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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 738318, 10 pages

http://dx.doi.org/10.1155/2014/738318

## Delay-Dependent Robust Exponential Stability and Analysis for a Class of Uncertain Markovian Jumping System with Multiple Delays

School of Mathematics Science, Liaocheng University, Shandong 252000, China

Received 26 November 2013; Accepted 23 January 2014; Published 17 March 2014

Academic Editor: Hao Shen

Copyright © 2014 Jianwei Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the problem of robust exponential stability and performance analysis for a class of uncertain Markovian jumping system with multiple delays. Based on the reciprocally convex approach, some novel delay-dependent stability criteria for the addressed system are derived. At last, numerical examples is given presented to show the effectiveness of the proposed results.

#### 1. Introduction

It is well known that time delay is usually the main reason for instability and poor performance of many practical control systems [1–5]. The stability results for delayed systems can be generally classified into two categories: delay-independent stability criteria and delay-dependent criteria. And the delay-dependent results are often less conservative than the delay-independent ones, especially when the time delays are small. Therefore, much more attention has been focused on study of the delay-dependent stability conditions in recent years. For example, the system transformation method in [6], the descriptor system method in [7], parameter-dependent Lyapunov-Krasovskii functional method in [8], Jensen inequality method in [9], Free-weighting matrix method in [10, 11], integral inequality method in [12], augmented Lyapunov functional method in [13], convex domain method in [14], interval partition method in [15, 16], reciprocally convex method in [17], and so forth. And those approaches have been widely used in the stability analysis for lots of delayed systems in recent years [18–20].

On the other hand, since Markovian jumping systems can model many types of dynamic systems subject to abrupt changes in their structures, such as failure prone manufacturing systems, power systems, and economics systems [21–27], a great deal of results related to stability analysis and synthesis for this class of systems with time delays has been reported in recent years. For example, for the delay-independent results, sufficient conditions for mean squares to stochastic stability were obtained in [28], while exponential stability conditions were proposed in [29]. The robust filtering problem was dealt with in [30]. For the delay-dependent ones, the stability and control results were presented by resorting to some bounding techniques for some cross terms and using model transformation to the original delay system in [31]. The control and Filtering problem were taken into account in [32] using the Free-weighting matrix method. The stability and analysis was proposed in [33] with the idea of delay partition. Filtering problem with a new index was considered in [34] using the reciprocally convex method. It is worth mentioned that inspite of the deep study for the delayed stochastic in recent years as mentioned above, there are few papers that consider the problem of stability analysis for uncertain stochastic systems with multiple delays, which motivates our study.

In this paper, the robust exponential stability and performance analysis for a class of uncertain Markovian system with multiple time-varying delays is investigated. Some new delay-dependent stability conditions are derived. And numerical simulation is given to demonstrate the effectiveness of the result.

*Notation.* Throughout this paper, for symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite); is the identity matrix with appropriate dimension; represents the transpose of the matrix ; denotes the expectation operator with respect to some probability measure ; is the space of square-integrable vector functions over ; refers to the Euclidean vector norm; stands for the usual norm; ) is a probability space with the sample space and is the -algebra of subsets of the sample space. Matrices, if not explicitly mentioned, are assumed to have compatible dimensions.

#### 2. System Description and Preliminaries

Consider the following uncertain Markovian jumping system with multiple time-varying delays: where is the state; is the noise disturbance which is assumed to be an arbitrary signal in ; is the signal to be estimated; is a homogenous stationary Markov chain defined on a complete probability space and taking values in a finite set with generator given by where and , is the transition rate from to if and . The scalar is the time-varying delay with , , , for any , where , and are positive scalar constants; is the initial function defined in with ; , , and are matrix functions with time-varying uncertainties described as , , where are known constant matrices, while uncertainties , are assumed to be norm bounded as where , , , and , in (2) are known constant matrices with appropriate dimensions. The unknown matrix functions are having Lebesgue-measurable elements and satisfying

*Remark 1. *When , the system with multiple time-varying delays (1)–(3) is actually deduced to the uncertain Markovian jumping system with interval delay, which have been deeply studied in recent years. That is, the obtained results of multiple delayed systems can be directly deduced to the interval delayed systems.

Throughout this paper, we will use the following Definitions and Lemmas.

*Definition 2. *The uncertain Markovian jumping system with multiple time-varying delays (1)–(3) is said to be robustly exponentially stable in mean square for all admissible uncertainties, if there exist scalars and such that for all ,
where is the trivial solution of systems (1)–(3) with .

*Definition 3. *Given a scalar , uncertain Markovian jumping system with multiple time-varying delays (1)–(3) is said to be robustly exponentially stable with a prescribed performance level if it is robustly exponentially stable, and under the zero initial condition, satisfies
for all admissible uncertainties and nonzero , where

Lemma 4 (see [9]). *For any constant matrix , , scalar , vector function such that the integrations in the following are well defined; then
*

Lemma 5 (see [35]). *Let , , be real constant matrices with appropriate dimensions; matrix satisfies . For any , such that ,
*

Lemma 6 (see [36]). *Consider system (1) with , , for any matrices and satisfying ; the following inequality holds
**
where , and
*

#### 3. Main Results

For simplicity, we define

##### 3.1. Robust Exponential Stability Analysis

The criteria of the robust exponential stability for the systems (1)–(3) are proposed in the following Theorem.

Theorem 7. *Systems (1)–(3) with is robustly exponentially stable if there exist positive matrices , , , , , , any matrices with appropriate dimensions satisfying , ; , , and positive scalars , such that the following LMI holds
**
where
**
And are defined in (3).*

*Proof. *On one hand, using Lemma 5 and Schur complement lemma to (15), we have

On the other hand, define a new process , . Choose a Lyapunov-Krasovskii functional
where

Let be the weak infinitesimal generator of the random process . Then for each , , we have

Applying Lemma 4 to results in
and applying Lemma 6 to , we have that there exists with , , such that
where and are defined in (13). Meanwhile, we note that
that is,

Then, we can deduce from (19)–(24) that
where is defined in (17). Therefore, by Definition 2 and the results in [37], we have that the system (1) is robustly stable. Now, we will prove the robust stochastic exponential stability in mean square for system (1). Setting , we have
Choose , where ; then
Integrating the above inequality (27), we get

From (19), it can be inferred that
Note that
We denote ; then

From (29) to (31), we obtain
where

By the similar method, we have
where . Therefore, by (28)–(34), we get

Choose such that
then

Since , it can be shown from (37) that
where
which implies that system (10) is robustly exponentially stable by Definition 2. This completes the proof.

##### 3.2. Robust Exponential Stability Analysis

The criteria of the robust exponential stability with performance for the systems (1)–(3) are proposed in the following Theorem.

Theorem 8. *Given a scalar , the systems (1)–(3) are robustly exponentially stable with a prescribed performance level if there exist matrices , , , , , , any matrices with appropriate dimensions satisfying , ; , , and positive scalars , such that the following LMI holds
**
where
*

*Proof. *Implying Lemma 5 and Schur complement lemma to (40), we obtain

Set
Then, it is easy to have
where is defined in (18). Similar to the proof of Theorem 7, we can obtain
where is given in (42) and is defined in (14). Then, it follows from (40) and (45) that
This implies that for any nonzero ,
Therefore, by Definition 3, the system is robustly exponentially stable with a prescribed performance level . This completes the proof.

#### 4. Numerical Example

In this section, we provide an example to demonstrate the effectiveness of the proposed method.

Let and ; consider the systems (1)–(3) with parameters as follows. Mode 1 Mode 2 And , , , , , . Then, by solving the LMI (15) with the constraints in Theorem 7, we obtain

If we fix the lower bound of and , that is, and , for the different , we can get the upper bounds of as in Table 1.

If we fix the lower bound of and , that is, and , for the different , we can get the upper bounds of as in Table 2.

#### 5. Conclusion

The robust exponential stability and performance analysis for uncertain Markovian jumping system with multiple time-varying delays has been investigated based on the reciprocally convex approach. Some new delay-dependent stability conditions are obtained in term of LMIs. Numerical example has been proposed to illustrate the effectiveness of result.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was supported National Natural Science Foundation of China under Grant nos. 61004046, 61104117, and 61304064.

#### References

- Y. He, G. Liu, and D. Rees, “New delay-dependent stability criteria for neural networks with yime-varying delay,”
*IEEE Transactions on Neural Networks*, vol. 18, no. 1, pp. 310–314, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Xu, J. Lam, and D. W. C. Ho, “A new LMI condition for delay-dependent asymptotic stability of delayed Hopfield neural networks,”
*IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 53, no. 3, pp. 230–234, 2006. View at Publisher · View at Google Scholar · View at Scopus - B. Zhang, S. Xu, and Y. Zou, “Relaxed stability conditions for delayed recurrent neural networks with polytopic uncertainties,”
*International Journal of Neural Systems*, vol. 16, no. 6, pp. 473–482, 2006. View at Publisher · View at Google Scholar · View at Scopus - B. Zhang, S. Xu, and Y. Li, “Delay-dependent robust exponential stability for uncertain recurrent neural networks with time-varying delays,”
*International Journal of Neural Systems*, vol. 17, no. 3, pp. 207–218, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. Xia, C. Sun, and B. Zhang, “New robust
*H*control for uncertain stochastic Markovian jumping systems with mixed delays based on decoupling method,”_{∞}*Journal of the Franklin Institute*, vol. 349, no. 3, pp. 741–769, 2012. View at Publisher · View at Google Scholar · View at Scopus - L. Xie and C. Souza, “Criteria for robust stability and stabilization of uncertain linear systems with state delay,”
*Automatica*, vol. 33, no. 9, pp. 1657–1662, 1997. View at Publisher · View at Google Scholar · View at Scopus - E. Fridman and U. Shaked, “A descriptor system approach to
*H*control of linear time-delay systems,”_{∞}*IEEE Transactions on Automatic Control*, vol. 47, no. 2, pp. 253–270, 2002. View at Publisher · View at Google Scholar · View at Scopus - Y. S. Lee, Y. S. Moon, W. H. Kwon, and P. G. Park, “Delay-dependent robust
*H*control for uncertain systems with a state-delay,”_{∞}*Automatica*, vol. 40, no. 1, pp. 65–72, 2004. View at Publisher · View at Google Scholar · View at Scopus - K. Gu, V. Kharitonov, and J. Chen,
*Stability of Time-Delay Systems*, Birkhauser, 2003. - Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties,”
*IEEE Transactions on Automatic Control*, vol. 49, no. 5, pp. 828–832, 2004. View at Publisher · View at Google Scholar · View at Scopus - S. Xu and J. Lam, “Improved delay-dependent stability criteria for time-delay systems,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 3, pp. 384–387, 2005. View at Publisher · View at Google Scholar · View at Scopus - Q.-L. Han and D. Yue, “Absolute stability of Lur'e systems with time-varying delay,”
*IET Control Theory and Applications*, vol. 1, no. 3, pp. 854–859, 2007. View at Publisher · View at Google Scholar · View at Scopus - Y. He, G. P. Liu, and D. Rees, “Augmented Lyapunov functional for the calculation of stability interval for time-varying delay,”
*IET Control Theory and Applications*, vol. 1, no. 1, pp. 381–386, 2007. View at Publisher · View at Google Scholar · View at Scopus - H. Shao, “New delay-dependent stability criteria for systems with interval delay,”
*Automatica*, vol. 45, no. 3, pp. 744–749, 2009. View at Publisher · View at Google Scholar · View at Scopus - Y. Zhang, D. Yue, and E. Tian, “Robust delay-distribution-dependent stability of discrete-time stochastic neural networks with time-varying delay,”
*Neurocomputing*, vol. 72, no. 4–6, pp. 1265–1273, 2009. View at Publisher · View at Google Scholar · View at Scopus - R. Yang, Z. Zhang, and P. Shi, “Exponential stability on stochastic neural networks with discrete interval and distributed delays,”
*IEEE Transactions on Neural Networks*, vol. 21, no. 1, pp. 169–175, 2010. View at Publisher · View at Google Scholar · View at Scopus - P. Park, J. W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,”
*Automatica*, vol. 47, no. 1, pp. 235–238, 2011. View at Publisher · View at Google Scholar · View at Scopus - T. Li, W. X. Zheng, and C. Lin, “Delay-slope-dependent stability results of recurrent neural networks,”
*IEEE Transactions on Neural Networks*, vol. 22, no. 12, pp. 2138–2143, 2011. View at Publisher · View at Google Scholar · View at Scopus - Z.-G. Wu, J. H. Park, H. Su, and J. Chu, “Stochastic stability analysis of piecewise homogeneous Markovian jump neural networks with mixed time-delays,”
*Journal of the Franklin Institute*, vol. 349, no. 6, pp. 2136–2150, 2012. View at Publisher · View at Google Scholar · View at Scopus - T. Li, X. Yang, P. Yang, and S. Fei, “New delay-variation-dependent stability for neural networks with time-varying delay,”
*Neurocomputing*, vol. 101, pp. 361–369, 2013. View at Publisher · View at Google Scholar - N. Krasovskii and E. Lidskii, “Analysis design of controller in systems with random attributes—part I,”
*Automation and Remote Control*, vol. 22, pp. 1021–1025, 1961. View at Google Scholar - H. Shen, X. Huang, J. Zhou, and Z. Wang, “Global exponential estimates for uncertain Markovian jump neural networks with reaction-diffusion terms,”
*Nonlinear Dynamics*, vol. 69, no. 1-2, pp. 473–486, 2012. View at Publisher · View at Google Scholar · View at Scopus - H. Shen, X. Huang, and Z. Wang, “Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback,”
*Journal of the Franklin Institute*, 2013. View at Publisher · View at Google Scholar - H. Shen, J. Park, L. Zhang, and Z. Wu, “Robust extended dissipative control for sampled-data Markov jump systems,”
*International Journal of Control*, 2014. View at Publisher · View at Google Scholar - S. He and F. Liu, “Adaptive observer-based fault estimation for stochastic Markovian jumping systems,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 176419, 11 pages, 2012. View at Publisher · View at Google Scholar - S. He and F. Liu, “Robust stabilization of stochastic Markovian jumping systems via proportional-integral control,”
*Signal Processing*, vol. 91, no. 11, pp. 2478–2486, 2011. View at Publisher · View at Google Scholar · View at Scopus - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Passivity analysis for discrete-time stochastic markovian jump neural networks with mixed time delays,”
*IEEE Transactions on Neural Networks*, vol. 22, no. 10, pp. 1566–1575, 2011. View at Publisher · View at Google Scholar · View at Scopus - K. Benjelloun and 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. View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - S. Xu, T. Chen, and J. Lam, “Robust H∞ filtering for uncertain Markovian jump systems with mode-dependent time delays,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 5, pp. 900–907, 2003. View at Publisher · View at Google Scholar · View at Scopus - E. K. Boukas, Z. K. Liu, and G. X. Liu, “Delay-dependent robust stability and
*H*control of jump linear systems with time-delay,”_{∞}*International Journal of Control*, vol. 74, no. 4, pp. 329–340, 2001. View at Publisher · View at Google Scholar · View at Scopus - S. Xu, J. Lam, and X. Mao, “Delay-dependent
*H*control and filtering for uncertain Markovian jump systems with time-varying delays,”_{∞}*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 54, no. 9, pp. 2070–2077, 2007. View at Publisher · View at Google Scholar · View at Scopus - X. Zhao and Q. Zeng, “New robust delay-dependent stability and H∞ analysis for uncertain Markovian jump systems with time-varying delays,”
*Journal of the Franklin Institute*, vol. 347, no. 5, pp. 863–874, 2010. View at Publisher · View at Google Scholar · View at Scopus - B. Zhang, W. Zheng, and S. Xu, “Filtering of Markovian jump delay systems based on a new performance index,”
*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 60, no. 5, pp. 1250–1263, 2013. View at Publisher · View at Google Scholar - Y. Zhang, Y. He, and M. Wu, “Delay-dependent robust stability for uncertain stochastic systems with interval time-varying delay,”
*Acta Automatica Sinica*, vol. 35, no. 5, pp. 577–582, 2009. View at Publisher · View at Google Scholar · View at Scopus - Z. Wu, J. Lam, H. Su, and J. Chu, “Stability and dissipativity analysis of static neural networks with time delay,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 23, no. 2, pp. 199–210, 2012. View at Publisher · View at Google Scholar - S. Xu and T. Chen, “Robust H∞ control for uncertain stochastic systems with state delay,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 12, pp. 2089–2094, 2002. View at Publisher · View at Google Scholar · View at Scopus