Further Result on Passivity for Discrete-Time Stochastic T-S Fuzzy Systems with Time-Varying Delays

Abstract

The passivity for discrete-time stochastic T-S fuzzy systems with time-varying delays is investigated. By constructing appropriate Lyapunov-Krasovskii functionals and employing stochastic analysis method and matrix inequality technique, a delay-dependent criterion to ensure the passivity for the considered T-S fuzzy systems is established in terms of linear matrix inequalities (LMIs) that can be easily checked by using the standard numerical software. An example is given to show the effectiveness of the obtained result.

1. Introduction

Fuzzy control offers an alternative control approach for certain nonlinear systems [1, 2]. Among various model-based fuzzy control approaches, the method based on Takagi-Sugeno (T-S) model is thought of as an effective way for the control of complex nonlinear systems, which is presented by a family of fuzzy IF-THEN rules that represent the local linear input-output relations of the system. Over the past decades, there have been significant research efforts on the stability for T-S fuzzy systems; for example, see [3–16] and references therein.

On the other hand, the passivity theory is another effective tool for the stability analysis of system. The reason is mainly twofold: (1) passivity is an expected system behavior, since the storage function induced by passivity is closely related to system energy and therefore serves as a natural candidate for Lyapunov functions and (2) stability and stabilization problems can be solved once the passivity property is assured. The passivity theory was first proposed in the circuit analysis [17] and has then been applied in many areas such as stability, signal processing, complexity, fuzzy control, chaos control, and synchronization [18–21].

Recently, some authors have studied the passivity of some systems and obtained sufficient conditions for checking the passivity of the systems that include linear systems with delays [22–24], delayed neural networks [25, 26], networked control systems [27], nonlinear discrete-time systems with direct input-output link [28], and T-S fuzzy systems [29–33]. In [29], the stability of fuzzy control loops is proven with the unique condition that the controlled plant can be made passive by zero shifting. For linear time-invariant plants, this approach leads to frequency response conditions similar to the previous results in the literature but which are more general and can include robust stability considerations. In [30], the passivity and feedback passification of T-S fuzzy systems with time delays were considered. Both delay-independent and delay-dependent results were presented, and the theoretical results were given in terms of LMIs. In [31], the contiguous-time T-S fuzzy systems with time-varying delays were investigated; several criteria to ensure the passivity and feedback passification were given. In [32], the passivity and feedback passification of T-S fuzzy systems with both discrete and distributed time-varying delays were investigated without assuming the differentiability of the time-varying delays. By employing appropriate Lyapunov-Krasovskii functionals, several delay-dependent criteria for the passivity of the considered T-S fuzzy systems were established in terms of LMIs. In [33], the stochastic T-S fuzzy system with both discrete and distributed time-varying delays was considered; several delay-dependent criteria to ensure the passivity and passification of the considered T-S fuzzy systems were established. In [34], discrete-time T-S fuzzy systems with delays were considered; some sufficient conditions to verify the passivity of the uncertain discrete-time fuzzy systems were obtained. In [35], the passivity of uncertain discrete-time T-S fuzzy systems with time delays was investigated; a sufficient condition on the existence of robust passive controller was established based on the Lyapunov stability theory. In [36], stochastic discrete-time T-S fuzzy systems with delay were considered; a sufficient condition in LMIs ensuring the passivity performance of the T-S fuzzy models was presented by utilizing the Lyapunov functional method, the stochastic analysis combined with the matrix inequality techniques. In this paper, we continue to study the passivity for stochastic discrete-time T-S fuzzy systems with time-varying delay. By employing appropriate Lyapunov-Krasovskii functionals and stochastic analysis technique, we obtain a new delay-dependent sufficient condition for checking the passivity of the addressed T-S fuzzy systems. As pointed out in [37–39], the delay-dependent criteria have less conservatism than the delay-independent ones.

Notations. The notations are quite standard. Throughout this paper, 𝐼 represents the unitary matrix with appropriate dimensions; ℕ stands for the set of nonnegative integers; ℝ^𝑛 and ℝ^𝑛×𝑚 denote, respectively, the 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices. The superscript “𝑇” denotes matrix transposition and the asterisk “*” denotes the elements below the main diagonal of a symmetric block matrix. The notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that 𝑋 and 𝑌 are symmetric matrices and that 𝑋−𝑌 is positive semidefinite (resp., positive definite). Also ‖⋅‖ is the Euclidean norm in ℝ^𝑛. And 𝜆_min(𝐴) (resp., 𝜆_max(𝐴)) denotes the least (resp., largest) eigenvalue of symmetric matrix 𝐴. For a positive constant 𝑎, [𝑎] denotes the integer part of 𝑎. For integers 𝑎, 𝑏 with 𝑎<𝑏, ℕ[𝑎,𝑏] denotes the discrete interval given by ℕ[𝑎,𝑏]={𝑎,𝑎+1,…,𝑏−1,𝑏}. Also 𝐶(ℕ[−𝜏,0],ℝ^𝑛) denotes the set of all functions 𝜙∶ℕ[−𝜏,0]→ℝ^𝑛. Let (Ω,ℱ,{ℱ}_𝑡≥0,𝒫) be a complete probability space with filtration {ℱ}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all 𝒫-null sets). 𝔼{⋅} stands for the mathematical expectation operator with respect to the given probability measure 𝒫. Denote by 𝐿²_ℱ0(ℕ[−𝜏,0],ℝ^𝑛) the family of all ℱ₀-measurable 𝐶(ℕ[−𝜏,0],ℝ^𝑛)valued random variables 𝜓={𝜓(𝑠)∶𝑠∈ℕ[−𝜏,0]} such that sup_{𝑠∈ℕ[−𝜏,0]}𝔼{|𝜓(𝑠)|}<∞. Δ𝑉(𝑘) denotes the difference of function 𝑉(𝑘) given by Δ𝑉(𝑘)=𝑉(𝑘+1)−𝑉(𝑘). Matrices, if not explicitly specified, are assumed to have compatible dimensions.

2. Model Description and Preliminaries

In this section, we consider a discrete-time T-S fuzzy system with stochastic disturbances and time-varying delay with the 𝑖th rule formulated in the following form.

Plant Rule 𝑖. If 𝑧₁(𝑡) is 𝑀_𝑖1 and … and 𝑧_𝑝(𝑡) is 𝑀_𝑖𝑝, then 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+𝑈_𝑖𝐽(𝑘)+𝜎_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))𝜔_𝑖(𝑘),𝑦(𝑘)=𝐶_𝑖𝑥(𝑘)+𝐷_𝑖𝑥(𝑘−𝜏(𝑘))+𝑉_𝑖𝐽(𝑘),𝑥(𝑘)=𝜑(𝑘),𝑘∈ℕ[−𝜏,0] (1) for 𝑘∈ℕ, 𝑖=1,2,…,𝑟, and 𝑟 is the number of IF-THEN rules, where 𝑥(𝑘)∈ℝ^𝑛 is the state vector, 𝐽(𝑘)∈ℝ^𝑚 is a deterministic exogenous input, 𝑦(𝑘)∈ℝ^𝑚 is the measurement output vector, 𝐴_𝑖, 𝐵_𝑖, 𝑈_𝑖, 𝐶_𝑖, 𝐷_𝑖, and 𝑉_𝑖 are system matrices with compatible dimensions, the positive integer 𝜏(𝑘) corresponds to the transmission delay and satisfies 𝜏≤𝜏(𝑘)≤𝜏 (𝜏≥0 and 𝜏≥0 are known integers), 𝜎∈ℝ^𝑛 is the diffusion coefficient vector, and 𝜔_𝑖(𝑘) is a scalar Brownian motion defined on (Ω,ℱ,𝒫) with 𝔼{𝜔_𝑖(𝑘)}=0, 𝔼{𝜔²_𝑖(𝑘)}=1 and 𝔼{𝜔_𝑖(𝑙)𝜔_𝑖(𝑘)}=0 (𝑙≠𝑘).

Let 𝜇_𝑖(𝑘) be the normalized membership function of the inferred fuzzy set 𝛾_𝑖(𝑘); that is, 𝜇_𝑖(𝑘)=𝛾_𝑖(𝑘)∑^𝑟_𝑖=1𝛾_𝑖(𝑘), (2) where 𝛾_𝑖(𝑘)=∏^𝑝_𝑗=1𝑀_𝑖𝑗(𝑧_𝑗(𝑘)) with 𝑀_𝑖𝑗(𝑧_𝑗(𝑘)) being the grade of membership function of 𝑧_𝑗(𝑘) in 𝑀_𝑖𝑗(𝑘). It is assumed that 𝛾_𝑖(𝑘)≥0 (𝑖=1,2,…,𝑟) and ∑^𝑟_𝑖=1𝛾_𝑖(𝑘)>0 for all 𝑘. Thus 𝜇_𝑖(𝑘)≥0 and ∑^𝑟_𝑖=1𝜇_𝑖(𝑘)=1 for all 𝑘. And the T-S fuzzy model (1) can be represented as 𝑥(𝑘+1)=^𝑟_𝑖=1𝜇_𝑖(𝑡)𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+𝑈_𝑖𝐽(𝑘)+𝜎_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))𝜔_𝑖(𝑘,𝑦(𝑘)=_𝑟𝑖=1)𝜇_𝑖(𝑡)𝐶_𝑖𝑥(𝑘)+𝐷_𝑖𝑥(𝑘−𝜏(𝑘))+𝑉_𝑖𝐽(𝑘,𝑥(𝑘)(𝑘)[−)=𝜑,𝑘∈ℕ𝜏,0]. (3)

In the literature, different definitions of passivity have been used. Taking into account the stochastic nature of the T-S fuzzy systems considered in this paper, we adopt the definition in [36].

Definition 1. System (1) is called passive in the sense of expectation if there exists a scalar 𝛾>0 such that 2^𝑇_𝑘=0𝔼𝑦^𝑇(𝑘)𝐽(𝑘)≥−𝛾^𝑇_𝑘=0𝔼𝐽^𝑇(𝑘)𝐽(𝑘) (4) for all integers 𝑇 and the solution of (1) with 𝜙(⋅)≡0.

Throughout this paper, we make the following assumption.

Assumption 2 (see [36]). There exist matrices 𝑊_𝑖 and 𝑆_𝑖 such that 𝜎^𝑇_𝑖𝑥,𝑦𝜎_𝑖𝑥,𝑦≤𝑊_𝑖𝑥²+𝑆_𝑖𝑦²,∀𝑥,𝑦∈ℝ^𝑛,𝑖=1,2,…,𝑟. (5)

In obtaining the main result of this paper, the following lemma will be useful for the proof.

Lemma 3 (see [40]). Suppose that matrices 𝑀_𝑖∈ℝ^𝑛×𝑚 (𝑖=1,2,…,𝑠) and a positive-semidefinite matrix 𝑃∈ℝ^𝑛×𝑛 are given. If ∑^𝑠_𝑖=1ℎ_𝑖=1 and 0≤ℎ≤1, then ^𝑠_𝑖=1ℎ_𝑖𝑀_𝑖^𝑇𝑃^𝑠_𝑖=1ℎ_𝑖𝑀_𝑖≤^𝑠_𝑖=1ℎ_𝑖𝑀^𝑇_𝑖𝑃𝑀_𝑖. (6)

3. Main Results

In this section, we will establish our main criterion based on the LMI approach. For presentation convenience, in the following, we denote that 𝛿=[(𝜏+𝜏)/2].

Theorem 4. Under Assumption 2, model (1) is passive in the sense of Definition 1 if there exist two scalars 𝛾>0 and 𝜆>0 and eight symmetric positive definite matrices 𝑃, 𝑄₁, 𝑄₂, 𝑄₃, 𝑄₄, 𝑅₁, 𝑅₂, and 𝑅₃ such that the following LMIs hold for 𝑖=1,2,…,𝑟: Π𝑃+𝑅<𝜆𝐼,_(𝑖)+Ω₁Π<0,_(𝑖)+Ω₂<0 (7) hold or Π𝑃+𝑅<𝜆𝐼,_(𝑖)+Ω₃Π<0,_(𝑖)+Ω₄<0 (8) hold, where Π^(𝑖)=Π^(𝑖)₁₁Π^(𝑖)₁₂000Π^(𝑖)₁₆∗Π^(𝑖)₂₂000Π^(𝑖)₂₆∗∗−𝑄₁000∗∗∗−𝑄₂00∗∗∗∗−𝑄₃0∗∗∗∗∗Π^(𝑖)₆₆,Ω₁=−𝑅₁00𝑅₁00∗−3𝑅₂2𝑅₂𝑅₂00∗∗−2𝑅₂−𝑅₃0𝑅₃0∗∗∗−𝑅₁−𝑅₂00∗∗∗∗−𝑅₃0,Ω∗∗∗∗∗0₂=−𝑅₁00𝑅₁00∗−3𝑅₂𝑅₂2𝑅₂00∗∗−𝑅₂−𝑅₃0𝑅₃0∗∗∗−𝑅₁−2𝑅₂00∗∗∗∗−𝑅₃0,Ω∗∗∗∗∗0₃=−𝑅₁00𝑅₁00∗−3𝑅₃𝑅₃02𝑅₃0∗∗−𝑅₂−𝑅₃𝑅₂00∗∗∗−𝑅₁−𝑅₂00∗∗∗∗−2𝑅₃0,Ω∗∗∗∗∗0₄=−𝑅₁00𝑅₁00∗−3𝑅₃2𝑅₃0𝑅₃0∗∗−𝑅₂−2𝑅₃𝑅₂00∗∗∗−𝑅₁−𝑅₂00∗∗∗∗−𝑅₃0,∗∗∗∗∗0 (9) in which 𝑅=𝜏²𝑅₁+(𝛿−𝜏)²𝑅₂+(𝜏−𝛿)²𝑅₃, Π^(𝑖)₁₁=𝐴^𝑇_𝑖(𝑃+𝑅)𝐴_𝑖−𝑃+𝑄₁+𝑄₂+𝑄₃+(1+𝜏−𝜏)𝑄₄−𝑅𝐴_𝑖−𝐴^𝑇_𝑖𝑅+𝑅+𝜆𝑊^𝑇_𝑖𝑊_𝑖, Π^(𝑖)₁₂=𝐴^𝑇_𝑖(𝑃+𝑅)𝐵_𝑖−𝑅𝐵_𝑖, Π^(𝑖)₁₆=𝐴^𝑇_𝑖(𝑃+𝑅)𝑈_𝑖−𝑅𝑈_𝑖−𝐶^𝑇_𝑖, Π^(𝑖)₂₂=𝐵^𝑇_𝑖(𝑃+𝑅)𝐵_𝑖−𝑄₄+𝜆𝑆^𝑇_𝑖𝑆_𝑖, Π^(𝑖)₂₆=𝐵^𝑇_𝑖(𝑃+𝑅)𝑈_𝑖−𝐷^𝑇_𝑖, and Π^(𝑖)₆₆=𝑈^𝑇_𝑖(𝑃+𝑅)𝑈_𝑖−𝑉_𝑖−𝑉^𝑇_𝑖−𝛾𝐼.

Proof. Defining 𝜂(𝑘)=𝑥(𝑘+1)−𝑥(𝑘), we consider the following Lyapunov-Krasovskii functional candidate for model (1) as 𝑉(𝑘)=⁴_𝑖=1𝑉_𝑖(𝑘), (10) where 𝑉₁(𝑘)=𝑥^𝑇(𝑘)(𝑘),𝑉𝑃𝑥₂(𝑘)=^𝑘−1_{𝑖=𝑘−𝛿}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)+^𝑘−1_{𝑖=𝑘−𝜏}𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)+^𝑘−1_{𝑖=𝑘−𝜏}𝑥^𝑇(𝑖)𝑄₃𝑥(𝑖),𝑉₃(𝑘)=^𝑘−1_{(𝑘)𝑖=𝑘−𝜏}𝑥^𝑇(𝑖)𝑄₄𝑥(𝑖)+^𝑘−𝜏_{𝑙=𝑘−𝜏+1}^𝑘−1_𝑖=𝑙𝑥^𝑇(𝑖)𝑄₄𝑥(𝑖),𝑉₄(𝑘)=𝜏^𝑘−1_{𝑙=𝑘−𝜏}^𝑘−1_𝑖=𝑙𝜂^𝑇(𝑖)𝑅₁𝜂(𝑖)+(𝛿−𝜏)^{𝑘−𝜏−1}_{𝑙=𝑘−𝛿}^𝑘−1_𝑖=𝑙𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)+(𝜏−𝛿)^{𝑘−𝛿−1}_{𝑙=𝑘−𝜏}^𝑘−1_𝑖=𝑙𝜂^𝑇(𝑖)𝑅₃𝜂(𝑖). (11)(12)(13)(14)
Calculating the difference of 𝑉_𝑖(𝑘)(𝑖=1,2,3,4) along the trajectories of model (1) and taking the mathematical expectation, we obtain that 𝔼Δ𝑉₁(𝑘)=𝔼^𝑟_𝑖=1𝜇_𝑖(𝑡)𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+𝑈_𝑖𝐽(𝑘)+𝜎_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))𝜔_𝑖(𝑘)^𝑇𝑃×^𝑟_𝑖=1𝜇_𝑖(𝑡)𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+𝑈_𝑖𝐽(𝑘)+𝜎_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))𝜔_𝑖(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)≤𝔼^𝑟_𝑖=1𝜇_𝑖(𝑡)𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+𝑈_𝑖𝐽(𝑘)+𝜎_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))𝜔_𝑖(𝑘)^𝑇𝑃×𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+𝑈_𝑖𝐽(𝑘)+𝜎_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))𝜔_𝑖(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝔼^𝑟_𝑖=1𝜇_𝑖(𝑡)𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃𝐴_𝑖−𝑃𝑥(𝑘)+2𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+2𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃𝑈_𝑖𝐽(𝑘)+𝑥^𝑇(𝑘−𝜏(𝑘))𝐵^𝑇_𝑖𝑃𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+2𝑥^𝑇(𝑘−𝜏(𝑘))𝐵^𝑇_𝑖𝑃𝑈_𝑖𝐽(𝑘)+𝐽^𝑇(𝑘)𝑈^𝑇_𝑖𝑃𝑈_𝑖𝐽(𝑘)+𝜎^𝑇_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))×𝑃𝜎_𝑖(𝑥(𝑘)((𝑘,𝔼,𝑥𝑘−𝜏)))Δ𝑉₂(𝑘𝑥)=𝔼^𝑇(𝑘)𝑄₁+𝑄₂+𝑄₃𝑥(𝑘)−𝑥^𝑇(𝑘−𝛿)𝑄₁𝑥(𝑘−𝛿)−𝑥^𝑇(𝑘−𝜏)𝑄₂𝑥(𝑘−𝜏)−𝑥^𝑇(𝑘−𝜏)𝑄₃𝑥(𝑘−𝜏,𝔼)Δ𝑉₃(𝑘)=𝔼^𝑘−𝜏_{()𝑖=𝑘+1−𝜏𝑘+1}𝑥^𝑇(𝑖)𝑄₄𝑥(𝑖)+^𝑘−1_{𝑖=𝑘−𝜏+1}𝑥^𝑇(𝑖)𝑄₄𝑥(𝑖)+𝑥^𝑇(𝑘)𝑄₄𝑥(𝑘)−^𝑘−1_{(𝑘)𝑖=𝑘−𝜏+1}𝑥^𝑇(𝑖)𝑄₄𝑥(𝑖)−𝑥^𝑇(𝑘−𝜏(𝑘))𝑄₄𝑥((𝑘+(𝑘−𝜏))𝜏−𝜏)𝑥^𝑇(𝑘)𝑄₄𝑥(𝑘)−^𝑘−𝜏_{𝑙=𝑘−𝜏+1}𝑥^𝑇(𝑙)𝑄₄𝑥(𝑙)≤𝔼(1+𝜏−𝜏)𝑥^𝑇(𝑘)𝑄₄𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏(𝑘))𝑄₄𝑥((𝑘,𝔼𝑘−𝜏))Δ𝑉₄(𝑘𝜂)=𝔼^𝑇(𝑘)𝑅𝜂(𝑘)−𝜏^𝑘−1_{𝑖=𝑘−𝜏}𝜂^𝑇(𝑖)𝑅₁𝜂(𝑖)−(𝛿−𝜏)^{𝑘−𝜏−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)−(𝜏−𝛿)^{𝑘−𝛿−1}_{𝑖=𝑘−𝜏}𝜂^𝑇(𝑖)𝑅₃𝜂(𝑖). (15)(16)(17)(18) In deriving inequalities (15) and (17), Lemma 3 and the condition 𝜏≤𝜏(𝑘)≤𝜏 have been used, respectively.
From the definition of 𝜂(𝑘) and (3) and applying Lemma 3, we have 𝔼𝜂^𝑇(𝑘)𝑅𝜂(𝑘)≤𝔼^𝑟_𝑖=1𝜇_𝑖(𝑡)𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑅𝐴_𝑖−𝑅𝐴_𝑖−𝐴^𝑇_𝑖𝑅+𝑅𝑥(𝑘)+2𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑅𝐵_𝑖−𝑅𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+2𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑅𝑈_𝑖−𝑅𝑈_𝑖𝐽(𝑘)+𝑥^𝑇(𝑘−𝜏(𝑘))𝐵^𝑇_𝑖𝑅𝐵_𝑖𝑥(𝑘−𝜏(𝑘))+2𝑥^𝑇(𝑘−𝜏(𝑘))𝐵^𝑇_𝑖𝑅𝑈_𝑖𝐽(𝑘)+𝐽^𝑇(𝑘)𝑈^𝑇_𝑖𝑅𝑈_𝑖𝐽(𝑘)+𝜎^𝑇_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))×𝑅𝜎_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘))). (19) It is easy to get −𝜏^𝑘−1_{𝑖=𝑘−𝜏}𝜂^𝑇(𝑖)𝑅₁𝜂(𝑖)≤−^𝑘−1_{𝑖=𝑘−𝜏}𝜂^𝑇(𝑖)𝑅₁^𝑘−1_{𝑖=𝑘−𝜏}𝜂(𝑖)=𝑥(𝑘)𝑥(𝑘−𝜏)^𝑇−𝑅₁𝑅₁∗−𝑅₁𝑥(𝑘)𝑥().𝑘−𝜏 (20)
When 𝜏≤𝜏(𝑘)≤𝛿, let 𝑎(𝑘)=(𝜏(𝑘)−𝜏)/(𝛿−𝜏); then 0≤𝑎(𝑘)≤1. It is easy to get that −(𝛿−𝜏)^{𝑘−𝜏−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)=−(𝛿−𝜏)^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)−(𝛿−𝜏)^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)=−(𝛿−𝜏(𝑘))^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)−(𝜏(𝑘)−𝜏)^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)−(𝛿−𝜏(𝑘))^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)−(𝜏(𝑘)−𝜏)^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)≤−(𝛿−𝜏(𝑘))^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)−𝑎(𝑘)(𝛿−𝜏(𝑘))^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)−(1−𝑎(𝑘))(𝜏(𝑘)−𝜏)^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)−(𝜏(𝑘)−𝜏)^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)≤−^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂(𝑖)−𝑎(𝑘)^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂^{𝑘−𝜏(𝑘)−1}_{𝑖=𝑘−𝛿}𝜂(𝑖)−(1−𝑎(𝑘))^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂^𝑇(𝑖)𝑅₂^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂(𝑖)−^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂^𝑇(𝑖)𝑅₂^{𝑘−𝜏−1}_{𝑖=𝑘−𝜏(𝑘)}𝜂(𝑖)=𝑥(𝑘−𝜏(𝑘))𝑥(𝑘−𝛿)𝑥(𝑘−𝜏)^𝑇−2𝑅₂𝑅₂𝑅₂∗−𝑅₂0∗∗−𝑅₂𝑥((𝑘𝑥()𝑥()(𝑘)𝑘−𝜏))𝑘−𝛿𝑘−𝜏+𝑎𝑥(𝑘−𝜏(𝑘))𝑥(𝑘−𝛿)^𝑇−𝑅₂𝑅₂∗−𝑅₂𝑥((𝑘𝑥()+((𝑘𝑘−𝜏))𝑘−𝛿1−𝑎))𝑥(𝑘−𝜏(𝑘))𝑥(𝑘−𝜏)^𝑇×−𝑅₂𝑅₂∗−𝑅₂𝑥((𝑘𝑥(),−(𝑘−𝜏))𝑘−𝜏𝜏−𝛿)^{𝑘−𝛿−1}_{𝑖=𝑘−𝜏}𝜂^𝑇(𝑖)𝑅₃𝜂(𝑖)≤𝑥(𝑘−𝛿)𝑥(𝑘−𝜏)^𝑇−𝑅₃𝑅₃∗−𝑅₃𝑥()𝑥(𝑘−𝛿𝑘−𝜏). (21)(22)
From the first inequality of condition (7) and Assumption 2, we get 𝜎^𝑇_𝑖(𝑥(𝑘),𝑥(𝑘−𝜏(𝑘)))(𝑃+𝑅)𝜎_𝑖(𝑥(𝑘)((𝑘𝑊,𝑥𝑘−𝜏)))≤𝜆_𝑖𝑥(𝑘)²+𝑆_𝑖𝑥(𝑘−𝜏(𝑘))². (23)
Denote that 𝛼(𝑘)=(𝑥^𝑇(𝑘),𝑥^𝑇(𝑘−𝜏(𝑘)),𝑥^𝑇(𝑘−𝛿),𝑥^𝑇(𝑘−𝜏),𝑥^𝑇(𝑘−𝜏),𝐽^𝑇(𝑘))^𝑇. It follows from (15) to (23) that 𝔼Δ𝑉(𝑘)−2𝑦^𝑇(𝑘)𝐽(𝑘)−𝛾𝐽^𝑇(𝑘)𝐽(𝑘)≤𝔼^𝑟_𝑖=1𝜇_𝑖(𝑡)𝛼^𝑇(𝑘)𝑎(𝑘)Π^(𝑖)+Ω₁+((𝑘×Π1−𝑎))_(𝑖)+Ω₂𝛼(𝑘). (24) From the second inequality and the third inequality of condition (7), we get 𝔼Δ𝑉(𝑘)−2𝑦^𝑇(𝑘)𝐽(𝑘)−𝛾𝐽^𝑇(𝑘)𝐽(𝑘)≤0 (25) and therefore we have 2^𝑇_𝑘=0𝔼𝑦^𝑇(𝑘)𝐽(𝑘≥_𝑇𝑘=0)𝔼{Δ𝑉(𝑘)}−𝛾^𝑇_𝑘=0𝔼𝐽^𝑇(𝑘)𝐽(𝑘=_𝑇𝑘=0)𝔼{𝑉(𝑇)−𝑉(0)}−𝛾^𝑇_𝑘=0𝔼𝐽^𝑇(𝑘)𝐽(𝑘)≥−^𝑇_𝑘=0𝔼{𝑉(0)}−𝛾^𝑇_𝑘=0𝔼𝐽^𝑇(𝑘)𝐽(𝑘)=−𝛾^𝑇_𝑘=0𝔼𝐽^𝑇(𝑘)𝐽(𝑘) (26) for all integers 𝑇≥0. From Definition 1, we know that (26) implies that the stochastic T-S fuzzy system (1) is globally passive in the sense of expectation.
When 𝛿≤𝜏(𝑡)≤𝜏, let 𝑏(𝑘)=(𝜏̄−𝜏(𝑘))/(𝜏̄−𝛿); then 0≤𝑏(𝑘)≤1. In the similitude of the proof of inequality (21), we have −(𝜏−𝛿)^{𝑘−𝛿−1}_{𝑖=𝑘−𝜏}𝜂^𝑇(𝑖)𝑅₃𝜂(𝑖)≤𝑥(𝑘−𝜏(𝑘))𝑥(𝑘−𝜏)𝑥(𝑘−𝛿)^𝑇−2𝑅₃𝑅₃𝑅₃∗−𝑅₃0∗∗−𝑅₃𝑥((𝑘𝑥(𝑘−𝜏))𝑘−𝜏)𝑥()(𝑘)𝑘−𝛿+𝑏𝑥(𝑘−𝜏(𝑘))𝑥(𝑘−𝜏)^𝑇−𝑅₃𝑅₃∗−𝑅₃𝑥((𝑘𝑥(𝑘−𝜏))𝑘−𝜏)+(1−𝑏(𝑘))𝑥(𝑘−𝜏(𝑘))𝑥(𝑘−𝛿)^𝑇×−𝑅₃𝑅₃∗−𝑅₃𝑥((𝑘𝑥(),−()𝑘−𝜏))𝑘−𝛿𝛿−𝜏^{𝑘−𝜏−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖)≤𝑥(𝑘−𝜏)𝑥(𝑘−𝛿)^𝑇−𝑅₂𝑅₂∗−𝑅₂𝑥()𝑥().𝑘−𝜏𝑘−𝛿 (27)(28) It follows from (15)–(20), (23), (27), and (28) that 𝔼Δ𝑉(𝑘)−2𝑦^𝑇(𝑘)𝐽(𝑘)−𝛾𝐽^𝑇(𝑘)𝐽(𝑘)≤𝔼^𝑟_𝑖=1𝜇_𝑖(𝑡)𝛼^𝑇(𝑘)𝑏(𝑘)Π^(𝑖)+Ω₃+((𝑘×Π1−𝑏))_(𝑖)+Ω₄𝛼(𝑘). (29) From the second inequality and the third inequality of condition (8), we get 𝔼Δ𝑉(𝑘)−2𝑦^𝑇(𝑘)𝐽(𝑘)−𝛾𝐽^𝑇(𝑘)𝐽(𝑘)≤0. (30) By using same method in (25) and (26), we know that the stochastic T-S fuzzy system (1) is globally passive in the sense of expectation. The proof is completed.

Remark 5. In [36], the passivity of discrete-time stochastic T-S fuzzy systems was investigated. Compared with the approach of [36], the method in this paper has two advantages: (i) different from the method of [36], the variation interval of the time-varying delays 𝜏(𝑘) is divided into two subintervals, and the variation of the Lyapunov-Krasovskii functional is checked for the cases when the time-varying delay varies in the subintervals; (ii) the terms −(𝛿−𝜏)∑^{𝑘−𝜏−1}_{𝑖=𝑘−𝛿}𝜂^𝑇(𝑖)𝑅₂𝜂(𝑖) and −(𝜏−𝛿)∑^{𝑘−𝛿−1}_{𝑖=𝑘−𝜏}𝜂^𝑇(𝑖)𝑅₃𝜂(𝑖) are both divided into four parts and two scalars 𝑎(𝑘) and 𝑏(𝑘) are introduced, respectively. From above-mentioned techniques, we know that more information of the time-varying delays 𝜏(𝑘) was employed in deriving the result of this paper. Thus, the obtained result in this paper is expected to have less conservatism than the result in [36].

4. Example

Example 1. Consider a T-S fuzzy system (1) with 𝑟=2, where 𝑟 is the number of IF-THEN rules. The time-varying delay 𝜏(𝑘) is assumed to have 𝜏=2 and 𝜏=5. Other parameters are given as follows: 𝐴₁=0.040.010.01−0.02,𝐵₁=,𝑈−0.020.010.010.03₁=−0.010.020.010.01,𝐶₁=,𝐷00.02−0.010.01₁=0.040.01−0.02−0.01,𝑉₁=,𝐴−0.01−0.0200.01₂=0.010.0300.01,𝐵₂=,𝑈−0.010.02−0.020.03₂=−0.010.020.010.01,𝐶₂=,𝐷0.0100.020.01₂=−0.01−0.010.010.02,𝑉₂=,𝑊0.01−0.010.010.02₁=−0.010.010.020,𝑆₁=,𝑊0.0200.01−0.01₂=0.0100.020.05,𝑆₂=.0.01−0.010.010.01 (31)
By using the Matlab LMI Control Toolbox, we can find a solution to the LMIs in (7) as follows: 𝑃=10⁴,𝑄1.0868−0.0007−0.00071.0878₁=10³,𝑄1.3480−0.0014−0.00141.3499₂=10³,𝑄1.3488−0.0011−0.00111.3501₃=10³,𝑄1.3489−0.0009−0.00091.3502₄=10³,𝑅1.3603−0.0037−0.00371.3639₁=,𝑅1.72000.82320.82320.5407₂=,𝑅0.21180.15250.15250.1121₃=,1.85630.87030.87030.5674𝛾=1.3546×10³,𝜆=1.2234×10⁴. (32) According to Theorem 4, the considered model (1) is passive in the sense of Definition 1.

5. Conclusions

In this paper, the passivity for discrete-time stochastic T-S fuzzy systems with time-varying delays has been investigated. By constructing appropriate Lyapunov-Krasovskii functionals and employing stochastic analysis method and matrix inequality technique, a delay-dependent criterion to ensure the passivity for the considered T-S fuzzy systems has been established in terms of linear matrix inequalities (LMIs) that can be easily checked by using the standard numerical software. An example is also given to show the effectiveness of the obtained result.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the editor and the reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China under Grants 61273021 and 11172247 and in part by the Natural Science Foundation Project of CQ cstc2013jjB40008.