Large Scale Systems: Decentralization, Structure Constraints, and Fixed Modes (Lecture Notes in Control and Iinformation Sciences)

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.VVyner

120 L. Trave, A. Titli, A. Tarras

Large Scale Systems: Decentralization, Structure Constraints and Fixed Modes

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Series Editors M. Thoma • A. Wyner

Advisory Board L D. Davisson • A. G. J. MacFarlane • H. Kwakernaak .I.L. Massey • Ya Z. Tsypkin. A. J. Viterbi Authors Louise Trave Andre Titli Ahmed Maher Tarras Laboratoire d'Automatique et d'Analyse des Systemes du Centre National de la Recherche Scientifique Toulouse France

ISBN 3-540-50787-6 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-50787-6 Springer-Verlag NewYork Berlin Heidelberg Library of Congress Cataloging in Publication Data Trave, L. (Louise) Large scale systems : decentralization structure constraints and fixed modes L. Trave, A. Titli, A. Tarras. (Lecture notes in control and information sciences ; 120) Bibliography: p. Includes indexes. ISBN 0-387-50787-6 (U.S.) 1. System theory. 2. Control theory. I. Titli, Andre. I1. Tarras, A. (Ahmed). II1. Title. IV. Series. Q295.T73 1989 88-35984 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1989 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: B, Helm, Bedin 2161/3020-543210

PREFACE

The

growing

dimensions and

c o m p l e x i t y of

the

present

day

technological,

e n v i r o n m e n t a n d societal p r o c e s s e s is one o f t h e f o r e m o s t c h a l l e n g e s to s y s t e m t h e o ry.

Determining a

solution

for

the

problems

arising

in l a r g e

scale

systems

may

become e i t h e r v e r y uneconomical o r e v e n impossible if u s i n g t h e c l a s s i c a l mathematical tools d e v e l o p e d f o r s y s t e m a n a l y s i s a n d c o n t r o l .

T h e main r e a s o n i s t h a t classical

t h e o r i e s a r e n o t b u i l t for d e a l i n g with h i g h d i m e n s i o n a l i t y models. Now, t h e e s s e n t i a l c h a r a c t e r i s t i c s of l a r g e scale s y s t e m s a r e a h u g e n u m b e r of i n p u t a n d o u t p u t v a r i a b l e s on s u b s y s t e m s w h i c h a r e g e n e r a l l y g e o g r a p h i c a l l y d i s t r i b u t e d . T h e s e new f e a t u r e s i n v o l v e l a r g e a n d complex m o d e l s , problem

may

not

be

solvable.

Moreover,

we m u s t

face

though the

modelling

economical a n d

reliability

p r o b l e m s r e l a t e d to t h e i n f o r m a t i o n t r a n s f e r b e t w e e n c o n t r o l s t a t i o n s . For t h e s e r e a s o n s , t h e d e c o m p o s i t i o n , a g g r e g a t i o n a n d model r e d u c t i o n t e c h n i c s h a v e r e c e i v e d c o n s i d e r a b l e a t t e n t i o n in t h e l a s t t e n y e a r s . A g r e a t deal of t h e o r e t i c a l a n d p r a c t i c a l r e s u l t s c o n c e r n i n g t h e i r a p p l i c a t i o n s h a v e b e e n o b t a i n e d in t h e a r e a o f stability and decentralized control.

In p a r t i c u l a r ,

t h e p r o b l e m of s t a b i l i z a t i o n a n d

pole p l a c e m e n t with d e c e n t r a l i z e d dynamic c o m p e n s a t i o n is of g r e a t p r a c t i c a l i n t e r e s t . Despite the numerous advances around

this problem,

which are

materialized by

a

l a r g e n u m b e r of p a p e r s , t h e r e is n o n e s y n t h e t i c a l s u r v e y work e x c l u s i v e l y c o n c e r n e d with t h i s p r o b l e m a n d t h e v a r i o u s o t h e r o n e s which a r e r e l e v a n t . The main o b j e c t i v e of t h i s book is to p r o v i d e s u c h global s u r v e y b y p r e s e n t i n g t h e p r e s e n t d a y r e s u l t s which can b e u s e d f o r : -the

a n a l y s i s of stabiHzability a n d pole p l a c e m e n t u n d e r d e c e n t r a l i z e d c o n s -

traints, - t h e d e t e r m i n a t i o n of a c o n t r o l policy s o l v i n g t h e p r o b l e m of s t a b i l i z a t i o n o r pole p l a c e m e n t w h e n d e c e n t r a l i z e d dynamic c o m p e n s a t i o n fails { p r e s e r v i n g a d e c e n t r a l i z e d s c h e m e of c o n t r o l o r minimizing t h e c o s t a s s o c i a t e d to t h e i n f o r m a t i o n t r a n s fer), - t h e d e s i g n of t h e s e p r e s p e c i f i e d c o n t r o l l a w s .

IV By t h i s w a y ,

t h i s b o o k s u p p l i e s t h e tools for b u i l d i n g a m e t h o d o l o g y w h i c h

b r i n g s a s o l u t i o n to t h e complete p r o b l e m of c o n t r o l in t h e c o n t e x t of l a r g e systems.

Moreover,

the last part

scale

of t h e w o r k t a k e s i n t o a c c o u n t p a r a m e t r i c a n d

structural robustness constraints. C h a p t e r I p r e s e n t s an o v e r v i e w of t h e w e l l - k n o w n r e s u l t s a r o u n d t h e p r o b l e m o f s t a b i l i z a t i o n a n d pole a s s i g n m e n t of l i n e a r t l m e - i n v a r i a n t dynamic s y s t e m s s u b j e c t e d to c e n t r a l i z e d c o n t r o l (no s t r u c t u r a l c o n s t r a i n t s ) . T h e f u n d a m e n t a l c o n c e p t s of c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y are i n t r o d u c e d a n d t h e y a r e e x t e n d e d to t h e c o n cepts of s t r u c t u r a l

controllability and

observability,

which a r e

of major p r a c t i c a l

i n t e r e s t in t h e s t u d y of l a r g e scale s y s t e m s . I n d e e d , t h e s e p r o p e r t i e s a r e e s t a b l i s h e d from t h e s y s t e m s t r u c t u r e a n d t h e y do n o t d e p e n d on t h e p a r t i c u l a r c o n f i g u r a t i o n o f the parameters I values.

In t h i s f r a m e w o r k ,

t h e p r o b l e m r e d u c e s to one of b i n a r y

n a t u r e t h a t allows f o r t h e a p p l i c a t i o n of g r a p h - t h e o r e t i c c o n c e p t s . T h i s a p p r o a c h is t h u s specially a d e q u a t e for l a r g e scale s y s t e m s . The n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e of a s o l u t i o n to t h e problem of stabilization and cases

pole a s s i g n m e n t a r e

presented

for

the

following two

:

- centralized state feedback - centralized output feedback T h e y a r e s t a t e d in t e r m s of t h e c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y p r o p e r t i e s of the system. It is c l e a r t h a t a good u n d e r s t a n d i n g of t h e

c o n c e p t s of c o n t r o l l a b i l i t y and

o b s e r v a b i l i t y is i n d i s p e n s a b l e b e f o r e p r o c e e d i n g to t h e s t u d y of t h e a b o v e p r o b l e m with s t r u c t u r a l c o n s t r a i n t s on t h e c o n t r o l . T h i s p r o b l e m i s i n t r o d u c e d in C h a p t e r II, In t h e

c o n t r o l of l a r g e s c a l e s y s t e m s w h o s e e s s e n t i a l c h a r a c t e r i s t i c is t h e i r

high dimensionality, conventional techniques

fail to

give r e a s o n a b l e s o l u t i o n s w i t h

r e a s o n a b l e c o m p u t a t i o n a l e f f o r t s . The classical c o n t r o l t h e o r y g e n e r a l l y s t a n d s on t h e assumption of a centralized information pattern ; i.e., s y s t e m is available at a g i v e n c e n t e r ,

all t h e i n f o r m a t i o n on t h e

g e n e r a l l y a g e o g r a p h i c a l p o s i t i o n , w h e r e all

t h e c a l c u l a t i o n s can b e c a r r i e d o u t . F o r most l a r g e scale s y s t e m s , t h i s c e n t r a l i z a t i o n a s s u m p t i o n d o e s n o t h o l d d u e to t h e

g e o g r a p h i c a l d i s t r i b u t i o n of t h e

information.

This new constraint leads

economical a n d r e l i a b i l i t y p r o b l e m s r e l a t e d to t h e i n f o r m a t i o n t r a n s f e r .

to

This implies

t h a t t h e c o n t r o l s y s t e m s h o u l d b e made of a n u m b e r o5 local c o n t r o l l e r s t h a t a r e only allowed to u s e p a r t o f t h e whole i n f o r m a t i o n in o r d e r to g e n e r a t e p a r t of t h e whole

V c o n t r o l . In p a r t i c u l a r ~ t h e d e s i g n of f e e d b a c k c o n t r o l l e r s r e q u i r e s r e s t r i c t i o n s on t h e p a r t i c u l a r s y s t e m o u t p u t - i n p u t p a i r s t h a t t h e c o n t r o l l e r can c o n n e c t . When no t r a n s f e r of i n f o r m a t i o n b e t w e e n t h e d i f f e r e n t local s t a t i o n s is allowed, t h i s y i e l d s to a d e c e n t r a l i z e d s c h e m e of c o n t r o l .

When some b u t

n o t all t r a n s f e r s

( t h o s e of minimum c o s t for example) a r e allowed we o b t a i n a n o n s t a n d a r d r e d u c e d information p a t t e r n . It is c l e a r t h a t t h e d e c e n t r a l i z e d c o n t r o l s c h e m e i s t h e most economically a d v a n t a g e o u s s i n c e no t r a n s f e r of i n f o r m a t i o n for one g e o g r a p h i c a l location to a n o t h e r is r e q u i r e d : system inputs are

a s s i g n e d to a g i v e n s e t of local c o n t r o l l e r s ( s t a -

t i o n s ) , w h i c h o b s e r v e only local s y s t e m o u t p u t s . This is t h e r e a s o n t h e f i r s t s t u d i e s for t h e p r o b l e m of s t a b i l i z a t i o n a n d pole a s s i g n m e n t w e r e i n v e s t i g a t e d w i t h i n a d e c e n t r a l i z e d c o n t r o l s c h e m e . T h e r e s u l t s r e f e r i n g to t h i s s t u d y a r e p r e s e n t e d in t h e f i r s t p a r t of t h e c h a p t e r . T h e s e r e s u l t s w e r e t h e n e x t e n d e d to t h e more g e n e r a l case of a r b i t r a r i l y s t r u c t u r a l l y c o n s t r a i n e d c o n t r o l w h i c h is c o n s i d e r e d in t h e s e c o n d p a r t of t h e c h a p t e r . The main c o n c e p t to deal with t h i s k i n d of p r o b l e m is t h e new n o t i o n of f i x e d modes w h i c h is

f u n d a m e n t a l in t h e

s t u d y o f s t a b i l i z a t i o n a n d pole p l a c e m e n t with

s t r u c t u r a l l y c o n s t r a i n e d dynamic c o m p e n s a t i o n . I n d e e d t

t h e e x i s t e n c e o f a solution

d e p e n d s c r i t i c a l l y on t h e p r o p e r t i e s of t h i s finite s e t of n u m b e r s . T h e p r e s e n c e o f u n s t a b l e f i x e d m o d e s i n d i c a t e s t h a t s t a b i l i z a t i o n is i m p o s s i b l e while t h e p r e s e n c e of a n y s o r t of f i x e d modes r u l e s out a r b i t r a r y pole p l a c e m e n t . Due t o t h e t h e o r e t i c a l a n d p r a c t i c a l i m p o r t a n c e o f t h e notion of f i x e d m o d e s , t h e whole C h a p t e r 3 is c o n c e r n e d with t h e i r c h a r a c t e r i z a t i o n . T h e n u m b e r of d i f f e r e n t c h a r a c t e r i z a t i o n s which can b e f o u n d in t h e s c i e n t i f i c l i t e r a t u r e i s i m p r e s s i v e . Moreover, e v e r y one of them is e x p r e s s e d in t e r m s of i t s own a u t h o r s d e f i n i t i o n s a n d i n t r o d u c e d in a d i f f e r e n t w a y . presented

in two g r o u p s ,

With t h e main o b j e c t i v e of classifieation~

t h e time-domain a n d t h e

they

are

frequency-domain characteriza-

t i o n s . A p a r t i c u l a r a t t e n t i o n is g i v e n to show t h e e x i s t i n g e q u i v a l e n c e s . The a n a l y s i s of e v e r y one of them allows u s to p o i n t o u t t h e c o n d i t i o n s for t h e e x i s t e n c e o f f i x e d modes a n d to g i v e a d e a p i n s i d e into t h e i r i n t e r p r e t a t i o n r e l a t e d to t h e i r o r i g i n s . The d i f f e r e n t t y p e s of f i x e d modes a r e o u t l i n e d : no s t r u c t u r a l l y f i x e d modes w h i c h n e e d a q u a n t i t a t i v e a n a l y s i s of t h e s y s t e m a n d s t r u c t u r a l l y f i x e d modes for w h i c h a s t r u c t u r a l a p p r o a c h is more s u i t a b l e ( r e p r e s e n t a t i o n of t h e s y s t e m b y a g r a p h , u s e o f g e n e r a l c o n c e p t s of g r a p h t h e o r y ) . From

the

above

analysis,

the

different

results

concerning

the

problem

of

d e c e n t r a l i z e d s t a b i l i z a t i o n a n d pole p l a c e m e n t a r e u n i f i e d a n d e x p r e s s e d in t e r m s o f t h e d i f f e r e n t t y p e s of f i x e d m o d e s .

VI Whereas structure

Chapter

2 makes clear

(decentralized

the existence

that

for example)

of fixed modes,

the

choice a priori

can generate

Chapter

of a feedback

some problems

3 provides

control

if it gives

all t h e n e c e s s a r y

rise

to

t o o l s to a n a l y s e

and explain the situation.

As

a

different

natural

methods

consequence,

the

following

chapters

which are available to determine

are

concerned

an acceptable

with

the

control policy such

that stabilization or pole placement is possible. In the tralized

context

control

of large

structure

scale

(Chapter

systems,

allow

to

4 presents

avoid

fixed

a

class

modes

An original approach trol.

Vibrational

thods

(based

control

compatible presence

with

decentralized

too

consider

or

constraints

fixed modes.

From another

when

fixed modes. the

based

on

Chapter

the

an

can

-

the

system

origin.

vibrational

conventional

as

it

solves

point of view,

the

me-

of lack

method

is

problem

in

a particular

a method

con-

control

because

a stabilization

that

for the

applicadesign

of

or non-linear

control laws appears

to

of structurally

fixed modes,

we m u s t

which minimizes the cost

which is of immense practical

be

constraints

of Chapter

interest,

is

5 is to present

feedback

appropriate

more

Roughly

o n t h e c o n t r o l s e e m s to b e t h e m o s t

or pole placement

characterizations

is physically

geographical

and

a new control structure

stabilization

appropriate

s i t u a t i o n we a r e d e a l i n g w i t h .

different

in u s i n g

where

constitutes

in p r e s e n c e

the structural

different

3 : one

structural

which

5.

The purpose of

that

This problem,

to s o l v e t h e

design

a

) do n o t a p p l y

control

of time-varying

we a r e

transfer.

In fact, relaxing way

from

controllers

control laws.

the problem of determining

convenient

not

cases

principles

vibrational

the subject of Chapter

are

such

5).

time-varying

: it c o n s i s t s

in the

control is presented

of the information

for

developed

that

time-varying

difficult

they

or feedforward

When the implementation be

that

decentralization

of unstable

tion of vibrational

decentralized

a decen-

a new control structure

is minimum (Chapter

can be useful

It is shown the

i s to p r e s e r v e

laws are examined and compared.

is then

on feedback

of measurements.

of

provided

Several kinds of time-varying

objective

4) o r t o d e t e r m i n e

that the cost of the information transfer Chapter

the

the

control of

fixed

than

speaking,

partitioned

locations of the inputs

different

structure. modes

another two types

in several and

problem

methods

are

presented

depending

on

of situations

due

In this

of

available methods

These

which

stations,

outputs.

in presence

the

type

are in of

can occur

:

for example

to

case,

it is clear

Vll

that the decentralized structure

would b e t h e m o s t a p p r o p r i a t e .

O u r goal i s t h u s to

d e t e r m i n e t h e minimal i n f o r m a t i o n e x c h a n g e s b e t w e e n s t a t i o n s w h i c h g e t r i d of f i x e d modes.

The

optimality

criterion

can

be

chosen

as

the

number

of f e e d b a c k

links

b e t w e e n two d i f f e r e n t s t a t i o n s o r a s t h e c o s t a s s o c i a t e d with t h e i m p l e m e n t a t i o n of these feedback Hnks,

-

present

either the system does not reflect a prespecified partitioning or the stations the

particularity

that

the

cost

of local

f e e d b a c k s is n o t n e g l e c t a b l e w i t h

r e s p e c t to t h e c o s t of f e e d b a c k s b e t w e e n two d i f f e r e n t s t a t i o n s . I n t h e s e c a s e s , w a n t to d e t e r m i n e t h e minimal c o n t r o l s t r u c t u r e s

(if s e v e r a l )

we

for which t h e s y s t e m

h a s n o f i x e d m o d e s . T h e y do n o t g e n e r a l l y i n v o l v e all t h e local f e e d b a c k s .

Note

that

the

problem resulting

from t h e

second

s i t u a t i o n is more g e n e r a l .

I n d e e d , we a r e b r o u g h t b a c k to t h e f i r s t p r o b l e m b y s e t t i n g to z e r o t h e c o s t s a s s o dated

to t h e local f e e d b a c k s .

Therefore,

all t h e m e t h o d s w h i c h a r e p r e s e n t e d i n t h i s

g e n e r a l f r a m e w o r k c a n also b e u s e d f o r t h e p a r t i c u l a r c a s e .

As a logical following to t h e d e t e r m i n a t i o n of a d e q u a t e f e e d b a c k c o n t r o l s t r u c tures,

Chapter

structural techniques.

6 c o n s i d e r s t h e p r o b l e m of t h e

constraints.

It p r o v i d e s

synthesis

of f e e d b a c k

an o v e r v i e w of a p p r o p r i a t e

C o n s i d e r a t i o n s on the r o b u s t n e s s

gains under

near-optimal design

o f s u c h c o n t r o l l e r s a r e also i n c l u d e d ,

in t h e s e n s e t h a t u n c e r t a i n t i e s d u e to p a r a m e t e r v a r i a t i o n s o r e x t e r n a l d i s t u r b a n c e s are considered.

T h e e f f e c t s of s t r u c t u r a l

t u a t o r s , line c u t s . . . )

Chapter robustness.

7,

perturbations

( f a i l u r e of s e n s o r s

ac-

are studied later.

and

the

last

one,

approaches

indeed

the

problem

of

structural

The chapter extends the results concerning decentralized or structurally

c o n s t r a i n e d c o n t r o l s y s t e m s to s y s t e m s s u b j e c t e d to s t r u c t u r a l f a i l u r e s of s e n s o r s

or

actuators

S e v e r a l w a y s to c o n c l u d e on t h e r o b u s t n e s s are presented.

perturbations,

mamely

or c u t s of l i n e s i m p l e m e n t i n g f e e d b a c k - l o o p s .

n o t i o n s of s t r u c t u r a l l y r o b u s t c o n t r o l a n d s t r u c t u r a l l y r o b u s t structure,

or

of a c o n t r o l ,

The

modes are introduced.

knowing its prespecifled

T h e y a p p e a r a s a n e x t e n s i o n of w e l l - k n o w n r e s u l t s d e r i v e d

from f i x e d m o d e s c h a r a c t e r i z a t i o n s . At l a s t , a g r a p h - t h e o r e t i c

algorithm is presented

to d e t e r m i n e t h e i n f o r m a t i o n p a t t e r n o f a r o b u s t r e g u l a t o r w i t h minimum c o s t . Through ples

an the book,

which make easier

collection of p a c k a g e s book

their

e v e r y r e s u l t is i l l u s t r a t e d b y small s i g n i f i c a t i v e e x a m understanding.

corresponding

( e v a l u a t i o n of f i x e d m o d e s ,

trained structure

Moreover,

the

appendices

contain

to some i m p o r t a n t a l g o r i t h m s p r e s e n t e d

d e t e r m i n a t i o n of t h e i r

feedback matrices...).

type,

a

in t h e

c a l c u l a t i o n of c o n s -

Vlll T h i s b o o k i s t h e c o n s e q u e n c e o f an i n t e n s i v e r e s e a r c h a c t i v i t y of s e v e r a l y e a r s in t h e a r e a o f a n a l y s i s a n d c o n t r o l o f l a r g e s c a l e s y s t e m s a n d more p a r t i c u l a r l y in d e c e n t r a l i z e d c o n t r o l in t h e

Lahoratoire

d'Automatique et

d'Analyse des

Syst~mes

(LAAS). T h e a u t h o r s would like to e x p r e s s t h e i r g r a t i t u d e to all t h e c o l l e a g u e s of t h e i r r e s e a r c h g r o u p a n d to t h e D i r e c t o r of t h e LAAS, P r o f e s s o r A.

COSTES,

for their

scientific and financial s u p p o r t . The a u t h o r s s i n c e r e l y t h a n k Miss C.

FABRE f o r t y p i n g

all t h e s e p a g e s a n d

Mr. E. LAPEYRE-MESTRE for t h e d r a w i n g s of t h i s b o o k ,

Toulouse, January 1989

Louise T R A V E Andr~ TITLI Ahmed TARRAS

TABLE

OF

CONTENTS

INTRODUCTION

CHAPTER

1. C E N T R A L I Z E D

CONTROL

: STABILIZATION

AND

POLE

ASSIGNMENT

I.I. - Introduction 1.2. - Controllahility a n d observablllty

Stability

1 . 2 . I.

-

1.2.2.

-

Controllability

1.2.3.

-

Observability

1.2.4.

- Kalman's canonical form

1.2.5. - Practical importance of the concepts of controllability a n d B

observability

8

1.2.6.

-

Stabilization a n d pole a s s i g n m e n t

1.2.7.

-

Origins of uncontrollable a n d u n o b s e r v a b l e

10

modes

14

1.3. - Structural controllability a n d observability

15

I. 3. I. - Structural controllability 1.3.2. - General results o n structural controllability a n d observability

19

I. 3.3. - Computational

20

considerations

31

I. 4. - Conclusion

CHAPITER POLE

2. S T R U C T U R A L L Y

CONSTRAINED

CONTROL

• STABILIZATION

AND

ASSIGNMENT

2.1. - Introduction

33

2.2. - Decentralized structural constraints

34

2.2. I. - P r o b l e m

formulation

35

2.2.2. - Decentralized fixed m o d e s

37

2.2.3. - Decentralized stabilization a n d pole a s s i g n m e n t

39

X 2.3. - A r b i t r a r y

50

structural constraints

53

2.4. - Evaluation of fixed m o d e s

2.4.1.

- By

the spectra

comparing

of t h e o p e n - l o o p

and closed-loop dynamic

matrix

53

2.4.2.

- By

calculation of the s y s t e m

2.4.3.

- Concluding

modes

sensitivity

remarks

60

2. S. - C o n c l u s i o n

CHAPITER

61

3. C H A R A C T E R I Z A T I O N

OF

FIXED

MODES

3. I. - Introduction

62

3.2. - Characterization

in t e r m s of transmission

3.2.1.

- T a r o c k ' s results

3.2.2.

- Hu

and

66

and

67

Wiswanadham

70

results

71

- Seraji's results

3.2.5. - D a v i s o n 3.2.6.

63

zeros

Jiang results

B.2.3. - V i d y a s a g a r 3.2.4.

and

Wang

72

results

75

- Comments

3.3. - Algebraic characterizations

3.3.1.

- Matrix rank

3.3.2.

- Recursive

3.3.3.

- Particular cases

75

: time d o m a i n

75

test characterization

80

characterization

83 87

3.3.4. - C o m m e n t s

3.4. - A l g e b r a i c

3.4.1.

characterizations

- Necessary

- Transfer distinct

3.5.

: frequency

88

function matrix for 88

of f i x e d m o d e s

function matrix characterization

for systems with 91

poles

3.4.3.

- Polynomial matrix rank

3,4.4.

- General transfer

3.4.5.

- Interpretation

- Structurally

domain

conditions on the transfer

the existence 3.4.2,

54

fixed modes

test characterization

function matrix characterization

94 95 102

104

XI 3.5.1,

- Preliminaries

3,5.2,

- Controllability

104

information 3,5.3,

-

3.5.4.

- Evaluation

observability

under

decentralized

111

structure

Characterization

of structurally

of structurally

structural 3.5.5.

and

sensitivity

fixed

fixed

of the

modes

modes

modes

by

of the

I19

calculation

of the

system

130

- Comments

135 137

3.6. - Graph-theoretic characterization of fixed m o d e s

3.6.1.

- Preliminaries

3.6.2,

- Frequency

3.6.3.

- Time

3.6.4.

- Comments

137 domain

domain

graph-theoretic

graph-theoretic

137

characterization

characterization

141 147

3.7. - Conclusion

CHAPTER

FIXED

4,1.

- Introduction

4.2.

-

4.3.

- Use

4.5.

STABILIZATION

4. D E C E N T R A L I Z E D

STRUCTURALLY

4.4.

148

Sample

IN P R E S E N C E

OF

NON

MODES

150

and

hold

of time-varying

T52 controllers

156

4.3.1. - Piecewise constant f e e d b a c k laws

156

4.3.2. - Sinusoidal f e e d b a c k laws

158

4.3.3. - C o n c l u d i n g r e m a r k s

160

- Vibrational

161

control

4.4. i. - Vibrational control principle

161

4.4.2. - Stabilization b y vibrational control

167

4.4.3. - Vibrational f e e d b a c k control laws

169

175

- Conclusion

CHAPTER

5. C H O I C E

OF

FEEDBACK

CONTROL

STRUCTURE

TO

AVOID

FIXED

MODES

5.1.

- Introduction

177

Xll 5.2.

5.3.

5.4.

- Relaxing

prespecified

5.2.1.

-

Preliminaries

5.2.2.

-

llang

5.2.3.

- Armentano

and

feedback

and based

5.2.5. - S p e c i f i e d

approach

- Choice

of minimal

179

procedure

Singh'

5.2.4. - A p p r o a c h

- Concluding

178 178

Davison'

5.2.6.

constraints

182

procedure

on the for

system

modes

structurally

186

sensitivity

fixed

modes

of type

(i)

197

remarks

control

190

197

structures

5.3.1.

- Preliminaries

5.3.2.

- Senning's a p p r o a c h

198

5.3.3.

- Locatelli

202

5.3.4.

-

5.3.5.

- Concluding

197

et al.

Specified

approach

approaches

for

structurally

fixed

modes

205

remarks

227

- Conclusion

CHAPTER

227

6. D E S I G N

6.1.

- Introduction

6.2.

- The

TECHNIQUES

- PARAMETRIC

ROBUSTNESS

229

optimization

6.2.1.

- Dynamic

6.2.2.

- Static

6.2.3.

- Necessary

problem

229

controllers

230

controllers

231

conditions

for optimality

- Gradient

matrix

233

calculation

6.3.

6.4.

- Decentralized

control

with

parameter

6.3.1.

- The

algorithm

of Geromel

6.3.2.

- The

algorithm

of Jamshidi

optimization

and

236

Bernussou

236 241

6.3.3.

- Iterative

procedure

of Chen

6.3.4.

- Iterative

procedure

of Geromel

et al.

6.3.5.

- Comments

- Design

of robust

6.4.1.

- Controllers

6.4.2.

- Optimal

242

and

Peres

245 247

decentralized

with

control

controllers

a prescribed with

247

degree

performance

of stability

index

sensitivity

248

reduction

249

XlII 6.4.3. - R o b u s t

control with respect to large perturbations

i n the

system dynamics

6.5. - R o b u s t

255

decentralized s e r v o m e c h a n i s m

260

problem

6.5.1. - P r o b l e m formulation

260

6.5.2. - Existence of a solution

262

6.5.3. - R o b u s t

264

decentralized controller design

267

6.5.4. - Sequentially stable robust controller design 6.5.5. - R o b u s t

decentralized controller for u n k n o w n

systems

270

273

6.6. - Decentralized control via hierarchical calculation

6.6.1. - Three-ievel calculation algorithms

273

6.6.2. - Two-level calculation algorithm

279

6.7. - Calculation m e t h o d s using a n interconnection m o d e l

6.7. I. - T h e

general interconnecfion

282 282

model

285

6.7.2. - Model-following m e t h o d

6.8. - Decentralized control for s y s t e m s with overlapping

6.8.1. - E x p a n s i o n ,

contraction,

6.8.2. - O v e r l a p p i n g

information set

a n d inclusion

289

decomposition

291

295

6.9. - Conclusion

CliAPTER

289

7. S T R U C T U R A L

ROBUSTNESS

7.1.

- Introduction

7.2.

- Structural

perturbations

affecting

the system

297

7.3.

- Structural

perturbations

affecting

the control system

299

296

7.3.1. - S t r u c t u r a l

perturbations

7.3.2.

- Structural

robustness

7.3.3.

- Characterization

7.3.4.

- Example

7.3.5.

- Structurally

characterization

299 303

of structurally

robust

modes

304 310

robust

information pattern

control design

- T h e c h o i c e of t h e 313

XIV 7.4.

- Conclusion

318

APPENDIX

I, Multivariable system zeros

319

APPENDIX

2. A Fortran subroutine to evaluate the fixed modes using open-loop

321

and closed loop system poles

APPENDIX

3. A Fortran routine to evaluate the fixed modes using their sensitivity 330

A P P E N D I X 4. A n d e r s o n

and Clements'

A P P E N D I X 5. D e t e r m i n a t i o n using variations

A P P E N D I X 6, A F o r t r a n

test package

of the gradient

for real modes

m a t r i x of t h e p e r f o r m a n c e

340 index by

calculus

routine

with possible robustness

346

to d e t e r m i n e requirements

an optimal constrained

feedback

matrix 349

REFERENCES

368

AUTHOR

379

SUBJECT

INDEX

INDEX

382

I

CHAPTER

CENTRALIZED

STABILIZATION

CONTROL

AND

POLE

:

ASSIGNMENT

I.I. - I N T R O D U C T I O N

The

g e n e r a l i n t r o d u c t i o n p o i n t e d out

control p r o b l e m s are

characterized by

that,

very

structurally

often,

large

scale systems

constrained feedback patterns.

Before t a k i n g i n t o a c c o u n t t h e s e new r e q u i r e m e n t s , t h i s c h a p t e r p r e s e n t s an o v e r view of t h e w e l l - k n o w n r e s u l t s c o n c e r n e d b y t h e p r o b l e m of s t a b i l i z a t i o n a n d pole a s s i g n m e n t of a l i n e a r t i m e - i n v a r i a n t dynamic s y s t e m s u b j e c t e d to c e n t r a l i z e d c o n t r o l (no s t r u c t u r a l c o n s t r a i n t s ) .

The f u n d a m e n t a l c o n c e p t s of c o n t r o l l a b i l i t y a n d o b s e r -

vability a r e i n t r o d u c e d a n d e x t e n d e d to t h e c o n c e p t s o f s t r u c t u r a l c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y which a r e

of a major p r a c t i c a l i n t e r e s t

in t h e

study

of l a r g e

scale

s y s t e m s . I n d e e d , t h e s e p r o p e r t i e s a r e e s t a b l i s h e d from t h e s y s t e m s t r u c t u r e a n d do not d e p e n d on t h e p a r t i c u l a r c o n f i g u r a t i o n of t h e p a r a m e t e r s ' v a l u e s .

In t h i s f r a -

mework, t h e p r o b l e m r e d u c e s to one of b i n a r y n a t u r e t h a t allows f o r t h e a p p l i c a t i o n of g r a p h - t h e o r e t i c c o n c e p t s .

This

a p p r o a c h is t h u s

especially adequate

for

large

scale systems.

T h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s for t h e e x i s t e n c e o f a s o l u t i o n to t h e problem of s t a b i l i z a t i o n a n d

pole

assignment are

presented

for t h e

following two

cases :

-

centralized state feedback

- centralized output feedback T h e y a r e s t a t e d i n t e r m s of t h e c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y p r o p e r t i e s o f the system. It is c l e a r t h a t a good u n d e r s t a n d i n g

of t h e c o n c e p t s of c o n t r o l l a b i l i t y a n d

o b s e r v a b i l i t y is i n d i s p e n s a b l e b e f o r e p r o c e e d i n g to t h e s t u d y of t h e a b o v e p r o b l e m with s t r u c t u r a l c o n s t r a i n t s on t h e c o n t r o l .

2 1.2. - C O N T R O L L A B I L I T Y

AND

OBSERVABILITY

(FOS-77)

(KAI-80)

C o n s i d e r t h e l i n e a r t i m e - i n v a r i a n t dynamic s y s t e m d e s c r i b e d b y t h e following s t a t e - s p a c e model : x(t) = A x(t) + B u(t)

y(t)

= c x(t)

where x ( t ) ~

(1.2.1)

R n, u(t) 6 R m a n d y(t) ~ R r are the state, input and output vectors

respectively, and A, B and C are invariant matrices of appropriate dimensions.

1 . 2 . 1 . - S t a b i l i t y (WIL-70) Definition I . I .

The a u t o n o m o u s s y s t e m ( 1 . 2 . 1 )

a n y g i v e n v a l u e e > 0, t h e r e e x i s t s a n u m b e r

]l X (to)]l (

($I ==> ]Ix(t) ] < E:

(i.e. ~l(e,

f o r all

with u ( t )

= 0) is s t a b l e if for

t0) > 0 s u c h t h a t :

t>t 0

The autonomous s y s t e m ( 1 . 2 . 1 ) is aymptotically s t a b l e if : (i) - it is s t a b l e

(ii)-~

x

(to),

x(t)

~ 0 t -==b0o

It is w e l l - k n o w n t h a t t h e s o l u t i o n of t h e e q u a t i o n s

(1.2.1)

with u ( t ) =

0 is

given by : x(t) = eA(t-t0 ) x 0 Given

{X1,

....

Xn}

t h e s e t of e i g e n v a l u e s o f A, s y s t e m ( 1 . 2 . 1 ) with u ( t ) = 0 is

a s y m p t o t i c a l l y s t a b l e if a n d o n l y if all t h e e i g e n v a l u e s of A h a v e a n e g a t i v e r e a l part. In the opposite case, the state space X can be split into the stable subspace X S which is generated b y the set of eigenvectors associated with the stable eigenvalues and the unstable subspace X U which is generated b y the set of eigenvectors

associated with the

unstable

c o n v e r g e s t o w a r d z e r o . For

eigenvalues.

For

x ( t 0) £ X S,

the system response

x ( t 0) ~ X U, the s y s t e m r e s p o n s e d i v e r g e s .

3 I. 2.2. - Controllability

Definition 1.2. A state x I is said to be controllable at time t O if for every initial state x 0 defined at time t0, there exists a control u(t) that transfers the system from the state x 0 to the state x I in a finite time If every said to be equivalent The are stated

state of the system that the pair

necessary

and

1.1.

The

system

following conditions

holds

(1.2.1)

generate 2. such

AB

-

conditions

Note that

is of rank

n

products n),

(1.2.1)

to b e c o n t r o l l a b l e

if a n d

only if either

of the

two

The columns of the controllability

matrix

:

rank

criterion <w i ,

~C = n . (KAI-80).

There

bj> a r e n o n z e r o ,

exists

j £

(1 . . . . .

m}

w h e r e bj i s t h e j t h c o l u m n

of A.

i s n o t U m i n i m a l ' . More o f t e n t h a n n o t ,

i t will t u r n

:

AB . . . . . for

system

are the left eigenvectors

KahnanWs c r i t e r i o n

out that the matrix

#C = (B,

....

for

is controllable

(KAL-62).

Popov-Belevitch-Hantus

(i=l,

is

this is

An-IB)

. . . . .

all t h e s c a l a r

of B and wi,

the system

(1.2.1),

:

a space of dimension n, i.e.,

that

t o may be, For system

:

1. - KalmanWs c r i t e r i o n ~C = ( B ,

controllable.

( A , B) i s c o n t r o l l a b l e .

sufficient

in the following theorem

Theorem

is controllable whatever

"completely controllable ~ or just to s t a t i n g

(tl-t0).

somev

A~-IB) less

than

n.

The

smallest

such

x),

say

~c'

will t h e n

be

called the controllability index.

Popov-Belevitch-Hantus may be restated The system

Rank

where

criterion

may be more convenient

in some cases since it

in the following form • (1.2.1)

(~I-A B) = n

is controllable if and only if :

V ~ E

o (A)

o (.) denotes the set of eigenvalues of (.).

(1.2.2)

In t h i s new formj t h i s c r i t e r i o n i n t r o d u c e s t h e d e f i n i t i o n o f a c o n t r o l l a b l e pole ( e i g e n v a i u e of A) as a pole f o r w h i c h c o n d i t i o n ( 1 . 2 . 2 ) h o l d s . The c o m p o n e n t s of e v e r y s t a t e x ~ X of t h e s y s t e m can b e p a r t i t i o n e d s u c h t h a t : x = x c ~ Xun c w h e r e x c G X C a n d Xun c ~ XUN C. X C (XuN C) is t h e c o n t r o l l a b l e ( u n c o n t r o l l a b l e ) s u b s p a c e g e n e r a t e d b y t h e e i g e n v e c t o r s a s s o c i a t e d to the controllable

(uncontrollable)

e i g e n v a i u e s o f A.

It

can t h e n b e s h o w n t h a t t h e

e q u a t i o n s ( 1 . 2 . 1 ) can t a k e t h e form :

L unc.j

[:llu

A22

k uncd w h e r e i t a p p e a r s t h a t t h e c o m p o n e n t s o f XUN C a r e n o t c o n n e c t e d to t h e i n p u t . From t h i s p o i n t of viewj P o p o v - B e l e v i t c h - H a n t u s c r i t e r i o n g i v e s a d e e p i n s i g h t i n t o t h e c o n t r o l l a b i l i t y p r o p e r t i e s of t h e s y s t e m .

K a h n a n ' s c r i t e r i o n is s u i t a b l e for

c h e c k i n g t h e global c o n t r o I l a b i l i t y o n l y .

1.2.3.

-

Observabllity

Definition 1 . 3 .

A s t a t e x ( t 0 ) = x 0 is said to b e o b s e r v a b l e at time t O if it can b e

d e t e r m i n e d from t h e k n o w l e d g e of t h e i n p u t u ( t ) a n d of t h e o u t p u t y ( t ) o v e r a finite i n t e r v a l of time ( t 0 , t 1) . I f e v e r y s t a t e of t h e s y s t e m is o b s e r v a b l e w h a t e v e r t o may b e , t h e s y s t e m i s s a i d to be "completely o b s e r v a b l e n o r j u s t o b s e r v a b l e .

For system (1.2.1),

t h i s is

e q u i v a l e n t to s t a t i n g t h a t t h e p a i r ( C , A ) is o b s e r v a b l e . The n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s for s y s t e m

(1.2.1)

to b e o b s e r v a b l e

a r e s t a t e d in t h e following t h e o r e m ." T h e o r e m 1.2.

T h e system ( 1 . 2 . 1 )

is o b s e r v a b l e if a n d o n l y i f e i t h e r

of t h e

following c o n d i t i o n s h o l d s : 1. - K a h n a n ' s c r i t e r i o n ( K A L - 6 2 ) . T h e r o w s of t h e o b s e r v a b i L i t y m a t r i x :

two

~D - C A

n-1

generate

a space of dimension n, i.e.,

2.

- Popov-Belevitch-Hantus

such that and vi,

criterion

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

(i=l,

ooo, n ) ,

¢O = n . (KAI-80).

There

a r e n o n z e r o ,

are the right

As for controllability,

rank

eigenvectors

i t will g e n e r a l l y

turn

exists

j ~ {1 . . . . .

r}

w h e r e cj i s t h e j t h r o w o f C

of A.

out that the matrix

:

EJ CA

~0

=

A v-

i s of r a n k

n

for

some ~ less

called the observability

than

n.

The

smallest

such

x~, s a y ~ , will t h e n 0

be

index.

In the observability

case,

Popov-Belevitch-Hantus

criterion

may be restated

in

the following form :

The system

(1.2.1)

is observable

if and only if :

IXI-A] = n

and an observable

Xun °

where

(unobservable) (unobservable) observability

of every

xo

subspace

E

state

XO and generated

eigenvalues puts

(1.2.3)

pole is defined as a pole for which condition

The components x0 ~

~4X~c (A)

equations

of A. (1.2.1)

of the system X u n ° ~"

can be partitioned

XUN O .

XO

by the eigenvectors The

(1.2.3)

decomposition

( X u N O) associated

of the

in the following form :

is

holds. such

that

x =

the

observable

to t h e

observable

system

with regard

to

:o]

rail

•u n ° /

LA21

y

[Oo] [] Xun

o

+

A22

= [C1

u

B2

O] [Xo0 Xun]

w h e r e it is clear t h a t t h e c o m p o n e n t s of XUN O a r e n o t c o n n e c t e d to t h e o u t p u t . The

obvious

analogy

between

theorems

between the concepts of controllability and

1.1

and

1.2 p o i n t s out

observability.

the

Two s y s t e m s a r e

duality called

dual if t h e y a r e d e f i n e d r e s p e c t i v e l y b y t h e e q u a t i o n s : = A x + B u

[x* =

A' x* + C' u*

S* :

S • y

C x

~y*

= B' x*

T h e s e s y s t e m s a r e s u c h t h a t , if S is c o n t r o l l a b l e , S* is o b s e r v a b l e a n d vice v e r s a . It is t h u s

p o s s i b l e to c h e c k t h e o b s e r v a b i l i t y of a s y s t e m b y e x a m i n i n g t h e c o n -

t r o l l a b i l i t y of t h e dual s y s t e m .

1.2.4. - K a l m a n ' s c a n o n i c a l form

(KAL-62)

In view of p a r a g r a p h s 1 . 2 . 2

a n d 1 . 2 . 3 , it follows t h a t t h e s t a t e - s p a c e X can

be decomposed into four s u b s p a c e s such that :

X = X1 • X2 • X3 • X4

where :

X 1 = X C n XUN O Xz = XC n XO x 3 = XUN C

n XUN O

X 4 = XUNC

n XO

(controllable and unobservable subspace) (controllable and observable subspace) (uncontrollable and unobservable subspace) (uncontrollable and observable subspace)

Kalman (KAL-62) s h o w e d t h a t t h e r e e x i s t s a r e a l , r e g u l a r t r a n s f o r m a t i o n m a t r i x s u c h that the system (1.2.1)

can be p u t in t h e following canonical form •

x2 x3

y

jail =

0 o 0

=[o

]

AI2

A13

A22

0

A2~ /

0

A33

A3~"[

o

0

A44J

C2

[Xl] x2

x3

B2

+

(1.2.4)

x4

I

x I x2 x 3

x4]

i l l u s t r a t e d b y f i g u r e 1.1 :

w

Fig. 1.1.

: C a n o n i c a l d e c o m p o s i t i o n of a l i n e a r t i m e - i n v a r i a n t s y s t e m

S t a r t i n g from t h e c a n o n i c a l f o r m , t h e t r a n s f e r

f u n c t i o n m a t r i x of t h e s y s t e m

is :

Y(p) W(p) = U(p) = C2 [pl- A22 ]-1 B2

(p : Laplace v a r i a b l e )

in w h i c h o n l y t h e s i m u l t a n e o u s l y c o n t r o l l a b l e a n d o b s e r v a b l e poles a r e p r e s e n t . Note t h a t t h e poles of t h e s y s t e m c o r r e s p o n d i n g to t h e e i g e n v a l u e s of A l l , a n d A44 ( t h e n o n s i m u l t a n e o u s l y u n c o n t r o l l a b l e a n d u n o b s e r v a b l e p o l e s ) condition :

rank

= n

easily d e r i v e d from t h e P o p o v - B e l e v i t c h - H a n t u s c r i t e r i o n ( 1 . 2 . 2 )

A22

verify the

and (1.2.3).

8 1 . 2 . 5 . - Practical importance of the c o n c e p t s of controllability a n d o b s e r v a b i l i t y (FOS-77) It is n o w i n t e r e s t i n g to e x a m i n e t h e c o n s e q u e n c e s o f t h e e x i s t e n c e of u n c o n t r o l l a b l e a n d u n o b s e r v a b l e m o d e s on t h e b e h a v i o u r of t h e s y s t e m .

T h e s e few follo-

wing r e m a r k s c o n s i d e r several cases and point out the practical importance of the c o n c e p t s of c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y .

Remark 1.1.

As s h o w n i n p a r a g r a p h

to t h e i n p u t .

p e n d e n t l y of t h e c o n t r o l i n p u t , Its

1 . 2 . 3 , a n u n c o n t r o l l a b l e mode i s n o t c o n n e c t e d

T h e r e s p o n s e a s s o c i a t e d w i t h s u c h mode will t h u s e v o l v e in time i n d e -

e v o l u t i o n will d e p e n d

w h e t h e r in o p e n - l o o p o r

closed-loop configuration.

o n l y on t h e mode d y n a m i c s a n d t h e c o r r e s p o n d i n g i n i t i a l

conditions.

Remark 1.2.

C o n s i d e r t h a t a n u n c o n t r o l l a b l e mode is u n s t a b l e .

t h e u n s t a b i l i t y will a p p e a r at t h e o u t p u t a n d will t h u s the

fact

that

the

If it is o b s e r v a b l e ,

be detectable.

Nevertheless,

mode is u n c o n t r o l l a b l e e x c l u d e s all p o s s i b i l i t y of s t a b i l i z i n g t h e

s y s t e m . What is r e q u i r e d is n o t a c o n t r o l law b u t a m o d i f i c a t i o n of t h e s y s t e m s t r u c ture.

Remark 1.3.

C o n s i d e r now t h e c a s e f o r w h i c h a n u n s t a b l e mode is n o t o b s e r v a b l e .

T h e u n s t a b l e d y n a m i c s of t h i s mode will n o t a p p e a r on t h e o u t p u t , s e e n in p a r a g r a p h e

s i n c e we h a v e

1 . 2 . 4 t h a t u n o b s e r v a b l e m o d e s a r e n o t c o n n e c t e d to t h e o u t p u t .

T h e s y s t e m m a y t h u s be o b s e r v e d a s s t a b l e . N e v e r t h e l e s s p t h e i n t e r n a l u n s t a b i l i t y of t h e s y s t e m may come to e i t h e r a b r e a k - u p

o f t h e s y s t e m o r t h e a p p e a r e n c e of a n o n

l i n e a r f u n c t i o n ( s a t u r a t i o n ) so t h a t t h e l i n e a r model is n o l o n g e r v a l i d . These

remarks

strength

the

importance

of

having

criteria

which

allow

the

d e t e c t i o n of u n c o n t r o l l a b l e a n d u n o b s e r v a b l e m o d e s w h e n c o n s i d e r i n g s y s t e m c o n t r o l

problems.

1.2.6. - Stabilization and pole assignment The

problem

system (1.2.1)

of

(i.e.,

stabilization

is

(BRA-70)

formulated

(WON-67) as

follows :

given

h a v i n g some p o l e s w i t h p o s i t i v e r e a l p a r t s ) ,

the

unstable

find a controller

o f t h e form : u = K y

(1.2.5)

such that the closed-loop system : x ( t ) = (A + B K C ) is s t a b I e ;

i.e.

(1.2.6)

x(t)

e v e r y e i g e n v a l u e of t h e c l o s e d - l o o p d y n a m i c m a t r i x (A + BKC) h a s a

negative re~ part.

1,2.6.a.

- State feedback control

First consider the case for which every

s t a t e of t h e s y s t e m

(1.2.1)

can b e

measured, what can be expressed by : C = I n ( I d e n t i t y m a t r i x of o r d e r n x n ) y=x T h e f e e d b a c k c o n t r o l t h e n t a k e s t h e form : u = K x

System

(1.2.1)

(1.2.7)

is s t a b i l i z a b l e u s i n g s u c h a c o n t r o l law if a n d o n l y i f t h e u n s t a b l e

s u b s p a c e X U ( s e e § 1 . 2 . 1 ) is i n c l u d e d in t h e c o n t r o l l a b l e s u b s p a c e X C ( s e e § 1 . 2 . 2 ) and every

pole of ( 1 . 2 . 1 )

can be arbitrarily

c o n t r o l l a b l e . T h i s is c l e a r l y u n d e r s t a n d a b l e However,

more o f t e n t h a n n o t ,

assigned

if a n d o n l y if

(1.2.1)

is

from t h e d e f i n i t i o n of c o n t r o l l a b i l l t y .

t h e s t a t e s a r e n o t d i r e c t l y a v a i l a b l e from t h e

measurements and additional conditions are required.

1.2.6.b.

- Output feedback control

With a c o n t r o l law of t h e (1.2.1)

(see

~2 ~3

form ( 1 . 2 . 5 ) ,

u s i n g t h e Kalman's c a n o n i c a l form of

1. Z. 4), t h e c l o s e d - l o o p s y s t e m is d e s c r i b e d b y : All

A12+BIKC 2

AI3

AI4+BIKC4 -

x1

o

A22 +B2K C 2

0

A24 +B2K C 4

x~

0

0

A23

A34

x3

0

o

0

A##

x~

I t is t h u s a p p a r e n t t h a t t h e c l o s e d - l o o p s y s t e m is s t a b l e if a n d o n l y if :

(i.z.8)

10 (i) t h e u n s t a b l e s u b s p a c e X U is i n c l u d e d in t h e c o n t r o l l a b l e s u b s p a c e X C a n d in t h e o b s e r v a b l e subspace X O ~ i . e . p

Stated in a different way,

the

e i g e n v a l u e s of A l l ,

A33 a n d

A44 a r e

stable.

this is equivalent to have all the unstable poles con-

trollable and observable. (ii) there exists a matrix K such that (A22 + B z K C z) is stable, what is expressed by : F2 = K C 2

and

rank (C2, F 2) = rank C 2

where F 2 is a stabilizing state feedback for the second subsystem that always exists since the second subsystem is controllable. W h e n considering arbitrary pole assignment, condition (i) is replaced by ." (i*) the system (1.2.1) is controllable and observable. When

condition (ii) cannot be verified, a dynamic output feedback control of

the form :

{

~(t) = S z(t) + R y(t)

(1.2.9)

u(t) = Q z(t) + K y(t) + v(t)

is required. Then, condition (i) is sufficient (and necessary) to stabilize the system (1.2.1) and arbitrary pole assignment is possible if and only if (1.2.1) trollable and observable (BRA-70)

showed

(condition

is con-

(i*)). In this latter case, Brasch and Pearson

that the minimal order of the required dynamic

achieve pole assignment is :min ( Vc-1 , ~o-1), where

x)c and

compensator to

~o are the controlla-

bility and observability indices, respectively.

1.2.7. - Origins of uncontrollable and unobservable modes In

order

to

show

the

mechanism

of

uncontrollability

and

unobservability,

c o n s i d e r t h e following simple example of a s i n g l e - v a r i a b l e s y s t e m t a k e n from (FOS-

77) : -2

0

-2

3

0 x +

I

l

1

1

1

0

0

(1.2.10) y

0

0

x

11 that can be r e p r e s e n t e d by t h e b l o c k - d i a g r a m of f i g u r e 1.2 :

xI

q +J

x2

x4

1.2.

Figure

Using conditions (1.2.2) and ( 1 . 2 . 3 ) , it can be easily c h e c k e d t h a t t h e mode ~1=1 is u n c o n t r o l l a b l e and ~2=-1 is u n o b s e r v a b l e :

o

0 3

o o

o

-2-1 3

-I

-1

0

0

0

-I

-1

o

2 rank [M-A

B]kl=l

:

rank

rank[1i-A

= rank

=-I

4

o

0

1

o

o

1

-1

-2

o

-1

-1

o

2

0

o

0,5

0.5

o 0 -

o

o

=3

<4

o

=3<4

This system is composed of two s u b s y s t e m s in cascade and can be r e w r i t t e n in the e q u i v a l e n t form of f i g u r e 1.3.

:

p+l

(p+~) (p-l)

(p+l) (p+2) F i g u r e 1.3.

~_4~ Y

12 where

it a p p e a r s

that

the

uncontroUability

cancellation of a pole of one transfer

and

the

unobservahiltty

result

from the

function by a zero of the other and vice versa.

In fact, if the cancellation is of the form : 1 - "zero upstream,

pole downstream",

2 - npole upstreamt

z e r o d o w n s t r e a m n, i t i n d u c e s

From this not appear

result,

it follows that

it induces

uncontrollability unobservability

the uncontrollable

as poles of the global transfer

a s f o r ~1=1. as for

and unobservable

function of the system

~,2=-1.

m o d e s do

:

l

Indeed, respect

the

uncontrollable

to t h e i n p u t - o u t p u t

and

relation

unobservable

since the

input and the second ones are disconnected It

is

clear

parameters'

that

this

situation

arises

first

¢ on the parameters

from

a31 a n d a 3 2 ,

the

special

of

two

with

from the

configuration

(1.2.10).

0

of

Introducing

the dynamic matrix becomes

-2

configuration

"transparent"

disconnected

from the output.

A

The

are

are

v a l u e s in t h e m a t r i c e s o f t h e m o d e l o f t h e s y s t e m

perturbation

figure

modes ones

the a

:

0

°°° 1

subsystems

in

1+¢

1

o

1

0

-3

cascade

is

1.4 :

x2

I p" I Figure

1.4

preserved

as

shown

in

13 or

e q u i v a l e n t l y in f i g u r e 1.5 :

u.__.~

(p+l)(p+2)

]

-"

[

(p+3)(p-l)

F i g u r e 1.5 In f i g u r e

1.5,

it is clear that

the p o l e - z e r o cancellation g i v i n g r i s e to the

u n o b s e r v a b i l i t y no l o n g e r o c c u r s for ~ ¢ 0. T h e r e f o r e ,

this u n o b s e r v a b i l i t y can be

avoided b y c h a n g i n g t h e v a l u e s of some a d e q u a t e components of the s y s t e m , C o n s i d e r now the case for which a31 and r e f l e c t i n g the i n t e r n a l i n t e r c o n n e c t i o n s t r u c t u r e entries

cannot be p e r t u r b e d

by

any

change

a41 in ( 1 . 2 . I 0 ) a r e fixed z e r o s , of t h e system ( i . e . , t h e s e zero

of the

components

values).

In

this

situation, the system is r e p r e s e n t e d b y f i g u r e s 1.6 and 1.7 below :

xI

x#

2 F i g u r e 1.6.

p+l ~ (p+3)(p- l)

u.l,+

Y

F i g u r e 1.7. and its t r a n s f e r function is :

Y=

where the pole ~2---1 is a b s e n t .

B(p+l) (p+2)(p+3)(p-- 1)

This r e s u l t is not s u r p r i s i n g since F i g u r e 1.6 shows

that the block - 2 / p + l is d i s c o n n e c t e d from the o u t p u t . s e r v a b l e mode.

X2=-I is t h e r e f o r e an u n o b -

14

The i n t e r e s t i n g p o i n t o f t h i s d i s c u s s i o n is t h a t t h e pole %2=-1 will n o t a p p e a r in t h e t r a n s f e r f u n c t i o n a n d will r e m a i n u n o b s e r v a b l e h o w e v e r t h e p a r a m e t e r s o f t h e system are changed.

This s i t u a t i o n a r i s e s from t h e i n t e r c o n n e c t l o n s t r u c t u r e o f t h e

s y s t e m . S u c h a pole is called s t r u c t u r a l l y u n o b s e r v a b l e . A similar d i s c u s s i o n could be

d o n e f o r u n c o n t r o l l a b i l l t y coming to t h e

c o n c e p t of s t r u c t u r a l l ) r

uncontroIlable

pole s . T h e s e c o n c e p t s p r e s e n t a g r e a t p h y s i c a l i n t e r e s t . I n d e e d , p r a c t i c a l l y , most of t h e e n t r i e s o f A, B a n d C in

(1.2.1)

are

k n o w n w i t h t h e a p p r o x i m a t i o n o f some

e r r o r s o f m e a s u r e m e n t s (it is e s p e c i a l l y t r u e f o r l a r g e scale s y s t e m s ) . Only some of t h e s e e n t r i e s a r e k n o w n w i t h 100 p e r c e n t p r e c i s i o n : t h i s h a p p e n s f o r t h e e n t r i e s that

are

equal

to

zero and

reflect

the

internal

interconnection structure

of t h e

s y s t e m . The c o n c e p t s of s t r u c t u r a l c o n t r o l l a b i l i t y a n d s t r u c t u r a l o h s e r v a b i l i t y t h e n makes t h e m e a n i n g of c o n t r o l l a b i l i t y and o b s e r v a b i l i t y more complete from t h e p h y sical p o i n t o f v i e w . T h e following p a r a g r a p h perties and

presents

t h e mathematical formulation o f t h e s e p r o -

some r e s u l t s e x p r e s s i n g t h e n e c e s s a r y a n d

s u f f i c i e n t c o n d i t i o n s for a

s y s t e m to b e s t r u c t u r a l l y c o n t r o l l a b l e a n d o b s e r v a b l e .

1.3. - S T R U C T U R A L

CONTROLLABILITY

These concepts were

AND

OBSERVABILITY

first introduced by

p r o v i d e t h e c o n d i t i o n s for s t r u c t u r a l

Lin in

1974

controllability (and by

(LIN-74).

His r e s u l t s

duality for s t r u c t u r a l

o b s e r v a b i l i t y ) of s i n g l e - v a r i a b l e s y s t e m s in a g r a p h - t h e o r e t i c a p p r o a c h .

They were

e x t e n d e d to m u l t i - v a r i a b l e s y s t e m s b y Shield a n d P e a r s o n in 1976 (SHI-76) b u t in a purely algebraic approach.

I n t h e same y e a r ,

Glover a n d Silverman (GLO-76) p r e -

s e n t e d a simple a l g e b r a i c solution a n d d e r i v e d a r e c u r s i v e algorithm f o r d e t e r m i n i n g structural

controllability

which

utilizes

Boolean

operations

only.

presents these results and establishes the equivalence between the

This

paragraph

graph-theoretic

and the algebraic approaches. C o n s i d e r t h e l i n e a r t i m e - i n v a r i a n t dynamic s y s t e m in t h e form :

{

~(t) = A x(t) + B u(t) y(t) C x(t)

(1.3.1)

where x C R n, u C R m and y ~ R r. A, B, and C are invariant matrices of appropriate dimensions in which the entries are either fixed zeros of undeterminate (arbitrary). Such a system is called a structured system and A, B and C are structured matrices (SHI-76). It is worth noting that the model class of systems.

(1.3.1) refers therefore to a

15 Definition

1.4.

The

triple

e q u i v a l e n t to) t h e t r i p l e

(~,

~,

~)

has

the

same

structure

as

(is s t r u c t u r a l l ~

(C, A, B) of t h e same d i m e n s i o n s if for e v e r y

e n t r y of t h e m a t r i x ( ~ ~ ~ ) ,

fixed zero

t h e c o r r e s p o n d i n g e n t r y of t h e m a t r i x (C A B) is also

a fixed zero, and vice versa. O b v i o u s l y , if t h e t r i p l e ( ~ , X , l~) is s t r u c t u r a l l y A, B ) ,

then the pairs

(~,

~) and

(~, g)

e q u i v a l e n t to t h e t r i p l e (C,

are structurally

e q u i v a l e n t to t h e p a i r s

(C, A) a n d (A, B ) , a n d t h e i n v e r s e i s also t r u e . Definition 1.5.

A pair

(A,

B)

((C,

A)) is s t r u c t u r a l l y

t h e r e e x i s t s a p a i r e q u i v a l e n t to (A, B) ( ( C ,

controllable

(observable)

if

A)) w h i c h is c o n t r o l l a b l e ( o b s e r v a b l e )

in t h e u s u a l s e n s e . 1.3.1.

-

1.3.1.a.

Structural controllability - A l g e b r a i c a p p r o a c h (SHI-76)

Definition 1 . 6 .

T h e m a t r i x (A B) h a s form I if t h e r e e x i s t s a p e r m u t a t i o n m a t r i x P

satisfying :

P' [A B]

F: 0] EAI 0 I

with A l l o f o r d e r t x t ,

1 .~ t ( n .

A21

A22

0] B2

We a s s u m e t h a t B 2 h a s a t l e a s t o n e n o n z e r o e l e -

merit. T h e m a t r i x (A B) h a s form II if : g r ( A B) ( n where gr (.)

d e n o t e s t h e g e n e r i c r a n k 1 of ( . ) .

1 C o n s i d e r a s t r u c t u r e d m a t r i x 1~ w i t h ~ arbitrary entries. Then the parameter s p a c e RV is a s s o c i a t e d w i t h ~ s u c h t h a t e v e r y d a t a p o i n t d ~ R v d e f i n e s a m a t r i x M=~(d). C o n v e r s e l y , a s t r u c t u r e d m a t r i x M is a s s o c i a t e d with e v e r y m a t r i x M s u c h t h a t M=~(d) for d ~ R V. T h e g e n e r i c r a n k of M o r of M is d e f i n e d as follows : g r ( M ) = g r ( ~ ) = m a x { r a n k ~ (d) } dew

16

T h e o r e m 1.3

(SHI-76).

The pair

( A , B) i s s t r u c t u r a l l y

controllable if and only if the

two f o l l o w i n g c o n d i t i o n s b o t h h o l d : 1 - t h e m a t r i x (A B) i s n o t o f f o r m I. 2 - t h e m a t r i x (A B) i s n o t o f f o r m I I ,

1.3.1.b.

- Graph-theoretic

The

structured

approach

nature

of the

(LIN-74) system

(1.2.5)

naturally

comes to

associate

a

digraph. A digraph edges

is a pair D=(V,E),

( v j , v i) d i r e c t e d

w h e r e V is a s e t o f v e r t i c e s

from t h e v e r t e x

vj to t h e v e r t e x

a n d E is a s e t o f

v i , I f ( v j , v I) E; E t h e n vj

is s a i d to b e a d i a c e n t to v i a n d v i a d j a c e n t from v j . T h e a d j a c e n c y r e l a t i o n d e f i n e s a square

binary

m a t r i x R=(~-ij) c a l l e d t h e a d i a c e n c y

a n d o n l y if ( ~ ,

vi)•

E0 A s e q u e n c e o f e d g e s

w h e r e all t h e

vertices

cides with Vl,

then

are distinct,

the path

(or interconnection)

{ ( v 1, v 2 ) ,

is called a path

is a cycle.

from v 1 to v k .

A digraph

m a t r i x ~..=11] i f

( v 2, v 3) . . . . .

(Vk_ 1, v k ) } When v k c o i n -

D is s a i d to b e

a c y c l i c i f it

contains no cycles.

If t h e r e i s a p a t h f r o m v i t o v i , t h e n we s a y t h a t v i is r e a c h a b l e

from vj.

a subset

vertex every

Similarly,

V£

c

V is r e a c h a b l e

from a subset

in V. i s r e a c h a b l e f r o m some v e r t e x in V . . We s a y t h a t V. 1

vertex

lity relation

,]

Vj c

J

c

V if every

V reaches

V. i f 1

i n V. r e a c h e s a t l e a s t o n e v e r t e x in V . . Like a d j a c e n c y , t h e r e a c h a b i ] 1 o n V c a n b e r e p r e s e n t e d b y a b i n a r y m a t r i x R=(r_.j)~ s u c h t h a t r .1j. = l i f

a n d o n l y if v. i s r e a c h a b l e f r o m v . . 1 j A pair of vertices c h a b l e from e a c h o t h e r .

of

D a r e s a i d to b e s t r o n g l y

A maximal s u b g r a p h

strongly connected is called a strong The

strong

components

components D1,...,DN, (i=l . . . . . k ) , performed

of

connected

if t h e y

are rea-

o f D in w h i c h e v e r y p a i r o f v e r t i c e s a r e

c o m p o n e n t o f D.

D are

uniquely

they can be ordered

i s r e a c h a b l e from a n y o t h e r

determined.

If

D has

N strong

s u c h t h a t f o r some 1 ~ k ~ N, n o n e Di ,

Dj, j>i. I f t h e c o r r e s p o n d i n g

o n t h e a d j a c e n c y m a t r i x o f D, i t a p p e a r s

permutation is

t h e n in a b l o c k - t r i a n g u l a r

form

where every block in the diagonal is the adjacency matrix of a strong component.

Let M be a rectangular associated adding

to M a s t h e

(q-p)

zero rows.

pxq structured

digraph

matrix and define the digraph

whose adjacency

m a t r i x is M', o b t a i n e d

D=(V,E)

from M by

17 Definition 1 . 7 .

A d i g r a p h D=(V,E)

c o n t a i n s a dilation if a n d only if t h e r e e x i s t s a

s e t S c V I (V I c V is t h e s e t of v e r t i c e s w h i c h a r e a d j a c e n t from at l e a s t o n e v e r t e x in V) of K v e r t i c e s s u c h t h a t t h e s e t T ( S ) c V

o f v e r t i c e s t h a t a r e a d j a c e n t to a v e r t e x

of S c o n t a i n s no more t h a n ( K - I ) e l e m e n t s . The dilation is d e n o t e d b y { S , T ( S ) } . This

d e f i n i t i o n i s a g e n e r a l i z a t i o n of t h e

c o n c e p t of dilation i n t r o d u c e d b y

(LIN-74) in t h e c o n t e x t of s t r u c t u r a l a n a l y s i s for s i n g l e - v a r i a b l e s y s t e m s . In p r a c t i c e ,

D c o n t a i n s a dilation if a s e t of K rows can be f o u n d in t h e

a d j a c e n c y m a t r i x of D s u c h t h a t t h e r e a r e no more t h a n ( K - l ) columns w i t h n o n z e r o e n t r i e s in t h e s u b m a t r i x formed b y t h e s e K r o w s . Given

the

triple

(C,A,B),

one

defines

{U= U l , . . . , u m} i s t h e s e t of i n p u t v e r t i c e s ,

a

d i g r a p h F = ( U v X u Y,E)

X={Xl,...,Xn}iS

where

t h e s e t of s t a t e v e r -

tices a n d Y={~/1 . . . . y r } is t h e s e t o f o u t p u t v e r t i c e s . E i s t h e s e t o f e d g e s s u c h t h a t (u i,

xj)£E

if a n d

only i f bji#0 i n

B , ( x i , x j) E: E i f a n d only if aji#0 in

A and

( x i , Y j ) ( E if a n d only if cji#0 i n C. In t h i s w a y , t h e d i g r a p h r c o m p l e t e l y d e s c r i b e s t h e s t r u c t u r e o f t h e s y s t e m . Definition 1.8 ( S I L - 7 8 ) . The s y s t e m ( C , A , B ) is s a i d to b e i n p u t r e a c h a b l e (or i n p u t c o n n e c t a b l e (DAV-77a)) if X is r e a c h a b l e from U a n d o u t p u t r e a c h a b l e

(or o u t p u t

c o n n e c t a b l e (DAV-77a)) if X r e a c h e s Y. A l t h o u g h Lints work (LIN-74) d e a l t only with s i n g l e - i n p u t s y s t e m s , it can b e d i r e c t l y e x t e n d e d to m u l t i - i n p u t s y s t e m s .

Consider the pair (A,B) and its associated digraph rl=(U v X,E I) obtained from r by deleting the set of output vertices and every edge from X to Y. Then, we h a v e t h e following r e s u l t : Theorem 1 . 4 .

The

pair

(A,B)

is s t r u c t u r a l l y

c o n t r o l l a b l e if and only if t h e

two

following c o n d i t i o n s b o t h hold : 1 - (A,B) is input reachable. 2 - F1 d o e s n o t c o n t a i n a d i l a t i o n .

1 . 3 . 1 . c . - E q u i v a l e n c e of t h e two a p p r o a c h e s It is now w o r t h v e r i f y i n g t h a t t h e o r e m s 1.3 a n d 1.4 a r e e q u i v a l e n t . C o n s i d e r t h e f i r s t p a r t s of t h e t h e o r e m s .

18

If matrix

(A B)

has

form I,

by

definition 1.6,

there

matrix P satisfying :

P' [A B]

with A l l o f o r d e r t x t ,

a

permutation

o]

=

i

exists

A22

LA21

B2

1 ~ t (n.

T h e f i r s t e q u a t i o n in ( 1 . 3 . 1 ) is t h u s r e w r i t e d v i t h a p a r t i t i o n e d s t a t e v e c t o r :

I17 [ °]Ix,][ °] =

)(2

+

A21

A22

X2

U

132

a n d t h e g r a p h a s s o c i a t e d with t h e p a i r ( A , B ) in t h i s form is g i v e n in F i g u r e 1.8.

F i g u r e 1.8. It is c l e a r t h a t t h e s t a t e v e r t i c e s in X 1 a r e n o t i n p u t r e a c h a b l e . The i n v e r s e also h o l d s : if t h e g r a p h of a p a i r

(A,B)

contains non-input-reachable state ver-

t i c e s , t h e m a t r i x (A B) can b e b r o u g h t to form I. T h e f i r s t p a r t s of t h e t h e o r e m s 1.3 a n d 1.4 a r e t h e r e f o r e e q u i v a l e n t . To p o i n t o u t t h e e q u i v a l e n c e of t h e s e c o n d p a r t s o f t h e t h e o r e m s , we n e e d t h e following r e s u l t due to Shield a n d P e a r s o n ( S H I - 7 6 ) . Theorem 1 . 5 .

(SHI-76).

Assume a r e c t a n g u l a r m a t r i x M of o r d e r p x q with p ~ q .

gr(M) ( t for some t , 1 ~ t ~ p , if for some k in t h e r a n g e q - t < k ~ q , M c o n t a i n s a z e r o s u b m a t r i x of o r d e r ( p + q - t - k + l ) x k . Theorem

1.5

following c o r o l l a r y :

a p p l i e d to M=(A B)

of o r d e r

nx(n+m)

obviously

leads

to t h e

lg Corollary

1.1.

m( k ~ n+m,

: gr(A

B)

Making the change gr(A

B)

submatrix

(n

(i.e,

(A B)

has

(A B) c o n t a i n s a z e r o s u b m a t r i x

: K=n+m-k+l,


of order

the digraph

of variables

Kx(n+m-K+l),

associated

range

(A,B)

if f o r s o m e k i n t h e

1 ~

contains

range

(n+m-k+l)xk.

corollary

1.1 becomes

K
which is equivalent,

with the pair

Figure 1.9 with an example

f o r m II)

of order

(A B)

contains

from definition

a dilation.

:

1.8,

a zero to h a v e

This is illustrated

by

: ¢.q.t..~K + 1

E

m

o

(

~

I

"/iv

o o iil

o

o

o

o

X

X

ii X

i

S = { x 1, x 2 , x 3 } =) K=3

T(S)=(u,

x 1} Figure

1.3.2.

- General results As far

study

as the

on structural

dual concept

controllability

of structural

could be done and the general

1.3.2.a.

1.9

results

Theorem 1.6.

The system

(1.2.5)

is concerned,

a similar

in the following theorems

represented

1 - ( A , B ) i s n o t o£ f o r m I. B)=n

(i.e,

=n ( i . e ,

by

the triple

(C,A,B)

:

is structurally

if a n d o n l y if t h e f o l l o w i n g c o n d i t i o n s h o l d :

2 - (A I, C I) i s n o t o f f o r m I.

4- gr

observability

are stated

- Algebraic approach

controllable and observable

3 - gr(A

and observability

(A B) i s n o t o f f o r m I I ) .

(A ~ C t) i s n o t o f f o r m I I ) .

20

1 . 3 . 2 .b. - Graph-theoretic Theorem

1.7. T h e

system

approach

(1.2.5)

controllable a n d observable

represented

by

i - (A,B)

is input reachable.

2 - (A,C)

is output reachable.

3 - The

digraph

associated with the pair (A,B)

4 - The

digraph

associated with the pair ( M ,

1.3.3.

- Computational

the triple

(C,A,B)

if a n d only if the following conditions hold

is structurally :

does not contain a dilation. G') does not contain a dilation.

considerations

T h e most p r a c t i c a l way to c h e c k s t r u c t u r a l

c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y of

a given system (C,A,B)

is to combine t h e a l g e b r a i c a n d g r a p h - t h e o r e t i c

Indeed,

output

the

input

and

reachability

can be easily tested

approaches.

a n d on t h e o t h e r

h a n d , a lot of e f f i c i e n t a l g o r i t h m s e x i s t to d e t e r m i n e t h e g e n e r i c r a n k of a m a t r i x .

l. 3.3. a. -Input and output reachability conditions T h e a d j a c e n c y m a t r i x of t h e d i g r a p h a s s o c i a t e d with ( C , A , B )

h a s t h e following

form : X

U

Y

o

o

U

C

o

y

(1.3.z)

:

where ~,B and ~ are obtained by replacing every undeterminate (arbitrary)

entry in

A , B a n d C b y 1. The reachability matrix,

w h i c h h a s t h e same d i m e n s i o n as ~ a b o v e is g i v e n

by :

I R =

X

U

Y

E

F

0

X

0

o

0

U

G

H

o

Y

(1.3.3)

21 R can be computed directly from ~ as s h o w n in (SIL-78). By the (SIL-78) output

;

sole observation

(CjA~B)

reachable

is

of R j

input

if a n d

- Generic rank

A number generic

rank

((SHI-76),

of authors

Among the (PRE-81)

the determination an

nxm

have

no zero columns j and

suggested

matrix.

no

zero

conditions rows,

it i s

it i s i n p u t - o u t p u t

rea-

we m e n t i o n

rank

the

were

the

one b y

of "nontaking

of

the

finding to

Prescott

(LIU-68).

of an nxm structured

zeros

for

shown

be

the

wrong

problem.

mathematics

of the maximum number where

algorithms

algorithms

a tricky

on combinatorial

of the generic

chessboard,

alternative

Several

revealing

algorithms,

based

that the evaluation

reachahiltty

if F h a s

zero rows nor zero columns.

(DAV-77a)),

efficient

(JAM-83)

on the

only

conditions

of a structured

(BUR-81),

if a n d

o n l y if G h a s

c h a b l e if a n d o n l y if H h a s n e i t h e r

1.3.3.b,

we c a n c o n c l u d e

reachable

matrix

and

Pearson

basic

idea is

matrix is equivalent

roots" are

The

to

that can be placed on

interpreted

as

forbidden

by Johnston

et al.

positions. Another

interesting

84) t h a t i s b a s e d matrix

is

disjoint rows

first

blocks

than

on the detection

brought appear

columns

simple procedure

algorithm was presented back

in

to

the

indicate

in a systematic a triangular

diagonal.

a possible

The rank

in ( E V A - 8 4 ) .

e q u a l to t h e

The algorithm

which

appropriate

t h a t we p r e s e n t

is specially reordering

suitable

that

have

deficiency,

fact that

of the

a greater

which

is

The initial

where

different number

determined

of

by

a

entries.

the

nonzero

refering

generic

diagonal

rank

of a matrix is

rank

of a matrix is

blocks

(see the next section)

to large

the global generic

to s m a l l e r d i m e n s i o n e d

way by

in ( J O H - 8 4 ) .

here

for application

of the matrix,

out in a sequential

of t h e a l g o r i t h m

blocks

form

to c o m p u t e t h e g e n e r i c

on the

dimensions

of any dilation.

(JOH-

constituting

the

matrix.

in t e r m s o f c o n d i t i o n s carried

It is based

sum of the

maximal permutation

86a)

manner

block-structure

examining the position of the nonzero

We m e n t i o n a l s o t h a t a n A P L r o u t i n e provided

recently

using,

at e a c h

scale systems.

rank

matrices.

step,

is the one in

(TRA-

Using

an

condition is expressed The checking

a slightly

is t h u s

modified version

22

1,3.3.c. - A sequential algorithm to conclude on structural controllability and observability of large scale systems (TRA-86a) When dealing with l a r g e scale s y s t e m s , c h e c k i n g t h e c o n d i t i o n s f o r s t r u c t u r a l c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y may c a u s e c o m p u t a t i o n a l d i f f i c u l t i e s a n d r e q u i r e h i g h calculation time. T h i s s e c t i o n p r e s e n t s t h e w o r k in (TRA-86a) which t a k e s a d v a n t a g e of t h e i n h e r e n t s t r u c t u r e of a c l a s s of l a r g e scale s y s t e m s which can be d e c o m p o s e d into hierarchically

ordered

interconnected irreducible

subsystems.

The

underlying

idea is to a c h i e v e computational a n d c o n c e p t u a l s i m p l i f i c a t i o n s b y s u b s e q u e n t l y s o l v i n g each subsystem.

This approach has been used to analyse systems stability in

(SEZ-81c) (MIC-78) and for estimators and controllers design in (PIC-83b). Although (PIC-83b) uses the property that each subsystem is structurally controllable and observable, the conditions guaranteing these properties have not been yet derived within this perspective.

This problem is considered in (TRA-86a)

and

sequential

conditions for structural controllability (observability) of the overall system

are

provided. A n iterative algorithm is proposed and its improvements with respect to an a l g o r i t h m u s i n g a global a p p r o a c h a r e d i s c u s s e d . An i l l u s t r a t i v e example is p r e s e n ted.

1. P r e l i m i n a r i e s In t h i s s e c t i o n , some new c o n c e p t s of g r a p h t h e o r y (TRA-86a) a r e i n t r o d u c e d , Definition 1.9. A G-dilation ( G e n e r a l i z e d dilation) i s d e f i n e d as a n y p a i r of s e t s { S c V ' , T ( S ) c V } s u c h t h a t e a c h v e r t e x of T ( S )

is a d j a c e n t to some v e r t e x o f S. The

o r d e r of a G-dilation is d = c a r d S - c a r d T(S)

w h e r e c a r d * d e n o t e s t h e n u m b e r of

s

e l e m e n t s in t h e s e t *. O b v i o u s l y , a dilation is a G - d i l a t i o n w h o s e o r d e r is p o s i t i v e . T h e n , it is c l e a r that

gr(M)=p

(full

generic rank)

if a n d

only if t h e

order

of all t h e

G-dilations

c o n t a i n e d in D i s - ~ 0 . We s a y t h a t a G - d i l a t i o n { S , T ( S ) }

of o r d e r

d s is maximal i f ,

for all p r o p e r

s u b s e t s L c ( V ' - S ) , t h e G-dilation {~¢=L,T(ScL)} is of o r d e r ~d s . T h e o r e m 1.8. M contains

( T R A - 8 6 a ) . I f t h e d i g r a p h D a s s o c i a t e d with a full g e n e r i c r a n k m a t r i x k

G-dilations {Si,T(Si)},

G-dilation of o r d e r 0 { ~ , T ( ~ ) } 0r i . e . ,

SiC fir T(Si) C T ( ~ ) ,

(i=l, . . . . k ) ,

of

order

0,

then

the

maximal

is u n i q u e a n d " i n c l u d e s " all t h e G - d i l a t i o n s of o r d e r

(i=l . . . . . k ) .

23

2. Seciuential c o n d i t i o n s for s t r u c t u r a l controllability C o n s i d e r the system

(1.2.1)

represented

b y the t r i p l e (C,A,B) and its a s s o -

ciated d i g r a p h F = ( U t K u Y , E) as d e f i n e d in section 1 . 3 . 1 . b . summarizes the r e s u l t s p r o v i d e d in (LIN-74) and (SHI-76) :

The following theorem

Theorem 1.9. The system (1.2.1) is s t r u c t u r a l l y controllable ( o b s e r v a b l e ) if and only if the two following conditions both hold : 1- ( A , B )

( ( C , A ) ) is i n p u t ( o u t p u t ) r e a c h a b l e (SIL-77)

2- gr(A B) = n (gr(A' C')= n)

From the computational p o i n t of v i e w , c h e c k i n g the above conditions r e q u i r e s the evaluation of the t e a c h a b i l i t y matrix of the d i g r a p h associated with ( A , B )

((C,

A)) and the d e t e r m i n a t i o n of the g e n e r i c r a n k of a matrix whose dimension is in the r a n g e of t h e o r d e r of the s y s t e m .

A l t h o u g h e f f i c i e n t algorithms a r e available

(see

(SIL-78) (BOW-76)) for t h e r e a c h a b i l i t y matrix and (JOH-84) (MOA-80) for t h e g e n e ric r a n k ) ,

some computational

difficulties may arise when dealing with l a r g e scale

systems. The s y s t e m is decomposed into t h o s e s u b s y s t e m s for which t h e s t a t e v a r i a b l e s c o r r e s p o n d to the s t r o n g components of F x = ( X , E x ) , o b t a i n e d from Yp and t h e c o r r e s p o n d i n g

edges.

If the s t r o n g

r b y deleting U,

components a r e a d e q u a t e l y o r d e r e d

and a f t e r t h e c o r r e s p o n d i n g p e r m u t a t i o n of the s t a t e v a r i a b l e s , system (1.2.1)

takes

the following form :

]A2L

A2~

Al ] I

~(t): / I:

",,,\,,

LA~, Y(t)=

where Aij ~ R n i x n j ,

[C l

B i E R nixm,

%,

C2 . . . .

x(t)+

2

LB~,J

u(t) (1.3.4)

Csl

Ci ~ R r x n i ,

(i,j=l . . . . . s t .

Elaborated p r o c e d u r e s

for

the d e t e r m i n a t i o n of the s t r o n g components of a d i g r a p h can be found in (HAR-65), (KAU -68).

24 2.i. - Reachabilit~, condition

Using

the above

decomposition,

the reachability condition of Theorem

1.9 is

very easily expressed at the level of each subsystem in (1.3.4).

Theorem 1.10 ( T R A - 8 6 a ) .

The s y s t e m (1.2.1)

(1.3.4)

is i n p u t r e a c h a b l e if and only

if : (B i Ail Ai2 . . .

(1.3.5)

(i=l ..... s)

Ai,i_ 1) ~[~ 0

This r e s u l t is s t r a i g h t f o r w a r d from the definition of s t r o n g components and the hierarchically

reordered

digraph

corresponding

o b s e r v a t i o n of the matrices in (1.3.4)

the

sole

allows to conclude on i n p u t r e a c h a b i l i t y .

to

(1.3.4).

Therefore,

The

dual r e s u l t for o u t p u t r e a c h a b i l i t y is o b t a i n e d b y r e p l a c i n g B i by C's_i+ 1 and Aij by A's_j+lps_i+ 1 in ( 1 . 3 . 5 ) .

2.2. - Generic r a n k condition Our main r e s u l t is p r e s e n t e d in this p a r a g r a p h .

C o n s i d e r t h e system

(1.3.4)

and define the matrices A*I and B*, (i=l . . . . . s) as follows :

A*I : A l l ' B * I : B I '

A~I']---

EAt1

[ --i Ai,i_ 1 t

....

s where A.*~Rni*xn~ and B.* C Rni*xm with n* = Y. i

I

j=l

I

B~i:

-

(1.3.6)

nj.

Theorem 1.11 ( T R A - 8 6 a ) . A n e c e s s a r y and s u f f i c i e n t condition to h a v e g r ( A B)=n is that gr

(A~ B~)

The problem

= n.*l for all (i=l . . . . . s) consists,

therefore,

(1.3.7) in

finding

the

conditions

to h a v e

gr(A* i

B~)=n~ assuming t h a t gr(A~_ 1 B~_ 1) =n~_ 1. Theorem 1.12 ( T R A - 8 6 a ) . Given the r e c t a n g u l a r p x q (p~q) matrix M and its a s s o c i a ted d i g r a p h D=(V,E), assume t h a t gr(M)=p and t h a t { ~ , T ( ~ ) )

is the maximal G-dila-

tion of o r d e r 0 in D. T h e n the n e c e s s a r y and s u f f i c i e n t condition to h a v e

25

gr

= gr (MN) =p+p', w h e r e

N1 N2 m a t r i x , is t h a t

:

I J M

gr

0

~'=V'-£

a n d ME

associated vertices

denotes

(1.3.8)

=p'+card ~"

T(~)

NT ( a )

where

N 1 is a p'xq matrix and N 2 i s a p ) x p )

N2

the submatrlx

c o m p o s e d b y t h e c o l u m n s of M w h o s e

do n o t b e l o n g to t h e s e t E.

T h i s s i t u a t i o n i s c l a r i f i e d w h e n t h e m a t r i x MN i s r e w r l t e d {£,T(~

)}

with respect

to

as follows -"

T( f~)

T(f~)

V2

I I

q£

V=T(~) u~(.~) u V2

I

MT(£)

V': £u~ u V2

I I

I

N

l~(a)t t

--

7(~)

T(n)

N1

I

N2

-i

V2

In o r d e r to a p p l y T h e o r e m 1.12 t o o u r s p e c i a l p r o b l e m ,

define

:

I'-'"-' o ] M l l = [ B I A l l ] , Mii = Ni,i-I

(1.3.9)

Nii

Ni,i_l=[Bi All Ai2 ... Ai,i_l], Nil = [A~I] Theorem

1.13

(TRA-86a).

The necessary

where A and B are given in (1.3.4),

and sufficient

is t h a t

conditions to have gr(A

B)=n

:

1 - gr[Mll]=nl

2 - gr

MT(g2 E-I) i-l,i-I

o

1 /

Ni,i-I

Nil

J

(1.3.10) :gr[F i]:ni +card"i_ l

(i=2,...,s)

26 where{~i_ 1, T ( ~ i _ l ) } Theorem

is t h e maximal G - d i l a t i o n of o r d e r 0 of Mi_l, i - l "

1.13 is

straightforward

1.12 to t h e m a t r i c e s Mii, ( i = l , . . . s ) ,

from t h e

successive

d e f i n e d in ( 1 . 3 . 9 ) .

a p p l i c a t i o n of T h e o r e m

The above result is streng-

t h e n e d b y t h e f a c t t h a t {ff2i, T ( ~ i ) } c a n also b e d e t e r m i n e d in a s e q u e n t i a l way. T h e o r e m 1.14 ( T R A - 8 6 a ) .

A s s u m e t h a t Mii h a s full g e n e r i c r a n k ,

G - d i l a t i o n of o r d e r 0 of Mii is { ~ i ' T (~i)} w i t h ~ i = ~i-1 u 6 i or(6i) w h e r e ( 6 i ' T ( d i ) } is t h e maximal G - d i l a t i o n of o r d e r

t h e n t h e maximal

a n d T ( ~ i ) = T ( ~ i _ 1) 0 of t h e m a t r i x F i

d e f i n e d in ( 1 . 3 . 1 0 ) . Dual r e s u l t s a r e easily o b t a i n e d for s t r u c t u r a l o b s e r v a b i l i t y b y r e p l a c i n g B i b y C's_i+ 1 a n d Aij b y A ' s _ j + l , s _ i + 1 in ( 1 . 3 . 9 ) . T h e a b o v e r e s u l t s p r o v i d e t h e b a s i s f o r an i t e r a t i v e a l g o r i t h m to c o n c l u d e o n structural

c o n t r o l l a b i l i t y ( o b s e r v a b i l i t y ) of l a r g e scale d y n a m i c a l s y s t e m s .

2.3. - Algorithm Since t h e

sequential

require any calculations, tions (1.3.10)

reachability

conditions

the subsequent

in T h e o r e m 1.13 o n l y .

(1.3.5)

in T h e o r e m

1.10 do n o t

a l g o r i t h m r e f e r s to t h e g e n e r i c r a n k c o n d i -

At e a c h s t e p i, we n e e d to v e r i f y w h e t h e r o r

n o t F., h a s a g e n e r i c r a n k d e f i c i e n c y a n d , if n o t , to d e t e r m i n e i t s maximal G - d i l a t i o n of o r d e r 0. T h i s c a n b e p e r f o r m e d b y u s i n g a s l i g h t l y modified v e r s i o n of t h e g e n e ric rank

algorithm in

(JOH-84).

The

first

step

is t h e

same as in

(JOH-84)

and

c o n s i s t s in a r e o r d e r i n g of t h e m a t r i x F i . Algorithm

REORDER

(M)

:

Step 1

i=0, j=0, d e l e t e t h e null c o l u m n s .

Step 2

F i n d t h e row with t h e minimal n u m b e r of n o n - d e l e t e d choice exists,

select

the

one with

entries

minimal n u m b e r of e n t r i e s

(say ~).

If

(deleted or

non-deleted). Step 3 Step 4

A s s o c i a t e t h i s row with i n d e x i a n d d e l e t e i t , i--i+l. A s s o c i a t e t h e columns w h i c h h a v e e n t r i e s i n t h i s row w i t h t h e i n d i c e s j + l , j+2 . . . . .

j + a a n d d e l e t e t h e m , j=j+a.

27 Step 5

If a n y row is l e f t , go to s t e p 2. O t h e r w i s e , w r i t e M o r d e r i n g t h e rows a c c o r d i n g to i n c r e a s i n g i n d i c e s from top to bottom a n d t h e columns a c c o r d i n g to i n c r e a s i n g i n d i c e s from l e f t to right, stop.

T h i s a l g o r i t h m p u t s t h e m a t r i x M in t h e following form :

I .....I-2-: R1

|

IRK Denote

the

number

of rows

and

columns

of

the

block

Ri

by

Pi

and

qi'

(i=l,... ,K). In a s e c o n d s t e p , t h e m a t r i x Fi is a n a l y z e d . If a dilation is d e t e c t e d , F i h a s a g e n e r i c r a n k d e f i c i e n c y . O t h e r w i s e , we d e t e r m i n e i t s maximal G-dilation of o r d e r 0. The p r o c e d u r e i s t h e same as i n {JOH-84), with t h e modification t h a t some r o w s a n d columns are memorized for t h e d e t e r m i n a t i o n of t h e maximal G-dilation o f o r d e r 0. Algorithm ANALYZE (M)

:

Step i

i=l.

Step 2

If p i - q i < 0 , go to s t e p 8. O t h e r w i s e , j = l , k=0.

Step 3

Tag

t h e qi+k f i r s t r o w s in Ri.

Also tag

all t h e

r o w s in

Ri w h i c h

have

e n t r i e s only in t h e same columns as t h e e n t r i e s of t h e t a g g e d r o w s . Step 4

Tag t h e columns w h i c h h a v e n o n z e r o e n t r i e s in t h e t a g g e d r o w s a n d

cal-

culate h i = n u m b e r of t a g g e d r o w s - n u m b e r of t a g g e d c o l u m n s . ] Step 5

If ~i > 0, M c o n t a i n s a d i l a t i o n , s t o p .

i

If 1~'=-0, memorize t h e t a g g e d r o w s a n d c o l u m n s . J Step 6

If all t h e r o w s a b o v e R i a r e t a g g e d , go to s t e p 7. The r o w s a b o v e

Ri w i t h e n t r i e s in t h e t a g g e d columns a r e now s c a n n e d .

Select t h e one with minimal n u m b e r of e n t r i e s in t a g g e d c o l u m n s . If choice e x i s t s , s e l e c t t h e one with minimal n u m b e r of e n t r i e s a n d tag i t , j=j+l, go to s t e p 4.

28 Step 7

If all t h e r o w s in t h e block Ri a r e t a g g e d , go to s t e p 8. O t h e r w i s e , d e l e t e all t h e t a g s , j=j+l, k = k + l .

Step 8

If i=K, s t o p , t h e maximal G - d i l a t i o n of o r d e r 0 of M i s g i v e n b y : : memorized r o w s

T(6)

. memorized columns

O t h e r w i s e , i----i+l, go to s t e p 2. A s s u m i n g t h a t (B A) h a s initially b e e n p u t in form ( 1 . 3 . 4 ) b y u s i n g a n y of t h e m e t h o d s p r o p o s e d in (SIL-78) (BOW-76), t h e global algorithm is t h e following : Step 1

i=l,

B = n u m b e r o f d i a g o n a l b l o c k s , M00=0.

Step 2

S u c c e s s i v e l y d e l e t e t h e r o w s with only one e n t r y

and

the

corresponding

columns. If no r o w is left, gr(B A)=n,

Step 3

stop.

If t h e row block i h a s b e e n d e l e t e d o r F. h a s only one row l e f t , go to s t e p 1

4. O t h e r w i s e , REORDER (Fi) , ANALYZE ( F i ) . If Fi c o n t a i n s a d i l a t i o n , g r ( A B ) < n , s t o p . Otherwise, Step 4

{6i, T( 6i)~

is r e t u r n e d b y t h e algorithm ANALYZE.

If i= B, g r ( A B ) = n , s t o p . Otherwise,

delete

the

columns

corresponding

to

T( 6i ) ,

delete

the

null

r o w s . Fi+ 1 i s o b t a i n e d b y a d d i n g t h e row b l o c k ( i + l ) . .Step 5

i=i+l, go to s t e p 2.

It is a well k n o w n r e s u l t t h a t a r e c t a n g u l a r p x q (p
{S 0,

T ( S 0) } d e f i n e s an i n d e p e n d e n t p e r m u t a t i o n

d e f i n e d b y S 0 h a v e n o n z e r o e n t r i e s only in the

submatrix.

Since t h e

~ columns d e f i n e d b y

g rows

T(S0),

the

n o n z e r o e n t r i e s of t h e final p e r m u t a t i o n m a t r i x (if any) a r e n e c e s s a r i l y l o c a t e d in t h e square

submatrix

d e f i n e d b y {S O,

T ( S 0) } .

It

is

then

clear

that

the

remaining

e n t r i e s in t h e columns T ( S 0) a r e n o t available a n y m o r e ( S t e p 4 ) . This

remark

has

also b e e n

used

to i m p r o v e

the

r e d u c t i o n of t h e whole matrix at e a c h i t e r a t i o n ( S t e p 2).

algorithm

by

a preliminar

29

2.4. - Example To i l l u s t r a t e form (1.3.4))

bI

b2

x x

F1

the algorithm,

described

by

1

let us consider

the system

(1.2.1)

8

II

(already

put in

(B A) a s f o l l o w s :

2

3

4

5

7

6

9

I0

12

13

14

ix

1

Ix

2 3 4

; Y/x× -

--'--'

.

.

.

.

×"

5

-'i

.

×

6

!

7

I

8

9

I I

x

xl

1!

Ix

x

I

,

This system

x

satisfies

x

the reachability

conditions

I

(1.3.5)

12

X

x IX

I

generic rank

10

X

of Theorem

i.I0.

algorithm is now applied.

* i=__!1, ~ = s . S t e p Z : R o w s 3, 4, 8, a n d c o l u m n s 2, 5, 9 a r e d e l e t e d .

* i--2 :

--

S t e p 3 : F 2 d o e s n o t n e e d to b e r e o r d e r e d .

ANALYZE F 2

( ~1={i,2}

--)

~T( Step 4 : These

rows and columns are deleted. b2

3

4

6

X

X

X

7

8

10

(B A) b e c o m e s 11

12

13

:

14 5 6 7 9 10 II

X X

12

X

X X

X

X ×

13 14

13 14

X

61)= { b1,1}

The

30 *

i=3

S t e p 2 : Rows 6, 7, 9, 10, 11 a n d columns b 2, 7, 8, 10, 11 are d e l e t e d . * i:5

:

(i=4 is j u m p e d s i n c e t h e row block 4 h a s b e e n d e l e t e d )

Step 3 : 3

~

6

X

X

X ,I

F5:

12 iI

13

3 5 REORD

X

I X I

18

12

X

13

12

4

X

l~ ~5= ( 12, 14}

I

13

X

X I

ix

a

6 12

x

S t e p 4 : Since i= 8 , s t o p : g r

14

X !

I

ANALYSE[F 5]

13 T(d5)={3,13} 5

I

(B A) = n .

T h e r e f o r e , t h i s s y s t e m is s t r u c t u r a l l y c o n t r o l l a b l e . Comments : It is c l e a r t h a t t h i s a l g o r i t h m d o e s n o t p r e s e n t a n y a d v a n t a g e f o r s t r o n g l y c o n n e c t e d s y s t e m s . Its a p p l i c a b i l i t y is t h e r e f o r e limited to l a r g e scale s y s tems w h i c h can be d e c o m p o s e d i n t o s e v e r a l h i e r a r c h i c a l l y o r d e r e d s u b s y s t e m s a n d it is i n t u i t i v e l y u n d e r s t a n d a b l e t h a t i t s e f f i c i e n c y i n c r e a s e s with t h e n u m b e r of s u b s y s t e m s . Some o t h e r a s p e c t s of t h e s y s t e m like t h e location of t h e dilations o r o f t h e i n d e p e n d e n t G - d i l a t i o n s of o r d e r 0 in (B A) m u s t also b e t a k e n i n t o c o n s i d e r a t i o n . It r e s u l t s t h a t t h e e f f i c i e n c y of t h e a l g o r i t h m may v a r y f o r e a c h p a r t i c u l a r c a s e so t h a t it is d i f f i c u l t to c a r r y out a n y m e a n i n g f u l q u a n t i t a t i v e c o m p a r i s o n w i t h o t h e r a l g o r i t h m s u s i n g a global a p p r o a c h . H o w e v e r , a q u a l i t a t i v e d i s c u s s i o n is p o s s i b l e . The a l g o r i t h m p r e s e n t e d h e r e m u s t be c o n s i d e r e d as a whole f o r t h e s t r u c t u r a l c o n t r o l l a b i l i t y p r o b l e m . If t h e g e n e r i c r a n k alone is of i m p o r t a n c e , t h e n a p p l y i n g t h e a l g o r i t h m in (JOH-84) to t h e global m a t r i x may b e a good c h o i c e . Solving t h e

structural

controllability problem without a decomposition scheme

requires : - T h e d e t e r m i n a t i o n of t h e r e a c h a b i l i t y m a t r i x of t h e

d i g r a p h of t h e

system

w h i c h is e q u i v a l e n t to t h e p r o b l e m of f i n d i n g all t h e p a t h s in t h e d i g r a p h of t h e s y s t e m . I f t h i s l a t t e r p r o b l e m is s o l v e d , it is c l e a r t h a t t h e s t r o n g c o m p o n e n t s a r e determined. Therefore, putting the system (1.2.1)

in form ( 1 . 3 . 4 )

i s an e q u i v a l e n t

t h e d e t e r m i n a t i o n of t h e g e n e r i c r a n k of (B A) b y u s i n g ,

for example, the

problem.

-

a l g o r i t h m of (5OH-84). T h e n , t h e f i r s t s t e p c o n s i s t s in r e o r d e r i n g (B A) i n t o a row

31 echelon form

(algorithm

REORDER),

which can

be

a

heavy

task

for

large

scale

s y s t e m s . This s t e p is a v o i d e d b y o u r algorithm w h i c h u s e s REORDER at t h e s u b s y s tem level. M o r e o v e r , w h e r e a s t h e algorithm in (JOH-84) c a r r i e s out t h e calculations on t h e whole r e o r d e r e d m a t r i x ,

t h i s algorithm p r o c e e d s to a r e d u c t i o n of (B A) at

each s t e p ( S t e p 2), w h i c h can c o n s i d e r a b l y r e d u c e t h e t a s k (see t h e e x a m p l e ) . O v e r t h e s c o p e of t h i s s e c t i o n , t h e algorithm may b e a d a p t e d for t h e s e q u e n tial d e t e r m i n a t i o n of minimal e s s e n t i a l i n p u t

(output)

s e t s which a r e f u n d a m e n t a l in

the p r o b l e m of d e t e r m i n i n g t h e minimal f e e d b a c k p a t t e r n s avoiding s t r u c t u r a l l y f i x e d modes (TRA-86b)

1.4.

-

( s e e C h a p t e r V, Section 5 . 3 . 4 b . 3 ) .

CONCLUSION This c h a p t e r p r o v i d e s an o v e r v i e w o f t h e i m p o r t a n t c o n c e p t s of c o n t r o l l a b i l i t y

and o b s e r v a b i l i t y t h a t play a f u n d a m e n t a l role in t h e p r o b l e m of s t a b i l i z a t i o n and pole a s s i g n m e n t of l i n e a r t i m e - i n v a r i a n t dynamic s y s t e m s . The

s o l u t i o n of t h i s

analysis of t h e

origins

p r o b l e m is e x p r e s s e d

of u n c o n t r o l l a b l e

and

in t e r m s of t h e s e

unobservable

concepts.

modes p o i n t s

i n t e r e s t for e x t e n d i n g t h e s e c o n c e p t s into a s t r u c t u r a l f r a m e w o r k .

out

An the

Structural con-

troUability a n d o b s e r v a b i l i t y a r e t h u s i n t r o d u c e d a n d a p p e a r as s t r o n g e r p r o p e r t i e s than c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y in t h e u s u a l s e n s e .

Using t h e s e c o n c e p t s , t h e

above p r o b l e m t u r n s to w h i c h of g e n e r i c s t a b i l i z a t i o n a n d g e n e r i c pole a s s i g n m e n t . The s t r u c t u r a l p r o p e r t i e s are e s t a b l i s h e d from t h e s t r u c t u r e o f t h e s y s t e m w i t h o u t any c o n s i d e r a t i o n of t h e p a r a m e t e r s ' v a l u e s of s y s t e m s .

Therefore,

this

approach

:

t h e y are in f a c t c o n c e r n e d w i t h c l a s s e s

allows d e a l i n g

with s y s t e m s w h o s e n o n z e r o

p a r a m e t e r s a r e k n o w n with i n c e r t a i n t i e s . It is w o r t h n o t i n g t h a t t h i s s i t u a t i o n o c c u r s f r e q u e n t l y in l a r g e scale s y s t e m s . The s t u d y

of t h e s e p r o p e r t i e s can be a c h i e v e d with a p u r e l y

p r o a c h or with a g r a p h - t h e o r e t i c a p p r o a c h .

s t r u c t u r a l c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y is b y Thus,

algebraic ap-

N e v e r t h e l e s s , t h e e a s i e s t way to c h e c k combinating the

two a p p r o a c h e s .

t h e c o n d i t i o n s to be e x a m i n e d involve b o t h t e a c h a b i l i t y of t h e g r a p h a s s o c i a -

t e d with t h e s y s t e m a n d g e n e r i c r a n k d e t e r m i n a t i o n . In s e c t i o n

1.3.3.c,

t h e a l g o r i t h m of (TRA-86a)

troUability a n d o b s e r v a b i l i t y is p r e s e n t e d . large scale s y s t e m s .

Indeed,

for c h e c k i n g s t r u c t u r a l c o n -

It is s p e c i a l l y s u i t a b l e f o r a p p l i c a t i o n in

b y u s i n g an a p p r o p r i a t e

decomposition of t h e s y s t e m

t h e global c o n d i t i o n s a r e e x p r e s s e d in t e r m s of s e v e r a l c o n d i t i o n s r e f e r i n g to smaller dimensioned s y s t e m s ( T R A - 8 6 a ) . tial way.

The c h e c k i n g can t h u s b e c a r r i e d o u t in a s e q u e n -

32 In the investigated require

next in

structural

chapter,

the

context

of large

constraints

We will s e e t h a t the introduction

the problem

the fact of taking control.

systems

and

pole assignment

whose

characteristics

these

new constraints

will b e generally

on the control.

of new concepts

the case of centralized

of s t a b i l i z a t i o n

scale

into account

which appear

as

an extension

of those

y i e l d s to defined

in

CHAPTER

STRUCTURALLY

2

CONSTRAINED

STABILIZATION

AND

POLE

CONTROL

•

ASSIGNMENT

2. I. - I N T R O D U C T I O N

In high

the

control

of

dimensionality,

reasonable assumption system

is

large

conventional

computational

efforts.

of a centralized available

For most large geographical

economical and

systems

techniques

whose fail

to

pattern

center,

control. particular

system

this centralization the

of

problems

information.

related

the design of feedback

When no transfer this yields

to

a

pairs

(those of minimum cost information pattern.

for

These

station 1

the

position,

2.1.a

with on the

on where

the all

example) situations

are

allowed

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

station i

- Decentralized

leads

This that

part

requires

to

implies are only

of the whole

restrictions

on the

can connect.

the different

are illustrated

transfer.

to g e n e r a t e

controllers

of control.

constraint

o f local c o n t r o l l e r s

in o r d e r

between

scheme

does not hold due

new

When

local s t a t i o n s

some but

we o b t a i n by Figure

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

not

structurally-constrained

is allowed, all t r a n s f e r s

a nonstandard 2.1 :

station S

L A R G E SCALE SYSTEM

Figure

stands

information

assumption This

that the controller

of information

decentralized

all

their

solutions

generally

to t h e i n f o r m a t i o n

of the whole information

output-input

reasonable

is

out.

scale systems,

In particular,

characteristic

a geographical

distribution

reliability part

give

~ i.e.,

generally

that the control system should he made of a number allowed to u s e

essential

The classical control theory

information

at a given

the calculations can be carried

to t h e

scale

control

reduced

34

l

i yll

[. . . .

station 1

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

Controller 1

Controlleri I. . . . station i

ControllerS [

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

station S

L A R G E SCALE SYSTEM

Figure

2.1.b.

It is clear vantageous

that

: system

which observe

for the

the

decentralized

since no transfer

is required tions),

- Example of arbitrarily

problem

centralized

inputs

second

paragraph

general

case of arbitrarily

of this

scheme

is the

are

to a

assigned and

refering

These

set

l o c a t i o n to a n o t h e r

o f local c o n t r o l l e r s

T h i s is t h e r e a s o n

pole assignment

The results chapter.

given

outputs.

results

the first

were investigated

to t h i s

structurally-constrained

control

most economically ad-

from one geographical

o n l y local s y s t e m

scheme.

control

of information

of stabilization

control

structurally-constrained

study

were

then

(stastudies

within

a de-

are presented extended

in the

to t h e

control which is considered

more in the

third paragraph.

2.2. - D E C E N T R A L I Z E D

Consider

a large

model. As pointed partitioning

The

scale

CONSTRAINTS

system

represented

o u t in t h e i n t r o d u c t i o n ,

of the

tion stations. particular

STRUCTURAL

control

and

reflect

a linear time-invariant

its geographical

the observation

models retained

by

this

in several situation

dynamic

decomposition results

in a

local c o n t r o l a n d o b s e r v a -

and

appear

in t h e

following

forms :

I-Frequency-domain model

Yl

"Wl I(P)

W I S(p)-

I I

I I I I I I

YS

u,! 1 i I

I I I !

wsl(P)

Wss(P)

uS

(2.2.1)

35 2-Time-domaln model

I

x(t) = A x(t)

+

S Z i=l

LYi(t) = C i x ( t ) where

x(t)

uiE:Rmi

~

Rn

(2.2.2)

(i=1 . . . . .

is

the

a n d Yi C R r i

observation station, R e m a r k 2.1.

Bi u i ( t ) S)

state.

The

input

are the input

specified

output

vectors

vectors

Note t h a t t h e s t a t e v e c t o r is n o t p a r t i t i o n e d ,

subsystems. for

are

partitioned

:

of the ith control and

(i=l . . . . . S t .

t h e s y s t e m i t s e l f is t a k e n a s a w h o l e I i . e . , results

and

and the output

The

following

"interconnected

systems"

reflecting

it is n o t d e c o m p o s e d

results

hold

will b e

in

this

presented

as

general

the

fact that

into several wellframework.

a particular

case

The in

a

further paragraph. Let u s d e f i n e : S m ~ iEI= m i

c, =

B = [131, ..., B S]

S r = i~l= I

,

.....

ri

%1

(2.2.37 y'

=

such that system

{

x(t)

u'= [u~ ,..., ~,]

[y~, ..., Y~]

(2.2.2)

= A x(t)

can be rewritten

in t h e following g l o b a l form :

+ B u(t)

(2.2.4)

y(t) = C x(t)

2.2.1.

- Problem

formulation

If a d e c e n t r a l i z e d with e a c h s t a t i o n .

Every

control structure

is desired,

local c o n t r o l l e r i s d e s c r i b e d

a local c o n t r o l l e r i s a s s o c i a t e d by the

following g e n e r a l m o -

del :

fui(t)

= Qi z i ( t ) + Ki Yi (t) + v i ( t ) (2.2.5)

! ~i(t)

S i z i ( t ) + R i Yi (t)

36

w h e r e z i ( t ) E R v i i s t h e s t a t e of t h e i t h c o n t r o l l e r a n d v i ( t ) ~ Rmi is t h e i t h e x ternal input. This control structure

associated with system

(2.2.2)

(2.2.4)

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

b y F i g u r e 2.2 :

I

SYSTEM

F i g u r e 2.2 Let u s d e f i n e : S = block-diag.

(S1,...,Ss)

R = block-diag.

(R 1 . . . . ,R S)

Q = block-diag.

( Q I ' . . . . QS )

K = block-diag.

(K1,...,Ks)

z ' ( t ) = (z' 1 . . . . . z' s ) v'(t) = (v' 1 ..... V's)

t h e n , t h e global c o n t r o l l e r is d e s c r i b e d b y • l u(t) = Q z(t) + K u(t) + v(t)

(2.2.6) ~.(t) = S z ( t ) + R y ( t ) where the decentralization

appears

in t h e b l o c k - d i a g o n a l

structure

of t h e m a t r i c e s

S , R , Q a n d K. We c o n s i d e r

t h e p r o b l e m of s t a b i l i z i n g

the

s i g n i n g i t s p o l e s ) w i t h a c o n t r o l law s p e c i f i e d b y

system

(2.2.4)

(or a s -

(2.2.6).

T h i s p r o b l e m is

e q u i v a l e n t to t h e p r o b l e m of e x i s t e n c e of a s e t of m a t r i c e s S , R , Q

and K such that

the closed-loop system :

(2.2.5)

(2.2.2)

37

I 'c :°1

z(t)J

RC

is a s y m p t o t i c a l l y s t a b l e ( i . e . ,

2.2.2.

- Decentralized

Even if t h e (2.2.2)

(2.2.4)

fixed

[:l

[z(t)

v(t)

all i t s p o l e s h a v e a n e g a t i v e r e a l p a r t ) .

modes

conditions

for

the

stabilization

(or pole

h o l d with a c e n t r a l i z e d i n f o r m a t i o n p a t t e r n

assignment}

is n o t always s t a b i l i z a b l e (pole a s s i g n a b l e } with a d e c e n t r a l i z e d c o n t r o l . 78b1 p r o v i d e d

the

example

of a completely

controllable

of

system

(see § 1 . 2 . 6 1 , t h e s y s t e m interconnected

Wang (WANsystem

for

which e v e r y s u b s y s t e m ( A i i , B i ) was also c o n t r o l l a b l e s u c h t h a t t h e c o n d i t i o n s f o r i t s s t a b i l i z a t i o n b y m e a n s of c e n t r a l i z e d s t a t e f e e d b a c k w e r e s a t i s f i e d .

Nevertheless,

showed t h a t

constraints

the

system

could not

be

s t a b i l i z e d if d e c e n t r a l i z e d

he

were

imposed on t h e c o n t r o l . This means that additional conditions

are required.

e x p r e s s e d in t e r m s of t h e new c o n c e p t of d e c e n t r a l i z e d

These

conditions

f i x e d modes.

can b e

This concept

was i n t r o d u c e d b y Wang a n d D a v i s o n i n 1973 (WAN-73bl. Definition 2.1 (WAN-73b). G i v e n t h e t r i p l e ( C , A , B ) a s s o c i a t e d w i t h t h e s y s t e m (2.2.41 w i t h A g R n x n , B ~ R nxm a n d C g: R r x n l e t u s d e f i n e ~d as t h e s e t of block-diagonal matrices :

~d={KlK=block-diag. ( K 1 , . . . , K s I

, K i ~ Rm x r ,

( i : l , . . . . S}}

(2.2.7)

T h e d e c e n t r a l i z e d f i x e d polynomial of s y s t e m ( 2 . 2 . 4 ) is t h e n d e f i n e d as : (p,C,A,B

fld ) = g . c . d

(2.2.8)

{dot ( p I - A - B K C ) }

K ~ ~d Definition 2.2 (WAN-73bl. T h e s e t of f i x e d modes of s y s t e m t h e c o n t r o l law ( 2 . 2 . 6 ) A (C.A,B ~d ) =

(2.2.4)

with r e s p e c t to

( d e c e n t r a l i z e d f i x e d modes1 is g i v e n b y : n o(A + B K C )

Kc~ d w h e r e a ( . ) d e n o t e s t h e s e t of e i g e n v a l u e s of ( . ) .

(2.2.9)

38

It is c l e a r t h a t t h e d e c e n t r a l i z e d f i x e d modes a r e t h e z e r o s of t h e d e c e n t r a l i z e d f i x e d polynomial a n d can t h u s b e d e f i n e d as : h ( C , A , B , ~d ) = { p e ~ Note t h a t

A (C,A,B,

/

~d) c o (A)

s i n c e K--0 i s an element of ~d" T h e d e c e n -

t r a l i z e d f i x e d modes of t h e s y s t e m ( 2 . 2 . 4 ) that

cannot be

(2.Z.10)

~ (p,C,A,B, ~d ) = 0 }

a r e t h e r e f o r e t h e modes of t h e s y s t e m

moved with a decentralized control

(2.2.6),

i n d e p e n d e n t l y of t h e

p a r a m e t e r v a l u e s a s s i g n e d to t h e c o n t r o l l e r ( m a t r i c e s S , Q , R a n d K ) . T h e following f i g u r e i l l u s t r a t e s t h e s i t u a t i o n :

O ( A + B K 3 c) O (A+BI~ IC)

(A) h (C,A,B, ~d )

0 (A+BK

K 1,K 2 ,K 3 C ~d

F i g u r e 2.3

It is now i n t e r e s t i n g to n o t e t h a t t h e s e t o f f i x e d modes d o e s n o t d e p e n d on t h e dynamical o r n o n - d y n a m i c a l n a t u r e of t h e c o n t r o l . If a d e c e n t r a l i z e d s t a t i c f e e d b a c k c o n t r o l is u s e d i n s t e a d of ( 2 . 2 . 6 ) zero,

the

~ i.e.,

the matrices S and R are identically

s e t of d e c e n t r a l i z e d f i x e d modes r e m a i n s t h e same (WAN-73b).

only d e p e n d s on t h e p a r t i c u l a r o u t p u t - i n p u t p a t t e r n

This s e t

specified by the s t r u c t u r e

of

t h e m a t r i x K. From t h e i r d e f i n i t i o n , t h e d e c e n t r a l i z e d f i x e d modes a p p e a r i n t u i t i v e l y as an e x t e n s i o n of t h e w e l l - k n o w n u n c o n t r o l l a b l e a n d u n o b s e r v a b l e modes d e f i n e d in C h a p t e r 1. The r e l a t i o n s b e t w e e n t h e s e c o n c e p t s a r e c l a r i f i e d b y f u r t h e r d i s c u s s i o n . Consider

the

triple

(C,A,B)

associated

to t h e

canonical form

(1.2.4)

of a

g e n e r a l s y s t e m ( 1 . 2 . 1 ) . If t h e s y s t e m is c o n t r o l l e d b y c e n t r a l i z e d o u t p u t f e e d b a c k of t h e form ( 1 . 2 . 9 ) with K e R m x r , Wang a n d Davison (WAN-73b) s h o w e d t h a t : A ( C , A , B , R mxr) = o ( A l l ) u o (A33) V o (A44).

39 T h i s r e s u l t p o i n t s o u t t h a t t h e s e t of c e n t r a l i z e d f i x e d m o d e s i s t h e u n i o n o f t h e u n c o n t r o l l a b l e a n d u n o b s e r v a b l e m o d e s . T h e a n a l o g y is c l e a r w h e n we t h i n k t h a t uncontrollable and unobservable modes are those that cannot be moved by the control : it w a s s h o w n i n C h a p t e r

1 t h a t t h e f i r s t o n e s a r e d i s c o n n e c t e d from t h e i n p u t

a n d t h e s e c o n d o n e s from t h e o u t p u t .

From a n o t h e r h a n d ,

i f we d e n o t e t h e s e t o f

c e n t r a l i z e d f i x e d m o d e s b y Ac p t h e following d e f i n i t i o n h o l d s : h

c

= A (C,A,B,R mxr) =

where the analogy with the

n o (A + BKC) K ( Rmxr

definition

(2.2.9)

(2.2.11)

of d e c e n t r a l i z e d

fixed modes is ob-

vious.

Clearly,

from the

definitions

(2.2.9)

and

(2.2.11),

and since

~d c R m x r ,

the

s e t of c e n t r a l i z e d f i x e d m o d e s i s i n c l u d e d i n t h e s e t o f d e c e n t r a l i z e d f i x e d m o d e s : Ac c Ad

where hd =

(2.2.12)

h

fld).

(C,A,B,

We u n d e r s t a n d

intuitively that the modes that

cannot be moved u s i n g a c e n -

tralized control c e r t a i n l y c a n n o t be moved u s i n g a d e c e n t r a l i z e d control.

2 . 2 . 3 . - D e c e n t r a l i z e d s t a b i l i z a t i o n a n d pole a s s i g n m e n t

Due tralization, tralized

to

the

the

output

great

economical a n d

problem of stabilization feedback

76a,b) (FES-79)

(FES-80)

has

been

reliability and

pole

investigated

(POT-79)

by

advantages assignment

provided by

many authors

by

decen-

of

decen-

(AOK-72)

(COR-

means

(WAN-73a,b).

Only t h e m o s t r e l e v a n t r e s u l t s

are presented

here.

T h e r e s u l t s of Wang a n d

D a v i s o n (WAN-73b) t h a t c l e a r l y p o i n t o u t t h e i m p o r t a n c e of d e c e n t r a l i z e d f i x e d m o d e s in t h i s p r o b l e m a r e Morse ( C O R - 7 6 ) , complete s o l u t i o n .

presented

first.

Then,

we p r e s e n t

the study

of C o r f m a t a n d

who h a d a d i f f e r e n t a p p r o a c h to t h e p r o b l e m a n d p r o v i d e d a m o r e

40 2 . 2 . 3 . a . - Wang a n d Davison r e s u l t s (WAN-73b) The following t h e o r e m a n d c o r o l l a r y s t a t e t h e c o n d i t i o n s f o r t h e e x i s t e n c e of a s o l u t i o n in t e r m s of d e c e n t r a l i z e d o u t p u t

f e e d b a c k in o r d e r to s t a b i l i z e t h e s y s t e m

( o r to a s s i g n i t s p o l e s ) . Theorem (2.2.4) and

2.1

(WAN-73b).

Given

the

triple

(C,A,B)

associated

a n d ~d t h e s e t of b l o c k - d i a g o n a l m a t r i c e s d e f i n e d in

s u f f i c i e n t c o n d i t i o n for t h e

with

(2.2.7),

e x i s t e n c e of a c o n t r o l law

(2.2.6)

the

system

a necessary

such that the

c l o s e d - l o o p s y s t e m ( 2 . 2 . 7 ) i s a s y m p t o t i c a l l y s t a b l e is t h a t : h d = A ( C , A , B ~cl) c where

~-

Corollary

~-

(2.2.13)

d e n o t e s t h e l e f t h a l f p a r t of t h e complex p l a n . 2.1.

Under

the

same a s s u m p t i o n s

as in t h e o r e m

s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e of a c o n t r o l law ( 2 . 2 . 6 )

2.1,

a necessary

and

s u c h t h a t all t h e poles

of t h e c l o s e d - l o o p s y s t e m a r e in ~) is t h a t : h d = h ( C , A , B ~d ) c ~ where

~

(2.2.14)

is a n a r b i t r a r y p r e s p e c i f i e d s y m m e t r i c r e g i o n of t h e complex p l a n .

As a r e s u l t of t h e a b o v e c o r o l l a r y , t h e p o l e s of t h e s y s t e m can b e a r b i t r a r i l y a s s i g n e d if a n d only if :

Ad= ~ The s i m i l a r i t y b e t w e e n t h e s e r e s u l t s and t h o s e p r e s e n t e d for c e n t r a l i z e d c o n t r o l in p a r a g r a p h e

1 . 2 . 6 e m p h a s i z e s t h e f a c t t h a t d e c e n t r a l i z e d f i x e d modes a r e t h e

g e n e r a l i z a t i o n of u n c o n t r o l l a b l e a n d u n o b s e r v a b l e m o d e s to d e c e n t r a l i z e d c o n t r o l .

[20o E'] II

Example 2.1. C o n s i d e r t h e following 2 - s t a t i o n l i n e a r t i m e - i n v a r i a n t d y n a m i c s y s t e m :

~(t) =

Yl(t) =

o

a

o

0

0

-I

~1

1

OJ

x(t) +

o

0

x(t)

ul(t)

+

u2(t)

41

where a is an a r b i t r a r y r e a l p a r a m e t e r . The d e c e n t r a l i z e d o u t p u t f e e d b a c k matrices K h a v e t h e following s t r u c t u r e

K =

F [ kl

L The

0

0

poles

:

k2 of t h e

system

det(kI-A)=(k.-2)(k+l)(),-a).

are

Since

the

zeros

~=2 is a z e r o ,

of

the

characteristic

polynomial :

the open-loop s y s t e m is u n s t a b l e

i n d e p e n d e n t l y of the s i g n of a. The poles of t h e closed-loop system are the z e r o s of the closed-loop c h a r a c teristic polynomial : det(kI-A-BKC)=(~-2-kl)

(k + l - k 2 ) (k-a)

From Definition 2.1, the d e c e n t r a l i z e d fixed polynomial is ()~-a) and t h e system has a d e c e n t r a l i z e d

fixed mode at

assigned by an a p p r o p r i a t e

k=a.

A l t h o u g h t h e pole

~=2 can be a r b i t r a r i l y

choice of k 1, the pole ;~ =a cannot be m o v e d , N e v e r t h e -

less, if a(0~ the system is stabilizable.

2.2,3.b.

- Corfmat and Morse r e s u l t s (COR-76)

The a b o v e problem was c o n s i d e r e d geometric a p p r o a c h

(COR-75)

by

Corfmat

and

Morse in

1976 u s i n g

a

(WON-74), The problem was a p p r o a c h e d by a n s w e r i n g

the following q u e s t i o n : "Is t h e r e a control law in the form of a s t a t i c d e c e n t r a l i z e d o u t p u t f e e d b a c k such that the system becomes controllable and o b s e r v a b l e by a single station ?" If the a n s w e r is p o s i t i v e ,

the s t a n d a r d

algorithms of c e n t r a l i z e d control can

then be u s e d for the design of an additional dynamic c o n t r o l l e r a s s o c i a t e d with this station s u c h t h a t t h e stabilization or pole a s s i g n m e n t is a c h i e v e d . For the system

(2.2.4),

this a p p r o a c h leads to t h e following control laws and

is i l l u s t r a t e d b y F i g u r e 2.4 :

I

t u i t) = Ki Yi (t) + v i ( t ) ~j(t) Sj z j ( t ) + Rj y j ( t )

/vj(t)

(i=l,...,s) j ¢

Qj zj(t) + Nj y j ( t )

{1 . . . . .

s)

(2.2.1s)

42

1 I I

vl+~ uj

[ I I [ I

~Yl ,] I UlLf

lY-

i

SYSTEM

vs

÷~'I~ ---~-~

I I I I I I

--~

I S I _.i

Observator F i g u r e 2.4 Of c o u r s e ,

the initial system

(2.2.4)

is s u p p o s e d

t o b e globally c o n t r o l l a b l e

and observable. Corfmat and (digraph)

Morse a n a l y s i s is b a s e d

to t h e s y s t e m .

on t h e a s s o c i a t i o n of a d i r e c t e d

graph

A n o d e is a s s o c i a t e d to e v e r y s t a t i o n a n d an e d g e e x i s t s

from t h e n o d e i to t h e n o d e j i f a n d o n l y if : Wij (p) = Cj ( p I - A ) - I Bi ~ 0 i.e.,

if t h e o u t p u t a t s t a t i o n j c a n b e a f f e c t e d b y t h e i n p u t at s t a t i o n t .

Example 2°2. C o n s i d e r t h e following s y s t e m d e s c r i b e d i n t h e f r e q u e n c y - d o m a i n b y :

w11(p) o oj W(p)= 21~p) w32Wz2(P)(p)wa3(P)o T h e a s s o c i a t e d g r a p h is t h e following :

43

3 Definition Z.3. A s y s t e m is s a i d to b e s t r o n g l y c o n n e c t e d if i t s a s s o c i a t e d d i g r a p h is strongly connected.

(Note t h a t t h i s does n o t n e c e s s a r i l y imply t h a t all t h e t r a n s f e r

matrices Wij(p) a r e # 0 ) . A non-strongly-connected

s y s t e m c a n always b e d e c o m p o s e d i n s e v e r a l s t r o n g l y

c o n n e c t e d subsystems c o n s t i t u t e d b y t h e s t a t i o n s a s s o c i a t e d with t h e n o d e s d e f i n i n g the s t r o n g c o m p o n e n t s of t h e d i g r a p h . In t h e e x a m p l e 2 . 2 ,

t h e s y s t e m is n o t s t r o n g l y c o n n e c t e d a n d c a n b e d e c o m -

posed in two s t r o n g l y c o n n e c t e d s u b s y s t e m s ,

Sl

- Yl = W l l ( P )

which are :

Ul

s2 Fy21222(P 2g(P ]

Definition 2.4 ( C O R - 7 6 ) .

A single station system given by the triple (C,A,B) is said

to be complete if :

i - C (pI-A) -I B # 0 ii - the remnant polynomial p (C,A,B) (COR-76) defined as the product of the n~ R (A nI.x n ) t o first invariant polynomials (ROS-70)of the matrlxl},IcA 0BI is equal

Remark 2 . 2 .

rank[

( C O R - 7 6 ) . C o n d i t i o n ii is e q u i v a l e n t to :

> n

for

Vk ~_o(A)

L

C

i.e.j

no p o l e of t h e s y s t e m is s i m u l t a n e o u s l y u n c o n t r o l l a b l e a n d u n o b s e r v a b l e

1.2.4).

(see §

44 Let u s i n t r o d u c e now a f u n d a m e n t a l d e f i n i t i o n in t h e

study

of m u l t i - s t a t i o n

s y s t e m s -" t h e c o n c e p t of c o m p l e m e n t a r y s u b s ) r s t e m s . Definition 2 . 5 . C o n s i d e r t h e S - s t a t i o n s y s t e m ( 2 . 2 . 2 ) . a n d the p a r t i t i o n

Ha= { i l , . . . , i k}

Define t h e s e t s ~ = { 1 , . . . , S }

, ~8 = {ik+l . . . . . i s } .

T h e s y s t e m is t h u s p a r d -

t i o n e d in two a g g r e g a t e d s t a t i o n s a a n d 8 • Define t h e following m a t r i c e s :

°o=

L % .....% ]

i,

C

58 =

(%

=

E B i k + l . . . . . BiS ]

Ii ik

L%J

H e n c e , t h e s e t of c o m p l e m e n t a r y s u b s y s t e m s of s y s t e m (2.2.2) is d e s c r i b e d by t h e t r i p l e s (C B , A , B a ) '

(k=l,...S-1).

T h e f i r s t r e s u l t of Corfmat a n d Morse d e a l s w i t h s t r o n g l y c o n n e c t e d s y s t e m s : Theorem 2.2 (2o2ol)

(COR-76b).

(2°2.2).

Consider the

globally c o n t r o l l a b l e a n d o b s e r v a b l e s y s t e m

T h e n , u s i n g t h e c o n t r o l laws s p e c i f i e d b y ( 2 . 2 . 1 5 ) ,

with a r b i t r a r y

j ¢ {1 .... s} * t h e s p e c t r u m of t h e s y s t e m c a n be a s s i g n e d a r b i t r a r i l y if a n d o n l y i f all i t s comp l e m e n t a r y s u b s y s t e m s a r e c o m p l e t e . T h i s is e q u i v a l e n t to :

i - C 6 (pI-A) -I B a #0

(k=l ..... S-1)

(2.2.1) (2.2.2) is strongly connected. ii - p ( C s , A , B a) = 1 (k=l ..... S-l) with 0 (C B ,A.Ba ) denoting the remnant polynomial of the complementary subsystem (C~ , A , B ). * t h e s y s t e m can b e s t a b i l i z e d if a n d o n l y if : - C o n d i t i o n (i) h o l d s . iii- p (C~ , A , B a) i s a s t a b l e polynomial, (k=l . . . . . S - l ) . Note

that

a necessary

c o n d i t i o n for t h e

e x i s t e n c e of a d e c e n t r a l i z e d s t a t i c

o u t p u t f e e b a c k t h a t makes t h e s y s t e m c o n t r o l l a b l e a n d o b s e r v a b l e b y a s i n g l e s t a t i o n , is t h a t t h e s y s t e m is s t r o n g l y c o n n e c t e d .

Example

2.3

(KAT-81).

Consider

the globally controllable and observable

described by the following matrices :

system

io,,

45

0 0 0

A:

o

o

I

BI=

0 0 0

'

~:[o

[

I

o

o

,

152= 0 I

o]

m a t r i x of t h e s y s t e m is :

o l p2(p_2)

Wll(p)

Wl2(P)]_

l

W21(P)

W22(P)J - p2 (p- l)(p-2)

Since W l Z ( P ) = C I ( P I - A ) - I B 2 strongly connected and condition

* Calculation of

p2 (p=2)

0

p2(p-2)

o

¢ 0,

t h e s y s t e m is

a r e ( C 2 , A , B I) a n d ( C 1 , A , B 2 ) .

O ( C 1 , A , B 2) :

I that can be p u t

p(p=l) _(p_l) 1

¢ 0 and W21(P)=C2(PI-A)-IB1 (i) of t h e o r e m 2.2 h o l d s .

The complementary subsystems

mials :

o

1

0

o

The transfer

=

0

oo oo 1 I o

CI =

W(p)

[, I°J

X

I=A Cl

B21 0

in Smith)s form

=

X

-I

-I

0

X

0

-I I 0

0 0

0 0

),-I 0

0 I 0 X-2 lj 1

l

0

0

i

0

0

(ROS-70)

0

0

0

0 0 1 0

I 0

0

0 0

to make a p p a r e n t

the invariant

polyno-

46 ! o I I 0

0

l !

0

1

I I I

0 0

-,1

=~

0 (CI,A,B 2) : 1

1

o

O.Q_

o

* Calculation o f

0(C2,A,B1)

:

C2

),

-I

-I

0

k

0

0

0

k-I

0

0

0

0

0

1

0 -I 0 k-2 0

a n d S m i t h ' s form is t h e following : i

E

I

1

9_

a

I

I

0 --

o

--

I I

l 0

I

=~

p (C2,A,131) : 1

k 2(I-2)

All t h e c o m p l e m e n t a r y s u b s y s t e m s a r e c o m p l e t e . A c c o r d i n g to T h e o r e m 2 . 2 , the s p e c t r u m of t h e s y s t e m can b e a r b i t r a r i l y a s s i g n e d b y u s i n g

first a static output

f e e d b a c k at s t a t i o n 2, w h i c h m a k e s t h e s y s t e m c o n t r o l l a b l e a n d o b s e r v a b l e b y s t a t i o n 1, a n d t h e n a p p l y i n g an a p p r o p r i a t e dynamic o u t p u t f e e d b a c k at s t a t i o n 1 f o r w h i c h s t a n d a r d d e s i g n t e c h n i q u e s can b e u s e d ( B R A - 7 0 ) . Example 2 . 4 . C o n s i d e r now t h e globally c o n t r o l l a b l e a n d o b s e r v a b l e s y s t e m d e s c r i b e d b y t h e following m a t r i c e s :

47 0

1

0

0

0

1

0

0

l

A =

Cl

0

O

1

0

0

0

0

1

I

o

o

o]

o

I

o

o

o

o

o

I

I

which is also s t r o n g l y

* C a l c u l a t i o n of

0

k-I

0

0

o

o

o

1

0

0

connected.

p (CI1,A,B2)

- 1

0

B2=

Bl=

=

-~,

[°1 il 0

:

o

o

0

0

X-I

0

o

o7

I I I I I I

~,-1

I I

I !

0 0

I

I

0

0

0

o

1

o

o So,

i

o

O

--~

tl J

o__

0 o_

p ( C . 2 ± A , B 1)

k-I

: 0

0

0

0 1 |

o

~.-I

o

o

o

o

o

X-I

O

o

1

l

0

P ( C 2 , A j B 1)

=X-i.

i

1

o

o

o

o

o

I [

o

1

--o

[I

o _

Consequentlyp

0 --

x-ll

-

i

l

Lj ]

0

o

not v e r i f i e d .

o

I

o

0 0 P(CI,~,B2) = 1

* Calculation o f

I I

I

I 1

I

o

I

_o

conditions

(ii)

x (x

and

(iii)

-t)

of Theorem

T h i s s y s t e m c a n n o t b e s t a b i l i z e d b y a c o n t r o l law o f t h e k i n d

2.2

are

(2.2.15)

since t h e mode k 1=1 c a n n o t b e m o v e d b y d e c e n t r a l i z e d c o n t r o l .

In the

case

cedure proposed

for which the

by Corfmat and

conditions of Theorem Morse leads

2.2 a r e

satisfied,

to all m e m o r i l e s s c o n t r o l l e r s ,

the

pro-

save one

48 of h i g h c o m p l e x i t y ( o r d e r n - l ) . s e n t s some d i s a d v a n t a g e s .

From a p r a c t i c a l p o i n t of v i e w , t h i s s i t u a t i o n p r e -

In p a r t i c u l a r ,

it g e n e r a l l y r e q u i r e s h i g h g a i n s : e v e n if

all t h e m o d e s of t h e s y s t e m a r e c o n t r o l l a b l e a n d o b s e r v a b l e a t o n e station~ some of them m i g h t h a v e a weak d e g r e e o f c o n t r o l l a b i l i t y o r o b s e r v a b i l i t y , t h a t n e e d s to be c o m p e n s a t e d b y h i g h g a i n s . It would be b e t t e r to s p r e a d t h e c o n t r o l c o m p l e x i t y more e q u a l l y among t h e local c o n t r o l l e r s , i . e . ,

to e n d o w e a c h local c o n t r o l l e r with some

d y n a m i c s . A r e s u l t in t h i s d i r e c t i o n was o b t a i n e d b y A n d e r s o n a n d L i n n e m a n n (AND84)

for

interconnected systems

(the

subsystems are

interconnected through

their

inputs and outputs). They showed that, t r o l l e r can b e t a k e n

for stabilization p u r p o s e ,

t h e o r d e r of t h e i t h local c o n -

at l e a s t e q u a l to t h e n u m b e r o f s t a b l e z e r o s of t h e i t h

sub-

s y s t e m , while t h e t o t a l o r d e r o f t h e d e c e n t r a l i z e d c o n t r o l l e r (sum of t h e o r d e r s o f t h e local c o n t r o l l e r s ) does n o t e x c e e d n - 1 . A l t h o u g h T h e o r e m 2.2 p r o v i d e s an i n t e r e s t i n g r e s u l t , t h e whole p r o b l e m is n o t solved yet.

The c o n d i t i o n s r e q u i r e d b y T h e o r e m 2.2

guarantee

t h e e x i s t e n c e of a

c o n t r o l law t h a t u s e s only one d y n a m i c c o n t r o l l e r at one s t a t i o n . T h e y a r e t h e r e f o r e much too c o n s t r a i n i n g

for

our

problem which

allows

dynamic

controllers

at

each

s t a t i o n . In t h e s e c o n d p a r t o f t h e i r w o r k , Corfmat a n d Morse (COR-76b) r e l a x e d t h e c o n s t r a i n t s of T h e o r e m 2.2 a n d t h e r e b y p r o v i d e d t h e complete s o l u t i o n of t h e p r o blem. Theorem

2,3.

Consider

the

globally

controllable

and

T h e n , with a d e c e n t r a l i z e d c o n t r o l s u c h t h a t ( 2 . 2 . 6 )

observable

system

(2.2.4).

:

* i t s s p e c t r u m can b e a r b i t r a r i l y a s s i g n e d if a n d o n l y i f t h e sum o f t h e d i m e n s i o n s of t h e s t r o n g l y c o n n e c t e d s u b s y s t e m s is e q u a l to t h e dimension of t h e whole system (2.2.4),

a n d if t h e s p e c t r u m o f e v e r y s t r o n g l y c o n n e c t e d s u b s y s t e m can b e

arbitrarily assigned. * the system spectra

of

the

(2.2.4)

strongly

c a n b e s t a b i l i z e d if a n d o n l y if t h e complement o f t h e connected

subsystems

with

the

spectrum

of . t h e

system

( 2 . 2 . 4 ) is s t a b l e , a n d if e v e r y s t r o n g l y c o n n e c t e d s u b s y s t e m can b e s t a b i l i z e d . Example matrix :

2.5.

Consider

the

2-station

system

described

by

the

following t r a n s f e r

49

W(p)=

[

1

p- 1

p-I-E

o

1 P

This s y s t e m is n o t s t r o n g l y c o n n e c t e d and the two s t r o n g l y c o n n e c t e d s u b s y s tems a r e , in t h i s c a s e , r e d u c e d to s l n g l e - s t a t i o n s y s t e m s .

T h e i r s p e c t r a can t h e r e -

fore be a r b i t r a r i l y a s s i g n e d . Both a r e of dimension 1. * Case 1 • E = 0. In this case,

the dimension o f the global system is 2 and it is

equal to t h e sum of the dimensions of the s t r o n g l y

connected subsystems.

Either

stabilization or pole a s s i g n m e n t a r e p o s s i b l e . * Case 2 : E~ 0. In this c a s e , the dimension of the global system is 3. The pole XI=I+E

c a n n o t be moved b y a d e c e n t r a l i z e d c o n t r o l .

of the system remains possible if Xl=l+c

2.2.3.c.

N e v e r t h e l e s s , the stabilization

is s t a b l e .

Comments

-

The r e s u l t s of Corfmat and Morse were c a r r i e d out using geometrical methods (WON-74) obtained

of system equivalent

theory. results

With the using

same a p p r o a c h ,

polynomial matrix

Fessas

methods

(FES-79) (ROS-?0)

(FES-80) (WOL-74).

Potter et al. (POT-79) also t r e a t e d the same problem in the case of 2 - s t a t i o n systems and p r o v i d e d e q u i v a l e n t conditions e x p r e s s e d in terms of a r a n k condition for the system matrix. T h e i r r e s u l t is s t a t e d in the following theorem :

Theorem 2 . 4 . .

C o n s i d e r a globally controllable and o b s e r v a b l e 2-station

system

(C 1

C 2, A, B 1 B 2) with C I ( P I - A ) - I B 2 ~ 0 and C 2 ( P I - A ) - I B 1 ~ 0 ( i . e . , the system is strongly c o n n e c t e d ) . T h e r e e x i s t s an r e a l - v a l u e d f e e d b a c k matrix K 2 of a p p r o p r i a t e dimension s u c h t h a t the system (C1,A + B2K2C2,B1) is controllable and o b s e r v a b l e if and only if :

I I-A

rank /

X

LC2

~ 11

n

~ % ~ o(A)

(2.2.16a)

50

rank [~I-A

~21

~ n

~ ~ ~ o (A)

(2.2.16b)

As it will b e s e e n in t h e n e x t c h a p t e r ,

conditions (2.2.16) present the interest

C 1

to b e t h e same as t h e c o n d i t i o n p r o v i d e d b y A n d e r s o n a n d Clements (AND-81a) characterize

decentralized

characterizations

fixed modes.

of f i x e d modes

The next

chapter

deals with t h e

to

different

in t h e t i m e - d o m a i n a n d in t h e f r e q u e n c y - d o m a i n .

T h e e q u i v a l e n c e b e t w e e n t h e r e s u l t s of Corrnat a n d

Morse a n d t h o s e of Wang a n d

D a v i s o n e x p r e s s e d in t e r m s of fixed modes will t h u s b e c l a r i f i e d . However,

a n immediate c o r r e l a t i o n c a n b e e s t a b l i s h e d from t h e following t h e o -

rem o b t a i n e d b y F e s s a s ( F E S - 8 0 ) . Theorem 2.5.

C o n s i d e r t h e globally c o n t r o l l a b l e a n d o b s e r v a b l e s y s t e m

suppose

i t is s t r o n g l y

that

connected.

This system

(2.2.4)

and

c a n b e made c o n t r o l l a b l e a n d

o b s e r v a b l e b y a s i n g l e s t a t i o n u s i n g a s t a t i c d e c e n t r a l i z e d c o n t r o l if a n d o n l y if t h e s y s t e m h a s no f i x e d m o d e s .

2.3. - A R B I T R A R Y

STRUCTURAL

CONSTRAINTS

As was p o i n t e d o u t in t h e i n t r o d u c t i o n , metimes allowed to s h a r e information pattern the controller structural

some i n f o r m a t i o n .

reflecting

can connect.

constraints

a particular

t h e d i f f e r e n t local s t a t i o n s a r e s o -

This yields

to a n o n - s t a n d a r d

c o n f i g u r a t i o n of o u t p u t - i n p u t

reduced pairs

that

T h e a n a l y s i s of t h e p r o b l e m of c o n t r o l with a r b i t r a r y

does not require

a model of t h e s y s t e m in w h i c h t h e p a r t i -

t i o n i n g in s p e c i f i e d s t a t i o n s is a p p a r a n t . T h e s y s t e m is now d e s c r i b e d b y t h e following g e n e r a l model : £(t)

= A x(t)

y(t)

C x(t)

+ B u(t)

(2.3.1)

w h e r e x ~ R n , u 6 R m, y ~ R r a n d : B = (b 1 . . . . . b m) E: R n x m CI= (c' 1 . . . . . c' r ) ~ R r x n The particular

information pattern

m a t r i x F of d i m e n s i o n m x r s u c h t h a t '

can b e a d e q u a t e l y

described

by a binary

51 f.. = 1 i f a f e e d b a c k Ij lf'j = 0 o t h e r w i s e . For every

input

is allowed from the output

ui(t),

(i=l,...,m),

j to t h e i n p u t i.

define the following indices set

:

. . . . . r } / fij = 1}

si = { j c { t

The dynamic

controller

(2.3.2)

corresponding

to t h i s i n f o r m a t i o n

pattern

is thus

des-

cribed as follows :

I ~_i(t) = Si zi(t) + ~ rij yj(t) j£3 i [ui(t )

where zi(t)• input.

q[ zi(t) ÷ Z

J~Ji

R ~i i s t h e

state

(i=l,...,m)

kij yj(t) + vi(t)

of t h e

ith controller

Si, qi a n d rij a n d kij a r e c o n s t a n t

matrices,

The concept of fixed modes introduced case of decentralized

Definition 2.6

constraints

(SEZ-81a).

Define the set

by Wang and

vi(t)

is

the

ith

external

and scalars. Davlson

( W A N - 7 3 b ) in t h e as follows :

:

f.. = 0} 1]

t h e n , t h e s e t o f f i x e d m o d e s of s y s t e m is

and

vectors

on the control can now be generalized

~* = {K ~ R m x r / k.. = 0 if 13

by ( 2 . 3 . 3 )

(2.3.3)

(2.3.1)

(2.3.4)

with respect

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

:

A (C,A,B,

~*)

= K ~ n/ *

o (A + B K C )

(2.3.5)

where a (°) denotes the set of eigenvalues of (.). This definition is the extension an a r b i t r a r y

Note t h a t i f we c o n s i d e r

K1

o f D e f i n i t i o n 2.2 t o a s e t o f m a t r i c e s

structure.

K2 c

.....

the output

feedback

matrices

:

K having

82 with t h e following i n c l u s i o n r e l a t i o n :

the corresponding sets of fixed modes of system (2.3.1) are related by the inclusion relation : A (C,A,B

fl{) c... c A (C,A,B

f~)

c

O(A)

It is c l e a r t h a t if t h e s y s t e m h a s u n c o n t r o l l a b l e o r u n o b s e r v a b l e m o d e s , t h e y will b e p r e s e n t in a n y of t h e p r e c e d i n g s e t s . This s i t u a t i o n is i l l u s t r a t e d b y t h e following f i g u r e :

k

1 F i g u r e 2.5

It was s h o w n b y S e z e r a n d Siljak (SEZ-81a) t h e e x i s t e n c e of a s e t of c o n t r o l laws ( 2 . 3 . 3 )

(SIL-82b) t h a t t h e c o n d i t i o n s for

such that the system (2.3.1)

can be

s t a b i l i z e d (or h a v e all its p o l e s a r b i t r a r i l y a s s i g n e d ) a r e t h e same as t h o s e in T h e o rem 2.1 a n d C o r o l l a r y 2.2 b u t r e p l a c i n g t h e s e t A ( C , A , B , ~ d ) b y Remark 2.3.

h (C,A,B,

~ *).

C o n s i d e r t h e p a r t i c u l a r c a s e for which t h e g e o g r a p h i c a l d i s t r i b u t i o n of

t h e s y s t e m d e f i n e s S s p e c i f i e d c o n t r o l a n d o b s e r v a t i o n s t a t i o n s s u c h t h a t t h e model of t h e s y s t e m is g i v e n b y

(2.2.4).

The i n f o r m a t i o n t r a n s f e r is d e f i n e d b y t h e p h y -

sical c o n n e c t i o n s b e t w e e n s t a t i o n s a n d it can t h e r e f o r e b e a g g r e g a t e d at t h e level of the

stations

(ui(t)

dimension mi a n d r i ) .

and

Yi(t),

(i=l . . . . , S ) ,

are

no l o n g e r s c a l a r s

but

v e c t o r s of

53

The structurally

constrained

control is now represented

a binary

by

square

matrix F o f d i m e n s i o n s S x S : f..x] = 1 i f a f e e d b a c k i s a l l o w e d f r o m s t a t i o n j t o s t a t i o n i f.. = 0 o t h e r w i s e 13 a n d t h e s e t s t h a t w e r e a s s o c i a t e d to e a c h s c a l a r i n p u t

in ( 2 . 3 . 2 )

are now associated

to e a c h c o n t r o l s t a t i o n :

zi = {J E h . . . . .

s} ! ~j = 1}

0=1 . . . . .

The dynamic controller corresponding

~i (t) = Si zi(t) +

where

s)

to t h i s i n f o r m a t i o n p a t t e r n

)~ Qij yj(t) J Jr 3i

ui(t) = H i zi(t)

+ Z

Si,Qij,Hi,Kij

are

j¢3 i

is described

by •

(i=l,...,S)

Kij yj(t) + vi(t)

matrices

of

dimensions

(~)t,vi),

(~)i,rj),

(mi,~)i),

(mi,r j),

r e s p e c t i v e l y a n d v i i s t h e i t h e x t e r n a l i n p u t v e c t o r o f d i m e n s i o n mi . Finally, t h e s e t

~*

f~* is d e f i n e d a s :

= { K 1~ R m x r

2.4. - E V A L U A T I O N

This paragraph

OF

/ K=block (Kij),

FIXED

presents

of a s y s t e m w i t h r e s p e c t routines are provided

(i,j = 1 .....

S) a n d Kij = 0 i f

fij = 0}

MODES

two algorithms for determining

to a s p e c i f i e d c o n t r o l s t r u c t u r e .

in Appendices

2.4.1. - By comparing the spectra

the set of fixed modes

The corresponding

Fortran

2 a n d 3.

of the open-loop and

closed-loop dynamic matrix This algorithm was proposed decentralized (2.2.2)

control.

It

is

based

by Davison and Ozguner on

the

definition

g i v e n b y Wang a n d D a v i s o n ( W A N . 7 3 b ) .

of

( D A V - 7 6 a ) in t h e c a s e o f decentralized

fixed

modes

54 A l g o r i t h m 2.1 : (DAV-76a) T h e s e t of f i x e d m o d e s o f t h e s y s t e m ( 2 . 2 . 4 ) control

whose

structure

d e f i n e d in ( 2 . 2 . 7 )

is

specified

by

the

set

w i t h r e s p e c t to a d e c e n t r a l i z e d

of

output

feedback

matrices

~d

c a n be c o m p u t e d a s follows :

1 - C o m p u t e t h e e i g e n v a l u e s o f m a t r i x A : o (A) 2 - Choose an a r b i t r a r y

f e e d b a c k m a t r i x Kd

C

~d

( b y g e n e r a t i o n of p s e u d o -

r a n d o m n u m b e r s for e x a m p l e ) . 3 - C o m p u t e t h e e i g e n v a l u e s of (A + BKdC) : o (A + B K d C ) . 4 - T h e f i x e d m o d e s a r e t h e e l e m e n t s of t h e i n t e r s e c t i o n of t h e two s e t s a (A) and

a (A + BKdC) ( w i t h a p r o b a b i l i t y 1 ) .

It is c l e a r t h a t t h i s a l g o r i t h m c a n e a s i l y b e e x t e n d e d to t h e c a s e o f a r b i t r a r y structural constraints by replacing the set

~d by

~* d e f i n e d in ( 2 . 3 . 4 ) .

Example 2.6 ( D A V - S 3 ) . C o n s i d e r t h e S - s t a t i o n s y s t e m g i v e n b y t h e following t r i p l e :

ii]l

o Joo i-1ol

_L _2 _i o,i,ooiO,

Following t h e s t e p s of a l g o r i t h m 1, we h a v e : 1 -

o(A) = ( - 1 ( o r d e r

2), - 2 , - 3 }

2 - We c h o o s e a r b i t r a r i l y 3 -

: Kd = diag. (0.13, - 0 . 1 7 , - 0 . 1 )

o ( A + BKdC) = { - 0 . 0 1 , - 2 . 1 5 9 , - 2 . 1 , - 3 )

4 - T h e s e t of f i x e d m o d e s is g i v e n b y : Although our

example considers

c(A)

n

o (A + B K d C ) = { - B }

a s y s t e m of small d i m e n s i o n ,

this

algorithm

a p p l i e s e f f i c i e n t l y f o r l a r g e s c a l e s y s t e m s . For i n s t a n c e , i t w a s u s e d f o r a s h i p s t e a m generator,

w h i c h is a s y s t e m w i t h 119 s t a t e v a r i a b l e s a n d B c o n t r o l s t a t i o n s (DAV-

76a).

2 . 4 . 2 . - By calculation of the s y s t e m modes s e n s i t i v i t y (TAR-84) (TAR-85)

The

preceding

system which remain

paragraphs invariant

showed under

that

fixed

structurally

modes

are

constrained

the

modes

feedback,

of

the

indepen-

55 d e n t l y of t h e n u m e r i c a l of t h i s p r o p e r t y

values of the nonzero

to define

feedback

fixed modes in terms

gains.

of the

We c a n t a k e

concept

advantage

of eigenvalue

sensi-

tivity.

Definition 2 . 7

(TAR-84).

control whose

structure

d e f i n e d in ( 2 . 3 . 4 ) of s e n s i t i v i t y )

The is

fixed

modes of the

specified

by

the

set

are the modes of the closed-loop

with respect

to v a r i a t i o n s

on

the

system of

(2.3.1)

output

system nonzero

with respect

feedback

to a

matrices

which are insensible elements

of the

~* (void

feedback

matrix.

Consider

an arbitrary

problem of determining

the

real matrix variation

v a r i a t i o n dD o n t h e m a t r i x D . the quantities

d~r and

Faddeeva (FAD-63) dk

r

= w'

Where v r a n d

(D-

r

let us

by the

following

analyze

formula due

. v

of order

to F a d d e e v

. v

right

and

left eigenvectors

of D corresponding

:

(2.4.2)

= 0

(2.4.3)

= 1

r

From (2.4.1),

1, and

(2.~+.1)

r

the normalized

; i.e.

- XrI)

the

from a

~rI) v r = 0

W'r ( D

w'

and

kr o f D r e s u l t i n g

F o r t h e c a s e i n w h i c h Xr i s a n e i g e n v a l u e

dD a r e r e l a t e d

wr are

to Xr' r e s p e c t i v e l y

nxn

:

. dD

r

D of dimension

dkr of any eigenvalue

(2.4.4)

it c a n b e s h o w n

(MOR-66)

(ROS-65b)

that

:

Tr {Q (X r.).dD } d~'r=

Tr { Q ( ~ r ) }

(2.4.5)

w h e r e Q(~) i s t h e a d j o i n t m a t r i x o f ( X I - D ) Although

Morgan

formula (2.4.5), (ROS-65b) where

(MOG-66)

proposed

it m a y b e m o r e c o n v e n i e n t Q( ~r ) i s r e p l a c e d

: Q(p)

= adj (kI-D)

a n a l g o r i t h m to e v a l u a t e

dk r d i r e c t l y

to u s e t h e f o r m u l a g i v e n

by its explicit form (ROS-65a)

by

from

Rosenbrock

(GAN-79)

:

56 Tr {[

ir (D-X il)].dD} i(i#r)

(2,#,6)

d~.r= i(ilr)

Another with respect that

the

order

approach

(~ r-

consists

in

Xi)

determining

right

and

gradient

of

the

e i g e n v a l u e Xr

d.. o f t h e m a t r i x D . S i n c e L a n c a s t e r ( L A N - 6 4 ) s h o w e d 1l e i g e n v e c t o r s v r a n d w r a s s o c i a t e d w i t h a n e i g e n v a l u e Xr of

left

1 are continuous

dD ( ~

a t di.,j we d e r i v e

(2.4.2)

dv I) v r + (D-)trI) ~

dXr = d(dij)

B y m u l t i p l y i n g on t h e l e f t b y wr a n d u s i n g

dX

r 'd(dij)

= w' r

a n d we o b t a i n

:

= 0

(2.4.3)

and

(2.4.4)

we o b t a i n

:

dD v d(dij) r

In our particular system

the

to t h e e l e m e n t s

case,

the matrix

(2.4.7)

D is the

dynamic

closed-loop

matrix

: D = A + B K C = A + .E. b i k . . c . 1,] 1] ]

where

bi and

concerned only.

(2.4.7)

J

becomes

tJ

(2.4.8), w e

:

tJ

have

a D _ bi.c.

a kij

(2.4.8)

c. a r e t h e i t h c o l u m n o f B a n d t h e j t h r o w o f C , r e s p e c t i v e l y .

by the variation

8~ r : w' aD ak.. r ~ Vr

From

of the

j

:

o f Xr w i t h r e s p e c t

to v a r i a t i o n s

on the elements

We a r e kij o f K

57 a n d f i n a l l y , we o b t a i n

~X r ak..

:w'

:

(2.~.9)

bic. v I r

r

|J

If the

feedback

matrix

has

some fixed zero elements,

it i s c l e a r t h a t

equal to z e r o f o r k.. = 0. 1]

Definition (2.3.4)

2.8

(TAR-84).

and a feedback

Consider

a

control

structure

m a t r i x K C ~ *. T h e s e n s i t i v i t y

eigenvalue X of order

1 with respect

r

defined

the

is

set

~* i n

matrix of a closed-loop

matrix

to t h i s c o n t r o l i s g i v e n b y

by

~ r ~k.. II

•

SK r = ( s k i j ) i=l,...,m j=l .....

1 (2.4.10)

with

{~

'r

skij =

b i cj v r

Theorem 2.6 (TAR-84). ~*' kr

k. = 0 q

if

Given the system

(2.3.1)

i s a f i x e d m o d e if a n d o n l y if e i t h e r

and the set of feedback

matrices

of the two following equivalent

condi-

tions holds : 1 - the sensitivity 2 - Sr = W r { [ i ( i ~ r )

m a t r i x SK r d e f i n e d i n ( 2 . 4 . 1 0 )

(D-hi)

. dD}

is identically

zero (2.4.11)

= 0

w h e r e D--A+BKC, K E f~*, d D = B d K C a n d dK E: ~ * . This result

is straightforward

from Definition

(2.2.7)

and the relations

(2.4.6)

and (2.4.10).

Using Theorem

2.6,

the fixed modes of a system

poles) can be computed by the following algorithm

:

(not necessarily

with distinct

58 Al$orithm 2.2 (TAR-84).

1

Select

an

system modes

arbitrary

feedback

o(A+BKC)

matrix

K ~ ~*

such

that

the

closed-loop

are distinct.

2 - Compute the sensibility

matrix

SK r in ( 2 . 4 . 1 0 )

o r Sr in ( 2 . 4 . 1 1 )

for

~r ~ o ( A + B K C ) 3 - T h e f i x e d m o d e s a r e t h o s e f o r w h i c h SK r i s i d e n t i c a l l y Remark

2 . 4 : It i s b e t t e r

distinct poles. the

system

systems

Indeed,

has

to r e s t r i c t

the first

multiple

fixed

with multiple poles.

Example

2.7.

described

Consider

application

of Algorithm

step of this algorithm modes.

Obviously,

the

globally

this

controllable

2 to s y s t e m s

situation

is only possible

tI°0 tli

1° o

0

1

I

and

observable

2-station

0

I

0

0

I

l

i

to t h e d e c e n t r a l i z e d

control

:

K = diag.(k I, k 2) = d i a g . ( I , 5 ) The results

obtained

by applying

Algorithm 2 are the following :

Closed-loop

S SK

eigenvalue

-2

: ~,

I

r

:

-o.7645 0

-

1.088

0

--o.49 10-15 o

~Xr ;)K

r

for dK=K

0

1

-29.9999

- @1529

o 0.2177

for

subsequently.

•

1

with

m a y n o t h a v e a s o l u t i o n when

T h i s c a s e will b e d i s c u s s e d

by the following triple

submitted

the

z e r o o r Sr i s z e r o .

I

J

o - %32 10 -17

19.9999

I

J

0.666 10 - 1 4

system

59

T h e r e f o r e Xr = 1 is a f i x e d mode with an a c c u r a c y ) 10 -14. This a l g o r i t h m h a s b e e n u s e d f o r a s h i p s t e a m g e n e r a t o r model in ( T A R - 8 5 ) . The p r e c e d i n g a n a l y s i s is r e s t r i c t e d to simple f i x e d modes a n d t h e a p p r o a c h is specially a d e q u a t e for s y s t e m s w i t h simple m o d e s , w h i c h g u a r a n t e e t h a t t h e f i r s t s t e p of the algorithm c a n b e a c h i e v e d . The g e n e r a l i z a t i o n of t h i s a p p r o a c h to s y s t e m s w i t h multiple modes is n o t e a s y (TAR-85). I n d e e d , if Xr i s a multiple e i g e n v a l u e of o r d e r q o f a real m a t r i x D, i t s variations r e s u l t i n g from v a r i a t i o n s of t h e e l e m e n t s di] of D a r e g i v e n b y t h e s o l u tions of the following a l g e b r a i c e q u a t i o n of o r d e r q (PAR-74) :

q I [d k-I QCk)] X 1_ q~ {d q-I [Tr Q ( k ) ] A =X r } . d X r : Tr {k :E--~-, l

r

(2.4,12)

.dD}

with Q (~) = adj (XI-D). G e n e r a l l y , a multiple e i g e n v a l u e ~ r of o r d e r q g i v e s r i s e , a f t e r t h e p e r t u r b a tion dD,

to

(d~r)i, (i=l

q

. . . . .

simple e i g e n v a l u e s : Xr+

(d~r) 1 .

. . . .

kr+(dkr) i . . . . . ~ r + ( d ~ r ) q ,

where

q ) , a r e t h e s o l u t i o n s of e q u a t i o n ( 2 . 4 . 1 2 ) . From Definition ( 2 . 2 . 7 ) ,

~r is a f i x e d mode o f o r d e r q if ( d k r ) i = 0,

(i=l . . . . . q ) .

So,

from

(2.4.12),

the

following c o n d i t i o n m u s t h o l d :

q 1 s r = Tr { r ~ k=l

[dk-I

Q(X)] X= Xr

.dD}

=0

(2.4./3)

Remark 2.5. A s p e c i a l c a s e may o c c u r f o r w h i c h n o t all t h e v a r i a t i o n s ( d k r ) i , are zero b u t o n l y some of t h e m ,

s a y q' ( q .

Though

condition

(i=l . . . . . q ) ,

(2.4.13)

does not

hold, ~r is a f i x e d mode o f o r d e r q~. Since D=A+BKC, t h i s means t h a t a n o t h e r matrix K ~ ~* can be f o u n d w h e r e b y ~ is an e i g e n v a l u e of o r d e r q ' o f D. In t h i s situationp all the v a r i a t i o n s ( d ~ r ) i ,

(i=1 . . . . q ' ) , a r e z e r o a n d we a r e b r o u g h t b a c k t o t h e e a s e

for which c o n d i t i o n ( 2 . 4 . 1 3 ) h o l d s . The e v a l u a t i o n of S r r e q u i r e s t h e analytical calculation of t h e ad]oint m a t r i x of (~I-D) and of i t s v a r i a t i o n s from t h e o r d e r 1 to ( q - l ) ,

in o u r c a s e r e s u l t i n g from

6O t h e v a r i a t i o n s of t h e e l e m e n t s k.. of t h e f e e d b a c k m a t r i x K. 1] very heavy task.

Obviously,

t h i s is a

Like in t h e case o f simple f i x e d m o d e s t a n o t h e r a p p r o a c h c o n s i s t s in evaluating the

g r a d i e n t s o f kr w i t h r e s p e c t

to t h e

e l e m e n t s kij o f K. We can t h u s

u s e the

following t h e o r e m due to L a n c a s t e r (LAN-64). Theorem 2.7

(LAN-64).

Given ~r a multiple e i g e n v a l u e of o r d e r q of a real matrix

D(R) p w h e r e R is a p a r a m e t e r p c o n s i d e r t h a t t h e r i g h t a n d l e f t e i g e n v e c t o r matrices Vq a n d Wq a r e c h o s e n s u c h t h a t WWqQq=I, T h e n ,

t h e q d e r i v a t i v e s o f ~ r with r e s -

p e c t to R a r e t h e e i g e n v a l u e s of t h e matrix W d D / d R V . (For t h e c o n d i t i o n s of q q e x i s t e n c e of Wq a n d Vq, t h e r e a d e r i s r e f e r e d to ( L A N - 6 4 ) ) . In o u r

case,

D=A+BKC a n d t h e p a r a m e t e r

R must be taken

s u c c e s s i v e l y as

e a c h e l e m e n t k.. o f K. M o r e o v e r , as we h a v e a l r e a d y s e e n : 1]

~D a kij

(A ÷ )~ bikijcj) i,j a kij

:

= bic.= 0 }

a n d we h a v e t h e following r e s u l t : Theorem 2.8 ( T A R - 8 5 ) . Given t h e s y s t e m ( 2 . 3 , 1 ) a n d t h e s e t of f e e d b a c k m a t r i c e s ~*'.;~r is a f i x e d mode of o r d e r q if a n d only if all t h e e i g e n v a l u e s of e v e r y matrix WqbXCjVq,

(i,j

s u c h t h a t ki..j ~ 0 ) ,

are

e i g e n v e c t o r m a t r i c e s c o r r e s p o n d i n g to k

zero.

W'q

and

Vq a r e t h e l e f t a n d right

c h o s e n s u c h t h a t W' V = I. r q q

Note t h a t t h e a b o v e t h e o r e m is not e a s y to u s e , e i t h e r .

This r e q u i r e s ,

first,

t h e d e t e r m i n a t i o n o£ t h e e i g e n v e c t o r m a t r i c e s W a n d V a n d , t h e n , t h e computation q q of t h e e i g e n v a l u e s of ~ m a t r i c e s of dimension q x q ( w h e r e U is t h e n u m b e r of nonzero e l e m e n t s in t h e

feedback matrices

K ~ ~*).

The

c o n c l u s i o n is t h a t

the

approach

b a s e d on t h e s e n s i t i v i t y p r o p e r t i e s of e i g e n v a l u e s is n o t s u i t a b l e to d e t e r m i n e multipie f i x e d m o d e s .

2.4.3

-

Concluding

remarks

T h e two p r e c e d i n g a l g o r i t h m s a r e easily i m p l e m e n t e d . It is i n t e r e s t i n g to note t h a t t h e y can also be u s e d to c o m p u t e t h e u n c o n t r o l l a b l e a n d u n o b s e r v a b l e modes of a system by replacing the set

~* b y R m x r . T h e only p r o b l e m , w h i c h may i n d u c e a

61

difficult i n t e r p r e t a t i o n of t h e r e s u l t s , gorlthm, one h a s equal a n d ,

in t h e s e c o n d a l g o r i t h m ,

computer z e r o .

is t h e c o m p u t e r a c c u r a c y .

For t h e

first al-

to d e c i d e in w h i c h limits two e i g e n v a l u e s can b e c o n s i d e r e d as

However,

t h e p r o b l e m is to d e c i d e t h e a c c u r a c y of t h e

Davison e t a l .

(DAV-78b)

showed that

if a c l o s e d - l o o p

eigenvalue is ' v e r y close' to an o p e n - l o o p e i g e n v a l u e , t h e t r a n s f e r m a t r i x c o m p u t e d when c o n s i d e r i n g t h i s e i g e n v a l u e as a f i x e d mode is a good a p p r o x i m a t i o n of t h e t r a n s f e r m a t r i x of t h e s y s t e m .

2.5.

-

CONCLUSION This c h a p t e r d e a l s with t h e p r o b l e m of s t a b i l i z a t i o n a n d pole a s s i g n m e n t w h e n

a specified r e s t r i c t e d i n f o r m a t i o n p a t t e r n is r e q u i r e d , w h i c h c o n s t r a i n t s t h e f e e d b a c k control s t r u c t u r e . The

ap-

p r o a c h e s of Wang a n d Davison (WAN-73b) a n d of Corfmat and Morse (COR-76)

The c a s e o f a d e c e n t r a l i z e d s c h e m e of c o n t r o l is p r e s e n t e d

are

both c o n s i d e r e d .

the

Their

results

give,

in

different

terms,

the

first.

conditions

for

existence of a s o l u t i o n to t h e a b o v e p r o b l e m . The e q u i v a l e n c e o f t h e s e r e s u l t s will b e d i s c u s s e d in t h e n e x t c h a p t e r a n d we will s e e how t h e r e s u l t s of Corfmat a n d Morse can be e x p r e s s e d in t e r m s of t h e c o n c e p t of f i x e d modes i n t r o d u c e d b y Wang a n d Davison. It will also b e p o i n t e d o u t how t h e s e r e s u l t s a r e r e l a t e d to t h e c o n c e p t of controllability u n d e r d e c e n t r a l i z e d i n f o r m a t i o n s t r u c t u r e . The c a s e o f a r b i t r a r y s t r u c t u r a l c o n s t r a i n t s on t h e c o n t r o l is t h e n c o n s i d e r e d and t h e c o n c e p t of f i x e d modes is e x t e n d e d to t h i s g e n e r a l f r a m e w o r k . Finally, two a l g o r i t h m s f o r t h e e v a l u a t i o n of f i x e d modes a r e p r e s e n t e d . This c h a p t e r p o i n t s out t h e

f u n d a m e n t a l i m p o r t a n c e of f i x e d m o d e s .

Indeed,

the e x i s t e n c e of u n s t a b l e f i x e d modes i n d i c a t e s t h a t s t a b i l i z a t i o n is i m p o s s i b l e , while the p r e s e n c e of a n y k i n d of f i x e d modes r u l e s out a r b i t r a r y pole p l a c e m e n t .

Fixed

modes a p p e a r t h u s as a g e n e r a l i z a t i o n of u n c o n t r o l l a b l e a n d u n o b s e r v a b l e modes in the case of n o n - c o n s t r a i n e d c o n t r o l

(centralized control).

i n t e r e s t e d in t h i s p r o b l e m ( f o r a s u r v e y , different characterizations chapter.

of

fixed

see (TRA-84a))

modes,

which

will be

Many a u t h o r s

have been

and their r e s u l t s provide presented

in

the

next

The a n a l y s i s of t h e s e c h a r a c t e r i z a t i o n s , b o t h in t h e time a n d in t h e f r e -

quency domain,

will allow u s

to p o i n t out

the

situations

that

give r i s e

modes, and to g i v e an i n t e r p r e t a t i o n of f i x e d modes r e l a t e d to t h e i r o r i g i n .

to f i x e d

CHAPTER

CHARACTERIZATION

3

OF

FIXED MODES

3. I. - I N T R O D U C T I O N

The last fixed modes.

chapter Indeed,

pointed

out

the

fundamental importance

t h e e x i s t e n c e of a s o l u t i o n

for the

p l a c e m e n t p r o b l e m of a l i n e a r i n v a r i a n t s y s t e m r e q u i r i n g

of t h e

c o n c e p t of

stabilization or

some s t r u c t u r a l

t h e pole

constraints

on t h e c o n t r o l d e p e n d s c r i t i c a l l y on t h e p r o p e r t i e s of t h i s f i n i t e s e t of n u m b e r s . p r e s e n c e of u n s t a b l e f i x e d m o d e s i n d i c a t e s t h a t

The

s t a b i l i z a t i o n i s i m p o s s i b l e while the

p r e s e n c e of a n y s o r t of f i x e d m o d e s r u l e s o u t a r b i t r a r i l y pole p l a c e m e n t , The

first

important

problem

u n d e r some p r c s p e c i f i e d s t r u c t u r a l

of e v a l u a t i n g

fixed

modes

for

a

given

system

c o n s t r a i n t s on t h e c o n t r o l h a s b e e n t r e a t e d in the

l a s t c h a p t e r a n d two e f f i c i e n t a l g o r i t h m s h a v e b e e n p r e s e n t e d .

Our

present

o b j e c t i v e is to o b t a i n some i n s i g h t i n t o t h e b e h a v i o u r of t h e fixed

m o d e s . For t h i s p u r p o s e ,

t h e p r e s e n t c h a p t e r d e a l s w i t h t h e c h a r a c t e r i z a t i o n of fixed

m o d e s a n d a t t e m p t s to p r o v i d e an u n d e r s t a n d i n g of t h e i r o c c u r e n c e .

We will s u c c e s s i v e l y c o n s i d e r t h e c h a r a c t e r i z a t i o n s of f i x e d m o d e s w i t h a time ° domain a n d

a frequency-domain representation

t h e c o m p a r i s o n of t h e d i f f e r e n t

of t h e

characterizations,

the

system.

From t h e

reasons

of t h e

study

and

o c c u r e n c e of

f i x e d m o d e s a n d t h e c o n d i t i o n s f o r t h e i r e x i s t e n c e will b e p o i n t e d o u t . It will a p p e a r t h a t f i x e d m o d e s m a y o r i g i n a t e f r o m two d i s t i n c t s o u r c e s

: they may have a structu-

ral or

a c l e a r c l a s s i f i c a t i o n of the

a parametric origin.

different types

We will a t t e m p t to p r o v i d e

of fixed modes and

a whole p a r a g r a p h

will deal w i t h

the

specific

c h a r a c t e r i z a t i o n s of f i x e d m o d e s a r i s i n g from s t r u c t u r a l p a r t i c u l a r i t i e s .

The last paragraph

of t h i s

t e r i z a t i o n s of f i x e d m o d e s . I n d e e d ,

chapter

i s an o v e r v i e w of t h e

g r a p h s p r o v i d i n g a n e f f i c i e n t w a y to e x p l o r e v a r i o u s s t r u c t u r a l t h e c o n c e p t s of g r a p h - t h e o r y .

graphical charac-

dynamic s y s t e m s can be r e p r e s e n t e d

by directed

properties by using

N u m e r o u s r e s u l t s h a v e b e e n o b t a i n e d for t h e problem

of c h a r a c t e r i z i n g f i x e d m o d e s in t h i s g r a p h - t h e o r e t i c f r a m e w o r k .

63 3.2. - CHARACTERIZATION IN TERMS OF TRANSMISSION ZEROS This c h a r a c t e r i z a t i o n g i v e s an i n t e r e s t i n g i n s i g h t into t h e r e a s o n s of o c c u r e n c e of fixed modes a n d p r o v i d e s a simple e x p l a n a t i o n f o r t h e i r e x i s t e n c e in t e r m s of t h e familiar c o n c e p t o f s y s t e m p o l e s a n d z e r o s . I t was p r o v i d e d i n d e p e n d e n t l y b y s e v e r a l authors (HUJ-84)

(TAO-84)

and f o r m u l a t e d u s i n g

(DAV-85)

(SER-82)

different methods.

(VID-83) w h o s e r e s u l t s a r e p r o v e d

This c h a r a c t e r i z a t i o n can e q u i v a l e n t l y b e

i n t e r p r e t e d a n d u s e d e i t h e r in t h e f r e q u e n c y - d o m a i n c o n t e x t o r in t h e t i m e - d o m a i n context s i n c e

transmission

zeros are

a c o n c e p t as

well d e f i n e d b y

m e a n s of t h e

system m a t r i x as b y means of t h e t r a n s f e r f u n c t i o n matrix (see A p p e n d i x 1). A l t h o u g h t h e r e s u l t s a r e d e v e l o p p e d in a t i m e - d o m a i n f r a m e w o r k in (DAV-85a) and in a f r e q u e n c y - d o m a i n f r a m e w o r k in

(SER-82)

(VID-83), t h e i r e q u i v a l e n c e a p -

p e a r s c l e a r l y in (HUJ-84) a n d ( T A O - 8 4 ) . In all t h e p a p e r s b u t expression relating

the

(DAV-85),

t h i s c h a r a c t e r i z a t i o n is o b t a i n e d

c l o s e d - l o o p c h a r a c t e r i s t i c polynomial to t h e

using

an

controller pa-

rameters a n d z e r o polynomials of c e r t a i n c o n s t i t u a n t s u b s y s t e m s . C o n s i d e r t h e c l a s s of l i n e a r t i m e - i n v a r i a n t s y s t e m s d e s c r i b e d b y t h e following state=space r e p r e s e n t a t i o n :

{

~:(t) = A x ( t ) + B u ( t ) y(t)

(3.2.1)

C x(t)

where x ( t ) ~ Rn ,

u ( t ) ~ R m, y ( t ) ~ R r a r e t h e s t a t e , i n p u t a n d o u t p u t ,

respecti-

vely and A, B, C a r e real m a t r i c e s of a p p r o p r i a t e d i m e n s i o n s . Many l a r g e scale s y s t e m s a p p e a r to b e g e o g r a p h i c a l l y d i s t r i b u t e d or composed of an i n t e r c o n n e c t i o n of s u b s y s t e m s . T h e s e c h a r a c t e r i s t i c s can be t a k e n i n t o a c c o u n t in the model o f t h e s y s t e m b y a p a r t i t i o n i n g of t h e i n p u t a n d o u t p u t v e c t o r s r e s u l ting in s e v e r a l c o n t r o l a n d o b s e r v a t i o n s t a t i o n s :

{

~(t) : A x ( t ) +

S £ g i ui(t) i=l

Yi(t) = C i x(t)

(3.2,2) ( i=l,...,S )

which is r e l a t e d to ( 3 . 2 . 1 ) b y :

B = (B 1 ..... B S) C'= (C' 1 .... ,C' S)

64

u'(t)

= (U'l(t) ..... U's(t))

y'(t)

= (y'l(t) ..... y's(t))

w i t h u i ( t ) ~ Rmi, Yi(t) E R r i , S m = i Zl_

and

(i=1 . . . . .

S r = i Zl_

mi ,

S),

r i, w h e r e S is t h e n u m b e r

of control and observation

stations. In order traints

to p o i n t o u t t h e p r o b l e m s

on t h e c o n t r o l ,

arising

s p e c i a l l y from t h e s t r u c t u r a l

we make t h e a s s u m p t i o n

that

system

(3.2.1)

cons-

(3.2.2)

is g l o -

bally controllable and observable. The

following

development

is

taken

from

(TAO-84)

in

which

the

chosen

n o t a t i o n s c a n b e s i m p l y m a n i p u l a t e d a n d do n o t l e a d to h u g e f o r m u l a . Consider by

the

system

(3.2.1)

for which the transfer

function matrix

is g i v e n

:

= C

W(p)

(pI-A) -I B =

where N(p)=C adj(pI-A) is t h e c h a r a c t e r i s t i c The called

set

function

matrix

subscripts

q,s

(3.2.3)

B is t h e r x m n u m e r a t o r

polynomial of the system

of i-input

i-dimensional

N(p) ~(p)

i-output

subsystems

subsystems

and

denoted

transfer

function matrix and

@ (p)

(3.2.1). of by

(3.2.1), (Cq,A,B s)

(i=l . . . . . r a i n ( r e , r ) ) , or

by

their

are

transfer

Wi ( p ) = C i ( p I - A ) - l B i = W . i ( p ) , where for notation convenience the q,s q s j h a v e b e e n r e p l a c e d b y j." C lq, ( q = l , . . . , q c ) , are the set of submatrices

f o r m e d from i r o w s o f t h e m a t r i x C a n d Bis, ( s = l , . . . . S b ) , a r e t h e s e t o f s u b m a t r i c e s f o r m e d from i c o l u m n s o f t h e m a t r i x B, w h e r e

r'

qc : ( r - i )

m:

:i:

Sb - (m-i) 'i'

Wlj(p), (j--1 . . . . . ji ) , a r e t h e s e t o f i x i s u b m a t r i c e s o f W(p) a n d j i = q c . Sb . As

was

independent

pointed of the

out

dynamic

in

Chapter or

static

2,

we k n o w

nature

of the

that

the

s e t o f f i x e d m o d e s is

controller.

Therefore,

we will

c o n s i d e r a s t a t i c c o n t r o l law o f t h e f o r m "-

u = K y + v

(3.2.4)

65

where v is the mxl external any a r b i t a r y

input

The closed-loop characteristic

*c(p) where

[ "l

vector

and the

constant

mxr matrix

K can take

structure. p o l y n o m i a l is t h u s o b t a i n e d a s :

(3.2.5)

*(p) stands

for det (.).

The determinant

in

(3.2.5)

+

Z

can be expanded

in terms of the principal

minors

of ( - W ( p ) K) to g i v e :

•c(p) where hi(p)

= ~

(p) (1

is t h e

Cauchy formula

r

i=l

hi(p))

(3.2.6)

sum of principal

(GAN-79),

these

ith order

minors

minors can be written

of

(-W(p)

K),

From

Binet-

in t e r m s o f d e t e r m i n a n t s

of

s u b m a t r i c e s o f W(p) a n d K t a s : Ji

hi(p) = j : [

This r e s u l t

can equivalently

b e s t a t e d in a t i m e - d o m a i n f r a m e w o r k .

tion of t h e z e r o p o l y n o m i a l o f a s y s t e m i-dimensional subsystems

is g i v e n b y

From the defini-

(KWA-72), the set of zero polynomials of the

:

B si

I PI-A

q

where the roots of Z((p) are the set of transmission

zeros of the i-dimensional sub-

systems. From ( 3 . Z . 6 ) , - Frequency

c(P)

(3.2.7)

and

(3.2.8),

t h e two f o l l o w i n g r e s u l t s

are equivalent

:

domain :

(p) +

r

Ji

z

z

i=lj=l

(-l)ilKjl } ~(p)lWi(p)l J

(3.2.9)

66

- Time domain ;

r

¢c(p)=

Ji

qb(p)+ r. Z (-I) i [K';I ZI(p)

(3.2. l O)

i:l j:l

When i rows a n d i columns have b e e n selected in K' to form K]wi,Cqi and Bis denote the submatrices of C and B obtained by selecting the same rows and columns. The

subsystem

of (C,A,B)

or W(p)

corresponding

to the submatrix

K'~ is thus

denoted b y (Cq, i A,Bis) or e q u i v a l e n t l y Wi(p). Now, c o n s i d e r that the controller (3.2.4) is c o n s t r a i n e d to have the following decentralized s t r u c t u r e :

(3.2.11)

K = b l o c k - d i a g . ( K 1 . . . . . KS) where K. ~

Rmixri, (i=1 . . . . . S).

1

The fixed modes are easily c h a r a c t e r i z e d u s i n g the above development.

3 . 2 . 1 . - T a r o c k ' s r e s u l t s (TAO-84) Theorem 3.1.

The n e c e s s a r y a n d sufficient condition for a pole ~0 of t h e system

( C , A , B ) or W(p) to be a fixed mode with r e s p e c t to the controller K in (3.2.11) is that : [~,0 i- A

BSi ]

.

rank

LC'q

<

n+i ,

(i=l,...,min(rn,r))

(3.2.12)

0

or equivalently :

~(~o )

,J wj(~o)J--, 0 ,

(i=1 . . . . . rain (re,r))

(3.2.13)

67 where (Cio,A,B s) or We(p) are the subsystems of (C,A,B) corresponding to non-singular submatrices of K I. The above

theorem

states

t h a t )~0 is a f i x e d

m o d e of

respect to K in (3.2.11) if and only if all. the subsystems responding to non-singular submatrices K~1

(C,A,B)

or

W(p)

with

(Cq,A,B s) or W~(p) cor-

of K ~ have a common

transmission zero

corresponding to the system pole ~0" This result is straightforward from formulae (3.2.9) or (3.2.10).

3.2.2. - H u and Jiang results (HUJ-84) Hu a n d J i a n g

state

their

results

using

some p r e l i m i n a r y

definitions

presented

below. From the integers m i and ri specifying the controller structure in (3,2.11), define : i

m 0 = 0,

~i = j-_~0 mj~,

r 0 = 0,

i Fi = jZ= 0

such that the input

U

=

(i=l . . . . . rj ,

and output

!m

--

m = ~S

Ii] -'-"

r = ~S vectors

U

i

are partitioned

in t h e following w a y :

u~i_l+l =

,, ,,

u~ i YYi] 1 1 y =

y = r

Let n o n n e g a t i v e

integers

i_~is

1

s i a n d qi" ( i = l , . . . . S), s a t i s f y

S

0-~s i . < ~ i ,

S)

S

0

, 0,
i_E1 q~ 0

a n d j for s i ) 0 a n d qi ) O, d e f i n e t h e p o s i t i v e i n t e g e r s

:

the condition

:

68 f i ) j ' (j=l . . . . . s i)

(3.2.14)

gi,j) (j=l . . . . . qi ) "~i-1 +1 "~ fi,1 < "'" < f i , s i

x< ~ i

~i-1 +1 ~ gi,1 < "'" < g i , q i

~ Ti

The following set of s u b s y s t e m s of (C,A)B) or W(p) is now defined : (~,A,~)

(3.Z.l~)

W(p) = : ( p i - A ) - l ~

or

~i : [hfi,l ..... bf~,~i ] ")i

L~SJ

gi,q i

with :

=~=

--i tl

[i]

I!,1 =

Luf u fi,si

-1

I

Yg,i,l

-yi =

=

Ygi,qi

and = b l ° c k - d i a g ' ( ~ l . . . . ' ~ S )) Note that when s.=0, ~ i 1

~i

~i

~

Rsixqi'

(i=l . . . . . S)

d i s a p p e a r and when qi=0, ~ i

(3.2.16) ~i d i s a p p e a r . In

both c a s e s , Ki d i s a p p e a r s .

Definition 3.1. The s u b s y s t e m s ( C , A , B ) or W(p) in (3.2.15) a s s o c i a t e d with the controller ~ in (3.2.16) are called the normal s u b s y s t e m s of ( C , A , B ) or W(p). In p a r t i c u l a r ) when si=qi) t h e y a r e called the n o n s i n g u l a r l y normal s u b s y s t e m s .

69 The set of normal subsystems is denoted by N.sub.(C,A,B)

and the set of

nonsingularly normal subsystems by N.N.sub. (C,A,B). With t h e s e d e f i n i t i o n s , Hu a n d J i a n g s t a t e t h e following r e s u l t s : Theorem 3.2. ~0 i s a f i x e d mode of ( C , A , B )

with r e s p e c t to K i n ( 3 . 2 . 1 1 )

if ~0 is a f i x e d mode of all t h e n o r m a l s u b s y s t e m s

(C,A,B)

if a n d o n l y

with r e s p e c t

to ~ i n

(3.2.16) ; i.e. :

A(C,A,B,K) Theorem 3 . 3 .

(U,A,~) ~

The necessary

D

N.sub.(C,A,B)

h (~,A,B,K)

a n d s u f f i c i e n t c o n d i t i o n f o r a pole ~0 of t h e

system

( C , A , B ) or W(p) to b e a f i x e d mode with r e s p e c t to K i n ( 3 . 2 . 1 1 ) is t h a t •

rank

I°:

< n +

0.2.is)

1 si

0

or e q u i v a l e n t l y • d~(~0)

det ~

(X0) = 0

(3.2.19)

for all t h e n o n s i n g u l a r l y n o r m a l s u b s y s t e m s of ( C , A , B )

or W ( p ) .

T h i s t h e o r e m s t a t e s t h a t 10 i s a f i x e d mode of ( C , A , B )

o r W(p) i f a n d o n l y i f

~0 is a common t r a n s m i s s i o n z e r o of all t h e n o n s i n g u l a r l y n o r m a l s u b s y s t e m s

('C,A,~}

or ~ ( p ) w h i c h c o r r e s p o n d s with a pole of t h e s y s t e m . T h e s i m i l a r i t y of t h i s r e s u l t w i t h t h e o n e of T a r o c k is o b v i o u s . I n d e e d , it is 5 clear t h a t ( C , A , B ) or ~ ( p ) of d i m e n s i o n Z s. c o r r e s p o n d to t h e s u b s y s t e m s i=! 1 ( Ci , A ) ,B n o o r W~(p)j of d i m e n s i o n i while t h e m a t r i c e s Ki in (3.2o16) w i t h 8i=cli correspond

to

the

non

singular

matrices

t h e r e f o r e i d e n t i c a l to formula ( 3 . 2 . 1 2 ) equivalent.

and

K!. Formula ( 3 . 2 . 1 7 ) a n d ( 3 . 2 . 1 8 ) a r e l ( 3 . 2 . 1 3 ) a n d T h e o r e m s 3.3 a n d 3.1 a r e

T h e a d v a n t a g e of Hu a n d J i a n g f o r m u l a t i o n is t h a t t h e s u b s y s t e m s to b e

c o n s i d e r e d a r e s y s t e m a t i c a l l y d e f i n e d b y t h e s e t of i n t e g e r s in ( 3 . 2 . 1 4 } . T h e same r e s u l t was also p r o v i d e d p r e v i o u s l y b y S e r a j i ( S E a - 8 2 ) a n d V i d y a s a gar

a n d Wiswanadham (VID-83) i n a f r e q u e n c y - d o m a i n c o n t e x t .

b u t i o n s of t h e s e

papers

Though the contri-

p r o v i d e t h e same a b o v e e s t a b l i s h e d r e s u l t u s i n g t h e same

e x p a n s i o n of t h e c l o s e d - l o o p c h a r a c t e r i s t i c polynomial g i v e n i n ( 3 . 2 . 9 ) they differ by the notations and the formulations which are used.

and (3.2.10),

70 Their results concepts

are briefly presented

are pointed

in the two next paragraphs.

out and the originalities

The equivalent

which can be advantageously

exploited

are shown.

3.2.3.

- Vidyasagar

and Wiswanadham results

The main result polynomial,

whose

of t h i s p a p e r

zeros are the

the control structure

is presented

as a characterization

f i x e d m o d e s of t h e

system

(3.2.1)

of t h e f i x e d

with respect

to

specified by K in (3.2o11)o

D e f i n e t h e s e t s M= {1 . . . . . m} of elements

(VID-83)

of the

set

(*).

a n d R={1 . . . . , r )

If I = J,

then

and denote by

W[I3] d e n o t e s

*

the number

the minor of W(p)formed

,rom ,ho

'n

way and Z

=

(p) W 3 "

Theorem

3.4.

The

fixed

(3.2.11)

is g i v e n b y

polynomial

F(p)

of s y s t e m

(3.2.1)

with respect

to

K in

:

F(p) = g.c,d

(p) ,

Z

~2

S

L3ilU Ji2 u

u

(3.2.19)

is_]

with : Iij c Rj = {'~]-1 + 1 . . . . . ~]} Jijc

(i=l . . . . .

Mj = { ~ j - 1 + 1 . . . . . ~ j }

and [l~ijl} --]lJijl

,

It is obvious Jiang stated

that

in T h e o r e m s

I I i l u Ii2 u . . .

S)

(3.2.20)

(j=~ . . . . . s ) .

this

result

is equivalent

3.1 and 3.3.

to those

of Tarock

and

Hu a n d

Indeed,

u Iis 1

Z

! / L J i l u 3i2 u . . . v J i s J is a polynomial whose roots

_S [[ Iij[[ c ° r r e s p ° n d i n g j-~l to some 5 ( p )

z e r o s of some s u b s y s t e m

of dimensiol

to some nonsingular m a t r i x K [ I ] . T h i s p o l y n o m i a l c o r r e s p o n d s 3 notations.

in Tarockts

The subsystems Jiang result

are the transmission

to b e c o n s i d e r e d

b y t h e s e t s of i n t e g e r s

are

Jil a n d

specified Ill in

J

sets of integers

{fi,1 . . . . .

fi,s[ }

and

(gi,1 .....

i n a s i m i l a r w a y a s i n Hu a n

(3.2.20)

which correspond

J

gi,q~

in (3.2.14),

respectively.

to t h e

71 Note t h a t

this

work appeared

one year

before

the

two other

ones

in the lite-

rature.

3.2.4.

- Seraji's results This

result

main r e s u l t to b e

is even

is given

considered

The result

very

back

controller

that the controller

a

to t h e o n e o f V i d y a s a g a r

easily

controllers

determinated.

the general is treated

is brough

can thus

Consider

anterior

for diagonal scalar

is

o r i g i n a l i t y to b r i n g decentralized

(SER-82)

Then,

applying

into a diagonal scalar

whose

structure

is

Wiswanadham.

The

set of subsystems

approach

presents

situation.

a transformation

be applied on the transformed

controller

the

case to this particular

by

back

and

for which the

Any arbitrary

on the

structure

the

system

such

in the new basis.

system. specified

by

the

output

feedback

matrix in the following way :

Ks = block-diag.(k

1.....

k S)

ki ~ R

Seraji gives the following result Theorem

3.5.

system (3.2.1) subsystems

The

necessary

with respect

of dimension

and

(3.2.21)

(equivalent

sufficient

to K s i s t h a t

to the three

condition

anterior

f o r ~0 to b e

:

a fixed

X0 i s a c o m m o n t r a n s m i s s i o n

( i = l , . . . . S) f o r m e d s e l e c t i n g

ones)

the same inputs

mode of

z e r o o f all t h e and outputs

of

the system. Now,

if the

controller

r i ~ 1, S e r a j i a p p l i e s

used for the transformed Consider

K

structure

a transformation

by

K as in such

(3.2.11)

that

where

Theorem

m i ~, 1,

3.5 can he

system,

the transformation

=PK

is given

on the system

matrix P :

s

(3.2.22)

W(p) = Wt(p) P P = K diag.

1

l

{F- .... ' k--} "l S

The fixed modes of the system fixed modes of system Theorem 3.5.

Wt(p)

W(p)

with respect

with respect

to K a r e

then

given

to Ks , which can be determined

by

the

by using

72 Despite

of the

termination,

originality

from the original

z e r o s h a v e to b e c h e c k e d

3.2.5.

-

system,

Wang

by

approach,

is

of the subsystems

clear

that

the

direct

de-

for which the transmission

(DAV-85)

the same characterization using

it

s e e m s to b e m o r e c o n v e n i e n t .

Davison and Wang results

Recently, and

of this

a more

direct

of fixed modes was established

method.

Their

result

is

by

Davison

straightforward

to t h e

application of a lemma taken from (AND-82).

Lemma 3.1

(AND-82).

G i v e n M0 ~

e R n , a n d l e t qi b e a r e a l s c a l a r

Rnxn,

d e t M0 ~ 0, l e t Mi = b i e'.x

where

b i, c i

all d i s j o i n t v a l u e s

o f i 1,

; then

S

d e t (M 0 + i~l qi Mi) = 0

~tqi~: R, ( i : l , . . . , s )

if a n d o n l y if t h e f o l l o w i n g c o n d i t i o n s

det

M0

bi I

bi 2

• . .

bit

c!

0

0

...

0

0

0

. . .

0

0

0

...

0

t!

c!

are satisfied

:

= 0

• [2

c~ L tt

(i 1 = 1 , . . . , s )

;

(i 2 = 1 . . . . s )

;

...

;

(i t = 1 . . . . .

s)

for

i 2 . . . . ,i t ; ( t = l , . . . , s ) . Now, t h e s y s t e m

{ and

(3.2.1)

can be rewritten

~(t)

= A x(t)

y(t)

= (Cl,C 2 ..... Cr)' x(t)

the

controller

matrix K in

+ (bl,b 2 .....

information

b m) u ( t )

flow c o n s t r a i n t

(3.2.11) c a n b e r e p r e s e t e n d

{(il,Jl),

(i2,J2) .....

as :

(is,is) }

represented

by

the

output

by the following pairs of integers

feedback :

(3.2.33)

73 where i k 6 {1 . . . . . m } , jk ~ { 1 . . . . . r ) s e n t s a n o n z e r o e l e m e n t i n K. T h e f i x e d modes of ( 3 . 2 . 1 ) the p r o p e r t i e s ~0 ~

o

, ( k = l . . . . . s)

: i.e.,

each pair (ik,Jk) repre-

with r e s p e c t to K i n ( 3 . 2 . 1 1 )

(3.2,23)

have thus

: (A)

(3.2.24) S

det {A- XoI ÷ k__£1qk bi k Cik }

=0

V q k ~ R , (k=l . . . . . s)

a n d from Lemma 3.1, t h e following r e s u l t is s t r a i g h t f o r w a r d : Theorem 3.6 ( D A V - 8 5 ) . T h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n for XOE fixed mode of t h e s y s t e m ( 3 . 2 . 1 ) with r e s p e c t to K in ( 3 . 2 . 1 1 ) a t r a n s m i s s i o n z e r o of all of t h e following s u b s y s t e m s

(3.2.23)

o

(A) to b e a is t h a t k 0 is

:

Ik1 (

lk2 Lc, ,

, A , (b i

, bi kl

..... k2

bi

)

(3.2.25)

kt

]k t where ( k t = l , 2 , . . . , s + l - t ) )

( k 2 = k l + l , k l + 2 . . . . . s+Z-t) ; ( k t = k t _ l + l , k t _ l + 2 . . . . , s )

( t = l , 2 . . . . . rain ( m ) r ) ) I t is c l e a r t h a t

: for

(3.2.26) the above result

is e q u i v a l e n t

to t h o s e

stated

before.

The

o r i g i n a l i t y of t h i s a p p r o a c h s t a n d s on t h e f a c t t h a t it a r i s e s d i r e c t l y from t h e a p p l i cation of Lemma 3 . 1 . Example 3 . 1 .

Consider the linear dynamic system described in the frequency-domain

by t h e fo]lowing t r a n s f e r m a t r i x : l

p(p_ i )

0

0

p-I

0

0

0

0

p-I

i p,-I

- -

I

w(p)

i 1~1

1

0

74 for which

q ( p ) = p ( p - 1 ) 3.

T h e c o n t r o l l e r s t r u c t u r e i s s p e c i f i e d b y t h e following o u t p u t f e e d b a c k m a t r i x :

!Ii K =

Adopting

the

k12

k13

0

0

0

k24

0

0

k34

notations

of

and

Vidyasagar

I(' =

and

kll

0

0

k12

0

0

k13

0

0

0

k24

k34

Wiswanadham,

the

]

J

polynomial

Z

g i v i n g t h e t r a n s m i s s i o n z e r o s of t h e s u b s y s t e m s c o r r e s p o n d i n g to t h e n o n s i n g u l a r s u b m a t r i c e s of K' a r e t h e following : * One-dimensional subsystems •

zI:]0

* Two-dimensional subsystems :

All t h e s e s u b s y s t e m s h a v e a common t r a n s m i s s i o n z e r o at ~0=1 which c o r r e s p o n d s to a pole of t h e s y s t e m . T h e r e f o r e , k0=l i s a f i x e d mode.

75 3.2.6.

- Comments

From the above characterization of fixed modes, several interesting results can immediately be deduced. T h e y are summarized in the following corollaries.

Corollary 3.1 ( T A O - 8 4 ) .

C o n d i t i o n s (i) a n d (ii) below a r e n e c e s s a r y for a s y s t e m to

have a fixed mode a t ~0 : i- The

entries of W(p)

corresponding

to nonzero

elements of K' must have

pole-zero cancellations at P=~0"

ii - For the c a s e m = r, P=~0 must be a transmission zero of the entire system.

Corollary

,3,2

(TAO-84).

If a n y

submatrices of K' is minimum

of

the

phase,

subsystems

corresponding

to nonsingular

the system has no unstable fixed modes.

(All

transmission zeros of m i n i m u m phase systems lie in the open left half plan).

I t is i n t e r e s t i n g

to n o t i c e t h a t f o r s i n g l e - i n p u t s i n g l e - o u t p u t s y s t e m s a t r a n s -

mission z e r o is t h e f r e q u e n c e Therefore,

for which the transfer

function numerator

vanishes.

a f i x e d mode c a n b e v i e w e d as t h e f r e q u e n c e w h i c h c u t s t h e i n f o r m a t i o n

flow b e t w e e n t h e i n p u t a n d o u p u t of some s u b s y s t e m s .

T h i s can p h y s i c a l l y e x p l a i n

why a d e c e n t r a l i z e d c o n t r o l may fail in t h e s t a b i l i z a t i o n o r pole a s s i g n m e n t t a s k .

3.3.

ALGEBRAIC

The

CHARACTERIZATIONS

following

characterizations

: TIME-DOMAIN

are stated

by using

a state-space

represen-

tation of t h e s y s t e m .

3 . 3 . 1 . - Matrix r a n k t e s t c h a r a c t e r i z a t i o n First,

let u s c o n s i d e r a p a r t i t i o n e d s y s t e m in t h e form ( 3 . 2 . 2 )

a n d t h e follo-

wing d e c e n t r a l i z e d c o n t r o l i n a c c o r d a n c e with t h e d e c o m p o s i t i o n of t h e s y s t e m : I u i ( t ) = Qi z i ( t ) + Ki Yi ( t ) + v i ( t ) (i=l ..... S)

(3.3.1)

~i(t) = S i z i ( t ) + Ri Yi (t) where z i ( t ) is t h e s t a t e of t h e i t h c o n t r o l l e r a n d v i ( t ) is t h e i t h e x t e r n a l i n p u t ,

76 In this context,

the following interesting algebraic characterization of de-

centralized fixed modes was provided by Anderson and Clements (AND-81a):

Theorem 3.7.

Let be the s e t ~ { 1 , . . . , S }

s u b s e t s na = { i l , . . . , i

k}

and

B(I = [Bil .... , Bik]

C

L

" Define also the matrices :

B B = [Bik+ 1 ""'Bi s ]

cB:

O~

and define a p a r t i t i o n of ~ into disjoint

~8 = ( i k + l ' ' " ' i s }

I i ik+l

Lis J

'k

Consider t h e system

(3.2.2)

a n d the decentralized control ( 3 . 3 . 1 ) .

n e c e s s a r y a n d s u f f i c i e n t condition for A0

(3.2.2) is that t h e r e exists at least one complementary s u b s y s t e m 2.6, § 2 . 2 . 3 b , C h a p t e r II) s u c h that :

rank

Then, a

o (A) to be a decentralized fixed mode of (see Definition

( n

(3.3.2)

cB Remark 3.1. Condition (3.3.2) is e q u i v a l e n t to 0 (C fl , A , B a )#1 so t h a t an immediate connection a p p e a r s with the r e s u l t s of Corfmat and Morse (COR-76b) (see Theorem 2.2, C h a p t e r I I ) . Example 3.2. C o n s i d e r the following p a r t i t i o n e d system ;

A=

i1o ]

CI= [1

C3=

1

0

0

1

0

0

0

0

0

0]

EOo o1 o1 0,1

B1 =

[i] Iil

B2 =

B3

77 for w h i c h we c o n c l u d e d Example 2 . 1 , This

§ 2.2.3a,

result

by using Chapter

is v e r i f i e d

t h e d e f i n i t i o n o f Wang a n d D a v i s o n

II) t h a t i t h a s a d e c e n t r a l i z e d

by

using

I n d e e d , i f we c h o o s e ~a = {3} a n d

rank

i -A C1

the characterization

(WAN-73)

(see

f i x e d m o d e at X0 = I . stated

in T h e o r e m

3.1.

~ = { 1 , 2 } , we h a v e t h e f o l l o w i n g :

= 3 < 4

C2 which is s u f f i c i e n t to v e r i f y c o n d i t i o n

(3.3.2).

Moreover,

for this example,

we h a v e

also :

I-A

B1

B3

rank C2

0

0

II-A

B2

B 3-

rank c I

o

A Fortran

= 3 < 4

for

~a = { 1 ' 3 }

and

~B = (2}

= 3 < 4

for

~a = (2,3}

and

~'8 = ( 1

o

routine for detecting the real decentralized

which is b a s e d on A n d e r s o n a n d C l e m e n t s ' t e s t , This result

gives some insight

I n d e e d , it p r o v i d e s system

(3.2.2)

stations

a

and

}

into the reasons

an immediate interpretation

if t h e r e

exists

~ such

stations and unobservable

simultaneously

by the other one.

in A p p e n d i x 4.

of occurence

of fixed modes.

• )'O i s a d e c e n t r a l i z e d

a disjoint partition

t h a t ;~0 is

fixed modes of a system,

is p r o v i d e d

of the

system

uncontrollable

fixed mode of

in two a g g r e g a t e by

one

T h i s s i t u a t i o n is i l l u s t r a t e d

3.1 : Aggregate station %0

a

A g g r e g a t e station

uncontrollable a

U,

lI

•.'

Uik

13

X0 unobservable ~

Yi I "" Yik

Uik+ 1 "'" Uis

SYSTEM

F i g u r e 3.1

Yik+ l "'" Yis

4L

of by

these Figure

78 This interpretation is

simultaneously

l e a d s to t h e following i n t e r e s t i n g

controllable and

observable

by one

c o n c l u s i o n ". if k 0 ~ o (A)

single

station

i~{

1 . . . . . S),

then ~ is not a decentralized fixed mode of the system.

U n f o r t u n a t e l y t t h i s c h a r a c t e r i z a t i o n does n o t help p a r t i c u l a r l y in t h e c o m p u t a t i o n of f i x e d modes

s i n c e it r e q u i r e s

t h e e v a l u a t i o n of t h e r a n k

of 2S-2 m a t r i c e s

whose d i m e n s i o n s a r e s u p e r i o r to t h e o r d e r of t h e s y s t e m . A m e t h o d to o v e r c o m e t h i s d e s a d v a n t a g e is p r e s e n t e d

b y P e t e l a n d Misra (PET-B4) in t h e c a s e for w h i c h A h a s

a s u p e r i o r H e s e n b e r g form. T h e y p r o v i d e a c o n d i t i o n w h i c h is n u m e r i c a l l y e q u i v a l e n t to c o n d i t i o n

(3.3.2)

certain transfer

a n d in w h i c h f i x e d modes a p p e a r

functions.

They

apply

thus

their

as t h e t r a n s m i s s i o n z e r o s of

algorithm

for t h e

e v a l u a t i o n of

transmission zeros. C o n s i d e r now t h e g e n e r a l c a s e of a n o n p a r t i t i o n e d s y s t e m some a r b i t r a r y

feedback structure

(3.2.1)

constraints have been specified.

for which

For e a c h i n p u t

u i , we d e f i n e t h e i n d e x s e t : :Ii = {j ~ {1 . . . . . r} / t h e f e e d b a c k is allowed from yj to u i} As we a l r e a d y saw in p a r a g r a p h

(3.3.3)

2.3 of C h a p t e r I I , t h e l i n e a r t i m e - i n v a r i a n t dynamic

controller associated with these feedback constraints is given by : I

Z ~'i( t ) = Si z i ( t ) + j e J

rij y j ( t ) (i=l . . . . . m)

[ui(t)

q i' z i ( t ) + j ~Y' l

(3.3.4)

kij y j ( t ) + v i ( t )

w h e r e z i ( t ) is t h e s t a t e of t h e c o n t r o l l e r a n d v i ( t ) is t h e i t h e x t e r n a l i n p u t .

Sip Cli

a n d r.. a n d k.. a r e c o n s t a n t matrices~ v e c t o r s a n d s c a l a r s of a p p r o p r i a t e d i m e n s i o n s . 11

xl

In t h e g e n e r a l c o n t e x t ,

an e x t e n t i o n of t h e p r e v i o u s c h a r a c t e r i z a t i o n of d e c e n -

t r a l i z e s f i x e d modes was s t a t e d b y Pichai e t al. ( P I C - 8 4 ) p r o v i d i n g a g e n e r a l m a t r i x r a n k t e s t c h a r a c t e r i z a t i o n of fixed m o d e s . T h e o r e m 3 . 8 . For a n a r b i t r a r y s u b s e t

:

I = ( i 1 . . . . . i k } c M = (1 . . . . . m} define I = i ~r M - t

~i = {Jl" - - - , l "q }

w i t h Ji as d e f i n e d in ( 3 . 3 . 3 ) . the

complementary

(k--1,...,m-1).

subsystems

T h e n ~ similarly to t h e c a s e of d e c e n t r a l i z e d c o n t r o l , of

system

(3.2.1)

are

given

by :

(Cj,

A,

BI) ,

79 Then, (3.2.1)

a

with

necessary

respect

subsystem such that

rank

to

and the

sufficient control

condition

(3.3.4)

is

f o r X0

that

to

there

be

exists

a

fixed

mode

of

a complementary

:

[:o I' 1

< n

(3.3.5)

J

Example 3 . 3 . C o n s i d e r t h e f o l l o w i n g s y s t e m w i t h 3 i n p u t s

-1

A=

-2

B=

-3

C=

0

0

1

1]

1

0

0

0

0

1

0

1

for w h i c h t h e f e e d b a c k s t r u c t u r e

Jl = {2}

constraints

1

0

1

1

I

0

0

1

0

0

1

are given by

:

:

; J 2 = {1,3} ; J3 = {I}

c o r r e s p o n d i n g to t h e f e e d b a c k m a t r i x

0 K =

0

and 3 outputs

k21 Lk31

k12

0

0 0

k23 0

:

]

T h e n , if we c o n s i d e r t h e s u b s e t

I = {1,2} , we o b t a i n 5 = {1}

a n d we h a v e ,

for

kO=-3 : I rank

XO I-A

[c I

T h e r e f o r e , X0=-3 is a f i x e d m o d e . In t h i s g e n e r a l case of d e c e n t r a l i z e d tioning of t h e

system

case,

the interpretation

control.

Indeed,

does not appear

so clearly as for the

we h a v e to t a k e i n t o a c c o u n t t h a t t h e p a r t i -

in t w o a g g r e g a t e d

stations

I and

J is not

disjoint.

The

in-

80 t e r p r e t a t i o n can b e g i v e n as follows : a n e c e s s a r y c o n d i t i o n f o r X0 ~

o (A) to b c a

f i x e d mode is t h a t t h e r e e x i s t s a s u b s e t of i n p u t s w h i c h c a n n o t c o n t r o l X0 a n d that t h e s u b s e t of o u t p u t s i n v o l v e d in t h e f e e d b a c k c o n t r o l of t h e c o m p l e m e n t a r y s u b s e t o f i n p u t s c a n n o t o b s e r v e X0" This s i t u a t i o n is i l l u s t r a t e d b y F i g u r e

I UI X 0 uncontrollable Inputs Ul [ XO controllable ....

3.2.

Yj

1

SYSTEM

X0 observable Y3

Outputs

X unobservable 0

. . . .

F i g u r e 3.2

with UI v U~ = {u 1 . . . . . u m}

a n d Y j u Y~ = {Yl . . . . . Yr }

a n d ~ I : C o n t r o l l e r o f t h e s u b s e t o f i n p u t s UI ~T : C o n t r o l l e r of t h e s u b s e t of i n p u t s UT

3.3.2.

-

Recursive

characterization

Consider again decentralized

feedback

a

partitioned controller

s y s t e m in (3.3.1).

In

the

form

this

context,

(3.2.2)

t e r i z a t i o n o f f i x e d modes p r o v i d e d b y Davison a n d O z g u n e r

the

and

its

associated

following c h a r a c -

(DAV-83) p r e s e n t s

the

i n t e r e s t to s h o w e x p l i c i t e l y t h a t t h e e x i s t e n c e of f i x e d modes for a S - s t a t i o n system always r e d u c e s to t h e e x i s t e n c e of f i x e d modes f o r a s e t of 2 - s t a t i o n s y s t e m s (by r e c u r s i v e a p p l i c a t i o n of T h e o r e m 3.9 w h i c h b r i n g s b a c k S to ( S - l ) ) .

Theorem 3 . 9 . I - Given t h e S - s t a t i o n s y s t e m ( 3 . 2 . 2 ) d e c e n t r a l i z e d f i x e d mode of

(3.2.2)

with S )/ 3, t h e n X0 ~

~ (A) is n o t a

if a n d only if ~0 is n o t a d e c e n t r a l i z e d fixed

mode of a n y of t h e following ( S - 1 ) - s t a t i o n s y s t e m s :

81

C2

(2)

, A,

[[BI, B2], B3,-..,B $] )

, A,

[B 1, [B 2, B3],--., BS] )

[BI,...,Bs_ 2, [BS_I, BS~)

(s)

Cs-2 Cs

, A,

[BI,..., BS_3, [Bs_ 2, BS], BS_1 ])

L Cs_13

II - Given the S-station system (3.2.2) with S=2, then X0 ~ o (A) is not a decentralized fixed mode of (3.2.2) if and only if the two following conditions both hold :

|

2

rank

I

X I-A

>.. n C2

rank I X01-A

LCl

BII 0 121 >..n

82 Now, i t is o f i n t e r e s t to a n a l y s e t h e m e a n i n g of t h e c o n d i t i o n s (1) a n d (2) of p a r t II of T h e o r e m 3.9.

T h e s e c o n d i t i o n s c o r r e s p o n d to t h e s i m u l t a n e o u s r e q u i r e -

ments that : S t a t i o n 1 (or 2) can c o n t r o l t h e mode ~0

-

- Station 2 (or 1) can o b s e r v e t h e mode ~0 -

10 is n o t a t r a n s m i s s i o n z e r o of c e r t a i n s u b s y s t e m s of t h e s y s t e m .

In f a c t ,

these conditions are

e q u i v a l e n t to t h e condition ( 3 . 3 . 2 )

of Theorem

3.7 ( s e e § 3 . 3 . 1 ) for a 2 - s t a t i o n s y s t e m . It was a l r e a d y o b v i o u s in c o n d i t i o n (3.3.2) t h a t t h e e x i s t e n c e of d e c e n t r a l i z e d f i x e d modes for a S - s t a t i o n s y s t e m is r e d u c e d to t h e e x i s t e n c e of f i x e d modes f o r a s e t of 2 - s t a t i o n s y s t e m s s i n c e t h i s c o n d i t i o n shows a partition

of t h e

s y s t e m in

Z aggregated

stations a and

B.

It

stated

that

the

s y s t e m h a s no f i x e d modes if a n d o n l y if t h e following condition h o l d s f o r every possible partition .

rank

>~ n C

V )'0 6_ o"(A)

0 6

This is e q u i v a l e n t to c h e c k i n g t h e e x i s t e n c e of f i x e d modes for e v e r y 2-station

system ( B a B 6 ,

A,

Ca

C6), ~ a c ~ ,

~ u~ 6 = ~.

Note t h a t f o r a S - s t a t i o n s y s t e m , t h e n u m b e r o f p o s s i b l e p a r t i t i o n s is e q u a l to ES-2. By u s i n g t h e p r o c e d u r e p r o p o s e d b y Davison a n d O z g u n e r a n d d e s c r i b e d in Theorem 3.9, we o b t a i n t h e same t e s t s .

Nevertheless, the recursive characterization

of t h e s y s t e m s l e a d s to some r e d u n d a n c i e s .

Indeed,

it r e s u l t s in

(S!/Z)

2-station

s y s t e m s and it is c l e a r t h a t some a r e r e p e a t e d . Example 3 . 4 .

C o n s i d e r a 4 - s t a t i o n s y s t e m (C 1 C 2 C 3 C4, A, B 1 B 2 B 3 B 4 ) . Condi-

tion ( 3 . 3 . 2 ) m u s t be t e s t e d for t h e 7 following 2 - s t a t i o n s y s t e m s :

(c I (c z c 3 c 4)

A

B 1 (B 2 B 3 B4))

(z)

(c 2 (c I C 3 c 4)

A

B 2 (B 1 B 3 B4))

(3)

(c 3 (c I c 2 c 4)

A

B 3 (B 1 B 2 B4))

(4)

( c 4 ( c 1 c 2 c 3)

A

B 1 B z (B 3 B4))

(5)

(C 1 C z (C 3 C 4)

A

B 1 B 3 (B 2 B4))

(6)

(c I C 3 (c z C 4)

A

B 1 B 3 (B 2 B4))

(7)

(c I c 4 (c z c 3)

A

B 1 B 4 (B 2 B3))

(i)

83 By using the p r o c e d u r e in Theorem 3.9, we o b t a i n 12 Z-station systems b u t 5 of them are r e d u n d a n t :

((C 1 C 2, C 3) cAt, A, (B 1 B2 B3) BAt ) (1) ((C 1 C 2) C 3 C 4, A, (B 1 B2) B3 B4) ((C 1 C 2) (C 3 C4), A, (B 1 B2) (B 3 BAt)) (2) ((C 1 C 2 C 4) C 3, A, (B 1 B2 B4) B 3 ) (3)

(C 1 (C 2 C 3) C/4, A, B 1 (B2 B3)B4)

((C 1 C 2 C 3) CAt, A, (B 1 B2 B3) B4)

(4)

(C 1 (C 2 C 3 CAt), A, B 1 (B2 B3 B4))

(5)

((C 1 Co) (C2 C3), A, (B 1 BAt) (B2 B3)) (6)

((C 1 C 2) (C 3 Co), A, (B 1 B2) (B 3 BO)) (7) (C 1 C 2 (C 3 CO), A, B 1 B2 (B3 B4)) (C 1 (C 2 C 3 C4), A, B I (B2 B 3 B4))

(8)

((C i C 3 C 4) C 2, A, (B 1 B3 B4 ) B2 )

(9)

((C 1 C 2 C/fl C3, A, (B 1 8 2 B~.) B3 )

(10)

(C 1 (C2 C 4) C 3, A, B I (B2 B/4) B3) (C 1 (C2 C 4 C3), A, B l (B2 B4 B3))

(ll)

((C 1 C 3) (C 2 C4), A, (B 1 B3) (B2 B4)) (/2)

Systems (1) a n d (4) ; (5), (8) and (11) ; (3) a n d (10) ; (2) a n d (7) are the same.

However, this r e c u r s i v e method p r e s e n t s the a d v a n t a g e to p r o v i d e a systematic way to determine all the p a r t i t i o n s .

3,3,3,

-

3,3.3,a,

Particular

-

cases

Diagonal

systems

Consider the diagonal :

following 2 - s t a t i o n system in which the

Xl ~(t)

=

\

N\\

x(t) +

dynamic matrix A is

1 k%J VB] ul(t) +

LB,J

u2(t)

(..3.3.6)

84 Yl(t)

= (C I , C I ) x ( t )

Y 2 ( t ) = (C~, C 2) xCt) where u 1 ~ are

ml,

Rml'

u2 ~

R m 2 ' Yl ~

m2 row vectors,

Rrl

respectively

' Y2 ~ R r 2

a n d k i ~* ~ ' ( i = l ' ' " n ) "

a n d C 1, CZ, a r e

B1, B2

r 1, r 2 c o l u m n v e c t o r s ,

res-

pectively. Let B l = [b I , ..., b ml l ]

B2 : [b~ , ..., b 2

m2 ]

cI

c

'I Izl

CI=

C2=

cI rI

and

assume

that ki'

LCr2

(i=l . . . . , n ) ,

are

all d i s t i n c t

and

occur

in

complex

conjugate

pairs. Then,

by applying

Theorem

manipulations,

t h e following r e s u l t

Theorem

(DAV-83).

3.10

3.7 to this particular is obtained

~'0 i s n o t a d e c e n t r a l i z e d

a n d o n l y if t h e f o l l o w i n g c o n d i t i o n s

c a s e a n d w i t h some matrix

: fixed

m o d e of s y s t e m

(3.3.6)

if

hold :

[_c2 i.e.,

)t 0 i s n o t a c e n t r a l i z e d

f i x e d m o d e (it i s c o n t r o l l a b l e a n d o b s e r v a b l e ) .

ii - T h e f o l l o w i n g c o n d i t i o n s *B

do n o t s i m u l t a n e o u s l y

hold :

=0

*C~=0 * XO i s subsystems

ci

a

transmission

zero

of

all

the

following

single-input

:

,

l

tn-i

V j ¢ { 1,2,...,mq}

q-~l,2

single-output

85

Note t h a t this theorem can easily be e x t e n d e d to the case for which t h e system has more than 2 stations b y applying T h e o r e m 3.9 (DAV-83)

(see also (PET-84)).

3.3.3.b. - Interconnected systems This paragraph deals with a particular class of systems of type (3,2.2) consisring of a n u m b e r

of subsystems interconnected together.

These systems are repre-

sented in the state space b y the following set of equations : S

t ~i(t) : A i i xi(t) + j=l r. Aij xj(t) + Bi ui(t) jti

(i=1..... S)

(3.3.7)

[ Y i ( t ) = Ci x i ( t ) x i £ Rni ' ui £ Rmi ' Yi ~ Rri A=

{Aij, (i=1, .... S), (j=l..... S)} c R n x n (B 1 ..... BS) c

R nxm

C -- block.diag. (GI,...,Cs) c

R rxn

B = block.diag.

Aii, Aij, Bi and Ci, (i=l . . . . ~S), j~i, a r e i n v a r i a n t matrices of a p p r o p r i a t e dimension.

1 - Characterization with c o n s t r a i n e d i n t e r c o n n e c t i o n s Consider the class of i n t e r c o n n e c t i o n s in the form : Aij = Bij Lij Cij

(i,j=l . . . . . S)

jCi

(3.3.8)

where Lij is the matrix of i n t e r c o n n e c t i o n gains a n d Bij a n d Cij are a r b i t r a r y . T h e n , the following r e s u l t was d e r i v e d b y Davison (DAV-83) : Theorem 3.11. Given the system (3.3.7) with s t r u c t u r e ( 3 . 3 . 8 ) , if (Ci,Afi,B i) is controllable a n d o b s e r v a b l e for ~/ i=1,2 . . . . . S t h e n ( 3 . 3 . 7 ) (3.3.8) has no d e c e n t r a lized fixed modes for almost all i n t e r c o n n e c t i o n gains Lij, (i=l, . . . . S), (j=l . . . . , S ) , i~j, i.e. the class of n o n z e r o gains L.. for which (3.3.7) (3.3.8) has fixed modes is 1] either empty or lies on a s u b s e t of a h y p e r s u r f a c e in the p a r a m e t e r space of Lij. A more i n t e r e s t i n g r e s u l t is p r o v i d e d if it is assumed that the system (3.3.7) is i n t e r c o n n e c t e d b y the o u t p u t s ; i . e . :

86

Aij = Bi Lij Cj Note t h a t

(i=l . . . . . S ) ,

e v e n if t h i s

(j=l . . . . . S ) ,

(3.3.9)

j~i

c l a s s o f s y s t e m s s e e m s to

be

very

restrictive

with

r e s p e c t to t h e c l a s s of g e n e r a l s y s t e m s ( 3 . 2 . 2 ) , a lot of p h y s i c a l s y s t e m s h a v e this particular structure.

I n d e e d , t h e d e c e n t r a l i z e d s t a b i l i z a b i l i t y s t u d y f o r t h i s t y p e of

s y s t e m s was t h e p r o b l e m w h i c h m o t i v a t e d t h e e x t e n t i o n to more g e n e r a l s y s t e m s like ( 3 . 2 . 2 ) or ( 3 . 3 . 7 ) . Theorem 3.12.

The s y s t e m ( 3 . 3 . 7 )

with s t r u c t u r e

( 3 . 3 . 9 ) h a s no d e c e n t r a l i z e d fixed

modes if a n d o n l y i f • ( C i , A i i , B i) c o n t r o l l a b l e a n d o b s e r v a b l e for all (i=l

. . . . .

S).

For t h i s t y p e of s y s t e m s , t h e s e t of d e c e n t r a l i z e d f i x e d modes is e q u a l to the s e t of c e n t r a l i z e d f i x e d modes ( u n c o n t r o l l a b l e o r u n o b s e r v a b l e modes) which is itself equal to t h e union of t h e s e t s of c e n t r a l i z e d f i x e d modes of e a c h d i s c o n n e c t e d system.

This r e s u l t was also d e r i v e d b y

Saeks

(SAE-79)

a n d s t a t e d in t h e following

way ; Theorem 3¢12bis.The s e t of c e n t r a l i z e d f i x e d modes of ( 3 . 3 . 7 )

with s t r u c t u r e

(3.3.9)

is g i v e n b y : 5 Ad(C,A,B) = ~ ( C , A , B ) :

u

i=l

Ac(Ci,Aii,8 i)

T h e r e f o r e , f o r t h i s c l a s s of s y s t e m s , a d e c e n t r a l i z e d c o n t r o l is e q u i v a l e n t to a c e n t r a l i z e d c o n t r o l as f a r as t h e pole a s s i g n m e n t p r o b l e m is c o n c e r n e d .

2 - C h a r a c t e r i z a t i o n u s i n G t h e p r o p e r t y of b l o c k - d i a g o n a l dominance C o n s i d e r t h e s y s t e m ( 3 . 3 . 7 ) a n d t h e following s e t of local c o n t r o l l e r s :

ui = Kii Yi

(i=l . . . . . S) s u c h t t h a t K = b l o c k . d i a g . ( K l l , . . . . KSS)

The dynamic m a t r i x of t h e c l o s e d - l o o p s y s t e m is •

!/~I! A + BKC=

AI2 ........

AI5

'~22

' ' . ". " . . . ASI

" ~SS

87 where ~ii = Aii + Bi Kii Ci ' ( i = l , . . . , S ) . If the diagonal s u b m a t r i c e s ~.. a r e non s i n g u l a r a n d i f : 11

S

<

i-z_ II A i(I

v

j/~

then {A+BKC) i s s t r i c t l y b l o c k - d i a g o n a l d o m i n a n t .

.....

s

*

d e n o t e s a norm of t h e m a t r i x

(*), for i n s t a n c e :

I[*i =

l

1

laijl

The following w e l l - k n o w n r e s u l t : Theorem 3.13.

I f t h e m a t r i x (A + BKC) is s t r i c t l y b l o c k - d i a g o n a l l y d o m i n a n t , t h e n

(A + BKC) is n o n s i n g u l a r . leads to t h e s u b s e q u e n t c h a r a c t e r i z a t i o n of fixed m o d e s . Corollary 3.3 (ARM-82). If k0 ~ o (A) i s a d e c e n t r a l i z e d f i x e d mode of ( 3 . 3 . 7 ) , there e x i s t s i C { 1 , . . . , S }

then

such that :

s II(~ii-x° I)-lll-l~ j-~l [iAijll

for

VKii

e~

Rmixri

(3.3.1o)

j/i The i n t e r e s t o f t h i s c h a r a c t e r i z a t i o n will a p p e a r l a t e r (in C h a p t e r 5) s i n c e it is used by A r m e n t a n o a n d S i n g h to d e t e r m i n e a c o n t r o l s t r u c t u r e s u c h t h a t f i x e d modes are avoided.

3.3.4. - C o m m e n t s The c h a r a c t e r i z a t i o n s p r e s e n t e d in t h i s p a r a g r a p h

a r e s t a t e d in a t i m e - d o m a i n

framework. It is c l e a r t h a t t h e most r e l e v a n t is t h e m a t r i x r a n k t e s t c h a r a c t e r i z a t i o n which was p r o v i d e d b y A n d e r s o n a n d Clements (AND-81a) a n d , as it h a s b e e n p o i n t e d out, all the o t h e r o n e s a r e e q u i v a l e n t . This c h a r a c t e r i z a t i o n allowed u s to i n t e r p r e t e t h e f i x e d modes in t e r m s of t h e concepts of

controllability

and

observability

and

the

s u b s y s t e m s . We will f i n d a g a i n t h i s p a r t i t i o n i n g of t h e

d e f i n i t i o n of c o m p l e m e n t a r y s y s t e m in

two a g g r e g a t e d

88

s t a t i o n s in t h e

f r e q u e n c y - d o m a i n c h a r a c t e r i z a t i o n s w h i c h will give us t h e

tools to

i n t e r p r e t e in a d e e p e r way t h e r e a s o n s for t h e o c c u r e n c e o f f i x e d m o d e s . D e s p i t e t h e t h e o r e t i c a l i n t e r e s t of t h e m a t r i x r a n k t e s t c h a r a c t e r i z a t i o n , it is c l e a r t h a t it d o e s n o t seem to b e v e r y e f f i c i e n t from t h e c o m p u t a t i o n a l

p o i n t of view

s i n c e it r e q u i r e s to t e s t all t h e c o m p l e m e n t a r y s u b s y s t e m s . An

interesting

result

has

been

obtained

for interconnected systems

whose

i n t e r c o n n e c t i o n s a r e made b y t h e out-puts s i n c e t h e f i x e d modes of t h e s e s y s t e m s are j u s t t h e i r u n c o n t r o l l a b l e a n d u n o b s e r v a b l e modes.

3.4. - A L G E B R A I C

CHARACTERIZATIONS

This paragraph

matrix $ i.e.

DOMAIN

d e a l s with t h e c h a r a c t e r i z a t i o n o f f i x e d modes from t h e i n p u t -

output relations describing the system. nomial m a t r i c e s

: FREQUENCY

The s y s t e m is r e p r e s e n t e d e i t h e r b y poly-

( " m a t r i x f r a c t i o n d e s c r i p t i o n n) or

by

a rational

transfer

function

;

U(p) = W(p) Y(p)

(3.4.1)

w h e r e U a n d Y a r e t h e i n p u t a n d o u t p u t v e c t o r s of dimension m a n d r r e s p e c t i v e l y , a n d W(p) is t h e t r a n s f e r f u n c t i o n m a t r i x of dimension m x r . Let s - l ( p ) T ( p )

be a l e f t coprime f r a c t i o n d e s c r i p t i o n of W(p), t h e n t h e system

can be d e s c r i b e d b y •

s(p) Y(p) = T(p) U(p)

(3.4.2)

w h e r e S ( p ) a n d T ( p ) are polynomial m a t r i c e s with r a n d m c o l u m n s , r e s p e c t i v e l y .

3 . 4 . 1 . - N e c e s s a r y c o n d i t i o n s on t h e t r a n s f e r f u n c t i o n m a t r i x f o r t h e e x i s t e n c e of f i x e d m o d e s Before

presenting

the

general

frequency-domain

characterizations

of

fixed

m o d e s , t h i s p a r a g r a p h p r o v i d e s some n e c e s s a r y c o n d i t i o n s for t h e i r e x i s t e n c e , which are i n t e r e s t i n g b e c a u s e

they can b e c h e c k e d b y t h e sole examination of t h e t r a n s f e r

matrix. C o n s i d e r t h e s y s t e m d e s c r i b e d b y ( 3 , 4 . 1 ) with :

89 p) w(p) = N¢ ( (p)

(3.4.3)

where N ( p ) = C a d j ( p I - A ) B is a polynomial m a t r i x a n d ¢ (p) is t h e c h a r a c t e r i s t i c p o l y nomial of t h e s y s t e m . Now,

if we c o n s i d e r

the

c o n t r o l law in

matrix K can t a k e a n y a r b i t r a r y

structure,

(3.2.4)

where

the output

the closed-loop transfer

feedback

m a t r i x is g i v e n

by : Wc(P,K) = [ I - W ( p ) K ] - I w ( p )

(3.4.4)

= C(pI-A-BKC)-IB

which can b e r e w r i t t e n as : Nc(P,K )

Wc(P,K) =

(3.4.5)

where N c ( P , K ) = C a d j ( p I - A - B K C ) B

is a polynomial m a t r i x a n d

~(p,K)

is t h e c l o s e d -

loop c h a r a c t e r i s t i c polynomial. If t h e

system

has

fixed

modes,

it is c l e a r

that

the

fixed

polynomial F ( p )

divides t h e c l o s e d - l o o p c h a r a c t e r i s t i c polynomial s u c h t h a t we can w r i t e : $c(P,K)

=

F(p).P(p,K)

If we d e r i v e

~bc(P,K) with r e s p e c t to K :

,a¢c (p,K) ~) K

~ p(p,K) =

F(p)

8 K

and it is c l e a r t h a t if P=;~0 is a f i x e d mode, t h e n :

a~c (~ o' K)

(3.~.6)

=0 8K

C o n s i d e r now t h e following t h e o r e m • Theorem 3.14

(BIN-78)

(BER-81).

T h e J a c o b i a n m a t r i x of t h e c l o s e d - l o o p c h a r a c t e -

ristic polynomial qbc(P,K) with r e s p e c t to K is g i v e n b y :

~¢c(pJ<) - - N c (p,K) aK

(3.4,7)

90

w h e r e N ' c ( P , K ) i s t h e t r a n s p o s e of N c ( P , K ) i n

(3.4.5).

T h e o r e m 3 . 1 4 a n d f o r m u l a ( 3 . 4 . 6 ) l e a d to t h e following r e s u l t :

Theorem respect

3.15. to

A necessary

K in

(3.2.4)

is

condition that

the

f o r ~0 to be projection

a

fixed

of t h e

mode o f

(3.4.1)

with

closed-loop

transfer

matrix

parameter

v a l u e s in

K, it

n u m e r a t o r N c ( P , K ) on K ~ i s z e r o f o r P=~0" Since this

result

must

hold i n d e p e n d e n t l y

of t h e

m u s t in p a r t i c u l a r hold for K=0. T h e r e f o r e t h e following c o r o l l a r y d i r e c t l y follows :

Corollary

3.4.

A necessary

r e s p e c t to K in ( 3 . 2 , 4 )

condition

f o r X0 to b e

a

f i x e d mode of

is t h a t the projection of the t r a n s f e r

(3.4.1)

with

m a t r i x n u m e r a t o r N(p)

o n K t i s z e r o f o r P=~0" It is i n t e r e s t i n g same as the condition

to n o t i c e t h a t

the

c o n d i t i o n in

Corollary

3 . 4 i s e x a c t l y the

(i) ha C o r o l l a r y 3.1 w h i c h w a s o b t a i n e d f r o m t h e c h a r a c t e r i -

zation of f i x e d m o d e s i n t e r m s of t r a n s m i s s i o n z e r o s . T h i s r e s u l t c a n also be e x p r e s s e d in t h e following f o r m : Corollary 3.5. order

A m u l t i p l e mode of o r d e r q c a n b e a f i x e d mode if it i s n o t a pole of

s u p e r i o r to ( q - l )

f o r a n y s u b s y s t e m f o r m e d f r o m a n y local s u b s y s t e m n e i t h e r

f o r t h e c h a r a c t e r i s t i c p o l y n o m i a l of t h e p r o j e c t i o n of t h e t r a n s f e r m a t r i x on K w.

Remark 3.2.

As a c o n s e q u e n c e of C o r o l l a r y

complementary

Note t h a t structure

3.5,

the

f i x e d m o d e s a r e p o l e s o f the

subsystems. Theorem 3.15,

C o r o l l a r i e s 3.4 a n d

3.5 a r e v a l i d f o r a n y a r b i t r a r y

of t h e f e e d b a c k m a t r i x K ( n o t n e c e s s a r i l y b l o c k - d i a g o n a l a s in ( 3 . 2 . 1 1 ) ) .

For e x a m p l e , in t h e p a r t i c u l a r

c a s e of c e n t r a l i z e d c o n t r o l ,

we f i n d a g a i n from Co-

r o l l a r y 3.5 t h a t a s i m p l e u n c o n t r o l l a b l e a n d / o r u n o b s e r v a b l e mode d o e s n o t b e l o n g to t h e c h a r a c t e r i s t i c p o l y n o m i a l (minimal r e a l i z a t i o n ) , w h i c h i s a w e l l - k n o w n r e s u l t . E x a m p l e 3 . 5 . C o n s i d e r t h e s y s t e m d e s c r i b e d b y t h e following t r a n s f e r m a t r i x :

91 for w h i c h ¢

(p)--p(p-l) (p+Z)

If we c o n s i d e r

(p-Z).

a decentralized

c o n t r o l in t h e f o r m :

K = b l o c k - d i a g . ( k 1 , k 2 , k 3) the projection

of

W(p)

on

a c c o r d i n g to C o r o l l a r y Note

that

the

K'

is indicated

by

the

non

shaded

Therefore,

blocks.

3 . 5 , ~=2 a n d k =-Z a r e n o t f i x e d m o d e s . conditions

determine a structure

for

derived

the

feedback

in

this

matrix

paragraph such

that

can the

be

easily

system

used

to

has

no fixed

and observation

stations

mo de s •

3.4.2. - Transfer

function matrix characterization

for

systems with distinct poles Consider

the

system

such that its transfer

W(p)

| t

(3.4,1)

A0 P-

w h e r e A0~0 a n d ~ i ~ 0

+

~0

in the following form :

i

wsL(p)

.........

rixm i,

(i,j=l .....

At

i=i

P-~i

In t h e c a s e o f 2 s t a t i o n s ,

S),

and

assume

that

all i t s p o l e s a r e

as •

n-I

(i=l . . . . .

(3.~.s)

ws's(p)

W(p) c a n b e f a c t o r i z e d

VC(p) --

in S control

wi(p]

. . . . . . . . .

w h e r e Wij(p) i s o f d i m e n s i o n distinct ; i.e.

partitioned

function matrix appears

(3.4.9)

n-l). the time domain representation

( 3 . 3 . 6 ) a n d W(p) c a n b e r e w r i t t e n

as follows :

of (3.4.9)

is given in

92

CI +

Using

(3.4.10),

Davison

and

1 ~1

0 [}51

1£_

C2

B 2]

(3,t4.10)

p-kn_l

Ozguner

(DAV-83)

p r o v i d e d in

the

frequency

domain t h e t r a n s p o s i t i o n of t h e r e s u l t e s t a b l i s h e d b y T h e o r e m 3.10 in a time-domain framework.

The c a s e

for w h i c h t h e s y s t e m h a s more t h a n 2 s t a t i o n s is considered

b y a p p l y i n g t h e r e c u r s i v e c h a r a c t e r i z a t i o n of f i x e d modes g i v e n in T h e o r e m 3.9. T h e o r e m 3.16. l 0 i s n o t a d e c e n t r a l i z e d f i x e d mode of ( 3 . 4 . 1 )

( 3 . 4 . 8 ) i f a n d only if

n o n e of t h e following c o n d i t i o n s o c c u r with r e s p e c t to t h e m a t r i c e s :

W(p)

A0 a n d

or their respective

P= XO

-

transposes

.

C a s e 1 : (S=2)

(i)

AO =

and

(p) -

=

0

p=X 0

Case 2 : (5=3) (i)

(ii)

Ao =

AO=

0

X

0

0

0

and

~

and

(p) -

0

(iii)

A0 =

= P - XO

-P = XO

x0] 0

0

X

0

and

(p) -

= P = XO

i

X

X

X

0

0

X

0

X

X

0

X

X

L xx1 X

0

X

X

°.

I X

X

X

X o X o

f X

I

X

X:

I

o

~

X X X X

I

X

o

X

I X

X

X

I

o

X

X

I

o

~

X

!

x

X

X

I

X X X X X

X

I

X

X

X

~

~

X

X X X X

X

X

o

o

X

X

X

X

~

o

X

o

I X

X

X

~

X

:;,< ,I

X

o

X

>¢

X

o

X

X

X

X

X

o

o

I

o |

X

o

o

o

Ii

i

o

11

X

o

X

o

o

,r,

I

J

IR '

I

X

u

<

o

o

cr~

o

X

o

o

o

I

0

M

o

"0 r-' r~

0

~

I0

~

0

0 ,I

II

X

0

I

0

X

o

0

i

o

i

0

0 0

0

I

X

J

0

X

X

0

ii

0

X

'

'

o

Ii

Li t~

0

o

I

~:)

o

ii

i


0

o

0

X

0

o

X

l

ii e~

i

x

o

0 I

o

I

X I

H

'l 1

o

o

I

o

II

I

o

x

o

I

ii

i

X

'

0

X

H

&

I

o

>(

X

~:

I o

c)

0

I

o

X

0

I H

I

-0

o

X

0

o

X

o

0

o

X o

0

0

o

o X

C)

0

I

X X

0

0

I

X o

0

0

''

"o

I

c~

I

c~

X o

0

0

o

o

o

0

0

o

o

0

0

o

0

0

L

0

I

0

I

0

I

(D

I

0

I

0

<

0

0

II

4:'

0

I

0

II

A

IW

u

¢)

¢)

"O

im

r~

O

o

a

(J

t-

o

¢,

94 trollable and observable by a single station.

All t h e e l e m e n t s of ~ 2 1 ( P )

h a v e a zero

a t X0 a n d t h o s e of W12(p) h a v e a pole a t k0, w h a t p r o d u c e s a p a r a m e t r i c cancellation w h e n one local f e e d b a c k is a p p l i e d . T h i s will b e e x p l a i n in d e t a i l s in Paragraph 3.4.4. It is c l e a r t h a t

this

c o n d i t i o n is s t r i c t l y

e q u i v a l e n t to t h e

conditions

in

Theorem

3.10. A similar interpration can be given for the case S)2. Example 3.6.

C o n s i d e r a g a i n t h e s y s t e m in Example 3.5 a n d a s s u m e t h a t we want to

c h e c k if X0 =-1 is a f i x e d mode w i t h r e s p e c t to K = b l o c k - d i a g ( k l , k 2 , k 3 ) W ( p ) w r i t t e n in t h e form ( 3 . 4 . 9 )

w(p)

ii

o 0

o

L-~

o

o

3 i>-2

0

1

0

BS:

0

p-2

O

0

0

0

0

0

A0

]

w(p)- p-~

1

p(p-2)

1

0

[

X

p+l p ( p - 2)

1

p-2

0

p+2

a n d we h a v e : l 0

can be

•

x p=- l

0]

0

X

X

0

X

0

which s a t i s f i e s c o n d i t i o n (ii) for t h e case 2 (S=3). T h e r e f o r e X0=-I i s a d e c e n t r a l i z e d fixed mode.

3.4.3.

- Polynomial

matrix

rank

test

characterization

C o n s i d e r a l i n e a r t i m e - i n v a r i a n t s y s t e m p a r t i t i o n n e d in S c o n t r o l a n d o b s e r v a t i o n s t a t i o n s s u c h t h a t i t s m a t r i x f r a c t i o n d e s c r i p t i o n in ( 3 . 4 , 2 ) t a k e s t h e form : ($1(P) ..... Ss(P))

U(p) = (TI(P) ..... s(p))

Y(p)

(3.4.11)

95 where Si(p) and T i ( P ) , (i=1 . . . . . S), have r i and mi columns, r e s p e c t i v e l y p c o r r e s ponding to the number of o u t p u t s and i n p u t s of the ith s t a t i o n . In this framework, A n d e r s o n (AND-81a) gives the following c h a r a c t e r i z a t i o n of decentralized fixed modes : Theorem 3.17. X0 is a d e c e n t r a l i z e d fixed mode of system (3.4.2) (3.4.11) with respect to K in (3.2.11) if and only if t h e r e e x i s t s some nonempty s u b s e t { i l , . . . i , k } of {1 . . . . . S} such t h a t : k [Sil (~0), ..., Sik (~0), Ttl (~0), ..., Tik (~0)] i -~1 r iJ (3.4.12) k It we set .E r.. = ~3 , then the d e g r e e of the fixed mode ~0 is given b y the ]:l 11 largest positive i n t e g e r d such t h a t all ~ x 8 minors of the matrix in the left hand side of (3.4.12) have a zero at )0 of o r d e r at least d. rank

Note that this r e s u l t was r e c e n t l y p r o v e d in a d i f f e r e n t way b y Zheng (ZHE84).

3.4.4.

- General

transfer

function

matrix

characterization

This general c h a r a c t e r i z a t i o n of fixed modes is the most complete one in the frequency domain. It was d e r i v e d b y A n d e r s o n in 1982 (AND-82) using the r e s u l t that he obtained t o g e t h e r with Clement in 1981 (AND-81a) and which was p r e s e n t e d in the p r e c e d i n g p a r a g r a p h .

3.4.4.a. - P a r t i c u l a r case : 2x2 t r a n s f e r function matrix with simple pole at ~0 Consider the system (3.4.1) where W(p) is given by :

Wll(P)

Wl2(P) 1

w(p) =

O.4.13)

W21(P)

Suppose t h a t W(p) = s - l ( p ) T ( p ) S(p) = [

W22(P)

with :

Sll(P)

Sl2(P)

s21 (p)

s22(P)

1

96

T(p) =

I

tl l(P)

tl2(P)

t21 (p)

t22(P)

t

(3.4.10)

and S(p) and T(p) are left coprime. If we apply Theorem 3.17 to this particular case, there is a fixed mode at )~0 ff and only if, by reordering the inputs and outputs if necessary, the following condition holds :

rank

I SllJ(XO) s21(2' 0)

i.e.,

tll(XO) 1

( 1

t2l()~ 0)

sl 1(~0) =s21 (k0) =tll (k0) =t2l (k0) =0.

Wij(p) in (3.4.13) can easily be expressed in terms of sij(p) and tij(p) for i,j:l,2 and the following result follows : Theorem 3.18 (AND-82). k 0 is a decentrelized fixed mode of system (3,4.13) if and only if W(p) or W'(p) has the following form : I entry with no pole at k0

entry with pole at ~'0

1 (3.4.1s)

entry with zero at k 0

entry with no pole at X0

J

This result can easily be interpreted. Indeed) consider system (3.4.13) as illustrated by Figure 3.4a and suppose that we set u 2 -- k 2 Y2' producing thus a system with input u I and output Yl as illustrated by Figure 3.4h.

97

uI

~[

wII(P) ]

+~

~l

y,

j'.

,,

j

kl F i g u r e 3.4a

÷ ~C~y÷I

L

F i g u r e 3.4b

The t r a n s f e r f u n c t i o n o f t h e n e w

Yl u'-~

= Wll

system is given by :

k 2 W21

+ WI2

l - W 2 2 k2

If condition ( 3 . 4 . 1 5 )

h o l d s for W ( p ) , t h e p o l e - z e r o c a n c e l l a t i o n a r k 0 in t h e p r o d u c t

Wl2. WE1, a s s o c i a t e d with t h e f a c t t h a t Wll and W22 h a v e no pole at )'0 (k0 is not simultaneously controllable a n d o b s e r v a b l e b y a s i n g l e s t a t i o n ) ,

makes t h a t ( 3 . 4 . 1 6 )

is u n c o n t r o l l a b l e .

system

unobservable.

If c o n d i t i o n

Of c o u r s e ,

(3.4.15)

holds

for W ' ( p ) ,

then

(3.4.16)

is

t h e a p p l i c a t i o n of f e e d b a c k at s t a t i o n 1 (u 1 = k 1 y l ) do

not c h a n g e t h i s s i t u a t i o n a n d , in b o t h c a s e s , ~0 is a f i x e d mode o f s y s t e m ( 3 . 4 . 1 3 ) . Note t h a t T h e o r e m 3.18 a g r e e s with t h e r e s u l t of Davison a n d O z g u n e r (DAV81) given in

T h e o r e m 3.16

for t h e c a s e

1 and

also with

the n e c e s s a r y

condition

e x p r e s s e d in Theorem 3.15. M o r e o v e r , t h i s p a r t i c u l a r c a s e is a p e r f e c t i l l u s t r a t i o n o f Theorem 2.5 ( C h a p t e r 2) (FES-80) t h a i s h o w s a f i r s t r e l a t i o n b e t w e e n t h e r e s u l t s of Corfmat a n d Morse ( C O R - 7 6 a , b ) a n d t h e c o n c e p t of f i x e d m o d e s . In t h e case f o r w h i c h ~0 is n o t a simple p o l e , t h e above r e s u l t can be e x t e n ded as follows :

98

Theorem

3.19

( A N D - 8 2 ) . X0 i s a d e c e n t r a l i z e d

o n l y if t h e o r d e r

of X0 a s a p o l e i n @ l l ( P ) ,

f i x e d mode of s y s t e m

W22(p)

(3.4.13)

if and

a n d W(p) is l e s s t h a n t h e order

it h a s as a p o l e in W12(P) o r W 2 1 ( P ) . In

this

case,

a

pole-zero

cancellation

also

occurs

in

the

transfer

function

(3.4.16).

3.4.4.b.

- General case : systems

multi-input Consider

multi-output

now a general system

W ( p ) as in ( 3 . 4 . 8 ) matrix fraction For (i 1 .

. . . .

a

with more than 2

stations

or,

equivalently,

description

matrix

M,

of W ( p ) ,

M Jl ""

ik) and the columns

(3.4.1)

described

by (3.4.11),

by a transfer

where

and the decentralized

Jk

denotes

the

function

s-l(p)T(p)

minor

matrix

is a l e f t coprime

c o n t r o l in ( 3 . 2 . 1 1 ) . of M f o r m e d

(Jl . . . . . j k ) of M. B y r e o r d e r i n g

of i n p u t s

by

the

rows

a n d out-puts,

t h e s y s t e m is p u t in t h e following f o r m :

W(p) =

I w~a (P)

wc~IS (P)

wsts

Wisct (p) W(p) : [ S c ( ( p ) Sis

(p) y i

i = a

(P) (3-t+-lS)

in t h e form •

g

(3.4.19)

A s s u m e t h a t rc~ ( r IS) is t h e n u m b e r Was

(3.4.17)

[Tc, (p) TIS (p) ]

a n d we a s s u m e c o n t r o l s t r u c t u r e s U. = K . Y . 1 1 1

1

o f r o w s a n d m cL ( m i s ) t h e n u m b e r

of c o l u m n s of

(p) (WISIS(p)),

The number

of z e r o s a t )~0 o f IY }l

Jk

#= 0

)~0 is n e i t h e r

]t ( 0

)~0 i s a pole o f o r d e r

l/=

t h e m i n o r is i d e n t i c a l l y

oo

is denoted

by with :

J

a pole nor a zero - # zero

We h a v e m = mc~ + mis a n d r = rc~ + r B " A m o n g t h e f i r s t rc~ r o w s , the number

Jk

of r o w s w h i c h do n o t b e l o n g to t h e c o n s i d e r e d

minor by :

let us define

99

I, ...

Or

dered minor by

first columns,

the number

consi-

Jk

denotes the number

of elements of the set

In t h i s f r a m e w o r k ,

Anderson

(S a (~0) T a ( ) , 0 ) )

and the fixed mode exists

(*).

(AND-82) showed the following result

The following two conditions

2 - there

to t h e

1[)1' ""' )k } n {1, ..., m~)JJ

: Jl

1 - rank

of columns which belong

:

@c

Theorem 3.20.

{I , .... r(, }II

i k } u { i 1 . . . . ,Jr_ k } = {1 . . . . . m }

and a m o n g t h e m a

*

i' i_k} n

Jk

I

w h e r e • {i 1 . . . . .

II { i' i .....

=

are equivalent

:

:

(3.4.20)

< r a

0 has degree

d (see Theorem 3.17)

d such that 0 < d < D and such that,

#

whenever

0r+0 c > r a

~/(d-D) ÷ (Or + Oc - r a )

(3.~.21)

Jl "'" Jk

for all m i n o r s o f W ( p ) ,

Remark

3.3.

The

certain minors

D is the degree

preceding

have

theorem

~0 a s a z e r o

o f XO i n t h e c h a r a c t e r i s t i c

states

polynomial

q~ ( p ) .

t h a t ~0 i s a f i x e d m o d e of d e g r e e

of certain

minimum order

or

as a pole

d if

of limited

m u l t i p l i c i t y , w h i l e ~0 i s a t t h e s a m e t i m e a p o l e o f W ( p ) . We r e c a l l t h a t are n o t in t h e the first m a 0c-r a

is

the number

scrutiny

and

o f r o w s a m o n g t h e f i r s t rc~

0c i n d i c a t e s

c o l u m n s w h i c h a r e in t h i s s a m e m i n o r .

associated

nonnegative,

@r i n d i c a t e s

minor under

the

which t h i s q u a n t i t y

with

minor

the is

position

constrained

is negative,

of the by

the

minor

the number

Therefore, in W ( p ) .

inequality

the quantity

@r +

When

0c-r a

(3.4.2).

the minor is not cons,reined.

rows which

of columns among

Or + In

the

case

is for

100

Example

3.7.

Consider

function matrix

2-station

the

system

described

by

the

following

transfer

: 1 p(p-l)

0

0

1 p-1

0

0

0

1 p-I

1 p-I

0

W(p) :

1 p-I

for which

dp ( p ) = p ( p - 1 ) 3 . -

The dotted

lines

denote

the partition

tralized control has the following structure

: ra

2 stations

and

0

u l ikll

0

0

k 24

UI3

0

0

0

k 34

=

We h a v e

in

the

associated

decen-

:

Vct

kl2 k3 0

= 3, r B = 1, m ct = 1, m B = 2 a n d D=3. T h e m i n o r s v e r i f i y i n g

Or+Oc ) 3 a r e t h e f o l l o w i n g

:

wI:] (:] f:] {: :1 {: :] wi: :]wE::] w[: :1 {: :] From theorem such that

/'t

3 . 2 0 , ~0=1 i s a f i x e d m o d e o f d e g r e e

:

[1] 1

=- 1 ~ ( d - 3 )

1t

[] l

=-1

d if a n d

~ (d-3)+l

only

if t h e r e

exists

d

101

#

#

E:] I 1

=-1 ~ (d-3)

#

=-I

~ (d-3)+l

2

= ~ ) (d-3)

4/

=-2

I:l

#

~ (d-3)

= ~

#

),. ((t-3),1

= ~ ~,, (d-3)

l

l

#

;-2 LI

These

~ (d-3)

#

2

twelve

1

inequalities

fixed m o d e of d e g r e e

Theorem

3.20

are

satisfied

in i t s g e n e r a l

3.21

d=l.

Therefore,X

0=1 i s a

decentralized

complex.

Therefore,

s i m p l e f o r m if we c o n s i d e r

it i s i n t e r e s -

the case for which

is easily obtained

from condi-

3.20 when d=D=l.

(AND-82).

(3.4.8)

for

The following result

Under

the

t h a t ~0 i s a s i m p l e z e r o o f • ( p ) . (3.4.1)

3

form is rather

X0 is a s i m p l e p o l e o f t h e s y s t e m .

Theorem

~ (d-3)

1.

ting to n o t e t h a t it t a k e s a p a r t i c u l a r tion 2 of t h e o r e m

=-2

if and

Then,

o n l y if t h e r e

s u c h t h a t W(p) o r W ' ( p ) a p p e a r s

same assumptions

as in Theorem

~fl i s a d e c e n t r a l i z e d

exists

a partition

3.20,

assume

fixed mode of system

of the system

a s in

(3.4.17)

in t h e f o l l o w i n g f o r m :

m s

r~

no entry

has a pole

X0 i s a s i m p l e z e r o o f t h e characteristic

a t X0

polynomial

of this block

(3.4.2z)

r~

every

entry

has a zero

no entry

a t X0

Example 3.8.

Consider

has a pole

a t X0

the same system

p u t in f o r m ( 3 . 4 . 1 7 )

taking

station

and 3 as the second,

W(p) is partitioned

a s in E x a m p l e 3 . 5 a n d 3 . 6 .

1 as the

first

aggregated

as follows :

station

If the system is and

stations

2

102 3

!

0

p+l p(p-2)

I . . . . . .

W(p)-

L.

I I

1

1

1

p+2 I ! )

l

p(p-2)

1

0

p+2

a n d it is c l e a r t h a t W ' ( p ) h a s t h e form i n T h e o r e m 3 . Z l w i t h ~0=-1.

Therefore,

we

f i n d a g a i n a s in E x a m p l e 3.6 t h a t k 0 = - I i s a f i x e d m o d e . Theorem

3.Zl

is

Davison and Ozguner

strictly

equivalent

(DAV-83)

to

(see § 3.4.2)

Theorem

t h e s y s t e m w h i c h l e a d to f o r m ( 3 . 4 . 2 2 ) d e d u c e d from t h e s t r u c t u r e s A0

W(p) -

which

was

derived

for the case of simple poles.

d i f f e r e n c e c o n s i s t s in t h e f a c t t h a t in T h e o r e m 3 . 1 6 , o f t h e s t a t i o n s in 2 a g g r e g a t e d

3.16

by

T h e only

all t h e p o s s i b l e p a r t i t i o n i n g s of

in T h e o r e m 3.21 a r e l i s t e d y r i t h o u t r e o r d e r i n g

stations.

T h e two a g g r e g a t e d

stations

c a n e a s i l y be

of t h e m a t r i c e s A 0 o r

as follows

:

P-~O * rows r a (rfl) at

X

(0)

or

P-}'O P=)~°

: rows of A0 (W(p) ---

rows

o f W(p)

-

P=~O (AO)

w h e r e t h e r e is a b l o c k s p e c i f i e d

where

there

is no

block

s p e c i f i e d at

0 (X).

* c o l u m n s ma

(m13)

: c o l u m n s o f A 0 (W(p) - - - - ~ 0

s p e c i f i e d at X (0) o r c o l u m n s o f W(p)

P=

w h e r e t h e r e is n o b l o c k

I ,>0

- A'-~'O01 ~ p:},o(AO) w h e r e t h e r e

is a b l o c k s p e c i -

fied at 0 ( X ) . Note also t h a t

t h e two d i a g o n a l b l o c k s

agree with the necessary

for w h i c h

no entry

has

a p o l e at ~0

c o n d i t i o n f o r k0 to b e a f i x e d mode in T h e o r e m 3.15 ( s e e §

3.4.1).

3.4.5.

- Interpretation I t is i n t e r e s t i n g

function

matrix

to n o t e t h a t i n t h e r e s u l t s

characterization

again the partitioning

of the

of fixed modes

system

obtained

(Theorems

in two a g g r e g a t e d

for the general

transfer

3.20

we find

stations

and

3.21),

which

was already

103 p r e s e n t in t h e s t a t e = s p a c e c h a r a c t e r i z a t i o n of A n d e r s o n a n d Clements (see § 3 . 3 . 1 , Theorem 3 . 7 ) . This s i t u a t i o n can

be

intuitively

controlled in a d e c e n t r a l i z e d w a y ,

interpreted

as

follows.

When a

s y s t e m is

a s t a t i o n i can t r a n s m i t some i n f o r m a t i o n to a

station j t h r o u g h the t r a n s f e r c h a n n e l C j ( p I - A ) - I B i = Wii(p)., A similar s c h e m e h o l d s for a set of s t a t i o n s to a n o t h e r . The lack of i n f o r m a t i o n at one s t a t i o n o r at one s e t of s t a t i o n s , which r e s u l t s from the d e c e n t r a l i z a t i o n c o n s t r a i n t (only local v a r i a b l e s a r e a v a i l a b l e ) , can t h e r e f o r e be c o m p e n s a t e d b y

the information t r a n s f e r t h r o u g h the t r a n s f e r

f u n c t i o n s of t h e

interconnectton t e r m s . Hence, f o r a S - s t a t i o n s y s t e m , if all t h e s u b s e t s of s t a t i o n s {il)...,ik}

c {1,...,S)

can t r a n s m i t some i n f o r m a t i o n to t h e i r c o m p l e m e n t a r y s u b s e t s

of s t a t i o n s { i k + l ) . . . , i S} a n d vice v e r s a ,

t h e i n f o r m a t i o n is s p r e a d e d among all t h e

stations of t h e s y s t e m a n d a d e c e n t r a l i z e d c o n t r o l will s u c c e e d e i t h e r f o r t h e s t a b i = lization t a s k or for t h e pole a s s i g n m e n t t a s k . As we a l r e a d y

saw,

the

information

transfer

can

be

prevented

by

special

parametric c o n f i g u r a t i o n s , which r e s u l t in t h e e x i s t e n c e of f i x e d m o d e s . It is also clear t h a t t h i s t r a n s f e r is i m p o s s i b l e if t h e t r a n s f e r f u n c t i o n b l o c k s c o r r e s p o n d i n g to the i n t e r c o n n e c t i o n s : C a (pI~l) -I B B = WaB (p) and C B (pl-A) -I B a

= WBa(p)

are identically z e r o . In o r d e r to i l l u s t r a t e t h i s r e m a r k ,

c o n s i d e r again t h e case of a s y s t e m with

two s i n g l e - i n p u t s i n g l e - o u p u t s t a t i o n s . We saw in P a r a g r a p h 3.4.4a t h a t if t h e f e e d back u 2 = k 2 Y2 is a p p l i e d a t s t a t i o n 2, t h e t r a n s f e r f u n c t i o n of t h e r e s u l t i n g s y s t e m is given b y ( 3 . 4 . 1 6 )

Yl Ul

:

k2 = WII + WI2

W21

I-W22 k2

Assume t h a t t h e pole of t h e s y s t e m k 0 is not s i m u l t a n e o u s l y c o n t r o l l a b l e and observable b y a s i n g l e s t a t i o n ( W l l ( p ) a n d W22(p) h a v e no pole at ~0) .

However, k0

is controllable a n d o b s e r v a b l e b y t h e global s y s t e m (W12(p) or W21(p) h a v e a pole at ~0' say W 1 2 ( p ) ) .

104 We saw in P a r a g r a p h 3 . 4 . 4 . a t h a t i f ~0 i s a z e r o of W21(P), t h e p o l e - z e r o cancellation in t h e p r o d u c t fixed mode.

W12. W21 l e a d s to t h e u n c o n t r o l l a b i l i t y of ( 3 . 4 . 1 6 )

Now, if W21(p) = 0, it i s o b v i o u s t h a t

system

(3.4.16)

: t 0 is 2

is also uncon-

t r o l l a b l e a n d t0 i s also a f i x e d m o d e .

In t h e l a s t c a s e

(W21(p) = 0), l 0 i s a f i x e d mode i n d e p e n d e n t l y of t h e para-

m e t e r v a l u e s : t h i s t y p e of f i x e d m o d e s is called a s t r u c t u r a l l y f i x e d m o d e . In t h e f i r s t c a s e (W21(p) # 0 ) , ~0 i s a f i x e d mode d e p e n d i n g of t h e parameter values : a non s t r u c t u r a l l y fixed mode.

If we c o n s i d e r now t h e s p e c i a l c a s e f o r w h i c h ~0 = 0, t h e c o n d i t i o n for k0 to b e a f i x e d mode is also m e t f o r g e n e r i c v a l u e s of t h e p a r a m e t e r s e v e n w h e n W21(p) # 0. T h e f i x e d m o d e s at t h e o r i g i n e m a y t h u s b e a p a r t i c u l a r c a s e of s t r u c t u r a l l y

fixed

m o d e s.

T h e e x t e n t i o n of t h e p r e c e d i n g d i s c u s s i o n to t h e c a s e of m u l t i - s t a t i o n systems is e a s i l y d e r i v e d . ~0 is a s t r u c t u r a l l y f i x e d mode if Wa8 (p) or WB~ (p) i s identically zero a n d t h e c o n d i t i o n s of T h e o r e m 3.20 the system) hold.

(or T h e o r e m 3.21 if ~0 i s a s i m p l e pole of

If A0 = 0, t h e c o n d i t i o n s in T h e o r e m 3.20 a n d 3.21 m a y also hold

f o r g e n e r i c v a l u e s of t h e p a r a m e t e r s a n d A0 is also a s t r u c t u r a l l y

f i x e d mode in this

case.

T h e a n a l o g y w i t h t h e c a s e of c e n t r a l i z e d c o n t r o l b e c o m e s now m o r e p r e c i s e . As we s a w in C h a p t e r II,

f i x e d m o d e s a p p e a r a s an e x t e n t i o n of t h e c o n c e p t s of uncon-

t r o l l a b l e a n d u n o b s e r v a b l e m o d e s a n d it i s c l e a r t h a t s t r u c t u r a l l y blish

the

relation

1.3, C h a p t e r I).

with

structurally

uncontrollable

f i x e d m o d e s esta-

unobservable

modes

(see §

B o t h a r e d e f i n e d f o r g e n e r i c v a l u e s of t h e p a r a m e t e r s of t h e system

and are therefore structural properties. ral controllability and s t r u c t u r a l for s t r u c t u r a l l y

and

fixed modes.

A s i m i l a r s t u d y to t h e o n e made f o r s t r u c t u -

o b s e r v a b i l i t y in c h a p t e r

The next

paragraph

I can t h u s be

presents

the results

c a r r i e d out w h i c h have

b e e n o b t a i n e d on t h i s p u r p o s e .

3.5.

- STRUCTURALLY

3 . 5 . I.

FIXED

MODES

- Preliminaries

It h a s b e e n p o i n t e d o u t in t h e l a s t p a r a g r a p h from two d i s t i n c t s o u r c e s :

t h a t a f i x e d mode may originate

105 - a perfect

matching

nonzero parameters

can

in system

eliminate

parameters

the

(in t h i s c a s e ,

fixed mode)

a perturbation

: it is then

called

of the

a non

struc-

turally fixed mode, - the special

structure

however the nonzero

within the system

parameters

values

(in t h i s

are perturbed)

case the fixed mode remains = it i s t h e n

called a structu-

rally f i x e d m o d e . It w a s a l s o p o i n t e d o u t t h a t different origines,

though

structurally

- the lack of interconnections - the

fixed modes themselves

both are of a structural between

nature.

may have

They may result

the different

two

from :

stations,

special location of a mode at the origin.

The following example illustrates

Example 3 . 9 .

Consider

the following 2-station

0

1

0

1

1

0

0

0

1

C1 =

~0

o

l'l

C2 :

~1

o

ol

A =

which is globally

the above situations

controllable

BI=

system

l] 0

0

and

observable.

:

:

B2=

El

If a decentralized

control

law o f t h e

form :

u = K y

ol k2

is a p p l i e d to t h i s matrix :

system,

the resulting

closed-loop

system

has the following dynamic

106

I 0l

AK=A+BKC=

k2

and

d e t ( p I - A K) = ( p - l )

is independent

where

c

0

0

1

(p2-p-l-klk

010]

if o n e o f t h e n o n z e r o

1

1

0

0

0

I+E

is an arbitrary

2)

where

it a p p e a r s

that the eigenvalue

A0=I i s a d e c e n t r a l i z e d

entries

small constant,

d e t ( p I - A K) = p ( p - 1 ) ( p - ] and

k1

of k I a n d k 2. T h e r e f o r e ~

However,

A =

1 I

in A i s s l i g h t l y p e r t u r b e d

we h a v e

a t ~0=1

fixed mode. a s follows :

:

e)-(p-l-c)-klk2(P-1)

t h e f i x e d m o d e d o e s n o t e x i s t a n y m o r e : t h e f i x e d m o d e at ~0=1 w a s t h e r e f o r e

non structurally

Consider

n o w t h a t t h e d y n a m i c m a t r i x of t h e s y s t e m i s g i v e n b y

A =

such that

fixed mode.

: det

L! ' °l 0

0

0

0

( p I - A K) = p ( p 2 - 1 - k l k 2 ) .

In this case, of any perturbation Consider

:

the system

h a s a f i x e d m o d e a t )~0=0 w h i c h will r e m a i n r e g a r d l e s s

of the non zero elements of the matrices

also the following system

:

[ 00] [1 0

c~=[l

-1

oo3

[°]

0

~-=Eo 2

A,B1,B2,C 1 and

1

,

l-I

C 2.

a

107 We have : det (pl-kK)* = p(p-l-kl)(P+l-k 2) and the system (B~I,B2,A~*,CI,C 2 ) ** has also a fixed mode at k 0=0 which will remain however the nonzero elements of the are perturbed.

matrices

Neverthless, in the two above cases, the origin of the

structurally fixed mode at 10=0 is not the same. This appears clearly in the associated t r a n s f e r f u n c t i o n m a t r i c e s . Case 1 :

I wj(p)

System {BI,B2,A~,CI,C2~

I p-l

:

1 _

P

l

I I

0

I I" I I

p+l

I

1

It is now c l e a r t h a t t h e f i x e d mode at A0=0 a r i s e s from t h e lack of t h e i n t e r connection t e r m b e t w e e n station~ 1 a n d 2 w h i c h is s h o w n b y C ; ( p I - A * ) - I B ~ --- 0 in Wl(p). In t h i s c a s e , k0=0 will b e called a s t r u c t u r a l l y f i x e d mode of t y p e ( i ) .

Case 2

:

W2(p)

System ( B 1 , B 2 , A , C 1 , C 2 )

=

[

0

(p+P1)(p-1)

1

0

In W2(P), n o n e of t h e i n t e r c o n n e c t i o n t e r m s b e t w e e n the s t a t i o n s a r e m i s s i n g . The fixed mode at k0=0 a r i s e s from i t s s p e c i a l location at t h e o r i g i n t h a t r e s u l t s in a pole-zero cancellation a t k0=0 b e t w e e n t h e i n t e r c o n n e c t i o n t e r m s . This f i x e d mode is called a s t r u c t u r a l l y fixe mode of t y p e (ii). In b o t h c a s e s , it is c l e a r t h a t no p e r t u r b a t i o n of t h e n o n z e r o e n t r i e s in t h e matrices can c h a n g e t h e s i t u a t i o n . mode in c a s e parameter,

1 and

N e v e r t h e l e s s , t h e d i f f e r e n t o r i g i n s of t h e f i x e d

2 can be s i g n i f i c a n t l y s h o w n b y a f f e c t i n g a value

which r e s u l t s

in

to a zero

s h i f t i n g t h e pole ~0=0 w i t h o u t a f f e c t i n g the i n t e r e o n -

nection s t r u c t u r e of t h e s y s t e m s •

Case I :

108

10o] 0

2

0

0

0

-1

Case 2 :

The pole at the origin is now replaced

[0101 1

0

0

0

0

2

b y a p o l e a t Xl=2. I n c a s e 1, w e h a v e n o w :

p-I

Wl(p) =

where

nothing

turally

°1

I__

IF_

(p-2)

p+l

is c h a n g e d

with respect

f i x e d m o d e a t 40=0 i s r e p l a c e d

ting that the origin of the structurally On the contrary,

to t h e z e r o i n t e r c o n n e c t i o n by a structurally

term.

The struc-

fixed mode at ~ I=2,

reflec-

fixed mode is a lack of interconnections.

i n c a s e 2~ we h a v e

:

p-2

W2(p) = P

0

(p-2)(p- i )

where

the pole-zero

no fixed mode.

cancellation

Therefore,

in

a t ~0---0 i s n o l o n g e r

case

2,

the origin

possible

of the

; i.e.

the system

fixed mode was its

has

special

location at the origin. The

next

paragraphe

modes can be characterized Refering not understand consider

will

show

in d i f f e r e n t

to t h e a b o v e

discussion,

that

these

two

types

of

we w o u l d l i k e to p o i n t o u t t h a t

t h a t all f i x e d m o d e s a t t h e o r i g i n a r e o f a s t r u c t u r a l

the following system

:

structuralIy

fixed

ways. one

nature.

should Indeed,

[1]

109

0 0 0

A=

0 0 1 a22 0 0 2

c~:[0

i

13

c~:[J

0

01

where az2 i s a n a r b i t r a r y The transfer

g j :-

[° 1

0

-l

0

2

real paramater.

f u n c t i o n m a t r i x is g i v e n b y : P+2(l-a22)

P

where it is c l e a r t h a t t h e s y s t e m h a s a f i x e d mode at k0=0 i f a n d o n l y if a22 = 1 | t h e r e f o r e , in t h i s c a s e ~0=0 i s n o t a s t r u c t u r a l l y Note a l s o t h a t t h e l a c k o f i n t e r c o n n e c t i o n is n e c e s s a r y

for

a

fixed

mode

to b e

I n d e e d , c o n s i d e r t h e following s y s t e m

A =

fixed mode. t e r m s in t h e t r a n s f e r

structural

(of type

(i))

but

function matrix not

sufficient.

:

[010][ l

0

0

0

0

2

gl=

C! =

c2~ E,

-,

for which d e t ( p I - A K) The t r a n s f e r

, l =

(p-2)(p-2kl-1)(p-2kz+l).

~1=2 i s

f u n c t i o n m a t r i x o f t h i s s y s t e m is g i v e n b y :

therefore

a fixed mode.

110 2 p-I

0

l p'-2

2 p+l

W(p) :

which exibits a zero block. mode of t y p e cellation.

(i).

One could t h u s

t h i n k t h a t ~1=2 i s a s t r u c t u r a l l y

fixed

H o w e v e r , it m u s t b e n o t i f y t h a t t h e z e r o b l o c k r e s u l t s from a can-

Indeed,

if one

replaces,

for

example,

the

first

entry

in

B2 by

2,

the

transfer function matrix becomes :

2 p-I

1 p-I

w(p) :

which has

1

3

p-2

p+l

no m o r e z e r o b l o c k . ~ 1=2 i s no l o n g e r

a f i x e d mode s i n c e t h e r e

is no

a n d s u f f i c i e n t c o n d i t i o n s f o r a pole to b e a s t r u c t u r a l l y

fixed

p o s s i b l e c a n c e l l a t i o n in t h e p r o d u c t

l

p-2

The necessary

:

1

X

p-1

mode will b e p r e s e n t e d in t h e s u b s e q u e n t p a r a g r a p h s . As w a s p o i n t e d o u t in P a r a g r a p h

3.4.5,

f i x e d m o d e s a r e t h e e x t e n t i o n of the

c o n c e p t s of u n c o n t r o l l a b l e a n d u n o b s e r v a b l e m o d e s e x i s t i n g in c e n t r a l i z e d c o n t r o l and structurally

fixed modes establish the

relation with

structurally

u n c o n t r o l l a b l e and

u n o b s e r v a b l e m o d e s . T h e p r o p e r t i e s of c o n t r o l l a b i l i t y , o b s e r v a b i l i t y a n d t h e i r s t r u c tural

counterparts,

structural

controllability

and

structural

observability9

were

d e f i n e d in C h a p t e r I. In t h e d e c e n t r a l i z e d c o n t r o l c a s e , t h e s e p r o p e r t i e s w e r e s t a t e d a n d s t u d i e d i n (KOB-78)

(KOB-82) a n d

controllability and observability under

(MOM-83) w h i c h i n t r o d u c e d t h e c o n c e p t s of decentralized information structure

a n d their

structural counterparts. T h e s e p r o p e r t i e s a r e p r e s e n t e d in t h e n e x t p a r a g r a p h fixed modes appear e v e n more o b v i o u s .

clearly.

The analogy with the

a n d t h e i r r e l a t i o n s with

centralized

control

c a s e is then

111 Some o f t h e d e f i n i t i o n s ter I a n d t h e r e a d e r

used

in the following were already

will b e r e f e r e d

to it w h e n e v e r

3.5.2. - Controllability and observability decentralized

information structure

Consider

S-station

the

t h e form ( 3 . 2 . 4 )

system

Definition (3.2.4)

(3.2.2)

and

a decentralized

3.4.

The

system

structure

(3.2.2)

if there

which

i n t e r v a l of time T , i . e . , 3.5.

The

transfers

lability

system

to b e

defined in

et

al

under

under

appears

(3.2.2)

different

with t h e d i f f e r e n t

to

be

in

(MOM-83).

controllable

decentralized initial

is

if there

state

output x(0)

under

decentralized

feedback

to t h e

(KOB-82) information

information

from which

3.5.

glance,

said

Though

to

be

of the

origine

structurally

exists a structurally

decentralized

(KOB-78)

decentralized

Definition

said a

any

decentralized

importance at first

Definition 3 . 6 .

feedback

form

in a f i n i t e

x(T)=0.

1.3) w h i c h is c o n t r o l l a b l e u n d e r Kobayashi

is

exists

decentralized control structure

controllable

control

(3.2.11).

(3.2.11),

Definition

in C h a p -

under

T h e f o l l o w i n g d e f i n i t i o n w a s g i v e n b y Momen a n d E v a n s

information

introduced

it will b e n e c e s s a r y .

the

under

system

(see §

structure. provided

also

structure.

structure

defined

controllable

equivalent

in

the

in D e f i n i t i o n

differences

a definition

for

Nevertheless, sense 3.4

do n o t

of

and

seem

we will s e e t h a t t h e i r i n t e r p r e t a t i o n

the

a system control-

Kobayashi even to

from

be

et

al

which

of a major

is strongly

connected

types of fixed modes.

The

information s t r u c t u r e

system

(3.2.2)

is

said

to

be

controllable

(in t h e s e n s e o f K o b a y a h s i e t al) if t h e r e

under exists

decentralized a decentralized

control of t h e f o r m :

ui(t) = Fi Ii(t)

(i=1 . . . . .

S)

(3.5.1)

where I . ( t ) is t h e s e t o f a v a i l a b l e d a t a a t s t a t i o n i a t t i m e t : 1

Ii(t) = {yi(p), which t r a n s f e r s i . e . , x (T)=O.

ui(q), p ~ (0,t),

any initial state

x(O)

q ~ (0,t)) to t h e o r i g i n e

in a f i n i t e i n t e r v a l

o f time T ;

112 T h e d i f f e r e n c e b e t w e e n D e f i n i t i o n s 3.6 a n d 3 . 4 can b e f o u n d in t h e different types

of d e c e n t r a l i z e d

(3.2.4)

(3.2.11)

control

laws

which are

used.

Undeed,

the

h a s n o t t h e d e g r e e of freedom of t h e one in ( 3 . 5 . 1 ) .

under decentralized information structure more similar to s t r u c t u r a l

control

law in

Controllability

in t h e s e n s e of K o b a y a s h i et al i s , in fact,

controllability under

decentralized

information

structure

d e f i n e d in D e f i n i t i o n 3 . 5 . We will s e e t h a t t h e r e q u i r e d c o n d i t i o n s f o r a s y s t e m to h a v e t h i s p r o p e r t y are also of a s t r u c t u r a l

nature.

However,

a slight

difference

still r e m a i n s ,

b a s e d on t h e e x i s t e n c e of two d i f f e r e n t t y p e s of s t r u c t u r a l l y p o i n t e d o u t in P a r a g r a p h

3.5.2.a.

which is

fixed modes,

as was

3 . 4 . 5 a n d Example 3 . 9 .

- K o b a y a s h i e t al. a p p r o a c h (KOB-78) (KOB-82)

T h e a p p r o a c h of K o b a y a s h i e t al. is b a s e d on t h e o b s e r v a t i o n t h a t ,

in decen-

t r a l i z e d c o n t r o l s y s t e m s , a c o n t r o l s t a t i o n can t r a n s m i t n e c e s s a r y i n f o r m a t i o n to other control stations

through

the

state

space

by

using

signaling

strategies.

That

is,

s t a t i o n j c a n t r a n s m i t i n f o r m a t i o n to s t a t i o n i if ( a n d o n l y if) C i ( P I - A ) - I B ] = #0. In t h i s c a s e , s t a t i o n j can e n l a r g e i t s c o n t r o l l a b l e s u b s p a c e b y s e n d i n g information

to s t a t i o n

i and

station

i can e n l a r g e

its observable

i n f o r m a t i o n from s t a t i o n j . K o b a y a h s i et al. r e p r e s e n t m a t r i c e s C~i a n d Bi,* ( i = l , . . . , S ) ,

subspaee

by

receiving

t h i s s i t u a t i o n b y d e f i n i n g the

which are interpreted

as t h e o b s e r v a t i o n a n d the

d r i v i n g m a t r i c e s of s t a t i o n i u n d e r t h e c o o p e r a t i o n of t h e o t h e r s t a t i o n s :

B~ : [Bi, Bjl . . . . . Bjk ]

if Cj(pI-A)-IBi f 0

~ j e {Jl' "" Jk"}

C.1 C C.~ =

if

Ci(PI-A)-lB j I 0

Vj ~ {ql ..... qv }

C' qv

T h e n t h e y d e v e l o p , t h e i r s t u d y b y a s s o c i a t i n g with t h e s y s t e m t h e same directed

graph

as

Corfmat

and

Morse

(COR-76a,b),

which

was

defined

in

Paragraph

2 . 2 . 3 b of C h a p e r I I . Like Corfmat a n d Morse, t h e y decompose t h e s y s t e m in s t r o n g l y connected subsystems. For a system (3.2.2)

with N s t r o n g l y c o n n e c t e d s u b s y s t e m s ,

t h e s t a t i o n s p u t t h e s y s t e m in t h e following form :

t h e r e o r d e r i n g of

113

~(t) =

A

x(t) + B1 ~1 (t) + "'" M BN ~N (t)

~i (t) = "Gi x ( t )

with

(i=l . . . . . N)

~i = [Bjl ' " "

B'i'i ]

6.. Rnxmi

~

=

Jl L

E R '

Ji

where i l , _ . . , i k areN the stations b e l o n g i n g to the s t r o n g component i. Note t h a t the matrices B. and

C, a r e composed of the

1

1

driving

,~,

and

o b s e r v a t i o n matrices of the

,~

station within the ith s t r o n g component. Bi and C i a r e also the d r i v i n g and o b s e r v a tion matrices of e v e r y station with in the s t r o n g component t u n d e r the cooperation of the remaining stations within this same s t r o n g component. The r e o r d e r e d t r a n s f e r function matrix of the system a p p e a r s t h u s in a b l o c k triangular form : N

w(p) :

I

WII

!

.~

I I

W21

N

W22

I i i _1

I l

2 WNI . . . . . . . .

r~ ....

IiWNN I

_

where ~..*] is the t r a n s f e r function matrix from the s t r o n g l y c o n n e c t e d s u b s y s t e m i to the strongly c o n n e c t e d s u b s y s t e m j. The s y s t e m p u t in the above form is called t h e q u o t i e n t s y s t e m . Kobayashi et el give then the following theorem : Theorem 5,2Z

(KOB-82).

information s t r u c t u r e

The

system

(3.2.2)

is controllable

(in the s e n s e of Kobayashi et a l . ,

0nly if the q u o t i e n t system has no d e c e n t r a l i z e d

under

decentralized

see Definition 3.6)

if and

fixed modes vrith r e s p e c t

to the

decentralized control law : U = ~ Y

= block-diag,

(K1 . . . . . KN )

Ki ~ Rmixri

(3.5.2)

(i=l . . . . .

The f i x e d modes of the q u o t i e n t system c o r r e s p o n d modes of t y p e

(i) of t h e

system

(3.2.2).

Indeed,

none

N)

to the s t r u c t u r a l l y partition

of the

fixed

quotient

114

s y s t e m can be f o u n d s u c h t h a t t h e i n t e r c o n n e e t i o n t e r m b e t w e e n t h e f i r s t aggregated station and

the

s e c o n d is n o t i d e n t i c a l l y z e r o .

As we saw in P a r a g r a p h

3.4.5, if

Theorem 3.20 is s a t i s f i e d in t h i s c a s e , t h e f i x e d modes a r e s t r u c t u r a l a n d a r i s e from t h e lack of i n t e r c o n n e c t i o n s . The n o n s t r u c t u r a l l y f i x e d modes a n d t h e structurally f i x e d modes of t y p e (it) of t h e s y s t e m ( 3 . 2 . 2 ) in ( 3 . 5 . 2 )

(if a n y ) a r e s e t a s i d e b y t h e fact that

t h e c o n t r o l law a s s o c i a t e d to each s t r o n g l y c o n n e c t e d s t a t i o n (which may

contain s e v e r a l s t a t i o n s of t h e s y s t e m ( 3 . 2 . 2 ) ) is c e n t r a l i z e d . This l e a d s to t h e following c o r o l l a r y : Corollary

3.6.

The

structure

(in t h e

system

(3.2.2)

is c o n t r o l l a b l e

s e n s e of K o b a y a s h i et a l . )

under

d e c e n t r a l i z e d information

if a n d only if it has no structurally

f i x e d modes of t y p e ( i ) . An

interesting

c o n s e q u e n c e is

that

a strongly

controllable u n d e r d e c e n t r a l i z e d i n f o r m a t i o n s t r u c t u r e

connected

s y s t e m is

always

(in t h e s e n s e of Kobayashi et

al) and c a n n o t h a v e s t r u c t u r a l l y f i x e d modes of t y p e ( i ) . If we e s t a b l i s h t h e c o n n e c t i o n with the r e s u l t s of Corfmat a n d

Morse (COR-

76a,b) p r e s e n t e d in P a r a g r a p h 2 . 2 . 3 b of C h a p t e r IIp t h e f i x e d modes can b e charact e r i z e d in t e r m s of t h e r e m n a n t polynomials of t h e c o m p l e m e n t a r y s u b s y s t e m s (see Definition 2 . 6 ,

C h a p e r II) and the r e s u l t s of Corfmat a n d Morse can b e r e s t a t e d in

t e r m s of f i x e d modes. Theorem 2.2 l e a d s to t h e following c o r o l l a r y : Corollary

3.7.

The

system

(3.2.2)

with

S stations,

controllable,

o b s e r v a b l e and

s t r o n g l y c o n n e c t e d can be s t a b i l i z e d b y a c o n t r o l law of t h e form =

u i ( t ) = Ki Yi(t) + v i ( t )

f

zj(t)

Sj z j ( t ) + Rj y j ( t )

vj(t)

Qj zj(t) + Nj y j ( t )

(i=l . . . . . S) j e (1 . . . . . S}

if a n d only if t h e s e t of f i x e d modes (which c a n n o t b e s t r u c t u r a l l y f i x e d modes of t y p e (i)) is s t a b l e . The a r b i t r a r y pole p l a c e m e n t is p o s s i b l e if a n d only if t h e s e t of f i x e d modes is e m p t y .

(Note t h a t t h e control law u s e s a dynamic c o n t r o l l e r only at

one s t a t i o n s i n c e s t r o n g l y c o n n e c t e d s y s t e m s w i t h o u t f i x e d modes can b e made controllable a n d o b s e r v a b l e b y a s i n g l e s t a t i o n u s i n g s t a t i c d e c e n t r a l i z e d f e e d b a c k ) , For t h i s s y s t e m , t h e s e t of d e c e n t r a l i z e d f i x e d modes

( i n c l u d i n g n o n struc-

t u r a l l y f i x e d modes a n d s t r u c t u r a l l y f i x e d modes of t y p e (it)) is c h a r a c t e r i z e d b y the f i x e d polynomial :

115 F(p) = l.c.m.

P ( G a , A,BI3 )

k=l,...,S-1 ~a = { i l . . . . . ik}C

~

= {1 . . . . . S}

If we c o n s i d e r now t h e g e n e r a l case of n o n s t r o n g l y c o n n e c t e d s y s t e m s , T h e o rem 2.3 can be r e s t a t e d as follows : Corollary 3.8. The c o n t r o l l a b l e a n d o b s e r v a b l e s y s t e m ( 3 . 2 . 2 ) can be s t a b i l i z e d u s i n g a dynamic d e c e n t r a l i z e d c o n t r o l of t h e form ( 2 . 2 . 6 ) modes ( s t r u c t u r a l l y a n d non s t r u c t u r a l l y )

if a n d only if t h e s e t of f i x e d

is s t a b l e . T h e a r b i t r a r y pole p l a c e m e n t is

possible if and only if t h e s e t of fixed modes is e m p t y . For t h i s s y s t e m ,

t h e s e t of s t r u c t u r a l l y

f i x e d modes of t y p e

(i) is g i v e n b y

the set of poles of t h e s y s t e m which a r e n o t pole of a n y s t r o n g l y c o n n e c t e d s u b s y s tem (fixed modes of t h e q u o t i e n t s y s t e m ) . T h e r e m a i n i n g f i x e d modes ( i n c l u d i n g non structurally f i x e d modes and s t r u c t u r a l l y fixed modes of t y p e (ii)) a r e g i v e n b y t h e union of t h e s e t s of f i x e d modes of e v e r y s t r o n g l y c o n n e c t e d s u b s y s t e m . For each strongly c o n n e c t e d s u b s y s t e m , t h e y can be c h a r a c t e r i z e d as follows :

Fi(P)

= l.c.m ~iac

P(~ia 'A , ~i~)-

h (~i,A,~i,R~nix~i )

1Ti ~i = {Jl . . . .

Ji }

The e q u i v a l e n c e b e t w e e n the r e s u l t s of Wang a n d 2.2.3.a,

Davison

(WAN-73)

C h a p t e r II) a n d t h o s e of Gorfmat a n d Morse is now clear.

(see §

Note t h a t

the

results of Corfmat a n d Morse a r e more complete s i n c e t h e y i n c l u d e t h e d i f f e r e n t i a t i o n between s t r u c t u r a l l y a n d non s t r u c t u r a l l y f i x e d m o d e s .

3.5.2.b. - Momen a n d Evans approach (MOM-83) The a p p r o a c h of Momen and E v a n s is p u r e l y s t r u c t u r a l .

The p r i n c i p l e remains

the same as in K o b a y a s h i e t al. a p p r o a c h s i n c e t h e i r r e s u l t s are also b a s e d on t h e determination of t h e e n l a r g e d s t r u c t u r a l l y

c o n t r o l l a b l e a n d o b s e r v a b l e s u b s p a c e s of

every station u n d e r t h e collaboration of t h e o t h e r o n e s . Let Li , (i=l . . . . . S) be t h e s t r u c t u r a l l y c o n t r o l l a b l e a n d o b s e r v a b l e s u b s p a c e of station i a n d L.* t h e e n l a r g e d s t r u c t u r a l l y c o n t r o l l a b l e a n d o b s e r v a b l e s u b s p a c e o f 1 station i u n d e r t h e c o o p e r a t i o n of t h e o t h e r s t a t i o n s . T h e n , we h a v e t h e following result :

116 Theorem 3,23

(MOM-83).

The system

decentralized information structure S u i-1

The condition

(3.2.2)

is s t r u c t u r a l l y

controllable under

L~ = R n

(3.5.3)

the

( s e e D e f i n i t i o n 3 . 5 ) if a n d o n l y if :

(3.5.3)

m e a n s t h a t t h e u n i o n of t h e e n l a r g e d s t r u c t u r a l l y

con-

t r o l l a b l e a n d o b s e r v a b l e s n b s p a c e s o v e r all t h e s t a t i o n s is e q u a l to t h e s t a t e s p a c e of the

system.

observable

Note

that

under

if

condition

decentralized

(3.5.3)

holds,

information

the

s y s t e m is a l s o s t r u c t u r a l l y

structure.

The

connection

with

fixed

m o d e s is t h e n made in t h e following t h e o r e m : T h e o r e m 5.24 vable) under

(MOM-83). T h e s y s t e m ( 3 . 2 . 2 )

is s t r u c t u r a l l y

decentralized information structure

controllable

(and obser-

i f a n d o n l y if it h a s no s t r u c t u r a l l y

fixed modes. Therefore,

Momen

decentralized structure

and

Evans

(MOM-83)

(see Definition 3.4)

define

a

controllable

system

m o d e s of a n y t y p e a n d e x t e n d t h i s r e s u l t in a s t r u c t u r a l

w a y in D e f i n i t i o n 3.5 a n d

T h e o r e m 3,24 (As w a s p o i n t e d o u t in t h e e x a m p l e s g i v e n in P a r a g r a p h s y s t e m with s t r u c t u r a l l y

under

as a s y s t e m without decentralized fixed 3.5.1,

for a

f i x e d m o d e s of t y p e (i) or ( i i ) , n o n e s t r u c t u r a l l y e q u i v a l e n t

s y s t e m without fixed modes can be f o u n d ) .

The

difference between

tion s t r u c t u r e

and

structural

controllability

s e n s e of K o b a y a s h i e t a l .

controllability under

under

decentralized

c a n be f o u n d i n t h e f a c t t h a t ,

rally fixed modes include both structurally Indeed,

the

d e t e r m i n a t i o n of t h e

structurally

s u b s p a c e s is b a s e d on t h e c o n d i t i o n s for s t r u c t u r a l ( s e e T h e o r e m s 1.6 a n d

1.7,

Chapter

I).

in T h e o r e m 3 . 2 4 ,

f i x e d m o d e s of t y p e

enlarged

decentralized informa-

information structure

in

(i) a n d of t y p e

controllable

and

the

structu(ii).

observable

controllability and observability

These conditions involve both reachability

conditions and generic rank conditions. The teachability conditions t u r n into c o n n e c tivity

conditions

strongly

in

the

connected and,

The generic rank

case

of

therefore,

decentralized

in d e c e n t r a l i z e d c o n t r o l ,

f i x e d m o d e s of t y p e Remember that subspaces

were

i.e.,

the

the

s y s t e m m u s t be

f i x e d m o d e s of t y p e

conditions take into account the particular

o r i g i n w h i c h m a y a l s o lead to s t r u c t u r a l way,

control :

it h a s no s t r u c t u r a l l y

u n c o n t r o l l a b i l i t y or u n o b s e r v a b i l i t y .

s y s t e m is a l s o r e q u i r e d

(i).

c a s e of a pole a t t h e By this

n o t to h a v e s t r u c t u r a l l y

(ii). in

Kobayashi et

determined using

al,

the

connectivity

enlarged

controllable

conditions only :

and

this

is

observable why

their

117 controllability under decentralized information structure to h a v e s t r u c t u r a l l y

only requires the system not

f i x e d m o d e s of t y p e ( i ) .

It is c l e a r now t h a t d e c e n t r a l i z e d f i x e d m o d e s c a n be d e f i n e d a s t h e m o d e s t h a t are

uncontrollable

under

the

decentralized control in the

decentralized form

(2.2.6).

information However,

structure

using

an i n t e r e s t i n g

a

dynamic

remark

c a n be

made. F r o m T h e o r e m 3.24 a n d C o r o l l a r y 3.6 a n d t h e D e f i n i t i o n s 3.5 a n d 3 . 6 , we c a n d e d u c e t h e following r e s u l t : Corollary 3.9.

Non s t r u c t u r a l l y

f i x e d m o d e s a n d s t r u c t u r a l l y f i x e d m o d e s of t y p e (ii)

are controllable u n d e r the d e c e n t r a l i z e d information s t r u c t u r e

using a dynamic time-

v a r y i n ~ d e c e n t r a l i z e d c o n t r o l l e r in t h e form ( 3 . 5 . 1 ) . Structurally information

f i x e d m o d e s of t y p e

structure

even

if

the

(i)

are not controllable

time-invarience

constraint

under on

the

decentralized decentralized

c o n t r o l is r e l a x e d . This point

will be d i s c u s s e d

a g a i n in C h a p t e r

t a t i o n of t h i s r e s u l t will b e p o i n t e d o u t . 3.9 e x p r e s s e s

IV a n d t h e p h y s i c a l i n t e r p r e -

It is i n t e r e s t i n g

to m e n t i o n t h a t

Corollary

a major d i f f e r e n c e w i t h r e s p e c t to t h e c a s e of c e n t r a l i z e d c o n t r o l s i n c e

uncontrollable and unobservable modes remain uncontrollable and unobservable

even

if a d y n a m i c t i m e - v a r y i n g c o n t r o l l e r i s u s e d .

A l t h o u g h t h e a b o v e s t u d y w a s c a r r i e d o u t in a d e c e n t r a l i z e d i n f o r m a t i o n s t r u c t u r e f r a m e w o r k , it is c l e a r t h a t t h e r e s u l t s

c a n be e x t e n d e d

to t h e g e n e r a l c a s e of

arbitrary constrained information structure.

3.5.2.c.

- P o t e n t i a l pole a s s i g n m e n t u s i n g d e c e n t r a l i z e d

static output feedback The issue

of c o n t r o l l a b i l i t y

information structure concepts

of

fixed

modes.

( a b s e n c e of s t r u c t u r a l l y potential paragraph using

a

pole

and

structural

controllability

under

decentralized

h a s b e e n d i s c u s s e d in t h e p r e c e d i n g s e c t i o n a n d r e l a t e d to t h e It

has

been

pointed

out

that

f i x e d m o d e s ) is t h e n e c e s s a r y

assignment

using

a

decentralized

structural

controllability

a n d s u f f i c i e n t c o n d i t i o n for

dynamic

controller.

The

present

g i v e s t h e n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r p o t e n t i a l pole a s s i g n m e n t decentralized

static

controller.

Of c o u r s e ,

a s s u r e t h a t t h e s y s t e m h a s no s t r u c t u r a l l y

this

condition

is

sufficient

f i x e d m o d e s b u t it is n o t n e c e s s a r y .

to

The

a p p r o a c h of E v a n s a n d K r u s e (EVA-84) i s o r i g i n a l in t h e s e n s e t h a t it d o e s n o t r e f e r e x p l i c i t l y to t h e

c o n c e p t s of c o n t r o l l a b i l i t y ,

t h e y a r e implied in t h e final r e s u l t .

observability or fixed modes,

although

118

The r e s u l t is c a r r i e d out b y a g r a p h - t h e o r e t i c m e t h o d u s i n g t h e work of Mason (MAS-56) w h i c h e s t a b l i s h e s t h e r e l a t i o n b e t w e e n t h e s t r u c t u r e of t h e d i g r a p h a s s o c i a t e d to a s q u a r e m a t r i x and i t s c h a r a c t e r i s t i c polynomial. Consider a nxn s u c h t h a t V=

matrix

Vl,...,v n

M=(mij)i,j= 1 . . . . n"

One

defines

the

digraph

D=(V,E)

is a s e t of n v e r t i c e s a n d E is a s e t of e d g e s from V to

V . ( v i , v j) E E if a n d only if mji#0 in M. T h e w e i g h t mji is a s s o c i a t e d with t h e e d g e ( v i , v j ) . The c h a r a c t e r i s t i c polynomial of M is g i v e n b y : ¢ (p) = d e t ( p I - M ) = p n + a n _ l P

n-l+

...

+

alP+a 0

then, the coefficients a 0 , . . . , a n _ 1 are given by :

an_ 1 =

Z ()~)

an_ z =

z [~

a1

= E In

n-1

(x)]

(x)]

a0 = r [~ (x)] in which

~[~

(l)]

d e n o t e s t h e sum o v e r t h e c y c l e s of l e n g t h j (or p r o d u c t

of disjoint c y c l e s w h o s e sum of l e n g t h s is j) of t h e p r o d u c t s of t h e w e i g h t s a s s o ciated to t h e e d g e s i n v o l v e d in e a c h c y c l e . The c o m p l e t e a s s i g n m e n t of all r o o t s o f ¢ (p) to some s e t of a r b i t r a r y v a l u e s can only b e a c h i e v e d if n o n e of t h e c o e f f i c i e n t s a o , . . . , a n _ 1 is zero. Consider the system (3.2.1)

and t h e c o n t r o l law : U=KY

w h e r e K is t h e o u t p u t f e e d b a c k matrix with an a r b i t r a r y s p e c i f i e d s t r u c t u r e .

Then,

from t h e a b o v e r e s u l t , p o t e n t i a l pole a s s i g n m e n t can be a c h i e v e d if : 1-All t h e c o e f f i c i e n t s in t h e c h a r a c t e r i s t i c polynomial of (A+BKC) can be a s s i g n e d i n d e p e n d e n t l y b y t h e e x i s t i n g cycle g a i n s of t h e a s s o c i a t e d g r a p h . 2 - S u f f i c i e n t cycle g a i n s can be a s s i g n e d i n d e p e n d e n t l y b y t h e f e e d b a c k g a i n s in K. U s i n g t h e a b o v e d i s c u s s i o n , E v a n s a n d K r u s e r ( E V A - 8 4 ) p r o v i d e d t h e following result :

119

Let ( k 1 . . . . , k q }

b e t h e s e t of f e e d b a c k g a i n s in K a n d {L1, . . . . L c} b e t h e s e t

of e x i s t i n g c y c l e s in t h e g r a p h a s s o c i a t e d with ( A + B K C ) . Define .

an_ 1

.-.

a0

L1 MLa= L c

s u c h t h a t m L a ( i , j ) = l if a n d only if Li is of l e n g t h j o r a p p e a r in a p r o d u c t

of d i s -

joint c y c l e s w h o s e sum of l e n g t h s is j, a n d m L a ( i , j ) = 0 o t h e r w i s e . Define also '

L 1 ....

Lc

MKL= i kq where

rnKL(i,j)=l

if a n d

only

if

k i appears

in

the

gain

of

the

cycle

Lj,

and

mKL(i,j)~0 o t h e r w i s e . Then,

the necessary

a n d s u f f i c i e n t c o n d i t i o n for p o t e n t i a l p01e a s s i g n m e n t is

that :

gr[MKL

* MLa] = n

w h e r e * d e n o t e s t h e b o o l e a n p r o d u c t of two b i n a r y m a t r i c e s . This result provides an interesting test, test

determination,

feedback.

f o r p o t e n t i a l pole

in t h e form of a simple g e n e r i c r a n k

assignment

using

F o r t h e p r a c t i c a l i m p l e m e n t a t i o n of t h e t e s t ,

decentralized

static output

Evans and Kruser

(EVA-84)

p r o v i d e d a n APL r o u t i n e w h i c h g i v e s t h e list of e x i s t i n g c y c l e s in a d i g r a p h .

3 . 5 . 3 . - C h a r a c t e r i z a t i o n of s t r u c t u r a l l y

f i x e d modes

Similarly to t h e case of c e n t r a l i z e d c o n t r o l , t h e c h a r a c t e r i z a t i o n of s t r u c t u r a l l y fixed modes ( a n d t h e r e f o r e of s t r u c t u r a l structure)

controllability under constrained information

can be carried out either within a pure algebraic framework or within a

graph-theoretic framework.

120

3,5,3,a.

Algebraic approach

-

Using (LIN-74)

the

results

(SHI-76)

characterization T h e o r e m 3.25

obtained

(see

§ 1.3,

of structurally (SEZ-81a).

for

Chapter

structural

controllability

I),

and

Sezer

Siljak

and

observability

derived

the

following

fixed modes :

The system

(3.2.2)

with S stations has

structurally

fixed

m o d e s if a n d o n l y i f o n e o f t h e f o l l o w i n g c o n d i t i o n s h o l d s : i

-

exists

There

matrix P such that A1 1

A31

L

P'g =

. . . .

I 0

Ct

o B

il - T h e r e e x i s t s a

permutation

2 Bet

(3.5.~)

~a o n =

{1 . . . . .

S}

fixed modes of type

can never

such that

n

(i) a r e c h a r a c t e r i z e d of the system

b e p u t in t h e form ( 3 . 5 . 4 ) ) .

m a t r i x A22. T h i s s i t u a t i o n i s i l l u s t r a t e d f i x e d m o d e s o f t y p e ( i ) . It a p p e a r s from t h e a g g r e g a t e d structure

by condition

(of course,

(i) w h i c h

a strongly

They are the eigenvalues

by Figure 3.5.

x = ( x l , x 2 , x 3) w h e r e x 2 a r e t h e s t a t e s

of the block-triangular

:

0

e x i b i t s a n o b v i o u s lack o f c o n n e c t i v i t y

fer is possible

a

g

C

3 subvectors

and

]

[A 0]

gr

ted system

( ~ a u ~B ---7)

I

g

Structurally

S}

133 C~

A32 i A33

[c 1

CP=

{1 . . . . .

II 0 0 t. . . . . I i .422 ~ 0 . . . . .

CP=

~a c ~ =

:

..... A21

P'AP :

a

connecof t h e

T h e s t a t e v e c t o r i s s p l i t e d in

associated with the structurally

c l e a r l y in t h e f i g u r e t h a t n o n e i n f o r m a t i o n t r a n s station

a to t h e

o f t h e m a t r i x A.

aggregated

station

B because

121

/

\

\

/

F i g u r e 3.5 S t r u c t u r a l l y f i x e d modes of t y p e (ii) a r e c h a r a c t e r i z e d b y c o n d i t i o n (ii) which is t h e s t r u c t u r a l c o u n t e r p a r t of c o n d i t i o n ( 3 . 3 . 2 )

( C h a r a c t e r i z a t i o n of f i x e d modes of

A n d e r s o n and Clement. see § 3.3.1~ C h a p t e r II) for a pole at t h e o r i g i n e . Note t h a t

a structurally

condition ( 3 . 5 . 5 )

f i x e d mode o f t y p e

(i)

at the o r i g i n s a t i s f i e s also

b u t it a r i s e s from t h e lack of c o n n e c t i v i t y of t h e s y s t e m a n d r e -

mains u n c o n t r o l l a b l e u n d e r dynamic t i m e - v a r y i n g d e c e n t r a l i z e d c o n t r o l . C l e a r l y , t h e i n v e r s e is n o t

true ; i.e.

if t0=0 s a t i s f i e s

(3.5.5),

condition

(3.5.4)

is

not

ne-

c e s s a r i l y satisfied. The similarity with s t r u c t u r a l l y u n c o n t r o l l a b l e a n d u n o b s e r v a b l e m o d e s a p p e a r s clearly s i n c e , as we saw in C h a p t e r I, t h e y may a r i s e from t h e lack of r e a c h a b i l i t y or from t h e i r s p e c i a l location at t h e o r i g i n .

The c o n n e c t i v i t y condition ( 3 . 5 . 4 )

cor-

r e s p o n d s to t h e r e a c h a b i l i t y c o n d i t i o n s ( C o n d i t i o n s 1 a n d 2 of T h e o r e m s 1.6 a n d 1.7 in C h a p t e r I) rank

and

conditions

the

generic rank

( C o n d i t i o n s 3 and

condition

(3.5.5)

4 of T h e o r e m s

c o r r e s p o n d s to t h e

1.6 a n d

generic

1.7 in C h a p t e r

I).

Of

c o u r s e , in t h e case o f d e c e n t r a l i z e d c o n t r o l , we have only t h e c o n c e p t of f i x e d mode ( i n s t e a d of t h e two dual c o n c e p t s of u n c o n t r o l l a b l e a n d u n o b s e r v a b l e mode in c e n tralized control) peration

since the

of s e v e r a l

stations

c o n t r o l l a b i l i t y of a mode is a c h i e v e d t h r o u g h by

using

the

feedback

from some o u t p u t

the

coo-

stations

in

which t h e o b s e r v a b i l i t y c o n c e p t is a l r e a d y i n v o l v e d . Note t h a t Theorem 3.25 can b e g e n e r a l i z e d to a r b i t r a r i l y c o n s t r a i n e d c o n t r o l : in t h i s c a s e , 3.3, § 3.3.1).

t h e s e t s nc~ a n d ~

are

replaced by

the

sets I and J

(see Definition

122

3.5.3.b.

-

Graph-Theoretic

approach

Similarly to t h e c a s e of c e n t r a l i z e d c o n t r o l , t h e c o n d i t i o n s f o r s t r u c t u r a l c o n trollability

under

decentralized information s t r u c t u r e

theoretic framework.

Several characterizations exist,

can

he

derived

d e p e n d i n g on t h e

in a g r a p h graph

that

one a s s o c i a t e s to t h e s y s t e m . The one w h i c h is p r e s e n t e d f i r s t is t h e e x a c t g r a p h i c a l c o u n t e r p a r t of t h e a l g r e b r a i e c h a r a c t e r i z a t i o n g i v e n in P a r a g r a p h 3 . 5 . 3 a .

1 - U s i n g t h e g r a p h of t h e c l o s e d - l o o p s y s t e m Given t h e s y s t e m ( 3 . 2 . l ) , U = {u 1 . . . . . ur~

vertices and Y = { y l , . . . , y r t h a t ( u i , xj)

one a s s o c i a t e s a d i g r a p h

is the set of input v e r t i c e s , }

r = ( u U X u Y, E), w h e r e

X = {x 1 . . . . . x }

is t h e s e t of s t a t e

t h e s e t of o u t p u t v e r t i c e s . E i s t h e s e t of e d g e s s u c h

E if a n d only if bji~0 in B , ( x i, xj) ~ E i f a n d only if aji ¢0 in A and

(xi,Yj) ~ E if a n d only if cji~t0 in C. Note t h a t t h i s d i g r a p h is t h e same as t h e one d e f i n e d in P a r a g r a p h

1.3.1.b

( C h a p t e r I) for the s t u d y of s t r u c t u r a l c o n t r o l l a b i l i t y

a n d o b s e r v a b i l i t y in a g r a p h i c a l f r a m e w o r k . N e v e r t h e l e s s , in o u r p r e s e n t s t u d y ,

we

add a s e t of e d g e s EK from Y to U which r e p r e s e n t s t h e p a r t i c u l a r s t r u c t u r e of t h e control.

(Yl'Uj) g E K i f a n d only if kji¢0 in t h e

a s s o c i a t e d to t h e s y s t e m a n d t h e c o n t r o l s t r u c t u r e

f e e d b a c k m a t r i x K.

The

digraph

is t h e r e f o r e I~K= (U u X u Y , E u

EK) • In t h i s g r a p h i c a l f r a m e w o r k , t h e following g r a p h i c a l c h a r a c t e r i z a t i o n of s t r u c t u r a l l y f i x e d modes was p r o v i d e d i n d e p e n d e n t l y b y L i n n e m a n n (LIE-83) a n d b y Pichai e t al. (PIC-84)

:

Theorem 3.26.

The s y s t e m ( 3 . 2 . 1 )

the arbitrarily

s t r u c t u r a l l y c o n s t r a i n e d control

h a s no s t r u c t u r a l l y (3.3.4)

f i x e d modes w i t h r e s p e c t to if a n d only if b o t h of t h e

following two c o n d i t i o n s hold : i - e a c h s t a t e v e r t e x x k c X is i n v o l v e d in a s t r o n g c o m p o n e n t of FK which i n c l u d e s an e d g e from EK. ii - t h e r e e x i s t s a s e t of d i s j o i n t c y c l e s t . k = ( V k , E k ) , ( k = l , . . , , c ) that

in FK

•

c

Xc (i.e.,

u Vk k=l

t h e whole s t a t e v e r t i c e s s e t is s p a n n e d b y a d i s j o i n t u n i o n of c y c l e s ) .

such

123

Condition condition (ii)

(i)

corresponds

to t h e

to t h o s e s of t y p e

(ii).

structurally

Obviously,

fixed

condition

e q u i v a l e n t to c o n d i t i o n (i) i n T h e o r e m 3.25 a n d F i g u r e c o r r e s p o n d i n g to x 2 a r e

not

modes of

c o n t a i n e d in a s t r o n g

type

(i)

(i) in T h e o r e m

3.5 s h o w s t h a t

component including

and

3.26 is

the

states

feedback

e d g e s w h e n f e e d b a c k is allowed from Ya to Ua a n d from Y~3 and U 6. The e q u i v a l e n c e of c o n d i t i o n (ii) in Theorem 3.26 and c o n d i t i o n (ii) in Theorem 3.25 i s n o t so o b vious. The p r o o f can he f o u n d in ( P I C - 8 4 ) . Example 3 . 1 0 . C o n s i d e r t h e two following c a s e s a l r e a d y t r e a t e d in Example 3.9 : Case 1 :

l . . . . . m

=

0

l

o 2

i

I.._ . . . . .

0

,=EI

o

J- . . . . .

0

!

o

I

0

Bl:

|1. . . . .

~

I

52-

-I

o ]

T h i s s y s t e m a l r e a d y a p p e a r s in t h e form ( 3 . 5 . 4 )

with B a =

B1, BB = B2, C a =

C 1 and C~3 = C 2. So, c o n d i t i o n (i) of T h e o r e m 3.25 is s a t i s f i e d . T h e r e f o r e , ), 0--2 is a s t r u c t u r a l l y f i x e d mode of t y p e

( i ) . It is e a s y to v e r i f y t h a t condition (ii) of T h e o -

rem 3.25 d o e s n o t h o l d I i . e . ,

the s y s t e m has no s t r u c t u r a l l y

(ii).

f i x e d modes of t y p e

The same c o n c l u s i o n s can b e s t a t e d b y c o n s i d e r i n g t h e d i g r a p h a s s o c i a t e d to

this s y s t e m :

ul /

klJ

~

Yi

~2 Y2

w h e r e it a p p e a r s c l e a r l y t h a t t h e v e r t e x x 2 is n o t c o n t a i n e d in a s t r o n g c o m p o n e n t i n c l u d i n g a f e e d b a c k e d g e . H o w e v e r , t h e s t a t e v e r t i c e s s e t is s p a n n e d b y a d i s j o i n t union o f c y c l e s as it is s h o w n below :

124

(~)

Ul ~

Yl

@

¢)

Therefore, condition (i) of Theorem 3.26 is satisfied but not condition (ii). Case 2 : I

0 1 0

1 0 0

0 0 0

1

Cl= E 0

0

l

]

A=

]'I:ll

No permutation matrix can be found to put the system in the form (3.5.4) i i . e . , the system has no structurally fixed modes of type (i). Nevertheless, condition (ii) of Theorem 3.25 holds since :

gr

A

BI ] =2<3

C2

0

I

The digraph of this system is the following :

uI

Yl

Y2

125

We

find that condition

(ii) of Theorem

contains no disjoint union of cycles such

3.26 does not hold since the graph

that all the state vertices are involved.

Therefore, this system has a structurally fixed mode of type (ii) at the origin. 2-Using t h e g r a p h of the open-loop This

characterization

was

system

also p r o v i d e d

by

P i c h a i et al

(the

it was p r e s e n t e d

same

characterization)

before.

from t h e c o m p u t a t i o n a l p o i n t of v i e w , it is m u c h

It will a p p e a r c l e a r l y t h a t ,

but

(PIG-83)

a u t h o r s as t h o s e of t h e p r e c e d i n g

one y e a r

l e s s i n t e r e s t i n g t h a n t h e p r e c e d i n g o n e . I n d e e d , it c o n s i s t s of a t e s t t h a t h a s to b e a p p l i e d to all t h e 2S-2 c o m p l e m e n t a r y s u b s y s t e m s of t h e s y s t e m while t h e c h a r a c t e r i z a t i o n g i v e n b e f o r e is s t a t e d in t e r m s of t h e whole s y s t e m . s e n t s t h e i n t e r e s t of c h a r a c t e r i z i n g b o t h s t r u c t u r a l l y

Nevertheless,

fixed modes of t y p e

it p r e -

(i) a n d of

t y p e (ii) in a s i n g l e t e s t . T h e d i g r a p h a s s o c i a t e d to t h e s y s t e m ( 3 . 2 . 1 ) is t h e s a m e as in t h e p r e c e d i n g p a r a g r a p h

For a given d i g r a p h ,

:

r=(uu

described by the triple

(C,A,B)

x u Y, E ) .

t h e foIlowing s p e c i a l s u b g r a p h s

are first defined :

Definition 3 . 7 ( P I C - 8 3 a )

Input

Stem : A p a t h

from a n i n p u t

vertex

(the root)

to a s t a t e v e r t e x

(the

to a n o u t p u t v e r t e x

(the

tip), or a single i n p u t v e r t e x . O u t p u t Stem : A p a t h from a s t a t e v e r t e x

(the root)

t i p ) , or a s i n g l e o u t p u t v e r t e x . Input-Output

Stem : A p a t h from a n i n p u t v e r t e x to a n o u t p u t v e r t e x .

S t a t e Stem : A p a t h b e t w e e n two s t a t e v e r t i c e s , o r a s i n g l e s t a t e v e r t e x . Cycle : Already defined (see § 1.3.1b,

Chapter I).

I n p u t c a c t u s : An i n p u t s t e m w i t h at l e a s t o n e s t a t e v e r t e x . tip of t h e i n p u t

s t e m a r e also t h e r o o t a n d

The root and the

t h e tip of t h e i n p u t c a c t u s .

An i n p u t

c a c t u s c o n n e c t e d to a c y c l e from a n y p o i n t o t h e r t h a n t h e t i p is a l s o an i n p u t c a c tus. O u t p u t C a c t u s : Similar d e f i n i t i o n to a n i n p u t c a c t u s . C h a i n : A g r o u p of d i s j o i n t c y c l e s c o n n e c t e d to e a c h o t h e r in s e q u e n c e ,

or a

single cycle. L i n k ~ An i n p u t s t e m c o n n e c t e d to t h e f i r s t c y c l e of a c h a i n other than the tip),

(from a n y p o i n t

t h e l a s t c y c l e of w h i c h is c o n n e c t e d to an o u t p u t

any point other than the root).

These special subgraphs

a r e i l l u s t r a t e d i n F i g u r e 3.6 :

stem

(from

126

(I) input stem

*--~---e---D,--*---~--O (2) output stem

(3) input=output stem

(4) state stem

© (5) cycle

(7) output cactus

(6) input cactus

(8) chain

(9) Link F i g u r e 3.6

A g e n e r a l i z e d c a c t u s is t h e n d e f i n e d as follows Definition

3.8

(PIC-83a).

Each

of t h e

following d i g r a p h s

is

called

a

generalized

cactus : - Disjoint u n i o n of one or more i n p u t - o u t p u t

s t e m s p l u s t h e same n u m b e r of

s t a t e s t e m s p l u s some (or n o n e ) c y c l e s , i n p u t s t e m s , o u t p u t s t e m s . -Disjoint

u n i o n o f a link with some

(or n o n e )

cycles, input

stems,

output

stems. - Disjoint u n i o n of i n p u t cacti and o u t p u t c a c t i . C o n s i d e r t h a t s y s t e m ( 3 . 2 . 1 ) is c o n t r o l l e d b y an a r b i t r a r i l y c o n s t r a i n e d c o n t r o l in

the

form

(3.3.4).

The

complementary

subsystems

( C j , A , B I)

r e s p e c t to ( 3 . 3 . 4 ) were d e f i n e d in Definition 3.3 ( P a r a g r a p h 3 . 3 . 1 ) .

of

(3.2.1)

with

127

Pichai e t al. (PIC-83) s t a t e d t h e following c h a r a c t e r i z a t i o n of s t r u c t u r a l l y f i x e d modes : Theorem 3.27.

The s y s t e m ( 3 . 2 . 1 )

the c o n t r o l law ( 3 . 3 . 4 )

h a s no s t r u c t u r a l l y

f i x e d modes with r e s p e c t to

if a n d only if e a c h d i g r a p h a s s o c i a t e d with a c o m p l e m e n t a r y

s y s t e m is s p a n n e d b y a g e n e r a l i z e d c a c t u s . Pichai e t aI.

(PIC-83a)

p r o v i d e d an algorithm to c h e c k w h e t h e r a d i g r a p h is

s p a n n e d b y a g e n e r a l i z e d c a c t u s , w h i c h i n v o l v e s only b i n a r y c o m p u t a t i o n s . T h e t e s t for t h e e x i s t e n c e of s t r u c t u r a l l y f i x e d modes r e q u i r e s to a p p l y t h i s algorithm to each "complementary s u b g r a p h " ,

i.e.

2S-2 times. This is d u e to t h e fact t h a t t h e s t r u c -

t u r e of t h e c o n t r o l is n o t i n v o l v e d in t h e g r a p h a s s o c i a t e d with t h e trarily

to t h e

case of t h e p r e c e d i n g

characterization),

system (con-

Consequently,

this charac-

t e r i z a t i o n r e q u i r e s an e n o r m o u s c o m p u t a t i o n a l t a s k .

3-Using t h e g e n e r i c r a n k of t h e p r o d u c t of s e v e r a l r e a l m a t r i c e s Recently,

Papadimitriou and

Tsitsiklis

(PAP-84) p r o v i d e d an i n t e r e s t i n g g r a -

phical r e s u l t to e v a l u a t e t h e g e n e r i c r a n k of t h e p r o d u c t of s e v e r a l r e a l m a t r i c e s . Its application to t h e d e t e r m i n a t i o n of s t r u c t u r a l l y f i x e d modes of t y p e

(ii) l e a d s to a

nice c o m p u t a t i o n a l a l g o r i t h m . Consider (i=l, . . . . k ) ,

the

sequence

of s t r u c t u r e d

matrices

(see

§ 1.3,

Chapter

I),

Mi,

a s s o c i a t e d to t h e s e q u e n c e of r e a l m a t r i c e s Mi & RPiXqi. We d e f i n e t h e

g r a p h G=(V,E) as follows : V = { v i : ( i = l . . . . . k ) , ( j = l . . . . . pi ) ; V~+l:

(j=l . . . . . qk+l)}

is a s e t of v e r t i c e s .

One v e r t e x is a s s o c i a t e d to e v e r y row of e v e r y matrix Mi, ( i = l , . . . , k ) .

The l a s t q k + l

v e r t i c e s a r e a s s o c i a t e d to t h e columns of t h e last m a t r i x Mk of t h e s e q u e n c e . T h e r e is an e d g e from t h e v e r t e x vlP to t h e v e r t e x viq+l if t h e ( p , q ) t h e n t r y of t h e matrix M. is n o n z e r o . 1

An i n f o r m a t i o n p a t h is d e f i n e d as a p a t h from a v e r t e x with s u b s c r i p t 1 to one with s u b s c r i p t k + l .

Two i n f o r m a t i o n p a t h s a r e s a i d to b e i n d e p e n d e n t if t h e y a r e

vertex-disjoint. T h e n , we h a v e t h e following r e s u l t : Theorem 3 . 2 7 b ( P A P - 8 4 ) . information p a t h s in G.

k gr[i~l

MiJ is e q u a l to t h e maximum n u m b e r of i n d e p e n d e n t

128

In o r d e r to a p p l y t h i s r e s u l t to t h e c h a r a c t e r i z a t i o n of s t r u c t u r a l l y f i x e d m o d e s of t y p e

(ii),

note that condition

(ii) in T h e o r e m 3.25 is e q u i v a l e n t to g r

n , w h e r e A, B, C a r e t h e s t r u c t u r e d

m a t r i c e s of s y s t e m ( 3 . 2 . 1 )

red matrix specifying the feedback information pattern.

(A+BKC) <

and K the structu-

Note m o r e o v e r t h a t (A+BKC)

can be r e w r i t t e n a s t h e p r o d u c t of t h r e e m a t r i c e s t

(A+BKC) with

:

= M I . M 2 . M 3

M 1 =

[I

B]

0

K

M2 =

I denotes the identity structured

173.5.6)

m a t r i x a n d 0 t h e n u l l m a t r i x of a p p r o p r i a t e d i m e n -

sion s.

T h e following c h a r a c t e r i z a t i o n of s t r u c t u r a l l y f i x e d m o d e s of t y p e (ii) r e s u l t s : T h e o r e m 3.28 with r e s p e c t

(PAP-84).

System (3.2.1)

to t h e f e e d b a c k p a t t e r n

information paths

in t h e

graph

h a s no s t r u c t u r a l l y specified by

associated

to

the

f i x e d m o d e s of t y p e

sequence

of s t r u c t u r e d

MI,M2,M 3 d e f i n e d in ( 3 . 5 . 6 ) is e q u a l to n . Example 3.11.

C o n s i d e r a g a i n t h e s a m e s y s t e m a s in t h e p r e c e d i n g e x a m p l e s :

0

1

0

1

0

0

0

0

0

=to

o

1]

0

0]

A =

C1

c211 and

K[kl0

BI=

(ii)

K i f a n d o n l y if t h e n u m b e r of

B2=

[°1 0

1

matrices

129

The s t r u c t u r e d matrix corresponding to

I

O X X

X 0 0

A+BKC is :

X 1 0 0

and can be written as the product M1.M2.M3 where :

[

o

X

L

0

0

0

0

0

0

0 0

X

0 ]

xoll

MI=0X

0 0 0 X

0

0

0

0

0 0

O0

LO0 0 0

0 0

X 0

M2~O0

J

0

x

0

0 0 X

x o

M3

L°xo

The graph associated with the sequence of matrices M1,M2,M3 is the following : l

[

v2

v3

2v

2

v

v# 3

V

l

~

5

__-- P~ v 3 ~ 5

--

v4

where there are only pairs of independent information paths : (P1,P2),

(P1,P3),

(P2,P4), (P3,P4). Therefore, the system has a s t r u c t u r a l l y fixed mode of type (li) at the origin. The i n t e r e s t of this characterization compared to the one given by condition (ii) in Theorem 3.26 is that the available algorithms to determine the vertex-disjoint information paths of a graph are more efficient than those to determine the v e r t e x disjoint cycles. Indeed, it is proved in (PAP-84) that the test can be carried out in time 0 (n 5/2).

130

3 . 5 . 4 . - Evaluation of s t r u c t u r a l l y f i x e d modes b y calculation o f t h e s t r u c t u r a l s e n s i t i v i t y of t h e modes o f t h e s y s t e m In P a r a g r a p h 2.4.2

( C h a p t e r I I ) , t h e f i x e d modes w e r e c h a r a c t e r i z e d u s i n g t h e

f a c t t h a t s u c h modes a r e t h e modes of t h e c l o s e d - l o o p s y s t e m which are i n s e n s i b l e to v a r i a t i o n s of t h e e l e m e n t s of t h e f e e d b a c k m a t r i x . A l t h o u g h t h i s c o n d i t i o n is v e r i f i e d e i t h e r by s t r u c t u r a l l y o r n o n s t r u c t u r a l l y f i x e d m o d e s , t h i s a p p r o a c h can b e u s e d to c h a r a c t e r i z e s t r u c t u r a l l y f i x e d modes if one a d d s t h e following s u p p l e m e n t a r y c o n d i tions

:

1 - S t r u c t u r a l l y fixed modes of t y p e (i) a r e only s e n s i b l e to v a r i a t i o n s of some p a r a m e t e r s of t h e s y s t e m : t h e s e p a r a m e t e r s a r e t h o s e of t h e matrix A22 d e f i n e d in ( 3 . 5 . 4 ) in Theorem 3.25. 2-Structurally

f i x e d modes

of t y p e

(ii)

are

i n s e n s i b l e to v a r i a t i o n s

of any

p a r a m e t e r of t h e s y s t e m . T h e s e c o n d i t i o n s r e f e r to t h e f a c t t h a t , t h o u g h t h e e x i s t e n c e of a s t r u c t u r a l l y f i x e d mode is i n s e n s i b l e to v a r i a t i o n s of t h e p a r a m e t e r s of t h e s y s t e m , t h e v a l u e of a structurally

f i x e d mode of t y p e

(i) is d e f i n e d b y t h e e i g e n v a l u e s of A22 and t h e

location of a s t r u c t u r a l l y fixed mode of t y p e (ii) is always at t h e o r i g i n . This s i t u a tion was a l r e a d y p o i n t e d out in Example 3.9. C o n s i d e r an

arbitrary

real

matrix

D of dimension n x n .

The

gradient

simple e i g e n v a l u e Xr of D with r e s p e c t to t h e e l e m e n t s o f D is g i v e n b y ( 2 . 4 . 7 ) § 2 . 4 . 2 , C h a p t e r II)

dXr d(dij)

where

wr

and

vr

of a (see

:

-W )

r

are

dD d-~

(2.~.7)

Vr

t h e left

and

right

e i g e n v e c t o r s o f D c o r r e s p o n d i n g to h r .

Assume t h a t t h e e l e m e n t s d.. a r e f u n c t i o n s of q p a r a m e t e r s and ~let P be one of t h e m . *j T h e n , we h a v e • 8D 8P ij

~..(P) 0

li, l. if d.. = 0 q

with

5ij(P) =

d..(P) q

(3,5,7)

131 w h e r e I t a n d L a r e t h e i t h column a n d j t h r o w o f t h e t i v e l y . So, ( 2 . 4 . 7 )

becomes '

i f d "": 0

0

U

i f d m": 0

q

w h e r e w. a n d v. a r e t h e i t h a n d j t h e n t r i e s j

sider now the

q parameters

respec-

5ij (P) w..v.t)

"

jij

matrix

{

¢$ (P) W'r "ll'l~v) r LaP

nxn identity

(PI,...,Pq),

of w

r

and v

the gradients

r

respectively.

I f we c o n -

of an eigenvalue

of D with

r e s p e c t to e a c h o f t h e m a r e •

[

: { 6i)(Pt) wi vj

a~, r l

8Pt. ] ij

0

if

i,j=l ..... n

d..=O U

(3.5.8)

t=l,...,q

S i n c e a t l e a s t o n e e l e m e n t dij i s f u n c t i o n o f e a c h p a r a m e t e r the derivatives

~j{Pt ) cannot be simultaneously

the variations of Pt if and only if wi.v.=0, ] following lemma : Lemma 3 . 1 .

The

necessary

real m a t r i x

D ~

Rn x n

and

sufficient

to b e i n s e n s i b l e

zero.

(i,j=l, .... n).

condition to t h e

Pt'

Therefore,

(t=l,...,q),

all

Xr i s i n s e n s i b l e

to

T h i s r e s u l t i s s t a t e d in t h e

for a simple eigenvalue k r of a

variations

of the

elements

o f D is

that : w..v.=0 1 l

(i,j=l

. . . . .

n)

w h e r e w., a n d v.] a r e t h e i t h a n d j t h e n t r i e s c o r r e s p o n d i n g to X , . D e f i n i t i o n 3.9

(TAR-84).

The structural

of the left and right eigenvectors

s e n s i t i v i t y m a t r i x SS

r

of D

of a simple eigenvalue

X r is d e f i n e d a s follows : SS r = ( s s i j ) i=l ..... n j=l,...,n with • ssij =

[ 1

i f w i . v ] t6 0

l

if d.. = 0 o r w . . v . = 0 11 i l

0

F r o m D e f i n i t i o n 3.9 a n d t h e a b o v e d i s c u s s i o n , structurally

f i x e d m o d e s follows :

(3.5.9)

t h e following c h a r a c t e r i z a t i o n

of

132

Theorem 3.29 ( T A R - 8 4 b ) .

Consider the system (3.2.1t

a n d an a r b i t r a r y c o n s t r a i n e d

f e e d b a c k p a t t e r n s p e c i f i e d b y K . A simple e i g e n v a l u e ~r o f t h e c l o s e d - l o o p f e e d b a c k m a t r i x (A+BKC) is a s t r u c t u r a l l y f i x e d mode of t h e s y s t e m if a n d only if one of t h e two following c o n d i t i o n s h o l d s : i - T h e s t r u c t u r a l s e n s i t i v i t y m a t r i x SS r

(with r e s p e c t to (A+BKC)) of ~r is

e q u a l to one of t h e s t r u c t u r a l s e n s i t i v i t y m a t r i c e s SS r of

{ ~l,...,~n

}

is the

s e t of e i g e n v a l u e s of D'

and

Xr' ( r = l , . . . , n ) ,

D' is s t r u c t u r a l l y

where

e q u i v a l e n t to

(A+BKC). ii - The s t r u c t u r a l s e n s i t i v i t y m a t r i x SS

r

(with r e s p e c t t o t (A+BKC)) of kr i s

identically zero. From t h e r e m a r k s

s t a t e d at

t h e b e g i n i n g of t h e p a r a g r a p h ,

it is clear that

c o n d i t i o n (it c h a r a c t e r i z e s t h e s t r u c t u r a l l y f i x e d modes o f t y p e (it a n d c o n d i t i o n (ii) t h o s e of t y p e ( i i ) . The f i x e d modes of s y s t e m ( 3 . 2 . 1 )

(with d i s t i n c t modest

with r e s p e c t to an

a r b i t r a r y f e e d b a c k p a t t e r n s p e c i f i e d b y K can be d e t e r m i n e d with d i s t i n c t i o n of t h e i r d i f f e r e n t t y p e s b y t h e following algorithm : Al$orith.m 3.1 (TAR-84) 1 - C h o o s e a n u m e r i c a l v a l u e for t h e n o n z e r o e n t r i e s of t h e f e e d b a c k m a t r i x K s u c h t h a t t h e e i g e n v a l u e s of (A+BKC) a r e d i s t i n c t . 2 - F o r all kr c a ( A + B K C ) , e v a l u a t e t h e s e n s i t i v i t y m a t r i c e s SK r ( s e e Definition 2.8, C h a p t e r I I ) . If SK r =0, t h e n ~ r is a f i x e d mode ; i . e . X r ~ h . 3-If

A = ~, STOP. T h e s y s t e m h a s no f i x e d m o d e s .

4 - F o r all ~

c A, c a l c u l a t e t h e

structural

sensitivity matrix

SS r .

If SSr=0,

t h e n kr is a s t r u c t u r a l l y f i x e d mode of t y p e (iit ; i . e . ~r ~ A $2"

s-fi

A-

2

If h 1 --At, go to 10. 5 - C h o o s e a m a t r i x D* s t r u c t u r a l l y e q u i v a l e n t to

(A+BKC)

and s u c h t h a t i t s

eigenvalues are distinct. ? - E v a l u a t e t h e s e t of f i x e d modes A~ of t h e new s y s t e m b y r e p l a c i n g (A+BKC) b y D' in s t e p 2. If

A* = ~ ,

ANS = A ] . G o to 10.

8 - F o r all ~ * q ~ h*, e v a l u a t e t h e s t r u c t u r a l s e n s i t i v i t y m a t r i c e s S S q . ?-For all~rEh

1, compare SS r

t h e r e e x i s t s one S S ' q e q u a l to SS r , (i) ; i . e . ,

~r • AS1"

with all t h e

SS*q d e t e r m i n a t e d at s t e p

t h e n )'r is a s t r u c t u r a l l y

8.

If

fixed mode of t y p e

133

10-STOP.

A : s e t of fixed modes. : set of s t r u c t u r a l l y fixed modes of t y p e (i). : set of s t r u c t u r a U y fixed modes of t y p e (ii). • set of non s t r u c t u r a l l y fixed modes.

AS1 AS2 ANS

Example 3.12. C o n s i d e r the following system :

Ii A=

03

00

00

01 ]

O 0 0

1 0 0

1 2 0

0 0 2

c 1 = (o

o

o

1

o)

Cz = (1

o

o

o

o)

c 3 = (o

1

1

o

o)

=

B2=

B3 =

i!oolj2

For K = d i a g . ( l , 2 , 1 ) ,

A+BKG

I] I]

gI =

4 0 0 0

we have :

1 1 0 0

O 1 2 0

0

The following r e s u l t s are o b t a i n e d : eigenvalues of (A+BKG)

SK

SS

1 ' _1

0_

0

0

0

0

0

0

0

0

0

0

0

0 0

1 0

0

0

0

0

0

0

0

0

0

0

134

-

-0.48 I

-0.24

-1

0.49 0.24 0

~

10-15

0 . 5 2

10-16

-0.39

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0 0 0 0 1

The fixed modes of the system

are therefore

v a l e n t m a t r i x to ( A + B K G ) is c h o s e n

:

I O 0 0 1.6 1.05

The eigenvalues tivity

matrix

of

0.65

0 l.Q 0 0

0 O.Q 0.6.5 0

0.2 0 0.7 1.9

0.25] 0 0 0

0

0

0

2.5

o f D* a r e is f o u n d

: {1.4, to

be

0.65)

The structural

0 0 0 0

zero.

A* -- {0.65}.

sensitivity

0 1 0 0

matrix of 0.65 is :

0 0 0 0

w h i c h i s t h e s a m e a s w h i c h o f 1. We o b t a i n t h e f o l l o w i n g r e s u l t s

:

h ={1,2}.

-0.245,

identically

D°O°I

modes of the new system is

:

2.62,

A structurally

2.024},

Therefore,

equi-

Only the sensithe

set

of fixed

135 = ( i , 2}

A

ANS =

~sl

(2 }

= { 1}

3.5.5. - Comments The

above

paragraph

was

concerned

with

the

concept

m o d e s . S t r u c t u r a l l y f i x e d m o d e s a r e r e l a t e d to t h e p r o p e r t y decentralized information structure

of s t r u c t u r a l l y

fixed

of c o n t r o l l a b i l i t y u n d e r

for which several definitions have been provided.

T h e d i f f e r e n c e s b e t w e e n t h e m h a v e b e e n c l a r i f i e d a n d r e l a t e d to t h e e x i s t e n c e of two t y p e s of s t r u c t u r a l l y f i x e d m o d e s .

S t r u c t u r a l l y f i x e d m o d e s of t y p e

(i) a r i s e f r o m t h e l a c k of c o n n e c t i v i t y of t h e

s y s t e m . T h e c h a r a c t e r i z a t i o n g i v e n b y S e z e r a n d Siljak t h e g e n e r a l i z a t i o n , in a s t r u c t u r a l ment (AND-81a)

(see § 3.3.1)

gated stations such that

(SEZ-81a)

f r a m e w o r k , of t h e r e s u l t s

( s e e § 3 . 5 . 3 a ) is

of A n d e r s o n

and Cle-

: t h e r e e x i s t s a p a r t i t i o n o f t h e s y s t e m in two a g g r e -

structurally

f i x e d modes of t y p e

c o n t r o l l a b l e b y o n e of t h e s t a t i o n s a n d s t r u c t u r a l l y

(i)

are

structurally

un-

unobservable by the other one.

It h a s b e e n p o i n t e d o u t t h a t t h e s e f i x e d m o d e s a r e n o t c o n t r o l l a b l e u n d e r

decentra-

lized i n f o r m a t i o n p a t t e r n e v e n if t h e time i n v a r i a n c e of t h e c o n t r o l l e r is r e m o v e d . Structurally origin.

f i x e d m o d e s of t y p e

(ii) a r i s e f r o m t h e i r

T h e y c a n be m a d e c o n t r o l l a b l e u n d e r

t i m e - v a r y i n g c o n t r o l l e r is allowed

special location at the

decentralized information pattern

( s i m i l a r l y to n o n

structurally

fixed modes).

if a The

s p e c i a l s i t u a t i o n of a f i x e d mode at t h e o r i g i n h a s b e e n c l a r i f i e d a n d c a n b e s u m m a rized a s f o l l o w s . G i v e n a s y s t e m w i t h a simple f i x e d mode a t t h e

origin,

two c a s e s m u s t b e

considered : Case 1 :

T h e t r a n s f e r f u n c t i o n m a t r i x c a n b e p u t in a b l o c k - t r i a n g u l a r form :

W(p)=

I We,c, (p) WaB (P)

o W88

] (p)

136 w h e r e t h e c h a r a c t e r i s t i c polynomial of Was (p) h a s a pole at t h e o r i g i n e . 1- T h e

z e r o b l o c k is d u e to t h e s p e c i a l s t r u c t u r e

of t h e s y s t e m : l 0=0 is a

s t r u c t u r a l l y f i x e d mode of t y p e ( i ) . 2- T h e z e r o b l o c k a r i s e s from a c a n c e l l a t i o n d u e to s p e c i a l v a l u e s of t h e s y s tem p a r a m e t e r s

Case 2

: ~ 0=0 is a n o n s t r u c t u r a l l y f i x e d mode.

:

The transfer

f u n c t i o n m a t r i x h a s no p a r t i c u l a r form ( t h e g r a p h of t h e s y s t e m

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

W(p) =

:

(P) Wag (P)

WSa(P) t WIBB(P)

I Waa

t h e c h a r a c t e r i s t i c polynomial of W Ba ( p ) h a s a pole a t p=0 a n d e v e r y e n t r y of Wcts(p) h a s a zero a t ~0=0. 1- One o r more e n t r i e s of Wa8 (p) h a v e a z e r o at p=0 b e c a u s e of p a r a m e t e r c a n c e l l a t i o n s ( a p + b - c with b=c for example)

: ~0=0 i s a n o n s t r u c t u r a l l y f i x e d mode.

2- T h e e n t r i e s of Was (p) h a v e a zero a t )~0=0 b e f o r e a n y c a n c e l l a t i o n : X0=0 is a structurally

f i x e d mode of t y p e

(ii).

T h i s f i x e d mode is i n d e p e n d e n t of t h e p a r a -

meter values but arises, as non structurally

fixed modes,

from a p o l e - z e r o c a n c e l l a -

t i o n of t h e i n t e r c o n n e c t i o n b l o c k s W B (p) a n d WBa ( p ) . Several

characterizations

of s t r u c t u r a l l y

fixed

modes

have

been

presented,

s p e c i a l l y in a g r a p h i c a l f r a m e w o r k s i n c e g r a p h s a r e v e r y a d e q u a t e f o r t h e s t u d y of structural properties. account

arbitrary

The graph-theoretic

structural

constraints

a p p r o a c h is more c o n v e n i e n t to t a k e into on t h e c o n t r o l ,

b u t o n l y i n t h e case

for

w h i c h t h e g r a p h is a s s o c i a t e d to t h e c l o s e d - l o o p s y s t e m . From t h e c o m p u t a t i o n a l p o i n t of v i e w , t h e a l g e b r a i c c h a r a c t e r i z a t i o n i s n o t v e r y convenient since it requires,

for structurally

f i x e d modes

of t y p e

(i),

to

find a

p e r m u t a t i o n m a t r i x s u c h t h a t t h e s y s t e m is p u t in t h e form ( 3 . 5 . 4 ) a n d , f o r s t r u c t u r a l l y f i x e d m o d e s of t y p e

( i i ) , to t e s t all t h e c o m p l e m e n t a r y s u b s y s t e m s .

t h e e x i s t e n c e of s t r u c t u r a l l y

f i x e d modes of t y p e

However,

(i) c a n e f f i c i e n t l y b e t e s t e d u s i n g

t h e g r a p h i c a l c o n d i t i o n g i v e n in T h e o r e m 3.26 : f i r s t , f i n d t h e s t r o n g c o m p o n e n t s of the graph state

FK,

vertices.

then check whether

t h e r e is a s t r o n g c o m p o n e n t c o m p o s e d only b y

T h i s can b e d o n e i n time O ( n 2)

(PAP-g4).

For structurally

fixed

modes of t y p e (fi), t h e c h a r a c t e r i z a t i o n u s i n g t h e g e n e r i c r a n k of s e v e r a l r e a l m a t r i -

137 ces seems

O(n 5/2)

to be

the

most

convenient

since the

test

can

be

carried

out in time

(PAP-84).

Finally, a c h a r a c t e r i z a t i o n in t e r m s of s t r u c t u r a l s e n s i t i v i t y h a s b e e n p r o v i d e d and t h e c o r r e s p o n d i n g algorithm for t h e d e t e r m i n a t i o n of f i x e d modes with d i s t i n c t i o n of t h e i r t y p e s h a s b e e n d e r i v e d . It is p a r t i c u l a r l y i n t e r e s t i n g f o r s t r u c t u r a n y f i x e d modes of t y p e

(i)

s i n c e it d o e s n o t only d e t e c t t h e i r e x i s t e n c e b u t

it d e t e r m i n e s

their value.

3.6. - G R A P H - T H E O R E T I C

CHARACTERIZATION

OF FIXED

MODES

3 . 6 . 1 . - Prel~min~'ies T h e g r a p h - t h e o r e t i c c h a r a c t e r i z a t i o n s r e f e r i n g to s t r u c t u r a l l y f i x e d m o d e s h a v e already b e e n

presented

in t h e

last

paragraph.

Graph-theoretic methods are

very

efficient f o r t h e a n a l y s i s of s t r u c t u r a l p r o p e r t i e s like t h a t s t r u c t u r a l c o n t r o l l a b i l i t y , structural observability and s t r u c t u r a l l y fixed modes. The a d v a n t a g e s of a g r a p h theoretic a p p r o a c h ,

s p e c i a l l y w h e n we h a v e to deal with l a r g e scale s y s t e m s ,

are

n u m e r o u s . I n d e e d , it allows t h e a p p l i c a t i o n of all t h e c o n c e p t s of g r a p h - t h e o r y a n d leads to c h a r a c t e r i z a t i o n p r o c e d u r e s w h i c h can b e e a s i l y i m p l e m e n t e d b y u s i n g existing set arbitrary

of a l g o r i t h m s .

control

structures.

Morever, It

is

it is

specially convenient when

now i n t e r e s t i n g

to

note

that

the

d e a l i n g with

graph-theoretic

methods can also be u s e d in t h e case for w h i c h t h e p r o p e r t i e s to a n a l y s e a r e n o t s t r u c t u r a l b u t p a r a m e t e r v a l u e d e p e n d e n t . The a n a l y s i s can b e c a r r i e d o u t b y a s s o ciating a w e i g h t to e v e r y e d g e of t h e g r a p h ; i . e . ,

a w e i g h t e d d i r e c t e d g r a p h is

a s s o c i a t e d to t h e s y s t e m . Two

general graph-theoretic

the distinction between

characterizations of fixed m o d e s exist, which allow

structurally a n d non structurally fixed modes.

is carried out in the frequency-domain ted to the transfer

~ i.e., a weighted

function matrix of the system.

study in the time-domain

The

; in this case, the weighted

The

directed graph second

first one is associa-

one results

from

a

directed graph is associated

with the state space representation of the system.

3.6.2. - Frequency-domain g r a p h - t h e o r e t i c characterization C o n s i d e r a Hnear t i m e - i n v a r i a n t s y s t e m with simple modes r e p r e s e n t e d b y i t s t r a n s f e r f u n c t i o n m a t r i x as in ( 3 . 4 . 1 )

:

138

u(p) = w(p) x(p) U ~ Rm with :

Y ~ Rr m

Yj(P) = i=l ~

wj'i(P) ui(p)

(j=l . . . . . r )

Wj pi(p) =cj (pI-A) - l b i The c o n t r o l s t r u c t u r e is d e f i n e d b y t h e following s e t of p a i r s : (j)i) ~ S

if

k . . # 0 in K 1,J

(3.6.11

w h e r e K is t h e o u t p u t f e e d b a c k m a t r i x . The g r a p h a s s o c i a t e d to (3.4.11 is D S = ( V s , L S) a n d it is d e f i n e d as follows :

V S i s a s e t of v e r t i c e s s u c h t h a t : V S = V1S u V2S V1S --- {i ] ( j ) i ) E S f o r some j} V2S = {j / ( j - m , i ) ~

S for some i}

L S i s a s e t of e d g e s s u c h t h a t : L S = L1SU L2S L1S = { ( i , j ) / (i,j) ~ V1S x V2S, Wj_m,i(p) # 0} L2S = { ( i , j ) / ( i - m , j ) ~ S t E v e r y v e r t e x r e p r e s e n t s e i t h e r an i n p u t or an o u t p u t of t h e s y s t e m w h i c h is i n v o l v e d at l e a s t in a f e e d b a c k loop a n d e v e r y e d g e r e p r e s e n t s

either a nonzero

t r a n s f e r function or a permitted feedback llnk. A t r a n s m i t t a n c e t i , j ( p ) is a s s o c i a t e d to e v e r y e d g e (i)j) ~ L S :

t i , j ( p ) = Wj_m)i(p)

f o r ( i , j ) E LIS

ti,j(p) = 1

for ( i , j ) ~ L2S

The t r a n s m i t t a n c e T p ( p ) of a n y p a t h P in D S i s d e f i n e d as t h e p r o d u c t of t h e t r a n s m i t t a n c e s a s s o c i a t e d with t h e e d g e s composing P . Using t h e a b o v e d e f i n i t i o n s ) Locatelli e t al.

(LOC-77)

give t h e following c h a -

r a c t e r i z a t i o n of f i x e d modes : Theorem 3.30. A pole X0 o f t h e s y s t e m ( 3 . 4 . 1 ) feedback structure

d e f i n e d b y S in ( 3 . 6 . 1 )

t r a n s m i t t a n c e a s s o c i a t e d with a c y c l e o f D S.

is a f i x e d mode with r e s p e c t to t h e

if a n d o n l y if X0 is n o t a pole of a n y

139

This

characterization

is

the

graph-theoretic

transcription

of

the

algebraic

c h a r a c t e r i s a t l o n o f fixed modes in t e r m of t r a n s m i s s i o n z e r o s p r e s e n t e d i n P a r a g r a p h 3.2. I n d e e d ,

a cycle of D S d e f i n e s a s u b s y s t e m

c o r r e s p o n d i n g s u b m a t r i x i n K' is n o n s i n g u l a r .

W! (p) (see § 3.2) for w h i c h t h e ] T h e s e t of r o w s s e l e c t e d in W(p) is

g i v e n b y t h e o u t p u t v e r t i c e s i n v o l v e d i n t h e cycle a n d t h e s e t of columns b y t h e s e t of i n p u t v e r t i c e s i n v o l v e d in t h e cycle.

T h e t r a n s m i t t a n c e a s s o c i a t e d w i t h t h e cycle

is t h u s e q u a l to t h e m i n o r W!(p). I t is c l e a r t h a t if t h e m i n o r W!(p) h a s no pole a t J J t h e n ¢ ( p ) . W { ( p ) h a s a zero at X0 a n d l 0 i s a t r a n s m i s s i o n zero of t h e c o n s i d e r e d ~0' J subsystem. Nevertheless, graph-theoretic

we h a v e to p o i n t o u t a f u n d a m e n t a l d i f f e r e n c e w h i c h m a k e s t h i s

characterization less powerfull than the algebraic one.

cycle a s s o c i a t e d with a s u b s y s t e m f o r w h i c h W~(p) h a s a t r i a n g u l a r c o r r e s p o n d s to a n o n s i n g u l a r s u b m a t r i x of K'.

T h e r e is n o

form e v e n if it

T h i s is w h y t h i s c h a r a c t e r i z a t i o n is

only v a l i d f o r t h e c a s e of s y s t e m s w i t h simple p o l e s .

Indeed,

consider the subsys-

tem :

W!(p)_IWi'i(P)0 1 }

LWi,j(p)

Wj,j(p)

and a s s u m e t h a t it c o r r e s p o n d s to a n o n s i n g u l a r s u b m a t r i x in K'. T h e n , it i s c l e a r t h a t Wi,i(p)

a n d Wj,j(p)

correspond

also to n o n s i n g u l a r s u b m a t r i c e s i n K'.

For a

s y s t e m w i t h simple p o l e s , two s i t u a t i o n s can a r i s e : * ~0 i s n e i t h e r

a pole of Wi,i(p)

n o r of Wj,j(p)

: consequently,

~0 is n o t a

pole of W~(p). * XO i s a pole of Wi,i(p) or of Wj,j(p) b u t n o t of b o t h : c o n s e q u e n t l y , ~ 0 is a pole of W~(p). Therefore,

all t h e i n f o r m a t i o n a b o u t WI(p) is c o n t a i n e d in Wi,i(p) a n d Wj,j(p)

and t h e s u b s y s t e m W~.(p) does n o t n e e d to b e t a k e n i n t o a c c o u n t . C o n s i d e r now t h a t ~0 i s n o t a simple pole b u t a pole of o r d e r t h a t it is a pole of o r d e r q' for W i , i ( p ) a n d of o r d e r q - q ' ,

~(h0 ) , Wi,i(¢0) = 0

and

q and assume

for W j , j ( p ) . T h e n :

qS(10). Wj,j{~ 0) = 0

i . e . , )~0 is a t r a n s m i s s i o n z e r o of t h e s u b s y s t e m s Wi,i(p) a n d W j , j ( p ) . B u t :

140

¢,(~o) wi< p)

$

o

a n d X0 i s n o t a t r a n s m i s s i o n z e r o o f W~(p). C o n s e q u e n t l y , X0 i s n o t a f i x e d mode. In t h i s c a s e , we n e e d to c o n s i d e r t h e s y s t e m wij(p) to d e r i v e t h e c o n c l u s i o n . Nevertheless, this characterization remains interesting.

In p a r t i c u l a r ,

it p r e -

s e n t s t h e a d v a n t a g e to t r e a t t h e c a s e s of an a r b i t r a r y c o n t r o l s t r u c t u r e v e r y e a s i l y . M o r e o v e r , it allows some d i s t i n c t i o n b e t w e e n t h e d i f f e r e n t t y p e s of f i x e d m o d e s . X 0 is a s t r u c t u r a l l y f i x e d mode of t y p e (i) if t h e r e is no e d g e w h o s e t r a n s m i t t a n c e h a s XO as a pole i n v o l v e d in t h e composition of a c y c l e . ~0 is a n o n s t r u c t u r a l l y f i x e d mode o r a s t r u c t u r a l l y f i x e d mode of t y p e (ii) i f at l e a s t o n e e d g e w h o s e t r a n s m i t t a n c e has ~0 as a pole c o m p o s e s one c y c l e a n d if ~0 is n o t a pole of t h e t r a n s m i t t a n c e of t h i s c y c l e d u e to a p o l e - z e r o c a n c e l l a t i o n . Note t h a t t h e d i s t i n c t i o n b e t w e e n n o n s t r u c t u r a l l y f i x e d m o d e s a n d s t r u c t u r a l l y f i x e d modes o f t y p e the

cancellations within the t r a n s f e r

(ii) is n o t p o s s i b l e s i n c e

f u n c t i o n s t h e m s e l v e s do not a p p e a r ,

b u t the

q u e s t i o n only a r i s e s f o r t h e c a s e of f i x e d modes at t h e o r i g i n . Example 3.13. C o n s i d e r t h e following s y s t e m with simple p o l e s : 3 p-2

W(p) :

0

_p+l p(p-2)

i

1

1

p-2

p+2

p(p-2)

l

[

p+ !

p+2

T h e p o l e s of t h e s y s t e m a r e { 0 , - 1 , - 2 , 2 }. The c o n t r o l s t r u c t u r e is g i v e n b y :

K =

f kll k22 k331

and S :

{ ( I , I ) ; (2,2) ; (3,3) }

The g r a p h DS= ( V s , L S) a s s o c i a t e d with t h i s s y s t e m is s h o w n in F i g u r e 3.7 :

141

3

6 F i g u r e 3.7

VlS = {1,2,3} VZS = {4,5,6} LIS = {(1,4);(1,5);(1,6);(2,5)1(2,6);(3,4);(3,5)} LZS = {(4,1);(5,2);(6,3)} T h e r e a r e 5 c y c l e s in D S w i t h t h e following a s s o c i a t e d t r a n s m l t t a n c e s :

l {

(1,o,1)

{ { )

(2,5,2)

{ 3 p-2

{ l

1

I

p~-2

I

(1,6,3,0,1) , (p+l)

l l

,..I

(p+~)

1

x p(p-2)

= p(p-2) - -

(2,6,3,5,2)

l

({,5,2,6,3,o,{)

I p(p-2)(p+2)

1 { I

p+l p(p+2) (p-2)2

l Among t h e p o l e s of t h e s y s t e m , X0 =-1 d o e s n o t a p p e a r as a pole of a n y t r a n s mittance.

This

(1,6,3,4,1).

situation

arises

T h e r e f o r e , X0=-I

is

from a non

the

pole-zero

structurally

cancellation f i x e d mode

for (it

cannot

s t r u c t u r a l l y f i x e d mode of t y p e (ii) s i n c e it is n o t l o c a t e d at t h e o r i g i n ) .

3.6.3. - Time-domain g r a p h - t h e o r e t i c c h a r a c t e r i z a t i o n The s y s t e m is now r e p r e s e n t e d b y i t s s t a t e - s p a c e model ( 3 . 2 . 1 )

the

:

cycle he

a

142 e(t)

= A x(t)

+ B u(t)

y(t) = C x(t) a n d we c o n s i d e r an a r b i t r a r i l y c o n s t r a i n e d c o n t r o l s p e c i f i e d b y t h e o u t p u t f e e d b a c k m a t r i x K° T h i s c h a r a c t e r i z a t i o n is i s s u e d from t h e w o r k of Mason (MAS-56) r e l a t i n g t h e s t r u c t u r e of t h e g r a p h a s s o c i a t e d w i t h a s q u a r e m a t r i x to its d e t e r m i n a n t . We c o n s i d e r an n x n m a t r i x M and we a s s o c i a t e a w e i g h t e d d i g r a p h c o n s i s t i n g o f n v e r t i c e s a n d a s e t of e d g e s .

T h e r e is an e d g e from t h e v e r t e x i to t h e v e r t e x j i f mji ~ 0

a n d t h e w e i g h t of t h i s e d g e is mji° T h e n , we h a v e t h e following :

det M :

[: m l q 1 m2q 2 ... mnq n even permutations

~: odd mlql permutations

m2q2

... m nqn

(3.6.2)

w h e r e ( q l ' . . . . q n ) is some p e r m u t a t i o n of (1 . . . . . n ) . Definition 3.10 (REI-83)

(REI-84a,b).

A cycle family is a s e t of d i s j o i n t c y c l e s . I t s

w e i g h t is g i v e n b y t h e p r o d u c t o f t h e

w e i g h t s o f all edges c o m p o s i n g t h e

cycle

family. Each t e r m ( m l q mzq . . .

mnq)

in

(3.6.2)

c o r r e s p o n d s t h u s to a c y c l e family

w h i c h t o u c h e s all n v e r t i c e s in t h e g r a p h . I f t h i s c y c l e family c o n s i s t s of f disjoint c y c l e s , t h e n t h e s i g n f a c t o r of i t s c o r r e s p o n d i n g s u m m a n d in ( 3 . 6 . 2 ) is (-1) n - f . The obtain a

approach

of R e i n s c h k e

graph-theoretic

(REI-84a)

c o n s i s t s in u s i n g

c h a r a c t e r i z a t i o n of t h e

polynomial of t h e c l o s e d - l o o p s y s t e m ,

(3.6.2)

coefficients of

the

in o r d e r to characteristic

Note t h a t t h i s a p p r o a c h was a l r e a d y u s e d b y

E v a n s a n d K r u s e r (EVA-84) to d e t e r m i n e t h e c o n d i t i o n s of p o t e n t i a l pole a s s i g n m e n t using decentralized static control (see § 3.5.2c)

a n d r e m e m b e r t h a t t h e y n e e d e d to

go t h r o u g h t h e calculation o f t h e c l o s e d - l o o p d y n a m i c m a t r i x (A+BKC).

This d i s a d -

v a n t a g e is a v o i d e d b y R e i n s c h k e b y u s i n g t h e following (n+m+r)x(n+m+r) m a t r i x :

M=

The

I! c 01

graph

A

B

0

0

associated

with

M

is

DK=(U u X u Y,

X = { X l , . . . . x n} i s t h e s e t of s t a t e v e r t i c l e s , Y = { y l p . . . , y ~ v e r t i c e s a n d U= { U l , . . . , u m} is t h e s e t of i n p u t v e r t i c e s .

E u E K)

where

is t h e s e t o f o u t p u t

E is a s e t o f e d g e s s u c h

143

that ( u i , x j) £ E if bji# 0 and bji is the weight of this e d g e , and aji is the weight of this e d g e ,

(xi,Y j) ~

edge. E K is a s e t of f e e d b a c k e d g e s s u c h t h a t (Yi' uj) ~ weight of this e d g e . Note

that

this

graph

is

the

same

(x i, xj) C E if aji¢0

E if cji~ 0 and cji is the weight of this

as

the

one

E K if kji#0 and kji is the

already

used

in

Paragraph

3.5.3.b-I.

Definition 3.11

(REI-84a).

The width of a cycle family in DK is d e f i n e d as the

number of s t a t e v e r t i c e s i n v o l v e d in it. Using

(3.6.2)

and Definitions 3.10 and 3.11,

Reinschke

states

the following

result : Theorem 3.31

(REI-84a).

The c o e f f i c i e n t s a i '

(i=] . . . . . n ) ,

of the closed-loop c h a -

racteristic polynomial : det ( p I - A - B K C ) = p n + a l p n - l +

. . . + an_ 2 p 2 + a n _ 1 p + a n

are given b y the sum of t h e w e i g h t s of all the cycle families of width i in D K multiplied by a sign factor (-1) d w h e r e d is the n u m b e r of disjoint c y c l e s in the cycle family u n d e r c o n s i d e r a t i o n . The c h a r a c t e r i z a t i o n Theorem 3.31,

of fixed modes is b a s e d on the following a r g u m e n t s .

t h e c h a r a c t e r i s t i c polynomial is g i v e n as a s e r i e s e x p a n s i o n

p=0 b u t a s e r i e s r e p r e s e n t a t i o n of t h e same polynomial is possible t0 ; i.e.

In

around

a r o u n d any value

:

det ( p I - A - B K C ) = d e t ( ( p - ~ O ) I - ( A - t o I ) - B K C )

=(p-10)n+(p-10)n-la l(10)+...+(p-10) an_l(10)+ an(l 0)

(3.6.3)

and the coefficientsai(l0) can be obtained using Theorem 3.31 where A is replaced by (A-101) when defining the graph DK, which will be denoted by D K (I0). E v e r y c o e f f i c i e n t a i ( t O) can be r e p r e s e n t e d as : ai(~0) : a ; ( t 0 ) wherea°(10),

+ gi (10'K)

(i=l,...,n),

(3.6.4)

a r e the coefficients of the open-loop c h a r a c t e r i s t i c p o l y -

nomial in the form of a s e r i e s e x p a n s i o n a r o u n d k 0.

144

P=~0 is a f i x e d mode if a n d only if it is a zero of t h e o p e n - l o o p a n d also of the closed-loop characteristic polynomials, i . e .

but

]-

a o ( x o) = 0

2-

an(),0 ) = 0

from

(3.6.4),

(2)

implies

ai(Ao)=O, (i=n . . . . . n - h + l ) ,

(1)

and

(2)

:

is

therefore

sufficient.

If

moreover,

b u t Oh_h(;~O)~ O, t h e m u l t i p l i c i t y of t h e f i x e d mode at P=~O

is of c o u r s e h . The g r a p h - t h e o r e t l c i n t e r p r e t a t i o n of t h e a b o v e d i s c u s s i o n l e a d s to t h e following r e s u l t : T h e o r e m 3.32

(REI-84a).

The s y s t e m

(3.2.1)

P=~0 with r e s p e c t to t h e c o n t r o l s p e c i f i e d b y

h a s a f i x e d mode of m u l t i p l i c i t y h at K i f a n d only if~ for j - - n p . . . , n - h + l

( b u t n o t f o r j - - n - h ) , one of t h e following two c o n d i t i o n s h o l d s : l - t h e r e a r e no cycle families o f w i d t h j in DK(R0). 2 - t h e r e e x i s t s two o r more cycle families of w i d t h j in DK(~,0) which c a n c e l each o t h e r n u m e r i c a l l y for all a d m i s s i b l e v a l u e s of t h e n o n z e r o e n t r i e s of K. It is c l e a r t h a t

condition

(2)

c o r r e s p o n d s to n o n s t r u c t u r a l l y

fixed modes.

F i r s t , n o t e t h a t t h e r e is a l e a s t one d i f f e r e n t e d g e b e l o n g i n g to E in two d i f f e r e n t c y c l e families. The cancellation of c o n d i t i o n (2) a r i s e s from t h e p a r t i c u l a r v a l u e s of t h e w e i g h t s a s s o c i a t e d to t h i s e d g e s .

It is c l e a r t h a t if t h e w e i g h t o f o n e of t h e

e d g e s i n v o l v e d i n one cycle family a n d n o t in t h e o t h e r is c h a n g e d , t h e c a n c e l l a t i o n will no l o n g e r o c c u r . T h e r e f o r e , t h i s f i x e d mode d e p e n d s on t h e v a l u e s of t h e p a r a m e t e r s of t h e s y s t e m a n d is n o n s t r u c t u r a l . satisfied,

the

conclusion r e q u i r e s

a deeper

In t h e case for w h i c h c o n d i t i o n (1) i s analysis.

Consider the

graph

DK(p)

a s s o c i a t e d with a s y s t e m of d i m e n s i o n n for a n o n s p e c i f i e d value of p a n d a s s u m e t h a t it c o n t a i n s N cycle families of w i d t h n : N is at l e a s t e q u a l to 1 s i n c e t h e r e is a s e l f - l o o p at e v e r y s t a t e v e r t e x with w e i g h t ( a f t - p ) . For c o n d i t i o n (1) to b e s a t i s f i e d , t h e a f f e c t a t i o n of a n u m e r i c a l v a l u e k0 to p m u s t eliminate some s e l f - l o o p s . T h i s will o c c u r w h e n k0 i s s e t to some aft. We o b t a i n t h u s a f i r s t i n t e r e s t i n g r e s u l t : Corollary 3 . ] 0 .

A

necessary

c o n d i t i o n for A0 to b e a f i x e d mode a r i s i n g from c o n d i -

tion (1) in T h e o r e m 3.32 is t h a t Since condition modes

)D i s e q u a l to some aii, i ~ ~I . . . . .

(2) of T h e o r e m

3.32 only characterizes non

(but not all of them), the following result directly follows :

n}. structurally fixed

145

Corollary 3.11.

A n e c e s s a r y c o n d i t i o n for a pole ~0 to b e a s t r u c t u r a l l y f i x e d mode

is t h a t i t s v a l u e a p p e a r s in t h e d i a g o n a l e l e m e n t s o f t h e d y n a m i c m a t r i x A. Now, c o n d i t i o n (1) of T h e o r e m 3.32 can b e s a t i s f i e d if t h e r e is a t l e a s t one s e l f - l ~ p with w e i g h t aii- p a n d afi=X0 in e v e r y cycle family of w i d t h n of D K ( P ) ° Two c a s e s can o c c u r : Case 1 : T h e r e is at l e a s t 1 s e l f - l o o p with w e i g h t ~ 0- p in e v e r y cycle family of w i d t h n of D K ( p ) w h i c h c o r r e s p o n d s to t h e same s t a t e v e r t e x . T h e n , w h e n we s e t p t o k 0, all t h e c y c l e families of w i d t h n become o f w i d t h { n - l ) • i n d e p e n d e n t l y of t h e p a r a meter v a l u e s of t h e s y s t e m ( w e i g h t s o f t h e o t h e r e d g e s ) . Case 2 : T h e r e is at l e a s t 1 s e l f - l o o p with w e i g h t k0- p in e v e r y cycle family of w i d t h n of DK(P)

a n d n o n e of them c o r r e s p o n d s to t h e same s t a t e v e r t e x .

p a r a m e t e r in t h e

T h e n , if a n y

diagonal of A c o r r e s p o n d i n g to t h e s e s t a t e v e r t i c e s is c h a n g e d ,

some c y c l e family of w i d t h n wilt r e m a i n . In t h i s c a s e , c o n d i t i o n (1) of T h e o r e m 3.32 d e p e n d s of t h e p a r a m e t e r v a l u e s , This d i s c u s s i o n can b e s u m m a r i z e d in t h e following c o r o l l a r y : Corollary 3.12,

~0 is a s t r u c t u r a l l y

f i x e d mode if a n d only if t h e r e is at l e a s t

1

self-loop with w e i g h t ~0- p in e v e r y cycle family of w i d t h n of D K ( p ) which c o r r e s p o n d s to t h e same s t a t e v e r t e x . The p a r t i c u l a r case of f i x e d modes at t h e o r i g i n d o e s n o t r e q u i r e a d i f f e r e n t a n a l y s i s : X0 i s s e t to 0 a n d t h e g r a p h to be c o n s i d e r e d is t h u s DK. T h e d i f f e r e n tiation b e t w e e n s t r u c t u r a l l y f i x e d modes of t y p e

(i) a n d

(ii) c a n b e c a r r i e d o u t b y

using c o n d i t i o n (i) of T h e o r e m 3,26 : k0 is a s t r u c t u r a l l y f i x e d mode o f t y p e (i) if it is not i n v o l v e d in a n y cycle family of a n y w i d t h i n c l u d i n g an e d g e of E K. This c h a r a c t e r i z a t i o n is of a g r e a t i n t e r e s t s i n c e if is t h e o n l y complete g r a p h t h e o r e t i c c h a r a c t e r i z a t i o n of f i x e d modes ( t h e one p r e s e n t e d in t h e p r e c e d i n g p a r a g r a p h is o n l y valid for s y s t e m s w i t h simple p o l e s ) . From (LIA-69)

the

computational p o i n t

(RAO-69)

of v i e w ,

the

well-known algorithms

for f i n d i n g p a t h s a n d c y c l e s can h e a d a p t e d

aided d e t e r m i n a t i o n of c y c l e families of p r e s c r i b e d w i d t h . Example 3.14. C o n s i d e r t h e following s y s t e m :

(KRO-67)

for t h e c o m p u t e r

[oooij A-=

3

0

0

0

0

Q

I

0

0

0

2

0

0

0

0

I

0)

C1 =

(0

0

0

1

C2 =

(1

0

0

0

0)

C3 =

(0

~

1

0

0)

146

I

I I 0

0

BI=

B2=

B3=

0 0

a n d a d e c e n t r a l i z e d c o n t r o l s p e c i f i e d b y : K=diag. ( k l , k 2 , k 3 ) .

The o p e n - l o o p c h a r a c -

t e r i s t i c polynomial is ,

det (pI-A)=(p-2)(p-4)(p-3)(pZ-p=l) The g r a p h DK(P) is s h o w n in F i g u r e 3 . 8 .

T h e r e a r e 6 cycle families of w i d t h 5 in

DK(P) w h i c h a r e g i v e n in F i g u r e 3 . 9 . -p

:'22

Fig. 3.8

Fig. 3 . 9 Case 1 : b = 4 In this case, there is no more cycle families of width 5 ; moreover, the selfloop with weight 4-p appears in every cycle family of width 5 in DK(p) and it corresponds to the same state vertex x3. Therefore X =4 is a structurally fixed mode 0 (of type ( i ) , of course). Case 2 : h0=2 In this case, the cycle families 1, 2, 3 and 4 are of width 4 and have not to be considered anymore.

The coefficient a5(hg=2) i s calculated from the cycle families 5 and 6 :

Therefore

3.6.4.

0

=2 is not a fixed mode.

- Comments The two graphical characterizations given in this paragraph present the advan-

tage of being complete in the sense that both structurally and non structurally fixed modes are characterized. This result is performed by using weighted directed graphs associated to the system, whose weights reflect the specific values of the system parameters.

The f i r s t one, which is stated in a frequency-domain

framework is,

unfortunately, only valid for systems with simple poles. Contrarily, the second one, stated in a time-domain framework, can be applied to the general case of systems

148 w i t h m u l t i p l e p o l e s a n d t h e m u l t i p l i c i t y of t h e f i x e d m o d e s is e a s i l y d e t e r m i n e d .

Both

c h a r a c t e r i z a t i o n s allow to c o n c l u d e on t h e t y p e of t h e f i x e d m o d e s i n a n e a s y w a y . From t h e c o m p u t a t i o n a l p o i n t of v i e w ,

they require

to d e t e r m i n e t h e

c y c l e s in t h e

d i g r a p h a n d , f o r t h e c a s e of l a r g e s c a l e s y s t e m s , e f f i c i e n t a l g o r i t h m s a r e n e c e s s a r y .

3.7.

- CONCLUSION

This chapter

presents

a n o v e r v i e w of all t h e d i f f e r e n t e x i s t i n g c h a r a c t e r i z a -

t i o n s of f i x e d m o d e s . T h e y c a n b e c l a s s i f i e d in f o u r g r o u p s :

- C h a r a c t e r i z a t i o n s in t e r m s of t r a n s m i s s i o n z e r o s of some s u b s y s t e m s of the s y s t e m : t h e s a m e r e s u l t c a n b e o b t a i n e d in a f r e q u e n c y o r in a t i m e - d o m a i n f r a m e w o r k s i n c e t h e c o n c e p t of t r a n s m i s s i o n z e r o s i s a s well d e f i n e d i n b o t h d o m a i n s . the

C h a r a c t e r i z a t i o n s in t h e t i m e - d o m a i n : t h e m o s t i m p o r t a n t o n e i s

f o r m of a m a t r i x r a n k

test which points out

complementary subsystems resulting gregated

stations.

the importance of the

s t a t e d in c o n c e p t of

from t h e p a r t i t i o n i n g of t h e s y s t e m in two a g -

This characterization reveals that fixed modes are simultaneously

uncontrollable by one aggregated station and unobservabie by the other one. -

Characterizations

in t h e

frequency-domain : these characterizations

d e e p i n s i g h t i n t o t h e r e a s o n s of o c c u r e n c e of f i x e d m o d e s . tioning of the

s y s t e m in two a g g r e g a t e d

representation

is

nection

between

terms

specially adequate

fixed modes may arise pattern.

the

aggregated

from d i f f e r e n t

out

the

stations. origins

This

importance approach

related

to

the

function matrix of t h e

intercon-

shows clearly

interconnection

that terms

T h e y c a n b e c l a s s i f i e d in two d i f f e r e n t t y p e s :

- structurally

fixed modes whose e x i s t e n c e is

v a l u e s of t h e s y s t e m a n d a r i s e f r o m t h e s t r u c t u r e -

We f i n d a g a i n t h e p a r t i -

stations and the transfer

to p o i n t

give a

non

structurally

fixed

modes

whose

independent

of t h e

parameter

of t h e s y s t e m .

existence

depends

on

the

parameter

v a l u e s a n d a r i s e from s o m e p e r f e c t p a r a m e t r i c c a n c e l l a t i o n s . T h e s e f i x e d m o d e s can be removed b y p e r t u r b i n g

t h e p a r a m e t e r s of t h e s y s t e m .

A differenciation between s t r u c t u r a l l y fixed modes t h e m s e l v e s is p o i n t e d out.

It is c l e a r t h a t s t r u c t u r a l l y than

non

leading

structurally

to n o n

fixed

structurally

fixed modes seem to h a v e a more p h y s i c a l reality

modes.

Indeed,

fixed modes have

the

perfect

a very

parametric

low p r o b a b i l i t y

cancellations in

practice.

M o r e o v e r , t h e y c a n b e c o n s i d e r e d a s e f f e c t i v e o n l y in t h e c a s e f o r w h i c h t h e m a t h e -

149

matical p a r a m e t e r s

involved

in

the

cancellations

correspond

to

the

same p h y s i c a l

p a r a m e t e r s . O t h e r w i s e , it i s c l e s r t h a t t h e l a c k o f a c c u r a c y i n t h e m o d e l i n g p r o c e s s makes i m p r o b a b l e

these

cancellations.

Nevertheless,

t h e s e c a n c e l l a t i o n s in a m a t h e m a t i c a l f r a m e w o r k . which close v a l u e s of t h e p a r a m e t e r s

it r e m a i n s i m p o r t a n t

They

reveal

may h a v e an a d v e r s e

controllability u n d e r c o n s t r a i n e d information s t r u c t u r e

modes a r e

to d e t e c t

s i t u a t i o n s in

e f f e c t on t h e d e g r e e o f

of a mode.

T h e i n t r o d u c t i o n of t h e c o n c e p t of c o n t r o l l a b i l i t y u n d e r tion s t r u c t u r e

critical

decentralized informa-

makes clearer t h e analogy with the case of centralized control.

the

Fixed

e x t e n s i o n of t h e c o n c e p t s of u n c o n t r o l l a b l e a n d u n o b s e r v a b l e m o d e s

and s t r u c t u r a l l y and u n o b s e r v a b l e

fixed modes establish the modes.

s t u d y of all s t r u c t u r a l

relation with

A graph-theoretic

properties,

approach

structurally

uncontrollable

is s p e c i a l l y a d e q u a t e

s o is f o r s t r u c t u r a l l y

for the

f i x e d m o d e s f o r w h i c h some

graph-theoretic characterizations are presented. -

Graph-theoretic

characterizations :

s t r u c t u r a l p r o p e r t i e s like s t r u c t u r a l l y titative d a t a if t h e y a r e w e i g h t e d .

graphs

can

U s i n g t h i s tool,

zati0ns of f i x e d m o d e s ( s t r u c t u r a l or n o t s t r u c t u r a l ) In

the

whole c h a p t e r , similar,

are u s e d to c a r r y

used

for

the

study

of

a particular

effort

two g r a p h - t h e o r e t i c

differing only by

o u t t h e final r e s u l t .

characteri-

are provided.

has

equivalences between the different characterizations, are p e r f e c t l y

be

f i x e d m o d e s b u t t h e y c a n a l s o deal w i t h q u a n -

been

made

to

point

out

the

It is s h o w n t h a t s o m e of t h e m

the notations and by the

definitions which

T h i s is t h e c a s e f o r t h e c h a r a c t e r i z a t i o n s in

term of t r a n s m i s s i o n z e r o s p r e s e n t e d i n P a r a g r a p h

3.2.

In P a r a g r a p h

3 . 3 , it a p p e a r s

t h a t all t h e t i m e - d o m a i n c h a r a c t e r i z a t i o n s a r e d e r i v e d from t h e s o - c a l l e d m a t r i x r a n k test characterization.

Finally,

those presented

in P a r a g r a p h

3.4

in

the

frequency

domain a r e all " i n c l u d e d " in t h e s o - c a l l e d g e n e r a l t r a n s f e r f u n c t i o n c h a r a c t e r i z a t i o n . I t h a s to b e n o t e d t h a t g r a p h - t h e o r e t i c c h a r a c t e r i z a t i o n s p r e s e n t to deal w i t h a r b i t r a r y

feedback structures

since t h e f e e d b a c k p a t t e r n is r e p r e s e n t e d

the advantage

as easily as with d e c e n t r a l i z e d s t r u c t u r e s in t h e g r a p h .

C h a p t e r II p o i n t e d o u t t h e p r o b l e m s a r i s i n g from t h e e x i s t e n c e o f f i x e d m o d e s in r e f e r e n c e to t h e p r o b l e m of s t a b i l i z a t i o n a n d pole a s s i g n m e n t . T h e two n e x t c h a p t e r s a r e c o n c e r n e d now w i t h t h e d i f f e r e n t m e t h o d s w h i c h c a n b e u s e d to a v o i d o r to eliminate f i x e d m o d e s p r e s e r v i n g

t h e o b j e c t i v e of a " r e d u c e d " i n f o r m a t i o n flow. T h e s e

m e t h o d s a r e b a s e d on t h e c h a r a c t e r i z a t i o n s p r e s e n t e d in t h i s c h a p t e r .

CHAPTER 4

DECENTRALIZED STABILIZATION

IN P R E S E N C E

4. I .

OF N O N

STRUCTURALLY

FIXED M O D E S

- INTRODUCTION

In the previous structurally structure.

chapter,

controllable

we h a v e s h o w n t h a t f i x e d m o d e s c a n b e c l a s s i f i e d in under

constrained

information

The controllable modes which are a consequence

of a p e r f e c t

m a t c h i n g of

system parameters

uncontrollable

modes

a r e w h a t we called n o n s t r u c t u r a l l y

f i x e d m o d e s of t y p e mation t r a n s f e r

and

(ii).

fixed modes and

structurally

T h e u n c o n t r o l l a b l e m o d e s w h i c h a r i s e f r o m a l a c k of i n f o r -

among the

stations

are

what

we c a l l e d

structurally

fixed

m o d e s of

type (i). In this chapter, information changing

structures

the

control

T h i s is a f u n d a m e n t a l system

are

even

I t is s h o w n

when

difference between

if

modes),

then

no matter

non dynamic,

distributed

if k0 i s s u c h

with controllable fixed modes under

considered.

nature

Indeed,

that

a

only systems

has

centralized

modes

w h a t c o n t r o l law is u s e d or finite-dimentional),

a mode,

the

decentralized

remain

modes

d i m e n s i o n a l c o n t r o l law i s u s e d

when

~ and

traint is removed without changing

in

constrained

eliminated by

are

decentralized or

(linear or non linear,

maintained. fixed modes. unobservable dynamic or

the fixed modes remain in the sense

arbitrary

(AND-81a)

the result.

can be

constraints

and

Wang a n d an

they

(uncontrollable

closed-loop response

to exp0~ 0 t ) .

78) a n d A n d e r s o n

structural

centralized

fixed

contains terms proportional fixed

the

that

for a suitable initial Davison linear, the

(WAN-73) invariant

condition

showed and

finite-dimensionality

Nevertheless,

Kobayashi et al.

a n d Moore ( A N D - 8 1 b ) p o i n t e d o u t t h a t b y u s i n g a g e n e r a l

that

finitecons(KOBcontrol

law of t h e f o r m :

ui(t) = K i (Ii(t)) with Ii(t) = {Yi (~)' u i(x)) ~ T~[0, t] ~ [ 0 ,

(4.1.I)

t] }

151

the controllable n a t e d in t h e

fixed modes

sense

that

under

such

constrained

controller

information

can bring

structure

any initial state

may be

elimi-

to t h e z e r o s t a t e

in a f i n i t e t i m e . Consequently, tralized

the

decentralized

fixed modes can be achieved

one of t h e p r o p e r t i e s

of linearity

An e a s y i n t e r p r e t a t i o n terization of Anderson

matrix

of t h i s r e s u l t

If t h e s y s t e m

(P)

~ I'

W12

t

W22 (

presence

of unstable

control

law,

decen-

provided

that

have been secrified.

is g i v e n

For simplicity,

(or its transpose)

I Wll W(p)

in

a decentralized

or time-invariance

(AND-82).

system with simple modes. its t r a n s f e r

stabilization by

by the frequency-domain let u s c o n s i d e r

has a decentralized

charac-

a 2-input

2-output

f i x e d m o d e a t t 0,

has the following form (AND-82)

then

:

(Pl

= !

entry

w i t h n o p o l e at k 0

entry

w i t h p o l e a t )'0

J

w i t h z e r o at )'0

entry

w i t h n o p o l e a t )'0

J

WCp) =

l entry Now, if t h e f e e d b a c k

c o n t r o l u 2 = k2Y 2 i s a p p l i e d at s t a t i o n

2, we h a v e

:

Yi(P) k2 Ul(P) = Wll (P) + Wl2 (P) • l_W22(P)k2 • W21 (P) If W12 ( p )

has

a p o l e at }~O ( a n d W21 ( p )

occurs resulting the

in a f i x e d m o d e .

cancellation

block

with

an

will

no longer

adjacent

(4.1.2)

h a s a z e r o at k 0 '

a pole-zero

B u t if k 2 i s a t i m e - v a r y i n g

occur

because

time-inveriant

one

block,

cannot

and

commute the

thereby

cancellation

or non linear element, juxtapose

time-varying a

cancelling

pole-zero pair.

Finally, originates

we n o t e t h a t if W12 ( p )

from the structure

o r W21 ( p )

of the system

lized c o n s t r a i n t s .

The nature

of the decentralized

the existence

such

mode.

of

relax the structural The system

fixed

constraints considered

In

this

is i d e n t i c a l l y

zero,

a n d it is u n c o n t r o l l a b l e case,

feedback the

only

is t h u s

the fixed mode under

without

possible

a n d is d e s c r i b e d

effect on

approach

on the control.

h e r e h a s S local s t a t i o n s

decentra-

by :

is to

152 5

l

;(t) = A x(t) +

~ B i ui i=l

tYi(t) = Cix(t)

where

x C Rn ,

(4.1.3)

(i=l,...,S)

u i ~ Rmi a n d Yi ~ R r i a r e

o u t p u t v e c t o r s o f t h e it h s t a t i o n .

the state

vector

a n d t h e local i n p u t

and

T h e g l o b a l d e s c r i p t i o n is g i v e n b y :

B LB1

Bs ]

with

(4.1.4)

Cx

C

~

.°.

4 . 2 . - SAMPLE AND HOLD Let

us

consider

the

system

(4.1.3)

exponential decay a and sampling period Then,

the resulting

and

put

a

general

sampler-hold,

T , in s e r i e w i t h e a c h i n p u t

d i s c r e t e - t i m e s y s t e m is d e s c r i b e d

with

of the system.

by :

S x ((k+l)T)=

r x(kT)

+ i-1 E

Gi u i ( k T )

(4.2.1) yi(kT)

= Ci x(kT)

(i = 1 . . . . .

S)

where F = e x p (AT) f T Gi = e x p ( - a T ) ~

[exp(A+aI)]

d t Bi

0 From ( W A N - g 2 ) , t h e s e t o f f i x e d m o d e s o f t h e d i s c r e t e - t l m e

system (4.2.1)

is defined

a s follows : S Ad = K. Cn Rr[xmO

(F + i=lZGiKIC i)

(4.2.2)

1

A sufficient

condition

for

the

existence

of a stabilizing

s e t o f local d i s c r e t e

time

d y n a m i c c o n t r o l l e r s is g i v e n b y t h e f o l l o w i n g t h e o r e m : Theorem

4.1

(WAN-82).

If t h e r e

exists

such that the fixed modes of (4.2.1) (4.2.1),

and hence

controllers.

(4.1.3),

a sampling

period

T and

a real number a

a r e c o n t a i n e d in t h e u n i t d i s c , t h e n t h e s y s t e m

c a n b e s t a b i l i z e d b y a s e t o f local d i s c r e t e - t i m e

dynamic

153 Wang (WAN-82) illustrated constructive

procedure

Consider the system

=

y=

this

to d e t e r m i n e

with

a simple example.

However,

no

a and T was proposed,

:

[0 oj io] -2

-3

0

0

0

1

O

0

11

-I

1

0

E

approach

x+

1

uI +

u2

0

x

which can also be represented

as follows :

i

sl

u 1_ _

p-I

p2 + p3 + 2

rI ,

[', S2

-~r

Yl

This system

has a fixed mode at

l,

F

which can be viewed

zero c a n c e l l a t i o n w h e n local f e e d b a c k i s a p p l i e d . each input

and apply

a unity

following c o n f i g u r a t i o n

vI _

_

feedback

u[

at s t a t i o n

as a result

2,

then the overall system

:

s a m p l e and LuI_DJ

p-I

[

. . . . . .

jI p2~p+~l

,o,d

1

S2

Y~

yl

77 /

of pole-

P u t a s a m p l e a n d h o l d in s e r i e w i t h

!

.old

~

Y2

has the

154

The pulse

transfer

functions

sampling period T are given by

o f S 1 a n d S 2 i n s e r i e w i t h a s a m p l e a n d h o l d of

:

hl(Z ) : z(-0.5 + 2 e = T - 1.5 e -2T) ÷ (- 1.5 e -T + 2 e - 2 T (z-

e-T) (z-

0.5 e -3T)

e-2T)

T h 2 /~z, ] _ e Z-

- 1 T e

and the pulse transfer has no pole-zero

f u n c t i o n b e t w e e n v 1 a n d Yl* i s e q u a l to h l ( 2 )

cancellation.

1 can easily be designed

Hence, a discrete-time

0) may b e n e c e s s a r y On systems

another with

dynamic compensator

so t h a t t h e o v e r a l s y s t e m is s t a b l e .

h o l d (a = 0) is s u f f i c i e n t in t h e a b o v e e x a m p l e ,

x h2(z),

which

at station

Although a zero-order

a sample and hold with decay

(a~

in some c a s e s .

hand,

distinct

the

poles.

validity

of

this

For example)

approach

the

seems

following system

to

be

restricted

provided

by

to

Wang

(WAN-82) c a n n o t b e s t a b i l i z e d u s i n g s a m p l e a n d h o l d t

1

0

x:

0

1

0

1

x +

uI +

0

u2

0

oo]x

y E:

1

0

0 -12~ x Y2 : E 0 This system has a simple fixed mode at 1 corresponding For the class of systems ...,

(4.1.3)

to t h e m u l t i p l e p o l e .

f o r w h i c h A h a s n d i s t i n c t e i g e n v a l u e s k i , i = 1, 2,

n and that can be transformed

as :

S

(4.2.3) Yi

= ~. 1

~

(i = 1 .....

w h e r e k = d i a g { k 1' ~ 2 . . . . .

s)

kn}

155

an i n t e r e s t i n g

result

der

system

that

the

sampling period

was provided (4.2.3)

T > 0 and

can b e d e s c r i b e d

by

latter

is

by Ozguner

controlled

a zero-order

by

hold.

and

Davison

a digital Then,

(OZG-85).

controller

the

with

resultant

Consi-

a constant

sampled

system

:

x ( t + T ) = e ~T ~ ( t )

S ~

+ r

Bi ~i

i:l

(4.2.4)

i = 1 ..... Yi ( t ) = ~ i ~ ( t ) w h e r e r = d i a g - ~ r 1, I"2 . . . . .

S

rn]

and

Fi = T

i f ~'i = 0

e TXi - l F =~

if ~i F 0

T h e n , we h a v e t h e f o l l o w i n g r e s u l t Theorem 4.2

(OZG-85).

Assume

:

that

the

a m o n g w h i c h t h e f i x e d m o d e s ~j (j = 1, (i). T h e n ,

the sampled system

(4.2.4)

system ....

(4.1.3)

(4.2.3)

has

ps ) are structurally

p

fixed

modes

fixed modes of type

h a s o n l y P s f i x e d m o d e s ek i T ,

j = 1.....

Ps'

for a l m o s t all T > 0.

The

interpretation

when fixed modes zero c a n c e l l a t i o n s sampling

has

on

of

(except

the

of a specific poles

results

structurally

and

kind

zeros

presented

in

this

fixed modes of type in

the

make

decentralized

that

the

section (i))

system.

cancellations

becomes

clearer

are viewed as poleThe

do n o t

effects occur

in

that the

model o f t h e s a m p l e d s y s t e m .

4 . 3 . - USE OF T I M E - V A R Y I N G C O N T R O L L E R S In

this

Purviance stabilize

section,

and linear

the particular

Tylee

I

the

that

systems

case of systems

with

results

use

a

of Anderson

decentralized

decentralized

+ B 1 ul(t)

and

Moore

tlme-varying

fixed

w i t h two c o n t r o l s t a t i o n s ,

a n d Moore to s y s t e m s

a controllable and observable

x (t) = A x(t)

Yi(t)

(PUR-82)

invariant

extension of Anderson Consider

we p r e s e n t

modes.

First

t h e n we d i s c u s s

(AND-81b), feedback

to

we p r e s e n t briefly

the

with S stations. two-station

system

described

by

•

+ B2 u2(t)

(4.3.1) Cixi(t)

(i = I , 2)

156 w h e r e x 6-R n ,

ui~

Rmi a n d YiE: Rri a r e t h e s t a t e v e c t o r a n d t h e local i n p u t and

o u t p u t of s t a t i o n i r e s p e c t i v e l y . dimensions.

Suppose

A, Bi a n d Ci a r e c o n s t a n t m a t r i c e s of a p p r o p r i a t e

also t h a t we a p p l y a p e r i o d i c

time-varying

control

law,

with

period T, at the second station : u 2 ( t ) = K z ( t ) Y2(t)

(4.3.2)

t h e n t h e r e s u l t i n g t i m e - v a r y i n g c l o s e d - l o o p s y s t e m is :

(t) =EA + BzKz(t) Ca] + BlUl(t) (4.3.3) Yl(t) = C I x(t) F o r t h i s s y s t e m , u n i f o r m c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y (KAI-80) m e a n s t h a t we can d e s i g n an o b s e r v e r a n d a l i n e a r s t a t e f e e d b a c k w h i c h will s t a b i l i z e t h e s y s t e m . d e n o t e b y ¢K2 ( t , T) t h e t r a n s i t i o n m a t r i x of s y s t e m

If we

(4.3.3),

then the observability

T @K2 (t,T) C 1 C 1 ¢K2(tJ) dt

(4.3.4)

grammian m a t r i x is :

OG(T , • + T) ~

fT+T

a n d t h e c o n t r o l l a b i l i t y grammian m a t r i x is

CG(~:, T+T) _/:+T

@K2 (T,t)

:

BIB l

T (T, t) dt

~2

(4.3.s)

T h e c o n d i t i o n of u n i f o r m c o n t r o l l a b i l i t y a n d o b s e r v a b i l i t y is s a t i s f i e d if t h e m a t r i c e s CG (~, z+T) a n d OG (~, T+T) a r e s t r i c t l y p o s i t i v e - d e f i n i t e .

4.3.1.

-

Piecewise

constant

feedback

laws

A n d e r s o n a n d Moore (AND-81b)

p r o p o s e d to u s e a p e r i o d i c p i e c e w i s e c o n s t a n t

f e e d b a c k at t h e s e c o n d s t a t i o n . Given t h e two following a s s u m p t i o n s : - Centralized controllability and observability, trollable -

i.e.

[(B1B2),

A,

(C 1' C2')w] is

con-

and observable.

Connectivity

assumptions,

identicaly zero :

i.e.

the

transfer

matrices

between

stations

are

not

157

B2 # 0

(4.3.6)

C 2 ( P I - A ) -1 B 1 # 0

(4.3.7)

W12(p) = C l ( P I - A ) - i w21(p)

t h e i r r e s u l t s a r e e x p r e s s e d in t h e following t h e o r e m : Theorem 4.2

(AND-81b).

Consider the controllable and observable

sense) system given by (4.3.1).

(in a c e n t r a l i z e d

A p p l y i n g a p e r i o d i c f e e d b a c k u 2 ( t ) = K 2 ( t ) Y2(t) at

the s e c o n d s t a t i o n p t h e s y s t e m ( 4 . 3 . 2 ) is u n i f o r m l y c o n t r o l l a b l e a n d o b s e r v a b l e if t h e connectivity assumptions (4.3.6)

and (4.3.7)

hold a n d if K 2 ( t ) is p i e c e w i s e c o n s t a n t

taking at l e a s t l + m a x ( m 2 , r 2) d i s t i n c t v a l u e s o v e r one p e r i o d . Remark 4.1 : T h e a s s u m p t i o n s r e q u i r e d

b y t h e a b o v e t h e o r e m a r e e q u i v a l e n t to t h e

a s s u m p t i o n of c o n t r o l l a b i l i t y a n d o b s c r v a b i l i t y u n d e r d e c e n t r a l i z e d i n f o r m a t i o n s t r u c ture. T h i s r e s u l t c a n b e a n a l y s e d as follows : if t h e s y s t e m h a s a f i x e d mode d u e to a lack of o b s e r v a b i l i t y of s t a t i o n 1, t h e n b y t h e a s s u m p t i o n of c e n t r a l i z e d o b s e r v a bility,

station 2 observes

t h i s mode a n d t r a n s m i t s

r e l a t e d i n f o r m a t i o n to s t a t i o n 1

t h r o u g h t h e t r a n s m i s s i o n c h a n n e l W12. A dual a n a l y s i s can b e made if t h e f i x e d mode is c a u s e d b y a lack of c o n t r o l l a b i l i t y of s t a t i o n 1. T h e a b o v e t h e o r e m c a n b e similarly d e r i v e d Moreover,

and Moore (AND-81b) mentary

discrete

systems

(JAM-83).

~ t h e y s h o w e d t h a t if t h e c o n n e c t i v i t y a s s u m p t i o n s of comple-

subsystems

hold

(system

decentralized constraints), observable

from

station

structurally

controllable

and

observable

under

t h e n t h e s y s t e m c a n b e made u n i f o r m l y c o n t r o l l a b l e a n d 1 by

f e e d b a c k c o n t r o l law u . ( t )

applying

= Ki(t)

successively

Yi(t),

i = 2,

...

a periodic

piecewise

constant

S at t h e o t h e r s t a t i o n s .

For

l

each s t a t i o n i , K i ( t ) m u s t t a k e a t l e a s t period.

for

t h e c a s e of s y s t e m s w i t h S c o n t r o l s t a t i o n s was c o n s i d e r e d b y A n d e r s o n

~ 1+max(re., r . ) d i s t i n c t v a l u e s o v e r o n e j:2 l J I t is c l e a r t h a t t h e n u m b e r of d i f f e r e n t v a l u e s i n c r e a s e s d a n g e r o u l s y with

the n u m b e r of s t a t i o n s making d i f f i c u l t t h e p r a c t i c a l i m p l e m e n t a t i o n of t h i s a p p r o a c h . E,.xamIAe 4.1

(AND-81b).

Consider

a controllable

and

observable

stations given by : 1

0

0

0

1

0

0

0

2

yl=[O

0

[1

~=

X +

1

0

u1

f0 0

1

u2

system

with

two

158 The system has a decentralized

rank

Applying

F L c,

f i x e d m o d e a t X0 = 1 s i n c e

XI-A 2 . . . . . .

= 2 < 3

10j

the following time-varying

u2(t)

= K2(t)

for X = ~ 0 : 1

c o n t r o l at t h e s e c o n d

station

:

Y2(t)

with

K2t fl °ljl 1

or2k t 2kl

0 •

for 2k+l

t < 2k+2

k = 0,

the

observability

the

range

(2k,

...

and

controllability

grammian

matrices

2k+l)

f o r all k= 0,

1,

are positive

number 1 approximately (piecewise constant) These properties

1, 2,

equal

2,

to 100 a n d

gain K2(t)

provides

can be improved

...

6,

(calculated

respectively.

reasonable

definite So,

the

controllability

by choosing another

analyticaly)

over

with a condition use

of a periodic

and observability.

kind of time-varying

feedback

l a w s a s it i s s h o w n i n t h e f o l l o w i n g s e c t i o n .

4.3.2.

- Sinusoidal feedback Purviance

put

controllable

prete ting

Tylee

to s t a t i o n the

(PUR-82)

and observable

the observability

through K2(t).

and

laws

1 the

transfer

problem value

consider

system

resulting

of the

function

This problem is a standard

WI2,

the particular

with a decentralized

fixed via

case of a 2-input

2-out-

fixed mode.

inter-

They

in the fixed mode as which of communicamode the

is

observable

by

station

time-varying

(which

feedback

law

with

o n e in c o m m u n i c a t i o n s y s t e m

2) gain

analysis and a good

1The condition number (CN) of a rectangular m a t r i x w i t h full r a n k i s g i v e n b y t h e ratio between the maximal and the minimal singular values of the matrix (MOO-8]). H e n c e , it i s a g o o d m e a s u r e o f t h e e f f e c t i v e r a n k o f t h e m a t r i x , a n d of c o u r s e , it is i n o u r i n t e r e s t to h a v e a c o n d i t i o n n u m b e r a s c l o s e to 1 a s p o s s i b l e ( i f CN = 1, t h e n the effective rank equals the actual rank).

159 solution is to u s e

sinusoldal modulation

(feedback)

at a f r e q u e n c y m a t c h e d to t h e

f r e q u e n c y r e s p o n s e of t h e communication c h a n n e l W12 ( V A N - 6 8 ) , U s i n g a simple e x a m p l e , t h e y show t h a t if t h e s y s t e m v e r i f i e s t h e c o n n e c t i v i t y assumptions, trollability

t h e n b y u s i n g a s i n u s o i d a l f e e b a c k law,

and

observability

is

higher

(decrease

the resulting

of t h e

d e g r e e of c o n -

corresponding

grammian

condition n u m b e r ) t h a n b y u s i n g t h e b i n a r y f e e d b a c k law p r o p o s e d b y A n d e r s o n a n d Moore ( A N D - 8 1 b ) . Example

4.2

(PUR-82).

Consider

a

2-input

2-output

controllable

and

observable

system d e s c r i b e d b y :

x=

yl=[0

-1

0

0

-2

x+

I

,Ix

uI + Ii

u2

y2=Ll 0 07 x The t r a n s f e r m a t r i x of t h i s s y s t e m is :

0

(p+t) (p+2)

W(p) ~-

0

and it is c l e a r

that the system has a decentralized structurally

fixed mode of t y p e

(fi) (x 0 = 0 ) .

A p p l y i n g a s i n u s o i d a l f e e d b a c k c o n t r o l at t h e s e c o n d s t a t i o n K 2 ( t ) = k sin 0~t, the c o n d i t i o n n u m b e r of t h e o b s e r v a b i l i t y grammian m a t r i x of t h e c l o s e d - l o o p s y s t e m is shown b y F i g u r e 4 . 1 - a a n d F i g u r e 4 . l - b , of k .

f o r d i f f e r e n t v a l u e s of ¢0 a n d two v a l u e s

160 500

sO0

0

~

2

C

3

0

4

Figure

4.1-a

: Condition number

for k = 0.05

Forc0=c0c = / 2 W12 a n d when

we c o u l d

w = coc ) ,

central

expect

m a x i m u m f o r k = 1. energy

the

result

points

However, Figure

system

the

k

and

by

~

2

3

¢

: Condition number

of the bandpass

can be explained This number of

the

for 0~=~ c,

by

between the

fact

is required

2 and

that

condition

for

with

1

"large"

and

the

destroys

observability

a good observability.

consideration

the

function

station

characteristics

and to achieve

energy

of the transfer

grammian is minimum for k =

system's

balance

of

for k = 1 (from (PUR-82))

in

control

number

law

This

design.

of OG(100,

0)

(see

is 72.7.

To m a k e a c o m p a r i s o n proposed

1

optimal communication

modes.

importance

= 0.05

.

of the observability

This

a small condition

out

with

4.l-a)

frequency

fixed mode dominates

between

g r a m m i a n to h a v e

,

4. l - b

OG(t,0)

to a c h i e v e

0.05 and

balance

Figure

the condition number

feedback the

of

(from (PUR-82))

(c0c i s t h e

il

FrequencY~o (rad/sec)

Frequencg~0o {rad/sec) OG(t,0)

,

Anderson

following binary

and

feedback

with the case of piecewise constant Moore

(AND-81b),

with period

o

o.lt<

1

l,~t<2

let us apply

feedback

at the second

c o n t r o l law station

the

2 :

i

K2(t) =

The condition number yields

better

obtained

4.3.3.

by

observabi]ity

0) is 1 3 2 . 9 .

about

using

54 ~.

less

This example shows that the condition

Thus,

control

it s e e m s t h a t

energy.

sinusoidal

A similar

conclusion

feedback

laws can

feedback can

be

for controllability.

In this with

decentralized the

of OG(100,

decreased

~ Concluding

systems

ving

number

has been

remarks

section, fixed

we h a v e modes

constraints.

decentralized

that

shown are

Although feedback

that

time-varying

structurally this approach

structure,

some

controllable presents

the

difficulties

and

advantage are

stabilize

observable

under

of preser-

encountered

when

161

considering of v i e w ,

the practical

we t h i n k

feedback laws,

that

but

feedback

can be made using

Vibrational zation a p p r o a c h

controllers.

From the practical

laws are easier

result

vibrational

point

to implement than

has not been

extended

We will s e e in t h e f o l l o w i n g s e c t i o n t h a t

4.4. - VIBRATIONAL

feedforward

sinusoldal

of s u c h

the corresponding

more t h a n 2 s t a t i o n s . this approach

implementation

binary

to s y s t e m s

a possible

with

extension

of

control.

CONTROL

control

was introduced

in c a s e s w h e r e

cannot

be

in

1973 b y

conventional

used

because

Meerkov

(MEE-73)

control principles

of a lack

such

of available

as a stabili-

that

feedback or

measurements

on

the

system. The principle tions

(with

properties

zero

value)

control consists

on

the

of the system in a desired

Indeed, control

of vibrational

mean

vibrational

signals,

so

control signals disturbances

control

that

it

(vibrations)

does

proach

to s t a b i l i z e

trained ments,

feedback

systems fails.

its application

pattern

Vibrational

depend

upon

any and

the

constraints

vibrational

with

vibra-

modify

the

for

does

constraints

The

or

upon

variable

t.

as a lack of measu-

which not

additive

feedback)

s e e m s to b e a s u i t a b l e

control

of the

or

principles.

of the independent

fixed modes

vibrational

is i n d e p e n d e n t

(not

can be interpreted

control

unstable

measurement

feedforward

states

but are just functions

control

structurally

require

imposed

any

on

the

ap-

cons-

measureinformation

control principle

control is a method of stabilization

zero m e a n v i b r a t i o n s properties

of the

vibrations

into

achieved by factors,

require

do n o t

Because

of s u c h

which

(TRA-85).

4.4. ], - Vibrational

require

not

feedback

local s t a t i o n ,

parameters

manner.

from

(not feedforward),

at each

in the introduction

system

differs

Since the control structure rements

dynamic

(oscillations)

system,

systems

on t h e

in p a r t i c u l a r

that

are

not

etc...

Unlike

control matrix son, the design

nor

conventional

of deviations observation

control

parameters

stability.

mechanical

oscillation of the technological

measurement

system

its

consisting

but

of an

which that

arbitrary

by varying

techniques,

vibrational

to a p p e a r

i.e.

in t h e

method does not differ for autonomous

may

the

parameters,

and disturbances,

matrix

Note

in the introduction

modify the

introduction

of

nature

be

the

can

amplification

control

does

it d o e s n o t r e q u i r e system

model.

of

For

not

neither

this

rea-

or non autonomous systems.

162 C o n s i d e r t h e following t i m e - i n v a r i a n t , (t) = A x(t)

finite-dimensional linear system

:

x ~R n (4.4,1)

A

(aijl i , j = l . . . . . n

and introduce

p e r i o d i c z e r o m e a n v i b r a t i o n s on t h e e l e m e n t s o f m a t r i x A a c c o r d i n g to

t h e law :

V(t) = (vij(t))

i,j=l,...,n

(4.4.2)

where v i i ( t ) = a'lj s i n t~j t mij ~ t0sk The resulting

f o r ij ~ s k

time-varying

( t / = [A + V ( t ) ] Before describing

s y s t e m is -" x(t)

(4.4.3)

t h e d e s i g n m e t h o d , l e t u s come u p w i t h t w o q u e s t i o n s

:

1. Which s y s t e m s a r e v i b r a t i o n a l l y s t a b i l i z a b l e ( i . e ° b y m e a n s o f v i b r a t i o n a l c o n t r o l ) ? 2. How to d e t e r m i n e t h e v i b r a t i o n p a r a m e t e r s

(gains and frequencies1

?

T h e a n s w e r to t h e s e q u e s t i o n s is g i v e n b y t h e following : D e f i n i t i o n 4.1

(MEE-801.

System (4.4.11

is v i b r a t i o n a l l y s t a b i l i z a b l e if t h e r e

periodic zero mean matrix V(t) such that system (4.4.3) Theorem 4.3

(MEE-80).

( c , A) i s o b s e r v a b l e ,

Assume that there

then a necessary

exists

a vector-row

c such

The extension

that the pair

and sufficient condition for system

b e v i b r a t i o n a l l y s t a b i l i z a b l e is to h a v e t h e t r a c e o f m a t r i x A n e g a t i v e Remark 4.2.

exists a

is asymptotically stable.

to t h e c a s e T r A ~ 0 will b e c o n s i d e r e d

(4.4.11

to

( T r A ( 01. in a s u b s e q u e n t

paragraph. The analysis of the consequences tions V(t)

can be achieved applying

73) to s y s t e m

(4.4.31.

resulting

For this purpose,

a . . a n d c0.. ( s e e f o r m u l a ( 4 . 4 . 2 ) ) 15 1]

of the vibra-

s c h e m e (VOL-621

certain conditions are required

- the matrix V(t) has a quasi-triangular -

from t h e i n t r o d u c t i o n

the Volosov's averaging

(MEE-

:

form

are sufficiently large (see Theorem 4.41.

163

Volosov's

averaging

scheme

consists

in

determining

a time-invariant

system

such t h a t i t s motions d e s c r i b e in " a v e r a g e " t h e motions of t h e t i m e - v a r y i n g s y s t e m (4.4.3).

T h i s s y s t e m a p p e a r s in t h e following form •

(4.4.4)

(t) = (k + V) z(t)

where V is a c o n s t a n t m a t r i x d e p e n d i n g on v i b r a t i o n a m p l i t u d e s a n d f r e q u e n c i e s .

So,

the s t a b i l i t y

the

conditions

for

the

system

(4.4.3)

can

be

established

s t a b i l i t y c r i t e r i a for t i m e - i n v a r i a n t s y s t e m s o n s y s t e m ( 4 . 4 . 4 ) .

by

using

I n d e e d , t h e e n t r i e s of

d e p e n d o n a i j ' s a n d 0~ijrs : if t h e y a r e c h o s e n to v e r i f y t h e s t a b i l i t y c o n d i t i o n s for the t i m e - i n v a r i a n t s y s t e m ( 4 . 4 . 4 ) ,

t h e y will also e n s u r e s t a b i l i t y to s y s t e m ( 4 . 4 . 3 ) in

v e r t u e of t h e p r o p e r t y of Volosov~s a v e r a g i n g s c h e m e . Now,

assume

that

the

vibration

matrix

has

the

following

lower

(or

upper)

q u a s i - t r i a n g u l a r form 2 :

m 0

(%1 ~i.

~o21 t)

(ec31 sin m31 t)(a32 v(t)

I I I i

=

(anl

With t h i s

0_.

o. sin m32 t ) " " . . . I I I I

sin Wnlt)(an2

assumption,

0

(4,4.5)

sinmn 2 t)

- ( a n l , n _ 1 sinw n l , n _ l t )

0

t h e d e t e r m i n a t i o n of V a n d so of t h e t i m e - i n v a r i a n t s y s t e m

(4.4.4) c a n b e p e r f o r m e d b y t h e following simple p r o c e d u r e .

C o n s i d e r t h e following

system of d i f f e r e n t i a l e q u a t i o n s : ~¢ ( t ) = V ( t ) x ( t ) where

V(t)

is

the

(4.4.6)

quasi-triangular

matrix

given

by

(4.4.5).

Suppose

the

initial

conditions of ( 4 . 4 . 6 ) to b e :

xi(t 0) = x 0

(i=l . . . . . n)

i

Determine t h e s o l u t i o n of t h e f i r s t two e q u a t i o n s of ( 4 . 4 . 0 )

2V(t) is c h o s e n in q u a s i - t r i a n g u l a r any o t h e r form (MEE-80),

:

form b e c a u s e t h e s y s t e m c a n n o t be a n a l y s e d w i t h

164

x l(t) = x 0 x 2 ( t ) = x0 + I F 2 1 (t) - F21(t0) ] x~ where

F21(t)

is a z e r o mean p e r i o d i c

function

t h i r d e q u a t i o n a s s u m i n g t h a t F21(t0) = 0) i , e .

of t .

Determine

of t h e

:

0

xl(t)

= x1

x 2 ( t ) = x~ +

F21 ( t ) x ~

A p p l y an a n a l o g o u s p r o c e d u r e to e a c h ( i - l ) th s t e p of t h e p r o c e d u r e we f i n d : k-l j_El [ F k j ( t )

0

Xk=

Xk+

- Fkj(t0) ]

equation

x~

of

system

k-1 j~l

Xk = Xk0 + where Fij(t)

F k j ( t ) x j0

(k=l

a r e almost p e r i o d i c f u n c t i o n s

matrix E such that

(4.4.6),

then

at

the

(k=l ..... i-l)

T h e s o l u t i o n o f t h e i th e q u a t i o n is s o u g h t b y s u b s t i t u t i n g

triangular

the solution

. . . .

:

i-l)

of t with a z e r o mean.

Define the quasi-

:

E = (eij)i,j= 1 ..... n with

T

e..,] = ] i m T ÷¢o

Fij2(t ) dt

F i n a l l y , t h e m a t r i x V of ( 4 . 4 . 4 ) = H~ij{[i,j=l . . . . . n where

O denotes

(4.4.7)

0

is g i v e n b y :

= -(A'

O

element-by-element

E)

(¢.¢.8)

m u l t i p l i c a t i o n of t h e m a t r i c e s ,

i.e.

~..1j = -a..p .

e... II T h e o r e m 4.4 (MEE-80). such thataij

If t h e r e

exist sufficiently large positive constants

) cx0 and coij >coo for all ij, t h e n s o l u t i o n s

x(t)

of ( 4 . 4 . 3 )

a 0 and coo (4.4.5)

and

165

z(t) of ( 4 . 4 . 4 )

(4.4.8)

( d e f i n e d with i d e n t i c a l initial c o n d i t i o n s x ( t 0) = z ( t 0 ) )

are

related by the e x p r e s s i o n s :

~(t) = [I + F ( t ) ]

z(t)

to[0,

~[ (4.4.9)

llx(t) where F ( t )

- ~(t)ll

,~ 1/%

is t h e m a t r i x c o m p u t e d t h r o u g h t h e above p r o c e d u r e .

If t h e t i m e - i n v a -

rlant s y s t e m ( 4 . 4 . 4 ) is a s y m p t o t i c a l l y s t a b l e , r e l a t i o n ( 4 . 4 . 9 ) h o l d s for all t E [ 0 , ~[~ in t h e o p p o s i t e c a s e , it h o l d s o n l y f o r t £ [

0, a 0 [

T h e o r e m 4.4 s h o w s t h a t if t h e a m p l i t u d e s a n d f r e q u e n c i e s of t h e v i b r a t i o n s a r e sufficiently l a r g e , t h e n e q u a t i o n s ( 4 . 4 . 4 )

(4.4.8)

the t i m e - v a r y i n g s y s t e m ( 4 . 4 . 3 )

In p a r t i c u l a r ,

stable,

then system

(4.4.B)

has

(4.4.5). also t h e

d e s c r i b e in a v e r a g e t h e motions o f if ( 4 . 4 . 4 )

same p r o p e r t y .

is a s y m p t o t i c a l l y

Therefore,

stability c r i t e r i o n for t i m e - i n v a r i a n t s y s t e m s on s y s t e m ( 4 . 4 . 4 ) ,

applying any

one can o b t a i n t h e

stability c o n d i t i o n s for t h e t i m e - v a r y i n g s y s t e m and d e t e r m i n e t h e v i b r a t i o n s . It is o b v i o u s t h a t v i b r a t i o n a l c o n t r o l r e m a i n s w i t h o u t e f f e c t if ~ = 0 ~ in t h i s case, t h e d y n a m i c s of t h e initial a n d a v e r a g e d e q u a t i o n s do not d i f f e r . This r e m a r k motivates t h e i n t r o d u c t i o n of t h e following c o n c e p t (see Definition 3 a n d Theorem 3 of (MEE-80)) : v i b r a t i o n a l l y c o n t r o l l a b l e a r e t h o s e e l e m e n t s aij o f t h e m a t r i x A f o r which t h e r e e x i s t s an almost p e r i o d i c matrix V ( t )

(with a z e r o mean value) s u c h t h a t

V.. ¢ 0. In view of ( 4 ° 4 . 8 ) , v i b r a t i o n a l l y controllable a r e t h e r e f o r e t h o s e e l e m e n t s a.. 1] 1} of the matrix A for which i > j a n d aji ~ 0. From a p r a c t i c a l p o i n t of v i e w , t h e i n t r o d u c t i o n of v i b r a t i o n s is i m p l e m e n t e d b y making t h e t e c h n o l o g i c a l p a r a m e t e r s oscillate ] h e n c e , t h e i n t r o d u c t i o n of v i b r a t i o n s on zero e n t r i e s of m a t r i x A is n o t p h y s i c a l l y p o s s i b l e .

Consequently p vibrationally

controllable a r e t h o s e e l e m e n t s aij ~ 0 of A f o r w h i c h i > j a n d aji ~t 0. Remark 4 . 3 . 1. To e s t a b l i s h

the

analogy

between

vibrational

control

and

time-varying

feedback c o n t r o l c o n s i d e r e d in t h e p r e v i o u s s e c t i o n , we mention t h a t

output

for t h e s y s -

tem :

I~ = Ax + Bu Cx the i n t r o d u c t i o n of v i b r a t i o n s V ( t ) i n t o t h e matrix A h a s a similar e f f e c t as a t i m e varying o u t p u t f e e d b a c k c h a r a c t e r i z e d b y :

166

K(t)

B

G = V(t)

or

K ( t ) = B "[" V ( t )

G1

where B f and C+ are the pseudo-inverses 2. I n c e r t a i n the desired

cases,

there

is no need

effect can be achieved

of B and C.

to i n t r o d u c e

vibrations

with large

amplitudes

;

by using only small oscillations of the parameters.

See ( M E E - 7 3 ) .

3.

The

not

design

differ,

method

but

in the

in the

h a s to b e c o n s i d e r e d .

case of autonomous

case of non-autonomous Let the system be given by

:< = A x + B u

x~-R n

and

non-autonomous

systems,

systems

a supplementary

does

condition

:

and u ¢-R m

Suppose that the control ui, i = 1 .... , m , a r e s l o w f u n c t i o n s o f time i n t h e s e n s e du. ~l is bounded ; then the supplementary condition on the vibrations frequendt cies is :

that

t0

> > m a x (sup ~ I]

4.

The determination

tions.

But

in

controllable 84b)

the

element

(TAR-85)

vij(t)

l~

~ij

needs

case

per

the

row,

we h a v e

c o s coij t

then ~ H..2

and finally

~

particular

= a i j s i n ~ij t

13

13

(i,j: l ,...m) (% = l , . . . , m )

of matrix

; indeed

F..(t) = -

e..

)

£

-

2 co;~ 2 LJ

: a i j2

Vij = - aij eij = - aij 2wij

for

a heavy which

calculation :

integration

matrix

A has

of V b e c o m e s

of t r i g o n o m e t r i c only

one

relatively

func-

vibrationally simple

(TRA-

167 4 . 4 . 2 . - S t a b i l i z a t i o n b y v i b r a t i o n a l c o n t r o l (TRA-85) In t h i s s e c t i o n , we c o n s i d e r a c l a s s of s y s t e m s with u n s t a b l e f i x e d modes with r e s p e c t to a s t r u c t u r a l l y c o n s t r a i n e d f e e d b a c k c o n t r o l . T h e dynamic m a t r i x is s u p p o sed to v e r i f y t h e

c o n d i t i o n of T h e o r e m 4.3 so t h a t

the

systems are

vibrationally

stabilizable. To i l l u s t r a t e t h e a p p l i c a t i o n of v i b r a t i o n a l c o n t r o l , let us c o n s i d e r t h e following example • Example 4.3

(TAR-85).

Given t h e t w o - s t a t i o n c e n t r a l l y c o n t r o l l a b l e a n d o b s e r v a b l e

system :

x=

1

1

~1

-l

Y l = [0

0

'3

y2=[3

-1

03

x +

-I 2

uI +

[] 0 0

u2

(4.4.10)

x

its t r a n s f e r m a t r i x is :

: I 2p+5

1>-2 1

(p+l)(p+2)

W(p)

(p-l)(p+l)(p+2)

( ~ '

3p+2 (p+l)(p+2)

It is c l e a r t h a t t h e s y s t e m h a s an u n s t a b l e d e c e n t r a l i z e d f i x e d mode at ;~0 = 1, t h e n any d e c e n t r a l i z e d o u t p u t f e e d b a c k fails to stabilize t h e s y s t e m . C o n s i d e r t h e autonomous s y s t e m a s s o c i a t e d with s y s t e m (4.4.10)

k=Ax~

:

[2, :,I 1

1

I

-I

x

-

This s y s t e m is v i b r a t i o n a l l y s t a b i l i z a b l e , s i n c e f o r c = ( i 0 0 ) ' , t h e p a i r

(c, A) is

o b s e r v a b l e a n d T r A = -2 ( 0. The v i b r a t i o n a l l y c o n t r o l l a b l e e l e m e n t s of A a r e a21

168

a n d a3z (or a12 and a23) for a lower ( u p p e r ) q u a s i - t r i a n g u l a r v i b r a t i o n matrix, These c o n s i d e r a t i o n s lead to the following matrix :

V(t)

0

1 sinm21 t

0J

0

0

0

~2 sin ~2 t

0

r e s u l t i n g in the t i m e - v a r y i n g system : = [A + V(t)]

x

(4.4.11)

The determination of V can be performed b y a p p l y i n g the a v e r a g i n g scheme described in the p r e v i o u s section. Matrix A has only one v i b r a t i o n a l l y controllable element in each row, t h e n in accordance with Remark 4.3 of the p r e v i o u s section, the "averaged" system is :

z : (A+V)z =

I

-2 1 +V 21

1

O 1

1

1

z

(4.4.12a) 1

- 1 +~32

- 1

where a212 V-2l : - a 1 2 - 2~212

~ 2l 2 =-

2t021

2

<0 (4.4.12b)

~2 -

v32

=

~

-a23 2t0322

~2 2 =-

2t032

2

<0

The c h a r a c t e r i s t i c equation of the system (4.4.12) is : det ( p I - A - V ) = p3 + 2p2 _ (1+V21 + V32) P _ 2 - ~21 - 2~32 a n d the stability conditions for this system, e s t a b l i s h e d by u s i n g R o u t h ' s criterion, are :

~21 < 0 V32 < 0 V21 < - 2(1+~32)

169

v21

v32 -l

/'

/

Fig. ~.2

/

The domain D of admissible s o l u t i o n s for t h e s e i n e q u a l i t i e s is g i v e n b y F i g u r e 4.2. By c h o o s i n g Y21 ¢- D a n d V3Z~--D, t h e t i m e - i n v a r i a n t s y s t e m ( 4 . 4 . 1 2 ) i s made asymptotically s t a b l e .

Then,

chosen a c c o r d i n g to ( 4 . 4 . 1 2 b )

if t h e

amplitudes

and

frequencies of vibrations

are

a n d w i t h s u f f i c i e n t l y l a r g e v a l u e s (cf. T h e r o e m 4 . 4 ) ,

the t i m e - v a r y i n g s y s t e m ( 4 . 4 . 1 1 ) is made a s y m p t o t i c a l l y s t a b l e .

4.4.3. - Vibrational f e e d b a c k control laws

The

stabilization method used

in

the

last

section

is

based

on

introducing

v i b r a t i o n s on t h e p a r a m e t e r s of t h e s y s t e m model. From a p r a c t i c a l p o i n t of v i e w , this can b e p e r f o r m e d b y a c t i n g on t h e t e c h n o l o g i c a l p a r a m e t e r s of t h e s y s t e m . T h i s implies, f i r s t ,

that

one k n o w s t h e e x p l i c i t e x p r e s s i o n s of t h e model p a r a m e t e r s as

functions of t h e t e c h n o l o g i c a l p a r a m e t e r s and s e c o n d , t h a t some p h y s i c a l e n t i t i e s of the s y s t e m can b e d e s i g n e d so t h a t t h e y h a v e a v i b r a t o r y v a l u e . The f i r s t r e q u i r e ment is n o t g e n e r a l l y s a t i s f i e d s i n c e t h e s y s t e m model is o f t e n o b t a i n e d b y i d e n t i f i c a tion m e t h o d s . N e i t h e r is t h e s e c o n d r e q u i r e m e n t : a l t h o u g h it may b e c o n s i d e r e d to give a v i b r a t o r y value to an e l e c t r i c a l e n t i t y , ments.

The

domain of a p p l i c a t i o n of t h e

t h i s is i m p o s s i b l e for mechanical e l e -

a b o v e m e t h o d is

therefore restricted

to

specific c a s e s . The a p p r o a c h p r e s e n t e d in t h i s s e c t i o n u s e s at t h e same time v i b r a t i o n a l a n d feedback c o n t r o l t h e o r y . The a b o v e d i f f i c u l t i e s a r e t h u s

encompassed by introducing

the v i b r a t i o n a l c o n t r o l action t h r o u g h t h e f e e d b a c k ( R U N - 8 5 ) . Consider the system

(4.1.3)

and,

for s i m p l i c i t y ,

case of S>2 s t a t i o n s will be c o n s i d e r e d s u b s e q u e n t l y ) . controllability

and

observability

assumption,

make

the

case

S=2

(the

general

In addition to t h e c e n t r a l i z e d

the

following

connectivity

as-

sumption :

ClB 2 $ 0 ,

C 2 B 1 *0

(4.4.13)

170 which guarantees 81a)

that the system has no structurally

(COR-76a,b),

fixed modes of type

(i)

((SEZ-

see § 2.2.3)

Apply the following dynamic controller to the system

(4.1.3)

:

ul(t ) =ae K I (t) Yl(t) + Vl (t) u2(t ) - K 2 (-~ Y2(t) (t)

=

(4.4.14)

F (--t 6) ~(t) + G (--~ Yl(t)

~l(,) + H (t) ~;(t) + L (t) y~(t)

where

all t h e m a t r i c e s

have

almost periodic

entries

and

~ is t h e

state

of the

con-

t r o l l e r o f d i m e n s i o n v. T h e n t h e c o n t r o l l e d s y s t e m is :

~-

Ax + -a- Bl K 1 (~)ClX + B2 K 2 ~ ) C 2 x E

Yi : Cix

t

With ~ (~-)

being

+ B1 v1

(4.4.15)

i=1,2

a fundamental

matrix

t

f o r ~- B1KI(~-)_

C 1, d e f i n e t h e t r a n s f o r m a -

tion

x(t) = ~ (~) z (t) Consider

(4.4.16)

the almost periodic,

C1BI Q(t))

differentiable

m1 x r 1 matrix

O (t)

such

that

(I +

i s n o n s i n g u l a r f o r all t a n d : ¢

I

= Q (t) (I + C I B I Q

T h e n we h a v e

:~-I A~z4[B l B2]

I Pc'K2"K2c2B' B1Q Qpc, c2K2"K21I'Iz -12 Bivl (~.#.17)

171

where

p ( tc)

= _ Q (I)

(z + ClB 1 Q ( t ) ) - I

The c o n t r o l l e r in ( 4 . 4 . 1 4 ) is chosen as

= F ~+ G (I + CIB 1 Q ( t ) ) - 1 Yl

(4.4.18)

v I = (I + p(t) CIBI)-I H ~+ (I + P(~) CIBI)-I L (I + CIB 1 Q(t))-I Yl where F, G, H and L are constant matrices of appropriate dimensions. After some manipulations, (4.4.17) and (4.4.18) can be written as [PCI B2 K 2 C2 S 1 Q z

L

PC'B2K 1 VC]z

÷ BILCIZ+BIH (4.4.19)

/ K2C2BI Q

~:F~+GCIZ

Taking a v e r a g e s 3 in (4.4.19) gives the following time-invariant system :

(4.4.20)

= Ag + B1LCI~ + B1H : F 3 + c c1~

where~ = ~ +

1 2LF21 F22j

C2

1

3The average of an almost periodic matrix M (t) is given b y : M = lira .j../TF T ÷ooT.,'O

M (T) d'~

172

~" = ~ - 1

F

A ~= A + BI"~CIA + ABIQC 1 + B I P C I A B I ( ~ C 1

l

l

= PC1B2K2C2B1Q

r12 = PC1B2K2

F21 = K2C2B1Q

F22 = K 2

T h e a v e r a g e r can b e a s s i g n e d b y c h o o s i n g K1, a n d h e n c e Q, a n d K 2. I t is shown in

(RUN-85)

that an appropriate

trollable and observable.

choice of F c a n make t h e t r i p l e

Therefore,

(~,

u s i n g t h e r e s u l t s of ( B R A - 7 0 ) ,

B 1, C 1) con-

t h e m a t r i c e s L,

H, G a n d F ( w h e r e ~ i s of d i m e n s i o n ~ = rain (~)o-1, ~)c-1), ~)o(~c) b e i n g t h e o b s e r vability

(controllability)

index

of

(4.4.19))

can

be

chosen

such

that

the

system

( 4 . 4 . 2 0 ) is made a s y m p t o t i c a l l y s t a b l e . T h e n , u s i n g t h e r e s u l t s of (MEE-73) a n d

(BOG-61),

t h e main r e s u l t s h o w n in

(RUN-85) is t h a t ¢0 c a n b e s e l e c t e d so t h a t t h e s y s t e m ( 4 . 4 . 1 9 )

also b e c o m e s s t a b l e

for all 0 < c < E 0 a n d b y ( 4 . 4 . 1 6 ) so does t h e o r i g i n a l s y s t e m ( 4 . 4 . 1 5 ) . T h e following t h e o r e m s u m m a r i z e s t h e a b o v e d i s c u s s i o n T h e o r e m 4.5 ( R U N - 8 5 ) . A s s u m e t h a t ( 4 . 4 . 1 3 ) h o l d s . T h e n t h e r e e x i s t s a n ¢0 ) 0 and a c o n t r o l l e r of t h e form ( 4 . 4 . 1 4 )

t h a t e n s u r e s t h e a s y m p t o t i c s t a b i l i t y of t h e s y s t e m

( 4 . 1 . 3 ) for all 0 ( E-~ E 0. The subsequent

algorithm

(RUN-B5)

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

for t h e d e s i g n of the

controller (4.4.14). S t e p 1 : Choose

feedback matrices

controllability any Kl(t)

and

observability

any K2(t),

Step 2 : Design a linear,

and

K 2 ( ~t)

properties

(almost

K1 (!¢)

that achieve the desired any r and,

consequently,

works(POT-79)).

time i n v a r i a n t ,

dynamic f e e d b a c k c o n t r o l l e r for t h e a v e -

raged system (4.4.20). S t e p 3 : S u b s t i t u t e t h e p a r a m e t e r s of t h e c o n t r o l l e r from s t e p 2 i n t o t h e c o n t r o l l e r for t h e o r i g i n a l s y s t e m u s i n g ( 4 . 4 . 1 8 ) .

With c s u f f i c i e n t l y small, t h i s c o n t r o l l e r

e n s u r e s t h e a s y m p t o t i c s t a b i l i t y of ( 4 . 4 . 1 5 ) . Example 4.4 ( R U N - 8 5 ) . C o n s i d e r t h e following s y s t e m :

A =

173

°l

0 0

0 0

which h a s a s t r u c t u r a l l y

2

B1 :C

f i x e d mode of t y p e

1

(ii) at X0: 0. T h e s y s t e m is c e n t r a l l y

controllable a n d o b s e r v a b l e a n d C1B 2 = C2B1 = 1 ( c o n d i t i o n (4.4o13) is s a t i s f i e d ) . In t h i s c a s e , t h e s y s t e m ( 4 . 4 . 1 9 ) is g i v e n b y :

2]vc1 z iLcz iN

K2 : ~ - 1 A ~ z + [ B i B 2 ]2l qF "

"2 LC2]

LqK2 ~:

(4.4.21)

F { +GCIZ

Choose q = ¢~-sin

t E

' K2 =

"r+/~" B sin __t and therefore c

K 1 : q(l + CIBIq)-I

: 1'2"----~ac o s t g c

Since

::~-IA~=

[ 101 0 0

and t a k i n g a v e r a g e s in ( 4 . 4 . 2 1 ) , - a6

1

~0

1

=

0 0

0 0

: A,

0

the system (4.4.20) is given by :

2 - ~'a

1

0

0

y

0

a6

+ B I L C I g + BIH (4.4.22)

~"= F ~+ G CI :

Since the

triple (~,

B 1, C I) is controllable

and

observable

choose for example a = 6 = y = 1. The dynamic compensator order a (Vo = Vc = 3). Set the matrices :

for alla,6

and "¢,

for (4.4.22) must be of

174

oI

F = 0 then

(4.4.22)

The

,L=I

13 =

,

2

can be rewritten

compensator

i + j, -1,

h 2]

f2

I:l -

H = [h I

-1,

as :

-1

1

1-1

hI

h2

1

0

0

0

0

1

0

1

0

0

0

0

gl

fl

0

0

0

g2

0

F2

coefficients

are

I:]

chosen

to

have

the

closed-loop

eigenvalues

at

-2 :

fl = -3 +/5 1=-12

=- 0,764

/5

f2 = - 3 -

[-90/5"-" 1122)/5 10-5]

h i = g l =L

= - 5,236

= 0,106

[ 10,+0'7/,11~ The resulting

time-varying

0

1

1

0

system

_ 1 $2_2 E:2 c o s - t

6,850,

0

0

0

0

0

0

= 0,764

0

0

0,146

0

0

6)850

Therefore,

in

the original system

(4.4.23)

spite

existence

I:J

(4.4.23)

-5,236

0

is asymptotically

of the

has been

is

0,1o,6

0

Simulations show that

(4.4.15)

s t a b l e if ~ < 0 , 1 ( R U N - 8 5 ) .

of a d e c e n t r a l i z e d

stabilized by decentralized

unstable

vibrational

fixed

feedback

mode,

control.

175 Remark 4 . 4 . 1.

Although

the

above

control, i t s f e a t u r e (Kl(t)

stabilization

is t h a t

approach

combines

vibrational

t h e d e s i g n of the c o r r e c t i n g

and

feedback

vibrational control

action

a n d K 2 ( Et )) a n d t h e f e e d b a c k c o n t r o l law (L, H, G, F) can b e c o n d u c t e d

independently.

The f e e d b a c k c o n t r o l d e s i g n is d o n e for a much simpler l i n e a r time-

invariant system.

However,

in r e g a r d

to t h e o r i g i n a l s y s t e m , t h e v i b r a t i o n a l c o n t r o l

action t a k e s place t h r o u g h t h e f e e d b a c k . 2. In t h e g e n e r a l case of S ) 2 s t a t i o n s ,

(4.4.13)

m u s t b e r e p l a c e d b y t h e following

condition : 3 1 X ( ix( S s u c h t h a t GiBj $ 0 a n d CjBi 4: 0,

1,,( jx( S,

j $ i

(4.4.13')

and the c o n t r o l l e r ( 4 . 4 . 1 4 ) becomes : Pi = c* ~- Ki(_tc) y i ,

l..< i K s

vi

i<

pj = K} (t) Yi ¢=FO

¢+GOY

j /i

i 4s

(4.4.14')

i

vi = H (~ ¢+ L {~) Yi

Then T h e o r e m 4.5 r e m a i n s t r u e with ( 4 . 4 . 1 3 ' ) a n d ( 4 . 4 . 1 4 ' )

in place of ( 4 . 4 . 1 3 ) a n d

(4.4.14), respectively.

4.5. - CONCLUSION In t h i s c h a p t e r ,

it is s h o w n t h a t s y s t e m s with u n s t a b l e non s t r u c t u r a l l y

modes can b e s t a b i l i z e d b y u s i n g t i m e - v a r y i n g or n o n - l i n e a r f e e d b a c k which p r e s e r v e

the

feedback

structure

constraints.

The

use

fixed

c o n t r o l laws

of sample

and

hold

(WAN-82) ( O Z G - 8 5 ) , t h e a p p l i c a t i o n of piecewise c o n s t a n t o r sinusoTdal f e e d b a c k laws (AND-81b)

(PUR-82),

system p a r a m e t e r s

and

the

introduction

of

almost

periodic

vibrations

on

the

(TRA-85) or t h r o u g h t h e f e e d b a c k (P, UN-~5) a r e d i f f e r e n t s t a b i l i -

zing a p p r o a c h e s w h i c h e r e p r e s e n t e d in t h i s c h a p t e r .

176 Obviously,

t h e s a c r i f i c e of l i n e r e r i t y o r time i n v a r i e n c e of t h e c o n t r o l b r i n g s

d i f f i c u l t i e s in t h e a n a l y s i s of s u c h c o n t r o l l e d s y s t e m s ,

e v e n of small d i m e n s i o n a l i t y .

T i m e - v a r y i n g s y s t e m s a n a l y s i s m e t h o d s o r a v e r a g i n g s c h e m e s m u s t be u s e d . On a n o t h e r uneonvenient. constraints

hand

the

Consequently

i m p l e m e n t a t i o n of s u c h it

is

the

authors

control

believe

(if p o s s i b l e ) is a more r e a l i s t i c a p p r o a c h .

that

laws m a y be p r a c t i c a l l y relaxing

the

structure

T h i s a p p r o a c h is i n v e s t i g a t e d

in t h e following c h a p t e r a n d s e v e r a l a l g o r i t h m s a r e p r o v i d e d .

CHAPTER

CHOICE

OF FEEDBACK

5

CONTROL

STRUCTURE

TO

AVOID

FIXED MODES

5.1. - INTRODUCTION

Chapter origines.

III p o i n t e d

Fixed

modes

out that

arising

such as non structurally controllable under prevent

for t h e

design

since they systems.

of s u c h they

require

using

Moreover,

encountered resulting

when

from

(decentralization

laws

option,

which

for example),

allows

of a time-invariant, relaxed.

been

existence of structurally tion t r a n s f e r

between

Therefore,

to

Note

also

than

linear that

fixed modes of type

stations

relaxing

is

the

under

constraints

on

systems nonlinear

problems

way

can be

the total cost

law f o r w h i c h

(i) w h i c h a r i s e

chapter.

or

structural

only

of

methods

dimension

Consequently,

the

and are uncontrollable

the structural

and

specifications

the cost associated

control

this

small

some physical

laws.

constraints

the property

for time-varying

preserve

may be higher

and implementation have

control

values

(ii) r e m a i n

in t h e p r e v i o u s

for

tools

point of view,

such

traints

even

either

approaches

presented

analytical

from the practical implement

been

parameter

the structural

Different

difficulties

the

to t h e i r

of such fixed modes does not

[towever,

sacrified.

have

analysis

in

according

fixed modes of type

The existence

be

appropriate

trying

this

configurations

to t h e s y s t e m .

must

control

involve

classified

e c o n t r o l law s a t i s f y i n g

beheviour

o r of ] i n e a r i t y

Unfortunately,

special

constraints.

of f i n d i n g

which gives a satisfactory time-invariance

from

be

fixed modes or structurally

structural

the possibility

fixed modes can

to t h e d e s i g n

structure

to

cope

cons-

with

the

from a lack of informa-

structural

the control

constraints. seems

to b e

the

most c o n v e n i e n t

w a y to s o l v e t h e s t a b i l i z a t i o n o r p o l e p l a c e m e n t p r o b l e m in p r e s e n c e

of f i x e d m o d e s .

The

purpose

methods for the design are based

of this

chapter

of an appropriate

on t h e d i f f e r e n t

characterizations

is

feedback

to

present

the

different

control structure.

These

of fixed modes which were

available methods

presented

in

178 chapter

3 : one

can

be

more

appropriate

s i t u a t i o n we a r e d e a l i n g w i t h .

than

- t h e s y s t e m is p h y s i c a l l y p a r t i t i o n e d different

geographical

clear that

the

locations

decentralized

another

Roughly speaking,

of t h e

structure

in s e v e r a l

inputs

depending

and

w o u l d be

stations,

the

the

by

the

existence

of

of f e e d b a c k

fixed links

modes. between

The

optimality

two d i f f e r e n t

due

outputs.

the

type

of

f o r e x a m p l e to

In this

most appropriate.

will t h u s be to d e t e r m i n e t h e minimal i n f o r m a t i o n e x c h a n g e s

number

on

two t y p e s o f s i t u a t i o n s c a n o c c u r :

it is

purpose

between stations required

criterion stations

case,

Our

or

can

be

chosen

as the

cost

as

the

associated

w i t h t h e i m p l e m e n t a t i o n of t h e s e f e e d b a c k l i n k s .

either

-

present

the

respect

to t h e

the system does not reflect a prespecified

particularity

that

the

c o s t of a f e e d b a c k

c o s t of a local between

partitioning or the stations

feedback

two d i f f e r e n t

w a n t to d e t e r m i n e t h e minimal c o n t r o l s t r u c t u r e ( s )

is

not

neglectable

stations.

In this case,

with we

f o r w h i c h t h e s y s t e m h a s no fixed

m o d e s . T h e y will g e n e r a l l y a p p e a r w i t h o u t all t h e local f e e d b a c k s .

Note Indeed, costs

the

resulting

from

the

to t h e

local

feedbacks.

Therefore,

PRESPECIFIED

FEEDBACK

situation

is

more

all

the

methods

for the p a r t i c u l a r

general.

to z e r o the

which

will be

case.

CONSTRAINTS

- Preliminaries

Since t h e e x i s t e n c e of f i x e d m o d e s a r i s e s control structure,

the most natural

constraints

to

;

i.e.

introduce

x (t) = A x ( t )

f"

+

S Y: i=l

Yi(t) = Ci x ( t ) :

B = EB 1 .....

BS3

C' =Ec/ . . . . . % 3 u,{t)

= Eff{t

y'(t)

=

.....

Ey{(t) . . . . .

y~(t)']

from t h e

constraints

i m p o s e d on the

w a y to eliminate t h e m is to p a r t i a l l y r e l a x t h e s e

additional

C o n s i d e r the following s y s t e m ( C , A , B )

where

second

in t h e g e n e r a l f r a m e w o r k c a n also be u s e d

- RELAXING

5.2.1.

problem

f i r s t p r o b l e m c a n be f o r m u l a t e d in t h e s a m e w a y b y s e t t i n g

associated

presented

5.2.

that

the

information

exchanges

between

stations.

:

Bi u i ( t ) (i = 1 .

. . . .

S)

(5.2.1)

179 with x ( t )

~n and ui(t)

S m = [~1- mi'

p, mi , Yi(t)

S r = iE i,

ri

Rri,

and

i -- 1 . . . . .

S is t h e

S.

number

of c o n t r o l e n d

observation

stetions. If t h e

control

matrix K a p p e a r s

is

subject

to a d e c e n t r a l i z e d

with a block diagonal structure

ciated with a local f e e d b a c k

(K

l~d).

structure,

the

output

feedback

w h e r e e a c h d i a g o n a l b l o c k is a s s o -

Suppose

that

the set of decentralized

fixed

modes is n o t e m p t y ( A ( C , A , B , ~d ) # 0) ; t h e p r o b l e m is t h u s to d e t e r m i n e a n e w s e t fl* s u c h

t h a t f2d

C ~*

and A(C,A,B,~*)

=

0.

The

b e t w e e n some s t a t i o n j a n d some s t a t i o n i a p p e a r s nal block in t h e f e e d b a c k m a t r i x

additional

information

transfer

b y t h e a d j u n c t i o n o f an o f f - d i a g o -

:

f i.

.

.

.

•

"F"

K

C

.

fld

.

i

.

/

4- --_72 1t_ _J .

.

.

'

"

The new structure

"

P--

Cfl*

K

A (C,A,B ~d ) / /

'

A (C,A,B,~*)

is d e t e r m i n e d

with r e s p e c t

can be e i t h e r t h e n u m b e r of s u p p l e m e n t a r y

1

--

to an o p t i m 2 l i t y c r i t e r i o n w h i c h

feedback links between different

stations

or t h e i r a s s o c i a t e d c o s t ( s p e c i f i e d from p h y s i c a l c o n s i d e r a t i o n s ) .

5.2.2. - Wang a n d D a v i s o n p r o c e d u r e Given t h e s y s t e m

(5.2.1),

mining a c o n t r o l s t r u c t u r e

(WAN-78e)

Wang a n d D a v i s o n c o n s i d e r e d

t r a n s m i s s i o n c o s t . C o n s i d e r t h e following s e t o f m a t r i c e s I K(rij, i , j = l . . . . .

the problem of deter-

allowing the stabilization of the system and minimizing the

S) = {K/K = b l o c k {Kij} --

K . . C R m i x r j , r a n k K.. = r.. } 11 11 1]

:

KII

...

K. I 5 1

KS1

...

KSSJ

(5.2.2)

180 and

define

as

station j per

z.. the 2j

cost

of transmitting

unit of time. Each feedback

a time function cost resulting

5

The problem

Z

j=l

i:l

can thus

trices

i to

the transmission

of

hi C , A , B , K ( r i j ,

(5.2.3)

r.. z..

1]

p

be formulated

i,j=l .....

S)]

a s follows :

:~ (rij, i,j=l

rnin (mi,r j)

w h e r e f ~ - is t h e h a l f l e f t h a n d

°I(A)

law u i = Kii Yi r e q u i r e s

from station

S

Z

Min 0 ..< rij 4

Every

time function

w i t h r.. c o m p o n e n t s f r o m s t a t i o n j to s t a t i o n i . T h e t o t a l t r a n s m i s s i o n 1] from the implementation of u = K y, K K ( r i j , i,j--1 . . . . . S) i s :

~(rij, i,j=l ..... S) =

under

a scalar

c

..... S)

~-

(5.2.4)

side complex plane.

sub-matrix

K.. o f r a n k r.. c a n b e w r i t t e n a s t h e p r o d u c t o f t w o s u b m a i} Il : Kij = Lij Mij, w h e r e Lij is o f s i z e m i x rij a n d Mij i s o f s i z e rij x r j . If

= {~1 . . . . . kq}

is the set of unstable

is s a t i s f i e d if a n d o n l y if t h e r e

det

(k i I - A - B [ b l o c k

eigenvalues

of A,

e x i s t Lij a n d Mij s u c h t h a t

then

condition

(5.2.4)

:

(5.2.5)

{Lij Mij}] C)

~¢~i ¢ °i(A) is n o t i d e n t i c a l l y Condition

zero. (5.2.5)

mode with respect At t h i s apart

of the feedback

step,

iteratively

Wang a n d

the worse

is only a finite number be solved testing be

0)

means that

n o n e of t h e u n s t a b l e control u =

Davison

solutions

(WAN-78a)

of steps.

in c h o o s i n g Their

L.. M.. lJ 11

did

from the others.

of possibi]ites

in a f i n i t e n u m b e r

block

eigenvalues

not

y,

give

any

the choice of the

set

(rij,

i,j=l .....

S),

begining

by

lowest cost. This is illustrated Example 5.1.

Consider

by the following example

:

t h e f o l l o w i n g s y s t e m w i t h two s t a t i o n s

to set

that

minization problem

procedure

consists

all t h e p o s s i b i l i t i e s g i v e n b y t h e c h o i c e o f t h e c o n t r o l s t r u c t u r e and

criteria

Given the argument

rij ms, t h e

search

o f A i s a fixed

:

those

there can

c o a r s e l y in

( s o m e Kij can involving

the

181

0

0

0

0

yl:EO

[ 01

x t

0

l

0

0

ul

+

E°] 0

u2

i

o l]×

o o] Y2 =

1

0

x

end d e f i n e Z l l = z22 = O, z12 = 1 a n d z21 = 2 . 1. O b v i o u s l y ,

we c o n s i d e r

first the decentralized

feedback

represented

by the set

:

0 03_j

K 1 = K (rll

=

r22

=

1,

rl2 : r2l = 0)

[-kLl ;, ~_~2-1--I,. . . . .o. . . { i

The corresponding

F(p;

fixed polynomial of (c:,h,-,)

C , A , t 3 , K 1) = g . c . d

{det

oF<

,k..~R} lj

k32

is :

(pI - A - r~:c)}

= p

K£K I T h e r e f o r e w i t h local f e e d b a c k s ,

2. Now, c o n s i d e r

t h e s y s t c m h a s a f i x e d m o d e at t h e o r i g i n .

t h e c a s e in w h i c h we h a v e e t r a n s f e r

K2 = K ( r l l = r22 = r21 = 1, r l 2 = 0) :

and t h e c o s t is ~ ( r l l

= r 2 ] = r 2 2 = 1,

{

rl2

from station

1 to s t a t i o n

kll k21

0 0

0 0

k31

k32

k33

= 0) = 1. T h e

2 :

1 , k i j C R}

f i x e d p o l y n o m i a l is a l s o

equal to p ; K 2 i s n o t a n a d m i s s i b l e s o l u t i o n . 3, F i n a l l y ,

consider

t h e c a s e in w h i c h we t r a n s m i t

o b s e r v a t i o n a t s t a t i o n 2 to s t a t i o n

] :

a single linear combination of the

182 kll K 3 = K ( r l l =r22 = rl2=l, r21 = 0) = | k a l L 0

and the associated cost is

k12 k22 k32

k23

k,3l

kij

rank kI2

G R,

k22

k33

k23

( r l l = r21 = r22 = 1, r21 = 0) = 2.

The r a n k c o n s t r a i n t can be e x p r e s s e d by :

[,2,2 ,3kll [2] g

1

I1

where f l ' f2' g l and g2 are a r b i t r a r y real n u m b e r s .

I g.c.d

{det

Ii' gi' kij

The fixed polynomial is now :

P-flgl

-fig2

-kl 1 1

-I2g 1

p-f2g 2

-k21

-k33

p+2

-k32

}

= 1

i=1,2 such that the system has no fixed modes anymore. If we choose :

f l = f2 = k l l = 1 : gl = k21 = k33 = 0 ; g2 = k32 = - 1 all the poles of the system are located at - l . T h o u g h in the case of the example the solution is obtained in 3 s t e p s ,

it is

clear that the number of possibilities to be t e s t e d i n c r e a s e s rapidly with the dimensions of the system. In consequence~

this procedure cannot be used for large scale

systems which are effectively those requiring

structural constraints in the contro|

and minimization of the information transmission cost.

5 . 2 . 3 . - Armentano and Singh ' p r o c e d u r e (ARM-82) Armenteno and (5.2.1)

Singh p r o c e d u r e

called i n t e r c o n n e c t e d

For s u c h s y s t e m s ,

systems

is limited to the p a r t i c u l a r class of systems and s a t i s f y i n g

the specifications

in

(3.3.7).

t h e y g a v e a c h a r a c t e r i z a t i o n of fixed modes b a s e d on the block-

diagonal dominance p r o p e r t y (see § 3 . 3 . 3 . b - 2 - ,

Chapter III).

183 Given the system u = K y where

with

(3.3.7),

consider

matrices,

i c {1 . . . . .

of

Corollary

S} s u c h t h a t

[[ (Aii - k 0

t h e n we h a v e

3.3

to

(A

to t h e t w o c o n t r o l s t r u c t u r e s +

BKC),

K ~ ~F~

shows

Chapter

II).

~d and flF' the

that

there

exists

:

[I-' 4 ~!l[IAii +.

I)-1

:

(see § 2.2.2.,

c h ( C , A , B , ~d)

If k *0 i s a f i x e d m o d e w i t h r e s p e c t application

law o f t h e f o r m :

and ~d ~- ~F

K ~ 12F

9d is t h e s e t o f b l o c k - d i a g o n a l

A(C,A,B,~ F)

a feedbeck

(5.2.5)

B i Kij Cj II

j,q V

K.. E R m i x r i , n

Therefore, (i,j=l . . . . . S ) . = "S =

~ K.. ~ R m i x r j 1]

as a particular

Define the sets

case,

(5.2.5)

is a l s o v e r i f i e d

f o r Kii :

0, Kij -- 0,

:

{i ~ {1 .....

S} / ( 3 . 3 . 1 0 )

{i6{1

S} [ ( 5 . 2 . 5 )

.....

a n d i$j.

is satisfied)} is satisfied)}

then w h e h a v e S c: S .

Consider know t h a t

that k 0 is a decentralized

if k 0 r e m a i n s

fixed mode.

a fixed mode with respect

From the above

to a n o t h e r

discussion,

control structure,

we the

~d

K.. 's c a n b e s e t to z e r o f o r i 6 S w i t h o u t a f f e c t i n g t h e s i t u a t i o n . ~ t o r e o v e r , ( 5 . 2 . 5 ) 1j is s a t i s f i e d f o r a s u b s e t o f i n d i c e s of ,~. In o r d e r to e l i m i n a t e t h e f i x e d m o d e , A r mentano and

Singh propose

indices i ~ ~.

s u b s e t o f {1 . . . . . fixed m o d e ,

S}.

corresponding

: Consider

0

if the

system

can be used

has

replacing

more than

one

decentraIized

~ b y t h e u n i o n o f t h e s e t s ~i

the following system l

0

0

0

0

0

0

l

0

0

l

x22/

o

o

o

o

X3l

0

0 ~J 0

i

x32]

o

o :o

o

x12 / =

,' 0

to t h e

can be applied only when ~ is a proper

to e a c h f i x e d m o d e k i .

-"711 ] x21 /

the off diagonal submatrices

this strategy

Note a l s o t h a t

the same approach

corresponding Example 5 . 2

to a d d

It is c l e a r t h a t

0

0

Xll Ix 12

J 0 l 0

0 1

I

0

0

x22

!t

0

1

x3i

', o

o

x32

I

i 0 i 0 J i J 0 ! 0

o-

x21

*

o

:l,,O

0

: 0

! l

184

The decentralized

and,

control laws are of the form

w h e n a p p l i e d to t h e s y s t e m ,

:

the following closed-loop

m a t r i x is o b t a i n e d

:

I

A+BK

0

I

kll

k12

1

0

0

0

I 0 0 I I I 0 0 $. . . . . . . . . . . . . . I i I I

=

0

0

0

0

The system

has thus

I I ~I

0

l

k21

k22

0

1

0

0

a decentralized

Using the following matricial norm

IIAI]

0 0

I I I I

0

1

0

0

I I t ,

0

l

k31

k32

f i x e d m o d e a t t h e o r i g i n e X 0 = O.

:

£1aij l

= m a~x

= 0 satisfies

I 0 I I I 0 e . . . . . . . . . . .

j

(3.3.10)

for S =

[] ~i~ -'ll-I

2,3

:

• k,al max

It c a n e a s i l y b e v e r i f i e d

:

-kil

i

,:l

i:2,3

+ -1 kil ) ,

t h a t ~ 0 = 0 r e m a i n s a s a f i x e d m o d e f o r K* w h i l e it is

e I i m i n a t e d f o r K** :

K11 0

K*=

Though that

K22

K23

K31

K32

K33

the

it is b a s e d

01

K21

efficiency on

of the

a sufficient

[11 KIaK,3 K22 0

above

(and

K**=

not

procedure necessary}

cannot

0

K33

be

condition

contested, for

the

the absence

fact of

185

fixed m o d e s m a k e s rows

(indices

i)

it r o u g h . where

the

Indeed,

it o n l y allows t h e

addition

of

elimination of the fixed modes. or on t h e

number

ding example, suboptimal. subset

The

o f {1 . . . . .

off-diagonal that

can

by

any

using

this

be

concluded

I ) , in w h i c h c a s e n o n e i n f o r m a t i o n

tion is i l l u s t r a t e d

the following system

0

I 0

d ]--o I

Lx2.1L o

o

, o

Ly2JL o

o

i

With a d e c e n t r a l i z e d

the

on t h e i n d i c e s j

If we c o n s i d e r

O in K**,

the

procedure

is

the prece-

fixed mode is thus

obviously

c a s e f o r w h i c h ~ is a p r o p e r

after

the

evaluation

can be drawn

out.

of

(3.3.10)

This situa-

feedback

0

Xl

1

-

:

Ul 0 IFt /oi ..... [°Io

Lol

o

o

c o n t r o l law K = b l o c k

i.jLo=,

(Kll,K22),

the closed-loop

: 1

0

0

1

kll

kl2

0

0

1

0

k21

1

0

0

k22

0

A+BKC:

and the system

block-

certifies

by the following example :

Consider

m a t r i x is g i v e n b y

of the

matrices

information

submatriees.

i s l i m i t e d to t h e

only

determination

off-diagonal

if K13 r e m a i n s

obtained

its application

S} ( w h i c h

for e a c h i = l , . . . , S

Example 5 . 3 .

solution

Moreover,

the

It d o e s n o t p r o v i d e

of necessary

it i s e a s i l y v e r i f i e d

also e l i m i n a t e d .

all

has a fixed mode at the origin.

Unfortunately,

since

:

186

IIAII-III-I

1

'

m~×{l,

<= l

I. I 'kl2

IIA22 1[1 i

+ kij)} k12

l

=,

m a x {__1_1 k22

= {1,2}

5.2.4.

is n o t a p r o p e r

- Approach

based

The procedure fixed modes using

subset

of

{1,2} a n d t h e p r o c e d u r e

on the system

proposed

their

, (1 + k2------L)} k22

in t h i s

sensitivity,

modes sensitivity paragraph

cannot be applied.

(TAR-84)

is b a s e d

which was presented

(TAR-85)

on t h e c h a r a c t e r i z a t i o n in Paragraph

2.4.2,

of

chap-

t e r II a n d a p p l i e d to t h e i r e v a l u a t i o n . Given

a prespecified

closed-loop

eigenvalue

structural

which

constraint

remains

matrix

(see

on

insensible

Definition

the

to

entries

of the

feedback

2.7).

sitivity

matrix

SK r o f a m o d e )~r ( s e e D e f i n i t i o n 2 . 8 ) ,

control,

any

a fixed

variations

From the

of

definition

it w a s s t a t e d

m o d e is a the

nonzero

of the

sen-

t h a t ;~r i s a fixed

m o d e if a n d o n l y if SK subjected nonzero

to

is i d e n t i c a l l y z e r o ( s e e T h e o r e m 2 . 6 ) . If t h e c o n t r o l is not r structural constraint (i.e., full f e e d b a c k is a l l o w e d ) , t h e n the

any

entries

of the

sensitivity

matrix

matrix which may affect the corresponding Theorem

5.1

feedback

pattern

the feedback

K~ r and

SK

indicate the elements of the feedback r m o d e ~r" T h e f o l l o w i n g t h e o r e m c o m e s :

(TAR-85).

;~ i s n o t a f i x e d m o d e o f t h e s y s t e m w i t h r e s p e c t to a r K ~ flF if e n d o n l y if a t l e a s t o n e e l e m e n t o f t h e s e t K;~r a p p e a r s in

m a t r i x K , w h e r e KAr is d e f i n e d b y

:

= {kij ! ( S k r ) i j ~ 0 }

(Skr)ij This

are the entries theorem

feedback

links

Consider

that

provides

which

performs

the system

l i n k s K* m u s t s a t i s f y

of the sensitivity

:

has

a

simple the

way

m a t r i x SK r . to

elimination

q fixed modes,

determine of

the

the

fixed

set modes

of supplementary of the

then the set of supplementary

system. feedback

187 Card where

(K*

n Kxk)

K Xk = { k i j

I

(SKk)ij

evaluated with respect Remark 5.1.

>i 1

(k=l

. . . . .

¢ 0} a n d

q)

SK k i s t h e

sensitivity

matrix

of the

m o d e Xk

to a f u l l f e e d b a c k .

We r e c a l l t h a t t h e e v a l u a t i o n

is d i f f i c u l t to p e r f o r m

(see

§ 2.4.2).

of the sensitivity

This approach

matrix of a multiple mode

should

thus

be limited

to s y s -

tems w i t h s i m p l e m o d e s . Note t h a t sets KXK.

K* c a n

Indeed,

(K* n K,•)

)

generally

if K)d

1 so that

c

be determined

KAj , i#j a n d

Card

,,Fhj c a n b e s u p p r e s s e d ,

without

taking

into

account

all t h e

(K* n K)i ) >j1, it follows t h a t C a r d q ~ q sets

need

to b e

considered

i n s t e a d oF q . T h e p r o b l e m is t h u s Problem 5 . 1 . Card

the following

F i n d K* s u c h t h a t (K* n KX ) >/ I

:

: (i=l . . . . .

~ < q)

1

If we c o n s i d e r

that

is c l e a r t h a t o u r i n t e r e s t the t o t a l c o s t r e s u l t i n g

Consider

K• .

the

a different

cost

is to d e t e r m i n e from the feedback

set

of

elements

is a s s o c i a t e d

with every

feedback

K*. solution of Problem 5.1,

link,

it

and minimizing

l i n k s i n v o l v e d in K * .

constituted

by

the

union

of

all

retained

sets

:

l

Z =

q E K~,i i=l

Card

Z = z ~<

Remark 5.2. an i n p u t .

Every

So f a r ,

m

x r

element of Z represents the notation

the input i. For convenience, Associate

a c o s t c i ~/

lowing b o o l e a n v e c t o r

:

a feedback

link between

an

output

w a s k.. f o r a f e e d b a c k l i n k b e t w e e n t h e o u t p u t 11 t h e e l e m e n t s o f Z a r e r e n a m e d zi, (i--1 . . . . , z ) .

0 with

every

feedback

link

zi of Z a n d

define

and j and

the

fol-

188 W -- (w I . . . . . with

= ~ 1

Wr)' if z i

£

K*

Wi

L0

otherwise

Define also the following matrix

: /,

L = (1..)i=l . . . . . ~ *J j=l, . . . . , z

The problem tem has gram

with

if

1.. = l l U

to

of finding

z. 1

K )~i

otherwise

the minimum information

no fixed modes can thus

be formulated

pattern

K* s u c h

that

the sys-

by the following boolean linear pro-

: Z

Problem 5.2.

min

Y. c. w_ j-I l J

Z under

J=ZI

lij wj ~ 1

Now,

(i=l . . . . .

it i s i n t e r e s t i n g

well-known

"covering

terms of graphs Consider

set

9)

to n o t i c e

problem"

of

that

Problem

graph

5.2

theory

appears

which

can

in t h e be

form of the

reformulated

in

a s follows : the unidirectional

Z = {z 1 . . . . . Kx = {KxI

graph

G = [ Z,Ks,

h]

where

:

z z} .....

K q}

: set of parts

o f Z.

A : u n i v o c a p p l i c a t i o n f r o m K)~ to Z A ( z i) = { K x

/ zieK,

j}

(i=1 . . . . .

z).

J end the costs ci associated P r o b l e m 5 . 2 is t h u s

Prohlem 5.3.

Find

Hc

minimizing This

g ziEH

problem

in t h e l i t e r a t u r e

:

with each vertex brought

Z /

z i,

(i=l . . . . .

z).

b e c k to t h e f o l l o w i n g c o v e r i n g

set problem

:

u k (z i) = K)~ zi¢.. H

c. 1

can be solved

by

using

any

of t h e

following algorithms

existing

189 - Method of the covering - Branch

set

(KAU-68)

and Bound procedure

(ROY-70)

(KAU-68)

(ROY-70)

- Gomory's method (KAU-68) - Thiriezls

Example 5 . 4 .

method

Consider

modes : A= {kl=l,k

(THI-71)

again

the

The

2=2}.

example

associated

3.12

in

which

sensitivity

the

system

matrices

with

has

two

respect

Ioo-1131Ioool

fixed to

full

feedback are given by

SK l =

Then, we have

0

0

-I/3

0

0

0

SK 2 :

1/3

0

1/6

0

0

0

:

KXl = { k 1 3 , k23}

K~2 = { k 2 1 , k 2 3 } and

Z = {k]3, k23,

k21} = { Z l ,

z 2 , z 3}

L{: °1 In this example,

1. All t h e

costs

are

number of feedback

the boolean linear program

equal links)

to

1,

to s o l v e t a k e s t h e f o l l o w i n g f o r m :

c. = 1 ( i = 1 , 2 , 3 ) 1

(minimization with respect

to t h e

:

rain (w 1 + w 2 + w 3) w 1 + w 2 >/ 1 under

wi =If w 2 + w 3 )/

The solution

(i = 1 , 2 , 3 )

1

i s w = (0

1 0)'

and

the

addition

of k23 is sufficient

fixed modes.

2. T h e c o s t s a r e g i v e n b y c I = 1, c 2 = 3, c 3 = 2 :

to e l i m i n a t e t h e

190

min (w 1 + 3w 2 + 2w 3) w 1 + w 2 >/ 1 under

w. = 1

w 2 + w3 ) 1

{:

(i = 1 , 2 , 3 )

I n t h i s c a s e t h e p r o g r a m g i v e s two s o l u t i o n s : W = ( 1 0 1 )' corresponding

to t h e e l e m e n t s k13 a n d k21

IV = ( 0 1 0 ) ' c o r r e s p o n d i n g

to t h e e l e m e n t k23

Remark

5.3.

It is i n t e r e s t i n g

to n o t i c e t h a t

general

case for which a prespecified

minimal f e e d b a c k c o n t r o l s t r u c t u r e for every

this approach

structure

can thus

c a n also b e u s e d

in t h e

f o r t h e c o n t r o l is n o t i m p o s e d .

he obtained

by evaluating

The

the sets K

mode o f t h e s y s t e m a n d a p p l y i n g t h e s a m e o p t i m i z a t i o n p r o c e d u r e .

5.2.5 - Specified approach

for structurally

fixed modes of type

(i)

(TAR-B5)

(TRA-

84b) This type

paragraph

concerns

(i) a n d c h a r a c t e r i z e s

to a p r e s p e c i f i e d

only

the

feedback

pattern

in o r d e r

is b a s e d on t h e a l g e b r a i c c h a r a c t e r i z a t i o n a n d Siljak ( S E Z - 8 1 a )

5.2.5a.

-

Use

Sezer

of

and

systems

with

structurally

fixed

modes

of

t h e s e t of s u f f i c i e n t f e e d b a c k l i n k s w h i c h m u s t b e a d d e d to e l i m i n a t e t h e m .

This

of fixed modes of type

characterization

(i) g i v e n b y S e z e r

(see § 3.5.32).

S e z e r a n d Siljak c h a r a c t e r i z a t i o n Siljak

(SEZ-Bla)

system with structurally

showed

that

fixed modes of type

the

state

space

form :

:

+

B2 c( B3

L A31 C I

A32

0

I

a

0] X

representation

of a

(i) c a n b e p u t in t h e f o l l o w i n g s p e c i a l

I

I 'l I

°][] 0

6~

Uct UB

(5.2.6)

191 where the control and observation

stations

are partitioned

in two a g g r e g a t e d

stations

and B • The fixed modes with respect K = block-diag.

are the eigenvalues

(Ka ,

by the other

one,

observable

by

the

A22. T h e s e

c~ S i n c e t h e

fixed

modes

aggregated

whose addition is sufficient

are

system

station

B.

Lo If matrix

the at

reduced Theorem

l

K

i

K

-21

structure

of

station B are

(see Chapter

stations,

is s u p p o s e d by

the

Consequently,

III) a r e s i m u l -

hereB , and inobservable to b e g l o b a l l y aggregated the

set

controllable

station a

of

to e l i m i n a t e t h e f i x e d m o d e s i s g i v e n b y

matrix becomes

FK

modes

controllable

KaB = { k i j / ij s u c h t h a t u i c U c ~ a n d and the feedback

•

(5.2.7)

b y o n e of t h e a g g r e g a t e d

here

the

pattern

Kfl)

of the submatrix

taneously uncontrollable and observable,

to t h e f e e d b a c k

feedback

and links

=

yj c YB}

(5.2.8)

:

l

(5.2.9)

BJ

the

taken

control into

matrix

account,

at

station a and

it c o m e s t h a t

the

of

the

sufficient

observation set

can be

a s follows : 5.2.

Given

t h e s e t of s u f f i c i e n t

the

system

supplementary

K s u f = {ki] / i E

(5.2.6)

with

structurally

links is given by

(i),

which are not identically

zero

I, ] C - J }

w h e r e I (J) i s t h e s e t o f i n d i c e s o f t h e c o l u m n s ( r o w s ) in t h e m a t r i c e s

fixed modes of type

:

B* (C*), w i t h

:

c~

192 Remark 5.4.

When t h e c o n t r o l

(5.2.7)

is a p p l i e d to t h e s y s t e m ( 5 . 2 . 6 ) ,

t h e closed-

loop d y n a m i c m a t r i x t a k e s t h e following form :

D =

_

A l l + B2a A32 . . . . . . .

[

which has the

DzDIII

,

Ca C1

0

ri I

a

3 C1 + B B

+B3c~ K

'=

Ka

l

K B CB

0

A22

~

-]

/

0

- ", - - ;~ - - - ~FB-3 .... 3- -,

DI2D22]

(5.2.10)

same b l o c k - t r i a n g u l a r

form as t h e o p e n - l o o p

dynamic matrix

A and

w h e r e t h e b l o c k A22 is n o t a f f e c t e d b y t h e c o n t r o l : it r e s u l t s t h a t t h e e i g e n v a l u e s of A22 a r e fixed m o d e s . It is c l e a r

that

the

fixed modes

can be e l i m i n a t e d b y

which d e s t r o y s t h e b l o c k - t r i a n g u l a r s t r u c t u r e Now,

consider

the

feedback

any

control

feedback

of D b y a f f e c t i n g t h e b l o c k D]2.

matrix

K' in

K c2!B ~.B.s.

1 K ~ ~

(5.2.9).

The

closed-loop

dynamic

m a t r i x is :

/

~_

cI ~B_B

I

a

aB

L

a

czB B I c~ c~B

F~t

K

~ B' ' ~

F-~2-~--cl-J~J"K B

a

"":

aB

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

c2-'Sr

E

B I cz q~B

c 3] B .I Ir -o-' ~

~ -I Bj

L D'21

'! D'12] - - - IF %2

J

with :

where

it a p p e a r s

that

the

block-triangular

structure

has been

destroyed.

KaB is

t h e r e f o r e a s u f f i c i e n t s e t of f e e d b a c k l i n k s to eliminate t h e f i x e d m o d e s . To show t h a t KRB can b e r e d u c e d to K s u f , n o t e t h a t D'12 can b e w r i t t e n as :

193

=

IB:]

:

+ 13-

C*

(5.2.11) D'I2

where

:

DI2

r. i,j

+

(bi) *

kij

(bl) * is t h e i - t h column of B* and

The e x p r e s s i o n

(5,2.11)

shows

that

(c.)* I

(c.)* is t h e j - t h row of C* and kij

if (hi) *J o r

does not a f f e c t D'I2 and can be e l i m i n a t e d .

(c~)* a r e i d e n t i c a l l y

zero,

then

KcxB. ki]

T h e r e m a i n i n g kij~s a r e t h o s e s p e c i f i e d in

Ksu f" Remark 5.5. 1. Note t h a t the s e t Ksu f is n o t e m p t y . I n d e e d , s i n c e t h e s y s t e m is globally~ c o n trollable a n d o b s e r v a b l e , the r e a c h a b i l i t y c o n d i t i o n s impose t h a t B 1 # 0 and C~^g 0. C~

2. If t h e r e

is no p r e s p e c i f i e d

control

proach t h a t t h e f e e d b a c k s t r u c t u r e

K"

0

,,

KI3~

tt

by T h e o r e m 5.2 a n d K

Ct

it can be s h o w n

by t h e same a p -

:

K B

z I

allows to a v o i d s t r u c t u r a l l y

KB

structure,

0

f i x e d modes of t y p e

(i).

In t h i s s t r u c t u r e ,

K ~B is g i v e n

by :

K•a = {kij / ij s u c h t h a t u i , z U• and yj ~ Y } can be i d e n t i c a l l y z e r o in the p a r t i c u l a r case

for which

U

c o n t r o l s t h e whole

Ct

space a n d YI3 o b s e r v e s

t h e whole s p a c e .

Example 5.5.

the Example 3.12 w h e r e a55=2 is c h a n g e d b y 4 (which a v o i d s

Consider

the e x i s t e n c e of a non s t r u c t u r a l l y matrix

fixed mode at 2).

•

p =

f 0l

0¢)

00

0¢)

01 t

0

0

0

l

0

0

0

1

0

0

0

l

0

O

0

Given t h e following p e r m u t a t i o n

194 the system takes

the form :

0

1

0

1

/4

0

0

0

2

[

I

I i

P'AP =

I I

/ °

0

-o---o---/--~ 5-~_ 0

0

0

['0

0

I

yiJl Y3

and

the

0

system

decentralized

I 0

I

O]

o o" , Io ,

I

o/~

0

I

l

0 I I

From Theorem k23}.

5.2,

Then

k22,

both

guaranty

5.2.5b.

the

Sezer

k23 /

0

k33j

of fixed

of feedback

the aggregated

the and

: I = 0.2}

k22

- Use of the sensitivity

way when

a n d J={3} a n d ,

state Siljak'

paragraph space

k31

to t h e

consequently

Ksu f =

modes

characterization stations showed

that

representation

characterization.

This

be applied.

of structurally the

can

then

Ksu f can be obtained

of the system

(i)

Indeed,

choice

be

made

with

to

Unfortunately,

directly

the characterization

the

~3

a and8

f i x e d m o d e s of t y p e

cannot

and

k32

links for example.

with structurally

§ 3.5.4).

at 1 w i t h r e s p e c t

0

procedure (see

(i)

k33]

absence

The preceding by

f i x e d m o d e of t y p e

we o b t a i n t h e s e t s

to t h e n u m b e r

determine

Y13

:

K '=K ~K suf

respect

B

:

K = block Ekl],

{k13,

°/

c~

a structurally

control

°

I--°--- °- L-° .I

3

has

u3

k °~-~° :J21 u u

I

I

u2

p'B= 1_o___2____o_]

i

0

uI

do n o t a p p e a r difficulty

fixed modes based

structure]

in

sensitivity

has the

in

the form general

e very (5.2.6) case,

in t h i s f o r m a n d

can bc

encompassed

on t h e i r s e n s i t i v i t y

matrix

(see

simple

Definition

given systems

the above by

using

(TAR-84) 3.9)

allows

195 the determination result ; i.e.

the

the eigenvalues thoses

of the entries entries

of the dynamic matrix

o f t h e b l o c k A22 i n

o f A22 a r e u n s e n s i b l e

belonging

to

A22 i t s e l f .

U a i f it r e a c h e s

(5.2.6).

Taking

X2 ant

This

variations

Consequently,

fixed modes can be determinated. gated station

to a n y

A from which the fixed modes

the

states

into account

that

that

yj b e l o n g s

comes from the fact that

of the parameters

except

to

X2 corresponding

to

ui belongs

aggre-

to t h e

to t h e a g g e g a t e d

station

the Y~ if

it c a n b e r e a c h e d b y X2, we c a n d e t e r m i n e t h e " m i n i m a l " a g g r e g a t e d s t a t i o n s U m a n d m YB b y u s i n g t h e t e a c h a b i l i t y m a t r i x R ( s e e § 1 . 2 ) o f t h e s y s t e m w h i c h h a s t h e following f o r m :

R =

The reason YB' b u t control

rather

so-called

the

set

of the

loop.

This approach

is r e d u c e d Um a

0

U

G

H

0

Y

(5.2.12)

the aggregated

may belong teachability

fixed

gives thus

a n d YB m

1. C o n s i d e r

2.

0

by

to U cx (YB) patterns

applying

modes

stations

stations, without

of the Theorem

s i n c e it g u a r a n t i e s

a better

themselves,

U m¢~ a n d

Ua and

YB'm is t h a t

reaching

(being

other

state

5.2

with

that

X2 is i n v o l v e d

solution since the number

a

rea-

variables. Nero m U a a n d Y• is

of feedback

in a links

(K'su f ¢ Ksuf).

Algorithm 5.1

PSI = {~" i '

0

K'su f obtained

s u f f i c i e n t to e l i m i n a t e t h e

Y

"minimal" aggregated

variabIe

X2 because

vertheless,

U

w h y we do n o t o b t a i n the

(observation)

ched by)

X

can be determined

:

(TAR-84).

the set of structurally (i=l . . . . .

Determine

by the following algorithm

the

fixed modes of type

(i) of t h e s y s t e m s

(C,A,B)

:

r ) .} structural

sensitivity

matrix

corresponding

to

the

set

of

fixed

m o d e s AS 1 : SS = SS 1 + . . . where

SS i is t h e

+ SS i + . . .

structural

+ SS r

sensitivity

matrix of the

m o d e Xi

AS1 and

"+'

denotes

t h e "logic OR" o p e r a t o r .

3. D e t e r m i n e t h e s e t of s t a t e v a r i a b l e s

x i ~ X 2 if t h e r e

X2 corresponding

exists at least one nonzero entry

to t h e f i x e d m o d e s

:

in t h e r o w o r c o l u m n i o f SS

196 4. Determine the r e a c h a b i l i t y matrix of the system ( C , A , B ) 5. umct -- {uj

(see § 1.2).

/ t h e r e e x i s t s i s u c h t h a t x i ~ X 2 and fij = 1}

6. Y$m = ( y j / t h e r e e x i s t s i such t h a t x i K X 2 and gij = 1} 7. The set of s u f f i c i e n t s u p p l e m e n t a r y links is g i v e n by : K' s u f = {k i.J ! u i £ Example 5.6 : C o n s i d e r

Uc~ m and Y~I ~Ym~} again

the

Example 3.12 with the

same modification

as in

Example 5.5. The system has a s t r u c t u r a l l y fixed mode of t y p e (i) at X1=1.

1"/~Sl = {~1 = 1} 2. T h e structural sensitivity matrix corresponding to ~ 1=1 is :

Ii ss(~=

3.

1) =

00

00

00

00

0

1

0

0

0

0

0

0

0

0

0

0

x 2 = {x 3 }

4. In the r e a c h a b i l i t y matrix the row and column c o r r e s p o n d i n g are

to x 3 in F and G

:

u1 x 3 ['0

u2

u3

l

0 "1-4--- 3-rd row of F J

Y2 I °0 Yl Y3

1

L_ 3-rd column of G 5. Um ={u2} C~ m

6. Y B

_--

{Y3 }

7. KIsu f ={ k23 } . This leads to the following feedback s t r u c t u r e s

;

197

kll K' =

Remark 5.6.

0

0

0

k22

k23

0

0

k33

K"=

I

O 0

0 0

0 1 k23

k31

k32

0

Note t h a t if one w a n t s to d e t e r m i n e t h e a g g r e g a t e d

s t a t i o n s Uct a n d YB

( i n s t e a d of ( U ~ a n d Y ~ ) , t h i s c a n be p e r f o r m e d b y r e p l a c i n g t h e t e a c h a b i l i t y m a t r i x of

the

open-loop

(C,A+BKC,B),

system

(C,A,B)

by

the

one

of

the

closed-loop

w h e r e K is t a k e n as t h e p r e s p e c i f i e d c o n t r o l s t r u c t u r e ,

system

and applying

then s t e p s 5 a n d 6 of t h e a l g o r i t h m . We w a n t

to p o i n t o u t

that

t h e a d d i t i o n of t h e

feedback

links

determined

Ksu f ( K ' s u f) p r o v i d e s a s u f f i c i e n t c o n d i t i o n to eliminate t h e s t r u c t u r a l l y of t y p e

(i).

However,

the a b o v e a p p r o a c h

some of them may b e r e d u n d a n t

f i x e d modes

and therefore unnecessary

does n o t give a n y i n f o r m a t i o n at t h i s p u r p o s e .

in -

The interest of

the p r o c e d u r e can be viewed in t h e e a s y way Ksu f is d e t e r m i n e d once t h e s y s t e m is p u t in form ( 5 . 2 . 6 ) . 5. Z. 6. - C o n c l u d i n g r e m a r k s This paragraph

presents

different approaches

to eliminate fixed modes b a s e d

on the idea t h a t t h e s t r u c t u r a l c o n s t r a i n t s must b e r e l a x e d . Wang a n d Davison a p p r o a c h tems b e c a u s e

of t h e

obtaining the solution. connected systems.

high

(WAN-78a) is n o t c o n v e n i e n t for l a r g e s c a l e s y s -

number

of p o s s i b i l i t i e s

which

Armentano and Singh approach

must

be

checked

before

(ARM-82) is limited to i n t e r -

It p r o v i d e s o n l y a r o u g h s o l u t i o n in t h e s e n s e t h a t t h e s o l u t i o n

is b a s e d on a s u f f i c i e n t c o n d i t i o n to eliminate fixed modes a n d is t h e r e f o r e timal. T h i s is also t h e c a s e of t h e p r o c e d u r e p r o p o s e d i n p a r a g r a p h

subop-

5 . 2 . 5 for s t r u c -

turally f i x e d modes of t y p e ( i ) . T h e o n l y a p p r o a c h i n c l u d i n g a r e e l o p t i m i z a t i o n p r o c e d u r e is t h e one b a s e d on t h e mode s e n s i t i v i t y a n d p r e s e n t e d (TAR-84).

B u t s i n c e it r e q u i r e s

in p a r a g r a p h

t h e c a l c u l a t i o n of s e n s i t i v i t y m a t r i c e s ,

5.2.4

it can b e

applied only w h e n t h e s y s t e m h a s simple modes.

5.3. - C H O I C E

5.3.1.

-

OF MINIMAL

CONTROL

STRUCTURES

Preliminaries

The

approaches

presented

in this section deal with systems

for which

a pro-

198

specified control structure when

no

partitioning

of

is n o t a p r i o r i a d v a n t a g e o u s . the

(like g e o g r a p h i c a l d i s t a n c e )

input

and

output

This situation occurs

arises

from physical

either

considerations

o r w h e n t h e c o s t s a s s o c i a t e d w i t h local f e e d b a c k s

a r e in

t h e same r a n g e a s t h o s e a s s o c i a t e d w i t h f e e d b a c k l i n k s b e t w e e n d i f f e r e n t s t a t i o n s .

In this cases, the

system

has

t h e p r o b l e m is t h u s to d e t e r m i n e t h e f e e d b a c k p a t t e r n

no fixed

modes

(i.e.,

d y n a m i c c o n t r o l law in a c c o r d a n c e cost criterion

based

such

that pole a s s i g n m e n t

with the specified

on t h e n u m b e r

for which

is p o s s i b l e w i t h a

structure(s))

and minimizing a

of f e e d b a c k l i n k s o r t h e s u m o f t h e i r a s s o c i a t e d

costs.

As w a s p o i n t e d o u t in t h e g e n e r a l i n t r o d u c t i o n of t h i s c h a p t e r , more g e n e r a l

than

problem

be

can

the one stated

formulated

in

the

with a prespecified same

way

by

structure.

setting

to

t h i s p r o b l e m is

Indeed,

zero

the

this

costs

latter o f the

f e e d b a c k l i n k s w h i c h a r e i n v o l v e d in t h e initial s t r u c t u r e .

5.3.2.

- Senning's

Senning's terizations

approach

(SEN-79)

a p p r o a c h is t h e o n l y o n e w h i c h is n o t b a s e d on o n e o f t h e

of f i x e d m o d e s g i v e n in C h a p t e r

t h e f r a m e w o r k of o p t i m a l c o n t r o l t h e o r y

Consider partitioned

the

class

in s e v e r a l

of

systems

stations

and

B. T h e p r o b l e m is r a t h e r

charac-

c o n s i d e r e d in

for linear systems with a quadratic criterion.

in

(5.2.1)

assume that

where

the

no f e e d b a c k

input pattern

and

output

are

seems a priori

advantageous.

T h e p r o b l e m is f o r m u l a t e d in t e r m s o f t h e d e t e r m i n a t i o n o f a " f e a s i b l y

decentralized"

control.

Definition

(SEN-79).

5.1

the system is stebitizable

A control

structure

is s a i d

with this control structure

to b e f e a s i b l y d e c e n t r a l i z e d and the

if

c o s t o f i n f o r m a t i o n is

minimal.

T h e p r o b l e m is s t a t e d

in

such

a way

that

two

problems

are

solved

simulta-

neously :

- the

classic

parametric

optimization

problem

based

on

the

traditional

quadratic

criterion for linear s y s t e m s . -

the determination of an optimal control s t r u c t u r e

with respect

to a c r i t e r i o n t a k i n g

i n t o a c c o u n t t h e p a r t i t i o n i n g of t h e s y s t e m a n d t h e c o s t s of t h e f e e d b a c k l i n k s .

The solution provides a feasibly decentralized control in the form :

199

S ui = Kii Yi + j Z1

K ij

yj

(5.3.1)

i=l,...,S

jli

The

extended

E.O.C. with

ft

:

0

second

first

S X li=l

(i=l .....

S).

one

term

goes

part

of

in the

into

local i n f o r m a t i o n . the

measure

control

S

the norm

Ki

2

(5.3.2)

performance by

as the

index

a weighted vector

appropriated

from station

m

:

measure

function

scalars

j to station

wij,

(P.I)

norm

while

the

of the

non-

of the

non-

penalizing

more

or

i.

= II K, ~, v II

(5.3.3)

j/i

of a matrix

II~[I ~ = t r and with

considerations

~" u"ll:ll gi ~"J K,,, II

j/i where

S X i--I

classic

is defined by

as follows

S

:[I iS,

m[

is t h e

weighted

of information

is d e f i n e d

ti ~ R i u i) d t + t

E.O.C.

structural

This

less the exchange

criterion

(x t Q x +

Q ) O, R i ) O, The

local

optimization

I0

t

is d e f i n e d

as below

:

~'(t)M(t)dt

: ~ [ K i l , ...,

Yil

Ki,i_ 1 ,

O,

Ki, i+ 1 , ..-,

Kis

lr I

=_0

"°

Y i,i- 1

0 .3.4)

]

I

ri_ 1

0

W. ~ 1

Yi,i+l 0

I

ri÷l • -.,,°

Y i,S Ir S (s.3.5)

200 I

stands

r.

for the identity

matrix of dimension

r.

1

Consequently,

the E . Q . C ,

= P.I

E.O.C.

becomes

S Z IlK i r i i=l

+

r.. 1

:

y[[2

a n d we h a v e t h e f o l l o w i n g o p t i m i z a t i o n t a s k Find the optimal matrices

x

1

K{ . . . . .

E.O.C. (K~ . . . . . K~) < E . o . c .

:

K~ s u c h t h a t

•

(~1 . . . . . KS)

f o r all a d m i s s i b l e m a t r i c e s K 1 , . . . . K S. A necessary criterion

condition

with respect

d E.O.C. Senning which both

of optimality

to t h e f e e d b a c k

! d K. = 0

(i=l . . . . .

1

gives the expression

satisfy

a Lyepunov

is

the

vanishing

m a t r i c e s Ki ,

(i=l . . . . .

of S)

the

gradient

of

the

:

S)

of this gradient

equation

in t e r m s o f t w o m a t r i c e s

and he provides

P and X

t h e s o l u t i o n to t h e p r o b l e m

a s follows : Theorem

5.3

equations

:

(SEN-79).

The

optimal

solution

1. R i K i C X C' + K i Fi C X C ' Fi + t~' 1 S >: i=l

2. / ~ P + P A 0 + Q + C'

Ki ,

(i=l,...,S)

P X C = 0

(i=l

P

satisfies

the

following

. . . . S)

( K ' R i K i + r i ~, K i Fi) C = 0

3, A 0 X + X A~ + X0 = 0 4. A0 = A +

S E i=l

B. K. C *

1

The value of the optimal extended

E.O.C.

(K 1 . . . . .

criterion is given by

:

K S ) = t r (P X0)

X 0 = x 0 x~ a n d x 0 = x ( t 0) i s t h e i n i t i a l s t a t e . In

the

first

system

with

system

(including

feedback tralized

step

a dynamic both

(for details,

of

his

work,

compensator the plant see

and

(SEN-79)).

dynamic compensator

Senning is

shows

equivalent

to

the compensator The problem

can therefore

that the

the

control

control

dynamics)

of determining

be solved by appying

of using

of a linear

an

augmented

static output

a feasibly Theorem

decen-

5 . 3 to t h e

201

extended

system,

which makes this approach

in t h e f o l l o w i n g e x a m p l e Example 5.7

(SEN-79).

even

Consider i

-2

I .-I

3

'

2

0

I

-6

I

0

3

I

-3

4

i

6

-7

1 I I I

' J

5

I

-

I

i

7

9

-1

2

0

0

,3

',0

0

',0

o3

c2:[0

l i

I

I0

03

'

0

,i i

03

The

E0

0

,

weightings

namic c o m p e n s a t o r s

for the

state

of the plant

B1 =

and

:

ii iol M

-4

Cl~Ei

C3=

T h i s is i l l u s t r a e d

t h e f o l l o w i n g s y s t e m in t h e f o r m ( 5 . 2 . 1 )

-20 I -------~ A =

more powerfull.

;

those

for the state

of the

dy-

are choosen as :

(I00)

Qplant = diag

Q c o m p = d i a g (1) and the weightings

for the inputs

Ri = diag (1),

i=1,2,3

The

information

non-local

decentralized

to t h e p l a n t a n d to t h e c o m p e n s a t o r

is w e i g h t e d

by

a factor

of

30,

favorizing

control.

T h e optimization yields t h e compensators as below : U,

I

81.6

-56.6

-1.76

0.13

2.06

0.62

Yl

Z, ---i

I

146

-107

0.75

-0.12

-1.02

-0.31

zl

U~ I

0

0

552

0.19

0

Y2

-0.32

0.23

=1884

3.1

0

z2

0.4

-0.22

-1.9

-0.1 "I -232

Z~

I

0.3

-0.2

0.1

0

I

I 592 I

as :

-71.5

Y3

lgl

z3

complete

202 The

optimal

compensator

is

not

completely

decentralized

(i.e.,

c a n n o t b e a c h i e v e d w i t h a completely d e c e n t r a l i z e d c o n t r o l s t r u c t u r e ) t u r e s h o w s t h e following i n f o r m a t i o n p a t t e r n

*

~ ' S t a t i o n

1

~

7-[S t a t i o n 31

i S t a t i ° n 21

Scnning's

a p p r o a c h is s p e c i a l l y a t t r a c t i v e b e c a u s e n o t only t h e optimal s t r u c -

t u r e b u t also t h e optimal p a r a m e t e r s can a p p l y

stabilization and its struc-

either

for

the

case

are returned

of s t a t i c o u t p u t

by the optimization. feedback

either

M o r e o v e r , it

for t h e

d e s i g n of

dynamic c o m p e n s a t o r s b y c o n s i d e r i n g an a u g m e n t e d s y s t e m . A n o t h e r p o i n t of i n t e r e s t is t h a t it does n o t r e q u i r e to c h e c k ,

in a f i r s t s t e p ,

w e t h e r o r n o t t h e s y s t e m has

u n s t a b l e fixed modes for t h e f a v o r i z e d c o n t r o l s t r u c t u r e decentralized

structure).

This

is

possible

because

(in o u r e x a m p l e ,

the

optimization

includes the quadratic performence index and the structural is c l e a r t h a t if t h e structure,

5.3.3.

system has

no u n s t a b l e

completely

criterion

both

optimization c r i t e r i o n . It

fixed modes for t h e f a v o r i z e d

control

t h e optimal c o n t r o l will c e r t a i n l y h a v e t h i s same s t r u c t u r e .

- Locatelli e t al. a p p r o a c h : (LOC-77) T h e a p p r o a c h of Locatelli e t al.

graphical characterization

(LOC-77)

is b a s e d on t h e i r f r e q u e n c y

of fixed modes w h i c h was p r e s e n t e d

in P a r a g r a p h

domain 3.6.2.

C o n s i d e r t h e c l a s s of l i n e a r t i m e - i n v a r i a n t s y s t e m s with simple modes r e p r e s e n t e d the general frequency stations.

Then,

domain model ( 3 , 4 . 1 )

Locatelli et el s t a t e t h e following p r o b l e m : find t h e minimal s e t of

f e e d b a c k l i n k s S* c S (S as d e f i n e d in ( 3 . 6 . 1 ) ) = {~'1. . . . .

~h }

by

w h e r e t h e r e is no p a r t i t i o n i n g in s e v e r a l

co(A)

mode with r e s p e c t

can be arbitrarily

s u c h t h a t e v e r y mode in t h e s e t A*

assigned ; i.e.,

to t h e c o n t r o l s t r u c t u r e

d e f i n e d b y S*.

no mode i n A *

is a fixed

T h e optimization is p e r -

formed with r e s p e c t to t h e following cost c r i t e r i o n :

R(S) =

):

r.. l,l

(i,9 £ s w h e r e r . . is a cost a s s o c i a t e d with t h e allowed f e e d b a c k link from o u t p u t i to i n p u t j l,l ( ( i , j ) ~ S ) . It is c l e a r t h a t t h i s p r o b l e m h a s a s o l u t i o n if a n d o n l y if no mode in A* is a f i x e d mode with r e s p e c t to t h e c o n t r o l s t r u c t u r e

d e f i n e d b y S. T h e s o l u t i o n is

t h e n o b t a i n e d b y s o l v i n g t h e following l i n e a r boolean p r o g r a m :

203 rain

under

E r i - m , j qi,j ( i , j ) ~ L2S

: for g=l,...,h

Z

(Cgl)

(i,j) E LIS

j/(i,j) ¢~ L S

(cg)

zg ,< i,j

vgj

zg

i)j

i,j

>/

i/{j,i) ~ L S

(i,j)

qi, j

j,l

L25

with :

f

qi,j =

O

if

(i-m,j) ~ / S *

if

(i-m,j) £ S *

(i, j ) 6 L 2 s

and for g=l . . . . ,h if the edge (i,j) is involded in a cycle whose t r a n s m i t t a n c e has pole and whose selection minimize the c r i t e r i o n .

z,g.

l,l

I:

=

otherwise.

f!

vg

i,]

Xg as a

=

if for (i,j) ~ L1S, zero),

Wj_m,i(Xg) # O,

¢o (Xg is n e i t h e r

a pole nor

a

if for (i,j) ~ LIS, Wj_m,i(lg) = Qo (Xg is a p o l e ) . f if for (i,j) E L1S, lira P"~Xg

wJ-m'i(p) (p- ~.gJ f

= O, ¢=

(kg is a zero of order f).

From the definition of rig j, the c o n s t r a i n t (Clg) g u a r a n t i e s that at least one edge (i,j) whose t r a n s m i t t a n c e has Xg as a pole is selected and the c o n s t r a i n t (C~) guaranties that this edge belongs to a cycle ( i . e . , from Theorem 3.30, ~.g is not a fixed mode). The c o n s t r a i n t (C~) s e t s a p a r t the selected e d g e s which do not c o r r e s pond to f e e d b a c k links and do not affect the c r i t e r i o n .

204

The interest

of t h i s p r o g r a m is e n f o r c e d b y t h e f a c t t h a t it c a n b e u s e d ,

with

s l i g h t m o d i f i c a t i o n s , to p r o v i d e t h e s o l u t i o n of s e v e r a l p r o b l e m s o t h e r t h a n t h e one it w a s stated f o r :

i - D e t e r m i n a t i o n of t h e f i x e d m o d e s w i t h r e s p e c t

to t h e c o n t r o l s t r u c t u r e

specified

b y S u s i n g a s o l v a b i l i t y t e s t s u c c e s s i v e l y a p p l i e d to A* = {k i }, Xi E o ( A ) . 2 -

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

to t h e

number

of f e e d b a c k

links by setting

r i , j = 1,

S.

(i,j)

3 - D e t e r m i n a t i o n o f t h e minimal f e e d b a c k p a t t e r n s

avoiding fixed modes : by setting

S = {(j,i) /(j=l ..... r) ; (i=l ..... m)}

h* =

c~ (A)

4 - Determination

of the minimal

initial control structure

S = {(j,i) ri, j = 0 Example 5.8.

set of feedback

specified b y

I (j=l . . . . . r )

links which

must

S O to eliminate the fixed m o d e s

be

added

b y setting

; (i=l . . . . . m) }

for (j,i) E S O C o n s i d e r t h e s y s t e m in t h e e x a m p l e 3.13 w h i c h h a s a f i x e d mode at

X 0 = -1 f o r t h e d e c e n t r a l i z e d

control structure

specified by SO = {(1,1)~(2,2)~(3,3)} .

We w a n t to d e t e r m i n e t h e s e t o f f e e d b a c k l i n k s to a d d t o t h i s initial p a t t e r n to e l i m i n a t e

the

supplementary

fixed

links.

program by setting

A* = S

mode.

The

optimization

:

-i

= {(i,j)

/ i=1,2,3

~ j=lp2,3}

ri, j = 0

f o r (i,j)~E S O

ri, j = 1

for (i,j) ~ S - SO

min q42 + q43 + q51 + q53 + q61 + q62

(c])

•

z]6 ) ]

criterion

is

The solution of this problem can be

T h e p r o g r a m to b e s o l v e d is t h e following

under

to an :

taken obtained

as

the

by

in o r d e r

number

the

of

previous

205

rZl4 + z15 + z16 = z41 + z51 + z61 z25 + z26 = z4z + z52 + z62 z34 + z35 = z43 + z53 + z63 (Cz)' z41 + z42 + z43 = z14 + z34 z51 + z52 + z53 = z15 + z25 + z35 z61 + z62 + z63 = z16 + z26 (C3)

zij x< qij

i=4,5,6

j=1,2,3

We o b t a i n two optimal s o l u t i o n s :

s~ = { (z,1)} which c o r r e s p o n d to t h e following f e e d b a c k s t r u c t u r e s

I kll

:

0

k13

kll

kl2

0

k22

0

0

k22

0

0

k33

0

0

k33

5.3.4. - S p e c i f i e d a p p r o a c h e s for s t r u c t u r a l l y f i x e d modes The p r o c e d u r e s

presented

in t h i s p a r a g r a p h

a r e b a s e d on t h e g r a p h - t h e o r e t i c

a p p r o a c h e s l e a d i n g to c h a r a c t e r i z a t i o n s of s t r u c t u r a l l y c o n s i d e r e d h e r e from a s t r u c t u r a l by t h e s u b s e q u e n t they

are

not

procedures

concerned

by

fixed m o d e s .

p o i n t of view a n d t h e c o n t r o l s t r u c t u r e s

g u a r a n t y t h e a b s e n c e of s t r u c t u r a l l y those

T h e p r o b l e m is

fixed

modes

which

arise

returned

fixed modes b u t

from p a r a m e t e r

value

considerations. We c o n s i d e r l i n e a r s y s t e m s in t h e g e n e r a l form : S(t) = A x(t) + B u(t)

I

y(t)

where x ( t )

C x(t) Rn , u ( t )

(5.3.6) R m, y ( t )

R r a n d A, B, C a r e real m a t r i c e s of a p p r o p r i a -

te d i m e n s i o n s . We c o n s i d e r t h e g e n e r a l f e e d b a c k p a t t e r n

:

206

u(t) =K

y (t)

F and FK are respectively,

5.3.4.a.

the

(5.3.7)

digraphs

- Procedures

presented

based

structurally

Theorem l.

-

type

open

loop

and

closed-loop

sys terns,

characterization

modes

b e l o w a r e all b a s e d

provided

in

on t h e g r a p h i c a l

(LIE-83)

and

(PIG-84)

characterization

and

formulated

in

3.26.

Determination (i)

:

of the

(TRA-87)

condition

(i)

in

control

Theorem

3.26.

associated

the desired

brought

back

algorithms

to

exist

feedback

This

to

modes

condition

w a y s to s t a t e

the

well-known literature

approach,

"covering and

set

which was

the problem

avoid of is

structurally

type first

(i)

are

fixed

modes

of

characterized

expressed

in

terms

in F . I n a s e c o n d s t e p ,

the optimization problem

control structure.

in t h e

In a second

fixed

to a s t a t e v e r t e x

l a t i o n i s u s e d i n two d i f f e r e n t provides

structure

• Structurally

concept of "loop-set"

5.2.3.

the

3.5.3.b-1

presented

fixed

to

3.5.3b-1.

on the graphical

in Paragraph

The procedures of

associated

a s d e f i n e d in P a r a g r a p h

by

of the

this formu-

whose solution

In a first approach,

t h e p r o b l e m is

problem"

some

for

already

which

encountered

is solved by using

in

a successive

efficient

Paragraph "elimina-

tion" procedure. Definition 5.2 by

(TRA-87).

The loop-set

associated

with the state vertex

x k is defined

•

=

Kxk The which,

loop-set

associated

and an output

from the

graph

With t h i s d e f i n i t i o n , Corollary

5.1

following

condition

modes of type

card where

Xk, x k reaches to

xk

is

y j

therefore

the

set

of

implemented one at a time, are such that the vertex

to an input either

kij / in r e a c h e s

Therefore,

either

from the

is

Consider

sufficient

a feedback for

system

:

(K* , K

the loop-sets the

x k

) /~ 1

(k=l .....

K* = { kij / kij i s a n o n z e r o

entry

n)

o f K}

pattern (5.3.6)

derived

matrix

to

links

connected

of the system.

from Theorem

in t h e f o r m not

feedback

can easily be determined

teachability

t h e f o l l o w i n g c o r o l l a r y is d i r e c t l y

(TRA-87). (i)

vertex.

itself

those

x k is strongly

have

(5.3.7),

3.26 : then

structurally

the fixed

207

The c o n d i t i o n e x p r e s s e d in Corollary 5.1 i s t h e same as t h e one a l r e a d y d e rived in T h e o r e m 5.1 of P a r a g r a p h

5.2.3.

The only d i f f e r e n c e c o n s i s t s in t h e s e t s

we are dealing with : in T h e o r e m 5.1 we w e r e c o n c e r n e d with t h e s e t s K~r a s s o c i a ted with t h e

s e n s i t i v i t y m a t r i x of t h e

present case,

t h e s e t s are

(k=l, . . . . n ) .

fixed modes Xr ,

(r=l . . . . . q ) ,

t h e l o o p - s e t s Kxk a s s o c i a t e d to t h e s t a t e

while, in t h e v e r t i c e s Xk,

T a k i n g into a c c o u n t the a b o v e r e m a r k , t h e p r o b l e m to b e s o l v e d is also

Problem 5 . 1 . The same c o s t c r i t e r i o n as in P a r a g r a p h 5 . 2 . 3 can be a d d e d to Problem 5.1 in t h e p r e s e n t c a s e . This would lead to the boolean l i n e a r p r o g r a m f o r m u l a t e d in Problem 5.2 w h i c h has b e e n s h o w n to be a w e l l - k n o w n " c o v e r i n g s e t problem" of graph t h e o r y . Example 5.9. C o n s i d e r t h e s y s t e m w h o s e a s s o c i a t e d g r a p h is t h e following :

~

~

~

Y2 9

The l o o p - s e t s are g i v e n b y :

Kx I Kx 2

{kll ) =

Kx3={

{k12 } k l 2 , k22}

Note t h a t K c K , therefore K can be eliminated. If we c o n s i d e r an x2 x3 x3 optimization c r i t e r i o n b a s e d on t h e n u m b e r of f e e d b a c k l i n k s , t h e optimization p r o blem, w h i c h is t r i v i a l in t h i s c a s e , r e t u r n s t h e s o l u t i o n :

K* = { k l l , Remark 5.7.

kl2}

The c o n d i t i o n p r o v i d e d b y Corollary 5.1 is only s u f f i c i e n t ; i . e . ,

all

the admissible s o l u t i o n s a r e not c o n s i d e r e d to find t h e optimal solution of Problem 5.1. This r e m a r k is clarified b y t h e following g r a p h i c a l c o n f i g u r a t i o n :

208

°,©

Cyx2

°2© f o r w h i c h we h a v e

:

Kxl = {k21} and the unique

{kll,

Kx2

= {kl2

solution for problem

Nevertheless,

i f we c o n s i d e r

k22} insures

also

and an output

vertex

that

}

5 . 1 i s • K* = {k21 , k l 2 } the condition

x 1 and

x 2 are

,

(i) o f T h e o r e m

strongly

3.26, the choice

connected

to a n i n p u t

vertex

in D K .

The lack of necessity for some vertex

-©y2

x k,

of the condition of Corollary

condition

tation of more than

one

(i) of T h e o r e m

feedback

link and

5 . 1 i s d u e to t h e f a c t t h a t ,

3.26 can be verified

this possibility

by

the implemen-

is not taken

into account

in our formulation. Because is s h o w n in

of this restriction, (SEZ-83)

t i o n s to s a t i s f y

that

Condition

this approach

the number

may provide

of necessary

(i) o f T h e o r e m

and

a suboptimal

sufficient

3.26 is given by

solution.

feedback

It

connec-

:

~r = max (u r, yr ) where

u r is t h e m i n i m a l n u m b e r

of inputs

b i l i t y ) a n d Yr is t h e m i n i m a l n u m b e r an

output

inspection

In

(output

of the reachability

our

formulation,

the minimal number Therefore,

teachability).

to r e a c h e v e r y

of outputs Therefore,

such that ~r

can

state vertex every

easily

be

state

(input

teacha-

vertex

reaches

determined

by

the

matrix of the system.

a solution

of input-output

s p e c i f i e s ~r f e e d b a c k

paths

connections,

where

to c o v e r t h e w h o l e s e t o f s t a t e

~r is

vertices

X.

a s o l u t i o n i s o p t i m a l if a n d o n l y if :

r -~ ~ r Unfortunately, = card

K* ;

conditions

i.e.,

this general after

condition can be checked

solving

c a n b e o f h e l p in c e r t a i n

the

optimization

cases

:

problem.

after obtaining However,

the

K* a n d a r following

209 A s o l u t i o n is o p t i m a l if t h e n u m b e r

-

is e q u a l to u r .

(This

"cover" one loop-set

-

comes

from t h e

of loop-sets

fact

that

one

involved in the optimization

feedback

links s u p e r i o r

to

; i . e . ~ r = ~ r )"

A s o l u t i o n is s u b o p t i m a l i f t h e n u m b e r o f i n d e p e n d e n t

the o p t i m i z a t i o n

l i n k is s u f f i c i e n t

is s u p e r i o r

or

equal

to

~r"

o r e q u a l to u r is n e c e s s a r y

(In

this

l o o p - s e t s i n v o l v e d in

case,

a number

to " c o v e r '~ t h e i n d e p e n d e n t

some a d d i t i o n a l f e e d b a c k l i n k s a r e r e q u i r e d

of feedback loop-sets

and

to " c o v e r " t h e r e m a i n i n g o n e s ~ i . e . a r

~r ) •

In t h e c a s e o f E x a m p l e 5 . 9 ,

the solutions provided

by the above procedure

are

optimal. Though

the

s o l u t i o n may s p e c i f y a s u b o p t i m a l n u m b e r

approach presents

the advantage

covering set problem.

The degree

pensated by the reduced The

second

to b e s p e c i a l l y

(TAR-85)

which t h e o p t i m i z a t i o n c r i t e r i o n p r o b l e m is c a r r i e d

out starting

applying a successive Considering (Yi,ui). s u c h satisfy

this

is t h e

there

condition.

this com-

by the procedure.

presented number

b e l o w is r e s t r i c t e d of feedback

links,

to t h e c a s e f o r The optimization

from a n i n i t i a l n o n minimal s e t o f f e e d b a c k l i n k s a n d

elimination procedure

Condition

that

links,

formulation as a

o f s u b o p t i m a l i t y of t h e s o l u t i o n is t h e r e f o r e

efforts required

approach

of feedback

s i m p l e d u e to i t s

(i)

w h i c h u s e s two r u l e s .

in T h e o r e m

is n o i n p u t - o u t p u t Consequently,

3.26, path

it i s

clear

that

a feedback

from u i to yj i s n o t n e c e s s a r y

we d e f i n e a n i n i t i a l s e t o f " u s e f u l l "

link to

feedback

links a s follows : K 1 = (kij / a p a t h from u i to yj e x i s t s i n F} Obviously,

this

several input-output To e v e r y

/

is n o t

minimal

since

some

state

vertices

paths.

kij in K 1, a s s o c i a t e t h e following b o o l e a n v e c t o r

zij = E z i j ( 1 ) ' " z..(t)=

set

(5.3.8)

:

z i j ( n ) 3'

1

if kij i s a n e l e m e n t o f t h e l o o p - s e t o f x t ,

0

otherwise

1l

and l e t u s s t a t e t h e following d e f i n i t i o n s :

(t=l . . . . . n )

may

belong

to

210 Definition 5.3

(TAR-85).

The state vertex

x t i s s a i d to b e d i s j o i n t i f a n d o n l y if :

zij(t) = 1 zij(t)

AND Z q r ( t )

Definition 5.4

= 0

(TAR-85).

f o r all q r ~ ij s u c h t h a t k q r E K 1

The vector

zij(t) = 1 implies Zqr(t) As an extention

zij(t)

= 1 for

f o r all ( t = l . . . . .

of D e f i n i t i o n

minate the set of vectors

(t=l .....

= 1

z.. i s s a i d to d o m i n a t e t h e v e c t o r II

5.4,

the

set

z

qr

if :

n) of vectors

z..c xj

Z 1 is said

to do-

Zqr c Z2 if :

some

zij a Z 1 i m p l i e s

=

Zqr(t)

1 for

some

Zq r c Z2,

for

all

the absence

of

n).

Using Definition 5.4 and Corollary Corollary

5.2 - A set of feedback

structurally

fixed modes of type

K* = { k i j / t h e

5.1,

links

the following result

K* i s s u f f i c i e n t

comes :

to g u a r a n t y

(i) if :

set of vectors

{z..}l] d o m i n a t e s t h e w h o l e s e t o f v e c t o r s

associated

to K1} Therefore,

this

approach

is

concerned

with

determining

v e c t o r s { zij } w h i c h d o m i n a t e s t h e w h o l e s e t o f v e c t o r s The two following rules are used K 1 to d e t e r m i n e Rule vector

1 :

The

associated

the

minimal

set

of

to K 1.

in t h e e l i m i n a t i o n p r o c e d u r e

which starts

from

K* : feedback

link

kij E K 1 c a n

be

eliminated

from

Kl

if its

associated

zij is d o m i n a t e d b y a t l e a s t o n e v e c t o r Z q r . It is c l e a r t h a t i f Zqr d o m i n a t e s

the input-output

path

Yr" T h e r e f o r e ,

the presence

ted components

than the presence

Rule 2 : The

feedback

zij, t h e n e v e r y

f r o m u i to yj b e l o n g s of kqr

involves

state

vertex

a l s o to t h e i n p u t - o u t p u t more state

vertices

which belongs path

to

f r o m Uq to

in s t r o n g l y

connec-

of k... 1]

l i n k kij ~ K 1 i s n e c e s s a r y

if f o r s o m e t C {1 . . . . .

n},

zij(t) =

1 a n d x t is a d i s j o i n t s t a t e v e r t e x .

I f x t is a d i s j o i n t which,

taken

alone,

vertex

and

makes x t belong

zij(t)

= 1,

to a s t r o n g l y

then

kij i s t h e

connected

only

component.

feedback

link

211 Using these

rules,

the

following e l i m i n a t i o n

procedure

is p r o p o s c d

in

(TAR-

85 ) to d e t e r m i n e K* :

1 - Using the digraph

Y a s s o c i a t e d to t h e s y s t e m o r i t s r e a c h a b i l i t y m a t r i x

:

1.1 - D e t e r m i n e t h e s e t K] d e f i n e d in ( 5 . 3 . 8 ) 1.2 - F o r e v e r y Kij ~. K 1 d e t e r m i n e t h e a s s o c i a t e d v e c t o r zij. 2 - Set K* =

{0}

3 - Determine the subset K2 corresponding

to t h e d i s j o i n t s t a t e v e r t i c e s

:

K 2 = {kij / z i j ( t ) = 1 a n d x t is a d i s j o i n t s t a t e v e r t e x } I f K 2 = 0, go to 4, e l s e : 3.1 - Set K* = K* u K 2 3.2 - I f f o r all ( t = l , . . . , n ) , 3.3

- Set

Zqr(t)

K2 and kqr

OR z i j ( t ) ,

(t=l . . . . . n)

f o r all ij # q r s u c h

t h a t kij

K 1.

3.4 - K 1 = 3.5-k=

t h e r e e x i s t s z i j ( t ) = 1 f o r some kij C K*, go to 7

= Zqr(t)

K (K1 n K 2 ) 1 1 0

3.6 - K 2 =

4 - Determine K 3 c

K l such that the set of k vectors associated to the k elements of

K 3 dominates the set of remaining vectors associated to the elements in K I.

If K 3 = 0, go to 5. 4.1 - K* = K ' o K 3 (If K 3 i s n o t u n i q u e ,

t h e s o l u t i o n K* is n o t u n i q u e ) .

Go to

7. 5 - Determine

K4c

K 1 such

that

the

vectors

associated

to t h e

e l e m e n t s in K 4 a r e

dominated b y k v e c t o r s . I f K 4 = 0, go to 6. 5.1 - K 1 = CK (K I n K 4) ! 6 - k = k + 1, go to 3. 7 - STOP : K* v e r i f i e s C o r o l l a r y 5 . 2 .

Example 5.10 - C o n s i d e r t h e same s y s t e m a s in t h e p r e v i o u s e x a m p l e 5 . 9 . is g i v e n b y

: K 1 ={ k l l ,

kl2,

k22} a n d

x1

x2

x3

Zll =

[

1

0

0

]

z12 =

[

0

I

i

]

z22 =

[

0

O

0

]

Since t h e s t a t e

vertices

the s o l u t i o n K* = { k l l ,

x I and k l 2 }.

x 2 are

the associated vectors are

disjoint,

the

procedure

The set K1

:

is trivial

and

gives

212

Note that K 1 since

the

they

procedure

are

set

aside

doe~ n o t p r o v i d e in

step

5.

all t h e m i n i m a l s o l u t i o n s

However,

one

could

jump

i n c l u d e d in

over

step

5 if

desired. Of c o u r s e , by

using

the

we o b t a i n t h e s a m e s o l u t i o n a s i n E x a m p l e 5 . 9 w h e r e

first

approach.

optimization criterion quently 2 -

a control

sufficient been

of the

: a two s t e p

mining

in t h e

shown

the

procedure

case

(TRA-87)

avoiding

for which

§ 3.5,

structure

the

Chapter

III)

that

In the other the absence

approach

approach

consists

(i) a n d C o n d i t i o n

in

(ii).

may not be optimal.

the procedure In t h e graph

to g u a r a n t y

it u s e d .

Indeed, if t h e r e e x i s t s

the

The reader

it is clear

any

fixed

type

of type

Conse-

alternative,

condition

separately

of c y c l e

This (ii)

problems

derived

family and

are

(ii) o f T h e r o e m 3.26

fixed modes of type

the

is

s i n c e it has

(ii).

corresponding

clear that the solutions obtained

of the two procedures

fixed

by deter-

(i).

fixed modes of type

of structurally

solving

is refered

can thus

same

of structurally

has no modes at the origin

to

w i t h this

b y t h e s i m p l i c i t y of

in t h e l a s t s e c t i o n .

its width

in a g i v e n

a di-

to D e f i n i t i o n s 3 . 1 0 a n d 3 . 1 1 .

Condition

(ii) of T h e o r e m

in FK a c y c l e f a m i l y o f w i d t h ) n i n v o l v i n g

following procedure

on the

the optimization.

modes

structurally

It is thus

concept

that

it w a s d e r i v e d

based

O n e m o r e t i m e , t h i s is c o m p e n s a t e d

which uses either following,

are

- The first section was concerned

system

must be satisfied This

approaches

to a v o i d

structurally

always located at the origin.

Condition

two

5.3 still apply here.

control

structure

(see

fact,

a n d d i f f e r o n l y b y t h e w a y to p r o c e e d

the comments of Remark

Determination

modes

In

be proposed

3.26 is satisfied all t h e s t a t e

i f a n d only

vertices.

The

:

1 - i=0 2 - Consider

the digraph

F.define

F A = ( X , E A) a n d d e t e r m i n e 1 t h e s e t F n _ i = { f l , . . . , f e r v

of cycle families of width n-i. 3 - If F n _ i , ~ 0 go to s t e p 5. 4 - i = i + l , g o to s t e p

2.

5 - If i=0, go t o s t e p 8. O t h e r w i s e ,

select one fk ~ Fn-i and consider

the i vertices

{ x j ,1 . . . . xj}~ w h i c h a r e n o t i n v o l v e d i n t h i s c y c l e f a m i l y .

lThe determination of cycle families has been regarded a s a s t a n d a r d p r o b l e m of applied graph theory for many years. T h e w e l l - k n o w n m e t h o d s a n d a l g o r i t h m s for f i n d i n g p a t h s a n d c y c l e s ( L I A - 6 9 ) ( K R O - 6 7 ) ( R A O - 6 9 ) m a y b e c o m p a r a t i v e l y easily adapted for computed-aided determination of cycle families of prescribed width.

213

6 - Apply problem,

either

i.e.

by

of the taking

two p r o c e d u r e s

derived

only into account

and a d d i n g a n e w c o n s t r a i n t

to g u a r a n t e e

in

the state

the

last

vertices

section

to a r e d u c e d

determined

at s t e p

5,

disjoint cycles :

rain c a r d K 2 c a r d (K 2 n K i) > 1 V il, iz ~ {1,...s,}, klk such that

(i=l . . . . . r)

w i ! . w.,2 = 0 if s.1 l a n d

st2 c o r r e s p o n d

respectively

to k s v a n d

:

s = 1 and v#k or

s # 1 a n d v=k

7 - Consider

the

edges resulting

digraph

obtained

from

F=

(V,E)

by adding

the

from s t e p 6 a n d d e t e r m i n e t h e s e t o f c y c l e families o f maximal w i d t h .

If t h e r e is n o n e o f t h e s e c y c l e f a m i l i e s i n v o l v i n g all s t a t e v e r t i c e s , 8 - Discard the state vertices K 2.

Apply

problem,

set of feedback

either

i.e.

by

whose loop-set contains

of the

two p r o c e d u r e s

taking

only

into

derived

account

the

go t o s t e p 5.

a feedback edge belonging

in t h e

last

remaining

section

state

to

to a r e d u c e d

vertices.

Let

the

solution b e K 1 . 9 - T h e g l o b a l s o l u t i o n is g i v e n b y : K* = K 1 u K 2. Example 5 . 1 1 .

Consider

the same system

a s in E x a m p l e s

5.9 a n d 5.10 w h i c h h a s a

s t r u c t u r a l m o d e at t h e o r i g i n s i n c e t h e g e n e r i c r a n k o f i t s d y n a m i c m a t r i x is e q u a l to n-l=2.

The

observation

of its associated

digraph

FA s h o w s

that

F3=0 a n d

one c y c l e family o f w i d t h 2. x 3 i s t h e o n l y v e r t e x w h i c h is n o t i n v o l v e d . S i n c e Kx3

K~

{kl2, k2z}

K 2= z

= {klz }

we o b t a i n two s o l u t i o n s at s t e p 6 :

{kzz }

which r e s u l t in t h e two following c y c l e families o f w i d t h 3 :

x2

k12

\x ~.

x3

X'~G3

~~

At t h i s s t e p ,

x2

"I

2y2

u2 Q~

~

0

"~ ~ 2 "~22

C o n d i t i o n (it) o f T h e o r e m 3.26 is s a t i s f i e d .

=-

~ Y2

there

is

214

a) c o n s i d e r t h e solution K 21 = {kl2 }

T h e r e d u c e d s e t of s t a t e s for which the

l o o p - s e t s do not c o n t a i n k l 2 is {Xl~ In t h i s c a s e t h e solution is : K12 = { k l l ) 2 and t h e global solution is : K 1

= { k l l , kl2}

2 b) c o n s i d e r t h e solution K 2

= {k22} , the r e d u c e d s e t of s t a t e s is in t h i s case 2 {x 1, x 2 }. The s o l u t i o n is now Kl2 = { k l l , k l 2 } a n d the global solution is : K 2 ={kll ' . k12, k22} w h i c h is clearly w o r s e t h a n K 1 s i n c e K~ c K~. T h e r e f o r e , t h e solution K 1 is r e t a i n e d , Note t h a t we o b t a i n t h e same solution as in Examples 5.9 and 5.10 w h e r e only structurally

f i x e d modes of t y p e

(i) were c o n s i d e r e d .

This is due to t h e f a c t that

t h e cycle family of w i d t h 2 c o n t a i n e d in FA is only composed of s e l f - c y c l e s . In one of the

first procedures,

cycle t o g e t h e r

w h e n Condition

with x 2.

(i) is s a t i s f i e d for x3,

B e c a u s e of o u r

special configuration,

x 3 is i n v o l v e d in a we o b t a i n

a cycle

family of width 3 composed b y this cycle a n d the s e l f - c y c l e at x 1. T h e r e f o r e , Condition (ii) is also s a t i s f i e d . As a m a t t e r of f a c t , when a cycle family c o m p o s e d o n l y b y s e l f - c y c l e s e x i s t s in the

s e t of cycle families of maximal w i d t h

(Step

2),

the

g e n e r a l p r o b l e m can be

s o l v e d b y u s i n g one of t h e f i r s t p r o c e d u r e s as well. This a l t e r n a t i v e may be advant a g e o u s s i n c e t h e optimization t a s k is p e r f o r m e d in one s t e p . Remark 5 . 8 .

The d e g r e e of s u b o p t i m a l i t y of t h e s o l u t i o n s can be e v a l u a t e d as it is

s h o w n below. Define U I , . . . , U q cardinality ur (yr) Bu[

(i=l . . . . , q )

(YI,...,Yt)

as t h e s u b s e t s of i n p u t s

(outputs)

of minimal

s u c h t h a t t h e s y s t e m is i n p u t r e a c h a b l e ( o u t p u t r e a c h a b l e ) . Let

(Cjs, j = l , . . . , t )

be t h e m a t r i c e s c o m p o s e d b y t h e columns of B (rows

of C) c o r r e s p o n d i n g J t o t h e i n p u t s in Ui ( o u t p u t s in Yj) a n d d e f i n e t h e i n t e g e r s di, A (i=l . . . . . q)

(~,

j=l . . . . . t)

as the

generic rank

Define

d e f i c i e n c y of (A Bu. !

)

( Cy

). ]

also dm=m!n d i a n d 6m=m!n 6j- T h e n , we h a v e t h e following p r o p o s i t i o n , L

P r o p o s i t i o n 1 (TRA-87)

)

: Given t h e s t r u c t u r a l l y c o n t r o l l a b l e , s t r u c t u r a l l y o b s e r v a b l e

s y s t e m ( 5 . 3 . 6 ) the minimal n u m b e r of f e e d b a c k links s u c h t h a t s y s t e m ( 5 . 3 . 6 ) h a s no s t r u c t u r a l l y f i x e d modes is g i v e n b y :

215

6m)

= m a x ( u r + d m , Yr +

and t h e f o l l o w i n g c o r o l l a r y c a n b e s t a t e d

Corollary (5.3.6)

2 (TRA-87

assume that

It

is

clear

calculations.

) : Given

that

the

should

global

the

fixed

better

modes.

procedure

returned

For

- Sezer's

Sezer's pattern(s) approach,

procedure

systems

with

(5.3.7)

essential" input

Definition

provides

5.5

define t h e

BI

For

(C j )

a greet

amount

Therefore,

the

way

(5.3.6)

without at

m o d e s at t h e o r i g i n

to guaranty the

origin,

for

the absence it

will

he

of

of

more

in t h e f o l l o w i n g p a r a g r a p h .

which

requires,

the

subset

determine

the

h a s no s t r u c t u r a l l y costs

any

as

to

with respect

different

procedure

(SEZ-83).

~ = ur.

optimal.

modes

sets and the "minimal essential"

matrix

system

requires

presented

also

for which system

step

observable

(SEZ-83)

allow to c o n s i d e r

It is a t w o

and

is not necessarily

section suffices

the optimization is proceeded

and it d o e s n o t tions.

procedure

then

to s y s t e m s

of the first

c o n v e n i e n t to u s e o n e o f t h e p r o c e d u r e s

5.3.4.b.

controllable

generally

solution

be restricted

which o n e o f t h e p r o c e d u r e s structurally

structurally

Ur = m a x ( U r , y r ) = m a x ( m , p ) ,

Moreover,

this a p p r o a c h

the

:

I

output

(J)

In this

of feedback

the different first,

feedback

fixed modes.

to t h e n u m b e r

for

optimal

feedback

to d e t e r m i n e

the

"minimal

sets as defined below :

of

the

s e t {1 . . . . .

m} ( 1 .....

of

B

(C)

consisting

of

indices

I

(J)

is

submatrix

links

connec-

the

r ),

columns

(rows) w i t h i n d i c e s i n I ( J ) .

A subset (A,B I)

of inputs

((Cj,A))

((Cj,,A))

is

(outputs)

structurally

(observable),

said but

to

be

not

essential any

if

(A,BI,)

i f I' c I ( J ' c J ) .

The essential

input

(output)

are c a l l e d m i n i m a l e s s e n t i a l i n p u t For the Paragraph

with

controllable

definitions

1.3

(observable)

and

sets

of structural

we r e m i n d

that

the

if and only if :

1 - (A,B) is input

(output)

having

(output)

reachable.

a minimal number

of inputs

(outputs)

sets.

controllabillity system

and observability,

(C,ApB)

is

structurally

we r e f e r

to

controllable

216

[A] (gr[cj:o,.

(AB)=n

2-gr

Given

the

system

(5.3.6),

the

determination

can be performed

by using

1 -

procedure

(REI-81).

Consider

proceeded

such

the matrix

Reinsehke's

columns has been

wing block-triangular

Reinschke's

procedure,

that

I

A~ 1

that

a permutation

A of system

set

below.

of the

(5.3.6)

rows

and

h a s t h e follo-

components

in

the

(5.3.9)

'ANN]

w h e r e t h e d i a g o n a l b l o c k s of A a r e i r r e d u c i b l e

Several algorithms

which is presented

_.0 1

A22 ,,

ANI . . . . . . . . . .

connected

(output)

form :

All

gly

of a minimal input

digraph

e x i s t in t h e l i t e r a t u r e

matrices

for

which

(HAR-65)

(corresponding A is

the

(KAU-68)

to t h e s t r o n -

adjacency

(KEV-75)

matrix).

to p u t

A in

the above form. D e f i n e Z(A)

(A)

as the

as the submatrix The

and dz,

submatrix

consisting

d e f i c i t of g e n e r i c

respectively,

consisting

(structural)

and are given by

Zd = z - gr

Z(A)

d z = z - gr

(A) z

in t h e

z last right

c o l u m n s of A a n d

of t h e z f i r s t r o w s o f A . rank

o f Z(A)

and

(A)

z

are

denoted

by

Zd

:

(z=l,..,,n)

It is c l e a r t h a t n d = Assume the property

that

d

n

= n - gr

0 d = do = O. T h e n ,

Zd = ( z - 1 ) d

+ 1, w h e r e

d e f i n e d in a s i m i l a r w a y u s i n g

d z.

(A) = d. we d e n o t e b y

z

{I .....

n}.

1z , . . . ,

d z the d indices

The indices

zi,

(i=1,

. . . .

having d),

are

217

Example

5.12. dz

l ,, --l--]

l

o

zd

:

°o

l

i

2

1

1

,

T 1

z

Therefore, 1

we have 2

z = 1,

z 1 = 1,

Using results

z =4

z2 = 4

these

5.4.

definitions,

Given

irreducible, (A,B)

1 - the

d entries

2 - the

entries

If first

It

(REI-81)

(REI-83)

(A,B)

Theorem which

= n,

5.5.

1 - the the

provides

the

following

blocks

then

that

Given

entries

If gr(A)

the

are

be

form

a minimal

not

allowed

not

be

input

the

diagonal

blocks

matrix

B which

makes

n x d with

to be

all zero

identically

where

input

is of dimension

minimal not

Condition

the

pair

ci, of

= n,

hypercolumn

same

(C,A)

:

zero.

if all the

off-diagonal

blocks

zero.

matrix

B is

n

x

= n

and

1 where

the

entries

of

all zero.

1 implies

a

matrix

gr(A,B)

,d)

are

hypercolumn

then

A as in Theorem

structurally

i z (i=l ....

of A corresponding

the last

d < n,

2 implies

that

the

pair

reachable.

the

entries

hyperrow

of B must

clear

makes

= n -

controllable

to this

hyperrow

is input

A in a block-triangular

gr(A)

b (i=l ..... d) are z i ,i of a hyperrow of B must

gr(A)

is

with

structurally

of A corresponding

2 -

Reinschke

a matrix

and

the pair

the

:

:

Theorem are

d = 2 and

the

of C must

not

of

to this

observable

C

allowed must

minimal not

be

output

to be

not

hypercolumn

are

matrix

all zero,

5.4,

a minimal

is of dimensions

be

output

matrix

d x 1 with

C

:

zero. all

zero

identically

C is

if

all

the

off-diagonal

zero.

1 x d where

the

entries

of

218

Example 5.13.

F o r t h e m a t r i x A a s in E x a m p l e 5 . 1 3 ,

we o b t a i n :

d Z

I

I

!

X

i

A =

X

,

- - - 7 - ~ [__

zd

2

.

C =

L

! .

.

l

B:

__

X

X

l

1

2 1

I

.

/.

:

where X stands

-7 __

I'

for a nonzero

entry

and where the

shaded

r o w of B m u s t h a v e

at

least one n o n z e r o e n t r y .

Note t h a t t h e a b o v e p r o c e d u r e

2 - Determination of the

does not provide a unique solution.

minimal c o n t r o l s t r u c t u r e

(SEZ-83) - The following definitions are n e c e s s a r y

Definition

5.6

(SEZ-83).

k.. = 0 (F r e p r e s e n t s 1] 1 -

A structure

Define the b i n a r y

the structure

F is s a i d

modes with respect

to b e

avoiding structurally

fixed modes

to o u t l i n e t h e p r o c e d u r e

matrix

F s u c h t h a t f.. = 1 if a n d o n l y if 1] of the feedback matrix).

favorable

if t h e

system

has no structurally

4 -

fixed

to t h i s f e e d b a c k c o n t r o l s t r u c t u r e .

2 - G i v e n F 1 a n d F 2, F l i s s a i d to i m p l y F 2 if f.! = 1 i m p l i e s f 2 = 1. 1] ~j 3 - A f a v o r a b l e s t r u c t u r e F is s a i d to b e e s s e n t i a l if t h e r e is n o o t h e r structure

:

favorable

w h i c h i m p l i e s F.

Among all t h e

essential

favorable

structures,

the

o n e s w i t h minimal n u m b e r

of

n o n z e r o e n t r i e s a r e s a i d to be m i n i m a l .

We r e m i n d t h a t a s y s t e m to a c o n t r o l s t r u c t u r e

IA gr(M F) = gr

(C,A,B)

represented

B

h a s no s t r u c t u r a l l y

b y F if a n d o n l y if :

0

0

Im

F

C

0

Ir

= n+m+r

fixed modes with respect

219

and each state vertex

in t h e d i g r a p h

~

belongs

c o n t a i n i n g at l e a s t o n e e d g e c o r r e s p o n d i n g In the rally

following,

controllable

we m a k e t h e

and

observable,

to a s t r o n g l y

connected

component

to a f e e d b a c k l i n k .

assumption that

which

implies

the system

that

if g r ( A )

(5.3.6) = n

-

is s t r u c t u d,

then

d <

min (m, r ) .

Sezer stated

Theorem 5.6.

t h e following r e s u l t

:

C o n s i d e r t h e s e t s of i n t e g e r s

I = (i l . . . . . i k }

d < k < m

J = {Ji . . . . . Jq }

d ( q ( r

sucht that the system SIj = (BI,A,Cj)

and s u c h that g r ( A , B i , )

= n and gr

:

is s t r u c t u r a l l y

[:]

controllable and observable,

= n.

Y

If F is a s t r u c t u r e

such

that

gr

(FI,j,)

= d a n d s u c h t h a t FI_I, j _ j , c o n t a i n s

at l e a s t o n e n o n z e r o e n t r y in e a c h r o w a n d c o l u m n , t h e n F is a f a v o r a b l e s t r u c t u r e .

A favorable

structure

F

satisfying

Theorem

5.6

is

not

necessarily

essential

u n l e s s t h e s e t s I a n d J a r e c h o s e n to be t h e m s e l v e s e s s e n t i a l .

Moreover,

be minimal if we w a n t to o b t a i n

Given the matrices B

a n d C, S e z e r p r o p o s e s

a minimal e s s e n t i a l s t r u c t u r e .

to u s e R e i n s c h k e ' s p r o c e d u r e

e s s e n t i a l i n p u t s e t a n d a minimal e s s e n t i a l o u t p u t

Unfortunately, is n o t p e r f e c t l y tive p r o c e d u r e

Sezer structure

provides

set.

Reinschke's

to t h i s p r o b l e m a n d t h a t it fails in s o m e c a s e s .

will b e p r o p o s e d ,

tems s i n c e it p r o c e e d s

must

in o r d e r to d e t e r m i n e a minimal

it will b e s h o w n in t h e n e x t s e c t i o n t h a t

adequate

they

w h i c h is s p e c i a l l y

appropriate

for large

procedure An a l t e r n a scale sys-

in a s e q u e n t i a l w a y .

thus

the

following p r o c e d u r e

to

determine

.

1 - D e t e r m i n e a minimal i n p u t s e t I a n d a minimal o u t p u t

set J.

a minimal

essential

220

2 - C h o o s e I' c I a n d J ' c

gr

J such that gr (A,BI,) = n and

= n

3 - Construct

FI,j,

such

that

it

contains

exactly

d nonzero

entries

located

in

dif-

ferent rows and columns. 4 - Construct

FI_I, j _ j , s u c h

t h a t it c o n t a i n s e x a c t l y m a x

(k,q)-

d nonzero

entries

l o c a t e d n o t to l e a v e a z e r o r o w o r c o l u m n . 5 - Set all the other entries of F to 0.

From

Theorem

5.6,

F is a f a v o r a b l e

structure.

I t is a l s o e s s e n t i a l

since

for

some f.. : 1l -

if i a_ I - I t a n d j ~ J - Y t h e n t h e l o s s o f t h e f e e d b a c k e d g e ( y j , u i) l e a v e s a 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 in r K w i t h o u t f e e d b a c k e d g e . - if i ~ I a n d j ~ J ,

s i n c e I a n d J a r e minimal, t h e s y s t e m w i t h o u t t h e i n p u t u i a n d

t h e o u t p u t yj is n o t s t r u c t u r a l l y

Moreover, number

F is

of f e e d b a c k

c o n t r o l l a b l e n e i t h e r o b s e r v a b l e a n d fij i s n e c e s s a r y .

minimal s i n c e links

I and

o f a minimal

J

are

w h e r e k a n d q a r e t h e minimal n u m b e r o f i n p u t s tural controllability and observability

[0 0] c=[: 01,

Example 5.14.

0

minimal.

therefore

and outputs

equal

Note t h a t to

the

max(k,q),

which guaranty

struc-

to t h e s y s t e m .

B--

[ °] 0

I

We h a v e I = J = { 1,2 } a n d unique solution F = diag.

I' = J ' = {1 }, t h e r e f o r e

Sezer's

procedure

provides

the

•

i s a l s o f a v o r a b l e a n d minimal. C o n s e q u e n t l y , above

l

(1,1).

It h a s to b e n o t i c e d t h a t

The

is

C o n s i d e r t h e following s y s t e m :

A=

solutions.

themselves

structure

example

shows

that

this procedure a minimal,

does not provide

essential,

favorable

ell t h e

structure

22

t

d o e s n o t n e e d to c o n t a i n a s e t o f d f e e d b a c k

l i n k s f r o m Yi' j

J ' to u i , i

I'.

This

r

situation occurs

when

:

gr

= n + rain ( k , q ) Cj,

in w h i c h c a s e , Example 5.15.

A =

C =

g r ( M F) = n + m + r f o r all F p r o v i d e d Consider

that

now the system described

0 0

X 0

0 1 X

0

X

0

0

and

I ) = J' = {1,2}.

following s o l u t i o n s

The

above

III

I !___

I

"~----

I

no feedback

Therefore,

the

section),

all t h e

two

I: ]

first,

Beside

calculations.

fact that

which are

necessary

to initialise

o f 2n s u b m a t r i c e s

procedure

performs

Sezerts

optimization

with

provided

gr(Fi,j,)

all t h e

Nevertheless,

it f a i l s in s o m e c a s e s

the block-triangularization

rank

the

the

g r ( M F) = n + m + r u n l e s s solutions

=

minimal

detected.

itself does not require

luation of the generic

a n d it i s n o t a d e q u a t e

F verifies

gives

sets have been

procedure.

it n e e d s

pattern

procedure

and output

Sezerls procedure on Reinschkels

the

I1------I--I -I I I I l

1

I

essential input

provides

II I I------ I.

,l FI=

In this case,

procedure

:

I!.

d = 2.

:

°xox ooOX 1

O X

We h a v e I = J = { 1 , 2 , 3 }

= rain ( k , q ) .

by the following structure

ix x o J

I

gr(Fij)

of matrix

of A in order

procedure. respect

to

A and

must

the

number

in t h e c a s e f o r w h i c h e a c h f e e d b a c k

then,

to d e t e r m i n e

One

also of

it i s b a s e d

(see the next the eva-

the sets I,J

notice that feedback

link has a different

this links cost.

222

3 - Sequential 86b).

The

d e t e r m i n a t i o n of t h e minimal e s s e n t i a l i n p u t

d e t e r m i n a t i o n of t h e minimal i n p u t a n d o u t p u t

Sezer's procedure.

For this p u r p o s e ,

Sezer proposed

and output

s e t s is t h e

sets first

(TRAstep

of

to u s e R e i n s c h k e ' s p r o c e d u r e

( p r e s e n t e d in s e c t i o n 1 ) . However,

this

procedure

was

p r o b l e m of d e t e r m i n i n g a m a t r i x (minimal n u m b e r of i n p u t s lable

(observable).

r e m s 5.4 a n d 5 . 5 ) .

B

developed (C)

(outputs))

for

solving

the

slightly

such that the

It was s h o w n t h a t d i n p u t s

s y s t e m is s t r u c t u r a l l y

(outputs)

are

control-

T h i s r e s u l t a p p l i e s b e c a u s e t h e n o n z e r o e n t r i e s of B (C) c a n be

q u i r e d to fulfil t h e g e n e r i c r a n k not,

(rows)

sufficient (see Theo-

a r b i t r a r i l y l o c a t e d a n d can a l w a y s b e c h o s e n s u c h t h a t t h e d i n p u t s condition.

different

with a minimal n u m b e r of c o l u m n s

In t h e p r e s e n t

case,

(outputs)

re-

c o n d i t i o n s a t i s f y at t h e s a m e time t h e c o n n e c t i v i t y matrices B and C are

known and,

more often t h a n

t h e i r e n t r i e s a r e n o t l o c a t e d i n t h i s optimal w a y . T h e r e f o r e , more t h a n d i n p u t s

(outputs) are generally necessary. T h e s e s l i g h t d i f f e r e n c e s make t h a t the p r o c e d u r e

of (REI-81)

is not perfectly

s u i t a b l e for s o l v i n g t h i s p r o b l e m a n d it m a y fail in some c a s e s a s it i s s h o w n in t h e following e x a m p l e .

Example 5 . 1 6 . C o n s i d e r t h e following s y s t e m ( A , B )

:

-

--4--] x L__4__. , X

X'

'

"--i I I

A = X

B

X

P X

X

XI

X

I

X

XIX

X

×,,

X X

w h e r e A is a l r e a d y in t h e r e q u i r e d form a n d w h e r e d z i s i n d i c a t e d o n t h e r i g h t s i d e . From T h e o r e m 5 . 4 ,

the connectivity (teachability)

condition

(2) is s a t i s f i e d b y u 1,

a n d C o n d i t i o n (1) s p e c i f i e s t h a t a n o n z e r o e n t r y is r e q u i r e d in t h e f i r s t a n d l a s t row of m a t r i x B.

Since t h e l a s t row is e m p t y ,

not structurally

the wrong conclusion that the

controllable could be s t a t e d .

s y s t e m is s t r u c t u r a l l y c o n t r o l l a b l e a n d t h a t

s y s t e m is

N e v e r t h e l e s s ) it can b e s h o w n t h a t t h i s Ul,U 2

and

Ul)U 3

a r e minimal e s s e n t i a l

input sets. In t h i s s e c t i o n , mine t h e

we p r e s e n t

minimal e s s e n t i a l i n p u t

the sequential procedure (output)

sets

of a s y s t e m

of ( T R A - 8 6 b ) (5.3.6)

by

to d e t e r identifying

223 first,

t h e minimal i n p u t

(output)

s e t s which s a t i s f y t h e c o n n e c t i v i t y condition and

t h e n t h e minimal i n p u t ( o u t p u t ) s e t s which e n s u r e t h a t t h e g e n e r i c r a n k condition is satisfied. In t h e

s u b s e q u e n t d e v e l o p m e n t , t h e p r o b l e m is

approached

from t h e

inputs

p o i n t of v i e w . Dual r e s u l t s can be s t a t e d for t h e o u t p u t s . Define I C 1 , . . . , I c h

as t h e s e t s of i n d i c e s c o r r e s p o n d i n g to t h e minimal i n p u t

s e t s which s a t i s f y t h e c o n n e c t i v i t y condition for s y s t e m

(5.3.6)

and II 1 .

ITg as

. . . .

t h e s e t s o f i n d i c e s c o r r e s p o n d i n g to the minimal i n p u t s e t s to h a v e t h e g e n e r i c r a n k condition s a t i s f i e d . Theorem 5.7 ( T R A - 8 6 b ) . UI is a minimal e s s e n t i a l i n p u t s e t for s y s t e m ( 5 . 3 . 6 ) if and only if I=IciUIIj, i {1 . . . . . h} , j {1 . . . . . g} a n d i t s c a r d i n a l i t y is minimal. The a b o v e r e s u l t means t h a t t h e s e a r c h of t h e minimal e s s e n t i a l i n p u t s e t s can be p e r f o r m e d in two i n d e p e n d e n t s t e p s . algorithm d e r i v e d in

(TRA-86a)

For t h i s p r u p o s e ,

to c o n c l u d e on s t r u c t u r a l

bflity) of a g i v e n s y s t e m ( 5 . 3 . 6 ) .

we u s e t h e r e s u l t s and controllability

This algorithm can be a p p l i e d ,

(observa-

with some a d d i -

tional o p e r a t i o n s , to solve o u r p r o b l e m . In an initial s t e p , we p r o c e e d to a decomp o s i t i o n of t h e s y s t e m s u c h t h a t t h e new matrix A p r e s e n t s t h e b l o c k - t r i a n g u l a r form in ( 5 . 3 . 9 ) , each diagonal block c o r r e s p o n d i n g to t h e s t r o n g c o m p o n e n t s of t h e g r a p h a s s o c i a t e d with t h e

system

( t h i s initial s t e p

is t h e

same as in

t h e p r o c e d u r e of

(REI-81)). To a v o i d

trivialities,

we make t h e a s s u m p t i o n t h a t

t u r a l l y c o n t r o l l a b l e (and o b s e r v a b l e ) .

system

(5.3.6)

In the o p p o s i t e case h o w e v e r ,

is s t r u c -

t h e algorithm

below would d e t e c t t h e u n c o n t r o l l a b i l i t y and s t o p . T h i s p r e s e n t s t h e a d v a n t a g e t h a t no p r e l i m i n a r y c o n t r o l l a b i l i t y c h e c k i n g is r e q u i r e d . With

the

proposed

decomposition,

the

sets

IC1 . . . . , I c h

can

be

determined

without a n y calculations b y u s i n g t h e following r e s u l t • Theorem 5.8 ( T R A - 8 6 a ) . The s y s t e m ( 5 . 3 . 6 ) is i n p u t r e a c h a b l e if and only if :

EB i Ail Ai2 . . .

Ai,i_1~ $ 0

Vi = 1 . . . . . N

(5.3.9)

w h e r e the m a t r i c e s B. and A.. are t h o s e c o r r e s p o n d i n g to ( 5 . 3 . 9 ) . 1

I]

In a s e c o n d s t e p , t h e s e t s I ' l , . . . , I ' g (TRA-86a) a n d p r e s e n t e d in P a r a g r a p h trollable,

the algorithm r e t u r n s

a r e i d e n t i f i e d b y u s i n g t h e algorithm of

1.3.c.

When t h e s y s t e m is s t r u c t u r a l l y c o n -

the fpXfq (fp
224

w h o s e columns c o r r e s p o n d to some i n p u t a n d s t a t e v a r i a b l e s . I n d e e d , it is s h o w n in (TRA-86a)

that

the

generic rank

g e n e r i c r a n k for all (i=l,o0 . , N ) .

c o n d i t i o n is s a t i s f i e d if a n d only if F. h a s 1

full

The i n d i c e s of all t h e i n p u t s w h i c h a r e n o t r e p r e -

s e n t e d in F N c o n s t i t u t e a s u b s e t o f I' i,

(i=l . . . . . g)

( T R A - 8 6 a ) . F N c o n t a i n s all t h e

n e c e s s a r y i n f o r m a t i o n to d e t e r m i n e t h e r e m a i n i n g o n e s .

Since we h a v e

gr(FN)=fp,

f q - f p columns of F N can be eliminated p r o v i d e d t h a t at l e a s t one n o n z e r o e n t r y by row a n d b y column is p r e s e r v e d FN correspond

to i n p u t

( c o n d i t i o n of full g e n e r i c r a n k ) .

variables,

it may be

that

If some columns of

some of them can b e d e l e t e d .

Among all p o s s i b l e s o l u t i o n s , t h o s e which d e l e t e a maximal n u m b e r of columns c o r r e s p o n d i n g to i n p u t v a r i a b l e s

are

r e t a i n e d : the i n d i c e s of t h e r e m a i n i n g

"input co-

lumns" c o n s t i t u t e the r e m a i n i n g i n d i c e s of a s e t I ' . . 1

In o r d e r to d e t e r m i n e all t h e s e t s It 1 . . . . . I t g , t h e p r o b l e m is t h u s to i d e n t i f y all t h e p o s s i b l e s e t s of columns of F N which can be d e l e t e d w i t h o u t a f f e c t i n g t h e full generic rank,

The a p p r o a c h c o n s i s t s in d e t e r m i n i n g all t h e s q u a r e s u b m a t r i c e s of F N

h a v i n g full g e n e r i c r a n k • This is p e r f o r m e d in a g r a p h - t h e o r e t i c f r a m e w o r k . C o n s i d e r a p x q (p
at t h e bottom of M s u c h t h a t we o b t a i n a q x q s q u a r e

M. A s s o c i a t e to M a d i g r a p h

D=(V,E),

w h e r e V = { v l , . . . . Vq} is t h e

s e t of

v e r t i c e s a n d E is t h e s e t of e d g e s s u c h t h a t ( v i , v j) E if a n d only if mji~0. T h e o r e m 5.9 ( T R A - B 6 b ) . C o n s i d e r t h e m a t r i x M a n d t h e d i g r a p h D as d e f i n e d a b o v e . T h e n u m b e r of ( p x p )

s q u a r e s u b m a t r i c e s h a v i n g full g e n e r i c r a n k of M is e q u a l to

t h e n u m b e r of c y c l e families of w i d t h q of D. C o n s i d e r a n y cycle family o f w i d t h q of D.

T h e n , t h e columns a n d r o w s of M w h o s e e n t r i e s m.. c o r r e s p o n d to e d g e s 1l ( v j , v i) i n v o l v e d in t h e cycle family c o n s t i t u t e a s q u a r e ( p x p ) s u b m a t r i x of M h a v i n g

full g e n e r i c r a n k . The

above

theorem provides

m a t r i x F N to M a n d b u i l d F N a n d

the

key

for

solving

its a s s o c i a t e d g r a p h

the problem. D F.

Then,

Identify the

all t h e s q u a r e

s u b m a t r i c e s of F N h a v i n g full g e n e r i c r a n k a r e d e t e r m i n e d b y t h e s e a r c h o f all t h e cycle families of w i d t h fq of D F. At t h i s s t e p ,

we h a v e d e t e r m i n e d I C 1 , . . . , I c h

and I'1,..

•

, I ' g. U s i n g Theorem

5.7, we can t h e r e f o r e d e d u c e all the minimal e s s e n t i a l i n p u t s e t s a n d know for each of them w h i c h i n p u t s stem from I c i or from I ' . . A similar a n a l y s i s m u s t t h e n be made ]

for t h e o u t p u t s in o r d e r to obtain JC1 . . . . . J c k ' output sets.

J'l''"'J'f

a n d t h e minimal e s s e n t i a l

This p r o b l e m is easily b r o u g h t b a c k to t h e i n p u t p r o b l e m b y u s i n g t h e

d u a l i t y b e t w e e n o b s e r v a b i l i t y a n d c o n t r o l l a b i l i t y (TRA-86a) 86a) is a p p l i e d on t h e m a t r i x

(C' A ' ) p u t

: the algorithm of (TRA-

in t h e r e o r d e r e d form, T h e o r e m 5.6 can

225 thus

be

to o b t a i n

applied

the

set

of ell minimal e s s e n t i a l f e e d b a c k

patterns

for

(5.3.6). Example 5.17. C o n s i d e r a s y s t e m ( C , A , B ) w h e r e A a n d B a r e t h e same as in Example 5.16 a n d

C =

li

0

0

0

0

0

X1

0

0

0

0

0

0

0

0

X

0

0

0

U s i n g T h e o r e m 5.8 a n d t h e dual o n e for o u t p u t r e a c h a b i l i t y (TRA-86a)

(Theo-

rem 5 . 9 ) , we d e t e r m i n e :

Icl= {i} and JCl= {I} JC2 = {2} Now, we a p p l y t h e algorithm of (TRA-86b) ( s e e § 1 . 3 . 3 . c ) to t h e m a t r i x (B A) and we obtain t h e m a t r i x F 4 in a r e o r d e r e d form as follows :

u3 X

u2

X

~ I I

__

x u x j" l_ _ . . ,

E ' X

X X

L --X / '

I X

X

I

]

X

Since u I is n o t r e p r e s e n t e d in F4, it is clear t h a t all the s e t s I' i c o n t a i n t h e i n t e g e r 1. The r e m a i n i n g o n e s a r e d e t e r m i n e d by u s i n g T h e o r e m 5.9 w h i c h p r o v i d e s the k e y to e x t r a c t all t h e s q u a r e s u b m a t r i c e s of F 4 h a v i n g full g e n e r i c r a n k . Note t h a t s i n c e t h e t h r e e l a s t columns of F 4 are i d e n t i c a l , t h e y can be a r b i t r a r i l y i n t e r c h a n g e d a n d two of them can t h e r e f o r e be d e l e t e d . In o u r c a s e , none of them c o r r e s p o n d s to i n p u t

v a r i a b l e s so t h a t t h e

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

In a n o t h e r

c a s e , it is c l e a r t h a t t h e o n e s c o r r e s p o n d i n g to i n p u t v a r i a b l e s s h o u l d be d e l e t e d . We h a v e : u3

u2

K xL,

L

1

x I~ i_- 1

F I

/4

x x __ x

X

X

X

uX_t___[ --;-~--I X

X]

|

226 The number

inspection

of e d g e s

of DF shows three

associated

with input

cycle families of width 5 involving

columns (denoted

by dotted lines)

a minimal

:

V

t

Consequently, minimal essential If t h e get

we

obtain

:

I'1

= {1,2}

I' 2 = { 1 , 3 } .

We h a v e

therefore

two

i n p u t s e t s I i = I ' 1 a n d I2=I' 2.

same

problem

is n o w

approached

from

the

outputs

point

of view,

we

:

Ixx' /P

X

x

Ft.2=

x

--~

I X

I

. . . .

X

1

Y3

-

2--' I,_

__

L

.

.

I X

.

.

X

i

This means that

all t h e s e t s

columns of F 4 are identical, n e e d to go t h r o u g h

the one corresponding

the cycle families procedure

From Theorem system

J'i contain the indices

5.6,

we o b t a i n

] a n d 2. S i n c e t h e two l e s t

to Y3 i s d e l e t e d

to c o n c l u d e

four minimal essential

and

we

do n o t

that J'i=Jl={1,2}. feedback

patterns

for this

:

F 1=

[oo] [ixoI FOil [iox] 0

X

0

0

0

0

F2

0

0

F3 = 0

0

0

0

0

0

F# -

0

0

0

0

227 5.3.5. - Concluding remarks

The algorithms presented

in t h i s s e c t i o n p r o v i d e

mining t h e f e e d b a c k p a t t e r n ( s )

the f i x e d m o d e s c h a r a c t e r i z a t i o n s of C h a p t e r n u m b e r of f e e d b a c k and,

a s y s t e m a t i c w a y of d e t e r -

w h i c h g u a r a n t y t h e a b s e n c e of f i x e d m o d e s b y u s i n g III.

An o p t i m i z a t i o n c r i t e r i o n

as

the

l i n k s or t h e s u m of t h e i r a s s o c i a t e d c o s t s is g e n e r a l l y a d o p t e d

among all t h e a d m i s s i b l e f e e d b a c k p a t t e r n s ,

t h e o n e s which minimize t h e

cri-

terion are selected. Senning)s procedure the

parametric

and

the

(SEN-79) structural

is t h e o n l y o n e w h i c h c o n s i d e r s s i m u l t a n e o u s l y optimization

by

using

an

extended

optimization

c r i t e r i o n t a k e n a s t h e s u m of t h e c l a s s i c q u a d r a t i c c r i t e r i o n of l i n e a r s y s t e m s a n d a w e i g h t e d m e a s u r e of t h e n o n - l o c a l i n f o r m a t i o n . rithm provides

in o n e

optimal s t r u c t u r e . and the

step

Therefore,

t h e optimal g a i n s of t h e

t h e s o l u t i o n of h i s a l g o -

dynamic compensator and

its

All t h e o t h e r s a l g o r i t h m s p e r f o r m o n l y t h e s t r u c t u r a l o p t i m i z a t i o n

g a i n s m u s t be e v a l u a t e l a t e r

by

u s i n g one

of t h e p a r a m e t r i c o p t i m i z a t i o n

p r o c e d u r e s p r e s e n t e d in t h e n e x t c h a p t e r .

The

different

approaches

are

not

equivalent

and

one

must

choose

the

w h i c h is t h e m o s t a d e q u a t e to i t s p a r t i c u l a r p r o b l e m a n d s i t u a t i o n . I n d e e d ,

one

Locatelli

et al. a p p r o a c h (LOC-77) is o n l y valid f o r s y s t e m s with simple m o d e s a n d t h e p r o c e dures presented

in P a r a g r a p h

fixed modes.

However,

specially

large

for

5 . 3 . 4 do n o t g u a r a n t y t h e a b s e n c e of n o n s t r u c t u r a l l y

note t h a t

scale

these

systems)

limitations are

since

they

set

not

apart

the

too m u c h cases

constraining,

for w h i c h some

u n p r o b a b l e p e r f e c t p a r a m e t r i c m a t c h i n g s could o c c u r .

5 . 4 . - CONCLUSION T h i s c h a p t e r p r e s e n t s d i f f e r e n t a l g o r i t h m s w h i c h c a n be u s e d f o r t h e d e s i g n of the feedback control structure n u m b e r of f e e d b a c k l i n k s situations are

s u c h t h a t t h e s y s t e m h a s no fixed m o d e s a n d t h a t t h e

(or t h e

considered.

In

s u m of t h e i r a s s o c i a t e d c o s t s )

the

first

one,

an

initial

control

is m i n i m i z e d . Two structure

a r i s e from t h e p a r t i t i o n i n g of t h e s y s t e m d u e to p h y s i c a l c o n s i d e r a t i o n s . system

has

fixed

m o d e s with

respect

to t h i s

control

structure,

the

naturally When t h e

a l g o r i t h m s of

Section 5.2 allow to d e t e r m i n e t h e s u p p l e m e n t a r y i n f o r m a t i o n e x c h a n g e s w h i c h eliminate fixed modes. and

the

In t h e s e c o n d s i t u a t i o n , no initial c o n t r o l s t r u c t u r e

algorithms

(Section

5.3)

provide

all t h e

control

structures

is c o n s i d e r e d for

which the

s y s t e m h a s no f i x e d m o d e s . Since t h e p r o b l e m s o l v e d in t h e s e c o n d s i t u a t i o n is more general than the

first one,

the

corresponding

p a r t i c u l a r f r a m e w o r k of t h e f i r s t c a s e .

a l g o r i t h m s can also be

used

in t h e

228

At t h i s s t e p ,

we a r e g u a r a n t e d t h a t t h e s y s t e m is pole a s s i g n a b l e b y u s i n g a

dynamic c o m p e n s a t o r w h o s e s t r u c t u r e h a s b e e n d e t e r m i n a t e d b y one of t h e a l g o r i t h m s p r e s e n t e d in t h i s c h a p t e r . 79) h a s b e e n u s e d , is

c o n s i d e r e d in

E x c e p t in t h e case f o r w h i c h S e n n i n g ' s a l g o r i t h m (SEN-

the g a i n s of t h e c o m p e n s a t o r remain to d e t e r m i n e . This p r o b l e m

the

next

chapter

and

several

p a r a m e t r i c optimization a l g o r i t h m s

under structural constraints are presented. If one w a n t s to a c h i e v e pole a s s i g n m e n t u s i n g s t a t i c o u t p u t f e e d b a c k , tional c o n d i t i o n s o t h e r t h a n t h e a b s e n c e of f i x e d modes a r e g e n e r a l l y r e q u i r e d . problem was c o n s i d e r e d in P a r a g r a p h 3 . 5 . 2 c in w h i c h t h e control s t r u c t u r e s static controller are characterized their design chapter.

was p r o v i d e d

and

(EVA-84).

addiThis of t h e

H o w e v e r , no s y s t e m a t i c algorithm for

t h i s is w h y t h i s

c a s e was not i n t e g r a t e d in t h i s

CHAPTER

6

DESIGN TECHNIQUES - PARAMETRIC ROBUSTNESS

6. I . - I N T R O D U C T I O N

The previous

chapters

showed

that

the existence

tralized stabilization

(or pole placement)

problem

fixed

a

structure

modes.

When

given

feedback

depends

structure.

i s to s p e c i f y This

This problem was considered

numerical values

chapter

structural

considers

constraints

techniques.

that

actuator,

now the on t h e

uncertainties

are taken into account. line cuts...)

in

generally

unstable

fixed

in c h o o s i n g

in the last chapter.

of

modes,

a new feed-

The remaining

task

for the gain matrices.

and provides

Considerations

in t h e s e n s e

for the decen-

critically on the absence

results

s t a b i l i z a t i o n is i m p o s s i b l e a n d t h e s o l u t i o n c o n s i s t s back

of a solution

problem

of synthesis

and overview robustness

due

of such

to p a r a m e t e r

The effects of structural will b e s t u d i e d

of feedback

of appropriate

controllers

variations

design

are also included,

or external

perturbations

gains under

near-optimum

disturbances

(failure on sensors

or

in t h e f o l l o w i n g c h a p t e r .

6.2. - THE OPTIMIZATION PROBLEM Consider

x(t)

Yi(t) where vectors priate

a linear time-invariant

= A x(t)

+

Ci x(t)

x {E R n is t h e of the

it h

dimension.

system with S local control stations

:

S £ Biui(t) i:l (i=l . . . . . state

S)

vector,

local c o n t r o l Define )

(6.2.1)

u i ~= Rmi a n d

station.

A,

Bi,

Yi E R r i Ci are

are

the

constant

input matrices

and

output

of appro-

230

(6.2.2) then the equations

(6.2.1) b e c o m e :

~(t)

= A x(t)

y(t)

= C x(t)

+ B u(t)

(6.2.3 .a)

where the global number

of inputs

5

5

m=i~ ~

The the

mi

1

.

problem

global

system

performance

r=

Z i= 1

r

and outputs

(6.2.3.b)

is to f i n d

S local o u t p u t asymptotically

criterion

:

1

(6.2.3)

is

are given by

(or state) stable

and

feedback

controls

such

that

classical

quadratic

the

that

J is minimized.

co

J =

f

(x' O x + u' R u) dt

(6.2.4)

0 where

Q and

R are

weighting

nite and positive definite, Only thesis

static

feedback

of a dynamic

controller

matrices

of appropriate

dimensions, positive-semi

defi-

respectively. control

controller

for an augmented

can

system

will b e be

considered.

brougth

(SEN-79).

back

The to

reason

the

is that

synthesis

the

of

syn-

a static

T h i s i s s h o w n in t h e f o l l o w i n g s u b s e c -

tion.

6.2. I. - D y n a m i c

controllers

Assume that the controller ~(t) = A u(t)

where are

Z

given by

:

z(t) + G y(t)

(6.2.5.2)

= H z(t) + E y(t)

(6.2.5.b)

z £ R n 0 is t h e s t a t e a n d

constant

matrices

verifying

Az ( n o x n o ) ,

G {no x r ) ,

the

constraints

structure

H(m x n o ) and (Az,

G,

H,

E (mxr)

E are

block

d i a g o n a l m a t r i c e s if t h e c o n t r o l i s c o m p l e t e l y d e c e n t r a l i z e d ) .

Applying becomes

:

the

control

(6.2.5)

to the

system

(6.2.3),

the

closed-loop

system

231

[] [A+BEcBl[x ] GC This equation x*(t)

Az

z

can be rewritten

=

(A*

+

as

:

B* K* C*) X * ( t )

where

rA,, olo

A*4___t___l

r !ol.

,

Lo l O]no n

system

y*(t)

= C* x * ( t )

this

the

controllability

n

dynamic

I'E I~lm

no

LG

no

compensator

to s p e c i f y i n g

a static

I AzJ n o

1

(6.2.5)

compensator

n

(i.e.

0

the

matrices

u * = K* y * f o r t h e

the

+ B* u * ( t )

determination

of c h o o s i n g

order

no

l

:

= A* x * ( t )

the problem

6.2.2.

specifying

x*(t)

Note t h a t

ver

m

a n d A z) is e q u i v a l e n t

augmented

c*-- / - - - ' - - I Lo ','.1

L o l' J"o

no

Therefore, E,H,G

r iol

B-d-_.,-_ /

must

of the

augmented

the compensator

be

at

least

and observability

equal

order to

system

may be

delicate

because

n o is not completly solved yet.

min

(Uo' Uc ) '

whereu °

Howe-

a n d ~)c a r e

the

indices of the system.

- Static controllers

Suppose

now that

back with time-invari2nt u(t)

= K y(t)

the

control

feedback

u(t)

gains,

K s u b j e c t to

J =

i.e.

Y~(x' 0

Q x + u' Ru)

by

linear

static

output

feed-

:

= K C x(t)

(6.2.6)

the optimization problem can be formulated

rain

is generated

dt

as follows :

232

(6.2.7)

= Ax + Bu y=Cx u---Ky Applying

~(t) and x(t)

the control (6.2.6)

= (A + B K C ) x ( t ) is given by

x(t)

where ¢(t)

= [exp

(6.2.3),

the closed-loop system is :

(6.2.8)

= D x(t)

:

(A+BKC)t ] x(0)

is the transition

in the criterion

to t h e s y s t e m

x(O)

= ¢(t)

m a t r i x of t h e s y s t e m

expression,

(6.2.8).

Substituting

x(t)

and u(t)

we o b t a i n

c~

J(K)

= x'(0)

{f 0

@'(t)

J(K)

= x'(0)

P x(0)

Q + C ' K ' R KC

@ (t)

dr} x ( 0 )

(6.2.9)

or

(6.2.10)

where co

P = ~ ~'(t)

QI(K)

@(t) d t

Q I ( K ) = Q + C ' K f R KC If t h e plan,

matrix

D = A + BKC

has

all i t s

then P satisfies the matricial Lyapunov

eigenvalues

equation

in

the

(LEV-?0)

left

half

complex

:

f = D' P + PD + Q1 (K) = 0 It

is well k n o w n

condition of the solution

the solution

of this

state

x(0)°

To o v e r c o m e

this

for

a set

of initial

valuable

random

variable

identity

matrix,

(LEV-70) if x ( 0 )

Then taking

ioe.,

= T r (P X0)

where

X0 = E [ x(0)

with

given

is s u p p o s e d

the expectation

of the trace operator,

J(K)

that

x'(0) ]

tr(b'a)

problem

difficulty

conditions covariance

we

is depending and

consider

x(0)

X0 = E[x(0)

uniformly distributed

around

of both sides of (6.2.10) = tr (abt),

to o b t a i n as

on the initial a suboptimal a

zero

x ' ( 0 ) ] (X = 1 / n

mean I,

the origin).

and using

the properties

we g e t : (5.2.11)

I

233 The optimization problem

can be reformulated

in the following static

equivalent

f or m :

min J(K)

= Tr[

P X 0]

K subject to : f = D'P + PD + Q I ( K )

The

constraints

on

feedback matrices set

K

c

={K/K

where F(K)

F(K)

= 0

the

control

C Rmxr such that

indicates

the feedback

= K - block diag.(K 1.

F(K)

rain

introduced

defining

the

admissible

= 0}

. . . .

(6.2.12)

KS)

control.

the optimization problem can be rewritten

J(K)

by

loops that are not allowed.

in t h e c a s e of c o m p l e t l y d e c e n t r a l i z e d

Finaly,

are

•

= Tr [P

as

:

X0 ]

K~K

c s u b j e c t to : f = D*P + PD + Q I ( K ) QI(K)

(6.2.13)

= 0

= C'K'RKC + Q

D = A + BKC X0 = E [x(0)

x'(0)]

I n t h e c a s e of f u l l s t a t e

min

J(K)

= Tr [P

feedback

the problem

(6.2.13)

becomes

:

X O]

K~K ¢ s u b j e c t to : f = D ' P + PD + Q + KIRK = 0

(6.2.14)

D=A+BK x 0 = F: [ x(O) x ' ( O ) ]

6.2.3.

- Necessary gradient

The

conditions for optimality :

matrix calculation

gradient

matrix

can

be obtained

analytically

by applying

the perturbation

234 theory

on the criterion

(6.2.9)

(LEV-70)

provided by Geromel and Bernussou

Let respect

f(K)

to i t s

be

a

scalar

arguments.

function By

( s e e A p p e n d i x 5) b u t a s i m p l e r m e t h o d w a s

(GER-79b, BER-81,

of

the

definition,

matrix

the

BER-82).

K w h i c h is

matrix

gradient

differentiable

df/dK

is

the

with

matrix

whose elements are ~ f/Sk... 11 The static case Theorem 6.1.

:

Let f ( x ,

to all i t s a r g u m e n t s

G(x))

b e a s c a l a r f u n c t i o n w h i c h is d i f f e r e n t i a b l e w i t h r e s p e c t

; x and G(x) are matrices.

Then

:

d_~_~ ((x), G(x)) : a__L (x, y, z) dX ~}x where

y,

z are

the

matrices

obtained

s t a t i o n a r i t y of t h e L a g r a n g i a n f u n c t i o n

L

= f (x, y) + Tr {z'(G(x)

by

solving

the

necessary

conditions

for the

:

- y) }

T h e d y n a m i c c a s e : Let t h e o b j e c t i v e f u n c t i o n b e : T J(K) =

f 0

f(x,

K) dt + g ( x ( T ) )

where f and g are scalar functions

(differentiable with respect

x, K are matrices linked across the differential system

]~ = F ( x ( t ) ,

K)

;

x(0)

=

x0

Theorem 6.2. T d3

where H (x,

:

0f

~H ~

(x, z, K) dt

z, K) is t h e H a m i l t o n i a n f u n c t i o n

I! = f ( x , K) + T r [ z ' F ( x ,

and x,

K)]

z satisfy the conditions of stationarity 8H £ = -~Z = _8 FI 8x

x(0) = x 0 z(T) =

:

d~ d x(T)

:

:

to t h e i r a r g u m e n t s )

;

235 Applying

Theorem

gian function L = Tr

(6.1)

to our problem in its static

with

= 2 (RKC

g(K,X0) QI(K)

= DS

tions.

+ Q1

= C'K'RKC

Hence,

for

the initial (HOS-77)

This

this

result

is o b t a i n e d

K,

the

gradient

matrix

(Lyapunov

is a reasonable

by using

can be

equations),

task

even

the method of varia-

obtained

by

solving

one of them depending

for large

scale

system

two on (see

states

are measurable

then

u = Kx a n d t h e n e c e s s a r y

condi-

become :

f(K) = D'P + P D g ( K , X 0) = D S

where

5,

(DAV-68)).

] ~--J= 2 ( R K - B ' P ) S with

(6.2.15c)

X 0 = E [ x(0) x(0)']

a given

i f all t h e s y s t e m tions (6.2.15)

(6.2.15a) (6.2.15b)

(K) = 0

matricial equations

conditions. and

:

+ Q

and

6.1 - In Appendix

linear uncoupled

conditions

+ SD' + X 0 = 0

D = A ¢ BKC

Remark

with the Lagran-

- B'P) SC' = 0

f(K) = D'P + P D

where

(6.2.13),

(P X 0) + T r [ S ' f ]

we o b t a i n t h e f o l l o w i n g n e c e s s a r y d3

form

:

= 0

(6.2.16a)

+ Q + K'RK

+ SD

= 0

(6.2.16b)

+ X0 = 0

(6.2.16c)

D = A + BK

The optimal feedback

matrix,

in t h i s c a ~ e , i s g i v e n b y

:

K = - R -1 B I P

(6.2.17)

where P is the symetric

definite positive solution of the Riccati equation

A'P + PA - PBR-1B'p

Observe not depend In the problem

that

the

:

+ Q = 0

equation

(6.2.16c)

(6.2.18) becomes

unusefull

and

the

control

does

on the initial conditions. general

case,

the otpimisation

problem

is thus

which can be solved by using one of the design

following section.

reduced

techniques

to a p a r a m e t r i c presented

in the

236 6.B.

- DECENTRALIZED

CONTROL

This section provides blems

6,2.15

Some

use

or

6.2.16.

gradient

PARAMETRIC

OPTIMIZATION

a s o l u t i o n to t h e n e a r - o p t i m u m

The

matrix

BY

presented

techniques

algorithms (GER-79a,b)

others

do n o t ( G E R - 8 4 ) .

6.3.1.

- The algorithm of Geromel and Bernussou Geromel and Bernussou

propose

are

decentralized

based

on

(JAM-83),

control pro-

iterative and

schemes,

(CHE-84))

and

(GER-79)

a gradient

method

(ZOU-70)

which is summari-

zed below : 1 - Initialization by an admissible

gain matrix,

diagonal feedback

that the closed-loop system is stable.

matrices),

such

2 - Calculation of the gradient

i.e.

K E: K d

s e t of b l o c k

matrix.

3 -

Determination

of a feasible

direction

G (for which

and

the

constraint

verified)

and

structure

(K d i s t h e

is

of the

the

cost

function

decreases

s t e p a of p r o g r e s s i o n

in

this

direction. 4 - Convergence

test.

If the test is satisfied

: stop

; otherwise,

update

K by K -aG

a n d go to 2.

6.3.1.a

- Initialization of the procedure

The problem

of f i n d i n g

authers

(AOK-73),

76a,b),

(GER-79a,b),

tano and Singh

: Algorithm of Armentano

an admissible

(WAN-78b),

(FES-79),

(ARM-81)...

stabilizing (IKE-79),

Among those,

(ARM-SI) which always provides

and Stngh

gain was considered (SEZ-81b),

we p r e s e n t a solution

by many

(DAV-76a},

(COR-

the algorithm of Armen(of course,

if no unstable

fixed modes exist). The approach the

closed-loop

with respect

v and

The

to t h e f e e d b a c k

Q= ~kij

where

consists in minimizing the real part

system,

-W' b

i

gradient

of an eigenvalue

gains is given by

ci v

of the dominant eigenvalue k 0 of the

(see Chapter

closed-loop

of

system

3) :

i=l,...,m j: l,...,r

0

ilk.

w are the right

b i is t h e i t h c o l u m n o f B , a n d

=

U

0

and left eigenvectors

associated

c i is the jth row of C.

with the eigenvalue

)~0'

237

The (kij)

gradient

of the

can be expressed

gi] = Re

real

part

o f X0 w i t h

as a matrix G = (gij),

respect where

to

the

feedback

gain

K =

:

]

t. akij

j

The algorithm is the following : Step 1 : Choose an arbitrary Step 2 :

Compute

Otherwise,

the

eigenvalue

compute the right

Xd" Step 3 : Compute

the gradient

S t e p 4 : Do a s t e e p e s t -aG,

K E Kd .

dominant

where

matrix

of

A+BKC.

If

it i s

negative,

stop.

v d a n d w t a s s o c i a t e d to

G d.

descent

a is t h e s t e p

kd

and left eigenvectors

search

size.

in the direction

Go to s t e p

of -G and update

K = K

2.

Remark 6.2. a) If t h e d o m i n a n t e i g e n v a l u e

with positive

local m i n i m u m , t h e a l g o r i t h m m u s t b e s t a r t e d b)

The unidimensional

like q u a d r a t i c

search

or null real part

turns

again from a different

may be performed

by

using

o u t to b e a

matrix. efficient

algorithms

interpolation.

c) I f , a t a n y i t e r a t i o n ,

the dominant eigenvalue

is multiple,

perturb

slightly

K

( W A N - 7 3 a ) in s u c h a w a y t h a t it b e c o m e s s i m p l e .

6.3.1.b.

- Feasible direction

Assume

that

matrix is not zero. existence

K ~ K d is an

admissible

initial condition

T h e p r o b l e m i s to d e t e r m i n e

of a step length a such

t h a t J (K - a G )

a matrix

that

the

gradient

to g u a r a n t e e

( J (K) f o r all 0 ( a ( a

a ) 0. It is e a s y to s e e t h a t t h i s m a t r i x c a n b e o b t a i n e d 7 9 b , 82)

such

G in o r d e r

the

and some

by solving the problem

(GER-

:

min ~I

Tr[ (d3 ~

- G ) ' ( ~d3

- G)]

(6.3.1)

s u b j e c t to : F (G) = 0 If t h e c r i t e r i o n thogonal projection in (GER-79b)

is written

of dJ/dK

explicitly~

we e n d

up

on the linear set defined

with

the

problem

by the constraint.

of the

or-

It is shown

that the solution is "

G = block-diag. ( ~d3 )

(6.3.2)

238 Therefore,

the

o r t h o g o n a l p r o j e c t i o n onto

the

decentralization

set

is

easily

o b t a i n e d b y s e t t i n g all t h e o f f - b l o c k - d i a g o n a l t e r m s of t h e g r a d i e n t m a t r i x to z e r o . This

result

can

be

generalized

to a r b i t r a r y

structure

constraints

and

the

p r o j e c t i o n of t h e g r a d i e n t m a t r i x is g i v e n b y :

G = (gij)

The

=

/

d~0~ H

if

kij

if

kij = 0

e x i s t e n c e of a feasible

I

0

direction

(6.3.3)

matrix

is

guaranted

by

the

following

lemma: Lemma 6.1 (GER-82). The optimal s o l u t i o n of t h e p r o b l e m ( 6 . 3 . 1 ) is s u c h t h a t : a) if

IIo~

0 then

~ > 0 such that J(K - a G )

< J(K)

0~

a

~

~-

b) if [[G[[ = 0 t h e n the m a t r i x K s a t i s f i e s t h e K u h n - T u c k e r c o n d i t i o n s of t h e problem min J ( K ) w h e r e Kd r e p r e s e n t s t h e d e c e n t r a l i z a t i o n c o n s t r a i n t . KEK d This

lemma

guarantees

the

e x i s t e n c e of a

step

sizes

such

that

(K - a G )

s t a b i l i z e s t h e s y s t e m , b u t it does n o t give a n y i n f o r m a t i o n a b o u t its c h o i c e . At e a c h i t e r a t i o n , ¢z may b e f i x e d b y s o l v i n g t h e following s i n g l e v a r i a b l e optimization p r o b l e m along t h e s e a r c h d i r e c t i o n : rain

J (K - a G )

(6.3.4)

a~/0 H o w e v e r , in p r a c t i c e , it is g e n e r a l l y a d v a n t a g e o u s to a d o p t a h e u r i s t i c method of a d a p t a t i o n of t h e s t e p d u r i n g t h e i t e r a t i o n . In (GER-79a), t h e following a d a p t i v e rule is p r o p o s e d : Let i be t h e i t e r a t i o n i n d e x , t h e n : i+l =~ i i i a = Va

if

J ( K i - a iGi ) < J ( K i) a n d P (Ki-C~G i) > 0

otherwise

w h e r e ~ > 1 a n d 0 ( v ( l0 The initial value of a is a r b i t r a r y .

(6.~.5)

239 6.3.1.c

- The properties

of the algorithm

The algorithm presented

above has the two following properties

1 - The stability of the overall system is guaranted

The step size a must be chosen i)

at each iteration.

-"

(K - ( ~ G ) ~ K s w i t h K s =[ K f A + B K C s t a b l e ]

ii) J (K - a G ) It i s s h o w n

x( J ( K ) in

positiveness

of P, i.e.

with respect The

nonlinear

that

algorithm

X0 the

by means of the definite-

> 0.

t o a local o p t i m u m

optimization

problem

is summarized

routine,

P(K)

Since the algorithm requires

(i) c a n b e p e r f o r m e d

to K b u t t h e c o n v e r g e n c e

Fortran

X0] x( T r

s u c h c~ e x i s t s .

P, the test

P(K -~G)

2 - Uniforme convergence The

or Tr [P(K-aG)

(GER-79a)

calculation of the matrices

ponding

such that

:

is

I

here

is

not

to a local o p t i m u m i s e n s u r e d

by

provided

considered

the

organigram

in A p p e n d i x

of Figure

generally

(GER-79a). 6.1.

6.

A, B, C, X 0 '0

E

c~

i L

c o m p u t e an initial gain ~

K

Solve t h e Lyapunov e q u a t i o n s to g e t P a n d S

i

convex

The

corres-

240

..,

i

Compute the gradient matrix d J/d< =,

Gradient projection [ ~ G = ~ kij

i~ kij ~ 0 if kij : 0

I, 0

K : K - aG

YES

I

°-

] I

STOP

]

Figure 6.1 Example 6.1 ( G E R - 7 % ) . -

:

Consider the system (composed of three subsystems

I

I 0~5

I

I

i 0.6

0

1

I

l

0

I

-2

-3 I 1

0

-o~- - i - T o o o~ / z

2

I J

05

3 0

I 0 I I 0 I -3

0 1

1

0

I

1

0

0.5 I 0,5 I

0

~0

I I

--F7 o] X+

-#

I

and the quadratic cost : J =f 0

I

(x'x + uIu) dt

Consider the case of decentralized state feedback :

IO

# I I 3 I

: S=3) :

241

(K : b l o c k d i a g o n a l m a t r i x )

u--Kx

w i t h X¢~ = I a n d t h e i n i t i a l g a i n m a t r i x

,l l 0.',

K1

I

3.41

1

I1

14.96

0.06

I

With a 1 = 0 , 0 1 ,

~= 0,5

•21

I -

K 59

:

a n d ~ = 1 , 2 , we h a v e

0.35 1 -

-

-[

~

3-

--o.-;-~iI

I 0,87

2.46 I ~1.58

L

0.24

I

which satisfies

the convergence

:

1

test with e = 0,001.

T h e c o s t g o e s f r o m J ( K 1) = 8 , 7 6

to J (K 59) = 3.

6.3.2.

- The algorithm of Jamshldi

Instead by

using

of using

the

(JAM-83)

the feasible direction

well-known

method,

Davidon-Fletcher-Powell

The algorithm is the following

the problem

variable

metric

(6.2.16)

is solved

method

(FLE-63).

:

S t e p 1 : S e l e c t a n i n i t i a l m a t r i x K = K 0 a n d s e t i = O. Step 2 : Solve

the

Lyapunov

equations

(6.2.16b

and

c)

using

A,

B,

Q,

R,

X0,

K

a n d t h e a l g o r i t h m o f D a v i s o n a n d Man ( D A V - 6 8 ) . Evaluate

the gradient

~J/~K and the criterion

Step 3 : Use the Fletcher-Powel K i - n e w = Ki - o l d + a D i w h e r e Di is t h e s e a r c h Step 4 : Check

whether

a prespecified

m e t h o d to u p d a t e

direction

during

the convergence

tolerance

value.

J(K). the gain matrix Ki:

the ith iteration.

is achievedp

e.g.,

If the convergence

II~ 3 / ~ K i l

is reached,

g o to s t e p 2. The following example illustrates

the use of this algorithm

:

(¢, stop,

where E is otherwise

242

000 1

Example 6.2 (JAM-83).

-l

-2

Consider

a fourth-order

I 1

-~

-

~

X+

0,0.

-o,~ I o

o.2~

and the cost function

:

l .

.

.

.

.

]

0, /

L

-o.2

s y s t e m w i t h two s u b s y s t e m s

I

I A

:

co

J = 1/2 where

f 0

(x' Q x + u' Ru) dt

Q = 21 a n d

R = 21, T h e p r o b l e m i s to f i n d a s t a t e

matrix

:

k12 LI . 0. . . . 0

kl I

K =

feedback

0

0

[

k23

k24

such that J is minimized : Take X 0 = I, and consider

KO:

.l

0.1~l 0 0 i 0.I

these two initial values

0 0.I

f o r K0:

0 - - 6 2 I 3"-

KO

I

Then,

the

respectively.

algorithm

The resulting

of J a m s h i d i averaged

converges

.414 0.168II 0

K*=

0

in 7 and

13 i t e r a t i o n s

o p t i m a l g a i n K* i s f o u n d to b e

(¢ = 0 , 0 1 ) ,

:

01

I 0.35

0,58

I

with J(K*) = 1.0985. 6.3.3.

Iterative

procedure

of Chen et al.

The algorithm of Geromel and gence

and

excessive

block-diagonal

computation

feedback

(CHE-84)

Bernussou per

cycle

(see § 6.3.1) due

matrix at each iteration.

to t h e

suffers

requirement

of poor converof retaining

a

243

T h e following i t e r a t i v e p r o c e d u r e was p r o p o s e d to overcome t h e s e d e s a d v a n rages.

It p r e s e r v e s

t h e p r o p e r t i e s of u n i f o r m c o n v e r g e n c e to a local minimum and

g u a r a n t e e s t h e s t a b i l i t y of t h e global s y s t e m at e a c h i t e r a t i o n . The p r o c e d u r e c o n s i t s in s t a r t i n g t h e i t e r a t i o n s with an optimal s t a t e f e e d b a c k matrix (full m a t r i x g i v e n b y (6.2.17) a n d s e t t i n g s u c c e s s i v e l y t h e o f f - d i a g o n a l b l o c k s to z e r o . At e a c h i t e r a t i o n , an optimal f e e d b a c k m a t r i x is f o u n d u s i n g a c o n j u g a t e g r a d i e n t m e t h o d . The p r o c e d u r e is s u m m a r i z e d below : Step 1 : D e t e r m i n e P b y s o l v i n g t h e a l g e b r a i c Riccati e q u a t i o n ( 6 . 2 . 1 8 ) . Compute t h e full gain matrix K g i v e n b y ( 6 . 2 . 1 7 ) .

Set j=l a n d Kj = K.

Step 2 : a) write K j as : K

KJ::

l

K21

Kl2 ....... KIS

K22 . . . . . . . r

KSI

,,.

KS2 . . . . . .

KSS

b) D i s c a r d one block from Kj, s a y K1S :

Kll

KI2 . . . .

KI,S_ 1

K21

K22 . . . . . .

0 K2S

Ki=

,

KSI

KS2 . . . . . . . .

i:

j+l

~.Ks$

c) Calculate t h e e i g e n v a l u e s of the m a t r i x (A-BK i) . If t h e m a t r i x is s t a b l e , go to s t e p 3. O t h e r w i s e , c o n s i d e r Ki as t h e initial g a i n m a t r i x for t h e algorithm of Armentano and

Singh

(see

§ 6.3.1a).

Find a gain m a t r i x K (with t h e same

s t r u c t u r e as Ki) w h i c h s t a b i l i z e s ( A - B K ) . Set Ki = K. Step 3 : Using t h e t e c h n i q u e of Hoskins e t al.

(HOS-77), solve t h e L y a p u n o v e q u a -

t i o n s ( 6 . 2 . 1 6 b and c) and compute t h e g r a d i e n t

(6.2.16a).

Then, a conjugate

g r a d i e n t r o u t i n e with a q u a d r a t i c i n t e r p o l a t i o n in t h e o n e - d i m e n s i o n a l s e a r c h is u s e d to o b t a i n t h e minimum of t h e p e r f o r m a n c e i n d e x . Step 4 : D i s c a r d a n o t h e r block from Ki, s a y KS1 :

244

K

K i+l =

K 12 . . . . . .

11

K22,. K21 ! Ks_l~ 1 I I

K l,S-I

0 - K2S

"-.

,

, 2 - . KSS 0 K 8, S e t i=i+l a n d go to s t e p 2. T h e p r o c e d u r e

g o e s o n u n t i l Ki h a s a b l o c k - d i a g o -

nal form. E x a m p l e 6.3 ( C H E - 8 4 ) . Consider again the Example 6.1. The procedure

begins with : -

!

0.9004

0.2487

i

I 0.1887

0.1823

t[ 0.3117

0.04

0.4902

! 0.4006

0.1202

I

-r 0.2761

0.29

[

1.143

I I

K1 = 0,4608

I [

I 0.6536

0.2684

2,219

i 0.1366

i

--

0.3873

0.1207

I

After six iterations the procedure

-0.031

I

I

0.6068

[ 0.2878 I

1,184

0.32

0

0

I

gives

•

I

1.145

0.3162

I

0

0

1.264

0.4981

0

0

0.7892

2.224

0

0

1.366

0,2269

[ I

I

I

0

0

I

K6=

I I

0

0

I i

l

o

o

I

o

o

I

I

I I

T h e c r i t e r i o n v a l u e s a r e J ( K 1) = 2.718 a n d J(K 6) = 3 . 0 0 7 . Remark 6.3. 1. S i n c e t h e r e

is no constraint

a p p l i c a b l e f o r an a r b i t r a r y 2. T h e p r o c e d u r e initial

gain

(6.2.18))

matrix

in e l i m i n a t i n g

control structure

can be used

for output

K = R - 1 B I P C -1

and replace the equation

(P

feedback.

being

(6.2.16)

off-diagonal

blocks,

the

procedure

is

without modifications. the

I f ~1 e x i s t s , solution

by (6.2.15).

of the

one can take the Riccati

equation

If C -1 d o e s n o t e x i s t ,

then

245 the initial gain matrix

can be found by using

the algorithm

of Armentano

and

Singh

(see § 6.3.1a). 3. This procedure was compared with the algorithm of Geromel and B e r n u s s o u 6.3.1)

(CHE-84).

Both algorithms were implemented on a PDP-10

(see §

and the recorded

C P U times for Example 6.3 were :

Geromel a n d B e r n u s s o u

algorithm

Initial block-diagonalization of Geromel and Bernussou

by the technique (GER-79a) .............................

53.2 sec

suboptimal hlock-diagona lization..................................

101.26 sac

Procedure

o f C h e n e t al

Initialization

(optimal full gain matrix)

Suboptimal block-diagonallzation The numbers

6.3.4.

show a significant

Iterative

procedure

The problem

K

c

under

structural

= {K / K ~ R m x r s u c h t h a t

constraint

F ( K ) = K [ I - C' ( C C ' ) -1 C in t h e e a s e o f c e n t r a l i z e d

F(K)

output

= K - block-diag.

in t h e c a s e o f d e c e n t r a l i z e d We r e c a l l

that

g i v e n in ( 6 . 2 . 1 7 )

by

the

(GER-84)

constraints

T h e s e t of a d m i s s i b l e f e e d b a c k

where F(K) is the structure

Z.56 s e c 35.81 s e c

computational improvement.

of Geromel and Peres

(6.2.16)

and Peres in (GER-84).

~......................

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

F(K)

was considered

= 0 }

given by

Geromel :

(6.2.11)

:

]

(6.2.12a)

feedback

(K1, K2,

by

matrices is defined by

,..,

control and

KN)

(6.Z.lZb)

control.

optimal

solution

for the

state

feedback

control

:

K* = R - 1 B ' p

where P is the symetric positive

solution of the Riccati equation

(6.2.18).

problem

is

246

A'P + P A

which can be,

- PBR-IB'p

+ Q = 0

for simplicity, written

as :

(P) = Q The iterative perty

procedure

of Geromel and

t h a t if a g a i n m a t r i x K s a t i s f i e s

Peres

(GER-84)

is based

on the pro-

:

K + L = R-1B'p

(6.3.6)

w h e r e P i s t h e p o s i t i v e d e f i n i t e s o l u t i o n of t h e R i c c a f i e q u a t i o n

:

(P) = Q + L ' R L

then,

the matrix

Since

(A-BK) is asymptotically

L is a r b i t r a r y ,

showed that L is given by

L = F

where

so

= (J-

chosen

such

that

K ~ K . Geromel and c

Peres

:

(6.3.8)

of suboptimality

J*)

J* i s t h e

(6.2.17)

it c a n b e

stable for any matrix L.

(R-IB'p)

The degree

d

(6.3.7)

o f t h e s o l u t i o n is d e f i n e d b y :

/ J*

cost value

corresponding

to t h e

optimal

full gain

matrix

given

by

step

5,

end J the cost value at the convergence.

The following procedure

was proposed

Step 1 : Set the iteration index

:

to z e r o : i=0.

Step 2 : Solve the Riccati equation

:

(Pi) = Q + Ll ' R L.l if i = 0, calculate also J* = 1[2 T r (P0) Step 3 : Calculate Li+l = F (R-1B'Pi).

S t e p 4 : If

|Li+ I

otherwise

Li~¢

(¢ b e i n g

a positive

s e t i = i+l a n d go to s t e p

S t e p 5 : C a l c u l a t e J = 1/2 T r The degree SO

of suboptimality

= (J - J*)/

J*

real

(Pi) a n d K ~ K c b y s e t t i n g

K = R - 1 B ' P i - Li+ 1 d

small

2.

is given by

:

•

number),

go

to

247

Remark 6 . 4 . 1. T h e p e r f o r m a n c e of t h e p r o c e d u r e

d e p e n d s m a i n l y on t h e n u m e r i c a l m e t h o d u s e d

to solve t h e R i c c a t i e q u a t i o n at s t e p 2. We p r o p o s e t h e a l g o r i t h m of K l e i n m a n (KLE-

68). 2. At t h e c o n v e r g e n c e we h a v e : (Pi) = Qi = Q + L'1 R L.1 Where Qi is a s y m m e t r i c s e m i - p o s i t i v e d e f i n i t e m a t r i x , viewed a s a l i n e a r - q u a d r a t i c

T h e m e t h o d c a n t h e r e f o r e be

d e s i g n which determines the weighting state

m a t r i x in

o r d e r to g e t K ~ K . c Because

Qi )

Q'

the

objective

function

increases,

and

the

control

will

be

n e a r - o p t i m a l . T h i s is t h e " p r i c e " for a n optimal c o n t r o l w i t h t h e d e s i r e d s t r u c t u r e . 3. T h e s o l u t i o n i s i n d e p e n d e n t

of t h e

initial c o n d i t i o n

X(0)

a n d is

such

that

the

closed-loop s y s t e m is a s y m p t o t i c a l l y s t a b l e . 4. It is e a s y to t a k e i n t o a c c o u n t a n a r b i t r a r y c o n t r o l s t r u c t u r e . 6.3.5. Comments

T h e d e s i g n t e c h n i q u e s p r e s e n t e d in t h i s s e c t i o n h a v e two d i s a d v a n t a g e s : - T h e y c o n v e r g e to a local minimum of t h e c r i t e r i o n .

T h e s o l u t i o n is t h e r e f o r e

su, o p t i m a l . - They are they are b a s e d

not a d e q u a t e for v e r y large scale s y s t e m s

on a n o p t i m i z a t i o n in t h e p a r a m e t e r

space

(e.g.

and

n=100).

require

Indeed,

calculations

i n v o l v i n g t h e global s y s t e m ( r e s o l u t i o n of m a t r i c i a l e q u a t i o n s ( L y a p u n o v o r R i c c a t i ) ) .

However,

for m a n y p r o b l e m of p r a c t i c a l i n t e r e s t ,

they represent

viable design

techniques.

6.4. - D E S I G N

OF

ROBUST

DECENTRALIZED

CONTROLLERS

T h i s s e c t i o n is c o n c e r n e d w i t h t h e s y n t h e s i s of d e c e n t r a l i z e d c o n t r o l l e r s w h e n u n c e r t a i n t i e s d u e to p a r a m e t e r v a r i a t i o n s m u s t b e t a k e n i n t o a c c o u n t . Three

aspects

of r o b u s t n e s s

cribed d e g r e e of s t a b i l i t y parameter variations

are

(AND-71),

(without

considered : control synthesis

and under

disturbances).

The

small ( T A R - 8 5 ) presence

with a p r e s -

or large

of e x t e r n a l

(PEK-83)

disturbance

s i g n a l s will b e c o n s i d e r e d in t h e n e x t s e c t i o n d e a l i n g w i t h t h e s t u d y of t h e " r o b u s t d e c e n t r a l i z e d s e r v o m e c h a n i s m p r o b l e m N.

248 6 . 4 . 1 . Controllers with a p r e s c r i b e d

degree of stability

T h i s s e c t i o n d i s c u s s e s t h e p r o b l e m of pole p l a c e m e n t in a p r e s c r i b e d the

complex

plane

which

p r a c t i c a l p o i n t of v i e w ,

is

more

it is n o t

closed-loop system but rather

general

than

the

stabilization

e s s e n t i a l to fix p r e c i s e l y

to e n s u r e

r e g i o n of

problem.

From a

t h e e i g e n v a l u e s of the

that these eigenvalues are within a certain

r e g i o n of t h e complex p l a n e . T y p i c a l d e s i r e d r e g i o n s a r e s h o w n in F i g u r e 6 . 2 . Im

Im

Re

Re t

Fig. 6.2.

Here,

we

are

v

J

j

"

/

•

Two c a s e s of p r a c t i c a l s t a b i l i t y m a r g i n s

concerned

by

regions

as

shown

in

Figure

6.2b

(For

details

c o n c e r n i n g F i g u r e 6 . 2 a , see ( K A W - 8 1 ) ) .

T h e o p t i m i z a t i o n p r o b l e m is : =o

rain

J =

K

f 0

{x'Qx + u'Ru) dt

s u b j e c t to x = Ax + Bu u=-Kx

a n d s u c h t h a t t h e e i g e n v a l u e s of t h e

c l o s e d - l o o p s y s t e m a r e l o c a t e d in t h e d a s h e d

r e g i o n of F i g u r e 6 . 2 b ( i . e . e i g e n v a l u e s h a v e r e a l p a r t s l e s s t h a n - a ) . A n d e r s o n a n d Moore (AND-71) s h o w e d t h a t t h i s p r o b l e m i s e q u i v a l e n t to : rain J* =

-]°~e2at ( x ' Qx + u ' R u ) dt

(6.4.1a)

s u b j e c t to x = A x + Bu U--Which c a n

itself

KX be

brought

following v a r i a b l e t r a n s f o r m a t i o n •

back

to

a standard

LQ-problem by

using

the

249 c~t

~=e x ~ = ec~t u

Then,

the problem becomes rain j r -

:

{ ~ ( ~ ' Q "~ + 91 RE) d t

K subject

to ~ = A ~ + B ~ ,

'I

A = A + c~I

~=-K~ Its solution must fulfill the following necessary

conditions

(6.4.1b)

(see § 6.Z).

~J = Z (RK - B' P) S ~K (A - B K ) ' S + S (A - BK) + K' RK + Q = 0 (A - B K ) P + P (A - BK) + X 0 = 0 The existence the pair

of a solution is guaranted

QIIz)

(A,

controllable and

(A,

is o b s e r v a b l e . Q1/2)

Anderson

is observable,

if t h e p a i r

( A , B) i s c o n t r o l l a b l e

and

showed

then

Moore

( A : B)

and

(A:

that

if

QI/2)

(A,

and B)

is

are also con-

trollable and observable.

the

solutions

U=-K~ ~ u = - Kx.

Observe

This

solution ensures

the original system,

and the corresponding

This

means

that

that

to

garanty

that

of

(6.4.1a)

the

closed-loop system is less than -a,

and

(6.4.1b)

with degree values

real

part

it s u f f i c e s

a,

are

of the criteria of the

same,

A by

i.e.

stability of

J and J'are

equal.

eigenvalue

of the

dominant

to r e p l a c e

the

the asymptotic

(A + a I) in t h e o r i g i -

nal o p t i m i z a t i o n p r o b l e m .

back

The above

result

control

the

region.

if

In this case,

to s o l v e e q u a t i o n s

6.4.2.

is still valid

system

has

for output

no

decentralized

(6.2.15)

after

changing

optimal

system

control

optimal f o r into a c c o u n t

parameters

determined

a different

for

set

the uncertainties

may

e set

arising

or

be

decentralized

modes

out

of

the

feeddesired

in t h e l a s t s e c t i o n

(A + a I ) .

reduction

uncertain

and

of nominal parameter This

for

presented

index sensitivity

vary

of values.

A by

or

fixed

we c a n u s e o n e o f t h e a l g o r i t h m s

Optimal control with performance

The

feedback

section

is

values

considers

from small parameter

it

(TAR-85)

that

the

not

generally

the problem

of taking

variations.

is

clear

250 In a s t o c h a s t i c approach Gaussian noises.

uncertainties

The control strategy

uncertainties but reduces variations.

the

are

m o d e l l e d in

general

by

white

p r o v i d e d in t h i s s e c t i o n d o e s n o t model the

t h e c r i t e r i o n s e n s i t i v i t y w i t h r e s p e c t to s y s t e m p a r a m e t e r

If a p a r a m e t e r d e v i a t e s from i t s n o m i n a l v a l u e , t h e r e d u c e d s e n s i t i v i t y of

t h e p l a n t p e r f o r m a n c e i n d e x e n s u r e s t h a t t h e p l a n t b e h a v i o u r will n o t c h a n g e d r a m a tically.

T h e m e a s u r e of t h i s s e n s i t i v i t y is o b t a i n e d b y

(TAR-85)

and alternate ap-

p r o a c h is in ( Y A H - 7 7 ) .

6 . 4 . 2 a . Formulation of the problem (TAR-85) Let q b e a s c a l a r p a r a m e t e r of t h e s y s t e m . I f q c h a n g e s to q + d q t h e p e r f o r m a n c e i n d e x c a n t h e n be a p p r o x i m a t e d b y : J (q + Aq) = J ( q ) + d J ( q ) Hence,

= J(q)

+ dq . dJ/dq

t h e v a r i a t i o n of J is p r o p o r t i o n a l

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

to q.

T h e s e n s i t i v i t y c a n be m e a s u r e d b y t h e n o r m of t h i s d e r i v a t i v e . C o n s i d e r t h e s y s t e m ( C , A , B ) of ( 6 . 2 . 3 )

and the optimization problem :

¢o

min

JI(A,B,C,K)

=

K

f ( x ' Q x + u ' R u ) dt 0

s u b j e c t to J¢ = (A - BKC) x = Dx

where Q and

R are

semi-positive and positive definite matrices, respectively.

It is

s h o w n ( s e e § 6 . 2 ) t h a t t h i s p r o b l e m c a n be w r i t t e n in t h e following s t a t i c form :

min

Jl

(A,B,C,K,P) = T r (PX 0)

s u b j e c t to : f (A,B,C,K)

(6.4.2)

= D'P + PD + Q1 = 0

w i t h Q1 = Q + C' K' RKC X0 = E (x(0) x ( 0 ) ' ) T h e s e n t i v i t i e s of J l c a l c u l a t e d u s i n g T h e o r e m 6.1 tions

(TAR-85),

this result

(A,B,C,K,P) w i t h r e s p e c t to s y s t e m p a r a m e t e r s c a n be (we o b t a i n t h e d e r i v a t i v e s u s i n g t h e m e t h o d of v a r i a is g i v e n i n A p p e n d i x

5).

T h e o r e m 6.1 a p p l i e d to t h e p r o b l e m ( 6 . 4 . 2 ) i s w r i t t e n :

LI(A,B,C,K,P,S)

= T r (PX0) + T r (S' f ( A , B , C , K ) )

The

L a g r a n g i a n f u n c t i o n of

251 T h e s e n s i t i v i t i e s of t h e p e r f o r m a n c e i n d e x c a n b e o b t a i n e d a n a l y t i c a l l y 831 3A = 8A a31 3B = a B

a L1 = 8A a L1 = a B

a Jl JC =

8C

= 2 PS

- - 2 PSC'K'

8I"1 = 8---C = 2 K ' ( R K C -

a 31 8L I JK = a K = a-K

:

B'P)S (6.4.3)

= 2 ( R K C - B'P)SC'

8 L1 a----5 = f (A,B,C,K,P) = D'P+PD + Q I ( C , K ) = 0 8 L1 8----P = g (A,B,C,K,S) =

S i n c e we w a n t to k e e p

theses

DS + SD + X 0 = 0

sensitivities

f o r m a n c e i n d e x J a s small a s p o s s i b l e ,

near

to z e r o a n d m a i n t a i n t h e

per-

we d e f i n e a n e w p e r f o r m a n c e i n d e x a s :

|

w h e r e L, M, E, F a r e s y m m e t r i c s e m i - d e f i n i t e m a t r i c e s o f a p p r o p r i a t e

Observing

that the equations

(6.4.3)

can be r e w r i t t e n

dimension.

as follows :

J A = 2 PS JB = - JA " C' K' w i t h JKC = (RKC-B'P)S

JC = K'. JKC JK = J K C " C'

we n o t e t h a t if JA = 0 ( o r 3 A < ¢ A ) a n d JKC = 0 ( o r J K C ~ E K C ) ties

are

zero

(or

sufficiently

small).

Therefore,

the

problem

can

c o n s i d e r i n g o n l y t h e s e n s i t i v i t i e s JAB a n d JKC o T h i s is e q u i v a l e n t

all t h e be

sensitivi-

simplified by

to a s s u m i n g

that

252

A a n d B in o n e h a n d , applying

Theorem

and K and

_

~'AB

~ 3

JAB

(JKc)

variations variations

of

is

A and

L and

(K

= 31 +~-

F are

and

= Tr

vary

simu taneously.

Then~

3A

B'P)$

of

C).

the

These

criterion results

with

are

1

weighting

Tr (JKC "

matrices.

respect

obtained

The new performance

Tr (3AB. L. 3',~ 3) +~-

appropriate

can then be formulated min J 3 ( K )

sensitivity

5 (TAR-85).

l

J3 (I<)

where

B

in Appendix

=

= 2 (RKC-

the

hand,

:

= 2 PS

3 A. ~ B.K.C

33 ~IKc = a (KC)

where

C, in the other

(6.1) we obtain

index J3'

F.

The

J'KC

to

by

simultaneous

the

method

is given by

of

:

)

global optimization

problem

a s follows : (PX 0) + T r

(SPLPS)

+ Tr

((RKC-B'P)SFS

(RKC-B'P)')

K subject

to f ( K ) = D ' P + PD + Q1 (K) = 0

(6.4.4)

g ( K , X 0) = DS + SD' + X 0 = 0 where

the matrices

is given by

A, B a n d

their nominal value.

The Lagrangian

function

:

L3 = J3 + T r Applying

C assume

( V ' f) + T r

Theorem

(6.1)

3L3 -

-

~K

~tL 3 ~

=

_

~ J3 3K _

(W' g )

to p r o b l e m

(6.4.4),

necessary

conditions of optimality are

2 [(RKC - B'P)V + R (RKC - B'P) SFS - B'WS]C'

: f(K,P) : D ' P + PD ÷ Ql (K) = 0

3L 3 ~---~ = g (K,S) : DS + SD' + X 0

0

(6.4.5a)

(6.t~-Jb)

(6.4.5c)

~L 3 ~---S = fl (K,P,S,W) ~ D'W + WD + Q2 (K,P,S) = 0

(6.t~.Sd)

3L 3 --= ~S

(6.zt.5e)

gl (K,P,S,V) : DV ÷ VD' + Q3 ( K ' P ' S ' X 0 ) ~ 0

:

253

whereD

= A - BKC

QI(K)

= Q + C' K' R K C

Q2(KjP,S)

= SPLP

+ PLPS

+ (RKC-B'P)'

(RKC-B'P)SF

+

F S ( R K C - B ' P ) '( R K C - B ' P ) Q3(K,P,S,X0)

= X 0 + SSPL

+ LPSS

- B(RKC-B'P)

SFS

- SFS

(RKC

- B'P)'B'

Particular case In

the

case of state

feedback

(C = I a n d

F = 0) t h e e q u a t i o n

(6.4.5)

beco-

mes : ~3 3

=

(RK

B'P) V - B ' W 5

~K

wheref(K,P)

= D'P + PD + Q + K ' R K = 0

g ( K , S , X 0) = DS + SD' + X 0 = 0 f l ( K p P , S p W ) = DIW + WD + SPLP + PLPS = 0 gl(K,P,S,V.X0)

= DW + VD' + X 0 + SSPL + LPSS = 0

T h e o p t i m a l f e e d b a c k m a t r i x , is g i v e n b y :

K o p t = R - I B ' p + R - I B ' w S V -I =

where K1

K 1

+

K 2

is t h e o p t i m a l s o l u t i o n f o r t h e n o m i n a l v a l u e s of A a n d

p a r t o f K o p t w h i c h minimizes t h e that Kopt depends

6.4.2b.

i.e.,

on t h e i n i t i a l c o n d i t i o n s t h r o u g h

B,

and

K2 is the

it minimizes (J3 - J1 ) '

Observe

X 0.

Solution of the problem (TAR-85)

The gradient admissible

solution of the technique feedback

system

• guess

an

matrices)

(6.4.5e),

and the gradient

of e q u a t i o n s

initial such

calculate P and S from ( 6 . 4 . 5 b ) and

sensitivttes,

and from

(6.4.5)

feedback

that

the

(6.4.5c)p

c a n be o b t a i n e d

matrix

closed-loop

K0 E K c system

(K c is

is

by the

stable.

using

a

set

of

We c a n

then calculate V and W from (6.4.5d)

( 6 . 4 . 5 a ) . The feedback matrix can be u p d a t e d 333 = ~ 0, g i v i n g a local m i n i m u m .

by successive iterations until the gradient

For this,

one of the algorithms presented

example, the algorithm of Geromel and Bernussou

in § 6.3 c a n b e u s e d . (see § 6.3.1)

:

Consider,

for

254 Step ] :

Find

K16

such that the c of Armentano and Singh

algorithm

K

Step 2 : Solve the Lyapunov

equations

(6.4.5b)

: f = 0 --'- P

(6.4.5c)

: g = 0 ---- S

(6.4.5d)

: fl = 0---

W

(6.4.5e)

: g l = 0 --~

V

and determine

Taking

The

K <-- K -aG

into account

nov equations

at step

gradient

(6.3.1b),

sou,

i.e.,

system tary

uniform

Lyapunov

projection

(6.4.5c),

G C K c . If ~ C ~ S T O P ,

by

using

the

(6.4.5d),

and

(6.5.4e)

:

otherwise

go t o 4.

s i z c ) a n d go to 2.

variations

on

in s e t t i n g

introduces

the

constraint

convergence

two supplementary

Ly~pu-

at Step

2.

Consider

stability

more CPU time because This

same as in section to k.. = 0 . " 11

as the algorithm of Geromel and Bernus-

is the

"price"

to t h i s a l g o r i t h m i s p r o v i d e d

(TAR-85).

3) i s t h e

corresponding

to a local m i n i m u m a n d

It requires

equations

(step

to z e r o all t h e t e r m s

has the same properties

corresponding

Example 6.4

stable

(see § VI.3.1a). (6.4.5b),

(u : step

parameter

at each iteration.

routine

is

2.

and consists

The algorithm

system

the matrix gradient

Step 3 : Find a feasible direction Step 4 : Update

closed-loop

closed-loop

for robustness.

in A p p e n d i x

again the Example 6.1,

of the

of the two supplemenA Fortran

6.

with the initial data

:

X0 = I , L = I, F = 0 a n d K1 g i v e n b y

(6.3.7). 1.549

E

K32 =

The

convergence

After 0,437

32 i t e r a t i o n s

we o b t a i n

l ~-I 1.5z6 jl 0.993

0.57 2.33

:

J

t 1t I 1.91

0.275

I

test

is satisfied

with

e=

0.001.

J

(K 1) = 8 . 9 3 5 ,

J1

(K32) =

3.049 and J3 (K32) = 3.232.

Comparing criterion

these

results

with

those

of

v a l u e J 3 ( K 32) i s 1 . 0 7 7 t i m e s s u p e r i o r

twice longer.

the

Example

to J

(K 59} a n d

6.1, that

we n o t e the

that

the

C P U time i s

255 Remark

6.5.

Performance

cribed degree

of stahility

index

sensitivity

If t h e f e e d b a c k c o n t r o l s t r u c t u r e tions

(6.4.1)

solved a f t e r

and

(6.4.2)

changing

reduction

can

be

achieved

with

a pres-

:

allows pole a s s i g n m e n t , t h e r e s u l t s of s e c -

can b e u s e d t o g e t h e r .

the

dynamic m a t r i x

A by

The

equations

A + aI,

(6.4.5)

w h e r e a is

the

must be desired

d e g r e e of s t a b i l i t y . 6 . 4 . 3 . R o b u s t c o n t r o l with r e s p e c t to l a r g e p e r t u r b a t i o n s in t h e s y s t e m d y n a m i c s This

s e c t i o n p r o v i d e s c o n d i t i o n s on t h e p e r t u r b a t i o n

bounds

such

that

the

s t a b i l i t y of d e c e n t r a l i z e d c o n t r o l s y s t e m is n o t a f f e c t e d . Consider again the system

(6.2.1)

with S c o n t r o l s t a t i o n s w h i c h is g i v e n in

compact form b y :

I

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

(S)

[y(t)

x ( t 0) = x 0 (6.4.6)

C x(t)

Let t h e d e c e n t r a l i z e d o u t p u t f e e d b a c k b e g i v e n b y : u(t) = K y(t)

(K = b l o c k - d i a g o n a l m a t r i x )

(6.4.7)

and s u p p o s e t h a t t h e c l o s e d - l o o p s y s t e m : £ ( t ) = (A + BKC) x ( t ) = D x ( t ) is s t a b l e . When t h e

decentralized control

(6.4.7)

is a p p l i e d to t h e s y s t e m

(6.4.6)

the

performance index : ff = 0f~(x ' QX + u' Ru) dt can be w r i t t e n as : J = x(0)' P x(0) where

P satisfies the matricial equation D'P + P D

with

:

+ Q1 = 0

(6.4.8)

D = A + BKC Q1 = C' K' R K C

+ Q

(6.4.9)

For c o n v e n i e n c e , it is a s s u m e d t h a t t h e matrix Q1 is a n o n s i n g u l a r m a t r i x ,

which

g u a r a n t e e s t h a t P is p o s i t i v e d e f i n i t e .

6 . 4 . 3 . a . Characterization of the p e r t u r b a t i o n s T h e model of t h e p e r t u r b a t e d modelling e r r o r s . . . ) ,

is t a k e n as :

s y s t e m a n d c o n t r o l (large p a r a m e t e r v a r i a t i o n s ,

256

P = A Xp + B Up + AP x P + B P Up p

(S*) u

x* (to) = x 0

(6.4.10a)

KC x p

w h e r e t h e m a t r i c e s A, B a n d C a r e t h o s e of t h e nominal s y s t e m ( S ) . All t h e p e r t u r b a t i o n s a r e lumped into t h e m a t r i c e s Ap a n d B p . One way to c h a r a c t e r i z e t h e p e r t u r b a t i o n s is (PEK-83, 84a,b)

to combine t h e

i n f o r m a t i o n c o n c e r n i n g t h e i r p h y s i c a l n a t u r e a n d t h e i r mathematical m o d e l i s a t i o n . One d e f i n e s t h e d i r e c t i o n s w h i c h a r e t h e most p r o b a b l e from t h e p h y s i c a l p o i n t of v i e w . The p e r t u r b a t i o n s a r e d e c o m p o s e d in two c o m p o n e n t s , one o f w h i c h lies along t h e g i v e n d i r e c t i o n in t h e s p a c e of all p e r t u r b a t i o n m a t r i c e s A A B

P

and B

:

P

= q ( t ) A* + dA

P

= q ( t ) B* + dB

P

where q(t)

is a s c a l a r

(6.4.10b) function,

and

dA a n d

dB r e p r e s e n t

the p e r t u r b a t i o n s

in

s y s t e m d y n a m i c s w h i c h lie o u t of t h e d i r e c t i o n s A* a n d B*. The

above

characterization

following his i n t u i t i o n a n d

can

experience,

be

motivated

has

enough

by

the

fact

that

a

designer,

i n f o r m a t i o n to s e l e c t t h e most

a p p r o p r i a t e d i r e c t i o n s in t h e s p a c e of all p e r t u r b a t i o n m a t r i c e s . O t h e r i n t e r p r e t a t i o n s can be g i v e n a n d f o r e x a m p l e , f o r weakly c o u p l e d s y s t e m s , the p e r t u r b a t i o n d i r e c t i o n s are d e f i n e d b y (PEK-79) : 0

A f t . . . . . . .AIN

A21

0 ......... : : ",

A ~':

0

BI2 . . . . . . . . Bib 0.. ..........

°'° :

ANI

,. •

BNI

AN2 . . . . . . 0

[0 0]

BN2 .......

:

a n d f o r s i n g u l a r l y p e r t u r b e d s y s t e m s b y (KOK-76) :

C* =

A21

A22

5* =

[:2]

A n o t h e r r e a s o n for c o n s i d e r i n g l a r g e p e r t u r b a t i o n s is t h a t it may be n e c e s s a r y to c h a n g e t h e p a r a m e t e r v a l u e s d u r i n g t h e o p e r a t i o n of t h e s y s t e m . In t h i s c a s e , t h e e x p l i c i t e x p r e s s i o n s of t h e m a t r i c e s A* a n d B* a r e g e n e r a l l y k n o w n .

257 6.4.3.b.

- Robustness

characterization

Before giving the general results, 1. - P a r t i c u l a r

case 1 (PEK-84a).

then the perturbation loop p e r t u r b e d

Xp

=

l e t u s c o n s i d e r some p a r t i c u l a r

Consider

the particular

matrices are given by : A

system becomes :

(A + BKC) Xp + (dA + dB.KC)

:

case for which q(t)

= dA a n d B

P

cases

P

= 0,

= dB a n d t h e c l o s e d -

Xp

(S*) = D x The

+ dD x

P

bounds

of t h e

= (D + dD) x

P

perturbations,

preserving

system are given by the following theorem Theorem 6.3.

If t h e p e r t u r b a t i o n

P the

stability

of the

perturbed

:

m a t r i c e s d A a n d dB s a t i s f y t h e i n e q u a l i t y

:

(6.4.11a)

Q1 - d D ' . P - P . dD > 0 w h i c h in t u r n is s a t i s f i e d if : d D . QI" dD' < 1/4 p - 1 Q1 p - 1 then the perturbed given by (6.4.8)

system

S* i s a s y m p t o t i c a l l y

and (6.4.9)

If the matrices dA and

Theorem

6.4.

dB a r e

unknown,

If the perturbation

matrices

lldDi S

n)

JldDl s~/[

<

one has

some k n o w l e d g e

about

dA a n d

dB s a t i s f y

one of the

following

Xrnin (Q 1)

2 lmax(P)

)'rain(p-l" QI" p-l) 4 ~max(Ql )

]I/2

f o r all t ~ [ 0, ~[, t h e n t h e p e r t u r b e d (6.4.8)

and

(6.4.9)

w h e r e U. R s , ; k m a x

(.)

eigenvalue of (.),

respectively.

Lemma

perturbation

6.4.

but

:

:

I)

are given by

(6.4.11b) The matrices P and Q] are

respectively.

their size, the following theorem can be used

inequalities

stable.

The

and~min

system remains asymptotically stable. DdD|s = ]dA|s

and

(.)

denote

bounds

defined

closed-loop pole of the nominal umperturbed JJdDJl s < - Re [X max ( D ) ]

2 The spectral norm is

IYl;Emax (W')]I/2.

spectral

in

P a n d Q1

+ UdB~Is - H K C | s n o r m , maximum

Theorem

6.4

and

system are linked through

and

the •

minimum

dominant

258 2. - P a r t i c u l a r c a s e 2 (PEK-84b)

: Suppose that the perturbation

m a t r i c e s Ap and

Bp lie a l o n g t h e d i r e c t i o n s A* a n d B*, i . e . dA = 0 a n d dB = 0. Now, t h e closed-loop perturbed (S~)

s y s t e m , s a y S~, c a n b e r e p r e s e n t e d b y :

xP = (A + BKC) x p + q ( t ) (A* + B*KC) x p = D = (D + q ( t ) D*) x

x

P

+ q ( t ) D* x

P

P

T h e n , t h e following t h e o r e m c a n b e e s t a b l i s h e d : T h e o r e m 6 . 5 . I f t h e following i n e q u a l i t i e s a r e s a t i s f i e d :

=£I

(H)

X ~ n (H) = qmin < q ( t ) < qmax max for all t ~ [ 0, co[, t h e n t h e p e r t u r b e d s y s t e m

(S*)

remains

asymptotically

stable.

q ( t ) is a m e m o r y l e s s , t i m e - v a r y i n g n o n l i n e a r i t y , a n d H = Q-1 (D,I p + p D*) P a n d Q1 a r e g i v e n b y ( 6 . 4 . 8 ) a n d ( 6 . 4 . 9 ) . C o r o l l a r y 6.1. If ~min (H) is n o t n e g a t i v e , t h e n t h e b o u n d qmin does n o t e x i s t a n d q ( t ) E: ] -co, q m a x ] - If Xmax (H) i s n o t p o s i t i v e , t h e n t h e b o u n d qmax d o e s n o t e x i s t a n d q ( t ) E [ q m i n ' + Qo[ Refering

to T h e o r e m 6 . 5 ,

Petkovski

(PEK-84b)

proposes

t h e following p r o c e -

d u r e to d e t e r m i n e t h e l a r g e s t p o s i t i v e n u m b e r qmax a n d t h e s m a l l e s t n e g a t i v e n u m b e r qmin s u c h t h a t t h e p e r t u r b e d

s y s t e m (S~) r e m a i n s s t a b l e :

S t e p 1 : U s i n g T h e o r e m 6.5 d e t e r m i n e ( q m i n ) 0 a n d (qmax)0 . Step 2 : C o n s i d e r t h e p e r t u r b e d s y s t e m (Sj) as u n p e r t u r b e d ,

i.e.

:

A = A + ( q m i n ) j _ l A* B -- B + (qmin )j -1 n* a n d d e t e r m i n e (qmin)j u s i n g T h e o r e m 6 . 5 . Step 3 • If t h e c l o s e d - l o o p s y s t e m (Sj) is s t a b l e , go to Step 2. O t h e r w i s e ,

J qmin - i=0 ( q m i n ) i " Step 4 : T h e s m a l l e s t qmin is g i v e n b y :

259 co

qmin = i-Z0 ( q m i n ) i Step 5 : c o n s i d e r t h e p e r t u r b e d

s y s t e m (~j) as u n p e r t u r b e d

i.e.

:

A = A + ( q m a x ) j _ l A* B = B + ( q m a x ) j _ l B* a n d d e t e r m i n e (qmax)] u s i n g T h e o r e m 6 . 5 . Step 6: If t h e c l o s e d loop s y s t e m (~]) is s t a b l e , go to s t e p 5. O t h e r w i s e ,

J --

Z

qmax = i= ~) ( q m a x ) i " Step 7 : T h e l a r g e s t q--max is g i v e n b y : co

q m a x = i=O Z (qmax)i T h e following lemma p r o v i d e s a n a l t e r n a t i v e e x p r e s s i o n for t h e b o u n d s of t h e scalar function q(t)

(observe the similarity with Theorem 6.4).

Lemma 6 . 5 . I f q ( t ) s a t i s f i e s t h e c o n d i t i o n : q(t)

.

for all t e l 0 ,

[[D* [ < ~ m i n (Q1) Is 2 kmhx (P) ~[ , t h e c l o s e d - l o o p p e r t u r b e d

s y s t e m is asymptotically s t a b l e . P and

Q1 a r e g i v e n b y ( 6 . 4 . 8 ) a n d ( 6 . 4 . 9 ) a n d

i[ D* lls --I[ A* ]Is + IIB* IIs like IIs T h e following lemma is t h e a n a l o g of Lemma 6 . 4 . Lemma 6 . 6 .

The perturbation

the nominal u n p e r t u r b e d

[q(t)]

.lID* ]Is • -

3. - G e n e r a l c a s e

P S*

Re[~max

(PEK-83).

closed-loop p e r t u r b e d

bounds

of q ( t )

a n d t h e d o m i n a n t c l o s e d - l o o p pole of

s y s t e m (S) a r e l i n k e d t h r o u g h

:

(D)]

In t h e

case of g e n e r a l p e r t u r b a t i o n s

s y s t e m S* is g i v e n b y :

= (A+BKG) x

+ q ( t ) (A* + B* KC) x + (dA + d B . K C ) x P P P = D x + q ( t ) D* x + dD . x P P P = (D + q ( t ) D* + dD) x P

T h e o r e m 6.6. If t h e p e r t u r b a t i o n m a t r i x dD s a t i s f i e s t h e i n e q u a l i t y ,

(6.4.10b),

the

260 Xmin[Ql - q(t) (D~'P + PD ~) 2 x (p) max

for all t g [ 0, ~ [ , t h e n t h e p e r t u r b e d

s y s t e m S* r e m a i n s a s y m p t o t i c a l l y s t a b l e . P and

Q1 a r e g i v e n b y ( 6 . 4 . 8 ) a n d ( 6 . 4 . 9 ) a n d IldD [Is = ]Id A Hs 4. -

R o b u s t n e s s c h a r a c t e r i z a t i o n with a p r e s c r i b e d

+ I]dB Hs

" ]]KC Hs

d e s r e e of s t a b i l i t y

(PEK-84b).

When t h e d e c e n t r a l i z e d c o n t r o l is d e t e r m i n e d s u c h t h a t t h e u n p e r t u r b a t e d

closed-loop

s y s t e m is a s y m p t o t i c a l y s t a b l e with d e g r e e a , 6.5, 6.6,

Lemmas 6 . 4 ,

6.5,

6.6,

the above results

a n d C o r o l l a r y 6.1)

( T h e o r e m s 6 . 3 , 6.4,

hold i f t h e m a t r i c e s P a n d Q1

are given by : (D+aI)' P + P (D + a I ) 01 = Q + C' K' R K C

+ Q + C' K' R K C

= 0

+ 2 aP

We can o b s e r v e t h a t i f t h e s y s t e m is s t a b l e with d e g r e e a , Therefore,

then

- Re IX max (D)]>~a

Lemma 6.4 (or Lemma 6.6) e s t a b l i s h e s a r e l a t i o n s h i p b e t w e e n t h e a c c e p -

table p e r t u r b a t i o n s

and

the

prescribed

degree

of

stability a.

Consequently,

the

p a r a m e t e r a , can be u s e d as a d e s i g n p a r a m e t e r .

6 . 5 . - ROBUST DECENTRALIZED SERVOMECHANISM PROBLEM This s e c t i o n g i v e s an o v e r v i e w of t h e r e s u l t s o b t a i n e d b y Davison in r e f e r e n c e to t h e so caUed " D e c e n t r a l i z e d s e r v o m e c h a n i s m p r o b l e m " , c o n s i d e r e d in v a r i o u s forms (DAV-76a,b,c,d,

77b,

78a, 79a, 82).

Our a t t e n t i o n f o c u s e s on t h e r e s u l t s o b t a i n e d

w i t h i n a r o b u s t c o n t r o l a p p r o a c h (DAV-76c, 77b, c o n s i d e r a t i o n can

be p e r t u r b a t e d

78a, 79a, 87).

The s y s t e m s u n d e r

b y l a r g e v a r i a t i o n s of t h e p l a n t p a r a m e t e r s and

d y n a m i c s a n d b y e x t e r n a l d i s t u r b a n c e s . The p r o b l e m c o n s i s t s in d e s i g n i n g a d e c e n tralized

controller

such that the closed-loop

perturbated

s y s t e m r e m a i n s s t a b l e and

that satisfactory tracking or regulation o c c u r s .

6 . 5 . 1 . - Problem formulation C o n s i d e r a l i n e a r t i m e - i n v a r i a n t s y s t e m , with S s t a t i o n s , d e s c r i b e d b y : S = A x + i=~i

Bi u i + E m

Yi = Ci x + Di u i + Fi t0 , y~-- C ~ x + D [ % + F['m , ei = Yi - YP

(i=1 . . . . . S) (i--1 . . . . . S) ,

(i=I . . . . .

S)

(6.5.1)

261

where x ~ Rn is the the output

to

disturbance reference

B

be

[

=

m

u i ~ R m i , Yi ~ R r i "

regulated,

vector

output

state,

and

at

which may or may not be measurable,

Yi

B I .....

measurable

-r. m t (rim~< r i) a r e

output

and the output

the

Yi E •

local e r r o r

station and

i.

the input, 00ERq i s t h e

ei are

the

desired

at station i. Define :

BS]

D = block-diag. (D I ..... D S) D m=

block-diag. (D~, .... D~S)

C =

Cm=

s

Fm :

F=

Los

and assume that ~ belongs

(6.5.2)

LFs

FFl IYll Ieli e =

yd =

to t h e f o l l o w i n g c l a s s o f s y s t e m s

:

Zl = A1 Zl (6.5.3)

= H1 z1 where

z 1 d= R n l

output arises

and

Zl(0)

may or may not

be

from the following class of systems

known,

and

the

desired

reference

:

~'2 = A2 z2 z y

where

d d

= H2 z 2

(6.5.4)

d

=Gz

z2 ~ Rn2

and

z2(0)

is known.

I t is a l s o a s s u m e d

without

loss of generality

that :

rank[El rank

and that

(H1,

tems (6.5.3)

The follows :

= rand

G = rank

A1) ,

and

"robust

H1 = q H 2 = dim (z d )

(H 2,

(6.5.4)

A 2) a r e

observable.

are unstable

decentralized

In addition,

we a s s u m e

that

the sys-

to a v o i d t r i v i a l i t y .

servomechanism

problem"

i s d e f i n e d in ( D A V - 7 6 c ) a s

262 Find a decentralized linear time-invariant controller

(S local c o n t r o l l e r s )

for

the system (6.5.1) - (6.5.4) such that • • The c l o s e d - l o o p s y s t e m is a s y m p t o t i c a l l y s t a b l e , • Asymptotic tracking,

in p r e s e n c e of d i s t u r b a n c e s , o c c u r s i n d e p e n d e n t l y of

all a r b i t r a r y p e r t u r b a t i o n s in t h e p l a n t model ( 6 . 5 . 1 ) or plant

dynamic i n c l u d i n g c h a n g e s in model o r d e r )

(e.g.

plant parameters

w h i c h do n o t a f f e c t the

s t a b i l i t y of t h e r e s u l t a n t c l o s e d - l o o p s y s t e m , i . e . lira e ( t ) = 0 V x ( 0 ) ~ R n , t->oo V z 1 (0) E R n l , V z 2 (0) E Rn2 a n d f o r all c o n t r o l l e r initial c o n d i t i o n s .

6.5.2.

-

Existence

of

a solution

The c o n d i t i o n s u n d e r w h i c h a r o b u s t d e c e n t r a l i z e d c o n t r o l l e r e x i s t s a r e p r o vided. 6.5.2.a.

- G e n e r a l c a s e (DAV-76c, 77b)

S r = i~ 1

Define

ri,

S = i~ 1

m

m i and

rm

S = i~ I

r[n, a n d

the

matric

Cm*

of

dimension ( r m + r ) x (n+r) as follows •

C*m :

c;.,, ct~, , "'"

%7

(6.S.Sa)

w h e r e t h e C~.'s are g i v e n b y : 1

"E:

Irl

0 ........

0

lr. ......

0

0 °

C2 =

0

........°1

C3=

0

....... i1 "'r

The minimal polynomials of A I a n d A 2 o f ( 6 . 4 . 3 ) a n d (P2(s).

The

(6.4.4)

l e a s t common multiple o f @1(s) a n d (pz(s)

a r e d e n o t e d b y %01(s)

(multiplicity i n c l u d e d )

is

given by : r i~l (s - l i ) = s g + p g s g - I + P g - I s g - 2 + " ' " + P2 s + P l where 11' !2 . . . . . ,

Ig a r e i t s z e r o s .

(6.5.6)

263

Theorem 6 . 7 (DAV-76c, 7 7 b ) . A s o l u t i o n to t h e r o b u s t d e c e n t r a l i z e d s e r v o m e c h a n i s m problem f o r t h e s y s t e m

(6.5.1)

-

(6.5.6)

e x i s t s i f a n d o n l y i f t h e following c o n d i -

t i o n s all h o l d : (i) T h e s y s t e m (C m, A, B) h a s n o u n s t a b l e d e c e n t r a l i z e d f i x e d m o d e s . (ii) T h e s e t of d e c e n t r a l i z e d f i x e d modes of t h e g s y s t e m s : respectively. (iii) T h e o u t p u t

Yi is c o n t a i n e d in y ~ ,

(i=l,...,S),

i.e.,

Yi is p h y s i c a l l y

measu-

rable. I n t h e c a s e f o r w h i c h mi = r i ,

(i=l,...,S),

C o r o l l a r y 6.2 ( D A V - 7 8 a ) . Assume t h a t mi = r i , tion to t h e

decentralized robust

we h a v e t h i s s i m p l e r c o n d i t i o n • (i=l,...,S),

servomechanism problem

then there exists a solufor t h e s y s t e m

(6.5.1)

-

( 6 . 5 . 6 ) if a n d o n l y if :

I

A = Xi I

rank

B1

C

=

n + r

(i:l,...,g)

D

T h e c o n d i t i o n of t h e a b o v e c o r o l l a r y m e a n s t h a t no e i g e n v a l u e )~] (]=1 . . . . . g) of ( 6 . 5 . 6 ) c o i n c i d e s with a t r a n s m i s s i o n zero of t h e s y s t e m (see A p p e n d i x 1).

6.5.2.b.

- P a r t i c u l a r c a s e of i n t e r c o n n e c t e d s y s t e m s

(DAV-76c,

79a).

The

c o n s i d e r e d h e r e is a composite s y s t e m , c o n s i s t i n g of i n t e r c o n n e c t e d s u b s y s t e m s

plant :

S

&i

=

Ai xi + Bi ui

+ Ei ~0 + i~ 1 /~ij xj

Yi = Ci xi + Di u i + Fi to

(6.5.7)

ym= C ~ x i + D~ i ui + F im d ei = Yi - Yi

( i = l . . . . S)

x i {~ R n*i is t h e s t a t e , a n d u i ' Yi' y~, Yid a n d m a r e d e f i n e d as i n t h e l a s t s e c t i o n . By a s s u m p t i o n t h e i n t e r c o n n e c t i o n m a t r i x is g i v e n b y t h e g e n e r a I model : where

A.. ~-H.. K . M..

1j

lj

ij

ij

( i , j = l . . . . . S)

i/j

(6.5.8)

264 w h e r e K,. lj d e n o t e s t h e i n t e r c o n n e c t i o n gain c o n n e c t i n g t h e s u b s y s t e m s i a n d j. i t h s u b s y s t e m is o b t a i n e d b y s e t t i n g Aij = 0, (j=l . . . . . S) a n d iCj, in ( 6 . 5 . 7 ) . T h e o r e m 6.8 (DAV-76c,

79a).

Assume that

there

exists

a solution

to

the

The

robust

c e n t r a l i z e d s e r v o m e c h a n i s m p r o b l e m (DAV-75) f o r e a c h s u b s y s t e m of ( 6 . 5 . 7 ) . (i) T h e n t h e r e e x i s t s a solution to t h e r o b u s t d e c e n t r a l i z e d s e r v o m e c h a n i s m problem f o r t h e c o m p o s i t e s y s t e m ( 6 . 5 . 7 ) if t h e i n t e r c o n n e c t i o n g a i n s K.. lj a r e "small e n o u g h " . (ii)

Assume,

in

(i=l,...,S),

addition,

that

(Cim, A,

Bi)

is

controllable

and

observable

for

t h e n t h e r e e x i s t s a solution to t h e r o b u s t d e c e n t r a l i z e d s e r v o m e c h a n i s m

p r o b l e m f o r t h e composite s y s t e m ( 6 . 5 . 7 ) f o r almost all i n t e r c o n n e c t i o n g a i n s Kij. (iii) Assume t h a t t h e i n t e r c o n n e c t i o n m a t r i c e s A.. o f ( 6 . 5 . 8 ) h a v e t h e p r o p e r t y t h a t 1] (i=l . . . . . S ) , t h e n t h e r e e x i s t s a solution to the

Hij = B i, Mij = Cj a n d Di = 0 f o r

r o b u s t d e c e n t r a l i z e d s e r v o m e c h a n i s m p r o b l e m for t h e composite s y s t e m ( 5 . 4 . 7 ) if and only if t h e r e e x i s t s a solution to t h e r o b u s t c e n t r a l i z e d s e r v o m e c h a n i s m p r o b l e m for e a c h s u b s y s t e m of ( 6 . 4 . 7 ) .

5.5.3. - Robust decentralized controller design 6 . 5 . 3 . a. - C o n t r o l l e r s t r u c t u r e Consider the system (6.5.1) lized c o n t r o l l e r ,

then any decentralized controller which regulates

following s t r u c t u r e

1

(6.5.1)

decentrahas the

(DAV-76c, 77b) :

u. = K. v. + K~. w. 1

a n d assume t h a t t h e r e e x i s t s a r o b u s t

1

1

1

(i=l . . . . . S)

(6.5.9)

w h e r e v i ~ Rris is t h e o u t p u t of a d e c e n t r a l i z e d s e r v o - c o m p e n s a t o r , and wi £ R is t h e o u t p u t of a d e c e n t r a l i z e d s t a b i l i z i n g c o m p e n s a t o r . Consider the system (6.5.1) for ( 5 . 5 . 1 )

- (6.5.4),

then a decentralized servo-compensator

(DAV-76c) is a c o n t r o l l e r with i n p u t e i E R r i

and output v i ~ R r (

given

by : ~r = ~. v. 1

1

+

~ . e. 1

(6.5.10)

1

~i = b l o c k - d i a g . ( ~ , , ~ , . . . .

~,)

i matrices

265

B--i = block-diag.

(~,, g, ..... ~,) w i matrices

~, and B , are the (rxr) companion matrix and the (qxl) matrix, defined below : .

0

1

0

0

I0 l

;

g. =

°'.°.

*

;

:

:

-Pl

-P2

o

O,

•

;

..

0

J

_l ]

-P3 . . . . . :'-Pg

Pi' (i=l . . . . . g), are given by (6.5.6). The decentralized stabilizing compensator and output wi, ( i = l , . . . , S ) , is given by :

(DAV-76c),

with inputs

Yi'm vi, ui

~'i = GO i zi + G1i Yim~ G2 vi (6.5.1H

wi = G~i zi + G~ yim+ Ghvii where Yi = Yim - Dmui"

mm

The controller s t r u c t u r e as described above is illustrated in Figure 6.3.

• he gain mat.ces "i" ~

• 00, old,

0[, ~:,

0: and. ~.~, can be determ'nated

through the decentralized stabilization scheme of Wang and Davison (WAN-73) in order to stabilize and give the desired behaviour to the following augmented s y s tern :

X

A

,}

FIC 1

;

0

.........

U 1 ........

0

X

0

V

= o •

_

%1 .J

•

BsC 5 0 . . . . . . . . . C S

4-

Iblock-Bd lag. (BI D I ' " ' g s D I ) ] "vD]

(6.5.12.a)

266 t0

! Ul ~ J

/ • ( ~i

iyf I I I

~ t i

Yl I I

"~

SYSTEM

~

\ y.

~Lc4;%%t°ri

'

' i~

t

~

!

t I

\

'---J

stabilizing compensator

1 I I

I

Yi

Fig. 6.3 : C o n t r o l l e r s t r u c t u r e

L c,x1 vi

The system

(i= 1 ,...,S)

(6.5.12b)

LVi

(6.5.12)

h a s d e c e n t r a l i z e d fixed modes e q u a l to t h e d e c e n t r a l i z e d

f i x e d m o d e s of (C m, A, B) (if a n y ) .

6.5.3. b. Controller optimization In g e n e r a l , guarentee :

t h e o p t i m i z a t i o n of t h e d e c e n t r a l i z e d

stabilizing compensator must

267 (i) f a s t r e s p o n s e (ii) low i n t e r a c t i o n in t h e s y s t e m , i . e . ,

when a r e f e r e n c e output signal c h a n g e s , the

o t h e r o u t p u t s s h o u l d remain as close as p o s s i b l e o f t h e i r p r e v i o u s v a l u e s . The p a r a m e t e r optimization m e t h o d p r o p o s e d b y Davison e t a l .

{DAV-73,

79a,

81, 825 minimizes a q u a d r a t i c p e r f o r m a n c e i n d e x of t h e form : J = E((x'

Q x + u' Ru) dt

w h e r e E d e n o t e s t h e e x p e c t a t i o n o p e r a t o r , s u b j e c t to any i m p o s e d e n g i n e e r i n g c o n s traints.

In

particular

Davison

and

Chang

(DAV-825

showed

that,

if

the

system

(6.5.15 is o p e n - l o o p s t a b l e a n d if Re (kit = 0, (i=l . . . . , g S , w h e r e t h e k.I1 s a r e g i v e n by (6.5.65,

(e.g.

we h a v e p o l y n o m i a l - s i n u s o i d a l t y p e of d i s t u r b a n c e s a n d r e f e r e n c e

s i g n a l s ) , t h e n t h e r e always e x i s t s an initial f e a s i b l e s t a r t i n g p o i n t f o r t h i s p a r a m e t e r optimization p r o b l e m .

6 . 5 . 3 . c . Some p r o p e r t i e s of t h e c o n t r o l l e r (DAV-76c) I. Using the robust controller described before, one can locate the eigenvalues of the dosed-loop

system

in any

(the decentralized

fixed modes

nonempty

of (Cm,

symmetric

A,B)

(if any)

region of the complex

plane

must be in the desired re-

gion). 2.

A robust

d e c e n t r a l i z e d c o n t r o l l e r e x i s t s g e n e r i c a l l y (WAN-73)

for

"almost

all" p l a n t s (6.5.15 p r o v i d e d t h a t : (i) m i )/ r i ( i = l . . . . .

S)

(ii) the output Yi is physically measurable at station i. If either (it or (lit do not hold, then a solution to the robust

decentralized

ser-

vomechanism problem never exists.

6.5.4. - Sequentialiy stable robust controller design A realistic

s i t u a t i o n is to c o n s i d e r t h a t

no

central

authority

is allowed f o r

c a l c u l a t i n g t h e local c o n t r o l l e r s , a n d t h a t a complete k n o w l e d g e of t h e mathematical model of t h e p l a n t is n o t n e c e s s a r i l y available at a n y c o n t r o l s t a t i o n . T h e p r o b l e m is thus

to

find

a

solution

to t h e

robust

decentralized

servomechanism problem

for

s y s t e m ( 6 . 5 . 1 ) u n d e r t h e two following c o n s t r a i n t s : (i)

The c o n t r o l l e r s y n t h e s i s must be c a r r i e d o u t in a s e q u e n t i a l s t a b l e way

(DAV-79bS, i . e . ,

t h e c o n t r o l l e r s can be c o n n e c t e d to t h e s y s t e m one a f t e r a n o t h e r

r e s u l t i n g a t a n y time in a s t a b l e c l o s e d - l o o p s y s t e m .

268 T h i s is m o t i v a t e d b y

physical

constraints

like time l a g s

c o n t r o l l e r s c o n n e c t i o n , lack of communication h a r d s t r u c t u r e . . , is a c h i e v e d w i t h a c o n n e c t i o n s e q u e n c e

41,2 . . . . . S ) ,

associated etc.

with the

If this property

t h e c o n t r o l l e r is s a i d to b e se__z-

cluentiall 7 s t a b l e w i t h r e s p e c t to c o n t r o l s t a t i o n o r d e r ( l p 2 p . . . , S ) . (ii) No c e n t r a l a u t h o r i t y m u s t b e u s e d in d e c e n t r a l i z e d d e c i s i o n m a k i n g ,

and

e a c h c o n t r o l s t a t i o n p o s s e s s e s o n l y a limited k n o w l e d g e of t h e m a t h e m a t i c a l model of the system

(typically,

e a c h s t a t i o n of a l a r g e

scale s y s t e m p o s s e s s e s

o n l y a local

model ( D A V - 8 2 ) ) .

6.5.4.a.

- E x i s t e n c e of a c o n t r o l l e r

C o n s i d e r t h e s y s t e m ( 6 . 5 . 1 ) withe0=0 a n d yd=0 g i v e n b y : = Ax + i=~l Bi u i Y~= Cir~x + I~i ui

(6.5.13)

Yi = Cix + D'mlu.1

(i=l . . . . . S )

Apply t h e c o n t r o l : Am ~ ~ o ui = Ki Yi + Ki v i ( i = l , . .o ,S) where a n d AmYi= y~n_ "~Ki R r i x r ? _ b~ ui,

(6.5.14a) a n d w h e r e t h e following c o n t r o l l e r s h a v e

a l r e a d y b e e n a p p l i e d to c o n t r o l s t a t i o n s (1,2 . . . . , i - 1 ) , i~2 : v °. = K. v. + K.~w. ]

J

J

J

J

(j=1,2 . . . . . i - l )

iE{2 ..... S)

(6.5.14b)

v. is t h e o u t p u t of a d e c e n t r a l i z e d s e r v o c o m p e n s a t o r g i v e n b y ( 6 . 5 . 1 0 ) a n d w. is t h e J ] o u t p u t of a d e c e n t r a l i z e d s t a b i l i z i n g c o m p e n s a t o r g i v e n b y ( 6 . 5 . 1 1 ) . T h e minimal s t a t e r e a h z a t i o n of t h e r e s u l t a n t applying the controller (6.5.14a,b)

closed-loop system obtained by

to t h e s y s t e m ( 6 . 5 . 1 3 )

for c o n t r o l s t a t i o n i (with

i n p u t v oi a n d o u t p u t y ? is c a l l e d t h e i t h s t a t i o n ' s local model of t h e s y s t e m . The

problem

of f i n d i n g

a robust

decentralized

servomechanism

control

with

s e q u e n t i a l s t a b i l i t y , w h e n e a c h s t a t i o n p o s s e s s e s o n l y a local model of t h e s y s t e m a n d w h e n t h e c e n t r a l d e c i s i o n m a k i n g a u t h o r i t y is n o t allowed i s called t h e local model robust decentralize d servomechanism problem. I t is a s s u m e d ces/reference criterion,

(DAV-79b,

signals poles, i.e.

here

stability

o r pole

82) t h a t e a c h c o n t r o l s t a t i o n k n o w s X] . . . . . ~g of ( 6 . 5 . 8 ) , assignability

modes, i f a n y ) of t h e c l o s e d - l o o p s y s t e m .

(except

the disturban-

a n d h a s t h e same p e r f o r m a n c e for

the

decentralized

fixed

269 Theorem 6.9

(DAV-82).

Consider the system (6.5.1)

in which A is a s s u m e d to b e

a s y m p t o t i c a l l y s t a b l e . T h e n t h e r e e x i s t s a s o l u t i o n to t h e local model r o b u s t d e c e n t r a l i z e d s e r v o m e c h a n i s m p r o b l e m if a n d only if t h e r e e x i s t s a solution to t h e r o b u s t decentralized s e r v o m e c h a n i s m p r o b l e m ( s e e T h e o r e m 6 . 7 ) .

6.5.4. b. - Controller synthesis A s s u m i n g t h a t T h e o r e m 6.9 h o l d s , t h e following algorithm p r o v i d e s a s y n t h e s i s procedure. Algortihm 6 . 1 . ( D e c e n t r a l i z e d s y n t h e s i s solution) (DAV-82). Step 1 : Apply t h e o u t p u t f e e d b a c k c o n t r o l : :

ui

~'m A~ ~ i Yi + Ki v °

(i=l

. . . . .

S)

A K. ~ R mi x r i,m K. ~: Rmixri

where

are

arbitrary

non

zero

I~i = r i, a n d w h e r e t h e Ki's a r e c h o s e n "small e n o u g h "

matrices

with

rank

so a s to maintain t h e

s t a b i l i t y of t h e c l o s e d - l o o p s y s t e m . Step 2 : Using a c e n t r a l i z e d s y n t h e s i s method (DAV-75) a n d t h e k n o w l e d g e of s t a tion l ' s local model of t h e s y s t e m , a p p l y t h e s e r v o c o m p e n s a t o r ( 6 . 5 . 1 0 )

with

i=l to t h e t e r m i n a l s of c o n t r o l s t a t i o n 1 a n d a p p l y t h e s t a b i l i z i n g c o m p e n s a t o r : O

v 1 = K 1 v I + K~ w 1 (v I is g i v e n b y (6.5.10) that

the resulting

c l o s e d - l o o p s y s t e m is

a n d w 1 is g i v e n b y ( 6 . 5 . 1 1 )

stable

and

has

a

desired

so

dynamic

r e s p o n s e . The r e s u l t i n g s y s t e m h a s t h u s t h e p r o p e r t y of h a v i n g Yl r e g u l a t e d , Step 3 : R e p e a t

sequentially step

2 for

(i=2,3,...,S)

until

all t h e

stations

have

regulated outputs. If pole a s s i g n m e n t is d e s i r e d ,

t h e a b o v e a l g o r i t h m can he modified as follow

(DAV-82). Algorithm 6.2. (Pole a s s i g n m e n t d e c e n t r a l i z e d s y n t h e s i s ) (DAV-82). Assume with no loss of g e n e r a l i t y t h a t t h e c o n t r o l s y n t h e s i s is p r o c e e d in t h e c o n t r o l s t a t i o n o r d e r 1, 2, . . . ,

S.

Step 1 : i=l Step 2 : Using a c e n t r a l i z e d s y n t h e s i s m e t h o d a n d t h e k n o w l e d g e of s t a t i o n i ' s local model of t h e s y s t e m ( i . e . t h e minimal r e a l i z a t i o n of t h e s y s t e m ( 6 . 5 . 1 3 ) a l r e a d y c o n t r o l l e d at s t a t i o n s ( 1 , 2 , . . . , i - 1 ) with r e s p e c t to t h e i n p u t u i a n d t h e o u t p u t ~m Yi ) ' a p p l y t h e s t a b i l i z i n g c o m p e n s a t o r ;

270

ui = Ki "~m Y i + K;zi m

(6.5.15)

;3 = Gi zi + Gi Yi to s t a t i o n i so t h a t : i

Z B.K .C. +j 1 j j ] G 1 C1

BIK 1 . . . . . . . GI: . . . . . . . . •

1

1

0

%.

•

G.*C.m

BjKT

*°.

0 ........

"LG

t

f:m

h a s all i t s e i g e n v a l u e s c o n t a i n e d in Cg ( e x c e p t t h e d e c e n t r a l i z e d f i x e d modes o f {

I • A , (B I . . . . . Bi))

which lie o u t s i d e of C~, if a n y ) . Cg is a s p e c i f i e d r e g i o n

LC] of ~-. This is always possible for almost all Kj, Gj, 0=1,2 .....i-l) (DAV-8Z). Step 3 : If i=S, s t o p , o t h e r w i s e , i=i+l, go to Step 2. Remark 6 . 6 . I.

If T h e o r e m 6.9 h o l d s , t h e n for almost all g a i n s c h o s e n in s t e p s I to 3 of

Algorithm 6 . 1 , it is always p o s s i b l e to c a r r y out t h e s y n t h e s i s (DAV-82). 2. If t h e s e q u e n t i a l s t a b i l i t y c o n s t r a i n t is r e l a x e d , t h e n Algorithms 6.1 and 6.2 are still a p p l i c a b l e for t h e case of u n s t a b l e o p e n - l o o p s y s t e m s • 3. Note t h a t t h e c o n t r o l l e r s o b t a i n e d b y A l g o r i t h m s 6.1 a n d 6.2 a r e , g e n e r a l l y , n o t u n i q u e with r e s p e c t to t h e c o n t r o l a g e n t s e q u e n c e . 4. I f Dral = 0, Di = 0 ( i = l , . . . , S ) ,

t h e n t h e r e s u l t s of t h i s s e c t i o n hold f o r t h e

g e n e r a l c a s e f o r w h i c h t h e i n f o r m a t i o n flow b e t w e e n c o n t r o l s t a t i o n s is a r b i t r a r i l y c o n s t r a i n e d ( n o t n e c e s s a r i l y d e c e n t r a l i z e d ) ( D A V - 8 2 ) . I n d e e d , as it is p o i n t e d o u t in (WAN-TBb), a r e o r d e r i n g of t h e o u t p u t s can always b e p e r f o r m e d to form an e q u i v a lent s t a n d a r d decentralized control problem•

6.5.5. - Robust decentralized controller for unknown systems In t h i s s e c t i o n , we c o n s i d e r t h a t t h e s y s t e m ( 6 . 5 . 1 )

t h a t we w a n t r e g u l a t e , is

n o t completly k n o w n . The only i n f o r m a t i o n on t h e s y s t e m is t h e following ; (i) The s y s t e m is d e s c r i b e d b y a finite dimensional l i n e a r f i m e - i n v a r i a n t model. (ii) T h e s y s t e m is o p e n - l o o p a s y m p t o t i c a l l y s t a b l e .

271

(iii) T h e d i s t u r b a n c e s a f f e c t i n g t h e s y s t e m a n d t h e t r a c k i n g r e f e r e n c e s i g n a l s are of polynomial/sinusoTdal t y p e , i . e . Re (Xi}=0, (i=l . . . . . g) in ( 6 . 5 . 6 ) . (iv) The s y s t e m i n p u t s can b e e x c i t e d , a n d t h e s y s t e m o u t p u t s to r e g u l a t e can m be m e a s u r e d , i . e . yi = Yi " With t h i s

sole i n f o r m a t i o n ,

it is

desired

to f i n d

a

decentralized controller

which

solves t h e r o b u s t s e r v o m e c h a n i s m p r o b l e m . T h e q u e s t i o n is to know w h e t h e r o r n o t t h e r e e x i s t s a f i n i t e s e t of e x p e r i m e n t s (taking into a c c o u n t n o i s y m e a s u r e m e n t s } to p e r f o r m on t h e p l a n t , necessary and

s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e of a

such that the

solution

to

the

above

problem can b e e x p r e s s e d in t e r m s of t h e s e e x p e r i m e n t s . If a solution e x i s t s , t h e following q u e s t i o n is to know w h e t h e r t h e r e e x i s t s a c o n t r o l l e r s y n t h e s i s p r o c e d u r e (using o n - l i n e t u n i n g

methods}

which satisfies the

decentralized controller tuning

s y n t h e s i s c o n s t r a i n t s above : (i) At a n y time, one c o n t r o l l e r can be i m p l e m e n t e d on one c o n t r o l s t a t i o n o n l y . (ii) A f t e r a c o n t r o l l e r h a s b e e n i m p l e m e n t e d on a g i v e n c o n t r o l s t a t i o n ,

this

c o n t r o l l e r is f i x e d a n d c a n n o t b e r e a c t u a l i z e d . (iii)

The

resultant

closed-loop system

must

remain

stable

any

time

of

the

controller s y n t h e s i s . This p r o b l e m is called t h e r o b u s t d e c e n t r a l i z e d s e r v o m e c h a n i s m p r o b l e m f o r u n k n o w n

systems.

6 . 5 . 5 . a . E x i s t e n c e of a s o l u t i o n Recall t h a t K d is t h e s e t of b l o c k - d i a g o n a l m a t r i c e s K d = { K]K = b l o c k - d i a g .

[K~ . . . . . Ks], K i e R m i = i

, (i=1 . . . . . S).

Definition 6.1 (DAV-78). 1. The s t e a d y - s t a t e t r a c k i n g gain p a r a m e t e r s T k ( i , j ) , of t h e s y s t e m ( 6 . 5 . 1 ) for t h e c a s e mi = r i , I Cj (Xk I - A) -1 Bi

(i=l,...,S),

(i,j=l . . . . . S ) , (k=l . . . . .

if i# j

Tk(i,j ) __a

(6.5.16) C i (Xk I - A } - I Bi + Di

2.

g)

are given by :

The

(k=l,...,g}, (i=l,...,S), given b y :

steady-state

tracking

of t h e s y s t e m (6.5.1} w h e r e r a n k Ki= r i ,

i f i=j gain

parameters

Tk ( i , j j K i ) ,

(i,]=l . . . . , S ) ,

w i t h r e s p e c t to t h e i n p u t m a t r i c e s K i E R m i x r i ,

(i=l . . . . . S ) ,

f o r t h e c a s e mi >/ r i,

(i=l . . . . . S ) , a r e

272 a [ C J (k k I - A ) - I Bi Ki

if i#j

=I

Tk(i'j;Ki)

(6.5.17)

Ci (k k I - A) -1 B i Ki + Di Ki

if i=j

It is clear that for the case Ki = I and mi = r i (i=l . . . . . S), T k ( i , ]) = T k ( i , j ; K i ) . It is worth noticing t h a t the s t e a d y - s t a t e t r a c k i n g gain p a r a m e t e r T k ( i , j) is equal to the t r a n s f e r function matrix between the i n p u t u i a n d the o u t p u t yj. Davison (DAV-76d, 78) s u g g e s t e d algorithms, called " e x p e r i m e n t s " , to evaluate the parameters T k ( i , j ) a n d T k ( i , j ; K i ) . Theorem 6.10 (DAV-78a). Consider the system (6.5.1) for which mi=ri,i {1. . . . . S), if i ~ ilp i 2 ) . . . , i d ,

a n d mi ) r i if i = i l , i 2 , . . . ) i d, a n d a set of mixr i i n p u t matrices Ki, with r a n k K.=r.. T h e n a n e c e s s a r y a n d sufficient condition for the

i=il,...,id,

1 1

existence of a solution to the r o b u s t decentralized servomechanism problem for u n k nown systems is t h a t t h e r e e x i s t s a list of d i s t i n c t i n t e g e r s (s 1, s 2 , . . . s S) (not n e c e s s a r i l y u n i q u e ) , si£ { 1 , . . . , S ) , such t h a t the following S successive r a n k conditions hold :

1.

rank[Tk(S 1, s I ;--Ksl )] = s I

2.

ran

[Tk(S , s2 ; ~ s 2 )

(k=l,...,g)

T k (s2, s I ;~'s2) ] [

= rsl = rs2

(k=l,...g)

~Tk(S I, s2 ;~'Sl) Tk(S1, s I ;~'Sl) J

rank[ Tk (Ss' sS ; ~ S s ) .... Tk (sS, C 1 ; ~ S s )

N.

S = ~ i=l

LTk (s I, s S ; ~ S l ) .... T k (s 1, s I ; ~ S l )

rs

l

(k=l,...g)

where

I

Irsi

s i ~ 11,...,, . .d

il

Ks i Ks.

if

. . si = ll)...,l d

I

Assuming t h a t carrying

out

the

Theorem 6.10 holds,

decentralized

controller

an algorithm is given in synthesis

T k ( i , j ) , u s i n g one dimensional o n - l i n e t u n i n g methods.

in

terms

(DAV-78)

of the

for

parameters

273

Remark 6 . 7 . 1.

I f mi

>/ r i,

(i=l . . . . . S),

Theorem

6.10

holds

for

almost all

(C~A,BtD)

s y s t e m s . On t h e o t h e r h a n d , if mi < r i f o r some i ~ {I . . . . . S}, t h e n T h e o r e m 6.10 does n o t h o l d , a n d no s o l u t i o n e x i s t s . 2. It is i n t e r e s t i n g to n o t e t h a t t h e local c o n t r o l l e r s s y n t h e s i s m u s t b e c a r r i e d out in s p e c i f i e d s e q u e n c e (not n e c e s s a r i l y u n i q u e ) . If t h i s s e q u e n c e is n o t r e s p e c t e d t h e n , in g e n e r a l , no c o n t r o l l e r s y n t h e s i s can b e p e r f o r m e d . H o w e v e r , t h i s is n o t t h e case if a s s u m p t i o n (it) is r e l a x e d in t h e t u n i n g s y n t h e s i s c o n s t r a i n t s ( D A V - 7 9 b ) .

6.6. - D E C E N T R A L I Z E D

CONTROL

BY

HIERARCHICAL

CALCULATION

T h i s s e c t i o n is c o n c e r n e d w i t h t h e h i e r a r c h i c a l calculation m e t h o d s of a d e c e n t r a l i z e d c o n t r o l for t h e c l a s s of l a r g e - s c a l e l i n e a r i n t e r c o n n e c t e d s y s t e m s . Two t y p e s of a l g o r i t h m s a r e p r e s e n t e d : t h r e e - l e v e l calculation a l g o r i t h m s (HAS-78a,b~ 79) a n d two-level c a l c u l a t i o n a l g o r i t h m s ( X I N - 8 2 ) .

6.6.1. - Three-level calculation algorithms

This subsection presents the algorithm of Hassan

a n d Singh

(HAS-78b)

a n d its

extension to the case of robust decentralized control (HAS-79).

6.6. l.a. - Decentralized near-optimal controller

(HAS-78b)

Consider the large-scale linear i n t e r c o n n e c t e d system described by : 5 xi = Ai xi + Bi ui + i--E1 A i j x j

(6.6.1)

orj in a compact form, b y :

I~ = Ax + Bu +

Cz

Lx

(6.6.2)

where A, B a n d C are appropriate block-diagonal

matrices with S blocks, a n d L is a

full matrix representing

the interconnections

trol t h e s y s t e m

b y d e c e n t r a l i z e d s t a t e f e e d b a c k minimizing a q u a d r a t i c p e r -

(6.6.2)

between

the systems.

formance i n d e x . The optimization p r o b l e m can b e w r i t t e n :

We

want to con-

274 7

rain

J = 1/2 f

( x ' Qx + u ' R u ) d t

K subject to : = Ax + Bu + Cz (6.6.3)

Z = Lx

u = -Kx where

Q a n d R are appropriate weighting matrices.

I t is s h o w n

in

(SIN-76)

that

the

solution of the

above problem has

the

fol-

lowing form •

u = - Gx - Tx where

G

equation,

is

a

block-diagonal

matrix

obtained

by

solving

the

decomposed

Riccati

a n d T is a full m a t r i x o b t a i n e d b y h i e r a r c h i c a l c a l c u l a t i o n .

Now, s u b s t i t u t i n g

(6.6.4)

into the criterion,

we o b t a i n

:

fT 0 ( x ' Qx + x ' W* x ) dt

Jopt = i/2 with

(6.6.4)

W* = ( G + T ) ' R ( G + T )

S i n c e it is d e s i r e d to o b t a i n a d e c e n t r a l i z e d matrix Td,

rain

control,

we c o n s t r a i n

T to b e a d i a g o n a l

and the optimization problem becomes : T J = 1/2 f0 ~ x ' Ox + x' Wx)]dt

Td s u b j e c t to £ = ( A - B G ) x + Cz - B T d x z = Lx

(6.6.5)

W = (G + T d ) ' R (G + T d ) w h e r e B is a n x n introduce

m a t r i x ( i f , in p r a c t i c e ,

B is o f l o w e r d i m e n s i o n t h a n n x n ,

we can

additional fictitious controls).

Let G d (Go) , A d ( A o ) ,

Qd(Q0),

a n d B d (B 0) be t h e m a t r i c e s c o m p o s e d o f t h e

d i a g o n a l ( o f f - d i a g o n a l ) e l e m e n t s o f t h e m a t r i c e s G, then the matrix W can be written

(A-BG),

:

W = (G d + T d ÷ GO)' R (G d + T d + G O) = (F + GO)' R (F + G O) w h e r e F = G d + T d is a d i a g o n a l m a t r i x .

The optimization problem can be rewritten

as :

Q, a n d B, r e s p e c t i v e l y ,

275

T = i/2 0/ [x' Q d x + x'F' R F x

rain J with

g (x,F,G0)

+ g (x,F,G o)] dt

= x' (Q0 + F' R G O + G O R F + G O ' R G O ) x

subject to : :~ = A d X with

(6.6.6.)

- B d T d x + y (x,z,T d)

y (x,z,T d) = A 0 x + Cz - B 0 T d X To s o l v e

which consists

this

problem,

in adding

the o p t i m i z a t i o n

problem

trajectories supplied a fixed point type

XCf

:

Hassan

certain into

and

additional

a number

by the second level.

algorithm.

Singh

(HAS-78b)

linear

constraints

of independent These

Let us introduce

use a prediction in o r d e r

subproblems

trajectories

to d e c o m p o s e for

are then

some fixed

improved

the additional linear constraints

using "

X

(6.6.7)

Td* = T d Substituting

(6.6.7)

rain J with

method

g (x*,

into (6.6.6),

the optimization problem becomes

:

T = 112 _~[ x ' Q d x + x * ' F' R F x * + g ( x * , F * , G O) ]

F * , G O) = x * ' (Q0 + F * ' R G O + G O' R F * + G O' R G O) x*

s u b j e c t to : ~¢ = AdX - B d T d x * + y ( x * , z , T d * ) z

= Lx

Td* = T d X*

with In order

---- X

y (x*,z, T d * ) = A 0 x * + Cz - B 0 T d * x * to s o l v e t h i s p r o b l e m ,

1 x' Qd x + ~1 x ~, H = ~-

+y ' [AdX-

let us write the Hamiltonian :

F' RF x * + ~1 g (x% F% G O) +

B d T d x ~ + y (x% z, Td~)] + ~ ' ( L x -

z) +

n

+ 13' ( x -

x ~) +

w h e r e ~ , B, v i a r e L a g r a n g e The necessary

Y: i:l

v[ (Td. - T~.) L

multipliers,

conditions

and y is the costate variable.

for optimality can be written

as :

dt

276 aH an

0

-

~

aH aT = 0 aH a6

z = Lx

--~

~ = C'

= 0

~

x* = x

= 0

---,,.

T*,.o

al-i aS'i

=

i

aH a

= 0 ~

T~

aH a x* = 0

(6.6.8) 5'

(6.6.9) (6.6.10) (6.6.11)

rd. ,

V = d i a g [(R G O x* - B'o 5" ) x * ' ]

(6.6.12)

B= ( F ' R F + Qo + F * ' R G o + C'o RG o) x* + (A o -

-'~

T~' B'o - T~j B~t ) ~" (6.6.t3)

Suppose then

the

now that x*,

Hamiltonian

can

be

Td*, 6 and V have been provided decomposed

such

that

each

b y t h e s e c o n d level,

subproblem

has

only

one

variable Tdi.

H aTd.

.-4,.

*2 x.t

or

Td"

_ Gd"

l

,

= 0

(Gd- + Td ) R i t i

Bd. Yi x~-t + vi = 0 t

l

-

-

-

1 .

R.

aaYiH = xi

= Ad i xi - Bd. [- Gd. l ~

I ~2 (vi - Bd. Y i xi)* ] x *i + Yi (x*, z, T *d) R-x. t gd" t i 2 Bd. % l

with

=-~.

i + vi)

X ~*

l

1

2 T I. = Ad i xi - Bd i Gd i x *i - - - -R.x.

3H 3xi

(- Bd.3'i

x- 2 1

v i - ' - ~i

Ti + Yi (x*, z, T d)

I

= Qd. xi * Ad.~'i + ki + 6 i l t

J

k. = L: ~. l

1

L e t Yi = Pi xi + h i '

l

t h e n a f t e r m i n o r m a n i p u l a t i o n s we o b t a i n

:

(6.6.1M

277 2 Bd.

Pi :

- 2

Ad. Pi ÷-'I~. P~l

Qi

wi'th Pi(T) = 0

(6.6• 15)

1

Bd. v i rli ) - Pi [Bdi Gd.1 x.*L + ' R I x ~•

P~. = (- Ad.l ÷

Ri

1

Hassan and singh

suggest

45

+ Yi (x , z, T d) ] - k i -

Bi(6.6.16)

1

the following three-level

algorithm.

Algorith m 6.3 (HAS-78b). Step 0 : G u e s s t h e i n i t i a l t r a j e c t o r i e s Step 1 :

Guess

the

initial

zh and k h at level 3 for the initial index h=l

trajectories

x *j,

T d * , B j,

vj

at

level

2,

and

set

the

iteration index j=l. Step 2 : U s i n g x *j, Td* J, B j, v j o b t a i n e d and (6.6.16), Step 3 : S u b s t i t u t e right the

sides

(BJ+I-BJ),

1, c a l c u l a t e P i ' n i f r o m ( 6 . 6 . 1 5 )

and

x and y obtained

a t l e v e l 1, Ir a n d

of ( 6 . 6 . 1 0 ) - ( 6 . 6 . 1 3 )

integral

from s t e p

x from ( 6 . 6 . 1 4 ) ,

of

and

the

norm

( ~ j + l _vj)

T (fromy=Px

to o b t a i n

of t h e

+ •).

x *]+1,

differences

C a l c u l a t e also T d .

z obtained

(x *iT1 -

are not sufficiently

small,

x'J),

the

decentralized

(k h + l

gain matrix,

- kh )

and

and

v j+l.

If

(Td*J÷l-Td*}),

go to s t e p

go to l e v e l 3 a n d c a l c u l a t e n e w k h + l a n d z h + l f r o m ( 6 . 6 . 9 ) n o r m of t h e d i f f e r e n c e s

at level 3 into the

Td~*J+l , B .3+1,

2.

and

Otherwise,

(6.6.8}.

If the

(z h + l - z h ) a r e s m a l l ,

otherwise

r e c o r d T d as h+l 1 using kh+l, z as the

go to s t e p

new guesses. Remark 6 . 8 . 1. T h e A l g o r i t h m be p r o v e d

using

for n o n l i n e a r

6.3 i s a p r e d i c t i o n

a similar technique

type

algorithm,

to t h e o n e u s e d

and its convergence

by Hassan

(HAS-76)

and only the decentralized

gains are

systems.

2. T h e e n t i r e

c a l c u l a t i o n is d o n e o f f - f i n e ,

u s e d o n - l i n e to c o m p u t e a n d i m p l e m e n t t h e o p t i m a l d e c e n t r a l i z e d 3.

The

desadvantage

a l t h o u g h it is n o t s e n s i t i v e

6.6.1.b.

- Robust

Hassan, to p r o v i d e prescribed and t a k e s

75,76).

can

and Singh

of

the

algorithm

to small v a r i a t i o n s

decentralized

near-optimal

S i n g h a n d Titli ( H A S - 7 9 )

a robust

decentralized

degree a (in the into account

control

sense

external

is

that

T d is

dependent

of t h e i n i t i a l c o n d i t i o n s .

controller

extended

(HAS-79).

the approach

which ensures

of A n d e r s o n

disturbances

control. initial-state

and and

of the above

exponential

Moore

structural

(AND-71),

stability see

perturbations

section with a § 6.4.1) (SIL-73,

278

Consider crlbed by

an interconnected

dynamical system

composed by

S subsystems

des-

: S

x i = Ai R'i * B i ~ i

*j

e.. 1J A.. D ~.J + ~.'

1

( i = I , . "" ,S)

where

t h e e. 2s a r e t h e e l e m e n t s o f t h e i n t e r c o n n e c t i o n m a t r i x E, w h i c h a r e i n t r o 1] d u c e d to i n c o r p o r a t e a n y s t r u c t u r a l p e r t u r b a t i o n w h i c h m a y o c c u r d u r i n g t h e o p e r a -

tion of the system.

E i s c o n t i n u o u s i n t i m e , w i t h 0 x< e l i ( t ) x( 1, ( i , j = l . . . . .

We s a w in s e c t i o n ( 6 . 4 . 1 ) a,

t h a t to e n s u r e

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

index

S).

t h a t t h e s y s t e m is s t a b l e w i t h d e g r e e :

5 i f = i__Z1 1/2 f0Te2c~t [ ( R i - ~ i ~ ' Qi ( ~ i -

~d)+

u--i, Ri ~ i ]

dt

w h e r e T e q u a l s a t l e a s t 4 t i m e s t h e time c o n s t a n t o f t h e s y s t e m . variable transformation, form, i.e.

"Linear Quadratic"

: 5

T

rain3 :

i

Qi

s u b j e c t to : with

With a n a p p r o p r i a t e

we c a n p u t t h i s p r o b l e m i n t o a s t a n d a r d

xi-

+ u i,

dt

S

~ i = Ai x i + Bi u i + jZ=l A. = ~. + a I l

e..H A.. i] x.] + d i ( t )

I

d i ( t ) = ~ii(t) e a t The optimal control for this system can be written as :

u1" G.

where

I

P. i s t h e I

system.

S Z

= - G,x 1

R7 ! I

e.. T., x, - s.

j=l

1]

E

]

1

B ) P. I

1

s o l u t i o n o f t h e local

(i.e.

decomposed)

Riccati equation

for the

it h

sub-

Let T =[e.. T . . ] , t h e n t h e o p t i m a l c o n t r o l , in i t s g l o b a l f o r m , b e c o m e s : 1] 11

u = - Gx - T x - S Now, w i t h t h e

same

approach

a n d n o t a t i o n s a s in t h e l a s t s e c t i o n ,

the optimi-

zation problem can be written T min J = 1]2 f0

(x-xd)'

Q(x-xd~

+ x'F' RFx + 2 X'Gd'RS +

+ 2 x ' T d ' RS + S' RS + g ( x , F , G O) s u b j e c t t o : ~ = AdX - B d T d X + y ( x , z , T d ) Z

---- L x

(6.6.17)

279 w h e r e g ( x , F , G 0) = (x-xd) ' O 0 ( x - x d )

+ x' F' R G 0 x + x'

+ x ' G O' R F x + x ' G O' RG0x y ( x , z , T d) = A0x + Cz - B 0 TdX + D D=d-BS

This p r o b l e m

is

similar

to

the

problem

(6.6.6)

and

can

be

solved

Algorithm 6.3 a f t e r c h a n g i n g t h e o p t i m a l i t y c o n d i t i o n s ( 6 . 6 . 8 ) - ( 6 . 6 . 1 6 )

by

using

the

by the appro-

priate ones.

6.6.2. - Two-level

calculation algorithm (XIN-82)

Xinogalas,Mahmoud

and

Singh

(XIN-82)

-

considered

the

following

optimization

problem : 1

rain Ki

S

3 =~- Z f (x! Qi xi + u! R i ui)dt i=l 0 l 1

subject

S xi : Aii xi + 5i ui + i=lZ Aij X.j

to

Ui = - K i x i

(i=l,...,S)

This p r o b l e m c a n b e w r i t t e n in a g l o b a l f o r m a s : co

min J = 1 / 2 f0

( x ' Qx + u ' R u )

dt

KEK d s u b j e c t to •

= Ax + Bu u=-

(6.6.19)

Kx

w h e r e B, Q a n d R a r e d i a g o n a l m a t r i c e s ,

and Kd is given by

K d = {K/K = b l o c k - d i a g . [ K 1 . . . . . K S ] , K i E R m i x r i

It is e a s y t o s h o w t h a t

min

(6.6.19)

can be b r o u g h t

:

, (i=l . . . . . S)}

b a c k to :

J = T r [ ( Q + K'RK) S ) ]

KCK d

subject to : g (S,x 0) = S (A-BK)' + ( A - B K ) S

+ X 0 =0

(6.6.20)

280

with

X0 = E [ x ( 0 )

Let

Ad = diag .(Aii)

x(0)']

= diag.

( x i)

A0 = A - A d An alternative

formulation of the optimization problem

(6.6.20)

is given by

:

rain J = T r [ (Q + K I R K ) S ] subject

K~K d to : + X0 +Z=0

g ( S , X 0) = S ( A d - n K ) t + ( A d - B K ) S Z = A O S + S A O'

The corresponding

Lagrangian

L = Tr [(Q + K'RK)S]

function can be formed as :

+ Tr[P

g (S,X0)]

For this static optimization problem,

tL --= '2 T

0

8L -0 8Z t___kL ~P

the necessary

--~

Z = A0P + PA"u

~

T-P

--4,-

(A d -

0

)__LL = ~ ~S

BK) S + S (A d -

(A d - BK)' T + T (A d -

D/., = 0 aK

~

K = R -1 B'

where

+ T r [T(AoS + SA 0' - Z ) ] conditions

for optimality are

:

(6.6.21)

BK)' + X0 ÷ Z = 0

(6.6.22)

BK) + Q + K ' R K + A~) P + PA 0 = 0

M d Sd 1

M d = diag. (T5) S d = diag.(S)

To s o l v e t h e 82) p r o p o s e

above optimality

conditions

Xinogalas,

Mahmoud and

Singh

(XIN-

the following tow-level algorithm.

Al~orithm 6.4 (XIN-82). Step 1 : Guess an initial value of the decentralized Step 2 : Compute have

negative

mentano

and

the eigenvalues

of the matrix

real parts,

step

Singh

g o to

(ARM-81)

(see

3.

g a i n m a t r i x Kq . (A d - B K q ) .

Otherwise,

§ 6.3.1.a)

use

to c o m p u t e

I f all t h e e i g e n v a l u e s the

algorithm

of A r -

a stabilizing

decen-

281

tralized

feedback

matrix

Kq ,

i.e.

such

that

(A d

-

BK q)

is

asymptoticaly

stable. Step 3 • S t a r t t h e t w o - l e v e l h i e r a r c h i c a l c o m p u t a t i o n s t r u c t u r e with g u e s s e d v a l u e s for t h e m a t r i c e s Zq a n d

Tq and send these values,

together with the

gain

m a t r i x K q , to t h e f i r s t level. Set q = l . Step 4 : At t h e f i r s t level, ( 6 . 6 . 2 1 ) Bartels and Stewart

(BAR-72).

a n d ( 6 . 6 . 2 2 ) a r e s o l v e d u s i n g t h e t e c h n i q u e of The m a t r i c e s S q a n d T q a r e c o n v e y e d to t h e

second level. Step 5 : New p r e d i c t i o n s of t h e m a t r i c e s Z, P a n d K a r e c a l c u l a t e d a c c o r d i n g t o :

Z q+l = A 0 S q + S q A 0' pq+l = Tq K q+] = R -I B' M q

(Sdq)-I

If the conditions : ~]HZ q+ll[- [[Zq[[ < ~Z k~llPq+l [[- ][Pq II < ep q [ [ K q + l ] [ - [[Kq[i <e K Kq+l are satisfied, regard

as

the

optimal

solution

and

finish the

iterative

s c h e m e . O t h e r w i s e , u p d a t e t h e m a t r i c e s Zq + l , p q + l a n d K q+l u s i n g t h e r u l e s : Zq+l = c 1 Zq + d I Zq+l = c2 p q + d2 p q + l Kq+l = c 3 K q + d 3 K q+l w h e r e t h e c o n s t a n t s c] a n d dj s a t i s f y cj + dj = 1 for j = 1 , 2 , 3 .

(The q u a n t i t i e s

eZ, ep, eK a r e small p r e s e l e c t e d t o l e r a n c e v a l u e s ) . Go to s t e p 2.

6.7. - CALCULATION METHODS USING AN INTERCONNECTION MODEL In t h i s s e c t i o n , we p r e s e n t an a l t e r n a t e a p p r o a c h to t h e optimization p r o b l e m , which u s e s

a

simple r e d u c e d

model

for

the

interactions

between

the

subsystems

(HAS-78a) (HAS-80) (CHE-81).

6.7.1. - The g e n e r a l i n t e r c o n n e c t t o n model (HAS-78a) The optimization p r o b l e m c o n s i d e r e d h e r e is : S

~o

min J = i/2 i__E1 0f [(xi - xd)' Qi (xl - xd) dt + u:,Ru.], dt subject to : ~i = Ai xi + Bi ui + Ci zi + di

(i=1 . . . . . S)

(6.7.1)

282

where

Qi ) 0,

Ri

tion vector

which

subsystems,

i.e.

z = Lx

0 are appropriate

>

is assumed

weighting

to b e

a linear

and

zi is the interconnec-

of the states

of the other

(L • f u l l m a t r i x )

The optimal control of each subsystem = _ ~1

ui

matrices,

combination

R'

is given

by

:

P. x . - R - I B . ' s. x 1 1 1

(6.7.2)

w h e r e P. i s t h e s o l u t i o n o f d e c o m p o s e d R i c c a t i e q u a t i o n ( i . e . R i c c a t i e q u a t i o n f o r the .th x 1 subsystem) a n d s i i s a s o l u t i o n o f a l i n e a r d i f f e r e n t i a l e q u a t i o n a n d d e p e n d s on the states

of the other

subsystems

(SIN-76)

:

S si

=

i~=l Wij xj + O i

If t h e i n t e r a c t i o n s

(j=l . . . . .

between

S)

the subsystems

(6.7.3) are ignored,

t h e c o n t r o l in

(6.7,2)

t

becomes

ui

= U l i = - R ~ I Bi' Pi xi - R i l =K . x. - w , 1

The control

I

(6.7.2)

Bi 0i (6.7.4)

I

is d e c o m p o s e d i n t o t w o c o m p o n e n t s

:

u i = Uli + u2i

(6.7.5)

where Uli is the control given by

(6o7.4),

a n d u 2 i will b e c o m p u t e d

as shown

in t h e

following. Now,

if

the

interconnections

optimization problem

are

can be written

as

considered

as

unknown

disturbances,

the

:

S "'-" J = subjecto

o

Qi ( " i -

+ *i

dt

to : ~ ' = A.* ~ . + I

with

(6.7.6)

C. ~ . + D.

I

1

I

I

A.* = A. - B,K. 1

1

1

1

D. = d. - B. w. 1

1

1

1

w h e r e ~i ( u 2 i , 5 2i ) i s a n a p p r o p r i a t e Let

the

v e c t o r Yi b e

Bi ui2 - CiY i, then

we h a v e

chosen :

function of the control ui2 and its derivative. to

minimize

the

Euclidean

norm

of

the

vector

283

Yi = (C'Ci)-1 Ci' Bi ui2 = ~ u2i Substituting

~i

B i u 2 i b y C i Yi' we h a v e

+ ci (B u2i + ~i ) + "~" = Ai* i + Ci Yi + Di Yi = B u2i + ~zi

with

=

(6.7.7)

Ai* ~i

where Axi is t h e a p p r o x i m a t e

Di (6,7.8) (6.7.9)

s o l u t i o n of t h e s t a t e s r e s u l t i n g

Let ~. b e a p p r o x i m a t e d

from these substitutions.

by the following linear differential

1

equation

:

~i = Azi Azi + dzi where Azi is t h e p a r t corresponds

of A G (A G i s t h e d y n a m i c m a t r i x o f t h e o v e r a l l s y s t e m ) ,

to t h e e l e m e n t s of 9 . . T h e d e r i v a t i v e

of ( 6 . 7 . 9 )

1

which

can then be written

-"

= Azi + IB fi2i - Azi IB u2i + dzi with

-- Azi Yi + vi + dzi

(6.7.10)

vi = ~ fi2i - Azi B u2i

(6.7.11)

D e f i n e t h e f u n c t i o n ~i in ( 6 . 7 . 6 )

as :

~i (u2i' u2i) = vi' Ri vi Then the optimal decentralized

control problem can be rewritten

S min J = 1]2 i=Z1 f0¢o(~'i - ~ d ) , Q i ( x~i _ .,sd .~i) + vi , R i v i s u b j e c t to : ~. = X. ~. + B. v. + D. 1

1

1

1

1

1

where

~i:

= Yi

[o:] ~L =

d

[: 0] aod

~i =

0

i

and t h e o p t i m a l s o l u t i o n of t h i s p r o b l e m is g i v e n b y

:

A. l

dt

as :

284

~7: B,

vi = - ~T1 ~i' ~i ~i = - Gil q

- Gi2 Yi -

~

~-ilB i.

~ i si

(6.7.12)

w h e r e ~ i is t h e s o l u t i o n of t h e R i c c a t i e q u a t i o n

a n d ~. is t h e s o l u t i o n of t h e l i n e a r - v e c t o r 1

=

-

Substituting

differential equation

I - Pi ~i

v i in ( 6 . 7 . 1 0 ) ,

6.7.11),

:

:

+ Qi ~d

we o b t a i n

:

~* = Ai* xi* + C i Yi* + D i

(6.7.13)

Y* = ( A z i - l G i 2 ) Yi* - Gil xi* + dzi* dzi* = - R~ ~ i ~ i + dzi

(6.7.14)

From t h e e q u a t i o n

(6.7.11),

we h a v e :

B u2i = v i + Azi IB u2i Again,

b y m a k i n g a similar a p p r o x i m a t i o n

calculating vectors

the

two v e c t o r s ai, ~3i w h i c h

to t h a t made in e q u a t i o n minimize t h e

Euclidean

(6.7.7),

norms

i.e.

of t h e

by two

•

IBa i - Azi ~B u2i and

BB i - v i

and substituting

them in e q u a t i o n

(6.7.8)°

we h a v e : (6.6.15)

u~i = Aui u~i - Hi xi* - Fi Yi* - Si with A

. = (13' ~ ) - I

B.' A I

Ul

. ~B. Zl

I

Hi = (13i' ]3) -1 ]Bi' Gil Fi

([Bi' B i ) - I 13i' Gi_~

Si

([Bil 13i)-i 13i' R i

and t h e d e c e n t r a l i z e d

[Bi' s i

c o n t r o l is g i v e n b y

(6.6.13)

g r a m of F i g u r e 6.5 s h o w s t h e c o n t r o l l e r s t r u c t u r e .

(6.6.14)

and

(6.6.15).

T h e dia-

285 Cizi*+Di

1

2 Fig.

6.7.2.

- Model-following method

(HAS-80)

Let the optimization problem be S

6.5

(CHE-81)

:

co

min J = 112 iZ__l Of (xi' Qi xi + ui' s u b j e c t to •

R i u i) d t

~c. = A. x. + B. u. + C. z. 1

~1

1

1

I

1

zi = j=~l

Lij xj

In order

to s o l v e t h i s

model o f i n t e r a c t i o n s

(6.7.16)

I

problem,

Hassan

and

."

z = Azi z i and the optimization problem can be rewritten 5

as :

¢o

rain J = I/2 iE__l Of (Yi' ~i Yi + ui' Ri ui) dt subject to : where

:

Singh

(HAS-80)

use the following

286

[xi] E:ic] [:ij I i]

Yi :

~i =

l

zi

~i :

and Qi =

Az i

The optimal solution of this problem is given by

:

ui = - R~ 1 ~ i Pi' Yi w h e r e P. is t h e s o l u t i o n o f t h e R i c c a t i e q u a t i o n . 1

T h i s c o n t r o l c a n b e w r i t t e n in the

form : (6.7.17)

u i = - K l i x i - K2i z i Now,

consider

the subsystem

m o d e l w h o s e i n p u t ~. is p r o v i d e d 1

by the inter-

connection model :

4.

1

= A. Ax. + B. u. + C. ~z. 1

1

1

w h e r e ~x.1 is t h e (from ( 6 . 6 . 1 7 ) ,

state

1

1

(6.6.18)

1

of subsystem

model.

Then,

a f t e r r e p l a c i n g z p a r Az) in ( 6 . 6 . 1 6 )

if we s u b s t i t u t e and (6.6.18)

u i by

its

we o b t a i n

value

:

-'xi = ~i ~i + Gi zi - Bi K2i~i

(6.6.19)

with ¢h1 = A i - B i K l i xi

= ~i~i

(6.6.20)

+ (Ci - Bi K2i) zi

w h e r e ~ is t h e i TM s u b s y s t e m 1

Substracting

Ai

,~

(6.6.20)

+

state resulting from ( 6 . 6 . 1 9 )

we g e t :

ci

where x i and zi are error vectors given by I

l

"~. = z . 1

I 1

I

-'~.

1

and the optimization problem is rewritten :

from t h e u s e o f ~z. i n s t e a d o f z . . 1

1

287 S

co

min J = 1/2 i=lZ °f subject to :

(x~i'Hi ~i + '~i'Si ~'i)dt

~". = ~. ~. + c. ~. 1

1

1

1

1

The optimal solution of this problem is given by : ~i* = = S~I C.' P.1 ×.1 1 where P.1 is the solution of the decomposed Riccati equation. Then we get : zi* = ~i - S~ 1C.' P. 1 1 1 and the decentralized control is given by :

~.i = <~i + Bi K2i s] I ci' Pi~ xi + ci "i - Bi K21 (~i + s~ I .~i = ~i Axi + (Ci - Bi Kzi) "~. = A .~z, I ml I

I

Figure 6.6 illustrates the control s t r u c t u r e . C i zi

[-Bi ,<,2! [

I C i - Bi K2i i z r~

4---

-I-

O

I _]

"1 , I_ s~l ci Pi IFig. 6.6

Cil Pi xi)

288 Choice of t h e i n t e r a c t i o n model It is c l e a r t h a t t h e choice of Azi p l a y s an i m p o r t a n t p a r t in t h e e x i s t e n c e of s u c h c o n t r o l l e r . H a s s a n a n d S i n g h (HAS-80) p r o p o s e d to choose A . a s t h e block of Zl

t h e global m a t r i x A c o r r e s p o n d i n g to t h e v e c t o r z i . To c l a r i f y t h i s i d e a , c o n s i d e r the following e x a m p l e . Let t h e o v e r a l l s y s t e m (with two s u b s y s t e m ) m a t r i x b e : -

1

all

a12

a21

a22

I

-

0

0

a15

a23

0

0

I

Then,

I I

A+CL=

a31

0

a41

o

0

0

I

I I

a33

a34

a35

a~3

a4~

a45

I I

a53

as~

a55

I

a c c o r d i n g with t h e p r o p o s i t i o n of Hassan a n d

S i n g h in

(HAS-80),

the

i n t e r a c t i o n model is :

ix I

zI =

x3

z2 = x I

,

Az2

AZl :

aa31

L a35

a33

= all

This r u l e for c h o o s i n g Azi t a k e s i n t o a c c o u n t t h e s p a r s i t y of t h e o v e r a l l d y n a mic matrix b y eliminating columns w h e r e t h e o f f - d i a g o n a l p a r t s a r e z e r o .

This p r o -

v i d e s a good model in t h e case f o r w h i c h t h e s e l e c t e d e l e m e n t s a r e t h e d o m i n a n t ones in t h e r o w s of A. H o w e v e r , if it is not t h e c a s e ,

t h i s choice may mean a l o n g e r

c o r r e c t i o n p e r i o d o n - l i n e a n d t h e Azi may be u n s t a b l e (MOO-81). To o v e r c o m e t h i s difficulty, Himn a n d S i n g h (CHE-81) p r o p o s e to choose Azi as a diagonal m a t r i x with n e g a t i v e e l e m e n t s , i . e . t h e y c h o o s e a s t a b l e A . o r , in o t h e r zl w o r d s , t h e y a s s i g n t h e p o l e s of t h e model in t h e l e f t h a l f complex p l a n e . H o w e v e r , t h e choice of t h e i n t e r a c t i o n model r e q u i r e s i n d e e d t h e j u d g m e n t of t h e d e s i g n e r . As a m a t t e r of f a c t , t h e r e is no simple, infallible way o f c h o o s i n g Azt. A n y w a y , for any choice of Azi ( s t a b l e ) , t h e e r r o r s b e t w e e n t h e s y s t e m s t a t e a n d t h e model s t a t e

will b e c o r r e c t e d to a l a r g e e x t e n t b y t h e f e e d b a c k c h a n n e l . Of c o u r s e ,

t h e a c c u r a c y will p a r t i a l l y d e p e n d on t h e f i g u r e s in Azi. F i n a l l y , n o t e t h a t t h e loop I

289 (see

Figure

6.6)

can

be

viewed

as

a

reference

input.

It

is

independent

of

any

control that could be applied.

6.8. - D E C E N T R A L I Z E D C O N T R O L F O R SYSTEMS WITH O V E R L A P P I N G I N F O R M A T I O N SET

In the previous tralized

systems

disjoint

information

control

stations,

regulation multiple

s e c t i o n , we h a v e c o n s i d e r e d

assuming sets,

overlapping

the

i.e.

there

tlowever9

(ISA-73).

control

that

power

systems

subsystems,

in

some s y n t h e s i s

a c t i o n s of t h e is

no

practice~

systems

sharing

many

(SIL-7g.

for

reliability

and

the

controllers

models,

CAr,-7g}.

of

carried

of information

system

enhpnc~,ment

sharing,

methods for decen-

are

among

for

information

are

among

using

the

example

economic systems (SIL-f~0}.

out

the

local traffic

(AOK-76),

constituted

of

controllers

is

absolutely essential.

O n e w a y of d e s i g n i n g n i q u e s of e x p a n s i o n

and

a c o n t r o l l e r f o r t h i s t y p e of s y s t e m s is to u s e t h e t e c h -

contraction.

The

idea

is

to e x p a n d

(under

certain

condi-

t i o n s ) t h e s t a t e s p a c e of t h e o r i g i n a l s y s t e m so t h a t t h e e x p a n d e d

s y s t e m c o n t a i n s all

the n e c e s s a r y

subsystems

as

disjoint.

troller

can

information about the original system Then,

be

conventional

used

for

the

techniques

expanded

for

the

system

and

and

that

design the

the

appear

of a d e c e n t r a l i z e d

resulting

controller

is

concon-

t r a c t e d f o r i m p l e m e n t a t i o n on t h e o r i g i n a l s y s t e m ,

6.8.1.

- Expansion,

contraction,

and inclusion

Consider the system S given by

S : ~ = Ax + Bu

w h e r e x ~ Rn , tant matrices.

x(0)

:

= x0

(6.g.1)

u ¢-R m are the s t a t e and

the input,

Associate the performance index

and

A, B a r e a p p r o p r i a t e

cons-

:

oo

a (x 0, u ) = 0-r

( x ' Ox + u ' R u ) dt

where Q ~ 0 and R > 0 are appropriate

Associate the

pair

(~,

~)

with

weighting matrices.

the

l,Mr

a s s o c i a t e d p e r f o r m a n c e i n d e x a r e g i v e n By :

{q,

D,

where

the

s y s t e m .'~ a n d i t s

290

:~=~+~u

~(o) :~x 0

o0

(z0, u~ = f0 (~' ~ ~ + u, ~u~ dt ~ 0 and R > 0 are appropriate weighting matrices, and T ~

R~ , ~ >~ n .

I n t r o d u c e t h e linear t r a n s f o r m a t i o n : = Tx where

T

~(t,~0,u)

is

(6.8.2) an

~xn

constant

matrix

with

full

column

rank.

Let

x(t,x0,u)

and

be t h e s t a t e s of t h e s y s t e m s S and ~ c o r r e s p o n d i n g to t h e initial condi-

t i o n s x o , ~0 a n d t h e f i x e d i n p u t u . The l i n e a r t r a n s f o r m a t i o n 46.8.2) c a n be u s e d to r e l a t e t h e p a i r s ( S , J ) a n d ( S , J ) in t h e c o n t e x t o f i n c l u s i o n as follows : Definition 6.2

(IKE-81).

The p a i r

(~,~)

includes the pair

(S,J)

if t h e r e e x i s t s a

m a t r i x T s u c h t h a t for any initial c o n d i t i o n x 0 of S, t h e choice ~0 = T x 0

(6.8.3)

of t h e initial s t a t e ~'0 o f ~ implies t h a t : f o r all t >/0

x ( t ; x 0, u) = T I x ( t I ~ 0 ' u)

J (x o, u) = ~ ' ( ~ 0 , for a n y i n p u t u ( t ) ,

u)

w h e r e T I is a g e n e r a l i z e d i n v e r s e of T.

If t h e p a i r (~',~) i n c l u d e s t h e p a i r ( S , J ) , sion of ( S , J ) , (S,J),

t h e n (~,~) is s a i d to be a n d e x p a n -

a n d ( S , J ) is called a c o n t r a c t i o n of ( ~ , ~ ) . Note t h a t if ( ~ , ~ ) i n c l u d e s

t h e n t h e optimization p r o b l e m c o r r e s p o n d i n g to ( ~ , ~ ) is e q u i v a l e n t to t h e one

for ( S , J ) p r o v i d e d t h a t 46.8.3) h o l d s . Now, u n d e r t h e t r a n s f o r m a t i o n

(6.8.2)

t h e m a t r i c e s of t h e e x p a n d e d s y s t e m

and the original system are related by ; = TAT I + MA,

N

B

= TB

+ NB,

= (TI) ' Q T I + MQ and ~ = R + N R w h e r e MA, NB, sion.

MQ a n d N R a r e c o n s t a n t c o m p l e m e n t a r y m a t r i c e s of p r o p e r

dimen-

291

Theorem 6.12 (IKE-81,

(i) MAT = 0,

or

The pair

N B = 0,

(~',~') i n c l u d e s

MQ MiA1 T = 0

MQ Mi - 1 N = 0 a n d N R = 0

(i=1,2 ..... ~)

are

Although must

be

such

that

out that

the conditions

two

different

the

choice of matrices

chosen

such

sets

that

it can be used

(6.8.4)

of conditions

the

T,

In

the

following,

on the original system Definition 6.3 tractible

for controller

system

we c o n s i d e r

(IKE-81,

expansion

no means unique design,

and

(IKE-81).

(IKE-80),

they

observable

(a d e t a i l e d s t u d y

and about

(MAL-85)).

design

problem

can be contracted

and

discuss

the

conditions

to u = - K x f o r i m p l e m e n t a t i o n

SIL-82a).

The control

u = - ~

(t; ~0'

u)

~t

of t h e e x p a n s i o n

~ is con-

S if ~ 0 = T x i m p l i e s t h a t

> 0

u.

6.13 ( I K E - 8 1 ) .

MAT = 0 a n d

If

NB = 0

then any control law u = - ~ given by

do n o t i m p l y e a c h o t h e r , and

S.

K x ( t ; x 0, u ) = ~

Theorem

the

(6.8.5)

is c o n t r o l l a b l e

or observer

:

(6.8.5)

to t h e c o n t r o l u = - K x f o r t h e o r i g i n a l s y s t e m

for any fixed input

(S,3) if either

(6.8.4)

contraction

MA, N B i s b y

expanded

which the control u = - ~

and

for

this choice is given by Malinowski and Singh

under

the pair

T I MQT = 0 a n d N R = 0

(il) MIMAT = 0, T MiA1 NB = 0,

We p o i n t they

SIL-SZa).

is contractible

to t h e c o n t r o l law u = - K x ,

and K is

:

K:~T

6 . 8 . Z. - O v e r l a p p i n g Consider

again

composed of three vely,

i.e.

:

decomposition the

system

subvectors

Xl,

(6.8.1), x 2 and

and

assume

that

x 3 of dimension

its nl,

state n 2 and

vector

x

is

n 3 respecti-

292

x = ( X l ' , x z' x3')'

and n = nI + nz + n3

Let t h e i n p u t b e decomposed i n t o two s t a t i o n s :

u = (u I' u2')' where u I E Rml,

u 2 ~ RmZ

and m = m I + m 2 .

With this representation the system S can be described as :

x2

I AI~21 L~A2"-2~; A23

x2

B2I

l B22 -- II

LA3,

×3

"~;1--

B32

',A~2

A33

t

(6.8.6)

w h e r e the submatrices correspond to the c o m p o n e n t s of the state and input vectors.

Let the state be zation to a n y n u m b e r

decomposed

of c o m p o n e n t s

into two

overlapping

is obvious)

components

(the generali-

non square

matrix defined

:

~1 = ( x l ' x 2 ' ) ' 9t2 = (x 2' x3')'

s u c h t h a t t h e new s t a t e v e c t o r is :

= (~'I' ~ 2 ' ) ' T h i s v e c t o r is r e l a t e d to x b y : ~=Tx where ~ ~ R~ a n d ~' = n 1 + 2n 2 + n 3 , by :

T=

I1 0 0 0

0 12 12 0

a n d T is a ~ x n

0}

0

0 13 •

293 i = 1,2,3.

where I i = i = 1,2,3 a r e u n i t y matrices of dimension n i x n i Define the e x p a n s i o n ~ :

where

~ = TAT I + M ,

~ = TB + N

Ikeda et al. (IKE-81) p r o p o s e the choice (not u n i q u e )

TI=

i o o o] I!

1

1

~- 12

~- x2

0

o

0

l

0

M =

: l

~ AI2

- ~ AI2

1

1

~ A22

0

- g A22

1

13

1

0 - ~ A22 0

which s a t i s f i e s the conditions ( g i v e n by p a r t

~ A22

I

I

- ~ A32

--~-A32

(i) of Theorem 6.12)

0 0

N=0

0 0

for ~ to be an

expansion of S. With the t r a n s f o r m a t i o n ( 6 . 8 . ? ) the e x p a n s i o n "S is g i v e n b y :

All

0

AI3

BII

II BI2

A22 11 0

A23

B21

1 B22

AI2 1I I

A21

I

w

x2J

A21

0

II

A22

A231

I

A31

0

iI

A32

By comparing the s y s t e m s S and ~

A33

23

(6.8.8}

I

B2i

I B22 I

331

1 B32

we see t h a t t h e o v e r l a p p i n g decomposition

of the system S r e s u l t s in a disjoint decomposition of the e x p a n s i o n ~ of S. S t a n d a r d d e c e n t r a l i z e d control t e c h n i q u e s can t h u s be u s e d to d e s i g n a c o n t r o l l e r for ~. The e x p a n s i o n ~ can be r e p r e s e n t e d as two i n t e r c o n n e c t e d s u b s y s t e m s :

: ;.Xl = ~1 ~1 + ~1 Ul + 12 ~2 + 12 u2 x2 ~2 ~2 + ~2 u2 + A21 ~1 + B21 Ul where t h e matrices of the d e c o u p l e d s u b s y s t e m s a r e g i v e n b y :

294

~

" Xl = AI'~I + L I Ul ,4, x2 ~2 ~2 + B2 u2

IF-AII

and •

Al2]

"~1

=

[ Bll

~l = . A21

A22.]

A22

A23

["

BI2

[] B22

A2 : A32 A33] and'~2 = 1332 The interconnection matrices are given by :

0

.%1

A23

0

[B22J

[B31]

Let us associate the following performance indices with the decoupled subsystems : (T10' Ul) =

(Xl ' ~1 ~1

Ul' ~1 Ul) dt

~ (xz0' u2) ~oo(~Z" Q2 ~2 + u2' ~2 u2) dt where ~10 and ~20 are the initial states of ~lD and ~SD2 and ~1' ~2' ~1 and '~2 weighting matrices of appropriate dimensions.

are

The global performance index can be written : N

00

J(%,u)=I 0 (~,~+u~.)dt

where

6 = diag. (~1' ~Z ) = diag. (~1' ~2 ) In virtue of part (i) of Theorem 6.12, the performance index ~ (~0' u) is a expansion of : co

J (x 0. u) = 0f (x' Qx + u' Ru) clt with Q = T I ~T

295

and J (Xo, u) is t h e p e r f o r m a n c e i n d e x a s s o c i a t e d to t h e o r i g i n a l s y s t e m . Now, t h e local c o n t r o l laws :

Ul = - K1 ~i u2 = - ~2 ~2 are c a l c u l a t e d to optimize t h e local p e r f o r m a n c e i n d i c e s f o r t h e local s u b s y s t e m s ~ p and ~

. The global c o n t r o l is t h e n w r i t t e n :

I

Ii

II "K23

K24

and t h e c o n t r o l to b e i m p l e m e n t e d on t h e o r i g i n a l s y s t e m is g i v e n b y t h e c o n t r a c tion : Kll

[

K

2

I K23

%v-----

K24 J

6.9. - CONCLUSION Decentralized

control

and

in

general

constrained

structure

control

induce

p a r a m e t r i c optimization p r o c e d u r e s for t h e s y n t h e s i s of a d e q u a t e c o n t r o l s t r u c t u r e s . In t h i s c h a p t e r , we h a v e p r e s e n t e d most o f t h e available t e c h n i q u e s f o r dealing with t h i s p r o b l e m , k i n d of p r o b l e m ,

In o r d e r to g i v e a more complete a n d r e a l i s t i c s o l u t i o n to t h i s we h a v e also c o n s i d e r e d t h a t t h e s y s t e m could b e p e r t u r b e d

small o r l a r g e p a r a m e t e r v a r i a t i o n s o r a f f e c t e d b y e x t e r n a l d i s t u r b a n c e s .

by

The s o l u -

tion p r o v i d e s a r o b u s t d e c e n t r a l i z e d c o n t r o l , Since

decentralized

control

remains

an

active

research

area

with

practical

impact in many fields of a p p l i c a t i o n , t h e p r e s e n t e d r e s u l t s s h o u l d b e c o m p l e t e d soon by other efficient techniques and algorithms.

CHAPTER

STRUCTURAL

7.1.

7

ROBUSTNESS

- INTRODUCTION The

problem

of designing

has no fixed modes and that Chapter

5.

Chapter

an

6 was then

which can be used

concerned

to d e t e r m i n e

disturbances

control

with

number which, every

the

studies

system have

in a d d i t i o n to p r o v i d e the

occurence

(ALB-83)

disturbances

to t h e

connection Siljak

of o n e

(SIL-75)

is concerned

In addressed thesizing

to

or

more

who

constrained

failures

number

(SIL-75)

(SIL-78)

concept

was

the

stability

importance A recent

of

(DAV-81)

sensor

(LOC-

structural

A structural

perturba-

as the

originally e

properties

system,

composite

class of structural

either

refering

regulators

of these

controlled

This

affecting

controller.

to t h e c o n t r o l l e d s y s t e m w h e n

a certain

The other

in

technics

constraints

synthesizing

of

for composite systems

considered

system

values.

perturbations.

of a n o p e r a t i n g

may be disconnected. controller

optimization

problem

perturbations

subsystems.

the

to s t r u c t u r a l

the

preserve

class is defined

that

it i s a l s o o f a r e a l p r a c t i c a l

some desirable properties working,

In presence

(SIL-78)

with

possible

dis-

introduced system

by when

perturbations

measurements,

actuator

or line cuts. (DAV-81), with

an

the

plant

the property (1,2 .....

j-l),

robust

additional

a decentralized

S-decentralized still has agents

first

subsystems

behaviour

parametric

can affect the plant itself or the control system.

tion related

certain

attention

of structure]

(ACK-84).

the

of the system parameter

can be subjected

paid

component is properly

under 83)

that

of

such

is m i n i m i z e d w a s c o n s i d e r e d

with the addition of robustness

and variations

When d e a l i n g w i t h l a r g e s c a l e s y s t e m s , to c o n s i d e r

structure

transfer

the gains of the structurally

This problem was also considered to e x t e r n a l

optimal

the information

decentralized reliability

controller

such

that

The

to s o l v e t h e r o b u s t

if t h e j - t h

that satisfactory Vj ~ {1,2 . . . . .

servomechanism

requirement. control

robust

problem

problem

servomechanism

agent

fails,

tracking/regulation

S} (sequential

reliability).

(DAV-76c)

consists

in

problem

the resulting occurs

is

synfor a

system

for control

297 Also in

(LOC-83),

t h e p r o b l e m of d e s i g n i n g

lator which performes robust preserves

zero-regulation

zero-error

of all t h e

chapter,

variables

the

consequences

using

of s t r u c t u r a l

structurally

fecting

the

sidered.

In

(Section

the

last

section,

which, besides preventing to h e

and the

subset)

is

a r e s p e c i f i e d in a d i f f e r e n t The study

stabilizability

control.

design

the i-th one when a failure

problem.

on

affecting

Both

the faced

of

conditions,

an

is f o c u s e d on

or pole a s s i g n a b i l i t y

structural

controller

some m o d e s of t h e s y s t e m

f i x e d m o d e s in n o r m a l o p e r a t i n g

another

structural

feedback

7.Z)

but

constraints

perturbations

constrained

plant

regu-

is d i s c u s s e d .

the robustness

way a n d we a r e c o n c e r n e d w i t h a p u r e

decentralized

r e g u l a t i o n in n o r m a l o p e r a t i n g c o n d i t i o n s a n d

error

o c c u r s in t h e i - t h f e e d b a c k loop ( r e l i a b i l i t y )

In t h e p r e s e n t

a completely

perturbations

(Section

optimal

7.3)

control

are

afcon-

structure

(the unstable ones for example)

guarantees

that

these

modes

(or

a r e n o t f i x e d m o d e s w h e n t h e s y s t e m is s u b j e c t e d to a p r e s p e c i f i e d

c l a s s of s t r u c t u r a l

perturbations

7.2. - S T R U C T U R A L

(affecting the controller).

PERTURBATIONS

AFFECTING

THE

SYSTEM

C o n s i d e r t h e following s y s t e m c o m p o s e d of S s u b s y s t e m s

(OZG-~2)

•

= Ax + Bu

(7.2.1)

All .........

AIS

A =

B =

1

132. `

O

I I i

ASI . . . . . . . . .

"

A55

where

A.. C R n i x n j a n d B. EE R n i x m i , ( i , j = l . . . . . *] 1 d e c e n t r a l i z e d s t a t e f e e d b a c k in t h e f o r m :

u = Kx

K = block-diag.[

K1 K2 ...

S).

This

BS

system

is

controlled

KS]

by

(7.2.2)

where K. ~ R m i x n i

, (i=l . . . . . S ) .

1

We a s s u m e

that

(7.2.1)

is s u b j e c t e d

to s t r u c t u r a l

in t h e d i s c o n n e c t i o n of o n e o r m o r e s u b s y s t e m s . system

(7.2.1)

c a n be

represented

by

perturbations

which consist

]'he interconnection structure

a digraph

G = (V,E).

Each

node

of the

v. ESV is 1

a s s o c i a t e d to a s u b s y s t e m

Si, i E

{1 . . . . . S }

and each edge

( e i , e j) 6- E r e p r e s e n t s

a

298 nonzero interconnection digraph

from Si to Sj w h i c h i m p l i e s t h a t

w a s a l s o u s e d in

G T = (V T .

(COR-76b),

see Chapter

ET) o f G is a c o l l e c t i o n of n o d e s

connecting

the nodes

V T in G.

t r i c e s of A a n d B c o r r e s p o n d i n g

2,

A.. $ 0 a n d v i c e v e r s a ( t h i s P Section 2.2.3.b). A subgraph

V T c_ V a n d

the set of edges

T h e m a t r i c e s AT a n d B T a r e d e f i n e d as t h e s u b m a to t h e s u b s y s t e m s

a s s o c i a t e d to V T .

T h e n t h e following r e s u l t p r o v i d e s s u f f i c i e n t c o n d i t i o n s u n d e r (7.2.1)

is p o l e a s s i g n a b l e u s i n g t h e c o n t r o l ( 7 . 1 . 2 )

Theorem Section

7.1

(OZG-825.

If for every

25 G T = (V T ,

system (7.2.1)

Remark 7.1.

ET),

(see

connected

subgraph are

T h e model ( 7 . 2 . 1 )

in

The results

2,

then

the

perturbations.

s o l u t i o n s to t h e

(OZG-825.

Chapter

controllable,

all s t r u c t u r a l

of d e c e n t r a l i z e d

( D A V - 7 6 c ) is a l s o d i s c u s s e d

~: = A x + B u

perturbations.

(A T , BT5

fixed modes under

which the system

structural

strongly

h a s no d e c e n t r a l i z e d

s e n t e d in t h i s r e m a r k .

under

the matrix pairs

T h e p r o b l e m of e x i s t e n c e

nism problem

E T c_ E

servomecha-

are briefly pre-

is m o d i f i e d a s follows :

+ Ew

(7.2.3)

y=Cx

ill[cc 0c201

E =

Es where

A

(i=l . . . . .

S).

and

Cs ] B

are

the

same

as

in

(7.1.15,

E. E Rni x~,

ei = Yi - Yi

ref

and

1

y= (Y'l Y ' 2 " " Y ' S )' is t h e o u t p u t to b e r e g u l a t e d

C. E: R P i x n i , 1

so t h a t t h e e r r o r s

(i=l,...,S)

t e n d to z e r o a s t -> co.

We a s s u m e t h a t t h e d i s t u r b a n c e

vector w satisfies :

~] = F 1 z 1 w where

(7.2.4)

= H1 z1

z 1 ~ R--1 a n d

where ref

reference input vector y

~2 = F2 z2

(H1,

F1)

satisfies

:

is o b s e r v a b l e

and

Zl(0)

is

not

known.

The

299

y

ref

= H2 z 2

(7.2.5)

/I

where

z 2 ~: Rn2

and

where

(H2,

F 2)

is o b s e r v a b l e a n d

y r e f is m e a s u r a b l e .

The

minimal polynomials of F 1 a n d F 2 a r e d e n o t e d b y h I (p) and h2 (p) a n d t h e i r l e a s t common multiple b y ~.(p). Let t h e z e r o s of A(p) (multiplicities included} b e g i v e n b y ~ 1 ' 1 2 ' . . . . Xq. A s y s t e m is said to b e d e c e n t r a l l y r e t u n a b l e u n d e r s t r u c t u r a l p e r t u r b a t i o n s if a f t e r any s t r u c t u r a l p e r t u r b a t i o n , d e c e n t r a l i z e d c o n t r o l l e r s can be d e s i g n e d so as to solve t h e s e r v o m e c h a n i s m problem for t h e p e r t u r b a t e d s y s t e m . We h a v e t h e following result : Theorem 4.2 (OZG-82}. If for e v e r y s t r o n g l y c o n n e c t e d s u b g r a p h G T = (VT, ET) (i) the m a t r i x p a i r s (A T , B T) a r e controllable (ii) the s u b s y s t e m s (CT, AT , B T) h a v e no t r a n s m i s s i o n zero coinciding with ), 1' )~2' . . . . ),q (C T is d e f i n e d in a similar way as AT a n d B T) t h e n t h e s y s t e m ( 7 . 2 . 3 ) is d e c e n t r a l l y r e t u n a b l e u n d e r all s t r u c t u r a l p e r t u r b a t i o n s .

7.3.

STRUCTURAL PERTURBATIONS AFFECTING THE CONTROL SYSTEM

-

(TRA-

84b ) In t h i s s e c t i o n , using

structurally

we a r e

constrained

s u p p o s e d to a f f e c t t h e

also c o n c e r n e d b y controllers

controller.

The

but

t h e p r o b l e m of pole a s s i g n a b i l i t y structural

perturbations

following s u b s e c t i o n

specifies the

p e r t u r b a t i o n s t h a t we c o n s i d e r a n d p r o v i d e s a model for the p e r t u r b a t e d

are type

now of

controlled

system.

7.3.1.

- S t r u c t u r a l p e r t u r b a t i o n s c h a r a c t e r i z a t i o n (TRA-84b) C o n s i d e r t h e c l a s s of l i n e a r t i m e - i n v a r i a n t s y s t e m s d e s c r i b e d b y t h e following

state-space representation

:

:~(t) = A x ( t ) + B u ( t ) y(t) = Cx(t) where x ( t ) ~

Rn ,

(7.3.]a) u ( t ) • Rm,

y ( t ) ~ Rr

are t h e s t a t e ,

input

and

r e s p e c t i v e l y a n d A, B, C are real m a t r i c e s of a p p r o p r i a t e d i m e n s i o n s . Define

B = [ b I . . . . . b m] C = [ c 1 . . . . . Cr ]l

output

vectors,

300

so that

the

written

:

equivalent

representation

of the

system

in t h e f r e q u e n c y

d o m a i n c a n be

(7.3.1b)

y(p) = w(p) u(p) with

m yi(p)

= i_E1 w j , i ( P )

w..(p)

= c (pI-A)-lbi

,1

Consider

(j=l . . . . . r )

ui(P)

j

the following feedback

c o n t r o l law f o r s y s t e m

(7.3.1)

:

u = K y

(7.3.2)

whose structure

is s p e c i f i e d b y t h e f e e d b a c k

K = (kij)i= 1

. . . . .

m

matrix K :

w i t h s o m e kii. c o n s t r a i n e d

to b e z e r o ,

j: 1,...,r We a s s u m e of t h e c o n t r o l l e r

that

the controlled

components

system

(sensors,

behaviour

actuators,

may be perturbed

lines).

These

by

failures

failures are specified

below : Definition 7.1. 1. If t h e i t h a c t u a t o r ,

2. If t h e i t h s e n s o r , The behaviour

ui(t) ;1 c~ i

~i(t)

=ct i ~ i ( t )

m} f a i l s a t t i m e x , t h e n u i ( t )

i ~{1 ..... p}

of t h e it h a c t u a t o r

if t h e a c t u a t o r

1 0 if

i ~" {1 . . . . .

f a i l s at t i m e T, t h e n can be expressed

= 0,

t ~,T

Y i ( t ) = 0, t ~/ z

b y "-

(i=l . . . . . m) is p r o p e r l y

(7.3.3) working

a failure occurs

is t h e c o n t r o l t h a t s h o u l d b e a p p l i e d to t h e s y s t e m

is e f f e c t i v e l y

and ui(t)

is t h e c o n t r o l t h a t

applied.

Similarly,

the behaviour

~i(t) =Bi Yi(t)

of t h e i t h s e n s o r

(i=l . . . . . r)

can be expressed

by

:

(7.3.4)

301

10 if t h e s e n s o r is p r o p e r l y

working

B i= if a f a i l u r e o c c u r s ~ i ( t ) is t h e m e a s u r e d v a l u e o f t h e r e a l o u t p u t Y i ( t ) . Line f a i l u r e s m u s t b e c o n s i d e r e d

differently

P r a c t i c a l c o n s i d e r a t i o n s l e a d to d i s t i n g u i s h 1. T h e p h y s i c a l

lines establishing

the

from a c t u a t o r

or sensor

t h e t w o following s i t u a t i o n s

feedback

from o n e o u t p u t

failures.

:

to o n e i n p u t

are

i s o l a t e d o n e from a n o t h e r . 2. T h e l i n e s e s t a b l i s h i n g inputs

(corresponding

unique

physical

tems

for

line.

which

s t a t i o n s Si,

c o n n e c t i o n s from a s e t o f o u t p u t s

to a g i v e n

geographical

station)

are put

This

situation

corresponds

to g e o g r a p h i c a l l y

is

a natural

partitioning

of inputs

there

(i=l . . . . .

the feedback

each

and

to a s e t o f

together

in a

distributed

sys-

outputs

in

several

S).

Definition 7.2. 1. I f t h e line a s s o c i a t e d to t h e f e e d b a c k c o n n e c t i o n b e t w e e n o u t p u t a n d i n p u t u i , i • { 1 . . . . . m} fails at time ~, t h e n ki] = O, 2. If t h e line a s s o c i a t e d S i, i , j ~ { 1 . . . . .

to t h e

feedback

S) fails at time t ,

then

(If a r e o r d e r i n g

of inputs

and outputs

t ~ ~ .

connection between

station

S. a n d J

station

:

k s v = 0 f o r all s , v s u c h t h a t u s e S i a n d Y v C

on t h e s y s t e m model ( 7 . 3 . 1 ) ,

Yi' ] C { 1 , . . . , r )

according

this corresponds

Sj.

to t h e s t a t i o n s h a s b e e n p e r f o r m e d

to s e t t i n g to z e r o t h e w h o l e b l o c k K.. q

in t h e f e e d b a c k m a t r i x K d e f i n e d in ( 7 . 3 . 2 ) ) . I n v i e w o f t h i s d e f i n i t i o n , we c a n d e f i n e a n e w f e e d b a c k m a t r i x ~ w h i c h t a k e s into a c c o u n t t h e line f a i l u r e s

1.~=

:

(lij kij) i=l ..... m j = 1 ..... r if there is a b r e a k

(7,3,5a) of the line j-i

lij = t °1 o t h e r w i s e 2, ~ . = b l o c k (Lij Kij) i , j = l Lij = / 0

.....

S

if t h e r e is a b r e a k o f t h e line b e t w e e n s t a t i o n Sj a n d s t a t i o n S i otherwise

(7.3.5b)

302 I f we d e f i n e

:

ct =

diag.[ a 1 . . . . .

=

diag.[ E 1 . . . . .

am ] E 8 p]

(7.3.6)

Rmxm

E Rrxr

we o b t a i n t h e f o l l o w i n g model f o r t h e p e r t u r b a t e d

system :

PLANT

State space

:

£(t)

= Ax(t)

+ B a ~(t)

,7(t) =~Cx(t) Frequency

(7.3.7a)

domain :

7(p) = 8w(p) ct~(p)

(7.3.7b)

CONTROL

~(t)

(7.3.8)

= ~ 7(0

The closed-loop s y s t e m ~7.3.7)

~ ( t ) = (A + B a ~

(7.3.8)

is given by

;

~ C) x ( t )

a n d i l l u s t r a t e d b y t h e following s c h e m e :

y(t) SYSTEM

Fig.

3. ]

Remark 7.2. 2. I n t h e c a s e o f d y n a m i c f e e d b a c k c o n t r o l a s :

= Sz + R y u = Qz + Ky + v

we s u p p o s e t h a t t h e s t r u c t u r e the

output

feedback

matrix

(7.3.9)

of t h e c o m p e n s a t o r is e o n d i t i o n n e d b y t h e s t r u c t u r e K (S,

R and

Q have

the

same s t r u c t u r e

than

K).

of The

303 existence

of a solution

tence of fixed be s o l v e d b y Chapter

7.3.2.

to t h e p r o b l e m

modes)

with

considering

the static

- Structural

liability...)

Therefore,

and

pole assignability

dynamic

feedback

of the

practical

considerations

compensation

same structure

(exis-

can

thus

((WAN-735,

see

robustness

make

a designer

perturbations

some

structural

generally

wants

which he considers

L e t Fa = {c~ 1 . . . . . c a } cx, a n d 8 a r e and

constrained

2, S e c t i o n 2 . 2 . 3 a 5 .

For a given controlled system, line

of stabilizability

structurally

defined

line failures.

in

Then

perturbations to r e s t r i c t

(like s e n s o r

more

the study

probable

technology, than

by specifying

others.

a class of

like t h e m o s t p r o b a b l e .

, F s = { 81 . . . . . 8 s} a n d FL = { ~ 1 . . . . . -~L}, w h e r e K',

(7.3.5)

and

P ={F a,

(7.3.6),

Fs,

represent

F L} s p e c i f i e s

a class of actuator,

a class

of structural

sensor, perturba-

tions. A controlled respect

system

(7.3.1)

within the

class

turally robust

P.

In this

with respect

case,

such

that

controller

perturbations

a 6F a, • ~ F s,

(see Chapter

Proposition

7.1.

The

controlled

respect

P

and

only

if

said

to

be

structurally

all p o s s i b l e (7.3.9)

robust

structural

with

perturbations

itself is said

P = {F a,

Fs,

to b e

F L} , we c a n

systems £p composed by the set of perturbated

m o d e s (WAN-73)

to

the

is under

struc-

to P .

To a c l a s s o f s t r u c t u r a l class of perturbated (7.3.85

(7.3.9)

to P if it r e m a i n s p o l e a s s i g n a b l e

and

K E F L.

Then

from

(7.3.9)

is

the

associate

systems

definition

a

(7.3.7) of

fixed

robust

with

2 ) , it c o m e s ;

if n o

system

(7.3.1)

perturbated

system

within

structurally the

class £p

has

fixed

modes. Introduce Definition turally,

7.3.

robust

the classFp

the following definition Given the controlled mode with respect

:

system

(7.3.15

(7.3.95,

k 0 ~ o (A)

to P if a n d o n l y if n o p e r t u r b a t e d

h a s X0 a s a f i x e d m o d e .

Using this definition,

Proposition

7.1 can be rewritten

is a s t r u c -

system

as follows :

within

304 Corollary respect

7.1.

The

controlled

system

(7.3.1)

(7.3.9)

to P if a n d o n l y if all t h e m o d e s o f ( 7 . 3 . 1 )

with respect

Remark

7.2.

I f we a r e

interested

the necessary

- Characterization In this section,

modes

(see

modes.

The

first

control,

i.e.

the

robustness

robust

of structurally

3)

two

to

provide

robust

(A) a r e s t r u c t u r a l l y

with robust

definitions

three

(7.3.9)

the

problem

of

7.1 must be replaced is s t a b l e " .

and the characterizations

characterizations

are

to

modes

given

in

matrix has a block-diagonal

feedback

reference

condition of Corollary

robust

characterizations

feedback

with

modes of (7.3.1)

we u s e t h e a b o v e

Chapter

be used for arbitrary

the

of

context

structure.

of fixed

structurally of

robust

decentralized

The third

one can

structures.

- In the state s p a c e

Consider ponding

by

and sufficient

by "the set of no structurally

7.3.3.a.

6o

structurally

to P .

stabilization,

7.3.3.

is

that

reordering

the of

system

(7.3.1)

inputs

and

is p a r t i t i o n e d

outputs

is

in

S stations.

performed,

we

If the

corres-

the

following

obtain

model : S £ = Ax + i~l

Bi u i

Yi = Ci x

with B = [B1, C. ~ R r i x n .

(i=l . . . . .

Bz .....

(7.3.10)

S)

BS 1

C = [ C ' 1,

C' 2 . . . . C ' S ] '

and

where

Bi ~ Rnxmi

and

1

The feedback

ui(t) The

structure

= Kii Y i ( t ) matrices

c~ a n d

failures are partitionned

is supposed

(i=l . . . . .

to b e d e c e n t r a l i z e d

:

(7.3.11)

S)

8 (defined

in

(7.3.6))

specifying

the

actuator

and

sensor

in the same way :

a = b l o c k - d i a g . [ XI,..., X S ]

X i = diag. [ Otil,... , aim. ]

i=l,...,S (7.3.12)

i

t3 = block-diag.[ rl,..., IS] F i = d i a g ' [ S i 1' " " ' 13i ] r. l

i=l,...,S

305 First, note that

in

t h e c o n t e x t of d e c e n t r a l i z e d c o n t r o l , t h e f a i l u r e of t h e line

a s s o c i a t e d to t h e f e e b a c k - l o o p at s t a t i o n i is e q u i v a l e n t to t h e elimination of s t a t i o n i in t h e s y s t e m model ( 7 . 3 . 1 0 ) remains identically zero). represented

( i n d e e d t h e r e is no more u s e made of Y i ( t ) ,

and ui(t)

A c o n f i g u r a t i o n of line f a i l u r e s K* E F L c a n t h e r e f o r e b e

b y d e f i n i n g t h e s e t Tr* =

{1 . . . . . S } -

{ i / L i i = 0} , Lii as d e f i n e d in

(7.3.5). T h e following c h a r a c t e r i z a t i o n is a s t r a i g h t f o r w a r d A n d e r s o n a n d C l e m e n t s (AND-82)

e x t e n s i o n of t h e r e s u l t of

(see C h a p t e r 3, Section 3 . 3 . 1 )

Proposition 7.2. Given the decentralized controlled system (7.3.10) ) , 0 ~ a (A) is a s t r u c t u r a l l y

: (7.3.11),

r o b u s t mode with r e s p e c t to P = { F a, F s ,

if a n d

F L}

only if : E

VB ~ F s ,

Fa,

I A

VIT* c o r r e s p o n d i n g to ~ * • F L,

X0I

-

BK a K l

rank

~ ~ - K Cw~-K

(7.3.13)

n

0

for all k s u c h t h a t K = { i l , . . . , i k }

c~*,

where : BK =[Bil ..... a K = Block-diag. 8~,_ K

C ~*-K =[C'i k+ l . . . . . C'is ] '

[×il , .., Xik ]

= block-diag. [rik+l

Proposition perturbated

Bi~

,..., riS]

7.2 means t h a t we u s e

system within

the matrix rank

test

(7.3.13)

for

every

P in o r d e r to c h e c k w h e t h e r o r not t0 is a d e c e n t r a l i z e d

fixed mode for some of t h e m . If we w a n t to c o n c l u d e w h e t h e r a d e c e n t r a l i z e d c o n t r o l is s t r u c t u r a l l y

robust,

a laborious task.

we m u s t c h e c k all t h e modes of t h e s y s t e m . T h i s is o b v i o u s l y

From a p r a c t i c a l p o i n t of view, t h e r e is no d o u b t t h a t t h e following

c h a r a c t e r i z a t i o n is more c o n v e n i e n t s i n c e t h e whole s e t of n o n s t r u c t u r a l l y

robust

modes (if a n y ) is d e t e r m i n e d in one s t e p . 7.3.3.b.

- In t h e f r e q u e n c y domain

T h i s c h a r a c t e r i z a t i o n is b a s e d on t h e f i x e d mode c h a r a c t e r i z a t i o n of V i d y a s a g a r

306

and Wiswanadham (VID-83) (see C h a p t e r 3, Section 3 . 2 . 3 ) . It p r o v i d e s a direct determination of the non s t r u c t u r a l l y

r o b u s t polynomial,

whose zeros are the non s t r u c t u r a l l y r o b u s t modes of the system. The same notations as in Section 3.2.3 are u s e d . C o n s i d e r the

partitioned

system

(7.3.10)

in a f r e q u e n c y

domain r e p r e s e n -

ration : y = [ Wll(p)

" ' " ~71s(P)]

u (7.3.14)

LWsI(P )

Wss(P) J

and the d e c e n t r a l i z e d feedback control ( 7 . 3 . 1 1 ) . We recall that the fixed polynomial c~(p) of the system are the d e c e n t r a l i z e d fixed modes of (7.3.14) , is g i v e n [ b; ; t e r i s t i c polynomial ~ ( p )

of (7.3.14)

(7.3.14),

the g . c . d ,

whose zeros of the c h a r a c -

and the minors W 1 of W(p) c o r r e s p o n d i n g

to

non s i n g u l a r s q u a r e submatrices of I( v (VID-83) : a(p):

g.od.

{ ¢ (P)'

W

[ f l u 12 U''" U I s ] } 3 IU 32u.,.

i-I I i c R i = {1£1.= r i + 1, ...,

3icMi

i-I Z :{j=l

IIliII

m: + 1, ..., /

u 3SJ

i=l j=lT' rj + r i }

i-I Z j:l

m + m } ! [

(7.3,15)

i=l,...,S

llJiU

Given a class of p e r t u r b a t i o n s P = {F a, F s, FL} , it is clear from Definition 7.3 that the non s t r u c t u r a l l y r o b u s t polynomial of (7.3.14) with r e s p e c t to P is equal to the 1.c.m.

of the

fixed polynomials of all the p e r t u r b a t e d

systems

withinEp.

Consider a s t r u c t u r a l p e r t u r b a t i o n a E F a, 8 ~ F s, and ~ ~ F L, (as defined in (7.3.6) and ( 7 . 3 . 5 ) ) t h e n the c o r r e s p o n d i n g p e r t u r b a t e d system is g i v e n b y ( ? . 3 , 7 ) (7.3.8)

:

7 ( p ) -- B w(p)

c~'(p)

307 The

matrix 8 W(p)a

responding

t o Bi =

j~{1

It

m}.

0,

is obtained

i~{I

follows

..... r}

that

the

f r o m W(p) and

any

minors

by setting

to z e r o a n y

columnr.1 J c o r r e s p o n d i n g

[BWc~] /11/

such

k~j

that

row i cortoc~j

i ~ I or

=

j~J

0, are

e q u a l to z e r o . From

another

hand,

~

b l o c k s Kii c o r r e s p o n d i n g the feedback

is

loop at s t a t i o n i ) .

K'

from there

K by

setting

to

zero

respect

7.3.

The

is s t r a i g h t f o r w a r d

non

to t h e s t r u c t u r a l ~(p) = g.c.d

singular.

JsJ

The following result Proposition

diagonal

submatrices

s u c h t h a t I i # O, Ji ~ fl a r e s t r u c t u r a l l y J1 u. • • uJiu. • .u

the

is a f a i l u r e o f t h e l i n e i m p l e m e n t i n g

It f o l l o w s t h a t t h e s q u a r e

IlU...UIiu...UIs ]

Ij=

obtained

to Lii = 0 ( i . e .

structurally

robust

perturbation

{ ~b ( p )

,

W

from the above discussion. polynomial

a , B , ~" i s g i v e n b y

[i:] ,

of

(7.3.14)

(7.3.11)

with

:

} (7.3.16)

I' = ( I ' l U

I' 2 u . . . u I ' S) -

J' = (J'lU

J ' 2 u ' ' ' U J ' S) - [ J"1 s u c h

I'.c

R'. = R. I

1

1

1

that

Lii = 0} L..11 = O}

{ k such that g k = 0 } "

3'. c M'. = M. - { k s u c h I

{ I' i s u c h t h a t

thatch.

K

1

= O} (i=1 . . . . .

II

s)

II = I1 i II

R. a n d M. a r e d e f i n e d i n ( 7 . 3 . 1 5 ) 1

1

Using respect

above proposition,

to a c l a s s of p e r t u r b a t i o n s

Proposition respect

the

7.4.

The

.....

robust

perturbations

polynomial P =

{ ~(p) }

As an example,

consider

the

problem of robustness

with

a s follows :

structurally

to t h e c l a s s of s t r u c t u r a l

~p(p) = 1 . c . m 7 P

i.e.

non

we c a n c o n s i d e r

of

(7.3.14)

{ F a , F s , F L}

(7.3.15)

is given by

with :

(7.3.17)

that

we a r e c o n c e r n e d

by the failure of one actuator,

P = {F a } , F a = { c x 1 = b l o c k - d i a g . [ 01...1] ..... a i = block-diag. [1..101..1] e ~x = b l o c k - d i a g . [ 1 . . . 1 0 ] } , t h e n t h e n o n s t r u c t u r a l l y r o b u s t p o l y n o m i a l is

given by

:

308

p(p)

= l.c.m.

{g.c.d.

¢ (r,), vl

}}

t

k=l .... m I P = I 1 u 12

I'. c R . 1

u ...

J, = 31 u J2 u . . . u 3S

IS

I

3'. c M'. = M. - ( k }

IIPi II = IIJ'i The problem of one sensor a similar

way

(TRA-84b).

o r o n e l i n e f a i l u r e c a n a l s o b e s i m p l y f o r m u l a t e d in

Although

the

with the number

of perturbations

interest

b a c k all t h e c a l c u l a t i o n s

to b r i n g

7 . 3 . 3 . c. - G r a p h - t h e o r e t i c This terization theoretic

characterization

decentralized).

t h a t We c o n s i d e r ,

of the

problem

grows

this characterization

obviously

presents

to t h e o r i g i n a l n o n p e r t u r b a t e d

the

system.

characterization derives

of Locatelli eta]. framework

complexity

from

(LOC-77)

a l l o w s to c o n s i d e r

The counterpart

the

(see

fixed

arbitrary

is t h a t

modes

Chapter

3,

feeback

the approach

graph-theoretic

Section

3.6.2).

structures

charac-

The

graph-

(not necessarily

is only applicable

for

systems

with simple modes.

The system

same

(7.3.1)

digraph

rS =

(V S .

and the same notations

LS)

as

in

R e f e r to t h e f i x e d m o d e s c h a r a c t e r i z a t i o n turally

robust

perturbation

mode

with

respect

does not result

to

Section

3.6.2

is

associated

to

the

a s in t h a t s e c t i o n a r e u s e d .

of Theorem

some structural

in t h e d c s a p p e a r a n c e

3.30.

Then~,0

perturbation

is a struc-

(a, 6,

o f all t h e e l e m e n t a r y

~)

if the

cycles of

FS f o r w h i c h ;k0 i s a p o l e . The perturbations ciated

to t h e o r i g i n a l

can be easily integrated system.

From Definitions

by 7.1

can be expressed

by the elimination of the vertex

be

the

expressed

line supporting

by

]" ~ 0

of t h e

kij c a n b e e x p r e s s e d

The following result Proposition

elimination

by

vertex

modifying and

7.2,

i ~ V1S,

the

digraph

the jth sensor

(j+m) ¢ . V 2 s ,

Fs a s s o -

t h e it h a c t u a t o r

and

the

the elimination of the edge

failure

failure can

failure

of the

(j+m, i ) ~

L2S.

comes.

7.5

is structurally

robust

with

respect

to t h e i t h a c t u a t o r

a n d o n l y i f ~0 i s a p o l e o f s o m e e l e m e n t a r y

(jth sensor)

f a i l u r e if

c y c l e o f t"S i n w h i c h t h e v e r t e x

i ~ VIS

(]+m ~ VS2) i s n o t i n v o l v e d . 2. X 0 i s s t r u c t u r a l l y

robust

with respect

to t h e

failure

of the line supporting

ki] if

309

and

only

if ~0 is

a

pole

of

some

elementary

cycle

of rS

in

which

the

edge

(j+m,i) ~ LS2 is n o t i n v o l v e d . For

a perturbation

(a, ~ ,

K')

involving

several

actuator,

sensor,

and

line

f a i l u r e s , it is c l e a r t h a t the s e t of c o n d i t i o n s of r o b u s t n e s s a r e g i v e n b y t h e i n t e r section of t h e c o n d i t i o n s c o r r e s p o n d i n g to e v e r y e l e m e n t a r y p e r t u r b a t i o n .

The same

is t r u e for a c l a s s of p e r t u r b a t i o n s , T h e following corollary p r o v i d e s some r e s u l t s r e f e r i n g to p a r t i c u l a r

c a s e s of

practical i n t e r e s t : Corollary 7.2. 1. k 0 is s t r u c t u r a l l y r o b u s t with r e s p e c t to one a c t u a t o r ( s e n s o r ) failure if and only if ~0 is a pole of at l e a s t two e l e m e n t a r y c y c l e s of FS s u c h t h a t t h e s e t s of v e r t i c e s V1S ( E : V 2 s ) i n v o l v e d in each cycle a r e d i s j o i n t . 2. k 0 is s t r u c t u r a l l y r o b u s t with r e s p e c t to one line f a i l u r e of if a n d only i f k 0 is a pole of at l e a s t two e l e m e n t a r y c y c l e s of FS s u c h t h a t t h e s e t s of e d g e s ~

L2S i n v o l -

ved in e a c h cycle a r e d i s j o i n t . 3. k0

is

structurally

robust

with

respect

to one

actuator,

s e n s o r or

line

failure

( o c c u r i n g one at a time) if and only if k0 is a pole of at l e a s t two e l e m e n t a r y disjoint cycles of FS. Remark 7 . 3 . T h i s g r a p h - t h e o r e t i c a p p r o a c h allows t h e d e t e r m i n a t i o n of t h e n a t u r e of the non s t r u c t u r a l l y r o b u s t

modes.

We c o n s i d e r t h e

d i g r a p h F' S a s s o c i a t e d to t h e

p e r t u r b a t e d s y s t e m (F~ is o b t a i n e d b y r e m o v | n ~ the v e r t i c e s a n d e d g e s c o r r e s p o n ding to t h e p e r t u r b a t i o n ) . ~0 is a non s t r u c t u r a l l y f i x e d mode (SEZ-81a)

for t h e p e r t u r b a t e d

s y s t e m if some

e l e m e n t a r y c y c l e s remain in rrS for which ~0 is n o t a pole d u e to a p o l e - z e r o c a n c e l lation in t h e c y c l e t r a n s m i t t a n c e s . k0 is a s t r u c t u r a l l y f i x e d mode (SEZ-81a) f o r t h e p e r t u r b a t e d s y s t e m if t h e a b s e n c e , due to t h e f a i l u r e , of e l e m e n t a r y c y c l e s for w h i c h k 0 is n o t a pole is not a c o n s e q u e n c e of p o l e - z e r o c a n c e l l a t i o n s , N e v e r t h e l e s s , some e d g e s ~ LIS f o r which k0 is a pole remain in FIS , k O is an uncontrollable or inobservable

mode

for the perturbated

system

(only for

actuator and sensor failures) if no edge ~- LIS for which ~0 is a pole remains in F~S. The following s c h e m e i l l u s t r a t e s t h e p o s s i b l e c o n s e q u e n c e s of a p e r t u r b a t i o n :

310

original

system 1

non fixed mode (~, I~) KV /

/

perturbatled system

X0 non structurally Iixed mode

l

1~,

- structurally fixed mode /

(a,•) /(a,13)

13~

/ ~0 uncontrollable or ~ ' /

unobservable

rood 1

Fig. 7.2

7.3.4.

- Example

Consider the B-station system described b y the following t r a n s f e r matrix : 3 p-2 W(p) :

0

p+l p(p'-2)

1

1

p-2

p+2

p+l

p+2

for which the c h a r a c t e r i s t i c

1

p(p-2)

polynomial is . ~ ( p )

= p(p+l)(p+2)(p-2).

Consider a

decentralized feedback s t r u c t u r e given b y the feedback matrix -" K = b l o c k - d i a g . [ k l l , k22,

k33 ]

Using one or the other of the fixed mode characterizations given in Theorem 3.4 or Theorem 3.30, we can determine that this system has a non s t r u c t u r a l l y fixed mode at X0 = -1. Now let us determine,

for example, the n o n s t r u c t u r a l l y r o b u s t modes with

r e s p e c t to one a c t u a t o r failure :

311 1. Using the f r e q u e n c y domain c h a r a c t e r i z a t i o n We h a v e : P1 = {1}

P2

M' I = { I } - { k }

(Proposition 7 . 4 ) .

= {2} P3 = {3 } M' z = {2}

- {k}

The non structurally robust polynomial is given by

I3 1

~p (p) : l.om.

{ g.c.d.

:,.,.m.

= 1.c.m.

{g.c.d.

{¢ (p), W

1

M' 3 = {3 } -{k} :

I2

13 ]

32

33

"t:J

}}

"[: '1',

{p(p+l)(p+2)(p-2);p(p+l)(p-2);0;(p+l)}

;

g . c . d . { p (p+l) (p+2) (p-2) ;3p (p+l) (p-2) ;0; (p+l) (p+2)} g.c.d. { p(p+l) (p+2) (p-2) ; 3p(p+l) (p-2) ;0; (p+l) (p+2) ; 3p (p+l) }} ;

= l.c.m.

{(p+l)

; (p+l)(p+2)

; p(p+l)}

~p(p)= p(p+l) (p+z) T h e r e f o r e the system has t h r e e non s t r u c t u r a l l y r o b u s t modes with r e s p e c t to one a c t u a t o r failure : ~0 -- - 1 ~'1 = 0, ~2 = -2. Obviously, the fixed mode ~0 = -1 a p p e a r s also as a non s t r u c t u r a l l y r o b u s t mode, 3. Using the g r a p h - t h e o r e t i c c h a r a c t e r i z a t i o n ciated to the system is the following :

(Corollary 7 . 2 ) .

The d i g r a p h r S a s s o -

312

f

..

2

/ ....

F S = (V S = { 1 , 2, 3 }

with

5

, L S)

Fig. 7.3.

VIS V2S ={4, 5, 6} LIS = {(1,4).(1,5),(1,6).(2.5),(2,6),(3,4),(3,5)} L2S = { ( 4 . 1 ) , ( 5 . 2 ) . ( 6 , 3 ) }

F S has five elementary

T

cycles for which the transmittances

I (p) = lY23

T

~'~ = (2,6,3,5,2) 1

It a p p e a r s

:

3 (p ) _ p +,_.l ' p(p--2) = p(p.-2) ... 1

~'5 = ( I , ~ , 2 , 6 , 3 , 4 , l ) p+l

=

T t~ (P) =p(p-2)(p+2)

consequence

T

1 2 (p) = p+2

are given below

T 5(p) p(p+2)(p_2)2

clearly that

of the pole-zero

~0 = - 1 i s a n o n s t r u c t u r a l l y c a n c e l l a t i o n in T

3(p),

there

fixed mode.

Indeed,

as a

is n o c y c l e f o r w h i c h

;~0=-1 i s a p o l e . N o w , we u s e t h e f i r s t r e s u l t robust

modes with respect

two e l e m e n t a r y disjoint is robust

in C o r o l l a r y 7 . 2 to d e t e r m i n e

to o n e a c t u a t o r

failure.

cycles such that the sets of vertices

X= 2.

These

modes with respect

two cycles

a r e "-~1 a n d ~4"

to o n e a c t u a t o r

the non structurally

T h e o n l y m o d e w h i c h i s a p o l e of

failure are

~ V l s i n v o l v e d i n e a c h c y c l e are Therefore, :

the non structurally

313 X0 = - 1

We

obtain

k l = 0

the

same

result

significant lower number

7.3.5.

k2=

- Structurally

as

- 2

using

the

frequency

domain

characterization

with

a

of calculations.

robust

control design

The choice of the information pattern A

significant

robustness

advantage

conditions

using binary

of

the

of Proposition

variables

associated

graph-theoretic

7.5

and

to t h e c o m p o n e n t s

way f o r s o l v i n g t h e p r o b l e m o f o p t i m a l s t r u c t u r a l l y Let

us

consider

the

system

(7.3.1)

characterization

Corollary

with

7.2

of the digraph. robust

the

is

that

can easily be

the

expressed

This provides

a

control design.

assumption

that

it

has

simple

poles. We s t u d y

the

same

which was presented

problem

in S e c t i o n

as

the

5.3.3

but

one

solved

by

L~catelli e t

we a d d r o b u s t n e s s

al,

(LOC-77)

constraints,

The

same

n o t a t i o n a s in S e c t i o n s 3 . 6 . 2 a n d 5 . 3 . 3 a r e u s e d . The problem consists of t h e

system

contained

defined as in (3.6.1)

(j,i)

E

by

r (i,j) ~

a minimal set

S* c S s u c h

that every

s e t A* = {)~1' . . . . ' ~ h * } i s s t r u c t u r a l l y

robust.

pole S is

:

S if k . . # 0 1,]

The optimization criterion

R(S*)

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

in t h e

(i=l . . . . .

remains

m)

; (j=l . . . . .

r)

-"

S* r i , j

where r.. is a c o s t a s s o c i a t e d 1,j

to t h e

feeback

connection

from the

output

i to t h e

element

o f A* i s

input j. Of course, structurally

robust

the

problem

If we w a n t to d e t e r m i n e a unique

perturbation,

Section 5.3.3

for

has

with respect

the

the

a solution if and

a structurally simplest

perturbated

by e l i m i n a t i n g t h e c o r r e s p o n d i n g

o n l y if e v e r y

to S . robust

approach

system feedback

control structure

i s to s o l v e

(7.3.7).

The

connections

with respect

the problem

line failures

from S.

presented

are

to in

considered

314

N o w , i f we w a n t to t a k e i n t o a c c o u n t a c l a s s o f p e r t u r b a t i o n s , the program

remains

the same but

new constraints

expressing

the structure

the robustness

of

requi-

rement must be added.

Our study -

will b e r e s t r i c t e d

one actuator

- one sensor -

-

to t h e f o l l o w i n g c l a s s e s o f p e r t u r b a t i o n s

:

failure failure

one line failure one actuator,

which correspond Consider line failure

sensor,

or line failure

to t h e e a s e s c o n s i d e r e d first

the

in C o r o l l a r y

class of perturbations

(case 3 of Corollary

The constraint

( o n l y o n e at a t i m e )

(G g ) i s r e p l a c e d

v.g.

7.2). by

z.g.

>i Z

t h a t two e d g e s

(i,j)

1,J

1,]

Then,

7.2.

specifying

one actuator,

the original program

sensor,

or

i s e a s i l y modified,

:

(iij) £ LI 5 which assures

f o r w h i c h ~,g * i s a p o l e will b e r e t a i n e d ,

T h e two f o l l o w i n g c o n s t r a i n t s

(cg)

must be added

:

(i,j) E LIS i ¢ VIS] (k,i) G L25

which elimines

the

possibility

of a unique

cycle

for

which kg*

is a p o l e

of order

tWO.

(cg5)

~

~.g. ~ l

j/(i,j) ¢ L S

which

guarantees

sufficient

the variables

Remark

that

to a s s s u r e

7.4.

separately.

certifies

i ~v s

l,l

the

that

two

cycles

do

not

the two cycles are

involve

the

disjoint because

same

vertices.

the boolean

This

is

nature

of

that the two cycles are not composed by the same edges.

A significant Consequently,

advantage we c a n

of this approach impose

to

is that every

a m o d e Xi* to

be

m o d e is t r e a t e d

structurally

robust

315

whereas a n o t h e r mode Xj* is r e q u i r e d

not to b e fixed o n l y

(we modify C~ a n d a d d

C~, C~ o n l y ) . In t h e case for which we c o n s i d e r t h e c l a s s of p e r t u r b a t i o n s actuator f a i l u r e o r t h e c l a s s of p e r t u r b a t i o n s Corollary

7.2),

the

established above.

corresponding The

programs

two c y c l e s

are

not

specifying

one

s p e c i f y i n g one s e n s o r f a i l u r e ( c a s e 1 of are

particular

required

cases

of t h e

program

to b e d i s j o i n t = for a c t u a t o r

( s e n s o r ) f a i l u r e , t h e y a r e n o t allowed to i n v o l v e t h e same v e r t i c e s of V1S ( V 2 s ) b u t some v e r t i c e s

of V2S ( V 1 s )

c a n b e u s e d twice.

Therefore,

the constraint

(C~) is

relaxed as follows ; Actuator failure r. j(/(i,j) ~" L S

Finally,

zg i,j ~

consider

Sensor failure

I

Z j/(i,j) (~L S

i ~ VIS

the

case

of t h e

c l a s s of p e r t u r b a t i o n s

zg. l,) ~ I

specifying

i E.V2s

one line

failure ( c a s e 2 of C o r o l l a r y 7 . 2 ) . T h e two c y c l e s c a n n o t b e composed to b e t h e same e d g e s of L2S b u t an e d g e from L1S c a n b e l o n g to t h e two c y c l e s . T h i s p r o b l e m can be s o l v e d b y c o n s i d e r i n g some no boolean v a r i a b l e s o r , if we want to p r e s e r v e

the advantageous

boolean n~.ture of t h e p r o g r a m ,

b y a d d i n g some

redundant boolean variables. 1. The v a r i a b l e s zig,j, a s s o c i a t e d with t h e e d g e s (i, i) ~ LIS a r e n o t b o o l e a n = t 0 if ( i , j ) does not b e l o n g to t h e r e t a i n e d cycles ( i , j ) ~ : L 1 s , zlSj =

1 if (i,j) b e l o n g s to one r e t a i n e d c y c l e 2 if ( i , j ) b e l o n g s to two r e t a i n e d cycles

T h e p r o g r a m to b e s o l v e d is t h e same as t h e o n e a l r e a d y e s t a b l i s h e d b u t t h e c o n s t r a i n t (C~) m u s t b e r e m o v e d . 2. Two boolean v a r i a b l e s zlgj a n d x g j a r e a s s o c i a t e d to e a c h e d g e (i, i) ~ L1S T h e c o n s t r a i n t s Clg, C~ a n d C~ a r e modified as follows :

316

(c~)

z

(C2g)

zg i,j

j/(i,j)~ LS

(C4g)

(zgj

vg i,j

(i,j) E LIS

2

+ xg )~ i,j

xg + i,j

Z(i,j ) ~ L 1 S

(z~j

=

E

zg

j/(j,i) ~ Ls

+ x~j)

v~j

j,i

+ x~,

i ~

i

VS

Zkg,i ~ 2

i ~ V i s / ( k , i ) ~ L2 S

Moreover,

the constraint

As a n e x a m p l e , consider

C g is r e m o v e d . given

the

t h e following p r o b l e m

Find the feeback tions such that

same

system

as i n t h e

example

of S e c t i o n

7.3.3,

:

structure

S* c S w i t h a minimal n u m b e r

of f e e d b a c k

connec-

:

-X 1" = - 1 i s n o t a f i x e d m o d e -X 2* = 2 is specifying

structurally

one actuator,

Only the feeback are allowed,

sensor,

connections

and the associated

r..

1,J (i,j) E S

--

robust

with

respect

to t h e

class

of p e r t u r b a t i o n s

o r l i n e f a i l u r e ( c a s e 3 of C o r o l l a r y 7 . 2 ) .

specified

by

S = {(1,1),(2,1),(3,1),(2,2),(2,3),(3,3)}

costs are :

1

T h e s o l u t i o n is o b t a i n e d

b y s o l v i n g t h e following b o o l e a n l i n e a r p r o g r a m

:

rain w4,1 + w5,1 + w6,1 + w5, 2 + w5, 3 + w6, 3 (C~)

Z1l , 6 >/ 1

(C?)

z2 + z2 + z 2 z2 1,4 1,5 3 , 4 + 3,5 >/ 2 (~2 4 + z2 5 ) ( z g ,

1 + z2

z2 + z2 + z 2 1,4 1,5 1,6 ~< I

+

+

+

z~,3) ~ z

317

"~.5 ÷ q.6 ,< ( 8 5) z24,1 x( 1

z2,1 + 4,2 + z2,3 ~< 1 z2,1 + z2,3x< i and for g = 1,Z

z~, 4 + z~, 5 + z~, 6 = z~, 1 + z~, 1 + z~, 1 z~, 5 + z~, 6 = zsg 2 z~, 4 + z395 = z~, 3 + z6g, 3 (C~) z4g, 1 = z~, 4 + zig,4

z~.I + zsg,2+ z5g,3= zlg,5+ z2g,5+ z3g.5 z6g,1+ z6g,3= zlg,6+zZg,6 (c~)

z~,l '.< "%,1

z~,3 ( w5,3

z~,l < w6,,

z#3 ,< w6,3

There is

a u n i q u e optimal solution :

s* = { ( 1 , 1 ) , corresponding

=

(3,1))

to the following f e e d b a c k s t r u c t u r e kll

K

(z,3),

0 0

0 0 k32

kl3 ] 0 0

:

318

7.4.

- CONCLUSION When a controlled

the

controller

or

the controlled

system

of the

system.

for a good pursuit

is o p e r a t i n g ,

system

itself

it m a y h a p p e n

fails resulting

Such structural

perturbations

of the operations.

that

s o m e c o m p o n e n t of

in a s t r u c t u r a l

m o d i f i c a t i o n of

may be dangerously

As an example,

consider

detrimental

that the perturbeted

system is unstable. Two approaches one consists proceeding

can be used

in i m p l e m e n t i n g

to p r e v e n t

a system

for

to a r e a l t i m e r e c o n f i g u r a t i o n

mics of the proceeding

system,

this

solution,

unefficient.

The

second

approach

perturbations

in the

such

controller

some desirable

Such focuses

controller

on

the

assignability consider systems

properties

using

controller.

be

They

turally

robust

Section

7.3,

in t a k i n g

design

of the

preserves,

to t h e c o n t r o l l e d

of

structurally

structural

disconnected. is

the design

mizes the cost associated

feedback the

Section

7.2,

stem from actuator, modes

introduced

sensor, and

under

and

then

on the dyna-

new

components, and

the

control system.

robust.

affecting In

account

therefore

eventuality

of

The synthesis

is

structural

In this

perturbations

constrained

perturbations

Depending

installing

into

T h e first

diagnosis

perturbations,

system.

is s a i d to b e s t r u c t u r a l l y

consequences

structural may

the

situations.

and

controller.

may require

consists

some structural

that

detection

may be too much time consuming

then

performed

of the

(which

to n e w m e a s u r e m e n t s . . . )

such inacceptable

failure

structural

the

In

the

Section

sense

that

control feedback

structure

7.1, affect

The concept are

study o r pole we

some sub-

perturbations

characterizations

to t h e i n f o r m a t i o n t r a n s f e r .

chapter,

stabilizability

control. in

or line failures.

some

is f a c e d o f a r o b u s t

plant

on

the

of struc-

provided.

In

w h i c h mini-

1

APPENDIX

MULTIVARIABLE SYSTEM ZEROS

This

appendix

multivariable system.

is

concerned

by

the

different

types

of

zeros

appearing

in

Each t y p e of zero is d e f i n e d a n d some r e l a t i o n s h i p s a r e o u t -

lined. C o n s i d e r t h e following t i m e - i n v a r i a n t m u l t i v a r i a b l e s y s t e m :

= y

where

Ax

= Cx

+

Bu

+ Du

x C Rn ,

output

vectors,

u ~ R m,

and

respectively.

y ~ Rr A,

B,

(max C,

( m , r ) x( n) D are

are

constant

the

state,

matrices

of

input,

and

appropriate

d i m e n s i o n s . T h e polynomial m a t r i x :

[~

I A

B1

-

P(p)=

( n + r , n+m)

D

is called t h e s y s t e m m a t r i x

(ROS-70).

If r a n k

P(p)= q,

t h e n t h e SmithVs form of

P ( p ) is g i v e n b y :

S*(p)q,q

0q,n+m_ q

0n+r_q, q

0 n + r - q , n+m-c

S(p)= w h e r e S * ( p ) = diag ( S l , s 2 . . . . , Sq) a n d s i, (i=l . . . . . q) (s i d i v i d e s S i + l ) , a r e t h e i n v a r i a n t polynomials of P ( p ) . If Mj(p) d e n o t e s t h e g r e a t e s t common d i v i s o r of all j t h o r d e r m i n o r s of P ( p ) ,

sj(p)

=

t h e n t h e polynomial s.] is g i v e n b y :

M;(p) (j=1,2 . . . . . q )

320 with M0 = I , The t r a n s f e r f u n c t i o n m a t r i x of t h e s y s t e m is :

G(p) = C ( p I - A ) -1 B + D =

u~p]

and i t s Smith-Mc Millan form is g i v e n b y "

I~

*(P)qxq

M(s)=

0q'm-q

l

0r - q , m - q

Lr-q,q where .

M

E l (p)

c n(P)

.....

(p) = diag.( 7

~

)

and e i is t h e ith i n v a r i a n t polynomial of N(p) nomiallof G(p),

i.e. ~(p),

and

q = rank

divided by the characteristic poly-

G(p).

Note t h a t ¢i d i v i d e s ci+ 1 a n d ~ i + l

d i v i d e s ~i o The f i r s t c l e a r c l a s s i f i c a t i o n of t h e

z e r o s of l i n e a r m u l t i v a r i a b l e s y s t e m s was

g i v e n b y R o s e n b r o c k ( R O S - 7 0 ) . We f i n d t h e following t y p e s of z e r o s : Element Zeros ( E . Z . )

:

An e l e m e n t z e r o is a n y value of p f o r which t h e n u m e r a t o r of an e l e m e n t gi~(p) of G(p) v a n i s h e s . This

type

of

zero h a s

no

special meaning

in

multivariahle

systems

theory

b e y o n d i t s role in m o n o - v a r i a b l e s y s t e m t h e o r y . Decoupling Zeros ( D . Z . )

:

The d e c o u p l i n g z e r o s , i n t r o d u c e d b y R o s e n b r o c k ( R O S - 7 0 ) , a r e a s s o c i a t e d with t h e e x i s t e n c e of u n c o u p l e d modes. T h e y a r e d e f i n e d as t h e v a l u e s of p f o r w h i c h t h e matrices (pI-A These modes.

They

B) a n d / o r (picA)-- a r e r a n k d e f i c i e n t .

z e r o s are are

commonly k n o w n as

a s s o c i a t e d with

the

uncontrollable and/or

a p o l e - z e r o cancellation a n d ,

t h e y do n o t a p p e a r in the c o r r e s p o n d i n g t r a n s f e r f u n c t i o n .

unobservable

as a c o n s e q u e n c e ,

321 Three types of decoupling zeros can be defined

the input-decoupling

-

zeros (I.D.Z.)

- the output-decoupling -

the

So,

we

input-output-decoupling

have

which are the uncontrollable modes

zeros (O.D.Z.) zeros

uncontrollable and unobservable

:

which are the unobservable

(I.O.D.Z.)

which are

= I.D.Z.

n

O.D.Z.

D.Z.

= I.D.Z.

u

O.D.Z.

Transmission Zeros (T.Z.)

These

zeros

are

: (ROS-70)

defined

as

the

roots

of

the

numerator

Smith-Me Millan f o r m of G ( p ) .

In t e r m s o f t h e m i n o r s of G ( p ) ,

the

of all t h e q t h o r d e r

of the numerators

A transmission others.

properties

The

zero appears

(q = rank

a s a pole in s o m e e n t r i e s o f G ( p ) a n d a s a z e r o in

are

physically

associated

of the system

(see

(MAC-76)).

Note t h a t R o s e n b r o c k c a l l s t h e s e z e r o s t h e

matrix

Zeros (I.Z.)

with

the

transmission-blocking

(ROS-70).

:

system

zeros.

I n t e r m s of t h e m i n o r s o f P ( p ) ,

r o o t s of the monic g . c . d ,

Physically, behaviour

of the

frequency

for

non-square

G(p)).

a s t h e i r common d e n o m i n a t o r .

The roots of the invariant polynomials of the system matrix P(p) invariant

of the

T.Z.

zeros of the transfer

Invariant

polynomials

t h e y a r e t h e r o o t s of

minors of G(p)

Note t h a t t h e s e m i n o r s m u s t b e a d j u s t e d to h a v e ~ ( s )

some

simultaneously

modes.

:

I.O.D.Z.

g.c.d,

the

modes

the

system

system.

which

systems,

of all t h e m i n o r s o f P ( p )

the

invariant

They system

zeros

correspond output

t h e s e t of i n v a r i a n t

are called the

the invariant

zeros are the

of m a x i m u m o r d e r .

are

to t h e

associated

with

particular

values

is i d e n t i c a l l y

zero.

In the

z e r o s is c o m p o s e d b y t h e

the

zero-output

of the

complex

general

c a s e of

set of transmis-

sion zeros p l u s some decoupling z e r o s .

System

Zeros

(S.Z.)

The system

1

: (ROS-74)

zeros are

t h e r o o t s o f t h e monic g . c . d ,

of t h e f o r m P , I = {1,2 . . . . . n , obtained by selecting 0 ~ k ~ min(m,r).

the

n+i 1,

rows and

n+i 2 . . . . .

n÷ik},

of all t h e m i n o r s o f P ( p )

l

w h e r e PI d e n o t e s

columns corresponding

the minor

to t h e s e t I in P ( p )

and

322

R o u g h l y s p e a k i n g , t h e s e t of s y s t e m z e r o s is t h e u n i o n of t h e s e t o f t r a n s m i s s i o n z e r o s a n d t h e s e t of d e c o u p l i n g z e r o s : S,Z°

S.Z. = T.Z. u D.Z~ T.Z.

Note a l s o t h a t t h e s e t of i n v a r i a n t z e r o s

D.Z.

is i n c l u d e d in t h e s e t of s y s t e m z e r o s . These relationships are illustrated by Figure A1.1.

Figure A1.1.

When t h e

s y s t e m is c o m p l e t e l y c o n t r o l l a b l e

S.Z.,

I.Z. and T.Z.

G(p)

or P ( p ) .

and

completely o b s e r v a b l e ,

the

s e t s of

c o i n c i d e b e c a u s e t h e s e t of d e c o u p l i n g z e r o s is e m p t y .

So f a r , t h e v a r i o u s t y p e s of z e r o s h a v e b e e n d e f i n e d in t e r m s of t h e m i n o r s of Some a u t h o r s

g i v e e q u i v a l e n t d e f i n i t i o n s in t e r m s of t h e p a r t i c u l a r

f r e q u e n c y v a l u e s for w h i c h G ( p ) a n d P ( p ) loose r a n k

:

Z1 : Wolowich (WOL-73a) T h e z e r o s of t h e c o n t r o l l a b l e a n d o b s e r v a b l e s y s t e m (A,

B,

C,

D) a r e t h o s e

P0 s u c h t h a t r a n k P (p0) < r a n k P ( p ) . ZZ : D a v i s o n a n d Wang (DAV-74 et 76e) T h e t r a n s m i s s i o n z e r o s of t h e s y s t e m (A, B, C, D) a r e t h o s e c o m p l e x n u m b e r s P0 w h i c h s a t i s f y t h e following i n e q u a l i t y •

rank

Therefore, of

all

the

n+min(m,r),

P(po ) < n + min

(m,r)

t h e t r a n s m i s s i o n z e r o s ( m u l t i p l i c i t y i n c l u d e d ) a r e t h e r o o t s of t h e g . c . d . (n+min(m,r)) th then every

s a i d to b e d e g e n e r a t e d The T.Z.

order

minors

of

P(p).

Note

that

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

if and

rank the

P(s)

<

s y s t e m is

(DAV-74).

as d e f i n e d h e r e a r e t h e r o o t s

(including multiplicities) of the poly-

nomial o b t a i n e d b y m u l t i p l y i n g all n u m e r a t o r p o l y n o m i a l s of t h e Smith-McMillan form of G ( p )

( ( D A V - 7 6 e } , T h e o r e m 1 ) . It is c l e a r t h a t t h e p r e s e n t d e f i n i t i o n c o i n c i d e s with

Rosenbrock's definition.

Note t h a t

f o r t h e s p e c i a l c a s e of n o n d e g e n e r a t e d

systems

323

with D = 0, t h e T . Z . a r e t h e r o o t s of t h e t r a n s m i s s i o n p o ] y n o m i a l s of t h e s y s t e m (A, B, C) d e f i n e d b y Morse ( M O R - 7 3 ) . Z3 : Wolovich (WOL-73b), D e s o e r a n d S c h u l m a n (DES-74) T h e t r a n s f e r m a t r i x of t h e s y s t e m c a n be f a c t o r i z e d a s : G ( p ) = C ( p I - A ) - I B+D = V ( p ) T - l ( p )

where V ( p )

and

T(p)

are

relatively

+ D

r i g h t prime polynomial matrices.

The

z e r o s of

the s y s t e m (A, B, C, D) a r e t h o s e c o m p l e x n u m b e r s P0 s u c h t h a t

r a n k V(Po) ( r a n k V ( p ) It

is o b v i o u s

that

Z1 a n d

Z3 a r e

equivalent

and,

except

for m u l t i p l i c i t i e s ,

t h e s e d e f i n i t i o n s a r e e q u i v a l e n t to R o s e n b r o c k ' s d e f i n i t i o n of t r a n s m i s s i o n z e r o s .

APPENDIX

A FORTRAN

SUBROUTINE

OPEN-LOOP

AND

TO

2

EVALUATE

CLOSED-LOOP

THE

FIXED MODES

SYSTEM

USING

POLES

PURPOSE The system

FORTRAN

described

=

Ax

+

by

IV

subroutine

DATFM e v a l u a t e s

the

set

of fixed

modes

of a

:

Bu

y = Cx with respect

or

where

to t h e o u t p u t

or state

feedback

control.

set of admissible

feedback

matrices

u=Ky

K~'K F

u=Kx

KCK

KF is the

F specifying

the

feedback

struc-

ture. The based set

subroutine

DATFM u s e s

the algorithm

on the Definition 2.2 of fixed modes.

of open-loop

system

poles (eigenvalues

poles

(eigenvalues

of A+BKC) for three

described

It c a l c u l a t e s o f A)

different

and

in S e c t i o n 2 . 4 . 1

the intersection the

w h i c h is

between

set of closed-loop

v a l u e s o f K.

UTILIZATION The subroutine

statement

is :

S U B R O U T I N E D A T F M (N, M, L, A , B ,

INPUT ARGUMENTS N

Order

of the system.

C,

AK, EPS,

Z, J J ,

A A , WK, Z F ,

ZV)

the

system

325 M

L

N u m b e r of s y s t e m i n p u t s . N u m b e r of s y s t e m o u t p u t s .

T h i s p a r a m e t e r is s e t e q u a l to N in t h e c a s e of

state feedback. A,B,C

S y s t e m m a t r i c e s of d i m e n s i o n ( N , N ) ,

AK

Real m a t r i x of d i m e n s i o n (MxN) d e s c r i b i n g t h e d e s i r e d f e e d b a c k s t r u c t u r e , i.e.

EPS

(N,M) a n d ( L X N ) , r e s p e c t i v e l y .

AK ~ K F.

A c c u r a c y p o s i t i v e p a r a m e t e r u s e d to c o m p a r e t h e e i g e n v a l u e s .

OUTPUT ARGUMENTS JJ

N u m b e r of f i x e d m o d e s . JJ=0 m e a n s t h a t t h e s y s t e m h a s no f i x e d m o d e s w i t h respect

to t h e g i v e n f e e d b a c k s t r u c t u r e

K F a n d f o r an a c c u r a c y e q u a l to

EPS. Complex v e c t o r with

N components containing

the

set

of f i x e d

modes

(if

a n y ) in t h e f i r s t J J p o s i t i o n s .

"WORK AREA

ARGUMENTS

AA

Work a r e a of d i m e n s i o n ( N x N ) .

WK

Work a r e a of d i m e n s i o n N.

ZF

C o m p l e x w o r k a r e a of d i m e n s i o n N.

ZV

C o m p l e x w o r k a r e a of d i m e n s i o n NxN.

CALLED S U B R O U T I N E S EIGRF

Eigenvalues calculation subroutine,

d e s c r i b e d in :

"IMSL L i b r a r y

Manual",

Edition 8, 1980. GAUSS

S t a n d a r d IBM s u b r o u t i n e f o r r a n d o m n u m b e r s g e n e r a t i o n d e s c r i b e d in : 1130 scientific

subroutine

Package

(ll30-Cm-02X),

P u b l . IBM H20-0252-3, 1968. MULT

Two r e a l m a t r i c e s m u l t i p l i c a t i o n ( s e e l i s t i n g ) .

MULTS

T h r e e real matrices multiplication (see l i s t i n g ) .

Programmer's

Manual,

IBM,

326 SUBROUTINE D A T F M ( N , M , L , A , B , C , A K , E P S , Z , J J , A A , W K , Z F , Z V ) IMPLICIT REAL*8 ( A - H , O - Y ) , C O M P L E X * 1 6 ( Z ) , I N T E G E R ( I - N ) DIMENSION A ( N , N ) , B ( N , M ) , C ( L , N ) , A K ( M , N ) , Z ( N ) , Z F ( N ) DIMENSION AA(N, N), WK(N), ZV (N, N) NIT=3 IX=675543 *** OPEN-LOOP POLES CALCULATION *** IJOB=0 D O 26 I=I,N D O 26 J=I,N 26

AA(I,J)=A (I,J) C A L L EIGRF (AA, N, N, IJOB, Z,ZV, N,WK, IER) DO 70 I I = l , 3 I F ( I I . E Q . 2 ) IX=975 I F ( I I . E Q . 3 ) IX=79861 ***

FEEDBACK

MATRIX

SELECTION

***

D O 38 I=I,M D O 38 J=I,L I F C A K ( I , J ) . E Q . 0 ) GO TO 38 CALL G A U S S ( I X , 0 . 3 3 , O,V) AK(I,J)=V 38

CONTINUE *** C L O S E D - L O O P

POLES CALCULATION ***

I F ( L . N E . N ) GO TO 44 CALL MULT(B ,N,M, AK, N, N, AA, N, N, N, M, L) G O T O 50 44

CALL MULT3 ( B , N , M , A K , M , N , C , L , N, N , M , L , N , AA,N)

50

DO 52 I = I , N DO 52 J = I , N

52

AA(I,J)=A(I,J}+AA(I,J) CALL EIGRF (AA, N, N, IJOB, ZF, ZV, N,WK, IER) *** INTERSECTION OF THE SET OF CLOSED-LOOP POLES AND ***

327 *** THE SET OF OPEN-LOOP POLES ***

J J=0 NN=N NV=0 IF ( I I . EQ. 1) LI=N DO 68 I = I , L I IF ( N V . E Q . 0 )

GO TO 62

NN=NN-1 NV=0 62

DO

66 J = I , N N

XX=REAL(Z ( I ) ) VV=REAL(ZF(J) ) I F ( D A B S ( X X - V V ) . G T . E P S ) G O TO 56 YY=AIMAG

(Z (I))

WW=AIMAG ( Z F ( J ) ) I F ( D A B S ( Y Y - W W ) . G T . E P S ) G O TO 66 NV=I JJ=JJ+l

Z ( J J ) = Z (I) NNN=NN-I DO

64 K I = J , N N N

64

ZF(K1)=ZF(KI+I)

66

CONTINUE

68

CONTINUE

GO TO 58

IF (JJ. E Q . 0) R E T U R N LI=JJ 70

CONTINUE RETURN END

SUBROUTINE

MULT(A, NA, MA, B, NB, MB, C, NC, MC, N , M , L )

************************************************ * TWO

REAL

MATRICES

MULTIPLICATION

*

*

C=A*B

*

*

A

*

*

B (M,L)~

(NB,MB)

*

*

C ( N , L ) x < (NC,MC)

*

(N,M) ~ ( ( N A , M A )

328 IMPLICIT REAL*8(A-H,O-Y),

INTEGER(I-N)

DIMENSION A ( N A , M A ) ,B ( N B , M B ) ,C ( N C , MC) DO 1 I = I , N DO 1 J--1,L C(I,J)=0.D0 DO 1 K = I , M

C (I, J)=C (I, J)+A(I,K)*B

(K, J)

RETURN END

* T H R E E REAL MATRICES M U L T I P L I C A T I O N

*

*

*

QQ = A . B . C

*WITH

*

*

A(N,M)

~ (NA,NA)

*

*

B(M,L)

x((NB,MB)

*

*

C(L,K)

X((NC,MC)

*

* QQ(N,K) x((NQ,NQ) * ************************************************

S U B R O U T I N E MULT3 ( A , N A , M A , B , NB, MB , C , N C , M C , N , M,L, K, QQ, N 0 )

I M P L I C I T REAL*8 ( A - H , O - Y ) DIMENSION A ( N A , M A ) , B ( N B , M B ) , C ( N C , M C ) , Q Q ( N Q , N O )

DO Z I = I , N DO 2 J = I , K S=O DO 1 I I = I , M DO 1 J J = I , L S=S+A ( I , I I ) * B ( I I , J J ) * C (J J , J) QQ(I,J)=S CONTINUE RETURN END

SUBROU T I N E GAUSS (IX , S, AM, V) ***************************************************** * CALCUL DE D I S T R I B U T I O N NORMALE V

*

* DE VALEUR MOYENNE AM ET DE V A R I A N C E S * *****************************************************

329 A=0.0 DO

1 1=1,12

CALL

RANGE(IX,IY,Y)

IX=IY A=A+Y V=(A-6.0)*S+AM RETURN END

SUBROUTINE

R A N G E ( I X , IY, YFL)

* CALCULS * RANDOM

OF A UNIFORM VARIABLE

IY=IX*65539 IF (IY)5,6,6 IY=IY+2147433647+I YFL=IY YFL=YFL*0.4656613E-9 RETURN END

DISTRIBUTION

BETWEEN

0 AND

1

* *

APPENDIX A FORTRAN EVALUATE

This

appendix

sensitivity approach

is

FIXED MODES

concerned

with

3

ROUTINE

TO

USING THEIR SENSITIVITY

the

(see § 2.4.2 and 3 . 5 . 4 ) .

evaluation

of

fixed

c o m p u t e t h e f i x e d m o d e s of a s y s t e m u s i n g v a r i a t i o n c a l c u l u s . p o n d s to t h e A l g o r i t h m 2.2 i n P a r a g r a p h

modes

using

the

T h e A p p e n d i x 3.1 g i v e s a r o u t i n e to

2.4.2.

This routine corres-

The routine provided by Appendix

3.2 u s e s t h e e i g e n v a l u e s g r a d i e n t c a l c u l a t i o n a p p r o a c h a n d c o r r e s p o n d s to t h e A l g o r i t h m 3.1 in P a r a g r a p h e 3 . 5 . 4 .

A p p e n d i x B. 1 ROUTINE BASED ON VARIATION CALCULUS The FORTRAN s u b r o u t i n e

STFM1 c o m p u t e s t h e

fixed modes of a s y s t e m with

simple m o d e s . B a s e d on t h e A l g o r i t h m 2 . 2 , it c o m p u t e s t h e v a r i a t i o n s of t h e m o d e s of t h e s y s t e m r e s u l t i n g from c h a n g e s in t h e f e e d b a c k m a t r i x .

If t h e v a r i a t i o n is z e r o ,

t h e c o r r e s p o n d i n g mode is of c o u r s e a f i x e d mode. T h e s y s t e m is d e s c r i b e d b y :

£ = Ax + B u y=Cx

and the feedback structure is described by the set of admissible matrices K F.

UTILIZATION

T h e s u b r o u t i n e s t a t e m e n t is SUBROUTINE STFMI

(A,N,B,M,C,L,AK,EPSpMM,Z)

331

INPUT ARGUMENTS O r d e r of t h e s y s t e m . N u m b e r of s y s t e m i n p u t s . Number of system o u t p u t s .

This p a r a m e t e r is s e t e q u a l to N in t h e c a s e of

state feedback.

A,B,C

S y s t e m m a t r i c e s of dimension ( N , N ) , (N,M) a n d ( L , N ) , r e s p e c t i v e l y .

AK

Real

feedback

matrix

of dimension

(M,L)

with admissible s t r u c t u r e .

The

v a l u e s of i t s e n t r i e s a r e s u c h t h a t t h e c l o s e d - l o o p s y s t e m h a s simple modes. EPS

A c c u r a c y p o s i t i v e small p a r a m e t e r .

OUTPUT ARGUMENTS N u m b e r of f i x e d modes (if a n y ) .

MM

MM=0 means t h a t t h e s y s t e m h a s no f i x e d

modes with r e s p e c t to t h e f e e d b a c k s t r u c t u r e

K F and for an a c c u r a c y e q u a l

to EPS. Complex v e c t o r of l e n g t h N c o n t a i n i n g t h e s e t of f i x e d modes (if a n y )

in

the first MM positions.

REQUIRED MEMORY If N > 20 t h e dimension s t a t e m e n t must be modified a c c o r d i n g to : DIMENSION WK(N), Z 0 ( N , N ) ,

Z2(N,N), Z3(N,N),

ZI(N,N,N)

CALLED SUBROUTINES EIGRF

E i g e n v a l u e s calculation s u b r o u t i n e ,

described in:

Edition 8, 1980.

LISTING SU[I~OUTINE STFM1 (A, N, B,M, C , L , A K , E P S , MM, Z)

IMPLICIT REAL*8(A-H,O-Y) ,COMPLEX*I6(Z) DIMENSION A(N,N),B (N,M), C(L,N),AK(M,L), Z(N) DIMENSION WK(20),ZI(20,20,20)

"IMSL Library Manual",

332 DIMENSION

Z0(20,20),Z2(20,20),Z3(20,20)

NA=20 IF(L.NE.N) G O T O 40 D O 37 I=I,N DO

37 J=I,N

DD=0 D O 35 K=I,M 35

D D = D D + B (I oK) *AK (K, J) A(I, J)=A(I, J)+DD ZZ(I,J)=A(I,J)+(0.,I.)*0.

37

Z3(l, J)=DD+(0. , 1 . ) * 0 . G O T O 48

40

DO 44 I = I , N DO 44 J = I , N DD=0 DO 42 II=I,M DO 42 J J = I , L

42

DD=DD+B(I,II)*AK(II,JJ)*C(JJ,J) A (I, J ) = A ( I , J)+DD ZZ(I,J)=A(I,J)+(0.,I.)*0.

44 48

Z3 (I, J ) = D D + ( 0 . , 1 . ) * 0 . I JOB=0 CALL EIGRF(A, N , N , I J O B , Z, Z0,NA,WK,IER) MM=O D O 60 K=I,N D O 60 I=I,N D O 58 J=I,N

58

Z1 ( I , J , K ) = Z 2 ( I , J )

60

Z1 (I, I, K)=Z2 (I, I ) - Z ( K ) DO 100 K=I,N DO 55 I I = I , N DO 54 J J = I , N

54 55

ZO(ll,JJ)=(O. ,0.) zo(II,II)=(l.,0.) DO 70 J = I , N IF(J.EQ.K)

G O T O 70

D O 65 II=I,N D O 65 JZ=I,N

333

65

Z2 (I2, J2)=Z2 (I2, J2)+Z0 (I2, K 2 ) * Z I ( K 2 , J2, J) DO 67 I3=I,N DO 67 J 3 = I , N

67

Z0(I3,J3)=Z2 (I3,J3)

70

CONTINUE

ZZ=(O. ,0.) DO 75 I4=I,N DO 75 J4=I,N 75

ZZ=ZZ+Z0 (I4,34) * Z3 (J4, I4) RR=CDABS(ZZ) I F ( R R . G T . E P S ) GO TO 100 MM=MM+I Z(MM)=Z(K)

100

CONTINUE RETURN END

A p p e n d i x 3.2 ROUTINE BASED ON GRADIENT CALCULATION The FORTRAN r o u t i n e STFM2 p e r f o r m e s t h e same t a s k as STFM1. B a s e d on t h e algorithm

(3.1),

it c o m p u t e s t h e g r a d i e n t of t h e s y s t e m modes

with r e s p e c t to t h e f e e d b a c k m a t r i x . If t h e g r a d i e n t is zero ( s u f f i c i e n t l y small), t h e n t h e c o r r e s p o n d i n g mode is a f i x e d mode. In a d d i t i o n ,

STFM2 d e t e r m i n e s t h e t y p e of

t h e f i x e d modes b y d e t e r m i n i n g t h e s t r u c t u r a l s e n s i t i v i t y m a t r i x (see § 3 . 5 . 4 ) . The d a t a r e q u i r e d b y STFM2 a r e t h e same as t h e i n p u t a r g u m e n t s of STFM1 (see A p p e n d i x 3 . 1 ) .

CALLED SUBROUTINES EIGRF

See A p p e n d i x 3.1.

FIKT

Left e i g e n v e c t o r i n d e x s e a r c h (see l i s t i n g ) .

MODF

Calculation of t h e s e n s i t i v i t y with r e s p e c t to t h e f e e d b a c k m a t r i x ting)

ECRIV

Writing of a complex v e c t o r (see l i s t i n g ) .

(see lis-

334

LISTING

PARAMETER

(N=I2,M=3,L=3)

PARAMETER

(IK=2*N+I ,N2=N-2)

IMPLICIT REAL*8(A-H,O-Y) ,COMPLEX*I6(Z) DIMENSION ZM(N),Z(N,N),ZT(N,N) DIMENSION ZMT(N),ZMS(N) DIMENSION A(N,N), B (N,M) ,C (L,N) ,AK (M, N) DIMENSION WK(IK),AA(N,N),AS(N,N),AB(N,N) I N T E G E R S(N,N,N2),SS(N,N) READ(Z3,*) EPS DO 4 I = I , N

READ(23,*) (A(I,J),J=I,N) DO 5 I = I , N

READ(23,*) (B(I,J),J=I,M) IF(L.EQ.N) G O T O 7 D O 6 I=I,L R E A D (23,*) (C(I,J),J=I,N) DO 8 I--I,M

R E A D (23,*) (AK (I,J),J=l ,L) AL=0 I F ( L . E Q . N ) GO TO 40 DO 37 I = I , N DO 37 J = I , N AL=AL+0.05 DD=0 DO 35 K=I,M

35

DD=DD+B ( I , K ) * A K ( K , J ) A S ( I , J) =AL*A (I, J)+DD AB ( I , J ) = A ( I , J ) + D D

37

AA(J,I)=AB (l,J) GO TO 48

40

DO 44 I = I , N DO 44 J = I , N AL=AL+0.05 DD=0 DO 42 II=I,M DO 4Z J J = I , L

335

42

DD=DD+B (I, II)*AK (II, J J ) * C ( J J , J) AS(I,J)=AL*A(I,J)+DD AB ( I , J ) = A (I, J)+DD

44

A A ( J , I ) = A B (I, J) *** CLOSED-LOOP EIGENVALUES AND *** *** EIGENVECTORS CALCULATION

48

***

IJOB=I CALL E I G R F ( A B , N , N , I J O B , Z M , Z , N , W K , I E R ) IJOB=I CALL EIGRF(AA,N,N ,IJOB,ZMT ,ZT ,N, WK,IER) IR=0

IR2=0 D O I00 K=I,N C A L L FIKT(ZM, ZMT,K,KT, N) C A L L M O D F (B,AK, C,N, M,L, Z, ZT, K,KT, SK) IF(SK.GT.EPS) G O T O I00 IR=IR+ 1

ZM(IR)=ZM(K) IS=0 DO 70 I I = I , N DO 70 J J = I , N S (II,JJ,IR)=0 IF(A(II,JJ).EQ.0)

GO TO 70

ZZ=ZT(II,KT)*Z(JJ,K) RS=CDABS(ZZ) I F ( R S . L E . E P S ) GO TO 70 S ( I I , J J , IR)=I IS=IS+l 70

CONTINUE

IF(IS.GT.0) G O T O 100 IR2=IR2+I ZMS (IR2)=ZM(K) IR=IR- 1

I00

CONTINUE IRT=IR+IR2 IF(IRT.NE.0) G O T O 120 WRITE(6, ii0)

336 ii0

FORMAT(/,5X,,'THE

SYSTEM

H A S N O FIXED MODES',//)

STOP 120

WRITE(6,125)

125

F O R M A T ( / , 5 X , ' T H E FIXED MODES ARE : ' , / ) IRT=IR*IR2 IF(IRT.EQ.O) CALL

G O T O 140

ECRIV(ZM,IR, N)

C A L L E C R I V (ZMS,IR2, N) WRITE(6,130) 130

FORMAT(/,5X,'THE

STRUCTURALLY

FIXED M O D E S

O F T Y P E II A R E

C A L L E C R I V (ZMS,IR2, N) G O T O 160 140

IF(IR.NE.0)

G O T O 155

C A L L E C R I V (ZMS, IR2, N) WRITE(6,150) 150

FORMAT(/,5X,'ALL

T H E FIXED M O D E S

ARE

STRUCTURALW,II)

'OF T Y P E II',/) STOP 155

CALL E C R I V ( Z M , I R , N )

160

CONTINUE *** CALCULATION OF CLOSED-LOOP EIGENVALUES

***

*** AND EIGENVECTORS OF AN EQUIVALENT SYSTEM *** DO 170 I = I , N DO 170 J = I , N 170

AA(J,I)=AS(I,J) IJOB=I CALL EIGRF(AS,N,N,IJOB,ZMS,Z,N,WK,IER) IJOB=I CALL EIGRF (AA, N, N, I JOB, ZMT, ZT, N, WK, IER) IS=0 DO 230 K = I , N CALL F I K T ( Z M S , Z M T , K , K T , N ) CALL M O D F ( B , A K , C , N , M , L , Z, Z T , K , K T , SK) I F ( S K . G T . E P S ) GO TO 230 JS=0 DO 190 I I = I , N DO 190 J J = I , N

:',])

337

SS(II,JJ)=0 IF(A(II,JJ).EQ.0)

G O T O 190

ZZ=ZT(II,KT)*Z (JJ,K) RS=CDABS(ZZ) I F ( R S . L E . E P S ) GO TO 190 SS(II, JJ)=l JS=JS+I 190

CONTINUE I F ( J S . E Q . 0 ) GO TO 230 DO Z20 K K = l , I R DO 200 II--1,N DO 200 J J = l , N IF(SS(II, JJ).NE.S(II,JJ,KK))

200

GO TO 220

CONTINUE IS=IS+I ZMS(IS)=ZM(KK)

220

CONTINUE

230

CONTINUE IF(IS.EQ.0)

GO

TO

250

WRITE(6,240) 240

FORMAT(/,SX,'THE

STRUCTURALLY

FIXED M O D E S

OF TYPE I ARE

C A L L E C R I V (ZMS,IS,N) IF(IR.NE.IS) G O T O 950 STOP 250 260

WRITE(6,260) FORMAT(I,5X,'THE

NON

STRUCTURALLY

IF(IS.NE.0) T O G O 265 C A L L ECRIV(ZM,IR,N) STOP 265

D O 300 I=I,IR D O 290 J=I,IS ZZ=ZM(I)-ZMS(J) RX=DREAL(ZZ) RX=DABS(RX) I F ( R X . G T . E P S ) G O TO 270 RY=DIMAG(ZZ) RY=DABS (RY) I F ( R Y . L E . E P S ) GO TO 290

270

WRITE(6,280)

FIXED M O D E S

ARE

",/)

.l,/)

338

280

FORMAT(SX,'(',FI2.6,' +J',FI2.6,' )') G O T O 300

290

CONTINUE

3O0

CONTINUE STOP END ***

LEFT

EIGENVECTOR

INDEX

SEARCH

***

SUBROUTINE F I K T ( Z A , Z B , K , K T , N ) IMPLICIT R E A L * 8 ( A - H , O - Y ) ,COMPLEX*16(Z) DIMENSION ZA(N) , Z B ( N ) EPS=I.E-10 D O 5 II=I,N ZZ=ZA(K)-ZB (II) VA=DREAL(ZZ) VB=DABS(VA) IF(VB.GT.EPS) G O T O 5 WA=DIMAG(ZZ) WB=CDABS(ZZ) IF(WB.GT.EPS)

GO

TO

5

KT=II RETURN CONTINUE RETURN END

* CALCULATION * RESPECT

OF TO

THE

THE

SENSITIVITY

FEEDBACK

WITH

*

MATRIX

*****************************************************

S U B R O U T I N E MODF(B, AK,C, N, M, L, Z, ZT,K,KT, SK) IMPLICIT REAL*8(A-H,O-Y) ,COMPLEX*I6(Z) DIMENSION B (N,M) ,C(L, N),AK(M,N) DIMENSION

SK=0 DO 50 I=I,M

Z(N,N),ZT(N,N)

339

DO

50 J=I,L

ZSK=(O.,O.) IF(AK(I,J).EQ.0)

GO TO

50

XW=0 YW=0 DO

10 LL=I,N

XW=XW+DREAL(ZT(LL,KT))*B(LL,I)

10

YW=YW+DIMAG

(ZT (LL, KT) )*B (LL, I)

ZW=XW+(0., I.)*YW IF(L.EQ.N)

GO TO

30

XV=0

YV=O DO 20 LL=I,N XV=XV+C (J,LL)*DREAL (Z (LL, K) ) 20

YV=YV+C ( J , LL) ~'DIMAG ( Z (LL, K) ) ZV=XV+(0.,1.)*YV GO

30 40

TO

40

ZV=Z(J,K) ZSK=ZW*ZV SKI=CDABS(ZSK) SK=DMAX

50

l (SK, SKI)

CONTINUE RETURN END

* COMPLEX * Z (N)

SUBROUTINE

COMPLEX*f6

DO

VECTOR

* *

E C R I V (Z, N, N M A X )

Z (NMAX)

5 I=I,N

5

WRITE(6,10) Z(I)

I0

FORMAT RETURN END

WRITTING

N ~ NMAX

(I 5X,'(',FI2.6, ' +J',FI2.6,' )')

APPENDIX

ANDERSON

AND

CLEMENTS

FORTRAN

W subroutine

TEST

4

PACKAGE

FOR

evaluates

the

REAL

MODES

PURPOSE

The

ACTFM

f i x e d m o d e s of a N S - s t a t i o n s y s t e m d e s c r i b e d b y

set

of real

decentralized

:

NS = Ax + i__~1 B i u i (A4.1) Yi = Ci x Defining

(i=I,...,NS)

:

B =

(B 1 . . . . , BNS ) (A4.2) !

C =

(C ..... C'NS)'

the s y s t e m can be w r i t t e n

= Ax

:

(A4,3)

+ Bu

y=Cx

The

subroutine

ACTFM

d e s c r i b e d in s e c t i o n 3 . 3 . 1 .

uses

the

algebraic

characterization

UTILIZATION

The subroutine

s t a t e m e n t is :

S U B R O U T I N E ACTFM

of

Note t h a t ACTFM e x a m i n e s r e a l p o l e s o n l y .

(A,N,B,M,C,L,NS,IM,IR,EPS,Z,JJ)

fixed

modes

341

INPUT ARGUMENTS

N

Order

M

Number of the system inputs.

L

Number of the system outputs.

A,B,C

System matrices of dimension

NS

Number of the control stations.

IM

of the system.

Integer

vector

of dimension

(N,N),

(N,M)

and

(L,N),

respectively.

NS c o n t a i n i n g

the

number

of inputs

of the ith

of outputs

of the ith

s t a t i o n in t h e i t h p o s i t i o n . IR

Integer

vector

of dimension

NS c o n t a i n i n g

the

number

s t a t i o n in t h e i t h p o s i t i o n . EPS

Small p o s i t i v e r e a l n u m b e r

IT

Option parameter IT = 1 writting

defining

the zero accuracy.

: of the open-loop

poles.

IT # 1 no writting.

OUTPUT

JJ

ARGUMENTS

Number of real decentralized system

XX

Real

vector

any)

in t h e f i r s t JJ p o s i t i o n s .

CALLED

of length

J J=0 m e a n s t h a t

the

fixed modes.

N containing

the

real

decentralized

fixed

modes

(if

SUBROUTINES

EIGRF

See A p p e n d i x

DSVD

Subroutine

computing

rectangular

matrix,

GARBOW B . S . ,

2.

J.M.

"Matrix Eigensystem Lecture RANK

fixed modes of the system.

has not real decentralized

the

described BOYLE, Routines

notes in computer

Subroutine

Singular

determining

Value

Decomposition

arbitrary

real

in : J.J.

DONGARRA,

C.B.

MOLER

- EISPACK Guide Extension".

s c i e n c e n ° 51, S p r i n g e r - V e r l a g ,

the

of an

rank

of

2 real

matrix

of

New-York, dimension

1977.

(M,N)

by

calling DSVD.

REQUIRED

MEMORY

If N T = N + m a x ( M p L ) > 2 0 o r ged according

to :

NS>5,

then

the

dimension

statement

must

be chan-

342 DIMENSION II(NS), ZV(NT,NT), AA(NT,NT) DIMENSION WK(NT), U ( N T , N T ) ,

V(NT,NT),

RVI(NT)

LISTING SUBROUTINE ACTFM (A, N, B , M , C , L , N S , I M , I R , EPS, Z , J J , I T ) IMPLICIT R E A L * 8 ( A - H , O - Y ) , I N T E G E R ( I - N ) ,COMPLEX*16(Z) DIMENSION

A(N,N) ,B(N,M), C (L,N),IM(NS),IR(NS), Z(N)

DIMENSION

II(5),ZV(20,20),AA(20,20)

DIMENSION

WK (20), U (20,20) ,V (20,20), RV1 (20)

NT=20 *** O P E N - L O O P 15

16

P O L E S CALCULATION ***

IJOB=I DO

16 I=I,N

DO

16 J=l,N

AA(I,J)=A(I,J) C A L L EIGRF (AA, N,NT, IJOB, Z, ZV, N T , W K , IER) IF(IT.NE.I)

G O T O 25

WRITE(6,18) 18

FORMAT(5X,'THE

OPEN-LOOP

POLES ARE

D O 20 l=l,N 20

WRITE(6,22)

22

F O R M A T (5X,EI2.6,2X,EI2.6,/)

25

JJ=O

Z(1)

EPSI=10D-8 DO 90 J I = I , N YY=AIMAG (Z ( J I ) ) I F ( D A B S ( Y Y ) . G T . E P S 1 ) GO TO 90 XX=REAL(Z ( J I ) ) IF(JI.EQ.])

GO TO 40

JII=JI-1 DO 30 I = l , J I I YY=AIMAG (Z (I)) I E ( D A B S ( Y Y ) . G T . E P S 1 ) GO TO 30 XX I=REAL(Z (I))

',])

343 IF(DABS(XX-XXI).LT.EPS1) 30

GO TO 9~

CONTINUE *** B U I L D I N G OF THE MATRIX B L O C K - D I A G . ( ( A - S I ) , O )

40

DO 44 I = I , N DO 42 J = I , N

42

AA(I,J)=A (I,,I)

44

AA(I, I)=AA(I,I)-XX N I=N+ 1 IK=N+M IKI=N+L DO 46 I = N I , I K 1 DO 46 J = N I , I K

46

AA(I,J)=O. *** COMPLEMENTARY S U B S Y S T E M S SEARCH ***

MAX=2**NS-2 D O 90 N B = I , M A X II=O

NU=NB D O 48 I=I,NS ND=NU/2 NR=NU-ND*2 IF (NR.EQ'.0) G O T O 48 II=Ii+l

II(II)=I 48

NU=ND

*** B U I L D I N G OF THE T E S T E D MATRIX ***

56

Ii=l IP=N JP=N D O 68 IS=I,NS IC=0 JC=0 IK2=IS-I IF (IK2.LT.I) G O T O 60 D O 58 K=I,IK2 IC=IC+IR (K)

***

344

58

JC=JC+IM(K)

60

IF(IS.EQ.II(I1))

GO T O 64

IK3=IP+I IK4=IP+IR (IS) DO 62 I = I K 3 , I K 4 IC=IC+I DO 62 J = I , N 62

A A ( I , J ) = C (IC,J) IP=IP+IR (IS) GO

64

TO

68

IKS=JP+I IK6=JP+IM (IS) DO

66 J=IK5,IK6

JC=JC+l DO 66

66 I=I,N

A A ( I , J ) = B (I, JC) JP=JP+IM (IS) II=Ii+l

68

CONTINUE CALL

RANK(AA,IP,

IF(IRANK.LT.N)

GO T O 78

JP, N T , W K ~U,V, R V I , I R A N K ,

GO

TO

EPS)

78

90

JJ=JJ+l Z (J J)=Z (JI)

90

CONTINUE RETURN END

S U B R O U T I N E R A N K ( A , M , N , NM, W, U, V, R V 1 , I R A N K , EPS) ************************************* * DETERMINATION * A

OF

THE

RANK

OF

~*********~.****** ~*** A REAL

MATRIX

: MxN

* NM

= A

* REAL

* *

~M,~N

MATRICES

*

:

*

* W ( N M ) , U ( N M , NM) ,V (NM, NM) , R V 1 (NM)

*

* R E S U L T IN I R A N K * **** ~ * * * * * * * * * * * * * $ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * REAL*8 DO

A ( N M , N ) , W ( N ) , U ( N M , N ) ,V ( N M , N ) ,RVI(N)

1 I=I,NM

W(I)=O. CALL

DSVD(NM,N,M,A,W,.FALSE.,U,.FALSE.,V,IERR,RVI)

345

IRANK=0 MM=N I F ( M . G E . N ) GO TO 2 MM=M DO 3 I=I,MM IF (W(I).LT.EPS) G O T O 3 IRANK=IRANK+I CONTINUE RETURN END

APPENDIX 5

D E T E R M I N A T I O N OF T H E G R A D I E N T M A T R I X OF T H E P E R F O R M A N C E I N D E X BY U S I N G V A R I A T I O N C A L C U L U S

This appendix the performance

uses

variation

calculus

(LEV-70)

to determine

the derivatives

of

index.

AS. 1. P r e l i m i n a r i e s

The three

Theorem

AS.1

represents

(BEL-70).

the unique

Theorem number.

following results

A5.2

If the integral

s o l u t i o n of

(BEL-70).

K l e i n m a n ' s lemma ( K L E - 6 6 ) .

f(x+ ehx)

then

A5.2.

Development

Consider

£(t)

: A x(t) = c x(t)

Let f(x)

+ Trace M(x)

=M'(x)

(TAR-85)

the system

y(t)

matrix

:

+ B u(t)

{

e At

C

e]~t

exists

for

all C,

e (A+CB)t, in ~ as

where ~ is

a

small

:

B e As ds

be a trace

:

= f(x)

: dr(x)

the

to the first order

= e AL + ¢ 0 / t e A ( t - s )

e - - > 0, we c a n w r i t e

: x = -

development.

it

Ax + xB = C.

Consider

It can be approximated

e (A+CB)t

are useful in the subsequent

-

hx

function.

If,

f o r a]] x a n d f o r

real

347 2nd the

performance

index

.'

co

J = ~

Let the

(x'Q

control

x + u' Ru)

be

given

x = (A - B K C )

which has

the

Substituting get

by u = - Ky,

so that

the

closed-loop

system

is

:

x = Dx

solution

in

dt.

the

: x = e Dt x 0 ,

expression

of

y

x 0 = x(0)

and

using

the

trace

function

properties,

we

:

J(D)

= Tr

[ f0 e D '

t QI(K,C ) eDt dt X 0 ]

where X 0 = E [ x(0) x(0)' ] QI(K)

= Q + C' K' R K C

Suppose

that the s y s t e m matrices

A changes

where

~A _N

to B + e B . A B

C changes

to C + e C . AC

D changes

to D + e D . A D

CA,

£]3' e C a n d £ D a r e "~ ¢C

x(t)

with and

~- e D

" Then

small real the

numbers

elosep-loop

of the

system

same order,

becomes

i.e.

:

= (D + e . A D ) x ( t )

AD = AA - AB. K C - B . A K . C the

subjected to small perturbations, i.e. :

to A +e A . AA

B changes

eB

are

criterion

- B.K.

AC

: oo

J(D+

cAD)

By

developing

and

the

trace

one obtains

= Tr

this

[ ~

e (D+eAD)'t

expression

function

property

to

the

Tr(AB)

QI(K+e~K,

first

C+c~C)

order,

= Tr(BA)

using

e (D+cAD)t dt

Theorems

= Tr(A'B')

= Tr(B'A'),

:

J(D+~AD)

= J(D) + e T r

D'P + P D

+ Q + C'K'R KC

[2S(C'K'R-PB).A

with = 0

(KC)

A5.1

+ 2SP. A ( A , B ) ]

X0 ]

and

A5.2,

348 DS + SD + X 0 = 0

AD =

A(A,B)

A(KC) =

- B.

A (KC)

+ K. AC

AK.C

A(A,B) = AA -AB.

Two

1 -

cases

of variations

Simultaneous

KC

are

considered

variations

on

In this case, w e h a v e A(KC) Then

= 0 and

AD

:

A and

B

:

= A(A,B)

one gets : J [D+E.

A(A,B)]

and by application ~J

= J(D) + e T r

[2SP. A ( A , B ) ]

of Kleinman's l e m m a

:

-2PS

(A,B)

This

case

combines

two

situations

:

• variations

on

A only,

~T we get z-~ = 2 PS. ~A

• variations

on

B only,

~T w e g e t ~--Z-~ : ~B

2 -

Simultaneous

In

this

case,

A (A,B) The

criterion

J[

and

by

and

becomes

• variations

combines

on

B

n(KC)

:

= J(D)

+

lemma

e Tr

[ (2SC'K'R

-

:

- B'P) S

two situations

K only,

C

:

of Kleinman's

- 2 (RKC (KC)

case

K and

AD = -

B. A (KC)]

application

on

we have

= 0

D - e.

8 J

This

variations

- PSC'K'

we get

:

-B J : 2 K ' ( R K C

- B'P)S

3C • variations

on

C only,

we get

~J ~K

= 2 (RKC

- B'P) SC'

2SPB)]

(KC)

APPENDIX 6

A FORTRAN

ROUTINE

CONSTRAINED

TO

ROBUSTNESS

This

appendix

constrained

provides

= Ax

+ BU

AN WITH

REQUIREMENTS

a routine

x ~ R n,

and its sensitivities

State feedback

• Output

OPTIMAL POSSIBLE

for the

determination

of an optimal

:

u~R m

:

feedback

u

with respect

(see § 6.4.2).

=

to the classical quadratic

criterion

T h e r o u t i n e c o n s i d e r s two c a s e s

:

Kx

: u = K C x = Ky

T h e o p t i m i z a t i o n p r o b l e m is :

min

J3(K)

= Tr

(P V 0) + T r

(SPLPS)

+ Tr

[ (RKC-B'P)SFS(RKC-B'P)']

KEK F

subject to D'S + S D + Q + C ' K ' R K C DP

where R,Q,L,F

+ PD' + V 0

= 0

= 0

a r e w e i g h t i n g m a t r i c e s of a p p r o p r i a t e

T h e s o l u t i o n of t h i s p r o b l e m is g i v e n b y

B J3 = 2

with

[(RKC

= 0

DP

+ PD' + V 0 = 0

Du

+ uD' + F 2 ( K , P , S ) + F 3 (K,P,S)

= A + BKC

dimension.

:

+ B'S)~ + B' ~ P + R ( R K C + B ' S ) P F P ]

D'S + S D + FI(K)

D' X + X D whereD

(local)

y ~ Rr

T h e o p t i m i z a t i o n is p e r f o r m e d

•

MATRIX

f e e d b a c k m a t r i x K ~ KF f o r t h e l i n e a r s y s t e m

y = Cx

6.3.1)

DETERMINE

FEEDBACK

= 0 = 0

C'

(see §

350 FI(K) = Q + C'K'RKC F2(K,P,S) = V0 + PPSL + LSPP + B(RKC+B'S) PFP + PFP(RKC+B'S)'B' F3(K,P,S) = PSLS + SLSP + (RKC + B'S)' (RKC+B'S)PF +

+ FP(RKC+B'S)'(RKG+B'S) This problem tion)

to s a t i s f y

is solved by using

the

structural

the feasible direction

constraints

and

method

an adaptative

step

(gradient size

as

projecgiven

in

(6.3.6). The required

data are

:

SYSTEM DATA

N

Order

M

Number of inputs.

L

Number of outputs.

A,B,C

System matrices of dimension

IES

of t h e s y s t e m .

Option parameter

(N,N),

(N,M) and

(L,N),

respectively.

:

IES = 0 i f s t a t e f e e d b a c k IES ~ 0 i f o u t p u t

feedback.

OPTIMIZATION DATA

AL

Initial step

size.

PI

Real positive number

superior

ANU

Real positive number

such that

EPS

Accuracy

NI

Allowed maximum number

IT

Option parameter I T = 1,

to 1. 0 < ANU < 1.

positive small number

considered

as zero.

of iterations.

;

writing of the intermediate II i t e r a t i o n

number

F

norm

gradient

CR criterion

results

:

value

AL s t e p s i z e "VPD d o m i n a n t c l o s e d - l o o p

eigenvalue

of the matrix S.

I T ~ 1, n o w r i t t i n g . IGB

Option parameter

:

IGB=I for minimizing the reduced This

6.3.1)

case

corresponds

to

the

criterion algorithm

: J3 = T r [ P V ~ . of

Geromel

and

Bernussou

(see §

351

IGB=2 for minimizing t h e c r i t e r i o n : 33 = T r [PV0] + T r [SPLPS ] IGB#I a n d IGB#2 for s o l v i n g t h e g e n e r a l p r o b l e m d e s c r i b e d a b o v e . AK

Initial s t a b i l i z i n g f e e d b a c k m a t r i x s u c h t h a t AK ~ KF.

VO Q

Initial s t a t e c o n d i t i o n m a t r i x V0 = E Ix(0) x ( O ) ' ] .

R

I n p u t w e i g h t i n g m a t r i x of dimension (M,M).

PL

Weighting m a t r i x of dimension

S t a t e w e i g h t i n g matrix of d i m e n s i o n (N,N) (NxN)

f o r t h e s e n s i t i v i t y with r e s p e c t to A

a n d B. T h i s matrix is n e c e s s a r y if IGB~I. Weighting m a t r i x of d i m e n s i o n (N,N)

PF

for the

s e n s i t i v i t y with r e s p e c t to K

a n d C.

CALLED

MULT

SUBROUTINES

Two real m a t r i c e s multiplication (see l i s t i n g ) .

MULT3

T h r e e r e a l m a t r i c e s multiplication ( s e e l i s t i n g ) .

F1KC

Calculation of F I ( K )

(see l i s t i n g ) .

F2KSPV Calculation of F 2 ( K , P , S ) ( s e e l i s t i n g ) . F3KSPV Calculation of F 3 ( K , P , S )

(see listing).

RKCBS Calculation of QQ = ( R . A A + B ' S ) a n d of AB=QQ.P.

S t a t e f e e d b a c k : AA=K,

o u t p u t f e e d b a c k : AA = KC (see l i s t i n g ) . MST

Calculation of F - (A+A') ( s e e l i s t i n g ) .

LYAPUN L y a p u n o v e q u a t i o n s o l v i n g ( s e e l i s t i n g ) . MATBF C l o s e d - l o o p matrix calculation ( s e e l i s t i n g ) . PGV

D e t e r m i n a t i o n of t h e

smallest

(or

biggest)

e l e m e n t of a real

vector

(see

listin g ) . MEV

D e t e r m i n a t i o n of t h e

smallest r e a l p a r t of t h e e i g e n v a l u e s of a real matrix

(see l i s t i n g ) . CRIT

C r i t e r i o n calculation ( s e e l i s t i n g ) .

GRADK G r a d i e n t calculation (see l i s t i n g ) . EIGRF

Eigenvalue

calculation

subroutine,

described

in :

"IMSL L i b r a r y

Manual",

"IMSL

Manual",

Edition 8, 1980. LINV2F Real

matrix

inversion

subroutine,

described

in

Library

Edition 8, 1980. PRINT

Writing of a real matrix (see l i s t i n g ) .

REQUIRED MEMORY The d i m e n s i o n s t a t e m e n t s m u s t be modified if N > 15, M > 5, o r L > 5 a c c o r ding to :

352

DIMENSION A ( N , N ) , B ( N , M ) , C ( L , N ) , A K ( M , N ) , WK(IK), R(M,M) The same for all t h e o t h e r m a t r i c e s of dimension ( N , N ) . NA,MA,LA a n d IK m u s t b e s e t to : NA=N, MA=M, LA=L a n d I g >/ N2 + 3N

LISTING IMPLICIT REAL*8 ( A - H , O - Y ) , COMPLEX*I6(Z) COMMON /MATSS/ A(15,15) ,B(15,5) ,C(5,15),AK(5,15) COMMON / M A T P I / Q(15,15),R(5,5) COMMON /MATP2/ PL(15,15) ,PF(15,15) COMMON /MATLI/ S(15,15),P(15,15),VO(15,15) COMMON /MATL2/ PMU(IS,15),SLA(I5,15) COMMON /GRADS/ GA(15,15),GK(IS,]5) DIMENSION AF(15 ,15),AFF(15,15) ,G (15,15) ,D (15,15)

DIMENSION E(15,15),AA(15,15),AC(15,15),WK(270) DIMENSION ZV(15),Z(15,15)

NA=15 MA=5 LA=5 IK=270 *** SYSTEM MATRICES READING *** READ (19,*) N , M , L , I E S DO 12 I = I , N 12

READ (19,*) ( A ( I , J ) , J = I , N ) DO 17 I = I , N

17

READ (19,*) (B ( I , J ) , J=I,M) NL=N I F ( I E S . E Q . 0 ) GO TO 24 NL=L DO 20 I = I , L

20

READ (19,*) ( C ( I , J ) , J = I , N ) *** OPTIMIZATION DATA READING ***

24

READ (19,*) A L , P I , A N U , E P S , N I , I T , I G B

Also t h e p a r a m e t e r s

353

DO 26 I=I,M 26

READ (19,*) ( A K ( I , J) , J = I , N L ) DO 28 I = I , N

28

READ ( 1 9 , * ) ( V O ( I , J ) , J = I , N )

D O 32 I=I,N 32

READ(19,*) (Q(I,J),J=I,N) D O 36 I=l,M

36

READ(I9,*) (R(I,J),J=I,M) IF(IGB.EQ.1)

GO T O 52

DO 44 I = I , N 44

READ(19,*) (PL(I,J),J=I,N) IF(IGB.EQ.2)

GO TO 52

DO 48 I = I , N 48

READ(Ig,*) (PF(I,J),J=I,N) *** I T E R A T I O N S

52

IF(IT.NE.I)

BEGINING

***

G O T O 58

WRITE(6,56) 56

F O R M A T (6X,'II', 10X,'F', 10X,'CR', 10X,'AL', 10X,'VPD',/)

58

II=0

60

CONTINUE

*** CLOSED-LOOP MATRIX DETERMINATION

CALL

MATBF(N,

M, NL, NA,MA,

***

LA, AF)

*** SOLVING THE 1ST LYAPUNOV EQUATION

***

***

***

(CALCULATION OF S)

DO 65 I = I , N DO 65 J = I , N 65

A F F ( I , J ) = - A F ( I , J) CALL F1KC(N,NA,M,MA,NL,LA,AA,AC) CALL LYAPUN(AFF,N,NA,AC,NA,S,NA,P,G,ZV,Z,WK,IK) IF(IGB.NE.1)

GO TO 72

CALL C R I T ( N , M , I G B , C R ) IF(II.NE.0)

G O T O 72

WRITE(6,71) CR

354

71

FORMAT(LX,'INITIAL C R I T E R I O N VALUE=',EI2.6,/)

72

IF(II.EQ.0) G O T O 75 IF(IGB.NE.I) G O T O 73 IF(CR.GE.Y) G O T O 135

73

C A L L M E V (S, N,NA, G, ZV, Z,WK, IK, VPD) IF(VPD.LE.0) G O T O 135 ***

SOLVING

***

75

THE

2ND

LYAPUNOV

(CALCULATION

OF

EQUATION P)

*** ***

D O 76 I=I,N D O 76 J=I,N AFF(I, J)=-AF(J,I)

76

E(I,J)=-VO(I, J) G A L L L Y A P U N (AFF, N,NA,E, NA,P, NA,AC,G, ZV, Z,WK, IK)

C A L L R K C B S (AA,M, N,NA,MA,LA, E,GK) IF(IGB.NE.I) GO T O 90 IF(IES.EQ.0) G O T O 83 D O 80 I=I,N D O 80 J=I,L 80

AC(I,J)=C(J,I) C A L L M U L T (GK,NA, NA, AC, NA, NA, G, NA, NA, M,N, L) G O T O II0

83

D O 85 I=I,M D O 85 J=I,NL

85

G(I,J)=GK(I,J) G O T O 110

90

CALL MULT (S, NA, N A , P , NA, NA, GA, NA, NA, N, N, N) *** SOLVING THE 3RD LYAPUNOV EQUATION *** ***

95

(CALCULATION OF MU)

***

CALL F2KSPV(M,N,IGB,G,AFF,AA,NA,MA,LA)

D O 97 I=I,N D O 97 J=I,N 97

AFF(I, J)=-AF(J,I) C A L L L Y A P U N (AFF,N, NA, AA, NA,PMU, NA, G, AC, ZV, Z,WK, IK)

355

*** SOLVING THE 4TH LYPAUNOV E Q U A T I O N ***

***

(CALCULATION

OF L A M D A )

***

C A L L FBKSP(E,M,N,IGB,AFF, G , A A , N A , M A , LA) D O 103 I=I,N D O 103 J=I,N I03

AFF(I, J)=-AF(I, J) CALL LYAPUN(AFF,N,NA,AA,NA,SLA,NA,G,AC,ZV,Z,WK,IK) *** C R I T E R I O N

CALCUATION

***

C A L L CRIT(N,M,IG, CR) IF(II.NE.0) G O T O 107 WRITE(6,105) C R 105

FORMAT(LX,'INITIAL C R I T E R I O N VALUE=',E12.6,/)

107

IF(CR.GT.Y) G O T O 135

109

CALL

G O T O 109

GRADK(E,AA,AC,N,M,NL,NA,MA,LA,G)

*** G R A D I E N T P R O J E C T I O N ***

ii0

F=0 D O 115 I=I,M D O 115 J=I,NL D(I,J)=0 IF(AK(I,J).EQ.0) G O T O 115 D(I,J)=G(I,J) F I = D A B S (G ( I , J) ) F=DMAX 1 ( F , F1)

115

CONTINUE

Y=CR 11=11+1 IF(IT.NE.1)

GO TO 118

W R I T E ( 6 , 1 1 7 ) I I , F , C R , A L , VPD 117

118

F O R M A T (5X,13, IX,El2.6, IX,El2.6, IX,El4.8, IX,E12.6,/) IF(F.LT.EPS) G O T O 140 IF(II.GT.NI) G O T O 160

356

AL=PI*AL KL=I 120

DO 122 I=I,M DO 122 J = I , N L

122

A K (I,J) = A K (I,J)-AL*D (I,J) G O T O 60

135

CONTINUE IF(IT.NE.I)

G O T O 538

WRITE(6,136)CR,AL,VPD 136

FORMAT

138

D O 139 I=I,M DO

139

(22X, E12.6, IX, El4.8, IX,El2.6,/)

139 J=I,NL

A K (I,J) = A K (I,J)+AL*D (I,J) IF(KL.EQ. l) AL=AL/PI AL=AL*ANU IF(AL.LT.IE-10)

G O T O 160

KL=0 GO TO 120 540

WRITE(6,150) II

150

FORMAT(5X,'THE CONVERGENCE IS OBTAINED A F T E R ' , I X , I 4 , 1 X , ' I T E R A T I O N S ' , / , 5 X , ' T H E OBTAINED FEEDBACK MATRIX I S ' , / ) GO TO 170

160

W R I T E (6,165)II

165

FORMAT(5X,'THE

CONVERGENCE

,'ITERATIONS',/,5X,'THE 170

IS N O T

OBTAINED

OBTAINED

FEEDBACK

AFTER',I4

MATRIX

IS',/)

CALL P R I N T ( A K , M , N L , M A , 1) WRITE(6,180) Y

180

FORMAT(5X,'THE CRITERION VALUE= ' , E 1 2 . 6 , / / ) WRITE(6,190) F

190

FORMAT(5X,'THE GRADIENT NORM VALUE EPS = ' , E 1 2 . 6 , / ) STOP END

* CLOSED-LOOP *

AA=A+B.AK.C

MATRIX

DETERMINATION

IF O U T P U T

FEEDBACK

* *

357

* A=A+B.AK IF STATE FEEDBACK * ************************************************ SUBROUTINE MATFB ( N , M , N L , NA,MA,LA, AA) IMPLICIT REAL*8 ( A - H , O - Y ) COMMON /MATSS/ A ( 1 5 , 1 5 ) , B ( 1 5 , 5 ) , C ( 5 , 1 5 ) , A K ( 5 , 1 5 ) DIMENSION

AA(NA,NA)

I F ( N L . N E . N ) GO TO 4 CALL MULT(B,NA, MA, AK, MA, LA, AA, NA, NA, N, M, NL) GO TO 6 CALL

M U L T 3 (B,NA, MA, AK, MA,LA, C, LA, NA, N,M, NL, N,AA, NA)

D O 8 I=I,N D O 8 J=I,N A A (I,J)=A(I,J)+AA(I,J) RETURN END ********************************************************** * CALCULATION OF THE MATRIX QQ : * Q Q = -(Q+CIK~RKC) * QQ

=

-(Q+K'RK)

SUBROUTINE

IF O U T P U T IF S T A T E

FEEDBACK

* (NL=L)

FEEDBACK(NL=N)

F I K C ( N , N A jM,MA,NL,LA, BB, QQ)

IMPLICIT REAL*8 ( A - H . O - Y ) COMMON /MATSS! A ( 1 5 , 1 5 ) , B ( 1 5 , 5 ) , C ( S , 1 5 ) , A K ( 5 , 1 5 ) COMMON / M A T P I / Q ( 1 5 , 1 5 ) , R ( 5 , 5 ) DIMENSION QQ(NA, NA) ,BB (NA,NA) IF(NL.EQ.N)

GO TO 2

CALL MULT (AK, MA,LA,C t LA, NA, BB, NA, NA, M, NL, N) GO TO 6 2

D O 4 I=I,M D O 4 J=IpNL

4

B B (I,J)=AK(I,J)

6

DO

10 I=I,N

DO

I0 J=I,N

S=0 D O 8 II=I,M D O 8 JJ=I,M 8

S=S+BB (II,I)*R (II,J J) *BB (JJ, J)

I0

QQ(I,J)=-S-Q(I,J)

* *

358

RETURN END **********

~*****************************************

*SOLVING *

THE

A'Q+QA+C

*INPUT *

MATRIX

*

:

*

=0

*

ARGUMENTS

REAL

EQUATION

**** **

:

*

A(N,N)

N ~< N A

*

C(N,N)

N ~< N C

*

*

RI(N,N)

N ~< NA

*

*

R2(N,N)

N ~< N A

*

WK(IK)

I K >/ N * N + 3 * N

*

COMPLEX

*

* OUTPUT

*

*

REAL

Z(N,N),ZI(N) ARGEMENTS

*

N x< N A

*

:

*

Q(N,N) N ~< N Q

REFERENCE

* *

:

*

"THE

NUMERICAL

*

W.D.

HOSKINS,

*

IEEE T R A N S .

*

N. 5, O C T .

SUBROUTINE

SOLUTION D.S.

AUT.

OF

A'Q+QA=-C"

MEEK AND D.J.

CONT.,

VOL.

WALTON

AC-22,

1977, 882-883.

LYAPUN(A,

IMPLICIT R E A L * 8

N,NA,

C,NC,Q,NQ,RI,R2,

(A-H,O-Y)

A (NA, NA), C (NC, NC), Q (NQ, NQ)

DIMENSION

R I ( N A , N A ) ,R2(NA,NA) ,WK(IK) ZI(NA),Z(NA,NA)

K=0 K=K+I IF(K.GT.1O0)

G O T O 30

D O 3 I=I,N D O 3 J=I,N RI (I,J)=A(I,J) IJOB=0 CALL

EIGRF(R], N, NA, IJOB, Z], Z, NA, WK, IER)

D O 5 I=I,N W K (I)= R E A L (Zl (T)) II=0 C A L L P G V (WK,N,IK,PV,II) IF(PV.GT.0)

G O T O 12

* *

DIMENSION COMPLEX*f6

* *

ZI, Z,WK,IK)

359

DO

10 I=I,N

11=1+1 DO

9 J=11,N

Q(J,I)=0 9

l0

Q(I,J)=0

Q(z,I)=-I RETURN

12

II=i CALL

P G V (WK,N,IK,GV,II)

XX=(PV*GV)**0.5 X=I/(PV+XX)**2 ALPHA=2*PV*X BETA=PV*GV*ALPHA EPSI=X* (PV-X X) **2 DO 14 I = l , N DO 14 J = l , N 14

R2(I,J)=A(I,J) IDGT=0

CALL LINV2F(R2, N , N A , R I , I D G T , W K , IER) DO 17 I = I , N DO 17 J = I , N 17

Q(I,J)=RI(J,I) DO 20 I = l j N DO 20 J = I , N

2O

A (I, J)=ALPHA*A (I, J ) + B E T A * R l ( I , J) CALL MULT3 (Q,NQ, NQ, C , N C , N C , R 1 , NA, NAp N, N, N, N, R2, NA)

22

DO 24 I = l , N DO 24 J = I , N

24

C (I, J) =ALPHA*C (I, J) +B ETA*R2 (I, J) IF(EPSI.GE.1.E-07)

GO T O

1

DO 25 I = I , N DO 25 J = I , N 25

Q(1,J)=-0.5*C(I,J)

30

WRITE(6,32)

32

FORMAT

RETURN

RETURN END

II

(RX, ' I T E R A T I O N S

N U M B ER ( L Y A P U N O V ) = ' , 14, l ])

360 ********************************************************** * FINDING * THE

THE

REAL

* INPUT

VECTOR

*

Y REAL

K OPTION

*

K=0

*

K#0

ELEMENT

Y OF DIMENSION

ARGEMENTS

*

* USED

SMALLEST/BIGGEST

X

OF

N x< N M A X .

*

PARAMETER

:

SMALLEST

FOR

BIGGEST

AS AN

OUTPUT,

* DESIRED

* *

:

VECTOR

FOR

*

ELEMENT,

* ELEMENT

ELEMENT

*

K IS T H E

.I.E.

INDEX

OF

THE

X--Y(K)

* *

**********************************************************

SUBROUTINE

IMPLICIT

PGV(Y,N,NMAX,X,K)

REAL*8

(A-H,O-Z)

DIMENSION

Y(NMAX)

IF(K.NE.0)

GO

TO

2

K=I X = Y (i) DO

1 I=2,N

IF(Y(I).GE.X)

GO TO

1

GO

3

x=v(i) K=I CONTINUE RETURN K=I X=Y(1) DO

3 I=2,N

IF(X.GE.Y(I))

TO

X=Y(I) K=I CONTINUE RETURN END ********************************************************** * FINDING

THE

* EIGENVALUES *

WORKING

*

REAL*8

SMALLEST OF A REAL

AREA

:

AA(N,N),WK(IK)

REAL

PART

MATRIX

VPD

A(N,N).

OF THE

* * * *

361

*

COMPLEX*f6

*

N ~< NA,

SUB

ROU

TINE

IMPLICIT

ZV(N),Z(N,N)

*

IK >I N

MEV

(A , N , NA

*

, AA,

ZV

, Z , WK

, IK , VPD)

REAL*8 ( A - H , O - Y )

COMPLEX*16 Z V ( 1 5 ) , Z ( 1 5 , 1 5 ) DIMENSION A ( N A , N A ) , A A ( N A , N A ) , W K ( I K )

DO

2 I=I,N

DO

2 J=I,N

AA(I,J)=A(I, J) IJ O B = 0 CALL DO

E I G R F (AA,N, NA, IJ O B ,ZV, Z, N A , W K , I E R )

4 I=I,N

W K (1)=REAL (ZV (I)) II=O CALL

P G V (WK ,N, IK ,V P D ,II)

RETURN END

*** C R I T E R I O N

CALCULATION ***

SUBROUTINE C R I T ( N , M , I G B , CR)

IMPLICIT

REAL*8

(A-H,O-Y)

COMMON

/MATP2/

PL(15,I5),PF(15,15)

COMMON

]MATLI]

S(15,15),P(15,15),V0(15,15)

COMMON

]GRADS]

GA(15,15),GK(15,15)

CR=0 DO

2 I=I,N

DO

2 J=I,N

C R = C R + S (I, J) *V0 (J, I) IF(IGB.EQ. I) R E T U R N TR=0 D O 6 I=I,N SD=0 D O 4 II=I,N DO

4 JJ=I,N

362 4

SD=SD+G A ( I. II ) * P L ( I I , J J) * GA ( I, J J)

6

TR=TR+SD CR=CR+TR IF(IGB.EQ.2)

RETURN

TR=0 DO 10 I = I , N SD=0 DO 8 I I = I , M DO 8 J J = ] , M 8 10

SD=SD+GK ( I I , I) * P F ( I I , J J) *GK (J J , I) TR=TR+SD CR=CR+TR RETURN END ********************************************************** * C A L C U L A T I O N OF T H E M A T R I C E S *

QQ = R.AA

*

AB = ( R . A A + B ' . S ) . P

* AA= K . C

:

+ B'.S

* *

IF STATE FEEDBACK (NL=N)

*

* AA(M,N)

*

N ~< NA, M ,,< N A

* AB(M,N)

S U B R O U T I N E R K C B S ( A A , M, N, N A , M A , L A , Q Q , A B )

COMMON ] M A T S S / A(15,15),B(lS,]5),C(5,15),AK(5,15) /MATPI/

COMMON ] M A L T I /

Q(I5,]5),R(5,5) S(15,15),P(15,15),V0(15,15)

DIMENSION Q Q ( N A , N A ) , A A ( N A , N A ) , A B ( N A , N A ) DO 12 I = I , M DO 8 J = I , N SD=0 DO 4 I I = I , M S D = S D + R (I, II) * A A (II,J) SS=0 DO 6 K = I , N SS=SS+B (K,I)*S(K,J) QQ(I, J ) = S S + S D

* *

IMPLICIT REAL*8 (A-H,O-Y)

COMMON

*

I F OUTPUT FEEDBACK (NL=L)

* AA= K

* QQ(M,N),

*

363

D O I0 J=I,N SS=0 D O i0 K=I.N

SS=SS+Q Q (I, K) *P(K ,J) I0

AB(I,J)=SS

12

CONTINUE RETURN END

*** CALCULATION OF THE MATRIX -F2(K,S,P) ***

SUBROUTINE F2KSPV(M,N, IGB, G,AFF,F2, NA,MA, LA) IMPLICIT REAL*8 (A-H,O-Y) COMMON /MATSS/ A(15,15),B(15,5),C(5,15),AK(5,15) COMMON /MATP2/ PL(15,15),PF(15,IS) COMMON /MATLI/ S(15,15),P(15,15),V0(15,15) COMMON ]GRADSf GA(15,15),GK(15,15) DIMENSION F2(NA,NA),AF(NA,NA),G(NA,NA)

CALL M U L T 3 (PL, NA, NA, GA, NA, NA, P, NA, NA, N, N, N, N, AFF,NA)

D O 2 I=],N D O 2 J=I,N F2(I,J)=-V0(I,J) C A L L MST(AFF,F2,N,NA) IF(IGB.EQ. 2) R E T U R N C A L L M U L T (B ,MA,NA,GK.NA, NA,G, NA,NA, N,M, N) C A L L MULT3(G, NA, NA,PF, NA,NA,P,NA, NA, N, N, N, N, AFF, NA) C A L L M S T (AFF,F2,N, NA) RETURN END ***************

* ~****~***************

* CALCULATION OF F= F-(A+A')

*

*

WHERE

*

*

A(N,N), F (N,N), N ~< N A

*

*************************************

S U B R O U T I N E MST(A,F,N,NA) IMPLICIT REAL*8 (A-H,O-Y) D I M E N S I O N A(NA,NA) ,F(NA,NA)

364

D O 2 I=I,N D O 2 J=I,N A(I,J)=A(I, J)+A(J,I) A(J,I)=A(I,J) D O 4 I=I,N D O 4 J=I,N F (1,J) =F (I,J)-A (I,J) RETURN END

*** CALCULATION OF - F 3 ( K , S , P ) SUBROUTINE

***

F3KSP(E, M,N,IGB,AFF,D, F3,NA,MA, LA)

IMPLICIT REAL*8 ( A - H , O - Y ) COMMON / M A T S S / A ( 1 5 , I S ) , B ( 1 5 , 5 ) , C ( 5 , 1 5 ) . A K ( 5 , 1 5 ) COMMON /MATP2/ P L ( I 5 , I 5 ) o P F ( I S , 1 5 ) COMMON /MATL1/ S ( 1 5 , 1 5 ) , P ( 1 5 , 1 5 ) , V 0 ( 1 5 , 1 5 ) COMMON / G R A D S / G A ( 1 5 , 1 5 ) , G K ( 1 5 , 1 5 ) DIMENSION AFF (NA, NA), E (NA, NA). D (NA, NA ). F3 (NA, NA) CALL MULT3(S,NA,NA,PL,NA,NA,GA,NA,NA,N,N,N,N,AFF,NA) DO 2 I = I , N DO 2 J = I , N F3(I,J)=0 CALL MST ( A F F , F 3 , N , NA) IF(IGB.EQ.2)

RETURN

DO 4 I = I , N DO 4 J = I , M D(I,J)=E(J,I) CALL MULT 3 (D, NA, NA, G K, NA, N A, PF, NA, NA, N, M, N, N, AFF, NA) CALL MST ( A F F , F 3 , N , NA) RETURN END

*** GRADIENT CALCULATION *** SUBROUTINE G R A D K ( E , D , A C , N, M,NL, N A , M A , L A , G )

365

IMPLICIT REAL*8 ( A - H , O - Y ) COMMON /MATSS! A ( 1 5 , 1 5 ) , B ( 1 5 , 5 ) , C ( 5 , 1 5 ) , A K ( 5 , 1 5 ) COMMON /MATP1/ Q ( 1 5 , 1 5 ) , R ( 5 , 5 ) COMMON /MATPZ/ P L ( 1 5 , 1 5 ) , P F ( 1 5 , 1 5 ) COMMON / M A T L I / S ( 1 5 , 1 5 ) , P ( 1 5 , 1 5 ) , V 0 ( 1 5 , 1 5 ) COMMON /MATL2/ P M U ( 1 5 , 1 5 ) , S L A ( 1 5 , 1 5 ) COMMON / G R A D S / G A ( 1 5 , 1 5 ) , G K ( 1 5 , 1 5 ) DIMENSION AC (NA, NA) ,E (NA, NA), G (NA, NA) ,D (NA, NA)

CALL MULT ( E , N A , NA,PMU,NA, NA, G, NA, NA, M, N,N) DO 2 I=I,M DO 2 J = I , N A C (I,J)=B(J,I) C A L L M U L T 3 (AC,NA, NA, SLA,NA,NA,P,NA, NA, M, N, N, N,D, NA) D O 4 I=l,M D O 4 J=l,N G(I,J)=G(I,J)+D(I,J) IF(IGB.EQ.2)

GO TO 6

CALL MULT3 (R,MA, MA, GK,NA, NA, PF, NA, NA, M, M, N, N , A C , NA) CALL M U L T ( A C , N A , N A , P , N A , N A , D , N A , N A , M, N,N) DO 5 I=I,M DO 5 J = I , N G(I,J)=G(I,J)+D(I,J) IF(NL.EQ.N) R E T U R N D O 8 I=I,N D O 8 J=I,NL AC(I,J)=C (I,I) CALL MULT(G,NA,NA,AC, NA,NA,D,NA,NA,M.N,NL) DO 10 I=I,M DO 10 J = I , N L 10

G(I,J)=D (I,J) RETURN END

SUBROUTINE MULT (A, N A , M A , B , N B , M B , C , NC, MC, N, M,L) ********************************* ******************* ****** * TWO REAL MATRICES MULTIPLICATION : C=A.B *

A (N,M),

N x( N A ,

M ~< MA

* *

366 *

B

(M,L),

M ,,< NB,

L 4 MB

*

*

C

(N,L),

N k< NC,

L x< MC

*

IMPLICIT

(A-H,O-Y)

REAL*8

DIMENSION

A(NA,NA),B(NB,NB),C(NC,NC)

D O I I=I,N D O I J=I,L C(I,J)=0.D0 DO

I K=I,M

C (I, J ) = C (1, J)+A (I, K)*B

(K, J)

RETURN END

*************************************************************** * THREE * THE

REAL

MATRICES

MATRICE

MULTIPLICATION

DIMENSION

ARE

: Q

= A.B.C

:

* *

*

A(NpM),

N,

M x< MA

*

*

B ( M , L ) , M,

L ~ MB

*

*

C(L,K),

*

QQ(N,K), N ~

SUBROUTINE

L x< NC, K ,,< MC

*

NQ, K < N O

*

MULT3(A,NA,MA,B,NB,MB,C,NC,MC,

N,M,L,K,QO,NQ)

IMPLICIT R E A L * 8 (A-H,O-Y) D I M E N S I O N A (NA, MA), B (NB,MB),

C (NC, MC),

D O 2 I=I,N D O 2 J=I,K S=0 D O I II=I,M D O 1 JJ=I,L S=S+A (I,11)*B (II,J J) *C (JJ. J) QQ(I,J)=S CONTINUE RETURN END SUBROUTINE PRINT (A, M, N, MMAX, IT)

QQ(NO_,

NQ)

367 ********************************************************* * REAL MATRIX *

A:MxN

WRITING

SUBROUTINE

*

* MMAX M a x i m a l d i m e n s i o n * A as specified * the

*

Mx<MMAX

calling

in the

of the

dimension

rows

of the

statement

program.

of

matrix

* * *

* IT=] FOR WRITING * ********************************************************* R E A L * 8 A ( M M A X , 1)

IF (IT.NE.I) G O T O 6 K=(N-I)/I0+i D O 3 KK=I,K NN=I0 D O 2 I=I,M IF (N.GT.10*KK) G O T O l NN=N-10* (KK-I) CONTINUE WRITE(6,5) (A (I, (KK-I)*I0+J) ,J=l ,NN) WRITE(6,4) FORMAT(/]/) F O R MAT ( 10 ( D 1 2 . 4 ) ) RETURN END

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MANGASARIAN

AUTHOR

INDEX

17, 21, 37, 38, 39,

ACKERMANN

296

ALBERT

296

40, 51, 53, 54, 61,

ALDERSON

212

63, 72, 73, 80, 82,

ALOS

296

84, 85, 92, 97, 102,

ANDERSON

39, 48, 50, 72, 76,

115, 150, 155, 175, 179,

87, 95, 98, 99,

180, 197, 235, 236, 260,

i01, 103, 135, 150,

265, 267, 2?0, 271, 2?2,

DAVISON

296, 297, 298, 299, 322

151, 155, 156, 157, 159, 160, 1?2, 175,

DESOER

323

247, 248, 249, 277,

DJOROVIC

289

305, 340

EVANS

21 110, 111, 115, 116, 117, 118, 119, 142, 228

AOKI

39, 236, 289

ARMENTANO

87, 182, 183, 197,

FADDEEV

55

236, 243, 245, 254,

FADDEEVA

55

280,

FERGUSON

97, 267, 296

ATHANS

234, 232, 346

FESSAS

49, 50, 236

BARTELS

281

FLETCHER

241

BARTON

21, 23, 26, 27, 30,

FOSSARD

31

GEROMEL

BOGLIVBOV

172

BRASCH

8, I0, 46, 172

GESING

2, 8, 10 233, 236, 237, 238, 239, 242, 245, 246, 253, 350 61, 267, 271, 272

BRISK

21, 23, 26, 27, 30,

HANAFUSA

110, 111, 112, 113, 150

31

HARARY

23

BURROWS

21

HASSAN

CALOVIC

289

CARTWRIGHT

23

HOSKINS

CHEN

242, 244, 245, 281,

HU

63, 67, 69, 70

285, 288

HUSEYIN

236

CHANG

260, 267, 270

IKEDA

236, 290, 291, 293

CLEMENTS

50, 76, 87, 95, 103,

ISAKSEN

289

135, 15o, 340

JAMSHIDI

20, 157, 236, 241

39, 41, 43, 44, 46,

JIANG

63, 67, 69, 70

48, 49, 50, 76, 77,

BERNUSSOU

233, 236, 237, 238, 239,

CORFMAT

235, 243

242, 245, 253, 380

87, 97, 112, 114, 115 170, 298, 305, 340

2?3, 275, 2??, 281, 282, 285, 288

JOHNSTON

21, 23, 26, 27, 30, 31

380 KAILATH

2, 5, 156

MORGAN

55

KALMAN

3, 4, 6

MORSE

39, 41, 43, 44, 46, 48,

KARCANIAS

321

KATTI

voir (KAT-81)

112, 114, 115, 170, 172,

KAUFMANN

189, 216

298, 305, 323, 340

KAWASAKI

248

MORARI

KERKOVIAN

216

MURTI

145, 212

KLEINMAN

247, 346, 348

NORMAN

23

KOBAYASHI

110, 111, 112, 113,

O'MALLEY

256

150

OZGUNER

80, 82, 84, 85, 92, 97,

49, 50, 76, 77, 87, 97,

23, 55

KOKOTOVIC

256

102, 155, 175, 297, 298,

KROFT

145, 212

299

KRUSER

21, Iii, 115, 117,

KWAKERNAAK LANCASTER

56, 60

PAYNE

289

LEVINE

234, 232, 346

PEARSON

8, I0, 14, 15, 18, Zl,

LI

236

LIN (C.T.)

14, 16, 17,23, 120

PERES

236, 245, 246

LIN (P.M.)

212

PETKOVSKI

247, 256, 257, 258

LINNEMANN

48, 122, 206

PETEL

78, 85

LIU

21

PICHAI

22, 52, 78, 122, 123,

LOCATELLI

138, 202, Z27, 296, 297, 308, 313

POTTER

39, 48, 172

321

POWELL

241

MACFARLANE

PALMAY

267

118, 119, 142, 228

PAPADIMITRIOU

127, 128, i36, 137

65, 248

PARASKEVOPOULOS

59

23, 46, 120, 172

125, 127, 206

242, 244, 245, 273

PURVIANCE

155, 158, 175

279, 280

PRESCOTT

21

MALINOWSKI

291

RAKIC

247

MAN

235

RAO

MAHMOUD

MASON

118, 142

RAV

MEEK

235, 243

REINSCHKE

MEERKOV

161, 162, 163, 164, 166, 169, 172, 174,

145, 212

267 142, 143, 144, 216, 217, 219, 222

ROSENBROCK

43, 45, 49, 55, 319, 320, 322, 323

175 MICHEL

22

ROY

MILLER

22

RUNOLFSSON

189 169, 172, 174, 175

MISRA

78, 85

SAHINKAYA

21

MITROPOLSKY

172

SANNUTI

MOMEN

110, III, 115, 116

SCATTOLINI

296, 297

MOORE

155, 156, 157, 158,

SCHIAVONI

138, 202, 227, 296, 297,

SCHULMAN

323

159, 160, 175, 247, 248, 249, 277, 288

308, 313

381 SEAKS

86

SENNING

199, 200, 202, 227, 228, 230

54, 55, 57, 58, 61, 131,

SERAJI

63, 69, 71

194, 197, 206, 212, 214,

SEZER

22, 51, 52, 78, 120, 122, 123, 125, 127,

233, 273, 274, 275, 277,

135, 170, 190, 206, 208, 215, 218, 219, 222, 236, 309

• TITLI

132, 161, 167, 175, 186, 215, 222, 223, 224, 225, 282

TRAVE

14, 15, 18, 21, 23, 120

SHIMEMURA

248

SILJAK

17, 21, 22, 23, 28, TSITSIKLIS 51, 52, 78, 120, 122, TSONIS TYLEE 123, 125, 127, 135, TZAFESTAS 180, 206, 236, 277, 289, 290, 291, 293, VAN TRESS VIDYASAGAR 296 14 VISWANADHAM WALTON 87, 182, 183, 197, WANG 236, 242, 242, 244,

SINGH

245, 254, 273, 274, 275, 277, 279, 280,

65, 248

SUNDARESHAN

277 STEPHANOPOULOS 23, 55 STEWART 281 TANG 22 TARANTANI 138, 202, 227, 308, 313 TARRAS 54, 55, 57, 58, 59, 60, 61, 131, 132, 161, 166, 167, 175, 186, 190, 194, 195, 197, 206, 209, 210, 211, 212, 214, 215, 247, 249, 250, 346 TAROKH 63, 64, 66, 69, 70, 75 THIRIEZ

189

190, 206, 212, 214, 215, 222, 223, 224, 225, 299 3O8 127, 128, 136, 137 59

155, 158, 175 59 159 63, 69, 70, 71, 74, 306 63, 69, 70, 71, 74, 306 235, 243 37, 38, 39, 40, 50, 51, 53, 61, 63, 72, 73, 115, 150, 162, 154, 175, 179, 180, 197, 236, 265, 267,

281, 282, 285, 288, 291 SIVAN

21, 22, 24, 25, 26, 31, 61, 161, 166, 167, 175,

SHIELDS

SILVERMAN

21, 22, 24, 25, 26, 31,

270, 271, 272, 303, 322 WILLEMS

2

WHTIE

290, 291, 293

WOLOVICH

49, 322, 323

WONHAM

8, 41, 49

XINOGALAS YAHAGI

273, 279, 280

YOSHIKAWA

110, 111, 112, 113, 150

250

ZHENG

95

ZOUTENDIJK

236

SUBJECT

cactus generalized

126, 127

input-

125

output-

125

INDEX

static-

64, 117, 228, 230. 231

time invariant

262

time v a r y i n g -

117, 155

cycle

16, 118, 122, 125, 129,

canonica] form

4. 6. 7

138, 139, 140, 148, 203,

chain condition number

125

213, 308, 309, 314

contraction

289, 290. 291, 295

158

control decentralized-

-family

142, 143, 144, 145, 212, 213, 214, 224

digraph

16, 17, 19, 20. 42, 118,

34, 35. 38, 40, 48.

122, 125, 126, 142, 148,

58 75. 78, 91. 98.

206, 211, 212, 216, 224,

I00. 103, 112, 116. 142. 151, 184, 194,

297. 298, 308, 313 dilation

236, 245, 255, 260, 274, 284, 289, 293, 296, 304 feasibly decen-

17, 19, 20, 21, 27, 30, 213

generalized-

22, 24, 26, 27. 28, 30

essential -input set

215, 220, 222, 223, 224

-output set

215, 220, 222, 223, 224

tra/ized

198

feedback-

111

expansion

289, 290, 291, 293, 294

optimal-

198, 278, 297

feedback control

38, 175, 177, 178, 206,

robust-

247, 255, 262, 264, 273, 313, 318

time-varyingvibrational-

218, 227, 230, 302, decentralized- 38, 44, iii, 114, 117,

117. 120, 156, 158,

151, 155, 181, 273, 297,

175

306

169, 175

gene~c rank

vibrational feedback-

15, 20, 21, 22, 23, 24, 26, 27, 28, 30, 31, 116,

169, 174, 175

118, 120, 127, 136, 214,

controllability

216. 222, 223, 224

-index

3

inclusion

structural-

14, 19, 22, 23, 24,

information

30, 31, 104. 110,

transfer

117, 119, 137, 220

link125, 178, 179, 186, 187,

controller dynamic-

33, 318 190, 191, 195, 198, 202,

41, 48, 51, 53, 64, 114, 117, 152, 230

robust-

289, 290

206, 210, 214, 221, 227 matrix

247, 264, 267, 270,

adjacency-

16, 216

277, 295

gradient-

233, 234, 236, 237, 238, 254, 346

383

reachability-

20

sensitivity-

57, 186, 187, 195, 197, 207

optimization -parametric-

236, 295

overlappin g

structural

-decomposition 291, 293

sensitivity-

131, 132, 333

structured-

14. 128

path

transfer

7, 61, 63, 64, 88,

pole4, 7, 10, 14, 63, 69, 91,

function-

91, 95, 98, 113. 135,

115, 116, 120, 138, 140,

137, 148, 320

148, 154, 177, 248, 288,

-subsystems

289 16, 208, 21fl

mode

308, 309

centralized

pole assignment

8, 10, 31, 34, 37, 41,

fixed-

39, 84, 86, 150

75, 86, 103, 117, 118,

decentralized

37, 39, 40, 50, 76,

142, 198, 228, 255, 269

fixed-

77, 78, 80, 81, 82, 84, 86, 87, 92, 155, 179, 249, 263, 298. 340

fixed-

decentralized- 39, 269 polynomial c h a r a c t e r i s t i c - 41, 63, 64, 65, 69, 89,

90, 118, 119 131, 142, 143, 144, 306, 320

37, 50, 51, 52. 53, 54, 57, 58. 59. 60.

decentralized

63, 64, 66. 67, 69,

fixed-

37, 38, 41

70, 72, 73, 75, 90,

fixed-

70, 89, 114, 181, 182, 306

138, 144, 151. 155, 177, 227, 297, 324,

invariant-

43, 45, 319, 320, 321

330, 333

-matrix

88, 94, 319

non s t r u c t u -

104, 105, i13, 115,

non s t r u c t u -

rally fixed-

150, ]77, 309

rally r o b u s t -

306, 307

structurally

31. 104, 105, 107,

remnant-

43, 44, 114

fixed-

113, 115, 117, 129,

teachability input-

20, 24, 208

137, 145, 155, 177,

-matrix

20, 195, 196, 197, 206,

output-

20, 24, 208, 225

190, 205, 214, 309 structurally

303. 304. 305. 308.

robust-

314, 318

model frequencydomain

120

120. 125. 127, 132,

208, 211 robustness

229

parametric-

229, 296

structural-

296

sample and hold

152, 153, 154

state-space-

34, 88 2, 142

sen sitivity

33O

time-domain-

34, 75

eigenvalue-

55

mode-

54, 186

observability -index

5

structural-

20, 104, 110, 137,

stability

2, 22

220

stabilization

8, 31, 34, 37, 39, 41,

structural-

130, 137, 333

48, 75, 177, 248, 304

384

decentralized-

229, 265

station local-

33 126

stem input-

125

input-output-

125

output-

125

strongly connected -components

210, 216, 220

-subsystems

43

structural -constraint

151

-perturbation

296, 297, 298, 299, 303, 306, 307, 308, 318

-robustness

303

subsystems complementary- 44, 76, 78, 79, 90, 114, 125, 136, 148, 157, normal-

68, 69

overlapping-

289

strongly connected-

30, 43, 44, 48, 112, 115

system complete-

43, 44

large scale-

24, 30, 31, 33, 34, 63, 137, 148, 219, 227, 247, 268, 296

structured-

14

quotient-

113, 115

decoupling-

320 320 320 63, 321 63, 65, 67, 69. 70. 71, 72, 73, 75. 78,

zero elementinvariantsystemtransmission-

82, 84, 139, 140, 148, 149, 299, 321, 323.