Synchronization control of switched linearly coupled neural networks with delay

ARTICLE IN PRESS

Neurocomputing 73 (2010) 858–866

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/neucom

Synchronization control of switched linearly coupled neural networkswith delay

Wenwu Yu a,b,�, Jinde Cao a, Wenlian Lu c

a Department of Mathematics, Southeast University, Nanjing 210096, Chinab Department of Electronic Engineering, City University of Hong Kong, Hong Kongc Laboratory of Nonlinear Mathematics Science, Institute of Mathematics, Fudan University, Shanghai, China

a r t i c l e i n f o

Article history:

Received 2 January 2009

Received in revised form

21 July 2009

Accepted 4 October 2009

Communicated by J. LiangA globally convergent algorithm involving convex optimization is also presented to construct such

Available online 18 November 2009

Keywords:

Complex network

Linearly coupled delayed neural networks

Lyapunov functional

Global synchronization

Linear matrix inequality

Switched system

12/$ - see front matter & 2009 Elsevier B.V. A

016/j.neucom.2009.10.009

esponding author. Tel.: +852 21942553.

ail address: [email protected] (W. Yu).

a b s t r a c t

In this paper, synchronization control of switched linearly coupled delayed neural networks is

investigated by using the Lyapunov functional method, synchronization manifold and linear matrix

inequality (LMI) approach. A sufficient condition is derived to ensure the global synchronization of

switched linearly coupled complex neural networks, which are controlled by some designed controllers.

controllers effectively. In many cases, it is desirable to control the whole network by changing the

connections of some nodes in the complex network, and this paper provides an applicable approach. It

is even applicable to the case when the derivative of the time-varying delay takes arbitrary. Finally,

some simulations are constructed to justify the theoretical analysis.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Synchronization and stability control of dynamical systems isan important topic in nonlinear system control [1–10,44–46] inthe past decades. Recently, arrays of coupled systems haveattracted much attention of researchers in different researchfields. The study of synchronization of coupled neural networks isan important step for both understanding brain science anddesigning coupled neural networks for practical use.

Networks of coupled connection have been widely investi-gated [11–26] since Wang and Chen introduced an array of N

linearly coupled connected complex network model [27,28].Consider a complex dynamical network consisting of N identicallinearly and diffusively coupled nodes, with each node being ann-dimensional dynamical system in [27,28] as follows:

_xiðtÞ ¼ f ðxiðtÞÞþcXN

j ¼ 1;ja i

GijGðxjðtÞ�xiðtÞÞ; i¼ 1;2; . . . ;N; ð1Þ

where xiðtÞ ¼ ðxi1ðtÞ; xi2ðtÞ; . . . ; xinðtÞÞT ARn ði¼ 1;2; . . . ;NÞ is the

state vector representing the state variables of node i, f :Rn�!Rn is continuously differentiable, the constant c is thecoupling strength, G¼ diagðg1; g2; . . . ; gnÞARn�n is a constant 0–1

ll rights reserved.

matrix linking the coupled variables with gi ¼ 1 for a specific i andgj ¼ 0 ðja iÞ, that is, there is only one 1 in the diagonal of matrix Gand all the other components of G are zeros, G¼ ðGijÞN�N is thecoupling configuration matrix representing the topological struc-ture of the network, in which Gij is defined as follows: if there is aconnection between node i and node j ðja iÞ, then the couplingstrength Gij ¼ Gji40; otherwise, Gij ¼ Gji ¼ 0 ðja iÞ, and the diag-onal elements of matrix G are defined by

Gii ¼�XN

j ¼ 1;ja i

Gij: ð2Þ

Then, in this case, the complex network (1) reduces to the model

_xiðtÞ ¼ f ðxiðtÞÞþcXN

j ¼ 1

GijGxjðtÞ; i¼ 1;2; . . . ;N: ð3Þ

Hereafter, suppose that the network (3) is connected in the sensethat there are no isolate clusters. Thus, the coupling configurationG is an irreducible matrix.

In the following, a brief introduction of recent works aboutsynchronization of linearly coupled complex networks are givenbased on the model (1)–(3).

For the case that the coupling matrix G is irreducible,symmetric, and all the off-diagonal elements of G are nonegativeand satisfies (2), local synchronization analysis via linearlizationtechnique was studied in [16–20], where the eigenvalues and

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Fig. 1. Architecture of the switched coupled neural networks.

W. Yu et al. / Neurocomputing 73 (2010) 858–866 859

Jacobian matrix are given in the criteria of ensuring the localsynchronization of the complex network. Wu and Chua [29,30]investigated the synchronization in an array of linearly coupleddynamical systems, and then a lot of works [11–13,15,21] weredevoted to investigating the global asymptotical synchronizationof the complex network by using the synchronization manifoldand Lyapunov method. In [29], the authors defined a distancebetween the collective states and the synchronization manifold,based on which the a methodology was proposed to discuss globalsynchronization of the coupled systems.

In [11,12,15], the following linearly coupled neural networkssatisfying (2) was studied:

_xiðtÞ ¼�CxiðtÞþAf ðxiðtÞÞþBf ðxiðt�tÞÞþ IðtÞþXN

j ¼ 1

GijGxjðtÞ; i¼ 1;2; . . . ;N;

ð4Þ

where C ¼ diagðc1; c2; . . . ; cnÞARn�n is a diagonal matrix withpositive diagonal entries ci40; i¼ 1;2; . . . ;n, A¼ ðaijÞn�n andB¼ ðbijÞn�n are weight and delayed weight matrices, respectively.IðtÞ ¼ ðI1ðtÞ; I2ðtÞ; . . . ; InðtÞÞ

T ARn is an external input vector,G¼ diagðg1; g2; . . . ; gnÞARn�n.f ðxiðtÞÞ ¼ ðf1ðxi1ðtÞÞ; f2ðxi2ðtÞÞ; . . . ; fnðxinðtÞÞÞ

T ARn corresponds to theactivation functions of neurons. Equivalently, system (4) can bewritten as

_xikðtÞ ¼�ckxikðtÞþXn

l ¼ 1

aklflðxilðtÞÞþXn

l ¼ 1

bklflðxilðt�tÞÞ

þ IkðtÞþXN

j ¼ 1

GijgkxjkðtÞ; i¼ 1;2; . . . ;N; k¼ 1;2; . . . ;n: ð5Þ

With the rapid development of intelligent control, hybridsystems have been widely investigated. It is found that manyphysical and biological models are governed by more than onedynamical system and these systems are changed depending ontime. Switched systems [31–34], a special case in hybrid systems,are regarded as nonlinear systems, which are composed of afamily subsystems and a rule that orchestrates the switchingbetween the subsystems. Recently, switched systems havenumerous applications in communication systems [6,35,36],control of mechanical systems, automotive industry, aircraft andair traffic control, electric power systems [37] and many otherfields. In [31–34], the stability of switching system was investi-gated, which is a combination of discrete and continuousdynamical systems.

The main contribution of this paper is of threefolds. Firstly, westudied global synchronization of coupled systems with time-varying delay by using LMI and distance function from collectivestates to the synchronization manifold [26]. A delay-dependentcondition is given to ensure the synchronization of coupledsystems in this paper based on free-weighting matrix approachand cone complementarity linearization algorithm [47–49]. It isnoted that the derivative of time delay can take any value.Secondly, the feedback matrix of the network is designed to adjustthe configuration matrix, i.e., the connections among the nodes.Thirdly, it is very difficult to design the feedback matrix due to thecomplexity of the systems. So, a globally convergent algorithminvolving convex optimization is presented.

The rest of the paper is organized as follows: In Section 2,preliminaries are given. In Section 3, the main results are derived.A sufficient condition is given to ensure the synchronization ofswitched coupled networks and a globally convergent algorithminvolving convex optimization is also presented to design suchcontrollers effectively. In Section 4, numerical simulations areconstructed to justify the theoretical analysis in this paper.Finally, the conclusion is drawn.

2. Preliminaries

A set of coupled complex neural networks is considered as theindividual subsystems of the switched system and the switchedcoupled neural network is described as follows:

_xiðtÞ ¼ �CaxiðtÞþAaf ðxiðtÞÞþBaf ðxiðt�tÞÞþ IaðtÞþXN

j ¼ 1

GaijDxjðtÞ; i¼ 1;2; . . . ;N;

ð6Þ

where D is inner coupling matrix and a is a switching signal whichtakes its value in the finite set I ¼ f1;2; . . . ;Ng. This means thatthe matrices ðCa;Aa;Ba; Ia;GaÞ are allowed to take values, atparticular time, in a finite set fðC1;A1;B1; I1;G1Þ; ðC2;A2;B2; I2;G2Þ;

. . . ; ðCN ;AN ;BN ; IN ;GN Þg. Throughout this paper, we assume thatthe switching rule a is not known priori and its instantaneousvalue is available in real time.

Since in most cases the time delay is not a constant, in thispaper, the coupled neural network with time-varying delay isstudied. Consider the state-feedback control law

uaiðtÞ ¼XN

j ¼ 1

KaijDxjðtÞ; i¼ 1;2; . . . ;N; ð7Þ

where

Kaii ¼�XN

j ¼ 1;ja i

Kaij: ð8Þ

It is useful to design a memoryless state-feedback controller uaiðtÞ

so that the coupled system (6) is globally synchronized. In thispaper, a linearly feedback controller (7) is added to the coupleddynamical system (6)

_xiðtÞ ¼ �CaxiðtÞþAaf ðxiðtÞÞþBaf ðxiðt�tðtÞÞÞþ IaðtÞ

þXN

j ¼ 1

ðGaijþKaijÞDxjðtÞ; i¼ 1;2; . . . ;N: ð9Þ

It is easy to see that one can control the synchronization ofcoupled neural network by adjusting the configuration couplingmatrix, that is, the network topology can be changed to achievesynchronization. The architecture for such switched coupledneural networks is shown in Fig. 1. Next, we focus on globalasymptotical synchronization of coupled feedback system (9).

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W. Yu et al. / Neurocomputing 73 (2010) 858–866860

Define an indicator function xðtÞ ¼ ðx1ðtÞ; x2ðtÞ; . . . ; xN ðtÞÞT ,

where

xkðtÞ ¼

1 when the switched system is described

by the kth mode ðCk;Ak;Bk; Ik;GkÞ;

0 otherwise;

8><>: ð10Þ

with k¼ 1;2; . . . ;N . The model of switched coupled neuralnetwork model (9) can be written as

_xiðtÞ ¼XN

k ¼ 1

xkðtÞ½�CkxiðtÞþAkf ðxiðtÞÞþBkf ðxiðt�tðtÞÞÞþ IkðtÞ

þXN

j ¼ 1

ðGkijþKkijÞDxjðtÞ�; i¼ 1;2; . . . ;N: ð11Þ

It follows thatPN

k ¼ 1 xkðtÞ ¼ 1 under any switching rules.We assume that the system (11) satisfies the following

initial conditions: xiðtÞ ¼fiðtÞACð½�r;0�;RÞ ði¼ 1;2; . . . ;NÞ withr¼maxtARftðtÞg, where Cð½�r;0�;RÞ denotes the set of all contin-uous functions from ½�r;0� to R.

In order to derive the main results, it is necessary to make thefollowing assumptions:

A1: The activation functions fiðxiÞ ði¼ 1;2; . . . ;nÞ are Lipschitzcontinuous, that is, there exist constants Fi40 such that

jfiða1Þ�fiða2ÞjrFija1�a2j; 8a1;a2AR: ð12Þ

A2: tðtÞ is a bounded differential function of time t, i.e.,r¼maxtARftðtÞg, and the following condition is satisfied:

0r _tðtÞrh; ð13Þ

where h is a positive real constant.A3: The coupling matrix Ga and the feedback gain matrix Ka are

defined by

Gaii ¼�XN

j ¼ 1;ja i

Gaij; i¼ 1;2; . . . ;N; a¼ 1;2; . . . ;N ; ð14Þ

and

Kaii ¼�XN

j ¼ 1;ja i

Kaij; i¼ 1;2; . . . ;N; a¼ 1;2; . . . ;N : ð15Þ

Let In be the n-dimensional identity matrix.

Definition 1 (Lu and Chen [11]). Let r¼maxtARftðtÞg, the setS¼ fx¼ ðx1ðsÞ; x2ðsÞ; . . . ; xNðsÞÞ : xiðsÞACð½�r;0�;RÞ; xiðsÞ ¼ xjðsÞ,i; j¼ 1;2; . . . ;Ng is called the synchronization manifold.

Definition 2. Synchronization manifold S is said to be globallyasymptotically stable, equivalently, the coupled system (11) isglobally asymptotically synchronized, if for any e40, for eachinitial data fiðsÞ; sA ½�r;0�; i¼ 1;2; . . . ;N, there exists T40, suchthat

JxiðtÞ�xjðtÞJre; ð16Þ

holds for all t4T ; i; j¼ 1;2; . . . ;N.

Definition 3 (Wu and Chua [29]). Let R denote a ring, and defineTðR;KÞ ¼ fthe set of matrices with entries R such that the sum ofthe entries in each row is equal to K for some KA R:g

Definition 4 (Wu and Chua [29]). Set of MN1 ð1Þ: MN

1 ð1Þ iscomposed of matrices with N columns. Each row (for instance,the i th row) of ~M AMN

1 ð1Þ has exactly one entry ai and one entry�ai, where aia0. All the other entries are zeros.

Definition 5 (Wu and Chua [29]). Set of MN1 ðnÞ: MN

1 ðnÞ arematrices M obtained by replacing entry mij in ~M AMN

1 ð1Þ with

mijIn, i.e., MN1 ðnÞ ¼ fM¼

~M � In : ~M AMN1 ð1Þg,where � is Kronecker

product.

Definition 6 (Wu and Chua [29]). MN2 ðnÞ �MN

1 ðnÞ: If MAMN2 ðnÞ,

then, for any pair of indices i and j, there exist indices j1; j2; . . . ; jl,where j1 ¼ i and jl ¼ j, and p1; p2; . . . ; pl�1 such that Mpq ;jq a0 andMpq ;jqþ 1

a0 for all 1rqo l.

Definition 7 (Kronecker product). For matrices A and B, the notationA� B stands for the matrix composed of submatrices AijB, i.e.,

A� B¼

A11B A12B � � � A1nB

A21B A22B � � � A2nB

� � � � � � & � � �

Am1B Am2B � � � AmnB

0BBBB@

1CCCCA;

where Aij; i¼ 1;2; . . . ;m; j¼ 1;2; . . . ;n stands for the ij th entry ofthe m� n matrix A.

Lemma 1 (Wu and Chua [29]). Let G be a N � N matrix in TðR;KÞ.Then the ðN�1Þ � ðN�1Þ matrix H is defined by H¼MGJ satisfying

MG¼HM, where M is the ðN�1Þ � N matrix

M¼

1 �1

1 �1

&

1 �1

0BBB@

1CCCA; ð17Þ

and J is the N � ðN�1Þ matrix

J¼

1 1 1 � � � 1

0 1 1 � � � 1

& 1

� � � 1 1

0 0 � � � 0 1

0 0 0 � � � 0

0BBBBBBBB@

1CCCCCCCCA; ð18Þ

in which 1 is the multiplicative identity of R.

Lemma 2 (Wu and Chua [29]). Let x¼ ðx1; x2; . . . ; xNÞT , where

xiARn; i¼ 1;2; . . . ;N. Then xAS if and only if

JMxJ¼ 0 ð19Þ

holds for some MAMN2 ðnÞ. We use dðxÞ to denote a nonegative real-

valued function that measures the distance between the various

nodes. In particular, dðxÞ is of the following form:

dðxÞ ¼ JMxJ2¼ xT MT Mx; MAMN

2 ðnÞ: ð20Þ

Because of the assumptions on M, the crucial property of dðxÞ is that

dðxÞ�!0 if and only if JxiðtÞ�xjðtÞJ�!0 for all i and j.

Lemma 3 (Chen and Chen [38]). By the definition of Kronecker

product, the following properties can be satisfied for appropriate

dimensions:

(1)

Lemma 4 (Schur complement [39]). The following linear matrix

inequality (LMI):

Q ðxÞ SðxÞ

SðxÞT RðxÞ

!40;

where Q ðxÞ ¼ Q ðxÞT ; RðxÞ ¼ RðxÞT , is equivalent to one of the following

conditions:

(i)

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W. Yu et al. / Neurocomputing 73 (2010) 858–866 861

Lemma 5 (Gu et al. [40]). For any constant matrix

WARm�m; W ¼WT , scalar r40, vector function o : ½0; r�ARm such

that the integrations concerned are well defined, then

r

Z r

0oT ðsÞWoðsÞdsZ

Z r

0oðsÞds

� �T

W

Z r

0oðsÞds

� �:

3. Main results

In this section, new criteria are presented for the globalsynchronization of system (11) based on Lyapunov functionalmethod and linear matrix inequality (LMI) approach.

First, some notations are given to simplify the proof. M isdefined in (17). Let M¼M � InAMN

2 ðnÞ be the matrix defined inDefinition 6.

Let A� B denote the Kroneker product of matrices A and B andalso let

Ck ¼ IN � Ck; C1k ¼ IN�1 � Ck; Ak ¼ IN � Ak; A1

k ¼ IN�1 � Ak;

Bk ¼ IN � Bk; B1k ¼ IN�1 � Bk; Gk ¼ Gk � D; Kk ¼ Kk � D;

xiðtÞ ¼ ðxi1ðtÞ; xi2ðtÞ; . . . ; xinðtÞÞT ; 8i¼ 1;2; . . . ;N;

xðtÞ ¼ ðxT1ðtÞ; x

T2ðtÞ; . . . ; x

TNðtÞÞ

T ;

fðxðtÞÞ ¼ ðf T ðx1ðtÞÞ; fT ðx2ðtÞÞ; . . . ; f

T ðxNðtÞÞÞT ;

IkðtÞ ¼ ðITk ðtÞ; I

Tk ðtÞ; . . . ; I

Tk ðtÞÞÞ

T ;

where kAI ¼ f1;2; . . . ;Ng.The linearly coupled dynamical system (11) can be rewritten

as

_xðtÞ ¼XN

k ¼ 1

xkðtÞ½�CkxðtÞþAkfðxðtÞÞþBkfðxðt�tðtÞÞÞþIkðtÞþðGkþKkÞxðtÞ�:

ð21Þ

Next, a theorem is established to ensure the global asymptoticalsynchronization of system (21).

Theorem 1. Under assumptions ðA1Þ2ðA3Þ, the dynamical system

(21) is globally asymptotically synchronized if there are positive defi-

nite matrices PARðN�1Þn�ðN�1Þn, Q ARðN�1Þn�ðN�1Þn, RARðN�1Þn�ðN�1Þn,TARðN�1Þn�ðN�1Þn, positive definite diagonal matrices R¼ diag ðS1;

S2; . . . ;SðN�1ÞnÞARðN�1Þn�ðN�1Þn, K¼ diagðL1;L2; . . . ;LðN�1ÞnÞARðN�1Þn�ðN�1Þn, a matrix W¼ ðWT

1 WT2 WT

3 WT4 WT

5ÞT A

R5ðN�1Þn�ðN�1Þn and the feedback gain matrix KkARN�N , for each

kAI ¼ f1;2; . . . ;Ng such that

Uk ¼

Uk11 Uk12 Uk13 Uk14 Uk15 Uk16

UTk12 Uk22 �WT

3 �WT4 �W2�WT

5 0

UTk13 �W3 �RþQ 0 �W3 A1T

k T

UTk14 �W4 0 �ð1�hÞQ�K �W4 B1T

k T

UTk15 �WT

2�W5 �WT3 �WT

4 �1

rT�W5�WT

5 0

UTk16 0 TA1

k TB1k 0 �

1

rT

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCAo0;

ð22Þ

where

Uk11 ¼ Pð�C1kþHkþUkÞþð�C1

kþHkþUkÞT PþRþFRFþW1þWT

1;

Uk12 ¼WT2�W1;

Uk13 ¼ PA1kþWT

3;

Uk14 ¼ PB1kþWT

4;

Uk15 ¼WT5�W1;

Uk16 ¼ ð�C1kþHkþUkÞ

T T;

Uk22 ¼�ð1�hÞRþFKF�W2�WT2;

F ¼ diagðF1; F2; . . . ; FnÞARn�n, F¼ IN�1 � F, Hk ¼MGkJ, Hk ¼

Hk � D;Uk ¼MKkJ, Uk ¼Uk � D, M and J are defined in (17) and

(18).

Proof. Consider the following Lyapunov functional:

VðtÞ ¼Xi ¼ 4

i ¼ 1

ViðtÞ; ð23Þ

where

V1ðtÞ ¼ xT ðtÞMT PMxðtÞ; ð24Þ

V2ðtÞ ¼

Z t

t�tðtÞfTðxðsÞÞMT QMfðxðsÞÞds; ð25Þ

V3ðtÞ ¼

Z t

t�tðtÞxT ðsÞMT RMxðsÞds; ð26Þ

V4ðtÞ ¼

Z 0

�rdyZ t

tþy_xTðsÞMT TM _xðsÞds: ð27Þ

Taking the derivative of VðtÞ along the trajectories of (21) and byLemma 5, one has

_V ðtÞjð21Þ ¼ 2xT ðtÞMT PM _xðtÞþfTðxðtÞÞMT QMfðxðtÞÞ

�ð1� _tðtÞÞfTðxðt�tðtÞÞÞMT QMfðxðt�tðtÞÞÞþxT ðtÞMT RMxðtÞ

�ð1� _tðtÞÞxT ðt�tðtÞÞMT RMxðt�tðtÞÞþr _xTðtÞMT TM _xðtÞ

�

Z t

t�r

_xTðyÞMT TM _xðyÞdy

rXN

k ¼ 1

xkðtÞ 2xT ðtÞMT PM½ð�CkþGkþKkÞxðtÞþAkfðxðtÞÞn

þBkfðxðt�tðtÞÞÞþIkðtÞ�þfTðxðtÞÞMT QMfðxðtÞÞ

�ð1�hÞfTðxðt�tðtÞÞÞMT QMfðxðt�tðtÞÞÞþxT ðtÞMT RMxðtÞ

�ð1�hÞxT ðt�tðtÞÞMT RMxðt�tðtÞÞþr _xTðtÞMT TM _xðtÞ

�1

r

Z t

t�tðtÞM _xðyÞ dy

� �T

T

Z t

t�tðtÞM _xðyÞdy

� �): ð28Þ

By the structure of M, following equalities are easy to verify:

MCk ¼ C1kM; MAk ¼A1

kM; MBk ¼ B1k M; MIkðtÞ ¼ 0:

Therefore, from (28) one obtains

_V ðtÞjð21ÞrXN

k ¼ 1

xkðtÞ 2xT ðtÞMT P½ð�C1kMþMGkþMKkÞxðtÞ

nþA1

kMfðxðtÞÞþB1k Mfðxðt�tðtÞÞÞ�þfT

ðxðtÞÞMT QMfðxðtÞÞ

�ð1�hÞfTðxðt�tðtÞÞÞMT QMfðxðt�tðtÞÞÞþxT ðtÞMT RMxðtÞ

�ð1�hÞxT ðt�tðtÞÞMT RMxðt�tðtÞÞþr _xTðtÞMT TM _xðtÞ

�1

r

Z t

t�tðtÞM _xðyÞdy

� �T

T

Z t

t�tðtÞM _xðyÞdy

� �): ð29Þ

By assumption A1, it is obvious that

fTðxðtÞÞMTRMfðxðtÞÞ ¼

XN�1

j ¼ 1

½f ðxjðtÞÞ�f ðxjþ1ðtÞÞ�TRj½f ðxjðtÞÞ�f ðxjþ1ðtÞÞ

rXN�1

j ¼ 1

½xjðtÞ�xjþ1ðtÞ�T FRjF½xjðtÞ�xjþ1ðtÞ�

¼ xT ðtÞMT FRFMxðtÞ; ð30Þ

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W. Yu et al. / Neurocomputing 73 (2010) 858–866862

where Rj ¼ diagðSðj�1Þnþ1; . . . ;SjnÞ, and

fTðxðt�tðtÞÞÞMTKMfðxðt�tðtÞÞÞrxT ðt�tðtÞÞMT FKFMxðt�tðtÞÞ:

ð31Þ

From Lemmas 1 and 3, we obtain

2xT ðtÞMT PMGkxðtÞ ¼ 2xT ðtÞMT P½ðM � InÞðGk � DÞ�xðtÞ

¼ 2xT ðtÞMT P½MGk � D�xðtÞ

¼ 2xT ðtÞMT P½HkM � D�xðtÞ

¼ 2xT ðtÞMT P½ðHk � DÞðM � InÞ�xðtÞ

¼ 2xT ðtÞMT PHkMxðtÞ; ð32Þ

and

2xT ðtÞMT PMKkxðtÞ ¼ 2xT ðtÞMT PUkMxðtÞ; ð33Þ

where Hk ¼MGkJ; Hk ¼ ðMGkJÞ � D, Uk ¼MKkJ; Uk ¼ ðMKkJÞ � D, M

and J are defined in (17) and (18).

From the Leibniz–Newton formula, the following equation is

true for any matrix W with appropriate dimensions:

2ZT ðtÞWM xðtÞ�xðt�tðtÞÞ�Z t

t�tðtÞ_xðsÞds

� �¼ 0; ð34Þ

where

ZðtÞ ¼ ðxT ðtÞMT xT ðt�tðtÞÞMT fTðxðtÞÞMT fT

ðxðt�tðtÞÞÞMT R tt�tðtÞ _x

TðsÞ

dsMTÞT . Let Pk ¼ ð�C1

kþHkþUk 0 A1k B1

k 0Þ, then we have

_xTðtÞMT TM _xðtÞ ¼ ZT ðtÞPT

k TPkZðtÞ: ð35Þ

Combining (29)–(35), we obtain

_V ðtÞjð21ÞrXN

k ¼ 1

xkðtÞ 2xT ðtÞMT Pð�C1kþHkþUkÞMxðtÞ

n

þ2xT ðtÞMT PA1kMfðxðtÞÞþ2xT ðtÞMT PB1

kMfðxðt�tðtÞÞÞþxT ðtÞMTFRFMxðtÞ�fT

ðxðtÞÞMTRMfðxðtÞÞ

þxT ðt�tðtÞÞMT FKFMxðt�tðtÞÞ�fTðxðt�tðtÞÞÞMTKMfðxðt�tðtÞÞÞ

þfTðxðtÞÞMT QMfðxðtÞÞ�ð1�hÞfT

ðxðt�tðtÞÞÞMTQMfðxðt�tðtÞÞÞþxT ðtÞMTRMxðtÞ�ð1�hÞxT ðt�tðtÞÞMT RMxðt�tðtÞÞ

þrZT ðtÞPTk TPkZðtÞ�

1

r

Z t

t�tðtÞM _xðyÞdy

� �T

T

Z t

t�tðtÞM _xðyÞ dy

� �

þ2ZT ðtÞWM xðtÞ�xðt�tðtÞÞ�Z t

t�tðtÞ_xðsÞds

� ��

¼XN

k ¼ 1

xkðtÞ ZT ðtÞðWkþrPTkTPkÞZðtÞ

n o; ð36Þ

where

Wk ¼

Uk11 Uk12 Uk13 Uk14 Uk15

UTk12 Uk22 �WT

3 �WT4 �W2�WT

5

UTk13 �W3 �RþQ 0 �W3

UTk14 �W4 0 �ð1�hÞQ�K �W4

UTk15 �WT

2�W5 �WT3 �WT

4 �1

rT�W5�WT

5

0BBBBBBBB@

1CCCCCCCCA:

ð37Þ

It is easy to see that WkþrPTk TPko0 is equivalent to the

condition (22) Uko0 by Lemma 4 (Schur complement). So by

Lemma 2 and from (36), we know that under the given condition

(22), _V ðtÞr0 and we obtain VðtÞrVð0Þ, namely, VðtÞ is a bounded

function. Thus, JMxðtÞJ�!0. This completes the proof. &

The term Kk is both involved in W11 and W16, so it is difficult tosolve this by Matlab LMI Toolbox. In order to solve the feedback

gain matrix Kk, a simple transformation is made to derive thefollowing theorem.

Theorem 2. Under assumptions ðA1Þ2ðA3Þ, the dynamical system

(21) is globally asymptotically synchronized if there are positive defi-

nite matrices PARðN�1Þn�ðN�1Þn, Q ARðN�1Þn�ðN�1Þn, RARðN�1Þn�ðN�1Þn,TARðN�1Þn�ðN�1Þn, positive definite diagonal matrices R¼ diagðS1;S2; . . . ;SðN�1ÞnÞARðN�1Þn�ðN�1Þn, K¼ diagðL1;L2; . . . ;LðN�1ÞnÞARðN�1Þn�ðN�1Þn, a matrix W¼ ðWT

1 WT2 WT

3 WT4 WT

5ÞT

AR5ðN�1Þn�ðN�1Þn and a matrix OkARðN�1Þn�ðN�1Þn, for each

kAI ¼ f1;2; . . . ;Ng such that

Uk ¼

Uk11 Uk12 Uk13 Uk14 Uk15 Uk16

UTk12 Uk22 �WT

3 �WT4 �W2�WT

5 0

UTk13 �W3 �RþQ 0 �W3 A1T

k P

UTk14 �W4 0 �ð1�hÞQ�K �W4 B1T

k P

UTk15 �WT

2�W5 �WT3 �WT

4 �1

rT�W5�WT

5 0

UT

k16 0 PA1k PB1

k 0 �1

rPT�1P

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCAo0;

ð38Þ

where

Uk11 ¼ Pð�C1kþHkÞþOkþð�C1

kþHkÞT PþOT

kþRþFRFþW1þWT1;

Uk16 ¼ ð�C1kþHkÞ

T PþOTk ;

F ¼ diagðF1; F2; . . . ; FnÞARn�n, F¼ IN�1 � F; Hk ¼MGkJ, Hk ¼Hk � D;

Uk ¼MKkJ, Uk ¼Uk � D, M and J are defined in (17) and (18).Moreover, the estimation gain matrix Uk ¼ P�1Ok.

Proof. Pre- and post-multiplying U in (22) by diagðI; I; I; I; I;PT�1Þ

and diagðI; I; I; I; I;T�1PÞ, respectively, and introducing a newvariable Ok ¼ PUk yield (38), where I is identical matrix withappropriate dimensions. &

It is noted that the resulting condition in Theorem 2 are nolonger LMI conditions due to term PT�1P in (38). As a result, wecannot solve (38) by using Matlab LMI Toolbox. However, thisnon-convex problem can be solved by using an iterative algorithmbased on the algorithms in [41–43].

First, we define a new positive definite matrix L such thatPT�1PZL and replace (38) with

Uk ¼

Uk11 Uk12 Uk13 Uk14 Uk15 Uk16

UTk12 Uk22 �WT

3 �WT4 �W2�WT

5 0

UTk13 �W3 �RþQ 0 �W3 A1T

k P

UTk14 �W4 0 �ð1�hÞQ�K �W4 B1T

k P

UTk15 �WT

2�W5 �WT3 �WT

4 �1

rT�W5�WT

5 0

UT

k16 0 PA1k PB1

k 0 �1

rL

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCAo0;

ð39Þ

and

PT�1PZL: ð40Þ

Since (40) is equivalent to P�1TP�1rL�1, it is expressed as

L�1 P�1

P�1 T�1

!Z0 ð41Þ

by Lemma 2 (Schur complement), then, by introducing newvariables X, Y and Z, the original condition (38) can be

ARTICLE IN PRESS

W. Yu et al. / Neurocomputing 73 (2010) 858–866 863

represented as (39) and

X Y

Y Z

� �Z0; X¼ L�1; Y¼ P�1; Z¼ T�1: ð42Þ

Using a cone complementary problem, this problem is convertedto the following LMI-based nonlinear optimization problem:

Minimize trðLXþPYþTZÞsubject to (39) and

X Y

Y Z

� �Z0;

L I

I X

� �Z0;

P I

I Y

� �Z0;

T I

I Z

� �Z0:

ð43Þ

The algorithm proposed in this paper is demonstrated as follows:

Algorithm 1. For a given precision d40, let N be the maximumnumber of iterations.

Step 1: Find a feasible set ðP0;Q 0;R0, T0;R0;K0, W0;O0k ; L

0,

X0;Y0;Z0Þ satisfying (39) and (43). Set j¼ 0.

Step 2: Solve the following LMI problem for variables

(P;Q ;R;T;R;K, W; L;Ok;X;Y;Z)

Minimize trðLjXþLXjþPjYþPYj

þTjZþTZjÞ

subject to (38) and (43).

Set Lj¼ L, P

j¼ P; T

j¼ T, X

j¼X; Y

j¼ Y; Z

j¼ Z.

Step 3: If the condition (38) is satisfied, then exit. If the

condition (38) is not satisfied, within a specified number of

iterations, i.e., j¼ N , then exit. Otherwise, set j¼ jþ1 and go to

Step 4.

Step 4: If jtrðLjXjþ L

jXjþPjY

jþ P

1YjþTjZ

jþ T

jZjÞ�2 trðLjXj

þPjYj

þTjZjÞjod then go to Step 2, else go to Step 5.

Step 5. Compute y�A ½0;1� by solving:

MinimizeyA ½0;1�tr½LjþyðL

j�LjÞ�½XjþyðX

j�XjÞ�þ½Pj

þyðPj�PjÞ�½Yj-

þyðYj�YjÞ�þ½Tj

þyðTj�TjÞ�½ZjþyðZ

j�Zj�:

Set Ljþ1¼ Ljþy�ðL

j�LjÞ, Xjþ1

¼Xjþy�ðX

j�XjÞ, Pjþ1

¼ Pjþy�ðP

j

�PjÞ, Yjþ1

¼ Yjþy�ðY

j�YjÞ, Tjþ1

¼ Tjþy�ðT

j�TjÞ, Zjþ1

¼ Zjþ

y�ðZj�ZjÞ, then go to Step 2.

Note that we obtain P and Ok from Algorithm 1, our mainpurpose is to choose feedback gain matrix Kk. As Uk ¼ ðMKkJÞ � D,so we cannot solve Kk directly from Theorem 2. Since it is easy tosee from Lemma 4 (Schur complement) that if (38) and

P½ðMKkJÞ � D�rOk ð44Þ

are satisfied, then (22) is satisfied. If (38) is solved by Algorithm 1,we can use the following algorithm to solve (44):

Algorithm 2. Minimize trðOk�P½ðMKkJÞ � D�Þ

subject to (44).

Note that under condition (44), the lower bound of theobject is zero, which means that gain matrix should not be verylarge.

Moreover, the adopted approaches here can also be used tocontrol the linearly coupled neural network (21) if there are noswitching signals ðN ¼ 1Þ, which is reduced to one coupledcomplex network. It is noted that an applicable method isproposed to control the synchronization of coupled networks bychanging the connection structure.

Corollary 1. Under assumptions ðA1Þ2ðA3Þ, the dynamical system

(21) ðN ¼ 1Þ is globally asymptotically synchronized if there are

positive definite matrices PARðN�1Þn�ðN�1Þn, Q ARðN�1Þn�ðN�1Þn,RARðN�1Þn�ðN�1Þn, TARðN�1Þn�ðN�1Þn, positive definite diagonal ma-

trices R¼ diagðS1;S2; . . . ;SðN�1ÞnÞARðN�1Þn�ðN�1Þn, K¼ diagðL1;L2;

. . . ;LðN�1ÞnÞARðN�1Þn�ðN�1Þn, a matrix W¼ ðWT1 WT

2 WT3 WT

4 WT5Þ

T

AR5ðN�1Þn�ðN�1Þn and the feedback gain matrix KkARN�N , for k¼ 1,

such that

U¼

Uk11 Uk12 Uk13 Uk14 Uk15 Uk16

UTk12 U22 �WT

3 �WT4 �W2�WT

5 0

UTk13 �W3 �RþQ 0 �W3 A1T

k T

UTk14 �W4 0 �ð1�hÞQ�K �W4 B1T

k T

UTk15 �WT

2�W5 �WT3 �WT

4 �1

rT�W5�WT

5 0

UTk16 0 TA1

k TB1k 0 �

1

rT

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCAo0;

ð45Þ

where

Uk11 ¼ Pð�C1kþHkþUkÞþð�C1

kþHkþUkÞT PþRþFRFþW1þWT

1;

Uk12 ¼WT2�W1;

Uk13 ¼ PA1kþWT

3;

Uk14 ¼ PB1kþWT

4;

Uk15 ¼WT5�W1;

Uk16 ¼ ð�C1kþHkþUkÞ

T T;

Uk22 ¼�ð1�hÞRþFKF�W2�WT2;

F ¼ diagðF1; F2; . . . ; FnÞARn�n, F¼ IN�1 � F, Hk ¼MGkJ, Hk ¼Hk � D,Uk ¼MKkJ, Uk ¼Uk � D, M and J are defined in (17) and (18).

Remark 1. To the best of our knowledge, there are few worksabout the synchronization control of switched linearly coupleddynamical systems. In this paper, we consider global synchroni-zation of switched linearly coupled delayed neural network. Inaddition, some controllers are designed to ensure the globalsynchronization of coupled dynamical system based on theconvex optimization algorithm.

Remark 2. It is noted that when there is only one node in theswitched coupled neural network, i.e., N ¼ 1, the obtained resultscan also be satisfied for the linearly coupled neural networks. It iseven applicable to the case that the derivative of the time-varyingdelay takes any value compared to the assumption _tðtÞo1 ofearlier works.

4. Numerical examples

In this section, simulation examples are presented to illustratethe utility of theoretical analysis in this paper.

Example 1. Consider the following linearly coupled neural net-work model:

_xiðtÞ ¼ �CaxiðtÞþAaf ðxiðtÞÞþBaf ðxiðt�taðtÞÞÞþ IaðtÞ

þXN

j ¼ 1

ðGaijþKaijÞDxjðtÞ; i¼ 1;2;3; a¼ 1;2; ð46Þ

where xiðtÞ ¼ ðxi1ðtÞ; xi2ðtÞÞT , f ðxiðtÞÞ ¼ ðtanhðxi1ðtÞÞ; tanhðxi2ðtÞÞÞ

T ,I1ðtÞ ¼ I2ðtÞ ¼ ð0;0Þ

T ,

C1 ¼1 0

0 1

� �; A1 ¼

2:0 �0:1

�5:0 3:0

� �; B1 ¼

�1:5 �0:1

�0:2 �2:5

� �;

G1 ¼

�0:2 0:1 0:1

0:1 �0:1 0

0:1 0 �0:1

0B@

1CA;

C2 ¼1 0

0 1

� �; A2 ¼

1:8 �0:1

�4:5 4:0

� �; B2 ¼

�1:6 �0:1

�0:2 �2:8

� �;

ARTICLE IN PRESS

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−5

−4

−3

−2

−1

0

1

2

3

4

5

x11

x 12

Fig. 3. Trajectories of one node in the coupled networks of mode 2.

0 50 100 150 200

0

5

10

15

t

err (

t)

Fig. 4. Error distance of the switched coupled networks without control.

W. Yu et al. / Neurocomputing 73 (2010) 858–866864

G2 ¼

�0:3 0:2 0:1

0:2 �0:2 0

0:1 0 �0:1

0B@

1CA;

D¼ ð 10:8

0:51 Þ, tðtÞ ¼ 1þ0:1sinð12tÞ. It is obvious that

0otðtÞo1:1¼ r, _tðtÞr1:2¼ h. Clearly, assumptions A12A3 aresatisfied (F ¼ I2). Note that many earlier works assume _tðtÞo1and assumption A3 is just needed in this paper. It is evenapplicable to the case that the derivative of the time-varyingdelay takes any value.

Choose the following initial conditions:

x1ðsÞ ¼0:1

�0:3

� �; x2ðsÞ ¼

0:5

�1

� �; x3ðsÞ ¼

1

�0:5

� �:

By Algorithms 1 and 2, the condition (22) in Theorem 1 issatisfied. The following coupling gain matrices are obtained:

K1 ¼

�15:3789 7:4186 7:9603

7:4186 �15:3088 7:8902

7:9603 7:8902 �15:8505

0B@

1CA;

K2 ¼

�14:0595 6:8074 7:2521

6:8074 �13:8852 7:0778

7:2521 7:0778 �14:3298

0B@

1CA:

If there is no switched rule, the trajectories of one node in mode 1ða¼ 1Þ and mode 2 ða¼ 2Þ are shown in Figs. 2 and 3 by choosingthe initial conditions:

x1ðsÞ ¼ 0:4; x2ðsÞ ¼ 0:6; 8sA ½�1;0�:

Next a random switching rule is used for the two coupled neuralnetworks. The error distance among the nodes of trajectories inthe coupled networks are

errðtÞ ¼X2

i ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½x1iðtÞ�x2iðtÞ�

2þ½x1iðtÞ�x3iðtÞ�2

q:

It is shown from Fig. 4 that the trajectory of error distance ofcoupled system (46) without control does not converge to zero,which means that the switched coupled systems without controlare not synchronized. The trajectories of the switched coupledneural networks with control are illustrated in Fig. 5, and thecorresponding trajectory of error distance is illustrated in Fig. 6. It

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−4

−3

−2

−1

0

1

2

3

4

x11

x 12

Fig. 2. Trajectories of one node in the coupled networks of mode 1.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5

−4

−3

−2

−1

0

1

2

3

4

5

x11

x 12

Fig. 5. Trajectories of one node in the switched coupled networks.

ARTICLE IN PRESS

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

1.5

2

t

err (

t)

Fig. 6. Error distance of the switched coupled networks with control.

W. Yu et al. / Neurocomputing 73 (2010) 858–866 865

is easy to see that the switched coupled system with control (46)is globally synchronized in Fig. 6.

5. Conclusions

In this paper, synchronization control of switched linearlycoupled delayed neural networks is considered based on Lyapu-nov functional method and linear matrix inequality (LMI)approach. The obtained results are easy to apply.

To the best of our knowledge, there are few works aboutsynchronization control of switched coupled delayed systems. Aglobally convergent algorithm involving convex optimization isalso presented to construct such controllers effectively. In manycases, we want to control the whole network by changing theweights of some nodes in the complex network, and this paperprovides an applicable approach.

Acknowledgements

The authors would like to thank Prof. Guanrong Chen forhelpful suggestions. This work was jointly supported by theNational Natural Science Foundation of China under Grant60574043, International Joint Project funded by NSFC and theRoyal Society of the United Kingdom, the Natural ScienceFoundation of Jiangsu Province of China under GrantBK2006093, the Hong Kong Research Grants Council under theCERG Grant CityU 1114/05E, and Tianyuan Fund for Mathematicsunder Grant No. 10826033.

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Wenwu Yu (S’07) received both the B.Sc. degree ininformation and computing science and M.Sc. degree inapplied mathematics from the Department of Mathe-matics, Southeast University, Nanjing, China, in 2004and 2007, respectively. Currently, he is working towardsthe Ph.D. degree at the Department of ElectronicEngineering, City University of Hong Kong, Hong Kong.

Mr. Yu held several visiting positions in China,Germany, Italy, the Netherlands, and USA. He is theauthor or coauthor of about 30 referred internationaljournal papers, and a reviewer of several journals. Hisresearch interests include multi-agent systems, non-

systems, neural networks, cryptography, and communications. Mr. Yu is therecipient of the Best Master Degree Theses Award from Jiangsu Province, China in2008.

Jinde Cao (M’07, SM’07) received the B.S. degree fromAnhui Normal University, Wuhu, China, the M.S.degree from Yunnan University, Kunming, China, andthe Ph.D. degree from Sichuan University, Chengdu,China, all in mathematics/applied mathematics, in1986, 1989, and 1998, respectively. From March 1989to May 2000, he was with Yunnan University. In May2000, he joined the Department of Mathematics,Southeast University, Nanjing, China. From July 2001to June 2002, he was a Post doctoral Research Fellow inthe Department of Automation and Computer-aidedEngineering, Chinese University of Hong Kong, Hong

Research Fellow or a Visiting Professor in the School of Information Systems,Computing and Mathematics, Brunel University, UK.

He is currently a TePin Professor and Doctoral Advisor at the South-east University. Prior to this, he was a Professor at Yunnan University from1996 to 2000. He is the author or coauthor of more than 160 journal papersand five edited books and a reviewer of Mathematical Reviews and Zentralblatt-Math. His research interests include nonlinear systems, neural networks,complex systems and complex networks, stability theory, and applied mathe-matics.

Professor Cao is an Associate Editor of the IEEE Transaction on Neural Networks,Journal of the Franklin Institute, Mathematics and Computers in Simulation,Neurocomputing, International Journal of Differential Equations, and DifferentialEquations and Dynamical Systems.

Wenlian Lu received the B.S. degree in mathematicsand the Ph.D. degree in applied mathematics fromFudan University, Shanghai, China, in 2000 and 2005,respectively. He is currently an Associate Professorwith the School of Mathematical Sciences, FudanUniversity, Shanghai, China. His research interestsinclude neural networks, nonlinear dynamical sys-tems, and complex systems.