# Delay-Dependent RobustH∞ Filtering of the Takagi-Sugeno Fuzzy Stochastic Systems

Authors: Ze Li; Xin-Hao Yang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the problem of the robustH∞ 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.


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## Body

## 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. TheH∞ 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-dependentH∞ 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. Section2 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). Rn denotes the n-dimensional Euclidean space, and Rm×n denotes the set of all m×n real matrices. I stands for an identity matrix of appropriate dimension, while In∈Rn 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 rulei:  IF s1(t) is μi1, and s2(t) is μi2, and ⋯ and sg(t) is μig, THEN
(1)dx(t)=[(Ai+ΔAi(t))x(t)+(Ad1i+ΔAd1i(t))x(t-τ1(t))+(Ad2i+ΔAd2i(t))x(t-τ2(t))+Biv(t)]dt+[(Hd2i+ΔHd2i(t))(Hi+ΔHi(t))x(t)+(Hd1i+ΔHd1i(t))x(t-τ1(t))+(Hd2i+ΔHd2i(t))x(t-τ2(t))]dω(t),(2)dy(t)=[Cix(t)+Cd1ix(t-τ1(t))+Cd2ix(t-τ2(t))+Div(t)]dt,(3)z(t)=Lix(t)+Ld1ix(t-τ1(t))+Ld2ix(t-τ2(t)),(4)x(t)=φ(t),t∈[-max(h1,h2),0],i=1,2,…,r,
where s(t)=[s1(t)s2(t)⋯sg(t)] are the premise variables, μij is the fuzzy set, and r is the number of IF-THEN rules; x(t)∈Rn is the system state; φ(t) is a given differential initial function on [-u2,0], and y(t)∈Rm is the measured output; z(t)∈Rl is a signal to be estimated; v(t)∈Rs 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,
(5)0≤τ1(t)<h1,0≤τ2(t)<h2.
In the considered system, Ai,  Ad1i,Ad2i,Bi,Hi, Hd1i,Hd2i,Ci,Cd1i,Cd2i,Di,Li,Ld1i, and Ld2i are known constant matrices with appropriate dimensions. ΔAi(t), ΔAd1i(t), ΔAd2i(t), ΔHi(t), ΔHd1i(t), and ΔHd2i(t) represent the unknown time-varying parameter uncertainties and are assumed to satisfy
(6)[ΔAi(t)ΔAd1i(t)ΔAd2i(t)ΔHi(t)ΔHd1i(t)ΔHd2i(t)]=[K1iK2i]Fi(t)[T1iT2iT3i],
where K1i, K2i, T1i, T2i, and T3i are known real constant matrices and the unknown time-varying matrix function satisfying
(7)Fi(t)TFi(t)≤I∀t.Now, the defuzzied output of the dynamic fuzzy stochastic model in (1)–(4) can be represented as follows:
(8)(Σ):dx(t)=∑i=1r‍ρi(s(t))ddddddddddd×[(Ai+ΔAi(t))x(t)dddddddddddddd+(Ad1i+ΔAd1i(t))x(t-τ1(t))dddddddddddddd+(Ad2i+ΔAd2i(t))x(t-τ2(t))dddddddddddddd+Biv(t)]dtddddddddddd+[(Hi+ΔHi(t))x(t)dddddddddddddd+(Hd1i+ΔHd1i(t))x(t-τ1(t))dddddddddddddd+(Hd2i+ΔHd2i(t))x(t-τ2(t))]dω(t),dy(t)=∑i=1r‍ρi(s(t))dddddddd×[Cix(t)+Cd1ix(t-τ1(t))dddddddddd+Cd2ix(t-τ2(t))+Div(t)]dt,z(t)=∑i=1r‍ρi(s(t))dddddddd×[Lix(t)+Ld1ix(t-τ1(t))dddddddddd+Ld2ix(t-τ2(t))],x(t)=φ(t),t∈[-max(h1,h2),0],
where
(9)ρi(s(t))=ω-i(s(t))∑j=1rω-j(s(t)),ω-i(s(t))=∏j=1g‍μij(sj(t));
using the fuzzy theory, it is easy to see that, for all t,
(10)ρi(s(t))≥0,i=1,2,…,r,∑i=1r‍hi(s(t))=1.Then, we consider the following fuzzy filters:(11)dx^(t)=∑i=1r‍ρi(s(t))[Afix^(t)dt+Bfidy(t)],z^(t)=∑i=1r‍ρi(s(t))[Lfix^(t)],
in which, the fuzzy rule has the same representation as in (1)–(4). Now we consider x^(t)∈Rn and z^(t)∈Rl. The matrixes Afi,  Bfi, and Lfi are the filters need to be determined.Letξ(t)=[x(t)Tx^(t)T]T,  z~(t)=z(t)-z^(t).For convenience, the filtering error dynamic system can be written as(12)(Σ~):dξ(t)=[(A~+ΔA~(t))ξ(t)ddddd+(A~d1+ΔA~d1(t))Gξ(t-τ1(t))ddddd+(A~d2+ΔA~d2(t))Gξ(t-τ2(t))+B~v(t)]dtddd+[(H~+ΔH~(t))ξ(t)+(H~d1+ΔH~d1(t))dddddd×Gξ(t-τ1(t))ddddd+(H~d2+ΔH~d2(t))Gξ(t-τ2(t))]dω(t),z~(t)=L~ξ(t)+L¯d1Gξ(t-τ1(t))ddddd+L¯d2Gξ(t-τ2(t)),
where
(13)A~=[A¯0B¯fC¯A¯f],A~d1=[A¯d1B¯fC¯d1],A~d2=[A¯d2B¯fC¯d2],H~=[H¯000],H~d1=[H¯d10],H~d2=[H¯d20],B~=[B¯B¯fD¯],ΔA~(t)=[ΔA¯(t)000],ΔA~d1(t)=[ΔA¯d1(t)0],ΔA~d2(t)=[ΔA¯d2(t)0],ΔH~(t)=[ΔH¯(t)000],ΔH~d1(t)=[ΔH¯d1(t)0],ΔH~d2(t)=[ΔH¯d2(t)0],A¯=∑i=1r‍ρi(s(t))Ai,A¯d1=∑i=1r‍ρi(s(t))Ad1i,A¯d2=∑i=1r‍ρi(s(t))Ad2i,H¯=∑i=1r‍ρi(s(t))Hi,H¯d1=∑i=1r‍ρi(s(t))Hd1i,H¯d2=∑i=1r‍ρi(s(t))Hd2i,C¯=∑i=1r‍ρi(s(t))Ci,C¯d1=∑i=1r‍ρi(s(t))Cd1i,C¯d2=∑i=1r‍ρi(s(t))Cd2i,B¯=∑i=1r‍ρi(s(t))Bi,D¯=∑i=1r‍ρi(s(t))Di,L~=[L¯-L¯f],G=[I0],A¯f=∑i=1r‍ρi(s(t))Afi,B¯f=∑i=1r‍ρi(s(t))Bfi,L¯f=∑i=1r‍ρi(s(t))Lfi,L¯=∑i=1r‍ρi(s(t))Li,L¯d1=∑i=1r‍ρi(s(t))Ld1i,L¯d2=∑i=1r‍ρi(s(t))Ld2i,ΔA¯(t)=∑i=1r‍ρi(s(t))ΔAi(t),ΔA¯d1(t)=∑i=1r‍ρi(s(t))ΔAd1i(t),ΔH¯(t)=∑i=1r‍ρi(s(t))ΔHi(t),ΔH¯d1(t)=∑i=1r‍ρi(s(t))ΔHd1i(t),ΔA¯d2(t)=∑i=1r‍ρi(s(t))ΔAd2i(t),ΔH¯d2(t)=∑i=1r‍ρi(s(t))ΔHd2i(t).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(ɛ)>0 for any ɛ>0 such that
(14)E(∥x(t)∥2)<ɛ,t>0,
when sup-h≤s≤0E(∥φ(s)∥2)<δ(ɛ), for any uncertain variables. In addition,
(15)limt→∞E(∥x(t)∥2)=0,
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
(16)∥z(t)∥E2<γ∥v(t)∥2,
for any uncertain variables, where
(17)∥z(t)∥E2=(E{∫0∞‍|z(t)|2dt})1/2.Lemma 3.
For the given matricesD,  E, and F with FTF≤I and positive scalar ɛ>0, the following inequality holds:
(18)DFE+(DFE)T≤ɛDDT+ɛ-1ETE.


## 3. Robust Stochastic Stabile

First, we derive the robust stochastic mean-square stochastic conditions and theH∞ conditions for system (Σ~). Defining the following variables for convenience:
(19)Φ(t)=(A¯+ΔA¯(t))ξ(t)+(A¯d1+ΔA¯d1(t))Gξ(t-τ1(t))+(A¯d2+ΔA¯d2(t))Gξ(t-τ2(t))+B¯v(t),g(t)=(H¯+ΔH¯(t))ξ(t)+(H¯d1+ΔH¯d1(t))Gξ(t-τ1(t))+(H¯d2+ΔH¯d2(t))Gξ(t-τ2(t)).Theorem 4.
For given scalars0≤h1,  0≤h2, the filtering error system (Σ~) is robust stochastic mean-square stables and (16) is satisfied if there exist matrices P=PT>0,  Qi=QiT>0,  Ri=RiT>0,  Ni,Mi,  i=1,2,  and Zj=ZjT>0,  j=1,…,4, such that the following matrix inequalities hold:
(20)Ψ=[ΩΨ12Ψ13*Ψ220**Ψ33]<0,
where(21)Ω=[Ω1100Ω14Ω1500PB~*Ω220N2M2000**Ω3300000***Ω440000****Ω55000*****-R1h100******-R2h20*******-γ2I],Ψ12=[N~1N~2h1N~1h1N~2M~1M~2h2M~1h2M~2],Ψ22=diag{-Z3,-Z3,-h1Z1,-h1Z1,-Z4,-Z4,-h2Z2,-h2Z2},Ψ13=[A˘GTU1H˘GTU2H˘PL˘],Ψ33=diag{-U1,-U2,-P,-I},Ω11=P(A~+ΔA~(t))+(A~+ΔA~(t))TP+GT(Q1+Q2+h1R1+h2R2)G,Ω14=P(A~d1+ΔA~d1(t))+GTN1T,Ω15=P(A~d2+ΔA~d2(t))+GTM1T,Ω22=-Q2-N2-N2T,Ω33=-Q2-M2-M2T,Ω44=-N1-N1T,Ω55=-M1-M1T,N~1=[000N1T0000]T,M~1=[0000M1T000]T,N~2=[0N2T000000]T,M~2=[00M2T00000]T,A˘=[A~T+ΔA~T(t)00A~d1T+ΔA~d1T(t)A~d2T+ΔA~d2T(t)00B~T]T,H˘=[H~T+ΔH~T(t)00H~d1T+ΔH~d1T(t)H~d2T+ΔH~d2T(t)000]T,L˘=[L~T00L¯d1TL¯d2T000]T,U1=h1Z1+h2Z2,U2=h1Z3+H2Z4.Proof.
Define the following Lyapunov-Krasovskii candidate for system (Σ~):
(22)V(ξ(t),t)=ξT(t)Pξ(t)+∫t-h1t‍ξT(s)GTQ1Gξ(s)ds+∫t-h2t‍ξT(s)GTQ2Gξ(s)ds+∫-h10‍∫t+θt‍ξT(s)GTR1Gξ(s)dsdθ+∫-h20‍∫t+θt‍ξT(s)GTR2Gξ(s)dsdθ+∫-h10‍∫t+βt‍ΦT(s)GTZ1GΦ(s)dsdβ+∫-h20‍∫t+βt‍ΦT(s)GTZ2GΦ(s)dsdβ+∫-h10‍∫t+βt‍gT(s)GTZ3Gg(s)dsdβ+∫-h20‍∫t+βt‍gT(s)GTZ4Gg(s)dsdβ.
Whenv(t)=0,
(23)dV(ξ(t),t)=LV(ξ(t),t)+2ξT(t)Pg(t)dω(t).
By the Newton-Leibnitz formula, we can get the following equations for any matricesNi,Mi,  i=1,2 with any appropriate dimensions:
(24)2ηT(t)N¯1G[ξ(t)-ξ(t-τ1(t))-∫t-τ1(t)t‍Φ(s)dsdddddddddd-∫t-τ1(t)t‍g(s)dω(s)]=0,2ηT(t)N¯2G[∫t-h1t-τ1(t)‍ξ(t-τ1(t))-ξ(t-h1)dddddddddd-∫t-h1t-τ1(t)‍Φ(s)ds-∫t-h1t-τ1(t)‍g(s)dω(s)]=0,2ηT(t)M¯1G[ξ(t)-ξ(t-τ2(t))-∫t-τ2(t)t‍Φ(s)dsdddddddddd-∫t-τ2(t)t‍g(s)dω(s)]=0,2ηT(t)M¯2G[∫t-h2t-τ2(t)‍ξ(t-τ2(t))-ξ(t-h2)dddddddddd-∫t-h2t-τ2(t)‍Φ(s)ds-∫t-h2t-τ2(t)‍g(s)dω(s)]=0,τ1(t)ηT(t)N¯1Z1-1N¯1Tη(t)-∫t-τ1(t)t‍ηT(t)N¯1Z1-1N¯1Tη(t)ds=0,τ2(t)ηT(t)M¯1Z2-1M¯1Tη(t)-∫t-τ2(t)t‍ηT(t)M¯1Z2-1M¯1Tη(t)ds=0,(h1-τ1(t))ηT(t)N¯2Z1-1N¯1Tη(t)-∫t-h1t-τ1(t)‍ηT(t)N¯1Z1-1N¯1Tη(t)ds=0,(h2-τ2(t))ηT(t)M¯2Z2-1M¯1Tη(t)-∫t-h2t-τ2(t)‍ηT(t)M¯2Z2-1M¯1Tη(t)ds=0,
where
(25)N¯1=[000N1T000]T,N¯2=[0N2T00000]T,M¯1=[0000M1T00]T,M¯2=[00M2T0000]T,
and η(t) is a new vector defined as follows: (26)ηT(t)=[ξT(t)ξT(t-h1)GTξT(t-h2)GTξT(t-τ1(t))GTξT(t-τ2(t))GT(∫t-h1t‍Tξ(s)ds)GT(∫t-h2t‍Tξ(s)ds)GT].
Using the above formulas (24) and Lemma 3, it can be seen that
(27)LV(ξ(t),t)=2ξT(t)PΦ(t)+gT(t)Pg(t)+ξT(t)GTQ1Gξ(t)+ξT(t)GTQ2Gξ(t)-ξT(t-h1)GTQ1Gξ(t-h1)-ξT(t-h2)GTQ2Gξ(t-h2)+h1ΦT(t)GTZ1GΦ(t)+h1ξT(t)GTR1Gξ(t)+h2ξT(t)GTR2Gξ(t)+h2ΦT(t)GTZ2GΦ(t)-∫t-h1t‍ξT(s)GTR1Gξ(s)ds-∫t-h2t‍ξT(s)GTR2Gξ(s)ds+h1gT(t)GTZ3Gg(t)-∫t-h1t‍ΦT(s)GTZ1GΦ(s)ds-∫t-h2t‍ΦT(s)GTZ2GΦ(s)ds+h2gT(t)GTZ4Gg(t)-∫t-h1t‍gT(s)GTZ3Gg(s)ds-∫t-h2t‍gT(s)GTZ4Gg(s)ds≤ηT(t)[Ω¯+N¯1Z3-1N¯1Tddddddddd+H^(GTh1Z3G+GTh2Z4G+P)H^ddddddddd+A^GT(h1Z1+h2Z2)GA^T+N¯2Z3-1N¯2Tddddddddd+h1N¯1Z1-1N¯1T+h1N¯2Z1-1N¯2T+h2M¯1Z2-1M¯1Tddddddddd+h2M¯2Z2-1M¯2T+M¯1Z4-1M¯1T+M¯2Z4-1M¯2T]η(t)-∫t-τ1(t)t‍[ηT(t)N¯1+ΦT(s)GTZ1]Z1-1ddddddddd×[Z1GΦ(s)+N¯1Tη(t)]ds-∫t-h1t-τ1(t)‍[ηT(t)N¯2+ΦT(s)GTZ1]Z1-1ddddddddd×[Z1GΦ(s)+N¯2Tη(t)]ds-∫t-τ2(t)t‍[ηT(t)M¯1+ΦT(s)GTZ2]Z2-1ddddddddd×[Z2GΦ(s)+M¯1Tη(t)]ds-∫t-h2t-τ2(t)‍[ηT(t)M¯2+ΦT(s)GTZ2]Z2-1ddddddddd×[Z2GΦ(s)+M¯2Tη(t)]ds+(∫t-τ1(t)t‍g(s)dω(s))TGTZ3G(∫t-τ1(t)t‍g(s)dω(s))-∫t-τ1(t)t‍gT(s)GTZ3Gg(s)ds+(∫t-h1t-τ1(t)‍g(s)dω(s))TGTZ3G(∫t-h1t-τ1(t)‍g(s)dω(s))-∫t-h1t-τ1(t)‍gT(s)GTZ3Gg(s)ds+(∫t-τ2(t)t‍g(s)dω(s))TGTZ4G(∫t-τ2(t)t‍g(s)dω(s))-∫t-τ2(t)t‍gT(s)GTZ4Gg(s)ds+(∫t-h2t-τ2(t)‍g(s)dω(s))TGTZ4G(∫t-h2t-τ1(t)‍g(s)dω(s))-∫t-h2t-τ1(t)‍gT(s)GTZ4Gg(s)ds,
where(28)Ω¯=[Ω1100Ω14Ω1500*Ω2200N2M20**Ω330000***Ω44000****Ω5500*****-R1h10******-R2h2],A^=[(A~+ΔA~(t))T00(A~d1+ΔA~d1(t))T(A~d2+ΔA~d2(t))T00]T,H^=[(H~+ΔH~(t))T00(H~d1+ΔH~d1(t))T(H~d2+ΔH~d2(t))T00]T.Note that
(29)E(∫t-τ1(t)t‍gT(s)GTZ3Gg(s)ds)=E((∫t-τ1(t)t‍TgT(s)dω(s))dddddddd×GTZ3G(∫t-τ1(t)t‍gT(s)dω(s))∫t-τ1(t)t‍T),E(∫t-h1t-τ1(t)‍gT(s)GTZ3Gg(s)ds)=E((∫t-h1t-τ1(t)‍TgT(s)dω(s))dddddddd×GTZ3G(∫t-h1t-τ1(t)‍gT(s)dω(s))∫t-h1t-τ1(t)‍T),E(∫t-τ2(t)t‍gT(s)GTZ4Gg(s)ds)=E((∫t-τ2(t)t‍TgT(s)dω(s))dddddddd×GTZ4G(∫t-τ2(t)t‍gT(s)dω(s))∫t-τ2(t)t‍T),E(∫t-h2t-τ2(t)‍gT(s)GTZ4Gg(s)ds)=E((∫t-h2t-τ2(t)‍TgT(s)dω(s))dddddddd×GTZ4G(∫t-h2t-τ2(t)‍gT(s)dω(s))∫t-h2t-τ2(t)‍T).
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:
(30)Ω¯+N¯1Z3-1N¯1T+H^(GTh1Z3G+GTh2Z4G+P)H^+A^GT(h1Z1+H2Z2)GA^T+N¯2Z3-1N¯2T+h1N¯1Z1-1N¯1T+h1N¯2Z1-1N¯2T+h2M¯1Z2-1M¯1T+h2M¯2Z2-1M¯2T+M¯1Z4-1M¯1T+M¯2Z4-1M¯2T<0.
From (27), (29), and (30), we can see that
(31)LV(ξ(t),t)<0,
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
(32)E(V(ξ(t),t))=E(∫0t‍LV(ξ(s),s)ds).
Now, set a functionalJT as
(33)JT=E(∫0T‍[z(t)Tz(t)-γ2v(t)Tv(t)]ds),
where t>0. From (32) and (33), it is easy to show that
(34)JT=E(∫0T‍[z(t)Tz(t)-γ2v(t)Tv(t)+LV(ξ(s),s)]ds)-E(V(ξ(t),t))≤E(∫0T‍[z(t)Tz(t)-γ2v(t)Tv(t)+LV(ξ(s),s)]ds),
for all t>0. By using the Schur complement to (3), there is
(35)z(t)Tz(t)-γ2v(t)Tv(t)+LV(ξ(s),s)≤η¯(t)Ψη¯(t)<0,
where (36)η¯T(t)=[ξT(t)ξT(t-h1)GTξT(t-h2)GTξT(t-τ1(t))GTξT(t-τ2(t))GT(∫t-h1t‍Tξ(s)ds)GT(∫t-h2t‍Tξ(s)ds)GTv(t)].
It follows from (33) and (34) that
(37)J(t)<0,
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. RobustH∞ Filter Design

In this section, a sufficient condition for the solvability of robustH∞ 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 ɛi>0 and matrices W>0,  X>0,  Qi>0,  Ri>0,  i=1,2; Zj>0,   j=1,2,3,4;   N1i,   N2i,   M1i,   M2i,   i=1,2,3,4;   {Γi=ΓiT,1≤i≤r},   {Πij,1≤i<j≤r}, such that the following LMIs hold:
(38)[Θii-Γi+ɛiΞiΞiTϱi*-ɛi]<0,(1≤i≤r),(39)[Θij+Θji-Πij-ΠijT+ɛiΞiΞiT+ɛjΞjΞjTϱiϱj*-ɛi0**-ɛj]<0,ddddddddddddddddddddddddddddd(1≤i<j≤r),(40)[Γ1Π12⋯Π1r*Γ2⋯Π2r⋮⋮⋱⋮**⋯Γr]<0,(41)X-W>0,
where(42)ϱiT=[K1iTWK1iTX01*15K1iTU1K2iTU2K2iTXK2iT],ΞiT=[T1iTT1iT00T2iTT3iT015*1],Θij=[Θ11Θ12Θ13*Θ2208*8**Θ33],Θ11=[Λ11Λ1200Λ15Λ1600Λ19*Λ2200Λ25Λ2600Λ29**Λ330Λ35Λ36000***Λ4400000****-N1-N1T0000*****-M1-M1T000******-R1h100*******-R2h20********-γ2I],Λ11=WAi+AiTW+Q1+Q2+h1R1+h2R2,Λ12=WAi+AiTX+CjTΦ1iT+Φ2iT+Q1+Q2+h1R1+h2R2,Λ15=N1T+WAd1i,Λ16=M1T+WAd2i,Λ19=WBi,Λ29=XBi+Φ1iDj,Λ22=XTAi+Φ1iCj+CjTΦ1iT+AiTX+Q1+Q2+h1R1+h2R2,Λ25=N1T+XTAd1i+Φ1iCd1j,Λ26=M1T+XTAd2i+Φ1iCd2j,Λ33=-Q1-N2-N2T,Λ35=N2,Λ36=M2,Λ44=-Q2-M2-M2T,Θ12=[000000000N20h2N2000000000M20h2M2N10h1N1000000000M10h2M1000000000000000000000000000000000],Θ22=diag{-Z3,-Z3,-h1Z1,-h1Z1,-Z4,-Z4,-h2Z2,-h2Z2},Θ13=[h1AiTZ1+H2AiTZ2h1HiZ3+h2HiZ4HiTXHiTLiT-Φ3iTh1AiTZ1+H2AiTZ2h1HiZ3+h2HiZ4HiTXHiTLiT0000000000h1Ad1iTZ1+H2Ad1iTZ2h1Hd1iZ3+h2Hd1iZ4Hd1iTXHd1iT0h1Ad2iTZ1+H2Ad2iTZ2h1Hd2iZ3+h2Hd2iZ4Hd2iTXHd2iT0000000000000000],Θ33=diag{-h1Z1-h2Z2,-h1Z3-h2Z4,[-X-I*-Φ4],-I}.When the LMIs (38)–(41) are feasible, there exist nonsingular matrices σ and β satisfying σβT=I-XW-1. Under this condition, a desired time-dependent filter can be chosen as
(43)Lfi=Φ3iW-1β-T,i=1,…,r.Proof.
From the demonstration in [21], we know that I-XWT-1 is nonsingular. Therefore, there always exist nonsingular matrices σ and β such that σβT=I-XW-1 holds. Then, we define the nonsingular matrices χ1 and χ2 as follows:
(44)χ1=[W-1IβTo];χ2=[IX0σT].
Let P=χ2χ1-1; there is
(45)P=[XσσTβ-1W-1(XW)W-1β-T],
which means that P>0.
Now, using Lemma3 and recalling (40), we can deduce that
(46)κ=∑i=1r‍ρi2(s(t))[Θii+ϱiFi(t)ΞiT+ΞiFi(t)ϱiT]+∑i=1r‍∑j>ir‍ρi(s(t))ρj(s(t))dddddddd×[Θij+ϱiFi(t)ΞiT+ΞiFi(t)ϱiTdddddddddddd+Θji+ϱiFj(t)ΞjT+ΞjFj(t)ϱjT]<∑i=1r‍ρi2(s(t))[Θii+ɛi-1ϱiϱiT+ɛiΞiΞiT]+∑i=1r‍∑j>ir‍ρi(s(t))ρj(s(t))[Θij+ɛi-1ϱiϱiT+ɛiΞiΞiTddddddddddddddddddddd+Θji+ɛj-1ϱjϱjT+ɛjΞjΞjT]<∑i=1r‍ρi2(s(t))Γi+∑i=1r‍∑j>ir‍ρi(s(t))ρj(s(t))[Πij+ΠjiT]=[h1(s(t))Ih2(s(t))I⋮hr(s(t))I]T[Γ1Π12⋯Π1r*Γ2⋯Π2r⋮⋮⋱⋮**⋯Γr][h1(s(t))Ih2(s(t))I⋮hr(s(t))I]<0.
Then, it is easy to see that
(47){diag(χ2-T[W-100I],I,…,I)}κ{diag(χ2-T[W-100I],I,…,I)}=[ΩΨ12Ψ13*Ψ220**Ψ33]<0,
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 asτ1(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:
(48)A1=[-1.90.40.2-1.1],Ad11=[-1.10.1-0.8-0.9],Ad21=[-0.020.010.05-0.05],H1=[-0.040.10.03-0.05],Hd11=[-0.010.020.01-0.05],Hd21=[-0.040.010.03-0.05],C1=[1-0.4],Cd11=[-0.40.1],Cd21=[-0.1-0.4],D1=0.3,L1=[1.0-0.5],Ld11=[1-0.5],Ld21=[1-0.5],A2=[-2.10.20.1-1.1],Ad12=[-0.90.1-1.1-1.2],Ad22=[-0.040.050.02-0.04],H2=[-0.40.010.03-0.05],Hd12=[-0.050.010.03-0.05],Hd22=[-0.060.020.01-0.05],C2=[0.2-0.5],Cd12=[-0.40.5],Cd22=[-0.3-0.4],L2=[-0.20.3],Ld12=[-0.2-0.5],Ld22=[-0.3-0.5]B1=[1.0-0.2],B2=[0.30.1],D2=-0.3,
and the parameter uncertainties are shown as
(49)K11=[1.0-0.5],K12=[-0.20.3],K21=[0.8-0.1],K22=[-0.10.4],T11=[0-0.3],T21=[-0.20.1]T31=[-0.50.1],T12=[0-0.3]T22=[0-0.1],T32=[-0.3-0.2].
The two membership functions are, respectively, described as
(50)h1(x1(t))=(1-11+exp(6x1(t)+2)),h2(x1(t))=1-h1.
In this example, the noise attenuation levelγ 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) with(51)[Af1Bf1Lf1Af2Bf2Lf2]=[-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].The simulation results of the state response of the plant and the filter are given in Figure1, where the initial condition is x0(t)=[0.42.5]T, x^0(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 ofx(t) and x^(t).Figure 2
Responses of the signalz~(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 prescribedH∞ 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.



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*Source: 696058-2013-11-28.xml*