WP3 3:OO. Bor-Sen Chen. Ik!pa~tmrl~i. (1f'. Klectrical Enginr:rrjr,g lintional Tsing Hua University. K~~ang. FII road, Hsinchu. Taipei, Taiwan, Republic of China.
Proceedings of the 26th Conference on Decision and Control Lor Angeles,CA December 1987
WP3
3:OO
Hung-Ching Lu
Bor-Sen Chen
Department of Electrical hngirlrrring Tatung Institute of Technology 4 0 Chungshan North Road. 3 r d . Sec Taipei, Taiwan, Republicof China
Klectrical Enginr:rrjr,g lintional Tsing Hua University K ~ ~ a nFII g road, Hsinchu Taipei, Taiwan, Republicof China
I k ! p a ~ t m r l ~( 1i f '
ttschniques
At!S'I'HA[:'l
can
he
easily
employed
to
treat
this
p r o b l em.
Since systems are generally often subject to some forms of perturhat i t r ~ ~ ,sucll a s ~~nmtrdcllcd rlynamics. environmental variat.ions. and parameter uncertainties, is O I I ~(11 tile most importarlt therobuststability things to be considered. Owine to its complexity and largeness,the large scalesystem is more easy to suffer uncertainties than t h e small one. Thus i n this paprr, we consider the robust stahjl i l y o f large s [ , a l e system with parametricperturbation.Thesufficient is dcrived via condition o f therobuststability comparison theorem and norm theory from the viewpoint o f t ime domain. Also, a design procedure is proposed to find a controller to satisfy the robuststability cond i t j o n . I . INTRODUCTION
'The remaining part of this paper is ol,):iinized tis follows. In section 1 1 . the problem we concerned with is described. In section 1 1 1 , the results of system wit:h nonlinearily parametric perturbation are presented. The systnm with I irlearily parametric pfrturbat.ion is given in section IV. And the design procedure is given in section V. Finally,w(: ronclude the paper with some discussions. The proofs of the theorems are given in appendix A , H , respectively.
PRELIMINARY MATHEMATIC Hrfc~rc! f u r t h ~ ranitlysis. matical tools. If we denote vector x€ Rn H S 8x1i , t h c . t i ' A 1 ~ of AGR'"~" 1s ctefined a s
interconnection and Because of complexity of largeness of dimension, the robust stability property is more important in the large scale system than in the single loop system. Siljak[l2] discr~ssed the structural perturbations in the large scale system:
, ,AxiI
IA;,
=
A.x.+B.u.+ X 1 1 1 1
eijAijxj where
j = 1 ,j+i
0
=
sup x40
-= :x(,
S I I ~
'
' A x !:
=
xi !=I
x€Rn
5 e ij -< 1
Over the recentyears, many methodshave been suggested for the control of perturbatlve large scale interconnected structure [Q], [IO] from the viewpoint of time domain. More recently there have also been many papers discussing the robust stability of perturhative large scale systems from the viewpoint of frequency domain. Limebeer and Hung [ I O ] used the concept of generalized block diagonal dominance to derive a robust stability criterion for linearily perturbative largescalesystems.Chen and Desoer[2]alsoderiveda necessary and sufficientconditionfortherobust stability of aninterconnected system made of many linear time--invariantmultivariable system i n which perturbations are assumed to be linear and stable.
/ A X ,1
sup
xi
the f~~llowiny, robust stabilif; condition hold
1
of states of the ith subsystem, for i-1.2.
where,u(X.
.)
'To 1,
1
I
1
.J. Davison, "Decentralizedrobustcontrol of unknown systems using tuninF: rcgulators," IEEE Trans. Automat. Contr. vol.AC-23, pp.276-288, 1978.
1.
Ai+LAiAi2+LAii,Ali+LA
N
L
lj
ANi +&A Si
M . F . Hnssan, M . G . Singh and R . Hurteau, "Stability, stabilisation and pcrformance of rnult.ilrvc~1 controllers under structural perturbations. Part I : cutting the links bet.ween co ordinator and subsystems," IEE, Proc. Part 0 , ~ 0 1 . 1 2 7 ,pp.207-213,1980. M . F . Hassan and M. G . Singh, "Stability, stabiliof multilev~l c:ontrollers satjon and performance under structural perturbations. Part 11: stabilisat i o n underanystructuralperturbat ion." IEE Proc.Fig.1 Part 0 , vcll. 127, pp.214-219, 1980.
-
PerturbativeLarge
Scale system
E. H i l l e ,
1,ect.ures on ordinary differential eauaAddison-.Wesley. New York, 1969.
-,
J. C . Hsu and A.11. Meyer, Modern control principles and applications, Mctiraw-Hill, 1968.
Y.
Hung and D. J . N . Limeheer, "Robust stability additivelyperturbedinterconnected systems," IEEE Trans. Automat. Contr. vol. AC-29, p p , 10691075, 1984. S.
of
N. Limebeer, and Y.S. Hung, "Robust. stability interconnected systems," IEEE Trans. Circuits Syst., V O ~ .CAS-30, p p . 397-403, 1983.
[ l o ] 1l.J.
of
1111 l ) , I ) . S i l j a k , 1.avp.e. scale dynamic: systems: stability
a n d structure,
Elscvier North-Holland. New
York,
1978. 1121 D . D . Sirjak. "On stability of l a r g e - s c a l f ! systems under structural perturbations," IEEE Trans. Syst. Man and Sybern. S M C - 3 , p p . 415-417, 1973. Vidpasagar, 1978. Prentice-Hall.
1131 M .
Nonlinear
systems analysis, Fig.2
Nonlinearily Parametric perturbation bounded by [ - - B X i 9 B X i 1
1141
w.
[15]
H.Bellman and K. L. Cooke, Differential-difference equations, New York : Academic Press. 1963.
A. Coppel, Stahllity and asymptotic behaviour of differential equations, Boston : U . C. 1965.
ACKNOWLEDGEMENT This work was supported by the National Council of the Republic of China.
Science
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