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Decentralized robust control for interconnected systems with time-varying uncertainties
1I, pp.
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Brief Paper
Decentralized Robust Control for Interconnected Systems with Time-varying Uncertainties* GUANG-HONG Key Words-Interconnected uncertainty.
YANG? and SI-YING ZHANGS
systems: decentralized
Abstract-This paper is concerned with decentralized robust control design for a class of interconnected systems with time-varying uncertainties. The interconnected systems under consideration contain time-varying uncertain parameters whose values are unknown but bounded in given compact sets. Sufficient conditions under which a considered system can be quadratically stabilized by a linear decentralized feedback control for all admissible variations of uncertainties are given in terms of the structures of uncertainties. For the interconnected systems satisfying the stabilizability conditions, a decentralized control design procedure is also provided. Copyright 0 Elsevier Science Ltd.
control:
robustness;
quadratic
stabilizability;
assumptions and definitions are laid out. In Section 4, conditions under which a considered system can be quadratically stabilized by a linear decentralized control for all admissible variations of uncertainties are presented in terms of the structures of uncertainties, and a design procedure for desired robust stabilizers is provided. Section 5 concludes the paper. 2. Preliminaries The following definitions will be used in the development to follow. Definition 1. An n X n uncertain matrix M(9) (9 E Q, Q t R” is a given set) is said to be diagonally quasi-stable if there exists a nonsingular diagonal matrix K such that M(9)K + KM’(q) is positive-definite for all 9 E Q. Definition 2. An n X n uncertain matrix M(9) (9 E Q) is said to be in strongly standard form if
1. Introduction In the past two decades, a great deal of attention has been paid to the problem of robust control of systems with parametric uncertainty; see Barmish (1983, 1985), Khargonekar et al. (1990), Wei (1989, 1990) and references therein. Also, the problem of decentralized robust control for interconnected uncertain systems has been investigated; see Chen ef al. (1991). Siliak (1991). Chen (1992) and references therein. But the results for the existence of robust decentralized controllers given in these works commonly involve the conditions under which the interconnection matrix in the considered system satisfies the matching condition or a group of Riccati equations corresponding to the system are solvable, which are strict or difficult to test for many interconnected systems. Wei (1989, 1990) derived conditions for robust stabilization of centralized control systems with structural uncertain parameters in terms of a geometric pattern with respect to the location of uncertain parameters. In the field of decentralized stabilization of interconnected systems, there have also been a number of decentralized stabilizability conditions described in terms of the structures of the interconnection matrices (Davison, 1974; Ikeda and Siljak, 1980: Sezer and Siliak. 1981: Ikeda et nf.. 1983: Shi and Gao. 1986). In this paper, we shall study the problem of decentralized robust control for interconnected uncertain systems using Wei’s method, and present a new decentralized stabilizability condition in terms of the structures of the interconnection matrices. The paper is organized as follows. Section 2 gives some preliminaries. In Section 3, the systems,
c.. X
E:X,
::: mp-,
x:x,
rnP
E:X, E:X, X X
.”
mP
: 1 :
X
.
ml (1)
wherem,,+m,+...+m,,=nandm,,~m,>...~m,,;each E is a diagonally quasi-stable uncertain matrix with proper dimension: each X or X, denotes a don’t-care submatrix. Definition 3. (Wei (1989).) A simplified representation M(9) of an uncertain matrix M(9) with a strongly standard form as in (1) is of the form * e * ___ * $
*.
*
e.
1* * * *
*
*
*
,_,
*
*
*
e *
where fl represents E; * represents X, and all don’t-care submatrices X, are cancelled. Definition 4. An uncertain matrix M(9) = {m,,} as in (2) is said to have an antisymmetric stepwise configuration when the following conditions are satisfied: if p 2 k + 2 and mk,, f 0, and ulk+l. An then ml,” = 0 for all UZU, usp-1 uncertain matrix M(9) is said to have a strongly antisymmetric stepwise configuration if it is in a strongly standard form and its simplified representation M(9) as in (2) has an antisymmetric stepwise configuration. In this case, m,, represents a submatrix X and we mean m,, 50 (or m,, f 0) if and only if X is (or is not) a zero matrix for all 9. Remark 1. In the above definition of a strongly antisymmetric stepwise configuration, a new technical condition under which each submatrix E in (1) is diagonally quasi-stable is given instead of the quasi-stable condition in the definition of an antisymmetric stepwise configuration (Wei, 1989) which is required in the decentralized control design procedure given in Section 4. It is easy to see from definition 1, that an
*Received 12 October 1994; revised 7 August 1995; received in final form 19 May 19%. This paper was not presented at any IFAC meeting. This paper was recommended for publication by Associate Editor Masao Ikeda under the direction of Editor Andrew P. Sage. Corresponding author Dr Guang-Hong Yang. Tel. +6.5 799 5472: Fax +65 792 0415; E-mail [email protected]. t School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 638798. On leave from Department of Automatic Control, Northeastern University, Shenyang, Liaoning, 110006, P.R. China. # Department of Automatic Control, Northeastern University, Shenyang, Liaoning, 110006, P.R. China. 1603
1604
Brief Papers
diagonal matrix M(q) = diag [m,(q) m*(q) . (q E Q) is diagonally quasi-stable if and only if all the uncertainties m,(q) (i = 1, . , n) are sign-invariant for all q E Q, from which it further follows that an uncertain matrix uncertain
m,(q)]
*.e*.0e. 0.0. 1
will be diagonally quasi-stable if its diagonal entries are sign-invariant and dominant for non-diagonal entries. By Definition 4, the following two class of matrices have strongly antisymmetric stepwise configuration:
* * i *
* * *
* * *
.. i, I 0
0
0 *
e 0
* e
b i, 0 *
... .,. _,,
e * *
0 e *
... ...
* *
* *
... ,,_
I
0 *
Ux, r) = x’[A’(q(r))P + PA(q(r))Lr + ‘WPB(q(t))Kx 5 -a /1xI12 (3)
(4)
e * e . *
...
0 *
1
Notation. In the sequel, I or I, will denote the identity matrix and the norm j\Mll of a real matrix M will be taken to be the square root of the largest eigenvalue of M’M. Also, m,nu,,axr[] will denote the operation of taking the smallest ;I ar‘ges t) et‘genvalue (defined for a symmetric matrix).
3. Systems, assumptions and definitions Consider an interconnected uncertain system Z that may be described as an interconnection of N subsvstems 2,. .I
z2,...,%by
= A&(t)M)
+ i
A&(t))x,(t)
In the following, in order to avoid complexity in symbols, we assume that n, = n, (i = 1, , s), n, = nN (j = s + l,..., N),andn,,>nN. Denote
i=l,...,N
b&(r)) 0
x.,,(r)]’
xdr) = T(n,,, nN, s, N)x(r).
&:
&(t) = &(q(r))x+)
= 7%
hi. s, N)A(q(t))T--‘(n,.
B(q) = diag [h(q) x2(t)]‘,
x(t) = [x;(t)
b&)1,
u(t) = [u,(t)
Throughout the remainder of this paper, assumptions are taken as standard.
1
0
0
0
1
0
0 *
0 *
I
+ [ i (7)
and B(.)
Assumption
set Q is
(Compactness).
The
Qr)
the following
A. (Continuity). The matrices A(.) depend continuously on their arguments. B.
ro
u,w(t)]‘. (8)
Assumption
bounding
nN,
= T(n.,, n,v, s, N)&?(r)).
s, NJ,
(13) (14)
01 01 IX1(r) *J
(6)
. . . A,&(q)
h(q)
(12)
Remark 2. The goal of applying the transformation defined in (10) and (11) to the system Z of (5) is to consider centrally the structure of the overall system C in terms of a strongly antisymmetric stepwise configuration. As an illustration, consider an interconnected system Z_ = (A,., B,) with the following structure:
L*
A&)
+ &(q(r))M,
where
a,(r) =
&v(q)
(11)
Then from (5) we have
is in standard form. Denote
A,,(q)
, n, - nN)
matrix T(n,, nN, s, N) be such that
Let the permutation
ro
Am&)
(i = 1,
(5)
1
An(q)
(10)
h--nN+,W = [xl n,mn,+,O) -h~mnN+,(f) x.,+,,(t) xN,(f)l’ (i = 1, , Q).
B&(r))
where x,(t) = [x,,(t) . xin,]’ E WI and ui(r) E R’ are the state and the input of ‘X, (i = 1, , N) respectively, q(t) E Q c W is the model uncertainty, which is restricted to a prescribed bounding set Q. For each i (1~ i 5 N), the matrix A&(t)) 0
-%,(W3
.G2(0
xri(f) = [x,,(r)
j=l.,Za
ts0,
= [x;,(t)
where
A&(t))
+ h(qWh(r)>
(9)
for all pairs (x, t) E IR’ X [0, m). Furthermore, the system X is said to be quadratically stabilizable via linear decentralized control (with respect to Q) if K = diag [K, Kz KN], where Ki is a 1 X n, constant matrix for i = 1, , N.
-4)
where 0 represents a diagonally quasi-stable uncertain submatrix and * represents a don’t-care submatrix. For a detailed discussion on antisymmetric stepwise configuration and the definition of an uncertain matrix in standard form, the reader is referred to Wei (1989, 1990).
W
exists an N X v (v = zz”=, n,) constant matrix K, a v Xv positive-definite symmetric matrix P and a constant OL>O such that the following condition is satisfied: given any admissible uncertainty q(.), for the Lyapunov function V(x) = x’Px, the Lyapunov derivative L(x, t), corresponding to the closed-loop system with the feedback control law u(x) = Kx, satisfies the inequality
=
i
i
!]x&)
1
0
01
0
0
1
0
0
0
0 1 IX2(r)
I
+ [ !]a&),
(15)
compact. Assumption q(.): [0, m)+
uncertainty The (Measurability). C. Q is required to be Lebesgue-measurable.
where
In this paper, we shall be concerned with the following notion of stabilizability (Barmish, 1983) for the system Z. each * represents
Dejinition 5. The system Z of (5) is said to be quadratically stabilizable via linear control (with respect to Q) if there
an uncertainty.
Let x,(t) = [xi xi]’ and
u,(r) = [u,(t) u*(r)]‘. Then, from (IO)-(12). we know that s = N = 2, n, = 4. xV7(t) = [xiTl(t) .&At) x&(r) x:,,(r)]‘,
Brief Papers with x,,(r) = [x,,(t) x,,(t)]’ (i = 1, 2, 3, 4) transformed system & of X, is as follows. G(r)
and
= A
the
The following preliminaries Theorem 1.
(17)
Definition
where A711 ATi2 Ai-,3 ACT =
A ATZ’ A
A714
rw 1
At-22
Am
An4
~7,
A-m
Am
An4
741
A ~2
~4743
A744
BIT =
1605
6. If an uncertain system & = (AT(q), of the form : x, 0 B,(9)= [ E(9)
B72
v
B
A 7(9) x
A:(9)=[
Bq E Iw’“’ (i,j = 1,2,3,4) and A,,, AT,,=A7,-l=AT2,=A722=AT24=BT,= A,4 = B74 = 12, BR = Bm = 0. By Definition 4, the matrix A,,,
12
0
0
A
713 I2
= AT23 =
0
0
0
0
A 732
A 733
12
0
A
A 7x2
A743
A744
12
0
0
0
0
A,Be 1
has a strongly antisymmetric ever, the matrix
stepwise configuration.
corresponding
0 [ 0 does not have this structural feature.
0.
results
A,,,,(4) AT,>+,(~) A
r,,,,(Y) 0
AT,,,(~) A,,,+,>(4) AT,,,(~) 0
BTI(9)
s
A.,,,,(9)
B,,(9)
s
An,,>(9) A m+l,,,(9)
Bm+,(4)
M9)
uncertain
+ 1, n,V + 1, s, N), CC,+= r(n,, + 1, flN, s, N)S‘+T’(n, + 1, rlv, s, N).
(20) (21)
uncertain 1. Consider an system &.= (A.(q), B7(9)) and its strongly down-augmented system Z; = ((A;, B;(q)) as in Definition 6 with k = j = N. If there exists a positive-definite symmetric matrix S = diag [S, Sz . SN] with S, E R”.~x”~(i = 1,. , s) and S,, E R”Nx”N(m = s + 1, . , N) such that Lemma
9) = @‘(A.(9)S7 + S7.4;(9))@
W79
Anq(9)
quasi-stable
(19)
is negative-definite for all 9 E Q then there exists a Sv+ = positive-definite symmetric matrix S+ E @n,+i)XW,+i) diag [ST ST SJ with 1,. , s) and SL E UP’~+t)x~n~+i)’ (m = s + 1,. , N) !tch that
s
N
z+($‘+, 9) = O,$‘(A;(q)S;+
+ Sr+Ar+‘(q))O&
A T#,,n,(9)
&ns(9)
N
is also negative-definite
0
0
N
N
N
Proof: Since the matrix E(9) is diagonally quasi-stable, it follows from Definition 1 that there exists a nonsingular constant matrix J = diag[J, . JN] E RNxN such that E(9)J +JE’(q) is positive-definite for all 9 E Q. Let
(18)
1. If the uncertain matrix M(9) of (18) has a strongly antisymmetric stepwise configuration then the system X of (5) is quadratically stabilizable via linear decentralized control. Theorem
If every isolated subsystem of the system Z of (5) has the structure described by (3) ((4)) and each interconnection submatrix A,-(q) also has the corresponding structure (3) ((4)) with 8 = 0, then it follows from Remark 1 and Theorem 1 that the system X is quadratically stabilizable, and many physical processes satisfy the structural conditions; see Siljak (1991) and references therein. 3.
Remark 4. Theorem 1 presents a new decentralized stabilizability condition for interconnected systems with uncertainties, which cannot be obtained from the results in Ikeda and Siljak (1980) Sezer and Siljak (1981) Ikeda er al. (1983) and Shi and Gao (1986). For example, the system given by equations (15) and (16) satisfies the condition of Theorem 1, so it is decentralized stabilizable, but it does not satisfy the conditions given by Ikeda and Siljak (1980) Sezer and Siljak (1981). Ikeda et al. (1983) and Shi and Gao (1986).
for all 9 E Q.
r
where h = n, - nh. Then the main result of the paper is as follows.
Remark
: x,
ST = T(n,, n,v, s, N)ST’(n,, nN, s, N), sg+ = T(n, + 1, 7rv + 1, s, N)Sv+T’(n,V
1 AT,,(~) A,>,(y)
f%(9)
]p
I
: x,
In the following, we use 0, 0; and @,c to denote an X (r - N) matrix [I,_, 0]‘, an (r + N) X r matrix [1 0)’ and an (r + s) X r matrix [I, 01’ respectively, where r = sn,$+ (N - s)n,.,. If the matrices S E W”‘, SN+ e R(r+N)W+N) and S’+ E R(‘+s)x(r+,‘) then the matrices S7, SF+ and S++ are defined as follows:
How-
Let
An,(q) An,(y)
=
x
0
r
to the system &
In the next section, we shall present a sufficient condition for the quadratic stabilizability of the system Z in terms of the structure of the system Z,. 4. Main
B;(9)
B7(9)
where E,,(9) is a j X j diagonally matrix.
A771
741
j-k
where E(9) is a k X k diagonally quasi-stable uncertain matrix with k, j = s or N and j 2 k and Xi is a don’t-care submatrix, then a strongly down-augmented system 2; = ((A;(q), Bf(9)) of x7 is defined as follows:
I I
0
is
’
LB741
FAi7 R;71 =
B,(q))
1 : XI
k
73
with
will be used in the proof of
#+ = A straightforward
loi
I”&1
(22)
computation yields
x+(S?+. 9) = A7(9P7 0
+ S7AX9)
0
[4Sn
= L0
I
r(E(9P +Je’(9) 9) m&,9) = m’(S7,9) Y(E(9)J + JE’(9)) + WS7,9) Since z(S,. 9) and E(q)./ + JE’(9) are negative-definite and positive-definite respectively for all 9 E Q, it follows from Assumptions A and B that one can choose Y
I.
Il~(S7.9)l12 - &JW7,9)1 IIW&, 9)11:9 E Q I &x]x(&, 9)1&ni”tE(9)J + JE’(9)l
1
and SN+lN++-m to guarantee
Y2 IIJII’
that n’(Sp+, 9) is negative-definite
for all
Brief Papers
1606
9 E Q and SF+ is positive-definite, respectively. From (ll), (19) (20) and (22), it is easy to see that the matrix SN+=T’(n.,+1,n,v+l,s,N)S~+7’(n,~+1,n,+1,s,N)=diag [ST S: S,$]is positive-definite, where
r
101
down or up), where A,,(q) = [0] or A,(q) = [Xl, and B,(9) is an N x N or s X s diagonally quasi-stable uncertain matrix. The lemma follows immediately from Definition 4 and the results in Wei (1989) and the proof is omitted. Lemma 4. If there exists a positive-definite SN+ = diag [ST S; S&] with s+
For the case where k = j = s or k = s and j = N, similar results can be derived, and the details are omitted. Definition 7. If the system & =(A,(q), form
B,(q))
is of the
=
I[
S’s,,1Efp,+lPe,+l) s,
s,2
(i = 1I...,
rZ
SL=[ili
~~:]cl~(n~+i)x(~~+r)
j
where E(9) is a j X j diagonally quasi-stable uncertain matrix with k, j = s or N, and k % j, then a strongly up-augmented system Z.r+ = ((A.+(q), B,+(q)) of & is defined as follows: E’;) k b.+(9)
Proof
A{9)],
k
quasi-stable
uncertain
Sh,S&.
(24)
Let S&S,
Sb&]],
x’]A(9)S + SA’(9)lx= y’]Wr.,, nv, s, N)A(9) X ST’@,, no, s, N) + 7%
an uncertain system &= (A.(q), B,(q)) and its strongly up-augmented system &+ = ((A,+(q), E,+(q)) as in Definition 7 with k =s and j = N. If there exists a positive-definite symmetric matrix S =diag[S, Sz S,] with S, E iw”~““~(i = 1,. ,s) and S,,, E R”N”“N(m = s + 1,. , N) such that
nN, s, N)SA’(9)T’(n.,, n,v, s, N)]Y
=Y{AA.(~)S, + S,A;i9) + B,(9) diag ]S;Z S;Z +
W,,
nN,
s, NJ
x(&s q) = @‘(AT(q)& + &A x9))@
rr+(S;‘, 9) = O:(A7+(9)S;+
diag
CL
sb2lT’h,
1
nN,
s, N)
&.,&%-(9)1~
S12
ST
is negative-definite for all 9 E Q then there exists a positive-definite symmetric matrix S” = diag [SC ST S,J with S,+ E R(n~+‘)x(n~+‘) (i = 1,. ,s) and SJz=S,,,(m=s+l ,._., N)suchthat
Proof
S;&’
Given any nonzero vector x E R’n~+(N~‘)n~, denote x = T’(n,, nN, n, N)y. Then it follows from (13) (14), (19) and (20) that
=
2. Consider
is also negative-definite
+ + S$‘+Ar’(q))O,$
A(9) = A(9) + B(9) diag (S;zSfi’ S = diag [S,, S,, Sv,].
where E,,(9) is a k X k diagonally matrix. Lemma
(m=s+l,...,N)
is negative-definite for all 9 E Q, where A%9) is given by (23) then the system Z = (A(9), B(9)) of (5) is quadratically stabilizable via linear decentralized control. In this case, a stabilizing decentralized state-feedback control law is given by u(x) = diag [S;,S;i’
AT+(9) = [I
s),
such that rr+(S$+, 9) = O;‘(A$(q)S:+
k
symmetric matrix
’ [ diag [S;,
r(n,, nN, s, N) diag [Si2 diag[S,, S,,]
diag [S;,
+ S;‘A.+,(q))O:
ShJT’(n,,, nN, s, N)
ST
Shz]r’(n,,
nN, s, N)
Vn.,, nN, s, N) diag
for all 9 E Q.
diagh
[.L
&I
S,
1 1
SM]
Let
rsN+IN+II, I YJ
01
=y’B&‘(A~q)S~+
+ S;+Av(q)]O;y
< 0.
Since the matrix S is positive-definite, the system Z is quadratically stabilizable via linear decentralized control. 0
(23)
Proof of Theorem 1. It follows from Lemma 3 that the system ZT.= (AKq), B$(q)) is generated from the simplest interconnected system Z,, = (A,,(9), B,,(q)) via a sequence of augmentations (either down or up), where A,,(9) = [0] or A,,(9) = [Xl, and B,,(q) is an N X N or s X s diagonally quasi-stable uncertain matrix. It is obvious that the positive-definite matrix S = IN or S = [, satisfies the condition of Lemma 1 or 2. Then, following the augmentation order and applying Lemmas 1 and 2 repeatedly, one can generate a positive-definite matrix S ,v+ = diag [ST ST . S,Z,] with S’ E ~(%+l)xl%+r) (i = 1,. ,s) a,,,, s; E @N+I)~(W+I) (m =s+l,... , N) such that the condttton of Lemma 4 holds. Hence it follows from Lemma 4 that the system Z is quadratically stabilizable via linear decentralized control. 0
system (either
Remark 5. From the above proof of Theorem 1, it is easy to see that the diagonally quasi-stable condition plays an
where J = diag [Ji .I,] E UP”” is a nonsingular constant matrix such that E,,(q)J +JE;,(q) is positive-definite for all 9 E Q. The rest of the proof is similar to the proof of Lemma cl 1. and the details are omitted. For the case where k = j = s or k = j = N, similar results can be derived. and the details are omitted. Lemma 3. If the uncertain matrix M(9) of (18) has a strongly antisymmetric stepwise configuration then the system ZK9). BR9)) with
1% ~%9)=
[l]
can be generated from the simplest interconnected X,, = (A,,(9), B,,(q)) via a sequence of augmentations
Brief Papers
1607
important role in the decentralized controller design for the system Z of (5) and the condition cannot be replaced by the weaker quasi-stable condition used in centralized controller designs (Wei, 1989). If the matrix M(9) of (18) has a strongly antisymmetric stepwise configuration then it follows from Definition 4 that the submatrices A mu+:) 6 = 1,. .I A’: i#h=n,-nN) and A’k,h+l(q)r Arhh+,(9) = quasi-stable, the P:,t~+1(9):~1, must be diagonally condition contains restrictions on the values of some parameters of the system 2. In fact, it follows from Definition 1 and Remark 1 that some entries of the system I: must be sign-invariant for all 9 E Q under the condition.
A,,(9) = A,,(9) = *
A,,(9)=
Remark 6. By taking advantage of the results in Wei (1989) it is easy to show that the condition of the strongly antisymmetric stepwise configuration is an extension of the matching condition (Chen et al., 1991) for decentrally stabilizability, and Theorem 1 can be extended to the case where some subsystems in the considered system are multi-input. The details are omitted. For the case where the condition of Theorem 1 is not satisfied but some parameter ranges in a given system are not sufficiently large, applying the results of this paper to the system is still possible; see Wei (1990) for a detailed discussion.
If the system I: = (A(9). B(9)) of (5) satisfies the condition under which M(9) given by (18) has a strongly antisymmetric stepwise configuration then a decentralized stabilizing controller for the system Z can be constructed using the following procedure. Design procedure
1.
Step 1. For the given system Z = (A(9), B(9))
of (5) find the transformed system Zr = (AT(q), Bz(9)) by the transformation Tof (11) and then determine if the matrix M(9) of (18) has a strongly antisymmetric stepwise configuration. If it does, then do Step 2. Step 2. Specify the augmentation x: = (A+(9), 8~9)) of (23).
order
of the system
Step 3. Construct a positive-definite
matrix S”‘+ by following the augmentation order and applying Lemmas 1 and 2 respectively, which satisfies the condition of Lemma 4.
Remark 7.
It must be pointed that the stabilizing controller
constructed by the above design procedure is non-unique when the condition of Theorem 1 is satisfied. Comparing with the robust decentralized control designs given by Chen et al. (1991) and Chen (1992) the gains of decentralized stabilizing controllers constructed by the above design procedure only depend on the system structure, and the given varying ranges of all uncertain parameters do not depend on any preassigned nominal system. Remark 8. If every isolated subsystem in an interconnected
system has an antisymmetric stepwise configuration and the condition of Theorem 1 is satisfied then a decentralized connectively stabilizing controller for the interconnected system can be constructed by Design procedure 1 for the notion of connective stabilizability, see Siljak, 1991).
Example 1. Let the considered
system Z be described by (5).
x,(r) = [x,,(r)
x&)
x,,(r)
x&)1’
(i = 1,2), ~(0 = [x,,(r) A,,(9) A(9) = A,,(9) i A,,(9) B(9) = diag P,(9)
A,,(9) AZ(~) A32(9) b,(9)
M9)l
(25)
a,*(9)
*
0
;
;
0”
@i(9)
*
0
;
;
;
0”
*
*
**
A13(9) = A&9) =
* 0 0’ [ 0 0 0 0
A,,(9) =A,,(9)
=
0
0
o-
0 0 0 0
0 *
0 0.
b,(q) =b,(q) = [O 0 0 II’,
b,(q) = [O 0 11’
and where each * represents a bounded uncertainty, from (lo)-(12) we know l9,2(9)1 s : and lv”zi93)cL - 2;. Then, = 4 that s = 2, n,=3, and xr(t) = with’ in(r) = [xii(r) xdt)l and [G, x;z G3 XL]’ x,+1(9) = [xi,+i(r) xZ,+i(r) x4’, and the transformed system Zr of Z is as follows: G(r) = AT(9)XT(r) + &(9)u(r)
(27)
where
A.(9)=
A,,,(9)
ATIZ(~) A,,,(9)
ATM(~)
AR,(~)
A,,(9)
A7249)
And9)
I
~(9)
A,,(9)
A,,(9)
A,,(9)
~473.49)
A74d9)
A,,(9)
A744(9).
rB,rd9n B,(9)= b74(9)J
with AT.,,(9) E FP2, AT,,(q) E Iw’“” (i = 2, 3.4) and ad9)
1 A,,(q)=A,,(q)=B,(q)=I,,
0
0I ’
.4711(9)=0
A,,(9)=0>
AT,~(~)=ATZ*(~)=AR~(~)=BT,(~)=B~~(~)=B~(~)=O.
Since
lalz(9)(~:
1IS
and
la,,(q)1 5 $, it follows from 1 9,2(q) diagonally c n,,(9) 1 quasi-stable. Thus, from Definition 4, the matrix Definition 1 that the matrix
M(9)=
=
[
A,(9) o
B,(9) 0. I
0
And(~) 0
And9) A,,;(9)
,47x(9) AT;(~)
I
19 (26)
xX2(t) x,,(r)], A13(9) A,,(9) A,,(9)
*
0 I *01:**1 [ 0 1 0 0 0001 ;
AX(~)=
In the following, we present an example to illustrate the results of this paper.
with
*
[
Step 4. Compute a decentralized
stabilizing controller for the system 2 by using the method given in Lemma 4.
*
0
A.$91
A7:$)) 0
0 0 13 A,(9) 0
0 0 0 4 01
(28)
has a strongly antisymmetric stepwise configuration. The corresponding system ZF= (A$(q), BKq)) of (23) is given by
and it is easy to see that the system Z$ can be generated from the simplest interconnected system & = (A,,(9), B,,(9)) with
1608
Brief Papers
A,(q) = 0 and B,(q) = Is, and its augmentation order is down-up-down-down. Thus it follows from Theorem 1 that the considered system 2 is quadratically stabilizable via linear decentralized control, and a decentralized stabilizing controller for the system can be conducted by Design procedure 1 and the above augmentation order if the bounds of all uncertainties are given; the computation is omitted. 5. Conclusions In this paper, we have studied the problem of designing a linear decentralized state feedback control to stabilize a class of interconnected systems with time-varying uncertainties. Sufficient conditions under which a considered system can be quadratically stabilized by a linear decentralized feedback control for all admissible variations of uncertainties have presented .in terms of the structures of uncertainties. The conditions require that all uncertainties in the transformed matrix corresponding to the system form a special geometrical pattern called a strongly antisymmetric stepwise configuration. For interconnected systems satisfying the stabilizability conditions, a control design procedure for desired robust stabilizers has also been provided. Acknowledgements-The authors are greatly indebted to the referees for many useful suggestions and corrections on the initial manuscript of the present work. This work was partly supported by the Chinese National Natural Science Grants Council. References Barmish, B. R. (1983). Stabilization of uncertain systems via linear control. IEEE Trans. Automat. Control, AC-28, 848-850.
Barmish, B. R. (1985). Necessary and sufficient conditions for quadratic stabilizability of an uncertain system. J. Optim. Theory Applic., 46,399-408. Chen, Y. H. (1992). Decentralized robust control for large-scale systems: a design based on the bound of uncertainty. J. Dyn. Syst., Measurement, Control, 114,1-9. Chen, Y. H., G. Leitmann and X. Z. Kai (1991). Robust control design for interconnected systems with timevarying uncertainties. ht. J. Control, 54, 1119-1142. Davison, E. J. (1974). The decentralized stabilization and control of a class of unknown nonlinear time-varying . systems. Automatica, 10, 309-316. Ikeda. M. and D. D. Siliak (1980). On decentrallv stabilizable large-scale systems. Automatica, 16, 331-334. ’ Ikeda, M., D. D. Siljak and K. Yasuda (1983). Optimality of decentralized control for large-scale systems. Automatica, 19,309-316. Khargonekar, P. P., I. R. Petersen and K. Zhou (1990). Robust stabilization of uncertain linear systems: quadratic stabilizability and H, control theory. IEEE Trans. Autom. Control, AC-35,356-361. Sezer, M. E. and D. D. Siljak (1981). On decentralized stabilization and structure of linear large-scale systems. Automatica, 17, 641-644. Shi, Z. C. and W. B. Gau (1986). Stabilization by decentralized control for large-scale interconnected systems. Large Scale Syst., 10, 147-155. Siljak, D. D. (1991). Decentralized Control of Complex Systems. Academic Press, San Diego. Wei, K. (1989). Robust stabilizability of linear time varying uncertain dynamical systems via linear feedback control. In Proc. IEEE ICCON or Control Applications, Jerusalem. Wei, K. (1990). Quadratic stabilizability of linear systems with structural independent time varying uncertainties. IEEE Trans. Autom. Control, AC-35,268-277.
Pergamon
Auromorrca. Vol. 32, No. lM)3-lf,O8, 19% Copyright 0 19% Elsevier Science Ltd Printed in Great Britain. All rights reserved ooO5-1098/% $15.00 + 0.00
PII: SOOO5-1098(%)00112-4
Brief Paper
Decentralized Robust Control for Interconnected Systems with Time-varying Uncertainties* GUANG-HONG Key Words-Interconnected uncertainty.
YANG? and SI-YING ZHANGS
systems: decentralized
Abstract-This paper is concerned with decentralized robust control design for a class of interconnected systems with time-varying uncertainties. The interconnected systems under consideration contain time-varying uncertain parameters whose values are unknown but bounded in given compact sets. Sufficient conditions under which a considered system can be quadratically stabilized by a linear decentralized feedback control for all admissible variations of uncertainties are given in terms of the structures of uncertainties. For the interconnected systems satisfying the stabilizability conditions, a decentralized control design procedure is also provided. Copyright 0 Elsevier Science Ltd.
control:
robustness;
quadratic
stabilizability;
assumptions and definitions are laid out. In Section 4, conditions under which a considered system can be quadratically stabilized by a linear decentralized control for all admissible variations of uncertainties are presented in terms of the structures of uncertainties, and a design procedure for desired robust stabilizers is provided. Section 5 concludes the paper. 2. Preliminaries The following definitions will be used in the development to follow. Definition 1. An n X n uncertain matrix M(9) (9 E Q, Q t R” is a given set) is said to be diagonally quasi-stable if there exists a nonsingular diagonal matrix K such that M(9)K + KM’(q) is positive-definite for all 9 E Q. Definition 2. An n X n uncertain matrix M(9) (9 E Q) is said to be in strongly standard form if
1. Introduction In the past two decades, a great deal of attention has been paid to the problem of robust control of systems with parametric uncertainty; see Barmish (1983, 1985), Khargonekar et al. (1990), Wei (1989, 1990) and references therein. Also, the problem of decentralized robust control for interconnected uncertain systems has been investigated; see Chen ef al. (1991). Siliak (1991). Chen (1992) and references therein. But the results for the existence of robust decentralized controllers given in these works commonly involve the conditions under which the interconnection matrix in the considered system satisfies the matching condition or a group of Riccati equations corresponding to the system are solvable, which are strict or difficult to test for many interconnected systems. Wei (1989, 1990) derived conditions for robust stabilization of centralized control systems with structural uncertain parameters in terms of a geometric pattern with respect to the location of uncertain parameters. In the field of decentralized stabilization of interconnected systems, there have also been a number of decentralized stabilizability conditions described in terms of the structures of the interconnection matrices (Davison, 1974; Ikeda and Siljak, 1980: Sezer and Siliak. 1981: Ikeda et nf.. 1983: Shi and Gao. 1986). In this paper, we shall study the problem of decentralized robust control for interconnected uncertain systems using Wei’s method, and present a new decentralized stabilizability condition in terms of the structures of the interconnection matrices. The paper is organized as follows. Section 2 gives some preliminaries. In Section 3, the systems,
c.. X
E:X,
::: mp-,
x:x,
rnP
E:X, E:X, X X
.”
mP
: 1 :
X
.
ml (1)
wherem,,+m,+...+m,,=nandm,,~m,>...~m,,;each E is a diagonally quasi-stable uncertain matrix with proper dimension: each X or X, denotes a don’t-care submatrix. Definition 3. (Wei (1989).) A simplified representation M(9) of an uncertain matrix M(9) with a strongly standard form as in (1) is of the form * e * ___ * $
*.
*
e.
1* * * *
*
*
*
,_,
*
*
*
e *
where fl represents E; * represents X, and all don’t-care submatrices X, are cancelled. Definition 4. An uncertain matrix M(9) = {m,,} as in (2) is said to have an antisymmetric stepwise configuration when the following conditions are satisfied: if p 2 k + 2 and mk,, f 0, and ulk+l. An then ml,” = 0 for all UZU, usp-1 uncertain matrix M(9) is said to have a strongly antisymmetric stepwise configuration if it is in a strongly standard form and its simplified representation M(9) as in (2) has an antisymmetric stepwise configuration. In this case, m,, represents a submatrix X and we mean m,, 50 (or m,, f 0) if and only if X is (or is not) a zero matrix for all 9. Remark 1. In the above definition of a strongly antisymmetric stepwise configuration, a new technical condition under which each submatrix E in (1) is diagonally quasi-stable is given instead of the quasi-stable condition in the definition of an antisymmetric stepwise configuration (Wei, 1989) which is required in the decentralized control design procedure given in Section 4. It is easy to see from definition 1, that an
*Received 12 October 1994; revised 7 August 1995; received in final form 19 May 19%. This paper was not presented at any IFAC meeting. This paper was recommended for publication by Associate Editor Masao Ikeda under the direction of Editor Andrew P. Sage. Corresponding author Dr Guang-Hong Yang. Tel. +6.5 799 5472: Fax +65 792 0415; E-mail [email protected]. t School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 638798. On leave from Department of Automatic Control, Northeastern University, Shenyang, Liaoning, 110006, P.R. China. # Department of Automatic Control, Northeastern University, Shenyang, Liaoning, 110006, P.R. China. 1603
1604
Brief Papers
diagonal matrix M(q) = diag [m,(q) m*(q) . (q E Q) is diagonally quasi-stable if and only if all the uncertainties m,(q) (i = 1, . , n) are sign-invariant for all q E Q, from which it further follows that an uncertain matrix uncertain
m,(q)]
*.e*.0e. 0.0. 1
will be diagonally quasi-stable if its diagonal entries are sign-invariant and dominant for non-diagonal entries. By Definition 4, the following two class of matrices have strongly antisymmetric stepwise configuration:
* * i *
* * *
* * *
.. i, I 0
0
0 *
e 0
* e
b i, 0 *
... .,. _,,
e * *
0 e *
... ...
* *
* *
... ,,_
I
0 *
Ux, r) = x’[A’(q(r))P + PA(q(r))Lr + ‘WPB(q(t))Kx 5 -a /1xI12 (3)
(4)
e * e . *
...
0 *
1
Notation. In the sequel, I or I, will denote the identity matrix and the norm j\Mll of a real matrix M will be taken to be the square root of the largest eigenvalue of M’M. Also, m,nu,,axr[] will denote the operation of taking the smallest ;I ar‘ges t) et‘genvalue (defined for a symmetric matrix).
3. Systems, assumptions and definitions Consider an interconnected uncertain system Z that may be described as an interconnection of N subsvstems 2,. .I
z2,...,%by
= A&(t)M)
+ i
A&(t))x,(t)
In the following, in order to avoid complexity in symbols, we assume that n, = n, (i = 1, , s), n, = nN (j = s + l,..., N),andn,,>nN. Denote
i=l,...,N
b&(r)) 0
x.,,(r)]’
xdr) = T(n,,, nN, s, N)x(r).
&:
&(t) = &(q(r))x+)
= 7%
hi. s, N)A(q(t))T--‘(n,.
B(q) = diag [h(q) x2(t)]‘,
x(t) = [x;(t)
b&)1,
u(t) = [u,(t)
Throughout the remainder of this paper, assumptions are taken as standard.
1
0
0
0
1
0
0 *
0 *
I
+ [ i (7)
and B(.)
Assumption
set Q is
(Compactness).
The
Qr)
the following
A. (Continuity). The matrices A(.) depend continuously on their arguments. B.
ro
u,w(t)]‘. (8)
Assumption
bounding
nN,
= T(n.,, n,v, s, N)&?(r)).
s, NJ,
(13) (14)
01 01 IX1(r) *J
(6)
. . . A,&(q)
h(q)
(12)
Remark 2. The goal of applying the transformation defined in (10) and (11) to the system Z of (5) is to consider centrally the structure of the overall system C in terms of a strongly antisymmetric stepwise configuration. As an illustration, consider an interconnected system Z_ = (A,., B,) with the following structure:
L*
A&)
+ &(q(r))M,
where
a,(r) =
&v(q)
(11)
Then from (5) we have
is in standard form. Denote
A,,(q)
, n, - nN)
matrix T(n,, nN, s, N) be such that
Let the permutation
ro
Am&)
(i = 1,
(5)
1
An(q)
(10)
h--nN+,W = [xl n,mn,+,O) -h~mnN+,(f) x.,+,,(t) xN,(f)l’ (i = 1, , Q).
B&(r))
where x,(t) = [x,,(t) . xin,]’ E WI and ui(r) E R’ are the state and the input of ‘X, (i = 1, , N) respectively, q(t) E Q c W is the model uncertainty, which is restricted to a prescribed bounding set Q. For each i (1~ i 5 N), the matrix A&(t)) 0
-%,(W3
.G2(0
xri(f) = [x,,(r)
j=l.,Za
ts0,
= [x;,(t)
where
A&(t))
+ h(qWh(r)>
(9)
for all pairs (x, t) E IR’ X [0, m). Furthermore, the system X is said to be quadratically stabilizable via linear decentralized control (with respect to Q) if K = diag [K, Kz KN], where Ki is a 1 X n, constant matrix for i = 1, , N.
-4)
where 0 represents a diagonally quasi-stable uncertain submatrix and * represents a don’t-care submatrix. For a detailed discussion on antisymmetric stepwise configuration and the definition of an uncertain matrix in standard form, the reader is referred to Wei (1989, 1990).
W
exists an N X v (v = zz”=, n,) constant matrix K, a v Xv positive-definite symmetric matrix P and a constant OL>O such that the following condition is satisfied: given any admissible uncertainty q(.), for the Lyapunov function V(x) = x’Px, the Lyapunov derivative L(x, t), corresponding to the closed-loop system with the feedback control law u(x) = Kx, satisfies the inequality
=
i
i
!]x&)
1
0
01
0
0
1
0
0
0
0 1 IX2(r)
I
+ [ !]a&),
(15)
compact. Assumption q(.): [0, m)+
uncertainty The (Measurability). C. Q is required to be Lebesgue-measurable.
where
In this paper, we shall be concerned with the following notion of stabilizability (Barmish, 1983) for the system Z. each * represents
Dejinition 5. The system Z of (5) is said to be quadratically stabilizable via linear control (with respect to Q) if there
an uncertainty.
Let x,(t) = [xi xi]’ and
u,(r) = [u,(t) u*(r)]‘. Then, from (IO)-(12). we know that s = N = 2, n, = 4. xV7(t) = [xiTl(t) .&At) x&(r) x:,,(r)]‘,
Brief Papers with x,,(r) = [x,,(t) x,,(t)]’ (i = 1, 2, 3, 4) transformed system & of X, is as follows. G(r)
and
= A
the
The following preliminaries Theorem 1.
(17)
Definition
where A711 ATi2 Ai-,3 ACT =
A ATZ’ A
A714
rw 1
At-22
Am
An4
~7,
A-m
Am
An4
741
A ~2
~4743
A744
BIT =
1605
6. If an uncertain system & = (AT(q), of the form : x, 0 B,(9)= [ E(9)
B72
v
B
A 7(9) x
A:(9)=[
Bq E Iw’“’ (i,j = 1,2,3,4) and A,,, AT,,=A7,-l=AT2,=A722=AT24=BT,= A,4 = B74 = 12, BR = Bm = 0. By Definition 4, the matrix A,,,
12
0
0
A
713 I2
= AT23 =
0
0
0
0
A 732
A 733
12
0
A
A 7x2
A743
A744
12
0
0
0
0
A,Be 1
has a strongly antisymmetric ever, the matrix
stepwise configuration.
corresponding
0 [ 0 does not have this structural feature.
0.
results
A,,,,(4) AT,>+,(~) A
r,,,,(Y) 0
AT,,,(~) A,,,+,>(4) AT,,,(~) 0
BTI(9)
s
A.,,,,(9)
B,,(9)
s
An,,>(9) A m+l,,,(9)
Bm+,(4)
M9)
uncertain
+ 1, n,V + 1, s, N), CC,+= r(n,, + 1, flN, s, N)S‘+T’(n, + 1, rlv, s, N).
(20) (21)
uncertain 1. Consider an system &.= (A.(q), B7(9)) and its strongly down-augmented system Z; = ((A;, B;(q)) as in Definition 6 with k = j = N. If there exists a positive-definite symmetric matrix S = diag [S, Sz . SN] with S, E R”.~x”~(i = 1,. , s) and S,, E R”Nx”N(m = s + 1, . , N) such that Lemma
9) = @‘(A.(9)S7 + S7.4;(9))@
W79
Anq(9)
quasi-stable
(19)
is negative-definite for all 9 E Q then there exists a Sv+ = positive-definite symmetric matrix S+ E @n,+i)XW,+i) diag [ST ST SJ with 1,. , s) and SL E UP’~+t)x~n~+i)’ (m = s + 1,. , N) !tch that
s
N
z+($‘+, 9) = O,$‘(A;(q)S;+
+ Sr+Ar+‘(q))O&
A T#,,n,(9)
&ns(9)
N
is also negative-definite
0
0
N
N
N
Proof: Since the matrix E(9) is diagonally quasi-stable, it follows from Definition 1 that there exists a nonsingular constant matrix J = diag[J, . JN] E RNxN such that E(9)J +JE’(q) is positive-definite for all 9 E Q. Let
(18)
1. If the uncertain matrix M(9) of (18) has a strongly antisymmetric stepwise configuration then the system X of (5) is quadratically stabilizable via linear decentralized control. Theorem
If every isolated subsystem of the system Z of (5) has the structure described by (3) ((4)) and each interconnection submatrix A,-(q) also has the corresponding structure (3) ((4)) with 8 = 0, then it follows from Remark 1 and Theorem 1 that the system X is quadratically stabilizable, and many physical processes satisfy the structural conditions; see Siljak (1991) and references therein. 3.
Remark 4. Theorem 1 presents a new decentralized stabilizability condition for interconnected systems with uncertainties, which cannot be obtained from the results in Ikeda and Siljak (1980) Sezer and Siljak (1981) Ikeda er al. (1983) and Shi and Gao (1986). For example, the system given by equations (15) and (16) satisfies the condition of Theorem 1, so it is decentralized stabilizable, but it does not satisfy the conditions given by Ikeda and Siljak (1980) Sezer and Siljak (1981). Ikeda et al. (1983) and Shi and Gao (1986).
for all 9 E Q.
r
where h = n, - nh. Then the main result of the paper is as follows.
Remark
: x,
ST = T(n,, n,v, s, N)ST’(n,, nN, s, N), sg+ = T(n, + 1, 7rv + 1, s, N)Sv+T’(n,V
1 AT,,(~) A,>,(y)
f%(9)
]p
I
: x,
In the following, we use 0, 0; and @,c to denote an X (r - N) matrix [I,_, 0]‘, an (r + N) X r matrix [1 0)’ and an (r + s) X r matrix [I, 01’ respectively, where r = sn,$+ (N - s)n,.,. If the matrices S E W”‘, SN+ e R(r+N)W+N) and S’+ E R(‘+s)x(r+,‘) then the matrices S7, SF+ and S++ are defined as follows:
How-
Let
An,(q) An,(y)
=
x
0
r
to the system &
In the next section, we shall present a sufficient condition for the quadratic stabilizability of the system Z in terms of the structure of the system Z,. 4. Main
B;(9)
B7(9)
where E,,(9) is a j X j diagonally matrix.
A771
741
j-k
where E(9) is a k X k diagonally quasi-stable uncertain matrix with k, j = s or N and j 2 k and Xi is a don’t-care submatrix, then a strongly down-augmented system 2; = ((A;(q), Bf(9)) of x7 is defined as follows:
I I
0
is
’
LB741
FAi7 R;71 =
B,(q))
1 : XI
k
73
with
will be used in the proof of
#+ = A straightforward
loi
I”&1
(22)
computation yields
x+(S?+. 9) = A7(9P7 0
+ S7AX9)
0
[4Sn
= L0
I
r(E(9P +Je’(9) 9) m&,9) = m’(S7,9) Y(E(9)J + JE’(9)) + WS7,9) Since z(S,. 9) and E(q)./ + JE’(9) are negative-definite and positive-definite respectively for all 9 E Q, it follows from Assumptions A and B that one can choose Y
I.
Il~(S7.9)l12 - &JW7,9)1 IIW&, 9)11:9 E Q I &x]x(&, 9)1&ni”tE(9)J + JE’(9)l
1
and SN+lN++-m to guarantee
Y2 IIJII’
that n’(Sp+, 9) is negative-definite
for all
Brief Papers
1606
9 E Q and SF+ is positive-definite, respectively. From (ll), (19) (20) and (22), it is easy to see that the matrix SN+=T’(n.,+1,n,v+l,s,N)S~+7’(n,~+1,n,+1,s,N)=diag [ST S: S,$]is positive-definite, where
r
101
down or up), where A,,(q) = [0] or A,(q) = [Xl, and B,(9) is an N x N or s X s diagonally quasi-stable uncertain matrix. The lemma follows immediately from Definition 4 and the results in Wei (1989) and the proof is omitted. Lemma 4. If there exists a positive-definite SN+ = diag [ST S; S&] with s+
For the case where k = j = s or k = s and j = N, similar results can be derived, and the details are omitted. Definition 7. If the system & =(A,(q), form
B,(q))
is of the
=
I[
S’s,,1Efp,+lPe,+l) s,
s,2
(i = 1I...,
rZ
SL=[ili
~~:]cl~(n~+i)x(~~+r)
j
where E(9) is a j X j diagonally quasi-stable uncertain matrix with k, j = s or N, and k % j, then a strongly up-augmented system Z.r+ = ((A.+(q), B,+(q)) of & is defined as follows: E’;) k b.+(9)
Proof
A{9)],
k
quasi-stable
uncertain
Sh,S&.
(24)
Let S&S,
Sb&]],
x’]A(9)S + SA’(9)lx= y’]Wr.,, nv, s, N)A(9) X ST’@,, no, s, N) + 7%
an uncertain system &= (A.(q), B,(q)) and its strongly up-augmented system &+ = ((A,+(q), E,+(q)) as in Definition 7 with k =s and j = N. If there exists a positive-definite symmetric matrix S =diag[S, Sz S,] with S, E iw”~““~(i = 1,. ,s) and S,,, E R”N”“N(m = s + 1,. , N) such that
nN, s, N)SA’(9)T’(n.,, n,v, s, N)]Y
=Y{AA.(~)S, + S,A;i9) + B,(9) diag ]S;Z S;Z +
W,,
nN,
s, NJ
x(&s q) = @‘(AT(q)& + &A x9))@
rr+(S;‘, 9) = O:(A7+(9)S;+
diag
CL
sb2lT’h,
1
nN,
s, N)
&.,&%-(9)1~
S12
ST
is negative-definite for all 9 E Q then there exists a positive-definite symmetric matrix S” = diag [SC ST S,J with S,+ E R(n~+‘)x(n~+‘) (i = 1,. ,s) and SJz=S,,,(m=s+l ,._., N)suchthat
Proof
S;&’
Given any nonzero vector x E R’n~+(N~‘)n~, denote x = T’(n,, nN, n, N)y. Then it follows from (13) (14), (19) and (20) that
=
2. Consider
is also negative-definite
+ + S$‘+Ar’(q))O,$
A(9) = A(9) + B(9) diag (S;zSfi’ S = diag [S,, S,, Sv,].
where E,,(9) is a k X k diagonally matrix. Lemma
(m=s+l,...,N)
is negative-definite for all 9 E Q, where A%9) is given by (23) then the system Z = (A(9), B(9)) of (5) is quadratically stabilizable via linear decentralized control. In this case, a stabilizing decentralized state-feedback control law is given by u(x) = diag [S;,S;i’
AT+(9) = [I
s),
such that rr+(S$+, 9) = O;‘(A$(q)S:+
k
symmetric matrix
’ [ diag [S;,
r(n,, nN, s, N) diag [Si2 diag[S,, S,,]
diag [S;,
+ S;‘A.+,(q))O:
ShJT’(n,,, nN, s, N)
ST
Shz]r’(n,,
nN, s, N)
Vn.,, nN, s, N) diag
for all 9 E Q.
diagh
[.L
&I
S,
1 1
SM]
Let
rsN+IN+II, I YJ
01
=y’B&‘(A~q)S~+
+ S;+Av(q)]O;y
< 0.
Since the matrix S is positive-definite, the system Z is quadratically stabilizable via linear decentralized control. 0
(23)
Proof of Theorem 1. It follows from Lemma 3 that the system ZT.= (AKq), B$(q)) is generated from the simplest interconnected system Z,, = (A,,(9), B,,(q)) via a sequence of augmentations (either down or up), where A,,(9) = [0] or A,,(9) = [Xl, and B,,(q) is an N X N or s X s diagonally quasi-stable uncertain matrix. It is obvious that the positive-definite matrix S = IN or S = [, satisfies the condition of Lemma 1 or 2. Then, following the augmentation order and applying Lemmas 1 and 2 repeatedly, one can generate a positive-definite matrix S ,v+ = diag [ST ST . S,Z,] with S’ E ~(%+l)xl%+r) (i = 1,. ,s) a,,,, s; E @N+I)~(W+I) (m =s+l,... , N) such that the condttton of Lemma 4 holds. Hence it follows from Lemma 4 that the system Z is quadratically stabilizable via linear decentralized control. 0
system (either
Remark 5. From the above proof of Theorem 1, it is easy to see that the diagonally quasi-stable condition plays an
where J = diag [Ji .I,] E UP”” is a nonsingular constant matrix such that E,,(q)J +JE;,(q) is positive-definite for all 9 E Q. The rest of the proof is similar to the proof of Lemma cl 1. and the details are omitted. For the case where k = j = s or k = j = N, similar results can be derived. and the details are omitted. Lemma 3. If the uncertain matrix M(9) of (18) has a strongly antisymmetric stepwise configuration then the system ZK9). BR9)) with
1% ~%9)=
[l]
can be generated from the simplest interconnected X,, = (A,,(9), B,,(q)) via a sequence of augmentations
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important role in the decentralized controller design for the system Z of (5) and the condition cannot be replaced by the weaker quasi-stable condition used in centralized controller designs (Wei, 1989). If the matrix M(9) of (18) has a strongly antisymmetric stepwise configuration then it follows from Definition 4 that the submatrices A mu+:) 6 = 1,. .I A’: i#h=n,-nN) and A’k,h+l(q)r Arhh+,(9) = quasi-stable, the P:,t~+1(9):~1, must be diagonally condition contains restrictions on the values of some parameters of the system 2. In fact, it follows from Definition 1 and Remark 1 that some entries of the system I: must be sign-invariant for all 9 E Q under the condition.
A,,(9) = A,,(9) = *
A,,(9)=
Remark 6. By taking advantage of the results in Wei (1989) it is easy to show that the condition of the strongly antisymmetric stepwise configuration is an extension of the matching condition (Chen et al., 1991) for decentrally stabilizability, and Theorem 1 can be extended to the case where some subsystems in the considered system are multi-input. The details are omitted. For the case where the condition of Theorem 1 is not satisfied but some parameter ranges in a given system are not sufficiently large, applying the results of this paper to the system is still possible; see Wei (1990) for a detailed discussion.
If the system I: = (A(9). B(9)) of (5) satisfies the condition under which M(9) given by (18) has a strongly antisymmetric stepwise configuration then a decentralized stabilizing controller for the system Z can be constructed using the following procedure. Design procedure
1.
Step 1. For the given system Z = (A(9), B(9))
of (5) find the transformed system Zr = (AT(q), Bz(9)) by the transformation Tof (11) and then determine if the matrix M(9) of (18) has a strongly antisymmetric stepwise configuration. If it does, then do Step 2. Step 2. Specify the augmentation x: = (A+(9), 8~9)) of (23).
order
of the system
Step 3. Construct a positive-definite
matrix S”‘+ by following the augmentation order and applying Lemmas 1 and 2 respectively, which satisfies the condition of Lemma 4.
Remark 7.
It must be pointed that the stabilizing controller
constructed by the above design procedure is non-unique when the condition of Theorem 1 is satisfied. Comparing with the robust decentralized control designs given by Chen et al. (1991) and Chen (1992) the gains of decentralized stabilizing controllers constructed by the above design procedure only depend on the system structure, and the given varying ranges of all uncertain parameters do not depend on any preassigned nominal system. Remark 8. If every isolated subsystem in an interconnected
system has an antisymmetric stepwise configuration and the condition of Theorem 1 is satisfied then a decentralized connectively stabilizing controller for the interconnected system can be constructed by Design procedure 1 for the notion of connective stabilizability, see Siljak, 1991).
Example 1. Let the considered
system Z be described by (5).
x,(r) = [x,,(r)
x&)
x,,(r)
x&)1’
(i = 1,2), ~(0 = [x,,(r) A,,(9) A(9) = A,,(9) i A,,(9) B(9) = diag P,(9)
A,,(9) AZ(~) A32(9) b,(9)
M9)l
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a,*(9)
*
0
;
;
0”
@i(9)
*
0
;
;
;
0”
*
*
**
A13(9) = A&9) =
* 0 0’ [ 0 0 0 0
A,,(9) =A,,(9)
=
0
0
o-
0 0 0 0
0 *
0 0.
b,(q) =b,(q) = [O 0 0 II’,
b,(q) = [O 0 11’
and where each * represents a bounded uncertainty, from (lo)-(12) we know l9,2(9)1 s : and lv”zi93)cL - 2;. Then, = 4 that s = 2, n,=3, and xr(t) = with’ in(r) = [xii(r) xdt)l and [G, x;z G3 XL]’ x,+1(9) = [xi,+i(r) xZ,+i(r) x4’, and the transformed system Zr of Z is as follows: G(r) = AT(9)XT(r) + &(9)u(r)
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where
A.(9)=
A,,,(9)
ATIZ(~) A,,,(9)
ATM(~)
AR,(~)
A,,(9)
A7249)
And9)
I
~(9)
A,,(9)
A,,(9)
A,,(9)
~473.49)
A74d9)
A,,(9)
A744(9).
rB,rd9n B,(9)= b74(9)J
with AT.,,(9) E FP2, AT,,(q) E Iw’“” (i = 2, 3.4) and ad9)
1 A,,(q)=A,,(q)=B,(q)=I,,
0
0I ’
.4711(9)=0
A,,(9)=0>
AT,~(~)=ATZ*(~)=AR~(~)=BT,(~)=B~~(~)=B~(~)=O.
Since
lalz(9)(~:
1IS
and
la,,(q)1 5 $, it follows from 1 9,2(q) diagonally c n,,(9) 1 quasi-stable. Thus, from Definition 4, the matrix Definition 1 that the matrix
M(9)=
=
[
A,(9) o
B,(9) 0. I
0
And(~) 0
And9) A,,;(9)
,47x(9) AT;(~)
I
19 (26)
xX2(t) x,,(r)], A13(9) A,,(9) A,,(9)
*
0 I *01:**1 [ 0 1 0 0 0001 ;
AX(~)=
In the following, we present an example to illustrate the results of this paper.
with
*
[
Step 4. Compute a decentralized
stabilizing controller for the system 2 by using the method given in Lemma 4.
*
0
A.$91
A7:$)) 0
0 0 13 A,(9) 0
0 0 0 4 01
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has a strongly antisymmetric stepwise configuration. The corresponding system ZF= (A$(q), BKq)) of (23) is given by
and it is easy to see that the system Z$ can be generated from the simplest interconnected system & = (A,,(9), B,,(9)) with
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A,(q) = 0 and B,(q) = Is, and its augmentation order is down-up-down-down. Thus it follows from Theorem 1 that the considered system 2 is quadratically stabilizable via linear decentralized control, and a decentralized stabilizing controller for the system can be conducted by Design procedure 1 and the above augmentation order if the bounds of all uncertainties are given; the computation is omitted. 5. Conclusions In this paper, we have studied the problem of designing a linear decentralized state feedback control to stabilize a class of interconnected systems with time-varying uncertainties. Sufficient conditions under which a considered system can be quadratically stabilized by a linear decentralized feedback control for all admissible variations of uncertainties have presented .in terms of the structures of uncertainties. The conditions require that all uncertainties in the transformed matrix corresponding to the system form a special geometrical pattern called a strongly antisymmetric stepwise configuration. For interconnected systems satisfying the stabilizability conditions, a control design procedure for desired robust stabilizers has also been provided. Acknowledgements-The authors are greatly indebted to the referees for many useful suggestions and corrections on the initial manuscript of the present work. This work was partly supported by the Chinese National Natural Science Grants Council. References Barmish, B. R. (1983). Stabilization of uncertain systems via linear control. IEEE Trans. Automat. Control, AC-28, 848-850.
Barmish, B. R. (1985). Necessary and sufficient conditions for quadratic stabilizability of an uncertain system. J. Optim. Theory Applic., 46,399-408. Chen, Y. H. (1992). Decentralized robust control for large-scale systems: a design based on the bound of uncertainty. J. Dyn. Syst., Measurement, Control, 114,1-9. Chen, Y. H., G. Leitmann and X. Z. Kai (1991). Robust control design for interconnected systems with timevarying uncertainties. ht. J. Control, 54, 1119-1142. Davison, E. J. (1974). The decentralized stabilization and control of a class of unknown nonlinear time-varying . systems. Automatica, 10, 309-316. Ikeda. M. and D. D. Siliak (1980). On decentrallv stabilizable large-scale systems. Automatica, 16, 331-334. ’ Ikeda, M., D. D. Siljak and K. Yasuda (1983). Optimality of decentralized control for large-scale systems. Automatica, 19,309-316. Khargonekar, P. P., I. R. Petersen and K. Zhou (1990). Robust stabilization of uncertain linear systems: quadratic stabilizability and H, control theory. IEEE Trans. Autom. Control, AC-35,356-361. Sezer, M. E. and D. D. Siljak (1981). On decentralized stabilization and structure of linear large-scale systems. Automatica, 17, 641-644. Shi, Z. C. and W. B. Gau (1986). Stabilization by decentralized control for large-scale interconnected systems. Large Scale Syst., 10, 147-155. Siljak, D. D. (1991). Decentralized Control of Complex Systems. Academic Press, San Diego. Wei, K. (1989). Robust stabilizability of linear time varying uncertain dynamical systems via linear feedback control. In Proc. IEEE ICCON or Control Applications, Jerusalem. Wei, K. (1990). Quadratic stabilizability of linear systems with structural independent time varying uncertainties. IEEE Trans. Autom. Control, AC-35,268-277.