Research Article | Open Access

# Finite-Time Filtering for Singular Stochastic Markovian Jump Systems with Time-Varying Delays

**Academic Editor:**P. Balasubramaniam

#### Abstract

The issue of finite-time filtering for singular stochastic Markovian jump systems with time-varying delays is concerned in this paper. filtering is designed for underlying closed-loop singular Markovian jump system and system state does not exceed a given bound over some finite-time interval. Considering the full information of underlying Markov process, sufficient conditions are obtained to guarantee that the described system is finite-time stability and filtering finite-time boundedness. By establishing the results of stochastic character and finite-time boundedness, the closed-loop singular Markovian jump system trajectory stays within the given bound. At last, a numerical example is supplied to show the efficiency of the proposed method.

#### 1. Introduction

As the fundamental problem in the field of control application and output measurement is used, the filtering problem has been attracting much attention [1, 2]. Among the existing robust control approaches, the traditional Kalman filtering is a well-known effective way to deal with the state estimation problems, which requires the exact knowledge of the statics of system model and external noise signal. The disturbances of stationary Gaussian noises with known statistics and system model are exactly known under the consideration. Over the past two decades, because of their applications in a variety of areas, the filtering problems have been widely studied in practical engineering. As mentioned above, filtering technique is employed in most applications to describe the input or output behavior of the controlled system. It is important to investigate filtering problem, and many results have been established [3–8].

On the other hand, as an important kind of hybrid systems, Markovian jump systems have a strong practical background in physical systems, which are subjected to abrupt variations in their structures. Moreover, Markovian chain in a finite mode set always determines the transitions models, which can describe a set of linear Markovian jump systems. In the past two years, Markovian jump systems have received much attention because they continue being a hot research area of mathematical models to represent many physical systems. It should be noted that, the existence of abrupt changes on the modes of operation in many practical systems, and the class of systems known as Markov jump systems has been proved to be useful to model a great number of them. Recently, a large number of results on the stability, estimation, and control problems related to such systems have been reported [9–16].

In many engineering systems, singular systems have the extensive applications in power systems, electrical circuits, and other fields [17, 18]. Singular systems are famous for its better performance in physical systems when compared with state-space ones. Recently, the filtering problems with theory of state-space ones have also found their wide application in the singular systems. Furthermore, as one special family of stochastic systems associated with time delays, the transition probabilities of Markovian jump systems are better to decide the singular system performance. Up to now, many results on filtering problem for singular Markovian jump systems with or without time delay have been reported [19–24], while there are some related issues that need to be solved.

It should be pointed out that the classical filtering problems concern the asymptotic performance over the infinite-time interval. However, the transient behavior of systems is always considered in practical problems and traditional asymptotical stability is not applicable; the finite-time stability was proposed [25]. In the finite-time interval, finite-time stability is investigated to address these transient behaviors of control systems when compared with classical asymptotical stability. Recently, the concept of finite-time stability has been revisited in the light of linear matrix inequalities (LMIs) and theory of Lyapunov function; many results are reported subject to finite-time stability and finite-time boundedness [25–33]. It should be noted that there are still some related issues to be solved; to the best of our knowledge, the finite-time filtering problems for singular Markovian jump systems have not been fully studied. The topic remains interesting and challenging, which motivates the present study.

This paper is concerned with the finite-time boundedness of filtering for singular Markovian jump systems. Finite-time stability can be affected by switching behavior significantly, which is an independent concept from traditional Lyapunov stability; thus, it deserves our investigation. In this paper, we choose the appropriate Lyapunov-Krasovskii functional; making full information of the underlying Markov process, the sufficient conditions are derived to guarantee the finite-time boundedness of the systems. The finite-time boundedness (FTB) criteria can be tackled in the form of LMIs and optimization algorithms. At last, an example is given to illustrate the effectiveness of the developed method.

#### 2. Preliminaries

A probability space is given, where , , and , respectively, represent the sample space, the algebra of events, and the probability measure defined on . Let the random form process be the Markov stochastic process taking values on a finite set with transition rate matrix , . Define the following transition probability from mode at time to mode at time aswhere , , , and with transition probability rates for , , and .

In this paper, we consider the singular systems as follows:where is the state vector of the system, is the measured output, is the controlled output, and is a singular matrix with . , , , , and are known mode-dependent constant matrices with appropriate dimension. is the external disturbance vector which satisfiesand time-varying delay is the continuous function satisfying

For system (2), we are interested in designing the filter described bywhere is the filter state, is the estimation of in system (2), and the matrices , , and are the unknown filter parameters to be designed.

Augmenting the model of (2) to include filter (6), we obtain the following filtering error system:where the state estimate error and the output error and

For brevity, in the sequel, , , and , for every , and the other symbols are similarly denoted.

Some definitions and lemmas should be introduced to facilitate the following discussion.

*Definition 1 (see [21]). *(i) The singular system (7) is said to be regular if is not identically zero for every .

(ii) The singular system (7) is said to be impulse-free if for every .

*Definition 2 (see [21]). *The closed-loop continuous-time singular Markovian jump system (SMJS) (7) satisfying (3) is said to be singular stochastic finite-time boundedness (SSFTB) with respect to with and , if the stochastic system is regular and impulse-free in time interval and

*Definition 3 (see [21]). *The closed-loop continuous-time singular Markovian jump system (SMJS) (7) satisfying (3) is said to be singular stochastic finite-time boundedness with respect to , if the closed-loop continuous-time SMJS (7) is with respect to and under the zero-initial condition the controlled output satisfiesfor any nonzero which satisfies (3), where is a prescribed positive scalar.

*Definition 4 (see [16]). *Consider as the stochastic Lyapunov function of the resulting system (7); its weak infinitesimal operator is defined as

*Definition 5 (see [27]). *The jump rates of the visited modes from a given mode are assumed to satisfy where and are known parameters for a given mode and represent the lower and upper bounds when all the jump rates are known; that is, and . Meanwhile, the number of the visited modes from a given mode is denoted by including the mode itself.

Lemma 6 (see [4]). *For the positive matrices the following integral inequality holds:*

#### 3. Finite-Time Performance Analysis

Theorem 7. *Consider the closed-loop continuous-time SMJS (7); there exist positive scalars and with , for all admissible subject to condition (3), if there exist symmetric positive definite matrices , , , and , , such that, for all , the following linear matrix inequalitieswherehold; the closed-loop SMJS (7) is SSFTB with respect to .*

*Proof. *At first, we will prove that the singular Markovian jump system (7) is regular and impulse-free.

Noting (16), we can have thatSince , there must exist two invertible matrices and such thatWe denoteAccording to (15), it can be obtained thatPre- and postmultiplying (22) by and , we can easily obtain . Therefore, is nonsingular. Otherwise, supposing is singular, there exists a nonzero vector which ensures that . And then we can conclude that , and this contradicts .

Then, it can be shown thatwhich implies that is not identically zero and . Therefore, by Definition 1, we can obtain that the closed-loop SMJS (7) is regular and impulse-free in time interval .

We consider the following Lyapunov-Krasovskii functional:Taking the time derivative of along the trajectory of system (7), one hasIt should be noted thatFrom Lemma 6, one hasMoreover, we haveThen, from (24)–(29) and Schur complement, one can obtainwhereFrom condition (15), we haveMultiplying (32) by , it yieldsIntegrating (33) from 0 to , then we can obtainNoting that , . Define , , , and ; by Dynkin’s formula, we haveOn the other hand, it follows from (24) thatThen, one hasTherefore, from (37) and by Definition 2, we conclude that . This completes the proof.

Theorem 8. *For a given constant and , system (4) is robust finite-time stability with respect to , for all admissible subject to condition (6), if there exist symmetric positive definite matrices , , , , such that the following linear matrix inequalities hold:*

*Proof. *We will show the performance of system (7); from Theorem 7, we haveCondition (38) yieldsDefineMultiplying (41) by , it follows thatAccording to condition (42), integrating this inequality during yieldsThen we haveThus it is concluded by Definition 3 that the closed-loop SMJS (7) is SSFTB with performance . The proof is completed.

#### 4. Finite-Time Filter Design

Theorem 9. *Consider the finite-time switched discrete-time system (2) and a given scalar . Then there exists a switched filter in the form of (6) such that the filtering error system (7) is finite-time boundedness with performance level , if there exist real matrixes , , , , and . , , and of appropriate dimensions and real matrices , such that andwherewith hold; then the closed-loop SMJS (7) is SSFTB with respect to with performance .**In this case, desired filter parameter matrices are given by*

*Proof. *The Lyapunov matrices can be parameterized as follows:where are symmetric positive definite mode-dependent matrices. We also parameterize asin which are nonsingular matrices. Note that are matrices and belong to . Also we will assume which excludes the existence of a permanent mode in the system.

First, we haveAccording to Schur’s complement, (37) can be rewritten aswhere With matrices , inequalities (51) could be decomposed intowhereSince the equalityholds for every , is equivalent toAlso we haveTaking , , and , we get Therefore, we can get (44) and the proof is complete.

#### 5. Illustrative Example

Consider a finite-time stabilization of switched system as follows:

Initial states satisfy ; the filtering objective is to find feedback filter parameter ensuring that system (5) is finite-time boundedness with minimum value of . Choosing , . According to Theorem 9, we can also obtain that feasible solution , , and performance level is . Furthermore, Figures 1 and 2 are given to illustrate the optimal value with different value of . By solving the matrix equalities in Theorem 9, we have the following filter parameters:

Two classes of filters are involved in this example; filter (4) serves in short-time singular Markovian jump systems to guarantee the error state bounded in the prescribed boundary.

#### 6. Conclusion

In this paper,we have examined the problems of filtering for short-time singular Markovian jump systems. Based on the analysis result, the static state feedback finite-time boundedness is given. Although the derived result is not in LMIs form, we can turn it into LMIs feasibility problem by fixing some parameters. At last, a numerical example is also given to illustrate the effectiveness of the proposed approach. Furthermore, the mode-dependent singular matrices and uncertain matrices will be studied in the future for the recurrent singular Markovian jump system.

#### 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 (Grant no. 61173121), the Postdoctoral Foundation of China (Grant no. 20100471665), the Youth Science Foundation of Sichuan Province, in China (Grant no. 2010JQ0032), and the Science Foundation of Chengdu City in China (Grant no. 60873092).

#### References

- A. Elsayed and M. J. Grimble, “A new approach to the ${H}_{\infty}$ design of optimal digital linear filters,”
*IMA Journal of Mathematical Control and Information*, vol. 6, no. 2, pp. 233–251, 1989. View at: Publisher Site | Google Scholar | MathSciNet - D. Zhang, W. J. Cai, and Q.-G. Wang, “Mixed ${H}_{\infty}$ and passivity based state estimation for fuzzy neural networks with Markovian-type estimator gain change,”
*Neurocomputing*, vol. 139, pp. 321–327, 2014. View at: Publisher Site | Google Scholar - B. Shen, Z. Wang, H. Shu, and G. Wei, “${H}_{\infty}$ filtering for nonlinear discrete-time stochastic systems with randomly varying sensor delays,”
*Automatica*, vol. 45, no. 4, pp. 1032–1037, 2009. View at: Publisher Site | Google Scholar | MathSciNet - P.-L. Liu, “Improved delay-dependent robust stability criteria for recurrent neural networks with time-varying delays,”
*ISA Transactions*, vol. 52, no. 1, pp. 30–35, 2013. View at: Publisher Site | Google Scholar - J. Liu, S. Hu, and E. Tian, “A novel method of ${H}_{\infty}$ filter design for time-varying delay systems,”
*International Journal of Innovative Computing, Information and Control*, vol. 7, no. 3, pp. 1299–1310, 2011. View at: Google Scholar - 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 Site | Google Scholar | MathSciNet - D. Zhang, W. J. Cai, L. H. Xie, and Q. G. Wang, “Non-fragile distributed filtering for T-S fuzzy systems in sensor networks,”
*IEEE Transactions on Fuzzy Systems*, 2014. View at: Publisher Site | Google Scholar - D. Yue, Q.-L. Han, and J. Lam, “Network-based robust ${H}_{\infty}$ control of systems with uncertainty,”
*Automatica*, vol. 41, no. 6, pp. 999–1007, 2005. View at: Publisher Site | Google Scholar | MathSciNet - S. Xu, J. Lam, and X. Mao, “Delay-dependent ${H}_{\infty}$ 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. 2070–2078, 2007. View at: Publisher Site | Google Scholar | MathSciNet - Y. Zhang, S. Xu, and B. Zhang, “Robust output feedback stabilization for uncertain discrete-time fuzzy Markovian jump systems with time-varying delays,”
*IEEE Transactions on Fuzzy Systems*, vol. 17, no. 2, pp. 411–420, 2009. View at: Publisher Site | Google Scholar - P. Balasubramaniam and G. Nagamani, “Global robust passivity analysis for stochastic interval neural networks with interval time-varying delays and Markovian jumping parameters,”
*Journal of Optimization Theory and Applications*, vol. 149, no. 1, pp. 197–215, 2011. View at: Publisher Site | Google Scholar | MathSciNet - S. Lakshmanan and P. Balasubramaniam, “New results of robust stability analysis for neutral-type neural networks with time-varying delays and Markovian jumping parameters,”
*Canadian Journal of Physics*, vol. 89, no. 8, pp. 827–840, 2011. View at: Publisher Site | Google Scholar - G. Wang, Q. Zhang, and V. Sreeram, “Design of reduced-order ${H}_{\infty}$ filtering for Markovian jump systems with mode-dependent time delays,”
*Signal Processing*, vol. 89, no. 2, pp. 187–196, 2009. View at: Publisher Site | Google Scholar - J. Cao and Z. Lin, “Bayesian signal detection with compressed measurements,”
*Information Sciences*, vol. 289, pp. 241–253, 2014. View at: Publisher Site | Google Scholar - G. Wang, Q. Zhang, and V. Sreeram, “Partially mode-dependent ${H}_{\infty}$ filtering for discrete-time Markovian jump systems with partly unknown transition probabilities,”
*Signal Processing*, vol. 90, no. 2, pp. 548–556, 2010. View at: Publisher Site | Google Scholar - Q. Ma, S. Xu, Y. Zou, and J. Lu, “Stability of stochastic Markovian jump neural networks with mode-dependent delays,”
*Neurocomputing*, vol. 74, no. 12-13, pp. 2157–2163, 2011. View at: Publisher Site | Google Scholar - L. Dai,
*Singular Control Systems*, Springer, Berlin, Germany, 1989. View at: MathSciNet - J. Cao, “Improved delay-dependent stability conditions for MIMO networked control systems with nonlinear perturbations,”
*The Scientific World Journal*, vol. 2014, Article ID 196927, 4 pages, 2014. View at: Publisher Site | Google Scholar - X. Sun and Q. Zhang, “Delay-dependent robust stabilization for a class of uncertain singular delay systems,”
*International Journal of Innovative Computing, Information and Control*, vol. 5, no. 5, pp. 1231–1242, 2009. View at: Google Scholar - A. Haidar and E. K. Boukas, “Exponential stability of singular systems with multiple time-varying delays,”
*Automatica*, vol. 45, no. 2, pp. 539–545, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Z.-G. Wu, J. H. Park, H. Su, B. Song, and J. Chu, “Reliable ${H}_{\infty}$ filtering for discrete-time-singular systems with randomly occurring delays and sensor failures,”
*IET Control Theory & Applications*, vol. 6, no. 14, pp. 2308–2317, 2012. View at: Publisher Site | Google Scholar | MathSciNet - J. Cao and L. Xiong, “Protein sequence classification with improved extreme learning machine algorithms,”
*BioMed Research International*, vol. 2014, Article ID 103054, 12 pages, 2014. View at: Publisher Site | Google Scholar - S. Xu and J. Lam, “Reduced-order ${H}_{\infty}$ filtering for singular systems,”
*Systems & Control Letters*, vol. 56, no. 1, pp. 48–57, 2007. View at: Publisher Site | Google Scholar | MathSciNet - Z. Wu, H. Su, and J. Chu, “${H}_{\infty}$ filtering for singular systems with time-varying delay,”
*International Journal of Robust and Nonlinear Control*, vol. 20, no. 11, pp. 1269–1284, 2010. View at: Publisher Site | Google Scholar | MathSciNet - F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,”
*Automatica*, vol. 37, no. 9, pp. 1459–1463, 2001. View at: Publisher Site | Google Scholar - J. Cheng, H. Zhu, S. Zhong, Q. Zhong, and Y. Zeng, “Finite-time ${H}_{\infty}$ estimation for discrete-time Markov jump systems with time-varying transition probabilities subject to average well time switching,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 20, no. 2, pp. 571–582, 2015. View at: Publisher Site | Google Scholar | MathSciNet - J. Cheng, S. Zhong, Q. Zhong, H. Zhu, and Y. Du, “Finite-time boundedness of state estimation for neural networks with time-varying delays,”
*Neurocomputing*, vol. 129, pp. 257–264, 2014. View at: Publisher Site | Google Scholar - S. He and F. Liu, “Finite-time boundedness of uncertain time-delayed neural network with Markovian jumping parameters,”
*Neurocomputing*, vol. 103, no. 1, pp. 87–92, 2013. View at: Publisher Site | Google Scholar - Y. Zhang, C. Liu, and X. Mu, “Robust finite-time stabilization of uncertain singular Markovian jump systems,”
*Applied Mathematical Modelling*, vol. 36, no. 10, pp. 5109–5121, 2012. View at: Publisher Site | Google Scholar | MathSciNet - F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems with jumps,”
*Automatica*, vol. 45, no. 5, pp. 1354–1358, 2009. View at: Publisher Site | Google Scholar | MathSciNet - J. Cheng, H. Zhu, S. Zhong, F. Zheng, and Y. Zeng, “Finite-time filtering for switched linear systems with a mode-dependent average dwell time,”
*Nonlinear Analysis: Hybrid Systems*, vol. 15, pp. 145–156, 2015. View at: Publisher Site | Google Scholar | MathSciNet - S. He and F. Liu, “Stochastic finite-time boundedness of Markovian jumping neural network with uncertain transition probabilities,”
*Applied Mathematical Modelling*, vol. 35, no. 6, pp. 2631–2638, 2011. View at: Publisher Site | Google Scholar | MathSciNet - X. Luan, F. Liu, and P. Shi, “Finite-time filtering for non-linear stochastic systems with partially known transition jump rates,”
*IET Control Theory & Applications*, vol. 4, no. 5, pp. 735–745, 2010. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2015 Bin Yan et al. 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.