcms, 1987) in terms of the infimM coml)lcmentary detectability subspacc (Willems and Commault, 1981) a.nd the infimal ahnost complemcntm'y obscrwd~ility suhspace (Willems, 1982). In closing this section, we lncntion that if in addition to (5.5), the condition

rankCiGi = rankGi

(5.14)

holds in t.he subsystem S i, there exists a local (minimal order) observer which comphrtcly rojccts tlm influence of the intcrconneetion input on the sta.te estimate (Kudva,

235

Viswaamdham, and Ramakrishna, 1980). Then, wc have no problem in implementing the local sta.tc fccdback (Viswanadham and Ramakrishna, 1982).

6

DECENTRALIZED OUTPUT FEEDBACK CONTI~OLLER.

It is well known that in the centralized control case, tile design of a stabilizing output feedback controller can be decomposed into thc stabilizing sta.tc feedback design and the stable observer design, which can bc carried out independently. This is not truc in the case of dcccntrMizcd control, and the two dcsign problems are coupled together mentioned in the previous section. Thcrcfore, it is not so easy to determine the state feedback gains and observer gains in local observer-based state-feedback controllers in order to stabilize the overall system. From this point of view, an appropriate approach Lo the design of decentralized output feedback controllers would bc that which is based on the factorization of transfer matriccs of givcn systems (Vidyasagar, 1985). The most fundamental and significant rcsult of this a.pproach is the paramctrizatiou of all (centralized) stabilizing controllers for a given system. In this section, wc outline how Lhc factorizatioa opproach can bc utilized in the design of stabilizing decentralized controllers. The underlining idea is as follows. Wc first define local stabilizing controllers for individual subsystems, which have unspecified parameters. Then, we tune the local parameters to stabilize the overall closed-loop system (Tan and Ikcda, 1987; Ikcda and Tan, 1989). In this section, wc say that a rationM matrix in s with rcM cociticicnts is stable if it is analytic in the closed right half complex plane C+ (excluding s = oo ). By I1~ and Rl,s wc dcnotc the sets of stable and proper stt~blc rational matrices, respectively. To apply the fimtorization approach, wc represent the subsystem Si of (2.6) by the trmlsfcr nmtrix

whcrc Z¢q(p, q = 1,2) arc dcfincd Z~, = H , ( s I - A , ) - ' G ' , ,

Z~,, = lI,(s_r - d , ) - ' B i

Z~, = C , ( s I - A , ) - ' G , ,

Z~.~ = C , ( s I - A , ) - ~ Bi

(o.2)

and we use the same notations ul, yl, vl, wl in the s domain as in the timc domain. Since wc havc assmncd stabilizability of (Ai, Bi) and detectability of (Ci, Ai), stabilizatiou of the subma.trix Z~2 implies stabilization of the whole Si. For this purpose, wc factorize the s~,rictly propcr Z~2 as Z; 2 = NiD~-I = ~ - 1 ~ / , (6.3)

236 where Ni, Di E Rps and Ni, Di E Rps satisfy

-N~

Di

N;

Q;

=

0 [

for somc Pi, Qi, Pi, Qi E Rps. This is callcd a doubly coprime factorization. Then, the sct of all stabilizing output fccdb~ck controllcrs for S i is given as

LCi:

ui = Iii(Ri)yi,

(6.5)

where I(i(Ri) is thc gain trtmsfer m~ttrix defincd by =

-(P, + D,B,)(O, -

(6.o)

a n d / 7 i E Rps is arbitrary (Vidyasagar, 1985). ~v\resclcct later the parameter matrix Ri a l)propriatcly to stabilize the overM1 system. When we apply the local stabilizing controller LCi of (6.5) to the subsystem Si of (6.1), the transfer matrix Ti fi'om the intcrconncction input vi to the intcrconncction output w~ of the rcsult¢mt closed-loop system is calculated using (6.3) and (6.4) T~(R;) = T ~ - T'~I~T~ ' i

(6.7)

which is an afEnc function of the para.mcter Ri, where T;• =

: T~ =

i ~ i Z~I - Z12PiDiZ21

' ZI2Di

(6.s)

-DiZ21.

The matriccs T~, i Tj,i and T~ belong to Rps because T/(/?i) is st~d)lc for may/?i E Rps. Then, we define the ana.trix V = [vii] as "~s =

1-

IIT,(/~,)I I IIE.II.

IIE; II,

i = j

i# j

(6.9)

whcrc thc norm is thc smuc onc as in (3.14). We employ Thcorcm 3.2 to state the following: L E M M A 6.1 When the local controllers LCi of (6.5) with the gain tran,~fer matrices of (6.6) arc applied to the overall system S of (2.5), the result, ant eloaed-loop ay.qtcm is stable if the ma, trix V of (6.9) is an M-matrix.

237

For the matrix V defined by (6.9) to be an M-matrix, we need to dmose the parameters Ri 6 Rps so that {[Ti(Ri)[I are sufficicntly small. If V is not an M-matrix cvcn for tl~c infimum of HT,.(RI)I] with respect to 12;, it can ncvcr bc made so by changing Ri. Therefore, for testing purpose, it might seem bcttcr to define V using the infimums instead of [IT~(R,)II in (6.9). In the ease where IIT,-(R,)]I is the function norm induced from thc L2 vcctor norm, the calculation of the infimum is reduced to thc Ho~ optimization problem (Vidyasagar, 1985; Francis, 1987). However, this requires some computation efforts. In the cases of norms induced from the Lx and Loo vector norms, wc know little about tile infimums at present. To present a practical, though morc conscrvativc, stabilizability condition, we no~e that if there exists an X i 6 Rs such that the equation T~X Ti

(6.10)

holds, then inf

R,£Rps

IIT (R,)II

=

o.

(6.11)

This is obvious in case X i is proper and wc c,-m sct Ri = X i. %Vhcn X i is not proper, wc use a proper approximation (Francis, 1987) to define R~ and conclude (6.11). Although tim matrices T[, T~ and TJ in (6.10) are dcfined by (6.8) using the factorization of Z~2, which is not unique (Vidyasagar, 1985), it can be shown that the cxistcncc of X i is independent of the choice of the factorization (Ikeda and Tan, 1989). Employing a particular factorization (Neff, Jacobson, and Balas, 19S4) Ni = Ci( s I - A~;)-' BI, N~i

P~

= =

Ci(sI

--"

4 Li ]~-IB. ',

K i ( s I -- A Li ) - , Li,

Pi = I(i(s$ - A~;)-' LI,

Di = K i ( s I - A~¢)-XBI + I Di = C i ( s I -

i -1 Li + I AL)

Qi = - I f d s I

- A~)-IBi + I

Qi = - C d s I - A ~ ) - ~ L i + I,

where A~c ~--- Ai + BiKi, A~ = Ai + LiCi and Ki, Li are xnatrices such that A i1¢, A Li arc stable, we can show the following lcmma (Ikcda and Tan, 1989). The lcmma is written ill terms of the matrices At, 131,el, Gi, Hi of the given open-loop subsystem Si, and implies that we do not need any factorization of Z~2 to see whether we can nmke the norm IIT,-(RI)I] arbitrarily small. L E M M A 6.2 The equation (6.10) haa a aolution X i in R~ if and only if the equation

238

The solution X ~of the equation (6.10) can be represented using the solution (]']~, ]]~, i~,, I ~, i ) of (6.13) in a simple form, if it exists (Ikcda and Tan, 1080). If

rank [ s I - A, gi ] 0 J t Ci r m f l ' [ s/-AIHi

B=i ]0

= f u l l column rank,

/ o r a l i a in Re s > O

(6.14) fullrowrank,

f or all s in Re s > O,

then (6.13) can obviously be solved using the pseudoinverses of these matrices. We note that the first condition of (6.14) is the same as (5.5) which guaranteed the existence of an almost disturbance dccoupling observer. The second condition of (6.14) has been known ,as the condition for pcrfcct regulation (Kimura, 1981). A way of investigating solvability of the general two-sided matrix equation (6.13) is transfommtion of the coefficient polynomial matrices in the left htmd side into the Smith form. This transformation reduces the matrix equation to a set of scalar equations, which is equivalent to the original equation. The scalar equations arc much more tractable, and the solutions can be computed rcadily. Now, wc recall the directed graph G defined in Section 4, which describes the intcrconncction pattern of the given large scale system S of (2.5). If the equation (6.13) is solvable in ILa, then we remove all tim branches which go into or go out of node i. We refer to this graph as G , and present a. graph theoretic condition for decentralized stabilizability (Ikcda and Tan, 1989). T H E O R E M 6.1 If there ia no directed loop in the graph G, t,hen the ayatcm S 0f(2.5) ia deccntrally stabilizablc. This tlmorem can be shown as follows. The k-th leading principal nfinor of thc matrix V defined by (6.9) can bc expressed as 1 - *, where * is composed of products o~ ]IT~(R,)II and lIE,,[] along tho directod loops in the subgraph of G containing the nodes 1 , 2 , . . . , k with branches mnong them. The condition of Theorem 6.1 means that there is at least one ][T,-(R~)[[ in each product, which can be made arbitrarily small by choosing Ri appropriately. Thus, * can be made small as well in order to make V to be an M-matrix. Then, Lennna 6.1 concludes this theorem. In this section, wc discussed whcthcr we can stabilize a subsystcm and at thc same time, can make the norm of the transfer matrix Ti(Ri) fl'om the intcrconnection input to thc intcrconncction output arbitrarily small as described by (6.11). If we consider tlm intcrconncction input as an external disturbance and the interconnection output as a controlled output in each decoupled subsystem, this stabilization problem is identical

239

~o the almost disturbance decoupling problem with stability by measurement fccdback (Wciland and Willcms, 1989). Actually, the condition (6.13) of Lcmma 6.2 h ~ bccn obtained for such disturbance decoupling (Shimizu and Ikcda, 1986). We can give an equivalent condition in terms of ahnost invariant subspaces, which has been derived in the same coatcxt (Wciland and Willcms, 1989).

7"

CONCLUDING

REMARKS

It does not seem that in the second half of the elghtics, a lot of researchers are iutcrcstcd in decentralized control problems for large scale systems. In the author's opinion, one of the reasons is that there are not many actual control problems which motivate the rcscarch of this particular arca at present. In the scventics, this arca was excited by a large number of actual problems in divcrse fields such as energy systems, transportation systems, socioeconomic systcms, water systems, etc.. The control object which may stimulatc this arca in thc near future would bc flexible large space structures. Another reason is that although some problems formulated in the scvcntics have bccn solved to somc extent, they wcrc tractablc oncs aald thc contributions of thc solutions to actual problems have not bccn much appreciated. To nmkc significant contributions, wc llccd to reformulate the decentralized control problems including the stagc of modeling. The modeling of large scale systems should bc diffcrcnt fi'om the ccntralizcd cascs, and the synthesis of decentralized controllers is not separable from thc modeling. In the case of large space structures, wc can obtain prccisc subsystcm modcls, but the collcction of them may not ncccssarily form a prccisc ovcrM1 model. Decentralized control strategies arc more suitable to such systems than centralized oncs. Without doubt, the necessity of dcccntralizcd control will incrcasc in the future as thc systcms wc deal with bccomc larger and more complcx. Dcvclopmcnt of mathcnlatical system theory for laxgc scale systems is rcaUy cxpcctcd.

ACKNOWLEDGEMENT Thc author is gratcful for many useful discussions on the topic of this paper with Prof. D. D. Siljak of Santa Clara University, Santa Clara, California, U.S.A., Prof. M. E. Sczcr of/3ilkcnt University, Ankara, Turkey, and Prof. K. Yasuda and Mr. H. -L. Tan of I(obc University, Kobe, Japan.

240 REFERENCES

[1] Araki, M. (1975). Applications of M-matrices to the stability problems of composite dynamic,'d systems, J. Mathematical Analysis and Applications, 52, 309-321. [2] Araki, M. (1978). Stability of large-scale nonlinear systems: Quadratic order theory of composite system method using M-matrices, IEEE Trans. Automatic Control, AC-23,1291,12.

[3] Araki, M., and B. Kondo (1972). Stability and transient behavior of composite nonlinear systems, IEEE Trans. Automatic Control, AC-17, 537-541. [4] Davison, E. J. (1974). The decentralized stabilization and control of a class of unknown nonlinear time-varying systems, Aulomalica, 10,309-316. [5] Davison, E. J. (1987). Control and stabilization of interconnected dynamical systems, Large-Scale Systems: Theory and Applications 1986, ed. by II. P. Gcering and M. Mansour, IFAC Proceedings Series, 1987, No.l l, Pergamon Press, 41-52. [6] Francis, B. A. (1987). A Course i~ Iloo Conlrol Thcory, Springer Vcrlag. [7] Geering, II. P., and M. Mansour, Eds. (1987). Large-Scale Systems: Theory aud Applica. lioJzs 1986, 1FAC Proceedings Series, 1987, No.11, Pergamon Press. [8] l[autus, M. L..I., and L. M. Silverman (1983). System structure and singular control, Li~,car Algebra and lls Applicalio1*s, 50,369-d02. [9] Ikeda, M., and D. D. Siljak (1979). Counterexaml)les to Fessas' conjecture, IEEE 7)'ans. Aulomalic Control, AC-2d, 670. [10] Ikeda, M., and D. D. Silj,~k (1980~). Overlapping decompositions, expansions and contractions of dynamic systems, Larye Sealc Syslcms, 1, 29-38. [11] Ikeda, hi., and D. D. Siljak (1980b). On decentrally stabilizable large-scale systems, Au. lomalica, 16,331-33.1. [12] lkcda, M., and D. D. Silj~Lk(1982). When is a linear decentralized control optimal?, Proc. 5lh 11,lcrltational Conference o7~ Analysis and Oplimizalion of Systems, ,119-,131. [13] Ikcda, hi., and D. D. Siljak (1987). Stability of reduced-order models via vector Liapunov functions, Proc. 1987 American Col,lrol Conference, 482-,189. [1,I] lkeda, M., D. D. Siljak, and D. E. White (1981). Decentralized control with overlapping information sets, J. Optimization Theory and Applications, 3,1,279-310. [15] lkeda, M., D. D. Siljak, and D. E. White (198,1). An inchlsion principle for dynamic systems, IEEE Trans. Automatic Control, AC-29, 24,1-2,19. [16] Ikcda, M., D. D. Siljak, and ],:. Yasuda (1983). O1)timality of decentralized control for large-scale systems, Automaliea, 19, 309-316. [17] Ikeda, M. and II.-L. Tan (1989). Decentralized control for connective stability: A factorization approach, Proc. ChiT~a-dcq~au doiT~t Symposium on Systcm Control Theory a~tl lls A pplicalio;zs, 15-20.

241

[18] Ikeda, hi., and O. Umefuji, and S. Kodama (1978). systems, Syslems, Compulcrs, and Control, 7, 34-41.

Stabilization of large-scale linear

[19] Ikcda, M., and J. C. Willems (1987). An observer theory for decentralized control of largescale interconnccted systems, Large-Scale Sys.tcms: Theory and Applications 1986, ed. by 11. P. Geering and M. Mansour, IFAC Proceedings Series, 1987, No.ll, I'ergamon Press, 329-333. [20] Jamshi(li, M. (1983). Large-Scale Systems: Modeling and Control, North-lIolland. [21} Kailath, T. (1980). Linear Syslcms, Prcntice-llall. [22] Kimura, 1I. (1981). A new approach to the perfect regulation and the bounded peaking in linear multivariable control systems, IEEE Trans. Automatic Control, AC-26, 253-270. [23] Kudvu, P., N. Viswanadham, and A. Ramakrishna (1980). Obscrvcrs for linear systems with unknown inputs, IEEE Trans. Automatic Co,tirol, AC-25, 113-115. [2,1] Luenbcrger, D. G. (1967). Canonical forms for lincar multivariable systems, IEEE Trans. A ulomalic Control, AC-12, 290-293. [25] Luenberger, D. G. (1971). An introduction to observers, IEEE Trans. Automatic Control, AC-16, 596-602. [26] Michel, A. N., and R. K. Miller ('1977). Qualitative A~mlysis of Large Scale Dynamical Systems, Academic Press. [27} Nett, C. N., C. A. Jacobson, and M. J. Balaz (198,1). A connection between state-space and doubly coprime fractional representations, IEEE Trans. Automatic Control, AC-29, 831-832.

[28] Ozguner, U. (1975). Local optimization in large scale composite dynamic systems, Proc. 9lh Asilomar Conference on Circuits, Syslcms, and Compulcrs, 87-91. [29] Patel, R. V., and M. Toda (1980). Quantitative nieasures of robustness for multivariable systems, Proc. 1980 Joint Automatic Control Coofcrcnce, TP8-A.

[30] Safonov, M. G., and M. Athans (1977). Gain and 1)h~e margin for multiloop LQG regulators, IEEE Trans. Automatic Conlrol, AC-22, 173-179. [31] Sezer, M. E., and D. D. Silja.k (1981). On decentralized stabilization and structure of linear large-scale systems, elulomalica, 17, 6,11-6+1. [32] Shi, Z.-C., and W.-B. Gao (1086). Stabilization by decentralized control for large-scale interconnected syst eros, Large Settle Systems, t0, 1.17-155. [33] Shimizu, II., and M. lkeda (1986). On the almost disturbance decoupling problem: A solval)ility condition in ca~e iuternal stability is required, l'roc. 151h SICE Symposium on (;o~lrol Thcor!l , 283-286 (In .lapa.nes(,). [:r,I] Si]jak, D. l). (19[J9). Nonli,,car .S),s/cm.% John Wiley. [35] Silj~tk, D. 1). (1978). Lmyc Scale Systcms: SlabiliQi and Slruct,rc, North-Ilolland.

242

[36] Siljak, D. D., an(l M. B. Vukcevic (1977). Decentrally stabilizable linear and bilinear large-scale systems, h~t. J. Control, 26, 289-305 [37] Sin{~h, M. G., and A. Titli (1978). Systems: Decomposition, Oplimizalion, and Control) Pergamon Press.

[38] Tamura, iI., and T. Yoshikawa, Eds. (1989). Large Scale Systems Control and Decision Making; Marcel Dekker.

[39] Ta~l, II.-L., and M. Ikeda (1987). Decentralized stabilization of large-scale interconnected systems: A stable factorization approach, Proc. 26lh IEEE Conference on Decision and Control, 2295-2300. [40] Tartar, L. (1971). Une nouvelle caracterisation des M matrices, RAIRO, 5, 127-128. [,11] Vidyasagar, M. (1985). Conl)vl Systcm Synlhcsis: A Factorizalion Approach, MIT Press. [,12] Viswanadham, N., and A. Ramakrishna (1982). Decentr~dized estimation and control for interconnected systems, Large Scale Syslcms, 3,255-266. [43] Wang, S. II., and E. J. Davison (1973). On the stabilization of decentrMized control systems, IEEE Trans. Aulomalic Control, AC-18, 473-478. [4,1] Weilan([, S., and J. C. Willems (1989). Ahnost disturbance deconpling with internal st~ bility, IEEE Trans. Aulomatic Control, AC-34,277-286. [45] Willems, J. C. (1981). Ahnost invariant subspaccs: An approach to high gain feedback design - Part 1: Ahnost controlled invariaJlt subspaces, IEEE T)uns. Automatic Conlrol, AC-26,235-252. [46] Willems, J. C. (1982). Ahnost invariant snbspaces: An approach to high gain feedback design - Part 2: Ahuost conditionaily inv,'u'iant subspaces, IEEE Trans. Automatic Conlrol) AC-27, 1071-1085. [47] Willems, J. C., and C. Commault (1981). Disturbance decoupling by measurement feedback with stability or pole placement, SIAM J. Control, 19,490-504.

[48] Willems, J. C., and M. Ikcd~ (198,1). Decentralized stabilization of large-scMc interconnected systems, Proc. 6lh International Conference on Analysis and Optimization of Systems, 236-2+1.

[,19] Yoshikawa, T. (1989). Decentralized control systems and fixed modes, Large Scale Systems Conlrol and Decision Making, ed. by II. Tamura and T. Yoshikawa, Marcel Dekker.

[50] Zames, G. (1966). On input-output stability of time-varying nonlinear systems, Parts I and II, IEEE T)mts. Aulo)natic Control, AC-11,228-238, ,165-,t76.

State Space Approach to the Classical Interpolation Problem and Its Applications H. Kimura Department of Mechanical Engineering for Computer-Controlled Machinery Osaka University 2-1, Yamada-oka, Suita 565, Japan

ABSTRACT Tile Pick-Nevanlinna interpolation theory in classical analysis plays all important role in the recent progress of linear system theory in the frequency domain. In this paper, we shall show how the classical interpolation theory which relies heavily on function theoretic properties is described in the algebraic framework of the state space. The notion of conjugation, or more specifically, of J-lossless conjugation is shown to be the state-space representation of the classical interpolation, or its modern versions. Thus, the notion of J-lossless conjugation provides a unified treatment of the H°°eontrol problem, as well as of robust stabilization and the model reduction.

1.

INTRODUCTION The transfer function G(s) of a linear time-invariant lumped-parameter system has two distinct aspects. If you regard it as a complex function, you are concerned with the analytic aspect of the system. If you regard it as a rational function, you are concerned with the algebraic aspect of the systems. These two aspects of linear systems are, of course, closely related and the interplay between them is one of the main sources of the rich structure in system theory. Let us take the stability criteria of linear systems as an example. The Nyquist stability test is essentially based on a

244

function-theoretic property of loop transfer functions, while tile Routh stability test is essentially based on the rationality of transfer functions, The analytic or topological procedure of the Nyquist test looks totally different from the algebraic procedure of the Routh test. Sometimes, it is hardly believable for the beginner of control theory that the stability can be checked in such different ways. If he is smart enough, he might sense the logical depth of system theory there. In classical system theory, the analytic theory of transfer functions was fertile. We can list some fundamental results concerning the analytic aspect of transfer functions, such as the Nyquist stability test, the realizability condition of Wiener-Paley, the Bode formula concerning the relation between the real and the imaginary parts of minimal phase transfer functions, the Fujisawa criterion for ladder networks, to name a few. These results, some of which are extended in the recent book by Freudenberg and Looze[15], did not receive much attention in the far-reaching progress of modern system theory, in which the state-space paradigm dominates. The revival of analytic system theory gradually took shape in the mid70's. Perhaps, the multivariable synthesis methods in the frequency domain developed mainly by the British school [32] [33] 138] paved the avenue for the more systematic and sophisticated theories in the frequency domain, such as Hankel-norm model reduction, H °° control and robust stabilization. In the early stage of the developement of these fields, the functiontheoretic aspect of transfer functions played a dominant role. It is remarkable that, in these new frequency domain theories, function theoretic properties of transfer functions manifest themselves as interpolation constraints. Thus, the classical interpolation theory in classical analysis, which dates back to the beginning of this century, gives the analytical basis of the new system theory in the frequency domain. Especially, the PickNevanlinna interpolation theory almost directly solves the robust stabilization problem. When writing [24], the author was amazed to notice that the result of 70 years ago can be used in a very straightforward way to solve a problem of contemporary technology. It was in the late 60's when the Pick-Nevanlinna interpolation theory was first brought into system theory. Youla and Saito gave a circuit theoretical proof of the Pick criterion [44]. It is worth mentioning that the celebrated paper on lattice filters by Itakura and Saito [20] appeared four years later which revived the interest in the moment theory developed from the Caratheodory-Fejer interpolation problem, another famous classical interpolation theory related closely to time series analysis. The paper by Youla and Saito did not seem to receive much attention in circuit theory at that time. However, approximately ten years later, Youla's paper on broad-band matching [43j which gave a motivation for [44] was given a new light by Helton [181 who used a more sophisticated operatortheoretic version of the Pick-Nevanlinna interpolation theory. In the field of control, we had a pioneering work by Tannenbaum who applied initially the Pick-Nevanlinna theory to the problem of robust stabilization [35]. This

245

work was extended later in [23]. The problem of robust stabilizability for SISO system turned out to be ahnost directly connected to the PickNevanlinna problem [24]. In the field of H ~* control, the paper by Chang and Pearson 14] solved tile so-called one-block problem based on the matrix version of PickNevanlinna theory [5]. Also, a multivariable extension of the robust stabilizability was given based oil the same technique in [42]. Ilowever, the matrix version of Pick-Nevanlinna theory developed in [5] is not suited for computation and the resultant controller is not minimal. A version of matrix interpolation problem was proposed in [26] to cover these drawbacks. If we confine our scope to linear time-invariant lumped-parameter systems, we can have another important property of systems in our bands. This is the rationality. In state-space theory, we can neglect cumbersome arguments on the existence of Laplace transform and many functiontheoretic properties like spectral factorization can be translated into the much more transparent algebraic properties of systems. The same route can be found for the classical interpolation theory. In this paper, we shall show how the classical interpolation theory, which relies heavily on function theoretic properties, is described in the algebraic framework of the state space. The notion of conjugation, or more specifically, of J-lossless conjugation is shown to be the state-space representation of the classical interpolation, or its modern versions. Thus, the notion of J-lossless conjugation provides a unified treatment of the I1'~ control problem, as well as of robust stabilization and model reduction. It should be noted that the operator-theoretic approaches to the interpolation problems have contributed substantially to the development of system theory in the frequency domain. Since Zames' paper, the first breakthrough in H ~ control theory was done by Francis and Zames [12] which was based on tile work of Sarason [39]. The geometric approach by Ball and Helton 12] [3] was extensively used in [13]. Dym [9] proposed another framework of interpolation theory, in which the matrix version was treated in a natural way. In Sections 2 and 3, the classical treatments are reviewed briefly as faithfully as possible. The lattice structure of the Nevanlinna algorithm is also described. In Section 4, some applications of classical interpolation are briefly discussed. In Section 5, the notion of conjugation is introduced. It is shown in Section 6 that the inner-outer factorization is regarded as a special class of conjugation, the lossless conjugation. The J-lossless conjugation is introduced in Section 7. The relation between the J-lossless conjugation and the interpolation problem is fully discussed. Section 8 is devoted to solving the model-matching problem based on J-lossless conjugation. The complete solution to the one-block problem of H** control is given.

Notations Pt" =

{f(z) ; analytic and Re f(z) > 0 in Izl < 1 }

246 P : = { f(z) ; analytic and Re f(z) > 0 in Re s > 0} B I : = { f ( z ) ; a n a l y t i c a n d [ f ( z ) [ < 1 inlzl--- 1) B

=

:

{f(s) ; analytic

anal f(s)I---

1 in Re s > 0}

R m x r : The set o f real matrices o f size m x r . a H m x r ; The set o f rational p r o p e r stable matrices o f size m x r . OO

OO

BHmx r ; The subset o f

RHmxr consisting

o f contractions

{A,B,C,D}: = D+C(sl-A)-IB G - ( s ) : = OT(-s) , G*(s) = ~l(s).

2. f(zi),

PICK-NEVANLINNA INTERPOLATION PROBLEM In 1916, Pick posed the following question [35] : Let f ( z ) /z i / < 1, i = 1, ...,n.

complex

numbers

What conditions

are i m p o s e d

e F 1

and w i =

on the n pairs oJ

(z i, wi), i = 1,...,n ?

Obviously, the condition Re wi > 0, i = 1,--.,n, is necessary due to the definition of P I . The crucial fact is that it is n o t sufficient. An additional condition is imposed which comes from the analyticity of f(z). In order to answer the a b o v e question, Pick derived an integral representation of f(z) in the disk I zl < r < 1 which is now known as Schwarz's formula [1, p.168]: f(z) = _ z~'flo

reJ° 4"Z R ~ f(re "j0)] dO + _~_f0 z* Im [f(reJ0)] dO j0

(2.1)

re - z

From this representation, f(z.~ + f(zj) _ 2 r

-

- z~zj

1 f' 2x.]

it follows that z~ o

Re[f(reJ0)] '

(re J ° - z-~(re

-jo

--

dO,

- zj)

where r is taken sufficiently close to 1 enough to guarantee that I zil < r for each i. Therefore, for all xi, i = 1, .-.,n, we have

2 ~ f(zi)+ 2 f(z))XiX-J = ~fo~ Re[f(reJ°)] I "~" j0Xi [ d0>0.

id=l

r - zizj

Since f ( z ) E P I, the integrand at tile right-hand r-o 1 verifies the inequality

i=1 re

- zj

side is non-negative.

Letting

247

W1 +

W1 -

1 -

P:

w I +

w n

-

i

z~zl

1

-

ZlZ n >

= Wn +

Wi -

Wn+

0

(2.2)

Wn

-

m

1 -z#l

1 -

Z~n

The matrix P defind in (2.2) is usually referred to as the Pick m a t r i x . N o w , it has been proved that P > 0 is a necessary condition for the existence of f(z) ~P1 satisfying the interpolation conditions wi = f(z'0,

i = 1,2,...,n.

(2.3)

At first glance, it seems obvious that P _> 0 is also sufficient for the existence of f(z) ~ P1 satisfying (2.3). Pick himself did not consider tiffs problem. Actually, it is far from trivial and an answer was given later by Nevanlinna [34] who derived an algorithm to construct such an f. B e f o r e p r o c e e d i n g to the N e v a n l i n n a c o n s t r u c t i o n , we note an important property of the Pick matrix (2.2) which was fully discussed by Pick himself in [35]. Consider a linear fractional transformation a + bX a - cz z ~, (2.4) c + d~.' -b + dz This maps the unit disk in the z-plane to a region in the k-plane. For instance, 1 -~. 1-z z ~, (2.5) I+L' l+z" maps the unit disk to the right half plane ~. + ~ > 0. Assume that k i is mapped to z i in (2.4). 1 - ZiZ'-j=

From the identity

K(~i'_ ~j)

(2.6)

(c + dX.O(c + ~X)

12 ( ) +

K( X~, N~) = ( I c - l a

+ ( c d - ab )~.j

( ca - tab )~.~

+ (]d[ 2- [b[ 2 )X~b

tile unit disk in tile z-plane is mapped to the region K( X , ~ ) > 0 in the k - p l a n e . Also, a linear fractional transformation

(2.7)

248 al + b1~3 W

~"

al [3

C 1 '4- dl13'

-

ClW

--

-bl + dlw

maps the right half plane Re z ~ 0 to a region

(2.8)

L( 1~,13 ) >- 0

L(~i, 13j)

in the [3-plane, where

is defined as

L( 13i,13j )

Wi+W j =

(2.9)

( Cl + dl13i )( cl + dll3 j ) L( [3i, ~'j ) = ( alcl + alcl ) + ( Clbl

+ aldl )~i

+( aldl + clbl )~j + ( bidl

+ bldl )~i~j.

From (2.6) and (2.9), it follows that w i + w--j _ 1 - ziz--j

( c + d)~ i )(c+

d-~j )

L( ~i,~j )

( c 1 + d113i )( c 1 + dl~ j )" K( Z.i, ~,--j )

Therefore, tile inequality (2.2) is equivalent to the inequality m

L( 131, 131)

L( ~i, [Bn)

~,] )

K( ;Z l, ~'n )

K( ~,i,

> 0. L( 13,,, ~, ) m

(2.10)

L([3~ ~.) m

Thus, if f(L) is analytic and satisfies L(f, ~ _> 0 in tile region (2.7), the interpolation data ~i = f( ki ), i = 1, ..., n, must satisfy the inequality (2.10). As an example, take the transformation (2.5) and

w = 1-13 1+~'

(2.11)

which maps the half plaqe Re w >_ 0 to the unit disk [l~l < 1. Since K( ki, ~J ) = 2( Xi + Xj ) and L( 13i, 13j ) = 2( 1 - [~il3j ), the inequality (2.10) becomes

249 m

1-

{~l~l

Xl + 21 p

I- {~,~n Xl + 2.

=

_> O.

1-

I]~l

1-

+ 2t

(2.12)

~,~n

X. + 2.

The inequality (2,12) is actually a necessary and sufficient condition for the existence of f(X)~ B satisfying the interpolation conditions 13i = f(Xi), Re k i >_. 0,

i=l, ...,n. 3.

NEVANLINNA A L G O R I T H M Three years after Pick's work [35], a paper by Nevanlinna [34] appeared, in which the converse of Pick's result was extensively discussed. He worked with B 1 instead of P1 and formulated the problem as follows : Find a necessary and sufficient condition on tile n pairs ( zi, ~i ), ]zi / ~- 1, i=l,...,n, which guarantees the existence of a function f e b I satisfying

I

~i = f ( z i ) ,

i = 1,-..,n,

(3.1)

Probably, Nevanlinna did not know the result of Pick at that time, because the Pick's paper [35] was not quoted in [34], and his approach was totally different from Pick's approach. Instead, Nevanlinna's paper seemed to be strongly influenced by the work of Schur [40] who gave all alternative proof of the Caratheodory-Toeplitz theorem for the Caratheodory-Fejer interpolation problem. The Nevanlinna's construction algorithm is essentially sequential, and is based on the Schwarz lemma at an arbitrary point. Schwarz L e m m a Then, for any z satisfying

Assume

Iz-zll = rll -ztzl, f(z) satisfies the inequality

that f(z)e B and 131=f(zl) with

[ z I [
0
If(z)-13 II -< r l l - 1 3 1 f ( z ) l. The special case z: = 13t = 0 in tile above lemma is usually referred to as tile Schwarz lemma in textbooks on complex function theory. A direct application of this lcmma proves that, if 131 = f(zl) and f(z) ~ B, then

250

fl(z ) : -

I

-

f(z)_- 131 ~ B1

ziz

z - zt

1-

(3.2)

[3~f(z)

Solving the above identity with respect to f(z) yields f(z) = _B l ( z ) f i ( z ) + 131 , [~tBl(Z)fl(z) + 1

BI(Z ) _

z - _zt 1 - ztz

(3.3)

Since B l ( z ) e B 1, f(z)e B l if and only if fl(z)c B 1. Moreover, f(zl)= 131 for any choice of fl(z), tlence, the first interpolation condition of (3.1) is always satisfied irrespective of the selection of fl(z). In order to satisfy the remaining n-I interpolation conditions, it is sufficient to choose fl(z) in (3.3) such that fl(zi)

[~ i(2)

= [312)," =

i = 2, ...,n

1 -- Z1Z i [3i --

[31

zi-

113i

zl "l -

(3.4)

Thus, the problem is reduced to a simpler one, in which interpolation conditions is less than the original problem. Since fl(z)e B1, the inequalities ]]3~2) I _< 1,

i

must be satisfied. If 113m(2)] = 1 in B 1 satisfying (3.5) due to the [32(2) = 133(2) . . . . . 13~(2) must hold interpolation problem. If the can apply the same procedure based on the representation

f2(z) : -

I - z2z

Z-

Z2

f1(z)

= 2,...,n

the number of

(3.5)

for some m, fl(z) -- lira(2) is the only function maximum modulus theorem. Therefore, in this case for the solvability of the original strict inequalities hold for each i in (3.5), we to the simpler interpolation problem (3.4)

-

]~2)

1 - 1°'(2)1(Z 132 )

and its inverse

fl(z) : =

B2(z)f2(z)

-

~2)

Z2 _ 1 - ZzZ Z

,

B2(z)

-

_(2)

[32 Bz(z)fz(z) + 1

-

Tile problem is reduced further to tile simpler one

f2(zi) = 1313),

i = 3,...,n

(3.6)

251

1~13)

1-Z3Zi

~ I 2) - 1 3 7 --(2)_ (i) " ' 1 - 1~2 152

Zi - Z3

In general, we have the recursion

fj(z) = _Bj(z)fj+,(z) + Oj , pjBj{z)fj+l(Z ) + 1

Bj(z)

=

1 - zjz z -zj

j = 1, . . . , n ,

(3.7)

where we write pj = 13j(j) which is computed by another recursion •

-

13i0+1) _

1 -

zjz i

z i -

zj

" 1 -

PJ

(3.8)

O3-

IliPj

The recursion can be continued as long as [pj [ < 1 holds. In that case, we satisfy the interpolation constraints at j = n, and therefore, we can choose any fn+l ~ B lNevanlinna showed that the original interpolation problem is solvable if and only if either

or

I, '1 < ,,

j = 1, 2, ..., n

(3.9a)

j

(3.9b)

= 1, 2, =

..., k - I

[~k+l =

Tile solvability criterion (3.9) looks totally different from thc Pick's criterion which is represented as i

i

1 - 13d31 i

i

-

Z1Z 1

1

-

ZlZ n

p =

> O. 1-

~in~,

1-z~l

1-

(3.10)

13n~n

1-Z~n i

in this case. Latcr, it turned out that the Nevanlinna algorithm is essentially identical to the Cholesky factorization of P in (3.10). The structure of the Nevanlinna algorithm can be more clearly viewed in the form of the fractional representation

252

fj(z) = nl{z). dj(z)

(3.11)

The recursion (3.7) is represented as

nj(z)] =

[nj+l(~) 1 ®J(z) [dj+l(Z) j

dj(z)J

,

Oj(z) = ~1--ipj[

2

(3.12)

[1p-j

o1 1"

(3.13)

Here, we assumed condition (3.9a) holds. The common term (1-Ipjl2)-l/2is a normalizing factor. Since Bj(z-1)Bj(z) = 1 , we have

--T-'I[ 10] ®j(z )

0

®j(z) =

[10] 0

-1

"

This implies that Oj(z)is a J-unitary matrix. Such matrices will be discussed extensively in Section 7. The relation between fj(z) and fj+l(z) is represented in the form of a generalized lattice section, as shown in Fig. 1.

.................. e

~1! B(z) Fig.1

._.- il-:i~ i~ ........

( 1 -ipl 2)

~

Lattice Structure of Nevanlinna Algorithm

253

Based on the recursion (3.12), the solution of the Pick-Nevanlinna interpolation problem f(z) = n(z)/d(z) is represented as (n(z)~ (n.+i(z)'~ = 01(z)O2(z)... On(z) ~.dn+l(z)j' d(z)] where f,+l ( z ) = formula f(z)

nn+l(z)/d,+l(z ) can be freely chosen front B I. Thus, the

=

0xl(z)S(z) + 012(z) 021(z)S(z) + 0/z(z)

(3.15)

-I

O(z)= [ 0H(z) 012(z)| : = Ol(z)Oz(z)---On(Z) 0zl(z) 022(Z) J

(3.16)

gives a. parameterization of all solutions to the Pick-Nevanlinna proble,n under the condition (3.9a), where S(z) represents an arbitrary element of B I. The factorization (3.16) was extensively studied in the classical work of Potapov [36].

4. A P P L I C A T I O N S 4.1 Broadband Matching Tile first serious application of the Pick-Nevanlinqa interpolation theory was probably the broadband matching. The problem itself is old, and Youla formulated it as a Pick-Nevanlinna problem as shown below [43]. Later, more general cases were treated by Helton [18][19]. Consider the simple circuit of Fig. 2. The power absorbed by the load is calculated to be P -

RI

.E 2 =

(Rg + R,) 2

(

1

(Rg -

Ri

/2/2 E

(4.1)

+ R,)14k,

This intplies that tile maximum power is absorbed by tile load when tile load is "matched" to the source, i.e., R 1 = Rg, and the maximum absorbed power is given by Pmax = E2/4Rg . In the case where the load impedance is frequency dependent, it is necessary to construct a network N which transfers the maximum power front the source to the load over a wide range of frequencies. This is the problem of broadband matching t101118]143], which is depicted schematically in Fig. 3. The network N is usually assumed to be lossless and is specified by the output impedance Z(s).

254

El Rt

Rg

Fig. 2

A Simple Electrical Circuit

Z~

I

N

Fig. 3

Broadband Matching

Tile ratio of tile power P absorbed by the load impedance Zl(S) to the maximum power Pmax absorbed by the ideal but not always realizable impedance at a frequency co is given by

[ z0o~) - zt(jo~) 1~

P P max -

1 -

which is a generalization of (4.1). u(s)

=

(4.2)

ZO °3) + zi(J°~)

Z(s) - zl(-s) Z(s) + zl(s)

See [43] for the derivation.

Let (4.3)

255

Our task is to choose a passive (hopefully lossless) impedance Z(s) such that l u(jc0)l is as small as possible for a given frequency range. It is wellknown that Z(s) represents an impedance of a passive network, if and only if Z~ P. Since Zl(-S ) is not in P , we cannot choose Z(s)=zl(-s). Let {-Itl, -It2, ...,-Itm } be the set of poles of Zl(S), and define

b(s)

=

( s - _lttl ) ( s - _~2)'"(s - ~t m_ ) . ( s q- ~ t l ) ( s q- ~ 2 ) . . . ( s -I- ~ m )

(4.4)

From (4.3), it is obvious that v(s) = b(s)u(s)

(4.5)

is stable. In other words, the unstable poles tai, i=l,.-., m, of u(s) which come from Zl(-S ) are cancelled out by the zeros of b(s). Since b(s)b(-s)=l,we have

1 - v(s)v(-s)=

(Z(s) + Z(-s) ) (zl(s) + zl(-s) ) (Z(s) + zj(s) )(Z(-s) + z~(-s) )

Since Z(s)e P and Zl(S)e P, we have Iv(jco) l -< 1. This implies that v(s)e B. Let ~.i, i = 1,--., n, be the numbers satisfying zl( ~.i ) + zl( -Ki ) = 0, Due to (4.3), u(~.i)=l. v(~.i)

Re ~.i > 0.

Hence,

= b(Xi) ,

i = 1, ..-, n

must be satisfied for any choice of Z(s). finding U e B which satisfies (4.6). This problem.

(4.6) Thus, the problem is reduced to is exactly the P i c k - N e v a n l i n n a

4.2

Robust Stabilization Consider tile c l o s e d - l o o p system of Fig. 4, where p(s) denotes tile transfer function of a single-input single-output plant and c(s) the transfer function of the controller. We say that p(s) belongs to a class A ( p o , r) corresponding to the nominal model P0(S) and the uncertainty band r(s), if (i) I p(jco) - po(jo) I --- I r(jco) I, v0~, r(s) : stable. (ii) p(s) and p0(s) have the same number of unstable poles.

Our purpose is to find a fixed controller c(s) such that the closed-loop system of Fig.4 is stable for any plant p(s) in the class A ( p 0, r). If such controller exists, we call it a robust stabilizer of A(po, r). The existence of a robust stabilizer will be shown to be reduced to the solvability of a PickNevanlinna interpolation problem.

256

C(s)

Fig. 4

t

I

._l p(s) I

-I

l

v

The unity feedback system

For a given controller, let q(s): =

c(s) 1 + po(s)c(s)

(4.7)

Using the Nyquist stability criterion, one can easily show that c(s) is a robust stabilizer of A(po,r), if and only if it is a stabilizer for the nominal plant p0(s) and satisfies [rOco)q(jm)l

< 1,

Vo~.

(4.8)

The inequality (4.8) was apparenty first derived by Doyle [8]. Due to Zames and Francis 146] , c(s) is a stabilizer for po(s) if and only if (i) q(s) is stable, ( i i ) I- po(s)q(s) has zeros at the unstable poles of po(s), multiplicities included. Let ~.1, 7V2, " , L n be unstable poles of po(s) satisfying Re 7vi > 0, i = 1,...,n. We assume that the ~-i s' are all distinct. Define b(s) =

( s - ~vl )( s - 2~2 ) ' " ( s - X.n )

(4.9)

( S+ ~l )( S+ ~2)"'( S+ ~n) From the assumption, u(s)" = b(s)p0(s) is also stable.

(4.10)

Since q(s) in (4.7) has unstable zeros at ~i , we see that

w(s) : = q(s)/b(s) is also stable.

Therefore, since r(s) is stable,

t0(s):= r(s)w(s)

(4.11)

is stable. Moreover, since Ib(joa) l = l for each to, it follows, from (4.8), that I q~(jto) l < 1. Hence, cp~ B.

257

The interpolation constraints come from the condition that l- p0(s)q(s) = 1- v(s)w(s) vanishes at s=X i, i=l,-.-,n. This implies that gJ(TLi) = r(~Li)w(;L.,) = 13i:

[3i,

i = 1,---, n

(4.12)

= r(;LO/v(L).

Thus, the problem is reduced to finding a function q~e B s a t i s f y i n g the interpolation conditions (4.12). This is again exactly the Pick-Nevanlinna problem, and was discussed in [24]. 4.3

ll°°-control and Directional Interpolation Problem Consider the closed-loop system of Fig.5, where (4.13) /PziP22]~u/

denotes the plant and u = C(s)y the controller. The closed-loop transfer function (1)(s) from the exogenous signal v to the controlled variable z is given by 4~ = l~ll + P I 2 C ( I - P22c)'tp21.

v

u

.J

~

Fig. 5.

P(s)

(4.14)

I

I

~z

y

H °° control scheme

The purpose of II°°control is to find a stabilizing controller C(s) satisfying the norm bound of • represented as

II ,~11 oo <~ for a given G > 0.

(4.15)

258

It is now standard that • in (4.14) can be transformed to a simple representation through the parameterization of all stabilizing controllers. Let -1

P22=NM

-1

=MlN1

be right and left coprime factorizations of P22, respectively, such that

iv, II l:I 0] -U~ -N1 MI

for some proper. Then, it system of

M N

I 0

(4.16)

'

U, V, Us and Vs. Here, all the matrices in (4.16) are stable and The definition of the coprimeness in this context is found in [14]. is well-known that any controller that stabilizes the closed-lo0p Fig. 5 is represented as C = (U+ MQ)(V +NQ)-I

for some stable Q. =

(4.17)

Substituting (4.17) in (4.14) and using (4.16) yield

T lT2QT3 T l : = Pll + Pt2UMsP21, T2: = Pl2 M, T3: = -MIP21

(4.18)

Thus, the problem is reduced to finding a stable Q such that liT 1 -

T2QT3lloo <

1,

(4.19)

where we normalized the norm bound 6 to unity. This problem is usually referred to as the model-matching problem. [14]. The model-matching problem is reduced to a version of the matrix Pick-Nevanlinna interpolation problem under the assumption that (A1) both T2-1 and T3-1 exist. If both Tz-S and T3-1 are stable, tile problem is trivially solved by taking Q = T2-STsT3-s. Let Xi , i = 1, 2, .-., n 2 be the unstable poles of T2-I. Since they are the unstable zeros of T z , we can find a left annihilator gi v of T2(X i ) SO that ~ i T T 2 ( ~ . i ) = 0. Therefore, each • of the form (4.18) must satisfy T

~i~(L~

T

=

aqi,

i = 1, 2,--., n2,

(4.20)

irrespective of the selection of Q, where rli r : = ~ i T T l ( ~ . i ). The condition (4.20) specifies ~ ( ~ i ) with respect to a given direction. Due to (4.19), must be a contraction. The problem of finding a contraction • satisfying

259

(4.20) was called a directional interpolation problem and was extensively discussed in [26]. It was shown in [11][25] that a contraction ~ s a t i s f y i n g (4.20) exists, if and only if T=-

T--

T.--_

~t~ 1-11 lrl i Xl+~t P:

T--

~ ~G-rl irl. '

Z.l+2. > 0

..~ T=-.

T--

T.--_

(4.21)

T--

G~-n°n~

GG-n.n.

x.+2t

x.+2°

Here we write n 2 = n for the sake of notational simplicity. This condition is obviously a generalization of (2.12). In the same way, at each unstable zero I.tj, j = 1, ..., n 3 of T3(s), • in (4.18) must satisfy q>(I.tj)~j = ~j,

j = 1, ..., n 3

(4.22)

where ~j is tile left annihilator of T3(Id.j ) and Wj: = Tl(kt j )~j. Tile existence condition o f a c o n t r a c t i o n ~ satisfying both (4.20) and (4.22) was first obtained by L i m e b e e r and Anderson [30] as m

p

°

>0

5.

CONJUGATION Multiplication by an all-pass function can r e p l a c e the poles o f the system by their conjugates. For example, let g(s) = (13s + 7 )/(s - ~). Tile multiplication by an all-pass function (s - cx)/(s + t~) yields

g(s) S " cc 8+(3(,

_

15s + 3' S +

(5.1)

(X

Here the pole at s = ct of g(s) is replaced by its conjugate s = -or . It is important to notice that the multiplication by an all-pass function does not

260

change the jc0-axis gain. This operation has been extensively used in various problems of system theory. We have actually observed that this operation is a crucial step to formulate the problem as an interpolation problem in Section 4. In the broadband matching problem, the unstable poles of u(s) in (4.3) are "conjugated" by b(s) in (4.4) in the definition of v(s) in (4.5). In the robust stabilization problem, the unstable poles of Po (s) are also "conjugated" by b(s) in (4.9) to yield a stable system u(s) whose gain on the jco-axis is same as that of the original po(S). Now we formulate this operation in the state space. Let G(s) = {A, B, C, D} be a state-space realization of a transfer function matrix G(s). transfer function V(s) such that T

G(s)V(s) =

{ - A , * , * , * },

(5.2) We seek a

(5.3)

where * denotes a matrix whose exact form is not rclevant. The identity (5.3) is a generalization of (5.1), in which the A-matrix of G(s) is replaced by its c o n j u g a t e -A T in in (5.3). A system V(s) which carries out this job is called a conjugator of G(s). To make the subsequent argument simpler, we make the following assumption : (A2) The matrix A ill (5.2) has no eigenvalue on the jo-axis. This assumption is a technical one and can be removed easily by extracting tile jo~-mode from A. In order to characterize the conjugator, we first note that the A-matrix of the conjugator of G(s) must be similar to -A"r because of (A2). Write T

V(s) = { - A , B e , C e , D c}

(5.4)

The well-known product rule is applied to yield

G(s)V(s) = { [

A

BC~

0

-ATJ 'L .jBc '

(5.5)

Let X be a matrix satisfying AX + XA

T

= -BCe

(5.6)

Note that this equation is always solvable uniquely due to the assumption (A2). Now, we choose Bc and D c as satisfying BDc = XBc.

(5.7)

261

Tile similarity transformation of the realization (5.4) with tile matrix

T=

[ I 0

"I]'

Tq = [ I 0

X]

verifies that the modes associated with A are cancelling out this uncontrollable portion, we get G(s)V(s) = {-A T B e, CX + D C c, DD c }

uncontrollable.

After

(5.8)

Now we have seen that V(s) in (5.4) with the conditions (5.6) and (5.7) satisfies (5.3). We regard (5.6) and (5.7) as the defining equations of the conjugator of G(s) in (5.2). Note that (5.6) and (5.7) depend only on A and B in (5.2). DEFINITION

5.1

A system V(s) given in (5.4) is said to be a

conjugator of G(s) in (5.2) (or of tile pair (A, B) ), if it satisfies (5.6) and (5.7). The operation performed by a conjugator is called a conjugation. Now, we shall discuss some properties of tile conjugator. Assume that there exists Dc I such that DcDcl" = I. From (5.7), it follows that B = XBcD,fl . The equation (5.6) then becomes AX + x ( A T + BcDcI"Cc) = 0.

(5.9)

From (5.7) and (5.9), A k B E Ira(X) for each k. Hence, X is iavertible if (A, B) is controllable. In this case, (5.9) implies that the eigenvalues of -A TB c D c t C o or the zeros of V(s), are identical to the eigenvalues of A. This implies that all the poles of G(s) are cancelled out by the zeros of V(s). This cancellation is obviously necessary for (5.3) to be satisfied. Next, we assume that both D -I and D¢ -t exist. Then, from (5.6) and (5.7), it follows that -AT-B,(DD¢)-I(CX + DC~) = X-I(A- BDqC)X. In view of (5.8), this idcntity establishes tile following important property of conjugators. L E M M A 5,2 If both O(s) and V(s) are invertible, tile zeros of G(s)V(s) are identical to those of G(s). 6.

LOSSLESS CONJUGATION From tile structure of tile defining equations (5.6) (5.7) of conjugators, it is clear that various classes of conjugators can be generated corresponding to

282 the various selections of C c in (5.6). This section is a short digression, in which a class of conjugations called a lossless conjugation (the conjugation by a lossless V(s) ) is shown to be actually equivalent to the inner-outer factorization of invertible systems. Let F ( s ) ~ R H m ~ which is invertible. The inverse F(s) 1 c a n in the state-space form

F(s)

=

{

A~

'

be represented

B1

where A o is anti-stable (all tim eigenvalues have positive real parts) and Al is stable. Now, we carry out a conjugation of the pair (A 0, B0) with the special selection of T

C c = - B o.

(6.2)

Tile equations (5.6) (5.7) become in this case T

T

A0X + X A 0 = BOB0,

BoDc = XB c

(6.3)

If we take an orthogonal matrix as De, the conjugator T

-1

T

V(s) = { - A 0 , X B0D c, -B o, D c}

(6.4)

is an inner matrix, i.e., V(s) is stable and satisfies V-(s)V(s)

= I.

(6.5)

Some manipulations using (6.3) and the product rule yield. K ( s ) : = F(s)'IV(s) =

0

-A 0 T

AIoX - BIB o

Ai

and K(s)q

:[

Ao 0 { Alo AI

BO D" C O Cl] B1

I

C 0 - DBoTX

1o'

C1

}

Since both -Ao T and A I are stable, so is K(s)-. Also, since the A-matrix of K ( s ) -~ is identical to the A-matrix of (F (s)-t) -l = F(s), K(s) -t is also stable. Therefore, the identity F(s)

=

V(s)K(s) "l

263

represents an inner-outer factorization solving a Riecati equation.

of F(s),

which

is obtained

without

7.

J-LOSSLESS CONJUGATION It is well-known that the interpolation problem is suitably treated in the framework of J- lossless systems [2] [3]. In this section, we shall show that the interpolation problem is reduced to the J-lossless conjugation, the conjugation by a J-lossless matrix, in the state space. A transfer function matrix ®(s) of size (m + r)x(m + r) is said to be J unitary, if ®-(s)J•(s)

= J,

(7.1)

where J is a signature matrix defined as

J: =

0

(7.2)

-I "

A J-unitary matrix O(s) is said to be J-lossless, if it is J-contractive in the right half plane, i.e., O*(s)JO(s)

<

J,

Re Is] >

0.

(7.3)

The lossless matrix is a special case of J-lossless matrices for which either m = 0 or r = 0, or equivalently, J = I. Let a controllable pair (Ao, Bo) be given with Ao~ Rnx n and Bo~ Rnx(m+r). In the preceding section it was shown that a lossless conjugator of ( A o , B 0 ) always exists under the condition that A o is anti-stable. This is no longer true for J-lossless conjugators. Though Ao is allowed to be unstable, the pair (A 0, Bo) must satisfy a strong condition in order that there exists a J-lossless conjugator (a conjugator which is J-lossless) of (A o, B0). Tile following result gives an existence condition of the J-lossless conjugator, as well as its statespace representation. The proof is found in [27]. TIlEOREM controllable pair

7.1 There exists a J-lossless conjugator of a given (A o, Bo), if and only if the equation

AoP O+ P0 AT = B o J B T

(7.4)

has a positive definite solution P0given by T

-1

In that case, a J-lossless conjugator is T

®(s) = {-A 0 , P o B o D c , - J B 0 , D c } , where D c is any constant J-unitary matrix.

(7.5)

264 In order to show the relevance of this theorem to the interpolation problem, assume that A0 is anti-stable and write B 0 = [L

M],

L E Rnx,n,

M ~ Rn×r

(7.6)

We seek ~(s) ~ R Hmx~such that L~(s) - M is "divisable" by sI - A0, i.e., L ~ ( s ) - M = (sI - A0qJ(s)

(7.7)

for some ~ ( s ) ~ R H m x r . This problem is essentially equivalent to the directional interpolation problem discussed in 4.3. To see this, let 7~i be an eigenvalue of A 0 with xi T as its left eigenvector. Writing ~iT= xiTL, rli T = xiTM, we have from (7.7) ~iT (~.i) = TliT, (7.8) which is identical to (4.20). By the pre-multiplication of equation (7.4) yields

xiT and the post-multiplication of xj, the

TI

"r

-

-

xiPoxj

=

Tli T~j

(7.9)

This implies that P0is congruent to the Pick matrix (4.21) under the condition that the eigenvalues of A0are distinct. Thus, Theorem 7.1 gives a condition for the existence of a contraction qb(s) satisfying (7.7). The significance of (7.7) becomes clear in the next section. A similar argument is found in [21] for a scalar interpolation problem. Now, we shall consider a factorization of a J-lossless system into the product of two J-lossless systems. Assume that a transfer function matrix Go(s) = {Ao. B0. Co. Do} is given in a form of spectral decomposition (7.10)

Also, assume that Go(s ) has a J-lossless conjugator @(s). O(s) is represented as

®(s)

=

v

-A2

where we take D c = Ira+r definite solution of

Due to Theorem 7.1,

•

L

(7.i1)

2J

for the sake of simplicity and P0

is the positive

265

[

o] [::] I T

A1 0

0 ] P o + P0 A2

A1 0

(7.12)

Az

Write

Po= P,IT P121 P12 P22J.

(7.13)

First, we conjugate tile pair (Al, B 1 ). (7.4) is satisfied by Pll, where we write

Since the equation corresponding to

T p~tlB1, _JBT,Im+r} 01 = {-A1, according to (7.5).

Here, we again take D c

G(s)Ol(s ) = {

0

A2-B~IBI

0

(7.14)

,

Bz

-A Tt

0

The product rule yields

= Ira+ r.

, C1 C 2 - D J B I ,

D}.

Pt IB I

Applying the similarity transformation given by

T =

[ J [ ] I

0

0 0

0 I I -Pl

"Ptl

.1

,

T

=

I

Pll

0 0

Pz2 I I 0

0

and cancelling out tile uncontrollable portion yield "

l

G(s)Oi(s) = {

Pi',Bt ]

0

A2

,

where Ct/= C1P1t + C2PI2 - DJBIT. Now we conjugate the remaining portion (A 2, B 2 -PI2PII-1B1). (7.12), it is not difficult to see that the Schur complement T

-!

P2 = P 2 2 - P12P~tP12 satisfies the equation

(7.15)

Based on

266 T

-I A2P2 + P2A2T = ( B 2 " PIxPnBI)J(B2 - PI2 P-1Bllt) which corresponds to (7.4) for tile pair (A 2, B 2 - P12PIt-IBI). conjugator of this pair is given by T

O2(S) = [A2,

9

Titus, a J-lossless

p~l(B 2 - Px2PIlBI), q -I -J (B 2 - PI2PttB1) T, Ira+r}. (7.16)

From the product rule, it follows that -A T1 ol(S)o2(s)

=

_p;llB tJuT

{

T -A 2

0 where U

= B 2 - PtzPtt-lBt.

P;';lout1

The similarity transformation given by

I T = 0 applied to ®l(s) and O2(s) shows that O(s) = OKs)O2(s). Thus, we have established that the original J-lossless conjugator ( 7 . 1 9 ) i s decomposed into the two consecutive J-lossless conjugators e l ( s ) a n d 02(s) given respectively by (7.14) and (7.16). This factorization clearly represents the factorization (3.16) associated with the Nevanlinna algorithm in the state space. A similar result was found in [16].

8.

MODEL MATCHING PROBLEM In Section 4, we showed that the H °~ control problem boils down to the model matching problem. It was further shown that the model matching problem was equivalent to a version of the matrix Pick-Nevanlinna problem. In this final section, we solve the model matching problem in the state space by reducing it to the problem of finding a J-lossless conjugator.

Let Tt~ R H mxt, T2e RIlm× m and T3~ R H ~ in the state space as Ti(s) = {A i, B i, C i, Di} , We assume that (A3) T2-1(s ) and

T3-t(s )

be given which are represented

i = 1, 2, 3.

exist and are anti-stable.

(8.1)

267 D2 -x and D3 d

This assumption implies that

A3 : -

A2: = A2 - B2D2"Ic2,

exist and (8.2)

A3 - B3D3"1C3

are anti-stable. The model matching problem is formulated as follows • (1) Determine whether there exists a stable Q such that

(2)

(8.3)

IIOI1~, < 1.

: = T1 - T2QT3,

I f a solution exists, characterize all such q~ and Q.

Let -1

T

L2" = B 2 D 2 ,

-1

L3"=

Due to the assumption stable, the equations

(8.4)

D3C3.

(A3), both A 2 and A 3 are anti-stable.

A 2 R 2 - R2AI = L2CI,

R 3 A 3 - AIR 3 = B1L T

have the unique solutions R 2 and R 3, respectively. T

M2:=

LzD t + RzB l,

From (8.1) and (8.3), it follows that P~2)(sl - Az)-IB2 .

M3:= L2T2(s )

(8.5)

Write T

(8.6)

CiR 3 + D I L 3. =

Since AI is

L2(D 2 + C2(SI - A2)-IB2) = ( s I -

Also, due to (8.5), it follows that L2TI(s) A

=Lz(DI+CI(sI - At)A

1BI)=Mz - (Rz(sI - A1)-A2 Rz+R2Ax)(sI - A1)-IB 1 = M 2 - (sI- A2)R2(sI - A t ) ' l B t . Therefore, if O(s) is of the form (8.3), it satisfies L 2 0 ( s ) - M2 = (sI - ,~2)qJ2(s), where tF2(s ) = -(sl - A2)-IB2Q(s)T3(s) - R2(sI - A l ) - l B l . the relation (7.7). Analogously, we obtain (D(s)L3- M3 = ~F3(s)(sI - ~,3),

(8.7) This corresponds to

(8.8)

where q'3(s) = -Tz(s)QC3(sI - A3)-I- Cl(sI - A1)-IR 3. This relation gives tile right interpolation constraints (4.22). The rest of this section is devoted to showing that there exists a stable Q satisfying (8.3), if and only if the pair

268

......T

A2

0

Lz

-M2

0

-A3

M3

-L3

(8.9)

has a J-lossless conjugator. In view of Theorem 7.1, this is equivalent to the condition that the solution P0 of the equation 2 0

A2 0 Po + Po

-A3J

0 -A

is positive definite. given by

L2

-M 2

L2

M2

-L 3

MT

=

L3J

(8.10)

In that case, a J-lossless conjugator of the pair (8.9) is

( wl0 (

-A2 0 e(s) = { , P ~, 0 -~3 J due to (7.5), where we took D c Let

)( M:)T

L2

_M 2

M3

-L3

,

Lz

, Ira+r}

M3

(8.11)

for simplicity.

= Ira+ r

o ]i

-I

0

T l(S)] T3(s) j '

(8.12)

and write

1-I2(s): = G2(s)®(s).

(8.13)

A lengthy but straightforward calculation yields

(8.14)

Fl2(s) = { A , B , C , D } ...T

A

=

i

-A2

0

0

A t

0

T BIM 2 T

-B3M 2

i

-D;' D;'D,

, D=

[ J 0

0

A3 _

D3

I°°l 0 -B1

B =

0

C

=

269

+

B3

0 0

-M 2

Pol

L2 1

M3

-L 3

I D2(C2P11 -1 + L2T - DIM2) T -D21(Cl + C2R2) 0 T -D3M 2

0

1

C3

If we represent O(s) and H2(s) in the partitioned forms

e2~(s)

e22(s)J '

n2(s) =

n21(s)

n22(s)J

which are consistent with (8.12), we have from (8.13), that ([I21S (OztS + O21) -l for each S. It follows that T t - T2(FIIIS for each S.

+

l-Ii2)(I-I21S + FI22)-1T3

=

+ I-I22)-1T3 =

( O i l s + Olz)(O2xS + 022) -1, (8.15)

It is well-known that •

=

(OILS + Olz)(021S + ®2z)

-1

(8.16)

is a contraction if O is J-lossless and S is contractive. (8.16) is of the form (8.3) with

From (8.15), • given by

°1

Q = ( H n S + Fllz)(I121S + I-I22) . In [27], it was shown that Q is stable.

(8.17)

Thus, • is a solution to tile model

matching problem (8.3) for each Se B Hm~. We have now established that the positive definiteness of P0 in (8.10) is sufficient for the solvability of the model matching problem. Conversely, assume that there exists a contraction • of the form (8.3). It was shown in [27] that the solution P 0 o f (8.10) can be represented as

Po =

where

o[

I Mt[:

if" K(jto) L2 2x _ M3

-L 3

- I

T L2 M3

o

(8.18)

270

[

sI- ,2 0 ]-t

K(s): =

.,.,T SI + A3

0

Since • is a contraction, the integrand of (8.18) is positive. Therefore, P0 > 0. Thus, we have established that P0 > 0 is necessary for the solvability of the model-matching problem. The above results are summarized as follows : TtIEOREM 8.1 There exists a stable Q satisfying (8.3) if and only if the solution P0 of (8.10) is positive definite. In that case, • and Q are parameterized as (8.16)(8.11) and (8.17)(8.14), respectively, where S is an arbitrary contraction in B H~,~. Finally, we shall discuss the application of Theorem 8.1 to the I10" control problem. Assume that the plant P(s) described in (4.13) has a statespace form

P(s) = {A,

BI

B

,

C

'

(8.19)

D21

with A~ Rn× n , DII~ Rmx r ,DI2~ Rrx r. We put D22 = 0 without loss of generality. Let Fe Rrx n and He Rnx r be such that AF:= A +B2F, are both stable. are given by TI

An:=

A+

HC 2

(8.20)

It is well-known that the matrices TI, T2 and T3 in (4.18)

2

= {

T3

0

AHJ

B t + IID21

C + Dt2F

-DI2F

0

C2

q ] [D2t rOll21}. (8.21)

The assumption (A3) implies that both D12-1 and D21-1 exist and -1 -I A2 = AF - B2D12(Ct + DI2F) = A - B2D12C l

(8.22a)

-I -1 ~'3 = AH - (Bl + HD21)D21C2 = A - B1D21Cz

(8.22b)

are anti-stable.

From (8.3),

it follows that

271 -I L 2 = B2D2I ,

T -I L3 = D21C 2.

It is obvious that R 2 = [-I from (8.6), it follows that

(8.23)

0] and R 3 = -[I

.i M 2 = B2DI2D11- B1,

Th ere fore,

l]Tsatisfy (8.4).

T M3 = DltD2tlc2- C l

(8.24)

The solution Po of (8.10) is represented as

P0 = [ P'tI where

Pll

(8.25)

p212]

and P22 are the solution of Lyapunov-type equations -1 (A - B2Dt2Ct)Ptl + PII(A - B2D]~,Ct) "~ -1 -T T -1 -1 = B2DI2DI2B 2 - (B 1 + B2DI2DIi)(BI + BzDI2Dll) T -1 -i (A - B1DzIC 2)TP22 + P22 (A - BID21C2)

(8.26a)

= C2TD21-TD2cIC2 - ( C I - D llDzl - l c 2 ) x ( C i -D 1t D 2 1 q e z )

(8.26b)

The system I-I 2 (S) in (8.14) is calculated to be

IT]I: -A2

rI2(s)

=

{

0

A21 Atl

0

1 [ olI I 0

+

B I + HD2]

C11

-F

T -D3M2

C3

pot

L2 M3

q

-L 3 '

-1 -1 ] -Dr2 D12D n } 0

D2x J

whcre A21 = -(B 1 + HDzl)(B l -BzDlz-lDll)T, CII = DI2-1((C1 + DI2F)PII + L2T DIITM2T)..Based on the above representation, the controller K(s) achieving the norm bound [~l~[L,o < 1 is givenby K(s) = Vii + VI2S(I - V22S)-IV21 V(s) = { A , B , C , D

=

~T

_

.I[-M21

}

T _ L3PiI )

0]

.IF_M2 ] -C3 I D M3 L2 ; ~ [ Po E

0]i

272 -1

ol

Dt2D11D21 C =

{

I

-

[M2J [L3J

Pl]}

D =

-1

-D12

-1

D21

0

"1

/

J

The extension to the case where T2d and/or T3-1 are no longer anti-stable is found in [22]

9.

CONCLUSION It has been shown that the classical function-theoretic treatment of the interpolation problem is transferred to the purely algebraic theory of conjugation in the state space. It is important to notice that, if we limit our scope to the systems with rational transfer functions, then the algebraic aspect always dominates which sometimes enables us to discard heavy and advanced mathematical tools. It is not our intension to deemphasize tile role of mathematics. On the contrary, as is seen from the short history of H°° control theory, mathematics plays a vital role at the initial stage of the problem formulation where the solvability is the most important issue. At the same time, it is of supreme importance for engineering theory to have a clear, elementary and self-contained framework of exposition, in order to penetrate into the modern technology. The theory of conjugation stated in this article is a state-space representation of the classical interpolation theory which gives the most elementary framework for H °° control theory for systems with rational transfer functions. From the space limitation, the more advanced theory of conjugation cannot be exposed which deals with the four-block problem in the most general way; neither can the discrete-time case. Here, we only mention some literature to appear along this line [25] [27]. Also the other applications of classical interpolation in system theory, specially in signal processing, have not been discussed [6][17].

ACKNOWLEDGEMENT Professor Jan Willems has been one of tile greatest leaders in system theory for many years, not only in terms of his numerous outstanding and original achievements, but also in terms of his attractive humanity. It is the author's great pleasure to be able to contribute to his birthday present in such a novel way. The author is grateful to Profs. Schumacher and Nijmeijer for giving him the possibility of making a contribution to this important volume.

273

REFERENCES L.V. Ahifors, Complex Analysis,.McGraw-Hill, 1979. J. A. Ball, "Nevanlinna-Pick interpolation: Generalizations and applications," Proc. of Special Year in Operator Theory, Indiana U n i v e r s y i t y 1985-1986. [3] J . A . Ball and J. W. Helton, "A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory, "J. Operator Theory, vol.9, pp. 107-142, 1983 [4] B. C. Chang and J. B. Pearson, "Optimal disturbance reduction in linear multivariable systems, " IEEE Trans. Automat. Contr., vol AC-29, pp. 880-887, 1984. [5] P. Delsarte, Y. Genin and Y. Kamp, "The Nevanlinna-Pick problem for matrix-valued functions," SIAM J. of Appl. Math., vol. 36, pp. 47-61, 1979. [6] P. Delsarte, Y. Genin and Y. Kamp, "On the role of the Nevanlinna-Pick problem in circuit and system theory," Circuit Th. and Appl., vol.9, pp.177-187, 1981. [7] P. Dewilde and H. Dym, "Lossless chain scattering matrices and optimal linear prediction : The vector case," Circuit Theory Appl., vol. 9, pp. 135-175, 1981. [8] J. C. Doyle, "Synthesis of robust controllers and filters, " Proc. IEEE Conf. Decision and Control, San Antonio, pp. 109-114, 1983. 191 It. Dym, "J-contractive matrix functions, reproducing kernel Hilbert spaces and interpolation,' Monograph, Dept. of Theoretical Math., The Weizmann Inst. Science, 1988. [10] R. M. Fano, "Theoretical limitations on the broadband matching of arbitary impedances," J. Franklin Inst., vol. 249, pp. 57-83, 1960. [11] I. P. Fedcina, "Solvability criteria of the Nevanlinna-Pick tangent problem," Mat. lssled. Kinshinev (in Russian) yp : 4, pp. 213-227, 1972. [12] B. A. Francis and G. Zames, "On H"°-optimal sensitivity theory for SISO feedback systems," IEEE Trans. Automat. Contr., vol. AC-29, pp. 880887, 1984. 113] B. A. Francis, J. W. Heiton and G. Zames, "II °~ optimal feedback controllers for linear multivariable systems, " IEEE Trans. Automat. Contr., vol. AC-29, pp. 888-900, 1984. [14] B. A. Francis, A Course in H 0° Control Theory, Springer Verlag, New York, 1987. [15] J. S. Freudenberg and D. P. Looze, Frequency Domain Properties oj Scalar and Multivariable Feedback Systems, Springer, New york 1988. [161 Y. Genin, P. van Dooren and T. Kailath, "On ~.-lossless transfer functions and related questions," Linear Algebra and Appl., vol. 50 pp. 251-275, 1983. [17] T. T. Georgiou and P.P. Khargonekar, "Spectral factorization and Nevanlinna-Pick interpolation," SIAM J. of Control & Optimiz., voi.25, pp.754-766, 1987.

[I] 12]

274 [18] J. W. Helton, "Broadbanding : Gain equalization directly from data," IEEE Trans. Circuit and Systems, vol. CAS-28, pp. 1125-1137, 1981. [19] J. W. Helton, "Non-Euclidean functional analysis and electronics, " Bull. Amer. Math. Soc., vol. 7, pp. 1-64, 1982. [20] F. ltakura and S. Saito, "Digital filtering techniques for speech analysis and synthesis," in Proc. 7th Int. Cong., Acoust., Budapest, Paper 25-c-1, pp. 261-264, 1971. [21] V. E. Katsnel'son "Methods of J-theory in continuous interpolation problems of analysis," Harihov, translated by T. Ando, 1982. [22] R. Kawatani and H. Kimura, "Synthesis of reduced-order H °° controller," to appear in Int. J. Control. [23] P.P. Khargonekar a,ld A. Tannenbaum, "Non-Euclidian metrics and the robust stabilization of systems with parameter uncertainty," IEEE Trans. Automat. Control, vol. AC-30, pp.1005-1013, 1985. 124] It. Kimura, "Robust stabilizability for a class of transfer functions, " IEEE Trans. Automat. Contr., vol. AC-29, pp. 788-793, 1984. [251 tI. Kimura, "Directional interpolation approach to H'~-optimization and robust stabilization," ibid., vol. AC-32, pp. 1085-1093, 1987. [26] H. Kimura, "Directional interpolation in the state space," Systems and Control Letters, vol. I0, pp. 317-324, 1988. [27] H. Kimura, "Conjugation, interpolation and model-matching in H'*," lilt. J. Control, vol. 49, pp. 269-307, 1989. [28] H. Kimura and R. Kawatani, "Synthesis of H "° controller based on conjugation, " Proc. IEEE Conf. on Decision and Control, Austin, pp. 7-13, 1988. [29] H. Kimura, "Conjugation of Hamilton systems and model-matching in H°°, '' under preparation. [30] D. J. Limebeer and B. D. O. Anderson,"An interpolation theory approach to H °° controller degree bounds," Linear Algebra and its Appl., vol. 98, pp.347-386, 1988. [31] K. Z. Liu and T. Mita, "Conjugation and H °° control of discrete-time systems, " to appear in Int. J. Control. [32] A. G. J. MacFarlane (ed.), Frequency-Response Methods in Control Systems, IEEE Press, NY., 1979. [33] D. Q. Mayne, "The design of linear multivariable systems," Automatica, vol. 9, pp. 201-207, 1973. [34] R. Nevanlinna, "0ber beschr~nkte Funktionen die in gegebencn Punkten vorgeschreibene Funktionswerte bewirkt werden," Ann. Acad. Sci. Fenn., Ser A, vol. 13, pp. 1-71, 1919. [35] G. Pick, "f.)ber die beschr~inkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden," Math. Ann.,voi. 77, pp. 7-23, 1916. [36] V. P. Potapov, "The multiplicative structure of J-contractive matrix functions, " Amer. Math. Soc. Transl., vol. 15, pp. 131-243, 1960.

275

[37] A. C. M. Ran, "State space formulas for a model matching problem, " Systems and Control Letters, vol. 12, pp. 17-21, 1989. [38J H. H. Rosenbrock, Computer-Aided Control System Design, A c a d e m i c Press, 1974. [39] D.Sarason, "Generalized interpolation in H "~, " Trans. Amer. Math. Soc., vol. 127, pp. 180-203, 1967. [40] I. Sehur, "Uber die Potenzreihen, die im Inneren des Einheitskreises beschrankt sind, " 1; 2, J. Reine Angew. Math., vol. 147, pp. 205-232, 1917; vol. 148, pp. 122-145, 1918. [41] A. Tannenbaum,"Modified Nevanlinna-Pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor," Int. J. Control, vol. 36, pp. 331-336, 1982. [42] M. Vidyasagar and H. Kimura, "Robust controllers for uncertain linear multivariable systems, " Automatica, vol. 22, pp. 85-94, 1986. [43] D. C. Youla, "A new theory of broadband matching, " IEEE Trans. on Circuit Th., vol. CT-I1, pp. 30-50, 1964. [44] D. C. Youla and M. Saito, "Interpolation with positive real functions," J of Franklin Inst., vol. 284, pp.77-108, 1967. [45] G. Zames, "Feedback and optimal sensitivity; model r e f e r e n c e transformations, multiplicative seminorms, and approximate inverses, " IEEE Trans. Automat. Control, vol. AC-23, pp. 301-320, 1981. [46] G. Zames and B. A. Francis, "Feedback, minimax sensitivity and optimal robustness," IEEE Trans. Automat. Contr., vol. AC-28, pp.585-601, 1983.

Generalized State-Space Systems and Proper Stable Matrix Fractions V. Ku(:era Czechoslovak Academy of Sciences Institute of Information Theory and Automation 182 08 Prague 8, Czechoslovakia

Abstract The

concept

applied class

here of

to

systems

state-space basic

properness

the

controllers stable

significance of

of

that will of specific

The

will

feedback. make be this

In

the

For of

a

all

closed

characterized result

control

will

systems.

The

context

will key

be

are

here

system in

then

natural

generalized and

their

of

internal

is

that

generalized

generalized loop

most

is

In detail.

addressed given

fractlon

defined

properties

be discussed is

matrix

systems.

thls

systems

which

family

rational

linear

consider These

problem by

system

stable

study

and stability main

linear

design

to

proper

reviewed.

stabilization

and

the

systems.

properties

The

of

state-space

state-space

linear

internally

proper

parametric be

of

illustrated

form. on

The the

277

H i s t o r i c a l Backgvo~nd The

first

controllers the

attempts

that

early

seventies.

single-lnput all stable

1975;

(1976}

obtained

controllers

a characterization

way

to express

these

was e x p r e s s e d c0-workers algebraic

in terms

setting

solved

of

systems

feedback

traced

the

problem

The

Jabr

matrix

solutions

for

Bongiorno

stabilizing

parameter.

of

of

in K u d e r a

and

of c o n t i n u o u s - t i m e

to

through

extension

appeared

Youla,

back

systems

equation.

rational all

led

and

the

This

Dlophantine

see A n t s a k l i s

This

This

idea was

(1986),

.two fractions the

use

a

of

of

adequate

and

general employed

on the

resulting

by V i d y a s a g a r

fractions are

system

Desoer

more

of fractions

elaborated

polynomial

the

by

course

stable related,

fractions

greatly

the analysis.

the we

properness

but

type

in

requirement

further

replacing These

fractional

studying

ultimate

of

fractions.

problem the

that

the

function

matrix

the

showed

to

transfer

of polynomial

with

fractions.

simplifies

the

formulated

system. and

rational

results,

(1980}

should be m a t c h e d

approach

generalized

are of

the

and

rational

fractions

not

system

impulsive

to

system E~(t}

be

used

this

is

but to

the proper

(1984;

1986}

useful In

in

these

also

with

avoid

both

and stable

to d e l i n e a t e

form.

Llnear

governed

+ Gu(t),

= Hx(t)

systems.

stability

Therefore

by K u ~ e r a

(E,F,G,H)

y(t)

linear

with

State-Space

= Fx(t}

particularly

designed;

in p a r a m e t r i c

Generalized

linear

proved

only

behavlour.

were

all such c o n t r o l l e r s

has

state-space

concerned

unstable

The

be

dlscrete-tlme

multi-output

of a stable

family

can

involved,

To o b t a i n

systems

(1974a)

linear

Independently

in terms

the

system

of a D i o p h a n t i n e

1979).

was an explicit

[1985)

Kudera

to m u l t i - i n p u t

[1974b;

feedback

characterize a given

single-output solutions

this idea

equation

to

stabilize

Systems

by the e q u a t i o n s

t ~ 0

{i}

278 is

called a generalized state-space

called

implicit,

Rosenbrock (1980),

(1974},

Verghese,

matrices n

E,

x n,

n

output

singular,

F,

LOvy and K a l l a t h and

and

x

the

H

p

are

x n,

It Is to be n o t e d

that

implicit

whose

are

treated

systems by

systems, If

the

(I}

is

said

are

obtained

Bernhard

however,

to

be

for

T(s) is the

transfer

The

free

may

exhibit

(sE

-

poles

F) -1

and

From

{sE matrix, the

integrators, equations

-

possibly

and,

improper

the

are

(1988}.

matrices.

rectangular Feedback

for

nonslngular unique

The

the

solutions

p x q

system of

(I}

matrix (2)

of

a regular with

the

associated

system finite

with

function

poles

the

(2}

(I} of

infinite

may

be

any

interconnection

of

or u n s t a b l e .

of

view,

and

conversely,

y

{I).

transfer

differentiators

The

n x n,

- F)-IG

modes

point

{I) a r e s q u a r e

case

associated

The

{1986). size

Input,

matrices

is

u{t}.

t ~ 0

impulsive F} -1 .

the

Campbell

understood.

F

this

of s y s t e m

modelling

(i}

-

= H(sE

modes

and Lewis

Grimm

well

and

x(t},

exponential

of

rational

function

response

and

sE In

F

(1978),

the s y s t e m .

F in

yet

matrix

x(0-}

of

and

s y s t e m s , see

respective

u denotes

and

(1982}

regular.

all

of

state

E

is not

polynomial

{1981)

and

E

seml-state

Verghese

real,

(generalized}

The

such

or

L u e n b e r g e r (1977},

G

x q

and

s y s t e m . Such s y s t e m s a r e als0

descriptor

any

scalors

can

equations

be

(i}

s u c h an I n t e r c o n n e c t i o n . As an e x a m p l e , a p u r e

described

always

by

represent

integrator

I can

be d e s c r i b e d by {I} w i t h E = i, a pure d l f f e r e n t i a t o r

and a s c a l e r

k

D

The o r d e r (i.e.,

of

G = i,

H = i

by

by E = O,

elements

F = O,

system

F = 1, {I)

integrators

G = -i, but

H = k.

Is

n

and

dlfferentiators)

the

number o f

dynamical

may be

lower.

279 The n u m b e r

of

of

(sE -

the

dynamical

integrators

F} - 1

given

is e q u a l

to

the

E.

For

and s h o w n

In

mode a n d n o condition

by

elements

by r a n k

deg

whose

total

Flg. I

the

1 has

det of

1

that

has

s

system

[sE -

(1}

s

total

poles

number

of

independent -1 F) glven

{sE by

having

one

ls

exponential

a joint

initial

xl=

F

Y

than d y n a m i c a l

elements

if

G] = n

and

ls

[g

G] = n .

observable

rank complex

finite

are

of

x2(O-)

less modes

rank

for every

while

that

s y s t e m {1) I s r e a c h a b l e

complex

We s a y t h a t

poles

xz

rank for every

The

described

two

Note

F).

the

D.

A system which

We s a y

-

of

conditions

the

order

u

Fig.

number

(sE

system

mode.

and

the

initial

number

example,

impulsive

for

equals

if

[ s E - F] H

=

n

and rank

['1 H

= n.

A dynamical i n t e r p r e t a t i o n of t h e s e n o t i o n s can be found In Cobb {1984).

A generalized the

form

reachable part,

that but

and

function

the of

a

Kalman

displays

decomposition the

unobservable unreachable regular

reachable part, and

system

the

will and

observable

unobservable (I)

bring

depends

system

observable but

part.

only

on

(i}

part,

to the

unreachable The

the

transfer reachable

280 and

observable

reachable

part.

and

transfer

observable

only

are

the

regular

completely

systems

that

characterized

are

by

their

function.

On

the

there

other

exists

hand,

a

realizations their

Hence

given

realization

there

smallest

is

For

rational

(E,F,G,H)

some

size

observable.

any

n,

in

such

see

Conte

x

q

matrix

satisfying

which

and

details

p

the

a

(2).

matrices

Among

E and

realization and

T{s},

is

Perdon

all

F have

reachable

{1982}

and

and

Grimm

(1988).

Properness

The

behaviour

considerable

proper

has

no

the

no

if

the

finite

Stability

at

say

t

0

a

and

We s a y

that

{sE a

the

(sE

F) -1

F} - 1

is

proper, {1)

is

right

of

(1)

is

system

-

closed

-~ co i s

system -

regular

matrix

in

t

regular

matrix

rational

poles

=

that

rational

poles.

stable

has

(1)

We

if

lnfinlte

(internally) I. e. ,

system

importance.

(internally) i.e.,

of

and

is

stable,

half-plane

Re s z O. This (1984).

definition The

comprises

no

condition free

free

as

The internal

external

a

and

for

x(O-).

The

system

which

is g i v e n

simply

for

are

visualized

1 0

to

not

the

to the

reflect be

strictly

viewed the

by

an

properness

An

internally

example

of

proper

by

F =

1 0

,

in

initial

tends

as

(2}.

(I)

standard:

to

amounts

[!00] [ 1i] [i] O 0

system

(1}

stability

but

Ku6era

stability

function

externally

by

every

is

system

and

and

transfer

0

and

system

latter

is

=

stability

properness

the

introduced of a p r o p e r

a stable

properness

system

t

of

every

of

the

at

of

of

was

t > 0

t > 0

the

observer.

stable

modes

notions

from

regular

E =

x(t),

of

x(t),

definition

properties

stability

and

The

t -* co

distinguished

and

impulsive

above

properness

response

x(O-).

response

origin

of

Fig.

2.

G =

,

Observe

-

that

H =

[1

0 0l

,

the

system

has

one

281

unreachable

mode

at

U

Fig.

2

s = 0

=

X2

X3

An e x t e r n a l l y

but

proper The

properness

effectively

It

and

is

not

and

well

stable

known

with

the

every real

rational

a

field matrix

where

Al(S)

and

Bl(S)

that

(I)

and

rational

system

is

stable

functions

field

rational

=

s = w.

and stable

system

proper

whose

T{s)

Tls}

at

Fractions

of

of

ring of

proper

Matrix

means

form

isomorphic

internally

stability

by

mode

X 1[ = y

and Stable

studied

fractions. proper

and one u n o b s e r v a b l e

of

functions.

which

are

fractions

is

As

can be f a c t o r i z e d

most matrix

a

result,

as

{s} BI(S)

are proper

(31

stable

rational

matrices,

and

also as T(s) where

A2(s)

well. one,

and

B2(s)

of

proper

We speak (4)

is a right

The units sometimes

and

a

is

b_~piroper

proper

{4)

stable

matrix

rational

fractions:

matrices

(3) is a

as

left

of p r o p e r and

stable

rational

bistable

stable

rational

proper

and

matrix stable

functions

rational whose

are

functions. inverse

exists

be

termed

will

and bistable. Al(s)

the

a sense

proper

A21(s)

one.

biproper

also

When (over

are

stable

of the rlng

called

Similarly

= B2(s}

ring unique

Bl{s)

and

proper

stable

and

B1(s)

of proper

(Vidyasagar,

A~(s),

B~(s)

rational

in

stable

are

(3)

1985}.

relatively functions),

More

two pairs

matrices

are

rational

precisely,

of r e l a t i v e l y

such that

left

prime

they

are

in

if

Al(S),

left

prime,

282

T{s)

= A~I(s)

BI(S)

= A~-I(s)

B~(s)

then A~{s}

for

a

blproper

result

holds

right

and

bistable

when

B~(s)

Al{S} ,

A2(s}

rational and

= Ul{S} B l ( S )

matrix

B2(s)

In

Ul{S}. {4)

A similar

are

relatively

prime. The

relatively

naturally det

= UI{S}

closely

A2(s)

rational

prime

matrix

related.

up

to

fractions

In particular,

multiplication

(3}

det

by

a

and

Al(S)

biproper

(4)

are

equals and

bistable

function.

Feedback Systems Let

where

us n o w

Ei

p x nI

consider

two

generalized

El~i(t)

= FlXl(t)

Yl(t)

= HlXl(t)

and

F1

are

+ GlUl{t},

n I x nl,

GI

E2~2(t)

= F2x2(t}

Y2(t)

= H2x2(t)

ul(t}

where v1 loop system

they

are

is

n I x q,

+ G2u2(t),

closed (5)

G2 to

(5}

III

is

to

study

loop

and(6}

completely

n2 x p

+ Yl(t)

the

inputs.

regularity, We s h a l l

regular,

characterized Tl(S)

is

= vl(t}

system. are

t z 0

- Y2{t)

and v2 are external i s shown i n F i g . 3.

Our aim i s this

t z 0

= v2(t)

u2(t}

systems

systems

and

w h e r e E 2 and F2 a r e n 2 x n2, q x n 2. We c o n n e c t them a c c o r d i n g

of

state-space

F1)-IG1

H2

Is

(7)

The

suppose

by t h e i r

= HI(SE 1 -

and

resulting

properness

reachable

(6}

that

and s t a b i l i t y the

and o b s e r v a b l e transfer

closed

component so

functions

that

283

and T2(s) The s p e c i a l then

structure

makes

stability

it

= H2{sE 2 - F z ) - l G 2 .

of

possible

the to

by means of

closed

study

Tl(S}

and

loop

its

system

regularity,

YZ

> +

To t h i s

3

I -1 +

+

u2

A prototype

effect

Al{S} ,

we w r i t e

of feedback

TiCs}

are

of proper

relatively

Pl{s),

Bl{S}

are

in

system

terms

is)

Ql{S)

relatively announced

prime.

matrix

=

= B2(s)

relatively

right

stable

T2{s)

are first

= A~l(s)

Bl{S}

terms

Q2{s} result

vl

of

proper

stable

fractions,

B2(s}

where

and

I System 21~

Tl{s} where

System I

,

-

Fig.

matrix

f

~

{7)

properness

Yl

i

by

T2{s).

ul

V2

implied

are

left

while

we w r i t e

A2(s), T2(s)

in

as

= Q2{s)

relatively

{8}

prime

Similarly

fractions

Ql(s)

A21{s)

{s),

left

(9)

prime

right prime. Then b y K u ~ e r a {1984}.

we

while

have

the

P2{s), following

Theorem 1. Let {5) and {6) be regular, reachable and observable systems

giving

respectively. {7}

is

rise Then t h e

regular,

to

the

closed

proper

and

Al(S}

P2(s}

transfer

functions

loop system stable

if

defined

and

only

(8) by (5}, if

the

and

{9),

{6} a n d rational

matrix

ts

biproper

and bistable

or,

+ Bl(s)

Q2{s)

equivalently,

(10) the

ratlonal

matrix

284

Pl(S)

A2(s)

+ Ql(S} B2(s}

i s b l p r o p e r and b i s t a b l e . P r o o f : To p r o v e t h e n e c e s s i t y ,

let

regular.

function

We c o n s i d e r

FA2 ( s } ] Tll{S}=[B2(s}

the transfer

the

(11) closed

loop

rP~ (

lAl{S)P2(s)

+ BI{S}Q2(s)

l-1

be

Pl{S)]

[PI{S)A2 (s} + q l ( S l B 2 ( s } ] - l [ - q l ( s )

which relates the inputs v l , v 2 and t h e o u t p u t s ul,Y 1 closed loop system, along with the transfer function

122{s)=LQ2(s)

system

[AIIS)

of the

BI(S)]

which relates the inputs Vl, v 2 and the outputs u 2 ' Y2 of t h e c l o s e d l o o p s y s t e m . I f t h e c l o s e d l o o p s y s t e m i s p r o p e r and stable then T l l ( S } and T22{s} are proper stable rational matrices. Since Pl{S), Ql(s} are relatively l e f t p r i m e and A 2 ( s ) , B2(s} are relatively r i g h t p r i m e , no c a n c e l l a t i o n s are possible in forming Tll(S} and [ P l { S ) A 2 ( s ) + Q l ( s } B 2 ( s ) ] -1 is p r o p e r and s t a b l e . Since Al(S), Bl(S) are relatively left prime and P2(s), Q2(s} are relatively right prime, no cancellations a r e p o s s i b l e when f o r m i n g T 2 2 ( s ) , either, and llence the [ A l ( S ) P 2 ( s ) + Bl(S} Q 2 ( s } ] - I i s p r o p e r and s t a b l e , m a t r i c e s (10) and (11) a r e b i p r o p e r and b i s t a b l e . To p r o v e t h e s u f f i c i e n c y , we r e p r e s e n t the dynamical action o f s y s t e m (6) on s y s t e m (5) t h r o u g h t h e a c t i o n o f t h e s t a t i c state feedback

u2{t } upon t h e e x t e n d e d

[:: When we d e n o t e

=

HI

x2(t

+ LVl( t

system

o {t 0 ~2(t}] = [:1 F2] [:12(t}'] + [:1 G2]

Lu2(tl}]ru
285

the

regularity,

properness

system are

coded

in

As

and

(6)

stable

(5}

rational

the

and { H a u t u s

B2siS)

FI}-IGI

{6}

and

(sE 2 - F2}-IG 2 = Q2s(S)

P21(s)

closed

loop

F - GK.

there

Q2s(S}

A21is}

exist

such

proper

that

(12)

1978}

= HIB2s(S) ,

are

the

sE -

systems

= B2siS)

and Heymann,

and

of

matrix

observable

matrices

B2is) As ( 5 )

stability

polynomial

are

(sE I -

and

reachable

q2{s}

= H2Q2s(S}.

systems,

the

proper

(13) stable

rational

matrices

P2{s) are r e l a t i v e l y

right

'

prime,

BsiS ) where

Ais},

are

Bis)

Bs{S)

=

0

Q2s(S)

(141

llence we can w r l t e

=

~sls )

relatively

right

prime

polynomial

matrices

and

Bs is is

column

while di,d 2 It

reduced

D(s) dp+q follows

is

with a

column

degrees

polynomial

and

D-iis)

from

i12)

dl,d 2 ..... dp+q,

matrlx

having

column

say, degrees

is stable.

through

(14)

that

isE - F} B S is) = GA(s}. By s u b t r a c t i n g

GKBs(S},

(sE - F - GK}

we o b t a i n Bs(S)

= G [A(s)

- KBs(S)].

ii6)

286 This

relation

A[s)

-

KB is).

Now

let

enable

either

matrix, from

blproper

(i0)

say

(9)

that The

or

sE -

F -

GK

through

(ii)

be

or

a

biproper

U2(s},

and

bistable

respectively.

Then

It

that

AI(S

-B2(s)

and b i s t a b l e .

A(s)

-

KBs(S)

relation

-

is

KBs(S)

=

U2(s)

we o b t a i n

02,s:]

_B2(s) and

~

P2(s bistable.

gives =

can

=

biproper

(15)

(16)

P2(s

Applying (18),

- KBs(S}

factorizatlon

A(s) and

study

01,s:][A2,s,°2,s:] [o 0]

A(s) so

to

Ul(S)

(8) a n d

BI(s} is

us

S

rational follows

will

[A(s)

be

- K§s(S)]

rewritten

(17)

D-I(s}

in

terms

of

polynomial

matrices, (sE We f i r s t

-

F -

observe

GK) B {s}

that

A{s}

X(s) is a r a t i o n a l

= G [A{s}

S

= B

solution

S

-

KB i s } s

is

(s)

[A(s)

- KB

of the

over

rank

the

(sE -

field F -

KB { s ) ] .

(18)

S

nonsingular. S

Is)]

Then

-1

equation

{sE - F - GK} Hence,

-

of r a t i o n a l

GK} = r a n k

X(s}

= G.

functions,

[sE -

F -

GK

G]

In = rank

[sE - F

G]

= rank

[sE - F

G]

+n 0 l z -K Ip+q

]

=n I +n 2 so

that

closed

sE loop

-

F

system

-

fiK

is

is r e g u l a r .

nonsingular.

This

implies

that

the

287 Then

(18)

can

be

(sE - F As

(5}

and

and G

(6) are

right a result,

s

where

c

ls

A(s) deg

so

-

a nonzero

det

Since

[A(s}

-

det

- KB

KB ( s )

is

(sE-F

S

reachable

-GK)

systems

(19)

= deg

D(s)

{5)

Ku~era

and

no i m p u l s i v e

modes

A(s)

- KB i s ) s and the c l o s e d

To in Fig.

Zagalak

Is

4

for

system

Theorem

internal v

Flg.

subsystem

4

z

An

is a s c a l o r

KB

S

is)

are and

ilK) by

(17),

(19)

yields

= rank

E,

that

the

L

so

det

we

closed

proper. (sE

-

is stable.

F -

loop

Furthermore, GK)

is

check

the

of

described

the

by

has

since

stable

and

feedback stability.

x!

+

system

by

i19)

///

properness

example

F -

-

(s}

GK

(6)

therefore

I,

-

s

F -

•

and

bistable,

loop

illustrate

is

and Ais}

(sE

-

|

i1988},

and

A{s)

sE

(sE -F -GK) det D

biproper, det

matrices

Consequently,

det

= c

p+q F. d t=1 see

matrices

constant.

P+q =Ed i=l For

the

= c det

is)]

KBs(S)]-i

the As

are

real

-

prime.

KBsls}]

[A(s)

[A(s)

systems,

left

prime,

det

form

= Bs(S)

reachable

relatively

(s).

the

GK)-IG

-

are

relatively As

given

feedback

system

system The

shown first

288 E1

and

the

second

=

0,

One I s

F1

=

a pure

1.

G1

=

-1,

H1

dlfferentlator

=

1

given

by

Hence AI(S)

= A2(s)

= 1,

Pl(S ) = P2(s ) _ for

an arbitrary

positive

BI{S)

1

s+a

real

= B2(s)

= 1

QI(S ) = Q2(s ) =

'

constant

s

s+~

a and

A l ( s ) P 2 ( s ) + B l ( s } Q 2 ( s } = ss ++al a

biproper

proper

and

bistable

rational

function,

When

feedback

subsystem second

(5}

one

control

(6}

is

a

closed

the

loop

admissible. described Theorem

to

be

The

any

{5}

be

a

gives

rlse

to

the proper

PI(s) is

transfer stable

is

Pl-l(s}

while

that and

the

the

the

overall requires

stable,

thus

modes.

that

renders

and

the

stable

will

controllers

resulting be

will

called now

be

1986). reachable function

rational

+ Ql{S)

nonslngular.

so

plant

One u s u a l l y

proper

admissible (1984;

regular,

A2(s)

found

exponential

purposes,

the

manner.

proper

of

Ku~era

Let

be

controller

family

control

called

internally

regular,

following

Pl(S}

and

and unstable

system

for

a desired

system

2.

that

given

In

plant,

arbitrary

used

controller

behaves

impulsive

Given

such

system

Controllers

are

usually

control

avoiding

be an

systems

is

system the

that

the

and stable.

Admissible

that

ttence

B2(s)

and (8}.

solution = Iq

observable Let

plant

Pl(S), of

the

Ql(s)

equation (20)

Then Ql{S}

(21)

289 range

all

over

blproper

in (20) or

Ul(S}

U2(s)

and

bistable

rational

matrlces,

say

in (22}. However,

P l ( S ) = Ul(S) P l ( S ) ,

Ql(S) = Ul(S) Q l ( S ) ,

Pz(S}

Q2(s)

and

define just and (23),

= Pz(s)

U2(s),

alternative

Pl-l(s}

relatively

ql(S)

= q2(s)

U2(s),

prime matrix

= ~1-1(s)

fractions

(21)

Ql(S)

and

Qz(S) P z - l ( s )

///

= Q2(s) P 2 - 1 ( s ) .

If P ~ ( s ) , O~(s) is a particular s o l u t i o n of (20) and P~(s), Q~(s) is a particular s o l u t i o n of (22) t h e n t h e g e n e r a l s o l u t i o n s of t h e s e e q u a t i o n s r e a d (Ku5era, 1979) Pl(S)

= Pi(s)

+ Bl(s)

Vl(s)

Ql(S}

= Q~(s)

- AI(S)

Vl(S)

and P 2 ( s ) = P~(s) + V2(s) B2(s) Q2(s) = Q~(s) - V2(s) A2(s) where Vl(S) and V2(s) range over proper stable rational m a t r i c e s of a p p r o p r i a t e s i z e . T h e r e f o r e we can c h a r a c t e r i z e the family

of

transfer

c o n t r o l l e r s for

the

functions given

T2(s)

plant

s t a b l e r a t i o n a l ) parameter matrix

T2(s) = [P~(s) + Bl(s) Vl(s)]-I = [Q~(s) - v2{s)

A2(s)]

(5)

of

in

Vl(S)

terms or

all of

V2{s)

admissible one

(proper

as f o l l o w s

[Q~(s) - Al(s) Vl(s)] [F~(s) + V2{s)

B2(s)] -I.

(24)

Finally we note that Theorem 2 specifies just the transfer functions of all admissible controllers and not their generalized state-space realizations. In fact any reachable and observable reallzatlon (6} of the transfer function (21) or (23) Is an admissible controller.

290 is the transfer function of an admissible controller plant

(5}. ALso let

P2(s), q2(s}

(6} for the

be an arbitrary proper

stable

rational solution of the equation AI{S) P2{s} + BI(S) Q2(s) = Ip such that

P2(s} is nonslngular. Q2 ( s )

is

the

transfer

plant

(5).

the

(6}

Since

ring

of

matrices.

stable Ql(S)

We s h a l l

(5)

are

the

end,

of

generated

in this

of

exists

P~(s),

all

prime

be

(over

(20}

stable

a solution

Q~(s}

the

way.

equation

proper

for

admissible

right

functions}, class

there

let

(6)

functions relatively

rational in

show t h a t

To t h i s

are

controller

with

any

has

rational Pl(S}

solution

of

Then

also

a solution

Since

both

real

so

are

transfer

B2(s7

Pl(s),

(23}

an a d m i s s i b l e

the

and

Then

P2 l ( s }

the plant

proper

nonsingular.

is

for

A2(s}

a solution

(20).

of

Furthermore,

controllers Proof:

function

(22)

Pl(s}

= P~(s}

+ Bl(s)

Ql(s}

= Q~(s)

- AI(S} V

of

Al{s}

(207

and

such

that

nonsingular.

Put

for

any real

A2(s)

the

are

constant

V

constant

q x p matrix

nonslngular,

matrices

there

A l ( S o}

exists

and

V. a

A2(So7

V = All(s o } Qi(So}. Then

Ql(So } = 0

Pl(S)

is

so

that

nonslngular.

established

along

Since

the

the

Pl(So}

The

ldentlty

matrices

from

Theorem

1 that

Pl(S)

and

P2(s}

nonsingular

plant.

function We

admissible or

of

an

show

controllers

whole famlly (207

shall

Is generated

P2(s},

Q2(s)

(Sol of

by

(201.

equation

(22)

Hence can

be

same l i n e s .

follows transfer

=

solvability are

any

biproper

solution

of

defines

via

admissible that

are

the

of

transfer

equation

in

or

for

this

way. Ql{s7

whose

it

(227

with

{23} the

functions

Pl(S}, (22}

bistable, or

(21)

controller

generated

by s o l u t i o n s

and (20}

the

given of

Indeed

all the

of equation

right-hand

sides

291 A Structured The plant

(5)

to be

Plant

controlled

is

often

structured

in

the

following way:

El~l(S) ~ FlXl(t} + GlUl(t) Yl(t} = HlXl(t) y~(t} = HiXl(t} where

E1

and

F1

are

n1 x n1

G I' is n I x q' and H1 interpret x I as the state, the disturbance controlled

input,

output.

Yl

The

is uI

t

+ O~u~(t),

(25)

matrices

while

G I is

n 1 x q,

p x nl, Hi is p' x n I. We as the control input, u' I as

as the measured output

closed

~o

loop

system

and

defined

y~ as the

by

(25)

and

(6),(7) Is then shown in Fig. 5.

l

Yl!

U ~

Y" Co-:rolle+

Fig.

We s h a l l this

closed

5

,

V I

--

+

+

A feedback system with structured

analyze

the

loop system.

lnternal

properness

The t r a n s f e r

and

function

of

plant

stability (25)

is

of given

by

Tl ( s )

=

Let t h e p r o p e r stable partitioned as f o l l o w s [Alll (s) TI(s) = [Al21(s )

Hi

(sE 1 - F 1) matrix

0 A122{s )

G1

fractions

ci]. defined

]-l[Blll(s) [B 21(s)

in

Bll2(s} ]

Bi22(s} j

{8)

be

292

-1 B221{s}

B222(s)

{26}

A222(s

where

Alll(s) is p x p, A122{s} I s p' x p' and A211(s) is q x q, A2 22 i s ) is q' x q ' . The z e r o blocks c a n a l w a y s be a c h i e v e d by b l p r o p e r and b i s t a b l e transformations. When the transfer function of (6), T2(s)

H2lsE - F 2 } - l G 2 ,

=

is augmented compatibly with Tl(S), f r a c t i o n s d e f i n e d I n {9) t a k e t h e f o r m

ITs(:, :]=

the

proper

stable

matrix

i:] [°C' ;] -1

-1 =

and one h a s t h e f o l l o w i n g

Corollary.

Let

(El,

F1,

0

corollary

[G 1 Gi'],

(27)

Iq,

0

o f Theorem 1.

FH1

IH~|)and

(E2,

F2,

G2,

H2)

give

rise

to

LJ1

be

regular,

reachable

and

observable

systems

that

the transfer functions (8) and (9) p a r t i t i o n e d as shown i n (26) and ( 2 7 ) . Then t h e c l o s e d l o o p s y s t e m {25), (6) and (7) is r e g u l a r , p r o p e r and s t a b l e i f and o n l y i f t h e r a t i o n a l m a t r i c e s Alll{s}

P2(s)

+

Bill{s}

Q2{s),

A122(s}

the are btproper and b l s t a b l e or, equivalently, matrices Pl(S) A211ls) + Ql(S) B21I{s}, A222 {s) are biproper To transfer

and b i s t a b l e .

illustrate function

this

rational

/// result,

consider

a

plant

(25}

with

293

i Tl(s) = a - - ~ where

a(s),

relatively

prime

be a n y

d{s}

gl(s),

be a n y

of

= bl(S)

greatest

e{s)

form

rational

divisor

bls)

c(

e(s)

and

and stable

common

= al(s)

g2(s)

c(s},

proper

greatest

a(s} Let

b(s},

b(s) d(s)

a

quintuple

functions,

a(s),

b(s),

gl{s),

c(s)

common d i v i s o r

Let

c(s)

gl(s}

and write

= Cl(S)

of

of

gl(s)-

a(s),

b{s},

d{s)

and w r i t e

a(s) = a2(s) g2(s), Let

g{s)

a{s)

b(s) = b2(s) g2(s),

d(s) = d2(s) g2(s).

be any greatest common divisor of

and

b(s) e(s) - c { s )

d(s)

and write

a(s) = f(s) g(s).

Then Alll(s} A211(s) so

that

(1)

a{s)

b(s}

an

= a2(s},

-

b{s) c(s}

bl(S),

=

B211(s)

admissible

and

e(s}

Blll(s}

= al(s),

= b2(s),

controller

are d(s)

the

of

if

and

gl{s)

= f(s)

exists

prime ring

= f{s)

A222{s}

(6}

relatively in

A122{s)

and

(il)

proper

g2{s) only

a(s} stable

if

divides rational

functions.

Model M a t c h i n g We s h a l l the

analysis

described

now i l l u s t r a t e and

synthesis

= FlXl(t)

y(t) z(t)

p x n1 is

of

use

of Theorem

control

1 and Theorem 2 in

systems.

Consider

a plant

by El~l(t)

where

the

E1 and F 1 and

reachable

Hi and

t

~ 0

(28)

= HlXl(t) = H~xl(t)

are is

+ GlU(t),

n1 x n1

m x n 1. observable.

It

matrices, Is The

assumed problem

G1

is

that of

n 1 × q, (El,

F1,

Hi

is

r ,11

GI,[t{~])

model matching

with

294 internal

properness

admissible

where

controller

the

= F2x2{t}

u(t)

= H2x2(t)

and

E2

F2

and n2 x r v to z from 6

of

equals

are

H2 in a

finding

in

+ G2Y{t}

+ G~v(t),

t ~ 0 (29}

matrices,

G2

is

n 2 x p,

q x n2, such that the transfer closed loop system {28), {29}

given

an

form

n2 x n2

Is the

consists

stability

E2~2(t)

is

Fig.

and

(proper

stable

rational)

G~

function s h o w n In

model

matrix

M{s).

v

u

z

IControl ler

r Y

Fig.

We d e f i n e

6

the

Block

diagram

matrix

H~(sg I -

matrices

A2(s}, such

Similarly

B2(s) that

= B2(s)

A2 l { s )

F1)-IG 1 = C2{s)

A2 l ( s )

and

are

C2{s}

A2(s),

are

C2(s

proper relatively

(30} stable right

rational prime.

we d e f i n e -t{2(sE 2 - F2)-IG 2 = Pl-l(s) t{2(sE 2 - F2)-IG~

for

system

fractions

HI(SE 1 - F1)-IG1

where

of model matching

some p r o p e r

T h e o r e m 3.

stable

The m o d e l

rational matching

(i)

A2(s)

and

C2(s)

is a left d i v i s o r

B2(s)

are

(31}

= pl-l(s)

Rl(s)

matrices

Pl(S),

ql{s)

and

solvable

If

and only

problem

(ii)

Ql{s)

is

relatively of

M(s}

right

prime

Rl(S}. if

295

over the ring Any

of proper

and

all

functions iS1} stable rational Pits}

such that Proof:

rational

functions.

controllers

possess

where Pits), Ql(s) and RI(S} solution triple of the equations A2(s)

Pl(S}

Let

stable

admissible

the form

transfer any

+ Ql(S}

B2ts)

= Iq

(32)

C2(s)

RltS)

= M{s)

(B3)

is nonslngular.

the

closed

loop

system

t28},

t29}

be

proper

stable. By T h e o r e m 1, t h e m a t r i x Plts} A2(s) + Ql(S} blproper and b i s t a b l e . Hence i t ) f o l l o w s . Let the function r e l a t i n g

v

and

H{s) = C2{s) and [Pits} This implies

A2(s} (ll}.

Conversely

stable

z

(1)

appropriately This

rational

matrix

generalized results (1975),

is

Rlts}

another

taken

such

that

controllers

from

systems

Ku~era the

literature

Among

(1981),

and

stable.

proper

stable

(33}

is

such of a

verified.

t32} a n d (33} model matching.

are

admissible

Consider

the

(1986)

plethora

of

and

when

extends

others,

see

Wolovlch

and K u ~ e r a

to

trasfer-functlon

on m o d e l m a t c h l n g

and Halabre

demonstrate

example.

exist

RltS} satisfying that effects the

these

Output We s h a l l

there

proper

///

In the

systems.

Pernebo

is

with Pl(s) nonstngular then entails the existence

and (29),

all

state-space

available

state-space

and

realized.

theorem

Then

Pl{S), Ql(S} Condition (il)

any

B2(s} I s transfer

H i s ) . Then

B2(s}l-lRl(S}

hold.

Any t r i p l e Pl(s), Ql{S) defines a controller, via By T h e o r e m 2,

equal

and

[P1(s) A2{s} + Ql{s) B 2 ( s ) ] - I R t ( s )

+ Qlts}

let

rational matrices t h a t {32) h o l d s . proper

proper

of ordinary

(1974},

Morse

(1984}.

Regulation use

a plant

of

Theorem

described

by

1 and

Theorem

2 on

296 El~i{t}

where

E1

and

is

p x n 1 and

is

reachable

= FlXl(t)

y(t}

= HlXI(t)

z(t)

= Hlxl{t)

F1

are

Hi

is

t

+ GlU(t),

n1 x n1

~ 0

matrices,

(34}

G1

is

n 1 x q,

H1

m x nl.lt Is assumed that (EI,F1,GI,~HI]}

and

observable.

Consider

a

also

reference

generator E2~2(t}

= F2x2(t)

w{t} where H2

E2 is

and

m x

properness

F2

n 2. and

controller

of

are

The

n 2 x n2,

problem

u(t)

= H3x3{t)

the

loop

system

transfer (34),

of

output

G2

is

n2 × r

regulation tn

with

finding

an

and

internal admissible

form = F3x3(t}

that

(35)

~ 0

matrices,

consists

E3~3(t)

such

t

= H2x2(t)

stability

the

+ G2v{t),

(36)

+ G3Y(t}

+ G~v(t),

t

~ 0 (36)

function

from

v

shown

Fig.

7 is

in

to

w -

z

proper

in

and

the

closed

stable.

w

z

Plant

Fig. 7 Denoting formulation and

unstable

and

x3{O-).

Block diagram for output regulation system e(t)

means modes

= w(t} that for

In particular

e(t},

every

z(t)

the t

z

0

initial z{t)

will

regulation is

to

be

error, free

conditions asymptotically

of

the

above

impulsive

Xl(O-}, follow

x2{O-} w(t}.

297 We d e f i n e

the matrix

fractions (37)

Hi(sE t - FI}-IGI = B2(s) A 2 i(s) H~(sE 1 - F1)-IG 1 = C2{s) A2-t{s} H2(sE 2 - F2j-IG 2 = Fl-l(s} GI(S}

are Fi(s}, Gl(S) are proper stable where A 2 ( s } , B 2 ( s ) , C2(s} rational matrices such that Fl(S), Gl(s) are relatively left prime

and

A2(s),

LC2(s)jare

relatively

right

prime.

Similarly

define

for

some

-H3(sE 3 - F3}-IG 3 = Pl-l(s}

QI(S)

H3(sE 3 - F 3 ) - I G ~ = P l - l ( s )

Rl{s)

proper

rational

stable

matrices

(38}

Pl(S},

Qt(s)

and

Rl(S). Theorem 4. if

The o u t p u t

regulation

problem

is

solvable

{ I)

A2{s)

and

B2(s)

are relatively right prime

{11)

C2{s)

and

Fl(S)

are

over t h e r i n g

of p r o p e r

stable

internally rational

if

and o n l y

skew p r i m e functions.

Any and a l l admissible controllers possess the transfer f u n c t i o n s (38) w h e r e P i { s ) , Q l ( S ) , R l ( S ) along with S2(s) form any p r o p e r s t a b l e rational solution quadruple of the equations

such t h a t PI(S) Proof: Let the

Pl(S)

A2(s} + q l ( S )

B2(s)

= Iq

(39}

C2(s)

RI(s)

FI(S)

= Im

(40)

+ S2(s)

is nonsingular. closed loop system

(34},

(36}

be

proper

and

s t a b l e . By Theorem 1, t h e m a t r i x P l ( S ) A2{s) + Q l ( S ) B2(s} is biproper and b i s t a b l e , ttence ( l ) follows. Let the transfer function f r o m v t o e be p r o p e r and s t a b l e . I t i s g i v e n by

298 e(s)

= { I p - C2(s)

[Pl(s)

A2(s) x

+ Ql{S)

Rllsl}

B2{s)]-I

x

(s} Ol(S) v ( s }

where Xl(S)

= [Pl(S)

I s p r o p e r and s t a b l e . prime, we c o n c l u d e Ip-

C2(s)

matrix

Xi(s),

Y2(s)

A2(s)

Since Fl(s) that Fl(S) i.e.,

there

B2(sll-lRl(S)

and G l ( S ) a r e r e l a t i v e l y Is a right divisor of

exists

a

proper

+ Y2(s}

Fl(S}

(il) u s i n g the t e r m i n o l o g y

Conversely

let

(i)

hold.

Then

of W o l o v i c h there

e(s) = S2(s) view

Pl(s), (88),

of

{89),

Ql(S}

and

a

2,

any

appropriately This to

and

theorem

all

is

effects these

(1977),

Pernebo

(i981)

Although reachable

exist

taken

for

reachable

all

from

stable.

and

(40)

output

their

stable

are

Hence any

define,

via

regulation.

By

admissible

(1986)

the

and

when

Wolovich

transfer

and

which

functions,

impulsive

Pearson

(1974},

Ferreira

(1979),

(1988).

i through

systems,

extends

transfer-function

including W o n h a m and

and V i d y a s a g a r

whose

and observable.

Ku~era

of Theorems

observable systems

and

(89)

the

systems

(1977),

and Francis

by

proper

controllers

state-space

the results

characterized

proper

Pl(S) nonsingular ( W o l o v l c h , 1978) the R l ( s ) , S2(s} such

///

Bengtsson

and

(1978).

v(s)

satisfying

that

of many researchers,

Francis

true

Rl(S}

realized.

generalized

results

{40) and it is

controller

Theorem

rational

= Ip.

rational matrices Pl(s) and Ql(S)'with s u c h t h a t (39) h o l d s . C o n d i t i o n ( i i ) i m p l i e s existence of proper stable rational matrices t h a t (40) I s v e r i f i e d . Then

in

stable

left

such that C2(s} Xl(s}

This proves

+ ql(S)

and

4 are are

these

stated

for

completely results

unstable

modes

held are

299 References

hntsaklls, P.J. (1986). Proper stable transfer matrix factorizations and i n t e r n a l system descriptions. IEEE T r a n s . A u t o m a t . C o n t r . , AC-31, 6 3 4 - 6 3 8 . B e n g t s s o n , G. frequency Bernhard, P. systems.

(1977). Output regulation and i n t e r n a l models; d o m a i n a p p r o a c h . A u t o m a t i c a , 18, 3 3 5 - 3 4 5 . (1982). On s i n g u l a r implicit linear SIAM J . C o n t r . O p t l m l z . , 20, 6 1 2 - 6 3 3 .

Cambell, S. L. (1980}. Singular E q u a t i o n s . P i t m a n , San F r a n c i s c o . Cobb,

Systems

of

dynamical

Differential

J . D. ( 1 9 8 4 ) . C o n t r o l l a b i l i t y , observablllty, and in singular systems. IEEE T r a n s . Automat. Contr., 1076-1082.

Conte, G. and realizations Syst. Contr.

A. Perdon {1982}. Generalized of non-proper rational transfer Lett., 1,4.

A

duality AC-29,

sLate-space functions.

Desoer, C . A . , R. W. L l u , J . M u r r a y and R. S a e k s ( 1 9 8 0 ) . F e e d b a c k system design: The f r a c t i o n a l representation approach to analysis and s y n t h e s i s . IEEE T r a n s . A u t o m a t . C o n t r . , AC-25, 899-412. F r a n c i s , B.A. {1977}. The m u l t i v a r i a b l e from the input-output viewpoint. C o n t r . , AC-22, 3 2 2 - 3 2 8 .

servomechanism problem IEEE T r a n s . Automat.

F r a n c l s , B.A. and H. V l d y a s a g a r ( 1 9 8 3 } . A l g e b r a i c aspects of the regulator p r o b l e m f o r lumped A u t o m a t i c a , 19, 8 7 - 9 0 . Grimm, J. (1988). Realization and s y s t e m s . SIAM J . C o n t r . O p t l m i z . ,

and t o p o l o g l c a l linear systems.

canonicity for 26, 1 8 3 1 - 1 3 4 7 .

implicit

Hautus, M. L. J . and M. Heymann ( 1 9 7 8 ) . Linear feedback, algebraic a p p r o a c h . SIAM J . C o n t r . O p t i m t z . , 16, 8 3 - 1 0 5 . Kudera, V. (1974a). Closed-loop stability of discrete single-variable systems. Kybernetika, 10, 1 4 6 - 1 7 1 .

an

linear

Ku~era, V. ( 1 9 7 4 b } . A l g e b r a i c t h e o r y o f d i s c r e t e optimal control for multlvariable systems. Kybernetika, vols. 10-12, pp. 1-240. Published in instalments.

300 K u ~ e r a , V. ( 1 9 7 5 } . S t a b i l i t y of discrete linear feedback systems. Preprlnts 6 t h IFAC W o r l d C o n g r e s s , Boston, vol. 1, p a p e r 44.1. Ku~era, V. (19791. Discrete Equation Approach. Wiley,

Linear Control: Chlchester.

The

Polynomial

Ku5era, V. ( 1 9 8 4 } . Design of i n t e r n a l l y proper and stable systems. P r e p r l n t s 9th IFAC World Congress, Budapest, vol. V I I I , pp. 94-98. K u ~ e r a , V. ( 1 9 8 6 ) . Internal properness systems. Kybernetika, 22, 1 - 1 8 .

and

stability

K u S e r a , V. a n d P. Z a g a l a k (1988}. feedback for singular systems.

Fundamental Automatlca,

Lewis, F.L. Circuits,

llnear singular 5, 3 - 3 6 .

(1986). Syst.,

A survey of Signal Prec.,

Luenberger, D.G. { 1 9 7 7 } . IEEE T r a n s . A u t o m a t .

Dynamic Contr.,

in

linear

theorem of state 24, 653-6~8. systems.

equations in descriptor AC-22, 312-321.

J.

form.

Malabre, M. a n d V. K u ~ e r a ( 1 9 8 4 } . Infinite structure and exact model matching problem: A geometric approach. IEEE T r a n s . Automat. Contr., AC-29, 266-268. Horse, A.S. (1975}. System tnvariants under feedback and control. Prec. Internat. Symp. M a t h e m a t i c a l System Udlne, Italy. Springer-Verlag, New Y o r k , 6 1 - 7 4 . Pernebo, L. {1981). controllers for Automat. Contr.,

An a l g e b r a i c theory linear multlvariable AC-26, 171-194.

Rosenbrock, H.H.(1974}. Structural systems. Int. J. Control, 20,

properties 191-202.

for the systems.

of

cascade Theory,

design of IEEE T r a n s .

linear

dynamical

Verghese, G. C. (1978). I n f l n l t e - f r e q u e n c y behaviour generalized dynamical systems. Ph. D. Thesis, Dept. E l e c t r i c a l Engineering, Stanford University. Vldyasagar, H. ( 1 9 8 5 } . C o n t r o l S y s t e m S y n t h e s i s : Approach. H.I.T. Press, C a m b r i d g e , HA. Wolovlch, W.A. (1974). Linear Sprlnger-Verlag, New Y o r k .

A Factorizatlon

Multlvarlable

Wolovlch, W.A. (1978}. Skew-prime polynomial Trans. Automat. Contr., AC-23, 880-887.

In of

Systems.

matrices.

IEEE

301

Wolovlch, W.A. and P. Ferrelra {1979). tracking in linear multlvarlable Automat. Contr., AC-24, 460-46~.

Output regulation and systems. IEEE Trans.

Wonham, W.H. and J. B. Pearson (1974}. Regulation and internal stabilization in linear multivarlable systems. SIAH J. Contr. Optimlz., 12, 5 - 1 8 . Y0ula, D.C., H.A. Jabr and J.J. Bongiorno (1976}. Hodern Wiener-Hopf design or optimal controllers, Part II. IEEE Trans. Automat. Contr., AC-21, 319-338.

Scattering Theory, Unitary Dilations and Gaussian Processes S. K. Mitter Department of Electrical Engineering and Computer Science and Laboratory for Information and Decision Systems Massachusetts Institute of Technology, Cambridge, MA 02139, U. S. A. and Scuola Normale Superiore, Pisa, Italy

Y. B. Avniel Department of Electrical and Computer Engineering Drexel University, Philadelphia, PA 19104, U. S. A.

I.

INTRODUCTION

Tile themes o f modelling and representation o f linear deterministic and stochastic systems have dominated much of systems theory in the last thirty years. In the critical period of this development attention was focussed on the r e c o n c i l i a t i o n between the i n p u t - o u t p u t (external) and state space (internal) points of view of systems. A central result in this development is the statement that a minimal (in the sense of dimension) state space realization of linear finitedimensional system is unique (up to isomorphism) and corresponds to one which is both controllable and observable. Ill recent years Jan Willems has forcefully argued that this input-output-state space view is narrow and inadequate to deal with models of dynamical systems arising out of physics, econometrics as well as stochastic processes (notably Markov processes). For one thing, there is no natural identification of what is an input and what is an output in these systems. For another, the need to fix the initial state as an equilibrium state in

(1)

This rcsearch has been supported by the Air Force Office of Scicntific Research under

Grant AFOSR-85-0227 and of the Army Research Office under Grant ARO-DAAL03-86-k-017 through the Center for Intelligent Control Systems.

303 conventional realization theory is unnatural and leads to conceptual difficulties. For a detailed exposition of this work see [231 . The other important theme in systems theory is one of optimization and approximation. Models of systems obtained from physical principles or data are often of high dimension. When these models are used for prediction and feedback control it is necessary to obtain approximate models of much lower order so that the algorithms for prediction and/or control are computationally tractable. There is an important question here as to what is the appropriate representation on which the approximation (reduction) process should be carried out. Zames [24] has argued that this approximation should be done on an inputoutput basis and an internal representation of the approximate input-output map could then be obtained for the purpose of prediction and control. This argument rests on the notion that two systems may be near each other in an input-output sense (for example in L°°-topology) and yet may drastically differ dimensionally in their internal representations. The two processes of approximation and internal representation do not, in general, commute and working with the inputoutput representation for the needs of approximation is a more stable operation. There is an apparent contradiction here since we have just argued that the input-output view so prevalent in the early days of systems theory is not an appropriate one. Fortunately, Scattering Theory as developed by Adamjan and Arov [I] and Lax and Phillips [15] and the theory of Abstract Hankel Operators comes to the rescue here. The work of Adamjan, Arov and Krein (for example, 12]) and Ball and Helton [61 provide a mathematical framework for dealing with representation and approximation issues in a Hilbert space setting in a rigorous manner. In tile systems theory context, Scattering Theory for Gaussian Processes was investigated in the doctoral thesis of Y. Avniel at M.hT. [cf. 5] and Scattering Theory and Approximation of Linear Systems has been investigated by Willems himself in his own framework for dynamical Systems [22]. A state space view of Hankel approximation has been provided by Glover [13] in an important paper. Intimately connected with tile theory of Scattering is the theory of minimal unitary dilations of contraction semigroups contracting strongly to zero. Indeed, the theorem of Nagy asserts that every contraction semigroup contracting strongly to zero has a (unique) minimal unitary dilation. This theorem has a physical interpretation of coupling a dissipative system to a heat bath so that the resulting composite system is conservative [cf. 17]. The dual of this question, namely, how certain observables of an infinite-dimensional conservative system can exhibit dissipative behavior has been investigated by Picci by using the theory of stochastic realization [cf. 20]. This semi-expository paper consists of two parts. In the first part we describe in an essentially self-contained manner the Scattering Theory associated with statio,mry Gaussian processes. This is done in discrete time to avoid certain

304

tcclmical difficulties.

The new contribution in this part of this paper is the result

which states that for completely non-deterministic stationary Gaussian processes the spectral density can be recovered (up to unitary isomorphism) from the Hankel operator induced by the scattering function associated with the process. This result to some extent justifies using this Hankel operator for model reduction in stochastic systems. We then show the relationship of this scattering view t0 the theory of Unitary Dilations and Markovian representations. These latter ideas are all contained in the work of Lindquist and Picci [cf.18] and Foias and Frahz0 [101. The exposition serves to show that if we restrict ourselves to a £2-theory then representation questions for stochastic (and deterministic) systems are nothing else but a version of Scattering Theory "~ la Adamjan, Arov and Krein. It is worth mentioning that the Scattering Function plays an important role in the parametrization of the unit ball of the quotient space L ~ ' / H °* and the corresponding extension problem for 111ankel operators. Tile second part of the paper is concerned with tile theory of minimal unitary dilations of contraction semi-groups, its relation to positive definite functions on a group and the theory of open s y s t e m s [8]. We also present a new construction of a unitary dilation of contraction semi-groups which makes evident the coupling to white noise (heat bath) which is implicit in the construction of the dilation.

2.1

NOTATION Z stands for tile set of integers, ~i(n) for tile indicator function of {0} c Z, C

tile complex numbers, and for a ~ C, ~-i denotes tile complex conjugate of a. matrix

A

(aij) .j=l we denote by A* the Hennitian conjugate of A :

For a

= (bij) j=l

bij = + aji, and by A' its transposition. For a family of subsets {Mj}j of Hilbert space H, we denote by V Mj the smallest closed linear manifold (subspace) that J

includes

each

Mj, and by .AMj the largest subspace contained .t i

(their intersection).

in each of them

I

Mj denotes the closure of Mj in I1. For subspaces M, N, of H,

M 6~ N denotes the orthogonal complement of N in M.

For a countable family {Mj}

of mutually orthogonai subspaces: Mi .L Mj i ~: j, we let Y~ @ Mj be their orthogonal J sum. PM stands for the orthogonal projection of H onto the subspace M. For a bounded linear operator A : H I ~ H 2 of Hilbert space H 1 into H 2, we denote by [A] the matrix of A with respect to specified orthonormal bases in 1I i, Ill2.

305 A* : H l ~

H2

denotes the adjoint of A.

AIM stands for the restriction of A to the

subspace M c H l . B(H l, H2) denotes the Banaeh space of all bounded linear operators from H l into H2 with B(H) = B(H,H). By .0.2(-*0,00; N) we denote the usual Ililbert space of sequences {hJ}7=_,,. with values in (tile Hilbert space) N for which Y Ilhjll2 <**. ~2(0,,,0; N), ~2(-oo, -1; N) are

J

oo

seen naturally as subspaces of .0.2(-,,*,oo; N). L2, L

will denote respectively the

Lebesgue spaces on the circle T = {eix : ~. ~ [-,,*, o0]} (with respect to the normalized d3Lebesgue measure ~ - o f square integrable, essentially bounded complex valued functions).

Each function can be viewed as defined on [-~, hi.

Similarly for the

spaces L2(CI~, L**(C~ of functions f taking values in (C p) for which IIf(+)llcp e Lp, IIf(+)llcp ~ Lo. respectively. Loo(B(cP)) measurable,

B(CI~ valued functions

f

is

defined

analogously

for

weakly

for which ess. sup {11 f(e i~.) II B(C 0) : 3.

+ I-n,n]} <~}. I!~ are tile subspaces of L2 defined by

+ 1 H2 = {f~ L2 :~'~- I f(ei~')e-in~') -/1;

d3- = 0 , n = - l , - 2 .... }

|

112= {f~ L2:~"~" I f(ei~')ein~" -/i:

d3- =0, n = 0 , 1 , 2 .... },

+ and we have tile orthogonal decomposition L2 = 1I2 @ 112-. Each f ~ H 2 Fourier series oo

f(ei~.) - ~ an ein~" 0 generates tile function oo

g(z) = ~ anz" 0

having a

306 belonging to the Hardy class H2 of functions g(z) holomorphie in I z I < I and such that

II g IIH2

= sup [~ o
Ig(rei~.)l

~. J / 2 < , , . .

-g

Moreover, the (a.e. existing) radial limit g(e i~.) of g(z) equals f(e i~.) a.e. and II f I1 L2 + = II g II . The function g(z) is seen as the analytic extension of f ~ H 2 to the unit I-I2 disc I z I < 1 and is denoted by f(z). We identify II 2 and tI 2 and denote them commonly by 1-I2.

Using the conjugation with respect to the unit circle (z ~ z

reflection principles for f ~ II2 c

L2 the function

an analytic extension to I z I > 1 : f ( z

i

1

) by the

T defined by f ( e i~-) = f(e i~.) has

) which we again denote by

f . The space

FI2 = f ~ L2 : f ~ H2 is the space of functions f ~ L2 having an analytic extension to the exterior of the disc I z I < 1 and we have

II f II = sup [ ~ L2 p>l

[

I f(pei~.)12dX] 1/2

f ~ H2 are called conjugate analytic. Analogously for the Banach space L

we have the subspaces H oo

of functions

f ~ L

= H+ ¢.o

c

L oo

having an analytic extension f(z) to I z I < 1 with oo

II f II L.o

= sup I f ( z ) l = l l f l l . Izl
Similarly, for the Ililbert space L2(C o)

we have the subspaces H2(CI~ =

H2(CI~ with the orthogonal decomposition L2(C~ = 1-I2(CI~ @ It2(C~. again Ho,,(B(C p) is defined as the subspaee

In Loo(B(cP),

of functions in L , . ( B ( C p)

negatively indexed (matrix valued) Fourier coefficients vanish.

I-I2(C p ,

For 0 e H

whose (B(C p)

307

the function 0* defined by 0*(e ix) = [0(elX)] * is identified with its analytic extension 0*

=[0

tolzl>l

.

A function f e H is called inner II**(B(CI~ if 0(e ix) is unitary a.e..

if I f(ei~.) I = I a . e . .

Similarly for 0

f e H2 is called outer if V {xnf} = II2 where Z n>0

denotes the function on T defined by z(e i~.) = e ix . For ~p ~ L**(B(Ct~ the Toeplitz operator T~ :H2(C~ ~ H2(CI~ whose matrix is block Toeplitz with respect to the standard basis {eik~-el, elk"e2 .... ' eik~'ep]k ~ 0 • {el, e2 ..... ep] being the standard basis in C p, is defined by T~f = g+(~f) where n+ is the Riesz projection of L2(C p) onto H2(ci~. H~ will denote the Itankel operator [with block Hankel matrix with respect to the standard bases in H2(CP), I-I~(C~I, H~: 1-12(CI~ ~ I-I~(C~ defined by T~f = n.(~f) , ~_ being the Riesz projection of L2(C p) onto H2(Ct~. The con-vention we

e,nploy regarding

a

llenkel operator

as acting from

I~(CI~ into 112(C1~ is

not in accordance with the one employed in systems theory, where t12(C~ into 1-12(CP): ItCf = n+(¢f).

we

act on

It, however, conforms to that employed by

Adamjan-Arov-Krein and enables us to use their results without modification, as well as to refer to them.

2.2

SCATTERING "ITIEORY Let H be a complex separable Hilbert space and let U be a unitary operator

on H. A subspace D+ is said to be outgoing

(i)

(2.2.0+

(ii)

UD+ a D+

~ U n D+ = {0} -oo

for (U,H) if it satisfies

308 (iii)

~ u n D+ = H -oo

A subspace D for which

(2.2.1)_

(i)

UD

(ii)

~ U n D = {0}

(iii)

cD

~U n D =H -oo

is said to be incoming 2.2.1 DEFINITION.

for (H,U). A quadruple (U,H,D+,D_) satisfying (2.2.1) is said to be a

scattering system.

2.2.2 TIIEOREM

(Translation

(U,II,D+) be outgoing.

Represcntation

Theorem

Then there exists a Hilbert space

[19, Th. II.1.1]).

Let

N+ and a unitary map r+

oft1 onto £2(-o,,,*o;N+) such that

(i)

r+ [D+] = P-2 (0,~,; N+)

(ii)

U+=r+Ur+ l

(2.2.2)

is tile right shift operator on £2 (-o0,~,,; N+). automorphism of N+. Proof

(Standard (ef. [15, p. 77]).

introduced later. unitary.

This representation is unique up to

We give tile proof to establish various quantities

By (2.2.1)+ - (ii) the operator UID+ is all isometry having no

By Wold's decomposition theorem [19, Th. 1.1.1] we may write uniquely

309 oo

(2.2.3)

D+= E(9 UnN+

N+=D+8

n--O

Since for any m > 0

U-mD+

=

u'm[D+ E) UmD+) (9 UmD+] =

U -m [(m~ I(9 u k N + ) (9 UmD+] = k=0 -

nl

( ~ (9 U k N+) (t) D+)

k=-I we obtain by (2.2.1 - iii)+ that oo

(2.2.4)

H = ~ (9 U n N+

It follows that for arbitrary h e H oo h = ~(9

2 U n PN+U "nh, II h Iltl

-o0

,,o

2

E" PN+O -n h ,.. -oo

llence the map r+ : H ~

£2 (-o~,.o; N+)

defined by oo

(2.2.5)

r+h = {PN+ U-nh}n=,~.

UD+

310

is isometric.

Since for {hn}7** E ~2 (-'~,*"; N+), h = ~ U n hn ~ H, the map r+ is onto -co

and thus unitary.

By (2.2.3) we obtain (i).

From (2.2.5)

r+Uh = PN+ {u-(n'D h}7=-~ = U+ (r÷h)

and (ii) follows. By (2.2.4) U and the uniqueness follows.

,

is a bilateral shift of multiplicity equal to dim N+

2.2.3 DEFINITION. The representation (U+, 9-2 (0,,,o; N+), £2 (-`0,00; N+)) is called an outgoing transition representation. For (U,H,D) incoming we similarly obtain

(2.2.6)

o

n

D = ~@U

N

,

N_=D

*

@UD

n~-oo

and oo

(2.2.7)

H= 2~

UnN

-oo

For the corresponding map r of H onto f-2 (-'0:*; N.) we define

(2.2.8)

r h = [PN u'(n+l) h}7=-00

Thus (i)

r_ [D ] = ~.2 (_,o, -1; N.)

(ii)

U. =

r

-I Ur_

is the right shift on .0_2(-,0,00; N ) .

The representation

(U_, ,0-2(-,,o,-1; N_),

£2 (_0o,`0; N_) is called an incoming translation representation.

311

2.2.4 DEFINITION ([1] , [15]). The operator

s = r., r+t: ~-2 (-0~,00; (C~) --, 9-2 (-0-,0-; (c °) )

is called the abstract scattering operator. Clearly S is unitary.

Denoting by

V the right shift on £2 (-°°,*°;(CP)) , we

readily obtain by the translation representation theorem (2.2.9)

SV = r_r+1 V = r. Ur+ 1= VS .

Let F : ,0-2 (-oo,,,,,; (C5) "~ L2(C5 be the Fourier transform operator.

Tile unitary

operator FSF -1 : L2(C0) ~

L2(C I')

thus commutes by (2.2.9) with L x , tile operator of multiplication by ;t. [19] that FSF "t

It follows

is a Laurent operator L S e Loo(B(CP)), such that S(e iz.)

a.e. is a unitary map on

C p.

2.2.5 DEFINITION ([1], [15]). S is called tile scattering matrix. It is clear from the translation representation theorem that to within right and left multiplication by unitary transformations on

S is determined C p.

The unitary maps F = F r . , F+ = Fr+ are called the incoming and outgoing spectral representation.

(2.2.9)

We have the following:

a)

F (D_=H2(CP).

b)

F÷(D÷)= SH (CO).

c)

F (Uh) = xFh, h ~ H.

Moreover, the operator

312 P . P + : D+ ~

(2.2.10)

D

is unitarily equivalent to tile Hankel operator I-Is,

where P_+ = PI)±, and the operator P D _-1-

(2.2.1 |)

2.3

P+: D+--~ D

-I-

is equivalent to the Toeplitz operator TS.

COMPUTATION OF THE SCATTERING FUNCTION FOR REGULAR, MAXIMAL RANK, STATIONARY GAUSSIAN SEQUENCES Let (f2, A, P) be a fixed probability space and let

{ y ( n ) : n e Z} , y ( n ) = i

pin)J be a centered stationary process with yj(n) ~ I..2 ( ~ , ~,1., P) j=l ..... p. Let fyy(X) = P (fkj(X))k,j=l, ~. E [-g, re]

(2.3.1)

I

2n

]

be its spectral density satisfying

log det fyy(~.) d~. > -** i

i

i.e., the process is regular and of maximal rank. Let H = Hy

nVz {Yl(n)' Y2 (n) ..... yp (n)] c

L 2 (~,A,P)

be tile space spanned by tile process and let U be tile unitary shift operator on H associated with the y process [21, p. 14]: Uyj(n) = yj(n+l)

We consider the past and future

j = I ..... p,

n~ Z

of [y(n)}?** defined by

313 D

= I ~_ (0) = V 0 { y l ( k ) ..... y p ( k ) } +

D+ = Hy (0) = _

V {yl(k) ..... yp(k)l

k~O

By (2.3.1) it follows [21, Th. II.6.11

~oo

We readily obtain that Now let

U n D =(01=

(U, H, D+, D_)

~ U n D+.

-o,o

is a scattering system.

(U, H, D+, D_) be the scattering system associated with the regular

maximal rank y

process.

The subspace N

= DOU

forward (backward) innovation subspace at n = 0. (U,H, D+ D )

D_(N_= DOU

D _ ) is the

Since for a scattering system

we have dim N_ = multiplicity U = dim N+

we can arrange the maps r+ to be onto We

{vl(0)

next Vp(0)}

compute

the

£ 2 (- ~ o, e, ; (CP)).

scattering

matrix

S

for

the

y

be an orthonormal basis for N . Let vj(n) = u n v j ( 0 )

t(n)]

v(n) =

,

process.

Let

and define

n ~ Z.

k.vpCn)J e,Q

By (2.2.4)tile process {v(n)}_~, is a (centered) white noise process with covariance Rvv(n) = 8(n)1

constituting the forward innovation process for the y process. Cp is determined up to a choice of basis in N . By (2.2.6) we may write

It

314

00 y (0)= ~ A ( k ) v ( k ) -

p A(k) = [0], k > 0 .

A(k) = (aij(k)) i,j=t

-oo

(Wold's representation).

It follows from (2.2.8) oo

(2.3.2)

r_yj(0) = { ~ Ctjm(k+l)vm(O) } k=-~ m=l

Identifying N_with

(C o) we readily obtain the representation

ajl(k+l)'~ r_yj(0) =

oo

".......

\ajp(k+ 1)) k=-*o Consider the function ¢.a

A(z) = ~ A'(k)z k -oo

Since Itx ij(k) 12 < k=-~

A(z) is analytic i n l z l >

i,j=l

Ilyj(0)ll tt j=l

1. ForA(z) we have,

J - A*(~)h(z) = fyy(X) 27:

By tile incoming properties

n~(c5

=

v

{ei"XA(o~x)~: ~, ~

( c p)}

n
i.e., A is coniugate outer [14, p. 121). Thus (2.3.3)

(Ft. yi(0) ..... Fr_ yp(0)) = ~A

Since the translates (in H y ) o f Yl(0) ..... yp(0) and their linear combinations are dense in

H y . Fr

is determined by the above expression.

We now consider the outgoing representation. ortlmnormal basis in

N + . We similarly obtain

Let el(0) ..... ep(0) be an

315 .o

p

y (0) = ~ B(k)r,_(k) _0.

-

This

representation

B(k) = ([3ij(k)) id=l

constitutes

the

B(k) = [01, k < 0 .

representation

of

y(0)

in

terms

of

the

q(n) backward innovation process {f..(n)}~.., c.(n)=

. W e define cp(n)ff

r(z) = ~

B'(k)z k

0

which

is a n a l y t i c

in

I z I <

i.

In a s i m i l a r fashion

we

obtain

by d i r e c t

computation

1

2x Also

with F being outer. (2.3.4)

r*(z)r(z) = fyy(~,)

z = e iL

( F r + y ! (0) .... Fr+yp(0)) = F

Combining (2.3.3) with (2.3.4), we obtain

S F = ~A and thus S = ~AI''-I One easily verifies that 2.3.1 T H E O R E M .

S(e i~.) is unitary a.c. k E [4t,~] .

W c thus obtain

For a regular maximal rank process {y(n)} we have S = ~ A F "1

w h e r e S is determined up to left and right multiplication by constant matrices. For the case p --- 1 we have 2.3.2 C O R O L L A R Y .

For a regular process [ y ( n ) } ~

unitary

316

s=~

,

and S is determined up to multiplication by a constant of unit modulus.

Proof. The outer function ~, satisfies I A I = 1 F I on T and thus is a constant of unit modulus.

/k = T F a.e. where'/

2.3.3 REMARK. Tile scattering matrix S was defined by outer and conjugate outer factors of the density fyy. Since those are determined up to left multiplication by -

-

i

a constant unitary matrix, we may wish to make a canonical choice (which amounts to choosing specific orthonormal bases in N+, N.) in the following fashion: For F(0) we consider its polar decomposition F(0) = KP (K unitary, P > 0) and define F'l(Z) = K'IF(z) . For F1 we have FI(0 ) > 0. This F1 is unique. Similarly for A.

In this way, the density fyy will have a unique S associated with it.

From the

viewpoint of seeing S as the phase function associated with f y y , this may be appealing.

2.4

COMPLETELY NON-DETERMINISTIC STATIONARY SEQUENCES, TIIEIR SCATI'ERING FUNCTIONS, AND INDUCED I IANKEL OPERATORS

The scattering function of stationary sequences plays tile role of the lleiscnberg S-matrix ill quantum mechanics. The physics of quantum systems is believed to be contained in the S-matrix and this object can in principle be dctermined experimentally. A natural question then is whether the scattering function of stationary Gaussian sequences, which measures the interaction between the past and future of the process, determines the spectral density of the process. To answer this question we introduce the class of completely nondeterministic processes. 2.4.1 DEFINITION [71. (2.4.1)

The process y is said to be completely non-deterministic if

+ I Iz-(0) n tly_(l) = {0). +

This condition states that no value in H y(1) can be predicted without error based I

on the information H~(0). w

This condition is more restrictive than regularity (cf.

317

BLOOMFIELD-JEWELL-IIAYASHI, loc.cit., for an example of a regular process which is completely non-deterministic). 2.4.2 THEOREM.

The scattering matrix

to the f o r m

K * fyy K w h e r e

completely

non-deterministic .

$ determines the spectral density fyy u p

K is a constant

p x p non-singular matrix iff

y_ i s

2.4.3 REMARK.

For p=l, this result was obtained by Levinson and McKean [161.

2.4.4 LEMMA.

The scattering matrix S determines the density

fyy(X) ttp to the

levitt

(2.4.2)

K* fyy (X) K

where K is a constant pxp non-singular matrix, iff

(2.4.3) Proof.

dim Ker TS = p . First note that for any representation of S S = ~YX "t

with the colunms of X in H2(C p) and those of belong to Ker "Is.

~Y in H2(CI~, tile columns of X

Moreover (on T) F+

Y*Y = (SX)*SX = X*X Assume (2.4.3) holds. (2.4.4)

It thus follows that

X(e ix) = r(eiX)K

where K is a pxp full rank constant matrix. 1 x • (z) X(z) =

1

Thus,

K*r*(z)r(z)K : K* fyy (~.) K

Z = e iL

318 proving the 'if' part. Now assume (2.4.3) not to hold, i.e., dim Ker "Is > p . We can thus find a pxp matrix X(e i~.) of full rank a.e. L such that the columns of X belong to Kcr TS and (2.4.4) does not hold. If we define Y = ~SX then the columns of

~Y are in I12 (C p) and S =

~YX -1 with Y*Y = X*X.

The result

follows. We next characterise condition (2.4.3) on a process level. 2.4.5 LEMMA.

We have *

4-

F+ [Ker TS] = H ; ( 0 ) A H y ( 0 ) i

Proof.

Let 0 C f ~

follows

that

Ker TS

i

From the well-known

identity l l s l l s + T s T s = I, it

HSH S f = f i.e., IIII S fll = Ilfll.

From (2.2.10) and (2.2.11) we obtain for ~ = F + f ~ I l y ( 0 ) l i P ~ll = Ja~fr,

and ~, ~ l ly (0). Thus F+* [Kcr TS] c Now let ~

I ~ (0) A H + (0). _

Z

if

It follows from (2.2.10) and (2.2.11)

Hs(F4- ~,) = E ~ • Let f = F+~E II2(CI~ .

(o>

We obtain

319

IIHsfll

=

I1~" ,~ll

=

II{ll

=

IIF+~ll

=

IIfll

and HSII S f = f . Thus f ~ Ker TS which implies

F+ {l~ (0) ^ Xiy_ (0)1 ~

KCr TS.

By the unitarity of I~ dim K e r T s = dim Ily(0) A l l y (0), arid since y is regular and of full rank we readily conclude

dim I-ly (0) A H y (0) = p Proof of Theorem 2.4.2.

iff

dim Hy (0) Al-ly (1) = 0 .

Combine Lemmas 2.4.4. and 2.4.5.

The converse question, namely, when is a function S E L . , ( B ( C p)

tile

scattering matrix of some full rank, p-dimensional completely non-deterministic process is of interest. We first observe that any S e Loo(B(cP)) which is unitary valued a.e. oil T

is

the scattering matrix of the canonical scattering system [I] u = L z , H = L2(C p) , D+ -- SI~:(C p) , O_ = lt~(C p)

The above question amounts to charactcrizing all scattering systcms (U, II, D+,D_) for which there exists a set {El, ..., ep} of linearly independent vcctors such that

H = span {un~j • j = l . . . . . p, n=0, 5:1 . . . . }

320 D+= span {un~j : j=l .....

and

such

cardinality

that

p.

any

n

D_ =

s p a n {U ~j : j = l

other

linearity

, and

the

spectral

.....

p, n < 0}

independent

Tile corresponding process will

{(n) =

p, n _> 0}

density

set

be

is

satisfying

{~(n)}_o °

tile a b o v e

where

~(0) =

is of

,

obtained by

P

(d(Ez~i,~i)ll) {EL : L ~ [-r~, ~]} being tile resolution of tile identity f ~ (~) = ~. d~ ) i,j=l ..... p for U.

Tile answcr is given in the following.

2.4.6 THEOREM. Let S ~ Loo(B(CI~) be such that

(i)

S(e iL) is a.e. X a unitary map on (C p)

(ii)

dim K e r ' I s = p

Then there exists a p-dimensional y whose scattering matrix is S . Proof.

Lel F I , 1-2 ..... Fp

,

full rank c o m p l e t e l y non-deterministic

span the kernel of TS and define F = [ F11 F21 ... 11-'p ] .

Lct A=SF.

process

321 Since Aj = (SFj) + ~+ (SFj) = rr_(SFj), j=l ..... p, the columns of A = [Al I A21 ... I Ap]

it.re

in I|2(C~ and by O) A*(z)A(z)

= F*(z)F(z)

z = mix. .

If we define 1

fy Y (Z.) = ~

r*(eiX) F(eiX)

_I

the theorem follows provided we show that F is outer aud zA conjugate outer. U = L~. and define : ^

~

^

n_<-I

.

D . = V {Z SF I .....

O = V {ZnAl ..... Ap] C 1 ((cP)) '

+

znsr v}

c

Let

srl2(cP).

n>0

Let

(2.4.5)

^

11 = ( neVz

U n ^

D_) V ( nVz

U n

A

D+).

It is easily verified that (2.2.1)_+ - (i), (ii) holds for (U,D+). show li, Th. 2.51 that a quadruple (U,It,D+,D_)

satisfying

In [1] Adamjan-Arov (2.2.1)_+

-(i),

(ii)and

(2.4.5) has a scattering matrix S which is unitary valued a.e. on T iff V U n D+ = II = ncZ

V Un D

n~Z

and, moreover, from their generalized functional model [1, Th. 2.1] we need have

= rt (c , D+ = A straightforward

computation

gives A

S=S and tile result follows.

rI2(C .

322 From the theorem of Nehari [3],and its vector generalization we know that for a bounded Hankel operator H with symbol (p ~ L , there exists a function (pp.e L °° such that IIH(pll = Iltpitlloo. The function (p~ is called a minifunction for llq). however

In general (Pit is not unique.

We

have

2.4.7 T H E O R E M . The Hankel operator ils determines S uniquely. Indeed, S is its unique minifunction.

Proof.

From L e m m a 2.4.4 we note that f e Ker TS is an eigenvector of HsI-I S c0r-1

responding to the eigenvalue IIHsll = 1. Since S = ZAF to

this

kernel.

Thus,

the

projection

coordinate in 12 (0,oo;(C~) spans F(0).

of

the

every column of F belongs

above

eigenspace

Now observe that for r ( 0 )

on

the

first

we have, because

of its outer property in H2(B(cP)),

log

Idet r ( 0 ) l (2~)p/2

1 4~

~

log det fyy (~) d~ > -

oo

,

_i

so that F(0) is of full rank.

We conclude

that the aforementioned

projection is

onto the firtst coordinate space. According to a result of Adamjan-Arov-Krein 12, Corollary 3.1] for Hankel operator I-1, to have a unique minifunction, it is sufficient that the projection of the eigenspace of Ilql14~ corresponding the first coordinate space be onto. 2.4.8 R E M A R K . deterministic

The result follows.

It is of interest

process,

to observe that since

the eigenvectors

Illlsll are only the columns of F,

to IIH~II on

for a c o m p l e t e l y

of H s H S corresponding

the projection

non-

to the eigenvalue

of this cigcnspace

on the first

coordinate is not only onto, but also 1-1.

In [2, Sec. 2] it is shown that for any

lhmkel

satisfying

operator

11 : I I 2 ( C p) ~

lninifuuction is of the form

II 2(C~

pS , p = IIIIII.

this

condition

its

unique

323 Thus, up to a constant multiple p > 0 all minifunctions of such Hankc! opcrators

are in

I-I

correspondence

with

regular,

full

rank,

completely

nondetcrministic

processes.

The case o f R a t i o n a l

Functions.

Let

{y(n)}7,,o have rational density

Ip(z)tZ fy y (~,) = iQ(z)l 2

z = eik ,

where the polynomials P,Q have no zeros in I z I < 1 and are relatively prime. Since fyy ~ L I , the polynomial Q has its zeros in I z I > 1. Write

P = PIP2

where PI of degree k has its zeros on T and P2 in I z I > 1. For P](z) =

k H (z-&j) , j=t

{ajl:k_l,j_ C T we have

P l(e i~-) - e- i k k ( - l ) k

P 1( e i x )

k ['I &j. j= I

Thus

(2.4.6)

s =

_k+l ~F..~.e

w

P2

w

where I y I = I and ~lte is outer.

=

v = (-l)

k

k

&j

In [31 A d a m j a n - A r o v - K r c i n show that (2.4.6) is

the general form of unimodular minifunctions and that in this case k+l is the dimension of the eigenspace corresponding to the singular value 1 = Illlsll. W e conclude that a regular p r o c e s s with rational spectral density is c o m p l e t e l y nondeterministic iff it has no zeros on T .

2.5 MARKOV PROCESSES AND UNITARY DILATIONS In this section we show how a Markov structure is intrinsically associated with unitary dilations and the resulting Scattering theory of Lax and Phillips. The results in this section are due to Lindquist and Picci [181 and Foias and Frahzo

324

110]. In some sense however tile results of this section arc essentially contained in Adamjan-Arov [1]. This is demonstrated in this section. In a Hilbert space setting, a centered stationary process {x(n)}~_,~ is said to be Markov if for all n _> s

PH](s) Px(s) h

h ~ lt2(n )

,

where X(s) = span {xj(s): j=l ..... m}. In our setting, all stationary processes will be generated by the shift U (on l l y ) associated with the y process. Thus, for a stationary process { x ( n ) } _ ~ (in Hy) we will have x(n) = Unx__(0).

It readily follows

i

from above that one can define the notion of a Markov subspace X c l l y X satisfies (see [18])

(2.5.1)

(for U) if

y

Ps m unx v U x

=

p

. Un x uSx

,

n>s,x~

X

Thus X is a Markov subspace (for U) iff the process {U n X} has tile (weak) Markov property.

In what follows a Markov process {U n X} will invariably arise in this

fashion. Markov subspaces X c lly which are

representations

for the process y , i.e.,

- -

I

for which {Yl (0) ..... yp (0)} c X , satisfy e,o

Ily =

n

v U X

,

- o o

and arc said to be of full range. There is a direct relationship between Markov processes of full range and unitary dilations (see also !10]). Recall 119] that a unitary operator U on a llilbert space II is said to be the minimal unitary (power) dilation of a contraction A oil X c I! if o o

A n = P X U nIX

n_> Oand II =

v U n X (mininudity).

325

2.5.1 PROPOSITION. X c Hy is a Markov subspace of full range iff

U (on H y ) is

tile minimal unitary (power) dilation of the state operator A = PxUIX : X ---)X

Proof.

From (2.5,1) we obtain for x,x' ¢ X and m,n > 0 ( u - m x , u n x ') = (U

- In

x,PxU

n

x).

Denoting A(n) = Px Un IX, we obtain

(x,A(m+n)x')

= (x,um+nx ') = (u-mx,unx ') = (u-mx,Px Un x')

= (x,PXU-m PX U nx')

=

(x,A(m)A(n)x') .

We infer that A(m+n) = A(m)A(n) and A(n) = An(l) = A n . Since X is of full range, we conclude that U in of A (in X). argument. Ilaving

This proves the 'only if' part. made the connection

between

Ily

is the minimal unitary dilation

The 'if' part follows by reversing the a Markov

process

{U n K} and

the

dilation property characterizing it, tile work of Adamjan-Arov [ll on the duality between dilation theory and the scattering operator model is directly applied. First, note that the process {U n X} is regular, i.e., satisfies

A

V

u k x = [0} =

n_>O k_
A

V

ukx

n>O k_>n

iff An

0

,

A*n ~ 0

(n

oo)

.

:326 Second, those Markov processes which in addition to being regular represent y (and are thus of full range) correspond to scattering systems according to a result of Adamjan-Arov [1, Th. 3.4] : 2.5.2 THEOREM. Let X c tly be a regular Markov subspace of full range. lly decomposes and, moreover, uniquely into the orthogonal sum

fly

Then

= D_ . x . ~ ,

where (U, IIZ ,D+,D)is a scattering system.

2.5.3 DEFINITION.

A scattering system (U,II,D+,D_)x for which

DcD+ is called a Lax-Phillips (L-P) scattering system. Let {Un X} be an arbitrary regular

Markov process

fly ,D+,D_)x its associated L-P scattering system. scattering matrix.

Let

of full range, and (U,

Ox(e ix.) be the corresponding

For the induced incoming spectral representation FX we obtain

Vx tD.l -- 10C°)

,

Fi: tD_l -- O×rI2~C5

Since D+ _1_ D

Ox~II~ d~). To each regular full range Markov process there is thus associated an inner function O×, which is the scatlering mat~

of the corresponding L-P system (U,

tly ,D+,D_ )X • From [1, Th. 3.3] it follows that the scattering matrices O associated i

with regular full range Markov processes

are precisely the inner functions O c

ilo,,B(CI~) which arc purely contractive [19, p. 188], i.e.; for which

327

IIO(0)11 < 1

2.6

FACTORIZATION OF THE SCATFERING MATRIX AND MODELLING The

1-I correspondence

x~(u,

IIy ,D+, D ) x

enables us to translatc the realization problem of finding all regular Markovian

representaions

for y to a covering problem ill L2(C p) via thc outgoing spcctral

rcpresentation for (U, lly ,D+,D_). Let X c

Hy

be a regular Markov subspace

rcprsenting y, (U, lI_y ,D+,D)x its L-P scattcring system, OX its scattering matrix, + and F X the corresponding outgoing spectral rcprcsentation.

Since {Yl(0) ..... yp)0)}

c X it follows

(2.6.1)

+ "+ 1" x IlF(0)I ~ F×

aud I,"+x tlC(0)l

is

~ full

[ k
÷ [D- ff) Xl = = FX

'

range left shift invariant subspace of 1-12(CI~ . Let V be the

corresponding inner function obtained from lhe Bcurling-Lax theorem, i.e., F÷x

l~(0)l = v*~(c p)

+ Bc the unitary of F X

Fx(Y(0)) = ~V*A. Thus { Y 1(0) ..... yp(0) } c X

:}28 +

translate under F X to tile equivalent condition

(2.6.2)

~V*A • t12(Cp)

@ e*I12(C ~ .

Conversely, if (9, V are inner functions for which (2.6.2) holds, the mapping y(0) ~ ~V*^ induces in a natural fashion a spectral representation FO,v for which

X = r o-t. v

OL~(c[') e

o*H~(d')

l

is a Markovian representation for y, and its corresponding L-P scattering system has its scattering matrix Ox

coinciding with O.

We have therefore proved the

following 2.6.1 TIlEOREM. Finding all models (realizations) of y is equivalent to finding all inner functions O l such that

e

o111;(C)

.for some inner function V. Each model c~, "responds to a pair

2.6.2 COROLLARY. All regular Markovi, , representations of by precisely those inner functions O l for vhich (2.6.3)

V*S = O 1 0 2

for some inner function Proof.

02

(OI,V). y are paramterized

11 (B(C p))

V.

(2.6.2) holds iff e l ~ A e II2(C~ iff O I V * S F ~ H2(C°).

latler holds iff @V*S ~ H,,.(B(cP)), i.e., iff (6.3) holds.

Since F is outer the

329

2.6.3 COROLLARY.

All regular Markov subspaces X c ~ ( 0 ) representing

parametrized by those and only those inner functions (2.6.4)

S = 0102

y are

O1 for which

0 2 e H (B(cP)).

Proof. X e ll-.(0)y implies D+ ~ X = V u n x n_<0

c

I1~.(0)~, combining with (6.1) we

conclude that V is a constant unitary matrix. The possibility of writing the scattering matrix X i,~ the form (2.6.4) has an interpretation on a process level. By the Beurling-Lax theorem, (2.6.4) holds iff (the invariant subspace for the left shift): H2(CI~

@

(range lls)

is of full range (for X)

which is equivalent to +

(2.6.5)

tly(0)

O

P~ly(0)Hy(0)

is of full range (for U).

A n L,,,(B(CI~) function satisfying (2.6.4) is called [91 strictly

non-cyclic,

and the

corresponding process - having a slrictly non-cyclic scattering matrix - is called strictly monocyclie. We thus obtain [18, Lemma 7.3 and Th. 7.6]: 2.6.4 COROLLARY. Let S = QIQ2 = P2PI be respectively the left, right coprime factorization of S. Then all mivimal regular Markov subspaces representing yare parametrized by those and only those inner functions O1 such that (2.6.6.)

V S = O10 2

where O 1 , 02 are left coprime and V is an arbitrary left divisor of I)2. Moreover we

]HI v ¢

330

det Ol Proof.

= detQi •

Combine Corollary (2.6.2) with [14, Lemma III. 5-8].

The general Fuhrmann degree theory for strictly non-cyclic functions [12, Ch. iii.5] now arises naturally - S playing a central role. All regular Markovian subspaces X c H_y representing y are parametrized by an inner function O I E H (B(C p ) ) will be of degree d(O1) = d e t O l

,

an inner function in H , and

d ( Q l ) divides d(O1). Thus, the degree of the minimal subspace is the lowest, in the sense that d ( Q l ) i s the weakest among the degrees of all other regular Markovian subspaces representing y. Applying [12,Th. I1. 14.111 we infer that two minimal regular Markov subspaces representing y are quasi-similar. 2.6.5 COROLLARY. For a p = 1 dimensional process y the minimal Markovian r e p r e s e n t a t i o n of y is parametrized by the inner function ql f o r which s = qtqz

is a coprime fizctorization.

III.

UNITARY DILATIONS OF IRREVERSIBLE EVOLUTIONS

In the previous section we have shown how we can associate a Markov semigroup with a Gaussian process via the associated Scattering System. Ill this section, we consider the dual view of associating a Unitary Dilation with a Markov semigroup. The evolution of a Hamiltonian (conservative) system is reversible while the evolution of real physical system is not. The real system returns to a state of thermal equilibrium at a temperature determined by its surroundings. This is the physical interpretation of the Poincar6 Recurrence Theorem. We may argue that the construction of a unitary dilation of a Markov semigroup is the abstract interpretation of coupling a physical system of a finite number of degrees of freedom to a heat bath thereby producing a Hamiltonian system of infinite

331

number of degrees of freedom. The ideas of this section are due to Ford, Kac, Mazur [11], Lewis and Thomas [17], Evans and Lewis [9] and the first author. We undertake the d e v e l o p m e n t of this section in continuous time since this is its natural setting.

3.1

N O T A T I O N AND PRELIMINARIES

<',"

Let It be a >1t and norm II •

Ilil.

separable Hilbert space with scalar product, When there is no confusion the subscript H will be

dropped. R denotes the set of real numbers , R + the non-negative real numbers and R - the nonpositive real numbers. For I c R , an interval, L2(I) denotes tile space of Borel m e a s u r a b l e s q u a r e - i n t e g r a b l e c o m p l e x - v a l u e d f u n c t i o n s and L2(I;It) the s p a c e of B o r e l - m e a s u r a b l e H - v a l u e d square integrable functions. W 1 , 2 ( R ; H ) denotes the Sobolev space of H-valued functions and w l , 2 ( I ; I I ) the Sobolev space obtained by restriction of W I , 2 ( R ; l l ) . Let (Tt)te R+ denote a oneparameter, strongly continuous, contractive semigroup on H, with To = 1, and l e t A be the infinitesimal generator of ( T t ) t e R + . We know Tt: = e -tA is a contractive semi-group iff, V x e D(-A), ~] x* e II with IIx*ll =1, (x*,x) = Ilxll and Re <x*, Ax> > 0. A semigroup of contractions

(Tt)te R+

is said

to contract strongly to zero if

V h e I1, we have lira IITthll = 0 . t.-...> oo

3.2

M I N I M A L UNITARY DILATION OF A CONTRACTIVE SEMIGROUP

3.2.1 D E F I N I T I O N . strongly continuous

Given a contractive semi-group (TOt e R+ on H, we say that a one-parameter

group (Ut)te R on H is a unitary dilation of (Tt,

tl) if there exists an isometry i: H ~ H such that the following diagram commutes: Ot H i

~H

[ 1"

i :=P

H

>l-I

Tt

The unitary dilation is said to be minimal if

332 t[,= W { U t ( i H )

l t ~

R} .

3 . 2 . 2 T H E O R E M [ L A X - P H I L L I P S ] . Let (Tt, H ) b e a strongly continuous, contractive, semigroup contracting strongly to zero. Then there exists a unitary dilation (Ut, ~ ) . The dilation (Ut, H ) h a s the canonical representation with H = L2(R; N), N a Hilbert space and (Utlt ~ R ) b e i n g the unitary group of right translations on L2(R; N): (3.2.1)

Proof.

(3.2.2)

(Utf)(s) = f(s-t). Since ( T 0 t e R +

is a contraction,

Q(h) < A h , l l > + < h ,

All>>0V

he

D(A).

Let NO = KerlQ(h)] and let P be the canonical projection of D(A) onto the quotient space D(A)/N0. On D(A)/N0 there exists a scalar product < • , • >A such that (3.2.3)

< Ph, Pk >A = < Ah, k > + < k, Ah > , V h,k ~ D(A).

Let N denote the Hilbert space completion of D(A)/N0 with respect to the norm induced by (3.2.3). Therefore 0 (3.2.4)

f IIPT_shll2ds = Ilhll2 - IITthll 2, V h e D(A) t > 0. -t

If we let t ---> ~,, since Tt contracts strongly to zero, there exists an isometric embedding i: H ---->L2(R; N), such that on D(A),

(ih)(s) = P T - s h , V s < 0 . Regarding L 2 ( R - ; N) as a subspace of L2(R; N), we have for V h ~ D(A) and t -> 0

(Utih)(s) = I pTt'sh

|

s --< t s > t

= (iTth)(s) + rh(S) where llt(s) ~ L2(R+; N) c

i(ll) -L. llcnce, V t < 0

333 ,

Tt = i Utl, and therefore Ut is a unitary dilation of Tt on 3t; = L 2 ( R ; N).

The unitary dilation we have constructed is in fact minimal. This is done by constructing a linear stochastic differential equation involving an operator-valued Brownian motion. W e first introduce positive definite kernels and consider their decomposition. 3.2.3 D E F I N I T I O N . such

maps is denoted

A map K: R x R ~ by K(R; 1I).

B(II) is said to be a kernel.

A kernel

K

is said to be

The set of all

positive

definite

if

V 111..... hn in II and Xl, ... , Xn in R. n

(3.2.5)

~ < K(x i, xj)hj, hi > > 0. i,j= 1

3.2.4 D E F I N I T I O N .

Let

K ~ K(R; I!).

Let II' be a Ililbert space and let V: R

B(H;Ir) be such that K(x,y) = V(x)*V(y). Then V is said to be a K o l m o g o r o f f decomposition of K. This decomposition is minimal if lI' = u { V ( x ) h l x e R, h~ II}. One c,'m prove that every positive definite kernel has a minimal K o l , n o g o r o f f decomposition. This is done with the aid of the reproducing kernel Itilbert space associated with K. With the notation of Theorem 3.2.2, let us introduce an operator-valued Brow^Jan motion as follows: Let W: R ~ B(N; H) be the map given by Zl0,tl(S)ri, t > 0 (3.2.6)

(Wt rl)(S) = -Xtt,0l(S)rt, t < 0,

where rl e N and Z(-) is the characteristic function. Consider (s,t) ~

the positive-definite

kernel:

(s ^ t)IN , where I denotes the identity operator.

Then

(s ^ t)lN = W t W s.

Ill tile sequel we denote by (D(A), I.I) the llilbcrt space D(A), with tile graph norm.

334

3.2.5 THEOREM. Let (Ut, H ) b e the dilation of (Tt, H ) g i v e n in Theorem 3.2.2. Then there exists a bounded linear operator B: (D(A), 1.1) ~ N and an operator. wtlued Brownian motion Wt: R ---> B(N; M), w h e r e M = u { W s ~ l s e R,al e N} and Wtsatisfies (3.2.6) such that t

(3.2.7)

(Uti - Usi)h = -.[ UriAhdr + (Wt - Ws)Bh, V h ~ D(A). S

P r o o f . The proof is constructed by verifying equation (3.2.7) for h e D(A 2) and then by density for h ~ D(A). For h e D(A 2) one can show that a solution is given

by t

Utih = e-A(t-S)Usih + f W ( d r ) B e ' A ( t ' r ) h S

where the last term is a Wiener integral, which can be defined by an integration by parts formula. The fact that Ut is a minimal unitary dilation follows from the fact that wt is a minimal Kolmogoroff decomposition. 3.2.6 REMARK.

The stationary solution of the equation is given by I

Utih=

fW(ds)Be-A(t-S)h. -¢.,o

We may verify that this Ut defines a regular stationary Gaussian process and there is a Lax-Phillips structure associated with it. We may also obtain an ordinary stochastic differential equation for the Markov semigroup attached to this Lax-Phillips system. 3.2.7 A NEW REPRESENTATION OF THE DILATION. Let

us

assume

that

tile semigroup

(Tt)te R +

on H is self-adjoint with

generator -A. Then A is a positive self-adjoint operator which we assume to be injective. In this case. one can show that there exists a minimal unitary dilation (Ut, H), where :H. = I[ @ L2(R; H). Let us write a vector ~0e L2(R; H) asq~=q~++~0- with ~0+ ~ L2(R+; H) andqY L2(R-; I1). Then one can write the unitary dilation for

(3.2.8)

Ut =

Tt A t / ' Bt St+Ct

where

t e R + as

335 t At: L2(R; D(AI/2)) ---> H: (p ---¢ (2A) 1/2 ~ T t _ s g ( s ) d s o Bt: D(At/2) ~ L2(R; tI): h ~ (Bth)(s) = X[O,tl(S)(-2A)l/2Tt_sh

Ct: L2(R; D(A)) ---) L2(R; It): q) ~ (Ct(p)(s) = X[o.tl(s)(-2A)l/2,dtt-s(p St: L2(R; H) ~ L2(R; H): q~ ~ (Stq0)(s) = tp(s-t)

At, Bt, Ct are densely defined contractions. Moreover written as

writing

(3.2.9)

Ut = e itx', on physical grounds the Hamiltonian K can be

K = K s • Kc (~ KR

Y-,R the Hamiltonian

of the reservoir

is the generator

of the shift of Brownian

motion. K,s, the Hamiltonian of the system is zero. Y-,c , the Hamiltonian of the coupling is of the form

where C: D(A 1/2) -~ L2(R; ll): h - ~ ~i0 ® (2A)l/2h, 50being the Dirac mass (this is formal and needs to be jusdfied) and C*: L2(R; 1t) ~ D(A1/2): q~ --~ (2A)l/2~p(o) (this is also fortnal). We now give some indications on how the new construction is arrived at. Sittce ( T O t e R + is contractive, the quadratic form F(x,x) = (Ax,x) + (x,Ax) = Ilxll2-11Texll 2 lira c$0

c

< 0 for x ~ D(A). We claim that there exists an operator C : D(A)

---) lI1 where II1 is a tIilbert space equal to CD(A) , such that IIC×II2 = F(x,x) V xc D(A). In a similar manner there exists a pair (C', 112) for t~ R . , such th.'lt IIC'II2 = (A*x,x) + (x,A*x) V x~ D ( A * ) .

The operators C and C' are to be thought of as

coupling operators. In the self-adjoint case C = (2A) ~n. The idea is to construct the dilation on the space ~f, = L 2 ( R _ ; H I ) @ H @ L2( R+; 112). Now since (Tt, 1I) is a Markov semigroup, we must have (Us o P0_L ° Ut) o i ( ~ ) i s

336

orthogoflal to i(H) where i : It --¢ H is tile injection ~ ~ ~ / '

P0:H

~

It is the

VJ , /

orthogonal projection and P0_L is the orthogonal

projection on the orthogonal

complement of H. This suggests picturing the unitary dilation as follows: Ct

,,'It 11

; 1-I

Tt The operators A t and :Bt "couple" (Tt, II) to (St , L2(R_; Ill)) and 4-

+

(S t ,L2(R+; 1-12)) where St and S t are right shifts.

-C+Tt-s~

For example :Bt is givcn by

if s ~ I0,t]

(l?,t~)(s) =

, ~ E D(A) 0

if s ~/ [0,tl)

It can be shown that 13t is a contraction. In the self-adjoint case there is a simplification and it is enough to couple It to L 2 ( R ; H) and we try to give aa intuitivc justification of (3.2.8). In a physical setting the shifts will correspond to the random behaviour of the heat bath and will be the flow of Brownian motion. We expect the coupling between the system and the heat bath to be instantaneous and this coupling will take place via the coupling operator (2A~1/2. For t > 0, we therefore expect that avector

(0)

(D(A) (L2(R+;H))"~ to be transformed into

( - A x d@t ( - ( 2 A ) l / Z x ) ) (.dbt

in time dr,

where bt denotes standard Brownian motion and dbt ® (-(2A)l/Zx) is an clement of L2(R+; H) = L2(R +) ® H (tensor product). The

second

component dbt ®

(-(2A)UZx) itl integrated form is essentially :Bt in (3.2.8). Finally, we can explain the form of of Ks, KR and Kc on physical grounds. Since the time evolution on H is self-adjoint, it does not contain a unitary part and

337

we expect Hs to be zero. The fact that Ke should be of the form (3.2.10) follows from the same argument given above.

RF__IT_RENCES !.

Adamjan, V.M. and Arov, D.Z.: On unitary couplings of semi-unitary operators, Amer. Math. Soc. Transl. 95(2)(1970), 75-129. 2. Adamjan, V.M., Arov, D.Z., and Krein, M.G.: Infinite Hankel block matrices and related extension problems, Amer. Math. Soc. Transl. 3(2)(1978), 133156. 3. Adamjan, V.M., Arov, D.Z., and Krein, M.G.: Infinite Hankel matrices and generalized Caratheodory-Fejer and Riesz problems, Functional Anal. Appl. 2. (1968), 1-18. 4. Adamjan, V.M., Arov, D.Z., and Krcin, M.G.: Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. 86(128)(1971), 34-75; Math. U.S.S.R. Sb. (1971), 31-73. 5. Avniel, Y.: Realization and approximation of stationary stochastic processes, Report LIDS-TH-1440, Laboratory for Information and Decision System, MIT, Cambridge, MA., February 1985. 6. Ball,J. and Helton, W.: Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parametrization of the set of all solutions, Integral Equations and Operator Theory, 9, 1986, 155-203. 7. Bloomfield, P.B., Jewell, N.P., and ltayashi, E.: Characterizations of completely nondeterministic stochastic processes, Pacific J. of Math. 107 (1983), 307317. 8. Davies, E.B.: Quantum Theory of Open Systems, Academic Press, New York, 1976. 9. Evans, D.E. and Lewis,J.T.: Dilations of irreversible evolutions in algebr:lic quantum theory, Comm. of Dublin Institute for Advanced Studies, No. 24, 1977. 10. Foias, C. and Frazho, E.: A note on unitary dilation theory and state spaces, Acta Sci. Math. 45 (1983), 165-175. 11. Ford, G.W., Kac, M., and Mazur, P.: Statistical mechanics of assemblies of coupled oscillators, J. of Math. Physics, 6, 1965, 504-515. 12. Fuhrmann, P.A.: Linear Systems and Operators in llilbert Space. McGrawIlill, New York, 1981. 13. Glover, K.: All optimal Ilankel-norm approximations of linear multi-variable systems and their L~-error bounds, Int.J. of Control, 39, 1984, 1115-1193. 1,1. llelson, 11.: I.cclurcs on Inv~lriant Subsp;Ices. Academic Press, New York, 1964.

338

15. 16.

17. 18. 19.

20.

21. 22.

23. 24.

Lax, P.D. and Phillips, R.S.: Scattering Theory. Academic Press, New York, 1967. Levinson, N. and McKean, H.P.: Weighted trigonometrical approximatioas on R' with applications to the Germ field of stationary Gaussian Noise, Acta Math., 112, 1964, 99-143. Lewis, J.T. and Thomas, L.C.: How to make a heat bath, Functional Integration, ed. A.M.Arthurs, Oxford, Clarendon Press 1974. Lindquist, A. and Picci, G.: Realization theory for multivariate stationary Gaussian processes, SIAM J. Control and Optimization 23 (1985), 809-857. Sz-Nagy, B. and Foias, C.: Harmonic Analysis of Operators on tlilbert Space. Amsterdam, North-Holland, 1970. Picci, G.: Application of Stochastic Realization Theory to a Fundamental Problem of Statistical Physics, in Modelling, Identification and Robust Control, (eds.: C.I. Byrnes and A. Lindquist), Elsevier Science Publishers B.V. (North-Holland), 1986. Rozanov, Y.A.: Stationary Random Processes. llolden-Day, San Francisco, 1963. Willems, J.C. and Heij, C.: Scattering Theory and Approximation of Linear Systems, in Proceedings of the 7th International Symposium on the Mathematical Theory 0f Networks and Systems MTNS-85, June 10-14, 1985, Stockholm, North-Holalnd, Amsterdam, 1985. Willems, J.C.: Models for Dynamics, to ,~ppear in Dynamics Reported. Zames, G.: Private Communication.

On the Theory of Nonlinear Control Systems H. Nijmeijer Dept. of Applied Mathematics, University of Twente P. O. Box 217, 7500 AE Enschede, the Netherlands

Abstract A review of theory is presented. A state feedback and its system. The relations invariance and a general

some of the recent advances in nonlinear control central theme in the discussion is the notion of use for altering the structural properties of a between various synthesis problems, controlled decomposition problem are investigated.

I. Introduction One of the most important new developments in nonlinear system theory in the last decade has been, without any doubt, controlled

invariant

distributions,

the introduction of invariant and [20,21,25].

This

(differential)

geometric approach, as it is often called, provides a mathematically elegant and effective

approach

for

solving

various

synthesis

problems.

In

this

regard, we mention the Disturbance Decoupling Problem and the Input-Output Decoupling

Problem.

sophisticated

Moreover,

theoretical

this

picture

approach for

provides

understanding

a

very

clear

nonlinear

and

control

systems, and therefore is valuable in the interpretation of classical system theoretic concepts

like obse~-vability, accessibility

(controllability)

and

invertibility. Invariant and controlled invariant distributions in nonlinear system theory play the same

role as - and in fact generalize - invariant

and controlled

340 invariant subspaces in linear system theory. For linear systems, the crucial features of such subspaces are well known; see, for instnnce,

the geometric

approach

of

Marro

[3].

approach

the

and

involve

Wonham

[51]

essential

and

the

tools

work

stem

of

from

Basile

linear

and

algebra

In this linear

mappings, subspaces and so on. For general nonlinear systems, Joe. a system having its dynamics of the form = f(x,u), various characterizations of controlled invariant distributions have heen obtained, ef. problems

still

From a control

[23,37,38],

though certainly many open mathematical

remain. theoretic point of view

involve the notion of state feedback.

the aforementioned concepts deeply

It is certainly one of the essential

contributions of the past decade in nonlinear control that a more advanced theory

of

examples, type

of

state

variable

feedback

has

the conference proceedings feedback

studied

been

initiated;

[16,9]. We note

is static,

though

some

see

that

for

numerous

the most common

interesting

results with

dynamic state feedback have also appeared recently. Another important topic in the theory of nonlinear control systems

is that of decomposition of a

system. The idea is to study how a nonlinear system can be decomposed, in a nontrivial thereby

manner,

reducing

as

the

the

intereonnection

control

complexity

of

of

lower

the

dimensional

original

system.

systems, A

basic

contribution on the (cascade) decomposition of a nonlinear system is due to the

Krener

notion

cf.

of

[30],

which

controlled

is

also

reminiscent

invariance.

However,

of

the

the

later

analysis

introduced of

system

decomposition is still far from its completion. The

purpose

controlled

of

this

invariance

paper and

is its

twofold. relations

Firstly with

the

static

main

results

about

and

dynamic

state

feedback problems are reviewed in sections 2 and 3. Secondly, we

formulate

discuss

in section 4,

the general decomposition problem for a nonlinear

the

interrelations

with

the

dynamic

state

feedback

system and problems

of

sections 2 and 3.

Acknowledgement This paper expresses day nonlinear

control

theory,

the author's personal view on present

and may not reflect

Since the time I was a graduate student, enthousiasm

and

inspiration

from

Jan

C.

the opinion of others.

I have appreciated Willems.

His

the continuing

fiftieth

birthday

serves as an excellent occasion to thank him with this paper. I also want to express my gratitude

to Arian van der Schaft

have had on this and other papers.

for the many

discussions we

341 2, C o n t r o l l e d 2.1 L i n e a r

First,

invariance

systems

we b r i e f l y

systems,

cf.

review

[3,51].

Z

:

V c •

static s t a t e

Consider

of

the

standard

the l i n e a r

geometric

theory

for

linear

system:

(2.1)

x - Ax + Bu

with x 6 ~ - ~n, subspaee

parts

u 6 ~ = ~m a n d A is

called

and

controlled

B matrices

invariant

of

if

appropriate

there

exists

sizes. a

feedback

u - Fx + Imv,

where

v E ~

closed-loop

A

linear

is

a

new

(2.2)

input

signal,

such

that

~

is

invariant

for

the

dynamics

~F

:

X -- ( A + B F ) x + B v:

(2.3)

that is, (A+BF) ~ c F

This is e q u i v a l e n t

(2.4)

to the r e q u i r e m e n t

AT/ c ~ + im B

Choosing a b a s i s a basis

for F,

(2.5)

[e I ..... e k ,ek+ l ..... e n ) f o r 9~, s u c h

the s y s t e m ~F c a n be w r i t t e n

that

{e I ..... e x ) f o r m s

as

Ixlx2t [F11o A+ A+F21IxI [B] +

(A+BF) z z

where x I a n d x z are k- a n d (2.6) y i e l d s

a linear

system

x2 = ( A + B F ) z z X z

In a m o r e as 7.F, on

abstract

the

(n-k)- d i m e n s i o n a l

way,

quotient

xz

vectors

v

(2.6)

Bz

respectively.

" m o d u l o V" as

(2.7)

+ B2v

the s y s t e m space

Clearly,

~-

(2.7)

is a l i n e a r

X (rood F).

Namely,

system,

to be

letting

~

: ~

denoted > ~

be

342 the A

+

projection

along

~

such

BF

:

> ~

F,

(2.4)

that

implies

the

existence

~(A+BF) - (A+BF)~;

the

of

a

linear

quotient

mapping

system

~

is

defined as ~F

where

: ~ -- (A+BF)x + B v

B - RB.

representation condition

Equation

(2.7)

of

This

(2.8).

(2.8)

is

therefore

explains

how

nothing

else

to obtain ~F

than

from Z,

a

matrix

given the

(2.4). To recover the system E from the quotient system ZF we need

to add to (2.7) the dynamics

X1

m

AIIx I

+

AI2x 2

+

(2.9a)

B1 u

together with the "inverse" feedback law

v-

-FIx I - Fzx 2

+

(2,9b)

Im u

where we have w r i t t e n F - [FIiF2] in the obvious way. Note

that

the

feedback.

equations

(2.9a,b)

For an arbitrary

constitute,

in

fact,

a

system ~ : x ~ Ax + Bv a dynamic

dynamic state

state

feedback

or precompensator is defined as

2 : I z - Pz + Qx + Ru (2.10) v where z q Z - ~ ,

Sz+Tx+Uu v 6 ~, and P, Q, R, S, T and U are matrices of appropriate

sizes. The integer v is the dimension of the precompensator 2. The system in

closed

preceding applied

loop

with

analysis to

the

the shows

system

dynamic that (2.7)

feedback

the

2

will

precompensator

precisely

reproduces

be P

denoted defined

the

(2.1). This may be represented as in Diagram i.

J

Z

)v

Diagram 1

X

:

Z - ~.Y

}x

as

~-P.

The

via

(2.9a,b)

original

dynamics

343 In geometric linear system theory, dccoupling problems are often phrased in terms of controlled

invariant subspaces.

For

instance,

in the Disturbance

Decoupling Problem (DDP), one considers the system

~ x - Ax y - Cx

+

Bu

+

Eq

(2.11)

where x and u are as before and q 6 8 - ~2 and y E ~P are the disturbances and outputs respectively.

In the DDP, one searches for a feedback law (2.2)

which isolates the disturbances

from the outputs;

the problem

is solvable

if, and only if, there exists a controlled invariant subspace ~ satisfying.

im E C V C ker C.

Provided (2.12) holds, satisfies

(2.4).

(2.12)

a solution of the DDP

Notice

that

(2.12)

in

is given by a feedback

Diagram

I

implies

that

that

the

disturbances only enter via ~ and that the output only depends upon x.

Remark Usually one solves rendering

a

solution

the DDP by searching for a feedback law u - Fx,

without

a

reference

input.

As

the

solvability

conditions are the same in both cases, we prefer to consider the class of feedbacks

(2.3),

because

the new

input v may be

used

to achieve

further

controller design goals.

2.2 Nonlinear systems

We now review some essentials from the differential

geometric approach

to

nonlinear systems. A nonlinear input-state system,

or shortly control system

Z is a 3-tuple

7.(M,B,f) where M is a manifold, B is a fiber bundle over M with projection : B

> M and f is a smooth mapping such that Diagram 2 commutes, where

~M denotes the natural projection of TM on M. B

f ,

)

TM

M

Diagram 2: A nonlinear system Z(M,B,f)

In this definition, M is considered as the state space of the system while

344 tile fibers of B r e p r e s e n t

the s t a t e - d e p e n d e n t

for the m o t i v a t i o n of this definition. we can l o c a l l y

identify B as the C a r t e s i a n p r o d u c t

the fiber space U. u denotes

input spaces;

C h o o s i n g local c o o r d i n a t e s

the c o o r d i n a t e s

see

[6,49,46,37]

Because ~ : B --> M is a fiber bundle,

for the fibers,

of the state space M and

x for M and (x,u) for B where

this d e f i n i t i o n

l o c a l l y reduces to

the e q u a t i o n

- f(x,u),

where

we

have

framework, =

: B

(2.13)

abused

static

notation

state

) B, i.e., into

letting

feedback

for

diffeomorphically

by

can

each

itself.

be

(x,u)

I

defined

) (x, f(x,u)). as

a

x in M, ~ maps

With

the

same

bundle

tile

abuse

of

In

this

isomorphism

fiber

notation

~-l(x)

as

before,

the f e e d b a c k ~ is locally d e s c r i b e d as

u - ~(x,v).

Next,

we

~(M,B,f).

introduce In

differential smooth, is

what

the

notion

follows

we

geometry

constant

a

(2.14)

linear

of

use

[4]. Let D be

dimensional

subspace

of

controlled some

standard

a regular

and involutive;

that is,

TxM

smoothly

depending

[XI, Xz] b e l o n g s

for

terminology

distribution

d i m e n s i o n - say k - and for each pair of v e c t o r XI, X z 6 D the Lie b r a c k e t

invariance

a

system

coming

on M,

from

i.e.

D is

for each x in M, D(x) on

x,

of

constant

fields X I and X z on M with

to D.

Let E be a n o n l i n e a r control system w h i c h is l o c a l l y d e s c r i b e d as in (2.13). We

say

exists

that

the

a static

regular state

distribution

D

f e e d b a c k ~, w h i c h

is

controlled

invariant

if

there

is l o c a l l y g i v e n as in (2.14),

such

that the c l o s e d loop dynamics - f(x,~(x,v))

~ f(x,v)

(2.15)

satisfies [f(.,v),

D] c D for e v e r y c o n s t a n t v ~ U,

w h e r e O x U is a local t r i v i a l i z a t i o n

(2.16)

of ~ : B - ~ M. Here

(2.16)

means

[f( ,v), X] 6 D for each v e c t o r f i e l d X b e l o n g i n g

to D. The c o n d i t i o n

implies,

(2.6),

similarly

decomposition. there

[4]).

exist

to

the

linear

case,

This can be seen as follows. local

coordinates

see

a

sort

of

that

(2.16) local

L o c a l l y a r o u n d each p o i n t x ° 6 M

x - (xl,....,xn) T

such

that

(Frobenius,

cf

345 D.span{

a P ax I

We obtain from (2.16)

. . . . .

P

ak } ax

(2.17)

"

that the system

(2.15)

in these coordinates

is of the

form

{ xl - f1(xl,xz,v) x2

(2.18)

f2 (Xz ,v)

where x I - (x l,...,xk) T and x 2 - (xk+1,..., xn) T. At that finding properly ~.

B

a suitable

chosen

feedback

fiber

~ can be

respecting

this

understood

coordinates

point as

for

we

the

the

emphasize

selection fiber

of

bundle

. >M,

A necessary feedback

and

sufficient

(2.14)

defined

conditon

around

invariance of the distribution

an

for

the

local

existence

arbitrary

point

D for the system

(2.15)

in

of

B

a

state

yielding

f.(~.-l(D)) C D + f.(A0), provided

that

the

distribution is

(2.19)

f. (A 0) O D

f. : TB

> T(TM)

~. : TB

> TM is defined similarly,

in the coordinates

the

mapping

n

(x l,...,x ,

~lj

the

is that (see [38])

defined

has as

constant

dimension.

D is a regular distribution

. . .

}[ere,

f.(b,f) - (f(b), Dfb(f)),

"n

,x ) for TM has

the form

on TM which

(compare

with

(2.17)) D_

span

{

a ax I

and finally general,

....

~

~

._a_a }

ax ~

ax 1

ax k

A 0 is the distribution

(2.19)

only assures

on B given by A 0 - {X E TBI~.X - 0}.

the local existence

of a static

a; to guarantee

that such an a can be globally defined,

are needed

[8,23,37]).

(cf.

state

further

In

feedback

assumptions

Remark For an affine nonlinear control system m

x - f(x) +i~i gl (X)Ui the condition

(2.19)

'

takes the more familiar

(2.20) form

[f, D] C D + span (gz ..... gin) (2.21) [gi,D] c D + span (gl ..... gin l, i - l,...m provided that the distribution

D n span {gl ..... gm} has constant

dimension.

346 Henceforth we consider a system which satisfies the condition

(2.19) and so

a suitably locally defined state feedback a brings the system into the form (2.18).

As

in

distribution

the

linear

D induces

case,

the

local

locally a nonlinear

controlled system on

invariance

of

the

the "manifold" M(mod

D), namely, see (2.18), : xz " fz(Xz,V) (Note

that M(mod

D)

(2.22)

locally

forms

a neighborhood

M(mod D) is not a Hausdorff manifold). from

the

system

system ~

Z

given

in

(2.13),

in general

This explains how we locally obtain

via

given in (2.22). To recover

in ~n-k, but

a

feedback

(2.14),

the

quotient

(2.13) again from (2.22), we need to

add the dynamics ½1 -- fI(XI'Xz, u)

(2.23a)

together with the "inverse" feedback law

v - a

-i

(x 1,x z,u)

,

where for each x - (xl,xz) T, We

observe

that

the

(2.23b)

~-l(x,.) is the inverse of the mapping e(x,.).

equations

dynamic state feedback.

(2.23a,b)

define

(locally)

a

particular

In general, for a system locally described by

= f(x, v) a dynamic state feedback is given as

z -

p
v

=(z,x,u)

(2.24)

where

the state

Z c ~v.

The

denoted

as ~.~.

-

of the precompensator

system ~ The

together with foregoing

z belongs

the

analysis

dynamic yields

to an open neighbourhood state that

feedback the

defined via (2.23a,b) applied to (2.22) precisely reproduces

2 will be

precompensator 2 (2.13) and thus

Diagram i is locally valid in this nonlinear situation too. Regular essential

distributions role

that

in nonlinear

are

(locally)

synthesis

controlled

problems.

As

invariant

an example

play

an

we briefly

discuss the local nonlinear Disturbance Decoupling Problem. Le t I x - f(x,u,q) (2.25) y

h(x)

347 be a nonlinear

control

system in local coordinates

outputs y. Let ~ be the projection ~ : (x,u,q) : (x,u,q)

~

u - ~(x,v)

> (x,u).

which

Then

isolates

there

the

locally

i

disturbances

q and

> x and ~ the projection

exists

disturbances

with

from

a static the

state

outputs

if,

feedback and

only

if, there is a regular distribution D on M satisfying

provided

f, (=,-I(D)) c D + f, (~,-i(0))

(2.26a)

f,

(~.-1(0)) C D

(2.26b)

D c ker dh

(2.26c)

the

distributions

f, (~,-i(0)) N D

and

f, (~,-I(0))

dimension, see [38]. Notice that (2.26a) precisely yields, q, the condition equations

(2.19).

(2.26a,b,c)

Again,

as

in the

linear

case,

imply that in the corresponding

nonlinear system (2.25),

have

constant

for each constant (see

(2.12)),

the

flow Diagram I of the

the disturbances enter via ~ and the outputs y only

depend on the state ~ = x 2.

3. Static and dynamic state feedback Consider

again

respectively

a

linear

(2.13).

or

In various

question of adding control system

satisfies

Disturbance

a

nonlinear

set

Decoupling

system

controller

E

design

of

the

problems

form one

faces

prescribed

Problem

forms

design a,

goals.

perhaps

The

aforementioned

naive

but

added are not randomly chosen but usually depend on the observations

for controller

assume

design - in general

this

v

[

)

J

may

not be

the

as for instance

loops now depend on the state x and we arrive

at a situation which is depicted in Diagram 3. ..

of the

that the state x

case and one has to resort to more restrictive alternatives, The control

simple,

Of course the control loops to be

system E. To simplify our discussion we henceforth

output feedback.

the

loops to the system E such that the closed loop of

illustration of such a synthesis problem.

of E is available

(2.1),

z

)

zlx

)u

x

(

Diagram 3: Closed loop system E.P

348 As in section 2, ~ is a s s u m e d linear or nonlinear)

where

to bQ another s~rstem (in analogy with ~ either

the

inputs consists

of a new set of reference

inputs v together with the state x of Z, and the control u appears as the output of ?. Thus, ~ typically is of the form (2.10) or (2.14) and forms a dynamic

state

feedback.

Notice

that

it also

includes

the static

state of

feedback (2.2) (respectivily (2.14)).

Remark 3.1 The closed loop system in Diagram 3 has the same structure as the system depicted in Diagram I. However a closer inspection shows the obvious differences:

in Diagram 1 the system Z appears as the precompensated system

of a system ~ whereas in the latter case the possible closed loop systems Z.~ are considered.

The

importance

of

overemphasized. formulated as: such

that

Depending

In

precompensator

the

closed the P

feedback

essence

Given

the on

the

almost

loop

be

all

system Z,

nature may

structure

of

Z-2

the

imposed.

controller

find;

system

above

design

if possible, has

problem In

given

the

some

problems

be

can be

a precompensator

prescribed

further last

cannot

properties.

requirements

decades

on

nonlinear

the

control

theory has focused on typical synthesis problems as described below.

(Note

that only a sample of such problems will be discussed here; various others have also been addressed in the literature). 3.1 The feedback stabilization problem One of the most widely studied problems stabilization problem. with

That

an equilibrium point

static

state

is given

the system ~

f(x0,u0) - 0,

feedback u = ~(x)

closed loop system

in ,:ontrol theory is the feedback

with

described as

the question

u 0 - ~(x0)

is whether

exists,

(locally) asymptotically stable.

in

which

(2.13)

or not a

renders

the

And

if it does exist,

can it be assumed to have certain smoothness properties,

such as C k, C~ or

analytic, see also [28]. Thus in our setting the precompensator T is static, with reference inputs v set equal to zero. Many

results

around the local stabilization problem have

appeared

through

the last decades, but in its full generality the question is still far from being solved. As we will not pursue this problem here, we refer the reader to

[2]

Byrnes

and

the

references

and

Isidori,

where

therein. a

We

slightly

also

refer

different

to

the

approach

recent work of to

the

global

stabilization problem is advocated; see [7] and the references therein.

349 3.2 The feedback llnearizatlon problem Consider

the nonlinear

problem is to find,

system

Z

locally

described

local coordinates

the system Z-Y is a controllable i.e.

solved completely. nonlinear

~

described

In fact,

systems

in

(2.15).

a precompensator • such

if possible,

state

feedbacks,

as

as

in

this

extending

the

For static

question

[27] and

the

in suitable

linear system.

(2.14),

this has been done in

(2.20) - thereby

that

Then

has

been

[21] for affine

results

of

[5]

and

[29] - and then for general systems the solution was given in [45]. To describe

this solution we introduce with

(2.13)

the so called

"extended"

system.

{

~

-

f(x,u)

(3.1)

G = ~

which is an affine

nonlinear

Now,

system

the

nonlinear

controllable

linear

system

(2.13)

system

if

since

is

and

the controls

locally only

feedback

if

the

u appear

linearly.

linearizable

corresponding

into

a

"extended"

system (3.1) is. It is therefore enough to give the solution of the feedback linearization problem (2.20)

we

[gl ..... gm} system

and

(2.20)

f(x0) = 0

for an affine control

introduce

if

define

is and

regular and dim

the

distributions

recursively

locally

feedback

only

the

(~-i)

if

system

(2.20).

With

the system

as

follows.

Let

40 - span

Ai

~k -- [f,Ak-1], linearizable

distributions

- dim M. Observe

an

algebraic

manner,

linearization problem

see

e.g.

in contact

[19] with

a

point

the aforementioned

x0

the with are

conditions

One can also formulate them in

and

the

around

Then

Ak, k - 0,i ....... n - 1

that

on the distributions Ak are purely geometric.

k - 1,2 ....

this

brings

classical

geometric theory of normal forms of Pfaffian systems,

the

(Cartan)

feedback

differential

see [19] and [17].

So far we have discussed the local linearization of a nonlinear system via a static state dynamic

state

feedback.

An

interesting

feedback

is

also

extension

allowed.

Some

of

this

question

preliminary

results

is when on

this

idea may be found in [26,33], but a complete solution is still far away.

3.3 Model Matching Problems Given

the

y = h(x),

system find,

Z if

as

in

(2.13),

possible,

a

matches a given reference model ~

together

dynamic

with

an

output

precompensator

described as

Y

mapping, such

that

say ~.~

35O { xm - fm(xm,v) (3.2) Ym " hm(Xm), Note

that the linear model matching problem,

was solved in the early seventies,

cf.

i.e.

~, ~M and P are linear,

[35,36]; see also

[32].

In the last

decade, several versions of the nonlinear problem have been studied. In

a

first

problem

of

this

type

(see

[24])

it

is

required

that

the

input-output behaviour of ~-~ matches the input-output behaviour of a linear model ~

(thus (3.2) forms a linear system).

k the Volterra kernels of @.~, V~.~, should coincide with the k Volterra kernels V M of a linear system ~M, k - 1,2 ..... and thus satisfy 1 k V H is independent of x and V M - 0 for k > i. In other words,

Notice that the matching of input-output behaviour between @.~ and ~ not

impose

conditions

on

the

0-order

Volterra

kernels;

the

does

autonomous

behavior may differ in principle. Sufficient - but

not

necessary - conditions

for

the

solvability

for

this

matching of a prescribed linear input-output behaviour are given in [24]. As a matter of fact in [24] the problem is solved for those systems E which do have a linear input-output behaviour after applying a suitable static state feedback,

and

this class

of systems

is

identified

in the same

reference.

Note that this forms a weaker linearization problem then the one discussed in section 3.2. A second, less restrictive, model matching problem has been studied in [12]. In

this

case

no

further

assumptions

on

the

model

~

are

made

and

the

question is to search for conditions on E which guarantee the existence of a precompensator ~ such identical. matching problem

So far,

problem

that

input-output behaviours

only sufficient conditions

are

known

to a Disturbance

Disturbance

the

Decoupling

and

these

Decoupling

Problem

in

are

of ~.E

and ~

for the solvability of this

based

on

the

reduction

Problem with Measurements, which

are

the

disturbances

of the

that is a

are

measured

(compare with the exposition on the DDP in section 2). 3.4 Decoupling Problems A large class of synthesis problems decoupling)

of

a

subset

of

the

essentially

inputs

from

the

involve output.

the isolation (or So

far

we

have

encountered the Disturbance Decoupling Problem with or without Measurements as a typical example. Another example of a problem of this type fitting in the general formulation is the input-output decoupling problem. Consider again the nonlinear system

351 locally

described

by

(2.13)

Yl - hi(x) ....... Ym " ha(x)-

together

Here

with

the number

a

set

of

output

of output-blocks

mappings

equals

the

number of inputs. In the input-output deeoupling problem one searches such that the closed loop system ?.Z with input-output deeoupled,

i.e.

several

versions

of

the above outputs Yl .... , Ym

the

i-th output Yl, i - i .... , m.

received a lot of attention the

is

the i-th input v i does not influence the j-th

output yj, j ~ i and v I "controls" deeoupling problem has

for a preeompensator

problem

have

been

in the

solved.

In

This

literature

and

particular

situation where it is required that ~ is a static state feedback,

the

a local

solution involving the notion of controlled invariant distributions has been obtained

for

affine

nonlinear

systems,

[40,41],

see

also

[45]

for

a

treatment of the problem for general nonlinear systems. For affine nonlinear systems with scalar outputs also the problem with a dynamic precompensator has been solved, dynamic state

see

[10,42]. The general block decoupling problem with a

feedback

being addressed

for

is

solved

decoupling

in

[11,18,34].

problems,

see

Stability

[22]

for

the

issues

are now

noninteracting

control problem and [48] for the disturbance decoupling problem.

Remark 3.2 A promising differential

recent

algebraic

tools

approach are

used

is given by in

~le

study

Fliess

in

of various

[13]

where

nonlinear

synthesis problems. Without doubt these methods will complement and enrich throughout the next decade the differential geometric approach to nonlinear control theory that h a s b e e n

described here.

4. Decomposition of systems

In the

previous

between Diagrams structure

of

section,

(see

Remark

i and

The

purpose

Diagram

3.

I

in

more

3.1), of

detail.

we

this

observed section

Basically

the

is

the

similarity

to pursue

the

configuration

of

Diagram i shows that the given system 7., either linear or nonlinear, appears as

the

closed

loop

system

~ - Y..~ for

a

precompensator T, both also linear or nonlinear. and f are unknown and the problem

certain

system

Of course,

is: under which

~

and

a

in general,

conditions

does

such a

factorization of Z exist. This can be viewed as a Decomposition Problem for 7.. The main motivation for looking at this pcoblem is that when Y. allows a factorization as in Diagram I, then 7. is built up from the simpler systems ~. and P. Here a "simpler system"

is used as a synonym for a system having a

352 smaller

state

space

control

of

cascade

of simpler

the

finite

automata,

decomposition

dimension

more

complex systems

cf.

(see

also

system

Z.

already

[1,31].

if E decomposes

A

[30]).

The

can

decomposltion

appeared system

This

useful

of

a system

in the Krohn-Rhodes

Z

is

said

in the

be

to

as a

theory for

admit

a

cascade

as in Diagram 4.

x

Diagram 4: Cascade

Note

that

a

cascade

decomposition given by nonlinear

or

systems

papers

have

is given.

On the other hand investigated way

we

deal

a

a rather

sequel,

we

nonlinear Consider

studied

explicit

take

special

in [14,15]

relation

the decomposition

Clearly

decomposition

controlled

with

indicate

invariant

one

other

in

[23,39]

[30,44] where

(controlled)

and

in the

invariant

such as parallel-

shown

in section

and

in Diagram

possible

i appears,

or distributions.

a faetorization way

of

2, one

This

is is

appears.

In the

decomposing

affine

systems. the affine nonlinear

system with a one dimensional

input

- f(x) + g(x) u

Assume

for

[44].

as was

subspaces such

the

as stated here has not yet been

as depicted

in which

of

decompositions

terms

and

see e.g.

problem

type

in the preeompensator

Also other decomposition-types

limited way

will

a

in Lie-algebraic

viewpoint

have been studied,

(local)

with

however

is

i; namely,

in its full generality.

in which

when

been

most

series-decompositions,

indeed

of Z and P

the linking map v = u. Cascade

geometric

the

distributions

by Diagram

(2.24)

from a differential latter

decomposition

described

(2.10)

decomposition

we

coordinates

work

around

a

point

(4.1)

x 0 with

x - (x I .... ,xn) ~ such

that

g(x0) ~ 0.

Then

g(x) - a/~x n ,

there

exist

(Frobenius'

local

Theorem,

353 see section 2). In these coordinates,

{~ i

Applying

-

fl (xl

j ....

x")

,

xn-1

f,-I (xl, .... , xn)

x"

fn(x ~ p

the

static

....

x")

p

state

the system (4.1) takes the form

(4.2)

+ u

feedback

u - - fn(X I,...., xn) + u

yields

the

system x

- fl (xl

i -I

xn)

- fn- I ( xl

(4.3)

xn)

A closer inspection of the last equation of (4.3) shows that the variable x n appears as an arbitrary smooth function taking values

in a neighbourhood of

x~ , and therefore can be interpreted as an input function. have obtained a decomposition of the system

In this way we

(4.1) into the system ~ locally

given by "I

n-I

fl (Xl

X

, ....

,

X

,

V)

n-1

j

i X" n - 1

(4.4) f._1 (X I ~ . . . . ,

--

~

V)

and the precompensator P as given by x .-i

x" - fn(X I

xn) + u (4.5)

n v

Note

that

--

this

x

procedure

Z : x = Ax + bu yields

applied

that the subspace

a linear system,

the above procedure

it follows

all

thereby x-

that

yielding

Ax + bu

can be

further properties system (4.1)

subspaces

that

any

factored

a

more

general

a

single

can be

almost out;

see

(4.4)

input

linear

system

im b can be factored out. For such iterated several

im b + ... + Akim

b

controlled also

on almost controlled

as the system

only one way of decomposing needs

to

[50,47]

invariance.

together with

can

be

times,

invariant for

the

and so

factored

out,

subspace

of

definition

and

Obviously,

writing

the precompensator

the

(4.5) is

(4.1). For a further decomposition of (4.1) one

approach

for

dealing

with

not

necessarily

affine

nonlinear systems as automatically appear in (4.4). One such result for the system (4.1) as

the

is as follows,

involutive

closure

ef. of

[43]. Define the

assume this distribution is regular,

vector

the involutive

fields

distribution D k g, adfg ...... adfg, and

i.e. has constant dimension.

Then,

if

354 rank if,D] the

distribution

nonlinear

system

preeompensator requirement

(mod D)

~ I,

D

can

"factored

~

on

be the

state

P of dimension

equal

(4.6) out", space

thereby •

locally

(mod D),

to the dimension

yielding

together

of D. Note

(4.6) is in the case of a linear system automatically

for distributions

(- subspaces)

D of the form im b + A i m

a

with

a

that the satisfied

b + ...+ Akim b.

The above results on single input nonlinear systems can be viewed as a first step

towards

the

Decomposition

general Decomposition this

direction

is

Problem

for

nonlinear

Problem is usually not addressed.

certainly

needed.

The

relation

system. Further

(or

Today

the

research in

duality)

with

the

problems of section 3 can perhaps be exploited fruitfully.

References

[I]

M.A. Arbib, The algebraic theory of machines, Academic Press New York (1968).

languages and semigroups,

[2]

A. Bacciotti, "The local stabilizability problem systems",IMA Jnl. Math. Control & Inform. 5, pp. 27-39,

[3]

G. Basile & G. Marro, "Controlled and conditioned invariant in linear system theory", Jnl. Optimiz. Theory Appl. 3, pp. (1969).

subspaces 306-315,

[41

W.M. Boothby, An introduction to differential geometry, Academic Press, New York, (1975).

[5]

R.W. Brockett, "Feedback invariants for nonlinear systems", Proe. Vllth IFAC Worlds Congress, Helsinki, pp. 1115-1120, (1978). R.W. Brockett, "Global descriptions of nonlinear control problems, vector bundles and nonlinear control theory", CBMS Conference Notes, (1979).

[6]

manifolds

for nonlinear (1988).

and Riemannian

[7]

C.I. Byrnes & A. Isidori, "IIeuristics for nonlinear control", in Modelling and Adaptive Control, C.I. Byrnes & A. Kurzhanski (Eds.), LNCIS i05, Springer, Berlin, pp. 48-70, (1988).

[8]

C.I. Byrnes & A.J. Krener, "On the existence of globally (f,g)-invariant distributions", in Differential Geometric Control Theory, R.W. Brockett, R.S. Millman & II.J. Sussmann (Eds.), Birkh~user, Boston, pp. 209-225, (1983).

[9]

C.I. Byrnes & A. Lindquist (Eds.), Theory and applications control systems, North Holland, Dordrecht, (1986).

[i0

J. Descusse & C.H. Moog, "Decoupling with dynamic compensation for strong affine nonlinear systems", Int. Jnl. Control 42, pp. 1387-1398, (1985).

of nonlinear

355

[Ii] M. Di Benedetto, J.W. Grizzle & C.ll. Moog, nonlinear systems", SlAM Jnl. Control & Optlmiz.,

"Rank invariants of to appear, (1989).

[12] M. Di Benedetto & A. Isidori, "The matching of nonlinear models dynamic state feedback", SIAM Jnl. Control & Optimiz. 24, 1063-1075, (1986). [13] M. Fliess, "Automatique appear, (1989).

et corps

diff~rentlels",

Forum.

Math.

i,

via pp.

to

[14] M. Fliess, "D~compositions en cascades des syst~mes automatiques et feuilletages invariants", Bull. Soc. Math. France 113, pp. 285-293, (1985). [15] M. Fliess, "Cascade decomposlt£ons of nonlinear systems, follations and ideals of transitive Lie algebras", Systems & Control Lett. 5, pp. 263-265, (1985). [16] M. Fliess & M. }|azewinkel (Eds.), Algebraic and geometric nonlinear control theory, Reidel, Dordrecht, (1986).

methods

in

[17] R.B. Gardner, "Differential geometric methods interfacing control theory", in Differential Geometric Control Theory, R.W. Brockett, R.S. Millman & H.J. Sussmann (Eds.), Birkh~user, Boston, pp. 117-180, (1983).

"~Computing the differential 26th CDC, Los Angeles, pp.

[18] J.W. Grizzle, M. Di Benedetto & C.H. Moog, output rank of a nonlinear system", Prec. 142-145, (1987).

[19] R. Hermann, "Perturbation and linearization of nonlinear control systems", in Systems, Information and Control, vol. II, L.R. Hunt & C.F. Martin (Eds.), Math. Sci. Press., pp. 195-238, (1984). [20] R.M. Hirschhorn, "(A,B)-invariant distributions and disturbance decoupling of nonlinear systems", SlAM Jnl. Control & Optimiz. 19, pp. 1-19, (1981). [21] L.R. Hunt, R. Su & G. Meijer, "Design for multi-input nonlinear systems", in Differential Geometric Control Theory, R.W. Brockett, R.S. Millmann & l].J. Sussmann (Eds.), Bir~:h~user, Boston, pp. 268-298, (1983). [22] A. Isldori & J.W. Grizzle, "Fixed modes and nonlinear noninteracting control with stability", IEEE Trans. Autom. Control 33, pp. 907-914, (1988). [23] A. Isidori, A.J. Krener, C. Gori-Giorgi & S. Monaco, "Nonlinear decoupling via feedback: a differential geometric approach", IEEE Trans. Automat. Control 26, pp. 331-345, (1981). [24] A. Isidori, "The matching of a prescribed linear input-output hehaviour in a nonlinear system", IEEE Trans. Automat. Control, 30, pp. 258-265, (1985). [25] A. Isidori, Nonlinear control systems: Springer-Verlag, Berlin, (1985).

an

introduction,

LNCIS

72,

356

[26] A. Isidori, A. de Luca & F. Nicolo, "Control of robot arm with elastic joints via nonlinear dynamic feedback", Proc. 24th CDC, Ft. Lauderdale (FI), pp. 1671-1679, (1985). [27] B. Jakubozyk & W. Respondek, "On linearization Bull. Acad. Polon. Sci. (Math.), 28, pp. 517-522,

of control (1980).

[28] M. Kawski, "Stabilization of nonlinear Control Lett. 12, pp. 169-176, (1989).

in

systems

systems",

the plane",

[29] W. Korobov, "Controllability, stability of some nonlinear Differencialnyje Uravnienje 9, pp. 466-469, (1973). [30] A.J. Krener, "A decomposition theory for differentiable Jnl. Control & Optimiz. 15, pp. 813-829, (1977).

Syst.

systems",

systems",

SIAM

[31] K.B. Krohn & J.L. Rhodes, "Algebraic theory of machines: I. The main decomposition Theorem", Trans. Amer. Math. Soc. 116, pp. 450-464,

(1965). [32] M. Malabre, "Structure ~ l'infini des triplets invariants. Application & la poursuite parfaite de module", in Analysis and Optimization of Systems, A. Bensoussan & J.L. Lions (Eds.), LNCIS 44, Springer Verlag, Berlin (1982). [33] R. Marino, B. Charlet & J. Levine, "Two Sufficient conditions for dynamic feedback linearization of nonlinear systems", in Analysis and Optimization of Systems, A. Bensoussan & JoL. Lions (Eds.) LNCIS iii, Springer-Verlag, Berlin, pp. 181-191, (1988). [34] C.H. Moog lin~aire",

& J.W. Grizzle, "D~couplage non lin~aire vu de l'alg~bre C.R. Acad. Sci. Paris, t.307, S~rie I, pp. 497-500, (1988).

[35] B.C. Moore & L.M. Silverman, "Model matching by state dynamic compensations", IEEE Trans. Automat. Contr. 17, (1972).

feedback and pp. 491-497,

[36] A.S. Morse, "Structure and design of linear model following IEEE Trans. Automat. Contr. 18, pp. 346-354, (1973).

systems",

[37] H. Nijmeijer & A.J. van der Sehaft, "Controlled invariance for nonlinear systems", IEEE Trans. Automat. Contr. 27, pp. 907-914, (1982). [38] H. Nijmeijer & A.J. van der Schaft, "The disturbance decoupling problem for nonlinear control systems". IEEE Trans. Automat. Contr. 28, pp. 621-623, (1983). [39] H. Nijmeijer & A.J. van der Schaft, "Partial symmetries systems", Math. Syst. Theory 18, pp. 79-96, (1985).

for nonlinear

[40] H. Nijmeijer & J.M. Schumacher, "Zeros at infinity for affine nonlinear control systems", IEEE Trans. Automat. Contr. 30, pp. 566-573, (1985). [41] H. Nijmeijer & J.M. Sehumacher, "The regular local control problem for nonlinear control systems", SIAM Optimiz. 24, pp. 1232-1245, (1986).

noninteracting Jnl. Contr. &

357 [42] H. Nijmeijer & W. Respondek, nonlinear control systems", 1065-1070, (1988).

"Dynamic input-output IEEE Trans. Automat.

decoupling Contr. 33,

of pp.

[43] H. Nijmeijer, "Preliminaries on almost controlled invariance of single input nonlinear systems", Proe. 27th CDC, Austin (TX) pp. 932-933,

(1988). [44] W. Respondek, "On decomposition of nonlinear Contr. Lett. i, pp. 301-308, (1982).

control

systems",

Syst.

[45] A.J. van der Schaft, "Linearization and input-output deeoupling for general nonlinear systems", Syst. Contr. Lett. 5, pp. 27-33, (1984). [46] A.J. van der Schaft, System theoretic description of physical systems, CWI Tract 3, Amsterdam, (1984). [47] H.L. Trentelman, Almost invariant suhspaces and high gain feedback, CWI Tract 29, Amsterdam, (1986). [48] L. van der Wegen & H. Nijmeijer, ~'The local disturbance decoupling problem with stability for nonlinear systems", Syst. Control Lett. 12, pp. 139-149, (1989). 149] J.C. Willems, "System theoretic models for the analysis systems", Ricerche di Automatica i0, pp. 71-106, (1979).

of physical

[50] J.C. Willems, "Almost invariant subspaees: an approach to high feedback design - part I, almost controlled invariant subspaces", Trans. Automat. Contr. 26, pp. 235-252, (1981). [51] W.M. Wonham, Linear multivariable Springer-Verlag, Berlin, (1979).

control:

a

geometric

gain IEEE

approach,

Aggregation of Linear Systems in a Completely Deterministic Framework G. Picci Universita di Padova Dipartimento di Elettronica e Informatica Via Gradenigo 6/A, 35131 Padova, Italy and LADSEB-CNR, Padova, Italy

ABSTRACT:

Although

Professor

WILLEMS'exhortation

to

theologians

and

cosmologists to get into infinite dimensional Systems Theory could hardly be dissented with, complete

it is suggested

Behaviours

may

make

a

in

this paper

quite

that

enlightening

the

study of non

exercise

also

for

finite-dimensional minded System Theorists.

1.

INTRODUCTION

In

this

problem

for

A~gregation.

paper

we

study

Dynamical

a

Systems

particular that

we

representation

would

like

to

or

name

"modelling"

Deterministic

In very general (and somewhat imprecise) terms the problem can

be described as follows. One

would

like

to generate

the output

trajectories,

t ~

y(t),

of a

"large" autonomous dynamical system like

~(t)

= FCzCt)) zCt) ~ M c ~N

y(t) = H(zCt))

Cl.1)

359 by m e a n s o f a s m a l l e r

system with inputs,

dimensional

say

x(t) = f(x(t)) + g(x(t))u(t)

(1.2) y(t) = h(x(t)) where x(t)

~ X c ~n.

and that u be WILLEMS system

locally

[1986, (I.1)

The crucial free,

1988].

should

Of

be

requirements are

i.e. an "input"

course,

assumed

to

make

variable,

the

irreducible

that n
in

in the sense of

problem the

(aggregation)

interesting,

sense

that

the

no

lower

u(.))

such

dimensional autonomous system can generate the same output signal. If for every initial state z(O) ~ M there are pairs

(x(O),

that the output of the large system (1.1) with initial state z(O) and the output of the system (1.2) with initial state x(O} and input u('), are the

sane f u n c t i o n o f time,

we s a y t h a t

(1.1)

is

aggregable and that

(1.2)

is

an

aggregation of (1.1). Our motivation for considering aggregation problems of the type roughly

defined

above

originally

came

from

Statistical

Mechanics.

Some

common

grounds and motivations probably also exist with the notion of aggregation considered in Dynamic Economic Theory. See AOKI [1976, 1980]. In PICCI [1988, 1989] a stochastic version of the idea of aggregation is introduced and aggregability

is studied

in detail

Systems.

frame,

some

In

the

stochastic

at

for Linear

stage

the

Hamiltonian

large

system

is

"randomized" by introducing an invariant probability measure for the state evolution

flow.

The

output

y(.)

becomes

then

a

stationary

stochastic

process defined on the phase space M. One has then to search for a smaller dimensional

stochastic Dynamical

System

like

(1.2) where

now u

is white

noise, producing an output process stochastically equivalent in some sense to the output process of

(I.i).

The problem

is essentially phrased

as a

Stochastic Realization Problem. The

particular

case

of

Linear

Hamiltonian

Systems

is

especially

instructive since it can be analyzed in detail and from several different points of view.

In particular,

Linear Hamiltonian

Systems

since

the natural

are Gaussian.

invariant

the aggregation

measures

problem

can

for be

rephrased as a stochastic Realization Problem for Gaussian processes. The deterministic problem that we shall with Linear Hamiltonian Systems. with (and complement)

consider here will

still deal

In fact we shall eventually make contact

the "deterministic" approach

tried at

the beginning

360 of our previous In

this

paper

paper

PICCI

we

[1986].

shall

use

the

WILLEMS'Theory of Dynamical Systems.

language

and

the

general

ideas

of

In this framework the problem can be

formulated very naturally as a realization problem for a certain concretely defined Behaviour.

In this respect, a non-secondary goal of this paper will

also be to clarify, by means of an example,

some of the relations existing

between WILLEMS'Theory and the ideas of Stochastic Modelling formulated in Stochastic

Realization

LINDQUIST-PICCI

Theory.

[1985])

As

a

noticed

while

ago

(compare

there are some non superficial points of contacts

of the two Theories. A surprise, for some, may come from the discovery that the point at which

they really come

to overlap

is the

(linear)

infinite

dimensional "non complete" case. This case, which unfortunately is largely yet

to be worked out,

has

a very

rich

structure

and potentially

covers

interesting applications. A reason why infinite dimensional Systems and random processes are so strictly

related

objects

may

be

traced

back

(at

linear-Gausslan case) to the equivalence proven in PICCI

2

-

LINEAR

We s h a l l

EAMILTONIAN

least

for

the

[1988b].

SYSTEMS

formulate

the

aggregation

problem

in

a

linear

time-invariant

setting. The structure of candidate aggregate models will be defined first. In

a

lineav-time-invariant

framework

the

natural

candidates

are

linear

finite dimensional models of the type

X = Ax + Bu

(s)

y=Cx

where

x(t)

belongs

m-dimensional output

to

some

n-dimensional

vector

(or "external") variable and u,

space

X,

y

is

the

the latent "driving"

input, ranges on a vector space H of p-dimensional vector functions closed

under

concatenation.

In

particular

~

contains

all

finite

linear

combinations of indicator functions. The natural topology to be given to H will be discussed later. For the moment it will just be assumed to contaln all measurable functions

u: R9 RP for which (S) has a global solution for

each

This ~

initial state

under concatenation.

xEX.

is clearly a linear vector space closed

36t We s h a l l

o n l y c o n s i d e r models (S) t h a t a r e i r r e d u c i b l e

the r e p r e s e n t a t i o n

map ZS: X x ~ ~ (Em)a o f t h e s y s t e m ,

Zs(X,u)(t) = C eAtx + I °t C e A ( t - s )

is

on

injective

shift-invariant,

the

product

space

Bu(slds

X x

everywhere defined,

in the sense that d e f i n e d as

(2.1)

~.

Note

that

ZS

is

a

linear

map. Comparing w i t h WILLEMS [1983,

p.

588] it is seen that the range space of Z S, ~ : = ~ ( E s) is just the Behaviour of the system (S). Obviously a necessary condition for irreducibility is that observable pair.

(A,C) be an

It is also immediate to check that if (S) is irreducible

and p>O then its Behaviour ~

is an infinite dimensional vector space.

The

obvious consequence is,

PROPOSITION 2.1 A necessary condition for an irreducible linear autonomous system

{

z = Fz

~N

(2.2)

z(t) e y = Hz

to a d m i t

aggregate

models

of

system

(2,2) can be aggregable

Proof:

Let N be flnite,

the

type

only

if

(S) i s it

is

that

N=~.

infinite

then the Behaviour of

In

other

words,

the

dimensional.

(2.2)

is N-dimensional.

It

can therefore be described by an irreducible model of the type (S) only if p=O. But then both models are autonomous and observable and so necessarily n=N. So no aggregation is possible. D

We

shall

justifications

abstain and/or

from

commenting

interpretations

of

about this

possible

condition.

The

"physical" interested

reader may find some speculations in PICCI [1989]. Due

to

different

the previous question

than

observation we are forced the one we originally

had

to consider in mind,

a slightly namely

what

interesting classes of (infinite dimensional) linear autonomous systems are aggregable. A candidate is defined below.

362 DEFINITION 2 . 1 A Linear space)

Hamiltonian

System

is

a

is a real separable Hilbert

triple

{H,~(t),h'}

space,

{#(t)}te ~

where

H (the

phase

(the HamiItonian

flow)

is a strongly continuous group of unitary operators on H an m-tuple of linear-bounded functionals

and h":

H ~ ~m is

(the observables of the system)

i

hk(-)

=
The state

or

to z(t)

or,

of

the

system,

symbolically,

k = I ..... m

z(t) is

~ H,

evolves

a solution

in

time

according

of

= F z(t)

F is

tile

operator).

(2.3)

generator

of

~(t)

(a

real,

self-adjoint,

densely

defined

Each scalar observable produces an output signal,

Yk(t)

The

h k e H,

"phase"

= ~(t)z(0)

z(t)

where

.>,

=
(column)

output

vector

variable of the system.

y(t)

=
w h e r e
t ~ N

z(t)>

z(t)>

y(t)=col[Yl(t)

k=l ..... m

.....

.

Y m ( t ) ] ~ ~m i s

the

external

It will be written as

=

z> : = c o l [ < h I ,

A Linear

,

gamiltonian

fixed vectors for ~(t),

z> . . . . .

System

(2.4)

] , is

zaH.

"nonsingular"

i.e. ~ ( t ) x

= x,

V

if

there

are

no nontrivial

(This is the same as F

t, ~ x=O.

having no zero eigenvalues). A Linear Hamiltonian System is "irreducible"

span

(~(t)hk;

i.e.

h I .....

~(t).

This

turn,

to

that

the

hm are is

k=l,...,m,

to

the non existence Linear

t ~ ~} = H

generators

equivalent

if,

for

the Hilbert

t h e map of

Hamlltonian

(2.5)

space

z~

a proper System

~(t)-invariant {Ho,~o(t),

~(t)

and h

i

to to

{H,

~(t),

h*}.

the

invariant

ltere subspace

•

0

(t)and H . 0

m

h ) m

behaviour)

h

H

with

being

0

respect

injective

subspace is

and, II

o

to in

c H such

equivalent

(same

0

are

the

restrictions

of

363 The terminology may seem arbitrary but is not. As reported in PICCI

[1988],

it can be shown that the phase space of any nonsingular Linear Hamlltonian System has a natural symplectic structure i.e. it can always be split into the orthogonal direct sum of two isomorphic real Hilbert spaces. II=ll ® }I P q and that there exists a symplectic bounded operator J on H, i.e. an o

operator satisfying j2 = -I, J = -J, having the matrix representation m

J

=

E

0

where E: =J[H : Hp ~ Hq i s u n i t a r y . P There i s an i n j e c t i v e positive operator)

self

on H f o r w h i c h t h e o r t h o g o n a l

commutes with E

(and E*)

adjoint direct

i.e. VE = EV and

operator

V (the

sum H e H i s P q

the generator

potential

reducing.

of ~(t)

V

can be

expressed as

[° vl

F

=

EV

0

The "differential equation"

(2.3) for z(t) = col(p(t),q{t))

then has

the

c a n o n i c a l structure

p e t } = -E°V qCt},

It is not hard equations of transform

(2.6)

qCt) = EV p e t )

to see that this is the general form which

a system with

into

under

the

a general linear

the canonical

nonsin£ular quadratic Hamiltonian,

transformation

which

normalizes

the

quadratic Hamiltonian to the "energy norm" I/2 Iz~ 2. The

coding

terms of Def.

of

infinite

2.1

is at

dimensional

linear

Hamiltonian

models

the roots of LAX-PHILLIPS scattering

in

the

theory for

hyperbolic systems. More examples are given in LEWIS and MAASSEN [1984]. There are plenty of interesting examples of infinite dimensional Linear Hamiltonian

System

problem.

may

We

for

add,

which on

the

aggregation

is

indeed

mathematical

side,

that

a

deep, evolution

relevant models

governed by a unitary group can always be seen as dilations of a very broad family

of

possible

linear

infinite

dimensional

autonomous

systems

(see

FUILRMANN [1981] for details on this). So the theory could really be given a much wider scope than the one we restrict to in this paper.

364 3 - ON THE B~VIOUR

SPACE

OF A L I N E A R

HAMILTONIAN

SYSTEM

The Behaviour space of a Linear Hamiltonian System is

~:

= {yz (-}

=
w h e r e we u s e y z ( . ) initial

state

z.

¢(-)z>;

to denote

Note that

has a natural

real

functions

multiplication

and

vector

(3.1)

z ~ H}

the dependence

}f i s space

of the output

trajectory

made o f R m - v a l u e d c o n t i n u o u s structure

by r e a l

with addition

scalars

defined

on the

functions

and

of Nm-valued vector in

the

obvious

way. t

llenceforth we shall assume that the Linear Hamiltonian System {If, ¢(t), h } is non singular

and irreducible.

This

in particular

means

that

the finite

sums

T @(tk)h k

Z=7-~

~k ~ ~m

(3.2)

are dense in H. We shall introduce


T:

yw(.)>2¢: z-)yz(.)

defined

eVo},

(S(t)f)(s)

= is

(3.3)

linear

and makes J{ i n t o

t h e n we o b t a i n S(t),

in J( the inner product

and a real

a system unitarily

where = f(s÷t),

S(t)

is

the

tnjective Hilbert

equivalent left

and e v ( f ) = f ( O ) , o

by

space. to

translation is

irreducibility,

(3.3)

By means o f

{H, @ ( t ) , group

the evaluation

on

the

T map

h }, d e n o t e d time

map a t

is

{H,

functions,

zero.

i

¢Ct)

. H T . . . . . ~ ~m

H T I

1 S(t)

We define

¢(t),

h (3.4} ev o

the covariance matrix

A(.)

of

ttamiltonian

S y s t e m (H,

h } as

Akj(t):

=

k , j = 1. . . . . m .

Using a self explanatory vector notation, as

the Linear

(3.5)

we may express h(.) directly also

365

A(t) = .

LEMMA 3.1 The Behaviour

of a n

space

irreducible Linear Hamiltonian

System

is

generated by the translates of the columns of its covariance matrix i.e.

= span {SCt)A(.)~

where

the closure

product

<.,.>~.

;

is with

Moreover

~ ~ R m, t ~ ~}

respect ~

is

a

to

(3.6)

the metric

reproducing

induced

kernel

by

the

Hilbert

scalar

space

with

hl,...,h m for H are mapped

into

reproducing kernel A(.).

Proof:

Under the unitary map T the generators Yhl(') ..... Yhm(')'

that is into the columns of the covariance

(compare (3.5)). Then the columns are generators group.

translation

T[~ (t)hj]= Aj(*-t)

The

reproducing

for J{ with respect

kernel

property

to the

follows

from

(~-th column) and therefore,

yz(t) = <~ (t)h, z> = col{<~ (t)hl,Z>...<~

= col {l{ .....
•

matrix A(.)

=
(tlhm, Z>}

(3.7)

yz>/t}

yz>~ []

Since

A

is

a

continuous

posltive

definite

kernel

it

has

a

Bochner

representation

ACt) =

+~

I

e

iXt

(3.8)

dN(iA)

where M is a finite positive Hermitian matrix valued measure, distribution

measure

imaginary axis.

of

the

system,

It is well known

defined

on

that M contains

the

the spectral

Borel

sets

all spectral

about the unitary group $(t) on H. In fact, by the spectral

of

the

information

representation

theorem, $(t) acting on H is unitarily equivalent to multiplication by iAt e on the Hilbert space L2(O, dH) of m-dimensional measurable functions m

366 on the imaginary axis, square integrable with respect to the matrix measure M

(see e.g.

FUHRMANN

corresponding to in

[1981]).

z e H

It

is customary

to write

the function fZ

as a row vector function so that the inner product

L2(@, dM) (~ L2(dM)) i s w r i t t e n as

m

+~ = f_ L2(dM) ~ f z 1 (-iA)dM (i~) f ~ (i~)

(3.9)

We now come to an important result which will be instrumental in connecting aggregation

to

"spectral

factorization",

as

explained

in

the

the

next section.

PROPOSITION 3.2 Yz ~ Jt

Every element

has a spectral representation

in LZ(dM)

of the

form

yzct : where f

ei dMCi ) (i )

is the spectral

(3101

representation of the initial phase z in L2(dM).

Z

Tile inner product jf is equal to the right hand member in (3.9).

The p r o o f of (3.10) f o l l o w s from the formula

Proof.

yz(t) = <@ (t)h, z>

after expressing the inner product in the right hand side by means of the representatives in

L2(dM).

D

For finite dimensional Linear Hamiltonian Systems finite number of jumps at points suppose

that

@(t)

t iw k

has absolutely

Ll-valued nonnegative matrix

M(iA)

has just a

k=l ..... N/2. On the opposite side,

continuous

spectrum.

Then there is an

@ such that

M(A) = F ~(iA)dA )& for every Borel set representation convolution.

& ¢ 8. It is not hard to imagine that in this case the

(3.10) may be made to look like the Fourier transform of a In

spite

of

this

encouraging

appearance

an

additional

367 difficulty

p o p s up i n

this

case however,

dimensional function s p a c e ,

as J{ b e h a v e s l i k e

"complete" in the sense of WILLF3~S [1986, be

in

a

sharp

a truly

in particular it is not finitely

contrast

with

the

1988].

famous

infinite

determined

or

This fact may be feared to

equivalence: , Completeness

realizability by a finite dimensional linear model of the type (S), so much advertised

in WILLEMS'work.

(Although

we

are

not

statement of this kind in the continuous-time the next

section

that

in

the

present

aware

case).

setting

the

of

any

explicit

It will be clear difficulty

is

in

indeed

present and so we shall have to do something about it at the very problem formulation

level.

Let V be a vector introduce

elements

space of Rq-valued

time functions

the subsets Y a+ and V-a of forward, f • V, say

V a+ = { f +a

:

and s i m i l a r l y

for

[a,

;

+~1 ~ ~ q

f:(t)

defined

and backward

= f(t),

tmO

on R. We

truncations

of

f e Y}

Y a- .

DEFINITION 3 . 3 The l i n e a r

finite-dimensional

x+=A+x++B+u

is a forward tile "forward"

Z+a : X x

model

y = C+x+

realization

o f Jf i f ,

representation

(S+}

for

every

finite

a • R the range space

of

map,

-> (Nm) [a, +-) I/+ a

where,

Z a+ (xCa) ' u)Ct) = C+e is

g a+ . Dually,

taa

we say that

x_ = A_x_ + 13_u ,

is a backward

A+(t-a) it A+(t-v) x(a) + a C+e B+u(~)d¢,

realization

y = C_x_

of

H if

(S_)

for

every

finite

a

the

range

space

of

368 the

"backward" representation map,

Z-: X x ~ - ~ (~m)[-~,a] a

a

where, _ A_(t-a) t A_Ct-¢) Za(X(a),u)(t) = C_e x(a) + ~ C_e B_u(~)A~,

t_
a

is the whole of

~-a

Clearly a forward or backward realizatlon of a complete system is just a bonafide realizations in the sense defined in the previous section. This is so

because

subspace.

of An

the very

definition

of

incomplete behaviour

a

may

complete

(=finitely

instead admit

forward

determined) or

backward

realization although not admitting finlte dimensional realizations.

REMARK In spelled

the

above

out

in

Definition sufficient

there

is

detail.

one

Of

technical

course

the

point

which

question

generate ~f as an (infinite dimensional) Hilber£ space.

is not

here

is

to

In order to do this,

+

Z- should be viewed as Hilhert space operators and, ~ (and X) should also a be given a Hilbert space structure. (In fact this should be done also for +

the appropriate truncation spaces ~ ,

+

+

+

~-etc.).a The equalities H-=~CZ-)a a are

+

then understood to mean that Z-a are bounded linear operators acting on the +

appropriate

Hilbert

spaces

X x

observation brings us back

~-a

+

with

closed

to the question of

range, equal the natural

to 9e-. This a topology

that

should be given to ~. It seems difficult

to give a general answer to this question.

It will

however be shown in the next section that for aggregable Linear Hamiltonian

Systems

(in

the

sense

that

H

admits

a

finite

backward realization of dimension strictly smaller

dimensional

forward

or

than N = dim ~f ) the

input space ~ must have the Lebesgue space L2(~;~ p) structure. The

study

realizations

of of

the

relations

JC will

have

existing to

be

between

delayed

forward

until

some

and more

backward precise

information on the spectrum of A+ an A_ will be available. See Theorem 4.7.

369 NECESSARY

4.

In this

CONDITIONS

section

Linear Hamiltonian

we

FOR A G G R E G A B I L I T Y

study necessary

System.

conditions

The class of candidate

class of finite dimensional

linear systems

sect.

however

2.

will

Aggregation

be

discussed at the end of the previous

for

aggregability

aggregate

models

of a is the

(S) defined at the beginning of

understood

section,

in

the

weakened

namely we shall

sense

say that an

(irreducible) Linear Hamlltonian System is aggregable if there is a forward

(S+) or a backward (S_) realization of its behaviour space J~, of dimension strictly smaller than N(=dimK).

Of course,

since N=~ has already been shown

£o be a necessary condition for aggregabllity we just require realizability by a finite dimensional

forward or backward model.

LEMMA 4.1 (S)

Let

be

an

irreducible

Behaviour space J{ of an

(forward

or

backward)

infinite dimensional

Linear

realization

of

the

Hamiltonian

System.

Consider the decomposition of the state space of (S), X=Xi®X ±, where X i is the purely imaginary (generalized) eigenspace of A and let xi=col(xi,Xz)

be

a representation of x relative to the above direct sum decomposition. Then,

i)

The

purely

imaginary

equivalent to

pXiB

Xi

eigenspace

is

uncontrollable.

This

X.

= 0 (P l=projection onto X i alon E X+) or to

is

~(B) c

c X±.

ii)

(S) has a direct sum decomposition,

~.1

=A.x. l

1

x+ = Atx± + Btu

y = Cix i + Ctx± where minimal

Ai=A]x i,

At=A]x ±,

triple,

in

(Ai,C i)

iii)

Jf a d m i t s a s h i f t

invariant

1

and is

(A+,C+,B+) is a controllable.

r e d u c i n g s u b s p a c e ~{. which i s r e a l i z e d

the subsystem xi=Aixi , Y=Cixi, trajectory y E ~.

observable

(A+,B+) column rank p.

particular

i r r e d u c i b i l i t y B± has a l s o f u l l

is

1

By

by

of the same dimension ni=dim X i. Every

is a linear combination of n. simple real harmonic i

370 oscillations.

Proof:

Note

that

each

trajectory

in ~

is an

uniformly

bounded

function

as

lYkCt)l Rhkll I111 by C2.4 . i)

Because of observability each imaginary mode shows in the output and therefore and)

it cannot

(have Jordan blocks

be controllable,

suitable

input u e ~

for otherwise producing

of

length greater

it would be possible

an unbounded

output.

We

than one to find a

stress here

that such an input u can be found even in ~=L2(R;RP) (')

ii)

Recall

that

Assume

X±

(A+,B+).

the eigenmodes of A t diverge as t goes is

strictly

Then

there

trajectories

(evolving

observability trajectories.

they

larger are on

show

than

the

initial

states

subspaces)

in

which

unaffected

the output

and

to +m or to -m.

controllable

produce

by

produce

subspace

the

for state

inputs.

unbounded

By

output

Hence the subsystem (C+,A+,B+) must be controllable.

iii) The eigenvalues of A. must also be eigenvalues of F. The corresponding 1

eigenspace

tI.

1

is

reducing

for

O(t)

by e l e m e n t a r y

spectral

theory,

so

the same holds for ~ = TH.. 1

By

the

above

result

nothing

really

interesting

(from

the

aggregation

point of view) is attached to the imaginary eigenspace of a realization of

Jr.

The purely oscillatory component,

x.=A.x.I 1 i' Y=Cixi'

of

(S) describes a

shift invariant subspace H.c H of the same dimension n.. In fact (assuming 1 1 irreducibility) this subsystem is isomorphic to a purely oscillatory i

component,

zi = Fizi,

y=h (zi), Fi=FIHi,

of the original Linear Hamiltonian

System. For this reason otherwise

in the following we shall feel free

explicitly

stated,

that

the

oscillatory

to assume,

subsystem

has

unless been

removed from any realization (S), tacitly assuming that the same operation

was done also on the given Behaviour space H.

Professor E. Fornasini ha 9 been very how to construct such an L ~ input.

helpful

with

this,

showing

to me

371 Recall now the notions of i n c o m i n g and outgolng subspaces group U(t) on a Hilbert letting St:=U(t)S,

for a unitary

space }|. A subspace S c H is incoming for U(t)

one has U (t] S c S for tzO

(i.e.

if

S t is increasing in

time) and V

St = H ,

~

tER

Dually,

S t = (0}

(4.1)

tER

S c II is outgoing for U(t) if U(t)S c S [i.e. St is decreasing for

t increasing)

and m

V St = H t~R

,

n S t : {0} t~R

(4.2)

(the wedge means closed vector sum). in (4.1) and

Note that the intersection

(4.2) really only regard

the tail behaviour

condltions

of {S t } at

t=-m

and of {S.} at t=+~. t THEOREM 4.2 Assume that the Behaviour space Jt of a Linear llamiltpnian System has a forward realization such

(S+),

V a e N, V z e tl 3 x ( a )

i.e.

e X

and

u

~ 11a÷'

that

yz(t) = C+e

A+(t-a) it A+(t-s) x(a) + C+e B+u(s)ds

(4.3]

a

for all

tza.

Then the translation group S(t) on H has an outgoing subspace.

Proof:

Define

A+(t-a) Sa:= {y e ~; 3 x e X such that y(t) = C+e

By assumption

Sa

is a

subspace

of J{.

(4.4)

x , for tza}

(Its elements

are

obtained

by

setting u=O in (4.3)). m

The family {Sa}aE N is defined by letting S a = S (a)S o. It is easy to see that S

~ S 0

for a ~ O, for if y ~ S a

y(t) = C+e

then, o

A+(t-a) x(a),

A+a x(a] = C+e

x(O)

372 for all

t m a. Moreover by a s s u m p t i o n each t r u n c a t e d space

and so for every

y e ~,

each segment Y[-~,a]

belongs,

S-a

c o n t a i n s ~ -a

for any

a e R,

to

the backward a-truncation of the space

S:

=

m

V ae~

S

This means that

S --O0: = ¢ ~ S

a

S

= H. Finally,

a

a

consists of those functions

y(t)

= C+e

y e ~ such that

A+(t-a) x(a)

for

t->a , A+ t

f o r e v e r y a ~ R. on

the w h o l e

This means that all y(,) e S

time axis.

irreducible.

Now, we may without

are of the form t-> C e

x,

loss of generality assume

(S+)

--O~

4-

Setting u=0 in (4.3) and invoking observability of

reach the conclusion that A + must be stable,

(C+,A+) we

i.e.

Re A(A+) < 0

(recall all

that

(4.5)

we f a c t o r e d

nonzero element of S

out

the

imaginary e i g e n s p a c e of

above

proves

that

S

is 0

translation S

0

group.

But t h e n

would have to be unbounded a s t g - ~ c o n t r a d i c t i n g

uniform boundedness of each y e ~. Therefore S

The

(S+)),

incoming for

= {0} .

the

adjoint

S (t)

of

the

^

Now d e f i n e Sa:

a ~ R. Then Sa = S ( a ) ~ o '

= S_a,

a e N and

is outgoing for S(t). []

REMARK A totally analogous statement holds in case H has a b a c k w a r d realization (S_). The stability of A+,

implied by the forward representation property,

is now replaced by a n t i s t a b i l i t y

of A_,

i.e.

Re A(A ) > 0 Then, that

by arguing

(4.6) exactly

as

in the proof

of Theorem

4. Z it can be

shown

373

a

: = {y ~ ~; 3x ~ X such that y(t) = C e

form a n i n c r e a s i n g ag+~

and

shrlnks

family to

of subspaces

the

S(t)

translation group

zero

A (t+a) x, for

which fills

function

when

t~a}

(4.7)

up t h e

ag-~.In

whole of

other

~ as

words,

the

admits an incoming subspace. []

Theorem

4.4

aggregable

describes

Linear

system-theoretical

a

key

structural

condition

Hamiltonian

Systems.

implications

w h i c h we s h a l l

This

to

condition explore

order to do this we first need a representation

be has

satisfied

by

a

of

number

in the following.

In

Lemma.

LEMMA 4.3 Let

+

Z :

L2(R) q

translation

~

g

be

a

linear

bounded

operator

commuting

with

i.e.

Z+S(t)f = S(t)Z+f

A s s u m e Z + maps L ~ ( - m , a ) T h e n Z+ r e s t r i c t e d

Vf ~ L2(~) q considered

'

to L2(-~,a) q

has

as a subspace

o f L2(~) q

'

onto

S a , Y a ~ R.

the representation,

(Z+f)(t) = ~a W+(t-s)f(s)ds,

t ~ N

(4.8)

A+ t where W+(t) = C+e

A dual

result

holds

N+ for

tzO

any

for

and is zero for

translation

invariant

incoming

subspace

t
Z-

:

L2 [IR)--)R q

"9

mapping

L~(a,q +~)

onto

the

9"a, ¥ a e IR.

Proof:

Pick f ~ L2(R y(t) = (Z+f)(t), for t~O, can be written as q - ), then A +t C+e x(O) for a unique x(O). There is then a well defined linear map f ~ x(O)e

X from

L~(R_)

( a s s u m i n g X= Nn w i t h o u t x(O) = ~0

onto

the

n-dlmensional

loss of generality)

Q(s)f(slds

linear

must n e c e s s a r i l y

space

X,

which

have the form

374 for

some nxq matrix Q with rows in

L2(~ q - ).

Now

S(-a)u

is

mapped into

S(-a)y ~ S a so that y(t-a) = C+e

A+(t-a)_a J_ Q(a,s)f(s-a)ds,

But S(t-a) also belongs to S

t~a

.

and therefore O

y(t-a) = C+e

A+(t-a) h+a e x(O)

t->a

. [3

Comparing the two expressions we get the conclusion.

Now, resp.

by a fundamental outgoing,

result

subspaces

Sa,

existence of unitary operators

Z+: L2(~) ~ R q

and

in Analysis,

S,

for-St(t)

the existence on

of

incoming,

If is equivalent

to

the

(called translation representations),

Z-: L2(~) ~ R q

commuting with translation and also inducing unitary maps,

Z+: L2(-~,a) ~ S a, q for

all

a E ~.

translation

Z-: L2(a,+~) 9 q

(I~re

S (t),

we have

as it

chosen

seems

to

a

to work

be

with

easier

the

to

adjoint

"left"

visualize).

The

o

S (t)-outgoing subspace S a is ~ - a with ~ a defined by (4.7).

multiplicity of S(t)

the

vectors

for

read-out H. Compare

~)., We

map

h

have

are

a

on

R

q~m as

(l.e. the

(generally

the

minimal

number

m representatives non

minimal)

set

of

generating

hi,..., mh of of

generators

the for

(3.4).

Using Lemma 4.3 we immediately obtain a representation for E + and a similar backward representation

(Z-f)Ct) = ~

W_Ct-s)fCs)ds, a

where

The number q is

At W (t)= C e - N

f ~ L2(a,+00)

q

of the type (4.8)

for Z-,

(4.9)

for t~O and zero for t>O. Since both Z + and Z-

are

375 Injective on L 2 the matrices N+,N q

both have full column rank q.

translation representations

are Just

~+,Z-,

the

Input-output

Now the

maps of

two

linear systems x + = A x+

y=Cx

+

+ N f+

+

cs+)

+

and =Ax

+Nf

y=Cx

both acting on the input space L~(~). and ( S )

It follows from unitarity that

are irreducible forward and backward realizations of R

as they have the same state spaces as the original (S_),

it must hold that B+=N+Q+,

matrices.

B =N Q

. Moreover,

realizations

where Qt are qxq

(S+)

(S+) and

nonslngular

So for an irreducible realization the number of inputs p must be

equal to q = multiplicity of S(t). Actually more is true.

COROLLARY 4.4 There i s j u s t one choice o f the

input

realization space of

alphabet)

by

which

the input space ~ (modulo isomorphisms any

irreducible

becomes a one to one norm preserving

isomorphism).

This choice

is ~ = L2(~), q

forward

(or

map onto J{ (i.e.

where

of

backward) a Hilbert

q is the m u l t l p l l c i t y

S(t}.

Proof:

The c o r o l l a r y i s j u s t a r e s t a t e m e n t of the uniqueness o f t h e t r a n s l a t i o n r e p r e s e n t a t i o n . There i s make (say) unitary

the

map

a technical detail

forward r e p r e s e n t a t i o n , Z+ : a

starting

from

Z+:

~ Jr+ into a

a

involves

~/ = L2(R).- For i r r e d u c i b l e systems p = q, and P realization can be identified with a translation

From now on we s h a l l f i x

irreducible

unitary

X x L2(a,+~)

L2(~) ~ J(. Thls q introducing a suitable norm on X, but we shall skip the details.

every

a

on how one should n a t u r a l l y

376

representation fact

has

of

representation realization fact the

H.

implicitly

It

is

theorem

o£ H

worth remarking at

already

been used)

there

is

no

as a r e p r e s e n t a t i o n of

entire trajectories

this

that,

need

point

because

to

think

of of

h o l d i n g o n l y on h a l f

(although the

this

translation

(say)

a

forward

lines

[a,+~].

If are generated by the input-output

In map

u g Z +u' u E L2(R), of the realization. This is equivalent to saying that q the system (S+) is started at the "boundary state" x(-m)=O at t = -m. Similarly,

any backward realization generates J{ starting from the

boundary

x(+~)=O,

state

All the preparatory

work done so far eventually

and neat characterization

leads to a very explicit

of aggregabillty.

THEOREM 4.5 If

a

Linear

Hamiltonian

System

is

aggregable

distribution measure M is absolutely continuous.

of

a

forward

realization.

Then

W+

its

is

an

spectral

@(iA) be the

In fact, let

and W+(s) = C+(sI-A+)-IB+

mxm spectral density matrix of M function

then

be the transfer

analytic

Re>O)

(in

spectral factor of • i.e.

e(i~) = w+(i~)w+(-i~) T Similarly,

the

transfer

(SF+)

.

function W (s)

= C (sl-A)-IB

realization is a coanalytic (analytic in Re
of

spectral factor,

@CiA) = W (iA)W C-iA) T .

a backward i.e.

(SF)

In particular, @ must be a rational function of IA.

Proof:

Let

(S+)

be a n

representation T-1 o f

(3.4),

irreducible

forward

c o r r e s p o n d i n g to yielding

II. If we rename

realization

W+(-s):= V+(s),

translation the equality

+m "

c a n be r e w r i t t e n

as

I

~.

The

(S+) c a n be composed w i t h t h e

a (unitary)

< • (t)h, z > = yz(t) =

of

W÷(t-s)u(s)ds

representation

translation unitary

map

T+: L~(~)

377 < ¢ " (t)h,

z > = <

S" ( t ) V+,

u

>L2(~ ) q

for

all

corresponding

pairs

(u,z)

under

T +.

This

clearly

implies

that

under

T+, h k corresponds to the k-th row W+,k(-.)T of W+(--). Note Incidentally, that W+,k(-.) is in L2(~). Hence the k-th column of the covarlance matrix q of the system has the expression

,hct)

W+(t-s)W+k(-s)Tds

=

--C0

for

k=l .....

m.

This

relation

is

equivalent

to

(SF+).

A totally

analogous

argument also works for (SF). Q

As we s e e , spectrum number of

of

the @(t)

existence which

imaginary

has

of to

aggregate be

eigenvalues

of

to

models

depends

ultimately

on

perhaps

a finite

Lebesgue type ( p l u s take

into

account

also

the

the

oscillatory

component which we decided to ignore in order to simplify the exposition) and on the observables

.

An

interesting question

in this respect

would be to "design" the observables of a Linear Hamiltonlan System (known to

have

an

evolution

operator

¢(t)

of

finite

multiplicity

and

with

"essentially" Lebesgue Spectrum as described above) in such a way to obtain rationality of the spectrum. Let us very explicitly point out that the condition of rationality of the Spectrum stated in the Theorem is also sufficient

for aggregabllity.

This has been shown (using a slightly different language) in PICCl [198S] and we shall not repeat the argument here. More then that, any full rank rational rational

analytic

solution

coanalytie

respectively,

of

solution

(SF+) of

(SF),

a backward representation

spectral representation (3. I0), which,

and,

respectively, provides

a

any

full

forward

rank and.

of J(. This is immediate from the,

given any such solution W of (say)

(SF+), can be rewritten as Yz (t) = [+~e iAt W(IA)u(IA)dA

where

(4.10)

378

u (ix): = w

This

u is

just

(_IA)Tf~(ix) the

(4.11)

drlvlng

input

function

phase z o f t h e L i n e a r t l a m i l t o n i a n System.

corresponding

Note t h a t

to

the

initial

by i r r e d u c i b i l i t y ,

the

functions

Rm

iAtk ,

fz(iA) = X a~ e span L2(dM)

(compare

mk ~

(3.2) and therefore

the inputs given by (4.11) span

exactly L 2. Here, of course, the number of inputs is determined by the full q rank condition on W. Recall that q = rank W = rank @ = multiplicity of s(t).

COROLLARY 4 . 6 The t r a n s f e r

W+ o f t h e i r r e d u c i b l e f o r w a r d r e a l i z a t i o n s of R modulo m u l t i p l i c a t i o n f r o m t h e r i g h t by a c o n s t a n t o r t h o g o n a l qxq

(defined matrix)

are

degree

(also

functions

the a n a l y t i c called

spectral minimal

factors stable

o f • w h i c h have m l n i m a l spectral

factors).

analogous statement holds for the transfer functions

W

A

HcMillan

completely

of the irreducible

backward models.

Proof: ^

We o n l y need t o p r o v e t h e e q u i v a l e n c e b e t w e e n m i n i m a l i t y o f W+ a s a f u l l rank spectral

factor

and i r r e d u c i b i l i t y

of

the c o r r e s p o n d i n g r e a l i z a t i o n .

Suppose W+ i s non m i n i m a l , then it has a c o p r i m e f a c t o r i z a t i o n Q a nonconstant rational

i n n e r m a t r i x and W, o u t e r .

"Coprime"

W+:W.Q w i t h means t h a t

the McHi I lan degrees of the factors add up to the HwMillan degree of W+. Now it easy to check that for any minimal realization of Q, say, XQ = AQXQ + BQu dim xQ(t) = nQ

(Q)

yQ = CQXQ + u

for every

X=XQ(O)

there is a (feedback)

input u(.) which makes yQ(t)=O,

Vt ~ N (i.e. V'= NnQ). This implles that the representation map ZQ:XQ x ~ 9 (~q)~ o f realization

the

(minimal) system

o f W has s t a t e

(Q)

space

is

certainly

not

injective.

A minimal

X = X. • XQ and hence i f nQ>O i t

cannot

379 be

irreducible

since

for

initial

states

x=col(O,XQ)

there

is always

an

input function producing Identically zero output.

We shall

finally

brin E out

the relation

realizations.

There

is a natural

of

in

certain

H

which

realization

a

is modified

sense

under

between

forward

and

pairing of forward-backvrard describes

"time

how

reversal".

a

given

backward

realizations (irreducible)

In algebraic

terms,

this

^

correspondence

ties

together

solutions

(W+,W_)

of

(SF+)

and

(SF_)

havln E

the same zero structure.

THEOREM 4.7 Let

(S+)

be

an

irreducible

forward

companion backward realization ( S ) in the backward direction,

realization

Ya = Za (x,v)

The realization ( S )

There

is

such that,

¥ a ~ ~ .

t = a f r o m t h e same i n i t i a l

state

as

(S+),

x_(a) = x+(a).

has parameters

A_ = -PA~P -I .

B_ = B+

C_ = C+

where P is the unique simmetric solution of the Lyapunov equation

A+P + PAl + B+BI = 0

The input trajectory

as

v

a

V a e

then there is a corresponding input v of ( S )

at

~.

i,e, if y is described as:

Ya+ = Z a+ (x.u)

(S_) i s s t a r t i n g

of

describing the same output trajectories

driving

the backward

the forward system)

systems

is given

(and producing

the same output

by

v(t) = u(t) - BTp-Ix+(t) n

380 6.

EPILOGUE

As t h e

picture

is

starting

to

become q u i t e

recognizable

now a v e r y good t i m e t o end t h e s t o r y .

The s t r i k i n g

present

setup

and t h e

Stochastic

exposed

e.g.

in

details,

at

we

should

Llnear-Caussian

LINDQUIST

the risk just

PICCI

of boring

llke

to

[1985]

point

out,

be

pushed

to death. very

Is probably

similarity

between the

realization

could

the reader

it

framework as to

Instead

briefly,

the

extreme

of doing this

some

unexpected

"deterministic" versions of phenomena which a priori might be tought to be peculiar only of "stochastic" modelling. One such thing is Irreversibility of the aggregate description, which we would llke to define as the capability of a model of describing the given behaviour H only in one direction of time. A forward model, starting in the state x(t ) only describes the trajectories of H for tat . If we want to o o look backward (i.e. "reverse time") and describe the trajectories of ~ for t~t

we

need

to

use

the

backward

companion

model

which

is

definitely

0

(A+ cannot be similar to -A~) from the forward one.

different

This irreversibility at the aggregate level, contrasts sharply with the reversibility of the "microscopic" Hamiltonian system F).

In

linear

a

fascinating puzzling

context,

paradox

physicists

of

the

present

Statistical

since

the

end

theory

Mechanics of

last

(where instead -F =

"explains" which

seems

century.

a

famous

to

WILLEMS'

have

and been

notion

of

completeness has been instrumental in this explaination.

REFERENCES

AOKI M.

[1976].

On fluctuations

in microscopic states of a large system. Y.C. Ho and S.K. Mitter eds.

Directions in Large Scale Systems,

Plenum Press New York. [1980]. Dynamics and control of a system composed of a of similar subsystems. Dynamic Optimization and Economics. Pan Tai Liu ed. Plenum Press New York.

FUIIRMANN P.A. [ 1 9 8 1 ] . L i n e a r S y s t e m s and O p e r a t o r s graw } t i l l New York.

large

number

Mathematical

Spaces.

Mc

LAX P.D., PHILLIPS R . S . [1967]. Scattering Theory. Academic Press, York

New

in llilbert

381 LINDQUIST A., PICCI G. [1985]. Realization theory for multlvarlable s t a t i o n a r y G a u s s t a n p r o c e s s e s . SIAM J. Control Optim. 23, 8 0 9 - 8 5 7 . LEWIS J . T . , MAASSEN 11. [ 1 9 8 4 ] . H a m i l t o n i a n models o f c l a s s i c a l and q u a n t u m s t o c h a s t i c p r o c e s s e s . Quantum P r o b a b i l i t y and Applications to the Quantum Theory o f I r r e v e r s i b l e P r o c e s s e s . L. A c e a r d i , A. F r l g e r l o and V. Gorlnl eds. Springer L.N. in Mathematics, lOSS, Springer Verlag. PICCI G. [ 1 9 8 6 ] . A p p l l c a t l o n s of stochastic realization theory to a fundamental problem of statistical physics. Hodelling Identification and Robust C o n t r o l , C . I . B y r n e s and A. L i n d q u i s t e d s . N o r t h H o l l a n d . [1988a]. H a m l l t o n t a n r e p r e s e n t a t i o n of s t a t i o n a r y p r o c e s s e s . Operator T h e o r y Advances and A p p l i c a t i o n s , 35, pp. 193-215. [ 1 9 8 8 b ] . S t o c h a s t i c a g g r e g a t i o n . L i n e a r C i r c u i t s S y s t e m s and Signal Processing, Theory and Applications, C.I. Byrnes, C. Martin, R. Saeks eds. North Holland. [1989]. Stochastic aggregation of dynamical systems. Submitted for publication. WILLEMS J . C . [ 1 9 8 3 ] . Input-output and s t a t e space representations of finite-dlmensional linear tlme-lnvariant systems, tin. Alg. Appl., 50, 5 8 1 - 6 0 8 . [1986]. From t i m e series to llnear systems. Part I. Finite D i m e n s i o n a l L i n e a r Time I n v a r l a n t S y s t e m s . Automatica, 22, 561-580. [ 1 9 8 8 ] . Models f o r Dynamics. Dynamics Reported, 2, W i l e y and T e u b n e r .

Linear System Representations J. M. Schumacher Centre for Mathematics and Computer Science (CWl) Kruislaan 413, 1098 SJ Amsterdam, the Netherlands and Department of Economics, Tilburg University P. O. Box 90153, 5000 LE Tilbur9, the Netherlands

1. INTRODUCTION The theory of system representations is concerned with tile various ways in which a 'system' (a dynamical relation between several variables) can bc describcd in mathcmatical terms. This paper will concentrate on the class of linear, time-invariant, deterministic, finite-dimensional systems, for which there exists indeed a variety of representations. The study of system representations is of interest for two reasons, which correspond to two different points of view. First of all, even when representation types (or 'modcl classes') are mathematically equivalent, the case with which a particular problem is handled may be quite representation-dependent. Also, it may happen for instance that a problem is best understood theoretically in one representation, but that another representation is most useful for the numerical solution. Thus, one should be able to switch from one representation to another. The study of the corresponding transformations belongs to representation theory. "llac second reason for interest in system representations is connected with the modcling problom. Often, a model for a physical system is built up by writing down equations for the components and for the connection constraints. In this way, one obtains a system representation. It may be useful, though, to rewrite the equations; the derivation of the Euler-Lagrangc equations of mechanics could be citcd as an example. Again, we have here a problem of transformation between system representations. Interest in the theory of system representations has been stimulated in rccent years by a series of papers by J. C. Willcms [64, 68-70, 72, 73]. In this work, the 'modeling' point of view has bccn emphasized. As noted by Willcms, even such raw data as an observed time series can already be takcn as a systcm representation, and the identification problem then becomes a problem of transformation of representations. In this paper, we shall concentrate on representations by equations rather than by mcasurcd data. A survey of system representations and transformations will be prcscntcd in thc spirit of [71]. We shall use the notion of "external equivalence', again following

383

Willcms. The next section contains a brief historical survey of system representations in connection with control theory, centered on the description of linear, finite-dimensional, deterministic systcms. After that, we shall attempt to give an up-to-date account of the results concerning the representation of this class under external equivalence. Section 4 will be devoted to an application of the theory to the idea of a factor system, and the paper will be dosed with conclusions and research perspectives. 2. SYSTEM REPRESENTATIONS" A HISTORICAL SKETCH

The birth of mathematical control theory is often dated 1868, the year of the publication of J. C. Maxwell's paper "On Governors" [42]. In this paper, Maxwell deals with a number of contrivances that in his time were in use to regulate the operation of steam engines. Maxwell uses second order equations to describe the motions of the engine itself and the regulators. He takes the coupling of the different parts into account and iincarizcs to obtain a coupled set of secondorder linear differential equations. As an example, the following equations appear for a steam engine rcgulatcd by a combination of "lllomson's governor with Jenkin's governor (in Maxwell's notation):

A d20 + X dO + K - ~ t + TqJ+J~p = P - - R dt 2 dt B d2q' + Y ~ -dt - z

K dO --7/-=o

c d-L¢- + z @ dt 2

- T , = o.

Here, P - R denotes the cffcctive driving torque. The main variable is O, which represents the deviation of the main shaft angle from its nominal value. The variables 'k and ~bcorrespond to the two governors. Maxwell then writes the general solution for 0, which, by the standard theory of ordinary diffcremial equations, involves a linear combination of exponential functions. These exponential functions are determined by the roots of a polynomial equation that can be derived readily from the givcn system. Maxwell writes n for the unknown, and obtains a fifth-dcgrcc equation by sctting

-

Bn+Y -T

0

=0

Cn 2 + Zn

(a factor n has been cancelled right away in the second row). I le is then confronted with the problem of determining conditions on the coefficients under which all solutions of this equation are located in the left half of the complex plane. This, of course, led to the work of Routh on conditions for the stability of polynomials of arbitrary degree. We see that Maxwcll's fifth-order equation arises from the application of a fourth-order controller to a second-order system, and that the conditions for stability arc given by him in terms of the zeros of a polynomial matrix that is obtained directly from a standard modeling procedure. Maxwell used second-order differential equations, but it gradually became standard in the nineteenth century to write differential equations in first-order form. The fact that a higher-order differential equation in one variable may bc replaced by a tirst-ordcr equation in several variables

384

was actually already known in Cauchy's time. The Lagrangian equations of mechanics were later put into a suitable first-order form by Hamilton; towards the end of the century, Poincar6 and Lyapunov used first-order vector representations systematically. Naturally, therefore, representations of this type (called s t a t e representations later on) have dominated control-fllcorctical work that was done in close connection with the theory of ordinary differential equations. This conccrned mainly linear stability theory at first, but later, in dac first decades of the twentieth century, attention shifted to nonlinear problems. This line of research was held up high especially in the USSI~ (see for instance the survey by Minorsky in [45]). The work in connection with differential equations had a natural tendency to emphasize closed-loop systems, obtained by combining a given system with a given controller. Indeed, for such systems one may readily apply the powerful methods from the theory of ordinary differential equations and allied disciplines, such as the theory of dilYerential-diffcrence (delay) equations. The analysis by Maxwell, as briefly described above, is an example of this approach. The closed-loop point of view is quite satisfactory for many problems in mechanical engineering. To the communications engineer, however, it is more natural to use an open-loop point of view, in which a system is viewed as an operation that acts on an input signal and produces an output signal. This "opcrationar point of view called for a representation which would express the output signal as the result of some operator acting on the input signal. Such a representation is provided, at least for linear systems, by the convolution integral. I lowever, competing representations were soon to appear. Indeed, the use of complex quantities for the representation of complex signals, the Fourier and Laplace transforms, and l lcaviside's Operational Calculus were all in principle available by the turn of the century. The value of these techniques was gradually recognized among electrical engineers, bc it certainly not without resistance (see for instance [46]). From the mathematical point of view, the use of operational methods led to the introduction of techniques quite different from the ones usually found in the theory of differential equations. Applications of complex function theory were limited at first to partial fraction expansions and computation of integrals, but the appearance of the Nyquist criterion [47] made engineers realize that full-fledged function theory was a natural tool to use in the analysis of linear systems [10, p.9]. Izunction-thcoretic tools, in particular Cauchy's theorem, were used extensively by Bode in his book [9], which incorporated the celebrated Bode gain-phase relation and the minimum phase concept. The developmcnt of the root locus method by Evans in 1948 [18] firmly established the view of the transfer function as a function defined on the complex plane rather than just on the real frequency axis. For a more extensive discussion of the development of frequency-domain methods, we refer to [40]. We will not at all review the developments in the area of stochastic systems. In connection with what just has been said, however, it is interesting to quote Wiener on some of the differences between his own work and that of Kolmogorov: ... my work, unlike the explicitly published work of Kolmogoroff, concerns the instrumentation which is necessary to realize the theory of prediction in automatic apparatus for shooting ahead of an airplane. This engineering bias leads me to emphasize more than does Kohnogoroff the problem of prediction in terms of linear operators in the scale of frequency, rather than in similar operators on the scale of time. [63, p. 308] While the communication engineers developed their own methods, work on the ODE-type approach to control systems was still continuing, in particular in the Soviet Union. During the Second World War, a research centre was formed in Kazan where work on applied problems was done by outstanding mathematicians such as L.S. Pontryagin, who had already acquired fame

385

bccause of his pre-war contributions to topological algebra. After the war, rescarch cfforts in control theory continued at various mathematical institutes in the USSR. One important research direction centered around 'Aizerman's conjecture' [1], a nonlinear generalization of the Nyquist criterion. This problem called for a representation of systems with an explicitly appearing input variable, unlike the setting that was mainly used before in the 'ODE' framework. Systems with one input were studied first, in line with the original work of Nyquist, but the extension to several inputs was a natural one. For instance, Letov [35] considered in 1953 the following system (in original notation): ~k = ~

~kaTQa ~- Ilkl~l "~11k2~2

(k = 1, " - - , 1 1 ) ,

a-I

a-I ~: = f:(a:),

a2 = ~ p2,n~ - r21~l -- rz:~z. ¢x-- 1

We recognize the first cquation (with hindsight, perhaps) as a linear state equation with two inputs. The early fifties saw the rise of modern optimal control theoo,. One of dae first problems to be studied was time-optimal control. In some applications, it is natural to consider control strategies in which one switches between full power in one direction and full power in the reverse direction. This motivated a study of differential equations with discontinuous forcing terms by D.W. Bushaw at Princeton University [1 l]. Bushaw noted that the switching instant could be optimized to obtain a transfcr from one state to another in minimal time. Subsequently, J. P. LaSalle observcd that 'bang-bang' policies would be optimal among all possible control policies which lead from a given state to another. LaSalle used a nonlinear formulation, but later on Bellman et al. considcrcd linear systems 17]. In this paper, Bellman and his co-authors required invertibility of thc input matrix (as we would now call it), so in particular they let the number of inputs bc cq,:al to the number of states. In indcpendent work, Gamkrelidze [25] considered shortest time problcms for linear systcms with n states and r inputs. He writes the following state equation 125, p.451]: ,~ :

A x - I - b t t , 1 q- " ' "

q - b r ur

which is practically the formula A: = Ax +Btt that has become ubiquitous in control theory. lly the end of the fifties, the time had come for an amplification of the notion of 'state' far bcyond its meaning as the vcctor that appears whcn dynamic equations arc written in a first-order form. This was duc to the role that this concept had to play in Bellman's dynamic programming method, but also to developments in the theory of automata (finite state machines; Nerode equivalence). In control theory, the announcement by Pontryagin of his Maximum Principle at the International Mathematical Congress in :Edinburgh in 1958 had a tremendous impact on rcsearch in optimal control. Bang-bang control problems, in which one seeks to steer from one state to another, naturally led to the formulation of the concept of controllabifit.y by R. E. Kalman. This concept, and the dual notion of observability, turned out to play a crucial role in what Kalman called the realization problem:

386

Given an (experimentally observed) impulse response matrix, how can we identify the linear dynamical system which generated it? [30, p. 153] The word 'realization' is used here in a sense that is different from the traditional usage in electrical engineering. There, one would look for realization of a given driving-point impedance as an actual or idealized electrical circuit (cf. also the use of the term 'realize' in the quotation from Wiener given above). Although Kalman did advertize the state space realization as a 'blueprint' which could serve as a basis for implementation in an analog network [28], this connection was hardly emphasized in subsequent research. In the newly founded SIAM Journal on Control, E.Gilbert argued that the transfer representation was misleading and could lead to erroneous results. His point was that unobservable and/or uncontrollable states could be created by system composition: Thus transfer-function matrices may satisfactorily represent all the dynamic modes of the subsystems but fail to represent all those of the composite system. Furthermore, the loss of hidden response modes is not easy to detect because of the complexity of the transfer-function matrices and matrix algebra. [27, p. 140] To develop linear control theory from the state space point of view, it had to be shown that the familiar concepts from the frequency domain could be translated to state space terms. For this, the new realization theory was an indispcnsiblc tool. Gilbert [27] used partial fraction expansion (much in the tradition of Heaviside, one might say) to obtain a state space realization for a transfer matrix having only simple poles. This method can be extended to the general situation (not necessarily simple poles), but then becomes somewhat involved (see [50]). A more elegant realization algorithm was published by Kalman and B.L. Ho in 1966 [28]. The algorithm was based on a new parametrization of the transfer matrix - - new at least to control theory: in 1894, A. A. Markov had already used essentially the same parametrization for a study of continued fractions [41]. The 'Markov paranaeters' arc the first (matrix) coefficients in the power series development around infinity of a proper rational matrix. For a while, 'realization theory' was, at least to the system theorist, practically equivalent to the determination of a state space representation from a transfer matrix given through its Markov paranacters. The seventies, however, brought a renewed interest in polynomial representations. An important impetus for this development came from the appearcnce of Rosenbrock's book [51] on multivariable systems. In this work, Roscnbrock considered input/output systems given in the form

T(s)l~ = U(s)u y = V(s)~ + W(s)u where all matrices are polynomial. Great emphasis was placed on the study of equivalence notions. Rosenbrock found a "lifting' of Kalman's system equivalence concept to the more general representation displayed above, which he called sttqct system equivaleno,. It seems safe to say that the systematic development of the theory of system representations, system equivalence and system transformations starts with [51 ]. From Rosenbrock's system matrix, the transfer matrix is represented as V(s)T l(s)U(s) + W(s), i.e., as a ratio of polynomial matrices. It is not difficult to see that, in fact, every rational matrix can be written in either of the two forms V(s)T-I(s) or T ~(s)U(s), where, moreover, a coprimeness condition may be imposed. These coprimefractional representa-

387 tions were used very successfully by Ku~zera [31,32] and by Youla et al. [79, 80] to give a parametr-

ization of all stabilizing controllers for a given plant. This is an example of a result that appears quite naturally in one representation but would be awkward to derive in some other representations. At the same time, fractional matrix representations were also used in work on infinitedimensional realization problems done at Harvard University by R.W. Brockett, J.S. Baras, and P.A. Fuhrmann (see for instance [6]). In the infinite-dimensional context, the available mathematical tools strongly suggested to replace polynomials by functions analytic on the unit disk (in the discrete-time case - - for continuous-time systems, the class to use would be the set of functions that arc analytic on the right half plane). This idea was picked up by researchers in finite-dimensional system theory, who discovered that some difficulties with the Ku~:era-Youla parametrization could be ironed out by using the ring of rational functions that have no poles in the dosed right half plane (including the point at infinity) rather than the ring of polynomials (see for instance [16]). The fractional representation over the ring of proper and stable rational functions was subscquently used extensively in the cmcrgin G 11 ~-thcory, which is in itself an example of an application of function-theoretic techniques to control problems in a way that would probably have been quite beyond the imagination of Nyquist and Bode. On the other hand, ll°~theory has also relied heavily on state space representations, since the representation in terms of constant matrices makes it possible to use standard numerical software. The cooperation between the two representations was facilitated by the discovery (attributed to D.Aplevich in [62]) that there is an easy way to pass from a state space representation to a fractional representation over the ring of proper stable rational functions. (Fractional representations over the rin G of polynomials cannot be obtained in a comparable way from a state space representation.) Nevertheless, polynomial representations were emphasized again in the mid-seventies when Fuhrmann worked out an elegant procedure to go from a polynomial matrix fraction representation to a state space representation [23]. The discovery of this procedure, now known as Ftdu'mann's reali-ation, spurred considerable research on the relation between state space conccpts, as developed in particular in the 'geometric approach' to linear systems [77], and polynomial or transfer matrix concepts. For an introduction to this, see for example Chapter I of [24]. Polynomial matrices, even when less suitable for a number of purposes than stable proper rational matrices, arc important in system theory because they arise nattxrally in modeling. Indeed, a polynomial matrix representation can be written down immediately from a set of linear differential and algebraic equations describing a given system. Maxwcll's equations for the controlled steam engine, as given above, may serve as an example. Of course, by the old trick of replacing higher-order derivatives by ncw variables, it is also possible to obtain a first-order represcntation. Instead of the Roscnbrock form discussed above, one then gets a representation in the form lz~ = A x + Bu y = Cx + Du,

where E, A, B, C, and D are constant matrices. The variable "x" which appears here was called the descriptor variable by Luenbcrger, who was first to make an cxtensivc study of this reprcscntation

in system theory [37, 381. Contrary to the standard state space representation, the descriptor form is capable of representing systems having a non-proper transfer function (also called 'non-causal systems' or 'sinGular systems'). ThrouGh the years, the term 'descriptor system' has come to be used almost exclusively for such systems, although this was certainly not Lucnbcrgcr's original

388

intention - - hc was trying to emphasize the modeling issue, rather than the question of causality. The descriptor form was used by Verghese [60] to define an equivalence concept which deals neatly with pole/zero cancellations at infinity. This cleared up a problem which had remained unsolved in Rosenbrock's work. Alternative solutions were given later by Anderson, Coppel and Cullen [2] and by Pugh, Hayton and Fretwell 148, 491. The fact that the notions of equivalence defined by these authors are indced the same was established by Fcrreira [19]. Further comments on descriptor systems will be given in thc next section. In recent years, the study of system representations has been stimulated by the work of J.C. Willems. There are several important points where his approach is different from other approaches discussed above. First of all, Willcms uses an intrinsic definition of system equivalence (i.e., one that does not depend on a specific representation). He does this by defining a 'system' simply as 'a family of trajectories of given variables' (such as the port voltages and currents of an electrical network, or forces and displacements in a mechanical system). The given variables which appear in the definition are also denoted as 'external variables', to distin~lish them from 'internal variables' which arc possibly used as auxiliary quantities in a description of the system. The external variables may consist of what arc usually called "inputs' and "outputs', but, as shown in section 4 of this paper, other interpretations can sometimes also be useful. The family of trajectories is also referred to as a 'behavior' ~. In this approach, there is some flexibility associated with the choice of the function space to which the trajectories that make up the system are supposed to belong. In the study of differential equations, one normally uses function spaces that allow for exponentially growing solutions (such as C °~, or the space of distributions). In the context of system theory, however, it also makes sense to consider for instance only those trajectories that are square integrablc. Different choices of function spaces lead, in this way, to different notions of system; put in another way, they lead to different equivalence relations on system descriptions. More on this will bc said below. Willems has shown [68] that, if one interprets 'external variablcs' as "inputs and outputs' and uses the classical function spaces alluded to above, the equivalence relation that emerges is in fact different from the equivalence relations that were mentioned above. It should be noted that the definition of a 'system' as a family of trajectories is not new. Compare, for instance, McMillan's definition of a 2n-pole: qllc constraints imposed by a general 2n-pole N on voltages and currents arc completely described by the totality of pairs [v, k] which N admits. We shall de.fine a general 2n-pole, therefore, as (i) a collection ofn oriented ideal branches, as in 4.11, and (ii) a list of pairs Iv2k] of voltages and currents admitted in these branches. 144, p. 2281 (The oriented ideal branches in §4.11 of McMillan's paper serve just to define the pairing of the terminals.) In recent work in system theory, the equivalence notion as used by Willems has in fact occurred in several places; see [4, p. 513] ('external equivalence') and 18, p. 92] ('input-output equivalence'). Nevertheless, there is no doubt that the consequences of the acceptance of this intrinsic definition of what a system is have bccn explored to the fullest in the work of Jan Willems.

389

3. A a o ^ o M^V Or rU~VmZSENTA'nONS In this section, we shall review the available representations for a specific class of systems, viz., the class of finite-dimensional, deterministic, time-invariant, real, linear systems in continuous time, without further special structure. (The addition 'without further special structure' refers to the fact that we shall not consider special properties that arise, for instance, for systems defined on a symplectic space.) This is the class that has served as sort of a standard in system theory during the last three decades, except that causality is often imposed as an additional requirement. This condition was not included in the list above for two reasons. First of all, we arc sometimes interested in external variables that are not to be considered as 'inputs' and 'outputs' (cf. Section 4 of this paper, for instance), and in such cases causality need not be a relevant issue. Secondly, even when wc do distinguish inputs and outputs, there are no simple ways to tell, at a general level of representation such as l~.osenbrock's system matrix, whether a given system is causal or not [51, p. 51]. Imposing causality as a constraint on such gcncral linear system rcprescntations would therefore be awkward.

3.1 Notioas of equivalence When discussing system representations, we will have to specify under which circumstances we shall say that two representations are equivalent in the sense that they correspond to the same system. There arc three main options. There is the notion of strong equivalence, which boils down to Kalman's concept of equivalence for causal input/output systems in standard state space form. Definitions of this equivalence (by specification of a list of allowed transformations) were given at the level of descriptor systems by Verghcse [60] and by Pugh et al. [48, 49], and by Anderson et at'. at the level of the P,osenbroek system matrix [2]. Secondly, for every class of representations that have a given input/output structure and that define a transfer matrix, one has the notion of transfer equivalence according to which two representations are equivalent if and only if they dcfme the same transfer matrix. Finally, if one considers representations that define a family of trajectories of the external variables (an 'external behavior' in the sense of [68]), then there is the notion of external equivalet~ce according to which two representations are equivalent if and only if they induce the same external behavior. As noted before, external equivalence can in fact be understood in various ways, depending on the choice of a function space for the trajectories, and on the choice of external variables. There is also some freedom that arises from the interpretation of the external variables. For exampie, if we allow only permutation transformations on the external variables, this means that these variables are intcq~rcted as quantities which each have there own meaning and are measured on a fixed scale. On the other hand, if we allow general invcrtible linear transformations, then the implication is that the vcctor of external variables is understood as an element of a general linear space, it goes without saying that, depending on the problem one has at hand, some of the external variables can bc interpreted in one way and others in another way. (The same might bc said about the choice of a function space.) The term 'external equivalence' will be used for what might be eallcd the "classical' interpretation: the function space is such that exponentially growing solutions are admitted (we shall use C °O to make life a little bit easier), and only permutation operations will be allowed on the external variables. We call this the 'classical' form because it would seem that the notion of equivalence that is used (often implicitly) in treatments of ordinary differential equations is of this type. If one uses an L2-space rather than a C ~ - s p a c e as a trajectory space, then (cf. [74]) the corresponding notion of external equivalence turns out to be an extension of transfer equivalence, in the sense that it coincides with transfer equivalence on the

390

class of systems that define a transfer matrix. Suppose now that one has a system of equations in the form (3.1)

ox = A x + Bu

(3.2)

y = Cx + Du.

One might propose to take u,y, and x as external variables following C~-trajcetorics, to interpret u a n d y in a 'classical' sense, and to interpret x as a variable in a general linear space. The resulting concept of equivalence is Kalman's equivalence. It may be suspected that a similar reinterpretation in terms of external equivalence is also possible for strong equivalence. To keep the presentation manageable, we shall consider transformations under "classical' external equivalence. For other types of equivalence, the picture will be different but similar. We will discuss special representations for systems equipped with an i/o structure, but the particular representations that are available only for causal systems will be omitted. 3.2 A catalog of rel)reaentations

We start by listing a number of representations. A number of basic types will be distinguished that are different by appearance; within these, we distinguish subtypes that do not differ notationally but that are subject to more or less severe constraints. The most unspecific type of representations wc shall take into consideration is the AIq/MA class. An AFI/MA representation is specified by two polynomial matrices P ( s ) and Q(s), which determine file external behavior consisting of all trajectories w of the external variables for which there exists a trajectory ~ of the internal variables such that p (,,).~ = o

(3.3)

w --- O(o)~.

In the continuous-time interpretation we use here, o stands for d / d t . The class is called A R / M A because of the discrete-time interpretation in which o is the shift: in this case, (3.3) implies that the external variables are expressed as a moving average of the internal variables, which themselves satisfy an autorcgrcssive equation. Every representation of this kind can trivially be rewritten as a 'systcm with auxiliary variables' [68] (later also called an ' A R M A ' representation by Willems [73]), which is defined by an equation of the form P'(o)~ + Q'(o)w = 0;

(3.4)

simply take P'(s) = LQ(s)j,

Q'(s) =

1"

On the other hand, it is also easy to write an AR/MA representation for a system with auxiliary variables, by extending the space of internal variaLlcs and writing P(s) = l/"(s)

Q'(s)],

Q(s) = [0

1 I.

(3.6)

We see that the AFI/MA representation is, in general, less parsimonious in the use of internal variables than the representation as a system with auxiliary variables. Since we are looking for an unspecific representation, this might be construed as an argument against the representation in the form (3.4). Actually, when dealing with systems described bypartial differential equations, one easily runs into clear-cut cases in which an AFt/MA representation appears much more naturally

391

than a representation with auxiliary variables as in (3.4). For systems with an i/o structure, another general representation is RSM (Rosenbrock system matrix 151]). An RSM representation is specified by four polynomial matrices T(s), U(s), V(s), W(s), where T(s) is square and invertiblc. The external behavior defined by an RSM representation consists of the set of all input trajctorics u and output trajectories), for which there exists an internal-variable trajectory ~ such that file following equations hold:

"1"@)~ = U@)u ) , = V(o)l~ + W(o)u.

(3.7)

The third polynomial representation we shall consider is the AR representation [69]. An AR rcpresentation is specified by a single polynomial matrix R(s), which should have as many columns as there are external variables. The external bchavior it defines is simply the set of all external-variable trajectories 1,, satisfying R (o)w = 0.

(3.8)

Wc shall always require R (s) to have full row rank; this simply means that the equations specified by tile rows of R (s) are independent. An AR representation given by R (s) will be said to be minflnal if the sum of tile row degrees of R (s) is minimal in the set of all AR representations of the samc system. One can show (see for instance [69, Thm.6]) that a matrix R (s) is minimal in this sense if and only if it is row proper. The class of minimal AR representations will be denoted by AR~,. If the external variable is partitioned into inputs and outputs, the defining matrix R (s) of an AR representation will be divided into two blocks R l(s) and R2(s), which correspond to outputs and inputs respectively. If R i(s) is square and nonsingular, the representation so obtained will be called an LMF representation ('left matrix fraction'). Iiy introducing new internal variables, it is easy to transform an AR/MA reprcsentation to a first-order form oG~ = F/j

w : tl~

(3.9)

(F, (7, and 11 are constant matrices). This representation, specified by the three matrices F, (7, and H, will be called thepencil representation ([33]; cf. also [4, 56]), and the corresponding class of representations will be denoted by P. To be complete, one should also indicate the spaces on which the various mappings arc defined, and so we shall sometimes also give a P representation as a sixtuplc (F, G, 11 ; Z, X, 14")where F and G are mappings from the 'internal variable space' Z to the 'equation space' X, and tl maps Z into the external variable space W. A descending chain of subclasses can bc formed by putting more and more strict requirements on the triple (F, G, tt). If G is surjective, the corrcsponding class will be denoted by Pdv, because this class is closely related to the DV representations that will be discussed below. The class of representations which in addition satisfy the condition that [G r I1T] r is injcctive will be denoted by Pio; in a representation of this type, one can easily see which partitionings of the external variables into inputs and outputs will lead to a causal i/o structure (cf. [33], Lemma 5.1 and Lcmma 6.1). Finally, pencil representations that also satisfy the requirement that [ s G T - F r /IT]T has full column rank for all s e C form a class that will be denoted by Pmin- It has been shown in [33] (Prop. 1.1) that a pencil representation is minimal under external equivalence if and only if it belongs to Pmin, Ncxt in our collection of representations is the DV (driving-variable) representation [4, 68, 69, 73], which, as already mentioned, is closely related to the Pdv class. A DV representation

392

is spccificd by four constant matrices A, B, C', and D', which determine an cxternal behavior by the equations o~ = A~ + Bo w = C'/j + 0 ' 7

(3.10)

(4 and ~qarc auxiliary variables). The class of DV representations for which the matrix D ' is injcctire will be denoted by DVio. If also the requirement is imposed that the 'system pencil'

has full column rank for all s, then we obtain a class of representations that will be denoted by DVmin. It has been shown in [68] that ~ DVmin representation is minimal in the class of DV representations, ill the sense that both the length of ~ and the length of 7/are minimal. For input/output behaviors, there are further special representations that may be used. A well-known form is the descriptor representation 137, 38]. The class of such representations will be denoted by D. A descriptor representation is specified by five constant matrices E, ,4, B, C, and D, and determines an input/output behavior by the equations aE~ = A~ + Bu y = C~ + Du.

(3.11)

Tile domain of the mappings E and A will be denoted by Xd (dcscriptor space), tile codomain will be written as Xc (equation spacc). Quite a few special properties have bcen used in the literature in connection with this representation (see for instance [5, 13, 36, 52, 6 i, 78]. We shall use the following conditions. The representation (3.11) is said to bc controllable at itt./inity if i m E + imB + A ( k e r E ) = X e.

(3.12)

It is said to be reachable at htfini(v if i m E + imB = X c.

(3.13)

It is called observable at infini O' in the sense o f Verghese if kerE n kcrC n A

I[imE] = {0)

(3.14)

and obselwable at iafilfity ia the sel,se o f Rosettbroek if k c r E n kerC = {0}.

(3.15)

The representation (3. I 1) is said to have no i~omtynamic variabh,s if A (kcrE) C imE.

(3.16)

These are all properties that relate to the point at infinity. We note that, for represem~fions that satisfy (3.16), thcrc is no diffcrcncc bctwcen controllability and reachability at infinity or between the two notions of obscrvability at infinity. In connection with the finite modes, we shall need the following condition: a representation of the form (3.11) is said to have nofinite unobservable modes if

393 In principle, a considerable number of descriptor representation types could be formcd by taking combinations of the six conditions mentioned above. We shall consider just four types, which together seem to present a reasonable hierarchy. The most unspecific type is the general descriptor form, for which the symbol D has already been introduced. The symbol Dri will be used for the class of descriptor representations that are reachable at infinity. Descriptor representations that have no nondynamic variables and that are both controllable and observable at infinity will be denoted as Dmi representations ('minimal at infinity'). Finally, the class of Omin representations consists of the D,ni representations that have no finite unobservablc modes. It is shown in [34] that a descriptor representation is minimal under external equivalence if and only if it belongs to this class. 3.3 The road map To indicate the connections bctwecn the somewhat vast number of representations introduced above, we shall now present a map. The following organizational principles have been applied:

O

polynomial representations are on the left, first-order representations on the right;

O

representations that do not distinguish between inputs and outputs are in the middle, i/o rcpresentations are on the extremes;

©

more specific representations are higher up in the diagram than less specific ones.

Moreover, arrows have been used to indicate known transformation procedures (including the trivial ones, which involve no transformation at all, and the very easy ones, such as the transformation from AR to AR/MA). "llae organization of the diagram is such that arrows going up represent the heaviest computational loads. The result is shown in Fig. i below.

i i

(

I A min I

Ir

1 'ME

Pmin ~

DVmin

Drain I

I, Pio ~

DVio

Dmi I

DV

Dri

[

°

I

)

Pay ~

Iy

RSM }-~ .--~,AR/MA

p

LI-

Y

-I

FIGURE 1. Rcprescntations and transformations of linear systems. The arrows going down in this diagram all correspond to trivial rcwritings or reinterprctations. For instance, an LMF representation is a special case of an RSM representation, obtained by taking V(s) = 1 and W(s) --=-0. The connection between LMF and AR is also quite clear. One gcts from an RSM representation to an AR/MA rcprcscntation simply by identifying the inputs with new internal variables. It is quite obvious how to transform the various types of

394 DV representations to the P representations on the same level, and vice versa. The transformation from AR/MA to P is by the standard trick of replacing higher-order derivatives by new variables. Most of d~e other transformations require more work, however, and some of the corresponding algorithms will be discussed below.

3.4 Algorithms We start with the transition from an ARtMA representation to an AR representation. For this, we have the following procedure. AI.GOR]THM 1 Let an AR/MA representation be given by (P(s), Q(s)). For instance by the algorithm of reduction to licrmite form ([39, pp. 32-33]; see also [29, pp. 375-376] or [ 12, p. 34]), find a unimodular matrix U(s) such that

u=,<s> u==(s)j LQ

,)j ---

where T(s) has ful[ iow rank. Let R (s) be a maximal selection of independent rows from U;2(s). Under these conditions, R(s) gives an AR representation that is externally equivalent to the AR/MA representation (P(s), Q (s)). For a proof of tiffs, see [68, Prop. 3.3] or [33, Lcmma 4.1]. The algorithm in [681 is actually based on the Smith form; from a computational point of view, this presents a considerable amount of overkill. In the algorithm given above, is is easy to sce that U22(s) will automatically have full row rank (so that we simply have R (s) = U22(s)) when P(s) has full row rank, which is a natural restriction to impose. The passage from AR to ARmin is just the reduction of a polynomial matrix to row proper form. The standard algorithm to do this is described for instance in [76, pp. 27-29] and in [29, p. 386]. This algorithm essentially requires only operations on constant matriccs, and the computational load involved is in general much less than in a transition from AR/MA to AR form. The steps leading from P to DV, from DV to DVio, and from DVio to DVmin are detailed in [56]. These steps can be 'lifted' to the level of P representations, and, in fact, it turns out that they can be derived quite naturally in this context. We shall now explain this in some detail. First, consider the transition from a general P representations to the Pdv representation. From the equation aG~ = F~, it follows that any/~-trajectory satisfying this cquation must belong to the subspace F l[imG]. This implies, of course, that G~ belongs to GF l[imG]. From that fact, it follows that any trajectory ~'(- ) satisfying aG~ = F~ must actually belong to the subspacc F - I [ G F -IlimG]], which obviously is contained in F I[imG]. We can go on in this way; a subspace recursion emerges which can be summarized as follows. Let the space on which G and F act be denoted by Z. Define Q0 = Z

(3.19)

Qk + I = F - IGQk.

(3.20)

and

We have Qk ~I c Qk for all k, and so a limit must be reached after finitely many (in fact, at most dina Z) steps. The limit subspace will be denoted by Q* (1, G) or simply by Q* if there is no risk of confusion. We arrive at the following algorithm to obtain a Pdv representation from a P representation.

395 ALGORITHM 2 Let (b, G, H ; Z, X, W) be a P representation. Compute the subspace Q* of z as the limit of the sequence of subspaces defined by (3.19-3.20). Take Z = Q*, ~" --: GQ*, and define /~, G, and H as the restrictions of the respective mappings to Z and ~'. (Note that, by the definition of Q*, F does indeed map Q* into GQ*.) Under these conditions, a Pov representation that is equivalent to the original P representation is given by (/7, (~, tt; Z, X, 140. Next, we consider the transformation from a Pdv to a Pio representation. Let (F, G, t t ; Z, X, W') be a Pov representation, and suppose that [G 1 H I ] I is not injective. We can then split up the internal variable space Z as Z = Z~ (9 Z2, where Z2 = k e r G A k e r H is nonzero. With respect to this decomposition, write G = [GI 0], F = [FI F2], H = [111 0]. The equations oG/~ = F/L w = H/2 then appear in the following form: oGI/J I = FII~ I + F21~2

(3.21)

w = Hl/~ I.

(3.22)

Sincc there are no restrictions on ~2, the above equations are equivalent to crTG 1~1 = TFI~I

(3.23)

w = H I~l

(3.24)

where T is any map satisfying ker T = im F 2. It is natural to let T be surjective, and we sec that the above transformation achieves a reduction of the dimension of the internal variable space and perhaps also a reduction of the dimension of the equation space. In more geometric terms, what we have done is the following. Define S 1 = k e r G fq k e r / l , and let Z j = Z / S 1, X I = X / F S I. With these definitions, the factor mappings G j : Z i --* X j, F i : Z 1 --9 X1, and t t 1: Z l ~ 14/are aU well-defined, and the representation ( F i, G i, H i ; Z i, X i, W) is equivalent to the original representation. There is no guarantee that, after this step, the reduced representation is of the Pio type, and in general the reduction will have to be repeated a number of times. For instance, the reduction in the second stcp is determined by the subspace k e r G I N k e r H i = {zmod S 1 [ Gz ~ F S j and t t z = 0 } = (G

IFS1 Iq k e r H ) m o d S 1.

(3.25)

The subspace rccursion that emerges is the following: S o = {0} Sk iI = G

(3.26) IFSk ¢q kerH.

(3.27)

We have S x t 1 D S k at every step, and so after finitely many (-<- dim Z) steps a limit is reached. The limit subspace will be denotcd by S* (F, G, !I) or simply by S* if the context is clear. The algorithm to go from a P0v to a Pio representation can now be formulated as follows. ALGORITHM 3 Lct (F, G, H ; Z, X, W) be a Pdv rcpresentat_ion. Define the subspace S* of Z as the limit of the sequence dcfin~.i by _(3._26-3.27)._ Define Z = Z / S * , X = Z / F S * . With these definitions, the factor mappings G : Z ~ X . F:Z~](, and [ I : - - , W are well-defined, and (F, G, 11; Z, X, IV) forms a Pio representation that is equivalent to the original Pov representation. The final transformation in this series is the one that leads from Pio to Pmin representations.

396 To achieve this reduction, we note that a redundancy in Plo descriptions is associated with subspaces N of the internal variable space Z that satisfy the two properties F N c GN

(3.28)

N c ker 1L

(3.29)

and

Indeed, if N is a nonzero subspace having these properties, then we can decompose the internal variable space Z and the equation space X i n such a way that H = [H l 0] and the mappings G and F take the form

0]

G =

G22

F =

F21

01

F22 "

(3.30)

Of course, both G I1 and G n must be surjective. The equations become oGit~ 1 = Fjtt;l

(3.31)

aG22~2 = F21,~l + Fn~2

(3.32) (3.33)

w = 1Ii1~1.

Because G n ~s surjectivc, the second equation can always bc satisfied by a suitable choice of ~2; therefore, no constraint is imposed on/~1. This means that the second equation as well as the variable ~2 may be removed without altering the external behavior. Speaking geometrically, this means that we replace Z by Z = Z~ N and X by X = X / G N , and that flae mappings F, G and H are replaced by the respective factor mappings. The reduction that is obtained in this way increases with N, and so we are interested in the largest clement of the set of subspaces satisfying both (3.28) and (3.29). (The fact that this set indeed has a largest element follows from the fact that the set is closed under subspace addition.) Let us denote this largest clement by N* (F, G, 11). The question is, how to compute this subspace. The answer to this is provided by the following equality, which expresses perhaps the most basic result in the geometric theory of linear systems: N*(F, G, t t ) = Q*([0G],

[//i]).

(3.34)

Indeed, this gives us an algorithm to compute N*. The proof of (3.34) is not diflacult, and may essentially be found in the standard reference [77, p. 91]. A considerable amount of translation of terms is needed, though, and the reader may find it easier to construct a direct proof. Rewriting the algorithm (3.19-3.20) a little bit to suit the special form which appears in (3.34), we finally obtain the following algorithm. ALGORITHM 4 Let (F, G, !1 ; Z, X, 14") bc a Pio representation. Define a sequence of subspaccs of Z by NO: Z N k~l = F

(3.35) 1GNk Q kerll.

(3.36)

Dcnotc the limit subspaceby N*, and define Z. = Z / N * , X -- X / G N * . With thcse dcfinitions, the factor mappings G : Z - - , ) ( , 1-:: }S --, )(, and [ I : Z - - , W (corresponding to G, F, and 11

397 respectively) are well-defined, and the representation (~, G,/-1; ;~, ,~', 140 is a Pmin representation that is equivalent to the given representation. It has been proved in [56] (using somewhat different terminology) that this algorithm does indeed lead to a minimal representation. There is a trivial way to pass from a general pencil representation to a general descriptor representation. If (F, G, 11) is a P representation, and 11 = 1//.~ H~] T is the decomposition of 11 associated with a given partitioning of the external variables into inputs and outputs, then an equivalent D representation is obviously given by

io,lu

(3.37)

y = !(,,¢

(3.38)

"Fhc main virtue of this transformation is that it docsn't require computation. A transformation that does a better job at preserving minimality properties is given by the following algorithm. ALGORrrHM 5 Let (F, G, l l ; Z , X, W) be a pencil representation, and let an i/o structure be given, so that 11 -- [i1.~ liT] T. Decompose the internal variable space Z as Z0 @ Z I f:t) Z2 where ZI = kerG n kerll,,, and Zi @ Z2 = kerG. Accordingly, write G=[G0

0

11.,.=11t,.o

0], I!,.1

F=[F H.,.2],

0

F]

F2] ,

11,,=[I!,, o

(3.39) 0

I/,2 ].

(3.40)

Thc matrix 11,,2 has full column rank, and by renumbering the u-variables if nccessary, we can write

1'.o

-_ r1114, L//2oj

1t,,2

f,,,4

= [H22J

(3.41)

where I122 is invertible (or empty, if ker G C ker tl,). Define descriptor parameters by

E =

,

A =

tllo - II]2H22]l12u

C = [i/to - 11"2112211120 I/r]],

,

B =

-I

D = 10 !!r21t221112o].

IIi211i2 ] '

(3.42)

These parameters define a D representation without nondynamic variables that is externally equivalent to the original P rcprcscntation. Moreover, if the given rcprcscntation is of the P0v (Pio, Pmin) type, then the obtained representation is of the Dri (Dmi, Drnin) type. The proofs of the statcments above arc given in [34]. At the 'Pdv' level and higher, it might be said that the algorithm in fact uses the driving-variable rcprcscntation as an intermediate stcp, so that the DV rcprcscntations fit into the picture as shown in Fig. 1. The converse transformation is obtained as follows. ALGORITHM 6 Lct a D,i reprcscntation bc given by (E, A, B, C, D) (so that [E Choose coordinates in such a way that

B ] is surjcctivc).

398 C =1C1

C2],

(3.43)

D = [ D 1 D2]

where B n is invcrtible. Define matrices A', B', C', and D' by C'

D'

=

--Bll 1 =

!

0

0

1

0

0

-D

Bn ~

/3221

]

Cj

C2

0

0

A 21

A 22

(3.44) B21

J

The DV rcprcscntation given by the four-tuplc (A ', B', C', D') is cxternally equivalent to the given D representation. Moreover, if the givcn descriptor representation is in the Dmi (Drain) class, the resulting driving-variable representation is in the DVio (DVmin) class. For a proof of these statements, see again [34]. More refined statements could be made; for instance, it is clear that to obtain a DVjo representation from the algorithm above, it is sufficient that the D representation we start with is reachable at infinity and observable in the sense of Verghesc. The corresponding reduction to minimal form in the 'DV' branch can bc thought of as a reformulation of the above in special coordinates. The details have been worked out in [56]. The reductions take a somewhat different form at the 'D" level. Verghese [60] already gave a simple algorithm to remove nondynamic variables. It has been shown in [33] (Lemma 7.3 and Lemma 7.4) how to reduce a given descriptor representation in case it does not satisfy either one of the conditions '[E B] surjcctivc' or [ E T c r ] T injective'. Clearly, by repeating these reduction steps if necessary, it is always possible to arrive at a situation in which these conditions are satisfied. Thc final passagc to Dmin comcs down to removing the finite unobscrvable modes. This might for instance bc done via rcduction to the Weicrstrass canonical form of thc pencil sE - A [26] foIlowed by an application of the well-known procedure to remove unobservable modes in standard state space systems. Finally, wc come to the transformation from AIq to Pmin. This is essentially the Fuhrmann realization 123, 24]. In [33], the transformation is given the following form. ALGORITJ~M 7 Let an AFI rcprcscntation be specified by R(s). Consider the following spaces of rational vector functions in a formal parameter h (# denotes projcction onto the proper rational functions, If' is the space of external variables, k is the number of rows of R (s)): X R = {w(;x)~X i i¥[[~. 1]] I ~r_R(h)w(~.) = 0}

(3.45)

XR = {p (X) ~ RkIh] I 3w (X) ~ X ~WIlX- l]] s. t.p (h) = R 0,)w (h))

(3.46)

N t¢ = {w(Jk)~ h- J WllX-~]]1 R (,X)w(h) = 0).

(3.47)

q11c following mappings (G and F f r o m XR/)x lNa to XR, H from Xa/)x 1N~ to IV) are welldefined: G: w(?x) mod}x--IN t¢ ~

R(h)w(~.)

F : w(X) m o d h 1N/¢ r--* R(X)'rr. (hw(h))

(3.48) (3.49)

399

ll:w(A)modh

INR

~

W

1.

(3.50)

With these definitions, (/7, G, II) is a minimal pencil representation that is externally equivalent to the AR representation given by R (s). This version differs from Fuhrmann's original one in two respects. First, the resulting representation is given in pencil form rather than in standard state space form, so that it becomes possible to consider noncausal systems. (The Fuhrmann realization has been used before in a noncausal context [14, 75], but only by separating finite and infinite frequencies, and under the assumption that a transfer matrix exists.) Secondly, the procedure is presented as one under external equivalence, rather than as one under transfer equivalence. The transformation algorithm given above is abstract, and may be used very well in theoretical considerations. However, a more computational form can also be given (see [33, §8]). This requires the given representation to be in A R m i n form, and produces a representation in DVmin form, which explains the arrow between the corresponding boxes in our map of linear system representations. 4. T H E FACTOR SYSTEM

In [67], J.C.Willcms has pointed out that there is a close connection between the notion of an 'almost controlled invariant subspace' and that of a 'factor system'. Before discussing the connection, let us briefly recall what these two notions mean. To define the factor systcm, following the development in [65], lct first X bc a finite-dimensional vector space ovcr R. Also, lct ,4 bc a linear mapping from X into itself, and let B be a linear mapping ranging in X. The smooth system ~(A, B) on Xdctermincd by .4 and B is the following sct of C°°-functions from R into X: X(A, B) -- {x(. ) e C ~ ( R ; X) ] .x(t) - `4x(t) e i m B for all t}.

(4.1)

Lct X bc a smooth systcm on X and Ict K be a subspacc of X. Consider the following set of trajcctorics on the factor space X~ K:

X/K

:=

{x(" )mod K I x ( - ) e X}.

(4.2)

If this set of trajectories is a smooth system on X~ K, then X / K is called the factor system determined by v and K. The notion of an almost controlled invariant subspacc can be defined in the same context. So let us assume that a state space A, a state mapping A, and an input mapping B have been given. A subspace K is said to be ahnost controlledmvariant [66] if for every ¢ > 0 and for every x0 in K there exists a trajectory x ( • ) in )_2(A, B) such that x(0) = x0 and dist(x(t), K) ~< ¢ for all t ~ 0. This concept has many applications in control theory, of which some arc rcviewcd in the contribution by J. L. Willems to this volume. Given a smooth system X(A, B), one would of course like to know under what conditions on K the set .X/K is a factor system. It is claimcd in [67J (Theorem A) that this will hold if and only if K is almost controlled invariant. In the cited paper, only a sketchy proof is provided for thc 'if' part of this statement, and the 'only if' part is given without proof. Later on, a detailed proof of the 'if' part has bccn provided in 159], but a complete proof of the reverse implication is still lacking in the literature. Our goal in this section is to provide a short proof of Theorem A of [67], using a result in [56]. This proof is essentially based on manipulation of representations. In the previous section, algorithms were presented for the removal of redundancies in pencil representations. In these algorithms, certain subspace rccursions played a key role. We will also

400 need these recursions below, as well as some related recursions which we will introduce now. To the sequence of subspaces S k defined by (3.26-3.27), another sequence ~k can be related by £.k = G - IFSk.

(4.3)

From (3.26-3.27), wc see that this sequence might also be defined by the recursion g.o = ker a

(4.4)

~k ÷I = G ][,.[~.k n kerH].

(4.5)

Denoting limS k by ,~*, we also see from the definitions tlmt S* = S" n kcrH. It is furthermore useful to introduce two subspace recursions that do not take place in the 'internal variable space' Z but in the 'equation space' X. The first of these is obtained if we define V k = GN k.

(4.6)

The corresponding recu rsion is v° = x

(4.7)

V k ~ 1 = G[F -I V k 1"3 kerH].

(4.8)

Similarly, we define •r~ = Gg'~ ( = FS k)

(4.9)

with the corresponding recursion To

= (01

T k ~l = F [ G

(4.10) ITk n kcrH].

(4.11)

The limit subspaccs resulting from these rccursions will bc denoted by V* and T*, respectively. The subspaccs that have now been introduced play a role in the characterization of some important system invariants in terms of the parameters in a Pdv representation. If (1, G, tl : Z, X, W) is a Pmin representation of a behavior .~, we define the degree of this behavior, to be denoted by dcg(~J), as dim X. Also, we define the order of 5~, to be denoted by ord(~d), as dim Z. Since a Prom representation is determined up to isomorplaisms of the internal variable space and the equation ~pace, the degree and the order are clearly independent of the choice of a particular Pmin representation. There are of course many other equivalent characterizations: for instance, the degrcc is also equal to the sum of the row degrees of the matrix R (s) in any ARmin representation of ~, and to the dimension of the state space in any minimal state space representation of any causal input-output behavior that can be obtained from ~ by partitioning the external variables in inputs and outputs. (For a catalog of such results, see [69], Thin. 6.) From the fact that the internal variable space in a Pmin representation is obtained from the internal variable space in a given P0v representation by successively factoring out the subspaccs S* and N*, it might bc suspected that the dcgrce is given in terms of a Pdv representation by codlin(N* + S*). It has been established in [56] (Thm.4.1) that dfis is indeed the case. The relevant result may be summarized, with some rephrasing, as follows. PROPOSITION4.1 Let a behavior ~¢ be given b.r a Pdv rq, resentation (F, G, 11; Z, X, W). D~fine subspaces S*, N*, and S* of Z, and subspaees V* and T* of X by the recursions (3. 26-3. 2 7), (3.35-3. 36), (4.4-4.5), (4. 7-4.8), and (4.10-4.11) r~l)ect iveO,. We then have the following equalities:

401 deg (.ad) = codim(N* + ,~*) = codim(V* + 7"*)

(4.12)

ord (a.q0') = codim(N* + S*).

(4.13)

In case ker t t contains ker G, an alternative formula f o r the order is

o r d ( ~ ) = codim(V* + (7"* N G(kerH))).

(4.14)

Our next concern is to charactcrize a 'smooth system' in terms of system invariants. This is described in the following lemma. LErctl~IA4.2 A linear time-invariant behavior ~ with external variable w has a rel)resentation in the 01711

ox = A x + Bu

w = x

(4.15) (4.16)

if and onlt, i f ~ has no static cot~trahlts (i. e. f o r all w o ~ 14/there exists a w ~ ~ such that w (0) --:- wo), and dim W is equal to ord (.~').

Prtool: Consider the 'if" part first. If dim IV equals ord (~J~'),then there exists a P,,,,, representation oG~ : 1:~ .. =

,'{~

(4.17)

(4. ]8)

in which the matrix 1I is square. From the requirement that ~ has no static constraints, it follows that !i must be nonsingular. Let G ~ denote a right inverse of G, and let F b e a mapping satisfying im 13, - kcr G. The equation (4.17) is then equivalent to o~ = G ' F/~ + FT!

(4.19)

where 7/is a new internal variable. Using a nonsingular transformation, of the ~-variable, we can replace t i by the identity mapping, and then the desired form is reached. For the "only if' part, we first note that fl~e behavior defined by (4.15-4.16) has no static constraints. To determine the order of the behavior represented by (4.15-4.16), we have to take into account the fact that this representation is not minimal. Let T be any surjective mapping such that ker T = imB; then (4.15) is equivalent to o7"r -- TAx.

(4.20)

Moreover, the representation (4.20-4.16) is minimal and we see that dim IV is equal to ord (.'~, as claimed. To obtain the main result of this scction we combine the above characterization of smooth systems, the result that gives the order in terms of a Pay representation, and a characterization of ahnost controlled invariant subspaces in terms of subspace recursions, taken from [66]. THF.OREM 4.3 Let a smooth .~),stetn ~(.4. B ; X) be given, and let K he a subspace of X. Under these conditions, the set of trajectories of ~ tmx4ulo h, Z / K, is a amooth s.ystem if and on O' if K is ahnost controlled htvariant.

I'R(Jo]: l.ct C : X --, X / K be the factor mapping. Obviously, a rcprcscntation of the behavior >2/ K is given by

402

trx = Ax + Bu

(4.21)

w = Cx

(4.22)

and so we have to find the conditions on K under which this is a smooth system. First of all, note that the behavior X / K can have no static constraints bccause otherwise the original system Ywould also have static constraints, which we know is not the case. Thercforc, from the above lemma and the proposition we see that X / K is a smooth system if and only if

d i m X / K = codlin(V* + (T* N G(kerH)))

(4.23)

where everything is taken with respect to the parameters G = [/

0l,

F = [A

B],

1t = [C

01.

(4.24)

(Note that indeed ker ti contains ker G, so that the above formula applies.) Rewriting the V*- and T*-algorithms for the above special values of the Pdv parameters while keeping in mind that k c r C = K, we obtain V° = X V ~-~1 = K N A - 1 ( V

(4.25) k+imB)

(4.26)

and 7`0 = {0}

(4.27)

T k¢l = A [ T ~ n K ] + imB.

(4.28)

The algorithm (4.25-4.26) is recognized as the invariam subspace algorithm [77, p. 91]. If wc define ~;-k = T ~. n K, then the associated recursion is .~0 = (0}

(4.29)

{-~- ~ i = K n (AT k + i m B )

(4.30)

and this is recognized as the controllability subspace algorithm [77, p. 107], also known as the ahnost controllability suhspace algorithm [66]. Noting that G [kcr 11] = ker C - K, wc see that wc always have K D I/* + (T" n K)

(4.31)

so that the condition (4.23) may be rewritten as K = V* + (T* N K) = V* + T*.

(4.32)

But this is exactly the condition given in [66] for a subspace K to be almost controlled invariant with respect to (A, B).

403

5. CONCLUSIOrCS It should be emphasized that our 'road map' of system representations covers only a small area in the large field of representation theory. We have only been looking at the 'classical' form of external equivalence, thereby excluding representations such as the matrix fractional form over the ring of proper and stable rational functions, which is one of the main tools in the latest developments in control theory 121,43]. A l s o , there arc many other classes of systems for which representation theory leads to useful results. This of course includes the generalizations to nonlinear and infinlte-dimensional systems, but important new aspects also arise if one considers systems with particular properties. A simple cxample is provided by the case of linear systems with a Hamiltonian or a gradicnt structure, such as appear in the modcling of mechanical structures and electrical networks. The problem of setting up state equations for such systems, starting from (higherorder) differential equations and algebraic constraint equations, is in fact a classical one. For a treatment following lines as presented here, see [57]. Of course, the Hamiitonian structure is important in the nonlinear context as well, and the problem of dealing with systems with mixed differential and algebraic equations comes up naturally for instance in setting up models for robots. For general nonlinear systems, the relations between systems of higher-order differential equations on the one hand and the standard state space form on the other have been widely discussed; an carly reference is [22]. and [15, 20, 55, 58} provide a sample of recent contributions. It has been shown in [54], a nonlinear system of algebraic and differential equations in a DV-type form can bc rcduccd to a minimal rcpresentation in standard state space form if and only if certain intcgrability conditions are satisfied. In the nonlinear case, the partitioning of external variables into inputs and outputs to obtain a causal i/o structure is, in general, a local construction. This could be one of tile reasons for interest in a nonlinear version of tile pencil form. Such a nonlinear i3cncil form might be specified by giving a submanifold of the tangent bundle of a manifold of internal variables, plus a mapping from that manifold to the manifold of external variables. Representation theory for stochastic systems is a very well developed subject. The richer structure of stochastic systems allows for a variety of representations, some of which arc discussed in the contribution by J. I 1. van Schuppcn to the present volume. However, it seems that not so much study has been made of questions concerning nonminimal representations, such as sometimes appear in modcling problems. As an example, consider an electrical network with linear elements containing some noisy resistors. Writing down network equations in the usual way, one could write down a representation in the form C~ = t.'~ + ~rn w = t1~

(5.1)

(5.2)

where 7/is 'white noise', and w represents the port variables. It requires proof to show that this can be rewritten in the standard form .~: ---- A x + B u + N v

(5.3)

y ~- C x + D u + M y

(5.4)

where v is white noise, and w has bccn partitioned into inputs u and outputs.),. Represcntation of stochastic systems is also the subject of debate in econometric circles (see for instance [3, 17]). Some aspects of the representation of infinite-dimensional linear systems are discussed in the contribution of R. F. Curtain to this volume. A great deal of effort has been spent by the infinitedimensional systems community on trying to fit into the standard (A, B, C, D) framework

404

equations like the following one (the normalized string equation with forces and displacemcnts at both ends as cxtcrnal variables): ~)2

0~x,

[,(o,,)l 02

t) = 0-~-: ~(x, t)

t) •

(5.5)

(5.6)

[,/~ (1, t) (The variable x is used here as the spatial variable, and the prime dcnotes differentiation with respect to x.) Such an equation would fit more naturally into representations of the pencil type. This advantage docsn't come without a price, however; whcrcas standard scmigroup theory is available for writing down solutions of the equations in (,4, B, C, D) form, another route will have to be taken for systems in pencil form. Nevertheless, it would seem to be worth the effort to pursue this direction. It should be noted that a representation which easily incorporates equations like thc string equation above has been proposcd by D. Salamon under thc name "boundary control systems' [53]; however, this class was introduced by Salamon for specific purposes, and the restrictions he imposes are consequently more severe than one would like to see in a pencil representation. The theory of system representations can be viewed as a theory of modeling. Systemtheoretic ideas may bc ,applied to modeling problems as well as to control problems, and it may cvcn bc that some problems that are now considered as control problems will eventually be looked at rather as representation problems (model matching might fall in this category). In the process, it may be necessary to abandon some conventional wisdom. This paper has been written as a tribute to Jan Wiilcms, one of the best abandoners of conventional thinking that I know. ACKNOWLEDGEMENT I would like to thank Margrcct Kuijpcr, Hcnk Nijmcijcr, and Arjan van dcr Schaft for their comments on an carlicr version of this paper. REFERENCES 1. 2. 3. 4. 5.

6.

M.A. AIZERMAN (1949). On a problem concerning the stability in the large of dynamic systems. Usp. Mat. Nauk. 4, 187-188. (In Russian.) B.D.O. ANDI-RSON,W.A. CoI'PEL, D.J. CULLEN 0985). Strong systcm cquivalcncc. J. Austral. Math. Soc. Ser. B 27, 194-222 (part I), 223-237 (part ll). M. AOKI (1988). Nonstationarity, cointcgration, and crror correction in economic modeling. J. Econ. Dyn. Contr. 12, 199-201. J.D. APLEVlCH (1981). Time-domain input-output representations of linear systems. A utomatica l 7, 509-522. V.A. ARMENTANO(1984). The pencil sE - A and controllability-obscrvability for generalized linear systems: a geometric approach. Proe. 23rd I E E E C o ~ Dec. Contr, IEEE, New York, 1507-1510. 3. S. BARAS, R.W. BROCKETT, P.A. FuH~MANN 0974). State space models for infinitedimensional systems. I E E E Trans. Automat. Contr. A C-19, 693-700.

40B 7. 8. 9. 10.

11. 12. 13, 14. 15.

16.

17. 18. 19. 20.

21. 22. 23. 24. 25. 26. 27.

R. BELLMAN, I.GLIcKSBURO, O.GRt.3SS (1956). On the 'bang-bang' control problem. Q. Appl. Math, 14, 11-18. H. BLOMI~EaG, R.YLINEN (1983). Algebraic Theory for Multivariable Linear Systems, Ac. Press. 1,ondon. H.W. BODE (1945). Network Analysis and Fet,dback Amplifier Design° Van Nostrand, Princeton, NJ. H.W. BODE (1960). Feedback - - The history of an idea. S.ymposimn on Active Networks and Feedback S.vstems (Polytechn. Inst. Brooklyn, April 19-21, 1960), Polytechnic Press, New York, 1-17. (Reprinted in IL BELLMAN,R. KALABA(cds.), Selected Papers on Mathematical Trends hz Control TheoQ,, Dover, New York, 1964.) D.W. BosHAw (1952). Differential Equations with a Discontinuous Forcing Term (Ph.D. thesis), Dept. of Marl1., Princeton Univ. F.M.CALHER, C. A. DESOER(1982). Multivariable Feedback Systems, Springer, New York. J.D.Conn (1984). Controllability, observability, and duality in singular systems. IEEE Trans. Automat. Contt; AC-29, 1076-1082. G. CoNTE, A.M. PEgDON (1982). Generalized state-space realizations for non-proper rational transfer functions. Svst. Cono: Lett. 1, 270-276. P.E.CRouCH, F. LAMNAm~I-LAGARRIGUE(1988). State space realizations of nonlinear systems defined by input-output differential equations. A. BENSOUSSAN, J.L. LIONS (eds.). Analysis and Optimi:ation ofSl,stem (Proc. INRIA Conf., Antibes, France, 1988), Lect. NGt~.s Contr. Inf. Sci. 11 I, Springer, Berlin, 138-149. C.A. DESOER, R.W. LIu, J. MURRAY, R. SAEKS (1980). Feedback system design: The fractional representation approach to analysis and synthesis. IEFE Trans. Automat. Contr AC25, 399-412. IL F. ENGLr, C. W.J. GRANGER(1987). Co-integration and error correction: Representation, estimation, and tcsting. Economeo'ica 55, 251-276. W.R. EvaNs (1948). Graphical analysis of control systems. AIEE Trans. 67, 547-551. P.M.G. FERREIRA (1987). On system equivalence. IEISE Trans. Automat. Contt: AC-32, 619-621. M. FmEss (1988). Nonlinear control theory and differential algebra. C.I. BYRNES, A. KURZHANSKZ(eds.). Modelling and Adaptive Control (Proc. IIASA Conf., Sopron, Hungary, 1986), Lect. Notes Contr. Inf. Sci. 105, Springer. Berlin, 134-145. B.A. FRANCiS (1987). A Com'se in t1~ Control Theoo,, Lect. Notes Contr. Inf. Sci. 88, Springcr, Berlin. M.I. lzREEDMAN, J.C. WtLLEMS (1978). Smooth representations of systems with differcntiatcd inputs. IEEE 7)'aas. Automat. Conn- 23, 16-22. P.A. FuHRMANN (1976). Algebraic system theory: an analyst's point of view. J. Franklin hzst. 301,521-540. P.A. FUHRMANN (1981). Linear Systems and 01)erators #z llitbert Space, McGraw-I-lill, New York. ILV. GAMKgELIDZE (1958). Theory of time-optimal processes for linear systems, lzvestia Akad. Nauk SSSR 22, 449-474. (In Russian.) F.R. GANTMACHER(1959). Mattqx Theot.'y (Part 11), Chelsea, New York. (Russian original: 1954.) E. GtLBERT (1963). Controllability and observability in multivariable control systems. S I A M J . Control I, 128-151.

406

28. B.L. I1o, R. E. KALMAN (1966). Effective construction of linear, state-variable modcls from input/output functions. Regelungstechnik 14, 545-548. 29. T. KAILATH(1980). Linear ,~ystenLv, Prentice-Hall, Englewood Cliffs, N.J. 30. R.E. KALMAN(1963). Mathematical description of linear systems. S I A M J . Control 1, 152192. 31. V. KO~ERA (1974). Algcbraic theory of discrete optimal control for muitivariablc systems. Kyhernetika 10-12, 1-240. (Published in instailmcnts.) 32. V. KU~EP.A (1975). Algebraic approach to discrete linear control. IEEE 7)'arts. Automat. Contr. 20, 116- ] 20. 33. M. KUIJPER, J. M. SCHUMACHER(1989). Realization of autoregressive equations in pencil and descriptorform, Report BS-R8903, CWI, Amsterdam. 34. M. KUIJPER, J. M. SCI-IUMACHER(1989). Transformations under extelvzal equivalence for descriptor s~,stenL,. (In preparation.) 35. A.M. LErov (1953). Stability of control systems with two actuators. Prikl. Mat. Mekh. 4. 36. F.L. LEwis (1986). A survey of linear singular systems. Circuits Systems Signal Process. 5, 3-36. 37. D.G. LuENDERGER (1977). Dynamic equations in descriptor form. 1EEL Trans. Automat. Conlr AC-22, 312-321. 38. D.G. LUENBER~Ert(1978). Timc-invariant dcscriptor systems. Automatica t4, 473-480. 39. C.C. MAcDurI~EE (1956). The TheoO, of Matrices, Chelsea, New York. (Reprint of original, 1933.) 40. A.G.J. MACFArtLANE(1979). The development of frequency-response methods in automatic control. IEEE Trans. Automat. Contr. AC-24, 250-265. (Reprinted in: A. G. J. MAcFARLANE (ed.), Frequeuct,-Resl)onse Methods in Control S),stems, IEEE Press, New York, 1979, pp. 116.) 41. A.A. MARKOV (1894). On functions obtained by converting series into continued fractions. Zap. Petersburg Akad. Nauk. (In Russian. Reprinted in: A.A. MARKOV, Collected Works, Moscow, 1948, pp. 78-105.) 42. J.C. MAXWELL(1868). On Governors. Proc. Royal Soc. London 16, 270-283. 43. D. MCFARLANE, K. GLOVER(to appear). Robust Controller Design Usiag Normalized Coprime Factor Platlt Descriptions. 44. B. McMILLAN (1952). Introduction to formal rcalizability theory. Bell Syst. Tech. J. 31, 217-279 (part !), 541-600 (part II). 45. N. MINORSKY(1947). h~troduction to Nonlinear Mechanics, Edwards Bros., Ann Arbor, Mich. (Originally published 1944-1946 as Restricted Reports by the David W.Taylor Modcl Basin, U. S. Navy.) 46. P.J. N AHIN (1988). Oliver lleaviside: Sage in Solitude, I EEL Press, New York. 47. t 1. NYQUtST (1932). Regeneration theory. Bell Syst. Tectm. d. 11, 126-147. (Reprinted in R. BELLMAN, R. KALABA (cds.), Selected Papers on Mathematical Trenda ia Cotttrol Theoo, , Dovcr, New York, 1964, and in A. G.J. MAcFARLANE(cd.), Frequency-Respot~se Methods in Control Systems, 1EEL Press, Ncw York, 1979.) 48. A.C.I'tJ~H, G. E. HAVTON, P. FRETWELL (1983). Somc transformations of matrix equivalence arising from linear systems theory. Proc. Automat. Contr. Co1~ (San Francisco, CA). 49. A.C. l'u~H, G. E. ItA~CTON, P. FRETWELL (1987). Transformations of matrix pencils and implications in linear systems theory, lut. J. Cotttt; 45, 529-548. 50. H.H. ROSENBROCK(1968). Computation of minimal rcprcscntations of a rational transfer-

407 function matrix. Proc. lEE 115, 325-327. 51. H. 1t. ROSEr~m~OCK(1970). State Space and Multivariable Theor),, Wiley, New York. 52. H.H. ROSUrqBROCK(1974). Structural properties of linear dynamical systems, hJt. J. Control 20, 191-202. 53. D.S^t.nMOtq (1987). Infinite dimensional systems with unbounded control and observation: a functional analytic approach. Trans. A M S 300, 383-431. 54. A.J. VAN DER SCHAFF(1987). On realization of nonlinear systems described by higher-order differential equations. Math. S.yst. Th. 19, 239-275. (Correction: Math. Syst. Th. 20 (1987), 305-306.) 55. A.J. VAN DER SCHAFr (1989). Representing a nonlinear state space system as a set of higher-order differential equations in the inputs and outputs. Syst. Contr. Lett. 12, 151-160. 56. J.M. ScHuMACHER (1988). Transformations of linear systems undcr external equivalence. Lin. AIg. AppL 102, 1-34. 57. J.M.SCHUMACItER (1988). State representations of linear s),stems l,qth output cottstraints, Report OS-R8807, CWI. Amsterdam. (To appear in MCSS.) 58. E.D. SoN'rAC, (1988). Bilinear realizability is equivalcnt to existence of a singular affine I/O equation. S.yst. Contr. Lett. 11, 181-187. 59. H.L.TREN'rELMAN (1986). Ahnost lnvariant Subspaces and lligh Gaht Feedback, CWI Tract 29, CWI, Amsterdam.' 60. G.C. VERGHESE(1978). h~finite Frequency Behavior in Generalized Dvnaniical Systems (Ph.D. thesis), Information Systems Lab., Stanford Univ. 61. G.C. VERGHESE, B. Lg:vv, T. KAILArrt (1981). A gcncralized state space for singular systems. IEEE Trans. A utomat. Contt: A C-26, 81 i-831. 62. M.VIDVASAr^R (1984). The graph metric for unstable plants and robustness estimates for feedback stability. I1-EE 7)'aJls. Automat. Contt; AC-29, 403-418. 63. N.WIENER (1950). Comprehensive view of prediction theory, t'roc. Int. Congr Math., Cambridge, Mass., 1950, vol. 2, 308-321. (Reprinted in: P. M^S^NI (cd.), Norhert IVienet; Collected Works with Commelltatqes, M IT Press, Cambridge, Mass., 1981, 109-122.) 64. J.C. WILLEMS(1979). System thcoretic models for the analysis of physical systems. Ricerche di A utomatica 10, 7 I- 106. 65. J.C. WILLEMS(1980). Topological classification and structural stability of linear systems. J. Diff. Eq. 35, 306-318. 66. J.C.WILLEMS (1980). Almost A (mod B)-invariant subspaces. Astdrisque 7.5-76, 239-248. 67. J.C.W1LLEMS (1981). Almost invariant subspaces: An approach to high gain feedback design. Part 1: Almost controlled invariant subspaces. IL:EE Zratts. Automat. Contr AC-26, 235-252. 68. J.C. WtLLEMS (1983). Input-output and state-space representations of finite-dimensional linear time-invariant systems. Un. Alg. AppL 50, 581-608. 69. J.C. WILLEMS (1986). From time series to linear system. Part I: Finite dimensional linear time invariant systems. Automatica 22, 561-580. 70. J.C. WILLEMS(1986). From time series to linear system. Part II: Exact modelling. Automatica 22, 675-694. 71. J.C. WILLEMS(1986). Deducing the input/output and the input/state/output structure from the external behavior. Proc. 25th IEEE Co~¢ Dec. Contr (Athens, Greece, Dec. 1986), IEEE l'rcss, New York, 1936-1937. 72. J.C. WiLLEt,~S (1987). From time series to linear system. Part III: Approximate modelling.

408 Automatica 23, 87-115. 73. J.C.WILLEMS (1988). Models for dynamics. U. KIRCHGRABER, I[.O. WALTHER (eds.). Dynamics Reported (Vol. 2), Wiley/Teubner, 17 !-269. 74. J.C.W1LLEMS, C.I-IEIJ (1986). Scattcrin8 theory and approximation of lincar systems. C.I. BVRNES, A. LINDQUIST (eds.). Modelling, ldel~tification and Robust Comrol, NorthHolland, Amsterdam, 397-411. 75. lt.K. WIMMER (1981). The structure of nonsingular polynomial matrices. Math. Systems Theos), 14, 367-379. 76. W.A. WOLOVICH(1974). Linear Multivariable Systems, Springer, New York. 77. W.M. WoNHAM (1979). Linear Mttltivariable Col~trol: a Geometric Approach (2nd cd.), Springer, Ncw York. 78. E.L. YIP, R. F. SINCOVEC(198 l). Solvability, controllability, and obscrvability of continuous descriptor systcms. IEEI'." Trans. A l¢t~,nat. Conts: A C-26, 702-707. 79. D.C. YOULA, J.J. BONGIORNO, I~. A. JABR (1976). Modern Wicncr-llopf dcsign of optimal controllcrs. Part l: The singlc-input-output case. IEEE Trat~s. Automat. Contr. 21, 3-13. 80. D.C. YOULA, J.J. BONGXORNO, H.A. JABR 0976). Modern Wicner-Hopf design of optimal controllcrs. Part 2: The multivariablc case. I E E E Trans. Automat. Contr. 21,319-338.

Optimal Control H. J. Sussmann Department of Mathematics, Rutgers University New Brunswick, NJ 08903, U. S. A.

§1. I n t r o d u c t i o n . Optimal Control Theory (OCT), in its modern sense, began in earnest in the 1950's, with the formulation and proof of the Pontryagin Maximum Principle (PMP), cf. [P], [Be], [LM]. Thc PMP is a far-reaching generalization of the Euler-Lagrange equations from the classical Calculus of Variations, and thc development of OCT has procccdcd, to a large extent, as an outgrowth of the Calculus of Variations, from which it has borrowed many of its themes, problcms and mcthods. More recently, techniques bascd oil ideas from "Geometric Control Theory" have becn applied, leading to new and exciting results. This paper is an attempt to survcy some of the recent developments in OCT, with a special emphasis on thc contribution made by these new techniques. We will deal exclusively with finite-dimensional, deterministic optimal control. In §2 we describe some of the basic problems of OCT. In §3 we discuss results on properties of optimal trajectories, including results on necessary conditions for optimality, and the important particular case of Local Controllability. ht §4 wc discuss results on optimal synthesis and the value function. * This work was aupported in p~rt by tl~e National Scie~ce Foundation under Grant DMS83-01678-01, and by the CA1P Center, Rutgers University, with funds provided by the New Jersey Commission on Science and Technology and by CAIP's industrial members.

410

§2. T i l e b a s i c p r o b l e m s

of optimal

control theory

.

T h r o u g h o u t this paper, we deal with control s y s t e m s =f(=,~,)

, xen

, ueU

(2.1)

where fl is an open subset of JR.", U is a set, and f : f l x U --* ILlTM is of class C 1 in z for each fixed u. A control for (2.1) is a function 77 : I --* U whose domain I = D o m ( 7 ) is an interval of the real line. T h e control 7 is f-admissible if the time-varying vector field f,, : F t x I --, ~'~ given by f,l(x, t) = f ( x , 7(t)) satisfies the conditions of the C a r a t h d o d o r y cxistcncc and uniqueness theorem of o r d i n a r y diffcrcntial cquations, that is: (i) fn is jointly measurable in x and t, and (ii) for each c o m p a c t interval J C I and each c o m p a c t K C ~ thcrc exists a Lebesgue integrable function ~ : J ---, IIZ such that

IIf,(x,t)ll + [[D=f,7(x,t)l[ <_ ~o(t) for all x 6 K , t 6 g .

(2.2)

A trajectory for a control 77 : I ~ U is an absolutely continuous curve "7 : I ~ such t h a t "~(t) = f ( 7 ( t ) , 7(t)) for a h n o s t all t 6 I. T h e C a r a t h 4 o d o r y t h e o r e m (cf., e.g., [CLef), implies the local existence and global uniqueness of trajectories, t h a t is: if 7 is f-admissible, then (i) given { 6 I , 2 6 fl, thcrc exists e > 0 with thc p r o p c r t y that thcrc is a t r a j e c t o r y 7 for thc restriction of 7 to the interval ( t - e, t + e) N I, such that 7 ( 0 = ~, and (ii) if 7 and 6 are trajectories for 77 such t h a t 7 ( 0 = 5(t-) for some { in I, then ~ - 6. If 7/is f-admissible and 7 is a t r a j e c t o r y for 7, then (7, 7]) is an f-admissible pair. It follows from our definitions t h a t every piecewise constant conlrol is admissible. Morcovcr, the following i m p o r t a n t Piecewise Constant Approximation Theorem ( P C A T ) holds: if (7, 7) is an admissible pair, D o m ( 7 ) is c o m p a c t , t- 6 D o m ( o ) , 2 = 7(t0, xj 6 ft, x i ---* 2 as j ---* + o o , then there exist piecewise c o n s t a n t controls 7i : D o m ( 7 ) ---, U and corresponding trajectories ~j- such t h a t ~i(t0 = x i and 7/ "-* 7 uniformly as j ---* oo.

(For a proof, cf. [Su14].) A control 7/starts at time to if to 6 D o m ( 7 ) and Dora(7) C [to, oo). In t h a t case, if "~ is a t r a j e c t o r y for ~, then the point x0 = 7(t0) is the initial point o f T . T h e definition of w h a t it means for a control to end at a timc tl, and for a point xl to be the terminal point of a trajcctory, is completely analogous. If Xo and xl arc thc initial and terminal points of a t r a j e c t o r y 7 t h a t corresponds to a control ~, thcn wc say t h a t 7 goes from Xo to xl, or t h a t 7/steers xo to xl. We want to consider more general terminal conditions, such as thc rcquircmcnt t h a t 7(t) ---, 0 as t -~ oo. So wc dcfinc a terminal condition on trajectories ( T C T ) to be, simply, a set 7" of gt-valucd curves with the p r o p e r t y that, if two curves 7, 5 coincidc from a certain time on, then 7 6 ~- iff 5 6 7-. If T is a T C T and 9' 6 ~-, then wc say t h a t 7 satisfies ~-. A particular cxamplc of T C T is the point terminal condition T~, i.c. thc set of all curves t h a t end at x. A target is a set 7" of pairwisc disjoint T C T ' s . A curve 7 rcachcs the target T (or hits T, or ends at T, or gets to T) if 7 satisfies some T 6 T . Wc considcr an optimal control problem P = (~, U, f, L, T , ~o), given by the specification of (i) a control s y s t e m (2.1), (ii) a function L : f l x U ~ IR, called the Lagrangian, (iii) a target T , and (iv) a function ~ : 7" ---, ~i.. We require that L be of class C 1 in x

411

for each fixed u. If 77 : I ---* U is an admissible control, and 3' : I --* ~ is a trajectory for T/that satisfies the T C T r E T, then wc would like to define the cost of (% 77) by

J(% zl) = f L(~/(t), ~(t)) dt + ~V(T) .

(2.3)

However, the function t --* L(7(t), 7l(t)) may fail to be intcgrable (or even measurable) unless some extra technical hypotheses hold. So we need to restrict further the set Dora(J) of those pairs (7, 7/) for which J(7, 7/) is defined. For instance, we may take Dom(J) to be the set of those f-admissible (7, ~?) such that 7 E ~- for some "r E 7" and the integrand h(t) of (2.3) has a well defined (but possibly infinite) Lebesgue integral (i.e. h is measurable and at least one of tile integrals f l h+, f l h_ is finite, where h+ = max(h, 0) and h_ = h + - h ) . If, as is often tile case, the Lagrangian is nonnegative, and the f-admissibility of (7, 77) suffices to imply that the function h is measurable, then this amounts to letting Dom(J) be the set of all f-admissible pairs (3',~/). A second possibility is to require 7/ to be (f,L)-admissible, i.e. admissible for the augmented control system obtained from (2.1) by adding a new variable y with equation 9 = L(x, u). In this case, the intcgrability of h is guaranteed if I is finite (e.g. if the target T consists entirely of point TCT's and 7/ has a starting time). If I is allowed to bc infinite, then wc restrict ourselves cvcn further and require (7, 77) to bc such that the integral exists. We shall refer to the above two possibilities as "Formulations I and II of the Optimal Control Problem P." For x E ~2 wc define V(x) to be the infimum of J(7,r/) taken over all (7, 77) E Dora(J) such that "7 starts at x and ends at the target 7". (If no such "7 exists then of course Y(x) = +c~.) The function V : 12 ~ IR, U {-oo, +c~} is the value function, or Bellman function, of the problem 7~. An optimal pair (often simply referred to as an "optimal trajectory") is a pair (-y,~/) that has a starting point x and is such that J('7,77) = V(x). Wc remark that the choice of formulation (i.e. the choice of Dora(J) as explained above) may make a fundamental difference. This is most dramatically illustrated by the "Lavrcnticff phcnomcnon" (cf., e.g., [BM]). One can exhibit a nonnegativc polynomial L : lR3 --, IR and real numbers x0, xi such that, for the problem of minimizing f(~ L(x(t),~(t),t)dt among all "curves" t --~ x(t) such that x(0) = x0 and x(1) = xl, the solution exists if by "curve" we mean "absolutely continuous function," and also if we mean "Lipschitz function." However, the values V1, V2 that correspond to these two interpretations satisfy V2 > V~. In control terms, this corresponds to the control system 5: = u, y = l, with an obvious choice of Lagrangian. Requiring x(.) to bc Lipschitz amounts to demanding that the control be measurable and bounded, in which case it is clearly (f,L)-admissible. Moreover, it follows easily from the PCAT (applied to the augmented system) that every trajectory x(.) for an (f,L)-admissible control can be approximated by Lipsehitz trajectories xj(.) in such a way that the costs of the xj converge to that of x. Therefore V2 is precisely the value for Formulation II. On the other hand, the condition that x(.) be absolutely continuous is precisely the requirement that the control bc f-admissible, so V1 is the value for Formulation I. The choice of fornmlation is important, in particular, for the problem of necessary conditions for optimalily. Indeed, the most basic of such conditions is the PMP, and the PMP is valid for Formulation II but may fail for Formulation I. For this reason, from

412

now on it will always bc understood that the optimal control problem 79 is interpreted in the sense of Formulation IL For a control system (2.1) and a target T , define a presynthcais to be a family F = {(7~, rl~) : x E S} of admissible pairs such that, for each x E S, 7~ starts at x and ends at the target. The set S is the domain of F. Call a presynthesis F memoryless if, whcncvcr x E S and t E Dom(r/=), it follows that y = 7~(t) E Dora(F) and r/u is precisely the rcstriction of r/= to the interval Dom(r/~) n [t, +c0). A mcmorylcss presynthesis F will bc referrcd to as a partial synthesis. If the domain of P is the largest possible (i.e. the set of all points x such that there exists a trajectory that goes from x to the target), then F will be catted a synthesis. If each pair (7~, r/~) is optimal for the control problem 79, then wc call F an optimal synthesis for 79. T h e following problems are of interest in Optimal Control Theory: 1. 2. 3. 4. 5. 6. 7.

cxistcnce of optimal trajectories, characterization of optimal trajectories, properties of optimal trajectories, existence of an optimal synthesis, characterization of optimal syntheses, properties of optimal syntheses, characterization and properties of the value function.

§3. O p t i m a l q_~-ajectories . The problem of the existence of optimal trajectories is very classical, and has been studied in great detail in the 1950's and 1960's, so we choose not to discuss it here, and refer the reader instead to the book [Be] by L.D. Berkovitz for a rather complete account of the results. Regarding the characterization of optimal trajectories, it has been clear for a long time that, except for very special problems such as linear systems with a convex Lagrangian, the question of sufficient conditions is essentially hopeless. On the other hand, the search for necessary conditions for optimality (NCO's) has been vigorously and fruitfully pursued since the early days of the PMP. Since the P M P is, roughly, a "tirst order condition," it was natural to look for "high order conditions." Some classical results of this endeavor are described in the papers [GK], [Go], [JS], the book [B J], and especially H.W. Knobloch's book [Kn], which gives a detailed and mathematically rigorous treatment based on a formalism of asymptotic expansions. More recently, several new techniques have been introduced, many of which are based on the systematic use of Lie brackets and Lie algebras of vector fields. In many cases, the purpose was to solve some of the other problems in our list. In particular, the synthesis problem has led --as explained below-- to an interest in specific properties of optimal trajectories, and new NCO's had to be developed in order to establish these properties. 2b illustrate why Lie brackets come into the picture, let us remark that, in order to obtain necessary conditions for a pair (7, 77) (with Dom(z/) assumed to be a compact interval [a,b]) to be optimal, the time-honored method is to make a perturbation of 77 --called a control variation-- and embed r/ in a one-parameter family {r/~ : 0 < e < g} of controls with Dom(rk)) = [a, b,]. These 71, will then give rise to trajectories 7~: [a, b,] ---, ~Z(with 7,(a) --- 7(a)). If the variation is smooth in some appropriate sense,

413

then the optimality of ('7,77) will imply that the derivative of J(7~,~Te) with respect to at e -- 0 has to be nonnegativc. (We are ignoring hcrc the cxtra difficulty that the perturbed trajectories must also satisfy the desired target conditions.) If wc assume that we are already dealing with the augmcntcd problem, as explained above, then the requirement becomes the condition that v • V~b(p) >_ 0, where p = ~/(b), ¢ is a function on the state space 12, and v is the variational vector d ~=0~e(b~ )

(3.11

So the vector v must belong to a particular subset of ~ ' . When onc applies this condition to a reasonably large class of variations, one obtains an NCO. In particular, the P M P is obtained by taking variations of a specially simple kind (cf., e.g. [P]), and more powerful NCO's arise by considering more sophisticated variations. Onc is thus naturally led to the study of the set of all possible variational vectors. And it can be seen in a number of ways that these vectors can be obtained as leading Lerms of asymptotic expansions that involve Lie brackets. (This idea, as well as its use for high order NCO's, was proposed by [Bro], fAG2] and [gr].) For a simple illustration of this situation let us consider a system of the form .4:.= f ( x ) -t- ug(x), where f and g are C °O vector fields and thc scalar control u takes values in U = [-1, 1]. (This is called a two vector field system, because the possible dircctions of motion at a point p are X ( p ) , Y(p), and their convex combinations, where X = f - 9 and Y -- f -t- g.) Assume moreover that a point p is an equilibrium, i.e. that f ( p ) = O. Then the constant trajectory "/(t) = p corresponds to the control 77(t) -- 0. One can prove that, if r/: [0,7'] ---, [-1,1] is a general control, and ")',7 is the corresponding trajectory with initial condition ")',~(0) = p, then the asymptotic expansion "r,(T) . . . .

e~3(')b~ e"=(')b2 e'~'(')b'p

(3.2)

holds, where (i) the bi arc vcctor fields obtained from ccrtain formal Lic brackcts B= in the two indcterminatcs F and G by plugging in f for F and g for G, (ii) the Bi arc a P. Hall basis (cf. [Sc], [Vi]) of the frcc Lie algcbra in the indeterminatcs F and G (for instance, one ca,, take B, -- F, B2 -- G, B3 = [F, C], B4 -- [F~ [F, C]], B5 -- [G, [F, G]], • ..), (iii) the a~(~/) are iterated integrals calculated from ~/ by explicit formulae (for instance, if the P. llall basis is as indicated above, then cq(7/) = T, a2(7/) = for 7/(t)dr, =

I [ t,7(t) t,

=

t:,7(t),tt,

= f0T Ig, t

(iv) the

exponential notation stands for the flow, i.e. l ---, etVp is the integral curvc of the vector ficld V that gocs through p at time t = 0, (v) " ~ " stands for "asymptotic cquality," in the scnsc that, if Pk dcnotes the right-hand sidc of (3.2) with all thc Bi of dcgrcc > k omitted, then "7,(T) - Pk = O ( T k+x) as T ---* 0, uniformly in 77. (The right-hand side of (3.2) is thc Chcn series. Cf. [Su2] for the asymptotic formula, and [Sull] for the fact that thc Chcn series is given as an infinitc product of exponentials.) If one now manages to choosc 77e : [0,Te] --* [-1, 1] so that all the integrals ak(~e) other than a particular am(7?~) are o(e 6(m)) (where 5(m) is the degree of the integral am, i.e. the dcgrcc of the formal Lie bracket Bin) whereas am(~?e) itself is ,.- ee 6(m), with c ¢ 0, then we will have exhibited a control variation in the direction of =t=b,-,(p), the sign being that of c. It is natural to choose :/~ = ¢. Thc iterated intcgrals associated to Lie monomiais

414

of degree > ~5(m) are then automatically O(E6(m)+I), so the only real difficulty is to make sure that the integrals of degree i _< 6(m), whose natural asymptotic behavior is .,~ c ~, are in fact nmch smaller. As an example, let us exhibit a variation in the direction of tile vector - [ 9 , [f,g]](P). We begin by picking a function r/ : [0, 1] ---, [-1, 1] which is L2-0rthogonal to thc functions 1, t and t 2 but not =- 0. (Onc can takc r/ to be a cubic polynomial.) Then define r/~(t) -- z/(~), with domain [0, el. An easy calculation shows that al(~TE) = e, a2(Ve) = a3(r/~) = a4(~Te) = 0, and ab(T/~) = - c ~ 3, whcre c = ~1 fo1 (fot ~?(s) ds) 2 dt, so that c > 0. It is clear that the brackets Bi of degree < 3 arc precisely B1, B2, B3, B4 and Bb. Since e=~(n)b*p = p for all 7/ (because bl = f and f(p) = 0), and b5 =- [9, [f, gl], we see that 7n,(e) = p - cz3[g, [f,g]](P)+ o(E3) • So -[9, [f, g]](p)is a variational vcctor. The prcccding example shows how a particular direction can be exhibitcd as a variational direction by "neutralizing" those directions that in principle might have given rise to larger terms in the asymptotic expansion. Moreover, the example makes it clear that there are "semialgcbralc" properties of the iterated integrals that make it possible to neutralize certain vectors but not others. For instance, the fact that as(r/) is negative follows from the equalities 0~2(?~) = OZ3(~) = O~4(71) = 0. Therefore the vector +[9,[f,g]](p) cannot be exhibited as a variational vector by this method. On the othcr hand, it may happen for a particular system that [9, [f, g]](p) is equal to some othcr vector (c.g. 9(p)) which is a variational vector for other reasons. In that casc, [g, [f, 9]](p) will b c a variational vector after all. Therefore, in order to understand thc variational vectors, one has to understand the semialgcbriac properties of iterated intcgrals that cause certain vectors such as -[9, [f, 9]] (p) (but not +[9, [f, 9]](p)) to show up automatically as variational vcctors, as well as the interplay between this algebra and thc spccial Lic-algcbralc properties of particular systems that pcrlnit more gcneral vectors such as [9, [f,y]](p) to bc variational in certain cases evcn though thcy are not variational in gcncral. The linc of thought outlincd above has been actively pursued in rcccnt ycars, leading to a series of papers on small-time locally controllable (STLC) systems. A system Z is STLC from a point p if for arbitrarily small T > 0 the set ~ ( < T,p) (i.e. the rcachable sct from p in timc < T) contains p in its intcrior. The study of the STLC condition is important because it is a very interesting special case of tim more general question of NCO's. Indeed, a system fails to be STLC from p ill the constant trajectory 7(t) - p --which for an equilibrium point of a two vector field system corresponds to the control r](t) _= 0 - - is a boundary trajectory, in the sense that 7(t) E 0~(_< t,p) for all sufficiently small t > 0. So a sufficient condition for the STLC property immediately yields a necessary condition for a particular 7 to be a boundary trajectory, i.e. an NCO of a special kind. The scarch for sulticient conditions for a system to be STLC has been guided by the above ideas. It is at least intuitively clear that, if a suitable definition of "variational vector" is given, then the set ~)(p) of all variational vectors at p should bc a convex cone, and a system should be STLC from p if ~)(p) is the whole space IR'*. To make these considerations rigorous one may proceed as in the classical theory of the PMP (cf. [I']), or one may follow the approach proposed by H. Frankowska (cf. [Fra]) and use open mapping theorems for set-valued maps. In either case, one is led to the question of sultlcient conditions for ])(p) to be the whole space, and this requires the

415

algebraic analysis outlined above. As expected, Lie brackets arise most naturally. The development of the theory was initiated by H. Hermes in a series of papers ([He1], [He2],[He3]). For the particular case of two vector field systems, tIermes pointed out that it was crucial to distinguish between those brackets with an even number ofg's and those with an odd number. Let us assume that we are considering real analytic syslems. Then we may assume that the "accessibility property" (AP) holds (i.e. that the Lie algebra A generated by f and g satisfies A(p) = IR/'), since this is known in any case to be necessary for the reachable set from p to have a nonvoid interior (of. [SJ]). Hermes singled out the property that Sk(f,g)(p) = Sk-1(f,g)(p) for all even k and conjectured that, together with the AP, this condition was sufficient for STLC. (Here Sk(f, 9) is thc linear span of •all the brackets of f ' s and g's with no more than k g's. Notice that the Hermes condition holds if p is an equilibrium and Sl(f,g) already equals IWL This is precisely thc case when the linearization at p is controllable, so the Hermes condition is weaker than the classical condition on the lincarization, and the corresponding sullicicncy theorem stronger.) In [Su3] it was provcd that the Hermes condition is indccd sufficient for tim STLC property. This was done by introducing a new technique for handling tim algebraic part of thc argument, namely, tile use of groups of symmetries. Using this technique in a more refincd way, a much stronger sufilcicncy thcorcm was established in [SuT]. Since then, new and more powerful sullicicnt conditions have bccn found by K. Wagner, R.M. Bianchini and G. Stcfani, and M. Kawski (cf. [Wa], [BS1], [Kat], [Stl], [St2], and especially the survey article [Ka2] by Kawski). Moreover, the work on the STLC condition has bccn extended to the study of local controllability about more gcncral --i.e. not neccssarily c o n s t a n t - - reference trajectories, that is to the gcncral NCO problem that had originally provided the motivation for studying this particular case. Work by A.A. Agrachcv and R.V. Gamkrelidze (lAG2]) dcvelopcd a general formalism for expressing, in terms of Lic brackcts, variational vcctors arising from perturbations of general controls (cf. also [Su4]). In various other papers (e.g. lAG1], lAG3]), these authors applied their formalism to the derivation of new NCO's. P. Crouch and F. Lamnabhi-Lagarriguc ([CLa]) have pursued the development of the algebra nccdcd to understand the perturbation expansions. Using their work, new NCO's have bccn obtained by Lamnabhi-Lagarriguc, Bianchini and Stcfani (cf. [LLa], [BS2], [LLS], [St3]). Another set of NCO's were obtained by A. Brcssan in [Brl] using methods from nonlinear functional analysis (the Mountain Pass'Lemma), and then applied by Brcssan himself in his work on synthesis (cf. [Br2]). A third approach to obtaining NCO's has been used by II. Sch/ittler in his synthesis work (cf. [Scl], [Se2] and [SSu]). The method here is to use the Campbcll-IIausdorff scrics to compare two controls dcfincd on small time intervals. Suppose, for instance, that wc arc dealing with the minimum time problem for a two-vcctor field system as above. Then we may want to study bang-bang controls of the form rh, ~'~..... ......L~, k i.e. controls obtained by letting u = - 1 during a time rl, then u = 1 d u r i n g a t i m c Q, thcn u = 1 during atimcT-2, and so on. Such a c o n t r o l steers a point p to a point q = e~kre * k x . . . e*2Ye'r2xe*tYe~'lXp

.

(3.3)

If we have two different controls of the above type that steer p to the same point q, then wc can use the asymptotic expansion of the right-hand side of (3.3) to conlpare

416

the total time ~ ti + ~ ri for both controls, and we may be able to conclude that one of the two cannot be optimal. Finally, we give an example of an NCO which is of a different type, in that it generalizes the theory of envelopes from the classical Calculus of Variations. Define a oneparameter field of extrcmals (1PFE) to be a family {(Te, r/e, ,ke) : 0 < ~ < ~} of triples such that the (%, rk) are extremals, defined on intervals [ac, be], the Ac : [ac, b~] --* IR.'~ are Pontryagin adjoint variables for the (7~,r/~), and the a~, be, 7e and ,k~ "depend smoothly on E." (Naturally, to make this precise we need a precise definition of smoothness. In [Sul2], the particular case of bang-bang extremals is considered, and "smoothness" simply means that the switching times of the controls and the initial conditions for the 7e and the Ac are continously differentiable functions of ~. A paper developing the general theory under minimal smoothness assumptions is in preparation.) if the a~ and the 7c(ac) do not depend on c, then we shall say t h a t we have a fixed initial condition (FIC) field. Now suppose that we have a Pontryagin extremal (7,r/) defined on an interval [a, b]. Define an envelope for (7, 7/) to be a curve 6 : [ - g , 0] ~ ft, with E > 0 with the property that there is a F I C 1 P F E {(%,r/~,Ac) : 0 < c < C}, defined on intervals [a, be], such that 6 ( - ~ ) = 7(bE) for 0 < e < g, and 6 is itself a trajectory, corresponding to some control 0 : [-E, 0] ~ U, in such a way that (A~(be),f(6(-¢),O(e))) = 0 for all ¢. One thcn proves that the trajcctory obtained by concatenating "re with 6 (with an obvious definition of thc controls) has exactly the same cost as (7, r/). In particular, if (7, r/) is optimal then it follows that (6, 0) is optimal as well. This can be used in many cases to prove that (% r/) cannot be optimal. For instance, suppose that the terminal point q of 7 lies in a region R such t h a t every optimal trajectory in R is bang-bang. Suppose also that wc managc to produce an envelope 6 which is not bang-bang. Then it follows that (7, 71) is not optimal. This method has been successfully applied in [Sul2], [Sell, [Sc2] and [SSu] to the study of synthesis problems in low dimensions. Wc now turn to the problem of properties of optimal trajectories. Naturally, there are no sharp boundaries separating this question from that of N C O ' s since, strictly speaking, an NCO is exactly the same as a statement t h a t an optimal trajectory must have some special property. IIowever, the question that will interest us now is that of proving theorems that say that optimal trajectories must have some interesting "regularity properties." For instance, if it was truc that every optimal control is picecwise smooth (assuming now that we are dealing with problems with U C lI~m and f reasonably smooth in u as well as in x), then this would bc an example of the kind of theorem we want. Unfortunately, no such theorem is true in full generality, and two natural questions arise. First, one wishes to know whether some weaker result (e.g. that optimal controls ncccssarily have countably many discontinuities, or that the set of points of discontinuity has measure zero) is truc in general. Second, one wishes to know whether strong theorems arc truc for interesting special classes of systems. Two classical examples of the latter situation are (i) the theorem that, for a classical Calculus of Variations problem whose Lagrangian L(x, x) is of class C ~ and has an everywhere nonsingular IIcssian with respect to x, all optimal trajectories are of class C °°, and (ii) the bang-bang theorem from Linear System Theory, according to which, for a linear system 5: = Ax + Bu with a polyhedral control constraint u E K = c o ( u l , . . . , u k ) , whenever there is a trajectory from a point p to another point q, there is a time-optimal trajectory from p to q which is bang-bang with a finite number N of switchings that

417

can actually be estimated to be <_ C + DT, where C, D arc constants and T is the optimal time for going from p to q. Notice that the second result does not say that a time-optimal trajectory necessarily has to be bang-bang, since the system may be such that, for instance, one of the state coordinates - - s a y x l - - satisfies 5:1 = 1, in which case every trajectory is time-optimal. W h a t the result does say is that "bang-bang trajectories sulIicc for time-optimality." More generally, let us say that a class C of trajectories su]ficcs for an optimal control problem P if, whenever there is an optimal trajectory from a point p to the target T , it follows that there is an optimal trajectory from p to the target which belongs to C, and let us say that C suffices for a class of problems if it suffices for each problem in the class. (So, for instance, bang-bang trajectories suffice for linear time-optimal control problems with polyhedral constraints.) Notice that this formulation includes the situation when the cost is identically zero and the targets arc points, in which case our question is just that of finding classes that are sufficicnl for rcachabilily, i.e. such that whenever it is possible to go from a point p to another point q by means of zomc trajectory, then this can be done by means of a trajectory in C. The problem of finding interesting sufficient classes of trajectories is essentially handled by the use of NCO's, i.e. by assuming that one has an optimal pair (% ~7) and trying to prove --using all available N C O ' s - - that ('7,7/) is "special" in some sense. There is, however, a significant twist that gives this line of research a slightly different technical flavor: wc arc now no longer restricted to working with given (%~1); we are allowed to modify (% 71) in order to make it "nicer," as long as we do not change the initial and terminal conditions. The work on sufficient classes has so far proceeded by a combination of analytic arguments based on the use of the P M P and other NCO's, and geometric reasoning. In many cases (such as the work by Schfi.ttler and Brcssan mentioned above) it was the nccd to prOVE suificicncy theorems that led to the discovery of new NCO's. In most of this work, Lic brackets have played an essential role. The simplest of the new sufficiency theorems is the result of [Sul], according to which bang-bang trajectories arc also sufficient for certain classes of two vector field systems. The specific condition on the systems can easily be understood as a particular consequence of lincarity: a two vector field system is linear iff all Lie brackets involving two g's vanish identically, whereas the property of [Sul] says that all brackets with two g's must be expressible in a special way as linear combinations of brackets with only one g (cf. [Sul] for the precise details). It then turns out that this weaker property is all that is required for the bang-bang theorem. The result of [Sul] only makcs use of the PMP. In [Su6], [Su8], [Su9], [Sul0], [Sul3], [Scl], [Sc2], [Br2]) a number of other results on sufficient classes were proved for various classes of systcms. Remarkably, the result of [Su8], [Su9] and [Sul0] shows that, for real analytic two vector field systems in two dimcnsions, trajectories of piccewise analytic controls (with a finilc number of points of nonanalyticity), arc suflicicnt for time-optimality. Unfortunately, this result fails, cvcn in two dimensions, as soon as more complicated cost functionais are considered. For instance, for the problem of reaching the origin with minimmn cost, subject to :i: = y, !) = u, and JuJ _< 1, the cost functional being J" X 2 , it can be shown that the optimal control from any initial condition p ~ (0, 0) is bang-bang with infinitely many switchings. (This is "Fuller's problem;" cf. [Ma] for a classical treatment, and [Ku] for a an analysis showing that FuUer-like

418

situations appear in most optimal control problems, in sufficiently high dimensions.) One can easily modify Fullcr's problem to obtain a minimum time problem in 114.3for which similar "pathology" occurs. One may then ask whether there arc any results at all that can be expected to hold in full generality. The answer turns out to be closely related to real analyticity. Indeed, it was shown in [Sul4] that, if an arbilrary measurable function 77 : [0, T] ---* [ - 1 , 1] is given, then one can construct a two vector field system of class Coo in ]1{3, and points p, q, such that 77 steers p to q and no other control docs. So no class of controls strictly smaller than that of all controls can bc sufficient for all C °o problems. On the other hand, it turns out that the above construction is not possible if one wants the vector fields to be analytic. As shown in [Sul3], the class [ t A D of controls 7/with the property that 77 is real analytic on an open dcnse subset of Dom(r/) is sufficient for rcachability and for time optimality for all real analytic two vector field problems. The problem of the general structure and regularity properties of optimal controls, cvcn for the case of two vector field systems (and cvcn in 1R.a) appears at the moment to be beyond the reach of currently existing methods. (ttowever, substantial progress has been made, in three dimensions, in the understanding of generic cases , cf [Scl], [Sc2], [Br2].) It is still a mystery where exactly, somewhere in between the very large class I~AD and the much smallcr class of piecewise analytic controls with at most countably many points of nonanalyticity, lies the frontier that characterizes the maximum possible pathology of optimal controls. §4. O p t i m a l S y n t h e s i s . The existence problem for optimal synthesis is closely related to the question of existence of optimal trajectories. Once one knows that an optimal trajectory exists for every initial point p, then an optimal presynthesis is obtained by just choosing one such trajectory for each p, which is certainly possible if one accepts the Axiom of Choice. However, to get a synthesis one has to make the presynthesis mcmoryless. Whether this is possible in general appears at thc moment to bc an open question. Regarding the characterization of optimal synthesis, the situation is quite remarkably diffcrcnt from what happens for optimal trajectories. Nice sufficient condilions can now be proved. Indeed, the question of optimal synthesis is intimately related to the study o1" the solutions V of the Hamilton-Jacobi-Bellman (IIJB) equation

fI(VV(x),x) = 0 .

(4.1)

The function If is the normal minimized ttamiltonian, given by H()t, x) = /-I(A, 1, x), wtmre /-I(A, Ao, x) = inf {7-/(A, Ao, x, u ) : u E U} , (4.2) and ~/(A, Ao, x, u) = ()~, f(x, u)) + AoL(x, u) .

(4.3)

(Ilere A C 11['~ and Ao E lI~.) Now, let F be a synthesis and let Vr be thc corresponding valuc function, i.e. Vr(x) = J('~,~7~). Let V be the vaIue function of the problem under consideration. Then of course F is optimal iff Vr = V. So the problem of deciding whether F is

419

optimal is precisely that of checking whether Vr = V. Moreover, for F to be optimal it is clearly necessary that each trajectory (7~, rh:) b c a Pontryagin extrcmal. (Recall that a Pontryagin extremal is an ( f , L ) admissible pair (%7/) with the property that that there exist (i) an absolutely continuous function t --, A(t), defined for t E Dora(r/), and (ii) a constant ,~o > O, such that the equalities = -

(4.4)

,7(t))

and =

= 0

(4.5)

hold for almost all/, E Dom(r/), and ()~(t),Ao) # (0,0) for s o m e - - a n d hence e v e r y - value of t. The P M P says that an optimM trajectory must be a Pontryagin extremal.) So an obvious necessary condition for F to be optimal is that it bc an extremal synthesis, i.e. a synthesis all whose trajectories are Pontryagin extremals. We shall use t= to denote the starting time of the control ~?,. It turns out that, "modulo some technicM conditions," the value function of the optimal control problem can bc characterized as the unique solution of the HJB equation that satisfies certain boundary conditions. So what we need to know is whether Vr is a solution of the IIJB equation and whether it satisfies the required boundary conditions. If one compares the IIJB equation with the statement of the P M P given above, it is easy to scc that the value function Vr will satisfy the tlJB equation provided we show that the adjoint variable A=(.) whose existence is asserted by the PMP can be chosen so that VVr(x) = A(/:), and the corresponding )~o can be taken to be equal to 1. It turns out that --again "modulo technical conditions"-- this is indeed possible. So an extremal synthesis is indeed optimal if suitable technical conditions hold. The preceding paragraph is intendedly vague, because of the repeated reference to unspecified "technical conditions." As written, the paragraph constitutes an accurate rellcction of the current state of our knowledge. It is "known" that in some sense the statements arc truc in general, but no satisfactory way has yet been found to translate them into theorems that are both precise and sufficiently general. Wc shall refer to the problem of finding such theorems as the Fundamental Problem of Synthesis Theory (FPST). Naturally, the main question is that of finding the appropriate technical conditions that make all the desired results true. The F P S T splits into three parts. First, there is the problem of rigorously defining what is meant for a function to bc a solution of the HJB equation. The difficulty hcre is that the value function typically is not everywhere diffcrcntiable, so one needs some concept of "weak solution." To illustrate the difficulties involved, consider the simple cxaml)lc of the system :/: = u on the interval [-1,1], with control constraint lul < 1. Assume that wc want to reach the boundary of the interval in minimum time. It is easy to sec that the value function V is just V(x) = i - [x[, the IIJB equation is IV'(x)[ = l, and the boundary comtition is V ( - 1 ) = V(I) = 0. H.cquiring that a solution of the IIJB equation bc diffcrcntiablc everywhere woukl exclude the function V. If wc only require ditfercntiability ahnost everywhere, or cvcn everywhere except at just one point, then wc obtain the dcsircd function V as well as inany undesirable functions, such as - V .

420

Second, there is the question of passing from the fact that Vr satisfies the HJB equation and the boundary conditions to the conclusion that Vr is the value function. It is clear that the value function V satisfies the Dynamic Programming Inequality (DPI) V(x) <_V(y) + / L(v(t), ~(t)) dt ,

(4.6)

whenever (% 7/) is a trajectory that goes from x to y and that, conversely, any function that satisfies this inequality plus the boundary conditions must be bounded above by tile value function. In particular, if Vr satisfies tile DPI and the boundary conditions then Vr is the value function. So the technical problem here is the passage from the HJB equation to the DPI. Formally, this is done by observing that, the HJB equation implics that (VV(v(t)),'~(t)) + L('y(t),~?(t)) > 0, from which the DPI follows by integration. The tcchnical problem is to handle the case when VV does not c:,:_ist cvcrywhcre, and therefore the proof by integration is not justified. Actually, all that is uscd in the formal derivation is the inequality H(VV(x),x) >_O. Let us call a function that satisfies this inequality in some sense a subsolution of HJB. Then the real problem is to find the adequate technical definition of "subsolution" so that it becomes a true theorem that a function is a subsolution iff it satisfies the DPI. Third, there is tile derivation of the HJB equation from the property that F is cxtrcmal. In this case, there is a formal proof which is just bascd on differentiating Vr at a point x in a direction v. Asssuming, for simplicity, that the target is just a single point, one writes Vr(x + hv) as an integral, and diffcrcntiatcs with rcspcct to h at h = 0. The result is precisely (A(t~),v) and, sincc v is an arbitrary direction, the desired conclusion follows. One is then left with the technical problem of justifying the differentiation under the integral sign. V.G. Boltyansky (cf. [Bo]) developed a theory of "regular synthesis" in order to tacklc the above problems. Roughly speaking, a rcgular synthesis in Boltyansky's sense is a synthesis in which the trajectories and the controls depend on x in a "pieccwise smooth" way. (The precise definition is quite technical, and we shall not repeat it here.) Boltyansky's result is that a regular synthesis in his sense is indeed optimal. However, it is not hard to give examples of optimal syntheses that arc not regular in Boltyansky's sense but are good enough so that Boltyansky's arguments work, possibly after some trivial modifications. So Boltyansky's theory has to be extended and generalized. Examples of such extensions have been proposed in [Bru2], [Su3] and [Su5], but none of these can bc regarded as definitive. The main reason why we do not have a good theory of "regular synthesis" is that we lack a good theory of how "nice" the optimal synthesis ha~ to bc for reasonably large classes of problems. Indeed, suppose we could identify a property P such that one could prove, for every optimal control problem in a sufficiently large class, that (a) an ot)timal synthcsis with Property P exists, and (b) if an extremal synthesis has Property P then it is optimal. Then wc could regard the F P S T as solved: one would consider P to bc the natural property to be required of a synthesis, in that an existence theorem holds, and there is a simple necessary and sullicicnt condition for optimality, namely, extremality. So wc arc naturally led to the question of properties of optimal sygthcscs. The study of this problem began with the work of P. Brunovsky [13rul], [Bru2], who showed,

421

for certain classes of problems (linear time optimal with a polyhedral control constraint) that a "regular synthesis" (in a sense slightly different from Boltyansky's) exists. Although the class of problems originally studied by Brunovsky was quite limited, his work made a far-reaching contribution, in that it introduced the idea of using the theory of subanalytic sets (cf. [Ha], [Hi], [Su15]) in optimal control. As was subsequently noticed (e.g. by Brunovsky himself in [Bru2], and also in [Su3]), the crucial point of his approach is that, thanks to the use of subanalytic sets, the problem of proving existence of a nice synthesis can bc reduced to that of proving that certain classes of trajectories are suMcicnt for optimality. This has led to a number of results on existence of regular synthesis (cf. Brcssan [Br2], Sch~ttlcr [Scl], [Sc2], Sussmann [Sul0]). The problem of the characterization of the value function is closely related to the FPST, except that now we are just given a function V~ and we ask for conditions under which this function will be the value function V. For the synthesis problem the function l~ already comes from a synthesis, so it automatically satisfies V9 _< V. IIcre, on the other hand, we need conditions that will imply both inequalities V9 _< V and Va >_ V. An elegant characterization of the value function along these lines is given by the theory of viscosity solutions duc to M.G. Crandall and P.L. Lions (cf. [CLi], [LS]). A continuous function V is a viscosity subsolution of the HJB equation if, whenever q9 is a C 1 function defined on a neighborhood of a point x, and such that V - ~o has a local maximum at x, it follows that II(V~o(x),x) >_ O. It is then easy to prove that, if V is continuous, then V is a viscosity subsolution if and only if it satisfies the DPI. Replacing "maximum" by "minimum" and "H _> 0" by "H _< 0" one obtains the definition of viscosity supcrsolutiou. Naturally, a viscosity solution is then defined to b c a function which is both a subsolution and a supersolution. It then turns out that for large classes of problems the value function can be characterized as the unique viscosity solution of the HJB equation that satisIies the appropriate boundary conditions (cf.[LS]). REFERENCES

lAG1]

Agrachev, A.A. and R.V. Gamkrelidze, "A second-order optimality principle for a time-optimal problem," Math. Sbornik I00, 142 (1976).

[hC2l

Agrachev, A.A. and R.V. Gamkrelidze, "The exponential representation of flows and the chronological calculus," Math. Sbornik 109, 149 (1978).

[AC3]

Agrachev, A.A. and R.V. Camkrelidze, "Tile Morse index and the Maslov index for smooth control systems," Doklady Akad. Nauk USSR 287 (1986). Ball, J.M. and V.J. Mizel, "One-dimensional variational problems whose minimizers do not satsify the Euler-Lagrange equation," Arch. Rational Mech. Anal. 63 (1985), pp. 273-29,1.

[~J]

Bell, D.J. and D.II. Jacobson, Singular Optimal Control Problems, Academic Press (1973).

[Sel

Bcrkovitz, L.D. Optimal Control Theory, Springcr-Vcrlag (1974).

[BSI] Bianchini, R.M., and G. Stefani, "Sufficient conditions for local controllability," in Proceedings 25lh IEEE Conference on Decision and Control (1986).

422

[BS1] Bianchini, R.M., and G. Stefani, "Local controllability about a reference trajectory," in Analysis and Optimization of Systems, A. Bensoussan and J.L. Lions eds., Springer-Verlag Lcct. Notes Contr. Inf. Sci. 83 (1986).

[Bo]

Boltyansky, V.G., "Sutficicnt conditions for optimality and thc justification of the Dynamic Programming Principle," SIAM J. Control 4 (1966), pp. 326-361.

Ira1]

Brcssan, A,, "A high-order test for optimality of bang-bang controls," SIAM J. Control Opt. 23 (1985), pp. 38-48.

[Br2]

Brcssan, A., "The gencric optimal stabilizing controls in dimension 3," SIAM J. Conlrol Opt. 24 (1986), pp. 177-190.

[tool

Brockctt, R.W., "Lie theory, functional expansions and necessary conditions in ot)timal control," in Mathematical Control Theory, W.A. Coppcl cd., SpringerVcrlag (1978), pp. 68-76.

[Brul] Brunovsky, P., "Evcry normal linear system has a regular synthesis," Mathematica Slovaca, 28 (1978), pp. 81-100.

[Bru2]

Brunovsky, P., "Existence of rcgular synthesis for gcncral problcms," J. Diff. Equations, 38 (1980), pp. 317-343.

[CLe] Coddington,

E.A. and N. Lcvinson, Theory of Ordinary Diffcrcnlial Equations, McGraw-Ifill (1955).

[CEil Crandall, M.G. and P.L. Lions, "Viscosity solutions of Hamilton-Jax:obi equations," Trans. Amer. Math. Soc. 277 (1983), pp. 1-42.

[tEal Crouch, P.E. and F. Lamnabhi-Lagarriguc, "Algebraic and multiplc intcgral identities," to appcar in Acta Applicandae Mathcmathicac.

[im]

Fleming, W.lI. and R.W. l~ishel, Deterministic and Stochastic Optimal Control, Springer-Verlag (1975).

[Fra] Frankowska, tl., "An open mapping principlc for set-valued maps," Ccntrc dc I~.cchcrchcs Mathdmatiqucs, Univ. de Montrdal, rapport CRM-1364 (1986).

[CK]

Gabasov, V., and F.M. Kirillova, "lIigh-ordcr ncccssary conditions for optimality," SlAM J. Control 10 (1972), pp. 127-168.

[Co] Goh, B.S., "The second variation for the Bolza problem," SIAM J. Control4 (1966), pp. 309-325. Hardt, R.M., "Stratifications of real analytic maps and images," Inventiones Math. 28 (1975), pp. 193-208.

[Itcl] tIcrmcs, II., "On local and global controllability," SIAM J. Control 12 (1974), pp. 252-261.

[11e2] Ilcrmcs, II., "Lie algebras of vector fields and local approximation of attainable sets," SIAM J. Conlrol Opl. 16 (1978), pp. 715-727. [ltc3] l lcrmcs, II., "ConLrol systems which generate decomposable Lie algebras," J. Diff Eqaalions 44 (1982), pp. 166-187. [lli] Hironaka, II., Subanalytie Sets, Lcct. Notes Istituto Matematico "Lconida ToneUi," Pisa, Italy (1973).

423

[JS]

Jacobson, D.H. and J.L. Speycr, "Necessary and sufficient conditions for optimality for singular control problems: a limit approach," J. Math. Anal. Appl. 34 (1971), pp. 239-266.

[Kal]

Kawski, M., "A ncw necessary condition for local controllability," to appear in Proceedings Conference on Differential Geometry, San Antonio, Texas, I988, Amcr. Math. Society Contemporary Mathematics Series.

[Ka2] Kawski, M., "Iligh order small-time local controllability," to appear in Nonlinear Oplimal Conlrol and Controllability, H.J. Sussmann ed., M. Dekkcr, Inc. [KKM] Kelley, II.J., R.E. Kopp and H.G. Moyer, "Singular cxtrcmals," in Topics in Optimization, G. Lcitman ed., Academic Press (1967).

[Knl

Knobloch, II.W., High Order Necessary Conditions in Optimal Control, SpringcrVcrlag (1975).

[Kr]

Krcncr, A.J., "The higher order maximum principle and its application to singular cxtrcmals," SIAM J. Control Opt. 15 (1977), pp. 256-293.

[Ka2]

Kupka, I., "The ubiquity of l?ullcr's phenomenon," to appear in Nonlinear Optimal Control and Controllability, II.J. Sussmann cd., M. Dekker, Inc.

ILLs] Lanmabhi-Lagarriguc, F., "Singular optimal control problems: oll the order of a singular arc," Systems aTtd Control Letters 9 (1987), pp. 173-182. ILLS] Lamnabhi-Lagarriguc, F., and G. Stcfani, "Singular optimal control problcms: on thc ncccssary conditions for optimality," to appcar in SIAM J. Control Opt.

[LM]

Lcc, E.B. and L. Markus, Foundations of Optimal Control Theory, J. Wiley, New York (1967).

[LS]

Lions, P.L. and T. Souganidis, "Differential games, optimal control and dircctional derivatives of viscosity solutions of Bellman's and Isaacs' equations," SIAM J. Control Opt. 23 (1985), pp. 566-583.

[Ma]

Marchal, C., "Chattcring arcs and chattering controls," J. Optim Theory Appl. 11 (1973), pp. 441-468.

[P]

Pontryagin, L.S., V.G. Boltyansky, R.V. Gamkrelidze and E.F. Mischenko, The Mathematical Theory of Optimal Processes, J. Wiley (1962).

[S l]

SchSttlcr, H., "On the local structure of time-optimal bang-bang trajectories in ]I~3,'' SLiM J. Control Opt. 26 (1988), pp. 186-204.

[Sc2]

Schattlcr, II., "The local structure of time-optimal trajcctorics in ]K3 under generic conditions," SL'IM J. Control Opt. 26 (1988), pp. 899-918.

[ss.]

SchS.ttlcr, tI., and [I.J. Sussmann, "On the rcgularity of optimal controls," J. Appl. lvIath. Physics (ZAAIP) 38 (1987), pp. 292-301.

[Se]

Scrrc, J.P., Lic algebras and Lie groups, W.A. Benjamin, New York (1975).

[Stl] Stc['ani, G., "Local controllability of nonlinear systems: an example," Systems and Conlrol Letters fi (1985), pp. 123-125.

424

[st2]

Stefani, G., "On the local controllability of a scalar-input control system," in Theory and Applications of Nonlinear Control systems, C.1. Byrnes and A. Lindquist eds., Elscvier (1986).

[st3]

Stcfaui, G., "A sufficient condition for extrcmality," in Analysis and Optimization of Systems, A. Bcnsoussan and J.L. lions cds., Springer-Verlag Lcct. Notes Contr. lnf. Sci. 111 (1988), pp. 270-281.

[SJ] Sussmann, H.3. and V. Jurdjcvic, "Controllability of nonlincar systcms," J. Diff. Equations 12 (1972), pp. 95-116.

[sul]

Sussmann, II.J., "A bang-bang thcorem with bounds on the numbcr of switchings," SIAM J. Control Opt. 17 (1979), pp. 629-651.

[Su2]

Sussmann, H.J., "Lic brackcts and local controllability: a sufficicnt condition for scalar input systcms," SIAM J. Control Opt. 21 (1983), pp. 686-713.

[Sn3l

Sussmann, II.J., "Lie brackcts, real analyticity and gcomctric control thcory," in Differential Geometric Conlrol Theory, R.W. Brockctt, R.S. Millman and H.J. Sussmann cds., Birkh~iuscr Boston Inc. (1983), pp. 1-115.

[Su]

Sussmann, H.J., "A Lic-Voltcrra expansion for nonlinear systcms," in Mathematical Theory of Networks and Systems, Proceedings of the MTNS-83 International Symposium, Bcer-Sheva, Israel, P.A. Fuhrmann Ed., Springcr-Verlag (1984), pp. 822-828. Sussnmnn, ll.J., "Lic Brackcts and real analyticity in control theory," in Mathcmatical Control Theory, C. Olcch ed., Banach Ccntcr Publications, Volumc 14, PWN-Polish Scicntific Publishcrs, Warsaw, Poland, 1985, pp. 515-542. [Su6] Sussmann, tI.J., "Rcsolution of singularities and linear time-optimal control," in Proceedings 23rd IEEE Conference on Decision and Control, Las Vegas, Nevada (Dcc. 1984), pp. 1043-1046.

[SuT] Sussmann,

H.J., "A gcncral theorcm on local controllability," SIAM J. Control Opt. 25 (1987), Pl)- 158-194.

[s 8]

Sussmann, II.J., "The structure of time-optimal trajectories for single-input systems in the plane : the C °O nonsingular case," SIAM J. Control Opt. 25 (1987), pp. 433-465.

[sug] Sussmann,

lI.J., "The structure of time-optimal trajectories for single-input systems in the plane: the general rcal-analytic case," SIAM J. Control Opt. 25 (1987) pp. 868-904.

[SulOl Sussmann,

II.J., "Rcgular synthesis for timc-optimal control of single-input realanalytic systcms in the planc," SIAM d. Control Opt. 25 (1987), pp. 1145-1162.

[Su1,]

Sussmann, tl.J., "A product expansion for thc Chert series," in Theory and Applications of Nonlinear Control Systems, C. Byrncs and A. Lindquist Eds., North]lolland (1986), pp. 323-335.

[su12]

Sussmann, H.J., "Envclopcs, conjugate points and optimal bang-bang extrcmals," in Algebraic and Geometric Mcthods in Nonlinear Control Theory, M. Fliess and M. llazcwinkcl Eds., D. Reidcl Publishing Co., Dordrecht, The Netherlands (1986), i)p.325-3.16.

425

[Sul3] Sussmann, II.J., "A weak regularity theorem for real analytic optimal control problems," Rcvisla Malcmdtica lbcroamcricana 2 (1986), pp. 307-317.

[Su141

Sussmann, H.J., "Rccent dcvclopmcnts in the regularity thcory of optimal trajectories," in Linear and Nonlinear Malhematical Control Theory, Rcndiconti del Scminario Matcmatico, Univcrsitk e Politecnico di Torino, Fascicolo Specialc 1987, pp. 149-182.

[Su15]

Sussmann, II.J., "Real analytic dcsingularization and subanalytic sets: an clemcntary approach," to appcar in "~ansaclions Amer. Math. Soc.

[vi] [Wa]

Vicnnot, G., Alg~brcs de Lie ct Monoi'des Libres, Springcr-Vcrlag (1978). Wagncr, K., "Ubcr dcn Stcucrbarkcitsbegriff bci nichtlincaren Kontroilsystcmcn," Arch. Math. 47 (1986), pp. 29-40.

System Theory and Mechanics A.J. van der Schaft Dept. of Applied Mathematics, University of Twente P. O. Box 217, 7500 AE Enschede, the Netherlands

Abstract. This paper discusses a system theoretic approach to mechanics, regarding Hamiltonian systems as conservative "mechanical m-ports". Recent results in the Hamiltonian realization problem are surveyed, and generalizations are being indicated. The potential use for control purposes of the Hamiltonian structure of nonlinear control systems is exemplified.

0. I N T R O D U C T I O N

Although

mechanical

application

for

(in particular an

system

and

always

scientific

cal network

passivity,

with

with

important

historically

area

of

level) between system and control

theory as

and mechanics

has

been

rather

weak.

or automata theory have had a much larger

of system and control

theory.

Especially

the framework of modern system

physical

concepts

reciprocity and duality.

of passivity

theory,

an

interplay

discipline

theory has enriched

the very beginning

constituted

the

areas such as electronics

impact on the development

have

control

on a theoretical

independent

Indeed,

systems

(input-output)

such as

energy,

electri-

theory from

external

power,

In particular the relation of the notion

stability

of systems

has been

one of the

first major themes of system theory. In the beginning

of the seventies

the notion of a passive

state space system having additional

(external

(or dissipative)

and internal)

symmetry

pro-

427 perties,

initially

motivated

by

electrical

formalized and extended by Willems

network

[WI,W2],

with

theory,

towards parts A systematic

such as

thermodynamics,

elasticity

and mechanics.

comparison

of

basic

of

theory

mechanics

was

notions

of

concepts

pioneered

by

Hamiltonian

established,

system

Brockett

and

[B].

Lagrangian

In

this

and

of

(analytical)

innovative

systems

control

and several problems were formulated,

further

an eye also

of physics

the

was

paper

were

the

basically

which have set the trend

for subsequent research in this area.

In the present opinion,

to survey

some

of the main

features,

in our

of the theory of Hamiltonian control systems as it has evolved over

the last

fifteen years.

Hamiltonian paper

paper we want

by

First

and Lagrangian

Brockett,

subsequent

and

sections

in Section

control

indicate

2,3,4

can

1 we motivate

systems, some

be

very much

open

read

the definition

of

in the spirit of the

problems

and

independently.

extensions.

Section

2

The

briefly

surveys the global coordinate free definition of Hamiltonian control systems using

tools

Hamiltonian

from

symplectic

realization

Hamiltonlan

geometry.

problem,

i.e.

In

Section

the

problem

system in terms of its external behavior,

towards the proble~ of mechanical synthesis. obtained which

in [CS],

fits

[W3,W4].

into

but

In Section

Hamiltonian

indicate

the

theory

as

4 it is illustrated

structure

of

purposes,

enhancing

attractive

theoretical

a

the

system

with

the

characterizing

a

which is a first step

case a more recently

be

of

computational

of

deal

general

developed

in three particular

can

robustness

and

we

We mainly survey the results as

in the linear

realization

3

profitably

the

Willems

cases how

used

controller

shortcuts.

approach

by

for

and

Finally

the

control

providing

in

Section

5

some concluding remarks are given.

Due to space limitations,

several other topics of interest

between system theory and mechanics we

like

to

mention

symmetries

dimensional

Hamiltonian

[BS,THC,S8],

Poisson control

[C,B],

and

systems, systems

and last but not least,

in the interplay

are not dealt with here; conservation quantum [San,KM],

the relations

in particular

laws

[$2,$3],

infinite

mechanical

control

systems

systems of a gradient nature between mechanics

and optimal

control theory (see also Example 6 in Section i).

Acknowledgements, the

occasion

of

It is a great pleasure his

fiftieth

birthday

for me to thank Jan C. Wlllems at for

putting

me

on

this

track

of

428 research,

and

scientific science.

most

of

research

all

and

for

his

conveying

neverending

his

stimulating

curiosity

enthusiasm

about

the

for

concepts

I also llke to thank Peter Crouch for a v e r y pleasant

of

and fruitful

c o o p e r a t i o n on the material of Section 3.

i. HAMILTONIAN CONTROL SYSTEMS. Let us b r i e f l y [G,Wh].

review some basic

Consider

a mechanical

represented

by

d'Alembert's

principle

(11>

n

system. part,

Let

remaining

one

i - i ..... n.

1

decompose

part

with

n

degrees

coordinates

_ a___T _ aql

that

of classical

forces

derivable

consisting

locally

Based

on

the equations of motion

Fi, from

a conservative

i - l,...,n,

into

a

energy

potential

external

and

tile LagranEian function L(q,q)

Defining

freedom,

and F i are the forces acting on the

dissipative

of

of

see e.g.

1 ..... n,

(co-)energy, the

is

mechanics,

q = (ql .... qn).

of virtual w o r k one obtains

is the kinetic us

i.e.

system

configuration

d (aT) dt aql

where T(q,q)

elements

V(q),

and

forces,

a

F~,

as T(q,q) - V(q),

one

arrives at the c e l e b r a t e d Euler-LaErange equations (1.2)

d (aL } d--{ - aq~

For F~ - 0,

aL aq i

e Fi,

i - l,...,n,

i.e.

i - I ..... n.

a closed

conservative

(1.2) also a;ise from e x t r e m i z i n g

equations

mechanical

the a c t i o n integral I Ldt.

is usually called Hamilton's principle [G,Wh].) Alternatively, that the r i g h t - h a n d let

the system

according

the (This

one could say

defines the extra forces F~ necessary to

side of (1.2)

evolve

system,

to an arbitrary

trajectory

q(t),

see

e.g.

IT]. From

(1.2)

forces

one

can

obtain

and interpreting

variables.

(Notice

a

control

the external

system

that this already constitutes

classical point of v i e w where F~ are u s u a l l y q,q and/or

time

t.) More

directly c o n t r o l l e d

(1.3) with

generally,

then one obtains

d aL aL ~ ui , d--t ( • ) - aq---~- [ 0 aqi , ul,...,u m

being

the

by

disregarding

dissipative

forces F~ in (1.2) as input or control a m a j o r departure

from the

regarded as given functions

if only some degrees

of

of freedom can be

the control system

i ~ i ..... m, i - m+l, .,n, .

(independent)

controls.

For

instance

robotic

429 manipulators torques while

are

of

this

corresponding

type,

to

the remaining,

the

not

if one

neglects

relative

directly

joint

controlled,

dissipation

angle

and

coordinates

degrees

takes as

of freedom

the

inputs,

may model

flexibility. Even more

generally

in vector notation d

aL

(1.4)

d-t { - ) aq

for

n x m

some

control

aL

matrix

Example

simple

i. Consider

where

as

already systems

Lagrangian

a linear mass-spring

d

(1.5)

is

OL

Lagrangian

This

Brockett

is illustrated

[B],

have

to

by the

[B]).

system attached

mechanical depending

system directly

energy)

the on

- m(q+u)

to a moving

of

n

frame. u,

frame,

and

The the

Lagrangian equations

of

- kq - O.

degrees

on control

the equations

[]

of

variables of motion,

freedom u

with

(through

a the

in the absence

are given as •

aL

•

~-~ ( . (q,q,u)) aql

- a-~l(q,q,u)

given function

of

q,q,t,

with

zero

Euler-Lagrange

We notice

aL • - ~(q,q,u)

the potential

of other forces,

by

[]

a

L(q,q,u)

kinetic and/or

argued

(see for a more involved example

)

for

described

do not necessarily

(as in (1.3) or (1.4)).

d aL • ~-~ (__(q,q,u)) aq

general,

equations

- 0,

then

i - i ..... n.

(1.5)

external

can

forces

be for

interpreted the

as

the

time-dependent

L(q,q,u(q,q,t)).) that equations

(1.5), by taking in (1.5)

(1.6)

However,

input u is the velocity of = ~i m(4+u)2 - ~ 1 kq 2 depends directly

motion are

systems

m

the

L(q,q,u)

control

q e ~ , u e ~ , m S n,

in mechanical

forces

example

v-u

(If u

B(q).

or input variables

following

mechanical

n

- ~q = B(q)u,

appear as external

In

one could consider as

L(q,q,u)

(1.3)

can always

the control

- L(q,q)

+

~. ujqj jIl

be regarded

dependent

as a special

Lagrangian

case of

430 On the other hand,

(1.4) are a special case of (1.5) only in case

equations

every J-th row of the transposed m a t r i x BT(q)

is

some

the

function

conditions

or,

equivalently,

if

the gradient of

(locally) following

integrability

are satisfied aBi ~ aBk 3 aq---~--(q)- a--~-1(q),

(1.7)

with

H 3 (q),

B~j(q)

the

(i,j)-th

i,k - I ..... n, j - i ..... m,

element

of B(q).

In this

case

one

recovers

(1.4)

from (1.5) for the L a g r a n g i a n

(1.8)

L(q,q,u)

- e(q,q) +

~ u3H 3(q) 3-I

M o t i v a t e d by this we finally define a Lagrangian control system as a system in

local

configuration

Lagrangian

L(q, q, u).

definition forces

are

(we

controls (1.7)

are

is

are

of

to

we

automatically

given

that

we

this

forces

do not

require

allow

the

of

form

(1.5),

for

some

limitations

in

this

for

as

that

this

in

last

theory

in

case

integrability (1.4),

corresponding

the

dissipative

internal

secondly

extra

if,

forces

and/or

the main

and

satisfied

Relaxation

quasi-coordinates

in

the

issue),

generalized

coordinates).

configuration use

in principle

the

q

conclude

later

external

(which

u I , . . . ,um

the

that

return

coordinates We

condition

the

controls

independent

to

condition

of

the

will

involve

non-holonomic

systems

[Wh]; h o w e v e r we will not go into this here.

Let

us

now

pass

control system

on

to

Hamilconian

the

For

Pl - --F-(q,q,u), aq~

(q,q,u)

mechanical defines

where partial zero,

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

system),

everywhere

H(q,p,u)

then p - (Pl ..... Pn) are independent

q and

p are

derivatives one

~

~. P i q i i-1 related

the case for a

functions,

and one

transform of L(q,q,u)

L(q,q,u),

by

the

equations

of the right-hand

immediately

(i,j)-th element

(which is generally

the Hamiltonian function as the Legendre

(1.10)

Lagrangian

i - i ..... n.

(Notice that Pl may depend on u.) If the n × n matrix with azL

the

(1.5) we define the generalized m o m e n t a in the usual way as

aL

(1.9)

formulation.

concludes

that

side

(1.9).

of (1.10)

H does

not

(Since

by

with respect depend

on

(1.9)

the

t o qi are q.)

It

is

431 well-known

that with

(1.9)

and

(I.i0)

the Euler-Lagrange

equations

(1.5)

transform into the 11amiltonian equations of motion aH

•

(l.lla)

q~

Op I (q,p,u)

i - l,...,n. (l.llb)

Pl

(In fact,

OH aq i (q,p,u)

(l.llb)

follows by

follows

from substituting

(I.I0).) We call

advantage of

(i.ii)

(1.9)

into

(1.5),

and

(l.lla)

(i.ii) a llamiitonian control system. The main

in comparison with

(1.5)

is that

(i. II) are

explicit

first-order differential equations and thus constitute a control system in state space form, with state (q,p)

(in physics usually called the phase).

Moreover the variables q and p are completely dual to each other; indeed it is well-known,

see

e.g.

[A,AM],

that

there

is

an

underlying

geometric

structure to equations (i.ii), called the sympleotic or canonical structure, (see

also

structure

Section

2).

invariant

The

are

state

called

space the

transformations

canonical

which

leave

transformations.

this

(Let

us

furthermore mention that the Hamiltonian formalism in physics does not only underly classical mechanics, but also statistical and quantum mechanics.) As a particular case of (I.Ii) we note that if the Lagrangian is of the form L0(q,q) +

~

uj}lj(q)

(as

in

(1.8)),

then

the

Hamiltonian

equals

H0(q,p) -

~ ujHj(q), with H0(q,p) the Legendre transform of L0(q,q). More j-i generally, a Hamiltonian control system with II(q,p,u) of the form

(1.12)

will

be

H(q,p,u) - H 0(q,p) -

~ ujHj(q,p) j=l

called

Hamiltonian

an

affine

control

system

with

in~ernal

(double

pendulum)

Hamiltonian }|0, and Hj being the interaction Hamiltonians.

Example

2.

Consider

a rigid

two-link

robot manipulator

with the relative joint angles ql,q2 being the configuration coordinates and as controls Ul,U z the torques at the joints (as delivered by actuators)• The i

l[amiltonian is of the form ~ pTM-1(q)p + V(q) - ulq I - uzq2, with M(q) (positive-definite) (gravity).

inertia

matrix

and

V(q)

the

potential

tlle

energy []

432 Example

3. Consider

obtain p - m(q+u) the

internal

the system of Example

i m(~+u)2 i where L - ~

and H(q,p,u) - H0(q,p) - up,

energy.

Notice

that

even

1 - ~ kq z. We

with H°(q'P) - Z'-~lpZ + ~i kq2

though

L

is

quadratic

in u,

H is

still of the form (1.12).

Example

4.

their

Consider

own

k

point

gravitational

V(ql .... q k ) -

~

masses

field

ml,

with

positions

corresponding

to

ql 6 ~3, i E _k, in

the

potential

the positions of

mlmj/~qi qj[[. Suppose

the

first

energy 2 masses

(2 s k) can be c o n t r o l l e d . We o b t a i n the Hamiltonian c o n t r o i system •i

OH

qJ -

OH

.i

i' PJ apj

i' 3qj

f - 2+i ..... k,

j - 1,2,3,

k

with ti(q/+l,.,q~ pl+l,. ,pk ul ' ' ' u 21 . V(u 1 , . , u i , q 2+i ,-,qk) +

~ I ~pi~Z. £]

Z i -,e+l

We

now

come

control

to

system

to (l.lla,b) (l.llc)

the

definition

aH @uj(q,p,u),

(Equivalently for the 0L yo - a-~(q,q,u), j e m.) then

Hamiltonians There

in

the

this

instance,

actuators.)

case

since the

Lagrangian We

natural

reasons

system

control

that if H(q,p,u)

note outputs

are

for adopting

duality between

way

generalized

structures

conservative j 6 m,

of

a

Hamiltonian

outputs associated

j - 1 ..... m.

simply

(1.5),

is

given

this definition inputs

if ul,...,u m are generalized

corresponding flexible

outputs

the natural

of

the

as

the

we

define

affine

form

interaction

Hj(q,p).

are several

First,

natural

the

we define

as

yj

(1.12),

of

(l.lla,b) o Indeed,

Indeed,

this

called

a Hamiltonian

"mechanical it forms external

forces,

configuration is

m-port"

a direct behavior

system

of

induced.

of

(In

can be

dual

by

the

For

will be the

the

co-located

theory sensors

regarded

variables

of an electrical

specified

is

then Yl,''',Ym

(l.lla,b,c) pairs

analogue is

case

of natural outputs.

outputs

coordinates.

the

with

and

m-port, evolution

of and

as a

(uj,yj), in which of

m

of

symmetry or reciprocity

between

the inputs and the natural outputs results,

as will become clear in

Section

3. We call

voltage/current

pairs.

Secondly

a strong

the total system

type

(l.lla,b,c)

system, or briefly a Hamiltonian system.

a Hamiltonian input-output

433 Example If,

on

2 (continued). the

other

The

hand,

force at the end-point Cartesian coordinates

the

natural

outputs

controls

are

would

of the manipulator,

the

be

the

relative

angles

horizontal

then the natural

and

ql,qz"

vertical

outputs

are the

of the endpoint.

Example 3 (continued).

The natural

Example

Natural

4 (continued).

O

output

is the momentum

p.

are the forces as experienced

outputs

by the

first 2 point masses.

Example

5 (Nonlinear

inductive

nl-port

satisfying

electrical

(a set of

Kirchhoff's

the current, Similarly

respectively

that

with

corresponding

respectively

i3 - - - ,

(~1,''.,~n,

j 6 n,

as --, ~in+ k

~n+k

the

by (ql ..... qn, v,+1 ..... v,z), ST , j 6 n, aq~

energy

~j

-

electric

Otl O~a

,

be

channel. and let channels.

i.e.

in+1 ..... inl)"

Then

L is

aT --, 8vn+ k

the

IBM].

respect

to

that C can be

k ~ nz-n

Legendre

energy

with

suppose

then C is given as

transform

Defining

a__E_ H aq3

~13

external

at the external

transform

S + T we obtain

(l.16a)

and ~j

(only capacitors),

IBM]. Analogously,

q=*k

T(q I ..... qn,Vn+ I ..... vnz ) the

in some way i3

k ~ nl-n

Legendre

parametrized

qn+1 .... ,qn z of

Let

J - I ..... n.

of the magnetic

with

to the j-th

nz-port

other

relation

S(~1,...,~n,in+1,...,inl )

v0 -

to each

channels).

charges,

~n.1,...,~nl

(1.15)

Let L be a nonlinear

of L and C (n S n I, n S n2),

L can be parametrized by

as 8~Oj

(1.14)

[S3]).

connected

capacitive

vj - ~j,

given by the constitutive

see

n I external

the first n channels

ij - - 43,

Suppose

with

flux,

the voltages,

We interconnect

LC m-port,

induetors

laws,

let C be a nonlinear

and q3 be

(1.13)

D

j - i ..... n,

the

with

respect

[{amiltonian

to H

as

434

(l.16b)

which

~n+k

8H @in+ k ,

k - l,...,nl-n,

qn+k

8H @vn+ k ,

k - 1 ..... nz-n,

is

a

Hamiltonian

system

with

state

(in+ I ..... inl,Vn+ I ..... Vn2 ) and outputs

Example

6 ([S3,S6]).

minimizing

Consider

with respect

(~i .... , ~ n , q l , " ' , q n ) ,

the (smooth and unrestricted)

to u(-)

inputs

(~n+1 ..... ~nl,qn+i ..... qnz)"

[]

Bolza problem of

the cost functional T

(l.17a)

J(x 0,u(-))

- K(x(T))

+ f L(x(t),u(t))dt, 0

(with x and u in ~ n (l.17b)

~m), under

x(t) - f(x(t),u(t)),

In order us

respectively

to solve

to consider

the co-state, (i 18a) •

constraints

x(0) - x 0.

this optimal

control

the pseudo-Hamiltonian and the Hamiltonian

x~

the dynamical

problem

the Maximum

H(x,p,u):-

control

pTf(x,u)

Principle

tells

- L(x,u),

with p

system

aH (x,p,u) ap i

x(0) - x 0 i - i .... ,n,

(l.iSb)

aH i(x,p,u) ax

Pl

where x(T)

is the solution of (l.18a) - (l.17b)

A necessary condition that

aK (x(T)) PI(T) - - @x--?

for every

for a control

t 6 [0,T],

at time T.

function u*(.) on [0,T]

to be optimal

H(x (t),p*(t),u (t)) - max H(x (t),p (t),u),

is

where

U

x*(-),p*(') problem: U

is

belongs

to ~m

j - l,...,m. Hamiltonian (i.18c)

the

solution

Find for every this

Hence

of

(1.18).

yields

the

the

and a necessary

we

Maximum

first-order

Principle

system given by (l.18a,b)

Y3

So

are

led

(x,p) a U* such that H(x,p,u*)

aH @uj (x'p'u)' condition

conditions

leads

in

to

the

following

- max H(x,p,u).

a

u aH ~3

natural

*

(x,p,u) way

balance.

major Assume

consequence that

to

- 0, the

together with the output equations

j - i ..... m,

for u (.) to be optimal

is that the outputs y3 of

this system are zero on [0,T].

Another

Since

of

H(q,p,u)

[]

definition is of

the

(l.llc) form

is

(1.12)

the (see

following for

the

energy general

435 [B,S3]),

case

then

dH 0

(1.19)

-

m

at

,

~ ujy~. 3-1

Hence the increase of the internal energy H 0 is a function of the inputs and time-derivatives

of the outputs;

indeed the right-hand side of (1.19) is the

total external work performed on the system by the controls u I ..... u m. From a mathematical the natural

structure

Hamiltonian that

point of view equations

H(q,p,u)

symplectic

the natural

manifold

with

see [SI,$3].

the same footing, and outputs

space

of

is a cotanEent vector

(u I ..... um)

u I ..... um),

the

inputs

([$3],

canonical

In this

(cf. Willems

[W3,W4]);

the adopted

energy H 0 is conserved

do possess

analysis

as

difficult to quantify,

see for details definition

able to incorporate

damping,

extension

equations,

[AS]).

of

a priori

For general is a general

between

at

inputs

[SI,$3]. or Hamiltonian

(By (1.19)

in practice

especially

a so

(y,u) - (Yl ..... ym,

of a Lagrangian

as

for

control

the internal

mechanical

weakly

for

purposes.

systems

damped mecha-

Damping

of the controlled

it is unsatisfactory

dissipation on a conceptual (Although

as an additional

of the definition

as possible

in case

idealization usually forms a natural starting

well

control system.

into account

that

is

often

and the actual presence of unmodelled damping will in

also Section 4. Nevertheless

mechanical

also

systems.

only improve the characteristics

many cases

suggest

since y and u are regarded

for u - 0.) Although

inherent

nical systems the conservative for

see

coordinates

last case,

system only covers conservative mechanical

point

outputs

space of inputs and outputs

one does not have to distinguish

As remarked before,

always

and

(1.19)

is a cotanEent bundle T'Y, with Y an output manifold,

(1.12)

Hamiltonians

for

(l.llc) and

the reciprocal

of course

feedback

term.)

of a Hamiltonian structure

of

see

level in our definition of a damping

can be always

A possible

system,

which

the Euler-Lagrange

is suggested by adding to equations

system,

that we have not yet been

(1.2)

taken

non-conservative retains

as much

or Hamiltonian

a dissipation

term in

the following way

(1.20) where

d [~__) aL + a R . d-t aq i - aq---? @ql R(q)

is

the

classical

F~,

i - I ..... n,

Rayleigh's

modelling for example viscous damping.

d~ssiparion

function

([G,Wh]),

436 2. G E O M E T R I C

In

this

DEFINITION

section

canonical

briefly

coordinates

coordinate Readers

we

free

not

given

way,

definition

as

using

familiar

Then (M,~)

is

called

exist

called

a

local

(i.e.,

dim M

A

to L is zero, It follows

(M,~)

be

can

skip

local

in a global (see [AM,A]).

this

definition

such

on M prolongs TM,

to

[Tu] ;

1.dim M

(q,p)

space

cf.[%13]).

of

(%1,~G) be

external

The product

symplectic

be the projections

Definition

2.1.

form,

are

~. (dpl A dql + dpl A dql) , with I-i nares for TM. let

manifold,

[A,AM]

variables*

for

M

coordinates

Lagrangian respect

denoting

coordinates). denoted

local

by ~,

canonical

(q,p,q,p)

a 2m-dimensional

manifold

theorem

Such

L is maximal w i t h

symplectic

-

Furthermore,

Darboux's

(d~ - 0).

if

to this

-- n.

and m o m e n t u m

a symplectic if

closed

X dpl A dql. i-I L c (M,~) is c a l l e d

and furthermore

configuration

By

and

that ~ -

submanifold

that dim L -

form ~

cf.

manifold.

~ 0),

(q,P) - (ql ..... qn,Pl ..... Pn)

a 2n-dimensional

(generalized

product

geometry

defined

geometry

in

for any tangent v e c t o r X at x 6 M there

sympleetic

is even)

bundle

the

can be

sympleetic

that ~(x)(X,Y)

coordinates

canonical.

restricted property. let

from

symplectic

v e c t o r Y at x such

(necessarily

space

(l.lla,b,e),

systems,

At the end of the section we will illustrate the

form if ~ is non-degenerate

Now

Hamiltonian

(see e.g. [A,AM]) that a two-form ~ on a m a n i f o l d M is a symplectic

is a tangent

are

hew

for linear H a m i l t o n i a n systems.

%1e recall

there

in

tools

with

w i t h o u t loss of continuity.

sketch

being

(inputs

TM x %1 is again

and

on

the state symplectie

the

natural

manifold,

outputs

then

coordi-

denoting

taken

a symplectic

form ~ • (-~') -: ~3. (More precisely,

tangent

coordinates

the

symplectic

The

together,

manifold,

let ~i,

with

resp.

~z,

from TN x %1 onto TM, resp. W, then ~ = ~I ~ - ~; o . )

[SI,S3]

Let

(TM x W,~)

be as above.

A Hamiltonian

system

with stare space M, and space of external variables W, is defined by a submanifold L C TM x W havin E ~ e

followin E two properties:

(2.1a)

L

(2.1b)

L is a Lagrangian submanifold of (TM x W,~),

can

be

parametrized

by

coordinates

for

M

toEether

with

m

coordinates for W.

*Apparently the use of the letter %1 for the space of external not due to N. %1iener.

variables

is

437 Let (q,p) - (ql,''',qn,P. ..... Pn) be local follows

from

ul,..,um)

(2.1a)

for

that we can

W with

exists

local

coordinates

coordinates

It

(y,u) - (Yl .... ym,

X c i du i A dyl, e i - ± i (such coordinates are I=i such that L is parametrized by (q,p,u). By (2.1b)

locally

generating

a

function

([A,AM,Wh])

H(q,p,u)

implying that L as a submanifold of TM x W in its natural (q,p,q,p,y,u)

for M.

~" -

called semi-canonical), there

find

canonical

for

L,

local coordinates

is given by the equations aH 0pi(q,p,u)

.

q~

i - l,...,n,

(2 2) •

pi

OH (q,p,u) aq i aH

YJ - - cj 8-~j(q,p,u) thereby recovering corresponds

to

what

representation; example,

(l.lla,b,e)

in

in a

j - I ..... m, if c$ - i, j E m. (The case of alternating ej

electrical

mechanical

network

context

theory

this

the inputs consist of force variables

the outputs

are

the complementary

position

We remark that in most mechanical

systems

is

called

situation

2.1

covers

two

extreme

the symplectic

cases.

if,

First

if M

see

manifolds forms

is

[$3].) M and W

[AM].

absent

(i.e.

then L C W defines a static reciprocal system (see [AM,p.412]

dynamics),

references).

Alternatively

if

W

is

absent

for

and position variables while

and force variables,

appear as cotangent bundles with their natural symplectic Definition

hybrid

a

arises

(no

external

variables)

no for then

L c TM defines a locally Hamiltonian vectorfield on M ([Tu]).

Definition 2.1 specializes form on

a

bilinear

form,

linear

By Darboux's in which

J

space

M-

to linear systems as follows. A linear symplectic ~zn is

simply

a

skew-symmetric

and thus is given by a skew-symmetric

matrix J.

theorem there exist linear canonical coordinates (q,p) for ~2n (0 -Inl the form tIn 0 ~" The prolonged symplectie form J on

takes

TM ~ ~2n X ~ 2 n

the

matrix

(0 j

( W - ~2=,j.) be linear symplectic

spaces.

A Lagranglan

is a (2n+m)-dimenslonal

of ~2n x ~2n X ~2m

the symplectic

(2.3)

non-degenerate

invertible

is

given

by

subspace

form given by the matrix

,J ~ ( - J ' )

0 J 0 - [ J 00] 00-J

J0) .

Let

now

(N - ~zn ,J)

subspace

with

and

L of TM × W

the property

that

438 is zero restricted subspace

to L,

of ~zn X ~

coordinates

and a linear

X ~,

of ~ .

parametrized

as

by

system

the first

is a Lagrangian

2n coordinates

It follows that we can take linear canonical

(y,u) - (yl,..,ym,ul,..,um) given

Hamiltonian

x - Ax + Bu,

for

~,

y - Cx,

such

that

satisfying

the

Hamiltonian

ATj + JA - 0

(A

and m

coordinates system

is

is

called

a

llamiltonian matrix) and BTJ - C.

3. R E A L I Z A T I O N

As

remarked

regarded

A N D SYNTHESIS

before,

as

the

electrical

m-port.

understood

(using

state-space

the

definition

conservative For

of a Hamiltonian

mechanical

linear

electrical

frequency-domain

methods

in

e.g.

analogue networks

methods

[AV,W2])

as

the

(i. II)

can be

definition

it has

in

which

system

of

been

[Be],

and

additional

of an

quite

well

subsequently

properties

the

transfer matrix of a linear system should have in order to be realizable as the

driving-point

impedance

consisting

of basic

specified

class.

Hamiltonian

(or

elements

(resistors,

s~thesis

Similar

systems,

admittance)

linear

as well

interest to know which (nonlinear)

of

an

electrical

capacitors .... ) in some

problems

could

as nonlinear.

be

For

a priori

formulated

instance,

input-output behaviors

m-port

for

it is of

can be synthesized

by certain types of robotic manipulators. Here we will not go into the synthesis problem itself (although some results

on linear mechanical

synthesis

as

Hamiltonian

a

preamble

characterize systems

the

those

the

secondly

to

Realization

input-output

(i. Ii). Two aspects

identify

have been obtained

out

which

come

amongst

possessed

all

by

for

controlling

to be Hamiltonian the

system

we

from

(nonlinear)

(see

4).

As

Hamiltonian

in Mechanics,

of classical

A

first

result

Brockett y - Cx,

see

(control)

& Rahimi

in

intrinsically

the

importance,

information discussed

in

For the relationship

with

and

since if

may be used [S3,CS],

a

Inverse

quantization

systems we refer to [Sa,THC,SS].

the [BR],

is Hamiltonian

satisfies

[Sa],[T].

first to

systems

special case of the Hamiltonian Realization problem is the classical Problem

to

systems,

then this extra Section

want

Hamiltonian

Hamiltonian ones. This last aspect is of some practical we know a system

but consider

i.e.,

of this problem can be distinguished:

properties

special single

behaviors

in [B,S3]),

problem;

G(s) - GY(-s).

Hamiltonian showing

that

Realization a minimal

problem linear

was

system

obtained

by

x - Ax + Bu,

if and only if its transfer matrix G(s) - C(Is-A)-IB Since GY(-s)

is the transfer matrix of the adjoint

439 system

z - - ATz - cTu,

Hamiltonian be

iff it is equivalent

self-adjointness.)

called

yields

y - BTz,

a

skew-symmetric

BTj ~ C, implying

this

implies

that

a

to its adJoint system.

Indeed

matrix

J

the

of

state

full

system

is

(This property will

space

rank

linear

uniqueness

such

theorem

that

ATj + JA - 0,

that the system is llamiltonian with respect

to the linear

symplectic form defined by J. This basic idea was generalized - f(x,u),

(3.1)

in [CS] to nonlinear systems

x E M,

y - h(x,u),

u e

with M a k-dimensional

~=,

a m,

y E

manifold.

Along any solution curve

of (3.1), with u, say, piecewise constant,

(~(t),~(t),y(t))

the variational system is defined

as the time-varying linear system

I ~af( ~ (-~ ) , U ( ~ ) ) V ( ~ )

~(~)

+ ~a(f X -( ~ ) , U ( ~ ) )

UV( ~ )

(3.2)

v e a k, ah -

yV(t) - ~(x(t),5(t))v(t) with

u v E a m,

yV G a m

the

ah -

+ ~(x(t),~(t))

variational

inputs,

u~(t)

resp.

variational

outputs.

the adjoinr variational system is given as

Furthermore

~(t)

-

af - {~(x(t),~(t)))Tp(t)

ah - {~(x(t),~(t)))

T

u~(t) p

(3.3)

y"(t) with

u a E a m,

(~(E(t),~(t)))Tp(t)

ya E a m

outputs.

variational

the

adjoint

(For

global

e

ak,

ah ,~(t))) T u'(t) + (~(x(t) variational

inputs,

coordinate-free

resp.

adjoint

considerations

consult

lCS].) Theorem

3.1

complete

[CS]

(Self-adjointness

system

(u(t),x(t),y(t)),

(3.1) t z 0,

is with

Hamiltonian x(O)

A

condition)

fixed

iff and

minimal, along

analytic

az~

u piecewise

and

trajectory

constant,

the

input-output maps of (3.2) and (3.3) for v(0) - 0 - p(0) are equal, i.e. all variational systems are self-adjoint. Remark

Furthermore,

minimal

canonical transformations,

Hamiltonian

realizations

are

unique

up

to

while also the internal energy is uniquely deter-

mined (modulo constants) by the input-output behavior

([S3,BG,CS]).

440 Note

that

Theorem

conditions

3.1,

as

for a system

it

stands,

does

to be Hamiltonian,

not

yield

since

easily

verifiable

in principle

it requires

the integration of the system. Up to now only in particular cases (including the Inverse

Problem

in Mechanics)

recast into a more constructive

the conditions

of Theorem

3.1 have been

form ([CL,Sa,T]).

A rather different line of characterizing Hamiltonian systems was pursued by Crouch

and

Irving

input-output

[CI1,CI2]

map of

(3.1)

and Jakubczyk

for x(0) - x0,

[JI,J2,J3],

expanded

by

considering

in a Volterra

the

series or

generating power series. Very elegant algebraic conditions on the kernels of these

series

condition

have

for

been

linear

obtained,

systems

vastly

interpreted

response matrix H(t) - C exp(At)B, A geometrically

in modified

as

a

the

condition

Brockett-Rahiml on

the

impulse

namely H(t) - - HT(-t).

appealing characterization

terms of their variational and proved

generalizing

input-output

form in [CS].

of Hamiltonian systems,

behavior,

This

solely in

was conjectured

characterization

in

starts

[$3],

from the

Let (u(t,~),x(t,e),y(t,~)),l~ I small, be a family of

following observations. trajectories

of (3.1) parametrized by ~, with u(t,0) - u(t), x(t,0) - x(t), a y(t,0) - u(t). Then the variational trajectory uV(t) - ~ u(t,0) -:6u(t), a a v(t) - ~ x(t,O) -:6x(t), yV(t) - ~ y(t,0) -:6y(t) is a solution of the variational in

system

this way.

Now

(3.2), let

and moreover

(u(t),x(t),y(t))

system, and let (61u,61x,61y), along

(u(t),x(t),y(t)).

Then

all solutions be

any

of

(3.2)

trajectory

are obtained

of a Hamiltonian

i - 1,2, be any two variational trajectories by

the

self-adjointness

property

we

have

in

canonical coordinates for any tl,t z

(3.4)

(&~p(ta)6zq(tz)-61p(tz)6xq(tz)) f~Z

=

(6~u(t)6zy(t)

[6~p(tl)6zq(tl)-6~p(tl)6xq(tl) )

-

~l=(t)~ly(t))dt

_

t1

where

Six - (61q,61p),

symplectic consider property implies

that that

Omitting of

forms

~

variational if for

on

i - 1,2. M

and

(This can be formulated e

on

trajectories

support t i - T i,

~m

m

x ~ , cf.

(6u,6x,6y)

(6u,6y) c [TI,T2] i - 1,2,

the

admissible

with

compact

variations

with

support. compact

Fully support

(see [CS] for details and other versions).

(u,x,y)

side

Now

of

let

satisfying

(3.4)

the us the This

zero.

is

are admissible variations

external were

2.)

using

6x(T1) - 0 - 6x(T2).

left-hand

the (x,6x)-part we will say that (6u,6y)

(u,y)

Section

of

then

globally

characterizations

given

in

[CS].

We

of

obtain

441 Theorem 3.2. [CS] complete system

(Variational (3.1).

condition)

Consider a minimal, analytic and

If for any input-output

trajectory

(u,Y),

wi~h

piecewise constant, all possible pairs of admissible variations (61u,61y),

i - 1,2, with compact support of (u,y) satisfy +~

(3.5>

I

(~u(t)~y(t)

-

61~
then every variational system i s

-

0

self-adjoint, and thus (Theorem 3.1)

the

system is Hamiltonian. Furthermore assume (3.1) is Hamiltonian. Let (Du,Dy) be any time function from ~ to ~m X R ~ Wl~h compact support. Assume that

(3.6)

I

(DTu(t)6y(t) - 6Tu(t)Dy(t))dt - 0

for all admissible variations (6u,6y)

with compact support of

(u,y).

Then

also (Du,Dy) is an admissible variation of (u,y) with compact support.

The above theorem implies that, at least formally, of

a

Hamiltonian

Lagrangian

system

"submanifold"

(cf.

output behaviors

of systems

symplectie

suggested

form

(i.ii)

is

the input-output behavior

geometrically

Section

2) of

the

characterized

"manifold"

of all

a

input-

(3.1) modelled on the state space M, with the by

the

left-hand

side

of

(3.5),

and with

admissible variations of compact support being the "tangent vectors". more precise

as

statement we refer to [CS];

in the linear case, however,

formulation can be easily given as follows.

the

For a the

Consider a linear Input-output

system, for example given in matrix polynomial form (3.7)

D(d~)Y(t ) - N[d~)U(t)

where the polynomial

matrices

u,y e R m D(s) and N(s)

strictly proper transfer matrix,

are such

that D-1(s)N(s)

is a

and D(s) and N(s) are loft co-prime (i.e.,

no pole-zero cancellations). Define

on

the

linear

(u,y):~ --9 ~m × ~m with

(3.8)

U1

U2

space

H

compact

~ [[yl),(yz) ) - ~

of

all

support

piecewise

the

linear

continuous (weak)

functions

symplectic

form

(u~(t)y2(t) - uI(t)y1(t))dt

Denote the linear space of all input-output

trajectories

(the input-output

behavior) of (3.7) by $, and define 2=:- 8 n N. Then Theorem 3.2 implies the

442

A

following:

linear

system

(3.7)

can

be

realized

by

a

minimal

linear

Hamiltonian system if and only ~f Z a is a LaEranEian subspace of (~,~). The main drawback of all the above c h a r a c t e r i z a t i o n s

of H a m i l t o n i a n

systems

is that basically they are c o n c e r n e d with input-output maps, and so they do not

fit

very

well

into

the

e s p e c i a l l y for linear systems, relies the

for the c o n s t r u c t i o n

nonlinear

[Sul].

At

minimal Let

least

for

w I - (ul,Yl),

uniqueness

linear

systems

- (u2,yz)

be

will

elements

subspace of (N,O). Define the equivalence -

(3.9)

+

for

initialized

now

developed,

show how

of

space M on system,

we

can

cf.

obtain

the

input-output behavior.

So,

with

relation

-

w I - w z < - > (wl.w

as

form on the state

theorem we

theory

In particular Theorem 3.1

[W3,W4].

directly from

realizations w2

realization

of the symplectic

state-space

Hamiltonian

general by Willems

Se

a

Lagrangian

(see [W3,W4])

+

6 S c <-> wz.w

e $c)

+ (Here W l . W

denotes

t - 0; wl denotes of

co-primeness

the

concatenation

of

D(s)

and

N(s)

r e a l i z a t i o n of (3.7) is isomorphic (3.10) Thus

of the

external

signals

w I and w in

the past of wl, and w + the future of w.) By the assumption in

(3.7)

state

the

a

minimal

t - 0 corresponding

to any

to the linear space

space

of

[W3,W4]

X - So/_

the equivalence

input-output

class

[w]

is

the state

at

trajectory w e [w]. Denoting x I - [(ul,yl)],

x 2 - [(u2,Y2)], we

define on X the function 0

(3.11)

J(xl,x2)

- f

(u~(t)yz(t)

- u~(t)y1(t)ldt -

This

is

correctly

defined.

thus,

since ~ is zero on ~c,

Indeed,

let

xI - 0

then

0

(3.12)

f

7 - uz(t)y1(t))dt

(u~(t)y2(t)

- - r j ~ [O.y2(t) -

p r o v i n g that J only depends on the equivalence C l e a r l y J is a bilinear, as follows.

skew-symmetric

Suppose x - [(u,y)]

f

(uY(t)y(t) .

Then define

.

.

.

~-

÷

ul(t).Oldt-

and

o

form. Non-degeneracy of J is proved

is such that for every x - [(u,y)]

- uY(t)y(t))dt - 0 +

-

classes xl,x z.

0 (3.13)

+

(u1.0 ,y1.0 ) 6 $c

+

(u,y):-(u .0 ,y .0 ). It follows

that

443 +co

for all that

(uT(t)y(t)

f

(3.14)

(u,y) G Zc.

(u,y) E Zc.

-- u T ( t ) y ( t ) ) d t - 0,

Since ~= is a L a g r a n g i a n subspace Clearly

proving n o n - d e g e n e r a c y

[(u,y)] - O,

and

hence

of (N,~)

this implies

x - [(u,y)] - O,

thus

of J.

In fact we did not only prove

that J is a n o n d e g e n e r a t e

skew-symmetric

bi-

linear form on X, but also

Proposition

Let ~= be a L a g r a n s i a n

3.3.

subspace of (N,fl). D e f i n e on Z c the

bilinear f o r m o

(3.15)

~(wl ,w2) - I

for

w I - (ul,yl),wz - (uz,yz) e Z c.

any

[u~(t)y2 (t) - u2 (t)y~ (t))dt Then

w I -w z

if

and

only

if

w I - w 2 e Ker J. H e n c e X - S c/Ker J.

Remark.

Notice

that the above

results

still make

sense

if X is not

finite-

dimensional.

From now on we write the bilinear form J(xl,x2) for X as x~Jxz, with J a skew-symmetric By taking

concatenations

at arbitrary

d e f i n e d b y (3.11)

in a basis

matrix of rank 2n (- dim X). times

tl,t2,

and by

time-invariance,

(3.11) implies that

x~
- x~(t,)Jx2(t~) - I t2 (u~(t)y~(t) -

uI(t)y~(t)ldt

t1 or in differential (3.16)

d

xT(t)Jxz(t ) _ uT(t)yz(t)

By co-primeness an observable in

(3.16)

form

of D(s) and N(s) in (3.7) it follows that $¢ is generated by

and controllable

then

immediately

system x - A x + Bu, y - Cx, x 6 X.

yields

which ensure that the r e a l i z a t i o n can be

also

T _ uz(t)y1(t)

directly

obtained

the

identities

is Hamiltonian.

from

the b e h a v i o r

ATj + J A - O,

However 2=.

In

Insertion BTJ - C,

the matrices A,B,C fact,

the

internal

energy for the system in state x - [(u,y)] at t - 0 can be d e f i n e d as 0

(3.17)

H(x) = ~

uT(t)y(t)dt.

444 Similar arguments equivalence

as employed above show that indeed H only depends

class x. Furthermore

can be represented

on the

in x, and s0

in a basis for X as

1

(3.18) with

it follows that H is quadratic

H(x) - ~ xTQx Q

a

symmetric

Furthermore

matrix.

C:X --9 ~"

is

Then

simply

A

is

defined

defined

by

by

the

setting

relation

C[(u,y)]

JA - Q.

- y(0)

for

(u,y) E 2=, and the definition of B follows from the relation BTj - C.

4. CONTROL OF HAMILTONIAN

SYSTEMS

The design of a controller

for a complex

be

available

based

on

all

knowledge

argued already by several authors

(nonlinear)

about

the

[B,C,S3,SI],

system

system.

ideally should

In particular,

in modern control theory more

emphasis should be put on exploiting the special physical properties by classes of systems. physics

Indeed it is claimed that a further insight

of a system can provide

both on the theoretical motivated

controller

engineering

will

methodology

tend

such

will now be illustrated before,

us with attractive

and computational to be

as

level,

more

dynamic

and rewarding

enjoyed into the

shortcuts

and that a more physically

robust,

analysis.

as This

on the class of Hamiltonian

in three examples.

as

well

as

general

systems,

closer

to

philosophy

as dealt with

(For other uses of the Hamiltonian

structure

for

control purposes we refer e.g. to [HS,M,NS,S4].)

The first example concerns stabilization by feedback of Hamiltonian systems. For clarity of exposition we will restrict ourselves (I.ii), with H(q,p,u)

(4.1)

Here

H(q,p,u) iT ~p G(q)p

energy,

and

configuration this

type.

is

the

to Hamiltonian

systems

of the form

= ~1

pTG(q) p + V(q) -

the

kinetic

controls

coordinates Equilibria

are

energy the

u = 0

(thus

are

q,p E

C(q) > 0),

generalized

yj = Hi(q),

for

~ ujHj(q), L 3-I

forces

V(q)

points

the

potential

corresponding

j 6 m. Most mechanical the

~n

(q0,O)

systems in

phase

to the are of space

satisfying dV(q0 ) - 0. Suppose now that V(q) has a strict local minimum in i q0- Then H0(q,p) = ~ pTG(q)p + V(q) has a strict local minimum in (q0,0). By conservation Lyapunov

of

energy

function,

never asymptotically

for u - 0

implying

(see (1.19)) dH° - 0, and thus H 0 is a dt that the uncontrolled system is critically (but

[AM]) stable.

445 Physically

it

is

evident

that

system we should add damping.

uj - - cjy~,

(4.2)

in

order

to

asymptotically

stabilize

the

Indeed define

cj > 0,

j 6 _m,

with yj - Hj (q), j E m, the natural outputs.

By (1.19)

the c o n t r o l l e d

system

satisfies

(4.3)

2 cjyj _%< O. @

d]lo - -

dt

j-1 We

now

have

Define

the

the

following

Po~sson

canonical coordinates

(4.4) and

{F,G}(q,p)

denote

[AM],

that

of

two

of

LaSalle's

functions

invariance

F(q,p)

and

G(q,p)

principle. in

local

as

i~l(aF aC , ~ Oq~

-

inductively {F,G)

application

bracket

equals

ad;Gthe

a~ ap~

G,

aF ) ( q , p ) Oqt

adFkG-

{F,adFk-iG),

time-derivative

of

the

k >_ I.

(Recall,

function

G

}|amiltonian v e c t o r f i e l d with H a m i l t o n i a n F.) Define for any (q,p)

along

cf. the

the linear

space (co-distribution) k P(q,p) - span {dH0(q,p) , d(adu0Hj)(q,p)} , j e _m, k > 0}

(4.5)

Theorem 4.1. Let V have

[$5]. a

Consider the Hamiltonian

strict

local

minimum

in

q0,

system with Hamiltonian and

q # q0 in some neiEhborhood of qo. Furthermore for

all

(q,p) , with

q # q0 ,

in

some

assume assume

neighborhood

chac

(4.1).

dV(q) # 0

for

that dim P(q,p) - 2n of

(q0,0).

Then

the

feedback (4.2) locally asymptotically stabilizes the Hamiltonian system.

(The c o n d i t i o n the d a m p i n g

on P(q,p)

(4.2) spreads

is an observability condition, through the whole system

which

The main r e s t r i c t i o n of Theorem 4.1 is that V is a s s u m e d local

minimum

in

q0-

On

the

other

hand,

feedback uj - - kjuj + v j, j e m, with vj

ensures

application

to have a (strict)

of

the

the new controls,

proportional

is easily

to result in another H a m i l t o n i a n system with m o d i f i e d H a m i l t o n i a n

[t(q,p,v) -

1 TG(q)p + ~'(q) ~.p

~

vj Hj (q)

j-l, where

(4.6)

V(q) - V(q) + ~

kjyj, J-l

yj - Hj(q),

that

([Jo],[S5]).)

j E m,

seen

446 is

the modified potential

The

energy.

possibilities

of

"shaping"

in this

manner the potential energy V(q) are indicated in

Proposition 4.2. Let dV(q0) - 0 and Hj ( % ) f o2v

1~(%)/

Hessian matrix

i~ p o s i t i v e

- 0, j e _m. Assume that the n x n

definSte

whe. res==~cted t o t h e s,,b-

m

space

N ker grad Hj(q0 ). Then there exists a feedback uj - - kjyj, k 3 _> 0, J-1 j e m_, such that V(q) defined by (4.6) satisfies the assumptions of Theorem 4.1,

i.e.,

V has

a strict

local

minimum

in

qo

dV(q) ~ 0 for all q

with

around q0, with q ~ q0"

Combination

of

consideration (4.7)

Theorem

4.1

and

Proposition

4.2

leads

naturally

to

the

of controllers of PD-type

uj - - kjyj - cjyj,

kj >_ 0,

cj > 0,

J e m.

Let us now assume that the conditions of Theorem 4.1 and Proposition 4.2 are indeed

satisfied,

so

that

feedback

(4.7).

Since

dashpots

it

immediately

is

PD-controller

there

(4.7)

clear

in

(4.1)

to

any

perturbations

of V(q),

strict

minimum

local

perturbations c3

in

(4.7)

stability,

the from

an

asymptotically

addition

of

physical

other

springs

considerations

definite

that

in

q0-

Also

matrix,

may become

arbitrarily

the

robustness

with

on u) is very good; large

while cj > 0 may be arbitrarily

without

the

of the matrix and

for

as long as V(q) defined in (4.6) continues

(static perturbations

and

Indeed the closed-

stable for any perturbation positive

stabilizing

linear

(4.7) has very good robustness properties.

loop system remains asymptotically g(q)

exists

mimics

all

to have a

respect

to

series

in particular ~

affecting

and

the asymptotic

small and kj only has to satisfy

a lower bound (see e.g.

[GI] for some appropriate concepts in this context).

Furthermore,

the robustness

unmodelled,

favorable. (4.7)

of. but

[SI], assumed

properties

to be physically

of

(4.7)

structured,

with

dynamics

respect seem

to

to be

A main designing task is to tune the gain parameters kj and cj in

(or to take suitable nonlinear functions of y and y), so as to achieve

"optimal"

transient

nonlinear analogue

as

the

of critical dan~in E for second order linear systems;

behavior

of

the

closed-loop

system

(such

see

[K2] for a discussion in a robotics context). Having

very

classical

physical

used in various contexts stabilization purposes

roots,

(satellite

the

feedback

control,

([ALMN],[Ba],[G],[Jo]).

scheme

(4.7)

has

been

flexible structures,

etc.) for

For point-to-point

control of

447 rigid

robot

[KI]);

manipulators

indeed,

its

use

has

been

recognized

in

for m - n and H3, j E ~, being independent,

[TA]

(see

also

the conditions

of

Proposition 4.2 and Theorem 4.1 are trivially satisfied.

A second example attractive

where

shortcuts

the use

of the Hamiltonian

is in the computation

system (I.ii).

For linear systems

zeros

important

is very

notion

of

zeros,

linear notion,

for

or better

of

structure

the

it is well-known

controller zero

design.

dynamics,

can

zeros

be

defined,

case

the

extending

the

as follows.

(i.12). First we define the clamped or constrained (i.ii)

constraints

some

that the location of the

In the nonlinear

For clarity of exposition we only consider Hamiltonians

system

provides

of a Hamiltonian

as

that

part

of

Yl - ... " Ym - 0,

the where

system y3,

H(q,p,u)

of the form

dynamics of a |[amiltonian

dynamics

j 6 m,

compatible

are

the

with

natural

the

outputs

(l.llc). Now define for any i e m the integer Pl as the smallest integer Z 0 such that

{H3,ad~

Hi) , 0 for some J 6 m. Throughout

Pl < ~, i 6 m. Define then A(q,p) (H,,ad~

we shall assume

as the m x m matrix with

that

(r,s)-th element

Hr} , r,s 6 m. We have

Theorem 4.3.

[S7,S9]

Consider a Hamiltonian

system

(1.11) with Hamiltonian

of the form (1.12). Assume that rank A(q,p) - m,

for every point (q,p) in

(4.8) N* -

Then N* is, codimension

((q,p)

e MIHi(q,p)

if non-empty,

-

ad.oHi(q,p)

a symplectic

-

..

submanifold

-

ad~ioHi(q,p)

-

0,

i ~ _m}

of the phase space H of

X (Pl + I), and the clamped dynamics are given by the Hamiltoni'l

ian vectorfield on N 8HQ(q,p)

a~ (4.9)

~HO(q,p)

where (q,p) are canonical coordinates for N , and H0(q,p) of the internal

energy H0(q,p)

Moreover,

under the assumption

correctly

called

Co N . (4.8) the clamped dynamics

the zero dynamics

(For linear systems

is the restriction

the assumption

(4.9) can be also

[BI] of the Hamiltonian (4.8)

implies

system

left-invertibility

[$7,$9]. of the

448 system.) We remark that for Hamiltonians of the form (4.1), assumption (4.8) is automatically satisfied, with p, - I, i 6 m systems

we

can

conjectured nonlinear

dispense

that

case

if

with

assumption

assumption

still

the

zero

(4.8)

[$7]. For linear Hamiltonian

(4.8)

([$7]),

not

satisfied

is

dynamics

will

be

and

in

then

given

by

[$9] also

a

it is in

the

Hamiltonian

veetorfield of the form (4.9). Summarizing,

the zero dynamics for a Hamiltonian system, at least under some

extra assumptions,

are given by a Hamiltonian vectorfield

on N

with

C M,

Hamilton~an Ho obtained by restricting Ho to N*. Since the free dynamics of a Hamiltonian with

system are simply given by

Hamiltonian

H0,

we

immediately

the Hamiltonian vectorfield

obtain

"poles" and "zeros" of a Hamiltonian system. definite

relations

strong

For example,

the

if H 0 is positive

(and thus the free dynamics are critically stable),

(and thus the zero dynamics are critically stable).

on M

between

then so is H0

In particular,

if H 0 is

also of the form (4.1), then in the slngle-input case the poles and zeros of the

system are interlacin E on the imaginary axis.

(linearized)

Furthermore

just by the fact that the free and zero dynamics are Hamiltonian it follows that neither

of them can be

asymptotically

stable,

and

that

the poles

as

well as the zeros of a linear or linearized nonlinear Hamiltonian system are located symmetrically with respect to the imaginary axis.

For some control

applications of these observations we refer to [HS,S7].

As

a third

example,

tonian system

we

(I.ii),

mention

the

study

e.g. with Hamiltonian

of

controllability

(1.12).

of

a Hamil-

It is well-known

that,

at least in the analytic case, the controllability properties are determined by

all

(repeated)

Lie

brackets

of

the

drift

and

the

input

vectorfields

[Su2]. Now the Lie bracket of two Hamiltonian vectorfields with Hamiltonian F,

respectively

G,

is

again

a

Hamiltonian

vectorfield,

with

H~niltonian

{F,G} (the Poisson bracket of F and G). Hence all Lie brackets in this case are

Hamiltonian

Hamiltonian

vectorfields,

system

are

l]0,HI .... H m. The theoretical at

this

related

moment;

at

least

to observability

this observation. of motion;

and

determined

the by

controllability the

(repeated)

properties Poisson

of

the

brackets

of

implications of this observation are not clear it

[$3].

follows

that

controllability

Notice also the computational

is

very

much

advantages of

It implies that we do not have to go through the equations

the knowledge of the Hamiltonian

valid for other problems,

see e.g.

the conditions of Theorem 4.1.)

(1.12)

suffices.

(This is also

the computation of the zero dynamics or

449 5. CONCLUDING REMARKS In the development of modern system and control theory over the last decades there has been a tendency to neglect the natural structures imposed by the physical character of the system.

Instead the emphasis has been on Eeneral

(state-space) systems. On the other hand,

in modelling physical systems the

underlying physical structure is often exploited in a crucial way

(this is

especially clear in the context of nonlinear mechanical

see also

systems,

Section i). Thus it seems that an appropriate system theoretic framework for modelling and representation issues should incorporate the relevant physical structure on a fundamental level. In Section 4 we have shown in some examples that also for control purposes it is advantageous to use the physical structure of the system under consideration in an explicit way,

especially

when dealing with complex (nonlinear, infinite-dimensional) systems. Already from the theory of dynamical systems (without inputs) it is well-known that one

cannot

hope

instead one

also

for

one

has

to concentrate

single

theory on

covering special

Hamiltonian vectorfields.

Furthermore,

many

modern

not

clear

applied

control

theory

to systems

do

with

have

a

a specific

of

all

the

for

constructions

physical

physical

nonlinear

subclasses,

systems; instance used

interpretation

structure.

Apart

in

when

from being

unsatisfactory from a theoretical point of view this could also be a drawback in applications. $u~mnarizing, a lot of work remains to be done, both in the system theoretic approach systems

to modelling with

of physical

physical

structure

systems,

as well

(Hamiltonian

as

systems,

in the control gradient

of

systems,

dissipative systems). Finally, not only system and control theory can gain from a closer study of physics,

but conversely,

could benefit

as already argued by Willems

from the consideration of fundamental

[W3,Wi,W2],

system

physics

theoretic con-

cepts (like (minimal) state, controllability and observability, realization).

References IA]

V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin (1978) (translation of the 1974 Russian edition). [ALMN] J.N. Aubrun, K. Lorell, T.S. Mast, J.E. Nelson, "Dynamic analysis of the actively controlled segmented mirror of the W.M. Keck ten-meter telescope", IEEE Contr. Syst.Mag., 7, no. 6 (1987). [AM] R.A. Abraham & J.E. Marsden, Foundations of Mechanics (2nd edition), Benjamin/Cummings, Reading, Mass. (1978). [AS] H. Abesser, J. Steigenberger, "On feedback transformations in hamiltonian control systems, Wiss. Z.TH Ilmenau 33 (1987), 33-42.

450 [AV]

B.D.O. Anderson, S. Vongpanitlerd, Network Analysis and Synthesis, Prentice Hall, Englewood Cliffs, N.J. (1973). [B] R.W. Brockett, "Control theory and analytical mechanics", in Geometric Control Theory (eds. C. Martin & R. Hermann), Vol VII of Lie groups: History, Frontiers and Applications, Math. Sci.Press, Brookline (1977), 1-46. [Ba] M. Balas, "Direct velocity feedback control of large space structures", J.Guid.Contr., 4 (1979), 480-486. [Be] V. Belevitch, Classical Network Theory, Holden-Day, San Francisco (1968). [BG] J. Baste GonGalves, "Realization theory for Hamiltonian systems", SIAM J.Contr.Opt., 25 (1987), 63-73. [BI] C.l. Byrnes, A. Isidori, "A frequency domain philosophy for nonlinear systems, with applications to stabilization and adaptive control", Proc. 23rd IEEE Cenf. Decision and Control (1984), 1569-1573. IBM] R.K. Brayton, J.K. Moser, "A theory of nonlinear networks", Quart. Appl. Math., 22 (1964), Part I, pp. 1-33, Part II, pp. 81-104. [BR] R.W. Brockett, A. Rahimi, "Lie algebras and linear differential equations", Ordinary Differential Equations (ed. L. Weiss), Academic, New York (1972). [BS] A.G. Butkovskii, Yu.I.Samoilenko, "Control of quantum systems", I & II, Autom.Rem. Contr.40 (1979), 485-502, 629-645. [C] P.E. Crouch, "Geometric structures in systems theory", Prec. IEE, 128, Pt.D (1981), 242-252. [CII] P.E. Crouch, M. Irving, "On finite Volterra series which admit Hamiltonian realizations", Math. Syst.Th., 17 (1984), 293-318. [CI2] P.E. Crouch, M. Irving, "Dynamical realizations of homogeneous Hamiltonian systems", SIAM J.Contr. Opt., 24 (1986), 374-395. [CL] P.E. Crouch, F. Lamnabhi-Lagarrigue, "State space realizations of nonlinear systems defined by input-output differential equations", Analysis and Optimization of Systems (eds. A. Bensoussan, J.L. Lions) Loci.Notes in Contr. Inf. Sc. iii, Springer, Berlin (1988), 138-149. [cs] P.E. Crouch, A.J. van der Schaft, Variational and Hamiltonian Control Systems, Lect.Notes in Contr. Inf. Sc. 101, Sprihger, Berlin (1987). [c] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Mass.

(1950). ICe] [Cl] IllS]

[J1] [J2] IJ3~ [Jo]

[Kl]

W.B. Gevarter, "Basic relations for control of flexible vehicles", AIAA Journal, 9 (1970). S.T. Glad, "Robustness of nonlinear state feedback - a survey", Automatica, 23 (1987), 425-435 }].J.C. Huijberts, A.J. van der Schaft, "Input-output decoupling with stability for Hamiltonian systems", University of Twente, Dept. Applied Mathematics, Memo 722 (1988), to appear in Math. of Control, Signals and Systems. B. Jakubczyk, "Poisson structures and relations on vectorfields and their Hamiltonians", Bull.Pol.Ac.: Math, 34 (1986), 713-721. B. Jakubczyk, "Existence of }[amiltonian realizations of nonlinear causal operators", Bull. Pol.Ac.: Math, 34 (1986), 737-747. B. Jakubczyk, "Hamiltonian realizations of nonlinear systems", Theory and Applications of Nonlinear Control Systems (C.I. Byrnes, A. Lindquist, eds.), North-Holland, Amsterdam (1986), 261-271. E.A. Jonckheere, "Lagrangian theory for large scale systems", preprint, University of Southern California, Dept. of Electrical Engineering (1981). D.E. Koditschek, "Natural motion for robot arms", IEEE Prec. 23rd Conf. Decision and Control (1984), 733-735.

451 D.E. Koditschek, "Robot-control systems", Encyclopedia of Artificial IntellIEence (Stuart Shapiro, ed.), Wiley (1987), 902-923. [KM] P.S. Krishnaprasad, J.E. Marsden, "Hamiltonian structures and stability for rigid bodies with flexible attachments", Arch.Rat.Mech.Anal., 98 (1987), 71-93. [M] R. Marino, "Hamiltonian techniques in control of robot arms and power systems", Theory and Applications of Nonlinear Control Systems (C.I. Byrnes, A. Lindquist, eds.), North-Holland, Amsterdam (1986), 65-73. [MB] N.H. McClamroch, A.M. Bloch, "Control of constrained |{amiltonian systems and applications to control of constrained robots", Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits (F.M.A. Salam, M.L. Levi, eds.), SIAM, 1988, 394-403. [NS] H. Nijmeijer, A.J. van der Schaft, "Input-output decoupling of Hamiltonlan systems: The nonlinear case", Analysis and Optimization (J.L. Lions, A. Bensoussan, eds.), Lect.Not. Contr.lnf. Sei. 83, Springer, Berlin (1986), 300-313. [SI] A.J. van der Schaft, "Hamiltonian dynamics with external forces and observations", Math. Syst.Th., 15 (1982), 145-168. [$2] A.J. van der Schaft, "Symmetries, conservation laws and time-reversibility for H~uiltonian systems with external forces", J.Math. Phys., 24 (1983), 2095-2101. [$3] A.J. van der Schaft, System theoretic descriptions of physical systems, CWI Tract 3, CWI, Amsterdam (1984). [$4] A.J. van der Schaft, "Controlled invarianee for Hamiltonian systems", Math. Syst.Th., 18 (1985), 257-291. [$5] A.J. van der Schaft, "Stabilization of Hamiltonian systems", Nonl.An. Th.Math.Appl., 10 (1986), 1021-1035. [$6] A.J. van der Schaft, "Optimal control and Hamiltonian input-output systems", AiEebraic and Geometric Methods in Nonlinear Control Theory (M.Fliess, M. Hazewinkel, eds.), Reidel, Dordrecht (1986), 389-407. [$7] A.J. van der Schaft, "On feedback control of Hamiltonian Systems", Theory and Applications of Nonlinear Control Systems (C.I. Byrnes, A. Lindquist, eds.), North-Holland, Amsterdam (1986), 273-290. [$8] A.J. van der Schaft, "Hamiltonian and quantum mechanical control systems", 4th Int. Sem.Mathenmtieal Theory of Dynamical Systems and Microphysics (A. Blaqui~re, S. Diner, G. Lochak, eds.), CISM Courses and Lectures 294, Springer, Wien (1987), 277-296. [$9] A.J. van der $chaft, "}|amiltonian control systems: decomposition and clamped dynamics", Control Theory & Multibody Systems, AMS Contemporary Mathematics, to appear (1989). [Sa] R.M. Santilli, Foundations of Theoretical Mechanics I, Springer, New-York (1978). [San] G. Sanchez de Alvarez, Geometric Methods of Classical Mechanics applied to Control Theory, Ph.D.Thesis, Dept.Mathematics, Univ. of California, Berkeley (1986). [SI] J.J.E. Slotine, "Putting physics in control - the example of robotics", IEEE Control Systems Magazine, December 1988, 12-18. [Sul] ||.J. Sussmann, "Existence and uniqueness of minimal realizations of nonlinear systems", Math. Syst. Th., i0 (1977), 263-284. [Su2] H.J. Sussmann, "Lie brackets, real analyticlty and geometric control", Differential Geometric Control Theory (R.W. Brockett, R.S. Millman, }|.J. Sussmann, eds.), Birkha~ser, Boston (1983), 1-116. [T] F. Takens, "Variational and conservative systems", Report ZW 7603, Univ. of Groningen (1976). [TA] M. Takegaki, S. Arimoto, "A new feedback method for dynamic control of Manipulators", Trans. ASME, J. Dyn. Syst.Meas. Contr., 103 (1981), 119-125. [K2]

452 [THC] [Tu] [WI] [W2] [W3] [W4]

[wh]

T.J. Tarn, G. Huang, J.W. Clark, "Modelling of quantum mechanical control systems", Math.Modelling, i (1980), 109-121. W.M. Tulczyjew, "Hamiltonian systems, Lagrangian systems and the Legendre transformation", Symp. Math. 14 (1974), 247-258. J.C. Willems, "Dissipative dynamical systems", Part I & Part II, Arch.Rat.Mech.Anal., 45 (1972), 321-392. J.C. Willems, "Realization of systems with internal passivity and symmetry constraints", J. Franklin Inst., 301 (1976), 605-621. J.C. Willems, "System theoretic models for the analysis of physical systems", Ricerche di Automatica, I0 (1979), 71-106. J.C. Willems, "From time-serles to linear system - Part I. Finite dimensional linear time invariant systems", Automatlca, 22 (1986), 561-580. E.T. Whittaker, A treatise on the analytical dynamics of particles and riEid bodies, 4th edition, Cambridge Univ. Press, Cambridge (1959).

The Impact of the Singular Value Decomposition in System Theory, Signal Processing, and Circuit Theory J. Vandewalle, L. Vandenberghe, M. Moonen ESAT (Electrical Engineering Department) Katholieke Universiteit Leuven Kardinaal Mercierlaan 94, 3030 Heverlee, Belgium L. Vandenbcrghe and M. Moonen ,are supportcd by the N. F. W. O. (Belgi0al National Ftmd of Scientilic Research).

Abstract

System theory is a discipline which applies mathematical methods in order to provide a unified approach for many application areas. Such a unillcation is not only useful for conmmnication between experts, but also iu order to be able to carry over concepts, methods and software between areas. Also intellectually such a unified approach is mandatory since the same mathematical results are reused attd the same derivations apply in the different application areas. The first aim of this contribution is to discover some conmmn grounds in signal processing, circuit theory and some other engineering areas. Second the ixnpact of singular value decomposition and its generMizations will be discussed. Third it. will be shown how system theory can provide a unified approach to the problems and open up new avenues in these fields.

1

Introduction

A n n i v e r s a r i e s a r e v e r y g o o d occasions to reflect a b o u t tile b r o a d e r scope of p a s t a c t i v i t i e s a n d p l a u n e d projects~ c e r t a i n l y in t i m e s when m o s t e n e r g y is d i s s i p a t e d in o r d e r to satisfy i m m e d i a t e nccds. It is a p l e a s u r e for us to s h a r e these reflections w i t h you at the o c c a s i o n of J a n W i l l e m s ' fiftieth a n n i v e r s a r y . It is the a i m of t h e c o n t r i b u t i o n to e x t r a c t from o u r p a s t e n g i n e e r i n g r e s e a r c h e x p e r i e n c e c o m m o n s y s t e m g r o u n d s a n d to p o i n t to s o m e fltturc o r i e n t a t i o n s . It is q u i t e n a t u r a l t h a t disciplines like c o n t i n e n t s drift a p a r t . However on t h e e a r t h

454

surface continents drift apart in one sense and converge in the other. Similarly system theory can bring together several remote disciplines and consolidate the nmthematical fi'amework. This unification of system theory is important not only because it provides a platform for mathematically correct reasoning but also because it leads to parsinlony of concepts, ideas, tools and software. Indeed it is much easier to teach, understand, extend and apply the same concepts, methods and even software. Such a unified system approach provides an analytical link between on the one hand classes of systems and models and on the other hand classes of behavior (stability, bifurcation, sensitivity, ... ). These unification efforts can be seen at the smnc level as the great unification efforts in mathematics with the advent of modern algebra. However from an engineering point of view design is more important than analysis. Since technical systems such as electrical, nlechanical or computational or the combination of these systems (often called meehatronie systems) become more and more complex, a global system approach is mandatory. In this paper we illustrate these ideas with certain concrete problems encountered during engineering research in mathcnmtical modeling, circuit theory and complexity and we extract from these the common system grounds. The vehicle for this tour is the singular value decomposition (SVD) which is an old concept from matrix theory (Autonne 1902 [30]) which was used extensively in numerical linear algebra since the sixties and which is now widely applied in signal processing, system theory and control under various generalizations. This contribution is organized as follows. In Section 2 some general thoughts are given on the role of mathematical modeling in engineering and education. In Section 3 an overview of singular value decomposition and its generalizations is presented. In Section 4 SVD is uscd in the study of static systems. The identification of dynamic systems from input-output measurements with SVD is discussed in Section 5. In the last section the conclusions are presented. In the whole paper special emphasis is placed on the rank reduction properties that can bc pcrfornmd based on the spectrum of singular values. This is especially relevant in order to compare the complexity of models [3], in at)proximate modeling, nmdel reduction and separation of signals and noise based on their strength.

2

On the role of m a t h e m a t i c a l m o d e l i n g in engineering and e d u c a t i o n .

When dealing with complex anMysis, design and control problenls, engineers more and more turn to nlathematlcal models and computer aids. In fact conlputers can only deal with nlathelnatical models of physical or technical systems. Hence one should not forget that a computer simulation or a computer aided design is based on an underlying model. It is then the responsibility of the user or engineer to verify whether the underlying asstunptions of the model (e.g. lincarity, time i n v a r i a n c e . . . ) arc satisfied in reality. This can bc a serious problem since certain types of bchaviour can bc undetected because a too restrictive class of models is used. So one can confuse in certain simulations [7,8]

455

periodic behavior (with a large period), chaotic behavior and noise. Of course there is always a trade-off in engineering and in other aplication areas between on the one hand the complexity of the model and on the other hand the accuracies of the model. The black box model as introduced by J. Willems [2,3] for a dynamical system is very adequate for many engineering applications like signal processing, circuit theory and identification. D e f i n i t i o n 1 A dynamical sltstem is defined by the time ~et 71' C R , the signal alphabet W and the behavior B E W T. Usually the time set T is R for the continuous time systems or the integer nlultiples r a of the sampling interval r for discrete time systems. This together with the alphabet (R,6' or discrete) is known in advance to the engineer. The most relevant part for the engineer however is the behavior set B which is in fact the collection of all admissible signals for the external variables. In this general framework [2,3] the external variables should reilect the interaction of the dynamical system with the rest of the world. They can be inputs or outputs. In general it may not be clear which one is an input or an output. Although this is an unusual situation in control theory it is a very realistic approach in circuit theory, many measurement and process control situations. Example : A diode is characterized by the time set R the alphabet R × R and the behavior

B = {v(t), i(t) I v(t)i(t) = 0, o(t) < 0, i(~) ~ 0}

(1)

This description of a diode corresponds with the traditional description of an ideal diode in circuit theory. On an unknown twoterminal element, called device under test, ill a complex electronic circuit one can do electrical measurements and verify that the voltage v(t) and the current i(t) satisfy v(t)i(t) = 0 and v(t) <_ O,i(t) > O. Thereby we assume that the only interaction with the outside world is via v(t) and i(t). When the diode can enfit light (LED) the model shonld be modified. It is also not possible to call v(t) or i(t) an input or an output. In circuit theory one can deal with diodes and many other components like transistors, resistors, capacitors, inductances, transformers, ... in a similar way without specifying inputs or outputs [10]. Such a black box approach is useful, handy and appropriate in circuit theory and IC design. However for those who are involved in device physics and IC technology the diode is described by equations in quantum physics. Although this physical description is useful because it explains much more of the internal mechanisms like the temperature dependence and the dependence on the geometry, for system design and evcn VLSI design the black box approach or a slight refinement of it are quite sufficient. Also in control theory and signal processing mathematical models are quite often used because they show how the system interacts with the external world via external variables. Of course one should not use such models in order to obtain complete information on the internal mechanisms. ttowever for the purpose of design (e.g. the buildup of a complex system with the black box) and for process control (e.g. quality control, ... ) the black box model is adequate for engineers. Itence we can conclude that the dcfinition of a dynamical system as given

456

in mathenlatical system theory, is quite uatural for engineers involved in system design, signal processing and control. It is at this level of abstraction that system theory provides a variety of concepts like time constants, transfer function, impulse response, state equations, transient, steady state, poles, convolution which are particularly suited to analyze, design and control linear systems. The fact that the~c concepts exist for any linear system implies that these can be applied in many individual engineering disciplines like on electrical, mechanical, hydraulic, thermodynamic systenls and even on more global systems which encompass several individual disciplines like ecology, mechatronics, sensors, transducers and actuators. At the didactical level, system theory can be a general course for all engineering students. At the industrial level a unified systems approach to problems has many advantages. Since the problems often have aspects at the material, iliformatioil and organization level, a global systems approach is mandatory. Even if there are only qualitative relations and not quantitative, they can be quite rewarding [29]. In a wider scope, a general systems approach can introduce to engineers a wealth of unifying concepts, e.g. convolution, chain fractions expansion, sensitivities, orthogonality, singular value decomposition, least squares, modulo calculus, recursion, fast Fourier transform, simulated annealing, stability, convergence, adaptivity, complexity theory, dynamic systems, ~veragc, standard deviation, yield, graph theory. In order to illustrate the convolution example, one c,~n observe that convolution is used in the study of the sum of two random variables, the multiplicatiou of polynomials, the nndtiplication of integers with a digital circuit, the relationship between input and output of a dynamical system . . . . Chain fractions expansion of rational numbers appear to be useful in gear boxes and electronic counters, whereas chain fractions expansions of ratios of polynomials are used in numerical analysis and lilter design. Quite often one system representation is more usefld for a certain aspect than another. Hence the conversion from one representation to another is important. Many objects can be described in a geometric form (plots, block diagram, flow charts, . . . ) as well as in a linguistic form (data, formulas, programs, ... ). For a given discrete time dynamical system for example one can find many representations by algebraic expressions like the transfer function, the impulse responsc . . . . Most often all engineer is more interested in a graphical or geometric representation. Students can train their understanding of the different representations and the relationships by varying some parameters and observing the difrerences between the plots as generated by a personal computer. It is not only in engineering but also in mathcntatics that one can observe an evolution towards the use of persona.1 computers and of graphical representations. The publications of the Visual Mathcmatics Library [11] arc quite interesting in this respect. An intportant contribution of systems theory is that it provides links between reality and computer, in fact the same computer infrastructure and simulation tools can be used often for many different systems. The set of available tools should be considered in a very wide sense and should include audiovisual tools, computer tools (spreadsheet, symbolic manipulation e.g. MACSYMA), simulation tools, optimization tools. Quite often engineering students and engineers fecl so confident about the simulations eu-

457

vironment (workstation or PC and a user friendly softwarepackage), tlmt they accept whatever results are obtained by the simulatiou. However computers need a mathematicM model of reality for simulation and optinfization and depending on the kind of model at hand one can observe certain phcaonmna. Hence engineers should be quite critical with respect to the results gcncratcd by thc computer. Hcre again a solid mathematicM understanding of the models, the phenomena as well as the numerical methods is important in order to makc the correct decisions. In this respect one can observe that the distinction between software and hardware becomes more and more artificial. Software is used to design hardware and hardware can help to accelerate the execution of software. Here again it is important that engineers have an integrated and broad view.

3

Singular concepts

value decomposition and generalizations

(SVD)

and SVD

based

In this section, the theorems stating the existence and the properties of singular value decomposition and its generalizations are presented. For a proof, software and computational requirements the reader is referred to the literature I44]. T h e o r e m 1 The s i n g u l a r v a l u e d e c o m p o s i t i o n . If A is an m × n real matviz of

rank r then there ezist real orghogonal matrices

v = {,,, ,,~... ~m], v = [~ ,~... ,,,,1

(2)

such that Ut'A'V=

[ diag(al'~'2'''''~)O

00 ] = ~

(3)

0

(4)

tohcre ~'~ >- ~'~ > . . . -> ~', > ~',+l . . . . .

The al arc the singular values of A and the vectors ui and v~ arc respectively the i.th left and the i-th right singular vector. The set {ui, cri, vi} is called the i-th singular triplet. The singular vectors (triplets) corresponding to large (smaU) singular values are called large (small) singular vectors (triplets). The SVD of A

A = UEV t reveals a great deal about the structure of a matrix as evidenced by the following well known corollaries : C o r o l l a r y 1 Let the SVD of A b c given an in theorem 1 then

(I). Rank property r(A) = r a,~a

N(A) = ~pa,~{v,,,...,v,,}

(5)

458

10

It

12

13

14

15

Numl~r

Figure t: Typical singular spectrum of a 15 x 15 matrix.

(2). Dyadic decomposition A =

"i'~i'v~

t

(6)

i=I

U)" Norms

(~[). Rank k approximation. Dcfine Ak by k a k = ~ui'(7"

i.73 it

with k < r

(8)

i=1

then r a i n ]]A

~(B)=k

-

BII~ = I[A

-

Ak[l~ = ,,k+~

r a i n I[A - B[[~. -- [[A - Ak[l~- ---- , ~ + , -t. . . .

,(n)=k

(9) 4- ~ '

(ao)

This i m p o r t a n t result is the basis of m a n y concepts and applications such as total linear least squares, d a t a reduction, image enhancement, dynamicM s y s t e m realization theory and in all possible problems where the heart of tile solution is tile approximation, measured in 2-norm or Frobeniusnorm, of a matrix by one of a lower rank. A typical singular value s p e c t r u m of a 15 x 15 matrix is depicted in Fig. 1. Such spectra typically occur in situations where the matrix A is generated with measurements of limited accuracy. Due to the limited accuracy of the m e a s u r e m e n t s (range of 0.1) the matrix is of full rank (15). lIowever with some small perturbations on the matrix A (2-norm in the range of 0.1v/]-5), the matrix can be a p p r o x i m a t e d by a rank 9 matrix. In most systems, control and signal processing applications where such matrices occur, the lower rank al)proxlmation of the matrix is much more relevant than the d a t a matrix A. In Figure 1 there is a clear dit[erence between the large and relevant singular values a l . . . or:, and the small singular values, which are due to the iuaccuracies in the measurements. Although in some applications the distinction may not be clear, it is usually

459

appropriate to approximate a data matrix by nlaking all singular values zero which are within the raugc of inaccuracies. This procedure is oftcn essential in order to come up with meaningful and practical models for systems (sce Section 4 and 5). The same arguments can be applied to the generalizations of the singular value dccomposition (see below) which involve more than one matrix and which also produce a generalized singular spcctrum. The SVD provides an important tool in the generalization and charactcrization of important geometrical concepts. Onc of these is the notion of a n g l e s b e t w e e n s u b spaces, which is a generalization of the anglc betwecn two vectors. D e f i n i t i o n 2 Let F and G bc subapaces in R " whose dimensions satisfy p = dim(if) _> dim(G) = q >_ 1 The t, rincit,al angles 0~, 0 ~ , . . . , Oq 6 [0, 7r/2] between F and G arc defined recursively by COS(O k ) =

lllO.X l n a x

uEF vcG

st.t,

=

?.gk.~ k

subject to IMI = 11.,11 = a;

0

i = 1,..., k - ]

Vt.Vi = 0

i = 1,..., k - 1

at.~ti :

The vectors 'Ui, Vi, (F,C).

i = 1 , . . . ,q arc called the principal vectors of the subapacc pair

If the columns of P (m × p) and Q (m × q) define orthonormal bases for the subspaces F and G respectively, then it follows from the minimax characterization of singular values that : [ul,...,%]

=

P.Y

[o~,...,~]

=

Q.Z

cos0k = ak

k = 1...q

where Y, Z and the crk are given by the SVD of the ('generalized inner') product p t . Q = Y.diag(o.t,. . . , (r,1).Z t

Fronl this, it is not dimcult to devise all algoritlun to compute tile intersection of subspaees that are for instance tile eohmm spaces of two given matrices A and B. This is prccisely the idea bchind the tcchnique of canonicM correlation, which appcars to be vcry fruitful in the idcntification of lincar dynamicM state spacc models fi'om noisy input-output measurements [25,26]. Therc arc several ways to compute thc canonical correlation structure of a matrix pair (A, B), roughly all possible ways of computing an orthonormal basis for the row spaces (e.g. two QR decompositions, two singular value dccompositions) followed by

460

an SVD of the generalized inner product of the orthonormal bases matrices. Another method is the computation of the right null space of the concatenated matrix

i A] However, it is expected that depending on the application at hand, one m e t h o d could be preferable with respect to the others. From now on we call the singular value decomposition the ordinary SVD or OSVD. A proposal [31] has been made to standardize the nomenclature for the generalizations of SVD. These are PSVD, QSVD, RSVD, SSVD and TSVD. We will briefly overview these different generalizations and explain in more detail the QSVD because it is needed in the next sections. However some other generalizations (like SSVD) have an important impact on control [341136] and more applications of the generalizations are expected. T h e o r e m 2 The q u o t i e n t s i n g u l a r v a l u e d e c o m p o s i t i o n (QS VD ). I~ A is a,t m × n matrix with n > ,n. and B is an nt × p matrix, then there exist orthogoual mat,'iccs Qa (n × n ) and QB (p × p) and an invcriiblc X (m × m) 3uch thai :

X t.A.Qa =Da =diag(a~) ai>_O i = l , . . . , m X t . B . QB = D~ = diag(fli) /3i > 0 i = 1 , . . . ,q = rain(re, p)

(11)

whcPc

/3, >/3~ _>. . .

_>/3~ _>/3r+, . . . . .

/3~ = 0 r = , - , , , k ( B )

(12)

Observe that the QSVD reduces to tile OSVD in the case that B = I,.. Tile elements of the set ,,(A, B) = { ~ l / f l , , - . . , ~r//3r} (13) arc referred to as the quotient singular values of A and B. The quotient singular values corresponding to the /3/ = 0 arc infinite. They are considered to be equal and come first. For our purposes it is more convenient to order the diltgonal elements in DA and DB according to decreasing quotient singular value rather than by (12). The recent introduction of the fundamentM concet)ts of o r i e n t e d e n e r g y and orle n t e d s l g n a l - t o - s i g n a l r a t i o [21] has provided a rational framework in which both the estimation of ranks and subspaccs can bc formalized in a rigorous way. Moreover cxtremal directions of oriented energy and oriented signal-to-signal ratio can be calculated with OSVD and QSVD. Let A and B contain measurement vector sequences (typicMly a nulnbcr of consecutive sample vectors from m mcasurcmcnts channels). The columns of A and B are dcnotcd by a1,,bk. D e f i n i t i o n 3 The oriented energy of the malrix A, mca~ttrcd in a direction q is defined a8

:

rt

Eq[A] = ~__.(qtak)2 = IlqtA]] 2 k=l

(14)

461

D e f i n i t i o n 4 The oriented signal-to-signal ratio of the two vector sequences A and B in the direction q is defined a~ : Eq[A,B] = Eq[A]/Eq[B]

(15)

There are straightforward generalizations of these definitions to oriented energy and signal-to-signal ratios in subspaces Q~. In [21] it is shown that the analysis tool for the oriented energy distribution of a matrix A is the singular value decomposition, while the analysis tool of the oriented signal-to-signal ratio of two vector sequcnces A and B is the quotient singular value decomposition of the matrix pair (A, B). These well understood matrix faetorizations allow to characterize the directions of extremal oriented energy and oriented signal-to-signal ratio. T h e o r e m 3 Extrcmal directions of oriented cncrgy. Let A be an m × n matri~ with OSVD A = U Z V t where Z = diag{o'i}. Then each direction of extrcmal orientcd energy of A is generated by a left singular vector ui with czlremal cncrgy equal to the corresponding singular value squared ~r~. T h e o r e m 4 Extrcmal dircctions of oricntcd signal-to-signal ratio. Let A (m × n.) and B (m x p) bc matrices with QSVD : a B

= =

X -t.D..Qt a X -~. D~. Qb

D~=diag{a,} D~ = d iag{fli}

(16)

where thc quotient singular vahtca (possible infinite) are ordered auch that (ax/fll) k (a~/fl2) > ... >_ 0. Then each direction of cxtremal signal.to-signal ratio of A and B is gencratcd by a column xl of the matrix X and thc corresponding c'xtremal signal-to.signal ratio is the quotient singular value squared (ai//Ji) 2. These two theorems are illustrated for two dimensions in figure 2. Observe that for the oriented energy the maximum and nlininlunl corresl)ond to the largest resp. smallest singular vectors while a saddle point would correspond to the intermediate singular vector. Observe that the extrcmal directions of oriented energy are orthogonal while this is not necessarily the casc for the signal-to-signal ratio. Now one can proceed by investigating in which directions of the ambient space the vector signal in the matrix A can be best distinguished from the vector signal in the matrix B. This leads to the delinition of lnaxinlal nlinilnal and minimal maximal signal-to-signal ratios of two vector sequences. D e f i n i t i o n 5 Maximal minimal and minimal maximal signal-to-~ignal ratio. The maximal minimal signal.to-signal ratio of two m-vector" scqucnccs contained in the matrices A and B over all possible r-dimensional subspace~ (r < m) is defined as : M m R I A , B, v] = max nfin Eq[A, B] Q ' c R " q~Q"

Similarly, the minimal maximal signal-to-signal ratio is dcfincd as : mMr~[a, B , r] = rain max E~[A, 13] Q'~CR m q E Q "

462 R}

,so

,I

100

u2

.. [^]

I

u1

50

0

-50

-I00

-I50 -150

-100

-50

0

50

100

150

b) 1

-1 -2

-3

-4

i

-2

i

0

;3

F i g u r e 2: a) O r i e n t e d e n e r g y of a 2 - v e c t o r s e q u e n c e A a n d b) o r i e n t e d s i g n a l - t o - s i g n a l distribution of two 2-vcctor sequences A and B.

463

Tile idea behind these definitions is tile following : for a given subspace Qr of the m-¢limensional ambient space (r < m) there is a certain direction q E Q" for which tile signal-to-signal ratio of the two vcctorsequcnccs A and B is minilnal. This direction corresponds to the worst direction q in the sense that in this direction the energy of A is ditlicult to distinguish from the energy of B. This worst case of course depends upon the precise choice of the subspace Qr. Among all r-dimensional subspaccs, at least one r-dimensional subspacc has to exist where the worst case is better than all other worst cases. This subspace is the r-dimensional subspace of maximal minimal signal-to-signal ratio. It comes as no surprisc that the QSVD allows to find this subspace : it is the r-dimensional subspucc generated by the first r columns of X, when the quotient singular values arc ordered as in theorem 4. Hence, the concept of oriented signal-to-signal ratio and the QSVD allow to formalize all model identification approaches, in which * the determination of a suitable rank r provides the complexity of the model. * the model parameters follow fi'Oln the corresponding subspacc of maximal lninilnal signal-to-signal ratio. Moreover, it can bc shown that whcn the vector sequence ]3 consists of an unobscrvablc stochastic vector signal with known first and second order statistics (as is the case in most cnginccring applications), the QSVD solution corresponds precisely to the 'classicM' Mahalanobis transformation that is commonly used in statisticM estimators as a kind of prewhitening filter. Still other extensions to the OSVD exist and thcse will now bc discussed briclly. 0nc unified gcncralization is the R e s t r i c t e d S i n g u l a r V a l u e D e c o m p o s i t i o n , introduced by Zha [4I], see also De Moor and Oolub [32]. The RSVD involves a matrix triplet (A, B, C) and allows to compute a lower rank al)proximatioa to the matrix A by subtracting a matrix which is restricted to have columns in the column space of B and rows in the row space of C. Given the matrices A E R '''×", B E R '''×p and C E R q×", the restricted singular values of the (A, B, C) are defined as o'k+I(A,B,C)=

rain {IEll2 I rank ( A + B E C ) < / + }

EE/~p×q

k=0...n-1

(17)

If no such matrix E exists, a~ is defined to be c~, if m < u and m. < k < n, ak = 0. Thus the interpretation of the restricted singular values is very similar to the ordinary singular values (see eqs. (8) a,~d (9) ,~,~d fig.l). A nunlber of special cases of this RSVD are itnlmrtant in their own right : The O S V D of A is obtained by choosillg B = / , , and G' = 1,,, The Q S V D of (A, B) is obtained obtained by taking C = L,, The P S V D (Product Singular Value Decomposition) of the pair (B t, C) is the RSVD of (L,,,B, C). The PSVD was introduced by Fernando and IIammarling in 1987 [37].

464 A different type of restriction leads to tile definition of tile S t r u c t u r e d S i n g u l a r V a l u e , introduced by Doyle in 1982 [34]. Consider a block partition of a matrix A an

Axx ...

Alq

]

and a matrix AA, partitioned in the same way as A, consisting of zero and nonzero blocks A A i j , with possibly some constraints AAI./ = AAm. D e f i n i t i o n 6 The structured singular raffle crss v is defined as :

crssv = rain I]AA)I2 such that rank(A + A A ) <

,'auk A.

Applications are mainly in If¢o control theory and some characterizations and algorithms are found in [36]. Finally, we mention the Takagi Singular Value DecompoMtion (TSVD) : T h e o r e m 5 If A is a symmetric complex matrix, there cxizls a unitary U and a real nonncgatioc matriz Z = d i a g ( c q , . . . , o , , ) ~uch that A = U.~,U ~. Thc columns of U are an orthonormal ~ct of cigcnvcctors for A A and the corresponding diagonal entries of arc thc nonncgalivc square ~vots of the corresponding cigcnt, aluc~ of A A . This was introduced by Takagi in 1925 [39].

4

S V D for static s y s t e m s

Static systems have no m e m o r y or state and are hence more easy to deal with. Here wc will study the use of OSVD for static models for electric components and the use of both OSVD and QSVD for signal processing. Quite often algcbraic methods of physics or algebraic approaches for identification lead to m a t h e m a t i c a l models that are very sensitive to the d a t a on which tile model is based. This is a situation which is well known in numerical algebra [13] and in filter design. T h e approach that is followed in both cases is that one should investigate the collection of algebraically equivalent approaches and select in that class the one that is least sensitive to the variations in the data. Often these arc orthogonal methods. When there exist m a n y algebraically equivalent models then one can also best use that model that is least sensitive. To make the point clear we consider a simple and hence a somewhat trivial example. Consider a two port resistor which is nothing but a parallel connection of the two ports. A strrdghtforward implicit description is given by 171

1 -1 00 011

0

.i/2

ix

0

465 Observe that F constrains the admissible values of vlv~iai2. If voltage and current measuring devices are used which have the samc precision (say 1%) then one can e ~ i l y check that any set of measured vari~tbles presents a solution to eq.(18) within the expected accuracy limits. By hook or by crook we obtain another implicit description with new matrix '1) 1

1 0

-1 0

10 ~ 1

10 6 1

0

il i~.

A set of measured vlv~ili2 will only satisfy eq. (19) with a rcason~blc precision when the cttrrent is measured very accurately. Algebraically equations (18) and (19) are however perfectly equivalent. So clearly it is desirablc to dispose of some tools to compare algebraically equivalent multiports in less trivial cases. The basic tool is OSVD.

(20)

F =

bet us now consider any implicit description of a resistive n-l)ort [10] i

= 0

(21)

where F is an m x 2n matrix. Tile constant offset has bceu dropped for simplicity. Such a description is very general and can arise from a physical device e.g. by a linearization, physical equations or from network equations. It may also bc the spccilication of a multiport to bc synthesized. In both situations the OSVD of F provides uscfiil information concerning the implicit description. The action of F on the exact part and the inaccuracy of the d a t a are easily analyzed with OSVD. First the orthonormal transformation V does not increase or decrease the sizes of both parts. Then the ditl'crent components of the exact part and the inaccuracy arc scaled differently with ~rl,~z,... (7~,0,0. Afterwards an o r t h o n o r m a l tr,~nsformation does not afl'cct the sizes again. The condition nttmbcr is then to(F) = al/(r, which is the largest increase in error of the measured data. Clearly the best situation which can occur is that crt = (~z . . . . a,. If the representation (21) is the exact specification for a design, the OSVD of F produces an algebraically equivalent but numerically most reliable description by i

where [ A B ] are tile top r rows in V =

(22)

=0

D

" Indeed by multiplying (21) with U

we obtain

v[v 1 i

=0

By scaling the equations with o'i-l,o'~-1,... ~;-l wc obtain cq. (22).

(23)

466 If eq. (21) is an inaccurate model of a physical device ghis scaling is dangerous and one has to stick with (23). The OSVD then allows us to determine in a meaningful way the dimension of the n-port~ which is by definition 2n. minus the rank of the matrix F (22). A straightforward application of this definition on an inaccurate model of eq. (21) would generically produce a dimension which is 27~,- m, if F is an m x 2n matrix and m _< 2n. In this way the inaccuracy of the parameters is used to generate restrictions oil the admissible v - i pairs, which is unacceptable. Rather the model should be modified by replacing by zero all singular values c r , , a , + l , . . . a ~ such that (r,/al is smaller than the accuracy in the parameters of F. This operation is justified by the fact that these modifications can be obtained while varying the 1)arametcrs within the prescribed inaccuracy intervals. Another problem is to classify the explicit representations for algebraic multiports and to generate if possible optimal explicit representations. To set the stage, classically one considers different representations for a resistive multil)ort (resistance, conductance, hybrid, transmission, scattering, ... ). Remember that existence as wcU as measurement issues have stimulated the use of hybrid parameters for transistors, eomnlon and differential nmdes in op amps and scattering parameters in high frequency applications. Though the existence of these parametrizations is not always trivial, a more important issue is whether the explicit representation as a relationship between variables can blow up iuaccuraeies ill the variables. In general using port coordinates x, y an explicit description is given [10] by

By choosing appropriate coordinate transformations f/ one can obtain the resistance, thc conductance, any hybrid or scattering or trausmission rcpresentation and a wealth of other explicit rcprescntations which have not reccived a namc. The condition number of A is then the upper linfit of the amplification in signal to noise ratio from x to y. If A is invcrtiblc this is also valid for thc rclation fronl y to x. Thus thc diffcrcnt reprcscntations can bc rankcd according to their condition number. This information brings the whole sct of existing parametrizations into perspcctivc. Clcarly thc bcst situation is when the condition numbcr is a(A) = 1 or A is orthonorlnal up to a scaling factor. One can now womlcr whcther any multiport rcsistor has an optimal reprcscntation i.c. is thcre an orthonormal ~2 such that ~;(A) = 1. Wc start from thc implicit represcntation of an n-dinlensional n-port resistor i.e. i

=0,

rank

=n

(25)

Applying the OSVD e.g. (20) of F, we obtain eq. (22) with 7" = n. orthogonal coordinate transformation

=

[

-1/,/2

lJ I[A 1/v~

6'

D

i

]

'

Choosing tile

,26,

467

we can express ~he implicit description eq. (22) with the orthogonMity of l]

l/v~

- 1 / ~ / 2 ] [ ~: ]

The conclusion is then that by using the orthogonal transfo~'mation o/cq. (26), which i~ obtained fi'om the OSVD o/cq. (20.), an explicit description ~! = :c is obtained which is numerically optimal. Of course the choice of the coordinates assumes already a certain description (25) or may be done approximately in advance but as long as it is close to that of eq. (26) the explicit description y = A.v will have a good condition nunaber i.e. ~(A) is close to 1. The point to be made here is that there is advantage to work and to compute with the x - y coordinates rather than with others. As a second application of SVD for static systems we consider the extraction of fetal ECG. The measurements are obtained from cutaneous electrodes placed at tim heart and the abdomen of the mother. If there are p measurement channels (typically 6 to 8), the sampled data are stored in ~ p × q matrix ~Ipq where q denotes the number 0f consecutive samples that are processed. The p observed signals m.i(t) (the rows of Mpq) are modeled as unknown linear combinations (modeled by a static p × r matrix T) of r source signals sj(t), corrupted by additive noise signals n~(t) with known (or experimentally verified) second order statistics. Hence the model has the well known factor-analysis-like structure : Mpq = Tp. • S,~ + IV~

where the rows of S~q are the source signals. The problem now consists of a rank decision to estimate 7" and of a subspaee determination problem to deternfine tile subspace generated by the columns of tile matrix T, which are tile so-called lead vectors. Since tile second order statistics are assumed known, the conccptual fl'amcwork of oriented signal-to-signal ratio could be applied, tlowever, it has been verified [18,24] that for this specific application with an appropriate position of the electrodes, the.subspace spanned by the lead vectors of the mother heart is three dixnensional and orthogonal to the three-dimensional subspace generated by the lead vectors of the fetal heart transfer. Moreover, the source signals of mothcr heart and fetal heart are orthogonal vectors if considered over a sufficiently long time wherein the coutributioll of the mother heart is much stronger than of the fetal heart. For all these reasons, one single OSVD suffices to idenl.ify the subspace corresponding to the fetal ECG and by projecting the measurements on this subspace, the MECG can be elinfinated almost completely. For more details on this separation based on the strength of the signals we refer to [18,24].

468

5

SVD

5.1

for dynamic

systems.

Total Linear Least Squares approach

In [1-3] a c o n c e p t u a l f r a m e w o r k is developed in which the m o d e l i n g probleln is translated into an a p p r o x i m a t i o n context based upon the p a r a d i g m of low complexity and high a c c u r a c y models. T h e key concepts in this a p p r o a c h are the c o m p l e x i t y of a model and the misfit between a m o d e l and the observations. Approxinaate nmdefing then consists of i m p l e m e n t i n g the principle t h a t either the desired o p t i m a l m o d e l is the least complex one in a given n m d e l class which a p p r o x i m a t e s the observed d a t a up to a preassigned tolerated misfit, or t h a t it is the nmst a c c u r a t e model within a preassigned tolerated c o m p l e x i t y level. A particulzLrly simple e x a m p l e is the total linear least a~luares app~vach [45,48] which essentially consists of fitting a linear s u b s p a c e to a finite n u m b e r of points. Consider an m x n m a t r i x A (n > m ) containing n m e a s u r e m e n t s on a m - v e c t o r signal. Denote by ai its i-th colunm. Let Q~ be an r - d i m e n s i o n a l s u b s p a c e of R ' " then, the complexity is defined as : c(,,) : Q" --+ C = [0, 1]: c(,-) = d i m ( Q " J / m = , ' / m T h i s r - d i m e n s i o n a l s u b s p a c e Q" call be considered as a lower r a n k ,~pl)roxlmation to tile range of A, with a misfit defined as

~±Q'

IMI

where E,a[A ] is tile oriented energy as defined in section 3, ~'hcn, we lutve the following t h e o r e m [3] : 6 Let ~i,i A = U.P..V t be the S V D of the m . x n malriz A of r a n k s (a <__m < n) with singular values ~rl >_ . . . >_ ~r, > 0 and left singMav vectors ui, i = 1 , . . . , m . The unique optimal a p p , v z i m a t e model Q" with compte'zitu c ( Q ' ) = ,~, and misfit c(A, Q~) = ,r,+l ia an r - d i m e n s i o n a l subspace where : Theorem

• I f Cadln is the m a x i m a l admissible complexity, then : - ifint[m.Cadm] = 0, r = 0 and Q~ = 0. -

ff i n t [ m . C a d m ]

_ s, r = .% Q " = s p a n c o l [ A ]

- if ak > ainti,,,.~adm]+l, r = k, Q~ : 5'kr • If q.ol is the maa:imal tolerated misfit, then : - i f e t o l : _" ~ a l ' r = 0

-

andQ" =0

' f ' t o l < ~'0," = s , Q ~ = s p a n c o t [ A l

- if ~rk > eto 1 >_ a k + t , r = k and Q~ = S~r Proof

: s e e [3].

r-1

469 In tiffs fi'amework of al)proximate modeling, tile appropriate rank r is thus determined from either an a priori fixed admissible complexity or a maximal tolerable nfisfit. As for the identification of state space models for dynamic systems, it will now be shown how these concepts apply to the determination of a suitable nmdel order. Both an OSVD approach for the white noise case, and a QSVD approach for the coloured noise case will be highlighted. Each time, the nmdel order n follows fi'om a r = 2mi + n -law, where r is tile appropriate rank choice, and m and i are certain constants yet to be defined.

5.2

OSVD-based

system

identification,

the

white

noise

case

For the time bcing, wc consider time invariant lincar, discrete time, nmltivariablc systems with state space representation xk+l Yk

:

A.xa + B.uk

=

C.xk + D.uk

where uk,yk and xk denote the input (m-vector), output (/-vector) and state vector at time k, the dimension of xk being the minimal system order n. A,/3, C and D are the unknown system matrices to be identified, making use only of recorded I/O-sequences tt k ~ 'lt,kq_ 1 ~ . . .

a l l ( [ Yk ~ Yk+ 1 ~ • • •

Let us first present the identification scheme. In [47] it was shown how a state vector sequence can bc c o m p u t e d from I / O - m e a s u r e m e n t s only, as follows. Let HI and H2 be dcfiaed as Uk

Hi

H2

Uk+l

.........

ut.+./-1 Yk+j-1

Yk

Yk+l

.........

ltk+l

Irk+2

.........

uk+j

Yk+l

Yk+2

.........

Yk+j

ltk+i-1

"ltk+i

.........

~tk+j+i- 2

Yk+i-I

Yk+i

.........

Y~+j+i-2

'Irk+ i

'ltk+i+ 1

.........

u~+i+j- I

yk+i

Yk+i+l

.........

yk+i+j-I

l/.k+i+l

ltk+i+2

.........

ltk+i+j

/Jk+i+l

IJk+i+2

.........

Y~.+i+.i

.,,

...

Irk+21-1

"Uk+2i

.........

~tk+21+j-2

Y],.+2i- 1

Yk4-2i

.........

yl,.+21+j-2

j > 2(m +

l)i

and let the state vector sequence .l" be defined as •l ' =

[xk+i zk+i+l . . . zk+,+~-l]

470 then, under certain conditions (see [47] ) spanrow(X" ) = s p a n r o w ( H ~) V/spanrow(H 2) so that any basis for tiffs intersection constitutes a valid state vector sequence A' with the basis vectors as the consecutive row vectors. Once A" = [xk+i xk+i+a -.. xk+i+j-1] is known, the system matrices can bc identified by solving an (overdetermined) set of linear equations:

~Ik+i

• • •

Yk+i+j-2

D

"

Irk+ i

...

ll.k+i+j_

2

T h e above results constitute tile heart of a two-step identification scheme. First a state vector sequence is realized as tile intersection of the row spaces of two block Hankel matrices, constructed with I / O - d a t a . Then the system matrices are obtained at once fi'om the least squares solution of a set of linear equations.

Let us now discuss the computational details. The following derivation (which is slightly different from the one in [47]), shows how these c o m p u t a t i o n s can be carried out quite easily, resulting in a consistent (see below) d o u b l e O S V D i d e n t i f i c a t i o n algorithm. In a first stcp, tile intersection of the row spaces spanned by t l l and H2, call be recovered from the OSVD of the concatenation H=

H

[.1] H~

=[,,11 Hz

0]., U12 "IlI U,)I U22 ]" [ S01 [

=

u,.s,.v/,

=

[ Uli

0

dim(U~l)

=

( m i + li) x (2mi + n)

(liln(U12)

=

(,,ti + li) x (2li - n)

din](U2x)

=

( m i + li) × (2mi + n)

dim(U22)

=

(n~i + li) x ( 2 l i -

dim(Sla)

=

(2mi + n) x (2mi + 77)

n)

(see [47] for dctails). From

U~2.I-I1 = -U~2.H~ it follows that the row space of U[z.H1 equals the required intersection. However, U~2.H1 contains 2li - n row v e c t o r s , only n of which are linearly independent (dimension of the

471

intersection). Thus, it remains to select n suitable combinations of these row vectors. As UI~ and U22 form an orthogonM matrix, they can be decomposed as follows [44, p.22]

=

.

C.×,,

~'2

=

[ U~) U~) U~} ] [ O(ti-'~)x(.i-'') .

]

.V:

O(li-n)x(li-, 0

]

S,,×.

.W

l(}i-,~)x(ti-n)

C = diag(cl,...,c.) S = diag(~,,...,s,) L,×, = C ~ + S ~ where U~'2) then constitutes the (It - n)-dimensional orthogoual complement of If,. Clearly, only U~ ) delivers useful combinations for the conlputation of the intersection, and we can take A" /r(2)' .HI The above expressions for U1~ and U2~ are in itself OSVD's of these matrices, and can be computed as such. It thus suffices to compute e.g. the OSVD of U12. The computation of the required intersection then reduces to the computation of two successive OSVD's (for H and U12 respectively). Up till lmW, we have assumed that the data were error-fi'ce. If there is some inacc u r a c y o n t h e m e a s u r e m e n t d a t a in HI and H:, generically the row spaces of these matrices do not intersect, and all singular values in SH are non-zero. Hence, one should approximate H by a matrix of lower rank by setting the smallest singular values equal to zero, in order to obtain the model that is least contlicting with the data. A suitable deternfination of this lower rank, is then to be carried out along the lines of the total leant squares approach to approximate modeling, as was detailed in the previous section. Note that the above derivation is nothing more than a d o u b l e O S V D a p p r o a c h to computing the QSVD of the matrix pair (HI, Itz), following fi'om the constructive QSVD-proof in [48]. From this last remark, one might be tempted to immediately apply a one stage QSVD-procedure to the matrix pair. This latter method would however compute the exact intersection of the row spaces, which in the presence of noise turns out to be completely absent (generically). The outcome of applying such an algorithm wouhl then be a zero dimensional intersection, as could bc guessed beforehand. The difference between these methods turns out to be the intermediate rank decision after the first OSVD in the first approach (double OSVD), tha.l, fixes the dimension of the approximate intersection to be computed next. Although this (possibly dill]cult) intermediate rank decision has been a main m o t i v e / o r developing a one stage QSVD algorithm, for our purpose it is somehow inevitable. In the second step, the system matrices are to be identilied from a set of linear equations. Much like it was done in [47], it can straightforwardly be shown that the

472 system matrices can be computed from tim following reduced set as well (obtained after discarding the common orthogonal factor VH ) [ u12""2)''.tJu~'n'+ l + l : ( i + l ) ( m + l ) , l : 2 m i + , t ) . S t , ] U H ( m i + li + m + i : ( m + I)(i + 1),1 : 2 m i + n).Sl~ t J =

[A C

/71 [ D

cr r12 l l ) '.~ r f Ht~ '" : m i + li, I : 2 m i + '~).5'tl Utt(mi+li+l mi+li+m,l:2mi+n).Sll

]

where U'H(r : s,v : w) is a submatrix of U1t at the intersection of rows r , r + 1 , . . . ,s and columns v , v + 1 , . . . , w.

The identification procedure is proven to be c o n s i s t e n t if the number of cohmms in H tends to infinity and if the input-output mcasurements arc corrupted with additive white measurement noise, or in othcr words, if the columns in H arc subject to independcntly and identically distributed errors with zero mean and common error covariance matrix equal to the identity matrix, up to a Factor of proportionality. For that case, it can indeed be shown [43] that the left singular basis UH can bc computed consisLcntly (as opposcd to the singular vahtes SH and the right singular basis VH). As the system matrices are next computed essentially from Utt oldy (scc the above set of equations1), the nmdel estimate is clearly consistent. The corresponding noise model is depicted in Figure 1.

5.3

QSVD-based

system

identification,

the

coloured

noise

case

Let us now proceed to the casc where the I / O - d a t a arc corrupted by colourcd noise. Assume that the columns ill the concatenatcd matrix H =

H2

are subject to independently and idcnticMly distributed errors with zero mean and common error covariancc real.fix A up to a factor of proportionality, where A = Ra.l~

is the Gholesky factorization of A (Rzx lower triangular). One cau easily verify that the cohmms in the transformed matrix R ~ I . H have an error covariance matrix equal to the identity matrix up to a Factor of proportionality. One way of carrying out the identification would then consist in having the identification based on the SVD of R~ 1.H (with ~ consistent computation of the left singular basis, tThe matrix SH in this set imposes weights on the different equations. This does not influence the outcome if the set of equations can be solved exactly, which is particularly the case under the assumed conditions.

473 see the previous section) instead of H , and including some kind of a re-transformation with R.A in ordcr to compensate for the tirst transformation with RX ~. The overall identification schelne would then deliver a consistent estimate. However, if Ila is singular or ill-conditioned, file matrix inverse RX 1 should not be computed explicitly. Instead, one should make use of the q u o t i e n t s i n g u l a r v a l u e d e c o m p o s i t i o n ( Q S V D ) of the matrix pair (H, Ra) (which in the non-singular case indecd reduces to the SVD of RX1.H). We can now show how a d o u b l e Q S V D ideutification scheme can be designed, analogously to the d o u b l e O S V D scheme for the white noise case (the latter being a special case of the former where every single QSVD reduces to an OSVD, as can easily be verified). The QSVD of (H, R a ) is defined as

X~.H.QH = EH Xt.RA.Q~a = ERa where

En = diag(c~l,...,ot2//+~,,,;) Ena = diag(/3,,...,/3zu+2,,,i) OLI

O~2

0121i+2,,,,

/3-7 > ~ >"" >/32,,+~,,,, Much like it was done for the white noise case, where the intersection of the row spaces of H1 all H2 was computed making use of the directions of minimM oriented signal energy

intersection making use of the directions of nfinimal oriented signal-to-noise ratio, viz. [ X' t'°2b1e d e f i n e d n e x t ' A ' 2 2 It is instructive to first consider the noise free ease (error eovarianee proportional to A, but with a zero factor of proportionality), and then demonstrate that the derivations still hold if fimre is a non-zero error contribution. If the data are noise free, then from the above QSVD definition, it follows that

It=[ = =

]112 Ha x-'.x~.c2;,

[x. x~]-' ['.Zl~01 X2, X22

"

0

0

.Q~t

dim(X,,)

= (mi +li) × (2mi + n)

di,,~(X,~)

=

(.,i + U) × (2U - .,~)

=

(mi + li) × (2mi + ,,)

dim(X22)

= (mi + li) x (21i - n)

dim(Ell)

=

dim(Xn)

(2mi + n) x (2m.i + n)

474

Again, from X~.Itt

= - X t , vg~.

it follows that the row space of X ~ v H 1 equals the required intersection . As X~vtIt contmns 2li - n row vectors , only n of which are linearly independent (dimension of the intersection), it remains to select n suitable combinations of these row vectors. Making use of a QSVD, one can easily show that

X'z = [X~12) X~22) X~)] . [ l(u-'0x(t'-'0

]

C.×,,

.T. t

O(II-~QxlH-,,) =

.

s.×,,

.7. ~

[(li-n)×(li-n) C

=

diag(cl,...,c,)

S

=

diag(sl,...,s,, )

L,×.

=

C 2 + S~

Clcarly, only X}~)dclivers uscful combinations for the computation of tile intersection, and we can take W = -'k12 Y(2)'.111

Note that in the white noisecase, this last QSVD reduced to a CS-decomposition and could then be computed from a single SVD, resulting in an overall double SVD scheme for the computationof the intersection. In the genera[ case, the conlputationof this intersection is carried out in a double QS VD scheme. In the second step, the system nlatrices can be computed fronl the following reduced set of equations (obtained after discarding the common orthogonal factor Q1i' ) [ X[:)'.X-t(m+l+l:(i+ 1)(m+/),1:2mi+,,).~i~ ] X - t ( m i + li + m + 1: ( m + l)(i + 1),1 : 2mi + n.).En _

[ ][ A C

B D

A12 .A t ~ : m i + l i , l:2mi+n).Zll X - t ( m i + li + 1 m i + li + m , 1 : 2 m i + n.).~H

]

where X - t ( v : s , v : w) is a submatrix of X -t at the intersection of rows r , r + 1 , . . . , s a n d c o h t l l l n s v , v -}- 1 , . . . , w. If thcre is some i n a c c u r a c y o n tile m e a s u r e l n e n t d a t a in H, and H2, all quotient singular wdues in Sn are non-zero. Again, one should set the smallest quotient singular values equal to zero, in order I.o obtain the ulodcl that is least conflictiug with the data and a suit.able deLerminafio,1 of this lower rank is to be carried out Mong t.he lines of the total least squares approach to approximate modeling. It remains to show that the above identification schemc delivers c o n s i s t e n t results if the number of colu,nns in H tends to iufiuity, and if the columns in H are subject to independent and identically distributed errors with zero mean and common error

475

covariance matrix equal to A, up to a factor of proportionality. For that case, it can again be shown [43] that the matrix X in the QSVD can be computed consistently. As the system matrices are next comI)uted essentially from X only (the matrix ZH in the above set of equations again imposes weights that do not influence the solution in the considered case, see section 2), the model estimate is clearly consistent.

6

Conclusions.

System theory can provide a unifying fi'amework for the study of many complex problems in and outside engineering. Thereby it creates a unique platform for the cxclmnge of concepts, tools, and software among different engineering disciplines. IIence it can stimulate the interaction and comnmnication among specialist and lead to a holistic approach of problems. It also leads to a parsimony of concepts, tools and software wl,ich is very valuable in educatio,t. The engineering approach to systems and models is usually not the deductive one from general to specific. On the contrary one usually starts with the simplest nmdel or the specification at a high level and one gradually refines alxd further specifies the systems. In this paper, it is claimed that the singular value decomposition and the quotient singular value decomposition have a great potential for systctn theory and signal processing in much the same way as the F F T had a great impact on digital signal processing in the seventies and eighties. Several applications were presented or referred to. The benefits of using the (quotient) singular value decomposition are most pronounced in those applications : • where essentially rank decisions and the computation of the corresponding subspaces determine the complexity and parameters of the nmdel • where numerical reliability is of crucial importance and the potential loss of numerical accuracy is to be avoided. • where a conceptual framework, such as the notion of oriented signal-to-signal ratio, may provide unrevealed additional insight, such as in factor-analysis-like problems. * where the problem can be stated directly in terms of the (quotient) singular value decomposition, which leads immediately to a reliable and robust solution, such as in a canonical correlation analysis environment. Moreover, in most engineering applications the number of measurements or the data acquisition poses only minor organisational problems (although the design of a measurement set up causes considerable efforts). The cost of the sensors however increases with higher accuracy and signal-to-noise requirements. In this enviromnent the (quotient) singular value dccomposition is the optimal bridge between linfitcd measuremcnt precision and robust modeling.

476

References [1] Willems J.C., "Dissipative dynamical systems, part I General Theory; part lI Linear systems with quadratic supply rates" Arch. Rational Mech. Analysis Vol. 45 pp. 321-351; pp. 352-393, 1972. [2] Willems J.C., "System theoretic models for the analysis of physical systems", Richerche di Automatica,Vol. 10, no. 2, pp. 71-106, 1979. [3] Willems J.C., "From times series to linear systems, parts I, II" Automatica, Vol. 21, 1986, pp. 561-580 and pp. 675-694, Vol. 23, 1987, pp. 87-115. [4] Anderson B.D.O. and Vongpanitlerd S., "Network analysis and synthesis", Prentice Hall, Englewood Cliffs 1973. [5] Willsky A.S., "Relationships between digital signal processing and control and estimation theory", Proc. IEEE, Vol. 66, no. 2, pp. 996-1027, 1978. [6] "Matlab Manual", The Mathworks, Mass. 1985. [7] Mees A.I. and Sparrow C.T., "Chaos", IEE Proc., Vol. 128, Pt. D, No. 5, Sept. 1981, pp. 201-205. [8] Sugarman R. and Wallich P. "The linfits to simulation." IEEE Spectrum, p. 36-41, April 1983. [9] Chua L.O., Komuro M., and Matsumoto T., "The double scroll family Part I, II", IEEE Trans. on Circuits and Systems, Vol. CAS-33, Nov. 1986, pp. 1072-1097, pp. 1097-1118. [10] Chua L.O., "Dynamic nonlinear networks : State-of-the-art", IEEE Trans. on Circuits and Systems, Vol. CAS-27, pp. 1014-1044, Nov. 1980. [11] Abraham R. and Shaw C., "Dynanfics - the geometry of behaviour, part 1,2,3" Tile visual mathematics library, Aerial Press, Santa Cruz, CA, 1985. [12] Van Dooren P., "Numerical linear algebra : All increasin~ interest in linear system theory.", Proc. ECCTD The Hague, 1981, pp. 243-251. [13] Staar J., Wemans M. and Vandewalle J., "Comparison of multivariable MBH realization algoritlnns ill the presence of multiple poles, ~nd noise disturbing the M arkov sequence", in "Analysis and Optimization of Systems", ed. by A. Bensoussan and J.L. Lions, Springer Verlag, pp. 141-160, Berlin, 1980. [14] Staar J. and Vandewalle J., "Numerical implications of the choice of a reference node in the nodal analysis of large circuits", Int. Journal of Circuit Theory and Applications, Vol. 9, pp. 488-492, 1981. [15] Staar J. and Vandewalle J., "Singular value decomposition : A reliable tool in tile algorithnfic analysis of linear systems.", Journal A, Vol. 23, pp. 69-74, 1982.

477 [16] Vandewalle J. and Staar J., "Modelling of linear systems : critical examples, problems of numerically reliable approaches.", Proc. IEEE Int. Syrup. on Circuits and Systems, ISCAS-82, Rome, pp. 915-918, 1982. [17] Vandcwalle J., Vanderschoot J. and De Moor B., "Source separation by adaptive singular value decolnposition.", Proc. IEEE ISCAS Conf. Kyoto 5-7 June 1985, pp. 1351-1354. [18] Vandcrschoot J., Callacrts D., Sansen W., Vandewallc J., Vantrappen G., Janssens J., "Two methods for optimal MECG elimination and FECG detection fl'om skin electrode signals.", IEEE Trans. on Biomedical Engineering, Vol. BME-34, no. 3, pp. 233:243, March 1987. {19] De Moor B., Vandew,'dle J., "Non-conventional matrix calculus in the analysis of rank deficient Hankel matrices of finite dimensions.", System and Contr. Lett., Vol. 9, pp. 401-410, 1987. [20] De Moor B., Vandewalle J., "An adaptive singular value dcconq)osition algorithm based on generalized Chebyshev recursions." in : Mathematics in signal processing, T.S. Durrani, J.B. Abbiss, J.E. Hudson, R.N. Madan, J.G. McWirther and T.A. Moore (ed.), Clarendon Press-Oxford, 1987, pp. 607-635. [21] De Moor B., Staar J. and Vandewalle, "Oriented energy and oriented signal-tosignal ratio concepts in the analysis of vector sequences aat time series.", in "SVD and Signal Processing" E. Dcpretterc (ed.), North Holland, ] 988, pp. 209-232. I22] Van Huffel S., Vandewalle J., "The total lea~t squares technique : computation, properties and applications.", in "SVD and Signal Processing" E. Dcprcttcrc (cd.), North Holland, 1988, pp. 189-207. [23] Vandcwalle J., De Moor B., "A variety of applications of singular value decomposition in identification and signal processing." in "SVD and Signal Processing" E. Deprettere (ed.), North Holland, 1988, pp. ,13-91. [2,1] Callaerts D., Vandewalle J., Sansen W. and Moonen M., "On-line algorithm for signal separation based on SVD." in "SVD and Signal Processing" E. Deprettere (ed.), North Holland, pp. 269-276, 1988. [25l De Moor B., Moonen M., Vandenberghc L. and Vandewalle J., "Identification of linear state space models with singular value decoml)OSil,ion using canonical correlation concepts.", in "SVD and Signal Processing" E. Dcprcttere (ed.), North Holland, pp. 161-169, 1988. [26] De Moor B., Vandewalle J., Moonen M., Van Mieghem P. and Vandenberghe L., "A geometrical strategy for tlle identification of state space models of linear multivariable systems with singular value decomposition.", Preprints 8th IFAC/IFORS Symposium o11 identification and system parameter estimation, Beijing, August 27-31, 1988, pp. 700-704.

478

[27] VandewMle 3 , "Trends in the need of mathematics for engineering and the impact on engineering education.", SEFI, Proc. 5th European Seminar on Mathematics in Engineering Education, Plymouth, March 23-26, 1988, pp. 94-105. [28] De Moor B., Mooncn M., Vandenberghe L. and Vandewalle J., "The application of the canonical correlation concept to the identification of linear state space nmdels', in A. Bensousan, J.L. Lions, (Eds) Analysis and Optinfization of Systems, Springer Verlag, Heidelberg, 1988, pp. 1103-1114. [29] Van Belle H. en Van Brussel H., "Inteiding tot de systeemtheorie_ Pleidooi voor een ruimere toepassing.", Ilet IngenieursbIad, no. 12, 1979. [30] Autonne L., "Sur les groupes lin~aires, rdelles et orthogonaux", Bull. Soc. Math., France, Vol. 30, pp. 121-133, 1902. [31] De Moor B., Golub G.H., "Generalized singular value decompositions : A proposal for a standardized nomenclature." Internal Report, Department of Computer Science, Stanford University, January 1989 (submitted for publication). [32] De Moor B., Golub G.H., "The restricted singular value decomposition : properties and applications." InternM Report, Department of Computer Science, Stanford University, March 1989 (subnfitted for publication). [33] Deprettere Ed. (Editor), "SVD and Signal Processing : Algorithms, Applications and Architectures", North Holland, 1988.

[34]

Doyle J.C. "Analysis of feedback systems with structured uncertainties." Proc. IEE, Vol. 129~ no. 6, pp. 242-250, Nov. 1982.

[35] Eckart G., Young G., "The approximation of one matrix by another of lower rank.", Psychometrika, 1 : 211-218, 1936. [36] Fan M.K.H., Tits A.L., "Characterization and efficient computation of the structured singular value.", IEEE Trans. Automatic Control, Vol. AC-31, no. 8, August 1986, pp. 734-743. [37] Fernando K.V., IIammarling S.3., "A Product Induced Singular Value Decomposition for two matrices and balanced realisation.", NAG Technical Report, TR8/87. [38] Paige C.C., Saunders M.A., "Towards a generalized singular value decomposition.", SIAM J. Nunler. Anal., 18, pp. 398-405, 1981. [39] Takagi T., "On an algebraic problem related to an analytic theorenl of Caratheodory and Fejer and on an allied theorem of Landau.", Japan. J. Math., 1, pp. 83-93, 1925. [40] Van Loan C.F., "Generalizing the singular vMue decomposition." SIAM J. Numer. Anal., 13, pp. 76-83, 1976.

479

[41] Zha II., "Restricted SVD for matrix triplets and rank determination of matrices." Scientific Report 89-2, Berlin (subufitted for publication). [42] Damen A.A.II, Van den Ilof P.M.J., Hajdasinski A.K., "Approxi,nate realization based upon an alternative to the Hankel matrix : the Page matrix.", Syste,ns and Control Letters, Vol.II, No. 4, pp. 202-208, 1982. [43] De Moor B., "Mathematical Concepts attd techniques for modelling static and dynanfie systems", Doct. Diss., K.U.Leuven, 1988. [44] Golub G.H. and Van Loan C.F., "Matrix computations.", North Oxford Academic Publishing Co., Johns Hopkins University Press, 1983.

[45] Golub G.H., Van Loan C.F., "An analysis of the total least squares problem.", SIAM J. Numer. Anal., Vol. 17, No. 6, pp. 883-893, 1980. [46] Kung S.Y., "A new identification and model reduction algorithm via singular value decomposition.", Proc. 12th Asilomar Conf. on C,ircuits, Systems and Computers. Pacific Grove, pp. 705-714, 1978. [47] Moonen M., De Moor B., Vandenberghe L., Vandewalle J., "On- and off-line identification of linear state space models", International Journal of Control, Vol. 49, No. 1, pp. 219-232, 1989. [48] Van IIuffel S., "Analysis of the total least squares problem and its use in parameter estimation.", Doct. Diss. K.U.Leuven, 1987.

[49]

Van Iluffel S., "Analysis and properties of the generalized total least squares problem A X .~ B when some or all columns in A are subject to error.", SIAM J. Matrix Anal. Appl., to appear 1989.

[50] Zeiger tt.P. and McEwen A.J., "Approximate linear realizations of given (limensions via IIo's algorithm.", IEEE Trans. Aut. Control, vol. AC-19, (pp. 153), 1974. [51] Vall den Hof P., "On residual-based parametrization and identification of multivariable systems", Doer. Diss. T.U.Eindhoven, 1989.

Stochastic Realization Problems J. H. van Schuppen C e n t r e f o r M a t h e m a t i c s a n d C o m p u t e r S c i e n c e ( CWl ) K r u i s l a a n 4 1 3 , 1 0 9 8 SJ A m s t e r d a m , t h e N e t h e r l a n d s

The stochastic reahzation problem asks for the existence and the classification of all stochastic systems Ior which the output process equals a given process in distnbution or almost surely. This is a lundamental problem of system and control tl',,~ory. The stochastic realization problem is of importance to mode/ling by stochastzc systems in engineering, biology, economics etc. Several slochastic systems are mentioned lor which the solution el the stochastic realization problem may be useful. As an example recent research on the stochastic realization problem for the Gaussian lacier model and a Gaussian factor system is discussed. This paper is dedicated to J. C. Wfllems on the occasion of his fiftieth birthday.

I. INTRODUCTION The purpose of this paper is to introduce the reader to stochastic re',dization theory. This will be done by presentation of a verbal introduction, a survey of Gaussian stochastic realization theory, formulation of open stochastic realization problems, and a discussion of the stochastic realization problem for Gaussian factor models. This tutorial and survey-like paper is written for researchers in system and control theory, but may also be of interest to researchers dealing with mathematical models in engineering, biolo~', and economics. The Kalman filter and stochastic control algorithms have proven to be very useful for those control and signal processing problems in which there is a considerable amount of noise in the observation processes. Examples of such problems are:" minimum variance control of a papcr machine, access control of communication systems, and prediction of water levels. The solution of stochastic control and filtering problems depends crucially on the availability of a model in the form of a stochastic system in state space form. There is thus a need for modelling and rcalization of noisy processcs by stochastic systems. Stochastic realization theory addresses this modelling problem. 8),stem and control theory is the subject within enginccring and mathematics that deals with modelling and control problcms for dynamic processes or phenomena. Such a phenomenon may

481

initially be described by specifying the observation process or trajectories, which description will bc termed the external deacription. For reasons of modelling and control it is often better to work with an internaldescription. The form of such an internal description depends on the properties of the observation process. For deterministic linear systems it may be a description in state space form. The state of such a system at any particular time contains all information from the past necessary to determine tile future behavior of the state and output process. For stochastic systems the internal description is a stochastic system in state space form. Here the state is that amount of information that makes the past and the future of the observations and the state process conditionally independent. For a vector valued random variable one may consider the internal description of a Gaussian factor model, see section 5. For models of images and spatial phenomena in the form of random fields, other internal descriptions are needed. The realization problem of system theory can then be formulated as how to determine an internal description of a model given an external description. Motivation for this problem comes from engineering, in particular from system identification and signal processing, from biology, and from econometrics. In these subject areas one may want to estimate parameters of the internal description from observations. The question should then be posed whether these parameters can be uniquely determined from the observations, that is whether they are identifiable. This question may bc resolved by solution of the realization problem. First one must impose the condition that the model is minimal in some sense. The concept of minimality will depend on the class of internal descriptions. Secondly, there is in general no unique internal description for a phenomenon given an external description. The realization problem therefore also asks for a classification of all minimal internal descriptions that correspond to a given external description. Such internal descriptions may be called equivalent. Once the equivalence class has been determined one may choose a canonical form for it. From that point on standard techniques from system identification and statistics may bc used to determine the internal description of the model. The part of system theory that deals with modelling questions is referred to as realization theoo,. It treats topics such as transformations between representations, parametrization of model classes, idcntifiability questions, and approximate modelling. A brief description of this paper's content follows. Section 2 contains a verbal introduction to the modelling procedure of system theory. In section 3 a tutorial is presented on Gaussian stochastic realization theory. Several examples of stochastic systems for which the stochastic realization problem is open and relevant for engineering and economics, are mentioned in section 4. As an example the stochastic realization problem for the Gaussian factor analysis model is discussed in section 5, and for Gaussian factor systems or error-in-variables systems in section 6. 2. MODELLING

AND SYSTEM THEORY

2.1. Introduction As identified in the previous section there is a need for stochastic models of engineering and economic phenomena. The purpose of this section is to describe the modelling procedure of system and control theory. Particular attention will be devoted to modelling of economic processes.

2.2. The modelliagprocedure It is assumed that data, possibly in the form of time series, arc available for the modeller. It is well-recognized that useful data are easy to obtain in the technical sciences but hard to obtain in economics. One reason is that economics is not a laboratory science: c-:i:criments arc often

482

impossible or if possiblc cannot be rcpcatcd. Also data gathcring is mudl morc cxpensive in economics than in the tcchnical sciences. The objective of modclling is to obtain a model for a phenomenon that is realistic and of low complexity. A model is called realistic if its observed behaviour is in close agreement with the phenomenon. A mcazure of fit for this agreement has to be formulated. The term low coJtplexit), should be considered as in ordinary use. A mathematical dcfinition of this term is very much model dependent. Models of high complexity are mathematically not well analyzable and compurationally not feasible. The two modelling objectives mentioned arc conflicting. Therefore a compromise or trade-off between these objcctives is necessary. The preferred modelling procedure consists of the following two steps: selection of a modcl class; selection of an elcment in the model class involving the above mcntioned trade-off. This procedure must be applied in an iterative fashion. If the selected element in the model class is not a realistic modcl then the model class may be adjusted, ql~e two steps of this procedure will now be discussed separately. -

2.3. Selection of a model class In the selection of a model class one has to kccp in mind the objectives of a realistic model and a model of low complexity. The selection procedure demands application of concepts and results both from the research area of the object to be modelled, and from system and control theory. The formulation of realistic economic models is difficult for several reasons. One reason is that economic transactions involve multiple decisionmakers compared with a single decisionmaker in most engineering problems. The appropriate mathematical models are therefore game and team models and their dynamic counterparts. The status of dynamic game and team theory is not yet at a level at which a body of results is available for applications. A second reason, closely related to the first, is that a dccisionmakcr must also model the decisionmaking process of the other decisionmakers. This remark is well-known in the literature on stochastic dynamic games. The discussion about rational expectation also illustrates this point. A third reason is that the rules of the cconomic process change quickly compared with the periods over which economic data are available. Assumptions of timc-invariance or stationarity are often unrealistic. In systcm theory a formalism has been developed for the formulation of mathematical models of dynamic phenomena and for a modelling procedure. For a dynamic phcnomcnon in the form of a time series a prcfcrrcd deterministic model is called a d),namic system in state space form. One distinguishes h~puts and outputs of such a system, and a state process. The state of a dynamic system at any particular time is that amount of information that together with the future inputs completely determines the future outputs. Tile trajectories of the input, output and state process arc tile basic objccts of a dynamical system. The reader is referred to [78] for material on linear systems. Stochastic systems have proven to be useful models in several areas of engineering such as signal processing, communications and control. Within economics they are used for example in connection with portfolio theory. In stochastic system theory, probability theory is used as a mathematical model for uncertainty. A stochastic system is specified by a measure on the space of trajectories. This is a fundamental difference betwcen deterministic and stochastic systems. For a stochastic system without inputs the state at any particular time makes the past and the future of the output and state processes couditionally independent. Despite the fact that a stochastic system is specified by a measure, tile representation in terms of trajectories, for example by a stochastic

483

differential equation, is crucial to the solution of control and filtering problems. Why are stochastic models realistic in certain cases? Within economics reasons for this are that such modelling involves: aggregation over many dccisionmakers; uncertainty over future actions of other decisionmakers; uncertainty in the measurement process, due to vague definitions and averaging. Remark that the costs involved often prevent the gathering of full information. Therefore aggregation must be used. The variability of the data then suggests a stochastic model. This author is not optimistic about the applicability of stochastic models to economic phenomena. Reasons for this are the relatively short time series and the frequent change in structural relations. Should one use a deterministic or a stochastic model class to model a certain phenomenon? What is needed is a criterion to decide whether for a specific phenomenon the class of deterministic systems or that of stochastic systems is the appropriate model class. A crucial observation from system theory is that the choice of model class is all-important. Of course, a model must be realistic and of low complexity. But within these constraints there is left some freedom in the mathematical formulation of the model. Given this freedom it is advisable to choose a model class for which the motivating control problem is analytically tractable. An example of such a choice is the Gaussian system that leads to the Kalman filter. Filtering theory was formulated by N. Wiener and A. N. Kolmogorov for stationary Gaussian processes. 1',. E. Kalman restricted attention to a particular class of stationary Gaussian processes, those generated by linear stochastic systems driven by white noise. For this class of systems the solution of the filtering problem has proven to be straightforward. That this class may be extended to include nonstationary processes is then a useful corollary. How is this observation to be used in economic modelling? As suggested by R. E. Kalman, a detailed study must be made of economic models that arc published in the literature to see whether changes in the mathematical formulation of these models are advantageous for the solution of control problems. The selection of the model class seems a creative process that involves knowledge of both the research area of the phenomenon to be modelled and of system theory. For stochastic processes indexed by the real line the model class of stochastic systems seems an appropriate model. See section 3.1 for a definition of this concept. F o r a vector of random variables the model class of Gaussian factor models may be useful, see section 5. For random ficlds it is not yet clear what the appropriate model class should be. Once the model class has been determined, the modelling procedure prescribes the solution of the stochastic realization problem. In section 3 this problem is formulated and the solution shown for the case of Gaussian processes. -

-

2.4. Selection o f an element in the model class Given the data and the model class, the problem arises of how to sclect an clcmcnt in the model class. As indicated earlier, the selection of a model is a trade-off between the objective of a realistic model and the objective of a model with low complexity. For deterministic dynamical systems results on the selection of an element in the model class arc reported in [35, 791. For stochastic systems a formalism for the selection of an clement in the class of stochastic systems is described below. Consider first a measure of fit between the observations of the phenomenon and the external bchaviour of a stochastic system. Recall that the observations consist of numbers while the external behaviour consists of a mcasure on the sample space of observation trajectories. The way to proceed is to use the observations, the numbers, to estimate the

484

measure on the sample space of observation trajectories. In case this measure is Gaussian and the observation process is stationary it suffices to estimate the mean and covariance function of this measure. One can define a measure of fit between the measure for the output trajectories estimated from observations and the measure associated with the external description of the system. Exampies of such a measure are the Kullback-Lciblcr measure and the Hcllingcr measure; scc section 3.7. For stochastic systems one also needs a measure of complexity. A stochastic complexity measure introduced by J. l,Lissanen [60-64] seems the appropriate tool for this purpose. Stochastic complexity is based on A. N. Kolmogorov's complexity theory. Since this subject is wcU covered elsewhere the reader is referred to the indicated references. The actual selection procedure given data, a model class, and measures of fit and complexity, consists then of a combination of analysis and numerical minimization. The details of this will not be discussed here. 3. GAUSSIANSTOCHASTICREALIZATION The purpose of this section is to present the modelling procedure for Gaussian processes. In this tutorial part of the paper results for the Gaussian stochastic realization problem arc summarized. For a reference on the weak Gaussian stochastic realization problem see the book [24] and for a shorter introduction in the English language [23]. For a survey of the strong Gaussian stochastic realization problem scc [471. Notation The following notation is used. N = { 0 , 1 , 2 , . - - }. Z ~ ={1,2, . . . }. Z = { - - - , - 1 , 0 , 1 , " - }. Zk = { 1,2, .. • ,k }. R denotes tim set of real numbers, and R ~ =[0, oo). For a probability space (~2,F , P ) consisting of a set f/, a a-algebra F and a probability measure P, denote L ~ (F) = {x :~2--*R ~ I x is a random vatqable measurable with respect to F}. x E G ( O , Q ) denotes that the random variable x has a Gaussian distribution with mean zero and variance Q. For a stochastic processy :~2× T o R k the following notation is used for the a-algebra's generated by the process P:~'= F~' = a(~), (s),Vs <~t }) and F}' ' = o({y (s),Vs ~ t }).

DEFINITION 3.0.1. 7he a-algebra's F i, F 2 are called conditionally indcpendcnt givo~ the o-algebra

c~f E [ z l z 2 I G] = E [ - l l G ] E [ - z I G] f o r all z i E L ' (I.'~). 7he notation (F1,F2 [ G ) G C I will be used to denote that F I, F2 are conditionally ind~Tendent given G amt CI will be called the conditional independence relation.

485

3.1. Stochastic s~,stems and GaussiaJ~ O,stems The purpose of this section is to define stochastic dynamic systems. Attention is restricted to discrete-time stochastic dynamic systems. Stochastic systems with inputs will not be considered here. A motivation for the definition of a discrete time stochastic dynamic system follows. Consider the object that is usually called a stochastic system, Xt ~ ] = A X t + M v t ,

(3.1.1)

x0,

(3.1.2)

.Yt ---- Ca't + Nvt,

where x o : ~ 2 - * R n , x o ~ G ( m o , Q o ) , v:f~×T--->R" is a Gaussian white noise process with vt ~ G (0, 1"3, 1'.xo , F ~,, arc independent o-algebras, A 62R n×", ~ . I ~ R n×m, C E R p×n, N ~ H P ×m, x:~2 × T--,W' and y :~2× T---,RP defined by the above equations. It may be shown that this object is equivalent with the object specified by: Xo ~ G ( m o , Qo);

(3.1.3)

IAx']

E[exp(iu'txt~,+iw'llv,'IFir'"

u

vF,'.-l]=cxp(i[,u]'r[Cx,j-½[w]TS[w]),

u

(3.1.4)

for all t G T and some S GR (n ~p)×qz ~p). Observe that the conditional characteristic function of (xt ~ I,y~) given (Fi~ vF~' 1) depends only on the random variable x,. It then follows that E[cxp(iuY'xl ~ l +iw'ryl)[Ft ~ v F ~ ' ~ l ] = E[exp(iurxt ~ i + i w t ) ' , ) l F ~'1

(3.1.5)

for all t E T. A stochastic dynamic system could now be defined as a state process x and an output proccssy such that for all t G 7" there is a map xt ~ distribution of(x, , j, )5) ]'his definition may be found in [42; p. 5]. Below a different definition will be adopted. It may be shown that (3.I.5) is cquivaIcnt with the condition that for all t G T ( ~ " v l - ' } ' ',F}'

ivr{

IF~')~CI,

where /'7 ' = o ( { x , , V s > ~ t } ) , I.~~ = o ( { x a , V s ~ t ) ) , and similar definitions for I.T~,F.} ' . The property that the past and future of the state and output process arc conditionally independent given the current state will be taken as the definition of a stochastic dynamic system. Dm:lNrrtoN 3.1.1. A discrete-time stochastic dynamic system is a collection o = {[2, I',P. 7, Y, B r , A , B x o , , x } , who'e {~2, !", P } is a comFlete flrobabilitv sflac~;" T --Z, to be calh'd the time index set; ( Y. B r) is a measurable space to be calh'd the output space; ( X, B x) is a measurable space, to be called the state space; y :~ × T--~ Y is a stochasticprocess, to be called the output process; x :[~ X T--~ X is a stochastic process, to be ealh, d the state process; such that for all t C T

486

(I':;'" vFi" + ,I"T::l v l : i ~- I F X ' ) e C L A stochastic dynamic system on T c 7 dctloted by SX.

(3.1.6)

is defined analogot~ly. The class o f stochastic systems is

The above definition of a stochastic dynamic system is based on related concepts given in [48, 52, 72, 731. From the definition of a stochastic dynamic system one obtains that the state process satisfies the condition (F~ ~,F't~

[F'~')ECI

for all t ET. This is equivalent with x being a Markov process. Markov processes are thus als0 stochastic dynamic systems, and the latter class thus contains the classical model of state processes. The defining condition of a stochastic dynamic system is more or less symmetric with respect to time in the past and future of the state and output process. This is an advantage over the asymmetric formulation given in the representation (3.1. I) and (3.1.2). The condition (3.1.6) is asymmetric with respect to the output process, q]ais is a convention. A priori thcre arc four possible conditions for a stochastic dynamic system which are listed below: (F~'~ v F - t r - ' , F J ' - v F ' t '-

IF~')~CI V t G T ;

(3.1.7.1)

(I':~"~1 v F t ~ ' ,F~t'L-1Vl:t~ - ]F'~') C C I V t E T ;

(3.1.7.2)

(F}' : v F [ ~,F~'_-i v F t ~- I F : " ) E C 1 V t E T ;

(3.1.7.3)

(F~' ,'I v l : i ~ ~ ,F~'- v F t ~- [ F " ) E C I

(3.1.7.4)

V t ~7:

Condition (3.1.7.1) and a property of conditional expectation imply that I':'" C(FJ' ' V / ' ~ ' - ) C F x'

which fact is not compatible with the intuitive concept of state in that the output is in gcncral not part of the state. Condition (3.1.7.2) is not suitable because it would allow examples that arc counter-intuitive to the concept of state, see example 3.1.6. The conditions (3.1.7.3) and (3.1.7.4) thus remain, of which condition 3 has been chosen. This is a convention. Condition (3.1.7.4) results in the representation xt ~ 1 -----Axt + Mvt, Yt ~1

=

C.~t + Nvt,

which form is inconsistent with the system theoretic convention of (3. I. I & 3.1.2). The option of taking condition (3.1.7.3) or (3.1.7.4) in the definition of a stochastic dynamic system is related to the option of considering Moore or Mealey machines in automata theory, see [50; I.A.2]. The definition of a stochastic system is formulated in terms of o-algebras rather than in terms of stochastic processes. This is a geometric formulation in wlfich emphasis is put on spaces and subspaccs rather than on the variables or processes that generate those spaces. DEFINITION 3.1.2. Given a stochastic dynamic system o --- (~LF, P,T, Y, B y , X , B x , y , x } G S E .

487 This system is called: a. stationary or time-invariant/f(x,y) is ajoint(y stationmyprocess; b. Gaussian/f Y=R/', X = R " for certainp, n~Tt ~, B r = B P and B x = B n are Borel o-algebras ml Y respectivel), X, and if (x,.v) is ajointl.v Gaussian process; by way of abbreviation, a GausMan stochastic d),namic system will be called a Gaussian system and the class of such systems is denoted by GSY; c. finite/f Y, X are finite sets and B r , B x are the o-algebras on Y, X generated b.e all subsets; by wa), of abbreviation a finite stochastic dynamic system will be called a finite stochastic system and the class of such sr,stents is denoted by FSE. PROPOSITION3.1.3. Consider a collection {~,F,P, T, Y,I~y ,X, B x.y,x } as defined in 3.1.1 but without condition (3.1.6). "I71efollowing statements are equivalent: a. for all t ~ T (F[ + v F i x ~ ,F}'FI VF?x- ]F~')~CI; b.

for all t G T ( F r ' v F x''',F}' I v F i ~

c.

Fx')ECI;

for all t G T

(/::~" vF-; '-,1;-'" 'vl.'"

lF'X')eCl.

The following result is a useful sufficient condition for a stochastic dynamic system. PROI'OSlTION3.1.4. Consider the collection o = {~2,F,P,T,Y, Bv, X, Bx,y,x } as defined in 3.1.1 but without condition (3.1.6). l f for all t E T 1.(F}",FL

vFI' 71 ]F'X')EC1;

2.(F~~ +,F}x v/':[ ~ I F ~ ' ) ~ C l ; then a E S E . Below two examples of stochastic dynamic systems are presented. EXAMPLE 3.1.5. Consider a Gaussian system representation xt ~1' = Ax t + Mvt,

(3.1.8)

.l"t = Cxt + Nvt,

(3.1.9)

with the conventions given below (3.1.1 & 3.1.2). As indicatcd thcrc tiffs representation is equivalent with E[exp(iuTxt i 1 +iwl)'t)l Fit'- VF}' I I

488

L xl for all t E T a n d x 0 GG. This and a property of conditional independence imply that (FX'4'vFY',F[ ' -1 vF'r~ - [FX') ~ C I , q t ~ T ,

and from 3.1.3 then follows that, with x,y specified by (3.1.8 & 3.1.9), o = {~,F,P, T, Re,BP,[~n,Bn:~,,x} E S Z .

From properties of Gaussian random variables follows that (x,y) is a jointly Gaussian process, hence o is a Gaussian system or o ~ G S Z . In the following (3.1.8 & 3.1.9) will be caUed a forward representation of a Gaussian system. EXAMI"LE 3.1.6. Let v :~ × T---~R be a standard Gaussian white noise process. Definey :~2× T ~ R , x :~ × T--,R by A'r "-~ vt -I, .)'t = Xt -[- I'r = vt. 1 -4- vt.

Then the following hold. a. For all t G 7" (F~' ?I,F[' !1 I N ) E CI, where N C2F is the trivial o-algebra. Thus file processy is the output process of a stochastic dynamic system according to (3.1.7.2) with a trivial state

spacc. b.

For all t E T E[exp(iuyl) I F.~' ~1]

is nondeterministic, indicating that the process), has some kind of memory. c. (1,:~'' vF~, ~ ,/~:~'-1 v F ~ * - I F x ' ) E C I for all t e T, hence o = {~,F,l',7, Y,B,X,B:y,x} EGSZ.

3. 2. Fo/~'ard and backward representations of Gauss&n s~,stems The purpose of this subsection is to show that a Gaussian system has both a forward and a backward representation, and to derive relations between these rcprcsentations.

PROPOSITION 3.2.1. Let o = {12,1",P,T,~P,Be,R",B",)',x} ~ G S ~ be a Gaussian s},~tem. Assume Q: T ~ n " ×", Q (t) = E [x,xlr] >0. a.

that

for

all

t GT

E[xt]=O, ED,t]=O

and

that

The Gaussian s.ystem has what will be called a forward representation given by xt ~ 1 : A f(t)xt + IL,lvf, Xo, yt = c f ( t ) x t -~ Nv(,

(3.2.1) (3.2.2)

489 where vf:12X T--,R n ' k is a Gaussian white noiseprocess with intettsity vf. Given o then ~If(t) = E[xt ~ lx1"]Q(t)

1,

C:(t) = ED,txf] Q (t)- l,

[Q(t+ I)E[x,+ty;r]]fAf
"f(t)= [E[j,tA-T~.I] E[YI.Yt T] J-

[cf(t)J~l" '

M = (I, O ) ~ R "x(n it'), N = (0 Ie)E~Rex(n bt'>.

b.

Conversely, given a forward representation with ALes, V f M, N functions anct x,y ct,~nect by the above forward representation (3.2.1 & 3.2.2), then o is a Gaussian system. The Rivets Gat~sian system has also a backward representation given b), xt

1 = At'(t)xt + Mv~, x o,

(3.2.3)

)'t

I = ct'(t)xt + Nv~',

(3.2.4)

where v t' :~25( T.-oR n +g is a Gaussian while noiselJt'ocess with in tet~sit): V t'. Given o

I,

(3.2.5)

ixlrlQ(t) " w,

(3.2.6)

Al'(t) = Eix, -ix~']Q(t) Ch(t) = Elf',

=

,I

j-

l "c,>J

'

M = (!, 0), N = (0 17).

c.

d.

Conversell,, given a backward representation with A t',C t', Vt',M,N attd x O, as defined b.v the above backward representation, lhen o is a Gaussian system. The relation between the forward and backward t'cTreseatation of a Gaussian system is given by A f ( t ) Q ( t ) = Q(t + I)(At'(t + 1)) ~',

(3.2.8)

Cb(t)Q(t) = c f ( t - 1)Q(t - l)(Af(t - l)) "r + N V f ( t - I)M r,

(3.2.9)

c f ( t ) Q ( t ) = Ct'(t + l)Q(t + 1)(At'(t + 1)) "r + NVb(t q- I)M 7'.

(3.2.10)

Assume that the given Gaussian s),stem is stationary. Thetz A f C f Vf, A t,, Ct,, Vt,, do not depend e.~ficitl.y on t ~ET" and Q ( t ) = Q G[t n×', Q = Q r > O . The relation betweo~ the forward and backward representatioJt is then g,iven by A f = Q(Ab)'rQ -I,

(3.2.11)

A t, = Q ( A f ) T Q --1,

(3.2.12)

C I, = C f Q ( A f ) ' r Q -I + N V f M ' r Q - - I .: CfA t, + N V f M T Q - I,

(3.2.13)

c f -_ ChQ(At,)TQ - 1 + N V b M ' r Q - t = Ct, A f + NVI, M r Q -l

(3.2.14)

in the following the superscriptsfand h will be omitted when it is clear from the context which representation is referred to.

490 3.3. Stodtastic obselwability and stochastic recotLstructibilitJ.,

The theorem on the characterization of minimality of a stochastic realization makes use of the concepts of stochastic observability and stochastic reconstructibility. Below these concepts are introduced. DEFINITION 3.3. !. Consider a stochastic system o = {~2,F,P,T, R P , B P , R n , B n , y , x } E S E . a.

This system is called stochastically observable on the interval {t, t + 1 , . . . , t + t ~} if the map t~

xr ~ E[cxp(i y. usl)'l +s) l/'x' ] a~0

from 3ct to the conditional characteristic function o f {Yl,.rt i ]. . . . . . t'1 ~t, } given x, is O~jective on b.

the support o f xt. Assume that the sl,stem a is stationa.ty. Then it is called stochastically observable i f there exists a t, t] E T , O < t l < ° % such that it is stochasticallr, observable on the htterval { t, t + I, . . . . t + t z } as defined above. Br' stationaritv tbis then holds for all t @ T.

The interpretation of a stochastically observable stochastic system is that if one knows thc conditional distribution of {3',0'1 ~] . . . . . . r't ~t, } given xt, then one can uniquely determine the value of x,. Note that the conditional distribution of 0', . . . . . . v, ~,, } given x, can in principle be determined from measurements. PROPOSITION 3.3.2. Consider the Ga~sian system o = {~2,F,P,T, R t ' , B P , R " , B " o , , x } ~GSV-, with forward representation Xt * I :

A (t)x t +

Mvt,

Yt = C(t)xt + Nvt, with vt EG(O, V(t)). a. The srstem a is stochastically obsetvalde on { t,t +1 . . . . . t q- t l } iff

(3.3.0 LC(t + t i)'l~(t + t l,t)J

iff tl

rank( ~_~ C(t +s),b(t +s,t)fb(t + s , t ) T c ( t + s ) T) = n. J--U

b.

Assume that the systenl is stotioua O' with forward representation Xt

, I =

Axt

-t-

Mvt,

)'t --- C& + Nvt,

491 with v t ~ G (0, V). Then this system is stochastically observable i f f

CA rank

(3.3.2)

= n. "tl --]

DEFINITION 3.3.3. Consider the stochastic system o = {~2,F,P, T, R e , B e , R ' , B ' o , , x a.

} ESZ.

This system is called stochastically rcconstructiblc on the interval {t - 1,t - 2 . . . . . t - t l} /f the map tl

x, ~ Elcxp(i :~] u'~t),t _AIF ~'1 ,i-=]

b.

is injective on the support of x t. Assume that the s))stem is stationary. Then it is called stochastically reconstructible tf there exist t, t l ~ T , 0
PROPOSITION 3.3.4. Cotlsider tile Gaussian system o = {[2,I,P,T, R e , B I ' , • ' , B ' , y , x }

EGS y

with backward representation Xt )'t-I

I : A (t)Xt + M v t , :

C ( t ) x t -F Nvt,

with vt ~ G ( O , g(t)). a. The system o is stochastically reconstructible on the interval { t - l,t - 2 . . . . . t - t i } iff

1 ~s.s)

[c(, -,, ),z,(, -, i,,)J iff Ii rank ( ~_~ C ( t -s)c~(t -s,t)dP(t - - s , t ) T c ( t --s) T) = n. s--I Assume that the sl,stem o is stationa O, with backward representation Xt

I ---- Axt + M y ,

.)'t

l = C x t "b Nl't,

492

with vt (~G(0, IO. Then it is stochasticall), reconstructible i f f

ral|k

. ..

=

It.

~.4)

Note that the condition (3.3.2) is expressed in terms of the matrices (A,C) of the forward representation of the Gaussian system and the condition (3.3.4) is expressed in terms of the matrices (A,C) of the backward representation. See section 3.2 for the way the matrices of the forward and backward representation are related. 3.4. The weak Gaussian stochastic reafization problem Attention is again directed to the problem of modelling by a stochastic system. So, one is given a measure on the observed process that has been estimated from the data. One is asked to determine a stochastic system in the model class such that the measure restricted to the observation process equals the given measure. PROBLEM 3.4.1. The weak Gaussian stochastic realization problem f o r a stationa O" Gaussianprocess is, given a stationa O, Gaussiau process on T = Z takh~g values bz (R e, B t') hav#~g mean value function zero and covariance function W: T--~RP xe, to solve the following subproblen~. a. Does there exist a stationart, Gaussian syMem o = (~,F,P,T, Re,BP,R',B",y,x} ~GSZ

b.

such that the output process.v o f this s)'stem equals the given process in disttqbutiou. This means that these processes have the same f a m i O, of finite dimeasional disttqb,ttios~s. Effectively this means that the covariance funetion o f the output process must be equal to the given covariance function W beeaase both processes are Gaussian. I f such a s.ystem exists, then one calls a a weak Gaussian stochastic realization o f the given process, m; i f the context is known, a stochastic realization. Classify all minimal stochastic realizations o f the given process. A weak Gaussian stochastic realization is called minimal if the dimension of the state sl)ace is miaimal. The following subprohlems mt~t be solved: I. characterize those stochastic realizatious that are mbffmal; 2. obtahz the classification as such; 3. h~dicate the relation between two minimal stochastic realizations; 4. produce an algorithm that constructs all mhfimal weak Gaussian stochastic realizations of the given process.

In problem 3.4.1 one is given a stationary Gaussian process with zero mean value function. Such a process is thus completely charactcrizcd by its covariance function. In part a. of this problem the question is whether the given process can be the output of a stationary Gaussian system. Because by definition such a Gaussian system has a finite-dimensional state space, not all stationary Gaussian processes can bc the output process of a Gaussian system. The question should therefore be interpreted as to determine a necessary and sufficient condition on the given process, or its

493 covariancc function, such that it can be the output process of a Gaussian systcm. In part b. of problem 3.4.1 a classification is asked for. This question arises because a stochastic realization, if it exists, is in general nonunique. This will be indicated below. The dimensions of the state space of two stochastic realizations may also be different in general. For system theoretic reasons, such as identifiability, one should restrict attention to those stochastic realizations for which the dimension of the state space is minimal. Such a realization is called minimal. In general minimal stochastic realizations are also nonunique. A classification of all minimal stochastic realizations is then useful for the solution of the identifiability question. The above defined problem is related to the problem of determining spectral factorizations of the spectral density of the given process. Below a notation is used for the parameters of a time-invariant finite-dimensional linear system of the form x ( t + I) = A x ( t ) + Bu(t), ),(t) : C x ( t ) + Du(t), with U = R"', X = I~", Y = Re, u : T---~U, x : T ~ X . y : T--, Y. The notation is then pls = {p,n,m,A,B,C,D} E L Z P . In the formulation of thcorem 3.4.2 use is made of the set Q ~ . The definition of this set is given in subsection 3.5. THEOREM 3.4.2. Consider the weak Gaussian stochastic realization l)roblem f o r a stationa O, Gauasian process asposed in 3. 4.1. Assume that lira W(t) = 0 and that W(O) >0. a.

There exists a weak Gaussian stochastic realization o f the given process i f f there exists apls = {l~,n,p, F, G, t t , J } E L Z P with J = J "r such that l t F t. IG, ift >0, W(t)--: ~2/, ift =0, G'r(F'r) t l t l r , / f t < O .

[

(3.4.1)

(a function having the form (3.4.1) will be called a discrete-time Bohl function; the right hand side of(3.4.1) will be called a covariance realization of the covariance functim~ W.)

iff

fV(a)=

~

IV(t)X-I,I

(3.4.2)

t~Z

b.

is a rational function. The dimension n in the eovariance realization (3.4.1) is also called the McMillan dcgrec o f the covariance fimction. A weak Gaussian stochastic realization is minimal iffit is stochastically obsetwable and stochasticatlr recottstructible. A mhffmal weak Gaussian stochastic realization is nonunique in two ways. I. l f pgs I = {p,n,m,A, C, M,N, V} E G S Z P are the paramaers of a forward rcTresentation o f a minimal stochastic realization, and i f S E a " ×'' is nons#tgular, then pgs z -- {p,n,m, S A S - J, CS - 1,SM, N, V} ~ G S Z P are also the iTarameters o f a forward repre~etatation ~f a minitnal stochastic realization. 2. Fix the parameters o f a minimal covariance realization as given in a. above,

494

pls = (fl, n,p,I~, G, ll,J } ELZPmin. Denote the parameters of a forward representation of a minimal Gaussian stochastic reali. zation by (p,n,A,C, V} and the set of such parameters by WGSRPmin. Define the classification map c#s: Q#s .-.9 WGSRPmin, cea(Q ) = ~,n,A,C, V},

(3.4.3)

~y X =r, C =H, V = V(Q)=

d.

rQ - F Q F r G - F Q H T ] [ G T _ I f Q F r 2 J _ H Q t l T j.

Then, for fixed pls CLYPmi n is Cpts a bijection. 77ms all minhnal weak Gaussian stochastic realizations are classified by the elements of Q~-~. Tbe stochastic realization algorithm as defined in 3.4.3 below is well defined and constructs all minimal weak Gaussian stochastic realizations.

ALGORITHM 3.4.3. The stocllastic realization algorithm for weak Gaussian stochastic realizatim~s of stationao' Gaussian l)rocesses. Data: given a stationa(T Gaussian process with zero mean value function and covariance function W: T--oRP xp. Assume that the condition of 3.4.2.a. holds. 1. Determine a minimal eovariance realization of W via a realization algorithm for thne-invariant finite-dimensional linear st,stems, or pls = {fl, n,p, F, G, H,J } E L Z P rain, such that

l i F t IG, tft ;> 0, W(t) = JZI, tft =0, GT(F ~') , IllT, ift
(3.4.4)

[

2.

For algorithms for this step see books on linear system theoo,. Determine a Q G Q ~ , or a Q E K" x,, satisfying Q = Q "r~0,

[Q

- F Q F T G - F Q I I "r ] G "r- IIQF "r 2.1 >i0.

3.

-tlQlfrJ

(3.4.5)

Let A = F, C = 11, M = (In 0 ) ~ H nx(n ~P), N = (0 I p ) ~ R px(n-~,°),

v

=

[Q-FQF T -~(Q)--- [G, nQF~

a - b ' Q n "c ] Z I _ H Q l l . r ] E R (" ~e)x("+P),

cottstruct a probability space by ~ = (R(,, ~p))T, F = IIT ® B (" ~e) v:~2×T~R~"~e), v(o~,t) = w(t), P:F-~[0,1] a probabilit], measure such that v is a Gaussian white noise process with httensity V, x : ~ X 1"---~ n y :~2X T--.R p defined by xt 41 = A x t + Mvt, x

)'t = Cxt + Nvv

~o = 0 ,

(3.4.6) 0.¢7)

495

The/! a = {~,F,P,T, R e , B e , R " , B " , y , x } E G S Z

(3.4.8)

is a minimal weak Gaussian stochastic realization of the given process, meaning that the output process y is a Gaussian process with covariance function equal to the given covariance function W.

A mistake that is sometimes made is the following. Consider the following forward representation of a Gaussian system Xt ~ 1 = A x t + M v t , yt = Cxt + Nvt,

with vt EG(0, V). A statement is that if the pair of matrices ( A , M V ~) is a reachable pair and if (A, C) is an observable pair, that then the stochastic realization described by the above system representation is a minimal realization of the output process. This statement is false as the following example shows. EXAMPLE3.4.4. Considcr the Gaussian systcm a = {~,F,P, 7; R,B, R , B , y , x } E G S ~

with forward representation )Q ~ 1 ~ a x t -]- bvt, .)'t = xt "~ Vt,

with vt EG(0, 1), a ~ ( - 1. + I), a=/=0, b : ( a 2 - l ) / a . a. Then (a,b) is a reachable pair and (a, 1) is an observable pair. b. The system o is a nonminim,I realization of its output process. It is possible to interpret certain stochastic realizations as a Kalman filter but this will not be done here. For a refcrencc see 1241. The implication of the weak Gaussian stochastic realization problem for the idcntifiability qucstion is illustratcd by the following example. EXAMPLE3.4.5. Considcr the time-invariant Gaussian system o = { ~ , F , P , T , R , B , R , B , y , x } ~GSY.

with forward representation xr ~ l = a x 1 + (l O)v,

(3.4.9)

C'X't +

(3.4.10)

)'t :

(0 l)vt,

with vt EG(0, IO, (3.4.11)

496

Consider the asymptotic Kalman filter for the Gaussian system (3.4.9 & 3.4.10)

.~t ~ 1 = aS'~ + k(rt-ca'~),

(3.4.121

-it = .vt - J,'t,

(3.4.13)

in which 7:f~ × T - o R is a Gaussian white noise process with ~t C G (0,r). This asymptotic Kalman tilter may be rewritten as (3.4.14)

;c, , ~ = a.~, + k-i, = a.~', + (10)vKt),

(3.4.15)

Yt = c~', + -vt = c.~', + (0 l)vl(t), in which v t:~2 × T ~ R z is a Gaussian white noise process with v l(t) @ G (0, V l), =

(3.4.16)

1 r(kl).

From these forward representations one deduces that (3.4.9 & 3.4.10) and (3.4.14 & 3.4.15) are both weak Gaussian stochastic realizations of the output processy. This may be verified by computing the covariance function of the output process. This example shows that one may not be able to uniquely determine the parameters of the noise process of a Gaussian system, here (3.4.1 I) and (3.4.16), from the covariance function of the output process. For results on the parametrization of Gaussian systems see [34]. Attention has also been devoted to the partial weak Gaussian stochastic realization problem in which one is not given a covariance function on all of T = Z but only on a finite time set, say T = ( -- t l, -- t I + 1. . . . . -- 1,0,1 . . . . . t x}- The motivation for this problem is that in practice one can estimate from a finite time series only the covariance function on a finite time set.

3.5. The dissipation matrix htequality In subsection 3.4 it has been stated that the minimal weak Gaussian stochastic realizations are classified by the set Qp~. In this section the set Qe)~ and its dual Qvt, will be considered. Throughout this section J = J r. The results of this subsection may be found in [23, 24]. DEtqNmON 3.5. I. Let pls = {p,n,p, I'; G, H,J } E L E P with J / > 0 and

[o_ - F ' Q r 1t Qets = { Q G n " x " I Q = Q

"r>~O, V ( Q ) =

r"Qo]

[ I I _ G , r Q F 2J_G,,.QG I>~O },

(3.5.1/

and for l~S = {p, n,p, F T, H 1",G T,j } E LY~P

Q~-u2 = { Q E N " X ' I Q = Q

"1">~0, V ( Q ) =

[O.-FO.F' [G.r

O-FO_N T ]

HQF. r Z / _ t / Q / / r j ~ 0

}.

(3.5.2)

I'ROI3L~M 3.5.2. GIVENpIS @LY~P AND Qpl~. a. Classify all elements of Q~t,. b. Determine an algorithm that constructs all elements of Qpt~I'ROPOSITION 3.5.3. Consider l)ls = (j),n,p,F, G, ll, J } ~ L ~ P m i n and Qels. Assume that Qpt,=/=O, Then QI'/~ is a convex, closed and bounded set, and there exists a Q , Q + EsQoI~

attd that J > 0 .

497

such that for any Q e Q et~, Q- ~ Q ~ Q +

DEnNmON 3.5.4. a. The regular part of Q?ta is defined as Qe'~.~ = {Q e Q # ' l 2 J - G r Q G > O } . b.

The set Q#~ will be called rcgular if Q#, = Qp&r. F o r Q 6 R " X " with Q = Q r a n d 2 J - G T Q G > O d e f i n e D(Q) = Q - F T Q F - [ t I r - F ' r Q G ] [ 2 J - G ' r Q G ]

c.

I[ItT--FTQG] "r.

(3.5.3)

Correspondinglr define

o~.,.

= (Q

D(Q)

=

eo,~ 122- t l Q I t T > O } ,

Q --FQF "r - [G--FQItT][22--ltQitTI--'[G--FQItT] "r,

and Qt-~ is regular if Q ~

(3.5.4)

= {~l~,r.

PitOPOStTION3.5.5. I_x'tl)ls = (p,n,IJ, F,G, II,J } CLY.P. Let Q ER nx', Q =QT. Assume that 2.1 - G TQG >0, and let

a.

(.+.5.5) Then

02 2 - G r O G] = T't V(Q)T,

(3.5.6)

and V ( Q ) = T T[ D(Q) 0 1 2.1 _ G T Q G T- ],

b.

(3.5.7)

where V(Q) is as defined it2 3.5. I. Assume that 2.1-G'rQG>O. Then V(Q)~O iff D(Q)>~O. Also V(Q)>0/ffD(Q)>0. fact, rank (V(Q)) = rank (D(Q)) + p.

In

e.

Qpt,.r = {Q E R ' x "

IQ =Q 'r>~O,2.I --GTQG>O, D(Q)~0}.

Notation for the boundary of Q#, will be needed. Tile following notation will be used in the sequel,

x'QTQx] j ,

(3.53)

B(Q,¢) -- {S EH"×" I IIS-QII2~<¢}.

(3.5.9)

110112 =

Is

UlL~e,.,~u

xT x

DEIqNITION3.5.6. l.ctph ~ l, Xl' and consider Qpls. Define the boundaly of Qt,/s as the set

498 OQat~ = ( Q E O p ~ IV c E R , c>O, ] S E B ( Q , Osuch that S = S "r, S@Q, S q~Oet, },

and the interior ofQpls as the set int (Op,~) = Qpt~ Fl(OQp/s)c.

PROPOSITION 3.5.7. Letpls = {p,n,p,F, G,H,J } ~ L E P . a. b.

Q ~ a O e t ~ i f f V ( Q ) issingutar. QEmt(Qpts) iffV(Q)>O. Assume that Qets is regular. Then Q E~Qt,t~ iff D ( Q ) is sh~gular; and Q Eint(Qpa) iff

D(Q)>0. DEFINITION 3.5.8. Let pls = {p,n,p,F, G,H,J } G L X P aad coasider Octa. Theset of singular boundary points of Q/,ts is defi, edas

a.

~Qet.,.~ = { Q E ~Qet, I rank ( V ( Q )) = rank (2.I --G T QG) }.

b.

The set of singular boundary points of the regular part of Ql,ls is defined as OOet~.r.~ = { Q EQt, t,.r A OOt,t, [rank(V (Q)) =p }.

TnEOga~ 3.5.9. Let pls = {p,n,p,F, G,H,J } ~LYPmi n. Assume that QetC/= ~ and that it is regular. Let F - = F -- G [ 2 J - G ' r Q

G] I [ H r - - F Q -G] r.

Then Q - + AQ EOlJt~ and AQ>O iff

1. 2.

AQ ER '~x", AQ >0,' (AQ) -~ - F-- ( A Q ) I ( F - ) T - G [ 2 J - G r Q -

3.

for some S ~ R sp(F-)CC-.

TM, S

GI- ZG r - S = O,

(3.5.10)

=s'r>~o;

3.6. 77re strong Gaussian stochastic reaK-atioa problem

PROBLEM 3.6.1. The strong Oaussian stochastic realization problem for a stationa O, Gaussian process is, given a probability space (~~,/', P), a tone index set T = Z and a stationary Gaussian process z: I2 >( T---,RP having zero mean value function and covariance function W: T - , R e xt', to solve the fob lowing subproblems. a. Does there exist a stationaQ, Gaussian s~'stem o = (~LF, P,T,~e,Be, R " , B ' , y , x } ~GSY~ with forward representation xt i ! ~ Axt + Mvt. Xo, )'t = C-x't + Nvt, such that

499

b.

1. y t = z t a.s.foralltET," 2. FX'cF~oforallt~T. I f such a system exists then one calls o a strong Gaussian stochastic realization of the given process, or, if the context is known, a stochastic realization. Classify all mhlimal stochastic realizations of the given process. A strong Gaussian stochastic realization is called minimal if the dimemion of the state space is mh~imal.

The difference between the weak and the strong Gaussian stochastic realization problems is that the given process and the output process of the Gaussian stochastic system are equal in the sense of the family of finite-dimensional distributions respectively equal in the sense of almost surely. For the strong Gaussian stochastic realization problem this requires flint the stochastic system is constructed on the same probability space as the givcn process. Therefore the state process has to be constructed from the given process, and this explains condition 2 of problem 3.6. l.a. For a survey of the strong Gaussian stochastic realization problem the reader is referred to the paper [47]. 3. 7. Pseudo-distances on the set of probability measures The purpose of this subsection is define distances on the set of probability measures as a preparation for the approximate stochastic realization problem to be discussed in the next subsection. DEFIN ITION 3.7.1. Let X be a set. A pseudo-distance is a function d: X X X - o R such that

I. 2.

d(x,y)>~Ofor allx,y EX; d(x:v) = 0 / f i x =y.

If a pscudo-distance is not symmetric then one may construct its symmetrized version. A pseudodistance need not satisfy the triangle inequality. DV.FINI'rION3.7.2. Let F2~ = {_f:R ¢ - o r I f c C 2, f ( 1 ) =0, Vx E(O, oo), f ' ( x ) > 0 } . DEFINITION3.7.3. Given a measurable SlmCe (P.,F), let P : (P:F--~R ~ ] P is aprobabili O, measure }. For f EFz~ define the pseudo-distance df:P X P---,R on the set ofprobability measures P on (f~, F) by t" 1

I" 1

d f ( P b P 2 ) = EQ[f (-~2 )rz] = E,,2If('~'-2)l wha'e Q is a o-finite measure ou (~2,F) such that dP I dP 2 l " j < O with ~ = ri, P 2 < O with ~ = r2. The pseudo-distance df is also called the f-information measure, the f-entropy or the f-divergence. A o-finite measure Q as mentioned above always exists, for example Q = P I -I-P2 will do. In case ( B , F ) = ( n , B ) one may sometimes take Q to be Lebesgue measure. Because r 2 > 0 a.s. P2 the

500 above expression is well defined. The above definition has been given in [ 1]. PROPOSITION 3.7.4. //].

a. b.

The function d/ defined in 3. 7.3. is a pseudo-distance. The pseudo-distance df does not depend on the choice of the o-finite measure Q.

DEFINITION 3.7.5. Tire Kullback-Lciblerpseudo-distance is defined as d/~ :P × P---~R with

x ~>0, f l : R +--->R, : l ( x ) = ]JO,In(x)' x = 0 ,

df,(PI,P2) :

r I

i" I

/,.'p:Lfl(--~2-2 )] :

r1

EQlfl(-~22)r21= EQ[rlln(~2)l(r:>O)].

DEFINITION 3.7.6. The Hcllingcr pseudo-distance is defined as df , :P × P--,R with

f 2 : R , - - * n , f2(x) = ( ~ x -

1)2,

d:,(P,,?2)= E,,,I(V7~/~2 -I) ~] = Zrel(V:~-,- V7~-2)21. The Hellinger pseudo-distance is symmetric. Consider the set of functions on T = Z with values in R k. Let P be the set of Gaussian measures on this space that make the underlying process a stationary Ga--ussian process with zero mean value function. An expression for the Kullback-Leibler pseudo-distance on this set was derived in I431. PROPOSITION 3.7.7. Let P I,P2 be m'o probabifi O, measures on the set of functioas &fined on T = Z with values in R k. Asstone that these measures are such that the underlying process is Gaussian, sta. tionaty, has zero mean value function, and covariance functiott~ W1, W2 respectively. Moreover, assume that these covariance functioas admit spectral densities W1, 14:2 respectivelr, and that they sati.~e condition C of[43]. Then the Kullback-Leibh, rpseudo-distaace is given b.y the e~q~ression

dx/.(Pl ,Pz) = ~

"=f[ tr( (VI l (h)[ 17V2( h ) - I~, (~)]) - In (~/i 1(~.)t~2(A)) ]dh.

3.8. The approximate weak Gaussian stochastic realization problem How to fit to data a model in the form of a Gaussian system'?. In engineering, in biology and in economics there are many modelling problcms for which an answer to this question is useful. As indicated in section 2, from data one may estinlate a measure on the set of observation trajectories. In case that one models the observations as a sample function of a Gaussian process, one may estimate its covariance function. Suppose further that one wants to model the observations as the output process of a stationary Gaussian system. Such a system has a finite-dimensional state space. In theorem 3.4.2 it has been shown that a covariance function has a stochastic realization as a Gaussian system only if it has a covariance realization as indicated or if it is rational. Now an arbitrary covariancc function obtained from data may not correspond to such a c0variance function. Therefore one has to resort to approximation.

501

The approximate stochastic realization problem is then to determine a stochastic system in a specified class such that the measure on the output process of this system approximates the measure on the same space determined from the data. Attention below will be restricted to the class of stationary Gaussian systems with dimension of the state space less or equal to n ~ Z +. As a measure of fit the Kullback-Leibler pseudo-distance will be taken as mentioned in subsection 3.7. A measure of complexity will not be considered here; it may be based on stochastic complexity as indicated in section 2. PROBLEM 3.8.1. Approximate wcak Gaussian stochastic realization problem. Let y r denote the set of thne series defined on T = Z with values #z R IJ, and let p ( y T ) denote the set o f probabili.tv measures on y T. Given is a Gaussian measure P o E P ( Y T) such that the underlying process corresponds to a stationa O, Gaussiau process with zero m~an function. Given is also an integer n E Z ~ and let GSX(n ) be the set of Gaussian systems with state space dimension <.u. Solve the optimization problem itfo E as'x(,,) dKL (P o, P(o)) where dKL is the Kulllmck-Leibler pseudo-distanee on the set o f probability measures on p ( y'r), and P (o) E l ' ( y T ) is the probability measure on y T associated with the Gaussian system o ~ GS 3~(n ). As indicated in 3.7.7, if the pseudo-distance on the set of Gaussian measures is the KullbackLcibler measure then the pseudo-distance may be expressed as a pseudo-distance on the set of covariance functions dKL(Po,P(o)) ---- d l ( W o , W(a)) where W0 is tile covafiance function associated with the Gaussian measure P0 and W(o) the covariance function associated with the Gaussian measure P(o). Note that tile ¢ovafiance function W(a) is a rational function with McMillan degree less or equal to n because it corresponds to a Gaussian system of state space dimension less or equal than n. The approximate weak Gaussian stochastic realization problem may therefore be considered as an approximation problem for a covariance function. In this problem the approximant W(o) has to be a rational function of McMillan degree at most n while the given covariance function W o may neither bc rational nor of finite McMillan degree. The approximate stochastic realization problem 3.8.1 is unsolved. Approaches along three different lines have bccn investigatcd. Approach 1. Given any pseudo-distance d j, problem 3.8.1 can be reformulated as an approximation problem for covariance functions with the criterion d i (Wo, W(o)) where W u is the covafiance function associated with the Gaussian measure Po and W(a) the covarJancc function associated with the Gaussian measure P(o) related to a ~ GSZ. PROBLEM 3.8.2. Given a covalqance function W o: T--oRP ×e solve info ~_aSX, d l ( Wo, W (a)).

502

The pseudo-distance d I on thc set of covariancc functions may be taken to be the Hankel norm or the H-infinity norm. Possibly the L2-norm is suitable. The above problem may be rephrased as, given a not necessarily rational covariancc function, to determine a rational covariance function that approximates the given covariance with respect to an approximation criterion. Note that a function is a covariancc function iff it is antisymmetric and a positive definite function. It seems that a Hankel norm approximation of a covariance function is not itself a covariance function. The positive definiteness of a covariance function is therefore an essential constraint. References on this approach are [28, 29, 31, 38, 51, 65]. There is a related approach in which one first determines a spectral factor of the given covariance function and then a rational approximation of the spectral factor. This approach seems too restrictive to start with, although it may be the solution to some approximation criterion. Of course, given any rational approximation of the covariancc function one will still have to determine a state space realization for it.

Approach 2.

By analogy with the approximate prediction problem for finite-dimensional Gaussian random variables, algorithms have been proposed for the approximate weak Gaussian stochastic realization problem.

ALGORI i'HM 3.8.3. LET BE GIVEN A COVARIANCE FUNCTION 14/0.

I.

Solve an approximate prediction problem. Fix t E 7". Let

y~(t)=

. ....

v-(t)=

....

The variance of the pair (~ ~ (t):y (t)) may be computed from the eovariance function W0. Let n E Z ~. Determine a matrix S ~ N n x~ such that with x(t) = Sy - (t) the following prediction criterion is minimized

ilfseu,*, tr( E l ( y ' 2.

(t)-E[),' t(t)[ F~(n])0,,

~(t)--E[y

' (t)lF'~(0])7' l ).

Determine a Gaussian system via regression by proceeding as follows,

y(t) J

x(t) + v(t), v(t)~G(O,V),

where fx (t + 1)) v(t)=

f[.xy( (t t+)l I -

[AI x(t)"

Finally, replace the Gaussian process v with a Gaussian white noise process w with variance V. The above algorithm in a somewhat different form appeared first in a paper of H. Akaike [3]. Other references arc [ ! l, 12, 44-46.75, 76]. These papers differ mainly in the way they perform step l of the above algoriflim. For canonical correlation analysis and the prediction problem see

503

[27,57]. It is not clear in what sense the Gaussian system determined in step 2 of the above algorithm is a good approximation to the given Gaussian process. In other words, the approximation criterion, although inspired by the static approximate prediction problem, is never mentioned. The replacement of the process v by a Gaussian white noise process is also unmotivated. Approach 3. Canonical correlation analysis for finite-dimensional Gaussian random variables has been generalized to infinite-dimensional Hilbert spaces in [36,37,49]. One has investigated approximate prediction problems for time series by canonical correlation analysis techniques. Approximation bounds have been derived I30]. It remains to be seen whether this approach is useful in practice. Approach 4. Inspired by the above mentioned second approach to the approximate weak Gaussian stochastic rcalization problem yet another approach has bccn formulated. This approach has been worked out by M. St6hr at the Centre for Mathematics and Computer Science. The following results up to the end of section 3 are due to M. St6hr and are as of yet unpublished.

NOTATION 3.8.4. Let k r,k 2, n E Z ~ , k = k r + k 2. Recall that G (0, Q) denotes a Gaussian measw'e, saF on R k, with zero mean and variance Q. For Q E R ~ x~ the deeon~osition

for, qi,] o = ion; Q,2J will be used in which Q i1 E R k ' xk,, Q22 ER~~ xk:, at,d Q 12 c r Y ' xk~, Let

O(,t) = {Q ~ R k X k l Q ---Q'r>~0, rank(Q~2)~n}. PROBLEM 3.8.5. The static approximate weak Gaussian stochastic realization problem. Given are k b k 2 , n E T a , k = k l +k2, and a Gaussian measure G(O, Qo) with Q 0 = Q ~ ' > 0 . lx't dh'L be tire Kullback-Leiblerpseudo-distance on the set o f Gaussian measures on R k. Solve infc(o.e4, e~co~,,) dKL(G(O, Qo),G(O, Qj)).

One may interpret the above problem in the light of approach 2 indicated above. Associate tile space R k' with the past of the observations, and the space R k' with the future of the observations. The Gaussian measure G (0, Qu) may then be associated with that derived from the data. In problem 3.8.5 one is asked to determine the measure G (0,Q I) with Q I 6~Q(n). The latter condition implies that the dimension of the state space associated with G(0,Q l) is less or equal to n. Therefore the essential constraint on the dimension of the state space is taken care of. PROPOSITION 3.8.6. Consider problem 3.8.5. The Kullback-Leibler measure o f two Gaussian meastires G (0, Q a) atrd G (0, Q 1) on • k is given by the expression dKL(G(O, Qo),G(O, QI)) = IA[ t r ( Q ( IQ0 ) - ln(det(Qi-rQ0)) - k ] k

= l/z[ ~ (~i(Qo,Ql) - ln(~i(Q0,Qi))) - k ], i-I

504

where {Ai(Q0, Q l),i ~Zk} are the generalized eigenvalues of Qo with respect to Q j, here d~fincd as the zeroes ofdet (Q 1~ - Q 0) = 0. It can bc shown that the gcneralizcd cigcnvalues arc real and satisfy ~i(Qo, Q 1)I>0, for i @Z k. NOTATION 3.8.7. For Qo ERkxk, Q o = Q ~ > 0 , n EZ+ let f~. E R 5 1:1 O ~EQ (n) such that ge, teralized eigenvalues } A(Q°'n) = ~ of Qo with respect to Q are (Xl . . . . . X,~.} "

andfor X E R ~] let Q ~O(n)lgeneraUzed eigenvalues Qs(O°'n'X) = ~ of O o with respect to a are (At . . . . . Xk}J-" k

f : N k~~ 0 t ~, f(X) = V21~] (X~-In(Xi))-k]. i=l

It may be shown that the functionfis convex. There are results on the structure of the matrices in the set Q,(Qo,n, ~). PROBLEM 3.8.8.

Considerproblem 3.8.5 and the notation 3.8. 7. Solve

i,f x ~_,",(cd~,,,) f (~). Suppose that there cxists a,k* EA(Q0,n ) such that

f(X*) = inf x~A((,o.,,~ f(A). The solution set of problem 3.8.5 is then given by Q~(Q0,n,),*). Note that problem 3.8.8 is the infimization of a convex function over the set A(Q0,n). The latter set is a cone. It is conjectured that it is a polyhedral cone. It may be shown that the optimal solution of problem 3.8.8 is such k

that ~. Ai = k. This property simplifies the functionf. If this constraint is taken into account then i--I

tile set A(Q0,n ) is rcduced to a shifted simplex. It is not yet known whether problem 3.8.8 admits an explicit expression as solution or whether one has to resort to numerical minimization. The hope is that tile solution of problem 3.8.5 provides information on the solution of the approximate weak Gaussian stochastic realization problem 3.8.1. 4. SPECIFIC OPEN STOCHASTIC REALIZATION PROBLEMS

The purpose of this section is to present several stochastic systems and processes for which the solution to the stochastic realization problem may be useful for engineering, economics etc. The prescntation of these models is brief. The tutorial and survey-like character of this paper may make it useful to mention these models.

505

Gaussian systems The approximate weak Gaussian stochastic realization problem, as describcd in subsection 3.8, is unsolved. For Gaussian systems there are unsolved problems for specific subclasses of systems that may be of interest to specific application areas. Some of these problems and models are described below.

The co-integration and the error correction model. As a model for economic processes that move about an equilibrium, C . W . J . Granger [32] has proposed a model that is known as the cointegratim~ model The components of a vector valued processy :[~ ×Z--->R t' are said to be co-hltegrated of order

1,1 if after dilrercncing once (~7),(t) =y ( t ) - y (t - 1)) the resulting process has a stationary invertible AutoRegressive-Moving-Average (ARMA) representation without deterministic component; 2. there exists a vector a E R k, a=/t:O, such that .'(t)=a't]V(t) has again a stationary invertible A R M A rcprcsentation without deterministic component. The interpretation of this model is that the economic process that is modelled consists of a trend and stationary fluctations, but is such that a linear combination of the process is stationary. The linear combination should be associated with some difference of economic processes, say income minus consumption. According to the model this differcnce fluctuates around some equilibrium value and it may be considered as forced towards this equilibrium by economic forces. A generalization of this model has bccn proposed, sce [22]. That paper also reports on the suitability of the co-integration model for economic processes. A vector valued proces y:12X T---,R k is said to have an error correction representation, see [22], if it can be expressed as: 1.

.4 ( a ) ( l - B ) y ( t ) =

- Vz (t - 1 ) + u ( t )

in which u is a stationary process representing a disturbance, A (.) is a matrix polynomial with

A(O)=I, B is the delay operator defined by B y ( t ) - - y ( t - 1 ) , there exists a a ~ R k such that z(t)=aT]y(t) and y E R k, 7:/=0. rl'hc intcrprctation of an crror corrcction model is that the disequilibrium of one period, = (t - I), is uscd to determine thc cconomic proecss in thc ncxt pcriod. F o r recent work on the co-integration and error correction model see a special issue of Join'hal ~fEconomic Dynamics and Control that is opcncd by the special cditor M. Aoki with the papcr [8]. In that issue there is another paper by M. Aoki [9] in which he shows that the co-integration model may be obtaincd from a Gaussian system representation under a condition on the poles of the system. In that approach a co-integration vector is not assumed, nor are assumptions needed on trends or periods. An approach to the stochastic realization problem for the co-integration model and the error correction model may be bascd on stochastic realization theory for a particular class of Gaussian systems.

Gaussian systems with iaputs. A timc-invariant Gaussian system with inputs has a forward representation of the form

x(t + 1) = Ax(t) + Bu(t) + My(t),

506

y ( t ) = Cx(t) + Du(t) + Nv(t), where u :~2× T ~ R " is an input process, and v :fl × T--,R k is a Gaussian white noise process. Such systems are used in stochastic control. The stochastic realization problem for this class of systems has not yet bcen treated. It is motivated by stochastic control theory. An unsolved question is whether such a stochastic system is a minimal realization of the measure on the observation processes of o u t p u t y and input u. The conditions for minimality should be related to the solvability conditions of the linear-quadratic-Gaussian stochastic control problem. For this class of systems one has also to investigate the stochastic realization problem associated with the solution to the linear-exponential-quadratic-Gaussian stochastic control problem [14, 77]. This solution is rclatcd to recent results in H-infinity theory.

The Gaussian factor model This model and the associated stochastic realization problem are discussed in section 5 of this paper. Factor s),stems These systems and the associated stochastic realization problem are discussed in section 6. Positive stochastic linear s),stems A stochastic systcm in which the state and obscrvations process take valucs in the vector space R + will be called apositive stochastic system. The gamma distribution is an example of a probability distribution on R ). Such systems may be appropriate stochastic models in economics, biology, and communication systems where the state variables arc economic quantities, concentrations etc. Examples from biology may be found in [56]. Several examples of such systems follow. Portfolio models. A portfolio model is a dynamic model for the growth of assets such as shares, bonds and money in savings accounts. After the fall of share prices in October 1987 there is a renewed interest in portfolio models. A stochastic portfolio model may be specified by dp(t) = ap(t)dt + p(t)dv(t),p(O), where p:I2XT--,R represents the price of the asset, a ~ R represents a growth trend and v:~2× T ~ R represents random fluctuations. More refined models can be defined to account for control of buying and selling, and for switch-over costs. A realistic portfolio model would require a realistic macro economic model for short-term and long-term economic growth, preferably on an international scale. The portfolio model should be seen as a special case of a growth model. In addition, growth models that exhibit saturation should be investigated in connection with market saturation effects. qlae rcalization problem for the stochastic portfolio model would have to deal with questions as whether the trends and variances of these models can be dctermincd from observed prices. This problem becomes more interesting if, for example, the price of a share is related to development of the markets in which the company is active, to its management structure, and to long-term growth of the economy.

507

The Gale model and a Lemztieff.9,stem. For production planning of firms a model proposed by D. Gale is used. For references on this model see the book by V. I. Arkin and I. V. Evstigncev [ 101. The classical Leonticll" model is a matrix relation between inputs and outputs of an economic unit. A dynanfic version of this model has been proposed, it will be called a Leontieffsvstem. The Gale model is specified by fx(t - 1)] J' x,),:T--,R'~

z(t) = ( ) ' ( t )

(4.1)

z (t) ~ Q (t),

(4.2)

y q) ~ x (t),

(4.3)

satisfying

where Q(t)~R2.," is a convex set. Here x(t - 1 ) is called the hynlt, and.y(t) the output in period (t - 1,t ], and z (t) the technologicalprocess at time t ~ 7". Condition (4.2) is a technological feasibility condition; condition (4.3) implies that the input at any time step cannot cxceed the output of the previous step. A parametric form of this model is given in subsection 1.1.8 of [10]. There is also a stochastic version of the Gale model, see the subsections 2.4.1 and 2.4.7 of U0]. Optimal control problems for the Gale model are treated in [10]. The results are maximum principles and turnpike theorems.

Finite stochastic sy.~tt,ms In section 3 a finite stochastic system has been defined. It consists of an output process taking values in a finite set and a finite-state Markov process. The stochastic realization problem for this class of systems is then to classify all minimal stochastic systems such that the output process of such a system equals a given process either in distribution or almost surely. The motivation of this problem comes from the use of finite stochastic systems as models for communication or computers systems. For such technical problems, stochastic models with discrete variables arise naturally or are useful approximate models. The stochastic realization problem was formulated in 1957 in a paper by Blackwcll and Koopmans [ 15]. During the 1960's several publications appeared that provide a necessary and sufficient condition for the existence of a finite stochastic realization. For references see [52]. Unsolved questions are the characterization of minimality of the state space and the classification of all minimal stochastic realizations. The main bottleneck is currently the characterization of the minimality of the state space. This question leads to a basic problem for positive linear algebra, that is, linear algebra over R ~.

Countingprocess systems An example of a counting process system is a continuous-time stochastic system of which the output process is a counting process with stationary increments and in which the intensity process of the counting process is a finite-state Markov process. The stochastic realization problem for this class of systems is unsolved. The motivation for this stochastic realization problem comes from the use of counting proccss models in communication, queueing theory, computer science, and biology. The observation process may oftcn be taken as a counting process with stationary increments. The above mentioned class of stochastic systems has been investigated in [68, 69]. The question of characterizing the minimal size of the state space is closely related to the same question for

508

the finite stochastic realization problem. Gaussian random fields

For this class of stochastic objects new mathematical models arc nccded. 5. FACTORANALYSIS In this section the stochastic realization problcm for the Gaussian factor analysis model will be formulated and analyzed. The factor analysis model was proposed early this century. For references on the factor analysis model see [7,74]. Factor analysis is used as a quantitative model in sociology and psychology. R. Frisch has suggested the factor analysis model as a way to determine relations among random variables 125]. R. E. Kalman has emphasized this modcl and formulated the associated stochastic realization problem 139-4 I]. Since then scvcral researchers have considered the stochastic realization problem for this model class. This problem is still unsolved. Below one finds a problcm formulation, questions, partial rcsults and conjectures for this stochastic realization problem. For recent publications on this problem see the special issue of J. of Econometrics that is opcncd by the paper [21. P1`obh,m formulatiot~

From economic data that exhibit variabilit 2" one may estimate a covariance. Suppose that this data vector may be modelled by a Gaussian random variable. Effectively one is thus given a Gaussian measure, say on R ~. The initial problem may then be stated as: how to represent this measure such that the dependencies between the components of the vector are exhibited? The factor analysis model will be used to describe these dependencies. DErZNIT~ON 5.1. A Gaussian factor analysis model or a Gaussian factor model is defined be the specification .y = H x + w,

(5.1)

Yi = l l i x 4- wi, i --- l,. . . ,k,

(5.2)

o1"

where x:f2---~R", x EG(O, Qa) is called the factor, w:[]--,R j', w G:G(0,Q,r ) is called the noise, y:~2--*Rk,) ' eG(O, Qy) is called the observation vector, H GR kx" is called the matrix of factor Ioadings, Q,. is a diagonal marl`ix, and (x, w) a1`e independent random variables.

The interpretation of tile Gaussian factor analysis modcl (5.2) is that each component of the observation vector consists of a systematic part Ittx and a noise part w,. Observe that the condition that Q,. is diagonal is equivalent to the condition that (wl . . . . . wk) arc independent random variablcs. A gcncralization of thc above definition may be given to the case in which Q,. is block diagonal. The Gaussian factor model in rudimentary form goes back to [67]. The Gaussian factor analysis model is equivalent to the cott[tuence analysis model introduced by R. l"risch [25]. In this model the representation of the observation vector is specified by y : u -f- w, Art --- O,

in which A E R (k ,,)x~- u,w arc independent random variables, and Q,. is a diagonal matrix. For

509

other references on tiffs approach see the publications of O. Reiersol [58, 59]. The Gaussian factor analysis model, or, equivalently, the confluence analysis model, has been suggested as an alternative to regression analysis. Strong pleas for this approach are the introduction of the book by R. Frisch [25], and the papers of R.E. Kalman [39-41]. Within economic and statistical literature the questions regarding regression and factor models have been recognized, sec for example [7, 66, 70, 80]. PROBLEM 5.2. The weak stochastic realization problem for a Gaussian factor model is given a Gausshm measure G (0, Q) ou It t" to solve the following sul~roblems. a. Determine a Gaussiatt factor model, say .y = l l x + w, such that the measure o f y equals the given measure or yG-G(O, Qv) = G(O,Q). l f such a Gaussian factor model exists then it is culled a weak stochastic realization o f the given measure.

b.

c.

Determine the minhnal dimension n*(Q) of the factor x in a weak stochastic realization of the given measure G (0, Q ). Call a weak stochastic realization minimal ([ the dimension o f the factor"s~,stems equals n * (Q ). Classify all minimal weak stochastic realizations of the given meastu'e.

Part a. of problem 5.2 is equivalent to: determine (n, Q.~,Q,,,, H) EI~ × R" x, × Rk xk × Rk ×n such that Q = I I Q . , I I T + Q,,,

where Q+,= Qx-10, Q , . - Q , . ~0, and Q,. is diagonal. Part a of the above problem is trivial, the hard parts of the problent are b and e. Corresponding to problem 5.2 there is a strong stochastic rt,lization problem f o r a Gaassian factor model. In this problem one is given a probability space (~2,b; P) and a Gaussian distributed random variable z ok-_G (0, Q). The problem is then to construct a Gaussian factor model y = l l x -t-- w

on the given probability space such that 7. = . y

a . s.

and to classify all minimal models of this type. This problem has been defined in [54], where a generalization of the Gaussian factor model for Hilbert spaces is introduced. The strong stochastic realization problem will not be discussed in detail here. What is the main characteristic of the Gaussian factor model? To answer this question one has to introduce the following concept. DEFINITION 5.3. 7he o-algebra's algebra G i f

F i,Fz .....

Fm

arc called conditionally indepcndcnt giveu the

lq:~... :,,, I 6] = E[:~ [ G ] . . . EIz,,IC]

a-

510

for all z i @L ~ (Fi). The notation (F1,F2...,I~;,IG)~CI will be used to denote that F1 . . . . . P,, are conditionally htdependent given G and CI will be called the multivariate conditional independence relation. The following elementary result then establishes the relation between the Gaussian factor model and the conditional independence relation. PROPOSITION 5.4. Lx't.l'i: f/--*H, i -----I, 2..... k, x : f2---,l~n. The following statements are equivalent: a. The random variables (v I...,Yk, x) are jointly Gaussian with zero mean and satis~,

(I":v'.... F'' ]F "~)C CL b.

The random variables y , x satisfy the conditions of the Gaussian factor aaalysis model of S.l with the representation y = l t t " + w.

The conditional indepcndcnce property of a Gaussian factor model is now scen to be its main characteristic. It will be called thefactorproper O, of a Gaussian factor model. It allows cxtensions to non-Gaussian random variables. Such extensions have been considercd in the literature, see for refcrcnecs 174]. Thc factor property is a generalization of the concept of state for a stochastic system. In such a system the future of the state and output process on one hand, and the past of the state and output process on the other hand arc conditionally independent given the present state. The analogy is such that the state corresponds to the factor and the output process to the observation vector of the factor model. The factor property or the conditional independence property occurs in many mathematical models in widely different application areas. Below the stochastic rcalization problem 5.2 will be discussed, first in terms of the external description and then in terms of the internal description.

"lhe stochastic realization problem in terms of the external description. In this subsection one is assumed to be given a Gaussian mcasure G(0,Q~.). The weak stochastic realization problem for a Gaussian factor model specializes in this case to the following question. QUESTION 5.5. Given a Gaussian measure G(0,Qy). a. What is the mminml dimension n* (Qr) of the factor h2 a stochastic realization of G (0, Q)? b. What is the classificatioJt o['all minimal stochastic reali-atiom of G (0, Q ), or all decompositions of the fo,'m Q.,, = Q l + Q,,

it, which Q I = Q~ >-0, Qw = Q II,;>~0 L~diagonal and rank (Q O=n*(Q,.). NOTATION 5.6. a. l f Q E R ~ ×k then

D(Q)C~

TM

511

b.

is a diagonal matrix with on the diagonal the elements of the diagonal of the matrix Q. l f Q E R k ×k then the matri.x OD (Q) ~ r k ×~, called the off-diagonal part of Q, is defined by OD (Q), = O, OD (Q)i4 = Q,.j, for all i,j E Zx., i=/=j.

C.

Q(Q,,,k,n) = ~ ( Q l ' Q ' ) e a k x k × r k × k l

"

~

QI=Q'~'~°' rank(Ql)=n"~ Ow =Qit, I>~0, Q.. diagonal, Q,. = Q I + Qw J

d.

n*(Qy) = min{n ~1%113 (Q l,Q,) EQ(Q.,.,/,',n)} It turns out to be useful to work with a standard form for the variance matrix, a canonical form. DEH NrrioN 5.7. One sa.),s that the matrices Q l, Q 2 E R ~ xk, that are assumed to be strict O,positive definite, are equivalent if there exists a diagonal matrix D @(0, oo)k ×k such that

QI -- DQ2D. A canonical form with respect to this equivalence relation is then such that D(Q) = I. An investigation should be made of another equivalence relation defined as in 5.7 in which negative elements are also admitted on the diagonal. Question 5.5.a is still unsolved. Characterizations of n*(Qv) are known in the two extreme cases of n*(Q,.)= 1 and n*(Qr)=k - I. These results are stated below. The characterization for n* (Qy)= 1 may go back to C. Spearman and co-workers. The formulation given here is from [13]. THEOREm 5.8. [131. Given Q,. E R ~ ×A, Q,. = Q.it.,>o. Assume that k I>4, Q,. @(0, oo)k×k, and that Qy is irred,cible. Then n*(Qi.)= 1 tff

qitqjm - qimqjt = 0, qitqji - qi, qjt ~ O, V i,j,l,m EZk, l=/=m,js~l,j~m,i~4=j, is/=l,i=/=m.

THEOREM 5.9. [13, 39, 581. Given Q.r CRxxk, Q,'= QIt,'>0. Then n*(Q,.)=k - i iffQ.;. ' has strictly positive dements, possib O, after sign changes of rows and cotv'esponding columns. What are the generic values of n*(Q),)? Below are stated the main results from a study by J. P. Dufour [20J on this question. DEFINITION 5.10. L e t

51 = ( Q ~ R b ; X t I Q = Q T } . Note that thc condition of positive definiteness is not imposed in the definition of the set 5~. In the following the Euclidcan topology is used on the vector space R n.

512

"I'm.ORZ~M5.1 I. [20]. a.

There exists an open and dense subset S C 5 ~ such that f o r all Qy G 5 ,t*(Q,,) >- ,~(2k + 1 - l x / i - 7 ~ - ) . 77tis inequalit.i, is known ~t~ the Lcdermann bound. Let Q c $. For eveO" Q ~ in a sufficiently small neighberh ood o f Q in $ the relation

b.

n*(Q) = n*(Ql) holds, For at{}' integerp suck t/tat

c.

'k(2k + 1 - X/1 + 8k ) <- p <~ k - 1 there exists a Q E $ such that n*(Q)=p. By way of illustration there follow characterizations on the value of n*(Q,) for variance matrices (~r E Rk ×k with several low values of k.

PRoPosrr[oN 5.12. Let Q,. ~ U 3×3, Q .= Qit.'>0, O(Q,.)=I. a. n*(Q,.) = 0 iff Q,. is diagonal. b. n*(Q.r) = 1 iff mw o f ;';efollowing cases applies. Case 1. l f ql2>O, q13:>0, q23:>0 attd qp~q13

q23

c.

q~2q23 qi3q2.t --t~10,11.

, - - ,

q 13

q 12

Case 2. l f q12>O, q13 =0, q23 =0. Other cases are derived fi'om the above be permutatioas of signs and indices. n* (Qy) = 2 iff othetwise.

For the special case in which Qy ~ C 4 x 4 and n* (Qy)= 1 a characterization is given in [6]. l'rtoposrrmN 5.13. Let Qr G (0, ~ ) a x 4. TheJl n* (Qr) = 1 iff tq~ to apermutatim, of indices, 1. c .

ql2ql3

.

q23

2. c > q ~ ,

.

ql2qla

ql3ql4

q24

q34

.

.

@(0, I];

c > '/~3, e ->- q~4-

Classification. In tiffs subsubscction the classification question 5.5.b will be discussed. Thus, given Qv CRk ×k, the question is to classify all decompositions of the form Q,.=QI

+Q,

in whichrank (Q I)--n* (Qr)- Geometry seems tile appropriate too[ for this classification, in particular polyhedral cones and convex analysis. For an approach along these lines see [19]. Below another approach is indicated that combines analysis and geometry. Remark that in the decomposition Q,. = Ql + Qw --- IIQ_~1t7"+ Qw

513

the off-diagonal elcmcnts of Q I are equal to the off-diagonal elements of Q,.. Moreover, by convention D (Q,.)= I. I lence the set Q(Q.r,k,n*(Qv)) may be classified by the diagonal of Q1. PaoJ'OS~TJON 5.14. Let

D(Qv,k,n) = { D ~ R k X ' , l) diago,~al, -OD(Q,.) <~ D <~ I, rank(D+OD(Q,.))=n}, f :D(Qr,k,n)--~Q(Qy,k,n ), f (D) = (D -4 0 D ( Q r ) , I - D ) . Thea f is a bijection. Remark that the set O(Qy,k,n) without the rank condition is a closed convex set. From 5.14 and some linear algebra one obtains the following result on the classification. "FtmOREM 5.15. l~,t Q,. _ Q,.T > 0 , • E ~ kxk, Q.,.-

D (Q.r)= t,

D I @R n ×" [ D 1 diagonal, 0 < D I ~1,

o~(~.,..k,.) = )h~.D,+OI)(A)>O, ~ 2 : = D q ~ + O 0 ( ~ l

'B-

(C) '

is diagonal and satisfies 0 <- D 2 ~ I g :D 1(Qj,,k,n)---*Q(Qy,k,n),

B.r

p, p T

p).

Zhell" a.

b. ¢.

g is well defined; g is sutjective; The diagonal matrix

;o]

is tmi,lue uI) to a flermutation. The proof of the above thcorcm is clcmcntary with the aid of thc following Icmma. LEMMA5.16. Letk, n C Z ~ , k > n ,

AER"×'~,B~

":'(t m , C ~ l '

rank --: i1,

Q = a.

B.r

•R

TM,

T =

Thea T'rQT =

C -BTA

IB "

I

n)x(k n ) , A = A ' r , C = C

T,

514

b. c. d.

rank (T) = k. rank(Q)=niffC-BrA -)B=0. Q>~Oiff C - B ' r A - 1B~O.

The study of the classification along the lines sketched above must proceed by an investioqtion of the following relations for the diagonal matrix D ] E R" x,,:

D l +OD(A)>O, D 2 : = BT"[DI+OD(A)]

I B - O D ( C ) , O<~D2<~l, D2isdiagonaL

For the cases n*(Qr)=k - 1 and n*(Qv)= 1 theorem 5.15 directly yields explicit classifications. The classifications of three low-dimensional examples are listed. I'RoPosvrtoN 5.17. l.'orthe casek =2, Q r G R z×2, n*(Qr)= 1 with

lhe classification, in the nolation o_['5.15, is given by D(Qr,2,1)= { d ] ~ R ~

Iq2~dl

~< 1 }

and q

Prol'osvrlon 5.18. For the case k =3, Qr @(0, ~)3x3, and n*(Qr)=2 the classificadon according to 5.15 is given by

fd 1 o I

/

dlq~3 +d2q~3--2q12q13q23 E'O 1" DI(Q, ',3,2) = ~ ~ ~-5-5----~--~_2- I , 1 " | dld2-ql2 [ and coaditioas obtainedbypermutation of indices

I ;'| J

PKOPOSlTION 5.19. For the case k = 3, Q). G(O, o0)3x3, tl*(Q.).) --= 1, the deconNosition is unique with

• q 12,1 )._____L

q23 Q1 =

q12 q13

ql2

q13

q12q23 q23 q13 q 13q23

q23

qJ2

515 PROPOSITION 5.20. For the case k =4, Qy E(O, oo)4×4, n , ( Q v ) = 2 the classification according to 5.15 is given b.e

[:0] d2

D 1 (Qy, 2) =

I d l ' d z ~ [ 0 ' l l ' dldz--q~z=/=O'

134 = d2q 13q J4 + d lqz3q24 -- q 12q 14q23 -- q 12q ]3q24, d2q~3 +dlq~3 -2q12q13q23 El0, i], d~ q~4 + d2q~4 - 2q 12q 14q24 ~[0, I], and conditions obtained by permuting the indices

The stochastic realization problem in terms o f the internal description The specification of the Gaussian factor model as given in 5.1 will be called the internal descril~tion. It is called internal because the specification is in terms of the matrices (It, Q.,,Q,.) rather than in terms of Qv. The questions for the internal desciption require one definition.

DEFINITION5.21. The Gaussian factor model with rel)rerentation y -- l l x + w is calh,d minimal/fn ---n*(Qr) in which x:fL--)R", Q~,>0 and

Q,. = H Q . , H I + Q,. Introduce the convention Q., = 1. The weak stochastic realization problem for a Gaussian factor model specializes in this case to the following question. QUESTION5.22. a. Which conditions on the matrices (II, Q.~.,Q,.) are equivalent w#h mininzality o f the Gaussian factor model? b. llow are two minimal Gaussian factor ttlodel$ related? The above questions are still open. The minimality question 5.22.a seems most interesting because its answer will involve a new system theoretic concept like stochastic observability. To hint at what may be needed a special case is considered. Consider a special Gaussian factor model of the form ,',

t'2

-_

,,,,x

+ w

in which the variance Q.. is required to be block-diagonal, in particular it consists of two blocks only

One says that this Gaussian factor model is slochasticalll, observable if the map

516 x ~ E [exp(iu 7), I ) I Fx ] is injcctivc on the support of x. Similarly onc says fl~at the Gaussian factor model is stochastically reconstructible if the map x ~ E[cxp(iu'r~,2)lF is injeetivc on the support of x. It may then be proven that the Gaussian factor model is minimal iff it is stochastically observable and stochastically rcconstructible ill" r a n k ( t l l ) = n = r a n k ( l l 2 ) 1711. Let's return to question 5.22.a, when is a Gaussian factor modcl minimal in case Qw is restricted to be diagonal. The following conjecture comes to mind first: A Gaussian factor model is minimal iff the map x ~ E[exp(itg,,) ] FX], f o r i ~ Z k , Is mjecttvc on the support of x for all i E,Et,. This conjecture is false, because the effective dimension n o f x may be larger than 1. liven i f n = I it is false, sce 5.23 below. The special case o f k =3 and n = 2 mentioned in 5.24 shows that the equivalent condition for minimality of a Gaussian factor system needs more thinking. The minimality characterizations for the following special cases may bc helpful in formulating conjectures for the general rcsulf. PROPOSITION 5.23. Consider a Gaussian factor model .)" = hx ~ w uqth k->-2, n = 1, h C R k. Then this model is minimal iff q i , j E T k , i=/=j, such thath,=/=O andhff/=O.

/'ROOF. The Gaussian factor model with a = I is minimal iff the dimension of the factor cannot be rcduccd. This is truc iff n* > 0 or iff Q,. is non-diagonal. Note that OD (Q.v) = OD(hh 1"). [] PROPOSITION 5.24.

Consider the Gaussianfactor model ~f5.1 with k ---3, n = 2,

fh,,] . -- lh l

z,(Q,.)--1.

Lh]j Assume thal It ~h 2 >0, h'(173 >0, h]h 3 >0. "fhe~t this Gaussia~ factor model is miaimal iff one of the fallowing conditions is satL~fied:

i.

(/7 ~/s 2)(/~ D7 3) ~lO, ll, (,'1~h 3)

2.

(h ~h 2)(h ]'h t) (h~h3) ~I0, I],

3.

(l~ ft, 3)(h ~1~ 3) (h~h2) ~[0,1].

517

PROOF. This follows from 5.12. []

Classification of internal description The motivating question here is whether the internal description of a Gaussian factor model is uniquely determined by the variance of the observation vector. In general such a model is not unique. This qucstion is related to question 5.5.b. For the classification of the internal description of factor analysis models with block-diagonal structure see 153]. To structure the discussion a definition is introduced. DEFINITION5.25. Two Gaussianfitctor models

y = 1Ix + w and .-y = 1t~ + w are called cquivalcnt tf llO., ll'r + Q.. : I-IQ-.J-~'r+ -Q.,. Note that tile two Gaussian factor models of 5.25 that are defined to be equivalent both have tile same variance matrix O,., since

Q,' = l t l Q x , l f ~ + O,', = lt2Qx, l t r + Q,'2" Therefore they cannot bc distinguished given Qv- It is well-known that if (n,/l, Qx, Q,.) are the parameters of a Gaussian factor system and if S •R n xn is an orthogonal matrix ( s s T = / ) , the two Gaussian factor modcls specified by (n,H, Qx,Q,) and (n, llS, Sl"QxS, Q,.) are cquivalcnt. Howcvcr, there may be othcr ways in which two Gaussian factor models are equivalent. In applications of Gaussian factor analysis it has been recognized that there may be many equivalcnt models. To reduce the class of equivalent models practitioners fix certain elements of the matrix of factor Ioadings, based on prior knowledge about the observation vector or arbitrarily. The question now is, given a Gaussian factor model, to describe the equivalence class of all Gaussian factor models that are equivalent with the given one. This question is still open. 6. GAUSSIANFACTORSYSTEMS The purpose of this section is to formulate the conccpt of a Gaussian factor system and to survey the preliminary rest,lts of the stochastic realization problem for this class of systems. A motivation for the study of this class of systems is the stochastic realization problcm for Gaussian systems with inputs. One would like to know whether it is possible to determine from an obscrvcd vector-valued process which components arc inputs and which are outputs of a Gaussian system. Another motivation for the study of this class of systems is the exploration of the extension of Gaussian factor models to dynamic systems. DEFINITION6.1. A Gaussian factor system, in discrete time, is an object specified kP the equations

B18

x ( t + 1) = A x ( t ) + Bu(t),

y ( t ) = lCx(t) + Du(t)] + w(t) O#"

y ( t ) = Y~ s/(t - s ) u ( s ) + w(t) .~C7"

where u :~ X T---~Rt' is a stations O' Gaussian process caUed the factor process, w : ~2X T---~Rk is a stationa O' Gaussiav process called the noise process, y :I2× T - o R ~ is called the observed process, it, ,v l . . . . , w k are bMependent processes, the stTeetral densities o f u, w l . . . . . w k are rational functions, and the Fourier transform o f the traasfer function 11 is rational and causal

A Gaussian factor systcm is said to have the factor property if the processes u, w l , . . . , w k are independent processes. This condition can also be rephrased in terms of conditional independence but this will not be done here. Note that the processes w l . . . . . wk need not be white noise processes.

Concepts similar to that of a Gaussian factor system have bccn introduccd in the literature. An elementary version of a Gat, ssian factor system with tl a constant matrix is introduced in [58]. In [26] a Gaussian factor system is defined without the rationality and causality conditions. In [21] one can find thc definition 6.1 and a generalization. In [54] a generalization of 6.1 is p,csentcd in which the spectral density of the process w is not diagonal but block-diagonal and in which the transfer function 11 not bc causal. The term dynamic errors-in-variables systems is uscd instead of Gaussian factor system in the publications of B. D. O. Anderson and M. Deistlcr [4-6, 16, 17]. An intcrprctation of this term follows. Consider a deterministic finite-dimensional linear system in impulse response rcprescntation .~(t) = ~_~ I I ( t - s ) u ( s ) . a~T

Suppose that the variables of input ~ and output ~. of this system are observed with errors or noise, say by

u(t) = i',(t) + w~(t),.),(t) =.[,(t) + w2(t), in which wi,w2 arc indcpendent Gaussian white noise processes. Combining these expressions one obtains

.<,>1

fw, ,>l = i

+

which is a Gaussian factor system except for the fact that the spcctral density of the noise is not a diagonal function but block-diagonal with two blocks. The interpretation of the above defined system of which the variables are observed with error, illustrates the term errors-in-variables model. PROnLEM 6.2. 771eweak stochastic realization problem for a Gaussian factor system is to solve the followDIg sttbl)roblems. Assmne given a stationaJy Gaussia~l proeess with zero mean fanction and covariance fimction Q or spectral delisity Q. a. Find conditiom under which there exists a Gaussian factor systenl

519

y ( t ) =- y. l t ( t - s ) u ( s ) + w(t) s c:T such that the spectral density, of)' equals the given spectral demity, or = ~r = llO,, 17T + 0,,.

b.

I f such a Gaussian factor s~,stem e~ists then it is called a weak stochastic rcalization of the given process. Classify all mbHmal weak stochastic realicatiom of the given process. A weak stochastic realiza^

~

"x'T

tion is called minimal f r a n k (ttQuH ) is mh#mal. A difficulty with the above defincd problem is the definition of minimality. In addition to the concept dcfined in 6.2, which is minimality of the dimension of the factor process u, one could ^

~

-~-T

also consider minimality of the degree of i1Q,1t . From a viewpoint of linear system theory the latter concept would be preferable. Possibly a mixture of both the dimension of the factor process and the degree has to be considered. Because of this difficulty the author of this paper is not yet convinced that a Gaussian factor system is a suitable model for economic and engineering practice. Howcver, what may be of interest is the spccial case in which the spectral density of the noise is block-diagonal with two blocks. The weak stochastic realization problem for Gaussian factor systems is unsolved. Only for low-dimcnsional cascs have results been published. For the case of an observed process with two componcnts see [4, 18, 33] and for the case with three components see 16, 18]. A discussion of the problcm may be found in [I 7]. Questions of identifiability and problems of parameter estimation for Gaussian factor systems have bccn discussed in [21,26]. A strong version of the weak stochastic rcalizationjgroblem of 6.2 has been proposed in [54]; see also [55]. The case in which the spectral density Q., of the noise consists of two diagonal blocks has been treated there. ACKNOWLEDGEMENTS The author acknowledges J. C. Willems for his inspiring conceptual approach to system and control theory. For the material on factor analysis and factor systems the attthor has benefited from discussions with L. Baratehart, M. Dcistlcr, R. E. Kalman, and G. Picci. REFERENCES 1. 2. 3.

4.

5. 6.

N.L.AGGARWAL (1974). Sur I'information de Fisher. Thdories de Fhforn~ation, Lecture Notes in Mathematics 398, Springer-Verlag, Berlin, i 1I-I 17. D.J. AIGNER and M. DmsTm~t (1989). Latent variables models - Editor's introduction. d. Econometrics 41, I-3. H. AKAmE (1976). Canonical correlation analysis of time series and the use of an information criterion. P,. K. MmiRA, D. G. LAINIOTIS(cds.). Sl'stem identification - Advances and case Mudit.s, Academic Press, New York, 27-96. B. D. O. ANDERSON(I 985). Identification of scalar errors-in-variables models with dynamics. A utomatica J. -ll:/I C 21,709-716. B.D.O. ANDI~RSON and IVI.DE[STEER (1984). Idcntifiability in dynamic errors-in-variables models. J. "lime Series A,aL 5, 1-13. B. D. O. ANDERSONand M. D~tSTLER (I 987). Dynamic errors-in-variables systems with three

520

7.

8. 9. 10. 11.

12.

13. 14.

15. 16.

17.

18. 19. 20. 21. 22. 23. 24. 25.

variables. A utomatica J.-11".4 C 23, 611-616. T.W.ANDERSON and H. RUBIN (1956). Statistical inference in factor analysis. J.NEYMAr~ (ed.). Proc. 77ffrd Berkeley Syn~osium on Mathematical Statistics and Probability, Volume V, University of California Press, Berkcley, I 11-150. M. AOKI (1988). Nonstationarity, cointegration, and error correction in economic modeling, J. Econ. l)yn. Control 12, 199-201. M. AoKt (1988). On altcrnative state space representations of time series models. J. Ecol~ Dt'n. Conu'o112, 595-607. V. 1. ARKIN and I. V. EVSTtGNEEV(1987). Stochastic models of control and economic dynamics, Academic Press, New York. K.S.ARuN, D.V. BttASKAit R^O, and S.Y. KUNG (1983). A new predictive efficiency criterion for approximate stochastic realization. 1983 Coltference on Decision and Control, 1EEE, 1353-1355. K.S.AaoN and S.Y. KONG (1984). State space modeling and approximate realization methods for AKMA spectral cstimation. I E E E hlternational CoiJference oll Systems, Meal and C.),bernetics, I ILEE. I'. A. BEKKERand J. DE LEEUW (1987). The rank of rcduccd dispersion matrices. Psvchometrika .52, 125-135. A. BENSOUSSANand J. It. V^N SCHtJPPEN (1985). Optimal control of partially observable stochastic systems with an exponential-of-integral performance index. S l A M 3. Control Optim. 23, 599-613. D. Bt.ACKWELL and L. KooI'MANS (1957). On the idcntifiability problem for fimctions of finite Markov chains. Antt. Math. Statist. 2& 1011-1015. M. DHsTLJ:4t (1986). Linear errors-in-variables models. S. BI1"rANaI (cd.). Time serq,s and liltetu'.~l'~tems, Lecture Notes in Control and Information Sciences, Springer-Vcrlag, Berlin, 37-67. M.I)EIST"LER and B.D.O. ANDERSON (1988). Linear dynamic errors in variables models: Some structure theory. A. BENSOUSSAN, J. L. LIONS (eds.). Anal~,sis and optimization of systems, Lecture Notes in Control and Information Sciences I 1 !, Springer-Verlag, Bcrl;,, 873883. M.I)~I.~JI.L:k and B.D.O. ANDf;RSON (1989). Linear dynamic errors-in-variables models Some structure theory. J. Econometrics 41, 39-63. B. DE Moo~. (1988). Mathematical concepts and techttiques for modelling of static and dl,namic sl'stems, Thesis, Katholiekc Univcrsiteit Leuven, Leuvcn. J.I'.DuI:ouR (1983), Resultats generiques en anal~,se factorielle, Departcment de Math~:matiqucs, U.S.T.L., Montpcllier. P,. ENGI.E and M. WATSON (1981). A one-factor multivariate time series model of mctropolitan wage rates. J. A m e r Statist. Assoc. 76, 774-781. R.F. ENGLF and C. W.J. GR^NGErt (1987). Co-integration and error correction: Representation estimation, and testing. Econometrica 55, 251-776. P. I:^URItE (1976). Stochastic realization algorithms. R.K. MEHR^, D.G. LMNIOTIS (eds.). ~).stem hh, Jltification - Advaltces alld Case Studies, Academic Press, New York, 1-25. I'.F^uRRI~, M.CLEROrr. and 17. f.3ERMAIN (1979). Ope'rateurs rationnels positifs, Dunod, Paris. I{. Ft~ISCH (I 934). Statistical col~tuence anaO,sis b v means of complete regression systems, Publ. no. 5, University of Oslo Economic Institute, Oslo.

521

26. J.F.GEWEKE and K.J. SINGLETON (1981). Maximum likelihood 'confirmatory' factor analysis of econometric time series. Int. Economic Rev. 22, 37-54. 27. R.D.GrrrENs (1985). Canonical analysis - A review with applications in ecologr,, SpringerVcrlag~ Berlin. 28. K. GLovER and E. JONCKltEERE (1986). A comparison of two llankel-norm methods for approximating spectra. C.I. BYRNES, A. LINDQUIST (eds.). Modelling, Identification m,d Robust Control, Elsevier Science Publishers B. V. (North-I lolland), Amsterdam, 297-306. 29. A. GOMBANIand M. PAVON (1985). On the Hankel-norm approximation of linear stochastic systems. S~'stems & Control Lett. 5, 283-288. 30. A.GOMBANI and M. PAVOr~ (1986). On approximate recursive prediction of stationary stochastic processes. Stochastics 17, 125-143. 31. A. GoMBANL M. PAVON, and B.CoPPo (1986). On Hankcl-norm approximation of stationary increment processes. C.i. BYrtNES, A. LINDOUXST (eds.). Modelling, Identification and Robust Coatrol, Elsevier Science Publishers B. V., Amsterdam, 307-323. 32. C.W.J. GRANGEa (1981). Some properties of time scrics data and their use in econometric model specification, d. Econometrics, 121-130. 33. M.Gru~EN and B.D.O. Ar~oEgsoN (1986). Identification of multivariablc errors in variable modcls with dynamics. IEEE Trans. Atttomatic Control 31,467-471. 34. E.J. I IANNANand M. DEXSTLER (t988). The statistical theoO, ofliaear s),stems, John Wiley & Sons, New York. 35. C. IIEIJ (1988). Determhfistic identification of d),aamical sl,stems, Thesis, University of Groningen, Groningcn. 36. N.P.JEWELL and P. BLOOMHELD(1983). Canonical correlations of past and future for time series: Definitions and theory. Amt. Statist. 11,837-847. 37. N.P.Jt:.WELt, P. BLOOM~ql:t.D, and F.C. IIAWrMANN (1983). Canonical correlations of past and future for time series: Bounds and computation. Am~. Statist. 11,848-855. 38. E.A.JoNcKm~I:.rtE and J.W. tlEt:roN (1985). Power spectrum rcduction by optimal llankcl norm approximation of the phase of the outer spectral factor. I E E E Trans. Automatic Control 30, 1192-1201. 39. R.E. KALMAN (1982). Systcm identification from noisy data. A.R. BEDr~MtEK, L.C~sAai (cds.). Dynamical Sir,stems II, A cadcmic Press, New York, 135-164. 40. R.E. KALMAN (1982). ldcntification from real data. M. IIAzEWXNKEL, A.H.G. RINNOOY KAN (eds.). Current develol;ments in the hztetface: Economics, Eco, ometrics, Mathentatics, D. Rcidel l'ublishing Company, Dordrecht, 161-196. 41. R.E. K^LMAN (1983). IdentiSability and modeling in econometrics. P.R. KRISlINAIAH(cd.). Developments hi Statistics 4, Academic Press, New York, 97-136. 42. R.E. KALMAN. I'. L. FAL}~, and M.A.ARB]B (1969). Tol;i~ in mathematical system theoo, , McGraw-Hill Book Co., New York. 43. D. KAZAKOSand 1'. i'AeAr~TOr~I-KAZAKOS(1980). Spectral distance measures between Gaussian processes. IEEE Trans. Automatic Comro125, 950-959. 44. S.Y. KUNG and K.S.ArtuN (1983). Approximate realization methods for ARMA spectral estimation. 1983 IEEE International ~),mpositun on Circuits and ,S),stems, IEEE, 105-109. 45. W.E. LAaIMOaE, S. M~,~tMOUt~, and R. K. ME~ltA (1984). Multivariate adaptive model algorithmic control. Proceedings 23rd Cotlference on Decision a~d Control, IEEE, 675-680. 46. W.E. LMtkIMORE (1983). System identification, reduced-order filtering and modeling via canonical variate analysis. Proceeditlgs 1983 American Control Coaferettce, 445-45 I.

522 47. A. LINDQUISTand G. PICCI (1985). Realization theory for multivariate stationary Gaussian processes. SIA 54 J. Control Optim. 23, 809-857. 48. A. LINDQUZSr, G. PIccl, and G. RUCKEBUSCH (1979). On minimal splitting subspaccs and Markovian representations. Math. Systems Tit. 12, 271-279. 49. M. PAVON (1984). Canonical correlations of past inputs and future outputs for linear stochastic systems. Syst. Cono'ol Lett. 4, 209-215. 50. A. PAZ (1971). Introduction toprobabilistic automata, Academic Press, New York. 51. V.V. PELLArt and S.V. KHaUSHCHEV (1982). Hankel operators, best approximations, and stationary Gaussian processes. Russian Math. Suive),s 37, 61-144. 52. G. I'xccl (1978). On the internal structure of finite-state stochastic processes. Proc. of a U. S.-lta O' Semh~ar, Lecture Notes in Economics and Mathematical Systems 162, SpringerVerlag, Berlin, 288-304. 53. G. PICCl (1989). Paramctrization of factor analysis models. J. Econometrics 41, 17-38. 54. G. P1ccl and S. PINZONI (1986). Factor analysis models for stationary stochastic processes. A. BENSOUSSAN, d. L. LIoNs (cds.). Analysis amt Optimization of Systems, Lecture Notes in Control and Information S~.;cnces 83, Springer-Verlag, Berlin, 412-424. 55. G. Plccl and S. PINZONJ (1986). Dynamic factor-analysis models for stationary processes. IMA J. Math. Control and lafo. 3, 185-210. 56. P. PuRDuE (1979). Stochastic compartmental models: A review of the maflaematicai theory with ecological applications. J.H. MA'rls, B.C. P^-rr~:N, G.C. WtlITE (eds.). Compartmental anaO,sis ~f ecos.;stem models, International Co-operative Publishing House, Fairland, MD, 223-260. 57. C.R. R^o (1964). The use and interpretation of principal component analysis in applied research. Sankh.va, Series A 26, 329-358. 58. O. R~IERSOL (1941). Conflt, encc analysis by means of lag moments and other methods of conltucnce analysis. Econometrica 9, 1-24. 59. O. RElEaser. (1945). Confluence analysis by means of instrumental sets of variables. Arkiv flit Mathematik, Astronomi och F3,sik 32A. 60. J. RIsshNEN (1978). Modeling by shortcst ~tata description. Automatica J. IFAC 14, 465471. 61. J. RISS^N~N (1983). A universal prior for integers and estimation by minimum description length. Ami. Statist. 11,416-431. 62. J. RISS^NEN (1985). Modeling by the mitfimum description lenghth principle - A surve),, Preprint. 63. J. RISShNEN(1986). Stochastic complexity and modeling. Am~. Statist. 14, 1080-1 I00. 64. J. RlssANEN (1986). l'rcdictivc and nonprcdictive minimum description length principles. S. BITrANTI (ed.). Time .series and linear systems, Lecture Notes in Control and Information Sciences 86, Springcr-Vcrlag, Berlin, 115-140. 65. J. RUCKI-;BUSCH(1978). Sur I'tq~proximation rationnelle des filtres, Ccntre de Mathe'matiques Applique'cs, Ecole l'olyteclmique, Palaiseau. 66. I'.A. SAMUELSON(1979). A note on alternative regressions. J. E.S'rIGLrrz (ed.). The collected scientific papers ~fPaulA. Samuelson, Ftfth Printing, M. I. T. Press, Cambridge, 694-697. 67. C.A. SPr:AaMAN (1904). General intelligence, objectively determined and measured. Amer. J. Psych. 15, 201-293. 68. P.J.C. SPaeu (1986). Self exciting counting process systems with finite state space, Report OSR8613, Centre for Mathematics and Computer Science, Amsterdam.

523 69. P.J.C. SPg~tr (1987). Comuing process systems - Identification attd stochastic realization, Doctoral Thesis, University of Twente, Enschede. 70. J.II.S'rEIGER (1979). Factor indeterminacy in the 1930's and the 1970's some interesting parallels. Psychometrika 44, 157,167. 71. C. VAN Pu'rrEN and J. II. VAN SCHUPPES (1983). The weak and strong Gaussian probabilistic realization problem. J. Multivariate Anal 13, 118-137. 72. J.H. VAN SCHUPPi~N(1979). Stochastic filtering theory: A discussion of concepts, methods and results. M. KOHLM^r~N, W. VOG~.L (eds.). Stochastic control theory and stodzastic differential systems, Lecture Notes in Control and Information Sciences 16, Springcr-Verlag, Berlin, 209-226. 73. J.H. VAr~ ScnuPP~:N (1982). The strong finite stochastic realization problem - Preliminary results. A. BI~NSOUSSAN, J.L. LXONS (eds.). Analysis and optimization of systems, Lecture Notes in Control and Information Sciences 44, Springer-Vcrlag, Berlin, 179-190. 74. J.H. VAN SC}~UPPEN (1986). Stochastic realization problems motivated by econometric modeling. C.I. BYRNES, A. LINDQUIST (eds.). Modelling, Identification and Robust Control, Elsevier Science Publishers B. V. (North-I lolland), Amsterdam, 259-275. 75. E. VERgtEST (1986). Projection Techniques for model reduction. C.I. BYm~ES, A. LINDQUIST (eds.). Modelling, Identification and Robtt~t Control, Elsevier Science Publishers B.V. (NorthHolland), Amsterdam, 381-396. 76. J.V.WmxE (1983). Stochastic state-space models from empirical data. Proceedings 1983 Coaference on Acoustics, Speechs and Signal Processing, 1EE~ 243-246. 77. I'.WHITTL~ (1981). Pdsk-sensitive linear/quadratic/Gaussian control. Adv. AppL Prob. 13, 764-777. 78, J.C.WILL~;MS (1986). From time series to linear systems - Part I. Finite dimensional linear time invariant systems. AutomaticaJ. 1FAC 22, 561-580. 79. J.C. WXLLEMS(1987). From time series to linear systems - Part 111. Approximate modelling. AutomaticaJ. 1FAC 23, 87-I 15. 80. E.B.WtLSON (1929). Review of'Crossroads in the mind of man: A study of dill'crentiable mental abilities' by T. L. Kelley. J. Get~. Psychology 2, 153-169.

Robust Stabilization of Uncertain Dynamic Systems J. L. Willems Engineering Faculty, University of Gent Gent, Belgium

i. I n t r o d u c t o r y R e m a r k s

It

gives

me

brother's

a

strange

fiftieth

own birthday!

Only

to c o n t r i b u t e

I began

be.

feeling

birthday. after

On the one h a n d

to

write

a

It is almost

accepting with

to realize

contribution

like w r i t i n g

great pleasure

how d i f f i c u l t

the

it is not p o s s i b l e to w r i t e

fessional

for

primarily

professional,

one's

twin

brother,

because

certainly

not!

On

for

the

task

S y s t e m Theory".

a s c i e n t i f i c w o r k for the m a t h e m a t i c a l It

took

me

a

long

time

to

reach

a

twin

for my

invitation

was

going to

s o m e t h i n g p u r e l y pro-

the

relationship

the

other

hand

would not be fit to w r i t e a c o n t r i b u t i o n with too p e r s o n a l a b o o k on " M a t h e m a t i c a l

my

a paper

The b o o k

I

is felt

not it

a flavor in

is indeed m e a n t to be

system t h e o r y community. conclusion.

I

finally

decided

to

o r g a n i z e my c o n t r i b u t i o n as follows:

a f t e r some c o m m e n t s on our profes-

sional

survey

relationship,

stochastic

dynamical

together.

This

very

to Jan:

dear

I will systems,

topic

high gain feedback.

give

the

indeed

a

on

robust

stabilization

of

a topic we w o r k e d on and p u b l i s h e d about embraces

Riccati

a

equation,

number

of

(almost)

concepts invariant

which

are

subspaces,

It is like m e n t i o n i n g t h r e e of his m i d d l e names!

525 As

is q u i t e

normal

for twin brothers,

school and the same h i g h school.

we a t t e n t e d

the

same e l e m e n t a r y

It is less c o m m o n t h a t a f t e r f i n i s h i n g

high school b o t h of us d e c i d e d to study e n g i n e e r i n g

at the university.

It is p r o b a b l y even much less c o m m o n that we b o t h c h o s e e l e c t r o m e c h a n i cal e n g i n e e r i n g as our m a j o r field.

We thus g r a d u a t e d t o g e t h e r in elec-

tromechanical

Engineering

engineering

sity of Gent, Gent, university,

from the

Belgium,

in J u l y 1963.

Faculty

time

our w a y s

ships

the

end

of

the

separated.

from d i f f e r e n t

transatlantic traveled

the

of

Indeed,

Jan

made

Rotterdam.

1963

left we

for

and w e r e s u c c e s s the

obtained

U.S.;

voyage

Since

most

aboard

people

the

in

at

graduate

we t r a v e l e d to the U.S.

the

a s s o c i a t e b o t h of us w i t h M.I.T.

we

because

foundations,

ships;

on

summer

Univer-

we b o t h a p p l i e d for

a f e l l o w s h i p for g r a d u a t e study in the U n i t e d States, At

the

During our final y e a r at the

p r o b a b l y thanks to our f a t h e r ' s advice,

ful.

of

on d i f f e r e n t

Queen

the

Mary,

control

and with R o g e r B r o c k e t t

that

fellowI

field

in particular,

it m a y s u r p r i s e m a n y that from that time o n w a r d s we n e v e r w o r k e d at the same i n s t i t u t e at the same time. ing

my

Master's

degree

Indeed I s t u d i e d at M.I.T.

during

the

academic

year

for obtain-

1963-1964,

and

then

r e t u r n e d to Belgium. Jan s t u d i e d at the U n i v e r s i t y of R h o d e Island from 1963

to

1965,

came

and

to M.I.T.

in

1965

to w o r k

towards

the

Ph.

D.

M.I.T.;

he

degree.

Both

of

us

were

supervised theory

my

in

happened

fortunate

Master's

general,

to

work

thesis

and

in

to Jan two years

with

and

stimulated

stability

later.

joint

research which

Brockett my

theory

at

interest

in

in

particular.

control The

same

B e c a u s e of our a s s o c i a t i o n w i t h R o g e r

B r o c k e t t we w o r k e d on r e l a t e d problems. do some

Roger

This g a v e us the o p p o r t u n i t y to papers.

The ex-

change of ideas t o o k p l a c e d u r i n g family v i s i t s and by letters,

led to a n u m b e r

of j o i n t

between

c o m m e n t s on the p r o g r e s s of the c h i l d r e n in school and on the h e a l t h of some

aunt

or

group

in

1970,

uncle.

We

when

Jan

leave of a b s e n c e fellow left

with

M.I.T.

was

succeeded

to

be

in

the

on the F a c u l t y at M.I.T.

same

and

research

I went

on a

from the U n i v e r s i t y of G e n t to w o r k as a p o s t d o c t o r a l

Roger and

almost

Brockett.

had

joined

However the

by

that

Division

time

of

Roger

Engineering

Brockett and

had

Applied

Physics at Harvard.

Our

first

criteria

two

joint

a r t i c l e s were

for n o n l i n e a r

feedback

of r a t h e r m o d e s t c o n t r i b u t i o n s ! tion'

at

M.I.T

we

did

some

discussion

systems;

T h a n k s to our

joint

work

notes

even Jan

on

[1,2]

on s t a b i l i t y

began with

a couple

'Lyapunov f u n c t i o n educa-

the

of

Lyapunov

f u n c t i o n s for the a n a l y s i s of t r a n s i e n t p o w e r s y s t e m s t a b i l i t y

generation

[3]. Our

526 larger

piece

robust

stabilization,

with by

of

joint

stochastic

white

with

research namely

first

noise

paper

the

coefficients. processes.

state-dependent [4]

we

work

This

concerned

robust

The

and/or

was

feedback

stochastic

leads

the

to the

ideas

a problem

control

of

coefficients

noise

of J a n ' s

systems

of

systems

[4, 5].

earlier

of

are modeled

consideration

control-dependent

combined

with

In

work

on

the pole

a s s i g n m e n t of d e t e r m i n i s t i c linear systems and on the a l g e b r a i c Riccati equation,

w i t h my p r e v i o u s

ity of s t o c h a s t i c the second p a p e r of a l m o s t is

in

centrates

on

approach,

and

details

[5] we d e s c r i b e d

(A,B)-invariant

discussed

the

the

the

r e s e a r c h results on the m e a n square

systems with s t a t e - d e p e n d e n t

of

development

on the

reader

this of

referred

contribution.

the

interpretation

is

In

further r e s u l t s a p p l y i n g the concept

s u b s p a c e s Jan developed.

sequel

stabil-

s t o c h a s t i c elements.

to

model,

on

T h i s r e s e a r c h topic The

the

motivation

of the results. the

original

discussion of

conthe

For the technical

papers,

cited

in the

bibliography.

2. S y s t e m M o d e l i n g A large fraction of the r e s e a r c h on control systems in the last quarter of a c e n t u r y

is c o n c e r n e d with the linear control

system,

d e s c r i b e d by

the set of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s

d x ( t ) / d t = Ax(t)

(i)

+ Bu(t)

where x d e n o t e s the state v e c t o r and u the control input vector,

with n

and

of

m

components

propriate

respectively.

dimension.

possibility

of

A

and

In p a r t i c u l a r

stabilizing

system

B are

very (i)

constant

complete or

matrices

results

arbitrarily

exist

It

is

real

obvious

that

physical

whether

results

in other w o r d s

(i)

system. obtained whether

is

only

It

is

for

an

approximate

therefore

(i) still hold for

properties,

such as the

pole a s s i g n a b i l i t y p r o p e r t y m e n t i o n e d above, perturbations been a v e r y

of

the

active

system

model.

r e s e a r c h area

physical

system

is

only

an

to

of

the

investigate

'neighboring'

systems,

stabilizability

or the

are r o b u s t w i t h respect to

Robustness

of

in recent years

important aspect of a design study; a

its

[6, 7].

representation

interesting

on the

assigning

e i g e n v a l u e s by means of state f e e d b a c k control s t r a t e g i e s

ap-

control [8].

systems

has

It is indeed an

the model u s e d for the a n a l y s i s of

approximate

representation

since

some

p h e n o m e n a are u n k n o w n or n e g l e c t e d and h e n c e not i n c l u d e d in the model.

527 The

exact

model

including

of

the

nonlinear

physical

and

system

time-varying

account

phenomena

effects

are u n p r e d i c t a b l e

which

are

would

in

phenomena

governed

by

most

as well

external

cases

require

as t a k i n g

effects.

The

from the p o i n t of v i e w of the s y s t e m

The model p e r t u r b a t i o n s can be m o d e l e d in d i f f e r e n t ways;

into

latter itself.

for a discus-

sion of this aspect the r e a d e r is r e f e r r e d to the e x t e n s i v e

literature

on robust control system design. The

model

previous

considered papers

in

[4, 5]

the

present

includes

'white

noise'

quires

the c o n s i d e r a t i o n

coefficients;

a

rigorous

in

along

described

mathematical

of It6 d i f f e r e n t i a l

of the a n a l y s i s and the d i s c u s s i o n the p l a n t m a t r i x A

contribution

perturbations

the by

line

description

equations.

of

stochastic re-

For s i m p l i c i t y

in this p a p e r only p e r t u r b a t i o n s of

(I) are considered,

but no p e r t u r b a t i o n s

of the

input m a t r i x B. This leads to the e q u a t i o n

(2)

dx = (Ax+Bu)dt + Z i aiFix dW i where

the

last

term

stochastic processes

(i =

I,

...

are a s s u m e d

, M)

denotes

ary n o r m a l i z e d s t a n d a r d W i e n e r processes.

E(dWi)

the

disturbances.

to be z e r o - m e a n u n c o r r e l a t e d

The

station-

Thus

= 0

E(dWi2 ) = dt

E(dWidWj)

= 0

for i / j

where E d e n o t e s the e x p e c t e d value. sities

of

the disturbances.

The

Intuitively

factors a i i n d i c a t e the inten(2) should

be r e g a r d e d

as the

d i f f e r e n t i a l e q u a t i o n with s t o c h a s t i c p a r a m e t e r s

d x ( t ) / d t = [A + Z i fi(t)Fi]x(t)

(3)

+ Bu(t)

where the p r o c e s s e s fi(t) are white noise s t o c h a s t i c processes. The

stabilizability

problem

considered

in

this

paper

is

the

question

w h e t h e r there exists a state feedback control law of the form

u

=

Kx

such that the c o n t r o l l e d s y s t e m

(4)

528 dx =

(s)

(A+BK)x dt + Z i a i F i x dW i

is stable. More

explicitly,

the

following

robustness

issues

are d i s c u s s e d

in the

present contribution: Robust

stabilizability.

Find

exists w h i c h s t a b i l i z e s

conditions

such

that

a

feedback

(5) for a g i v e n range of the n o i s e

control

intensities

ai• Robust that

stabilizability

for

any range

which stabilizes Perfect

robust

for all noise

of the noise

intensities.

intensities

Find c o n d i t i o n s

a feedback control

such

exists

(5) for that range of noise i n t e n s i t i e s a i.

stabilizability.

Find c o n d i t i o n s

a f e e d b a c k control w h i c h s t a b i l i z e s

such that t h e r e exists

(5) for all noise intensities.

The s t a b i l i t y p r o p e r t y c o n s i d e r e d is m e a n s q u a r e a s y m p t o t i c stability: Definition 1 System

(2)

is said to be mean

initial s t a t e s x(0)

square

asymptotically

stable

the second m o m e n t m a t r i x E[x(t)xT(t)]

if for all

tends to zero

as t tends to infinity.

The e x p l i c i t d e f i n i t i o n s t i o n e d above,

of the robust s t a b i l i z a b i l i t y p r o p e r t i e s men-

are h e n c e as follows.

Definition 2

System

(2)

intensities that

is

said

to

be

robustly

(Sl, s2,...,SM)

stabilizable

for

a

set

of

if there exists a f e e d b a c k c o n t r o l

noise

(4) such

(5) is m e a n square a s y m p t o t i c a l l y stable for the n o i s e intensities

satisfying

ai ~ si

(i = 1 ..... M)

Definition 3

System

(2)

is

said

to

be

robustly

stabilizable

for

all

sities if it is r o b u s t l y s t a b i l i z a b l e for all bounds Sl,

noise ...

inten-

, s M.

529 Definition 4 System

(2)

exists

is

said

to

be

a feedback control

perfectly (4) s u c h

robustly

that

stabilizable

if

there

(5) is mean square asymptoti-

cally stable for all noise intensities. The

property

expressed

by

expressed by Definition the bounds

si;

some

Definition

3

is

weaker

than

the

property

4 in that the feedback matrix K may depend on

entries

of K may hence

increase

without

bound

as

the bounds s i tend to infinity. The analysis of robust stabilizability uses properties of the algebraic Riccati

equation

ized by means

[9].

of

bility

for all noise

almost

invariant

ii].

Hence

by

Perfect robust stabilizability

(A,B)-invariant intensities

(A,B)-invariant

discussing

this

vance of many of Jan Willems'

subspaces

[i0,

can be character-

ii]. Robust stabiliza-

is closely related to the concept subspaces

problem

and high gain feedback

area,

the

importance

of

[i0,

and rele-

research results can be illustrated.

3. Preliminary results For the

linear

stability

deterministic

system

of the uncontrolled

it

system

is well

known

that

(I) can be related

asymptotic

to properties

of the Lyapunov equation ATp + PA = -Q

(6)

with the superscript T denoting matrix transposition. feedback

stabilizability

of the controlled

system

On the other hand

(i) is connected

to

the Riccati equation ATp + PA - PBR-IBTp = -Q Indeed that

asymptotic

(7)

stability

of

the

uncontrolled

system

(i)

requires

(6) has a positive definite solution P for a given positive defi-

nite matrix Q. Stabilizability by state feedback requires the existence of a positive

definite

solution

P of

(7)

for given positive

matrices Q and R. In both cases it can be shown that, is satisfied positive

for some positive

definite matrix Q, and in

definite matrix R, then it is also satisfied

definite

if the condition (7) for some

for all positive

530 matrices

Q and R. M o r e o v e r

the s t a b i l i z i n g

feedback

d e r i v e d from the p o s i t i v e d e f i n i t e s o l u t i o n of

control

of

(I) is

(7):

u = -R-IBTpx

Similarly,

(B)

mean

square

asymptotic

s t a b i l i t y of the u n c o n t r o l l e d

(2) can be a n a l y s e d by m e a n s of a linear m a t r i x e q u a t i o n w h i c h close

to

the

means

of

an

Lyapunov equation

From the e q u a t i o n uncontrolled

equation, which

is

and

mean

close

to

square

the

system is very

stabilizability

matrix

Riccati

by

equation.

for the second m o m e n t s of the s t a t e v a r i a b l e s of the

system

(2),

a s y m p t o t i c s t a b i l i t y of

we

derive

the

condition

(2) w i t h o u t control

that

mean

square

is implied by the existence

of a p o s i t i v e d e f i n i t e s o l u t i o n P of the e q u a t i o n

[12]

(9)

ATp + PA + Z i u i 2 F i T p F i = -Q for a p o s i t i v e d e f i n i t e m a t r i x

Q. The

stabilizability

of

(2) by means

of state f e e d b a c k is e q u i v a l e n t to the e x i s t e n c e of a p o s i t i v e definite s o l u t i o n P of the e q u a t i o n

(io)

A T p + PA - P B R - I B T p + Z i o i 2 F i T p F i = -Q

for

some

positive

feedback closed An

control

realizing

loop s y s t e m

noise

system

(2)

is

Q

and

square

(8),

with

R.

Also

asymptotic

it

the

has

asymptotically

same

a

suitable

stability

of

P the said s o l u t i o n of

of the above d i s c u s s i o n square

here,

property

the (i0).

is t h a t

if the uncon-

stable

for a set of

for

all

smaller

noise

The same is true for s t a b i l i z a b i l i t y .

immediately

e q u a t i o n and

mean

is m e a n

intensities,

intensities.

matrices

is g i v e n by

immediate consequence

trolled

It

definite

clear

that

(9)

can

be

associated

with

a

Lyapun0v

(i0) w i t h a R i c c a t i e q u a t i o n if

(11)

Q - z i ai2 F i T p F i > 0 w h e r e P is the s o l u t i o n of out

how

the

techniques

analysis

available

d e t e r m i n i s t i c systems.

By m e a n s square

of

(9)

for the

and

(I0) respectively. (i0)

Lyapunov

can be

This r e m a r k points

carried

or R i c c a t i

out

equations

by means

of

for linear

This is further e l a b o r a t e d in the next sections.

of A - i n v a r i a n t

asymptotic

(9) or

subspaces

stability

of the

a criterion

can be d e r i v e d

uncontrolled

stochastic

for mean

system

(2)

531 for all noise to derive

intensities

robust

a i. This result

stabilizability

defined by the following

we

: =

Wj

: = < n i Fi-Iwj_ 1 I A>

where

in the next sections

Let

the

subspaces

Wj

be

recursive algorithm:

{0)



subspace

denotes

(12)

the maximal

F. Note that Wj

are nested.

denoted by W*,

Criterion

1

uncontrolled

all noise

A-invariant

is a subspace

subspace

of Wj+I,

intensities

in the

leads to a limiting

in a finite number of steps.

system

the n-dimensional

contained

hence the subspaces Wj

This shows that the recursive a l g o r i t h m

subspace,

The

is used

conditions.

(2)

is mean

if and only

square

asymptotically

stable

if the m a t r i x A is Hurwitz

for

and W* is

space.

Remarks

(i)

The

conditions

of the above c r i t e r i o n

the s o l v a b i l i t y matrices

A,

FI,

the Hurwitz (ii)

The

above

of the matrix Lie algebra g e n e r a t e d ...

, FM,

the n i l p o t e n c y

condition

is

also

FM,

where

blocks

the

matrices of

mean

by the set of

of the m a t r i c e s

equivalent

block-triangularizing

form

[5] to F i and

c h a r a c t e r of A.

simultaneously

is

are also e q u i v a l e n t

the

the

F

and

system

square

on

the

Hurwitz

diagonal

representation

zero

for

clearly

stable

the

possibility

matrices

are

submatrices

asymptotically

to

the

for

the

shows all

A,

FI,

submatrices matrix why

noise

the

of

...

A.

,

for This

system

intensities

[5].

Example 1 An

example

all noise

of a system which intensities

is

is mean

square

asymptotically

stable

for

532

dW

where

~

and

@ are

the p e r t u r b a t i o n

negative

only

constants.

affects

d i a g o n a l e l e m e n t s ~ and @

Intuitively

the o f f - d i a g o n a l

the

reason

is that

element

6, but

not the

(the e i g e n v a l u e s of the s y s t e m matrix).

4. R o b u s t s t a b i l i z a b i l i t y

The

discussion

ness

criteria

of the p r e v i o u s can

be

section a l r e a d y p o i n t s

obtained.

The

feedback

control

out how (8),

robust-

with

P the

p o s i t i v e d e f i n i t e s o l u t i o n of (7), yields a closed loop s y s t e m w h i c h is mean

square

asymptotically

stable

for a l l n o i s e

intensities

(ii). }iowever this only y i e l d s s u f f i c i e n t conditions;

satisfying

d i f f e r e n t choices

of the m a t r i x Q and R may lead to m o r e or less c o n s e r v a t i v e conditions. From the p r o p e r t i e s of the s o l u t i o n s of the a l g e b r a i c R i c c a t i

equation

it is c l e a r t h a t a s m a l l e r R y i e l d s less c o n s e r v a t i v e conditions. therefore

interesting

to

express

R as ~R

and

to c o m p u t e

the

It is

limiting

v a l u e of P for ~ d e c r e a s i n g to zero. A l s o b e t t e r b o u n d s can be found by choosing

Q = Q1 + eQ2 where

the

matrix

semi-definite,

Ker(Q1)

Q2

is p o s i t i v e

a is positive,

definite,

the

matrix

Q1

is

(13)

= n i Ker(F i)

w i t h Ker d e n o t i n g

the

positive

and

kernel

or null space of a matrix.

P is computed

for the l i m i t i n g case for ~ d e c r e a s i n g to zero. This a l g o r i t h m leads to a n e c e s s a r y and s u f f i c i e n t condition, to the

exact

cial case:

*

M=

I,

maximum

allowable

noise

intensity,

in the

and hence

f o l l o w i n g spe-

533 F 1 has

rank one:

F 1 = blc I w h e r e

b I is a c o l u m n v e c t o r and c I a

row vector, the

s y s t e m has

a single

input:

B = b,

w h e r e b is a c o l u m n v e c -

tor.

Then the s y s t e m e q u a t i o n has the f o l l o w i n g form

dx = The

(14)

(Ax+bu)dt + a blC 1 dW above

discussion

leads

to the

necessary

and s u f f i c i e n t

conditions

for s t a b i l i z a b i l i t y :

(i)

(A,b)

(ii)

a2blTPob I < 1

Here zero,

Po

is

is s t a b i l i z a b l e

the

(15)

limiting

value,

as

the

positive

constant

@

tends

to

of the u n i q u e p o s i t i v e s e m i - d e f i n i t e s o l u t i o n of

ATp + PA - (I/~)PbbTp = clTCl which

is such

that

A-(I/@)bbTp

is a Hurwitz

matrix.

Another

inter-

p r e t a t i o n of b l T P o b I is

f°

blTPobl = inf k J0

(ClX)2 dt

where the output ClX

(16)

in the integral

is to be c o m p u t e d a l o n g the solu-

tions of the d e t e r m i n i s t i c s y s t e m e q u a t i o n

d x ( t ) / d t = Ax(t)

(iv)

+ bu(t)

with x(0)

= bI

the above

expression

and with state

f e e d b a c k control

is to be t a k e n o v e r the

u = kx. The i n f i m u m in

set of s t a b i l i z i n g

state

f e e d b a c k control strategies.

If b l T P o b l noise

is

zero,

intensities.

then

This

the

system

special

case

is r o b u s t l y shows

stabilizable

clearly

tween p e r f e c t robust s t a b i l i z a b i l i t y on the one hand,

all

and r o b u s t stabi-

l i z a b i l i t y for all noise intensities on the o t h e r hand. require

for

the d i f f e r e n c e be-

Both p r o p e r t i e s

534 blTPobl = 0

Moreover

(18)

perfect

robust

infimum at the right

side of

be

for

realizable

responds gain

to

by

the

(high g a i n

stabilizable,

some

stabilizability (16)

feedback

limiting

is

required

a c t u a l l y be a minimum,

control.

value

feedback),

it

for

If

however

increasing

then s y s t e m

(14)

i.e.

the

values

that

the

that it

infimum

cor-

of the

feedback

is not p e r f e c t l y

robustly

but o n l y r o b u s t l y s t a b i l i z a b l e for all n o i s e intensities.

5. P e r f e c t r o b u s t s t a b i l i z a b i l i t y

The results of S e c t i o n 3 can be u s e d to g e n e r a t e c o n d i t i o n s for perfect robust

stabilizability,

such that

that is for the e x i s t e n c e of a f e e d b a c k matrix

the c l o s e d loop system s a t i s f i e s the c o n d i t i o n s of Criterion

I. The

stabilizability

matrix

K

should

s a t i s f y the c o n d i t i o n s of c r i t e r i o n I. This can be converted to

an

such

explicit

that

condition

(A,B)-invariant

system

on

the

subspaces

invariant

subspace

state

be

can

criterion

the

is

plant

system

and

such

transferred

is that there s h o u l d e x i s t a feedback

with

data

that

by an

for

any

These

concepts

Wonham [13].

required

[7]; Many

an

that

were

the

in

discussion

developments

are

due

A+BK,

means

of

instead the

states

input

in

from

An

it

the

system

the

one

to the

system

given

Willems

subspace

stabilizes

control

also

to Jan

of

(A,B)-

the input can be chosen to be

control

the

was

of A,

concept

subspaces.

For a s t a b i l i z a b i l i t y

feedback

introduced

early

two

appropriate

loop or a f e e d b a c k control.

is m o r e o v e r

by

stabilizability

other state w i t h o u t leaving the subspace; an open

matrix

by

literature

Basile

[ii].

To

it

the system.

and

by

Marro

formulate an

e x p l i c i t c r i t e r i o n the following formal d e f i n i t i o n s are needed:

Definition 5

Let

S

be

a

subspace

of

R n.

Then

a

subspace

there exists a m a t r i x K such that V is (A+BK)V(V. and

V*(S)

denotes

Let V(S) the

the

d e n o t e the set of

largest

largest

(A,B)-invariant

(A,B)-invariant

V

is

(A,B)-invariant

(A+BK)-invariant,

i.e.

if

such that

( A , B ) - i n v a r i a n t s u b s p a c e s in S, subspace

subspace

contained

in

S.

Vg(S)

in S w i t h the

additional

Vg(S) : = s u p ( V 6 V(S) J ( A + B K ) V C V and a(A+BK) < Cg for some K}

(19)

r e s t r i c t i o n t h a t the m a t r i x A + B K is Hurwitz:

535 with

a(M)

denoting

the

spectrum

of the matrix

M,

and

Cg d e n o t i n g

the

left half complex plane Cg : = {s ~ C I Re(s) It can be proved

< O)

that these

exist s t r a i g h t f o r w a r d

concepts

algorithms

are well

defined,

and that there

to compute them.

Definition 6 The recursive

Vg,o

:

=

algorithm

{0)

, o o

Vg,j

: = Vg(n i Fi-lVg,j_l) . ° °

defines a limiting subspace Vg* in a finite number of steps. Criterion 2 System

(2) is perfectly

n-dimensional

robustly stabilizable

if and only if Vg*

is the

v e c t o r space.

Remarks (i)

It can readily be proved that Vg* = R n

is equivalent

Vg,k ~

to

n i im F i

(20)

for some k, where the sum denotes

the direct sum of v e c t o r spaces

and im denotes the image or range space of a matrix. (ii)

Suppose

there

matrix equation in

at

is only

F 1 has

rank

one

one,

stochastic such

that

element the

and the a s s o c i a t e d

system

is

described

by

(14). It can be shown that for this case Vg* is obtained

most

equivalent

two to

steps.

IIence

the

condition

of

Criterion

2

is

536 V g ( K e r Cl) This

Dim

(21)

bI

implies the e x i s t e n c e of a f e e d b a c k m a t r i x K s u c h that A+BK

is a H u r w i t z m a t r i x and is

equivalent

with

stability

ClX.

This

functions sary

and

can

to

the

from be

associated

(14)

condition the

of

with

vanishes

disturbance

disturbance

expressed

sufficient

b i l i t y of

c(Is-A-BK)-ib

input

explicitly

the

condition

identically. decoupling

im b I to

in terms

the

of the

It

[14]

output

transfer

original

system

model:

the

for the

perfect

robust

stabiliza-

neces-

is that the ratio of t r a n s f e r f u n c t i o n s

Cl(Is-A)-Ibl/Cl(IS-A)-ib

is s t r i c t l y after

proper

and

has

only poles

with

negative

Note

the

real parts,

cancellation

of common

factors.

relation

of this

c o n d i t i o n w i t h the

(stronger)

p r o p e r t y that the s y s t e m is minimum

phase w i t h respect to the output ClX.

Example 2 The system w i t h s t o c h a s t i c p e r t u r b a t i o n s

IdxllE° :IExll =

|dx2|

dt

0

-

+

dt

+

a

[i

x2

dW

2]

2

can be p e r f e c t l y r o b u s t l y s t a b i l i z e d by means of the f e e d b a c k control

u = -~Xl-(2~-O.5)x 2 where

a is an a r b i t r a r y p o s i t i v e constant.

W i t h this

f e e d b a c k the con-

t r o l l e d s y s t e m is m e a n square a s y m p t o t i c a l l y s t a b l e for any intensity a of the noise.

The c o n s t a n t a can be u s e d

for p o l e a s s i g n m e n t purposes.

The s y s t e m indeed s a t i s f i e s the above c r i t e r i o n since

c l ( I s - A ) - I b l / c l(Is-A) b = i/(2s+l)

537 6. R o b u s t s t a b i l i z a b i l i t y for all n o i s e i n t e n s i t i e s

In this s e c t i o n the q u e s t i o n is c o n s i d e r e d to w h a t e x t e n t the c r i t e r i o n of the

previous

s e c t i o n can be r e l a x e d

if only s t a b i l i z a b i l i t y

of the

system is r e q u i r e d for all u i. This m e a n s that for any a i a s t a b i l i z i n g feedback m a t r i x K s h o u l d exist,

dM/dt = (A+BK)TM + M(A+BK) is a s y m p t o t i c a l l y matrices.

Since

stable

the

such that

(22)

+ Z i ai2FiMFi T in

the

matrix

cone

K may

of

nonnegative

depend

on

the

definite

noise

(n x n)

intensities

some e l e m e n t s of K m a y go to infinity as the noise i n t e n s i t i e s

ai,

increase

without bound. T h e n t h e r e does not exist a s i n g l e f e e d b a c k m a t r i x which stabilizes

(4)

in

the

mean

square

for

all

noise

intensities

and

the

system is h e n c e not p e r f e c t l y r o b u s t l y s t a b i l i z a b l e . It has

been

shown

[5] that

for s t a b i l i z a b i l i t y

for all

ties a v e r y e l e g a n t r e l a x a t i o n of the c o n d i t i o n s

noise

intensi-

of c r i t e r i o n

2 can be

obtained by m e a n s of the clever c o n c e p t of a l m o s t

(A,B)-invariant

sub-

spaces

introduced

(A,B)-invariant

sub-

space

is such t h a t

input

such

that

other w h i l e However, quired input

by Jan

for any two

the

the

Willems

system

may

or an

states

state

trajectory

if the m a x i m a l input

larger

infinite gain

An

almost

in the s u b s p a c e

is

remains

distance

become

[ii].

transferred

arbitrarily

becomes and

and

f e e d b a c k control.

with

the

open

left

half

complex

plane

and

the

one

to the

to

an

the

s u b s p a c e s Vg,j

replaced

by

an the

re-

impulsive

in the p r e v i o u s

Cg

to

subspace.

smaller,

tend

Let the

defined in the same way as the s u b s p a c e s Vg,j but

close

smaller

larger

t h e r e exists

from

be

section,

the

closed

left half p l a n e

Cg

: =

{s e c

[ Re(s)

Let R b * ( S ) d e n o t e this m e a n s which the

any

that two

Ll-norm

small.

of

~ 0)

the largest L l - a l m o s t

R b * ( S ) is the

states the

can

largest

(A,B)-invariant subspace

be t r a n s f e r r e d

distance

of the

state

to one

of the another

trajectory

subspace state while

in S;

space

in

keeping

to S a r b i t r a r i l y

For a formal d e f i n i t i o n the r e a d e r is r e f e r r e d to the l i t e r a t u r e

[ii]. Then we o b t a i n

538 Criterion 3 System

(2)

is robustly

is s t a b i l i z a b l e

stabilizable

for all noise

intensities

if

(A,B)

and

Vg(n i FiVg* ) + Rb*(n i Fi-IVg* ) ~

Z i im F i

where Vg* has the same meaning as in the previous

section.

Remark For the special

case of system

sary and sufficient

(14) the above criterion yields a neces-

condition

Vg(Ker Cl) + Rb*(Ker Cl)

~

im b I

(23)

This can elegantly be expressed as a frequency domain criterion. (23)

is equivalent

to the condition

zable and that the ratio of transfer

that the system

(A,b)

Indeed

is stabili-

functions

Cl(IS-A)-ibl/Cl(IS-A)-Ib has

no

poles

factors.

The

with

positive

condition

bance d e c o u p l i n g

real

parts,

is equivalent

[ii] with stability

after

cancellation

to the property

of

of almost

common distur-

from the input im b I to the output

ClX. Example 3 The system with stochastic p e r t u r b a t i o n s

LJ:dXII IEXII I: I:l i:l IXll =

dt +

|dx2|

can

be

-

robustly

intensities

dt + a

[a

x2

stabilized

dW

2]

x2

by

if a is nonnegative.

means

of

state

It is readily

feedback

for

all

noise

seen that the system is

539 not

perfectly

robustly

stabilizable

if

a~l.

Indeed

one

of

the

condi-

tions such that

u = - ~X -- ~X s t a b i l i z e s the s y s t e m in the m e a n square sense,

is

3;] > 4 ( a - 1 ) 2 a 2 This

shows

that,

gain m u s t

except

for the

case

where

increase without bound with

a

equals

the n o i s e

a g r e e m e n t w i t h the f r e q u e n c y d o m a i n condition.

i,

the

intensity.

feedback

This

is in

We o b t a i n

c l ( I s - A ) - I b l / c l ( I s - A ) - I b = [2(a-1)s+a]/(2s+a) The p o l e of this rational then

the

stochastic

intensities. only

The

f u n c t i o n is n o n - p o s i t i v e

system

function

in t h a t case

is

robustly

if a is n o n n e g a t i v e ;

stabilizable

is s t r i c t l y p r o p e r

only

for

all

if a is equal

noise to

i;

is the s t o c h a s t i c s y s t e m p e r f e c t l y r o b u s t l y s t a b i l i -

zable.

7. F u r t h e r R e m a r k s

i.

A similar

analysis

can be p e r f o r m e d on robust s t a b i l i z a b i l i t y

of

d i s c r e t e - t i m e systems with s t o c h a s t i c p a r a m e t e r s

x(t+l) where

= Ax(t) the

+ Bu(t)

processes

time processes. noise time

is

vious paper 2.

It was

fi(t)

are

The a n a l y s i s

technically

systems.

+ Z i aifi(t)Fix(t )

Some

more

results

zero

mean

white

of d i s c r e t e - t i m e

straightforward and e x a m p l e s

noise

discrete-

systems w i t h w h i t e

than

for

continuous-

are d i s c u s s e d

in a pre-

[5].

shown

that

perfect

robust

stabilizability

corresponds

to

the p o s s i b i l i t y of s i m u l t a n e o u s l y b l o c k t r i a n g u l a r i z i n g the plant matrix

A+BK

of the c l o s e d

that the d i a g o n a l b l o c k s zero

matrices.

robustness tions:

is

This also

loop s y s t e m and the m a t r i c e s

of A + B K are H u r w i t z

readily valid

for

shows

that

nonlinear

in

Fi,

such

and t h o s e of F i are this

case

time-varying

perfect

perturba-

540 dx(t)/dt It

can

the

= Ax(t) be

+ Bu(t)

shown

criteria

that

some

of S e c t i o n

(24)

+ 7 i fi(x(t),t)Fix(t) (minor)

5 valid

changes

are

required

to make

for such s y s t e m s w i t h nonlinear

t i m e - v a r y i n g perturbations.

The

3.

condition

if

the

for

perfect

stochastic

system

c h a s t i c processes,

robust (3)

and also

stabilizability

is

considered

with

remains

valid

non-white

sto-

if other m o m e n t s t a b i l i t y properties

are considered.

8. C o n c l u s i o n

In

this

contribution

work w i t h tems. been

Conditions derived

bilization. original the very

a

Jan Willems on

which It

qualitative

the

been

shown

that

given

control

admissible

levels

of

stability,

the

of

some

concept

of

of

the

almost

joint

of u n c e r t a i n

analysis

the

sys-

perturbations

have

stabilizability, relies

algebraic

research

dynamic

heavily

results d e v e l o p e d by Jan Willems,

properties

interesting

has

do not d e s t r o y

was

research

review

on the robust

earlier

such as results on

Riccati

(A,B)-invariant

or sta-

on

equation

or a l m o s t

and the control-

l a b i l i t y subspaces.

References

[i]

Willems,

J.L.

nonlinear ters), [2]

J.C.

Willems,

nonautonomous

system",

vol.

Willems,

and

56, p. 244-245,

J.L.

and

nichtlinearer Steuern, [3]

Willems, methods

J.L.

vol.

and

the

[4]

Willems,

vol.

J.L.

PAS-89,

and

stochastic

systems

Automatica,

vol.

J.C. with

criterion

for a

of

the

IEEE

"Untersuchung

der

Stabilit~t

im

Frequenz-bereich",

114-116,

Willems,

computation

m u l t i m a c h i n e p o w e r systems", and Systems,

Proceedings

Willems,

ii, pp.

J.C.

stability

(Let-

1968.

Regelungssysteme

Regeln, to

J.C.

"A

of

Messen,

1968.

"The

application

transient

stability

of

Lyapunov

regions

for

IEEE T r a n s a c t i o n s on P o w e r Apparatus

pp. 795-801, Willems, state

12, pp. 277-283,

1970.

"Feedback and

control

1976.

stabilizability dependent

for

noise",

541 [5]

Willems, tain

J.L.

and J.C.

systems",

SIAM

21, pp. 352-374, H.

Willems,

Journal

"Robust

of

stabilization

Control

and

of uncer-

optimization,

vol.

1983.

[6]

Kwakernaak,

and

[7]

Wonham, W.M., Linear Multivariable Control: A Geometric Approach,

Wiley-Interscience,

R.

Sivan,

New York,

Linear

Curtain,

R.F.

(ed.), Modelling,

tion in Control Systems, Workshop, [9]

AC-16, [i0]

J.C.,

IEEE

pp. 621-634,

Willems,

J.C.,

Willems, gain

[12]

"Almost design

invariant

- Part

235-252,

1981.

Willems,

J.L.,

Basile,

" Mean

Problems G.

subspaces Willems,

and the algebraic Control,

subspaces",

subspaces:

vol.

Ast~risque,

an approach

I: almost controlled

on Automatic Control,

square

of

stability

Control

and

vol.

criteria

Information

to high

invariant AC-26,

subpp.

for stochastic Theory,

vol.

2,

1973. and

in

G.

Marro,

linear

J.C.

and

C.

"Controlled

system

Theory and Applications, [14]

control

on Automatic

1980.

IEEE Transactions

pp. 199-217, [13]

optimal

1971.

spaces",

systems",

1987.

"Almost A(modB)-invariant

J.C.,

feedback

Berlin,

Transactions

vol. 75-76, pp. 239-248, [ii]

1972.

Robustness and Sensitivity Reduc-

"Least squares

equation",

Systems,

Proceedings of a NATO Advanced Research

Springer-Verlag,

Willems, Riccati

Control

1972.

2nd Ed., Springer Verlag, New York, [8]

Optimal

theory",

and conditioned Journal

vol. 3, pp. 306-315,

Commault,

"Disturbance

of

1969. decoupling

surement feedback with stability or pole placement", on Control and Optimization,

vol.

invariant

Optimization

19, pp. 490-504,

by mea-

SIAM Journal

1981.

Acknowledgment

The

author

gratefully

acknowledges

Belgian Fund for Scientific Research

partial

research

support

(F. K. F. O. Grant).

from

the

On the Control of Discrete-Event Systems W. M. Wonham Systems Control Group, Dept. of Electrical Engineering University of Toronto Toronto, Ontario, Canada M5S 1A4

INTRODUCTION A discrete-event system (DES) is a dynamic system whose behavior is characterized by the abrupt occurrence, at possibly unknown irregular intervals, of physical events. For example, an event may correspond to the arrival or departure of a customer in a queue, the breakdown or restoration to service of a machine, or the transmission or reception of a message packet. Thus DES arise in service and logistic systems, manufacturing, and communications, as well as in many other domains such as vehicular traffic, and robot and process control at the level of task coordination. Abstractly the distinguishing features of DES are that they are discrete (in time and state space), asynchronous (event- rather than clock-driven), nondeterministic (generative and capable of internal choices), and modular (composed of quasi-independent component DES down to some level of primitives). In addition DES may be equipped with various means of control and intercommunication, notably for the enablement/disablement of selected controllable events and the signaling of observable events from one module to another. Control and communication are to be coordinated so that the flow of events within the system takes place in accordance with designer specifications.

543

The increasing complexity of man-made DES made possible by computer technology underlines the need for formal theories of and systematic design approaches to DES control. Standard control theory and design, though finding application to DES at the level of smallsystem optimization [Ho87, Co85], is linear-space-based and is ill-equipped to address the larger structural issues, which cannot be framed in a linear setting; while modelling approaches originating in computer programming theory (e.g. [Hr85]) neither capture the crucial properties of DES relating to their degree of controllability and observability nor formalize the issues of control and communication architecture. A control paradigm adapted to the DES area was introduced in [RW82] and has since been actively developed by those authors, their coworkers, and others (see [RW89] for an extensive bibliography). We refer to this framework as RW. While based on the broad concepts of feedback control and communication, RW incorporates the distinguishing features of DES itemized above, via constructs from automata, formal language and formal logic. While unspecific as to applications, RW has already been exploited by workers in communication protocols [Ci88], database management [La87], and flexible manufacturing [Ma86]. In this paper we provide a summary overview of RW, referring the reader for most technical details to the literature.

REPRESENTATION OF DES IN RW In RW a DES is represented either by a formal language or, more concretely, by the generator of a formal language. For the application of control-theoretic techniques it is convenient to take as the generator a state transition structure (automaton). A typical primitive example, that we shall call MACII, is the 'machine' displayed in Fig. 1. The three states are labelled I ('idle'), W ('working') and D ('broken down'); the corresponding transitions, or

events, then have the obvious interpretations. Tile events are labelled by symbols from an event alphabet, in this case the set 6={o~,~,)~,~t}. In the absence of any control, MACII may be thought of as spontaneously generating strings of symbols a~ E in accordance with the graph, starting from I as the initial state. Depending on the purposes of analysis one may consider all possible infinite strings so generated, or focus attention only on all possible strings of finite length. For simplicity we restrict attention for now to the finite strings. In standard notation the set of finite strings formed from Z is denoted by Z*; the strings (or

words) that can be generated by MACII thus make up a subset of Z* that we call the closed behavior L(MACII). It may be useful to select from L(MACII) those words that correspond to completed cycles of the type ~13 or 00~ta, or finite sequences of completed cycles: this may be done by marking suitable states of tile transition graph (not to be confused with 'marking'

544

of Petri nets). In this example I is marked and the resulting subset of marked strings in L(MACH) is called the marked behavior of M A C H , denoted by Lra(MACH). A feature of RW is that the foregoing more-or-less standard description is augmented by a control function as follows. We select a subset Xc ~ Z of events to be controllable; the complementary subset 2;u is uncontrollable. In M A C H the controllable subset is Yc = {cz,p} and is distinguished in Fig. 1 by a 'tick' on the event arrow. Controllable events have the interpretation that they can be disabled (prevented from occurring) or enabled (allowed but not forced to occur) by some control agent, for the moment unspecified; while an uncontrollable event can never be directly prevented from occurring if the DES happens to be at the appropriate state, hence is always enabled. Which events are declared to be controllable is a matter of modelling; for instance in M A C H it may be not unreasonable to assume that the transition I ~ W (initiation of a work cycle) can be disabled, and that D --~ I can be disabled (by witholding servicing in the case of breakdown), but that W ---~I (successful completion of a work cycle) or W --->D (breakdown) will occur uncontrollably, in accordance with underlying physical mechanisms which the controller is unable to access directly. In this scenario control by an extemal agency is permissive in the sense that no event is 'forced', except possibly by disabling all the alternatives and thus forcing by default. Extensions of the model to accommodate forced events explicitly have been proposed ([GR87], [BH88]) but will not be considered here. Several constructions exist for combining primitive DES like M A C I I into more elaborate structures of the same type: the simplest is shuffle, which creates a product structure DES from components over disjoint alphabets, and models the situation in which the generating actions of the components proceed independently from and asynchronously with one another. More generally one can bring in the synchronous product, allowing a priori synchronization of events having common labels in distinct components, and the concurrency

product (cf. [LiW88a]), which allows for the possibility of unsynchronized (i.e. unforced) simultaneity of events in distinct components. Finally, there is no compelling requirement that the state sets be finite; nor is it necessary to consider the models simply as "raw" transition structures. For instance the algebraic structure of vector addition systems (as in Petri nets) can be adjoined to the foregoing model and exploited to advantage in situations where the system state, or one of its factors in a product structure, can be modelled on the nonnegative integers, as for the occupancy number of I, IV or D in a group of machines, or the content of a buffer (cf. [LiW88b]).

545

C O N T R O L L A B L E LANGUAGES AND C E N T R A L I Z E D SUPERVISION OF DES

The type of control problem for which the foregoing modelling approach is natural is that of manipulating the controllable events, in the light of system past history, in such a way that the closed and marked behaviors actually generated by the DES under control satisfy designer specifications. Let G denote the DES to be controlled, S denote the controller (its style of representation is not important at the moment) and L (S/G) resp. Lm(S/G) the closed rcsp. markcd behavior of 'G under control of S'. Let A, E be sublanguages of Lm(G) corresponding respectively to 'minimal acceptable' and 'maximum pemfissible' marked behavior. Then a possible specification on S is that A ~Lm(S/G)~E. The questions now are whether any such S actually exist; if so, whether or not some notion of optimality can be attached to make the selection; and finally whether or not the whole approach can be made constructive in principle and computationally feasible in practice. To examine these problems we note first that a physically realizable controUer can do no more than map strings s ~ L ( G ) to subsets Y." of controllable events, the interpretation bcing that only events cr~Y.'u Eu are candidates for the event immediately following tile generation of s; namely events in E ' ~ E c are enabled, while events in Ec-E" are disabled. Under this constraint, the next event ~ (if one is possible) is generated in accordance with the transition structure of G, and the process is repeated. What sublanguages of Lm(G) can be generated by a controller, or supervisor, acting in this manner? To answer this question we nccd a concept of controllability. For any language K ~E*, denote by K the prefix-closure of K, namely K togcther with all the prefixes (initial segments), including the empty prefix, of strings in K. Then K is controllable provided

KZ. nL (G) g h"; namely the next occurrence of an uncontrollable event in O can never cause a string already in K to exit from K. (Here ~'E,, denotes the set of strings of the form so, with s ~ K and OEZu). Controllability can be thought of as "invariance with respect to the occurrence of uncontrollable events". The answer to our question is now immediate: K =Lra(S/G) for some supervisor S if and only i f K ~Lm(G) and K is controllable. Examination of the controllability condition reveals two properties that can be used to settle on a definition of optimality. The first is that the empty language is controllable: setting K = O we get that K=O, hence KZu=O and the claim is proved. The second is that the condition is closed under arbitrary unions, a claim that is easily verified from the convenient fact that the prefix-closure of an arbitrary union of languages is the tmion of the prefixclosures. Thus if E ~Lm(G) and C(E) is the collection of controllable sublanguages of E, then C(E)#O since ~

C(E) and so, taking the union of members of C(E), we find that the

fi46

supremal element Kmp := sup C(E) exists and belongs to C(E). Since Ksup is controllable and belongs to E, it is the natural candidate for the 'optimal' solution of the problem Lm(S/G) ~ E; a supervisor S that implements, or 'synthesizes' Ksup, is maximally permissive with respect to the constraint E. Finally we have an abstract solution to our supervisory existence problem: it is solvable if and only if Ksup ~ A It can be shown that Ksup can be characterized as the largest fixpoint of a certain mapping on Pwr(E*), the set of sublanguages of E*. In general this provides an approach to the explicit computation of Ksup by successive approximation. In the regular case (when all the given languages are representable by finite state generators) this computation converges to Ksup in a finite number of steps of worst case order IIEIt'IILm(G)II,where I1-11denotes Nerodc index (state size) of the indicated language. In practice the convergence is much faster. We have now shown that satisfactory solutions are at hand to the problems raised at the beginning of this section, at least in the regular case. Software that implements the approach is available; examples and computational detaiIs can be found in [Wo881. We now turn to architectural issues.

M O D U L A R SUPERVISION OF DES

As indicated in the Introduction, DES are often built up from modular elements, and so it makes sense to carry over the idea of modularity to control itself (cf. [WR88]). Very often a control task will consist of several specialized subtasks: for instance the group of machines making up a work cell may be subject to one control specification in respect to the prevention of overflow and underflow of buffers, and another that establishes priorities of repair whe,~ one or more machines are down. Controllers dealing with specialized subtasks may often be designed rather easily from tile subtask specifications; and these subcontrolIers can then be run concurrently to implement a modular solution of the original problem. In addition to being more easily synthesized, such a modular supervisor should ideally be more readily modified, updated and maintained. For example, if one subtask is changed, then it should only be necessary to redesign the corresponding subcontroller: in other words, the overall modular supervisor should exhibit greater flexibility than its 'monolithic' counterpart. Unfortunately these advantages are not always to be gained without a price. The fact that the individual control modules are simpler implies that their control action must be based on a partial or 'local' version of the global system state; in linguistic terms, a

547 subcontroller processes only a

projection of the behavior of the DES to be controlled. A

consequence of this relative insularity may be that different subsupervisors, acting quasiindependently on the basis of local information, come into conflict at the 'global' level, and the overall system thereby exhibits

blocking (inability to complete the global task) or even

deadlock (inability to continue operation). Thus a fundamental issue that always arises in the presence of modularity is how to guarantee the nonblocking property of the final synthesis. We can focus more sharply on the blocking issue through the following definition: languages L 1 and L2 are nonconflicting if Llt-,,L z = L t c~L2; namely any string that is both a prefix of L t and a prefix of L 2 can be completed to a common word o f L 1 and L 2. It can be shown that two individually nonblocking subsupervisors St and $2 acting concurrently will yield a nonblocking individual

languages

conjunction SIr, S 2 just in case tile

Lm(SI/G) and L~(S/G) are nonconflicting.

Furthermore,

if

E 1,E2 _cL,n(G) and if sup C(E I), sup C(E2) are nonconflicting, then sup C(E I h E 2 ) = sup C(E 1) n sup C(E2) While these results are more-or-less immediate consequences of the definitions, they lead to direct computational procedures for validating any proposed modular design in respect to nonblocking and optimality, as well as to analytical methods of inferring these desirable properties from an examination of specific modular structure in special cases. The reduction in complexity gained by exploiting modularity can be dramatic. As a simple example consider three 'machines' of the type of MACH together with a buffer, arranged as in Fig. 2. The buffer, of capacity 3, serves as output (sink) for MACII1 and MACII2 and as input (source) for MACH3. The specifications are (i) the buffer must not overflow or underflow, (ii) MACH1 and MACH2 are repaired in order of breakdown, and (iii) MACII3 has priority of repair over MACII1 and MACII2. As the DES to be controlled we take tile shuffle BIGMACII of MACHI, MACII2 and MACII3, consisting of 27 states and 108 transitions (written (27,108)). Expressing tile specifications as languages and combining these into their intersection, we obtain the 'monolithic' specification as a generator BIGSPEC (32,248). The optimal 'monolithic' supervisor is computed as an automaton BIGSUP (96,302), evidently a rather cumbersome structure to implement directly. By contrast the same behavioral result can be obtained by inspection using the conjunction of 4 subsupervisors based rather directly on the given individual specifications; the largest of these modular components has only 4 states. The details may be found in [Wo88].

548

I t l E R A R C I t I C A L SUPERVISION OF DES

Hierarchical structure is a familiar feature of the control of complex dynamic systems, where the controlled system may be thought of as executing some overall high-level task. It may be described generally as a division of control action and the concomitant information processing according to scope. Commonly, the scope of a control action is defined by the extent of its temporal horizon, or by the depth of its logical dependence in a task decomposition. Generally speaking, the broader the temporal horizon of a control and its associated subtask, or the deeper its logical dependence on other controls and subtasks, the higher it is said to reside in the hierarchy. Frequently the two features of broad temporal horizon and deep logical dependency are found together. Hierarchical structure in tile control of DES can be investigated in RW by means of a mild extension of the framework already introduced. While different approaches to hierarchical control might be adopted even within this restricted framework, the theory to be summarized in this section does capture the basic feature of scope already mentioned, and casts some light on an issue that we call hierarchical consistency. Our account follows [ZW88]. In outline our setup will be the following. Consider a two-level hierarchy consisting of a low-level plant GIo and controller Cio, along with a high-level plant

Ghl and controller Chl.

These are coupled as shown in Fig. 3. Our viewpoint is that Glo is the actual plant to be controlled in the real world by Cio, the operator; while Ghi is an abstract, simplified model of Gio that is employed for decision-making in an ideal world by Cht, the manager. The model Gl,i is refreshed or updated every so often via the information channel (or mapping) labelled Inflom (information low-to-high) to Gm from Glo. Alternatively one can interpret InfJohi as can-ying information sent up by the operator Cio to the manager Cht: in our model the forreal result will be the same. Another information channel, Inflo (low-level information), providcs conventional feedback from Gio to its controller CIo, which in turn applies conventional control to Glo via the control channel labelled Conb (low-level control). Returning to the high level, we consider that Ghi is endowed with control structure, according to which it makes sense for Cht to attempt to exercise control over the behavior of Ghl via the control channel COnhl (high-level control), on the basis of feedback received from Ghl via tile information channel Inf m (high-level information). In actuality, the control exercised by Chl in this way is only 'virtual', in that the behavior of Ght is determined entirely by the behavior of Gb, through the updating process mediated by Inftohi. The structure is, however, complctcd by the command channel COmhl b linking Chi to Czo. The function of Comhizo is to convey the manager's high-level control signals as commands to the operator Cj~,, which must translate these commands into corresponding low-level signals which will actuate Gio via Con~o. State changes in G b will eventually be conveyed in summary foma to Ghl via

549 Inflohi. Ghi is updated accordingly, and then provides appropriate feedback to Chl via Infl, t. In this way the hierarchical loop is closed. The forward path sequence COmhtlo; Conto is conventionally designated "command & control", while the feedback path sequence Inftohi; Infht will be referred to as "report & advise". As a metaphor, one might think of the command center of a complex system (e.g. electric power distribution system,...) as the site of the high-level plant model Gt,l, where a high-level decision-maker or manager Ch! is in command. The external real world and those operators coping with it are embodied ill GIo, CIo. The questions addressed by the theory concern the relationship between the behavior required, or expected, by the manager Chl of the high-level model Ghl, and the actual behavior implemented by the operator Cio in Gto in the manner described, when Gzo and Inflohl are given at the start. It will turn out that hierarchical consistency between these behaviors imposes rather stringent requirements on Inflohi and that, in ge,~eral, it is necessary to refine the information conveyed by this channel before consistent hierarchical control structure can be achieved. This result accords with the intuition that for high-level control the information sent up by the operator to the manager must be timely, and sufficiently detailed for various critical low-level situations to be distinguished. As usual we model GIo as the generator of a language Lto :=L (Gjo)~ E*, with the partition E = EcuY'.. as before. For M ~ L (G) we use the abbreviated notation M T := sup C(M) Now let T be a new alphabet of 'significant event labels'. T may be thought of as the events perceived by the manager which will enter into the description of the high-level plant model Gl~l, of which the derivation will follow in a moment. First, to model the information channel lnflot~i we postulate a map rl:Lto ~ T* with the properties rl(e) = e (e denotes the empty string over any alphabet)

~e

ither rl(s ) rl(sc) = [ o r rl(s)'r, some z ~ T for s~E*, t e E .

Thus rl is causal in the sense that it is prefix-preserving: if s.<_s" then

rl(s)
To := TU{Xo}, where Xo is a new symbol (~ 73 interpreted as the 'silent output symbol'. The

55O

abstract construction of

GIo,new from

Gio and 1"1is a routine exercise that we omit; and we

now rename Gin, new as simply Gio. The states of Gto are either vocal (state-output in T) or silent (output xo); a silent path in Gto is a path in the transition graph of Gio that starts at a vocal state and whose subsequent states are all silent. At this stage we temporarily define Ghl. For this we note that, in the absence of any control action, Glo generates the uncontrolled language Lto (unchanged from before). For now, Gm will be taken as the canonical recognizer for the image of Llo under rl: L(Ghl) = rl(Llo) ~ T * and we write L (Ghi) =: Lhi. As yet, however, the event label alphabet T of Ghi needn't admit any natural partition into controllable and uncontrollable subalphabets; that is, Ghi needn't admit any natural control structure. This defect can be remedied by refining the state structure of Glo and by splitting the elements of T into 'siblings'. Thus each ~e T is split into a pair (Zc,X,); in the extended state structure Gio,ext, say, a state-output '~ at a vocal state q is replaced by either "~c or x u according as q can or cannot be made unreachable along all silent paths in Gto,ext leading to q, by disablement of suitable ~ e Zc. It can be shown that the state size of Glo,ext is at most double that of Glo; in random examples the factor is usually much less. Now T is rephtced by Texs, say, in this way, and To by To.ext :=TextU{xo}. The corresponding extended map rl=t will induce an extended high-level m o d e l (~hi,ext o v e r Tex t. Since each element in Tcxt iS now unambiguously controllable or uncontrollable, the structure Gio,ext is said to be output-

control-consistent (OCC). Ghi,ext will be a DES having standard RW control structure, tO which the usual methods of Sects. 2 and 3 can be applied. Henceforth we assume that this extension procedure has been carried out (it has been implemented in the regular case), and drop the subscript 'ext'. Despite the fact that Glo is now OCC, it needn't be true that the controllability property i~ mapped in either direction between Gio and Ghl: that is, Klo ~L(GI,,) may be controllable with respect to Gio, yet Khi :=~(KIo)~L(Ghi) not controllable with respect to Gja; converscly Khi controllable does not imply that Kto :=r1-1 (Khl) is controllable.* The property that Gio is OCC is exploited as follows. High-level supervisory control is determined by a selection of high-level controllable events to be disabled, on the basis of high-level past history. That is, Cl~l is defined by a map

~[hi : Lhi x T --> {O, 1} such that %i(t,'c)=l for all t~Lhi and z~Tu. As usual, if )'hi(t,'~)----0the event labelled "¢is said *This remark corrects an inessential but annoyingerror in lZW88], where Proposition3.1 should be deleted.

551 to be disabled; otherwise x is enabled; of course, only controllable events ('c~ Tc) can be disabled. The result of applying this control on the generating action of Ghl would amount to the construction of a suitable supervisor over T as input alphabet. However, in the hierarchical control loop direct implementation of Chl is replaced by command & control: the action of Chz on Gta must be mediated via Coml~llo and Conto as already described. With Th/given, it turns out to be possible to construct a corresponding low-level disabled event map

Tto :Lto ×Z---> {O,1} that matches the command and control structural constraint, by which the operator can execute a command of the form "disable x" received from the manager. Now suppose that a nonempty closed specification language Ehi ~Lhi is established by the manager. It may be assumed that Eh/ is controllable; otherwise the manager simply replaces Ehi by E~i. Next Thi is determined in such a way that the corresponding high-level controlled language,

L ('Yhi,Ghi), say, is Ehi (or would be Ehi if direct control of Gja by Cl~l in the sense of Sect. 1 were possible). Define Eto as the preimage in Lto of Ehi under the map r I corresponding to Inl'lohl: Eto := 11-I (Ehi) ~ Lto In general, as we know, Eto is not controllable. Let Tto be detemlincd from Eto as described above. The main consequence of output-control-consistency is that by use of Tto the closedloop language L(Tto,Gio) synthesized in Gio by command & control is as large as possible subject to the constraint Eto just defined: L

=

Obviously the transmitted high-level behavior will satisfy the required specification constraint:

rl(L('~to,Gj,,)) ~ Ehi but in general the inclusion will be proper. That is, while the 'expectation' of the high-level controller Chl on using the control "Yhi might ideally be the synthesis in Gt~l of the controllable behavior Ehi, only a proper subset of this behavior can in general actually be realized. The reason is simply that a call by Cm for the disablement of some high-level event x~ Tc may require Cio (the control Tto), as an undesired side effect, to disable paths in GIo that lead directly to outputs other than x. However this result is the best that can be achieved under the current assumptions about Gjo. The main result above will be called low-level hierarchi-

cal consistency. Intuitively it guarantees that the updated behavior of Ghl will always satisfy the high-level specification constraint, and that the 'real' low-level behavior in Gio is as large as possible subject to this constraint.

Nevertheless, the situation from the manager's

viewpoint is still unsatisfactory: the high-level behavior he expects may be larger than what

552 the operator of Glo can optimally report. The desirable situation would be that, whenever Ehi is controllable, then TI((Ti-I ( Ehi) ) "r) = Ehi The foregoing property will be called high-level hierarchical consistency. In that case, the command and control process defined for Ehi will actually synthesize Ehi in Ghl. Achieving this property in general requires a further refinement of the transition structure of Glo: in other words, the possibly costly step of enhancing the information transmitted by Gto to'Ght. Suffice it to say here that the appropriate construction can be carried out effectively (at least in the regular case), resulting in the property for the refined version of Gto that it is now

strictly output-control-consistent (SOCC). That is, with Glo now SOCC, high-level hierarchical consistency is achieved for arbitrary high-level specification languages. Two conclusions that may be drawn from this rather involved discussion are that, first, RW supports a plausible hierarchical control architecture; but secondly, the design of consistent hierarchical supervisory controls can demand quite refined consideration of low-level system structure and of the definition of high-level significant events. The theory will be illustrated by developing a high-level hierarchical supervisor for Transfer Line, consisting of two machines M1, M2 plus a test unit TU, linked by buffers B1, B2 in the sequence: MI, B1, M2, B2, TU (Fig. 4). State transition diagrams of M1, M2, and TU are displayed in Fig. 5. TU either "passes" or "fails" each processed workpiece, signaling its decision with events 60, 80 respectively. In case of "pass test", the workpiece is sent to the system output (event 62); in case of "fail test", it is returned to BI (event 82) for reprocessing by M2. There is no limit on the number of failure/reprocess cycles a given workpiccc may undergo. For ease of display we consider only the simplest case, where B1 and B2 each has capacity 1. Initially an optimal low-level supervisor is designed by any of the methods of previous sections, to ensure that neither of the buffers is subject to overflow or underflow. In detail, let PL = shuffle(M1,M2,TU); and let B1SP, B2SP be the buffer specification generators (Fig. 6)*. Then we set BSP = meeI(B1SP, B2SP), and

* Boldface function names refer to computingprocedures of the softwarepackage TCT, available from the author.

553 P L S U P = supcon(PL, BSP) as displayed in Fig. 7. With P L S U P as the starting point for the development of hierarchical structure, we must first assign the "significant" events to be signaled to the "manager". Let us assume that the manager is interested only in the events corresponding to "taking a fresh workpiece" (low-level event 1, signaled as high-level event xl, say), and to "pass test" (lowlevel event 60, signaled as z2) or "fail test" (low-level event 80, signaled as x3). If too many failures occur the manager intends to take remedial action, which will start by disabling the failure/reprocess cycle. To this end the uncontrollable event 80 is now replaced in the lowlevel structure by a new controllable event 81. Furthermore, the meaning of the signaled events "el, x2, x3 must be unambiguous, so a transition entering state 1 like [8,62,1] must not be confused with the "significant" transition [0,1,1]; namely a new state (say, 12) must be introduced, transition [8,62,1] replaced by [8,62,12], and a new transition [12,2,2] inserted. The final Moore structure, GLO, is displayed in Fig. 8. Here the vocal [state, output] pairs are [l,xl], [8,xl], [7,'t:2] and [6,~:3]. We are now ready to carry out the procedures of the theory. By inspection of Fig. 8, it is clear that each of xl, x2, x3 is unambiguously controllable, that is, G L O is already output-control-consistent. Tile corresponding high-level model GItI is displayed in Fig. 9. However, for the manager to disable x2 will require the operator to disable low-level event 5, which in turn disables the high-level event 'r3 as an undesired side effect; thus GLO is not strictly-output-control-consistent (SOCC). To improve matters it is enough to vocalize the low-level state 5 with a new high-level output x4, signaling the new "significant" event that "TU takes a workpiece". This step incidentally converts the status of x2 from controllable to uncontrollable. With this the construction of a SOCC model, say C G L O , from G L O is complete (Fig. 10). The corresponding high-level model CGIII is displayed in Fig. 11, where "~1,~2, 'r3, 'r4 have been coded respectively as 11, 20, 31, 4I. The simple model CGIII can be supervised by the manager to achieve his objective of "quality control". A possible high-level specification might be: "If two consecutive test failures (31) occur, allow TU to operate just once more, then shut down the system"; this is modeled by IIISP as displayed (Fig. 12). The resulting supervisor CGIIISUP = supcon(CGIll, tllSP) is shown in Fig. 13. On termination of CGIIISUP at state 7, it can be easily verified that C G L O will have halted at its marker state 0.

554

OTHER DEVELOPMENTS The foregoing sections suffice to convey the flavor of RW control theory for DES. A number of important topics not touched on here have been discussed in the literature. Decentralized control based explicitly on local models of the global DES is investigated in [LnW88a], while supervision based on partial observations (i.e. observation of a subset of the event alphabet) is considered in [LnW88b]. Both points of view are combined in a concept of coordination explored in [LnW88c]. We comment briefly on an extension of the theory to supervisor synthesis subject to infinite-string specifications. While the framework of languages in Y.* employed in previous sections may be adequate for supervisor synthesis subject to most practical 'safety' specifications, nothing good is ever guaranteed to happen. One way to address such liveness issues is to bring in languages with infinite strings -- so-called E°~-tanguages. In this framework an event can be required to occur "eventually", without specifically stating when. While the theory becomes more technical, it is hoped that the final results will be natural and simply expressible. In addition the new framework ought to provide a semantics for the use of temporal logic as a convenient specification language. Appropriate definitions of controllability and nonblocking in the %co setting have been provided [TW87, TW88]. The supervisory synthesis problem has the same formal appearance as in Sect. 1, except that E* is replaced by y~0. Under technical conditions, a unique optimal solution will exist. An effective solution is possible at least when the languages A, E and L ( G ) have representations as finite automata (generators) over infinite strings, a situation that can be formalized in terms of so-called Muller automata. In the solvable case, a finite, nonblocking supervisor that solves ti~e synthesis problem can be effectively constructed. Finally we mention a generalization of RW that may well be a promising approach to real-time control of DES, although as yet no formal synthesis methods have been developed for the systematic computation of 'optimal' supervisors. Extended state machines (ESMs) in the setlse of [OW87], building on IMP83] and [Hr85], model DES as extended transition structures, involving a structured state space defined as the product of an automaton-like state set for activity variables and a state space (e.g. Z n) of conventional type for (e.g. numerical) process variables. Transitions are structured to include not only a transition label but also a boolean guard, program step (variable assignment or synchronous communication), and lower and upper time bounds. The closed loop system becomes a suitably defined concurrency product of a clock ESM, plant ESMs and controller ESMs. These component ESMs interact via shared and communicating transitions. Semantically, ESMs are interprctcd as gencrators of trajectories, namely infinite sequences of states and transitions. Trajectories are established by initialization, followed by consistency with the guards, variable

555

assignments and time bounds of the transitions. Specification of the behavior of a controlled system of this kind can be carried out in a version of temporal logic that includes the realtime feature; specifications may include properties of safety, priority and real-time liveness. Important current research problems in this area revolve around such issues as supervisor verification by effective decision procedures, criteria for supervisor existence, and the distributed control problems of modular design.

CONCLUSIONS In this paper we have provided an overview of one trend among others in the development of a control theory for discrete-event systems. In view of the relatively long history of prior approaches to discrete-event control design (notably discrete-event system simulation, and analysis via Petri nets, starting in the 1960s; and investigations via queueing theory and its variants, including perturbation analysis, from the early 1970s) it is perhaps surprising that attempts to evolve a synthetic, control-theoretic overview of the problem area, especially in its qtmlitative, logical aspects, have been both few in number and recent in appearance. In any case, it can fairly be said that control of DES is now an established branch of control theory. The current studies of control of DES in its qualitative aspects highlight the thesis that control science is defined in terms of problems and concepts, not in terms of techniques. In general control science may be described as the study of how information and dynamics are brought into purposeful interaction. Stimulated by the demands of technology and by developments in computer science, control science has entered a new phase, where discreteness, modularity and communication are fundamental. Alongside the traditional mathematics of control theory like differential equations and operator theory, new techniques are entering the field from automaton theory, formal language and formal logic; while developments in computer programming methodology, as for instance abstract data structures and the object-oriented paradigm, may strongly influence the way this new mathematics (new in control theory) will be put to work. For both researchers and educators in the control field, the challenges are plentiful.

REFERENCES

[BH88]

Y. Brave, M. Heymann. Formulation and control of real time discrete event processes. Proc. 27th IEEE Conf. on Decision and Control, IEEE Control

556

Systems Society, New York, Dee. 1988, pp. 1131-1132.

[Ci881

R. Cieslak, C. Desclaux, A. Fawaz, P. Varaiya. Supervisory control of discrete event processes with partial observations. IEEE Trans. Aut. Control 33 (3), 1988, pp. 249-260.

[Co85]

G. Cohen, D. Dubois, J.P. Quadrat, M. Viot. A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans. Aut. Control AC-30 (3), 1985, pp. 210-220.

[GR87]

C.H. Golaszewski, P.J. Ramadge. Control of discrete event processes with forced events. Proc. 26th IEEE Conf. on Decision and Control, IEEE Control Systems Society, New York, Dec. 1987, pp. 247-251.

[Ho87]

Y.C. Ho. Perturbation analysis explained. Proc. 26th IEEE Conf. on Decision and Control, IEEE Control Systems Society, New York, Dec. 1987, pp. 243-246.

[Hr85]

C.A.R. Hoare. Communicating Sequential Processes. Prentice-Hall, Englewood Cliffs, 1985.

[La87]

S. Lafortune. Modeling and analysis of transaction execution in database systems. Report CRL-TR-06-87, Computing Research Laboratory, The University of Michigan, Ann Arbor, Aug. 1987.

[LiW88a] Y. Li, W.M. Wonham. On supervisory control of real-time discrete event systems. Information Sciences 46 (2), 1988, pp. 159-183. [LiW88b] Y. Li, W.M. Wonham. A state-variable approach to the modeling and control of discrete-event systems. Proc. 26th Annual Allerton Conference on Communication, Control, and Computing, University of Illinois, 1988, pp. 1140-1149. [LnW88a] F. Lin, W.M. Wonham. Decentralized supervisory control of discrete-event systems. Information Sciences 44 (2), 1988, pp. 199-224. [LnW88b] F. Lin, W.M. Wonham. On observability of discrete-event systems. Information Sciences 44 (2), 1988, pp. 173-198. [LnW88c] F. Lin, W.M. Wonham. Decentralized control and coordination of discrete-event systems. Proc. 27th IEEE Conference on Decision and Control, IEEE Control Systems Society, New York, Dec. 1988, pp. 1125-1130. [Ma86]

O. Maimon, G. Tadmor. Efficicnt low-level control of flexible manufacturing systems. MIT LIDS Rpt. No. LIDS-P-1571, Cambridge, MA., 1986.

[MP831

Z. Manna, A. Pnueli. Verification of concurrent programs: A temporal proof system. Foundations of Computer Science IV, Mathematics Center Tracts, Amsterdam 1983, pp. 163-225.

557

[ow871 J.S. Ostroff, W.M. Wonham. Modelling, specifying and verifying real-time embedded computer systems. Proc. Eighth Real-Time Systems Symposium, IEEE Computer Society, New York, Dec. 1987, pp. 124-132. [RW821

P.J. Ramadge, W.M. Wonham. Supervision of discrete-event processes. Proc. 21st IEEE Conf. on Decision and Control, IEEE Control Systems Socicty, New York, Dec. 1982, pp. 1228-1229.

[RW89]

P.J. Ramadge, W.M. Wonham. Control of discrete-event systems. Proc. IEEE, Special Issue on Discrete Event Dynamic Systems, 77(1), Jan. 1989, pp. 81-98.

[TW871

J.G. Thistle, W.M. Wonham. Supervisory control with infinite-string specifications. Proc. Twenty-Fifth Annual Allerton Conference on Communication, Control and Computing, University of Illinois, 1987, vol. 1, pp. 327-334.

[TW88]

J.G. Thistle, W.M. Wonham. On the synthesis of supervisors subject to ¢0language specifications. Proe. 1988 Conference on Infornaation Sciences and Systems, Dept. of Electrical Engineering, Princeton University, 1988, pp. 440444.

[Wo88]

W.M. Wonham. A control theory for discrete-event systems. In M.J. Denhmaa, A.J. Laub (Eds.), Advanced Computing Concepts and Techniques in Control Engineering, NATO ASI Series, vol. F47, Springer-Verlag, Berlin, 1988; pp. 129-169.

[WR88] W.M. Wonham, P.J. Ramadge. Modular supervisory control of discrete event systems. Maths. of Control, Signals & Systems 1 (I), 1988, pp. 13-30. [ZW88]

H. Zhong, W.M. Wonham. On hierarchical control of discrete-event systems. Proc. 1988 Conference on Information Sciences and Systems, Dept. of Electrical Engineering, Princeton University, 1988, pp. 64-70.

558 I

0

~0

w

D

Fig. 1

'Machine' MACH

_1 _[ BUF (3) Fig. 2 'Factory'

I~ Infhl

]

~' ~" (Zon,~-~ COmhllo

G hi

Inf|ola

In o_ Conio -I Fig. 3 Two-Level Hierarchy

559

Fig. 4 Transfer Line

M1

1

2

M2

3

~

4

TU

62 ~ 60

82 80

Fig. 5 State Diagrams - TransferLine

~lSP

~--,~_

2,82

"~

selfloop{1,4,5,60,62,80}

3 4 B2SP

selfloop{1,2,3,60,62,80,82} 5 Fig. 6

q

f

t~

r~

L~

l~

St-.L

0

Q

,--,(

I

C,~C

"-4C

4~

~

t~

t',.)

C

4::I

tJ

tJ

OQ

(.,"1 0~ O

561

Fig. 9 GIII

60 6

81

82

14

y~'9

F

3

4

4

-10

4

I1

/r

3

8 Fig. 10 CGLO

5

I

562

31

-I

~

41

v

-'-t

2O

Fig. 11 CGHI

31

31

2

20

3

selfloop {11,41 } Fig. 12 IIISP

fail 0

fail

11 1 41 "2 31 3 41 4 31 5 41 6 20 7

20 pass Fig. 13 CGtIISUP