The stability of input-output dynamical systems

The Stability of Input-Output Dynamical Systems

This is Volume 168 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

The Stability of Input-Outpui Dynamical Systems

C. J. HARRIS Department of Electrical and Electronic Engineering Royal Military College of Science Shrivenham U.K .

J. M. E. VALENCA Matematica Universidade do Minho Braga Portuga1

1983

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British Library Cataloguing in Publication Data Harris, C. J. The stability of input-utput dynamical system(Mathematics in science and engineering ISSN 00765392) 3. matrices 1. System analysis 2. Stability 1. Title 11. Valenca, J.M.E. 111. Series 5 15'.35 QA402 ISBN 0- 12-327680-2 Filmset and printed in Northern Ireland at The Universities Press (Belfast) Ltd., and bound at William Brendon & Son, Ltd. Colchester.

The determination of the stability of dynamical feedback systems from open loop characteristics is of crucial importance in control system design, and its study has attracted considerable research effort during the past fifty years. Those stability criteria which have a simple graphical interpretation such as the Nyquist criterion have become popular with design engineers because extensions that include nonlinearities in the feedback loop are readily incorporated in the same graphical method via frequency domain methods such as the circle o r Popov criteria. Until the early 1960s almost all these methods were for scalar input-output feedback systems; however, the rapid developments in the state-space representation of dynamical systems and their realizations from transfer functions led to an equally important development in stability criteria for multivariable feedback systems. Much of this early work attempted to establish generalizations of the Nyquist, Popov and circle criteria by utilizing an extended version of the mathematical structures used for establishing scalar results. It is becoming increasingly clear to the authors that such system representations are inadequate for the analysis of generalized multivariable operators in feedback systems, and an alternative mathematically rigorous approach based upon the systems input-output spaces is required. Thus the definitions of system, system stability used in this book are based entirely upon input-output properties. The only systems representation admissible a priori is the description of the properties of the input-output map which defines the system. The existence of every other representation (including the representation by a transfer function) are therefore deduced from these properties. For this reason this book develops the necessary and sufficient conditions for inputoutput stability (which are implementable in a graphical form similar to the classical Nyquist criterion) by establishing the existence of a space of transfer functions which are in a one-to-one correspondence with the family of input-output maps which define stable systems. Various spaces of transfer functions in the complex domain (with their respective Principle of the Argument and Nyquist criterion) which are isomorphic with the algebra of multipliers in the time domain (for example, for linear timeinvariant operators defined on I$) are developed in this book. This radical approach to the representation of multivariable feedback systems V

vi

PREFACE

forms the basis of the development of readily implementable graphical stability criteria for both linear and nonlinear systems in the remainder of the book. Many results of the book are entirely new, some integrate previously derived stability criteria, and some previously known results are included to both unify the theory and place in context the most significant results. The mathematical approach adopted in the book is that of topological vector spaces, and Chapter One contains the necessary mathematical preliminaries of topology to ensure that the book is essentially self contained. This introductory chapter, which may be omitted by readers conversant with elements of functional analysis and measure theory, contains only those proofs which are germane to the subject development. In the development of frequency domain stability criteria for multivariable feedback systems, the single most important result in establishing generalized type Nyquist criteria is the Principle of the Argument. In Chapter Two a variety of local and generalized Principles of the Argument for complex polynomials with singular points (such as poles and zeros) and the associated encirclement conditions are derived for multivariable systems. The concepts of this chapter rely o n the properties of Riemann surfaces and the concept of homotopic triviality as found in algebraic topology. These basic concepts are also established in Chapter Two. In Chapter Three the fundamental question of the representation of linear time invariant operators is addressed such that a one-to-one correspondence between the concept of a multiplier (a linear, continuous-or sequentially continuous time-invariant operator) and the concept of a transfer function is established. For the spaces Lz and X ; we derive a space of transfer functions which are isomorphic with the space of multipliers; this is essentially a representation theory for linear timeinvariant operators and forms the basis of stability studies of linear feedback systems. For the important spaces L;, L2, X ; and X 2 a full representation theory is shown not to be possible, however, we established a multiplier in these spaces which has a transfer function in some space of complex functions. Chapter Three also contains the theory of multipliers and the conditions for invertibility of linear time-invariant operators, which are essential in establishing closed loop stability conditions from open loop considerations. Chapter Four establishes a family of generalized Nyquist stability criteria for linear feedback systems with operators defined on the spaces L,"and X,"by utilizing the results of Chapters Two and Three. The lack of a full representation theory for systems defined upon L ; , LE and X E spaces ensures that it is not possible to state, in similar terms to L; and X ;

PREFACE

vii

spaces, necessary and sufficient conditions for closed loop stability. However, by imposing some structural conditions at least necessary conditions for closed loop stability for these spaces can be determined. The concepts of input-output stability are essentially based upon the existence of operators defined on Banach spaces. However, many dynamical systems are frequently open loop unstable and function spaces which grow without bound are not contained within Banach spaces, so some mathematical description of unstable operators is necessary if feedback stability is to be interpreted from open loop system descriptions. This is achieved in Chapter Five by setting up the problem in extended spaces which contain well behaved as well as asymptotically unbounded functions. These generalized extended spaces contain all functions that are integrable or summable over finite intervals. A close relationship is established between extended spaces and locally convex spaces equipped with a projective limit type of topology. The concepts of causality, passivity, positivity and sectoricity are introduced in this chapter as a preliminary to their practical application in Chapters Six and Seven. Finally an introduction is given to the theory of multipliers and the multiplier factorisation theorem, which are utilized in the development of the various off-axis multivariable circle stability criteria. The final two chapters are concerned with establishing graphical conditions for the stability of nonlinear multivariable feedback systems. In Chapter Six we develop a series of interconnected multivariable on-axis circle type stability criteria that are based upon the various small gain theorems and the loop transformation theorem. Chapter Seven utilizes the concepts of sectoricity, multipliers and passivity to derive a series of new and powerful off-axis circle stability criteria and a Popov criterion for nonlinear multivariable feedback systems. This book is the result of a collaborative effort between the authors at the University of Oxford, University of Minho and the Royal Military College of Science, and the authors wish to acknowledge their debt to these institutions for their support and the provision of facilities to carry out this work. A major debt of gratitude is owed to the Science and Engineering Research Council for the financial support of this work. A special vote of thanks must be given to Dr R.K. Husband of Oxford University whose inspiration, and collaboration with the authors, resulted in the multivariable off-axis circle criteria of Chapter Seven. Finally, personal thanks are given to Mrs J.D. Swann whose excellent typing turned an untidy and complex manuscript into the final version of this book and to Mrs S.J. Prescott for the tracing of the figures.

February 1983

J.M.E. Valenca C.J. Harris

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Contents Preface

V

Chapter 1 Mathematical Preliminaries 1.1 Introduction 1.2 Basic topological notions 1.2.1 Linear topological spaces 1.2.2 Sequences, nets and continuity 1.2.3 Convex and compact sets 1.3 Topological vector spaces 1.3.1 General topological vector spaces 1.3.2 Metric spaces 1.3.3 Inductive and projective limits of normed spaces 1.3.4 Banach spaces and Hilbert spaces 1.3.5 Operators in Banach spaces 1.4 Fixed point theorems 1.5 Measures and function spaces 1.5.1 Measures and integrals on the positive real line 1.5.2 The function spaces L, 1.5.3 Real and complex measures 1.5.4 Indefinite integrals 1.6 Dual spaces 1.7 Notes References

1

Chapter 2 Riemann Surfaces and the Generalized Principal of the Argument 2.1 Introduction 2.1.1 Paths in topological spaces 2.1.2 Homotopy 2.2 Complex integration and Cauchy’s theorem 2.3 Riemann surfaces 2.4 Minimum contours and algebroid Riemann surfaces 2.5 Analytic functions and integration in algebroid Riemann surfaces 2.6 Singular points

ix

1 1 1 6 7 9 9 13

IS

21 23 24 26 26 29 31 32 34 39 39 40 40 41 42 45 48 55 58 63

CONTENTS

X

2.7

The Generalized Principle of the Argument References

Chapter 3 Representation of Multipliers 3.1 Introduction 3.2 Representation of multipliers in L2 and L; 3.3 Convolution algebra A(R+) 3.4 Representation of multipliers i n L , and L ; 3.5 Representation of multipliers in L, 3.6 Representation theory in X,-spaces 3.6.1 Sequential convergence vector spaces 3.6.2 Sequential inductive limit spaces 3.6.3 The spaces X , 3.6.4 Multipliers in X,, X2, X, 3.6.5 Space of transfer functions 92 References Appendix 3.1: The theory of multipliers in L, Chapter 4 Linear Input-Output Stability Theory 4.1 Introduction 4.2 General analytic formulation of stability 4.3 Graphical stability criteria for L,-systems 4.4 Graphical stability criteria for multivariable systems 4.5 Notes References Chapter 5 Extended Space Theory in the Study of System operators Introduction 5.1 5.2 Fundamental results Operators in extended spaces 5.3 Well posedness and feedback systems 5.4 Passivity in feedback systems 5.5 5.6 Theory of multipliers 5.7 Sectoricity 5.8 Notes References Chapter 6 Stability of Nonlinear Multivariable Systems-Circle Criteria 6.1 Introduction 6.2 Small gain theorems 6.2.1 Boundedness small gain theorem 6.3 Intermediate small gain theorem 6.4 Incremental gain theorem

72 80 82 82 83 87 92 97 97 98 99 104 105 108 111 112 121 121 123 129 135 140 141 143 143 144 147 151 153 161 165 170 170 172 172 173 178 184 191

CONTENTS

6.5 6.6 6.7

An M-matrix stability criterion System diagonaliszation and design Notes References

Chapter 7 Stability of Nonlinear Multivariable Systems-Passivity Results 7.1 Passivity stability theorems 7.2 Off-axis circle criteria 7.2.1 The linear operator & w ) is normal 7.2.2 The linear operator & w ) is non-normal 7.2.3 Determination of the sectors of Q(F-@) and Q 7.3 Off-axis circle criteria-multiplier factorization 7.4 Multivariable Popov criterion 7.5 Notes References

xi 203 207 212 213 215 215 223 223 227 23 1 235 248 254 255

Bibliography

259

Subject index

265

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Chapter One

Mathematical Preliminaries 1.1 INTRODUCTION This chapter provides an introduction to the fundamental topological and functional analytic concepts used extensively in this book. Input-output stability is essentially a topological concept defined on spaces of time functions which have various properties including those of linear topological spaces. Thus in Section 1.2 the concept of linear spaces and various topological notions associated with linear spaces are introduced. In Section 1.3 we develop generalized topological vector spaces with particular structural forms of metric spaces, inductive and projective limits, and Banach and Hilbert spaces. Various results in Chapters 5 and 6 depend upon the fixed-point theorems introduced in Section 1.4. The majority of time functions of interest in this book are L,, spaces and are reviewed as part of Section 1.5 on measures and function spaces. Finally in Section 1.6 the concepts of functionals and dual spaces are developed.

1.2 BASIC TOPOLOGICAL NOTIONS 1.2.1 Linear topological spaces A set E is called a linear space over a field K when:

(i) An operation + is defined on E and with this operation E is an abelian group; that is whenever x, y, z E E (a) x + y = y + z , (b) ( x + y ) + z = x + ( y + z) (c) There exists an identity 0 E E such that 0 + x = x + 0 = x (d) There exists an xf E E such that x + x f = 0 (ii) A scalar multiplication is defined which assigns any pair (a,x) E K x E to a x E E ; furthermore the following properties are verified

1

2

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

for (e) (f) (g)

all a,p E K and x, y E E a ( x + y) = a x +a y (a+ p ) x = a x + px and (a.p ) x = a ( p x ) 1 . x = x where 1 denotes the unit element of the multiplication in K

Throughout K is either the field of real numbers R ' , in which case E is said to be a real linear space, or the field C' of complex numbers, when E is said to be a complex linear space. Frequently the designation of vector space will be used as synonymous of linear space.

Example 1.1 Consider the linear space C ( R + )defined as the space of all continuous time functions r Hf(r) defined on the half real line, R , = [0, x ) , with addition and scalar multiplication defined by (f+g)(t) = f ( t ) + g ( t ) ( a f ) ( l= ) af(t)

The degree of proximity between two points in a linear space E can be expressed in terms of sets which are generalizations of the familiar balls in R". Let x be arbitrary in E and let B(x) = { U } be a family of subsets of E satisfying

~ . 1Every element of B(x) contains x 8.2 The intersection of two elements of B(x) is an element of B(x) 8.3 For every U E B ( ~ and ) every Y E U there exists a V € B ( y ) contained in U The collection B(x) is called a base of open neighbourhoods of the point

x. Any set W c E which contains some U EB(x) is called a neighbourhood of x. Axioms 8 . 1 - ~ . induce 3 the following properties in the family N ( x )

of all neighbourhoods of x.

~ . 1Every U E N ( x ) contains x ~ . 2Every set which contains an element of N ( x ) is itself an element of N ( x ) ~ . 3The intersection of a finite collection of elements of N(x) is an element of N ( x ) . ~ . 4Each U E N ( x ) contains some V E N ( x ) such that U is a neighbourhood of every point of V

Example 1.2 In C ( R , ) define

Each B ( 5 ) is called an open ball in C ( R + ) .It is trivial to show that the

1.

MATHEMATICAL PRELIMINARIES

3

collection B(f) = {f+ B(.$)}*,() is a base of neighbourhoods of the point f~ C ( R + ) .In particular, to show 8.3, consider g ~ f B+( 6 ) ; then the quantity 6' = 6-sup I f ( [ ) - g(t)l is non-zero positive. Clearly g + B(5') is re=

contained in f + B(6). When a class of neighbourhoods is defined for every point of a space E, then a topology is defined in E. Let T represent such a topology; the pair (E, T ) , called a topological space, is represented by E [ T ] . Frequently, when E is a linear space, the neighbourhoods of an arbitrary point x take the form x + U, where U is a neighbourhood of the origin. In these circumstances, the base of neighbourhoods of the origin defines completely the topology of the space. Such a space is called a linear topological space. Given a topological space E, whose topology is defined by families of a base of neighbourhoods of its points, an open set is defined as any set which contains a neighbourhood of each of its points. Note that 8.3 implies that every set in a base of neighbourhoods is an open set. A closed set is defined as the complement of an open set. The following properties of open sets are readily proved: 0.1 The empty set and the set E are open sets 0.2 An arbitrary union of open sets is an open set 0 . 3 An arbitrary finite intersection of open sets is an open set

The concept of topology has been defined through axioms 8.1-8.3 and the concept of a neighbourhood; alternatively it is possible to define topology through a definition of open sets which takes 0.1-0.3 as axioms. In this alternative definition, a neighbourhood of a point x is defined as .4 any set containing an open set which contains x. Properties ~ . 1 - ~ can easily be established. (The equivalence of these two alternative definitions of topology is given in Kothe (1969).) It is possible to define several topologies on the same space, the question now arises of relating these topologies. Let T , and T~ be two topologies on the same space E. The topology T~ is said to be stronger (or finer) than the topology T ~ when , for any X E E the collection of all neighbourhoods of x defined by T , contains the collection of all neigh, T~ is weaker (or coarser) than T , . bourhoods of x defined by T ~ here Clearly T , coincides with T~ when it is both weaker and stronger than T ~ .

Example 1.3 The topology C ( R + )defined by the collection of open balls (1.1) is called the uniform topology. Other topologies are sometimes useful in C ( R + ) :consider a finite collection of points O < T , < T 2 < .. . < T, and a collection of positive numbers 0 < t 1 <12<.. .<&. Let U be

4

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

f

-5

t

Ibl

I

I

Frci. 1.1

-5,

Illustration of Example 1.8 for extended space topologies

defined as

The elements of an open ball B ( t ) can be visualized as functions which lie on an infinite strip bounded by +[ and -6 (Fig. 1.la). The elements of U are functions which are bounded by *ti only in the interval [0, T,]. Therefore for t > T,,, the set of all possible values for f ( t ) (with f~ U ) is not bounded (Fig. 1.lb). For any X E E, the collection { x + v), where the totallity of sets U is obtained by taking in (1.2) all possible finite selections of points TI< T2< . . . < T,, and numbers 6, < t2< . . .< t,,,satisfies axioms B. 1 - ~ . 3and therefore defines a base of open neighbourhoods of x. The topology thus defined is called the topology of the extended space; we now relate this topology to the uniform topology.

1.

5

MATHEMATICAL PRELIMINARIES

We can state that the uniform topology is stronger than the extended space topology. In fact, given a neighbourhood V of a point x in the extended space topology, it must contain a set of the form x + U (for U given by (1.2)); but U always contains an open ball B ( 5 ) provided that ,$
+

hood of x in the uniform topology. Conversely we can prove that the topology of the extended space is always weaker than the uniform topology because we can find neighbourhoods of x in the uniform topology which are not neighbourhoods of x in the extended space topology. For example the neighbourhood x + B ( e ) is not a neighbourhood of x in the extended space topology because it is not possible to find any U of the form (1.2) contained in B ( 5 ) ;in fact, even if 5 > 6, = max ti, it is always possible to find some f~ U such that sup If(t)l>,$. r>T,

I

We

conclude that the uniform topology is stronger than the extended space topology and that the two topologies do not coincide.

Of particular interest are those topologies which separate points; a topology is said to separate points in the space E when two distinct points of E lie in disjoint open sets. Equivalently a topology separates points when the intersection of all closed neighbourhoods of a point x coincides with x itself. The topology that satisfies this property is called a Hausdorff topology.

Example 1.4 We can show that the extended space topology and the uniform topology in C ( R + )are both Hausdorff topologies. Taking the extended space topology, consider the intersection of all closed neigh~ We need to show that an arbitrary g#fo bourhoods of a point f ( ,C(R+). is not a point in this intersection. Since g and fo are distinct there exists a T>O and ,$>O such that

then fo+ U is a closed neighbourhood of fo which does not contain g. The extended space topology is therefore an H a u s d o d topology. Since the uniform topology is stronger than the extended space topology (see Example 1.3) then it must also be a Hausdorff topology. If F is a subspace of a topological space E [ T ]a topology is defined in F if we consider as neighbourhoods of the point x E F all the sets of the form U n F, where U is a neighbourhood of x in E, such a topology is called the topology induced in F by the topology of E.

6

THE STARlLlTY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

1.2.2 Sequences, nets and continuity Possibly the most important of all topological concepts are the concepts of limit and continuity. A sequence in a topological space E is defined as any map from the set of natural numbers Z into E; it will be represented by { x , , } ~or simply as {x,,}. If r is an arbitrary totally ordered set, we define a net in E as any map of r into E. A subset Zc Z is called a cofinal set when for each n E Z, there exists Z that m r n . Given a sequence { x , , } ~its restriction to a an ~ E such cofinal set Z is represented by {x,,}, and is called a cofinal subsequence of { x , , } ~Identically we can define a cofinal subnet. A point xo is said to be an adherent point of a net { x , } , when any neighbourhood of xo contains a cofinal subnet. The point x, is said to be the limit of the net {x,},., when for each neighbourhood U of xo there exists an ~ E such T that the subnet {x,},?, is contained in U. In a Hausdorff space the limit of a net is unique (provided that it exists); moreover it is also the unique adherent point. Of special interest in linear topological spaces are the Cauchy nets; a net { x , } is a Cauchy net when given any neighbourhood of the origin U there exists a y such that for all a,p L y, x , - x p E U. A linear topological space in which every Cauchy sequence has a limit is said to be complete. Given a subset F of a topological space E, we define the closure of F (represented by f i as the set of all limits of convergent nets of F. The closure F is the intersection of all closed sets which contain F. If F coincides with E then F is said to dense in E.

Example 1.5 Consider the space C,, of all functions f~ C ( R + )which satisfy lim suy I f ( ? ) \ = 0. Consider also the space Cooof all functions in T-r

t z

C ( R + )which vanish outside some finite interval [0, TI. We wish to show of C,,, coincides with C,. that the closure Consider first an arbitrary convergent sequence cf,} in C,, and let f be its limit. For any 5 > 0 there must exist an n such that for all m r n , sup If( t ) - f,(t)l < 5. Let T, be such that fm vanishes outside [0, T,]. Then

c,,,

t
sup

If( r)l <5, which illustrates that f lies in C,. Thus Co

I Z T ~

Conversely let f be arbitrary in C, and let satisfying:

3

coo.

{ A }be a sequence in C,,,,

1.

MATHEMATICAL PRELIMINARIES

7

Define f n ( t ) = f(t)&(t),then

5

lim sup If(t)l= 0

n-=-=

Consequently with Ccl.

t>n

coo C,, therefore the closure 2

c(,,,, of Coo must coincide

Given two topological spaces E and F and a map A :E + F ; A is said to be continuous in a point X,E E when, given any neighbourhood U of A(x,), there exists a neighbourhood V of x, such that A ( V ) c U. If A is continuous at every point of E, then A is continuous. Equivalently, A is continuous when, given any open set 0 in F, A-'(O) is open and, given any closed set C in F, A - ' ( C ) is closed. If given any convergent sequence {x,} in E we have lim A(xn)= n A(lim x n ) , then A is said to be sequentially continuous; every continuous n

map is sequentially continuous. Let A be a one to one map from a topological space E into a topological space F; then if both A and A-' are continuous, we say that A is a homeomorphism.

1.2.3 Convex and compact sets A subset F of a linear topological space E is absoruenr when, for any x E E, there exists a p > 0 such that px E E The set F is circled, when for any complex number a, ax E F whenever x E F and la1 5 1. F is said to be absolutely convex, when for any a, b E C ' which satisfy lal-tlblsl, we have ax+ by E F whenever x, y E F. The set F is bounded if, given any neighbourhood of the origin V,there exists a p > O such that F c pU.

Example 1.6 Consider an arbitrary opeii ball B( 5 ) in C(R+).For any f~ C ( R + ) either , f = 0, in which case f~ B ( [ ) ,o r ff 0, in which case for

we have p f ~B ( 5 ) . Thus B ( 5 ) is absorvent. Moreover, given any f, g E B ( 5 ) we have SUP l a m + bg(t)ls S(lal+ lbl) I<-=

8

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Thus, provided that l a l + l b l S l , then a f + b g E B ( [ ) , and B ( [ ) is absolutely convex. Taking the special case b = 0 we see that B ( 5 ) is circled. Finally given any neighbourhood U of the origin in the uniform topology, let B ( [ ' ) be an open ball contained in U. For any p > [ / [ ' we have B(,$)cpB(['), thus B ( 5 ) is bounded. A subset M of a Hausdorff topological space E is compact when every net in M has an adherent point in M. Compactness can be recognized in various alternative forms. The equivalence between some of these forms is stated in:

PROPOSITION 1.1 If M is a subset of a Hausdorff space E, then the following statements are equivalent:

(i) M is compact (ii) Every cover of M by open sets contains a finite subcover (iii) Every collection of closed sets in M with empty intersection contains a finite subcollection with empty intersection Compact spaces have special significance in the study of topological spaces of time functions, we therefore state some of its properties:

PROPOSITION 1.2 Let M be a compact subset of a topological space E. Then : (i) M is closed and bounded (ii) Every closed subset of M is compact, and every open subset of M is relatively compact (that is, its closure is compact) (iii) Any continuous image of M is also compact (iv) The union of finitely many compact sets is compact A Hausdorff space is locally compact if every neighbourhood of an arbitrary point x is relatively compact. Classical examples of locally compact spaces which are no; ,ulllpact are R" and C". A very important property of these spaces is expressed in:

THEOREM 1.1 : ALEXANDROFF'S THEOREM Every locally compact space E which is not compact can be enlarged with the addition of one point to give a compact space, called the one-point compactification of E. Moreover such a compactification is unique up to a homeomorphism. Example 1.7 Consider the space R' = (-03~30). With the addition of one point at infinity we can enlarge R' into a compact space R' =[-? m], where the points +m, --oo are the same point. The neighbourhoods of infinity are sets of the form (T,+w] or [-m, T). The space C(R') of all

1.

9

MATHEMATICAL PRELIMINARIES

continuous bounded functions in R' can be obtained from C ( R ' ) by taking all functions f E C ( R ' )for which lim f ( t ) and lim f ( t ) exist and I-+==

,---m

coincide; the value of f ( - ) in the point of compactification is defined as this limit. Identically we can enlarge the complex plane C' with a compact space C' by the addition of one point at infinity. The neighbourhoods of infinity are sets of the form U = { C J : ICJ~> 5). We can define a space C(c')of all continuous bounded functions in c' by taking the limit of C ( C )formed with all functions which converge uniformally at infinity to same limit. We have seen that every compact space is bounded and closed; the converse is not in general true. As an example, consider a bounded closed ball in C ( R + )

B

= {f:SUP If(t)l r<-

stl

Although we cannot show that B is compact, the following result (a consequence of the Ascoli-Arzlea theorem) recognizes a compact subset of B.

PROPOSITION 1.3 A subset B ' c B is compact if it lies in C(R:) where R: is the one-point compactification of R,, and is equicontinuous, that is, given any increasing homeomorphism p :[0,~3 + [0,1] and any 5 > 0 , there exists a 6 > 0 such that p(lt - t'l) < 6 implies sup I f ( t ) - f ( t ' ) l < t forall fcB'

t,

r'ER:.

1.3 TOPOLOGICAL VECTOR SPACES 1.3.1 General topological vector spaces Let E be a Hausdorff linear topological space; if the maps (x, y ) t+ x + y from E 2 into E, and (x, a)H a x from E XC' into E, are continuous then E is called a topological vector space. The following theorem identifies a linear space as a topological vector space.

THEOREM 1.2 Let E be a linear space. The following statements are equivalent: A E is a topological vector space B There exists a family N = { U,} of absorvent circled subsets of E which satisfy (i) the intersection of any two elements of N is an element of N

10

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

n

(ii) The intersection U, of all sets in N coincides with the singleton {@ a (iii) For every U EN there exists a V EN such that V + V c N When a topological vector space has a base of neighbourhoods of the origin consisting of absolutely convex sets, the space is said to be a locally convex space. This type of space is the more general form of topological vector space used in this book; it can be identified through the following theorem. THEOREM 1.3 Let E be a linear space. Then the two following statements are equivalent: A E is a locally convex space. There exists a family N = { U,} of absolutely convex absorvent subsets B of E satisfying; (i) The intersection of any two elements of N is an element of N (ii) The intersection Uaof the totality of elements of N coincides

n a

with the singleton {0} (iii) For any UEN and any p>O the set pU lies in N Example 1.8 Consider the space C ( R + )equipped with the extended space topology, and let us consider the collection N = { U } of all sets of the form U = { f :sup If (t)l< ti,i = 1, . . . , n } with

rST,

T,,> T,,-l >. . . > TI> O

and

5, >

>. . . > > O

Clearly every U is absolutely convex and absorvent. Conditions B(i) and

B(iii) of Theorem 1.3 are obvious; condition B(ii) was proved in Example

1.4. Thus C ( R + )with the extended space topology is a locally convex space. Identically we can show that equipped with the uniform topology, C ( R + )is still a locally convex space.

Locally convex spaces are closely related to finite semi-normed spaces which are defined as follows: Let E be a linear space and p : E - + R : a positive function in E. The function p ( is called a semi-norm when a )

s.1 For all x E E and complex numbers a,

p ( a x )= Ialpb) s.2 For all x,

Y E

E P(X

+ Y) 5 P ( X ) + P ( Y )

1. s.3

11

MATHEMATICAL PRELIMINARIES

If in addition the following is satisfied p(x)=O if and only if x=O then p ( . ) is called a norm and is usually represented by

11-(1.

A finite semi-norm space is a space where the range of every seminorm is restricted to R,, that is p ( x )<m for all x E E. In similar manner we define a finite normed space. The relationship between circled, absolutely convex subsets of E and semi-norms is explored in the following result. 1.4 PROPOSITION A

B

A finite semi-norm defines a family N = { U , } of absolutely convex absorvent subsets through U, = { x E E : p ( x )< 5,5> 0) Moreover the intersection of any two elements of N is an element of N A n absorvent absolutely convex set M defines a finite semi-norm through p ( x )= inf { k > 0 : x E k M )

The relationship between finite semi-normed spaces and locally convex spaces is established in:

THEOREM 1.4 Let E be a linear space. Then the two statements are equivalent: A E is a locally convex space. B There exists a collection {pa(*)}of finite semi-norms in E with the property that for each non-zero X E E , there exists one semi-norm p a ( . ) such that p,(x) # 0. A stronger result can be proved for finite normed spaces:

THEOREM 1.5 If E is a linear space equipped with a firrite norm (1.11, then E is a locally convex space and its topology is generated by a family of open balls, B ( [ ) = { ~ E E : I I x I I O}. &

Example 1.9 Consider in C ( R + )the positive function defined by

llfll= SUP If(t)l I
Clearly (1.11 is a finite norm, moreover the open balls generated by this norm coincide with the open balls (1.1) in Example 1.2. Then the topology generated by this norm coincides with the uniform topology. For the extended space topology (Example 1.3) we consider the family of positive functions defined for all T>O PTWPSUP If(t)l IST

12

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Clearly every pT( - ) is a semi-norm and given any f # 0 there exists some T > 0 such that sup If(t)l> 0; consequently given any f # 0 there exists a isT

semi-norm pT(.) such that pTCf>#O. Thus the family of semi-norms {pT( * )) define in E a locally convex space topology (Theorem 1.4). To see that this topology coincides with the extended space topology we note that the generic neighbourhood of the origin contains a set of the form

U = c f : p , C f > < & ; i = 1 ,..., , 2 n) which coincides with the generic set (1.2) in Example 1.3. Let p ( be an arbitrary finite semi-norm in a linear space E. Although p ( - ) is not necessarily a norm we can introduce a space where p ( - ) induces a norm. Let N c E be the null space of p ( - ) , that is N = {x E E :p ( x ) = 0). It is simple to show that N is a linear proper subspace of E, and unless p ( is a norm, N is distinct from the singleton (0). It is now possible to define the quotient space (Kothe, 1969), EIN. To each X,,E E there is assigned an element [XJE EIN, called the N-coset of xor which is the set of all x E E such that x - xo E N. Thus we can represent [xo] as the set xo + N. The map A :xo + [xu] is called the canonical mapping. The zero element of E / N is the N-coset of zero and is therefore in the set O + N = N. EIN can be identified with a linear space by defining [XI+ [y] A [x + y] and a[x] = [ax]; although these definitions seem to depend on the particular values of x and y chosen, its coherence results from the fact that given any X’E x + N and Y’E y + N, we have a )

0

)

(x’+ y ’ ) - ( x + y) = ( x ’ - x ) + ( y ’ -

Y)E

N

(ax’)- ( a x )= a ( x ’ - x) E N The canonical mapping A, is as a consequence of this definition a linear map. We can define a positive map 11 * 11 : E / N + R , through ll[xlll = P h )

This definition is also independent of the point x E [x] chosen, since given any X ’ E x + N, we have Ip(x’)- p(x)l Ip ( x ’ - x) = 0, which implies that p ( x ’ ) = p ( x ) . It is straightforward to show that II-II is a semi-norm; moreover if ll[x]ll=O then p ( x ) = 0 for all XE[X], which means that [XI= N = [ O ] . Therefore II.\I is a norm in EIN. We have now shown:

THEOREM 1.6 Let p(.) be a finite semi-norm in a linear space E. If N represents its null space, then EIN is a finite normed locally convex space. Moreover, every open ball o c E / N is of the form fi = A( U ) where A is the canonical mapping and U = {x E E :p ( x ) < 5,t> 0).

1.

MATHEMATICAL PRELIMINARIES

13

If E and F are two normed spaces and A :E + F is a linear, onto map such that llAxll= llxll for all x E E, then we say that E and F are isometrically isomorphic spaces. The map A is obviously a homeomorphism. As an example of isometrically isomorphic spaces we consider the following:

Example 1.10 Let C,(R+),or simply C,,be the space of all continuous, but not necessarily bounded, functions with domain in [0, m). Let p T ( - ) be the semi-norm defined in Example 1.9

Let NT be the null space of this semi-norm. We wish to show that the space C([O,TI), or simply C, of all continuous functions defined in the interval [0, TI, is isometrically isomorphic with C J N , we consider CT equipped with the uniform norm. Let P T : C,+ CT be the linear map defined as (PTf)(t)=f(t)

tEIO,

PTf=O if and only if f € N T ; moreover for all f~ C,,llPTfll=pT(f). Let @ : Ce/NT+ CT be defined as PTf. The coherence of these defini-

@[a’

tions result from the fact that, given any f’€f+NT, PTf’= PTf+PT(f-f’)=PTf.The map @ is clearly linear and onto; moreover Hence

@ is an

Il[flll = b ( f ) = llpTfll = Ils[flll isometric isomorphism.

1.3.2

Metric spaces

As a generalization of the concept of distance in a space E let d ( . , . ) be a positive real function in EX E which satisfies: M.1

d ( x , Y ) = d(Y, x)

~

2d(x,

.

y)=O

~ . 3d ( x , y ) 5 d ( x , z ) + d(z, y )

Vx,y€E if and only if x = y V x, y, z E E

The function d ( . , is called a metric in E, and the space E is called a metric space. A metric space is a topological space if we define as base of neighbourhoods of a point x E E, the family of open balls 0

)

q ( x ) = { y E E : d ( y , x) < 595 > 0 )

Form axiom ~ . 2this , topology is a Hausdorff topology. One of the most usual examples of a metric is defined in finite normed spaces through d ( x , y)AIlx- yII. Thus we can state that every finite

14

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

normed space is metrizable. In these cases the topology defined by the metric coincides with the topology defined by the norm. Some other locally convex spaces can also be metrizable, consider the space C,(R+) (Example 1.lo) of all continuous but not necessarily bounded functions with domain in R+= [0, s).Define the following family of semi-norms PT(f)=sup If(t)l, t5T

T>O

As was shown in Example 1.9, this family of semi-norms defines a locally convex topology in C J R , ) . We call C,(R+),the extended version of C ( R + ) .Although it is not possible to define a finite norm in C,(R+) we can define a metric as follows. For an unbounded increasing sequence TI < T2< . . . < T, < . . . and a sequence of positive numbers c, > c2 > . . . > c, >. . . such that

f

Ci<W

i= I

then

2 CiPT,(f-g)(l+PT,(f-g))-’ Cc

d(f, g ) =

i= I

defines a metric in C,(R+). The basic topological notions take a more familar appearance when expressed in terms of metrics, thus (i) A map T :(E, d ) + ( E , d ’ ) between two metric spaces is continuous at a point X,,E E when, given any ( > O , there exists a S > O such that d ( x , xo) < 6 implies d ’ ( T ( x ) ,T(x,,))< (. (ii) A sequence {x,} in metric space ( E , d ) is said to converge to a point xo when, given any ( > O , there exists an order N such that, for all n 2 N, d(x,, x,) < 5. (iii) A point xo is said to be a closure point of a subset F of a metric space ( E , d ) when, for all ( > O , there is a point x E F such that d ( x , xo) < 5. (iv) A subset F of a metric space ( E , d ) is compact if, given any ( > O , there exists a finite collection of points xI,x2, . . . x,, such that for all x E F, min d ( x , xi)< (. icn

We now introduce a new concept; a map T from a topological space ( E , d ) into a topological space (F, d’) is uniformally continuous when, given any 5 > 0, there exists a 6 > 0 such that d’( T ( x ) ,T(y))< ( whenever d ( x , y) < 6, x, y E E. If E is compact, then every continuous operator 7’:( E , d ) (F, d ’ ) is uniformly continuous.

-

1.

15

MATHEMATICAL PRELIMINARIES

Two metrics d and d‘ in the same space E are equivalent when the identity operator as a map on (E, d ) into ( E , d ’ ) and as a map on (E, d’) into ( E , d ) , is uniformally continuous. Consequently, if d and d’ are equivalent metrics, given any 5>0, there exists a S > O such that and

(i) d(x, y ) < 6 implies

d‘(x, y ) < 5

(ii) d’(x, y ) < S

d(x, y ) < 5.

implies

Clearly, equivalent metrics define the same topology in the space E.

1.3.3 Inductive and projective limits of normed spaces Although normed spaces are the most frequent of locally convex spaces used in stability studies, our studies will require more general topological vector spaces which can be seen as limits of normed spaces. Suppose that to each a > O we associate a finitely normed space E,; let ((.l(, represent the norm in E,. Let us also assume that E, =) Eb if b > a. We can define a vector space E as E=

u E,

a>O

which is called the inductive limit of the spaces E, (for a more general definition see Kothe section 19 (1969)). From this definition we see that for b > a > O E =) E, 3 Eb Let Z, : E, + E represent the embedding of E, in E, and let l a b :Eb 4 E, represent, for b > a, the embedding of Eb in E,, clearly = I,&,. The inductive limit E is frequently represented by

We can define a base of absolutely convex neighbourhoods of the origin in E by considering the convex covers of sets of the form U = U U, a XI

where U, is an absolutely convex neighbourhood of the origin in E, (the convex cover of U is the intersection of all absolutely convex sets containing U ) . If such a topology, denoted by T, is Hausdorff we say that E [ T ] is the topological inductive limit of the spaces E,.

THEOREM 1.7 The space E

=

U E, ,is the topological inductive limit of the a

spaces E, if and only if, for b > a , the topology induced in Eb by the topology of E, is weaker than the original topology of Eb. In these circumstances, the absolutely convex neighbourhoods of the origin in E are

16

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

the sets U = U U,, with U, an absolutely convex neighbourhood of the origin in E,.

a

In a family of normed spaces {E,},,o where E, 3 E h , for b > a, the requirements of Theorem 1.6 are equivalent to the requirement that IIxII, 5 I(xl(,,for all x E Eb. We now consider the relationship between E and the inductive limit E generated when we consider a cofinal subset of (0, m). Let r be a subset o f (0, m) such that, for all a > 0, there exists a y E r such that y < a, and let the inductive limit be generated by E = U E,. Then ,€I-

PROPOSITION 1.5 Under the above conditions, if E is a topological inductive limit, so is E and the two spaces are topologically isomorphic. As a consequence of Proposition 1.5, consider any sequence {a,} in (0,m) such that lim a, = 0, and if En represents E4, then n

E l c E z ~. .. c E , c . . . and the topological inductive limit E can be identified with the limit

CJEn.

n-l

Example 1 . 1 1

Let

1(.1 ,

a >0, be a norm defined in C ( R + )by

Ilfll,

= SUP exp ( a t )If(t)l I
Clearly Ilfll, ~Ilfll,,, for b >a and all f E C ( R + ) .Let C, represent the set of all f E C ( R + )for which Ilfll, is finite, then the space c ( R + )= C, can be

u>o

U

identified with the topological inductive limit of the spaces C, (Theorem 1.6). Frequently the properties of topological inductive limits are insufficient for stability studies; a stronger concept is required. Let E be the topological inductive limit of a countable family of spaces,

El c E , c E , . . .cEn c . .. and assume that the topology induced in En by the topology of En+, coincides with the topology of En. Then E is called the strict inductive limit of the spaces En.The equivalence between the normed topologies of En+, and En exists if and only if there exist positive constants 8, p > 0 such that

1.

17

MATHEMATICAL PRELIMINARIES

The two following results establish the raison d’ztre of strict inductive limits z

THEOREM 1.8 If E = IJ En is the strict inductive limit of the spaces En, n= I

then a set B is bounded in E if and only if there exists some n such that B is contained in En and is bounded in the topology of En. THEOREM 1.9

If E =

cc

U En is the strict inductive limit of the spaces

n=I

En,

then a sequence { x k } converges in E to a limit x , , ~E if and only if there exists some n such that both the sequence and the limit are contained in En and {x,} converges to xg in the topology of En. Although we can define strict inductive limits when we have a noncountable family of spaces {Eu},,o, it is not possible to prove results of the nature of Theorems 1.8 and 1.9 for these spaces. Therefore, as such, they are of little practical value in stability studies. As a consequence of Theorems 1.8 and 1.9 the following result can be proved:

1.6 Let T be a continuous linear mapping from a finitely PROPOSITION z

normed space F into a strict inductive limit E some n such that En contains T(F).

=

U En. Then there exists

n=l

Consider again a family {E,} of finitely normed spaces defined for all a 2 0. Suppose there exists a family {Pa,,} of linear onto maps, Pub: E,, + E, defined for all b > a such that

v C >b >a Consider the topological product space 6 = n E, ; g is the space of all functions x ( a )= xu E E,. Each 2 E E is represented as (x,) and x, E E, is called the component of f in E,. Let k, :E + E, be the map which assigns i ~ tog its component xu in E,. The mappings k, are called P,~P,,. Pa,

U

projections. The space is a linear space under the obvious definitions of sum and product by a scalar: ( x u ) + (y,) = ( x u + y,), a ( x , ) = ( a x , ) . We can define in € a topology, called the product topology, by constructing a base of absolutely convex neighbourhoods of the origin which are finite intersections of sets of the form PL-I’B,, where B, is an open ball in E,.

u=iri= l ~ ~ ~ l ) E{Exau, :,/ / X a , l l < t t } With such a topology,

E is a locally convex space.

(1.4)

18

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

The projective limit of the spaces E,, represented by E = 1 5Pa&,, is defined as the subspace E c satisfy

&

formed with all elements i

X, = P a & ,

= (x,)

v a
which (1. S )

In E the families of maps {Pab} and {@,} are related by

@,

t/ a

= pah@h


(1.6)

When E is equipped with the topology induced by the product topology we say that E is the topological projective limit of the spaces E,.

Example 1.12 Consider the space C, (see Example 1.10) equipped with the extended space topology (Example 1.9), and the family {CT}T>O(see also Example 1.10). We now show that C, can be identified with the topological projective limit of the spaces C,. Let P,: C,+ C, be the family of maps defined in Example 1.10. Given S > T and any fs, we see that PTf= PTf’ for any pair f, f’ E Pi-’)fs (see Fig. 1.2). Hence a linear onto map Pm : C, + C, is defined for all S > T, as pmA p,pg

i)

Obviously PTR =PmPsR for all R > S > T. It is therefore possible to define the topological projective limit of the spaces C,. Let be such a limit. We need to establish that C, and = 1@1 PmCs are topologically isomorphic. Let p: C,+ represent the map which assigns f~ C, to (f,) E when fT = PTf, since fT = the element (fT) lies in k. The map @ is onto, since given any ( f , ) ~C it is possible to define an f~ C, such that @,f=(f,), indeed it is only necessary to set

e

e

c

f(t)=fT(f)

for any

S

T

FIG.1.2

T rt

t

c

1.

MATHEMATICAL PRELIMINARIES

19

The coherence of this definition follows from the equivalence f s ( t ) = f T ( f ) for all S > T. Finally @ is one-to-one since @f = 0 means that PTf = 0 for all T>O and therefore f ( t ) - 0 . As vector spaces we can then conclude are isomorphic. Let us now compare topologies. that C, and In C, the generic element of the base of neighbourhoods of the origin is a set U of the form

e

U={fEC,:PT,(f)<~l,i= , .l. . , r } which can be rewritten as

U=

f?

{fECe:IIPT,fll<&)

I='

n C, n p g 1 ) ~ , , ( t I ) n

=

I=I

where B,,(&) is the open ball in C ,of radius

6,. Then

i=I

Noticing that ,@, the projection of and that @(C,) =

e:

k into C,,

can be expressed as PT@(-')

P(u)=en fi @ K ' ) B , , ( ~ ~ ) i- I

The second member above is the generic set in the base of neighbourhoods of the origin which define the topology of the projective limit, and C are topologically isomorphic. consequently

e

The above example introduced two important concepts: first, it introduced an alternative means of defining the projective limit; secondly it established the equivalence between a locally convex space and a projective limit. These results are generalized in the next two theorems.

THEOREM 1.10 Let {E,} be a family of finitely normed spaces and assume the existence of a linear complete topological space F and a family of linear operators {Pa}such that (i) P a F 3 E, and the map Pa :PL-"E, + E, is continuous (ii) PL-')Ea 3 Pi-')& for all a < b (iii) Pb(x)= Pb(x') implies P,(x) = P,(x') for all a < b and all x, x ' E F (iv) Pax = 0 for all a, implies x = 0

Then it is possible to define the projective limit of the spaces Ea, and the space E = PL-')Ea can be identified with such a limit.

n

a>O

20

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Proof The proof follows similarly to the technique used in Example 1.12. Define Pab :Eb-+E, as Pab A P , P ~ - ” ; this is possible since, from conditions (ii), (iii), given any xb E Eb and any pair x, X ’ E Pi-’)xb we have Pax = Pax’ E E, Clearly pa,,pbc= P,PL-l)P,,P(;’) = Pa,, hence it is possible to define the projective limit E = I @ pa&,. Let @, :E +. E, be the projection which assigns i E E to its complement x,. Consider now the map @: E +.

a=(&)

n E,, which assigns x n P:-”E, E

a >O

a

to

with xbApbx. Applying pab to xb We have Pa&, =p,Pi-’)PkX= Pax = x,, for any a < b. Hence x, = Pa&, fvr all a < b and therefore Px is always an element of the projective limit E. The operator P maps E into E, and from condition (iv) we see that @ is one to one. To show that it is and the set 0 = 0,, with 0, = Pb-”x,. onto, consider any i = (x,) E

n a

Notice that every 0, =P:-’)x, is a closed set as Pa is continuous. Moreover, since x, = p,P‘,-”xb we have x, = and therefore o , o b~. As a consequence, O = 0, is the intersection of a decreasing net of

n a

closed complete non-empty sets, hence 0 is a closed non-empty set. Furthermore, for any x E 0 we have Pax = x, for all a > 0; hence 0 c P(-’)x and is onto, thus is an isomorphism. Applying the projection @, and the definition of i we have on noting that Px = Pas = x, = P a x = P,Px (1.7)

a

Pa = Pa@

* Pa

A

A

= Pa@(-’)

The topology in fi is defined by a base of neighbourhoods of the origin whose generic element is

ir= E n in P:;’)B,, =l n

where B,, is an open ball in E,,. Hence

c=P ( E ) n A @PL;”B,,

(1.8)

i=l

We can therefore identify

ir with the set P(v> where n

It is natural to define the topology in E =

n P:-”E, a

by a base of

1.

MATHEMATICAL PRELIMINARIES

21

absolutely convex neighbourhoods of the origin whose generic element has the form (1.9). In these circumstances P is a topological isomorphism. 0

In each application of Theorem 1.10 it is important to see how large the space E = Pi-')E, is; it may happen that E is reduced to the single

n

a 20

point (0). Consider for example a space F of left continuous functions, defined in R+, which have at most a countable number of discontinuities: assume that F is equipped with the extended uniform topology. Let E, = C([O,a ] ) , a > 0, and let Pa be the truncation operator, f ( t ) for t s a for t > a Clearly the conditions of Theorem 1.10 are all satisfied. We can now see PL-"E, is the subspace of F formed by all continuous functions. that

n

a 20

A case of special importance arises when the mappings Paare into E,; that is, Pa maps F into E,. Under these conditions the maps Pa are also onto (condition (i), Theorem 1.10) and consequently P;+')E, = F. ThereP,-')E, coincides with F, and the space F is isomorphic with fore E =

n

a >0

the projective limit l@ PabEb. This result is used in the following theorem: THEOREM 1.11 Every complete locally convex space is topologically isomorphic with a topological projective limit of complete finitely normed spaces.

The proof follows the arguments used in Example 1.12. Whilst not intending to repeat the arguments here, we want to show how to construct the complete finitely normed spaces referred in the theorem. Take the family {Pa of finite semi-norms which define the topology in E and order it by taking a 5 b when P,(x) s P b ( x )for all x E E. Let N, be the null space of Pa.Then E, E/N, is a finitely normed space (Theorem 1.6). It is a simple exercise t o show that the family of canonical mappings A,: E +-E/N, satisfies each of the conditions of Theorem 1.10. (a)}

1.3.4 Banach spaces and Hilbert spaces A complete finitely normed space is called a Banach space. Because of their simple topological structure, Banach spaces are frequently utilized in stability studies. Example 1.13 The space C ( R + ) is a Banach space. To prove this statement we need only show the completeness of C ( R + )under the

22

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

uniform topology. Consider a Cauchy sequence cf,} in C ( R + )clearly, for each t E R+, the complex numbers { f n ( t ) } form a Cauchy sequence in C'. A function t Hf ( t ) can then be defined where f ( t ) is the pointwise limit of Cfn(t)}. For any t E R , and any fm, fn

llfn ( t ) - f(t)llsllfn - f m I1+ llfm ( t ) - f(t)II Taking the limit as m +m and then the supremum for all

t

Therefore since {fn} is a Cauchy sequence

It follows that {fn} converges to f in the uniform topology and therefore f is an element of C ( R + ) . A Banach space is called a Hilbert space, HS, when there exists a complex function (. , :HS2 + C', called an inner product, which satisfies a )

(i) llx112 = (x, x) (ii) (x, y) = (y, x)*

V x E HS

V x, y E HS

where x* denotes the complex conjugate of x. (iii) ( a y + p z , x ) = ( ~ ( y , x ) + p ( z , x )

V a , p ~ C l , x , yZ, E H S

PROPOSITION 1.7: SCHWARTZINEQUALITYFor all x, y E HS, I(x, y)I 5 IIxlI . IIY 11. Two elements x, y of a Hilbert space are said to be orthogonal when (x, y)= 0. Given a subset M EHS, we define MI as the orthogonal

complement of M, given by the set of all vectors x E HS which are orthogonal to every vector in M.

THEOREM 1.12 If HS is a Hilbert space and M is a linear closed subspace of HS, then HS can be decomposed in the direct sum HS = MCT3M'. This theorem illustrates the closeness between an arbitrary Hilbert space and the well-studied cases of R" and C". The theorem says that every x E HS has a unique decomposition x = x'+ x"

with X'E M and

X"E

MI.

1.

MATHEMATICAL PRELIMINARIES

23

THEOREM 1.13 Let f :HS + C' be an arbitrary linear, continuous, complex valued function in a Hilben space HS. Then there exists a unique y E HS such that V x E HS (i) f ( x ) = (x, y)

An important aspect of Hilbert spaces is illustrated in Theorem 1.13; in Section 1.6 it is shown that a linear, continuous, complex valued function that satisfies the conditions of Theorem 1.13 is called a functional and that the space of all functionals of a space E is called the dual space of E. Theorem 1.13 says in this context that the dual space of a Hilbert space coincides with the Hilbert space itself.

1.3.5 Operators in Banach spaces

Let E and F be Banach spaces equipped with norms II.IIE and II-IIF respectively. A linear continuous operator A : E + F is said to be compact or completely continuous when, given a sphere S in E, the closure of its image A(S), under A, is compact in F. One of the most frequent uses of compact operators is in the characterization of integral operators. For example, consider an arbitrary compact interval Z of the extended real line and the space C ( I ) . Given a complex, continuous valued function g :Z2+ C', then the operator G : C( I )+ C(Z) defined as ( ~ f i ( t= )

J g(t, s ) f ( s )ds I

can easily be shown to be compact. An operator A : E +. F is Lipschitz continuous when there exists a positive real y > 0 such that llA(x)-A(Y)llF

Y

[Ix - YllE

(1.10)

for all x, y E E. Obviously any Lipschitz continuous operator is also continuous but the converse is not generally true. The infimum of all y which satisfy (1.10) is called the Lipschitz norm or the incremental norm (or gain). It is important to note that, except in special circumstances, the incremental norm is not a norm in the strict sense; this term is, however, used for convenience. If there exists a y>O such that IIA(x)(IFSy llxllE for all x E E, the operator A is said to be bounded. The infimum over all values of y which satisfy this inequality is called the bound of A. If in addition A(0) = 0 (that

24

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

is, the operator A continuous, and the Let us represent operators A : E + F

is unbiased) then A is bounded if it is Lipschitz bound on A is inferior to the incremental norm. by B ( E , F) the space of all Lipschitz continuous which satisfy A(0)= 0; under these conditions:

THEOREM 1.14 In the space B ( E , F ) the incremental norm is a norm in the strict sense, and with this norm the space B ( E , F ) is a Banach space. A Banach algebra is a Banach space where the operation of multiplication is defined such that 1Ix. y 11 II(x(I. llyll for all x, y in the algebra. If we consider a Banach space E, the space B ( E , E ) can be seen as an algebra if multiplication is defined as the composition of operators. Moreover:

COROLLARY 1.14 B ( E , E ) is a left distributive Banach algebra.

If we take the subspace L ( E , F) formed by all linear operators which are elements of B ( E , F), then THEOREM 1.15

L ( E , F ) is a Banach space.

COROLLARY 1 . 1 5 L ( E , E ) is a linear Banach algebra.

1.4 FMED-POINT THEOREMS Fixed-point theorems are widely used in the study of equations where it is necessary to establish the existence of a solution, and as such they have a particular significance in determining the stability of dynamical systems. Given a subset M of a locally convex space E and an operator T: M - , E, we say that a point X,E M is a fixed point of T in M if it is invariant under the application of T; that is, T(x,) = x,,. The subset M is said to have the fixed-point property when every continuous operator which maps M into itself has, at least, one fixed point in M. The following two theorems allow us to identify those spaces with the fixed-point property. THEOREM 1.16: BROUWER FIXED-POINT THEOREM The closed unit bull in R" has the fixed-point property. A more general result which includes the above as a special case is

THEOREM 1 . 1 7 : SCHAUDER-TIKHONOV THEOREM Any absolutely convex compact subset of a locally convex space has the fixed-point property.

1.

MATHEMATICAL PRELIMINARIES

25

COROLLARY 1.17 Every compact operator T :E + E which maps a sphere S c E into itself has at least one fixed point in S. An operator T :E + E is said to be afine when it can be written as T ( x )= A ( x ) + T ( 0 ) where A :E + E is a linear operator. THEOREM 1.18: MARKOV-KATUTANAI THEOREM Let E be a topological vector space and M an absolutely convex compact subset of E. Let { T,} be a family of afine mappings satisfying:

(9 Ta(M)c M,

Va

(ii) T,Tb = TbT,,

V a, b

Then there exists a point Xo E M which is a fixed point of every map T,. A Lipschitz continuous operator in a Banach space, A :E + E is called a contraction when its incremental norm IlAll is inferior to the unit. Given a subset M of E such that A ( M ) cM, the operator A is, called a local contraction in M, if IIA(x)- A(y)llSyllx - yII for all x, y E M, for some y < 1. Local contractions provide the method frequently used for establishing the existence of a fixed point.

THEOREM 1.19: CONTRACTION MAPPING THEOREM Let E be a Banach space and let M be a closed subset of E. Assume that T :E + E is a local contraction in M, then there exists a unique fixed point of T in M. Moreover such a fixed point can be calculated as the limit of the sequence {x,} defined recursively as x,+~= T(x,) with X o E M. Finally, for all n

where

Proof

Consider the sequence {x,} defined recursively as xn+1=

Uxn),

M

X ~ E

Clearly as T ( M ) cM, the sequence is contained M, we have Hence

Il&+l-&ll=II~(&)- ~ ~ & - l ~ lIlx, l ~-&-Ill ll~l~ IIx,+i

- x ~ l l ~ l l q lIIT(xo)-xoll fl

26

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

For any rn > n,

Since

11711< 1, then

lim I1xm- xnI(= 0, hence {x,} is a Cauchy sequence. m,n

Because the space E is complete, the sequence has a limit X; moreover since M is closed, X must lie in M. Taking the limit in m in inequality (1.11)

To show that i is a fixed point of T we use the continuity of T(.) and have

T ( i )= T(lim x,) = lim T(x,)= lim (xn+,) = X. n

n

n

Finally we can show that i is unique in M by considering some point x E M such that x = T ( x ) :

I l i - XI1 = IIT(2)

-

T(x)lls 117111

f

l l i - XI1

but since 11Tl[<1 this inequality can only hold if x = i.

0

1.5 MEASURES AND FUNCTION SPACES

1.5.1

Measures and integrals on the positive real line

A a-algebra is a collection of subsets of a space X,which includes X and the null set 0, and is closed under countable unions of its elements. In this work we restrict our study to the smallest a-algebra of subsets of the extended real line, Re = [-m, 303, or the extended half real line R', = [0, m], which contain every open set. Let 6 represent such a a-algebra; its elements are called Borel sets. It can be shown that every bounded interval on the real line is a Borel set and that every compact set is also a Borel set. Let A be a a-algebra of subsets of the space X.A function p : A + R',

1.

27

MATHEMATICAL PRELIMINARIES

is called a finitely additive, positive measure, or simply a measure, if

u A k )= c p ( A k )for any finite collection of pairwise disjoint AkE A.

/ n

p(

k=l

\

n

k=l

If in addition, p

U Ak

(k11

Co

1 p ( A k ) ,for )-k=l -

any countable collection of

painvise disjoint AkE A , then p ( * ) is said to be a countably additive measure. The triplet (X, A, p ) is called a measure space. Throughout this work, X is restricted to either R', or Re and A is the a-algebra of Borel sets. Also we restrict measures to those defined in B, which take finite values when applied to an arbitrary compact set C c R ; such measures are called Borel measures. A Borel measure p is regular when for each set E E B for which p ( E )< m, we have

p ( E )= inf p ( 0 ) =sup p ( C ) where the infimum is taken over all open sets 0 containing E, and the supremum is taken over all compact sets C contained in E.

Example 1.14 Let F : R, + R , be a monotone increasing positive function, continuous everywhere except perhaps in a countable set which is only left continuous. Consider the collection of all semi-open intervals of the form [a, b ) and define ~ ( [ ab ), ) = F ( b ) - F ( a )

For an arbitrary Borel set E define

2 p(lk) n

p(E)=inf

k=l

where the infimum is taken over all finite unions I =

u n

k=l

zk

of semi-

open intervals I k = [ a k ,b k ) such that I z E . It can be shown (Barra, Chapter 9, 1974) that p ( - ) is a regular Borel Measure, p ( is called the Lebesgue-Stieltjes measure with respect to F( .). In particular the measure generated by the function F ( x ) = x is called the Lebesgue measure. a )

A function f : R , + R is said to be Borel measurable, or simply measurable, when for each c > 0, the set { t :R, :f(t) > c} is a Borel set. Clearly if f is measurable, then If1 is also measurable; moreover the space of all measurable functions is a linear space. A linear space of functions which, like the space of all measurable functions, is closed under the operation of taking absolute values, is called a linear lattice. Given two elements 4, J, of a linear lattice of functions we can define

28

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

the operations v and

4v 4A

A

+ +

through

+ M 4 ++)+t 14 - $1 = min(4, +Pi(+ + +) -f I4- +I

= max (4,

Clearly a linear lattice is closed under the operations v and A . A subspace of the space of all measurable functions which is itself a linear lattice is the space of simple functions in B, represented by S(B). Let xE represent the characteristic function of a set E

A simple function is any function 4 of the form 4 = c I x E+, c2xE,+ . . . + cnxE,where the ci are all real, non-zero constants and the Ei are pairwise disjoint relatively compact Borel sets. It is obvious that S ( 6 ) is a linear lattice, since measurable. Note that { t :+ ( t ) concentrated in

n

U Ei.

n

141= C IciJxEI,and that every 4 is i=l

n

# O}= U E i ; we can therefore say that 4 is i= I

i=l

PROPOSITION 1.8 Every positive measurable function f :R , 4R , can be expressed as the limit everywhere of an increasing sequence of positive simple functions. Proposition 1.8 is fundamental in the following definition of an integral. Let p be a Borel measure; we can define in S ( B ) a positive linear functional fi : S(B)+ R through for

fi(4)= CIP(EI)+ CZP(EZ)+. .+ c,p(En) *

4 = c 1 x E I + . . *+cnxE,

Clearly if 4 2 0 , then fi(4)rO;therefore fi is a positive functional. , czxE,+. . .+ cnxE, be such that C$ 2 0 and Conversely, let 4 = c l x E + f i ( $ ) = O . Since every ci is non-zero, positive, and every p ( E i ) is also positive, then p ( E i )= 0 for all i = 1,2, . . . , n. I;($)= 0 implies that 4 is concentrated in a set whose measure p is zero; such a function and associated set are called respectively a p-null function and a p-null set. We can now extend the positive functional fi to a positive measurable function f. Let be an increasing sequence of positive simple functions converging pointwise to f everywhere, such a sequence, by Proposition 1.8, always exists. Then the positive numbers @(&) form an increasing

+,,

1.

29

MATHEMATICAL PRELIMINARIES

sequence; if such a sequence is bounded, we say that f is p-integrable and we define k(f) as limfi(&). The positive functional 6 is called the n

p-integral and GCf) is more usually represented as J f dp. Although there may be an infinity of different increasing sequences of simple functions converging to f, it can be shown that the limit lim @(&) n

is independent of the particular sequence chosen and hence the above definition of the p-integral is consistent. If f is a positive p-null function (that is, the set {t:f(t)>O} is p-null) then every simple function 4 s f must also be p-null; consequently f dp = 0. Conversely if f r o and j f dp = 0, then for any 4 ~ fj 4, dp = 0. Thus every simple function which satisfies 4 l f must be p-null, and therefore { t : f ( t ) > O } must be a p-null set. Hence if f is a p-integrable positive function, we have j f d p = 0 if and only if f is p-null.

1.5.2 The function spaces Lp Define, Linc(p)as the space of all positive p-integrable functions. If two positive p-integrable functions f and g satisfy f + nf = g + ng,where nf and nR are positive p-null functions, then we can associate f and g with the same element of Linc(p). Under these conditions the integral J d p vanishes only on the zero elements of L'"'(p). Define LI(p)as the space of all measurable functions f : R , + R which can be written as f=g-h

(1.12)

with g, h E L'"'(p). We extend the integral j dp to LI(p) by defining

I f dP =

I

(1.13)

g dP - h dP

There are an infinity of pairs g, h E Linc(p)which satisfy (1.12); however, the value of the integral (1.13) is independent of the pair chosen. Of all partitions one has special importance: define f + = f v O and f-= (-f) v 0. Clearly f = f+- f- and If1 = f++ f-, with f+,f- E Linc(p).Therefore LI(p) is a linear lattice and

(1.14) Since the p-integral vanishes only in the zero of Linc(p),then if we must have f+= f- = f = 0.

If1

dp = 0

30

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

It can now be seen that the function

/

)I. 11: L , ( p )+ R,

defined by

llfll = IfI dP is a norm in Ll(p) and that with the topology defined by such a norm, the space L l ( p ) is a Banach space. A more general definition of an integral can be made in an arbitrary measure space ( X , A , p ) . The space of all p-integrable real valued functions thus defined is represented by L , ( X ,A, p ) , we see that L , ( p ) is an abbreviation of L , ( R + ,B, p ) . In particular, when p is the Lebesgue measure we shall represent this space simply as L , . For any 1 5 p < w we define L ( ) as the space of all measurable functions f : R , + R such that Iflp EP L'in= ( p ) . As in L , ( p ) and LinC(p)we can identify with some element of L , ( p ) any two functions which differ by a p-null function. The norm in L p ( p )is defined as

With the topology defined by such a norm, L p ( p )is a Banach space. As with L I ( p )we represent by Lp the particular case when p is the Lebesgue measure, and by L p ( X ,A, p ) in the more general case when we consider integration in the general measure space ( X ,A, p ) . To define the L J p ) space, we introduce the set V(f) for any positive measurable function f defined by V(f) = {a2 0 :p ( {t :f ( t ) > a})> 0) If V(f) is unbounded, we say that the essential supremum of f , represented by ess supf, is infinite. If Vcf) is bounded, the essential supremum of f is the least upper bound on V ( f ) . We shall define L , ( p ) as the space of all measurable functions f : R, + R such that ess sup If1
't(lfI2+Id')

hence the integral f g d p is well defined. It is easy to show that

i

(f?

/

g>2 f g dlL,

defines an inner product and so the space L 2 ( p )is also a Hilbert space.

1.

MATHEMATICAL PRELIMINARIES

31

So far we have only considered real valued functions, but our definition of an integral can be readily extended to complex valued functions, so f : R+ + C' is measurable (respectively, integrable) if both its imaginary and real parts are measurable (respectively, integrable) functions. The space L , ( p ) , 1 5p 5 0 3 can be defined as the space of all measurable functions f : R , + C' such that Ifl p lies in Linc(p).The inner product in L 2 ( p )is now defined by

1.5.3 Real and complex measures Until now we have considered only positive measures. The concept of a measure can be extended to real valued measures, also called signed measures, by considering set functions of the form p = o - y with o,y positive measures,

p(E)Lo(E)-y(E)

V E EB

Clearly there exists an infinity of pairs y, o which generate the same signed measure p. One of these pairs of particular importance: if we define p + ( E )= SUP p ( A ) ACE

/ - - ( E l = SUP

ACE

then p+ and p- are measures p+ and pimportant property of p+ + p- is the smallest

-p(A)

positive measures such that p = p + - p - . The are called the Jordon decomposition of p. An this decomposition is that the positive measure measure which satisfies

V EE B Ip(E)I 5 p + ( E )+ p - ( E ) The measure p+ + p- is called the total uan'arion of p and is represented by (PI. We then have I p ( E ) l + 4 (El VEEB Complex measures are set functions of the form p ( E ) pR(E)+jpI(E)

where p R and are signed measures. The total variation of p, represented by IpI, is defined as lpl= lpR I +

32

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Moreover the total variation is the smallest measure which satisfies - l P ( m I ~ l P I ( E ) VEE B 1.5.4

Indefinite integrals

Let A represent the Lebesgue measure, and let p be a finite measure that vanishes at every A-null set. The measure p is then said to be absolutely continuous with respect to the measure A. If in addition p is regular and countably additive, then a famous theorem specifies an explicit form for P*

THEOREM 1.20: RADON-NYKODYM THEOREM there exists a unique f E L , ( A ) such that p ( E ) = I fdA

Under the above conditions

VEEB

(1.15)

E

Conversely any integral of the form of (1.15) defines an absolutely continuous, regular, countably additive Bore1 measure. Note

For an alternative statement of this theorem, see Section 3.3.

Consider now some positive f

E L,,

F ( x )=

I

then for any x L 0 we can define

[O, X I

f dA + F(0)

(1.16)

A function F ( . ) of this form is called an indefinite integral. Clearly F ( x ) = p([O, x ] ) + F ( O ) ,where p ( * )is defined by (1.15). For any interval [ a , b) we have

d [ a , b ) )= F(b)- F ( a ) therefore p ( . ) is a Lebesgue-Stieltjes measure with the characteristic that F( is both left and right continuous everywhere. As p ( .) is absolutely continuous, then for any (> 0 there exists a 6 > 0 such that 1 (b,- a i )< 6 I implies that a )

(1.17) for [ai, bi) disjoint intervals. Notice that ure of the set

1 (bi- a i ) is the Lebesgue measi

U [ a , bi) and that 1 ( F ( b , ) - F ( a , ) )is the p-measure of i

i

the

same set. A function F ( - ) which satisfies (1.17) is called an absolutely continuous function. Because f E L 1 was defined to be positive, F ( . ) is

1.

33

MATHEMATICAL PRELIMINARIES

monotonically increasing. For a more general real valued function x H F(x) we say that F(x) is absolutely continuous when, for any .$ > 0, there exists a S > 0 such that C (bi - ai)< S implies i

C IF(bi) - F(ai)I<6 i

for [ai, bi) disjoint intervals. Consider then a generic real valued f~ L,(A) and define

F(x)= F(0)+ for. any b > a

F ( b ) - F(a)= [ah)

I

Lo. X I

(1.18)

fdA

fdA = p ( [ a , b ) )

where p ( * )is defined as in (1.15). Hence IF(b) - F(a)l= Ip([a, b ) ) l s Ipl([a, b ) )

If p, defined by (1.15), is absolutely continuous, so is its measure lpl and so Hence if

I= Therefore

i=l

u[ a , bi) n

i=l

and

A ( l ) + 0, then

IF(bi)- F(ai)15I p . ( ( I+ ) 0 as A ( l )

I p I ( I )+ 0

+0

and so F( .) is abso-

lutely continuous. We have seen how every indefinite integral generates an absolutely continuous function. The converse is also true; however, before we state this result we require some additional definitions. Let F: R, + R be a continuous function and let C c R, be a compact set. The total variation of F in C, represented by V,(F) is defined by n

C

V ~ ( F ) = S U P IF(ai)-F(bi)I i=l

where the supremum is taken over all finite unions

n

U [ai,bi) of

i=l

disjoint

intervals contained in C. The total variation of F in R,, represented by V,(R+), is V,(R+)=sup V,(F), where the supremum is taken over all compact sets C c R,.

34

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Every absolutely continuous function has finite total variation in any compact set but not necessarily in R,. If F is absolutely continuous and has finite total variation in R + , then F is represented by an indefinite integral of the form F(x)= F(0)+

(1.19)

fdh [O,Xl

where f is an element of L,. Moreover V J F ) = I If1 dh

Every absolutely continuous function F (even when it does not have finite total variation in R + ) ,can be represented by an integral of the form of (1.19), where f is now a function such that for every compact set C c R,, the function fc’ f . xc is an element of L,. In these circumstances Vc(F)= I l f c l dh.

1.6 DUAL SPACES In Section 1.5 it was shown that integrals are special forms of a more general concept-a functional. Given a locally convex space E, a functional in E is any linear, continuous, complex valued function in E. The zero functional is identified with the function which assigns zero to every x E E; two functions f, g, such that f(x) = g ( x ) for all x E E, are considered to be the same functional in E. The space of all distinct functionals in E is called the dual space of E and is represented by E’. Under the definition (af+P g ) ( x )

af(x) + P g ( x ) ,

a, P E C ’ ;

f, g E E’

the space E’ is clearly a linear space.

PROPOSITION 1.9 Let { p a } be a collection of semi-norms defining the topology in a locally convex space E. For any f E E’ there exists a finite collecof these semi-norms and a collection {pi}:=] of positive tion numbers such that If(x)I s m a x PiPa,(X),

V x EE

(1.20)

Proof Given any t > O , let {pa,}be a finite collection of semi-norms and let {ti} be a collection of positive numbers such that p,,(y)<&, i = 1,2, . . . , n implies If(y)I < 5; such collections must exist since f ( . ) is continuous. Defining pi = &/tand p(y) = max pipa,(y);the above stateI

1.

35

MATHEMATICAL PRELIMINARIES

ment can be written as If(y)l? 1 implies p ( y ) B 1. For any x in the null space of f(.), (1.20) is obviously true, if f(x) # O let x = I f ( x ) l i where i x/lf(x)l. Note that If(i)l= 1, so p ( i ) 2 1. And clearly since p ( is a semi-norm then p ( x ) = If(x)l p ( i ) and therefore, p ( x ) 2 If(x)l for all x E E. Consequently 0

If(x)l s m a x pipa,(x),

)

V X E E-

l

In a finitely normed space E, the topology is defined by a unique semi-norm (which is the norm itself); then as a consequence of Proposition 1.5. given f~ E’ there exists a p > 0 such that If1 5 pllxll for all x E E. The infimum of all such p is represented by llfll and we have

=

SUP I f ( 4

IIxII= 1

It is straightforward to establish:

PROPOSITION 1.10 If E is a finitely normed space, the positive function defined in E’ by (1.21) is a finite norm and, equipped with the topology thus defined, the space E‘ is a Banach space. The topology defined in E’ by the norm (1.21) is called the strong topology; E’ equipped with this topology is called the strong dual and is generally represented by E*. Let B be the unit closed ball in the finite normed space E; then (1.21) can be written as llfll= sup If(x)l. A similar relationship can be obtained XEB

even when E is an arbitrary locally convex space. Consider an arbitrary bounded set B in the locally convex space E and let f ( .) be a functional in E. Since f ( .) is continuous f ( B ) is bounded. A positive finite valued function pB :E’+ R , can be defined as

which is a semi-norm in E’. Consider now a B spanning all the bounded sets of E; we then obtain a family of finite semi-norms { p B ( * ) }in E‘. These semi-norms define a topology in E’. To see this consider Theorem 1.4: it is only necessary to show that for any f # O in E’ there exists a B such that p B ( - ) # O . This is true since f # O implies that for some X,E E, f(x,)#O; as the singleton {xo} is a bounded set, the conditions of Theorem 1.4 are satisfied. The topology defined in E’ by this family of semi-norms is called the strong topology. When E is a finitely normed

36

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

space, the topology generated by the semi-norms {pB( coincides with the topology generated-by the norm (1.21). As in the case of the normed space, the dual space equipped with the strong topology is called the strong dual and is represented by E*. Consider now arbitrary finite collections of single points in E, I = { x , , x 2 , . . . ,x"}. These are particular cases of bounded sets, and they include the single point sets. Therefore the family of semi-norms { p I } , generated when I spans all the finite collection of points in E, also defines a locally convex topology in E'. It is called the weak topology. Since the collection {p,(.)} is a sub-collection of {pB(.)}, the weak topology is weaker than the strong topology: hence its name! The generic elements of the bases of neighbourhoods of the origin in the strong dual and in the weak topologies are then, respectively, a)}

V,={fEE':sup[f(x)I<&, i = 1 , . . . , n ; ti> O XGB,

V, = { f E~' : If(xi)l < 4, i = 1 , 2 , . . . n ; ti> O

and Bi bounded} and

xi E E }

Let E and F be locally convex spaces and A : E F a linear continuous operator; for any f~ E' define a linear continuous map e E E' through

e(x)=f(A(x))=f@A(x) V X E E

(1.22)

where @ represents the composition of operators. Hence a map A* :F + E' is defined which assigns f~ F to e E E' through (1.22). The map A*, which is linear and continuous, is called the strong dual of A (if E' and F' are equipped with strong topologies). If E and F are finitely normed spaces, we also have llAll= IIA*II (Yosida, 1971). Having defined general dual spaces, we now consider explicit forms for the duals of some spaces of time functions used in this work. Consider the L,, spaces. Theorem 1.13 shows that the strong dual of any Hilbert space, HS, is isometrically isomorphic with itself. In particular, if p is any positive Bore1 measure, the space L 2 ( p )is a Hilbert space and therefore L T ( p )= L z ( p ) .Any f~ L f ( p ) can be written as

f(4)= If4 dA

v4 E L2h.L)

for some f~ L 2 ( p ) . The duals of L p ( p )spaces, 1< p < m, are spaces of the same type, and it can be shown that:

PROPOSITION 1.11 "he space L r ( p ) , 1 s p < m , is isometrically isomorphic with the space L q ( p ) , where q satisfies p - ' + q - * = l . Any f ~ L r ( p is)

1.

MATHEMATICAL PRELIMINARIES

37

uniformally determined by some f E L q ( p ) such that

j

f ( 4 )= 4f d p

v 4 E LAPI

The dual of Lm(p)cannot be expressed as one of the spaces L p ( p ) .In the appendix to Chapter 3 we refer to results which show that Lz contains a space of the form L,(A) but also contains a space of measures which are strictly not countably additive. An important space which is loosely related with L, is the space C(X) of all continuous bounded functions in the set X c R,. Assume that X is a compact space, then PROPOSITION 1.12 If X c R, is compact then C * ( X )can be identified with the space of all countably additive, regular, bounded measures in the a-algebra B n X, (where B denotes the class of all Borel sets in R+). A n y f E C*(X) is uniquely represented by such a measure p and

fW=j$dA This Theorem, called the Riesz’s representation theorem, illustrates how the measure p ( - ) is constructed from a functional f . We have assumed that f is real and positive; however, the extension to arbitrary complex functionals is straightforward. Let 0 = X be any open set and define

d o )= SUP f(4)

(1.23)a

where the supremum is extended to all 4 E C(X) which satisfies 4 5 xo. For any set A c X we define p ( A ) = inf p ( 0 )

(1.23)b

where the infimum is extended to all open sets 0 containing A. The set function defined by (1.23) is not necessarily a measure if we consider the a-algebra formed by all subsets of X. However, if we restrict the applicability of p ( . ) to Borel sets contained in X (that is, sets of the form B n X), it can be demonstrated that p ( - ) is a countably additive, regular, bounded measure. It is important to note that p ( .) is such a well-behaved measure only in the a-algebra B n X . It may happen that p ( * )can be extended to a set function in the whole B which has neither of the attributes of p ( - ) in B n X .

PROPOSITION 1.13 Let E =

m

U En be the topological strict inductive limit of

n=l

38

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

a countable family {E,,}of finitely normed spaces. Then the strong dual E* can be identified with a topological projective limit of the strong duals of the spaces E,,. Proof For each f E E' define f,, E E:, as the restriction of f to E,, let R,, : E'+ E:, be the map which represents such a restriction. Identically, for m > n, let R,, : EA + Ek represent the map which assigns f, E E h to its restriction in E,,; clearly R,, = R,,,R, for m > n For any f,,E E:, we have for some k, > 0, v x E Efl Iffl(X)II kfl IIxllfl, Since the topology induced in E,, by the topology Em, m > n, coincides with the topology of E,,, then the norms Il.ll,, and Il-l ,,, are equivalent in E,, and there exists a 6 > 0 such that llxlln 5 6 IlXllm

Vx E E n *

So If,,(x)I 5 k,,S llxllm for all x E E,,. Utilizing the Hahn-Banach theorem (Kothe, section 17.6, 1969), the functional f,, can be extended to a functional fm E EL such that R,,Rmfm= f,,. As a consequence the family of maps {Rflm} are always onto. Moreover if f E E' is such that Rnf = 0, for all n, then for any X E Ethere exists E,, which contain x and therefore f(x)=(R,f)(x)=O; thus RJ=O, for all n, implies f =O. It now follows from Kothe (section 19.7(8), 1969) that the maps R,, :E'+ E:, are onto. Thus using Theorem 1.10, it follows that as a vector space E'=l@ Rflm(E&)

To compare the topology of E* with the topology of the projective limit, consider an arbitrary semi-norm ps( of the family which defines the topology in E*. B is a bounded set in E so, utilising Theorem 1.8, there exists some E,, which contains B and in which B is a bounded set, thus B is contained in some closed ball B,, = {x E E,, :llxll,, I 6,). Therefore a )

pB(f) = sup If(x)l xtB

sup

xeB,

IRnf(X)I

= 6fl

IIRnfll:,

where II-II: denotes the norm in E:. Thus any neighbourhood U = c f E~* : pB(f)< 5) contains a neighbourhood of the form R',"'cf~ E: :Ilfll: < 66;'); therefore U is a neighbourhood in the topology of the projective limit. Conversely, any neighbourhood in the topology of the projective limit is also a neighbourhood in the strong topology of E*, thus the two topologies coincide.

1.

MATHEMATICAL PRELIMINARIES

39

1.7 NOTES The basic topological concepts introduced in Section 1.2 can be found in Kothe (1969) sections 1-3, together with the associated proofs. The definitions and results produced in Section 1.3.1 are as given in Valenca (1978). An exhaustive study on the properties of inductive and projective limits can also be found in Kothe, section 19 (1969). The example in Section 1.3.3 and Theorem 1.10 were introduced as a preliminary to the study of extended spaces in Chapter 5, the operators Pa (referred to in Theorem 1.10) are in the majority of applications truncation operators. The results of Sections 1.3.4 and 1.3.5 on Banach and Hilbert spaces can be found in Yosida (1971) and Valenca (1978). The proofs of the Brouwer and Schauder-Tikhonov fixed-point theorems can be found in Dunford and Schwartz (1957); the proof of the Markov-Katutanai theorem can be found in Larsen (1973). The approach to measure and integration theory adopted in this book is similar to that of Halmos, (1950), Hewitt and Ross (1979), Hewitt and Stromberg (1965) and Barra (1974), the presentation of these concepts follows that of Weir (1974). The analysis of the spaces L, of indefinite integrals is based upon the approach of Hewitt and Stromberg (1963, where the reader may also find the detailed proofs of Propositions 1.11 and 1.12.

REFERENCES

Barra, G. (1974). “Introduction to Measure Theory”. Van Nostrand Reinhold, New York. Dunford, N. and Schwartz, J. (1957). “Linear Operators” Part I. Interscience, New York. Halmos, P. R. (1950). “Measure Theory”. Van Nostrand, New York. Hewitt, E. and Ross, K. A. (1979). “Abstract Harmonic Analysis”, I, Band 115, 2nd Edition. Springer Verlag, New York. Hewitt, E. and Stromberg, K. (1965). “Real and Abstract Analysis”, Graduate Texts in Mathematics, 25. Springer Verlag, New York. Kothe, G. (1969). “Topological Vector Spaces”, I, Band 159. Springer Verlag, New York. Larsen, R. (1973). “Functional Analysis”. Marcel Dekker Inc, New York. Valenca, J. M. E. (1978). “Stability of Multivariable Systems”. D.Phi1. Thesis, Oxford University. Weir, A. J. (1974). “General Integration and Measure”. Cambridge University Press, Cambridge. Yosida, K. (1971). “Functional Analysis”, Band 123. Springer Verlag, New York.

Chapter Two

Riemann Surfaces and the Generalized Principle of the Argument 2.1 INTRODUCllON In the development of frequency domain stability criteria for multivariable feedback systems, the single most important result in establishing generalized Nyquist criteria is the Principle of the Argument. In this chapter a variety of local and generalized Principles of the Argument for complex polynomials with singular points (including multiple poles and zeros) and the associated encirclement conditions are derived. The concepts of this chapter rely o n the properties of Riemann surfaces and the concept of homotopic triviality as found in algebraic topology (Massey, 1967; Hilton and Wylie, 1960). Associated with every analytic function there exists a Riemann surface which has the property that the image of a simply connected region in the complex plane X is simply connected on the Riemann surface. The Principle of the Argument and Nyquist criterion in such cases are trivial. However, for algebraic functions whose Riemann surfaces are multiple connected, the extension of the Principle of the Argument is non-trivial, since the Principle of the Argument must hold for a region of the Riemann surface which is non-simply connected and whose boundary surface is composed of distinct closed contours. Riemann surfaces were originally introduced to solve the problem of a multivalued complex function such as s”’. The extension of the Principle of the Argument for general multiple valued analytic functions has only been considered by Evgrafov (1978), Postlethwaite and MacFarlane (1979), and Valenca and Harris (1980). The approach in Section 2.5 consists in replacing a complex region X with a surface Z (whose topological properties are closely related to X) and multivalued functions on X by single valued functions on Z. By this mechanism Cauchy’s theorem can be restated for homotopically trivial loops in some subset of Z. Cauchy’s residue theorem 40

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

41

is directly instrumental in establishing the Principle of the Argument in Section 2.6. To derive the generalized Principle of the Argument we present the theory of covering spaces (Section 2.3) and minimum contours (Section2.4). The essence of this approach is that Nyquist criteria are shown to be just aspects of homotopy theory. Homotopy is introduced in Section 2.1.2 as a relationship of equivalence in quotent spaces or in the fundamental group of a topological space. Indeed it is shown that a path r in a complex region X does not encircle the -1 point if and only if r is homotopic to a point in X-{-l}; such a path r is called homotopically trivial. Conversely r encircles -1 if and only if r cannot continuously be deformed to a point in X-{-l}.This simple analysis shows that the Nyquist encirclement condition is fundamentally a homotopy concept (see Decarlo and Seaks, 1977). Multiple singular points of complex polynomials are considered in Section 2.6; by identifying these points with punctures on simple regions we can define an algebroid Riemann surface C over simple regions with punctures. Finally by defining a minimum contour generated in C by the boundary loop of a bounded simple region of the complex plane the generalized Principle of the Argument and encirclement conditions are established.

2.1.1 Paths in topological spaces Let Z and Zl be arbitrary intervals of the extended real line of the form [a, b], - m r a < b s m , and let X be a topological space. Identify any continuous function f :Z + X with any function f l :ZI + X when there exists an increasing isomorphism h :Z 4ZI such that both h and h-’ are absolutely continuous and f(r) = f , ( h ( t ) )for all t E Z, that is f = f l 0 h. It is clear that such an identification defines an equivalence relationship in the space of X valued continuous functions of a real variable with connected compact support. Each equivalence class is called a path in X. When no ambiguity arises the same symbol will be used to represent both a continuous function f : Z + X and the class of equivalence to which it belongs. When in addition X is a region in the complex plane and f :Z + X is absolutely continuous, the path defined by f is rectifiable or that it is an arc. The variation of f in Z, represented by I ( f ) is called the length of f. It is well known (Hewitt and Stromberg, 1965, p. 283) that r -f’(t) can be identified with an element of L,(Z) so that the variation of f in I coincides with JI If’(t)l dt. Because a path is defined as a class of equivalence in a space of continuous functions, it is necessary to establish the consistency of the

42

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

definition of an arc and its length. Consider then f l = f h with h an absolutely continuous homeomorphism of Zl into Z. Then f,(s) is also differentiable almost everywhere such that 0

dflo= ds

($9

f=h(s)

dh dS

From Corollary (20.5) of Hewitt and Stromberg (1965) we have

-

and therefore f,(.) is also absolutely continuous and its variation in Zl coincides with the variation of f in Z. Given a path f : Z X it is always possible to define a homeomorphism h :[0, 13- X such that f and f 0 h are equivalent. Hence it is always possible to think of a path in X as being defined by a function whose domain is [0, 11. In this work, unless otherwise stated, it is assumed that a path is defined by a function f(-)whose domain is [0, 13. The point f(0) is called the origin of the path (represented by O(f)) and the point f(1) is called the extremity (represented by e(f)). If in a path f the origin and the extremity coincides then the path is closed. A closed rectifiable path is called a loop. If a path is defined by a constant function it is said to be trivial. Finally a contour is a finite sequence cf,}, i = 1 , 2 , . . . ,n, of closed paths, clearly any finite sequence of contours is also a contour.

2.1.2

Homotopy

Definition 2.1 Let X be a topological space and f and g be two paths in X. The two paths are said to be homotopic to each other (that is f-g) when f(0)= g(O), f(1)= g( 1) and there exists a continuous function H :[0,1] x [0,1] +-X such that: (a) f(0) = g(0) = H ( 0 , s) and f ( 1)= g(1)= H(1, s) (b) f ( t ) = H(t, 1) and g(t) = H(t, 0) V t E [0, 11

V s E [0, 11

A closed path in X is said to be homotopically triuial when it is homotopic to a trivial loop. Any path f has an inverse (represented by f) and defined by the function t H f(1- t ) . Given two paths f and g which satisfy g(O)=f(l) a third path h can be defined by the function

h ( t )=

for

t~[O,4]

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

43

The path h is said to be the concatenation of f and g and is represented by

h=f@g

Consider for example the path f @ f7 the function

demonstrates that f @ f is always homotopically trivial. An important result on homotopically trivial paths is:

PROPOSITION 2.1 Let f be a path in X and let g, and g2 be homotopically trivial closed paths which satisfy, respectively, gl(1)= f(0) and g2(0)= f ( 1). Then both g, @ f and f @ g2 are homotopic to f. Proof Let GI(?,s) and G 2 ( fs), be the continuous functions which establish respectively the homotopy of g , to g,(l) and the homotopy of g, to g2(0). The functions

and

establish then respectively the homotopy between f and g, @ f and the 0 homotopy between f and f @ g 2 .

Definition 2.2 A contour C = c f , } , i = 1,2, . . . , n, in a topological space X is said to be homotopically trivial when there exists a point X,,E X and arcs 4i in X such that (i) O ( + i ) = x o and e(bi)=O(fi) V i = 1 , 2 , . . . ,n (ii) C A (4, @ f l @ 61)@ (42@f2 @ 82)@ . . . @ (4" @fn@ 4")

(2.1)

is a homotopically trivial loop in X. Any loop 6 of the form (2.1) is called an associated loop of the contour C. Let a ( X , xg) represent the set of all closed paths in X with origin xo. It is now easy to show:

PROPOSITION 2.2 Let f,. g,, f, g E a(X, x,,) satisfy f - f ,

f @g -f1@ Proof

and g

- g,,

then

g,

Let F(t, s) and G(t,s) be the functions defining the homotopy

44

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

between f and f , and between g and g , respectively; then the function

{

H(t, s) = F(2r' s, G(2t - 1, s)

t E [O,

$1,

t E [0,4],

establishes the homotopy between f @ g and

fl @

s EW , 1 1 s E [O, 13 g,.

The concept of homotopy clearly defines a relation of equivalence in a ( X , x,). The quotient space thus defined is called the fundamental group of X and is represented by a(X,x,). The group operation is the operation induced in a ( X , x , ) by the concatenation of loops in a ( X , x , ) ; that is if f, g E a ( X , x g ) and Lf], [ g ] represent respectively the class of equivalence containing f and g , L f ] + [ g ] is defined as the class of equivalence which contains f @ g. Proposition 2.2 illustrates the mathematical consistency of this definition. The identity element (represented by 0) is defined as the class of all homotopically trivial loops. Proposition 2.1 shows that O + [ f ] = Lf] and that L f ] + 0 = L f ] , proving that 0 is a unit element. Since f @ f is always homotopically trivial, the inverse of Lfl, represented by -Lf], can be defined as the class of equivalence which contains f-; in this case [ f ] + ( - L f l ) = 0. Finally it is clear that f @ ( g @ h ) = (f @ g ) @ h for any f, g , h E a ( X , x,). Consequently the operation + is associative and is therefore a group operation. There exists sets, X , for which the fundamental group has a particularly simple and useful form. Consider, for example, the unit disc in the complex plane D {z :121Il}and a collection {z,, z2, . . . , z}, of n distinct interior points of 0.A region, X , of the complex plane which is homeomorphic with D -{zl, z2,. . . , z,} is called a simple region with n punctures. If n = 0, X is called a simple region, for these regions the following holds:

THEOREM 2.1 (Hilton and Wylie, p. 242, (1960)) Let X be a simple region with n punctures and let X , E X , then a ( X , xo) can be identified with a free abelian group of n generators. Theorem 2.1 establishes that v ( X , x,) is isomorphic with Z", where Z represents the abelian group defined by the integers, furthermore in a simple region every loop is homotopically trivial. Finally to complete this section it is convenient to present a result which relates the fundamental groups of the two topological vector spaces X and Y .

PROPOSITION 2.3 (Hilton and Wylie, pp. 234-235, 1960) Let h :X + Y be a continuous map, then h induces a homomorphism h* : a ( X , x,)

+ a(Y,h(x,))

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

45

by assigning the class of equivalence, which contains the path f in X to the class of equivalence which contains h f. Moreover if h(.) is a homeomorphism, h is a n isomorphism. 0

2.2

COMPLEX INTEGRATION AND CAUCHY’S THEOREM

Let X be a region in the complex plane and let H(X ) represent the set of all complex functions which are analytic in some open set containing X. Let z HF ( z ) be an element of H(X) and f an arc in X. Since t F(f(t)) is bounded and t Hf(t) is absolutely continuous, the integral

is well defined. Note that t I+ f’(t) can be identified with an element of L,[O, 11. Suppose now that g:[a, b]-X is equivalent to f and that the homeomorphism h :[a, b] +. [0,1] establishes such an equivalence, then g = f 0 h. By hypothesis h and h-’ are both absolutely continuous, so from Hewitt and Stromberg (corollary 20.5 (1965)),

I

b

ljo1(F0f)(f)f’(f)dt =

a

=r

{ F a f 0 h)(s)cf’o hl(s)h’(s) ds

( F g)(s)g’(s) ds 0

Hence the value of (2.2) is independent of the particular function f : I += X which defines the arc, and depends only o n F ( . ) and on the arc. We represent (2.2) by

Consider now two arcs f and g in X such that f ( l ) = g(0); the elementary properties of the Riemann integral (Hille, (1959)) and the definition of concatenation of paths produces immediately:

PROPOSITION 2.4

Now follows one of the most important theorems in analysis:

THEOREM 2.2 (Fundamental theorem of plane topology)

Let X be a

46

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

region in the complex plane and f and g two homotopic arcs in X . For any z H F ( z ) in H ( X ) F ( z )d z

=

F ( z ) dz

Proof For proof see Springer ( 1 9 5 7 , ~157). . A direct consequence of this theorem is Cauchy's theorem:

THEOREM 2.3 (Cauchy's theorem in the plane) Let X be a simple region and z H F ( z ) a complex function analytic in an open set containing X . I f C is a loop in X , then F ( z )d z

=0

LEMMA 2.1 Let X be a simple region, xo an interior point of X and f a path in X - x o . There then exists an arc f* which is homeotopic to f in x - x,.

Proof

Since t -f(t)

is continuous and [0,1] is closed

d p inf lf(t)-x,l 1

d o . 11

is non-zero. Moreover, since [0, 11 is compact, t -f(f) continuous and there exists a 6 > 0 such that

Consider the finite partition of [0, 11, o = to< t l < . . . < t, ti - tiP1< 6, i = 1 , 2 , . . . , n, and define for

is uniformly

=1

satisfying

f*(t) = f k l ) + ( t - ti-i)(ti - ti-i)pl(f(ti)-f(ti-J

(2.4)

t E [ti, tiP1].

then

If*(t)-X,l

2 If(ti-i)-xoI-If(ti)-f(ti-i)I

>d

-4

so t ~ f * ( f ) defines a path in X-x,. Furthermore t H f*(t) is differentiable everywhere except in a finite set, therefore f * is an arc. Consider now the function

H ( t , s)=sf(t)+(l-s)f*(r), for ) H ( t ,S> -f(ti-JJ

5 s If(t) -f(ti-i)l+

t E [ t i , ti-l]

(1 - s) It-

ti-11

(ti - ti-i)-'

x lf(G)-f(G-I)l

< sd + (1 - s)d = d.

(2.5)

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

47

Consequently H(t, s) lies in X - x o , for any (t, S ) E [0,1]'. Moreover H ( . , .) satisfies the remaining conditions of definition 2.2 of homotopy, therefore f and f * are homotopic. We can now establish the following theorem:

THEOREM 2.4 Let X be a simple region and xo, zo interior points of X , then any F E H ( X - zo) defines a homomorphism m :r ( X - zo, xo)+ C' as

m:f-

I

F(z)dx

(2.6)

where f is an arbitrary loop in the class of equivalence -f . Moreover either m = O or m(.) is one-to-one. Proof From Lemma 2.1, any closed path has a loop with origin xo to which it is homotopic every equivalence class f E r ( X - zo, xo) contains a loop. For such a loop If F ( z ) dz is well defined, furthermore from the fundamental theorem of plane topology the value of jfF(z) d z is independent of the particular loop chosen in f . In consequence^ the map (2.6) is well defined and a homomorphism (see Proposition 2.4). Finally suppose that for some f o # O we have m(fo)=O. Since r ( X - zo, xo) can be identified with a free abelian group of one generator (Theorem 2.1) then there exists a non-zero integer no such that f o = nos, where g represents the generator of the group. Then m(fo)=no&(g)=O and hence m(g)=O; consequently m(f)=O for a n arbitrary f E v ( X - z o , xo) since f = ng for some n. Clearly m(.) is identically zero, therefore if m P 0, then i(f) = 0 if and only if f = 0. 0 Example 2.1 Consider the function

For an arbitrary circle C with centre zo, then

.J,

F(Z)d z = 1# O

Consequently, using Theorem 2.4, it follows that

defines an isomorphism between r ( X - zo, xo) and m { r ( X - zo, xJ}. Given an arc f (not necessarily closed) which does not contain zo, it is

48

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

well known (Springer, 1957) that

represents the increase in the argument of (2 - z,) when z describes f and that A(f) is called the index of f with respect t o z,. When f is closed, A m gives the number of anticlockwise encirclements of zo by the loop f. Therefore the contradomain of m ( . ) is a subset of 2. For E > O sufficiently small the loop

f&)

= zo + 5 exp (j2mt)

lies in X - x,. If g is the class of equivalence containing fE then m ( g )= 1. Since m : m ( X - zo, x,) ++ 2 is obviously onto (given n E 2 there always exists an f = ng such that m(f) = n ) and is one-to-one, then m ( - )establishes the isomorphism between r(X-z,, x,) and 2. Moreover since m ( g ) = 1, g must be the generator of r ( X - z , , x,).

2.3 R I E M A " SURFACES This introduction to Riemann surfaces is based upon the concept of a covering space :

Definition 2.3 A topological space E is a covering space of a topological space X when a continuous onto map p :E + X exists such that every x E X has an open neighbourhood which satisfies (a) p - ' ( X ) is a disjoint union of open sets si in E (b) For each i, p defines a homeomorphism between si and X

Isa

The sets si are called the sheets of X , and p is called a covering; X is said to be evenly covered. Throughout this section it will be assumed that the space X is pathwise connected, that is, given two points x,, x1E X there exists a path in X with origin x, and extremity xl. Our use of the concept of covering space is based upon the following two theorems:

THEOREM 2.5 (Unique Lifting Theorem, Greenberg (1967)) Let p :E + X be a covering, f a path in X , and e, a point of p-'{OCf)}. If there exists a path f in E with origin e, satisfying p f = f, such path is unique. 0

The path

f

is called a lifhng of f.

THEOREM 2.6 (Hornotopy Lifting Theorem, Greenberg (1967)) Let p :E + X be a covering, f a path in X,and e, a point of p-'(OCf)}. I f f has

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

49

a lifting f and g is a path in X homotopic to f, then g has a lifting $jwhich is homotopic to f. An immediate consequence of Theorem 2.6 is that p : E + X induces a monomorphism p*: T(E,eo)+ r ( X , p(eo)).To prove this we need only note that p* defines a homomorphism (Proposition 2.3) and that p 0 f is homotopically trivial if and only if f is homotopically trivial (Theorem 2.6).

Remark Given a closed path of f in X , it is important to note that even if a lifting f of f exists, such a lifting is not necessarily closed (as in the case of Riemann surfaces). However, the point e m lies in p-'{O(f)}, as 0 0 = eCf> and f = p f . Moreover this point depends only on the class of equivalence to which f belongs (Theorem 2.6). Suppose now that E and X are such that every path in X has a lifting in E, then a map 6 : T ( X ,xo) + p-'(xo) is well defined as 0

2 :Ifl= eo, where of f.

cf] is the class of

equivalence which contains f and\f is the lifting

Consider now a connected complex region X with non-empty interior and let H ( X ) represent the ring of all complex functions which are analytic in some open set containing X . Let 4[A] represent an irreducible monic polynomial of degree n over H ( X ) such that

+ [ A ] = b , + b , A + . . . +b,-,A"-'+A" with b i E H ( X ) ,i = O , l , . . . , n - 1 . Let &[A] be the polynomial over C' obtained from 4[A] by replacing each coefficient bi E H ( X ) by the complex number b , ( s ) ;finally let &A; -) represent the element of H ( X ) obtained from 4[A] by assigning the complex number A to the indeterminate variable of the polynomial. It is assumed that the following holds: HYPOTHESIS 2.1

The region X is such that for each s E X the.polynomia1

4,[A] has n distinct roots.

Now let so be an interior point of X and A. one of the roots of +so[A], then for appropriate ai E H ( X ) . ~ ~ [ ~ ] = ( Y O ( S ) + ( Y ~ ( S ) ( A - A " ) + . . . +a,_,(A-A,)"-'+(A-Ao)"

with ( Y , , ( S ) ~ = 0 and a l ( s o )# 0. Selecting a sufficiently small po, the disc Is-so(<po is contained in X and every ai(.)has a Taylor's series expansion in this disc. From Theorem 4.5.1 (Hille, 1959), there exists a

50

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

p 5 po and a power series

which is convergent in the disc {s :1s -sol < p } such that 4 s [ 8 ( s ) ] = 0 for every s in this disc and such that 8(so)= a, = Ao. A power series of the form (2.8) is called a function element. The point so is called the centre of the function element 8 and is represented by z ( 8 ) .The largest value of p for which (2.8) converges in the open set 1s -sol < p is called the radius of Convergence of 8 and is represented by R ( 8 ) . The function element 8 ( . ) is well determined if the point so and the root of &,[A] which coincides with 8(s0) are well specified (any one of the remaining roots of &+rh] could have been specified). Therefore each so€X determines n function elements 8, all with centre so, each function element being associated with a particular root. Two function elements 8 and 8' are not distinct when z ( O ) = z ( 8 ' )and when in some disc with centre z(f3), 8(s) = 8'(s). Thus every so€X determines n distinct function elements; moreover these n function elements are the only function elements which have centre so, consequently the set formed by these elements can be represented as z-'(s0). By forcing so to describe the whole of X,C can be defined as the set of all function elements thus defined, that is C = U z-'(s0). The map S"EX

z :C + X is now an onto map and for each s EX,z - ' ( s ) is formed with

exactly n points of 2. Having defined C, the next step is to define a topology in 2 ; to begin with, the concept of immediate continuation needs to be developed. Let Oo be arbitrary in C and let so be the centre of O0. For any s' # so in the disc Is - sol < R ( 8 ) ,

c

c m

m

8&)=

ak(S-SO)k

=

ak[(s-s')+(s'-sO)]k.

k =O

k =O

By expanding [(s - s') + ( s ' - s O ) l k via the binomial expansion the above becomes m

=z="

m

The power series 8,(s) a ; ( ~ - s ' ) ~ has centre s' which is distinct from so. However, in the intersection of the regions of convergence, {s :1s -sol < R ( 8 ) } and {s :1s -s'I < R(8')}, the two power series take the same values (note that R(8') cannot be inferior to R ( 8 0 ) - ~ s o - s ' ~ )

2.

51

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

Definition 2.4

Any function element

OEX

which like

el satisfies

(a) Iz(e)-z(e,)l
Definition 2.5 The base of the neighbourhoods of the point defined as the collection of all sets U of the form

u = { 8 E s(e,)

:

IZ

OoEZ

is

(e)- Z( e,)) < t}

for some O < t < R(Bo).U is called the parametric disc with centre 8,, and radius 5. It is a straightforward exercise to demonstrate that all parametric discs satisfy the axioms used in the definition of the base of neighbourhoods (Section 1.2.1).

THEOREM 2.7

z :Z + X is a covering

Proof It has already been established that z ( * )is onto. Let V be an open disc in X . with centre so and radius O < (< min R ( 8 ) ,such that

v =(s

t k z -%,J

: 1s -sol


Let el, 0 2 , . . . , 0, be the elements of z-’(so) and consider the n parametric discs

ui= { 8 E s(ei) :( Z(e)- sol < 5) Let Ai be defined as ei(s0), that is A,, A * , . . . ,A, form the collection of the n roots of +JA]. By defining

d AminlAi -A,I>O if;

t can be chosen sufficiently small such as to ensure that d max sup lei(s)-Ai(
So for an arbitrary s E V

10,(s) - 6,(s)! > IAi - A, I - (0,(s) - hi I - 10,(s) - Aj 1

therefore for any i # j for any s E V, &(s) # O,(s). Let 8 be arbitrary in Ui and 8’ be arbitrary in Up If z ( e ) # z ( O ’ ) it follows that e# 8’. Suppose that s = z ( 0 )= z ( 0 ’ ) ;since 8 is an immediate

52

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

continuation of 8, and 8' is an immediate continuation of 8,, then 8(s) = 8,(s) and N ( s ) = O,(s). So 8 # 8' and therefore Ui n U, = 4, i # j . Consider now an arbitrary s E V. For each i = 1 , 2 , . . . , n it is always possible to find an immediate continuation of 0, with centre s. Let di be such a function element, then di E Vi and since Vi r l U, = 4 it follows that # 0, for i # j . In conclusion given any s E V there are n distinct elements of z-'(s), each of which are contained in its associated parametric disc U,. Since z-'(s) contains exactly n-distinct elements, it follows that z-'(s) is contained in

n

U

i=l

Vi ; moreover for each i, z ( Ui) = V and so 2-I(

V) =

u Vi n

i=l

Finally it is necessary to show that the map z restricted to Ui is an homeomorphism. It has already been shown that given an arbitrary s E V there exists one and only one function element in Ui whose centre is s, consequently z : Ui + V is an isomorphism. Let 0 be an arbitrary neighbourhood in Ui of some 8 E Ui, such that

0 = {CU E s(e)n ui:IZ(CU)

- z(e)l<

6)

for some S<,$-lz(e)-z(ei)l. Then

z(O)={sE V:Is-z(8)l<S} and z ( 0 ) coincides with a neighbourhood of z(8) in V, proving that z li and its inverse are continuous. In conclusion it was first established that there exists a neighbourhood V of a point so and a collection of n parametric discs Ui such that (a) z - ' ( v ) =

n

U ui

i=l

(b) U i n U i = 4 , i # j (c) z : U, + V is a homeomorphism. Consequently z is a covering.

0

Definition 2.6 The topological space C =

U

zw1(s0) is called the

SOEX

algebroid Riemann surface generated by the region X and by the polynomial

4 [ h ]=

f bihi

i =O

bi E H ( X )

A direct consequence of Theorem 2.7 and the homotopy lifting

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

53

theorem is:

THEOREM 2.8

The map z :C +. X induces a monomorphism Z*

: .rr(Z,e ) +.

~ ( x~ (,6 ) )

COROLLARY 2.1 If X is a simple region with punctures, then T ( C , 6 ) is an abelian group. Proof T ( X , z ( 0 ) ) is a free abelian group and T ( C , 0) can be identified with a subgroup of v ( X , z ( O ) ) ,hence it must be an abelian group. 0 It has been shown that C is a set of power series, as a consequence all properties of the space (including its topology) are defined locally, that is, they are defined in a small region around each point of the space. An important problem that now arises is how to find a global structure for the space? To achieve this the concept of analytic continuation is used. Let 8, be arbitrary in C and let f be a path in X whose origin coincides with z(0,). Due to the continuity of t -f(r) there exists a t l such that f ( t l ) lies in the circle of convergence of OO. Therefore there exists a unique 8,~ z - ' C f ( t ~which )} is an immediate continuation of O0. For sufficiently small t l .

-f(O)l<

If(t1)

m80)

But since R ( B l ) r R(8,)-If(tl)-f(0)(, O0 lies in the disc of convergence of el and must therefore be an immediate continuation of el. We now proceed by induction; consider a sequence of points in [0, 11 O'fo
and a sequence of points in 2, O0,

..< f k < . . . 5 1 e l , .. . , e k , . . . such that

(a) z(@)= f ( t i ) (b) 0, is an immediate continuation of both

eiPland of

Oi+l.

Suppose that the sequence to< tl < . . . < 4 < . . . reaches 1 in a finite number of steps; that is there exists a partition of [0, 11 such that

.. < t m = 1 function elements O,, e l , .. . , 0, O=t,
and a sequence of which satisfy conditions (a) and (b) above. Then it is said that the analytic continuation of 8, along f is possible and that 0, is the analytic continuation of 8, along f. Such an operation on 8, is represented as

em =

f

e,

Note that although 0, was obtained for a particular partition of [0, 13,

54

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

it can be shown (Bers, 1958) that the function element 8, is independent of the particular partition used. Let W(8,) represent the set of all analytic continuations of a function element 8,) along paths in X with origin ~ ( 8 , ) .W(8,) is a subset of Z and is called a complete analytic function in the sense of Weierstrass. Notice that W(8,) is defined in a space of power series which are not necessarily derived from a space of roots of a polynomial over H ( X ) . Furthermore through the device of analytic continuation, function elements which are not immediate continuation of each other can be related (that is W(8,) has a global structure). An important question with respect to the global structure of C is whether W(8,) is a proper subset of C..

THEOREM 2.9 (Bers, 1957) Let 4[A] be an irreducible polynomial of degree n over X such that for every s E X , &[A] has n distinct roots. Then the algebroid Riemann surface C thus generated is a complete analytic function in the sense of Weierstrass. Conversely, given any complete analytic function W i n thesenseof Weierstrasssuchthatforall 8 E W,z-'{z-'{z(8)} has exactly n function elements, then there exists an irreducible polynomial 4[A] over H ( X ) of degree n such that the algebroid Riemann surface generated by X and d[A] coincides with W.

-

Let f be an arbitrary path in X . For an arbitrary T E [ O ,13 let f r ( t ) A f ( t T ) ,then f T defines a path such that O ( f T ) = O ( f ) and e ( f T ) - f ( T ) . Let f ( T )E C be defined as 8. Using the definition of analytic continuation fT

it follows immediately that T + f(T)is continuous. So f defines a path in C, furthermore (2

[ I

f)(T= ) Z u ( T ) }= Z fT- 8 = eCfT}= f ( T )

that is z f =f . Consequently f is a lifting of f , and because z : Z + X is a covering (Theorem 2.7), f is the unique lifting with origin 8 (Theorem 2.5). We have now established the first part of 0

THEOREM 2.10 Let f be an arbitrary path in X and 8 an arbitrary element of z-'{O(f)}. Then there exists a unique path f in C which is a lifting o f f and whose origin is 8. Moreover if g is homotopic to f , the lifting of g which has origin 8 is homotopic to f.

Proof The second part of the theorem follows from the homotopy lifting theorem. 0 COROLLARY 2.2 Let f and g be two homotopic paths in X and let 8 E z-'{O(f)), then 7 8 = 7 8.

2.

55

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

Proof Let f, 6 be the liftings of f and g respectively; since f and g are homotopic, f and g are also homotopic and consequently 7 6 = e @ = e ( g )= 7 6. 0

2.4 MINIMUM CONTOURS AND ALGEBROID R E M A " SURFACES In the remainder, X is assumed to represent a simple region with punctures. Let 4 [ h ] be an irreducible polynomial of degree n over H ( X ) satisfying Hypothesis 2.1, and let Z be the algebroid Riemann surface thus generated. Let f be an arbitrary closed path in X: the lifting f of f, already noted, is not necessarily closed. In general we can only state that the extremity of f has the same centre as the origin of f, that is

e f i E z-"z{O(f)11

-

However, if f is concatenated with itself a certain number of times, the lifting of the resultant path is closed. Formally setting ~

f @ f @ . .. @ f = f " m limes

If [f]E m(X, O(f)) is the homotopy class containing f, then, as m(X, O(f)) is mu],and so a free abelian group (Theorem 2.1), If"]=

PROPOSITION 2.5 Let f be a closed path in X and 8 be arbitrary in z-'{O(jf)}. Then there exists an m In such that 6 = 6, where n is the degree of 4 [ h ] .

f*

Proof Let 6 ' # 6 be another element of z-'{Ocf)}. The following 6 must be true, otherwise 7 {YO'} = 8, and since f" @ f" is homotopically trivial, it is necessary that 6'= 6. If for all rn In, 6 # 0, then the n + 1 function elements

7 6'# 7

6,--+6,-6 f

f2

) . . . )+ 6 f"

would all be distinct and there would be n + 1 distinct function elements in z-'{Ocf)} (which is not possible). 0

-

Similarily the following may be derived:

-

PROPOSITION 2.6 Let f be a path in X , and let 6, O'E z-'{Ocf)} be such that 6' # 6 for any integer k. Then for any integers m, k, 6 # +6. fk

f"

fk

56

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Now consider an arbitrary closed path f in X. Let 8, be arbitrary in zp'{O(f)}, and represent the lifting of f with origin 8, as+@,. Let m , 5 n be the smallest integer which satisfies 01 =

Then the path

701

r, defined by

is closed. If m , = n, define a contour C in C as define 0, as

rl. Suppose that

m l < n and

This is the set of all function elements whose centre coincides with the centre of O,, which can be reached from 8, by analytic continuation along some multiple of the closed path f. Since m , < n, ~ - ' { O c f ) } - ~ ~is non-empty. Let O2 be an element of this - = 02. Another closed set and m2 be the smallest integer satisfying f " ' ~O2 path r2can be defined as

o2 o2

-

On defining = {02,7 02,. . . , - lo2}, Proposition 2.6 shows that the sets 0, -and are disjoint. Continuing in this manner form N closed paths ra --+ 0, until every function element of z-'{Ocf)} is contained in f

f'"Z

-8

some loop. Therefore the union of all sets @li = {Oi, p Oi, . . . ,J Oi} contains z-'{O(f)}. The contour C in C defined by the sequence of closed paths {ri}F=, is called a minimum contour generated in Z by the path f. A simple analysis shows that the minimum contours generated by the same closed path differ only in the choice and order of origins O(ri). Let C={fi}y=l be a contour in X and F, be a minimum contour generated in C by fi. The contour e={F,}Ll is said to be a minimum contour generated by the contour C.

THEOREM 2.11 Let C = H } y = , be a homotopically trivial contour in X . Then every minimum contour generated in 2 by C is homotopically trivial. The proof of the theorem is a consequence of the following lemmas: LEMMA2.2 Let f be a closed path in X and

F={ri}F=,a

minimum

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

contour generated in Z by f. Then if z 0 I; is homotopic with f ".

57

I; is any associated closed path of F,

Proof Given the contour F = { T i } E V =let , , Oi = O ( T i ) . For an arbitrary 8 E z-'{z(8i)} let #+ be the closed path in X which satisfies

ei = and let

-

c$i

be defined as

F

An associated path

__+

4,

8, i = 1,2,. . . , N

38 is constructed as N

I;= @d+ @ri@ & so

i=l

-

Since Ti = f Oi, z Ti = Noting that 4i and & are both closed paths with origins z(8) and that T(X, z ( 8 ) ) is a commutative group (it is an abelian group since by hypothesis X is a simple'region'with punctures), then di @ z 0 Ti @ c$~ is homotopic with z 0 Ti. And so z 0 ,F is homotopic OU)) is a commutative group, with @Elf",. Utilizing the fact that T(X, then z 0 I; is homotopic, with , however, in constructing a minimum contour m , + m2 + . . . + mN = n, and so z _F is homotopic with 0

fmi.

'"0

Cf)m~+m2+...tm~

0

f".

0

Proof of Theorem 2.11 Let E , i = 1,2, . . . , m be an associated loop of a minimum contour F, generated by fi, also let 8, be the origin and suppose that z(O,)=O(f). Let s be arbitrary in X , 8 arbitrary in z-'(s), and let q%i be a path in X satisfying ei = 7 8

-

Finally let CLi be the lifting of q%i ; then an associated contour of C is of the form

c

so that

so

i=l

-

Jli

c=@q%i

@ ;l;i @ lji

m

z

From Lemma 2.2, z

m

=@

0

I;I

0

i=l

@z

0

1;1 @lji

-f y, but

++@fin@& -(Gi@fi@&)n z

0 g - & Ci =L li @ h @ l j i ) n

58

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

And since the fundamental group of X is commutative, then

m

Now, @ +,t i=l

@fi @

is an associated loop of the contour C = u}El

which by hypothesis is homotopically trivial. S o z C is homotopically trivial and therefore C is also homotopically trivial (see Theorem 2.8). 0 0

2.5 ANALYTIC FUNCTIONS AND INTEGRATION IN ALGEBROID RIEMANN SURFACES Riemann surfaces were originally introduced (Springer, 1957) to solve the problem of multivalued complex functions such as s"*. The approach consists in replacing a complex region X with a surface C (whose topological properties are closely related with X ) and the multivalued function h :X + C' by a single-valued function fi : C + C'. The function h can be made single valued if its domain is restricted to a sufficiently small region U, for example sl'* can be made single valued if its domain is restricted to a region which does not contain the origin. The function fi must retain this characteristic and for a sufficiently small region 0 in C, and in some sense the two functions h, h must coincide. To formalize these considerations, consider an algebroid Riemann surface C generated by a region X and a polynomial 4[A] of degree n over H [ X ] . Let G be an open set in C, and let F; G + C' be a complex valued function in G. If U is an arbitrary parametric disc

u = ( 8 E s(8,): Iz(

8)- z(8,)l < 5)

then the map z :Z -+ X restricted to U ( z lu) defines a homeomorphism between U and z ( U ) : { s : l s - z ( 8 , ) l < 5 } . The function F is said to be analytic in G when for any 8,EG and for any parametric disc U with centre O0, the complex function s HF{(z lu)-'(s)} is analytic in the disc z ( U )= {s :1s - z(O,)l< 5). An analytic function of special importance in algebroid Riemann surfaces is the centre function 8 H z(8). Now, any such 8 can be expressed as a power series

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

59

FIG.2.1

so that by defining the so called value function V:C+ C' as

v(e)= a,

and recalling the definition of 2, then a, is the root of the complex polynomial &[A] around which the power series for 8 was constructed. Let Y be the complex region formed with all roots of the complex polynomial c$,[A] when s described X. Graphically the relationships between the spaces X , C and Y are represented in Fig. 2.1. The multivalued function which assigns each s E X to the roots of 4 , [ h ] can be written as V z-'. The multivalued character of the roots function is then absorbed in z-' leaving V(-) as a single valued function which will be shown to be analytic. The practical advantage of this approach is not immediately clear, since the usual problem is to relate a complex region X with the image of X under the roots function, and the difficulty of this problim lies in the multivalued character of such a map. It is not apparent how this difficulty is resolved by introducing the single valued function V(.), since it requires the introduction of an abstract space C and furthermore the ,multivalued character is simply transformed to 2-'. The answer to these questions lies in: 0

(a) The multivalued character is associated with z-' for many multivalued maps in X , that is, a large number of multivalued maps in X can be written as F o z-' where F : C + C' is a single valued function. Had Z and z not been introduced, the complex process of investigating each multivalued function in turn would be necessary. (b) Although C is an abstract surface, its topological properties are very closely related with the topological properties of X (see

60

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Sections 2.3 and 2.4). Therefore, to relate the topological properties of X and Y,it is natural to achieve it through C, since the topological properties of X and C are closely related, and because V : C - Y is single valued, it is likely that there exists a simple relationship between the topological properties of C and Y.

-

Following this digression it is now shown that the value function V : C C ' is an analytic function. Let U be a parametric disc with centre 00, for any S E z ( U ) , ( 2 I u ) - ' ( s ) is the function element with centre x, which is an immediate continuation of O0. Let so= z(8,) and suppose that O0 is the power series m

Since 8 is an immediate continuation of 8(),then

C ~ ; ( w - s =) ~c ~ , { ( w - s ) + ( s - ~ , ) ) ~ k m

m

e(w)=

k =O

=(I

for any w in the intersection of the disc of convergence of both series. In particular for w = s, m

= eO(s) =

1

k =O

a k b

-

(2.9)

Consequently for any s E z ( U ) , s H V{(z lu)-'(s)} is given by the series

c m

k =I)

ak(S

- so)k

and therefore must be analytic. Let F: C .-+ C' be an analytic function in some open set G c C, then for any parametric disc U the complex function s H F[(z l u ) - ' ( s ) ] is differentiable. Let f'(s) be such a derivative, then s-f'(s) is analytic and furthermore

The derivative of an analytic function in C can now be defined as

(2.10)

2.

61

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

This derivative is well defined and is such that

Hence 8 H F'(8) is also an analytic function in G. Identically derivatives of higher order can be defined, for example from (2.9) and (2.11)

V'k'(8,,)= k ! ak Let f be a path in C; f is said to be rectifiable when the path z f is rectifiable. Also the length of f is defined as the length in X of the path z f. Similar to the definitions for paths in the complex plane, an arc in 2 is a rectifiable path and a loop is a closed arc. Given an open set G in C, an analytic function F : G + C' and an arc f in G, the integral If F ( 8 ) d z ( 8 ) is defined as 0

0

(2.12)

THEOREM 2.12 (Springer, 1957) Let F be an analytic function in open set G of 2. I f f and f ' are homotopic in G, then

I,

F ( 8 )d z ( 8 ) =

I],

F ( 8 )d z ( 8 )

COROLLARY 2.3 (Cauchy's theorem) If F : G + C' is an analytic function in an open set G of C and f is a homotopically trivial loop in G, then

Let C = {h}r=,be a contour in an open set G c C formed by loops. Let 8 H F ( 8 ) be analytic in G and define

Also set

C be

an associate loop of C, then for appropriate arcs

C = @ ( 4 i efie4i) N

Hence

i=l

4i,

62

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

But since

then

I,

w)

~ ( 8 )

=

~ ( 0d)z ( 8 )

And so the integral of F along any associated loop of the contour C coincides with the integral along the contour.

2.6

SINGULAR POINTS

Consider a compact simple region R in the complex plane and an irreducible polynomial +[A] over H(R). It has been assumed throughout that for each s E R, +,[A] has n distinct roots. This restriction is now removed by allowing points so, s l , . . . ,sk, . . . in R for which the complex polynomials

..+a,-~(wk)A"-'+A" have multiple roots. The points so, s l , . . . , s k , . . . are called singular +,,[A]=a"(Wk)+a](W,)h+.

points. Suppose that for each s E R the n roots A i ( s ) , i = 1 , 2 , . . . , n (not necessarily distinct) of +s[A] are known. And define the complex number D ( s ) as (2.13) which is clearly symmetrical in the roots Ai(s). It can be shown (Hille, 1962) that s ~ D ( s defines ) an analytic function in some open set containing R.The function D ( s ) is called the discriminant of +[A]. Now, since s H D ( s ) is analytic in an open set containing the compact set R,there are at most a finite number of zeros of s I+ D ( s ) and at most a finite number of singular points of +[A] in R. Let sl, s2,. . . , s, be the collection of all such singular points; the region X = R -{sl, s2, . . . , s,} is now a simple region with m punctures and satisfies Hypothesis 2.1. Hence an algebroid Riemann surface C can be generated by X and by +[A]. The relationships between the function elements 8 E C, whose centres z ( 8 ) are in a neighbourhood of the singular points sk. are now explored. Consider an arbitrary point so in the collection of singular points. Let

2.

63

RIEMANN SURFACES A N D THE GENERALIZED PRINCIPLE

A, be a multiple root of +JA] and let N < n be its multiplicity. For appropriate analytic functions s H( Y k ( s ) , " +s[Al=%(s)+ ak(s)(A-hdk (2.14) k=N

with a , , ( ~ o ) = a I ( ~. ,. ). == ( Y ~ - ~ ( s ~ ) = O( Y, ~ ( s ~ ) # and O a , ( s ) = l . The function s - a o ( s ) coincides with s H&(s) and is shown in the following section to be of particular significance. Let M be the multiplicity of the ) be written zero so of such a function; in a neighbourhood of so, ( ~ ~ ( scan as

(2.15) where O [ x m ] represents an infinitesimum in x of order at least m. In a region D x U of C x C' where

D ={s : O < 1s - S O ( 5 5 ) u = {A :o < Ih -A,[ I6) The function (s, A ) I+ +s[A]

(2.16)

can be written as

where A(A, s) is of the form A(A - Ao, s - SO)

= O[(S -

+ O[(A

- Ao)N+l] + O ( S - so)O[(A - A J N ]

If + , [ A ] = 0, then (A -

= a ( s - s , ) ~ --

where

1

A(A

-

Ao, s - S O )

aN ( S O )

(2.18)

From Theorem 9.4.2 (Hille, 1959) the following proposition holds:

PROPOSITION 2.7 For sufficientlysmall 5 and 6 in 2.16 and for each s E D there exisrs exactly N distinct roots A,(s), A2(s),. . . , A&) of 2.14 in U. Each root tends to A, when s tends to so. Moreover A,(s), h2(s),. . . ,AN(s) are also roots of a polynomial A,)+. + [ h ] = b ( ) ( s ) + b , ( s ) (- A

. . + b N - I ( S ) ( A -&))+(A

(2.19)

64

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

of degree N whose coefficients s I+ bk(s) are analytic functions in D and tend to zero when s -so. For sufficiently small &, every singular point of 4[A] can be excluded from D. Therefore each s E D and for each of the N roots of Ak(s), there exists a unique function element 8 k E z-'(s) for which v(8k) = hk. Let C0 be the subspace of C formed with all such function elements, that is, Co is the set of all OEC such that z ( 8 )D~ and V ( ~ ) U, E or in simple terms, Co is the set of C defined by the roots A(s) of (2.14) which are close to A. when s is close to so. Heuristically (2.18) can be written as (A-Ao)N=a(s-so)M

(2.20)

Hence for a given s E D, the N function elements 8 with centre s which lie in CO, are precisely those for which

v(e)-A = [ a ( ~ - s ~ ) ~ ] ~ - '

each one being specified by one of the N determinations of [a(sM N-' so) 1 * Various questions arise concerning the global structure of C. Is it connected? If not, what are the relationships between its elements? To begin with, consider the second part of Proposition 2.7: suppose initially that the polynomial $[A] is irreducible in D ; then D and $[A] will generate an algebroid Riemann surface C' which is connected and a complete analytic function in the sense of Weierstrass (Theorem 2.9). Let z ' : C ' + D and V':C'+ U be respectively the centre function and the value function in 2'. From Theorem 2.8 it can be shown that for each s E D and 8 E z-'(s) there exists one and only one O'E (z')-'(s) such that V'(8)- V(8). From which it follows PROPOSITION 2.8 If the polynomial $[A] (defined by (2.19)) is irreducible over H ( D ) , then 2, can be identified with the algebroid Riemann surface generated by D and $[A]. If the polynomial $[A] is not irreducible over H ( D ) , it can be factored into a number of irreducible polynomials

$[A1

= $l[AI$2[Al.

. . $,[A1

of degrees N , , N 2 , . . . ,N, respectively. Each polynomial $i[A] will generate with D an algebroid Riemann surface Xi. Repeating the arguments of Proposition 2.8, the following proposition holds: PROPOSITION 2.9 If the polynomial $[A]

can be factored into p irreducible

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

65

polynomials

+[A1

=

+r,r~l+*[~l. . . +,,[A1

then C,, is the disjoint union of p components X I , Cz, . . . , X,,such that each Xican be identified with an algebroid Riemann surface generated by D and $i[AI*

The components X i can now be considered in more detail; it has been shown that for each s E D there are Ni function elements of Co in C i (Ni is the degree of +i). It is now demonstrated that this number of function elements is the same for each component Xi, that is, every polynomial +i[A] has the same degree. Consider a loop u describing the boundary of D u(t)= so + exp ( j 2 7 ~ t ) ,

t E [0, 13

Let s1 = s,,+t be the origin of u. Under these conditions PROPOSITION 2.10 Let Ci be one of the components of CO and let 8, 6’ be arbitrary in (z/Ci)-’(s,).Then there exists an integer k such that 8 = -8‘ v k

Proof Because Zi is a complete analytic function in the sense of Weierstrass (Theorem 2.9), there exists a path f in D such that 8 = t 8‘. Also since z(O)-z(O’), f is a closed path. Because D is a simple region with one puncture, the fundamental group r ( D , s,) can be identified with a free abelian group of one generator. Moreover the homotopy class containing u can be identified with the generator of the group. Hence for some integer k, f is homotopic with uk, and so

PROPOSITION 2.11 Let ci be a minimum contour generated in Ci by u and &[A]. Then c, is a closed path.

Proof Let be arbitrary in (z/Ci)-’(s1) and m be the smallest integer such that Or,= 7Or,. Then m must coincide with Ni otherwise the set

is non-empty and each 8 in such a set is distinct from every y Or,, k = 1 , 2 , . . . (which contradicts Proposition 2.10). Now, since m = Ni, ci must coincide with ,r’8, as every 8 E (z/2,)-’(s1)lies in such a path. 0

66

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Let 0, E Z,, have centre sl, and let m be the smallest integer for which

el = +0, Urn

From the previous proposition, m = Ni (Xi being the component of Zo which contains 0,). And from (2.18) (2.21) where A is any root of (2.14). For sufficiently small 5 and S we can set

Hence when s describes the circle a, M times, the corresponding root A(s) (defined by analytic continuation of 6,) describes a closed path which encircles the point Ao, N times. Moreover this encirclement condition is independent of the point s1 and of the function element el, which defines the origin of the path described by A(s). The question now arises concerning the relationship between the integers M and N (the multiplicity of the root A. of the complex and M (the multiplicity of the zero s =so of the polynomial &[A]) analytic function s H&[s]). The integer m is the least number of times the circle a need be described in order to close the path described by A(s). Considering (2.19) it is observed that m is the smallest integer for which m M is a multiple of N. Note that if s describes a, then s = so+( exp (j27rr) and from (2.18).

where q is some Nth root of unity. In conclusion the least integer m for which

holds is the smallest integer for which mM is a multiple of N. Moreover it has been shown that m is independent of the origin s1 of a and of the function element el E (z/Zo)-'(s1) that defined the origin of the path described by A. So m is independent of the component Zi which contains el. Since m = N i , it follows that the degrees N1,Nz,.. . , N , of the polynomials +,[A], t,hz[A], . . . ,&,[A] all coincide with m, and because

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

67

N 1 + N 2 +. . . + N , = N P=;

N

(2.22)

It is now clear that the components Xi are constructed in a similar manner to that used in the definition of minimum contour. Let s1 be the origin of a,and let O1 be arbitrary in (z/CJ'. For k = 0 , 1 , 2 , . . . , m - 1 the function elements -pO1 are contained in the same component of C, since they are reached through analytic continuations of O1 along path in D; let Cl be such a component. The remaining function elements O E C 1 can also be defined as being the result of analytic continuations of O1 along suitable paths in D. A function element 0 2 ~ C whose , centre is s1 and which does not coincide with any one of p 01, k = 0 , 1 , . . . , m - 1, cannot be reached from O1 by analytic continuation along a path in D. Therefore define a new component of Co, say C2, and the remaining elements of Cz are all analytic continuations of O2 along paths in D. The p components of Co can be constructed in this manner. Each component will contain exactly rn of the N function elements in each (z/Z,)-'(s), ED. For simplicity let zi(.)and Vi(.) represent the respective maps z ( - ) restricted to Ci and V(-) restricted to Xi. It has just been shown that z;'(s) has exactly m distinct function elements, therefore s t+ V , ( z ; ' ( s ) ) does not define a single valued function. It will be interesting to be able to define an analytic complex function which has the same values as the multivalued map s H V,(z;'(s)). To achieve this, consider the disc W = {w :0 < ) w )5 t"-'} in the complex w-plane and consider the map p : W + D defined by p(w) = so+ wm

t, then = s, + wo" = S" + t =

(2.23)

Let wo be the positive real mth root of p ( w,)

s1

Consider now the closed paths Ci, i = 1 , 2 , . . . , p, which according to Proposition 2.11 coincide with the minimum contours generated in Ciby the polynomials I,~~[A]and by m, and let 0, E z;'(sl) be the origin of Ci. If

in the power series 0, =

m

1

k =O

ak(S-sl)k,

s is replaced by s 0 + w m =

s,+[(w - w,)+ wOlm and s1 by so+ w r , then a power series in w is obtained with centre w,. Such a power series can be analytically continued along a path f in W. However, the derived power series in w is simply the power series obtained from 30, on replacing s by so+ [(w - w,)+ wOlmand s1 by so+ w?.

68

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Consider the particular case when f is a closed path in W. Let y be a circular loop describing the boundary of W y(t) = wg exp (j277t)

Since W is also a simple region with a single puncture, the closed path f must be homotopic with y k for some integer k, so

-e, P Q f from the definitions of

=

CT

poy*

and y it is clear that

--- p0y=CTm

and so

POf

as

e, =

-,

un8k

e, = 8,

8.’ = 8.’

ul”

It then follows that the analytic continuation of the w-power series 0, by an arbitrary closed path W coincides with the proper 0,. Consequently, if w is any point of W and f any path in W with origin wo and extremity w, the function element 8 = P ~ f > 8 , is independent of the particular path used. Two important conclusions can now be made: the first concerns the power series 0, when s is replaced by so+ w m . Because its analytic continuation for a certain point w does not depend upon the path used but only on the extremity of the path, the series must define a holomorphic function in W. The value of this function at a point W ’ E W is then given by the sum for w = w ’ of m

8=

s=so+wrn

ak(S-p(W’))k, k =O

where 8 = P O f * e i with O(f)= w o and e(f)= w ’ , that is, the value of the Formally we can then define a function at w’ coincides with Vi(e). homomorphic function in W, w H F,(w) as E(w) =

”,(-+ 0,) Par

(2.24)

where f is any path satisfying O(f) = wg and e(f) = w. The second conclusion (implicit in the first) is that a single valued function hi :W * Ci can be defined as hi(w) =

-

e,

P O f

(2.25)

2.

69

RIEMANN SURFACES A N D THE GENERALIZED PRINCIPLE

where, again, f is any path in W with origin wo and extremity w. Furthermore there is a very close relationship between W and C.,, expressed in:

PROPOSITION 2.12 homeomorphism.

The map hi : W + C defined by

(2.25) is a

Proof Suppose that for w, W ' E W, h , ( w ) = hi(w'); let 8 = hi(w)-hi(w'). Since z ( 0 )= p(w)-p(w') (see ( 2 . 2 5 ) ) ,then so+ wm = S ( ) + ( W ' ) m

This ensures that there exists an integer k such that Ikl< m and such that

(

3

w ' = w ex p j 2 ~ -

Let g be the arc in W

That is, g is a circular arc with origin w and extremity w'. Since the value of hi(w') is independent of the path used, then

(2.26) Let f and g be paths in D defined respectively as p f and p (2.26) it foilows that 0

-) 8,=-B

so 8, =-

r

(

-fe?

0

g. From

f

ei =-

f@@f

e,

(2.27)

We note that g is a closed path with origin p(w) which encircles the point so, k times, g- ( t ) = so+ g(t)" = s o + wm exp ( j 2 ~ k t )

Therefore the path f @ g @ f is also a closed path which encircles the f @ g- @ 7- is the point origin the same k iim&; hbwever, the origin _ s = S,) + 5. Alternatively from (2.27), f @ g @ f must be homotopic with a closed path of the form for s o m e h e g e r q, since they are the only paths which satisfy

,

70

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

That is, k = qm for some integer q. However, (kl< m, so q = 0 and k = 0 and therefore w = w'. We have shown that w I+ h ( w ) is one to one; it is also an onto map, since given any s E D, m different values of w can be found which satisfy p ( w ) = s. Defining 8 = h i ( w )we obtain, as above, m distinct function elements whose centre is s. These m function elements constitute the totality of all function elements in z ; ' ( s ) , since there are only m function elements in z ; ' ( s ) . Therefore every 8 g z ; ' ( s ) is of the form 8 = h ( w ) for some w ~ p - ' ( s ) .So in conclusion hi : W + Z i is an isomorphism. Consider now an arbitrary parametric disc U with centre in 8 = lq(w). The map zIu = zilu defines a homeomorphism between U and a disc U in D with centre z ( 8 ) (see Theorem 2.7). For a sufficiently small disc 0 around w, p defines a homeomorphism between 0 and p ( 0 ) c U,. So hi = ( ~ 1 ~ ) o ~p ' defines a local homeomorphism between 0 and h(0). Consequently both h i ( - )and h ; ' ( . ) are continuous, completing the proof that hi is an homeomorphism. 0

lo

COROLLARY 2.4 A n y two components Xi and Ci of C,, are homeomorphic to each other. Proof

Both Ci and Ci are homeomorphic to W.

0

It is now possible to establish the main result of this section:

THEOREM 2.13: LOCAL PRINCIPLE OF THE ARGUMENT FOR SINGULAR POINTS In the context of the definitions of this section, let 6 = { Ci}r= be the minimum contour generated in Zcoby cr. Then for sujlciently small 5

Proof

The integral

can be evaluated as

f

where

i=l

Ii

2. In

71

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

4 the integral is calculated along a closed path in Zi. Furthermore

Utilizing the homeomorphism hi : W -+Ci, given that the homeomorphic function w I+ F , ( w ) (see (2.24)) is defined as F, = V, hi and that Ci = hi y (where y ( t ) = wo exp 0 ' 2 ~ ~ defines )) the boundary of W, we obtain 0

0

(2.28) The function w w F , ( w ) is defined and analytic in the region W = { w :O < I wI 5 t".'}.Is it possible to extend the domain of such a function to the whole disc W = { W : ~ W \ I ~ " - ' } ? From Proposition 2.7 every root A(s) of &,[A]=O which defines an element 8 E C0 tends to A. when s tends to so. Therefore when w tends to the origin, s = p ( w ) tends to so and therefore F , ( w ) , as a root of &,[A], tends to A". In conclusion lim F , ( w ) = A, for all i = 1 , 2 , . . . ,p. Using now W-0

the Riemann extension theorem (Kendig, 1977) w H & ( w ) can be extended to a holomorphic function defined in the whole disc W = ( W : I W ~ ~ & if" ~C(0) ' } is defined in A". The classical Principle of the Argument (Hille, 1959) can now be utilized in evaluating the integral (2.28)). Since w H F , ( w ) - A0 is a holomorphic function in W,the integral (2.28) coincides with the number of zeros in W of the function w H F , ( w ) - A", each zero counting as many times as its multiplicity. For sufficiently small l, the origin is the unique zero of w - 8 ( w ) - A , in W,we have only to determine the multiplicity of the zero. It has been shown that m is the smallest integer for which m M is a multiple of N. Then let q be an integer satisfying m M = qN. From (2.21)

I N-'

1

But since lim w - " ~A ( C ( w ) - h , , w-0

sm)=O,

then lim I w - ~ ( F , ( w ) - A ~ ) ~ = w 4 0

la1 f 0 . The multiplicity of the zero w = O of the function w - F , ( w ) - A , is therefore q. So

=q,

for

i = 1 , 2 ,..., p.

72

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

In conclusion 1

= P9.

However, from (2.22) p = N / m and q e m M / N so pq == M.

0

2.7 THE GENERALIZED PRINCIPLE OF THE ARGUMENT In this final section we establish the main result of this chapter-the Generalized Principle of the Argument-which is of crucial importance in establishing the stability of linear and nonlinear multivariable feedback systems. We establish first a simple and local theorem for the Principle of the Argument. Let X be a simple region in the complex plane with punctures the set of all and 4[A] an irreducible polynomial of degree n over H(X), complex functions which are analytic in some open set containing X. Assume that X is free of all singular points of +[A] and let Z be the algebroid Riemann surface generated by Z and +[A]. Let so be a point of X and A,, one of the n distinct roots of the complex polynomials &[A]. Finally let u be a loop in X defined by u(t)= so+

6 exp (j27rt)

and let 6 be a minimum contour generated in conditions we have;

(2.29)

Z by

u. Under these

THEOREM 2.14: LOCALPRINCIPLE OF THE ARGUMENTFor sufficiently small

6

-

where M is the multiplicity of the zero s = s O of the complex function s

4*,,b).

Proof

By selecting

such that 0 < 5 1 min R ( 8 ) ,the circular loop u is B E z -'(s,,)

contained in the region of convergence of every function element 8 E z-'(s0). Let Oi, i = 1 , . . . , n be the elements of Y1(s0) and Ui, i= 1 , 2 , . . . , n, the parametric disc with centre 8, and radius 6. The minimum contour 6 generated by (T is then formed with n distinct loops yl, y z , . . . , y,, and these loops are such that yi lies in the parametric disc

ui. so

1

dV(8)

1

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

73

It was shown in Section 2.5 that the function s H F , ( s ) V{(zlu,)-'(s)} is well defined and holomorphic in the disc {s : 1s - sol 5 (} and coincides with the sum of the function elements Oi; furthermore zlq defines an So homeomorphism between {s : 1s - s o [ 5 (} and Ui.

(2.30) Clearly since F,(s,,) = V(0,) there exists a unique function s H F,(s) for F , ( s ) is such a function, which f i ( s , ) = Ao. For simplicity assume that s and hence the function s F , ( s )- A. has a zero at s = so. Moreover ., so is the unique zero of such a function in the for sufficiently small $ interior of u and every other function s ~ f i ( s ) - A ( )i, = 1,. . . , n has no zeros in the interior of u. Using now the classical Principle of the Argument (Hille, 1959) t o evaluate (2.30).

and that

where M# is the multiplicity of the zero s = so of the function s HF,(s)= A(). Now, &[A] can be written as

-

~ s [ A ] = a , ) ( ~ ) + a l ( s ) ( A - A o ).+.+(A-A,)"' .

with ao(so)= 0 and a I ( s o#) 0. The function s with s &(s). Then

(2.31)

aO(s)coincides exactly

~ , [ F , ( S ) ]0=can be written as Setting a = -a~~'(s,,)[a,(s,)M!]-', F , ( s ) = Ao = a ( s -

+ A(F,(s)- A,,

s - so)

where A(*, .) is of the form

orb -

+ O ( S - S,)O(FI(S)

- 4,)+ "Fib) -

Rearranging (2.32) and taking the limit as s

so gives

l i m ( s - ~ ~ ) - ~ ( F , ( s ) - A ~a#O )=

" S ' S

M I

(2.32)

74

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

So the multiplicity M# of the zero s = s, of the function s ++ F , ( s ) - A, coincides with M. 0 Only those polynomials $[A] whose coefficients are holomorphic functions in a prescribed simple region have been considered. In dynamical stability problems, polynomials with meromorphic coefficients are frequently encountered. A function s a(s) is meromorphic in a region fl if it is holomorphic everywhere in rR except at a finite set of points {sl, s2,. . . , sq} (called poles); for each pole si there exists an integer mi (called the multiplicity of the pole) and disc Di with centre si such that s ~ ( - si)"'Ca(s) s is holomorphic and nonzero everywhere in Di. Therefore given a simple region fl and an irreducible polynomial $[A]

+ a,(s)A"+ . . .+ A"

= ao(s)

with meromorphic coefficients; define a simple region with punctures, R'= fl-{s,, s2, . . . , sq} such that &[A] is an irreducible polynomial over H[R'] for {sl - s 2 , . . . ,sq} the collection of all poles of all coefficients s ++ ai(s).Consequently R' and $[A] can generate an algebroid Reimann surface C. Consider now an arbitrary pole sp and an arbitrary complex number A,, and let m, be the multiplicity of the pole sp of the function s ~ & " ( s ) . Let D be a disc with centre sp and radius 5 > 0 which is sufficiently small as to exclude from D every other pole or zero of s H $,(s) and every singular point of $[A]. Finally let a be a loop in R' defined by

a(t ) = sp + 5 exp (j277t) and let 6 be a minimum contour generated in C by a. Under these conditions the following local Principle for the Argument holds for poles:

THEOREM 2.15 1 V(B)-A,

Proof

Write $[A]

- -m,.

in D as

$[A]=a,(s)+a,(s)(A-A,)+.

. .+a,(s)(A-A,)'+. . . + ( A - A , ) "

with a , ( ~ ) = $ ~ ~ (Let s ) . mi be the multiplicity of the pole sp of the function s - a i ( s ) , and let m =max mi. In the punctured disc Do= I D-{sp}we can write n

$(A, S) = (S -

s ~ ) - ~~,(s)(s " C - S ~ ) ~ ( " - ~ ) [ ( -A A,)(s i=O

- s,)"]'

2.

RIEMANN SURFACES A N D THE GENERALIZED PRINCIPLE

75

Defining a:(s)= a,(s)(s w = (A

-

(2.33)

A,)(s - s,)"

+(w, S ) = a ; ( s ) + a : ( s ) w + . . .+sn,

we can see that every coefficient of the polynomial $(w, s) is an analytic function in D and therefore defines with D a Riemann surface (say 2,). Since 4 ( A , s) = (s - s ~ ) - ~ " $w,( s) and w = (A - A,)(s - s ~ ) every ~ , function element q E 2 , with centre so€ Do can be written as - A,,)c(~- so)+ (so - S J I ~

q(s)=

(2.34)

where 0 is a function element of 2. Then and

so (2.35) Note that &(s) = al(s) = a,(s)(s - s,)"" = a,(s)(s - s,)'"~l(s - s,)"'~-"~~. And because a,(s)(s - s,)"~~ is a holomorphic non-zero function in D then s H +(,(s) has in s, a zero of multiplicity mn - mo. Let (+ be a minimum contour generated in C, by (+.Then from (2.35)

Gl 1

we)

- 1 W r l ) dz(s) --m- 1 V(0)-Ao-GIw V(q) 2 - 1 ~ 1 z1( q ) - - s ,

The last term above can be written as 1

ds

--

and using Lemma 2.2 it follows that z ( a ) is homotopic to a" and therefore 1

=n

Invoking Theorems 2.13 or 2.14 (depending whether s, is a singular point of $ [ w ] ) , it follows that

mn - m,

76

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Consequently

The generalized Principle of the Argument can now be established as a consequence of Cauchy's theorem (Corollary 2.3) and the three local Principles of the Argument (Theorems 2.13, 2.14, 2.15). Let R be a bounded simple region of the complex plane and +[A] an irreducible polynomial of degree n whose coefficients are meromorphic in R. In general R contains both poles and singular points; as a result of Proposition 2.8 there are at most a finite collection {s,, s 2 , .. . , sq} of singular points in 0.Identically we can define the finite collection { z , , z 2 , .. . , z,} of all poles of +[A] in R. Let dR be the boundary loop of R encircling R in the anticlockwise direction and assume that such a boundary is free of all poles or singular points. The region

X = R - { s , , . . . , s q } - { z l )...,z p }

(2.36)

is now a simple region with q + p punctures. X and +[A] both satisfy Hypothesis 2.1 and therefore an algebroid Riemann surface C can be generated by X and +[A]. Finally, consider an arbitrary complex number A" distinct from any complex root of the polynomials +,[A] when s describes dR. Under the above conditions we have: THEOREM 2.16: GENERALIZED PRINCIPLE OF THE ARGUMENTLet d f l be a minimum contour generated in C by aR, then

where N, and N, represent respectively the number of zeros and poles in R of the complex function s H &,(s), each zero or pole counting as many times as its multiplicity. Proof Let w , , w 2 , .. . , wk be the zeros of S H C $ ~ ~ ( S )in X and let m i , m 2 , .. . , mk be their respective multiplicities. Let n , , n 2 , .. . , np be the multiplicity of the poles z,, z 2 , .. . , zp of s H +Ao(s)(if zi is a pole of +[A] but not of s-&,(s) we simply choose ni = O ) . Let -yi, i = 1 , 2 , . . . , r, be a small circular loop in R with centre wi described in the clockwise direction and with radius sufficiently small to exclude from its interior every zero (other than w i ) and every other singular point and every pole. Let a,,i = 1,2, . . . , q and Si, i = 1,2, . . . , r

2.

77

RIEMANN SURFACES A N D THE GENERALIZED PRINCIPLE

be similar loops around respectively each of the singular points and poles Of

Finally consider the Riemann surface

u ZP1(Wi)

C’=C-

i=l

where C’ is the algebroid Riemann surface generated by +[A] X ’ =x-{w,,. . . , w,}. The contour

aR’={aR, y , , . . . , 7” cr,, . . . , crq, 61,. . . , Sp}

and

(2.37)

is clearly homotopically trivial in X ’ (note that dR is described in the anticlockwise direction and every other loop is described in the clockwise direction). On utilizing Theorem 2.11, the minimum contour ah’ generated in C’ by an’ is also homotopically trivial. Moreover the function V ’ ( 0 ) / V ( 0 ) - A 0is analytic in 2’ as all singularities of such a function correspond to zeros or poles of s H &,(s), which by construction, are excluded from X’.Using Cauchy’s theorem (Corollary 2.3)

I‘

dV(W =O V(0)-Ao

2.rrj

(2.38)

The minimum contour dfl’ must have the form

. . ,Gq,&, . . . , &} where f i is a minimum contour generated in C by yi, Gi is generated by

ah’= {ah,fl,. . . ,y” .

and

ii is generated by

.

A

crl,.

Si. From (2.38) we have

cri

1

(2.39) Noting that yi is orientated in the clockwise direction, Theorem 2.14 yields (2.40) Consider

if the singular point si is not a zero of s H 4,(s), then the integral is zero

78

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

by Theorem 2.13. Assume that si is a zero of s H $*,,(s) with multiplicity Mi.The minimum contour bi can be divided into two parts bi ={&, &:} where 6;is generated by CT in the set of all function elements 8’ for which V(8’) is close to A, and when z ( 0 ’ ) is close to si, and similarly 6: is generated in the set of all function elements 8” for which z ( V ) is close to si and V(8”) is close to a root of $,,[A] distinct from A,. And so from Theorem 2.13

Combining the above with (2.40) gives

k

where the sum C Mk is extended to all singular points of s H $ h , , ( ~ ) . Consequently the above becomes

sk

which are zeros

Finally, consider

On noting that Si is described in the clockwise direction, using Theorem 2.14 gives

so (2.42)

Combining (2.39), (2.41), and (2.42) we obtain (2.43)

0 COROLLARY 2.5 Under the conditions of Theorem 2.16, N, - N, coincides with the number of anticlockwise encirclements of A, in the A-plane by the contour ail). Proof Since 8 I+ V(8) is analytic in C, every closed path in 2 is mapped

2.

RIEMANN SURFACES AND THE GENERALIZED PRINCIPLE

79

conformally by V(-) into a closed path in the complex A-plane. Setting A = V(8) then

I'

dV(8)

2 4 afi

We)-&

=-I

1 dA 2.rrj vcafi) A - A o

The latter integral is simply the index function with respect to A, of the contour V(a6), that is, the number of anticlockwise encirclements of A. by the contour V(d6). The first integral coincides with Nz-Np as demonstrated in Theorem 2.15. 0

Remark Although the definition of V(a6) requires the mechanism of minimum contours, its practical application is straightforward: it is only necessary to compute the root paths pi,i = 1 , 2 , . . . , n, by numerically evaluating the roots of $,[A] when s describes aR and utilizing the continuity of the roots to determine which numerical value corresponds to a particular path. These n paths are generally open, but they can be joined together by concatenation to form the contour V(d6). Frequently the polynomial $[A] is of the form

4[A] = det (AZ- A ) where A is a ( n x n) matrix whose coefficients are meromorphic complex functions in a given region R. Under these conditions the roots of &[A] are exactly the eigenvalues of A(s). The paths p i ( i = 1 , 2 , . . . , n ) are called eigenvalue paths and are numerically determined by computing for R eigenvalues of A(s). The contour V(a6) formed by the each S E ~the concatenation of the eigenvalue paths is represented by r = {pi}:=,and is called the eigenvalue contour generated by A and dR. Notice that Theorem 2.16 and Corollary 2.5 were established under the assumption that the polynomial involved was irreducible over H(R). The polynomial +[A] = det (AZ- A )

does not necessarily satisfy this property. However, it is always possible to factorize 4[A] into a number of polynomials irreducible over H[R]

4[A] = 4("[A]4'2)[A]. . . 4('")[A] Let z and let surface

where

k

be the algebroid Riemann surface generated by 4'k'[A]and R + C' be the respective value function. To each Riemann Theorem 2.16 can be applied from which it follows that

v k

ck

z k

:c,

and Pk represent respectively the number of zeros and poles in

80

-

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

R of the complex function s c$it)(s). Thus the number of anticlockwise encirclements of the point A, by the contour v k ( a R ) coincides with

z k - p k . The eigenvalue contour r coincides with the contour {vi(aR))r=:=,, since a zero of c$,[A] is necessarily zero in some c$jk’[A]. Therefore the index function of r with respect to the point A, coincides with the sum of the index functions of the contours v k ( d n ) with respect to the same point A(). Hence

The sums Z =

rn

1

k=l

z k

and P =

number of zeros and poles of s

-

m

C

p k

k=l

&(s)

coincide, respectively, with the

in R. We have now proved:

PROPOSITION 2.13 Given a simple region R, a matrix A whose elements are meromorphic in R and a complex number A, distinct from any eigenualue of A(s) when s describes aR, then if aR is free of singular points, the number of zeros of the complex function s det (A,,-A(s)) in R coincide with the number of poles in R of the same function added to the number of anticlockwise encirclements of A,, by the eigenvalue contour r of A(s) generated when s describes an.

-

-

PROPOSITION 2.14 Let A E H(R)”x“.The simple region R is free from zeros of the complex function s det (A(,- A(s)) if and only if the eigenn does not contain or encircle value contour r generated when s describes a the point A().

REFERENCES Bers, L. (1958). “Riemann Surfaces”. Courant Institute of Mathematical Science, New York. Decarlo, R. and Saeks, R. (1977). “The encirclement condition-an approach using algebraic topology.” Znt. J. Control 26, 179-287. Dunford, N. and Schwartz, J. T. (1957). “Linear Operators” Part I. Interscience Publishers, New York. Evgnafov, M. A. (1978). “Analytic Functions”. Dover, New York. Greenberg, M. H. (1967). “Lectures on Algebraic Topology”. Benjamin, New York. Hewitt, E. and Ross, K. A. (1963). “Abstract Harmonic Analysis”, Vol I, Band 115. Springer Verlag, Berlin. Hewitt, E. and Stromberg, K. (1965). “Real and Abstract Analysis”. Springer Verlag, Berlin.

2.

RIEMANN SURFACES A N D THE GENERALIZED PRINCIPLE

81

Hille, E. (1959). “Analytic Function Theory”, Vol I, Ginn & Co, New York. Hille, E. (1962) “Analytic Function Theory”, Vol 11, Ginn & Co., New York. Hille, E. and Phillips, R. S. (1957). “Functional Analysis and Semigroups”, Amer. Math. SOC.Colloquium Pub. Vol. 31. Hilton, P. J. and Wylie, S. (1964). “Homology Theory”. Cambridge University Press, Cambridge. Kendig, K. (1977). “Elementary Algebraic Geometry”. Springer Verlag, Berlin. Kothe, F. (1969). “Topological Vector Spaces”, Band 159. Springer Verlag, Berlin. Larsen, R. (1971). “An introduction to the theory of multipliers”, Band 175. Springer Verlag, Berlin. Massey, W. S. (1967). “Algebraic Topology: An Introduction”. Harcourt, Brace, World New York. Postlethwaite, I. and MacFarlane, A. G. J. M. (1979). “A Complex Variable Approach to the Analysis of Linear Multivariable Feedback Systems”, Lecture notes in Control and Information Sciences, 12, Springer Verlag, Berlin. Valenca, J. M. E. and Harris, C. J. (1981). “The Generalised Principle of the Argument and the Stability of Multivariable Systems”, 3rd IMA Int. Conf. Control Theory, Sheffield, 1980. Academic Press, London and New York.

Chapter Three

Representation of Multipliers 3.1 INTRODUCTION In this chapter we introduce a series of results which establishes a correspondence between the concept of a multiplier (a linear, continuous, or sequentially continuous, time-invariant operator) and the concept of transfer functions. For two important spaces, L, and X, (and their multivariable extensions, L," and X;) we derive a space of transfer functions which are in a one-to-one correspondence with the space of multipliers; this is essentially a representation theory for linear timeinvariant operators and forms the basis of any study on the stability of feedback systems described by linear time-invariant operators. For the important spaces L , , L,, XIand X,, it is not possible to develop a representation theory of the above form; we shall be able to show that every multiplier in these spaces has a transfer function which is an element of a certain space K of analytic complex functions, but we shall not prove that every element h E K is assigned to a multiplier in one of the above spaces which has h as a transfer function. Specifically for the spaces L , and L, we introduce another type of multiplier representation: the representation by convolution with a measof all countably additive, bounded regular ure in the space A(C(R+) complex Bore1 measures. We establish that there is a one-to-one correspondence between &(I?+) and the space M ( L , ) of all linear timeinvariant operators of L , into itself, and that any element of A(R+) defines a multiplier in L,. Finally it is shown that the representation of multipliers in La (and Xa) remains unsolved, since it is not possible to establish a one-to-one correspondence between M(L,) and a space of convolution measures or a space of transfer functions. However, there exist several weaker results (Larsen, 1971), but none is of use in stability studies of linear feedback systems. 82

3.

REPRESENTATION OF MULTIPLIERS

83

The fundamental problem of the representation of multipliers can be stated thus: given a space of time functions E we wish to establish an explicit representation for the algebra M ( E ) of all linear, time invariant and continuous operators of E into itself. Essentially we seek an algebra Y of complex functions such that each A E M ( E ) is assigned to some hE Y (the so-called transfer function of A ) such that

% A f ) ( s ) = h(s)(=ms) for all f E E and all s in some region of the complex plane, where Y represents the Laplace transform operator. 3.2 REPRESENTATION OF MULTIPLIERS IN L, AND L; Let M(L,) represent the algebra of all multipliers in L2 (that is, linear, continuous and time-invariant operators). Clearly M(L,) is a Banach algebra with topology induced by the topology of L,. Consider now the space K ( 0 ) of all complex valued functions SH h ( s ) of the complex variable s, bounded and holomorphic in the open right half plane { s :R e ( s )>O}. K ( 0 ) is normed algebra under pointwise multiplication of functions and under the norm

llhll=

SUP

Re ( s ) > O

Ih(s)l

There exists an isomorphism THEOREM 3.1 : L2-REPRESENTATION THEOREM of rings such that to each A E M(L,) there is assigned a transfer function h E K ( 0 ) satisfying: (i) 2 ( A f ) ( s ) = h ( s ) 2 ( f ) ( s ) for all f

Proof

E

L, and all R e ( s )> O

(ii) IlAll = llhll Let S - , :L2+ L,, T r 0, represent the operator

f ( t + T ) for t r O for t < O Similarly let S, represent the adjoint operator of SF Since L, is reflexive, S*,= (ST,) = SpF A linear continuous operator A : L2 + L2 is said to be time invariant when it commutes with S,, for all T 2 0 . Given any f E L,, the Laplace transform of f is defined as the function s w F ( s ) with domain in R e ( s ) > O such that ~ ( s= )f

( ~ , R e ( s )>0

84

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

where

and

f ( * )is the

element of LT associated with f~ L,. Then

WAncs) = (Af)(A)= ( A * 4 J f ) where A*: L:-. L: is the adjoint of A. But since L, is a Hilbert space, then L ; = L, and hence A* is a linear continuous operator on L, into itself and (Yosida, p. 195, 1971) IlA*Il = IlAll

(3.1)

Obviously A is time invariant if and only if A* commutes with S - , for all TrO. Let +ks E L, be defined as A*4,. Using the time invariance of A

(sTAf)(4s,) = (ASTf)(4s) = $s(%f> (sTAf)(4s) = (Af)(s-4 s ) For all T r 0, S - d S = 4s(T)4s.Hence (sTAf)(4s)= 4 s ( T ,4 s (f) Then for all f~ L,, all Re(s)>O and all TrO (3.2)

where @ represents convolution. Such a sequence is called a weak convolution unit and always exists (Hewitt and Stromberg, p. 400, 1965). By definition of (6,) l i p 4s(6fl@f) = 4Af) Alternatively by the definition of convolution where

4s(~,@f, = f(&)

(3.3)

3.

REPRESENTATION OF MULTIPLIERS

85

Using (3.2)' &, = $s(6n)+s,so

B,O= f ( & n

(3.4) Select an f o c L , such that S H ~ ~ ) ( + ~ ) is bounded away from zero in Re(s)> 0. Then 4 s (6,

= 4 s (6, ) f ( + s )

= 4s(Sn ~fo)[f"(+s)I-'

4s(Sn)

Taking the limit in n and using (3.3) lim

4s

(an

) = fn(+s

)I-'

)IYo(+s

= ( 2Afo)(s~r(~fo~(s~l-'

(3.5)

So that lim &(a,,) is well defined for all Re(s)>O. Let n

h ( s )Plim n $s(~n)

(3.6)

Since Afo and fo are elements of L, their Laplace transform is an holomorphic function in R e ( s )> 0 (Zamamian, 1968). Hence, from ( 3 3 , it follows that h ( s ) is holomorphic in Re(s)>O. From (3.4)

f(W = h ( s ) h )

v f E L,

(3.7)

Hence, in the sense of L,, ~J!,I = h(s)&

h(s)+s

= A*+,

I W I l l + s l l ~ l l ~ * l l ll+J *

*

Since IIA*ll= IlAll then

Therefore s ~ h ( s is) also bounded in Re(s)>O and is therefore an element of K(0). Moreover llhlls llAll. Utilizing Parseval's inequality

=supL[

m

a>n2r

5

Therefore

-=

Ih(a+jo)12( Y f ( a + j w ) l ' d o

I

1 " sup Ih(s)l'sup I2f(a + j w ) I 2 dw a>n2r

Re(s)>O

IIAfllt,

-"

llhllk") IlfIl'L,

86

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

and hence IlAll IllhllK(O,.And so from (3.8) Consider now the map Y: M(L,) +- K ( 0 ) which assigns A E M(L,) to its Laplace transform h E K(0). From (3.4) (ZAf)(s)= h ( s ) ( Z f ) ( s )

(3.10)

for all f~ L2 and all R e ( s ) > O . Utilizing the linearity of the Laplace transform and equation (3.10), it follows that Y is a morphism of rings. From (3.9) this morphism is one to one; it is now only necessary to show that it is also onto to complete the proof. Let h be arbitrary in K(O), then a linear continuous operator on the Hardy-Lebesgue class H'(0) into itself is well defined as

( A F ) ( s )= h ( s ) F ( s )

V F E H2(0), V R e ( s )> 0

The Laplace transform operator defines an isomorphism between L, and w(0)(Yosida, p. 163, 1971), hence a linear, continuous operator A : L, +- L2 is well defined as A = Z-'AZ

Clearly A is time invariant and its transfer function coincides with h ( s ) .0 The isomorphism established between M(L,) and K ( 0 ) transfers certain properties from one space to the other. For example K ( 0 ) is a commutative algebra, hence M(L,) must also be a commutative algebra. Also since M(L,) is a Banach space, K ( 0 ) is a Banach space. The representation of multipliers in L, can be readily extended to LZ. The algebra M(L,")can be identified with the algebra of all n x n matrices over M ( L , ) ; that is, M(L,")-M(L,)""".Since M(L,) is isomorphic with K ( 0 ) (Theorem 3.1), M(L,)""" is isomorphic with K(0)""". THEOREM 3.2: LZ-REPRESENTATION THEOREM The ring of all multipliers in L," is isomorphic with the ring of all n x n matrices over K(O), in such a way that to each A E M(L,") there is assigned d E K(O)""", called the matrix transfer function of A such that (6pAf)(s)= A(s)(L?f)(s)

Vf

E L;,

V R e ( s )> O

The isomorphism between K(0)""" and M(L;) ensures that an operator A E M ( L Z )is invertible if and only if its matrix transfer function AIZ K(0)"x"is invertible in K(0)""".Since both M(L,) and K ( 0 ) are commutative rings, the result of MacLane and Birkoff (p. 304, 1967), which states that a matrix M in the space R""" of all matrices over the commutative ring R is invertible in R""" if and only if its determinant is

3.

REPRESENTATION OF MULTIPLIERS

87

invertible in R, can be applied to operators A E M ( L ; ) . Hence A E M(L,")is invertible in M ( L ; ) if and only if det A is invertible in K(0). Now, since addition and multiplication in K ( 0 ) are performed pointwise then (det

a)(s)= det [A(s)]

An arbitrary s I+ h ( s ) is invertible in K(O), that is, the function s - h ( s ) - ' exists in K ( 0 ) if and only if s ~ h ( s has ) no zeros and is bounded away from zero in { s :R e ( s )> 0). Hence A E K(0)"x"is invertible in K(0)""" if and only if

In conclusion we have:

3.3 A n element A E M ( L ; ) is invertible in M(L,")i f and only if THEOREM

3.3 THE CONVOLUTION ALGEBRA &(R+) Let R+ represent the half real line [O,w) equipped with the topology induced by the usual topology of R'. Let R+=[O,w] represent the one-point compactification of R + . The a-algebra of all Borel sets in R+ will be represented by B. Finally let &(I?+) be the space of all countably, bounded, regular complex Borel measures (Hewitt and Ross, pp. 118127, 1963). The space A ( R + ) is a normed space under the norm ~ ~ p ~ ~ ~ where [ p ~ lpl ( Rrepresents + ) the total variation of the measure p. One of the most important properties of &(R+) is its identification with the strong dual of a space of continuous functions. The space C,(R+) is defined as the space of all continuous, bounded complex valued functions t t + f ( t ) such that for every ( > O there exists a compact set B (which may depend upon f and () such that SUP V ( t ) l < & ld R

The space Co(R+)is equipped with the uniform topology. Under the above conditions we have THEOREM3.4: RIESZ'S REPRESENTATION THEOREMThe strong dual CE(R+)of the space C,(R+) can be identified with &(I?+) in such a way

88 that if

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

fi E C:(R+)is assigned

to p E A ( R + ) then: ,

(a)

fi(f) = l f d p

(b)

Ilfill= SUP Ifi(f)l=l CLll= IPl(R+)

tl f E C"(R+)

Ilfll='

Proof Hewitt and Ross (011, 1963).

0

Given two measures p, A E A ( R + )the convolution of these measures can be defined severally as: (i) p @ A is given by the element G E C:(R+)defined by

G(f) = fi@)

where f(t)

for t c O {ok(s-,f)for t r o

and fi. k E C:(R+)are the functionals defined by and A, f is arbitrary in CO(R+)and KT:CO(R+) + C,(R+) is the operator defined as

(ii) p @ A is given by (p@A)(E= ) ( p x A)(&')

where

E R+ X

R+ is defined as

fi = { ( t , a ): t +

E E}

and p x A is the product of the two measures p and A. (iii) ( p @ A ) ( E )= ImA(E,)d p ( t )= Imp(E,) dA(t) where

0

0

E, =A { a r O : a + t E E ) It can easily be shown (Hewitt and Ross (019, 1963)) that the above definitions of convolution are equivalent. With multiplication defined by convolution it follows that A ( R + )is a commutative Banach algebra-note that C:(R+)is a Banach space since C,(R+)is a Banach space. Some subspaces of A ( R + ) are particularly important; let A a ( R + ) represent the subspace of A ( R + )formed with all absolutely continuous measures; that is, every p E&(R+) vanishes on null sets. (Note: in the sequel a null set denotes any set with a Lebesgue measure of zero.) Also

3.

REPRESENTATION OF MULTIPLIERS

89

let Ad(R+)be the subspace formed with all measures of &(R+) which are represent the space of concentrated on countable sets. Finally let As(R+) A ( R + ) formed with all measures of A ( R + ) concentrated on noncountable null sets. The elements of &(R+) are called discontinuous are called continuous, singular measmeasures; the elements of As(R+) ures. THEOREM 3.5 The spaces A a ( R + ) ,A ( R + )and As(R+) are closed linear subspaces of A ( R + ) ;moreover every p E A ( R + )can be uniquely written as p =p a +p d

where

+ ps

The spaces &(R+) and &(R+) have special representations as Banach spaces. In particular the elements of A a ( R + )obey the conditions of Theorem 1.20 which can be rewritten as:

THEOREM 3.6: RADON-NYKODYM THEOREM There exists an isometric isomorphism between Aa ( R+ )and L,(R+)such that to each Y E A a ( R + ) there is assigned an f E L , ( R + ) which satisfies (a) for any E E B Y ( E )= (b) for any

4 E Co(R+)

I E

f ( t )dt

%4)= \m4(t)f(t) dt Moreover for each p E A a ( R + ) the convolution p @ y is an element of L,(R+)and hence (cL@y)(t)=I

f(f-4

0

dP(c+)

A direct consequence of this result is that A,(R+)(and consequently L,(R+)is an ideal of the algebra A ( R + ) . As an example of an element of &(R+) we can consider the measure S, defined for some T E R +as 1 if T E E (3.11) = 0 if T $ E THEOREM3.7 The measure 6, defined by (3.11) lies in .&d(R+) and corresponds to the functions $(4) = $ ( T ) , V 4 E Co(R+).Moreover any

[

90

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

non-zero p E A d ( R + )can be written uniquely as

c m

P=

n=O

%ST,

where { a , } is a sequence of non-zero complex numbers such that m

1 la,l<@~and n =O

{T,} is a sequence of distinct elements of R+. The

Variation (pI of the measure p is given by m

and in particular

n =(I m

IIPII=

C

n=n

Proof

Ian1

Hewitt and Ross (019, 1963).

0

To complete this introduction to the algebra A ( R + ) some conditions concerned with the invertibility of elements of A ( R + )are presented. A maximal ideal of a commutative algebra is any ideal which is not a proper subset of any other ideal other than the algebra itself. It is known (Hille and Phillips, 04.14, 1-57) that the set of all non-invertible elements in an algebra, coincides with the union of all its maximal ideals. For example, none of the elements of L , can be invertible on A ( R + ) since L , is an ideal of A ( R + )and therefore lies in some maximal ideal. More generally it can be shown (Hewitt and Ross, 1963):

PROPOSITION 3.1 The direct sum of the spaces A a ( R + ) and A s ( R + ) (represented by A a ( R + ) @ A S ( R +is) )a closed ideal of A ( R + ) . A multiplicative functional m in A ( R + )is a linear continuous map rn : A ( R + ) - +C'

which in addition satisfies m ( p @ y )= m ( p ) m ( y )

v p, v Y E W R + )

Consider the set

(3.12)

I,,, = { p : m ( p )= 0 } (3.13) Clearly I,,, is an ideal which is also maximal and that every maximal ideal in A ( R + ) has the form (3.13). Therefore seeking the non-invertible elements of A ( R + ) is equivalent to finding the multiplicative linear functions in A ( R + ) . Consider the map m,:P H P ( { O } )clearly , m, is linear continuous functional. Moreover m o b €3 Y ) = F c 3Y({O}) =P

x 746)

3. where

REPRESENTATION OF MULTIPLIERS

91

6 = ((a, t ) E R 1 x R+ :a+ t = 0).

Then

6 = ((0,O)I

and P x

Hence

r(6)= P x rK0,O))= cL({Ol)Y({O)). ~ O ( P @ ' Y )

= mo(~)mo(r)

Thus m,, is a multiplicative linear functional, and so the set

I o = bm : o(~)=O)~{~:~({O))=O) EA~(R+)@A,(R+) is a maximal ideal. In particular every measure vanishes in the singleton { 0 }thus , it cannot be invertible. It follows that for a measure to be invertible in A ( R + ) it must have a non-zero component concentrated at the origin. Therefore every invertible p E A ( R + )must be of the form

+

p = a6,, A,

with (Y a non-zero complex number and A, an element of lo,6,) is clearly the unit of the algebra A ( R + ) .Hence if p-' exists it must be of the form where Some maximal ideals of special interest are directly associated with the Laplace transform. Consider the continuous linear functional ms(P) = b(4s)

4 is as defined in the Proof of Theorem 3.1. The function is the Laplace transform of the measure p ; since 4, E Co(R+) for all Re(s)>O and l14sll=1, then SH.C;(+~) is bounded and analytic in where

SH@(~,)

R e ( s )>O. Moreover definition (i) of convolution implies that

X(P@'Y)(S)= (P63y)(4,)= b ( 4 M 4 , )= ( X P ) ( S ) ( . 9 Y ) b )

Hence m, is also a multiplicative linear functional, the corresponding maximal ideal is

4 = { P : b(4s)= 0) In conclusion, any measure whose Laplace transform has a zero in the open right half complex plane lies in some maximal ideal and therefore cannot be invertible. The converse is generally not true; however, the

92

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

necessary and sufficient conditions for invertibility in terms of Laplace transforms can be established if restricted t o the space JGI,(R+)@Ad(R+).

THEOREM 3.8: HILLEAND PHILLIPS (1957) The space A a ( R + ) @ A d ( R + ) is a closed regular subalgebra of A ( R + ) .Moreover a measure p in this space is invertible if and only if

3.4 REPRESENTATION OF MULTIPLIERS IN L1 AND L ; A representation theorem for operators on L , of the type derived in Section 3.2 for L2 is not available; however, it is possible to establish a full representation of multipliers in L I in terms of a convolution algebra. Initially we review some properties of the Laplace and Fourier transforms of function f~ L , . The space LT can be identified with L,. Since 4s lies in L, for all R e ( s )2 0 , then for any f e L 1 ,

(3.14) defines a complex analytic function in { s : R e ( s ) > O } and a bounded function in { s : R e ( s ) r O } Moreover . the Fourier transform w ~ f i ( j w is) a continuous function which satisfies lim f i ( j w ) = o

Ibl-==

Thus wt+&~) can be identified with an element of C ( R ) (where R denotes the compactification of the real line). It follows (Hille, 1962) that the family of functions w w @ ( a+ j w ) converges uniformly to ~ H & w ) when a +O+. Therefore S H ~ ( S ) is bounded and holomorphic in { s : Re(s)> 0}, bounded and continuous in {s :Re(s)2 0) and satisfies

It was shown in Section 3.3 that L , is an ideal of A ( R + ) , so any p E &(I?+) defines a linear, time-invariant and bounded operator T, on L , into L,(T,: L , + L,) by

T, :f HCL 63 f We now wish to show that every multiplier in L1 can be defined in the same manner.

3.

93

REPRESENTATION OF MULTIPLIERS

THEOREM 3.9 Let A be arbitrary in M ( L , ) then: (a) There exists a function s H h ( s ) bounded and holomorphic in R e ( s )> 0 and continuous in R e ( s )2 0 such that

V f~ L,, V R e ( s )2 0

(6PAf)(s)= h(s)(6Pf)(s)

(b) The map A w h defines an injective morphism of rings (c) There exists p E A ( R + )such that c(i) A f = p @ f V f E L I c(ii) IIAII= IIPII c(iii) The map A Hp defines an isometric isomorphism of rings

Proof Let A * : LT += LT be the strong dual operator of A. Notice that LT = L,, hence for any Re(s)?O 4% = A*+s

is an element of L,. A We have (6PAf)(s) = (Af)(&) = f(&) for any f~ L1. Utilizing the time invariance of A and proceeding as in Theorem 3.1,

+s(T)f(&) =

CGhk)

(3.15)

for all f e L 1 and R e ( s ) r O . The space L1 as a convolution algebra contains an approximate convolution unit (Hewitt and Stromberg, 1965), this is a sequence (8,) in L1 such that (i)

Il8nllL,

=1

Vn

(ii) limI18,,@f-fllL,=o

Vf€L1

.n

Consider now the complex number (8,,&f)(&),from the definition of {an)

lim C S , , & ~ > C += ~ )f&)

(3.16)

Alternatively from the definition of convolution where

(8, 6f)(+s ) =

f(

+n

)

for t S O for t < O

(3.17)

94

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Selecting a particular foeL1 such that zero in R e ( s ) r O we--have

S H ~ ~ ( + is ~ )bounded

away from

&(+J = (863 fo)(+s)[fo(s)l-l Taking the limit in n and using (3.16) lim

=fo(+s)Uo(4s)~-1

(3.18)

Therefore lim&(+s) exists for all R e ( s ) r O , let h ( s ) represent such a n

limit. Then from the previous equation it follows that h ( s ) is holomorphic in {s : Re(s)> 0) and continuous in {s :R e ( s )2 0). Furthermore from (3.17)

= f ( k )= h ( s ) f ( h )

(3.19)

for all f € L 1 and all Re(s)rO. And so h ( s ) = lim n

-

Ih(s)lslim IIAlI . 116nlILI I l 4 s l l L But since

l+sllL=

1 for all R e ( s ) z O and (JSnl(LI = 1 for all n, thence

(3.20) The proof of (a) is complete; (b) results immediately from (3.19) which shows that AH h is a morphism of rings and from the uniqueness of the Laplace transform. To establish (c), consider the space 9 defined as the space of all functions 4 E L, of the form

where ai are complex numbers and Re(si)>O for all i. Thus 4'' is the space of all finite linear combinations of functions 4s with Re(s)>O. Notice that q is a linear subspace of C,(R+).Consider the map pn :q +. C' defined as

p n ( 4 )= &(A*4) =(A>,)(+) The map p n ( * )is clearly linear and continuous; moreover IPn(4)I5 IlAll ll6nllL, * IPn(4)l

IlAll ll4llL *

11411L

3. For any

REPRESENTATION OF MULTIPLIERS

n

4= C

95

in ? we have

i=l

i= 1

Hence

i=l n

lim ~

( 4= )C

i=l

aih(Si)

In conclusion, a linear functional can be defined as fi(4)= lim p,,($) n since the limit always exists. Furthermore Ifi(4)ls l i m lp,(4)1 S(IAIl.l1411L, therefore f i ( - ) defines a n functional in ?. In conclusion: it is possible to define a linear functional I;: :?+ C' such that (9

Ifi(4)l~llAllll4llL, *

(ii) fi :

v d€*

2 ai4.,H 2 a i h ( s i ) i=l

i=l

From the Hahn-Banach theorem it follows that a functional I;€

C z ( R + )exists such that

(3.21) i=l

i=l

Now since CE(R+)is identified with A ( R + )(Theorem 3.4) there exists a measure p E A ( R + )which defines a functional I;. The Laplace transform of such a measure is ( Z p ) ( s )= I;(4s) = h ( s )

Moreover, for any f e L ,

N P 60f)(s)= ( Pc% f)(4A= I;(4s)f(4s) = hWf(4A Using (3.19) Z(Af)(s)= (P6fN4A = ( Z P60 fNs) = (2Af)(s) the uniqueness of the Laplace transform in L, now implies that Af = p

60f

V ~ LE,

This proves c(i). We have

ll Afll IlPll llfll *

(3.22)

96

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

and so IlAll IIlpII. Alternatively, from (3.22) llpll5 A, thus 11p11= IlAll, proving c(ii). This last -result also establishes that the map which assigns A to p is one to one, and as it is also onto it must be an isometric isomorphism of rings. 0 We have shown that every multiplier A E M ( L , )is represented by a transfer function h E K ( 0 ) which is in addition continuous along the imaginary axis. However, it cannot be shown that every such function defines a multiplier in L,. Therefore a one-to-one correspondence between a space of transfer functions and the space M ( L , ) cannot be established in order to identify regular elements of M ( L , ) as those which have invertible transfer functions. Although for a sub-algebra of M ( L , ) it is possible to express the necessary and sufficient conditions for invertibility in terms of transfer functions (we show that M ( L , ) is isomorphic with A ( R + ) ) .Remembering from Theorem 3.8 that the space Aa(R+)@ Ad(R+) is a regular sub-algebra of A ( R + )and that an element p in such a space is invertible if and only if

If we now define LA+ (Willems, 1971) as the algebra of all linear continuous and time-invariant operators which can be expressed as the convolution with measure p of the form

with

fnE

L , and

CD

Iakl

k=O


THEOREM 3.10. The algebra LA+ is a regular sub-algebra of M ( L , ) and an element A E LA+ is invertible in M ( L , ) if and only if its transfer function S H h ( s ) inf Ih(s)l>O Re(s)>O

The multivariable extension of this result is almost immediate, if &(R+)"""represents the algebra of all ( n x n ) matrices over A ( R + )then clearly M ( L ; ) is isomorphic with A(R+)"x".If LAYx" represents the sub-algebra of M(L7) formed with all elements which are defined by the convolution with an ( n x n ) matrix whose entries are elements of LA+, then operator A E LA:x" is invertible in LA+ if and only if its matrix transfer function s ~ H ( ssatisfies )

inf

Re(s)>O

ldet H(s)l> 0

(3.23)

3.

97

REPRESENTATION OF MULTIPLIERS

The algebra L A + can be identified directly with the commutative Banach algebra d of Desoer and Vidyasagar (1975), although LAYX"= d n X "is not a commutative Banach algebra. For any fEL,(R+) (1' p IW) and gE LA+, it is easy to see that f @ g E L p ( R + )and llf@ gll, sllfllp llgllLA+; in particular for p = 2, I l f @ ~ l l ~ ~ lsup l f l Ig(jo)l. l~ w

3.5

REPRESENTATION OF MULTIPLIERS IN L,

The basic definitions and properties of the Lmspace and its dual are given in Appendix 3.1 together with the representation theorem for operators A :L, + L,. From the appendix we obtain the following:

THEOREM 3.11 The convolution algebra A ( R + ) can be identified with a sub-algebra of M(L,) by assigning to each p

E A ( R + )the

operator f

Hp

@f

THEOREM 3.12 Let A be arbitrary in M(L,). Then there exists a complex function S H h ( s ) in K ( 0 ) such that (a) ( Y A f ) ( s=) h ( s ) ( Y f ) ( s )

V f~ Lm, V Re(s)> 0

(b) sup Ih(s)l5 IlAll Re(s)>O

Moreover the map which assigns A to h E K ( 0 ) is a continuous, injective morphism of rings. Another useful result is PROPOSITION 3.2 (Larsen, 1971) For an arbitrary A E M(L,) there exists a unique p E A ( R + )such that for all f E C,(R+)

Af=p@f It is important to note that Theorem 3.11 and Proposition 3.2 cannot be combined in a full representation theorem, since Proposition 3.2 states that A f = p @ f for any fEC,(R+); it does not state the result for an arbitrary f E L, .

3.6 REPRESENTATION THEORY IN Xp-SPACES It was shown in Section 3.2 that every multiplier in L, can be represented by a transfer function which is an element of K ( 0 ) . Unfortunately a

98

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

generic h E K ( 0 ) is not necessarily defined everywhere along the imaginary axis, so it is not possible to establish a Nyquist stability criterion which can be applied to L, feedback systems. To establish such a criterion we introduce a family of spaces X, derived from the more familiar L,-spaces. The X, spaces are examples of sequential inductive spaces whose properties are now established prior to the consideration of multipliers in X,.

3.6.1 Sequential convergence vector spaces The vector space E is called a sequential convergence space (Zemanian, 1968) if there exists a rule which identifies certain sequences, called the convergent sequences, and assigns to such a sequence an element of E, called the limit, and in addition satisfies the following axioms: A. 1 If {x,} and { y , } are convergent sequences, then for each pair of scalars a,p the sequence {ax,,+py,} is also convergent and satisfies lirn { a x , + P y , } = a lirn {x,}+ p lim { y , } n

n

fl

The limit of any convergent sequence is unique Every cofinal sequence of a convergent sequence is also convergent and has the same limit A.4 The constant sequence x,, = x, for all n, is convergent with limit x A.5 For any convergent sequence {x,} in E and any convergent sequence of scalars {a,} A.2

A.3

lirn { a , x , } = lirn {a,}lirn {x,} n

n

?I

Any topological vector space is an example of a sequential convergence space. Given two sequential convergence spaces E and F and an operator T :E + F, T is said to be sequentially continuous, when for each convergent sequence {x,} in E the sequence {T(x,)}is convergent in F and

J.

satisfies lirn T(x,,)= T(lim x,

A subset B c E is said to be sequentially bounded in E when, for each sequence { x , } in B and each sequence of scalars {a,} converging to the origin, the sequence {a,x,} converges to the origin.

PROPOSITION 3.3 If T :E + F is linear and sequentially continuous and B c E is sequentially bounded, then T ( B ) is sequentially bounded. Proof Let T(x,) be an arbitrary sequence in T ( B ) .If {a,}+ 0 as n 4 m, then lim a,T(x,) = lirn T(a,x,) = T n

n

3.

REPRESENTATION OF MULTIPLIERS

99

due, respectively, to the linearity of T, the sequential continuity of T, the sequential boundedness of B, and finally the linearity of T. 0 If E is a sequential convergence, complex vector space and p :E -+ C' is a linear, sequentially continuous operator, then p is called a sequential functional in E. The space of all distinct functionals in E is called the sequential dual and is represented by E'. For any B, sequentially bounded in E, and any PEE',the set p(B) is bounded in C' (Proposition 3.3). Hence a real, positive, finite function can be defined in E' as: (3.24) It is trivial to show that p 1 3 ( - )is a finite semi-norm in E'. Given any BcE such that p 1 3 ( p ) # 0 ;this derives from the fact that for any p # 0 there always exists some x c E such that p ( x ) # O . Consider now the family {pB(.)}of all semi-norms defined as in (3.24) when B spans all sequentially bounded sets in E. From Theorem 1.4 (Kothe, 1969), this collection defines a locally convex topology in E'. The locally convex topological vector space thus generated is called the strong sequential dual of E and is represented by E*. p # 0, we can state that there exists some sequentially bounded set

3.6.2 Sequential inductive limit spaces Let {E,}, a > 0, be a family of topological vector spaces such that L. 1 For b > a, E, 3 Eb ~ . 2For b > a, the topology induced in Eb by the topology of T, of E, is weaker than the original topology Tb of Eb Let X represent the vector space U E,. The space X can be made a >fl

into a sequential convergence space if the following rule of convergence is used: ~ . 3A sequence {x,} in U E, converges in this space to a point x a >O

when there exists a , > 0 such that the sequence and the limit both lie in E%,and {x,} converges to x in the topology of Eao It is simple to show that the rule of convergence ~ . satisfies 3 all the axioms ~ . 1 - ~ . 5A. space X , defined by a family of spaces which satisfy conditions ~ . 1~, . 2 equipped , with the rule of convergence ~ . 3is, called the sequential inductive limit of the spaces E,. PROPOSITION 3.4 B c X is sequentially bounded if and only if there exists a > 0 such that B is a bounded subset of E,.

100

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Proof The propositions conditions are clearly sufficient. Conversely, assume that B is sequentially bounded and not contained in any E,. Then a sequence {x,} in B can be defined such that x, $ Earn,where a, is a sequence in R , converging to the origin. Let {a,} be a sequence of non-zero positive reals converging to the origin. The definition of sequential boundedness now implies that {a,x,} is a convergent sequence X converging to the origin. Condition ~ . now 3 implies that there exists some a > 0 such that E, contains every element of the sequence. In particular for any a, < a

Ea. 3 Ea 3 a n x n which contradicts the hypothesis. Therefore B lies in some E,. The rule of convergence ~ . now 3 implies that for any sequence {x,} in B and any sequence {a,}+ 0, the sequence {a,x,} converges to the origin in E, in the topology of E, ; thus B is bounded in E,. 0 PROPOSITION 3.5 Let E be a finitely normed space and X =

U Fa a

a >O

sequential inductive limit of finitely normed spaces. If T :E + X is sequentially continuous and linear, then there exists some a > 0 such that T ( E )c Fa * Proof Assume the converse of the proposition; let a, + 0 be a sequence in R , converging to the origin. Then, for each n there exists x, E E such that T(x,)# Fa.. The sequence {x,} cannot be bounded otherwise {T(x,)} would have to be a sequentially bounded set (Proposition 3.3) and so lie in some Fa (Proposition 3.4) which contradicts the hypothesis. Furthermore every x, must be non-zero as the origin is common to every Fa.The sequence {~~x,\~-'} is therefore well defined and converges to the origin as n + cc. Let a, = I I X , ~ ~ - ~ > 0, the sequence {a,x,} converges to the origin as - ' Hence T(a,x,) must also converge to the lim \la,x,l[ = lim ~ ~ x , ~=~0. n

n

origin. By ~ . there 3 exists some a > O such that Fa contains every term of the sequence. For any a, < a Fa. 2 Fa 2 T(anxn) = anT(xn) Finally as a, # 0, then T(x,)E Fan which contradicts the hypothesis.

0

Let X and Y be topological spaces (respectively sequential convergence spaces) and let L ( X , Y ) be the vector space of all linear, continuous (respectively, sequentially continuous) operators on X into Y. THEOREM 3.13 If E is a finitely normed space and Y =

u Fa is a

a >O

3.

101

REPRESENTATION OF MULTIPLIERS

sequentially inductive limit of finitely normed spaces, then as a vector space, L(E, Y )can be identified with the inductive limit of the spaces L(E , Fa). Proof For b > a, Fb =I Fa hence TbE L(E, Fb) can be uniquely extended to some T,EL(E, Fa) by embedding Tb(E) in Fa. It then follows that L(E, Fb) c L ( E , Fa). Let l a b :L(E, Fb)+ L(E, Fa) be the embedding of L(E, Fb) in L(E, Fa). From the definition of inductive limit (Section 1.3.3) it is now possible to define a space L which is the inductive limit of the spaces L ( E , Fa), that is

L = lim zabL(E,Fb) 4

It is now only necessary to show that L(E, Y )coincides with Given any T =

i=l

T,, T,

E L ( E , F%) in

t,T is clearly

L.

an element of

L(E, Y).Conversely given any T EL(E, Y )there exists some a > O such that T ( E )c Fa (Proposition 3 . 3 , hence T E L. COROLLARY 3.1 Under the conditions of Theorem 3.13, L ( E , Y)can be associated with the topological inductive limit of the spaces L(E, Fa), equipped with uniform topologies.

Proof For b > a, the topology induced in L(E, Fb) by the topology of L(E, Fa) is weaker than the original topology of L ( E , Fb). Utilizing Theorem 1.7 (Kothe, 1969) it follows that L(E, Y ) is the topological inductive limit of the spaces L(E, Fa). THEOREM 3.14 Let X =

U

a >O

E, be a sequential inductive limit of finitely

normed spaces and Y any sequential convergence space. The space L ( X , Y )can be identified with the projective limit of the spaces L(E,, Y ) . Proof

For a > 0, let Pa : L ( X , Y )+ L(Ea, Y )represent the projection

V T EL ( X , Y ) (PaT)(x) A T ( x ) , V x E E,, Identically we can define projections P b a :L ( E a ,Y )4L(Eb,Y )for b > a, as

(pbaTa)(x) Ta(x), v x e Eb, v TaEL(Ea,y) The following relationships are readily verified for all b > a > c > 0 (3.25) From the definition of projective limit (Section 1.3) it is now possible to define t =l@ Pb,L(Ea, Y)of the spaces L ( E a ,Y).i is the subspace of

102

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

fl L(E,, Y )formed a

by all elements

f ={T,} which satisfy

Tb=PbaTa,

V b>a

(3.26)

We wish to show that i coincides with L ( X , Y).Given any T E L ( X , Y),T can be associated with an element f = { T , } ~ fby defining T,=P,T. From (3.25) f satisfies (3.26). Conversely any { T J E can ~ be associated with an element T EL(X,Y) through T ( x ) T,(x) for all x E E,. The condition (3.26) makes this definition coherent for any b > a and for any x E Eb. Hence

i=

T ( x )= Tb(x)= (pbaTa)(x)= Ta(x) pb,L(E,, Y )= (Pb,T,)(X) = T,(X).

0

THEOREM 3.15 If F is a finitely normed space and X = U E, is a sequential inductive limit of finitely normed spaces, then:

a>O

(i) For any sequentially bounded set B c X the function (3.27)

is a finite semi-norm in L ( X , F ) . (ii) When B spans all sequentially bounded sets in X the family of semi-norms {pB(.)} define L ( X , F) as a locally conuex space. (iii) The space L ( X , F) equipped with the topology defined by the family of semi-norms {pB(.)} coincides with the topological projective limit of the spaces L(E,, F), each equipped with the uniform topology. Proof (i) There exists some a > 0 such that B is a bounded set in E,, in the topology of E, (Proposition 3.4). Hence (3.27) establishes one of the semi-norms which defines the uniform topology of L ( E , , F ) (Yosida, IV.7, 1971). (ii) For any non-zero T EL ( X , F) there exists some X " E X such that T(xo)# 0. Since a singleton is an example of a bounded set, there exists one element of the family {pB(.)} such that p B ( T ) # O . Utilizing Theorem 1.4 (Kothe 1969), the family of semi-norms { p B ( - ) }defines a locally convex topology in L ( X , F). (iii) Theorem 3.14 states that L(X, F) and lim Pb,L(E,, Y )= i coinc cide as vector spaces. We now compare topologies. An, arbitrary absolutely convex neighbourhood of the origin (n.0.) in L is of the

3.

REPRESENTATION OF MULTIPLIERS

103

form ((1.4), Section 1.3.3). (3.28) where Vai is an n.0. in L(Eai,F).

u,,P{T,€L(E,,,F):PB,,,(Ti)<Si.k, k = l , 2, . . . , mi} (3.29) with {Bi,k},k = 1 , 2 , . . . , m i , a family of bounded sets in E,, and { & k } , k = 1 , 2 , . . . , mi, a family of non-zero positive numbers. A general n.0. in L ( X , F) is

u={ T E L(x,F):p

~( T , )< & , k = 1, . . . , m }

(3.30)

with { B k } ,k = 1 , 2 , . . . , m, a family of sequentially bounded sets in X,and {&}, k = 1 , 2 , . . . , m, a family of non-zero positive numbers. Every Bk in (3.30) is a bounded set in some E,, (Proposition 3.4). Hence U can be written as

with Bk a bounded set in E,,. Thus

Therefore U is the set of the form of (3.28) and (3.29). Hence every absolutely convex n.0. in L ( X , F)is an n.0. in i.Conversely any set of the form defined by (3.28) and (3.29) has the form (3.30) because every bounded set Bi,k in E,, is a sequentially bounded set in X.Thus the two topologies coincide. 0 The topology defined in L ( X , F) by the family of semi-norms {pB(.)} is called the strong convergence topology. COROLLARY 3.2 The strong sequential dual X* of a sequential inductive limit X = U E, of finitely normed spaces can be identified with the as0

topological projective limits of the strong duals E: of the spaces E,.

X*= lim Pb,E: c

Proof Direct consequence of Theorem 3.15 with F = C ' .

0

THEOREM 3.16 Given two sequential inductive limit of finitely normed E,, Y = Fb, then as a vector space L ( X , Y) can be spaces, X =

u

a 20

u

b>O

104

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

identified with the projective limit, when a 40 of the spaces

u L(&Fb)*

La=

b >(I

Proof L ( X , Y ) is the projective limit of the spaces L(Ea,Y )(Theorem 3.14). Alternatively L(Ea, Y )coincides with La (Theorem 3.13). 0

3.6.3

The spaces Xp

Let Lp represent the usual space of complex valued time functions t H f ( f ) with domain R , , such that tHlf(t)lP is integrable with respect to the Lebesgue measure. Lp is a Banach space with topology defined by the norm r

For 1 Ip <m, the strong dual L: of L, coincides with Lq where p-' q-' = 1, and the action of p E L ; is defined as P(f) =

J-+

CL(f)f(f)

+

dt

The space L, is also a Banach space equipped with the norm

For any a>O, let Lp,,represent the set of all functions t ~ f ( t such ) that fHexp(at)f(t) is an element of L,. Let E, :LP,,+ L P be the operator which assigns twf(t) to twexp(at)f(t); clearly E, is an isomorphism. In L,,,, we shall adopt the norm

llfllb,~ IIEJllb Under such a definition, Lp., and Lp are seen to be isometrically isomorphic spaces. It is straightforward to show that for any b > a, Lp,,contains L p , b and that the topology induced in L p , b by the topology of Lp,, is weaker than the original topology of Lp,b. Thus with the rule of con, space vergence ~ . 3the x p

U

a>O

Lp,a

(3.31)

is a sequential inductive limit space.

THEOREM 3.17 For 1~ p < m the , strong sequential dual XE of Xp can be identified with the topological projective limit lim Pb,Lq,-,, where p-' + t q-' = 1.

3.

105

REPRESENTATION OF MULTIPLIERS

Proof Utilizing Corollary 3.2, X; = lim pbaL;,a, we therefore need to c demonstrate that L;,a = Lq.pa. Let fi be arbitrary in LEa, for any f~ &.,, F(f) = fi(E-,E,f); Eaf is arbitrary in Lpwhen f is arbitrary in L,,,, so FE-, defines a functional in Lp. Let pa E L; represent such a functional. We have shown that every functional in LE,a can be written as paEa, with pa E L ; . For q = (1+ p - ' ) - ' , let t++pa(t)represent the functional pa. For any f~ Lp,a

p(f) = pa(Eaf)=[R+

pa(?)exp ( a t ) f ( t )dt

Let p ( f )A p a ( ?exp ) ( a t ) ; tt+p(t) is an element of Lq,-a.Thus we have shown that every fi E L;,a can be represented by some p E Lq,-asuch that

F ( f )= Topologically

p.(t)f(t)dt, R+

IlccIIL:..

= =

Hence

SUP

IUIL.,= SUP

I

IIEdlCp- I

ICLIIL:.

V f~ Lp,a

(3.32)

I@(f)l [lp(f) exp ( - a t ) [ . lexp (at)f(t)ldt = IlE-a~IlL,= IIPlI&.-.

Conversely given any p E Lq,-a the integral (3.32) defines a functional in Lp,a.Thus L;,, = Lq,-a. 0

u

THEOREM 3.18 The space L(X,, X,) can be identified with 1 9 Pba c >o L(Z-P.ULP.C)

Proof

Direct consequence of Theorem 3.16. 3.6.4

0

Multipliers in X,,X,, and X ,

In the determination of graphical stability criteria the spaces X , , X,,and X , are particularly significant. PROPOSITION 3.6 For b > a, the space of all linear, continuous and timeinvariant operators A : Lp,a+ Lp,b is just the singleton (0).

Proof Assume that there exists an operator A ( A # 0), which is linear, continuous and time-invariant operator Let

/i

A Lp,a Lp,b represent the operator EbAE-,, then

/i maps Lp and into L,.

106

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

It is straightforward to show that for any T>O and any a 2 0 sTEa = 4 a ( T ) E a S T

where +a(t)

exp(-at) A (0

and

for for

t r O

t
As A is time invariant

Since b - a > 0, the norm IlAf 11 can be made as small as desired, since T can be selected as large as necessary. Therefore IlAfll= 0 which contradicts the condition llAll# 0 and consequently llAII # 0. Thus the null operator is the only linear, time-invariant operator on Lp,ainto Lp,b when b > a. 0

Definition 3.1 For any real a the space K ( a ) is the space of all complex valued functions SH h ( s ) of the complex variable s, which are bounded and analytic in the half-plane { s : Re(s)> a}. Definition 3.2 The space K, is the space of all complex functions st+h(s), analytic in { s : R e ( s ) > a } and such that for any ? > a , sup Ih(s)l is finite. Re(s)zy

Notice that K, is not contained in K ( a ) since there are functions h E K, for which SUP SUP Ih(s>l

y>u Re(s)zy

may not be finite. THEOREM 3.19 The space M(L,,,, L2.c),c < a of all linear, continuous and time-invariant operators in L2,a into L,,, can be identified with the space K - , in such a way that every A E M(L,,,, L2,c)is assigned to an element h E K - , such that

( 9 A f ) b )= h(s)(Sf)(s) for all f E L2,aand all Re(s)> -c.

3.

107

REPRESENTATION OF MULTIPLIERS

Proof Consider an arbitrary A E M(L,,,, L2,c) and introduce A 4 EcAE-, ; the operator A maps L2 into L , . Furthermore the time invariance of A implies sTA =

+,-,(T)As~ v T

~

O

Following the proof of Theorem 3.1, introduce IL, E L2 defined as Then

(3.33)

A*&

(YAf)(s)= (Af)(+s)= f(W for all f~ L, and Re(s)>O. Also from the proof of Theorem 3.1 we conclude that ( S d ) ( + 5 )= +s+a-c(T)f($s)

V Re(s)> O

Using this result and the arguments of Theorem 3.1, it follows that

f(4s)= h ( s ) f ( h + a - c )

(3.34)

for all Re(s)>O and for some s w h o ( s ) analytic in { s : R e ( s ) > O } which satisfies

so s ~ h ( ) ( slies ) in K O . Finally but and

(2’Af)(s) = ( 2 ’ E CAE- J ) ( s=) ho(s)(2’E,E-J)(s+ u - C) ( Y E , A E - J ) ( s )= ( Y A E - J ) ( s - C )

(2’EaE-,f)(s+ u - C) = ( Y E _ $ ) ( s+ u - c - U ) = ( Y E - , f ) ( s- C)

Since E-J is arbitrary in L,,, when f is arbitrary in L , it follows that there exists an hoe KO such that for all g E L,,, and all R e ( s )> O

Or The function assigned to a Conversely multiplier on where

(2’AgNs - c ) = h , ( s ) ( 3 g ) ( s- c ) ( L A g ) ( s )= h0(s + c)(zg)(s)

V R e ( s ) > -c

h ( s ) 4 h,,(s+ c ) is an element of K - , . So that A can be transfer function K - , . it is straightforward to show that any h E K - , defines a L,,, into L,,, through

108

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

THEOREM 3.20 The space M ( X , ) can be identified with the projective limit lim P b a U K c .Moreover any A E M(X,)is assigned to a transfer function

- h ( s ) analytic in some R e ( s )> C C a

SH

-min {a, y}

- y such that for all f E L2,aand R e ( s )>

(ZAf)(s)= h b ) ( Z f ) ( s )

Proof

Direct consequence of Theorems 3.18, 3.19.

0

With minimal alterations to Theorems 3.9 and 3.12 we can establish theorems similar to Theorem 3.20 for the spaces X,and X,. THEOREM 3.21 Let A be arbitrary in M(X,) (respectively M(X,))then there exists a complex function SH h ( s ) .in lim Pba U K - , and a real c csa

number y > O such that for all f E Ll,a (respectively L a )and all R e ( s )> -min {a, y} we have (ZAf)(s)= h ( s ) ( Z f ) ( s )

Moreover the map which assigns A to its transfer function is an injective morphism of rings. Let E a A ( R + )represent the space of all measures p of the form p with p O ~ A ( R +that ) , is for any Bore1 set B CLW

=

J

B

= Eapo,

exp (at)dcco(t).

With minimal alterations to Theorems 3.9 and 3.19 the following theorem can easily be derived:

THEOREM 3.22 The space M(X,)of all multipliers on X , into itself is isomorphic with the projective limit when a + 0 of the spaces Qa defined by Qa

=

U E-cJIC(R+)

csa

The generic element A EM(X,) is assigned to a measure p ~ lci m pbaQa where that

Pba

: Qa --., Qb,b > a are the appropriate projections in such a way

V f E XI.

A(f) = CL @ f

3.6.5 The space of transfer functions 3 Represent by 93 the space lim Pba c

u K c .It

csa

is important to note the

difference between the frequency representation of multipliers in Lp and

3.

109

REPRESENTATION OF MULTIPLIERS

X,. We saw in Sections 3.2, 3.4 and 3.6 that multipliers in L,, L , and L, were associated with complex functions whose domain of analyticity excluded the imaginary axis. Therefore it is not possible to develop general Nyquist stability criteria for feedback systems represented in these spaces. Although the space R appears complex, its elements have an important property: they are analytic in an open region which includes the closed right half complex plane. This property is crucial to the development of Nyquist type stability criteria. In order to analyse 3,represent by R, the space U K - = . Thus csa R = lim PbaRa. c If a function S H ~ ( S ) lies in %, then there exists a ya E (0, a] such that S H h ( s ) is bounded and holomorphic in the half-plane {s: R e ( s )> -y,}. The domain of analyticity of the function always includes the imaginary axis. The space 3 is the projective limit of the spaces 3,.Hence any h E $39 can be represented by a family of functions { s ~ h , ( s )and } a family of positive reals (7,) which satisfy (i) a 2 ya for all a > 0 (ii) 7 / b ” y a for b > a (iii) s ~ h , ( s is ) an element of K - , (iv) h,(s)= h b ( s ) for all b > a and all R e ( s ) r - y a

U K-,

PROPOSITION 3.6 The space R coincides with the space

7’0

Proof Let 6 be an arbitrary element of R and let {ha} be the family of functions satisfying conditions (i) to (iv) above, which represent h. Let y > 0 be chosen such that y s s u p ‘ya. A function h ( s ) can be a>O defined for any R e ( s ) > - y as

h ( s )= ha(s) where a is such that R e ( s ) z - y , . The coherence and unity of this definition follows from condition (iv). Clearly h ( . ) is an element of K-,. Thus any k E 92 can be identified with an element of U K,. Conversely, given any h~

U

Y<0

K - , , let y > 0 be such that K - , contains

Y’O

Define for any a > 0, a positive number ya as y,

min {a, y )

Also define a set of functions {sH~,(s)}

through

h a ( s ) =h ( s ) for all R e ( s ) > - y ,

SH~(S).

110

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Clearly the two sets {ya} and {ha} satisfy conditions (i) to (iv) and therefore represent an-element of R. This demonstrates that U K-, is contained in R and therefore the two spaces coincide.

7’0

0

PROPOSITION 3.7 A n element h E W is invertible in W if and only if there exists some y > 0 such that inf

Re(s)>-v

Proof

If the condition

inf

Re(s)>-.v

Ih(s)l> 0.

lh(s)[>Oholds, then h(.) is invertible in

K-, and therefore in R by Proposition 3.6. PROPOSITION 3.8 A n element of h €3bounded away from zero at infinity is invertible in R if and only if inf

Re(s)zO

Proof

Ih(s)l>O

The only if condition is obvious. Assume that the condition inf

Re(s)z=O

holds. Since h(.) is an element of

(3.35)

Ih(s)l>O

U K-,

it is possible to find an a > 0

Y

such that st+h(s) is bounded and analytic in an open set containing the half plane R e ( s )1-a.

Since h(.) is bounded away from zero at infinity, there exists a p>0 such that inf Ih(s)l > O ISlZP

-

-a

R e ( s )L -a

Hence all possible zeros of h ( s ) in the half plane R e ( s ) L - a exist in the compact region D(-a; p ) = {s : R e ( s )2 -a, I s / 2 p } . As SH h ( s ) is analytic in an open region containing D ( - a ; p ) , there are at most a finite number of such zeros, let z,, . . . , z, denote these zeros.

3.

REPRESENTATION OF MULTIPLIERS

111

Because condition (3.35) is satisfied then max R e ( z i )< 0 i l n

Select a y > 0 such that y < - max R e ( z , ) , then isn

Therefore using Proposition 3.7, PROPOSITION 3.9

SH

h ( s ) must be invertible in 9.

0

The projective limit

coincides with the space JCC(R+) Proof

Identical to the proof of Proposition 3.6.

3.6.6

0

The space Xi

The multivariable extension of the space X, is the nth Cartesian power of X,, represented by X;.The algebra L(X;,X;) is clearly identified with of all n x n matrices over the ring L(Xp,X,). the algebra L(Xp,Xp)nxn In particular for p = 2 and for the algebra M(X;)

M ( X ; )= M ( X J n x n Since M ( X J is isomorphic with 9 (Theorem 3.20), it follows that M(X;) and anxn must be isomorphic. The ring 9 is commutative and therefore THEOREM 3.23 A multiplier A E M(Xg)is invertible if and only if det invertible in 9, where € 9 l n Xisn the matrix transfer function of A.

A

A is

REFERENCES Hewitt, E. and Ross, K. A. (1963). “Abstract Harmonic Analysis”, Vol I, Band 115. Springer Verlag, Berlin. Hewitt, E. and Stromberg, K. (1965). “Real and Abstract Analysis”, Graduate Series in Maths. Springer Verlag, Berlin. Hille, E. (1962). “Analytical Function Theory”, Vol 2, Ginn, Aylesbury. Hille, E. and Phillips, R. S. (1957). “Functional Analysis and Semi-Groups”. Amer. Math. SOC.Colloquium Pub., Vol. 31. Kothe, G. (1969). “Topological Vector Spaces”, I, Band 159. Springer Verlag, Berlin.

112

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Larsen, R. (1971). “An Introduction to the Theory of Multipliers”, Band 175. Springer Verlag, Berlin. MacLane, S. and Birkhoff, G . (1967). “Algebra”. Macmillan, London. Willems, J. C. (1971). “The Analysis of Feedback Systems”, MIT Research Monograph, 62. Massachusetts Institute of Technology. Yosida, K. (197 1). “Functional Analysis”. Springer Verlag, New York. Zemanian, A. H. (1968). “Generalised Integral Transforms”. Pure and Applied Mathematics, 18, Interscience, New York.

APPENDIX 3.1 THE THEORY OF MULTIPLIERS IN L, Let w([-f,TI)= T - - f represent the usual Lebesgue measure and let A(*) be a positive countably additive measure satisfying, (i) 0 5 A(E)5 1 for any Lebesgue measurable set E E R, A(R+)= 1 . (ii) A(E)= 0 if and only if w ( E )= 0.

. Moreover,

Let N represent the collection of all A (null sets) and therefore the collection of all w null sets. Given a positive valued Lebesgue measurable function f : R+ + R+ we shall represent by V(f) the set

V(f) =(a 2 0 :( t :f(t) > a)$?! N)

(A. 1)

If V(f) is bounded, then the least upper bound of Vcf) is the essential supremum of f ; otherwise V(f) is unbounded and ess sup f = m. Let B denote the family of all Bore1 sets in R,. We shall represent by L,(R+, B, N), or L,(R+, B,A) (or more simply L,) the space of all measurable complex valued functions f : R+ + C’ such that ess sup If1 < m . Using the norm

llfll A!

ess S U P If1 the space L, is a Banach space (Hewitt and Stromberg, 20.14, 1963). As usual we identify the two functions f and g with the same element of L, when {t:f(t)-g(t)#O)EN

Let b.a.(R+, B, N) (or b.a. more simply) represent the space of all bounded additive measures p ( * ) in the measure space ( R + ,B) with the property p ( E )= 0 for every E E N and equipped with the norm,

IIPII = IPI(R+)

(A.2)

3.

113

REPRESENTATION OF MULTIPLIERS

We can now state the first result concerning the dual space L: of L , : LEMMA ~ . (Yosida 1 and Hewitt, 1952) The space b.a. and the dual space LZ of L, are isometrically isomorphic spaces. To each GEL: there is assigned a p E b.a. such that for all f E L,

This result can be considerably sharpened: consider n to be the subspace of b.a. formed by all purely finite additive measures. That is, 7~ E II if and only if there exists a sequence of sets El zE , z. . . zE,, z. . . such that lim p(E,,)= 0 for any countably additive measure p and such n

that .rr(En)= .rr(R+)for all n. Then, LEMMA~ . (Dempster, 2 1975) The space L2 can be identified with the direct sum Li(R+,B,A)@n in such a way that an arbitrary I; E L2 is uniquely represented as

THEOREM ~ . 1Let t e 4 ( f ) be a bounded function which satisfies lim I 4 ( t ) ( = 0 ,4 ( ~ ) + 0 f o r u lOl Z Z T < ~and ~ ( ~ + T ) = + ( ? ) C $ ( T )L. e t r b e a I-= purely finite additive measure satisfying r ( S T E ) = + ( T ) n - ( E )for all T 2 0 and E E B. Then T = 0. Proof Consider the interval E T = [ T , m ) . Since E T = S T R + , we have r ( E T )= ~ ( T ) T ( R + )Hence . ~ T ( E , ) I S ( ~ ( T1 )1 I~ 1. 1 and lim IT(ET)I = T-=

Assuming that 1 1 ~ 1 1 f 0, then given any 1> [> 0, there exists a T > 0 such that JrTT((ET) < [ (17~ll.Defining E k = [O, T) = R , - ET, we then have Hence

11~ll=14 ( E T ) + l 4 (Ek)stll4l+l~l (Ekh Il.rrll~(1 -Or'14 (E;)

(A.6)

Let pT be the countably additive measure defined as p T ( E ) = w ( E ; n E). Given any a < T there exists (by definition of purely finite

114

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

additive measures) a set E, c B such that (9 14 (E,) = 1 1 d (ii) pT(Ea) < Now, let EL be defined as E, r l E ; ; then pT(EL)< a and

II..II=I.rrI(Ea)=I.rrl(EL)+lrrl(E,nET) 5

And so

14 (EL)+ 5 1141.

Il.rrlla -5I-l 14 (EL)

(A.7)

We can conclude that EL is non-empty, since it is assumed that 1 1 ~ 1 1 > 0. But since EL is a non-empty Bore1 set, there exists a nonvoid union of pairwise disjoint intervals

m

U

k=l

I k , I k = [ a , . , b k ) such

that

m

m

(ii)

1 pT(lk)
k=l

Since EL is contained in [0, T), we can assume without loss of generality that every I k is contained in [0, TI. Let us order the intervals in such a way that a k + l ? b k , then the limits { b k } form a bounded increasing sequence. Let b , s T be the limit of such a sequence. Now, given any

(ij I k ) < a and

6 > 0 there exists n such that b, - b, < 6a. Therefore p T pT[bfl,b,]< 6a. Considering now the finite union

[il

k=l

lk]u[bn,

have a finite union of intervals containing E L (as [b,, b o ) z

b O ) , we k=n+l

&)

whose p T ( * )measure is inferior to ( l + S ) a .Since 6 is arbitrary, (1+6)a can be made less than T. Therefore without loss of generality we can assume that E& is defined by a finite union of pairwise disjoint closed intervals EL=

fl

U

k=l

I k , Ik

= [ a k ,b k ) ;also without loss of generality we can

assume that a k + l > bk for all k and that 1 ~ ( I1k ) # 0 for all k. Let EI: be the complement to [0, TI of the set EL.Because a < T and pT(EL)= w(EL)< a,the set ELI is necessarily non-empty. Moreover, EE is given by a finite union of pairwise disjoint intervals, fl-1

3.

REPRESENTATION OF MULTIPLIERS

115

Let d > O be defined as min w ( J ) , where J is any of the non-empty intervals in ( ~ 3 )Let . us now consider an arbitrary interval Ic [0, TI such that w ( I ) < d . We can consider separately the cases where I C E : and InE:#$. 0

Case 1: E : x I. \TI ( I ) I1 ~ (E:), 1 since E: is contained in the complement to R , of E,, and since \TI ( E a ) = \ l ~ lwe \ , must have I T I ( E ~ ) = O . Therefore \TI ( I ) = 0. Case 2: InEL#$. In this case, I n & is non-empty for some k. But since w ( I ) < d , then k is unique. Thus

\ TI ( I ) = 1 T I ( & n I ) ITI ( E i f-II ) =

(Ik

I)

(Ik).

Let Z = [ a , b ) , then b - a = w ( I ) < d . Let T > O be such that a k + , - b > 7 > bk - a. Such a T exists as ak+ I - bk > d > b - a. For any such T, b + T > ak+l and a + 7 > bk. Therefore

[ a -k 7,b + 7)C (bk, ak+l)G Eb: by hypothesis

d S T I )= $ ( T ) d I ) . But since S J C E: we have, as in Case 1, T(SJ = 0 , and therefore ~ (= l 0, ) as $(T) # 0. The above two cases demonstrate that there exists a number d > 0 such that every interval I contained in [0, TI which satisfies w ( I )< d also satisfies ~ ( 1= )0. The set EL can be written as a finite union (EL)= 0. Finally from ( ~ . 7 )we conof such intervals, and therefore 0 clude that T = 0.

THEOREM ~ . 2If T is a purely additive finite measure, then for all T Z O the measure r T ( * defined ) as is also purely finite additive. Proof

Let us assume that

T T ( E )= d W 3 T

is a real positive measure. Let El 2 E , 2

. . . =I En 1.. . be a sequence of sets such that T(E,) = T ( R + = ) 1 1 ~ 1 for 1 all

n and p ( E J 4 0 for a countably additive measure p . Define I = [0, T ) and let I’ = [T, m) be its complement. Finally, let EA and EL be defined respectively as En r l I and En fl 1’. Then

IITII= d E 3 + d E 3 IITl(=4 1 )+ dn

116

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

From the above we have

[T ( I )- T(EL)]+ [T(1’)- T(E:)] = 0 Since ELGZ and E ~ E Zwe , have .rr(EL)~.rr(I) and m(E;)sm(Z’).Both terms .rr(I) - .rr(EL) and ~ ( 1-‘ ).rr(E:) are therefore positive. Since their sum is zero, they must both be zero. Hence .rr{[O,

TI1 = .rr(EL)

T, m).))= .rr(EI)

However, since EE is contained in [T, m), is is possible to find a set

B, c R , such that

STB, = E: Therefore, rT(Bn)= m(STBn)= .rr(E:)= .rr{[T,m)}. Noting that [T, w) = STR+, thus

TT(B,)= dS&+) = T A R + ) (A.9) Any countably additive measure p is defined by a L, function p ( t ) . Then p(Bn) =

I

B.

Hence

p ( t )dt-

I

p(f-

T)dt

STB,

lim p(B,) = lim (STp)(E,)= 0. n

n

(A.10)

Consequently .rrT satisfies the conditions of the definition of a purely finite measure. If .rr is an arbitrary complex measure, then = (7#y -

where .rrIR’,

dR), and

m!!)

+ j(.rry)- &))

are positive measures. Then

.rrT( E ) = (rip)( STE)- d R ) ( S T E )+) j (

TY)(STE)- d!)(STE))

clearly defines a purely finite additive measure.

0

THEOREM ~ . 3 L,-REPRESENTATION : THEOREM Let A :L, L, be a linear, bounded time-invariant operator. Then there exists a bounded holomorphic complex function s H h ( s ) with domain in R e ( s )> 0 such that (i) (L?Af)(s)= a(s)(L?f)(s), for all

f~ L, and R e ( s )> O

3.

117

REPRESENTATION OF MULTIPLIERS

(ii) SUP Ih ( 44lA11 Re(,)>()

where 9 represents the Laplace transform operator. Proof For any f~ L, and Re(s)>O, the time invariance of A implies that (zASTf)(s)= (ySTAf)(s) (A.11) for all T r 0. Also for any f~ L, we have

(9fb)= &(f)7 where &(.) represents the element of L z given by the L, function r H + s ( t ) which is defined as exp(-st) for t r O for t < O Then (A.12) (yAf)(s)= d,(Af) = A*&(f) where A*: L z + Lz represents the dual operator of A. Let defined as A*+,. From ( ~ . 1 2 )we have Since

& ( t + T ) for for

we obtain S - 4 or

= +,(

4, E L2

be

tZO

t
T ) & Therefore

$s(sTf) = $

s ( ~ ) $ s ( ~ ~ )= $

~ ( ~ ) ~ * 4 s ( f ) 7

(STf)= 4 s ( T ,$(f> for all f~ L, and all TLO. Using Lemma ~ . 2the , element written as

(~.13)

4 s

= f i s + ;i,

$s

E L z can be (A.

14)

where fi, is a functional defined by a countably additive measure p , and is a functional defined by a purely finite additive measure 7rs. From ( ~ . 1 3 and ) (~.14)

+,

f i s ( s T f ) + +s(sTf) = + ~ ( ~ ) f i s ( f ) ++

S ( ~ ) + S ( ~ )

and we can conclude from Theorem ~ . that 2 the measure 7rj defined by .rr:(E)= n,(STE) is purely finite additive. Consequently (A.15)

118

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

From Theorem ~ . we 1 can see that GS= 0, and therefore defined by an element of L , : let t H & be such an element. Let (6,) be a sequence of elements L,rlLmwhich satisfy

4,= I;, is

v 12

(9 118nllL,= 1

(ii) lim I16Xf-fllLc = 0 V f e L n

Such a sequence is called an approximate convolution unit (Hewitt and Stromberg, 1965). For any EL, the number &(f) can be written as f(+,),where f ( - )is the element of LT defined by f, as is d e f i n c b y an element of L , . Now, consider the complex numbers $,,(s*,f) = (sXf)(+,). From condition (ii) above,

4

li?

(%)(+A

=

f(+J

Alternatively, by definition of convolution

(a)(+s) f(+6"9 =

where

Using ( ~ . 1 3 ) +LS"'(t) = + s ( t ) ~ n ( + s ) ,

therefore

( 8 3 )JI,~) = 6, ( 4 s )f(&s)

(A.

16)

The function S H ~ ( + , ) represents the Laplace transform of f(.). Choose f such that S H ~ ( + ~ ) is bounded away from zero in Re(s)>O, then

$"(+A

=

(a)(+s)m)}-l

On taking limits, we see that lim $,,(+,) n

l i p &(+s) Let

SH~(S)

exists such that

=f(+s)U(+s)IF1

(~.17)

be defined as lim in(&). From ( ~ . 1 6it) is clear that n

4Lf) = h ( s ) + s ( f )

(~.18)

for all Re(s)>O and all EL,. Equivalently (ZAf)(s)= h ( s ) ( W ) ( s )

(~.19)

3.

REPRESENTATION OF MULTIPLIERS

119

From ( ~ . 1 7 )s, + h(s) is holomorphic in R e ( s ) > O as s H( Y A f ) ( s )and s H (Lff)(s) are both holomorphic functions; moreover, from ( ~ . 1 8 )

f ( 4 - h(s)&s) = 0

for all ~ E L TConsequently, . in the sense of L, and so

hb)4=

$7

= A*A

la(s)l . I l 4 S l I L I -=lIA*ll. l14%llLl But since IIA*\I=llAll (Yosida, p. 195) SUP

Re(s)>O

IW5 IlAll

(A.20)

In conclusion, sw h ( s ) is bounded and holomorphic in R e ( s )> 0, bounded by IlAll and satisfies (LfAf)(s)= h b ) ( ~ f ) ( s )

for all f~ L, and all R e ( s )> 0.

0

Define K ( 0 ) as the space of all complex functions s H h ( s ) bounded and holomorphic in R e ( s ) > O equipped with the norm

llhll = Re(s)>O sup

Ih(s>l

Under pointwise sum and multiplication of functions, K(0) is clearly a normed algebra. The space M(L,) of all multipliers in L, is also a normed algebra when we define multiplication as composition of operators. Theorem ~ . then 3 establishes a relationship between a subspace of K ( 0 ) and M(L,) and we can formally state:

THEOREM ~ . 4The map 8 : M(L,) + K ( 0 ) which assigns a multiplier A E L, to its transfer function S H h ( s ) is continuous and is an injective morphism of rings. Proof An immediate consequence of the relationship is that 8 is continuous. In addition, as a result of ( ~ . 1 9 and ) of the uniqueness of the Laplace transform in L,, 8 is a morphism of rings, that is B(A + A') = 8(A) + 8(A') and 8(A A') = 8(A) . 8 ( A ' ) 0

Finally, 8 is an injective map derived similarly from the uniqueness of the Laplace transform. 0

120

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Convolution, invertibility and multipliers in L, Let A ( R + )represent the convolution algebra of all countably additive, bounded, regular, complex Bore1 measures (Hewitt and ROSS,1963). Each p E A@+)defines a linear time-invariant continuous operator A : L, + L, through Af = p*f. Moreover, we have I(A(1 5 )pl (R+). Finally, the uniqueness of the Laplace transform shows that two distinct elements p , p ’ A~( R + )define two distinct multipliers. We can therefore conclude that A ( R + )can be identified with a subclass of the space M(L,) of all multipliers in L,. Let A A ( R +and ) &d(R+) represent the sub-algebras of A@+)formed respectively with all measures absolutely continuous with respect to the Lebesgue measure and all measures concentrated on countable sets. The direct sum A A ( R + ) @ A d ( R + is)a regular sub-algebra of A ( R + ) (Hewitt and Ross, 1963); moreover we have the following important result in feedback stability theory (Hille and Phillips, 1957).

THEOREM ~ . 5 An element p € A a ( R + ) @ A d ( R +is) invertible in this space if and only if

where

S H ~ ~ ( S represents )

the Laplace transform of the measure p .

REFERENCES Dempster, M. A. H. (1975). “Abstract Optimization and Its Applications”, Lectures School of Mathematic Science. Melbourne University, Melbourne. Desoer, C. A. and Vidyasagar, M. (1975). “Feedback Systems: Input-Output Properties”. Academic Press, New York and London. Harris, C. J. and Valenca, J. M. E. (1980). “Extended Space Theory in the Study of System Operators”. RMCS Tech. Report, E. & E.E. Dept. Hewitt, E. and Ross, K. A. (1963). “Abstract Harmonic Analysis”, Vol. I, Band 115. Springer Verlag, Berlin. Hewitt, E. and Stromberg, K. (1965). “Real and Abstract Analysis”, Graduate Series in Mathematics. Springer Verlag, Berlin. Valenca, J. M. E. and Harris, C. J. (1980) “Nyquist criterion for input/output stability of multivariable systems.” Int. J. Control 31, 917-935. Yosida, K. and Hewitt, K. (1952). “Finitely additive measures.” Trans. Amer. Math. SOC. 72, 46-66. Hille, E. and Phillips, R. S. (1957). “Functional Analysis and Semi-Groups”. Amer. Math. SOC.

Chapter Four

Linear Input-Output Stability Theory 4.1 INTRODUCTION In this chapter we combine the Principle of the Argument of Chapter 2 and the Representation Theory of Chapter 3 to obtain graphical criteria of the Nyquist type which are applicable in determining the stability of linear multivariable distributed feedback systems. T h e generalization of the Nyquist stability criterion for multivariable feedback systems derived from sets of differential equations has been proposed by MacFarlane and Postlethwaite (1977, 1978, 1979); their study of the Principle of the Argument is based upon the concept of algebraic functions defined on Riemann surfaces. A more fruitful approach was initiated by DeCarlo and Saeks (1977), who based their study on covering map theorems. This approach, to the best of our knowledge, has not been pursued elsewhere, but it has convinced us of the potential value of a topological study of Riemann surfaces and the consequent Principles of the Argument of Chapter 2. Although none of the above criteria can accommodate distributed or infinite dimensional systems, Callier and Desoer (1973,1976) and Desoer and Wang (1980) introduced criteria for the stability of systems represented by a class of transfer functions which included some distributed systems. In the following, the necessary and sufficient conditions for inputoutput stability of linear multivariable feedback systems are developed. It follows that the definitions of a system and of stability must be based entirely on input-output properties. The only dynamical systems representation admissible a priori comes from the properties of the inputoutput maps which define the system, and so the existence of any other representation (including the representation by transfer functions) must be deduced from these properties. Based on these arguments a series of transfer function type representations for various input-output maps was developed in Chapter 3. Only for dynamical systems defined on the L2

121

122

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

and X , spaces was it possible to establish a full representation, that is, a space of transfer functions which are isomorphic with the space of input-output maps can be defined explicitly. For general L2 and X , feedback systems it is therefore possible to state necessary and sufficient conditions for closed loop stability in terms of the open loop transfer function. The lack of a full Representation theorem for systems defined upon the L1, L, and X , spaces makes it impossible to state, in similar terms to L, and X , systems, necessary and sufficient conditions for closed loop stability. However, departing from the above philosophy of input-output systems, we can impose restrictions on the explicit representation of these systems and obtain for these restricted systems some necessary and sufficient conditions for input-output stability. To achieve this end, systems defined o n L , and L, are restricted to those explicitly represented by an element of the algebra LA+ (which coincides with the algebra d ( 0 ) of Callier and Desoer (1978)), and similarly systems defined on X,and X , are restricted to those represented by an element of the algebra LA(equivalent to d - ( O ) of Callier and Desoer (1978)). For a feedback system whose open loop transfer function is included in these restricted classes, a result of Hille and Phillips (1957) enables us to obtain necessary and sufficient conditions for the existence of the feedback system in the algebra LA+ or LA-; this result does not by itself establish necessary and sufficient conditions for input-output stability. Trying to identify LA, (respectively LA-) stability with input-output stability is equivalent to stating the regularity of the sub-algebra LA, (respectively LA-) in M ( L , ) and M(L,) (respectively M ( X , ) and M(X,)). In Chapter 3 it was shown that M ( L , ) and M ( X , ) are respectively isomorphic to A ( R + ) and A - ( R + ) ;since LA+ and LA- are respectively regular sub-algebras of A ( R + )and A - ( R + ) ,we see that through the representation theorems the equivalence between LA+ and LA- stability and input-output stability can be established. In the study of linear feedback systems defined on L, and X , a different approach is adopted, by which the sufficient conditions of input-output stability provided by a result of Hille and Phillips (1957) are combined with the necessary conditions provided by the representation theorems of Chapter 3 for these spaces. Throughout this chapter we note that the various representation theorems of Chapter 3 for the above spaces are essential t o the establishment of the equivalence between LA+ or LA- stability and input-output stability. The stability criteria of this chapter will assume that the open loop system is stable. This limitation may at first seem strange, because the

4.

LINEAR INPUT-OUTPUT STABILITY THEORY

123

generalized Principle of Argument established in Chapter 2 considers systems described by matrix transfer functions with meromorphic coefficients and therefore systems with open loop unstable poles can easily be considered. The reason for imposing this apparently severe limitation on open loop dynamic behaviour lies in our inability to establish a representation for unstable input-output systems. It is not possible to provide the necessary and sufficient conditions for the stability of open loop unstable systems based on transfer function concepts when the structure or existence of a transfer function for unstable input-output systems remains unknown. It is possible to extend the Nyquist criterion to unstable systems simply by adding unstable poles to the usual stable transfer functions and considering the appropriate number of encirclements of the critical point through the generalized Principle of the Argument, but this is a mathematically unsatisfactory solution. Although the theory of extended spaces (see Chapter 5) has allowed a rigorous mathematical treatment of general input/output systems whose response is unbounded in finite time, these studies have unfortunately not been linked with the theory of linear systems and in particular with the existence of transfer functions. Future developments in the problem of determining the stability of open loop unstable systems will probably provide this link. In Section 4.2 we introduce several analytical Lp- and Xp-stability criteria (p = 1 , 2 , . . . ,w); the Xp-space and the associated Xp-stability theory is developed to overcome the difficulties associated with the behaviour along the imaginary axis of a transfer function representing a general Lp-system. In Section 4.3 we consider a graphical stability criteria for a restricted class of L, systems. Finally in Section 4.4 we introduce general Nyquist type multivariable stability criteria for Xp-systems and for a restricted class of Lp-systems.

4.2 GENERAL ANALYTIC FORMULATION OF STABILITY Consider a linear multivariable time-invariant dynamical system modelled by y=Gu (4.1) where u lies in some space E of C"-valued time functions and G is an operator that maps E into itself. Some restrictions must be placed on E and G so that the model (4.1) accurately describes physical systems that exhibit stability: (i) The elements of E must be in some manner bounded, since the

124

THE STABILITT OF INPUT-OUTPUT DYNAMICAL SYSTEMS

output of a stable physical system cannot be unbounded. The spaces Lp and X,,(or their multivariable versions) are good candidates for the space E. G must therefore be a bounded operator in one of these spaces. (ii) The outputs of a stable physical system must exhibit a degree of insensitivity, or robustness, to small variations in both the inputs to the system and system parameters. Such a constraint has two implications. First there must be a mechanism in E which allows the degree of proximity between two distinct elements of E to be determined; this implies that E must be a topological space on a sequential convergence space (see Chapter 3). Secondly the operator G must be continuous or sequentially continuous. These considerations justify the following definition of stability:

Definition 4.1 Let E be one of the spaces Lp, Lp“,X p or Xp“. The system y = Gu, is said to be E-stable when G is a continuous bounded operator on E into itself (when E = L p or Lp“) or when G is a sequentially continuous, sequentially bounded operator on E into itself (when E = X, or X;)for U E E . Consider now the structured feedback system represented by the functional equation

GEM(E), UEE (4.2) with u arbitrary in E and G a linear time-invariant bounded operator on E into itself. The fundamental problem of stability theory lies in deducing the conditions for the stability of the feedback system (4.2) in terms of the properties of the open loop map G. To solve this problem some generic results are established first: y=G(~-y)

LEMMA4.1 The feedback system y = G(u-y), G E M ( E ) ,U E Eis E stable if and only if the operator ( I + G ) is invertible in the algebra M ( E ) of all multipliers on E into itself. Proof The “if” condition of the lemma is obvious. To prove the “only if” condition assume that (4.2) is a stable system according to Definition 4.1. By this definition, for any u E E there exists a unique solution y E E to the Equation (4.2). A map 6 :E + E is now well defined by assigning u to the solution of (4.2); y = Gu. Since u = ( I + G ) y , G = ( I + G ) - ’ . The operator ( I + G ) is then invertible in the algebra of all operators of E into itself. Moreover ( I + G ) - ’ is clearly linear and time invariant. Using the bounded inverse theorem (Bachman and Narici, pp. 265, 271) and the

4.

125

LINEAR INPUT-OUTPUT STABILITY THEORY

fact that ( I + G ) - ' must be an onto map, it follows that (Z+G)-' is also bounded. Thus (I+G)-' is an element of the algebra M ( E ) . 0 Lemma (4.1) is equivalent to determining the stability of the feedback system (4.2). An immediate consequence of this lemma and Theorem 3.3 for E = L ; is: THEOREM 4.1: L;-STABILITY The feedback system represented by the functional equation

y=G(u-y)

is L,"-stable if and only if

GEM(L;),

UEL;

inf ldet (I+6(s))l>0, where

Re(s)>O

6~K(0)"x" is

the matrix transfer function of G. Equally, as a direct consequence of Lemma 4.1, Theorem 3.22 and Proposition 3.7, a similar result for X,-stability can be given: THEOREM 4.2: X;-STABILITY The feedback system represented by the functional equation

y=G(u-y)

GEM(X;),

UEX,"

is X;-stable if and only if there exists some y > 0 such that

inf

Re(s)>-y

where

E 5tnx"is

ldet (I+6(s))l> 0

the matrix transfer function of G.

Consider now analytic stability conditions in L , and L, systems. LEMMA4.2 A multiplier G E M ( L , ) (respectively M(L,)) is invertible in M ( L , ) (respectively M(L,)) only if its transfer function s I.+ h(s) satisfies

Proof Assume that G is invertible in M ( L , ) (respectively M(L,)). Using Theorem 3.9 (respectively Theorem 3.12) the inverse operator G-' has a transfer function s H k ( s ) bounded and analytic in Re(s)>O, which by definition of a transfer function must coincide with s H h-'(s). It therefore follows that inf Ih(s)l>O; otherwise s ~ k ( s )could not be Re(s)>O

bounded and analytic. Figure 4.1 illustrates the relationships between the spaces involved in the stability of L,, L , and L, systems. Note that the space LA, (Willems, 1971) coincides with the space A(0) of Callier and Desoer (1978), and for this space, Theorem 3.10 specifies the necessary and sufficient conditions

126

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

FIG.4.1 Relationship between stability spaces

for invertibility of the open loop operator in terms of its frequency description. Note also that LA+=A(O),defined as A a ( R + ) @ A d ( R +is) a regular subalgebra of A ( R + )= M ( L , ) . LEMMA4.3 A multiplier G E M ( L , ) which lies in the subalgebra LA+ is invertible in M ( L , ) if and only if inf Ih(s)l>O, where s + h(s) is the Re (s) S O transfer function of G. Proof

Direct consequence of Theorem 3.10.

LEMMA 4.4 A multiplier G E M(L,) which lies in the subalgebra LA+ is invertible in M(L,) if and only if the transfer function s H h(s) satisfies Proof

G is invertible in LA+ if and only if

3.8), thus the condition

inf

Re(s)>O

Ih(s)l>O (Theorem

inf Ih(s)l>O is sufficient to guarantee inverti-

Re(s)>O

bility in M(L,). The “only if” part of the lemma was established in Lemma 4.2. 0 Lemmas 4.2 to 4.4 are now utilized to establish some analytic stability criteria for L , and L, systems.

THEOREM 4.3 The feedback system represented by the functional equation y = G ( u - y ) with G E M(Ly) (respectively G E M(L:) and u E L ; (respec-

4.

tively u E L:) is L,-stable (respectively L,-stable) only if

e(s))l > 0 where

127

LINEAR INPUT-OUTPUT STABILITY THEORY

6 E K(0)""" is the matrix

inf ldet ( I +

Re(s)>O

transfer function of G.

Proof Since M ( L ; )= M(LI)""" (respectively M(L:) = M(L,)""") the operator ( I + G) is invertible in M ( L ; ) (respectively M ( L z ) )if and only if its determinant is invertible in M ( L , ) (respectively M(L,)). Using Lemma 4.2, det (Z+G) is invertible in the space of multipliers only if inf det (Z+ G(s))l> 0.

0

Re(s)>O

Note that the above theorem is only a necessary condition for L , and L,-stability, but by restricting the operator G to the sub-algebra LAY"" we have the following necessary and sufficient conditions for L ; and L z stability: THEOREM 4.4 Zf G is an element of M(L7) (respectively M ( L : ) ) contained in the sub-algebra LA:"", then the functional equation y = G ( u - y ) with u E L ; (respectively L z ) is LT-stable (respectively Lz-stable) if and only if inf ldet (Z+e(s))l>O Re(s)>O

Proof Using Lemma 4.1, the system y = G(u - y ) is stable if and only if ( I + G) is invertible in the algebra of multipliers M ( L ; ) (respectively M(L:)). The operator (I+G) is invertible in the algebra of multipliers if and only if its determinant is invertible in M ( L , ) (respectively M(L,)). Since the operator G lies in LA:"" by definition, det(Z+G) is an element of LA,. Now, utilizing Lemma 4.3 (respectively Lemma 4.4), it follows that the feedback system is stable if and only if inf ldet (I+

&))I

>o.

Re(sG-0

0

To complete this section, the stability conditions for feedback systems defined on XIand X , spaces are considered in detai!: LEMMA4.5 A multiplier G E M(X,)(respectively M ( X , ) ) is invertible in M(X,)(respectively M(X,))only if there exists some y > O such that inf Ih(s)l>O, where h E 92 is the transfer function of the operator G. R e ( s ) > -y

Proof If G is invertible, its inverse G-' has a transfer function (see Theorem 3.21) s &(s), which is an element of $32, and by definition of a transfer function must coincide with s~-+h-'(s) in the intersection of the domains of both functions. Finally, from Proposition 3.7, a scalar y > 0 must exist such that inf Ih(s)l>O. 0 Re(s)>-y

Theorem 3.22 and Proposition 3.9 showed that the space M(X,) can be

128

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

identified with the algebra of measures

A - ( R + )=

U E-,A(R+)

0'7

where E-$(R+) represents the space of all measures p of the form p = E - , po with p E A ( R + )(see Section 3.6.4). In the study of feedback systems defined on L , a sub-algebra LA+ of A ( R + ) was selected for which it is possible to state necessary and sufficient conditions for invertibility. Similarly for the study of XI-feedback systems we introduce the sub-algebra LA- of M(X,) defined by LA - =

U E-JAa(R+)@JIIci(R+)I

Y>O

(4.3)

We recognize in LA- the space A-(O) of Callier and Desoer (1978).

LEMMA4.6 LA- is a regular sub-algebra of A - ( R + ) and ~ E L A isinvertible in LA- if and only if there exists some a>O for which inf I f i ( s ) l > 0, where ii represents the Laplace transform of p. Re(s)>-a

Proof Consider an arbitrary p E LA- invertible in A ( R + ) ;then there exists some a > 0 and some y1 E LA + such that p E E -, 7,. Identically there exists some 0 < b 5 a and some To€ A ( R + )such that p-l= E-, qo. Defining yo = E+-,)Y1 we see that yo lies in E-(,-,,LA+ c L A + . So p and p-I can be written as p=E-bYO? p-I

YOELA+

?OEA(R+)

= E-b?O,

We have p @I p-' = 6, so (Zp)(s) * (Yp-I)(s) = 1

( Y Y o ) ( s + b ) . ( Y ? , ) ( s + b ) =1

V R e ( s )> 0

=1

(ZY")(S)* W ? " ) ( S )

VRe(s)>-b V Re(s)>-b

The unicity of the Laplace transform in A ( R + )implies that in the sense of A ( R + ) ,qO=7;.' Since yo€ LA + and LA+ is a regular sub-algebra of A ( R + )we must have Po€ L A + ; hence p-' = E+qo lies in EPbL A + and therefore in L A - . This proves that LA- is a regular subalgebra of &-(I?+). Moreover, as yo is invertible in LA - , we must have by Theorem 3.8 inf

Re(s)>O

I(~Yo)(S)l>0

and thus inf

Re(s)>-b

I;(s)l>O

0

4.

LINEAR INPUT-OUTPUT STABILITY THEORY

129

THEOREM 4.5 The feedback system represented by the functional equation y = G ( u- y ) , u E X ; (respectively X L ) with G E M ( X ; ) (respectively G E M ( X L ) ) is X;-stable (respectively XL-stable) only if there exists some a > 0 such that

where

b E 9"""is the transfer function of

operator G.

Proof Is a direct consequence of Lemmas 4.1 and 4.5. Note that like Theorem 4.3, the above X ; (XL)-stability theorem provides only necessary conditions. However, by restricting the feedforward operator G to the sub-algebra LA!!""" we have the following necessary and sufficient 0 conditions for X ; ( X z ) stability. THEOREM4.6: XY(X2) STABILITY If G is an element of M ( X y ) (respectively M ( X Z ) ) contained in the sub-algebra LA!!""",then the functional equation

y

= G ( u- y

u E X ; (respectively X z )

)

represents an X;-stable (respectively X2-stable) feedback system if and only if there exists some a > 0 such that inf

Re(s)>-a

where

b~9"" is " the matrix

ldet ( I + b(s))l>O

transfer function of G.

Proof The "only if" part was proved in Theorem 4.5;the "if" part results from Lemmas 4.1 4.6. 0

4.3 SOME GRAPHICAL STABILITY CRITERIA FOR L,-SYSTEMS The problems associated with graphical L,-stability criteria considered in this section are independent of number of inputs and outputs, and therefore it is assumed the feedback system y=G(u-y)

GEM(L;),

UEL,"

(4.4)

is single input-single output and that the qpen loop system is represented by the transfer function s H h ( s )= G(s).In Section 4.2 it was shown that this feedback system is L,-stable if and only if inf

Re(s)>O

ldet ( I + b(s))l= inf 1(1+h(s))l> O Re(s ) > O

130

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Therefore the stability of system (4.4) consists of finding the zeros of the complex function s ~ ( l h+( s ) ) in the closed right half plane Re(s)2 0. The problem of determining these zeros can be divided into three subproblems: (i) Finding the zeros of ( l + h ( s ) ) in the open right half plane {s:Re(s)>O}. (ii) Finding the points on the imaginary axis where (1 + h ( s ) ) has an infinium of zero. These are points of the form s = jo such that for any 6 > 0 , inf [ ( I +h(s))l= 0 Is-iol<€

keis j>o

(iii) Finding the zeros of (1+ h ( s ) ) at infinity. For practical implementation these zeros can be determined by a Nyquist type stability criterion. Such a criterion can only solve subproblems (i) and (ii) and therefore can only provide necessary and sufficient conditions for closed loop stability when (1 + h ( s ) )has no zeros at infinity. It is therefore assumed that there exists some method of establishing the existence of zeros of (l+h(s)) at infinity. Usually only functions h ( s ) which are such that (l + h (s)) is bounded away from zero at infinity are considered. For arbitrary a, p define

D ( a ;p ) p { s : R e ( s )> a, Is1 5 p }

(4.5)

Assuming that SH h ( s ) is bounded away from zero at infinity, then there exists some p > O such that all possible zeros of ( l + h ( s ) ) in the closed right half plane are in D(0;p ) . The classical Principle of the Argument (see Section 2.7) is conventionally used to determine the number of zeros of a function SHF(S)in a compact simple region such as D(0;p ) . However, this approach can only detect these zeros if the domain of analyticity of the function contains that region. Notice that in our case the region D ( 0 ; p ) contains points of the imaginary axis and the function SH h ( s ) is not necessarily analytic on the imaginary axis. It is well known (Hille, 1962) that an element h E K(0) can be extended to the imaginary axis in the complex plane through a function o ~ h ( j o ) defined almost everywhere as

h ( j o ) = lim h ( a + j o ) a-0+

The question now arises: is it possible by considering the behaviour of SH h ( s ) on the boundary of D(0;p ) to determine the number of zeros of

4.

LINEAR INPUT-OUTPUT STABILITY THEORY

131

(1 + h ( s ) )in D(0;p ) ? Along the imaginary axis O H h(jw) is defined (only) almost everywhere, and therefore its locus does not necessarily define an arc. Even when W w h ( j u )is defined everywhere, the answer to the above question for a general h E K ( 0 ) would still be no, as seen in the following example: Let h ( s ) = a exp(-Us), with a > 1. Clearly h ( . ) lies in K ( 0 ) and therefore defines a multiplier in L, with itself. The function O H h ( j o ) is defined everywhere as

ro h ( j o )= a lim -O+ h ( a + j o ) = I a e x p ( - i )

if

w=O

if

w+o

If the Principle of the Argument were applicable the number of zeros of (1+ h ( s ) ) in D(0;p ) could be determined by the number of encirclements of the -1 point by the locus of h ( s ) when s describes the boundary dD(0;p) of D ( 0 ; p ) . The first difficulty arises when it is noted that + ( s ) )has a zero although (1 + h(jw))# 0 everywhere, the function S H ( ~ h at the origin. Indeed, using the concept of a zero it can easily be shown that for any ( > O . inf 1(1+ h ( s ) ) l =0

lS1'5

Re(s)>O

A second difficulty is that O H h(jw) is not continuous so it cannot define an arc through its locus. Another question now arises: is there any arc which encircles the -1 point the same number of times as there are zeros of (1+h ( s ) ) in D(0;p ) ? This question is equivalent to another: is there any function s ~ h ( s )continuous and with finite total variation on aD(0;p ) , whose locus, when s describes aD(0;p ) , encircles the -1 point the same number of times as there are zeros of s-(l+ h ( s ) ) in D(0;p)? For the above example the totality of the zeros of S H ( ~ + h ( s ) ) in Re(s)> 0 are the points S" =

1

for

( h a+ j ( 2 n + 1 ) ~ )

n = 0, * l , * 2 , . .

Notice that the number of such points are infinite. The locus of any s ~ h ( s )in the circumstances of the above questions would have to encircle the -1 point an infinite number of times without containing this point. This property is incompatible with the required continuity and finite total variation of s ~ h ( s in ) dD(0; p). Hence no arc under the conditions of the above questions can exist in D(0;p).

132

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

In utilizing the Principle of the Argument (Section 2.7) to detect the zeros of (1+ h ( s ) ) in D(0;p ) the class of functions to which the Principle of the Argument is applicable must be restricted in such a way that every h ( . ) in that class satisfies at least the following properties: (p.1) The domain of h ( s ) must be extended to every point on the imaginary axis. (p.2) When s describes dD(0; p ) the function S H ~ ( S ) must be continuous with finite total variation. (p.3) The total variation of zeros of h ( s ) in dD(0; p ) must be finite. (p.4) The integral

must be defined and coincide with the number of zeros of h ( s ) in D(0;P I . Definition 4.2 The space K,(O) is the space of all h E K ( 0 ) for which the function WH lim h ( a + j w ) defined everywhere is absolutely continuous w-0+

in any compact interval of the imaginary axis and has finite total variation on the whole axis. It follows from Hewitt and Stromberg (1963) that for every h E K,(O), the function w ~ h ( j w=) lirn h(cr+jw) is continuous differentiable almost w-0+

everywhere and its total variation is given by

And so wH(d/dw)h(jw) defines an element of L1(-m,

00).

PROPOSITION 4.1 Given a n h E K,(O) there exists u number h(m) such that h ( s ) tends uniformly to h(m) when Is1 + and lirn h(jw) = h ( a ) .

c

O'fm

Proof Since w H h(jw) is absolutely continuous in any compact interval h(jw)= h(jwo)+

H ( h ) dh

where H ( w )= (d/dw)h(jw) (Hewitt and Stromberg, 1965). Also, since H(-)EL , ( - m , m), lirn J; H ( s ) ds is well defined. Let h(m)=h(jwo)+ UJ-=

I:,H ,( h ) dh. Using Theorems 18.3.5 and 18.3.6 of

Hille (1962), it follows that h(m)= lim h ( s ) uniformly in the sector larg (s)l I7r/2. 0 ISl--

4.

133

LINEAR INPUT-OUTPUT STABILITY THEORY

A consequence of the above result is that the function w H h(jw) can be extended to a function uniformly continuous and absolutely continuous on the compact interval [-m, t.3. Furthermore from Hille (1962):

PROPOSITION 4.2 For any h E K,(O) the function w I-+ h(cr + j w ) converges uniformly in the compact interval [-m, m] to w Hh(jw) as (+ +-0'. Also for the sub-algebra K,(O) of K ( 0 ) the following two important propositions can be established:

PROPOSITION 4.3 A n y h E K,(O) can be extended to a function S H ~ ( S ) bounded and analytic in R e ( s )> 0 and continuous in the compactification of the closed right half plane {s :R e ( s )2 0, (sI5 m}. Proof

Define

h(s)= The continuity of

[

h(s) h(m)

for all Re(s)>O, I s l < w for I s ( = m

h(jw)

for s = jw

h(.)results immediately from Propositions 4.1 and 4.2.

0

Since the function OH h(jw) is continuous in [-w, +w] and absolutely continuous in the same interval it defines an arc through its locus. But since h(jm)= h(-jm) then the arc is closed; let yo represent this loop. Identically let y,, a>O be the loop described by w H h ( a + j w ) when w describes [-m, +w].

PROPOSITION 4.4 Let hEK,(O) be such that w ~ h ( j w has ) no zeros. Under the above definitions for ya and yo, there exists some 6 > 0 such that for all a 5 6 the loops yo and ya are homotopic in the punctured plane C'-(0). Proof The fact that OH h(jw) is free of zeros and continuous in Proposition 4.2 implies that there exists an 6 > 0 such that

inf

+w] 4

m];

Jh(s)l>O

O=sRe(s)+

Let Ha :[0,1] X [-w,

[-m,

C' be defined for any a 5 6 as

H,(t, w ) = h(at+jw) for t E [ O , 11, w ~ [ - w +w] , Since a 1 5 the range of H a ( - , . ) is contained in C'-(0); moreover Proposition 4.3 implies that Ha is continuous. We also have (a,

0

)

H,(t, fw)= h(w) w HHa (0, w ) = w Hh(jw)

and

V

rE[O,

11,

w HHa (1, w ) = w Hh( a

+j w ) .

Therefore

134

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

by the definition of homotopy (see Section 2.1.2), yo and ya are 0 homotopic loops.

THEOREM 4.7 Let h E K,(O) be such that W H h ( j o ) is free of all zeros in the interval [-to, +..I. Then the zeros of SH h ( s ) in the compact right half s-plane are finite in number and coincide with the number of clockwise encirclements of the origin by the loop described by h ( j o ) when w describes [-m,

+m].

Proof From Proposition 4.4, there exists some 5>0 such that inf lh(s)I>O. Since h ( s ) converges uniformly when I s I - + w to

0sRe(s )c6

h(w)#O then there exists a p > O such that inf Ih(s)l>O. Then all possible ISlZP

zeros of S H h ( s ) in the compact right half s-plane are interior points of the simple region

m5;P ) = {s :W s )

2

5,Is15 P I

But since SH h ( s ) is analytic in an open set which contains D(5;0) these zeros are at most finite in number. Equally the classical Principle of the Argument (Section 2.7) can be utilized to obtain

where Z is the number of zeros of s I+ h ( s ) in D(5;p ) . In particular

where r p is the arc of the circle {s: R e ( s )2 5,Is[= p}. The integral

represents the index function with respect to the origin of h(T,). When

p + m, h(r,,) tends to a single point h(m)# 0, therefore the index function

tends to zero. Thus

This integral is the index function with respect to the origin of the loop ye Proposition 4.4 demonstrated that ye is homotopic to yo in C ' - { 0 } , and

4.

LINEAR INPUT-OUTPUT STABILITY THEORY

135

so 2 coincides with the number of clockwise encirclements of the origin by the locus of W H h(jw) when w describes [-w, +MI. 0 As a direct consequence of Theorems 4.1 and 4.7 we are able to state the following Nyquist stability criterion for transfer functions h E Ka(0).

THEOREM 4.8: NYQUISTSTABILITY CRITERION FOR Ka(0)SYSTEMS feedback system defined by the functional equation

The

~ = G ( u - Y )U , EL, with G E M(L,) represented by a transfer function h E Ka(0)is L,-stable if and only if the locus described by h(jw) when w describes [ - w , +03] does not contain nor encircle the -1 point.

4.4 GRAPHICAL STABILITY CRITERIA FOR MULTIVARIABLE SYSTEMS In this section graphical stability criteria of the Nyquist type applicable to open loop stable multivariable distributed feedback systems are introduced. These criteria are based on the results of Section 4.2 and on the generalized Principle of the Argument and its consequences (see Theorem 2.16 and Propositions 2.14 and 2.15). Given the linear feedback system y=G(u-y)

(4.6)

the analysis of Lz, L , or L, closed loop systems is based upon the determination of the zeros of the complex function A(s) = det (I+&s)) in the compact right half s-plane (see Theorems 4.1, 4.3 and 4.4). The analysis of X,,X,and X, closed loop systems is based upon the existence of some a > 0 such that A(s) has no zeros in Re(s) > -a (see Theorems 4.2, 4.5 and 4.6). Thus in using the generalized Principle of the Argument to determine these zeros it is important to note two conditions which limit the class of dynamical systems to which this result can be applied: (i) The Principle of the Argument determines zeros of a function in a simple region, hence it cannot be used to determine the zeros of A(s) at infinity. (ii) The Principle of the Argument requires the function A(s) to be analytic or meromorphic in an open set containing the simple region to which it is applied. Consequently it cannot be used in a general L,-stability criterion, since the transfer functions involved are not necessarily analytic on the imaginary axis.

136

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

The transfer function associated with a multiplier in X,, XI or X, is an element of 92 and is therefore always analytic in some open set which contains the right half s-plane. A problem still remains: the analytic stability criterion of Theorems 4.2, 4.5 and 4.6 searches for an a > 0 such that A(s) = det (I+&s)) has no zeros in Re(s)> -a, therefore a precise simple region to which the Principle of the Argument could be applied is not defined. This difficulty is associated with the fact that Nyquist-type criteria normally use the behaviour of transfer functions along the imaginary axis. However, this difficulty can be resolved, keeping compatibility with the usual Nyquist type criteria, if the condition that SHA(S) is bounded away from zero at infinity is imposed. Under these conditions, Proposition 3.8 which states that an element of 92 bounded away from zero at infinity is invertible in 92 if and only if inf Ih(s)(>O, can be Re(s)zO

utilized. This condition also solves the difficulty associated with the determination of zeros at infinity. We can then define a simple region

R = { s :Re(s)2 0 , Is1 5 p }

(4.7)

contained in the domain of analyticity of the transfer function such that the stability of feedback systems can be assessed by the presence or absence of zeros of A(s) in R. In conclusion, the Principle of the Argument can be used to establish X,-stability criteria (bu: not a general &-stability criterion) if the condition that st+det ( I + G ( s ) )is bounded away from zero at infinity is imposed. Another condition that this condition helps to resolve is the possibility of the existence of an infinite number of branch points along the imaginary axis (see also Desoer and Wang, 1980). The generalized Principle of the Argument requires that the boundary of the simple region to which it is applied must be free of branch points; these branch points are the zeros of the polynomial $[A] = det (A - &(s)). Because we are dealing with simple regions R of the form (4.7) and the discriminant D ( s ) (see Equation (2.13)) is an analytical function in an open set containing R, there are at most a finite number of such points in R and in particular in the region of the imaginary axis which lies in the simple region R. Thus along the imaginary axis a finite number of indentations can be constructed (as is usually done with Nyquist D loops) which frees the boundary aR of R of all branch points. It might be thought that the difficulties associated with an infinite number of branch points on the imaginary axis will persist if the case of an eigenvalue contour r generated by describing the whole imaginary axis is considered; this happens when lim b(s)exists uniformly. Note that the contour I' is the limit when I*l--

4.

LINEAR INPUT-OUTPUT STABILITY THEORY

137

p+m of the eigenvalue contour r p generated when s describes the boundary loop dR, of the region { s : I s l s p , R e ( s ) r O } . The Principle of the Argument is not applied to r but to r,, which is legitimately constructed since the number of possible branch points in dR, is at most finite. The index function of r with respect to the -1 point is just a convenient method of determining the limit when p + m of the index function of the contours r,, and this limit must exist since A(s) is bounded away from zero at infinity. In the sequel consider a Nyquist D region of the form

R,

= { s : R e ( s )I0, Is1 5 p }

(4.8)

indented in the usual manner along the imaginary axis so as to avoid all branch points and zeros of A(s) which may lie on the imaginary axis. The anti-clockwise orientated boundary loop of R, is represented by dR,. It is also assumed that transfer functions SH&(S) in W X nsatisfy at least one of the following conditions: c(i) There exists a p > O such that inf ldet (Z+&(s))l>O. blr,

c(ii) There exists in disjoint circles D,, D 2 , .. . , D,,,, m 5 n, none of which contains the -1 point, and there exists a p > O such that for any I s l r p , every eigenvalue of &(s) is contained in one of the circles Di. c(iii) The matrix transfer function &(s) tends to a constant matrix uniformly as Is1 403. Condition c(i) is just a restatement of the requirement that s w d e t (Z+ must be bounded away from zero at infinity; condition c(iii) contains condition c(ii), which in turn contains condition c(i). The multivariable Nyquist stability criteria for X;,X; and X : systems can now be given. &(s))

THEOREM 4.9

The feedback system y = G ( u- y),

G E M(X;), u EX;

(4.9)

is X;-stable if and only i f : (i) When G satisfies condition c(i), the eigenvalue contour r pgenerated when s describes dR, does not encircle or contain the -1 point, and none of the possible branch points on the imaginary axis corresponds to eigenvalues with value - 1 ; or (ii) When G satisfies condition c(ii), any contour formed by the concatenation of the eigenvalue paths generated when s describes the region of the imaginary axis - p 5 o 5 p and closed by arcs contained in the

138

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

circles D , , D2,. . . , Dm do not contain or encircle the -1 point, and in addition none of the possible branch points on the imaginary axis corresponds to an eigenvalue of -1; or (iii) When G satisfies condition c(iii), the eigenvalue contour r generated when s describes the whole imaginary axis do not encircle or contain the -1 point, and none of the possible branch points on the imaginary axis corresponds to an eigenvalue of -1. Proof In any of the conditions c(i)-c(iii), A(s) = det (I+&s)) is bounded away from zero at infinity. Thus, using Theorem 4.2 and Proposition 3.8, the feedback system (4.9) if and only if A(s) has no zeros in the simple region Or Proposition 2.15 can now be used to determine these zeros: if the eigenvalue contour r,, generated when s describes KIP,does not contain or intersect the -1 point, then there are no zeros of A(s) in the indented region R,. All possible zeros in the closed right half s-plane are therefore branch points. However, by hypothesis, none of these points corresponds to an eigenvalue of -1, so none can be a zero of A(s); this establishes part (i) of Theorem 4.9. Parts (ii) and (iii) of the theorem differ only from part (i) in the manner in which the encirclements of the -1 point by I', are counted. In part (ii) only the section of r pwhich corresponds to s describing the region of aR, on the imaginary axis and its indentations is used. The remainder of rP is contained, by hypothesis, in the circles D,, D2,. . . ,Om.Therefore the conditions of part (ii) of the theorem are satisfied if and only if the conditions in part (i) are satisfied; this establishes part (ii). In part (iii), consider the distinct eigenvalues , A l , AZ, . . . ,A, of G(m), the theorem hypothesis guarantees that they are all distinct from -1. Since &s) converges uniformly to &(m) when lsl+m, it is possible to find a p > 0 and circles D,, D2,. . . , Dm with centres A,, A*, . . . , A, which do qot contain the -1 point, such that I s ( 2 p implies that all eigenvalues of G(s) (including the eigenvalues which correspond to possibly infinite branch points on the imaginary axis) are contained in the union of the circles. Then, in this case, condition (iii) of the theorem is equivalent to conditions (ii); this completes the proof. 0 THEOREM 4.10 The feedback system

with W G X :(respectiuely U E X ~and ) G represented by an element of LA""", is XY-stable (respectively X2-stable) if and only i f : (i) When G satisfies condition c(i), the eigenvalue contour

r, generated

4.

139

LINEAR INPUT-OUTPUT STABILITY THEORY

when s describes ail,, does not encircle or contain the -1 point and none of the possible branch points on the imaginary axis corresponds to an eigenvalue of -1; or (ii) When G satisfies condition c(ii), any contour formed by the concatenation of the eigenvalue paths of G generated when s describes the region of aR, on the imaginary axis, and closed by arcs contained in the union of the circles D,, D,, . . . , D,, does not contain or encircle the -1 point, and in addition none of the possible branch points on the imaginary axis corresponds to an eigenvalue of -1. Proof Identical to the proof of Theorem 4.9 but utilizing Theorem 4.6 instead. 0 THEOREM 4.1 1 The feedback system

Y=G(~-Y)

with G E M ( X T ) (respectively G E M ( X 2 ) ) is X;-stable (respectively X 1 stable) only if the conditions of Theorem 4.10 are satisfied for U E X ; (respectively u E X z ) . Proof

Identical to Theorem 4.9 but utilizing Theorem 4.5 instead.

0

We now consider graphical stability criteria for L;, L ; and L1 feedback systems. Following the discussion made at the beginning of this section and in Section 4.3, we can only consider systems which are represented by transfer functions in K,(O)""". THEOREM 4.12

The feedback system

y=G(u-y), UE L; with G E M ( L ; ) represented by an element of K,(O)""" is L,"-stable if and only if the eigenvalue contour generated when s describes the compact imaginary axis does not contain or encircle the -1 point. Proof Direct consequence of Theorems 4.7 and 4.1 (see also Valenca and Harris, 1980). 0 THEOREM 4.13

The feedback system

Y =G(u-y) with u E L ; (respectively u E L 2 ) and G E LA:""such that converges uniformly as' Is1 + m to a constant matrix, is L;-stable (respectively L2stable) if and only if the eigenvalue contour generated when s describes the compact imaginary axis does not encircle or contain the -1 point.

e(s)

Proof Under the conditions of this theorem, there exists a p > O such

140

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

that inf Jdet(I+&s))l> lslrp

0. Moreover since A(s) = det (I+@s))

is an

element of LA,, SHA(S) is bounded and analytic in the interior of the region flp= {s: Is(Ip, Re(s)2 0 ) and continuous in the whole region. Moreover the properties of the Laplace transform of the elements of LA, show that SHA(S) must have total variation in the region of the imaginary axis - p I w s p . Using the arguments of Proposition 4.4 and Theorem 4.7, the zeros of A(s) in the closed right half s-plane are finite in number and can be calculated as the index function with respect toLhe origin of the loop described by A(s) when s describes an,. Since G ( s ) tends to a constant matrix as Is[ + 03, this loop coincides with the index function of the loop described by A(s) when s describes the imaginary axis. Finally, utilizing the fact that SHA(S) has no zeros in Re(s)rO if and only if the eigenvalue contour r does not encircle or contain the -1 point (Valenca and Harris, 1980), we complete the proof.

4.5

NOTES

Rosenbrock (1969, 1974), in investigating the stability of linear multivariable feedback systems, attempted a reduction of the stability problem to a series of simultaneous scalar problems whereby classical frequency domain design techniques could be utilized. In this Inverse Nyquist Array (INA) technique the aim was to diminish loop interaction by achieving diagonal dominance of the rational matrix transfer function of the open loop system by elementary matrix operations through a cascade compensator. This approach was employed by other workers whose similar aim was to reduce a multivariable feedback control problem to a series of single loop problems by iteration algorithms such as the sequential return difference approach of Mayne (1973). A more fundamental and general approach to the stability of multivariable feedback systems is through a generalization of the scalar control design techniques of poles, zeros, root-loci, and Nyquist and Bode diagrams. An initial attempt to generalize the Nyquist criterion to multivariable systems was made by Bohn and Kasvard (1963); however, complex variables based proofs of the Nyquist criterion followed with increasing generality by Barman and Katzenelson (1974), Postlethwaite and MacFarlane (1979), Desoer and Wang (1980), and Valenca and Harris (1980). In parallel with these recent generalizations of the Nyquist stability criterion complementary generalizations of root loci or characteristic frequency loci techniques based upon algebraic function theory were developed by Postlethwaite (1977), MacFarlane et al. (1977) and Postlethwaite and MacFarlane (1979). These natural and fundamental approaches to the multivariable feedback problem have been accompanied by an increasing collection of specifically multivariable systems approaches such as the dyadic method of Owens (1978), matrix transfer factorization of Sain (1973, frequency domain compensation, decoupling

4. LINEAR INPUT-OUTPUT

STABILITY THEORY

141

and pole placement of Wolovich (1974), and the related state space approach of Davison (1976) which highlights the importance of system insensitivity or robustness to parameter variations. An outstanding source of original and historically important frequency domain control design methods for multivariable systems is MacFarlane (1979).

REFERENCES Bachman, G. and Narici, L. (1966). “Functional Analysis”. Academic Press, New York and London. Barman, J. F. and Katzenelson, N. (1974). “A generalized Nyquist type stability criterion for multivariable feedback systems.” Int. J. Control 20, 593-622. Bohn, E. V. and Kasvand, T. (1963). “Use of matrix transformations and system eigenvalues in the design of linear multivariable control systems.” Proc. IEE 110, 989-997. Callier, F. M. and Desoer, C. A. (1973). “Necessary and sufficient conditions for stability of n-input n-output convolution feedback systems with a finite number of unstable poles.” IEEE Trans. AC-18, 295-298. Callier, F. M. and Desoer, C. A. (1976). “Open loop unstable convolution feed-back systems, with dynamic feedback.” Automatica 12, 507-518. Callier, F. M. and Desoer, C. A. (1978). “An algebra of transfer functions for distributed linear time invariant systems”. IEEE Trans. Circuit and Systems 25, 651-662. Davison, E. J. (1976). “The robust decentralized control of a general servomechanism problem”. IEEE Trans. AC-21, 14-24. DeCarlo, R. and Saeks, R. (1977). “The Encirclement condition: an approach using Algebraic Topology”. Int. J. Control 26, 279-287. Desoer, C. A. and Wang, V. T. (1980). “On the generalized Nyquist stability criterion”. IEEE Trans. AC-25, 187-196. Hille, E. (1962). “Analytic Function Theory” Vol 2. Ginn & Co, Aylesbury. Hille, E. and Phillips, R. S. (1957). “Functional analysis and semi-groups.’’ Amer. Math. SOC. Colloquium Pub. 31. Hewitt, E. and Stromberg, K. (1965). “Real and Abstract Analysis”, Graduate Series in Mathematics. Springer Verlag, Berlin. MacFarlane, A. G. J. (1979). “Frequency Response Methods in Control Systems”. IEEE Press, J. Wiley, New York. MacFarlane, A. G. J., Kouvaritakis, B., and Edmunds, J. M. (1977). “Complex variable methods for multivariable feedback systems analysis and design”, Alternatives for Linear Multivariable Control. National Eng. Consortium, Chicago, USA, 189-228. MacFarlane, A. G. J. and Postlethwaite, I. (1977). “Characteristic frequency functions and characteristic gain functions.” Int. J. Control 26, 265-278. MacFarlane, A. G. J. and Postlethwaite, I. (1977). “The generalized Nyquist stability criterion and multivariable root loci.” Int. J. Control 25, 81-127.

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MacFarlane, A. G. J. and Postlethwaite, I. (1978). “Extended Principle of the Argument.” Znt. J . Control 27, 49-55. Mayne, D. Q. (1973). “The design of linear multivariable systems.” Awtomatica 9,201-207. Owens, D. H. (1978). “Feedback and Multivariable systems”. Peter Peregrinus, London. Postlethwaite, I. (1977). “The asymptotic behaviour, the angles of departure and the angles of approach of the characteristic frequency loci.” Znt. J. Control 25, 677-695. Postlethwaite, I. and MacFarlane, A. G. J. (1979). “A Complex Variable Approach to the Analysis of Linear Multivariable Feedback Systems”, Lecture Notes in Control and Information Sciences, 12, Springer Verlag, Berlin. Rosenbrock, H. H. (1969). “Design of multivariable control systems using the inverse Nyquist array.” Proc. ZEE. 116, 1929-1936. Rosenbrock, H. H. (1974). “Computer Aided Control System Design.” Academic Press, London and New York. Sain, M. K. (1975). “A free modular algorithm for minimal design of linear multivariable systems.” Proc. 6th IFAC World Congress, part lB, 9.1.1-9.1.7. Sandberg, I. W. (1966). “On generalizations and extensions of the Popov criterion”. ZEEE Trans. CT-13,(l), 117-118. Valenca, J. M. E. and Harris, C. J. (1980) “Nyquist criterion for input/output stability of multivariable systems.” Znt. J. Control 31, 917-935. Willems, J. C. (1971) “The Analysis of Feedback Systems”, Research Monograph 62. MIT Press, Cambridge, Mass. Wolovich, W. A. (1974). “Linear Multivariable Systems”. Springer, New York. Zames, G. (1966). “On the input-output stability of time varying nonlinear feedback systems”. Parts I and 11. ZEEE Trans. AC-11,228-238, 456-476.

Chapter Five

Extended Space Theory in the Study of System Operators 5.1 INTRODUCllON The concept of input-output stability is based on the existence of an operator representing the system which maps a Banach space of inputs into a Banach space of outputs. In Chapter 3 a representation theory for stable operators was developed; however, for control system studies in considering the stability of feedback systems the open loop operator is frequently unstable or at least oscillatory. Function spaces that grow without bound are not contained in Banach spaces, and some mathematical description of unstable operators is necessary if feedback stability is to be interpreted from open loop system descriptions. This is achieved by establishing the stability problem in an extended space which contains well-behaved as well as asymptotically unbounded functions or those which oscillate; essentially these extended spaces are developed as an extension of an associated normed linear space (say, &) and which contain the unextended space (say, 15,) as subsets. The generalized extended space therefore contains all functions which are integrable or summable over finite intervals which yet may diverge asymptotically or oscillate. In this chapter we consider in detail extended spaces and operators in extended spaces, not only because of their importance in determining conditions for the stability of feedback systems but also because of their significance in general mathematical systems theory. In Section 5.2 a close relationship between extended spaces and locally convex spaces equipped with a projective limit type of topology is established. In particular, for practical frequency domain stability criteria, the extended Lz, space (whose topology is given by a system of positive semi-norms) can be identified with the projective limit of L,[O, TI; in addition the extended space Lze is shown to be complete. Supporting material for this section can be found in Section 1.3. The relationship between a unique fixed

143

144

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

point of an operator and the concept of generalized causality is developed in Section 5.3, and it is shown that strong causality effectively requires the presence of an infinitestimal delay in the operator. To ensure the existence and uniqueness of solution to mathematical models of dynamical systems, it is shown in Section 5.4 that it is necessary to invoke a continuity condition (such as Lipschitz continuity) on the system operators. One of the most powerful mechanisms for developing frequency domain stability is that of passivity; in Section 5.5, passivity (or positivity for causal operators) and the concepts of extended Hilbert spaces are developed and related to the invertibility of operators and well-posedness of feedback systems. It is demonstrated that incremental passivity for operators in feedback systems ensures not only the boundedness of solution but also its uniqueness, existence and continuous dependence as well as the invertibility of the return difference operator. The passivity theorem is shown to be quite stringent in its operator requirements for closed loop stability; however, by introducing a multiplier into the feedback loop it is shown in Section 5.6 that a variety of transformed dynamical systems are input-output equivalent. Additionally the factorization of such multipliers into causal and non-causal components allows a &-stability condition to be expressed in terms of the composition of the multiplier and the loop operators. This technique is used effectively in Chapter 7 to generate the multivariable Popov and off-axis circle stability criteria. Finally, the concept of conicity or equivalently sectoricity, which is closely related to passivity notions, for causal operators is introduced in Section 5.7 as a prelude to developing circle-type stability criteria for non-linear multivariable feedback systems in Chapters 6 and 7.

5.2 FUNDAMENTALRESULTS

Consider a linear space E equipped with a system of finite semi-norms SN = {pa(.)} satisfying: sn(i) pa(x) 5 p b ( x ) whenever 6 > a for all x E E sn(ii) For each non-zero x E E there exists at least one a E r such that

P a b ) # 0.

where a is in a given totally ordered set r. Condition sn(ii) is sufficient to guarantee that the system of semi-norms defines a topology, k f d , in E (Chapter 1, Theorem 1.4), and with such a topology, E [ M d ]is a locally convex space. Example 5.1 Consider the set of all measurable functions f(t) which are

5 . EXTENDED

145

SPACE THEORY

square integrable in any bounded interval [0, TI, i.e. joTf(t)2 d t < m

V T>O

If our totally ordered set is the positive real axis R , T > 0 a semi-norm can be defined by

= [0, m),

for any

It is obvious that this system of semi-norms satisfy both conditions sn(i) and sn(ii). Every semi-norm p a ( . ) can define a relationship of equivalence 52, in E by x = y(mod 52,) if p , ( x - y ) = 0 (5.2) If N, is the null space of p a ( . ) , given by

N, = {x E E :p , ( x ) = 0)

(5.3) then clearly N, is a linear closed subset of E and the quotient space E/N, is a normed locally convex space (Theorem 1.6). Using E, to represent the quotient space E/N, and A, to represent the canonical mapping from E into E,, then ~ I A , x ~ I = p , ( x and ) llXll=p,(x) for an arbitrary x in Using Theorem 1.11 it is now possible to define the equivalence class i. linear continuous onto maps Aab:Eb + E,, b > a as Aab

=&A:-’’

With such maps the space E[Md] is topologically isomorphic with the topological projective limit lim Aab(Eb)with Eb equipped with the norm t

topology. As we shall see, due to this isomorphism, the space E[Md] inherits some of the properties of the quotient space E,. Consider the subspace Eo of E formed with all x E E for which the net { p , ( x ) } is bounded; because the net is non-decreasing it must have a limit. Define a positive function in Eo P b ) = lim P , ( X ) which clearly satisfies p ( x ) 2 p , ( x ) , V a E r. The space E [ k f d ]is said to be the extended version of E o .

PROPOSITION 5.1 The space Eo is a normed locally convex space equipped with the norm (\XI(= p ( x ) = lim p , ( x ) . Moreover, the norm topology of Eo is a stronger than the topology induced in Eo by the topology of E.

146

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Proof EOis obviously linear and p(-) a semi-norm. Moreover, if x E E, is such that p ( x ) = 0, then- p , ( x ) = 0, for all a E r, which implies that x = 0; thus llxll= p ( x ) defines a finite norm. The norm topology of E, is stronger 0 than the induced topology as p ( x ) ? p , ( x ) for all a E r .

Example 5.2: Extended Lp spaces, L, Consider the Banach spaces Lp[O,TI, (1 Ip 100) consisting of all Lz, functions (see Section 1.5) with support contained in [0, TI. Also consider the space of all measurable functions f : R , + R for which PTfE&[O, TI for any TrO (the truncation operator PT is defined in Example 1.10); this space is represented by L,. The space L, is clearly linear and contains Lp. Moreover since {IIPTfll}is not necessarily a bounded net, the space L, is not finitely normed. The positive functions define a family of semi-norms which clearly satisfy conditions sn(i) and sn(ii) of this section. Thus L, can be equipped with a topology defined by these semi-norms. The subspace of L, formed by all feL, for which {h(f)} is a bounded net coincides precisely with L p ; thus L, is the extended version of Lp. Consider now Theorem 1.10; replacing in this theorem F by L,, E, by Lp[O, TI and Paby PT, and noticing that

by definition of L,, then we can say that L, can be identified with the topological projective limit of the spaces Lp[O,TI.

L, = lim PTP(T”&[O, T] c From the same result we conclude that L, can be identified with the topological projective limit of the spaces L I N T , where NT is the null space of p T ( * ) , this is not surprising due to the isomorphism between L,/NT and Lp[O, TI which can easily be established by following the arguments of Example 1.10. Of particular importance in the sequel is the case when p = 2 and the resultant stability criterion can be interpreted in the frequency domain (see Chapter 6). One of the most important properties that E[Md]can inherit from the spaces E, is completeness.

PROPOSITION 5.2 If every E, is complete then E[Md]is also complete. Proof

Kothe (1969), Section 19.10.

0

5.

147

EXTENDED SPACE THEORY

The above proposition and the completeness of every L,[O, TI space gives: COROLLARY 5.1 The extended space L, is complete.

5.3 OPERATORS IN EXTENDED SPACES In the study of feedback systems through input-output operators we are primarily interested in the mapping of operators from the extended space E [ M d ]into itself. In the sequel we will assume that r is the half line R , = [0, m).

Definition 5.1 Consider a system of operators {G,}, where G, maps E, into itself such that for all 6 > a GaAab

(5.4)

=AabGb

for A a b :Eb + E, as defined in Proposition 5.2. Such a system is said to satisfy the generalized causality condition. If in addition for each 5 > 0 there exists a 6 > 0 such that for all a E r and for all x E E, there exists b > a + 6 such that for all y, z E AhT’’x

IIGbY - GbZllEb

5 IIY

(5.5)

- zlIEb

then the system of operators is said to satisfy the strong generalized causality condition.

PROPOSITION 5.3 If {G,} is a system of operators satisfying the generalized causality condition, then a unique operator G mapping E [ M J into itself can be defined such that A,Gx = G,A,x Proof

for all

x EE

and

aEr

Let E denote the topological Cartesian product n E , and let a

B

denote the projective limit of the spaces E,, which (as we saw in Section 5.2) can be identified with E. For any x E E let (x,) be the corresponding vector in B. A vector can be defined in @ through A y a = Gaxa

The generalized causality condition now implies, that for any b > a, A a b Y b = A a b G b X b = G,A&b

= G,X, = y ,

Thus f is an element of B and, consequently, of E. An operator G :E + E can now be defined as G :( x , ) ~ ( G , x , ) ;thus by definition,

A,Gx

= Gax, = G,A,x

148

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

In this study we consider only operators which satisfy the generalized causality condition. In the extended space Lze, the generalized causality condition is equivalent to the classical definition of causality [J. C. Willems, 1969, 19713 or non-anticipatory operator G, which satisfies

PTG = PTGPT ,

(5.6)

and the operators {PTG}form the system of operators {Ga}of Proposition 5.3. We note that the truncation operator PT is a continuous linear operator on G and its operator norm is at most 1 (that is PT is the resolution of identity on LZe).

PROPOSITION 5.4 Let operators G, F : LZe+ LZebe causal: if the (i) Domain of G A D ( G )= D(F), then G + F is causal. (ii) Range of G 4 R ( G )c D ( F ) , then G F is causal.

Proof ( i ) Let x, y E D ( G ) and PTx =PTy, then PTGx=PTGy and PTFx = PTFy. SO that PTGx + PTFx = PTGy + PTFy 3 PT(G + F ) x= PT(G + F ) y hence G + F is causal. (ii) Now, since Gx, Gy E R ( G )c D(F) then PTF(Gx) = (PTFG)x = PTF(Gy)= (P,FG)y, hence FG is causal. 0 An equivalent definition to the above is the mapping G :LZe+ LZeis LZewith PTul = PTuZ,the said to be causal if for all T r O and all u l , equality P T G u ~= PTGuz holds. The operator G is said to be anti-causal if the operator (Z-P,)G commutes with (Z-PT) on Lze.It is clear that a memoryless operator is one which is both causal and anti-causal, that is, G is an instantaneous non-anticipatory operator. The concept of causality allows us to make a general definition of a dynamical system as one which has a causal or non-anticipatory mapping G : U + Y,that is, future inputs do not effect past outputs.

Example 5.3 Consider the mapping G : Lz(-m, by the convolution integral

03)

+ Lz(-a, m)

defined

m

(Gu)(t)=j

-m

g(t-T)U(T)

d7

for g E L1(-w, m), u E Lz(-m, 00). The mapping G is causal if and only if g ( t ) = O for r < O . The proof of the following proposition is obvious from the above definitions:

PROPOSITION 5.5 Zf {G,} is a system of operators satisfying the generalized

5.

EXTENDED SPACE THEORY

149

causality condition and G :E + E is defined by this system, then (i) G is continuous if every G, is continuous. (ii) G is onto if and only if every G, is onto. (iii) G is one-to-one if every G, is one-to-one. ( i v ) Two operators G and H are distinct if and only if for some a E r, Gaf Ha.

A more important result relates the fixed points of G with the fixed points of {G,}:

PROPOSITION 5.6 If G is defined by the system of operators {G,} satisfying the generalized causality condition then x E E is a fixed point of G if and only if A,x is a fixed point of G, for every a E r.

Proof If x = (x,) is a fixed point of g, then x, = G,x, and x, is a fixed point of G,. Conversely if every x, is a fixed point of G,, then x = (x,) = (G,x,) = Gx and 5 is a fixed point of G. 0

PROPOSIT~ON 5.7 If G is defined by a system of operators {G,} satisfying the strong generalized causality condition, i f every E, is complete, and if for some c, G, has a fixed point x,, then G has a fixed point x E E which satisfies A,x = x,. Furthermore, if x, is the unique fixed point of G, in E,, then x is the unique fixed point of G in E. Proof For every a < c G,A,x, = A,G,x, = A,,x, ; therefore x, as defined by A,x, is a fixed point of G, . We now seek possible fixed points of Gb for b > c. Using Proposition 5.6, if a fixed point x E E of G exists and satisfies A,x = x,, then Abx is a fixed point of Gb and is contained in Alix,. Furthermore, if x, is a unique fixed point of G, in E,, then all possible fixed points iE E of G satisfy A@E A ; d X c Therefore every possible fixed point of Gbmust lie in A,-dx,. Since &, is linear and continuous, the space ALdx, is a closed linear subspace of Ebr and since Eb is complete, the space A,-bx, is also complete. Let y be arbitrary in A;:xc, then &b(GhY)=

Gc(f%bY)=Gcxc= X c

Therefore GbyEA,-bx,, and so Gb maps AL,,’x, into itself. Consider now some real number 0 < .$<1 and the strong causality condition; then there exists a 6 > 0 and a b > c + 6 such that IIGby-GbZI1s.$IIY-XII

150

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

for all y , z E A,-dx,. Therefore Gb is a contraction mapping the complete space A,-dx, into itsdf, and by the contraction mapping theorem, Theorem 1.19 (Section 1.4), a unique fixed point xb of Gb exists in A;;x,. Moreover, if x, is the unique fixed point of G, in E,, then xb is the unique fixed point of Gb in Eb, as A i i x , contains all possible fixed points of Gb.Replacing c by b and repeating the above argument we can now show that for all a < b, x, = Adxb is a fixed point of G, in E, ; moreover x, is unique if x, is unique, otherwise there would be more than one fixed point of Gb.Finally, repeating the above arguments we can find a fixed point of Tb,&> b + S > c +2S (note that 6 is only a function of 5 and not of c or x,). And so by induction we can define a sequence {b,} in r, with bo = c, b, = b such that b n +I

> bn + 6

and such that a fixed point xbmE Ebn of Gbnexists satisfying &,.xbn = x,. Moreover, if x, is unique, Xbn is also unique. But since {b,} is an unbounded sequence, it is possible to define uniformly an x E E such that Abnx= xbm,n = 1,2, . . . , and &x = x,. The point x is a fixed point of G in E. Finally, if x, is unique, x is also unique, as every xb, is unique. 0

Example 5.4 In the extended space Lzc the strong causality condition can be expressed as: if PJ = Ply, then for any 5 > 0 there exists a dt > 0 such that IIpt+dt(Gx

-

5 IIpt+dt(x - Y)ll

So that the strong causality condition effectively requires the presence of an infinitesimal delay in the operator G.

PROPOSITTON 5.8 If {G,} is a system of linear continuous maps satisfying the generalized causality condition and G is the operator defined by such a system, then: (i) a(G,) c a(Gb)

V a
@

(ii) a ( G ) = U a a(G,)

-_-

where a(.) represents the spectrum of an operator.

Proof For any y € E b , there exists a unique solution, x € E b of the equation Ax = Gbx + y , for A arbitrary in the resolvent set of Gb.Projecting this equation E,, with a < b A A ~= x A a b G b X + Ady

= G,(Adx)

+ Ady

Since y is arbitrary in Eb, Aabyis arbitrary in E,, and therefore A also

5.

151

EXTENDED SPACE THEORY

belongs to the resolvent set of G,. Hence p ( G b ) c p ( G a and ) taking . it can be proven that for complements gives a ( G a ) c a ( G b )Identically any a E r, a ( G a )c a ( G ) and U a ( G a )c a ( G ) (5.7) a

Conversely suppose that A is an arbitrary point in the intersection

n p(Ga).Then for an arbitrary y = ( y , ) E E, the equation a

Ax = G,x + ya

(5.8)

has a unique solution for each a E r. Let x'") be such a solution then, r\AabX(b)=A

+ A a b Y b = Ga (AabX"))

a b Gb X C b )

Ya

Since the solution of (5.8) is unique, we have xCa)= AabX'b'

v a < b,

and consequently the vector ( x ' ~ )E) E, and A belongs to the resolvent set

of G. Hence

n p(Ga)c p(G), and taking complements gives a ( G )c n a

a

a ( G a ) ;taken with the above results, (5.8), condition (ii) of Proposition (5.4) follows. 0 Example 5.5 Suppose that G is a causal mapping of LZeinto itself; then the operators GT defined by GT

= PTG

are maps of LJO, TI into itself and the system of operators {GT} satisfies the generalized causality condition. Moreover from Proposition 5.8 we conclude

PROPOS~ION 5.9 If G is a causal operator of LZeinto itself defined by the system of operators {GT}, and ( c - G)-' exists as an operator from Lz, into itself, then (i) For all T 2 0, ( c - GT)-l exists as an operator from Lz[O,TI into itself. (ii) (c-G)-l is causal. (iii) PT(c - G)-' = ( c - GT)-',i.e. ( c - G)-' is defined by the system of operators { ( c- GT)-'}. 5.4

WELL POSEDNESS AND FEEDBACK SYSTEMS

To ensure the existence and uniqueness of solution to mathematical models of dynamical systems it is usual t o invoke some continuity

152

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

condition upon system operators. Although linear extended spaces are not normed, it is possible to define the concepts of continuity and boundedness. For simplicity we consider L ; spaces in the sequel. If G is a causal mapping of the extended space L> into itself, then G is said to be locally Lipschitz continuous on L k if PTGPT=PTG=GT is Lipschitz continuous on L: for all TrO, i.e.

Furthermore if the operator G is unbiased (i.e. G(0) = 0) and for all T r 0 PTGPT is bounded on LF, then the operator G is said to be locally bounded. Since GT is Lipschitz continuous for each TrO, then the Lipschitz norm (or gain) IlGTll rllGTtllwhenever T > T'. That is, llGTll= llPTGPTll is a monotone non-decreasing function of T. If the nondecreasing sequence of real numbers {llGTll}is bounded, then a limit exists such that llGll = llGTll (5.9)

In addition if G is a causal unbiased operator on L:, (or L k ) and is Lipschitz continuous, then its Lipschitz gains or norms on L$ and L: are equivalent. Consider the feedback system y=F(u-Gy),

UEL;

(5.10)

which has solution

y

= (I+FG)-'FU

(5.11)

provided that the operator (I+ FG)-' exists. Assume the following conditions on the system operators: (i) the feedforward operator F is causal, F : L E - + L $ , unbiased and locally Lipschitz continuous on L E ; (ii) the feedback operator G is causal, G : L$ + LE and locally Lipschitz continuous on L k . In addition, either G or F is strongly causal (see example 5.4); this is equivalent to the generalized causality condition defined through the projective limit of normed spaces.

Definition 5.2 The feedback system model y = F(u -Gy) is said to be physically well posed if the system feedback error e = u - Gy and output solutions y are unique and Lipschitz continuous upon the inputs, and are causally dependent o n finite intervals. As we shall see, the above definition is essentially satisfied if the operator (I+ FG) is invertible on LE X L k and the inverse (I+ FG)-' is

5.

EXTENDED SPACE THEORY

153

causal and Lipschitz continuous on LLX Lk. The condition that F or G is strongly causal is motivated by the fact that all physical systems exhibit some form of delay in the system loop. The above conditions ensure that the feedback system (5.10) is a suitable approximation to a physical system. To see this, the map A ( y )= F(u - Gy),

A :LE + L k

(5.12)

is strongly causal, since F or G is strongly causal. If 6 > O is the delay element present in operator A ( . ) ,then for all T < S , the null function y ( t ) = 0 is a fixed point of PTA in L;[O, TI. Since the space L;[O, TI is complete, we can use Proposition 5.7 to establish that there exists a fixed point y EL; of A ( . ) .Also since the null function is the unique fixed point of PTA in L;[O, TI for T<6, then the proof of Proposition 5.7 shows that y is a unique fixed point of A ( - ) . In conclusion the conditions (i) and (ii) imposed upon the system operator, F, G, are sufficient to guarantee that the feedback system (5.10) has a unique solution in L k . 5.5

PASSIVITY IN FEEDBACK SYSTEMS

A special Banach space, the so-called Hilbert space, which possesses the additional structure of an inner product enables us to generate positive definite quadratic functions that can be associated with the energy of physical systems. The Hilbert space approach is also the basis of the orthogonal projection theorem which has played such an important role in control and estimation (Luenberger, 1969) and in the stability of ordinary differential equations via Lyapunov’s direct method (Curtain and Pritchard, 1977). In the following we introduce the concepts of extended Hilbert spaces, passivity, positivity and their interrelationships with previous concepts such as the invertibility of operators ( Z + F G ) and wellposedness of feedback systems (see Section 5.4).

Definition 5.3: Inner Products An inner product (-;) that maps the .) : HS x HS + F ) is linear space HS over the complex or real field F such that ((a,

(i> ( a x + PY, 2) = d x , z)+

(ii) (x, Y ) = ( Y , x)* (iii) (x, x ) > O e x # 0

P ( Y , 2) (5.13)

for all a, P E F ;x, Y E HS. Utilizing the inner product, the function llxll= (x, x)”* is a norm on HS

154

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

and the normed linear space (HS, 11.11) if complete is a Hilbert space. This norm and the inner product have the following properties:

(9 (Y, y ) = O + Y = o (ii) I(x, y ) JI ( I x I ( . llyll (Schwartz inequality)

(iii) IIx + y1I2+(1x

-

y1I2 = ~ ( ( I x I ( ~+llylr)

(5.14)

(parallelogram law)

for all x, y E H. The adjoint G* of a bounded operator G on a Hilbert space is defined by V x, y E H. (Gx, y ) = (x, G * y )

Example 5.6 Consider the space HS = L2[[0,TI, R"] of n-vector valued functions x :[0, TI x R " + R", with the inner product

I,

T

( x , Y) =

X'(t)Y 0)dt,

this space is a Hilbert space. If G(t,T ) is an ( n X n) continuous valued matrix on [0, TI x [0, TI, then the adjoint of the operator G x :HS + HS,

( G x ) ( t )= is

6'

G(t,T ) X ( T ) dT

T

( G * y ) ( t )=

t

G ' ( T ,t)Y(T) dT.

In addition, the operator G is said to be self-adjoint on the Hilbert space HS if G* = G ; an obvious self-adjoint operator is the truncation operator PT.

Definition 5.4: Positivity A self-adjoint operator G ( G E (HS, 11-11)) positive if (Gx,x)>O

is

(5.15)

for all X E H S

and strictly positive if

(Gx, x ) 2 p I(x(I2 for p > 0. Self-adjoint and positive operators occur frequently in control theory and have the following properties: (i) If G E (HS, 1).11) is self-adjoint and x E H, then IlGl(= sup I(Gx, x)l. IIxII= 1

Also for any G E (HS, 11.11) GG* is clearly self-adjoint and 1(GG*(I= (IG1I2=sup (Gx, G x ) Ilxll= 1

5.

155

EXTENDED SPACE THEORY

(ii) There exist constants a,p such that a ~(x(~~I(Gx, x ) s p llx112 for all

x E HS,

If G is a positive operator, then a 2 0 , but if G is strictly positive, then a > 0 and G has a bounded linear inverse G-' such that

p-' 11~11~1(G-'x, x

) ~ a - '(lxI/* for all

x E HS,

also G has a unique strictly positive square root G"2. Note that if the operator G can be decomposed such that G = N*N, then G is self-adjoint and (Gx, x) =I(Nx1I2 so that G is also a positive operator. If in addition N*N = NN*, then I(Nx(l=(IN*xl(and the bounded linear operator N o n the Hilbert space HS is said to be normal. Selfadjoint operators are generalizations of real symmetrical matrices, and positive operators are generalizations of positive definite matrices.

THEOREM 5.2 Let G :HS + HS be a linear operator such that inf ( G x , x ) = y > O ,

IlxII= 1 XEHS

then G is bounded, one to one, and onto, and the inverse G-' is bounded with 11G-'116 y - ' . Proof The proof in Phillips (1959) requires the assumption that G is bounded; we can see that this is unnecessary because if G is bounded, then for each x E HS with I(x(I= 1, ((Gx, x)l SllGxll. llxllI11G11, that is, -11G11s(Gx, x ) I l l G l l and so if inf (Gx, x) > --c4

IlxII= 1 XEHS

which of course includes the condition of the theorem. Conversely if inf (Gx, x) > -03, then for A > 0 sufficiently large inf ((AZ

IIxII= 1

+ G)x, x) <0

xeHS

Then by the above theorem (AZ+G) is bounded and consequently G is bounded. 0

COROLLARY 5.2 Let G :HS + HS be a linear operator; then G is bounded if

inf (Gx, x) > --oo

IlxIl= 1 XEHS

156

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

If {PT :T ER} is the resolution of identity on L2[0,TI, then a continuous operator G :L2[0,TI-. L2[0,T ] is such that

(Gx - GY, PT(X - Y ) ) 2 Y IIPT(X- Y>11* for all x , y E L,[O, T] and T ER. If > 0, then Theorem 5.2 and the above condition are sufficient to establish that the operator G-' is causal. All of the above well-known results have been stated for bounded linear operators mapping a Hilbert space into a Hilbert space, that is, for causal operators on normed linear spaces with an inner product norm. To deal with non-causal or anticipative operators we now introduce the extended Hilbert space, HS,, which is the space of all x E F such that for all T E R , , X, = PTx has finite norm ll~,ll= (xT, xT) <m. The scalar products (., in HS, have the following properties: a)

(i) From Schwartz inequality (5.14) (XT,YT)=(XY T ,> = ( xY, T ) = ( xY,

VX, Y EHS, and T E R (5.16)

)T<~

(ii) ~ ~is x monotonically , ~ ~ increasing with T and lim

T-

IIx,~~= llxll

Vx

E

(5.17)

HS,

Having introduced the concept of positive operators on Hilbert spaces, we now introduce the notion of passivity for operators on HS, (Sandberg, 1964; Zames, 1966):

Definition 5.5 (i) An operator G :HS, -+ HS, is said to be passive if and only if (Gx, x ) 2~a, for some constant a and for all x E HS,, T ER. In addition G is said to be strictly passive if and only if (Gx, x ) ~ ? P )l~,11~+ a, for P > 0 and for all x E HS,, T ER. (ii) An operator G :HS, -.HS, is incrementally passive if and only if (Gx - Gy, x - Y ) 2~ 0, for all x, y E HS,, T ER. In addition G is strictly incrementally passive if and only if there exists a S > 0 such that (GX - Gy, x - y)T 2 8 (Ix- YII:

v s, y E Hs,,

T E R.

The concept of passivity has been used to establish the boundedness or stability of open loop unstable feedback systems of the form (5.10); however, no assumptions concerning the existence, uniqueness and continuity of solutions can be made except in the case of incremental passivity. (This is directly analogous to the small gain and incremental gain theorems. See Chapter 6.)

5.

EXTENDED SPACE THEORY

157

Note that if the operator G :HS, += HS, is linear then G is passive if and only if it is incrementally passive, and G is strictly passive if and only if it is strictly incrementally passive (for G linear p may be taken as zero). Let G E Ynx",the space of all linear time-invariant and bounded operators of Ly into itself, and define the convolution operator

G ( t- T ) U ( T ) dT

(Gu)(t)=

V u E L;,[O, TI

Since G is causal, then (Gu) :Lye -+ Lj;, and (4Gu), =

( h - 9

GUT)

LEMMA 5.1 The operator ( G ) is passive i f and only i f { ~ ( j o ) + ~ ( j ois ) * } positive definite for all w E R, and ( G ) is strictly passive if and only i f for some 6 > 0 the least eigenvalue of {Gis 2 6 for all o 2 0. Proof Now, (Gu, u ) = (u, Gu)* = (u, Gu) V u E L; since the functions are real valued. Therefore

(Gu, u ) = $ [ ( u ,Gu)+(Gu,u ) ] m

[(G @ u)(t)Tu(t)+u(t),(G 8 u(t)l dt

where Parseval's theorem has been used. But G(jo)+ &jo)* is Hermitian, and so

ii(jo)*[G(jo) + G(jo)*]ii(jw)= h [ G ( j w )+ G(jo)*]ii(jw)*ii(jo)

And so

Example 5.7

Consider the feedback system y

2

N G ( u - y),

u E L,"

where G is an operator of L," onto L; such that m

(Gx)(t)=S G ( ~ - T ) x (dT T) -m

with G ( t ) an n x n matrix on R whose elements are on L;[o,w)n~ ~ ( o , w )

158

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

and N : R" + R " is a continuous function which is such that N ( 0 )= 0, -[IN(x)ll<cu IIxI(,

and

V x E R", cu > 0

We note that if the non-linear operator N is incrementally passive (incrementally strictly passive), then this implies that yN 2 0 (yN > 0). Let

k

=

4 lIxII= inf

1

x T ( G ( j w )+ &jw)*)x

xcR"

Clearly k 2 0 if [ & j w ) + G ( j w ) *is ] positive semi-definite (and then by Lemma 5.1 operator G is passive). Similarly let = sup A[

A

W€R

6(jw )]

where A[A] is the square root of the largest eigenvalue of A*A. And so

llGll2 A.

It now follows that YG + Y N

lIBI122k + y N A 2 > 0

If yG > O , then operator G is invertible. And since

( I + N G ) = (G-' + N ) G then But since yG-l+N

2 yN

(I+NG)-'

= G-'(G-'

+ N)-'.

+ yG I(G(I-22 k + y N X 2> 0

it follows that the operator ( I + N G ) is invertible and Lipschitz continuous. Under these conditions if the operators N, G are causal, then (1 + NG)-' is also causal and the feedback system y = N G ( u - y) has a unique solution y = (I+ NG)-'NGu E L," and the mapping u H y is Lipschitz. (Note: we have not required that N be Lipschitz.) If instead, the operators N , G are only defined on L;,, then the condition k +yNA2>0 is sufficient to establish that for any u E L,",there exists a unique y EL,", that satisfies the feedback equation. The notions of passivity and positivity are connected through causal and non-causal operators. Suppose that G : HS + HS is a causal operator, then for any u E HS, and T ER, u,E HS.

(u,G u ) T = ( h ,( G u ) T ) = ( + , ( G + ) T ) = ( u T , Gu,)

(5.18)

5.

159

EXTENDED SPACE THEORY

By (strict) passivity

(u,Gu)T = PIIuTII' + a, P > O

(P 20)

(5.19)

But if addition u E HS, then let T+m in (5.19) and noting that

Iim

T-

(5.19) becomes (u, Gu) 2

IlxTll = lIxlI

Pbl12+ a, P > 0 ( P 2 0 )

(5.20)

which is the condition for (strict) positivity. This demonstrates that for causal operators G :HS + HS, G is (strictly) positive if and only if G is (strictly) passive; in other words positivity and passivity are equivalent for causal operators. If it is not difficult to show that if G :HS += HS is linear and its adjoint G* is causal then G is anticausal (or anticipative).

THEOREM 5.3 Consider the feedback system y = F(u - Gy), where F : HS, + HS,, G :HS, + HS, . Assume that for u E HS, then there exists a solution y E HS,. Suppose that (i)

--

a

IIFxl\T

(ii) (x, Fx>T

llXllT+ P I y IlXll$+ P2

(5.21) (5.22)

-

(iii) (Gx, x ) 5~6 11G~11$+03

(5.23)

Under these conditions, if (6 + y ) > 0, then u E H, y, u - G y = e, Fe, E HS, the map u Fe is passive, and for PI = P2= P3= 0 then the maps u e, u H y, u Fe,EL;. Proof

Now,

(U - GY,F(u - GY))T= (u,F(u - G

Y ) ~

(5.24)

Substituting (5.21-23) into the above gives

(Iu But since then

- Gyll$+6 IlGyll$5 a

llUllT I \ u

-GyllT+PI

llullT-@3-P2

(5.25)

IIU - GYll$~llull$+llGYIIZ-2 llull . IlGYll

( Y + 6 ) I I G ~ I K ~ ~ I I G Y I I{II~IIT T (y+;))+~i

(

;)I

s 2 ~ ~ G Y ~ \ UIlT T { ~ ~y f -

II~II+-P~-P~

I I ~ I I T - ~

.Ilull$+P1

llullT-P2-P3 (5.26)

160

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

where

+P1 l l u r l l - P 2 - P 3 ) 9

are monotonically increasing with T if ( y + 6 ) > O and both tend to finite constants a, 6 as T + m, since u E HS. Hence for all T ER, I l G y [ \ , ~ a ( T ) + l a ( T ) ~ +6(T)211'2

(5.27)

sa+(a2+b2)"',

Therefore G y E HS, u - G y E HS, and by (5.26) F(u - G y ) = y E HS. Now (u , Fe)T=(e, Fe)T+(Gy,

' Y IIeII++P2+6

Y>T

IIGYII++P~,

But G y = u - e and I(Gy(lZ,~llull~+lle11+. Therefore (4Fe),

2 (Y

+ 6) Ilell++ P 2 + P 3 2 P 2 + P 3

and the mapping u ~ F ise passive. For PI = P2+ P3 = 0 the coefficientsa ( T ) ,6(") in (5.26) are homogeneous polynomials in llull of order one and two respectively, so that IIGyII 5 A max (Ilull) and thence for u E L;, e = u - G y E L,". Finally the finite gain condition (5.21) gives llFellT s a IlellT, hence y = F ( e ) E L ; . We note that when 6 = 0, the above theorem holds if F is strictly passive with 0 finite gain and G is passive. Suppose now that the operators F and G are respectively strictly incrementally passive and incrementally passive, i.e. (FX

o< y1 <@J

-FZ, x - 2 ) ~2 y1 IIX - Z11$,

(5.28) (5.29)

(Gx-Gz, x - z ) T ~ O ,

for all x, z E HS,,, T ER. Also the operators F and G are unbiased (F(O)=O, G(O)=O) and F has finite incremental gain such that IIFx - FzllT 5 a1 Ib -

ZIT

(5.30)

for all x, z E HS,, T ER. The above finite gain condition on F implies that the operator F is causal, and the map F: HS, + HS, is uniformly Lipschitz continuous in PTHSe with Lipschitz constant a I . Setting z = 0 in (5.28-30) and P1= P2 = P3 = 0, y = yl, and a = a1 in (5.21-23), we conclude from Theorem 5.3 that e, y E HS; utilizing (5.25) in the proof of Theorem 5.3 for pairs of inputs ( u l , u2) and associated

5.

EXTENDED SPACE THEORY

161

feedback errors ( e l ,e l ) , Equations (5.28-30)gives

Y1 llel

-e2112

aI

lIul

- U211T 111. - e211T

(5.31)

This implies that if u 1= u 2 , then e l = e 2 . y1 = y2 and the solution y to the feedback system is unique. Clearly from (5.31)since yl. a,>O then Ilel - e2115 4 T )

where a ( T )= y ; ' a 1 llul - u2((<m increases monotonically with T. Therefore as u 1+ u 2 , llul - u211 0 and Jle,- e,l( + 0, so that mapping u H e is uniformly continuous on HS. We have briefly shown that incremental passivity conditions on the operators F, G ensure not only boundedness of solution but also its uniqueness, existence and continuous dependence. But since the solution to y = F(u - Gy) is y = (I+FG)-'Fu, then the above incremental passivity conditions and finite gain (or equivalently the mapping PTF: PTHSe+ PTHS, is continuous) condition upon operators G and F are also sufficient for the invertibility of the operator ( I + F G ) . In the special case when the operator ( I + F G ) is a linear causal map from HS, t o itself, then if ( I + F G ) is strictly passive, the mapping (I+ FG)-' : HS, + HS, is causal and strictly passive. Also, from Section 5.4, the feedback system y = F(u - Gy) is well posed. --j

5.6

THE THEORY OF MULTIPLIERS?

Lemma 5.1 establishes the passivity condition for a convolution operator G u : L Z + L 2 , u e L 2 as Re{&jw)}?O for all o or equivalently arg{G(jw)}E (-~/2, ~/2). For those feedback systems (see Theorem 5.3) whose loop operators d o not satisfy the stringent passivity condition the loop transformation Theorem 6.1 can be employed by introducing a multiplier Q ( s ) (Willems, 1971) such that Q @ G is passive or arg{Q(s)G(s)}e (-~/2, ~/2). A well-known example of a multiplier in stability theory is the Popov stability criterion Re[(l + q j w ) G ( j w ) ]2 0 for all o,whereby the multiplier Q(s) = 1 + q s has been used. A variety of stability criteria with simple graphical interpretations for non-linear feedback systems can be derived if the multiplier can be factorized into the product of a causal and an anticausal component. Let B be a Banach algebra with identity I, let P + :B + B be a linear projection on B (that is P + P , = P + ) . Define, P - A I - P , , assume that llP+ll, IIP-11 I1, and let B, ,B- be sub-algebras of B. t Multipliers here refer to loop operators and should not be confused with multipliers used in the representation theory of Chapter 3.

162

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Let { Q k } , {Q:} and {Q!} be sequences in B, B+ and B- respectively and define the associated power series

Q=z+

m

C

k=l

Qkrk,

(5.32)

m

Q + = z + C &rk,

(5.33)

k=l

Q-=z+

c m

(5.34)

Q!rk,

k=l

which converge for some r,rlrl ( r e C ' , r o E R + ) . Assuming that these series are related by Q = Q+Q-, substituting these series and equating equal powers of r gives Q: =

and so and

Q:

=P+@,

@=Qc+Q!!+

Q:+ Q 1

61= P - Q l

c QtQY-",

n-1

k=l

so

n = 2 , 3 , ...

n-I

(5.35)

(5.36)

k=l n-I

(5.37)

for n = 2,3, . . . . Thus the 6:, Q! are successively uniquely determined by the Q k k Consider now the equations, for I r l I l and some Z E B such that llZll< 1

6,

N+(Q+)= I + rP+(ZQ+)

(5.38)

Q-

N-(Q-) = I + rP-(Q-Z)

(5.39)

For any Q, Q'E B, the above definitions give - Q:>>ll~ r IIZII .11Q+- Q 1:1 Ilrp+
llrP-(Q- - Q:>(I 5 r ( I z ( I .11Q- - 6 111

and since r llZll< 1, then from the above inequalities the contraction mapping Theorem 1.19 shows that (5.38) and (5.39) have unique solutions 6+,Q-E B for all Irl I1; moreover by the functional iteration

5.

EXTENDED SPACE THEORY

163

method the convergent series (5.33), (5.34) give the solutions Q+E B-031, Q - E B - ~ ~ I . From (5.38) and (5.39)

( I - rZ>Q+= I - rP-(ZQ+)

Q-(I

Therefore

- r z ) = I - rP+(Q-Z)

(5.40)

(5.41)

the same upper bound holds for llrP+(Q-Z)(l. And so I-rP-(ZQ+), I-rP+(Q-Z), and ( I - r Z ) E B are all invertible in B for lr1<611211-'. Furthermore these inverses are representable by unique, convergent power series of the form (5.32),(5.33) in the sphere of radius 411Z11-'. Consequently (5.40) and (5.41) yield

( I -r ~ ) - = ' Q+(I - rP-(ZQ+))-' = ( I - rP+(Q-Z))-'QFrom which

6,= ( I - rP+(Q-Z))-' 6-= ( I - rP-(ZQ+))-' ( I - rZ)-'

= Q+Q-

But since (I- rZ)-', Q + , 6- are each representable by a convergent ~ can ~ - set l r = 1 and so power series for Irl< ~ ~ Zwe

Z -Z

= (Q+Q-)-' 9 Q-Q+= 0.

We have now established the multiplier factorization theorem:

THEOREM 5.4 Let Q = I - Z then

QE

B. If there exists a Z EB such that )IZI(<1 and

(i) Q = I- Z = Q-Q+ with Q-, Q+E B where Q-, Q, are inuertible in B (ii) Q-,(Q-)-*EICBB-

Q+,(Q+)-'E I 03 B + where 003B+ denotes the subspace of all elements in B of the form a I + Q,

a E R, Q E B , etc.

We note that the Banach algebra need not be commutative; this is particularly significant for the multivariable case. Consider the Banach algebra B = LAYx" (see Section 3.4) which represents the sub-algebra of

164

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

M ( L ; ) formed with all elements which are defined by the convolution with an ( n x n ) matrix whose entries are elements of LA,. If (LA:""), consists of all causal impulse responses in LA:"" and (LA:"")-consists of all anticausal impulse responses in LA:"", it is not difficult to establish the corollary to Theorem 5.4: COROLLARY 5.4 Let Q E LA:"". If there exists a Z ELA:"" such that I(Z(lLA+ < 1 and Q = I - Z , then there exists Q-, Q, E LA:"" that admit the factorization Q=Q-Q, where Q-, Q, are invertible in LA:"" and Q-, 01' E (LAY"")_, Q,, Q;' E (LAY""),. Consider the special case of a Banach algebra of the class of all linear continuous maps from the Hilbert space HS into itself. Consider the feedback system

y=Fe,

e=u-Gy

(5.42)

where u E HS, e E HS, and F, G are causal maps from HS to HS. We now establish the important result that the three feedback systems in Fig. 5.1 are input-output equivalent and L,-stability for one system implies L2-stability of the others. Assume that a non-causal map Q:HS+HS is factored such that Q = Q-Q, ; Q-Q,: HS + HS where Q-', Q;', QZ' are well defined

I

FIG.5.1 Input-output equivalence through multipliers

5.

165

EXTENDED SPACE THEORY

mappings of HS into HS each with finite gain (that is r ( K ' ) < m etc), Qis linear and hence Q? and (Q?)-' are well defined. Let x and z E HS and such that x = QTz; from the properties of the multiplier IIxI(I -y(Q?)( 1 ~ 1 1 and I(z(IIy[(Q?)-'] 1Ix(I. Consider some operator H : HS + HS, then

(z/QHz) = (z/Q-Q+Hz) = (Q?z/Q+Hz) = (x/Q+H(Q?)-'x) That is the strict positivity condition on operator QH

(z/QHz)Zal11z11*

V Z E HS, a l r O

(x/Q+H(QT)-'x) 2 a2Ilxlr,

V x E HS,

is equivalent to

(5.43)

a22 0

By similar reasoning the positivity condition

(z/HQ-'z) 2 0

V z E HS

on operator HQ is equivalent to

(x/Q?HQ;'x) 2 0 V x EHS (5.44) Given that the feedback system operators F, G :HS -+ HS are causal, then by definition, QF and GQ are non-causal and the loop operators of system I11 (Fig. 5.1), Q+F(QT)-' and QTGQ;' are causal and by inequalities (5.43) and (5.44) respectively strictly positive and positive. The passivity Theorem 5.3 can be utilized to establish L,-stability since r[Q+F(Q?)-'I 5 rEQ- Q+F(Q~)-'Q_*lr(Q~')r[(Q?)-'l 5

rEQ~'lr[QFlrE(Q~)-'l <m

provided that y[ QF] <03.

-

THEOREM 5.5 The feedback system y = F(u - Gy), F, G :HS HS and causal, u E HS, is L2-stable i f a non-causal operator Q = Q-Q+ :HS + HS exists such that (i) Q+, Q;', QT, (QT)-' are causal operarors. (ii) QF is strictly positive and r[QF]<w (iii) GQ-' is positive

5.7

SECTORICITY

We now introduce the concept of conicity or equivalently sectoricity for causal unbiased operators G :L; +L,"

166

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Definition 5.6 (i) G is said to be inside (respectively outside) the sector S{a, b } for scalars a, b E R, b r a if and only if ((G - d)*, (G - b 1 ) X ~5) 0

(respectively,

20 )

holds for all t 2 0 and x E L;. Or equivalently (ii) G is said to be interior (respectively, exterior) conic with parameters c and r if and only if - cr)xTII

(respectively,

r lbTll 5r

\I+\()

holds for all t 2 0 , x E L; and c = $ ( a+ b ) , r = $(b -a). The equivalent definitions of conic relations and sectoricity degenerate into the Definition 5.4 of positive operators if a = 0 and b + m. A natural extension of the above definitions to multivariable sectors is possible if diagonal matrices A = diag (a,), B = diag (bi);a,, biE R, bi 2 ui for i = 1 , 2 , . . . , n, are introduced:

Definition 5.7 (i) G is said to be inside (respectively, outside) the sector S{A, B} if and only if ((GX- A X ) ,(GX- Bx)T)5 0

(respectively,

5 0)

or equivalently (ii) G is said to be interior (respectively, exterior) conic if and only if II(Gx-%A + B ) x ) T II~IHB-A)XTII (respectively, ~ l l $ <-A)*\() B

holds for all x E L; and TrO.

Definition 5.8 G is said to be inside (respectively outside) the complex scalar sector S{a, b} for a, b~ C', a - a * = b - b * R e ( a ) s R e ( b ) if and only if

<(G(jo)- a ~ ) x(G(jo) , -b ~ ) x ) o (respectively,

2 0)

for all XEL; and W E R , . Definitions 5.6 and 5.7 encompass non-linear operators, whereas Definition 5.8 is purely a linear definition. These definitions can easily be extended to frequency-dependent sectors:

Definition 5.6a G is said to be inside the frequency-dependent sector S { a ( j o ) ,b(jw)}, for a ( j w ) , b ( j o ) E C', a ( j o ) - a ( j o ) *= b(jw)-b(jo)*,

5.

167

EXTENDED SPACE THEORY

R e ( a ( j w ) ) s R e ( b ( j w ) )V, w if and only if ((G(jw)-a(jw)I)x,(6(jw)-b(jw))Zx)sO

V X E L ; and ~ E R .

Definition 5.7a G is said to be inside the frequency-dependent sector S { A ( j w ) , B ( j o ) } , for A(jw)=diag{a,(jw)}, B ( j w ) =diag{bi(jo)}; ai(jw), bi(jw) E C', ai(jw)- q ( j w ) * = bi(jw) - bi(jw)*; Re(a,(jw))5 Re(bi(jw)) V w E R, i = 1, 2, . . . , n, if and only if ( ( G ( j w ) - A ( j w ) ) x ,( G ( j w ) - B ( j w ) ) x ) s O

VXEL;,

WE

R.

These frequency-dependent sector definitions can be equivalently restated in terms of the Euclidean norm, and definitions for outside sectors can be obtained by reversing the inequality sign and simultaneously ensuring that the Nyquist plot of G(s) does not encircle the point (c,O) for c = $ ( a ( j w )+ b(jw)), V w E R.

LEMMA 5.2 If G1 is inside the sector S { a , , b,} and G2 is inside sector S{a2, b2} for b l , b 2 > 0 , then (i) (ii) (iiiA) (iib)

aG, is inside S { a a l ,ab,}. (G, + G2) is inside S { a , + a , , b , + b2} Zf al>O then G;' is inside S{b;', a;'} If al
Proof (i) (((YG1X)T-aaIXT9(aG,x)T-ab,xT) = ( Y ~ ( ( G , X ) ~ - (G1X)T~ ~ X T b,XT)sO

(5.45)

for any a. An important special case is when a = -1, then -G1 is inside S { - a , , - b J if GI is inside S{al, b J . (ii) Now, by the triangle inequality and Definition 5.6 for G I and G2

II[(Gi+G2)Xl~-t(bi+b2+a i + a d x ~ I I sll(GixT)-f(ai + bi)xTll+ II(GZx)T-t(a2+

b2)XTll

s & b i -ad II+II+t(bz-az) I I ~ T I I (5.46) = $(bi + b2 - ai - a2) ~ T I I (iii) Suppose that y = G;'x and x = Gly, a, # 0 and b, > 0. Then

((GylX)T-b;'xT, ( G ; l X ) ~ - a ; ' X ~ ) = ( y ~b;'(GiY)T, Yy -~;'(GIY)T) = (aibi)-'((GiY)T-aiY, (G1U)T-biYT) (5.47) The sign of the above inner product depends upon a, S O , and so part 0 (iii) of Lemma 5.2 follows.

168

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Note that throughout Lemma 5.2 the words inside and outside are fully interchangable, since by Definition 5.6 only a sign change is necessary. The following lemma enables inside and outside sectors to be manipulated in a mixed manner.

LEMMA 5.3 If operator G1 is outside the sector S{cl - rl ,c1+ rl} and G2 is inside the sector S{c2- r2, c2+ r2}, rl > r2 ; then ( G ,+ G2) is outside the sector S{cl - rl +c2+ r2, c1+ rl +c2- r2}. Proof

Now, G1 satisfies the inequality ll(G1- C l I ) X l l 2

rl IIXII,

v x E L,"

and G2 satisfies the inequality - ll(G2- c21>xll2 r2 IIxII,

V x E L,"

which taken together imply that ll(G1- clI)xll- ll(G2- c21)xll2 (rl - r2) Ilxll, and from the triangle inequality ll{G1+ G2- (cl + c21>xll2 (rl - r2)Ilxll,

V x E L," V x E L,"

(5.48)

0

A similar result can be readily established for frequency-dependent sectors by employing a generalization of Lemma 1 of Zames (1966).

LEMMA 5.4 Suppose that operators G1, G 2 €LA?"" have transfer function matrices Gl(jo),G2(io).If (i> II(Gl(io)- cl(loO)I)x(iw)II2 IIrl(jo)x(jm)II, 0 E R, v x(io) and the NYquist plot of G , ( j o ) does not encircle (cl, 0), and (ii) 11(G2(i0)- c2(io>I>x(io)ll~Ilr2(io)x(b)ll, w E R, v x(iw); then and Il(Gl(b)+ G2&) - (cl + c2))x(jw)ll2 Il(rl(io) - r2(io))x(iw)ll ( G ,+ G2) is outside the sector S{cl - rl + c2+ r2, c1+ rl + c2- r2} for r2(jo)> rl(jw), o E R, and for all x(jo). Conditions (i) and (ii) above are respectively equivalent to: G2 is outside sector S{cl - rl. c1+ rl} and G2 is inside sector S{c2- r2, c2+ r2}. A somewhat simpler result for convolution operators can be obtained for symmetric frequency-independent sectors:

LEMMA 5.5 Let G a g C3 u ; u E L,",g = { g i i } € LA?"",gi E LA+. Also let n

pij

1 iZi(iw)gij(jo)

k=l

Then if either of the following two conditions are satisfied, G is inside the

5.

169

EXTENDED SPACE THEORY

sector S{-r, r}: (i) Condition

A:

sup {lpiil+

W€R+

(ii) Condition

lpiil]sr2;

2 ,. .., n

i

=~

i

= 1 , 2 , .. . , n

B:

sup {lpiil+

W€R+

Proof

f

j=1

C Ipiil}sr2;

j=l

G is inside the sector S{-r, r } if and only if IIGxIls r Ilxll,

V x E L2n

A sufficient condition for the above can be written in terms of the induced norm as

But by Parseval’s theorem to be inside S{-r, r } is

lllGlll= 1116111,and so a sufficient condition for G IIIG(jw)III

(5.49)

r

However, it is well known (Desoer and Vidyasagar, 1975) that the induced L; norm IllG(jw)lll= max A i ( 6 * ( j w ) 6 ( j w ) } ” 2 ,

w E R,

and so Condition (5.49) can be rewritten as max ~ ~ ( G * ( j w ) G ( j wr2) ) ~

i.weR+

(5.50)

The lemma is now established by identifying the matrix G * ( j o ) G ( j w ) with P = {pii} and applying Gershgorin’s theorem for the column and row 0 conditions. In order to establish input-output stability as well as the existence and uniqueness of solutions to feedback systems it is necessary to introduce incremental sectors.

Definition 5.9 G is said to be incrementally inside (respectively, incrementally outside) the sector S{a, b}, if b 2 a and ((Gx - GY)T - a(x - Y)T,(Tx- GY), - b(x - y)T) 5 0 (respectively, 2 0) for all x, y E L; and T 2 O .

170

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Consider for example operator G :L,, + L2,, where G is incrementally inside S{a, 6 ) . Then G satisfies the Lipschitz condition b ( x - y ) r G(x)- G(y) 2 Q ( X - y). So that operator G not only lies in a sector in the plane, its slope has upper and lower bounds. 5.8

NOTES

The concept of extended spaces was first suggested by Sandberg (1964, 1965) and Zames (1966). who used it to develop input-output stability criteria for time varying scalar non-linear feedback systems. Treatment of extended spaces and causality can be found in many texts on control and network theory (Desoer and Vidyasagar, 1975; Dolezal, 1979; Mees, 1981). The supporting mathematics for Section 5.2 o n topological projective limits and generalized extended spaces can be found in Section 1.3; the origins of this material can be found in Kothe (1969) and Valenca (1978). An extensive treatment of the concept of causality and well-posedness of feedback systems can be found in the excellent text of Willems (197 1) and in Willems (1969); the generalization to strong generalized causality can be found in Valenca (1978). The various passivity theorems have been developed by many research workers, including Sandberg (1964), Zames (1966), Cho and Narendra (1968) and Harris and Husband (1981). The associated concept of multipliers can be found in OShea (1966), Narendra and Taylor (1973), Willems (1971), and Desoer and Vidyasagar (1975). The development of sectoricity and conicity was originally due to Zames (1966) and extended to frequency-dependent multivariable sectors by Husband and Kouvaritakis (198 1).

REFERENCES Cho, Y. and Narendra, K. S. (1968). “An off-axis circle criterion for the stability of feedback systems with a monotonic nonlinearity.” ZEEE Trans AC-13(4), 4 13-4 16. Curtain, R. F. and Pritchard, A. J. (1977). “Functional Analysis in Modern Applied Mathematics”. Academic Press, London and New York. Desoer, C. A. and Vidyasagar, M. (1975). “Feedback Systems: Input-Output Properties”. Academic Press, London and New York. Dolezal, V. (1979). “Monotone Operators and Applications in Control and Network Theory”. Elsevier, Amsterdam. Harris, C. J. and Husband, R. K. (1981). “An off-axis multivariable circle stability criterion”, R o c , ZEE 128, 5, 215-218. Husband, R. K. and Kouvaritakis, B. (1981). “The use of sectors in the derivation of stability criteria for nonlinear systems”. OUEL Rept No 1359/81, Oxford University. Kothe, G. (1969). “Topological Vector Spaces”, Band 159. Springer Verlag, Berlin.

5.

EXTENDED SPACE THEORY

171

Luenberger, D. G. (1969). “Optimization by Vector Space Methods”. Wiley, New York. Mees, A. I. (1981). “Dynamics of Feedback Systems”. Wiley, Chichester. Narendra, R. S. and Taylor, J. H. (1973). “Frequency Domain Criteria for Absolute Stability”. Academic Press, New York and London. Phillips, R. S. (1959). “Dissipative operators and hyperbolic systems of partial differential equations.” Trans. Amer. Math. SOC.90, 193-254, Sandberg, I. W. (1964). “On the L2 boundedness of solutions of nonlinear functional equations.” Bell Syst. Tech. J. 43, 1581-1599. Sandberg, I. W. (1965). “Some results on the theory of physical systems governed by nonlinear functional equations.” Bell Syst. Tech. J. 44, 439453. Valenca, J. M. E. (1978). “Stability of Multivariable Systems”, D Phil Thesis. Oxford University. Willems, J. C. (1969). “Stability, instability, invertibility and causality.” SZAM J. Control 7, 645-671. Willems, J. C. (1971). “The Analysis of Feedback Systems”, Research Monograph 62. MIT Press, Cambridge, Mass. Zames, G. (1966). “On the input-output stability of time varying nonlinear feedback systems. Part I: Conditions derived using- concepts of loop gain conicity and positivity.” ZEEE Trans AC-11, (No. 2), 228-238. Zames, G. (1966). “On the input-output stability of time varying nonlinear feedback systems. Part 11: Conditions involving circles in the frequency plane and sector nonlinearities.” ZEEE Trans AC-11, (No. 3), 465-475.

Chapter Six

Stability of Nonlinear Multivariable SystemsCircle Criteria 6.1 INTRODUCTION In order to specify a dynamical system mathematically a suitable space of input and output functions needs to be defined. Frequently it is necessary to represent either dynamical systems which are open loop unstable or function spaces which grow without bound with increasing time. Such functions are clearly not contained in Banach spaces, which are usually used to model dynamical systems (such as &,-spaces), but by establishing the problem in extended spaces (see Chapter 5 ) which contain both well-behaved as well as asymptotically unbounded functions, rigorous system analysis can be made on finite time intervals. Consider a finite interval Ic R = (-03, w) and a normed space V with norm (1.11. Let Z be a linear function space mapping I into V, that is, Z: I + V. For each T EI, let PTbe the truncation operator (or projection operator) such that

T>O and f e z . By introducing a norm of the linear space Z.

11-11 on Z, we define a normed linear subspace E

Definition 6.1 E is the space of all functions f E Z such that (9 Ilf1l-J (5) PTf E E for all T EI (iii) The mapping T ~ l l P ~ fisl lfor every f E E an increasing bounded function of T such that lim IIPTfll. T-m

For example the spaces which consist of all measurable functions 172

6.

173

CIRCLE CRITERIA

g ( t ) : I+ R such that

[Tlg(f)lp dt < m

V T E I, p E [l,m)

satisfy conditions (i)-(iii) of Definition 6.1. Definition 6.2 The extension of the normed linear space E is the extended space E,, which is the space of all ~ E such Z that (i) PJEE for all T>O (ii) If the lim llPTfll exists, then it coincides with T--

llfll

We note that the above definition of the extended version E, of E equipped with a family of semi-norms is consistent with the theory of locally convex spaces with a projective limit type of topology covered in Section 5.2. A mathematical model of an input-output system can be derived through a relationship G on E, or a set of pairs (u, y) of functions in E,, where U E E ,we denote as the input and YEE,as the, output, that is, G : u + y. However, this definition allows the possibility of multivalued relationships in which an input (or output) is paired with several outputs. If G is restricted to a single valued mapping with domain ( G )= E,, then G is an operator that maps E, into itself. Consider an input-output system G with input signals u ( t ) and output signals y ( t ) (which may belong to different spaces) related by Y = G(u)

(6.1)

These functions are assumed to be sufficiently well behaved for their norms to exist, and functions which differ only on sets of measure zero are assumed as equivalent. All such systems will be assumed to causal or non-anticipative, that is YT = (G(uT))T V T EI (6.2) Definition 6.3 An operator G: E, + E, is said t o be input-output stable if there are non-negative constants 6, y such that I((G(u))Tlls Y I l ~ T l l +6, T EI (6.3) for uT E E. In the sequel we concentrate on mappings on L,"spaces for m-input-noutput dynamical systems, that is, G : L Z + L,",; so that the above definition of input-output stability specializes to: Definition 6.4 The mapping G is L,-stable if (i) G ( u ) EL," for u E L;, and

174

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

(ii) there exist non-negative constants y, 6 such that llG(411p5 y

ll4lP+ 6,

for u E Lp"

(6.4)

Effectively this definition of input-output stability requires that the system is Lp-stable if, whenever the input belongs to the unextended space Lp", the resulting output y = G ( u ) belongs to L ; and the norm of the output is less than y times the norm of the input plus a bias 6. The gain of the operator G, assuming that G is unbiased (G(O)=O), is given by y ( G )= inf [ y E R+ :

5y

IIuT(~~

+ 6, Vu, VT1

(6.5)

Clearly, condition (ii) of Definition 6.4 can be alternatively written as y( G) < m. Frequently the bias F is taken as zero for physical operators G and the gain of G is then given by

That is the supremum is taken over all possible input-output pairs and over all truncations. These gains are not distinguished from function norms, and truncated functions are used in the definition, since they are not known a pn'ori to be finite. If p = m, the above concept of L,-stability is called bounded input-output stability, that is, a bounded input produces a bounded output. The above definition of Lp-stability is further weakened by some authors (Valenca and Harris, 1979) by dropping the requirement of part (ii) in Definition 6.4. Our prime concern is in determining the stability of feedback systems with m-inputs and n-outputs defined by dynamical relationships of the form

y = G(u, y),

u E L;,

(6.7)

If G(u,y ) : L ; X L:e+ L$ with G(O)=O is strongly causal with a delay element 6 greater than zero, then Proposition 5.7 shows that there exists a fixed point y E L$ of G(u,y). And since y = 0 is the unique fixed point of PTG in L,"[O,T ] for T < 6 , then the proofs of Propositions 5.3, 5.7 show that for each U E L , "there exists a unique solution EL,"^ of (6.7). The operator G(u, y ) is too general for detailed study or for the derivation of practical stability criteria. A more tractable and representative feedback system can be obtained by decomposing the operator G(u,y) such that y=F(u-Gy),

UEL;

(6.8)

6.

CIRCLE CRITERIA

175

which, if F is a linear operator, has solution

y = (I+FG)-'FK (6.9) provided that the operator (I+FG)-' exists. The feedback system (6.8) is well posed or equivalently a suitable approximation to a physical system with solutions y, e = K - Gy, which are unique and Lipschitz continuous (see Section 5.4) if (i) the feedforward operator F: LPm,+ LFe is causal, unbiased, and Lipschitz continuous on LPm, (ii) the feedback operator G : LFe + LPm, is causal and locally Lipschitz continuous on L ; (iii) either F or G is strongly causal (iv) the operator (I+FG) is invertible on LPm,x Lp"eand the inverse is causal and Lipschitz continuous on LPm,x LFe. These rather severe restrictions on operators F and G can be dropped in the determination of feedback stability. However, the question of existence uniqueness and continuity of solution cannot be ascertained witho u t some constraints upon operators F and G. The feedback system (6.8) can be further structured by factorizing it in such a way that all nonlinearities are represented by a memoryless operator N and the memory of the system is contained in a linear time-invariant operator H, such that (6.10) y = F(u - HN(y)) where N : L;+ L ; is a Lipschitz continuous map with N ( 0 )= 0. Although the nonlinearities are contained in the feedback loop, other system configurations can be transformed into representation (6.10). For example if y = N ( H u-W ) ) ,

(6.1 1)

so that the nonlinearity is contained in the feedforward path. On defining x = F ( K- Hy) then y = N ( x ) and x = F ( K- H N ( x ) )

(6.12)

This is identical in structure to (6.10), so if x E Lp",then y = N ( x )E Lp"and the feedback system (6.11) is Lp"-stable if and only if system (6.12) is L ;-stable. The feedback system (6.10) can be interpreted as a condition for the existence of a unique fixed point in Lp" of the operator G(K,y). The contraction mapping theorem can provide sufficient conditions for the

176

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

existence of such a unique point. If G(.) is a contraction, that is (6.13) then G(-) has a unique fixed point in L;. The Lipschitz norm of G(-)can be bounded by (6.14) Then the contraction condition becomes IlGll

IIFHII."I

<1

(6.15)

so that the feedback system (6.10) is Lystable if F:L,"+ L," and IIFH(1.llAll< 1 . In this case the feedback system y = Su is bounded in its gain by (6.16) A frequency domain interpretation of the loop gain condition (6.15) can be made if p = 2, since if operator FH is a stable linear time-invariant Li-map into itself, it is representable by a matrix transfer function FH(s)E K(0)"x" (see Section 3.2 and the L,-Representation Theorem 3.2), where K(0)nxnis the space of all bounded and holomorphic complex functions-with domain {s : R e ( s )> 0). And by Parseval's theorem for u E L;, F H ( s )E K(0)nx" CT

llFHullZ = (27r-'/--

u * ( ~ o ) ~ * ( j o ) ~ ( ~ od) ou(jo)

For each w E R, the matrix m * ( j w ) e ( j w ) is positive definite, therefore if A [ F x ( j o ) ]is the maximum eigenvalue of this matrix then I(FHI(,= ~

~gA''2[~(i~)l

(6.17)

So that the feedback system (6.10) is L;-stable if F:L,"+ Lg and su A"2[%(jo)]

. llAll< 1

(6.18)

This stability criterion is not particularly useful because (i) it requires the open-loop system to be stable (ii) it has no simple graphical interpretation (unless FH(jw) is a normal matrix) h

6.

CIRCLE CRITERIA

177

(iii) it effectively requires that the open-loop gain be small at all frequencies, which is inconsistent with closed-loop dynamic performance. In the following we develop &-stability criteria that have a simple graphical interpretation and which incorporate open-loop unstable operators. In the remainder of this section we establish the loop transformation theorem which effectively eliminates the above stringent loop gain requirement. Consider the general feedback system

y = F(u - HNY)) where F and H are linear operators that map E," into itself, and N : E,"+ E l is a memoryless nonlinear map. Let u E E : and define an arbitrary operator N,, : E,"+ E," such that the above feedback system can be transformed into (I+ FHN0)y = FU - FH(N - No)(y ) or (6.19) y = ( I + FHNJ'Fu - (I+ FHN,)-'EH( N - Nn)( y) provided that (I+FHNJ' :E : + E,". THEOREM 6.1 : LOOPTRANSFORMATION THEOREM Let linear operators F, H, N,,, (I+FHN,,)-' map E : into E,", then Y E E : is a solution to y = F(u - HN(y)) if and only if it is a solution to (6.19) for u E E:.

The loop gain condition (6.15) can now be rewritten as

(I(I+FHNJ'MII. llN-Noll<1

(6.20)

Clearly by approximating N as closely as possible by a linear timeinvariant matrix gain N,,ER""" the stringent open-loop gain (6.15) has been dramatically reduced. Note that if we set G = (I+F.FfN,,)-'FH then

G-'= (FH-' + N;'

(6.21)

which is of particular significance in deriving graphical stability criteria when E : = L;, and FH is representable by a matrix transfer function. The loop transformation theorem holds when the operator N is nonlinear and when the space E : is replaced by E".

6.2 SMALL GAIN THEOREMS There are essentially three small gain theorems that establish the inputoutput stability of nonlinear feedback systems. The strictest assumes that

178

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

a Lipschitz condition is known for the nonlinearity and makes strong predictions concerning. stability and the existence and uniqueness of solution. The second or intermediate gain theorem assumes only uniform continuity and bounded total gain of the nonlinearity and establishes intermediate results concerning stability and existence of solution but makes no prediction on uniqueness or continuity. The third small gain theorem (the subject of this section) is the least conservative as well as most general; it drops the uniform continuity condition and predicts boundedness of solution-if they are known to exist.

6.2.1

Boundeduess small gain theorem

The boundedness small gain theorem is a very general theorem which gives sufficient conditions for bounded input-output stability. The important questions of existence, uniqueness and continuity of solution are divorced from the stability question and are usually assumed a priori. Consider the feedback system y=F(u-Ny),

UEE,

(6.22)

Let operators F, H :E, + E,, and system error e = u - Ny E E,. Then from (6.22) u=e+Ny y = Fe

(6.23)

Suppose that the operators F, G are such that II(Fe)Tll ll(NY)Tll

Il(e)TII+ P F yN ll(Y)Tll+ PN YF

(6.24) (6.25)

For all T EI, e E E,, y E E, ; yF,yN 2 0 and constants pF, pN. Note that it has been assumed a priori that e E E , and u is defined according to the system equations (6.23). This reversal of the usual definition of feedback system signal dependences is made to avoid the question of uniqueness and existence of solution to (6.22). The constants Pi and gains yi are given by

for each i. The constants Pi are included in the problem formulation so that the following small gain theorem applies to nonlinearities with discontinuities and hysteresis. Now (6.26) Y = Fe 3 llYTll= II(Fe)Tll y F lleTll + PF

6.

CIRCLE CRITERIA

And

e = u - NY3 Ile-rll5 l I ~ T l l +II(Ny)TII5 l l ~ T l l +YN l l ~ T l l +P N Combining (6.26) and (6.27) yields IleTll 5 (1 - ~F7/N)-'(lluTll+YNPF + P N )

and

Y F (~ YFYN)-~(IIuTII + YNPF + PN) + PF

ll~Tll5

179 (6.27) (6.28) (6.29)

provided that yF7/N < 1 for all T EI. The gain product yFyN and term yN&+PN can be replaced in inequalities (6.28), (6.29) by -yFN and PFN respectively if the bound is utilized. This substitution of individual operator gains by the loop gain yFN follows from the inequality

YFN5 YFYN

(6.30)

THEOREM 6.2: BOUNDED INPUT-OUTPUT STABILITYIf the feedback system y = F( u - N( y)) has system operators that satisfy inequalities (6.24) and (6.25) for all T E I and IIuII<M then the system is bounded input-ourput stable if the loop gain yFyN is less than unity, and the norms of system error and output satisfy llell5 (1 - YFYN)-~(IIuII + YNPF + P N )

IIYIl

YF

Ilell

+PF

This theorem holds for continuous time or discrete time systems as well as multivariable and infinite dimensional dynamic systems; indeed the operators F, N can be generalized multivalued relations which include hysteresis and saturation effects. An alternative and more useful bounded input-output stability results on utilization of the loop transformation Theorem 6.1. The feedback system (6.22) can be transformed into

y

=( I

+ FN,)-'Fu- (I+ FN,)-'F( N - No)(y )

(6.31)

provided that the inverse operator (I+FN,)-' exists in F, for arbitrary operator No : E, + E,. Now, setting

G = (I+FN,)-'F as a closed-loop "transfer function", then y = GU- G ( N - No)(y)

(6.32)

180

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Supposing that the arbitrary operator NO is such that NO)YTlls Y N llYTll+ P N ' for all TEZ,YEE,,and real constants y, given by

llY115(11G11 . ~

(6.33)

P, thence the bound for y is YN')-'

~ u ~ ~ + ~ N ' ) (* l - ~ ~ G ~ ~

(6.34)

Clearly the condition (6.35) I(G(I= Y N l l ( z + m O ) - ' f l I < is sufficient for bounded input-output stability provided that ( I + nV,,)-' E E. YN'

Example 6.1 Consider the spaces E = L ; , p ~ [ l , m ] for the feedback system (6.22) with UEL!. F is a linear time-invariant operator and is represented by the convoiution

(Fe)(t)=F@e=

I'

F ( t - T ) e ( t ) dt

(6.36)

We note that the norm on the convolution operator is bounded by (6.37) And in particular when p = 2 (6.38) where fi(s) is the Laplace transform matrix of operator F, which we assume for simplicity to be analytic in Re(s)2 0. The subsystem N is a memoryless, but not necessarily time-invariant, operator from R""" into itself and satisfies the bound condition (6.33). Under these conditions we have THEOREM 6.3 The feedback system y = F ( u - N ( y ) ) is L;-stable ( u E L ; + e , Y E L ; ) f o r p = [ l , w ] if YN'

mO)-lqlp

<

for N,ER""" and inf

Re(s)>O

ldet ( I + fi(s)N,)I > 0

The last condition follows from the Nyquist condition (see Sections 3.2, 4.2) and ensures that the closed-loop operator (Z+FN,)-'F is stable. The various nonlinear multivariable stability results derived in this

6.

CIRCLE CRITERIA

181

chapter are of little practical value unless they can be interpreted graphically in a similar manner to the Nyquist criterion for linear multivariable systems. The merit of such an approach is that it can easily be integrated with the vast numbers of computer-aided design methods that have been established (Rosenbrock, 1974) for the control system design of linear multivariable systems. To obtain a frequency domain interpretation of Theorem 6.3, we set p = 2 and consider those F : L , ” , + L , ” , which are linear time-invariant operators characterized by a matrix transfer function P(s) such that each element ij(s) is a meromorphic function in R e ( s )> O and satisfies the condition lim fii(s)= 0. Under these conditions Theorem bI+-

6.3 established that if U E L , ”then e, EL; if inf

Re(s)>O

ldet ( I + N ( , f i ( s ) ) l > O for N o € R”””

and

for all y N 8 €R. Where A [ Z ] denotes the largest eigenvalue of Z*Z. These conditions establish the conditions for boundedness of solutions y, e to the feedback system y = F ( u - N(y)) and assume a priori that such solutions exist. Questions concerning existence, uniqueness, and continuity of solution can only be established if the operator N is Lipschitz continuous. A graphical interpretation of the’akove L,” stability result can only be made if the operator 6 = ( I + h J 1 F is normal, since this means tcat at each frequency w the eigenspaces of the matrix transfer function G ( j w ) form a mutually orthogonal set spanning the whole space C”. And the norm for a normal operator G :L ; + L ; is given by lIGIl,=suP

max

lhl

o s R A~u[&(iw)l

where w [ 6 ] represents the set of eigenvalues of G. Otherwise the above stability criterion has to be numerically evaluated via the norm

Suppose that the nonlinearity N is such that a ( z T z ) ( z T ( N z ) ( t ) ~ P ( z T z ) ,RV+‘ ,fV~Z E R”

where a and p are scalars. If we select No = $(a+ @ ) I , the norm condition

182

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

is minimized by the gain yN,=&3-a). Then if the linear operator F: Lye+ L;, is normal, the condition (6.39) simplifies to $(p-a)sup

max

w c R AedF(iw)l

I

(1 +AcA) l k l

~

(6.41)

where c =;(a + p). The n-eigenvalues Ai(jo)( i = 1,2, . . . ,-n) can be defined such that they take on the values of the eigenvalues of F ( j o ) at each o ;in terms of these eigenvalues the condition (6.41) can be rewritten as Ic+A,(w)-'[>$(p-a)

(6.42)

for i = 1,2, . . . , n and for all w E R. That is, to satisfy stability condition (6.42) every loci A,(@)-' must be exterior to the circle C[-p, -a] in the complex plane with centre at -c = -;(a + p ) and radius &(c- a)= @ -a), I (Fig. 6.1). Since A,(o)-' takes on the values of the eigenvalues of lF(jo)l-' for each o,there is a close connection between this criterion and the Inverse Nyquist Array (Rosenbrock, 1974). An alternative graphical interpretation of inequalities (6.41), (6.42) can be obtained by transforming the inequalities through the conformal mapping w = z-'. Under such a mapping a circle C[a,b ] becomes a circle C[a-', b-'I, provided that both a and b are non-zero. Moreover, if the open interval (a, b) does not include the origin, then the interior of C[a,b ] is transformed into the interior of the circle C[a-', b-'1; if, however, b > 0 > a, the interior of C[a,b ] is mapped into the exterior of C[a-', b-'I. In making the transform o = z-' the position of the origin relative to the limits (a,p ) of the nonlinearity is clearly crucial. Thence if a and p have the same sign and are non-zero, conditions (6.41), (6.42)

-P

FIcj. 6.1 Graphical interpretation of inverse boundedness theorem

6.

Itlp> a>o

183

CIRCLE CRITERIA

111)

p>O>a

FIG.6.2 Graphical interpretation of boundedness theorem

are satisfied if the Nyquist plots of Ai(w) are exterior to the circle C[-a-', - p - ' ] (Fig. 6.2(i)). But when p > O > a, condition (6.42) is satisfied when the plots of Ai(w) are interior to the circle C[-p-', -.-'I (Fig. 6.2($). The special cases of a = 0 and p = 0 can be obtained from the above by taking the limits a + 0, p + 0, respectively. Theorem 6.3 required that, in addition to conditions (6.41) or (6.42) being satisfied by the above circle criterion, the following are simultaneously satisfied: inf ldet (I+$(s)NJl> 0 or (I+h " ) - ' P €K(o)flxfl (6.43) In the following, two separate cases of the forward transfer function matrix are considered: an open-loop stable plant, and where fi has poles with positive real part. Consider first the feedback system y = F(u - N(y)) with fi(s) open loop stable: linear condition (6.43) is satisfied if and only if the linear feedback system (6.44) y = F(u-Nay), u E L," is stable. The Nyquist encirclement theorem ensures that system (6.44) is L,"-stable if and only if the plots of the eigenvalues of fi(jw) do not encircle the point -c-' = -2(a + p)-'. Consider now the second case when P(s)has p-poles in the right half s-plane. Then from the Nyquist encirclement theorem (I+h")-'fi~ K(0)"x"when the total number of clockwise encirclements of -c-' by the

184

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

eigenloci, A i ( w ) , of P ( j o ) is ( n X p). We are now able to state the small gain circle theorem: THEOREM 6.4 Consider the feedback system y = F(u - N ( y ) ) for u E L i , F : L;, + L;, a normal operator with matrix transfer function P(s) and N a nonlinear operator such that ~(z'z)'

z'(Nz)(t)SP(z'~)

for all t E R , and z E R", a and p scalars. Then y , e = u - N ( t )E L ; if (a) fi is a stable normal operator and the eigenloci A i ( w ) of F ( j o ) (i) do not encircle or intersect the circle C[-p-', -a-'] for a, p of the same sign and non-zero or (ii) if p >O>a, the plots h i ( w ) lie inside the circle C[-p-', -.-'I or (iii) if p > a = 0, the eigenloci A i ( o ) lie to the right of the abscissa - p - ' , that is Re(A,(w))> -ppl,

or

(b)

V w E R, i = 1 , 2 , . ..,n

P is a

normal operator with p-right half poles, and the eigenloci of -.-'I but encircle it ( n x p ) times in the clockwise direction for a, p of the same sign and non-zero.

f i ( j w ) do not intersect the circle C[-p-',

Note (i) In the above case (b) for p > O > a it is not possible to satisfy conditions (6.42) and (6.43) simultaneously, since the point -c-' is exterior to the circle C[-p-', -.-'I. (ii) When p + a > 0, the nonlinearity N ( y )+ a l y and the critical circle C[-pp', -.-'I converges to the critical point (-ap', 0) in the complex plane and the above theorem becomes a multivariable Nyquist stability criterion. (iii) The restriction that P(s)should be a normal operator is quite severe, but by using the loop transformation Theorem 6.1 to approximate F ( s ) by a normal operator, a modified circle criterion can be readily derived. 6.3 INTERMEDIATE SMALL GAIN THEOREM The small gain Theorem 6.2 predicts boundedness of solution to feedback systems if the solutions are already known to exist, by utilizing the boundedness of system operators. This sector-based theorem holds under very weak system assumptions and is of a very general nature, but it is not easily implemented for practical use. If we now make the additional requirement that the nonlinearity in the loop be uniformly continuous

6.

CIRCLE CRITERIA

185

with bounded norm, we can utilize the Brouwer and Schauder-Tikhonov fixed point theorems 1.16 and 1.17 to establish existence as well as boundedness of solution, although uniqueness and continuity of solution cannot be predicted without the further assumption that the system operators be Lipschitz continuous. The question of existence of solution is crucial in ensuring that the system model equations are well posed, since a physical system must have a response to some stimulus. The main results in this section on the boundedness and existence of solution to feedback systems depend upon fixed point theorems for compact operators. In the following we utilize the fixed point theorems for compact operators to generate bounded response theorems for dynamic feedback systems. Fixed point theorems in linear vector spaces are essential in establishing the existence and uniqueness of solution under certain structural conditions to nonlinear differential or integral equations that represent dynamical systems. Let X be a normed vector space and define a space E on X by

E = { x :R ,

-+ X/llxllE< 03)

where the norm ll.llE is given. If M is a subset of E such that every continuous mapping of M onto itself has at least one fixed point, then M has the fixed point property. The conditions necessary for M c E to have the fixed point property are contained in the Schauder-Tikhonov fixed point theorem 1.17, which shows that for any mapping F: E 4E, with F M c M and compact, F has a fixed point in M. We note that there is no requirement that the vector space E be a Banach space. All that is required for F to have a fixed point is that the operator F is continuous and FM is compact. A general property of compact sets is that a compact subset of M of a normed space E is closed and bounded; the converse of this result is only true when the space E is finite dimensional. A compactness criterion is given in t h e following:

THEOREM 6.5 Let X and Y be normed spaces and F : X + Y a linear operator. Then F is compact if and only if it maps every bounded sequence {x,}E X onto a sequence {Fx,} E Y which has a convergent subsequence. Proof

(Dunford and Schwartz, Part 1, 1957).

0

In the special case of finite dimensional vector spaces the above simplifies

to:

THEOREM 6.6 The operator F is compact if dim ( X )< CQ or if F is bounded and dim ( F ( x ) )< CQ

186

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

An important operator F : E + E in systems theory is the convolution operator (Fx)(t)= f @ x =

I,'

~ ( ~ - T ) x ( Td ) ~

,

tE[O,

T]

(6.45)

where for each r, f(t) is a compact linear operator on X and J&lf(r)I2 d t < m. We note from Theorem 6.6 that F is automatically compact if dim (E)
(6.46)

where F is a linear operator on E and N: E, + E, is a bounded nonlinear continuous operator. The above system can be rewritten via the loop transformation Theorem 6.1 as Y = ( I + ~N,)-'FU-(I+~N")-'~(N-N,)(~)-(I+ 4NJ1(F-4)N(y) (6.47)

where No is an arbitrary time invariant operator on R""" and 4 : E + E is a linear operator which is such that (I++No)-'+ G is a bounded linear map on E. Clearly if G :E + E, then (I+4NJ-I = (I- GN,) is also a bounded linear map on E, and system (6.47) can be rewritten as y = ( I - GN0)Fu - ( I - GN,)( F - 4 ) N (y ) - G(N- No)(y )

(6.48)

Suppose now that the operator No is selected such that IIYTIl+PN, y N > 0 , PN20

(6.49)

- GNO)(F-4)N(yT)II

(6.50)

~~(N-NO)YT~~5~N

and define

K ( + ) = (lIYT\l)-'

as a measure of the error of approximation of operator F by 4. Then on taking norms of transformed feedback system (6.48) and applying the

6. above definition, a solution y satisfies

187

CIRCLE CRITERIA EE

exists to (6.46) and every solution y E E, (6.5 1)

provided that 3".IlGll . (1 - K ( $ ) ) - '

= Y N . (](I+~ N o ) - ' ~ ' K(+))-' II(~- <1

(6.52)

Obviously if F = 4 then K ( 4 )= 0 and the above loop gain condition simplifies to YN

[((I+nvO)-'41

<

(6.53)

but there is now much less flexibility in weakening the loop gain requirement (6.53) through the loss of choice in operator 4. In addition, if F: E, + E,, it may no longer be possible to ensure that (I+FNO)-' F is a Iinear operator on E. Suppose now that a solution y* E E to (6.48) is such that y* = My* where the operator M is given by

M y = ( I - G N J F u - [ ( I - G N O ) ( F - 4 ) N + G ( N - N J ] ( y ) (6.54) is well defined and continuous on E through the assumptions made on operators F, N, No and 4. Clearly every fixed point y* is contained in the ball

B ={Y : I I Y I I ~ ( ~ - K ( ~ ) IIGII)-'(KI-YN GNJFul\+PN)} (6.55) So for any My E B or M B c B and MB compact, then by the Schauder fixed point theorem 1.17, M has a fixed point in B. THEOREM 6.7: BOUNDED RESPONSE THEOREM Let E, be a space on which the composition of a linear convolution operator and a bounded continuous operator is compact. Let G = ( I + +NO)-'4 be a bounded linear map on E for N,, E R""" and 4 :E -+ E. Let N be a continuous nonlinear operator on E, such that l\(N-NO)YTl15YN

llYTll+PN,

Then every solution y* E E to y solution y E E, satisfies llyl15(1-K(4)-YN

vyEE,

7/N>O, P N L o

= F(u - N ( y ) ) for

llG1l)-'(ll(z-

u E E exists and every

GNO)FuII+PN)

provided that (1- K($))-'yN llGll< 1. In practice we select operator NO such that ll(N- NO)II is minimized

188

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

(effectively minimizing y N ) and simultaneously ensure that (I+ +No)-' exists in E. This guarantees the minimum gain condition (6.49) with the least stringent requirement on loop gain (6.52) for stability. Also K ( 4 ) is minimized by approximating the operator F by 4 as closely as possible, while ensuring that G :E + E. This is trivial in the scalar case since we can always set F = 4, but for multivariable systems we select in the following 4 to be a normal operator (although F is rarely normal in practical systems) so that Theorem 6.7 has a simple graphical interpretation.

COROLLARY 6.7 Suppose that E = L : and U E L : ( p = [ l , ~ ) )F, is a linear time-invariant operator defined by the convolution F @ e such that F E L:"" with Laplace transform matrix P(s),and that N : R"""+ R""" is a time-varying memoryless nonlinear operator that satisfies ll(N - N0)yTJJ I y N IlyTII. Then the feedback system y = F(u - N ( y ) ) is L,"-stable if there exist operators N o € R""", 4 such that yN(l -

K(4))-'~ \ ( z + ~ N O ) - ' ~ ~ \ <1

inf ldet (Z+&s)No)I>O

Re(s)>O

and for

K ( 4 )l l ~ \ l =11(I- G N d F - 4)N(y)lJ ( K ( 4 )< 1)

6 = ( I + &NO)-'c$analytic in R e ( s )2 0

A graphical interpretation of this boundedness and existence result for feedback systems can be readily seen if E = L ; and the nonlinear operator is bounded such that a(zTz)( zT(N(z))(t)SP(zTz)

for all t E R,, z E R" and P > a scalars. For the above limits on the nonlinearity the optimum choice for No is i(a+P)Z, and the norm on ( N - No) is minimized by yN = &3 - a).The necessary loop gain condition for stability of (6.46) can be written as SUP max

ocR

where and

A~[&(io)l

I

I

I2 ( 1 + cA( 4 ) A ) ( P ( 4 ) - a ( 4 ) ) <1

(6.56)

4 4 )= ;(a($)+ P(4)) a ( 4 )= tca + P ) - t ( P - a ) ( l - M 4 ) l - I P(4)= tca + PI + f(P - a M - W4))-'

We note that condition (6.56) is identical to (6.41) for the small gain

6.

CIRCLE CRITERIA

189

theorem and therefore has the same graphical interpretations, except that now f ( j w ) replaces &w) and a(4),p ( 4 ) replace a and p respectively. And so that condition (6.56) is satisfied if every loci A;'(w) ( A , ( w )are is exterior to the circle C[-p(+), -a(+)]in the the eigenvalues of &(jo)) complex plane. Equally the conformal mapping w = z-' can be used such that condition (6.56) is satisfied when the eigenloci Ai(w) are exterior to the circle C [ - a ( 4 ) - ' , -p(4)-'] if a(+) and p ( 4 ) are of the same sign. The closed-loop approximation gain condition (6.56) is insufficient to ensure L,"-stability for the feedback system (6.46), since in addition K ( 4 )< 1 and (I+4N0)-'F and ( I + c#dV0)-'4must both exist in K(0)"x". Clearly if the linear forward operator F is stable, then the normal operator 4 is selected from the class of stable linear operators such that K ( $ ) < 1, and in addition (Z+I#&-'E K(0)"x"if and only if the linear feedback system

(6.57) y = 4 ( u - NnY) is L,"-stable. The Nyquist encirclement theorem ensures that (6.57) is stable if the eigenloci of &(jo)do not encircle the point - c ( $ ) - ' = -(+(44)+ If, however, F ( s ) has p-poles in the right half s-plane and admits the decomposition P(s)= F0(s)d ( s ) - ' , where d ( s ) is a monk polynomial in s with all its zeros in the right half s-plane. Then the appro2imation operator f ( s ) is selected such that 4(s) = f o ( s ) d(s)-' where &o(S) is a normal matrix operator of analytic functions in the right half s-plane. So (I+4Nn)-'4 E K(0)"x" and ( I + ~ N J ' F E K(0)"x" when the total ) number of encirclements of the point - c ( 4 ) - ' by the eigenloci of ~ $ ( j w is ( n x p ) . It is not difficult to see that we have not only satisfied all the conditions of the small gain Theorem 6.3 but also established an existence condition for the solutions to (6.46) for a non-normal open-loop linear operator F which is compact.

Pw-'.

THEOREM 6.8 Consider the feedback system y = F(u - N ( y ) ) ;for u E L,", F : L,",+ L;, is a compact operator with matrix transfer function p(s) and feedback nonlinear operator N( y ) which is bounded and continuous such that

a(zTz)szT(Nz)(t)sP(zTz), VtER,, ZER" and a and /3 are scalars. Solutions to this feedback system exist such that Y E L;, e = u - N ( y ) L; ~ if (a) @(s) is a stable open-loop operator, and the eigenloci A i ( j o ) of the normal approximation $(jo)to P ( j w ) (i) do not encircle or intersect the circle C[-p-'(+), -0-'(4)] for a ( 4 ) ,p ( 4 ) the same sign and

190

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

1111 @IQl=-O>al01

FIG.6.3 Circle criterion for non-normal operators

non-zero, (ii) if p ( 4 ) > 0 > a ( 4 ) the plots hi@) lie inside the circle C[-p-'(4), -a-'(+)] or (iii) if p(4)> a ( 4 )= 0 , the plots of Ai(jw) lie to the right of the abscissa -p(4)-' for i = 1,2, . . . , n and for all w€R or

(b)

R(s)

has p-right half poles and admits the decomposition p(s)= where d ( s ) is a monic polynomial with all zeros in do not intersect the circle Re(s)>O, and the Nyquist plots of A($@)) C[-p-'(4), -a-'(4)] but encircle it ( n x p ) times in the clockwise direction for a ( 4 ) ,p(4) of the same sign

po(s)d(s)-'

6.

CIRCLE CRITERIA

191

This theorem is crucially dependent on the error of approximation K ( 4 ) between the compact linear operator F and its normal representation 4 being less than unity. The choice for 4 depends upon the structure of E For example, (i) if F is diagonally dominant, then select 4 = diag (F);(ii) if F is nearly self adjoint, then select 4 = &(F+F*); identically if F = -F*, then select 4 = $ ( F - F * ) ; (iii) if F is block diagonal, then each block can be approximated by a normal operator in the above theorem. For further discussion on the choice of 4 see Section 7.3.

6.4 THE INCREMENTAL GAIN THEOREM The incremental gain theorem has its origins in the contraction mapping theorem and ensures existence, uniqueness, boundedness, and continuity of solution to nonlinear feedback systems. The fact that the spaces on which the operators and inputs are defined are complete normed linear vector spaces (that is Banach spaces) is exploited in the following to establish the existence, uniqueness and continuity of solution as well as stability. Definition 6.5 Consider two Banach spaces B and D with respective D is said to be Lipschitz continuous if norms 11. I ( B , 11. IID. The map G : B a scalar +G > O exists such that

(6.58) IlG(xT)- G(YT)IID5 +G l l ~ T -YTIIB for all x, y E B and T EZ. Obviously a Lipschitz continuous operator is also a continuous mapping. The infimum of all y satisfying (6.58) is called the Lipschitz norm of the operator G, or more simply the incremental norm of G. That is

IlGllr = inf {+G > O I(G(x)- G(y)llD5 +ci IIx - y ( l B , Vx7 JJ E B) or

3ti

192

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

If G(O)=O and if G is Lipschitz continuous, then

Definition 6.6 Let G be a Lipschitz continuous operator from a Banach space B into itself. Then if llGllr < 1, the mapping G : B + B is called a contraction. The Banach fixed point theorem 1.19, showed that if B is a Banach space and G a contraction in B, then G has a unique fixed point in B. Consider some Banach space B and define Lipschitz continuous operators G, F that map B into itself; assume that G is invertible and its inverse G-' is Lipschitz on B and that ~ ~ GllFll, ~ < ' 1. ~ ~ , THEOREM 6.9 Under these conditions, the operator M = F + G is invertible in B, M-' is Lipschitz continuous on B, and

lIM-'lL 5 (lIG-'lL -lI~l,)-' Proof Clearly M = (I+FG-')G, since G-' is invertible. Under the conditions of the theorem the operator N = (I+FG-') is invertible, consequently H = I + F is invertible by definition. Define xi = (I+FG-')-' zi for zi E B, then

zi= (I+FG-')xi or xi = FG-'x,

+ zi

and hence by taking norms llG-'Il~)-'

l l ~ 1 - ~ 2 l l = l l ~ - ' ~ 1 - ~ - ' ~ 2 l l ~ ~ ~ - l l ~ l , IIZ1-~211.

Hence N-' is Lipschitz on B and so by definition of M, M-' Lipschitz, also l\N-'\ll5 (I- 11Fl1, l[G-'ll,)-'.

= G-'N-'

is

0

If we take the special case of G = I , the identity operator we have the obvious corollary which is useful in studying feedback systems. COROLLARY 6.9 Let F: B + B be Lipschitz on B with 11Fl1, < 1, then the operator H = I + F is invertible and its inverse is Lipschitz on B with llH-'Il, 5 (1-llflIl)-'. Suppose that we make further restrictions on the operator G such that its a Lipschitz operator mapping a Hilbert space HS onto itself and it is incrementally passive, that is (Gx - Gy, x - y)?S

IIx - ~ ( 1 ' VX, y E HS, S > 0

THEOREM 6.10 Under these conditions the operator G :HS + HS is invertible, G-' is Lipschitz on HS with ~ ~ G - ' ~ ~ , and for all x, y E HS ( G-'X - G-' y, x - y ) 2 S llG1(;'

IIx - ~ 1 1 '

6.

193

CIRCLE CRITERIA

Proof Utilizing the definition of incremental passivity, the Schwartz inequality shows that the operator G is invertible by the fixed point theorem. For some x, y E HS, putting x = G-'q, y = G-'( we have 77, 5~ HS and so equally

1 1 6 7 7- Gtll

Ilc-'x-

IlGllr llrl - 511

G-'Yll4lGIl~'I I X - Y l l

Using this result in the incremental passivity condition we have (G-'x - G-'y, x - y)'

6 IIG-'x - G - ' Y ~ \ ~ 26

G l 11;2

IIX

0

- Y1I2

Consider now the feedback system y=Fe,

(6.59)

e=u-Gy

where u E Be ; F, G : Be + Be and for each T EI, P,B, is a Banach space. Suppose that the operators F, G are Lipschitz continuous o n Be with respective incremental gains qF, qG. For this feedback system to be well posed, the operators F, G are required to be causal and at least one must be strongly causal. The strong causality requirement on F or G is motivated by the fact that all physical systems exhibit some form of delay in the system dynamics. Clearly for some T EZ YT

= {F(u - Gy))T A M(yT)

The incremental norm conditions on F and G applied to operator M ( y ) for yT, y T , € Be for all T E Z give

II{F(~T(GY')T)I-{F(~T - (GY )T)III5 ?F II(GY+)T -(GYT)TII ?F?G

1IYT'-

YTll

Hence operator M is a contraction of PTBe if qFqG
(6.60)

But yT = (FeT), implies that

IIYT' - YTII5 % IleT - eTll

(6.61)

Therefore (6.60) becomes o n rearrangement IleT' - eTl( ( 1- '?F'?G

I-' OUT' - UTll

(6.62)

194

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

provided that

+F+G

< 1. And hence from (6.61)

We are now able to state: THEOREM 6.11: INCREMENTALGAIN THEOREM Assume that the space PTBe is a Banach space for each T EI. Let F, G be causal operators on Be with associated incremental gains +F, +G respectively. Then if +F+G < 1, there exists for the feedback system

Y=F(u-GY),uEB, unique solutions y, e = u - Gy E Be, and the map u w e i s uniformly continuous on Be and B. I n addition if U E B ,then e, ~ E B . Note that if the operators F, G are linear, then the small gain and incremental gain theorems are identical if PG = PF = 0. The loop gain conditions (6.62), (6.63) can be further strengthened by application of the loop transformation theorem to both operators F and G. That is, following the approach introduced in the intermediate gain theorem 6.7, approximate the linear operator F by an operator 4 E B and the memoryless operator G by Go€R""".Therefore (6.59) transforms to y = (I- DGo)Fu - (I- DGo)(F- 4)G( y ) - D( G - Go)(y ) &M(Y)

provided that D = (I+4Go)-'4 is a bounded linear map on B. If the Lipschitz continuous operator M(y) :B +-B is a contraction on the Banach space B,then there exists a unique solution y E B which is a fixed point of M(y). That is

IK-DGO)V-~)II. IIGII+IIo(G- GoN< 1

(6.64)

and operators (I+4Go)-'F, (I+4Go)-'4:B + B. If we define the error of approximation of F by 4 as

K ( 4 ) = llU+ 4 G J ' W - 4111 . IlGll

(6.65)

then the sufficient loop gain condition (6.64) for stability can be written as

TG-Go(1-K(4))-'

11D11<1

(6.66)

if K ( 4 ) < 1 .

Example 6.2 Consider the Banach spaces B = L i , p = [l, w] for the feedback system (6.59) with U E L ~F, a causal linear time-invariant convolution operator described by matrix transfer function @(s), and G a

6.

CIRCLE CRITERIA

195

memoryless nonlinear operator that satisfies

where Go€ R”””.Then, similarly to the small gain theorem, application of the loop transformation theorem shows that the feedback system (6.59) is L;-stable for p = [l, m] if and

IKZ + F G J ’ q I i ?G < 1

In addition, for each u E L;, each solution e, y E L ; is unique and depends continuously upon u. We have now established for the space €3 = L ; three gain theorems which provide sufficient conditions for input-output &-stability with varying conclusions concerning the existence, uniqueness, and continuity of solution, each dependent o n the structural assumptions imposed upon loop operators. In all cases it was necessary to compute’the norms 1 141, and/or ll(I + F’Go)-’dlp to ascertain stability. For p = 2 we have shown that

llf;1I2 = sup A”’[&w)] W€R

where A[p] is the largest eigenvalue of fi(jo)fi(jw)*;otherwise only the lower bound

IIdI, 2 Ul€R SUP Ifi&)I is available for the norm IlflI, in terms of the frequency response of the convolution operator F. It is therefore not surprising that the majority of graphically interpreted stability criterion are for L2 spaces.

Example 6.3 Consider the space B = L ; and a linear convolution operator F : L,”+ L,”which is normal (that is commutes with its complex conjugate), then (6.67) where a [ A ] represents the set of eigenvalues of some matrix A. Suppose now that the operator G in the feedback system (6.59) is such that G :R ” + R “ and satisfies a ( Y ‘- Y ) 5 G(Y’)- G(Y)5 ( P - O(Y’- Y 1

196

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

for y, y ' L~z , a and P scalars and 6 some arbitrarily small positive constant. By setting F = 4 , K ( + ) = O in (6.65) and the gain condition (6.66) is minimized by selecting

Go= ;(a+ @ ) I in which case ?G-G,,

=

i(P - a )

and the incremental gain condition (6.67) (which is sufficient for L;stability of the feedback system (6.59)) simplifies to

S(P - a )II(I+i(a + P ) W ' F ( I , < 1 The above norm condition is readily evaluated through (6.67), since the operator ( I + i ( a+ @ ) F ) - ' F is also a normal operator. For a graphical interpretation of the incremental gain theorem, consider the following structured feedback system y = F(u - GN(y)) for

(6.68)

u E L;

where N is a nonlinear memoryless Lipschitz continuous map of L ; into itself such that there exist numbers ai, Pi satisfying ai(Yi-YI)sN(y)i -N(y')i sPi(Yi -

~l)

(6.69)

for all y, y ' L~; and i = 1 , 2 , . . . , n. G is a linear time-invariant operator of L ; into itself and as such is representable by a matrix transfer function G ( S ) E K ( O ) "F~ "is. a linear time-invariant mapping of L ; ~ into L;e, characterized by a matrix transfer function P(s) such that each element &(s) is a meromorphic function in the half-plane Re(s)>O and satisfies the condition lim fii(s)= 0. ISI'ICI

+

By approximating the linear operator FG by a normal operator and the nonlinear operator N by some N()ER""" and by applying the loop transformation theorem to (6.68)

y = - ( I + W J ' ( F G - +)N(Y) -(I+ W~)-'~(N-NO)(Y) +(Z++N,,)-'Fu (6.70) Applying the contraction mapping theorem to (6.70) shows that for

u E L:, the solution y E L ; is unique and continuously dependent upon t if

the linear operators (I++No)-'+, (I++No)-' (FG- +) and (I++No)-'F are all elements of K(0)""" (that is, they are Ly stable operators) and

(1- W4))-' [ \ ( I +W o ) - ' + ( N - No)ll< 1

Where

(6.71)

K ( 4 )= )I(Z+ +NO)-'(FG - +) I 1 . llNll is the error of approximation

6.

197

CIRCLE CRITERIA

of FG by 4 and must be less than unity for (6.71) to hold. The effective loop gain condition (6.71) can be minimized by appropriate selection of No as diag ( c i ) . And so (IN- Nol s m a x {lai- C i I , IPi - C i I I By selecting ci = ;(ai+ P i ) ( i = 1 , 2 , . . . , n ) , the norm \IN- Noll is minimized to max &3, - ai). When ai = a and pi = p for all i, we have the I

same optimum choices of N,, = &(a+ @ ) I , and (IN- Noll = t ( p - a) as in the small gain and intermediate gain theorems. Suppose that the scalars a S m i n ai, p z m a x pi, then the stability I

condition (6.71) can be rewritten as (6.72) This has the same graphical interpretation as inequality (6.56) for the intermediate gain theorem, and by similar reasoning we are able to state:

THEOREM 6.12 Consider the feedback system y N : L;-+ L ; is Lipschitz continuous such that ai(yi

= F(u - G N ( y ) ) , where

- ~9 5 N(y)i - N(y')i 5 P i ( y i -

YI)

for all y , y' E L ; and i = 1 , 2 , . . . , n. G :L ; + L ; is linear time invariant operator with matrix transfer function G ( s ) E K(0)"X",and F : Lze -+ I,& is a linear time-invariant operator with matrix transfer function E(s)= {fij(s)} which is meromorphic in R e ( s )> 0 with lim &(s) I+=

= 0.

Then the solution y E L ; is unique and continuously dependent upon t if u E L ; and there exists a normal operator 4 that approximates FG such that (a) E(s)is stable and (i) if P(4) and a ( 4 ) are non-zero and have the same sign, the eigenloci A i ( w ) of c$(jw) do not encircle or intersect the circle C[-P(4)-', -a(4)-'],or (ii) if P(c$)>O>a(4),the plots of A,(@) lie inside the circle C[-P(+)-', -a(4)-']. or (b) P(s) has p-poles in Re(s)>O, and P(4) and a(4) have the same sign and are non-zero, the eigenloci A i ( w ) of &jw) do not intersect the circle C [ - P ( $ ) - ' ,-a(4)-'] and encircle this circle in the clockwise direction ( n x p ) times where

a(4) = $(a+ P ) - - P N 1 - K(4))-' P(4)=+(a+ P)+t(a - @ ) ( I - W4))-'

198

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

a = min ai, p N 0 =I( 2 a+P)I. I

y pi, K ( 4 )= ll(I+ 4No)-'(FG- 4)II.IlAll and

=m

An obvious practical restriction of this Nyquist type result is that it requires the computation of the eigenvalues of the normal operator & j w ) . However, when & j w ) is diagonal, take No = diag ( q ) , ci = +(ai+ Pi) and condition (6.71) can be written as (6.73)

where

and

4 = diag (4ii)

This is satisfied if the Nyquist plots of &ii(jw)-' are exterior to the circle C[-Pi(4), - a i ( + ) ]and hence generates an alternative graphical st:bility interpretation of Theorem 6.12 directly in terms of the elements of 4 ( j w ) . An alternative approach which directly uses the elements d i j ( j w ) of $ ( j w ) and avoids the requirement that 4 be diagonal is based upon Gershgorin's theorem. It is well known that the union of the Gershgorin circles with centres pii(jw)and radius

c c n

ri(w)=

or

k#i

n

lPik(jw)I

or

k#i

lPki(jw)I

n

k#i

!dlPik(jw)l

+ IPki(jw)l)

contains the ith eigenvalue of the matrix T(jw)={p,,(jw)} for i = 1,2,. . . , n: So if & j ( j w ) are the generic elements of the matrix transfer function 4 ( j w ) , then the sufficient conditions for Lg stability of the feedback system (6.68) have the following altern$ve graphical interpretation: if the envelopes of all circles with centres +ii(jw) and radii r i ( w )do not encircle or intersect the circle C[-P($)-', -a(4)-'] for P(q)? a ( 4 ) > 0 or lie inside C[-P(4)-', -a(4)-'] if P ( $ ) > O > a ( + ) for F ( s ) stable. If, however, P(s) has p-right-half poles, then the feedback system (6.68) is stable if P(4)> a ( 4 )> 0 and the Gershgorin circles & j w ) do not intersect the circle C[-P(+)-', -a($)-'] but encircle it in the clockwise direction ( n x p) times (see Fig. 6.4).

( i d P I @ )> 0 > al@l

I

FIG. 6.4 L;-stability criterion for non-normal operators using Gershgorin’s theorem. (i) P(4)?a(+)>O (ii) P(+)>O>a(4)

199

200

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Returning to the stability condition (6.73) for the case when $ ( j w ) = diag {&(jw)} and No = diag {ci},ci = +(ai+pi),this sufficient stability condition can be rewritten as (6.74)

where Ri =$(Pi - a i ) ( l -K(+))-'. If now the Nyquist plots of $ii(jw) are replaced by plots of C ~ & ~ < ~ W then ) , the critical circle for the plot of c,$,,(w) is C[-ciP;', -cia;']. This latter circle is called an M-stability circle with M-value of

From the plot of ci&(jw), a value of M,(c&~) can be determined that corresponds to a maximum value of M (Fig. - 6.5), and so the stability inequality (6.74) can be rewritten as max M,(cibii)M;' < 1

I

-ci

pilol-'

I

I

FIG.6.5

-c, a

, d

M circle stability criterion

(6.75)

6.

CIRCLE CRITERIA

201

which is the familiar M-circle stability criteria. The parameter

can be considered as the effective gain margin of the feedback system (6.68). The effect of plant or output disturbances q upon the feedback system

Y =F(u-GN(y))+q

(6.76)

by setting u=O, since now llyll is a measure of the sensitivity of the feedback system to disturbances. By application of the loop transformation theorem to (6.76) with u = 0, N,, = diag ci, C#I = diag +ii y = (I+ 4Nnl-l T - [(I+ 4 N J 1 ( F G- + ) N + (1+ 4NJ14(N- Nn)(y)l

provided that

I K I + + N J 1 ( F G - + ) I l . llNI+lK1+WJ14(N-Nn)Il< 1 the contraction mapping theorem gives

-K(4))-' Il(I+ 4Nn)F'qll

= e(1

(6.77)

If the disturbance n-vector q = {qi} has independent elements with known power spectral density then

Clearly for disturbance rejection or minimum system sensitivity, it is w ) set as far as possible from desirable that the Nyquist plots c ~ & ~ ( ~are (-1,O). For practical evaluation it is necessary to know those qi which are important and the bandwidth of significant frequencies. In the control system design the linear operator FG usually contains some free design elements. A possible design strategy might be: (i) consider the initial FG from the class of operators that ensure a wellposed system with approximation normal operator & equal to, say, diag (FG); (ii) compute K(&) and then test closed-loop stability via the circle criterion; (iii) from the graphical stability criterion design for a new +1 and larger gain margin 8; (iv) for this new value of change the free elements of FG to an operator closer to 41and repeat process iteratively until 1(y(I< 5, some prescribed error criterion. It has been assumed throughout that the plant dynamics are known

202

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

exactly. This is rarely the case in practical systems, and feedback is incorporated not just to improve stability but also to reduce system sensitivity either to parametric variability or to external disturbances. The design objective in this case is to evaluate a compensator in the feedback loop which guaranteed that output performance specifications are satisfied over the range of plant uncertainty or set of disturbances. Consider the feedback system

u E L," y = F(u - G N ( y ) ) , By considering the feedback signal z = GN(y) to the comparator this dynamical system can be equally expressed as z = FGN(u - 2)

(6.79)

Assume that F, G, N are continuous operators from L,"eto Lie. The operators, F, G, and N are assumed to be Lipschitz continuous and operator FA= [fij(A)], with parameter vector A, known only to belong to a set A; the problem is to select feedback operators G N (or G and N separately) so that the system response is contained within the set of desired responses Z (assumed known a pnori) irrespective of the values of A E A . Provided that (I+FGZV-' :LLe+ L;, then (6.79) can be rewritten as z = (Z+F,GN)-'F,GNu

= TAu,u E Lp"

(6.80)

Since Z is known a prion the closed loop operator TAcan be defined to belong to a desired set V(A, z) of stable operators dependent upon the parameter set A and 2. Set P = G N and LA = FAP = FA&'FoP = FAK'Lo where Fo is arbitrary and invertible. The problem is now given some F,EFA find P = G N = Lo&' that satisfies (6.80) for any z E Z. Define QA= &'FA then

T , = ( I + QAL,,)-'QAL, =(I-(L,'+ QA)-'Li')

(6.81)

Given some initial L:, (6.81) gives T:, we now select Lh such that lim Th = Toc V, and hence the required Lo can be established. Equation I

(6.81) can be written as an iterative equation

T,"=[I-((L,"-')-'+QA)-'(L,")-'][I+ (L,"-')-']T: = (I--An-'(L,")-')Bn-'Tg for n = 1 , 2 , . . .

(6.82)

6.

CIRCLE CRITERIA

203

where A"-' = [(L,"-')-'+ QA]-' and B"-' = I+(L,"-')-' with L,' = 0. The convergence of this algorithm can be proven in two cases. (i) V is a compact set. By assumption ( I + F,GN)-' exists and FA, G, N, (I+F,GN)-' are operators mapping LFe+ Lk for all A E A , then TA:L k - . Lie for all A E A . The algorithm requires that { T ; } c V, but V is compact so the sequence {T;} is also compact. Therefore by the ArzelaAscoli theorem {T?} is a convergent sequence. The limit To generates the transmission operator Lo, which in turn provides the compensator solution G N = Lo&'. (ii) V is a bounded set. Select L? such that the sequence {T?} is monotone in the following sense: let { Vi} be a sequence of closed spheres in the space L k such that V I V, I V2 . . . = V,, . . . and T i + ' # V,, and T;+]E V,,,, with T ~ S.EThen by the nested sphere theorem, the sequence {T?} converges to To. It has been assumed throughout that the operator TAis LF-stable. This can be tested for p = 2 via the previous circle criterion for a given N. The above proofs also hold for time-varying operators, provided they are continuous with respect to the time argument.

6.5

AN M-MATRJX STABILlTY CRITERION

For feedback systems with a particular algebraic structure the following graphical stability criterion based upon the properties of M-matrices may succeed when previous criteria prove unsuccessful. Consider an arbitrary n vector x = { x i } € L," with norm

In the previous graphical L,"-stability studies the norm

has been used. But since llxlll SIIX~~~S n1'2IIxIII, then the above two norms are equivalent and define the same topology in L," and L,"-stability by one norm implies L,"-stability by the other. RY be the vector whose ith element is IIxillL. For any XEL,",let i~ Similarly for x E L,",, i ,R: ~ denotes the vector whose ith element is IIPTxillL, for PT a truncation operator. Let B(L,",L,")represent the space of all Lipschitz continuous operators F : L,"+ L," satisfying the unbiased condition F(0) = 0; the Lipschitz norm on this space is a norm in the usual sense. Let B,(L,",L ; ) c B(L,",L,")be

204

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

the space formed with all causal operators F whose entries are elements of Bc(L,, L,). For a Lipschitz continuous operator F : L ; + L ; , represented by a matrix cfij} of causal Lipschitz continuous mappings f i i : L, + L,, let F= { I l f i j l l l } E RYx". Similarly if F : L;,+ L;, is a causal mapping such that for T 2 0 , PTF:L;[O, T]+ L,"[O,TI is a Lipschitz continuous operator then ~ T = { ~ ~ PR?"", T ~ and i ~ ~ for, any } ~ T'> T, p T , r FT and lim FT
Consider now the feedback system

~=-FY+u

(6.83)

with FEBc(L;,L;). If F is well posed for each u E L,",,then there exists a unique solution y E L;, to the feedback system (6.83). Moreover, since F is causal

for i = 1,2,. . . , n. Taking

ll.lll

norms of the above gives yT 5 FTyT

+ fiT

(6.84)

If the spectral radius r,(F)= sup Ihl of the operator F is such that AEU(F)-

r,,(F)FT, r u ( F T ) < r , ( n < l and so (I--FT)-'e RY"", and inequality (6.84) becomes

But (Z-FT)-'<(I-&', since r,(pT)
fi,

FT,E E R : ~ "So .

v TL 0

and y E L;. We have now established the fundamental L,"-stability result: THEOREM 6.13 Given that the feedback system y = - F y + u with F E Bc(L,",L ; ) is well posed and u E L;, then if r , ( n < 1 there exists a unique solution y E L;.

The condition ru(F)< 1 is equivalent to the condition on the matrix M = { m i i } = ( Z - F ) E R " x "that , each m , s O for i # j and M - ' E R Y ~ " or equivalently every principle minor of M is positive non-zero (that is, M is an M-matrix-a generalization of the Minkowski matrix). Again consider the structured feedback system y=F(u-GN(y)), UEL;

(6.85)

6.

205

CIRCLE CRITERIA

where N is a memoryless Lipschitz continuous map of L ; into itself such that there are numbers ai, pi (i = 1 , 2 , . . . , n) which satisfy c r i ( y i - ~ : ) I N ( ~ ) ; - N ( y ' )~iP i ( y i - y l )

for all y, Y ' E L ; ; G : L ; + L ; is a linear time-invariant operator represented by a matrix transfer function G ( s ) EK(0)"X",F : L;,+ L;, is a linear time-invariant operator with matrix transfer function k(s)such that ;~ each f ( s ) i i are meromorphic functions in R e ( s )> 0 and lim f ( ~ =) 0. ISI--

For some arbitrary matrix NOE R""",the loop transformation theorem 6.1 allows the feedback system (6.85) to be written as Y = (1+4No)-'Fu-(Z+ ~ N ~ ) - ' ( F G - ~ ) N ( Y ) - ( ~ + ~ N ( , ) - ' ~ ( N - N (6.86) provided that (I+ +No)-'F, (I+ c#dVO)-'c$E K(0)"x".In this case the variab!e

v

= ( I + q5NJ'

Fu E L;

if u E L ; . So (6.86) can be rewritten as y = u - VNY)- UN,(y)

(6.87)

where V=(Z+q5N0)-'(FG-q5), U=(Z+q5No)-'q5 and N,(y)= N(Y)-NoY. Consider first the situation a;2 a,and P 2 Pi for all i and FG a normal operator such that 4 = FG and hence V = 0. In this case N,, = ;(a+ P)Z, and utilizing the 11.11, norm equation, (6.87) becomes jk6+EIjj

(6.88)

/v

UN, is bounded by

(6.89) where i r ( j w ) = (Z+$(a+ p ) F G ( j w ) ) - ' Z ( j w ) . Clearly each and Q = E RYx".Hence

{aii}

Qii is

positive

P(Z-Q)56 and by Theorem 6.13, provided that r,(Q) < 1 and u E L ; , the solution y to (6.85) is unique and y E L;. Under these conditions: THEOREM 6.14

The feedback system y = F ( u - GN(y)) is L;-stable with a

206

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

unique solution Y E L;, if U E L;, FG is a normal operator, U = ( I + $ ( ~ + P ) F G ) - ' F GK(0)"x" E and r,(Q)
The requirement that FG is a normal operator is clearly restrictive for practical considerations; suppose now that the normal approximation operator is set equal to diag (FG) and N,, = diag (ci)where ci = $(ai+ pi), minimizes the norm llN-Noll. In this case

i f j, i, j = 1 , 2 , . . . , n, and LijA{FG},. And so for the choice diag ( F G )

-

4=

(6.90)

UN,(Y)5 py'

where P = diag (Pii)E RYx". Similarly

where

A sup Iiij(jw)l . (I1 + ciiii(jw)l)-'max {lpil,[ail} for i f j , i, j =

Qij

weR

1 , 2 , . . . , n, and Qii= O for all i = j . Clearly hr

(6.91)

V N ( y )5 Qy'

where Q = { Q i i ) ~ R " " " . Taking 11.11, norms of (6.87) and utilizing inequalities (6.90) and (6.91) gives y'(I-Q-P)< O (6.92) Suppose that r,(Q + P) < 1. Then ( I - Q- P) is an M-matrix and therefore (I- Q- P)ERYx" and inequality (6.92) can be rewritten as y' ( ( I - Q- P)-'

P

(6.93)

and by Theorem (6.13) EL; if i j L;. ~ Under the above conditions: THEOREM 6.15 The feedback system y = F(u - G N ( y ) )is L;-stable with a unique solution y E L ; if ( I + c#")-'+E K(O)""", 4 = diag ( F G ) and No = diag ($(ai+ p i ) and r,(S) < 1 or equivalently ( I - S ) is an M-matrix where sii= Piifor i = j and sii = Qiifor i # j .

6.

CIRCLE CRITERIA

207

The above stability result has a simple graphical interpretation if Gershgorin's theorem is invoked. The feedback system y = F(u - GN(y)) is L; stable if the envelope of Gershgorin circles of radii (pi-ail-1

max{IPil,

Iail}t

j=I j# 1

l%(iw)lij

centred on the Nyquist plot of { F G ( ~ w )do } ~not ~ encircle or intersect the for i = 1 , 2 , . . . , n. circle C [ - & -ai], A necessary condition for the establishment of the M-matrix theorem 6.13 was that (I+4No)-'+ E K(0)"""; if the operator 4 = diag (FG) is stable, then this condition is equivalent to requiring the L,"-stability of the linear system Y = 4 ( u - N"Y 1 But since No = diag ( c i ) , the condition

( I + 4N")-I4 E K(0)"""

(6.94)

h

is satisfied if the Nyquist plot of {FG(jo)),' does not encircle the point -ci. But since the point -ci is contained within the circle C[-& - a i ] , condition (6.94) is automatically satisfied by the above M-matrix graphical interpretation via Gershgorin's theorem. Suppose now that F contains p poles in the right half s-plane and admits the representation P(s)= fio(s)&'(s) where d ( s ) is a monic polynomial with p zero in the right-half s-plane and fio(s) is a matrix transfer function which is analytic in the right-half s-plane. Under these conditions the multivariable Nyquist theorem enLbIes us to state that condition (6.94) is satisfied if the Nyquist plot of {FG(jw)};' encircles the circle C [ - p , -ai](n x p ) times.

6.6 SYSTEM DIAGONALIZATION AND DESIGN The use of input-output or frequency domain methods in the analysis and design of linear feedback systems represented by matrix transfer functions of rational polynomials in s is now well established (Harris and Owens, 1979; McFarlane, 1979). These apparently distinct linear multivariable design techniques have been essentially based upon the mechanism of generating a series of pseudo-scalar design problems. Although these approaches to multivariable design have a fundamentally different basis, a theoretical unit does exist through the use of permissible input-output

208

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

transformations. The question now arises of whether such linear multivariable design methods be applied to the design of controllers or compensators for nonlinear multivariable feedback systems of the form y=F(u-GN(y)),

UEL;

(6.95)

where F, G : L ; + L ; are linear time-invariant operators with matrix E K(0)""" and N : L;-, L ; is Lipschitz contransfer functions fi(s), tinuous such that

e(s)

ai(yi - ~

3 N(y)i 5 - N(y')i 5 Pi(yi - Y:)

(6.96)

Furthermore we assume that the open-loop transfer function fi(s) can be decomposed as fi(s) = P,(s)K(s) where K(s)E K(0)""" is the forward system controller or compensator that is to be designed in order to achieve closed loop stability and adequate system dynamic behaviour. All of the graphical stability criteria introduced in this chapter are readily implemented if the loop transfer function is diagonally dominant; in addition essentially scalar control or compensation design methods are appropriate. We therefore seek transformation methods that achieve diagonal dominance over the operating bandwidth of the system (6.95). Introduce the transformation P1(s)t+fi2(s) where fi,(s) = P1fi2(s)P2;for PI and P2 square non-singular matrices such that f i 2 ( s )= P;'F,(s)P;'. Such transformations are introduted to achieve a form of diagonal dominance and compensation of F,(s) while not introducing any new dynamics (other than desired control performance) into the feedback system. Special cases of this transformation are: (i)

Precompensation

The transformation pair (I,,, K - I ) where K E R""" can be regarded as the map f i , ( s )fi2(s)K ~ and describes the methods of constant precompensation whereby the high frequency eigenvectors of the characteristic loci (Kouvaritakis, 1979) and the approximate commutative control (MacFarlane and Kouvaritakis, 1977) are realigned and the pseudodiagonalization of f i ( s ) associated with the Inverse Nyquist Array of Rosenbrock (1974). Essentially the problem of precompensation is to find a K E R""" such that at some frequency of interest s = j w , fi(jw,) ={fij(jw)}'fi,(jw,)~

(6.97)

is diagonally dominant. We note that the ith column of fi is determined by the ith column of K, and we can consider the independent choice of

6.

209

CIRCLE CRITERIA

the n columns of K when attempting to satisfy

(6.98) for column (row) dominance for i = 1 , 2 , . . . , n. Clearly we require

for all i and s for diagonal dominance. Consequently the diagonal dominance of a complex matrix is essentially a min-max problem, that is, we seek the solution of min max J i ( j o ) K

WPAW~

where Am, is the frequency bandwidth of interest. One way of achieving diagonal dominance is via a permutation-matrix K,E R""" in which we seek to permute the row (column) of F , ( s ) to obtain the best overall column (row) dominance of the resultant transfer function matrix. Each row permutation affects two column dominance (similarly for permutation upon row dominance), so what is required is the least possible maximum dominance measure via an appropriate permutation matrix. This method determines the dominance of the permuted matrix transfer function column by column (row by row) by solving

In its simplest form K, is a matrix in which each row and column contains only a single entry of unity, all other entries being zero. The effect of permutation matrix K , is to renumber all inputs, essentially deciding on the optimum pairing of inputs and outputs of the compensated matrix transfer function to achieve dominance. Practical numerical algorithms for implementing various diagonal dominance schemes can be found in Bryant and Yeung (1981). (ii)

Re- and post-compensation

Here it is usual for PI, R""", although Rosenbrock (1974) has introduced a complex matrix K(s)EK(O)"""in cascade with an

210

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

elementary permutation matrix K ER""" such that

o.....

n n

K ( s )=

K(r)(s),

0

1 0 . . laiJ'(s)0 . . 0

r=l

1 and det K ( s )= 1. That is, K"'(s) is a unit diagonal matrix with the addition of a single entry a $ ) ( s )E K(O)(i# j ) which is a rational function. The matrix K ( s ) has the effect of adding to column j of $',(s)K a multiple of a$"(s)of column i. This approach is particularly appropriate to the Inverse Nyquist Array design method.

(iii)

Permissible matrices Pz,P2

In their most general form PI, C""" which must also be restricted to the class of non-singular realizable matrices for practical implementation. This complex pair (PI,P2) are restricted to the class of permissible matrices defined by

where the columns (rows) of PI,P2 are real or exist in complex conjugate pairs and bT = bl(i)whenever a: = for 1sj In. (The map j I+ 10') is an isomorphic map of integers (1,. . . , n ) onto itself by the structure of PI and so l ( l ( j ) ) = j for 1sjs n ) . It is straightforward to show for a permissible pair (PI,P2) that the matrix product PIP2is real and nonsingular and that the pairs (P1, P;'), (Pi1, P2) and (P;', Pi1)are also permissible. The power of permissible matrices in achieving diagonal dominance over a frequency interval is illustrated in the following theorem due to Owens (1981): THEOREM 6.16 Zf det $,(jo,) # 0 and $,(jol)fil(jo1)-'has a complete set of eigenvect:rs, then there exists a permissible pair (PI,P2) such that {g2(s)= P;lFl(s)Pil}s=jw, is diagonal and diagonally dominant in an open frequency interval that contains wl.

6.

21 1

CIRCLE CRITERIA

The system (6.95) can be rewritten by utilizing the loop transformation theorem as

Earlier stability studies (e.g. Theorem 6.8) indicated that closed-loop stability depended upon the linear operator (I++NO)-'4 and hence the eigenvalue plots of & j w ) around some critical circle in the complex plane. Where 4 is a normal approximation to FG and No is selected such that " ( N - N,,)(y)l( is minimized. It is straightforward to show that

(I+

= P;'(I+ +,N;)--1+,P;1

and that det (I+4N0)= det (I+4 ; N ; ) where 4'= P;'4P;', and N ; = P;'N,P;'. These identities indicate that the permissible transformation ( P , , P2) on P(s) is also a similarity transformation on the closed loop stability question. For finite dimensional systems Fi and K have rational polynomials as elements and for physical realizability these polynomials must have real coefficients. These realizability requirements hold for the permissible pair ( P , , P2) if and only if {p;*(s))jk + & ( s * ) } l ( j ) [ ( k ) * {12?(s)>jk

G{~l(s*)h(j)~(k),

v j, k v j, k

(6.99) (6.100)

We note that the set of all transfer function matrices satisfying (6.99), (6.100) are closed under multiplication and inversion. For approximate commutative control ( P , , P2)= ( V i ' , W;') so that K ( s )= W, diag{k,(s)}V, and k,(s)= P;'KP;' = diag { k i ( s ) } for j = 1,2, . . . , n. Where W, (respectively V,) is a real approximation to the eigenvector (inverse eigenvector) of {Pi(s)), =j w l . The method of dyadic expansions (Harris and Owens, 1979) utilizes complex permissible transformations such that $'i(s) is diagonal at some desired frequency w , and diagonally dominant over a bandwidth that includes s = j w , , and K,(s) = diag { k j ( s ) } . To achieve adequate closed-loop control a controller transfer matrix K,(s) is usually placed in cascade with the diagonalization matrix K. K,(s) is selected as a non-singular diagonal matrix with all its singularities in Re(s)< 0 and so that each entry provides the appropriate independent loop control. Additionally the gain of each controller is adjusted to overcome the problem that dominance methods are not usually scale invariant and some preadjustment of each loop is necessary before control.

212

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

6.7

NOTES

The functional analysis approach in determining input-output stability is intuitively more appealing and general than the Lyapunov approach (Willems, 1970; La Salle and Lefeschetz, 1961; Venkatesh, 1977). It also serves to provide a unified approach t o several different kinds of stability and yet provides solutions to problems that could have not been provided by any other means. W e have seen in Sections 6.1, 6.2 that the definition of input-output stability depends on the type of norm used and that different norms give rise to nonequivalent definitions. Moreover it is clear a priori that any of them will be equivalent to stability in the sense of Lyapunov with respect t o a given state space representation of the system (Araki, 1978). It has, however, been shown by Willems (1969) that under fairly weak assumptions a certain type of input-output stability does ensure the existence of Lyapunov functions. In addition Willems has established the equivalence of &-stability and global asymptotic stability of state space representation for finite dimensional systems. The study of input-output stability of nonlinear feedback systems was originated (Zames, 1966; Sandberg, 1964) as an alternative to Lyapunov's direct method. In this early work the now-classical circle stability criterion for nonlinear single input-single output systems was established (Holtzman, 1970; Narendra and Taylor, 1973). Sandberg (1964-1966) introduced the concept of extended spaces (see Chapter 5 for a detailed study) t o establish the stability conditions for multivariable systems defined on the spaces L;, L ; , and LE; unfortunately these conditions did not have a simple graphical interpretation. T h e first multivariable circle criterion was established by Falb et al. (1969) whereby the system

y=FN(e),

e=y-u

with N(.)=diag{N,(.)} is L;-stable if all Ni(e,) are contained within the sector S { a , p}, the linear element of the feedback system is normal for all frequencies and the eigenloci of P(s) d o not encircle o r intersect the on axis circle C [ - a - ' , - @ - ' ] . In 1973 Rosenbrock and Cook extended these results to the more general class of nonlinearities, Ni(ei)E S { a i , pi}.However, Rosenbrock and Cook's results were dependent upon various types of diagonal dominance of the linear operator,-so that a Gershgorin circle envelope centred along the Nyquist plots of diag ( F ( s ) ) could be utilized rather than the computationally difficult eigenvalues in determining stability. These results were extended by Araki (19761, using more general sector conditions instead of the concept of positivity; further extension of these methods to include compact operators and slope restricted nonlinearities have been made by Mees (1976) and Cook (1976). Mees and Rapp (1978) and Valenca and Harris (1979) independently extended the earlier work of Falb et al. (1969) for multivariable feedback systems t o account for non-normal linear operators and more general types of nonlinearities; in these methods the linear operator is approximated by a normal operator such that the critical circle diameter in the Nyquist plane is dependent upon the norm of the error of approximation.

6.

CIRCLE CRITERIA

213

Similar results to those of Rosenbrock (1973) and Cook (1973) for multivariable feedback systems consisting of a linear composite operator and a diagonal operator can be obtained using M-matrices (Araki, 1976, Harris and Valenca, 1981) (See also Section 6.5.)

REFERENCES Araki, M. (1976). “Input-output stability of composite feedback systems.” ZEEE Trans AC-21(3), 254-259. Araki, M. (1978). “Stability of large scale nonlinear systems-quadratic order theory of composite systems method using M-matrices,” ZEEE Trans AC-23, 129- 142. Bryant, G. F. and Yeung, L. F. (1981). “Dominance optimisation in multivariable design.” IEE Conf: “Control and its Applications”, Pub. No. 194, 28-32. Cook, P. A. (1973). “Modified multivariable circle theorems.” Zn “Recent Mathematical Developments in Control” (Ed. D. J. Bell), 367-372, Academic Press, London and New York. Cook, P. A. (1976). “Stability of systems containing slope restricted nonlinearities.” Zn “Recent Theoretical Developments in Control’’ (Ed. M. J. Gregson), 161-174. Academic Press, London and New York. Cook, P. A. (1976). “Conditions for the absence of limit cycles.” ZEEE Trans. AC-21(4), 339-345. Dunford, N. and Schwartz, J. T. (1957). “Linear Operators”, Part I. Interscience, New York. Falb, P. L., Freedman, M. I. and Zames, G. (1969). “Input/output stability-a general viewpoint”. Proc. 4th IFAC Congress, Warsaw, 41, 3-15. Harris, C. J. and Owens, D. H. (1979). “Multivariable control systems.” IEE Control and Science Record (Proc ZEE 126). Harris, C . J. and Valenca, J. M. E. (1981). “A circle stability criterion for large scale systems.” IEE Conf: “Control and its Applications”, Pub. No. 194, 272-275. Holtzman, J. M. (1970). “Nonlinear System Theory-a Functional Analysis Approach”. Prentice Hall, Englewood Cliffs, New Jersey. Kouvaritakis, B. (1979). “Theory and practice of the characteristic locus design method.” Proc. ZEE 126, 542-548. LaSalle, J. P. and Lefeschetz, S. (1961). “Stability by Lyapunov’s direct method with applications”. Academic Press, London and New York. MacFarlane, A. G. J. (1979). “‘Frequency Response Methods in Control Systems”. IEE Press, Wiley, New York. Mees, A. I. (1976). “On using the circle criterion to predict existence of solutions”. Zn “Recent Theoretical Developments in Control” (Ed. M. J. Gregson), 175-192. Academic Press, London and New York. Mees, A. I. and Rapp, P. E. (1978). “Stability criteria for multiloop nonlinear feedback systems.” 4th IFAC Symp. Multivariable Tech. Sys., Fredricton, Canada, 183-188. Pergamon Press, Oxford.

214

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Narendra, K. S. and Taylor, J. H. (1973).“Frequency domain criteria for absolute stability”. Academic Press, New York and London. Owens, D. H. (1981).“Some unifying concepts in multivariable feedback design”. Znt. J. Control 33, 701-711. Rosenbrock, H. H. (1973). “Multivariable circle theorems”, In “Recent Mathematical Developments in Control” (Ed. D. J. Bell), 345-365.Academic Press, New York and London. Rosenbrock, H. H. (1974): “Computer-Aided Control System Design”. Academic Press, London and New York. Sandberg, I. W. (1964): “A frequency domain condition for the stability of feedback systems containing a single time varying nonlinearity.” Bell System. Tech. J. 43, 1601-1608. Sandberg, I. W. (1964). “On the L, boundedness of solutions of nonlinear functional equations.” Bell Syst. Tech. J. 43, 1581-1599. Sandberg, I. W. (1965).“Some results on the theory of physical systems governed by nonlinear functional equations.” Bell Syst. Tech. J. 44, 439-453. Valenca, J. M. E. and Harris, C. J. (1979).“Stability criteria for nonlinear multivariable systems.” Roc IEE 126, 623-627. Venkatesh, Y. V. (1977):“Energy methods in time varying systems stability and instability analyses”, LNI Physics No. 68.Springer Verlag, Berlin. Willems, J. C. (1969).“Stability, instability, invertibility and causality.” SZAM J. Control 7 , 645-671. Willems, J. L. (1970).“Stability Theory of Dynamical Systems.” Nelson, London. Zames, G. (1966).“On the input-output stability of time varying nonlinear feedback systems. Part I: Conditions derived using concepts of loop gain conicity and positivity”. IEEE Trans. AC11(2), 228-238. Zames, G. (1966).“On the input-output stability of time varying nonlinear feedback systems. Part 11: Conditions involving circles in the frequency plane and sector nonlinearities”. ZEEE Trans. AC11(3), 465475.

Chapter Seven

Stability of Nonlinear Multivariable SystemsPassivity Results 7.1 PASSIVlTY STABILlTY THEOREMS The loop transformation theorem (6.1) results in a reduction in the stringent loop gain condition for closed-loop stability associated with the three small gain theorems of Chapter 6 (or equivalently, results in a shift in the conicity of the feedback and feedforward operators). This transformation technique can also result in a feedback system with passive operators so that the passivity theorem can be applied directly to determine closed loop stability. A second and significant transformation associated with the passivity theorem is the introduction of multipliers in cascade with the system loop operators (see Section 5.6). Usually the loop transformation theorem is used to ensure that the feedback operator is passive through a conicity transformation, and then a compensating multiplier is introduced to produce a passive operator in the feedforward operator without affecting the passivity of the feedback operator. The three small gain theorems required a normed linear vector space structure; however, the passivity theorem (5.3) requires in addition an inner product space (which if complete is a Hilbert space HS). The passivity theorem showed that the feedback system y = F(u - N ( y ) ) is stable for operators N, F: HS, + HS,, if the operator F is strictly passive with finite gain and N is passive. If in addition the operators F and N are both unbiased and are respectively strictly incrementally passive and incrementally passive, then the solutions e, y E HS are unique and uniformly continuous on HS for e = u - N ( y ) . Consider the class of feedback systems y = F(u- N ( y ) ) ,

u E L,”

(7.1)

with F: L,”+ L,”a linear time-invariant operator such that Fu kF@u for F E L A ! ! ~Also ~ . the nonlinearity N is assumed to be both causal and 215

216

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

lb)

la1

111

where A =

and

K,

a1 ,

= I1

B

= PI

, K,

A F I 11

+

= (6 - N

BFI-'

IB

I-'a-' -

A 1 F I1 t A F I - '

FIG.7.1 Equivalent systems through loop transformations

anticausal, unbiased and incrementally sector bound by the inequality: ( N ( x - Y ) - ( +~~ > ( x - Y ) N , ( x - Y ) - ( ~ - ~ ) ( x - Y ) ) T I O (7.2)

for all x , y E L;=, T E R and a,p, 5 constant scalars such that 5 > 0, p > a. Introduce the multiplier

VUEL;, QELA!!~"

Qu=Q@u,

(7.3)

where k =O

and bk

>ak

>o

for all k

The multiplier 6(s) is defined such that its poles and zeros alternate along the negative real axis and (arg &jo>lI r / 2 and as such its inverse d-'(s)

7.

217

PASSIVITY RESULTS

(and Q-'(s)) exists. By repeated operation of the loop transformation theorem (6.1), and introduction of the multiplier Q, the feedback system (7.1) of Fig. 7.l(a) is equivalent to Fig. 7.l(d) (see Section 5.6 for system equivalence), and so from the passivity theorem (5.3) we have:

THEOREM 7.1: PASSIVITY THEOREM (MULTIPLIER)The feedback system y = F(u - N ( y ) ) , u E L," is L,"-stable if s(i) Q ( I + BF)(Z+ AF)-' is strictly passive

s(ii) F(I + AF-' and ( I + BF)-' :L,"+ L ;

where F ELA?"" is a convolution operator and N : R" +-R" is a memoryless Lipschitz continuous nonlinearity with sector bounds given by (7.2). Proof The passivity theorem demonstrates that the input-output system u , w y l of Fig. 7.1 is Lz stable if conditions s(i) and s(ii) hold and the feedback operator s(iii) (N- A ) ( B- N)-'Q-' is passive We now show that condition s(iii) is automatically satisfied by virtue of the sector constraint on N ( y ) and by the existence of the inverse Q-' in L,". Let + A ( N - A ) ( B - N ) - ' , by the sector inequality N and (N-A) are incrementally inside the sectors S{a + 6, p - 6 ) and S{&p - a - 5) respectively. This implies that (B-N) and ( B - N ) - ' are strictly increasing So inside the respective sectors S{&, p - a - and S { ( p - a is incrementally inside the positive sector S{&(p- a g-'(p - a or is strictly incrementally passive. An equivalent condition to condition s(iii) is

e}

+

or

e)-',

e)-', e-'}.

e)},

(JIQ-'x', x') 2 0 (JIx, Q x ) r O ,

Vx, X ' E L,"

(7.4)

Now, Q ( s ) can be expanded in partial fractions as i=l

where $ ( s ) = k,s(s+ bi)-', k, > O and k , r O . Therefore i=l

Clearly (k,x, Jlx) is positive, and it only remains to show that terms

218 (@,

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

kx) are positive.

Let x

= i +biz

for all (i+ b i z ) € Lz. Then

(4x9 kx)= (+(i + biz17 kix)

ki (+(i +biz), biz) bi

=-

+,

But for strictly increasing (+(y), x ) = 0 implies that ( + ( y + x), x ) 20. Therefore (+x, k , x ) r O if ($(biz),b i i ) r O which is guaranteed by a theorem due to Zames and Falb (1968). Having shown that (B- N)-’ is strictly increasing inside the sector S { ( @- a - E)-’, .$-I} and Q-’ E L,”,if y1 E L,”, then also y = ( B - N)-’ Q - ’ y , E L,”. Finally condition s(ii) of the theorem and the positivity of (B- A ) ensure that the operator K1:L,”+ L;, and if u E L,”then also u1E L,”. 0 If we drop the restriction on the multiplier and set Q = I, Theorem 7.1 simplifies to the well-known results of Rosenbrock (1973) and Cook (1973) whereby the feedback system (7.1) is L,”stable if the operator (I+ BF)(I+ An-’ exists on L,” and is positive real. Although this result is somewhat simpler, it is far more conservative than the passivity result in determining stability via the nonlinearity sector constraints. An interesting connection between the passivity and the incremental small gain theorem can be demonstrated through the identity

Y=(I+BF)(I+AF-’ (7.5)

=(I+Z)(I-z)-l

It is not difficult to see that

Z = $ ( B- A ) [ $ [ B+ A ] + F ’ ] - ’ Assuming that Y is finite and unity is not an eigenvalue of Z , then ( I - Z ) is nonsingular and Z is finite. As Y + Y * is positive definite from the passivity theorem, then

x*(Y+Y*)x>O, Setting y

= (I- Z)-’x,

’

VXEC”f0

(7.6)

it follows that

x * ( Y + Y*)X = y * ( ( I - Z * ) ( I + Z ) + ( I + Z * ) ( I - Z ) ) y = 2y*[I- Z * Z ] y =

N Y 1’

- llZY 1)’

2211y1(’ (I-(IZ(I’)>O if

y#O

Hence 11Z(1< 1, which implies that unity cannot be an eigenvalue of Z, and (I- Z ) is nonsingular with Z finite. Recalling the definitions of operators

7.

PASSIVITY RESULTS

219

A, B, then $(I3 - A ) = $.(p- a)I, and $(I3 -A) =$(a+ @ ) I , and so the above stability condition llZll< 1 is equivalent to

M-a)ll(I+4(a + p)F)-'F)II < 1,

(7.7)

which is the incremental small gain theorem condition for L;-stability. We note that the converse of the above result holds, that is, if llZll< 1, then Y = (I+ Z ) ( I - Z)-' is finite and ( Y + Y*)is positive definite. The norm condition llZ((<1 can be seen as essential in establishing the various stability results. The question now arises: can we establish that llZll< 1 by considering only the diagonal elements of the matrix transfer function Z ( s ) or its inverse Z ( s ) ? Suppose that the complex matrix Z ( s )= {Zii(s)} is weighted mean dominant, that is n

I&(s)~- C A i h ; ' $ . [ l Z i i ( ~ ) I + ( ~ i ~ ( ~1) ( ] > i=1

(7.8)

j#i

for hi > O and i = 1 , 2 , . . . , n. Define the Hermitian matrix

x = &D*z + Z*@) where @Adiag{Ziil.Zii1-'} is a unitary matrix, so that the diagonal elements of X are real, and by (7.8) X ~ ~ - ~ A ~ A ; ' ~ X , ~ ~i = > l~ , ,2 , . . . , n j=1 j#i

And so by Gershgorin's theorem the eigenvalues of X are greater than unity, therefore

but since then

so

lZY1I . IlY II = 112Y II . I P Y II2 Y *XY

1141< 1

In conclusion any complex matrix Z ( s ) that satisfies the dominance with spectral norm less than unity. condition (7.8) has an inverse

z(s)

220

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Now for the feedback system (7.1)a complex matrix Z ( s ) exists such that Z ( S= ) i ( B - A ) [ I + ; ( B +A)E(s>-']-' where B

=

PI, A = aI,hence

THEOREM 7.2 (Rosenbrock and Cook (1973)) Given the feedback system y = F(u -AN(y)), u E L ; with F ELA!!"" characterized by a matrix transfer function F ( s ) and N : R" + R" a sector bound nonlinearity that satisfies

dY'- Y) 5 MY')

- N(Y1

5(P - O ( Y ' - Y)

for y, y' E L;, 6 > 0 and a, P scalars. It is'Lg-stability if n

-a)+ f i i ( s ) I -

C %Ifij<s>I+ I&(s>II>i(P

j=1 jZi

+a)

(7.9)

for i = 1,2,. . . , n, and each s E r',provided that each P(s) exists for all s on the contour r(r'2 r ) .

Note The contour r consists of the imaginary axis from s = -ir to s = +ir together with a semicircle of radius r, the contour r is indented into the left half s-plane to avoid imaginary poles of A i ( s ) , and r is large enough to include in r all right half poles of Ai(s)(i = 1,2,.. . , n ) . Now if F is a causal linear time-invariant stable operator FELA!!"" with matrix transfer function E(s) and the nonlinearity N: R" +. R" satisfies the incremental sector condition (7.2)for 6 > 0 and P 2 a, then the concepts of passivity and positivity are equivalent (see Sections 5.5 and 5.7) and stability criteria based upon sectoricity arguments can be derived for the feedback system y=F(u-N(y)), UEL; (7.10) Apply the loop transformation theorem 6.1 to this system with feedback/feedforward operator A = +(a+ @ ) I . In this case, N- A = N-$(a +@)I, (I+AF-'F=(F'+A)-'=(F'+~(a+ p ) I ) - ' , (Fig. 7.1(b)).From Definition 5.9 and Inequality (7.2),N is inside the sector

S[a + ,:

P

}-:

for P 2 a and 6 >0, and so from the summation property

of sectors (Lemma 5.2),N - i ( a + @ ) I is inside the sector S { i ( a + 6 - P ) , Definition 5.6,in terms of Euclidean norms

i ( P - a - 6). Or from

I J ( N x - ~ ( a + p ) I x ) I I ~ ~ ( P - a - ~ ) I l xfor ll, X E L ; However, the induced norm is defined by

(7.11)

7.

221

PASSIVITY RESULTS

Therefore the condition that ( N - + ( a + @ ) I ) is inside S { t ( a + 6- p), be rewritten as

t ( P - a - 5) can

lll(N-t(a + P)I)III

St(@- a 0 -

(7.12)

by taking sums over all x E L; in Inequality (7.11) Suppose now that the linear operator F is outside the sector S { - a - ' , -p-*} for p 2 a > 0, so that F-' is outside S { - a , -p}. From the summation properties of sectors (Lemma 5.2), the transformed linear operator (F' +$(a+ @ ) I )is outside S{-$(p -a),t ( p -a)}or equivalently (F-' ++(a+ p))-' is inside the sector S{-2(p -a)-', 2(p -a)-'}. As for the nonlinear operator ( N - $ ( a + p ) I ) , the induced norm equivalent condition for sectoricity can be employed,

lIl(F-' +$(a+ p)I)-'lllS 2(p - a)-'

(7.13)

for (F-' ++(a+ p ) I ) - ' inside S{-2(p -a)-',2(p - a)-'}. Now, multiplying inequalities (7.12), (7.13) gives

for N , = $ ( a + p ) and E > O . Which by the small gain theorem 6.2, is a sufficient condition for L ; input-output stability of the feedback system (7.10). THEOREM 7.3 Given the feedback system y = F ( u -N(y)), u E L; with FELA!"" characterized by matrix transfer function P(s). Then if

{

: I:

(i) N :R" + R" is inside the sector S a +-, p -- for 5 > 0 (ii) p 2 a > 0 ; F is outside the sector S { - a - ' , -p-'},

or for p > 0 > a ; F is inside the sector S{-p-', -ap'} the feedback system is L ; input-output stable. The proof for O>a follows identically that for a>O, except that inside sector conditions are utilized. This theorem was originally derived by Zames (1966). To generate graphical interpretations of the passivity theorem we assume that the linear time-invariant operators F, Q are causal so that the concepts of passivity and positivity are equivalent.

THEOREM 7.4: POSITIVITY STABILITY THEOREM The feedback system y F(u - N ( y ) ) is L;-stable if the following conditions are satisfied s(i) ( I - A B - ' ) [ { B N - ' - I + B ( F - ~ ) } - ' - A B - ' ] Q - '

=

Table 7.1 Proof of theorem 7.4 Consider the following set of loop transformations to the feedback system (7.1): Operation (Referred to the forward operator)

-

+B-', feedforward +F-@, negative feedforward (A-1-B-1

)-I,

negative feedback Extracting constants Rearranging and further extraction Introducing a Multiplier

Q

Input compensation

Forward operator

( F + B-')-'F

F F+B-'

(@+ B-')-'(G

output compensation

+ B-

-

~ - 1 - ~ - 1

A

( I + B@)(I +A @)-I Q ( I + B@)(I+ A@)-'

B-'

Q-I

Feedback operator

7.

is positive

PASSIVITY RESULTS

223

s(ii) Q(Z+B@)(Z+A@)-'is strictly positive s(iii) (@+ B-')-' :L," -+ L;.

Where matrices A and B E R " " " and @ is an arbitrary linear timeinvariant operator which satisfies the conditions: ( a ) @ has an impulse response, @ in LA!!++" and ( b ) &(jo)is normal for all frequencies w . Proof

For proof of Theorem 7.4, see Table 7.1 opposite.

Theorem 7.4 is now verified by applying the passivity theorem for positive operators to the above transformed feedback and feedforward elements, and noting that the input and output compensators are operators o n L ; by virtue of the conditions of the theorem. In the proof of this theorem the approximation or bounding matrices A, B are only required to be elements of R""".

7.2 OFF-AXIS CIRCLE CIUTERIA 7.2.1

The linear operator &o) is normal

If the linear portion of the feedback system is normal, that is, &w) commutes with its complex conjugate &o)* for all w, then Theorem 7.4 has a simple off-axis graphical interpretation by selecting @ = F. The conditions of Theorem 7.4 then reduce to: s'(i) Q - ' ( N - A ) ( B-N)-' is positive s'(ii) (Z+BF)(I+ AF-'Q is strictly positive s'(iii) ( F + B-')-':L,"+= L," (reversing the position of the multiplier). By selecting A = al and B = P I we have:

THEOREM 7.5: OFF-AXISCIRCLECRITERION FOR NORMAL OPERATORS (i) a > O

The feedback system y = F(u - N ( y ) ) ,u E L ; is L;-stable if the eigenloci of &o), w 2 0 do not encircle or intersect the critical of-axis circle passing through the points (-ap'- y, 0 ) and (-P-' + y, 0 ) in the complex plane for some y > 0 (Fig. 7.2(i)). (ii) a < O The feedback system y = F(u - N ( y ) ) , u E L," is L,"-stable i f the eigenloci of &w), o 2 0 lie entirely inside a critical of-axis circle passing through the points (-P-' + y, 0 ) and (-a-'- y, 0 ) for some y > 0 (Fig. 7.2(ii)).

224

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

FIG.7.2 Off-axis circle criterion for normal operators

Proof Case (i) a > O . Let N ’ = ( N - A ) ( B - N ) - ’ . Then it follows by standard sector arguments that N’ is incrementally inside the sector S{O, ( P - a - at-’). To establish condition s’(i) it is sufficient to show that (Q-’N’x,x)?O

VXEL;

7.

225

PASSIVITY RESULTS

If (N')-'Q is inside a positive sector, it follows that Q-'N' is also inside a positive sector. Therefore to establish s'(i) it suffices to show that

(("'Ox,

x )20

V x E LT

(7.14)

a},that is, Now, (N')-' is incrementally inside the sector S { ( ( p - a - ( ) - I , an incrementally positive operator. By a simple vector generalization of Zames theorem (1966, pp. 4 7 5 4 7 6 ) it follows that (N')-'Q is positive, hence condition s'(i) is satisfied by the above choice of operators A = al, B = 61, and a,p >O. To establish Theorem 7.5, it only remains to show that the graphical conditions of the theorem ensure that conditions s'(ii) and s'(iii) are satisfied. It has been shown (Valenca, 1978) that since fi(jo) is normal, (F+ B-')-':L; + L; if and only if the eigenloci of fi(jo)d o not encircle the point -p-' for B = 61. Q ( I + B @ ) ( l +A@)-' can be identified with F, provided condition s(iii) is satisfied. As &(jo)is a normal operator, a unitary matrix exists at each frequency to decompose &(jo)into diagonal form. It can also be shown that the same similarity transformation decomposes P ( j w )+ &a)* into diagonal form. Hence from Lemma 5.1 condition s'(ii) is satisfied if and only if

which can be recognised as a multivariable version, of Cho and Narendra's (1968) off-axis circle result for scalar feedback systems.

Proof Case (ii) a
f i ( s )=

1 s ( s + l ) '( s + 6 )

rsT:)

The eigenvalues of f i ( s ) are given by l , ( s ) = (s(s +6)(s + l))-' and i 2 ( s ) = (s(s + 1)2)-'and their Nyquist plots are shown in Fig. 7.3. If we select an arbitrary lower limit on the nonlinearity N of a = 0.64, the on-axis circle stability criterion (Theorem 6.4) yields an upper limit of p = 1.54, whereas Theorem 7.5 gives the less conservative value of = 2.63 for closed-loop stability (Fig. 7.3). Noting that f i ( s ) is not strictly diagonally dominant, and so the envelope of the Gershgorin circles of p(s)centred on '(jo)and fzz(jw)(Fig.

PI

226

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

On-axis circle

FIG.7.3 Off-axis circle criterion based upon eigenvalue plots

7.4) will lead to far more conservative values for the upper limits for N than for the eigenloci of Fig. 7.3. Figure 7.4 shows that the off-axis circle criterion based upon Gershgorin circles is almost as good as the on-axis circle criterion based upon eigenvalue plots for a system which is marginally diagonally dominant (a considerable practical computational advantage in determining closed-loop stability).

7.

227

PASSIVITY RESULTS

I

On-axis circle

\.

/ /

/

/

\.

\

I

/

I I

Q22

-1

I I

FIG.7.4 Off-axis circle criterion based upon Gershgorin’s theorem (-.-.Gershgorin’s circle envelope)

7.2.2

The linear operator &w)

B non-normal

The above graphical interpretation of Theorem 7.4 loses some of its simplicity when the operator &o) is non-normal. Additional information is required concerning the nature of the multiplier (its phase and frequency range etc) so that the multiplier can be located in a known sector. Fortunately the required information concerning the multiplier can be deduced from the eigenloci of &w). The graphical interpretation of the stability theorem is considered for two cases each dependent on the sign of the lower bound a of the system nonlinearity.

228

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

THEOREM 7.6: OFFAXISCIRCLECRITERION FOR NON-NORMAL OPERATORS Case (i) a > 0. The feedback system y = F(u - N ( y ) ) ,u EL';is L;-stable if the eigenloci of &(jw), w L 0 do not encircle or intersect a critical off-axis circle passing through the points (-K' + y, 0 ) and (-a-' - y , 0 ) for some y > 0 in the complex plane. Where a A p/(r + qa-'). b 4( q +()/(m + pp-') and ( > O , m >-pp-', and Q ( F - @ ) is assumed to lie inside the sector S{m, r } and Q is assumed to lie inside the sector S{p, q } (Fig. 7.5(a)). Case (ii) a < 0. The feedback system y = F(u - N ( y ) ) ,u E L'; is L';-stable if the eigenloci of &(jw), w 2 0 lie entirely within a critical of-axis circle that passes through the points (-b-' + y, 0 ) and (-a-' - y, 0 ) for some y > 0 in the complex plane. Where a A q / ( m + q a - ' ) , m<-qa-' and b p ( p + ( ) / ( r + p p - ' ) , r > - p p - ' , (>O. (See Fig. 7.5(b).)

Proof To establish Theorem 7.6 it is sufficient to show that the selected values of matrices A and B guarantee that condition s(i) of Theorem 7.4 is satisfied. Now, the operator E

'

[ ( I - A B -'){BN-' - Z + B ( F - a)}- - AB-']Q-'

(7.16)

can be rewritten as

(Z-AB-'){QBN-' -Q+QB(F-@)}-'-AB-'Q-'

(7.17)

We now make the following assumptions: A(i) Q ( F - @ ) is inside the sector S{m, r} A(ii) Q is inside the sector S{p, q}; p r O A(iii) A = aZ, ;a E R A(iv) B = bZ,; b E R ,

N is by definition incrementally inside the sector Case (i) a>O(a>O). Then QBN-' is S{bpp-', bqa-'},-O is by definition inside the QB(F-@) is inside the sector S{bm, br}. By the sum rule Q B N - ' - Q + Q B ( F - @ ) S{bpp-' - q + bm, bqa-' - p + br}. We now make the assumption,

S{a, p}. inside the sector sector S{-q, -p} and is inside the sector

A(V) pbp-' - q + bm >0,

so that [QBN-' - Q + QB(F-@)]-' is inside the sector or

S{(bqa-'-p+br)-', (bpp-'-q-brn)-') (Z-AB-')(QBN-'-

Q + QB(F-@))-'

7.

229

PASSIVITY RESULTS

\

lil a > 0 , az-0

t"' liil a < 0. a < O

I FIG.7.5 Off-axis circle criterion for non-normal operators

is inside the sector

'{ b( bqa

b-a -p

+ br) ' b( bp0-I

-q

+ bm)

This implies that E has a lower sector bound of ( b - a)b-'(bq-'a - p + br)-' -a(bp)-'.

230

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

To ensure that operator E is positive, it is sufficient to satisfy the following inequalities ( b - a)b-'(bq-'a - p + br)-' - a(bp)-' 2 0 (7.18) The first inequality is satisfied if b is selected as b = ( q + ( ) / ( m+pep') for 5 > 0 and m > -pp-'. And the second inequality is satisfied by selecting a = p / ( r + qa-'). Case (ii) a < O ( a < O ) . The analysis for the case a > O was relatively straightforward. The problem is now more complex because outside sector concepts are required. A result which might appear useful in the manipulation of outside sectors is the Zames (1966, p. 234 property (v) (iii)) sum rule; unfortunately it is wrong, and numerous counter examples can be constructed. The proof fails due to the lack of symmetry of the triangle inequality. The only conditions under which this sum rule holds are if both the upper bounds are positively infinite or if both the lower bounds are negatively infinite. In this context Lemma 5.3 enables inside and outside sectors to be manipulated in a mixed manner. Let us return to the proof of Theorem 7.6 for a (0. QBN-' is outside the sector S{bqa-', bpp-I}. And by Lemma 5.2 Q B W ' - Q Q B ( F - @ ) is outside the sector S{bqa-' - q + bm, bpp-' p + br} provided that b p - ' - q + bm < bpp-' - p + br. Now, introduce assumption

+

A(Vi) bqa-' - qbr < 0. This implies that operator {QBN-'- Q + QB(F-@)}-' is inside the sector S{(bqa-'-q+br)-', ( b p p - ' - p + b r ) - ' } . Therefore (I-AB-'1 { Q B N - ' - Q + A B ( F - @ ) } - ' is inside the sector

b-a s L ( b q a - l - q + b m ) ' b ( b p pb- '--ap + b r )

I

which implies that the operator E has a lower sector bound of

( b - a ) b - ' ( b q a - ' - q + bm)-'-a(bp)-'. Hence the sufficient conditions for the positivity of E are:

( b - a ) b-'(bqa-'- q + bm)-' bqa-' - q + bm (0 bpp-' - p + br > O

- a(bp)-' 2

0

These inequalities are satisfied if a and b are chosen as

a = q ( m +q a - ') - ',

b = (p + ( ) ( r + p@-')-',

m < -qa-'

(> 0.

0

7.

PASSIVITY RESULTS

23 1

7.2.3 Determination of the sectors of Q ( F - Q ) and Q For Theorem 7.6 to be of practical use, a method has to be devised to determine the sectors of Q ( F - Q ) and Q. Since Q is inside the sector S { p , q}, p 2 0 , then, if (F-Q) is inside S{-p-', rq-'}, Q ( F - Q ) is inside the symmetric sector S{-r, r}. Lemma 5.5 provides a frequency domain method of determining the symmetric sector of a convolution operator; this lemma together with the following information on Q enables us to determine the sector of Q ( F - Q ) . Case (i) Q is an RC multiplier, then p = 1, q = prc and c = 181 (7r/2)-' for 8 the angle in radians between the tangent to the circle on the real axis and the real axis (Fig. 7.6). Two further pieces of information concerning the critical circle are now required: first, the required tolerance 5 of 8 in radians (allowable deviation in 8 consistent with stability); second, the frequency range over which the multiplier is required to operate. This is easily determined as the smallest frequency w, at which the eigenvalue points h,{&(jw)} (w E [us, w]) are not contained in the family of off-axis circles generated as 8 takes on values in the interval [-7~/2,0]. p and r are now chosen such that for w E (0, w,]

+ 2 4 1 - c) log p < 5

(7.19)

where a>O, p > l and rER+. Case (ii) Q is an RC multiplier, then p = P - ~ ,q = 1. 8 now lies in the

FIG.7.6 Sector determination

232 interval

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

[0

, that is, the circle centre lies above the real axis c =

9

3

t9(7~/2)-'and the analysis now follows that for the RL case for determining r and c and hence p. It should be noted that the conditions of Theorems 7.5 and 7.6 do not coincide when @ = F. This apparent disparity follows from the more involved approach necessary in establishing Theorem 7.6 to allow for the presence of a composite linear time operator. Lemma 5.5 enables the general non-normal operator Theorem 7.6 to be readily implemented, although the end result will be more conservative in its stability estimates. If the computation of the eigenvalues of F is possible, a tighter result can be obtained by using the condition for operator F to be inside sector S{-r, r} of

to calculate r. A more conservative set of results with the benefit of minimal computation is possible by applying Gershgorin's theorem to &Go)in order to obtain bounds upon its eigenvalues. An on-axis circle interpretation of Theorems 7.5, 7.6 can be readily made by setting the multiplier Q = I in the above.

THEOREM 7.7: F A NORMAL OPERATOR (i) a >O. The feedback system y = F(u - N ( y ) ) is L,"-stable if the eigenloci of &o), o 2 0 do not encircle or intersect the critical on-axis circle passing through the points (-a-'- y, 0 ) and (-p-' + y, 0 ) in the complex plane for some y > 0. (ii) a 0. THEOREM 7.8: F A NON-NORMAL OPERATOR (i) a >O. The feedback system y = F(u - N ( y ) ) is L;-stable if the eigenloci of &(jo), o 2 0 do not encircle or intersect a critical on-axis circle passing through the points (-b-' + y, 0 ) and (-a-' - y, 0 ) for some y > 0 in the complex plane. Where a A l / ( r + a - ' ) , b P ( l + C ) / ( m + P - ' ) and & > O , m+P.'>O and ( F - @ ) is assumed to lie inside the sector S{m, r}. (ii) a
7.

233

PASSIVITY RESULTS

(-b-' + y, 0 ) and ( - a p 1- y, 0 ) for some y > 0 in the complex plane. Where a 4 l / ( m +a-'), m < -a-' and b (1+ e ) / ( r + p-'), r > -p-', (> 0. The proofs of these theorems follows exactly that of Theorems 7.5 and 7.6 with Q = I, p = 1, q = 1 and r = m. These theorems are simpler to implement than either the respective off-axis circle theorems or the associated incremental gain theorem of Section 6.4 for non-normal operators F, since the norm K ( @ )of the error of approximation of F by @ does not have to have computed in Theorem 7.8.

Example 7.2 To illustrate Theorem 7.8 consider the feedback system y = F(u - N ( y ) ) with linear transfer function matrix (s + 1) 1 F(s)=--(2s + 1) - 0 3 s - 1) (0.5s + 1) L

(s+ 1) (3s + 1) ( 0 5 s + 1)J

Select as an approximation operator the normal transfer functions matrix

(2s + 1) whose eigenvalues are obviously k,(s) = ((2s + l)(s + l)}-' and kJs) = ((2s + 1)(0-5s+ l)}-'; the associated Nyquist plots are given in Fig. 7.7. Lemma 5.5 gives the symmetric sector limits of (fi-4) as rs0.707. Typical limits on the nonlinearity N(y) are: (i) for p 2 a > 0: p = 1.24, a = 0-64; (ii) for /3 > 0 > a : p = 3.41, a = -0.59. A less conservative on-axis circle stability criterion can be derived if frequency-dependent sectors for the operators F and ( F - @ ) are employed (See Section 5.7). Given that the operators F, ( F - @ ) , @ € LA!!"" have transfer function matrices &o), (&) -&(io)), &(jw) and that operator ( F - @ ) lies inside S{m(jw), r ( j o ) } , for m ( j o ) , r(jw) real scalars, and the normal operator @ lies outside S{-a-' - r ( j o ) , - @ - I - m ( j o ) } ; Lemma 5.4, with the following identities, c1 = -;(ap1+ r + p-' + m), rl = ;(a!-' + r - p-' - m), c2 = $(r + m), r2 = -&m - r ) states that: LEMMA 7.1 If F is a stable operator such that &w) satisfies the Nyquist criterion with respect to the point (-$(a-'+p-'). 0 ) then F lies outside the sector S{-a-', - @ - I } i f

II(Rjo)- @ j w )

-t(r(jw) +m(jo))~)x(jo)II s t ( r ( j w ) - m&))

Ilx(io)ll

(7.20)

234

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

FIG.7.7 Example 7.2, graphical illustration of Theorem 7.8

and

+ + p-' + m(iw))I)x(iw)ll ?$(ap'+ r(jw>- p-' - m(jw)Ilx(jw)ll

Il(&(jw) +$(a-' r(jw)

(7.21)

for all w E R and x(jw)E L,". As 660) is a normal operator, a unitary matrix exists at each frequency to decompose &(jw) into diagonal form, and so condition (7.21) can be written as

I l ~ ~ { & ( j w ) } + f ( a - ' + r ( i o ) + ~m(jw))ll -'+ >&a-1

+ r(iw) - p-'

- m(jw))

(7.22)

for i = 1 , 2 , . . . , n, and for all w E R. Comparing this inequality with (6.42) we have the following graphical interpretation:

LEMMA7.2 A normal operator &(jw) is outside the sector S{-a-'-r(jw),--BPI- m(jw)}if and only if the eigenloci hi{&(jw)},( i = 1,2, . . . , n ) are exterior to the circle C[-a-' - r(iw), -p-' - m(iw)l. Combining the conditions of Lemmas 7.1 and 7.2 it is clear that the eigenvalues of P ( j w ) are exterior to the circle C[-a-', -p-']. Similarly these two lemmas taken together with Theorem 7.3 enable us to state a generalized on-axis circle criterion that does not involve the approximation of F by @.

7.

PASSIVITY RESULTS

235

THEOREM 7.9 ( p 2 a! >O) The feedback system y = F(u -N(y)) is L;stable i f the eigenloci of &(jo), o E R do not intersect nor encircle a critical on-axis circle C[-a!-' - r ( j o ) , -p-' - m ( j o ) ] in the complex plane. Where m(jw) p-' > 0 and (&a) - &Go))is assumed to lie inside the sector S { m ( j o ) , r(io)}.

+

Note that Theorem 7.8 contains the above as a special case when m = min m ( j o ) , r = max r ( j o ) . We see that the cost in utilizing the ap0

0

proximation of non-normal operator F by normal @ is an increase in the diameter of the critical circle by the sector of ( F - @ ) (compare circle diameters in Theorems 7.7, 7.8 and 7.9). An obvious corollary to Lemma 7.2 for normal operators that are inside sectors S{rn(jw), r ( j o ) } is:

LEMMA 7.3 A normal operator &Go)is inside the sector S { m ( j o ) ,r&)} if and only if the Nyquist plots of Ai(&jo)) ( i = 1,2, . . . , n ) lie inside or on the circle C [ m ( j o ) ,r(jo)l. Clearly the smallest on-axis circle C [ m ( j o ) ,r ( j o ) ] that contains all the eigenvalues hi of & ( j w ) defines the minimum sector S { m ( j o ) ,r ( j o ) } .

7.3 OFF-AXIS CIRCLE CRITERIA-MULTIPLIER FACTORIZATION While the off-axis circle criterion for normal operators (Theorem 7.5) has a simple and readily implementable graphical interpretation, the equivalent result (Theorem 7.6) for non-normal operators crucially depends upon the determination of the sectors of the multiplier Q and the product of the multiplier and the degree of approximation of the linear loop operator F and some normal approximating operator @. The direct involvement of the sector of Q and its associated conservative estimation via Lemma 5.5, effectively reduce the practical advantages of this multivariable off-axis circle stability criterion over those of the on-axis circle criterion of Theorems 7.8 and 7.9 (also those of Section 6.4). We show in this section that by utilizing the multiplier factorization theorem of Section 5.6 for a multiplier with an RL,R C realization or Popov structure that a series of generalized multivariable off-axis circle stability criteria independent of the sector of Q can be readily generated. To accommodate non-normal linear operators, F, some aspects of optimal decomposition of F into ( F - @ ) + @ , for 0 a normal operator, are given together with conditions for estimating the minimum sector width of ( F - @ ) .

236

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Consider the feedback system (7.23)

Y =F(u-N(y))

that satisfies the conditions of equations (7.1) and (7.2). Utilizing the multiplier factorization theorems of (5.4) and (5.5) in the passivity Theorem 7.1 yields the following conditions for L,"-stability of (7.23).

, 20 s'(i) ( Q * ) - ' Q + ( ~ + B F ) ( I + A F ) - ' xx)T

(7.24)

for all x E L,",, T ER, and there exists a 6 > 0 such that s'(ii) ((N- A ) ( B- N)-'(Q+)-'Q?x,

x)T

2S

11+112

(7.25)

for all x E L2nerT ER, and Q+, Q?, (Q?)-', (a:)-' :L,"+ L,". The multiplier Q = Q-Q+ is selected such that (7.26) (7.27) which are respectively anti-causal and causal operators. This choice of multiplier, although having a real time domain description, can be made to appear as a pure complex number in the frequency domain (see also Cook, 1976) and can be realized similar to the multipliers of Zames (1966) and Cho and Narendra (1968) as either RL or RC type realizations, depending on whether Q- and Q+ are defined by (7.26) and (7.27) or in their inverse forms. The nonlinear operator was defined in Section 7.1 to be incrementally inside the sector S{a + 5, p - 6). .$>0; by selecting A = al, B = PI we showed by standard sector arguments in Section 7.1 that the operator = (N- A ) ( B- iV-' is strictly incrementally passive. By Definitions U or an (7.26) and (7.27) the causal multiplier (Q+)-'Q? is either an I R C multiplier, and the composition operator +(Q+)-'Q? is causal and satisfies the strict positivity condition of (7.25) automatically via the system definitions of N and Q. L; stability of the feedback system y = F(u - N ( y ) ) therefore depends only upon the satisfaction of the condition

+

s'(i) (( Q*)-' Q+(l + BF)(I + AF'-'x, x)T

2 0,

VXE

VTER

As with condition s'(ii) all the above operators are causal and so a positivity condition may be utilized to infer the equivalent passivity condition.

7.

237

PASSIVITY RESULTS

LEMMA 7.4 The passivity condition

((Q"-'Q+(I+BF)(Z+AF)-'x, x ) ~ ~ O , V x EL;,,

V T ER

is equivalent to

@+(I+ B F ) ( I +AF)-'Q_Y, y)'o,

where

y=QI'x,

Proof Since

(QT)-'Q+

v y E L;

(7.28) (7.29)

XEL;

is causal, Inequality (7.24) is equivalent to

((Q*)-' Q+(Z+ BF)(Z+ A F - ~ X ,

X ) 2 0,

v xE L;

which in turn is equivalent to

(Q+(Z+BF)(Z+AF)-'x, Q I ' x ) r O , Defining y

=

VXEL;

Q I ' x the above becomes (Q+(Z+BF)(Z+AF)-'Q-y, y ) ? O ,

0

V y EL;

Provided that (I+ AF-': L; + L;, Inequality (7.28) can be written equivalently as: or

( Q + Q _ ( I + BF)x, ( I + A F ) x 2 0, (Q+Q_(B-' + F ) x , ( A - ' + F ) x 2 0 ,

V x E L; V x E L;

(7.30)

provided that a # 0. By Parseval's theorem the above inequality is equivalent to:

-'+ P(jw))x(jw))Edw} 2 0 (7.31) for all x ( t )L; ~ and where ( x , Y ) p x~ * y . A sufficient condition for Inequality (7.31) is

E }0 Re{(Q+Q-(B-' +l%w))x(jw), ( A .'+ E ( j ~ ) ) x ( j w ) ) 2

(7.32)

for all x(jw), w E [0,m]. Close examination of the multiplier Q ( j w ) reveals that it shares the same phase characteristics as the multiplier utilized by the originators of the off-axis circle theorem (Cho and Narendra, 1968). By a suitable choice of the multipliers parameters pi and qi, the multiplier Q ( j w ) can be made arbitrarily close to the complex number exp ( j e ) for w E [a, b ] , a subinterval of (0, m) and a constant 8 E (-7d2, d 2 ) . Strictly, the inverse multiplier Q - ' ( j w ) should be considered in parallel with Q ( j w ) to enable

238

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

8 to encompass all (-7r/2, 7r/2), but only a redefinition of QQw) as Q-'(jw) is necessary to accommodate this requirement. ) produces The restriction of w to a closed subinterval of ( 0 , ~only difficulties if an off-axis disc is required in the vicinity of the frequency points w = 0 and w = 03. This frequency constraint and its implications will be dealt with in the sequel. The frequency inequality (7.32) can now be seen to be satisfied if the following two conditions hold:

) )0,~ s;(i) Re (exp (jO)(B-' + P(jw))y(jw),( P ( j w )+ A - ' ) y ( j ~ 2

(7.33)

LEMMA 7.5 The condition s;(i) is equivalent to

( ( P ( j w ) + c - j r tan e ) y ( j w ) , ( P ( j w ) + c - j r tan 8)y(iw))E

For all y(jw), and w E [a, b ] , 8 E (--7r/2,d 2 ) . Where c = ~ ( c x - + ' @-') and r =t ( ( y - 1 - p - 7 . Proof For the same system conditions Inequality (7.33) is equivalent to (omitting arguments etc.) Re(cos 8 + j s i n 8 ) ( ( P y , P ~ ) ~ + a - ' @ - 'Y() yE ,+ C X - ' ( P Y , On setting ( y , f i y ) ,

= p +ja, the

cos e { ( P y , P y ) E + (Y-'P-'(Y,

y)E

Y ) E } ~ O

above inequality becomes

+ p(@-'+ CX-')} +sin 8 ( a ( u - ' - a @ - ' ) r O

Defining r = $(a-'- @-I) and c = &a-' on dividing throughout by cos 8 to:

+ @-I),

Inequality (7.36) simplifies


( P y , P ~ ) ~ + ( c - tan j r e ) ( p + j a ) + ( c + j rtan e ) ( p - j a ) r2 + c2+r2tan28-)(Y. Y>E2 0

(

C O S ~8

(7.36)

7.

239

PASSIVITY RESULTS

or alternatively as

(Fy, F Y )+~(c - jr tan 8)(y, Fy), + (C + jr tan e)(Fy,Y

) ~

or equivalently as:

0 From Definition (5.6a) we see that Inequality (7.37) is a frequency domain outside sector condition on F with a complex centre -c + jr tan 8 and radius rlcos 8. The second sufficient condition (7.34) for L,"-stability of the feedback system (7.23) can now be eliminated if caution is taken at the frequency limits w = 0 and w = w. Condition (7.34) is governed by the behaviour of Arg { Q ( j w ) }=

k =O

{tan-

1

(:)-tan-'

(E)}

in the vicinity of the frequency points w = 0, w = w (qk and pk are as defined by Cho and Narendra, 1968). Hence it can be easily shown that Arg { Q ( j w ) }-+ 0 as w -+ 0 or w + w. The effect of condition (7.23) upon the off-axis circle criterion to be established in the sequel can be ignored provided that certain eigenvalue loci (to be defined) in an arbitrarily small region of w = 0 and w = w, do not intersect the family of off-axis discs which tend to an on-axis disc as w + 0 and w + w. This constraint will be assumed to be implicit in the various circle stability criteria derived in this section that depend upon the multiplier Q defined by (7.26) and (7.27). To generate graphical stability for non-normal linear operators F, we utilize the decomposition &)

=&(jw)+(p-&)(jw),

for ~ E L A ! ! ~ "

(7.38)

where the approximation operator is normal. Before establishing the graphical interpretations of stability condition s'(i), (7.37), we define off-axis circles:

Definition 7.1 The off-axis circle Co[-I - a-', - m - p-'3 is defined by drawing any desired circle C through the points (-ap',0) and (-p-', O), the centre of the circle C, is then located by adding (-+
240

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

FIG.7.8 Definition off-axis circles C,,

THEOREM 7.10 The feedback system y = F(u - N ( y ) ) is L;-stable i f the eigenvalue plots, hi(&(jw)) do not encircle or intersect an of-axis circle C , , [ - l - a - ' , - m - @-'I, where ( F - a ) is inside the sector S{m, l}, for o 2 0 and 1 and m real scalars. Proof Following the same arguments used in deriving Lemma 7.1, based on the normality of &(jo),it can be shown that &(jo)satisfies

(7.39) for all w E [a, b ] , and x(io) and 11.11 denotes the Euclidean vector norm. From the conditions of Theorem 7.10, ($-&) satisfies the inequality

for all x ( j w ) , and o E R. Adding Inequalities (7.39) and (7.40) and employing the triangle inequality, it can be deduced that

r lI[$(jo)+cz-jr tan 0 ] x ( j o ) l l 2 - Ilx(jo)ll cos 8 for all o E [a, b ] and x ( j w ) . This is equivalent to the sufficient L;-stability condition of (7.35) for the feedback system (7.23). 0

7.

24 1

PASSIVITY RESULTS

t

Img

FIG. 7.9 Graphical interpretation of Theorem 7.10 (frequency independent sectors)

Theorem 7.10 is based upon a frequency-independent sector condition to establish an off-axis circle criterion for non-normal operators (Fig. 7.9). Frequency-dependent sectors (see Definitions 5.6a, 5.7a) can be readily incorporated within the same framework of Theorem 7.10: THEOREM 7.11 The feedback system y = F(u - N ( y ) ) is L;-stable if the do not encircle or intersect an of axis circle eigenvalue plots, A(&@)) C o [ - l ( j w ) - a - ' , - m ( j w ) - P - ' ] , where (P-6) is inside the sector S{rn(jw), l(jw)} for w r O and rn(jw), l ( j w ) ~ R .

Proof Proof follows identically to that of Theorem 7.10, except that 0 frequency-dependent sectors are utilized (Fig. 7.10). The above off-axis circle criteria offer simple geometric means of assessing closed-loop stability; unfortunately, the generality of the underlying theory is sacrificed for graphical clarity-despite the fact that the criteria are less conservative than currently known o n axis circle criteria. Very general circle criteria with tighter stability bounds can be readily derived, the cost being increased analytic effort by the designer. To illustrate the non-uniqueness of the multiplier choice, a Popov-type multiplier is now selected instead of an RL or RC multiplier, thus generalizing to the multivariable setting the scalar result of Hsu and Meyer (1968).

242

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

I

Real

FIG.7.10 Graphical interpretation of Theorem 7.11 (frequency dependent sectors)

By operations analogous to those outlined in the derivation of conditions (7.24) and (7.25), sufficient conditions for the L;-stability of the feedback system y = F(u - N ( y ) ) can be expressed as s"(i) ( Q ( I + BF)(I + A F - ' x , x ) 2~0 for all x E Lze, T ER. And there exists a 6 > 0 such that s"(ii) ((N- A ) ( B- N)-'Q-'x, x)=

26

llx#

for all x E L;e, T ER, and Q, 0-' :L; 4L'; By defining Q as the Popov multiplier Q ( j o ) = ( l + j q ) I , 4 2 0 , the above condition s"(ii) is automatically satisfied by standard sector arguments for A = aI,B = PI. And so by utilizing the same derivation used in establishing Lemma 7.5 a sufficient condition for condition s"(i) is

for all x ( j o ) , o r O , and ( F + A - ' ) - ' : L , " + L ; . Where c, r and O ( j o ) are defined by c = &a-1+ P-'), r = ;(ap1 - P-') and O(jo) = tan-' [(qo)-'].

Definition 7.2 The Popov circles C,[-l(jw) - a-', -rn(jw) - @-'I, are defined by the family of circles determined by selecting some q 2 0 (and

7.

PASSIVITY RESULTS

243

FIG.7.11 Definition of Popov circles C,

hence O(jw) = tan-' [(qw)-']) for w L 0. The Popov circles are constructed from an off-axis circle, C, drawn through the points (-a-',O) and ( - p - ' , 0) with a tangent at the point ( - @ - I , 0) at an angle of O(iw) to the real axis. The centre of the Popov circles C, are located by adding ( i ( m ( j w ) + l ( j w ) )0) , to the centre of circle C and by increasing the diameter of C by ( l ( j w ) - m ( j w ) )for w r O (Fig. 7.11). By analogy to Theorem 7.11 we have:

THEOREM 7.12 The feedback system y = F ( u - N ( y ) ) is L,"-stable i f the eigenualue plots, A(&(jw)) do not encircle or intersect the Popov circles C,[-l(jw)-a-', -m(iw)-p-'] where (k-6) is inside the sector S{m(jw),l(jw)}for w L O and m(jw), l(jw)E R (Fig. 7.12). To implement Theorems 7.10-7.12 a systematic mechanism for selecting the approximating normal operator CP must be considered. To achieve stability the radius of the sector of (fl-&) must be minimized with respect to CP or equivalently the matrix (F-CP) should be sparse with small entries. To achieve this, the largest normal structured operator CP contained within F should be selected. As already observed in Section 6.3, suitable choices for CP are diag (k),i(k-P*) or i(fl+fl*), or as unitary matrices. By reversing the signs used in the proof of Lemma 7.2 we are able to state: LEMMA 7.6 The normal operator &(jw) is inside the sector S{-m(jw), -l(iw)} if the circle C[-m(iw), -l(jw)] contains all the eigenvalue plots ~ ~ ( & ( i w()i )= I, 2 , . . . , n ) .

244

THE STABILITY OF INPUTOUTPUT DYNAMICAL SYSTEMS

FIG.7.12 Graphical interpretation of Theorem 7.12 (Popov circle criteria)

Clearly, then, the bounds of the smallest sector of the Hermitian matrix &Go) are m ( j w ) = min {Ai(&(io))} and I(iw) = max {Ai(6(jo))}. This sug1

1

gests that if (3- 6)can be selected as Hermitian, then its smallest sector can be easily determined. Select 6 ( j w ) = $(&) - P ( j o ) * )+ a(jw)Z for a E C', then ( F - & ) ( j o ) = $ ( F ( j o ) + 3 ( j w ) * ) -a ( j o ) z is also normal. And since the eigenvalues, h i @ ) , of the Hermitian matrix real, then the eigenvalues of (P-&)(iw) are { A i ( j o ) - a ( j o ) } .So that the sector bounds of (P-6) are m ( j o ) = min Ai(jw)- a ( j o ) and I(jo)= max A i ( j o ) - a ( j w ) . Although the sector

&l'(jo)+k(jo)*) are I

I

width of ( F - 6 ) is independent of a ( j o ) , the sector may be made symmetrical by selecting a ( i o ) = f(min h i ( j o )+max Ai(iw)); in this case I

I(jo)= $(max hi(jo)-min h i ( j o ) )= $ m ( j o ) . L

1

I

A more generalized technique for establishing the minimum sector (3-0) can be derived using the notion of the measure and numerical range of complex matrix defined on LA!!"". S { m ( j o ) ,I ( / @ ) } of operator

7.

245

PASSIVITY RESULTS

Defining

It is not difficult to see that max A2(jw)2 p 2 ( j w ) , X(lW)

w 2 0.

By using the Euclidean norm interpretation of Definition 5.6a, the above definitions for A(jw) and p ( j w ) give the alternative conditions for (fi-&) inside sector S{m(jw), I(jw)} as A 2 ( j w ) - ( m ( j w ) +I ( j w ) ) p ( j w ) + m ( j w ) I ( j w ) l O ,

V x ( j w )# O

(7.43)

Explicit determination of m ( j w ) and I(jo)from Inequality (7.43) is complicated by the fact that A(jw) and p ( j w ) are in general maximized at different values of x ( j w ) ( # O ) . However, if we let the eigenvalues of the Hermitian matrix f[(fi-&) +(fi-&)*] be ordered such that A,(jw) > A,-,(jw)>.. . .>Ai(jw), then clearly p ( j w ) ~ [ A , ( j w ) , A,(jw)] for any x ( j w ) # 0. If p,(jw) is the value for p(jw)which maximizes the left hand side of (7.43) when A(jw) is bounded from above by the maximum principal gain Arnp(jw)of (fi-&), then a sufficient condition to satisfy (7.43) which gives an optimum choice for sector bounds I(jw) and m ( j o ) is A k ( j w ) - ( m ( j w ) + I(jw))prn(jw)+ m(jw)l(jw)= 0

(7.44)

by minimizing the sector width I(jw)- m ( j w )= -

A kp(jw>+ m (jw)2- 2 m (jo)prn ( / w ) prn (b) - m(jw) A Lp(/w) + I(jo)2 - 2 l(jw)Frn( j w ) 1( j w )- prn ( j w )

(7.45)

Minimization of the sector widths with respect to I(jw) and m(jw) yields I(jw)= p r n ( j ~ ) + ( A k p ( i ~ ) - ~ k f , ( j ~ ) ) ” 2 (7.46)

m ( j o )= prn(jw)-(Akp(jo)-pk(/w))1’2

(7.47)

~. with an associated minimum sector width of 2(Aip(jw)- p k ( j ~ ) ) ”All that remains is to select an appropriate p , ( j w ) ~ [ A , ( j w ) , Al(jw)] to satisfy the sector condition (7.43). Three possibilities arise for the eigenvalues of

246

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

a[(P-&)+(fi-&)*], (i) A,(jw)>O (ii) A,(jw)O, (ii) p,,,('jw)=A,(jw) for h,(jw)
THEOREM 7.13 If N is diagonal and lies incrementally inside the sector S{A + SI, B - S I ) for 6 > 0 , A = diag (q),B = diag (Pi) and Pi > ai > 0, and (fi-6) is inside sector S{M(iw),L(iw)}, where M(iw) = diag { rq (jw)}, L ( j w )= diag { 4 ( j w ) } ; 4, rq E R, and &(iw) = diag Gi(iw)}. Then the feedback system y = F(u - N ( y ) ) is L;-stable if the Nyquist plot of fii(jw)do not encircle or intersect an of-axis circle C,,[-&(jw) -a;', - r q ( j w ) - P;'] for i = 1,2, . . . , n. L and M are chosen such that ( M L ) = k ( A - ' - B - ' ) G , for G Ad ia g { ( co sei)-'}, e, E ( - w / ~ , d2). Proof Some modifications to the proofs of Theorems 7.10, 7.11 is necessary; the multiplier Q is redefined to allow differing phase for i = 1 , 2 , . . . , n, whilst still remaining diagonal. Real, constant, and diagonal scaling factors are removed from the forward and feedback system operators. These operations do not invalidate the modified versions of conditions of s'(ii) (7.25) and the analagous condition to (7.24) is: ((Q*)-'Q+(B-' +F ) ( A- '+F)-' x , x),20 (7.48) for all x E L;=, and T ER. Provided that (F+A-')-' :L,"+ L;, (implicit in Theorem 7.12) a sufficient condition for (7.48) to hold is II[fi(jw) + C - j z ~ l x ( i w ) lIlZGx(iw)ll l~ for all w E [ a , b ] , and x(jw). Where O A d i a g (tan ei), 0, E (-d2, d 2 ) ; ZAdiag['(a;'-P;')], Cadiag[$(a;'+@;')]; GAdiag[(cos ei)-'], for i = l , 2,..., n. The operator a,by the conditions of Theorem 7.13, satisfies II[Njw) +

c+%+ ~ ) ( j w-j~OIx(i~)ll )

r ( I [ Z G+a(L -M )(j w )]~ (i w )l l(7.49) for all w E [a, b ] , and x(jw). By assumption the operator ( F - a ) satisfies, -II[(E-&)(jw) - &M +L) ( jw ) ] x ( iw ) ( l2-ll%(L-M)(iw)x(jw)ll (7.50)

7.

PASSIVITY RESULTS

247

FIG.7.13 Graphical interpretation of Theorem 7.13

Again, combining Inequalities (7.49) and (7.50) through the triangle inequality and invoking the definition M - L = k(A-' - B-')G, we conclude that

Il[fi(jo) + C -JZ@lx(jo>llz IIZGx(jo)Il for all o E [a, b], and x ( j o ) . This is a sufficient condition for inequality (7.48) to hold (Fig. 7.13). Analagous results to the above can be simply derived for the Popov multiplier. 0 For the special case of nonlinearities confined to the sector S(0,p } the above generalizations lead to alternative Popov-type stability criteria. All the results in this section relate to direct Nyquist-type interpretations; by utilizing a conformal mapping for p > a > 0, inverse Nyquist-type stability criteria can be readily derived. The decomposition of F into ( F - @ ) + @ yields criteria that can easily accommodate additive perturbations AF in F, by just adding A F to the operator ( F - a ) in the above stability criteria and in the estimation of the optimum sector bounds of ( F - @ ) through Inequalities (7.46) and (7.47) (see Kouvaritakis and Husband, 1981).

248

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

MULWARIABLE POPOV CIU’IERION

7.4

Consider the feedback system y = F(u - G N ( y ) ) ,

u E L;

(7.51)

with N a nonlinear Lipschitz continuous map on L; to L; such that a i ( y i - y l ) ~ N ( y ) i - N ( y ’ )~i

P i ( y i - ~ f )

(7.52)

for all y , Y ’ E L; and i = 1 , 2 , . . . , n, N(O)=O. The loop transformation theorem applied to the system (7.5 1) for some N o € R “ ” “gives

z + FGNJ

y =(

FU

-

z+ FGNJ

(

FG (N - N ~ )y )(

Suppose that G E K ( O ) ” ~but “ F contains some poles in the right half s-plane; the Nyquist encirclement theorem for the linear system y

= (I

+ FGN,,)-~FGU= M U

shows that (I+ F.?-ZNJ’FG exists so that y

(7.53)

= u’-MN’(y)

where u’ = (Z+FGN,)-’Fu, and N = ( N - N , ) ( y ) . Suppose that No = diag(ai), then Inequality (7.52) can be rewritten as O
or (Pi

-ai)(yi -Yi)(”(Y)i

-ai)(Yi

-Y;)

- N f ( y ’ ) i ) z ( N f ( y )-N’(y’)i)*ZO i

This last inequality implies that

where L = diag (Pi- a i )is a positive definite matrix. From (7.53) u’=y+M”(y)

If u EL;, assume that u’, u’ E L ; and that y EL;, and let M, D M : I$,+ L;, where D M : x + ( d / d t ) ( M x ( t ) )Then . u‘=y+DMN’(y)

7.

249

PASSIVITY RESULTS

But

T

(QY/N’(y))T

=

N’(Y)~QY dt vrr)

=

( N ’ ( Z ) ) T Qdz

=

PI

Y(0)

which may be made positive by suitable choice of Q. If Q = Z , this ~Q condition is restrictive since the path independence of J ~ N ( z )dz essentially requires N to be a gradient of some scalar valued function. This is always the case if N i ( z )depends only on zi, which may not be true, hence the requirement that Q be non-diagonal for a generalized N . Therefore if for some 6 > 0 ((u’+QD)Mx + L - ’ x / x ) ~ IlxIIF ?~

then

(7.54)

((u’+Q i r l x ) ) z S IIxIK-Pi

(7.55)

and u‘, u’EL; implies that N’(y)€L;; if in addition M, D M : L ; - + L , ” , then also y, Qy E L;. The L,” stability criterion (7.54) is a positivity condition, which is equivalent to the requirement that the matrix p ( w ) = a ( j c o ) + a * ( j w ) is positive definite for all w where a ( j w ) = (I+j w Q ) & f ( j o ) + L - ’ and M = { mii}= ( I + FGNJ’FG. Note that aii= (1 + j w q i i ) m i i ( j w ) for i # j then

aii= ( 1 + j w q i i ) m i i ( j w ) + ( p i-ai)-’ for p.. = ”

{

i =j

( 1 + j w q i i ) h i i+ (1 - j w q i i ) h F , i # j

2[Re ((1 + j w q i i ) h i i } + ( P i - a i ) - ’ ] ,i

=j

(7.56)

for i, j = 1 , 2 , . . . , n. Define the complex Popov function h l i ( j w ) by R e { hli(j w ) } = Re { hii( j w ) } Img{h&(jw)}= w Img { h i i ( j w ) }

(7.57)

Clearly, from (7.56) if the loci of h ; ( j w ) lies to the right of the straight

250

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

A

f

w ~ m g{rniiIjwI)

slope qu’

FIG.7.14 Multivariable Popov criterion

line of scope 4;’ through the point (-(pi -ai)-*,0 ) then pii(o) is positive for all w (Fig. 7.14); this is the scalar Popov stability criterion. Define a matrix Z = {zij(w)} such that

then lzijl2 lpijl for all w E R , and i, j = 1,2, . . . , n. The eigenvalues of the matrix Z are by Gershgorin’s theorem contained in the union of circles centred on pii of radii

n

1 lzijI for all

j=l j#i

j=1 j#i

j=1 j#i

i. But since

j=1 j#i

the eigenvalues of P are contained within the Gershgorin circles of Z. If in addition n

n

j=1 jZi

j=1 j#i

and is positive it follows that the eigenvalues of Z and P are positive and hence P is positive definite (a condition for L;-stability of feedback

7.

25 1

PASSIVITY RESULTS

system (7.51)). We are now able to state for the above definitions of qii, rhji and rhl

THEOREM 7.14 The feedback system y = F ( u - G N ( y ) ) , U E L ; is L;stable if the envelope of the Gershgorin circles of radii

i#i

i#i

centred on the Nyquist plots of m&(jw) lie to the right of the straight line of slope q;' through the point (-(pi-ai)-',0) for all i ; assuming that M E K(o)nxn. I f qii = 0 for all i the above multivariable Popov stability criterion is identical to Theorem 7.2 for a = 0, based upon the incremental or passivity approach. This is not surprising, since in the above derivation a combination of the incremental and positivity approaches have been used. However, the presence of the term qii(i= 1 , 2 , . . . , n) in the above adds another degree of flexibility and improved estimate of stability margin over that provided by the incremental gain theorem based upon Gershgorin's theorem. In Theorem 7.14 the positive definiteness of P was assured by constructing a second and related matrix Z. However, more directly P is positive definite if n

pii -

1Ipii(>0

j=1 j#i

for

i = 1 , 2 , . . . ,n

or from (7.56) n

1Ipii(>O

(7.60)

R e [(l+ j 6 ~ q ~ ~ ) 4 ~ ( + j w(pi ) ] - ai)-'- ri (0)> 0

(7.61)

Re[(l+jwqii)rit,i+(pi-ai)-l]-i

j=1 jfi

But since

then (7.60) can be written as n

for all ~ E R i = , l , 2 ,..., n, and r i ( w ) = $ C 1 . ~ ~ ~ 1 . j=l

And so the feedback system (7.51) is L;-stable if the modified Nyquist

-0.1

Real

252

- 0.1

Real

I&:,- r, I ijwI

tI Img

- 0.15

+Real

ii%',,-r,,~ I J W I

1

liil

q,,

= 0.5 , q2,= 0

FIG. 7.16 Illustration of Popov multivariable stability criterion ( q , ,= 0.5, q22

=0)

253

254

THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

plot of &;(io)displaced by r i ( o ) in the negative direction, is to the right of the straight line of slope 4;' passing through the point (-(pi -ai)-', 0). Also since n

ri(W)z+C i=l

Ipii(io)I

(i = 1,2,. . . , n)

the above displaced loci is always contained in the Gershgorin envelope of Theorem 7.14 and hence the above direct approach is less conservative. To establish y, y E L," the condition (I+FGNo)-'FG E K(0)nxnmust hold; this is essentially a linear multivariable stability problem. If the open loop operator $(s) = g0(s)d(s)-' has p-poles in the right half-s plane (all contained in the monic polynomial d ( s ) ) , then the multivariable Nyquist criterion shows that (I+ FGNo)-'FG E K ( 0 ) n x ifn the envelope of Gershgorin circles of radii

n

1 I{=(S)},~I

i = l,

centred on the Nyquist plots of

{%(s)},, encircle the points (--a;')p-times. To implement Theorem 7.14 values for qii (i = 1,2,. . . , n) have to be selected, these in turn influence the Popov loci and there appears to be no straightforward method of selecting the optimum value of qii for the least conservative stability limits. Example 7.3 Consider the feedback system (7.51) with linear portion given by E ( s ) =

Since the diagonal elements of the matrix transfer function F z ( s ) are different the Popov loci for both { @ ( s ) } ~ ~and { = ( s ) } ~ ~ have to be drawn to determine the limits on (pi -a,),i = 1, 2 for closed-loop stability. The Popov loci for q I 1= q 2 2 = 0 are shown in Fig. 7.15 from which (p1-a1)=13.9 and (p2-a2)=5.6, whereas for q l l = 0 . 5 , q Z 2 = 0 . Figure 7.16 gives the nonlinearity limits of ( & + a l ) = 250, and (p2-a2) = 7.14 for stability. 7.5 NOTES The notion of passivity has been utilized by many workers in the determination of closed-loop system stability. The definitive papers of Zames (1966) on circle criteria established the basic systems tools which later authors (Narendra and Taylor, 1973; Desoer and Vidyasagar, 1975; and the present authors) exploited to generate increasingly generalized input-output stability criteria. Cho and Narendra (1968) extended the earlier work of Zames (1964, 1966), Sandberg (1964) and Narendra and Goldwyn (1964) on the scalar on-axis circle

7.

PASSIVITY RESULTS

255

criterion to a less conservative off-axis circle criterion by the use of multipliers. Interest for multivariable feedback systems has concentrated upon the on-axis circle criteria and recently Mees and Rapp (1978), Valenca and Harris (1979), Harris and Valenca (1981), Savfanov (1981), Husband and Kouvaritakis (1981), Kouvaritakis and Husband (1981) and Mees (1981) have established a variety of stability criteria for non-normal operators. Until recently the problem of the generalization of the off-axis criterion remained unsolved. Although Cook (1976) demonstrated that an off-axis criterion based on the diagonal elements of the linear loop operator guarantees the absence of limit cycles, but the inference of stability remained elusive. However, Harris and Husband (1981) have recently established such a criterion for normal linear time-invariant operators. The generalization of the off-axis rnultivariable circle criterion and its implementation is given in Sections 7.2 and 7.3. Although the present authors restrict the choice of multiplier to an RL or RC realization, the authors conjecture that a much larger class of multipliers exist to satisfy an off-axis interpretation. Clearly, using the techniques of this chapter, many more graphical stability criteria can be generated for feedback systems with a particular structure (Husband and Harris, 1982). Jury and Lee (1965), using Lyapunov stability ideas, obtained a Popov-like stability criterion for non-linear multivariable feedback systems

y=EN(e),

e=y-u

with N(.) =diag{Ni(.)} with sector bounds S ( 0 , (pi -q)}.They showed that this system is stable if a real diagonal matrix Q can be found such that A ( S )= ( I + SQ)P(S) + K-’

(7.62)

is positive real for K =diag (pi - a i ) . This result has been extended by Moore and Anderson (1968) to have a simple graphical interpretation which is included as a special case of Theorem 7.13. Related to Cook’s (1973) work on Gershgorin banded Nyquist plots is that of Shankar and Atherton (1977), who obtained a Popov-type graphical interpretation for feedback stability, based upon the diagonal matrix Q used in the definition of the positive real matrix A(s) in (7.62). In this approach the nonlinearities are confined to the sectors N i ( e i )E S(0, pi} for Q = 0; this result is identical to Cook’s (1973) result for a circle criterion. More recently Mees and Atherton (1981) have extended the Popov criterion for use with Rosenbrock’s (1974) diagonal dominance technique and eigenvalue methods (see Mac Farlane, 1979) for near normal linear operators. The Popov criteria are more restrictive than circle criteria, since they require the nonlinearities to be stationary; however, they are less conservative than circle criteria, since they have the added flexibility of the free variable Q = diag {qii} and are equally simple to implement.

REFERENCES Cho, Y. S. and Narendra, K. S. (1968). “An off-axis circle criterion for the stability of feedback systems with an monotonic nonlinearity.” ZEEE Trans. AC-13, 413-416.

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THE STABILITY OF INPUT-OUTPUT DYNAMICAL SYSTEMS

Cook, P. A. (1973). “Modified multivariable circle theorems.” In “Recent Mathematical Developments in Control” (Ed. D. J. Bell). Academic Press, London and New York. Cook, P. A. (1976). “Conditions for the absence of limit cycles.” IEEE Trans. AC-21, 339-345. Desoer, C . A. and Vidyasagar, M. (1975). “Feedback Systems: Input-Output Properties”. Academic Press, New York and London Harris, C. J. and Husband R. K. (1981). “An off-axis multivariable circle stability criterion.” Roc IEE 128, 215-218. Harris, C. J. and Valenca, J. M. E. (1981). “A circle stability criterion for large scale systems.” IEE Conf “Control and its Applications.” Warwick, March 23-26, 1981. Publ. NO. 194 IEE. Hsu, J. C. and Meyer, A. U. (1968). “Modern Control Principles and Applications.” McGraw-Hill, New York. Husband, R. K. and Harris, C. J. (1982). “Stability multipliers and the multivariable circle criteria.” Znt. J. Control 36, 755-774. Husband, R. K. and Kouvaritakis, B. (1981). “The Use of Sectors in the Derivation of Stability Criteria for Nonlinear Systems.” OUEL Rept NO. 1359/81, Oxford University. Jury, E. I. and Lee, B. W. (1965). “The absolute stability of systems with many nonlinearities.” Automn. Remote Control 26, 943-961. Kouvaritakis, B. and Husband, R. K. (1981). “A generalisation of the circle criterion.” OUEL Rept No. 1355/81, Oxford University. MacFarlane, A. G. J. (1979). “Frequency Response Methods in Control Systems.” IEEE Press, Wiley, New York. Mees, A. I. (1981). “Dynamics of Feedback Systems.” Wiley, Chichester. Mees, A. I. and Atherton, D. P. (1981). “The Popov criterion for multi-loop feedback systems.” IEEE Trans. AC-25, 924-928. Mees, A. I. and Rapp, P. E. (1978). “Stability criteria for multi-loop non-linear feedback systems.” In Proc. 4th IFAC Symp. Multivariable Technological Systems (Ed. D. P. Atherton). Pergamon Press, Oxford, 183-188. Moore, J. B. and Anderson, B. D. 0. (1968). “A generalization of the Popov criterion.” J. Franklin Znst. 285, 488-492. Narendra, K. S. and Goldwyn, R. M. (1964). “A geometric criterion for the stability of certain nonlinear, non-autonomous systems.” IEEE Trans. CT-11, 406-408. Narendra, K. S. and Taylor, J. H. (1973). “Frequency domain criteria for absolute stability.” Academic Press, New York and London. Rosenbrock, H. H. (1973). “Multivariable circle theorems” In “Recent Mathematical Developments in Control” (Ed. D. J. Bell). Academic Press, London and New York. 345-365. Rosenbrock, H. H. (1974). “Computer Aided Control System Design.” Academic Press, London and New York. Sandberg, I. W. (1964). “A frequency domain condition for the stability of feedback systems containing a single time varying nonlinearity.” Bell Systems Tech. J. 43, 1601-1608.

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257

Safonov, M. G. (1979). “Stability and robustness of multivariable feedback systems.” MIT Press, Cambridge, Mass. Shankar, S. and Atherton, D. P. (1977). “Graphical stability of nonlinear multivariable systems.” Int. J. Control 25, 375-388. Valenca, J. M. E. and Harris, C. J. (1979). “Stability criteria for nonlinear multivariable systems.” Proc IEE 126, 623-627. Zames, G.(1964): “On the stability of nonlinear time varying feedback systems.” Proc NEC 20, 725-730. Zames, G. (1966). “On the input-output stability of time varying nonlinear feedback systems.” IEE Trans AC-11,Pt I, 228-238, Pt I1 465-475. Zames, B, and Falb, P. L. (1968). “Stability conditions for systems with monotone and slope restricted nonlinearities.” S I N J. Control & Optimization 6, 89108.

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Subject Index

Abelian group, 44 Absolute convex set, 7, 16 Absorbent subsets, 10 Adherent point, 6 Adjoint operator, 83, 154 Affine operator, 25 Alexandroffs theorem, 8 Algebroid Riemann surface, 52, 55, 58, 64 Analytic continuation, 53 functions, 58 Anti-causal operator, 148 Arzela-Ascoli theorem, 203 Arc, 41 Associated loop, 43 Banach algebra, 24, 83, 161 space, 21, 24 Base of neighbourhoods, 2, 51 Bore1 measure, 26, 32 set, 26 Bounded input-output stability, 174, 179 operator, 23 operator theorem, 155 response theorem, 187 set, 7 Brouwer fixed point theorem, 24 Canonical mapping, 12 Cauchy sequence, 6, 22 theorem, 45, 46, 61 Causality, 147 strong, 144, 150 Characteristic function, 28

265

Circle theorems functional methods, 184, 188, 197 off axis, normal operators, 223 non-normal operators, 228, 240, 246 on-axis normal, operators, 232 non-normal operators, 232 M-circle stability theorem, 200 M-matrix stability, 206 Popov criteria, 243 Closed set, 3, 6 Cofinal set, 6 Commutative Banach algebra, 88 Compact operator, 185 space, 8 Complete analytic function, 54 space, 6, 146 Complex integration, 45 measures, 31 Composition of operators, 36 Concatenation of paths, 43 Conicity, 165 Continuous singular measures, 88 immediate, 50 Lipschitz, 23, 152, 175, 191 Continuity, sequential, 6 uniform, 14 Contour integration, 42 Contraction mapping theorem, 25, 192 Convergent sequences, 98 Convolution algebra .M(R+),87 in L,, 120 integral, 148 measures, 88 operators, 168, 186 unit, 84

266

SUBJECT INDEX

Covering space, 48 C ( R + )space, 2, 14 Co(R+)space, 87 Dense set, 6 Direct space sum, 22 Discriminant, 62, 136 Discontinuous measure, 88 Diagonal dominance, 207, 220 Diagonalization, 207 Disturbances, plant, 201 Dual space, 34, 113 E-stability, 124 Eigenvalue contours, 80, 136 paths, 79, 137 Encirclements, 78 Essential supremum, 30 Extended Hilbert space, 156 L, space, 146 space operators, 147 space topology, 4, 11, 143, 173 Finite norm, 35 semi-norms, 11, 102 Fixed point theorems, 24 Frequency dependent sectors, 167 Function centre, 50 element, 50 Functional, 23, 34 Fundamental group, 44 theorem of plane topology, 45 General measure space, 30 Generalized causality, 144, 147 Generalized Principle of the Argument, 40, 72, 76 Gershgorin’s theorem, 198 Hahn-Banach theorem, 95 HausdorfT topology, 5, 13

Hilbert space, 21, 154 Homeomorphism, 7, 13, 44 Homotopic paths, 54 Homotopically trivial, 42, 56 Homotopy, 42, 134 lifting theorem, 48 Immediate continuity, 50 Incremental norm (gain), 23, 191 gain theorem, 191, 194 circle theorem, 197 sectors, 169 Indefinite integrals, 32 Induced topology, 5 Index function, 137 Inner product, 22, 153 Input-output stability, 121, 124, 173, 221 Inside sectors (conic), 166 Intermediate gain theorem, 184 circle theorem, 189 Invertible operators in HS, 192 LA-, 128 LA+, 126 LL, 152 L;, 86 L,, 120 M ( E ) , 124 M(Li), 125 M(L,), 125 M(Xi), 127 M ( X A 111 M ( X . 4 , 127 A ( R + ) ,92 %, 110 Irreducible polynomials, 64 Isometrically isomorphic spaces, 13, 36 Jordon decomposition, 31 K(0)-space, 83

SUBJECT INDEX

K,(O)-space, 132 &-space, 106

267

Metric space, 13 Multiplier factorization, 235 theorem, 163 operators, 161, 215 Multipliers in L,, 92 Lz, 83 L,, 97, 112, 120 X,,97, 104, 111 XI, X z , X,, 105 p-null, 28

LA,, algebra, 96 invertibility, 164 norm, 96 stability, 127 LA-, algebra, 128 invertibility, 128 L,-representation theorem, 92 L2-graphical stability, 125, 129, 220 representation theorem, 83, 86 L,, norm, 104 multipliers, 97 representation theorem, 116 &-spaces, 29 stability, 173, 180 Lebesgue measure, 27 Lebesgue-Stieltjes measure, 32 Left distributive Banach algebra, 24 Linear lattice, 27 topological space, 1, 3 Lipschitz continuity, 23, 152, 175, 191 norm, 23, 176, 191 Local contraction, 25 Principle of the Argument, 70, 72, 74 Locally bounded, 152 compact space, 8 convex space, 10, 11, 21, 102, 144 Loop gain condition, 176, 180, 187, 194 transformation theorem, 177, 216

Off-axis circle stability criteria, 223 non-normal operators, 228, 240, 246 normal operators, 223 On-axis circle stability criteria, functional, 184, 188, 197 M-circle, 200 normal operators, 232 Operator, adjoint, 154 gain, 174 spectrum, 150 Open balls, 11 Orthogonal space, 22 Outside sectors (conic), 166, 239

M-circle stability criterion, 200 matrix stability criterion, 203 M(E)-algebra, 83 Markov-Katutanai theorem, 28 Maximal ideal, 90 Measure, Borel, 26 singular, 88 Memoryless operator, 148 Meromorphic functions, 74

Parametric disc, 51 Parseval’s inequality, 85 theorem, 176, 237 Path length, 61 Passivity, 153, 156 stability theorems, 215, 217 strict, 156 Path extremity, 42 origin, 42

Non-anticipatory operator, 148 Normal operator, 155, 181, 195 Null space, 12, 145 Nyquist stability criterion, 135 in K,(O), 135 in L1, Lz, L,, 139 in XI, X,,X,, 137

268

SUBJECT INDEX

Permissible matrices, 210 Poles, singular points, 74, 76 Popov circle stability criteria, 243 Popov multivariable stability criteria, 248 Positivity, 153, 156 stability theorem, 221, 223 strict, 154 Pre- and Post-compensation, 208 Principle of the Argument, 41, 70, 72, 76, 132, 135 Product topology, 17 Projective limit, 18, 101 Projection operators, 17, 172 Purely additive measures, 111

Sequential inductive limit spaces, 99, 104 Simple region, 44 Singular points, 62, 70 Small gain theorems, 177, 184 Spectral radius, 204 Strict incremental passivity, 156 inductive limit, 16 positivity, 154 Strong causality, 144, 150 convergence topology, 103 dual, 36 generalized causality, 147 sequential dual, 99 topology, 3, 35 System diagonalization, 207

Quotient spaces, 12, 145 Radon-Nykodym theorem, 32, 89 Radius of convergence, 50 Real measures, 31 Rectifiable path, 61 Regular measure, 27 Riemann surfaces, 40, 48 Riesz’s representation theorem, 37, 87 Representation of multipliers, 82 in L , , 92 in L2, 83 in L,, 97 in A ( R + ) ,87 in X,, 97 %-space, 108 Schauder-Tikhonov theorem, 24, 185 Schwartz inequality, 22, 154 Sector determination, 231 Sectoricity, 165 Self-adjoint operator, 154 Semi-norm, 10, 12, 144 Sequential convergent sequences, 98 bounded, 98 dual, 99 functional, 99

inductive limit, 15 paths, 41 projective limit, 18, 21, 38, 102 uniform, 3, 11 vector space, 9 weak, 3, 36 Trivial path, 42 Truncation operator, 146, 172 Uniform topology, 3, 11 Uniformally continuous space, 14 Value function, 59 Weak topology, 3, 36 Well posedness, 151, 175 Xp-space, 97, 104, 111 X;-stability, 125 X;-stability, 129 Xktability, 129 Zero, singular point, 71, 76