# Delay-Range-Dependent Global Robust Passivity Analysis of Discrete-Time Uncertain Recurrent Neural Networks with Interval Time-Varying Delay

Authors: Chien-Yu Lu; Chin-Wen Liao; Hsun-Heng Tsai
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

This paper examines a passivity analysis for a class of discrete-time recurrent neural networks (DRNNs) with norm-bounded time-varying parameter uncertainties and interval time-varying delay. The activation functions are assumed to be globally Lipschitz continuous. Based on an appropriate type of Lyapunov functional, sufficient passivity conditions for the DRNNs are derived in terms of a family of linear matrix inequalities (LMIs). Two numerical examples are given to illustrate the effectiveness and applicability.


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## Body

## 1. Introduction

Recurrent neural networks have been extensively studied in the past decades. Two popular examples are Hopfield neural networks and cellular neural networks. Increasing attention has been draw to the potential applications of recurrent neural networks in information processing systems such as signal processing, model identification, optimization, pattern recognition, and associative memory. However, these successful applications are greatly dependent on the dynamic behavior of recurrent neural networks (RNNs). On the other hand, time delay is inevitably encountered in RNNs, since the interactions between different neurons are asynchronous. Generally, time delays, both constant and time varying, are often encountered in various engineering, biological, and economic systems due to the finite switching speed of amplifiers in electronic networks, or to the finite signal propagation time in biological networks [1]. The existence of time delay could make delayed RNNs be instable or have poor performance. Therefore, many research interests have been attracted to the stability analysis for delayed RNNs. A lot of results related to this issue have been reported [2–6].The theory of passivity plays an important role for analyzing the stability of nonlinear system [7] and has received much attention in the literature from the control community since 1970s [8–14]. It is well known that the passivity theory plays an important role in both electrical network and nonlinear control systems, provides a nice tool for analyzing the stability of system [15], and has found applications in diverse areas such as signal processing, chaos control and synchronization, and fuzzy control. The passivity condition for delayed neural networks with or without time-varying parametric uncertainties using LMIs [16] has been proposed in [17]. By constructing proper Lyapunov functionals and using some analytic techniques, sufficient conditions are given to ensure the passivity of the integro-differential neural networks with time-varying delays in [18]. In [19, 20], the authors studied the delay-dependent robust passivity criterion for the delayed cellular neural networks and the delayed recurrent neural networks. It should be pointed out that the aforementioned results are the continuous-time neural networks.Recently, the stability analysis problems for discrete-time neural networks with time delay have received considerable research interests. For instance in [21], global exponential stability of a class of discrete-time Hopfield neural networks with variable delays is considered. By making use of a difference inequality, a new global exponential stability result is provided. Under different assumptions on the activation functions, a unified linear matrix inequality (LMI) approach has been developed to establish sufficient conditions for the discrete-time recurrent neural networks with interval variable time to be globally exponentially stable in [22]. Delay-dependent results on the global exponential stability problem for discrete-time neural networks with time-varying delays were presented in [23, 24], respectively. However, no delay-range-dependent passivity conditions on discrete-time uncertain recurrent neural networks with interval time-varying delay are available in the literature and remain essentially open. The objective of this paper is to address this unsolved problem.The purpose of this paper is to deal with the problem of passivity conditions for discrete-time uncertain recurrent neural networks with interval time-varying delay. The interval time-varying delay includes both lower and upper bounds of delay, and the parameter uncertainties are assumed to be time varying but norm bounded which appear in all the matrices in the state equation. It is then established that the resulting passivity condition can be cast in a linear matrix inequality format which can be conveniently solved by using the numerically effective Matlab LMI Toolbox. In particular, when the interval time-delay factor is known, it is emphasized that delay-range-dependent passivity condition yields more general and practical results. Finally, two numerical examples are given to demonstrate the effectiveness.Throughout this paper, the notationX≥Y(X>Y) for symmetric matrices X and Y indicates that the matrix X-Y is positive and semidefinite (resp., positive definite); ZT represents the transpose of matrix Z.


## 2. Preliminaries

Consider a discrete-time recurrent neural network with interval time-varying delay described by(2.1)x(k+1)=(A+ΔA(k))x(k)+(W+ΔW(k))f(x(k))+(W1+ΔW1(k))f(x(k-τ(k)))+u(k),
where x(k)=(x1(k),x2(k),…,xn(k))T is the state vector, A=diag(a1,a2,…,an) with |ai|<1, i=1,2,…,n, is the state feedback coefficient matrix, Wn×n and W1n×n are the interconnection matrices representing the weighting coefficients of the neurons, f(x(k))=[f1(x1(k)),…,fn(xn(k))]T∈Rn is the neuron activation function with f(0)=0, τ(k) is the time-varying delay of the system satisfying
(2.2)τ1≤τ(k)≤τ2,k∈N,
where 0≤τ1≤τ2 are known integers. Let y(k)=f(x(k)) be the output of the neural networks. u(k) is the input vector. ΔA(k), ΔW(k), and ΔW1(k) are unknown matrices representing time-varying parameter uncertainties, which are assumed to be of the form
(2.3)[ΔA(k)ΔW(k)ΔW1(k)]=MF(k)[N1N2N3],
where M, N1, N2, and N3 are known real constant matrices, and F(·):R→Rk×l is unknown time-varying matrix function satisfying
(2.4)FT(k)F(k)≤I,k∈N.
The uncertain matrices ΔA(k), ΔW(k), and ΔW1(k) are said to be admissible if both (2.3) and (2.4) hold.In order to obtain our main results, the activation functions in (2.1) are assumed to be bounded and satisfy the following assumption.Assumption 1.
The activation functionsgi(·), i=1,2,…,n, are globally Lipschitz and monotone nondecreasing; that is, there exist constant scalars αi such that for any ς1,ς2∈R and ς1≠ς2,
(2.5)0≤gi(ς1)-gi(ς2)ς1-ς2≤αi,i=1,2,…,n.Definition 2.1 ([25]).
System (2.1) is called passive if there exists a scalar β≥0 such that
(2.6)2∑k=0kpyT(k)u(k)≥-β∑k=0kpuT(k)u(k)
for all kp≥0 and for all solutions of (2.1) with x(0)=0.


## 3. Mathematical Formulation of the Proposed Approach

This section explores the globally robust delay-range-dependent passivity conditions of the discrete-time recurrent uncertain neural network with interval time-varying delay given in (2.1). Specially, an LMI approach is employed to solve the robust delay-range-dependent passivity condition if the system in (2.1) is globally asymptotically stable for all admissible uncertainties ΔA(k), ΔW(k), and ΔW1(k) satisfying (2.6). The analysis commences by using the LMI approach to develop some results which are essential to introduce the followingLemma 3.1for the development of our main theorem.Lemma 3.1.
LetA, D, S, F, and P be real matrices of appropriate dimensions with P>0 and F satisfying FT(k)F(k)≤I. Then the following statements hold.(a)
For anyε>0 and vectors x, y∈Rn(3.1)2xTDFSy≤ε-1xTDDTx+εyTSTSy.(b)
For vectorsx, y∈Rn(3.2)2xTDSy≤xTDPDTx+yTSTP-1Sy.For any matrices Ei, Si, Ti, and Hi(i=1,2,…,8) of appropriate dimensions, it follows from null equations that
(3.3)Φ1=2[(x(k-τ(k)))xT(k)E1+xT(k-τ(k))E2+xT(k-τ2)E3+xT(k-τ1)E4+fT(x(k))E5+fT(x(k-τ(k)))E6+xT(k+1)E7+uT(k)E8]×[x(k)-x(k-τ(k))-∑j=k-τ(k)+1k(x(j)-x(j-1))]=0,(3.4)Φ2=2[(x(k-τ(k)))xT(k)S1+xT(k-τ(k))S2+xT(k-τ2)S3+xT(k-τ1)S4+fT(x(k))S5+fT(x(k-τ(k)))S6+xT(k+1)S7+uT(k)S8]×[x(k-τ(k))-x(k-τ2)-∑j=k-τ2+1k-τ(k)(x(j)-x(j-1))]=0,(3.5)Φ3=-2[(x(k-τ(k)))xT(k)T1+xT(k-τ(k))T2+xT(k-τ2)T3+xT(k-τ1)T4+fT(x(k))T5+fT(x(k-τ(k)))T6+xT(k+1)T7+uT(k)T8]×[x(k-τ1)-x(k-τ(k))-∑j=k-τ(k)+1k-τ1(x(j)-x(j-1))]=0,(3.6)Φ4=-2[(x(k-τ(k)))xT(k)H1+xT(k-τ(k))H2+xT(k-τ2)H3+xT(k-τ1)H4+fT(x(k))H5+fT(x(k-τ(k)))H6+xT(k+1)H7+uT(k)H8]×[(x(k-τ(k)))x(k+1)-(A+ΔA(k))x(k)-(W+ΔW(k))f(x(k))-(W1+ΔW1(k))f(x(k-τ(k)))-u(k)]=0,(3.7)Φ5=2fT(x(k))R1f(x(k))-2fT(x(k))R1f(x(k))+2fT(x(k-τ(k)))R2x(k-τ(k))-2fT(x(k-τ(k)))R2x(k-τ(k))=0.To study the globally robust delay-range-dependent passivity conditions of the discrete-time uncertain recurrent neural network with interval time-varying delay, the following theorem reveals that such conditions can be expressed in terms of LMIs.Theorem 3.2.
Under Assumption1, given scalars 0≤τ1<τ2, system (2.1) with interval time-varying delay τ(k) satisfying (2.2) is globally asymptotically robust stability, if there exist matrices P>0, Q1>0, Q2>0, Z1>0, Z2>0, diagonal matrices R1>0, R2>0, and matrices Ei, Si , Ti and Hi(i=1,2,…,8) of appropriate dimensions, a positive scalar ε>0 and a scalar β≥0 such that the following LMI holds:
(3.8)[Ωτ2Eτ21Sτ21THMεNτ2ET-τ2Z10000τ21ST0-τ21(Z1+Z2)000τ21TT00-τ21Z200MTHT000-εI0εNT0000-εI]<0,
where
(3.9)Ω=[Ω11Ω12Ω13Ω14Ω15Ω16Ω17Ω18Ω12TΩ22Ω23Ω24Ω25Ω26Ω27Ω28Ω13TΩ23TΩ33Ω34Ω35Ω36Ω37Ω38Ω14TΩ24TΩ34TΩ44Ω45Ω46Ω47Ω48Ω15TΩ25TΩ35TΩ45TΩ55Ω56Ω57Ω58Ω16TΩ26TΩ36TΩ46TΩ56TΩ66Ω67Ω68Ω17TΩ27TΩ37TΩ47TΩ57TΩ67TΩ77Ω78Ω18TΩ28TΩ38TΩ48TΩ58TΩ68TΩ78TΩ88],Ω11=-P+(τ21+1)Q1+τ2Z1+τ21Z2+E1+E1T+H1A+ATH1T+Q2,Ω12=E2T-E1+S1+T1+ATH2T,Ω13=E3T-S1+ATH3T,Ω14=E4T-T1+ATH4T,Ω15=E5T+H1W0+ATH5T+ΓTR1,Ω16=E6T+H1W1+ATH6T,Ω17=E7T-τ2Z1-τ21Z2-H1+ATH7T,Ω18=E8T+H1+ATH8T,Ω22=-Q1-E2-E2T+S2+S2T+T2+T2T,Ω23=-E3T-S2+S3T+T3T,Ω24=-E4T-T2+S4T+T4,Ω25=-E5T+S5T+T5T+H2W0,Ω26=-E6T+S6T+T6T+H2W1+R2T,Ω27=-E7T+S7T+T7T-H2,Ω28=-E8T+S8T+T8T+H2,Ω33=-Q2-S3-S3T,Ω34=-S4T-T3,Ω35=-S5T+H3W0,Ω36=-S6T+H3W1,Ω37=-S7T-H3,Ω38=-S8T+H3,Ω44=-T4-T4T,Ω45=T5T+H4W0,Ω46=-T6T+H4W1,Ω47=-T7T-H4,Ω48=-T8T+H4,Ω55=-R1-R1T+H5W0+W0TH5T,Ω56=H5W1+W0TH6T,Ω57=-H5+W0TH7T,Ω58=H5+W0TH8T-I,Ω66=-R2Γ-1-(R2Γ-1)T+H6W1+W1TH6T,Ω67=-H6+W1TH7T,Ω68=H6+W1TH8T,Ω77=-H7-H7T+P+τ2Z1+τ21Z2,Ω78=H7-H8,Ω88=H8+H8T-βI,E=[E1TE2TE3TE4TE5TE6TE7TE8T]T,S=[S1TS2TS3TS4TS5TS6TS7TS8T]T,T=[T1TT2TT3TT4TT5TT6TT7TT8T]T,H=[H1TH2TH3TH4TH5TH6TH7TH8T]T,N=[N1T000N2TN3T00]T,
in which Γ=diag(α1,α2,…,αn), τ21=τ2-τ1. Then system (2.1) satisfying (3.8) with interval time-varying delay is robust delay-range-dependent passivity condition in the sense of Definition 2.1.Proof.
Choose the Lyapunov-Krasovskii functional candidate for the system in (2.1) as
(3.10)V(k)=V1(k)+V2(k)+V3(k)+V4(k)=xT(k)Px(k)+∑i=-τ2-1∑j=k+i+1k(x(j)-x(j-1))TZ1(x(j)-x(j-1))+∑j=k-τ(k)k-1xT(j)Q1x(j)+∑i=-τ2-1-τ1∑j=k+i+1k-1xT(j)Q1x(j)+∑j=k-τ2k-1xT(j)Q2x(j)+∑i=-τ2-1-τ1∑j=k+i+1k(x(j)-x(j-1))TZ2(x(j)-x(j-1)).
Then, the difference ΔV(k)=V(k+1)-V(k) of V(k) along the solution of (2.1) gives
(3.11)ΔV1(k)=xT(k+1)Px(k+1)-xT(k)Px(k),ΔV2(k)=∑i=-τ2-1∑j=k+i+2k+1(x(j)-x(j-1))TZ1(x(j)-x(j-1))-∑i=-τ2-1∑j=k+i+1k(x(j)-x(j-1))TZ1(x(j)-x(j-1))≤τ2(x(k+1)-x(k))TZ1(x(k+1)-x(k))-∑j=k-τ2+1k-τ(k)(x(j)-x(j-1))TZ1(x(j)-x(j-1))-∑j=k-τ(k)+1k(x(j)-x(j-1))TZ1(x(j)-x(j-1)),ΔV3(k)≤xT(k){Q2+[(τ2-τ1)+1]Q1}x(k)-xT(k-τ(k))Q1x(k-τ(k))-xT(k-τ2)Q2x(k-τ2),ΔV4(k)=(τ2-τ1)(x(k+1)-x(k))TZ2(x(k+1)-x(k))-∑j=k-τ2+1k-τ(k)(x(j)-x(j-1))TZ2(x(j)-x(j-1))-∑j=k-τ(k)+1k-τ1(x(j)-x(j-1))TZ2(x(j)-x(j-1)).
Defining the following new variables:
(3.12)η(k)=[xT(k)xT(k-τ(k))xT(k-τ2)xT(k-τ1)fT(x(k))fT(x(k-τ(k)))xT(k+1)uT(k)]T,E=[E1TE2TE3TE4TE5TE6TE7TE8T]T,S=[S1TS2TS3TS4TS5TS6TS7TS8T]T,T=[T1TT2TT3TT4TT5TT6TT7TT8T]T,H=[H1TH2TH3TH4TH5TH6TH7TH8T]T,N=[N1T000N2TN3T00]T,
and combining null equations (3.3)–(3.7), it yields
(3.13)ΔV(k)-2yT(k)u(k)-βuT(k)u(k)=ΔV1(k)+ΔV2(k)+ΔV3(k)+ΔV4(k)-2yT(k)u(k)-βuT(k)u(k)+Φ1+Φ2+Φ3+Φ4+Φ5≤xT(k+1)Px(k+1)-xT(k)Px(k)+τ2(x(k+1)-x(k))TZ1(x(k+1)-x(k))-∑j=k-τ2+1k-τ(k)(x(j)-x(j-1))TZ1(x(j)-x(j-1))-∑j=k-τ(k)+1k(x(j)-x(j-1))TZ1(x(j)-x(j-1))+xT(k){Q2+[(τ2-τ1)+1]Q1}x(k)-xT(k-τ(k))Q1x(k-τ(k))-xT(k-τ2)Q2x(k-τ2)+(τ2-τ1)(x(k+1)-x(k))TZ2(x(k+1)-x(k))-∑j=k-τ2+1k-τ(k)(x(j)-x(j-1))TZ2(x(j)-x(j-1))-∑j=k-τ(k)+1k-τ1(x(j)-x(j-1))TZ2(x(j)-x(j-1))+2ηT(k)E[x(k)-x(k-τ(k))-∑j=k-τ(k)+1k(x(j)-x(j-1))]+2ηT(k)S[x(k-τ(k))-x(k-τ2)-∑j=k-τ2+1k-τ(k)(x(j)-x(j-1))]-2ηT(k)T[x(k-τ1)-x(k-τ(k))-∑j=k-τ(k)+1k-τ1(x(j)-x(j-1))]-2ηT(k)H[(x(k-τ(k)))x(k+1)-(A+ΔA(k))x(k)-(W+ΔW(k))f(x(k))-(W1+ΔW1(k))f(x(k-τ(k)))-u(k)]+2fT(x(k))R1f(x(k))-2fT(x(k))R1f(x(k))+2fT(x(k-τ(k)))R2x(k-τ(k))-2fT(x(k-τ(k)))R2x(k-τ(k))-2yT(k)u(k)-βuT(k)u(k).
Moreover,
(3.14)-2ηT(k)E∑j=k-τ(k)+1k(x(j)-x(j-1))≤τ2ηT(k)EZ1-1ETη(k)+∑j=k-τ(k)+1k(x(j)-x(j-1))TZ1(x(j)-x(j-1)),(3.15)-2ηT(k)S∑j=k-τ2+1k-τ(k)(x(j)-x(j-1))≤(τ2-τ1)ηT(k)S(Z1+Z2)-1STη(k)+∑j=k-τ2+1k-τ(k)(x(j)-x(j-1))T(Z1+Z2)(x(j)-x(j-1)),(3.16)2ηT(k)T∑j=k-τ(k)+1k-τ1(x(j)-x(j-1))≤(τ2-τ1)ηT(k)TZ2-1TTη(k)+∑j=k-τ(k)+1k-τ1(x(j)-x(j-1))TZ2(x(j)-x(j-1)).
Using Assumption 1 and noting that R1>0 and R2>0 are diagonal matrices, one has
(3.17)2fT(x(k))R1f(x(k))≤2fT(x(k))R1Γx(k),(3.18)-2fT(x(k-τ(k)))R2x(k-τ(k))≤-2fT(x(k-τ(k)))R2Γ-1f(x(k-τ(k))),
where Γ=diag(α1,α2,…,αn).
Following from Lemma3.1(a) results in
(3.19)2ηT(k)H(ΔA(k)x(k)+ΔW0(k)f(x(k))+ΔW1(k)f(x(k-τ(k))))=2ηT(k)HMF(k)NTη(k)≤ε-1ηT(k)HMMTHTη(k)+εηT(k)NNTη(k).
Substituting (3.14)–(3.19) into (3.13), it is not difficult to deduce that
(3.20)ΔV(k)-2yT(k)u(k)-βuT(k)u(k)≤ηT(k)[Ω+τ2EZ1-1ET+τ21S(Z1+Z2)-1ST+τ21TZ2-1TT+ε-1HMMTHT+εNNT]η(k).
Using the Schur complement to the (3.20) and in view of LMI (3.8), it follows that
(3.21)ΔV(k)-2yT(k)u(k)-βuT(k)u(k)<0.
It follows from (3.21) that
(3.22)2∑k=0kpyT(k)u(k)≥V(x(kp))-V(x(0))-β∑k=0kpuT(k)u(k),
for x(0)=0, one has V(x(0))=0, so (2.6) holds, and hence the system is robust delay-range-dependent passivity condition in the sense of Definition 2.1. This completes the proof of Theorem 3.2.Remark 3.3.
Theorem3.2 provides a sufficient passivity condition for the globally robust stability of the discrete-time uncertain recurrent neural network with interval time-varying delay given in (2.1) and proposes a delay-range-dependent criterion. Even for τ1=0, the result in Theorem 3.2 may lead to the delay-dependent stability criteria. In fact, if Z2=ε1I, with ε1>0,  being sufficiently small scalar, Ti=0, i=1,2,…,8, E4=0, S4=0, H4=0, Theorem 3.2 yields the following delay-dependent passivity criterion.Corollary 3.4.
Under Assumption1, given scalars τ2>0, τ1=0, system (2.1) with time-varying delay satisfying (2.3) is globally asymptotically robust stability, if there exist matrices P>0, Q1>0, Q2>0, Z1>0, diagonal matrices R1>0, R2>0, and matrices Ei, Si, and Hi(i=1,2,…,8) of appropriate dimensions and E4=S4=H4=0, a positive scalar ε>0, and a scalar β≥0 such that the following LMI holds:
(3.23)[Ωτ2Eτ2SHMεNτ2ET-τ2Z1000τ2ST0-τ2Z100MTHT00-εI0εNT000-εI]<0,
where
(3.24)Ω=[Ω11Ω12Ω13Ω15Ω16Ω17Ω18Ω12TΩ22Ω23Ω25Ω26Ω27Ω28Ω13TΩ23TΩ33Ω35Ω36Ω37Ω38Ω15TΩ25TΩ35TΩ55Ω56Ω57Ω58Ω16TΩ26TΩ36TΩ56TΩ66Ω67Ω68Ω17TΩ27TΩ37TΩ57TΩ67TΩ77Ω78Ω18TΩ28TΩ38TΩ58TΩ68TΩ78TΩ88],Ω11=-P+(τ2+1)Q1+τ2Z1+E1+E1T+H1A+ATH1T+Q2,Ω12=E2T-E1+S1+ATH2T,Ω13=E3T-S1+ATH3T,Ω15=E5T+H1W0+ATH5T+ΓTR1,Ω16=E6T+H1W1+ATH6T,Ω17=E7T-τ2Z1-H1+ATH7T,Ω18=E8T+H1+ATH8T,Ω22=-Q1-E2-E2T+S2+S2T,Ω23=-E3T-S2+S3T,Ω25=-E5T+S5T+H2W0,Ω26=-E6T+S6T+H2W1+R2T,Ω27=-E7T+S7T-H2,Ω28=-E8T+S8T+H2,Ω33=-Q2-S3-S3T,Ω35=-S5T+H3W0,Ω36=-S6T+H3W1,Ω37=-S7T-H3,Ω38=-S8T+H3,Ω55=-R1-R1T+H5W0+W0TH5T,Ω56=H5W1+W0TH6T,Ω57=-H5+W0TH7T,Ω58=H5+W0TH8T-I,Ω66=-R2Γ-1-(R2Γ-1)T+H6W1+W1TH6T,Ω67=-H6+W1TH7T,Ω68=H6+W1TH8T,Ω77=-H7-H7T+P+τ2Z1,Ω78=H7-H8,Ω88=H8+H8T-βI,E=[E1TE2TE3TE5TE6TE7TE8T]T,S=[S1TS2TS3TS5TS6TS7TS8T]T,H=[H1TH2TH3TH5TH6TH7TH8T]T,N=[N1T00N2TN3T00]T.
Therefore, the discrete-time recurrent neural network with time-varying delay in (2.1) (i.e., the lower bounds τ1=0 and the given upper bounds τ2) approaches globally robustly delay-dependent passivity condition in the sense of Definition 2.1.Remark 3.5.
In the stochastic context, robust delay-dependent passivity conditions are studied in [26] for discrete-time stochastic neural networks with time-varying delays. In this paper, however, robust delay-range-dependent passivity conditions are studied in the deterministic context. It should be noted that deterministic systems and stochastic systems have different properties and need to be dealt with separately. The results given in Theorem 3.2 provide an LMI approach to the robust delay-range-dependent passivity conditions for deterministic discrete-time recurrent neural networks with interval time-varying delay, which is new and represents a contribution to recurrent neural networks systems.Two numerical examples are now presented to demonstrate the usefulness of the proposed approach.


## 4. Examples

Example 4.1.
Consider the following discrete-time uncertain recurrent neural network:(4.1)x(k+1)=(A+ΔA(k))x(k)+(W+ΔW(k))f(x(k))+(W1+ΔW1(k))f(x(k-τ(k)))+u(k),
where
(4.2)A=[0.2000.14],W=[0.20.250.250.1],W1=[-0.20.10.10.3],M=[0.20.10.10.2],N1=[0.10.20.10.2],N2=[0.30.100.1],N3=[0.20.10.10.1].
The activation functions in this example are assumed to satisfy Assumption 1 with α1=0.5633, α2=0.0478. For interval time-varying delay, the best approached values of β by Theorem 3.2, for the given upper bound τ2=10 and the various lower bounds τ1, are listed inTable 1by the Matlab LMI Control Toolbox. Therefore, using Theorem 3.2, the discrete-time uncertain recurrent neural network with interval time-varying delay (4.1) satisfies robustly delay-range-dependent passivity conditions in the sense of Definition 2.1 for various β levels.Table 1
The various lower boundsτ1 and β levels for the given upper bound τ2=10.
τ1β05.918915,922825.814535.983245.801455.952265.974476.012686.018696.1851Example 4.2.
Consider the discrete-time uncertain recurrent neural network with the following parameters:(4.3)A=[0.10000.10000.2],W=[0.1-0.10.20.1-0.20.3-0.10-0.1],W1=[-0.10.20.1-0.10.40.20.2-0.10.4],M=[0.100.10.10.300.20.10],N1=[00.020.02-0.10.10.1-0.10.10.1],N2=[0-0.020.020.1-0.100.10-0.1],N3=[0.010.020-0.10.30.10.100.2].
The activation functions in this example are assumed to satisfy Assumption 1 with α1=0.034, α2=0.429, α3=0.508. By the Matlab LMI Control Toolbox, it can be verified that Corollary 3.4 in this paper is feasible solution for all delays 0≤τ(k)≤15 (i.e., the lower bound τ1=0 and the upper bound τ2=15) as follows:
(4.4)P=[2.3189-0.2657-0.0023-0.26572.2981-0.0574-0.0023-0.05742.3612],Q1=[0.0565-0.00560.0005-0.00560.0679-0.00520.0005-0.00520.0657],Z1=[0.0199-0.00380.0008-0.00380.01500.00020.00080.00020.0144],Z2=[0.0215-0.00420.0009-0.00420.01580.00020.00090.00020.0153],R1=[1.89980001.62980001.8378],R2=[1.29910000.29800000.2635],ε=3.4160,β=5.0169.
Thus, by Corollary 3.4, the discrete-time uncertain recurrent neural network with time-varying delay in (2.1) (i.e., the lower bound τ1=0 and the given upper bound τ2=15) attains globally robustly delay-dependent passivity condition in the sense of Definition 2.1.


## 5. Conclusions

This study has investigated the problem of globally robust passivity conditions for a discrete-time recurrent uncertain neural network with interval time-varying delay. A sufficient condition for the solvability of this problem, which takes into account the range for the time delay, has been established that the passivity conditions can be cast in linear matrix inequalities format. It has been shown that the bound for the time-varying delay in a range which ensures that the discrete-time recurrent uncertain neural network with interval time-varying delay attains globally robust passivity conditions can be obtained by solving a convex optimization problem. Two numerical examples have been presented to demonstrate the effectiveness of the proposed approach.



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*Source: 430158-2009-10-19.xml*