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Lecture note in control and information scienses

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

for e v e r y

is d e f i n e d

=

pe

by

i0

of

p, we h a v e

is a s u b s e t

that we need

Let

f • Lpe

for e a c h

belongs

does

[i,~].

the

[0,T]

= t

but

p •

to a n y of

since

largest

fl(t)

[i,-],

f, g E L 2 e
by

, the


T

ii

T

I

= =

f(t)

g(t)

p E

[I,~]

dt

0

Note
the q u a n t i t y
every

for e v e r y

I If| |Tp

T < ~

belongs

that,

is a w e l l - d e f i n e d

, though

I Ifl Ip

to the u n e x t e n d e d

and every
finite

is d e f i n e d

space

L

only

f E L

real number
if

Moreover,

f

pe '
for

actually

we have

P
12

Lemma

Let

is a n o n d e c r e a s i n g
unextended
c

as

P

IIfl |Tp

In o r d e r

Then

n-tuples

the s e t

f(.)~ =

is d e f i n e d

[fl(.)

Lpe)

¥i

.

! |" | 1

L

I

L n2

systems

I]fIITp +

having multiple

Ln
p

[1,~]

and

and

inputs

Ln
pe

let

n > 1

Lpe)
n

b e an in-

consists

... fn(.)] ' , w h e r e

of the v e c t o r

space,

with

I Ip d t } i / P

ass.

sup

t e

[O,~)
n o r m on

of all

fi(.)
f(.)

• Lp
6 L nP

is

p < - ,

if

p :-

I I-|Ip

to d e n o t e

Ln .
P

This usage

is

any c o n f u s i o n .

norm

of the n o r m
{ I-If

the c h o i c e

by

if

Rn

as the n o r m on

in

on
(14),

and the i n n e r p r o d u c t

is d e f i n e d

]If(t) IE

the same s y m b o l

as w e l l

P
to c a u s e

However,

product
in

In this case,

0

In the d e f i n i t i o n
choice

(the
P
constant

a n d is l e f t to the r e a d e r .

i o I If(t)

t h a t w e use

the n o r m on

not expected

ry.

f e L

a finite

T h e n o r m of a f u n c t i o n

is the E u c l i d e a n

Note
both

I If| |Tp

by

I lf(_) t l p :

where

exists

(respectively

f2(.)

{

14

p 6

L np

Then

from below.

the s p a c e s

Let

f C Lp

Furthermore,

< -

to d e a l w i t h

we introduce

(respectively

VT

is o b v i o u s

Definition
teger.

T

if t h e r e

, monotonically

The proof

13

_< c

and let

of

if a n d o n l y

T ÷ ~

and o u t p u t s ,

[i,~]

function

space)

such that

IIfll

p 6

I I-I Ip

Rn

on

L np,

is e s s e n t i a l l y

the s p a c e


L n2

the
arbitra-

is an inner-

of two e l e m e n t s

f,g


~

15



5

=

0

where
and

fi(.)
g(.)

and

f, (t) g(t)
~

gi(.)

are the c o m p o n e n t

, respectively.

~

The

truncated

and the t r u n c a t e d

inner product

fined

analogous

in a m a n n e r

to

(i0)

16

S

Definition
{x(i)}~= 0 .
in

S

For

The

set

set

: L n ÷ R+
pe
are de-

we introduce

the

subsets.

consists

1 ~ p < - , the

I I.I IT

f(.)

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

systems,

S

of

: L ~e × L~ e ÷ R

and

and its v a r i o u s

functions

norm

<. , ">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 <

equivalently,

26

(Gf) T =

27

Lemma

(GfT) T

whenever
gT

f

and

for some

operator

28

(24)

g

, we have

Proof

For

G :Lle

+Lle

÷Lle

is c a u s a l

if the f o l l o w i n g

(Gf) T =

in

Lle

has property

(Gf)T =

and that

in the

is t r u e

Such t h a t

fT =

let us say t h a t an

(s)

if

(Gg)T
in the s e n s e of D e f i n i t i o n

(s)

T o s h o w this,

fT = gT

for s o m e

T

suppose

(24)

first

Then by

is
that

(25), w e

have

29

(Gf) T =
so t h a t
property

G

(GfT) T =

has property
(s)

Since

(GgT) T =

(s)
fT =

:

(Gg) T

the sake of c l a r i t y ,

to p r o p e r t y

is c a u s a l ,

Vf E Lle

G :Lle

if and o n l y

show that causality

equivalent

YT < - ,

are two f u n c t i o n s

T < ~

fT = gT ~
We must

,

An operator

s e n s e of D e f i n i t i o n

G

the c o n c e p t

Rn

if

25
or,

n o r m on

(Gg) T

Conversely,
(fT)T

suppose

%~f, w e h a v e f r o m

G
(28)

has
that


30

(Gf) T = (GfT) T
so that

G

is causal.
It is clear

ators on

Lle

well define
Lqm
e

that there

is nothing

as far as causality

causality

or from

Sn

with respect

to

Sm

where



goes,

special

about oper-

and that one can equally

to operators
p, q 6

[i•~]

from

L pe
n

to

and

n,m

are



positive

integers.
We conclude

which plays

this section by introducing

an important

the set

role in the study of linear

A,

time-invari-

ant operators.
31

Definition
f(.)

The set

A

consists

of all distributions

of the form
f(t)

32

=~

0,

[

t < 0
fi 6 (t-t i) + fa(t)

,

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 <_ kTI lu(1)-u(2) IITp

14

fly(1)-y(2)[ITp<_kT[ lu(1)-uC2~lITp
The above d e f i n i t i o n
as it implies

(i) e x i s t e n c e

system equations,
(iii)

however,

Note that

we list

light it.
served w h e n

and u n i q u e n e s s

(W2)

continuity
(W2)

Gi, Hij

Gi, Hij

of solutions

of solutions

as functions

implied by

condition

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

are p e r t u r b e d

in [Wil.2]

slightly.
(Wl)

(W3) ;

in order

to high-

requires
be pre-

We do not m a k e

of w e l l - p o s e d n e s s .

5 that p r o p e r t i e s

to the

on inputs,

n a m e l y that all of the above p r o p e r t i e s

this a p a r t of the d e f i n i t i o n
shown in C h a p t e r

is quite broad,

of solutions

is a c t u a l l y

as a separate

The d e f i n i t i o n

something more:

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

(ii) causal d e p e n d e n c e

global L i p s c h i t z

of the input.

if each

in

are the c o r r e s p o n d i n g

13

and

eT

kT

u (2) are two sets of inputs

, {e(2)

of

In other words,

- (W3)

is r e p l a c e d by another o p e r a t o r

However,

it is

are p r e s e r v e d
of the same or


18

higher

"class"

Notice
assume

that in a d o p t i n g

that each

subsystem

of a m u l t i - v a l u e d
output

Yi

from

equations.

The

ator m e a n s

that this

in a n o t h e r

sense.

questions.
nection

Now,

Definition

Rather,

we a s s u m e

for i n s t a n c e

the error

ei

fact

G i : e i ~ Yi

that

system

(1), we

equations

subsystem

instead

that

by s o l v i n g

(9) is not c o n c e r n e d
that each

by an operator,

of the o v e r a l l

description

by an o p e r a t o r ,

suppose

set of d i f f e r e n t i a l

is r e p r e s e n t e d

posedness

is r e p r e s e n t e d

relation.

is o b t a i n e d

differential

the s y s t e m

the

a set of
is an oper-

is w e l l - p o s e d
with

such

and i n t e r c o n -

and ask a b o u t

from an i n p u t - o u t p u t

the w e l l point

of

view.

It is i m p o r t a n t
posed

in the sense

finite
over,

escape

are L i p s c h i t z

for all
ion

time b e c a u s e

in a w e l l - p o s e d

PTy

finite

(15)

and

However,

defined

Definition
merely

stable

if

ing p r o p e r t i e s

to

e,y

in

from

stable

this does

L n [0,T]
into itself,
P
in the sense of D e f i n i t -

not mean

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

The

that the maps

u ~ e

(15).

of the s y s t e m

(1),

to stability.

system

is o b v i o u s

For each
Ln
pe

(1)

is said

to be L - s t a b l e

from the context)

if the

(or

follow-

such

set of inputs
that

u 6 L n , we have that
P
(i) is satisfied, a c t u a l l y b e l o n g

Ln
P
($2)

that,

There

whenever

solutions

16

p

is w e l l -

from h a v i n g

hold:

(SI)
any

maps

in the sense of D e f i n i t i o n

we n o w turn our a t t e n t i o n

15

that

is p r e c l u d e d

u 6 Ln
implies
e,y 6 L n
. Morepe
pe
the m a p s
PT u ~ PT e
and
PT u ~

and are h e n c e

are stable

Having

that a s y s t e m
(9)

system,

continuous

T,

below.

u ~ y

to note

of D e f i n i t i o n

of

exist

u 6 Ln
and
P
(i), we have

Ileilp

~

finite

constants

e,y E L n
P

Ypiluilp + bp

are

yp

and

bp

such

some c o r r e s p o n d i n g


19

IIyilp pllullp÷bp

17

If

b

= 0, we say that the s y s t e m

(1) is ~ - s t a b l e

w i t h zero

P
bias.

It is i m p o r t a n t to note that in D e f i n i t i o n

(15), we do

not a s s u m e e x i s t e n c e and/or u n i q u e n e s s of s o l u t i o n s to

(i).

If

for a p a r t i c u l a r
satisfied,

u 6 L n , no
e,y 6 L n
e x i s t such t h a t (i) is
p
pe
then c o n d i t i o n s (Sl) and (S2) are s a t i s f i e d v a c u o u s l y .

In this way,

the s t a b i l i t y issue is d i v o r c e d from the issue of

well-posedness.

Also,

note that the s y s t e m

and only if the r e l a t i o n m a p p i n g
in the sense of D e f i n i t i o n

u

(3.1.1).

into

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

(e,y)

Similarly,

if

has finite gain,
the s y s t e m

(i) is

L - s t a b l e w i t h zero b i a s if and only if the r e l a t i o n m a p p i n g
u
P
into
(e,y)
has finite g a i n w i t h zero bias, the sense of Definition(3.1.1).

Note that,

in order for the system

in the sense of D e f i n i t i o n
either

(i) to be L -unstable
p
(15), one of two things m u s t h a p p e n :

(i) there exist a set of inputs

u 6 L n and a set of
P
e,y • L n
such t h a t (1) is satisfied, but e i t h e r
P
does not b e l o n g to L pn ' or (ii) there exists a sequ-

errors/outputs
e

or

y

i

ence of input sets

u (j) • L n and a c o r r e s p o n d i n g s e q u e n c e of
P
e r r o r / o u t p u t sets e ( ~ ) E L n , y ( J ) e L n such that one of the sequence
P
P
{I le(J)llp/l Is(J) llp}
or
{I IY(J)IIp/l lu(J) lip}
is unbounded.
(In this case,
in t h a t
fy

the s y s t e m

u 6 LP
n

~

(i) m a y still have a f o r m of "stability"

e 6 LP
n , y e L Pn ; however,

it does not satis-

(16) - (17)).

We now p r e s e n t and d i s c u s s an a l t e r n a t i v e
of the s y s t e m d e s c r i p t i o n and s t a b i l i t y definition.
that,

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

operators,

(Hij).

It is clear

(i), there are two types of

n a m e l y the s u b s y s t e m o p e r a t o r s

nection operators

formulation

(G i) and the i n t e r c o n -

This d i s t i n c t i o n serves a u s e f u l

p u r p o s e in d e r i v i n g the results of C h a p t e r s

6-9, w h e r e

the k i n d s

of c o n d i t i o n s i m p o s e d on the s u b s y s t e m o p e r a t o r s are q u i t e
d i f f e r e n t in n a t u r e from those i m p o s e d on the i n t e r c o n n e c t i o n
operators.
techniques

However,

in C h a p t e r 4 and 5, w h e r e we apply some

from g r a p h theory to study the d e c o m p o s i t i o n and w e l l -


20

posedness

of

large-scale

between

subsystem

irely.

In these

and

system

description

system

and

interconnection

situations,
that

described

this

systems,
operators

it is t h e r e f o r e

also makes

interconnection

To m e e t
systems

interconnected

the d i s t i n c t i o n
disappears

logical

no d i s t i n c t i o n

ent-

to a d o p t

between

a

sub-

operators.

objective,

in C h a p t e r

4 and

5 we

study

by
m

18

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

i=l,...,m

n.

where

ei,

u i 6 L p el

for some

n., and
S.. : L nj + L ni .
l
13
pe
pe
m
summing junctions, where
consist
other

of an e x t e r n a l

summing

Figure

junction

fixed

We

see

the

input

that

(u i)

outputs

this

positive

system

to e a c h

and

(Sij

some

consists

summing

interaction

ei).,

This

integer
of

junction

signals

depicted

from

in

2.4

19

Definition
following
(W1)

ui

+f--~

Sil

eI

The

For

each

set of e r r o r s

(18)

satisfied.

are

(W2)
causal;
such

i.e.,
that

The

2.4
(18)

is s a i d

to be w e l l - p o s e d

hold:
set

of i n p u t s

e 6 Ln
pe

dependence

whenever

u (I)

for

T > 0

some

ei

Sire em

system

conditions

a unique

Ln
P

and

inputs

FIGURE

if the

p

such

of
and

that

e
u (2)

we h a v e

u e L pe
n
the

and
are

' there

system

{Sij
two

ej}
input

exists

equations

on

u

sets

is
in


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