Elsevier

Neurocomputing

Volume 309, 2 October 2018, Pages 62-69
Neurocomputing

Exponential synchronization of discrete-time impulsive dynamical networks with time-varying delays and stochastic disturbances

https://doi.org/10.1016/j.neucom.2018.04.070Get rights and content

Abstract

In this paper, based on the discrete-time delayed impulsive system theory, exponential synchronization of discrete-time complex networks with both time-varying delays and stochastic disturbances is investigated. By applying an iterative Lyapunov function in combination with the linear matrix inequality (LMI) technique, a new synchronization criterion based on topology matrices and impulsive conditions is developed. Some numerical simulations are provided to verify the theoretical results.

Introduction

Since the discoveries of the small-world property [1] and scale-free feature [2] in complex networks, the notion of network science has attracted more and more attention from scientific and engineering communities. In the past decade, great progress has been made on complex networks, for example, regarding modelling and statistical analysis, epidemics and rumors spreading, big-data diagnosis on complex networks, dynamics and evolution patterns, control and synchronization, various applications of complex networks, and so on.

Synchronization, as an interesting collective behavior in a complex dynamical network, is receiving increasing research endeavor due to its frequent occurrence and broad applications in natural as well as manmade systems. Some recent advances on network synchronization have been reported in [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. In the earlier years, most research focused on the synchronization of continuous-time networks with or without delays. In [10], the sampled-data synchronization problem was formulated as an exponential mean-square stabilization problem for a class of dynamical networks, which involve both multiple probabilistic interval delays and sector-bounded nonlinearities. In [11], the synchronization of a complex dynamical network was investigated by pinning a small portion of nodes, instead of controlling all the nodes. Thereafter, the impulsive control method has also been widely utilized to control and synchronize complex networks [12], [13], [14], [15], [16], [17], [18]. It guarantees stability and synchronizability of complex networks by sending small impulses to the received systems at some discrete impulsive instants, which can reduce the information redundancy in the transmitted signals and increase the robustness against disturbances. Generally, for a continuous-time system or network, the stability is analyzed by constructing a Lyapunov function and then computing its derivative in impulsive intervals, with jumps at discrete impulsive instants. However, for a discrete-time system or network, analyzing its stability and convergence is often much more difficult because one has to compute the iterative Lyapunov function rather than its derivative. Therefore, the synchronization problem for discrete-time networks with different delays and impulses has so far received limited attention mainly due to its mathematical complexity. In fact, the main difficulties would be: (1) how to handle the error dynamics in terms of time-varying delays, couplings, nonlinearities and stochastic external disturbances; and (2) how to design a set of easy-to-implement impulsive controllers in order to realize the exponential synchronization of the dynamical network.

In [19], stability of some typical discrete-time dynamical networks under pinning control was studied. In [20], discrete-time complex dynamical networks subject to impulsive control were investigated. Combining these two issues, in [21], some criteria about impulsive pinning synchronization of stochastic discrete-time networks were derived. Two kinds of consensus problems for discrete-time linear multi-agent systems under switching network topologies were investigated in [22]. However, all the above-mentioned references did not consider time delays in the models and processes.

Recently, in [23], [24], global exponential stability of discrete-time neural networks and synchronization as well as state estimation for discrete-time complex networks were studied, respectively. In [25], the uniform asymptotic stability of impulsive discrete-time systems with time delays was investigated. Moreover, in [26] and [27], some results on the exponential stability of impulsive discrete-time systems with time delays were obtained. More recently, in [28], some sufficient conditions for synchronization of discrete-time impulsive switched systems with time delays were established.

Note that the delays in the aforementioned studies were assumed to be constants. In practice, however, delays may be time-varying, on which there are some results reported in the literature. For example, in [29], [30], the stability of discrete-time neural networks with time delays, possibly subject to stochastic effects, was studied. In [31], some criteria were derived for the exponential stability of discrete-time stochastic neural networks with time-varying delays subject to impulsive control. Very recently, in [32], some results about the exponential synchronization of discrete-time impulsive networks with time-varying delays were presented, based on the concept of “average impulsive interval”.

To the best of our knowledge, very little work has been done on the impulsive synchronization of discrete-time complex dynamical networks with both time-varying delays and stochastic disturbances. Motivated by the lack of the study on this important issue, in this paper, the exponential synchronization of such networks by impulsive control is investigated. By constructing an iterative Lyapunov function in combination with the linear matrix inequality (LMI) technique, a new synchronization criterion is developed for the presented impulsively controlled network based on Lemma 1. Finally, some numerical simulations are presented to illustrate the effectiveness of the proposed approach.

The main contributions of this paper are summarized as follows:

  • (1)

    The presented network model is more comprehensive and practical than the models used in [21], [31] and [32], for which it is more difficult to derive an easily-verifiable synchronization criterion.

  • (2)

    The less-conservative synchronization conditions derived in this paper are expressed by some relatively simple inequalities involving the inner connection matrix, adjacency matrix, impulsive intervals and impulsive feedback gain matrices, which give some design guidelines for selecting the controller parameters, and the new conditions are easier to verify and implement.

  • (3)

    This paper extends the study of stability for bidirectional associative memory (BAM) neural networks developed in [31] to synchronization of general complex networks. Because complex networks widely exist in the real world, this extension is significant with foreseeable potential applications.

Notation: For a real symmetric matrix P, P > 0 (P ≥ 0) means that the matrix P is positive definite (semi-positive definite); the symmetric terms below the main diagonal of a symmetric matrix are marked by *; for any matrix A, λmin (λmax) denotes its minimum (maximum) eigenvalue; I is the identity matrix with appropriate dimension; N denotes the set of nonnegative integers; Z+ denotes the set of positive integers; R denotes the set of real numbers; R+ denotes the set of positive real numbers; ‖ · ‖ stands for the Euclidean norm; E(·) denotes the mathematical expectation.

Section snippets

Problem formulation and preliminaries

Suppose s(k)=(s1(k),s2(k),,sn(k))TRn is a solution of an isolated node with time-varying delay: {s(k+1)=As(k)+f˜(s(k))+g˜(s(kτk)),s(τ)=ϕ(τ),τJ={τ¯,τ¯+1,,2,1,0},where A=diag{a1,a2,,an}, f˜=(f˜1,f˜2,,f˜n)T and g˜=(g˜1,g˜2,,g˜n)T are n-dimensional vector-valued functions satisfying f˜(0)=0 and g˜(0)=0, and τk is a bounded time-varying delay with k, which is an integer satisfying 0τkτ¯, with τ¯ being a positive integer. ϕ(τ): J → Rn is a discrete vector-valued initial function.

In the

Main results

In this section, the stability of the origin of the error systems (3) will be analyzed in detail.

Definition 1

The discrete-time stochastic impulsive network (2) is said to achieve exponential synchronization, if there exist constants M¯>0,μ>0 such that E{i=1Nei(k)2}M¯φ¯eμk,kZ+,where φ¯=maxkJ,1iN{φiT(k)φi(k)}.

For notational convenience, denote L1=diag{l1l1+,,lnln+},L2=diag{l1+l1+2,,ln+ln+2},Σ1=diag{σ1σ1+,,σnσn+},Σ2=diag{σ1+σ1+2,,σn+σn+2},Υ1=diag{ν1ν1+,,νnνn+},Υ2=diag{ν1+ν1+2,,νn+νn

Numerical simulations

For convenience in simulations, impulsive gain matrices are as usual chosen as a constant matrix Binl=B, and impulsive distances are set to be a positive constant Δl=4, namely impulse controllers are acted at k=4,8,12.

Example 1

Let the dynamical matrix in Eq. (1) be A=diag{0.2,0.3}, f˜1(s)=f˜2(s)=tanh(0.6s), g˜1(s)=g˜2(s)=0.3tanh(s), τk=mode(k,3)(kZ+). Then, the discrete time series of this isolate node is shown in Fig. 1.

Let the coupling function h˜1(s)=h˜2(s)=0.1s, the inner coupling matrix Γ=I

Conclusions

In this paper, the exponential synchronization of discrete-time complex networks, with both timevarying delays and stochastic disturbances, is studied. Based on the delayed impulsive control systems theory, by using the linear matrix inequalities and an iterative Lyapunov function, an easily verified criterion is derived based on rigorous mathematical analysis, for the exponential synchronization of this kind of complex networks. Some numerical simulations are demonstrated for verifying the

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant nos. 61304022, 11271295 and 61573011, the Hong Kong Research Grants Council under the GRF Grant CityU 11234916, the Educational Commission of Hubei Province under Grant nos. D20131602, Q20141609 and Q20151609, and the Science and Technology Innovation Program of Wuhan Textile University (No. 153027).

Qunjiao Zhang received the Ph.D. from Wuhan University, Wuhan, China, and Post-Doctoral Fellow in the Department of Applied Mathematics at Le Havre University in Normandy, France. Currently, she is a Professor in the College of Mathematics and Computer Sciences, at Wuhan Textile University, Wuhan, China. She is also the author or coauthor of more than 30 journal papers. Her research interests include complex systems and networks, chaos control and synchronization.

References (35)

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Qunjiao Zhang received the Ph.D. from Wuhan University, Wuhan, China, and Post-Doctoral Fellow in the Department of Applied Mathematics at Le Havre University in Normandy, France. Currently, she is a Professor in the College of Mathematics and Computer Sciences, at Wuhan Textile University, Wuhan, China. She is also the author or coauthor of more than 30 journal papers. Her research interests include complex systems and networks, chaos control and synchronization.

Guanrong Chen    received the M.Sc. degree in Computer Science from Sun Yat-sen University, Guangzhou, China in 1981 and the Ph.D. degree in Applied Mathematics from Texas A&M University, College Station, Texas in 1987. He has been a Chair Professor and the Founding Director of the Centre for Chaos and Complex Networks at the City University of Hong Kong since year 2000, prior to that he was a tenured Full Professor at the University of Houston, Texas, USA. He was elected IEEE Fellow in 1997, awarded the 2011 Euler Gold Medal, Russia, and conferred Honorary Doctorate by the Saint Petersburg State University, Russia in 2011 and by the University of Le Havre, Normandy, France in 2014. He is a Member of the Academia of Europe and a Fellow of The World Academy of Sciences, and is a Highly Cited Researcher in Engineering as well as in Mathematics according to Thomson Reuters.

Li Wan    received the Ph.D. from Nanjing University, Nanjing, China, and Post-Doctoral Fellow in the Department of Mathematics at Huazhong University of Science and Technology, Wuhan, China. Currently, he is a Professor in the College of Mathematics and Computer Sciences at Wuhan Textile University, Wuhan, China. He is also the author or coauthor of more than 40 journal papers. His research interests include nonlinear dynamic systems, neural networks, control theory.

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