Automatica 45 (2009) 1799–1807

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Automatica

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

Structure identification of uncertain general complex dynamical networks withtime delayI

Hui Liu a, Jun-An Lu a, Jinhu Lü b,∗, David J. Hill ca School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Chinab Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, Chinac Research School of Information Sciences and Engineering, The Australian National University, Canberra, ACT 0200, Australia

a r t i c l e i n f o

Article history:Received 13 April 2008Received in revised form29 December 2008Accepted 27 March 2009Available online 12 May 2009

Keywords:Complex networksStructure identificationParameters estimationTime delayAdaptive observer

a b s t r a c t

It is well known that many real-world complex networks have various uncertain information, such asunknown or uncertain topological structure and node dynamics. The structure identification problemhas theoretical and practical importance for uncertain complex dynamical networks. At the same time,time delay often appears in the state variables or coupling coefficients of various practical complexnetworks. This paper initiates a novel approach for simultaneously identifying the topological structureand unknown parameters of uncertain general complex networks with time delay. In particular, thismethod is also effective for uncertain delayed complex dynamical networkswith different node dynamics.Moreover, the proposed method can be easily extended to monitor the on-line evolution of networktopological structure. Finally, three representative examples are then given to verify the effectivenessof the proposed approach.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, there are numerous natural or man-made complexnetworks. Typical examples are the World Wide Web, Internet,communication networks, social networks, food webs, metabolicnetworks, and so on (Albert & Barabási, 2002; Barabási, 2002;Dorogovtsev & Mendes, 2002; Guimera, Arenas, Guilera, Redondo,& Cabrales, 2002; Jeong, Tombor, Albert, Oltvai, & Barabási, 2000;Lü & Chen, 2005; Lü, Yu, & Chen, 2004; Lü, Yu, Chen, & Cheng, 2004;Sorrentino, Bernardo, Garofalo, & Chen, 2007; Strogatz, 2001; Wu,2006; Yang, Cao, Wang, & Li, 2006). All the above networks can berepresented in terms of nodes and edges indicating connectionsbetween nodes. It is well known that complex networks pervadethrough almost all scientific and technological fields, includingmathematics, physics, engineering, biological sciences, ecology,and social sciences.In real-world complex networks, there exists various uncertain

information, such as unknown or uncertain topological structureand node dynamics (Lu & Cao, 2005; Wu, 2008; Yu, Righero, &Kocarev, 2006; Yu & Cao, 2007; Yu, Chen, Cao, Lü, & Parlitz, 2007;

I This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Derong Liu underthe direction of Editor Miroslav Krstic.∗ Corresponding author. Tel.: +86 10 62651447; fax: +86 10 62587343.E-mail addresses: uphui@126.com (H. Liu), jalu@whu.edu.cn (J.-A. Lu),

jhlu@iss.ac.cn (J. Lü), David.Hill@anu.edu.au (D.J. Hill).

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.03.022

Zhou & Lu, 2007; Zhou, Lu, & Lü, 2006). Moreover, time delay oftenappears in various complex networks, such as communicationnetworks, neural networks, and metabolic networks, in eitherthe state variables or the coupling coefficients (Chen, Zhou, &Liu, 2004; Lu & Chen, 2004; Pyragas, 1998; Zhang, Lu, Lü, & Tse,2008). Time delay is often caused by finite signal transmissionspeeds or memory effects (Chen et al., 2004; Pyragas, 1998; Yu& Cao, 2006, 2007). Therefore, the issue of network structure andparameter identification is of theoretical and practical importancefor uncertain complex dynamical networks with time delay. Thatis, can we estimate the uncertain topological structure and systemparameters of a specific complex networks by using its dynamicalbehaviors? In fact, in numerous real-world complex networks, theidentification of network structure becomes a key problem, suchas the protein-protein or protein-DNA (Deoxyribonucleic Acid)interactions in the regulation of various cellular processes. It iswell known that the protein-DNA interactions often play pivotalroles inmany cell processes, such as DNA replication,modification,repair and RNA (Ribonucleic Acid) transcription (Goto, Nishioka,& Kanehisa, 1998; Horne, Hodgman, Spence, & Dalby, 2004; Jin,Marsden, Chen, & Liao, 1999; Zhu, Lai, Hoppensteadt, & He, 2005).Another typical example is biological neural networks. In additionto modeling the neurons, the exact topological structure of abiological neural network often plays an important role and is theprime research interest (Yu&Cao, 2006; Zhu et al., 2005). However,due to the nonlinear, complex, and high dimensional nature of thepractical complex networks, it is very difficult to exactly identifyits topological structure by using the traditional approaches.

1800 H. Liu et al. / Automatica 45 (2009) 1799–1807

Recently, some new advances have been reported in thestructure identification of complex networks (Huang, 2006; Wu,2008; Yu et al., 2006; Zhou & Lu, 2007). However, most of theaforementioned approaches are only valid for several kinds oftypical complex networks with known system parameters (Lu &Cao, 2005; Wu, 2008; Yu et al., 2006; Zhou & Lu, 2007). In fact,in many practical settings, it is often difficult to exactly know allsystem parameters beforehand. Therefore, it is very necessary todevelop an effective method to identify the network topologicalstructure and system parameters for complex networks together.In this paper, by using network synchronization theory and

adaptive control techniques (Huang, 2006; Lin & Ma, 2007; Lü,Yu, & Chen, 2004; Lü, Yu, Chen, & Cheng, 2004; Zhou, Lu, & Lü,2008), an effective approach is then proposed to identify unknownnetwork topological structure and system parameters for generaluncertain delayed complex dynamical networks. Furthermore, thisdeveloped method is also valid for general uncertain delayedcomplex dynamical networks with different node dynamics.Finally, the proposed approach can be used to monitor the on-lineevolution of network topological structure and system parametersfor complex networks.The paper is organized as follows. Section 2 gives some useful

preliminaries. The identification of network topological structureand system parameters is then further explored for generaluncertain complex dynamical networks with coupling delay andnode delay in Sections 3 and 4, respectively. Section 5 uses threerepresentative examples to verify the effectiveness of the proposedapproach. Finally, some concluding remarks are then drawn inSection 6.

2. Preliminaries

Consider uncertain dynamical systems

xi(t) = fi(t, xi(t), αi), i = 1, 2, . . . ,N. (1)Rewrite systems (1) in the following formxi(t) = fi(t, xi(t))+ Fi(t, xi(t))αi, i = 1, 2, . . . ,N, (2)where xi(t) ∈ Rn are state vectors, αi ∈ Rmi are unknown systemparameter vectors for i = 1, 2, . . . ,N , inwhichmi are nonnegativeintegers. For every i, fi(t, xi(t)) is an n × 1 matrix, and Fi(t, xi(t))is an n×mi matrix.

Assumption 1 (A1). Suppose that there exist nonnegative con-stants Li (i = 1, 2, . . . ,N), satisfying

‖fi(t, x(t), αi)− fi(t, y(t), αi)‖ ≤ Li‖x(t)− y(t)‖, (3)

where x(t), y(t) ∈ Rn are time-varying vectors, and αi is theparameter vector of function fi(·). In the following, the norm ‖ · ‖of vector x is defined as ‖x‖ = (xTx)

12 .

If a time-varying delay τ(t) is considered, the above systems(1) can be rewritten as xi(t) = gi(t, xi(t), xi(t − τ(t)), βi), i =1, 2, . . . ,N . And they can be recast as followsxi(t) = gi(t, xi(t), xi(t − τ(t)))+ Gi(t, xi(t), xi(t − τ(t)))βi, (4)where xi(t), xi(t − τ(t)) ∈ Rn are state vectors, βi ∈ Rqi are theunknown parameter vectors for i = 1, 2, . . . ,N , in which qi arenonnegative integers. For every i, gi(t, xi(t), xi(t−τ(t))) is an n×1matrix, and Gi(t, xi(t), xi(t − τ(t))) is an n× qi matrix.

Assumption 2 (A2). Assume that there exists a nonnegativeconstantM satisfying

‖gi(t, x(t), x(t − τ(t)), βi)− gi(t, y(t), y(t − τ(t)), βi)‖

≤√M(‖(x(t)− y(t))‖2 + ‖(x(t − τ(t))

− y(t − τ(t)))‖2)12 , i = 1, 2, . . . ,N, (5)

where x(t), y(t), x(t − τ(t)), y(t − τ(t)) ∈ Rn are time-varyingvectors.

Assumption 3 (A3). Denote Fi(t, xi(t)) = (F (1)i (t, xi(t)), F(2)i (t,

xi(t)), . . . , F(mi)i (t, xi(t))). Suppose that F

(j)i (t, xi(t)) ∈ Rn for j =

1, 2, . . . ,mi, and {{F(j)i (t, xi(t))}

mij=1, {Axj(t − τ)}

Nj=1} are linearly

independent on the orbit {xi(t), xi(t−τ(t))}Ni=1 of synchronizationmanifold for any given i ∈ {1, . . . ,N}.

Assumption 4 (A4). Denote Gi(t, xi(t), xi(t − τ(t))) = (G(1)i (t,xi(t), xi(t − τ(t))),G

(2)i (t, xi(t), xi(t − τ(t))), . . . ,G

(qi)i (t, xi(t), xi

(t − τ(t)))). Assume that G(j)i (t, xi(t), xi(t − τ(t))) ∈ Rn forj = 1, 2, . . . , qi, and {{G

(j)i (t, xi(t), xi(t − τ(t)))}

qij=1, {Axj(t)}

Nj=1}

are linearly independent on the orbit {xi(t), xi(t − τ)}Ni=1 ofsynchronization manifold for any given i ∈ {1, . . . ,N}.

Assumption 5 (A5). Suppose that time-varying delay τ(t) isdifferentiable and satisfies τ (t) ≤ µ < 1, where µ is a constantfor µ < 1.

Obviously, this assumption holds for constant τ(t).In the following, a well known lemma will be introduced.

Lemma 1. For any vectors x, y ∈ Rn, the matrix inequality 2xTy ≤xTx+ yTy holds.

3. Structure identification of uncertain general complex dy-namical networks with coupling delay

Consider a complex dynamical networkwith time-varying cou-pling delay consisting of N different nodes, which is described by

xi(t) = fi(t, xi(t))+ Fi(t, xi(t))αi +N∑j=1

cijAxj(t − τ(t)), (6)

where xi(t) = (xi1(t), xi2(t), . . . , xin(t))T ∈ Rn for i = 1, . . . ,Nis the state vector of the i-th node, and delay τ(t) is time-varying.C = (cij)N×N ∈ RN×N is an unknown or uncertain coupling con-figuration matrix. If there exists a link from nodes i to j (j 6= i),then cij 6= 0 and cij is the weight or coupling strength; otherwise,cij = 0. A : Rn → Rn is an inner-coupling matrix which determinesthe interaction variables.Hereafter, the coupling configuration matrix C need not to be

symmetric, irreducible, or diffusive. Of course, it is necessary toensure the boundedness of complex dynamical networks in thispaper. The main goal is to identify these unknown or uncertaincoupling strengths, namely its network topological structure, andall unknown system parameter vectors αi (i = 1, 2, . . . ,N) of itsnode dynamical systems.Consider another complex dynamical network which will be

referred to as the response networkwith coupling delay and actualcontrol, as follows:˙xi(t) = fi(t, xi(t))+ Fi(t, xi(t))αi

+

N∑j=1

cijAxj(t − τ(t))+ ui, i = 1, . . . ,N, (7)

where xi(t) =(xi1(t), xi2(t), . . . , xin(t)

)T∈ Rn is the response

state vector of the i-th node, ui ∈ Rn is its controller, cij is theestimated value of weight cij, and vector αi is the estimated valueof the unknown parameter vector αi.Denote xi = xi − xi, cij = cij − cij, αi = αi − αi. The systems

(6) and (7) achieve asymptotical synchronization if xi → 0 ast → ∞. Hereafter, the synchronization manifold is defined byM , {xi(t) = xi(t)}. Then the error system is given by˙xi(t) = fi(t, xi(t))+ Fi(t, xi(t))αi − fi(t, xi(t))− Fi(t, xi(t))αi

+

N∑j=1

cijAxj(t − τ(t))−N∑j=1

cijAxj(t − τ(t))+ ui.

H. Liu et al. / Automatica 45 (2009) 1799–1807 1801

That is,˙xi(t) = fi(t, xi(t), αi)− fi(t, xi(t), αi)+ Fi(t, xi(t))αi

+

N∑j=1

cijAxj(t − τ(t))+N∑j=1

cijAxj(t − τ(t))+ ui,

i = 1, 2, . . . ,N. (8)

Theorem 1. Suppose that (A1), (A3), and (A5) hold. Then theuncertain coupling configuration matrix C and parameter vectors{αi}

Ni=1 of uncertain general delayed complex dynamical network (6)

can be identified by the estimated values C and {αi}Ni=1, respectively,via the response network

˙xi = fi(t, xi(t))+ Fi(t, xi(t))αi

+

N∑j=1

cijAxj(t − τ(t))+ ui

ui = −kixi(t), ki = di‖xi(t)‖2˙αi = −F Ti (t, xi(t))xi(t)˙c ij = −δijxi(t)TAxj(t − τ(t)),

(9)

where i, j ∈ {1, 2, . . . ,N} and di, δij are any positive constants.

Proof. Construct the Lyapunov candidate

2V (t) =N∑i=1

xi(t)Txi(t)+N∑i=1

N∑j=1

1δijc2ij +

N∑i=1

αTi αi

+

N∑i=1

1di(ki − k∗)2 +

∫ t

t−τ

11− µ

N∑i=1

xTi (θ)xi(θ)dθ,

where k∗ is a sufficiently large positive constant to be determined.Then, one has

V |(8) (9) =N∑i=1

xTi (t) ˙xi(t)+N∑i=1

N∑j=1

1δijcij ˙c ij +

N∑i=1

αT ˙α

+

N∑i=1

1di(ki − k∗)ki +

12(1− µ)

N∑i=1

xTi (t)xi(t)

−1− τ (t)2(1− µ)

N∑i=1

xTi (t − τ(t))xi(t − τ(t))

=

N∑i=1

xi(t)T{fi(t, xi(t), αi)− fi(t, xi(t), αi)}

+

N∑i=1

xTi (t)Fi(t, xi(t))αi

+

N∑i=1

N∑j=1

cijxi(t)TAxj(t − τ)−N∑i=1

ki‖xi(t)‖2

+

N∑i=1

N∑j=1

cijxi(t)TAxj(t − τ)+N∑i=1

N∑j=1

1δijcij ˙c ij

+

N∑i=1

αTi˙αi +

N∑i=1

(ki − k∗)‖xi(t)‖2

+1

2(1− µ)

N∑i=1

xTi (t)xi(t)

−1− τ (t)2(1− µ)

N∑i=1

xTi (t − τ(t))xi(t − τ(t)).

According to (A1), one obtains

V |(8) (9) ≤N∑i=1

Lixi(t)T xi(t)

+

{N∑i=1

xTi (t)Fi(t, xi(t))αi +N∑i=1

αTi˙αi

}

+

{N∑i=1

N∑j=1

cijxi(t)TAxj(t − τ)+N∑i=1

N∑j=1

1δijcij ˙c ij

}

+

N∑i=1

N∑j=1

cijxi(t)TAxj(t − τ)

−

N∑i=1

k∗‖xi(t)‖2 +1

2(1− µ)

N∑i=1

xTi (t)xi(t)

−1− τ (t)2(1− µ)

N∑i=1

xTi (t − τ(t))xi(t − τ(t)).

Denote that L = max{ 1 ≤ i ≤ N | Li}. From Theorem 1, one gets

V |(8) (9) ≤ LN∑i=1

xi(t)Txi(t)+N∑i=1

N∑j=1

cijxi(t)TAxj(t − τ(t))

−

N∑i=1

k∗‖xi(t)‖2 +1

2(1− µ)

N∑i=1

xTi (t)xi(t)

−1− τ (t)2(1− µ)

N∑i=1

xTi (t − τ(t))xi(t − τ(t)).

Denote e(t) = (xT1(t), xT2(t), . . . , x

TN(t))

T∈ RnN and P =

C⊗A, where

⊗is Kronecker product. Thus one has

V ≤ LeT(t)e(t)+ eT(t)Pe(t − τ(t))− k∗eT(t)e(t)

+1

2(1− µ)eT(t)e(t)−

1− τ (t)2(1− µ)

eT(t − τ(t))e(t − τ(t)).

According to Lemma 1, one deduces

V ≤ LeT(t)e(t)+12eT(t)PPTe(t)

+12eT(t − τ(t))e(t − τ(t))− k∗eT(t)e(t)

+1

2(1− µ)eT(t)e(t)−

1− τ (t)2(1− µ)

eT(t − τ(t))e(t − τ(t))

=

(L− k∗ +

12(1− µ)

)eT(t)e(t)+

12eT(t)PPTe(t)

+τ (t)− µ2(1− µ)

eT(t − τ(t))e(t − τ(t)).

From (A5), one obtains τ (t)−µ2(1−µ) ≤ 0. Therefore, one gets

V ≤(L− k∗ +

12(1− µ)

+ λmax

(12PPT

))eT(t)e(t).

Let k∗ = L + 12(1−µ) + λmax(

12PP

T) + 1. Thus one obtainsV |(8) (9) ≤ −eT(t)e(t).Therefore, we have supt≥0 V (t) ≤ V (0). According to the

Lyapunov candidate, V (0) is bounded. Thus, V (t) is also boundedsince 0 ≤ V (t) ≤ V (0). From eT(t)e(t) ≤ −V (t), then one has

limt→∞

∫ t

0eT(s)e(s)ds ≤ − lim

t→∞

∫ t

0V (s)ds

= V (0)− limt→∞

V (t).

1802 H. Liu et al. / Automatica 45 (2009) 1799–1807

Thus, e(t) ∈ L2. Obviously, e(t) is bounded, i.e., e(t) ∈ L∞.According to its error system (8), e(t) exists and is bounded fort ∈ [0,+∞). Therefore, one obtains limt→∞ e(t) = 0 by usingBarbalat Lemma (Lee & Jiang, 2005; Popov, 1973).Suppose that limt→∞ ˙xi(t) exists, one has limt→∞ ˙xi(t) = 0

since e(t) converges to a constant as t → ∞. According to (8),x(t) = (xT1(t), x2(t), . . . , x

TN(t))

T converges to

Ei = {x(t) : Fi(t, xi(t))αi +N∑j=1

cijAxj(t − τ(t)) = 0} as t →∞,

where i = 1, 2, . . . ,N . From (A3), one has cij → 0 and αi →0 as t → ∞ (Khalil, 1996). It implies that the unknown oruncertain coupling matrix C and system parameter vectors αi canbe successfully identified by using updating law (9) when its errorsystem achieves synchronization. �

Remark 1. Barbalat Lemma guarantees the asymptotical stabilityof its error system (8) at e(t) = 0. Moreover, the linearlyindependent condition (A3) indicates that cij → cij and αi → αi.

Remark 2. It should be especially pointed out that the couplingconfigurationmatrix C need not to be symmetric, irreducible, evendiffusive. The constants di and δij can control the synchronizationor convergence speed.

Corollary 1. If a complex network with coupling delay consists of Nidentical nodes, which can be described by

xi(t) = f (t, xi(t))+ F(t, xi(t))α +N∑j=1

cijAxj(t − τ(t)),

i = 1, . . . ,N. (10)

Under assumptions similar to (A1), (A3), and (A5), the unknown oruncertain coupling configuration matrix C and parameter vector α ofnetwork (10) can be identified by using the estimated values C and α,respectively, via the following response network:

˙xi = f (t, xi(t))+ F(t, xi(t))α

+

N∑j=1

cijAxj(t − τ(t))+ ui

ui = −kixi(t), ki = di‖xi(t)‖2

˙α = −

N∑i=1

F T(t, xi(t))xi(t)

˙c ij = −δijxi(t)TAxj(t − τ(t)),

(11)

where i, j ∈ {1, 2, . . . ,N} and di, δij are any positive constants.

Remark 3. Some recent research work (Wu, 2008; Yu et al.,2006; Zhou & Lu, 2007) on network topological identificationneglects the linearly independent condition mentioned in (A3). Asamatter of fact, it is a very essential condition for guaranteeing thesuccess of identification. Without this condition, it may cause thefalse identification results. This phenomenon was firstly furtherinvestigated in parameter identification of chaotic systems byseveral recent works of literature (Lin & Ma, 2007; Yu & Cao,2007). In this paper, a more general result is achieved for structureidentification of uncertain general complex dynamical networkswith time delay.

Remark 4. Our investigation shows that the synchronization ofthe drive network is harmful to the identification of topologicalstructure and system parameters since it is difficult to satisfy (A3)

in this case. In particular, for a network with identical nodes,its synchronizability should be firstly considered. As a matterof fact, it is often difficult to realize network synchronizationfor a delayed complex dynamical network with identical nodes.Furthermore, some recent research shows that it can achievenetwork synchronization only for a small time delay (Wu & Jiao,2007).

4. Structure identification of an uncertain general complexdynamical network with node delay

Consider an uncertain general complex dynamical networkconsisting ofN different nodeswith time-varying delay τ(t), calledthe drive network, which is described by

xi(t) = gi(t, xi(t), xi(t − τ(t)), βi)+N∑j=1

cijAxj(t),

i = 1, . . . ,N. (12)

The node dynamics can be rewritten as follows

gi(t, xi(t), xi(t − τ(t)), βi)= gi(t, xi(t), xi(t − τ(t)))+ Gi(t, xi(t), xi(t − τ(t)))βi, (13)

where βi (i = 1, 2, . . . ,N) are unknown or uncertain systemparameter vectors.Construct another controlled general complex dynamical

network, called response network, which is given by˙xi(t) = gi(t, xi(t), xi(t − τ(t)), βi)

+

N∑j=1

cijAxj(t)+ ui, i = 1, . . . ,N, (14)

where xi(t) = (xi1(t), xi2(t), . . . , xin(t))T ∈ Rn is the responsestate vector of the i-th node, ui ∈ Rn is its control input, cij andβi are the estimated values of cij and βi, respectively.Thus the error system is described by

˙xi(t) = gi(t, xi(t), xi(t − τ(t)), βi)− gi(t, xi(t), xi(t − τ(t)), βi)

+

N∑j=1

cijAxj(t)−N∑j=1

cijAxj(t)+ ui.

That is, one has˙xi(t) = gi(t, xi(t), xi(t − τ(t)), βi)− gi(t, xi(t), xi(t − τ(t)), βi) + Gi(t, xi(t), xi(t − τ(t)))βi

+

N∑j=1

cijAxj(t)+N∑j=1

cijAxj(t)+ ui, i = 1, . . . ,N. (15)

Theorem 2. Suppose that (A2), (A4), and (A5) hold. Then theunknown or uncertain coupling configuration matrix C and systemparameter vectors {βi}Ni=1 can be identified by using the estimatedvalues C and {βi}Ni=1, respectively, via the response network

˙xi = gi(t, xi(t), xi(t − τ(t)))

+Gi(t, xi(t), xi(t − τ(t)))βi +N∑j=1

cijAxj(t)+ ui

ui = −kixi(t), ki = di‖xi(t)‖2˙β i = −G

Ti (t, xi(t), xi(t − τ(t)))xi(t)

˙c ij = −δijxi(t)TAxj(t),

(16)

where i, j ∈ {1, 2, . . . ,N}, di and δij are any positive constants.

H. Liu et al. / Automatica 45 (2009) 1799–1807 1803

Proof. Construct the Lyapunov candidate

2V =N∑i=1

xi(t)Txi(t)+N∑i=1

N∑j=1

1δijc2ij +

N∑i=1

βTi βi

+

N∑i=1

1di(ki − k∗)2 +

∫ t

t−τ

M1− µ

N∑i=1

xTi (θ)xi(θ)dθ,

where k∗ is a sufficiently large positive constant to be determined.Then, one has

V |(15) (16) =N∑i=1

xTi (t) ˙xi(t)+N∑i=1

N∑j=1

1δijcij ˙c ij +

N∑i=1

βTi˙β i

+

N∑i=1

1di(ki − k∗)ki +

M2(1− µ)

N∑i=1

xTi (t)xi(t)

−M(1− τ (t))2(1− µ)

N∑i=1

xTi (t − τ(t))xi(t − τ(t)).

According to Eq. (15), one obtains

V |(15) (16) =N∑i=1

xi(t)T{gi(t, xi(t), xi(t − τ(t)), βi)

− gi(t, xi(t), xi(t − τ(t)), βi)}

+

{N∑i=1

xi(t)T Gi(t, xi(t), xi(t − τ(t)))βi +N∑i=1

βTi˙β i

}

+

{N∑i=1

N∑j=1

xi(t)TcijAxj(t) +N∑i=1

N∑j=1

1δijcij ˙c ij

}

+

N∑i=1

N∑j=1

xi(t)TcijAxj(t) −N∑i=1

ki‖xi(t)‖2

+

N∑i=1

(ki − k∗)‖xi(t)‖2 +M

2(1− µ)

N∑i=1

xTi (t)xi(t)

−M(1− τ (t))2(1− µ)

N∑i=1

xTi (t − τ(t))xi(t − τ(t)).

From Lemma 1 and (A2), one getsxi(t)T{gi(t, xi(t), xi(t − τ(t)), βi)− gi(t, xi(t), xi(t − τ(t)), βi)}

≤12xi(t)Txi(t) +

12‖gi(t, xi(t), xi(t − τ(t)), βi)

− gi(t, xi(t), xi(t − τ(t)), βi)‖2

≤12xi(t)Txi(t)+

M2(‖xi(t)‖2 + ‖xi(t − τ(t))‖2).

According to the updating laws of Theorem 2, one has

V ≤1+M2

N∑i=1

xi(t)Txi(t)+M2

N∑i=1

‖xi(t − τ(t))‖2

+

N∑i=1

N∑j=1

cijxTi (t)Axj(t)

− k∗N∑i=1

‖xi(t)‖2 +M

2(1− µ)

N∑i=1

xTi (t)xi(t)

−M(1− ˙τ(t))2(1− µ)

N∑i=1

xTi (t − τ(t))xi(t − τ(t)).

Denote e(t) = (xT1(t), xT2(t), . . . , x

TN(t)) ∈ R

nN and P = C⊗A.

Thus one gets

V ≤{1+M2− k∗ +

M2(1− µ)

}eT(t)e(t)+ eT(t)PeT(t)

+M(τ − µ)2(1− µ)

eT(t − τ(t))e(t − τ). (17)

From (A5), one hasτ (t)− µ2(1− µ)

≤ 0.

Therefore, one attains

V ≤{1+M2− k∗ +

M2(1− µ)

+ λmax

(PT + P2

)}eT(t)e(t).

Let k∗ = 1+M2 +

M2(1−µ) + λmax(

PT+P2 )+ 1. And one gets

V |(15) (16) ≤ −eT(t)e(t).

Similar to the proof in Theorem 1, one can easily deduce

e(t) ∈ L2⋂L∞.

Furthermore, according to error system (15), e(t) exists and isbounded for t ∈ [0,+∞). Then, from Barbalat Lemma, one haslimt→∞

e(t) = 0.

Suppose that limt→∞ ˙xi(t) exists, one has limt→∞ ˙xi(t) = 0since e(t) converges to a constant as t → ∞. According to (15),x(t) = (xT1(t), x2(t), . . . , x

TN(t))

T converges to

Fi = {x(t) : Gi(t, xi(t), xi(t − τ(t)))βi +N∑j=1

cijAxj(t) = 0}

as t →∞,

where i = 1, 2, . . . ,N . From (A4), one has cij → 0 and βi →0 as t → ∞ (Khalil, 1996). It indicates that the unknown oruncertain coupling configuration matrix C and system parametervectors {βi}Ni=1 can be easily estimated by using updating law(16) when drive the network and response system achievesynchronization. �

Corollary 2. If a complex network with node delay consists of Nidentical nodes, which is described by

xi(t) = g(t, xi(t), xi(t − τ(t)))+ G(t, xi(t), xi(t − τ(t)))β

+

N∑j=1

cijAxj(t), i = 1, . . . ,N. (18)

Under assumptions similar to (A2), (A4), and (A5), then the uncertainor unknown coupling configuration matrix C and system parametervector β of network (18) can be identified by using the estimatedvalues C and β , respectively, via the following response network

˙xi = g(t, xi(t), xi(t − τ(t)))

+G(t, xi(t), xi(t − τ(t)))β +N∑j=1

cijAxj(t)+ ui

ui = −kixi(t), ki = di‖xi(t)‖2

˙β = −

N∑i=1

GT(t, xi(t), xi(t − τ(t)))xi(t)

˙c ij = −δijxi(t)TAxj(t),

(19)

where i, j ∈ {1, 2, . . . ,N} and di, δij are any positive constants.

5. Numerical simulation examples

In this section, several representative examples are thengiven to verify the effectiveness of the proposed structure

1804 H. Liu et al. / Automatica 45 (2009) 1799–1807

c1j

c11

c12

c13

c14

c15

–8

–6

–4

–2

0

2

4

6

8

20 40 60 80 100 120 140

Time

0 160

α12

α22

α32

α42

α52

–5

0

5

10

15

20

25

30

20 40 60 80 100 120 140

Time

0 160

(a) Identification of network structure. (b) Identification of unknown parameters {αi2}5i=1 .

Time

–10

Time

α i3

10

20

30

α i1

0

40

0

10

20

0 20 40 60 80 100 120 140 160

0 20 40 60 80 100 120 140 160

(c) Identification of uncertain parameters αi1 and αi3 for i = 1, . . . , 5.

Fig. 1. Adaptive identification of network structure and system parameters for an uncertain network (6) with 5 nonidentical nodes and coupling time delay.

identification and parameters estimation approach based onnetwork synchronization in Sections 3 and 4.

5.1. Nonidentical nodes and coupling time delay

Consider network (6) consisting of N = 5 different Lüoscillators (Lü & Chen, 2002), whose coupling configurationmatrixis described by−5 1 2 2 02 −2 0 0 00 1 −2 1 00 0 1 −1 02 0 0 1 −3

. (20)

It is well known that Lü system (Lü & Chen, 2002; Lü, Chen,Cheng, & Celikovsky, 2002) is given byxi(t) = fi(t, xi(t))+ Fi(t, xi(t))αi, (21)where xi(t) = (xi1(t), xi2(t), xi3(t))T is state vector, fi(t, xi(t)) =(0,−xi1(t)xi3(t), xi1(t)xi2(t))T, Fi(t, xi(t)) = diag{xi2(t) − xi1(t),xi2(t),−xi3(t)}, and αi = (αi1, αi2, αi3)T, for i = 1, . . . , 5.In the following numerical simulations, let αi = (αi1, αi2, αi3)T

= (36, 20+ i, 3)T for i = 1, . . . , 5. Here, time delay τ(t) = 0.2and network inner-coupling matrix A = diag{1, 1, 1}.Since Lü attractor is bounded for any given system parameters,

(A1) holds. Obviously, (A5) holds. Moreover, many complexnetworkswith nonidentical nodes naturally satisfy (A3). Therefore,it is easy to verify that all conditions of Theorem 1 are satisfied.According to Theorem 1, the coupling configuration matrix C

–30 –20 –10 0 10 20 30

–50

0

50

0

10

20

30

40

50

Fig. 2. Delayed Lü system with parameter vector (36, 20, 3)T .

and system parameter vectors αi with i = 1, . . . , 5, of complexnetworks (6) can be identified by using adaptive control laws (9)in Theorem 1.Fig. 1(a) shows the identification of network structure. After

80 s, c11 switches from −5 to −6, c12 switches from 1 to 0,c14 switches from 2 to 3, and c15 switches from 0 to 1. It isvery clear that the identification of network structure is verysuccessful. Fig. 1(b) shows the identification of unknown systemparameters αi2. Fig. 1(c) shows the identification of uncertainsystem parameters αi1 = 36 and αi3 = 3 for i = 1, . . . , 5. Allnumerical simulations illustrate the effectiveness of Theorem 1.

H. Liu et al. / Automatica 45 (2009) 1799–1807 1805

Time

c1j

c11

c12

c13

c14

c15

–10

–5

0

5

10

15

0 50 100 150 200 250 300

erro

r xi

j

–25

–20

–15

–10

–5

0

5

10

15

20

Time

0 50 100 150 200 250 300

(a) Identification of network structure. (b) Synchronous errors xij for 1 ≤ i ≤ 5 and 1 ≤ j ≤ 3.

Fig. 3. Adaptive identification of network structure and system parameters for uncertain network (12) with 5 nonidentical nodes and node time delay.

1

2

3

4

56

7

8

9

10

11

12

13

14

151617

18

19

20

Pajek Time

c1j c11

c12

–8

–6

–4

–2

0

2

4

0 2 4 6 8 10

(a) Topological structure of an uncertain delayed directed dynamical networkwith 20 identical nodes. Here, the weights of blue and green lines are 1 and 2,respectively. And the sienna dot line is the link after switching.

(b) Identification of network structure.

Time

β1

β2

β3

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

(c) Identification of system parameters.

Fig. 4. Adaptive identification of network structure and system parameters for uncertain delayed dynamical network (10) with 20 identical nodes and coupling time delay.

5.2. Nonidentical nodes and node time delay

Consider network (12) consisting of 5 nonidentical delayed Lüsystems. According to (21), the delayed Lü system is describedby{xi1(t) = βi1(xi2(t − τ)− xi1(t − τ))xi2(t) = −xi1(t − τ)xi3(t − τ)+ βi2xi2(t − τ)xi3(t) = xi1(t − τ)xi2(t − τ)− βi3xi3(t − τ),

(22)

where βi = (βi1, βi2, βi3)T = (36, 20+ i, 3)T for i = 1, . . . , 5. LetA = diag{1, 1, 1} and τ = 0.002. Here, the coupling configurationmatrix C is also defined by (20).Fig. 2 shows that the attractor of delayed Lü system (22) is

bounded. It is easy to verify that all conditions of Theorem 2are satisfied. According to Theorem 2, the unknown or uncertaincoupling configuration matrix C and system parameter vectors βican be well estimated by using C and βi, respectively, via responsenetwork (16). Fig. 3(a) shows the dynamical evolution of 5 coupling

1806 H. Liu et al. / Automatica 45 (2009) 1799–1807

coefficients in the first row of matrix C . Fig. 3(b) shows theirsynchronous errors. All numerical simulation results illustrate theeffectiveness of Theorem 2.

5.3. Identical nodes and coupling time delay

Consider network (10) consisting of N = 20 identical Lüoscillators. Fig. 4(a) shows its network topological structure.For simplicity, suppose that the systemparameters of Lü system

are given by α = (36, 20, 3)T. Also, assume that the couplingcoefficients c11, c12 of matrix C are unknown beforehand.In the following numerical simulations, let τ = 0.1 and the

inner-coupling matrix A = diag{1, 1, 1}. Moreover, the couplingconfiguration matrix C is asymmetric, diffusive, and to be partiallyidentified: c11 switches from−6 to−8 and c12 switches from 0 to2. Fig. 4(a) shows the topological structure of an uncertain directeddynamical network. Fig. 4(b) shows the identification of networkstructure. Fig. 4(c) shows the identification of system parameters.All numerical simulation results indicate the effectiveness ofthe proposed structure identification and parameters estimationapproach.

6. Conclusion

We have proposed a novel adaptive feedback control approachto simultaneously identify the unknown or uncertain networktopological structure and system parameters of uncertain delayedgeneral complex dynamical networks together. Based on Lyapunovtheory and Barbalat Lemma, several useful identification criteriaare then attained. In particular, the proposed methods are alsovalid for the structure identification of complex networks withnonidentical nodes. Several representative numerical simulationsare then given to verify the effectiveness of the proposed adaptiveidentification schemes. It sheds light on the future possible real-world applications.

Acknowledgments

This workwas supported by the National Natural Science Foun-dation of China under Grants 70771084, 60574045, 60772158, and60821091, the National Basic Research Program (973) of China un-der Grants 2007CB310805, the Important Direction Item of Knowl-edge Innovation Project of Chinese Academy of Sciences underGrant KJCX3-SYW-S01, and the Scientific Research Foundation forthe Returned Overseas Chinese Scholars, State Education Ministry.

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Hui Liu received the B.Sc. Degree and M.S. Degree inmathematics, both fromWuhanUniversity,Wuhan, China,in 2004 and 2007, respectively.Currently, she is a Ph.D. candidate with the School

of Mathematics and Statistics, Wuhan University, Wuhan,China. Her main research interests are complex networks,nonlinear systems, chaos control and synchronization, andmulti-agent systems.

Jun-An Lu received the B.Sc. Degree in geophysics fromPeking University, Beijing, China, and the M.Sc. Degreein applied mathematics from Wuhan University, Wuhan,China, in 1968 and 1982, respectively.He is currently a Professor with the School of Math-

ematics and Statistics, Wuhan University, Wuhan, China.His research interests include nonlinear systems, chaoscontrol and synchronization, complex networks, and sci-entific and engineering computing. He has publishedmorethan 150 journal papers in the above fields. He received theSecond Prize of the Natural Science Award from the Hubei

H. Liu et al. / Automatica 45 (2009) 1799–1807 1807

Province, China in 2006, the First Prize of the Natural Science Award from the Min-istry of Education of China in 2007, the Second Prize of the State Natural ScienceAward from the State Council of China in 2008.

JinhuLü received the Ph.D. Degree in appliedmathematicsfrom the Chinese Academy of Sciences, Beijing, China in2002.Currently, he is an Associate Professor with the

Academy of Mathematics and Systems Science, ChineseAcademy of Sciences, Beijing, China. He held severalvisiting positions in Australia, Canada, France, Germany,Hong Kong andUSA, andwas a Visiting Fellow in PrincetonUniversity, USA from 2005 to 2006. He is the author oftwo research monographs and more than 70 internationaljournal papers published in the fields of nonlinear circuits

and systems, complex networks and complex systems. He served as a member inthe Technical Committees of many international conferences and is now servingas a member of IFAC Technical Committee on Large Scale Complex Systems and aVice-Chair of Technical Committee of Complex Systems and Complex Networks ofCSIAM. He is also an Associate Editor of IEEE Transactions on Circuits and SystemsII, Journal of Systems Science and Complexity, ARI — the Bulletin of the IstanbulTechnical University and DCDIS-A. Dr. Lü received the prestigious PresidentialOutstanding Research Award from the Chinese Academy of Sciences in 2002, theNational Best Ph.D. Theses Award from the Office of Academic Degrees Committeeof the State Council and theMinistry of Education of China in 2004, the First Prize ofthe Science and Technology Award from the Beijing City of China in 2007, the First

Prize of the Natural Science Award from theMinistry of Education of China in 2007,the Lu Jiaxi Youth Talent Award from the Chinese Academy of Sciences in 2008, theSecond Prize of the State Natural Science Award from the State Council of Chinain 2008. He is the co-author of the Most Cited SCI Paper of Chinese Scholars in thefield of mathematics during the periods of 2001–2005 and 2002–2006. He is alsoan IEEE Senior Member.

David John Hill received the B.E. and B.Sc. Degrees fromthe University of Queensland, Australia, in 1972 and 1974,respectively. He received the Ph.D. Degree in ElectricalEngineering from the University of Newcastle, Australia,in 1976. He is currently a Professor and AustralianResearch Council Federation Fellow in the ResearchSchool of Information Sciences and Engineering at TheAustralian National University. He is also Deputy Directorof the Australian Research Council Centre of Excellencefor Mathematics and Statistics of Complex Systems. Hehas held academic and substantial visiting positions

at the universities of Melbourne, California (Berkeley), Newcastle (Australia),Lund (Sweden), Sydney and Hong Kong (City University). He holds honoraryprofessorships at the University of Sydney, University of Queensland (Australia),South China University of Technology, City University of Hong Kong, WuhanUniversity and Northeastern University (China). His research interests are innetwork systems science, stability analysis, nonlinear control and applications. Heis a Fellow of the Institution of Engineers, Australia, the Institute of Electrical andElectronics Engineers, USA, the Society for Industrical and Applied Mathematics,USA and the Australian Academy of Science; he is also a Foreign Member of theRoyal Swedish Academy of Engineering Sciences.