Since filtering plays an important role in control systems, signal processing and communication, there has been a rapidly growing interest in filter designing due to its advantages over the traditional Kalman filtering. In the past few years, many contributions on filtering for stochastic systems can be found in the literature  -  , because it is an important research topic and has found many practical applications. In  , a H∞ filter was designed for nonlinear stochastic systems. H∞ filtering problems for discrete-time nonlinear stochastic systems were addressed in  . Delay-dependent H∞ filtering for discrete-time singular systems and fuzzy discrete-time systems were reported respectively in     . In  , a H∞ filter was designed for discrete-time systems with stochastic incomplete measurement and mixed delays. Optimal filter was studied for Itô-stochastic continuous-time systems in  . Dissipativity-based filtering and H∞ filtering were presented for fuzzy switched systems respectively in    . Fault detection filtering and distributed filter were proposed for fault detection filtering for nonlinear stochastic systems in   . In  , event-based variance-constrained H∞ filter was reported for stochastic parameter systems.
As is well known, the previously mentioned literature was based on Lyapunov asymptotic stability which focuses on the steady-state behavior of plants over an infinite-time interval. But in many practical systems, it is only required that the system states remain within the given bounds. In these cases, the introduction of finite-time stability or short-time stability was needed, which has caused extensive attention  -  . The problem of finite-time stability and stabilization for a class of linear systems with time delay was addressed in  . In  , the sufficient conditions were achieved for the finite-time stability of linear time-varying systems with jumps. The problem of robust finite-time stabilization for impulsive dynamical linear systems was investigated in  . In  , fuzzy control method was adopted to solve finite-time stabilization of a class of stochastic system. A robust finite-time filter was established for singular discrete-time stochastic system in  . Finite-time H∞ filtering was proposed respectively for T-S fuzzy systems, switched systems, nonlinear singular systems, Itô stochastic Markovian jump systems in     . Motivated by the contributions mentioned above, we investigated the mode-dependent finite-time filtering problems for stochastic nonlinear systems, which could be used to detect generation of residuals for fault diagnosis problems.
This paper will study the H∞ filtering problem for a class of Markov Jump stochastic systems with Lipschitz nonlinearlity. The main purpose of this study is to construct a H∞ filter such that the resulting filter error augmented system is FTSS. The sufficient condition for FTSS of the filter error system is obtained by constructing the Lyapunov-Krasovskii functional candidate combined with LMIs. We present an approach to design the desired FTSS filter.
This paper is organized as follows. Some corresponding definitions and lemmas and the problem formulation are introduced in Section 2. In Section 3, we give a sufficient condition for FTSS of the mentioned filtering error system in terms of LMIs. Moreover, an approach of a finite-time H∞ filter is presented. Some conclusions are drawn in section 4.
We use to denote the n-dimensional Euclidean space. The notation (respectively, ), where X and Y are real symmetric matrices, means that the matrix is positive definite (respectively, positive semi-definite). I and 0 denote the identity and zero matrices with appropriate dimensions. and denotes the maximum and the minimum of the eigenvalues of a real symmetric matrix Q. The superscript T denotes the transpose for vectors or matrices. The symbol * in a matrix denotes a term that is defined by symmetry of the matrix.
Consider a class of Itô stochastic nonlinear system with Markovian switching, which can be described as follows:
where, , , , are state vector, measurement, external disturbance, and controlled output respectively, where satisfies the constraint condition with respect to the finite-time interval
and is a standard Wiener process satisfying , , which is assumed to be independent of the system mode . The random form process is a continuous-time discrete-state Markov process taking values in a finite set . The set N comprises the operation modes of the system. The transition probabilities for the process are defined as
where , and for is the transition probability rate from mode i at time t to mode j at time and .
For each possible value of in the succeeding discussion, we denote the matrices with the ith mode by
, , , , , ,
, , , , ,
where for any are known constant matrices of appropriate dimensions.
Assumption 1. The nonlinear functions and satisfy the following quadratic inequalities:
, . (6)
We now consider the following filter for system (1) - (3):
where is the filter state, are the filter parameters with compatible dimensions to be determined.
Define and then we can obtain the following filtering error system:
, , , , ,
, , , .
We introduce the following definitions and lemmas, which will be useful in the succeeding discussion.
Definition 1 (  ): The filtering error system (9) (10) with
is said to be finite-time stochastic stable (FTSS) with respect to
Definition 2: The filtering error system (9) (10) with is said to be finite-time stochastic stable (FTSS) with respect to if it is stochastic finite-time stable in the sense of definition 1 for all nonzero satisfying the constraint condition (4) for all under the zero-initial condition.
Definition 3: Given a disturbance attenuation level , the filtering error system (9) (10) with satisfying (4) is said to be H∞ finite-time stochastic stable (FTSS) with respect to with a prescribed disturbance attenuation level , if it is stochastic finite-time stable in the sense of Definition 1 and
Lemma 1 (Gronwall inequality  ): Let be a nonnegative function such that
for some constants , then we have .
Lemma 2 (Schur complement   ) Given a symmetric matrix , the following three conditions are equivalent to each other:
2) , and ;
3) , and .
Lemma 3 (Itô formula  ) Let x (t) be an n-dimensional Itô process on with the stochastic differential
where and , . Then is a real-valued Itô process with its stochastic differential
where the weak infinitesimal operator
3. Main Results
Theorem 1: Suppose that the filter parameters in (7) (8) are given. The filtering error system (9) (10) is FTSS with respect to , if there exist scalars and symmetric positive definite matrices satisfying
such that the following LMIs hold
where , and “*” denotes the transposed elements in the symmetric positions.
Proof: Define the following stochastic Laypunov-Krasovskii functional candidate:
By Itô formula, we have the weak infinitesimal operator of as follows:
Applying (6) and the following well-known fact:
it follows that
Let , from (21) - (24), it follows
Applying Schur complement, we have the following inequality by taking (17) into consideration:
Multiplying the above inequality by and by Gronwall inequality (12), we obtain the following inequality
Then, we have
Taking (29)-(31) into account, we obtain
Therefore, it follows that condition (18) implies . The filtering error system is finite-time bounded with respect to . This completes the proof.
Theorem 2: The filtering error system (9) (10) is FTSS with respect to and satisfies the condition (11), if there exist positive constant and symmetric positive definite matrices such that (16) (18) hold and
Proof: For the filtering error system (9) (10), consider the same stochastic Laypunov functional as in (19). Obviously, condition (33) implies that
where is given in (26).
By theorem 1, conditions (17) and (18) guarantee that system (9) (10) is FTSS with respect to .
Therefore, we only need to prove that (11) holds.
Noting that (27) and (34), we obtain
Then using the similar proof as Theorem 1, condition (11) can be easily obtained.
Theorem 3: The filtering error system (9) (10) is FTSS with respect to and satisfies the condition (11), if there exist positive constant and symmetric positive definite matrices and matrices such that (16) (18) hold and
In addition, the suitable parameters of the filter (7) (8) are given as follows:
, , . (36)
Proof: By theorem 2, let , , ,
Apply Surch complement for (33), then pre- and post-multiply and respectively, we can get inequality (35) from (33).
In this paper, we deal with the finite-time H∞ filter designing problem for a class of stochastic nonlinear systems with Markovian switching. The sufficient conditions for FTSS of the filtering error system have been presented and proved by the Lyapunov-Krasovski approach. The designed filter is provided to ensure the filtering error system FTSS and satisfies a prescribed H∞ performance level in a given finite-time interval, which can be reduced to feasibility problems involving restricted linear matrix equalities.
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