A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays☆
Introduction
In the past few decades, delayed recurrent neural networks (especially delayed Hopfield neural networks, delayed cellular neural networks and delayed bidirectional associative memory neural networks) have found successful applications in many areas such as signal processing, pattern recognition, associative memories, and optimization solvers. Many important results have been reported on the existence, uniqueness, and global asymptotic or exponential stability of the equilibrium point for recurrent neural networks with constant delays or time-varying delays, see [1], [2], [7], [8], [10], [17], [18], [22], [23], [24], [25] and the references therein for some recent publications.
It is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillation, bifurcation, and chaos. In many applications, the properties of periodic solutions are of great interest, which have been successfully applied in, for example, learning theory [21] since effective learning usually requires repetition. In addition, an equilibrium point can be viewed as a special periodic solution of neural networks with arbitrary period. In this sense, the analysis of periodic solutions of neural networks can be considered to be more general than that of equilibrium point. Recently, the existence and stability of the periodic solution have been studied for recurrent neural networks with constant delays or time-varying delays, see e.g. [13], [19], [27] and references therein.
It is worth noticing that, up to now, most recurrent neural networks have been assumed to act in a continuous-time manner. However, when implementing the continuous-time recurrent neural network for computer simulation, for experimental or computational purposes, it is essential to formulate a discrete-time system that is an analogue of the continuous-time recurrent neural network. To some extent, the discrete-time analogue inherits the dynamical characteristics of the continuous-time recurrent neural network under mild or no restriction on the discretization step-size, and also remains functional similarity to the continuous-time recurrent neural network and any physical or biological reality that the continuous-time recurrent neural network has [15]. Unfortunately, as pointed out in [16], the discretization cannot preserve the dynamics of the continuous-time counterpart even for a small sampling period. Therefore, there is a crucial need to study the dynamics of discrete-time neural networks.
Recently, the dynamics analysis problem for discrete-time recurrent neural networks with or without time delays has received considerable research interest, see for example [6], [9], [11], [12], [14], [15], [16], [20], [28], [29], [30], [31] and references therein. In [12], [14], [15], [16], the global exponential stability has been investigated for discrete-time delayed recurrent neural networks, and several sufficient conditions for checking the global exponential stability of the equilibrium point have been obtained. In [9], the global robust stability problem has been considered for a general class of discrete-time interval neural networks that contain time-invariant uncertain parameters with their values being unknown but bounded in given compact sets, and three sufficient conditions ensuring the global robust stability have been given. In [28], [29], the authors have studied the stability and bifurcation problems for a class of discrete-time neural networks. In [31], the stability and convergence of the periodic solution have been studied for discrete-time neural network of two neurons. In [6], [20], several sufficient conditions have been derived for checking the existence and global exponential stability of the periodic solution for discrete-time recurrent neural networks with constant delay. In [11], the existence of a unique almost periodic sequence solution has been studied for discrete-time neural networks without delay. In [30], the authors have investigated the existence and global exponential stability of the periodic solutions for discrete-time BAM neural networks with periodic coefficients and distributed delays,
It should be pointed out that, the given criteria in [6], [20] have been based upon certain diagonal dominance or M-matrix conditions on weight matrices of the networks, which only depend on absolute values of the weights and ignore the signs of the weights. Therefore, the conditions are somewhat conservative. In [14], a linear matrix inequality (LMI) approach has been developed to deal with the analysis problem of exponential stability for a class of discrete-time recurrent neural networks (DRNNs) with time delays. Unfortunately, the construction of Lyapunov functional used in [14] appeared to be conservative, and there is much room to reduce the possible conservatism. Motivated by the above discussions, the objective of this Letter is to study the existence and stability of the periodic solution for discrete-time recurrent neural network with time-varying delays by employing a new Lyapunov–Krasovskii functional as well as a unified LMI approach. Under more general description on the activation functions, we utilize the latest free-weighting matrix method [3], [4], [5], [7], [8] and obtain several less conservative conditions, which can be checked numerically using the effective LMI toolbox in MATLAB. Two simulation examples are given to show the effectiveness and less conservatism of the proposed criteria.
Notations: The notations are quite standard. Throughout this Letter, and denote, respectively, the n-dimensional Euclidean space and the set of all real matrices. The superscript “T” denotes matrix transposition. The notation (respectively, ) means that X and Y are symmetric matrices, and that is positive semidefinite (respectively, positive definite). is the Euclidean norm in . If A is a matrix, denote by its operator norm, i.e., , where (respectively, ) means the largest (respectively, smallest) eigenvalue of A. For integers a, b, and , denotes the discrete interval given . denotes the set of all functions . Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.
Section snippets
Model description and preliminaries
In this Letter, we consider the following neural network model for , where , is the state of the ith neuron at time k; , denotes the activation function of the jth neuron at time k; , represents the external input on the ith neuron at time k; the positive integer corresponds to the transmission delay and satisfies
Main result
In this section, we shall establish our main criteria based on the LMI approach.
For presentation convenience, in the following, we denote
Theorem 1 Under assumptions (H1) and (H2), there exists exactly one ω-periodic solution of model (1), (3) and all other solutions of model (1), (3) converge exponentially to it as , if there exist five symmetric positive definite matrices P, Q, R, and , two positive diagonal matrices
Examples
Example 1 Consider a discrete-time recurrent neural network (1), where It can be verified that assumptions (H1) and (H2) are satisfied with , , , , , , . Thus, By the Matlab LMI Control Toolbox, we find a solution to the LMI in (5) as follows:
Conclusions
In this Letter, the existence and stability of periodic solution have been investigated for discrete-time recurrent neural network with time-varying delays. The description of the activation functions was more general than the recently commonly used Lipschitz conditions. By employing appropriate Lyapunov–Krasovskii functional and the free-weighting matrix method, a new LMI criterion is established to ensure the existence, uniqueness, and global exponential stability of periodic solution for the
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This work was supported in part by the National Natural Science Foundation of China under Grant 50608072, an International Joint Project sponsored by the Royal Society of the UK and the National Natural Science Foundation of China, and the Alexander von Humboldt Foundation of Germany.