DECENTRALIZED ROBUST CONTROL FOR A CLASS OF UNCERT...
14th World Congress of IFAC
G-2e-12-2
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing P.R. China J
DECENTRALIZED ROBUST CONTROL FOR, A CLASS OF UNCERTAIN INTER.CONNECTED SYSTEMS
Xinyu Liu "',1 Liqun Gao
li<
'"" Dept. of A uton~atic Crontrol; ]'y~ortheastern University Shenyan9 110006) L.iaoning, P.R. China E-m.a-il: a.s.ln.xi'l1YH.lill@Q63. net
Abstract: In this paper~ the problern of decentralized contrOl for a class of uncert.ain linea.r interconnected systems is considered. 'The uncertainties of each decolnposed system, \vhieh do not satisfy matching conditions, are assumed to be structured and value bounded. A robust decentralized scheme is proposed V\ hich guarantees t.he overall uncertain systerll to be quadratic stable. Copyright © 1999 IFAC 7
KeYVi.~ords:
Decentralir,ed control, Interconnected systems, Linear systen1S, Quadratic
stability, Robust control
I Ji.j 1< d1:j) v/here ~ is uncertainty; D is known constant rnatrix with non-negative entries~ 6ij and d ij denote the ij-entries of ~ and 1) respectively. This representation \vould be used in a practical situation. Based on t.hi~ representation: a sufficient condition for stabilization in terln of existence of the symmetric definite solutions of certain R.iccati-type equations is presented.
1. TNT'R,ODlJCTION
In recent years, decentralized robust control of uncertain systen1S has attracted great attention of rnan.y researchers. By decolnposing the large-sale syst.em into several lo\\~er-order subsystems) one can design the controller of each su bsystelll independently. So, the design procedure will be sinlplificd. lar~e-scale
'T'he so-called matching conditions (Gutmal1 l 1979) have been playing an important role in t.he robust. control of uncertain systerns. It is well known that, an uncertain systelTI is stabilizable ",~hen the matching conditions are satisfied. For linear large-Bcale :::;ysterns, many nice solutions have been given in the literature by assuming that matching conditions are satisfied (Ikeda and Siljak 19S0; Ni and Cben, 1996). In tbjs paper~ tllc Jnatching conditions are relexed for a. class of linear uncertain interconnected systerns. The uncertainties of each subsystem in this paper are structured and value bounded (Mehdi et aI., 1996L represented as J ~A 1-< D. This representation nleans that
The paper is organized as f01ln\/I/5. In sect.ion 2 the systen1. is described. Section;) gives a decentralized robust controller design for an uncertain interconnected tirne-varying syst~m which does not satjsfy luatching conditions. Section 4 and 5 give an illustra.tive cxanlplc and a brief conclusion rcspecti vcly.
l
2. SYSTEJ\1S AIVD,.\SSlJMPTIONS
Consider an uncertain large-scale
SystCIll
which is
cOInposed of L~ linear int.€r~onnect.ed subsystems
described by
Xi
==
(~'1ii
+ 6. A ii)Xi + (B i + ABi)Ui(t) S
+
This ,,,ark was supported in part by Nat.ional N at-ure Science Fund, National Educatlon COIIunitt.ee Fllnd and in part. by Nature Science Fund of Liaoning province. 1
L
CA.ij + ~.i-1ij)XJ;
(1)
j=;: 1 ,j?i=i
3450
Copyright 1999 IFAC
ISBN: 0 08 043248 4
DECENTRALIZED ROBUST CONTROL FOR A CLASS OF UNCERT...
where i == J, 2, .__ , ~9. Xi E R,n. and Ui (t) E R m are~ respectively, t.he state and control of the 1th subsystem. Aii, Bi and ~A.i.i represent the system and interconnection matrices~ respectjvely, and -4i i E R nxn : A~:j E R nxn and Bi E R,nxm. The pair (~4ii, B i ) is controllable. ~-A..ii, AA ij and .6..Bi, are the SyStCll1, interconnection and control uncerta.inties respectively.
14th World Congress of IFAC
\\~here
X'
= [xf
diag[B 11
...
' ... , x~]T: 'U ::::::: [uT, ... , ul]T: B ~B == diag[~Bl, .-. ,~8 5-)1
==
,Bs],
-/t 12
All
..4 1 S
The following further assumptions are rrlade on systen1( 1).
=
~A
tlAiJ
.21.4 i2
(2) i~j::=
~AS8
1,2, ... ~S.
Denote that . 4 == diag[~411: .4. 22 :", AS8] and (1\f1] )ki is with ns x ns dirnension having the form l
The nlatrlCei'j D ij and E i are assumed to be known ,vith positive entries.
o
In the seqllel, severallelTlnlas are given Virhich win be essential for t he proof in the next sect.ion.
o
o
o ...
2.1 For any syrrnnetrical non-negative TIlatrix R == (rij) E Rn xn, it holds
k
Le£Illlla
ndiag(R.)
o
o
o
2: R.
\vhere diag( R) ~ diag(rl1l ... , l'nn)' Lemma 2.2 If ~ E R Hxrn satisfies 16 1-< lJ 1 then (3)
Vi here
Alij E R,n x n
.
rewrite (7) as
NOVl J
5
X ::= [..4.
\vhere
s
+L
L
+ i
(J1 ij )ii (T)ij
.J= l,jf:.i
(8)
(~Aij)ii(I)ij]x
+ (B + ~B)u(t);
-i 107=l
where 1I . 11 denotes the standard Euclidean nornl. Proof: It is obvious t,hat 11
~D..T
III 2:
11 DJ)T H~II
\\t~hcrc I is the identity matrix; J E Rn Xn . It is obv lOllS that (( ~4ij )ii)
~L1T,
,6.T ~
H.
(( Li. -"4 i j ) i i ) ( ( ':)'.,4 i j
Hence) one has
By utiing lCffilna 2.1: it follo~ws that )
~ n.diag(~~T)
== ((I)jj),
L
== diag[
2
2 ~L3.T.
~~4iJ·~~4~.,
s
(A ij )ii {(~lij )ii)T
s
L
-Al.i~4rj
j=l .J;t: 1
ij~lj#i
s
L
~ASjj1~jL
j=lj;tS
FraIl1. lenmla 2.2 a.nd assunlption (2) ,one obtains
ll(D ij )
)ii )T
YG (J) ij )
((I
ndiag(DD T
(( --lii) )ii)T
s
L
(5)
((J)l~(I)ij)
== (s - l)diag[I, ... , I],
i l j;;;;:;.l j;ti j
IT(Ei )
2: ~Bi~Br·
S
(6)
L
5
(~~4ij)ii((~~4ij)ii)T :::: diag[I: ~Al.7L\j'tIj j==1
i,j==l
8
3. Iv1i\IN RESlrLTS
, ··'l
L
~ASj~~4'~j],
j=l 5
Re\\-Tite the system (1) as
x ==
(.4
+ ~A)x + (R + D.R)u(t),
L
(7)
((I)~.(I)i.i)
==
diag[sI] .__ ,51].
i,j~l
3451
Copyright 1999 IFAC
ISBN: 0 08 043248 4
DECENTRALIZED ROBUST CONTROL FOR A CLASS OF UNCERT...
14th World Congress ofIFAC
vVrite G F == :Si~j:=lj;ti ((A ij )ii (I)ij)
+ L!.i=l((~Ai.i)ii(I)ij)
, where G is a ns x ns(s'2 + s(s - 1)) -dimensional rnatrices formed by ((Aij)ii) and ((~Aij)ii); F is a ns(s2 + s(.5 1» x ns -dinlensiollal Inatrices formed by (( I)ij). 'rhen, it is easy to see t.hat () and F satisfy
L
GOT == diag[(
Alj
~ ~
(
+
+ PiAii
A'[;Pi
8
'L--~ASj~ASj)]' "' T
- Pi [BiRi
8
== diag[(2s - ]) l, .. " (2s - 1 )I].
(9)
'T'herefore, the system (8) can be "'ritten as
+ 2: rr (Dij)]Pi
Aij ..4~
+ c- 1 (28 (10)
==
l
}'(2: X i,
i
==
where !{
=::
+ G[i' + (B + ~B)I{].L~
and apply the follo\\ring decentralized controller to systen1 (1),
(11)
(14) the resultant closed-loop system will be quadratic stable. Proof:
and P
!{ =: diag[I{l~
Denote
diag[I1(E 1 ):
+ ~A + (8 + ~B)I{JT P + P[_1t + AA
=:
...
1
2
diag[P1 ,
... ,
Illatrix P
xT(PGF
+
x
T
{diag[Al 1 ,
"-J
~'lsJ - LP(B
== 2x T PGr'x
::; -O:)! where
]lvfi
X 11
2
s
A1j ..4ij + 2: n (D 1J))'
j==l,j;tl
)=1
S
== 1
+ 11~T aT P)x
=::
1)1- Pi[BiR;l
n:r -
,i
and FT F arc block diagonal n1atrices; it follows from Lemlua 3.1 that x T diag[lV 1 , ... , ]\ls]x
4I j + 2:11 (D s .i ))]Px. j
(16)
== 1, 2, __ .~ ST. Because _P A+ _ 4T p~ PCCT P
2:
ASj ..
j =.1 ,j;;t 5
~B)
= A~Pi+Pi . r1ii+EPiCLjS'=1,j~i-4ijAI;+
bil1(Ei)J~'
s
+
,
E%=l Il(Dij)]Pi + (-1(28 -
L
+
(H+~.H)Tp}X
pTaT P)x
... ,( L
(1.5)
From (12) and noting that LP(B + ~B)(B AB) T P i~ a block diagonal n1.atrix~ one has
.
For any f. > 0 and syn1.metric E R,'ru xns , the follo\ving equrl.tion hold8.
S
E
Ps]. It is obvious that
]{ == -L.n T P.
Lemll1a 3.1
+£xTPdiag[(
... l!{S],
II(Es)], L == diag[ol I) ... ,8 8 I]
+ (B + ~H)l{J}x If X 11
+ Qi
(13)
diag[I{J, !(2, ... \ Ifs].
::; -0:
1) J
1
1,2, ... , S,
Definition The systenl (11) is said to be qua.dratic stable if there exist a positive scalar c and a sYll1n1etric positive definite matrix P such that x T {[~4
(12)
j=l
"vllr~re
vI/here, Pi are ~ymnletric positive definite 111atrices. 'Ihis yields a closed-loop systenl as
X ::::: [A
0
:==
Apply the follovving decentralized controller to the system (10) 1.ii
BT S
L
+ tPi [
,j=l jf:.i
x == C4 + GF)x + (B + l113)u(t).
1
- 5i il(Ei)]Pi
.1==1
j= 1,ji:S
},T F
IIletric.: posit.ive definite solution Pi of the Riccati equation
j= 1
AsjA Ts ;,
0,
Theorenl. 1: If there exist a positive scalar f} a sylun1etric nonnegative matrix Qi and a syn'l-
Aij + 2:A~41j ~ ..4rj)
j= l,j;tl S .1·'"
~
and ( B), one can irnlllediatcly have Leulma 3.1.
s
5
~FX 11 2
T
= 11 ,ftC Px -
xT(P ..4+~4Tp+cPGGl'p+f-lFTF)x
Proof: It easily seen that x T (PGF
2.r T PGFx_
In the view of the fact that ~xT PGeT Px - 2x T PGFx
+ E-1x T pT Fx
+ ~-,A)T P + P(A + ~A)Jx, ¥lhere j\li == AT:Pi + PiA ii + (Pi[I=:.~=l,j;ti AijAl; + I::=1 II(Dij )]Pi + t - (2s - 1) I. ::=
xT[(_A
1
3452
Copyright 1999 IFAC
ISBN: 0 08 043248 4
DECENTRALIZED ROBUST CONTROL FOR A CLASS OF UNCERT...
l\1ehdj D., M. AL, Haulid and F. Pcrrin(1996), Robustness and optimality of linear quadratic controller for uncertain systen1.s. Automatica, 32, l081-10g3. Ni M. and Y. Chen(lgg6). Decentralized stabilization and output tracking of large-scale uncertain systems. Automatica~ 32, 1077-1080.
Furt her nlore ~
x T {diag[M1 , ... , Ms] - LP{B
+ D.-B)(B + ~B)T P}x
+ ~B)(B + ~B)T P}x
- LP(B
+ ~A)Tp + P(Jl + Ai1)
~ x T {(~4
- LP[BB T
x T {{~4
P[.A.
-
E]?
(17)
+ !lB){B + ~B)T P}l~
- LP(B
==
14th World Congress of IFAC
+ ~/1 -
L(B
+ ~B)BT PIT p+
+ AA -
L(B
+ ~B)BT P]}x
\Vit.h (5) in mind and noticing the block diagonal property of L, E and tJ.B~BT, one has
It follows frolll (15)l (17), (18) and (lfi) that xT
{[A
+ ~i1 + (B + ~H)I{lT P+
P(.4
~
+
- 0 11 X 11
2
~jt
+ (B + t1H)J{]}x
.
By Definition, it conclude~ that the systen1 (11) is q uadratic stable.
1. CONC:LlJSION In this paper, a robust decentralized control ~cheIne for uncertain interconnected systerns is proposed lAoThere the uncertainties are value bounded a.nd appear both in the subsystenlS and in the interconnections between the subsystelns, The pro~ posed controller is based on a solution of a RJiccati~ type equation and it guarantees the controlled systems to be quadratic stable. 1
[{,EFE:R,ENCES
Gutlnan, 5.(1979). (Jncertain dynanlical systenl: Lyapunov min~nlax approach. IEEE Trans. AutOIn. Control, AC-24, 437-443. Ikeda: 1\1. and D~ D. Siljak(1980). Decentralized stabilization of linear tilne-varying syst.en1S. IEEE TraIls. ~4utom. COlltrol~ ~J\.C~- 25, 106-107.
3453
Copyright 1999 IFAC
ISBN: 0 08 043248 4