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Research Article | Open Access
Delay-Dependent Robust Filtering of the Takagi-Sugeno Fuzzy Stochastic Systems
This paper is concerned with the problem of the robust filtering for the Takagi-Sugeno (T-S) fuzzy stochastic systems with bounded parameter uncertainties. For a given T-S fuzzy stochastic system, this paper focuses on the stochastically mean-square stability of the filtering error system and the performance level of the output error and the disturbance input. The design method for delay-dependent filter is developed based on linear matrix inequalities. Finally, the effectiveness of the proposed methods is substantiated with an illustrative example.
As an efficient technique to linearize the nonlinear differential equations, the T-S fuzzy model  has been an active research area over the past three decades. This model is capable of representing linear input-output relations of nonlinear systems by appropriate fuzzy sets, such as the stirred tank reactor system  and the truck trailer system . And it has been proved that this model can accurate to a compact set by a family of IF-THEN rules. This way, the stability analysis and synthesis of the nonlinear system turn into the analysis of linear systems, where the linear system theory can be conveniently applied. In recent years, the researches of T-S fuzzy have grown into a great number. To mention a few, the stability and control problem of T-S fuzzy systems have been investigated in [4–14] and the references therein.
Moreover, as a branch of state estimation theory, the filtering problem has become an important research field for many kinds of systems. The filtering problems for the T-S fuzzy systems have been addressed in [15–19]; references [20, 21] have considered the filtering problem for delayed T-S fuzzy systems with different methods.
On the other hand, stochastic system has received considerable attention because uncertain factors are unavoidable in most of the physical systems, for example, signal processing, engineering, finance, economics; biological movement systems, and so forth. Stochastic modeling has become important in many branches of engineering applications . And many results on the study of stochastic systems can be found in the literature. References [23, 24] study the problem of designing delay-dependent controllers and output feedback controller for nonlinear stochastic time-delay system with method, respectively. The sliding mode control for the time-delay nonlinear Itô stochastic systems was proposed in [25, 26]. References [26, 27] have investigated the stability of the time-delay stochastic neutral networks. The problem of filtering is considered in [29–31]. Fault detection problem of the stochastic system has been addressed in [32, 33]. And the problem of model reduction in stochastic framework is investigated in .
Through the above analysis, T-S fuzzy model could be used to represent the nonlinear stochastic system with several subsystems that could easily be analyzed. There have been a few works in this manner;  deals with the robust fault detection problem for T-S fuzzy stochastic systems. And  considers the stabilization for the stochastic fuzzy systems with delays. However, to the best of the authors knowledge, few results on filtering problem for TCS fuzzy stochastic systems are available which still remains challenging.
Inspired by the above discussions, this paper will focus on the robust fuzzy delay-dependent filter design for a T-S fuzzy stochastic system with time-varying delays and norm-bounded parameter uncertainties. The problem we consider here is to make sure that the fuzzy filters we design could ensure both the robust stochastic mean-square stability and a prescribed performance level of the filtering error system. During the proof of the theorems, some useful free-weighting matrices are introduced to reduce the potential conservatism as much as we can. By using the Lyapunov-Krasovskii functional technique, a linear matrix inequality (LMI) approach is proposed to solve the problem.
The remainder of the paper is organized as follows. Section 2 formulates the filter design problem. Section 3 gives the delay-dependent conditions for the stochastic stability problem of the T-S fuzzy stochastic systems. And the solvability of the filtering design problem is obtained in terms of LMIs, which are presented in Section 4. In Section 5, a numerical example is shown to illustrate the effectiveness of the proposed methods. Finally, we conclude the paper in Section 6.
Notation. The notation used in this paper is fairly standard. The superscript “” stands for matrix transposition. Throughout this paper, for real symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite). denotes the -dimensional Euclidean space, and denotes the set of all real matrices. stands for an identity matrix of appropriate dimension, while denotes a vector of ones. The notation * is used as an ellipsis for terms that are induced by symmetry. stands for a block-diagonal matrix. denotes the Euclidean norm for vectors, and denotes the spectral norm for matrices. represents the space of square-integrable vector functions over . stands for the mathematical expectation operator. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.
2. Problem Formulation and Preliminaries
Consider a T-S fuzzy stochastic time-delay system with time-varying parameter uncertainties.
Plant rule : IF is , and is , and and is , THEN where are the premise variables, is the fuzzy set, and is the number of IF-THEN rules; is the system state; is a given differential initial function on , and is the measured output; is a signal to be estimated; is the noise signal which belongs to ; and are continuous differentiable functions representing the time-varying delays, which are assumed to satisfy for all , In the considered system, , ,,,, ,,,,,,,, and are known constant matrices with appropriate dimensions. , , , , , and represent the unknown time-varying parameter uncertainties and are assumed to satisfy where , , , , and are known real constant matrices and the unknown time-varying matrix function satisfying
Let , .
For convenience, the filtering error dynamic system can be written as where
The purpose of this work is to design a sets of fuzzy filters in the form of (11) such that for any scalar , and a prescribed level of noise attenuation , the filtering error system () is mean-square stable, and the error system () satisfies performance.
Now, we introduce the following definitions and lemmas, which help to complete the proof of the main results.
Definition 1. The system () is said to be robust stochastic mean-square stable if there exists for any such that when , for any uncertain variables. In addition, for any initial conditions.
Definition 2. The robust stochastic mean-square stable system () is said to satisfy the performance; for the given scalar and any nonzero , the system () satisfies for any uncertain variables, where
Lemma 3. For the given matrices , , and with and positive scalar , the following inequality holds:
3. Robust Stochastic Stabile
First, we derive the robust stochastic mean-square stochastic conditions and the conditions for system (). Defining the following variables for convenience:
Theorem 4. For given scalars , , the filtering error system () is robust stochastic mean-square stables and (16) is satisfied if there exist matrices , , , , , and , , such that the following matrix inequalities hold: where
Proof. Define the following Lyapunov-Krasovskii candidate for system ():
By the Newton-Leibnitz formula, we can get the following equations for any matrices , with any appropriate dimensions: where and is a new vector defined as follows:
Using the above formulas (24) and Lemma 3, it can be seen that whereNote that So, the mathematical expectation of the last eight parts of (27) equals 0. And applying the Schur complement to (3), we can derive the following inequality with : From (27), (29), and (30), we can see that which, by Definition 2.1 [37, 38], ensures that system () with is robustly stochastically stable. By Itô’s formula, we can derive
Now, set a functional as where . From (32) and (33), it is easy to show that for all . By using the Schur complement to (3), there is where
It follows from (33) and (34) that for all , which implies that (10) is satisfied. This completes the proof.
Remark 5. The Lyapunov functional (22) contains the information of the uppers bound of the delays; by such a choice, delay-dependent results are obtained.
4. Robust Filter Design
In this section, a sufficient condition for the solvability of robust filter problem for uncertain T-S fuzzy stochastic time-delay system is investigated. The main result is given in the following theorem by LMI form.
Theorem 6. Consider the uncertain T-S fuzzy stochastic time-delay system (). The robust filtering problem is solvable with disturbance attenuation if there exist scalars and matrices , , , , ; , ; , , , , ; , , such that the following LMIs hold: where
Proof. From the demonstration in , we know that is nonsingular. Therefore, there always exist nonsingular matrices and such that holds. Then, we define the nonsingular matrices and as follows:
Let ; there is
which means that .
Now, using Lemma 3 and recalling (40), we can deduce that Then, it is easy to see that
which is equivalent to (3). Therefore, the condition in Theorem 4 is satisfied when the LMIs in (38)–(41) hold. Finally, it can be concluded that the filtering error system is stochastically stable with performance level .
Remark 7. There is more than one time delay appear in the electric system, the network system, and so on. So, it makes sense to investigate the system containing two terms of time delays as shown in this paper. The model we consider here contains two terms of time-varying delay as and in the state . The method we use can be easily drawn back to the system that has only one time delay. Meanwhile, the result we get can be extended to the system that contains more delays at the same time by developing the Lyapunov function with integral terms in the same way.
5. Numerical Example
In this section, we provide a numerical example to show the effectiveness of the results obtained in the previous section.
Example 1. Consider the T-S fuzzy stochastic system () with model parameters given as follows: