Elsevier

Chaos, Solitons & Fractals

Volume 42, Issue 3, 15 November 2009, Pages 1299-1304
Chaos, Solitons & Fractals

Synchronization of cellular neural networks of neutral type via dynamic feedback controller

https://doi.org/10.1016/j.chaos.2009.03.024Get rights and content

Abstract

In this paper, we aim to study global synchronization for neural networks with neutral delay. A dynamic feedback control scheme is proposed to achieve the synchronization between drive network and response network. By utilizing the Lyapunov function and linear matrix inequalities (LMIs), we derive simple and efficient criterion in terms of LMIs for synchronization. The feedback controllers can be easily obtained by solving the derived LMIs.

Introduction

During the last two decades, synchronization of chaotic systems has been extensively studied due to its potential applications in many different areas including secure communication, chemical and biological systems, information science, optics and so on [1], [2], [3], [4]. Carroll and Pecora [5] propose the drive-response concept, and use the output of the drive system to control the response system so that the state synchronization is achieved. As is well known, time delay may occur in the process of information storage and transmission in neural networks. The existence of time delays is often a source of oscillation and instability [6], [7], [8], [9]. Recently, there has been increasing interest in potential applications of the dynamics of delayed neural networks (DNNs) in many areas [10], [11], [12], [13], [14], [15], [16]. It has been revealed that if the network’s parameters and time delays are appropriately chosen, the DNNs can exhibit some complicated dynamics and even chaotic behaviors. Hence, it has attracted many scholars to study the synchronization of chaotic DNNs [17], [18], [19]. In the literature, all the control scheme for synchronization of the networks is based on static state-feedback controller. In some real control situations, there is a strong need to construct a dynamic feedback controller instead of a static-feedback controller in order to obtain a better performance and dynamical behavior of the state response. The dynamic controller will provide more flexibility compared to the static controller and the apparent advantage of this type of controller is that it provides more free parameters for selection [20].

In this paper, cellular neural networks of neutral type is considered. Due to the complicated dynamic properties of the neural cells in the real world, the existing neural network models in many cases cannot characterize the properties of a neural reaction process precisely. It is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions [21], [22]. To date, the synchronization problem of neural networks of neutral type has not been fully investigated. In this work, synchronization phenomena between drive neural network of neutral type and response one is investigated. For the synchronization, a dynamic feedback control scheme such that the dynamics of the closed-loop error system is globally stable is proposed. By constructing a suitable Lyapunov function and utilizing LMI framework, a novel criterion for the existence of proposed controller is given in terms of LMIs. The advantage of the proposed approach is that resulting criterion can be used efficiently via existing numerical convex optimization algorithms such as the interior-point algorithms for solving LMIs [23].

Notation

Rn denotes the n dimensional Euclidean space, and Rn×m is the set of all n×m real matrices. I denotes the identity matrix with appropriate dimensions. The superscript “T” represents the transpose of given vector or matrix. denotes the elements below the main diagonal of a symmetric block matrix. We denote the positive (nonnegative) definiteness of by A>0(A0) and the negative (nonpositive) definiteness of A by A<0(A0). A>B means A-B is a positive-definite matrix. diag{} denotes the block diagonal matrix.

Section snippets

Problem statement and main result

In this paper, consider a neural networks described by the following state equation:x˙(t)=-A0x(t)+A1f(x(t))+A2f(x(t-h(t)))+A3x˙(t-τ(t))+J,where x(t)=[x1(t),,xn(t)]TRn is the neuron state vector associated with n neurons, f(x(t))=[f1(x1(t)),,fn(xn(t))]TRn is the neuron activation functions, J=[J1,J2,,Jn]T is the external input vector at time t, h(t) and τ(t) corresponds to finite speed of axonal signal transmission delays satisfying 0h(t)h¯, 0τ(t)τ¯ and h˙(t)hd<1,τ˙(t)τd<1, A0=diag{ai}

Concluding remarks

In this paper, the synchronization problem via dynamic feedback controller has been studied for cellular neural networks of neutral type. A novel criterion for the existence of the controller has been presented for the network by using the Lyapunov method.

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