# New Results on Passivity Analysis of Delayed Discrete-Time Stochastic Neural Networks

Authors: Jianjiang Yu
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

The problem of passivity analysis for a class of discrete-time stochastic neural networks (DSNNs) with time-varying interval delay is investigated. The delay-dependent sufficient criteria are derived in terms of linear matrix inequalities (LMIs). The results are shown to be generalization of some previous results and are less conservative than the existing works. Meanwhile, the computational complexity of the obtained stability conditions is reduced because less variables are involved. Two numerical examples are given to show the effectiveness and the benefits of the proposed method.


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## Body

## 1. Introduction

Over the past decades, Neural Networks (NNs) have attracted considerable attention because of their extensive applications in pattern recognition, optimization solvers, model identification, signal processing, and other engineering areas [1]. Meanwhile, time delays are frequently encountered in various engineering, biological, and economic systems due to the finite switching speed of amplifiers and the inherent communication time of neurons. It has been revealed that time delay may cause instability and oscillation of the neural networks [2, 3]. For these reasons, the stability problem of NNs with delays has been extensively studied, for example, see [2–10]. It is well known that delay-dependent stability conditions are generally less conservative than delay-independent conditions, especially when the size of the delay is small. Therefore, considerable attention has been focused on the derivation of delay-dependent stability results and many effective approaches have been provided to reduce the conservatism of stability results for further improving the quality of delay-dependent stability criteria.Most NNs studied are assumed to act in a continuous-time manner; however, in implementing and applications of neural networks, discrete-time neural networks become more and more important than their continuous-time counterparts [11]. So, the stability analysis problems for discrete-time neural networks have received more and more interests, and some stability criteria have been proposed in literature; for example, see [11–15] and the references cited therein. Moreover, stochastic disturbance usually appears in the electrical circuit design of neural networks; a neural network could be stabilized or destabilized by certain stochastic inputs. The delay-dependent stability problems of stochastic neural networks are studied in some works, such as, [16, 17] and the reference cited therein.On the other hand, the passivity theory plays an important role in the analysis and design of linear and nonlinear delayed systems. Recently, the passivity of linear systems with delays [18–20] and the passivity of neural networks with delays [21–23] have been studied. Based on the Lyapunov-Krasovskii method and LMI framework, the passivity properties for delayed NNs were firstly studied in [21]. In [23], the problem of passivity and robust passivity for a class of discrete-time stochastic neural networks with time-varying delays was studied.In this paper, by constructing a new Lyapunov-Krasovskii functional, the improved delay-dependent passivity and robust passivity criteria of DSNNs are obtained in the form of linear matrix inequalities (LMIs). It is shown that the obtained conditions are less conservative and more efficiency than those in [23]. Two numerical examples are also provided to show the effectiveness of the proposed stability criteria.Notation 1.
Throughout this paper,ℕ+ stands for the set of nonnegative integers; ℝn and ℝn×m denote the n dimensional Euclidean space and the set of all n×m real matrices, respectively. Real symmetric matrix P>0(≥0) denotes P being a positive definite (positive semi-definite) matrix. The notation X≥Y (resp., X>Y) means that X and Y are symmetric matrices, and that X-Y is positive semi-definite (resp., positive definite). I is used to denote an identity matrix with proper dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetric terms in a symmetric matrix are denoted by *. The superscript “T” represents the transpose. We use λmin(·) and λmax(·) to denote the minimum and maximum eigenvalue of a real symmetric matrix, respectively. Let (Ω,ℱ,P) be a complete probability space with a filtration {ℱt}t≥0satisfying the usual conditions (i.e., it is right continuous and {ℱ}0 contains all P-pull sets); let Lℱ0p([-τ,0];ℝn) be the family of all ℱ0-measurable {𝒞([-τ,0];ℝn)}-valued random variables {ξ=ξ(θ):-τ≤θ≤0} such that sup-τ≤ϑ≤0𝔼{|ξ(θ)|}p<∞, where 𝔼{·} stands for the mathematical expectation operator with respect to the given probability measure P.


## 2. Problem Formulation and Preliminaries

Consider the uncertain DSNNs with time-varying interval delay described by(2.1)x(k+1)=C(k)x(k)+A(k)G(x(k))+B(k)G(x(k-τ(k)))+u(k)+σ(x(k),x(k-τ(k)),k)ω(k),k=1,2,…,y(k)=G(x(k)),
where x(k)=[x1(k),…,xn(k)]T∈ℝn is the neural state vector; G(x(k))=[g1(x1(k)),…,gn(xn(k))]T, and G(x(k-τ(k)))=[g1(x1(k-τ(k))),…,gn(xn(k-τ(k)))]T denote the neuron activation functions; y(k)=G(x(k)) is the output of the neural network; u(k)=(u1(k),u2(k),…,un(k))T is the input vector; τ(k) denotes the time-varying delay satisfying(2.2)0≤τm≤τ(k)≤τM,
and τm,τM are known positive integers. C(k)=C+ΔC(k), A(k)=A+ΔA(k), and B(k)=B+ΔB(k), where C=diag(c1,c2,…,cn) with |ci|<1 describing the rate with which the ith neuron will reset its potential to the resting state in isolation when disconnected from the networks and external inputs, A=(aij)n×n is the connection weight matrix, and B=(bij)n×n is the delayed connection weight matrix. ΔC(k),ΔA(k), and ΔB(k) represent the time-varying parameter uncertainties and are assumed to satisfy the following admissible condition:(2.3)[ΔCΔAΔB]=EF(k)[GCGAGB],
where E,GC,GA, and GB are known real constant matrices of appropriate dimensions; F(k) is the unknown time-varying matrix-valued function satisfying(2.4)F(k)TF(k)≤I,∀k∈ℕ+.
In system (2.1), ω(k) is a scalar Wiener process (Brownian Motion) on (Ω,ℱ,P) with(2.5)𝔼[ω(k)]=0,𝔼[ω2(k)]=1,𝔼[ω(i)ω(j)]=0(i≠j),
and there exist two positive constants that ρ1 and ρ2 such that(2.6)σT(x(k),x(k-τ(k)),k)σ(x(k),x(k-τ(k)),k)≤ρ1x(k)Tx(k)+ρ2x(k-τ(k))Tx(k-τ(k)),The initial condition of system (2.1) is given by(2.7)x(s)=φ(s),s∈[-τM,0].The activation functions in (2.1) satisfy the following assumption.Assumption 2.1.
Activation functionsgi(·) in (2.1) are bounded and satisfy gi(0)=0:
(2.8)σi-≤gi(x)-gi(y)x-y≤σi+,∀x,y∈ℝ,x≠y,i=1,2,…,n,
where σi+,σi- are constants. Denote Γ=diag(σ1-,…,σn-), and
(2.9)Σ1=diag(σ1+σ1-,…,σn+σn-),Σ2=diag(σ1++σ1-2,…,σn++σn-2).Remark 2.2.
As pointed out in [11], the constants σi+,σi- in Assumption 2.1 are allowed to be positive, negative, or zero. So the assumption is weaker in comparison with those made in [2, 13], and so forth.Definition 2.3.
The delayed DSNNs (2.1) are said passive if there exists a scalar γ>0 such that
(2.10)2∑j=0k0𝔼{yT(j)u(j)}≥-γ∑j=0k0𝔼{uT(j)u(j)},∀k0∈ℕ.
The purpose of this paper is to find the maximum allowed delay boundτM for the given lower bound τm such that the system described by (2.1) with uncertainties (2.3) is robust passivity.In obtaining the main results of this paper, the following lemma will be useful for the proofs.Lemma 2.4 (see [3]).
Given constant matricesP,Q,andR, where PT=P,QT=Q, then the LMI   [PRRT-Q]<0 is equivalent to the following condition: Q>0,P+RQ-1RT<0.Lemma 2.5 (see [11]).
Given matricesP,Q, and R with P=PT, then
(2.11)P+QF(k)R+(QF(k)R)T<0
holds for all F(k) satisfying FT(k)F(k)≤I if and only if there exists a scalar ϵ>0 such that
(2.12)P+ϵ-1QQT+ϵRRT<0.


## 3. Main Results

In this section, we present a delay-dependent criterion guaranteeing the passivity of DSNNs with time-varying delay:(3.1)x(k+1)=Cx(k)+AG(x(k))+BG(x(k-τ(k)))+u(k)+σ(x(k),x(k-τ(k)),k)ω(k).Theorem 3.1.
Under Assumption2.1 given two scalars τm>0 and τM>0, for any delay τ(k) satisfying (2.2), system (3.1) is  passive in the sense of Definition 2.3 if there exist two scalars γ>0,λ*>0, five positive definite matrices P,Q1,Q2,Q3,Z>0, three diagonal matrices K>0,U1>0,U2>0, and six matrices L1,L2,M1,M2,N1,N2 with appropriate dimensions, such that the following LMIs hold:
(3.2)τMZ+P<λ*I,(3.3)Ψ=[Ψ1Ψ2*Ψ3]<0,
where
(3.4)Ψ1=[Ψ11Ψ12Ψ13Ψ14Ψ15M1-N1-L1*Ψ22Ψ23Ψ24000**Ψ33Ψ34Ψ3500***Ψ44000****Ψ55M2-N2-L2*****-Q10******-Q3],Ψ2=[τM-τmLτM-τmMτMN],Ψ3=diag{-Z,-Z,-Z},
with
(3.5)Ψ11=-P+Q1+τQ2+Q3-τ(ΓTK+KTΓ)+N1+N1T+CTPC-U1Σ1+ρ1λ*+τM(C-I)TZ(C-I),Ψ12=τK+Σ2U1+CTPA+τM(C-I)TZA,Ψ13=τMCTZB+CTPB,Ψ14=τM(C-I)TZ+CTP,Ψ15=L1+N2T-M1,Ψ22=-U1+τMATZA+ATPA,Ψ23=τMATZB+ATPB,Ψ24=τMATZ+ATP-I,Ψ33=-U2+τMBTZB+BTPB,Ψ34=τMBTZ+BTP,Ψ35=-K+U2Σ2,Ψ44=τMZ+P-γI,Ψ55=-Q2-U2Σ1+L2+L2T-M2-M2T+ρ2λ*+ΓTK+KTΓ,τ=τM-τm+1,L=[L1000L200],M=[M1000M200],N=[N1000N200].Proof.
Choose a new Lyapunov-Krasovskii functional candidate as follows:(3.6)V(k)=V1(k)+V2(k)+V3(k)+V4(k)+V5(k),
where
(3.7)V1(k)=xT(k)Px(k),V2(k)=∑i=k-τmk-1xT(i)Q1x(i)+∑i=k-τMk-1xT(i)Q3x(i),V3(k)=∑i=k-τ(k)k-1xT(i)Q2x(i)+∑j=k-τM+1k-τm∑i=jk-1xT(i)Q2x(i),V4(k)=∑j=k-τMk-1∑i=jk-1ηT(i)Zη(i),V5(k)=2∑i=k-τ(k)k-1[G(x(i))-Γx(i)]TKx(i)+2∑i=k-τM+1k-τm∑j=ik-1[G(x(j))-Γx(j)]TKx(j),η(i)=x(i+1)-x(i).
DefiningΔV(k)=V(k+1)-V(k), calculating the difference of V(k) along the system (3.1), and taking the mathematical expectation, we have
(3.8)𝔼{ΔV(k)}=𝔼{ΔV1(k)}+𝔼{ΔV2(k)}+𝔼{ΔV3(k)}+𝔼{ΔV4(k)}+𝔼{ΔV5(k)},
where,
(3.9)𝔼{ΔV1(k)}=𝔼{[Cx(k)+AG(x(k))+BG(x(k-τ(k)))+u(k)]TP×[Cx(k)+AG(x(k))+BG(x(k-τ(k)))+u(k)]+σT(x(k),x(k-τ(k)),k)Pσ(x(k),x(k-τ(k)),k)-xT(k)Px(k)[Cx(k)+AG(x(k))+BG(x(k-τ(k)))+u(k)]T},𝔼{ΔV2(k)}=𝔼{xT(k)(Q1+Q3)x(k)-xT(k-τm)Q1x(k-τm)-xT(k-τM)Q3x(k-τM)},𝔼{ΔV3(k)}=𝔼{∑i=k+1-τMk-τmxT(k)Q2x(k)-xT(k-τ(k))Q2x(k-τ(k))+∑i=k+1-τmk-1xT(i)Q2x(i)+∑i=k+1-τ(k+1)k-τmxT(i)Q2x(i)-∑i=k+1-τ(k)k-1xT(i)Q2x(i)+(τM-τm)xT(k)Q2x(k)-∑i=k+1-τMk-τmxT(i)Q2x(i)}≤𝔼{xT(k)Q2x(k)-xT(k-τ(k))Q2x(k-τ(k))+(τM-τm)xT(k)Q2x(k)xT},𝔼{ΔV4(k)}=𝔼{τMηT(k)Zη(k)-∑i=k-τMk-1ηT(i)Zη(i)}=𝔼{∑i=k-τmk-1τMxT(k)(C-I)TZ(C-I)x(k)+2τMxT(k)(C-I)TZAG(x(k))+2τMxT(k)(C-I)TZBG(x(k-τ(k)))+2τMxT(k)(C-I)TZu(k)+τMGT(x(k))ATZAG(x(k))+2τMGT(x(k))ATZBG(x(k-τ(k)))+2τMGT(x(k))ATZu(k)+τMGT(x(k-τ(k)))BTZBG(x(k-τ(k)))+2τMGT(x(k-τ(k)))BTZu(k)+τMuT(k)Zu(k)+τMσT(x(k),x(k-τ(k)),k)Zσ(x(k),x(k-τ(k)),k)-∑i=k-τMk-τ(k)-1ηT(i)Zη(i)-∑i=k-τ(k)k-τm-1ηT(i)Zη(i)-∑i=k-τmk-1ηT(i)Zη(i)},𝔼{ΔV5(k)}=𝔼{2∑i=k+1-τ(k+1)k-1[G(x(i))-Γx(i)]TKx(i)+2[G(x(k))-Γx(k)]TKx(k)+2∑i=k-τM+2k+1-τm∑j=ik[G(x(j))-Γx(j)]TKx(j)-2∑i=k+1-τ(k)k-1[G(x(i))-Γx(i)]T×Kx(i)-2[G(x(k-τ(k)))-Γx(k-τ(k))]TKx(k-τ(k))-2∑i=k-τM+1k-τm‍∑j=ik-1[G(x(j))-Γx(j)]TKx(j)}≤𝔼{2(τM-τm+1)(G(x(k))-Γx(k))TKx(k)-2GT(x(k-τ(k)))Kx(k-τ(k))+2xT(k-τ(k))ΓTKx(k-τ(k))(G(x(k))-Γx(k))T}.
Define ξT(k)=[xT(k)GT(x(k))GT(x(k-τ(k)))uT(k)xT(k-τ(k))xT(k-τm)xT(k-τM)], and it is easy to see
(3.10)2ξT(k)L[x(k-τ(k))-x(k-τM)-∑i=k-τMk-τ(k)-1η(i)]≡0,2ξT(k)M[x(k-τm)-x(k-τ(k))-∑i=k-τ(k)k-τm-1η(i)]≡0,2ξT(k)N[x(k)-x(k-τm)-∑i=k-τmk-1η(i)]≡0.
From Assumption 2.1, if follows that for i=1,2,…,n(3.11)(gi(xi(k))-σi+xi(k))(gi(xi(k))-σi-xi(k))≤0,(gi(xi(k-τ(k)))-σi+xi(k-τ(k)))×(gi(xi(k-τ(k)))-σi-xi(k-τ(k)))≤0.
For any diagonal matrices U1≥0,U2≥0, and Σi(i=1,2) in Assumption 2.1, it is easy to get
(3.12)0≤{[x(k)G(x(k))]T[-U1Σ1U1Σ2*-U1][x(k)G(x(k))]+[x(k-τ(k))G(x(k-τ(k)))]T×[-U2Σ1U2Σ2*-U2][x(k-τ(k))G(x(k-τ(k)))]}.
On the other hand, using (3.2)
(3.13)𝔼{σT(x(k),x(k-τ(k)),k)[P+τMZ]σ(x(k),x(k-τ(k)),k)}≤𝔼{ρ1λ*xT(k)G1Tx(k)+ρ2λ*xT(k-τ(k))x(k-τ(k))}.
Since y(k)=G(x(k)), combining (3.9)–(3.13), we get
(3.14)𝔼{ΔV(k)-2GT(x(k))u(k)-γuT(k)u(k)}≤𝔼{∑l=k-τ(k)k-1ξT(k)[Ψ1+(τM-τm)LZ-1LT+(τM-τm)MZ-1MT+τMNZ-1NT]ξ(k)-∑l=k-τMk-τ(k)-1[LTξ(k)+Zη(l)]TZ-1[LTξ(k)+Zη(l)]-∑l=k-τ(k)k-τm-1[MTξ(k)+Zη(l)]TZ-1[MTξ(k)+Zη(l)]-∑l=k-τ(k)k-1[NTξ(k)+Zη(l)]TZ-1[NTξ(k)+Zη(l)]}≤𝔼{ξT(k)[Ψ1+(τM-τm)LZ-1LT+(τM-τm)MZ-1MT+τMNZ-1NT]ξ(k)}.
Applying Lemma 2.4 to (3.3) yields
(3.15)𝔼{Ψ1+(τM-τm)LZ-1LT+(τM-τm)MZ-1MT+τMNZ-1NT}<0.
From (3.15), we obtain
(3.16)2∑j=0k𝔼{yT(j)u(j)}≥∑j=0k𝔼{ΔV(k)}-γ∑j=0k𝔼{uT(j)u(j)}
for all k0∈ℕ. By the definition of V(k), we know that
(3.17)∑j=0k𝔼{ΔV(k)}=𝔼{V(k0+1)-V(0)}=𝔼{V(k0+1)}≥0
for all k0∈ℕ. Thus,
(3.18)2∑j=0k𝔼{yT(j)u(j)}≥-γ∑j=0k𝔼{uT(j)u(j)}.
then by Definition 2.3, it completes the proof of Theorem 3.1.Remark 3.2.
In Theorem3.1, the free-weighing matrices L1,L2,M1,M2,N1,andN2 are introduced so as to reduce the conservatism of the delay-dependent results.Remark 3.3.
Since the activation functions satisfy Assumption2.1, we know that
(3.19)σi-≤gi(x(k))x(k)≤σi+,∀x(k)∈ℝ,i=1,2,…,n.
So,
(3.20)0≤gi(x(k))x(k)-σi-≤σi+-σi-,
then
(3.21)(gi(x(k))-σi-x(k))x(k)≥0,∀x(k)∈ℝ.
So, V5(k) is nonnegative definite. Including the term V5(k) in the Lyapunov functional is to reduce the conservatism further, and it is illustrated by the later example. If V5(k) is not included in (3.6), we can get Corollary 3.4.Corollary 3.4.
Given two scalarsτm>0 and τM>0. Then, for any delay τ(k) satisfying (2.2), system (3.1) is passive in the sense of Definition 2.3 if there exist two scalars γ>0,λ*>0,  five positive definite matrices P,Q1,Q2,Q3,Z>0, two diagonal matrices U1>0,U2>0, and six matrices L1,L2,M1,M2,N1,N2 with appropriate dimensions, such that (3.2) and the following LMI hold:
(3.22)Ψ=[Ψ̂1Ψ2*Ψ3]<0,
where
(3.23)Ψ̂1=[Ψ̂11Ψ̂12Ψ13Ψ14Ψ15M1-N1-L1*Ψ22Ψ23Ψ24000**Ψ33Ψ34U2Σ200***Ψ44000****Ψ̂55M2-N2-L2*****-Q10******-Q3],
with
(3.24)Ψ̂11=-P+Q1+τQ2+Q3+N1+N1T+CTPC-U1Σ1+ρ1λ*+τM(C-I)TZ(C-I),Ψ̂12=τK+Σ2U1+CTPA+τM(C-I)TZA,Ψ̂55=-Q2-U2Σ1+L2+L2T-M2-M2T+ρ2λ*.Next, we provide the delay-dependent robust passivity analysis for uncertain DSNN (2.1).Theorem 3.5.
Given two scalarsτm>0 and τM>0. Then, under Assumption 2.1, for any delay τ(k) satisfying (2.2), system (3.1) is robust passive if there exist two scalars γ>0,ϵ>0, five positive definite matrices P,Q1,Q2,Q3,Z>0, three diagonal matrices K>0,U1>0,U2>0, and six matrices L1,L2,M1,M2,N1,N2 with appropriate dimensions, such that (3.3) and the following LMI hold:
(3.25)Γ=[Γ1Γ2Γ3Γ4Γ5Γ60ϵΓ7*-Z000000**-Z00000***-Z0000****-P0PE0*****-ZτMZE0******-ϵI0*******-ϵI]<0,
where
(3.26)Γ1=[Γ11Γ1200Γ15M1-N1-L1*-U10-I000**-U20Γ3500***-γI000****Γ55M2-N2-L2*****-Q10******-Q3],Γ2=[τM-τmL1000τM-τmL200],Γ3=[τM-τmM1000τM-τmM200],Γ4=[τMN1000τMN200],Γ5=[CTPATPBTPP000],Γ6=[τM(C-I)TZτMATZτMBTZτMZ000],Γ7=[GCTGATGBT0000],
with
(3.27)Γ11=-P+Q1+τQ2+Q3-τ(ΓTK+KTΓ)+N1+N1T-U1Σ1+ρ1λ*,Γ12=τK+Σ2U1,Γ15=L1+N2T-M1,Γ35=-K+U2Σ2,Γ55=-Q2-U2Σ1+L2+L2T-M2-M2T+ρ2λ*+ΓTK+KTΓ.Proof.
Assume that inequality (3.25) holds, according to Lemma 2.4, we have
(3.28)[Γ1Γ2Γ3Γ4Γ5Γ6*-Z0000**-Z000***-Z00****-P0*****-Z]+ϵ-1[0000PEτMZE][0000ETPτMETZ]+ϵ[Γ700000][Γ7T00000]<0.
Then, from Lemma 2.5, we know that LMI (3.28) is equivalent to the following inequality:
(3.29)[Γ1Γ2Γ3Γ4Γ5Γ6*-Z0000**-Z000***-Z00****-P0*****-Z]+[0000PEτMZE]F(k)[Γ7T00000]+[Γ700000]FT(k)[0000ETPτMETZ]<0.
It can be verified that the above inequality is exactly the left-hand side of (3.3) when C, A, and B are replaced with C+EF(k)GC, A+EF(k)GA, and B+EF(k)GB, respectively. The result then follows from Theorem 3.1.
We now consider the DSNNs without stochastic term. In this case, the system (2.1) reduces to
(3.30)x(k+1)=C(k)x(k)+A(k)G(x(k))+B(k)G(x(k-τ(k)))+u(k).
From Theorem 3.5, we can easily obtain the following corollary.Corollary 3.6.
Given two scalarsτm>0 and τM>0. Then, for any delay τ(k) satisfying (2.2), system (3.30) is robust passive if there exist three scalars γ>0,λ*>0,ϵ>0, five positive definite matrices P,Q1,Q2,Q3,Z>0, three diagonal matrices K>0,U1>0,U2>0, and six matrices L1,L2,M1,M2,N1,N2 with appropriate dimensions, such that the following LMI holds:s
(3.31)Γ=[Γ̂1Γ2Γ3Γ4Γ5Γ60ϵΓ7*-Z000000**-Z00000***-Z0000****-P0PE0*****-ZτMZE0******-ϵI0*******-ϵI]<0,
where
(3.32)Γ̂1=[Γ̂11Γ1200Γ15M1-N1-L1*-U10-I000**-U20Γ3500***-γI000****Γ̂55M2-N2-L2*****-Q10******-Q3],
with
(3.33)Γ̂11=-P+Q1+τQ2+Q3-τ(ΓTK+KTΓ)+N1+N1T-U1Σ1,Γ̂55=-Q2-U2Σ1+L2+L2T-M2-M2T+ΓTK+KTΓ.


## 4. Numerical Examples

This section presents two numerical examples that demonstrate the validity of the method described above.Example 4.1.
Consider a delayed DSNNs (3.1) with the following parameters:
(4.1)C=[0.04000.01],A=[0.010.08-0.050.02],B=[-0.050.010.020.07],ρ1=0.01,ρ2=0.02, and the activation functions satisfy Assumption 2.1 with σ1-=-0.1, σ2-=-0.2, σ1+=0.1, and σ2+=0.2.The activation  functions satisfy Assumption2.1 with σ1-=1, σ2-=2, σ1+=2, and σ2+=4.From Tables1 and 2, it is easy to see that the results in this paper are superior to those in [23].Table 1
Comparisons of allowable upper bound (τM) for given τm.
Methodsτm=1τm=3τm=10τm=20NumberofvariablesBy [23]4850576781By Corollary3.44850576745Table 2
Comparisons of allowable upper bound (τM) for given τm.
Methodsτm=1τm=3τm=10τm=20NumberofvariablesBy [23]79162681By Corollary3.479162645By Theorem3.1810172747Example 4.2.
Consider a delayed uncertain DSNNs (3.1) with the following parameters:
(4.2)C=[0.2000.4],A=[0.20.60.50.5],B=[-0.250.10.20.5],E=[1001],GC=[1001],GA=GB=[0.5000.5],ρ1=ρ2=0.01, and the activation functions satisfy Assumption 2.1 with σ1-=-0.1,σ2-=0.1, σ1+=0.1,andσ2+=0.2.
If the time-varying delays satisfy2≤τ(k)≤15, by the Matlab LMI Toolbox, we can find a solution to the LMI (3.25) as follows:
(4.3)P=[558.5625-139.9896-139.9896113.5136],Q1=[101.6690-27.6867-27.68677.8060],Q2=[12.4051-2.9092],Q3=[101.7661-27.7596-27.75967.7780],Z=[9.7355-2.4709-2.47091.0820],K=[0.3724000.0937],U1=[930.709900621.0133],U2=[135.327000190.4640],L1=[0.00020.00120.0002-0.0004],L2=[-0.73980.19040.1898-0.0785],M1=[-0.03490.00280.0030-0.0140],M2=[0.7406-0.1891-0.18880.0806],N1=[-4.48731.17741.1714-0.4350],N2=[0.0358-0.0020-0.00290.0135],
and γ=8845.9,andϵ=11.9457. Therefore, by Theorem 3.5, we know that system (3.1) with the above given parameters is robust passive.


## 5. Conclusions

In this paper, we have considered the problem of passivity and robust passivity analysis for a class of DSNNs with time-varying delay. By choosing a new Lyapunov-Krasovskii functional, the improved delay-dependent criteria have been proposed. Finally, two numerical examples have been provided to illustrate the effectiveness of the obtained results.



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*Source: 139671-2009-12-23.xml*