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Finite-Time Filtering for Singular Stochastic Markovian Jump Systems with Time-Varying Delays
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.
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 . 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.
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.
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 2 (see ). 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 ). 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 5 (see ). 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 ). 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.
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.
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).
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