# Stability Analysis for Discrete-Time Stochastic Fuzzy Neural Networks with Mixed Delays

Authors: YaJun Li; Quanxin Zhu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper is concerned with the stability problem of a class of discrete-time stochastic fuzzy neural networks with mixed delays. New Lyapunov-Krasovskii functions are proposed and free weight matrices are introduced. The novel sufficient conditions for the stability of discrete-time stochastic fuzzy neural networks with mixed delays are established in terms of linear matrix inequalities (LMIs). Finally, numerical examples are given to illustrate the effectiveness and benefits of the proposed method.


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## Body

## 1. Introduction

Such applications of neural networks heavily depend on the dynamical behaviors of the networks. Because neural network systems and their various generalizations have strong self-adaptive ability, learning ability, robustness, and fault tolerance, they have been extensively applied to intelligent signal processing, speech recognition, data mining, robot control, earthquake exploration, and so on.As is well-known, the existence-uniqueness and stability constitute a hot topic in the field of dynamics systems (e.g., see [1–23] and references therein). Just as dynamics systems, the main task of the neural networks designers is to ensure the stability of the equilibrium point of the designed system. At the same time, time-delay and stochastic disturbance are often considered as two main resources to affect the work performance of different neural networks systems. So in theory research, in order to keep the reliability of systems, the researchers have taken stability theory, Lyapunov functional technique, line matrix inequality and free weight matrix method, time-delay decomposition methods to guarantee stability and low conservatism of neural networks systems. In recent many decades, the stability and passivity problem for stochastic neural networks with different time-delays were investigated widely, and many important results were reported (see, e.g., [24–37] and references therein).Furthermore, most physical systems in the real world are nonlinear ones; fuzzy model has been widely used as an efficient method to deal with nonlinear systems because the controlled plant does not need exact mathematical model. When establishing dynamical systems, fuzzy systems in the form of Takagi-Sugeno (T-S) model, being described by a set of IF-THEN rules and nonlinear membership functions, have been extensively adapted. By a T-S fuzzy model, a nonlinear system can be described as a weighted sum of some simple linear subsystems. Recently, it has been used as an efficient tool when approximating the complex nonlinear systems. During the past many years, many results on the research of fuzzy neural networks were reported [38–43]. In [43], the authors studied the stability of fuzzy Markovian jump neural networks. In [42], by utilizing the inequality technic and free-weighting matrices, constructing novel Lyapunov functions, some new results about the robust exponential stability issue of the fuzzy uncertain neural networks were obtained, but adding on the free weight matrices methods has increased computation load and the conservatism of system. Also, a lot of literature published new research results about stability analysis of fuzzy neural networks with different time-delays [38, 39] and many new ideas were introduced. In [39], mean square exponential stability of the fuzzy neural networks with mixed delays was studied, but time-delay was only the constant one, which is not practical in the real applications. In particular, when implementing the delayed continuous-time neural networks for computer simulation, it becomes essential to formulate discrete-time neural network that is an analogue of the continuous-time delayed neural network.Motivated by the above discussion, in this paper we investigate the stability problem for a class of discrete-time stochastic fuzzy neural networks system with mixed time-varying delays. The mixed delays consist of both discrete and distributed delays. One of the main contributions in this paper is that our obtained results reduces the conservatism than the existing achievement. In order to reduce the conservatism, many inequalities technic are utilized and new Lyapunov-Krasovskii functions are proposed. Novel stability criteria are derived in terms of LMIs. Finally, numerical examples are given to show the new established results are less conservative than the existing ones.Throughout this paper,Rn denotes the n-dimensional Euclidean space, and Rn×m is the set of n×m real matrices. I is the identity matrix. · denotes Euclidean norm for vectors. (Ω,F,Ftt≥0,P) is a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions. MT stands for the transpose of the matrix M. For symmetric matrices X and Y, the notation X>Y (respectively, X≥Y ) means that the X-Y is positive definite (respectively, positive semidefinite). ∗ denotes a block that is readily inferred by symmetry. E· stands for the mathematical expectation operator with respect to the given probability measure P.


## 2. Problem Description

Consider the following discrete stochastic fuzzy neural networks with mixed delays:Rulei: IF θ1(k) is μi1 and θ2(k) is μi2 and ⋯ and θp(k) is μip, THEN(1)xk+1=Cixk+Aifxk+Bifxk-dk+D∑m=1+∞μmgxk-m+σk,xk,xk-dkwk(2)xk=χk,k∈-d2,0,i=1,2,…,rwhere μi1,…,μip are the fuzzy sets, x(t)=[x1(k),x2(k),…,xn(k)]∈Rn is the state vector of neural networks, f(x(k))=[f(x1(k)),f(x2(k)),…,f(xn(k))]T and g(x(k))=[g(x1(k)),g(x2(k)),…,g(xn(k))]T are the neural activation functions, Ci=diagci1,ci2,…,cin are positive diagonal matrices representing the self-feedback term with cil<1, Ai∈Rn is the connection weighting matrix, Bi,D∈Rn are the time-delay connection matrices, r is the number of IF-THEN rules, d(k) is the time-varying delay and satisfies d1≤d(k)≤d2, and ω(k) is a scalar Wiener process (Brownian motion) on (Ω,F,P) with(3)Eωk=0,Eωk2=1,Eωiωj=0i≠j.Then, the final model of discrete-time stochastic fuzzy neural networks is described as(4)xk+1=∑i=1rhiθkCixk+Aifxk+Bifxk-dk+D∑m=1+∞μmgxk-m+σk,xk,xk-dkwkwhere hi(θ(k))=νi(θ(k))/∑i=1rνi(θ(k)), vi(θ(k))=∏j=1pμij(θj(k)), and μij(θj(k)) is the grade of membership of θj(k)) in μij. Suppose νi(θ(k))≥0, i=1,2,…,r, and ∑i=1rνi(θ(k))>0 for all k. Therefor, hi(θ(k))≥0 for i=1,2,…,r and ∑i=1rhi(θ(k))=1 for all k.Assumption 1.
The functionσk,xk,xk-dk:R×Rn×Rn→Rn is Borel measurable and is locally Lipschitz continuous, satisfying the following assumption:There exist two positive constantsρ1, ρ2 such that(5)σTk,x,yσk,x,y≤ρ1xTx+ρ2yTy,∀k≥0,x,y∈Rn.Assumption 2.
For anyξ1,ξ2∈R,ξ1≠ξ2, i=1,2… n,(6)Fi-≤fiξ1-fiξ2ξ1-ξ2≤Fi+,Gi-≤giξ1-giξ2ξ1-ξ2≤Gi+,and, for presentation convenience, we define(7)T1=diag⁡F1-,…,Fn-,T2=diag⁡F1-+F1+,…,Fn-+Fn+,G1=diag⁡G1-,…,Gn-,G2=diag⁡G1-+G1+2,…,Gn-+Gn+2,F1=diag⁡F1-F1+,…,Fn-Fn+,F2=diag⁡F1-+F1+2,…,Fn-+Fn+2.The following Lemmas will be essential in the proof of main results.Lemma 3 (see [44]).
For any constant positive-definite matrixM∈Rn×n, M=MT, two positive scalars d2, d1 such that the sums concerned are well defined, then(8)∑i=1d1xiTM∑i=1d1xi≤d1∑i=1d1xTiMxi,∑i=k-d2k-d1-1∑j=ik-d1-1xjTM∑i=k-d2k-d1-1∑j=ik-d1-1xj≤d2-d1d2-d1+12∑i=k-d2k-d1-1∑j=ik-d1-1xTjMxj,∑i=-d2-d1-1∑j=k+ik-1xjTM∑i=-d2-d1-1∑j=k+ik-1xj≤d2-d1d2+d1+12∑i=-d2-d1-1∑j=k+ik-1xTjMxj.Lemma 4 (see [45]).
Suppose thatΛ, Λ1, Λ2 are constant matrices of appropriate dimensions, a∈[0,1]; then(9)Λ+aΛ1+1-aΛ2<0holds, if the following inequalities hold simultaneously (10)Λ+Λ1<0,Λ+Λ2<0.Lemma 5 (see [46]).
For the symmetric appropriate dimensional matricesR>0, Ξ, constant matrix Γ, the following two statements are equivalent:
(i)Ξ-ΓTRΓ<0,
(ii) There exists a matrixΛ of appropriate dimension such that Ξ+ΓTΛ+ΛTΓΛT∗-R<0.Lemma 6 (see [46]).
Letf1,f2,…,fn: Rm→R have positive values in an open subset D of Rm. Then, the reciprocally convex combination of fi over D satisfies (11)minαi∣αi>0,∑iαi=1⁡∑i1αifit=∑ifit+maxgi,jt⁡∑i≠jgi,jtsubject to (12)gi,j:Rm→,gj,it≜gi,jt,fitgi,jtgi,jtfjt≥0.Lemma 7 (see [47]).
LetM∈Rn×n be a positive semidefinite matrix, xi∈Rn×n, and ai≥(i=1,2,…). If the series concerned are convergent, the following inequality holds: (13)∑i=1+∞aixiTM∑i=1+∞aixi≤∑i=1+∞ai∑i=1+∞aixiTMxi.


## 3. Main Results

Theorem 8.
System (4) is said to be asymptotically stable, if there exist symmetric positive matrices P, Q1, Q2, Q3, R1, R2, S1, S2, Z, diagonal matrices U1, U2, U3, H1, H2, matrices V, T, W, Λ=[Λ1Λ2] of appropriate dimensions, and a scalar λ such that the following LMIs hold:(14)P≤λI,(15)Φ=Ω+ΓiTΛ+ΛTΓiΛ1TΛ2T∗-S1-T∗∗-S1<0,i=1,2(16)R2V∗R2>0,(17)S1T∗S1>0,where(18)Γ1=I00000000-I000000000000,Γ2=00000000000I000000000-I,and(19)Ω11=CiTPCi-P+λρ1+1+d12Q1+Q2+Q3-R1-2d12+1-T1H1+T2H2-F1U1,-G1U3,Ω12=R1,Ω15=Ci-ITW,Ω16=CiTPAi+F2U1+d12+1H1-H2,Ω17=CiTPBi,Ω18=G2U3,Ω19=CiPD,Ω22=-Q2-R1-R2-S2,Ω23=R2-V,Ω24=V,Ω210=S2,Ω33=-Q1-2R2+V+VT-F1U2+λρ2+2T1H1-T2H2-S2,Ω34=-V+R2,Ω37=F2U2-H1+H2,Ω311=S2,Ω44=-Q3-R2,Ω55=d12R1+d122R2+τ2S1+d12d12+14S2-W-WT,Ω56=WTAi,Ω57=WTBi,Ω59=WTD,Ω66=AiTPAi-U1,Ω67=AiTPBi,Ω69=AiTPD,Ω77=BiTPBi-U2,Ω79=BiTPD,Ω88=-U3+μ¯Z,Ω99=-Zμ¯+DTPD,Ω1010=-S2,Ω1111=-S2,d12=d2-d1,τ=d2-d1d2+d1+12,μ¯=∑i=1+∞μm.Proof.
Choose the following Lyapunov functional(20)V1k=xTkPxk,(21)V2k=∑i=k-dkk-1xTiQ1xi,(22)V3k=∑j=-d2+1-d1∑i=k+jk-1xTiQ1xi,(23)V4k=∑i=k-d1k-1xTiQ2xi+∑i=k-d2k-1xTiQ3xi,(24)V5k=∑i=-d1+10∑j=k-1+ik-1d1yTjR1yi+∑i=-d2+1-d1∑j=k-1+ik-1d12yTjR2yj.(25)V6k=∑i=-d2-d1-1∑j=i-1∑l=k+jk-1τyTlS1yl+12∑i=k-d2k-d1-1∑j=ik-d1-1∑l=jk-1yTlS2yl,(26)V7k=2∑i=k-dkk-1fxi-T1xiTH1xi+2∑i=-d2+1-d1∑j=k+ik-1fxj-T1xjTH1xj+2∑i=k-dkk-1T2xi-fxiTH2xi+2∑i=-d2+1-d1∑j=k+ik-1T2xj-fxjTH2xj,(27)V8k=∑i=1+∞∑j=k-ik-1fTxjZfxj.By defining ΔVi(k)=Vi(k+1)-Vi(k), calculating the difference of Vi(k) along the trajectory of neural networks (4), and taking the mathematical expectation, we get(28)EΔV1k=EΔV1k+1-ΔV1k=ExTk+1Pxk+1-xTkPxk,(29)EΔV2k=EΔV2k+1-ΔV2k=ExTQ1xT-xTk-dkQ1xk-dk+∑i=k+1-dk+1k-1xTiQ1xi-∑i=k+1-dkk-1xTiQ1xi≤ExTkQ1xk-xTk-dkQ1xk-dk+∑i=k-d2+1k-d1xTiQ1xi,(30)EΔV3k=EΔV3k+1-ΔV3k=E∑-d2+1-d1xTkQ1xk-xTk+jQ1xk+j=Ed2-d1xTkQ1xk-∑i=k-d2+1k-d1xTiQ1xi,(31)EΔV4k=EΔV4k+1-ΔV4k=ExTkQ2xk-xTk-d1Q2xk-d1+xTkQ3xk-xTk-d2Q3xk-d2,(32)EΔV5k=EΔV5k+1-ΔV5k=Ed12yTkR1yk-∑i=k-d1k-1d1yTiR1yi+d122yTkR2yk-∑i=k-d2k-d1-1d12yTiR2yi.By Jensen’s inequality ([48]), it is observed(33)E-∑i=k-d1k-1d1yTiR1yi≤E-∑j=k-d1k-1yTjTR1∑j=k-d1k-1yTj=E-xk-xk-d1TR1xk-xk-d1,and from convex reciprocal Lemma 6, we get(34)E-∑i=k-d2k-d1-1d12yTiR2yi=E-∑i=k-dkk-d1-1d12yTiR2yi-∑i=k-d2k-dk-1d12yTiR2yi≤E-d2-d1d2-dkγ1TR2γ1T-d2-d1dk-d1γ2TR2γ2Twhere γ1=x(k-d(k))-x(k-d2), γ2=x(k-d1)-x(k-d(k)). If there exists matrix V such that R2V∗R2>0 holds, then(35)E-∑i=k-d2k-d1-1d12yTiR2yi≤E-γ1γ2TR2V∗R2γ1γ2,which leads to(36)E-∑i=k-d2k-d1-1d12yTiR2yi≤E-xk-d1xk-dkxk-d2T-R2R2-VV∗-2R2+V+VT-V+R2∗∗-R2xk-d1xk-dkxk-d2,(37)EΔV6k=EΔV6k+1-ΔV6k=Eτ2yTkS1yk+d12d12+14yTkS2yk-τ∑i=-d2-d1-1∑j=k+ik-1yTjS1yj-12∑i=k-d2k-d1-1∑j=ik-d1-1yTjS2yj.By utilizing Lemma 3, we can get(38)E-τ∑i=-d2-d1-1∑j=k+ik-1yTjS1yj=E-τd2-dkd2+dk+1d2-d1d2+d1+1∑i=-d2-dk-1∑j=k+ik-1yTjS1yj-τdk-d1dk+d1+1d2-d1d2+d1+1∑i=-dk-d1-1∑j=k+ik-1yTjS1yj≤E-α∑i=-d2-dk-1∑j=k+ik-1yjTS1∑i=-d2-dk-1∑j=k+ik-1yj-β∑i=-dk-d1-1∑j=k+ik-1yjTS1∑i=-dk-d1-1∑j=k+ik-1yj,where(39)α=d2-dkd2+dk+1d2-d1d2+d1+1,β=dk-d1dk+d1+1d2-d1d2+d1+1α+β=1.By Lemma 6 again, we have the following.(40)E-τ∑i=-d2-d1-1∑j=k+ik-1yTjS1yj≤E-∑i=-d2-dk-1∑j=k+ik-1yjTS1∑i=-d2-dk-1∑j=k+ik-1yj-∑i=-dk-d1-1∑j=k+ik-1yjT2T∑i=-d2-dk-1∑j=k+ik-1yj-∑i=-dk-d1-1∑j=k+ik-1yjTS1∑i=-dk-d1-1∑j=k+ik-1yj≤E-μkxk-∑i=k-d2k-dk-1xiTS1μkxk-∑i=k-d2k-dk-1xi-νkxk-∑i=k-dkk-d1-1xiT2Tμkxk-∑i=k-d2k-dk-1xi-νkxk-∑i=k-dkk-d1-1xiTS1νkxk-∑i=k-dkk-d1-1xi≤E-xkψkϕkTνkInμkIn-νkIn00-μkInS1T∗S1νkInμkIn-νkIn00-μkInTxkψkϕkBy Lemma 3 again, we have(41)E-12∑i=k-d2k-d1-1∑j=ik-d1-1yTjS2yj=E-12∑i=k-d2k-dk-1∑j=ik-dk-1yTjS2yj-12∑i=k-d2k-dk-1∑j=k-dkk-d1-1yTjS2yj-12∑i=k-dkk-d1-1∑j=ik-d1-1yTjS2yj≤E-1d2-dkd2-dk+1∑i=k-d2k-dk-1∑j=ik-dk-1ykTS2∑i=k-d2k-dk-1∑j=ik-dk-1yk-1dk-d1dk-d1+1∑i=k-dkk-d1-1∑j=ik-d1-1ykTS2∑i=k-dkk-d1-1∑j=ik-d1-1yk≤E-d2-dkd2-dk+1xk-dk-1d2-dk∑i=k-d2k-dk-1xiTS2×xk-dk-1d2-dk∑i=k-d2k-dk-1xi-dk-d1dk-d1+1xk-d1-1dk-d1∑i=k-dkk-d1-1xiTS2×xk-d1-1dk-d1∑i=k-dkk-d1-1xi≤E-xk-d1-ϕkTS2xk-d1-ϕk-xk-dk-ψkTS2xk-dk-ψk,where(42)ϕk=1dk-d1∑i=k-dkk-d1-1xi,ψk=1d2-dk∑i=k-d1k-dk-1xi,μk=d2-dk,νk=d1-dk,(43)EΔV7k=EΔV7k+1-ΔV7k=E2d12+1fxk-T1xkTH1xk+2d12+1T2xk-fxkTH2xk-2fxk-dk-T1xk-dkTH1xk-dk,-2T2xk-dk-fxk-dkTH2xk-dk(44)EΔV8k=EΔV8k+1-ΔV8k=Eμ¯gTxkZgxk-∑i=1+∞μigTxk-iZgxk-i.Also, from Lemma 7 we can get that(45)-∑i=1+∞μigTxk-iZgxk-i≤-1μ¯∑m=1+∞μmgxk-mTZ∑m=1+∞μmgxk-m.Since y(k)=x(k+1)-x(k), we have(46)Eyk=Exk+1-xk=ECi-Ixk+Aifxk+Bifxk-dk+D∑m=1+∞μmgxk-m.Then for any matrix W, we have(47)E2yTkWTCi-Ixk+Aifxk+Bifxk-dk+D∑m=1+∞μmgxk-m-yk=0.From Assumption 2, we have(48)fixik-Fi-xikfixik-Fi+xik≤0,i=1,2,…,n,gixik-Gi-xikgixik-Gi+xik≤0,i=1,2,…,n,which lead to(49)xkfxkTFi-Fi+eieiT-Fi-+Fi+2eieiT∗eieiTxkfxk≤0,where ei denotes the unit column vector having the element 1 on its rth row and zeros elsewhere. Let U1=diag{d1,d2,…,dn}, U2=diag{h1,h2,…,hn}, U3=diag{l1,l2,…,ln}, then(50)∑i=1nxkfxkTFi-Fi+eieiT-Fi-+Fi+2eieiT∗eieiTxkfxk≤0⇒xkfxkTF1U1-F2U1∗U1xkfxk≤0.Similarly, we can get(51)xk-dkfxk-dkTF1U2-F2U2∗U2xk-dkfxk-dk≤0(52)xkgxkTG1U3-G2U3∗U3xkgxk≤0.Also, from Assumption 1, we get that(53)σk,xk,xk-dkTPσk,xk,xk-dk≤λρ1xTkxk+ρ2xTk-dkxk-dk.Combining (28)-(53), we have(54)EΔVk=EξTkΩ-ΓTkS1T∗S1Γkξk,where(55)Γk=μkI00000000-μkI0νkI000000000-νkI,(56)ξk=xTk,xTt-d1,xTt-dk,xTt-d2,yTk,fTxk,fTxk-dk,gTxk,∑m=1+∞μmgTxk-m,ϕTk,ψTkT.From LMI (15), we can get that Ω+ΓiTΛ+ΛΓiT<0 for i=1,2. Then by Lemma 4, the terms Ω+ΓiTΛ+ΛΓiT<0(i=1,2) guarantee(57)Ω+ΓTkΛ+ΛΓTk<0,which makes the following inequality true:(58)Φ=Ω+ΓTkΛ+ΛΓTkΛ1Λ2∗-S1-T∗∗-S1<0.By Lemma 5 and (58), we can get that(59)Ω-Γk-S1-T∗-S1ΓTk<0.Then, there must exist a positive scalar γ>0 such that(60)EΔVk≤-γExk2,∀xk≠0.By Lyapunov stability theory, the neural network systems is globally asymptotically stable; this completes the proof.Remark 9.
If we have the following discrete-time fuzzy neural network system without the distributed delays:(61)xk+1=∑i=1rhiθkCixk+Aifxk+Bifxk-dk+σk,xk,xk-dkwkfor the neural network system (61), we have the following stability results.Corollary 10.
System (61) is said to be exponentially stable, if there exist symmetric positive matrices P, Q1, Q2, Q3, R1, R2, S1, S2, Z, diagonal matrices U1, U2, H1, H2, matrices V, T, W, Λ=[Λ1Λ2], and scalar λ such that the following LMIs hold:(62)P≤λI,(63)Φ=Π+ΓiTΛ+ΛTΓiΛ1TΛ2T∗-S1-T∗∗-S1<0,(64)R2V∗R2>0,(65)S1T∗S1>0,where(66)Γ1=I000000-I0000000000,Γ2=000000000I0000000-I,Π11=CiTPCi-P+λρ1+1+d12Q1+Q2+Q3-R1-2d12+1-T1H1+T2H2-F1U1,Π12=R1,Π15=Ci-ITW,Π16=CiTPAi+F2U1+d12+1H1-H2,Π17=CiTPBi,Π22=-Q2-R1-R2-S2,Π23=R2-V,Π24=V,Π28=S2,Π33=-Q1-2R2+V+VT-F1U2+λρ2+2T1H1-T2H2-S2,Π34=-V+R2,Π37=F2U2-H1+H2,Π39=S2,Π44=-Q3-R2,Π55=d12R1+d122R2+τ2S1+d12d12+14S2-W-WT,Π56=WTAi,Π57=WTBi,Π66=AiTPAi-U1,Π67=AiTPBi,Π77=BiTPBi-U2,Π88=-S2,Π99=-S2.Proof.
The proof is similar to that of Theorem8 and hence is omitted here.Remark 11.
If we have the following deterministic neural network system without the distributed delays:(67)xk+1=Cxk+Afxk+Bfxk-dk+σk,xk,xk-dkwkfor the neural network system (67), Corollary 12 can be obtained.Corollary 12.
The neural network (67) is said to be exponentially stable if there exist symmetric positive matrices P, Q1, Q2, Q3, R1, R2, S1, S2, Z, diagonal matrices U1, U2, H1, H2, matrices V, T, W, Λ=[Λ1Λ2], and a scalar λ such that the following LMIs hold:(68)P≤λI,(69)Φ=Ξ+ΓiTΛ+ΛTΓiΛ1TΛ2T∗-S1-T∗∗-S1<0,(70)R2V∗R2>0,(71)S1T∗S1>0,where(72)Γ1=I000000-I0000000000,Γ2=000000000I0000000-I,Ξ11=CTPC-P+λρ1+1+d12Q1+Q2+Q3-R1-2d12+1-T1H1+T2H2-F1U1,Ξ12=R1,Ξ15=C-ITW,Ξ16=CTPA+F2U1+d12+1H1-H2,Ξ17=CTPB,Ξ22=-Q2-R1-R2-S2,Ξ23=R2-V,Ξ24=V,Ξ28=S2,Ξ33=-Q1-2R2+V+VT-F1U2+λρ2+2T1H1-T2H2-S2,Ξ34=-V+R2,Ξ37=F2U2-H1+H2,Ξ39=S2,Ξ44=-Q3-R2,Ξ55=d12R1+d122R2+τ2S1+d12d12+14S2-W-WT,Ξ56=WTA,Ξ57=WTB,Ξ66=ATPA-U1,Ξ67=ATPB,Ξ77=BTPB-U2,Ξ88=-S2,Ξ99=-S2.Proof.
The proof of this corollary is similar to that of Theorem8 and hence is omitted here.Remark 13.
If we have the following fuzzy neural network system with mixed delays:(73)xk+1=∑i=1rhiθkCixk+Aifxk+Bixk-dk+D∑m=1+∞μmgxk-m+Eixk+Fifxkwkfor the neural network system (73), Corollary 14 can be obtained.Corollary 14.
System (73) is said to be exponentially stable if there exist symmetric positive matrices P, Q1, Q2, Q3, R1, R2, S1, S2, Z, diagonal matrices U1, U3, H1, H2, matrices V, T, W, Λ=[Λ1Λ2] of appropriate dimensions, and a scalar λ such that the following LMIs hold:(74)P≤λI(75)Φ=Δ+ΓiTΛ+ΛΓiTΛ1Λ2∗-S1-T∗∗-S1<0(76)R2V∗R2>0,(77)S1T∗S1>0,where(78)Γ1=I0000000-I00000000000,Γ2=0000000000I00000000-I,Δ11=CiTPCi-P+λρ1+1+d12Q1+Q2+Q3-R1-EiTPEi+F1U1,Δ12=R1,Δ13=CiTPBi,Δ15=Ci-ITW,Δ16=CiTPAi+F2U1+EiTPFi,Δ17=G1U3,Δ18=CiTPD,Δ22=-Q2-R1-R2-S2,Δ23=R2-V,Δ24=V,Δ29=S2,Δ33=-Q1-2R2+V+VT+λρ2-S2+BiTPBi,Δ34=-V+R2,Δ35=BiTW,Δ36=BiTPAi,Δ38=BiTPD,Δ310=S2,Δ44=-Q3-R2,Δ1010=-S2,Δ55=d12R1+d122R2+τ2S1+d12d12+14-W-WT,Δ56=WTAi,Δ58=WTD,Δ66=ATPA-U1+FiTFi,Δ68=AiTPD,Δ77=μ¯Z-U3,Δ88=-Zμ¯+DPD,Δ99=-S2.The proof is similar to that of Theorem 8 and hence is omitted here.Remark 15.
In [39], the stability problem for the discrete-time stochastic fuzzy neural networks with constant time-delay is studied. Compared with the stochastic fuzzy neural network system in [39], system (73) in this paper has taken the time-varying delay into consideration. Time-varying delays exist in most practical applications and the constant delay is a special case of it. From this point of view, it is more practical to take the time-varying delays into consideration when studying the fuzzy neural networks.


## 4. Numerical Examples

Example 1.
Consider the neural networks (61) with (79)C1=0.4000.7,C2=0.5000.2,A1=0.100.010.05,A2=0.20.2-0.10.1,B1=0.10.10.1-0.1,B2=-0.20.100.1.The activation functions are described byf1(x)=tanh(0.8x) and f2(x)=tanh(-1x). Then from Assumption 2 we can get F1-=0,F1+=0.8,F2-=0,F2+=-1 and we have T1=diag{0,0}, T2=diag{0.8,-1}, F1=diag{0,0}, F2=diag{0.4,-0.5}. Let r1=r2=0.1, by solving the LMIs in Corollary 10, the allowable upper bounds of the time-delay for different d1 are shown in Table 1.Table 1
Maximumd2 for given d1 in Corollary 10.
d1 2 4 6 8 10 [42] 5 7 9 11 13 Corollary10 8 10 12 14 16When the initial condition isx(0)=[2-1]T with 2≤d(k)≤8, and the fuzzy membership functions are taken as (80)h1θk=1+sin⁡xk2,h2θk=1-sin⁡xk2,the simulation results of neural networks (61) are plotted in Figure 1. It is clearly observed from the simulation result that the neural networks system (61) is globally asymptotically stable.Figure 1
State trajectory of the neural network (61).Example 2.
Consider the neural networks (67) with (81)C=0.8000.9,A=0.001000.005,B=-0.10.01-0.2-0.1.The activation functions are described byf1(x)=f2(x)=tanh(0.5x). Then from Assumption 2 we can get that F1-=0,F1+=0.5,F2-=0,F2+=0.5 and T1=diag{0,0}, T2=diag{0.5,0.5}, F1=diag{0,0}, F2=diag{0.25,0.25}. Let r1=r2=0.01, by solving the LMIs (68)-(71) in Corollary 12, we can obtain the allowable upper bounds of the time-delay for different d1 in Table 2.Table 2
Calculated maximumd2 for given d1 for Corollary 12.
d1 2 4 6 8 10 [34] 11 12 13 14 16 Corollary12 16 17 18 19 21When the initial condition isx(0)=[2-1]T with 2≤d(k)≤16, the simulation results of neural networks are plotted in Figure 2. It is clearly observed from the simulation results that the neural networks system (67) is globally asymptotically stable.Figure 2
State trajectory of the neural network (67).Example 3.
Consider the neural networks (4) with (82)C1=0.4000.7,C2=0.5000.2,A1=0.100.010.05,A2=0.20.2-0.10.1,B1=0.10.10.1-0.1,B2=-0.20.100.1,D=0.30.1-0.10.6.The activation functions are described by f1(x)=tanh(0.8x),f2(x)=tanh(-1x),g1(x)=tanh(0.4x),g2(x)=tanh(-2x) and μm=2-(m+3). Then from Assumption 2 we can get that F1-=0,F1+=0.8,F2-=0,F2+=-1 and G1-=0,G1+=0.4,G2-=0,G2+=-2, then we have T1=diag{0,0}, T2=diag{0.8,-1}, F1=diag{0,0}, F2=diag{0.4,-0.5} and G1=diag{0,0}, G2=diag{0.2,-1}, μ¯=∑m=1+∞=μm=2-3. Let r1=r2=0.3, by solving the LMIs in Theorem 8, the allowable upper bounds of the time-delay for different d1 are listed in Table 3.Table 3
Calculated maximumd2 for given d1 for Theorem 8.
d1 2 4 6 8 10 Theorem8 4 6 8 10 12When the initial condition is set byx(k)=[2-1]T with 2≤d(k)≤4 and the fuzzy membership functions are taken as (83)h1θk=1+sin⁡xk2,h2θk=1-sin⁡xk2,the state trajectory of the neural network system (4) is plotted in Figure 3. It is clearly shown from the simulation result that the neural networks system (4) is asymptotically stable.Figure 3
State trajectory of the neural network (4).


## 5. Conclusions

The stability problem for a class of discrete-time stochastic fuzzy neural networks system with mixed time-delays has been investigated. With the purpose of reducing the conservatism and obtaining larger time-delay bounds, some new Lyapunov-Krasovskii functions and free-weighting matrices are introduced. The sufficient condition which makes the studied system asymptotically stable has been established in terms of LMIs. Numerical examples and simulations are given to illustrate the effectiveness of our presented method compared with the existing result. It should be mentioned that the proposed method in this paper can be extended to the passivity analysis, robustness research, andH∞ filter design of fuzzy neural network systems.



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