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

Abstract

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 the 𝐻_∞ 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.

1. 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 [4–14] 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 𝐻_∞ filtering problems for the T-S fuzzy systems have been addressed in [15–19]; references [20, 21] have considered the 𝐿₂−𝐿_∞ 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 𝐻_∞ 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 𝐻_∞ 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 𝐻_∞ 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 𝐻_∞ 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 “𝑇” stands for matrix transposition. Throughout this paper, for real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite). 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space, and 𝐑^𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. 𝐼 stands for an identity matrix of appropriate dimension, while 𝐼_𝑛∈𝐑^𝑛 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. 𝐋₂[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.

2. Problem Formulation and Preliminaries

Consider a T-S fuzzy stochastic time-delay system with time-varying parameter uncertainties.

Plant rule 𝑖: IF 𝑠₁(𝑡) is 𝜇_𝑖1, and 𝑠₂(𝑡) is 𝜇_𝑖2, and ⋯ and 𝑠_𝑔(𝑡) is 𝜇_𝑖𝑔, THEN 𝑑𝑥(𝑡)=𝐴_𝑖+Δ𝐴_𝑖(𝑡)𝑥(𝑡)+𝐴_𝑑1𝑖+Δ𝐴_𝑑1𝑖(𝑡)𝑥𝑡−𝜏₁(𝑡)+𝐴_𝑑2𝑖+Δ𝐴_𝑑2𝑖(𝑡)𝑥𝑡−𝜏₂(𝑡)+𝐵_𝑖𝑣(𝑡+𝐻)𝑑𝑡_𝑖+Δ𝐻_𝑖(𝑡)𝑥(𝑡)+𝐻_𝑑1𝑖+Δ𝐻_𝑑1𝑖(𝑡)𝑥𝑡−𝜏₁(𝑡)+𝐻_𝑑2𝑖+Δ𝐻_𝑑2𝑖(𝑡)𝑥𝑡−𝜏₂(𝑡)𝑑𝜔(𝑡),𝑑𝑦(𝑡)=𝐶_𝑖𝑥(𝑡)+𝐶_𝑑1𝑖𝑥𝑡−𝜏₁(𝑡)+𝐶_𝑑2𝑖𝑥𝑡−𝜏₂(𝑡)+𝐷_𝑖𝑣(𝑡)𝑑𝑡,𝑧(𝑡)=𝐿_𝑖𝑥(𝑡)+𝐿_𝑑1𝑖𝑥𝑡−𝜏₁(𝑡)+𝐿_𝑑2𝑖𝑥𝑡−𝜏₂(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈−maxℎ₁,ℎ₂,0,𝑖=1,2,…,𝑟, (1)(2)(3)(4) where 𝑠(𝑡)=[𝑠₁(𝑡)𝑠₂(𝑡)⋯𝑠_𝑔(𝑡)] are the premise variables, 𝜇_𝑖𝑗 is the fuzzy set, and 𝑟 is the number of IF-THEN rules; 𝑥(𝑡)∈𝐑^𝑛 is the system state; 𝜑(𝑡) is a given differential initial function on [−𝑢₂,0], and 𝑦(𝑡)∈𝐑^𝑚 is the measured output; 𝑧(𝑡)∈𝐑^𝑙 is a signal to be estimated; 𝑣(𝑡)∈𝐑^𝑠 is the noise signal which belongs to ℒ₂[0,∞); 𝜏₁(𝑡) and 𝜏₂(𝑡) are continuous differentiable functions representing the time-varying delays, which are assumed to satisfy for all 𝑡≥0, 0≤𝜏₁(𝑡)<ℎ₁,0≤𝜏₂(𝑡)<ℎ₂. (5) In the considered system, 𝐴_𝑖,  𝐴_𝑑1𝑖,𝐴_𝑑2𝑖,𝐵_𝑖,𝐻_𝑖, 𝐻_𝑑1𝑖,𝐻_𝑑2𝑖,𝐶_𝑖,𝐶_𝑑1𝑖,𝐶_𝑑2𝑖,𝐷_𝑖,𝐿_𝑖,𝐿_𝑑1𝑖, and 𝐿_𝑑2𝑖 are known constant matrices with appropriate dimensions. Δ𝐴_𝑖(𝑡), Δ𝐴_𝑑1𝑖(𝑡), Δ𝐴_𝑑2𝑖(𝑡), Δ𝐻_𝑖(𝑡), Δ𝐻_𝑑1𝑖(𝑡), and Δ𝐻_𝑑2𝑖(𝑡) represent the unknown time-varying parameter uncertainties and are assumed to satisfy Δ𝐴_𝑖(𝑡)Δ𝐴_𝑑1𝑖(𝑡)Δ𝐴_𝑑2𝑖(𝑡)Δ𝐻_𝑖(𝑡)Δ𝐻_𝑑1𝑖(𝑡)Δ𝐻_𝑑2𝑖(𝑡)=𝐾_1𝑖𝐾_2𝑖𝐹_𝑖(𝑡)𝑇_1𝑖𝑇_2𝑖𝑇_3𝑖, (6) where 𝐾_1𝑖, 𝐾_2𝑖, 𝑇_1𝑖, 𝑇_2𝑖, and 𝑇_3𝑖 are known real constant matrices and the unknown time-varying matrix function satisfying 𝐹_𝑖(𝑡)^𝑇𝐹_𝑖(𝑡)≤𝐼∀𝑡. (7)

Now, the defuzzied output of the dynamic fuzzy stochastic model in (1)–(4) can be represented as follows: (Σ):𝑑𝑥(𝑡)=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡×𝐴))_𝑖+Δ𝐴_𝑖(𝑡)𝑥(𝑡)+𝐴_𝑑1𝑖+Δ𝐴_𝑑1𝑖(𝑡)𝑥𝑡−𝜏₁(𝑡)+𝐴_𝑑2𝑖+Δ𝐴_𝑑2𝑖(𝑡)𝑥𝑡−𝜏₂(𝑡)+𝐵_𝑖𝑣(𝑡+𝐻)𝑑𝑡_𝑖+Δ𝐻_𝑖(𝑡)𝑥(𝑡)+𝐻_𝑑1𝑖+Δ𝐻_𝑑1𝑖(𝑡)𝑥𝑡−𝜏₁(𝑡)+𝐻_𝑑2𝑖+Δ𝐻_𝑑2𝑖(𝑡)𝑥𝑡−𝜏₂(𝑡)(𝑡),(𝑡)=𝑑𝜔𝑑𝑦^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡×𝐶))_𝑖𝑥(𝑡)+𝐶_𝑑1𝑖𝑥𝑡−𝜏₁(𝑡)+𝐶_𝑑2𝑖𝑥𝑡−𝜏₂(𝑡)+𝐷_𝑖𝑣(𝑡𝑧(𝑡)=)𝑑𝑡,_𝑟𝑖=1𝜌_𝑖(𝑠(𝑡×𝐿))_𝑖𝑥(𝑡)+𝐿_𝑑1𝑖𝑥𝑡−𝜏₁(𝑡)+𝐿_𝑑2𝑖𝑥𝑡−𝜏₂(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈−maxℎ₁,ℎ₂,0, (8) where 𝜌_𝑖(𝑠(𝑡))=𝜔_𝑖(𝑠(𝑡))∑^𝑟_𝑗=1𝜔_𝑗(𝑠(𝑡,))𝜔_𝑖(𝑠(𝑡))=^𝑔_𝑗=1𝜇_𝑖𝑗𝑠_𝑗(𝑡); (9) using the fuzzy theory, it is easy to see that, for all 𝑡, 𝜌_𝑖(𝑠(𝑡))≥0,𝑖=1,2,…,𝑟,_𝑟𝑖=1ℎ_𝑖(𝑠(𝑡))=1. (10)

Then, we consider the following fuzzy filters: 𝑑𝑥(𝑡)=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐴_𝑓𝑖𝑥(𝑡)𝑑𝑡+𝐵_𝑓𝑖(𝑡,𝑧(𝑡)=𝑑𝑦)_𝑟𝑖=1𝜌_𝑖(𝑠(𝑡))𝐿_𝑓𝑖𝑥(𝑡), (11) in which, the fuzzy rule has the same representation as in (1)–(4). Now we consider 𝑥(𝑡)∈𝐑^𝑛 and 𝑧(𝑡)∈𝐑^𝑙. The matrixes 𝐴_𝑓𝑖,  𝐵_𝑓𝑖, and 𝐿_𝑓𝑖 are the filters need to be determined.

Let 𝜉(𝑡)=[𝑥(𝑡)^𝑇𝑥(𝑡)^𝑇]^𝑇, 𝑧(𝑡)=𝑧(𝑡)−𝑧(𝑡).

For convenience, the filtering error dynamic system can be written as Σ:(𝑡)=𝐴(𝑡)𝜉(𝑡)+𝐴𝑑𝜉𝐴+Δ_𝑑1+Δ𝐴_𝑑1(𝑡)𝐺𝜉𝑡−𝜏₁(𝑡)+𝐴_𝑑2+Δ𝐴_𝑑2(𝑡)𝐺𝜉𝑡−𝜏₂(𝑡)+(𝑡+𝐻(𝑡)𝜉(𝑡)+𝐻𝐵𝑣)𝑑𝑡𝐻+Δ_𝑑1+Δ𝐻_𝑑1(𝑡)×𝐺𝜉𝑡−𝜏₁(𝑡)+𝐻_𝑑2+Δ𝐻_𝑑2(𝑡)𝐺𝜉𝑡−𝜏₂(𝑡)(𝑡),𝑧(𝑡)=(𝑡)+𝑑𝜔𝐿𝜉𝐿_𝑑1𝐺𝜉𝑡−𝜏₁(𝑡)+𝐿_𝑑2𝐺𝜉𝑡−𝜏₂(𝑡), (12) where 𝐴=𝐴0𝐵_𝑓𝐶𝐴_𝑓,𝐴_𝑑1=𝐴_𝑑1𝐵_𝑓𝐶_𝑑1,𝐴_𝑑2=𝐴_𝑑2𝐵_𝑓𝐶_𝑑2,𝐻=,𝐻𝐻000_𝑑1=𝐻_𝑑10,𝐻_𝑑2=𝐻_𝑑20,𝐵=𝐵𝐵_𝑓𝐷𝐴(𝑡)=Δ,Δ𝐴(𝑡)0,Δ𝐴00_𝑑1(𝑡)=Δ𝐴_𝑑1(𝑡)0,Δ𝐴_𝑑2(𝑡)=Δ𝐴_𝑑2(𝑡)0,Δ𝐻(𝑡)=Δ𝐻(𝑡)0𝐻00,Δ_𝑑1(𝑡)=Δ𝐻_𝑑1(𝑡)0,Δ𝐻_𝑑2(𝑡)=Δ𝐻_𝑑2(𝑡)0,𝐴=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐴_𝑖,𝐴_𝑑1=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐴_𝑑1𝑖,𝐴_𝑑2=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐴_𝑑2𝑖,𝐻=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐻_𝑖,𝐻_𝑑1=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐻_𝑑1𝑖,𝐻_𝑑2=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐻_𝑑2𝑖,𝐶=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐶_𝑖,𝐶_𝑑1=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐶_𝑑1𝑖,𝐶_𝑑2=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐶_𝑑2𝑖,𝐵=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐵_𝑖,𝐷=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐷_𝑖,𝐿=𝐿−𝐿_𝑓,𝐺=𝐼0,𝐴_𝑓=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐴_𝑓𝑖,𝐵_𝑓=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐵_𝑓𝑖,𝐿_𝑓=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐿_𝑓𝑖,𝐿=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐿_𝑖,𝐿_𝑑1=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐿_𝑑1𝑖,𝐿_𝑑2=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))𝐿_𝑑2𝑖,Δ𝐴(𝑡)=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))Δ𝐴_𝑖(𝑡),Δ𝐴_𝑑1(𝑡)=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))Δ𝐴_𝑑1𝑖(𝑡),Δ𝐻(𝑡)=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))Δ𝐻_𝑖(𝑡),Δ𝐻_𝑑1(𝑡)=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))Δ𝐻_𝑑1𝑖(𝑡),Δ𝐴_𝑑2(𝑡)=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))Δ𝐴_𝑑2𝑖(𝑡),Δ𝐻_𝑑2(𝑡)=^𝑟_𝑖=1𝜌_𝑖(𝑠(𝑡))Δ𝐻_𝑑2𝑖(𝑡). (13)

The purpose of this work is to design a sets of fuzzy filters in the form of (11) such that for any scalar 0≤ℎ₁, 0≤ℎ₂ and a prescribed level of noise attenuation 𝛾>0, the filtering error system (Σ) is mean-square stable, and the error system (Σ) satisfies 𝐻_∞ 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 𝛿(𝜀)>0 for any 𝜀>0 such that 𝖤‖𝑥(𝑡)‖²<𝜀,𝑡>0, (14) when sup_{−ℎ≤𝑠≤0}𝖤(‖𝜑(𝑠)‖²)<𝛿(𝜀), for any uncertain variables. In addition, lim_𝑡→∞𝖤‖𝑥(𝑡)‖²=0, (15) for any initial conditions.

Definition 2. The robust stochastic mean-square stable system (Σ) is said to satisfy the 𝐻_∞ performance; for the given scalar 𝛾>0 and any nonzero 𝑣(𝑡)∈𝐿₂[0,∞), the system (Σ) satisfies ‖𝑧(𝑡)‖_𝐸2<𝛾‖𝑣(𝑡)‖₂, (16) for any uncertain variables, where ‖𝑧(𝑡)‖_𝐸2=𝖤^∞₀|𝑧(𝑡)|²𝑑𝑡^1/2. (17)

Lemma 3. For the given matrices 𝐷,  𝐸, and 𝐹 with 𝐹^𝑇𝐹≤𝐼 and positive scalar 𝜀>0, the following inequality holds: 𝐷𝐹𝐸+(𝐷𝐹𝐸)^𝑇≤𝜀𝐷𝐷^𝑇+𝜀⁻¹𝐸^𝑇𝐸. (18)

3. Robust Stochastic Stabile

First, we derive the robust stochastic mean-square stochastic conditions and the 𝐻_∞ conditions for system (Σ). Defining the following variables for convenience: Φ(𝑡)=𝐴+Δ𝐴(𝑡𝜉(𝑡)+)𝐴_𝑑1+Δ𝐴_𝑑1(𝑡)𝐺𝜉𝑡−𝜏₁(𝑡)+𝐴_𝑑2+Δ𝐴_𝑑2(𝑡)𝐺𝜉𝑡−𝜏₂(𝑡)+(𝑡),𝑔(𝑡)=𝐵𝑣𝐻+Δ𝐻(𝑡)𝜉(𝑡)+𝐻_𝑑1+Δ𝐻_𝑑1(𝑡)𝐺𝜉𝑡−𝜏₁(𝑡)+𝐻_𝑑2+Δ𝐻_𝑑2(𝑡)𝐺𝜉𝑡−𝜏₂(𝑡). (19)

Theorem 4. For given scalars 0≤ℎ₁,  0≤ℎ₂, the filtering error system (Σ) is robust stochastic mean-square stables and (16) is satisfied if there exist matrices 𝑃=𝑃^𝑇>0,  𝑄_𝑖=𝑄^𝑇_𝑖>0,  𝑅_𝑖=𝑅^𝑇_𝑖>0,  𝑁_𝑖,𝑀_𝑖,  𝑖=1,2,  and 𝑍_𝑗=𝑍^𝑇_𝑗>0,  𝑗=1,…,4, such that the following matrix inequalities hold: Ψ=ΩΨ₁₂Ψ₁₃∗Ψ₂₂𝟎∗∗Ψ₃₃<0, (20) whereΩΩ=¹¹00Ω₁₄Ω₁₅00𝑃𝐵∗Ω₂₂0𝑁₂𝑀₂000∗∗Ω₃₃00000∗∗∗Ω₄₄0000∗∗∗∗Ω₅₅𝑅000∗∗∗∗∗−¹ℎ₁𝑅00∗∗∗∗∗∗−²ℎ₂0∗∗∗∗∗∗∗−𝛾²𝐼,Ψ₁₂=𝑁₁𝑁₂ℎ₁𝑁₁ℎ₁𝑁₂𝑀₁𝑀₂ℎ₂𝑀₁ℎ₂𝑀₂,Ψ₂₂=diag−𝑍₃,−𝑍₃,−ℎ₁𝑍₁,−ℎ₁𝑍₁,−𝑍₄,−𝑍₄,−ℎ₂𝑍₂,−ℎ₂𝑍₂,Ψ₁₃=𝐴̆𝐺^𝑇𝑈₁𝐻̆𝐺^𝑇𝑈₂𝐻̆𝑃𝐿̆,Ψ₃₃=diag−𝑈₁,−𝑈₂,Ω,−𝑃,−𝐼₁₁=𝑃𝐴+Δ𝐴(𝑡)+𝐴+Δ𝐴(𝑡)^𝑇𝑃+𝐺^𝑇𝑄₁+𝑄₂+ℎ₁𝑅₁+ℎ₂𝑅₂Ω𝐺,₁₄=𝑃𝐴_𝑑1+Δ𝐴_𝑑1(𝑡)+𝐺^𝑇𝑁^𝑇₁,Ω₁₅=𝑃𝐴_𝑑2+Δ𝐴_𝑑2(𝑡)+𝐺^𝑇𝑀^𝑇₁,Ω₂₂=−𝑄₂−𝑁₂−𝑁^𝑇₂,Ω₃₃=−𝑄₂−𝑀₂−𝑀^𝑇₂,Ω₄₄=−𝑁₁−𝑁^𝑇₁,Ω₅₅=−𝑀₁−𝑀^𝑇₁,𝑁₁=000𝑁^𝑇₁0000^𝑇,𝑀₁=0000𝑀^𝑇₁000^𝑇,𝑁₂=0𝑁^𝑇₂000000^𝑇,𝑀₂=00𝑀^𝑇₂00000^𝑇,𝐴̆=𝐴^𝑇+Δ𝐴^𝑇(𝑡)00𝐴^𝑇_𝑑1+Δ𝐴^𝑇_𝑑1(𝑡)𝐴^𝑇_𝑑2+Δ𝐴^𝑇_𝑑2(𝑡)00𝐵^𝑇𝑇,𝐻̆=𝐻^𝑇+Δ𝐻^𝑇(𝑡)00𝐻^𝑇_𝑑1+Δ𝐻^𝑇_𝑑1(𝑡)𝐻^𝑇_𝑑2+Δ𝐻^𝑇_𝑑2(𝑡)000^𝑇,𝐿̆=𝐿^𝑇00𝐿^𝑇_𝑑1𝐿^𝑇_𝑑2000^𝑇,𝑈₁=ℎ₁𝑍₁+ℎ₂𝑍₂,𝑈₂=ℎ₁𝑍₃+𝐻₂𝑍₄. (21)

Proof. Define the following Lyapunov-Krasovskii candidate for system (Σ): 𝑉(𝜉(𝑡),𝑡)=𝜉^𝑇(𝑡)𝑃𝜉(𝑡)+^𝑡_𝑡−ℎ1𝜉^𝑇(𝑠)𝐺^𝑇𝑄₁(𝑠)+𝐺𝜉𝑑𝑠_{𝑡𝑡−ℎ2}𝜉^𝑇(𝑠)𝐺^𝑇𝑄₂(𝑠)+𝐺𝜉𝑑𝑠_{0−ℎ1𝑡𝑡+𝜃}𝜉^𝑇(𝑠)𝐺^𝑇𝑅₁(𝑠)+𝐺𝜉𝑑𝑠𝑑𝜃_{0−ℎ2𝑡𝑡+𝜃}𝜉^𝑇(𝑠)𝐺^𝑇𝑅₂(𝑠)+𝐺𝜉𝑑𝑠𝑑𝜃_{0−ℎ1𝑡𝑡+𝛽}Φ^𝑇(𝑠)𝐺^𝑇𝑍₁(𝑠)+𝐺Φ𝑑𝑠𝑑𝛽_{0−ℎ2𝑡𝑡+𝛽}Φ^𝑇(𝑠)𝐺^𝑇𝑍₂(𝑠)+𝐺Φ𝑑𝑠𝑑𝛽_{0−ℎ1𝑡𝑡+𝛽}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₃𝐺𝑔(𝑠)𝑑𝑠𝑑𝛽+⁰_−ℎ2^𝑡_𝑡+𝛽𝑔^𝑇(𝑠)𝐺^𝑇𝑍₄𝐺𝑔(𝑠)𝑑𝑠𝑑𝛽. (22)
When 𝑣(𝑡)=0, 𝑑𝑉(𝜉(𝑡),𝑡)=𝐋𝑉(𝜉(𝑡),𝑡)+2𝜉^𝑇(𝑡)𝑃𝑔(𝑡)𝑑𝜔(𝑡). (23)
By the Newton-Leibnitz formula, we can get the following equations for any matrices 𝑁_𝑖,𝑀_𝑖,  𝑖=1,2 with any appropriate dimensions: 2𝜂^𝑇(𝑡)𝑁₁𝐺𝜉(𝑡)−𝜉𝑡−𝜏₁(𝑡)−^𝑡_{𝑡−𝜏1(𝑡)}Φ(𝑠)−𝑑𝑠_{𝑡𝑡−𝜏1(𝑡)}𝑔(𝑠)𝑑𝜔(𝑠)=0,2𝜂^𝑇(𝑡)𝑁₂𝐺𝜉𝑡−𝜏₁(𝑡)−𝜉𝑡−ℎ₁−^{𝑡−𝜏1(𝑡)}_𝑡−ℎ1Φ(𝑠)𝑑𝑠−^{𝑡−𝜏1(𝑡)}_𝑡−ℎ1𝑔(𝑠)𝑑𝜔(𝑠)=0,2𝜂^𝑇(𝑡)𝑀₁𝐺𝜉(𝑡)−𝜉𝑡−𝜏₂(𝑡)−^𝑡_{𝑡−𝜏2(𝑡)}Φ(𝑠)−𝑑𝑠_{𝑡𝑡−𝜏2(𝑡)}𝑔(𝑠)𝑑𝜔(𝑠)=0,2𝜂^𝑇(𝑡)𝑀₂𝐺𝜉𝑡−𝜏₂(𝑡)−𝜉𝑡−ℎ₂−^{𝑡−𝜏2(𝑡)}_𝑡−ℎ2Φ(𝑠)𝑑𝑠−^{𝑡−𝜏2(𝑡)}_𝑡−ℎ2𝑔(𝑠)(𝑠𝜏𝑑𝜔)=0,₁(𝑡)𝜂^𝑇(𝑡)𝑁₁𝑍₁⁻¹𝑁^𝑇₁𝜂(𝑡)−^𝑡_{𝑡−𝜏1(𝑡)}𝜂^𝑇(𝑡)𝑁₁𝑍₁⁻¹𝑁^𝑇₁𝜂(𝑡)𝜏𝑑𝑠=0,₂(𝑡)𝜂^𝑇(𝑡)𝑀₁𝑍₂⁻¹𝑀^𝑇₁𝜂(𝑡)−^𝑡_{𝑡−𝜏2(𝑡)}𝜂^𝑇(𝑡)𝑀₁𝑍₂⁻¹𝑀^𝑇₁𝜂(𝑡)ℎ𝑑𝑠=0,₁−𝜏₁(𝑡)𝜂^𝑇(𝑡)𝑁₂𝑍₁⁻¹𝑁^𝑇₁𝜂(𝑡)−^{𝑡−𝜏1(𝑡)}_𝑡−ℎ1𝜂^𝑇(𝑡)𝑁₁𝑍₁⁻¹𝑁^𝑇₁𝜂(𝑡)ℎ𝑑𝑠=0,₂−𝜏₂(𝑡)𝜂^𝑇(𝑡)𝑀₂𝑍₂⁻¹𝑀^𝑇₁𝜂(𝑡)−^{𝑡−𝜏2(𝑡)}_𝑡−ℎ2𝜂^𝑇(𝑡)𝑀₂𝑍₂⁻¹𝑀^𝑇₁𝜂(𝑡)𝑑𝑠=0, (24) where 𝑁₁=000𝑁^𝑇₁000^𝑇,𝑁₂=0𝑁^𝑇₂00000^𝑇,𝑀₁=0000𝑀^𝑇₁00^𝑇,𝑀₂=00𝑀^𝑇₂0000^𝑇, (25) and 𝜂(𝑡) is a new vector defined as follows: 𝜂^𝑇(𝑡)=^{𝜉𝑇(𝑡)𝜉𝑇𝑡−ℎ}₁^{𝐺𝑇𝜉𝑇𝑡−ℎ}₂^{𝐺𝑇𝜉𝑇𝑡−𝜏}₁^{(𝑡)𝐺𝑇}_𝜉^𝑇_{𝑡−𝜏2(𝑡)𝐺}^𝑇_∫^𝑡_𝑡−ℎ1^𝑇_{𝜉(𝑠)𝐺𝑑𝑠}^𝑇_∫^𝑡_𝑡−ℎ2^𝑇_{𝜉(𝑠)𝐺𝑑𝑠}^𝑇. (26)
Using the above formulas (24) and Lemma 3, it can be seen that 𝐋𝑉(𝜉(𝑡),𝑡)=2𝜉^𝑇(𝑡)𝑃Φ(𝑡)+𝑔^𝑇(𝑡)𝑃𝑔(𝑡)+𝜉^𝑇(𝑡)𝐺^𝑇𝑄₁𝐺𝜉(𝑡)+𝜉^𝑇(𝑡)𝐺^𝑇𝑄₂𝐺𝜉(𝑡)−𝜉^𝑇𝑡−ℎ₁𝐺^𝑇𝑄₁𝐺𝜉𝑡−ℎ₁−𝜉^𝑇𝑡−ℎ₂𝐺^𝑇𝑄₂𝐺𝜉𝑡−ℎ₂+ℎ₁Φ^𝑇(𝑡)𝐺^𝑇𝑍₁𝐺Φ(𝑡)+ℎ₁𝜉^𝑇(𝑡)𝐺^𝑇𝑅₁𝐺𝜉(𝑡)+ℎ₂𝜉^𝑇(𝑡)𝐺^𝑇𝑅₂𝐺𝜉(𝑡)+ℎ₂Φ^𝑇(𝑡)𝐺^𝑇𝑍₂𝐺Φ(𝑡)−^𝑡_𝑡−ℎ1𝜉^𝑇(𝑠)𝐺^𝑇𝑅₁(𝑠)−𝐺𝜉𝑑𝑠_{𝑡𝑡−ℎ2}𝜉^𝑇(𝑠)𝐺^𝑇𝑅₂𝐺𝜉(𝑠)𝑑𝑠+ℎ₁𝑔^𝑇(𝑡)𝐺^𝑇𝑍₃𝐺𝑔(𝑡)−^𝑡_𝑡−ℎ1Φ^𝑇(𝑠)𝐺^𝑇𝑍₁(𝑠)−𝐺Φ𝑑𝑠_{𝑡𝑡−ℎ2}Φ^𝑇(𝑠)𝐺^𝑇𝑍₂𝐺Φ(𝑠)𝑑𝑠+ℎ₂𝑔^𝑇(𝑡)𝐺^𝑇𝑍₄(𝑡)−𝐺𝑔_{𝑡𝑡−ℎ1}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₃(𝑠)−𝐺𝑔𝑑𝑠_{𝑡𝑡−ℎ2}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₄𝐺𝑔(𝑠)𝑑𝑠≤𝜂^𝑇(𝑡)Ω+𝑁₁𝑍₃⁻¹𝑁^𝑇₁+𝐻𝐺^𝑇ℎ₁𝑍₃𝐺+𝐺^𝑇ℎ₂𝑍₄𝐻+𝐺+𝑃𝐴𝐺^𝑇ℎ₁𝑍₁+ℎ₂𝑍₂𝐺𝐴^𝑇+𝑁₂𝑍₃⁻¹𝑁^𝑇₂+ℎ₁𝑁₁𝑍₁⁻¹𝑁^𝑇₁+ℎ₁𝑁₂𝑍₁⁻¹𝑁^𝑇₂+ℎ₂𝑀₁𝑍₂⁻¹𝑀^𝑇₁+ℎ₂𝑀₂𝑍₂⁻¹𝑀^𝑇₂+𝑀₁𝑍₄⁻¹𝑀^𝑇₁+𝑀₂𝑍₄⁻¹𝑀^𝑇₂𝜂(𝑡)−^𝑡_{𝑡−𝜏1(𝑡)}𝜂^𝑇(𝑡)𝑁₁+Φ^𝑇(𝑠)𝐺^𝑇𝑍₁𝑍₁⁻¹×𝑍₁𝐺Φ(𝑠)+𝑁^𝑇₁𝜂(𝑡−)𝑑𝑠_{𝑡−𝜏1(𝑡)𝑡−ℎ1}𝜂^𝑇(𝑡)𝑁₂+Φ^𝑇(𝑠)𝐺^𝑇𝑍₁𝑍₁⁻¹×𝑍₁𝐺Φ(𝑠)+𝑁^𝑇₂𝜂(𝑡−)𝑑𝑠_{𝑡𝑡−𝜏2(𝑡)}𝜂^𝑇(𝑡)𝑀₁+Φ^𝑇(𝑠)𝐺^𝑇𝑍₂𝑍₂⁻¹×𝑍₂𝐺Φ(𝑠)+𝑀^𝑇₁𝜂(𝑡)𝑑𝑠−^{𝑡−𝜏2(𝑡)}_𝑡−ℎ2𝜂^𝑇(𝑡)𝑀₂+Φ^𝑇(𝑠)𝐺^𝑇𝑍₂𝑍₂⁻¹×𝑍₂𝐺Φ(𝑠)+𝑀^𝑇₂𝜂(𝑡+)𝑑𝑠_{𝑡𝑡−𝜏1(𝑡)}𝑔(𝑠)𝑑𝜔(𝑠)^𝑇𝐺^𝑇𝑍₃𝐺^𝑡_{𝑡−𝜏1(𝑡)}𝑔(𝑠)(𝑠)−𝑑𝜔_{𝑡𝑡−𝜏1(𝑡)}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₃(𝑠)+𝐺𝑔𝑑𝑠_{𝑡−𝜏1(𝑡)𝑡−ℎ1}𝑔(𝑠)𝑑𝜔(𝑠)^𝑇𝐺^𝑇𝑍₃𝐺^{𝑡−𝜏1(𝑡)}_𝑡−ℎ1𝑔(𝑠)(𝑠)−𝑑𝜔_{𝑡−𝜏1(𝑡)𝑡−ℎ1}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₃(𝑠)+𝐺𝑔𝑑𝑠_{𝑡𝑡−𝜏2(𝑡)}𝑔(𝑠)𝑑𝜔(𝑠)^𝑇𝐺^𝑇𝑍₄𝐺^𝑡_{𝑡−𝜏2(𝑡)}𝑔(𝑠)(𝑠−𝑑𝜔)_{𝑡𝑡−𝜏2(𝑡)}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₄(𝑠)+𝐺𝑔𝑑𝑠_{𝑡−𝜏2(𝑡)𝑡−ℎ2}𝑔(𝑠)𝑑𝜔(𝑠)^𝑇𝐺^𝑇𝑍₄𝐺^{𝑡−𝜏1(𝑡)}_𝑡−ℎ2𝑔(𝑠)(𝑠)−𝑑𝜔_{𝑡−𝜏1(𝑡)𝑡−ℎ2}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₄𝐺𝑔(𝑠)𝑑𝑠, (27) whereΩΩ=¹¹00Ω₁₄Ω₁₅00∗Ω₂₂00𝑁₂𝑀₂0∗∗Ω₃₃0000∗∗∗Ω₄₄000∗∗∗∗Ω₅₅𝑅00∗∗∗∗∗−¹ℎ₁0𝑅∗∗∗∗∗∗−²ℎ₂,𝐴=𝐴+Δ𝐴(𝑡)^𝑇00𝐴_𝑑1+Δ𝐴_𝑑1(𝑡)^𝑇𝐴_𝑑2+Δ𝐴_𝑑2(𝑡)^𝑇00^𝑇,𝐻=𝐻+Δ𝐻(𝑡)^𝑇00𝐻_𝑑1+Δ𝐻_𝑑1(𝑡)^𝑇𝐻_𝑑2+Δ𝐻_𝑑2(𝑡)^𝑇00^𝑇. (28)Note that 𝖤^𝑡_{𝑡−𝜏1(𝑡)}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₃𝐺𝑔(𝑠)𝑑𝑠=𝖤^𝑡_{𝑡−𝜏1(𝑡)}^𝑇𝑔^𝑇(𝑠)𝑑𝜔(𝑠)×𝐺^𝑇𝑍₃𝐺^𝑡_{𝑡−𝜏1(𝑡)}𝑔^𝑇(𝑠)(𝑠),𝖤𝑑𝜔_{𝑡−𝜏1(𝑡)𝑡−ℎ1}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₃𝐺𝑔(𝑠)𝑑𝑠=𝖤^{𝑡−𝜏1(𝑡)}_𝑡−ℎ1^𝑇𝑔^𝑇(𝑠)𝑑𝜔(𝑠)×𝐺^𝑇𝑍₃𝐺^{𝑡−𝜏1(𝑡)}_𝑡−ℎ1𝑔^𝑇(𝑠)(𝑠),𝖤𝑑𝜔_{𝑡𝑡−𝜏2(𝑡)}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₄𝐺𝑔(𝑠)𝑑𝑠=𝖤^𝑡_{𝑡−𝜏2(𝑡)}^𝑇𝑔^𝑇(𝑠)𝑑𝜔(𝑠)×𝐺^𝑇𝑍₄𝐺^𝑡_{𝑡−𝜏2(𝑡)}𝑔^𝑇(𝑠)(𝑠),𝖤𝑑𝜔_{𝑡−𝜏2(𝑡)𝑡−ℎ2}𝑔^𝑇(𝑠)𝐺^𝑇𝑍₄𝐺𝑔(𝑠)𝑑𝑠=𝖤^{𝑡−𝜏2(𝑡)}_𝑡−ℎ2^𝑇𝑔^𝑇(𝑠)𝑑𝜔(𝑠)×𝐺^𝑇𝑍₄𝐺^{𝑡−𝜏2(𝑡)}_𝑡−ℎ2𝑔^𝑇(𝑠)𝑑𝜔(𝑠). (29) 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 𝑣(𝑡)=0: Ω+𝑁₁𝑍₃⁻¹𝑁^𝑇₁+𝐻𝐺^𝑇ℎ₁𝑍₃𝐺+𝐺^𝑇ℎ₂𝑍₄𝐻+𝐺+𝑃𝐴𝐺^𝑇ℎ₁𝑍₁+𝐻₂𝑍₂𝐺𝐴^𝑇+𝑁₂𝑍₃⁻¹𝑁^𝑇₂+ℎ₁𝑁₁𝑍₁⁻¹𝑁^𝑇₁+ℎ₁𝑁₂𝑍₁⁻¹𝑁^𝑇₂+ℎ₂𝑀₁𝑍₂⁻¹𝑀^𝑇₁+ℎ₂𝑀₂𝑍₂⁻¹𝑀^𝑇₂+𝑀₁𝑍₄⁻¹𝑀^𝑇₁+𝑀₂𝑍₄⁻¹𝑀^𝑇₂<0. (30) From (27), (29), and (30), we can see that 𝐋𝑉(𝜉(𝑡),𝑡)<0, (31) which, by Definition  2.1 [37, 38], ensures that system (Σ) with 𝑣(𝑡)=0 is robustly stochastically stable. By Itô’s formula, we can derive 𝖤(𝑉(𝜉(𝑡),𝑡))=𝖤^𝑡₀𝐋𝑉(𝜉(𝑠),𝑠)𝑑𝑠. (32)
Now, set a functional 𝐽_𝑇 as 𝐽_𝑇=𝖤^𝑇₀𝑧(𝑡)^𝑇𝑧(𝑡)−𝛾²𝑣(𝑡)^𝑇𝑣(𝑡)𝑑𝑠, (33) where 𝑡>0. From (32) and (33), it is easy to show that 𝐽_𝑇=𝖤^𝑇₀𝑧(𝑡)^𝑇𝑧(𝑡)−𝛾²𝑣(𝑡)^𝑇𝑣(𝑡)(𝜉(𝑠)(𝑉(𝜉(𝑡)+𝐋𝑉,𝑠)𝑑𝑠−𝖤,𝑡))≤𝖤^𝑇₀𝑧(𝑡)^𝑇𝑧(𝑡)−𝛾²𝑣(𝑡)^𝑇𝑣(𝑡)+𝐋𝑉(𝜉(𝑠),𝑠)𝑑𝑠, (34) for all 𝑡>0. By using the Schur complement to (3), there is 𝑧(𝑡)^𝑇𝑧(𝑡)−𝛾²𝑣(𝑡)^𝑇𝑣(𝑡)+𝐋𝑉(𝜉(𝑠),𝑠)≤𝜂(𝑡)Ψ𝜂(𝑡)<0, (35) where 𝜂^𝑇(𝑡)=^{𝜉𝑇(𝑡)𝜉𝑇𝑡−ℎ}₁^{𝐺𝑇𝜉𝑇𝑡−ℎ}₂^{𝐺𝑇𝜉𝑇𝑡−𝜏}₁^{(𝑡)𝐺𝑇𝜉𝑇𝑡−𝜏}₂^{(𝑡)𝐺𝑇}_∫^𝑡_𝑡−ℎ1^𝑇_{𝜉(𝑠)𝐺𝑑𝑠}^𝑇_∫^𝑡_𝑡−ℎ2^𝑇_{𝜉(𝑠)𝐺𝑑𝑠}^𝑇_𝑣(𝑡). (36)
It follows from (33) and (34) that 𝐽(𝑡)<0, (37) for all 𝑡>0, 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 𝐻_∞ Filter Design

In this section, a sufficient condition for the solvability of robust 𝐻_∞ 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 (Σ). The robust 𝐻_∞ filtering problem is solvable with disturbance attenuation 𝛾>0 if there exist scalars 𝜀_𝑖>0 and matrices 𝑊>0,  𝑋>0,  𝑄_𝑖>0,  𝑅_𝑖>0,  𝑖=1,2; 𝑍_𝑗>0,   𝑗=1,2,3,4;   𝑁_1𝑖,   𝑁_2𝑖,   𝑀_1𝑖,   𝑀_2𝑖,   𝑖=1,2,3,4;   {Γ_𝑖=Γ^𝑇_𝑖,1≤𝑖≤𝑟},   {Π_𝑖𝑗,1≤𝑖<𝑗≤𝑟}, such that the following LMIs hold: Θ_𝑖𝑖−Γ_𝑖+𝜀_𝑖Ξ_𝑖Ξ^𝑇_𝑖𝜚_𝑖∗−𝜀_𝑖<0,(1≤𝑖≤𝑟),Θ_𝑖𝑗+Θ_𝑗𝑖−Π_𝑖𝑗−Π^𝑇_𝑖𝑗+𝜀_𝑖Ξ_𝑖Ξ^𝑇_𝑖+𝜀_𝑗Ξ_𝑗Ξ^𝑇_𝑗𝜚_𝑖𝜚_𝑗∗−𝜀_𝑖0∗∗−𝜀_𝑗,<0,1≤𝑖<𝑗≤𝑟Γ₁Π₁₂⋯Π_1𝑟∗Γ₂⋯Π_2𝑟⋮⋮⋱⋮∗∗⋯Γ_𝑟<0,𝑋−𝑊>0, (38)(39)(40)(41) where𝜚^𝑇_𝑖=𝐾^𝑇_1𝑖𝑊𝐾^𝑇_1𝑖𝑋𝟎_1∗15𝐾^𝑇_1𝑖𝑈₁𝐾^𝑇_2𝑖𝑈₂𝐾^𝑇_2𝑖𝑋𝐾^𝑇_2𝑖,Ξ^𝑇_𝑖=𝑇^𝑇_1𝑖𝑇^𝑇_1𝑖00𝑇^𝑇_2𝑖𝑇^𝑇_3𝑖𝟎_15∗1,Θ_𝑖𝑗=Θ₁₁Θ₁₂Θ₁₃∗Θ₂₂𝟎_8∗8∗∗Θ₃₃,Θ₁₁=Λ₁₁Λ₁₂00Λ₁₅Λ₁₆00Λ₁₉∗Λ₂₂00Λ₂₅Λ₂₆00Λ₂₉∗∗Λ₃₃0Λ₃₅Λ₃₆000∗∗∗Λ₄₄00000∗∗∗∗−𝑁₁−𝑁^𝑇₁0000∗∗∗∗∗−𝑀₁−𝑀^𝑇₁𝑅000∗∗∗∗∗∗−¹ℎ₁𝑅00∗∗∗∗∗∗∗−²ℎ₂0∗∗∗∗∗∗∗∗−𝛾²𝐼,Λ₁₁=𝑊𝐴_𝑖+𝐴^𝑇_𝑖𝑊+𝑄₁+𝑄₂+ℎ₁𝑅₁+ℎ₂𝑅₂,Λ₁₂=𝑊𝐴_𝑖+𝐴^𝑇_𝑖𝑋+𝐶^𝑇_𝑗Φ^𝑇_1𝑖+Φ^𝑇_2𝑖+𝑄₁+𝑄₂+ℎ₁𝑅₁+ℎ₂𝑅₂,Λ₁₅=𝑁^𝑇₁+𝑊𝐴_𝑑1𝑖,Λ₁₆=𝑀^𝑇₁+𝑊𝐴_𝑑2𝑖,Λ₁₉=𝑊𝐵_𝑖,Λ₂₉=𝑋𝐵_𝑖+Φ_1𝑖𝐷_𝑗,Λ₂₂=𝑋^𝑇𝐴_𝑖+Φ_1𝑖𝐶_𝑗+𝐶^𝑇_𝑗Φ^𝑇_1𝑖+𝐴^𝑇_𝑖𝑋+𝑄₁+𝑄₂+ℎ₁𝑅₁+ℎ₂𝑅₂,Λ₂₅=𝑁^𝑇₁+𝑋^𝑇𝐴_𝑑1𝑖+Φ_1𝑖𝐶_𝑑1𝑗,Λ₂₆=𝑀^𝑇₁+𝑋^𝑇𝐴_𝑑2𝑖+Φ_1𝑖𝐶_𝑑2𝑗,Λ₃₃=−𝑄₁−𝑁₂−𝑁^𝑇₂,Λ₃₅=𝑁₂,Λ₃₆=𝑀₂,Λ₄₄=−𝑄₂−𝑀₂−𝑀^𝑇₂,Θ₁₂=000000000𝑁₂0ℎ₂𝑁₂000000000𝑀₂0ℎ₂𝑀₂𝑁₁0ℎ₁𝑁₁000000000𝑀₁0ℎ₂𝑀₁0,Θ00000000000000000000000000000000₂₂=diag−𝑍₃,−𝑍₃,−ℎ₁𝑍₁,−ℎ₁𝑍₁,−𝑍₄,−𝑍₄,−ℎ₂𝑍₂,−ℎ₂𝑍₂,Θ₁₃=ℎ₁𝐴^𝑇_𝑖𝑍₁+𝐻₂𝐴^𝑇_𝑖𝑍₂ℎ₁𝐻_𝑖𝑍₃+ℎ₂𝐻_𝑖𝑍₄𝐻^𝑇_𝑖𝑋𝐻^𝑇_𝑖𝐿^𝑇_𝑖−Φ^𝑇_3𝑖ℎ₁𝐴^𝑇_𝑖𝑍₁+𝐻₂𝐴^𝑇_𝑖𝑍₂ℎ₁𝐻_𝑖𝑍₃+ℎ₂𝐻_𝑖𝑍₄𝐻^𝑇_𝑖𝑋𝐻^𝑇_𝑖𝐿^𝑇_𝑖ℎ0000000000₁𝐴^𝑇_𝑑1𝑖𝑍₁+𝐻₂𝐴^𝑇_𝑑1𝑖𝑍₂ℎ₁𝐻_𝑑1𝑖𝑍₃+ℎ₂𝐻_𝑑1𝑖𝑍₄𝐻^𝑇_𝑑1𝑖𝑋𝐻^𝑇_𝑑1𝑖0ℎ₁𝐴^𝑇_𝑑2𝑖𝑍₁+𝐻₂𝐴^𝑇_𝑑2𝑖𝑍₂ℎ₁𝐻_𝑑2𝑖𝑍₃+ℎ₂𝐻_𝑑2𝑖𝑍₄𝐻^𝑇_𝑑2𝑖𝑋𝐻^𝑇_𝑑2𝑖0,Θ000000000000000₃₃=diag−ℎ₁𝑍₁−ℎ₂𝑍₂,−ℎ₁𝑍₃−ℎ₂𝑍₄,−𝑋−𝐼∗−Φ₄,−𝐼. (42)

When the LMIs (38)–(41) are feasible, there exist nonsingular matrices 𝜎 and 𝛽 satisfying 𝜎𝛽^𝑇=𝐼−𝑋𝑊⁻¹. Under this condition, a desired time-dependent filter can be chosen as 𝐿_𝑓𝑖=Φ_3𝑖𝑊⁻¹𝛽^−𝑇,𝑖=1,…,𝑟. (43)

Proof. From the demonstration in [21], we know that 𝐼−𝑋𝑊^𝑇−1 is nonsingular. Therefore, there always exist nonsingular matrices 𝜎 and 𝛽 such that 𝜎𝛽^𝑇=𝐼−𝑋𝑊⁻¹ holds. Then, we define the nonsingular matrices 𝜒₁ and 𝜒₂ as follows: 𝜒₁=𝑊⁻¹𝐼𝛽^𝑇𝑜;𝜒₂=𝐼𝑋0𝜎^𝑇. (44) Let 𝑃=𝜒₂𝜒₁⁻¹; there is 𝜎𝑃=𝑋𝜎_𝑇𝛽⁻¹𝑊⁻¹𝑋_𝑊𝑊⁻¹𝛽^−𝑇, (45) which means that 𝑃>0.
Now, using Lemma 3 and recalling (40), we can deduce that 𝜅=^𝑟_𝑖=1𝜌²_𝑖(𝑠(𝑡))Θ_𝑖𝑖+𝜚_𝑖𝐹_𝑖(𝑡)Ξ^𝑇_𝑖+Ξ_𝑖𝐹_𝑖(𝑡)𝜚^𝑇_𝑖+^𝑟𝑟_{𝑖=1𝑗>𝑖}𝜌_𝑖(𝑠(𝑡))𝜌_𝑗(𝑠(𝑡×Θ))_𝑖𝑗+𝜚_𝑖𝐹_𝑖(𝑡)Ξ^𝑇_𝑖+Ξ_𝑖𝐹_𝑖(𝑡)𝜚^𝑇_𝑖+Θ_𝑗𝑖+𝜚_𝑖𝐹_𝑗(𝑡)Ξ^𝑇_𝑗+Ξ_𝑗𝐹_𝑗(𝑡)𝜚^𝑇_𝑗<^𝑟_𝑖=1𝜌²_𝑖(𝑠(𝑡))Θ_𝑖𝑖+𝜀_𝑖⁻¹𝜚_𝑖𝜚^𝑇_𝑖+𝜀_𝑖Ξ_𝑖Ξ^𝑇_𝑖+^𝑟𝑟_{𝑖=1𝑗>𝑖}𝜌_𝑖(𝑠(𝑡))𝜌_𝑗(𝑠(𝑡))Θ_𝑖𝑗+𝜀_𝑖⁻¹𝜚_𝑖𝜚^𝑇_𝑖+𝜀_𝑖Ξ_𝑖Ξ^𝑇_𝑖+Θ_𝑗𝑖+𝜀_𝑗⁻¹𝜚_𝑗𝜚^𝑇_𝑗+𝜀_𝑗Ξ_𝑗Ξ^𝑇_𝑗<^𝑟_𝑖=1𝜌²_𝑖(𝑠(𝑡))Γ_𝑖+^𝑟𝑟_{𝑖=1𝑗>𝑖}𝜌_𝑖(𝑠(𝑡))𝜌_𝑗(𝑠(𝑡))Π_𝑖𝑗+Π^𝑇_𝑗𝑖=ℎ₁ℎ(𝑠(𝑡))𝐼₂⋮ℎ(𝑠(𝑡))𝐼_𝑟(𝑠(𝑡))𝐼^𝑇Γ₁Π₁₂⋯Π_1𝑟∗Γ₂⋯Π_2𝑟⋮⋮⋱⋮∗∗⋯Γ_𝑟ℎ₁(𝑠(𝑡𝐼ℎ))₂(𝑠(𝑡𝐼⋮ℎ))_𝑟(𝑠(𝑡))𝐼<0. (46) Then, it is easy to see that diag𝜒₂^−𝑇𝑊⁻¹0𝜅𝜒0𝐼,𝐼,…,𝐼diag₂^−𝑇𝑊⁻¹0=0𝐼,𝐼,…,𝐼ΩΨ₁₂Ψ₁₃∗Ψ₂₂𝟎∗∗Ψ₃₃<0, (47)
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 𝐻_∞ performance level 𝛾.

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 𝜏₁(𝑡) and 𝜏₁(𝑡) in the state 𝑥(𝑡). 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 ℎ₃,ℎ₄,… 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 (Σ) with model parameters given as follows: 𝐴₁=−1.90.40.2−1.1,𝐴_𝑑11=,𝐴−1.10.1−0.8−0.9_𝑑21=−0.020.010.05−0.05,𝐻₁=,𝐻−0.040.10.03−0.05_𝑑11=−0.010.020.01−0.05,𝐻_𝑑21=,𝐶−0.040.010.03−0.05₁=1−0.4,𝐶_𝑑11=,𝐶−0.40.1_𝑑21=−0.1−0.4,𝐷₁𝐿=0.3,₁=1.0−0.5,𝐿_𝑑11=,𝐿1−0.5_𝑑21=1−0.5,𝐴₂=,𝐴−2.10.20.1−1.1_𝑑12=−0.90.1−1.1−1.2,𝐴_𝑑22=,𝐻−0.040.050.02−0.04₂=−0.40.010.03−0.05,𝐻_𝑑12=,𝐻−0.050.010.03−0.05_𝑑22=,𝐶−0.060.020.01−0.05₂=0.2−0.5,𝐶_𝑑12=,𝐶−0.40.5_𝑑22=−0.3−0.4,𝐿₂=,𝐿−0.20.3_𝑑12=−0.2−0.5,𝐿_𝑑22=𝐵−0.3−0.5₁=1.0−0.2,𝐵₂=0.30.1,𝐷₂=−0.3, (48) and the parameter uncertainties are shown as 𝐾₁₁=1.0−0.5,𝐾₁₂=,𝐾−0.20.3₂₁=0.8−0.1,𝐾₂₂=,𝑇−0.10.4₁₁=0−0.3,𝑇₂₁=𝑇−0.20.1₃₁=−0.50.1,𝑇₁₂=𝑇0−0.3₂₂=0−0.1,𝑇₃₂=−0.3−0.2. (49) The two membership functions are, respectively, described as ℎ₁𝑥₁(𝑡)=11−1+exp6𝑥₁(𝑡),ℎ+2₂𝑥₁(𝑡)=1−ℎ₁. (50)
In this example, the noise attenuation level 𝛾 is set to 𝛾=0.42. From Theorem 6, it is to see that the robust 𝐻_∞ filtering problem is solvable for any 0<ℎ₁(𝑡)≤0.8, 0<ℎ₂(𝑡)≤0.9. Thus, a desired fuzzy 𝐻_∞ filter can be constructed as in the form of (11) with𝐴_𝑓1𝐵_𝑓1𝐿_𝑓1𝐴_𝑓2𝐵_𝑓2𝐿_𝑓2=−69.05748.86076.64153.611−0.5140.427−24.7515−26.076846.823−131.78158.857−161.087−0.0830.116.−76.7602−93.953 (51)

The simulation results of the state response of the plant and the filter are given in Figure 1, where the initial condition is 𝑥₀(𝑡)=0.42.5^𝑇, 𝑥₀(𝑡)=00^𝑇. Figure 2 shows the simulation results of the signal 𝑧(𝑡)−𝑧(𝑡), and the exogenous disturbance input 𝑣(𝑡) is given by 𝑣(𝑡)=12/(5+3.1𝑡),𝑡≥0, which belongs to ℒ₂[0,∞).

Figure 1 

State responses of 𝑥(𝑡) and 𝑥(𝑡).

Figure 2 

Responses of the signal 𝑧(𝑡)=𝑧(𝑡)−𝑧(𝑡).

6. 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 mean-square stability but also a prescribed 𝐻_∞ 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant nos. 61203048 and 61203047.