Delay-dependent stability and Hâ control of uncertain ... - CiteSeerX
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Systems & Control Letters 52 (2004) 361 – 376
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Delay-dependent stability and H∞ control of uncertain discrete-time Markovian jump systems with mode-dependent time delays Wu-Hua Chena , Zhi-Hong Guana;∗ , Pei Yub a Department
of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China b Department of Applied Mathematics, University of Western Ontario, London, Ont., Canada N6A 5B7 Received 12 August 2002; received in revised form 25 November 2003; accepted 19 February 2004
Abstract This paper considers the problems of stability, stabilization and H∞ control via memoryless state feedback for uncertain discrete-time Markovian jump linear systems with mode-dependent time delays. Based on linear matrix inequalities, delay-dependent solutions are obtained by using a descriptor model transformation of the system and by applying a new bounding technique for cross terms. Numerical examples are presented to illustrate the usefulness of the theoretical results. c 2004 Elsevier B.V. All rights reserved. Keywords: Discrete-time Markovian jump linear system; H∞ control; Linear matrix inequality (LMI); Delay-dependent criteria; Timedelay system
1. Introduction In practice, many dynamical systems can have diAerent structures due to random abrupt changes. It may be caused by random failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, modiBcation of the operating point of a linearized model of a nonlinear system, etc. Hybrid systems, which involve both time-evolving and event-driven mechanisms, may be employed to model the complex problem. A special class of hybrid systems is the so-called Jump Linear System (JLS). JLS is a hybrid system with many operation modes. Every mode corresponds to a deterministic dynamical behavior. A mode switching of the system is governed by a Markov process. When the mode is Bxed, the system’s state evolves according to the corresponding deterministic dynamics. The control of JLS has been a research subject and attracted many researchers since the mid 1960s. The optimal regulator, controllability,
Supported by the National Natural Science Foundation of China (NSFC No. 60074009 & 60274004) and the Natural Sciences and Engineering Research Council of Canada (NSERC No. R2686A02). ∗ Corresponding author. Tel.: +86-278-754-2068; fax: +86-278-754-2145. E-mail address: [email protected] (Z.-H. Guan).
c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2004.02.012
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observability, stability and stabilization problems have been extensively studied by many authors, see [10–12,16,18,21] and references therein. It has been recognized that time delays, which are the inherent features of many physical process, are the big sources of instability and poor performances. For continuous-time JLSs with time delay, the problem of stability has been studied in [1,19]. In [3,4,6,9,15], the stochastic Lyapunov functional approach was used to successfully investigate robust stabilization and H∞ control, and both delay-independent and delay-dependent methods were proposed for controller design. For discrete-time JLSs with time delay, using linear matrix inequality (LMI) approach or modiBed Riccati inequality approach, the problems of robust stability, stabilization and H∞ control have been studied in [2,8,22] for the constant time-delay case and in [5] for the mode-dependent time-delays case. However, the results in [5] depend on the diAerence between the largest and the smallest time-delay, while the results given in [8,22] are delay-independent. When the time-delays of all the system’s modes are identical, the results presented in [5] are also delay-independent. In general, delay-independent results are conservative since they cannot handle the systems whose stability or stabilization depends on the size of time-delay. In [2], with the system argumentation approach, a JLS with time-delay is Brst transformed into a high-order delay-free JLS, and then the existing stability and H∞ control theory for discrete-time delay-free JLSs is applied to obtain the delay-dependent conditions. However, in this way the problems of stabilization and H∞ control via a memoryless state feedback for discrete-time JLSs with delays cannot be solved, since the augmented states contain the original time-delayed state variables. In addition, as noted in [2], this scheme is only applicable to systems with small time delays because the dimension of the resulting delay-free JLS increases substantially as the time-delay increases. Therefore, for discrete-time state-delayed JLSs whose stability or stabilization depends on the size of delay, the problems of stability, stabilization and H∞ control via a memoryless state feedback have not been well studied. This paper is concerned with the problems of robust stability, stabilization and H∞ control via memoryless state feedback for uncertain JLSs with mode-dependent delays. In comparison with the results in [2,5,8,22], the focal point of this paper is on developing methods for the concerned problems based on LMIs and which depend on the size of the time-delay. Firstly, we present a new delay-dependent condition for the robust stability problem in terms of linear matrix inequalities (LMIs). Secondly, new delay-dependent suLcient conditions using memoryless state feedback controllers for robust stabilization and H∞ control are established in terms of LMIs, which are applicable to delay-dependent stabilizable systems. The desired state feedback controllers for stabilization and H∞ control can be constructed by using interior point algorithms for solving LMIs [7]. Numerical examples are given which show that our results are eLcient in solving the concerned problems for JLSs whose delays are comparatively large and the stability or stabilizability depends on the size of delays. Therefore, the method proposed in this paper overcomes the deBciency of the approaches developed in [2,5,8,22]. The main idea of our method is inspired by Fridman’s recent work [13,14,17], where a descriptor system approach is introduced for stability analysis and controller synthesis of deterministic delay systems. We extend this approach to investigate the problems of robust stability, stabilization and H∞ control for uncertain JLSs with mode-dependent delays. The advantage of the descriptor system approach lies in that it considerably reduces the conservatism entailed in the previously developed transformation methods since it is based on an equivalent “descriptor form” representation of the system and requires bounds for fewer cross terms. The bounds can now be made tighter by using the new bounding technique of [20]. Based on this approach, a new type of stochastic Lyapunov functional is introduced, and thus eLcient solutions are obtained in terms of delay-dependent LMIs. The rest of the paper is organized as follows. The problem is formulated in Section 2. Sections 3 deals with delay-dependent conditions for the robust stability and robust stabilization. Section 4 is developed to derive the LMI-based conditions for robust H∞ control. Numerical examples are presented in Section 5 to illustrate the theoretical results, and the conclusions are drawn in Section 6.
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2. Model description Throughout the paper, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation M ¿ 0 (¡ 0) is used to denote a positive (negative) deBnite matrix. min (·) and max (·) represent the minimum and maximum eigenvalues of the corresponding matrix, respectively. I denotes an identity matrix with appropriate dimension. · stands for the Euclidean norm of vectors or the spectral norm of matrices. Z and E[ · ] denote the set of non-negative integer numbers and the mathematical expectation. Let {rk ; k ∈ Z} be a Markov chain with state space S = {1; 2; : : : ; N }. Denote the state transition matrix by P = (pij )i; j∈S , i.e., the transition probabilities of {rk ; k ∈ Z} are given by Pr{rk+1 = j | rk = i} = pij for i; j ∈ S N with pij ¿ 0 for i; j ∈ S, and j=1 pij = 1 for i ∈ S. Consider a discrete-time, hybrid system with N modes. Suppose that the system’s mode switching is governed by {rk ; k ∈ Z}, and the system parameters contain norm-bounded uncertainties. Let the system dynamics be described by the following equations: xk+1 = A(k; rk )xk + Ad (k; rk )xk−(rk ) + B(k; rk )uk + B1 (rk )wk ; zk = C(k; rk )xk + Cd (k; rk )xk−(rk ) + Bc (k; rk )uk + C1 (rk )wk ; (1) xs = ’(s); s = −; P : : : ; −1; 0; where k ∈ Z, xk ∈ Rn is the system state, uk ∈ Rm the control input, wk ∈ Rq the disturbance input which ∞ belongs to l2 = {{ak ; k ∈ Z}| k=0 a2k ¡ ∞}, and zk ∈ Rp is the controlled output. (rk ) is a positive integer, denoting the time delay of the system involved in mode rk . =max{(i); P i ∈ S}. ’ : {−; P −(−1); P : : : ; 0} → Rn represents the initial condition. For each i ∈ S, we have A(k; i) = A(i) + QA(k; i);
Ad (k; i) = Ad (i) + QAd (k; i);
C(k; i) = C(i) + QC(k; i);
B(k; i) = B(i) + QB(k; i);
Cd (k; i) = Cd (i) + QCd (k; i);
Bc (k; i) = Bc (i) + QBc (k; i):
Here, all matrices A(i); B(i); C(i), etc. in Eq. (1) and above equations are constant matrices with appropriate dimensions. QA(k; i); QAd (k; i); QB(k; i); QC(k; i); QCd (k; i) and QBc (k; i) are unknown matrices, denoting the uncertainties in the system. We assume that the uncertainties are norm-bounded and may be described as QA(k; i) QAd (k; i) QB(k; i) E1 (i) = F(k; i)[H1 (i) H2 (i) H3 (i)]; QC(k; i) QCd (k; i) QBc (k; i) E2 (i) where E1 (i) ∈ Rn×nf ; E2 (i) ∈ Rp×nf , H1 (i); H2 (i) ∈ Rnf ×n , H3 (i) ∈ Rnf ×m are known constant matrices, and F(k; i) ∈ Rnf ×nf are unknown matrix functions satisfying F T (k; i)F(k; i) 6 I;
i ∈ S:
The aim of this paper is to establish the delay-dependent robust stability, robust stabilization and robust H∞ -control for system (1) using the LMI technique. For convenience of analysis, we introduce the following notations: p = min{pii ; i ∈ S};
= min{(i); i ∈ S};
a (i) =
N
pij j ;
j=1
$ = 1 + (1 − p)(P − );
%i = a (i) +
1− p (P − )(P + − 1): 2
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W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
3. Robust stability and stabilization In this section, the discussion is restricted to the case wk = 0, namely, xk+1 = A(k; rk )xk + Ad (k; rk )xk−(rk ) + B(k; rk )uk ; xs = ’(s);
(2)
s = −; P : : : ; −1; 0:
First, we present some delay-dependent suLcient conditions for the robust stochastic stability of the uncertain jump linear system (2) with uk = 0. Denition 1. The uncertain jump linear system (2), with uk = 0, is said to be robust stochastically stable, if for every initial state (’; r0 ) and all possible uncertainties F(k; i) 6 1 the following condition:
∞ 2 E xk (’; r0 ) |’; r0 ¡ ∞ k=0
is satisBed. The result obtained for the robust stability is given in the following theorem. Theorem 1. The uncertain jump linear system (2) with uk = 0 is robust stochastically stable, if for each i ∈ S, there exist a positive scalar &i , and matrices Pi ¿ 0; Pi1 ; Pi2 , Wi1 ; Wi2 ; Wi3 , Mi1 ; Mi2 ; S ¿ 0 and Q ¿ 0, satisfying the following coupled matrix inequalities: T T *i1 *i2 Pi1 Ad (i) − Mi1 + &i H1T (i)H2 (i) Pi1 E1 (i) T T ∗ *i3 Pi2 Ad (i) − Mi2 Pi2 E1 (i) (3) ¡0 T ∗ H (i)H (i) − S 0 ∗ & i 2 2 and
∗
∗
Wi1
Wi2
T Wi2 Mi1T
Wi3 Mi2T
0 Mi1
−&i I
Mi2 ¿ 0;
(4)
Q
where T (A(i) − I ) + (AT (i) − I )Pi1 + &i H1T (i)H1 (i) + (i)Wi1 + Mi1 + Mi1T + *i1 = Pi1
*i2 = AT (i)Pi2 − Pi2 +
N
N
pij Pj − Pi + $S;
j=1 T pij Pj − Pi1 + (i)Wi2 + Mi2T ;
j=1 T *i3 = −Pi2 − Pi2 + (i)Wi3 +
N
pij Pj + %i Q:
j=1
To prove the theorem, we need the following two matrix inequalities. Lemma 1 (Wang et al. [23]). Let D; E and F be matrices with appropriate dimensions. Suppose F T F 6 I , then, for any scalar & ¿ 0, we have DFE + E T F T DT 6 &DDT + &−1 E T E:
W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
365
Lemma 2 (Moon et al. [20]). Assume that , ∈ Rna , - ∈ Rnb and N ∈ Rna ×nb . Then, for any matrices X ∈ Rna ×na ; Y ∈ Rna ×nb and Z ∈ Rnb ×nb , the following inequality: T X Y −N , , −2,T N- 6 Z Y T − NT holds, where X Y YT
Z
¿ 0:
Proof of Theorem 1. First we write (2) with uk = 0 in the equivalent descriptor form xk+1 − xk = yk ;
yk = (A(k; rk ) − I )xk + Ad (k; rk )xk−(rk )
which can be further rewritten in the form with delay in the variable yk : xk+1 = xk + yk ; k−1
0 = −yk + (A(k; rk ) + Ad (k; rk ) − I )xk − Ad (k; rk )
yl :
(5)
l=k−(rk )
Then, construct the stochastic Lyapunov functional V (Xk ; Yk ; k; rk ) with Xk = (xk ; xk−1 ; : : : ; xk−(k) ), Yk = (yk ; yk−1 ; : : : ; yk−(k) ) as follows: V (Xk ; Yk ; k; rk ) =
3
Vi (Xk ; Yk ; k; rk )
(6)
i=0
with V0 (Xk ; Yk ; k; rk ) = xkT Prk xk ;
V1 (Xk ; Yk ; k; rk ) =
0
k−1
ylT Qyl ;
4=−(rk )+1 l=k−1+4
V2 (Xk ; Yk ; k; rk ) =
k−1
xlT Sxl ;
l=k−(rk )
V3 (Xk ; Yk ; k; rk ) = (1 − p)
− k−1
[ylT Qyl (l − k − 4 + 1) + xlT Sxl ]:
4=−+1 P l=k+4
Let the mode at time k be Ni, that is, rk = i. Recall that at time k + 1, the system may jump to any mode rk+1 = j. By letting PP i = j=1 pij Pj , one can then obtain E[V0 (Xk+1 ; Yk+1 ; k + 1; rk+1 ) | Xk ; Yk ; rk = i] − V0 (Xk ; Yk ; k; i) T =E[xk+1 Prk+1 xk+1 | Xk ; Yk ; rk = i] − V0 (Xk ; Yk ; k; i)
=E[(xk + yk )T Prk+1 (xk + yk ) | Xk ; Yk ; rk = i] − V0 (Xk ; Yk ; k; i) N = pij (xk + yk )T Pj (xk + yk ) − xkT Pi xk j=1
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W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
=(xk + yk )T PP i (xk + yk ) − xkT Pi xk =ykT PP i yk + 2xkT PP i yk + xkT (PP i − Pi )xk =25Tk GiT [ykT 0]T + xkT (PP i − Pi )xk + ykT PP i yk ; where
5Tk
=
[xkT
ykT ]
and
Gi =
(7)
PP i
0
Pi1
Pi2
;
Pi1 and Pi2 are constant matrices with appropriate dimensions. Thus, due to relation (5), we get k−1 y y 0 k k − = 25Tk GiT yl : 25Tk GiT −yk + [A(k; i) + Ad (k; i) − I ]xk Ad (k; i) l=k−(i) 0 Next, by Lemma 2, we Bnd k−1 k−1 0 0 −25Tk GiT yl yl = −2 5Tk GiT Ad (k; i) l=k−(i) Ad (k; i) l=k−(i) k−1
6
l=k−(i)
5k
T
yl
Wi
Mi − GiT [0 ATd (k; i)]T
∗
Q
k−1
=(i)5Tk Wi 5k +
5k
yl
ylT Qyl + 25Tk (Mi − GiT [0 ATd (k; i)]T )(xk − xk−(i) );
l=k−(i)
where
Wi =
Wi1
Wi2
Wi2T
Wi3
;
Mi =
Mi1
Mi2
and Q are constant matrices, satisfying (4). Therefore, k−1 yk yk T T T T 25k Gi + (i)5Tk Wi 5k + 6 25k Gi ylT Qyl 0 −yk + (A(k; i) − I )xk l=k−(i) +25Tk Mi (xk − xk−(i) ) + 25Tk GiT [0
ATd (k; i)]T xk−(i) :
For V1 (Xk ; Yk ; k; rk ), we can establish E[V1 (Xk+1 ; Yk+1 ; k + 1; rk+1 | Xk ; Yk ; rk = i)] − V1 (Xk ; Yk ; k; i) 0 k =E ylT Qyl | Xk ; Yk ; rk = i − V1 (Xk ; Yk ; k; i) 4=−(rk+1 )+1 l=k+4
(8)
W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
=
N
0
pij
j=1
k
4=−( j)+1 l=k+4
0
=pii
4=−(i)+1 0
=pii
k
k−1
k−1
N
ylT Qyl +
N
j=1; j=i
6
N
T (ykT Qyk − yk−1+4 Qyk−1+4 ) +
0
pij
j=1; j=i
l=k−1+4
N
k
−
4=−( j)+1 l=k+4
0
pij
k−1
4=−( j)+1 l=k+4
pij (j)ykT Qyk
k−1
−
j=1
0
−
k−1
k−1
ylT Qyl
0
pij (j)ykT Qyk −
T yk−1+4 Qyk−1+4
4=−(i)+1
ylT Qyl
4=−(i)+1 l=k+4
ylT Qyl
+
N
pij
j=1; j=i
l=k−(i)
6 a (i)ykT Qyk −
k−1
0
4=−(i)+1 l=k−1+4
j=1; j=i
4=−(i)+1
+
ylT Qyl
4=−(i)+1 l=k−1+4
−
l=k+4
0
ylT Qyl −
367
−
ylT Qyl + (1 − p)
l=k−(i)
0 4=−+1 P
k−1
0
−
4=−+1
k−1
ylT Qyl
l=k+4
ylT Qyl :
(9)
4=−+1 P l=k+4
Similarly, E[V2 (Xk+1 ; Yk+1 ; k + 1; rk+1 | Xk ; Yk ; rk = i)] − V2 (Xk ; Yk ; k; i) T 6 xkT Sxk − xk−(i) Sxk−(i) + (1 − p)
k−
xlT Sxl :
(10)
l=k+1−P
Moreover, a direct computation gives E[V3 (Xk+1 ; Yk+1 ; k + 1; rk+1 ) | Xk ; Yk ; rk = i] − V3 (Xk ; Yk ; k; i)
− k− k−1 6 (1 − p) 12 (P − )(P + − 1)ykT Qyk − ylT Qyl + (P − )xkT Sxk − xlT Sxl : (11) 4=−+1 P l=k+4
l=k+1−P
Now combining (7)–(11) results in E[V (Xk+1 ; Yk+1 ; k + 1; rk+1 ) | xk ; yk ; rk = i] − V (Xk ; Yk ; k; i) 6 7Tk 8i (k)7k ; T where 7Tk = [5Tk xk−(i) ], and
8i (k) =
9i
GiT [0 ATd (i)]T − Mi
∗
−S
T + EP 1 (i)F(k; i)H12 (i) + H12 (i)F T (k; i)EP T1 (i)
(12)
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W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
in which EP T1 (i) = [[0 E1T (i)]Gi 0 I T 9i = Gi A(i) − I −I PP i − Pi + $S + 0
0]; H12 (i) = [H1 (i) 0 H2 (i)], and T 0 AT (i) − I Mi Gi + (i)Wi + [Mi 0] + + I −I 0 0 : PP i + %i Q
(13)
Using Lemma 1, for any &i ¿ 0, we get 9i GiT [0 ATd (i)]T − Mi T 8i (k) 6 + &i−1 EP 1 (i)EP T1 (i) + &i H12 (i)H12 (i) := 8i : ∗ −S By Schur complement, (3) implies 8i ¡ 0. Let 0 = min{−8i ; i ∈ S}, then 0 ¿ 0 due to (3). Finally, from (12) we obtain that for any k ¿ 0, E[V (Xk+1 ; Yk+1 ; k + 1; rk+1 ) | Xk ; Yk ; rk = i] 6 V (Xk ; Yk ; k; rk ) − 0 xkT xk :
(14)
Setting k = 0 and k = 1 in (14) yields E[V (X1 ; Y1 ; 1; r1 ) | X0 ; Y0 ; r0 ] 6 V (X0 ; Y0 ; 0; r0 ) − 0 x0T x0
(15)
E[V (X2 ; Y2 ; 2; r2 ) | X1 ; Y1 ; r1 ] 6 V (X1 ; Y1 ; 1; r1 ) − 0 x1T x1 :
(16)
and
Taking expectation E[ · |X0 ; Y0 ; r0 ] on both sides of (16), with the aid of (15), leads to E[V (X2 ; Y2 ; 2; r2 ) | X0 ; Y0 ; r0 ] 6 V (X0 ; Y0 ; 0; r0 ) − 0
1
E[xlT xl | X0 ; Y0 ; r0 ]:
l=0
Then, one can continue the iterative procedure (14) to obtain E[V (XT +1 ; YT +1 ; T + 1; rT +1 ) | X0 ; Y0 ; r0 ] 6 V (X0 ; Y0 ; 0; r0 ) − 0
T
E[xlT xl | X0 ; Y0 ; r0 ];
l=0
implying that ∞
E[xlT xl | X0 ; Y0 ; r0 ] 6
l=0
1 V (X0 ; Y0 ; 0; r0 ) ¡ ∞: 0
This indicates that system (2) with uk = 0 is robustly stochastically stable, and thus the proof of Theorem 1 is completed. The remaining of this section is devoted to design a robust, memoryless state feedback controller in the form of uk = K r k xk such that the resulting closed-loop system is stochastically stable.
(17)
W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
369
Theorem 2. If for each i ∈ S, and for some prescribed scalars
Systems & Control Letters 52 (2004) 361 – 376
www.elsevier.com/locate/sysconle
Delay-dependent stability and H∞ control of uncertain discrete-time Markovian jump systems with mode-dependent time delays Wu-Hua Chena , Zhi-Hong Guana;∗ , Pei Yub a Department
of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China b Department of Applied Mathematics, University of Western Ontario, London, Ont., Canada N6A 5B7 Received 12 August 2002; received in revised form 25 November 2003; accepted 19 February 2004
Abstract This paper considers the problems of stability, stabilization and H∞ control via memoryless state feedback for uncertain discrete-time Markovian jump linear systems with mode-dependent time delays. Based on linear matrix inequalities, delay-dependent solutions are obtained by using a descriptor model transformation of the system and by applying a new bounding technique for cross terms. Numerical examples are presented to illustrate the usefulness of the theoretical results. c 2004 Elsevier B.V. All rights reserved. Keywords: Discrete-time Markovian jump linear system; H∞ control; Linear matrix inequality (LMI); Delay-dependent criteria; Timedelay system
1. Introduction In practice, many dynamical systems can have diAerent structures due to random abrupt changes. It may be caused by random failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, modiBcation of the operating point of a linearized model of a nonlinear system, etc. Hybrid systems, which involve both time-evolving and event-driven mechanisms, may be employed to model the complex problem. A special class of hybrid systems is the so-called Jump Linear System (JLS). JLS is a hybrid system with many operation modes. Every mode corresponds to a deterministic dynamical behavior. A mode switching of the system is governed by a Markov process. When the mode is Bxed, the system’s state evolves according to the corresponding deterministic dynamics. The control of JLS has been a research subject and attracted many researchers since the mid 1960s. The optimal regulator, controllability,
Supported by the National Natural Science Foundation of China (NSFC No. 60074009 & 60274004) and the Natural Sciences and Engineering Research Council of Canada (NSERC No. R2686A02). ∗ Corresponding author. Tel.: +86-278-754-2068; fax: +86-278-754-2145. E-mail address: [email protected] (Z.-H. Guan).
c 2004 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2004.02.012
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W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
observability, stability and stabilization problems have been extensively studied by many authors, see [10–12,16,18,21] and references therein. It has been recognized that time delays, which are the inherent features of many physical process, are the big sources of instability and poor performances. For continuous-time JLSs with time delay, the problem of stability has been studied in [1,19]. In [3,4,6,9,15], the stochastic Lyapunov functional approach was used to successfully investigate robust stabilization and H∞ control, and both delay-independent and delay-dependent methods were proposed for controller design. For discrete-time JLSs with time delay, using linear matrix inequality (LMI) approach or modiBed Riccati inequality approach, the problems of robust stability, stabilization and H∞ control have been studied in [2,8,22] for the constant time-delay case and in [5] for the mode-dependent time-delays case. However, the results in [5] depend on the diAerence between the largest and the smallest time-delay, while the results given in [8,22] are delay-independent. When the time-delays of all the system’s modes are identical, the results presented in [5] are also delay-independent. In general, delay-independent results are conservative since they cannot handle the systems whose stability or stabilization depends on the size of time-delay. In [2], with the system argumentation approach, a JLS with time-delay is Brst transformed into a high-order delay-free JLS, and then the existing stability and H∞ control theory for discrete-time delay-free JLSs is applied to obtain the delay-dependent conditions. However, in this way the problems of stabilization and H∞ control via a memoryless state feedback for discrete-time JLSs with delays cannot be solved, since the augmented states contain the original time-delayed state variables. In addition, as noted in [2], this scheme is only applicable to systems with small time delays because the dimension of the resulting delay-free JLS increases substantially as the time-delay increases. Therefore, for discrete-time state-delayed JLSs whose stability or stabilization depends on the size of delay, the problems of stability, stabilization and H∞ control via a memoryless state feedback have not been well studied. This paper is concerned with the problems of robust stability, stabilization and H∞ control via memoryless state feedback for uncertain JLSs with mode-dependent delays. In comparison with the results in [2,5,8,22], the focal point of this paper is on developing methods for the concerned problems based on LMIs and which depend on the size of the time-delay. Firstly, we present a new delay-dependent condition for the robust stability problem in terms of linear matrix inequalities (LMIs). Secondly, new delay-dependent suLcient conditions using memoryless state feedback controllers for robust stabilization and H∞ control are established in terms of LMIs, which are applicable to delay-dependent stabilizable systems. The desired state feedback controllers for stabilization and H∞ control can be constructed by using interior point algorithms for solving LMIs [7]. Numerical examples are given which show that our results are eLcient in solving the concerned problems for JLSs whose delays are comparatively large and the stability or stabilizability depends on the size of delays. Therefore, the method proposed in this paper overcomes the deBciency of the approaches developed in [2,5,8,22]. The main idea of our method is inspired by Fridman’s recent work [13,14,17], where a descriptor system approach is introduced for stability analysis and controller synthesis of deterministic delay systems. We extend this approach to investigate the problems of robust stability, stabilization and H∞ control for uncertain JLSs with mode-dependent delays. The advantage of the descriptor system approach lies in that it considerably reduces the conservatism entailed in the previously developed transformation methods since it is based on an equivalent “descriptor form” representation of the system and requires bounds for fewer cross terms. The bounds can now be made tighter by using the new bounding technique of [20]. Based on this approach, a new type of stochastic Lyapunov functional is introduced, and thus eLcient solutions are obtained in terms of delay-dependent LMIs. The rest of the paper is organized as follows. The problem is formulated in Section 2. Sections 3 deals with delay-dependent conditions for the robust stability and robust stabilization. Section 4 is developed to derive the LMI-based conditions for robust H∞ control. Numerical examples are presented in Section 5 to illustrate the theoretical results, and the conclusions are drawn in Section 6.
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2. Model description Throughout the paper, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation M ¿ 0 (¡ 0) is used to denote a positive (negative) deBnite matrix. min (·) and max (·) represent the minimum and maximum eigenvalues of the corresponding matrix, respectively. I denotes an identity matrix with appropriate dimension. · stands for the Euclidean norm of vectors or the spectral norm of matrices. Z and E[ · ] denote the set of non-negative integer numbers and the mathematical expectation. Let {rk ; k ∈ Z} be a Markov chain with state space S = {1; 2; : : : ; N }. Denote the state transition matrix by P = (pij )i; j∈S , i.e., the transition probabilities of {rk ; k ∈ Z} are given by Pr{rk+1 = j | rk = i} = pij for i; j ∈ S N with pij ¿ 0 for i; j ∈ S, and j=1 pij = 1 for i ∈ S. Consider a discrete-time, hybrid system with N modes. Suppose that the system’s mode switching is governed by {rk ; k ∈ Z}, and the system parameters contain norm-bounded uncertainties. Let the system dynamics be described by the following equations: xk+1 = A(k; rk )xk + Ad (k; rk )xk−(rk ) + B(k; rk )uk + B1 (rk )wk ; zk = C(k; rk )xk + Cd (k; rk )xk−(rk ) + Bc (k; rk )uk + C1 (rk )wk ; (1) xs = ’(s); s = −; P : : : ; −1; 0; where k ∈ Z, xk ∈ Rn is the system state, uk ∈ Rm the control input, wk ∈ Rq the disturbance input which ∞ belongs to l2 = {{ak ; k ∈ Z}| k=0 a2k ¡ ∞}, and zk ∈ Rp is the controlled output. (rk ) is a positive integer, denoting the time delay of the system involved in mode rk . =max{(i); P i ∈ S}. ’ : {−; P −(−1); P : : : ; 0} → Rn represents the initial condition. For each i ∈ S, we have A(k; i) = A(i) + QA(k; i);
Ad (k; i) = Ad (i) + QAd (k; i);
C(k; i) = C(i) + QC(k; i);
B(k; i) = B(i) + QB(k; i);
Cd (k; i) = Cd (i) + QCd (k; i);
Bc (k; i) = Bc (i) + QBc (k; i):
Here, all matrices A(i); B(i); C(i), etc. in Eq. (1) and above equations are constant matrices with appropriate dimensions. QA(k; i); QAd (k; i); QB(k; i); QC(k; i); QCd (k; i) and QBc (k; i) are unknown matrices, denoting the uncertainties in the system. We assume that the uncertainties are norm-bounded and may be described as QA(k; i) QAd (k; i) QB(k; i) E1 (i) = F(k; i)[H1 (i) H2 (i) H3 (i)]; QC(k; i) QCd (k; i) QBc (k; i) E2 (i) where E1 (i) ∈ Rn×nf ; E2 (i) ∈ Rp×nf , H1 (i); H2 (i) ∈ Rnf ×n , H3 (i) ∈ Rnf ×m are known constant matrices, and F(k; i) ∈ Rnf ×nf are unknown matrix functions satisfying F T (k; i)F(k; i) 6 I;
i ∈ S:
The aim of this paper is to establish the delay-dependent robust stability, robust stabilization and robust H∞ -control for system (1) using the LMI technique. For convenience of analysis, we introduce the following notations: p = min{pii ; i ∈ S};
= min{(i); i ∈ S};
a (i) =
N
pij j ;
j=1
$ = 1 + (1 − p)(P − );
%i = a (i) +
1− p (P − )(P + − 1): 2
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W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
3. Robust stability and stabilization In this section, the discussion is restricted to the case wk = 0, namely, xk+1 = A(k; rk )xk + Ad (k; rk )xk−(rk ) + B(k; rk )uk ; xs = ’(s);
(2)
s = −; P : : : ; −1; 0:
First, we present some delay-dependent suLcient conditions for the robust stochastic stability of the uncertain jump linear system (2) with uk = 0. Denition 1. The uncertain jump linear system (2), with uk = 0, is said to be robust stochastically stable, if for every initial state (’; r0 ) and all possible uncertainties F(k; i) 6 1 the following condition:
∞ 2 E xk (’; r0 ) |’; r0 ¡ ∞ k=0
is satisBed. The result obtained for the robust stability is given in the following theorem. Theorem 1. The uncertain jump linear system (2) with uk = 0 is robust stochastically stable, if for each i ∈ S, there exist a positive scalar &i , and matrices Pi ¿ 0; Pi1 ; Pi2 , Wi1 ; Wi2 ; Wi3 , Mi1 ; Mi2 ; S ¿ 0 and Q ¿ 0, satisfying the following coupled matrix inequalities: T T *i1 *i2 Pi1 Ad (i) − Mi1 + &i H1T (i)H2 (i) Pi1 E1 (i) T T ∗ *i3 Pi2 Ad (i) − Mi2 Pi2 E1 (i) (3) ¡0 T ∗ H (i)H (i) − S 0 ∗ & i 2 2 and
∗
∗
Wi1
Wi2
T Wi2 Mi1T
Wi3 Mi2T
0 Mi1
−&i I
Mi2 ¿ 0;
(4)
Q
where T (A(i) − I ) + (AT (i) − I )Pi1 + &i H1T (i)H1 (i) + (i)Wi1 + Mi1 + Mi1T + *i1 = Pi1
*i2 = AT (i)Pi2 − Pi2 +
N
N
pij Pj − Pi + $S;
j=1 T pij Pj − Pi1 + (i)Wi2 + Mi2T ;
j=1 T *i3 = −Pi2 − Pi2 + (i)Wi3 +
N
pij Pj + %i Q:
j=1
To prove the theorem, we need the following two matrix inequalities. Lemma 1 (Wang et al. [23]). Let D; E and F be matrices with appropriate dimensions. Suppose F T F 6 I , then, for any scalar & ¿ 0, we have DFE + E T F T DT 6 &DDT + &−1 E T E:
W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
365
Lemma 2 (Moon et al. [20]). Assume that , ∈ Rna , - ∈ Rnb and N ∈ Rna ×nb . Then, for any matrices X ∈ Rna ×na ; Y ∈ Rna ×nb and Z ∈ Rnb ×nb , the following inequality: T X Y −N , , −2,T N- 6 Z Y T − NT holds, where X Y YT
Z
¿ 0:
Proof of Theorem 1. First we write (2) with uk = 0 in the equivalent descriptor form xk+1 − xk = yk ;
yk = (A(k; rk ) − I )xk + Ad (k; rk )xk−(rk )
which can be further rewritten in the form with delay in the variable yk : xk+1 = xk + yk ; k−1
0 = −yk + (A(k; rk ) + Ad (k; rk ) − I )xk − Ad (k; rk )
yl :
(5)
l=k−(rk )
Then, construct the stochastic Lyapunov functional V (Xk ; Yk ; k; rk ) with Xk = (xk ; xk−1 ; : : : ; xk−(k) ), Yk = (yk ; yk−1 ; : : : ; yk−(k) ) as follows: V (Xk ; Yk ; k; rk ) =
3
Vi (Xk ; Yk ; k; rk )
(6)
i=0
with V0 (Xk ; Yk ; k; rk ) = xkT Prk xk ;
V1 (Xk ; Yk ; k; rk ) =
0
k−1
ylT Qyl ;
4=−(rk )+1 l=k−1+4
V2 (Xk ; Yk ; k; rk ) =
k−1
xlT Sxl ;
l=k−(rk )
V3 (Xk ; Yk ; k; rk ) = (1 − p)
− k−1
[ylT Qyl (l − k − 4 + 1) + xlT Sxl ]:
4=−+1 P l=k+4
Let the mode at time k be Ni, that is, rk = i. Recall that at time k + 1, the system may jump to any mode rk+1 = j. By letting PP i = j=1 pij Pj , one can then obtain E[V0 (Xk+1 ; Yk+1 ; k + 1; rk+1 ) | Xk ; Yk ; rk = i] − V0 (Xk ; Yk ; k; i) T =E[xk+1 Prk+1 xk+1 | Xk ; Yk ; rk = i] − V0 (Xk ; Yk ; k; i)
=E[(xk + yk )T Prk+1 (xk + yk ) | Xk ; Yk ; rk = i] − V0 (Xk ; Yk ; k; i) N = pij (xk + yk )T Pj (xk + yk ) − xkT Pi xk j=1
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W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
=(xk + yk )T PP i (xk + yk ) − xkT Pi xk =ykT PP i yk + 2xkT PP i yk + xkT (PP i − Pi )xk =25Tk GiT [ykT 0]T + xkT (PP i − Pi )xk + ykT PP i yk ; where
5Tk
=
[xkT
ykT ]
and
Gi =
(7)
PP i
0
Pi1
Pi2
;
Pi1 and Pi2 are constant matrices with appropriate dimensions. Thus, due to relation (5), we get k−1 y y 0 k k − = 25Tk GiT yl : 25Tk GiT −yk + [A(k; i) + Ad (k; i) − I ]xk Ad (k; i) l=k−(i) 0 Next, by Lemma 2, we Bnd k−1 k−1 0 0 −25Tk GiT yl yl = −2 5Tk GiT Ad (k; i) l=k−(i) Ad (k; i) l=k−(i) k−1
6
l=k−(i)
5k
T
yl
Wi
Mi − GiT [0 ATd (k; i)]T
∗
Q
k−1
=(i)5Tk Wi 5k +
5k
yl
ylT Qyl + 25Tk (Mi − GiT [0 ATd (k; i)]T )(xk − xk−(i) );
l=k−(i)
where
Wi =
Wi1
Wi2
Wi2T
Wi3
;
Mi =
Mi1
Mi2
and Q are constant matrices, satisfying (4). Therefore, k−1 yk yk T T T T 25k Gi + (i)5Tk Wi 5k + 6 25k Gi ylT Qyl 0 −yk + (A(k; i) − I )xk l=k−(i) +25Tk Mi (xk − xk−(i) ) + 25Tk GiT [0
ATd (k; i)]T xk−(i) :
For V1 (Xk ; Yk ; k; rk ), we can establish E[V1 (Xk+1 ; Yk+1 ; k + 1; rk+1 | Xk ; Yk ; rk = i)] − V1 (Xk ; Yk ; k; i) 0 k =E ylT Qyl | Xk ; Yk ; rk = i − V1 (Xk ; Yk ; k; i) 4=−(rk+1 )+1 l=k+4
(8)
W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
=
N
0
pij
j=1
k
4=−( j)+1 l=k+4
0
=pii
4=−(i)+1 0
=pii
k
k−1
k−1
N
ylT Qyl +
N
j=1; j=i
6
N
T (ykT Qyk − yk−1+4 Qyk−1+4 ) +
0
pij
j=1; j=i
l=k−1+4
N
k
−
4=−( j)+1 l=k+4
0
pij
k−1
4=−( j)+1 l=k+4
pij (j)ykT Qyk
k−1
−
j=1
0
−
k−1
k−1
ylT Qyl
0
pij (j)ykT Qyk −
T yk−1+4 Qyk−1+4
4=−(i)+1
ylT Qyl
4=−(i)+1 l=k+4
ylT Qyl
+
N
pij
j=1; j=i
l=k−(i)
6 a (i)ykT Qyk −
k−1
0
4=−(i)+1 l=k−1+4
j=1; j=i
4=−(i)+1
+
ylT Qyl
4=−(i)+1 l=k−1+4
−
l=k+4
0
ylT Qyl −
367
−
ylT Qyl + (1 − p)
l=k−(i)
0 4=−+1 P
k−1
0
−
4=−+1
k−1
ylT Qyl
l=k+4
ylT Qyl :
(9)
4=−+1 P l=k+4
Similarly, E[V2 (Xk+1 ; Yk+1 ; k + 1; rk+1 | Xk ; Yk ; rk = i)] − V2 (Xk ; Yk ; k; i) T 6 xkT Sxk − xk−(i) Sxk−(i) + (1 − p)
k−
xlT Sxl :
(10)
l=k+1−P
Moreover, a direct computation gives E[V3 (Xk+1 ; Yk+1 ; k + 1; rk+1 ) | Xk ; Yk ; rk = i] − V3 (Xk ; Yk ; k; i)
− k− k−1 6 (1 − p) 12 (P − )(P + − 1)ykT Qyk − ylT Qyl + (P − )xkT Sxk − xlT Sxl : (11) 4=−+1 P l=k+4
l=k+1−P
Now combining (7)–(11) results in E[V (Xk+1 ; Yk+1 ; k + 1; rk+1 ) | xk ; yk ; rk = i] − V (Xk ; Yk ; k; i) 6 7Tk 8i (k)7k ; T where 7Tk = [5Tk xk−(i) ], and
8i (k) =
9i
GiT [0 ATd (i)]T − Mi
∗
−S
T + EP 1 (i)F(k; i)H12 (i) + H12 (i)F T (k; i)EP T1 (i)
(12)
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W.-H. Chen et al. / Systems & Control Letters 52 (2004) 361 – 376
in which EP T1 (i) = [[0 E1T (i)]Gi 0 I T 9i = Gi A(i) − I −I PP i − Pi + $S + 0
0]; H12 (i) = [H1 (i) 0 H2 (i)], and T 0 AT (i) − I Mi Gi + (i)Wi + [Mi 0] + + I −I 0 0 : PP i + %i Q
(13)
Using Lemma 1, for any &i ¿ 0, we get 9i GiT [0 ATd (i)]T − Mi T 8i (k) 6 + &i−1 EP 1 (i)EP T1 (i) + &i H12 (i)H12 (i) := 8i : ∗ −S By Schur complement, (3) implies 8i ¡ 0. Let 0 = min{−8i ; i ∈ S}, then 0 ¿ 0 due to (3). Finally, from (12) we obtain that for any k ¿ 0, E[V (Xk+1 ; Yk+1 ; k + 1; rk+1 ) | Xk ; Yk ; rk = i] 6 V (Xk ; Yk ; k; rk ) − 0 xkT xk :
(14)
Setting k = 0 and k = 1 in (14) yields E[V (X1 ; Y1 ; 1; r1 ) | X0 ; Y0 ; r0 ] 6 V (X0 ; Y0 ; 0; r0 ) − 0 x0T x0
(15)
E[V (X2 ; Y2 ; 2; r2 ) | X1 ; Y1 ; r1 ] 6 V (X1 ; Y1 ; 1; r1 ) − 0 x1T x1 :
(16)
and
Taking expectation E[ · |X0 ; Y0 ; r0 ] on both sides of (16), with the aid of (15), leads to E[V (X2 ; Y2 ; 2; r2 ) | X0 ; Y0 ; r0 ] 6 V (X0 ; Y0 ; 0; r0 ) − 0
1
E[xlT xl | X0 ; Y0 ; r0 ]:
l=0
Then, one can continue the iterative procedure (14) to obtain E[V (XT +1 ; YT +1 ; T + 1; rT +1 ) | X0 ; Y0 ; r0 ] 6 V (X0 ; Y0 ; 0; r0 ) − 0
T
E[xlT xl | X0 ; Y0 ; r0 ];
l=0
implying that ∞
E[xlT xl | X0 ; Y0 ; r0 ] 6
l=0
1 V (X0 ; Y0 ; 0; r0 ) ¡ ∞: 0
This indicates that system (2) with uk = 0 is robustly stochastically stable, and thus the proof of Theorem 1 is completed. The remaining of this section is devoted to design a robust, memoryless state feedback controller in the form of uk = K r k xk such that the resulting closed-loop system is stochastically stable.
(17)
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369
Theorem 2. If for each i ∈ S, and for some prescribed scalars
