Delay-Dependent Robust H ∞ Filtering of the Takagi-Sugeno Fuzzy Stochastic Systems

This paper is concerned with the problem of the robust H ∞ 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 theH ∞ performance level of the output error and the disturbance input.Thedesignmethod for delay-dependent filter is developed based on linear matrix inequalities. Finally, the effectiveness of the proposed methods is substantiated with an illustrative example.


Introduction
As an efficient technique to linearize the nonlinear differential equations, the T-S fuzzy model [1] 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 [2] and the truck trailer system [3].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][5][6][7][8][9][10][11][12][13][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][16][17][18][19]; references [20,21] have considered the  2 −  ∞ 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 [22].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][30][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 [34].
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; [35] deals with the robust fault detection problem for T-S fuzzy stochastic systems.And [36] 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 freeweighting 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).R  denotes the -dimensional Euclidean space, and R × denotes the set of all  ×  real matrices. stands for an identity matrix of appropriate dimension, while   ∈ R  denotes a vector of ones.The notation * is used as an ellipsis for terms that are induced by symmetry.diag(⋅ ⋅ ⋅ ) stands for a block-diagonal matrix.| ⋅ | denotes the Euclidean norm for vectors, and ‖ ⋅ ‖ denotes the spectral norm for matrices.L 2 [0, ∞) represents the space of square-integrable vector functions over [0, ∞).E(⋅) stands for the mathematical expectation operator.Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

Problem Formulation and Preliminaries
where  1 ,  2 ,  1 ,  2 , and  3 are known real constant matrices and the unknown time-varying matrix function satisfying Now, the defuzzied output of the dynamic fuzzy stochastic model in (1)-( 4) can be represented as follows: where using the fuzzy theory, it is easy to see that, for all , Then, we consider the following fuzzy filters: in which, the fuzzy rule has the same representation as in ( 1)-( 4).Now we consider x() ∈ R  and ẑ() ∈ R  .The matrixes   ,   , and   are the filters need to be determined.
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 0 ≤ ℎ 1 , 0 ≤ ℎ 2 and a prescribed level of noise attenuation  > 0, the filtering error system ( Σ) is mean-square stable, and the error system ( Σ) satisfies  ∞ performance.Now, we introduce the following definitions and lemmas, which help to complete the proof of the main results.Definition 1.The system (Σ) is said to be robust stochastic mean-square stable if there exists () > 0 for any  > 0 such that when sup −ℎ≤≤0 E(‖()‖ 2 ) < (), for any uncertain variables.In addition, lim for any initial conditions.
Definition 2. The robust stochastic mean-square stable system ( Σ) is said to satisfy the  ∞ performance; for the given scalar  > 0 and any nonzero V() ∈  2 [0, ∞), the system ( Σ) satisfies for any uncertain variables, where Lemma 3.For the given matrices , , and  with    ≤  and positive scalar  > 0, the following inequality holds:
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.

Robust 𝐻 ∞ 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.
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 ( Σ) 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  1 () and  1 () 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 ℎ 3 , ℎ 4 , . . .integral terms in the same way.

Numerical Example
In this section, we provide a numerical example to show the effectiveness of the results obtained in the previous section.

Conclusion
This paper has investigated the filter design problem for the uncertain T-S fuzzy stochastic system with time-varying delays.An LMI approach has been developed to design the fuzzy filter ensuring not only the robust stochastic meansquare stability but also a prescribed  ∞ performance level of the filtering error system for all admissible uncertainties.A numerical example has been provided to show the effectiveness of the proposed filter design methods.