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New robust H∞ control for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method
Available online at www.sciencedirect.com
Journal of the Franklin Institute 349 (2012) 741–769 www.elsevier.com/locate/jfranklin
New robust HN control for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method Jianwei Xiaa,b, Changyin Suna,n, Baoyong Zhangc a
School of Automation, Southeast University, Jiangsu 210096, PR China School of Mathematics Science, Liaocheng University, Liaocheng 252000, PR China c School of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR China b
Received 14 July 2011; received in revised form 28 August 2011; accepted 12 September 2011 Available online 22 September 2011
Abstract This paper considers the problems of robust stochastic stabilization and robust H1 controller design for a class of stochastic Markovain jumping systems with mixed time delays and polytopic parameter uncertainties. Both the interval time-varying delay and distributed time delay are simultaneously considered. Some new delay-dependent sufficient conditions, which differs greatly from the most existing results, are obtained based on the decoupling method and some advanced techniques. A numerical example is provided to illustrate the effectiveness of the proposed criteria. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction During the past decades, considerable attention has been paid to the study of stochastic dynamical systems modeled by Ito’s formula [1–7]. Since time delays are frequently encountered in various physical and engineering systems, and they are often the sources of instability, oscillations and poor performances [8–11], therefore, the study of stability analysis and controller synthesis for stochastic systems with time delays has been extensively investigated in the past years, and a great deal of results related to such systems have been reported in the literatures [12–15]. To mention a few, the problem of exponential n
Corresponding author. E-mail addresses: [email protected] (J. Xia), [email protected] (C. Sun), [email protected] (B. Zhang). 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.09.009
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J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
stability in mean square for stochastic time-delay systems was proposed in [5,12]. When the parameter uncertainties were considered, the robust stochastic stabilization problem was solved in [13]. More recently, the problems of robust H1 control and H1 filtering were investigated in [14,15], respectively. It is noted that all these results are delay-independent, that is there is no information on the size of delays was considered. It is well known that delay-dependent results are generally less conservative than delay-independent ones, especially when time delays are small. This motivates to develop delay-dependent conditions for stochastic delay systems. To improve the performance of delay-dependent stability criteria, much effort has been devoted in recent works. For instance, by the methods of improved augmented Lyapunov–Krasovskii functional and (or) free-weighting matrices, delay-dependent results of stability analysis, robust stabilization, robust H1 control, robust H1 filtering, L2 L1 filtering, H1 reduction were studied for stochastic delay systems in [2,16–25], respectively. On the other hand, one of the important class of stochastic systems is the so-called stochastic systems with Markovian jump parameters, a great deal of attention has been devoted to the stochastic Markovian jumping systems in recent years, see, for example, [26–33]. The problems of delay-independent and delay-dependent exponential stability for stochastic systems with Markovian jump parameters were dealt with in [26,27], respectively. In [28,29], the problem of stochastic stabilization in mean square for several stochastic Markovian jumping systems with time delay was investigated. When both time delays and parameter uncertainties appear in a Markovian jump system, some results on the robust H1 control problem were reported in [30]. Moreover, the problems of delaydependent H1 and robust H1 filtering for stochastic Markovian jumping systems with mode-dependent time delays were considered in [31,32], respectively. It is worth mentioning that the parameter uncertainties in the existing literatures are almost timevarying norm bounded [30,32]. When parameter uncertainties in a polytopic type arise, however, the problem of robust H1 control for stochastic Markovian jumping systems with time delays has not been fully investigated. The problem of parameter-dependent robust stability analysis was proposed for uncertain Markovian jump systems with polytopic parameter uncertainties and time-varying delays in [34], but no robust stabilization criterion was developed. One of the reason is that the state feedback controller design approach is difficult to be achieved using the parameter-dependent Lyapunov–Krasovskii functional method, but a common Lyapunov–Krasovskii functional is used to ensure stabilization for all admissible uncertainties will usually leads to be conservative, which motivates this study. In this paper, we focus on the problems of delay-interval-dependent robust stochastic stabilization and robust stochastic H1 control for stochastic Markovian jumping systems (SMJS) with mixed time delays and parameter uncertainties. The parameter uncertainties are of the polytopic forms. By introducing a parameter-dependent Lyapunov–Krasovskii functional with some free-weighting matrices and decoupling matrices, new delaydependent robust stochastic stabilization and robust H1 control criterions are obtained in terms of LMIs. An example is given to show the effectiveness of the proposed method. Notation: Throughout this paper, for symmetric matrices X and Y, the notation X ZY (respectively, X 4Y ) means that the matrix XY is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension. MT represents the transpose of the matrix M. Efg denote the expectation operator with respect to some probability measure P. L2 ½0,1 is the space of square-integrable vector functions over
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
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½0,1. 9 9 refers to the Euclidean vector norm while J J2 stands for the usual L2 ½0,1 norm. ðO,F ,P) is a probability space with O the sample space and F the s-algebra of subsets of the sample space. Matrices, if not explicitly mentioned, are assumed to have compatible dimensions. 2. Problem formulation Fix a probability space ðO,F ,PÞ and consider the following a class of uncertain stochastic Markovian jumping systems (SMJS) with mixed time delays described by Z t dxðtÞ ¼ Aða,rðtÞÞxðtÞ þ A1 ða,rðtÞÞxðttðtÞÞ þ B0 ða,rðtÞÞuðtÞ þ B1 ða,rðtÞÞ xðsÞ ds tdðtÞ
þB2 ða,rðtÞÞvðtÞ dt þ ½Eða,rðtÞÞxðtÞ þ E1 ða,rðtÞÞxðttðtÞÞ þ B3 ða,rðtÞÞvðtÞ dwðtÞ, ð1Þ zðtÞ ¼ Cða,rðtÞÞxðtÞ, xðtÞ ¼ jðtÞ,
ð2Þ
8t 2 ½h,0, h ¼ maxftM ,dg,
n
m
ð3Þ p
where xðtÞ 2 R is the state; uðtÞ 2 R is the control input; vðtÞ 2 R is a disturbance input which belongs to L2 ½0,1, zðtÞ 2 Rq is the controlled output, and w(t) is a one-dimension Brownian motion satisfying EfdwðtÞg ¼ 0 and EfdwðtÞ2 g ¼ dt; frðtÞ,tZ0g be a rightcontinuous Markov process on the probability space which takes values in the finite space S ¼ 1,2,y,N} with generator P ¼ ðlm,n Þðm,n 2 SÞ given by ( lm,n D þ oðDÞ if man, Pfrðt þ DÞ ¼ n9rðtÞ ¼ mg ¼ ð4Þ 1 þ lm,m D þ oðDÞ if m ¼ n, where D40 P and limD-0 oðDÞ=D ¼ 0, lm,n Z0 is the transition rate from m to n if man and lm,m ¼ man lm,n . The bounded functions tðtÞ and d(t) represent interval time-varying delays and distributed delays satisfying 0rtm otðtÞrtM ,
t_ ðtÞrt, dðtÞrd,
ð5Þ
where tm , tM , t and d are real known constant scalars, jðtÞ is an initial function defined on [h,0]. The uncertain matrices of the system are assumed to be with the polytopic type described by ½Aða,rðtÞÞ A1 ða,rðtÞÞ B0 ða,rðtÞÞ B1 ða,rðtÞÞ B2 ða,rðtÞÞ Eða,rðtÞÞ E1 ða,rðtÞÞ B3 ða,rðtÞÞ Cða,rðtÞÞ
¼
r X
ai ½Aði,rðtÞÞ A1 ði,rðtÞÞ B0 ði,rðtÞÞ B1 ði,rðtÞÞ B2 ði,rðtÞÞ Eði,rðtÞÞ
i¼1
E1 ði,rðtÞÞ B3 ði,rðtÞÞ Cði,rðtÞÞ,
ð6Þ
and r X
ai ¼ 1,
ai Z0,
ð7Þ
i¼1
where Aði,rðtÞÞ, A1 ði,rðtÞÞ, B0 ði,rðtÞÞ, B1 ði,rðtÞÞ, B3 ði,rðtÞÞ, Eði,rðtÞÞ, E1 ði,rðtÞÞ B3 ði,rðtÞÞ, Cði,rðtÞÞ are known real constant matrices.
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
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Define the following two scalars which are with respect to the variation range of time delay tm þ tM tM tm , d¼ : ð8Þ t0 ¼ 2 2 For convenience, we set yðtÞ ¼ Aða,rðtÞÞxðtÞ þ B0 ða,rðtÞÞuðtÞ þ A1 ða,rðtÞÞxðttðtÞÞ Z t þB1 ða,rðtÞÞ xðsÞ ds þ B2 ða,rðtÞÞvðtÞ, tdðtÞ
gðtÞ ¼ Eða,rðtÞÞxðtÞ þ E1 ða,rðtÞÞxðttðtÞÞ þ B3 ða,rðtÞÞvðtÞ then, system (1) becomes dxðtÞ ¼ yðtÞ dt þ gðtÞ dwðtÞ:
ð9Þ
Throughout the paper the following concepts and lemmas will be needed. Definition 1 (Xu and Chen [35]). For system (1) with u(t) ¼ 0 and v(t)¼ 0, the equilibrium point is said to be robust stable mean square if for any e40, there is a dðeÞ40 such that 2
E9xðtÞ9 oe,
t40
for all admissible uncertainties, when 2
sup E9xðtÞ9 odðeÞ: mrsr0
If, in addition, 2
lim E9xðtÞ9 ¼ 0
t-1
for any initial conditions and all admissible uncertainties, then system (1) with u(t)¼ 0 and v(t)¼ 0 is said to be robustly asymptotically stable in mean square. Definition 2 (Xu and Chen [35]). Given a scalar g40, the stochastic system (1)–(3) is said to be robustly stochastically stable in mean square with disturbance attenuation level g, if it is stochastically stable and with zero initial condition, the inequality JzðtÞJE2 ogJvðtÞJ2
ð10Þ
holds for all admissible uncertainties, where Z 1 1=2 9zðtÞ2 9 : JzðtÞJE2 ¼ E
ð11Þ
0
Lemma 1 (Zhang et al. [36]). For any positive definite matrix M40, scalar g40, vector function o : ½0,g-Rn such that the integrations concerned are well defined, the following inequality holds: Z g T Z g Z g oðsÞ ds M oðsÞ ds rg oðsÞT MoðsÞ ds : ð12Þ 0
0
0
Lemma 2 (Gao et al. [37], Yang et al. [38]). For any two vectors x and y, positive matrix Q40 with compatible dimensions, the following matrix inequality holds: 2xT yrxT Qx þ yT Q1 y:
ð13Þ
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
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Lemma 3 (Zhang et al. [39]). For any constant matrices C1 and C2 with appropriate dimensions and a symmetric matrix Oo0, scalars d1 rd2 , and a function dðtÞ : Rþ -½d1 ,d2 , then ðdðtÞd1 ÞC1 þ ðd2 dðtÞÞC2 þ Oo0 holds, if and only if ðd2 d1 ÞC1 þ Oo0, ðd2 d1 ÞC2 þ Oo0:
ð14Þ
In this paper, our aim is to develop criteria of robust stochastic stabilization and robust H1 control for uncertain stochastic time-delay system (1)–(3). More specifically, we are concerned with the following two problems: (1) Robust stochastic stabilization problem: Design a state feedback controller uðtÞ ¼ F ðrðtÞÞxðtÞ
ð15Þ
for system (1)–(3) with v(t)¼ 0 such that the resulting closed-loop system is robustly asymptotically stable in mean square for all admissible uncertainties. In this case, the system (1)–(3) with v(t) ¼ 0 is said to be robustly stochastically stabilizable. (2) Robust H1 control problem: Given a constant scalar g40, design a state feedback controller in the form of Eq. (15) such that, for all admissible uncertainties, the resulting closed-loop system is robustly asymptotically stable in mean square and for any nonzero vðtÞ 2 L2 ½0,1, JzðtÞJE2 ogJvðtÞJ2
ð16Þ
is satisfied under the zero-initial condition. In this case, the system (1)–(3) is said to be robustly stochastically stabilizable with disturbance attenuation level g. 3. Main results 3.1. Robust stochastic stabilization In this subsection, an LMI approach will be utilized to solve the robust stochastic stabilization problem described in the previous section. To this purpose, we consider the system (1)–(3) with v(t) ¼ 0 as following: dxðtÞ ¼ ½Aða,rðtÞÞxðtÞ þ A1 ða,rðtÞÞxðttðtÞÞ þ B0 ða,rðtÞÞuðtÞ Z t þB1 ða,rðtÞÞ xðsÞ ds dt þ ½Eða,rðtÞÞxðtÞ þ E1 ða,rðtÞÞxðttðtÞÞ dwðtÞ, tdðtÞ
ð17Þ xðtÞ ¼ jðtÞ,
8t 2 ½h,0, h ¼ maxftM ,dg:
ð18Þ
Then, we have the following result. Theorem 1. The system in Eqs. (17) and (18) is robustly stochastically stabilizable, if there exist matrices Z, G(m), Y(m), P(i,m), X ði,i,m,lÞ, X ði,j,m,lÞ, Lk, with Z40, Pði,mÞ40, X ði,i,m,lÞ ¼ X ði,i,m,lÞT , for joi, i ¼ 1,2,y,r, m ¼ 1,2,y,N, k ¼ 1,2,y,6, l¼ 1,2,3,4,
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
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satisfying the following LMIs: 2 3 j11 ði,i,lÞ j12 ði,i,lÞ j13 ði,i,lÞ 6 n j22 ði,i,lÞ j23 ði,i,lÞ 7 4 5oX ði,i,m,lÞ, n n j33 ði,i,lÞ 2 6 4
j11 ði,j,lÞ
j12 ði,j,lÞ
n
j22 ði,j,lÞ
n
n
2 6 þ4
j13 ði,j,lÞ
ð19Þ
3
j23 ði,j,lÞ 7 5
j33 ði,j,lÞ
3 j12 ðj,i,lÞ j13 ðj,i,lÞ T j22 ðj,i,lÞ j23 ðj,i,lÞ 7 5oX ði,j,m,lÞ þ X ði,j,m,lÞ , n j33 ðj,i,lÞ
j11 ðj,i,lÞ n n
ð20Þ
½X ði,j,m,lÞrr o0,
ð21Þ
where 2 6 6 6 6 j11 ði,i,pÞ ¼ 6 6 6 6 4
Oði,i,mÞ
0
0
0
A1 ði,mÞ
B1 ði,mÞPði,mÞ
n
L2 þ LT2
n
n
0 L3 þ LT3
0 L3
L2 LT1
0 0
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
0 0
n
n
n
n
n
Z
2
0
6 6 0 6 6 0 6 j12 ði,i,1Þ ¼ j12 ðj,i,1Þ ¼ 6 6 0 6 pffiffiffi 6 dL 4 1 0 2 6 6 6 6 6 j13 ði,i,lÞ ¼ 6 6 6 6 4
0
0
0
0
0 0
L2 0
0 pffiffiffi dL3
0 L3
0 L1
0 0
0 0
0 0
0
0
0
0
ZðPði,mÞÞ Pði,mÞE T ði,mÞ
7 7 7 7 7, 7 7 7 5
3 pffiffiffi 2Pði,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
pffiffiffiffiffi 2dPði,mÞE T ði,mÞ
pffiffiffiffiffi 2dPði,mÞAT ði,mÞ
0 0
0 0
0 0
0 0
0 0
0 E1T ði,mÞ
0 pffiffiffiffiffi T 2dE1 ði,mÞ
0
0
0
0 pffiffiffiffiffi T 2dA1 ði,mÞ pffiffiffiffiffi T 2dB1 ði,mÞ
j22 ði,i,lÞ ¼ j22 ði,j,lÞ ¼ j22 ðj,i,lÞ ¼ diagfI,I,I,I,I,Ig, j23 ði,i,lÞ ¼ j23 ði,j,lÞ ¼ j23 ðj,i,lÞ ¼ 0, j33 ði,i,lÞ ¼ j33 ðj,i,lÞ ¼ diagfZ0 ðPði,mÞÞ Pði,mÞ I I G T ðmÞ þ GðmÞg,
3
D
3
7 07 7 07 7 7, 07 7 07 5 0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
n
0 0
0 0
A1 ði,mÞ L2
n
n
L3 þ LT3
n
n
n
L3 I
LT1 0
n
n
n
n
ð1tÞI þ L1 þ LT1
n
n
n
n
n
Z
Oðj,i,mÞ
0
0
0
A1 ðj,mÞ
B1 ðj,mÞPði,mÞ
n
L2 þ LT2
n
n
0 L3 þ LT3
0 L3
L2 LT1
0 0
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
0 0
n
n
n
n
n
Z
6 6 6 6 j11 ðj,i,pÞ ¼ 6 6 6 6 4
2
0 0
6 6 6 6 0 6 j12 ði,j,1Þ ¼ 6 6 0 6 pffiffiffi 6 dL 4 1 0 2 6 6 6 6 6 j13 ði,j,lÞ ¼ 6 6 6 6 4
2
3 B1 ði,mÞPðj,mÞ 7 0 7 7 7 0 7, 7 0 7 7 5 0
0 L2 þ LT2
Oði,j,mÞ
6 6 6 6 j11 ði,j,pÞ ¼ 6 6 6 6 4
2
747
0 0
0 L2 0
0 0 pffiffiffi dL3
0 0 L1 0
L3
0 0
0 0
0 0
0
0
0
ZðPðj,mÞÞ Pðj,mÞE T ði,mÞ
0 0
pffiffiffiffiffi 2dPðj,mÞAT ði,mÞ
0 0
0 0
0 0
0 0
0 0
0 E1T ði,mÞ
0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T tM tm E1 ði,mÞ
0
0
0
0 pffiffiffiffiffi T 2dA1 ði,mÞ pffiffiffiffiffi T 2dB1 ði,mÞ
Pði,mÞE T ðj,mÞ 0
pffiffiffiffiffi 2dPði,mÞE T ðj,mÞ 0
pffiffiffiffiffi 2dPði,mÞAT ðj,mÞ 0
0 0
0 0
0 0
ZðPði,mÞÞ 6 0 6 6 6 0 6 j13 ðj,i,lÞ ¼ 6 0 6 6 6 0 4 0
7 7 7 7 7, 7 7 7 5
3 pffiffiffi 2Pðj,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
pffiffiffiffiffi 2dPðj,mÞE T ði,mÞ
E1T ðj,mÞ
pffiffiffiffiffi T 2dE1 ðj,mÞ
0
0
pffiffiffiffiffi T 2dA ðj,mÞ pffiffiffiffiffi T1 2dB1 ðj,mÞ
j33 ði,j,lÞ ¼ diag½Z0 ðPðj,mÞÞ Pðj,mÞ I I G T ðmÞ þ GðmÞ,
3
D
3
7 07 7 07 7 7, 07 7 07 5 0 3 D 7 07 7 07 7 7, 07 7 07 5 0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
748
2
0 6 pffiffiffi 6 dL2 6 6 0 6 j12 ði,i,2Þ ¼ j12 ðj,i,2Þ ¼ 6 6 0 6 6 0 4 2
0 6 pffiffiffi 6 dL2 6 6 0 6 j12 ði,j,2Þ ¼ 6 6 0 6 6 0 4 0 2
0
0
0 0
L2 0
0 pffiffiffi dL3
0 L3
0 L1
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
L2
0
0 0
0 0
0 pffiffiffi dL3 0
L3 0
L1 0
0 0
0 0
0 0
3 pffiffiffi 2Pði,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
3 pffiffiffi 2Pðj,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0 3 B1 ði,mÞPði,mÞ 7 0 7 7 7 0 7, 7 0 7 7 5 0
n
0 L6
0 0
A1 ði,mÞ 0
n
n
L5 þ LT5
n
n
n
0 I
L5 LT4
n
n
n
n
ð1tÞ þ L4 þ LT4
n
n
n
n
n
Oði,j,mÞ n
0 L6 þ LT6
0 L6
0 0
A1 ði,mÞ 0
n
n
L5 þ LT5
n
n
n
0 I
L5 LT4
n
n
n
n
ð1tÞ þ L4 þ LT4
n
n
n
n
n
Z
Oðj,i,mÞ
0
0
0
A1 ðj,mÞ
B1 ðj,mÞPði,mÞ
n
L6 þ LT6
n
n
L6 L5 þ LT5
0 0
0 L5
0 0
n
n
n
I
n
n
n
n
LT4 ð1tÞ þ L4 þ LT4
0 0
n
n
n
n
n
Z
6 6 6 6 j11 ði,j,3Þ ¼ 6 6 6 6 4 2
0
0 L6 þ LT6
Oði,i,mÞ
6 6 6 6 j11 ði,i,3Þ ¼ 6 6 6 6 4 2
0
6 6 6 6 j11 ðj,i,3Þ ¼ 6 6 6 6 4
2
0 0
6 6 6 6 0 6 j12 ði,j,3Þ ¼ 6 6 0 6 pffiffiffi 6 dL 4 4 0
0 0
0 0
0 pffiffiffi dL6
0 L6
0 0
L5 0
0 0
0 0
L4
0
0
0
0
0
0
0
3 pffiffiffi 2Pðj,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
Z 3 B1 ði,mÞPðj,mÞ 7 0 7 7 7 0 7, 7 0 7 7 5 0 3 7 7 7 7 7, 7 7 7 5
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
0
6 6 0 6 6 0 6 j12 ðj,i,3Þ ¼ 6 6 0 6 pffiffiffi 6 dL 4 4 0
0
0 0 L5
0 pffiffiffi dL6 0
0 0 0 L4
L6 0
0 0
0 0
0 0
0 0 0 2 0 0 6 6 0 0 6 pffiffiffi 6 dL 0 5 6 j12 ði,i,4Þ ¼ j12 ðj,i,4Þ ¼ 6 6 0 0 6 6 0 L 4 4 0 0 2 0 0 0 0 pffiffiffi 6 6 0 0 0 dL6 6 pffiffiffi 6 dL 0 L5 0 5 6 j12 ði,j,4Þ ¼ 6 6 0 0 0 0 6 6 0 L4 0 0 4 0
0
0
0
0
0 0 0 L5 0 0 0 0 L6 0 0 0 0
749
3 pffiffiffi 2Pði,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0 3 pffiffiffi 0 0 2Pði,mÞ pffiffiffi 7 7 dL6 L6 0 7 7 0 0 0 7 7, 7 0 0 0 7 7 0 0 0 5 0
0
3 pffiffiffi 2Pðj,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
0
p ¼ 1; 2, l ¼ 1; 2,3; 4
and D ¼ B0 ði,mÞY ðmÞ þ GT ðmÞPði,mÞ, Oði,i,mÞ ¼ Aði,mÞPði,mÞ þ Pði,mÞAði,mÞT þ B0 ði,mÞY ðmÞ þY ðmÞT B0 ði,mÞT þ lm,m Pði,mÞ þ d 2 Z, Oði,j,mÞ ¼ Aði,mÞPðj,mÞ þ Pðj,mÞAði,mÞT þ B0 ði,mÞY ðmÞ þY ðmÞT B0 ði,mÞT þ lm,m Pðj,mÞ þ d 2 Z, Oðj,i,mÞ ¼ Aðj,mÞPði,mÞ þ Pði,mÞAðj,mÞT þ B0 ðj,mÞY ðmÞ þY ðmÞT B0 ðj,mÞT þ lm,m Pði,mÞ þ d 2 Z, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ZðPði,mÞÞ ¼ ½ lm,1 Pði,mÞ lm,m1 Pði,mÞ lm,mþ1 Pði,mÞ lm,N Pði,mÞ, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ZðPðj,mÞÞ ¼ ½ lm,1 Pðj,mÞ lm,m1 Pðj,mÞ lm,mþ1 Pðj,mÞ lm,N Pðj,mÞ, Z0 ðPði,mÞÞ ¼ diag½Pði,1Þ Pði,m1Þ Pði,m þ 1Þ Pði,NÞ, Z0 ðPðj,mÞÞ ¼ diag½Pðj,1Þ Pðj,m1Þ Pðj,m þ 1Þ Pðj,NÞ:
ð22Þ
In this case, the overall state feedback controller is represented by uðtÞ ¼ F ðrðtÞÞxðtÞ,
F ðrðtÞ ¼ mÞ ¼ Y ðmÞGðmÞ1 :
ð23Þ
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
750
Proof. From Eqs. (19)–(21), set l ¼ 1,2, then, we can easily get 2 6 4
3 2 j12 ðaÞ j13 ðaÞ j11 ði,i,1Þ r X 6 7 2 j22 ðaÞ j23 ðaÞ 5 ¼ n ai 4 i ¼ 1 n j33 ðaÞ n
j11 ðaÞ n n
þ
02
r X X i ¼ 1 joi
2 6 þ4
B6 ai aj @ 4
j11 ðj,i,1Þ
r X
j11 ði,j,1Þ n
n
3 j13 ði,i,1Þ j23 ði,i,1Þ 7 5 j33 ði,i,1Þ
n
3 j12 ði,j,1Þ j13 ði,j,1Þ j22 ði,j,1Þ j23 ði,j,1Þ 7 5
n
n o
j12 ði,i,1Þ j22 ði,i,1Þ
j33 ði,j,1Þ
n
j12 ðj,i,1Þ j13 ðj,i,1Þ
31
C j22 ðj,i,1Þ j23 ðj,i,1Þ 7 5A n j33 ðj,i,1Þ
a2i X ði,i,m,1Þ þ
r X X ai aj ðX ði,j,m,1Þ þ X ði,j,m,1ÞT Þo0, i ¼ 1 joi
i¼1
ð24Þ 2 6 4
3 2 j012 ðaÞ j13 ðaÞ j ði,i,2Þ r X 6 11 7 2 j22 ðaÞ j23 ðaÞ 5 ¼ n ai 4 i ¼ 1 n j33 ðaÞ n
j11 ðaÞ n n
r X X
þ
i ¼ 1 joi
02 B6 ai aj @ 4
n
2
j11 ðj,i,2Þ j012 ðj,i,2Þ 6 n j22 ðj,i,2Þ þ4
o
r X
j22 ði,i,2Þ
n
a2i X ði,i,m,2Þ þ
j13 ði,i,2Þ
3
j23 ði,i,2Þ 7 5
n
j11 ði,j,2Þ j012 ði,j,2Þ j13 ði,j,2Þ n
n
j12 ði,i,2Þ
jði,iÞ
3
j22 ði,j,2Þ j23 ði,j,2Þ 7 5 n j33 ði,j,2Þ
31 j13 ðj,i,2Þ C j23 ðj,i,2Þ 7 5A
j33 ðj,i,2Þ
r X X
ai aj ðX ði,j,m,2Þ þ X ði,j,m,2ÞT Þo0,
ð25Þ
i ¼ 1 joi
i¼1
where 2 6 6 6 6 j11 ðaÞ ¼ 6 6 6 6 4
n
0 L2 þ LT2
0 0
0 0
A1 ða,mÞ L2
n
n
L3 þ LT3
n
n
n
L3 I
LT1 0
n
n
n
n
ð1tÞI þ L1 þ LT1
n
n
n
n
n
Oða,mÞ
3 B1 ða,mÞPða,mÞ 7 0 7 7 7 0 7, 7 0 7 7 5 0 Z
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
0 0
6 6 6 6 0 6 j12 ðaÞ ¼ 6 6 0 6 pffiffiffi 6 dL 4 1 0
0 0
0 L2 0
0 0 pffiffiffi dL 3
0
0 0 L3
0 L1
0 0
0 0
0 0
0
0
0
0
2
ZðPða,mÞÞ Pða,mÞE T ða,mÞ 6 0 0 6 6 6 0 0 6 j13 ðaÞ ¼ 6 0 0 6 6 T 6 ða,mÞ 0 E 1 4 0 0
3 pffiffiffi 2Pða,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
j22 ðaÞ ¼ j22 ði,iÞ,
751
j23 ðaÞ ¼ 0,
pffiffiffi dPða,mÞE T ða,mÞ 0
pffiffiffiffiffi 2dPða,mÞAT ða,mÞ 0
0 0
0 0
pffiffiffi T dE1 ða,mÞ 0
pffiffiffiffiffi T 2dA ða,mÞ pffiffiffiffiffi T1 2dB1 ða,mÞ
j33 ðaÞ ¼ diagfZ0 ðPða,mÞÞ Pða,mÞ I I GT ðmÞ þ GðmÞg, 3 2 pffiffiffi 0 0 0 0 0 2Pða,mÞ 7 6 pffiffiffi 7 6 dL2 0 L2 0 0 0 7 6 p ffiffi ffi 7 6 0 0 0 dL3 L3 0 7 6 0 j12 ðaÞ ¼ 6 7 7 6 0 0 0 0 0 0 7 6 7 6 0 L1 0 0 0 0 5 4 0
0
0
0
0
3 D 7 07 7 07 7 7, 07 7 07 5 0
ð26Þ
0
and D ¼ B0 ði,mÞY ðmÞ þ GT ðmÞPði,mÞ, Oða,mÞ ¼ Aða,mÞPða,mÞ þ Pða,mÞAða,mÞT þ B0 ða,mÞY ðmÞ þ Y ðmÞT B0 ða,mÞT þlm,m Pða,mÞ þ d 2 Z, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi lm,mþ1 Pða,mÞ lm,N Pða,mÞ, ZðPða,mÞÞ ¼ ½ lm,1 Pða,mÞ lm,m1 Pða,mÞ Z0 ðPði,mÞÞ ¼ diag½Pða,1Þ Pða,m1Þ Pða,m þ 1Þ Pða,NÞ:
ð27Þ
Noting the controller in Eq. (23), pre- and post-multiplying inequalities (24) and (25) by 2
I 0 60 I 6 6 6^ ^ 6 6 40 0
0 0
^ 0
&
0 0
0
3 0 Bða,mÞF ðmÞ 7 0 0 7 7 7 ^ ^ 7 7 0 0 5 I 0 1617
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
752
and its transpose, we obtain 2 3 w11 ðaÞ w12 ðaÞ w13 ðaÞ 6 n w22 ðaÞ w23 ðaÞ 7 4 5o0, n n w33 ðaÞ 2 6 4
w11 ðaÞ w012 ðaÞ n
w22 ðaÞ
n
n
w13 ðaÞ
ð28Þ
3
w23 ðaÞ 7 5o0, w33 ðaÞ
ð29Þ
where 2 6 6 6 6 w11 ðaÞ ¼ 6 6 6 6 4
O1 ða,mÞ
0
0
0
A1 ða,mÞ
B1 ða,mÞPða,mÞ
n
L2 þ LT2
n
n
0 L3 þ LT3
0 L3
L2 LT1
0 0
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
0 0
n
n
n
n
n
Z
w12 ðaÞ ¼ j12 ðaÞ,
w22 ðaÞ ¼ j22 ðaÞ,
2
ZðPða,mÞÞ 6 0 6 6 6 0 6 w13 ðaÞ ¼ 6 0 6 6 6 0 4 0
Pða,mÞE T ða,mÞ 0 0 0 E1T ða,mÞ 0
w23 ðaÞ ¼ 0,
3 7 7 7 7 7, 7 7 7 5
w012 ðaÞ ¼ j012 ðaÞ,
pffiffiffiffiffi pffiffiffiffiffi 3 2dPða,mÞE T ða,mÞ 2dPða,mÞAT ða,mÞ 7 0 0 7 7 7 0 0 7 7, 0 0 7 7 pffiffiffiffiffi T pffiffiffiffiffi T 7 2dE1 ða,mÞ 2dA1 ða,mÞ 5 pffiffiffiffiffi T 0 2dB1 ða,mÞ
w33 ðaÞ ¼ diagfZ0 ðPða,mÞÞ Pða,mÞ I Ig,
ð30Þ
O1 ða,mÞ ¼ ½Aða,mÞ þ Bða,mÞF ðmÞPða,mÞ þPða,mÞ½Aða,mÞ þ Bða,mÞF ðmÞT þ lm,m Pða,mÞ þ d 2 Z:
ð31Þ
and
Define ^ Pða,mÞ ¼ P1 ða,mÞ,
^ Zða,mÞ ¼ P1 ða,mÞZP1 ða,mÞ
pre- and post-multiplying diagðP1 ða,mÞ,I,I,I,I,P1 ða,mÞ,I,I,I,I,I,I,I,P1 ða,mÞ,I,IÞ to Eqs. (28) and (29) leads to 2
3 w^ 11 ðaÞ w^ 12 ðaÞ w^ 13 ðaÞ 6 n w^ 22 ðaÞ w^ 23 ðaÞ 7 4 5o0, n n w^ 33 ðaÞ
ð32Þ
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2 6 4
n
w^ 012 ðaÞ w^ 22 ðaÞ
n
n
w^ 11 ðaÞ
753
3 w^ 13 ðaÞ w^ 23 ðaÞ 7 5o0, w^ 33 ðaÞ
ð33Þ
where 2 6 6 6 6 6 w^ 11 ðaÞ ¼ 6 6 6 6 4
^ 1 ða,mÞ O
0
0
0
^ Pða,mÞA 1 ða,mÞ
^ Pða,mÞB 1 ða,mÞ
n
L2 þ LT2
n
n
0 L3 þ LT3
n
n
n
0 L3 I
n
n
n
n
L2 LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
0 0 0 0 Z^
2
0
6 6 0 6 6 0 6 w^ 12 ðaÞ ¼ 6 6 0 6 pffiffiffi 6 dL 4 1 0 2
0
0
0
0
0
L2
0
0 0
0 0
0 pffiffiffi dL 3 0
L3 0
L1 0
0 0
0 0
0 0
^ ZðPða,mÞÞ 6 0 6 6 6 0 6 w^ 13 ðaÞ ¼ 6 0 6 6 6 0 4 0
pffiffiffi 3 2I 7 0 7 7 0 7 7 7, 0 7 7 0 7 5 0
w^ 22 ðaÞ ¼ w22 ðaÞ,
3 7 7 7 7 7 7, 7 7 7 5
w^ 22 ðaÞ ¼ 0,
pffiffiffiffiffi T pffiffiffiffiffi T 3 2dE ða,mÞ 2dA ða,mÞ 7 0 0 7 7 7 0 0 0 7 7, 0 0 0 7 7 p ffiffiffiffiffi p ffiffiffiffiffi ^ E1T ða,mÞPða,mÞ 2dE1T ða,mÞ 2dAT1 ða,mÞ 7 5 pffiffiffiffiffi T 0 0 2dB1 ða,mÞ
^ E T ða,mÞPða,mÞ 0
^ ^ w33 ðaÞ ¼ diagfZðPða,mÞÞ Pða,mÞ I Ig,
ð34Þ
where ^ 1 ða,mÞ ¼ Pða,mÞ½Aða,mÞ ^ O þ Bða,mÞF ðmÞ ^ ^ ^ þ lm,m Pða,mÞ þ d 2 Z, þ½Aða,mÞ þ Bða,mÞF ðmÞT Pða,mÞ ^ ZðPða,mÞÞ ¼½
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi lm,1 I lm,m1 I, lm,mþ1 I lm,N I,
1 ^ Z0 ðPði,mÞÞ ¼ diag½Pða,1Þ Pða,m1Þ1 ,Pða,m þ 1Þ1 Pða,NÞ1 :
ð35Þ
For systems (17), choose the following Lyapunov–Krasovskii function candidate: V ðxt ,t,rðtÞÞ ¼
5 X
Vi ðxðtÞ,t,rðtÞÞ,
ð36Þ
2hrtr0
ð37Þ
i¼1
where xt ðsÞ ¼ xðt þ sÞ,
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
754
and ^ V1 ðxt ,t,rðtÞÞ ¼ xT ðtÞPða,rðtÞÞxðtÞ, Z
Z
t
xðsÞT xðsÞ ds þ
V2 ðxt ,t,rðtÞÞ ¼ ttðtÞ
Z
tm
Z
t
yðsÞT yðsÞ ds dy,
tþy
tM
tm
Z
t
gðsÞT gðsÞ ds dy,
V4 ðxt ,t,rðtÞÞ ¼ tþy
tM
Z
0
Z
t
V4 ðxt ,t,rðtÞÞ ¼ d d
^ where Pða,rðtÞÞ ¼
Pr
i¼1
xðsÞT xðsÞ ds,
ttM
V3 ðxt ,t,rðtÞÞ ¼ Z
t
^ ds dy, xðsÞT ZxðsÞ
ð38Þ
tþy
^ Then, for every m ¼ rðtÞ 2 S, by Ito’s formula, we have ai Pði,rðtÞÞ.
^ ½ðAða,rðtÞÞxðtÞ þ B0 ða,rðtÞÞF ðmÞÞxðtÞ þ xðtÞ þ A1 ða,rðtÞÞxðttðtÞÞ LV1 ðxt ,t,mÞ ¼ 2xT ðtÞPða,mÞ
Z
t
þB1 ða,rðtÞÞ
^ xðsÞ ds þ g ðtÞPða,mÞgðtÞ þ xT ðtÞ T
! X lmj Pj xðtÞ,
tdðtÞ
j2S
LV2 ðxt ,t,mÞ ¼ 2xT ðtÞxðtÞð1tÞxT ðttðtÞÞxðttðtÞÞxT ðttM ÞxðttM Þ, Z ttm LV3 ðxt ,t,mÞ ¼ ðtM tm ÞyT ðtÞyðtÞ yðsÞT yðsÞ ds dy ttM Z tt0 Z ttm T T yðsÞ yðsÞ ds dy yðsÞT yðsÞ ds dy, ¼ ðtM tm Þy ðtÞyðtÞ ttM
LV4 ðxt ,t,mÞ ¼ ðtM tm ÞgT ðtÞgðtÞ
Z
ttm
M Z tt tt0
T
¼ ðtM tm Þg ðtÞgðtÞ
tt0
gðsÞT yðsÞ ds dy gðsÞT gðsÞ ds dy
Z
ttM
^ LV5 ðxt ,t,mÞrd x ðtÞZxðtÞ 2 T
Z
t
t
xðsÞ ds :
T
T
T
ð39Þ
tdðtÞ
" T
gðsÞT gðsÞ ds dy,
tt0
T Z xðsÞ ds Z^
tdðtÞ
Set
ttm
T
T
Z
T #
t
x ðtÞ ¼ x ðtÞ x ðttm Þ x ðtt0 Þ x ðttM Þ x ðttðtÞÞ
xðsÞ ds
,
tdðtÞ
I1T ¼ ½0 0 0 0 I 0,
I2T ¼ ½0 I 0 0 0 0,
I3T ¼ ½0 0 I 0 0 0,
ð40Þ
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
755
we will discuss the LV ðxt ,t,rðtÞÞ in the following two cases: Case I: when t 2 ½tm ,t0 , we have Z ttm Z ttðtÞ Z ttm yðsÞT yðsÞ ds dy ¼ yðsÞT yðsÞ ds dy yðsÞT yðsÞ ds dy, tt0
Z
ttðtÞ
tt0
ttm
gðsÞT gðsÞ ds dy ¼
Z
ttðtÞ
gðsÞT gðsÞ ds dy
Z
tt0
tt0
ttm
gðsÞT gðsÞ ds dy,
ð41Þ
ttðtÞ
By Newton–Leibuniz formula, for any matrices Li, i¼ 1,2,3 with appropriate dimensions, we obtain Z ttðtÞ Z ttðtÞ T 2x ðttðtÞÞL1 ðxðttðtÞÞxðtt0 Þ yðsÞ ds gðsÞ dwðsÞ ¼ 0, Z T 2x ðttm ÞL2 ðxðttm ÞxðttðtÞÞ
tt0
Z
ttm
yðsÞ ds
Z
ttðtÞ
Z
tt0
ttM
tt0
yðsÞ ds
2x ðtt0 ÞL3 ðxðtt0 ÞxðttM Þ
gðsÞ dwðsÞ ¼ 0,
ttðtÞ T
tt0 ttm
gðsÞ dwðsÞ ¼ 0,
ð42Þ
ttM
According to Lemmas 1 and 2, it can be easily verified that Z ttðtÞ T 2x ðttðtÞÞL1 yðsÞ dsrxT ðttðtÞÞðt0 tðtÞÞL1 LT1 xðttðtÞÞ tt0
Z
ttðtÞ
yT ðsÞyðsÞ ds
þ tt0
¼x
T
Z
T
T Z gðsÞy dwðsÞ
tt0
¼x
T
ðtÞI1 L1 LT1 I1T xðtÞ Z
T
Z
gðsÞy dwðsÞ tt0
T
Z þ
ttðtÞ
ðttm ÞðtðtÞtm ÞL2 LT2 xðttm Þ
ttðtÞ
Z
ttm
yT ðsÞyðsÞ ds,
ttðtÞ
gðsÞ dwðsÞrxT ðttm ÞL2 LT2 xðttm Þ
ttm
T Z
ttm
gðsÞ dwðsÞ
gðsÞ dwðsÞ ttðtÞ
gðsÞy dwðsÞ ,
tt0
¼ xT ðtÞðtðtÞtm ÞI2 L2 LT2 I2T xðtÞ þ ttm
ttðtÞ
gðsÞy dwðsÞ tt0
ttðtÞ
2x ðttm ÞL2
T Z
ttðtÞ
yðsÞ dsrx
Z
ttðtÞ
þ
ttm
2x ðttm ÞL2
T
yT ðsÞyðsÞ ds, tt0
gðsÞ dwðsÞrxT ðttðtÞÞL1 LT1 xðttðtÞÞ
tt0
ttðtÞ
ttðtÞ
þ
ttðtÞ
2x ðttðtÞÞL1 Z þ
Z
ðtÞðt0 tðtÞÞI1 L1 LT1 I1T xðtÞ
Z
ttm
þ ttðtÞ
yT ðsÞyðsÞ ds
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
756
¼ xT ðtÞI2 L2 LT2 I2T xðtÞ þ 2xT ðtt0 ÞL3
Z
Z
T Z
ttm
ttm
gðsÞ dwðsÞ ttðtÞ
gðsÞ dwðsÞ ,
ttðtÞ
tt0
yðsÞ dsrxT ðtt0 ÞðtM t0 ÞL3 LT3 xðtt0 Þ þ ttM Z tt0 T yT ðsÞyðsÞ ds, ¼ x ðtÞðtM t0 ÞI3 L3 LT3 I3T xðtÞ þ
Z
tt0
yT ðsÞyðsÞ ds
ttM
ttM
2xT ðtt0 ÞL3 Z
Z
tt0
ttM
tt0
þ
gðsÞ dwðsÞr2xT ðtt0 ÞL3 LT3 xðtt0 Þ
T Z gðsÞ dwðsÞ
ttM
¼ xT ðtÞI3 L3 LT3 I3T xðtÞ þ
tt0
gðsÞ dwðsÞ
ttM tt0
Z
T Z
tt0
gðsÞ dwðsÞ ttM
gðsÞ dwðsÞ :
ð43Þ
ttM
Define G1 ¼ 2xT ðttðtÞÞL1 ðxðttðtÞÞ2xT ðttðtÞÞL1 xðtt0 Þ þ xT ðtÞðt0 tðtÞÞI1 L1 LT1 I1T xðtÞ Z ttðtÞ yT ðsÞyðsÞ ds þxT ðtÞI1 L1 LT1 I1T xðtÞ þ Z
tt0
T Z
ttðtÞ
ttðtÞ
gðsÞy dwðsÞ
þ tt0
gðsÞy dwðsÞ ,
tt0
G2 ¼ 2xT ðttm ÞL2 xðttm Þ2xT ðttm ÞL2 xðttðtÞÞxT ðtÞðtðtÞtm ÞI2 L2 LT2 I2T xðtÞ Z ttm yT ðsÞyðsÞ ds þxT ðtÞI2 L2 LT2 I2T xðtÞ þ Z
ttðtÞ ttm
T Z
ttm
gðsÞ dwðsÞ
þ ttðtÞ
gðsÞ dwðsÞ ,
ttðtÞ
G3 ¼ 2xT ðtt0 ÞL3 ðxðtt0 Þ2xT ðtt0 ÞL3 xðttM Þ þ xT ðtÞðtM t0 ÞI3 L3 LT3 I3T xðtÞ Z tt0 þxT ðtÞI3 L3 LT3 I3T xðtÞ þ yT ðsÞyðsÞ dsxT ðtÞ ttM
Z
tt0
þ
T Z gðsÞ dwðsÞ
ttM
tt0
tt0
E
ttm
ttðtÞ
ð44Þ
ttM
Noting that ( Z Z ttðtÞ E gðsÞT gðsÞ ds ¼ E Z
gðsÞ dwðsÞ :
ttðtÞ
T Z gðsÞ dwðsÞ
tt0
T
gðsÞ gðsÞ ds ¼ E
(Z
gðsÞ dwðsÞ tt0
T Z
ttm
ttm
gðsÞ dwðsÞ ttðtÞ
)
ttðtÞ
ttðtÞ
) gðsÞ dwðsÞ ,
,
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
E
Z
tt0
( Z gðsÞT gðsÞ ds ¼ E
ttM
T Z
tt0
)
tt0
gðsÞ dwðsÞ
757
gðsÞ dwðsÞ
ttM
ð45Þ
ttM
we can easily get ( EfdV ðxðtÞ,tÞg ¼ EfLV ðxðtÞ,tÞgrE
6 X
LVi ðxðtÞ,tÞ þ
i¼1
3 X
) Gi
¼ EfxðtÞT YxðtÞg,
ð46Þ
i¼1
where Y ¼ Y1 þ Y2 þ Y3 þ Y4 þ ðt0 tðtÞÞI1 L1 LT1 I1T þ I1 L1 LT1 I1T þðtðtÞtm ÞI2 L2 LT2 I2T þ I2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T
ð47Þ
and 2 6 6 6 6 6 Y1 ¼ 6 6 6 6 4
2
n
0 L2 þ LT2
0 0
0 0
^ Pða,mÞA 1 ða,mÞ L2
n
n
L3 þ LT3
n
n
n
L3 I
n
n
n
n
LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
D1 ða,mÞ
E T ða,mÞ
3
2
E T ða,mÞ
3T
7 7 6 6 0 0 7 7 6 6 7 7 6 6 7 7 6 6 0 0 7Pða,mÞ ^ 7 , 6 Y2 ¼ 6 7 7 6 6 0 0 7 7 6 6 7 7 6 T 6 T 4 E1 ða,mÞ 5 4 E1 ða,mÞ 5 0
2
E T ða,mÞ
3
3 ^ Pða,mÞB 1 ða,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 Z^
2
3T
7 7 6 6 0 0 7 7 6 6 7 7 6 6 7 7 6 6 0 0 7ðtM tm ÞI 6 7 , Y3 ¼ 6 7 7 6 6 0 0 7 7 6 6 7 7 6 T 6 T 4 E1 ða,mÞ 5 4 E1 ða,mÞ 5
0
3 3T 2 T AT ða,mÞ A ða,mÞ 7 7 6 6 0 0 7 7 6 6 7 7 6 6 7 7 6 6 0 0 7 7 6 6 Y4 ¼ 6 7ðtM tm ÞI 6 7 , 0 0 7 7 6 6 7 7 6 6 6 AT ða,mÞ 7 6 AT ða,mÞ 7 5 5 4 1 4 1 BT1 ða,mÞ BT1 ða,mÞ
0
2
þ Bða,mÞF ðmÞ
E T ða,mÞ
^ D1 ða,mÞ ¼ Pða,mÞ½Aða,mÞ
0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
758
^ þ þ½Aða,mÞ þ Bða,mÞF ðmÞT Pða,mÞ
X
^ þ d 2 Z^ þ 2I: lmj Pða,mÞ
ð48Þ
j2S
Applying Schur complement to Eqs. (32) and (33), we obtain Y1 þ Y2 þ Y3 þ Y4 þ ðt0 tm ÞI1 L1 LT1 I1T þ I1 L1 LT1 I1T þ I2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T o0,
ð49Þ
Y1 þ Y2 þ Y3 þ Y4 þ I1 L1 LT1 I1T þ ðt0 tm ÞI2 L2 LT2 I2T þ I2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T o0:
ð50Þ
By using Lemma 3, Eqs. (49) and (50) are equivalent to Yo0:
ð51Þ
Combining Eqs. (46) and (51), we can conclude that there exists a scalar c1 40, such that 2
EfLV ðxt ,t,rðtÞÞgoc1 9xðtÞ9 :
ð52Þ
Case II: when t 2 ½t0 ,tM . Similar to the proof of case I, when l¼ 3,4, it can be checked with the same Lyapunov– Krasovskii function candidate denoted in Eqs. (36)–(38) that there exists a scalar c2 40, such that 2
EfLV ðxt ,t,rðtÞÞgoc2 9xðtÞ9 :
ð53Þ
From Case I and Case II, it can be seen that if Eqs. (52) and (53) hold, 2 EfLV ðxt ,t,rðtÞÞgominfc1 ,c2 g9xðtÞ9 . By using Lyapunov stability theorem in [14] that the dynamics of the systems (17) and (18) is robustly stochastically stabilizable. The proof is completed. & Remark 1. In the existing methods [40–43] which deal with the interval-time delays, the term (Eq. (47)) Y ¼ Y1 þ Y2 þ Y3 þ Y4 þ ðt0 tðtÞÞI1 L1 LT1 I1T þ I1 L1 LT1 I1T þ ðtðtÞtm ÞI2 L2 LT2 I2T þI2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T o0 is usually enlarged as Y1 þ Y2 þ Y3 þ Y4 þ ðt0 tm ÞðI1 L1 LT1 I1T þ I2 L2 LT2 I2T Þ þI1 L1 LT1 I1T þ I2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T o0, which may lead to considerable conservatism to the stability analysis. The idea of convexity is introduced by Lemma 3 can effectively deduce the possible conservatism of the achieved results. Remark 2. Note that Theorem 1 presents a new delay-interval-dependent stabilization criterion for SMJS with mixed time delays and polytopic uncertainties by using a new mode-dependent and parameter-dependent Lyapunov–Krasovskii functional in Eq. (38). On one hand, some weighting matrices (Lk,, k ¼ 1,2,y,6) are introduced to reduce the conservatism of the results, on the other hand, the decoupling matrices (G(m), m ¼ 1,2,y,N) are proposed to decouple the state feedback matrix B0 ða,rðtÞÞ and
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
759
Lyapunov–Krasovskii functional Pða,rðtÞÞ, which solve effectively the difficulty of state feedback controller design with the new parameter-dependent Lyapunov–Krasovskii functional. 3.2. Robust H1 control In this section, a sufficient condition for the solvability of the robust H1 control problem is proposed and an LMI approach for designing the desired state feedback controllers is developed. Theorem 2. Given a scalar g40, then the uncertain stochastic time-delay system in Eqs. (1)–(3) is robustly stochastically stabilizable with disturbance attenuation g, if there exist matrices Z, G(m), Y(m), Pði,mÞ, X ði,i,m,lÞ, X ði,j,m,lÞ, L1, L2, L3, with Z40, Pði,mÞ40, X ði,i,m,lÞ ¼ X ði,i,m,lÞT , for joi, i ¼ 1,2,y,r; m ¼ 1,2,y,N, l¼ 1,2,3,4, satisfying the following LMIs: 2 3 j^ 11 ði,i,lÞ j^ 12 ði,i,lÞ j^ 13 ði,i,lÞ 6 7 n ð54Þ j^ 22 ði,i,lÞ j^ 23 ði,i,lÞ ði,lÞ5oX ði,i,m,lÞ, 4 n n j^ 33 2 6 4
3 2 j^ 11 ðj,i,lÞ j^ 12 ðj,i,lÞ j^ 12 ði,j,lÞ j^ 13 ði,j,lÞ 7 6 n j^ 22 ði,j,lÞ j^ 23 ði,j,lÞ 5 þ 4 j^ 22 ðj,i,lÞ n n n j^ 33 ði,j,lÞ
j^ 11 ði,j,lÞ n n
3 j^ 13 ðj,i,lÞ T j^ 23 ðj,i,lÞ 7 5oX ði,j,m,lÞ þ X ði,j,m,lÞ , j^ 33 ðj,i,lÞ
ð55Þ ½X ði,j,m,lÞrr o0,
ð56Þ
where 2 6 6 6 6 6 6 j^ 11 ði,i,pÞ ¼ 6 6 6 6 6 4
Oði,i,mÞ
0 0 L3
A1 ði,mÞ L2 LT1
n
0 L2 þ LT2
n
n
0 0 L3 þ LT3
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
2
0 0
6 6 6 6 0 6 6 j^ 12 ði,i,1Þ ¼ j^ 12 ðj,i,1Þ ¼ 6 0 6 pffiffiffi 6 dL 1 6 6 4 0 0
0 0
0 L2 0 0
0 0 pffiffiffi dL 3 0
0 0
0 0 L3 0
L1 0
0 0
0 0
0 0
0
0
0
0
3 B1 ði,mÞPði,mÞ B2 ði,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ði,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
760
2
ZðPði,mÞÞ Pði,mÞE T ði,mÞ 6 6 0 0 6 6 0 0 6 6 0 0 j^ 13 ði,i,lÞ ¼ 6 6 T 6 ði,mÞ0 0 E 6 1 6 6 0 0 4
0 0
pffiffiffiffiffi T 2dE1 ði,mÞ 0
BT3 ði,mÞ
0
3 D 7 07 7 07 7 7 0 7, 7 07 7 7 07 5
pffiffiffiffiffi pffiffiffiffiffi 2dPði,mÞE T ði,mÞ 2dPði,mÞAT ði,mÞ 0 0
BT3 ði,mÞ
0 0
pffiffiffiffiffi T 2dA ði,mÞ pffiffiffiffiffi T1 2dB ði,mÞ pffiffiffiffiffi 1T 2dB2 ði,mÞ
0
j^ 22 ði,i,lÞ ¼ j^ 22 ði,j,lÞ ¼ j^ 22 ðj,i,lÞ ¼ diagfI,I,I,I,I,I,Ig, j^ 23 ði,i,lÞ ¼ j^ 23 ði,j,lÞ ¼ j^ 23 ðj,i,lÞ ¼ 0, j^ 33 ði,i,lÞ ¼ j^ 33 ðj,i,lÞ ¼ diagfZ0 ðPði,mÞÞ Pði,mÞ I I G T ðmÞ þ GðmÞg, 2
Oði,j,mÞ n n
n
0 0 L3 þ LT3
n
n
n
0 0 L3 I
n
n
n
n
ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
Oðj,i,mÞ
0
0
0
A1 ðj,mÞ
6 6 6 6 6 6 j^ 11 ði,j,pÞ ¼ 6 6 6 6 6 4
2
A1 ði,mÞ L2 LT1 0
0 L2 þ LT2
6 6 6 6 6 6 ^ j 11 ðj,i,pÞ ¼ 6 6 6 6 6 4
2
LT2
3 B1 ði,mÞPðj,mÞ B2 ði,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I
B1 ðj,mÞPði,mÞ B2 ðj,mÞ
n
L2 þ
n
n
0 L3 þ LT3
n
n
n
0 L3 I
n
n
n
n
L2 LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
0 0 0 0 Z
n
n
n
n
n
n
0
6 6 0 6 6 0 6 6 j^ 12 ðj,i,1Þ ¼ 6 0 6 pffiffiffi 6 dL 1 6 6 4 0 0
0
0
0
0
0
L2
0
0 0
0 0
0 pffiffiffi dL 3 0
L3 0
L1 0
0 0
0 0
0 0
0
0
0
0
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ðj,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
0 0 0 0 0 g2 I
3 7 7 7 7 7 7 7, 7 7 7 7 5
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
ZðPðj,mÞÞ Pðj,mÞE T ði,mÞ 6 6 0 0 6 6 0 0 6 6 0 0 j^ 13 ði,j,lÞ ¼ 6 6 T 6 ði,mÞ 0 E 6 1 6 6 0 0 4
2 6 6 6 6 6 6 j^ 13 ðj,i,lÞ ¼ 6 6 6 6 6 6 4
761
pffiffiffiffiffi 2dPðj,mÞE T ði,mÞ 0
pffiffiffiffiffi 2dPðj,mÞAT ði,mÞ 0
0 0
0 0
pffiffiffiffiffi T 2dE1 ði,mÞ
0
BT3 ði,mÞ
BT3 ði,mÞ
pffiffiffiffiffi T 2dA ði,mÞ pffiffiffiffiffi T1 2dB ði,mÞ pffiffiffiffiffi 1T 2dB2 ði,mÞ
ZðPði,mÞÞ
Pði,mÞE T ðj,mÞ
pffiffiffiffiffi 2dPði,mÞE T ðj,mÞ
pffiffiffiffiffi 2dPði,mÞAT ðj,mÞ
0 0
0 0
0 0
0 0
0 0
0 E1T ðj,mÞ
0 pffiffiffiffiffi T 2dE1 ðj,mÞ
0 pffiffiffiffiffi T 2dA1 ðj,mÞ pffiffiffiffiffi T 2dB ðj,mÞ pffiffiffiffiffi 1T 2dB2 ðj,mÞ
0
0
0
0
0
BT3 ðj,mÞ
BT3 ðj,mÞ
j^ 33 ði,j,lÞ ¼ diagfZ0 ðPðj,mÞÞ Pðj,mÞ I I G T ðmÞ þ GðmÞg, 2
0 6 pffiffiffi 6 dL2 6 6 0 6 6 j^ 12 ði,i,2Þ ¼ 6 0 6 6 0 6 6 4 0 0 2
0 6 pffiffiffi 6 dL2 6 6 0 6 6 j^ 12 ðj,i,2Þ ¼ 6 0 6 6 0 6 6 4 0 0
0
0
0
0
pffiffiffi 2Pði,mÞ
Pði,mÞC T ði,mÞ
0
L2
0
0
0
0 0
0 0
0 pffiffiffi dL 3 0
L3 0
0 0
0 0
L1 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0 0
L2 0
0 pffiffiffi dL3
0 L3
0 L1
0 0
0 0
0 0
0
0
0
0
0
0
0
0
3 7 7 7 7 7 7 7, 7 7 7 7 5
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ðj,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
3 D 7 07 7 07 7 7 0 7, 7 07 7 7 07 5 0
D
3
7 07 7 07 7 7 0 7, 7 07 7 7 07 5 0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
762
2
0 6 pffiffiffi 6 dL 2 6 6 0 6 6 j^ 12 ði,j,2Þ ¼ 6 0 6 6 0 6 6 4 0 0 2
0
0
pffiffiffi 2Pðj,mÞ
Pðj,mÞC T ði,mÞ
0 0
L2 0
0 pffiffiffi dL 3
0 L3
0 0
0 0
0 L1
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
n n
n
0 0 L3 þ LT3
n
n
n
0 0 L3 I
n
n
n
n
A1 ði,mÞ L2 LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
Oði,j,mÞ n
0 L2 þ LT2
n
n
0 0 L3 þ LT3
0 0 L3
A1 ði,mÞ L2 LT1
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
Oðj,i,mÞ n
0 L2 þ LT2
n
n
0 0 L3 þ LT3
n
n
n
0 0 L3 I
n
n
n
n
A1 ðj,mÞ L2 LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
6 6 6 6 6 6 j^ 11 ði,j,3Þ ¼ 6 6 6 6 6 4
2
0
0 L2 þ LT2
Oði,i,mÞ
6 6 6 6 6 6 j^ 11 ði,i,3Þ ¼ 6 6 6 6 6 4
2
0
6 6 6 6 6 6 j^ 11 ðj,i,3Þ ¼ 6 6 6 6 6 4
2
0 0
6 6 6 6 0 6 6 j^ 12 ði,i,3Þ ¼ 6 0 6 pffiffiffi 6 dL4 6 6 4 0 0
0 0
0 0
0 pffiffiffi dL6
0 L6
0 0
L5 0
0 0
0 0
L4
0
0
0
0 0
0 0
0 0
0 0
3 7 7 7 7 7 7 7, 7 7 7 7 5
3 B1 ði,mÞPði,mÞ B2 ði,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I 3 B1 ði,mÞPðj,mÞ B2 ði,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I 3 B1 ðj,mÞPði,mÞ B2 ðj,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ði,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
0
2
0 L5
L6 0
0 L4
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 pffiffiffi dL6
L6
0 0
L5 0
0 0
0 0
L4 0
0 0
0 0
0 0
0
0
0
0
0
0 Pði,mÞC T ði,mÞ
6 6 0 6 6 0 6 6 j^ 12 ði,j,3Þ ¼ 6 0 6 pffiffiffi 6 dL4 6 6 4 0 0 2
0
0
pffiffiffi 3 2Pðj,mÞ Pðj,mÞC T ði,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0
0
0 L5
0 pffiffiffi dL 6 0
L6 0
0 0
0 0
0 L4
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 pffiffiffi dL6
L6
0 0
L5 0
0 0
0 0
L4 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0 0
0 L5
0 pffiffiffi dL6 0
L6 0
0 L4
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
6 6 0 6 pffiffiffi 6 dL5 6 6 j^ 12 ðj,i,4Þ ¼ 6 0 6 6 0 6 6 4 0 2
0 0
0
pffiffiffi 2Pði,mÞ
6 6 0 6 pffiffiffi 6 dL5 6 6 j^ 12 ði,i,4Þ ¼ 6 0 6 6 0 6 6 4 0 2
0
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ðj,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0
0 pffiffiffi dL6 0
6 6 0 6 6 0 6 6 j^ 12 ðj,i,3Þ ¼ 6 0 6 pffiffiffi 6 dL4 6 6 4 0
0
6 6 0 6 pffiffiffi 6 dL5 6 6 j^ 12 ði,j,4Þ ¼ 6 0 6 6 0 6 6 4 0 0
0
0
0 0
763
3 7 7 7 7 7 7 7, 7 7 7 7 5
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ðj,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
pffiffiffi 3 2Pðj,mÞ Pðj,mÞC T ði,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
764
p ¼ 1; 2, l ¼ 1; 2,3; 4
ð57Þ
and D, Oði,i,mÞ, Oði,j,mÞ, Oðj,i,mÞ, Z(P(i,m)), Z(P(j,m)), Z0 ðPði,mÞÞ, Z0 ðPðj,mÞÞ are given in Eq. (22). In this case, a suitable stabilizing state feedback controller can be chosen by uðtÞ ¼ F ðrðtÞÞxðtÞ,
F ðrðtÞ ¼ mÞ ¼ Y ðmÞGðmÞ1 :
ð58Þ
Proof. For t40, we set Z t T T 2 JðtÞ ¼ E ½zðsÞ zðsÞg vðsÞ vðsÞ ds:
ð59Þ
0
Then, it is easy to get that Z t T T 2 ½zðsÞ zðsÞg vðsÞ vðsÞ þ LV ðxs ,s,rðsÞÞ ds JðtÞ ¼ E 0
Z
t
½zðsÞ zðsÞg vðsÞ vðsÞ þ LV ðxs ,s,rðsÞÞ ds, T
EfV ðxt ,t,rðtÞÞgrE
2
T
ð60Þ
0
where V ðxt ,t,rðtÞÞ is defined in Eq. (38). From Eqs. (54)–(56), similar to the proof of Theorem 1, we can easily deduce that there exist scalar c1 40 and c2 40 such that 2
zðtÞT zðtÞg2 vðtÞT vðtÞ þ LV ðxt ,t,rðtÞÞrzðtÞT Y1 zðtÞoc1 9xðtÞ9 ,
ð61Þ
2
zðtÞT zðtÞg2 vðtÞT vðtÞ þ LV ðxt ,t,rðtÞÞrzðtÞT Y2 zðtÞoc2 9xðtÞ9 , where
"
zðtÞT ¼ xT ðtÞ xT ðttm Þ xT ðtt0 Þ xT ðttM Þ xT ðttðtÞÞ
Z
ð62Þ T
t
xðsÞ ds
# vT ðtÞ ,
tdðtÞ
and Y1 and Y2 are omitted. Then, it follows from Eqs. (59) and (62) that 2
JðtÞ ¼ EfLV ðxt ,t,rðtÞÞgominfc1 ,c2 g9xðtÞ9 o0: This implies that for any nonzero vðtÞ 2 L2 ½0,1, JzJE2 ogJvJ2 : Therefore, by Definition 2, the uncertain stochastic Markovian jumping system in Eqs. (1)–(3) is robustly stochastically stable with disturbance attenuation level g. This completes the proof. & Remark 3. The results of Theorems 1 and 2 can be easily extended to the cases to deal with the problem of robust stabilization and robust H1 control in which there are no stochastic terms and/or there are no distributed time delays.
4. Illustrative example In this section, we provide an example to demonstrate the effectiveness of the proposed method.
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
765
Consider uncertain stochastic Markovian jumping systems in (1)–(3) with parameters as follows: Mode 1 3:3234 1:2351 1:2845 1:3245 Að1; 1Þ ¼ , Að2; 1Þ ¼ , 1:2234 1:2253 1:2354 3:2834 1:3234 2:2383 A1 ð1; 1Þ ¼ , 1:2835 1:2235 1:2893 2:2834 1:2934 1:7383 A1 ð2; 1Þ ¼ , B0 ð1; 1Þ ¼ , 1:2834 1:2383 2:2345 1:8345 1:2834 1:3923 B0 ð2; 1Þ ¼ , 1:2893 3:2322 1:2333 2:4560 1:3846 1:8346 B1 ð1; 1Þ ¼ , B1 ð2; 1Þ ¼ , 2:2845 1:2894 2:2389 1:2234 3:3343 7:4785 B2 ð1; 1Þ ¼ , 8:2245 5:5643 3:5751 3:2346 1:3464 1:2394 B2 ð2; 1Þ ¼ , Eð1; 1Þ ¼ , 1:2322 1:3345 2:4396 3:5896 1:3853 2:2895 Eð2; 1Þ ¼ , 1:3956 1:2389 2:3525 0:3764 3:4546 0:2367 E1 ð1; 1Þ ¼ , E1 ð2; 1Þ ¼ , 0:2563 0:1235 0:2356 2:3456 1:8237 2:2355 B3 ð1; 1Þ ¼ , 3:2353 2:4334 3:4557 3:2466 2:3533 2:2846 B3 ð2; 1Þ ¼ , Cð1; 1Þ ¼ , 2:3346 1:3456 2:9567 0:6393 0:1243 2:1234 Cð2; 1Þ ¼ , 2:1938 0:1234 a1 ¼ a2 ¼ 0:5,
l11 ¼ 0:5,
l12 ¼ 0:5:
Mode 2
1:8434 1:4167 1:3946 5:2345 , Að2; 2Þ ¼ , 2:2543 1:4563 1:3034 1:6424 2:3254 2:4345 , A1 ð1; 2Þ ¼ 0:2935 2:3235 4:2934 2:2034 3:2934 1:6213 A1 ð2; 2Þ ¼ , B0 ð1; 2Þ ¼ , 1:1234 0:2934 2:9383 2:1332
Að1; 2Þ ¼
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
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B0 ð2; 2Þ ¼
0:3044
1:0383
1:3345 1:2342
,
1:8373 2:2832 1:4272 , B1 ð2; 2Þ ¼ 0:2837 1:3242 2:9323 2:1292 1:2345 B2 ð1; 2Þ ¼ , 2:2235 0:5343 3:2346 1:2226 1:3235 B2 ð2; 2Þ ¼ , Eð1; 2Þ ¼ 1:8341 1:2321 2:4234 0:3323 2:2539 Eð2; 2Þ ¼ , 1:8956 1:2355 2:2032 0:2392 3:8333 E1 ð1; 2Þ ¼ , E1 ð2; 2Þ ¼ 1:2532 0:3345 0:9383 1:8237 2:2832 B3 ð1; 2Þ ¼ , 3:2353 2:4334 3:9327 1:6263 2:2134 B3 ð2; 2Þ ¼ , Cð1; 2Þ ¼ 1:3637 1:4645 2:5332 1:4942 2:1234 Cð2; 2Þ ¼ , 1:1293 0:1341 B1 ð1; 2Þ ¼
a1 ¼ a2 ¼ 0:5,
l21 ¼ 0:2,
1:2345 , 4:9383
1:2324
2:3234
1:1832 2:2343
,
,
1:4341 , 1:3242
l22 ¼ 0:2:
And t ¼ 0:2,
tm ¼ 0:1,
tM ¼ 2,
d ¼ 0:3,
g ¼ 2:
Then, using the Matlab LMI Control Toolbox to solve the LMIs in (54)–(56), we obtain a set of solutions as follows: 2:0492 0:0537 2:5371 0:0221 , Pð1; 2Þ ¼ 106 n Pð1; 1Þ ¼ , 0:0537 1:0997 0:0221 3:3059 0:2530 0:1622 Pð2; 1Þ ¼ , 0:1622 0:2762 2:5865 0:0530 5:6202 3:2910 Pð2; 2Þ ¼ 106 n , L1 ¼ , 0:0530 2:4773 3:0809 0:7932 0:9944 0:8644 L2 ¼ , 0:8373 0:7302 3:3187 0:1050 2:0814 1:8051 L3 ¼ , L4 ¼ , 0:2350 3:6801 0:1707 3:4377 11:5580 0:9557 L5 ¼ , 0:4495 10:7719
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
L6 ¼
3:6235
0:0246
,
Y ð1Þ ¼
0:0977 3:9628 2:6929 3:7166 Y ð2Þ ¼ , 14:2998 1:5234 25:6535 3:2543 Gð1Þ ¼ , 4:6243 2:3400 23:7880 19:6530 Z¼ : 19:6530 29:6351
19:1578
4:4426
5:4868
2:2321
Gð2Þ ¼
73:6063 24:9279
767
,
10:9179 , 4:7678
Then, the desired state feedback controller can be chosen as 0:5399 1:1477 , F ð1Þ ¼ Y ð1ÞG 1 ð1Þ ¼ 0:5149 1:6700 1:0131 3:0993 1 F ð2Þ ¼ Y ð2ÞG ð2Þ ¼ : 0:3834 0:5584 5. Conclusion The problems of robust stochastic stabilization and robust stochastic H1 control for a class of uncertain stochastic Markovian jumping systems with mixed delays have been considered in this paper. Based on the decoupling method, some improved delay-intervaldependent criteria have been proposed. These stability conditions have been obtained in terms of LMIs, which can be easily solved by using standard software. A numerical example has been provided to demonstrate the effectiveness of our results. Acknowledgment This work was supported National Natural Science Foundation of China under Grant 61004046, China Postdoctoral Science Foundation under Grant 20110491336, Postdoctoral Science Foundation of Jiangsu Province under Grant 1001007C, Young and Middle-Aged Scientists Research Awards Fund of Shandong Province under Grant 2009BSB01450. References [1] H. Gao, C. Wang, J. Wang, On H1 performance analysis for continuous-time stochastic systems with polytopic uncertainties, Circuits, Systems, and Signal Processing 24 (2005) 415–429. [2] H. Gao, J. Lam, C. Wang, Robust energy-to-peak filter design for stochastic time-delay systems, Systems and Control Letters 55 (2006) 101–111. [3] H. Gao, J. Lam, Z. Wang, Discrete bilinear stochastic systems with time-varying delay: stability analysis and control synthesis, Chaos, Solitons & Fractals 34 (2007) 394–404. [4] V.B. Kolmanovskii, A.D. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992. [5] X. Mao, N. Koroleva, A. Rodkina, Robust stability of uncertain stochastic differential delay equations, Systems and Control Letters 35 (1998) 325–336. [6] H. Zhang, H. Yan, Q. Chen, Stability and dissipative analysis for a class of stochastic system with time-delay, Journal of the Franklin Institute 347 (5) (2010) 882–893.
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Journal of the Franklin Institute 349 (2012) 741–769 www.elsevier.com/locate/jfranklin
New robust HN control for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method Jianwei Xiaa,b, Changyin Suna,n, Baoyong Zhangc a
School of Automation, Southeast University, Jiangsu 210096, PR China School of Mathematics Science, Liaocheng University, Liaocheng 252000, PR China c School of Automation, Nanjing University of Science and Technology, Nanjing 210094, PR China b
Received 14 July 2011; received in revised form 28 August 2011; accepted 12 September 2011 Available online 22 September 2011
Abstract This paper considers the problems of robust stochastic stabilization and robust H1 controller design for a class of stochastic Markovain jumping systems with mixed time delays and polytopic parameter uncertainties. Both the interval time-varying delay and distributed time delay are simultaneously considered. Some new delay-dependent sufficient conditions, which differs greatly from the most existing results, are obtained based on the decoupling method and some advanced techniques. A numerical example is provided to illustrate the effectiveness of the proposed criteria. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction During the past decades, considerable attention has been paid to the study of stochastic dynamical systems modeled by Ito’s formula [1–7]. Since time delays are frequently encountered in various physical and engineering systems, and they are often the sources of instability, oscillations and poor performances [8–11], therefore, the study of stability analysis and controller synthesis for stochastic systems with time delays has been extensively investigated in the past years, and a great deal of results related to such systems have been reported in the literatures [12–15]. To mention a few, the problem of exponential n
Corresponding author. E-mail addresses: [email protected] (J. Xia), [email protected] (C. Sun), [email protected] (B. Zhang). 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.09.009
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stability in mean square for stochastic time-delay systems was proposed in [5,12]. When the parameter uncertainties were considered, the robust stochastic stabilization problem was solved in [13]. More recently, the problems of robust H1 control and H1 filtering were investigated in [14,15], respectively. It is noted that all these results are delay-independent, that is there is no information on the size of delays was considered. It is well known that delay-dependent results are generally less conservative than delay-independent ones, especially when time delays are small. This motivates to develop delay-dependent conditions for stochastic delay systems. To improve the performance of delay-dependent stability criteria, much effort has been devoted in recent works. For instance, by the methods of improved augmented Lyapunov–Krasovskii functional and (or) free-weighting matrices, delay-dependent results of stability analysis, robust stabilization, robust H1 control, robust H1 filtering, L2 L1 filtering, H1 reduction were studied for stochastic delay systems in [2,16–25], respectively. On the other hand, one of the important class of stochastic systems is the so-called stochastic systems with Markovian jump parameters, a great deal of attention has been devoted to the stochastic Markovian jumping systems in recent years, see, for example, [26–33]. The problems of delay-independent and delay-dependent exponential stability for stochastic systems with Markovian jump parameters were dealt with in [26,27], respectively. In [28,29], the problem of stochastic stabilization in mean square for several stochastic Markovian jumping systems with time delay was investigated. When both time delays and parameter uncertainties appear in a Markovian jump system, some results on the robust H1 control problem were reported in [30]. Moreover, the problems of delaydependent H1 and robust H1 filtering for stochastic Markovian jumping systems with mode-dependent time delays were considered in [31,32], respectively. It is worth mentioning that the parameter uncertainties in the existing literatures are almost timevarying norm bounded [30,32]. When parameter uncertainties in a polytopic type arise, however, the problem of robust H1 control for stochastic Markovian jumping systems with time delays has not been fully investigated. The problem of parameter-dependent robust stability analysis was proposed for uncertain Markovian jump systems with polytopic parameter uncertainties and time-varying delays in [34], but no robust stabilization criterion was developed. One of the reason is that the state feedback controller design approach is difficult to be achieved using the parameter-dependent Lyapunov–Krasovskii functional method, but a common Lyapunov–Krasovskii functional is used to ensure stabilization for all admissible uncertainties will usually leads to be conservative, which motivates this study. In this paper, we focus on the problems of delay-interval-dependent robust stochastic stabilization and robust stochastic H1 control for stochastic Markovian jumping systems (SMJS) with mixed time delays and parameter uncertainties. The parameter uncertainties are of the polytopic forms. By introducing a parameter-dependent Lyapunov–Krasovskii functional with some free-weighting matrices and decoupling matrices, new delaydependent robust stochastic stabilization and robust H1 control criterions are obtained in terms of LMIs. An example is given to show the effectiveness of the proposed method. Notation: Throughout this paper, for symmetric matrices X and Y, the notation X ZY (respectively, X 4Y ) means that the matrix XY is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension. MT represents the transpose of the matrix M. Efg denote the expectation operator with respect to some probability measure P. L2 ½0,1 is the space of square-integrable vector functions over
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½0,1. 9 9 refers to the Euclidean vector norm while J J2 stands for the usual L2 ½0,1 norm. ðO,F ,P) is a probability space with O the sample space and F the s-algebra of subsets of the sample space. Matrices, if not explicitly mentioned, are assumed to have compatible dimensions. 2. Problem formulation Fix a probability space ðO,F ,PÞ and consider the following a class of uncertain stochastic Markovian jumping systems (SMJS) with mixed time delays described by Z t dxðtÞ ¼ Aða,rðtÞÞxðtÞ þ A1 ða,rðtÞÞxðttðtÞÞ þ B0 ða,rðtÞÞuðtÞ þ B1 ða,rðtÞÞ xðsÞ ds tdðtÞ
þB2 ða,rðtÞÞvðtÞ dt þ ½Eða,rðtÞÞxðtÞ þ E1 ða,rðtÞÞxðttðtÞÞ þ B3 ða,rðtÞÞvðtÞ dwðtÞ, ð1Þ zðtÞ ¼ Cða,rðtÞÞxðtÞ, xðtÞ ¼ jðtÞ,
ð2Þ
8t 2 ½h,0, h ¼ maxftM ,dg,
n
m
ð3Þ p
where xðtÞ 2 R is the state; uðtÞ 2 R is the control input; vðtÞ 2 R is a disturbance input which belongs to L2 ½0,1, zðtÞ 2 Rq is the controlled output, and w(t) is a one-dimension Brownian motion satisfying EfdwðtÞg ¼ 0 and EfdwðtÞ2 g ¼ dt; frðtÞ,tZ0g be a rightcontinuous Markov process on the probability space which takes values in the finite space S ¼ 1,2,y,N} with generator P ¼ ðlm,n Þðm,n 2 SÞ given by ( lm,n D þ oðDÞ if man, Pfrðt þ DÞ ¼ n9rðtÞ ¼ mg ¼ ð4Þ 1 þ lm,m D þ oðDÞ if m ¼ n, where D40 P and limD-0 oðDÞ=D ¼ 0, lm,n Z0 is the transition rate from m to n if man and lm,m ¼ man lm,n . The bounded functions tðtÞ and d(t) represent interval time-varying delays and distributed delays satisfying 0rtm otðtÞrtM ,
t_ ðtÞrt, dðtÞrd,
ð5Þ
where tm , tM , t and d are real known constant scalars, jðtÞ is an initial function defined on [h,0]. The uncertain matrices of the system are assumed to be with the polytopic type described by ½Aða,rðtÞÞ A1 ða,rðtÞÞ B0 ða,rðtÞÞ B1 ða,rðtÞÞ B2 ða,rðtÞÞ Eða,rðtÞÞ E1 ða,rðtÞÞ B3 ða,rðtÞÞ Cða,rðtÞÞ
¼
r X
ai ½Aði,rðtÞÞ A1 ði,rðtÞÞ B0 ði,rðtÞÞ B1 ði,rðtÞÞ B2 ði,rðtÞÞ Eði,rðtÞÞ
i¼1
E1 ði,rðtÞÞ B3 ði,rðtÞÞ Cði,rðtÞÞ,
ð6Þ
and r X
ai ¼ 1,
ai Z0,
ð7Þ
i¼1
where Aði,rðtÞÞ, A1 ði,rðtÞÞ, B0 ði,rðtÞÞ, B1 ði,rðtÞÞ, B3 ði,rðtÞÞ, Eði,rðtÞÞ, E1 ði,rðtÞÞ B3 ði,rðtÞÞ, Cði,rðtÞÞ are known real constant matrices.
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744
Define the following two scalars which are with respect to the variation range of time delay tm þ tM tM tm , d¼ : ð8Þ t0 ¼ 2 2 For convenience, we set yðtÞ ¼ Aða,rðtÞÞxðtÞ þ B0 ða,rðtÞÞuðtÞ þ A1 ða,rðtÞÞxðttðtÞÞ Z t þB1 ða,rðtÞÞ xðsÞ ds þ B2 ða,rðtÞÞvðtÞ, tdðtÞ
gðtÞ ¼ Eða,rðtÞÞxðtÞ þ E1 ða,rðtÞÞxðttðtÞÞ þ B3 ða,rðtÞÞvðtÞ then, system (1) becomes dxðtÞ ¼ yðtÞ dt þ gðtÞ dwðtÞ:
ð9Þ
Throughout the paper the following concepts and lemmas will be needed. Definition 1 (Xu and Chen [35]). For system (1) with u(t) ¼ 0 and v(t)¼ 0, the equilibrium point is said to be robust stable mean square if for any e40, there is a dðeÞ40 such that 2
E9xðtÞ9 oe,
t40
for all admissible uncertainties, when 2
sup E9xðtÞ9 odðeÞ: mrsr0
If, in addition, 2
lim E9xðtÞ9 ¼ 0
t-1
for any initial conditions and all admissible uncertainties, then system (1) with u(t)¼ 0 and v(t)¼ 0 is said to be robustly asymptotically stable in mean square. Definition 2 (Xu and Chen [35]). Given a scalar g40, the stochastic system (1)–(3) is said to be robustly stochastically stable in mean square with disturbance attenuation level g, if it is stochastically stable and with zero initial condition, the inequality JzðtÞJE2 ogJvðtÞJ2
ð10Þ
holds for all admissible uncertainties, where Z 1 1=2 9zðtÞ2 9 : JzðtÞJE2 ¼ E
ð11Þ
0
Lemma 1 (Zhang et al. [36]). For any positive definite matrix M40, scalar g40, vector function o : ½0,g-Rn such that the integrations concerned are well defined, the following inequality holds: Z g T Z g Z g oðsÞ ds M oðsÞ ds rg oðsÞT MoðsÞ ds : ð12Þ 0
0
0
Lemma 2 (Gao et al. [37], Yang et al. [38]). For any two vectors x and y, positive matrix Q40 with compatible dimensions, the following matrix inequality holds: 2xT yrxT Qx þ yT Q1 y:
ð13Þ
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745
Lemma 3 (Zhang et al. [39]). For any constant matrices C1 and C2 with appropriate dimensions and a symmetric matrix Oo0, scalars d1 rd2 , and a function dðtÞ : Rþ -½d1 ,d2 , then ðdðtÞd1 ÞC1 þ ðd2 dðtÞÞC2 þ Oo0 holds, if and only if ðd2 d1 ÞC1 þ Oo0, ðd2 d1 ÞC2 þ Oo0:
ð14Þ
In this paper, our aim is to develop criteria of robust stochastic stabilization and robust H1 control for uncertain stochastic time-delay system (1)–(3). More specifically, we are concerned with the following two problems: (1) Robust stochastic stabilization problem: Design a state feedback controller uðtÞ ¼ F ðrðtÞÞxðtÞ
ð15Þ
for system (1)–(3) with v(t)¼ 0 such that the resulting closed-loop system is robustly asymptotically stable in mean square for all admissible uncertainties. In this case, the system (1)–(3) with v(t) ¼ 0 is said to be robustly stochastically stabilizable. (2) Robust H1 control problem: Given a constant scalar g40, design a state feedback controller in the form of Eq. (15) such that, for all admissible uncertainties, the resulting closed-loop system is robustly asymptotically stable in mean square and for any nonzero vðtÞ 2 L2 ½0,1, JzðtÞJE2 ogJvðtÞJ2
ð16Þ
is satisfied under the zero-initial condition. In this case, the system (1)–(3) is said to be robustly stochastically stabilizable with disturbance attenuation level g. 3. Main results 3.1. Robust stochastic stabilization In this subsection, an LMI approach will be utilized to solve the robust stochastic stabilization problem described in the previous section. To this purpose, we consider the system (1)–(3) with v(t) ¼ 0 as following: dxðtÞ ¼ ½Aða,rðtÞÞxðtÞ þ A1 ða,rðtÞÞxðttðtÞÞ þ B0 ða,rðtÞÞuðtÞ Z t þB1 ða,rðtÞÞ xðsÞ ds dt þ ½Eða,rðtÞÞxðtÞ þ E1 ða,rðtÞÞxðttðtÞÞ dwðtÞ, tdðtÞ
ð17Þ xðtÞ ¼ jðtÞ,
8t 2 ½h,0, h ¼ maxftM ,dg:
ð18Þ
Then, we have the following result. Theorem 1. The system in Eqs. (17) and (18) is robustly stochastically stabilizable, if there exist matrices Z, G(m), Y(m), P(i,m), X ði,i,m,lÞ, X ði,j,m,lÞ, Lk, with Z40, Pði,mÞ40, X ði,i,m,lÞ ¼ X ði,i,m,lÞT , for joi, i ¼ 1,2,y,r, m ¼ 1,2,y,N, k ¼ 1,2,y,6, l¼ 1,2,3,4,
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
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satisfying the following LMIs: 2 3 j11 ði,i,lÞ j12 ði,i,lÞ j13 ði,i,lÞ 6 n j22 ði,i,lÞ j23 ði,i,lÞ 7 4 5oX ði,i,m,lÞ, n n j33 ði,i,lÞ 2 6 4
j11 ði,j,lÞ
j12 ði,j,lÞ
n
j22 ði,j,lÞ
n
n
2 6 þ4
j13 ði,j,lÞ
ð19Þ
3
j23 ði,j,lÞ 7 5
j33 ði,j,lÞ
3 j12 ðj,i,lÞ j13 ðj,i,lÞ T j22 ðj,i,lÞ j23 ðj,i,lÞ 7 5oX ði,j,m,lÞ þ X ði,j,m,lÞ , n j33 ðj,i,lÞ
j11 ðj,i,lÞ n n
ð20Þ
½X ði,j,m,lÞrr o0,
ð21Þ
where 2 6 6 6 6 j11 ði,i,pÞ ¼ 6 6 6 6 4
Oði,i,mÞ
0
0
0
A1 ði,mÞ
B1 ði,mÞPði,mÞ
n
L2 þ LT2
n
n
0 L3 þ LT3
0 L3
L2 LT1
0 0
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
0 0
n
n
n
n
n
Z
2
0
6 6 0 6 6 0 6 j12 ði,i,1Þ ¼ j12 ðj,i,1Þ ¼ 6 6 0 6 pffiffiffi 6 dL 4 1 0 2 6 6 6 6 6 j13 ði,i,lÞ ¼ 6 6 6 6 4
0
0
0
0
0 0
L2 0
0 pffiffiffi dL3
0 L3
0 L1
0 0
0 0
0 0
0
0
0
0
ZðPði,mÞÞ Pði,mÞE T ði,mÞ
7 7 7 7 7, 7 7 7 5
3 pffiffiffi 2Pði,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
pffiffiffiffiffi 2dPði,mÞE T ði,mÞ
pffiffiffiffiffi 2dPði,mÞAT ði,mÞ
0 0
0 0
0 0
0 0
0 0
0 E1T ði,mÞ
0 pffiffiffiffiffi T 2dE1 ði,mÞ
0
0
0
0 pffiffiffiffiffi T 2dA1 ði,mÞ pffiffiffiffiffi T 2dB1 ði,mÞ
j22 ði,i,lÞ ¼ j22 ði,j,lÞ ¼ j22 ðj,i,lÞ ¼ diagfI,I,I,I,I,Ig, j23 ði,i,lÞ ¼ j23 ði,j,lÞ ¼ j23 ðj,i,lÞ ¼ 0, j33 ði,i,lÞ ¼ j33 ðj,i,lÞ ¼ diagfZ0 ðPði,mÞÞ Pði,mÞ I I G T ðmÞ þ GðmÞg,
3
D
3
7 07 7 07 7 7, 07 7 07 5 0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
n
0 0
0 0
A1 ði,mÞ L2
n
n
L3 þ LT3
n
n
n
L3 I
LT1 0
n
n
n
n
ð1tÞI þ L1 þ LT1
n
n
n
n
n
Z
Oðj,i,mÞ
0
0
0
A1 ðj,mÞ
B1 ðj,mÞPði,mÞ
n
L2 þ LT2
n
n
0 L3 þ LT3
0 L3
L2 LT1
0 0
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
0 0
n
n
n
n
n
Z
6 6 6 6 j11 ðj,i,pÞ ¼ 6 6 6 6 4
2
0 0
6 6 6 6 0 6 j12 ði,j,1Þ ¼ 6 6 0 6 pffiffiffi 6 dL 4 1 0 2 6 6 6 6 6 j13 ði,j,lÞ ¼ 6 6 6 6 4
2
3 B1 ði,mÞPðj,mÞ 7 0 7 7 7 0 7, 7 0 7 7 5 0
0 L2 þ LT2
Oði,j,mÞ
6 6 6 6 j11 ði,j,pÞ ¼ 6 6 6 6 4
2
747
0 0
0 L2 0
0 0 pffiffiffi dL3
0 0 L1 0
L3
0 0
0 0
0 0
0
0
0
ZðPðj,mÞÞ Pðj,mÞE T ði,mÞ
0 0
pffiffiffiffiffi 2dPðj,mÞAT ði,mÞ
0 0
0 0
0 0
0 0
0 0
0 E1T ði,mÞ
0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T tM tm E1 ði,mÞ
0
0
0
0 pffiffiffiffiffi T 2dA1 ði,mÞ pffiffiffiffiffi T 2dB1 ði,mÞ
Pði,mÞE T ðj,mÞ 0
pffiffiffiffiffi 2dPði,mÞE T ðj,mÞ 0
pffiffiffiffiffi 2dPði,mÞAT ðj,mÞ 0
0 0
0 0
0 0
ZðPði,mÞÞ 6 0 6 6 6 0 6 j13 ðj,i,lÞ ¼ 6 0 6 6 6 0 4 0
7 7 7 7 7, 7 7 7 5
3 pffiffiffi 2Pðj,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
pffiffiffiffiffi 2dPðj,mÞE T ði,mÞ
E1T ðj,mÞ
pffiffiffiffiffi T 2dE1 ðj,mÞ
0
0
pffiffiffiffiffi T 2dA ðj,mÞ pffiffiffiffiffi T1 2dB1 ðj,mÞ
j33 ði,j,lÞ ¼ diag½Z0 ðPðj,mÞÞ Pðj,mÞ I I G T ðmÞ þ GðmÞ,
3
D
3
7 07 7 07 7 7, 07 7 07 5 0 3 D 7 07 7 07 7 7, 07 7 07 5 0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
748
2
0 6 pffiffiffi 6 dL2 6 6 0 6 j12 ði,i,2Þ ¼ j12 ðj,i,2Þ ¼ 6 6 0 6 6 0 4 2
0 6 pffiffiffi 6 dL2 6 6 0 6 j12 ði,j,2Þ ¼ 6 6 0 6 6 0 4 0 2
0
0
0 0
L2 0
0 pffiffiffi dL3
0 L3
0 L1
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
L2
0
0 0
0 0
0 pffiffiffi dL3 0
L3 0
L1 0
0 0
0 0
0 0
3 pffiffiffi 2Pði,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
3 pffiffiffi 2Pðj,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0 3 B1 ði,mÞPði,mÞ 7 0 7 7 7 0 7, 7 0 7 7 5 0
n
0 L6
0 0
A1 ði,mÞ 0
n
n
L5 þ LT5
n
n
n
0 I
L5 LT4
n
n
n
n
ð1tÞ þ L4 þ LT4
n
n
n
n
n
Oði,j,mÞ n
0 L6 þ LT6
0 L6
0 0
A1 ði,mÞ 0
n
n
L5 þ LT5
n
n
n
0 I
L5 LT4
n
n
n
n
ð1tÞ þ L4 þ LT4
n
n
n
n
n
Z
Oðj,i,mÞ
0
0
0
A1 ðj,mÞ
B1 ðj,mÞPði,mÞ
n
L6 þ LT6
n
n
L6 L5 þ LT5
0 0
0 L5
0 0
n
n
n
I
n
n
n
n
LT4 ð1tÞ þ L4 þ LT4
0 0
n
n
n
n
n
Z
6 6 6 6 j11 ði,j,3Þ ¼ 6 6 6 6 4 2
0
0 L6 þ LT6
Oði,i,mÞ
6 6 6 6 j11 ði,i,3Þ ¼ 6 6 6 6 4 2
0
6 6 6 6 j11 ðj,i,3Þ ¼ 6 6 6 6 4
2
0 0
6 6 6 6 0 6 j12 ði,j,3Þ ¼ 6 6 0 6 pffiffiffi 6 dL 4 4 0
0 0
0 0
0 pffiffiffi dL6
0 L6
0 0
L5 0
0 0
0 0
L4
0
0
0
0
0
0
0
3 pffiffiffi 2Pðj,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
Z 3 B1 ði,mÞPðj,mÞ 7 0 7 7 7 0 7, 7 0 7 7 5 0 3 7 7 7 7 7, 7 7 7 5
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
0
6 6 0 6 6 0 6 j12 ðj,i,3Þ ¼ 6 6 0 6 pffiffiffi 6 dL 4 4 0
0
0 0 L5
0 pffiffiffi dL6 0
0 0 0 L4
L6 0
0 0
0 0
0 0
0 0 0 2 0 0 6 6 0 0 6 pffiffiffi 6 dL 0 5 6 j12 ði,i,4Þ ¼ j12 ðj,i,4Þ ¼ 6 6 0 0 6 6 0 L 4 4 0 0 2 0 0 0 0 pffiffiffi 6 6 0 0 0 dL6 6 pffiffiffi 6 dL 0 L5 0 5 6 j12 ði,j,4Þ ¼ 6 6 0 0 0 0 6 6 0 L4 0 0 4 0
0
0
0
0
0 0 0 L5 0 0 0 0 L6 0 0 0 0
749
3 pffiffiffi 2Pði,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0 3 pffiffiffi 0 0 2Pði,mÞ pffiffiffi 7 7 dL6 L6 0 7 7 0 0 0 7 7, 7 0 0 0 7 7 0 0 0 5 0
0
3 pffiffiffi 2Pðj,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
0
p ¼ 1; 2, l ¼ 1; 2,3; 4
and D ¼ B0 ði,mÞY ðmÞ þ GT ðmÞPði,mÞ, Oði,i,mÞ ¼ Aði,mÞPði,mÞ þ Pði,mÞAði,mÞT þ B0 ði,mÞY ðmÞ þY ðmÞT B0 ði,mÞT þ lm,m Pði,mÞ þ d 2 Z, Oði,j,mÞ ¼ Aði,mÞPðj,mÞ þ Pðj,mÞAði,mÞT þ B0 ði,mÞY ðmÞ þY ðmÞT B0 ði,mÞT þ lm,m Pðj,mÞ þ d 2 Z, Oðj,i,mÞ ¼ Aðj,mÞPði,mÞ þ Pði,mÞAðj,mÞT þ B0 ðj,mÞY ðmÞ þY ðmÞT B0 ðj,mÞT þ lm,m Pði,mÞ þ d 2 Z, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ZðPði,mÞÞ ¼ ½ lm,1 Pði,mÞ lm,m1 Pði,mÞ lm,mþ1 Pði,mÞ lm,N Pði,mÞ, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ZðPðj,mÞÞ ¼ ½ lm,1 Pðj,mÞ lm,m1 Pðj,mÞ lm,mþ1 Pðj,mÞ lm,N Pðj,mÞ, Z0 ðPði,mÞÞ ¼ diag½Pði,1Þ Pði,m1Þ Pði,m þ 1Þ Pði,NÞ, Z0 ðPðj,mÞÞ ¼ diag½Pðj,1Þ Pðj,m1Þ Pðj,m þ 1Þ Pðj,NÞ:
ð22Þ
In this case, the overall state feedback controller is represented by uðtÞ ¼ F ðrðtÞÞxðtÞ,
F ðrðtÞ ¼ mÞ ¼ Y ðmÞGðmÞ1 :
ð23Þ
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
750
Proof. From Eqs. (19)–(21), set l ¼ 1,2, then, we can easily get 2 6 4
3 2 j12 ðaÞ j13 ðaÞ j11 ði,i,1Þ r X 6 7 2 j22 ðaÞ j23 ðaÞ 5 ¼ n ai 4 i ¼ 1 n j33 ðaÞ n
j11 ðaÞ n n
þ
02
r X X i ¼ 1 joi
2 6 þ4
B6 ai aj @ 4
j11 ðj,i,1Þ
r X
j11 ði,j,1Þ n
n
3 j13 ði,i,1Þ j23 ði,i,1Þ 7 5 j33 ði,i,1Þ
n
3 j12 ði,j,1Þ j13 ði,j,1Þ j22 ði,j,1Þ j23 ði,j,1Þ 7 5
n
n o
j12 ði,i,1Þ j22 ði,i,1Þ
j33 ði,j,1Þ
n
j12 ðj,i,1Þ j13 ðj,i,1Þ
31
C j22 ðj,i,1Þ j23 ðj,i,1Þ 7 5A n j33 ðj,i,1Þ
a2i X ði,i,m,1Þ þ
r X X ai aj ðX ði,j,m,1Þ þ X ði,j,m,1ÞT Þo0, i ¼ 1 joi
i¼1
ð24Þ 2 6 4
3 2 j012 ðaÞ j13 ðaÞ j ði,i,2Þ r X 6 11 7 2 j22 ðaÞ j23 ðaÞ 5 ¼ n ai 4 i ¼ 1 n j33 ðaÞ n
j11 ðaÞ n n
r X X
þ
i ¼ 1 joi
02 B6 ai aj @ 4
n
2
j11 ðj,i,2Þ j012 ðj,i,2Þ 6 n j22 ðj,i,2Þ þ4
o
r X
j22 ði,i,2Þ
n
a2i X ði,i,m,2Þ þ
j13 ði,i,2Þ
3
j23 ði,i,2Þ 7 5
n
j11 ði,j,2Þ j012 ði,j,2Þ j13 ði,j,2Þ n
n
j12 ði,i,2Þ
jði,iÞ
3
j22 ði,j,2Þ j23 ði,j,2Þ 7 5 n j33 ði,j,2Þ
31 j13 ðj,i,2Þ C j23 ðj,i,2Þ 7 5A
j33 ðj,i,2Þ
r X X
ai aj ðX ði,j,m,2Þ þ X ði,j,m,2ÞT Þo0,
ð25Þ
i ¼ 1 joi
i¼1
where 2 6 6 6 6 j11 ðaÞ ¼ 6 6 6 6 4
n
0 L2 þ LT2
0 0
0 0
A1 ða,mÞ L2
n
n
L3 þ LT3
n
n
n
L3 I
LT1 0
n
n
n
n
ð1tÞI þ L1 þ LT1
n
n
n
n
n
Oða,mÞ
3 B1 ða,mÞPða,mÞ 7 0 7 7 7 0 7, 7 0 7 7 5 0 Z
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
0 0
6 6 6 6 0 6 j12 ðaÞ ¼ 6 6 0 6 pffiffiffi 6 dL 4 1 0
0 0
0 L2 0
0 0 pffiffiffi dL 3
0
0 0 L3
0 L1
0 0
0 0
0 0
0
0
0
0
2
ZðPða,mÞÞ Pða,mÞE T ða,mÞ 6 0 0 6 6 6 0 0 6 j13 ðaÞ ¼ 6 0 0 6 6 T 6 ða,mÞ 0 E 1 4 0 0
3 pffiffiffi 2Pða,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 0
j22 ðaÞ ¼ j22 ði,iÞ,
751
j23 ðaÞ ¼ 0,
pffiffiffi dPða,mÞE T ða,mÞ 0
pffiffiffiffiffi 2dPða,mÞAT ða,mÞ 0
0 0
0 0
pffiffiffi T dE1 ða,mÞ 0
pffiffiffiffiffi T 2dA ða,mÞ pffiffiffiffiffi T1 2dB1 ða,mÞ
j33 ðaÞ ¼ diagfZ0 ðPða,mÞÞ Pða,mÞ I I GT ðmÞ þ GðmÞg, 3 2 pffiffiffi 0 0 0 0 0 2Pða,mÞ 7 6 pffiffiffi 7 6 dL2 0 L2 0 0 0 7 6 p ffiffi ffi 7 6 0 0 0 dL3 L3 0 7 6 0 j12 ðaÞ ¼ 6 7 7 6 0 0 0 0 0 0 7 6 7 6 0 L1 0 0 0 0 5 4 0
0
0
0
0
3 D 7 07 7 07 7 7, 07 7 07 5 0
ð26Þ
0
and D ¼ B0 ði,mÞY ðmÞ þ GT ðmÞPði,mÞ, Oða,mÞ ¼ Aða,mÞPða,mÞ þ Pða,mÞAða,mÞT þ B0 ða,mÞY ðmÞ þ Y ðmÞT B0 ða,mÞT þlm,m Pða,mÞ þ d 2 Z, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi lm,mþ1 Pða,mÞ lm,N Pða,mÞ, ZðPða,mÞÞ ¼ ½ lm,1 Pða,mÞ lm,m1 Pða,mÞ Z0 ðPði,mÞÞ ¼ diag½Pða,1Þ Pða,m1Þ Pða,m þ 1Þ Pða,NÞ:
ð27Þ
Noting the controller in Eq. (23), pre- and post-multiplying inequalities (24) and (25) by 2
I 0 60 I 6 6 6^ ^ 6 6 40 0
0 0
^ 0
&
0 0
0
3 0 Bða,mÞF ðmÞ 7 0 0 7 7 7 ^ ^ 7 7 0 0 5 I 0 1617
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
752
and its transpose, we obtain 2 3 w11 ðaÞ w12 ðaÞ w13 ðaÞ 6 n w22 ðaÞ w23 ðaÞ 7 4 5o0, n n w33 ðaÞ 2 6 4
w11 ðaÞ w012 ðaÞ n
w22 ðaÞ
n
n
w13 ðaÞ
ð28Þ
3
w23 ðaÞ 7 5o0, w33 ðaÞ
ð29Þ
where 2 6 6 6 6 w11 ðaÞ ¼ 6 6 6 6 4
O1 ða,mÞ
0
0
0
A1 ða,mÞ
B1 ða,mÞPða,mÞ
n
L2 þ LT2
n
n
0 L3 þ LT3
0 L3
L2 LT1
0 0
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
0 0
n
n
n
n
n
Z
w12 ðaÞ ¼ j12 ðaÞ,
w22 ðaÞ ¼ j22 ðaÞ,
2
ZðPða,mÞÞ 6 0 6 6 6 0 6 w13 ðaÞ ¼ 6 0 6 6 6 0 4 0
Pða,mÞE T ða,mÞ 0 0 0 E1T ða,mÞ 0
w23 ðaÞ ¼ 0,
3 7 7 7 7 7, 7 7 7 5
w012 ðaÞ ¼ j012 ðaÞ,
pffiffiffiffiffi pffiffiffiffiffi 3 2dPða,mÞE T ða,mÞ 2dPða,mÞAT ða,mÞ 7 0 0 7 7 7 0 0 7 7, 0 0 7 7 pffiffiffiffiffi T pffiffiffiffiffi T 7 2dE1 ða,mÞ 2dA1 ða,mÞ 5 pffiffiffiffiffi T 0 2dB1 ða,mÞ
w33 ðaÞ ¼ diagfZ0 ðPða,mÞÞ Pða,mÞ I Ig,
ð30Þ
O1 ða,mÞ ¼ ½Aða,mÞ þ Bða,mÞF ðmÞPða,mÞ þPða,mÞ½Aða,mÞ þ Bða,mÞF ðmÞT þ lm,m Pða,mÞ þ d 2 Z:
ð31Þ
and
Define ^ Pða,mÞ ¼ P1 ða,mÞ,
^ Zða,mÞ ¼ P1 ða,mÞZP1 ða,mÞ
pre- and post-multiplying diagðP1 ða,mÞ,I,I,I,I,P1 ða,mÞ,I,I,I,I,I,I,I,P1 ða,mÞ,I,IÞ to Eqs. (28) and (29) leads to 2
3 w^ 11 ðaÞ w^ 12 ðaÞ w^ 13 ðaÞ 6 n w^ 22 ðaÞ w^ 23 ðaÞ 7 4 5o0, n n w^ 33 ðaÞ
ð32Þ
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2 6 4
n
w^ 012 ðaÞ w^ 22 ðaÞ
n
n
w^ 11 ðaÞ
753
3 w^ 13 ðaÞ w^ 23 ðaÞ 7 5o0, w^ 33 ðaÞ
ð33Þ
where 2 6 6 6 6 6 w^ 11 ðaÞ ¼ 6 6 6 6 4
^ 1 ða,mÞ O
0
0
0
^ Pða,mÞA 1 ða,mÞ
^ Pða,mÞB 1 ða,mÞ
n
L2 þ LT2
n
n
0 L3 þ LT3
n
n
n
0 L3 I
n
n
n
n
L2 LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
0 0 0 0 Z^
2
0
6 6 0 6 6 0 6 w^ 12 ðaÞ ¼ 6 6 0 6 pffiffiffi 6 dL 4 1 0 2
0
0
0
0
0
L2
0
0 0
0 0
0 pffiffiffi dL 3 0
L3 0
L1 0
0 0
0 0
0 0
^ ZðPða,mÞÞ 6 0 6 6 6 0 6 w^ 13 ðaÞ ¼ 6 0 6 6 6 0 4 0
pffiffiffi 3 2I 7 0 7 7 0 7 7 7, 0 7 7 0 7 5 0
w^ 22 ðaÞ ¼ w22 ðaÞ,
3 7 7 7 7 7 7, 7 7 7 5
w^ 22 ðaÞ ¼ 0,
pffiffiffiffiffi T pffiffiffiffiffi T 3 2dE ða,mÞ 2dA ða,mÞ 7 0 0 7 7 7 0 0 0 7 7, 0 0 0 7 7 p ffiffiffiffiffi p ffiffiffiffiffi ^ E1T ða,mÞPða,mÞ 2dE1T ða,mÞ 2dAT1 ða,mÞ 7 5 pffiffiffiffiffi T 0 0 2dB1 ða,mÞ
^ E T ða,mÞPða,mÞ 0
^ ^ w33 ðaÞ ¼ diagfZðPða,mÞÞ Pða,mÞ I Ig,
ð34Þ
where ^ 1 ða,mÞ ¼ Pða,mÞ½Aða,mÞ ^ O þ Bða,mÞF ðmÞ ^ ^ ^ þ lm,m Pða,mÞ þ d 2 Z, þ½Aða,mÞ þ Bða,mÞF ðmÞT Pða,mÞ ^ ZðPða,mÞÞ ¼½
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi lm,1 I lm,m1 I, lm,mþ1 I lm,N I,
1 ^ Z0 ðPði,mÞÞ ¼ diag½Pða,1Þ Pða,m1Þ1 ,Pða,m þ 1Þ1 Pða,NÞ1 :
ð35Þ
For systems (17), choose the following Lyapunov–Krasovskii function candidate: V ðxt ,t,rðtÞÞ ¼
5 X
Vi ðxðtÞ,t,rðtÞÞ,
ð36Þ
2hrtr0
ð37Þ
i¼1
where xt ðsÞ ¼ xðt þ sÞ,
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
754
and ^ V1 ðxt ,t,rðtÞÞ ¼ xT ðtÞPða,rðtÞÞxðtÞ, Z
Z
t
xðsÞT xðsÞ ds þ
V2 ðxt ,t,rðtÞÞ ¼ ttðtÞ
Z
tm
Z
t
yðsÞT yðsÞ ds dy,
tþy
tM
tm
Z
t
gðsÞT gðsÞ ds dy,
V4 ðxt ,t,rðtÞÞ ¼ tþy
tM
Z
0
Z
t
V4 ðxt ,t,rðtÞÞ ¼ d d
^ where Pða,rðtÞÞ ¼
Pr
i¼1
xðsÞT xðsÞ ds,
ttM
V3 ðxt ,t,rðtÞÞ ¼ Z
t
^ ds dy, xðsÞT ZxðsÞ
ð38Þ
tþy
^ Then, for every m ¼ rðtÞ 2 S, by Ito’s formula, we have ai Pði,rðtÞÞ.
^ ½ðAða,rðtÞÞxðtÞ þ B0 ða,rðtÞÞF ðmÞÞxðtÞ þ xðtÞ þ A1 ða,rðtÞÞxðttðtÞÞ LV1 ðxt ,t,mÞ ¼ 2xT ðtÞPða,mÞ
Z
t
þB1 ða,rðtÞÞ
^ xðsÞ ds þ g ðtÞPða,mÞgðtÞ þ xT ðtÞ T
! X lmj Pj xðtÞ,
tdðtÞ
j2S
LV2 ðxt ,t,mÞ ¼ 2xT ðtÞxðtÞð1tÞxT ðttðtÞÞxðttðtÞÞxT ðttM ÞxðttM Þ, Z ttm LV3 ðxt ,t,mÞ ¼ ðtM tm ÞyT ðtÞyðtÞ yðsÞT yðsÞ ds dy ttM Z tt0 Z ttm T T yðsÞ yðsÞ ds dy yðsÞT yðsÞ ds dy, ¼ ðtM tm Þy ðtÞyðtÞ ttM
LV4 ðxt ,t,mÞ ¼ ðtM tm ÞgT ðtÞgðtÞ
Z
ttm
M Z tt tt0
T
¼ ðtM tm Þg ðtÞgðtÞ
tt0
gðsÞT yðsÞ ds dy gðsÞT gðsÞ ds dy
Z
ttM
^ LV5 ðxt ,t,mÞrd x ðtÞZxðtÞ 2 T
Z
t
t
xðsÞ ds :
T
T
T
ð39Þ
tdðtÞ
" T
gðsÞT gðsÞ ds dy,
tt0
T Z xðsÞ ds Z^
tdðtÞ
Set
ttm
T
T
Z
T #
t
x ðtÞ ¼ x ðtÞ x ðttm Þ x ðtt0 Þ x ðttM Þ x ðttðtÞÞ
xðsÞ ds
,
tdðtÞ
I1T ¼ ½0 0 0 0 I 0,
I2T ¼ ½0 I 0 0 0 0,
I3T ¼ ½0 0 I 0 0 0,
ð40Þ
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
755
we will discuss the LV ðxt ,t,rðtÞÞ in the following two cases: Case I: when t 2 ½tm ,t0 , we have Z ttm Z ttðtÞ Z ttm yðsÞT yðsÞ ds dy ¼ yðsÞT yðsÞ ds dy yðsÞT yðsÞ ds dy, tt0
Z
ttðtÞ
tt0
ttm
gðsÞT gðsÞ ds dy ¼
Z
ttðtÞ
gðsÞT gðsÞ ds dy
Z
tt0
tt0
ttm
gðsÞT gðsÞ ds dy,
ð41Þ
ttðtÞ
By Newton–Leibuniz formula, for any matrices Li, i¼ 1,2,3 with appropriate dimensions, we obtain Z ttðtÞ Z ttðtÞ T 2x ðttðtÞÞL1 ðxðttðtÞÞxðtt0 Þ yðsÞ ds gðsÞ dwðsÞ ¼ 0, Z T 2x ðttm ÞL2 ðxðttm ÞxðttðtÞÞ
tt0
Z
ttm
yðsÞ ds
Z
ttðtÞ
Z
tt0
ttM
tt0
yðsÞ ds
2x ðtt0 ÞL3 ðxðtt0 ÞxðttM Þ
gðsÞ dwðsÞ ¼ 0,
ttðtÞ T
tt0 ttm
gðsÞ dwðsÞ ¼ 0,
ð42Þ
ttM
According to Lemmas 1 and 2, it can be easily verified that Z ttðtÞ T 2x ðttðtÞÞL1 yðsÞ dsrxT ðttðtÞÞðt0 tðtÞÞL1 LT1 xðttðtÞÞ tt0
Z
ttðtÞ
yT ðsÞyðsÞ ds
þ tt0
¼x
T
Z
T
T Z gðsÞy dwðsÞ
tt0
¼x
T
ðtÞI1 L1 LT1 I1T xðtÞ Z
T
Z
gðsÞy dwðsÞ tt0
T
Z þ
ttðtÞ
ðttm ÞðtðtÞtm ÞL2 LT2 xðttm Þ
ttðtÞ
Z
ttm
yT ðsÞyðsÞ ds,
ttðtÞ
gðsÞ dwðsÞrxT ðttm ÞL2 LT2 xðttm Þ
ttm
T Z
ttm
gðsÞ dwðsÞ
gðsÞ dwðsÞ ttðtÞ
gðsÞy dwðsÞ ,
tt0
¼ xT ðtÞðtðtÞtm ÞI2 L2 LT2 I2T xðtÞ þ ttm
ttðtÞ
gðsÞy dwðsÞ tt0
ttðtÞ
2x ðttm ÞL2
T Z
ttðtÞ
yðsÞ dsrx
Z
ttðtÞ
þ
ttm
2x ðttm ÞL2
T
yT ðsÞyðsÞ ds, tt0
gðsÞ dwðsÞrxT ðttðtÞÞL1 LT1 xðttðtÞÞ
tt0
ttðtÞ
ttðtÞ
þ
ttðtÞ
2x ðttðtÞÞL1 Z þ
Z
ðtÞðt0 tðtÞÞI1 L1 LT1 I1T xðtÞ
Z
ttm
þ ttðtÞ
yT ðsÞyðsÞ ds
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
756
¼ xT ðtÞI2 L2 LT2 I2T xðtÞ þ 2xT ðtt0 ÞL3
Z
Z
T Z
ttm
ttm
gðsÞ dwðsÞ ttðtÞ
gðsÞ dwðsÞ ,
ttðtÞ
tt0
yðsÞ dsrxT ðtt0 ÞðtM t0 ÞL3 LT3 xðtt0 Þ þ ttM Z tt0 T yT ðsÞyðsÞ ds, ¼ x ðtÞðtM t0 ÞI3 L3 LT3 I3T xðtÞ þ
Z
tt0
yT ðsÞyðsÞ ds
ttM
ttM
2xT ðtt0 ÞL3 Z
Z
tt0
ttM
tt0
þ
gðsÞ dwðsÞr2xT ðtt0 ÞL3 LT3 xðtt0 Þ
T Z gðsÞ dwðsÞ
ttM
¼ xT ðtÞI3 L3 LT3 I3T xðtÞ þ
tt0
gðsÞ dwðsÞ
ttM tt0
Z
T Z
tt0
gðsÞ dwðsÞ ttM
gðsÞ dwðsÞ :
ð43Þ
ttM
Define G1 ¼ 2xT ðttðtÞÞL1 ðxðttðtÞÞ2xT ðttðtÞÞL1 xðtt0 Þ þ xT ðtÞðt0 tðtÞÞI1 L1 LT1 I1T xðtÞ Z ttðtÞ yT ðsÞyðsÞ ds þxT ðtÞI1 L1 LT1 I1T xðtÞ þ Z
tt0
T Z
ttðtÞ
ttðtÞ
gðsÞy dwðsÞ
þ tt0
gðsÞy dwðsÞ ,
tt0
G2 ¼ 2xT ðttm ÞL2 xðttm Þ2xT ðttm ÞL2 xðttðtÞÞxT ðtÞðtðtÞtm ÞI2 L2 LT2 I2T xðtÞ Z ttm yT ðsÞyðsÞ ds þxT ðtÞI2 L2 LT2 I2T xðtÞ þ Z
ttðtÞ ttm
T Z
ttm
gðsÞ dwðsÞ
þ ttðtÞ
gðsÞ dwðsÞ ,
ttðtÞ
G3 ¼ 2xT ðtt0 ÞL3 ðxðtt0 Þ2xT ðtt0 ÞL3 xðttM Þ þ xT ðtÞðtM t0 ÞI3 L3 LT3 I3T xðtÞ Z tt0 þxT ðtÞI3 L3 LT3 I3T xðtÞ þ yT ðsÞyðsÞ dsxT ðtÞ ttM
Z
tt0
þ
T Z gðsÞ dwðsÞ
ttM
tt0
tt0
E
ttm
ttðtÞ
ð44Þ
ttM
Noting that ( Z Z ttðtÞ E gðsÞT gðsÞ ds ¼ E Z
gðsÞ dwðsÞ :
ttðtÞ
T Z gðsÞ dwðsÞ
tt0
T
gðsÞ gðsÞ ds ¼ E
(Z
gðsÞ dwðsÞ tt0
T Z
ttm
ttm
gðsÞ dwðsÞ ttðtÞ
)
ttðtÞ
ttðtÞ
) gðsÞ dwðsÞ ,
,
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
E
Z
tt0
( Z gðsÞT gðsÞ ds ¼ E
ttM
T Z
tt0
)
tt0
gðsÞ dwðsÞ
757
gðsÞ dwðsÞ
ttM
ð45Þ
ttM
we can easily get ( EfdV ðxðtÞ,tÞg ¼ EfLV ðxðtÞ,tÞgrE
6 X
LVi ðxðtÞ,tÞ þ
i¼1
3 X
) Gi
¼ EfxðtÞT YxðtÞg,
ð46Þ
i¼1
where Y ¼ Y1 þ Y2 þ Y3 þ Y4 þ ðt0 tðtÞÞI1 L1 LT1 I1T þ I1 L1 LT1 I1T þðtðtÞtm ÞI2 L2 LT2 I2T þ I2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T
ð47Þ
and 2 6 6 6 6 6 Y1 ¼ 6 6 6 6 4
2
n
0 L2 þ LT2
0 0
0 0
^ Pða,mÞA 1 ða,mÞ L2
n
n
L3 þ LT3
n
n
n
L3 I
n
n
n
n
LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
D1 ða,mÞ
E T ða,mÞ
3
2
E T ða,mÞ
3T
7 7 6 6 0 0 7 7 6 6 7 7 6 6 7 7 6 6 0 0 7Pða,mÞ ^ 7 , 6 Y2 ¼ 6 7 7 6 6 0 0 7 7 6 6 7 7 6 T 6 T 4 E1 ða,mÞ 5 4 E1 ða,mÞ 5 0
2
E T ða,mÞ
3
3 ^ Pða,mÞB 1 ða,mÞ 7 7 0 7 7 0 7 7, 7 0 7 7 0 5 Z^
2
3T
7 7 6 6 0 0 7 7 6 6 7 7 6 6 7 7 6 6 0 0 7ðtM tm ÞI 6 7 , Y3 ¼ 6 7 7 6 6 0 0 7 7 6 6 7 7 6 T 6 T 4 E1 ða,mÞ 5 4 E1 ða,mÞ 5
0
3 3T 2 T AT ða,mÞ A ða,mÞ 7 7 6 6 0 0 7 7 6 6 7 7 6 6 7 7 6 6 0 0 7 7 6 6 Y4 ¼ 6 7ðtM tm ÞI 6 7 , 0 0 7 7 6 6 7 7 6 6 6 AT ða,mÞ 7 6 AT ða,mÞ 7 5 5 4 1 4 1 BT1 ða,mÞ BT1 ða,mÞ
0
2
þ Bða,mÞF ðmÞ
E T ða,mÞ
^ D1 ða,mÞ ¼ Pða,mÞ½Aða,mÞ
0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
758
^ þ þ½Aða,mÞ þ Bða,mÞF ðmÞT Pða,mÞ
X
^ þ d 2 Z^ þ 2I: lmj Pða,mÞ
ð48Þ
j2S
Applying Schur complement to Eqs. (32) and (33), we obtain Y1 þ Y2 þ Y3 þ Y4 þ ðt0 tm ÞI1 L1 LT1 I1T þ I1 L1 LT1 I1T þ I2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T o0,
ð49Þ
Y1 þ Y2 þ Y3 þ Y4 þ I1 L1 LT1 I1T þ ðt0 tm ÞI2 L2 LT2 I2T þ I2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T o0:
ð50Þ
By using Lemma 3, Eqs. (49) and (50) are equivalent to Yo0:
ð51Þ
Combining Eqs. (46) and (51), we can conclude that there exists a scalar c1 40, such that 2
EfLV ðxt ,t,rðtÞÞgoc1 9xðtÞ9 :
ð52Þ
Case II: when t 2 ½t0 ,tM . Similar to the proof of case I, when l¼ 3,4, it can be checked with the same Lyapunov– Krasovskii function candidate denoted in Eqs. (36)–(38) that there exists a scalar c2 40, such that 2
EfLV ðxt ,t,rðtÞÞgoc2 9xðtÞ9 :
ð53Þ
From Case I and Case II, it can be seen that if Eqs. (52) and (53) hold, 2 EfLV ðxt ,t,rðtÞÞgominfc1 ,c2 g9xðtÞ9 . By using Lyapunov stability theorem in [14] that the dynamics of the systems (17) and (18) is robustly stochastically stabilizable. The proof is completed. & Remark 1. In the existing methods [40–43] which deal with the interval-time delays, the term (Eq. (47)) Y ¼ Y1 þ Y2 þ Y3 þ Y4 þ ðt0 tðtÞÞI1 L1 LT1 I1T þ I1 L1 LT1 I1T þ ðtðtÞtm ÞI2 L2 LT2 I2T þI2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T o0 is usually enlarged as Y1 þ Y2 þ Y3 þ Y4 þ ðt0 tm ÞðI1 L1 LT1 I1T þ I2 L2 LT2 I2T Þ þI1 L1 LT1 I1T þ I2 L2 LT2 I2T þ ðtM t0 ÞI3 L3 LT3 I3T þ I3 L3 LT3 I3T o0, which may lead to considerable conservatism to the stability analysis. The idea of convexity is introduced by Lemma 3 can effectively deduce the possible conservatism of the achieved results. Remark 2. Note that Theorem 1 presents a new delay-interval-dependent stabilization criterion for SMJS with mixed time delays and polytopic uncertainties by using a new mode-dependent and parameter-dependent Lyapunov–Krasovskii functional in Eq. (38). On one hand, some weighting matrices (Lk,, k ¼ 1,2,y,6) are introduced to reduce the conservatism of the results, on the other hand, the decoupling matrices (G(m), m ¼ 1,2,y,N) are proposed to decouple the state feedback matrix B0 ða,rðtÞÞ and
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
759
Lyapunov–Krasovskii functional Pða,rðtÞÞ, which solve effectively the difficulty of state feedback controller design with the new parameter-dependent Lyapunov–Krasovskii functional. 3.2. Robust H1 control In this section, a sufficient condition for the solvability of the robust H1 control problem is proposed and an LMI approach for designing the desired state feedback controllers is developed. Theorem 2. Given a scalar g40, then the uncertain stochastic time-delay system in Eqs. (1)–(3) is robustly stochastically stabilizable with disturbance attenuation g, if there exist matrices Z, G(m), Y(m), Pði,mÞ, X ði,i,m,lÞ, X ði,j,m,lÞ, L1, L2, L3, with Z40, Pði,mÞ40, X ði,i,m,lÞ ¼ X ði,i,m,lÞT , for joi, i ¼ 1,2,y,r; m ¼ 1,2,y,N, l¼ 1,2,3,4, satisfying the following LMIs: 2 3 j^ 11 ði,i,lÞ j^ 12 ði,i,lÞ j^ 13 ði,i,lÞ 6 7 n ð54Þ j^ 22 ði,i,lÞ j^ 23 ði,i,lÞ ði,lÞ5oX ði,i,m,lÞ, 4 n n j^ 33 2 6 4
3 2 j^ 11 ðj,i,lÞ j^ 12 ðj,i,lÞ j^ 12 ði,j,lÞ j^ 13 ði,j,lÞ 7 6 n j^ 22 ði,j,lÞ j^ 23 ði,j,lÞ 5 þ 4 j^ 22 ðj,i,lÞ n n n j^ 33 ði,j,lÞ
j^ 11 ði,j,lÞ n n
3 j^ 13 ðj,i,lÞ T j^ 23 ðj,i,lÞ 7 5oX ði,j,m,lÞ þ X ði,j,m,lÞ , j^ 33 ðj,i,lÞ
ð55Þ ½X ði,j,m,lÞrr o0,
ð56Þ
where 2 6 6 6 6 6 6 j^ 11 ði,i,pÞ ¼ 6 6 6 6 6 4
Oði,i,mÞ
0 0 L3
A1 ði,mÞ L2 LT1
n
0 L2 þ LT2
n
n
0 0 L3 þ LT3
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
2
0 0
6 6 6 6 0 6 6 j^ 12 ði,i,1Þ ¼ j^ 12 ðj,i,1Þ ¼ 6 0 6 pffiffiffi 6 dL 1 6 6 4 0 0
0 0
0 L2 0 0
0 0 pffiffiffi dL 3 0
0 0
0 0 L3 0
L1 0
0 0
0 0
0 0
0
0
0
0
3 B1 ði,mÞPði,mÞ B2 ði,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ði,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
760
2
ZðPði,mÞÞ Pði,mÞE T ði,mÞ 6 6 0 0 6 6 0 0 6 6 0 0 j^ 13 ði,i,lÞ ¼ 6 6 T 6 ði,mÞ0 0 E 6 1 6 6 0 0 4
0 0
pffiffiffiffiffi T 2dE1 ði,mÞ 0
BT3 ði,mÞ
0
3 D 7 07 7 07 7 7 0 7, 7 07 7 7 07 5
pffiffiffiffiffi pffiffiffiffiffi 2dPði,mÞE T ði,mÞ 2dPði,mÞAT ði,mÞ 0 0
BT3 ði,mÞ
0 0
pffiffiffiffiffi T 2dA ði,mÞ pffiffiffiffiffi T1 2dB ði,mÞ pffiffiffiffiffi 1T 2dB2 ði,mÞ
0
j^ 22 ði,i,lÞ ¼ j^ 22 ði,j,lÞ ¼ j^ 22 ðj,i,lÞ ¼ diagfI,I,I,I,I,I,Ig, j^ 23 ði,i,lÞ ¼ j^ 23 ði,j,lÞ ¼ j^ 23 ðj,i,lÞ ¼ 0, j^ 33 ði,i,lÞ ¼ j^ 33 ðj,i,lÞ ¼ diagfZ0 ðPði,mÞÞ Pði,mÞ I I G T ðmÞ þ GðmÞg, 2
Oði,j,mÞ n n
n
0 0 L3 þ LT3
n
n
n
0 0 L3 I
n
n
n
n
ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
Oðj,i,mÞ
0
0
0
A1 ðj,mÞ
6 6 6 6 6 6 j^ 11 ði,j,pÞ ¼ 6 6 6 6 6 4
2
A1 ði,mÞ L2 LT1 0
0 L2 þ LT2
6 6 6 6 6 6 ^ j 11 ðj,i,pÞ ¼ 6 6 6 6 6 4
2
LT2
3 B1 ði,mÞPðj,mÞ B2 ði,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I
B1 ðj,mÞPði,mÞ B2 ðj,mÞ
n
L2 þ
n
n
0 L3 þ LT3
n
n
n
0 L3 I
n
n
n
n
L2 LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
0 0 0 0 Z
n
n
n
n
n
n
0
6 6 0 6 6 0 6 6 j^ 12 ðj,i,1Þ ¼ 6 0 6 pffiffiffi 6 dL 1 6 6 4 0 0
0
0
0
0
0
L2
0
0 0
0 0
0 pffiffiffi dL 3 0
L3 0
L1 0
0 0
0 0
0 0
0
0
0
0
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ðj,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
0 0 0 0 0 g2 I
3 7 7 7 7 7 7 7, 7 7 7 7 5
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
ZðPðj,mÞÞ Pðj,mÞE T ði,mÞ 6 6 0 0 6 6 0 0 6 6 0 0 j^ 13 ði,j,lÞ ¼ 6 6 T 6 ði,mÞ 0 E 6 1 6 6 0 0 4
2 6 6 6 6 6 6 j^ 13 ðj,i,lÞ ¼ 6 6 6 6 6 6 4
761
pffiffiffiffiffi 2dPðj,mÞE T ði,mÞ 0
pffiffiffiffiffi 2dPðj,mÞAT ði,mÞ 0
0 0
0 0
pffiffiffiffiffi T 2dE1 ði,mÞ
0
BT3 ði,mÞ
BT3 ði,mÞ
pffiffiffiffiffi T 2dA ði,mÞ pffiffiffiffiffi T1 2dB ði,mÞ pffiffiffiffiffi 1T 2dB2 ði,mÞ
ZðPði,mÞÞ
Pði,mÞE T ðj,mÞ
pffiffiffiffiffi 2dPði,mÞE T ðj,mÞ
pffiffiffiffiffi 2dPði,mÞAT ðj,mÞ
0 0
0 0
0 0
0 0
0 0
0 E1T ðj,mÞ
0 pffiffiffiffiffi T 2dE1 ðj,mÞ
0 pffiffiffiffiffi T 2dA1 ðj,mÞ pffiffiffiffiffi T 2dB ðj,mÞ pffiffiffiffiffi 1T 2dB2 ðj,mÞ
0
0
0
0
0
BT3 ðj,mÞ
BT3 ðj,mÞ
j^ 33 ði,j,lÞ ¼ diagfZ0 ðPðj,mÞÞ Pðj,mÞ I I G T ðmÞ þ GðmÞg, 2
0 6 pffiffiffi 6 dL2 6 6 0 6 6 j^ 12 ði,i,2Þ ¼ 6 0 6 6 0 6 6 4 0 0 2
0 6 pffiffiffi 6 dL2 6 6 0 6 6 j^ 12 ðj,i,2Þ ¼ 6 0 6 6 0 6 6 4 0 0
0
0
0
0
pffiffiffi 2Pði,mÞ
Pði,mÞC T ði,mÞ
0
L2
0
0
0
0 0
0 0
0 pffiffiffi dL 3 0
L3 0
0 0
0 0
L1 0
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0 0
L2 0
0 pffiffiffi dL3
0 L3
0 L1
0 0
0 0
0 0
0
0
0
0
0
0
0
0
3 7 7 7 7 7 7 7, 7 7 7 7 5
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ðj,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
3 D 7 07 7 07 7 7 0 7, 7 07 7 7 07 5 0
D
3
7 07 7 07 7 7 0 7, 7 07 7 7 07 5 0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
762
2
0 6 pffiffiffi 6 dL 2 6 6 0 6 6 j^ 12 ði,j,2Þ ¼ 6 0 6 6 0 6 6 4 0 0 2
0
0
pffiffiffi 2Pðj,mÞ
Pðj,mÞC T ði,mÞ
0 0
L2 0
0 pffiffiffi dL 3
0 L3
0 0
0 0
0 L1
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
n n
n
0 0 L3 þ LT3
n
n
n
0 0 L3 I
n
n
n
n
A1 ði,mÞ L2 LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
Oði,j,mÞ n
0 L2 þ LT2
n
n
0 0 L3 þ LT3
0 0 L3
A1 ði,mÞ L2 LT1
n
n
n
I
n
n
n
n
0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
Oðj,i,mÞ n
0 L2 þ LT2
n
n
0 0 L3 þ LT3
n
n
n
0 0 L3 I
n
n
n
n
A1 ðj,mÞ L2 LT1 0 ð1tÞI þ L1 þ LT1
n
n
n
n
n
n
n
n
n
n
6 6 6 6 6 6 j^ 11 ði,j,3Þ ¼ 6 6 6 6 6 4
2
0
0 L2 þ LT2
Oði,i,mÞ
6 6 6 6 6 6 j^ 11 ði,i,3Þ ¼ 6 6 6 6 6 4
2
0
6 6 6 6 6 6 j^ 11 ðj,i,3Þ ¼ 6 6 6 6 6 4
2
0 0
6 6 6 6 0 6 6 j^ 12 ði,i,3Þ ¼ 6 0 6 pffiffiffi 6 dL4 6 6 4 0 0
0 0
0 0
0 pffiffiffi dL6
0 L6
0 0
L5 0
0 0
0 0
L4
0
0
0
0 0
0 0
0 0
0 0
3 7 7 7 7 7 7 7, 7 7 7 7 5
3 B1 ði,mÞPði,mÞ B2 ði,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I 3 B1 ði,mÞPðj,mÞ B2 ði,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I 3 B1 ðj,mÞPði,mÞ B2 ðj,mÞ 0 0 7 7 7 0 0 7 7 0 0 7 7, 7 0 0 7 7 Z 0 7 5 n g2 I
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ði,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
2
0
2
0 L5
L6 0
0 L4
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 pffiffiffi dL6
L6
0 0
L5 0
0 0
0 0
L4 0
0 0
0 0
0 0
0
0
0
0
0
0 Pði,mÞC T ði,mÞ
6 6 0 6 6 0 6 6 j^ 12 ði,j,3Þ ¼ 6 0 6 pffiffiffi 6 dL4 6 6 4 0 0 2
0
0
pffiffiffi 3 2Pðj,mÞ Pðj,mÞC T ði,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0
0
0 L5
0 pffiffiffi dL 6 0
L6 0
0 0
0 0
0 L4
0 0
0 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 pffiffiffi dL6
L6
0 0
L5 0
0 0
0 0
L4 0
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0 0
0 L5
0 pffiffiffi dL6 0
L6 0
0 L4
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
6 6 0 6 pffiffiffi 6 dL5 6 6 j^ 12 ðj,i,4Þ ¼ 6 0 6 6 0 6 6 4 0 2
0 0
0
pffiffiffi 2Pði,mÞ
6 6 0 6 pffiffiffi 6 dL5 6 6 j^ 12 ði,i,4Þ ¼ 6 0 6 6 0 6 6 4 0 2
0
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ðj,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0
0 pffiffiffi dL6 0
6 6 0 6 6 0 6 6 j^ 12 ðj,i,3Þ ¼ 6 0 6 pffiffiffi 6 dL4 6 6 4 0
0
6 6 0 6 pffiffiffi 6 dL5 6 6 j^ 12 ði,j,4Þ ¼ 6 0 6 6 0 6 6 4 0 0
0
0
0 0
763
3 7 7 7 7 7 7 7, 7 7 7 7 5
pffiffiffi 3 2Pði,mÞ Pði,mÞC T ðj,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
pffiffiffi 3 2Pðj,mÞ Pðj,mÞC T ði,mÞ 7 0 0 7 7 7 0 0 7 7 0 0 7, 7 7 0 0 7 7 5 0 0 0
0
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
764
p ¼ 1; 2, l ¼ 1; 2,3; 4
ð57Þ
and D, Oði,i,mÞ, Oði,j,mÞ, Oðj,i,mÞ, Z(P(i,m)), Z(P(j,m)), Z0 ðPði,mÞÞ, Z0 ðPðj,mÞÞ are given in Eq. (22). In this case, a suitable stabilizing state feedback controller can be chosen by uðtÞ ¼ F ðrðtÞÞxðtÞ,
F ðrðtÞ ¼ mÞ ¼ Y ðmÞGðmÞ1 :
ð58Þ
Proof. For t40, we set Z t T T 2 JðtÞ ¼ E ½zðsÞ zðsÞg vðsÞ vðsÞ ds:
ð59Þ
0
Then, it is easy to get that Z t T T 2 ½zðsÞ zðsÞg vðsÞ vðsÞ þ LV ðxs ,s,rðsÞÞ ds JðtÞ ¼ E 0
Z
t
½zðsÞ zðsÞg vðsÞ vðsÞ þ LV ðxs ,s,rðsÞÞ ds, T
EfV ðxt ,t,rðtÞÞgrE
2
T
ð60Þ
0
where V ðxt ,t,rðtÞÞ is defined in Eq. (38). From Eqs. (54)–(56), similar to the proof of Theorem 1, we can easily deduce that there exist scalar c1 40 and c2 40 such that 2
zðtÞT zðtÞg2 vðtÞT vðtÞ þ LV ðxt ,t,rðtÞÞrzðtÞT Y1 zðtÞoc1 9xðtÞ9 ,
ð61Þ
2
zðtÞT zðtÞg2 vðtÞT vðtÞ þ LV ðxt ,t,rðtÞÞrzðtÞT Y2 zðtÞoc2 9xðtÞ9 , where
"
zðtÞT ¼ xT ðtÞ xT ðttm Þ xT ðtt0 Þ xT ðttM Þ xT ðttðtÞÞ
Z
ð62Þ T
t
xðsÞ ds
# vT ðtÞ ,
tdðtÞ
and Y1 and Y2 are omitted. Then, it follows from Eqs. (59) and (62) that 2
JðtÞ ¼ EfLV ðxt ,t,rðtÞÞgominfc1 ,c2 g9xðtÞ9 o0: This implies that for any nonzero vðtÞ 2 L2 ½0,1, JzJE2 ogJvJ2 : Therefore, by Definition 2, the uncertain stochastic Markovian jumping system in Eqs. (1)–(3) is robustly stochastically stable with disturbance attenuation level g. This completes the proof. & Remark 3. The results of Theorems 1 and 2 can be easily extended to the cases to deal with the problem of robust stabilization and robust H1 control in which there are no stochastic terms and/or there are no distributed time delays.
4. Illustrative example In this section, we provide an example to demonstrate the effectiveness of the proposed method.
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
765
Consider uncertain stochastic Markovian jumping systems in (1)–(3) with parameters as follows: Mode 1 3:3234 1:2351 1:2845 1:3245 Að1; 1Þ ¼ , Að2; 1Þ ¼ , 1:2234 1:2253 1:2354 3:2834 1:3234 2:2383 A1 ð1; 1Þ ¼ , 1:2835 1:2235 1:2893 2:2834 1:2934 1:7383 A1 ð2; 1Þ ¼ , B0 ð1; 1Þ ¼ , 1:2834 1:2383 2:2345 1:8345 1:2834 1:3923 B0 ð2; 1Þ ¼ , 1:2893 3:2322 1:2333 2:4560 1:3846 1:8346 B1 ð1; 1Þ ¼ , B1 ð2; 1Þ ¼ , 2:2845 1:2894 2:2389 1:2234 3:3343 7:4785 B2 ð1; 1Þ ¼ , 8:2245 5:5643 3:5751 3:2346 1:3464 1:2394 B2 ð2; 1Þ ¼ , Eð1; 1Þ ¼ , 1:2322 1:3345 2:4396 3:5896 1:3853 2:2895 Eð2; 1Þ ¼ , 1:3956 1:2389 2:3525 0:3764 3:4546 0:2367 E1 ð1; 1Þ ¼ , E1 ð2; 1Þ ¼ , 0:2563 0:1235 0:2356 2:3456 1:8237 2:2355 B3 ð1; 1Þ ¼ , 3:2353 2:4334 3:4557 3:2466 2:3533 2:2846 B3 ð2; 1Þ ¼ , Cð1; 1Þ ¼ , 2:3346 1:3456 2:9567 0:6393 0:1243 2:1234 Cð2; 1Þ ¼ , 2:1938 0:1234 a1 ¼ a2 ¼ 0:5,
l11 ¼ 0:5,
l12 ¼ 0:5:
Mode 2
1:8434 1:4167 1:3946 5:2345 , Að2; 2Þ ¼ , 2:2543 1:4563 1:3034 1:6424 2:3254 2:4345 , A1 ð1; 2Þ ¼ 0:2935 2:3235 4:2934 2:2034 3:2934 1:6213 A1 ð2; 2Þ ¼ , B0 ð1; 2Þ ¼ , 1:1234 0:2934 2:9383 2:1332
Að1; 2Þ ¼
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
766
B0 ð2; 2Þ ¼
0:3044
1:0383
1:3345 1:2342
,
1:8373 2:2832 1:4272 , B1 ð2; 2Þ ¼ 0:2837 1:3242 2:9323 2:1292 1:2345 B2 ð1; 2Þ ¼ , 2:2235 0:5343 3:2346 1:2226 1:3235 B2 ð2; 2Þ ¼ , Eð1; 2Þ ¼ 1:8341 1:2321 2:4234 0:3323 2:2539 Eð2; 2Þ ¼ , 1:8956 1:2355 2:2032 0:2392 3:8333 E1 ð1; 2Þ ¼ , E1 ð2; 2Þ ¼ 1:2532 0:3345 0:9383 1:8237 2:2832 B3 ð1; 2Þ ¼ , 3:2353 2:4334 3:9327 1:6263 2:2134 B3 ð2; 2Þ ¼ , Cð1; 2Þ ¼ 1:3637 1:4645 2:5332 1:4942 2:1234 Cð2; 2Þ ¼ , 1:1293 0:1341 B1 ð1; 2Þ ¼
a1 ¼ a2 ¼ 0:5,
l21 ¼ 0:2,
1:2345 , 4:9383
1:2324
2:3234
1:1832 2:2343
,
,
1:4341 , 1:3242
l22 ¼ 0:2:
And t ¼ 0:2,
tm ¼ 0:1,
tM ¼ 2,
d ¼ 0:3,
g ¼ 2:
Then, using the Matlab LMI Control Toolbox to solve the LMIs in (54)–(56), we obtain a set of solutions as follows: 2:0492 0:0537 2:5371 0:0221 , Pð1; 2Þ ¼ 106 n Pð1; 1Þ ¼ , 0:0537 1:0997 0:0221 3:3059 0:2530 0:1622 Pð2; 1Þ ¼ , 0:1622 0:2762 2:5865 0:0530 5:6202 3:2910 Pð2; 2Þ ¼ 106 n , L1 ¼ , 0:0530 2:4773 3:0809 0:7932 0:9944 0:8644 L2 ¼ , 0:8373 0:7302 3:3187 0:1050 2:0814 1:8051 L3 ¼ , L4 ¼ , 0:2350 3:6801 0:1707 3:4377 11:5580 0:9557 L5 ¼ , 0:4495 10:7719
J. Xia et al. / Journal of the Franklin Institute 349 (2012) 741–769
L6 ¼
3:6235
0:0246
,
Y ð1Þ ¼
0:0977 3:9628 2:6929 3:7166 Y ð2Þ ¼ , 14:2998 1:5234 25:6535 3:2543 Gð1Þ ¼ , 4:6243 2:3400 23:7880 19:6530 Z¼ : 19:6530 29:6351
19:1578
4:4426
5:4868
2:2321
Gð2Þ ¼
73:6063 24:9279
767
,
10:9179 , 4:7678
Then, the desired state feedback controller can be chosen as 0:5399 1:1477 , F ð1Þ ¼ Y ð1ÞG 1 ð1Þ ¼ 0:5149 1:6700 1:0131 3:0993 1 F ð2Þ ¼ Y ð2ÞG ð2Þ ¼ : 0:3834 0:5584 5. Conclusion The problems of robust stochastic stabilization and robust stochastic H1 control for a class of uncertain stochastic Markovian jumping systems with mixed delays have been considered in this paper. Based on the decoupling method, some improved delay-intervaldependent criteria have been proposed. These stability conditions have been obtained in terms of LMIs, which can be easily solved by using standard software. A numerical example has been provided to demonstrate the effectiveness of our results. Acknowledgment This work was supported National Natural Science Foundation of China under Grant 61004046, China Postdoctoral Science Foundation under Grant 20110491336, Postdoctoral Science Foundation of Jiangsu Province under Grant 1001007C, Young and Middle-Aged Scientists Research Awards Fund of Shandong Province under Grant 2009BSB01450. References [1] H. Gao, C. Wang, J. Wang, On H1 performance analysis for continuous-time stochastic systems with polytopic uncertainties, Circuits, Systems, and Signal Processing 24 (2005) 415–429. [2] H. Gao, J. Lam, C. Wang, Robust energy-to-peak filter design for stochastic time-delay systems, Systems and Control Letters 55 (2006) 101–111. [3] H. Gao, J. Lam, Z. Wang, Discrete bilinear stochastic systems with time-varying delay: stability analysis and control synthesis, Chaos, Solitons & Fractals 34 (2007) 394–404. [4] V.B. Kolmanovskii, A.D. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1992. [5] X. Mao, N. Koroleva, A. Rodkina, Robust stability of uncertain stochastic differential delay equations, Systems and Control Letters 35 (1998) 325–336. [6] H. Zhang, H. Yan, Q. Chen, Stability and dissipative analysis for a class of stochastic system with time-delay, Journal of the Franklin Institute 347 (5) (2010) 882–893.
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