Large Scale Systems: Decentralization, Structure Constraints, and Fixed Modes

398 Pages·1989·9.78 MB·English

by  L. Trave, A. Titli, A. Tarras

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 -359 8 4
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© Springer-Verlag Berlin, Heidelberg 1989
Printed in Germany
Offsetprinting: Mercedes-Druck, Berlin
Binding: B, Helm, Bedin
2161/3020-543210
PREFACE
The growing dimensions and complexi ty of t he p r e s e n t day technologica l ,
env i ronmen t and societal p r o c e s s e s is one of the foremost cha l lenges to sys t em t h e o -
r y . Determining a solut ion for the problems a r i s ing in la rge scale s y s t e m s may
become e i t h e r v e r y uneconomical or even impossible if u s i n g the c lass ical mathematical
tools deve loped for sys t em ana lys i s and cont ro l . The main r e a s o n is t h a t c lassical
theor i e s are not bui l t for deal ing with h igh dimensional i ty models. Now, the e s sen t i a l
c h a r a c t e r i s t i c s of l a rge scale s y s t e m s a re a huge n u m b e r of i n p u t and o u t p u t v a r i a -
b les on s u b s y s t e m s which are gene ra l ly geograph ica l ly d i s t r i b u t e d .
These new f e a t u r e s involve l a rge and complex models , t h o u g h the modelling
problem may not be so lvable . Moreover , we must face economical and re l iabi l i ty
problems r e l a t e d to the informat ion t r a n s f e r be tween cont ro l s t a t i ons .
For t h e s e r e a s o n s , the decomposi t ion , a g g r e g a t i o n and model r e d u c t i o n t echn ic s
have r e c e i v e d cons ide rab le a t t en t ion in the las t t en y e a r s . A g rea t deal of theore t i ca l
and p rac t i ca l r e s u l t s conce rn ing t h e i r appl ica t ions have been obta ined in t he area of
s tab i l i ty and d e c e n t r a l i z e d con t ro l . In p a r t i c u l a r , t he problem of s tabi l iza t ion and
pole p lacement with decen t r a l i zed dynamic compensat ion is of g r e a t p rac t ica l i n t e r e s t .
Despi te the n u m e r o u s a d v a n c e s a r o u n d th is p rob lem, which a re mater ia l ized by a
large number of p a p e r s , t h e r e is none syn the t i ca l s u r v e y work exc lus ive ly c o n c e r n e d
with th is problem and the var ious o t h e r ones which a re r e l e v a n t .
The main objec t ive of th is 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
the p r e s e n t day r e s u l t s which can be u s e d fo r :
- t h e ana lys i s of s tabiHzabil i ty and pole p lacement u n d e r d e c e n t r a l i z e d c o n s -
traints,
- the de te rmina t ion of a cont ro l pol icy solv ing the problem of s tabi l iza t ion or
pole p lacement when decen t r a l i zed dynamic compensa t ion fails { p r e s e r v i n g a d e c e n -
t r a l i zed scheme of cont ro l o r minimizing the cos t a s soc ia t ed to the informat ion t r a n s -
f e r ) ,
- the d e s i g n of t h e s e p r e s p e c i f i e d contro l laws.
IV
By th i s way, th is book supp l i e s the tools for bu i ld ing a methodology which
b r i n g s a solut ion to the complete problem of control in the con tex t of l a rge scale
s y s t e m s . Moreover , t he l a s t p a r t of t he work t a k e s in to account pa r ame t r i c a n d
s t r u c t u r a l r o b u s t n e s s c o n s t r a i n t s .
C h a p t e r I p r e s e n t s an ove rv i ew of t he wel l -known r e s u l t s a round the problem
of s tabi l iza t ion and pole a s s ignmen t of l inear t lme - inva r i an t dynamic s y s t em s s u b -
j ec t ed to cen t r a l i z ed contro l (no s t r u c t u r a l c o n s t r a i n t s ) . The fundamenta l c o n c e p t s of
control labi l i ty and obse rvab i l i t y are i n t r o d u c e d and t h e y are e x t e n d e d to the con -
c e p t s of s t r u c t u r a l control labi l i ty and o b s e r v a b i l i t y , which are of major p rac t i ca l
i n t e r e s t in the s t u d y of l a rge 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 re e s t a b l i s h e d
from the sy s t em s t r u c t u r e and t h e y do no t d e p e n d on the p a r t i c u l a r conf igura t ion of
t he p a r a m e t e r s I va lues . In th i s f ramework , the problem 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 fo r the appl ica t ion 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 special ly adequa te for l a rge scale s y s t e m s .
The n e c e s s a r y and su f f i c i en t condi t ions fo r t he e x i s t e n c e of a solut ion to the
problem of s tab i l iza t ion and pole a s s ignmen t a re p r e s e n t e d for the following two
cases :
- cen t r a l i zed s t a t e f e edback
- cen t ra l i zed ou t pu t f eedback
They a re s t a t e d in te rms of t h e contro l labi l i ty and obse rvab i l i t y p r o p e r t i e s of
the s y s t e m .
It is c l ea r t h a t a good u n d e r s t a n d i n g of the concep t s of cont ro l labi l i ty and
obse rvab i l i t y is i n d i s p e n s a b l e be fo r e p r o c e e d i n g to the s t u d y of t he above problem
with s t r u c t u r a l c o n s t r a i n t s on the con t ro l . This problem is i n t r o d u c e d in C h a p t e r II ,
In the cont ro l of l a rge scale s y s t e m s whose e s sen t i a l cha r a c t e r i s t i c is t h e i r
h igh d imens ional i ty , convent iona l t e c h n i q u e s fail to give r easonab le so lu t ions wi th
r ea sonab le computat ional e f f o r t s . The classical cont ro l t h e o r y genera l ly s t a n d s on the
a s sumpt ion of a cen t ra l i zed informat ion p a t t e r n ; i . e . , all the informat ion on the
sys t em is available at a g iven c e n t e r , gene ra l ly a geograph ica l pos i t ion , w h e r e all
t he ca lcula t ions can be c a r r i ed ou t .
For most l a rge scale s y s t e m s , t h i s cen t ra l i za t ion assumpt ion does no t hold due
to the geograph ica l d i s t r i bu t i on of t he informat ion . This new c o n s t r a i n t l eads to
economical and re l iabi l i ty p rob lems r e l a t e d to the informat ion t r a n s f e r . This implies
t ha t t he cont ro l sys tem should be made of a number o5 local con t ro l l e r s t ha t a re only
allowed to use p a r t of t he whole informat ion in o r d e r to g e n e r a t e p a r t of the whole
V
contro l . In par t icular~ the des ign of f eedback con t ro 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 the
pa r t i cu 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 he con t ro l l e r can c o n n e c t .
When no t r a n s f e r of informat ion b e t w e e n the d i f f e r e n t local s t a t i ons is al lowed,
th is y ie lds to a decen t r a l i z ed scheme of con t ro l . When some b u t no t all t r a n s f e r s
( those of minimum cos t for example) a re allowed we obta in a n o n s t a n d a r d r e d u c e d
informat ion p a t t e r n .
I t is c l ea r t h a t the d e c e n t r a l i z e d control scheme is the most economically a d -
van tageous s ince no t r a n s f e r of informat ion for one geograph ica l location to a n o t h e r
is r e q u i r e d : sys tem i n p u t s a re a s s i g n e d to a g iven se t of local con t ro l l e r s ( s t a -
t i o n s ) , which o b s e r v e only local sys t em o u t p u t s . This is t he r ea son the f i r s t s t u d i e s
for the problem of s tabi l iza t ion and pole a s s ignmen t were i n v e s t i g a t e d wi th in a d e -
cen t ra l ized cont ro l scheme. The r e s u l t s r e f e r i n g to th i s s t u d y a re p r e s e n t e d in the
f i r s t p a r t of the c h a p t e r . These r e s u l t s were t h e n e x t e n d e d to the more genera 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 cont ro l which is c o n s i d e r e d in the s econd pa r t
of the c h a p t e r .
The main concep t to deal with th i s k ind of problem is the new not ion of f ixed
modes which is fundamenta l in the s t u d y of s tabi l iza t ion and pole p lacement 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 compensa t ion . I n d e e d t the ex i s t ence of a solut ion
d e p e n d s cr i t ical ly on the p r o p e r t i e s of th i s f ini te s e t of n u m b e r s . The p r e s e n c e of
uns t ab l e f ixed modes ind ica te s t h a t s tabi l iza t ion is imposs ible while t he p r e s e n c e of
any s o r t of f ixed modes r u l e s out a r b i t r a r y pole p lacement .
Due to t he theore t i ca l and p rac t i ca l impor tance of the not ion of f ixed modes ,
the whole C h a p t e r 3 is c o n c e r n e d with t he i r c h a r a c t e r i z a t i o n . The number 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 be found in the sc ient i f ic l i t e r a t u r e is 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 rms of i t s own a u t h o r s def in i t ions and
i n t r o d u c e d in a d i f f e r e n t way. With the main objec t ive of classifieation~ t h e y are
p r e s e n t e d in two g r o u p s , the t ime-domain and the f r e q u e n c y - d o m a i n c h a r a c t e r i z a -
t ions . A p a r t i c u l a r a t t en t ion is g iven to show the e x i s t i n g equ iva l ences . The ana lys i s
of e v e r y one of them allows us to point out t he condi t ions for t he e x i s t e n c e of f ixed
modes and to give a deap ins ide into t he i r i n t e r p r e t a t i o n re l a t ed 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 ixed modes a re out l ined : no s t r u c t u r a l l y f ixed modes which
need a quan t i t a t i ve analys is of t he sys t em and s t r u c t u r a l l y f ixed modes for which a
s t r u c t u r a l a p p r o a c h is more su i tab le ( r e p r e s e n t a t i o n of t he sys tem by a g r a p h , use
of genera l c o n c e p t s of g r a p h t h e o r y ) .
From the above ana lys i s , t he d i f f e r e n t r e s u l t s conce rn ing t h e problem of
decen t r a l i z ed s tabi l iza t ion and pole p lacement a re uni f ied and e x p r e s s e d in t e rms of
the d i f f e r e n t t y p e s of f ixed modes .
VI
W h e r e a s C h a p t e r 2 m a k e s c l e a r t h a t t h e c h o i c e a p r i o r i o f a f e e d b a c k c o n t r o l
s t r u c t u r e ( d e c e n t r a l i z e d f o r e x a m p l e ) c a n g e n e r a t e s o m e p r o b l e m s i f i t g i v e s r i s e to
t h e e x i s t e n c e o f f i x e d m o d e s , C h a p t e r 3 p r o v i d e s all t h e n e c e s s a r y t o o l s to a n a l y s e
a n d e x p l a i n t h e s i t u a t i o n .
A s a n a t u r a l c o n s e q u e n c e , t h e fo l l owing c h a p t e r s a r e c o n c e r n e d w i t h t h e
d i f f e r e n t m e t h o d s w h i c h a r e a v a i l a b l e to d e t e r m i n e a n a c c e p t a b l e c o n t r o l p o l i c y s u c h
t h a t s t a b i l i z a t i o n o r p o l e p l a c e m e n t i s p o s s i b l e .
I n t h e c o n t e x t o f l a r g e s c a l e s y s t e m s , t h e o b j e c t i v e i s to p r e s e r v e a d e c e n -
t r a l i z e d c o n t r o l s t r u c t u r e ( C h a p t e r 4) o r to d e t e r m i n e a n e w c o n t r o l s t r u c t u r e s u c h
t h a t t h e c o s t o f t h e i n f o r m a t i o n t r a n s f e r i s m i n i m u m ( C h a p t e r 5 ) .
C h a p t e r 4 p r e s e n t s a c l a s s o f d e c e n t r a l i z e d t i m e - v a r y i n g c o n t r o l l e r s w h i c h
al low to a v o i d f i x e d m o d e s p r o v i d e d t h a t t h e y a r e n o t f r o m a s t r u c t u r a l o r i g i n .
S e v e r a l k i n d s o f t i m e - v a r y i n g l a w s a r e e x a m i n e d a n d c o m p a r e d .
A n o r i g i n a l a p p r o a c h i s t h e n d e v e l o p e d : i t c o n s i s t s in u s i n g v i b r a t i o n a l c o n -
t r o l . V i b r a t i o n a l c o n t r o l c a n b e u s e f u l in t h e c a s e s w h e r e c o n v e n t i o n a l c o n t r o l m e -
t h o d s ( b a s e d on f e e d b a c k o r f e e d f o r w a r d p r i n c i p l e s ) do n o t a p p l y b e c a u s e o f l a c k
o f m e a s u r e m e n t s . I t i s s h o w n t h a t v i b r a t i o n a l c o n t r o l a s a s t a b i l i z a t i o n m e t h o d i s
c o m p a t i b l e w i t h t h e d e c e n t r a l i z a t i o n c o n s t r a i n t s a n d t h a t i t s o l v e s t h e p r o b l e m i n
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 . F r o m a n o t h e r p o i n t o f v i e w , a p a r t i c u l a r a p p l i c a -
t i o n o f v i b r a t i o n a l c o n t r o l i s p r e s e n t e d t h a t c o n s t i t u t e s a m e t h o d f o r t h e d e s i g n o f
d e c e n t r a l i z e d t i m e - v a r y i n g c o n t r o l l a w s .
When t h e i m p l e m e n t a t i o n o f t i m e - v a r y i n g o r n o n - l i n e a r c o n t r o l l a w s a p p e a r s to
b e too d i f f i c u l t o r w h e n we a r e in p r e s e n c e o f s t r u c t u r a l l y f i x e d m o d e s , we m u s t
c o n s i d e r t h e p r o b l e m o f d e t e r m i n i n g a n e w c o n t r o l s t r u c t u r e w h i c h m i n i m i z e s t h e c o s t
o f t h e i n f o r m a t i o n t r a n s f e r . T h i s p r o b l e m , w h i c h i s o f i m m e n s e p r a c t i c a l i n t e r e s t , i s
t h e s u b j e c t o f C h a p t e r 5.
I n f a c t , r e l a x i n g t h e s t r u c t u r a l c o n s t r a i n t s 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
c o n v e n i e n t w a y to s o l v e t h e s t a b i l i z a t i o n o r po le p l a c e m e n t p r o b l e m in p r e s e n c e o f
f i x e d m o d e s . T h e p u r p o s e o f C h a p t e r 5 i s to p r e s e n t t h e d i f f e r e n t a v a i l a b l e m e t h o d s
f o r t h e d e s i g n o f a n a p p r o p r i a t e f e e d b a c k c o n t r o l s t r u c t u r e . T h e s e m e t h o d s a r e
b a s e d o n t h e 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 o f f i x e d m o d e s w h i c h a r e p r e s e n t e d in
C h a p t e r 3 : o n e c a n b e m o r e appropriate t h a n a n o t h e r d e p e n d i n g o n t h e t y p e o f
s i t u a t i o n we a r e d e a l i n g w i t h . R o u g h l y s p e a k i n g , two t y p e s o f s i t u a t i o n s c a n o c c u r :
- t h e s y s t e m i s p h y s i c a l l y p a r t i t i o n e d i n s e v e r a l s t a t i o n s , d u e f o r e x a m p l e to
d i f f e r e n t g e o g r a p h i c a l l o c a t i o n s o f t h e i n p u t s a n d o u t p u t s . I n t h i s c a s e , i t i s c l e a r
Vll
tha t t he d e c e n t r a l i z e d s t r u c t u r e would be the mos t a p p r o p r i a t e . O ur goal i s t h u s to
de t e rmine t h e minimal i n fo rma t ion e x c h a n g e s b e t w e e n s t a t i o n s which g e t r i d of f ixed
modes . T h e op t imal i ty c r i t e r i o n can be c h o s e n as t h e n u m b e r of f e e d b a c k l inks
be tween two d i f f e r e n t s t a t i o n s o r a s t he cos t a s s o c i a t e d with t h e imp lemen ta t ion of
t h e s e f e e d b a c k H n k s ,
- e i t h e r t h e s y s t e m does n o t r e f l e c t a p r e s p e c i f i e d p a r t i t i o n i n g o r t h e s t a t i o n s
p r e s e n t t h e p a r t i c u l a r i t y t h a t t he cos t of local f e e d b a c k s is no t n e g l e c t a b l e with
r e s p e c t to t he cos 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 . In t h e s e c a s e s , we
wan t to d e t e r m i n e t h e minimal con t ro l s t r u c t u r e s ( if s e v e r a l ) for wh ich t h e s y s t e m
h a s no f ixed m o d e s . T h e y do n o t g e n e r a l l y i nvo l ve all t h e local f e e d b a c k s .
Note t h a t t he p rob lem r e s u l t i n g from the s e c o n d 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 he f i r s t p rob lem by s e t t i n g to zero t h e c o s t s a s s o -
d a t e d to t he local f e e d b a c k s . T h e r e f o r e , all t h e m e t h o d s wh ich a r e p r e s e n t e d in t h i s
g e n e r a l f r a mewo r k can also be u s e d for t he p a r t i c u l a r c a s e .
As a logical fol lowing to t h e de t e r mi n a t i o n of a d e q u a t e f e e d b a c k con t ro l s t r u c -
t u r e s , C h a p t e r 6 c o n s i d e r s the p rob lem of t h e s y n t h e s i s of f e e d b a c k g a i n s u n d e r
s t r u c t u r a l c o n s t r a i n t s . It p r o v i d e s an o v e r v i e w of a p p r o p r i a t e n e a r - o p t i m a l d e s i g n
t e c h n i q u e s . C o n s i d e r a t i o n s on t he r o b u s t n e s s of s u c h c o n t r o l l e r s a re 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 due to p a r a m e t e r v a r i a t i o n s or e x t e r n a l d i s t u r b a n c e s
a re c o n s i d e r e d . T he e f f e c t s of s t r u c t u r a l p e r t u r b a t i o n s ( fa i lu re of s e n s o r s or a c -
t u a t o r s , l ine c u t s . . . ) a re s t u d i e d l a t e r .
C h a p t e r 7, a n d t he l a s t one , a p p r o a c h e s i n d e e d t h e p rob lem of s t r u c t u r a l
r o b u s t n e s s . The c h a p t e r e x t e n d s t he r e s u l t s c o n c e r n i n g d e c e n t r a l i z e d o r s t r u c t u r a l l y
c o n s t r a i n e d con t ro 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 p e r t u r b a t i o n s , mamely
f a i l u re s of s e n s o r s o r a c t u a t o r s or c u t s of l ines imp lemen t ing f e e d b a c k - l o o p s . The
no t i o n s of s t r u c t u r a l l y r o b u s t con t ro l a n d s t r u c t u r a l l y r o b u s t modes a r e i n t r o d u c e d .
Severa l wa y s to c on c lu d e on t h e r o b u s t n e s s of a c o n t r o l , k n o wi n g i t s p r e s p e c i f l e d
s t r u c t u r e , a re p r e s e n t e d . T h e y a p p e a r as an e x t e n s i o n of we l l -known r e s u l t s d e r i v e d
from f ixed modes 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 a lgo r i t hm i s p r e s e n t e d
to d e t e r m i n e t h e i n fo rma t ion p a t t e r n of a r o b u s t r e g u l a t o r wi th minimum c o s t .
T h r o u g h an t he book, 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 -
p le s which make e a s i e r t h e i r u n d e r s t a n d i n g . Moreover , the a p p e n d i c e s con ta in a
col lect ion of p a c k a g e s c o r r e s p o n d i n g 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 in t h e
book ( e v a l u a t i o n of f i xed m o d e s , d e t e r m i n a t i o n of t h e i r t y p e , ca lcu la t ion of c o n s -
t r a i n e d s t r u c t u r e f e e d b a c k m a t r i c e s . . . ) .
Vlll
This book is the c o n s e q u e n c e of an i n t e n s i v e r e s e a r c h ac t iv i ty of s e v e r a l y e a r s
in t he a rea of ana lys i s and cont ro l of l a rge scale s y s t e m s and more p a r t i c u l a r l y in
decen t r a l i z ed con t ro l in t h e Lahora to i re d 'Automat ique et d 'Ana lyse des Syst~mes
(LAAS).
The 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 the col leagues of t h e i r
r e s e a r c h g roup and to the Di rec to r of the LAAS, P r o f e s s o r A. COSTES, for t h e i r
sc ient i f ic and f inancia l s u p p o r t .
The a u t h o r s s incere ly t h a n k Miss C. FABRE for t y p ing all t h e s e p a g e s and
Mr. E. LAPEYRE-MESTRE for the d r a w in gs of th is book ,
Toulouse, January 1989
Louise TRAVE
Andr~ TITLI
Ahmed TARRAS
TABLE OF CONTENTS
INTRODUCTION
CHAPTER 1. CENTRALIZED CONTROL : STABILIZATION AND POLE ASSIGNMENT
I.I. - Introduction
1.2. - Controllahility and observablllty
1.2. I. - Stability
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 and
B
observability
1.2.6. - Stabilization and pole assignment 8
10
1.2.7. - Origins of uncontrollable and unobservable modes
1.3. - Structural controllability and observability 14
I. 3. I. - Structural controllability 15
19
1.3.2. - General results on structural controllability and observability
20
I. 3.3. - Computational considerations
I. 4. - Conclusion 31
CHAPITER 2. STRUCTURALLY CONSTRAINED CONTROL • STABILIZATION AND
POLE ASSIGNMENT
2.1. - Introduction 33
2.2. - Decentralized structural constraints 34
2.2. I. - Problem formulation 35
37
2.2.2. - Decentralized fixed modes
2.2.3. - Decentralized stabilization and pole assignment 39
X
2.3. - Arbitrary structural constraints 50
2.4. - Evaluation of fixed modes 53
2 . 4 . 1 . - By comparing t h e s p e c t r a of t h e o p e n - l o o p a n d c l o s e d - l o o p d y n a m i c
matrix 53
2.4.2. - By calculation of the system modes sensitivity 54
2.4.3. - Concluding remarks 60
2. S. - Conclusion 61
CHAPITER 3. CHARACTERIZATION OF FIXED MODES
3. I. - Introduction 62
3.2. - Characterization in terms of transmission zeros 63
66
3.2.1. - Tarock's results
3.2.2. - Hu and Jiang results 67
70
B.2.3. - Vidyasagar and Wiswanadham results
71
3.2.4. - Seraji's results
3.2.5. - Davison and Wang results 72
3.2.6. - Comments 75
3.3. - Algebraic characterizations : time domain 75
3 . 3 . 1 . - M a t r i x r a n k t e s t c h a r a c t e r i z a t i o n 75
3 . 3 . 2 . - R e c u r s i v e c h a r a c t e r i z a t i o n 80
83
3.3.3. - Particular cases
3.3.4. - Comments 87
3.4. - A l g e b r a i c c h a r a c t e r i z a t i o n s : f r e q u e n c y d o m a i n 88
3 . 4 . 1 . - N e c e s s a r y c o n d i t i o n s on t h e t r a n s f e r f u n c t i o n m a t r i x fo r
88
t h e e x i s t e n c e of f i x e d m o d e s
3 . 4 . 2 , - T r a n s f e r f u n c t i o n m a t r i x c h a r a c t e r i z a t i o n fo r s y s t e m s w i t h
d i s t i n c t p o l e s 91
3 . 4 . 3 . - P o l y n o m i a l m a t r i x r a n k t e s t c h a r a c t e r i z a t i o n 94
3 , 4 . 4 . - G e n e r a l t r a n s f e r f u n c t i o n m a t r i x c h a r a c t e r i z a t i o n 95
3 . 4 . 5 . - I n t e r p r e t a t i o n 102
3 . 5 . - S t r u c t u r a l l y f i x e d m o d e s 104