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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 934134, 8 pages

http://dx.doi.org/10.1155/2014/934134

## Nonfragile Control for Stochastic Systems with Markovian Jumping Parameters and Random Packet Losses

^{1}School of Electrical and Information Engineering, Anhui University of Technology, Ma’anshan 243002, China^{2}School of Information Science and Engineering, Northeastern University, Shenyang 110819, China

Received 11 December 2013; Accepted 6 January 2014; Published 12 February 2014

Academic Editor: Shuping He

Copyright © 2014 Jing Wang and Ke Zhang. 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 is concerned with the nonfragile control problem for stochastic systems with Markovian jumping parameters and random packet losses. The communication between the physical plant and controller is assumed to be imperfect, where random packet losses phenomenon occurs in a random way. Such a phenomenon is represented by a stochastic variable satisfying the Bernoulli distribution. The purpose is to design a nonfragile controller such that the resulting closed-loop system is stochastically mean square stable with a guaranteed performance level *γ*. By using the Lyapunov function approach, some sufficient conditions for the solvability of the previous problem are proposed in terms of linear matrix inequalities (LMIs), and a corresponding explicit parametrization of the desired controller is given. Finally, an example illustrating the effectiveness of the proposed approach is presented.

#### 1. Introduction

During the past several decades, stochastic systems have been the main focus of research receiving much attention since realistic models of most engineering systems involve random exogenous disturbances [1, 2]. As a simple yet significant mathematical model, stochastic systems have come to play a key role in many branches of science and engineering [3]. For this reason, many fundamental issues have been extensively addressed for stochastic systems, and consequently fruitful results have been presented in the literature; see, for example, [2, 4, 5] and the references therein.

In addition to stochastic systems, there have been great efforts in the research of the modeling of dynamic systems subject to random abrupt changes in their parameters [6–8]. Such random abrupt changes may be caused by various factors, including the switching between economic scenarios, abrupt changes in the operation point for nonlinear plant, and actuator/sensor failure or repairs, to name just a few. Fortunately, Markov jump systems provide a natural framework for modeling these practical systems subject to random abrupt changes. Since the pioneering work on Markov jump systems was introduced in [9], considerable research results related to Markov jump systems or system with Markovian jumping parameters have been presented in terms of a variety of methods. For more details, we refer to the literature [10–17]. When Markovian jumping parameters appear in stochastic systems, many control issues have been studied recently by researchers. For example, robust stability and stabilization problems were investigated in [18], passivity-based control problem was addressed in [19], optimal control problems were studied in [12, 20–22], and the sliding-mode control problem was solved in [23].

It is worth noting that the controller design methods proposed in the previous literature require two critical assumptions. One is that the controller can be implemented exactly, and the other is that the communication between the physical plant and controller is always perfect. Such two assumptions, however, may not be unreasonable in practice. Firstly, in the implementation of a design controller, uncertainties or inaccuracies do occur because of round-off errors in numerical computation. Some existing control synthesis methods have proven to be sensitive, or fragile, with respect to small perturbations in controller parameters. Therefore, it is an important question to design a controller, which guarantees that the controller is insensitive to some amount of errors with its gain, that is, the nonfragile or resilient control problem [24]. Secondly, a modern control system can hardly work without the help of the networks and the computers and their intercommunication. They bring a lot of advantages, but the existence of network-induced phenomena is unavoidable [25]. For instance, packet losses may occur due to the unreliability of the network links. These may limit the scope of the applications of the existing results related to stochastic systems with Markovian jumping parameters. The main purpose of this paper, therefore, is to shorten such a gap.

In this paper, we make the first attempt to deal with the nonfragile control for stochastic systems with Markovian jumping parameters and random packet losses. The packet losses phenomena are assumed to exist in communication links between the physical plant and controller. Attention is focused on the design of a nonfragile controller such that the resulting closed-loop system is stochastically mean square stable, and meanwhile a prescribed performance is satisfied. Sufficient conditions for the existence of such a controller are given in terms of LMIs. By solving a convex optimization problem, a desired nonfragile controller can be constructed based on the use of standard numerical algorithms [26].

*Notation.* Throughout this paper, for symmetric matrices , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite); and represent the identity matrix and zero matrix with appropriate dimension. The notation represents the transpose of the matrix stands for a block-diagonal matrix. In symmetric block matrices or complex matrix expressions, we employ an asterisk to represent a term that is induced by symmetry; denotes the expectation operator with respect to some probability measure is a probability space; is the sample space, is the -algebra of subsets of the sample space, and is the probability measure on is the space of square-summable infinite vector sequences over refers to the Euclidean vector norm; stands for the usual norm. Matrices, if not explicitly stated, are assumed to have compatible dimensions. represents .

#### 2. Problem Formulations

Consider the following discrete-time stochastic Markov jump system over a probability space : where is the system state vector; is the controlled input; is the controlled output; is the exogenous disturbance input that belongs to . For each , , , , , , , , and are real constant matrices with appropriate dimensions. is a one-dimensional zero mean Gaussian white noise sequence on a probability space with In system , the system mode switching is governed by a discrete-time homogeneous Markov chain , which takes values in a finite state space with transition probability matrix given by with , for any , and

In practice, it is usually of importance to require very accurate controllers to achieve given engineering specifications. However, the resulting closed-loop systems are sensitive to changes in controller gain. In this case, once there are some small perturbations in the controller parameters, the existence of these perturbations may cause a serious deterioration of system performance. Hence, it is imperative to consider the design of nonfragile controllers [24]. Consequently, in this paper, we are interested in designing the controller in the following form: where is the input of the controller; is the output of the controller; are the gain matrices of the controller, which will be determined; are real-valued unknown matrices denoting the additive gain variations as follows: where and , , are the known real constant matrices of appropriate dimensions and are unknown time-varying matrix functions, which satisfy the following constraint:

*Remark 1. *Normally, under the implicit assumption that the communication between the plant and the controller is perfect, one can readily get that the controlled input is equivalent to the the output of controller and the measurement state of the plant is also equivalent to the input of the controller . As noted in the previous section, such an assumption is sometimes unpractical especially under networked environments because of the existence of the packet losses.

Therefore, in this paper, the packet losses phenomena are considered in communication links. As a result, one has and , and the relations between them are modeled by using a stochastic method as follows: Here, and are two independent Bernoulli processes. As shown in (8), models the unreliable communication link from the sensor to the controller and models the unreliable communication link from the controller to the actuator. Inspired by [27], a natural assumption on and can be made as follows: where either or is a known constant satisfying and . Clearly, for , when (resp., ), it means that the communication link from the sensor to the controller fails (resp., successful transmission), and also has a similar inference. Throughout this paper, we also assume that the sequences , and are mutually independent. Clearly, one can get

In order to address the considered problem, before presenting further results, let us introduce a new Bernoulli process satisfying . Then, simple computation yields which implies that Under the control law (12), the resulting closed-loop system can be obtained as where

Before formulating the problem to be investigated, we first introduce the following definition for system .

*Definition 2. *The closed-loop system in (13) with is said to be stochastically mean square stable (SMSS) if there exists a such that
for any initial condition .

*Definition 3. *System is said to be SMSS with a guaranteed performance level , if system is SMSS according to Definition 2, and the prescribed disturbance attenuation level is made small in the feasibility of
for all nonzero under zero initial conditions, where

Now, let us state the problems concerned in this paper, which are listed as follows.

*Problem I*. Consider the stochastic system , suppose that the controller gain matrices and the additive gain variations are given, and determine under what condition the system is SMSS with a guaranteed performance level .

*Problem II*. Consider the system , and design a nonfragile controller in the form of (5) such that the resulting closed-loop system is SMSS with a guaranteed performance level , in spite of the presence of packet losses phenomena.

#### 3. Main Results

In this section, we will give an LMI approach to solving the nonfragile control problem formulated in the previous section. Before proceeding further, we shall introduce the following lemmas, which will be used in the proof of the main results.

Lemma 4 (see [28]). *Given constant matrices and of appropriate dimensions, then
**
if and only if
**
or, equivalently,
*

Lemma 5 (see [29]). *Let , , , , and be real matrices of appropriate dimensions with . Then one has*(1)*for any scalar and vectors , ,
*(2)*for any scalar , such that ,
*

Now, we first establish the following performance analysis criterion, which will play a key role in derivation of the solution to the nonfragile control problem.

Theorem 6. *Let the controller parameters in the filtering error system , scalars , be given. Then, system is SMSS with a guaranteed performance level , if there exist positive matrices , such that, for each ,**where .*

*Proof. *Firstly, we need to establish the stochastic mean square stability criterion of system . For this purpose, we consider system with and choose a stochastic Lyapunov function for system as follows:
where are the positive matrices to be determined for each . Then, we have that, for each and ,
On the other hand, it can be deduced from (23) that
By applying the Schur complement formula (i.e., Lemma 4) to (26), for system with , one can readily obtain that
that is, system with is SMSS according to [14, 16]. Next, we will show the performance analysis of system . To this end, we also obtain that each and ,
Note that
It can be verified that
Similar to the derivation of (27), we apply the Schur complement to (23) and get
For , summing up both sides of (31) under zero initial condition and noticing , it can be verified that
or, equivalently, condition (16) is satisfied. This completes the proof.

In the following, we will present a solution to the nonfragile controller design problem for system based on Theorem 6. The following theorem proposes a sufficient condition for the existence of such a controller for system .

Theorem 7. *Consider system , let scalars , be given, and let matrices , , and be fixed. Then, there exists an admissible controller in the form of (5) such that the resulting closed-loop system is SMSS with a guaranteed performance level , in spite of the presence of packet losses phenomena if there exist matrices such that the following LMIs hold for each :**where
**
In this case, a suitable nonfragile controller in the form of (5) is given by
*

*Proof. *Introduce the new variables ; then one can find that
which implies that
Note that
It follows from (37) that
In view of Lemma 5, one can get that
whereUsing Lemma 4 and combining (33), (39), and (40) result inwhere
Then, by pre- and postmultiplying (42) by diag and its transpose, one has that inequality (23) holds. Therefore, in light of Theorem 6, we can conclude that the resulting closed-loop system is SMSS with a guaranteed performance level . This completes the proof.

#### 4. An Illustrative Example

In this section, an example is used to illustrate the effectiveness of the presented nonfragile controller design method. Consider the discrete-time stochastic Markov jump system over a probability space with two modes () and the following parameters: Here, our aim is to design a nonfragile controller in the form of (5) such that the resulting closed-loop system is SMSS with a guaranteed performance level , in spite of the presence of packet losses phenomena. To this end, we suppose that , and , and choose the transition probability matrix as . Then, by applying Theorem 7, one can get feasible solutions as follows: Thus, the desired controller gains (,) can be given by

#### 5. Conclusions

In this paper, we have studied the problem of nonfragile control for stochastic systems with Markovian jumping parameters and random packet losses. An LMI approach has been developed to design a nonfragile controller which ensures both the stochastic mean square stability and a prescribed performance level for the resulting closed-loop systems in the presence of random packet losses. The proposed approach has been illustrated to be effective by an example. It should be pointed out that the states of the system are assumed to be precisely known, but this is difficult to achieve in practice [30–32]. Therefore, one of our further research topics is to develop nonfragile output feedback controller design methods for stochastic Markov jump systems.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61304066, 61104221, and 61104007, the Natural Science Foundation of Anhui Province under Grant 1308085QF119, and the Key Foundation of Natural Science for Colleges and Universities in Anhui province under Grant KJ2012A049.

#### References

- L. Arnold,
*Stochastic Differential Equations: Theory and Applications*, John Wiley & Sons, New York, NY, USA, 1974. View at MathSciNet - L. Wu, W. X. Zheng, and H. Gao, “Dissipativity-based sliding mode control of switched stochastic systems,”
*IEEE Transactions on Automatic Control*, vol. 58, no. 3, pp. 785–791, 2013. View at Publisher · View at Google Scholar - T. C. Gard,
*Introduction to Stochastic Differential Equations*, vol. 114 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1988. View at MathSciNet - H. Gao, J. Lam, and C. Wang, “Robust energy-to-peak filter design for stochastic time-delay systems,”
*Systems & Control Letters*, vol. 55, no. 2, pp. 101–111, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Dissipativity analysis for discrete-time stochastic neural networks with time-varying delays,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 24, no. 3, pp. 345–355, 2013. View at Publisher · View at Google Scholar - H. Shen, S. Xu, X. Song, and G. Shi, “Passivity-based control for Markovian jump systems via retarded output feedback,”
*Circuits, Systems, and Signal Processing*, vol. 31, no. 1, pp. 189–202, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Shen, Z. Wang, X. Huang, and J. 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. Zhao, Q. Chen, and S. Xu, “${H}_{\infty}$ guaranteed cost control for uncertain Markovian jump systems with mode-dependent distributed delays and input delays,”
*Journal of the Franklin Institute*, vol. 346, no. 10, pp. 945–957, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - N. N. Krasovskiĭ and È. A. Lidskiĭ, “Analytical design of controllers in systems with random attributes—Part I,”
*Automation and Remote Control*, vol. 22, pp. 1021–1025, 1961. View at MathSciNet - E.-K. Boukas and Z.-K. Liu,
*Deterministic and Stochastic Time-Delay Systems*, Birkhäuser, Boston, Mass, USA, 2002. - O. L. V. Costa, M. D. Fragoso, and M. G. Todorov,
*Continuous-Time Markov Jump Linear Systems*, Probability and Its Applications, Springer, Heidelberg, Germany, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. He and F. Liu, “Unbiased ${H}_{\infty}$ filtering for neutral Markov jump systems,”
*Applied Mathematics and Computation*, vol. 206, no. 1, pp. 175–185, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. He and F. Liu, “Robust finite-time stabilization of uncertain fuzzy jump systems,”
*International Journal of Innovative Computing, Information and Control*, vol. 6, no. 9, pp. 3853–3862, 2010. View at Scopus - H. Shen, S. Xu, J. Zhou, and J. Lu, “Fuzzy ${H}_{\infty}$ filtering for nonlinear Markovian jump neutral systems,”
*International Journal of Systems Science*, vol. 42, no. 5, pp. 767–780, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Asynchronous ${l}_{2}\text{\u2013}{l}_{\infty}$ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities,”
*Automatica*, 2013. View at Publisher · View at Google Scholar - Z.-G. Wu, H. Su, P. Shi, and J. Chu,
*Analysis and Synthesis of Singular Systems with Time-Delays*, vol. 443 of*Lecture Notes in Control and Information Sciences*, Springer, Heidelberg, Germany, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - B. Zhang, W. X. Zheng, and S. Xu, “Filtering of Markovian jump delay systems based on a new performance index,”
*IEEE Transactions on Circuits and Systems I*, vol. 60, no. 5, pp. 1250–1263, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - H. Shen, S. Xu, X. Song, and J. Luo, “Delay-dependent robust stabilization for uncertain stochastic switching systems with distributed delays,”
*Asian Journal of Control*, vol. 11, no. 5, pp. 527–535, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Shen, S. Xu, J. Lu, and J. Zhou, “Passivity-based control for uncertain stochastic jumping systems with mode-dependent round-trip time delays,”
*Journal of the Franklin Institute*, vol. 349, no. 5, pp. 1665–1680, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - O. L. V. Costa and A. de Oliveira, “Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises,”
*Automatica*, vol. 48, no. 2, pp. 304–315, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. He and C.-L. Liu, “Finite-time ${H}_{\infty}$ fuzzy control of nonlinear jump systems with time delays via dynamic observer-based state feedback,”
*IEEE Transactions on Fuzzy Systems*, vol. 20, no. 4, pp. 605–614, 2012. View at Publisher · View at Google Scholar - H. Ma and Y. Jia, “${H}_{2}$ control of discrete-time periodic systems with Markovian jumps and multiplicative noise,”
*International Journal of Control*, vol. 86, no. 10, pp. 1837–1849, 2013. View at Publisher · View at Google Scholar - Y. Niu, D. W. C. Ho, and X. Wang, “Sliding mode control for Itô stochastic systems with Markovian switching,”
*Automatica*, vol. 43, no. 10, pp. 1784–1790, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - G.-H. Yang and W.-W. Che, “Non-fragile ${H}_{\infty}$ filter design for linear continuous-time systems,”
*Automatica*, vol. 44, no. 11, pp. 2849–2856, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Dong, Z. Wang, D. W. C. Ho, and H. Gao, “Robust ${H}_{\infty}$ fuzzy output-feedback control with multiple probabilistic delays and multiple missing measurements,”
*IEEE Transactions on Fuzzy Systems*, vol. 18, no. 4, pp. 712–725, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, vol. 15 of*Studies in Applied and Numerical Mathematics*, SIAM, Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - R. Yang, P. Shi, and G.-P. Liu, “Filtering for discrete-time networked nonlinear systems with mixed random delays and packet dropouts,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 11, pp. 2655–2660, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Xu and J. Lam,
*Robust Control and Filtering of Singular Systems*, vol. 332 of*Lecture Notes in Control and Information Sciences*, Springer, Berlin, Germany, 2006. View at MathSciNet - S. Xu and T. Chen, “Robust ${H}_{\infty}$ 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 MathSciNet · View at Scopus - H. Shen, X. Song, and Z. Wang, “Robust fault-tolerant control of uncertain fractional-order systems against actuator faults,”
*IET Control Theory & Applications*, vol. 7, no. 9, pp. 1233–1241, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Zhang, S. Xu, and H. Shen, “Reduced-order observer-based output-feedback tracking control of nonlinear systems with state delay and disturbance,”
*International Journal of Robust and Nonlinear Control*, vol. 20, no. 15, pp. 1723–1738, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z. Zhang, H. Shao, Z. Wang, and H. Shen, “Reduced-order observer design for the synchronization of the generalized Lorenz chaotic systems,”
*Applied Mathematics and Computation*, vol. 218, no. 14, pp. 7614–7621, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus