# Delay-Dependent Stability Criteria of Uncertain Periodic Switched Recurrent Neural Networks with Time-Varying Delays

Authors: Xing Yin; Jun Li; Weigen Wu; Qiranrong Tan
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

This paper deals with the problem of delay-dependent stability criterion of
uncertain periodic switched recurrent neural networks with time-varying delays. When uncertain discrete-time recurrent neural network is a periodic system, it is expressed as switched neural network for the finite switching state. Based on the switched quadratic Lyapunov functional approach (SQLF) and free-weighting matrix approach (FWM), some linear matrix inequality criteria are found to guarantee the delay-dependent asymptotical stability of these systems. Two examples illustrate the exactness of the proposed criteria.


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## Body

## 1. Introduction

Recurrent neural networks (RNNs) are a very important tool for many application areas such as associative memory, pattern recognition, signal processing, model identification, and combinatorial optimization. With the development of research on RNNs in theory and application, the model is more and more complex. When the continuous-time RNNs are simulated using computer, they should be discretized into discrete-time RNNs [1–3]. Simultaneously, in implementations of artificial neural networks, time-varying delay may occur due to finite switching speeds of the amplifiers and communication time [4, 5]. Therefore, researchers have considered that discrete-time RNNs with time-varying delay are incorporated in the processing and/or transmission parts of the network architectures [6–9]. Parameter uncertainties and nonautonomous phenomena often exist in real systems due to modeling inaccuracies [4]. Particularly when we consider a long-term dynamical behaviors of the system and consider seasonality of the changing environment, the parameters of the system usually will change with time [10–14]. In order to model those systems with neural networks, the uncertain (or switched or jumping) neural networks with time-varying delay appear in many papers [6, 15–24]. So in this paper we consider the stability of the following discrete-time recurrent neural networks with time-varying delay:
(1.1)u(k+1)=(A+ΔA(k))u(k)+(W1+ΔW1(k))g(u(k))+(W2+ΔW2(k))f(u(k-d(k)))+(I+ΔI(k)),
where u(k)={u1(k),u2(k),…,un(k)}∈Rn is the state vector associated with n neurons, A=diag{a1,a2,…,an} is a diagonal matrix with positive entries, W1 and W2 are, respectively, the connection weight matrix and the delayed connection weight matrix, I is input vector, g(u)={g1(u),g2(u),…,gn(u)} and f(u)={f1(u),f2(u),…,fn(u)} are the neuron activation function vectors, and d(k) is nonnegative differential time-varying functions which denote the time delays and satisfy(1.2)0≤d1≤d(k)≤d2.In most literatures it is required that parameter uncertainty matrices, such asΔA(k), ΔW1(k), ΔW2(k), and ΔI(k), should be in the form(1.3)[ΔA(k)ΔW1(k)ΔW2(k)ΔI(k)]=DF(k)[EaEw1Ew2Ei],
where Ea, Ew1, Ew2, and Ei are given constant matrices of appropriate dimensions and F(k) is an uncertain matrix such that(1.4)FT(k)F(k)≤I.
In practice, however, ΔA(k), ΔW1(k), ΔW2(k), and ΔI(k) are generally difficult to have the decomposition of matrices for D, Ea, Ew1, Ew2, and Ei. In addition, periodic oscillation in recurrent neural networks is an interesting dynamic behavior as many biological and cognitive activities require repetition [7, 10, 11, 25]. Simultaneously, periodic oscillations in recurrent neural networks have been found in many applications such as associative memories, pattern recognition, machine learning, and robot motion control [25]. So, if (1.1) is an uncertain periodic neural network in which the period is less than a constant b, then (1.1) can be expressed as switched neural network for the finite switching state, that is, if (1.1) is a neural network with period a (0<a≤b), then ΔA(k)=ΔA(k+a),ΔW1(k)=ΔW1(k+a),ΔW2(k)=ΔW2(k+a), and ΔI(k)=ΔI(k+a), which is corresponding to a switched neural network set: Ω=⋃a=1bUa, where Ua={Sai=(Aai,W1ai,W2ai,Iai)∣0<i≤a}, Aai=A+ΔA(k+i), W1ai=W1+ΔW1(k+i), W2ai=W2+ΔW2(k+i), and Iai=I+ΔI(k+i). Suppose N is the number of elements of Ω; then (1.1) is actually modified by(1.5)u(k+1)=Ar(k)u(k)+W1r(k)g(u(k))+W2r(k)f(u(k-d(k)))+Ir(k),
where r(k) is a switching rule defined by r(k):N→Ω with Ω={S1,S2,…,SN}. Moreover, r(k)=j means the sub-recurrent neural network (sub-RNN) Sj, which is corresponding to (Aj,W1j,W2j,Ij), is active.The dynamic behaviors of those models are foundations for applications. Under (1.5) most papers discuss the stability of uncertain neural networks with the common Lyapunov function approach [5, 6, 15–23, 25]. To the best of the authors’ knowledge, up to now, there is scarcely any paper that studies the uncertain periodic neural networks using the SQLF. This situation motivates this research.Motivated by the above discussions, the authors intend to study a problem of the delay-dependent stability criterion of uncertain discrete-time recurrent neural networks with time-varying delays that the uncertain recurrent neural networks have a finite number of sub-RNNs, and the sub-RNNs may change from one to another according to arbitrary switching and restricted switching. The contributions of this paper are the following. (1) Using a switching graph, uncertain periodic recurrent neural networks with time-varying delays are transformed into switched recurrent neural networks; (2) the derivative of the SQLF (3.7) of the literature [8] is improved in (3.11), please see Remark 4.3 and Table 3; (3) based on the switching graph, the delay-dependent stability criteria of switched recurrent neural networks are studied by FWM and SQLF. Then an effective LMI approach is developed to solve the problem.This paper is organized as follows. In Section2, we give some basic definitions. We analyze the stability of the system (2.2) with the SQLF and FWM in Section 3. Some examples are given in Section 4. Section 5 offers the conclusions of this paper.


## 2. Preliminaries

In many electronic circuits, nonmonotonic functions can be more appropriate to describe the neuron activation in designing and implementing an artificial neural network [7]; hence, we have the following assumption.For anyj∈{1,2,…,n}, there exist constants gj-, gj+, fj-, and fj+ such that
(2.1)gj-≤gj(θ1)-gj(θ2)θ1-θ2≤gj+,∀θ1,θ2∈R,fj-≤fj(θ1)-fj(θ2)θ1-θ2≤fj+,∀θ1,θ2∈R.Under the assumption, the equilibrium points of UDNN (1.1) exist by the fixed point theorem [1]. In the following, let u∘={u1∘,u2∘,…,un∘} be the equilibrium point of (1.1); then x(·)=u(·)-u∘. The systems (1.1) and (1.5) are, respectively, shifted to the following form:(2.2)x(k+1)=(A+ΔA(k))x(k)+(W1+ΔW1(k))g(x(k))+(W2+ΔW2(k))f(x(k-τ(k))),x(k+1)=Ar(k)x(k)+W1r(k)g(x(k))+W2r(k)f(x(k-d(k))),r(k)∈Ω.
For convenience, the switching graph is defined.Definition 2.1.
LetΓ=(Ω,W) be a switching graph, where Ω is the set of sub-RNNs Si and W is the set of weighted arcs wij∈{0,1}. wij=1 (or 0) represents the sub-RNN Si switches (or does not switch) to the sub-RNN Sj.Remark 2.2.
Whenwij=1, if ∑l=1Nwjl=0, that is sub-RNN Sj cannot switch to any other sub-RNN, we suppose that the uncertain neural networks will always stay in the sub-RNN Si that means wii=1 and wij=0.Throughout this paper, the superscriptT stands for the transpose of a matrix, P>0 means that the matrix P is positive definite, and the symmetric terms in a symmetric matrix are denoted by *, for example,
(2.3)[MO*N]=[MOOTN],G1=diag⁡{g1-g1+,g2-g2+,…,gn-gn+},G2=diag⁡{-g1-+g1+2,-g2-+g2+2,…,-gn-+gn+2},F1=diag⁡{f1-f1+,f2-f2+,…,fn-fn+},F2=diag⁡{-f1-+f1+2,-f2-+f2+2,…,-fn-+fn+2}.


## 3. Asymptotical Stability of Uncertain Periodic Switched Recurrent Neural Networks

Theorem 3.1.
Letd1 and d2 be positive integers such that 0≤d1≤d2. Based on a switching graph Γ, the system (2.2) is asymptotical stable if, when wij=1(Si,Sj∈Ω), there exist the corresponding symmetric matrices Pi=PiT>0, Q1=Q1T>0, Q2=Q2T>0, R=RT>0, Z1=Z1T>0, Z2=Z2T>0, Xij=(Xij)T≥0, Uij=(Uij)T≥0, Hij=diag{h1ij,h2ij,…,hnij}≥0, Oij=diag{o1ij,o2ij,…,onij}≥0, and any appropriate dimensional matrices Nij, Mij, and Tij such that the following LMIs hold:
(3.1)Φij=[Φ11ijΦ12ijΦ13ijΦ14ijΦ15ijΦ16ij*Φ22ijΦ23ijΦ24ij00**Φ33ijΦ34ij00***Φ44ij0-F2Oij****Φ55ijΦ56ij*****Φ66ij]<0,(3.2)Λ1ij=[XijNij*Z1]>0,(3.3)Λ2ij=[UijTij*Z2]>0,(3.4)Λ3ij=[Xij+UijMij*Z1+Z2]>0,
where
(3.5)Φ11ij=AiTPjAi-Pi+Q1+Q2+(d2-d1+1)R+(Ai-E)T(d2Z1j+(d2-d1)Z2j)(Ai-E)+(N1ij)T+N1ij+d2X11ij+(d2-d1)U11ij-G1Hij,Φ12ij=(N2ij)T+T1ij+d2X12ij+(d2-d1)U12ij,Φ13ij=(N3ij)T-M1ij+d2X13ij+(d2-d1)U13ij,Φ14ij=-N1ij+(N4ij)T+M1ij-T1ij+d2X14ij+(d2-d1)U14ij,Φ15ij=AiTPjW1i+(Ai-E)T(d2Z1+(d2-d1)Z2)W1i-G2Hij,Φ16ij=AiTPjW2i+(Ai-E)T(d2Z1+(d2-d1)Z2)W2i,Φ22ij=-Q1-R+(T2ij)T+T2ij+d2X22ij+(d2-d1)U22ij,Φ23ij=-M2ij+(T3ij)T+d2X23ij+(d2-d1)U23ij,Φ24ij=-N2ij+(T4ij)T+M2ij-T2ij+d2X24ij+(d2-d1)U24ij,Φ33ij=-Q2-R-M3ij-(M3ij)T+d2X33ij+(d2-d1)U33ij,Φ34ij=-N3ij-(M4ij)T+M3ij-T3ij+d2X34ij+(d2-d1)U34ij,Φ44ij=-N4ij-(N4ij)T+M4ij+(M4ij)T-T4ij-(T4ij)T+d2X44ij+(d2-d1)U44ij-F1Oij,Φ55ij=W1iTPjW1i+W1iT(d2Z1+(d2-d1)Z2)W1i-Hij,Φ56ij=W1iTPjW2i+W1iT(d2Z1+(d2-d1)Z2)W2i,Φ66ij=W2iTPjW2i+W2iT(d2Z1+(d2-d1)Z2)W2i-Oij.Proof.
Suppose thaty(l)=x(l+1)-x(l); then we have x(k+1)=x(k)+y(k) and x(k)=x(k-d(k))+∑i=k-d(k)k-1y(l).
We consider the following SQLF:(3.6)Vr(k)(k)=V1r(k)(k)+V2r(k)(k)+V3r(k)(k)+V4r(k)(k),V1r(k)(k)=xT(k)Pr(k)x(k),V2r(k)(k)=∑l=k-d1k-1xT(l)Q1r(k)x(l)+∑l=k-d2k-1xT(l)Q2r(k)x(l),(3.7)V3r(k)(k)=∑θ=k-d2k-d1∑l=θk-1xT(l)Rr(k)x(l),(3.8)V4r(k)(k)=∑θ=-d2-1∑l=k+θk-1yT(l)Z1r(k)y(l)+∑θ=-d2-d1-1∑l=k+θk-1yT(l)Z2r(k)y(l).
It is clear that the following equations are true:(3.9)∑l=k-d2k-1yT(l)Z1r(k)y(l)=∑l=k-d2k-d(k)-1yT(l)Z1r(k)y(l)+∑l=k-d(k)k-1yT(l)Z1r(k)y(l),∑l=k-d2k-d1-1yT(l)Z2r(k)y(l)=∑l=k-d2k-d(k)-1yT(l)Z2r(k)y(l)+∑l=k-d(k)k-d1-1yT(l)Z2r(k)y(l).
Firstly, we prove that underwij=1 the SQLF is less than 0. Suppose that r(k)=i and r(k+1)=j, that means the sub-RNN Si switches to the sub-RNN Sj; we obtain
(3.10)ΔV1i(k)=V1j(k+1)-V1i(k)=(Aix(k)+W1ig(x(k))+W2if(x(k-d(k))))T×Pj(Aix(k)+W1ig(x(k))+W2if(x(k-d(k))))-xT(k)Pix(k),ΔV2i(k)=V2j(k+1)-V2i(k)=xT(k)(Q1j+Q2j)x(k)-xT(k-d1)Q1ix(k-d1)-xT(k-d2)Q2ix(k-d2)+∑l=k+1-d1k-1xT(l)(Q1j-Q1i)x(l)+∑l=k+1-d2k-1xT(l)(Q2j-Q2i)x(l),(3.11)ΔV3i(k)=V3j(k+1)-V3i(k)≤(d2-d1+1)xT(k)Rjx(k)-xT(k-d1)Rix(k-d1)-xT(k-d2)Rix(k-d2)-xT(k-d(k))Rix(k-d(k))+∑l=k+1-d1k-1xT(l)(Rj-Ri)x(l)+∑θ=k+1-d2k-d1∑l=θk-1xT(l)(Rj-Ri)x(l),(3.12)ΔV4i(k)=V4j(k+1)-V4i(k)=d2yT(k)Z1jy(k)+(d2-d1)yT(k)Z2jy(k)-∑l=k-d(k)k-1yT(l)Z1iy(l)-∑l=k-d(k)k-d1-1yT(l)Z2iy(l)-∑l=k-d2k-d(k)-1yT(l)(Z1i+Z2i)y(l)+∑θ=-d2-1∑l=k+1+θk-1yT(l)(Z1j-Z1i)y(l)+∑θ=-d2-d1-1∑l=k+1+θk-1yT(l)(Z2j-Z2i)y(l).
In order to strictly guaranteeΔV2i(k)<0,∑l=k+1-d1k-1xT(l)(Q1j-Q1i)x(l) should be less than 0. In the switching graph Γ if there exists sub-RNN Si(Si∈Ω), which satisfied ∑j=1Nwij=0, which means the Si cannot switch to any other sub-RNN, the equation Q1i=Q1j can be grounded, otherwise the switching sequence must be β={Sα1→Sα2→,…,→Sαj→⋯,→SαL→SαL+1}(Sαj∈Ω), and there exist l such that SαL+1=Sαl(1≤l<L). Because the affection of the sub-RNNs  S1,S2…,Sl-1 and Sl on the whole system is before time αl, after αl the β changes to a periodic sequence β′={Sαl→⋯,→SαL→SαL+1}. Suppose that i=αl=αL+1; then in switching sequence β′ the following LMIs all hold:
(3.13)Q1αl-Q1αl+1≥0(Q1i-Q1(l+1)≥0),Q1αl+1-Q1αl+2≥0,⋮Q1αL-Q1αL+1≥0(Q1(L)-Q1i≥0);
then the solution of (3.13) is Q1i=Q1αl=Q1αl+1=⋯=Q1αL. Thus, we suppose that
(3.14)Q1=Q1i=Q1j.
Similar to ∑l=k+1-d1k-1xT(l)(Q1j-Q1i)x(l), together with ∑l=k+1-d2k-1xT(l)(Q2j-Q2i)x(l), ∑l=k+1-d1k-1xT(l)(Rj-Ri)x(l), ∑θ=k+1-d2k-d1∑l=θk-1xT(l)(Rj-Ri)x(l), ∑θ=-d2-1∑l=k+1+θk-1yT(l)(Z1j-Z1i)y(l) and ∑θ=-d2-d1-1∑l=k+1+θk-1yT(l)(Z2j-Z2i)y(l), we suppose that
(3.15)Q2=Q2i=Q2j,R=Ri=Rj,(3.16)Z1=Z1i=Z1j,(3.17)Z2=Z2i=Z2j.
On the other hand, for any appropriately dimensioned matrices Nij, Mij, and Tij the following equations are true:
(3.18)2ξT(k)Nij[x(k)-x(k-d(k))-∑l=k-d(k)k-1y(l)]=0,(3.19)2ξT(k)Mij[x(k-d(k))-x(k-d2)-∑l=k-d2k-d(k)-1y(l)]=0,2ξT(k)Tij[x(k-d1)-x(k-d(k))-∑l=k-d(k)k-d1-1y(l)]=0,
where ξ(k)=[xT(k),xT(k-d1),xT(k-d2),xT(k-d(k))]T.
In addition, for any semipositive definite matrixXij=(Xij)T and Uij=(Uij)T, the following equations hold:
(3.20)d2ξT(k)Xijξ(k)-∑l=k-d2k-d(k)-1ξT(k)Xijξ(k)-∑l=k-d(k)k-1ξT(k)Xijξ(k)=0,(d2-d1)ξT(k)Uijξ(k)-∑l=k-d2k-d(k)-1ξT(k)Uijξ(k)-∑l=k-d(k)k-d1-1ξT(k)Uijξ(k)=0.
From the assumption, we have
(3.21)(gl(x(k))-gl-xl(k))×(gl(x(k))-gl+xl(k))≤0,l=1,2,…,n,(fl(x(k-dk))-fl-xl(k-dk))×(fl(x(k-dk))-fl+xl(k-dk))≤0,l=1,2,…,n.
Similar to the conclusion in [8], for Hij=diag{h1,h2,…,hn}≥0,Oij=diag{o1,o2,…,on}≥0 the following inequalities are also true:
(3.22)-[x(k)g(x(k))]T[G1HijG2Hij⋆Hij][x(k)g(x(k))]≥0,(3.23)-[x(k-d(k))f(x(k-d(k)))]T[F1OijF2Oij⋆Oij][x(k-d(k))f(x(k-d(k)))]≥0.
Then we add the terms on the right side of (3.16)–(3.23) to yield
(3.24)ΔVi(k)=ΔV1i(k)+ΔV2i(k)+ΔV3i(k)+ΔV4i(k)≤η1T(k)Φijη1(k)-∑l=k-d(k)k-1η2T(k,l)Λ1ijη2(k,l)-∑l=k-d(k)k-d1-1η2T(k,l)Λ2ijη2(k,l)-∑l=k-d2k-d(k)-1η2T(k,l)Λ3ijη2(k,l),
where η1T(k)=[ξT(k),gT(x(k)),fT(x(k-d(k)))]T,η2T(k,l)=[ξT(k),yT(l)]T.
Andξ(k) is defined in (3.18). Φij, Λ1ij, Λ2ij, and Λ3ij are defined in (3.1)–(3.4). Therefore, when the corresponding LMIs satisfy Φij<0, Λ1ij≥0, Λ2ij≥0, and Λ3ij≥0,Si,Sj∈Ω, ΔVi(k)≤0.
Secondly, based on the switching graphΓ, when wij=1(Si,Sj∈Ω), all corresponding ΔVi(k) are less than 0 that means the system (2.2) is asymptotical stable. This completes the proof of Theorem 3.1.Remark 3.2.
Using the method in [8], it is easily to know that the system is the globally exponentially stable.Remark 3.3.
InV3i(k) of the literature [8], -∑l=k-d2k-d(k)xT(l)Rix(l)≤-xT(k-d(k))Rix(k-d(k)) may lead to considerable conservativeness. Then, it is improved as -∑l=k-d2k-d(k)xT(l)Rix(l)≤-xT(k-d2)Rix(k-d2)-xT(k-d(k))Rix(k-d(k))-xT(k-d1)Rix(k-d1). Please see Table 3.
Combined with Theorem3.1, we consider the common Lyapunov function approach; then we have the following.Corollary 3.4.
Letd1 and d2 be positive integers such that 0≤d1≤d2. The system (2.2) is asymptotical stable if there exist symmetric matrices P=PT>0, Q1=Q1T>0, Q2=Q2T>0, R=RT>0, Z1=Z1T>0, Z2=Z2T>0, X=XT≥0, U=UT≥0, H=diag{h1,h2,…,hn}≥0, O=diag{o1,o2,…,on}≥0, and any appropriate dimensional matrices N, M, and T such that the following LMIs hold:
(3.25)Φi=[Φ11Φ12Φ13Φ14Φ15Φ16*Φ22Φ23Φ2400**Φ33Φ3400***Φ440-F2O****Φ55Φ56*****Φ66]<0,Si∈Ω,Λ1=[XN*Z1]>0,Λ2=[UT*Z2]>0,Λ3=[X+UM*Z1+Z2]>0,
where
(3.26)Φ11=AiTPAi-P+Q1+Q2+(d2-d1+1)R+(Ai-E)T(d2Z1+(d2-d1)Z2)(Ai-E)+N1T+N1+d2X11+(d2-d1)U11-G1H,Φ12=N2T+T1+d2X12+(d2-d1)U12,Φ13=N3T-M1+d2X13+(d2-d1)U13,Φ14=-N1+N4T+M1-T1+d2X14+(d2-d1)U14,Φ15=AiTPW1i+(Ai-E)T(d2Z1+(d2-d1)Z2)W1i-G2H,Φ16=AiTPW2i+(Ai-E)T(d2Z1+(d2-d1)Z2)W2i,Φ22=-Q1-R+T2T+T2+d2X22+(d2-d1)U22,Φ23=-M2+T3T+d2X23+(d2-d1)U23,Φ24=-N2+T4T+M2-T2+d2X24+(d2-d1)U24,Φ33=-Q2-R-M3-M3T+d2X33+(d2-d1)U33,Φ34=-N3-M4T+M3-T3+d2X34+(d2-d1)U34,Φ44=-N4-N4T+M4+M4T-T4-T4T-F1O+d2X44+(d2-d1)U44-R,Φ55=W1iTPW1i+W1iT(d2Z1+(d2-d1)Z2)W1i-H,Φ56=W1iTPW2i+W1iT(d2Z1+(d2-d1)Z2)W2i,Φ66=W2iTPW2i+W2iT(d2Z1+(d2-d1)Z2)W2i-O.


## 4. Examples

Example 4.1.
Consider the discrete-time recurrent neural network (2.2) with
(4.1)A1=[0.70000.20000.5],W11=[0.1-0.20.10.3-0.100-0.1-0.4],W21=[-0.30.1-0.20.2-0.10.10-0.020.07],A2=[0.10000.20000.4],W12=[0.4-0.200.2-0.20-0.3-0.1-0.5],W22=[-0.30.1-0.20-0.0300.5-0.20.5],A3=[0.10000.40000.4],W13=[0.200.30.04-0.30-0.1-0.6-0.1],W23=[-0.080.1-0.20.2-0.4-0.050-0.2-0.1],Γ=[010001100],g1(x)=tanh(0.4x),g2(x)=tanh(0.2x),g3(x)=tanh(-0.8x),f1(x)=tanh(-0.6x),f2(x)=tanh(0.4x),f3(x)=tanh(0.2x).
Then we have
(4.2)G1=F1=[000000000],G2=[0.20000.1000-0.4],F2=[0.3000-0.2000-0.1].
Employing the LMIs in Theorem3.1 yields upper bounds on d2 that guarantee the stability of system (1.1) for various lower bounds d1, which are listed in Table 1. When d1=1 and d2=3, it can be seen from Figure 1 that all the state solutions corresponding to the 10 random initial points are convergent asymptotically to the unique equilibrium x*={0,0,0}, and according to Theorem 3.1, LMIs (3.1)–(3.4) are solvable in Matlab 7.0.1.Table 1
Allowable upper bound ofd2 with given d1.
d10246820Theorem3.191113151729Figure 1
Global convergence of statesx1, x2, and x3 in Example 4.1, when d1=1 and d2=3.Example 4.2.
Consider the discrete-time recurrent neural network (2.2) with
(4.3)A=[0.10000.20000.3],W1=[0.5-0.20.20.1-0.10-0.4-0.1-0.3],W2=[-0.20.10.20-0.30.10-0.30.03],g1(x)=f1(x)=tanh(0.6x),g2(x)=f2(x)=tanh(0.4x),g3(x)=f3(x)=tanh(0.2x).
Then we have
(4.4)G1=F1=[000000000],G2=F2=[0.30000.20000.1].
Employing the LMIs in [8] and those in Corollary 3.4 yields upper bounds on d2 that guarantee the stability of system for various lower bounds d1, which are listed in Table 2. It is clear that the obtained upper bounds of this paper are better than those of [8]. It can be seen from Figure 2 that, when d1=1 and d2=3, all the state solutions corresponding to the 10 random initial points are convergent asymptotically to the unique equilibrium x*={0,0,0}.Table 2
Allowable upper bound ofd2 with given d1.
d1024681020Reference [8]20222426283040Corollary3.438404244464858Table 3
Allowable upper bound ofd2 with given d1.
d1024610Reference [8]1113151721Corollary3.41214161822Figure 2
Global convergence of statesx1, x2, and x3 in Example 4.2, when d1=1 and d2=3.Remark 4.3.
Employing the LMIs in [8] and those in Corollary 3.4 yields upper bounds on d2 that guarantee the stability of system (1.1) of the Example  1 of [8] for various lower bounds d1, which are listed in Table 3.


## 5. Conclusions

This paper was dedicated to the delay-dependent stability of uncertain periodic switched recurrent neural networks with time-varying delay. A less conservative LMI-based globally stability criterion is obtained with the switched quadratic Lyapunov functional approach and free-weighting matrix approach for periodic uncertain discrete-time recurrent neural networks with a time-varying delay. One example illustrates the exactness of the proposed criterion. Another example demonstrates that the proposed method is an improvement over the existing one.



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*Source: 325371-2011-12-21.xml*