# New Results on Passivity Analysis for Uncertain Neural Networks with Time-Varying Delay

Authors: Xueying Shao; Qing Lu; Hamid Reza Karimi; Jin Zhu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The paper investigates the stability and passivity analysis problems for a class of uncertain neural networks with time-delay via delta operator approach. Both the parameter uncertainty and the generalized activation functions are considered in this paper. By constructing an appropriate Lyapunov-Krasovskii functional, some new stability and passivity conditions are obtained in terms of linear matrix inequalities (LMIs). The main characteristic of this paper is to obtain novel stability and passivity analysis criteria for uncertain neural networks with time-delay in the delta operator system framework. A numerical example is presented to demonstrate the effectiveness of the proposed results.


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## Body

## 1. Introduction

Recently, neural networks have attracted considerable attention due to their applications in wide areas such as associative memory [1, 2], pattern recognition [1, 3], and optimization problems [4–7]. Recently, some stability conditions [8–12] and passivity analysis for neural networks [13–15] have been reported in the literature. The effect of time-delays [16–25] cannot be ignored in the real systems due to the facts that the delays can lead to instability [26–29], oscillation, or chaos. Recently, the stability results of time-delay neural networks have been presented in [30–36]. In addition, many results on passivity of neural networks with time-delay have been proposed [37–41].It is well known that the discrete systems are often used for computer realization and continuous systems are frequently applied to theoretical analysis, respectively. Sampling continuous systems can lead to considerable discrete systems. When the sampling is fast using the traditional shift operator, the poles are located in the stable boundary. Then, the discrete systems will lose stability in finite word length computer. Goodwin proposed delta operator approach in [42] which is used to replace the aforementioned operator with sample continuous systems, which can unify some previous related results of the continuous and discrete systems into the framework of the delta operator systems. The delta operator is defined by
(1)δx(t)={dx(t)dt,T=0;x(t+T)-x(t)T,T≠0,
where T is a sampling period. More recently, much attention has been focused on the stability and stabilization problems for some delta operator systems [43–47]. However, it should be mentioned that there are few achievements about passivity analysis for uncertain discrete neural networks with time-varying delay via delta operator approach; instability exists in the applications of neural networks when the sampling rate is high, which motivates this research.In this paper, the stability and passivity problems are investigated for uncertain neural networks with time-varying delay via delta operator approach. Both the parameter uncertainty and the generalized activation functions are considered in this paper. By choosing a new type of Lyapunov functional in delta domain and employing some novel methods to handle the delays, some stability and passivity criteria are proposed. The proposed conditions are expressed in terms of linear matrix inequalities (LMIs), which are dependent on the sampling period. The main characteristic of this paper is to obtain some stability and passivity analysis criteria for uncertain neural networks with time-varying delay in the delta operator system framework. Finally, a numerical example is given to demonstrate the effectiveness of the developed results.Notation. Throughout this paper, for the sake of convenience, we use tk to denote kT, where T is the sampling period. ℝn denotes the n-dimensional Euclidean space. The superscript T stands for matrix transposition. The notation diag⁡{⋯} denotes a block-diagonal matrix. For real symmetric matrices X and Y, the notation X≥Y (resp., X>Y) means that the X-Y is positive semidefinite (resp., positive-definite). I is the identity matrix with appropriate dimensions. The symbol “*” stands for the symmetric term in a matrix. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions.


## 2. Problem Formulation

Consider the following uncertain neural network with time-varying delay:(2)δx(tk)=-(W1+ΔW1(tk))x(tk)+(W2+ΔW2(tk))f(x(tk))+(W3+ΔW3(tk))f(x(tk-dk))+u(tk),(3)y(tk)=f(x(tk)),
where x(tk)∈ℝn stands for the network state at time tk; f(x(tk))∈ℝn denotes the activation at time tk; u(tk)∈ℝn is the external input at time tk; y(tk) is the output vector; W1 is a positive diagonal matrix; ΔW1(tk), ΔW2(tk), and ΔW3(tk) are unknown matrices; dk is a time-varying delay 0≤dm≤dk≤dM, with dm=nmT and dM=nMT; nm and nM are two known positive and finite integers; T is the sampling period. The time-varying parameter uncertainties ΔW1(tk), ΔW2(tk), and ΔW3(tk) are assumed to be in the following form:
(4)[ΔW1(tk)ΔW2(tk)ΔW3(tk)]=HF(tk)[E1E2E3],
where H, E1, E2, and E3 are known constant matrices; F(tk) is an unknown time-varying matrix satisfying
(5)FT(tk)F(tk)≤I.
The activation function f(x(tk)) satisfies the following condition:
(6)Ki-≤fi(α1)-fi(α2)α1-α2≤Ki+,i=1,2,…,n,
with α1≠α2,
(7)K-=diag⁡[K1-K2-⋯Kn-],K+=diag⁡[K1+K2+⋯Kn+].Before ending this section, some preliminaries are recalled which are used to prove the main results in the next section.Definition 1 (see [48]).
A delta operator system is asymptotically stable, if the following conditions hold:(i)
V(x(t))≥0 with equality if and only if x(t)=0;(ii)
δV(x(t))=[V(x(t+T))-V(x(t))]/T<0,where V(x(t)) is a Lyapunov function in the delta domain.Definition 2.
The neural network (2) and (3) is called passive if there exists a scalar γ≥0 such that
(8)∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)]≤0.Lemma 3 (see [49]).
The property of delta operator for any time functionx(t) and y(t) can be represented as
(9)δ(x(t)y(t))=δ(x(t))y(t)+x(t)δy(t)+Tδ(x(t))δy(t),
where T is a sampling period.Lemma 4 (see [50]).
For any constant positive semidefinite symmetric matrixW, two positive integers r and r0 satisfy r≥r0≥1; the following inequality holds:
(10)(∑i=r0r‍x(i))TW(∑i=r0r‍x(i))≤(r-r0+1)∑i=r0r‍xT(i)Wx(i).Lemma 5 (see [51]).
LetU, V, W, and M be real matrices of appropriate dimensions with M satisfying M=MT; then M+UVW+WTVTUT<0 for all VTV≤I, if and only if there exists a scalar ɛ>0 such that M+ɛ-1UUT+ɛWTW<0.Lemma 6 (see [52] Schur complement).
Given constant matricesΩ1, Ω2, and Ω3 with appropriate dimensions, where Ω1T=Ω1, and Ω2T=Ω2>0, then Ω1+Ω3TΩ2-1Ω3<0, if and only if
(11)[Ω1Ω3T*-Ω2]<0or[-Ω2Ω3*Ω1]<0.


## 3. Main Results

In this section, the stability and the passivity results for discrete-time uncertain neural network with time-varying delay via delta operator are given. Firstly, the stability conditions are given in the following part.


### 3.1. Stability Analysis

In order to consider the stability condition for uncertain neural networks with time-delay (2), we define ΔWi(tk)=0(i=1,2,3) and u(tk)=0 in (2). Then, we can have the following neural networks with time-delay:
(12)δx(tk)=-W1x(tk)+W2f(x(tk))+W3f(x(tk-dk)).
For neural networks with time-delay in (12), the stability criterion is obtained in the following theorem.Theorem 7.
For given scalars0≤dm≤dk≤dM, neural network (12) with (6) is asymptotically stable, if there exist P>0, Qi>0(i=1,2,3), R>0, and S>0, positive definite diagonal matrices L1, L2, X1, and X2, such that the following LMI holds:
(13)Ξ=[Θ11Θ12*Θ22]<0,
where
(14)Θ11=[Ψ11-SW10*Ψ220**-Q2],Θ12=[0SW2SW31nMTRPW2Ψ2500X2K+],Θ22=[-Q1-1nMTR00*Ψ550**Ψ66],Ψ11=TP+TnMR-S-ST,Ψ22=-2L1K-+2L2K++Q1+Q2+T(d+1)Q2-1nMTR-PW1-W1TP,Ψ25=L1-L2+X1K++PW2,Ψ55=-2X1+Q3,Ψ66=-2X2-Q3,L1=diag⁡{l11l12⋯l1n},L2=diag⁡{l21l22⋯l2n},X1=diag⁡{λ11λ12⋯λ1n},X2=diag⁡{λ21λ22⋯λ2n}.Proof.
Choose a Lyapunov-Krasovskii functional in delta domain as follows:(15)V(tk)=∑i=15‍Vi(tk),
with
(16)V1(tk)=xT(tk)Px(tk),V2(tk)=2T∑j=1n‍l1j∫0xj(tk-T)‍(fj(s)-Kj-s)ds+2T∑j=1n‍l2j∫0xj(tk-T)‍(Kj+s-fj(s))ds,V3(tk)=T∑i=1nM‍xT(tk-iT)Q1x(tk-iT)+T∑i=1nk‍xT(tk-iT)Q2x(tk-iT)+T∑i=1nk‍fT(x(tk-iT))Q3f(x(tk-iT)),V4(tk)=T2∑i=nmnM∑j=1i‍xT(tk-jT)Q3x(tk-jT),V5(tk)=∑i=1nM‍∑j=1i‍eT(tk-jT)Re(tk-jT),
where
(17)e(tk)=x(tk)-x(tk+T)=-Tδx(tk).
Applying Lemma 3, the delta-domain form of V1(tk) can be obtained as
(18)δV1(tk)=δTx(tk)Px(tk)+xT(tk)Pδx(tk)+TδTx(tk)Pδx(tk)=-xT(tk)W1TPx(tk)+xT(tk)W2TPf(x(tk))+xT(tk)W3TPf(x(tk-dk))-xT(tk)PW1x(tk)+xT(tk)PW2f(x(tk))+xT(tk)PW3f(x(tk-dk))+TδTx(tk)Pδx(tk).
From (6), for the scalars λ1i≥0,λ2i≥0, one can have
(19)2∑i=1n‍λ1ifi(xi(tk))[fi(xi(tk))-Ki+xi(tk)≤0],2∑i=1n‍λ2ifixi(tk-nkT)ddd×[fi(xi(tk-nkT))-Ki+xi(tk-nkT)]≤0,
which can be equivalently denoted as
(20)2fT(x(tk))X1f(x(tk))-2fT(x(tk))K+X1x(tk)≤0,2fT(x(tk-nkT))X2f(x(tk-nkT))-2fT(x(tk-nkT))K+X2x(tk-nkT)≤0.
Then, we obtain the following inequality:
(21)δV2(tk)=2∑j=1n‍l1j[∫0xj(tk)‍(fj(s)-Kj-s)dsdddddddd-∫0xj(tk-T)‍(fj(s)-Kj-s)ds]+2∑j=1n‍l2j[∫0xj(tk)‍(Kj+s-fj(s))dsdddddddddd-∫0xj(tk-T)‍(Kj+s-fj(s))ds]≤2∑j=1n‍l1jxj(tk)[fj(xj(tk))-Kj-x(tk)]+2∑j=1n‍l2jxj(tk)[Kj+x(tk)-fj(xj(tk))]≤2xT(tk)L1[f(x(tk))-K-x(tk)]+2xT(tk)L2[K+x(tk)-f(x(tk))]-2fT(x(tk))X1f(x(tk))+2fT(x(tk))K+X1x(tk)-2fT(x(tk-nkT))X2f(x(tk-nkT))+2fT(x(tk-nkT))K+X2x(tk-nkT).
Taking the delta operator manipulations of V3(tk), we can obtain that
(22)δV3(tk)=1T×T[∑i=1nM‍xT(tk-iT+T)Q1x(tk-iT+T)dddddddd-∑i=1nM‍xT(tk-iT)Q1x(tk-iT)]+1T×T[∑i=1nk‍xT(tk-iT+T)Q2x(tk-iT+T)dddddfddd-∑i=1nk‍xT(tk-iT)Q2x(tk-iT)]+1T×T[∑i=1nk‍fT(x(tk-iT+T))ddddddddddd×Q3f(x(tk-iT+T))ddddddddd-∑i=1nk‍fT(x(tk-iT))Q3f(x(tk-iT))]≤xT(tk)Q1x(tk)-xT(tk-nMT)Q1x(tk-nMT)+xT(tk)Q2x(tk)-xT(tk-nkT)Q2x(tk-nkT)+T∑i=nmnM‍xT(tk-iT)Q2x(tk-iT)+fT(x(tk))Q3f(x(tk))-fT(x(tk-nkT))×Q3f(x(tk-nkT)).
Taking the delta operator manipulations of V4(tk), the following results can be obtained:
(24)δV4(tk)=1T×T2[∑i=nmnM‍∑j=1i‍xT(tk-jT+T)ddddddddddddddd×Q3x(tk-jT+T)ddddgdddd-∑i=nmnM‍∑j=1i‍xT(tk-jT)Q3x(tk-jT)]=T[∑i=nmnM‍xT(tk)Q3x(tk)ddddddd×∑i=nmnM‍-xT(tk-iT)Q3x(tk-iT)]=T(d+1)xT(tk)Q3x(tk)-T∑i=nmnM‍xT(tk-iT)Q3x(tk-iT),
where d=nM-nm.
Taking the delta operator manipulations ofV5(tk) and using Lemma 4, it can be found that
(25)δV5(tk)=1T[∑i=1nM‍∑j=1i‍eT(tk-jT+T)gggggggggggg×Re(tk-jT+T)gggggg-∑i=1nM‍∑j=1i‍eT(tk-jT)Re(tk-jT)]=nMTeT(tk)Re(tk)-1T∑i=1nM‍eT(tk-iT)Re(tk-iT)≤TnMδTx(tk)Rδx(tk)-1nMT[x(tk-nMT)-x(tk)]T×R[x(tk-nMT)-x(tk)].
For a given positive definite matrix P1 with appropriately dimensions, one has that
(26)r1(tk)=2δTx(tk)S[-W1x(tk)+W2f(x(tk))ddddfdddd+W3f(x(tk-dk))-δx(tk)]=0.
Combining (18) and (21)–(26), the following inequality holds:
(27)δV(tk)=∑i=15‍δVi(tk)≤ηT(x(tk))Ξη(x(tk)),
where
(28)ηT(x(tk))=[(x(tk-nkT))δTx(tk),xT(tk),xT(tk-nkT),dxT(tk-nMT),fT(x(tk)),dfT(x(tk-nkT))].
It can be seen from Theorem7 that Ξ<0, which means that δV(tk)<0. Then, based on Definition 1, neural network (12) is asymptotically stable. The proof is completed.In the subsection, we continue to consider the robust stability problem of neural network (2) without inputs u(tk); equivalently, we have
(29)δx(tk)=-(W1+ΔW1(tk))x(tk)+(W2+ΔW2(tk))f(x(tk))+(W3+ΔW3(tk))f(x(tk-dk)).
For neural network (29), the robust stability criterion is given in the following theoremTheorem 8.
For given scalars0≤dm≤dk≤dM, neural network (29) under the conditions (4)–(6) is robustly asymptotically stable, if there exist P>0, Qi>0(i=1,2,3), R>0, S>0 positive definite diagonal matrices L1, L2X1, and X2, such that the following LMI holds:
(30)Ξ^=[Θ^11Θ^12*Θ^22]<0,
where
(31)Θ^11=[Ψ11SW100*Ψ2201nMTR**-Q20***-Q1-1nMTR],Θ^12=[SW2SW3SH0Ψ25PW3PH-ɛE1T0X2K+000000],Θ^22=[Ψ5500ɛE2T*Ψ660ɛE3T**-ɛI0***-ɛI],
with Ψ11, Ψ22, Ψ25, Ψ55, Ψ66, L1, L2, X1, and X2 defined in Theorem 7.Proof.
We choose the same Lyapunov-Krasovskii functional as Theorem7. According to neural network (2), we have the following equation:
(32)r2(tk)=2δTx(tk)×S[-(W1+ΔW1(tk))x(tk)dddd+(W2+ΔW2(tk))f(x(tk))dddd+(W3+ΔW3(tk))f(x(tk-dk))dddd-δx(tk)(W1+ΔW1(tk))]=0.
Similar to the proof of Theorem 7, we can have that
(33)δV(tk)=∑i=15‍δVi(tk)=∑i=15‍δVi(tk)+r2(tk)≤ηT(x(tk))Ξ˘η(x(tk))=ηT(x(tk))×(Ξ+φF(tk)ψ+ψTFT(tk)φT)η(x(tk)),
where
(34)Ξ˘=[Θ˘11Θ˘12*Θ˘22],Θ˘11=[Ψ11-SW1k(tk)0*Ψ˘220**-Q2],Θ˘12=[0SW2k(tk)SW3k(tk)1nMTR2PW2k(tk)Ψ˘2500X2K+],Θ˘22=[-Q1-1nMTR00*Ψ550**Ψ66],φT=[HTSTHTPT0000],ψ=[0-E100E2E3],Ψ˘22=-2L1K-+2L2K++Q1+Q2+T(d+1)Q2-1nMTR-PW1k(tk)-W1kT(tk)P,Ψ˘25=L1-L2+X1K++PW2k(tk),
with W1k(tk)=W1+ΔW1(tk), W2k(tk)=W2+ΔW2(tk), W3k(tk)=W3+ΔW3(tk), and Ξ, Ψ11, Ψ55, Ψ66, L1, L2, X1, and X2 have been defined in Theorem 7.
Applying Schur complement to (30), we have
(35)Ξ+ɛ-1φφT+ɛψTψ<0.
By Lemma 5, from the inequality (35), we can easily obtain
(36)Ξ+φF(tk)ψ+ψTFT(tk)φT<0.
Consequently, δV(tk)<0; from Definition 1, neural network in (29) is robustly asymptotically stable. The proof is completed.


### 3.2. Passivity Analysis

In this subsection, the passivity analysis results are given in the following part. We first consider (2) and (3) without the parameter uncertainties ΔWi(tk)=0, (i=1,2,3). Then, the following neural network can be obtained:
(37)δx(tk)=-W1x(tk)+W2f(x(tk))+W3f(x(tk-dk))+u(tk),y(tk)=f(x(tk)).Theorem 9.
For given scalars0≤dm≤dk≤dM, neural network is passive in (37), if there exist P>0, Qi>0(i=1,2,3), R>0, S>0 positive definite diagonal matrices L1, L2, X1, and X2, such that the following LMI holds:
(38)Ξ~<0,
where
(39)Ξ~=[Θ~11Θ~12*Θ~22],Θ~12=[0SW2SW3S1nMTRΨ25PW3P00X2K+0],Θ~22=[-Q1-1nMTR000*Ψ550-I**Ψ660***-γI],
and Θ~11=Θ11, Ψ25, Ψ55, and Ψ66 have been defined in Theorem 7.Proof.
In order to present the passivity condition for neural network (37), we choose the same Lyapunov-Krasovskii functional as Theorem 7. By following the same line of proof of Theorem 7 and considering the following inequality:
(40)∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)]=∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)+δV(tk)]-V(∞)+V(0)≤∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)+δV(tk)],
it can be seen from the LMI condition (40) that
(41)∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)]≤0,
which means that neural network in (37) is passive. This finishes the proof.In the following theorem, the passivity condition for uncertain neural network with time-varying delay in (2) and (3) is presented.Theorem 10.
For given scalars0≤dm≤dk≤dM, neural network in (2) and (3) is passive, if there exist P>0, Qi>0(i=1,2,3), R>0, S>0 positive definite diagonal matrices L1, L2, X1, and X2, such that the following LMI holds:
(42)[Θ¯11Θ¯12*Θ¯22]<0,
where
(43)Θ¯12=[SW2SW3SSH0Ψ25PW3PPH-ɛE1T000000X2K+000],Θ¯22=[Ψ550-I0ɛE2T*Ψ6600ɛE3T**-γI00***-ɛI0****-ɛI].
and Θ¯11=Θ^11, Ψ25, Ψ55, and Ψ66 have been defined in Theorem 8.Proof.
In order to present the passivity condition for neural network in (2) and (3), we choose the same Lyapunov-Krasovskii functional as Theorem 7. By following the same line of proof of Theorems 8 and 9, Theorem 10 can be proved.


## 3.1. Stability Analysis

In order to consider the stability condition for uncertain neural networks with time-delay (2), we define ΔWi(tk)=0(i=1,2,3) and u(tk)=0 in (2). Then, we can have the following neural networks with time-delay:
(12)δx(tk)=-W1x(tk)+W2f(x(tk))+W3f(x(tk-dk)).
For neural networks with time-delay in (12), the stability criterion is obtained in the following theorem.Theorem 7.
For given scalars0≤dm≤dk≤dM, neural network (12) with (6) is asymptotically stable, if there exist P>0, Qi>0(i=1,2,3), R>0, and S>0, positive definite diagonal matrices L1, L2, X1, and X2, such that the following LMI holds:
(13)Ξ=[Θ11Θ12*Θ22]<0,
where
(14)Θ11=[Ψ11-SW10*Ψ220**-Q2],Θ12=[0SW2SW31nMTRPW2Ψ2500X2K+],Θ22=[-Q1-1nMTR00*Ψ550**Ψ66],Ψ11=TP+TnMR-S-ST,Ψ22=-2L1K-+2L2K++Q1+Q2+T(d+1)Q2-1nMTR-PW1-W1TP,Ψ25=L1-L2+X1K++PW2,Ψ55=-2X1+Q3,Ψ66=-2X2-Q3,L1=diag⁡{l11l12⋯l1n},L2=diag⁡{l21l22⋯l2n},X1=diag⁡{λ11λ12⋯λ1n},X2=diag⁡{λ21λ22⋯λ2n}.Proof.
Choose a Lyapunov-Krasovskii functional in delta domain as follows:(15)V(tk)=∑i=15‍Vi(tk),
with
(16)V1(tk)=xT(tk)Px(tk),V2(tk)=2T∑j=1n‍l1j∫0xj(tk-T)‍(fj(s)-Kj-s)ds+2T∑j=1n‍l2j∫0xj(tk-T)‍(Kj+s-fj(s))ds,V3(tk)=T∑i=1nM‍xT(tk-iT)Q1x(tk-iT)+T∑i=1nk‍xT(tk-iT)Q2x(tk-iT)+T∑i=1nk‍fT(x(tk-iT))Q3f(x(tk-iT)),V4(tk)=T2∑i=nmnM∑j=1i‍xT(tk-jT)Q3x(tk-jT),V5(tk)=∑i=1nM‍∑j=1i‍eT(tk-jT)Re(tk-jT),
where
(17)e(tk)=x(tk)-x(tk+T)=-Tδx(tk).
Applying Lemma 3, the delta-domain form of V1(tk) can be obtained as
(18)δV1(tk)=δTx(tk)Px(tk)+xT(tk)Pδx(tk)+TδTx(tk)Pδx(tk)=-xT(tk)W1TPx(tk)+xT(tk)W2TPf(x(tk))+xT(tk)W3TPf(x(tk-dk))-xT(tk)PW1x(tk)+xT(tk)PW2f(x(tk))+xT(tk)PW3f(x(tk-dk))+TδTx(tk)Pδx(tk).
From (6), for the scalars λ1i≥0,λ2i≥0, one can have
(19)2∑i=1n‍λ1ifi(xi(tk))[fi(xi(tk))-Ki+xi(tk)≤0],2∑i=1n‍λ2ifixi(tk-nkT)ddd×[fi(xi(tk-nkT))-Ki+xi(tk-nkT)]≤0,
which can be equivalently denoted as
(20)2fT(x(tk))X1f(x(tk))-2fT(x(tk))K+X1x(tk)≤0,2fT(x(tk-nkT))X2f(x(tk-nkT))-2fT(x(tk-nkT))K+X2x(tk-nkT)≤0.
Then, we obtain the following inequality:
(21)δV2(tk)=2∑j=1n‍l1j[∫0xj(tk)‍(fj(s)-Kj-s)dsdddddddd-∫0xj(tk-T)‍(fj(s)-Kj-s)ds]+2∑j=1n‍l2j[∫0xj(tk)‍(Kj+s-fj(s))dsdddddddddd-∫0xj(tk-T)‍(Kj+s-fj(s))ds]≤2∑j=1n‍l1jxj(tk)[fj(xj(tk))-Kj-x(tk)]+2∑j=1n‍l2jxj(tk)[Kj+x(tk)-fj(xj(tk))]≤2xT(tk)L1[f(x(tk))-K-x(tk)]+2xT(tk)L2[K+x(tk)-f(x(tk))]-2fT(x(tk))X1f(x(tk))+2fT(x(tk))K+X1x(tk)-2fT(x(tk-nkT))X2f(x(tk-nkT))+2fT(x(tk-nkT))K+X2x(tk-nkT).
Taking the delta operator manipulations of V3(tk), we can obtain that
(22)δV3(tk)=1T×T[∑i=1nM‍xT(tk-iT+T)Q1x(tk-iT+T)dddddddd-∑i=1nM‍xT(tk-iT)Q1x(tk-iT)]+1T×T[∑i=1nk‍xT(tk-iT+T)Q2x(tk-iT+T)dddddfddd-∑i=1nk‍xT(tk-iT)Q2x(tk-iT)]+1T×T[∑i=1nk‍fT(x(tk-iT+T))ddddddddddd×Q3f(x(tk-iT+T))ddddddddd-∑i=1nk‍fT(x(tk-iT))Q3f(x(tk-iT))]≤xT(tk)Q1x(tk)-xT(tk-nMT)Q1x(tk-nMT)+xT(tk)Q2x(tk)-xT(tk-nkT)Q2x(tk-nkT)+T∑i=nmnM‍xT(tk-iT)Q2x(tk-iT)+fT(x(tk))Q3f(x(tk))-fT(x(tk-nkT))×Q3f(x(tk-nkT)).
Taking the delta operator manipulations of V4(tk), the following results can be obtained:
(24)δV4(tk)=1T×T2[∑i=nmnM‍∑j=1i‍xT(tk-jT+T)ddddddddddddddd×Q3x(tk-jT+T)ddddgdddd-∑i=nmnM‍∑j=1i‍xT(tk-jT)Q3x(tk-jT)]=T[∑i=nmnM‍xT(tk)Q3x(tk)ddddddd×∑i=nmnM‍-xT(tk-iT)Q3x(tk-iT)]=T(d+1)xT(tk)Q3x(tk)-T∑i=nmnM‍xT(tk-iT)Q3x(tk-iT),
where d=nM-nm.
Taking the delta operator manipulations ofV5(tk) and using Lemma 4, it can be found that
(25)δV5(tk)=1T[∑i=1nM‍∑j=1i‍eT(tk-jT+T)gggggggggggg×Re(tk-jT+T)gggggg-∑i=1nM‍∑j=1i‍eT(tk-jT)Re(tk-jT)]=nMTeT(tk)Re(tk)-1T∑i=1nM‍eT(tk-iT)Re(tk-iT)≤TnMδTx(tk)Rδx(tk)-1nMT[x(tk-nMT)-x(tk)]T×R[x(tk-nMT)-x(tk)].
For a given positive definite matrix P1 with appropriately dimensions, one has that
(26)r1(tk)=2δTx(tk)S[-W1x(tk)+W2f(x(tk))ddddfdddd+W3f(x(tk-dk))-δx(tk)]=0.
Combining (18) and (21)–(26), the following inequality holds:
(27)δV(tk)=∑i=15‍δVi(tk)≤ηT(x(tk))Ξη(x(tk)),
where
(28)ηT(x(tk))=[(x(tk-nkT))δTx(tk),xT(tk),xT(tk-nkT),dxT(tk-nMT),fT(x(tk)),dfT(x(tk-nkT))].
It can be seen from Theorem7 that Ξ<0, which means that δV(tk)<0. Then, based on Definition 1, neural network (12) is asymptotically stable. The proof is completed.In the subsection, we continue to consider the robust stability problem of neural network (2) without inputs u(tk); equivalently, we have
(29)δx(tk)=-(W1+ΔW1(tk))x(tk)+(W2+ΔW2(tk))f(x(tk))+(W3+ΔW3(tk))f(x(tk-dk)).
For neural network (29), the robust stability criterion is given in the following theoremTheorem 8.
For given scalars0≤dm≤dk≤dM, neural network (29) under the conditions (4)–(6) is robustly asymptotically stable, if there exist P>0, Qi>0(i=1,2,3), R>0, S>0 positive definite diagonal matrices L1, L2X1, and X2, such that the following LMI holds:
(30)Ξ^=[Θ^11Θ^12*Θ^22]<0,
where
(31)Θ^11=[Ψ11SW100*Ψ2201nMTR**-Q20***-Q1-1nMTR],Θ^12=[SW2SW3SH0Ψ25PW3PH-ɛE1T0X2K+000000],Θ^22=[Ψ5500ɛE2T*Ψ660ɛE3T**-ɛI0***-ɛI],
with Ψ11, Ψ22, Ψ25, Ψ55, Ψ66, L1, L2, X1, and X2 defined in Theorem 7.Proof.
We choose the same Lyapunov-Krasovskii functional as Theorem7. According to neural network (2), we have the following equation:
(32)r2(tk)=2δTx(tk)×S[-(W1+ΔW1(tk))x(tk)dddd+(W2+ΔW2(tk))f(x(tk))dddd+(W3+ΔW3(tk))f(x(tk-dk))dddd-δx(tk)(W1+ΔW1(tk))]=0.
Similar to the proof of Theorem 7, we can have that
(33)δV(tk)=∑i=15‍δVi(tk)=∑i=15‍δVi(tk)+r2(tk)≤ηT(x(tk))Ξ˘η(x(tk))=ηT(x(tk))×(Ξ+φF(tk)ψ+ψTFT(tk)φT)η(x(tk)),
where
(34)Ξ˘=[Θ˘11Θ˘12*Θ˘22],Θ˘11=[Ψ11-SW1k(tk)0*Ψ˘220**-Q2],Θ˘12=[0SW2k(tk)SW3k(tk)1nMTR2PW2k(tk)Ψ˘2500X2K+],Θ˘22=[-Q1-1nMTR00*Ψ550**Ψ66],φT=[HTSTHTPT0000],ψ=[0-E100E2E3],Ψ˘22=-2L1K-+2L2K++Q1+Q2+T(d+1)Q2-1nMTR-PW1k(tk)-W1kT(tk)P,Ψ˘25=L1-L2+X1K++PW2k(tk),
with W1k(tk)=W1+ΔW1(tk), W2k(tk)=W2+ΔW2(tk), W3k(tk)=W3+ΔW3(tk), and Ξ, Ψ11, Ψ55, Ψ66, L1, L2, X1, and X2 have been defined in Theorem 7.
Applying Schur complement to (30), we have
(35)Ξ+ɛ-1φφT+ɛψTψ<0.
By Lemma 5, from the inequality (35), we can easily obtain
(36)Ξ+φF(tk)ψ+ψTFT(tk)φT<0.
Consequently, δV(tk)<0; from Definition 1, neural network in (29) is robustly asymptotically stable. The proof is completed.


## 3.2. Passivity Analysis

In this subsection, the passivity analysis results are given in the following part. We first consider (2) and (3) without the parameter uncertainties ΔWi(tk)=0, (i=1,2,3). Then, the following neural network can be obtained:
(37)δx(tk)=-W1x(tk)+W2f(x(tk))+W3f(x(tk-dk))+u(tk),y(tk)=f(x(tk)).Theorem 9.
For given scalars0≤dm≤dk≤dM, neural network is passive in (37), if there exist P>0, Qi>0(i=1,2,3), R>0, S>0 positive definite diagonal matrices L1, L2, X1, and X2, such that the following LMI holds:
(38)Ξ~<0,
where
(39)Ξ~=[Θ~11Θ~12*Θ~22],Θ~12=[0SW2SW3S1nMTRΨ25PW3P00X2K+0],Θ~22=[-Q1-1nMTR000*Ψ550-I**Ψ660***-γI],
and Θ~11=Θ11, Ψ25, Ψ55, and Ψ66 have been defined in Theorem 7.Proof.
In order to present the passivity condition for neural network (37), we choose the same Lyapunov-Krasovskii functional as Theorem 7. By following the same line of proof of Theorem 7 and considering the following inequality:
(40)∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)]=∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)+δV(tk)]-V(∞)+V(0)≤∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)+δV(tk)],
it can be seen from the LMI condition (40) that
(41)∑k=0∞‍[-γuT(tk)u(tk)-2yT(tk)u(tk)]≤0,
which means that neural network in (37) is passive. This finishes the proof.In the following theorem, the passivity condition for uncertain neural network with time-varying delay in (2) and (3) is presented.Theorem 10.
For given scalars0≤dm≤dk≤dM, neural network in (2) and (3) is passive, if there exist P>0, Qi>0(i=1,2,3), R>0, S>0 positive definite diagonal matrices L1, L2, X1, and X2, such that the following LMI holds:
(42)[Θ¯11Θ¯12*Θ¯22]<0,
where
(43)Θ¯12=[SW2SW3SSH0Ψ25PW3PPH-ɛE1T000000X2K+000],Θ¯22=[Ψ550-I0ɛE2T*Ψ6600ɛE3T**-γI00***-ɛI0****-ɛI].
and Θ¯11=Θ^11, Ψ25, Ψ55, and Ψ66 have been defined in Theorem 8.Proof.
In order to present the passivity condition for neural network in (2) and (3), we choose the same Lyapunov-Krasovskii functional as Theorem 7. By following the same line of proof of Theorems 8 and 9, Theorem 10 can be proved.


## 4.  A Numerical Example

In this section, the following numerical example is presented to demonstrate the effectiveness of the proposed results.Example 1.
Consider discrete-time neural networks (2) and (3) with the following parameters:
(44)W1=[2002],W2=[11-1-1],W3=[0.88111],H=0.1×I,E1=0.1×I,E2=0.2×I,E3=0.3×I,K-=diag⁡{-1,-1},K+=diag⁡{0.9,0.9}.In order to illustrate the effectiveness of the obtained results, we chooseT=0.05, nm=10, and nM=20. Then, using the Matlab LMI toolbox to solve the LMI in (42), we obtain a solution as follows:
(45)P=[5.20802.09842.09849.7576],S=[1.24750.17760.17762.9572],Q1=[0.2133-0.0219-0.02191.2999],Q2=[0.46840.27520.27521.7084],Q3=[7.32906.06546.065412.4586],R=[0.62110.06210.06211.5979],L1=[0.0313000.2507],L2=[2.7406000.7734],X1=[9.48190031.8737],X2=[0.6547003.2283],γ=48.7332,ɛ=3.1900.


## 5. Conclusions

In this paper, the problems of stability and passivity analysis for discrete-time neural networks with time-varying delay have been studied via delta operator approach. This paper has considered the parameter uncertainty and the generalized activation functions. By constructing appropriate Lyapunov-Krasovskii functional, some novel stability and passivity criteria have been proposed in the delta operator system framework. The obtained conditions have been expressed in terms of LMI, which can be easily solved by standard software. A numerical example has been given to illustrate the effectiveness of the proposed results.



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*Source: 303575-2014-03-24.xml*