LMI-based criteria for global robust stability of bidirectional associative memory networks with time delay

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Abstract

In this paper, several new sufficient conditions are given to ensure existence, uniqueness and globally exponential robust stability of the equilibrium point for bidirectional associative memory (BAM) networks with delays. This novel approach, based on the Linear Matrix Inequality (LMI) technique, removes some existing restrictions on the system’s parameters, and the derived conditions are easy to verify via the LMI toolbox. In addition, two examples are given to show the effectiveness and the advantage of the proposed results.

Introduction

Kosko in [1], [2], [3] proposed a new class of networks called bidirectional associative memory (BAM) networks. This class of networks has been successfully applied to pattern recognition due to its generalization of the single-layer auto-associative Hebbian correlator to a two-layer pattern-matched heteroassociative circuit. However, in [1], [2], [3], the author discussed the BAM neural networks without any delay. In fact, time delay inevitably occurs in electronic neural networks owing to the unavoidable finite switching speed of amplifiers. It is desirable to study the BAM neural networks with delays. It is known that the BAM network with axonal signal transmission delays has been studied, see [14], [15], [13], [17], [8], [9], [10], [5], [6], [11], [22], [4], [12], [16]. Most of these studies involved various dynamics behaviors such as stability, (almost) periodic oscillation, chaos and bifurcation [19]. A set of delay-independent criteria have been obtained to ensure the asymptotic or exponential stability, and the existence and attractivity of (almost) periodic solution. The used methodology is mainly based on topological degree theory, fixed point theorem, M-matrix theory. In [20], [21], the LMI approach is proposed to deal with the stability problem of neural network with delays. In the applications and designs of neural networks, there are often some unavoidable uncertainties such as modelling errors, external perturbations and parameter fluctuations, which can cause the network to be unstable. Furthermore, in designing a neural network, one is concerned not only with the stability of the system but also with the convergence rate. That is to say, one usually desires a fast response in the network, therefore it is important to determine the exponential stability and to estimate the exponential convergence rate. Owing to the above reasons, it is essential to investigate the globally exponential robust stability of the network with errors and perturbations. To the best of our knowledge, only some limited works are done on this problem [7], [23], [24], [25], [26], [27], [28]. In [24], several criteria are presented for global robust stability of neural networks without delay. In [25], [26], global robust stability of delayed interval Hopfield neural networks is investigated with respect to the bounded and strictly increasing activation functions. Several M-matrix conditions to ensure the robust stability are given for delayed interval Hopfield neural networks. In [23], the authors viewed the uncertain parameters as perturbations and gave some testable results for robust stability. In [7], the authors derived a novel result based on a new matrix-norm inequality technique and Lyapunov method, and gave some comments on those in [27].

Motivated by the above works, the aim of this paper is to consider further global robust stability for BAM neural networks. In this paper, we present several novel criteria to guarantee existence, uniqueness and globally exponential robust stability of the equilibrium point for interval BAM neural networks with delays by using Lyapunov functions and linear matrix inequality (LMI) technique. It is worth noting that, since a novel approach based on LMI is used in discussing this robust problem, the obtained results are completely new which are expressed in matrix form. Moreover, the given conditions are easy to verify via the LMI toolbox. Besides, as a by-product, for the conventional BAM neural networks with delays, we can also derive some criteria for checking the global exponential stability.

The rest of this paper is organized as follows. In Section 2, problem formulation and preliminaries are given. In Section 3, several novel sufficient criteria are derived for checking globally exponential robust stability and the uniqueness of the equilibrium point of interval BAM neural networks with time delays. In Section 4, two examples are given to show the effectiveness of the proposed results. Finally, conclusions are given in Section 5.

Section snippets

Problem formulation and preliminaries

In this paper, we consider the BAM neural network model described by the following differential equations: ui(t)=aiui(t)+j=1mwjigj(vj(tτ))+Iii=1,2,,n,vj(t)=bjvj(t)+i=1nvijgi(ui(tσ))+Jjj=1,2,,m, in which u=(u1,u2,,un)TRn, v=(v1,v2,,vm)TRm, ai>0, bj>0, τ0, σ0 are constants. The activate functions gi satisfy the following properties:

(A1): gi are bounded on R, i=1,2,,max{m,n}.

(A2): gi() (i=1,2,,max{m,n}) are Lipschitz continuous and monotonically nondecreasing, i.e., there exist

Robust stability criteria

In this section, by using the Lyapunov function method and the Linear Matrix Inequality (LMI) techniques, several new sufficient conditions are given to ensure the globally exponential robust stability of the equilibrium point for system (3). Globally exponential stability of the equilibrium point for system (3) is obtained as a by-product which imposes less restrictions on the system’s parameters.

Theorem 1

Under the assumptions (H1)(H3) , model(3)is globally exponentially robust stable, if there exist

Two illustrative examples

We will give two examples to illustrate the effectiveness of our results and compare our results with those proposed in a previous paper.

Example 1

Consider the following BAM networks xi(t)=aixi(t)+j=12wjifj(yj(t0.1)),yj(t)=bjyj(t)+i=12vijfi(xi(t0.1)). Take fi(xi)=12(|xi+1||xi1|), i=1,2, τ=0.1, σ=0.1, we have Mi=Mj=1 (i=j=1,2). Let A¯=(3003),A¯=(2002),B¯=(4004),B¯=(3003),W¯=(0.160.130.130.16),W¯=(0.180.150.150.18),V¯=(0.150.120.120.15),V¯=(0.170.140.140.17), then we can easily calculate

Conclusions

In this paper, a novel approach based on LMI has been used in the discussions of robust stability, and several new sufficient criteria have been given, which are expressed as matrix inequalities, for the globally exponential robust stability of BAM neural networks with time delays. The obtained results generalize the earlier works, and are easily verified in practice via the LMI toolbox. In the end, two illustrative examples are given to show the effectiveness of the proposed results.

Acknowledgements

The authors would like to thank the reviewers and the editor for their helpful comments and constructive suggestions. This work was supported by the National Natural Science Foundation of China under Grants 60574043 and 60373067, and the Foundation for Excellent Doctoral Dissertation of Southeast University.

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