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

Authors: Ting Lei; Qiankun Song; Zhenjiang Zhao
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## 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.


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## Body

## 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; ℝn and ℝn×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 ℝn. And λmin⁡(A) (resp., λmax⁡(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],ℝn) denotes the set of all functions ϕ:ℕ[-τ,0]→ℝn. 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 Lℱ02(ℕ[-τ,0],ℝn) the family of all ℱ0-measurable C(ℕ[-τ,0],ℝn)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 theith rule formulated in the following form.Plant Rule
i
. If z1(t) is Mi1 and … and zp(t) is Mip, then
(1)x(k+1)=Aix(k)+Bix(k-τ(k))+UiJ(k)+σi(x(k),x(k-τ(k)))ωi(k),y(k)=Cix(k)+Dix(k-τ(k))+ViJ(k),x(k)=φ(k),k∈ℕ[-τ¯,0]
for k∈ℕ, i=1,2,…,r, and r is the number of IF-THEN rules, where x(k)∈ℝn is the state vector, J(k)∈ℝm is a deterministic exogenous input, y(k)∈ℝm is the measurement output vector, Ai, Bi, Ui, Ci, Di, and Vi 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∈ℝn is the diffusion coefficient vector, and ωi(k) is a scalar Brownian motion defined on (Ω,ℱ,𝒫) with 𝔼{ωi(k)}=0, 𝔼{ωi2(k)}=1 and 𝔼{ωi(l)ωi(k)}=0 (l≠k).Letμi(k) be the normalized membership function of the inferred fuzzy set γi(k); that is,
(2)μi(k)=γi(k)∑i=1rγi(k),
where γi(k)=∏j=1pMij(zj(k)) with Mij(zj(k)) being the grade of membership function of zj(k) in Mij(k). It is assumed that γi(k)≥0 (i=1,2,…,r) and ∑i=1rγi(k)>0 for all k. Thus μi(k)≥0 and ∑i=1rμi(k)=1 for all k. And the T-S fuzzy model (1) can be represented as
(3)x(k+1)=∑i=1r‍μi(t)[(x(k),x(k-τ(k)))Aix(k)+Bix(k-τ(k))+UiJ(k)+σi(x(k),x(k-τ(k)))ωi(k)],y(k)=∑i=1r‍μi(t)[Cix(k)+Dix(k-τ(k))+ViJ(k)],x(k)=φ(k),k∈ℕ[-τ¯,0].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
(4)2∑k=0T‍𝔼{yT(k)J(k)}≥-γ∑k=0T‍𝔼{JT(k)J(k)}
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 matricesWi and Si such that
(5)σiT(x,y)σi(x,y)≤∥Wix∥2+∥Siy∥2,σiT(x,y)∀x,y∈ℝn,i=1,2,…,r.In obtaining the main result of this paper, the following lemma will be useful for the proof.Lemma 3 (see [40]).
Suppose that matricesMi∈ℝn×m (i=1,2,…,s) and a positive-semidefinite matrix P∈ℝn×n are given. If ∑i=1shi=1 and 0≤h≤1, then
(6)(∑i=1s‍hiMi)TP(∑i=1s‍hiMi)≤∑i=1s‍hiMiTPMi.


## 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 thatδ=[(τ+τ-)/2].Theorem 4.
Under Assumption2, 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:
(7)P+R<λI,Π(i)+Ω1<0,Π(i)+Ω2<0
hold or
(8)P+R<λI,Π(i)+Ω3<0,Π(i)+Ω4<0
hold, where
(9)Π(i)=[Π11(i)Π12(i)000Π16(i)*Π22(i)000Π26(i)**-Q1000***-Q200****-Q30*****Π66(i)],Ω1=[-R100R100*-3R22R2R200**-2R2-R30R30***-R1-R200****-R30*****0],Ω2=[-R100R100*-3R2R22R200**-R2-R30R30***-R1-2R200****-R30*****0],Ω3=[-R100R100*-3R3R302R30**-R2-R3R200***-R1-R200****-2R30*****0],Ω4=[-R100R100*-3R32R30R30**-R2-2R3R200***-R1-R200****-R30*****0],
in which R=τ2R1+(δ-τ)2R2+(τ¯-δ)2R3, Π11(i)=AiT(P+R)Ai-P+Q1+Q2+Q3+(1+τ¯-τ)Q4-RAi-AiTR+R+λWiTWi, Π12(i)=AiT(P+R)Bi-RBi, Π16(i)=AiT(P+R)Ui-RUi-CiT, Π22(i)=BiT(P+R)Bi-Q4+λSiTSi, Π26(i)=BiT(P+R)Ui-DiT, and Π66(i)=UiT(P+R)Ui-Vi-ViT-γI.Proof.
Definingη(k)=x(k+1)-x(k), we consider the following Lyapunov-Krasovskii functional candidate for model (1) as
(10)V(k)=∑i=14‍Vi(k),
where
(11)V1(k)=xT(k)Px(k),(12)V2(k)=∑i=k-δk-1‍xT(i)Q1x(i)+∑i=k-τk-1‍xT(i)Q2x(i)+∑i=k-τ¯k-1‍xT(i)Q3x(i),(13)V3(k)=∑i=k-τ(k)k-1‍xT(i)Q4x(i)+∑l=k-τ¯+1k-τ‍∑i=lk-1‍xT(i)Q4x(i),(14)V4(k)=τ∑l=k-τk-1‍∑i=lk-1‍ηT(i)R1η(i)+(δ-τ)∑l=k-δk-τ-1‍∑i=lk-1‍ηT(i)R2η(i)+(τ¯-δ)∑l=k-τ¯k-δ-1‍∑i=lk-1‍ηT(i)R3η(i).
Calculating the difference ofVi(k)(i=1,2,3,4) along the trajectories of model (1) and taking the mathematical expectation, we obtain that
(15)𝔼{ΔV1(k)}=𝔼{(∑i=1r‍μi(t)[Aix(k)+Bix(k-τ(k))+UiJ(k)+σi(x(k),x(k-τ(k)))ωi(k)]∑i=1r‍μi(t))TP×(∑i=1r‍μi(t)[Aix(k)+Bix(k-τ(k))+UiJ(k)+σi(x(k),x(k-τ(k)))ωi(k)]∑i=1r‍μi)-xT(k)Px(k)(∑i=1r‍μi(t)[Aix(k)+Bix(k-τ(k))+UiJ(k)}≤𝔼{∑i=1r‍μi(t)[Aix(k)+Bix(k-τ(k))+UiJ(k)+σi(x(k),x(k-τ(k)))ωi(k)]TP×[Aix(k)+Bix(k-τ(k))+UiJ(k)+σi(x(k),x(k-τ(k)))ωi(k)]-xT(k)Px(k)∑i=1r‍}=𝔼{∑i=1r‍μi(t)[xT(k)(AiTPAi-P)x(k)+2xT(k)AiTPBix(k-τ(k))+2xT(k)AiTPUiJ(k)+xT(k-τ(k))BiTPBix(k-τ(k))+2xT(k-τ(k))BiTPUiJ(k)+JT(k)UiTPUiJ(k)+σiT(x(k),x(k-τ(k)))×Pσi(x(k),x(k-τ(k)))]∑i=1r‍},(16)𝔼{ΔV2(k)}=𝔼{xT(k)(Q1+Q2+Q3)x(k)-xT(k-δ)Q1x(k-δ)-xT(k-τ)Q2x(k-τ)-xT(k-τ-)Q3x(k-τ-)},(17)𝔼{ΔV3(k)}=𝔼{∑i=k+1-τ(k+1)k-τ‍xT(i)Q4x(i)+∑i=k-τ+1k-1‍xT(i)Q4x(i)+xT(k)Q4x(k)-∑i=k-τ(k)+1k-1‍xT(i)Q4x(i)-xT(k-τ(k))Q4x(k-τ(k))+(τ¯-τ)xT(k)Q4x(k)-∑l=k-τ¯+1k-τ‍xT(l)Q4x(l)}≤𝔼{(1+τ¯-τ)xT(k)Q4x(k)-xT(k-τ(k))Q4x(k-τ(k))},(18)𝔼{ΔV4(k)}=𝔼{ηT(k)Rη(k)-τ∑i=k-τk-1‍ηT(i)R1η(i)-(δ-τ)∑i=k-δk-τ-1‍ηT(i)R2η(i)-(τ¯-δ)∑i=k-τ¯k-δ-1‍ηT(i)R3η(i)}.
In deriving inequalities (15) and (17), Lemma 3 and the condition τ≤τ(k)≤τ¯ have been used, respectively.
From the definition ofη(k) and (3) and applying Lemma 3, we have
(19)𝔼{ηT(k)Rη(k)}≤𝔼{∑i=1r‍μi(t)[xT(k)(AiTRAi-RAi-AiTR+R)x(k)+2xT(k)(AiTRBi-RBi)x(k-τ(k))+2xT(k)(AiTRUi-RUi)J(k)+xT(k-τ(k))BiTRBix(k-τ(k))+2xT(k-τ(k))BiTRUiJ(k)+JT(k)UiTRUiJ(k)+σiT(x(k),x(k-τ(k)))×Rσi(x(k),x(k-τ(k)))]∑i=1r‍μi}.
It is easy to get
(20)-τ∑i=k-τk-1‍ηT(i)R1η(i)≤-∑i=k-τk-1‍ηT(i)R1∑i=k-τk-1‍η(i)=[x(k)x(k-τ)]T[-R1R1*-R1][x(k)x(k-τ)].
Whenτ≤τ(k)≤δ, let a(k)=(τ(k)-τ)/(δ-τ); then 0≤a(k)≤1. It is easy to get that
(21)-(δ-τ)∑i=k-δk-τ-1‍ηT(i)R2η(i)=-(δ-τ)∑i=k-δk-τ(k)-1‍ηT(i)R2η(i)-(δ-τ)∑i=k-τ(k)k-τ-1‍ηT(i)R2η(i)=-(δ-τ(k))∑i=k-δk-τ(k)-1‍ηT(i)R2η(i)-(τ(k)-τ)∑i=k-δk-τ(k)-1‍ηT(i)R2η(i)-(δ-τ(k))∑i=k-τ(k)k-τ-1‍ηT(i)R2η(i)-(τ(k)-τ)∑i=k-τ(k)k-τ-1‍ηT(i)R2η(i)≤-(δ-τ(k))∑i=k-δk-τ(k)-1‍ηT(i)R2η(i)-a(k)(δ-τ(k))∑i=k-δk-τ(k)-1‍ηT(i)R2η(i)-(1-a(k))(τ(k)-τ)∑i=k-τ(k)k-τ-1‍ηT(i)R2η(i)-(τ(k)-τ)∑i=k-τ(k)k-τ-1‍ηT(i)R2η(i)≤-∑i=k-δk-τ(k)-1‍ηT(i)R2∑i=k-δk-τ(k)-1‍η(i)-a(k)∑i=k-δk-τ(k)-1‍ηT(i)R2∑i=k-δk-τ(k)-1‍η(i)-(1-a(k))∑i=k-τ(k)k-τ-1‍ηT(i)R2∑i=k-τ(k)k-τ-1‍η(i)-∑i=k-τ(k)k-τ-1‍ηT(i)R2∑i=k-τ(k)k-τ-1‍η(i)=[x(k-τ(k))x(k-δ)x(k-τ)]T[-2R2R2R2*-R20**-R2][x(k-τ(k))x(k-δ)x(k-τ)]+a(k)[x(k-τ(k))x(k-δ)]T[-R2R2*-R2][x(k-τ(k))x(k-δ)]+(1-a(k))[x(k-τ(k))x(k-τ)]T×[-R2R2*-R2][x(k-τ(k))x(k-τ)],(22)-(τ¯-δ)∑i=k-τ¯k-δ-1‍ηT(i)R3η(i)≤[x(k-δ)x(k-τ¯)]T[-R3R3*-R3][x(k-δ)x(k-τ¯)].
From the first inequality of condition (7) and Assumption 2, we get
(23)σiT(x(k),x(k-τ(k)))(P+R)σi(x(k),x(k-τ(k)))≤λ[∥Wix(k)∥2+∥Six(k-τ(k))∥2].
Denote thatα(k)=(xT(k),xT(k-τ(k)),xT(k-δ),xT(k-τ),xT(k-τ¯),JT(k))T. It follows from (15) to (23) that
(24)𝔼{ΔV(k)-2yT(k)J(k)-γJT(k)J(k)}≤𝔼{∑i=1r‍μi(t)αT(k)[a(k)(Π(i)+Ω1)+(1-a(k))×(Π(i)+Ω2)]α(k)∑i=1r‍}.
From the second inequality and the third inequality of condition (7), we get
(25)𝔼{ΔV(k)-2yT(k)J(k)-γJT(k)J(k)}≤0
and therefore we have
(26)2∑k=0T‍𝔼{yT(k)J(k)}≥∑k=0T‍𝔼{ΔV(k)}-γ∑k=0T‍𝔼{JT(k)J(k)}=∑k=0T‍𝔼{V(T)-V(0)}-γ∑k=0T‍𝔼{JT(k)J(k)}≥-∑k=0T‍𝔼{V(0)}-γ∑k=0T‍𝔼{JT(k)J(k)}=-γ∑k=0T‍𝔼{JT(k)J(k)}
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.
Whenδ≤τ(t)≤τ¯, let b(k)=(¯τ-τ(k))/(¯τ-δ); then 0≤b(k)≤1. In the similitude of the proof of inequality (21), we have
(27)-(τ¯-δ)∑i=k-τ¯k-δ-1‍ηT(i)R3η(i)≤[x(k-τ(k))x(k-τ¯)x(k-δ)]T[-2R3R3R3*-R30**-R3][x(k-τ(k))x(k-τ¯)x(k-δ)]+b(k)[x(k-τ(k))x(k-τ¯)]T[-R3R3*-R3][x(k-τ(k))x(k-τ¯)]+(1-b(k))[x(k-τ(k))x(k-δ)]T×[-R3R3*-R3][x(k-τ(k))x(k-δ)],(28)-(δ-τ)∑i=k-δk-τ-1‍ηT(i)R2η(i)≤[x(k-τ)x(k-δ)]T[-R2R2*-R2][x(k-τ)x(k-δ)].
It follows from (15)–(20), (23), (27), and (28) that
(29)𝔼{ΔV(k)-2yT(k)J(k)-γJT(k)J(k)}≤𝔼{∑i=1r‍μi(t)αT(k)[b(k)(Π(i)+Ω3)+(1-b(k))×(Π(i)+Ω4)]α(k)∑i=1r‍}.
From the second inequality and the third inequality of condition (8), we get
(30)𝔼{ΔV(k)-2yT(k)J(k)-γJT(k)J(k)}≤0.
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)∑i=k-δk-τ-1ηT(i)R2η(i) and -(τ¯-δ)∑i=k-τ¯k-δ-1ηT(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:
(31)A1=[0.040.010.01-0.02],B1=[-0.020.010.010.03],U1=[-0.010.020.010.01],C1=[00.02-0.010.01],D1=[0.040.01-0.02-0.01],V1=[-0.01-0.0200.01],A2=[0.010.0300.01],B2=[-0.010.02-0.020.03],U2=[-0.010.020.010.01],C2=[0.0100.020.01],D2=[-0.01-0.010.010.02],V2=[0.01-0.010.010.02],W1=[-0.010.010.020],S1=[0.0200.01-0.01],W2=[0.0100.020.05],S2=[0.01-0.010.010.01].
By using the Matlab LMI Control Toolbox, we can find a solution to the LMIs in (7) as follows:
(32)P=104[1.0868-0.0007-0.00071.0878],Q1=103[1.3480-0.0014-0.00141.3499],Q2=103[1.3488-0.0011-0.00111.3501],Q3=103[1.3489-0.0009-0.00091.3502],Q4=103[1.3603-0.0037-0.00371.3639],R1=[1.72000.82320.82320.5407],R2=[0.21180.15250.15250.1121],R3=[1.85630.87030.87030.5674],γ=1.3546×103,λ=1.2234×104.
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.



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