Elsevier

Neurocomputing

Volume 379, 28 February 2020, Pages 227-235
Neurocomputing

Global exponential stability analysis of discrete-time BAM neural networks with delays: A mathematical induction approach

https://doi.org/10.1016/j.neucom.2019.10.089Get rights and content

Abstract

The problem of global exponential stability analysis for discrete-time bidirectional associative memory (BAM) neural networks with time-varying delays is investigated. By using the mathematical induction method, a novel exponential stability criterion in the form of linear matrix inequalities is firstly established. Then stability criteria depending upon only the system parameters are derived, which can easily checked by using the standard toolbox software (e.g., MATLAB). The proposed approach is directly based on the definition of global exponential stability, and it does not involve the construct of any Lyapunov–Krasovskii functional or auxiliary function. For a class of special cases, it is theoretical proven that the less conservative stability criteria can be obtained by using the proposed approach than ones in literature. Moreover, several numerical examples are also provided to demonstrate the effectiveness of the proposed method.

Introduction

In recent years, the dynamics of neural networks have been extensively studied because of their applications in pattern recognition, artificial intelligence, optimization, etc [1], [2], [3]. Bidirectional associative memory (BAM) neural networks were first introduced by Kosko [4], [5], which generalizes the single-layer autoassociative Hebbian correlator to a two-layer patternmatched heteroassociative circuits, and comes up with a complete and clear pattern stored in memory from an incomplete or fuzzy pattern. So the BAM neural networks possess better abilities of information memory and information association.

In hardware realization of neural networks, time delays are inevitably introduced because of the finite switching speed of the amplifiers and communication time among neurons. It is known that the time delays are a source of undesired dynamics like oscillation and instability. Therefore, it is more important to determine sufficient conditions for the asymptotic or exponential stability of delayed BAM neural networks [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Here we mention that closely related to this paper. Using the M-matrix methods, Liang and Cao [7] studied the exponential stability of continuous-time BAM neural network with constant delays and its discrete analogue. Liang et al. [8] obtained delay-dependent and delay-independent stability criteria of discrete-time BAM neural networks with variable time delays. However, the delay-dependent stability criteria above are based on certain constraints on the delays. By constructing an appropriate Lyapunov–Krasovskii functional and by using the technique of linear matrix inequality (LMI), Liu and Tang [10] further investigated the global exponential stability of discrete-time BAM neural networks without the constraints on delays in [8]. Gao and Cui [14] generalized the results in [8], [10] by establishing the exponential stability criteria for discrete-time BAM neural network with interval delays. However, it should be noted that in [11], in order to obtain a less conservative stability criterion, a more general Lyapunov–Krasovskii functional than the traditional one in [8], [10], [14] is constructed, and some slack matrices are introduced. Raja [15] deals with the problem of the global exponential stability analysis for a class of discrete-time BAM neural networks by employing the LMI approach. The problem of globally exponential stability analysis for discrete-time BAM neural networks is further investigated in [16] by constructing a modified Lyapunov–Krasovskii functional and by employing reciprocally convex and free-weighting matrix techniques. Some exponential stability criteria are derived, which can provide a larger admissible maximal delay bound than some existing results.

Motivated by the aforementioned discussions, constructing auxiliary function or Lyapunov–Krasovskii functional is too difficult and requires more skills. So it is necessary to find a new approach to analyze stability of delayed discrete-time BAM neural networks. The problem of globally exponential stability analysis for discrete-time BAM neural networks will be further investigated in this paper. By using the mathematical induction method, we derive several novel global exponential stability criteria in the simple form, which can easily verified via the standard tool software (e.g. MATLAB). It will be theoretically proven that the obtained global exponential stability criteria is less conservative than ones in [6]. In addition, several numerical examples are also provided to illustrate the effectiveness of theoretical results. On the one hand, compared with the existing results in [6, Theorems 3.1 and 3.2], the proposed approach does not construct any Lyapunov–Krasovskii functional and auxiliary function. On the other hand, compared with the linear matrix inequality-based stability criteria proposed in [8], [9], [11], [13], [14], [15], [16], the obtained stability criteria possess more simpler form, which reduces the computational complexity.

The organization of this paper is as follows. The problem considered in this paper will be formulated in Section 2. The novel criteria of global exponential stability criteria for discrete-time BAM neural networks with time-varying delays will be presented in Section 3. In Section 4, we will theoretically compare the criteria of the global exponential stability obtained in this paper and the ones in literature. Finally, numerical examples are provided in Section 5 to illustrate the effectiveness of the method proposed.

Notation. Suppose Z, R and C are sets of all integers, real numbers and complex numbers, respectively. Let Z[a,b] be the subset of Z consisting of all integers between a and b, and let Z[a,) be the limit case of Z[a,b] when b → ∞. For given positive integers p and q, let Rp×q denote the set of all p × q matrices over R. Set Rp=Rp×1.

A matrix MRn×n is called a Metzler matrix if all off-diagonal elements of M are nonnegative. Let In be the identity matrix in Rn×n and λ(M)={zC:det(zInM)=0}. The spectral abscissa of M is defined by s(M) ≔ max{Reλ: λ ∈ λ(M)}, and the spectral radius of M is defined by ρ(M) ≔ max {|λ|: λ ∈ σ(M)}.

For A=[aij]Rp×q and B=[bij]Rp×q, the matrix [aijbij], denoted by AB, refers to the Hadamard product of A and B, and the symbol AB (or BA) means that aij ≥ bij for all iZ[1,p] and jZ[1,q]; in particular, if aij > bij for all iZ[1,p] and jZ[1,q], then we write AB (or BA) instead of AB (or BA). Let |A|=[|xij|]. Then |XY|⪯|X||Y| for all XRp×q and YRq×r. Denote by Rp×q and Rp×q the sets of all p × q nonnegative and positive matrices, respectively. Similar notations are adopted for vectors.

Section snippets

Problem formulation

Consider the following discrete-time BAM neural network with time-varying delays, which can be described by Raja and Anthoni [15]:xi(k+1)=aixi(k)+j=1ncijfj(yj(k))+j=1neijgj(yj(khij(k)))+Ji,iZ[1,n],kZ[0,],yj(k+1)=bjyj(k)+i=1ndjif˜i(xi(k))+i=1nwjig˜i(xi(kτji(k)))+J˜j,jZ[1,n],kZ[0,], where xi(k) and yj(k) are the states of the ith neuron from the neural field FX and jth neuron from the neural field FY at time k, respectively; ai,bj(1,1) describe the stability of internal neuron

Global exponential stability criteria

In this section we will investigate sufficient conditions under which the zero equilibrium of system (4) is globally exponentially stable, that is, BAM neural networks (1) has a unique equilibrium which is globally exponentially stable.

To this end, we first introduce the result given in [17].

Lemma 1

[17]

Let A0Rn×n be a Metzler matrix and B0,C0,D0Rn×n. Then the following statements (i)–(iii) are equivalent:

  • (i)

    ρ(D0) < 1 and s(A0+B0(InD0)1C0)<0.

  • (ii)

    A0x+B0y0 and C0x+D0yy for some x,yRn.

  • (iii)

    s(A0) < 0 and ρ(C0(A0)

Theoretical comparisons

In this section, we will give theoretical comparisons of the global exponential stability criteria for the equilibrium of BAM neural network (1) presented in the previous section and [6], [7].

When C=D=0, BAM neural network (1) becomes:xi(k+1)=aixi(k)+j=1neijgj(yj(khij(k)))+Ji,iZ[1,n],yj(k+1)=bjyj(k)+i=1nwjig˜i(xi(kτji(k)))+J˜j,jZ[1,n], where kZ[0,). System (15) can be viewed as a discretized version of continuous-time neural network, if its parameters possess the following special form

Illustrative examples

In this section we will present the effectiveness of the proposed induction method by three numerical examples.

Example 1

Consider the discrete-time BAM neural network (1) with the following parameters:A=diag(0.05,0.05),B=diag(0.04,0.03),C=[0.030.010.020.02],D=[0.040.030.030.01],E=[0.050.050.20.01],W=[0.030.060.20.01],J=col(0.05,0),J˜=col(0,0.2),fj(s)=gj(s)=f˜i(s)=g˜i(s)=tanh(s),s[0,),i,jZ[1,2],

Clearly, Assumption 2 is satisfied. If we choose Γ1=Γ2=Γ˜1=Γ˜2=I2, then Assumption 1 is satisfied.

Conclusion

In this paper, the problem of global exponential stability for the discrete-time BAM neural networks with time-varying delays has been addressed. Firstly, the exponential stability criterion in the form of linear matrix inequality is established by using mathematical induction method. Then, the stability criteria which just depend on the system parameters are deduced, and their check can be carried out easily by using standard tool software (such as MATLAB). This method is directly based on the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported in part by the Natural Science Foundation of Heilongjiang Province (No. LH2019F030). The authors would like to thank the associate editor and the anonymous reviewers for their very helpful comments and suggestions, which greatly improves the original version of the paper.

Er-yong Cong received the B.S. and M.S. degrees in School of Mathematical Science from Heilongjiang University in 2003 and 2009, respectively. Since 2003 he has been working at Harbin University, where he is currently a Lecturer with the Department of Mathematical. His research interests include neural networks and stability analysis of delayed dynamic systems.

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    Er-yong Cong received the B.S. and M.S. degrees in School of Mathematical Science from Heilongjiang University in 2003 and 2009, respectively. Since 2003 he has been working at Harbin University, where he is currently a Lecturer with the Department of Mathematical. His research interests include neural networks and stability analysis of delayed dynamic systems.

    Xiao Han received Ph.D. degree in the School of Mathematics from Jilin University in 2007. Since 2007 she has been working at Jilin University, where she is currently a Lecturer in the School of Mathematics. Her current research interests include computational mathematics, numerical solutions of differential equations, and actuarial mathematics.

    Xian Zhang (M’10–SM’17) received Ph.D. degree in Control Theory from Queen’s University of Belfast in UK in 2004. Since 2004 he has been at Heilongjiang University, where he is currently a Professor in the School of Mathematical Science. His current research interests include neural networks, genetic regulatory networks, mathematical biology and stability analysis of delayed dynamic systems. He has received the Second Class of Science and Technology Awards of Heilongjiang Province in 2005 and the Three Class of Science and Technology Awards of Heilongjiang Province in 2015. He is a senior member of the IEEE, and a Vice President of Mathematical Society of Heilongjiang Province. Since 2006, he served as an Editor of the Journal of Natural Science of Heilongjiang University. He has authored more than 100 research papers.

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