Pinning Control of Coupled Networks with Time-Delay.pdf

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14 The Open Electrical & Electronic Engineering Journal,2012, 6, 14-20

1874-1290/12 2012 Bentham Open

Open Access

Pinning Control of Coupled Networks with Time-Delay

Kun Yuan1, Gang Feng

2and Jinde Cao

3,*

1

Southeast University, School of Automation, China2City University of Hong Kong, Department of Manufacturing Engineering and Engineering Management, Hong Kong

3Southeast University, Department of Mathematics, China

Abstract: This paper studies the global synchronization for undirect and direct coupled networks with delay by

employing Lyapunov functional method and Kronecker product technique. A novel pinning scheme is proposed and some

sufficient conditions are given to ensure the global synchronization of coupled networks. Finally, an example is given to

show the effectiveness of the proposed pinning scheme.

Keywords:Coupled networks, pinning control, synchronization, delayed neural networks, lyapunov functional.

1. INTRODUCTION

Various large-scale and complicated systems can bemodelled by complex networks, such as the Internet, geneticnetworks, ecosystems, electrical power grids, the socialnetworks, and so on. A complex network is a large set ofinterconnected nodes, which can be described by the graphwith the nodes representing individuals in the graph and theedges representing the connections among them. The mostremarkable recent advances in study of complex networksare the developments of the small-world network model [1]and scale-free network model [2], which have been shown tobe very closer to most real-world networks as compared withthe random-graph model [3, 4]. Thereafter, small-world andscale-free networks have been extensively investigated.

The dynamical behaviors of complex networks have

become a focal topic of great interest, particularly thesynchronization phenomena, which is observed in natural,social, physical and biological systems, and has been widelyapplied in a variety of fields, such as secure communication,image processing and harmonic oscillation generation. It isnoted that the dynamical behavior of a complex network isdetermined not only by the dynamical rules governing theisolated nodes, referred to as self-dynamics, but also byinformation flow along the edges, which depends on thetopology of the network. For a given network with identicalnode dynamics and diffusive coupling, two key factorsinfluencing the network synchronization are the networkcoupling strength and the coupling configuration matrix.Synchronization in an array of linearly coupled dynamical

systems was investigated in [8]. Later, many results on local,global and partial synchronization in various coupledsystems have also been obtained in [9-15]. As a special caseof coupled systems, coupled neural networks have also beenfound to exhibit complex behaviors and theirsynchronization has been investigated by many researchers,e.g. [16-23].

*Address correspondence to this author at the Southeast University Depart-

ment of Mathematics, China; Tel: +86-25-83792315; Fax: 86-25-83792316;

E-mail: [email protected]

In the case that the whole network cannot synchronize byitself, some controllers should be designed and applied to

force the network to synchronize. A special control strategycalled pinning control [24] is designed to achievesynchronization of complex networks by controlling only afraction of the nodes or even a single node [29] over thewhole network, which has become a common technique focontrol, stabilization and synchronization of coupleddynamical systems [25-32]. In [30], both specific andrandom pinning schemes were studied, where specificpinning scheme of the nodes with large degrees is shown torequire a smaller number of controlled nodes than randompinning scheme. In [26], an adaptive pinning controller wasproposed, and it was found that the nodes with very lowdegrees should be pinned first when the coupling strength issmall. However, the design of pinning control in those

references was for the undirected network, in other wordsthe interaction topology is bidirectional, thus the couplingmatrix is symmetrical. In practice, the directed network isimportant and unidirectional communication exists inapplication. This paper aims to investigate the followingissues for undirected and directed networks:(a) What kind opinning schemes should be chosen for the given undirectedand directed network to achieve synchronization? (b) Whatype of controllers should be designed to ensure the networksynchronization? (c) How to choose the coupling strength fora network with a fixed topological structure to achievesynchronization?

The rest of the paper is organized as follows. In section 2

some preliminary definitions and lemmas are brieflyoutlined. Some synchronization criteria are given and aselective scheme is proposed in Section 3. An illustrativesimulation is given to verify the theoretical analysis inSection 4. Conclusions are finally drawn in Section 5.

Notation. Throughout this paper, for real symmetric

matrices X and ,Y the notation X Y (respectively, X >Y

means that the matrix XY is positive semi-definite

(respectively, positive definite). The superscript `` T

represents the transpose. For a matrix A , A denotes the


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Pinning Control of Coupled Networks with Time-Delay The Open Electrical & Electronic Engineering Journal, 2012, Volume 6 19

Let twenty nodes are selected for pinning control and

gains ki= 40,(i = 1,2,20) . The fact I+ c(G K) < 0 is easily

verified, where and the matrix

P =23.7709 0

0 27.9348

can be obtained from solving the

LMI (8). Therefore, the coupled network with the pinning

controllers is globally synchronized. The closed-loop

response and the synchronization state response are shown in

Fig. (2) and the synchronization error is illustrated in Fig.

(3).

5. CONCLUSIONS

In this paper, synchronization of undirected and directednetworks has been investigated via pinning control. Somecriteria for ensuring coupled networks synchronization havebeen derived, and some analytical techniques have beenproposed to obtain appropriate coupling strength and controgain matrix for achieving network synchronizationFurthermore, the effective pinning schemes has been

designed for networks with fixed structure and couplingstrength. Finally, an numerical example has also been givento illustrate the theoretical analysis.

APPENDIX

Barabasi and Albert (BA) scale-free model. Asignificant discovery in the field of complex networks is theobservation that a number of large-scale complex networksincluding the Internet,WWW and metabolic networks, arescale free and their connectivity distributions have a powerlaw form. To explain the origin of power-law degreedistribution, BA proposed another network model [2], thealgorithm of which is given as follows:

(i) Growth. Start with a small number ( m0 ) of nodes; aevery time step, a new node is introduced and is connected tom m

0already-existing nodes. (In this paper, we use m = m

- 3for convenience).

(ii) Preferential attachment. The probability i that a

new node will be connected to node i (one of the m already-

existing nodes) depends on the degreei

k of node i , in such

a way that i =ki / jkj .

CONFLICT OF INTRREST

None declared.

ACKNOWLEDGEMENT

This work was jointly supported by a grant from RGC oHong Kong (CityU:113708), the National Natural ScienceFoundation of China under Grants No. 11072059 and No61004043, and the Specialized Research Fund for the Doc-toral Program of Higher Education under Grants No20110092110017 and No. 2009092120066.

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Fig. (1).The orbit of chaotic trajectory of (28).

Fig. (2).The state of the scale-free network.

Fig. (3).Error states ei(t) in a scale free network of (28).


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Received: December 14, 2011 Revised: February 11, 2012 Accepted: February 21, 2012

Cao et al.; LicenseeBentham Open .

This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided thework is properly cited.

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Pinning Control of Coupled Networks with Time-Delay.pdf