# New Improved Exponential Stability Criteria for Discrete-Time Neural Networks with Time-Varying

Authors: Zixin Liu; Shu Lv; Shouming Zhong; Mao Ye
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

The robust stability of uncertain discrete-time recurrent neural networks with time-varying delay is investigated. By decomposing some connection weight matrices, new Lyapunov-Krasovskii functionals are constructed, and serial new improved stability criteria are derived. These criteria are formulated in the forms of linear matrix inequalities (LMIs). Compared with some previous results, the new results are less conservative. Three numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed method.


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## Body

## 1. Introductionn

In recent years, recurrent neural networks (see [1–7]), such as Hopfield neural networks, cellular neural networks, and other networks have been widely investigated and successfully applied in all kinds of science areas such as pattern recognition, image processing, and fixed-point computation. However, because of the finite switching speed of neurons and amplifiers, time delay is unavoidable in nature and technology. It can make important effects on the stability of dynamic systems. Thus, the studies on stability are of great significance. There has been a growing research interest on the stability analysis problems for delayed neural networks, and many excellent papers and monographs have been available. On the other hand, during the design of neural network and its hardware implementation, the convergence of a neural network may often be destroyed by its unavoidable uncertainty due to the existence of modeling error, the deviation of vital data, and so on. Therefore, the studies on robust convergence of delayed neural network have been a hot research direction. Up to now, many sufficient conditions, either delay-dependent or delay-independent, have been proposed to guarantee the global robust asymptotic or exponential stability for different class of delayed neural networks (see [8–13]).It is worth pointing out that most neural networks have been assumed to be in continuous time, but few in discrete time. In practice, the discrete-time neural networks are more applicable to problems that are inherently temporal in nature or related to biological realities. And they can ideally keep the dynamic characteristics, functional similarity, and even the physical or biological reality of the continuous-time networks under mild restriction. Thus, the stability analysis problems for discrete-time neural networks have received more and more interest, and some stability criteria have been proposed in literature (see [14–25]). In [14], Liu et al. researched a class of discrete-time RNNs with time-varying delay, and proposed a delay-dependent condition guaranteeing the global exponential stability. By using a similar technique to that in [21], the result obtained in [14] has been improved by Song and Wang in [15]. The results in [15] are further improved by Zhang et al.  in [16] by introducing some useful terms. In [17], Yu et al. proposed a new less conservative result than that obtained in [16] via constructing a new augment Lyapunov-Krasovskii functional.In this paper, the connection weight matrixC is decomposed, and some new Lyapunov-Krasovskii functionals are constructed. Combined with linear matrix inequality (LMI) technique, serial new improved stability criteria are derived. Numerical examples show that these new criteria are less conservative than those obtained in [14–17].Notation 1.
The notations are used in our paper except where otherwise specified.∥·∥ denotes a vector or a matrix norm; ℝ,ℝn are real and n-dimension real number sets, respectively; ℕ+ is nonnegative integer set. I is identity matrix; * represents the elements below the main diagonal of a symmetric block matrix; Real matrix P>0<(0) denotes that  P is a positive definite (negative definite) matrix; ℕ[a,b]={a,a+1,…,b}; λmin(λmax) denotes the minimum and maximum eigenvalue of a real matrix.


## 2. Preliminaries

Consider a discrete-time recurrent neural network with time-varying delays [17] described by
(2.1)Σ:x(k+1)=C(k)x(k)+A(k)f(x(k))+B(k)f(x(k-τ(k)))+J,k=1,2,…,
where x(k)=[x1(k),x2(k),…,xn(k)]T∈ℝn denotes the neural state vector; f(x(k))=[f1(x1(k)),f2(x2(k)),…,fn(xn(k))]T, f(x(k-τ(k)))=[f1(x1(k-τ(k))),f2(x2(k-τ(k))),…,fn(xn(k-τ(k)))]T are the neuron activation functions; J=[J1,J2,…,Jn]T is the external input vector; Positive integer τ(k) represents the transmission delay that satisfies 0<τm≤τ(k)≤τM, where τm,τM are known positive integers representing the lower and upper bounds of the delay. C(k)=C+ΔC(k), A(k)=A+ΔA(k), B(k)=B+ΔB(k). C=diag(c1,c2,…,cn) with |ci|<1 describes 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; C,A,B∈ℝn×n represent the weighting matrices; ΔC(k),ΔA(k),ΔB(k) denote the time-varying structured uncertainties which are of the following form:
(2.2)[ΔC(k),ΔA(k),ΔB(k)]=KF(k)[Ec,Ea,Eb],
where K,Ec,Ea,Eb are known real constant matrices with appropriate dimensions, F(k) is unknown time-varying matrix function satisfying FT(k)F(k)≤I,forallk∈ℕ+.The nominalΣ0 of Σ can be defined as
(2.3)Σ0:x(k+1)=Cx(k)+Af(x(k))+Bf(x(k-τ(k)))+J,k=1,2,…,To obtain our main results, we need to introduce the following assumption, definition and lemmas.Assumption 1.
For anyx,y∈ℝ, x≠y,
(2.4)σi-≤fi(x)-fi(y)x-y≤σi+,i=1,2,…,n,
where σi-,σi+ are known constant scalars.
As pointed out in [16] under Assumption 1, system (2.3) has equilibrium points. Assume that  x*=[x1*,x2*,…,xn*]T is an equilibrium point of (2.3) and let yi(k)=xi(k)-xi*, gi(yi(k))=fi(yi(k)+xi*)-fi(xi*). Then, system (2.3), can be transformed into the following form:
(2.5)y(k+1)=Cy(k)+Ag(y(k))+Bg(y(k-τ(k))),k=1,2,…,
where y(k)=[y1(k),y2(k),…,yn(k)]T, g(y(k))=[g1(y1(k)),g2(y2(k)),…,gn(yn(k))]T, g(y(k-τ(k)))=[g1(y1(k-τ(k))),g2(y2(k-τ(k))),…,gn(yn(k-τ(k)))]T. From Assumption 1, for any x,y∈ℝ, x≠y, functions gi(·) satisfy σi-≤(gi(x)-gi(y))/(x-y)≤σi+,i=1,2,…,n, and gi(0)=0.Remark 2.1.
Assumption1 is widely used for dealing with the stability problem for neural networks. As pointed out in [13, 14, 16, 17, 26, 27], constants σi-,σi+(i=1,2,…,n) can be positive, negative, and zero. Thus, this assumption is less restrictive than traditional Lipschitz condition.Definition 2.2.
The delayed discrete-time recurrent neural network in (2.5) is said to be globally exponentially stable if there exist two positive scalars  α>0 and 0<β<1 such that
(2.6)∥y(k)∥≤α·βksups∈ℕ[-τM,0]∥y(s)∥,∀k≥1.Lemma 2.3 (Tchebychev Inequality [28]).
For any given vectorsvi∈ℝn,i=1,2,…,n, the following inequality holds:
(2.7)[∑i=1nvi]T[∑i=1nvi]≤n∑i=1nviTvi.Lemma 2.4 (see [29]).
For given matricesQ=QT,H,E and R=RT>0 of appropriate dimensions, then
(2.8)Q+HFE+ETFTHT<0,
for all F satisfying FTF≤R, if and only if there is  an ε>0, such that
(2.9)Q+ε-1HHT+εETRE<0.Lemma 2.5 (see [16]).
If Assumption1 holds, then for any positive-definite diagonal matrix D=diag(d1,d2,…,dn)>0, the following inequality holds:
(2.10)g(y(k))TDg(y(k))-yT(k)D(∏1+∏2)g(y(k))+yT(k)∏1D∏2y(k)≤0,k∈ℕ+,
where ∏1=diag(σ1-,σ2-,…,σn-), ∏2=diag(σ1+,σ2+,…,σn+).Lemma 2.6 (see [30]).
Given constant symmetric matricesΣ1,Σ2,Σ3 where Σ1T=Σ1 and 0<Σ2=Σ2T, then Σ1+Σ3TΣ2-1Σ3<0 if and only if
(2.11)(Σ1Σ3TΣ3-Σ2)<0,or,(-Σ2Σ3Σ3TΣ1)<0.Lemma 2.7 (see [13]).
LetN and E be real constant matrices with appropriate dimensions, matrix F(k) satisfying FT(k)F(k)≤I, then, for any ϵ>0, EF(k)N+NTFT(k)ET≤ϵ-1EET+ϵNTN,k∈ℕ+.


## 3. Main Results

To obtain our main results, we decompose the connection weight matrixC as follows:
(3.1)C=C1+C2.
Then, we can get the following stability results.Theorem 3.1.
For any given positive scalars0<τm<τM, then, under Assumption 1, system (2.5) without uncertainty is globally exponentially stable for any time-varying delay τ(k) satisfying τm≤τ(k)≤τM, if there exist positive-definite matrices P1,Q1,Q2,Q3, positive-definite diagonal matrices D1,D2,Q4,Q5,Q6, and arbitrary matrices Pi,Hi,i=2,3,…,21 with appropriate dimensions, such that the following LMI holds:
(3.2)Ξ≜(Ξ11Ξ12Ξ13Ξ14Ξ15Ξ16Ξ17Ξ18Ξ19Ξ1,10*Ξ22Ξ23Ξ24Ξ25Ξ26Ξ27Ξ28Ξ29Ξ2,10**Ξ33Ξ34Ξ35Ξ36Ξ37Ξ38Ξ39Ξ3,10***Ξ44Ξ45Ξ46Ξ47Ξ48Ξ49Ξ4,10****Ξ55Ξ56Ξ57Ξ58Ξ59Ξ5,10*****Ξ66Ξ67Ξ68Ξ69Ξ6,10******Ξ77Ξ78Ξ79Ξ7,10*******Ξ88Ξ89Ξ8,10********Ξ99Ξ9,10*********Ξ10,10)<0,
where
(3.3)Ξ11=2C1TP1C1-C1TP12C2-C2TP12TC1-2P1+H12C2+C2TH12T+Q2+Q3+(τM-τm+1)Q1+(1+τm)Q4+(1+τM)Q5+(τM-τm)Q6-2∏1D1∏2,Ξ12=C2T(H13-P13)T,Ξ13=C1T(P1+P12)-H12+C2T(H14-P14-P12)T+C1TP1T,Ξ14=C1TP2-H2+C2T(H15-P15)T,Ξ15=-C1TP2+H2+C2T(H16-P16)T,Ξ16=-C1TP12A+H12A+C2T(H17-P17)T+D1(∏1+∏2),Ξ17=-C1TP12B+H12B+C2T(H18-P18)T,Ξ18=-C1TP2+H2+C2T(H19-P19)T,Ξ19=C1TP2-H2+C2T(H20-P20)T,Ξ1,10=C1TP2-H2+C2T(H21-P21)T,Ξ22=-Q1-2∏1D2∏2,Ξ23=P13-H13,Ξ24=P3-H3,Ξ25=-P3+H3,Ξ26=-P13A+H13A,Ξ27=-P13B+H13B+D2(∏1+∏2),Ξ28=-P3+H3,Ξ29=P3-H3,Ξ2,10=P3-H3,Ξ33=P12+P14-H14+P12T+P14T-H14T+2P1,Ξ34=P2+P4-H4+P15T-H15TΞ35=H4-P2-P4+P16T-H16T,Ξ36=(H14-P12-P14)A+P17T-H17T,Ξ37=(H14-P12-P14)B+P18T-H18T,Ξ38=H4-P2-P4+P19T-H19T,Ξ39=P4+P2-H4+P20T-H20T,Ξ3,10=P4+P2-H4+P21T-H21T,Ξ44=P5-H5+P5T-H5T-Q3,Ξ45=H5-P5+P6T-H6T,Ξ46=H15A-P15A+P7T-H7T,Ξ47=H15B-P15B+P8T-H8T,Ξ48=H5-P5+P9T-H9T,Ξ49=P5-H5+P10T-H10T,Ξ49=P5-H5+P11T-H11T,Ξ55=H6-P6+H6T-P6T-Q2,Ξ56=H16A-P16A+H7T-P7T,Ξ57=H16B-P16B+H8T-P8T,Ξ58=H6-P6+H9T-P9T,Ξ59=P6-H6+H10T-P10T,Ξ5,10=P6-H6+H11T-P11T,Ξ66=H17A-P17A+ATH17T-ATP17T-D1-D1T,Ξ67=H17B-P17B+ATH18T-ATP18T,Ξ68=H7-P7+ATH19T-ATP19T,Ξ69=P7-H7+ATH20T-ATP20T,Ξ6,10=P7-H7+ATH21T-ATP21T,Ξ77=H18B-P18B+BTH18T-BTP18T-D2-D2T,Ξ78=H8-P8+BTH19T-BTP19T,Ξ79=P8-H8+BTH20T-BTP20T,Ξ7,10=P8-H8+BTH21T-BTP21T,Ξ88=H9-P9+H9T-P9T-(1+τM)-1Q5,Ξ89=P9-H9+H10T-P10T,Ξ8,10=P9-H9+H11T-P11T,Ξ9,9=P10-H10+P10T-H10T-(1+τm)-1Q4,Ξ9,10=P10-H10+H11T-P11T,Ξ10,10=P11-H11+P11T-H11T-(τM-τm)-1Q6.Proof.
Construct a new augmented Lyapunov-Krasovskii functional candidate as follows:(3.4)V(k)=V1(k)+V2(k)+V3(k)+V4(k)+V5(k)+V6(k),
where
(3.5)V1(k)=2YT(k)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)10n×10nY(k),YT(k)=[yT(k),yT(k-τ(k)),ηT(k),yT(k-τM),yT(k-τm),gT(y(k)),gT(y(k-τM)),  ∑i=k-τMkyT(i),∑i=k-τmkyT(i),∑i=k-τM+1k-τmyT(i)]T,η(k)=y(k+1)-C1y(k); 0 is zero matrix with appropriate dimensions:
(3.6)V2(k)=∑i=k-τ(k)k-1yT(i)Q1y(i),V3(k)=∑i=k-τmk-1yT(i)Q2y(i)+∑i=k-τMk-1yT(i)Q3y(i),V4(k)=∑j=k-τmk-1∑i=jk-1yT(i)Q4y(i)+∑j=k-τMk-1∑i=jk-1yT(i)Q5y(i),V5(k)=∑j=k-τM+1k-τm∑i=jk-1yT(i)Q1y(i),V6(k)=∑j=k-τM+1k-τm∑i=jk-1yT(i)Q6y(i).
Set ỸT(k+1)=[yT(k+1),yT(k-τ(k)),ηT(k),yT(k-τM),yT(k-τm),gT(y(k)),gT(y(k-τM)),∑i=k-τMkyT(i), ∑i=k-τmkyT(i), ∑i=k-τM+1k-τmyT(i)]T=[yT(k)C1T+ηT(k),ηT(k),yT(k-τM), yT(k-τm),  gT(y(k)),gT(y(k-τM)),∑i=k-τMkyT(i), ∑i=k-τmkyT(i),∑i=k-τM+1k-τmyT(i)]T. Define ΔV(k)=V(k+1)-V(k). Then along the solution of system (2.5) we have
(3.7)ΔV1(k)=2YT(k+1)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)Y(k+1)-2YT(k)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)Y(k)=2ỸT(k+1)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)Ỹ(k+1)-2YT(k)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)Y(k)≜2I1-2I2,(3.8)I1=YT(k)(C1T0000000000I00000000I0I0000000000I0000000000I0000000000I0000000000I0000000000I0000000000I0000000000I)(P1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)Ỹ(k+1)=YT(k)(C1T0000000000I00000000I0I0000000000I0000000000I0000000000I0000000000I0000000000I0000000000I0000000000I)(P1P2P120P3P130P4P140P5P150P6P160P7P170P8P180P9P190P10P200P11P21)(y(k+1)00).
On the other hand, since η(k)-C2y(k)-Ag(y(k))-Bg(y(k-τ(k)))=0, ∑i=k-τmky(i)-∑i=k-τMky(i)+∑i=k+1-τMk-τmy(i)-yT(k-τm)+yT(k-τM)=0, we have
(3.9)(y(k+1)00)=(C1y(k)+η(k)∑i=k-τmky(i)-∑i=k-τMky(i)+∑i=k+1-τMk-τmy(i)-yT(k-τm)+yT(k-τM)η(k)-C2y(k)-Ag(y(k))-Bg(y(k-τ(k))))(3.10)=(C10I0000000000I-I00-III-C20I00-A-B000)Y(k),(3.11)I2=YT(k)(P1H2H120H3H130H4H140H5H150H6H160H7H170H8H180H9H190H10H200H11H21)(y(k)00)=YT(k)(P1H2H120H3H130H4H140H5H150H6H160H7H170H8H180H9H190H10H200H11H21)(I000000000000I-I00-III-C20I00-A-B000)Y(k).(3.12)ΔV2(k)≤yT(k)Q1y(k)-yT(k-τ(k))Q1y(k-τ(k))+∑i=k+1-τMk-τmyT(i)Q1y(i),(3.13)ΔV3(k)=yT(k)(Q2+Q3)y(k)-yT(k-τm)Q2y(k-τm)-yT(k-τM)Q3y(k-τM).
From Lemma 2.3 we can obtain
(3.14)ΔV4(k)=∑j=k+1-τmk∑i=jkyT(i)Q4y(i)-∑j=k-τmk-1‍∑i=jk-1yT(i)Q4y(i)+∑j=k+1-τMk‍∑i=jkyT(i)Q5y(i)-∑j=k-τMk-1‍∑i=jk-1yT(i)Q5y(i)=∑j=k-τmk-1‍∑i=j+1kyT(i)Q4y(i)-∑j=k-τmk-1‍∑i=jk-1yT(i)Q4y(i)+∑j=k-τMk-1‍∑i=j+1kyT(i)Q5y(i)-∑j=k-τMk-1‍∑i=jk-1yT(i)Q5y(i)=∑j=k-τmk-1(yT(k)Q4y(k)-yT(j)Q4y(j))+∑j=k-τMk-1(yT(k)Q5y(k)-yT(j)Q5y(j))≤(1+τm)yT(k)Q4y(k)-∑j=k-τmkyT(j)Q4y(j)+(1+τM)yT(k)Q5y(k)-∑j=k-τMkyT(j)Q5y(j)≤(1+τm)yT(k)Q4y(k)-11+τm[∑j=k-τmky(j)]TQ4[∑j=k-τmky(j)]+(1+τM)yT(k)Q5y(k)-11+τM[∑j=k-τMky(j)]TQ5[∑j=k-τMky(j)],(3.15)ΔV5(k)=∑j=k+2-τMk+1-τm‍∑i=jkyT(i)Q1y(i)-∑j=k+1-τMk-τm‍∑i=jk-1yT(i)Q1y(i)=∑j=k+1-τMk-τm‍∑i=j+1kyT(i)Q1y(i)-∑j=k+1-τMk-τm‍∑i=jk-1yT(i)Q1y(i)=∑j=k+1-τMk-τm(yT(k)Q4y(k)-yT(j)Q1y(j))=(τM-τm)yT(k)Q1y(k)-∑j=k+1-τMkyT(j)Q1y(j).
Similarly,
(3.16)ΔV6(k)=(τM-τm)yT(k)Q6y(k)-∑j=k+1-τMk-τmyT(j)Q6y(j)≤(τM-τm)yT(k)Q6y(k)-1τM-τm[∑j=k+1-τMk-τmy(j)]TQ6[∑j=k+1-τMk-τmy(j)].
From Lemma 2.5, for any positive diagonal matrix D1,D2, it follows that
(3.17)2yT(k-τ(k))D2(∏1+∏2)g(y(k-τ(k)))-2gT(y(k-τ(k)))D2g(y(k-τ(k)))-2yT(k-τ(k))∏1D2∏2y(k-τ(k))≥02yT(k)D1(∏1+∏2)g(y(k))-2gT(y(k))D1g(y(k))-2yT(k)∏1D1∏2y(k)≥0.
Combining (3.7)–(3.17), we get
(3.18)ΔV(k)≤YT(k)ΞY(k).
If the LMI (3.2) holds, it follows that there exists a sufficient small scalar ε>0 such that
(3.19)ΔV(k)≤-ε∥y(k)∥2.
On the other hand, it can easily to get that
(3.20)V(k)≤2λm(P1)∥y(k)∥2+λmax(Q1)∑i=k-τ(k)k-1∥y(i)∥2+λmax(Q2)∑i=k-τmk-1∥y(i)∥2+λmax(Q3)∑i=k-τMk-1∥y(i)∥2+λmax(Q4)∑j=k-τmk∑i=jk-1∥y(i)∥2+λmax(Q5)∑j=k-τMk∑i=jk-1∥y(i)∥2+λmax(Q1)∑j=k-τMk-τm∑i=jk-1∥y(i)∥2+λmax(Q6)∑j=k+1-τMk-τm∑i=jk-1∥y(i)∥2≤2λmax(P)∥y(k)∥2+λ∑i=k-τMk-1∥y(i)∥2,
where λ=λmax(Q1)+λmax(Q2)+λmax(Q3)+(1+τm)λmax(Q4)+(1+τM)λmax(Q5)+(1+τM-τm)(λmax(Q1)+λmax(Q6)). Choose a scalar θ>1 such that -εθ+2(θ-1)λmax(P1)+(θ-1)λ·τMθτM=0. Then by (3.19) and (3.20), we get
(3.21)θk+1V(k+1)-θkV(k)=θk+1ΔV(k)+θk(θ-1)V(k)≤ε1θk∥y(k)∥2+ε2θk∑i=k-τMk-1∥y(i)∥2,
where ε1=-εθ+2λmax(P)(θ-1), ε2=λ(θ-1). Therefore, for arbitrary positive integer N≥τM+1, summing up both sides of (3.21) from 0 to N-1, we can obtain
(3.22)θNV(N)-V(0)≤ε1∑k=0N-1θk∥y(k)∥2+ε2∑k=0N-1∑i=k-τMk-1θk∥y(i)∥2≤ε2τM(τM+1)θτMsupi∈ℕ[-τM,0]∥y(i)∥2+(ε1+ε2τMθτM)∑k=0N-1θk∥y(k)∥2.
Noting that
(3.23)V(N)≥λmin(P1)∥y(N)∥2,V(0)≤(λτM+2λmax(P1))supi∈ℕ[-τM,0]∥y(i)∥2.
It follows that ∥y(N)∥≤α·βNsupi∈ℕ[-τM,0]∥y(i)∥, where β=(θ)-1, α=(λτM+2λmax(P)+ε2τM(τM+1)θτM)/λmin(P). By Definition 2.2, system (2.5) is globally exponentially stable, which completes the proof of Theorem 3.1.Remark 3.2.
By constructing the new augmented Lyapunov functional, free-weighting matricesPi,Hi,i=2,3,…,21 are introduced so as to reduce the conservatism of the delay-dependent result. Moreover, the decomposition of matrix C=C1+C2 makes the conservatism of the stability criterion reduce further, since the elements of matrices C1,C2 are not restricted to (-1,1) any more.Remark 3.3.
Since Theorem3.1 holds for arbitrary matrices C1,C2 satisfying C1+C2=C, then, when C1=0 or C2=0, respectively, we can easily obtains the following simplified useful corollaries.Corollary 3.4.
For any given positive scalars0<τm<τM, then, under Assumption 1, system (2.5) is globally exponentially stable for any time-varying delay τ(k) satisfying τm≤τ(k)≤τM, if there exist positive-definite matrices P1,Q1,Q2,Q3, positive-definite diagonal matrices D1,D2,Q4,Q5,Q6, and arbitrary matrices Pi,Hi,i=2,3,…,21 with appropriate dimensions, such that the following LMI holds:
(3.24)Ξ̃≜(Ξ̃11Ξ̃12Ξ̃13Ξ̃14Ξ̃15Ξ̃16Ξ̃17Ξ̃18Ξ̃19Ξ̃1,10*Ξ22Ξ23Ξ24Ξ25Ξ26Ξ27Ξ28Ξ29Ξ2,10**Ξ33Ξ34Ξ35Ξ36Ξ37Ξ38Ξ39Ξ3,10***Ξ44Ξ45Ξ46Ξ47Ξ48Ξ49Ξ4,10****Ξ55Ξ56Ξ57Ξ58Ξ59Ξ5,10*****Ξ66Ξ67Ξ68Ξ69Ξ6,10******Ξ77Ξ78Ξ79Ξ7,10*******Ξ88Ξ89Ξ8,10********Ξ99Ξ9,10*********Ξ10,10)<0,
where
(3.25)Ξ̃11=-2P1+H12C+CTH12T+Q2+Q3+(τM-τm+1)Q1+(1+τm)Q4+(1+τM)Q5+(τM-τm)Q6-2∏1D1∏2,Ξ̃12=CT(H13-P13)T,Ξ̃13=CT(H14-P14-P12)T-H12,Ξ̃14=CT(H15-P15)T-H2,Ξ̃15=H2+CT(H16-P16)T,Ξ̃16=H12A+CT(H17-P17)T+D1(∏1+∏2),Ξ̃17=H12B+CT(H18-P18)T,Ξ̃18=H2+CT(H19-P19)T,Ξ̃19=CT(H20-P20)T-H2,Ξ̃1,10=CT(H21-P21)T-H2.Corollary 3.5.
For any given positive scalars0<τm<τM, then, under Assumption 1, system (2.5) is globally exponentially stable for any time-varying delay τ(k) satisfying τm≤τ(k)≤τM, if there exist positive-definite matrices P1,Q1,Q2,Q3, positive-definite diagonal matrices D1,D2,Q4,Q5,Q6, and arbitrary matrices Pi,Hi,i=2,3,…,21 with appropriate dimensions, such that the following LMI holds:
(3.26)Ξ̂≜(Ξ̂110Ξ̂13Ξ̂14Ξ̂15Ξ̂16Ξ̂17Ξ̂18Ξ̂19Ξ̂1,10*Ξ22Ξ23Ξ24Ξ25Ξ26Ξ27Ξ28Ξ29Ξ2,10**Ξ33Ξ34Ξ35Ξ36Ξ37Ξ38Ξ39Ξ3,10***Ξ44Ξ45Ξ46Ξ47Ξ48Ξ49Ξ4,10****Ξ55Ξ56Ξ57Ξ58Ξ59Ξ5,10*****Ξ66Ξ67Ξ68Ξ69Ξ6,10******Ξ77Ξ78Ξ79Ξ7,10*******Ξ88Ξ89Ξ8,10********Ξ99Ξ9,10*********Ξ10,10)<0,
where
(3.27)Ξ̂11=2CTP1C-2P1+Q2+Q3+(τM-τm+1)Q1+(1+τm)Q4+(1+τM)Q5+(τM-τm)Q6-2∏1D1∏2,Ξ̂13=CT(P1+P12)-H12+CTP1T,Ξ̂14=CTP2-H2,Ξ̂15=-CTP2+H2,Ξ̂16=-CTP12A+H12A+D1(∏1+∏2),Ξ̂17=-CTP12B+H12B,Ξ̂18=-CTP2+H2,Ξ̂19=CTP2-H2,Ξ̂1,10=CTP2-H2.Remark 3.6.
It is worth pointing out that Theorem3.1 and Corollary 3.4 can be easily extended to robust exponential stability conditions. As for the stability of system (2.1), according to Lemma 2.4, we can obtain the following robust stability results.Theorem 3.7.
For any given positive scalars0<τm<τM, then, under Assumption 1, system (2.1) is robustly, globally, exponentially stable for any time-varying delay τ(k) satisfying τm≤τ(k)≤τM, if there exist positive-definite matrices P1,Q1,Q2,Q3, positive-definite diagonal matrices D1,D2,Q4,Q5,Q6, arbitrary matrices Pi,Hi,i=2,3,…,21 with appropriate dimensions, and a positive scalar ϵ such that the following LMI holds:
(3.28)Ξ′≜(Ξξ1ϵξ2T-ϵI0*-ϵI)<0,
where ξ1T=[KT(H12-C1TP12)T,KT(H13-P13)T,KT(H14-P12-P14)T,KT(H15-P15)T,KT(H16 -P16)T,KT(H17-P17)T,KT(H18-P18)T,KT(H19-P19)T,KT(H20-P20)T,KT(H21-P21)T], ξ2=[Ec,0,0,0,0,Ea,Eb,0,0,0].Proof.
ReplacingA,B,C2 in inequality (3.2) with A+KF(t)Ea, B+KF(t)Eb and C2+KF(t)Ec respectively, inequality (3.2) for system (2.1) is equivalent to Ξ+ξ1F(t)ξ2+ξ2TFT(t)ξ1T<0. From Lemmas 2.6 and 2.7, we can easily obtain this result, this complete the proof. Similarly, we have.Theorem 3.8.
For any given positive scalars0<τm<τM, then, under Assumption 1, system (2.1) is robustly, globally, exponentially stable for any time-varying delay τ(k) satisfying τm≤τ(k)≤τM, if there exist positive-definite matrices P1,Q1,Q2,Q3, positive-definite diagonal matrices D1,D2,Q4,Q5,Q6, arbitrary matrices Pi,Hi,i=2,3,…,21 with appropriate dimensions, and a positive scalar ϵ such that the following LMI holds:
(3.29)Ξ̃′≜(Ξ̃ξ1ϵξ2T*-ϵI0**-ϵI)<0.Theorem 3.9.
For any given positive scalars0<τm<τM, then, under Assumption 1, system (2.1) is robustly, globally, exponentially stable for any time-varying delay τ(k) satisfying τm≤τ(k)≤τM, if there exist positive-definite matrices P1,Q1,Q2,Q3, positive-definite diagonal matrices D1,D2,Q4,Q5,Q6, and arbitrary matrices Pi,Hi,P¯j,H¯j,i=2,3,…,21,j=1,2,…,6 with appropriate dimensions, such that the following LMI holds:
(3.30)Ξ′′≜(Ξ11′′Ξ12Ξ13Ξ14Ξ15Ξ16Ξ17Ξ18Ξ19Ξ1,10Ξ1,11′′*Ξ22Ξ23Ξ24Ξ25Ξ26Ξ27Ξ28Ξ29Ξ2,10Ξ2,11′′**Ξ33Ξ34Ξ35Ξ36Ξ37Ξ38Ξ39Ξ3,10Ξ3,11′′***Ξ44Ξ45Ξ46Ξ47Ξ48Ξ49Ξ4,10Ξ4,11′′****Ξ55Ξ56Ξ57Ξ58Ξ59Ξ5,10Ξ5,11′′*****Ξ66′′Ξ67Ξ68Ξ69Ξ6,10Ξ6,11′′******Ξ77′′Ξ78Ξ79Ξ7,10Ξ7,11′′*******Ξ88Ξ89Ξ8,10Ξ8,11′′********Ξ99Ξ9,10Ξ9,11′′*********Ξ10,10Ξ10,11′′**********Ξ11,11′′)13n×13n<0,
where
(3.31)Ξ11′′=Ξ11+EcTEc,Ξ66=Ξ66′′+EaTEa,Ξ77′′=Ξ77+EbTEb,Ξ1,11′′=(H12-C1TP12)K¯+C2T(H¯23-P¯23)T,Ξ2,11′′=(H13-P13)K¯,Ξ3,11′′=(H14-P14-P12)K¯+P¯23T-H¯23T,Ξ4,11′′=(H15-P15)K¯+P¯22T-H¯22T,Ξ5,11′′=(H16-P16)K¯-P¯22T-H¯22T,Ξ6,11′′=(H17-P17)K¯-ATP¯23T+ATH¯23T,Ξ7,11′′=(H18-P18)K¯-BTP¯23T+BTH¯23T,Ξ8,11′′=(H19-P19)K¯-P¯22T+H¯22T,Ξ9,11′′=(H20-P20)K¯+P¯22T-H¯22T,Ξ10,11′′=(H21-P21)K¯+P¯22T-H¯22T,Ξ11,11′′=(H¯23-P¯23)K¯+K¯T(H¯23-P¯23)T-I¯,K¯=[KKK],P¯22=(P¯1P¯2P¯3),P¯23=(P¯4P¯5P¯6),H¯22=(H¯1H¯2H¯3),H¯23=(H¯4H¯5H¯6),I¯=(I000I000I).Proof.
ReplacingA,B,C in system (2.5) with A+KF(t)Ea, B+KF(t)Eb and C+KF(t)Ec, respectively. Then, system (2.5) can be transformed into the following equivalent form:
(3.32)y(k+1)=Cy(k)+Ag(y(k))+Bg(y(k-τ(k)))+KF(t)Ecy(k)+KF(t)Eag(y(k))+KF(t)Ebg(y(k-τ(k)))=C1y(k)+C2y(k)+Ag(y(k))+Bg(y(k-τ(k)))+K¯Υ(k),
where
(3.33)Υ(k)=(F(t)Ecy(k)F(t)Eag(y(k))F(t)Ebg(y(k-τ(k))))=diag(F(t),F(t),F(t))diag(Ec,Ea,Eb)(y(k)g(y(k))g(y(k-τ(k)))).
Constructing a new augmented Lyapunov-Krasovskii functional candidate as follows:
(3.34)V(k)=V¯1(k)+V2(k)+V3(k)+V4(k)+V5(k)+V6(k),
where
(3.35)V¯1(k)=2Y¯T(k)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)13n×13nY¯(k),Y¯T(k)=[yT(k),yT(k-τ(k)),ηT(k),yT(k-τM),yT(k-τm),gT(y(k)),gT(y(k-τM)), ∑i=k-τMkyT(i),∑i=k-τmkyT(i),∑i=k-τM+1k-τmyT(i),ΥT(k)]T,  η(k)=y(k+1)-C1y(k); V2(k),V3(k),…,V6(k) are the same as in Theorem 3.1.
SetY¯¯T(k+1)=[yT(k+1),yT(k-τ(k)),ηT(k),yT(k-τM),yT(k-τm),gT(y(k)),gT(y(k-τM)),∑i=k-τMkyT(i),∑i=k-τmk  yT(i),∑i=k-τM+1k-τmyT(i),ΥT(k)]T  =[yT(k)C1T+ηT(k),ηT(k),yT(k-τM),yT(k-τm),gT(y(k)),gT  (y(k-τM)),∑i=k-τMkyT(i),∑i=k-τmkyT(i),∑i=k-τM+1k-τmyT(i),ΥT(k)]T.  Define ΔV(k)=V(k+1)-V(k). Then along the solution of system (3.32) we have
(3.36)ΔV¯1(k)=2Y¯T(k+1)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)Y¯(k+1)-2Y¯T(k)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)Y¯(k)=2Y¯¯T(k+1)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)Y¯¯(k+1)-2Y¯T(k)(P10⋯000⋯0⋮⋮⋮⋮00⋯0)Y¯(k)≜2I¯1-2I¯2.(3.37)I¯1=Y¯T(k)(C1T00000000000I000000000I0I00000000000I00000000000I00000000000I00000000000I00000000000I00000000000I00000000000I00000000000I¯)(P1P2P120P3P130P4P140P5P150P6P160P7P170P8P180P9P190P10P200P11P210P¯22P¯23)(y(k+1)00).
On the other hand, since η(k)-C2y(k)-Ag(y(k))-Bg(y(k-τ(k)))-K¯Υ(k)=0, ∑i=k-τmky(i)-∑i=k-τMky(i)+∑i=k+1-τMk-τmy(i)-yT(k-τm)+yT(k-τM)=0, we have
(3.38)(y(k+1)00)=(C1y(k)+η(k)∑i=k-τmky(i)-∑i=k-τMky(i)+∑i=k+1-τMk-τmy(i)-yT(k-τm)+yT(k-τM)η(k)-C2y(k)-Ag(y(k))-Bg(y(k-τ(k)))-K¯Υ(k))=(C10I00000000000I-I00-III0-C20I00-A-B000-K¯)Y¯(k),(3.39)I¯2=Y¯T(k)(P1H2H120H3H130H4H140H5H150H6H160H7H170H8H180H9H190H10H200H11H210H¯22H¯23)(I0000000000000I-I00-III0-C20I00-A-B000-K¯)Y¯(k).
Noting that
(3.40)ΥT(k)Υ(k)≤[yT(k),gT(y(k)),gT(y(k-τ(k)))]T×(EcTEc000EaTEa000EbTEb)(y(k)g(y(k))g(y(k-τ(k)))).
Combining (3.12)–(3.17), (3.36)–(3.40), similar to the proof of Theorem 3.1, one can easily obtain this result, which completes the proof.Remark 3.10.
Compared with the augmented Lyapunov functional constructed in Theorem3.1, this new augmented Lyapunov functional include the term Υ(k), which makes the conservatism of the stability criterion be reduced further (details for more, see Example 4.2).


## 4. Numerical Examples

In this section, three numerical examples will be presented to show the validity of the main results derived above.Example 4.1.
For the convenience of comparison, let us consider a delayed discrete-time recurrent neural network in (2.5) with parameters given by
(4.1)C=(0.8000.9),A=(0.001000.005),B=(-0.10.01-0.2-0.1).
The activation functions are given by g1(x)=g2(x)=tanh(x). It is easy to see that the activation functions satisfy Assumption 1 with σ1-=σ2-=0, σ1+=σ2+=1. For τm=2,4,6,8,10,20, references [15–17] gave out the allowable upper bound τM of the time-varying delay, respectively. Decompose matrix C as C=C1+C2, where
(4.2)C1=(0.4000.5),C2=(0.4000.4).
Table 1 shows that our results are less conservative than these previous results.Table 1
Allowable upper boundsτM for given τm.
Casesτm=2τm=4τm=6τm=8τm=10τm=20By [15]111112131421By [16]111213141623By [17]131317192131By Theorem3.1τM>0τM>0τM>0τM>0τM>0τM>0Example 4.2.
Consider an uncertain delayed discrete-time recurrent neural network in (2.1) with parameters given by
(4.3)C=(0.25000.1),A=(0.120.24-0.150.2),B=(-0.250.10.020.09),K=(0.2000.3),Ec=(0.150.10-0.7),Ea=(0.10.3-0.20.05),Eb=(0.130.06-0.050.15),J=(00).
The activation functions are given by f1(x)=tanh(0.55x)+sin(0.45x), f2(x)=tanh(0.65x)+sin(0.45x). It is easy to see that the activation functions satisfy Assumption 1 with σ1-=0.1,σ2-=0.2, σ1+=1,σ2+=1.1. For τm=2,4,6,8,10, references [16, 17] gave out the allowable upper bound τM of the time-varying delay, respectively. Decompose matrix C as C=C1+C2, where
(4.4)C1=(0.2-10.0120.05),C2=(0.051-0.0120.05).
Set ϵ=50, by using the MATLAB toolbox, the allowable upper bounds τM for given τm are showed in Table 2. Obviously, our results are less conservative than these previous results.Table 2
Allowable upper boundsτM for given τm.
Casesτm=2τm=4τm=6τm=8τm=10By [16]68101214By [17]1012141618By Theorem3.71719212325By Theorem3.92224262830Example 4.3.
Consider an uncertain delayed discrete-time recurrent neural network in (2.1) with parameters given by
(4.5)C=(0.8000.9),A=(0.070.100.05),B=(-0.10.01-0.2-0.1),K=(0.02000.03),Ec=(0.150.10-0.7),Ea=(0.10.3-0.20.05),Eb=(0.130.06-0.050.15),J=(00).
And the activation functions are the same as given in Example 4.2. Decompose matrix C as C=C1+C2, where
(4.6)C1=(0.4000.5),C2=(0.4000.4).
Set ϵ=50, by using the MATLAB toolbox, the allowable upper bounds τM for given τm are showed in Table 3.
The free-weighting matrices are obtained as follows whenτm=2,τM=60:
(4.7)Q1=(0.0074-0.0002-0.00020.0007),Q2=(0.0627-0.0191-0.01910.0094),Q3=(0.0627-0.0191-0.01910.0094),D1=(0.8703000.2672),Q4=(0.0079000.0012),Q5=1.0e-003(0.2326000.0303),Q6=1.0e-003(0.2493000.0320),P1=(6.3936-0.9854-0.98540.2599),P5=(-304.8096-16.0366-131.0952-930.9992),D2=(0.0612000.0083),P6=1.0e+003(1.19310.10970.33650.0269),P9=1.0e+003(-1.4764-0.1509-0.07911.0626),P10=(-350.0434-852.4925-26.8470904.2654),P11=1.0e+003(-0.5682-0.0257-0.04131.1255),P12=1.0e+004(-0.6548-1.62950.93994.7692),P13=1.0e+003(2.2001-3.7370-0.3034-0.3479),P14=1.0e+004(0.31860.7851-0.4972-2.3726),P17=(273.4789-200.5789-165.3251-31.3854),P18=1.0e+003(0.0888-3.84541.60804.3515),H5=(-304.4622-16.0335-131.0922-930.6433),H6=1.0e+003(1.19270.10970.33650.0265),H9=1.0e+003(-1.4768-0.1509-0.07911.0622),H10=(-349.6840-852.4933-26.8478904.6230),H11=1.0e+003(-0.5678-0.0257-0.04131.1259),H12=1.0e+004(-0.2616-0.65220.46982.3832),H13=1.0e+003(2.2001-3.7370-0.3034-0.3479),H14=1.0e+004(-0.3344-0.84430.44222.4005),H17=1.0e+004(274.5315-200.9016-167.3579-33.2092),H18=1.0e+003(0.0893-3.84401.59704.3553),P2=P3=P4=P7=P8=P15=P16=P19=P20=P21=H2=H3=H4=H7=H8=H15=H16=H19=H20=H21=0.Table 3
Allowable upper boundsτM for given τm.
Casesτm=2τm=4τm=6τm=8τm=10τm=20By Theorem3.78890929496105


## 5. Conclusion

By decomposing some connection weight matrices , combined with linear matrix inequality (LMI) technique, some new augmented Lyapunov-Krasovskii functionals are constructed, and serial new improved sufficient conditions ensuring exponential stability or robust exponential stability are obtained. Numerical examples show that the new criteria derived in this paper are less conservative than some previous results obtained in the references cited therein.



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*Source: 874582-2009-07-22.xml*