Improved Results on Fuzzy 𝐻∞ Filter Design for T-S Fuzzy Systems

Abstract

The fuzzy 𝐻∞ filter design problem for T-S fuzzy systems with interval time-varying delay is investigated. The delay is considered as the time-varying delay being either differentiable uniformly bounded with delay derivative in bounded interval or fast varying (with no restrictions on the delay derivative). A novel Lyapunov-Krasovskii functional is employed and a tighter upper bound of its derivative is obtained. The resulting criterion thus has advantages over the existing ones since we estimate the upper bound of the derivative of Lyapunov-Krasovskii functional without ignoring some useful terms. A fuzzy 𝐻∞ filter is designed to ensure that the filter error system is asymptotically stable and has a prescribed 𝐻∞ performance level. An improved delay-derivative-dependent condition for the existence of such a filter is derived in the form of linear matrix inequalities (LMIs). Finally, numerical examples are given to show the effectiveness of the proposed method.

1. Introduction

During the last decades, the filtering problem has attracted many researchers to study through various methodologies, see, for example, [1–20] and the references therein, in which these methods mostlyconsist of two main approaches, namely, the Kalman filtering approach [1–3] and the 𝐻∞ filtering approach [4–17]. In contrast with the Kalman filtering, the 𝐻∞ filtering approach does not require the exact knowledge of the statistics of the external noise signals and it is insensitive to the uncertainties both in the exogenous signastatistics and in dynamic models. This advantage renders the 𝐻∞ filtering approach very appropriate to some practical applications. Recently, the filter design contains two cases of filtering technique, that is, 𝐿₂−𝐿_∞ filtering technique [18–20] and the 𝐻∞ filtering technique [4–17].

On the other hand, Takagi-Sugeno (T-S) fuzzy model can provide an effective way to represent a complex nonlinear system into a weighted sum of some simple linear subsystems [8, 21, 22], which has been an increasing interest in the study of T-S fuzzy systems. In recent years, T-S fuzzy model approach has been extended to 𝐻∞ filter or controller design [4–6, 9, 10, 12, 15–21, 23–35]. For instance, the stability analysis and stabilization synthesis problems of T-S fuzzy systems were studied in [21, 29, 30, 33–35], while fuzzy controllers were designed in [23–28]. One set of fuzzy 𝐻∞ filters for a class of T-S fuzzy systems was designed in [32]. However, the above-mentioned works use common Lyapunov-Krasovskii functional, and the results under a common Lyapunov method are quite conservative. To reduce the conservatism, a fuzzy weighting-dependent Lyapunov method has been proposed in [6], which is effective in reducing conservatism of previous results on fuzzy systems. More recently, Lin et al. [4] and Su et al. [5] have concerned with 𝐻∞ filtering of nonlinear continuous-time state-space models with time-varying delays via T-S fuzzy model approach. However, some negative semidefinite terms are ignored and the lower bound of time delay is restricted to be zero, see, for example, [4–6] and the references therein. Qiu et al. [36] investigated the problem of delay-dependent robust stability and 𝐻∞ filtering design for a class of uncertain continuous-time nonlinear systems with time-varying state delay represented by T-S fuzzy models. However, there is room for further investigation to reduce the conservativeness of the filter design. This motivates the current research.

In this paper, we discuss the fuzzy 𝐻∞ filter design problem for T-S fuzzy systems with interval time-varying delay. Our aim is to design a suitable fuzzy filter, which ensures both the fuzzy stability and a prescribed performance level of the filter error system. By constructing a Lyapunov-Krasovskii functional, estimating the time derivative of the Lyapunov-Krasovskii functional less conservatively, and adopting convex optimization approach, an improved delay-derivative-dependent condition for the solvability of fuzzy 𝐻∞ filter design problem is proposed in terms of linear matrix inequalities (LMIs). Two examples are used to compare with the previous literatures and demonstrate the effectiveness of the proposed method.

The rest of this paper is organized as follows: The fuzzy 𝐻∞ filtering problem is formulated in Section 2; the fuzzy 𝐻∞ performance analysis is derived in Section 3; and fuzzy 𝐻∞ filter design is addressed in Section 4. Numerical examples are provided in Section 5, and Section 6 concludes this paper.

2. Problem Formulation

Consider a nonlinear system with interval time-varying delay which could be approximated by a class of T-S fuzzy systems with interval time-varying delays. The T-S fuzzy model with 𝑟 plant rules can be described by:

Plant rule𝑖: IF 𝜃₁(𝑡) is 𝑁_𝑖1 and⋯and 𝜃_𝑝(𝑡) is 𝑁_𝑖𝑝, THEN𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐴_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐵_𝑖𝑤(𝑡),𝑦(𝑡)=𝐶_𝑖𝑥(𝑡)+𝐶_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐷_𝑖𝑤(𝑡),𝑧(𝑡)=𝐿_𝑖𝑥(𝑡)+𝐿_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐺_𝑖𝑤(𝑡),𝑥(𝑡)=𝜙(𝑡),∀𝑡∈[−ℎ_𝑏,0], (2.1) where 0≤ℎ_𝑎≤𝜏(𝑡)≤ℎ_𝑏, and 𝑥(𝑡)∈ℝ^𝑛 is the state vector; 𝑦(𝑡)∈ℝ^𝑚 is the measurements vector; 𝑤(𝑡)∈ℝ^𝑞 is the disturbance signal vector which belongs to 𝐿₂[0,∞); 𝑧(𝑡)∈ℝ^𝑝 is the signal vector to be estimated; 𝜙(𝑡) is the continuous initial vector function defined on [−ℎ_𝑏,0]; The system coefficient matrices are constant real matrices with appropriate dimensions, where 𝑖=1,2,…,𝑟 and 𝑟 is the number of IF-THEN rules; 𝜃_𝑗(𝑡),(𝑗=1,2,…,𝑝) are the premise variables; 𝑁_𝑖1,𝑁_𝑖2,…,𝑁_𝑖𝑝 are the fuzzy sets. For the sake of convenience, we denote 𝛿_ℎ=ℎ_𝑏−ℎ_𝑎.

The time-varying delay 𝜏(𝑡) is assumed to be either differentiable with𝑑₁≤𝜏̇(𝑡)≤𝑑₂, (2.2) where 𝑑₁ and 𝑑₂ are given bounds, or fast-varying (with no restrictions on the delay derivative).

The fuzzy system (2.1) is supposed to have singleton fuzzifier, product inference and centroid defuzzifier. The final output of the fuzzy system is inferred as follows: 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐴_𝑖𝑥(𝑡)+𝐴_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐵_𝑖𝑤(𝑡)],𝑦(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐶_𝑖𝑥(𝑡)+𝐶_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐷_𝑖𝑤(𝑡)],𝑧(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐿_𝑖𝑥(𝑡)+𝐿_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐺_𝑖𝑤(𝑡)],𝑥(𝑡)=𝜙(𝑡),∀𝑡∈[−ℎ_𝑏,0], (2.3) where for 𝑖=1,2,…,𝑟,ℎ_𝑖𝜇(𝜃(𝑡))=^𝑖(𝜃(𝑡))∑^𝑟_𝑖=1𝜇_𝑖(𝜃(𝑡)),𝜇_𝑖(𝜃(𝑡))=^𝑝_𝑗=1𝑁_𝑖𝑗(𝜃_𝑗(𝑡)) (2.4) and 𝑁_𝑖𝑗(𝜃_𝑗(𝑡)) is the membership function of 𝜃_𝑗(𝑡) in 𝑁_𝑖𝑗. Here 𝜇_𝑖(𝜃(𝑡))≥0. Here, we assume that 𝜇_𝑖(𝜃(𝑡))>0, and ∑^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))=1.

Our aim is to design the following fuzzy filter.

Rule 𝑖: IF 𝜃₁(𝑡) is 𝑁_𝑖1 and⋯and 𝜃_𝑝(𝑡) is 𝑁_𝑖𝑝, THEN𝑥̇(𝑡)=𝐴_𝑓𝑖𝑥(𝑡)+𝐵_𝑓𝑖𝑦(𝑡),𝑥(0)=0,𝑧(𝑡)=𝐶_𝑓𝑖𝑥(𝑡)+𝐷_𝑓𝑖𝑦(𝑡),(𝑖=1,2,…,𝑟), (2.5) where 𝑥(𝑡)∈ℝ^𝑛 is the filter state, 𝑧(𝑡)∈ℝ^𝑝 is the estimation of 𝑧(𝑡) in fuzzy system (2.1), the constant matrices 𝐴_𝑓𝑖∈ℝ^𝑛×𝑛, 𝐵_𝑓𝑖∈ℝ^𝑛×𝑚, 𝐶_𝑓𝑖∈ℝ^𝑝×𝑛, 𝐷_𝑓𝑖∈ℝ^𝑝×𝑚 are the filter matrices to be determined. The final fuzzy filter of fuzzy system (2.1) is thus inferred as follows𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐴_𝑓𝑖𝑥(𝑡)+𝐵_𝑓𝑖𝑦(𝑡)],𝑥(0)=0,𝑧(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐶_𝑓𝑖𝑥(𝑡)+𝐷_𝑓𝑖𝑦(𝑡)]. (2.6)

Defining the augmented state vector 𝑥(𝑡)∶=col{𝑥(𝑡)𝑥(𝑡)}, 𝑒(𝑡)∶=𝑧(𝑡)−𝑧(𝑡), from (2.3) and (2.6), we can then obtain the following filtering error system:𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐴_𝜏𝐶(𝑡)𝐸𝑥(𝑡−𝜏(𝑡))+𝐵(𝑡)𝑤(𝑡),𝑒(𝑡)=𝐶(𝑡)𝑥(𝑡)+_𝜏𝜙(𝑡)𝐸𝑥(𝑡−𝜏(𝑡))+𝐷(𝑡)𝑤(𝑡),𝑥(𝑡)=^𝑇(𝑡)0^𝑇,∀𝑡∈[−ℎ_𝑏,0], (2.7) where 𝐴(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))^𝑟_𝑗=1ℎ_𝑗𝐴(𝜃(𝑡))^𝑗0𝐵_𝑓𝑖𝐶_𝑗𝐴_𝑓𝑖𝐵∶=𝐴(𝑡)0_𝑓(𝑡)𝐶(𝑡)𝐴_𝑓,𝐴(𝑡)_𝜏(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))^𝑟_𝑗=1ℎ_𝑗𝐴(𝜃(𝑡))^𝜏𝑗𝐵_𝑓𝑖𝐶_𝜏𝑗𝐴∶=^𝜏𝐵(𝑡)_𝑓(𝑡)𝐶_𝜏,(𝑡)𝐵(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))^𝑟_𝑗=1ℎ_𝑗𝐵(𝜃(𝑡))^𝑗𝐵_𝑓𝑖𝐷_𝑗𝐵∶=𝐵(𝑡)_𝑓,(𝑡)𝐷(𝑡),𝐸=𝐼0𝐶(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))^𝑟_𝑗=1ℎ_𝑗(𝜃(𝑡))[𝐿_𝑗−𝐷_𝑓𝑖𝐶_𝑗−𝐶_𝑓𝑖]∶=𝐿(𝑡)−𝐷_𝑓(𝑡)𝐶(𝑡)−𝐶_𝑓,𝐶(𝑡)_𝜏(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))^𝑟_𝑗=1ℎ_𝑗(𝜃(𝑡))[𝐿_𝜏𝑗−𝐷_𝑓𝑖𝐶_𝜏𝑗]∶=𝐿_𝜏(𝑡)−𝐷_𝑓(𝑡)𝐶_𝜏(𝑡),𝐷(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑡))^𝑟_𝑗=1ℎ_𝑗(𝜃(𝑡))[𝐺_𝑗−𝐷_𝑓𝑖𝐷_𝑗]∶=𝐺(𝑡)−𝐷_𝑓(𝑡)𝐷(𝑡). (2.8)

So far, the fuzzy 𝐻∞ filter design problem for fuzzy system (2.3) can be stated as follows. Given a scalar 𝛾>0, design a suitable fuzzy filter in the form of (2.5) such that the filtering error system (2.7) has a prescribed 𝐻∞ performance 𝛾, and the following two purposes are satisfied: (i)the system (2.7) with 𝑤(𝑡)=0 is asymptotically stable;(ii)the 𝐻∞ performance ‖𝑒‖₂<𝛾‖𝑤‖₂ is guaranteed for all nonzero 𝑤(𝑡)∈𝐿₂[0,∞) and a prescribed 𝛾>0 under the condition 𝑥(𝑡)=0, for all 𝑡∈[−ℎ_𝑏,0]. If this is the case, we say that the fuzzy 𝐻∞ filter design problem is solved.

3. Fuzzy 𝐻∞ Performance Analysis

In this section, we propose the sufficient criterion for the filter error system (2.7) satisfying a prescribed 𝐻∞ performance level for fuzzy system (2.1) or (2.3).

Theorem 3.1. Given scalars 0≤ℎ_𝑎≤ℎ_𝑏, 𝑑₁≤𝑑₂ and 𝛾>0, the 𝐻∞ filter error system (2.7), for all differentiable delay 𝜏(𝑡)∈[ℎ_𝑎,ℎ_𝑏] with 𝑑₁≤𝜏̇(𝑡)≤𝑑₂, is asymptotically stable and has a prescribed 𝐻∞ performance level 𝛾 if there exist real symmetry matrices 𝑅₀>0, 𝑅_𝛿>0, 𝑄₀>0, 𝑄_𝛿>0, 𝑃=[^𝑃1𝑃2_∗𝑃3]>0, 𝑅_𝜏≥0, 𝑃_𝜏≥0, and real matrices 𝑋_𝑖𝑗(𝑡), (𝑖=1,2;𝑗=1,2,…,6) with appropriate dimensions such that the two LMIs (3.1) where 𝜏̇(𝑡)=𝑑₁,𝑑₂, are feasible. Ξ_𝑖(𝑡)∶=Ω(𝑡)+[−I^𝑇_𝑖𝑄_𝛿I_𝑖+𝛿_ℎ𝑋_𝑖(𝑡)I_𝑖+𝛿_ℎI^𝑇_𝑖𝑋^𝑇_𝑖(𝑡)]ℎ_𝑎Γ^𝑇₁(𝑡)𝑄₀𝛿_ℎΓ^𝑇₁(𝑡)𝑄_𝛿𝛿_ℎ𝑋_𝑖(𝑡)Γ^𝑇₃(𝑡)∗−𝑄₀000∗∗−𝑄_𝛿00∗∗∗−𝑄_𝛿0∗∗∗∗−𝐼<0,(𝑖=1,2), (3.1) where 𝜑Ω(𝑡)=¹¹𝜑₁₂𝑄₀0𝜑₁₅𝜑₁₆∗𝜑₂₂00𝜑₂₅𝜑₂₆∗∗𝑅_𝜏−𝑅₀+𝑅_𝛿−𝑄₀−𝑄_𝛿0𝑄_𝛿0∗∗∗−𝑅_𝛿−𝑄_𝛿−𝑃_𝜏𝑄_𝛿0∗∗∗∗−(1−𝜏̇(𝑡))(𝑅_𝜏−𝑃_𝜏)−2𝑄_𝛿0∗∗∗∗∗−𝛾²𝐼,𝜑₁₁=𝑃₁𝐴(𝑡)+𝐴^𝑇(𝑡)𝑃₁+𝑃₂𝐵_𝑓(𝑡)𝐶(𝑡)+𝐶^𝑇(𝑡)𝐵^𝑇_𝑓(𝑡)𝑃^𝑇₂+𝑅₀−𝑄₀,𝜑₁₂=𝑃₂𝐴_𝑓(𝑡)+𝐴^𝑇(𝑡)𝑃^𝑇₂+𝐶^𝑇(𝑡)𝐵^𝑇_𝑓(𝑡)𝑃₃,𝜑₂₂=𝑃₃𝐴_𝑓(𝑡)+𝐴^𝑇_𝑓(𝑡)𝑃₃,𝜑₁₅=𝑃₁𝐴_𝜏(𝑡)+𝑃₂𝐵_𝑓(𝑡)𝐶_𝜏(𝑡),𝜑₂₅=𝑃^𝑇₂𝐴_𝜏(𝑡)+𝑃₃𝐵_𝑓(𝑡)𝐶_𝜏𝜑(𝑡),₁₆=𝑃₁𝐵(𝑡)+𝑃₂𝐵_𝑓(𝑡)𝐷(𝑡),𝜑₂₆=𝑃^𝑇₂𝐵(𝑡)+𝑃₃𝐵_𝑓𝐼(𝑡)𝐷(𝑡),₁=000−𝐼𝐼0,𝐼₂=,𝑋00𝐼0−𝐼0_𝑖(𝑡)∶=𝑐𝑜𝑙𝑋_𝑖1(𝑡)𝑋_𝑖2(𝑡)𝑋_𝑖3(𝑡)𝑋_𝑖4(𝑡)𝑋_𝑖5(𝑡)𝑋_𝑖6(𝑡),(𝑖=1,2),Γ₁(𝑡)∶=𝐴(𝑡)000𝐴_𝜏,Γ(𝑡)𝐵(𝑡)₂(𝑡)∶=𝐵_𝑓(𝑡)𝐶(𝑡)𝐴_𝑓(𝑡)00𝐵_𝑓(𝑡)𝐶_𝜏(𝑡)𝐵_𝑓,Γ(𝑡)𝐷(𝑡)₃(𝑡)∶=𝐿(𝑡)−𝐷_𝑓(𝑡)𝐶(𝑡)−𝐶_𝑓(𝑡)00𝐿_𝜏(𝑡)−𝐷_𝑓(𝑡)𝐶_𝜏(𝑡)𝐺(𝑡)−𝐷_𝑓(𝑡)𝐷(𝑡). (3.2)(3.3) where 𝑋_𝑖𝑘(𝑡)∶=^𝑟_𝑗=1ℎ_𝑗(𝜃(𝑡))[𝑋^𝑗_𝑖𝑘𝑋(𝑡)],𝑖=1,2;𝑘=1,3,…,6,_𝑖2(𝑡)∶=^𝑟_𝑗=1ℎ_𝑗(𝜃(𝑡))[𝑋^𝑗_𝑖2Γ(𝑡)],₁(𝑡)∶=^𝑟_𝑚=1ℎ_𝑚(𝜃(𝑡))[Γ^𝑚₁],Γ₃(𝑡)∶=^𝑟_𝑚=1ℎ_𝑚(𝜃(𝑡))[Γ^𝑚₃]. (4.7) Thus, from (2.8) and the above definition, we have Π_𝑖(𝑡)=^𝑟_𝑚=1ℎ²_𝑚(𝜃(𝑡))[Π_𝑖(𝑚,𝑚)]+^𝑟_𝑚<𝑛ℎ_𝑚(𝜃(𝑡))ℎ_𝑛(𝜃(𝑡))[Π_𝑖(𝑚,𝑛)+Π_𝑖(𝑛,𝑚)],(𝑖=1,2), (4.8) where Π_𝑖(𝑡)∶=Ψ(𝑡)+[−𝐼^𝑇_𝑖𝑄_𝛿𝐼_𝑖+𝛿_ℎ𝑋_𝑖(𝑡)𝐼_𝑖+𝛿_ℎ𝐼^𝑇_𝑖𝑋^𝑇_𝑖(𝑡)]ℎ_𝑎Γ^𝑇₁(𝑡)𝑄₀𝛿_ℎΓ^𝑇₁(𝑡)𝑄_𝛿𝛿_ℎ𝑋_𝑖(𝑡)Γ^𝑇₃(𝑡)∗−𝑄₀000∗∗−𝑄_𝛿00∗∗∗−𝑄_𝛿0∗∗∗∗−𝐼<0(𝑖=1,2) (4.9) with ΨΨ(𝑡)=¹¹Ψ₁₂𝑄₀0Ψ₁₅Ψ₁₆∗Ψ₂₂00Ψ₂₅Ψ₂₆∗∗𝑅_𝜏−𝑅₀+𝑅_𝛿−𝑄₀−𝑄_𝛿0𝑄_𝛿0∗∗∗−𝑅_𝛿−𝑄_𝛿−𝑃_𝜏𝑄_𝛿0∗∗∗∗−(1−𝜏̇(𝑡))(𝑅_𝜏−𝑃_𝜏)−2𝑄_𝛿0∗∗∗∗∗−𝛾²𝐼,Ψ₁₁=𝑃₁𝐴(𝑡)+𝐴^𝑇(𝑡)𝑃₁+𝑁₂(𝑡)𝐶(𝑡)+𝐶^𝑇(𝑡)𝑁^𝑇₂(𝑡)+𝑅₀−𝑄₀,Ψ₁₂=𝑁₁(𝑡)+𝐴^𝑇(𝑡)𝑈+𝐶^𝑇(𝑡)𝑁^𝑇₂(𝑡),Ψ₂₂=𝑁₁(𝑡)+𝑁^𝑇₁Ψ(𝑡),₁₅=𝑃₁𝐴_𝜏(𝑡)+𝑁₂(𝑡)𝐶_𝜏(𝑡),Ψ₂₅=𝑈𝐴_𝜏(𝑡)+𝑁₂(𝑡)𝐶_𝜏Ψ(𝑡),₁₆=𝑃₁𝐵(𝑡)+𝑁₂(𝑡)𝐷(𝑡),Ψ₂₆=𝑈𝐵(𝑡)+𝑁₂(𝑡)𝐷(𝑡). (4.10) Next, based on Theorem 3.1, we calculate the feasibility of the LMIs Π_𝑖(𝑡)<0,(𝑖=1,2).
Due to 𝑈>0, there exist a nonsingular real 𝑛×𝑛 matrix 𝑃₂ and a real 𝑛×𝑛 matrix 𝑃₃>0 such that 𝑈=𝑃₂𝑃₃⁻¹𝑃^𝑇₂.Let us define 𝐽∶=diag𝐼𝑃₂^−𝑇𝑃₃𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 (4.11) left- and right-multiply Π_𝑖(𝑡), 𝑖=1,2 defined in (4.8) by 𝐽^𝑇 and 𝐽, respectively, and take 𝑋_𝑖2(𝑡)∶=𝑃₃𝑃₂⁻¹𝑋_𝑖2(𝑡), 𝑖=1,2 and 𝐴(𝑡)=𝑃₂⁻¹𝑁₁(𝑡)𝑈⁻¹𝑃₂,𝐵(𝑡)=𝑃₂⁻¹𝑁₂(𝑡),𝐶(𝑡)=𝑁₃(𝑡)𝑈⁻¹𝑃₂,𝐷(𝑡)=𝑁₄(𝑡), (4.12) By replacing (𝐴_𝑓(𝑡),𝐵_𝑓(𝑡),𝐶_𝑓(𝑡),𝐷_𝑓(𝑡)) in Ξ_𝑖(𝑡),𝑖=1,2 defined in (3.1) with (𝐴(𝑡),𝐵(𝑡),𝐶(𝑡),𝐷(𝑡)), one yields Ξ_𝑖(𝑡)=𝐽^𝑇Π_𝑖(𝑡)𝐽,𝑖=1,2 (4.13) Note that if LMIs (4.1) and (4.2) hold, from (4.8), we arrive at Π_𝑖(𝑡)<0, then Ξ_𝑖(𝑡)<0.
On the other hand, from (4.1), notice that 𝑃₁−𝑈=𝑃₁−𝑃₂𝑃₃⁻¹𝑃^𝑇₂>0, applying Schur complement yields [^𝑃1𝑃2_∗𝑃3]>0.
So far, we conclude from Theorem 3.1 that the filter, that is, 𝑥̇‌(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑦(𝑡),𝑥(0)=0,𝑧(𝑡)=𝐶(𝑡)𝑥(𝑡)+𝐷(𝑡)𝑦(𝑡), (4.14) with (𝐴(𝑡),𝐵(𝑡),𝐶(𝑡),𝐷(𝑡)) defined in (4.12), guarantees that the 𝐻∞ filter error system (2.7) is asymptotically stable and has a prescribed 𝐻∞ performance level 𝛾.
And, performing an irreducible linear transformation 𝑥(𝑡)=𝑃₂𝑥(𝑡) in (4.14) yields 𝑥̇(𝑡)=𝑁₁(𝑡)𝑈⁻¹𝑥(𝑡)+𝑁₂(𝑡)𝑦(𝑡),𝑥(0)=0,𝑧(𝑡)=𝑁₃(𝑡)𝑈⁻¹𝑥(𝑡)+𝑁₄(𝑡)𝑦(𝑡). (4.15)
Therefore, the desired filter (2.5) with the filter matrices in (4.4) is readily obtained from (4.15). This completes the proof.

Similar to Corollary 3.2, when 𝑑₁ is unknown, by substituting 𝜏̇(𝑡)=𝑑₂ into (4.2), the following result is then obtained.

Corollary 4.2. Given scalars 0≤ℎ_a≤ℎ_b,d₂ and 𝛾>0, the fuzzy H∞ filter design problem, for all differentiable delay 𝜏(𝑡)∈[ℎ_a,ℎ_b] with 𝜏̇(𝑡)≤𝑑₂, is solvable if there exist matrices 𝑅₀>0, 𝑅_𝛿>0, 𝑄₀>0, 𝑄_𝛿>0, 𝑃₁>0, 𝑈>0, R_𝜏≥0, P_𝜏≥0, and real matrices 𝑁_1𝑖,𝑁_2𝑖,𝑁_3𝑖,𝑁_4𝑖, (𝑖=1,2,…,𝑟), 𝑋^𝑘_𝑖∶=col {𝑋^𝑘_𝑖1𝑋^𝑘_𝑖2𝑋^𝑘_𝑖3𝑋^𝑘_𝑖4𝑋^𝑘_𝑖5𝑋^𝑘_𝑖6}, 𝑖=1,2; 𝑘=1,2,…,𝑟 with appropriate dimensions such that the LMIs: (4.1) and (4.2) where 𝜏̇(t)=d₂, are feasible. Meanwhile, a desired filter in the form of (2.5) is given by the filter matrices in (4.4).
Moreover, if the above LMIs are feasible with 𝑅_𝜏=0,𝑃_𝜏=0, then the fuzzy H∞ filter design problem, for all fast-varying delay 𝜏(𝑡)∈[ℎ_a,ℎ_b], is solvable in which a desired filter in (2.5) is given by the filter matrices in (4.4).

Remark 4.3. Notice that for any scalar 𝜎, if (𝜎𝑍−𝑃)𝑍⁻¹(𝜎𝑍−𝑃)≥0, then −𝑃𝑍⁻¹𝑃≤−2𝜎𝑃+𝜎²𝑍. The fact played a key role in the existing results in [4, 5], respectively. But there existed some coupled matrix variables in the LMIs in [4, 5].Therefore, to solve filter design problem, [4, 5] must use decoupling technique similar to [42] to convert the conditions in [4, 5] into another form, respectively. These decoupling approaches were shown as [4, 5], respectively. Furthermore, because of a scalar being predescribed, the constraint may lead to considerable conservativeness of these results. Examples below show that for different 𝛿 yields different 𝛾_min. From simulation results in Table 2, we can see that if 𝛿=0.7 or 𝛿=20, the conditions in [4, 5] are unsolvable when ℎ_𝑏=1.0, while our result works. Meanwhile, the scalar is not needed in this paper. Examples 5.1 and 5.2 below show that our approach yields less conservative results.

5. Numerical Examples

In this section, three examples are given to show the effectiveness of the proposed method in this paper.

Example 5.1. Consider the following fuzzy system borrowed from [4, 5]: 𝑥̇(𝑡)=²_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐴_𝑖𝑥(𝑡)+𝐴_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐵_𝑖𝑤(𝑡)],𝑦(𝑡)=²_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐶_𝑖𝑥(𝑡)+𝐶_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐷_𝑖𝑤(𝑡)],𝑧(𝑡)=²_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐿_𝑖𝑥(𝑡)+𝐿_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐺_𝑖𝑤(𝑡)], (5.1) where 𝐴₁=−2.10.11−2,𝐴₂=−1.90−0.2−1.1,𝐴_𝜏1=−1.10.1−0.8−0.9,𝐴_𝜏2=,𝐵−0.90−1.1−1.2₁=1−0.2,𝐵₂=,𝐶0.30.1₁=10,𝐶₂=0.5−0.6,𝐶_𝜏1=−0.80.6,𝐶_𝜏2=𝐷−0.21₁=0.3,𝐷₂𝐿=−0.6,₁=1−0.5,𝐿₂=−0.20.3,𝐿_𝜏1=0.10,𝐿_𝜏2=,𝐺00.2₁=0,𝐺₂=0. (5.2) For 𝑑₂=0.3 and 𝛾=0.5, choosing 𝑑₁ and ℎ_𝑎 in Table 1 and applying Theorems 4.1, the results are 𝑑₁-dependent (see Table 1). Moreover, for unknown 𝑑₁ and 𝑑₂, that is, fast-varying delay case, according to Corollary 4.2, by setting 𝑅_𝜏=0,𝑃_𝜏=0,ℎ_𝑎=0,andℎ_𝑏=0.5, we get the optimal attenuation level 𝛾_opt=0.230 after 38 iterations.
For ℎ_𝑎=0, 𝑑₁unknown and 𝑑₂=0.2, to compare with the recently developed fuzzy 𝐻∞ filter, it is worthwhile to point out that a given scalar 𝛿 is needed in [4, 5] while the scalar 𝛿 is any value in our results. Thus, we consider different ℎ_b and 𝛿 to find the minimum index 𝛾. The results obtained by various methods in the literature and in this paper are listed in Table 2. Moreover, for the case of no additional prescribed scalar, in order to demonstrate the advantages of the proposed approach over the existing results, a detailed comparison between the minimum 𝐻∞ performance levels obtained by the methods in [4, 36, 37] and in this paper for different cases is summarized in Table 3. From Tables 2 and 3, it can be seen that stability conditions obtained in this paper are less conservative than the existing ones.
As an example, for given ℎ_𝑎=0,ℎ_𝑏=0.5,𝑑₁=0,ans𝑑₂=0.3, according to Theorem 4.1, solve LMIs in (4.1) and (4.2), and get the minimum performance level 𝛾_opt=0.206 after 32 iterations, and then compute the fuzzy 𝐻∞ filter matrices from (4.4) as follows 𝐴_𝑓1=−7.1207−5.3463−0.7273−4.5289,𝐵_𝑓1=,𝐶−0.19320.2146_𝑓1=−6.3345−2.6742,𝐷_𝑓1𝐴=0.2486,_𝑓2=−3.5662−1.3711−7.4183−10.6811,𝐵_𝑓2=,𝐶−0.17650.1956_𝑓2=−2.3625−5.2980,𝐷_𝑓2=0.2498. (5.3)

Table 1 

Maximum values of ℎ_𝑏 for 𝑑₂=0.3.

Table 2 

Minimum index 𝛾 for 𝑑₂=0.2 (𝑑₁ unknown and ℎ_𝑎=0).

Table 3 

Minimum index 𝛾 for different cases (𝑑₁ unknown and ℎ_𝑏=1.25).

In order to further show the merit of our method, let us consider the following numerical example.

Example 5.2. Consider the following fuzzy system with interval time-varying delay: 𝑥̇(𝑡)=²_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐴_𝑖𝑥(𝑡)+𝐴_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐵𝑤(𝑡)],𝑦(𝑡)=²_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐶_𝑖𝑥(𝑡)+𝐶_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐷𝑤(𝑡)],𝑧(𝑡)=²_𝑖=1ℎ_𝑖(𝜃(𝑡))[𝐿_𝑖𝑥(𝑡)+𝐿_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝐺_𝑖𝑤(𝑡)], (5.4) where 𝐴₁=−1000−0.900−0.5−1,𝐴₂=−0.90.20−0.2−0.500−0.1−0.8,𝐴_𝜏1=,𝐴−0.80.2−0.10.1−0.80−0.40.25−1_𝜏2=00−10.50.10.5−10−0.80.9−0.25,𝐵=0.5,𝐶₁=,𝐶0.50.40₂=0.5−10,𝐶_𝜏1=1−0.50.5,𝐶_𝜏2=10.1−0.5,𝐷=0.25,𝐿₁=0.500,𝐿₂=,𝐿1−0.50_𝜏1=0.10.50.5,𝐿_𝜏2=0.100.5,𝐺₁=0,𝐺₂=0. (5.5) To compare with the ones existing in [4, 5], we assumed that 𝑑₁is unknown and ℎ_𝑎=0. According to Corollary 4.2, choose 𝑑₂=0.2 and the simulations are run for two cases. In the first case, we compute the minimum index 𝛾 for the given different ℎ_𝑏 and 𝛿 in [4, 5] or any 𝛿 in this paper. In the second case, we compute the maximum values of ℎ_𝑏 for the given different 𝛾 and 𝛿 in [4, 5] or any 𝛿 in this paper. The simulation results are shown by Tables 4 and 5, respectively. It can also be clearly seen that our approach has less conservative results than the results in the literatures.
As an example, we assume that ℎ_𝑎=0,ℎ_𝑏=1.0,𝑑₁=0,𝑑₂=0.2,𝛾=0.5, the solutions can be obtained after 20 iterations in which the fuzzy 𝐻∞ filter in the form of (2.5) is given by the following filter matrices as 𝐴_𝑓1=−3.35793.2252−0.32865.2755−12.386312.6979−0.4419−0.1659−7.8342,𝐵_𝑓1=,𝐶4.17412.2921−0.2958_𝑓1=0.25970.05472.2850,𝐷_𝑓1𝐴=0.4963,_𝑓2=−1.6047−1.0030−4.28611.0643−4.94690.2980−0.24510.1274−3.3907,𝐵_𝑓2=,𝐶5.11352.0674−0.4810_𝑓2=−0.71620.2062−6.4570,𝐷_𝑓2=−0.2080. (5.6)

Table 4 

Minimum performance level 𝛾.

Table 5 

Maximum values of ℎ_𝑏.

Next, we will give another example to illustrate that our methods are reduced more conservative than the existing results.

Example 5.3. Consider the linear system (3.24) with the following parameters 𝐴=01−1−2,𝐴_𝑑=.00−11 (5.7) To compare with those results in the previous literatures, assume that 𝑑₁ is unknown. For ℎ_𝑎=1.0, 𝑑₂ unknown or 𝑑₂=0.3, the result of Corollary 3.3 coincides with the one in [41] (the latter are less conservative than those of [39]). Comparison with various existing methods in the literature for the admissible upper-bound ℎ_𝑏, which guarantee the stability of the system (3.24) is listed in Table 6. It is clear that our results are much less conservative than those in [37–39].

Table 6 

Maximum values of ℎ_𝑏 for ℎ_𝑎=1.0 (𝑑₁ unknown).

6. Conclusion

This paper deals with the problem of fuzzy 𝐻∞ filter design for T-S fuzzy systems with interval time-varying delay through T-S fuzzy models. By constructing a novel Lyapunov-Krasovskii functional and estimating the time derivative of the Lyapunov-Krasovskii functional less conservatively, an improved 𝐻∞ filter design scheme is proposed. Three numerical examples are used to illustrate the design procedure and the merit of the proposed method.

Acknowledgment

This work was supported in part by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (no. 708067), the Special Scientific Research Fund for Doctor Subjects of Universities (no. 200805320022), and in part by the Program for Changjiang Scholars and Innovative Research Team in University (no. 531105050037).