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### Documents

INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES

Volume 3, Number 2, Pages 231242

c 2007 Institute for Scientic Computing and Information

STOCHASTIC STABILITY, PASSIVITY ANALYSIS AND SYNTHESIS OF MARKOVIAN JUMP SYSTEMS OF NEUTRAL TYPE

XUYANG LOU AND BAOTONG CUI Abstract. Stochastic stability, passivity analysis and synthesis problems for Markovian jump systems of neutral type with state time-varying delay are investigated. Based on the denition of passivity extended from deterministic systems, in terms of a linear matrix inequality, a sucient condition on passivity for the solvability of the above problem is proposed. The passive controller via memoryless state-feedback is given to guarantee the stability and passivity of appropriate closed-loop system. Two examples are carried out to illustrate the sucient conditions. Key Words. passive control, Markovian jump system, linear matrix inequality, memoryless state-feedback.

1. Introduction Passivity plays an important role in system and control theory, and also arises naturally in many areas of science and engineering, see e.g. [2]. Extensive literatures deal with the analysis and synthesis related to positive real system, see [3, 29-30] and the references therein. The passivity theory intimately related to the circuit analysis methods [4] has received a lot of attention from the control community since 1970s [14-15, 26]. In addition, the passivity theory is benecial to analyze the stability of systems, and has found applications in diverse areas such as stability [15, 17-19], complexity like in [6], signal processing [32], chaos control and synchronization [31, 34], and fuzzy control [5]. However, the theory of passivity more generally studied for about three decades is from a deterministic setting and has rarely been extended to Markovian jump systems [1]. On the other hand, it should be argued that many physical systems are stochastic in nature and so the extension of the above theory to the stochastic case is long overdue. However, in this initial attempts we focus on the class of stochastic systems with Markov jump disturbances. Markovian jump systems (MJS) introduced in [16], are the hybrid systems with two components in the state. The rst one refers to the mode which is described by a continuous-time nite-state Markovian process, and the second one refers to the state which is represented by a system of dierential equations. Recently, many researchers have made a lot of progress in Markovian jump control theory [7-11, 20-25, 28, 33] and references therein. However, to the best of our knowledge, few authors study the passivity of Markovian jump systems of neutral type (MJSNT).

Received by the editors January 1, 2004 and, in revised form, March 22, 2004. 2000 Mathematics Subject Classication. 34K20, 34D20. This research was supported by the National Natural Science Foundation of China (No.10371072) and the Science Foundation of Southern Yangtze University (No.103000-21050323).

#### X. Y. LOU AND B. T. CUI

This paper contributes to the development of the passivity of MJSNT and extend the results in [20, 26], and address our study under a special kind of stochastic system setting named Markovian jump system of neutral type. Based on the denition of passivity under state-space representation for a class of MJSNT, the passivity condition is rstly obtained using stochastic Lyapunov functionals technique. The condition also shows that the passivity of MJSNT guarantees the stability. Assume that the mode of MJSNT is available, a kind of mode-dependent controller is proposed via state feedback to be ensured the passivity of the resulting closed-loop system. The layout of this paper is as follows. In Section 2, the problem to be studied is stated and some preliminaries are presented. Based on the Lyapunov stability theory and the linear matrix inequality (LMI) approach, the stochastic stability, passivity analysis and synthesis for the MJSNT are obtained in Section 3. Section 4 focuses on the passive control problems for the MJSNT. Numerical examples are presented in Section 5, and some conclusions are drawn in Section 6. The notations in this paper are quite standard. Rn and Rmn denote, respectively, the n-dimensional Euclidean space, and the set of all m n real matrices. E[] stands for the mathematical expectation. AT and A1 denote the transpose and the inverse of any (square) matrix A. A > 0 (or A < 0) denotes a positive(negative-) denite matrix. diag[] denotes a block diagonal matrix. The symbol within a matrix represents the symmetric term of the matrix. 2. System Description and Preliminaries In this paper, we consider the Markovian jump systems of neutral type described by the following stochastic dierential equations: x(t) = A(t )x(t) + Ad (t )x(t d(t)) +Ah (t )x(t h(t)) + Bw (t )w(t) +Bu (t )u(t), (1) z(t) = C(t )x(t) + Cd (t )x(t d(t)) +Dw (t )w(t) + Du (t )u(t), x(t) = 0 , t [, 0], where x(t) Rn is the state vector, d(t) > 0, h(t) > 0 are the unknown time varying delays satisfying 0 d(t) d+ , 0 h(t) h+ , = sup max[d(t), h(t)] > 0, where the bounds , d+ < 1, h+ < 1 are known constants, u(t) Rm is the control input vector, z(t) Rq is the performance output, w(t) Rq is the exogenous input. The initial vector 0 is a given continuous function on [, 0]. A(t ), Ad (t ), Ah (t ), Bw (t ), Bu (t ), C(t ), Cd (t ), Dw (t ), Du (t ) are mode-dependent matrices with appropriate dimensions, and represents a continuous-time discrete state Markov process; these coecients are given matrices for each value of t in a nite set S = {1, 2, , N }. Given a probability space (, , P), where is the sample space, is the algebra of events and P is the probability measure dened on. Let the random form process {t , t [0, +)} be a homogeneous, nite-state Markovian process with right continuous trajectories with generator = (ij ) and transition probability from mode i at time t to mode j at time t + , i, j S : (2) pij = ij + o(), 1 + ij + o(), if i = j, if i = j,

STOCHASTIC STABILITY, PASSIVITY ANALYSIS AND SYNTHESIS OF MJSNT N

with transition probability rates ij 0 for i, j S, i = j and ii = j=1,j=i ij , where > 0 and lim o()/ = 0. Note that the set S comprises the various operational modes of the system under study. In the sequel, for simplicity, while t = i, the matrices A(t ), Ad (t ), Ah (t ), Bw (t ), Bu (t ), C(t ), Cd (t ), Dw (t ), Du (t ) are represented by Ai , Adi , Ahi , Bwi , Bui , Ci , Cdi , Dwi , Dui. Some denitions, assumptions and lemmas used in this paper are listed as follows. Assumption 1. The mode t is available at time t. Denition 1. The Markovian jump system (1) of neutral type with w(t) 0 and u(t) 0, is said to be stochastically stable, if for all initial state 0 and mode 0 , the following relation holds,

#### x(t, 0 , 0 )

dt 0 , 0

From [13] and above denition, it follows that Denition 2. The Markovian jump system (1) of neutral type with u(t) 0, is said to be strictly passive, if for every T > 0 in zero initial condition 0 = 0, it satises

#### wT (t)z(t)dt > 0.

Furthermore, it is said to strictly passive (SP) with dissipation r if

(wT (t)z(t) rwT (t)w(t))dt > 0.

Remark 1. If the notation > is replaced by , above strictly passive denition is referred to passive denition. In this paper we focus on strictly passive problems, and often the strictly passivity is referred to as passivity wherever no confusion arises. T Assumption 2. Dwi + Dwi > 0, i = 1, , N. Lemma 1 (Schur complement). Given constant matrices 1 , 2 , 3 , where 1 = T and 0 < 2 = T then 1 + T < 0 if and only if T < 0 or 2 T 1 < 0.

3. Asymptotical Stability and Passivity In this section, we consider the problem of the asymptotical stability of the unforced disturbance-free Markovian jump system of neutral type (w(t) 0 and u(t) 0) and the passivity of the unforced Markovian jump system of neutral type (u(t) 0). Theorem 1. System (1) with w(t) 0 and u(t) 0 is stochastically stable, if there exist positive constants , , a set of positive-denite symmetric matrices Pi , i = 1, 2, , N and positive-denite symmetric matrices Qdi , Qhi such that (1, 1) Pi Adi Pi Ahi AT i (2, 2) 0 AT di < 0, i = 1, , N, (6) (3, 3) AT hi (4, 4)

where (1, 1) = AT Pi + Pi Ai + i

#### (2, 2) = (1 d+ )Qdi

ij Pj + Qdi + ij Qdj , ij Qhj ,

#### ij Qdj ,

(3, 3) = (1 h+ )Qhi (4, 4) =

#### j=1 N j=1

ij Qhj + Qhi

Proof. For mode i S, lets construct the following Lyapunov functional: V (x(t), t, i) (7) = xT (t)Pi x(t) +

#### t t td(t)

xT ()Qdi x()d

#### xT (s)Qhi x(s)ds.

The weak innitesimal operator L[] of the process {x(t), i, t 0} for the system (1) at the point {x(t), t, i} is given by [13] LV (x(t), t, i) = V (x(t), t, i) V (x(t), t, i) + xT (t) t x

#### ij V (x(t), t, i, j).

According to the Its rule, along the solution of the system (1) with w(t) 0 o and u(t) 0, using the similar method in Theorem 3.1 (see, [20]) the innitesimal generator is LV (x(t), t, i) = xT (t)Pi x(t) + xT (t)Pi x(t)

#### +xT (t)

j=1 N t

ij Pj x(t) xT ()Qdj x()d xT (s)Qhj x(s)ds

#### td(t) t

+x (t)Qdi x(t) + xT (t)Qhi x(t) T (1 d(t))x (t d(t))Qdi x(t d(t)) (1 h(t))xT (t h(t))Qhi x(t h(t)). From integral mean value theorem, there exists > 0 such that

#### j=1 N td(t)

xT ()Qdj x()d

#### xT (t)

ij Qdj x(t) xT (t d(t))

#### ij Qdj x(t d(t)).

STOCHASTIC STABILITY, PASSIVITY ANALYSIS AND SYNTHESIS OF MJSNT

Similarly, there exists > 0 such that

#### j=1 N th(t)

xT ()Qhj x()d

#### xT (t)

ij Qhj x(t) xT (t h(t))

#### ij Qhj x(t h(t)).

Thus we obtain LV (x(t), t, i) xT (t) AT Pi + Pi Ai x(t) i

#### ij Pj x(t)

+2xT (t)Pi Adi x(t d(t)) +2xT (t)Pi Ahi x(t h(t))

#### +xT (t)

ij Qdj x(t)

#### xT (t d(t))

ij Qdj x(t d(t))

#### xT (t h(t))

ij Qhj x(t h(t))

+xT (t)Qdi x(t) (1 d+ )xT (t d(t)) Qdi x(t d(t)) + Ai x(t) +Adi x(t d(t)) + Ahi x(t h(t))

#### Ai x(t)

+Adi x(t d(t)) + Ahi x(t h(t)) (1 h+ )xT (t h(t))Qhi x(t h(t)) (8) = T (t)i (t), where T (t) = xT (t) xT (t d(t)) xT (t h(t)) , (1, 1) Pi Adi Pi Ahi (2, 2) 0 i = (3, 3) T Ai N Ai Adi ij Qhj + Qhi + AT di j=1 AT hi (1, 1) = AT Pi + Pi Ai + i

#### ij Pj + Qdi +

236 N j=1 N j=1

(2, 2) = (1 d+ )Qdi (3, 3) = (1 h+ )Qhi Since

#### ij Qdj , ij Qhj.

ij Qhj + Qhi > 0, using Lemma 1, i < 0 implies (6) holds and

LV (x(t), t, i) < 0. On the other hand, since V (x(t), t, i) > 0 for x(t) = 0 and V (x(t), t, i) = 0 for x(t) = 0. On the basis of Denition 1 from LV (x(t), t, i) < 0 and V (x(t), t, i) > 0 for x(t) = 0, it follows that the free Markovian jump system (1) of neutral type with w(t) 0 and u(t) 0, is stochastically stable. Theorem 2. For real number r 0, system (1) with u(t) 0 is strictly passive with dissipation r, if there exist a set of positive-denite symmetric matrices Pi , i = 1, , N and positive-denite symmetric matrices Qdi , Qhi such that

#### (9)

(1, 1)

T Pi Bwi Ci T 2rI Dwi Dwi

#### Pi Adi Cdi (3, 3)

Pi Ahi (4, 4)

AT i T Bwi AT di AT hi (5, 5)

#### < 0,

(3, 3) = (1 d+ )Qdi

(4, 4) = (1 h+ )Qhi (5, 5) =

Proof. For mode i S, lets construct the following Lyapunov functional:

#### V (x(t), t, i) (10)

= xT (t)Pi x(t) +

#### t td(t)

According to the Its rule, along the solution of the system (1) with u(t) 0, the o innitesimal generator is LV (x(t), t, i) 2wT (t)z(t) + 2rwT (t)w(t)

= xT (t)Pi x(t) + xT (t)Pi x(t) + xT (t)

xT ()Qdj x()d xT (s)Qhj x(s)ds

+xT (t)Qdi x(t) + xT (t)Qhi x(t) T (1 d(t))x (t d(t))Qdi x(t d(t)) (1 h(t))xT (t h(t))Qhi x(t h(t)) 2wT (t)Ci x(t) 2wT (t)Cdi x(t d(t)) 2wT (t)Dwi w(t) + 2rwT (t)w(t)

#### x (t)

AT Pi i

#### + Pi Ai x(t) + x (t)

+2xT (t)Pi Adi x(t d(t)) + 2xT (t)Pi Ahi x(t h(t))

#### x (t d(t))

+2xT (t)Pi Bwi w(t) + xT (t)Qdi x(t) (1 d+ )xT (t d(t))Qdi x(t d(t)) + Ai x(t) + Adi x(t d(t)) + Ahi x(t h(t)) +Bwi w(t)

+Adi x(t d(t)) + Ahi x(t h(t)) + Bwi w(t) (1 h+ )xT (t h(t))Qh x(t h(t)) 2wT (t)Ci x(t) 2wT (t)Cdi x(t d(t)) (11)

2wT (t)Dwi w(t) + 2rwT (t)w(t).

In terms of the augmented state vector (t) = xT (t) wT (t) xT (t d(t)) xT (t h(t)) , we express LV (x(t), t, i) 2wT (t)z(t) + 2rwT (t)w(t) T (t)i (t),

#### where i =

AT i N BT + wi ij Qhj + Qhi T Adi j=1 T Ahi (1, 1) = AT Pi + Pi Ai + i

#### N j=1 N

Pi Ahi 0 0 (4, 4) Bwi

#### . where

ij Pj + Qdi + ij Qdj , ij Qhj.

(4, 4) = (1 h+ )Qhi Since

ij Qhj + Qhi > 0, using Lemma 1, i < 0 implies (9) holds and

LV (x(t), t, i) 2wT (t)z(t) + 2rwT (t)w(t) < 0, and from which it follows that

(wT (t)z(t) rwT (t)w(t))dt

1 V (x(T ), T, i) V (x(0), 0, i). 2

Since V (x(t), t, i) > 0 for x(t) = 0 and V (x(t), t, i) = 0 for x(t) = 0. It follows from Denition 2 that as T that system (1) with u(t) 0 is SP. 4. Passive Control Synthesis In this section, we consider passive synthesis problems for a class of Markovian jump system of neutral type. Recalling Theorem 2, it is easy to get following proposition, which gives the sucient condition for the existence of a mode-dependent feedback controller such that the resulting closed-loop system is SP. Proposition 1. Given a Markovian jump system (1) of neutral type and real number r 0, there exists a mode-dependent feedback controller such that the resulting closed-loop system is SP with dissipation r, if there exist a set of Pi , i = 1, , N and Qdi , Qhi satisfying condition (9). Denition 3. The mode-dependent feedback controller introduced in Proposition 1, is said to be a mode-dependent passive controller with dissipation r, or simply to be a passive controller. In the rest of this section our main goal is to design the passive controller via state-feedback while mode t = i, (13) u(t) = Ki x(t),

the resulting closed-loop system is then obtained x(t) = (Ai + Bui Ki )x(t) + Adi x(t d(t)) +Ahi x(t h(t)) + Bwi w(t), z(t) = (Ci + Dui Ki )x(t) + Cdi x(t d(t)) (14) +Dwi w(t), x(t) = 0 , t [, 0],

Theorem 3. For the system (14) and real number r 0 there exist a set of Qdi and Qhi satisfying condition (9), if and only if there exist a set of positive-denite symmetric matrices Xi Rnn , Yi Rmn and satisfying (1, 1) (1, 2) Adi Xi Ahi Xi AT Ri (X) i T (2, 2) Cdi 0 Bwi 0 T (3, 3) 0 Adi 0 < 0, (15) T (4, 4) Ahi 0 (5, 5) 0 Si (X) where T (1, 1) = Xi AT + Ai Xi + Bui Yi + YiT Bui i ij Qdj Xi , j=1 T T Bwi Xi Ci YiT Dui T 2rI Dwi Dwi N (1 d+ )Qdi ij Qdj , j=1 N (1 h+ )Qhi ij Qhj , j=1 N 1

#### +ii Xi + Xi

(1, 2) = (2, 2) = (3, 3) = (4, 4) =

#### (5, 5) = Ri (X) =

ij Qhj + Qhi

#### (i1)i Xi

(i+1)i Xi

#### N i Xi ,

Si (X) = diag{ X1 Xi1 Xi+1 XN }. Furthermore the mode-dependent gain matrix of the state-feedback passive con1 troller (13) can be obtained by Ki = Yi Xi. Then the system (1) is passive by the state-feedback passive controller (13) while mode t = i. Proof. For mode i S, lets construct the same Lyapunov functional (10) in the proof of Theorem 2. Then similar to the proof of Theorem 2, we can easily obtain that if the following LMI (1, 1) (1, 2) Pi Adi Pi Ahi AT i T (2, 2) Cdi 0 Bwi (3, 3) 0 AT < 0 (16) di (4, 4) AT hi (5, 5) holds, where (1, 1) = (Ai + Bui Ki )T Pi + Pi (Ai + Bui Ki ) +

#### ij Pj +

ij Qdj + Qdi ,

(1, 2) = Pi Bwi (Ci + Dui Ki )T , T (2, 2) = 2rI Dwi Dwi , (3, 3) = (1 d+ )Qdi

#### 240 N j=1

(4, 4) = (1 d+ )Qhi (5, 5) =

#### ij Qhj ,

System (14) is said to be strictly passive with dissipation r, i.e., the system (1) is passive by the state-feedback passive controller (13) while mode t = i. Now if we take Pi = Xi , and Ki = Yi Xi , it is easy to follow that (16) is equivalent to (15). The proof is completed. 5. Examples In this section, we present two numerical examples to illustrate our results. Example 1. We consider system (1) with n = 2. For the two operating conditions (modes), the associated data are: A(1) = Ad (1) = Ah (1) = 0.4 0.2 0.3 0.2 0.5 0.4 0.2 0.3 , A(2) = , Ad (2) = , Ah (2) = , , ,

0.5 0.3 0.4 0.2 0.3 0.4 0.5 0.2

Let r = 0.5, = 0.9, d+ = h+ = 0.5. In order to solve the solution simply, for i = 1, 2, we select Qdi = diag(1, 1), Qhi = diag(2, 2), = = 4. So using MATLAB LMI Toolbox we solve the LMI condition (6) and obtain that P (1) = P (2) = 7.6486 5.7558 5.7558 16.7135 > 0,

10.0384 5.7536 > 0. 5.7536 8.7832 Hence, applying Theorem 1 to this example shows that the system (1) with w(t) 0 and u(t) 0 is stochastically stable. Example 2. We consider system (1) with n = 2. For the two operating conditions (modes), the associated data are: A(1) = Ad (1) = Ah (1) = Bw (1) = Bw (2) = 0.4 0.2 0.3 0.2 0.5 0.4 0.2 0.3 , A(2) = , Ad (2) = , Ah (2) = , , ,

, C(1) = Cw (1) = , C(2) = Cw (2) =

, Dw (1) = 1, , Dw (2) = 1.

Let r = 0.5, = 0.9, d+ = h+ = 0.5. In order to solve the solution simply, for i = 1, 2, we select Qdi = diag(1, 1), Qhi = diag(2, 2), = = 4. So using MATLAB LMI Toolbox we solve the LMI condition (9) and obtain that P (1) = 1.2708 0.1259 0.1259 3.0847 > 0,

#### P (2) =

8.7595 3.8110

#### 3.8110 6.7514

> 0.

Hence, applying Theorem 2 to this example shows that the system (1) with u(t) 0 is strictly passive with dissipation r. 6. Conclusions This paper provides the stochastic stability, passivity analysis and synthesis for Markovian jump systems of neutral type with state delay. The sucient condition on passivity of jump system of neutral type is presented by stochastic Lyapunov functionals technique. Using the solution of LMI, the passive controller via statefeedback is obtained which guarantee the stability and passivity. Two illustrative examples are given to demonstrate the eectiveness of the obtained results. Acknowledgments This work is supported by the National Natural Science Foundation of China (No.10371072) and the Science Foundation of Southern Yangtze University (No.10300021050323). References

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[19] Lou, X. Y., & Cui, B. T., On passivity analysis of bi-directional associative memory neural networks with time delay. Proceedings ICIEA 2006, Marina, Singapore, 1, 2006. [20] Mahmoud, M. S., & Shi, P., Robust stability, stabilization and H control of time-delay systems with Markovian jump parameters. Int. J. Robust Nonlinear Control, 13, 755-784, 2003. [21] Mao, X., Stability of stochastic dierential equations with Markovian switching. Stochastic Processes and their Applications, 79, 45-67, 1999. [22] Mao, X., Matasov, A., & Piunovskii, A. B., Stochastic dierential delay equations with Markovian switching. Bernoulli, 5(5), 1-18, 1999. [23] Mao, X., Robustness of stability of stochastic dierential delay equations with Markovian switching. Stability and Control: Theory and Applications, 3(1), 48-61, 2000. [24] Mao, X., & Shaikhet, L., Delay-dependent stability criteria for stochastic dierential delay equations with Markovian switching. Stability and Control: Theory and Applications, 3(2), 88-102, 2000. [25] Mariton, M., Jump linear control systems. New York: Marcel-Dekker, 1990. [26] Niculescu, S. I., & Lozano, R., On the passivity of linear delay systems. IEEE Transactions on Automatic Control, 46(3), 460-464, 2001. [27] Shaikhet, L.E., Stability of stochastic hereditary systems with Markov Switching. Theory of stochastic processes, 2(3-4), 180-184, 1996. [28] Shaikhet, L.E., Numerical simulation and stability of stochastic systems with Markovian switching. Neural, Parallel and Scientic Computations. Atlanta, Dynamic Publishers, 10(2), 199-208, 2002. [29] Sun, W., Khargonekar, P. P., & Shim, D., Solution to the positive real control problem for linear time-invariant systems. IEEE Transactions on Automatic Control, 39, 2034-2046, 1994. [30] Wen, J. T., Time domain and frequency domain conditions for positive realness. IEEE Transactions on Automatic Control, 33, 988-992, 1988. [31] Wu, C. W., Synchronization in arrays of coupled nonlinear systems: passivity, circle criterion, and observer design. IEEE Trans. Circuits Syst. I, 48(10), 1257-1261, 2001. [32] Xie, L., Fu, M., & Li, H., Passivity analysis and passication for uncertain signal processing systems. IEEE Trans. Signal Processing, 46(9), 2394-2403, 1998. [33] Yuan, C. G., & Mao, X. R., Robust stability and controllability of stochastic dierential delay equations with Markovian switching. Automatica, 40(3), 343-354, 2004. [34] Yu, W., Passive equivalence of chaos in Lorenz system. IEEE Trans. Circuits Syst. I, 46(7), 876-878, 1999. Research Center of Control Science and Engineering, Southern Yangtze University, 1800 Lihu Rd.,Wuxi, Jiangsu 214122, P.R.China

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