Neurocomputing 74 (2011) 805–811

Contents lists available at ScienceDirect

Neurocomputing

0925-23

doi:10.1

$This

under G� Corr

E-m

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

Synchronization for general complex dynamical networks withsampled-data$

Nan Li �, Yulian Zhang, Jiawen Hu, Zhenyu Nie

College of Electromechanical Engineering, Zhejiang Ocean University, Zhoushan 316004, China

a r t i c l e i n f o

Article history:

Received 6 September 2010

Received in revised form

25 October 2010

Accepted 6 November 2010

Communicated by Z. Wangproblem of stability analysis for a differential equation with multiple time-varying delays. Then, by

Available online 15 December 2010

Keywords:

Complex networks

Exponential synchronization

Jensen’s inequality

Linear matrix inequalities (LMIs)

Sampled-data control

12/$ - see front matter & 2010 Elsevier B.V. A

016/j.neucom.2010.11.007

work was supported by the Research Project

rant X09Z4.

esponding author.

ail address: Linan3008@163.com (N. Li).

a b s t r a c t

In this paper, the sampled-data synchronization control problem is investigated for a class of general

complex networks with time-varying coupling delays. A rather general sector-like nonlinear function is

used to describe the nonlinearities existing in the network. By using the method of converting the

sampling period into a bounded time-varying delay, the addressed problem is first transformed to the

constructing a Lyapunov functional and using Jensen’s inequality, a sufficient condition is derived to

ensure the exponential stability of the resulting delayed differential equation. Based on that, the desired

sampled-data feedback controllers are designed in terms of the solution to certain linear matrix

inequalities (LMIs) that can be solved effectively by using available software. Finally, a numerical

simulation example is exploited to demonstrate the effectiveness of the proposed sampled-data control

scheme.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Complex dynamical networks are comprised a large set of nodesevolving according to their respective dynamical equations. Someof these nodes are usually coupled according to the networktopology. In the real world, a large number of practical systemscan be represented by models of complex networks, such asInternet, World Wide Web, food webs, electric power grids, cellularand metabolic networks, scientific citation networks, social net-works, etc. So far, the complex dynamical network has already beenone of the most popular topics in the areas of scientific research. Aswe all know, the synchronization of all dynamical nodes is one ofthe most significant and interesting properties in a complexnetwork. Therefore, the synchronization problem for complexdynamical networks has received increasing research attention,and a great deal of results have been available in the literature[1–3,7,9,11–15,18,19,21,22,26,29]. For example, the dynamical net-work model in terms of a differential equation with a coupling termhas been proposed in [22], where the synchronization phenomenon ofall dynamical nodes has been investigated for the considered net-works. Subsequently, some synchronization criteria have been estab-lished for a class of complex dynamical networks in [7,13], where thecoupling delays have been taken into account. In [15], the synchro-nization problem has been investigated for an array of coupled

ll rights reserved.

of Zhejiang Ocean University

complex discrete-time networks with the simultaneous presence ofboth the discrete and distributed time delays. In [14], the exponentialsynchronization problem has been studied for a class of stochasticdelayed discrete-time complex networks and the correspondingsynchronization criteria have been derived. Very recently, in [26],the global synchronization problem has been fully investigated for aclass of discrete-time stochastic complex networks, where bothrandomly occurred nonlinearities and mixed time delays have beentaken into account.

Note that it is often the case that, for a complex network, all thebehaviors of the dynamical nodes do not evolve synchronously. Assuch, it would be more interesting to study the condition that canguarantee the synchronization of all the nodes in complex net-works and how to achieve synchronization for an asynchronouscomplex network. A natural idea is to introduce a set of controllersto adjust the behaviors of the network nodes and accordinglyachieve the network synchronization. This is referred to as thesynchronization control problem. Some efforts have been made onthis topic and various control schemes have been proposed. Forexample, in [31], an impulsive control scheme has been proposed toachieve synchronization for complex dynamical networks withunknown coupling, and the synchronization strategy considers theinfluence of all nodes in the dynamical network. In [28], thesynchronization problem via pinning control has been studiedand the derived pinning condition with controllers given in a high-dimensional setting can be reduced to a low-dimensional conditionwithout the pinning controllers involved. In [33], the locally andglobally adaptive synchronization control problems have beenconsidered for an uncertain complex dynamical network and a set

N. Li et al. / Neurocomputing 74 (2011) 805–811806

of adaptive controllers has been designed for the networksynchronization.

On the other hand, as the rapid development of computerhardware, the sampled-data control technology has shown moreand more superiority over other control approaches. The sampled-data control theory also has rapidly developed. It is worthwhile tomention that, in [5,6], a new approach to dealing with the sample-data control problems has been proposed by converting the samp-ling period into a time-varying but bounded delay. By followingthis idea, in [30], the exponential synchronization sampled-datacontrol problem has been studied for neural networks with time-varying mixed delays, while the robust sampled-data H1 controlproblem for vehicle active suspension systems has been investi-gated in [8]. To the best of our knowledge, so far, the sampled-datasynchronization control problem has not been considered for thecomplex networks yet. Therefore, in this paper, we aim to design aset of sampled-data feedback controllers to achieve the synchro-nization of a class of general asynchronous complex networks withtime-varying coupling delays. The contributions of this work can besummarized as follows: (1) we make the first attempt to addressthe sampled-data synchronization control problem for a class ofgeneral complex networks with time-varying coupling delays; (2) asynchronization criterion depending on the sampling period is obtai-ned by constructing an appropriate Lyapunov functional and usingJensen’s inequality; and (3) a set of sampled-data feedback controllersis designed that can achieve the synchronization of the consideredcomplex network.

The remainder of this paper is organized as follows. Theexponential sampled-data synchronization control problem forcomplex networks with time-varying coupling delays is formu-lated in Section 2. In Section 3, a exponential synchronizationcriterion is derived and a set of sampled-data feedback controllersis designed for the considered complex network. An illustrativeexample is provided to show the effectiveness of the proposedsampled-data control scheme in Section 4. Finally, conclusions aregiven in Section 5.

Notation: The notation used here is fairly standard except whereotherwise stated. Rn denotes the n-dimensional Euclidean space.JAJ refers to the norm of a matrix A defined by JAJ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitraceðAT AÞ

p.

The notation XZY (respectively, X4Y), where X and Y are realsymmetric matrices, means that X�Y is positive semi-definite(respectively, positive definite). MT represents the transpose of thematrix M. I denotes the identity matrix of compatible dimension.

diagf� � �g stands for a block-diagonal matrix and the notation

diagnf�g is employed to stand for diagf�, . . . ,�zfflfflfflffl}|fflfflfflffl{n

g. Moreover, theasterisk � in a matrix is used to denote term that is induced bysymmetry. Matrices, if they are not explicitly specified, areassumed to have compatible dimensions.

2. Problem formulation and preliminaries

Consider a complex dynamical network consisting of N identicalcoupled nodes as follows:

_xiðtÞ ¼ f ðxiðtÞÞþcXN

j ¼ 1

GijAxjðt�tðtÞÞþuiðtÞ, i¼ 1,2, . . . ,N, ð1Þ

where xiðtÞARn and uiðtÞARn are, respectively, the state variableand the control input of the node i. f : Rn-Rn is a continuousvector-valued function. The scalar tðtÞ denotes the time-varyingdelay satisfying

0rtðtÞrm, _tðtÞrn, ð2Þ

where m and n are known positive constants. c40 is the couplingstrength, A¼ ðaijÞn�nARn�n is a constant inner-coupling matrix of

the nodes, and G¼ ðGijÞN�N is the outer-coupling matrix of thenetwork, where Gij is defined as follows: if there is a connectionbetween node i and node j ðja iÞ, then Gij¼Gji¼1; otherwise,Gij¼Gji¼0, and the diagonal elements of matrix G are defined by

Gii ¼�XN

j ¼ 1,ja i

Gij ¼�XN

j ¼ 1,ja i

Gji, i¼ 1,2, . . . ,N:

We denote the synchronization error by ei(t)¼xi (t)�s(t) wheresðtÞARn is the state trajectory of the unforced isolate node_sðtÞ ¼ f ðsðtÞÞ. Then, the error dynamics of complex network (1)can be obtained as follows:

_eiðtÞ ¼ gðeiðtÞÞþcXN

j ¼ 1

GijAejðt�tðtÞÞþuiðtÞ, i¼ 1,2, . . . ,N, ð3Þ

where gðeiðtÞÞ ¼ f ðxiðtÞÞ�f ðsðtÞÞ.In this paper, the following feedback controllers are adopted:

uiðtÞ ¼ KieiðtÞ, i¼ 1,2, . . . ,N,

where Ki is the feedback gain to be determined, and uiðtÞARn is acontinuous signal that will be sampled before it enters the complexnetwork.

For every i (i¼1,2,y,N), the sampled control input ui(t) isassumed to be generated by a zero-order hold function with asequence of hold times 0¼ t0ot1o � � �otko � � �

uiðtÞ ¼ uiðtkÞ ¼ KieiðtkÞ, tkrtotkþ1, ð4Þ

where tk denotes the sampling instant and satisfies limk-1tk ¼1.Moreover, the sampling period under consideration is assumed tobe bounded by a known constant p40, that is, tkþ1�tkrp for kZ0.

Remark 1. Note that the sampling periods considered here areonly required to be bounded, which means that the sampled-datafeedback controllers (4) can effectively deal with the case when thesampling periods are time varying. In this paper, we assume thatthe control inputs from all nodes are sampled by one sampler. It isworth pointing out that, in the case of multiple samplers, theproposed control approach is still feasible.

Define a sawtooth function as follows:

dðtÞ ¼ t�tk, tkrtotkþ1:

It is easily seen that

0odðtÞop, ð5Þ

and the control input of node i can be rewritten as

uiðtÞ ¼ Kieiðt�dðtÞÞ:

Consequently, the closed-loop error system of the complexnetwork (1) is obtained as follows:

_eiðtÞ ¼ gðeiðtÞÞþcXN

j ¼ 1

GijAejðt�tðtÞÞþKieiðt�dðtÞÞ, i¼ 1,2, . . . ,N,

ð6Þ

which can be further rewritten as the following compact form:

_eðtÞ ¼ ~g ðeðtÞÞþcðG� AÞeðt�tðtÞÞþKeðt�dðtÞÞ, ð7Þ

where

eðtÞ ¼ ½eT1ðtÞ eT

2ðtÞ � � � eTNðtÞ�

T , K ¼ diagfK1,K2, . . . ,KNg,

~gðeðtÞÞ ¼ ½gT ðe1ðtÞÞ gT ðe2ðtÞÞ � � � gT ðeNðtÞÞ�T :

Throughout this paper, the following assumption is made.

Assumption 1. The nonlinear function f : Rn-Rn satisfies

½f ðxÞ�f ðyÞ�Uðx�yÞ�T ½f ðxÞ�f ðyÞ�Vðx�yÞ�r0, 8x,yARn, ð8Þ

where U and V are constant matrices of appropriate dimensions.

N. Li et al. / Neurocomputing 74 (2011) 805–811 807

Remark 2. The nonlinearity description in (8) is quite general thatincludes the commonly Lipchitz condition and norm-boundedcondition as special cases. The examples of the application of sucha sector-like nonlinearity include the active function in geneticregulatory networks [25] and the quantization function in net-worked control systems [20].

We are now in a position to introduce the notion of exponentialsynchronization for the complex network (1).

Definition 1. The complex network (1) is said to be exponentiallysynchronized if the closed-loop error system (7) is exponentiallytable, i.e., there exist two constants a40 and b40 such that

JeðtÞJ2rae�bt sup�rryr0

JfðyÞJ2,

where r¼maxfm,pg and fð�Þ is the initial function of system (7)defined as eðtÞ ¼fðtÞ, tA ½�r,0�.

The main purpose of this paper is to design a set of controllerswith the form (4) to achieve the exponential synchronization of thecomplex network (1). In other words, we are interested in finding aset of feedback gain matrices Ki such that the error system (7) isexponentially stable.

Remark 3. Note that the coupling delays in (1) are time-varying,and hence the network model (1) generalizes the existing ones in[7,13]. Moreover, it should be mentioned that, to the best of ourknowledge, the sampled-data synchronization control problem forsuch a complex network with time-varying coupling delays has notbeen well studied yet.

3. Main results

In this section, we first analyze the stability and deriving thecorresponding exponential stability condition for the error sys-tem (7). Then, the sampled-data feedback controllers are desig-ned to achieve the exponential synchronization of the complexnetwork (1).

The following lemma will be used to derive our main results.

Lemma 1 (Jensen’s inequality [10]). For any constant matrix RARn�n, R¼ RT 40, scalar b40, and vector function x : ½0,b�-Rn,one has

b

Z b

0xT ðsÞRxðsÞ dsZ

Z b

0xðsÞ ds

!T

R

Z b

0xðsÞ ds

!

provided that the above integrals are well defined.

In the following theorem, a sufficient condition is given underwhich the error system (7) is exponentially stable.

Theorem 1. Let the controller gain Ki (i¼1,2,y,N) be given. The error

system (7) is exponentially stable if there exist matrices P40, Qi40,Zi40 (i¼1,2,3) and a scalar l40 such that

Z2þð1�nÞZ340 ð9Þ

and

O11 PKþ1pZ1 0 O14 0 P�l ~VL 0

� �2pZ1

1pZ1 0 0 0 KT

� � �Q1�1pZ1 0 0 0 0

� � � O441mZ2 0 cðG� AÞT

� � � � �Q2�1mZ2 0 0

� � � � � �lI I

� � � � � � �R�1

26666666666664

37777777777775o0,

ð10Þ

where

O11 ¼ Q1þQ2þQ3�1

pZ1�

1

m ðZ2þð1�nÞZ3Þ�l ~UL,

O14 ¼ cPðG� AÞþ1

m ðZ2þð1�nÞZ3Þ,

O44 ¼�ð1�nÞQ3�1

m ðZ2þð1�nÞZ3Þ�1

m Z2,

R¼ pZ1þmZ2þmZ3,

~UL ¼UTLVLþVT

LUL

2, ~VL ¼�

UTLþVT

L2

,

UL ¼ diagNfUg, VL ¼ diagNfVg: ð11Þ

Proof. Construct the following Lyapunov functional:

VðetÞ ¼ eT ðtÞPeðtÞþ

Z t

t�peT ðsÞQ1eðsÞ dsþ

Z t

t�meT ðsÞQ2eðsÞ ds

þ

Z t

t�tðtÞeT ðsÞQ3eðsÞ dsþ

Z 0

�p

Z t

tþy_eTðsÞZ1 _eðsÞ ds dy

þ

Z 0

�m

Z t

tþy_eTðsÞZ2 _eðsÞ ds dyþ

Z 0

�tðtÞ

Z t

tþy_eTðsÞZ3 _eðsÞ ds dy

where P, Qi and Zi (i¼1,2,3) are the solutions to matrix inequalities(9)–(10).

Calculating the time derivative of V(et) along the trajectory of

system (7) and considering (2), one has_V ðetÞr2eT ðtÞP _eðtÞþeT ðtÞðQ1þQ2þQ3ÞeðtÞ�eT ðt�pÞQ1eðt�pÞ�eT ðt�mÞQ2eðt�mÞ

�ð1�nÞeT ðt�tðtÞÞQ3eðt�tðtÞÞþ _eTðtÞðpZ1þmZ2þmZ3Þ _eðtÞ

�

Z t

t�p

_eTðsÞZ1 _eðsÞ ds�

Z t

t�m_eTðsÞZ2 _eðsÞ ds�ð1�nÞ

Z t

t�tðtÞ_eTðsÞZ3 _eðsÞ ds:

ð12Þ

By Lemma 1, it follows from (2), (5) and (9) that

�

Z t

t�p

_eTðsÞZ1 _eðsÞ ds¼�

Z t

t�dðtÞ

_eTðsÞZ1 _eðsÞ ds�

Z t�dðtÞ

t�p

_eTðsÞZ1 _eðsÞ ds

r�1

pðeðtÞ�eðt�dðtÞÞÞT Z1ðeðtÞ�eðt�dðtÞÞÞ

�1

pðeðt�dðtÞÞ�eðt�pÞÞT Z1ðeðt�dðtÞÞ�eðt�pÞÞ

ð13Þ

and

�

Z t

t�m_eTðsÞZ2 _eðsÞ ds�ð1�nÞ

Z t

t�tðtÞ_eTðsÞZ3 _eðsÞ ds

¼�

Z t

t�tðtÞ_eTðsÞðZ2þð1�nÞZ3Þ _eðsÞ ds�

Z t�tðtÞ

t�m_eTðsÞZ2 _eðsÞ ds

r�1

m ðeðtÞ�eðt�tðtÞÞÞT ðZ2þð1�nÞZ3ÞðeðtÞ�eðt�tðtÞÞÞ

�1

m ðeðt�tðtÞÞ�eðt�mÞÞT Z2ðeðt�tðtÞÞ�eðt�mÞÞ: ð14Þ

By setting

xðtÞ ¼ ½eT ðtÞ eT ðt�dðtÞÞ eT ðt�pÞ eT ðt�tðtÞÞ eT ðt�mÞ ~gTðeðtÞÞ�T ,

it can be obtained from (12)–(14) that

_V ðetÞrxTðtÞðFþFT RFÞxðtÞ, ð15Þ

N. Li et al. / Neurocomputing 74 (2011) 805–811808

where

F¼

O11 PKþ1pZ1 0 O14 0 P

� �2pZ1

1pZ1 0 0 0

� � �Q1�1pZ1 0 0 0

� � � O441mZ2 0

� � � � �Q2�1mZ2 0

� � � � � 0

2666666666664

3777777777775

,

O11 ¼ Q1þQ2þQ3�1

pZ1�

1

mðZ2þð1�nÞZ3Þ,

F ¼ ½0 K 0 cðG� AÞ 0 I�,

and O14, O44 and R are defined in (11).

From (8), it is not difficult to verify that the nonlinear function

~gðeðtÞÞ satisfies

eðtÞ

~gðeðtÞÞ

" #T ~UL~VL

� I

" #eðtÞ

~gðeðtÞÞ

" #r0, ð16Þ

where ~UL and ~VL are also defined in (11).

Consequently, for any scalar l40, it follows readily from (15)

and (16) that

_V ðetÞrxTðtÞðFþFT RFÞxðtÞ�l

eðtÞ

~g ðeðtÞÞ

" #T ~UL~VL

� I

" #eðtÞ

~g ðeðtÞÞ

" #

¼ xTðtÞðFþFT RFÞxðtÞ,

where

F ¼

O11 PKþ1pZ1 0 O14 0 P�l ~VL

� �2pZ1

1pZ1 0 0 0

� � �Q1�1pZ1 0 0 0

� � � O441mZ2 0

� � � � �Q2�1mZ2 0

� � � � � �lI

2666666666664

3777777777775

,

and O11 is defined in (11).

By using the Schur complement formula, it can be seen that the

inequality (10) is equivalent to FþFT RFo0, which implies_V ðetÞr�eJeðtÞJ2 where e¼�lmaxðFþFT RFÞ. Then, using a method

similar to the proof of Theorem 1 in [16], we can show the

exponential stability of system (7). This completes the proof of

Theorem 1. &

Remark 4. In Theorem 1, the stability analysis has been conductedby constructing an appropriate Lyapunov functional and usingJensen’s inequality (Lemma 1). It should be mentioned that somenewly developed approaches to dealing with time-delays can beemployed to further reduce the conservatism of the stabilitycondition. Research on deriving some less conservative resultsfor the sample-data control problem by using the delay-fractioningapproach [4,32] would be one of our future topics.

In the following, we will deal with the controller designproblem, and derive the explicit expression of the controller gainmatrices in terms of LMIs. The following theorem can be obtainedfrom Theorem 1.

Theorem 2. The complex network (1) is exponentially synchronized

by the sampled-data feedback controllers (4) if there exist matrices

P¼ diagfP1,P2, . . . ,PNg40, Qi40, Zi40 (i¼1,2,3), X ¼ diagfX1,X2, . . . ,XNg and a scalar l40 such that

Z2þð1�nÞZ340 ð17Þ

and

O11 Xþ1pZ1 0 O14 0 P�l ~VL 0

� �2pZ1

1pZ1 0 0 0 XT

� � �Q1�1pZ1 0 0 0 0

� � � O441mZ2 0 cðG� AÞT P

� � � � �Q2�1mZ2 0 0

� � � � � �lI P

� � � � � � �2PþR

266666666666664

377777777777775o0,

ð18Þ

where O11, O14, O44, R, and ~VL are defined in Theorem 1. Moreover, if

the LMIs (17) and (18) are solvable, the desired controllers gain

matrices are given as

Ki ¼ P�1i Xi, i¼ 1,2, . . . ,N: ð19Þ

Proof. By considering the inequality�PR�1Pr�2PþR, the LMI (18)implies

O11 Xþ1pZ1 0 O14 0 P�l ~VL 0

� �2pZ1

1pZ1 0 0 0 XT

� � �Q1�1pZ1 0 0 0 0

� � � O441mZ2 0 cðG� AÞT P

� � � � �Q2�1mZ2 0 0

� � � � � �lI P

� � � � � � �PR�1P

266666666666664

377777777777775o0:

ð20Þ

Then, noting the relation X¼PK and performing a congruence

transformation of diag{I, I, I, I, I, I, P�1} to the LMI (20), we

obtain the LMI (10). The rest of the proof follows directly from

Theorem 1. &

To this end, the sampled-data synchronization control problemaddressed in previous section has been solved in terms of a solution tothe LMIs (17) and (18). According to Theorem 2, the controllers givenby (20) can achieve the exponential synchronization of the complexnetwork (1) with sampled data. It should be pointed out that the mainresults obtained in this paper can be extended to complex networkswith various time-delays such as mixed time-delays [17,23] anddistributed delays [24,27]. In next section, the effectiveness of thedeveloped sampled-data control approach is shown by a numericalsimulation example.

4. An illustrative example

In this section, a simulation example is presented to demon-strate the effectiveness of the proposed synchronization controlscheme for the complex network (1).

Consider a complex network (1) with three nodes. The outer-coupling matrix is assumed to be G¼ ðGijÞ3�3 with

Gij ¼

�1 0 1

0 �1 1

1 1 �2

264

375:

The inner-coupling matrix and coupling strength are given asA¼diag{1,1,1} and c¼0.5, respectively. The time-varying delay ischosen as tðtÞ ¼ 0:2þ0:05sinð10tÞ. Accordingly, we have m¼ 0:25and n¼ 0:5.

The nonlinear function f ð�Þ is taken as

f ðxiÞ ¼�0:5xi1þtanhð0:2xi1Þþ0:2xi2

0:95xi2�tanhð0:75xi2Þ

" #, i¼ 1,2,3:

N. Li et al. / Neurocomputing 74 (2011) 805–811 809

It can be easily verified that f ð�Þ satisfies (8) with

U ¼�0:5 0:2

0 0:95

� �, V ¼

�0:3 0:2

0 0:2

� �: ð21Þ

In this example, we set the sampling period as p ¼ 0.02. By usingthe Matlab (with YALMIP 3.0 and SeDuMi 1.1), the LMIs (17) and(18) can be solved and a feasible solution is given by

P1¼0:8117 �0:0670

�0:0670 0:4922

� �, P2¼

0:8118 �0:0670

�0:0670 0:4922

� �,

P3¼0:8029 �0:0684

�0:0684 0:5016

� �,

Q1 ¼

0:4120 �0:0226 �0:0236 0:0018 �0:0809 0:0046

�0:0226 0:2545 0:0018 0:0066 0:0022 �0:1007

�0:0236 0:0018 0:4121 �0:0226 �0:0809 0:0046

0:0018 0:0066 �0:0226 0:2545 0:0022 �0:1007

�0:0809 0:0022 �0:0809 0:0022 0:4142 �0:0205

0:0046 �0:1007 0:0046 �0:1007 �0:0205 0:2668

2666666664

3777777775

,

Q2 ¼

0:4095 �0:0262 �0:0100 �0:0007 �0:0718 0:0093

�0:0262 0:2433 �0:0007 �0:0044 0:0093 �0:0562

�0:0100 �0:0007 0:4095 �0:0263 �0:0717 0:0093

�0:0007 �0:0044 �0:0263 0:2433 0:0093 �0:0562

�0:0718 0:0093 �0:0717 0:0093 0:4285 �0:0315

0:0093 �0:0562 0:0093 �0:0562 �0:0315 0:2450

2666666664

3777777775

,

Q3 ¼

0:4462 �0:0331 0:0164 �0:0065 �0:1164 0:0175

�0:0331 0:2271 �0:0065 �0:0027 0:0182 �0:0594

0:0164 �0:0065 0:4462 �0:0331 �0:1164 0:0175

�0:0065 �0:0027 �0:0331 0:2271 0:0182 �0:0594

�0:1164 0:0182 �0:1164 0:0182 0:5115 �0:0469

0:0175 �0:0594 0:0175 �0:0594 �0:0469 0:2255

2666666664

3777777775

,

0 1 20

50

100

T

x 11 a

nd s

1

0 1 2−100

0

100

T

x 21 a

nd s

1

0 1 20

50

100

T

x 31 a

nd s

1

Fig. 1. State xi1 of the uncontrolle

Z1 ¼

0:0189 0:0010 0:0012 0:0001 0:0011 0:0002

0:0010 0:0348 0:0001 0:0033 0:0003 0:0034

0:0012 0:0001 0:0189 0:0010 0:0011 0:0002

0:0001 0:0033 0:0010 0:0348 0:0003 0:0034

0:0011 0:0003 0:0011 0:0003 0:0163 0:0010

0:0002 0:0034 0:0002 0:0034 0:0010 0:0307

2666666664

3777777775

,

Z2 ¼

0:1658 0:0006 0:0093 �0:0009 �0:0120 0:0021

0:0006 0:1851 �0:0010 0:0119 0:0020 �0:0139

0:0093 �0:0010 0:1658 0:0006 �0:0120 0:0021

�0:0009 0:0119 0:0006 0:1851 0:0020 �0:0139

�0:0120 0:0020 �0:0120 0:0020 0:1906 �0:0022

0:0021 �0:0139 0:0021 �0:0139 �0:0022 0:2162

2666666664

3777777775

,

Z3 ¼

0:3918 �0:0146 �0:0168 �0:0003 �0:0187 0:0114

�0:0146 0:3137 �0:0003 �0:0399 0:0076 0:0294

�0:0168 �0:0003 0:3918 �0:0146 �0:0187 0:0114

�0:0003 �0:0399 �0:0146 0:3137 0:0076 0:0294

�0:0187 0:0076 �0:0187 0:0076 0:4214 �0:0230

0:0114 0:0294 0:0114 0:0294 �0:0230 0:3317

2666666664

3777777775

,

X1 ¼�0:5196 �0:0137

�0:0137 �0:8443

� �, X2 ¼

�0:5197 �0:0136

�0:0136 �0:8442

� �,

X3 ¼�0:2854 �0:0245

�0:0245 �0:6364

� �,

l¼ 1:5020: ð22Þ

Therefore, the gain matrices of the desired controllers can beobtained as follows:

K1 ¼ P�11 X1 ¼

�0:6497 �0:1602

�0:1162 �1:7369

� �,

K2 ¼ P�12 X2 ¼

�0:6498 �0:1602

�0:1162 �1:7369

� �,

3 4 5 6ime (t)

3 4 5 6ime (t)

3 4 5 6ime (t)

d complex network (i¼1,2,3).

N. Li et al. / Neurocomputing 74 (2011) 805–811810

K3 ¼ P�13 X3 ¼

�0:3638 �0:1402

�0:0984 �1:2877

� �: ð23Þ

In the simulation, we set the initial values of the consideredthree nodes and the unforced isolate node as x1ð0Þ ¼ ½4 3�T ,x2ð0Þ ¼ ½�2 �1�T , x3ð0Þ ¼ ½2 5�T and Sð0Þ ¼ ½1 0�T , respectively.Simulation results are presented in Figs. 1–4. Figs. 1 and 2 plotthe state trajectories of the complex network in the case of controlinput-free while the state trajectories of the controlled complexnetwork are shown in Figs. 3 and 4. From the simulation results, itcan be seen that the designed sampled-data feedback controllersachieve the exponential synchronization of the considered com-plex network well.

0 1 20

500

T

x 12 a

nd s

2

0 1 2−500

0

500

T

x 22 a

nd s

2

0 1 20

500

T

x 32 a

nd s

2

Fig. 2. State xi2 of the uncontrolle

0 5 10 15−4

−3

−2

−1

0

1

2

3

4

5

Time (t)

s1(t)

x11(t)

x21(t)

x31(t)

Fig. 3. State xi1 of the controlled complex network (i¼1,2,3).

5. Conclusions

In this paper, we have studied the sampled-data synchroniza-tion control problem for a class of general complex networks withtime-varying coupling delays. The addressed problem has firstbeen transformed to the problem of stability analysis for adifferential equation with multiple time-varying delays by con-verting the sampling period into a bounded time-varying delay.Then, by constructing an appropriate Lyapunov functional andusing Jensen’s inequality, an exponential stability criterion hasbeen obtained for the resulting delayed differential equation. Basedon that, the desired sampled-data feedback controllers have beendesigned in terms of the solution to certain LMIs. Finally, theeffectiveness of the proposed sampled-data control scheme hasbeen demonstrated by a numerical simulation example.

3 4 5 6ime (t)

3 4 5 6ime (t)

3 4 5 6ime (t)

d complex network (i¼1,2,3).

0 5 10 15−4

−3

−2

−1

0

1

2

3

4

5

Time (t)

s2(t)

x12(t)

x22(t)

x32(t)

Fig. 4. State xi2 of the controlled complex network (i¼1,2,3).

N. Li et al. / Neurocomputing 74 (2011) 805–811 811

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Nan Li received her master degree in Control Theoryand Control Engineering from Zhejiang University ofTechnology in 2009. Now she is a lecturer at ZhejiangOcean University. Her research interests include neuralnetworks, complex networks, nonlinear systems andbioinformatics.

Yulian Zhang received her master degree in Engineer-ing Mechanics from Xi’an Jiaotong University in 2002.Now she is a professor at Zhejiang Ocean University. Herresearch interests include complex networks, fluiddynamics and structural optimization.

Jiawen Hu received his bachelor degree in MechanicalDesign, Manufacturing and Automation from ZhejiangOcean University in 2003. Now he is a technician atZhejiang Ocean University. His research interests includemicrocontroller technology and electronic technology.

Zhenyu Nie received his master degree in PowerElectronics from Nanchang University in 2005. Now heis a lecturer at Zhejiang Ocean University. His researchinterests include the development of power electronicsdevice of shipping.