Delay-dependent Stability Analysis for Markovian Jump Systems with Interval Time-varying-delays

International Journal of Automation and Computing 7(2), May 2010, 224-229 DOI: 10.1007/s11633-010-0224-2 Delay-dependent Stability Analysis for Markovian Jump Systems with Interval Time-varying-delays Xu-Dong Zhao Qing-Shuang Zeng Space Control and Inertial Technology Center, Harbin Institute of Technology, Harbin 150001, PRC Abstract: This paper proposes improved stochastic stability conditions for Markovian jump systems with interval time-varying delays. In terms of linear matrix inequalities (LMIs), less conservative delay-range-dependent stability conditions for Markovian jump systems are proposed by constructing a different Lyapunov-Krasovskii function. The resulting criteria have advantages over some previous ones in that they involve fewer matrix variables but have less conservatism. Numerical examples are provided to demonstrate the efficiency and reduced conservatism of the results in this paper. Keywords: Stochastic stability, Markovian jump systems, linear matrix inequality (LMI). 1 Introduction Jumping linear systems, as an abstraction of hybrid automata where continuous dynamic system is affected by discrete actions, have received a great deal of interest due to their abundant and effective application in numerous areas such as chemical processes, automotive systems, and electrical circuit systems, etc. 1 In the past decades, the researches have focused on the stability of switched systems and designing stabilizing controllers in both continuoustime domain and discrete-time domain 2, 3. In particular, it has now been well recognized that the delay-dependent stability analysis of switched systems, which is an inherent feature of many physical processes, is worthy (see, for example 4, 5). On the other hand, in recent years, much attention has been devoted to the time-delayed jump linear systems with Markovian jumping parameters 6, where the parameters usually jump among finite modes, and the mode switching is limited by a Markov process 7. Recently, stability analysis problems for Markovian systems with time delays have been considered. Sufficient conditions for mean square stochastic stability were obtained in 8, while exponential stability conditions were proposed in 9. It is noted that all these results are delay independent, that is, they do not include any information on the size of delays. Therefore, delay-independent results are conservative. It is known that delay-dependent results are less conservative than delayindependent ones, especially in the case when the size of the delay is small. This motivates us to develop delaydependent conditions for delayed Markovian jump systems. It is worth pointing out that recent research effort in the study of delay systems are towards developing less conservative delay-dependent results. For the constant delay, in 10, delay-dependent stability conditions were obtained based on a first-order model transformation. Since additional eigenvalues were introduced, the transformed system was not equivalent to the original system. By different model transformations, the less conservative results of stability for the jump linear systems with constant delay were presented in 11 13. In 14, by using some zero equations, neither model transformation nor bounding for cross terms was required to obtain the delay-dependent results. However, to the authors knowledge, the delay-dependent stability results for Markovian jump systems with time varying delays are fewer (stability results for Markovian jump systems with interval time varying delays were much fewer). It is wellknown that there are systems which are stable with some nonzero delay, but are unstable without delay. In that case, if there is a time-varying perturbation on the nonzero delay, it is of great significance to consider the stability of systems with interval time-varying delay. Other typical example of systems with interval time-varying delays is networked control systems. Most recently, a less conservative stability criterion for Markovian jump systems with time varying delays was established in 15 with the free weighting matrix method, and a new stability condition for this class of systems was obtained in 16 by introducing some improved integral-equalities. Nevertheless, the criteria still leave some room for improvement in accuracy as well as complexity due to the method used. In this paper, we revisit the delay-dependent stability for systems with time-varying delays. With a different Lyapunov-Krasovskii function, its time derivative is estimated with less conservatism. The derived stability criteria turn out to be less conservative with fewer matrix variables than some recently reported ones. Last, some numerical examples are presented to illustrate the effectiveness of the proposed technique. In this paper, E stands for the mathematical expectation. denotes the Euclidean norm for vector or the spectral norm of matrix. M > 0 is used to denote a symmetric positive-definite matrix. When r(t) = i S = {1,, N}, we mark A i = A(r(t)). Manuscript received April 24, 2009; revised July 29, 2009

X. D. Zhao and Q. S. Zeng / Delay-dependent Stability Analysis for Markovian Jump Systems with 225 2 System description and preliminaries Consider the following stochastic system with Markovian switching: ẋ(t) = A(r(t))x(t) + A d (r(t))x(t d(t)) (1) x(t) = ϕ(t), t h 2, 0 where x(t) R n is the states, r(t) is a homogenous stationary Markov chain defined on a complete probability space {Ω, F, P} and taking values in a finite set S = {1,, N}. The state transition rate matrix Ξ = (µ ij) N N has the following form: µ ij +o( ); if j i P{r(t + ) = j r(t) = i} = (2) 1 + µ ii +o( ); if j = i where µ ij 0, j i, µ ii = N,j i µij. In system (1), d(t) denotes the time-varying delay which satisfies: h 1 d(t) h 2 and d(t) µ. In (1), ϕ(t) is a vector-valued initial continuous function defined on interval h 2, 0. Lemma 1 17. For any symmetric positive definite matrix W > 0, scalar a > 0, and vector function x : 0, a R n such that the integrations concerned are well defined, the following inequality holds: ( a 0 x(s)ds) T W ( a 0 x(s)ds) a a 0 x T (s)w x(s)ds. (3) Definition 1. The Markovian jump system (1) is said to be stochastically stable, if for finite ϕ(t) defined on h 2, 0 and r(0) S, the following is satisfied: lim t E{ 0 xt (t, ϕ, r(0))x(t, ϕ, r(0))d(s)} <. 3 Main results In this section, we first present a new delay-dependent stochastic stability condition for Markovian jump system (1) in the following Theorem 1. An improved delaydependent stochastic stability condition will be developed, and a corollary will be derived from Theorem 1. Theorem 1. Given scalars h 1, h 2, and µ. Then, for any delay d(t), the Markovian jump system (1) is stochastically stable if there exist n n matrices P i > 0, Q 1i > 0, Q 2i > 0, Q 3i > 0, Q 1 > 0, Q 2 > 0, Q 3 > 0, Z 1 > 0, and Z 2 > 0 such that the following LMIs hold for any i = 1,, N: Σ i I T 1 Z 2I 1 < 0 (4) Σ i I T 2 Z 2I 2 < 0 (5) µ ijq 1j Q 1 (6) µ ijq 2j Q 2 (7) where with µ ijq 3j Q 3 (8) I 1 = 0 0 I I I 2 = 0 I I 0 Σ i = Θ 1i Z 1 Λ 1i 0 Θ 2i Z 2 0 Θ 3i Z 2 Θ 4i Θ 1i = P ia i + (P ia i) T + Q 1i + Q 3i + h 1Q 1+ ()Q 2 + h 2Q 3 Z 1+ A T i (h 2 1Z 1 + () 2 Z 2)A i + Θ 2i = Q 1i + Q 2i Z 1 Z 2 µ ijp j Θ 3i = (1 µ)q 2i 2Z 2 + A T di(h 2 1Z 1 + () 2 Z 2)A di Θ 4i = Q 3i Z 2 Λ 1i = P ia di + A T i (h 2 1Z 1 + () 2 Z 2)A di. Proof. First, in order to cast our model involved in the framework of the Markov processes, we define a new process x t(s) = x(t + s), s 2h 2, 0. We choose a Lyapunov- Krasovskii function: V (x t, t, r(t)) = V 1(t) + V 2(t) + V 3(t) + V 4(t), where V 1(t) = x T (t)p (r(t))x(t) V 2(t) = h1 V 3(t) = x T (s)q 1(r(t))x(s)ds+ x T (s)q 2(r(t))x(s)ds+ x T (s)q 3(r(t))x(s)ds 0 h 1 h1 t+θ h 2 t+θ 0 V 4(t) = h1 h 2 t+θ h 1 h 1ẋ T (s)z 1ẋ(s)dsdθ+ ()ẋ T (s)z 2ẋ(s)dsdθ t+θ x T (s)q 1x(s)dsdθ+ 0 x T (s)q 2x(s)dsdθ+ x T (s)q 3x(s)dsdθ h 2 t+θ where P i, Q 1i, Q 2i, Q 3i, i = 1, 2,, N, Q 1, Q 2, Q 3, Z 1, and Z 2 are positive definite matrices with appropriate dimensions, Let L be the weak infinitesimal generator of the

226 International Journal of Automation and Computing 7(2), May 2010 random process x t, t 0. Then, for each r(t) = i, i S, it can be shown that LV 1(x t, t, i) = 2x T (t)p i(a ix(t) + A di x(t d(t)))+ µ ijx T (t)p jx(t) LV 2(x t, t, i) = x T (t)q 1ix(t) x T (t h 1)Q 1ix(t h 1)+ x T (t h 1)Q 2ix(t h 1) x T (t h 2)Q 3ix(t h 2)+ x T (t)q 3ix(t) (1 d(t))x T ()Q 2ix()+ x T (s)( µ ijq 1j)x(s)ds+ h1 x T (s)( µ ijq 2j)x(s)ds+ x T (s)( µ ijq 3j)x(s)ds LV 3(x t, t, i) = h 2 1(A ix(t) + A di x(t d(t))) T Z 1(A ix(t)+ A di x(t d(t))) + () 2 (A ix(t)+ A di x(t d(t))) T Z 2(A ix(t) + A di x(t d(t))) h 1ẋ T (s)z 1ẋ(s)ds h1 ()ẋ T (s)z 2ẋ(s)ds LV 4(x t, t, i) = h 1x T (t)q 1x(t) + ()x T (t)q 2x(t)+ h 2x T (t)q 3x(t) x T (s)q 1x(s)ds h1 Using Lemma 1, we have h 1ẋ T (s)z 1ẋ(s)ds On the other hand, h1 x T (s)q 2x(s)ds x T (s)q 3x(s)ds. (x(t) x(t h 1)) T Z 1(x(t) x(t h 1)). ()ẋ T (s)z 2ẋ(s)ds = d(t) h1 ()ẋ T (s)z 2ẋ(s)ds+ ()ẋ T (s)z 2ẋ(s)ds = (9) (10) d(t) d(t) h1 h1 d(t) h1 d(t) h 1 h 2 d(t) (h 2 d(t))ẋ T (s)z 2ẋ(s)ds+ (d(t) h 1)ẋ T (s)z 2ẋ(s)ds+ (d(t) h 1)ẋ T (s)z 2ẋ(s)ds+ (h 2 d(t))ẋ T (s)z 2ẋ(s)ds (h 2 d(t))ẋ T (s)z 2ẋ(s)ds+ (d(t) h 1)ẋ T (s)z 2ẋ(s)ds+ d(t) h1 Therefore, Lemma 1 gives h1 ()ẋ T (s)z 2ẋ(s)ds (h 2 d(t))ẋ T (s)z 2ẋ(s)ds+ (d(t) h 1)ẋ T (s)z 2ẋ(s)ds. (x(t d(t)) x(t h 2)) T Z 2(x(t d(t)) x(t h 2)) (x(t h 1) x(t d(t))) T Z 2(x(t h 1) x(t d(t)))+ d(t) h 1 (x(t d(t)) x(t h 2)) T Z 2(x(t d(t)) x(t h 2)) + h2 d(t) (x(t h 1) x(t d(t))) T Z 2(x(t h 1) x(t d(t))). Using this and combining (6) (10), we have LV (x t, t, i) = LV 1(x t, t, i) + LV 2(x t, t, i) + LV 3(x t, t, i)+ LV 4(x t, t, i) 2x T (t)p i(a ix(t)+a di x(t d(t)))+ µ ijx T (t)p jx(t)+ x T (t)q 1ix(t) x T (t h 1)Q 1ix(t h 1) + x T (t)q 3ix(t)+ x T (t h 1)Q 2ix(t h 1) x T (t h 2)Q 3ix(t h 2) (1 µ)x T (t d(t))q 2ix(t d(t)) + h 1x T (t)q 1x(t)+ ()x T (t)q 2x(t) + h 2x T (t)q 3x(t) (x(t) x(t h 1)) T Z 1(x(t) x(t h 1))+ h 2 1(A ix(t) + A di x(t d(t))) T Z 1(A ix(t)+

X. D. Zhao and Q. S. Zeng / Delay-dependent Stability Analysis for Markovian Jump Systems with 227 A di x(t d(t))) + () 2 (A ix(t)+ A di x(t d(t))) T Z 2(A ix(t) + A di x(t d(t))) (x(t d(t)) x(t h 2)) T Z 2(x(t d(t)) x(t h 2)) (x(t h 1) x(t d(t))) T Z 2(x(t h 1) x(t d(t))) d(t) h 1 (x(t d(t)) x(t h 2)) T Z 2(x(t d(t)) x(t h 2)) h2 d(t) (x(t h 1) x(t d(t))) T Z 2(x(t h 1) x(t d(t))) = η T (t)σ iη(t) d(t) h1 (x(t d(t)) x(t h 2)) T Z 2(x(t d(t)) x(t h 2)) h 2 d(t) (x(t h 1) x(t d(t))) T Z 2(x(t h 1) x(t d(t))) = η T d(t) h1 (t) (Σ i I1 T Z 2I 1)+ h 2 d(t) (Σ i I2 T Z 2I 2)η(t) where η(t) = x T (t) x T (t h 1) x T (t d(t)) x T (t h 2) T. Because 0 (h 2 d(t))/(h 2 h 1), (d(t) h 1)/(h 2 h 1) 1, we can see that (4) (8) are equivalent to LV (x t, t, i) < 0. Then, by Definition 1, the stochastic stability is established 15. Remark 1. Compared with some recent stability results for Markovian jump systems with time delays, the results in this paper reduce the conservatism (which will be illustrated by two examples) by not using model transformation to the original delay system, nor resorting to the free weighting matrix method. By this method, it can be found that the results in 15 are conservative for omitting d(t) ẋ T (s)zẋ(s)ds in the process of estimating the derivative of Lyapunov-Krasovskii function. In the proof of Theorem 1 of this paper, a smaller upper bound of the derivative of Lyapunov-Krasovskii function was given by introducing different Lyapunov-Krasovskii functions. Furthermore, the stability criteria of this paper turn out to be less conservative with fewer matrix variables so that it will be easier to investigate the synthesis problems for Markovian jump systems with time delays. When the information of the time derivative of delay is unknown, by elimination, we can obtain the following result from Theorem 1. Corollary 1. Given scalars h 1 and h 2, for any delay d(t), the Markovian jump system (1) is stochastically stable if there exist n n matrices P i > 0, Q 1i > 0, Q 3i > 0, Q 1 > 0, Q 3 > 0, Z 1 > 0, and Z 2 > 0 such that the following LMIs hold for any i = 1,, N: where with Σ i = Σ i I T 1 Z 2I 1 < 0 (11) Σ i I T 2 Z 2I 2 < 0 (12) µ ijq 1j Q 1 (13) µ ijq 3j Q 3 (14) Θ 1i Z 1 Λ 1i 0 Θ 2i Z 2 0 Θ 3i Z 2 Θ 4i Θ 1i = P ia i + (P ia i) T + Q 1i + Q 3i + h 1Q 1 + h 2Q 3 Z 1+ A T i (h 2 1Z 1 + () 2 Z 2)A i + Θ 2i = Q 1i Z 1 Z 2 µ ijp j Θ 3i = 2Z 2 + A T di(h 2 1Z 1 + () 2 Z 2)A di. When h 1 = 0, Theorem1 reduces to Corollary 2. Corollary 2. Given scalars h 2 and µ, for any delay d(t), the Markovian jump system (1) is stochastically stable if there exist n n matrices P i > 0, Q 2i > 0, Q 3i > 0, Q 2 > 0, Q 3 > 0, and Z 2 > 0 such that the following LMIs hold for any i = 1,, N: where Σ i ĪT 1 Z 2 Ī 1 < 0 (15) Σ i ĪT 2 Z 2 Ī 2 < 0 (16) µ ijq 2j Q 2 (17) µ ijq 3j Q 3 (18) Ī 1 = 0 I I Σ i = Θ 1i Λ1i 0, Ī2 = Θ2i Z 2 Θ 3i I I 0

228 International Journal of Automation and Computing 7(2), May 2010 with Θ 1i = P ia i + (P ia i) T + Q 2i + Q 3i + h 2Q 2 + h 2Q 3+ A d2 = 2.8306 0.4978 0.8436 1.0115 A T i h 2 2 Z 2A i + µ ijp j Z 2 Θ 2i = (1 µ)q 2i 2Z 2 + A T dih 2 2 Z 2A di Λ 1i = P ia di + A T i h 2 2 Z 2A di + Z 2. 4 Numerical examples When the time delays are constant (say, time delays are time-invariant), to show the reduced conservatism of the delay-dependent stability criterion in Theorem 1, we provide the following example. Example 1. Consider a Markovian jump system in (1) with two modes and the following parameters 12 : A 1 = A 2 = A d1 = A d2 = 0.5 1 0 3 5 1 1 0.2 0.5 0.2 0.2 0.3 0.3 0.5 0.4 0.5 with transition rates matrix 7 7 Ξ =. 3 3 To compare the stochastic stability result in Theorem 1 with those in 10, 12 14, we assume µ = 0 and h 1 = 0. By Theorem 1 in this paper, the upper bound on the time delay obtained is listed in Table 1. It is clear that the obtained results are significantly better than those in 10, 12 14. Table 1 Comparison of maximum allowed h 2 for Example 1 By 10 By 12 By 13 By 14 By Theorem 1 h 2 0.84 1.23 0.40 0.73 1.33 When the time delays are time-variant, we consider the following example. Example 2. Consider a Markovian jump system in (1) with two modes and the following parameters 15 : A 1 = A d1 = A 2 = 3.4888 0.8057 0.6451 3.2684 0.8620 1.2919 0.6841 2.0729 2.4898 0.2895 1.3396 0.0211 with transition rate matrix 0.1 0.1 Ξ =. 0.8 0.8 To compare the stochastic stability criterion in Theorem 1 with those in 15,16, we also assume h 1 = 0. For a given µ, the maximum h 2 that satisfies the LMIs in Theorem 1, can be calculated by solving a quasi-convex optimization problem. The comparison results for different µ are given in Table 2, which shows that Theorem 1 is less conservative than those in 15,16. Table 2 Comparison of maximum allowed h 2 for Example 2 µ = 0.6 µ = 0.8 µ = 1.6 h 2 by Theorem 1 of 16 0.4428 0.3795 0.3469 h 2 by Theorem 2 of 16 0.4492 0.4341 0.4314 h 2 by 15 0.4927 0.4261 0.3860 h 2 by Theorem 1 of this paper 0.5159 0.4814 0.4789 Furthermore, by the method in Corollary 1, we can obtain that the maximum h 2 allowed is 0.4789 when the information of the time derivative of delay is unknown. When the lower bound h 1 of the time-varying delays is larger than zero, the following example can demonstrate the effectiveness of Theorem1. Example 3. Consider a Markovian jump system in (1) with two modes and the system parameters are described as in Example 1. We assume µ = 0, h 1 = 0.5, and µ 11 = 7. For a given µ 22, the comparison results between Theorem 1 of this paper and the method of 18 are given in Table 3, which shows that Theorem 1 is less conservative than that in 18 when h 1 > 0. Table 3 Comparison of maximum allowed h 2 for Example 3 µ = 1 µ = 2 µ = 3 h 2 by 18 0.6898 1.1077 1.2455 h 2 by Theorem 1 0.6976 1.1384 1.5091 5 Conclusions By using a new Lyapunov-Krasovskii function, improved delay-dependent stability criteria for a class of Markovian jump systems with interval time-varying-delays have been obtained in terms of linear matrix inequalities (LMIs). The results of this paper turn out to be less conservative with fewer matrix variables. Three examples have been given to show that the criteria performance is feasible and effective.

X. D. Zhao and Q. S. Zeng / Delay-dependent Stability Analysis for Markovian Jump Systems with 229 Acknowledgement We thank the editor and anonymous reviewers for their valuable comments and suggestions that have helped us in improving the paper. References 1 R. A. Decarlo, M. S. Branicky, S. Pettersson, B. Lennartson. Perspective and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, vol. 88, no. 7, pp. 1069 1082, 2000. 2 X. D. Zhao, M. X. Ling, Q. S. Zeng. Robust D-stability and H control of uncertain discrete switched systems. Journal of Harbin Institute of Technology, vol. 41, no. 1, pp. 38 43, 2009. (in Chinese). 3 X. Y. Lou, B. T. Cui. Delay-dependent criteria for robust stability of uncertain switched hopfield neural networks. International Journal of Automation and Computing, vol. 4, no. 3, pp. 304 314, 2007. 4 R. Wang, J. Zhao. Exponential stability analysis for discrete-time switched linear systems with time-delay. International Journal of Innovative Computing, Information and Control, vol. 3, no. 6, pp. 1557 1564, 2007. 5 E. Fridman. New Lyapunov-Krasovskii functionals for stability of linear retard and neutral type systems. Systems & Control Letters, vol. 43, no. 4, pp. 309 319, 2001. 6 W. H. Chen, Z. H. Guan, X. M. Lu. Passive control synthesis for uncertain Markovian jump linear systems with multiple mode-dependent time-delays. Asian Journal of Control, vol. 7, no. 2, pp. 135 143, 2005. 7 G. Guo, B. F. Wang. Kalman filtering with partial Markovian packet losses. International Journal of Automation and Computing, vol. 6, no. 4, pp. 395 400, 2009. 8 K. Benjelloun, E. K. Boukas. Mean square stochastic stability of linear time-delay system with Markovian jumping parameters. IEEE Transactions on Automatic Control, vol. 43, no. 10, pp. 1456 1460, 1998. 9 H. Fujisaki. On correlation values of M-phase spreading sequences of Markov chains. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 12, pp. 1745 1750, 2002. 10 E. K. Boukas, Z. K. Liu, P. Shi. Delay-dependent stability and output feedback stabilisation of Markov jump system with time-delay. IEE Proceedings: Control Theory and Applications, vol. 149, no. 5, pp. 379 386, 2002. 11 W. H. Chen, J. X. Xu, Z. H. Guan. Guaranteed cost control for uncertain Markovian jump systems with modedependent time-delays. IEEE Transactions on Automatic Control, vol. 48, no. 12, pp. 2270 2277, 2003. 12 Y. Y. Cao, L. S. Hu, A. K. Xue. A new delay-dependent stability condition and H control for jump time-delay system. In Proceedings of American Control Conference, IEEE, Boston, USA, vol. 5, pp. 4183 4188, 2004. 13 Y. Y. Cao, J. Lam, L. S. Hu. Delay-dependent stochastic stability and H analysis for time-delay systems with Markovian jumping parameters. Journal of the Franklin Institute, vol. 340, no. 6 7, pp. 423 434,2003. 14 J. Wu, T. W. Chen, L. Wang. Delay-dependent robust stability and H control for jump linear systems with delays. Systems & Control Letters, vol. 55, no. 11, pp. 939 948, 2006. 15 S. Xu, J. Lam, X. Mao. Delay-dependent H control and filtering for uncertain Markovian jump systems with timevarying delays. IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 54, no. 9, pp. 2070 2077, 2007. 16 J. Wang, Y. Luo. Further improvement of delay-dependent stability for Markov jump systems with time-varying delay. In Proceedings of the 7th World Congress on Intelligent Control and Automation, IEEE, Chongqing, PRC, pp. 6319 6324, 2008. 17 K. Gu. An integral inequality in the stability problem of time-delay systems. In Proceedings of the 39th IEEE Conference on Decision and Control, IEEE, Sysdney, Australia, vol. 3, pp. 2805 2810, 2000. 18 H. Guan, L. Gao. Delay-dependent robust stability and H control for jump linear system with interval time-varying delay. In Proceedings of the 26th Chinese Control Conference, IEEE, Zhangjiajie, PRC, pp. 609 614, 2007. Xu-Dong Zhao received the B. Sc. degree in automation from Harbin Institute of Technology, PRC in 2005. He is currently a Ph. D. candidate in control science and engineering at Space Control and Inertial Technology Center, Harbin Institute of Technology. His research interests include Markovian jump systems, H control, and switched systems. E-mail: Zxd7777777@126.com (Corresponding author) Qing-Shuang Zeng received the B. Sc. and M. Sc. degrees in control science and engineering from Harbin Institute of Technology, PRC in 1987 and 1990, respectively, and Ph. D. degree in control science and engineering from Harbin Institute of Technology in 1997. He is currently a professor at the Space Control and Inertial Technology Center, Harbin Institute of Technology. His research interests include switched systems, control theory, H control, and inertial technology. E-mail: zqshuang2000@yahoo.com.cn