A Novel Approach to ℓ₂-ℓ_∞ Filter Design for T-S Fuzzy Systems with Multiple Time-Varying Delays

Abstract

This paper investigates the ℓ₂-ℓ_∞ filtering problem of T-S fuzzy systems with multiple time-varying delays. First, by the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a delay-dependent sufficient condition on ℓ₂-ℓ_∞-disturbance attenuation is presented, in which both stability and prescribed ℓ₂-ℓ_∞ performance are required to be achieved for the filtering-error systems. Then, based on the condition, the full-order and reduced-order delay-dependent ℓ₂-ℓ_∞ filter design schemes for T-S fuzzy multiple time-varying delays systems are developed in terms of linear matrix inequality (LMI). Finally, an example is given to illustrate the effectiveness of the result.

1. Introduction

Time delay arises frequently in many engineering areas of the real world, which is usually a source of instability. Therefore, the stability analysis and synthesis for time-delay system have been one of a most hot research area in the control community over the past years [1–9]. To research the nonlinear time-delay system, the scholars considered the Takagi-Sugeno (T-S) fuzzy time-delay model which is a kind of effective representation, and many analysis and synthesis methods for T-S fuzzy time-delay systems have been developed over the past years [10–13].

Since the state variables in control systems are not always available, filtering or state estimation of a dynamic system through an available measurement state is one of the fundamental problems in signal processing, communications, and control application [14–22]. There are many works that appeared to cope with the nonlinear filtering problem for T-S fuzzy systems with time delays [23–31]. For example, in [23, 24], a delay-dependent ℋ_∞ filter design via continuous-time T-S fuzzy model approach is proposed in terms of linear matrix inequalities. In [25], by using a basis-dependent Lyapunov function, a delay-dependent result on the ℋ_∞ performance of the discrete filtering error system is presented. Based on the similar Lyapunov function combined with Finslers Lemma, [26] researched the delay-dependent robust ℋ_∞ filtering problem for a class of uncertain discrete-time T-S fuzzy systems with interval-like time-varying state delay. Reference [27] proposed the delay-dependent approach to robust ℋ_∞ and ℓ₂-ℓ_∞ filtering for a class of uncertain nonlinear time-delayed systems. References [28, 29] investigated delay-dependent ℋ_∞ filter design problems for discrete-time T-S fuzzy time-delayed systems and continuous-time T-S fuzzy time-delayed systems, respectively, which were both based on a delay-dependent piecewise Lyapunov-Krasovskii functional.

Several ℋ_∞ filtering approaches for T-S fuzzy systems with multiple delays have been developed over the past few years [32–36]. For instance [32] studied the ℋ_∞ filter design problem for discrete-time T-S fuzzy systems with multiple time delays. In [33], a robust mixed ℋ₂/ℋ_∞ filtering problem for continuous-time T-S fuzzy systems with multiple time-varying delays in state variables was addressed.

While the time-varying delay functions above mentioned were all assumed slow-varying (the derivative of delay function is less than one) or fast-varying (the derivative of delay function is unknown). Reference [34] dealt with the fuzzy ℋ_∞ filter design problem for discrete-time T-S fuzzy systems with multiple time delays in the state variables. Reference [35] introduced a decentralized ℋ_∞ fuzzy filter design for nonlinear interconnected systems with multiple constant delays via T-S fuzzy models. Reference [36] addressed the problem of ℓ₂-ℓ_∞ filter design for T-S fuzzy systems with multiple time-varying delays, but the derivative of delay functions must be less than one.

To the best of our knowledge, the problem of ℓ₂-ℓ_∞ filter design for T-S fuzzy systems with multiple time-varying delays has not been fully investigated in the literature. As is well known, time delays usually exist in many physical systems and result in unsatisfactory performance, and the derivative of delay function may vary from −∞ to +∞. So, the research on T-S fuzzy systems with multiple time-varying delays is of great practical and theoretical significance. This motivates the research in this paper.

In summary, the purpose of this paper is to develop an ℓ₂-ℓ_∞ filter for T-S fuzzy systems with multiple time-varying delays. Based on the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a delay-dependent sufficient condition on ℓ₂-ℓ_∞-disturbance attenuation is presented. Then, the full-order and reduced-order delay-dependent ℓ₂-ℓ_∞ filter design schemes for T-S fuzzy multiple time-varying-delays systems are developed in terms of LMI. The example illustrates the effectiveness of the result.

This paper is organized as follows. In Section 2, the T-S fuzzy model and corresponding filter are formulated. In Section 3 we give the sufficient condition to assure asymptotic stability and the ℓ₂-ℓ_∞ noise-attenuation level bound for the T-S fuzzy filtering-error systems. Based on the condition in Section 3, we present a stable fuzzy filter in terms of LMIs. Section 4 provides illustrative examples to demonstrate the effectiveness of the proposed method. Conclusions are given in Section 5.

Notations. The notations used throughout this paper are fairly standard. The superscript “𝑇” stands for matrix transpose, and the notation 𝑃>0  (𝑃≥0) means that matrix 𝑃 is real symmetric and positive (or being positive semidefinite). 𝐼 and 0 and are used to denote identity matrix and zero matrix with appropriate dimension, respectively. The notation ∗ in a symmetric matrix always denotes the symmetric block in the matrix. The parameter diag{⋯} denotes a block-diagonal matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. System Descriptions and Preliminaries

Consider the nonlinear system with multiple state delays that is described by the following T-S model.

2.1. Plant Form

Rule 𝑖. IF 𝑠₁(𝑡) is 𝐹_𝑖1, 𝑠₂(𝑡) is 𝐹_𝑖2, and 𝑠_𝑛(𝑡) is 𝐹_𝑖𝑛, then 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+^𝑞_𝑘=1𝐴_𝑑𝑖𝑘𝑥𝑡−𝑑_𝑘(𝑡)+𝐵_𝑖𝜔(𝑡),𝑦(𝑡)=𝐶_𝑖𝑥(𝑡)+^𝑞_𝑘=1𝐶_𝑑𝑖𝑘𝑥𝑡−𝑑_𝑘(𝑡)+𝐷_𝑖𝜔(𝑡),𝑧(𝑡)=𝐸_𝑖𝑥(𝑡),𝑥(𝑡)(𝑡)[],ℎ=𝜙,𝑡∈−ℎ,0ℎ=max_𝑘,𝑘=1,2,…,𝑞, (1) where 𝑠₁(𝑡),𝑠₂(𝑡),…,  and  𝑠_𝑛(𝑡) are the premise variables that are measurable, and each 𝐹_𝑖𝑗  (𝑗=1,2,…,𝑛) is a fuzzy set. 𝑥(𝑡)∈𝑅^𝑛 is the state variables. 𝑦(𝑡)∈𝑅^𝑚 is the measured output of the system. 𝑧(𝑡)∈𝑅^𝑝 is the signal to be estimated. 𝜔(𝑡)∈𝑅^𝑏 is the disturbance input. 𝑟 is the number of IF-THEN rules. Also 0≤𝑑_𝑘(𝑡)≤ℎ_𝑘 is the time-varying delay in the state, and it is assumed that 𝑑̇_𝑘(𝑡)≤𝑑_𝑘. That is, the derivative of time-varying delay function is continuous and bounded. 𝑞 is the number of time delays. 𝜙(𝑡) is a vector-valued initial continuous function.

By using center-average defuzzifier, product inference, and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model: 𝑥̇(𝑡)=^𝑟_𝑖=1𝑢_𝑖(𝑠(𝑡))𝐴_𝑖𝑥(𝑡)+^𝑞_𝑘=1𝐴_𝑑𝑖𝑘𝑥𝑡−𝑑_𝑘(𝑡)+𝐵_𝑖𝜔(𝑡(𝑡)𝑥(𝑡)+)=𝐴^𝑞_𝑘=1𝐴_𝑑𝑘𝑥𝑡−𝑑_𝑘(𝑡)(𝑡)𝜔(𝑡),𝑦(𝑡)=+𝐵_𝑟𝑖=1𝑢_𝑖(𝑠(𝑡))𝐶_𝑖𝑥(𝑡)+^𝑞_𝑘=1𝐶_𝑑𝑖𝑘𝑥𝑡−𝑑_𝑘(𝑡)+𝐷_𝑖𝜔(𝑡(𝑡)𝑥(𝑡)+)=𝐶^𝑞_𝑘=1𝐶_𝑑𝑘𝑥𝑡−𝑑_𝑘(𝑡)(𝑡)𝜔(𝑡),𝑧(𝑡)=+𝐷_𝑟𝑖=1𝑢_𝑖(𝑠(𝑡))𝐸_𝑖𝑥(𝑡(𝑡)𝑥(𝑡))=𝐸 (2) with 𝑢_𝑖(𝑠(𝑡=𝛼))^𝑖(𝑠(𝑡))∑^𝑟_𝑖=1𝛼_𝑖(𝑠(𝑡)),𝛼_𝑖(𝑠(𝑡))=^𝑛_𝑗=1𝐹_𝑖𝑗𝑠_𝑗(𝑡), (3) in which 𝐹_𝑖𝑗(𝑠_𝑗(𝑡)) is the grade of membership of 𝑠_𝑗(𝑡) in 𝐹_𝑖𝑗. It is assumed that 𝛼_𝑖(𝑠(𝑡))≥0,  𝑖=1,2,…,𝑟,   ∑^𝑟_𝑖=1𝛼_𝑖(𝑠(𝑡))>0 for all 𝑡. Therefore, 𝑢_𝑖(𝑠(𝑡))≥0 and ∑^𝑟_𝑖=1𝑢_𝑖(𝑠(𝑡))=1 for all 𝑡. In this paper, we study the following filter form of order 𝑙 (𝑙=𝑛 for full-order filter, and 1≤𝑙<𝑛 for reduced-order filter): 𝑥̇_𝑓(𝑡)=^𝑟_𝑖=1𝑢_𝑖(𝑠(𝑡))𝐴_𝑓𝑖𝑥(𝑡)+𝐵_𝑓𝑖𝑦(𝑡,𝑧_𝑓)(𝑡)=^𝑟_𝑖=1𝐶_𝑓𝑖𝑥_𝑓(𝑡), (4) where the 𝐴_𝑓𝑖,𝐵_𝑓𝑖,and𝐶_𝑓𝑖 are the filter parameters to be designed. Combining (2) and (4) and defining 𝜉(𝑡)=[𝑥^𝑇(𝑡),𝑥^𝑇_𝑓(𝑡)]^𝑇, 𝑒(𝑡)=𝑧(𝑡)−𝑧_𝑓(𝑡), and 𝐾=[𝐼_𝑛×𝑛0_𝑛×𝑙], we have the filtering-error system: 𝜉̇(𝑡)=𝐴(𝑡)𝜉(𝑡)+^𝑞_𝑘=1𝐴_𝑑𝑘(𝑡)𝐾𝜉𝑡−𝑑_𝑘(𝑡)+𝐵(𝑡)𝜔(𝑡),𝑒(𝑡)=𝐸(𝑡)𝜉(𝑡), (5) where 𝜉(𝑡)≐[𝜙^𝑇(𝑡),0]^𝑇 for 𝑡∈[−ℎ,0] and 𝐴(𝑡)=𝐴(𝑡)0𝐵_𝑓(𝑡)𝐶(𝑡)𝐴_𝑓(𝑡)=^𝑟𝑟_𝑖=1𝑗=1𝑢_𝑖(𝑠(𝑡))𝑢_𝑗(𝑠(𝑡𝐴))^𝑗0𝐵_𝑓𝑖𝐶_𝑗𝐴_𝑓𝑖,𝐴_𝑑𝑘(𝑡)=𝐴_𝑑𝑘(𝑡)𝐵_𝑓(𝑡)𝐶_𝑑(𝑡)=^𝑟𝑟_𝑖=1𝑗=1𝑢_𝑖(𝑠(𝑡))𝑢_𝑗(𝑠(𝑡𝐴))^𝑑𝑗𝑘𝐵_𝑓𝑖𝐶_𝑑𝑗,𝐵(𝑡)=𝐵(𝑡)𝐵_𝑓(𝑡)𝐷(𝑡)=^𝑟𝑟_𝑖=1𝑗=1𝑢_𝑖(𝑠(𝑡))𝑢_𝑗(𝑠(𝑡𝐵))^𝑗𝐵_𝑓𝑖𝐷_𝑗,𝐸(𝑡)=𝐸(𝑡)−𝐶_𝑓(𝑡)=^𝑟𝑟_𝑖=1𝑖=1𝑢_𝑖(𝑠(𝑡))𝑢_𝑗(𝑠(𝑡))𝐸_𝑗−𝐶_𝑓𝑖. (6)

Before ending this section, we introduce the following definition, which will be used in the derivation of our main results.

Definition 1 (ℓ₂-ℓ_∞ performance). Given a scalar 𝛾>0, the system (1) is said to be with ℓ₂-ℓ_∞ performance if the system (1) is asymptotically stable and the output 𝑧(𝑡) satisfies ‖𝑧(𝑡)‖_∞≤𝛾‖𝜔(𝑡)‖₂ (7) for all nonzero 𝜔∈ℓ₂[0,+∞] under zero-initial condition, where, ‖𝜔(𝑡)‖₂=^∞₀𝜔^𝑇(𝑡)𝜔(𝑡),‖𝑧(𝑡)‖_∞=sup_𝑡𝑧^𝑇(𝑡)𝑧(𝑡). (8)

Here, we want to design a suitable filter (4) for the system (1) with a ℓ₂-ℓ_∞ performance.

3. Main Results

In this section, the conditions to assure the system (1) asymptotically stable with ℓ₂-ℓ_∞ performance 𝛾 for the T-S fuzzy filtering-error systems are presented (Lemmas 2 and 3). Then, based on the conditions, a filter is given in terms of LMIs.

Lemma 2. Given 𝛾>0, if there exist common matrices 0<𝑃∈𝑅^{(𝑛+𝑙)×(𝑛+𝑙)}, 0<𝑄_𝑘∈𝑅^𝑛×𝑛, 0<𝑅_𝑘∈𝑅^𝑛×𝑛, 𝑌_𝑘(𝑡)∈𝑅^𝑛×𝑛, 𝑇_𝑘(𝑡)∈𝑅^𝑛×𝑛, 𝑈_𝑘∈𝑅^𝑏×𝑛, 𝑘=1,2,…,𝑞, and 𝐴_𝑓(𝑡), 𝐵_𝑓(𝑡), and 𝐶_𝑓(𝑡) satisfying Ξ₁₁Ξ₁₂Ξ₁₃∗−Ξ₂₂0∗∗−Ξ₃₃<0,𝑃𝐸^𝑇(𝑡)∗𝛾²𝐼>0, (9)(10) where

Ξ₁₁=Π₀Π₁⋯Π_𝑞Π_𝑞+1∗−1−𝑑₁𝑄₁−𝑇₁(𝑡)−𝑇^𝑇₁(𝑡)00−𝑈^𝑇₁(𝑡)∗∗⋱⋮⋮∗∗∗−1−𝑑_𝑞𝑄_𝑞−𝑇_𝑞(𝑡)−𝑇^𝑇_𝑞(𝑡)−𝑈^𝑇_𝑞(𝑡),Π∗∗∗∗−𝐼₀=𝐴^𝑇(𝑡)𝑃+𝑃𝐴(𝑡)+^𝑞_𝑘=1𝐾^𝑇𝑄_𝑘𝐾+𝐾^𝑇𝑌_𝑘(𝑡)𝐾+𝐾^𝑇𝑌^𝑇_𝑘(𝑡)𝐾,Π₁=𝑃𝐴_𝑑1(𝑡)−𝐾^𝑇𝑌₁(𝑡)+𝐾^𝑇𝑇^𝑇₁(𝑡),Π_𝑞=𝑃𝐴_𝑑𝑞(𝑡)−𝐾^𝑇𝑌_𝑞(𝑡)+𝐾^𝑇𝑇^𝑇_𝑞(𝑡),Π_𝑞+1=𝑃𝐵(𝑡)+𝐾^𝑇𝑈^𝑇_𝑞(𝑡),Ξ₁₂=ℎ₁𝐴^𝑇(𝑡)𝐾^𝑇𝑅₁⋯ℎ_𝑞𝐴^𝑇(𝑡)𝐾^𝑇𝑅_𝑞ℎ₁𝐴^𝑇_𝑑1(𝑡)𝐾^𝑇𝑅₁⋯ℎ_𝑞𝐴^𝑇_𝑑1(𝑡)𝐾^𝑇𝑅_𝑞⋮⋮⋮ℎ₁𝐴^𝑇_𝑑𝑞(𝑡)𝐾^𝑇𝑅₁⋯ℎ_𝑞𝐴^𝑇_𝑑𝑞(𝑡)𝐾^𝑇𝑅_𝑞ℎ₁𝐵^𝑇(𝑡)𝐾^𝑇𝑅₁⋯ℎ_𝑞𝐵^𝑇(𝑡)𝐾^𝑇𝑅_𝑞,Ξ₁₃=ℎ₁𝐾^𝑇𝑌₁(𝑡)⋯ℎ_𝑞𝐾^𝑇𝑌_𝑞(𝑡)ℎ₁𝑇₁(𝑡)⋯0⋮⋱⋮0⋯ℎ_𝑞𝑇_𝑞(𝑡)ℎ₁𝑈₁(𝑡)⋯ℎ_𝑞𝑈_𝑞(𝑡),Ξ₂₂=Ξ₃₃=diag𝑅₁,…,𝑅_𝑞, (11)

then the system (5) is asymptotically stable with an ℓ₂-ℓ_∞ performance 𝛾.

Proof. Choose a Lyapunov-Krasovskii functional candidate as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (12) where 𝑉₁(𝑡)=𝜉^𝑇(𝑡)(𝑡),𝑉𝑃𝜉₂(𝑡)=^𝑞_𝑗=1^𝑡_{𝑡−𝑑𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_𝑗(𝑠)𝑉𝐾𝜉𝑑𝑠,₃(𝑡)=^𝑞_𝑗=1⁰_−ℎ𝑗^𝑡_𝑡+𝜃𝜉̇^𝑇(𝑠)𝐾^𝑇𝑅_𝑗𝐾𝜉̇(𝑠)𝑑𝑠. (13)
The time derivative of 𝑉(𝑡) along the solution of (5) is computed as follows: 𝑉̇₁(𝑡)=𝜉̇^𝑇(𝑡)𝑃𝜉(𝑡)+𝜉^𝑇(𝑡)𝑃𝜉̇(𝑡)=𝐴(𝑡)𝜉(𝑡)+^𝑞_𝑘=1𝐴_𝑑𝑘𝐾𝜉𝑡−𝑑_𝑘(𝑡)+𝐵(𝑡)𝜔(𝑡)^𝑇𝑃𝜉(𝑡)+𝜉^𝑇(𝑡)𝑃𝐴(𝑡)𝜉(𝑡)+^𝑞_𝑘=1𝐴_𝑑𝑘𝐾𝜉𝑡−𝑑_𝑘(𝑡)+𝐵(𝑡)𝜔(𝑡),𝑉̇₂(𝑡)≤^𝑞_𝑘=1𝜉^𝑇(𝑡)𝐾^𝑇𝑄_𝑘(𝑡)−𝐾𝜉1−𝑑_𝑘𝜉^𝑇𝑡−𝑑_𝑘(𝑡)𝐾^𝑇𝑄_𝑘𝐾𝜉𝑡−𝑑_𝑘(𝑡),𝑉̇₃(𝑡)=^𝑞_𝑘=1ℎ_𝑘𝜉̇^𝑇(𝑡)𝐾^𝑇𝑅_𝑘𝐾𝜉̇(𝑡)−^𝑞_𝑘=1^𝑡_{𝑡−𝑑𝑘(𝑡)}𝜉̇^𝑇(𝑠)𝐾^𝑇𝑅_𝑘𝐾𝜉̇(𝑠)𝑑𝑠. (14)(15)(16)
Applying free-weighting matrix method [37], using the Newton-Leibniz formula that 0=2𝜉^𝑇(𝑡)𝐾^𝑇𝑌_𝑘(𝑡)+𝜉^𝑇𝑡−𝑑_𝑘(𝑡)𝐾^𝑇𝑇_𝑘(𝑡)+𝜔^𝑇(𝑡)𝑈_𝑘(𝑡×(𝑡))𝐾𝜉−𝐾𝜉𝑡−𝑑_𝑘(𝑡)−^𝑡_{𝑡−𝑑𝑘(𝑡)}𝐾𝜉̇(𝑠)𝑑𝑠,𝑘=1,2,…,𝑞, (17)
and defining 𝜂^𝑇_𝑘(𝑡)=𝜉^𝑇(𝑡),𝜉^𝑇𝑡−𝑑_𝑘(𝑡)𝐾^𝑇,𝜔^𝑇(𝑡,𝜂)𝑘=1,2,…,𝑞,_𝑇(𝑡)=𝜉^𝑇(𝑡),𝜉^𝑇𝑡−𝑑₁(𝑡)𝐾^𝑇𝜉,…,_𝑇𝑡−𝑑_𝑞(𝑡)𝐾^𝑇,𝜔^𝑇(𝑡,𝑀_𝑇)(𝑡)=𝑌^𝑇(𝑡),𝑇^𝑇(𝑡),𝑈^𝑇(𝑡), (18)
we can know that −^𝑡_{𝑡−𝑑𝑘(𝑡)}𝜉̇^𝑇(𝑠)𝐾^𝑇𝑅_𝑘𝐾𝜉̇(𝑠)𝑑𝑠=−^𝑡_{𝑡−𝑑𝑘(𝑡)}𝜉̇^𝑇(𝑠)𝐾^𝑇𝑅_𝑘𝐾𝜉̇(𝑠)𝜉𝑑𝑠+2^𝑇(𝑡)𝐾^𝑇𝑌_𝑘(𝑡)+𝜉^𝑇𝑡−𝑑_𝑘(𝑡)𝐾^𝑇𝑇_𝑘(𝑡)+𝜔^𝑇(𝑡)𝑈_𝑘(𝑡×(𝑡))𝐾𝜉−𝐾𝜉𝑡−𝑑_𝑘(𝑡)−^𝑡_{𝑡−𝑑𝑘(𝑡)}𝐾𝜉̇(𝑠)𝑑𝑠≤ℎ_𝑘𝜂^𝑇_𝑘𝑀(𝑡)𝑅_𝑘⁻¹𝑀^𝑇(𝑡)𝜂_𝑘−^𝑡_{𝑡−𝑑𝑘(𝑡)}𝜂^𝑇_𝑘𝑀(𝑡)+𝜉̇^𝑇(𝑠)𝐾^𝑇𝑅_𝑘×𝑅_𝑘⁻¹𝑀^𝑇(𝑡)𝜂_𝑘+𝑅_𝑘𝐾𝜉̇(𝑠𝜉)𝑑𝑠+2^𝑇(𝑡)𝐾^𝑇𝑌_𝑘(𝑡)+𝜉^𝑇𝑡−𝑑_𝑘(𝑡)𝐾^𝑇𝑇_𝑘(𝑡)+𝜔^𝑇(𝑡)𝐾^𝑇𝑈_𝑘(𝑡×(𝑡))𝐾𝜉−𝐾𝜉𝑡−𝑑_𝑘(𝑡). (19)
Consider the index 𝐽≜𝑉(𝑡)−^𝑡₀𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠. (20) Then for any nonzero 𝜔∈ℓ₂[0,+∞] under zero-initial condition, 𝐽≜^𝑡₀𝑉̇(𝑠)−𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠. (21)
After substitution of 𝜉̇(𝑡) into (16) with (5). and taking into consideration (19), one has from (14), (15), and (16) that 𝐽≤𝜂^𝑇(𝑡)Ξ₁₁+Ξ₁₂Ξ⁻¹₂₂Ξ^𝑇₁₂+Ξ₁₃Ξ⁻¹₃₃Ξ^𝑇₁₃𝜂(𝑡). (22)
Applying the Schur complement to (22), we know that (9) guarantees 𝐽<0, which implies that 𝜉^𝑇(𝑡)𝑃𝜉(𝑡)≤𝑉(𝑡)<^𝑡₀𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠. (23)
On the other hand, using the Schur complement to (10), we can know that 𝐸^𝑇(𝑡)𝐸(𝑡)<𝛾²𝑃. Then it can be easily got that for all 𝑡≥0𝑒^𝑇(𝑡)𝑒(𝑡)=𝜉^𝑇(𝑡)𝐸^𝑇(𝑡)𝐸(𝑡)𝜉(𝑡)<𝛾²𝜉^𝑇(𝑡)𝑃𝜉(𝑡)<𝛾^2𝑡₀𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠<𝛾^2∞₀𝜔^𝑇(𝑠)𝜔(𝑠)𝑑𝑠. (24)
Taking the supremum over 𝑡≥0 yields ‖𝑒(𝑡)‖²_∞<𝛾²‖𝜔(𝑡)‖²₂ for all nonzero 𝜔∈ℓ₂[0,+∞].
Next, we prove the asymptotic stability of system (5) when 𝜔=0. Choose a Lyapunov-Krasovskii functional 𝑉(𝜉_𝑡) as in (12), where 𝜉_𝑡=𝜉(𝑡−𝛼), 𝛼∈[−ℎ,0]. It is easy to find that there exist two scalars 𝑐₁>0 and 𝑐₂>0 such that 𝑐₁𝜉²≤𝑉≤𝑐₂sup_{[]𝛼∈−ℎ,0}𝜉_𝑡². (25)
Similar to the above deduction, we can know from (9) that the time derivative of 𝑉 along the solution of (5) with 𝜔=0 satisfies 𝑉̇<0. This proves the asymptotic stability of system (5) with 𝜔=0 according to the same method of [19]. This completes the proof.

Lemma 2 is the sufficient condition for the ℓ₂-ℓ_∞ filter design which contains the coupled matrix variables in the matrix inequality. Using the decoupling technique as follows, we can transform Lemma 2 into another form.

Lemma 3. Given 𝛾>0, if there exist common matrices 0<𝑃∈𝑅^{(𝑛+𝑙)×(𝑛+𝑙)}, 0<𝑄_𝑘∈𝑅^𝑛×𝑛, 0<𝑅_𝑘∈𝑅^𝑛×𝑛, 𝑌_𝑘(𝑡)∈𝑅^𝑛×𝑛, 𝑇_𝑘(𝑡)∈𝑅^𝑛×𝑛, 𝑈_𝑘∈𝑅^𝑏×𝑛, 𝑘=1,2,…,𝑞, and 𝐴_𝑓(𝑡), 𝐵_𝑓(𝑡), and 𝐶_𝑓(𝑡), such that (9), (10) hold if and only if there exist matrices 0<Ω, 0<𝐹, 0<𝑄_𝑘, 𝑌_𝑘(𝑡), 𝑇_𝑘(𝑡), 𝑈_𝑘(𝑡), 𝑘=1,2,…,𝑞, and 𝐴_𝑓(𝑡), 𝐵_𝑓(𝑡), and 𝐶_𝑓(𝑡) such that the following inequalities hold: Σ(𝑡)<0,Ω𝐸𝐹𝐸^𝑇(𝑡)∗𝐹−𝐶^𝑇_𝑓(𝑡)∗∗𝛾²𝐼>0, (26)(27) where

Σ(𝑡)=Σ₁₁Σ₁₂Σ₁₃∗−Σ₂₂0∗∗−Σ₃₃,Σ₁₁=Φ₀Φ₁⋯Φ_𝑞Φ_𝑞+1∗−1−𝑑₁𝑄₁−𝑇₁(𝑡)−𝑇^𝑇₁(𝑡)00−𝑈^𝑇₁(𝑡)∗∗⋱⋮⋮∗∗∗−1−𝑑_𝑞𝑄_𝑞−𝑇_𝑞(𝑡)−𝑇^𝑇_𝑞(𝑡)−𝑈^𝑇_𝑞(𝑡),Φ∗∗∗∗−𝐼₀=Ω𝐴(𝑡)+𝐸𝐵_𝑓(𝑡)𝐶(𝑡)𝐸𝐴_𝑓(𝑡)𝐹𝐸^𝑇𝐴(𝑡)+𝐵_𝑓(𝑡)𝐶(𝑡)𝐴_𝑓(𝑡)+Ω𝐴(𝑡)+𝐸𝐵_𝑓(𝑡)𝐶(𝑡)𝐸𝐴_𝑓(𝑡)𝐹𝐸^𝑇𝐴(𝑡)+𝐵_𝑓(𝑡)𝐶(𝑡)𝐴_𝑓(𝑡)^𝑇+^𝑞_𝑘=1𝐾^𝑇𝑄_𝑘𝐾+𝐾^𝑇𝑌_𝑘(𝑡)𝐾+𝐾^𝑇𝑌^𝑇_𝑘(𝑡)𝐾,Φ₁=Ω𝐴_𝑑1(𝑡)+𝐸𝐵_𝑓(𝑡)𝐶_𝑑1(𝑡)𝐹𝐸^𝑇𝐴_𝑑1(𝑡)+𝐵_𝑓(𝑡)𝐶_𝑑1(𝑡)−𝐾^𝑇𝑌₁(𝑡)+𝐾^𝑇𝑇^𝑇₁(𝑡),Φ_𝑞=Ω𝐴_𝑑𝑞(𝑡)+𝐸𝐵_𝑓(𝑡)𝐶_𝑑𝑞(𝑡)𝐹𝐸^𝑇𝐴_𝑑𝑞(𝑡)+𝐵_𝑓(𝑡)𝐶_𝑑𝑞(𝑡)−𝐾^𝑇𝑌_𝑞(𝑡)+𝐾^𝑇𝑇^𝑇_𝑞(𝑡),Φ_𝑞+1=Ω𝐵(𝑡)+𝐸𝐵_𝑓(𝑡)𝐷(𝑡)𝐹𝐸^𝑇𝐵(𝑡)+𝐵_𝑓(𝑡)𝐷(𝑡)+𝐾^𝑇𝑈^𝑇_𝑞(𝑡),Σ₁₂=ℎ₁𝐴^𝑇(𝑡)𝐾^𝑇𝑅₁⋯ℎ_𝑞𝐴^𝑇(𝑡)𝐾^𝑇𝑅_𝑞ℎ₁𝐴^𝑇_𝑑1(𝑡)𝑅₁⋯ℎ_𝑞𝐴^𝑇_𝑑1(𝑡)𝑅_𝑞⋮⋮⋮ℎ₁𝐴^𝑇_𝑑𝑞(𝑡)𝑅₁⋯ℎ_𝑞𝐴^𝑇_𝑑𝑞(𝑡)𝑅_𝑞ℎ₁𝐵^𝑇(𝑡)𝑅₁⋯ℎ_𝑞𝐵^𝑇(𝑡)𝑅_𝑞,Σ₁₃=ℎ₁𝐾^𝑇𝑌₁(𝑡)⋯ℎ_𝑞𝐾^𝑇𝑌_𝑞(𝑡)ℎ₁𝑇₁(𝑡)⋯0⋮⋱⋮0⋯ℎ_𝑞𝑇_𝑞(𝑡)ℎ₁𝑈₁(𝑡)⋯ℎ_𝑞𝑈_𝑞(𝑡),Σ₂₂=Σ₃₃=diag𝑅₁,…,𝑅_𝑞,𝐸=𝐼_𝑙×𝑙0_{𝑙×(𝑛−𝑙)}, (28)

then the system (5) is asymptotically stable with an ℓ₂-ℓ_∞ performance 𝛾.

Proof. Necessity. Suppose (9), (10) hold. Partition as 𝑆𝑃=Ω𝐸𝑆_𝑇𝐸^𝑇𝑊, (29) where Ω>0, 𝑊>0, and 𝑆 is invertible. Let 𝐻=𝐼00𝑆𝑊⁻¹. (30)
We pre- and postmultiply 𝐷 and its transpose to (9) and (10), respectively, where 𝐷=diag{𝐻,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼}. (31)
Apply the changes of variables such that 𝐹=𝑆𝑊⁻¹𝑆^𝑇,𝐴_𝑓(𝑡)=𝑆𝐴_𝑓(𝑡)𝑊⁻¹𝑆^𝑇,𝐵_𝑓(𝑡)=𝑆𝐵_𝑓(𝑡),𝐶(𝑡)=𝐶_𝑓(𝑡)𝑊⁻¹𝑆^𝑇. (32)
Then we obtain (26) and (27).
Sufficiency. Suppose that (26) and (27) hold for Ω>0, 𝐹>0,   𝑄_𝑘>0,  𝑌_𝑘(𝑡), 𝑇_𝑘(𝑡), 𝑈_𝑘(𝑡), 𝑘=1,2,…,𝑞, and 𝐴_𝑓(𝑡), 𝐵_𝑓(𝑡), and 𝐶_𝑓(𝑡). Choose two matrices with 𝑊>0 and 𝑆 being invertible such that 𝐹=𝑆𝑊⁻¹𝑆^𝑇. Let 𝑃 and 𝐻 be defined as in (29) and (30). Then 𝑃>0 is concluded from (27). Pre- and postmultiply 𝐷⁻¹ and 𝐷^−𝑇 to (26) and (27), respectively. We can get (9) and (10) with the changes of variables as 𝐴_𝑓(𝑡)=𝑆⁻¹𝐴_𝑓(𝑡)𝑆^−𝑇𝑊,𝐵_𝑓(𝑡)=𝑆⁻¹𝐵_𝑓(𝑡),𝐶_𝑓(𝑡)=𝐶_𝑓(𝑡)𝑆^−𝑇𝑊. (33)
This completes the proof.

Theorem 4. Given 𝛾>0, if there exist common matrices Ω>0, 𝐹>0, 𝑄_𝑘>0, 𝑇_𝑘𝑖, 𝑌_𝑘𝑖, 𝑈_𝑘𝑖, 𝑘=1,2,…,𝑞, and 𝐴_𝑓𝑖, 𝐵_𝑓𝑖, and 𝐶_𝑓𝑖 such that the following inequalities hold: Θ_𝑖𝑗+Θ_𝑗𝑖<0,𝑖≤𝑗≤𝑟,Ω𝐸𝐹𝐸^𝑇_𝑖∗𝐹−𝐶^𝑇_𝑓𝑖∗∗𝛾²𝐼>0, (34)(35)
then ℓ₂-ℓ_∞ filter parameters in (4) are given by 𝐴_𝑓𝑖=𝐹⁻¹𝐴_𝑓𝑖,𝐵_𝑓𝑖=𝐹⁻¹𝐵_𝑓𝑖,𝐶_𝑓𝑖=𝐶_𝑓𝑖, (36) where

Θ_𝑖𝑗=Ψ₁₁Ψ₁₂Ψ₁₃∗−Ψ₂₂0∗∗−Ψ₁₃,Ψ₁₁=Φ₀Φ₁⋯Φ_𝑞Φ_𝑞+1∗−1−𝑑_𝑘𝑄₁−𝑇_1𝑖−𝑇^𝑇_1𝑖00−𝑈^𝑇_1𝑖∗∗⋱⋮⋮∗∗∗−1−𝑑_𝑞𝑄_𝑞−𝑇_𝑞𝑖−𝑇^𝑇_𝑞𝑖−𝑈^𝑇_𝑞𝑖,Φ∗∗∗∗−𝐼₀=Ω𝐴_𝑖+𝐸𝐵_𝑓𝑗𝐶_𝑖𝐸𝐴_𝑓𝑗𝐹𝐸^𝑇𝐴_𝑖+𝐵_𝑓𝑗𝐶_𝑖𝐴_𝑓𝑗+Ω𝐴_𝑖+𝐸𝐵_𝑓𝑗𝐶_𝑖𝐸𝐴_𝑓𝑗𝐹𝐸^𝑇𝐴_𝑖+𝐵_𝑓𝑗𝐶_𝑖𝐴_𝑓𝑗^𝑇+^𝑞_𝑘=1𝐾^𝑇𝑄_𝑘𝐾+𝐾^𝑇𝑌_𝑘𝑖𝐾+𝐾^𝑇𝑌^𝑇_𝑘𝑖𝐾,Φ₁=Ω𝐴_𝑑1𝑖+𝐸𝐵_𝑓𝑗𝐶_𝑑1𝑖𝐹𝐸^𝑇𝐴_𝑑1𝑖+𝐵_𝑓𝑗𝐶_𝑑1𝑖−𝐾^𝑇𝑌_1𝑖+𝐾^𝑇𝑇^𝑇_1𝑖,Φ_𝑞=Ω𝐴_𝑑𝑞𝑖+𝐸𝐵_𝑓𝑗𝐶_𝑑𝑞𝑖𝐹𝐸^𝑇𝐴_𝑑𝑞𝑖+𝐵_𝑓𝑗𝐶_𝑑𝑞𝑖−𝐾^𝑇𝑌_𝑞𝑖+𝐾^𝑇𝑇^𝑇_𝑞𝑖,Φ_𝑞+1=Ω𝐵_𝑖+𝐸𝐵_𝑓𝑗𝐷_𝑖𝐹𝐸^𝑇𝐵_𝑖+𝐵_𝑓𝑗𝐷_𝑖+𝐾^𝑇𝑈^𝑇_𝑞𝑖,Ψ₁₂=ℎ₁𝐴_𝑖𝐾^𝑇𝑅₁⋯ℎ_𝑞𝐴_𝑖𝐾^𝑇𝑅_𝑞ℎ₁𝐴^𝑇_𝑑1𝑖𝑅₁⋯ℎ_𝑞𝐴^𝑇_𝑑1𝑖𝑅_𝑞⋮⋮⋮ℎ₁𝐴^𝑇_𝑑𝑞𝑖𝑅₁⋯ℎ_𝑞𝐴^𝑇_𝑑𝑞𝑖𝑅_𝑞ℎ₁𝐵^𝑇_𝑖𝑅₁⋯ℎ_𝑞𝐵^𝑇_𝑖𝑅_𝑞,Ψ₁₃=ℎ₁𝐾^𝑇𝑌_1𝑖⋯ℎ_𝑞𝐾^𝑇𝑌_𝑞𝑖ℎ₁𝑇_1𝑖⋯0⋮⋱⋮0⋯ℎ_𝑞𝑇_𝑞𝑖ℎ₁𝑈_1𝑖⋯ℎ_𝑞𝑈_𝑞𝑖,Ψ₂₂=Ψ₃₃=diag𝑅₁,…,𝑅_𝑞,𝐸=𝐼_𝑙×𝑙0_{𝑙×(𝑛−𝑙)}. (37)

Proof. By considering (6),we know that Σ(𝑡) in (26) and the Θ_𝑖𝑗 in (34) satisfy Σ(𝑡)=^𝑟𝑟_𝑖=1𝑗=1𝑢_𝑖(𝑠(𝑡))𝑢_𝑗(𝑠(𝑡))Θ_𝑖𝑗=^𝑟_𝑖=1𝑢²_𝑖(𝑠(𝑡))Θ_𝑖𝑖+^𝑟_𝑖<𝑗𝑢_𝑖(𝑠(𝑡))𝑢_𝑗(𝑠(𝑡))Θ_𝑖𝑗+Θ_𝑗𝑖. (38)
Based on Lemmas 2 and 3, the ℓ₂-ℓ_∞ filter matrices are given by (33). Under the transformation 𝑆^−𝑇𝑊𝑥_𝑓(𝑡), the filter matrices functions can be of the following forms: 𝐴_𝑓(𝑡)=𝑆^−𝑇𝑊𝑆⁻¹𝐴_𝑓(𝑡)𝑆^−𝑇𝑊𝑊⁻¹𝑆^𝑇,𝐵_𝑓(𝑡)=𝑆^−𝑇𝑊𝑆⁻¹𝐵_𝑓(𝑡),𝐶_𝑓(𝑡)=𝐶_𝑓(𝑡)𝑆^−𝑇𝑊𝑊⁻¹𝑆^𝑇. (39)
Hence, the filter in (4) can be got by (36). This completes the proof.

4. Simulation

In this section, we give a numerical example to illustrate the use of the present method. Consider the system of the form (1) with two plants (𝑟=2) and two delays (𝑞=2), where 𝐴₁=−2.10.11−2,𝐴_𝑑11=,𝐴−1.10.1−0.8−0.9_𝑑21=,𝐴−0.10.1−0.1−0.3₂=−1.90−0.2−1.1,𝐴_𝑑12=,𝐴−0.90−1.1−1.2_𝑑22=,𝐵−0.10−0.2−0.2₁=1−0.2,𝐶₁=,𝐶10_𝑑11=,𝐶−0.80.6_𝑑21=−0.10.2,𝐷₁𝐸=0.3,₁=,𝐵1−0.5₂=0.30.1,𝐶₂=,𝐶0.5−0.6_𝑑12=,𝐶−0.10.1_𝑑22=−0.10.1,𝐷₂𝐸=−0.6,₂=−0.20.3. (40)

Here, we only consider the full-order filter design. First, we set ℎ₁=0.5, ℎ₂=0.3. Figure 1 shows the ℓ₂-ℓ_∞ gain bound got from Theorem 4 in pointwise manner, where the derivatives of time-varying delay function bound 𝑑₁∈[01.9], 𝑑₂∈[01.9]. We can find that the maximum ℓ₂-ℓ_∞ gain is 𝛾=0.3714 at the point 𝑑₁=1.9, 𝑑₂=1.9. The minimum ℓ₂-ℓ_∞ gain is 𝛾=0.3358 at the point 𝑑₁=0, 𝑑₂=0.

Figure 1 

ℓ2-ℓ∞ gain bound from Theorem 4 with ℎ1=0.5 and  ℎ2=0.3.

Then, we set 𝑑₁=0.4, 𝑑₂=0.4. Figure 2 shows the ℓ₂-ℓ_∞ gain bound got from Theorem 4 in pointwise manner, where the time-varying delay function bound ℎ₁∈[01.1], ℎ₂∈[01.1]. We can find that the maximum ℓ₂-ℓ_∞ gain is 𝛾=0.5257 at the point ℎ₁=1.1, ℎ₂=1.1. The minimum ℓ₂-ℓ_∞ gain is 𝛾=0.0055 at the point ℎ₁=0, ℎ₂=0.

Figure 2 

ℓ2-ℓ∞ gain bound from Theorem 4 with 𝑑1=0.4  and 𝑑2=0.4.

Now, We choose both time-varying delays to be 0.4sin(𝑡)+0.7 which gives ℎ₁=1.1, ℎ₂=1.1, 𝑑₁=0.4, and 𝑑₂=0.4, and we get a set of feasible solutions to Theorem 4 with the ℓ₂-ℓ_∞ gain 𝛾=0.5257 and ,𝐴𝐹=3.8096−1.2607−1.26070.6116_𝑓1=,𝐴−12.90711.85165.1902−1.1454_𝑓2=,𝐵−10.02764.26533.0519−1.5986^𝑓1=,𝐵−5.02661.9199_𝑓2=,𝐶−4.19591.8120_𝑓1=0.8524−0.3426,𝐶_𝑓2=−0.26290.1428. (41)

Therefore, we can solve the corresponding filter from (36) as 𝐴_𝑓1=,𝐴−1.8241−0.42064.7259−2.7398_𝑓2=,𝐵−3.08580.8012−1.3706−0.9622_𝑓1=,𝐵−0.88291.3191_𝑓2=−0.38062.1781,𝐶_𝑓1=,𝐶0.8524−0.3426_𝑓2=−0.26290.1428. (42)

To illustrate the performance of the designed filter, we assume zero initial condition and the disturbance 𝜔(𝑡) as follows: 𝜔(𝑡)=1,0.5≤𝑡≤1−1,1.5≤𝑡≤20,otherwise. (43)

The simulation result of signal 𝑒(𝑡) is given in Figure 3.

Figure 3 

The signal 𝑒(𝑡).

The resulting output ℓ_∞-norm of the filtering-error system is about 0.15, while ‖𝜔(𝑡)‖₂=1. Simulation result for the ratio of the output ℓ_∞-norm to the disturbance ℓ₂-norm is 0.15, and 0.15<𝛾=0.5257 with ℎ₁=1.1, ℎ₂=1.1, 𝑑₁=0.4, and 𝑑₂=0.4.

5. Conclusion

The problem on ℓ₂-ℓ_∞ filter design has been addressed for a class of TCS fuzzy-model-based systems with multiple time-varying delays. Based on the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a sufficient condition for the existence of ℓ₂-ℓ_∞ filter, which stabilizes the T-S fuzzy-model-based filtering-error systems and guarantees a prescribed level on disturbance attenuation, has been obtained in terms of LMI form. The numerical example has shown the effectiveness of the proposed method. In addition, the basis-dependent Lyapunov-Krasovskii functional approach for filtering problems of T-S fuzzy delayed systems is also challenging, and could be our further work.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (61203005), in part by Harbin Engineering University Central University Foundation Research Special Fund (HEUCFR1024), and in part by Postdoctoral Science-Research Developmental Foundation of Heilongjiang Province (LBH-Q12130) and Heilongjiang Province Natural Science Foundation (F201221).