Mathematical Problems in Engineering

VolumeΒ 2012Β (2012), Article IDΒ 431576, 16 pages

http://dx.doi.org/10.1155/2012/431576

## Robust Reliable Control for Nonlinear Stochastic Markovian Jump Systems

^{1}Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan University of Science and Technology, Wuhan 430081, China^{2}Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Received 29 November 2011; Accepted 2 April 2012

Academic Editor: WeihaiΒ Zhang

Copyright Β© 2012 Guici Chen and Yi Shen. 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

The robust reliable control problem for a class of nonlinear stochastic Markovian jump systems (NSMJSs) is investigated. The system under consideration includes ItΓ΄-type stochastic disturbance, Markovian jumps, as well as sector-bounded nonlinearities and norm-bounded stochastic nonlinearities. Our aim is to design a controller such that, for possible actuator failures, the closed-loop stochastic Markovian jump system is exponential mean-square stable with convergence rate and disturbance attenuation . Based on the Lyapunov stability theory and ItΓ΄ differential rule, together with LMIs techniques, a sufficient condition for stochastic systems is first established in Lemma 3. Then, using the lemma, the sufficient conditions of the solvability of the robust reliable controller for linear SMJSs and NSMJSs are given. Finally, a numerical example is exploited to show the usefulness of the derived results.

#### 1. Introduction

In the past few decades, Markovian jump systems (MJSs) have been considerably studied since this kind of hybrid systems consists of a number of subsystems and a switch signal, which includes applications in safety-critical and high-integrity systems (e.g., aircraft, chemical plants, nuclear power station, robotic manipulator systems, large-scale flexible structures for space stations such as antenna, and solar arrays) typically systems, which may experience abrupt changes in their structure, see, for example, [1] and the references therein. And now, some results of stability and stabilization for ItΓ΄ type stochastic Markovian jump systems are also available in many papers, see, for example, [2β4] and the references therein.

The analysis and synthesis problems of Markovian jump systems (MJSs) or stochastic Markovian jump systems (SMJSs) have attracted plenty of attention from many researchers. Many important and remarkable achievements reasonable have obtained. If the control systems possess integrity against actuator and sensor failures, we called reliable control systems or fault-tolerant control systems [5]. Recently, the robust reliable control and filtering problems for time-delay systems or Markovian jump systems (MJSs) have attracted considerable attention, and several approaches have been developed, see, for example, [6β11] and the references therein. Via linear matrix inequalities (LMIs), the authors designed the robust reliable controller for uncertain nonlinear systems [6]. In [7], for admissible uncertainties as well as actuator failures occurring among a prespecified subset of actuators, Zhang et al. studied the reliable dissipative control of Markovian jump impulsive systems. The reliable control problem for discrete-time piecewise linear systems with infinite distributed delays have been investigated in [8]. Recently, the study of stochastic filtering for the systems governed by stochastic ItΓ΄-type equations has attracted a great deal of attention, and Zhang and Chen [9] firstly solved the nonlinear stochastic delay-free filtering problem by means of a stochastic bounded real lemma derived in [10]. The reliable filtering problems for discrete time-delay systems with randomly occurred nonlinearities [11] and discrete time-delay Markovian jump systems with partly unknown transition probabilities [12] also has been studied, respectively. The reliable control problem for a class of Markovian jump systems with interval time-varying delays and stochastic failure is studied in [13]. In recent years, the research begins to focusing on robust reliable control problems for stochastic systems or stochastic switched nonlinear systems, see, for example, [14β16] and the references therein.

However, all the aforementioned results are mainly focusing on the reliable control and filtering problems of discrete-time-delay systems and Markovian jump systems. Up to now, to the best of the authorsβ knowledge, the robust reliable control problem for nonlinear stochastic Markovian jump systems (NSMJSs) has not been fully investigated, which is an open problem and gives the motivation of our present investigation. In this paper, our aim is to design a robust reliable controller for NSMJSs, such that the NSMJSs are globally mean exponential stable with convergence rate and disturbance attenuation .

##### 1.1. Notations

Throughout this paper, for symmetric matrices and , the notation (resp., ) means that the Matrix is positive semidefinite (respectively, positive definite). is an identity matrix with appropriate dimensions; the subscript ββ represents the Transposition. denotes the expectation operator with respect to some probability measure . is the space of square integrable vector functions over ; let be a complete probability space which is relative to an increasing family of algebras , where is the samples space, is algebra of subsets of the sample space, and is the probability measure on ., while stands for the usual norm, and denote the dimensional Euclidean space and the set of all real matrices, respectively. In this paper, we provide all spaces with the usual inner product and its corresponding 2-norm. Let denote the space of square-integrable -valued functions on the probability space . For any , we write for the closure of the open interval in and denote by the space of the nonanticipative stochastic processes with respect to satisfying . , , .

#### 2. Problem Formulation and Failure Model

In this paper, we mainly consider the following nonlinear stochastic Markovian jump systems (NSMJSs) with actuator failures: where is the system state, is the control input of actuator fault, is the exogenous disturbance input of the systems which belong to , is the system control output, is a zero mean real scalar Weiner processes on a probability space relative to an increase family of algebras . are the known real constant matrices with appropriate dimensions. Morever, we assume that

Let , be a right-continuous Markovian chain on the probability space taking values in a finite state space with generator given by where . Here is the transition rate from manner to manner , if while . We assume that the Markovian chain is independent of the Wienner process . It is well known that almost every sample path of is a right-continuous step function with a finite number of simple jump in any finite subinterval of .

is a unknown nonlinear function which describes the system nonlinearity satisfying the following sector-bounded conditions: also is a unknown nonlinear function which describes the stochastic nonlinearity satisfying the following: where are known real constant matrices with approximate dimensions.

*Remark 2.1. *The nonlinearities are bounded by sectors, which belong to , and are very general that include the usual Lipschitz conditions as a special case which is considerable investigated and includes several other classes well studied nonlinear systems [17β19]. The nonlinearities satisfy the norm-bounded conditions.

When the actuator experiences failure, we use to describe the control signal form actuators. Consider the following actuator failure model with failure parameter : where is the actuator fault matrix with

In which the variables quantify the failures of the actuators. means that th actuator completely fails, and means that the th actuator is normal.

Define the following: and hence, the matrix can be rewritten as

In this paper, our aim is to design the controller , such that the closed-loop systems satisfy the following conditions:(i)without the exogenous disturbance input (i.e., ), the closed-loop control systems (2.1) are globally exponentially stable with convergence rate ;(ii)with zero initial condition (i.e., ) and nonzero exogenous disturbance input (i.e., ), the following inequality holds:

If the above two conditions hold, we also called the systems that are exponential mean-square stable with convergence rate and disturbance attenuation .

#### 3. Main Results

Lemma 3.1 (Schur complement lemma [20]). *For a given matrix with , the following conditions are equivalent:*(1)*,*(2)*,*(3)*.*

Lemma 3.2 (see [21]). *Let and . Then, for any positive scalar , we have
*

##### 3.1. Robust Reliable for LSMJSs

To obtain our main results, we first consider the following linear stochastic Markovian jump systems (LSMJSs) without control input:

Lemma 3.3. *Suppose that is continuously differentiable, then the systems (3.2) are exponential mean-square stable with convergence rate and disturbance attenuation if and only if the following matrix functional inequalities hold:
**
where .*

*Proof. *At first, let , and defining the following Lyapunov function:

By ItΓ΄ formula, we get the following:
the matrix function inequalities (3.3) imply that , and let , , where means the maximum eigenvalue of matrix , and we have

Hence
where , and . Integrating the both sides of above inequality from to and taking expectation, we obtain that

Set , and the following inequality is obtained:
which implies that

That is to say that the stochastic systems are globally exponentially stable with convergence rate .

Then, considering the stochastic performance level for the resulting systems (3.2) with nonzero exogenous disturbance input (), for any, we define that

By general ItΓ΄ formula, we get he following:
where , From (3.3) we know that , which implies that

Therefore, the inequality holds. The proof is completed.

In the following time, we consider the following linear stochastic Markovian jump systems (LSMJSs) under the state feedback controller:

Theorem 3.4. *If there exist the positive matrices , and the constant matrices with approximate dimensions, such that the following LMIs hold
**
where ,
**
then the LSMJSs (3.14) are exponential mean-square stable with convergence rate and disturbance attenuation . In this case, the desired controllers are given as follows:
*

*Proof. *Defining the following Lyapunov function:

By Lemma 3.3, and similar to the proof of Lemma 3.3, we can get the following:
where .

Using Schur complement lemma together with contragredient transformation, we know that LMIs (3.15) imply that . So we have
Therefore, the inequality holds. The proof is completed.

Theorem 3.5. *If there exist the positive matrices , the positive diagonal matrices , and the constant matrices with approximate dimensions, such that the following LMIs hold:
**
where , , Then the LSMJSs (3.14) are exponential mean-square stable with convergence rate and disturbance attenuation . In this case, the desired controllers are given as follows:
*

*Proof. *Noticing (2.10), we can see that in (3.15) can be rewritten as
where .

By Lemma 3.2, we have
by Schur complement, we know that imply that . Therefore, we can know from Theorem 3.4 that the LSMJSs (3.14) are stabilizable with convergence rate and disturbance attenuation. This completes the proof.

##### 3.2. Robust Reliable for NSMJSs

In this section, we consider the following nonlinear stochastic Markovian jump systems (NSMJSs) under the state feedback controller:

Theorem 3.6. *If there exist the positive matrices , and the constant matrices with approximate dimensions, for the positive constant and the given scalar , such that the following LMIs hold:
**
where ,, then the NSMJSs (3.25) are exponential mean-square stable with convergence rate and disturbance attenuation. In this case, the desired controllers are given as follows:
*

*Proof. *Defining the following Lyapunov function:

by ItΓ΄ formula, we get the following:
where .

By Lemma 3.2, it follows that
from (2.4) which are equivalent to

Considering the stochastic performance level for the resulting systems (3.25) with nonzero exogenous disturbance input , for any, we define that

By general ItΓ΄ formula, for a given positive scalar , we get the following:
where

By Schur complement lemma, we see that is equivalent to the following matrix inequalities:
which is implied in LIMs (3.26). Hence .

Therefore, the inequality holds. The proof is completed.

Similar to the proof of Theorem 3.5, we can get the following theorem without proof immediately.

Theorem 3.7. *If there exist the positive matrices , and the constant matrices with approximate dimensions, for the positive constant and the given scalar , such that the following LMIs hold
**
then the NSMJSs (3.27) are exponential mean-square stable with convergence rate and disturbance attenuation. In this case, the desired controllers are given as follows:
*

#### 4. Numerical Example with Simulation

In this section, we will give an example to show the usefulness of the derived results and the effectiveness of the proposed methods (Figure 1).

Consider linear SMJSs (3.14) with , and the system parameters are given as follows:

The actuator failure parameters are as follows:

From Figure 2, we can see that the uncontrolled LSMJSs are not stable, according to Theorem 3.5. By using the LMI toolbox, the controller parameters can be calculated as follows:

Figures 3 and 4 give the simulation results of the response for the closed-loop LSMJSs, which confirm that the closed-loop LSMJSs are exponential mean-square stable with convergence rate and disturbance attenuation .

#### 5. Conclusions

In this paper, we have studied the robust reliable control problems for a class of NSMJSs. The system under study contains ItΓ΄-type stochastic disturbance, Markovian jumps, sector-bounded nonlinearities, and norm-bounded stochastic nonlinearities. Based on the Lyapunov stability theory and ItΓ΄ differential rule, sufficient condition which ensures exponential mean-square stable with convergence rate and disturbance attenuation for SMJSs has been established in Lemma 3.3. By the lemma, together with the LMIs techniques, the sufficient conditions for the designation of the robust reliable controller of linear SMJSs and NSMJSs have been obtained in terms of LMIs. Finally, a numerical example has been given to show the usefulness of the derived results and the effectiveness of the proposed methods. It is possible to extend our main results to the NSMJSs with time delay by using delay-dependent techniques, which is one of the future research topics.

#### Acknowledgments

This work is supported by National Science Foundation of China (NSFC) under Grant 61104127, 60904060, and 61134012 Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) under Grant Y201101.

#### References

- O. L. V. Costa, M. D. Gragoso, and R. P. Marques,
*Descrete-Time Markov Jump Linear Systems, Probability and Its Applications*, Springer, New York, NY, USA, 2005. - P. V. Pakshin, βRobust high moment stabilizing control for stochastic systems with jumps,β
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 47, no. 4, pp. 2473β2484, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X. Mao, βStability of stochastic differential equations with Markovian switching,β
*Stochastic Processes and their Applications*, vol. 79, no. 1, pp. 45β67, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Dong and J. Sun, βOn hybrid control of a class of stochastic non-linear Markovian switching systems,β
*Automatica*, vol. 44, no. 4, pp. 990β995, 2008. View at Publisher Β· View at Google Scholar - G.-H. Yang, J. L. Wang, and Y. C. Soh, βReliable ${H}_{\mathrm{\beta \x88\x9e}}$ controller design for linear systems,β
*Automatica*, vol. 37, no. 5, pp. 717β725, 2001. View at Google Scholar Β· View at Zentralblatt MATH - C.-H. Lien, K.-W. Yu, Y.-F. Lin, Y.-J. Chung, and L.-Y. Chung, βRobust reliable ${H}_{\mathrm{\beta \x88\x9e}}$ control for uncertain nonlinear systems via LMI approach,β
*Applied Mathematics and Computation*, vol. 198, no. 1, pp. 453β462, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Zhang, Z.-H. Guan, and G. Feng, βReliable dissipative control for stochastic impulsive systems,β
*Automatica*, vol. 44, no. 4, pp. 1004β1010, 2008. View at Publisher Β· View at Google Scholar - W. Zhang, G. Wei, and G. Feng, βReliable ${H}_{\mathrm{\beta \x88\x9e}}$ control for discrete-time piecewise linear systems with infinite distributed delays,β
*Automatica*, vol. 45, no. 12, pp. 2991β2994, 2009. View at Publisher Β· View at Google Scholar Β· View at Scopus - W. Zhang and B.-S. Chen, βState feedback ${H}_{\mathrm{\beta \x88\x9e}}$ control for a class of nonlinear stochastic systems,β
*SIAM Journal on Control and Optimization*, vol. 44, no. 6, pp. 1973β1991, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - W. Zhang, B.-S. Chen, and C.-S. Tseng, βRobust ${H}_{\mathrm{\beta \x88\x9e}}$ filtering for nonlinear stochastic systems,β
*IEEE Transactions on Signal Processing*, vol. 53, no. 2, pp. 589β598, 2005. View at Publisher Β· View at Google Scholar - Y. Liu, Z. Wang, and W. Wang, βReliable ${H}_{\mathrm{\beta \x88\x9e}}$ filtering for discrete time-delay systems with randomly occurred nonlinearities via delay-partitioning method,β
*Signal Processing*, vol. 91, no. 4, pp. 713β727, 2011. View at Publisher Β· View at Google Scholar Β· View at Scopus - Y. Liu, Z. Wang, and W. Wang, βReliable ${H}_{\mathrm{\beta \x88\x9e}}$ filtering for discrete time-delay Markovian jump systems with partly unknown transition probabilities,β
*International Journal of Adaptive Control and Signal Processing*, vol. 25, no. 6, pp. 554β570, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Gu, D. Yue, D. Wang, and J. Liu, βStochastic faulty actuator-based reliable control for a class of interval time-varying delay systems with Markovian jumping parameters,β
*Optimal Control Applications & Methods*, vol. 32, no. 3, pp. 313β327, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Liu, Z. Wang, and W. Wang, βRobust reliable control for discrete-time-delay systems with stochastic nonlinearities and multiplicative noises,β
*Optimal Control Applications & Methods*, vol. 32, no. 3, pp. 285β297, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. R. Xiang, C. H. Qiao, and R. H. Wang, βRobust ${H}_{\mathrm{\beta \x88\x9e}}$ reliable control of uncertain stochastic switched non-linear systems,β
*International Journal of Advanced Mechatronic Systems*, vol. 3, no. 2, pp. 98β108, 2011. View at Google Scholar - Z. Xiang, R. Wang, and Q. Chen, βRobust reliable stabilization of stochastic switched nonlinear systems under asynchronous switching,β
*Applied Mathematics and Computation*, vol. 217, no. 19, pp. 7725β7736, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Wang, Y. Liu, and X. Liu, β${H}_{\mathrm{\beta \x88\x9e}}$ filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities,β
*Automatica*, vol. 44, no. 5, pp. 1268β1277, 2008. View at Publisher Β· View at Google Scholar - Q.-L. Han, βAbsolute stability of time-delay systems with sector-bounded nonlinearity,β
*Automatica*, vol. 41, no. 12, pp. 2171β2176, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Lam, H. Gao, S. Xu, and C. Wang, β${H}_{\mathrm{\beta \x88\x9e}}$ and ${L}_{2}/{L}_{\mathrm{\beta \x88\x9e}}$ model reduction for system input with sector nonlinearities,β
*Journal of Optimization Theory and Applications*, vol. 125, no. 1, pp. 137β155, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, vol. 15 of*SIAM Studies in Applied Mathematics*, SIAM, Philadelphia, Pa, USA, 1994. - L. Xie, C. De Souza, and M. Fu, β${H}_{\mathrm{\beta \x88\x9e}}$ Bestimation for discrete time linear uncertain systems,β
*International Journal of Robust and Nonlinear Control*, vol. 1, no. 2, pp. 111β123, 1991. View at Google Scholar