Synchronization of hybrid-coupled heterogeneous networks: Pinning control and impulsive control schemes

https://doi.org/10.1016/j.jfranklin.2013.12.017Get rights and content

Abstract

In this paper, the synchronization control problem is considered for the delayed hybrid-coupled heterogeneous networks, i.e., complex networks with nonidentical dynamical nodes. Some effective control schemes are designed under which the whole network is globally asymptotically synchronized to an arbitrary objective trajectory. By imposing the open-loop control on the whole network together with the feedback control only on a small fraction of the nodes, an easy-to-verify sufficient condition is derived to guarantee the asymptotic synchronization of the complex network under study. Furthermore, to decrease the feedback control gains, the idea of adaptive control scheme is combined together, and the verified conditions are further weakened. Finally, by introducing the impulses to the open-loop network and using the improved Halanay inequality, other novel synchronization criteria are developed for the complex network. Comparisons of the obtained theoretical results as well as the detail pinning schemes are also given. Numerical examples of the undirected scale-free network and the directed small-world network are illustrated to demonstrate the applicability and efficiency of the proposed theoretical results.

Introduction

During the past few decades, synchronization phenomena known as typical collective behaviors have received notable attentions, especially phase synchronization of the coupled-oscillator systems and the chaotic synchronization of the drive-response (or the master–slave) systems [1], [2]. With the discoveries of the small-world networks [3] and the scale-free networks [4], complex networks have witnessed unprecedented developments in various fields, and the discussions of synchronization problems for complex networks have been extensively launched.

In the literature, the dynamical networks under synchronization studies are usually coupled linearly and instantaneously, and all the nodes in the network are governed by the same dynamical model when decoupled. These restrictions obviously do not match the practical cases in the real world, where the network nodes may evolve in different dynamical equations, and the delays do exist either in the individual node dynamics (called decoupling delays) or during the signal transmission processes from nodes to nodes (called coupling delays) in the form of constants, time-varying or distributed ones [5], [6]. Under proper coupling strengths, the synchronization phenomenon occurs spontaneously if the nodes of the complex dynamical network have a common synchronization manifold. However, in some networks such as network with nonidentical nodes, such a kind of synchronization manifold does not exist. In order to achieve the expected synchronization characteristic, extra controllers have to be added on the network nodes.

Recently, great efforts have been devoted to the investigation of synchronization control problem for the nonidentical coupled networks (or heterogeneous networks). For example, by a simple pinning control technique, cluster synchronization has been considered in [7] for the community networks with nonidentical nodes. The relationship between the cluster synchronization and the unweighted graph topology has been discussed in [8]. Via combining the open-loop control and the adaptive strategy as well as the impulsive effects, a dynamical network with nonidentical nodes has been synchronized to any given smooth goal dynamics [9]. By using only one impulsive controller [10], the authors proved that the network can be pinned to any prescribed state if the underlying graph of the network has spanning trees. In [11], the authors studied the synchronization control problem of impulsive dynamical networks under a single impulsive controller or a single negative state-feedback control controller. Based on the decentralized adaptive pinning control scheme, the authors in [12] considered the cluster synchronization of the undirected dynamical networks. By introducing multiple V-Lyapunov functions, the global asymptotic stability of the networks with nonidentical nodes has been studied [13]. Based on the construction of reasonable corresponding impulsively controlled response networks, the authors [14] investigated the generalized function synchronization of continuous and discrete complex networks by impulsive control. For more related works see [15], [16], [17] and the references cited therein. It should be noticed that in all the abovementioned literatures, dynamics of the isolated nodes are governed by the systems independent of time delays.

The synchronization problem of the complex network with nonidentical delayed nodes has been considered by pinning control in [18], [19] under adaptive coupling strengths. By proposing a high gain integral controller [20], the synchronization problem of complex dynamical networks with unknown nonidentical nodes was converted into a singular perturbation form. For the coupled dynamical networks with nonidentical Duffing-type oscillators without/with coupling delays, the impulsive synchronization problem was investigated in [21]. In [22], the synchronization for time-delayed complex networks with adaptive coupling weights and feedback gains was studied under pinning strategy. For more related works see [23], [24] and the references cited therein. While in [25], [26], stochastic perturbation cases have been taken into account, where the impulsive adaptive control method has been resorted to.

In this contribution, the synchronization control problem of a hybrid-coupled network with nonidentical nodes is considered. Each decoupled node is governed by a different time delay dynamical system, and coupling delays in the discrete-time/distributed forms are also considered. The main contributions of this paper can be summarized from the following four aspects. Firstly, by designing the pinning (adaptive) controller and the impulsive controller, the hybrid-coupled complex networks are synchronized asymptotically/exponentially to the objective system. To the best of our knowledge, the pinning synchronization control of such nonidentical hybrid-coupled network has not been explored up to date. Secondly, reduced-order matrix conditions are derived for the hybrid-coupled network which are different from the previous works with LMIs or full-order matrix verified conditions when dealing with time delays and pinning control problems. Thirdly, based on the improved Halanay inequality, the synchronized impulsive control becomes more efficiency and less conservative. Finally, comparisons of the obtained theoretical results are carried out in the end of the paper.

The remainder of the paper is organized as follows. In Section 2, the synchronization problem is formulated for the complex network with nonidentical dynamical nodes and some useful preliminaries are stated. Section 3 gives the main results of this contribution, meanwhile the specific pinning scheme is introduced both for the undirected networks and for the digraphs. In Section 4, numerical simulations are given to illustrate the effectiveness of the proposed approaches. Finally, conclusions are presented in Section 5.

Notation: Throughout this paper, Rn and Rn×m denote the n-dimensional Euclidean space and the set of all real n×m matrices, respectively. The notation P>Q, where P and Q are the symmetric matrices, means that the matrix PQ is positive-definite. The superscript ‘T’ stands for matrix transposition and diag{} denotes a block diagonal matrix. The Kronecker product of matrices A and B is denoted as AB and · is the Euclidean norm of a vector or its induced norm of a matrix. The abbreviation As of matrix A represents the matrix 12(A+AT). For any matrix MRN×N, Ml denotes the minor matrix of M by removing its first l(1lN) row–column pairs from M [27]. Let λmin(A) and λmax(A) be the minimal and maximal eigenvalues of matrix A, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for the algebraic operations.

Section snippets

Preliminaries and model description

Consider a general class of complex networks consisting of N nonidentical dynamical nodes with diffusive couplings and hybrid delays described by the following equations:ẋi(t)=fi(t,xi(t),xi(tσi(t)))+c0j=1NGij(0)Γ0xj(t)+c1j=1NGij(1)Γ1xj(tτ1(t))+c2j=1NGij(2)Γ2tτ2(t)txj(s)ds+ui(t),t0where i=1,2,,N and xi(t)=(xi1,xi2,,xin)TRn is the state variable of the ith node at time t. The function fi(·):R+×Rn×RnRn, denoting the local dynamics of the ith node, is continuous and capable of

Main results

In this section, we will investigate the global synchronization of the hybrid-coupled network (1) via different control schemes: pinning (adaptive) control and impulsive control which will be stated below in different subsections.

Numerical simulations

This section provides two simulation examples to verify the effectiveness of the proposed pinning (adaptive) synchronization control and the impulsive synchronization control schemes for the hybrid-coupled complex networks.

Example 1

Consider an undirected scale-free network consisting of 200 nonidentical nodes described by the delayed Lorenz-like system:ẋi(t)=fi(t,xi(t),xi(tσi(t)))+c0j=1200Gij(0)Γ0xj(t)+c1j=1200Gij(1)Γ1xj(tτ1(t))+c2j=1200Gij(2)Γ2tτ2(t)txj(s)dswhere i=1,2,,200, xi(t)=(xi1(t),xi2(t

Conclusions

This paper has investigated the synchronization control problem of the hybrid-coupled complex networks with nonidentical nodes, where the linear couplings include both the discrete time-varying case and the distributed delay form. The inner coupling matrices are not necessary to be diagonal, and the outer coupling matrices are just to be diffusive without restrictions on their symmetry or irreducibility. Several kinds of control schemes are utilized to synchronize the whole dynamical network to

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 61174136 and 61272530, the Natural Science Foundation of Jiangsu Province of China under Grants BK2011598 and BK20130017, and the Fundamental Research Funds for the Central Universities under Grant 2242012155.

References (47)

  • G. Zhang et al.

    Exponential synchronization for delayed chaotic neural networks with nonlinear hybrid coupling

    Neurocomputing

    (2012)
  • X. Guo et al.

    A new synchronization algorithm for delayed complex dynamical networks via adaptive control approach

    Commun. Nonlinear Sci. Numer. Simul.

    (2012)
  • X.F. Wang et al.

    Pinning control of scale-free dynamical networks

    Physica A

    (2002)
  • L. Pan et al.

    Anti-periodic solution for delayed cellular neural networks with impulsive effects

    Nonlinear Anal.Real World Appl.

    (2011)
  • W. Yu et al.

    Distributed control gains design for consensus in multi-agent systems with second-order nonlinear dynamics

    Automatica

    (2013)
  • W. Yang et al.

    Fast consensus seeking in multi-agent systems with time delay

    Syst. Control Lett.

    (2013)
  • R. Cepeda-Gomez et al.

    Exact stability analysis of second-order leaderless and leader–follower consensus protocols with rationally-independent multiple time delays

    Syst. Control Lett.

    (2013)
  • L. Ding et al.

    Network-based leader-following consensus for distributed multi-agent systems

    Automatica

    (2013)
  • M. Porfiri et al.

    Criteria for global pinning-controllability of complex networks

    Automatica

    (2008)
  • T. Opsahl et al.

    Node centrality in weighted networksgeneralizing degree and shortest paths

    Soc. Netw.

    (2010)
  • T.L. Carroll et al.

    Synchronizing chaotic circuits

    IEEE Trans. Circuits Syst.

    (1991)
  • M.G. Rosenblum et al.

    Phase synchronization of chaotic oscillators

    Phys. Rev. Lett.

    (1996)
  • D.J. Watts et al.

    Collective dynamics of ‘small-world’ networks

    Nature

    (1998)
  • Cited by (66)

    View all citing articles on Scopus
    View full text