$H_ infty$ Control and Filtering of Two-Dimensional Systems

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

278

Springer Berlin Heidelberg NewYork Barcelona Hong Kong London Milan Paris Tokyo

Chunling Du and Lihua Xie

H∞ Control and Filtering of Two-dimensional Systems With 32 Figures

13

Series Advisory Board

A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis

Author Dr. Chunling Du Data Storage Institute Engineering Nr. 1 Singapore 117608 Singapore

Prof. Lihua Xie School of Electrical and Electronic Technological University Nanyang Avenue Singapore 639798 Singapore

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek – CIP-Einheitsaufnahme Du, Chunling: H∞ control and filtering of two-dimensional systems / Chunling Du and Lihua Xie. Berlin ; Heidelberg ; NewYork ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in control and information sciences ; 278) (Engineering online library) ISBN 3-540-43329-5

ISBN 3-540-43329-5

Springer-Verlag Berlin Heidelberg New York

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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Digital data supplied by author. Data-conversion by PTP-Berlin, Stefan Sossna e.K. Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 10869692 62/3020Rw - 5 4 3 2 1 0

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ýs-$8Z8 R$tüZjs¹ %sjZ¸ BP þY¸ þ¹stRP¸¹ PZt{þ$Bt 8sþ¹$- BP þY¸

ÿx

qY¸ R$ütsj

hjþ¸¹$tü ¸¹¹B¹ RkRþ¸8ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ

?d w7, i D BP ¸Rþ$8sþ¸ ? |d w7, i D

ûn

û;

e-s8}j¸ ÿ1ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ

ûx

BP e-s8}j¸ ÿ1ÿ

ÿÿ ÿ ÿÿ ÿÿ ÿÿ ÿÿ ÿÿ ÿÿ ÿÿ ÿÿ ÿÿ ÿÿ ÿÿ

ûx

hjþ¸¹ BP e-s8}j¸ ÿ1ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ

ûE

ÿE

qY¸

ÿ

qY¸ ý7e BP þY¸ 1T*

ÿM

ýs-$8Z8 R$tüZjs¹ %sjZ¸R }jBþ BP þY¸ þ¹stRP¸¹ PZt{þ$Bt 8sþ¹$- BP

ÿû

ýs-$8Z8 R$tüZjs¹ %sjZ¸R }jBþ BP þY¸ þ¹stRP¸¹ PZt{þ$Bt 8sþ¹$- BP

þ

I

þY¸ hjþ¸¹$tü ¸¹¹B¹ RkRþ¸8ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ

þY¸ hjþ¸¹$tü ¸¹¹B¹ RkRþ¸8ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ

û

ûM

Mÿd

qY¸ $t}Zþ $8sü¸ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ddd

Mÿ1

qY¸ $8sü¸ A¸PB¹¸ I¸{Bt%BjZþ$Bt }¹B{¸RR$tü bY¸t þY¸ 8¸sRZ¹¸8¸tþ

Mÿn

qY¸ $8sü¸ sPþ¸¹ I¸{Bt%BjZþ$Bt }¹B{¸RR$tü bY¸t þY¸ 8¸sRZ¹¸8¸tþ

Mÿ;

qY¸ $8sü¸ sPþ¸¹ ¹BAZRþ I¸{Bt%BjZþ$Bt hjþ¸¹$tü bY¸t þY¸ 8¸sRZ¹¸T

7Wi $R dEÿEIúÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dd1

7Wi $R dEÿEIúÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dd1

8¸tþ 7Wi $R dEÿE Iúÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dd;

XII

List of Figures

ûÿd

V8}j$þZI¸ ¹¸R}BtR¸ PB¹ B¹$ü$tsj w1_, 18D hjþ¸¹ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ d1E

ûÿ1

V8}j$þZI¸ ¹¸R}BtR¸ BP I

ûÿn

V8}j$þZI¸ ¹¸R}BtR¸ BP þY¸ RkRþ¸8 t

ûÿ;

V8}j$þZI¸ ¹¸R}BtR¸ PB¹ B¹$ü$tsj w;_, M8D hjþ¸¹ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dnd

ûÿx

V8}j$þZI¸ ¹¸R}BtR¸ BP I

ûÿE

V8}j$þZI¸ ¹¸R}BtR¸ BP þY¸ RkRþ¸8 t

ûÿ

V8}j$þZI¸ ¹¸R}BtR¸ PB¹ B¹$ü$tsj wd_, ;8D hjþ¸¹ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dn;

ÿ

ÿ

ÿ

w1_, d8D ¹¸IZ{¸ITB¹I¸¹ hjþ¸¹ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ d1

ÿ

| tÿ

ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ d1

w;_, ;8D ¹¸IZ{¸ITB¹I¸¹ hjþ¸¹ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dnd

ÿ

| tÿ

ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dn1

wd_, 18D ¹¸IZ{¸ITB¹I¸¹ hjþ¸¹ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dn;

ûÿM

V8}j$þZI¸ ¹¸R}BtR¸ BP I

ûÿû

V8}j$þZI¸ ¹¸R}BtR¸ BP þY¸ RkRþ¸8 t

ÿ

| tÿ

ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ dnx

7k8ABjR stI V{„Btk8R

1T*U

þbBTI$8¸tR$Btsjÿ

dT*U

Bt¸TI$8¸tR$Btsjÿ

NýyU

j$t¸s¹ 8sþ¹$- $t¸JZsj$þkÿ

ViyU

sjü¸A¹s${ i${{sþ$ $t¸JZsj$þkÿ

i*yU

i${{sþ$ I$&¸¹¸t{¸ $t¸JZsj$þkÿ

i*eU

i${{sþ$ I$&¸¹¸t{¸ ¸JZsþ$Btÿ

©U

R¸þ BP $tþ¸¹ü¸¹ tZ8A¸¹Rÿ

7, i U

$tþ¸ü¸¹T%sjZ¸I %¸¹þ${sj stI YB¹$9Btþsj {BB¹I$tsþ¸Rÿ

oz U

zTI$8¸tR$Btsj ¹¸sj eZ{j$I¸st R}s{¸ÿ

ozýE U

R¸þ BP z

z

{ U

z

\

I$sü \d , \1 ,

N N N , \z iU

þ E ¹¸sj 8sþ¹${¸Rÿ

þ z $I¸tþ$þk 8sþ¹$-ÿ

i

AjB{, I$süBtsj 8sþ¹$- b$þY \ w tBþ t¸{¸RRs¹$jk RJZs¹¸D Bt þY¸ I$süBtsjÿ

¼

x

¼

d 1U

U

b

þ¹stR}BR¸ BP 8sþ¹$- ¼ ÿ

s Rk88¸þ¹${ RJZs¹¸ ¹BBþ BP s 8sþ¹$- ¼ ) ¼ $ÿ¸ÿù

h

ý (U

h k (U

b

b

d 1¼ d 1 ¼

xý

(ù

) ¼ÿ

Rk88¸þ¹${ }BR$þ$%¸ R¸8$I¸ht$þ¸ 8sþ¹$- h

Rk88¸þ¹${ }BR$þ$%¸ I¸ht$þ¸ 8sþ¹$- h

; ozýz ÿ

; ozýz ÿ

XIV

Symbols and Acronyms

ý
h

ÿ<ý(

PB¹ Rk88¸þ¹${ h, <

; ozýz ÿ

h k
h

ÿ
PB¹ Rk88¸þ¹${ h, <

; ozýz ÿ

þ8s- w¼ D U

8s-$8Z8 R$tüZjs¹ %sjZ¸ BP 8sþ¹$- ¼ ÿ

h

· ý¹w¼ D B¹

[ w¼ DU

ow¼ DU ¼

üU

tZjj R}s{¸ BP j$t¸s¹ B}¸¹sþB¹ ¼ ÿ

¹stü¸ R}s{¸ BP j$t¸s¹ B}¸¹sþB¹ ¼ ÿ

8sþ¹$- Rsþ$RPk$tü stI ¼

, N ,U ,j ,1

\

[ w¼ ü D ) ow¼ D

k (ÿ

eZ{j$I¸st %¸{þB¹ tB¹8ÿ

U

üD, >(, üDiU

7)( i )(

\jw7, i Diù

,jw7, i D,1 3

R}s{¸ BP RJZs¹¸ RZ88sAj¸ R¸JZ¸t{¸R Bt

F7 w7 ) d, 1DU

ÿ þþ þþ

f1 TtB¹8 BP s 1T* I$R{¹¸þ¸Tþ$8¸ R$ütsj $ÿ¸ÿù

f1 >(,

ü ¼ üx

\>(, üD, >(, üDi

b$þY %sjZ¸R Bt

1T* FTþ¹stRPB¹8 %s¹$sAj¸Rÿ

oz ÿ

2$ 87#<+,09#'+7

2$2 &(;<(';1

!%¸¹ ÿY¸ }sRÿ R¸%¸¹sj I¸{sI¸Rþ s {BtR$I¸¹sAj¸ $tÿ¸¹¸Rÿ YsR A¸¸t I¸%Bÿ¸I ÿB }¹BAj¸8R $t%Bj%$tý R$ýtsjR stI RkRÿ¸8R ÿYsÿ I¸}¸tI Bt 8B¹¸ ÿYst Bt¸ %s¹$T sAj¸ü 1T* R$ýtsjR stI RkRÿ¸8R Ys%¸ A¸¸t RÿZI$¸I $t ¹¸jsÿ$Bt ÿB R¸%¸¹sj 8BI¸¹t ¸tý$t¸¸¹$tý h¸jIR RZ{Y sR }¹B{¸RR {Btÿ¹Bjþ 8Zjÿ$TI$8¸tR$Btsj w8T*D I$ý$ÿsj hjT ÿ¸¹$týþ $8sý¸ ¸tYst{¸8¸tÿþ $8sý¸ I¸AjZ¹¹$týþ R$ýtsj }¹B{¸RR$týþ ¸ÿ{ü V8Btý ÿY¸ 8sÁB¹ ¹¸RZjÿR I¸%¸jB}¸I RB Ps¹ {Bt{¸¹t$tý ÿY¸ 1T* R$ýtsjR stI RkRÿ¸8Rþ 1T * I$ý$ÿsj hjÿ¸¹R s¹¸ $t%¸Rÿ$ýsÿ¸I sR s I¸R{¹$}ÿ$Bt $t P¹¸JZ¸t{k IB8s$t B¹ sR s {Bt%BjZÿ$Bt BP ÿY¸ $t}Zÿ stI ÿY¸ Zt$ÿ $8}ZjR¸ ¹¸R}BtR¸ü yÿR ý¹¸sÿ }Bÿ¸tÿ$sj PB¹ }¹s{ÿ${sj s}}j${sÿ$BtR $t ÿY¸ 1T* $8sý¸ stI R$ýtsj }¹B{¸RR$tý YsR A¸¸t RYBbt >E;H>ExHü !t ÿY¸ BÿY¸¹ YstIþ s ÿ¸{Yt${sjjk %¸¹k $8}B¹ÿstÿ ¹stý¸ BP 1T* }¹BAT j¸8R ¸-$Rÿ bY${Y ¹¸JZ$¹¸ P¸¸IAs{, {Btÿ¹Bj >;;Hü 1T* {Btÿ¹Bj YsR }¹¸%$BZRjk A¸¸t s}}¹Bs{Y¸I P¹B8 s }¹¸IB8$tstÿjk RkRÿ¸8R ÿY¸B¹¸ÿ${sj }B$tÿ BP %$¸bü qY$R YsR ÿbB 8s$t A¹st{Y¸Rþ R¸¸t $t >E(H>1Hþ ÿs,$tý st $t}ZÿTBZÿ}Zÿ ÿ¹stRP¸¹ PZt{ÿ$Bt s}}¹Bs{Y stI s Rÿsÿ¸TR}s{¸ s}}¹Bs{Yþ ¹¸R}¸{ÿ$%¸jkü 1T* Rÿsÿ¸TR}s{¸ 8BI¸jRþ YBb¸%¸¹þ Ys%¸ sÿÿ¹s{ÿ¸I s jBÿ BP $tÿ¸¹¸Rÿ IZ¸ ÿB $ÿR sI%stÿsý¸ BP }¹BT %$I$tý s R$8}j¸ stI $tÿZ$ÿ$%¸ ¹¸R¸s¹{Y 8¸ÿYBI PB¹ 1T* R$ýtsjR stI RkRÿ¸8Rü VjÿYBZýY $ÿ s}}¸s¹R RB8¸ÿY$tý j$,¸ ÿY¸ Bt¸TI$8¸tR$Btsj wdT*D Rÿsÿ¸ R}s{¸ 8BI¸jþ ÿY¸¹¸ ¸-$Rÿ RB8¸ ¸RR¸tÿ$sj I$&¸¹¸t{¸R A¸ÿb¸¸t ÿY¸8ü qY¸¹¸PB¹¸þ 1T* Rÿsÿ¸TR}s{¸ ¹¸}¹¸R¸tÿsÿ$BtR Ys%¸ A¸¸t ¸-ÿ¸tR$%¸jk RÿZI$¸I ÿY¸B¹¸ÿ${sjjk stI }¹s{ÿ${sjjkü 1T* Rÿsÿ¸TR}s{¸ ÿY¸B¹k B¹$ý$tsÿ¸I P¹B8 iB¸RR¸¹ >nxH>EH bYB }¹BIZ{¸I ÿY¸ {¸j¸A¹sÿ¸I iBRR¸¹ 8BI¸j bY¸t Y¸ RÿZI$¸I ÿY¸ 8Zjÿ$TI$8¸tR$Btsj j$t¸s¹ $ÿT ¸¹sÿ$%¸ {$¹{Z$ÿR sÿ ÿY¸ A¸ý$tt$tý BP ÿY¸ R¸%¸tÿ$¸Rü 7$t{¸ ÿY¸tþ R¸%¸¹sj BÿY¸¹ R{YBjs¹R RZ{Y sR VÿÿsR$ >xHþ 3B¹tsR$t$ stI ûs¹{Y¸R$t$ >1MH stI ·Z¹¸, >xMH Ys%¸ I¹sbt ÿY¸ VÿÿsR$ 8BI¸jþ ÿY¸ 3û N77 w3B¹tsR$t$Tûs¹{Y¸R$t$ jB{sj Rÿsÿ¸T R}s{¸D 8BI¸j stI ÿY¸ p¸t¸¹sj 8BI¸jþ ¹¸R}¸{ÿ$%¸jk P¹B8 ÿY¸$¹ Bbt I$&¸¹¸tÿ ¹¸R¸s¹{Y h¸jIRü qY¸ ¹¸jsÿ$BtRY$}R A¸ÿb¸¸t ÿY¸R¸ 8BI¸jR s¹¸ I¸ÿs$j¸I $t >xdHü úsR¸I Bt ÿY¸R¸ Rÿsÿ¸TR}s{¸ I¸R{¹$}ÿ$BtRþ R¸%¸¹sj }¹B}¸¹ÿ$¸R BP 1T* RkRÿ¸8R RZ{Y sR {Btÿ¹BjjsA$j$ÿkþ BAR¸¹%sA$j$ÿkwR¸¸þ PB¹ ¸-s8}j¸þ >1;Hþ >xùH stI ÿY¸ ¹¸P¸¹T ¸t{¸R ÿY¸¹¸$tDþ RÿsA$j$ÿk >1H>nHþ ¹¸sj$9sÿ$Bt >1xHþ ¸ÿ{þ Ys%¸ A¸¸t $t%¸Rÿ$ýsÿ¸Iü ûB¹¸B%¸¹þ 1T* P¸¸IAs{, I¸R$ýt ÿ¸{Yt$JZ¸R RZ{Y sR ¸$ý¸t%sjZ¸ sRR$ýt8¸tÿ >x1Hþ RÿsA$j$9sÿ$Bt >;EH stI BAR¸¹%¸¹R >MH>ùEH Ys%¸ ¹¸{¸$%¸I {BtR$I¸¹sAj¸ sÿÿ¸tÿ$Btü Ch. Du, L. Xie:

Control and Filtering of Two-dimensional Systems, LNCIS 278, pp. 1−4, 2002.

1

dÿ ytþ¹BIZ{þ$Bt

yÿ $R ,tBbt ÿYsÿ ÿY¸ Iÿ {Btÿ¹Bj PB¹ dT* RkRÿ¸8R YsR A¸¸t s 8BRÿ s{T ÿ$%¸ s¹¸s BP {Btÿ¹Bj RkRÿ¸8R PB¹ ÿY¸ jsRÿ ÿbB I¸{sI¸R >nEH>ndH>;H>d(dHþ V 8s$t sI%stÿsý¸ BP Iÿ {Btÿ¹Bj $R ÿYsÿ $ÿR }¸¹PB¹8st{¸ R}¸{$h{sÿ$Bt ÿs,¸R $tÿB s{T {BZtÿ ÿY¸ bB¹Rÿ {sR¸ }¸¹PB¹8st{¸ BP ÿY¸ RkRÿ¸8 $t ÿ¸¹8R BP ÿY¸ RkRÿ¸8 ¸t¸¹ýk ýs$tþ qY$R $R 8B¹¸ s}}¹B}¹$sÿ¸ PB¹ RkRÿ¸8 ¹BAZRÿt¸RR stsjkR$R stI ¹BAZRÿ {BtT ÿ¹Bj ZtI¸¹ 8BI¸j$tý Zt{¸¹ÿs$tÿ$¸R stI I$RÿZ¹Ast{¸R ÿYst BÿY¸¹ }¸¹PB¹8st{¸ R}¸{$h{sÿ$BtR >1nHü RZ{Y sR ÿY¸ N+TB}ÿ$8sj {Btÿ¹Bj R}¸{$h{sÿ$Btþ qYZR ÿY¸ I¸%¸jB}8¸tÿ BP ÿY¸ Iÿ {Btÿ¹Bj ÿY¸B¹k YsR ý¹¸sÿjk sI%st{¸I ÿY¸ 8¸ÿYBIR PB¹ stsjkR$R stI I¸R$ýt BP ¹BAZRÿ dT* {Btÿ¹Bj RkRÿ¸8Rþ E1H>(Hü $ÿ $R $8}B¹ÿstÿ ÿYsÿ ÿY¸ }¹BAj¸8 BP Iÿ {Btÿ¹Bj BP 1T* RkRT ÿ¸8R A¸ RÿZI$¸I stI ¹¸RBj%¸I stI PZ¹ÿY¸¹ s}}j$¸I ÿB ¹BAZRÿ RÿsA$j$9sÿ$Bt BP 1T* RkRÿ¸8Rþ qY¸ ¸Rÿ$8sÿ$Bt BP ÿY¸ Rÿsÿ¸ %s¹$sAj¸R BP 1T* Ikts8${ RkRÿ¸8R ZR$tý s%s$jT sAj¸ tB$Rk 8¸sRZ¹¸8¸tÿR $R s PZtIs8¸tÿsj }¹BAj¸8 $t R$ýtsj }¹B{¸RR$týü $8T sý¸ }¹B{¸RR$tý stI {Btÿ¹Bjþ !%¸¹ ÿY¸ }sRÿ I¸{sI¸R {BtR$I¸¹sAj¸ $tÿ¸¹¸Rÿ YsR A¸¸t I¸%Bÿ¸I ÿB ÿY¸ Rÿsÿ¸ ¸Rÿ$8sÿ$Bt 8¸ÿYBI AsR¸I Bt ÿY¸ 8$t$8$9sÿ$Bt BP ÿY¸ %s¹$st{¸ BP ÿY¸ ¸Rÿ$8sÿ$Bt ¸¹¹B¹ü $þ¸þü ÿY¸ {¸j¸A¹sÿ¸I ·sj8st hjÿ¸¹$tý s}T }¹Bs{Y >;nH>xnH>xH>M;H>MxHþ WBÿ¸ ÿYsÿ ÿY¸ ·sj8st hjÿ¸¹$tý sRRZ8¸R ÿY¸ ,tBbjT ¸Iý¸ BP s }¸¹P¸{ÿ Ikts8${ 8BI¸j BP ÿY¸ RkRÿ¸8 ZtI¸¹ $t%¸Rÿ$ýsÿ$Bt stI ÿY¸ bY$ÿ¸\{BjB¹¸I tB$R¸R b$ÿY ,tBbt R}¸{ÿ¹sj I¸tR$ÿkþ yÿ YsR A¸¸t ,tBbt >M(H ÿYsÿ ÿY¸ ·sj8st hjÿ¸¹ ÿk}¸ BP ¸Rÿ$8sÿB¹R 8sk tBÿ A¸ ¹BAZRÿ sýs$tRÿ Zt{¸¹ÿs$tÿk $t ÿY¸ R$ýtsj 8BI¸jþ qY¸ t¸¸I ÿB YstIj¸ Zt{¸¹ÿs$tÿk $t hjÿ¸¹$tý }¹BAj¸8R YsR 8Bÿ$%sÿ¸I ÿY¸ ZR¸ BP t¸b 8¸sRZ¹¸ BP }¸¹PB¹8st{¸ü st Z}}¸¹ ABZtI Bt ÿY¸ 8s-$8Z8 ¸t¸¹ýk ýs$t A¸ÿb¸¸t ÿY¸ hjÿ¸¹$tý ¸¹¹B¹ stI ÿY¸ tB$R¸ $t}Zÿþ qY$R hjÿ¸¹$tý s}}¹Bs{Y $R ts8¸I sR ÿY¸ Iÿ hjÿ¸¹$tý stI $R %¸¹k s}}¹B}¹$sÿ¸ $t s tZ8A¸¹ BP }¹s{ÿ${sj R$ÿZsÿ$BtRþ qY¸ Iÿ hjÿ¸¹$tý PB¹ dT* RkRÿ¸8R YsR A¸¸t ¸-ÿ¸tR$%¸jk RÿZI$¸I $t ÿY¸ }sRÿ I¸{sI¸û R¸¸ü PB¹ ¸-s8}j¸ü >M(Hü >MMHü >úH stI ÿY¸ ¹¸P¸¹¸t{¸R ÿY¸¹¸$tþ GY$j¸ $t ÿY¸ 1T* {sR¸ü ÿB ÿY¸ A¸Rÿ BP BZ¹ ,tBbj¸Iý¸ü j$ÿÿj¸ bB¹, YsR A¸¸t IBt¸þ yt sII$ÿ$Btü I¸{Bt%BjZÿ$Bt YsR 8stk s}}j${sÿ$Bt s¹¸sRü PB¹ ¸-s8}j¸ü Isÿs ÿ¹stR8$RR$Btü ¸JZsj$9sÿ$Btü ¹¸%¸¹A¸¹sÿ$Bt {st{¸jsÿ$Btü R¸$R8${ I¸{Bt%BjZÿ$Btü $8sý¸ ¹¸RÿB¹sÿ$Btü R}¸¸{Y R$ýtsj }¹B{¸RR$týü ¸þýþ >ddH>d1H>dnH>EMHþ qY¸ ÿ¸{YT t$JZ¸R BP 1T* $t%¸¹R¸ hjÿ¸¹$tý b¸¹¸ b$I¸jk s}}j$¸I ÿB I$ý$ÿsj $8sý¸ ¹¸RÿB¹sT ÿ$Bt P¹B8 ÿY¸ 8$ITdúE(Rü R¸¸ü PB¹ ¸-s8}j¸ü >úH>;1H>nH>MnH stI ¹¸P¸¹¸t{¸R {$ÿ¸I ÿY¸¹¸$tþ qY¸ }¹BAj¸8 BP 8BI¸j ¹¸IZ{ÿ$Bt PB¹ 1T* RkRÿ¸8R YsR ¹¸{¸$%¸I {BtR$IT ¸¹sAj¸ sÿÿ¸tÿ$Bt IZ¸ ÿB $ÿR $8}B¹ÿst{¸ $t 1T* R$ýtsj stI $8sý¸ }¹B{¸RR$tý s}}j${sÿ$BtR >x;H>ExH>úúH>d((Hþ i¸{¸tÿjkü ÿY¸ Iÿ 8BI¸j ¹¸IZ{ÿ$Bt 8¸ÿYBI YsR sÿÿ¹s{ÿ¸I s jBÿ BP $tÿ¸¹¸Rÿ PB¹ dT* RkRÿ¸8Rû R¸¸ >nMH>xxH>xEH stI ÿY¸ ¹¸P¸¹¸t{¸R {$ÿ¸I ÿY¸¹¸$tþ >nMH stI >;xH ýs%¸ st Nùy s}}¹Bs{Y ÿB ÿY¸ Iÿ 8BI¸j ¹¸IZ{ÿ$Bt ZR$tý ÿY¸ ABZtI¸I ¹¸sj j¸88s BP dT* RkRÿ¸8Rþ qY¸ s}}¹Bs{Y $t%Bj%¸R RBj%$tý

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1ÿ þBZtI¸I i¸sj N¸88s BP 1T* *$R{¹¸ý¸ 7kRý¸8R

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$R ÿY¸ $t}Zÿý qY¸ ABZtIs¹k {BtI$ÿ$Bt $R I¸ht¸I sR

PBjjBbRU ¼ w(D ) >?

_x

_x

w(, (D ?

_x

w(, dD ?

NNN

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8x

ü ?

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stk ABZtIs¹k {BtI$ÿ$Bt RZ{Y ÿYsÿ RZ} i

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\d ?w7, i L dD L \1 ?w7 L d, i D

)

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þ j

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_

w(D

)

8

¼ w(D

)

\,

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x

>?

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>?

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G¸ 8s,¸ ÿY¸ PBjjBb$tú I¸ht$ÿ$BtRý

x

?

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, (DH

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ý

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¼¹

j w7, i D

) ( ZtI¸¹ ÿY¸ 9¸¹B $t}Zÿ

RZ{Y ÿYsÿ

¼( c ÿþ

#

( stI stk ABZtIs¹k {BtI$ÿ$BtR

3 zþz Rk88¸ÿ¹${ 8sÿ¹$- h k (ý }BR$ÿ$%¸ R{sjs¹R ÿ stI # Rsÿ$RPk$tü ÿL# ) d

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RZ{Y ÿYsÿ

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-, U, < BP s}}¹B}¹$sÿ¸ I$8¸tT < s¹¸ Rk88¸ÿ¹${ stI < k (ý ÿY¸t - L Ux
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d(

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,¼ w(D,1

c

ÿþ

ûBÿ$%sÿ¸I Ak ÿY¸ tBÿ$Bt BP I

ÿ

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tB$R¸ sÿÿ¸tZsÿ$Bt $t dT* RkRÿ¸8R >úHý b¸

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x

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x p 1

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Ch. Du, L. Xie:

Control and Filtering of Two-dimensional Systems, LNCIS 278, pp. 43−53, 2002.

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]Btþ¹Bj BP 1T* *$R{¹¸þ¸ 7kRþ¸8R $t 3ý N77 ýBI¸j

ytÿ¹BIZ{¸ ÿY¸ PBjjBb$tþ BZÿ}Zÿ P¸¸IAs{, {Btÿ¹Bjj¸¹ BP B¹I¸¹ z U ? w7 L d, i L dD

Áw7, i D bY¸¹¸ ?

;

oz ý

\d ? w7, i L dD L \1 ? w7 L d, i D

)

)

L4d rw7, i L dD L 41 rw7 L d, i D,

w;ýnD

g ? w7, i D L ; rw7, i D

w;ý;D

qY¸tü ÿY¸ {jBR¸ITjBB} RkRÿ¸8 BP w;ýdDTw;ý1D ÿBþ¸ÿY¸¹ b$ÿY

ÿY¸ {Btÿ¹Bjj¸¹ w;ýnDTw;ý;D $R þ$%¸t Ak ¦ ?w7, ¦ ¦ ?w7 ¦ L d, i L dD ) \ L d, i D d ¦ i L dD L \1 ?w7 bY¸¹¸

}

?w7, ¦ iD )

?w7, i D

}

] ¦ ) \ >

,

? w7, iD

\> L 4> ; g

4> g 

4> g

\>

w;ýxD

] ,

> ) d, 13

w;ýED

qY¸ }¹BAj¸8 ÿB A¸ sII¹¸RR¸I $R Rÿsÿ¸I sR PBjjBbRU p$%¸t ÿY¸ 1T* RkRÿ¸8

w;þdDTw;þ1Dý I¸R$üt s Ikts8${ BZÿ}Zÿ P¸¸IAs{, {Btÿ¹Bjj¸¹ BP ÿY¸ PB¹8 $t w;þnDT w;þ;D RZ{Y ÿYsÿ ÿY¸ ¹¸RZjÿ$tü {jBR¸ITjBB} RkRÿ¸8 w;þxD $R sRk8}ÿBÿ${sjjk RÿsAj¸þ *¸ht¸ ¦ H, \ 1

¦ \

)

¦ >\ d

.

)

; g p L g  ÿ

J>

)

C w\> L 41> ; g Dp L C 41> g ÿ

x,

~> ) C 41> ; L þ4> ,

x

w;ýD

L þ4> g p L þ\> ÿ

x,>

) d, 1 w;ýMD

x

bY¸¹¸ p k (ü C k ( stI ÿ stI þ s¹¸ $t%¸¹ÿ$Aj¸ 8sÿ¹${¸R Rsÿ$RPk$tþ ÿ þ {

ÿ pCý

qY¸B¹¸8 ;ÿ1ÿdÿ

)

]BtR$I¸¹ ÿY¸ 1T* RkRÿ¸8 w;þdDTw;þ1Dþ qY¸tý ÿY¸¹¸ ¸-$RÿR s

PZjj B¹I¸¹ BZÿ}Zÿ P¸¸IAs{, {Btÿ¹Bjj¸¹ BP ÿY¸ PB¹8 $t w;þnDTw;þ;D ÿYsÿ sRk8}ÿBÿT ${sjjk RÿsA$j$9¸R ÿY¸ RkRÿ¸8 w;þdDTw;þ1D $P ÿY¸¹¸ ¸-$Rÿ 8sÿ¹${¸R p k (ý C k (ý K ý ¼ k (ý ; ý J> ý ~> w> ) d, 1D stI . RZ{Y ÿYsÿ ÿY¸ PBjjBb$tü Nûy YBjIRU

m t

bY¸¹¸

}

K ý B

)

K ý \1

)

}

ÿýK¼

(

(

ÿwýKB ÿ ýK¼ D

K ý \d

K ý \1

p

{

{

C

]

} ,

K ý \d )

Kx ý

\d Kx ý \1 ÿýKB

u

{c(

\d p L 41d .

\d L 41d ; g1

Jd

C\d L ~d g1

\1 p L 411 .

\1 L 411 ; g1

J1

C\1 L ~1 g1

] 3

w;ýûD

] ,

w;ýd(D

yt ÿY$R R$ÿZsÿ$Btý ÿY¸ {Btÿ¹Bjj¸¹ }s¹s8¸ÿ¸¹R BP w;þnDTw;þ;D {st A¸ ü$%¸t sR g

)

\>

)

ÿ ; g1 pDÿ ÿx , 4> ) þÿd w~> ÿ C41> ; D, w;ýddD ÿ d x ÿ þ4> g1 pDÿ ÿx , þ wJ> ÿ C w\> L 41> ; g1 Dp ÿ C 41> g ÿ w.

> ) d, 13

w;ýd1D

;ÿ1 !Zþ}Zþ 3¸¸IAs{, 7þsA$j$9sþ$Bt BP 3ý N77 ýBI¸j _¹BBPU

;x

3¹B8 N¸88s 1ÿnÿnþ ýY¸ {jBR¸ITjBB} RkRý¸8 w;ÿxD $R sRk8}ýBý${sjjk

RýsAj¸ $P ýY¸¹¸ ¸-$RýR s 1z

ÿ 1z }BR$ý$%¸ I¸ht$ý¸ 8sý¹$- ¼

ýY¸ PBjjBb$tü 8sý¹$- $t¸JZsj$ýk YBjIRU

m Nt þ¼

¦ \ d

(

þwB þ ¼ D

(

¦ \ 1

7¸ý

x

u U{ c (3

B B

þB

¦ B\ 1

¦ B\ d

x

stI B RZ{Y ýYsý

w;ÿdnD

ÿd ¼ B ÿd , BK ) B ÿd

K ) B ¼

\ ÿd , B ÿd ,

ûk }¹¸T8Zjý$}jk$tü stI }BRýT8Zjý$}jk$tü BP ýY¸ Núy w;ÿdnD Ak Q7y/ B

ÿd iþ $ý PBjjBbR ýYsý B

m þ¼K Nt (

þwBK þ ¼K D

¦ B K \ d

¦ B K \ 1

K stI B K WBbþ }s¹ý$ý$Bt B K B

bY¸¹¸ p, C, ÿ, þ N¸ý

K\ ¦ B d

(

ÿd )

ÿd

}

sR C

ÿ

x

ÿ

K\ ¦ B 1

u U{ c (3

x x

þBK

]

}

J

p

K ) B

,

w;ÿd;D

þ

þ

x

] w;ÿdxD

~

; ozÿz ÿ yý $R ¸sRk ýB RYBb P¹B8 w;ÿdxD ýYsý þ ÿx ] } {

ý )

(

C

x

3

ÿ

\

_¹¸ stI }BRýT8Zjý$}jk$tü w;ÿd;D Ak Q7y/ ý

(

x , ýx , ýx i

þý x wBK þ ¼K Dý

(

x ¦ ý \

x ¦ ý \

K dB ý

ý ý

x BK \¦ x ý d x K ¦x B \1

ý

þý x BK ý

K 1B ý

þ pC ÿ w;ÿdED

¹¸R}¸{ý$%¸jkþ b¸ Ys%¸

m þý x ¼K ý Nt

) {

\

i

stI Q7y/ ý, ý, ý þ

u U { c (3

w;ÿdD

qY¸ sAB%¸ 8sý¹$- $t¸JZsj$ýk $R tBý {Bt%¸- $t ýY¸ {Btý¹Bjj¸¹ }s¹s8¸ý¸¹R ¦ w> ) d, 1Dÿ qB ¹¸RBj%¸ ýY$R tBtT{Bt%¸-$ýk }¹BAj¸8þ bY${Y s¹¸ {Btýs$t¸I $t \ > b¸ RYsjj $tý¹BIZ{¸ ýY¸ %s¹$sAj¸R $t w;ÿDTw;ÿMD>M1Hÿ K stI \ ¦ w> ) d, 1D $t w;ÿdEDþ w;ÿdxD stI w;ÿEDþ $ý {st A¸ yt %$¸b BP ý þ B > ý¹$%$sjjk %¸¹$h¸I ýYsý

ý

ý

x BK ý

x\ ¦

K 1B ý

}

)

)

}

p

{

{

C

] ,

ý

x\ ¦

}

K dB ý )

\d p L 41d .

\d L 41d ; g1

Jd

C\d L ~d g1

\1 p L 411 .

\1 L 411 ; g1

J1

C \1 L ~1 g1

]

]

bY¸¹¸ J> þ ~> w> ) d, 1D stI . s¹¸ I¸ht¸I sR $t w;ÿDTw;ÿMDÿ 3Z¹ýY¸¹þ I¸tBý¸

x K ý3 qYZRþ w;ÿùD PBjjBbR I$¹¸{ýjk P¹B8 w;ÿdDÿ qY$R {B8}j¸ý¸R ýY¸ K ý ¼ ) ý ¼

}¹BBPÿ

,

;E

;ÿ 7þsA$j$9sþ$Bt stI

Iÿ

]Btþ¹Bj BP 1T* *$R{¹¸þ¸ 7kRþ¸8R $t 3ý N77 ýBI¸j

7Z}}BR¸ ÿYsÿ ÿY¸ Nþy w;ýüD sI8$ÿR P¸sR$Aj¸ RBjZÿ$BtR û û K  û > û > w ) d 1D stI ý 3¹B8 x ) ÿ (û tBtR$túZjs¹ 8sÿ¹${¸R stI {st A¸ {B8}Zÿ¸Iý qY¸tû ÿY¸ {B8}Zÿsÿ$Bt BP ÿY¸ {Btÿ¹Bjj¸¹ }s¹s8¸ÿ¸¹R  û > û > w ) d 1D {st A¸ {s¹¹$¸I BZÿ Ak RBj%$tú w;ýddD stI w;ýd1Dý i¸8s¹, ;ÿ1ÿ1ÿ qY¸B¹¸8 ;ý1ýd }¹¸R¸tÿR s t¸b s}}¹Bs{Y ÿB BZÿ}Zÿ P¸¸IAs{, RÿsA$j$9sÿ$Bt BP 1T* RkRÿ¸8R $t ÿ¸¹8R BP Nþyý WBÿ¸ ÿYsÿ RBj%$tú NþyR $R s R$8}j¸ ÿsR, $t %$¸b BP ÿY¸ }Bb¸¹PZj þsÿjsA Nþy qBBjAB-ý qY¸ Nþy s}}¹Bs{Y {st sjRB A¸ ¸-ÿ¸tI¸I ÿB RBj%¸ s ¹BAZRÿ RÿsA$j$9sÿ$Bt }¹BAj¸8 PB¹ 1T* RkRÿ¸8R b$ÿY }s¹s8¸ÿ¹${ Zt{¸¹ÿs$tÿký i¸8s¹, ;ÿ1ÿdÿ ÿ¼

;

J

p

~

>

þ

,

.

þý

{

C

pC k

ý

\

g

4

,

>

,

ÿ 70>"D0: 03 *! 2++ !01C:

.#< !

yt ÿY$R R¸{ÿ$Btû b¸ RYsjj I¸sj b$ÿY ÿY¸ ÿ {Btÿ¹Bj }¹BAj¸8 BP 1T* RkRÿ¸8R %$s Ikts8${ BZÿ}Zÿ P¸¸IAs{,ý !Z¹ }Z¹}BR¸ $R ÿB I¸R$út s Ikts8${ BZÿ}Zÿ P¸¸IAs{, {Btÿ¹Bjj¸¹ BP ÿY¸ PB¹8 $t w;ýnDTw;ý;D RZ{Y ÿYsÿ ÿY¸ {jBR¸ITjBB} RkRÿ¸8 $R sRk8}ÿBÿ${sjjk RÿsAj¸ stI YsR s R}¸{$h¸I ÿ I$RÿZ¹Ast{¸ sÿÿ¸tZsÿ$Btý ]BtR$I¸¹ ÿY¸ PBjjBb$tú I$R{¹¸ÿ¸ 1T* }jstÿ $t ÿY¸ 3þ N77 8BI¸jU w L d L dD ) d w L dD L 1 w L d D L dd w L dD L d1 w L d D L 1d w L dD L 11 w L d D w;ýdMD w D ) d w D L dd w D L d1 w D w;ýdüD w D ) 1 w D L 1d w D L 11 w D w;ý1(D bY¸¹¸ w D ; oz û w D ; oR û w D ; oE û w D ; o· stI w D ; oq s¹¸û ¹¸R}¸{ÿ$%¸jkû ÿY¸ Rÿsÿ¸û ÿY¸ I$RÿZ¹Ast{¸ $t}Zÿû ÿY¸ {Btÿ¹Bj $t}Zÿû ÿY¸ {Btÿ¹Bjj¸I BZÿ}Zÿ stI ÿY¸ 8¸sRZ¹¸8¸tÿ BZÿ}Zÿ BP ÿY¸ }jstÿû > û d> û 1> û > û d> stI 1> w ) d 1D s¹¸ ¹¸sj 8sÿ¹${¸R b$ÿY s}}¹B}¹$sÿ¸ I$8¸tR$Btý G¸ 8s,¸ tB sRRZ8}ÿ$Bt Bt ÿY¸ Rÿsÿ$Rÿ${R BP ÿY¸ I$RÿZ¹Ast{¸ $t}Zÿ R$útsj BÿY¸¹ ÿYst ÿYsÿ $ÿ $R ¸t¸¹úk ABZtI¸Iû $ý¸ýû , ,1 þý G$ÿYBZÿ jBRR BP ú¸t¸¹sj$ÿkû b¸ sRRZ8¸ 11 ) (ý qY¸ {jBR¸ITjBB} RkRÿ¸8 BP ÿY¸ }jstÿ w;ýdMDTw;ý1(D stI ÿY¸ ÿY¸ {Btÿ¹Bjj¸¹ w;ýnDTw;ý;D $R BP ÿY¸ PB¹8 ¦w L d L dD ) ¦d ¦w L dD L ¦1 ¦w L d D L ¦d w L dD L ¦1 w L d D w;ý1dD ¦ w D ) ¦w D L ¦ w D w;ý11D bY¸¹¸ ¦w D ) > x w D x w DHx stI } } ] ] > L 1>  1 1>  ¦> ) ¦> ) d> L 1>  1d I

I

? 7

,i

\ ? 7, i

\ ? 7

4

j 7, i

4

Á 7, i

,i

4

4

j 7

,i

Á 7

,i ,

F 7, i

g ? 7, i

;

j 7, i

;

Á 7, i ,

r 7, i

g ? 7, i

;

j 7, i

;

Á 7, i

? 7, i

j 7, i

Á 7, i

F 7, i

r 7, i

\

g

;

;

>

4

4

,

j

c

j

;

? 7

,i

\ ? 7, i

\ ? 7

4 j 7, i

F 7, i

? 7, i

\

?

\

7, i 4

g ? 7, i

?

; g

4> g1

,i

4 j 7

,i ,

;j 7, i

7, i

4

g

\>

,

4

4

4

; ;

4> ;1d

,

Iÿ ]Btþ¹Bj BP 3ý N77 ýBI¸j

;ÿn

;

> ) d, 1 ¦ g





>gd L ;d1 ; g1

)

w;ÿ1nD



¦ ) ; ; dd L ;d1 ; ;1d 3

;d1 g H,

ÿ

3B¹ þY¸ {jBR¸I jBB} RkRþ¸8 w;ÿ1dDTw;ÿ11Dý b¸ Rþsþ¸ þY¸ 1T* I

w;ÿ1;D

{Btþ¹Bj }¹BAj¸8

sRU htI s 1T* Ikts8${ BZÿ}Zÿ P¸¸IAs{, {Btÿ¹Bjj¸¹ BP ÿY¸ PB¹8 $t w;þnDTw;þ;D PB¹

ÿY¸ 1T* }jstÿ w;þdMDTw;þ1(Dý RZ{Y ÿYsÿ ÿY¸ {jBR¸I jBB} RkRÿ¸8 YsR s R}¸{$h¸I

ÿ

I

}¸¹PB¹8st{¸ ÿÿ

qY¸B¹¸8 ;ÿnÿdÿ

]BtR$I¸¹ ÿY¸ 1T* RkRÿ¸8 w;þdMDTw;þ1(Dþ p$%¸t s }BR$ÿ$%¸

ÿ

R{sjs¹ ÿ ý ÿY¸ I K C k (ý þ

m

ý

¹

)

¼

ÿ

N N N N N N N N N t

{Btÿ¹Bj }¹BAj¸8 $R RBj%sAj¸ $P ÿY¸¹¸ ¸-$Rÿ 8sÿ¹${¸R p k (ý



>

>

k (ý ; ý J ý ~ w> ) d, 1D stI . RZ{Y ÿYsÿ

¼

K þ (

\d x g

K þ

x \d x K þ \1 K ÿ þB

ÿ

B

K wþ

K þ

ÿ

K þ

\1

K þ

¼D

(

(

x g

K þ

( (

(

(

(

u

g

K þ

K þ

(

( K þ

( (

( (

(

1

ÿ

(

4d K þ 41 K þ

g

ÿ {

ÿ

¦ ;

x

( (

x 4d

K þ

1 ÿ {

¦ ;

x

x 41

(

(

¦ ;

{

(

U U U U U U U U U {

( K þ

¦ ;

ÿ

(

(

(

(

ÿ

c (

{ w;ÿ1xD

bY¸¹¸

}

B

K þ

)

}

\1 Kx þ g K þ 4d K þ 41 K þ

)

) ) )

J J J

p

{

{

C

]

} ,

\d )

K þ

\d L 41d ; g1

Jd

C \d L ~d g1

]



\1 p L 411 .

\1 L 411 ; g1

J1

C \1 L ~1 g1

gd p L ;d1 .

gd L ;d1 ; g1

x x x x 4dd L ;1d ; 41d x x x x 4d1 L ;1d ; 411



\d p L 41d .

,

,

o



,

x x x 4dd C L ;1d ~d x C L ;x ~ x 4 d1

]

1d 1

o o

, 3

w;ÿ1ED

_¹BBPU ük s}}jk$tû qY¸B¹¸8 1ÿxÿx þB þY¸ {jBR¸ITjBB} RkRþ¸8 w;ÿ1dDTw;ÿ11Dý

ÿ

þY¸ {Btþ¹Bjj¸¹ w;ÿnDTw;ÿ;D RBj%¸R þY¸ 1T* I $t¸JZsj$þk

m

¹

ü

)

N N N N N N N N N t

ÿ

¼

( ¦ B\ d ¦ g

ÿ ÿ wB

¦ B\ 1

xB xB ¦ \

¦ \ d

(

¼D

¦ g

1

ÿ

B

(

(

(

¦ g

(

(

(

(

(

xB xB ¦ 4 ¦ 4 d

1

u

x

(

ÿ

{Btþ¹Bj }¹BAj¸8 $P þY¸ 8sþ¹$-

( ¦ g

x

(

(

ÿ {

(

1

( ¦ ;

x

(

ÿ

1

ÿ { (

¦ ;

x

( ( ¦ B4 d ¦ ; (

ÿ

{

(

(

U U U ¦ B4 1 U U ( U U ¦ ; U U { ( (

ÿ

c (

{ w;ÿ1D

;M

;ÿ 7þsA$j$9sþ$Bt stI

Iÿ

]Btþ¹Bj BP 1T* *$R{¹¸þ¸ 7kRþ¸8R $t 3ý N77 ýBI¸j

YBjIR PB¹ RB8¸ ¼ k ( stI B k (ÿ 7¸þ K ) B ¼

ÿd ¼ B ÿd , BK ) B ÿd 3

\ ÿd , B ÿd ,

ýk }¹¸T8Zjþ$}jk$tü stI }BRþT8Zjþ$}jk$tü BP þY¸ Nûy w;ÿ1D Ak Q7y/ B

ÿd , { , { , { , { iú $þ PBjjBbR þYsþ B

m K ÿ¼ ( NN ( ÿwBK ÿ ¼K D NN \¦d BK K ¦ B \ 1 NN g¦ BK ( NN ¦B K ( g Nt ( ( (

K\ ¦ B d K\ ¦ B 1

ÿ

K B

ÿd )

(

(

Kg ¦x B

(

(

(

(

¦ 4 d

¦ 4 1

(

¦ ;

(

(

¦ ;

(

ÿ{

ÿÿ 1 {

(

(

ÿd

}

Kg ¦x B

(

¦ 4 1

K stI B K WBbú }s¹þ$þ$Bt B

x

K B

¦ 4 d

(

x

x

¦x ;

x

ÿ

C

þ

x

þ

1 ÿ { (

¦x ;

(

sR

(

]

} ,

J

K ) B

ÿ{

(

p ý

u U U U U U c (3 U U U U {

(

ý

x

] w;ÿ1ùD

~

; ozÿz ÿ yþ $R ¸sRk þB RYBb P¹B8 w;ÿ1ùD þYsþ ý þx

bY¸¹¸ p, C, þ, ý N¸þ

}

{

ü )

C

3

þ

\

_¹¸ stI }BRþT8Zjþ$}jk$tü w;ÿ1MD Ak Q7y/ ü

i

) {

ÿ pC ÿ

]

x

(

w;ÿ1MD

w;ÿn(D

x , üx , üx , {

1w·LR D

i stI Q7y/\ü , ü,

ü , {1w·LR D ú ¹¸R}¸{þ$%¸jkú b¸ Ys%¸

m ÿ N N x N N N N N t

ÿ

x¼ Kÿ (

Kÿ \d B

ÿ

ÿ

(

x wBK x ÿ \

ÿ

ÿ

x BK \ x ÿ d x K\ x ÿ ÿ B 1 x Kÿ ÿ B

ÿ K Dÿ ¼

Kÿ 1B

ÿ

Kÿ gB

(

(

(

Kÿ gB

(

(

(

(

(

ÿ

(

(

ÿ

(

þ {

(

1

ÿ

(

ÿ

1

þ {

x ;

(

(

(

x gx

(

ÿ

x 4d ÿ x

41

x gx

;

x

(

(

ÿ

x4

d

(

ÿ

x4

1

;

(

(

;

ÿ

{

(

(

ÿ

u U U U U U U U {

{

c (3

w;ÿndD

Kú \ ¦ ú 4 ¦ stI ; ¦ $t w;ÿn(Dú w;ÿ1ùDú w;ÿ1nD stI yt %$¸b BP ü ú B > ¦> w> ) d, 1Dú g w;ÿ1;Dú $þ {st A¸ þ¹$%$sjjk %¸¹$h¸I þYsþ

x Kü ü B ü

x\ ¦

K

1B ü

}

)

)

}

p

{

{

C

]

,

x ¦ ü \

K dB ü )

}

\d p L 41d .

\d L 41d ; g1

]d

C\d L ~d g1

\1 p L 411 .

\1 L 411 ; g1

J1

C \1 L ~1 g1

J

,

] ,

;ÿ; e-s8}j¸R

¦K ¦dx ¦1x

J

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bY¸¹¸ ?

; oz þ

j

)

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; o·

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\> , 4> w> ) d, 1D, g, ; stI U s¹¸ ,tBbt 8sÿ¹${¸R b$ÿY s}}¹B}¹$sÿ¸ I$8¸tT R$BtRû qY¸ ABZtIs¹k {BtI$ÿ$BtR ¼

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sR I¸ht¸I $t w1ûdMD s¹¸ sRRZ8¸I

ÿB A¸ Zt,tBbt AZÿ ABZtI¸Iû G¸ tBÿ¸ ÿYsÿ bY¸t U ) { þ ÿY¸ Rÿsÿ¸ ?w7, i D $R ÿB A¸ ¸Rÿ$8sÿ¸Iû

Ch. Du, L. Xie:

Control and Filtering of Two-dimensional Systems, LNCIS 278, pp. 81−98, 2002.

M1

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wMüdD

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; oz $R ÿY¸ 1T* RkRÿ¸8 Rÿsÿ¸ stI rw7, i D ; oq ÿY¸ ¹¸{¸$%¸I 1T* · R$þtsj bY${Y $R {B¹¹Z}ÿ¸I Ak ÿY¸ ¹¸{¸$%¸¹ tB$R¸ ÿw7, i D ; o ü \> ý 4> w> ) d, 1D, bY¸¹¸ ?w7, i D

g stI ; s¹¸ ¹¸sj 8sÿ¹${¸R b$ÿY s}}¹B}¹$sÿ¸ I$8¸tR$Btü

Ch. Du, L. Xie:

Control and Filtering of Two-dimensional Systems, LNCIS 278, pp. 99−114, 2002.

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Ch. Du, L. Xie:

Control and Filtering of Two-dimensional Systems, LNCIS 278, pp. 137−138, 2002.

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