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On the Stabilization and Pole-assignment of Decentralized Singular Systems
Copvrighl © (F.\C Large S,ale (.lerlin. (;f)R. J ~IH(I
S\Slt'IllS.
ON THE STABILIZATION AND POLE-ASSIGNMENT OF DECENTRALIZED SINGULAR SYSTEMS Hu Yangzeng* and Chen Shuzhong** *RI'.\I'(lI"("h Iw/i//I/I' (1/ A/I/(lIIIII/i( (;(lI//ml.
'if C/ll'llIiml TI'r//I/lJI(lg'''. Sh{lIIglllli. PRC (1/ .\la/hl'lI/a/in. f ."{/.\/ Chil/a .\'(11'11/111 L ·1/i;'l'ni/y. SllIlIIglllli. PIU :
I:."{/.\ / ("Ill/Ill CI/il 't' ni/.\"
** f)l'/HI 1'/11/1'11/
Abstract. ~he problems of stabilization and pole-assignment for decentralized singular systems are studied i~ this paper. In order to solve these problems. the concepts of decentralized fixed exponential modes and decentralize d fixed pulse modes a~e applied. Under a given assumption. called the matching assumption. we g ive necessary and sufficient conditions for stabilization and arbitrarily approximate pole- a ssig nment of decentralized singular systems. In the proof. we give a way to construct such controllers. Keywords. Singular systems; large-scale systems; linear systems; stabilization; pole-assig nment; fixed pulse modes; fixed exponential modes.
N
INTRODUCTION
-B=[B -" "" nxr ,r=L-r. ~ 1 ,D 2 ' ...• D N)ER i=1 1
Consider the continuous linear decentralized system given by .
EX(t )=Ax(t )+L 'S.u (t). x(O-)=x 1=1
{
Yi ( t ) x~R
where
-T -T -T T mxn N C=[C 1 ,C 2 ,··· ,CN) fiR • m=5 mi
N 1
i
O
(1 )
~(E.A)={s: det(sE-A)=O}. ~(A)~(I.A)
Let
=c\x(t ), i =1 • 2 , ••• , N
n
C-={s: Re(s)
r. is the state,
ui~R
1
m. and YitR
1
It is well known that there are matrices ~ such that the system (2) is r.s.e. (rest~icted system equivalent) to the singular system
p and
are the input and output, respectively, of the ith local control station (i=1 ,2, ••• , nxn N). EfR , A, E. and C. (i=1.2, •.. , N) aTe 1
1
i'1 (t) =HX 1 (t)+FU (t) . x1( O-)=x10
real matrice s of appro~riate sizes. Assume that det(sE-A ) t O, rank(E)=q. deg(det(sE -A) ) =p, ana p
j
Ji 2 (t ) =x 2 (t)+Gu(t) ,
when N=1, the system (1) is a centralize d singular system. For centralized singular systems. Cobb(1981). Lewis and OzcaldiTan (1985) studied the problem of pole assi gnment by state feedback. Some investigators (Cobb,1q83; Bender and Laub,1985; Pandolfi,1980) studied the problem of optimal regulation. Shayman and Zhou(1Q S7 J . Shayman(1988) studied the control problem by constant ratio proportional and derivative feedback.
x ( O-)=x 2 20
y(t )=Dx(t)
n. here XiER 1(i=1 ,2). H.F.G.J and Dare mama trices of appropriate size, and J is nilpotent with index of nilpotency k, and
Because there a~e many large-scale systems which are sin~l a r in practice, so we have to study the problems of stabiliza:ion and pole-assignment fo~ decentralized singul ~ r systems. This paper is devoted to studying these problel!ls.
The state solution of the system (2) is
P"l.OBLEM STA'i.'EHE:-l'T The system (1) can be written as
{
Ei(t)=~:(t)+BU ( t). x(O-)=xO
(2 )
y(t)= Cx( t) (. ) where u 1 (t) denotes the ith de~ivative
where 1-1:\
HlI Yangzeng and Chcn Shllzhollg
144
of u(t). and S(i)(t) de~otes the ith vative of the unit i~pulse function.
de~i
F~om (4).
we see that besides (stahle andl or unstable) exponential motion. the state i(t) and output y(t) of the system (2) contain impulse motion if and only if k~2. It can easily be shown that k=' if and only if p=q. So when p
rhe set of local output feedback control laws applied to the system (2) is
realizations of the IS K*. It is obvious that the controller (6) is under t~e IS K*, i.e .• the input o~ the ith c o ntrol station is produced only from tl-J.e output y.(t) observe d on the ith stati o n. 1 Definition 1. If the matrices \2Jsatisfy
E and A in
p= c e g (det(sE-A))
2:i (t )=8 i zi (t )+Riy i (t) ui(t)=Qizi(t)+KiYi(t)+vi(t) { i=' .2, ••• ,N
n.
here zi(t)~R 1.is the state of the ith r.
feedback controller. v.(t)tR 1 is the ith 1
local external input, Si' T{i' Q i and ;(i are real constant matrices of appropriate size. The controller cribed by
(5) can compactly be des-
Z(t)=SZ(t)+Ry(t) (6 )
{ u(t)=Qz(t)+Ky(t)+v(t) here
z(t)=[z;(t),z~(t),
•••
,z~(t)lTE:
Rn
R=block diag(R, .H 2 ,··· ,RN)
1. Wh i ch conditions should be satisfied for that the decentralized singular system (2) is stahilizable or arbitrarily approximately pole assignable under the IS K*.
Q=block diag(Q, 'Q2' ...• QN) K=block diag(K, .K 2 ,··· ,K N ) Ap~lying the controller (6)
to the system (2), we obtain the closed-loop system
RC
BQJ [X(t)j+[B]V(t) S z(t) 0
(7 )
Let K* be the information structure (IS) of the decentralized singular system (2) (Hu Yangzeng and Jiang Weisun.'984 ) . and R(K*) be the set
1[
R(K*)- K=
K,
",
1 KN
I
r.~m.
"~ ~R 1
l\.i
1
Definition ~. ro~ a given nonempty s ymmetric open set G in the com~lex plane, if tl-J.ere exists a controller (6) such that the closed-loop system (7) has no pulse mo d e and all exponential modes are in G, t l-J.en the syste~ (2 ) is said to be pole assi ~nable to G under the IS K*. If for any nonempt y symmetric open set G in the complex plane, the s y stem (2 ) is pole assi g nable to G under the IS K*. then the system (2) is said to be arbitrarily approximately pole assignable under the IS K*. This paper studies the p~oblems of stabilization and pole assignment for decentralized singular systems. The problems consist of two subproblems:
S=block diag(8, 'S2 ••.• 'SN)
[ E~(t)]=[A+~KC z(t)
Definition . If the system (2 ) has pulse ~ode s o r unstable exponential mode(s ) . then t ~ e systen (2) i~ said to be unstable. o:he~wise. stable. IT the system (2) is u~stable. an~ tl-J.ere is a controller (6) such that the closed-loop system (7 ) is stable. th e n tl-J.e singula~ system (2 ) is said to be stabilizable under the IS K*.
1
2 . If the decentralized singular s ystem (2) is stabilizable or arbitrarily a pproximately pole a ssignable under the IS K*, h ow we ~ ps j gn a decentralized controller (6 ) suc~ t~at the closed-loop syste~ (7) is stable. o~ the s yste~ (7 ) h 9 s no puls e ~oce ana all its exponential modes are i!1 G. ?itE L Ir~INA l.Y
IJEV3LUPMENT
Since rank(E)=q
(8)
det(sE-A-BKC)to
R(K*) is said to be the set of all regular
(9 )
Decentralized Singular Systems
where I q is the unit matrix of order q. It is obvious that there are many P and Q such that (g) holds. Let
{
W= (P,Q)
145
deg det(sE-A)=deg det(P(sE-X)Q) =de g det(
P, QtRnXn, det(p)~o.}
I
det(Q)~O,
[
sI -All q -A 21
(14 )
(10)
(9) holds
It is obvi ous that W i s nonempty. Takin~ (P,Q)tW, let x(t)=[xf(: ) x~(t)JT=~-li(t),
and rank(E)=rank(PBQ) =q . Thus the system (2) has no pulse mode if and onl y i~ the system (12) has no pulse mode. 2) ~ 3). Since
xl~nq, x2~~n-q. The system (1) is ~.s.e. (15 )
to the decentralized singular system given by
b Bl i u i (t) N
X1 (t )=A 11 xl (t )+A 1 2 x 2 (t )+
I
if and only if A22 is nons ingular.
N
O=A 21 xl (t )+A 22 x 2 ( t)+ EB 2i u i (t)
(11 )
Yi (t)=C i1 xl (t )-C i2 x 2 (t), i=l ,2, ... ,N
here All' A12 , A21 , A22 , B'i and Cij (j=l, 2; i =1,2, ... ,N ) are matric~s of appropriate size, and
The system (11) can be written as
Lemma 3. The following asserti ons are equivalent to each other. 1) . The system (2) is stable: 2 ). For s ome/any (p , Q) ~W, the s ystem (12) which is r .s.e. to the system (2) with respect to (p,~) is stable; 3). For some/any (p, Q)tW, A?2 is nonsingu-
l~f and ~(Iq,All )cC-, here Al1=Al1~A12 A A , A , 1,j=1,2, a~e the matrlces ij 22 21 in the system (12) which is r.s.e. to the system (2) with respect to (p,Q). ~. This lemma can be proved from Lemma 1 and Lemma 2. The details are omitted.
Xl ( t )=A 11 Xl (t )+A 12 X 2 (t )+B 1u( t)
I
O=A21 xl (t)+A 22 x 2 (t)TB 2u(t )
DELETI ON OF ( 12)
First, we give the following result: Theorem 1. If the system (2) has fUlse models), and there is a controller 6) under the IS K* such that the closed-loop system (7) has no pulse mode, then the static output feedback controller
y(t)=C 1x 1 (t)+C 2 x 2 (t)
here Bj=[Bjl,Bj2, ... ,BjN]'
Cj=[C~j,C~j"",C~jlT,
PULSE MODE
j=l,2.
u(t)=Ky( t)
Denote B=[B; B~JT, c=lc 1 C2 ], the system (12) can further be written as Ex(t)=Ax(t)+Bu(t), y(t)=Cx(t)
(16 )
can also delete the pulse models) of the s ystem (2), i.e .. the closed-loop sys t em ( 17 )
(13) has
In the following , the system (12 ) and (13) is sai d to be r.s.e. to the system (2) with respect to (p,Q). Lemma 1. For any (p , Q) tW, the system (2) and the system (1, ) which is r.s.e. to (2) with respect to (p,Q) have the same set of exponential ~odes, i.e., g(E,A)=v( E,A). Pr oof. ~(E,A ) ={s:det(sE-A ) = O J ={s:det( ~(s~-A )~)=~ }
'10 pulse mode. Proof. Since the singular system (7) has no pulse mode, hence
BQ] S )=rank(E)+n On the other hand, we can prove that deg det( [
= {s :det ( sB-A ) =O}=r-(E,A) Lemma 2. The fol lowi ~l .g assertions a re equi vale'1t to each other. , ). The system (2) has no pulse mode; 2). For some / any (P,~)EW, t he system (12) which is ~.s.e. to the system (2) wi t h r espect to (p, ~ ) has no pulse mode; 3 ) . For some/any (p,Q)tW, the matrix A22 in the s ystem (12) which is r.s.e. to the s y stem (2) wi th respect to (p, Q) is nonsingular. ~. 1) ~ 2 ) . Because P and Q a~e nonsin~ lar, hence
SE-(A+i3KC)
-qC
-BQ s1--S
1
)
n
So we have deg det(sE-(A+BKC»=rank(E)
(18)
This means that the singular system (17) has no pulse mode. Theorem 1 tells us that if we want to delete the pulse models) of Singular system via output feedback c ontroller, it is sufficient only to use static output feed-
HlI Yangzc ng a nd Che n Shllzhong
146
back controller. The fo llowing theorem gives one condi t ion fo r the deletion of pulse mod els }. Definition
~.
If
is sai d to be the se t of fixed exponential modes of the singular s y stem (2 ) with r espec t t o the I S K*, o r is said to be t he set of decent r a li zed f ixed exponenti ql modes of the system (2) .
max de g det(sE- ( A+BKc} }
Rema~k 1. Fo r the re ~ la~ s ystem (A. B. C) . t he s et of decent~a l ized fixed modes is denoted by FM(A.B. C; K* }. So we have FE~((I .A). B.C; K*)=FM(A. B.C; K* }.
Theorem 2. The s ys tem ( 2) ha s decen t ralized fixed pu lse mo de i f and onl y if f o r some/any ( p. Q} tW. 0 is a de c ent~alized fixed mode of the s yst em (A 22 .B 2 . C2 ). here A • B • C occur i n t ~e syst em (1 2) whi ch 22 2 2 i s 1.". s. e. to th e s .v s tem ( 2 ) with respect to (p.Q ) . Proof. From Definition 5. the system (2) has decentralized f. ixed pulse mode(s} if and only if for a ny Kt R(K* )
C A22 B2 =bl ock di qg(D 1 . D •...• DN) (22 ) 2 2 m.xr . 1 1 where DitR A22 • B2 • C2 occur in the s ys t em (12 ) which is r .S.e. to the s y stem (2 ) wi t h re s pec t cO ( p .Q ) .
Obviously. (19) holds if and only if for some/any (P. Q}EW.
he r e
p=
deg det(P(sE-(A+BKC) }Q} =(
here
(20)
q
-A 21
AlJ .. =A l. J. +B.KC 1_ j
. Obviously.
fo~
system
-1
Lemma 4. If there i s s ome ( p.Q ) ~W s uc h that A22 is nonsingul a r and (2 2) holds. then fo~ any (P.Q)EW.
C2A2~B2=C2 A2 ~ B2
A22 =LPAQR. B2=LPB.
C 2=CQR
( 23)
L= {Of I n-q ]. R= [oT. I n- q ) T
P and Q are nonslngular. and PEQ=E. It is obvious that there are nonsin~l a r matrices Rand L such that P=LP. ~=Q R. Let
P~of. Since (ELg }EW. so
sI -A11 [
I n this section. we assume that (2 ) . there is (P .Q )EW such t hat
(2 0) holds
if and only if A22=A22+B2KC2 is singular. This means that the theorem holds. Theorem 3. If the pulse mode (s ) of t he sin~ lar system (2) can be deleted via t he static ou t put feedback controller (16). t hen for almos t all Kt R(K*). when t he controller is coupled with t he s ystem (2). the closed-loop sy stem (17) has no pulse mode. Proof. Taking (P. Q}EW. let the system (12 ) is r.s.e. to the system (2) with respect to (p.Q). F~om Theorem 2. there is KOt R(K*) such that de t (A 22+B2KoC2 }1 0 . It is obvi ou s that
n .Xn.
here Lij.RijER
1
J . n 1 =o . n 2 =n-q. Since
So. L11R11=Iq. L 11 R12 =0 . L21 R11 =0. L21 R12 = =0 . hence, L11 • R11 • L22 and R22 a~e nonsingular. and L21 =0 . R12 =0 . From
{K: de6 net (sE- ( A+BKC})
(2 1 )
here A22=A22+B2KOC 2' Since det (A22 )1 0 • so the class of Kt R( K*) which satisfies ( 21) is either empty or lies on a hvpersurface in the parameter space of K ~. So for almost all KER(K* ) . t he sy stem (17 ) has no pulse mode. STABI LI ZATIO N OF UNSTABLE DECENTRALIZED SINGULAR SYSTEMS The following conc e pt is r equired in t his section. Defini t ion 6. The set FEM( ( E.A }.B.C; K* ) defined by FEM( (E.A ) .B.C; K*)=
n cr(E.A+BKC) KER( K* }
B=PB=LPB=LB=l: :J. C=CQ=CQR=CR=[C 1 we have A22=L22A22R22' SO. we have
C2 ]
B2=L 22 B2 • C2 =C2 R22 ,
Le~a 5. If there is some ( p. Q)(W such that A22 is nonsingular and ( 22 ) holds. tQen for_any (P. Q) EW a nd KER (K* ) . when ( A22 +B 2 KC2 ) is nonsingular. we have
C2 ( A22+B2KC2 } -1B2~blO C k diag(D 1 •...•DN)
Decent ralized Singular S\'stcms ~.
and
Since
- ~ - - -1- --1-1~ --1C2(A22+B2KC2) B2=(I+C2A22B2K) C2A22 B2 =block diag(D 1 ,15 2 "" ~
here
Di=(I+DiK i )
-1
,DN)
mixr i
Di~R
Definition . If there is some/any (p,Q)~W, such that 22) holds, where A22 , B2 and C2 occur in the system (12) which is ~.s.e. to the system (2) with ~espect to (P,Q), then we say that the system (2) satisfie s the matching condition. Lemma 6. For any K~R(K*) FEM((E,A+BKC),B,C;K*)=FEM((E,A),B,C;K* ) Proof. It is obvious that FEM((E,A+BKC),E,c;K*)
n
~(E,A+B(K+K)C)
KER(K*) ~(E,A+BKC)=FEM((E,A),B,C;K*).
n
147
KtR(K* ) Lemma 7. For any
(p,Q)~W,
here E, A, Band C occu~ in the system (12) which is ~.s.e. to the system (2) with respect to (p,Q).
n
Lemma g. Assume that the system (12) is ~.s.e. to the system (2) with respect to some (p, Q) ~W, A22 in (12) is nonsingular, the system (2) satisfies the matching condition and G is a ~iven nonempty open set in the complex plane, then the system (2) is pole-assi~nable to G unde~ the IS K* if and only if the system
X1 (t ) ~~ 1 1x 1 (t l+131u ( t ) _ { y ( t ) =C , x 1 (t )
(25)
is pole-assignable to G, where A11 , E1 . C, occur in (24). Proof. Necessity. Suppose that there is a controller (6) under the IS K* constructed for that when it is coupled with the system (2), the closed-loop system (7) has no pulse mode and all its poles are in G. 1). If (A +B KC ) is nonsingular, using 22 2 2 Lemma 1, we can prove that when the controller Z(t)=SZ(t)+Ry(t) { u(t)=Qz(t)+Ky(t)
FEM((E,A),B,C;K*)=FEM ( (~,A),B,C;K*)
Proof. FEM((E,A),B,C;K*)=
So /Lemma 8 holds. 7he details are omitted.
(26)
is coupled with the system (25), the closed loop system (27)
cr(E,A+BKC)
KE: R(K*) =
n
is: det(sE-(A+BKC) )=O}
n
{s: det(sE-(A+BKC))=O }
n
u (E,A+BKC)=FEM((E,A),B,C;K*).
KtR(lC* )
=
has also no pulse mode and all its poles are in G, here
Kt R(K*) K~R(K*)
Lemma 8. Assume that the system (12) is ~.s.e. to the system (2) with respect to some (p,Q)~W, A22 in (12) is nonsingular and the system (12 ) satisfies the matching condi tion, then FEM( (E, X), 13, C; K* )=FM(A'1 ,13 1 'C1 ;K*)
(24)
T=
n
KER( ., *)
cr(E,A+BKC)
det(A22+B2KC2)~O
Using Lemma 1, Lemma 2, Lemma 6 , LemMa 7 and the assumptions given in this lem~a we can prove that FEM((E,A),B,C;K*)=T
It is obvi ous that due to the matching condition, the contro ' l e ~ (26) is under the IS K* . 2). If (A 22 +B 2 KC 2 ) is singular, since A22 is~nonsingular, such K is on a hypersurface in,parameter space of the IS K*. We can slightly perturb K under the IS K* so that (A 22 +B 2 KC 2 ) is nonsingular and the matching condition is still satisfied (Lemma 5). Sufficiency . Suppose that there is a cont~oller (26) under K* so that when (26) is coupled with the system (25), the closedloop system (27) has no pulse mode and all its poles are in G. For the same reason ~ iven in the proof of necessity , we can assume that (A 22 +B 2 KC ) is nonsingular. 2 we c a n construct the controller z(t)=Sz(t )+Ry( t), u(t)=Qz(t)+Ky(t)
(28)
so that when (28) is coupled with the sys-
Hu Yangzeng and Chen Shuzhong
148
tem (2), the closed-loop system has no pulse mode and all its poles are in G. here _ -1 -1K=(I-KC 2A22 B2 ) K -1-1 ""')-1 -1 Q~(I-KC2A22B2) Q, R=R(I-C 2 A22 B2 K - -1"" -1 -1 ?( S=S+R(I-C 2A22 B2K) C2A22B2~ Due to the matching condition, the controller (28) is under the IS K*. Theorem 4. Let G be a given nonempty symmetric open set in the complex plane and the system (2) satisfy the matc " ing condition, then the system (2) is pole assignable under the IS K* if and only if the following two conditions are satisfied: 1). The system (2) has no fixed pulse mode ~i~h re~ect to the IS K*; 2). FEM((E,A),'B,C;K*)<=G. (29) Proof. Necessity. Suppose that the system ~s pole assignable to G. 1). From Theorem 1 and Definition 5, the system (2) has no fixed pulse mode with respect to the IS K*. 2). Taking any (P,Q)~W, let the singular system (13) be ~.s.e. to the system (2) with respect to (p,Q). From 1) of this theorem, there is Ko~R(K*) so that (A 22 + B2KOC2 ) is nonsingular. From Lemma 9, the decentralized singular system x1(t)=A11x1(t)+B1u(t), y(t)=C 1x 1(t) (30) is pole aSSignable to G, where -1
""
A11 =F 11 -
.,.
F12F22F21, B1=B1-F12F22B2' C1=C1-C2F22F21, Fij=Aij+BiKOCj' From the result ~iven by Wang and Davison(1973), the system (30) is pol~ as~iguable to G if and only if FM(A ,E ,C1;K*)cG. From Lemma 7, 6 and 8, 11 1 FEM((E,X),B,C;K*)=FEM((E,A),B,C;K*) =FEM((E,(A+BKOC),B,C;K*) =FM(A 11
,B 1 'C1 ;K*)
hence (2Q) holds. Sufficiency. Arbitrarily taking (p,Q)~W, let the system (13) be r.s.e. to the system (2) with respect to (p,Q). Since the system (2) has no pulse mode with respect to the IS K*,' so there is KO~R(K*) so that (A22!B2KQC2) i~*nonSingular: From (31), FM(A 11 ,B 1 '~1;K leG. AccoroJ.ng to the method given by Wang and Dqvison(1Q73) we can const~ct the controller given by z(t)=Sz(t)+Ry(t), u(t)=Qz(t)+Ky(t) so that the all poles of the closed-loop system
are in G. It can be proved that there is a controller given by Z(t)=SZ(t)+Ry(t) { u ( t ) =Q z ( t ) + Ky ( t )
is coupled with the system (2), the closed loop system has no pulse mode, and the set of all exponential modes is the same with the set of poles of the system (32). Theorem 5. Let the system (2) satisfies the matching condition, then the system (2) is stabilizable under the IS K* if and only if the following two conditions are satisfied: 1). The system (2) has no pulse mode; 2). FEM((E,X),B,C;K*)cC-. Proof. Taking G as the left open half complex plane in Theorem 4, we obtain this theorem. Theorem 6. Let the system (2) satisfy the matching condition, then the system (2) is arbitra~ily approximately pole assignable under the IS K* if and only if the following two conditions are satisfied: 1). The s~stem (£) has no pulse mode; 2). FEM((E,A);B,C;K*)=~. Proof. Takin~ two disjoint nonempty symmetric open sets in the complex plane, and using Theorem 4, we can prove this theorem. The details are omitted.
REFERENCES Bender, D.J. and A.J. Laub (1985). The linear-quadratic optimal regulator for descriptor systems. in Proc. 24th IEEE Conf. Decision Contr., Ft. Lauderdale, FL, Dec. 1985, 957-962. Cobb, J.D. (1983). Descriptor variable systems and optimal state regulation. IEEE Trans. Automat. Contr., Vol. AC28, 601-611. Cobb, J.D. (1981). Feedback and pole placement in descriptor variable systems. Int. J. Contr., Vol. 33, 1135-1146. Hu, Y.Z. and Jiang W.S. (1984). New characterization of decentralized fixed modes and their applications. Proc .. of the Ninth Triennial World Congress of IFAC, Vol. 3, 11 83-1188. Lewis, F.L. and K. Ozcaldiran (1985). On the eigenst~cture assignment of singular systems. in Proc. 24th IEEE Conf. Decision Contr., Ft. Lauderdale, FL, Dec. 1985, 179-182. Pandolfi, L. (1980). Controllability and stabilization for linear systems of algebraic and differential equations. J. Optimiz. Theory Appl., Vol. 30, 601-620. Shayman, M.A. (1Q ~ 8). Pole placement by dynamic compensation for descriptor systems. Automatica, Vol. 24, 279-282. Shayman, M.A. and Z. Zhou (1987). Feedback control and classification of generalized linear sY' cems. IEEE T~ans. Autmat. Contr., Vol. AC-32, 483-494. Wang, S.H. and E.J. Davison (1973). On the stabilization of decentralized control system. IEEE T-ans. Automat. Contr., Vol. AC-18, 473-478.
S\Slt'IllS.
ON THE STABILIZATION AND POLE-ASSIGNMENT OF DECENTRALIZED SINGULAR SYSTEMS Hu Yangzeng* and Chen Shuzhong** *RI'.\I'(lI"("h Iw/i//I/I' (1/ A/I/(lIIIII/i( (;(lI//ml.
'if C/ll'llIiml TI'r//I/lJI(lg'''. Sh{lIIglllli. PRC (1/ .\la/hl'lI/a/in. f ."{/.\/ Chil/a .\'(11'11/111 L ·1/i;'l'ni/y. SllIlIIglllli. PIU :
I:."{/.\ / ("Ill/Ill CI/il 't' ni/.\"
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Abstract. ~he problems of stabilization and pole-assignment for decentralized singular systems are studied i~ this paper. In order to solve these problems. the concepts of decentralized fixed exponential modes and decentralize d fixed pulse modes a~e applied. Under a given assumption. called the matching assumption. we g ive necessary and sufficient conditions for stabilization and arbitrarily approximate pole- a ssig nment of decentralized singular systems. In the proof. we give a way to construct such controllers. Keywords. Singular systems; large-scale systems; linear systems; stabilization; pole-assig nment; fixed pulse modes; fixed exponential modes.
N
INTRODUCTION
-B=[B -" "" nxr ,r=L-r. ~ 1 ,D 2 ' ...• D N)ER i=1 1
Consider the continuous linear decentralized system given by .
EX(t )=Ax(t )+L 'S.u (t). x(O-)=x 1=1
{
Yi ( t ) x~R
where
-T -T -T T mxn N C=[C 1 ,C 2 ,··· ,CN) fiR • m=5 mi
N 1
i
O
(1 )
~(E.A)={s: det(sE-A)=O}. ~(A)~(I.A)
Let
=c\x(t ), i =1 • 2 , ••• , N
n
C-={s: Re(s)
r. is the state,
ui~R
1
m. and YitR
1
It is well known that there are matrices ~ such that the system (2) is r.s.e. (rest~icted system equivalent) to the singular system
p and
are the input and output, respectively, of the ith local control station (i=1 ,2, ••• , nxn N). EfR , A, E. and C. (i=1.2, •.. , N) aTe 1
1
i'1 (t) =HX 1 (t)+FU (t) . x1( O-)=x10
real matrice s of appro~riate sizes. Assume that det(sE-A ) t O, rank(E)=q. deg(det(sE -A) ) =p, ana p
j
Ji 2 (t ) =x 2 (t)+Gu(t) ,
when N=1, the system (1) is a centralize d singular system. For centralized singular systems. Cobb(1981). Lewis and OzcaldiTan (1985) studied the problem of pole assi gnment by state feedback. Some investigators (Cobb,1q83; Bender and Laub,1985; Pandolfi,1980) studied the problem of optimal regulation. Shayman and Zhou(1Q S7 J . Shayman(1988) studied the control problem by constant ratio proportional and derivative feedback.
x ( O-)=x 2 20
y(t )=Dx(t)
n. here XiER 1(i=1 ,2). H.F.G.J and Dare mama trices of appropriate size, and J is nilpotent with index of nilpotency k, and
Because there a~e many large-scale systems which are sin~l a r in practice, so we have to study the problems of stabiliza:ion and pole-assignment fo~ decentralized singul ~ r systems. This paper is devoted to studying these problel!ls.
The state solution of the system (2) is
P"l.OBLEM STA'i.'EHE:-l'T The system (1) can be written as
{
Ei(t)=~:(t)+BU ( t). x(O-)=xO
(2 )
y(t)= Cx( t) (. ) where u 1 (t) denotes the ith de~ivative
where 1-1:\
HlI Yangzeng and Chcn Shllzhollg
144
of u(t). and S(i)(t) de~otes the ith vative of the unit i~pulse function.
de~i
F~om (4).
we see that besides (stahle andl or unstable) exponential motion. the state i(t) and output y(t) of the system (2) contain impulse motion if and only if k~2. It can easily be shown that k=' if and only if p=q. So when p
rhe set of local output feedback control laws applied to the system (2) is
realizations of the IS K*. It is obvious that the controller (6) is under t~e IS K*, i.e .• the input o~ the ith c o ntrol station is produced only from tl-J.e output y.(t) observe d on the ith stati o n. 1 Definition 1. If the matrices \2Jsatisfy
E and A in
p= c e g (det(sE-A))
2:i (t )=8 i zi (t )+Riy i (t) ui(t)=Qizi(t)+KiYi(t)+vi(t) { i=' .2, ••• ,N
n.
here zi(t)~R 1.is the state of the ith r.
feedback controller. v.(t)tR 1 is the ith 1
local external input, Si' T{i' Q i and ;(i are real constant matrices of appropriate size. The controller cribed by
(5) can compactly be des-
Z(t)=SZ(t)+Ry(t) (6 )
{ u(t)=Qz(t)+Ky(t)+v(t) here
z(t)=[z;(t),z~(t),
•••
,z~(t)lTE:
Rn
R=block diag(R, .H 2 ,··· ,RN)
1. Wh i ch conditions should be satisfied for that the decentralized singular system (2) is stahilizable or arbitrarily approximately pole assignable under the IS K*.
Q=block diag(Q, 'Q2' ...• QN) K=block diag(K, .K 2 ,··· ,K N ) Ap~lying the controller (6)
to the system (2), we obtain the closed-loop system
RC
BQJ [X(t)j+[B]V(t) S z(t) 0
(7 )
Let K* be the information structure (IS) of the decentralized singular system (2) (Hu Yangzeng and Jiang Weisun.'984 ) . and R(K*) be the set
1[
R(K*)- K=
K,
",
1 KN
I
r.~m.
"~ ~R 1
l\.i
1
Definition ~. ro~ a given nonempty s ymmetric open set G in the com~lex plane, if tl-J.ere exists a controller (6) such that the closed-loop system (7) has no pulse mo d e and all exponential modes are in G, t l-J.en the syste~ (2 ) is said to be pole assi ~nable to G under the IS K*. If for any nonempt y symmetric open set G in the complex plane, the s y stem (2 ) is pole assi g nable to G under the IS K*. then the system (2) is said to be arbitrarily approximately pole assignable under the IS K*. This paper studies the p~oblems of stabilization and pole assignment for decentralized singular systems. The problems consist of two subproblems:
S=block diag(8, 'S2 ••.• 'SN)
[ E~(t)]=[A+~KC z(t)
Definition . If the system (2 ) has pulse ~ode s o r unstable exponential mode(s ) . then t ~ e systen (2) i~ said to be unstable. o:he~wise. stable. IT the system (2) is u~stable. an~ tl-J.ere is a controller (6) such that the closed-loop system (7 ) is stable. th e n tl-J.e singula~ system (2 ) is said to be stabilizable under the IS K*.
1
2 . If the decentralized singular s ystem (2) is stabilizable or arbitrarily a pproximately pole a ssignable under the IS K*, h ow we ~ ps j gn a decentralized controller (6 ) suc~ t~at the closed-loop syste~ (7) is stable. o~ the s yste~ (7 ) h 9 s no puls e ~oce ana all its exponential modes are i!1 G. ?itE L Ir~INA l.Y
IJEV3LUPMENT
Since rank(E)=q
(8)
det(sE-A-BKC)to
R(K*) is said to be the set of all regular
(9 )
Decentralized Singular Systems
where I q is the unit matrix of order q. It is obvious that there are many P and Q such that (g) holds. Let
{
W= (P,Q)
145
deg det(sE-A)=deg det(P(sE-X)Q) =de g det(
P, QtRnXn, det(p)~o.}
I
det(Q)~O,
[
sI -All q -A 21
(14 )
(10)
(9) holds
It is obvi ous that W i s nonempty. Takin~ (P,Q)tW, let x(t)=[xf(: ) x~(t)JT=~-li(t),
and rank(E)=rank(PBQ) =q . Thus the system (2) has no pulse mode if and onl y i~ the system (12) has no pulse mode. 2) ~ 3). Since
xl~nq, x2~~n-q. The system (1) is ~.s.e. (15 )
to the decentralized singular system given by
b Bl i u i (t) N
X1 (t )=A 11 xl (t )+A 1 2 x 2 (t )+
I
if and only if A22 is nons ingular.
N
O=A 21 xl (t )+A 22 x 2 ( t)+ EB 2i u i (t)
(11 )
Yi (t)=C i1 xl (t )-C i2 x 2 (t), i=l ,2, ... ,N
here All' A12 , A21 , A22 , B'i and Cij (j=l, 2; i =1,2, ... ,N ) are matric~s of appropriate size, and
The system (11) can be written as
Lemma 3. The following asserti ons are equivalent to each other. 1) . The system (2) is stable: 2 ). For s ome/any (p , Q) ~W, the s ystem (12) which is r .s.e. to the system (2) with respect to (p,~) is stable; 3). For some/any (p, Q)tW, A?2 is nonsingu-
l~f and ~(Iq,All )cC-, here Al1=Al1~A12 A A , A , 1,j=1,2, a~e the matrlces ij 22 21 in the system (12) which is r.s.e. to the system (2) with respect to (p,Q). ~. This lemma can be proved from Lemma 1 and Lemma 2. The details are omitted.
Xl ( t )=A 11 Xl (t )+A 12 X 2 (t )+B 1u( t)
I
O=A21 xl (t)+A 22 x 2 (t)TB 2u(t )
DELETI ON OF ( 12)
First, we give the following result: Theorem 1. If the system (2) has fUlse models), and there is a controller 6) under the IS K* such that the closed-loop system (7) has no pulse mode, then the static output feedback controller
y(t)=C 1x 1 (t)+C 2 x 2 (t)
here Bj=[Bjl,Bj2, ... ,BjN]'
Cj=[C~j,C~j"",C~jlT,
PULSE MODE
j=l,2.
u(t)=Ky( t)
Denote B=[B; B~JT, c=lc 1 C2 ], the system (12) can further be written as Ex(t)=Ax(t)+Bu(t), y(t)=Cx(t)
(16 )
can also delete the pulse models) of the s ystem (2), i.e .. the closed-loop sys t em ( 17 )
(13) has
In the following , the system (12 ) and (13) is sai d to be r.s.e. to the system (2) with respect to (p,Q). Lemma 1. For any (p , Q) tW, the system (2) and the system (1, ) which is r.s.e. to (2) with respect to (p,Q) have the same set of exponential ~odes, i.e., g(E,A)=v( E,A). Pr oof. ~(E,A ) ={s:det(sE-A ) = O J ={s:det( ~(s~-A )~)=~ }
'10 pulse mode. Proof. Since the singular system (7) has no pulse mode, hence
BQ] S )=rank(E)+n On the other hand, we can prove that deg det( [
= {s :det ( sB-A ) =O}=r-(E,A) Lemma 2. The fol lowi ~l .g assertions a re equi vale'1t to each other. , ). The system (2) has no pulse mode; 2). For some / any (P,~)EW, t he system (12) which is ~.s.e. to the system (2) wi t h r espect to (p, ~ ) has no pulse mode; 3 ) . For some/any (p,Q)tW, the matrix A22 in the s ystem (12) which is r.s.e. to the s y stem (2) wi th respect to (p, Q) is nonsingular. ~. 1) ~ 2 ) . Because P and Q a~e nonsin~ lar, hence
SE-(A+i3KC)
-qC
-BQ s1--S
1
)
n
So we have deg det(sE-(A+BKC»=rank(E)
(18)
This means that the singular system (17) has no pulse mode. Theorem 1 tells us that if we want to delete the pulse models) of Singular system via output feedback c ontroller, it is sufficient only to use static output feed-
HlI Yangzc ng a nd Che n Shllzhong
146
back controller. The fo llowing theorem gives one condi t ion fo r the deletion of pulse mod els }. Definition
~.
If
is sai d to be the se t of fixed exponential modes of the singular s y stem (2 ) with r espec t t o the I S K*, o r is said to be t he set of decent r a li zed f ixed exponenti ql modes of the system (2) .
max de g det(sE- ( A+BKc} }
Rema~k 1. Fo r the re ~ la~ s ystem (A. B. C) . t he s et of decent~a l ized fixed modes is denoted by FM(A.B. C; K* }. So we have FE~((I .A). B.C; K*)=FM(A. B.C; K* }.
Theorem 2. The s ys tem ( 2) ha s decen t ralized fixed pu lse mo de i f and onl y if f o r some/any ( p. Q} tW. 0 is a de c ent~alized fixed mode of the s yst em (A 22 .B 2 . C2 ). here A • B • C occur i n t ~e syst em (1 2) whi ch 22 2 2 i s 1.". s. e. to th e s .v s tem ( 2 ) with respect to (p.Q ) . Proof. From Definition 5. the system (2) has decentralized f. ixed pulse mode(s} if and only if for a ny Kt R(K* )
C A22 B2 =bl ock di qg(D 1 . D •...• DN) (22 ) 2 2 m.xr . 1 1 where DitR A22 • B2 • C2 occur in the s ys t em (12 ) which is r .S.e. to the s y stem (2 ) wi t h re s pec t cO ( p .Q ) .
Obviously. (19) holds if and only if for some/any (P. Q}EW.
he r e
p=
deg det(P(sE-(A+BKC) }Q} =(
here
(20)
q
-A 21
AlJ .. =A l. J. +B.KC 1_ j
. Obviously.
fo~
system
-1
Lemma 4. If there i s s ome ( p.Q ) ~W s uc h that A22 is nonsingul a r and (2 2) holds. then fo~ any (P.Q)EW.
C2A2~B2=C2 A2 ~ B2
A22 =LPAQR. B2=LPB.
C 2=CQR
( 23)
L= {Of I n-q ]. R= [oT. I n- q ) T
P and Q are nonslngular. and PEQ=E. It is obvious that there are nonsin~l a r matrices Rand L such that P=LP. ~=Q R. Let
P~of. Since (ELg }EW. so
sI -A11 [
I n this section. we assume that (2 ) . there is (P .Q )EW such t hat
(2 0) holds
if and only if A22=A22+B2KC2 is singular. This means that the theorem holds. Theorem 3. If the pulse mode (s ) of t he sin~ lar system (2) can be deleted via t he static ou t put feedback controller (16). t hen for almos t all Kt R(K*). when t he controller is coupled with t he s ystem (2). the closed-loop sy stem (17) has no pulse mode. Proof. Taking (P. Q}EW. let the system (12 ) is r.s.e. to the system (2) with respect to (p.Q). F~om Theorem 2. there is KOt R(K*) such that de t (A 22+B2KoC2 }1 0 . It is obvi ou s that
n .Xn.
here Lij.RijER
1
J . n 1 =o . n 2 =n-q. Since
So. L11R11=Iq. L 11 R12 =0 . L21 R11 =0. L21 R12 = =0 . hence, L11 • R11 • L22 and R22 a~e nonsingular. and L21 =0 . R12 =0 . From
{K: de6 net (sE- ( A+BKC})
(2 1 )
here A22=A22+B2KOC 2' Since det (A22 )1 0 • so the class of Kt R( K*) which satisfies ( 21) is either empty or lies on a hvpersurface in the parameter space of K ~. So for almost all KER(K* ) . t he sy stem (17 ) has no pulse mode. STABI LI ZATIO N OF UNSTABLE DECENTRALIZED SINGULAR SYSTEMS The following conc e pt is r equired in t his section. Defini t ion 6. The set FEM( ( E.A }.B.C; K* ) defined by FEM( (E.A ) .B.C; K*)=
n cr(E.A+BKC) KER( K* }
B=PB=LPB=LB=l: :J. C=CQ=CQR=CR=[C 1 we have A22=L22A22R22' SO. we have
C2 ]
B2=L 22 B2 • C2 =C2 R22 ,
Le~a 5. If there is some ( p. Q)(W such that A22 is nonsingular and ( 22 ) holds. tQen for_any (P. Q) EW a nd KER (K* ) . when ( A22 +B 2 KC2 ) is nonsingular. we have
C2 ( A22+B2KC2 } -1B2~blO C k diag(D 1 •...•DN)
Decent ralized Singular S\'stcms ~.
and
Since
- ~ - - -1- --1-1~ --1C2(A22+B2KC2) B2=(I+C2A22B2K) C2A22 B2 =block diag(D 1 ,15 2 "" ~
here
Di=(I+DiK i )
-1
,DN)
mixr i
Di~R
Definition . If there is some/any (p,Q)~W, such that 22) holds, where A22 , B2 and C2 occur in the system (12) which is ~.s.e. to the system (2) with ~espect to (P,Q), then we say that the system (2) satisfie s the matching condition. Lemma 6. For any K~R(K*) FEM((E,A+BKC),B,C;K*)=FEM((E,A),B,C;K* ) Proof. It is obvious that FEM((E,A+BKC),E,c;K*)
n
~(E,A+B(K+K)C)
KER(K*) ~(E,A+BKC)=FEM((E,A),B,C;K*).
n
147
KtR(K* ) Lemma 7. For any
(p,Q)~W,
here E, A, Band C occu~ in the system (12) which is ~.s.e. to the system (2) with respect to (p,Q).
n
Lemma g. Assume that the system (12) is ~.s.e. to the system (2) with respect to some (p, Q) ~W, A22 in (12) is nonsingular, the system (2) satisfies the matching condition and G is a ~iven nonempty open set in the complex plane, then the system (2) is pole-assi~nable to G unde~ the IS K* if and only if the system
X1 (t ) ~~ 1 1x 1 (t l+131u ( t ) _ { y ( t ) =C , x 1 (t )
(25)
is pole-assignable to G, where A11 , E1 . C, occur in (24). Proof. Necessity. Suppose that there is a controller (6) under the IS K* constructed for that when it is coupled with the system (2), the closed-loop system (7) has no pulse mode and all its poles are in G. 1). If (A +B KC ) is nonsingular, using 22 2 2 Lemma 1, we can prove that when the controller Z(t)=SZ(t)+Ry(t) { u(t)=Qz(t)+Ky(t)
FEM((E,A),B,C;K*)=FEM ( (~,A),B,C;K*)
Proof. FEM((E,A),B,C;K*)=
So /Lemma 8 holds. 7he details are omitted.
(26)
is coupled with the system (25), the closed loop system (27)
cr(E,A+BKC)
KE: R(K*) =
n
is: det(sE-(A+BKC) )=O}
n
{s: det(sE-(A+BKC))=O }
n
u (E,A+BKC)=FEM((E,A),B,C;K*).
KtR(lC* )
=
has also no pulse mode and all its poles are in G, here
Kt R(K*) K~R(K*)
Lemma 8. Assume that the system (12) is ~.s.e. to the system (2) with respect to some (p,Q)~W, A22 in (12) is nonsingular and the system (12 ) satisfies the matching condi tion, then FEM( (E, X), 13, C; K* )=FM(A'1 ,13 1 'C1 ;K*)
(24)
T=
n
KER( ., *)
cr(E,A+BKC)
det(A22+B2KC2)~O
Using Lemma 1, Lemma 2, Lemma 6 , LemMa 7 and the assumptions given in this lem~a we can prove that FEM((E,A),B,C;K*)=T
It is obvi ous that due to the matching condition, the contro ' l e ~ (26) is under the IS K* . 2). If (A 22 +B 2 KC 2 ) is singular, since A22 is~nonsingular, such K is on a hypersurface in,parameter space of the IS K*. We can slightly perturb K under the IS K* so that (A 22 +B 2 KC 2 ) is nonsingular and the matching condition is still satisfied (Lemma 5). Sufficiency . Suppose that there is a cont~oller (26) under K* so that when (26) is coupled with the system (25), the closedloop system (27) has no pulse mode and all its poles are in G. For the same reason ~ iven in the proof of necessity , we can assume that (A 22 +B 2 KC ) is nonsingular. 2 we c a n construct the controller z(t)=Sz(t )+Ry( t), u(t)=Qz(t)+Ky(t)
(28)
so that when (28) is coupled with the sys-
Hu Yangzeng and Chen Shuzhong
148
tem (2), the closed-loop system has no pulse mode and all its poles are in G. here _ -1 -1K=(I-KC 2A22 B2 ) K -1-1 ""')-1 -1 Q~(I-KC2A22B2) Q, R=R(I-C 2 A22 B2 K - -1"" -1 -1 ?( S=S+R(I-C 2A22 B2K) C2A22B2~ Due to the matching condition, the controller (28) is under the IS K*. Theorem 4. Let G be a given nonempty symmetric open set in the complex plane and the system (2) satisfy the matc " ing condition, then the system (2) is pole assignable under the IS K* if and only if the following two conditions are satisfied: 1). The system (2) has no fixed pulse mode ~i~h re~ect to the IS K*; 2). FEM((E,A),'B,C;K*)<=G. (29) Proof. Necessity. Suppose that the system ~s pole assignable to G. 1). From Theorem 1 and Definition 5, the system (2) has no fixed pulse mode with respect to the IS K*. 2). Taking any (P,Q)~W, let the singular system (13) be ~.s.e. to the system (2) with respect to (p,Q). From 1) of this theorem, there is Ko~R(K*) so that (A 22 + B2KOC2 ) is nonsingular. From Lemma 9, the decentralized singular system x1(t)=A11x1(t)+B1u(t), y(t)=C 1x 1(t) (30) is pole aSSignable to G, where -1
""
A11 =F 11 -
.,.
F12F22F21, B1=B1-F12F22B2' C1=C1-C2F22F21, Fij=Aij+BiKOCj' From the result ~iven by Wang and Davison(1973), the system (30) is pol~ as~iguable to G if and only if FM(A ,E ,C1;K*)cG. From Lemma 7, 6 and 8, 11 1 FEM((E,X),B,C;K*)=FEM((E,A),B,C;K*) =FEM((E,(A+BKOC),B,C;K*) =FM(A 11
,B 1 'C1 ;K*)
hence (2Q) holds. Sufficiency. Arbitrarily taking (p,Q)~W, let the system (13) be r.s.e. to the system (2) with respect to (p,Q). Since the system (2) has no pulse mode with respect to the IS K*,' so there is KO~R(K*) so that (A22!B2KQC2) i~*nonSingular: From (31), FM(A 11 ,B 1 '~1;K leG. AccoroJ.ng to the method given by Wang and Dqvison(1Q73) we can const~ct the controller given by z(t)=Sz(t)+Ry(t), u(t)=Qz(t)+Ky(t) so that the all poles of the closed-loop system
are in G. It can be proved that there is a controller given by Z(t)=SZ(t)+Ry(t) { u ( t ) =Q z ( t ) + Ky ( t )
is coupled with the system (2), the closed loop system has no pulse mode, and the set of all exponential modes is the same with the set of poles of the system (32). Theorem 5. Let the system (2) satisfies the matching condition, then the system (2) is stabilizable under the IS K* if and only if the following two conditions are satisfied: 1). The system (2) has no pulse mode; 2). FEM((E,X),B,C;K*)cC-. Proof. Taking G as the left open half complex plane in Theorem 4, we obtain this theorem. Theorem 6. Let the system (2) satisfy the matching condition, then the system (2) is arbitra~ily approximately pole assignable under the IS K* if and only if the following two conditions are satisfied: 1). The s~stem (£) has no pulse mode; 2). FEM((E,A);B,C;K*)=~. Proof. Takin~ two disjoint nonempty symmetric open sets in the complex plane, and using Theorem 4, we can prove this theorem. The details are omitted.
REFERENCES Bender, D.J. and A.J. Laub (1985). The linear-quadratic optimal regulator for descriptor systems. in Proc. 24th IEEE Conf. Decision Contr., Ft. Lauderdale, FL, Dec. 1985, 957-962. Cobb, J.D. (1983). Descriptor variable systems and optimal state regulation. IEEE Trans. Automat. Contr., Vol. AC28, 601-611. Cobb, J.D. (1981). Feedback and pole placement in descriptor variable systems. Int. J. Contr., Vol. 33, 1135-1146. Hu, Y.Z. and Jiang W.S. (1984). New characterization of decentralized fixed modes and their applications. Proc .. of the Ninth Triennial World Congress of IFAC, Vol. 3, 11 83-1188. Lewis, F.L. and K. Ozcaldiran (1985). On the eigenst~cture assignment of singular systems. in Proc. 24th IEEE Conf. Decision Contr., Ft. Lauderdale, FL, Dec. 1985, 179-182. Pandolfi, L. (1980). Controllability and stabilization for linear systems of algebraic and differential equations. J. Optimiz. Theory Appl., Vol. 30, 601-620. Shayman, M.A. (1Q ~ 8). Pole placement by dynamic compensation for descriptor systems. Automatica, Vol. 24, 279-282. Shayman, M.A. and Z. Zhou (1987). Feedback control and classification of generalized linear sY' cems. IEEE T~ans. Autmat. Contr., Vol. AC-32, 483-494. Wang, S.H. and E.J. Davison (1973). On the stabilization of decentralized control system. IEEE T-ans. Automat. Contr., Vol. AC-18, 473-478.