Input-Output Analysis of Large-Scale Interconnected Systems: Decomposition, Well-Posedness and Stability (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Edited by A.V. Balakrishnan and M, Thoma

29 M. Vidyasagar

Input-Output Analysis of Large-Scale Interconnected Systems Decomposition, WelI-Posedness and Stability

Springer-Verlag Berlin Heidelberg New York1981

Series Editors

A. V. Balakrishnan - M. Thoma Advisory Board L. D. Davisson • A. G. J. MacFarlane • H. Kwakernaak J. L. Massey • Ya. Z. Tsypkin • A. J. Viterbi Author

Prof. M. Vidyasagar Dept. of Electrical Engineering University of Waterloo Waterloo, Ontario Canada

ISBN 3-540-10501-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10501-8 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to 'Verwertungsgesellschaft Wort', Munich. © Springer-Vedag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2061/3020-543210

This book is intended to be a fairly comprehensive treatment of large-scale interconnected

systems from an input-

output viewpoint.

Prior to treating the question of stability

(and instability),

we study both the decomposition

posedness of such systems.

It is not necessary

and the well-

for the reader

to have studied feedback stability before tackling this book, as we develop results concerning feedback systems as special cases of more general results pertaining to large-scale systems. However,

the reader should know some elementary

analysis

(e.g. Lebesgue spaces,

and have some general knowledge

(e.g. Perron-frobenius

The first chapter is introductory, background material;

after that,

functional

contraction mapping theorem), and chapters

theorem).

2 and 3 contain

the remaining chapters are

essentially independent and can be read in any order. I thank Peter Moylan for his careful reading of the manuscript and for several constructive ShakUnthala

for her support.

suggestions,

and my wife

Virtually all of my research

reported in this book was carried out, and most of the book was written, while I was employed by Concordia University,

Montreal.

I would like to acknowledge research support from the Natural Sciences and Engineering Research Council of Canada, lesser extent from the U.S. Department of Energy.

and to a

Finally,

thanks to Monica Etwaroo and Jane Skinner for typing the manuscript.

Waterloo September 29, 1980

M. Vidyasagar

my

TABLE OF CONTENTS

PAGE

PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . .

v

CHAPTER

1

i:

CHAPTER 2:

INTRODUCTION

~THEMATICAL PRELIMINARIES . . . . . . . . . . 2.1 2.2

CHAPTER 3:

3.2 3.3

4.2 4.3

5.2 5.3 5.4

2~ 26 42 46

Some Results From the Theory of Directed Graphs . . . . . . . . . . . . . Decomposition into Strongly Connected Components . . . . . . . . . . . . . . . . Results on Well-Posedness and Stability

Weakly Lipschitz, Smoothing and Strictly Causal Operators . . . . . . . . . . . . . Single-Loop Systems . . . . . . . . . . . Continuous-Time Systems . . . . . . . . . Discrete-Time Systems . . . . . . . . . .

s7 57

.

73 81

88 88 94 95 103

Single-Loop Systems . . . . . . . . . . . Criteria Based on a Test Matrix ..... C r i t e r i a B a s e d o n an E s s e n t i a l S e t Decomposition . . . . . . . . . . . . . .

105 107 126

DISSIPATIVITY-TYPE CRITERIA FOR L2-STABILITY . 133 7.1 7.2 7.3

CHAPTER 8:

12

SMALL-GAINTYPE CRITERIA FOR Lp-STABILITY.. • lO5 6.1 6.2 6.3

CHAPTER 7:

4

Gain, Gain with Zero Bias, and Incremental Gain . . . . . . . . . . . . . Dissipativity and Passivity . . . . . . . Conditional Gain and Conditional Dissipativity . . . . . . . . . . . . . .

WELL-POSEDNESS OF LARGE-SCALE I~TERCO~NECTED SYSTEMS. . . . . . . . . . . . . . . . . . . . 5.1

CHAPTER 6:

Truncations, Extended Spaces, Causality . . . . . . . . . . . . . . . . Definitions of Well-Posedness and Stability . . . . . . . . . . . . . . . .

DECOMPOSITION OF LARGE-SCALE INTERCONNECTED SYSTEMS. . . . . . . . . . . . . . . . . . . . 4.1

CHAPTER5:

4

GAIN AND DISSIPATIVITY . . . . . . . . . . . . 3.1

CHAPTER 4.

. . . . . . . . . . . . . . . . .

Single-Loop Systems . . . . . . . . . . . 134 General Dissipativity-Type C r i t e r i a . . . 139 Special Cases: Small-Gain and Passivity-Type Criteria . . . . . . . . . 144

L2-1NSTABILITY CRITERIA. . . . . . . . . . . .

164

8.1 8.2 8.3

164 168 175

Single-Loop Systems . . . . . . . . . . . Criteria of the Small-Gain Type ..... Dissipativity-Type Criteria . . . . . . .

Vl

TABLE OF CONTENTS CONT'D. . . . .

CHAPTER 9:

L~-STABILITY AND L~-INSTABILITY USING EXPONENTIAL WEIGHTING. . . . . . . . . . . . .

189

9.1 9.2 9.3

190 198 205

General Special General

Stability Result . . . . . . . . . Cases . . . . . . . . . . . . . . Instability Result . . . . . . . .

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . .

213

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . .

218

CIIAPTER i: INTRODUCTION D u r i n g the p a s t decade or so, there has b e e n a great deal of i n t e r e s t in the study of l a r g e - s c a l e systems as a s e p a r a t e d i s c i p l i n e in itself. many factors,

p h y s i c a l systems circuits,

This i n t e r e s t is t r a c e a b l e to

i n c l u d i n g the g r o w i n g r e a l i z a t i o n that many

etc.)

(e.g. power networks,

several s i m p l e r subsystems, and "structure"

large-scale

integrated

can in fact be v i e w e d as i n t e r c o n n e c t i o n s of and that m u c h v a l u a b l e

information

is lost if the m e t h o d of a n a l y s i s does not take

into a c c o u n t the i n t e r c o n n e c t e d nature of the s y s t e m at hand. Moreover,

several s u b j e c t s d e a l i n g w i t h

reached m a t u r i t y ,

"small"

systems have

so that in order to expand the h o r i z o n s of

k n o w l e d g e by t a c k l i n g new and c h a l l e n g i n g p r o b l e m areas,

re-

searchers have set their sights on l a r g e - s c a l e

Some

systems.

prime e x a m p l e s of this are o p t i m a l c o n t r o l theory, s t a b i l i t y t h e o r y of s i n g l e - l o o p

and the

f e e d b a c k systems.

It is as yet too soon to c l a i m that there e x i s t s a comprehensive

theory of l a r g e - s c a l e systems.

stability theory of l a r g e - s c a l e

Nevertheless,

systems is a w e l l - d e v e l o p e d

in w h i c h a large v a r i e t y of results is available. effect two m e t h o d o l o g i e s

in s t a b i l i t y theory,

methods and i n p u t - o u t p u t methods.

While

the area

T h e r e are in

namely Lyapunov

there are some con-

n e c t i o n s b e t w e e n L y a p u n o v s t a b i l i t y and i n p u t - o u t p u t stability, the actual t e c h n i q u e s used to e s t a b l i s h the two types of s t a b i l i t y are r a t h e r different; of l a r g e - s c a l e systems.

Lyapunov

systems are w e l l - d o c u m e n t e d Miller [Mic.

this is e s p e c i a l l y methods

so in the case

for l a r g e - s c a l e

in the r e c e n t books by M i c h e l and

i] and S i l j a k [Sil.

i] .

contains come i n p u t - o u t p u t results,

However,

though [Mic.

i]

there is not at p r e s e n t a

c o m p r e h e n s i v e book on the i n p u t - o u t p u t a n a l y s i s of l a r g e - s c a l e systems.

In the same vein,

Desoer and V i d y a s a g a r [Des.

the books by W i l l e m s [Wil.

2] and

2] cover f e e d b a c k systems quite

t h o r o u g h l y from an i n p u t - o u t p u t viewpoint,

and it is n a t u r a l to

attempt a s i m i l a r t r e a t m e n t of l a r g e - s c a l e

systems.

This b o o k is i n t e n d e d to be a h i g h - l e v e l r e s e a r c h m o n o g r a p h t h a t sets forth m o s t of the a v a i l a b l e results on the decomposition,

well-posedness,

s t a b i l i t y and i n s t a b i l i t y of large-

scale systems,

that can be o b t a i n e d by i n p u t - o u t p u t methods.

Since m a n y r e s u l t s

for f e e d b a c k systems can be o b t a i n e d as

special cases of those given here for l a r g e - s c a l e systems, not n e c e s s a r y to have read [Wil. book.

2] or [Des. 2|

it is

to follow this

T h o u g h the e m p h a s i s h e r e is on i n p u t - o u t p u t stability, we

note that i n p u t - o u t p u t m e t h o d s can be u s e d to e s t a b l i s h L y a p u n o v s t a b i l i t y as well. is i n p u t - o u t p u t stable,

In particular,

also g l o b a l l y a s y m p t o t i c a l l y (see [Wil.

3] , [Moy.

if a n o n l i n e a r system

r e a c h a b l e and detectable,

then it is

stable in the sense of L y a p u n o v

4] ) .

T h r o u g h o u t this book,

the e m p h a s i s

is on t r e a t i n g the

l a r g e - s c a l e s y s t e m at h a n d as an i n t e r c o n n e c t e d system, sisting of several s u b s y s t e m s c o n n e c t i o n operators. 2.2).

con-

i n t e r a c t i n g through various inter-

(For a p r e c i s e d e s c r i p t i o n ,

It is of course p o s s i b l e to "aggregate"

s y s t e m o p e r a t o r s and the v a r i o u s

see S e c t i o n

the v a r i o u s sub-

i n t e r c o n n e c t i o n operators,

so

that the l a r g e - s c a l e s y s t e m at h a n d is r e c a s t in the f o r m of a "single-loop"

f e e d b a c k system.

W i t h this r e f o r m u l a t i o n ,

the s t a n d a r d s i n g l e - l o o p f e e d b a c k s t a b i l i t y results, those in [Des.

2] and [Wil.

2] b e c o m e applicable.

w h e t h e r a given s y s t e m is a "single-loop" connected"

all of

such as

Therefore,

s y s t e m or an "inter-

s y s t e m depends on the m e t h o d of a n a l y s i s u s e d to

tackle it.

However,

it can be e a s i l y shown that c o n v e r t i n g the

s y s t e m into a "single-loop" conservative

f o r m u l a t i o n gives u n n e c e s s a r i l y

s t a b i l i t y c r i t e r i a and w e l l ' p o s e d n e s s

Therefore,

criteria.

in this b o o k we only p r e s e n t results that

p e r t a i n to i n t e r c o n n e c t e d systems, w h e r e b y the a n a l y s i s

is

c a r r i e d out in terms of the s u b s y s t e m o p e r a t o r s and the interc o n n e c t i o n operators;

we avoid t r e a t i n g the s y s t e m as a w h o l e .

For this reason, we e x c l u d e linear t i m e - i n v a r i a n t systems f r o m our study.

The r e a s o n is that,

and s u f f i c i e n t c o n d i t i o n s interconnected conditions

though one can derive n e c e s s a r y

for the s t a b i l i t y and w e l l - p o s e d n e s s

linear t i m e - i n v a r i a n t systems,

(of necessity)

of

the n e c e s s a r y

involve t a c k l i n g the s y s t e m as a whole.

A s u b s y s t e m level a n a l y s i s can p r o d u c e s u f f i c i e n t c o n d i t i o n s s t a b i l i t y and s u f f i c i e n t c o n d i t i o n s n e c e s s a r y and s u f f i c i e n t conditions.

for instability, b u t not

for

The book is organized as follows:

In Chapter 2, we

introduce the concepts of truncations and extended spaces, which provide the mathematical

setting for input-output analysis, we

then give precise definitions of well-posedness

and stability.

In Chapter 3, we introduce the concepts of gain and dissipativity, which play an important role in the various criteria for stability and instability,

and give explicit methods for com-

puting gains and testing dissipativity. In Chapter 4, we present a few graph-theoretic niques for the efficient decomposition of large-scale connected systems.

Specifically,

tech-

inter-

we show that by identifying

the so-called strongly connected components

(SCC's) of a given

system, we can determine the well-posedness

and stability of the

original system by studying only the SCC's. present some sufficient conditions system.

These criteria are graph-theoretic

given a very nice physical

In Chapter 5, we

for the well-posedness

interpretation.

In Chapter 6, we give some generalizations single-loop

of a

in nature and can be

of the

"small gain" theorem to arbitrary interconnected

systems, while in Chapter generalizations

7, we state and prove several

of the single-loop

"passivity"

Chapter 8, we derive several L2-instability scale systems.

Finally,

theorem.

In

criteria for large-

in Chapter 9, we show how the technique

of exponential weighting can be used to study L -stability and L -instability using the results of Chapters

6 to 8.

CHAPTER 2: MATHEMATICAL PRELIMINARIES 2.1

TRUNCATIONS,

In this notation

section,

and terminology

particular

notation

Let functions

X

R+ =

here

and

As

introduce

the m a t h e m a t i c a l

is f r o m

this book.

[Vid.

4] and

the set of all r e a l - v a l u e d

into

[0,~),

measure.

we briefly

employed

R+

SPACES r CAUSALITY

t h a t is u s e d t h r o u g h o u t

denote

mapping

numbers, Lebesgue X

EXTENDED

R, w h e r e

R

denotes

the m e a s u r a b i l i t y

is c u s t o m a r y ,

The

[Des.

measurable

the s e t of r e a l

is w i t h r e s p e c t

we define

2].

various

to the

subsets

of

as f o l l o w s :

1

Definition

For

p 6

[i,~),

the s e t

L P

notes

the s e t of all

functions

tion

t +

is i n t e g r a b l e

f(.)

E L

[If(t) I]P

for a f i x e d

P

2

p e

f(.)

[i,~)

in

over

X

such

[0,~).

if a n d o n l y

= L [0,~) deP t h a t the f u n c -

In o t h e r w o r d s , if

If(t) Ip dt < 0

Similarly, in

X

[0,-)

L

= L

such that •

If

p 6

[0,~) f(.)

[i,~)

denotes

the

set of all

is e s s e n t i a l l y we d e f i n e

,

bounded

the f u n c t i o n

functions

over I'

.

f(.)

the i n t e r v a l

Ilp : Lp

÷

R+

by

I tfI1p = [

If(t) lp dt] 1/p , vf e Lp 0

If

p = -

, we define

II-I I~ : L

I IfEl. = e s s ° t 6

= inf

where

p e

~[.]

[1,~],

space.

denotes

sup

÷ R+

by

IfCt) j

[0,~)

{r

: ~ [ t : I f ( t ) I > r] = 0}

the Lebesgue

measure

~f 6 L

of a set.

I t is w e l l - k n o w n

[Dun.

i, p. 146]

the o r d e r e d

(Lp

, I.I I . , Ip) .

pair

,

t h a t for e a c h

constitutes

a Banach

In o r d e r can

study

to h a v e

"unstable"

the c o n c e p t

of

as w e l l

truncated

Definition is d e f i n e d

a mathematical as

"stable"

functions

and

T < ~

; then

Let

For b r e v i t y , refer

the

we use

the

XT(.)

as

to

interval

sense

that

introduce

spaces.

the o p e r a t o r

PT

: X + X

Vx•

Note

that

that

PT

denotes

fT(. ) e L p

belong

to

of the

space

X

t > T notation the

xT

to d e n o t e

truncation

the

of a g i v e n

function

function

PT x, x(.)

[0,T].

the

PT

For

the YT

Lp).

operator

" PT =

Definition L pe [0,~)

we

we

t • [0,T]

0

to the

systems,

extended

whereby

by setting

(PTx)(t) = { x(t)

and

framework

a fixed

s e t of all < ~

The

PT

p •

Lpe

[i,~],

functions

(though

space

is a p r o j e c t i o n

on

X

in

symbol

L

=

"

f(.)

the

f(.) itself

is r e f e r r e d

in may

pe such

X

or m a y

to as the

not

extension

L P

Example e_~d s p a c e s

Lpe

The

for

the u n e x t e n d e d

spaces

tan

t

does

C X

.

Moreover,

all

finite

T

is t h e

Then

for

p • Lp

not belong

It is c l e a r

Lle

function

all

for

that, Lp

, it is c l e a r

Definition every

set

that

p E

, the

The

unextended

fixed

Lpe

[i,-]

c Lle to u s e

be

truncated

to

the e x t e n d -

not belong

to an[

function spaces

f2(t) L

of Vp •

[i,~].

in this

fixed, norm

L c L p pe [0,T] for

L1

and

Thus

book.

let

IIfl ITp

T < is d e f i n e d

IIf11 p = IIfTlIp= llpTfllp Let

p = 2, a n d

truncated

inner

let

T < ~

product

Then

T

I

= T

To study discrete-time s p a c e of s e q u e n c e s

n [ i=l

dt =

£p

of all

sequences

consists

of all {x (i) }

such that

I

17

Ix(i) Ip < "

i=O

The

set

[i,~)

,

£

consists

we define

of all b o u n d e d

the

function

II-I Ip :

£p

~

in R+

S

For p 6

.

by

Ix(i) Im) I/p

llxllp = ¢

18

sequences

i=0 We also define

II-I I. : £= + R+

[Ixllo--

19

by

sup Ix¢i>l i

W e can a l s o d e f i n e present

Definition is d e f i n e d

For each

i ~ 0 , the o p e r a t o r

Finally,

x (j)

0 < j < i

0

j > i

Sn

(rasp.

£~)

-

we define

Definition set

in the

Pi

: S + S

by

(Pix) (j) = {

22

of t r u n c a t i o n s

context.

20

21

the c o n c e p t

Let

n

-

the s p a c e s

S n a n d £n P

be a p o s i t i v e

is d e f i n e d

integer.

Then

as the set o f all s e q u e n c e s

the of

n-tuples 6 S

{x} (i) ~

(resp.

=

Zp)

[x~ i) Vj

.

,

x 2(i)

•

.... x n(i)] ,

The norm

If-lip

{" x ij~ ( ) "

such t h a t

: %n ÷ R+ P

is d e f i n e d

by

( ~

r I EXllp = 4 [

23

T1~(i) IIP) I/p

i--1 sup I Ix(i~ll

iz

p < -

if

p = -

i

I I-I ]

where

denotes

We next

24

the E u c l i d e a n

introduce

Definition causal

An operator

PT G = PT G P T

of c a u s a l i t y .

G :Lle

÷Lle

is s a i d

to be

'

~T
_ 0

i=0 where

6(.)

< ...

are real constants,

norm

denotes

If. If A

33

on

the unit impulse

A

is defined

I If(.) I IA =

The product

~ i=0

-

(f,g)

distribution,

{fi } q £i '

f(.)

f0

Remarks by delayed

subset of Moreover•

and

g (.)

in

A

is defined

i.e.,

(t) =

()tf(t-T)

g(T)

dT =

A, and that if pair

(Jtf(T)

g(t-T)

dT

0

Basically, impulses.

the ordered In

The

Ifa(t) I dt

0

mented

0 ~ tO < t1

fa(. ) G L 1 .

by

Ifil +

of two elements

as their convolution;

34

and

the set

A

consists

It is easy tO see that f(.) (A•

(34), one should

6 LI•

then

II.l IA) interpret

of L1

L1

aug-

is a

IIf(.) Ill = l]f(.)llA-

is a Banach

space.

10 35

(t-t a) * ~(t-t b) = ~ (t-ha- ~ )

36 Thus,

if

~(t-ta)

* fa(t) = fa(t-ta)

f

g

and

are of the form

37

f(t) =

~ fi 6(t-ti) i=0

38

g(t) =

~ i=0

+ fa (t)

gi ~(t-Ti)

+ ga (t)

then 39

(f,g) (t) =

+

~ ~ i=0 3

fi gj ~(t-ti-Tj)

~ gj fa(t-~j) j=0

+

fa(t-T)

and right-

IIf*gltA

40

Also, we see from 41

!

~ fi ga i=0 ga(T)

(t-ti)

dT

0

It is routine to verify from commutative, leftition, and that

+

(39) that convolution

is

distributive with respect to add-

IIfllA • IlglIA

(39) that

f*~ = ~*f = f ,

Vf • A

Hence the set A is a Banach algebra with a unit, with the norm, * as the product, and ~ as the unit. Given any

I I.I IA as

f(.) 6 A, the integral ~

f(s)

42

=

f

f(t)

~st dt

0

is well-defined whenever

Re s > 0,

and in fact,

43 where Laplace

C+ = {s: Re s ~ 0}. transformable,

Thus every element

f(.)

and the region of convergence

of

A

of the

is

Laplace transform C+

f(.)

i n c l u d e s the c l o s e d r i g h t h a l f - p l a n e

For n o t a t i o n a l c o n v e n i e n c e ,

44

Definition

The set

forms of the e l e m e n t s of

we i n t r o d u c e the set

A .

A c o n s i s t s of the L a p l a c e

trans-

A .

Since c o n v o l u t i o n in the time d o m a i n is e q u i v a l e n t to p o i n t w i s e m u l t i p l i c a t i o n in the s-domain, p r o d u c t s of e l e m e n t s of

A

can be shown q u i t e e a s i l y that any every

s E C+

f 6 A

, and a n a l y t i c at e v e r y

{s: Re s > 0}

A

A .

Also,

is c o n t i n u o u s

s ~ C+o

(where

C+).

Finally,

d e n o t e s the interior of

that e v e r y e l e m e n t of

we see that sums and

once again b e l o n g to

is b o u n d e d over

it

at

C+o

=

(43) shows

C+

^

A n×m of

A, d e n o t e d

45

by

such that

The set

fT(.) ~ A,

A

e VT ~ 0

N o t e that D e f i n i t i o n inition

We next define

the extension

c o n s i s t s of all d i s t r i b u t i o n s

(45) is e n t i r e l y a n a l o g o u s

to Def-

(7).

The set G

A , we can also d e f i n e

A e

Definition f(.)

if

and

Once we have d e f i n e d A and ~nxm in an o b v i o u s way.

Ae

is i m p o r t a n t b e c a u s e

it can be shown that,

is a linear c o n v o l u t i o n o p e r a t o r of the type (Gf) (t) = J'g(t-~) f

46

f(T)

dT

Lpe

into itself

0 then

G

is causal and m a p s

and only if the k e r n e l

(or "impulse response")

yp 6

[1,-],

if

g(.)

e Ae .

The

proof of this i m p o r t a n t f a c t can be o b t a i n e d by a d a p t i n g [Des. 2, T h e o r e m IV.7.5]. that we e n c o u n t e r

Thus,

Ae

(or, m o r e generally,

multivariable Thrm. 6.5.37]

g(')

system. that,

if

(the u n e x t e n d e d space) g(.) e A .

This

all linear c o n v o l u t i o n o p e r a t o r s

in this m o n o g r a p h

can be a s s u m e d to be of the form

(even the "unstable"

(46), w h e r e

the k e r n e l

ones) g(.)

E

~ An×me , in the case of a

Similarly, G

that of

it can be shown

is of the f o r m

into itself

shows that the set

Vp e A

(46), then [I,~],

[Vid. 4, G

maps L

if and o n l y if

e s s e n t i a l l y c o n s i s t s of

P

12

all "stable"

2.2

impulse r e s p o n s e s

(see D e f i n i t i o n 3.1.1).

D E F I N I T I O N S OF W E L L - P O S E D N E S S AND S T A B I L I T Y

In this section, we d e l i n e a t e interconnected

the class of l a r g e - s c a l e

systems u n d e r study in this book,

and we give pre-

c i s e d e f i n i t i o n s of w h a t is m e a n t by such a s y s t e m b e i n g w e l l p o s e d or stable.

T h r o u g h o u t this book, we shall be c o n c e r n e d w i t h analysis of a l a r g e - s c a l e

interconnected system

(LSIS)

d e s c r i b e d by the

set of e q u a t i o n s m

la

ei = ui -

[ j =i

H

ij

yj i = l,...,m

ib

Yi = Gi ei n.

where

ui' ei' Yi

fixed

p 6

[1,-]

all b e l o n g

Lpel

to the e x t e n d e d space

and some p o s i t i v e integer

n i , the o p e r a t o r G i

n.

maps n. l

n.

L i pe

into itself,

and the o p e r a t o r

H.. 13

maps

L 3 pe

into

.

Lpe

We can refer to

and output,

y

ui' ei' Yi

respectively.

to d e n o t e the m - t u p l e and

for a

to d e n o t e

(Ul,

(YI'

as the i-th input, error,

W h e r e convenient, ..., Um),

..., ym ) .

e

we use the symbol u

to d e n o t e

N o t e that

m Ln , where n = [ n. pe i=l i spirit, we s o m e t i m e s use the symbols G and H

to the p r o d u c t space

ators f r o m

Ln pe

G =

(el,

u, e, y

..., em),

all b e l o n g

In the same to d e n o t e o p e r -

into itself d e f i n e d by

I°J i.

G

*To a v o i d a p r o l i f e r a t i o n of symbols, we a s s u m e that the s y s t e m Gi

has an equal n u m b e r of inputs and outputs.

is e n t i r e l y d i s p e n s a b l e .

This a s s u m p t i o n

13

H =

IHll Hml

W i t h these definitions,

the system e q u a t i o n s

(1) can be c o m p a c t l y

e x p r e s s e d as

4a

e = u - Hy

4b

y = Ge

The system d e s c r i p t i o n able of r e p r e s e n t i n g think of

several

(i) as r e p r e s e n t i n g

subsystems,

(1) is quite g e n e r a l and is cap-

types of p h y s i c a l systems. several

"isolated"

c o r r e s p o n d i n g to the o p e r a t o r s

One can

or "decoupled"

GI,...,G m

, such that

the input to ui

G. is a linear c o m b i n a t i o n of an e x t e r n a l i n p u t l and several "interaction" signals Hij yj This is d e p i c t e d

in Figure

2.1

.

Yi

Gi

Hil Yl

Him Ym F I G U R E 2.1

For this reason, we refer G I, .....G m

to

m

as the n u m b e r of subsystems,

as the s u b s y s t e m operators,

and

Hll,...,Hmm

as the

i n t e r c o n n e c t i o n operators.

In some cases, p a r t i c u l a r l y

in p r o v i n g d i s s i p a t i v i t y -

type theorems for s t a b i l i t y and instability,

(Chapters 7 and 8)

we assume that for all i,j, the i n t e r c o n n e c t i o n o p e r a t o r Hij: n. n. Lpe3 ÷ L pez can be r e p r e s e n t e d by an nixn j m a t r i x ~ij of c o n s tant real numbers,

i.e.

that

14 n.

(Hij yj)(t) Actually, ality,

= H..~13 yj(t)

this a s s u m p t i o n

because

,

Vt,

Vyj e Lpe3

does not result in any loss of gener-

this a s s u m p t i o n

ing the number of subsystems

can always be satisfied by increas(m)

if necessary.

(If a particu-

lar o p e r a t o r

H.. cannot be r e p r e s e n t e d by a c o n s t a n t matrix, 13 m by one and include H.. among the operators 13 If all i n t e r c o n n e c t i o n operators can be r e p r e s e n t e d by

then increase G i) .

c o n s t a n t matrices,

then we refer

to the c o n s t a n t

n×n

matrix

H

defined by

H

l

=

LEml as the i n t e r c o n n e c t i o n

mmj

matrix.

uI

u2

FIGUR~ The standard 2.2

and studied

2.2

feedback

in detail

in

configuration,

[Des. i] and

is a special case of the system d e s c r i p t i o n system of Figure

2.2 is d e s c r i b e d by

7a

el = Ul - Y2

7b

e2 = u2 + Yl

shown in Figure

[Wil. i] among others, (I)

The feedback

15 7c

Yl = G1 el

7d

Y2 = G2 e2 where p •

Ul' u2' el

[i,~]

e2' YI' Y2

•

and some p o s i t i v e integer .

into itself.

To put the s y s t e m

(two subsystems),

H

where

0 ~%)

order

~×9

all b e l o n g to

L~ pe

~ , and

GI,G 2

(7) in the form

n I = n 2 = ~, n = 2~, Gl~ 2

for some fixed map

(i), let

as in

m = 2

(7), and

=

and

I~ ~

denote

respectively.

the null m a t r i x and i d e n t i t y m a t r i x of N o t e that the i n t e r - c o n n e c t i o n opera-

tors can be r e p r e s e n t e d by c o n s t a n t m a t r i c e s in this case• that the i n t e r c o n n e c t i o n m a t r i x ible.

L pe ~

H

and

is s k e w - s y m m e t r i c and invert-

T h e s e p r o p e r t i e s are i m p l i c i t y u s e d in m u c h of f e e d b a c k

s t a b i l i t y theory.

Comparing

the g e n e r a l l a r g e - s c a l e

(1) w i t h the f e e d b a c k s y s t e m d e s c r i p t i o n a g g r e g a t e the e q u a t i o n s are v e r y similar.

(I) into the form

In fact,

system description

(7), we see that if we (4), then

(4) is a s p e c i a l case of

(4) and

(7), w i t h

u I = u, u 2 = 0, G 1 = G, G 2 = H, e I = e, and Yl = y " shown in F i g u r e 2.3

.

Thus,

g i v e n an LSIS,

r e s e n t it in the d e c o m p o s e d form system level,

(7)

T h i s is

one can e i t h e r

rep-

(i) and a n a l y z e it at the sub-

or one can r e p r e s e n t it in the a g g r e g a t e d

and a n a l y z e it as a s i n g l e - l o o p system.

form

(4)

If one chooses the latt-

er option•

one can i m m e d i a t e l y apply all of the s t a n d a r d r e s u l t s

d e r i v e d in

[Des.

main emphasis

2] and

[Wil.

2] for f e e d b a c k systems.

in this m o n o g r a p h is on a n a l y z i n g a g i v e n LSIS at

the s u b s y s t e m level,

taking full a d v a n t a g e of the fact that the

system at h a n d is an i n t e r c o n n e c t i o n of several ler)

(presumably simp-

subsystems.

*Actually• and

T h u s the

Ul, el, Y2

u2' YI' e2

all n e e d to b e l o n g to the same space

all need to b e l o n g to the same space

in g e n e r a l we could have

P # q' 91 ~ ~2

"

92 Lqe

Lpe , but

The e x t e n s i o n of the

r e s u l t s p r e s e n t e d here to this s i t u a t i o n is transparent.

16

u

y

FIGURE

W i t h regard tions

(i)

to the system d e s c r i b e d by the set of equa-

(or, e q u i v a l e n t l y ,

pes of questions.

2.3

(4)), one can ask b a s i c a l l y two ty-

The first type of q u e s t i o n takes the following

form: Does the s y s t e m

(1) h a v e a u n i q u e set of s o l u t i o n s

e,y

in

Ln c o r r e s p o n d i n g to each set of inputs u e L n ? If so, is pe pe the d e p e n d e n c e of e,y on u causal, and g l o b a l l y L i p s c h i t z continuous? the s y s t e m

The d e f i n i t i o n and study of the w e l l - p o s e d n e s s of (i) takes into a c c o u n t such c o n s i d e r a t i o n s .

second type of q u e s t i o n takes the f o l l o w i n g form:

The

G i v e n a set of

inputs

u • Ln (the u n e x t e n d e d space) and a s s u m i n ~ that the P s y s t e m e q u a t i o n s (i) have one or m o r e s o l u t i o n s for e,y in L pe' n do these s o l u t i o n s in fact b e l o n g to L n ? If so, does the relaP tion m a p p i n g u into (e,y) have "finite gain"? The d e f i n i tion and study of the s t a b i l i t y of the s y s t e m a c c o u n t such c o n s i d e r a t i o n s

as the above.

(1) takes into

The r e a s o n for sep-

a r a t i n g the two types of q u e s t i o n s is that u s u a l l y the c o n d i t i o n s t h a t imply w e l l - p o s e d n e s s n a t u r e from the c o n d i t i o n s seen b y c o m p a r i n g C h a p t e r

are quite d i s t i n c t and d i f f e r e n t in that imply stability.

This can be

5 w i t h C h a p t e r s 6 to 9

We now turn to the d e f i n i t i o n s .

Definition

The s y s t e m

the f o l l o w i n g c o n d i t i o n s hold:

(i) is said to be w e l l - p o s e d

if

17

u e Ln there exists a pe ' unique set of errors e e Ln and a set of outputs y E L n such pe pe that the system equations (i) are satisfied.

i e.

(WI)

For each set of inputs

(W2)

The d e p e n d e n c e

whenever

u (I)

and

of

u (2)

e

and

y

on

u

is causal;

are two input sets in

L n such pe

•

that for some

T > 0

10

we have

:

then the c o r r e s p o n d i n g Y (2) }

solution

sets

, y(1) }

{e (I)

and

{e (2)

satisfy

ii

=

y(1)

12

(2) YT

=

(W3) YT

on u T

For each finite

for each

T < = , there exists

whenever

u (I)

{e(1)

• y(1)}

sets of

T, the d e p e n d e n c e

is g l o b a l l y L i p s c h i t z and

continuous.

a finite constant

, y(2)}

and

such that, Ln and pe solution

(i), we have l]e(1)-e(2) ]ITp - 0, in

p 6

[i,~],

~pCG~ = sup ~ 0

incremental

GAIN

that

n, m

are posit-

G

sup

Then we

I IGxl ITp ~ kl Ixl ITp + b,

Vx 6 Lne } p we set

G, d e n o t e d by

sup xT ~ 0

is d e n o t e d by

sup

T ~ 0 x T ~ YT

yp(G)

= ®

We define

~p(G), by

{IGxIIT~ llx11Tp

supremum does not exist,

vain of

np(G) =

such that

(2) is empty,

the vain with zero bias of

If the indicated

deriv-

systems.

of V a r i o u s Types of Gain

= inf {k:~b < -

k

are im-

criteria

is a given operator. G:L n ÷ L TM pe pe the vain of the operator G, denoted by yp(G), by

If the set of

operators,

these constants

and that

yp(G)

operat-

are couched in

calculate

can a c t u a l l y be applied

Definition ive integers, define

linear

constants

the results

GAIN WITH ZERO BIAS,

dissipativity.

Since almost all of the

to show that the various

ed in this m o n o g r a p h 3.1

Thus,

gain

passivity,

to the two m o s t common-

chapters

to k n o w how to a c t u a l l y

portant in order

that are

These include gain,

gain, passivity, strict

We also discuss how these concepts

criteria

concepts

we set

np(G),

~p(G)

= ~ . The

and is defined by

IIGx-GYIITp_ IIx-yIITp

R7 If the supremum in (4) does not exist, we set In the above definitions, stants yp(G),

~p(G), and ~p(G)

np(G)

we recognize

= ~

that the con-

depend not only on the operator G,

but also on the value of p . Note that, in general,

yp(G)

! ~p(G), and

~p(G)

~ np(G)

whenever G(0) = 0. Also, if yp(G) is finite, then G maps the unextended space L n into the unextended space L TM . (However, P P the converse is not true; the operator G:L e ÷ L ~e defined by (Gx) (t) = x2(t) maps

L

is easy to show that if

into G

L

, but

is linear,

y (G) = ~).

then

yp(G)

Finally,

it

= ~p(G) =

np(G). It is routine to verify that, given operators G2

defined on appropriate

G1

and

spaces, we have

yp(G 1 G 2) < yp(G I)

yp(G 2)

~p(G 1 G 2) -0

(18), suppose

first of all that

Ig(t,T) I dt
l(~)

-- 0

Ig(t,~)i

d= < ®

T

holds in

(24)

is f a c i l i t a t e d by a bit of

using the unit impulse d i s t r i b u t i o n

the basic idea of the proof

is demonstrated,

that the a r g u m e n t can be made m a t h e m a t i c a l l y sequence ~(.)

25

of

Ll-functions

x(t)

for a fixed

26

Once

it will become precise,

Ll-norm

clear

by using a

and converge

Suppose we let

= ~ ( t - T 0)

T0 ~ 0 .

Then we have

(Gx) (t) = g(t,T 0) Since

~(t-T 0)

can be a r b i t r a r i l y

sense of distributions, we conclude

27

that have unit

in the sense of distributions.

6(.)

that

g(.,~0 ) • L 1 and m o r e o v e r

by an

closely approximated,

Ll-function

having u n i t

in the Ll-norm,

31 ~

28

S

yl(G) = ~l(C) _> Ilg(-,T0) ll i =

l g ¢l t ,,T o ) ,

~t =

0

=

Ig(t,~o) I dt T0

where

in the last step we use the fact that

t < •

Since

(28) holds

for

every

g(t,T)

= 0 whenever

T O ~ 0 , we have

~

29

Yl(G)

Now,

(24) and

~

(29)

Next,

f

sup • 0~ 0

Ig(t,T0) I dt

TO

together

we prove

prove (19).

(18). Suppose

first of all that

t 30

I

sup t and let

Ig(t,T)l

x 6 L

£

d T + < x , R X > T + T > 0 , ~ T -

> 0 , Yx 6 L n 2e

43

Before we

first

state

to L e m m a

discussing

a general

Lemma

and only

Suppose

G

semidefinite.

for causal

of d i s s i p a t i v i t y ,

operators,

analogous

: Ln ÷ Ln is causal, and that Q 2e 2e G is (Q,R,S)-dissipative if

Then

if



Proof

+ <x,Rx>

First

(Q,R,S)-dissipative), T ~ m

Then

"

~Gx,Sx>

of all,

and

let

n Yx 6 L 2

>_ 0 ,

suppose

(2) h o l d s

x E L n2 .

since

suppose

xT 6 L 2



However,

+

Then

(4)

(i.e.,

G

follows

by

is

.

Conversely, x 6 Ln 2e

result

implications

(3.1.8).

is n e g a t i v e

letting

the

Q

G

satisfies

for all

+ <XT,RXT>

is n e g a t i v e

T

(4),

and by

and

let

(4) w e h a v e

+ > 0

semidefinite

and

G

is c a u s a l ,

we have

< G X T , Q G X T > T

Similarly,

by

the

causality

of

G

=

T

, we have

t

= T

Clearly,

Substituting

<XT,RXT>

= <x,Rx> T

into

gives

(5)

T + <x,Rx> T + T ~ 0

which

establishes

(2).

As with lies

in the

fact

[]

Lemma

that,

(3.1.8),

the

for a causal

significance

operator

G

of L e m m a

, one

can

(3)

44 determine

whether

is n e g a t i v e

or n o t

G

semidefinite)

o n the u n e x t e n d e d

is

(Q,R,S)-dissipative

by examining n L2 .

space

o n l y the b e h a v i o u r

W e n o w t u r n to the i m p l i c a t i o n s Note

that,

denote

in the i n t e r e s t s

il.ll 2

i0

Lemma where

Q

Proof

ll

of

~ =

.

Then

12

-

12

l lGxJ IT which

shows

13

that

respect

Lemma

15 respect

to

J IxJ IT < =


0

-

.

and

on n o t i n g

'

Then

I

¥ x 6 L n2e

G

is d i s s i p a t i v e

n

denotes

0n

denotes

the i d e n t i t y

G

that

ilxII

V T >- 0 ,

2e

: Ln + Ln is d i s s i p a t i v e w i t h 2e 2e ~2(G) < ~ . T h e n it is a l s o d i s s -

(Q-~In,R+~

Obvious,

~

'

.

2(G) In,S) , for a l l 2

by combining

The next definition

studied

(Q,R,S)

(-In,~2(G) In,0n), w h e r e

a n d the

which was historically

to

. o

E 2(G) 3 2

Suppose

respect

Proof

n×n

-

(Q,R,S)~

ipative with

~

Obvious

lIGxli

14

is ~2(G)

By completing

of d i m e n s i o n

Proof

Then

denote

Suppose

of d i m e n s i o n

J I-I I

(2) b e c o m e s

to the t r i p l e t

the null matrix matrix


0 •

(2) and

(14).0

concept to b e

45

16

Definition

An o p e r a t o r

G : Ln ÷ L n is said to be 2e 2e passive if it is d i s s i p a t i v e w i t h r e s p e c t to (0n,0n,In), i.e. if

<x,Gx> T > 0 ,

17

¥T > 0 ,

-

G

Vx 6 L n 2e

-

is said to be s t r i c t l y p a s s i v e if it is d i s s i p a t i v e w i t h

r e s p e c t to

(0n,-ZIn,I n) <x,Gx> T ->

18

for some

Ilxll~

~

¢ > 0 , i.e., VT

' ' ' T

'

> 0 -

if

VX E L n 2e

'

We now c o n s i d e r how these v a r i o u s c o n c e p t s apply to linear c o n v o l u t i o n o p e r a t o r s and m e m o r y l e s s n o n l i n e a r operators. Due to shortage of space, we do not d i s c u s s the d i s s i p a t i v i t y of i n t e g r a l o p e r a t o r s of the type is r e f e r r e d

to

[Wil.

o p e r a t o r s of the type

Lemma

19

(3.1.16).

Instead,

the r e a d e r

4] for a d i s c u s s i o n of the p a s s i v i t y of (3.1.16).

Consider a convolution operator

G : Ln2e ÷ Ln2e

of the form

Gx(t)

20

f

=

t G(t-T)

X(T)

dT

0 where

G(.) 6 A n×n

Then

G

is d i s s i p a t i v e w i t h r e s p e c t to

(0n,-~In,I n) , w h e r e 21

= Proof Lemma

22

(3).

Let

inf ~ 6R

Imi n

[G+(j~)

Clearly n

x e L2

<x,Gx> = - 2~ -

1 2~

2~

G

+ G(j~)]/2

is causal.

; then

by

H e n c e we can a p p l y

Parseval's

theorem,

Re

I" ~+

(jm)

Ix(je)

[G+(J~) + G ( ~ ) ] 2

d~

=

~

Iixll

x(J~)

2

d~

46

Hence G

(4) h o l d s

is

with

R = -6In,

By Lemma

S = In

(3) ,

(0n,-~In,In)-dissipative.

23

Lemma form

(3.1.85),

G(.,.) to

Q = 0n,

Suppose

where

satisfies

G

: L n2e + L n2e

G

belongs

(3.1.99).

is an o p e r a t o r

to the s e c t o r

Then

G

[e,8],

is d i s s i p a t i v e

of the

i.e.,

with

respect

( - I n , - ~ S I n , (~+B)In) . Proof

24

From

(3.1.99),

one

can easily

show

that

T ~ 0

which

can be

3.3

readily

manipulated

CONDITIONAL

GAIN

In this operators,

i.e.

AND

section,

operators •

CONDITIONAL

we

M

(G) =

The

conditional

to

Suppose

~ain

of

result.

D

DISSIPATIVITY

be

concerned

with

"unstable"

: Ln ÷ Ln w h i c h do n o t m a p pe pe i n t r o d u c e the b a s i c d e f i n i t i o n s , and

{x e L n P

P

shall

the d e s i r e d

G

Ln into L m . We first P P then discuss how they pertain

Definition

to y i e l d

G

linear

G

convolution

: L n ÷ Lm pe pe

operators.

Then we

define

: Gx • L n} P , denoted

by

~cp(G),

is d e f i n e d

by

II Gx IEp ~cp (G) =

If

n = m,

x E ~p(G)/{o}

p = 2, the

(Q,R~S)-dissipative

that

~cp(G)

with

zero

bias."

use

Ycp (G)

operator

with



Note

sup

respect

+ <x,Rx>

should However,

G

[I x lip is s a i d

+

properly this

to b e c o n d i t i o n a l l y

to a g i v e n

is

be too

triplet

(Q,R,S)

>_ 0 ,

Vx e M2(G )

called

"conditional

cumbersome,

if

gain

a n d we n e v e r

47 The f o l l o w i n g simple lemmas b r i n g out some r e l a t i o n ships b e t w e e n the c o n c e p t s "unconditional"

concepts

Lemma ~cp(G)

i n t r o d u c e d in D e f i n i t i o n

Suppose

~p(G) +

Since

Mp(G)

= ~p(G) Lemma

G

(1) and the

i n t r o d u c e d earlier.

G

is causal,

n Vx E L 2

~ 0

it follows by L e m m a

(3.2.3)

that

G

is

(Q,R,S)-dissipative.o

Lemma ally

Suppose

G

is causal,

( Q , R , S ) - d i s s i p a t i v e b u t not n Then M2(G) ~ L 2 .

that

G

is c o n d i t i o n -

( Q , R , S ) - d i s s i p a t i v e , and suppose

Q ~ 0 .

ally

Proof

T h i s is just the c o n t r a p o s i t i v e of L e m m a

Lemma

Suppose

G

is causal,

(Q,R,S)-dissipative where

definite);

suppose

M2(G)

Q < 0

~ Ln 2 .

Then

that

(i.e., G

G Q

is not

(6).o

is c o n d i t i o n is n e g a t i v e (Q,R,S)-dissipa-

tiv~.

Proof (3.2.10), w e h a v e Lemma

(5).

Since

If

G

is

(Q,R,S)-dissipative,

~2(G)< ~ , w h i c h shows that n M2(G) ~ L 2 , G c a n n o t be

then by L e m m a = Ln 2 , by

M2(G)

(Q,R,S)-dissipative.o

We next i n t r o d u c e a class of o p e r a t o r s t h a t are u n s t a b l e in a p a r t i c u l a r way.

Such o p e r a t o r s play an i m p o r t a n t

role in the i n s t a b i l i t y theorems of C h a p t e r s

i0

Definition b e l o n g to class U if

An o p e r a t o r

8 and 9.

G : Ln + Ln 2e 2e

is said to

48

such

(i)

G

(ii)

There

is l i n e a r is an u n b o u n d e d

family

of c o n s t a n t s

aT

that

ii

I IGxl IT2

= 0

D(t-T)

z(T)

dT d t

0

I"

I

z' (T)

0

D' (t-T)

x(t)

d t dT

x(T)

dT = 0

~t

>_ 0}

x(r+t)

dr = 0

Vt

>_ 0}

so t h a t

(D'x) (T) =

43

Hence,

by

44

D'(t-T)

x(t)

dt

(40) ,

M1

(G) = {x 6 L n2 :

D' (T-t) t

= { x 6 L 2n :

D'(T) 0

which

proves

(38).

a

Theorems characterizations somewhat which

(32) of

abstract.

are especially

the The

and

(37),

sets next

useful

though

Mp(G) result

they give

and gives

in C h a p t e r s

M1 more

(G),

complete are

concrete

8 a n d 9.

still results,

54

45 has

Theorem Suppose G : Ln + Ln ^ ~ 2e 2e an r . c . f . (N,D), and b e l o n g s to c l a s s

a pole

of

G(.).

(i)

Under

if

so

such

these

that

(ii)

D' (s O)

if

is real,

D' (sb)

belongs

Proof by Lemma

Hence NOW

there

46

so

x(t)

Clearly

x(.)

a nonzero

= v exp(-Sot)

~ D'(~)

a vector

Then

x(t)

, where

let

v

.

to

be

M1

x(t)

Cn

=

*

denotes

a vector

Then

in

(G);

in

Rn

such

= v exp(-Sot)

s o • C +o

is a p o l e

= 0 , so t h a t d e t vector then

v

such

D + ( s O)

of

D ' ( s O)

that

G(), = 0 also.

D ' ( s o)

v* = 0.

v = 0. Let

+ v* e x p ( - s ~ t )

because

x(~+t)

Let

be

belongs

v = 0

is c o m p l e x ;

I

47

.

M I (G)

d e t D ( s o)

• L n2 ,

v

v = 0 .

of all , if

First

(28),

exists

suppose

to

let

exp(-s~t)

conjugate,

that

then

+ v*

complex

So^

U

(32),

s o • C+o be

conditions.

is c o m p l e x ,

v exp(-Sot)

is as in T h e o r e m

so E C + °

(i.e.

Re

So>0) .

Also,

dT

0

=

D'(~)

v exp(-SoT),

D'(T)

V*

exp(-So£)

d~

0 ~

S

+

exp(-s~r),

exp(-s:t)

dT

0

=

D' ( S o ) V e x p ( - S o t )

+ D+ (So)V *e x p ( - s * t )

Wt

which s

o

shows

is r e a l

± space

M

that

x • M 1 (G),

follows

Theorem (G),

similarly.

(45)

in the

by Theorem

The

> 0

case

where

[]

demonstrates special

(37).

= 0 ,

some

case where

elements G

of

belongs

the

sub-

to

U

and

55

^ G(.)

C +o •

has a pole in

e l e m e n t s of

M I (G)

However,

it does not say that all

are of the f~rm

(46).

W e close out this s e c t i o n by giving a l t e r n a t e characteri z a t i o n s of c o n d i t i o n a l g a i n and c o n d i t i o n a l d i s s i p a t i v i t y

for a

linear convolution operator.

48

Lemma

Suppose

49

~c2(G)

=

Proof

The proof is e n t i r e l y a n a l o g o u s to that of

(3.1.83). whenever

50

In fact

G

is as in T h e o r e m

sup [Ima x

(3.1.84)

[G+(j~)

applies,

(45).

Then

~(j~)]}i/2

n Gx ~ L 2

if we note that

x e M2(G ) . D

Lemma be any r.c.f,

Suppose

G

G(.).

Then

of

is as in T h e o r e m G

w i t h r e s p e c t to a g i v e n t r i p l e t

(45), and let

(N,D)

is c o n d i t i o n a l l y d i s s i p a t i v e (Q,R,S)

if and only if the

m a t r ix 51

P(~)

=

[D+(j~)S N(j~)

+ N+(jm) S ' D ( 9 ~ ) ] / 2

+ D + ( j ~ ) R D(j~) is

positive

Proof dissipative

By D e f i n i t i o n

(1), G is

(Q,R,S)

conditionally

if and only if



52

+ N + ( j ~ ) Q N(j~)

s e m i d e f i n i t e for all ~ .

+ <x,Rx> +

~ 0

Vx 6 M 2 ( G ) ^

By T h e o r e m

(32),

x 6 M

(G)

if and o n l y if

^^

x = Dz for some

2^^

z E L n2 , in w h i c h case ly by u s i n g and d e f i n e

53

D

Gx =

Nz

If we abuse n o t a t i o n slight-

to d e n o t e the o p e r a t o r m a p p i n g

Nz = N,z

, then

z

into

D~z

(52) is e q u i v a l e n t to

+ +

>_ 0 ,

n Yz 6 L 2

i.e., w

54

,

~

n

where

* d e n o t e s the a d j o i n t operator.

(54) holds all

>_ 0

~ .

if and only if

P(~)

Vz 6 L 2

By P a r s e v a l ' s

is p o s i t i v e

equality,

semidefinite

for

,

56

NOTES AND R E F E R E N C E S The utility of the c o n c e p t of gain in studying stability [Zam.

arises from the work of S a n d b e r g

3].

The gain calculations

many sources,

including

[San.

of Section

3.1 are taken from

the work of S a n d b e r g

D e s o e r [Wu l] and W i l l e m s

[Wil.

of course w e l l - e s t a b l i s h e d

i].

[San.

The c o n c e p t

of p a s s i v i t y

c o n c e p t of d i s s i p a t i v i t y

[Wil.

3,4,5]

and e x p l o i t e d by Hill and M o y l a n [Hil.

[Moy.

2].

The concept of c o n d i t i o n a l and Bergen [Tak.

pativity was introduced M o y l a n [Hil.

3] , [Moy.

are i d e n t i f i e d

in ]Tak.

i], while c o n d i t i o n a l

operator

calculation

can be found in [Vid.

in the dissi ~

by Hill and

of a class U operator

The c a l c u l a t i o n

range of an unstable of M±(G)

1,2],

gain is implicit

The properties

i].

is

The

was i n t r o d u c e d by Willems

(under a d i f f e r e n t name) 3].

3], Wu and

in c i r c u i t and s y s t e m theory.

more general

w o r k of Takeda

feedback

2] and Zames

of the domain and

are found in [Vid. 7].

5], while

the

CHAPTER 4: DECOMPOSITION OF LARGE-SCALE INTERCONNECTED SYSTEMS In this chapter,

we show how the study of a g e n e r a l

large-scale interconnected

system

first d e c o m p o s i n g the LSIS

into its s o - c a l l e d s t r o n g l y c o n n e c t e d

components

(SCC's).

(LSIS) can be s i m p l i f i e d by

By using this procedure,

one n e v e r has to

deal w i t h a m o r e c o m p l e x s y s t e m than the one w i t h w h i c h he began, and in m a n y cases,

the o r i g i n a l s y s t e m is r e p l a c e d by several

smaller systems w h i c h t o ~ e t h e r are s i m p l e r than the o r i g i n a l system.

T h e p r i m a r y tool u s e d to e f f e c t this d e c o m p o s i t i o n

the t h e o r y of d i r e c t e d graphs,

or digraphs,

is

and we b e g i n w i t h a

d e v e l o p m e n t of the results that we need from this theory.

It

should be m e n t i o n e d at the o u t s e t that the f o l l o w i n g d i s c u s s i o n is h i g h l y selective;

the reader is r e f e r r e d to

[Baa. i] for a

m o r e c o m p l e t e discussion.

4.1 4.1.1

SOME RESULTS F R O M THE THEORY OF D I R E C T E D GRAPHS Basic Concepts Definition

(V,E), w h e r e subset of vertices,

V × V and

A directed ~raph

V = { V l , . . . , v n}

E

.

The set

(or digraph)

is a finite set V

is r e f e r r e d

and

is a pair E

is r e f e r r e d to as the set of to as the set of edges.

(vi,vj) e E , then we say that there is an edge from

A p a r t from the above a b s t r a c t definition, a p i c t o r i a l r e p r e s e n t a t i o n of a d i g r a p h in the plane are l a b e l l e d as from

vi

to

vj

is a

If vi

we o f t e n use

(V,E), w h e r e b y

w i t h an a r r o w h e a d d i r e c t e d towards

Consider

n

points

v i , . . . , v n , and an arc is d r a w n vj

(vi,vj) 6 E . Example

t_~o vj .

the d i g r a p h

(V,E), w h e r e

V = {Vl, v2, v3, v 4}

E = {(Vl,V2),(Vl,V4),(V2,V2),(V2,V3),(V3,Vl), (V3,V4),(V4,V2)} The p i c t o r i a l r e p r e s e n t a t i o n of this d i g r a p h is shown in

whenever

S8

Figure

4.1

.

vI

V3 v

~

v

FIGURE

Definition from

vi 6 V

with

A vi0 = v i

to

Given

vj 6 V and

4.1

V4

a digraph

(V,E),

is a n o r d e r e d

Vik

= vj

set

a p a t h of l e n g t h k

{ v i 0 , V l.l.,. .

, such that

(vi£,vi£+l)

,Vik } e E

for

£ = 0,...,k-i

equivalent

One can

state

the a b o v e

manner:

A p a t h of l e n g t h

sequence

of v e r t i c e s

V i k = vj

, such that there

£ = 0,...,k-i

is

from

vi

to

vj

is a

vi0 = v i

vi£

for all

from

from

of F i g u r e

to e ~ e r y o t h e r

Given

reachable

is a p a t h

vi R vi

is an e d g e

In the d i g r a p h

Definition

there

from

{vi0,Vil,...,Vik) , with

from every vertex

vj E V

k

in the f o l l o w i n g

and

__t° v.l£+l

for

.

Example path

definition

vi

a digraph vi E V

to

vj

4.1,

(V,E),

(denoted by

~

i , e v e n if t h e r e

there

is a

vertex.

we

say t h a t v i R vj)

By c o n v e n t i o n , is no p a t h

if

we t a k e

from

vi

to

itself.

Note obvious imply vi from

to

that

that vj

R

that

R

defines

is t r a n s i t i v e ,

v i R vk

a binary i.e.,

(in o t h e r w o r d s ,

and a path

to

relation

v i R vj

on

and

V

if t h e r e is a p a t h

from

vj

v k , then there

Given

a digraph

.

It is

vj R v k from

is a p a t h

v i to Vk). Definition

of v e r t i c e s

(vi,v j)

is s t r o n g l y

(V,E), w e say t h a t a pair

connected

if

v i R vj

and

58

vj R v i . If

i ~ J , then

there is a p a t h from By convention,

vi

a pair

(vi,v j)

to

vj

(vi,v i)

Definition

4.1.2.

vi

A digraph

(vi,vj) 6 V × V

vj

to

vi

is taken to be s t r o n g l y c o n n e c t e d

e v e n if t h e r e _ i s no p ~ t h frn~

every pair

is s t r o n g l y c o n n e c t e d if

and a path from

to itself.

(V,E)

is s t r o n g l y c o n n e c t e d if

is s t r o n g l y connected.

T e s t i n g f o r Strong C o n n e c t i v i t y

We n e x t d i s c u s s two c o m p u t e r a l g o r i t h m s

for d e t e r m i n i n g

w h e t h e r or not a g i v e n d i g r a p h is s t r o n g l y connected.

N o t e that

the a l g o r i t h m s g i v e n here are o n l y two of several p o s s i b l e ones; we s e l e c t these p a r t i c u l a r a l g o r i t h m s b e c a u s e they r e q u i r e v e r y little b a c k g r o u n d . is ~ e f e r r e d to

For a m o r e c o m p l e t e discussion,

the r e a d e r

[Baa. 1].

The a l g o r i t h m s g i v e n here m a k e use of B o o l e a n v a r i a b les, w h i c h are defined next.

Definition

i0

{o,1}

The B o o l e a n set

(B,+,-)

is the set

t o g e t h e r w i t h the b i n a r y o p e r a t i o n s + and

B =

" on B d e f i n e d

by

Ii

0+0

=

0,

12

0'0

=

i'0

0+i

=

i+0

=

0"i

=

=

I+i

=

1

0,

i'i

=

i

T h o s e f a m i l i a r w i t h a b s t r a c t a l g e b r a can v e r i f y that (B,+,-)

is a c o m m u t a t i v e algebra.

For the rest,

note that the b i n a r y o p e r a t i o n s + and commutative,

it s u f f i c e s to

• are a s s o c i a t i v e and

and that + is left - and r i g h t - d i s t r i b u t i v e over • ;

in o t h e r words,

for all

13

a+b = b+a

14

a+ (b+c) =

a,b,c

(a+b) +c

in

B, we have

60

15

a-b = b-a

16

a-(b-c)

=

(a.b).c

17

(a+b)-c

=

(a.c)+(b-c)

18

a- (b+c)

=

(a-b)+(a-c)

Sometimes "and".

+ is r e f e r r e d

Also,

(B,+,'). rows

and

this

by

defined also

we

can

Specifically, m

columns

D 6 B n×m in the

belongs

19

.

B n×m

"or",

of

and

Boolean

"-"

such

is r e f e r r e d

matrices

if D is an a r r a y

over

consisting

to as

the

algebra

of

n

that

Matrix

usual

to

to as

speak

way; and

d.. • B for all i,j, we d e n o t e 13 addition and multiplication are

i.e.,

E • B nxm

is d e f i n e d

by

,

then

F = D + E ~

~

f.. = d.. + e.. 13 13 13 where and

Similarly, a d d i t i o n is a c c o r d i n g to (ii). ×£ B n×£ and E 6 B , then F = D'E b e l o n g s to the

if

D 6 B n×m

is

defined

by m

20

bij

21

Definition Vn} , we d e f i n e

=

[ k=l

its

dik'ekj

Given

a digraph

adjacency

matrix

(V,E)

A 6 B nxn

1

if

(vi,vj)

0

if

(vi,v j) ~ E

with

V = {v I, ....

by

E E

22 aij

23

=

Definition Vn} , we d e f i n e

rij

Given

a digraph

its r e a c h a b i l i t y

=

{ 1

if

matrix

(V,E)

R E B nxn

v i R vj

24 0

Note

that,

otherwise

for

any

digraph,

with

we have

by

V = {Vl,...,

61

25

r.. = 1

for a l l

ll

i

It is clear that a d i g r a p h is s t r o n g l y c o n n e c t e d and o n l y i f 1 .

a l l e l e m e n t s of its t e a c h a b i l i t y m a t r i x are equal to

It is also c l e a r that the a d j a c e n c y m a t r i x of a d i g r a p h can

be formed by inspection.

In order to d e v e l o p an a l g o r i t h m to

test for s t r o n g - c o n n e c t e d n e s s , R

given

A

26

.

A

we d e r i v e a m e t h o d

for c o m p u t i n g

T h i s is the p u r p o s e of the next s e v e r a l lemmas.

Lemma ing

if

C o n s i d e r the m a t r i x

by itself

£

times

(for

A £ , o b t a i n e d by m u l t i p l y -

£ ~ i)

Then

if there is a p a t h of l e n g t h £ from Z ij

{ 1

v i to vj

27 0

Proof

otherwise

The proof

is by induction.

note that a path of length (vi,vj).

Hence

(27)

true for

£ = k-i

1

T h e n for

£ = i,

v. to v. is just an edge i ] £ = 1 . N o w suppose (27) is

is true for

.

First, w h e n

from

£ = k, we have

n

28

(Ak) ij = "

~

(Ak-l)im

m=l

F r o m the d e f i n i t i o n s of + and to see that

29

(Ak).. = 1 13 m

such that

" amj

"

(namely

(ii) and

(12)),

it is e a s y

if and o n l y if

(Ak-l) im = 1

By the i n d u c t i v e hypothesis,

(27)

and

is true

amj = 1

for

£ = k-i

.

Thus

(29) is e q u i v a l e n t to

30

m to

vm

Clearly

31

such t h a t there is a path of length k-I from

and an edge

(Vm,V j) .

(30) is e q u i v a l e n t to

T h e r e is a path of length k from

vi

to

vj

.

v.

l

62

Thus

(Ak) o. = 1

if a n d o n l y

if

for

~ = k

, and

completes

32

Lemma

Given

a digraph,

33

R =

13

(27)

is t r u e

~

Proof identity

i ~ j, some

we

Since

matrix),

see

that

Z)

from

Lemma

35

R =

shows by

that

induction. 0

we have

R = I + A + A2 +

we

vi

34

This

the p r o o f

see t h a t -

ro.

...

= 1

ll

(where

for a l l

I

denotes

i .

For

r.. = 1 if a n d o n l y if (A£).. = 1 for 13 13 if and o n l y if t h e r e is a p a t h (of some

£ h 1 , i.e.,

length

holds.

Ai ~

i=0

the

(31)

to

vj

Given

n-i ~

.o

a digraph

with

n

vertices,

we have

if w e c a n

show

Ai

i=O Proof whenever of

there

length

The

is a p a t h

at m o s t

n-i

there

is a p a t h

{viol

v i , ...,Vlk_l.

defining

this

of

sequence

of v e r t i c e s

latter

vj

to

quantity

argument to

vi

until

at

two

0 < ~ < m -

larger

vj

, and

sequence

the l i s t elements < k -

.

length

than

of

to

k+l

that

are

we

vertices the

clearly

k-(m-£). can

at m o s t

suppose

of v e r t i c e s

of

Then

n-l,

length

is a path

let

. ,...,Vik} a l s o ,VXm+l

it has

a path

~ v i , there Accordingly,

the

, then

{vi0 ,.. .,vi

find

vi be

least

, and

vj vj

that,

same.

the

defines If this

repeat

n-i

the

from

. []

36

Lemma

37

R =

that

Giwen

(7 + A) S

Proof clear

from

k [ n

is a g a i n we

to to

=~ vj}

where

vj

vi vi

k

, Vik

contains

v. = v. i£ im

from

from

If

Suppose

is e s t a b l i s h e d

from

length

path.

(vi0,...,Vik)

path

lemma

in f a c t

First,

a digraph

fox

from

with

every

n

vertices,

we have

s >_ n-I

the p r o o f

of L e m m a

(35),

it is

vi

a

63

s

R =

38

~ i=0

~

We

now claim

Ai

for

every

s >_ n - i

that S

Ai = "

39 i=0

Clearly,

if

We prove

(39)

since true

both for

(39)

(I + A ) s ~

for e v e r y

can be established,

by induction.

sides

equal

I

then

First, in t h i s

s = £, a n d o b s e r v e

s >

(39)

the lemma is t r u e

case.

that

0

Next,

M + M = M

is p r o v e d .

for

s = 0,

suppose

(39)

is

for all matrices

M.

Thus °

(I + A) £ + I =

40

(I + A) £

(I + A)

[ Ai÷ i=0

=

~+i ~

=

( [

A I)

(I + A)

[ A i÷l

"

i=0

"

Ai

i=O This

proves

the

The algorithm

for

lemm~, o

discussion computing

Algorithm

41

t

2s

M2s

M

.

given

suggests A

for computing

Given

A

step

2

select

an integer

Step

3

Calculate

Note

that step

M2s

, let

R

the

Reachability

M = I + A

as

3 can be

t

M 2t

such that

for

some

s < t,

then

R

2 t [ n-i

.

.

accomplished

are

Matrix

.

(M.M = M 2, M 2 . M 2 = M 4 , e t c . ) . multiplications

following

.

The

1

matrix =

R

to n o w

Step

multiplications than

up

by

t

matrix

Sometimes

fewer

required.

For

instance,

clearly

R = M

. Thus

if

if

64

2S

the squaring new matrix

process at

some

(M

. M 2s = M 2s+l

stage,

then

It is o b v i o u s log 2 n

matrix

Boolean

matrices

row additions Thus,

of o r d e r

(i.e.,

standard

42

of

R

Now,

n×n

can

for

R

computing and

notation of t h i s

is g i v e n

.

Warshall's

This

Algorithm

for

R + A

Step

2

For

k +

1

to n, do

For

i ÷

1

to n,

If

rik

matrix

two

(n2/log2 n)

226-231].

[Baa.

as W a r s h e l l ' s 2 requires n

t h a t we u s e ~i'"

i, pp.

the

The

222-223].

R

do

then r o w of

R i = R i + Rk R)

R + R+I

Example adjacency

replaces

in

computing

= 1

also

"~2

1

3

known

Note

found

that

using

proof.

Step

Step

requires

row additions.

algorithm

to m e a n

c a n be

2

in a

.

shown

I, pp.

algorithm,

(R i = i - t h

43

n

without

"~I ÷ ~2"

algorithm

algorithm

[Baa.

result R

be m u l t i p l i e d

requires

another

not

it c a n be

operations)

We now present

row additions,

details

"or"

does

is the m a t r i x

the a b o v e

multiplications.

computation

algorithm,

that

that

)

Consider

the

digraph

of Figure

4.2

.

Its

is

VI

V7

FIGURE

4.2

65

44

A

=

0

1

0

1

0

0

0

0

0

1

0

1

0

0

1

0

0

1

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

1

0

0

0

0

0

0

0

0

1

0

Using either A l g o r i t h m

(41) or A l g o r i t h m

(42), we can c o m p u t e

R

as

45

R

Since

R

=

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

0

0

1

1

1

1

0

0

0

1

1

1

1

0

0

0

1

1

1

1

0

0

0

1

1

1

1

does not c o n t a i n all l's,

1

1

the d i g r a p h is not s t r o n g l y

conneeted~

4.1.3.

D e c o m p o s i t i a n into S t r o n g l y C o n n e c t e d C o m p o n e n t s S u p p o s e that a g ~ v e n d i g a a p h is not s t r o n g l y connected.

Can we d e d u c e some f u r t h e r i n f o r m a t i o n a b o u t its s t r u c t u r e ? answer is p r o v i d e d by i d e n t i f y i n g the s o - c a l l e d components

46

(SCC's)

of t_he digraph.

Definition S

on

Given a digraph

V is d e f i n e d by

is s t r o n g l y c o n n e c t e d

v i S vj

V

.

{i.e., v i R vj

Since

(without ambiguity)

S

(V,E), the b i n a r y r e l a t i o n

if and o n l y if the pair

It is e a s y to s h o w that r e l a t i o n on

and

S

(vi,v j)

vj R vi).

is in fact an e q u i v a l e n c e

is an e q u i v a l e n c e r e l a t i o n we can say

that "a set of v e r t i c e s

is s t r o n g l y connected,"

w h e n w h a t we m e a n is that the v e r t i c e s are p a i r w i s e connected.

The

s t r o n g l y connected

strongly

66

Since partition digraph

V

S

is an e q u i v a l e n c e

into

(V,E)

its e q u i v a l e n c e

is s t r o n g l y class under

classes

under

of

V

reachability 48

matrix

Lemma denote

S R

Proof

(V,E)

Otherwise,

matrix.

"if"

rij

Clearly V

a

itself

is

the e q u i v a l e n c e

identified

Then

the

v i S vj

J-th

Suppose

row

Ri = Rj . .

since

S if

be a g i v e n d i g r a p h

denmtes

rii = I, so the h y p o t h e s i s Similarly,

under

V, we c a n

using

the

.

its r e a c h a b i l i t y Ri

S .

on

if and o n l y

c a n be e a s i l y

Let

R i = R j , where

classes

connected

an e q u i v a l e n c e

relation

implies

a n d let

R

if a n d o n l y

of

if

R .

By definition,

~

that

rji = i, i.e.,

= rjj = I, w e h a v e

v i R vj

vj R v i .

Hence

v i S vj "only vj R v i .

Now,

rik = i; this together

if"

Suppose

whenever

is b e c a u s e

with

v i R vj

implies

argument, Ri = Rj .

Example From S

are

{ V l , V 2 , V 3}

and

In g e n e r a l ,

classes

(41) or

under

i n g rows.

S

This

be efficiently

S , and

can order v b • Vj need

some

once

comparison

R

is c o m p u t e d

it is e a s y

, which

i.e., k

rik = 1 .

implies

of F i g u r e

is c a l l e d

V

rjk =i.

4.2

classes

R

either

under

see

~ith

[Ba~.

of the

the e q u i v a l e n c e its

succeed-

"string matching,"

and can

I, Ch. 4].

i n t o its e q u i v a l e n c e

Vl,...,V k . classes

using

to d e t e r m i n e

e a c h r o w of

w e par_tition

equivalence

in s u c h a w a y t h a t if

(Vb,V a) g E.

classes

We n o w s h o w t h a t w e

In o r d e r

v a•

to do this,

Vl, we

concepts.

Definition of l e n g t h g r e a t e r

for some

o u t by c o m p ~ t e r ~

i < j , then

further

vj R v k

v i R Vk,

{v4,v5,v6,v7}.

l a b e l t h e m as

these and

and

k, w e a l s o h a v e

a g a i n the d i g r a p h

by comparing

Suppase under

some

see t h a t t h e e q u i v a l e n c e

(42),

carried

that

rik = 1

Consider

(45), w e i m m e d i a t e l y

Algorithms

50

rjk = 1 implies

In o t h e r

49

i.e., v i R vj

for

By a s y m m e t r i c a l words,

v i S vj,

rjk = 1

Given a digraph

than one

from a vertex

(V,E),

a cycle

is a p a t h

to itself. A n e d g e of the

87 form

(vi,vi)

cycle.

is called a s e l f - l o o ~ and is not c o n s i d e r e d

(Note that

itself).

(vi,v i)

The digraph

contain any cycles

(V,E)

is a p r e d e c e s s o r if

v i + vj

,

and

vj 6 V , and (vi,vj)

Suppose

there exists a v e r t e x Proof

acyclic

Given a d i g r a p h

of

Lemma

52

is a path of length one from is

in

Assume

has a predecessor,

to

self-loops).

(V,E), we say that v i e V

vj

is a successor

of

vi ,

6 E .

the d i g r a p h V

vi

if it is does not

(it m a y however c o n t a i n

Definition

51

to be a

(V,E)

is acyclic.

Then

that does not have a predecessor.

the contrary,

i.e.,

suppose every vertex

and c o n s t r u c t a sequence in V

as follows:

Select

v. e V arbitrarily, and select v. to be a prede±0 ik+l cessor of v. for k > 0 . By assumption, this sequence can be ik c o n s t r u c t e d indefinitely, and since V contains only a finite number of elements, sequence.

...,v.1£+m = vi£ } Clearly

an e l e m e n t of

V

must occur

In other words we can c o n s t r u c t such that

V.lk+l

twice in the

a sequence

is a p r e d e c e s s o r

{v i

{vi£,vi£+l , of

Vik Vk.

,v i ,. . . . ~+m £+m-i "''v1£+l'V1~ (V,E), which c o n t r a d i c t s the h y p o t h e s i s

v. } is a cycle in l£+m that (V,E) is acyclic.

Hence the original

[]

53

a s s u m p t i o n is false.

Proposition digraph

(V,E).

Let

{Vil .... ,vi£ = Vil}

Then the set of v e r t i c e s

be a cycle

{Vil,...,vi£_l}

in a is

strongly connected. The proof 54

is obvious.

Definition denote

the e q u i v a l e n c e

ivity).

such that

of

Then the reduced d i g r a p h

The vertex an edge

Given a d i g r a p h classes

set

V = {Vl,...,Vk},

(Vi,V j) (Va,Vb)

V

(V,E), under

(V,E)

let

S

VI,...,V k

(strong c o n n e c t -

is defined

and the edge set

if and only if there exist

E

as follows: contains

Va e v i , v b 6 Vj

e E

The reduced digraph has a very simple

interpretation.

88 Suppose we modify the original digraph vertices in

V.

(V,E)

into a single vertex, for

resulting digraph is the reduced digraph. strongly connected,

by collapsing

i = l,...,k . Note that if

all

The (V,E) is

then its reduced digraph consists of a single

vertex and a self-loop.

55

Lemma

For any digraph

(V,E), its reduced digraph is

Proof

Assume the contrary, namely that

acyclic.

Vim,Vim+l = Vil}

is a cycle in

contains the edges of

(V,E).

This means that

(Vi2'Vil)''''' (Vim'Vil)"

By the definition

E , this implies that there exist vertices

v!ljI) ' v!lj2) in

Vi. , j = l,...,m , having the following property. edge set

E

contains the edges

(v(2) (i) , (v(2),v(1) im_l ,Vim im il ). class under

(v! 2) ± l 'v

Now, since

Vij

S , there is a path from

J = l,...,k .

(Vil'Vi2'''''

The original

)),(v(2) (i)) ..... i 2 'vi 3 is an equivalence

vlj ~i) to

v(2) ij

, for

So what we have shown is that there is a cycle in

the original digraph

(V,E)

containing the vertices

{v!l),v (2) lI iI '

"''' v(1) i ,v.~2) }. By Proposition (53), this implies that all these m m vertices are strongly connected. However, this is a contradiction, because from

{v!l),v! 2)} belongs to a distinct equivalence class 11 l1

(1),v(2)} {vi2 i2

(for example).

This contradiction shows that

our original assumption is false, and that the reduced digraph is acyclic. D We can now state a procedure for renumbering the equivalence classes VI,...,V k in the manner described before Definition (50). Given a digraph (V,E), first construct its reduced digraph

(V,E).

Now, identify all vertices in

do not have pre~ecessors~ and ]abe] this set as the vertices in

V1

as

all vertices in

91

and all edges leaving vertices in

all edges of the form

WI,W2,...,Wnl

Vl "

(Vi,.)

with

9

that

Renumber

Next, remove from (V,E)

V i e ~i).

91 (i.e.,

The resulting

69

digraph

is a g a i n a c y c l i c .

Identify

t h a t do n o t h a v e p r e d e c e s s o r s , to see t h a t Renumber

until

V

Vi

WI,...W k

of s y m b o l s ,

lie a m o n g

i < j, t h e n

original

digraph

then

set

V (i)

as

It is e a s y to

.

To

now denote

i.e.,

{ V l , . . . , V i _ I } , for ~ E

equivalence

Vl "

i = 2,..,k

t h a t if

form

of

avoid a the v e r t i c e s

such that all predecess-

With reference

this m e a n s

classes)

t h a t all p r e d e c e s s o r s

i = 2,...,k

Vl,...,V k

Given

a digraph

Vl,...,V k

VI,...,V k under

(ii)

"

belong

.

v a 6 V.3

the d e f i n i t i o n

This means

to the and

i < j,

of E).

Thus

the f o l l o w i n g

Theorem vertex

(i.e.,

in s u c h a w a y

(this f o l l o w s

V2 must

,... W Repeat this nl+ 1 ' n2 V are e x h a u s t e d . A t this

in

let

it as

of v e r t i c e s

W

the v e r t i c e s

(Vj,Vi)

(V,E),

(Vb,V a) ~ E

56

as

in the p r o p e r o r d e r ,

that if

we have proved

92

and l a b e l

its p r e d e c e s s o r

{ W I , . . . , W i _ I } , for

numbered

ors of

then

all v e r t i c e s

as

lie a m o n g

proliferation of

in

we h a v e r e n u m b e r e d

Vl,...,V k Wi

V i 6 V2'

the v e r t i c e s

procedure stage,

if

its c o l l e c t i o n

if

are

strong

(V,E), o n e c a n p a r t i t i o n

the

in s u c h a w a y t h a t the e q u i v a l e n c e

classes

of

V

connectedness;

v a e V i , v b • Vj

, and

(vb,v a) • E , t h e n

i > j

't/

v2

FIGURE

4.3

v4

v

v7

%

v5

70

57

Example adjacency

58a

matrix

A

=

Consider

the digraph

of Figure

4.3,

whose

is

1

1

1

1

0

0

0

0

0

1

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

1

0

1

0

1

0

1

0

0

0

0

0 i

Using

either

58b

R

Using

Lemma

vertices

59

Algorithm

=

(48),

(41)

or

(42),

we can compute

1

1

1

1

1

0

i"

0

1

1

1

1

0

1

0

1

1

1

1

0

1

0

0

0

1

0

0

0

0

1

1

1

1

0

1

0

1

1

1

1

1

1

0

1

1

1

1

0

1

we can determine

that

the reduced

digraph

is a s

shown

v

digraph and

V4

V 3 = {v4} , V 4 = { v 6}

in F i g u r e

4.4

.

v4 FIGURE

V1

classes

½

v3 This

the equivalence

are

V 1 = {Vl} , V 2 = { v 2 , v 3 , V s , V 7 } , Hence

that

4.4

is a c y c l i e ,

as e x p e c t e d .

Further,

do not have

predecessors.

So we

the v e r t i c e s

let

W1 = V1 ,

of

71 W2 = V4 .

If we remove these vertices,

ing these vertices, Clearly

V2

we get the digraph

does not have a predecessor,

V3

W4 = V3

appropriate 60a

so we let

leav-

.

W3 = V2 .

4.5

Hence the e q u i v a l e n c e

order,

4.5

V2 FIGURE

Finally,

as well as the edges shown in Figure

classes,

numbered

in the

are

V 1 = {Vl}, V 2 = {v6} , V 3 = {v2,v3,v5,v7}, Note that this o r d e r i n g

is not unique;

V 4 = {v 4}

for instance,

we can also

take 60b

v I = {v6}, V 2 = {Vl} , V 3 = {v2,v3,v5,v7},

V 4 = {v 4}

We close

introducing

out this s u b s e c t i o n by formally

concepts of a strongly

connected

component

and an i n t e r c o n n e c t i n g

subgraph. 61

Definition denote

the e q u i v a l e n c e

way that connected (V i U Vj subgraph

classes of

(Vb,V a) ~ E

Then the digraph

63

Given a digraph whenever

(vi,

component , (Vi×Vj) (IS), for Example

(Vi×Vi)

(SCC), N E)

(V,E),

V

under

let

Vl,...,V k

S, ordered

Va E V i , V b E Vj, and n E)

for

in such a i < j

is called the i-th s t r o n g l y

i = l,...,k

is called the

.

The digraph

ij-th i n t e r - c o n n e c t i n g

1 ~ i ~ j ~ k . Consider

once again the digraph of Figure

Its strongly c o n n e c t e d c o m p o n e n t s 63a

SCCI

~

({Vl},

(Vl,Vl) }

63b

scc2

=

({v6},

~)

63c

SCC3

=

({V2,V3,V5,V7}, (v7,v2))

4.3

are

(v2,v 3) , (v3,v 5) , (v5,v 7) ,

.

72 63d

SCC4 where

@

=

({v4} , ~)

denotes

the empty set.

Its i n t e r c o n n e c t i n g

64a

ISl2 =

({Vl,V6} , @)

64b

IS13 =

({Vl,V2,V3,V5,V7},

64c

IS14 =

({Vl,V 4},

64d

IS23 =

({v6,v2,v3,v5,v7},

64e

IS24 =

({v6,v4},

64f

IS34 =

({v2,v3,v5,v7,v4},

subgraphs

are

The i m p o r t a n t SCC's

a proper

~) (v3,v4),

connected, E .

(v5,v4))

(V,E) ; that is, the v e r t e x E .

the union of all the

subgraph of

subset of

SCC's of

(V,E),

in that its edge set is a for the digraph of Figure

SCC's

has only five edges.

the p r o b l e m of analyzing

simpler

than analyzing

4.1.4.

D i r e c t e d Trees Definition

the original

to be a m a x i m a l

(V,E)

that contains

neither

cycles

nor self-loops.

graph

(V,E t)

creates

66

in

a directed

tree of

neither cycles

that the i n c l u s i o n

(V,E)

(V,E),

subgraph

of

is a sub-

nor self-loops, of any edge from

with E-E t

either a cycle or a self-loop.

Example (v5,v 7)

the

c o n n e c t e d digraph

(V,E)

property

4.3,

digraph.

Given a strongly

that contains

is

Thus,

tree of

In other words,

(V,E)

(V,E)

all of the SCC's is still

we define a d i r e c t e d

the a d d i t i o n a l

set of the Unless

For instance,

union of the four

65

(v6,v 3) , (v6,v 5) , (v6,v7))

V, and its edge set is a subset of

is strongly

general,

(Vl,V4))

point to note is that the union of all the

is a subgraph of

union is

(Vl,V2) , (Vl,V3))

If we remove

the edges

from the digraph of Figure

Alternatively,

we can remove

4.2

(Vl,V 2)

(v3,vl) , (v4,v I)

, we get a d i r e c t e d and

(v7,v 6)

and get

and tree.

73 another directed 67

Lemma (V,E s)

tree. Given a strongly c o n n e c t e d

be a subgraph of

self-loops. (V,E t)

Then

(V,E)

there is a s u p e r s e t

is a d i r e c t e d

tree of

This lemma states

prove

DECOMPOSITION

system

arrangement with those discuss and

(LSlS)

"below"

it.

whereby

in S e c t i o n

tree. to

5 . SUBSYSTEMS inter-

into a h i e r a r c h i c a l

each s u b s y s t e m

The advantages

b e g i n n i n g with

Consider

in a d i r e c t e d

we show how a given large-scale

in turn each of a l t e r n a t i v e

(2.2.18),

such that

This lemma is needed

can be d e c o m p o s e d

are d i s c u s s e d

let

cycles nor

that contains

INTO STRONGLY C O N N E C T E D

of subsystems,

decomposition

Es

can be imbedded

results of C h a p t e r

In this section, connected

of

that any subgraph

and is omitted.

the w e l l - p o s e d n e s s

4.2

Et

(V,E),

neither

(V,E).

neither cycles nor self-loops The proof is obvious

digraph

that contains

interacts

only

of carrying out such a 4.3.

In what follows,

system d e s c r i p t i o n s

we

(2.2.1)

(2.2.1).

a large-scale

interconnected

system d e s c r i b e d

by m la

ei = ui = j~l Hij yj

} i = l,...,m

ib

Yi = Gi ei n,

where

ui' ei' Yi

belong

to

L pe~

for some fixed n.

some p o s i t i v e

integer

Given the system as follows: (vj,v i)

then it means

m

Hij ~ 0 that

joint subsystems

vertices

as

Vl,...,v m ,

If the r e s u l t i n g

(i) actually

H.. 13

: L pe3 + L pe" l

constructed

is not connected,

a collection

that do n o t i n t e r a c t with each other.

case each c o n n e c t e d

component

can be a n a l y z e d

we can safely assume that the d i g r a p h

n.

and draw an edge

digraph

represents

and

n.

and

(i), we associate with it a digraph

Label

if

[i,-]

n.

: Lpei ÷ L pei '

ni ' Gi

p •

of disIn this

separately.

associated with

Hence

(i) (referred

74 to h e r e a f t e r

as the s y s t e m

digraph)

If the s y s t e m d i g r a p h there

is n o t h i n g

not strongly vertices

as i n d i c a t e d

a renumbering into

further

connected,

to be done.

in S e c t i o n

l+l,...,Vn.}, --

no e d g e Now,

(Va,V b)

(uj, vj e Vi)

zi =

(yj, vj 6 Vi)

di =

(ej, vj 6 Vi)

(Gj,

t h i s is done, set

is the

w e have

V = { V l , . . . , v m}

, in s u c h a w a y

that

v a 6 V i , v b 6 Vj

and

i > j.

(i) in a c o r r e s p o n d i n g

z. = F .

xi,

d.

l

=

R.. = 0 w h e n e v e r 13 into SCC's.

the s y s t e m e q u a t i o n s

i < j.

(i) can

as i-i ~ R.. z. j=l 13 3

1

z i E L p e1 %).

Rij

that

d.

%).

and

v s 6 Vj)

we have

R.. z . 11 1

I

di,

i = l,...,k

definitions,

expressed

7b

(Uni_l +I' .... Un')l

of the d e c o m p o s i t i o n

the a b o v e

d. = x . i i

I

=

j E Vi)

of t h e r e n u m b e r i n g ,

is t h e o b j e c t i v e

7a

8a

i = l,...,k

(Hrs , v r e Vi,

be e q u i v a l e n t l y

L peI '

Once

the s y s t e m e q u a t i o n s

xi =

With

where

4.1.

and number

then

Define

Ri j = Because

If the s y s t e m d i g r a p h the S C C ' s

of the v e r t e x

exists whenever

F i = Diag

This

connected,

1

we partition

manner.

is a l s o s t r o n g l y

then identify

and partitioning

V i = {Vni

is c o n n e c t e d .

: Lp

x.

1

-

for s o m e p o s i t i v e

integer

%).

÷

R..

ll

pe

z.

1

.

L e t us d e f i n e

~i' Fi

: L pel ÷

75 8b

z. = F . d. 1 1 1 as

(Si), or the i-th isolated

equivalently isolated that

subsystems

(S i)

isolated

(S I) thru

system interacts

are a r r a n g e d

(Sj)

In this connection, system,

approach on a given system,

subscript.

has not lost anything.

On the other hand,

connected,

The and the

it is i m p o r t a n t is in general

if one tries this

then the w o r s t that can happen

is strongly

is not strongly

each

all the i n t e r c o n n e c t i o n

Therefore,

that the system digraph

k

Thus the

whereby

subsystems

because

have been omitted.

(or

property

system can be deduced by studying

alone.

than the original Rij

(i)

of the

is that the w e l l - p o s e d n e s s

to note that the union of the isolated simpler

i > j

only with those having a higher

subsystems

operators

if

in a hierarchy,

of such an a r r a n g e m e n t

stability of the overall isolated

Then the LSIS

(Sk) but with the a d d i t i o n a l

does not interact with

subsystems

advantage

subsystem.

(7)) can be viewed as an i n t e r c o n n e c t i o n

connected,

is

in w h i c h case one

if the system digraph

then c o n s i d e r a b l e

savings

in complex-

ity can result.

Example (i.e.,

Consider

5 subsystems),

a system of the form

and the following

(i), w h e r e

interconnection

m = 5

operators

are nonzero: H21, H25 , H32 , H42 , H43 , H51 , H53 . The r e m a i n i n g H.. , i ~ j are assumed to be zero. (Note that we need not a3 bother about operators of the form Hii , b e c a u s e they r e p r e s e n t self-loops

in the system digraph,

the d e t e r m i n a t i o n Figure

v1

4.6

of the SCC's).

and therefore do not enter into The system digraph

is shown in

One can easily verify that there are three SCC's,

v2

~

FIGURE

_

4.6

v

5

76 and

that their vertex

V 1 = {vi} , V 2 = {v2,

sets

(arranged

in the p r o p e r

v3, v5} , V 3 = {v4}.

Accordingly,

i0

x I = Ul, x 2 =

[u 2, u 3, u5]'

, x3 = u4

ii

d I = el,

d2 =

[e2, e3,

e5]'

, d3 = e4

12

Zl = YI'

z2 =

[Y2' Y3'

Y5 ] ' ' z3 = Y4

El 0 G2

13

F1 = G1 , F2

=

G3 0

14a

Rll = Hll

14b

R22

, RI2

= H12

I~ 22

=

LH52

The new system description

It is c l e a r

t h a t is u s e d

is w h e t h e r

can cause unnecessarily Example

(9) above,

ily c h e c k

if

o t h e r hand,

Chapter

H41

to s a f e l y

H23 H33

0 51 H3

0

H55j

' ~3=

[~4H3~ 0] ~ = •

"'33

(7).

decomposition

is a

t h a t the o n l y i n f o r m a t i o n

H.. is z e r o or n o n z e r o . S o m e t i m e s • this z3 conservative results. F o r i n s t a n c e , in were

would

to b e n o n z e r o ,

t h e n o n e c a n eas-

be s t r o n g l y

connected,

r e s u l t b y this p r o c e d u r e .

be very

ignore

it.

"small" Such

a n d it m a y

issues

O n the

therefore

are discussed

in

6.

Now we consider

a system described

equations m

15

F3 = G4

, RI3 = 0

in the s e n s e

H41

might

we d e f i n e

G

t h a t the s y s t e m d i g r a p h w o u l d

a n d no s i m p l i f i c a t i o n possible

,

t h a t the p r o p o s e d

decomposition,

are

ii

is n o w g i v e n b y

"structural"

order)

e. = u. - [ S.. e. , 1 ~ j=l 13 3

i

~

l•°,,•m

b y the s e t of

be

77 n,

where

ei,

u i 6 L p ez

for some f i x e d n,

integer

ni ,

associated manner

with

entirely

Vl,...,v m done,

and

i > j.

m

in a

vertices

Sij M 0 .

components

as

(Va,V b)

(uj

, vj • V i)

17

di =

(ej

, vj 6 Vi)

18

z.. z] =

(Sk£ e£

19

Rij =

(Sk£

Once

of the s y s t e m

V i = {Vni_l+l ,...,vni} exists whenever

to n o t e

m.m.xl

v a • V.l ,

z

]

that the

vector,

m.m. c o m p o n e n t s of z.. z 3 z] w h e r e a s the m.m. components z 3 matrix. With these definit-

R.. a r e a r r a n g e d in an m . x m 0 z3 z 3 ions, the s y s t e m e q u a t i o n s c a n be r e w r i t t e n

as

i d. = x. - ~ R.. d. l 1 j=l z3 3

20

because,

by construction,

Rij = 0

whenever

i < j.

We refer

to t h e s y s t e m

21

d. = x. - R.. 1

1

ll

as the i - t h i s o l a t e d

d.

1

subsystem.

We can further modify From

(18) w e see t h a t

zij = Rij

is

, v k • V i , v~ • V~)3

of

22

as

this

, Vk 6 V i , v£ • Vj)

it is i m p o r t a n t in an

s[stem digraph is c o n s t r u c t e d

We then define

xi =

are a r r a n g e d

if

connected

16

where

and some positive

We label

(vj,v i)

the v e r t i c e s

t h a t no e d g e

, and

(15)

to b e f o r e .

the s t r o n g l y

and r e n u m b e r

in s u c h a w a y v b ~ Vj

The

the s e t of e q u a t i o n s analogous

[1,~]

n.

: L p e3 ÷ L pez

Sij

a n d d r a w an e d g e

we find

digraph,

p •

dj

the s y s t e m d e s c r i p t i o n

(21).

78

where 23

Rij = Diag [Columns of

Rij]

Moreover, we have 24

Rij = Kij Rij where the operator

25

K.. ~3

is represented by the

m.×m.m, l • 3

matrix

Kij = [Im. I --IIm.] 1 l and

Im. denotes the identity matrix of dimension mi×m i. If 1 (I+Rii)-l exists for all i , we can express the system equations (20) solely in terms of the z. 's , as follows: 13

26

zij = Rij dj = Rij (I+Rjj)-I "

) -i

-- Rij(I+Rjj

=

Rij (I+Rjj)-I

(xj - k=l j~l Rjk dk)

j-i

(xj (xj

~ Kjk Rjk d k) k=l

j[l - k=l Kjk zjk)

Hence the final form of the system equations is 27

zij = Rij(I+Rjj) -I (xj - 311 Kjk Zjk) k=l where it is important to observe that the operator represented by a constant matrix.

28

Example

Kik

is

Consider a system described by (15), where

the following operators are nonzero: Sll, S13, $21, S22, $32, $41, $43, $46, $52 , $53, $54 , $67 , $75 , S77 The associated system digraph is shown in Figure 4.7. (Note that the self-loops corresponding to the operators SII , $22 , and $77 are not shown in Figure 4.7 because self-loops do not figure in the determination of the strongly connected subsystems).

79

VI

V7

FIGURE

From Figure of v e r t i c e s ,

4.7, w e see t h a t t h e e q u i v a l e n c e

dI

=

order,

Accordingly,

e2

,

d2

classes

[e4]

in the a p p r o p r i a t e

V 2 = {v 4, v5, v6, VT).

29

4.7

=

are

V 1 = {v l, v2,

v3},

we define

e5

e3

e6 e7

ii° !i

30

R I I --

[~

21

-0

R22 =

a n d of c o u r s e

s41° s4

$22

'

R21 =

$32

0

$46

0

S54

0

0

0

0

0

0

S6.

0

$75

0

$77

R I 2 = 0.

Moreover,

we have

0

$52

0 0

0 0

5

80

~ii

el ~

$21

e1

IS31 e 1 iSl2 e 2 31

Zll = !$22 e 2 i

$32 e 2 IS13 e 3 iS23 e 3

~ and

33 e3

z12 , z21, z22

are similarly defined.

In (31), we display

e.g. S31 e I instead of 0 (note that~ S31 = 0) to make the pattern clearer. Next, the operator Rll is defined by

32

Rll

while

R21 , R22

KII RII,

33

=

Sll

0

0

S21

0

0

s31

0

0

0

S12

0

0

$22

0

0

S32

0

0

0

S13

0

0

S23

0

0

S33

are similarly defined.

Note that

i

0

0

1

0

0

1

0

)I

0

1

0

0

1

0

0

1

0

0

1

0

0

1

0

0

where

Kn

=

I

;J

RII =

81 The m a t r i c e s

~21

' ~22

are o b t a i n e d

The operators column subsystems

Rij

(I+Rjj) -I

corresponding

As shown in the next section, in the d e c o m p o s i t i o n 4.3

system

(LSIS)

only the isolated

AND S T A B I L I T Y

subsystems,

system can be a s c e r t a i n e d The advantages

and

by studying

of such results

because

the union of all isolated

is in general

simpler

than the original

we relate the p r o p e r t i e s

inter-

into an i n t e r c o n n e c t -

the w e l l - p o s e d n e s s

apparent,

TO b e g i n with,

(27).

play a key role

we show that once a large-scale

subsystems.

the

systems.

has been d e c o m p o s e d

stability of the overall

systems

to the system d e s c r i p t i o n

RESULTS ON W E L L - P O S E D N E S S

ion of s t r o n g l y c o n n e c t e d

are readily

are said to r e p r e s e n t

column subsystems

of l a r g e - s c a l e

In this section, connected

similarly.

sub-

system.

of the system

i-i la

d i = x i - Rii z i -

j=l

R.. z. ~3 3 I

ib

(S)

i = l,...,k

zi = F i d i to those of the systems

2a

d i = x i - Rii z.1

2b

z. = F . d. l l l Theorem i > j , (i) finite

Consider

the operator

T , there exists

incremental

(Si)

gain of

the system

(S).

Suppose

that for all

R.. is causal, and (ii) for each ~3 a finite constant kij T such that the

PT Rij

is less than or equal to

kij T .

Under these conditions,

the system

(S) is w e l l - p o s e d

if each of the isolated

subsystems

(S i) is well-posed.

Proof posed. set

"if"

Suppose

each of the systems

We show first of all that,

x = (Xl,...,x k) E L n pe

'

corresponding

there exist a unique

if and only

(S i) is w e l l -

to each input d e Ln pe

and

82 n z £ Lpe

a unique actually tion.

a

such that

collection

First of all,

of

(I) is satisfied.

k

for

equations,

j = i,

Since

we prove

(i) is

this by induct-

(i) is

zI = F I d I which is the same as the isolated and u n i q u e n e s s

subsystem

follows by the h y p o t h e s i s

are well-posed.

N o w suppose

exactly one solution

for

that for

(S l)

; hence e x i s t e ~ e

that all subsystems

j = l,...,i-l,

dl,...,d j

and

(Si)

(i) has

Zl,...,z j .

For

j =i,

we have i-i 5a

di = xl•

5b

zi = F i d i

-

Ril.

z I. -

j~

=i

R..

z.

13

3

:

x'

-

i

R..

ix

z.

i

where i-i x! A x i - ~ R.. z. i = j=l 13 I

N o w note that

x~ l

is u n i q u e l y determined,

hypothesis,

and that

x! E L i i pe

determines

d i 6 Lpei

and

the inductive exhibits and

z

process

existence dj,

zj,

and u n i q u e n e s s

X l , . . . , x i , because on

Rij

x I! and hence on

ity of way.

(6) that

(S i)

PT d

and

PT z

Hence the system

well-posed

(5)

This shows that

Hence the system

of solutions.

(i)

To prove that

depend causally on x!

depends

d

x l,...,x i . as functions

di,

Xl,...,xi_ 1 .

c a u s a l l y on

is causal w h e n e v e r

i > j zi

Next,

depend causally

The global Lipschitz of

PT u

continu-

is p r o v e d in the same

(1) is well-posed.

"only if"

Suppose is not.

not well-posed.

uniquely.

is well-posed,

but

(S i)

is well-posed,

x , we again proceed by induction.

j = l,...,i-I

Then it is clear form

since the system

z i E Lpei

(S i)

can be continued.

depend c a u s a l l y on

Suppose

Since

by the inductive

that systems

(Sl),...(Si_l)

We show that the system

are

(i) is als0

83

To be specific, because

suppose that

it v i o l a t e s c o n d i t i o n

solution.

If

(S i)

is not w e l l - p o s e d

(WI) of D e f i n i t i o n

(2.2.9),

i.e.,

xi0 • LpeI , (2) does not have a

that for some s p e c i f i c i n p u t unique

(S i)

v i o l a t e s either

a r g u m e n t s b e l o w are e a s i l y modified.

(W2) or

(W3), the

x. = 0 for j = i,..., 3 i-l; by the "if" part of the proof above and the a s s u m p t i o n that (SI),...,(Si_ I)

dl,...,di_ 1

are w e l l - p o s e d ,

and

i = l,...,i-i

.

Zl,...,zi_ 1

W i t h this input,

such that

(I) is s a t i s f i e d

for

i-I [ Rij z. j=l 3

the i-th e q u a t i o n in

8a

d i = xi0 +

8b

z. = F . d. 1 1 1

(i) b e c o m e s

i-I i-i [ R.. z. - R.. z. j=l z3 3 ii 1 j=l Rij

(8) does not have a u n i q u e solution.

(i) is not well-posed.

Remarks that,

we see that there e x i s t u n i q u e

N o w let

x i = xi0 +

By assumption,

Let

(i)

zj = xi0 - R i i z i

Hence

system

D

It is clear from

the proof of T h e o r e m

of s o l u t i o n s to

(I), we can state the f o l l o w i n g result:

"(S)

e x h i b i t s e x i s t e n c e and u n i q u e n e s s of s o l u t i o n s c o r r e s p o n d i n g each

(3)

if we are o n l y i n t e r e s t e d in the e x i s t e n c e and u n i q u e n e s s

x 6 Ln pe

if and o n l y if e a c h s y s t e m

(S i)

to

has e x a c t l y one

n. s o l u t i o n c o r r e s p o n d i n g to e a c h x i e L i ,, In o t h e r words, we pe do not n e e d to m a k e any a s s u m p t i o n s about Rij if we are o n l y i n t e r e s t e d in e x i s t e n c e and u n i q u e n e s s of solutions.

(ii) that

Rij

Definition

The h y p o t h e s i s on

Rij

in T h e o r e m

is a w e a k l y L i p s c h i t z o p e r a t o r w h e n e v e r

(3) imply i > j

(see

(5.1.1)).

Theorem

(i) s o l u t i o n for

W i t h r e s p e c t to the s y s t e m

For each input set d, z

in

Ln pe

x e Ln pe

'

(S), suppose t h a t

(i) has a u n i q u e

84

(ii) Under

Y(Ris)

< ~

Vi > j

these conditions,

the system

if each of the i s o l a t e d

subsystems

Proof (I) to denote

"if"

To avoid confusion,

quantities

associated

Suppose

each of the subsystems

ki, b i

are finite

constants

xi' d(I)i , zi(I)

arbitrary

input to the system

from

subsystems.

and suppose

< -

ki!Ixill p + b. 1

satisfy

(2) .

(S).

Now let

Then

x e L np

d I = d I)

be an =

Zl

(I) Zl

(10), we get

Ii

lldlIl p, For

is Lp-Stable,

such that

whenever

Hence

(3.1.1)).

we use the superscript

with the isolated

(S i)

I d(I) l!p,!Iz(I) I I i i ''p

i0

(recall Definition

(S) is L -stable if and only P (S i) is Lp-Stable.

]]ZlI] p

i = 2, we have

from

~

(i) that

l!d21rp, T1~211p

12

k I llxl!l p + b I

_< ~2(Irx2- R21 h(I) l!p) ÷ b 2 k2CI!x2[rp ÷ k I lIXlllp + b 1) + b 2

The proof by induction "only if" theme exist

finite

is obvious. Suppose

com_s%ants

the system

(1) is Lp-stable.

k

such that

b

11dIIp , IIZllp 0 , there

for every (0,T)

and let

[0,T-~]

~ > 0 , there

89 Remarks

i) Vt •

[0,T].

as f o l l o w s :

Hence

pendence made

If

R

~ k T Vt e is c a u s a l ,

= 0 Y s e [0,t].

s ~

of c a u s a l i t y ,

we can also define

np(PtR)

2)

ever

because

a weakly

np(PtR)

~ ~ p ( P T R)

Lipschitz operator

R : L n + Lm is w e a k l y L i p s c h i t z if it pe pe a n d for e v e r y T > 0 there exists a finite constant

such that

RPt)x](s)

that,

an o p e r a t o r

is c a u s a l , kT

Note

[t,t+~]. of

(Rx)

Thus, (s),

small

It,t+6]

small

Suppose

For every

R

on

is s m o o t h i n g x(s),

incremental

(see E x a m p l e s ( 9 )

Example ing c o n d i t i o n :

it is e a s y to see t h a t

[(RPt+ ~ -

{[Pt+~(RPt+~-RPt)]x}(s)

an o p e r a t o r

s •

to h a v e a r b i t r a r i l y

sufficiently

Hence

[0,T].

and

s •

= 0

when-

if the de-

[t,t+~]

gain by making

can be 6

(21) b e l o w .

# : R+xR n ÷ R n

satisfies

T > 0, t h e r e e x i s t s

the f o l l o w -

a constant

kT

such

that

l]¢(t,x)

Then

the m e m o r y l e s s

(RlX) (t)

is w e a k l y y(.)

e Ln pe

Lipschitz.

- ¢(t,y) I I _< k T

operator

R1

I Ix-yl I, V x , y E R n , Y t E [0,T]

n ÷ L pe n : L pe

defined

by

= ¢ (t,x(t)) T O see this,

observe

that,

whenever

, we have

IIRlX-h YlITp ~ kT llx-ylITp , This

shows that

n p ( P T R I) ~ k T

Example R2 : Ln pe

Ln pe

Let

defined

f

, VT.

8 [ 0.

Then

the d e l a y o p e r a t o r

by

0 ,

t < 8

x(t-%)

t > 8

(R2x) (t)

is w e a k l y

Lipschitz,

because

np(PTR)

~ 1

%r9 .

x(.),

90

Example Then

the o p e r a t o r

Suppose

F : R+×R+ + R m x n

R3 : L n ÷ Lm pe pe

is continuous.

defined by

t i0

(R3x) (t) =

F(t,T)

x(T)

dT

0 is smoothing.

To see this,

observe

that

IS ii

I F(S,T) t

( [Pt+~ (R3Pt+6-R3Pt) Ix} (S) =

0

Hence,

whenever

12

t 6

[0, T-6],

dr, s 6

X(T)

[t,t+~]

otherwise

we have

I IPt+~ (R3Pt+ ~-R3P t) x] ]p

II

= { t Now,

following

(3.1.15), 13

F(S,T)

x(~)

dT I Ip ds}i/P

t

the same r e a s o n i n g

as in the proof of Lemma

we get

] IPt+6 (R3Pt+~-R3Pt) x] I p ~

C1l/p

c~/q I]xll

p

where t+~

14a

cI =

sup T 6

[t,t+6]

IT

l]F(s,~)ll

ds

rs 14b

c~ =

sup s E

[t,t+~]

j . II F(s,T ) II dr t

If we now define

15

--

sup 0 < s, • < T

I fF(s,~) f

it is easy to see that

c I ~ MT~

in (13), and o b s e r v i n g

that since

its i n c r e m e n t a l

, c® ~ MT~.

Substituting

Pt+6(R3Pt+~-R3Pt)

gain is the same as its gain,

gives

this

is linear,

gl

16 If we let

17

~p[Pt+~(R3Pt+~-R3Pt)]

0

Lipschitz,

Vt 6

since both to show that

and let np(PTH)

[0,T].

Now pick

~ > 0

~ nT 6 E

is (0,T)

such that 19

np[Pt+ ~ (GPt+~-GP t) ] < (~/n T) , Vt E This

is always possible

repeated 20

H

we have

Proof and

(16)

then so are G ± H.

are both weakly Lipschitz,

Lemma Suppose G : L n ÷ Lm L£ Ln pe pe H : ÷ is weakly Lipschitz. Then pe pe

GH

while

is smoothing.

because

use of causality,

G

[0,T-~]

is smoothing.

we get

~p [Pt+~ (GHPt+~ -GHPt) ] = ~p[Pt+~ (GPt+~HPt+6-GPtHPt)

]

= ~p[Pt+6 (GPt+~HPt+$-GPtHPt+6)

]

= np[{Pt+ ~ (GPt+6-GP t) }. Pt+6HPt+6] 0

smoothing,

we

is w e a k l y

Lipschitz

(4.2.18),

the operator

be can

arbitrary find

a

~ E

and

let

(0,T)

and

G

u < 1 such

is

z ~ GH(u,z) .

that

Since

96

~p[Pt+~(GH(u,Pt+~.) Now,

by causality,

-GH(u,Pt.))]

(3) is e q u i v a l e n t

[ dj b33 i=l

Id i bij I = -

i@j One can also express

d I ..... d n

is s t r i c t l y r o w - d o m i n a n t ,

[ d. b.. i=l 1 x3 '

Yj

i~j

(5) e q u i v a l e n t l y as

n

i=l (iv)

d i b.. > 0 13 '

Vj

There exist positive constants that

B C

Cl,...,c n

is s t r i c t l y c o l u m n - d o m i n a n t ,

C = Diag {c l, .... Cn}. exist positive constants

In other words, Cl,...,c n

such

where there

such that

n

[

bij cj > 0 ,

Vi

j=l In effect, sufficient conditions

(ii)-(iv) for

(i) .

are all e q u i v a l e n t n e c e s s a r y and The p r o o f of F a c t

(4), plus

several o t h e r e q u i v a l e n t n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s for (i), can be found in p. 396].

[Fie. i] or m o r e a c c e s s i b l y in

[Sil. i,

109

Lemma

Suppose

Then

p(A)

I -A n

are all positive.

< 1

.

Hence,

positive, p(A)

"if"

By Fact

(2),

l-p(A)

entries.

it follows

from Fact

(4) that

of

is an e i g e n v a l u e

if the leading p r i n c i p a l m i n o r s of

In-A

of

are all

l-p(A)

> 0 , i.e.

Then

IRe X I < 1

< 1 . "only if"

ever

X • Sp(A),

fore

1-Re X > 0

have p o s i t i v e

Suppose

p(A)

by the d e f i n i t i o n YX • Sp(A),

real parts.

Lemma (In-A)-i

Suppose

Proof positive, inverse,

p(A)

Fact elements. X n} with

whenThereIn-A that all

are p o s i t i v e . o

has all n o n n e g a t i v e elements

elements.

if the leading

are all positive.

< 1

by Lemma

(8).

Hence

In-A

In-A

are all

has an

is given by

which shows that ii

A E R nxn

In-A

(in_A)-i =

i0

In-A

of

(4), it follows

If the leading p r i n c i p a l minors of

then which

of

by Fact

has all n o n n e g a t i v e

p r i n c i p a l minors of

< 1 .

of the spectral radius.

so that all eigenvalues

Hence,

the leading p r i n c i p a l - m i n o r s

Then

has all n o n n e g a t i v e

if and only if the leading p r i n c i p a l minors

Proof In-A

A 6 R nxn

~ i=0

(A) i

(In-A) -I Suppose

has all n o n n e g a t i v e

B E R n×n has all n o n p o s i t i v e

T h e n there exists a diagonal m a t r i x Xi > 0 V i

elements,

such that

B' A + A B

if and only if the leading p r i n c i p a l minors of

off-diagonal

A = Diag

is p o s i t i v e B

o

{XI,...,

definite,

are all

positive. Proof

See [Ara.

6.2.2

BASIC

i].

"TEST-M/~TRIX"

TYPE C R I T E R I A

With the aid of these results, stability criteria

for the system

(i)

we can derive

several

110

Theorem

12

for all

Suppose all operators

i , and define the test matrix

13

Gi

have finite gains

Q1 e R m×m

by

qlij = 7p(Hij Gj) Assume that

qlij

is finite for all

i,j

Then the system

is Lp-stable

if all the leading principal minors of

Im-Q 1

(i) are

positive. Remarks equivalent

By Lemma

to requiring Proof

(8), the hypotheses on

that

bij

are

p(Ql ) < 1 .

By the definition of

finite constants

Q1

7p(Hij

Gj)

, there exist

such that n,

14

I Ixl ITp+bij , YT > 0, Vx E Lpe3

I IHijGjxl ITp < 7p(HijGj) Substituting equations

from

(ib) into

(la), we can recast the system

in the form m

15

ei = ui -

~ H.. G. e. , j=l a3 3 3

Taking norms in (15) and applying

i = l,...,m

(14), we get m

16

IIeilIT p ~ IluillT p + j=l [

IIH i3.G.e 3 J IITp m

T Then

illxll

the s y s t e m

÷

~i + ~2 > 0

7b

a2 + ~i > 0

T

F r o m the s y s t e m e q u a t i o n s

+ T

= T

= T

+ T

Cl]lelllT2 + ~211Y211~ Substituting

2 + si

for

'

(i) is L 2 - s t a b l e , p r o v i d e d

7a

Proof

6iltGiXllT

eI

(i), w e h a v e

+ T

+ 6 1 ] l Y l l i 2T + ~i + £ 2 1 1 e 2 1 1 ~ + ~2

from

(la)

and e 2 from

(ib) g i v e s

i=l,2

136

T + _ 0

,

Vx e

L~ 2e

13

!IGlX'! T -< E IIxl

T

'

~

'

vx~

n ~2 e

14

<x'G2X>T

T2

,

%~2

,

Vx

Then

the

system

15

~ ~ IIxl

(I)

is L 2 - s t a b l e ,

> 0

-

> -

0

E

L ~

2e

provided

e + 6 > 0

Proof

16

Since

(15)

holds,

pick

8 > 0

such

that

e - e + ~ > 0

NOW,

17

e, ~,

(12)

and

T _

>

(13)

together

~ I1~11~ = (~-e)

apply

that

IlxllT

IT,,x~, T2 + (e/~2) r[

(~-e) Now

imply

Theorem

(5) w i t h

2 + e Ilxl IIGlXlIT

e I = e-e,

2 IT

2

VT

61 = e/~

2 ,

> 0

,

-

~xE

L~ 2e

e 2 = 0 , and

~2 = 6 .o It is a l s o relaxed

to a n o t h e r

_
0

generally

L -stability,

the

VT

more

is n o t as b

Corollary

form:

,

+ b

a slightly

which

in C o r o l l a r y

6 = 0 , then we get

of

IlXllT

constant of

diminishes

If,

>

case,

needed,

the a d d i t i v e in the

possible

condition

loss

(5). present

of g e n e r a l i t y

(ii).

e

>

0

and

is h i s t o r i c a l l y

138 19

Corollary and

~

S u p p o s e there exist p o s i t i v e c o n s t a n t s

c

such that i

20

<X,Gl.>

T

>-

~

i

i

i ~

,,~llxltZ

vT

'

>-

0

'

L~

vx

2e

21 -

Suppose

G2

22

'

-

'

28

satisfies

<x,G2x> T > 0 , ~T > 0 , Vx 6 L v 2e U n d e r these conditions,

the s y s t e m

An o p e r a t o r operator,

G2

(i) is L 2 - s t a b l e .

satisfying

w h e r e a s an o p e r a t o r

G1

is c a l l e d a s t r i c t l y p a s s i v e operator. states the following:

Suppose

finite gain w i t h zero bias, system

is called a passive (28) w i t h

Hence Corollary

G2

is s t r i c t l y p a s s i v e and has

and that

G2

both Corollaries

i n t e r c h a n g e d throughout.

"symmetric"

is passive;

c o n d i t i o n s on

G1

and

(ii) and

are w o r t h m e n t i o n i n g .

23

Corollary passive;

N o t e that T h e o r e m

i.e.,

(5) imposes

G2

(5) that

Both are easy to prove.

Suppose both

G1

and

G2

are s t r i c t l y

suppose there e x i s t p o s i t i v e c o n s t a n t s

<x'G~X>T~ >- eiIIxIl~" • T Then the s y s t e m

25

26

G1

eI

and

such that

24

~2

'

%~f > 0 , Yx 6 L v2e ' i = 1,2

(i) is L 2 - s t a b l e .

Corollary and

then the

(19) hold w i t h

T h e r e are two o t h e r c o r o l l a r i e s of T h e o r e m

~2

e > 0 (19)

(i) is L2-stable.

Actually, and

G1

(22)

satisfying

Suppose

there e x i s t p o s i t i v e c o n s t a n t s

such that

<x,Gix> T ~ ~±lIGixll

v T2 , WT > _ 0 , VX E L2e

, i = 1,2

~i

139

T h e n the system

(i) is L 2 - s t a b l e .

N o t e that there is no a s s u m p t i o n of finite g a i n in Corollary

(23), and that there is no a s s u m p t i o n of strict passivity

in C o r o l l a r y

(25).

Also,

the results of s e c t i o n 3.2 are v e r y

useful for d e t e r m i n i n g the v a r i o u s c o n s t a n t s c o r r e s p o n d i n g to a g i v e n o p e r a t o r

7.2

E

and

6

G .

GENERAL D I S S I P A T I V I T Y - T Y P E C R I T E R I A

In this section we state and p r o v e some g e n e r a l d i s s i p a t i v i t y - t y p e c r i t e r i a for the L 2 - s t a b i l i t y of l a r g e - s c a l e i n t e r c o n n e c t e d systems.

S p e c i a l cases of the g e n e r a l results

g i v e n here, w h i c h are easier to apply,

are p r o v e d in the n e x t

section.

T h r o u g h o u t this section,

and i n d e e d t r o u g h o u t the rest

of this chapter, we shall be c o n c e r n e d w i t h a s y s t e m d e s c r i b e d by m

la

ei = ui -

~.

j=l

Hij Yj i = l,...,m

ib

Yi = Gi ei where Gi

u i , ei" Yi

maps

dimension

all b e l o n g to

n. L 2 e l into itself, nixn j .

and

Then matrix

n. L 2ez Hi3.

=

[zlil

is c a l l e d the i n t e r c o n n e c t i o n matrix.

We b e g i n w i t h an obvious

ni

is a c o n s t a n t m a t r i x of

H 6 R n×n

defined by

H

for some i n t e g e r

(where

n =

m ~ n i) i=l

140

3

Lemma i = l,...,m

.

4

Suppose

Then

is

Q = Diag

{QI'''''Qm }

5b

R = Diag

{R 1 ..... R m}

5c

S = Diag

{S I , . . . , S m}

Apply

Definition

The next result t h a t of the i n t e r c o n n e c t e d

Lemma (4) a n d Then

Consider

(3.2.1).

relates system

the dissipativity

Suppose

(1) is d i s s i p a t i v e

of

G

with

(1).

the s y s t e m

(2), r e s p e c t i v e l y .

the s y s t e m

by

where

5a

6

defined

for

{G 1 ..... G m}

(Q,R,S)-dissipative,

Proof

(Qi,Ri,Si)-dissipative

G : L n2e + Ln2e

G = Diag is

Gi

(i), a n d d e f i n e G

with

is

G,H

by

(Q,R,S)-dissipative.

respect

to the t r i p l e t

(Q,R,K) , w h e r e

7a

Q = Q + H'RH

7b

R = R

7c

S = S - 2H 'R

Proof written

9

The system equations

(i) c a n b e c o m p a c t l y

as

8

e

Since

1 - ~ (SH + H'S')

G

is

=

u

-

Hy

,

y

Ge

(Q,R,S)-dissipative,

T

Substituting

=

for

e

from

we have

+ T + < u - H y ' R u - R H Y > T

+ T + T + T However,

since

u

e+Hy

(14).

This gives

in

15

and

y

and c o l l e c t i n g

T

(12).

Then

Yu, y e F

(1), we can replace

(e+Hy)> T

terms in

+ <e,~

< ~

~ 0 ,

satisfy

-_ 0

is L 2 - s t a b l e

if

of

¥u E R the

A , the n * . Under

test marrix

is p o s i t i v e , so is

Suppose system

H

(i)

XI,...,X m

then

M

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

that

+ AR]

+

(P'AS

that

N = P'MP

so t h a t

. []

is L 2 - s t a b l e such

M { M o,

the

and

if t h e r e

let exist

test matrix

+ S'AP)/2

definite.

Proof M > 0

we r e s e r v e matrices.

with

that,

i.e.,

definite,

N = -[P'AQP

if

by

(7a)

1

to the

+ H'ARH]

If

is p o s i t i v e

25

only

of

is L 2 - s t a b l e

section

system

Corollary

positive

Q

from

definite.

Proof

24

this

Suppose

M o = -[AQ

P = H -I

varies respect

are defined

that

(i)

is p o s i t i v e ,

conditions,

Mo

i with

(18).

Corollary

22

if

Q,R,S

we get

system

We conclude

these

as

o

of T h e o r e m

matrix

summing

is d i s s i p a t i v e

where (12),

Hence

, and

(4)

the

Note

, so t h a t

N > 0

if a n d

.D

term

"positive

semidefinite"

for

symmetric

144 26

Corollary A, and that

H

S u p p o s e that

is nonsingular.

ASH

is p o s i t i v e

T h e n the system

for some

(i) is L2-stable

if the test m a t r i x

27

N

= -[P'AQP + AR]

o

is positive.

P

Proof

o n l y if

7.3

Note that

N o = P'MoP,

so that

No > 0

if and

M > 0 .~

S P E C I A L CASES:

Theorem

S~LL-GAIN

(7.12.18)

AND P A S S I V I T Y - T Y P E C R I T E R I A

is a p o w e r f u l general r e s u l t that can

be applied

to a wide v a r i e t y of situations,

operators

Gi

are s t r i c t l y passive,

etc.

can o b t a i n several useful, criteria. always,

e.g.

w h e n some of the

have finite gain, others are passive, By s p e c i a l i z i n g T h e o r e m and r e a d i l y applicable,

Some such r e s u l t s are p r e s e n t e d

still others (7.2.18), one

stability

in this section.

As

we study a system d e s c r i b e d by m

la

e i = u i = j=l [

H ij Yi } i = l,°..,m

Ib

Yi = Gi e i

where and

ui' ei Hij

'

n. Yi e L 2 ei

for some integer

is a c o n s t a n t m a t r i x of d i m e n s i o n

First, we p r e s e n t a "small gain"

n. n. 1 ÷ L 1 n i ' G i : L 2e 2e

'

n i x nj

type c r i t e r i o n based

on the d i s s i p a t i v i t y approach.

Theorem ¥i

By

.

C o n s i d e r the s y s t e m

Define the test m a t r i x

A [ B

W 6 R n×n

(A > B), we m e a n that

(positive definite)

A-B

(I), and suppose ~2(Gi) T + <x,~ i x> T ~ 0, where

Gi

we use

~i

= ~2(Gi)

is dissipative

Theorem

=

~

~ 0,

in the i n t e r e s t s

with respect

(7.2.18),

M

if o n e c a n

A - W'AW

definite,

A

Hij

to

of b r e v i t y .

2

(-In.,g i In ,0). 1 1

we g e t t h a t the t e s t m a t r i x

A

-

n. l

Nx e L2e

M

of

Thus

Applying (7.2.19)

H'ARH

where

R = Diag N o w n o t e that,

since

2 {~i In I'

2 "'~m I n m }

R is d i a g o n a l ,

we h a v e

AR = R I / 2 A

R I/2

Hence M

i0

Thus

=

the s y s t e m

is p o s i t i v e

A - H ' R I/2 A R I/2 H = A- W ' A W =

(i) is L 2 - s t a b l e definite,

o

if

A

c a n be f o u n d

such that

is

146

N o t e that, theorem

[Fre. I,

if and o n l y if

by the d i s c r e t e - t i m e v e r s i o n of Liapunov's

p.166],

one can find a

p(W)< i.

However,

not be able to find a d i a @ o n a l

A > 0

even if

A > 0

such that

p (W) < 1 ,

such that M > 0 .

try to give a m o r e e x p l i c i t c r i t e r i o n than T h e o r e m

M > 0

one may If we

(2), i.e.,

one that does not d e p e n d on b e i n g able to find some u n k n o w n constants

(with no s y s t e m a t i c p r o c e d u r e

r e s u l t that is very similar 11

Theorem ¥i.

to T h e o r e m

Consider

the s y s t e m

D e f i n e the test m a t r i x

12

nij = where

I IHijl I

N E Rm × m

~2(Gi ) IIHijll is the

to find them), we get a (6.2.71).

(i), and suppose ~2(Gi) 0

such that

Im-A

there exists a

A - A'A A > 0

As b r o u g h t out in Lemma aij [ 0

has all non-

and that the leading p r i n c i p a l m i n o r s of

(6.2.8),

the fact that

and the n o n n e g a t i v i t y of the leading p r i n c i p a l m i n o r s of

implies that

there exists a p r o p e r t y of

p(A)

~ > 0

< 1 .

Thus,

by L i a p u n o v ' s

such that ~ - A ' A A > 0 .

A, n a m e l y

a.. > 0 13 -

¥i,j,

theorem,

The s p e c i a l

allows us to select

to be d i a g o n a l as well.

Proof of T h e o r e m h y p o t h e s e s of T h e o r e m diagonal

~ > 0

(ll)

By L e m m a

(II) are satisfied,

such that

(13), if the then there exists a

~ - N' A N > 0 .

the d i a g o n a l e l e m e n t s of this

A , and define

claim that, w i t h this choice,

M

of

To e s t a b l i s h this claim,

Let A by

ll,...,Xm (6).

We

(5) is p o s i t i v e definite.

note that

A-N'AN

> 0

be

147

implies

14

that there

exists

an

e > 0

such that

m

m

m

;. ~ v~ _

~ ~

Jlxill 2--~ x,x, by Cz4~

i=l

This

shows

that

M > 0.

Thus,

by Theorem

(2),

the s y s t e m

(i) is

L 2 - s t a b l e . [] The r e s t of this passivity-type

stability

t h a t e a c h of the o p e r a t o r s following

conditions:

section

criteria. Gi

is d e v o t e d In w h a t

satisfies

(i) t h e r e e x i s t

to d e r i v i n g

follows,

we a s s u m e

o n e or the o t h e r

constants

ci

and

of the ~i

such that

16

2

Ixr T

< x , G i x > T ~ ci

n.

VT > 0 , Vx 6 L 1 2e

lIGiXllT ~ ~i Ix'

17

or

(ii)

there

exists

T

a constant

~i

such t h a t n.

18

<x'GiX>T

Throughout, positive,

-> ~i

I IGi xl 12 ' %~f -> 0 , V x 6 L2el

we a l s o a s s u m e

i.e.,

t h a t the i n t e r c o n n e c t i o n

matrix

H

is

148

x' Hx > 0

19 If

H

is positive,

then from (1) we get

m [ T = i=l

20

Vx ~ R n

m ~ T i=l

m ~ i=l

m ~ T j=l

m

0,

< x , G i x > T > ~i I Ixl 12 ,

Yx6L

1 2e

-

i=l ..... k 30

l IGixl i~ T ~

~illxl[~

2 + ~ llxl 2 = (~l-~) IlxlIT iT

(ci~ llxI1~+ (~z~ llGixll~, ~ 0 , n.

Vx6

Hence,

for

respect

i=l,...,k

, the o p e r a t o r

Gi

L ~ 2e

is d i s s i p a t i v e

39

Qi =

-

(~/~2)

in . , Ri

=_

(ei_~)

in . , Si

1

Similarly, operator

40

from Gi

Let

is

41

(31),

we

case,

us n o w

see

applicable.

M0 = -

that

for

with

respect

, R . = 0 ni

apply

ASH = H

corollary

is p o s i t i v e ,

Forming

[Q + H'

in .

=

1

is d i s s i p a t i v e

Qi = - 61• In i

In t h i s

with

to

the

RH]

i=k+l,...,m

1

, the

to

, S i = I ni

(7.1.22) so t h a t

test matrix

with

corollary

M 0 , we get

A = In

.

(7.1.22)

153

=

+ 0

=

H'

Db

MI-

0

A

0

0

0

~Q +

where 42

= Diag

{a/~

Inl,''-,u/~

We claim that this m a t r i x the first term, Next,

namely

its s u b m a t r i x

Finally,

A

is positive

Gi

satisfying

Theorem

l)

that

Ei

passive

is positive).

Similarly,

condition

One of the i n t e r e s t i n g

This trade-off

p o s i t i v e as possible

(iii)

~i

(Theorem

Note

and

in order

the s t a b i l i t y 3)

and

an o p e r a t o r Clearly

passive

operators

aspects of T h e o r e m

between

it is clear from T h e o r e m

choose the c o n s t a n t s satisfying

theorem

there is

operators.

the constants

is one of the n o t e w o r t h y

passivity

G i satisfy-

for the L 2 - s t a b i l i t y

of some p s e u d o - s t r i c t l y

is that it allows a "trade-off"

context,

Mo

the system

("pseudo!' b e c a u s e

w i t h finite gain and some p s e u d o - p a s s i v e

~i "

(7.2.22),

Let us refer to an o p e r a t o r

(28) gives a s u f f i c i e n t

single-loop

(21),

(31) is said to be pseudo-passive.

of an i n t e r c o n n e c t i o n

2)

is p o s i t i v e definite.

Hence by Lemma

whence by C o r o l l a r y

(29) as p s e u d o - s t r i c t l y

no a s s u m p t i o n

Hab + D b

definite.

F i r s t of all, semidefinite.

[]

Remarks ing

H'ab (Ea-eIr)

definite,

definite.

M 1 - u Q , is p o s i t i v e

is positive

(i) is L2-stable.

is positive

Ink}

features

(7.1.2)).

(28)

Ei

and

of the

In this

(28) that one should try to

6i

(as appropriate)

to be as

to have the b e s t chance of

conditions.

that if

are a u t o m a t i c a l l y

M1

is positive

satisfied.

definite

then

(ii)

154 4) the

Note that the h y p o t h e s e s

submatrix

H'aa E a H aa

5)

to be p o s i t i v e

The submatrices

the stability

conditions

(iv)

the r e q u i r e m e n t

H

(namely,

be positive).

stability

Thus,

Hab

and

Hbb

do not figure in

(28), except in condition

that the i n t e r c o n n e c t i o n

for all p r a c t i c a l

purposes,

among the p s e u d o - s t r i c t l y

(the i n t e r a c t i o n

t~o the p s e u d o - s t r i c t l y thru

Hba

of T h e o r e m

(iii) of T h e o r e m

passive

subsystems).

(28) are satisfied,

and if

Hba

Hbb

H

subsystems G1

Consider

represented

pseudo-strictly

are

0 H

H aa is to adjust

=

of three

G1 , G2 , G3 ,

and has finite gain,

Let the i n t e r c o n n e c t i o n

1

0

0

-i

1

0

el =ir-]li

and

where

G2 , G3

m a t r i x be

Y5

FIGURE

This system is shown in Figure matrix

M1

(i)

and thereby

an interconnection

by the o p e r a t o r

passive

pseudo~passive.

44

subsystems)

subsystems

the system. Example

43

positive,

(the

If conditions

add "compensation"

so as to make

the

passive

then one can always

and

matrix

Haa

from the p s e u d o - p a s s i v e

positive, stabilize

(28) require

semidefinite.

criteria depend only on the submatrices

"self-interaction" and

of T h e o r e m

in

(35) is

7.1

.

For this system,

7.1

the

155

45

M1 =

I! ° 1 el+62 0

Hence all the conditions system at hand

46

63

of Theorem

is L2-stable

el+~ 2 > 0 ,

(28) are satisfied

if

63 > 0

We now show that Theorem a special

case.

and suppose

Consider

G1 , G2

'

48

llGlXTl T -< ~ IIXIIT

'

49

<x,G2x> T -> 6 TllG2xt12

Then the interconnection

Hence

matrix

M1

C0~ M1

=

It is easy to verify satisfied

if

system

(7.1.2) (7.1.1),

'

~

> 0

~

-

'

> 0

~

'

> 0

-

'

vx ~

~

vxS

~ 2e

2e

vx ~

~2e

is

=

the test m a t r i x

51

Theorem

feedback

satisfy

< x , % x > T -> ~ Irxll~

H

(28) contains

the single-loop

47

50

and the

of

0~ (e+6)

(35) becomes

~I I

that all conditions

e+6 > 0 .

of Theorem

(12) are

156

We now present another stability criterion, which also contains the single-loop Theorem (7.1.2) as a special case, but is in general more restrictive than Theorem (28). However, it has the advantage that it has an instability counterpart (see Theorem (8.3.42)). 52

Theorem Consider the system (i), and suppose G 1 .... ,Gm satisfy (29)-(31), as appropriate. Suppose H is nonsingular and positive, and let P = H -I . Partition P as

r aa

n-r Pa bl

r

LPba

PbaJ

n-r

IP p =

53

Define E a and D b as in (33) and (34), respectively. Under these conditions, the system (i) is L2-stable if (i)

The matrix =FEa + Pba Db Pba

54

N1

ba Pbb P' Db 1

! Db Pbb

LPbb Db Pba

is positive definite. (ii)

we have

55

rank

N 1 = rank

56

rank

Pab = n-r

Proof

By Le~ma

an

e>0

I [N 1 I r Qnn_r ]

(24), if (55) holds,

such that the matrix

then there exists a

157

57

N2 =A NI

is p o s i t i v e

0 ]

0

0

semidefinite.

(30) we have,

58

-I 2~I r

as before,

Let

~>0

that for

be so chosen.

From

(29) -

i=l,...,k

<x, Gix> T _> ¢~i-oo

I lxl I~ + (o~/~.)

I IGixl I~

>_ ¢~i-oo

I lxll~ + ¢~/~2)

i iGixll~

,

~_>

n.

Vx 6 L 1 2e where = max i

59

Hence we define,

~ J-

as in the proof of T h e o r e m

A = Im e

(28),

S = I n , and

60

61

R = -

ITa r

QI If we apply C o r o l l a r y

:] :I

(7.2.26),

the test m a t r i x

No

becomes

o

158

-~I 62 N

o

= - (P'QP+R)

0

Pba Db Pba

Pba Db Pbb

0

+ L P'bb Db Pba

Pbb Db Pbb

= L 0a

I! PeaPab] aa Paa

+ "~--

Pab PabJ

ab Pea

i PP ill J aa

= N2 +

aa

aa Pab

Pab Paa

P'ab Pab

r

0

N3

Now,

N3

is clearly

positive

semidefinite.

Also,

the lowest

(n-r)×(n-r) submatrix of N 3 is positive definite, because P'ab Pab is positive definite (recall that Pab 6 R r×(n-r) has rank n-r). Hence by Lemma (21) , N o is positive definite, whence by Corollary

(7.2.26),

Theorem Theorem

r > n-r

have rank

, i.e. r > n/2

subsystems

not; moreover, (7.1.1)

Theorem

(49).

system of

In this case,

is nonsingular,

and

the latter (6.1.1),

implies

claim,

consider

the interconnection

that

Theorem

(28) does

the single-loop

as does Theorem

and suppose

than

is

that

However,

which Theorem

(52) also contains case,

H

at least half of the

passive.

counterpart,

matrix

the requirement

; in other words,

as a special

To prove feedback

(ii)

n-r automatically

m u s t be p s e u d o - s t r i c t l y

(52) has an instability Theorem

and

D

scope of application

(i~ the interconnection

to be non-singular,

Pab 6 R r×(n-r)

(i) is L2-stable.

(52) has a narrower

(28), because

required

the system

the single-loop

G1 , G2 matrix

result

(28).

satisfy H

(47)

given by

-

(50)

159

63

I9 The test matrix

N1

O,a of (54) is given by

64

It is easy to verify that all conditions of Theorem (52) are satisfied if e+d > 0 . Thus Theorem (52) contains Theorem (7.1.2) as a special case. Next, we further specialized the results of Theorem (28), and in the process, obtain some generalizations of Corollary (7.1.7). Though the results that follows are quite conservative, they have the advantage that they involve only the interconnection matrix H , and are therefore quite easy to apply. 65

Corollary some integer and

Suppose that for

k ~ m , there exist positive constants

~l,...,~k

66

Consider the system (i).

el,...,e k

such that

<x, T . ,Gix> ..

_> Eillxl ~ . , . .I

,

VT _> 0

,

Wx 6 Lnin.2e I i=l,...,k

67

IIGixl IT _< ~il Ixll T ,

Suppose

Gk+l,...,Gm

VT _> 0

,

Vx 6 L2el

satisfy n,

68

<x,Gix> T -> 0

,

%~f -> 0

, Vx 6 L 2el

'

i=k+l,...,m

k Let

r =

Z

i=l follows:

ni

and partition the interconnection matrix

H

as

160

r

69

H

Under

n-r

Haa

Hab

Hab

HbbJ

=

these conditions, (i)

H

(ii)

rank

the system

is positive,

Proof

n-r

(1) is L2-stable

if

and

Hab = n-r

Apply Theorem

(28) with

Then we have

Db = 0

[HAa] 70

M1

=

Fa

[Haa

Hab]

L";bJ Since

Ea

is positive

is positive Finally,

to verify

Hab

n-r and

is

H'ab Ea Hab

special

is positive

is positive

case.

it is clear

from

(36) holds,

by inspection.

Also,

(ii) of Theorem Ea

Remarks matrix

definite,

semidefinite.

(28), observe definite,

(54) that

M1

that since rank

we have

that

definite.

l) Corollary

To see this,

for the single-loop

(65)

observe system

contains

Corollary

(7.1.7)

that the interconnection

(7.1.1)

is

71 I~

which

satisfies 2)

questions:

09

all the hypotheses Corollary

(65) provides

under what conditions

passive

subsystems

finite

gain result

of Corollary

to the following

does an interconnection

and some strictly in an overall

an answer

(65).

passive

system

subsystems

of some with

that is L2-stable?

The

161

answer

is that the o v e r a l l system is L 2 - s t a b l e p r o v i d e d

i n t e r c o n n e c t i o n m a t r i x is positive, from the p a s s i v e s u b s y s t e m s have the f o l l o w i n g property: signals from the p a s s i v e subsystems,

and

(i) the

(ii) the i n t e r c o n n e c t i o n s

to the s t r o n g l y p a s s i v e s u b s y s t e m s If we k n o w the i n t e r c o n n e c t i n g

s u b s y s t e m s to the s t r o n g l y p a s s i v e

then we can u n i q u e l y d e t e r m i n e the o u t p u t s of the

p a s s i v e subsystems.

Roughly

speaking,

condition

(ii) m e a n s

that any e r r a t i c b e h a v i o u r at the o u t p u t s of the p a s s i v e s u b s y s t e m s can be d e t e c t e d t h r o u g h the i n t e r c o n n e c t i o n

signals

at the s t r o n g l y p a s s i v e subsystems.

3) have rank Hence,

Since

n-r

Hab • R r×(n-r)

implies that

, the r e q u i r e m e n t

r ~ n-r

in o r d e r to a p p l y C o r o l l a r y

, i.e.

that

that

Hab

r ~ n/2

(65), at least h a l f the sub-

systems m u s t be s t r o n g l y p a s s i v e w i t h finite gain.

Theorem

(72)

S u p p o s e all the h y p o t h e s e s of C o r o l l a r y

(65)

b e l o w r e m o v e s this r e s t r i c t i o n . 72

Theorem hold,

e x c e p t that

(ii')

(ii) is r e p l a c e d by the f o l l o w i n g condition:

whenever

v • R (n-r)

is a n o n z e r o

s o l u t i o n of

Hab v = 0 , we have

73

v' Hbb v > 0 Under these conditions,

Proof

74

for

Since

the system

(66) and

i=l,...,k

(67) hold,

that

e < ci

Then,

u s i n g the f a m i l i a r argument,

<X,~lX> T ~

(~i-~)

,

(i) is L2-Stable.

and let

choose

~ = max

~ > 0

such

{ ~ l , . . . , ~ k }.

we have

IlxllT 2 + (~/~2)

llGixll~

,

n,

VT > 0 , -

NOW apply Theorem

(7.1.18), w i t h

Vx6L

l 2e

i=l,...,k '

A = I n , S = In

162

(e/~ 2 ) I r Q = -

75

where in

E a is defined (7.1.19) becomes

(u/~2)

76

M

=

Then the test m a t r i x

in (33).

ir

L

0

R

denote

(Ea-eIr)Haa

H'aa (Ea-~Ir)Hab

ab

(Ea-eIr)Haa

H ab' (Ea-UIr) Hab

+ 0

(H + H')/2

the sum of the last two matrices.

positive

semidefinite,

positive

definite

defined

Ii ia

+

Let

M

we have by Lemma

if the b o t t o m

Since

(21) that

(n-r) x(n-r)

M

R

is

is

submatrix of

R ,

i.e. 77

Rbb =A H'ab is positive have

definite.

v' Rbb v ~ 0

is p o s i t i v e implies

(E a - ai r ) Hab + Rbb

Vv E R n-r

definite,

v = 0 .

Since

(Hbb + H'bb )/2

is p o s i t i v e So,

in order

semi-definite, to prove that

it is e n o u g h to show that

Accordingly,

suppose

we ~b

v' Rbb v = 0

v' Rbb v = 0 .

Then from

(77) we get 78

v' Rbb v = (Hab v)' Since (78)

E a - si r implies

is p o s i t i v e

that

79

Hab v = 0

80

V'

Hbb V

= 0

(E a - ai r ) (Hab v) + v' Hbb v = 0

definite

and

Hbb

is p o s i t i v e ,

163

N o w by c o n d i t i o n Hence Lemma

Rbb

(ii'),

(79) and

(80) t o g e t h e r

is p o s i t i v e definite,

M

imply t h a t

v = 0 .

is p o s i t i v e d e f i n i t e by

(21), and the system at hand is

L 2 - s t a b l e by T h e o r e m

(7.1.18). []

81

Example m = 3, and and

ni = 1

for

(67), and suppose

82

H

Then

H

.

.

Suppose

satisfy

but C o r o l l a r y

r < n-r , so that However,

Theorem

i=i,2,3 G2 , G3

G1

(68).

(I), w i t h

satisfies

(66)

Suppose

=

is positive,

because n-r

C o n s i d e r a system of the form

Theorem

Hab

(65) can not be applied,

can not p o s s i b l y h a v e rank

(72) has no such r e s t r i c t i o n .

Applying

(72) to the system at hand, we note first of all that

is positive. satisfied.

Next,

it is e a s y to v e r i f y that c o n d i t i o h

H

(ii')

is

Thus the g i v e n s y s t e m is L 2 - s t a b l e .

NOTES AND REFERENCES

The p a s s i v i t y t h e o r e m for f e e d b a c k systems was given by S a n d b e r g [San.

2] and Zames [Zam.

3].

The c r i t e r i o n a l l o w i n g a

"trade-off" b e t w e e n the forward and f e e d b a c k s u b s y s t e m s is due to Cho and N a r e n d r a [Cho 1 and 2].

The g e n e r a l d i s s i p a t i v i t y

c r i t e r i a are due to M o y l a n and Hill [Moy. e a r l i e r w o r k in [Sun.

I] and [Vid.

3].

general r e s u l t s can be found in [Vid.

2]; these g e n e r a l i z e

The s p e c i a l i z a t i o n s of the 8].

A n o t h e r e x t e n s i o n of

the p a s s i v i t y t h e o r e m to l a r g e - s c a l e systems is g i v e n by S a n d b e r g [San.

4].

As yet,

there are no s a t i s f a c t o r y g e n e r a l i z a t i o n s

the "multiplier" m e t h o d s [Zam.

4] to l a r g e - s c a l e

systems.

of

CHAPTER 8: L2-1NSTABILITY CRITERIA In this Chapter,

we p r e s e n t several c r i t e r i a

l a r g e - s c a l e i n t e r c o n n e c t e d s y s t e m to be L 2 - u n s t a b l e .

for a These

c r i t e r i a c o n t a i n the i n s t a b i l i t y c o u n t e r p a r t s of both the "small gain"

type s t a b i l i t y c r i t e r i a of C h a p t e r 6 and the "dissipativit~'

type s t a b i l i t y c r i t e r i a of C h a p t e r 7.

M a n y of the results here

are b a s e d on an o r t h o g o n a l d e c o m p o s i t i o n of the input space.

In

C h a p t e r 9, we show how these results can be e x t e n d e d to L i n s t a b i l i t y u s i n g the t e c h n i q u e of e x p o n e n t i a l w e i g h t i n g .

We b e g i n by d i s c u s s i n g the s i n g l e - l o o p case in Section 8.1.

In S e c t i o n 8.2, we p r e s e n t i n s t a b i l i t y c r i t e r i a of the

"small gain"

type, w h i c h are the i n s t a b i l i t y c o u n t e r p a r t s of the

results in C h a p t e r 6.

Finally,

in S e c t i o n 8.3, we p r e s e n t

i n s t a b i l i t y c r i t e r i a of the " d i s s i p a t i v i t y " instability yield both

type, w h i c h are the

c o u n t e r p a r t s of the results of C h a p t e r 7. "small gain"

as special cases.

and "passivity"

Throughout,

U.I 2 and U.IT2 , r e s p e c t i v e l y ,

8.1

These

type i n s t a b i l i t y criteria

we use a.~ and 11.1]T to denote b e c a u s e we deal only w i t h L2-spaces.

SINGLE-LOOP

SYSTEMS

In this section, we p r e s e n t the b a s i c results concerning the L 2 - i n s t a b i l i t y of s i n g l e - l o o p f e e d b a c k systems.

To f a c i l i t a t e the discussion, d e f i n i t i o n s and facts from S e c t i o n Definition

we restate here some

3.3.

An o p e r a t o r G: L v + L ~ is said to belong 2e 2e

to class U if

(i)

G is linear.

(ii)

The set M(G)

M(G)

v c L2 d e f i n e d by

= {x • L 2 : Gx e L }

is a p r o p e r subset of L 2.

185

(iii) 3 (iv)

There exists

a finite constant

UGxB

llxil, Yx e M(G)

~ ~c(G)

There exists (~T,T6R+)

4

"(GX)~T

~c(G)

a family of finite constants

such that

-< s T [[X[[ T , YX 6 L ~ 2e

A useful property of class U operators in the following 5

lemma

Lemma

such that

(see Lemma

(3.3.12)

Let G: L ~ L v belong 2e ÷ 2e

is b r o u g h t out

for the proof).

to class U

Then M(G)

v is a closed subspace of L 2. As stated in Lemma the property

that its "set of s t a b i l i z i n g

proper closed subspace c o m p l e m e n t MI(G) 6

(5), an operator

MI(G)

of L 2.

M(G)

is a

its o r t h o g o n a l

defined by = {z6L~:

= 0

VxeM(G)}

at least one nonzero element.

transfer

function m a t r i x G(-),

Recall

~n e x p l i c i t

inputs"

As a result,

contains

factorization

G of class U has

that,

if G is a linear c o n v o l u t i o n

(N(-), D(-))

and if G(.)

in ~n×n,

characterization

o p e r a t o r with

has a r i g h t - c o p r i m e

then it is p o s s i b l e

of the set M(G) (Theorem

to give

(3.3.32).

^

Further,

if G(-)

has a pole in the open right half-plane,

one can d e m o n s t r a t e Theorem instability

and ~c2(G)

of Ml(G)

(7) below is the basic

theorem

Theorem

* Throughout

some elements

for single-loop Consider

this chapter,

then

(Theorem 3.3.45).

"small gain"

type L 2-

systems.

a system d e s c r i b e d by

we use M(G)

in the interests

and

of brevity.

~c(G)

instead of M2(G)

166

8a

el = Ul - Y2

8b

e2 = u2 + Yl

8c

Yl = G1 el

8d

Y2 = G2 e2

where

Ul, u2' el' e2' Yl ' Y2 6 L 2e v for some positive

and G I, G 2 map LV2e into itself. and ~(G 2) < ~.

Suppose,

(G2) In particular,

~,

to Class U,

to each Ul, u 2 in L~

in L2e for e I , e 2, Yl' Y2"

the system

~c(GI)

G 1 belongs

that corresponding

(8) has at least one solution these conditions,

Suppose

integer

(8) is L2-unstable

Under

if

-< 1

we have that Yl g L2 whenever

u 2 = 0, and

u I E M±(GI)/{0}.

Note

that,

u I is a nonzero It is shown adapted

element

in Chapter

to encompass Note

Theorem

(6.1.1).

Roughly

the gain of the stable

theorem

belongs

system

system

(7) states

an unstable

is itself

unstable,

gain of the unstable

feedback

that,

if

forward provided system and

instability

systems.

the system

Suppose

of

one.

is the basic passivity-type

Consider

to Class U.

Theorem

counterpart

around

system does not exceed

for single-loop Theorem

10

MI(GI ) .

can be readily

(7) is the stability speaking,

feedback

(10)

whenever

L -instability.

of the conditional

Theorem

results

complement

9 that such results

then the overall

the product

(7), instability

of the orthogonal

that Theorem

we place a stable system,

in Theorem

that,

(8), and suppose

G1

for each u I, u 2 ~ L~,

~8) has

167

Suppose

at least one s o l u t i o n in L v 2 for el, e2, YI' Y2 .

in

a d d i t i o n that

(i)

T h e r e exists a c o n s t a n t e such that

ii

<X,Sl,X> ~ e Ilxll2 ,

(ii)

¥x E M(GI)

There exists a c o n s t a n t ~ such that

12

<x,G2x> ~ 6 IIG2xll2, (iii)

¥x • L 2

We have

13

G2x = 0

U n d e r these conditions,

14

= x = 0

the s y s t e m

(8) is L 2 - u n s t a b l e

if

~+~ > 0

Specifically,

if u I = 0 and u 2 • M I ( G I ) / { 0 } ,

we have that e i t h e r

Yl or Y2 does not b e l o n g to L 2. An i n t e r e s t i n g feature of T h e o r e m n o n z e r o input is applied,

not to the u n s t a b l e s y s t e m G I, but to

the p o s s i b l y stable s y s t e m G 2. w h e t h e r the s y s t e m

In T h e o r e m

It is still an o p e n q u e s t i o n

(8) can be m a d e L 2 - u n s t a b l e w i t h u 2 con-

s t r a i n e d to be zero,

inputs,

(I0) is that the

if

(14) holds.

(i0), we show that a p a r t i c u l a r c h o i c e of

e i t h e r Yl or Y2 does not b e l o n g to L 2.

By a d d i n g an

e x t r a assumption, we can show that Yl ~ L2 for a p a r t i c u l a r choice of inputs. 15

Corollary hold,

S u p p o s e all the h y p o t h e s e s

and that in addition,

conditions,

of T h e o r e m

G 2 maps L 2 into itself.

we have that Yl g L2 w h e n e v e r u I = 0,

u 2 • M I(G I)/{0}.

(i0)

U n d e r these

168

8.2

C R I T E R I A OF THE SMALL GAIN TYPE

In this section,

we present several L 2 - i n s t a b i l i t y

c r i t e r i a of the "small gain" systems.

These results

those of Chapter

6.

type for large-scale

are the instability

Note that all of the criteria

are based on selecting

some nonzero elements

c o m p l e m e n t MI(Gi) , and can therefore stability

(see Chapter Throughout

interconnected

counterparts

of

given here

from an orthogonal

be extended

to L=-in-

9).

this section,

we c o n s i d e r

systems described

by m

la

ei = ui -

lb

Yi = Gi ei

z Hij yj j=l i = l,...,m

n,

n.

where u i, e i, Yi 6 L2el for some positive n.

n.

integer n i , Gi:

L

2el

n,

L z and H : L 3 + L i 2e ' ij 2e 2e" time, we assume that,

Without

stating

corresponding

it e x p l i c i t y

every

to every set of inputs

n. U i

6 L2z Vi,

the system equations

(i) have at least one solution

n.

for ei, Yi in L2e.Z the system

This is a w e a k e r a s s u m p t i o n

(i) to be well-posed,

the system

(i) is well-posed.

conditions

for the w e l l - p o s e d n e s s

given in Chapter

and is certainly

Hij(0)

if

of systems of the form

(i) are

5. counterpart

of

(6.2.50). Theorem

belongs

satisfied

Recall that some sufficient

The first result is an i n s t a b i l i t y Theorem

than requiring

Consider

the s y s t e m

to class U for all i, and = 0) ¥i,j.

negative)

constants

(i), and suppose

(ii) H.. zj is unbiased

Suppose there exist

(not n e c e s s a r i l y

(i) G i (i.e. non-

sij such that n.

<ei'-HijGjej>

< ~ij

]leill Ilejll Ye i ~ L21, ¥ej 6 M(Gj)

169

Under these conditions, can find positive

the system

constants

(i) is L 2 - u n s t a b l e

ll,...,lm

if one

such that the m a t r i x

S E R m x m defined by

sij = li$ij

is positive

-

definite.

(liuij + ljuji)/2'

In particular,

6ij = K r o n e c k e r

delta

whenever

u i E MI(Gi ) Vi n. n. and u i ~ 0 for some i, we have that either e i ~ L21or Yi ~ L21 for some i. Proof contradiction

Let u i 6 MI(Gi) Vi, and suppose by way of n. n. that e i 6 L21 , Yi 6 L21 Vi. Since G i belongs

class U for all i, this implies the system equations

that e i 6 M(Gi)

vi.

Now,

to

from

(i), we get

m ei = ui -

Z H. • G. e. j=l 13 3 3

m <ei,ei > = <ei,ui>

m = - j = l~

where we use

<e i, Hij Gj ej>

j=l

m <ei, H..x3 G.3 e.>3 -< j=lZ ~ij IIeill IIejII

(3) and the fact that <ei,ui > = 0 because

u i ~ Ml(Gi ) . positive

-

Next,

definite.

m

let l l , . . . , I m be chosen such that S is Then

(6) implies

m

m

z li IIei{I2 z z i=l i=lj=l

m

However,

li ~ij

IIeill IIejll -< 0

m

E z i=l j=l sij

IIeill IIejll ~ 0

since S is p o s i t i v e

e i = 0 Vi.

that

definite,

G i e i = 0 Vi, and since Hij is unbiased, ¥i.

(8) implies

Since G i (being in class U) is linear,

This leads to a contradiction,

that we have

we have H i j Gj e 3. = 0

since this

implies

170

m

u. = e. + Z H.. G. e. = 0, Vi l l J -"=i 13 ] ]

whereas

u. was chosen to be n o n z e r o for at least some i. This 1 assumption is false, i.e. c o n t r a d i c t i o n shows that our o r i g i n a l n. ni either e i ~ L21 or Yi ~ L2 for some i .

i0

Corollary hold,

Suppose

and in addition,

conditions,

the system

that the hypotheses of T h e o r e m (2) n. n. Hij maps L21 into L21 , vi,j. Under these (2) is L2-unstable.

In particular,

w h e n e v e r u i q MI(Gi ) ¥i and u.l ~ 0 for some i, we have that n. Yi g L21 for some i. Proof contradiction we have

n. (2a) that e i e L21 Vi.

from

contradiction, original

Let u i q MI(G i) Vi, and suppose by way of n. nj n. that Yi 6 L21 Vi. Then, since Hij(L 2 ) c L21 ¥i,j, This now leads to a

just as in the proof of T h e o r e m (i). Hence our ni is false, i.e. Yi ~ L2 for some i.

assumption

The d i f f e r e n c e

between Theorem

(2) and C o r o l l a r y

(I0)

is that in T h e o r e m

produce errors n. not in L21. n.

In Corollary

Corollary

~(Hij)

is an i n s t a b i l i t y

Consider

the system

to class U for all i. < ~ ¥i,j.

L2-unstable

12

condition

that

counterpart

of

(6.2.43). Theorem

belongs

by a d d i n g t h e

C L21 ¥i,j, we draw a better conclusion: n a m e l y that n. i n p u t s p r o d u c e o u t p u t s t h a t a r e n o t i n L21. The next result

ii

(10),

n.

Hij(L23) certain

(2), we show that c e r t a i n inputs either ni t h a t a r e n o t i n L2 o r p r o d u c e o u t p u t s t h a t a r e

Suppose

(I), and suppose G i in addition

Under these conditions,

if the test m a t r i x P E R m x m defined by

Pij = ~(Hij ) ~c (Gj)

that

the s y s t e m

(i) is

171

has a spectral

radius

less than or equal to one.

ui • MI(Gi)/{0}

whenever

Proof

¥i,

In particular, n. Yi g L2 ~ f o r some i .

we h a v e t h a t

Let u i e M ± ( G i ) /

0

Vi, and suppose by way of

n. c o n t r a d i c t i o n that Yi 6 L21 Vi. Since ~(Hij) < - Vi,j, this n. n. n. implies by (la) that e i e L21 Vi. Next, e i E L21 , Yi ~ L21 implies

that e i E M(Gi)

~i.

N o w define

m

13

zi = u i - e i =

E Hij yj j=l

Since u i E M±(Gi ) and e i E M(Gi),

we have

llzill2 = lluill2 + IIeill2 > lleill2 Vi since u01 @ 0

14

On the other hand,

since

m 15

• =

zz

m

E

j=l

Hi"3

y "3

=

E j=l

H..

z3

G.

3

e.

3

we also get 16

IlzilI _
E v. IIeill i=l i=l l

the Perron-

is an eigenv can be

(and of course,

Let v be so chosen.

we get 17

~c(Gj ) llejll

Then

at from

(14)

172

On the other hand,

from

(16) we get

m 18

m

m

m

i=iE vi IIzill ~ i=lj=l~ l v i Pij

llejll = p(P)

where we use the fact that v is a row eigenvector since p(P) original

E i,

(17) and

assumption

of P.

Now,

(18) contradict

is false,

The following (2) and

j=iz v.3 fleeII

each other. Hence our ni i.e. Yi ~ L2 for some i.

result

is a corollary

of both Theorems

(Ii).

19

Theorem

Consider

the system

belongs

to class U for all i.

~(Hij)

< ~ Vi,j.

L2-unstable

Suppose

(1), and suppose in addition

Under these conditions,

if p(P)

that

the system

< I, where P • R mxm is defined

G. 1

by

(i) is (12).

particular,

whenever u i • Ml(G i) ¥i and u. ~ 0 for some i l n. have that Yi ~ L2Z for some i.

In we

Proof

Let u i 6 MI(Gi ) Vi, and suppose by w a y of n. contradiction that Yi e L21 ¥i. Then (la) implies that n. ni n. e i 6 L2z Vi; also, e i e L 2 , Yi e L21 implies that e i 6 M(Gi). Now define

z i as in

(13).

Since

u i 6 Ml(Gi ) and e i 6 M(Gi) ,

we have 20

IIzilI > IIeilI vi,

because

IIzilI > IIeilI for some i

u. ~ 0 for some i. 1

Also,

as in

(16), we get

m

21

llzill ~ j=iE Pij

By assumption,

p(P)

IIejll

< i, and P has all nonnegative

Also note that the gains that they can therefore increase

the gains

~(Hij)

and ~c(Gi)

be increased.

~(Hij)

and ~c(Gi)

elements.

are upper bounds,

So it is possible such that

and

to

(i) all elements

173

of P are p o s i t i v e ,

and

(ii)

t h e o r e m [Gan.

i, p.

corresponding

row e i g e n v e c t o r

positive

66]

p (P) < i.

elements.

p(P)

is an e i g e n v a l u e v c a n be c h o s e n

L e t v be so c h o s e n .

m

22

By the P e r r o n - F r o b e n i u s

Then

of P, a n d the such that v has from

all

(14) we g e t

m

Z v. llzill > Z v i lleilI i=l i i=l

because

of

(20) and the

j u s t as in

m

23

i.e.

As m e n t i o n e d (2) as w e l l

24

previously,

as T h e o r e m

of T h e o r e m

a corollary

so t h a t

25

if p(P)

To see t h a t T h e o r e m

~c(Gj)

(19)

< ~ Vi,j,

of is

then

lleill llejll

(6.2.8));

constants

definite,

A = Diag

ll,...,Im

Observe

minors

by F a c t

of I m - P a r e

(6.2.11)

s u c h t h a t A(Im-P)

+

there

(Im-P) 'A

where

is r e q u i r e d

is a c o r o l l a r y

the h y p o t h e s i s

therefore

{ll ..... Im }

is e x a c t l y w h a t (19)

~c(Gj ) = Pij

(by L e m m a

exist positive is p o s i t i v e

Theorem

is a c o r o l l a r y

t h a t if ~(Hij)

< i, t h e n the l e a d i n g p r i n c i p a l

all positive

This

(19)

is false,

(3) is s a t i s f i e d w i t h

~ij = ~(Hij)

26

(ii) o

< ~(Hij)

assumption

Theorem

(2), o b s e r v e

<ei,-HijGjej>

Thus,

7 v. llejll ==i j x 3

c o n t r a d i c t s (22). Hence our original ni Yi ~ L2 for s o m e i. o

Theorem

hand,

m

7. v. llzilI _< p(P) i i=l

which

On the o t h e r

fact t h a t v. > 0 Vi. l

(18) we g e t

for T h e o r e m

of T h e o r e m

(2) to apply.

Thus

(2).

a l s o t h a t if p (P) < 1 t h e n p (P) -< i, so t h a t

of T h e o r e m

(ii)

is s a t i s f i e d .

In this

sense,

174

Theorem

(19) is a c o r o l l a r y

of inputs that produce in the two theorems.

"unstable" In T h e o r e m

I, then instability

p(P) Theorem

results

then i n s t a b i l i t y

(ii).

outputs (ii),

However,

is slightly

the set different

it is shown that if

if u i 6 MI(Gi)/{0}

(19), it is shown that if p(P)

condition), inputs,

of T h e o r e m

¥i.

In

< 1 (which is a stronger

is p r o d u c e d by a b r o a d e r

class of

namely

other words,

6 MI(Gi ) ¥i and u i ~ 0 for at least one i; in ui instability can be e x c i t e d from anyone of the

inputs. Up to now, we have only c o n s i d e r e d of several

subsystems

of class U.

an interconnection

We now study interconnections

of some systems of class U and other systems h a v i n g Theorem

27

each i, either

Consider

the system

~(G i) < ~ or G i belongs

least one of the G.'s belongs 1 ~(Hij)

< ~ for all i,j.

28

Pij = ~(Hij)

that for

to class U.

Also,

suppose

~c(Gj )' ¥i,j the system

In particular,

(I) is L 2 - u n s t a b l e

if

u i e MI(Gi ) for all i and ni u i ~ 0 for some i, we have that Yi g L2 for some i.

M(Gi)

< i.

(i), and suppose

to class U, and that at

Define the test m a t r i x PqR m x m by

under these conditions, p(P)

finite gain.

whenever

Remarks Note that, if ~(G i) < ~ for a certain i, then ni = L 2 and ~c(Gi) = ~(Gi). Also, for this value of i,

MI(Gi ) = {0}, so that u i e MI(Gi ) implies u i = 0 w h e n e v e r ~(Gi)

< ~.

in T h e o r e m

Hence

the set of "destabilizing"

(27) consists

of:

inputs d e m o n s t r a t e d

u i = 0 if ~(G i) < ~, u i ~ MI(Gi )

if G. b e l o n g s to class U, and u. ~ 0 for some i. In 1 1 particular, i n s t a b i l i t y can be e x c i t e d from anyon e of the inputs to the class U subsystems. Proof can be increased

Since

~(Hii),

~c(Gi)

are all upper bounds,

in such a way that P.. > 0 ~i,j 13

and p(P)

they < i.

Let the inputs u. be selected as above, and suppose hy way of l ni c o n t r a d i c t i o n that Yi E L 2 ¥i. Then (la) implies that

175 n.

n0

e i E L21 ¥i, which e i E M(Gi)

together

whenever

with Yi • L21 implies

G i belongs

to class U.

that

As before,

define

m

29

zl• Now,

=

E j=l

H.

=

lj Yi

u-

-

1

i = l,...,m

e-,

1

if ~(G i) < -, then u i = 0 so that

belongs

to class U, then

llzill2 = lluill2 + IIeill2 since

ui • MI(Gi ) and e.l • M(G i) . 30

llzilI = IIeill; if G i

Hence

Ilzill >_ lleill ¥i Also, since u i ~ 0 for some i, strict inequality for some i. Now let v 6 R m be a row eigenvector corresponding v. > 0 ¥i.

to the eigenvalue

p(P),

Then on the one hand,

1

m

chosen

we have

holds of P

in

(30)

in such a way that

from

(30) that

m

31

v i IlzilI > i=l

E V i IleilI i=l

On the other hand, m 32

m

m

i=l ~ vi IIzill ~ i=l~ v.l P''l 3 lle911 = p (P) i=lZ v.3 llejll

Hence

(31) and

original

(32) contradict

assumption

8.3

is false;

DISSIPATIVITY-TYPE

In this section, criteria

the general

decomposition

parts of those

dissipativity

while

7.

type

the rest make use of

of the input space.

The results in Chapter

several

Some of these are based on the

dissipativity,

dissipating-type

type criteria.

CRITERIA

we derive

for L2-instability.

idea of conditional an orthogonal

each other, which shows that our ni i.e., Yi g L2 for some i. []

criteria,

By specializing

we can obtain passivity-

given here are instability

counter-

176

T h r o u g h o u t this section, we shall be c o n c e r n e d with systems d e s c r i b e d by m

la

ei = ui -

z

i=l

Hij Yj i -- i, . . . ,m

lb

Yi = Gi ei n. n. n. w h e r e ei, ui' Yi E L i for some i n t e g e r ni, Gi: L i ÷ L z and 2e 2e 2e' Hij

is a c o n s t a n t m a t r i x of d i m e n s i o n ni×n''3

assume that,

ThroughoUt,n. we

c o r r e s p o n d i n g to each set of inputs u i • L21 Vi,

the s y s t e m e q u a t i o n s

n. (i) have a unique s o l u t i o n set e i • L l 2e '

n. Yi • L2el ¥i, w h i c h depends in a causal way on the input set {Ul,...,Um}. well-posed

This is c e r t a i n l y the case if the s y s t e m

(i) is

(see C h a p t e r 5).

8.3.1

CONDITIONAL DISSIPATIVITY APPROACH

We b e g i n by r e c a l l i n g a d e f i n i t i o n from S e c t i o n

Definition

Suppose G: Ln2e ÷ L2e'n and let Q , R , S e R n×n

be given, w i t h Q and R symmetric.

Then G is said to be

c o n d i t i o n a l l y d i s s i p a t i v e w i t h r e s p e c t to

+ <x,Rx> + ~ 0, It is shown in S e c t i o n 3.3 that,

(Q,R,S)

if

YxEM(G) if G is a linear

c o n v o l u t i o n o p e r a t o r and its t r a n s f e r f u n c t i o n m a t r i x G(-) right-coprime

3.3.

factorization,

then it is p o s s i b l e to give a

n e c e s s a r y and s u f f i c i e n t c o n d i t i o n for G to be conditionally dissipative

has a

(see Lemma

(Q,R,S)

-

(3.3.50)).

We lead up to the m a i n r e s u l t of the s u b s e c t i o n t h r o u g h a series of lemmas~ Lemma form as

C o n s i d e r the s y s t e m

(i), d e s c r i b e d in aggregated

177

e

where

=

u

-

Hy,

y

e, u, y 6 L2e ,n

Ge

=

G: L n2e ÷ L2e,n a n d H e R n n

conditionally

dissipative,

then the system

conditionally

dissipative,

where

6a

= Q + H'RH

-

(SH + H ' S ' ) / 2

We wish

to s h o w that,

(5) is

If G is

(Q,R,S)

(Q,R,S)

-

6b

6c

= S - 2H'R

Proof

whenever

u 6 L n2 a n d

n y ~ L2, we h a v e

Now,

+

+

~ 0

n if u e L n2 a n d y e L n2, t h e n e e L2,

s i n c e G is whenever

(Q,R,S) - c o n d i t i o n a l l y n n e 6 L2, y 6 L2,



If w e r e p l a c e details (7.2.6).

+ <e,Re>

+

e by u- Hy in

are entirely

Further,

w e h a v e that,

~ 0

(8) b e c o m e s

to t h o s e

(7).

in the p r o o f

The

of L e m m a

D

Lemma

Consider

the

system

(i), and s u p p o s e

(Q,R,S)

- conditionally

dissipative,

(Q,R,S)

- dissipative.

Then the system

Q-
T + T + <e'Re>T + T

= T

+ <xi,RiXi>T

+ T



so t h a t

- <e,S'Pe>

= -<e,S'Pe>

w h e r e we use the f a c t t h a t

32

<e,S'Pu>

On t h e o t h e r h a n d , dissipative,

since

G. is 1

~ -<e,Re>

<e,Re> since Q1 34

0.

= 0 because

eleM(Gl),

(Qi,Ri,Si)

veMl(Gl )

-conditionally

we have



33

= <el,v>

Note

-

-

n o w that,

since y = P(u-e),

we have

Y2 = P 2 1 ( u l - e l ) + P 2 2 ( u 2 - e 2 ) = - ( P 2 1 e l + P 2 2 e 2 )

184

because

P21Ul

35

+ P22u2

Combining

(31)

= 0 from

>_ - < e , R e >

and

(35)

36

-<e,S'Pe>

37

0 >_ - < e , R e >

A

implies

from

(34).

this

contradicts

This

contradiction

i.e.,

that

>_ - < e , R e >

either

38

=

is r e p l a c e d

)>

-

<e,Ae>

we have

(30),

Theorem

becomes

-

Yl = G l e l

Finally,

(33)

- 0 by h y p o t h e s i s ,

this

Hence

gives

+ <e,S'Pe>

Since

(29).

is false;

if c o n d i t i o n

(iii)

by I

39

(iv)

40

(v)

be easily el=0.

The

rest

Just

shown,

it f o l l o w s

from

e 2 = 0.

It is of

(19). assumed ~c(Gl) Theorem

Form

in T h e o r e m < ~. (19).

Also,

from

we

n1 [Al--x--] = r a n k A P ~

this

follows

the

(19),

we arrive (37)

we

to c o m p a r e

assume

be both

that

that

e 2 = 0 as w e l l .

the

(25). o

constructive

criterion ~c(Gl)

(19)

criterion

of T h e o r e m

< ~, w h i c h

u Implies

in T h e o r e m

Theorems

as c a n and

= - P 2 2 e 2 ' i.e.,

G 1 e class

linear

Now,

that Ae=0

t h a t of T h e o r e m

because

G 1 must

Similarly,

(37).

(39)

implies

nonconstructive

(25),

at

and

g e t Y2 = G 2 e 2

(40),

interest

(25) w i t h

In T h e o r e m

as b e f o r e ,

(34),

of the p r o o f

of T h e o r e m

rank

( G 2 + P 2 2 ) e 2 = 0 ~ e 2 = 0.

Proof

Hence,

(G2+P22)

A ~ 0, a n d

and

(25) (25)

is a l s o

that but

not

assume

in

185

that Q1 ~ 0.

Thus,

the only extra a s s u m p t i o n s

are that S 1 and H are n o n s i n g u l a r . been discussed previously. a t e c h n i c a l condition,

(25)

The n o n s i n g u l a r i t y of H 1 has

The n o n s i n g u l a r i t y of S 1 seems to be

whose significance

note that the h y p o t h e s i s

in T h e o r e m

is not clear.

"A > 0" in T h e o r e m

s t r o n g e r than "Q < 0" in T h e o r e m

Finally,

(25) is s l i g h t l y

(19) because,

as can be e a s i l y

verified,

41

A = -P' (Q -

In o t h e r words,

[:001 )P

if A > 0, then Q < 0; also,

A > 0 if and only if Q < 0.

To summarize,

if Q1 = 0, then it is not n e c e s s a r y

to tack on too m a n y u n n a t u r a l c o n d i t i o n s to turn the nonconstructive

c r i t e r i o n of T h e o r e m

c r i t e r i o n of T h e o r e m

(25).

(19) into the c o n s t r u c t i v e

One only has to

(i) a s s u m e that

G e class U, i n s t e a d of just s a t i s f y i n g

~c(GI)

(ii) s t r e n g t h e n "Q < 0" to "A > 0", and

(iii) assume that SI, H

< ~,

are nonsingular.

By s p e c i a l i z i n g T h e o r e m "passivity-type"

Theorem

(25), one can o b t a i n various

i n s t a b i l i t y criteria.

(42) b e l o w is the m o s t general i n s t a b i l i t y

r e s u l t of the p a s s i v i t y type, and is an i n s t a b i l i t y c o u n t e r p a r t of T h e o r e m

42

(7.3.52).

Theorem

C o n s i d e r the s y s t e m

(i), and suppose that

G I , . . . , G k b e l o n g to class U for some integer k S m. there e x i s t real c o n s t a n t s

Suppose

el,...,e k and ~ k + l , . . . , ~ m such that

43

<x,G.x>l > Ei NxlI2 YX 6 M(Gi) , i = l,...,k

44

<x,Gix> ~ 6 i llGixll2, Vx 6 L 2e l ' i = k+l,...,m

n.

S u p p o s e H is n o n s i n g u l a r , conditions,

the s y s t e m

(i)

and let P = H -I

(i) is L 2 - u n s t a b l e

the m a t r i x

U n d e r these if

186

45

Al

is p o s i t i v e

= E + P'

D P

semidefinite,

where

46

E = Diag

_{~IInl , . . . , e k I n k , 0, .... 0}

47

D = Diag

{0,...,0,

(ii)

48

r =

I [M 21~]

k Z n ii=l

(iii) is p a r t i t i o n e d

Let in

P = H -I be p a r t i t i o n e d

(41).

m G.e. + ~ i i j=k+l

49

Ink+l'''''~mIm}

M 2 satisfies

rank M 2 = rank

where

6k+l

(iv) In p a r t i c u l a r ,

in the

same way

as H

Then

P..e. = 0 for 13 3

i = k+l,...,m

for

i = k+l,...,m

~ e. = 0 l

H is p o s i t i v e . if we

select

inputs

of the

form

k 50

Z

Hi

vj E MI(Gj) n. yj g L 2 1 for

for

ui where then

=

j=l

Proof

51

conditions

satisfied

because

i

=

1

,m

j = 1 ..... k a n d vj ~ 0 for at l e a s t

some

Apply

Q = -D,

Then

v

J j. . . . .

i.

Theorem

R = -E,

(i) and

(25) w i t h

S = I

(ii)

Q1 = 0, a n d

n

of T h e o r e m

(25)

S 1 = I r.

Also,

are the

A becomes

52

A = E + P'DP

one

+

(P'+P)/2

= A1 +

(P'+P)/2

automatically "test

matrix"

J,

187

If P is p o s i t i v e , positive Theorem

t h e n A ~ A I.

definite, (25).

we have

Everything

However,

to s l i g h t l y is the

same

since

modify unt~l

A 1 may

n o t be

the p r o o f (37),

which

of now

becomes

53

0 a <e,Ae>

Now,

(53)

and

show

that

e2 = 0

rest

of the p r o o f

54

(48)

~ <e,Ale>

show

(just as follows

Corollary GI,...,G k belong there

exists

55

positive

Next,

in the p r o o f

the

U for

constants

_> eiIIxll2,

system

some

e I = 0 plus

of C o r o l l a r y

t h a t of T h e o r e m

Consider

to c l a s s

<X,GlX>

and

t h a t e I = 0.

(25).

(I),

integer

V x 6 M(Gi) ,

(38)).

The

o

and

suppose

k ~ m.

Cl,...,c k such

(49)

Suppose

that

i = 1 ..... k

suppose n.

56

<x,Gix ~

Under

these

0,

conditions,

YT 6 L2e1 ,

i = k+l,...,m

the

(i)

system

is L 2 - u n s t a b l e

if

(49)

holds.

Proof

In t h i s

case,

we have

D = 0, so t h a t A 1 of

becomes

57

A1 =

0a

where

58

E a = Diag

Clearly

{ ~

A 1 is p o s i t i v e

Inl ..... ek Ink}

semidefinite

and

satisfies

(48).

x

(45)

188

NOTES AND R E F E R E N C E S

The o r t h o g o n a l d e c o m p o s i t i o n a p p r o a c h to the L 2i n s t a b i l i t y of f e e d b a c k systems was i n t r o d u c e d by T a k e d a and Bergen [Tak. [Vid.

i] , and e x t e n d e d to l a r g e - s c a l e

2,3,7,10].

The n o n d i s s i p a t i v i t y a p p r o a c h is g i v e n by

M o y l a n and Hill [Moy. [Vid.

10].

in [Mic.

systems in

2] and some further r e s u l t s are in

A d d i t i o n a l results on L 2 - i n s t a b i l i t y can be found

i] .

CHAPTER 9: L.-STABILITY AND L.-I!!STABILITY USING EXPntIE~ITIAL WEIGHTItIG In Chapters 6 and 7, we have presented two distinct approaches to obtaining stability criteria for large-scale interconnected

systems,

"dissipativity"

namely the "small gain" approach and the

approach.

Certain features of these two

approaches can now be stated.

The small gain approach can be

applied to systems set in an arbitrary space Lpe, p • [i,~].

In contrast,

for @ny

the dissipativity approach requires an

inner product structure on the input spaces, be directly used only to study L2-stability. favour of the small gain approach.

and can therefore This is a point in

On the other hand, the small

gain approach invariably leads to "weak interaction" results, which are sign-insensitive vative.

In contrast,

and are therefore conser-

the dissipativity

results that are sign sensitive,

type of

approach leads to

and less conservative than

those obtained by the small gain approach.

Unfortunately,

the

dissipativity approach of Chapter 7 can not be directly applied to study L -stability,

because L

is not an inner product space.

So, what we propose in this chapter is a method of studying the L -stability of a system by studying instead a property resembling the L2-stability of an associated system.

Since both

dissipativity as well as small gain methods can be used to study this property of the associated system,

it is clear that the

results given here in effect extend the scope of applicability of the dissipativity approach to include L -stability studies. There is also another point worth mentioning. Generally speaking,

the criteria for L -stability obtained by

directly using the small gain approach involve calculations the time domain.

in

The reason for this is that the L -induced

norm of a convolution operator is calculated in the time domain (see Lemma

(3.1.51)).

In contrast,

the L2-stability criteria

obtained by using either the small gain or dissipativity approaches usually involve frequency domain calculations.

The

reason is that the L2-induced norm of a convolution operator is calculated in the frequency domain,

and the matrices Q,R, and S

encountered throughout Chapter 7 are also calculated in the frequency domain

(see (3.1.55)

and Section 3.2).

Thus, as a

190 consequence obtain are

of

the r e s u l t s

frequency

very

useful

experimental frequency

domain because

data,

structure

on

the

"exponential system

the

input

they

space. we

readily often

is able

Such

applied

obtained

results

too make

to

results

to

by m e a n s

of C h a p t e r

use

By m a k i n g

show

whenever

GENERAL

general

"exponential

of

of

chapter

an a s s o c i a t e d

8 pertain

an i n n e r

appropriate

in this

STABILITY

result

weighting",

Definition n, :

one

product

use of

that

system

a given

is

L 2-

.

The

W

quite

instability

weighting",

9.1

of

chapter,

for L - s t a b i l i t y .

c a n be

are

because

is L - u n s t a b l e

unstable

in this

tests.

Finally, to L 2 - i n s t a b i l i t y ,

they

which

response

given

criteria

given which

Given

RESULT

here

depends

is d e s c r i b e d

a real

number

on the

technique

briefly.

=, the

operator

n.

L i + L i is d e f i n e d 2e 2e

by n.

(W x)(t)

= x(t)

exp(~t)

Yx 6 L l 2e

•

It is c l e a r

that W

merely

"weights"

each

function

n.

x(')

E L 1 by the 2e ~, W~ 1 = W _ ~ m a p s

exponential L ni 2e o n t o

If ~ is c l e a r by x

.

Similarly,

if

w

G. = W 1

Note

that,

from

* Yi = We Gi el• W G.

1

the

n. L 2e I

context,

n. _~ L 2 e 1

. W -I

* e.

1

=

Also,

then

W

~

G.

l

we

for

usually

, we e m p l o y

l

if Yi = Gi ei'

=

exp(~t).

each

itself.

Gi:

. G. ~

term

W -I e* ~ l

the

denote notation

W x

191

In p a r t i c u l a r , with

impulse response

suppose

matrix

G i is a c o n v o l u t i o n

Gi(-);

i.e.,

operator

suppose

t Yi(t)

= I o

Gi(t-T)

ei(~)dT

Then t Yi(t)

=

exp(~t)

Gi(t-T)

ei(~)dT

o t = I Gi(t-~)

exp(~(t-T))

. ei(~)

exp(~)

dr

o t =

where

the a s t e r i x

I G ; ( t - ~ ) e * i (3) dT O denotes

e x p o n e n t i a l w e i g h t i n g and not m a t r i x , transposition. I n o t h e r w o r d s , G. i s a l s o a c o n v o l u t i o n o p e r a t o r 1, w h o s e i m p u l s e r e s p o n s e m a t r i x i s G. ( . ) . Equivalently, 1 ^, G (j~) : G(~ + j~). In the same vein, linearity

in the sector

Yi(t)

where

¢i:

=

suppose

G i is a m e m o r y l e s s

[a,b] ; i.e.,

(Gie i) (t) = #i(t'

suppose

e i(t))

n. n. R+ x R i ÷ R i and

[ ¢i(t,a)

- aa] ' [ ¢i(t,u)

- b~]

_< 0, Yt>_0, Vc 6 R

Then Yi(t)

= ~i(t,ei(t))

where i0

¢i(t,~)

= exp(st)

~i(t,

exp(-~t) u)

Hence ii

non-

[ ¢i(t,o)-a~]

' [ ~i(t,~)

- bs]

n. 1

192 = [exp

at ¢ i ( t , e x p ( - e t ) ~ )

[ e x p at ¢i(t,

- ac] '.

exp(-~t)~)

= exp(2et)[¢i(t,exp(-at)o)

[ ~i(t'exp(-~t]°)

- bo]

- a exp(-at) a]'

- b e x p (-at) o]

n°

-< 0, ¥t >_ 0, %; o 6 R 1

which

shows

that

~i a l s o b e l o n g s

In a d d i t i o n

to the c o n c e p t

we a l s o n e e d t h a t of " d e c a y i n g 12

Definition decaying

Ll-memory

nonincreasing

An operator

if t h e r e

function

of e x p o n e n t i a l

Gi:

exists

m i(-)

[a,b].

weighting,

Ll-memory". n. n. L 2ei ÷ L 2ei is s a i d to have

a nonnegative-valued,

e L 1 such t h a t

t II(Gix) (t) ll2 < [ mi(t-~)

13

to t h e s e c t o r

Ilx(T)ll2 at

-

¥t>0 '

'

n. Vx e L 1 2e

o 14

L e m.... ma with

impulse

decaying

n. ni L 2ez ÷ L 2 e be a c o n v o l u t i o n

L e t Gi:

operator

response

Ll-memory

m a t r i x G. (.). T h e n the o p e r a t o r G. has 1 i n. ×n i if the f u n c t i o n t + G i(t) exp(~t) 6 L2X

for some e > 0.

Proof

We h a v e t

15

(Gix) (t) = I Gi(t-T)

X(T)

dT

o =

It

Gi (t--T) exp (~ (t-T))

x (T) e x p (-a (t--T)) d~

O Hence

by Schwarz's i n e q u a l i t y , t

16

II(Gix) (t)}12 _ < I o

[IGi(t-T)11 2 e x p ( 2 a ( t - T ) ) a t

193

t I Ilx(~)II2 exp(-2~(t-T))d~ o t

=

II~i (~) II2 exp ( 2 ~ ) d T o

fix (~) 112exp (-2c~ ( t - T ) ) d~ o

t --- I mi0 e x p ( - 2 ~ ( t - T ) ) H x ( T ) 1 1 2 O

where

17

llGi(r)l) 2 exp(2sr)d~

mio


0

llW~e illT2 -< ))vII~ exp(~T) , ¥T TM_ 0 n,

Then G.e.ll • L~I,

and

IIGieilI~ is bounded

by a constant

times

IIvlI.. Proof

(20) can be rewritten

as

t 21

I IIei(~)II 2 exp(-2~(t-T))

dT 0, we have

llerl~ II W e lIT2

28

- 0

(2~) n.

Hence,

if Gi:

n°

L 2ez + L 2ei

has decaying Ll-memory,

~ ( G i) < ~, in view of Lemma

(19).

then

195

We now present L=-stability deduced

of a given

by studying

associated

result,

large-scale

a property

whereby

interconnected

resembling

the s y s t e m can be

the L 2 - s t a b i l i t y

of an

system.

Theorem

29

a general

Consider

a system

described

by

m

30a ei = ui 30b

j--[lHij yj

}

i = 1 ..... m

Yi = G i e i n.

ui'

where

ei'

Yi a l l

b e l o n g t o L2eZ f o r some i n t e g e r

n. n. L i + L I , a n d H..: 2e 2e l]

Gi:

associated

n. n. L 3 + L l 2e 2e

D

n i,

Now consider

the

system m

31a

d. = u. [ H.. z. z i j=l z3 z

31b

z. = G . d. I 1 l

i = l,...,m

where

G I. W_u

Gi = W

Suppose number

~(Hij)

and H.. 13 = W u H ij W_s,

< = for a l l

u > 0 a n d an i n t e g e r

(i) (ii) (iii) (iv)

For i = l,...,k,

k s m such that

we have

H.. = 0 for k + l z3

~ i,j

the s y s t e m

satisfies,

(31)

previously.

i,j, and s u p p o s e we c a n find a

G i has d e c a y i n g

For i = k+l,...,m,

as d e f i n e d

Ll-memory.

~(Gi) 0, V v j 6 L 2 e~ , j=l the s y s t e m

Multiplying

system

(30)

i=l ..... k

is L = - s t a b l e .

both

sides

of

description,

which

is e q u i v a l e n t

(30) by W u, we g e t to

(30):

196

m

e:l = u*l

33a

H* * ij Yj

j=l i=l .... ,m

33b

Yi* Now,

(33)

=

G *i

ei*

is of the form

of d i , vi,

and

zi

(31), w i t h e i, u i, Yi p l a y i n g

the roles

~ ¥i. respectively. S u p p o s e now that uieL ni n. We w i l l show t h a t ei, Y i a L 1 Vi, and that [leiH~ , IIyill~ are m b o u n d e d by a c o n s t a n t t i m e S n ~ =I'[ lluill . * To nid° this, o b s e r v e ,

f i r s t of all that

if u i • L

' , then

u i 6 L2e

, and m o r e o v e r ,

T 34

IIUiHT2 =

-2

IIIui(t) II2

exp(2~t)

dt

o

T -< IluiIl2 I exp(2~t)

dt

o _< l]uill2 e x p ( 2 ~ T ) / 2 a so that 35

]]Ui]]~

l]uilIT2 -< c

where

c =

(i/2a) I'2. /

exp(eT),

Since

(32)

¥i

is a s s u m e d

to hold,

it f o l l o w s

that ,

36

m

[lujlI=l e x p (sT) , i=l ..... k

lleiIIT2 -< C [ j=l Since Lemma exists

G i has a d e c a y i n g L l - m e m o r y for i=l ..... k, we h a v e from n. (19) that Yi E L 1 for i=l ..... k, and m o r e o v e r , t h e r e a finite

constant

6

0

such

that

m

]]Yi ]]- -< ~0

37

Since

38

3

[

]]uj[]~,

i=l .....

H.. = 0 for i,j = k + l , . . . , m , 13

ei = ui -

k [ Hij yj, j=l

k

we have

i = k + l ..... m

197

n.

Since

~ ( H i j) < ~

(38) shows that e i 6 L i for

¥i,j,

i = k+l ..... m, and that there exists a finite c o n s t a n t

61 such

that lleiN~ ~ ~i j=l llujll. ,

39

i=k+l ..... m n.

Since

~ ( G i) < ~ Vi, we next have that Yi e L I for i=k+l ..... m,

and also the finite gain p r o p e r t y from

analogous

to

(37).

Finally,

(30a), we get m

40

ei = ui -

Since

~(Hij)

~ Hij yj, j=l

i=l ..... k

n. (40) shows that e i e L l for i=l,...,k,

< ~,

we also have the finite gain p r o p e r t y the system

(29).

First of all, consider

Conditions

and i n t e r c o n n e c t i o n

(i) and

operators

when viewed as o p e r a t o r s Condition

don't directly crucial

(iii)

"L2-stable"

the h y p o t h e s e s

have finite gain with

Hence

of

zero bias

on L -spaces of a p p r o p r i a t e

dimension,

have d e c a y i n g

states that the r e m a i n i n g

interact among themselves.

condition.

subsystems

Condition

It states that the a s s o c i a t e d

L l-

(iv) is the

system

(31) is

in a special

It is clear from

sense, namely that it satisfies (32). ni (32) that if v i E L~ i Vi, then d i e L 2 for

i=l,...,k.

However,

requiring

the relation

L2-stable.

and

(39).

(ii) state that all s u b s y s t e m

and that some of the s u b s y s t e m operators memory.

to

(30) is L -stable. [] Remarks

Theorem

analogous

in general,

(32) is not the same as

from v I ..... v m to dl,...,d k to be

To e s t a b l i s h

this equivalence,

we require

a few

extra conditions. 41

Lemma corresponding

With regard to the system. (31), suppose

to each set of input v i e L2~

that,

, there exists

a

n.

unique set d i e L2el • i=l,...,k satisfied. set

Suppose

(Vl,...,v m) into

further

such that the eauations ( 3 1 ) _

that the operator m a p p i n g

(dl,...,d k) is casual.

and only if the operator has finite L2-gain with

taking

Then

(Vl,...,v m) into

zero bias.

are

the input

(32) holds (dl,...,d k)

if

198

Proof

The "only if" part is obvious.

To prove the

"if" part, let (vl,...,v m) be a set of inputs in L n2e, and let T < = be specified. Consider the system of equations 42a

d0i = ViT - j~l= Hij z0i

42b

z0i = G i d0i

}

i=l,...,m

.

By uniqueness and causality, we have (d0i) T = diT, for i=l,...,k. Also, since the operator taking (vI .... ,vm) into (dl,...,d k) has finite L2-gain with zero bias, there exists a finite constant ~ such that m

43

Ifd0ifl~ -< ~

X

llvjTJl2 ,

i=l .....

k

j=l Finally,

from (43) we get m

44

;IvjlIT2,

IIdillT2 = IId0irrT2 ~ IId0ill2 ~

i=l .....

k

j=l and the lemma is proved. D 9.2

SPECIAL CASES

In this section, we present some specific criteria for L -stability, based on exponential weighting. obtained by applying Theorem

These criteria are

(9.1.29) in conjunction with the

results of Chapters 6 and 7. First, we consider a system described by m

la ib

e i = u i - HiY i -

~

j=l

Bij yj } i = l , . . . , m

Yi = G i e i n. where for all i we have ui,ei,Y i 6 L2em ' and moreover (i)

G.x is a linear convolution operator with

impulse r e s p o n s e m a t r i x Gi ( . ) .

199

(ii)

H. is a m e m o r y l e s s z [c i -

(l-~i)ri,

6i,ri.

For some ~ > 0, the function n.xn. E L21 l , and

(iv)

in the sector

ci+(l-~i)r i] for some real c i

and some p o s i t i v e (iii)

nonlinearity

sup Ema x [F~l (J~) F i(j~)l

t ÷ Gi(t)

exp(~t)

~ r?21

W

where Emax(-) + denotes

denotes the largest

eigenvalue,

the conjugate transpose,

and

A

Fi(J~)

= [I + c i Gi

One can think of the system loop feedback

systems

(i) as a c o l l e c t i o n

that are i n t e r a c t i n g

Bij, which may be n o n l i n e a r Our o b j e c t i v e for the L = - s t a b i l i t y

(~ + j~)]-i Gi(~+j~) "

and time-varying.

is to derive some s u f f i c i e n t

of the s y s t e m

that if B.. = 0 ¥i,j, x3

collection

of m isolated

it is i n t u i t i v e l y (17) below makes

then the system

this idea precise.

Suppose

Then

llqllT2 < ni llfIIT2 where

=

(9.1.29).

Hence

B.. are z3 (i) is L -stable. Theorem

Before p r o c e e d i n g

f, q ~ L2e satisfy

q = f - Hi Gi q

ni

(i) is just a

each of which can be shown

(2) and T h e o r e m

we give a p r e l i m i n a r y Lemma

(9.1.29)

As a first step,

clear that if the i n t e r a c t i o n s

"sufficiently weak", theorem,

6.

then the system

subsystems,

to be L -stable by virtue of

conditions

(i), using T h e o r e m

in c o n j u n c t i o n w i t h the results of Chapter observe

of m single-

through the operators

(Ici}ri I + l)/~i

to the

200

Proof

(4) is e q u i v a l e n t

q = f - H*i ql

to

ql = Gi* q

"

N o w define

q2 = q + Then

in terms

c.a. l"l

of q2'

(7) b e c o m e s W

q2 = f -

W

(Hi -

ciI)ql'

ql

*

= Giq2

-

ciGiql

or, e q u i v a l e n t l y , *

10

q2 = f where

(Hi - c.I) l

the i n v e r t i b i l i t y

of I + c.G. 1

that ^

Gi

g(~+s)).

is

a linear

Now

from

ql =

ql'

convolution

-1

(I + ciG i)

is g u a r a n t e e d

*

Giq 2 by

(2)

(note

1

operator

with

transfer

function

(10) we get

ii

Ilq211T2 ~< IIfIIT2 + ~2 [ (Hi-c;I)]

12

IIqlllT2 ~< ~2 [ (I + ciG

IIqlIIT2

w

Next,

observe

transfer

that

function

)-i Gi ] ilq211T2

. is a linear

G ~ 1

matrix

convolution

is G i(~+j~) ; hence

operator (2) implies

whose that

*-i * -< r -I (see L e m m a (3.1.83)) ~2 [ (I+ciGi) Gi] l , Also, H i b e l o n g s to the same s e c t o r as H i , n a m e l y

[c i -

(l-6i)r i, c i +

H i - ciI b e l o n g s

(l+~i)ri],

to the sector

~2(Hi

- ciI)

-< (i + ~i)ri

these

bounds

into

(ii) and

because [-(l~i)ri,

(see Lemma (12)

of

and d o i n g

Ilq211T2 ~< [IfIIT2 +

(l-~i)

Ilq211T2

hence

(l+~i)r i] , and

(3.1.103)).

we get

13

(9.1.11);

a ifttle

Substituting manipulation,

201

14

Nq211T2 ~ ~l j]fllT2

15

ilq2;IT2 -~ r~.l~iI JlfllT2 where the last step follows by substituting Finally,

16

(14) into

(12).

we have

llqllT2 ~ IIq211~2 + Icil IIq111T2 S (6~l+Ici 16ilr~ I) which is the same as

llfllT2

(5). []

Now we present the main stability criterion system

Theorem

17

for the

(i). Consider

the system

(i), and define the test

matrix P 6 Rm×m by 18

Pij = ~ij - ni 52 (Bij Gj) where

19

~ij denotes the Kronecker delta,

, , ~2(BijGj)

ni is given by

(5), and

IIBijGjXlIT2 = sup

Under these conditions,

sup

XT~0

the system

IIxIIT2 (i) is L -stable if the

leading principal minors of P are all positive. Proof

In order to apply Theorem

the system equations

(9.1.29), we rewrite

(i) as m

20a

ei = ui

20b

ei+m = Yi

20c

Yi = Gi ei

j~l Bij yj - Yi+m

~i=l,...,m

202

20d

Yi+m = Hi ei+m It is now routine to verify that the system conditions

(i)-(iii)

"associated

of Theorem

(9.1.29).

system" corresponding

to

(20) satisfies Further,

the

(20), as given by

(9.1.31),

is m 21a

d±•

21b

di+ m = z i

=

v I.

911=

-

Bi~ j

Zj - zi+ m

i=l,...,m 21c

Z.

1

=

G,

1

d.

l

w

21d

zi+ m = H i di+ m

It is easy to eliminate this gives m di = vi -j[l=

22

Applying Lemma

(3) to

z I ..... Z2m and dm+l,...,d2m

B..G.d.1] ] 3 - H.G.d.,I i i

from

(21);

i = 1 ..... m

(22) gives m

23

lldillT2 0, V x e L 2 e

where ^

28

= min Re g(~+j~) ~ER

Proof

This is a special case of L e m m a

29

Lemma

Let ¢:

30

0 ~ ~¢(~)

and let G: 31

(3.2.19).

R ÷ R b e l o n g to the sector [ 0,k] , i.e.

~ k~ 2,

V~ E R

L2e + L2e be d e f i n e d by (Gx) (t) = # (x(t)) ,

L e t ~ > 0 and let G

32

<x,G x> T ~

= W G W -I.

(l/k)

IIGx

Then we have

2 '

%~f ~ 0,

Vx ~ L2e

204

Proof

We have T

< x , G x> T =

33

x(t)

exp(c~t)

~[x(t)

exp(-c~t)]dt

OT P

= ~ exp(2at)

• x(t)

(-at) ] dt

exp(-c~t)~[x(t)exp

o T >

(l/k)

I exp(2~t) ( ¢ [ x ( t ) e x p ( - a t ) ] ) 2 d t o

=

T

(l/k)

J [ (G'x) (t)] 2at o

= so that

(32)

(l/k)

is proved.

IIG*XlIT2

[]

34

Theorem

Consider

the s y s t e m

35a

e i = u i - j=l hiJYJ

m i=l,...,m 35b

Yi = Gi ei

where Gi:

ui, ei, Yi • L2e for all i, hij L2e + L2e.

convolution

operator

in addition, invariant

Suppose with

that

for some

impulse

for i = k + l , . . . , m ,

nonlinearity

are real c o n s t a n t s , integer

response

k 5 m, G i is a

gi (-) for i=l,...,k;

G i is a m e m o r y l e s s

in the s e c t o r suppose

[0,ki]. that

for k+l ~ i,j ~ m.

Finally,

functions

t + qi(t)

e x p ( ~ t ) 6 L 2 for i=l,...,k.

36

E. = rain 1 ~ER

gi(~+j~),

37

~i = i/ki"

38

E a = Diag

{e 1 .... ,e k}

39

D b = Diag

{dk+l,...,6m}

^

i=l .... ,k

i = k+l ..... m

and

time-

Suppose

for some

hij = 0 f

~ > 0, the

Now define

205

Haa 40

H =

(hij) =

M =

Under

m-k

Haa

H'aa

Ea

Hab

Hab

Ea

H'aa

H'ab

Ea

Hab

(i)

the system

M is positive

(iii) (iv)

Proof

9.3

(35) is L -stable

Db

if

semidefinite

H'ab Ea Hab + D b is positive

definite

H is positive

Apply

Theorem

GENERAL

(9.1.29)

INSTABILITY we derive

one to deduce

system by establishing Theorem

+

H' [M I aa] I H' ab

In this section, that permits

1

we have

r a n k M = rank

In this sense,

m-k

Ea

(ii)

sult,

0

IH'aa

these conditions,

42

k

Hba k

41

Hab

(7.3.28).

a general

instability

the L -instability

re-

of a given

of a associated

is an instability

counterpart

system. of

(9.1.29). We begin by establishing Lemma

Suppose

G(.)

a preliminary

result.

is an nxn-dimensional

transformable

distribution

with

corresponding

operator

L n2e + L n2e by

G:

o

RESULT

the L 2 - i n s t a b i l i t y

this result

with Theorem

support

in [0,~),

Laplace

and define

the

206

rt (Gx) (t) =

~G(t--T)

x(r)

dz

O

Suppose

G belongs

to c l a s s

U

(see D e f i n i t i o n

^

(3.3.10)),

has a right-coprime

Definition D a(t)

,

(3.3.15))

D a(.)

in

(ii) o C+ of

G(-)

has

poles that

sup

factorization

where

e L- n1 x n

(i)

D(.)

Under

D(.))

fact t h a t

of G.

Otherwise, whence

= Do~(t)+

singularities

singularities

in

(i) f o l l o w s

of G(.)

Moreover, the

readily

in C+ are

there

strip

~c2(G)

< ~.

To p r o v e

0 ~ Re s ~ ~

from Lemma (ii),

let

isolated

is a ~ o > 0 such O

(3.3.48) , a n d ^

(N,D) be the r.c.f.

Then

inf

I d e t D(J~)I

there exists

G ( j ~ i) b e c o m e s

(3.3.28)).

Now,

f(a,M)

(4), f(0,M)

> 0

a sequence

unbounded,

consider

Idet D

= sup

> 0.

Since

> 0

{~i } s u c h t h a t D e t

in v i o l a t i o n

f(.,M)

is a Oo(M)

(3)

D(J~i)÷0, (see L e m m a

Also,

by the R i e m a n n - L e b e s g u e

(a+j~) I

is c o n t i n u o u s ,

s u c h t h a t f(o,M)

I~I ÷ ~; m o r e o v e r ,

of

the f u n c t i o n

there

and

(see

conditions

^

By

that

IIG[j~)II < ~

The only

no

(N(-),

is of the f o r m D(t)

these

f i n i t e order.

Proof the

and

in A n × n

^

G(')

Lemma,

d e t D O ~ 0 by

we see t h a t t h e r e e x i s t s a ~

for e v e r y M

> 0 whenever

D(~+j~) (4).

+ D

o

0~O~Oo(M).

whenever

Combining

these

> 0 such that O

^

inf

inf

Idet D(a+je) I > 0

0 ~ 0

*

Recall

t h a t C+ = {s:

Re s >- 0} and C +° = {s%

Re s > 0}

~0 facts,

207

This shows that G(.)

has no singularities in the strip ^ 0 ~ Re s S o O, i.e. that all singularities^ of G must lie in the

half-plane

Re s > ~o"

singularities

Since D is analytic

o all of these in C+,

must be isolated poles of finite order. []

Lemma

(i) shows that if G belongs

to class U and G has

an r.c.f,

in A n×n , then it is reasonable to assume that G has a ^ o Further, if G has a pole at s o 6 C +° ' pole at some s O 6 C+. then one can e x p l i c i t l y

calculate

Theorem

is the key p o i n t

(3.3.45).

(13) below,

This

some elements

of M±(G),

using

in the proof of T h e o r e m

w h i c h is the main result of this section.

C o n s i d e r now a large-scale

system described

by

m 7a ei = ui - j~l HiJYJ 7b

}

i=l ..... m

Yi = Gi ei n. n. n. L 2ez + L 2eI ' and where e i, ui' Yi 6 L2el for some integer ni, Gi: n. n. Hi3.: L 23e + L2el , ¥i,j. To set the stage for a general L~instability

result

of L2-instability. instability

There features:

8.1,

if a p p r o p r i a t e

instability

occurs

criteria

for L 2-

8.2, and 8.3.2, but they all

(i) they assume that some of the

say G I, .... G k, b e l o n g

instability

(7), we begin with a d i s c u s s i o n

are several diverse

given in Sections

share two common operators,

for the system

to class U;

criteria

whenever

(ii) they state that,

are satisfied,

the input u. in 1

then L 2-

(7) are s e l e c t e d to

be of the form k ui = j~l Fijvj'

vj 6 MI(Gj) /{0}

where the F.. are c o n s t a n t matrices 13 criteria used. We now introduce

.

n.

i.e.,

that depend on the p a r t i c u l a r

the e x p o n e n t i a l

weighting

n.

either e i g L21 or Yi g L2Z for some i.

used to

208

establish

the L - i n s t a b i l i t y

L2-instability Definition

of the s y s t e m

of an a s s o c i a t e d

(9.1.1)

system.

t h a t the o p e r a t o r

(7) by s t u d y i n g

Given

WI:

the

I > 0, r e c a l l

from

Ln ÷ Ln is d e f i n e d 2e 2e

by

(WAx) (t) = x(t) clearly system

exp(At)

W~ 1 = W_A a l s o m a p s L n into itself. 2e (7), w e c a n r e w r i t e (7) as

Now,

g i v e n the

m

10a

W_A e i = W_A u.l - j=l [ (W - AHijWA) . W _ l y j i=l,...,m

10b

W-k Yi =

Note

that,

weighting the

(W-IGiWI)

if i > 0, t h e n

ii

technique

of t h a t p r o p o s e d

of e x p o n e n t i a l

in S e c t i o n

9.1.

For

let us d e f i n e

G# i = W - xGiWx Then

ei

the a b o v e

is the o p p o s i t e

sake of b r e v i t y ,

W-I

the " a s s o c i a t e d

'

H# lj = W _ x H i j W ~

system"

corresponding

to

(7) is

m 12a

d.

12b

z. = G# d. 1 l 1

l

=

r.

l

-

H#. z. m]

[

j=l

i=l,...,m

where new

r i is the n e w

"output".

a n d H l#j• :

Moreover

n. n. L2 e + L 2ei

L -instability associated

following

system

2e

•i,j.

~heorem system

" e r r o r " , and z i is the G# : n. n. L I ÷ L I and ' l 2e 2e

(13) b e l o w

relates

the

(7) to t h a t of the

(12). Consider

conditions

(cl) F o r satisfy

'

d i is the n e w n. ri, di, z i e L i

of the original

Theorem

13

"input",

the s y s t e m

(7), and s u p p o s e

the

are satisfied:

some integer

the h y p o t h e s e s

that

k ~ m,

the o p e r a t o r s

(i) G i is a c o n v o l u t i o n

G i, i = l , . . , k operator

209

~n. ×n. l 1 , and G. in A l

^

belonging

to class U,

(ii) G i has an r.c.f,

has a pole at some s i e C+o , for i=l,. .. ,k . (c2) Select I > 0 such that ^

14

sup fIGi(l+j~)II < ~,

i=l ..... k

and such that G i has a pole Sio = ~io + J~io with ~io < 2~ Remark

(i) below).

associated

For this choice of l, c o n s i d e r the

system

and further,

(12).

Suppose

the system

there exist matrices

that L 2 - i n s t a b i l i t y i

(see

results

(12) is L2-unstable,

i=l .... ,m, j=l .... ,k such

Fi~,

i.e. d i ~g L 2ni or z i ~ L 2ni for some

whenever k

15

ri =

• Fij vj, j=l

~ J vj 6 MI(G~)/{0},

Under these conditions,

j=l

the original

.... k, i=l ..... m

system

(7) is

L -unstable. Remarks to satisfy

(i) It is always possible

(14).

In view of

(cl) and Lemma

to select

I so as

(i), there exists a

^

$o > 0 such that none of the functions Gi(-) in the strip [0,~o/2].

has a s i n g u l a r i t y

0 ~ Re s ~ ~o; one can choose any I in the interval

However,

other choices

for ~ may also be possible.

^

Note that Gi(-)

is only r e q u i r e d to have one pole with r e a l ^ p a r t

greater than 2~; there is no r e g u i r e m e n t have real parts

greater than 2~.

(ii) instability

As m e n t i o n e d

of

(12) results

not at all restrictive, Sections

8.1,

that a l l poles of Gi(-)

8.2,

Proof

earlier,

the r e q u i r e m e n t

for inputs r i of the form

since all of the i n s t a b i l i t y

that L 2(15) is criteria

and 8.3.2 meet this requirement.

F i r s t of all,

note that if G. is a c o n v o l u t i o n l

operator w i t h kernel Gi(.), operator with Gi(.) Also,

G~ belongs

that G~Xis

then G#l is also a c o n v o l u t i o n

exp(-l(.)),

so that

to class U, for i=l,...,k.

linear,

of

and satisfies

(3.3.11)

(s) = Gi(s+l). To see this, (because W_l,

note

G i, Wl

210

individually ~c (G[)

satisfy

(3.3.11)) ; finally,

(14) guarantees

that

< ~. ^

It is easy to see that if

^

^

(Ni,D i) is an r.c.f,

of G i,

^

then

(Ni(s+l) , Di(s+l)

Gi has a pole at Sio, (3.3.45),

is an r.c.f,

Similarly,

then G.# has a pole at s. -I. 1 io

MI(G #) contains

exp[ (-Oio+l)t]

of Gi(s+l).

a nonzero

cos(eiot).

element

if

By Theorem

of the form v i

Now, by condition

(c2), if we select

k 16

r i(t)

in

= j=l~ F.13. v.3 exp[ (-Ojo+~)t]cos(~jo t), i=l ..... m

(12), then either

(i0) and

n. n. z i ~ L21 or d i ~ L21 for some i.

Comparing

(12), we see that if we select k

17

W_lu i (t) =

i.e.,

j=l

F..v. expl (-~-'o+~)3 t] cos(~_.o t ) ] , i=l ..... m 13 J

if we select

18

ui(t)

k ~ F..v. exp[ (-~jo+2~)t] j=l 13 ]

=

cos

n,

in

(~jo t)

i=l ..... m

n.

(7), then either W_ly i ~ L21 or W_le i g L21 for some i.

In p a r t i c u l a r , t h i s means t h a t , w i t h t h e c h o i c e of i n p u t s u i ( . ) n. n. Yi g L ~ l o r e . ~ L 1 f o r some i ( n o t e t h a t , i f of ( 1 8 ) ' n e i t h e r f(.) E L i, then W _ X f ( . ) e L21 Zn' ) . S i n c e t h e i n p u t u i d e f i n e d b y n.

(18) belongs

to L 1 , we see that the s y s t e m

It is obvious obtain L -instability the L2-instability is unnecessary "small gain" 19

20

criteria

criteria

using Theorem corresponding

of sections

8.1,

to list of all of these.

type result

Theorem described

that,

for the purposes

Consider

a single-loop

(7)

is L -unstable.m

(9.3.13), to almost 8.2,

However,

e2 = u2+Yl'

all of

and 8.3.2. we state a

of illustration. feedback

system

by

el = ul-Y2'

one can

Yl = Glel'

Y2 = G2e2

It

211

where

G 1 is a c o n v o l u t i o n

in An×n;

operator

G 2 is a m e m o r y l e s s

21

of class

operator

U, a n d has an r.c.f.

of the f o r m

(G2e 2) (t) = ~(t,e 2(t)) and ~ belongs

22

to the s e c t o r [~,8],

[ ~(t,v)

under

- ~v] ' [ ~(t,v)

these conditions,

i.e.,

- 8v]

the s y s t e m

~ 0,

(20)

c a n find a I > 0 s u c h t h a t the f o l l o w i n g

~ t ~ 0, ¥ v • R n

is L - u n s t a b l e conditions

if one

are

satisfied:

23

(i)

24

(ii)

sup E m a x [ H% (~+j~)

H(-)

H (l+j~) ] < 6 -1

has at l e a s t one p o l e w i t h r e a l p a r t g r e a t e r

t h a n 21, w h e r e

25

^ (s) = G(s)

26

y = [ S+~]/2,

a n d Emax(-) % denotes

the

the

the c o n j u g a t e

former

involves

denotes

^ [I + yG(s)] -I

~ = [ 8-~]/2

largest

is s i m p l e

Comparing

Theorem

Gi(-~+j~)

of a m a t r i x ,

transpose.

The p r o o f

involves

eigenvalue

a n d is t h e r e f o r e

(19) w i t h T h e o r e m

G i ( l + j ~ ) for s o m e for s o m e ~ > 0.

omitted.

(9.2.17),

I > 0, w h i l e

the

we

see

latter

212

NOTES AND R E F E R E N C E S

The e x p o n e n t i a l w e i g h t i n g a p p r o a c h to study L s t a b i l i t y is due to S a n d b e r g [San.

2] and Zames [Zam.

R e l a t e d results can be found in [Ber. e x t e n d e d to l a r g e - s c a l e

i] .

2].

This a p p r o a c h was

systems by L a s l e y and M i c h e l [Las.

2].

The g e n e r a l s t a b i l i t y t h e o r e m given here is taken from [Vid. w h i l e s p e c i f i c r e s u l t s are f r o m [Las. instability

is c o n t a i n e d in [Vid.

8].

2].

6],

The a p p r o a c h to L -

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INDEX A A,9 Ae,ll A,II Acyclic digraph,67 Adjacency matrix,58

C Causality,8 for discrete-time systems,9,92 strict...,92 Class U operator,47 Connective stability,l16,122 Coprime factorization,49 Cycle,66

D Decaying Ll-memory,192 Directed graph,57 acyclic...,67 strongly connected...,59 system...,73,77 Directed tree,72 Dissipativity,42 conditional...,46 of a convolution operator,45 verification of conditional...,55 Dissipativity theorems for single-loop systems,135 for instability,179,182 for interconnected systems,141,142

E Essential set,127 Exponential weighting,190 instability theorems using...,208,210 stability theorems using .... 195,201,204 Extended space,5

219

G Gain,26 conditional...,46 incremental...,26 of a convolution

operator,34,38,39

of a linear integral operator,29,33 of a memoryless

nonlinearity,40

With zero bias,26

! Instability

theorems

for single-loop

systems,165,166

of dissipativity of passivity of small-gain Interconnection Isolated

type,179,182

type,185 type,165,168,170,172,174 matrix,14

subsystem,75,77

t Loop transformation for single-loop

systems,106,107

for interconnected Lossless

systems,123

interconnections,148

Lpe,5 L -stability,18,21 P with zero bias,18,21

M Mp (G) ,45 characterization M 1 (S)

of...,51

characterization

of...,52

some elements

in...,54

N Nonnegative theorems

matrices for...,108

P Passive

interconnections,148

220

Passivity,45 of a convolution operator,45 strict...,45 Passivity theorems for instability,185 for interconnected systems,150,156,159,161 for single-loop systems,137,138 Perron-Frobenius theorem,107

R Reachability,58 algorithms for testing .... 63,64 matrix,60

S Section graph,126 Sector,41 Self-loop,67 Small-gain type theorems for instability 165,168,170,172,174 for interconnected systems,l10,115,117,121,129,146 for single-loop systems,105,106,107 Smoothing operator,88 Stability,definition of,18,21 Strict passivity,45 Strong connectedness algorithms for testing...,63,64 of a digraph,59 of a pair of vertices,58 Strongly connected component,71 d~graph,59 System digraph,73,77

T Tree,directed,72 Truncated inner product,5 norm, 5 Truncation,5

221

W Warshall's algorithm, 64 Weak interaction,ll7 Weakly Lipschitz operator,88 Well-posedness definition,16,20 of continuous time systems,97 of decomposed systems,81,86 of discrete-time systems,103