adaptive output synchronization of complex delayed dynamical networks with output coupling

Adaptive output synchronization of complex delayed dynamicalnetworks with output coupling

Jin-Liang Wang a,b, Huai-Ning Wu b,n

a School of Computer Science & Software Engineering, Tianjin Polytechnic University, Tianjin 300387, Chinab Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

a r t i c l e i n f o

Article history:Received 22 January 2014Received in revised form18 March 2014Accepted 14 April 2014Communicated by Guang Wu Zheng

Keywords:Complex networksOutput synchronizationAdaptive controlOutput coupling

a b s t r a c t

In the present paper, two kinds of adaptive output synchronization problems for a complex delayeddynamical network with output coupling are investigated, that is, the cases with positive definite outputmatrix and with semi-positive definite output matrix. For the former, by using adaptive control method, asufficient condition is obtained to guarantee the output synchronization of the complex dynamical network.In addition, a pinning adaptive output synchronization criterion is also derived for such network model.Then we extend these results to the case when the output matrix is semi-positive definite. Finally, twonumerical examples are provided to illustrate the effectiveness of the proposed results.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

In the real world, complex networks can be seen everywhere, andhave been considered as a fundamental tool to understand dynamicalbehavior and the response of real systems such as food webs,communication networks, social networks, power grids, cellular net-works, World Wide Web, metabolic systems, disease transmissionnetworks, and many others [1,2]. The topology and dynamicalbehavior of various complex networks have been extensively studiedby researchers [3,4]. Especially, as one of the most significant andinteresting dynamical properties of the complex networks, synchro-nization has received much of the focus in recent years. So far, a greatmany important results on synchronization have been obtained forvarious complex networks such as time invariant, time-varying, andimpulsive networkmodels; see [5–20] and relevant references therein.

It should be noticed that the state synchronization of complexnetworks with state coupling was considered in these articles (seealso the above-mentioned references). Practically, there are twokinds of coupling forms in complex networks: state coupling andoutput coupling. As we know, many phenomena in nature can bemodeled as complex networks with output coupling [21,22].Nevertheless, there are very few works on complex dynamicalnetworks with output coupling [21–23]. To our knowledge, Jianget al. [21] first introduced a complex network model with outputcoupling. Some conditions for synchronization were established

based on the Lyapunov stability theory. In [22], Chen proposed acomplex network model with output coupling and random sensordelay. A sufficient synchronization condition was given to ensure thatthe proposed network model is exponentially mean-square stable.One should note that the state synchronization was investigated in[21,22]. It is well known that the node state in complex networks isdifficult to be observed or measured, even the node state cannot beobserved or measured at all. Moreover, in many circumstances, onlypart states are needed to make the synchronization to come true. Forthese phenomena, it is more interesting to study the output synchro-nization of complex networks [23,24]. For instance, Wang andWu [24]discussed the output synchronization of a class of impulsive complexdynamical networks with time-varying delay. By constructing suitableLyapunov functionals, some useful conditions were obtained toguarantee the local and global exponential output synchronization ofthe impulsive complex networks. Unfortunately, very few authorshave considered the output synchronization for complex networkswith output coupling [23]. So, it is essential to further study the outputsynchronization of complex dynamical networks with time-varyingdelay and output coupling.

Recently, the synchronization control problem of complex dyna-mical networks has become a very hot topic in both theoreticalresearch and practical applications. Among the existing results, someresearchers focused on adaptive control [5,6,25–32] on complexdynamical networks by applying state feedback controllers. Forinstance, Zhou et al. [5] investigated the locally and globally adaptivesynchronization of an uncertain complex dynamical network. In [26],Ji et al. proposed an adaptive control method to achieve the lagsynchronization between uncertain complex dynamical network

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n Corresponding author. Tel.: þ86 10 82317301; fax: þ86 10 82317332.E-mail address: [email protected] (H.-N. Wu).

Please cite this article as: J.-L. Wang, H.-N. Wu, Adaptive output synchronization of complex delayed dynamical networks withoutput coupling, Neurocomputing (2014), http://dx.doi.org/10.1016/j.neucom.2014.04.050i

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having delayed coupling and a nonidentical reference node. Wanget al. [32] proved that the states of a weighted complex dynamicalnetwork with time-varying delay can globally asymptotically syn-chronize onto a desired orbit under the designed controllers, and theadaptive controllers have strong robustness against asymmetriccoupling matrix, time-varying weights, delays, and noise. However,the node state often cannot be observed or measured, which makescontroller design very difficult. Moreover, these adaptive controlstrategies [5,6,25,27,28,30–32] are based on a special solution of anisolate node of the networks, which may be difficult to obtain insome engineering applications. To overcome these difficulties, someadaptive output feedback controllers are proposed in this paper.

Motivated by the above discussions, in this paper, we propose acomplex delayed dynamical network with output coupling. Theobjective of this paper is to study the adaptive output synchronizationof the proposed network model. By constructing appropriate Lyapu-nov functionals and utilizing adaptive control technique, some suffi-cient conditions on output synchronization are derived for theproposed network model.

The rest of this paper is organized as follows. In Section 2, ourmathematical model of complex delayed dynamical network ispresented and some preliminaries are given. The main results of thispaper are given in Sections 3 and 4. In Section 5, two numericalexamples are provided to illustrate the effectiveness of the theore-tical results. Finally, Section 6 concludes the investigation.

2. Network model and preliminaries

Let Rn be the n-dimensional Euclidean space and Rn�m be thespace of n�m real matrices. PARn�n⩾0 ðPARn�n⩽0Þ means thatmatrix P is symmetric and semi-positive (semi-negative) definite.PARn�n40 ðPARn�no0Þ means that matrix P is symmetric andpositive (negative) definite. In denotes the n�n real identity matrix.BT denotes the transpose of matrix B. � denotes the Kroneckerproduct of two matrices. λmð�Þ and λMð�Þ denote the minimum andthe maximum eigenvalue of the corresponding matrix, respectively.Cð½�τ;0�;RnÞ is a Banach space of continuous functions mapping theinterval ½�τ;0� into Rn with the norm JϕJ τ ¼ sup� τr θr0 JϕðθÞJ ,where J � J is the Euclidean norm.

In this paper, we consider a complex network consisting of Nidentical nodes with diffusive and output coupling, in which eachnode is an n-dimensional dynamical system. The mathematicalmodel of the network can be described as follows:

_xiðtÞ ¼ f ðxiðtÞÞþa ∑N

j ¼ 1Gijyjðt�τðtÞÞþuiðtÞ;

yiðtÞ ¼ CxiðtÞ

8><>: ð1Þ

where i¼ 1;2;…;N. The function f ð�Þ, describing the local dynamics ofthe nodes, is continuous and capable of producing various richdynamical behaviors; xiðtÞ ¼ ðxi1ðtÞ; xi2ðtÞ;…; xinðtÞÞT ARn is the statevariable of node i; yiðtÞ ¼ ðyi1ðtÞ; yi2ðtÞ;…; yinðtÞÞT ARn is the outputvector of node i; uiðtÞARn is the control input; τðtÞ is the time-varyingdelay with 0rτðtÞrτ; a is a positive real number, which representsthe overall coupling strength; the output matrix C ¼ diagðc1; c2;…; cnÞis a semi-positive definite matrix; G¼ ðGijÞN�N represents the topolo-gical structure of network and coupling strength between nodes,where Gij is defined as follows: if there exists a connection betweennodes i and j, then Gij ¼ Gji40; otherwise, Gij ¼ Gji ¼ 0 ðia jÞ, and thediagonal elements of matrix G are defined by

Gii ¼ � ∑N

j ¼ 1ja i

Gij; i¼ 1;2;…;N:

In this paper, we always assume that complex network (1) isconnected. Since network (1) is connected in the sense of having

no isolated clusters, which means that the coupling matrix G isirreducible. The initial condition associated with the complexnetwork (1) is given in form

xiðsÞ ¼ΦiðsÞ; yiðsÞ ¼ CΦiðsÞ

where i¼ 1;2;…;N;ΦiðsÞACð½�τ;0�;RnÞ.

Remark 1. In [21], Jiang, Tang and Chen first introduced acomplex network model with output coupling without time delay.However, time delays always exist in complex networks due to thefinite speeds of transmission and/or the traffic congestion, andmost of the delays are notable. So it is crucial for us to take thedelay into the consideration when we study complex networks.Moreover, absolute constant delay may be scarce and delays arefrequently varied with time. Therefore, a complex network modelwith output coupling and time-varying delay is considered in thispaper. In addition, another issue of importance deserving attentionis the fact that many existing complex network models with statecoupling can be represented by (1) through an appropriate choiceof the parameters, e.g., see also [33–37].

Remark 2. In recent years, a lot of efforts have been made to studythe adaptive state synchronization of complex networks with statecoupling [25–32]. Unfortunately, the node state in complex networksis difficult to be observed or measured, even the node state cannot beobserved or measured at all. Moreover, in many circumstances, onlypart states are needed to make the synchronization to come true.Therefore, it is more interesting to study the output synchronizationof complex networks. To the best of our knowledge, this is the firstpaper to consider the adaptive output synchronization of complexdelayed dynamical networks with output coupling, which is a veryimportant and interesting problem.

In what follows, we introduce some useful definitions andlemmas.

Definition 2.1. The complex network ð1Þ is said to achieve outputsynchronization if

limt-þ1

JyiðtÞ�1N

∑N

j ¼ 1yjðtÞJ ¼ 0 for all i¼ 1;2;…;N:

Lemma 2.1 (See Yu et al. [38]). Suppose that G¼ ðGijÞN�N is a realsymmetric and irreducible matrix, where

GijZ0 ðia jÞ; Gii ¼ � ∑N

j ¼ 1ja i

Gij:

Then,

(1) 0 is an eigenvalue of matrix G with multiplicity 1 and all theother eigenvalues of G are strictly negative.

(2) The largest nonzero eigenvalue λ2ðGÞ of the matrix G satisfies

λ2ðGÞ ¼ maxxT1N ¼ 0;xa0

xTGxxTx

:

(3) For any η¼ ðη1; η2;…; ηNÞT ARN ,

ηTGη¼ �12

∑N

i ¼ 1∑N

j ¼ 1Gijðηi�ηjÞ2:

3. Adaptive output synchronization of complex delayeddynamical network with positive definite output matrix

This section discusses the adaptive output synchronization ofcomplex delayed dynamical network (1) with positive definite out-put matrix. Several output synchronization criteria are obtained by

J.-L. Wang, H.-N. Wu / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





constructing appropriate Lyapunov functionals and utilizing thedesigned adaptive controllers.

For convenient analysis, we let

f ðxðtÞÞ ¼ Cf ðC�1xðtÞÞwhere xðtÞARn. Then we can get from (1) that

_yiðtÞ ¼ f ðyiðtÞÞþa ∑N

j ¼ 1GijCyjðt�τðtÞÞþCuiðtÞ ð2Þ

where i¼ 1;2;…;N.In order to obtain our main results, we make the following

assumption:(A1) There exist positive definite diagonal matrix P ¼ diag

ðp1; p2;…; pnÞ and diagonal matrix Δ¼ diagðδ1; δ2;…; δnÞ such thatf satisfies the following inequality:

ðx�yÞTP½ðf ðxÞ� f ðyÞÞ�Δðx�yÞ�⩽�ηðx�yÞT ðx�yÞfor some η40 and all x; yARn.

3.1. Adaptive output synchronization of complex network withoutput coupling

Let V ¼ f1;2;…;Ng and E� V � V respectively denote the set ofnodes and the set of undirected links in the network (1).GðtÞ ¼ ðGijðtÞÞN�N is a time-varying matrix, where GijðtÞ is definedas follows: if there is an edge between nodes i and j at time t, thenthe element GijðtÞ ¼ GjiðtÞ40; otherwise, GijðtÞ ¼ GjiðtÞ ¼ 0 ðia jÞ,and the diagonal elements of matrix GðtÞ are defined by

GiiðtÞ ¼ � ∑N

j ¼ 1ja i

GijðtÞ; i¼ 1;2;…;N:

In this paper, the topological structure of network (1) is fixed. Thatis, if there is no connection between nodes i and j, thenGijðtÞ ¼ GjiðtÞ ¼ 0 for all t.

Theorem 3.1. Let (A1) hold and _τðtÞrso1. Then, complex network(1) with positive definite output matrix achieves output synchroniza-tion under the adaptive controllers

uiðtÞ ¼ a ∑N

j ¼ 1GijðtÞyjðtÞ; i¼ 1;2;…;N ð3Þ

and updating laws

_GijðtÞ ¼ βijðyiðtÞ�yjðtÞÞTPCðyiðtÞ�yjðtÞÞ;Gijð0Þ ¼ Gjið0Þ40; ð4Þði; jÞAE, where βij ¼ βji are positive constants.

Proof. Let yðtÞ ¼ ð1=NÞ∑Nj ¼ 1yjðtÞ. Then, we have

_y ðtÞ ¼ 1N

∑N

j ¼ 1f ðyjðtÞÞ: ð5Þ

Define eiðtÞ ¼ yiðtÞ�yðtÞ, then the dynamics of the output errorvector ei(t) is governed by the following equation:

_eiðtÞ ¼ f ðyiðtÞÞ�1N

∑N

j ¼ 1f ðyjðtÞÞþa ∑

N

j ¼ 1GijðtÞCejðtÞþa ∑

N

j ¼ 1GijCejðt�τðtÞÞ;

ð6Þwhere i¼ 1;2;…;N.

Construct a Lyapunov functional for the system (6) as follows:

VðtÞ ¼ 12

∑N

i ¼ 1eTi ðtÞPeiðtÞþ ∑

N

i ¼ 1∑N

j ¼ 1ja i

a4βij

ðGijðtÞ�bijÞ2þa2

∑N

i ¼ 1∑N

j ¼ 1

jGijj1�s

Z t

t� τðtÞeTj ðsÞPCejðsÞ ds;

ð7Þwhere bij ¼ bji are nonnegative constants, and bij ¼ 0 if and only ifGijðtÞ ¼ 0.

Calculating the time derivative of V(t) along the trajectory ofsystem (6), we can get

_V ðtÞ⩽ ∑N

i ¼ 1eTi ðtÞP _eiðtÞþ ∑

N

i ¼ 1∑N

j ¼ 1ja i

a2βij

ðGijðtÞ�bijÞ _GijðtÞ

þa2

∑N

i ¼ 1∑N

j ¼ 1

jGijj1�se

Tj ðtÞPCejðtÞ

�a2

∑N

i ¼ 1∑N

j ¼ 1Gij eTj ðt�τðtÞÞPCejðt�τðtÞÞ��

¼ ∑N

i ¼ 1eTi ðtÞP f ðyiðtÞÞ�

1N

∑N


N

j ¼ 1GijðtÞCejðtÞ

"

þa ∑N

j ¼ 1GijCejðt�τðtÞÞ

#

þa2

∑N

i ¼ 1∑N

j ¼ 1

jGijj1�se

Tj ðtÞPCejðtÞþ

a2

∑N

i ¼ 1∑N

j ¼ 1ja i

ðGijðtÞ�bijÞðyiðtÞ

�yjðtÞÞTPCðyiðtÞ�yjðtÞÞ

�a2

∑N

i ¼ 1∑N

j ¼ 1jGijjeTj ðt�τðtÞÞPCejðt�τðtÞÞ

⩽ ∑N

i ¼ 1eTi ðtÞP f ðyiðtÞÞ�

1N

∑N

j ¼ 1f ðyjðtÞÞ� f ðyðtÞÞ

"

þ f ðyðtÞÞþa ∑N

j ¼ 1GijðtÞCejðtÞ

#þa2

∑N

i ¼ 1∑N

j ¼ 1

jGijj1�se

Tj ðtÞPCejðtÞ

þa2

∑N

i ¼ 1∑N

j ¼ 1ja i

ðGijðtÞ�bijÞðeiðtÞ�ejðtÞÞTPCðeiðtÞ�ejðtÞÞ

þa2

∑N

i ¼ 1∑N

j ¼ 1Gij eTi ðtÞPCeiðtÞ:�� ð8Þ

Since ∑Ni ¼ 1e

Ti ðtÞ ¼ 0, we can obtain

∑N

i ¼ 1eTi ðtÞP f ðyðtÞÞ� 1

N∑N

j ¼ 1f ðyjðtÞÞ

" #¼ 0: ð9Þ

According to (A1), we have

∑N

i ¼ 1eTi ðtÞP½f ðyiðtÞÞ� f ðyðtÞÞ�⩽ ∑

N

i ¼ 1eTi ðtÞð�ηInþPΔÞeiðtÞ: ð10Þ

Define the matrix B¼ ðbijÞN�N , where bii ¼ �∑Nj ¼ 1ja i

bij; i¼ 1;2;…;N.Then we can get

∑N

i ¼ 1∑N

j ¼ 1ja i

ðGijðtÞ�bijÞðeiðtÞ�ejðtÞÞTPCðeiðtÞ�ejðtÞÞ

¼ �2 ∑N

i ¼ 1∑N

j ¼ 1ðGijðtÞ�bijÞeTi ðtÞPCejðtÞ: ð11Þ

It follows from (8) to (11) that

_V ðtÞ⩽ ∑N

i ¼ 1eTi ðtÞ �ηInþPΔþað2�sÞjGiij

1�s PC� �

eiðtÞ

þa ∑N

i ¼ 1∑N

j ¼ 1bijeTi ðtÞPCejðtÞ

¼ eT ðtÞ �ηðIN � InÞþðIN � ðPΔÞÞþaðB � ðPCÞÞ½

það2�sÞ1�s ðG � ðPCÞÞ

�eðtÞ; ð12Þ

where G ¼ diagðjG11j; jG22j;…; jGNN jÞ; eðtÞ ¼ ðeT1ðtÞ; eT2ðtÞ;…; eTNðtÞÞT .According to Lemma 2.1, there obviously exists a unitary matrix

ϕ¼ ðϕ1;ϕ2;…;ϕNÞARN�N such that ϕTBϕ¼ Λ with Λ¼ diagðλ1;λ2;…; λNÞ. λi; i¼ 1;2;…;N, are the eigenvalues of B and0¼ λ14λ2Zλ3Z⋯ZλN . Let zðtÞ ¼ ðzT1ðtÞ; zT2ðtÞ;…; zTNðtÞÞT ¼ðϕT � InÞeðtÞ. Since ϕ1 ¼ ð1=

ffiffiffiffiN

pÞð1;1;…;1ÞT , one has z1ðtÞ ¼

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ðϕT1 � InÞeðtÞ ¼ 0. Then, we can derive from (12) that

_V ðtÞ⩽eT ðtÞ��ηðIN � InÞþðIN � ðPΔÞÞþaðϕ � InÞðΛ � ðPCÞÞðϕT � InÞ


�eðtÞ

¼ eT ðtÞ��ηðIN � InÞþðIN � ðPΔÞÞ


�eðtÞþazT ðtÞðΛ � ðPCÞÞzðtÞ: ð13Þ

Since P and C are positive definite diagonal matrices, we have

zT ðtÞðΛ � ðPCÞÞzðtÞrλ2zT ðtÞðIN � ðPCÞÞzðtÞ:Therefore,

_V ðtÞ⩽eT ðtÞ �ηðIN � InÞþðIN � ðPΔÞÞþað2�sÞ1�s ðG � ðPCÞÞ

� �eðtÞ

þaλ2zT ðtÞðIN � ðPCÞÞzðtÞ

¼ eT ðtÞ��ηðIN � InÞþðIN � ðPΔÞÞ


�eðtÞ

þaλ2eT ðtÞðϕ � InÞðIN � ðPCÞÞðϕT � InÞeðtÞ¼ eT ðtÞ �ηðIN � InÞþðIN � ðPΔÞÞ½

það2�sÞ1�s ðG � ðPCÞÞþaλ2ðIN � ðPCÞÞ

�eðtÞ: ð14Þ

By selecting bij sufficiently large such that

aλ2cþað2�sÞg c

1�s þδr0; ð15Þ

one obtains

ðIN � ðPΔÞÞþað2�sÞ1�s ðG � ðPCÞÞþaλ2ðIN � ðPCÞÞr0;

where

c ¼ mini ¼ 1;2;…;N

fpicig; c ¼ maxi ¼ 1;2;…;N

fpicig;

g ¼ maxi ¼ 1;2;…;N

fjGiijg; δ ¼ maxi ¼ 1;2;…;N

fpiδig:

Therefore,

_V ðtÞr�ηeT ðtÞeðtÞ: ð16ÞSince V(t) is monotonically decreasing and bounded, we canconclude that V(t) asymptotically converges to a finite non-negative value. From (16), we have

‖eðtÞ‖2⩽�_V ðtÞη

: ð17Þ

We can derive from (17) that limt-þ1R t0 ‖eðsÞ‖2 ds exists. Since

0rτðtÞrτ, we can get limt-þ1R tt� τðtÞ e

Tj ðsÞPCejðsÞ ds¼ 0; j¼ 1;

2;…;N. On the other hand, since GijðtÞ is monotonically increasing

and bounded [see (4)], one can conclude that GijðtÞ ðði; jÞAEÞasymptotically converges to a finite positive value. By the defini-tion of V(t), we can derive that limt-þ1∑N

i ¼ 1eTi ðtÞPeiðtÞ exists and

is a non-negative real number. In what follows, we shall prove that

limt-þ1

∑N

i ¼ 1eTi ðtÞPeiðtÞ ¼ 0:

If this is not true, we have

limt-þ1

∑N

i ¼ 1eTi ðtÞPeiðtÞ ¼ μ40:

Then, there obviously exists a real number M40 such that

∑Ni ¼ 1e

Ti ðtÞPeiðtÞ4μ=2 for tZM. From (16), we can get

_V ðtÞ⩽�η∑N

i ¼ 1eTi ðtÞPeiðtÞ

λMðPÞo� ημ

2λMðPÞ; t⩾M: ð18Þ

By integrating (18) with respect to t over the time period M toþ1, we can obtain

�VðMÞ⩽Vðþ1Þ�VðMÞ ¼Z þ1

M

_V ðtÞ dto�Z þ1

M

ημ

2λMðPÞdt ¼ �1:

This yields a contradiction, and so limt-þ1∑Ni ¼ 1e

Ti ðtÞPeiðtÞ ¼ 0.

Then, we can easily obtain limt-þ1 JeðtÞJ ¼ 0. Therefore, the com-plex network (1) achieves output synchronization under the adaptivecontrollers (3) and updating laws (4). The proof is completed. □

Remark 3. In recent years, many adaptive control [5,6,25,27,28]were proposed to guarantee the synchronization of complexnetworks. However, these adaptive control strategies are basedon a special solution of an isolate node of the networks, whichmay be difficult to obtain in some engineering applications. In thispaper, we remove this restriction.

3.2. Pinning adaptive output synchronization of complex networkwith output coupling

Based on the outputs of all neighboring nodes, an adaptivecontrol scheme is proposed to guarantee the output synchroniza-tion of the complex dynamical network (1) in the previoussubsection. Here, only a portion of neighboring nodes are neededto design the adaptive controllers.

Let E denote a subset of E� V � V . AðtÞ ¼ ðAijðtÞÞN�N is a time-varying matrix, where Aij(t) is defined as follows:

AijðtÞ ¼

AijðtÞ ¼ AjiðtÞ40 if ði; jÞA E ;

AiiðtÞ ¼ � ∑N

j ¼ 1ja i

AijðtÞ if i¼ j;

0 otherwise:

8>>>><>>>>:

In this subsection, we assume that complex network (1) isconnected through the undirected edges E .

Theorem 3.2. Let (A1) hold and _τðtÞ⩽so1. Then, complex network(1) with positive definite output matrix achieves output synchroniza-tion under the adaptive controllers

uiðtÞ ¼ a ∑N

j ¼ 1AijðtÞyjðtÞ; i¼ 1;2;…;N ð19Þ

and updating laws

_AijðtÞ ¼ βijðyiðtÞ�yjðtÞÞTPCðyiðtÞ�yjðtÞÞ;Aijð0Þ ¼ Ajið0Þ40; ð20Þði; jÞA E , where βij ¼ βji are positive constants.

Proof. Let yðtÞ ¼ ð1=NÞ∑Nj ¼ 1yjðtÞ and eiðtÞ ¼ yiðtÞ�yðtÞ. Then, the

dynamics of the output error vector ei(t) is governed by thefollowing equation:

_eiðtÞ ¼ f ðyiðtÞÞ�1N

∑N


N

j ¼ 1AijðtÞCejðtÞþa ∑

N

j ¼ 1GijCejðt�τðtÞÞ;

ð21Þwhere i¼ 1;2;…;N.

Define the following Lyapunov functional for the system (21):

VðtÞ ¼ 12

∑N

i ¼ 1eTi ðtÞPeiðtÞþ ∑

N

i ¼ 1∑

ði;jÞA E

a4βij

ðAijðtÞ�bijÞ2

þa2

∑N

i ¼ 1∑N

j ¼ 1

jGijj1�s

Z t

t� τðtÞeTj ðsÞPCejðsÞ ds; ð22Þ






where bij ¼ bji are nonnegative constants, and bij ¼ 0 if and only ifAijðtÞ ¼ 0.

Then, the following similar arguments as in the proof ofTheorem 3.1, we can obtain the desired result immediately. □

4. Adaptive output synchronization of complex delayeddynamical network with semi-positive definite output matrix

In this section, we extend the above-obtained results to thecase when the output matrix C is semi-positive definite.

Without loss of generality, rearrange the order of the elementsof node state and output and let the first r ð1rronÞ diagonalelements of matrix C be positive real numbers. That is,C ¼ diagðc1; c2;…; cr ;0;…;0Þ.

For convenience, we denote

C ¼ diagðc1; c2;…; crÞ; C ¼ ðC 0ÞARr�n;

x1i ðtÞ ¼ ðxi1ðtÞ; xi2ðtÞ;…; xirðtÞÞT ARr ;

x2i ðtÞ ¼ ðxiðrþ1ÞðtÞ; xiðrþ2ÞðtÞ;…; xinðtÞÞT ARn� r ;

xiðtÞ ¼ ððx1i ðtÞÞT ; ðx2i ðtÞÞT ÞT ; y1i ðtÞ ¼ Cx1i ðtÞ;yiðtÞ ¼ ððy1i ðtÞÞT ;0;0;…;0ÞT :Then, we can get from (1) that

_y1i ðtÞ ¼ Cf ðxiðtÞÞþa ∑

N

j ¼ 1GijCyjðt�τðtÞÞþCuiðtÞ; ð23Þ

where i¼ 1;2;…;N.In order to obtain our main results, an assumption is

introduced.(A2) Assume that

f ðxiðtÞÞ ¼f 1ðx1i ðtÞÞf 2ðxiðtÞÞ

!

where f 1ðx1i ðtÞÞARr ; f 2ðxiðtÞÞARn� r .According to (23) and (A2), we can obtain

_y1i ðtÞ ¼ f 1ðy1i ðtÞÞþa ∑

N

j ¼ 1GijCy1j ðt�τðtÞÞþCuiðtÞ; ð24Þ

where i¼ 1;2;…;N; f 1ðy1i ðtÞÞ ¼ C f 1ðC�1

y1i ðtÞÞ.Next, we make the following assumption:(A3) There exist positive definite diagonal matrix Z ¼ diag

ðz1; z2;…; zrÞ and diagonal matrix Q ¼ diagðq1; q2;…; qrÞ such thatf 1 satisfies the following inequality:

ðx�yÞTZ½f 1ðxÞ� f 1ðyÞ�Q ðx�yÞ�r�θðx�yÞT ðx�yÞfor some θ40 and all x; yARr .

4.1. Adaptive output synchronization of complex network withoutput coupling

For convenient analysis, we let

Z ¼ diagðz1; z2;…; zr ;0;…;0ÞARn�n:

Theorem 4.1. Let (A2) and (A3) hold, _τðtÞrso1. Then, complexnetwork (1) achieves output synchronization under the adaptivecontrollers

uiðtÞ ¼ a ∑N

j ¼ 1GijðtÞyjðtÞ; i¼ 1;2;…;N ð25Þ

and updating laws

_GijðtÞ ¼ βijðyiðtÞ�yjðtÞÞT ZCðyiðtÞ�yjðtÞÞ;Gijð0Þ ¼ Gjið0Þ40; ð26Þ

ði; jÞAE, where βij ¼ βji are positive constants. Here E and GijðtÞ havethe same meaning as in Section 3.1.

Proof. Firstly, we can derive from (24) and (25) that

_y1i ðtÞ ¼ f 1ðy1i ðtÞÞþa ∑

N

j ¼ 1GijðtÞCy1j ðtÞþa ∑

N

j ¼ 1GijCy

1j ðt�τðtÞÞ ð27Þ

where i¼ 1;2;…;N.Then, by a minor modification of the proof of Theorem 3.1, we

can easily get the conclusion. □

4.2. Pinning adaptive output synchronization of complex networkwith output coupling

In this subsection, E and AijðtÞ denote the same meaning as inSection 3.2. In addition, we also assume that complex network (1)is connected through the undirected edges E .

Theorem 4.2. Let (A2) and (A3) hold, _τðtÞrso1. Then, complexnetwork (1) achieves output synchronization under the adaptivecontrollers

uiðtÞ ¼ a ∑N

j ¼ 1AijðtÞyjðtÞ; i¼ 1;2;…;N ð28Þ

and updating laws

_AijðtÞ ¼ βijðyiðtÞ�yjðtÞÞT ZCðyiðtÞ�yjðtÞÞ;

Aijð0Þ ¼ Ajið0Þ40 ði; jÞA E ; ð29Þ

where βij ¼ βji are positive constants, Z ¼ diagðz1; z2;…; zr ;0;…;0ÞARn�n.

Proof. By the similar proof of Theorem 3.2, we can obtain theconclusion. Here we omit its proof to avoid the repetition. □

Remark 4. In this paper, some simple adaptive control schemes areproposed to synchronize the outputs of complex delayed dynamicalnetwork with output coupling. Here, only node output is needed todesign the adaptive controllers. The designed controllers are totallydifferent from some existing ones [5,6,25–32]. Obviously, comparedwith these existing results, our sufficient conditions for networksynchronization are very simple and the controllers can be realizedmore easily.

5. Numerical examples

In this section, we give two examples and their simulation toshow the effectiveness of the above-obtained theoretical results.

Example 1. Consider a complex dynamical network consisting of5 identical Chua's circuits, in which each node equation is [17]

_xi1_xi2_xi3

0B@

1CA¼

10ð�xi1þxi2�gðxi1ÞÞxi1�xi2þxi3�14:87xi2

0B@

1CA

where i¼ 1;2;…;5; gðxi1Þ ¼ �0:68xi1þ0:5ð�1:27þ0:68Þðjxi1þ1j�jxi1�1jÞ. Take a¼ 0:3; τðtÞ ¼ 0:4�0:4e� t ;C ¼ diagð0:5;0:5;0:5Þ.Then we have _τðtÞr0:4o1 and

f ðxÞ ¼10ð�x1þx2� gðx1ÞÞ

x1�x2þx3�14:87x2

0B@

1CA

where x¼ ðx1; x2; x3ÞT AR3; gðx1Þ ¼ �0:68x1þ0:5ð�1:27þ0:68Þðjx1þ0:5j�jx1�0:5jÞ. It is easy to verify that (A1) is satisfied with






P ¼ I3;Δ¼ 11I3 and η¼ 1. The matrix G is chosen as

G¼

�1:5 0:5 0:4 0 0:60:5 �0:9 0:4 0 00:4 0:4 �1:5 0:7 00 0 0:7 �1:2 0:50:6 0 0 0:5 �1:1

0BBBBBB@

1CCCCCCA:

Case 1: Choose β12 ¼ β21 ¼ 0:3; β13 ¼ β31 ¼ 0:4; β15 ¼ β51 ¼ 0:2;β23 ¼ β32 ¼ 0:3; β34 ¼ β43 ¼ 0:4; β45 ¼ β54 ¼ 0:5. By using the pro-posed adaptive controllers (3) and updating laws (4) for fivenodes, the simulation results are shown in Fig. 1.

Case 2: Choose β12 ¼ β21 ¼ 0:3; β13 ¼ β31 ¼ 0:4; β15 ¼ β51 ¼0:2; β34 ¼ β43 ¼ 0:4. Under the adaptive controllers (19) and updat-ing laws (20), the evolutions of the output variables of five nodesare shown in Fig. 2.

Next, we analyze the adaptive output synchronization ofcomplex delayed dynamical network with semi-positive definiteoutput matrix.

Example 2. Consider a complex dynamical network consisting of5 identical nodes, in which each node is a 4-dimensional linearsystem described by

_vi1

_vi2

_xi1_xi2

0BBBB@

1CCCCA¼

00vi1vi2

0BBBB@

1CCCCA

where i¼ 1;2;…;5. Take a¼ 0:4; τðtÞ ¼ 0:3�0:3e� t ;C ¼ diagð1;1;0;0Þ. Then we have _τðtÞr0:3o1 and f 1ðviðtÞÞ ¼ 0; viðtÞ ¼ðvi1; vi2ÞT . It is easy to verify that (A2) and (A3) are satisfied withZ ¼Q ¼ I2 and θ¼ 1. The matrix G is chosen as

G¼

�1:7 0:3 0:4 0:4 0:60:3 �0:7 0:4 0 00:4 0:4 �0:8 0 00:4 0 0 �0:7 0:30:6 0 0 0:3 �0:9

0BBBBBB@

1CCCCCCA:

0 2 4 6 8 10−5

0

5

t

y i1, i

= 1,

2, .

.., 5

0 2 4 6 8 10−2

0

2

t

y i2, i

= 1,

2, .

.., 5

0 2 4 6 8 10−5

0

5

t

y i3, i

= 1,

2, .

.., 5

Fig. 1. The change processes of the output variables of complex dynamical network(1) in time interval [0, 10].

0 2 4 6 8 10−5

0

5

t

y i1, i

= 1,

2, .

.., 5

0 2 4 6 8 10−1

0

1

t

y i2, i

= 1,

2, .

.., 5

0 2 4 6 8 10−5

0

5

t

y i3, i

= 1,

2, .

.., 5

Fig. 2. The change processes of the output variables of complex dynamical network(1) in time interval [0, 10].

0 2 4 6 8 10−3

−2

−1

0

1

t

v i1, i

= 1,

2, .

.., 5

0 2 4 6 8 10−2

−1

0

1

t

v i2, i

= 1,

2, .

.., 5

Fig. 3. The change processes of vi1ðtÞ and vi2ðtÞ; i¼ 1;2;…;5.

0 2 4 6 8 10−3

−2

−1

0

1

2

t

v i1, i

= 1,

2, .

.., 5

0 2 4 6 8 10−1

0

1

2

3

t

v i2, i

= 1,

2, .

.., 5

Fig. 4. The change processes of vi1ðtÞ and vi2ðtÞ; i¼ 1;2;…;5.






Case 1: Choose β12 ¼ β21 ¼ 0:2; β13 ¼ β31 ¼ 0:5; β14 ¼ β41 ¼ 0:3;β15 ¼ β51 ¼ 0:4; β23 ¼ β32 ¼ 0:2; β45 ¼ β54 ¼ 0:3. With adaptive con-trollers (25) and updating laws (26), the evolutions of the outputvariables of five nodes are shown in Fig. 3.

Case 2: Choose β12 ¼ β21 ¼ 0:2; β15 ¼ β51 ¼ 0:4; β23 ¼ β32 ¼ 0:2;β45 ¼ β54 ¼ 0:3. By using the proposed adaptive controllers (28)and updating laws (29) for five nodes, the simulation results areshown in Fig. 4.

6. Conclusion

In this paper, a new complex delayed dynamical network withoutput coupling and semi-positive definite output matrix has beenintroduced. The adaptive output synchronization of the proposednetwork model has been investigated, and some sufficient condi-tions have been obtained to guarantee output synchronization ofthe complex dynamical network. Illustrative simulations havebeen provided to verify the correctness and effectiveness of theobtained results. In future work, we shall study the outputsynchronization of complex delayed dynamical networks withoutput coupling and directed topologies under adaptive control.

Acknowledgments

The authors would like to thank the Associate Editor andanonymous reviewers for their valuable comments and suggestions.

This work was supported in part by the National NaturalScience Foundation of China under Grant 61121003 and in partby the General Research Fund project from Science and Technol-ogy on Aircraft Control Laboratory of Beihang University underGrant 9140C480301130C48001.

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Jin-Liang Wang was born in Hebei, China, in 1984. Hereceived the M.S. degree in applied mathematics fromthe Department of Mathematics, Chongqing NormalUniversity, Chongqing, China, in 2010, and the Ph.D.degree in control theory and control engineering fromthe School of Automation Science and Electrical Engi-neering, Beihang University, Beijing, China, in 2014.

In January 2014, he joined the School of ComputerScience & Software Engineering, Tianjin PolytechnicUniversity, Tianjin, China. His research interests includecomplex networks, neural networks, multi-agent sys-tems, distributed parameter systems.

Dr. Wang is the recipient of “2011 Excellent MasterDegree Thesis Award of Chongqing city”, and Scholarship Award for ExcellentDoctoral Student granted by Ministry of Education (2012).



http://refhub.elsevier.com/S0925-2312(14)00672-9/sbref1








































































































Huai-Ning Wu was born in Anhui, China, on November15, 1972. He received the B.E. degree in automationfrom Shandong Institute of Building Materials Industry,Jinan, China, and the Ph.D. degree in control theory andcontrol engineering from Xi'an Jiaotong University,Xi'an, China, in 1992 and 1997, respectively.

From August 1997 to July 1999, he was a PostdoctoralResearcher in the Department of Electronic Engineeringat Beijing Institute of Technology, Beijing, China. InAugust 1999, he joined the School of AutomationScience and Electrical Engineering, Beihang University(formerly Beijing University of Aeronautics and Astro-nautics), Beijing. From December 2005 to May 2006,

he was a Senior Research Associate with the Department of Manufacturing

Engineering and Engineering Management (MEEM), City University of Hong Kong,Kowloon, Hong Kong. From October to December during 2006–2008 and from Julyto August in 2010, he was a Research Fellow with the Department of MEEM, CityUniversity of Hong Kong. From July to August in 2011, he was a Research Fellowwith the Department of Systems Engineering and Engineering Management, CityUniversity of Hong Kong. He is currently a Professor with Beihang University. Hiscurrent research interests include robust control and filtering, fault-tolerantcontrol, distributed parameter systems, and fuzzy/neural modeling and control.

Dr. Wu serves as an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS,MAN AND CYBERNETICS: SYSTEMS. He is a member of the Committee of TechnicalProcess Failure Diagnosis and Safety, Chinese Association of Automation.






adaptive output synchronization of complex delayed dynamical networks with output coupling

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