Synchronization of singular complex dynamical networks with time-varying delays

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Abstract

This paper considers delay dependent synchronizations of singular complex dynamical networks with time-varying delays. A modified Lyapunov–Krasovskii functional is used to derive a sufficient condition for synchronization in terms of LMIs (linear matrix inequalities) which can be easily solved by various convex optimization algorithms. Numerical examples show the effectiveness of the proposed method.

Introduction

In recent years, complex dynamical networks have attracted much attention because they occur widely in various fields in the real world. A complex dynamical network is a set of interconnected nodes, in which a node is a basic unit with specific contents or dynamics. Examples of complex dynamical networks include the internet, which is a network of routers or domains, an electricity distribution network and so on. There have been a series of important research topics, especially about synchronization. Many researchers developed various efficient synchronization techniques for complex dynamical networks [1], [2], [3], [20], [21]. Wang and Chen [4] introduced a uniform dynamical network model as well as its synchronization and control. Wang and Chen [5] further investigated the synchronizability against random removal of nodes. Although these works describe the network structure well, they do not reflect time delays due to finite speed of transmission and traffic congestions. Li and Chen [6] presented a complex network model with coupling delays between the network nodes. Recently, the time-varying or uncertain models of complex networks have also been studied [7], [17], [18], [19]. Zhou et al. [8] proposed an adaptive synchronization method for uncertain complex dynamical networks and Zhang et al. [9] studied their synchronization through adaptive feedback control. Li et al. [10] considered the time-varying delays in the coupling networks of complex dynamical networks and proposed their synchronization criterion. Xiong et al. [11] studied the synchronization of the complex dynamical networks which have singular and coupled nodes without time delays.

This paper proposes a synchronization criterion for the singular complex dynamical networks with time-varying delays. Modified Lyapunov–Krasovskii functionals are used in searching for the maximum allowable time delays for synchronization. The projection lemma is utilized for handling the equality constraint. The derived sufficient condition is formulated in terms of LMIs that are easily solvable using various numerical methods.

Notations. Rn denotes the n-dimensional Euclidean space. Rn×m is the set of all n × m real matrices. For a real matrix X, X > 0 and X < 0 mean that X is a positive and negative definite symmetric matrix, respectively. I is an identity matrix with appropriate dimension and 0 is a null matrix with appropriate dimension. For a given matrix ARn×m such that rank(A) = r, we define ARn×(n-r) as the right orthogonal complement of A by AA. diag(·) represents a block diagonal matrix.

Section snippets

Problem statement

Consider the following complex dynamical network consisting of N coupled nodes with time-varying delays:Ex˙i(t)=Axi(t)+f(xi(t),t)+cj=1NdijGxj(t-τ(t)),t>0,i=1,2,,N,where ERn×n is a singular matrix (i.e., 0 < rank(E) = r < n), xi(t)=[xi1,xi2,,xin]TRn is the state vector of node i, ARn×n is a constant matrix, f(xi(t), t) is a vector-valued time-varying nonlinear function, c > 0 is a positive constant which represents coupling strength, G=diag(γ1,,γn)Rn×n is a constant diagonal inner-coupling

Main results

In this section, we derive an LMI condition for delay-dependent stability for an error dynamical network (9) with time-varying delays. Using (2), the Lipschitz condition for the nonlinear term gi is derived asgi(t)=k=1N[f(xk(t),t)-f(s(t),t)]ukik=1Nf(xk(t),t)-f(s(t),t)·|uki|k=1Nlkxk(t)-s(t)=k=1Nlkek(t)=k=1Nlkz(t)ukTk=1Nl¯zk(t)=k=2Nl¯zk(t),where uki is kth element of Ui and l¯=max(lk).

Time delays are defined as τb = βτd and τc = (1  β)τd where β  (0, 1) is an arbitrary constant.

Numerical examples

In this section, numerical examples are presented to illustrate the effectiveness of the proposed method.

Example 1

Consider the following singular network system (N = 6) with time delays [11]:Ex˙i(t)=Axi(t)+f(xi(t),t)+cj=16dijGxj(t-τ(t)),i=1,2,,6.where xi(t) = [xi,1(t)xi,2(t)]T,E=8000,A=-211-2,f(xi(t),t)=115[tanh(xi,1(t),t),tanh(xi,2(t),t)]T,D=-5111111-4111011-4101111-4101101-4110101-3,G=I,li=115 for i = 1, 2,  , 6 and l¯=115. Choosing the parameter β = 0.51, LMI solutions are obtained. The LMI solutions for node

Conclusion

This paper proposed an synchronization criterion for singular complex dynamical networks with time-varying delays. The delay-dependent sufficient condition for stability is derived in terms of LMIs. A numerical example is given to show the effectiveness of the proposed method.

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