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

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,

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 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
-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

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
~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 = /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)


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.