Topology identification of complex dynamical networks

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Topology identification of complex dynamical networks

Citation for published version (APA):

Zhao, J., Li, Q., Lu, J. A., & Jiang, Z. P. (2010). Topology identification of complex dynamical networks. Chaos, 20(2), 023119-1/7. [023119]. https://doi.org/10.1063/1.3421947

DOI:

10.1063/1.3421947

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Topology identification of complex dynamical networks

Junchan Zhao,1,2,a兲 Qin Li,3,b兲 Jun-An Lu,2,c兲 and Zhong-Ping Jiang4,d兲

1_{Research Center of Nonlinear Science, Wuhan Textile University, Wuhan 430073, China} 2_{School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China}

3_{Department of Mechanical Engineering, System Engineering Group, Eindhoven University of Technology,}

Eindhoven 5600 MB, The Netherlands

4_{Department of Electrical and Computer Engineering, Polytechnic Institute of New York University,}

Brooklyn, New York 11201, USA

共Received 17 November 2009; accepted 12 April 2010; published online 17 June 2010兲

Recently, some researchers investigated the topology identification for complex networks via La-Salle’s invariance principle. The principle cannot be directly applied to time-varying systems since the positive limit sets are generally not invariant. In this paper, we study the topology identification problem for a class of weighted complex networks with time-varying node systems. Adaptive identification laws are proposed to estimate the coupling parameters of the networks with and without communication delays. We prove that the asymptotic identification is ensured by a persis-tently exciting condition. Numerical simulations are given to demonstrate the effectiveness of the proposed approach. © 2010 American Institute of Physics.关doi:10.1063/1.3421947兴

The topology identification, as an inverse problem, is a significant issue in the study of complex networks. For example, if a major malfunction occurs in a communica-tion network, power network, or the Internet, it is very important to quickly detect the location of the faulty line. This paper proposes a novel adaptive identification ap-proach for the topology identification of the weighted complex dynamical networks. We show that the concept of persistent excitation plays a key role in the process of topology identification. Our result overcomes the limita-tion of previous methods, which rely on the use of La-Salle’s invariance principle, and is applicable to networks with time-varying node systems and diverse time-varying coupling delays.

I. INTRODUCTION

Today, complex networks have become an important part in our daily life. Examples of such networks include transportation and phone networks, Internet, wireless net-works, and the World Wide Web, to name a few. A complex dynamical network can be described by a graph in math-ematics. In such a graph, each node represents a basic ele-ment with certain dynamics, and edges represent interactive topology of the network. Analysis and control of the behav-iors of complex networks consisting of a large number of dynamical nodes have attracted wide attention in different fields in the past decade.1–5In particular, special attention has been focused on the control and synchronization of large-scale complex dynamical networks with certain types of topology.6–15 Another attractive topic on complex networks is to develop systematic schemes for the topology

identifica-tion. Applications can be found in various scientific and en-gineering fields. For instance, if a major malfunction occurs in a communication network, power network, or the Internet, it is very important to quickly localize the faulty spot or the failing edge. There are also other works that involve network identification such as the understanding of proteDNA in-teractions in cell processes.16Note that the information trans-mission delay 共called communication delay in this paper兲 is ubiquitous in complex networks and should be regarded as a critical issue in both the control and identification problems.17–19

Recently, many researchers have applied LaSalle’s in-variance principle in the topology identification of complex networks.20,21 However, for a general network with time-varying node systems, the principle is not applicable as the positive limit sets may not be invariant.22–24 On the other hand, it has been reported in Refs. 25–27 that the earlier results neglected a crucial condition, which requires that the inner function should be mutually linearly independent of the synchronization manifold. However, it is difficult to verify the linear independence condition, especially on the synchro-nization manifold.

In this paper, we address the topology identification for weighted time-varying dynamical networks with nonlinear inner coupling. An adaptive identification rule is first pro-posed for the networks without communication delay. Unlike previous results, the asymptotic identification of the topology is guaranteed under a persistency excitation condition, which is widely used in dealing with parameter identification and adaptive control problems.22–24,28–31 Based on similar idea, an identification rule is further presented for the networks with diverse and time-varying communication delays, which brings our approach closer to practical applications.

The rest of this paper is organized as follows. Section II introduces some mathematical preliminaries used in this work. The system description and main results on the

topol-a兲_{Electronic mail: junchanzhao@163.com.} b兲_{Electronic mail: q.li@tue.nl.}

c兲_{Electronic mail: jalu@whu.edu.cn.} d兲_{Electronic mail: zjiang@control.poly.edu.}

共2010兲

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ogy identification are presented in Secs. III and IV, respec-tively. In Sec. V, illustrative simulations are provided by tak-ing the chaotic system in Ref.35as the node systems in the network. Finally, some concluding remarks are given in Sec. VI.

II. PRELIMINARIES

In this section, we present some mathematical prelimi-naries, which will be used in this work. Throughout the pa-per,储A储 is used to denote the spectral norm of matrix A, and ␭max共A兲关␭min共A兲兴 represents the maximum 共minimum兲

ei-genvalue of A. L2_{denotes the square integrable space, and I}

n

is the identity matrix on the order of n.

Let G共V,E兲 be a directed graph with the vertex set V =兵1,2, ... ,N其 and the directed edge set E=兵共i, j兲:i, j苸V其. The adjacency matrix A =共aij兲N_⫻N of a directed graph with

weighted edge G共V,E兲 is a non-negative matrix defined as aij= w if and only if 共i, j兲 is an edge with weight w. The

out-degree of a vertex v is the sum of the weights of the edges emanating from v.32 See Ref. 33 for more basics of graph theory.

A matrix is reducible if it can be written as

P

冋

A B 0 C

册

P

T

,

where P is a permutation matrix. A matrix is irreducible if it is not reducible.32

Definition 1:共Reference28兲 A function␸: R+→Rn⫻mis persistently exciting 共PE兲 if there exist T0,␦1,␦2⬎0 such

that ␦1Inⱕ

冕

t t+T0 ␸共␶兲␸T_共_␶_兲d_␶_ⱕ_␦ 2In 共1兲

holds for all tⱖ0.

Remark 1: The PE condition requires that␸ rotates suf-ficiently in space that the integral of the matrix␸共␶兲␸T_共_␶_{兲 is}

uniformly positive definite over any interval of some length T0. The upper bound in Eq. 共1兲is satisfied whenever␸共t兲 is

bounded.

Lemma 1:共Reference30兲 Given a system of the follow-ing form:

再

e˙₁= g共t兲e₂+ f₁共t兲, e₁苸 Rp_, _e 2苸 Rq e˙2= f2共t兲

冎

共2兲 such that

共i兲 limt_→⬁储e1共t兲储=0, limt_→⬁储f1共t兲储=0, and limt_→⬁储f2共t兲储

= 0;

共ii兲 g共t兲 and g˙共t兲 are bounded and gT_{共t兲 is PE;}

then limt→⬁储e2共t兲储=0.

III. TOPOLOGY IDENTIFICATION OF NONDELAY DYNAMICAL NETWORKS

Consider a dynamical network consisting of N coupled oscillators, with each node being an m-dimensional dynami-cal system described by

x˙i= fi共t,xi兲 +⑀

兺

j=1 N

cijHij共t,xj兲, i = 1,2, ... ,N, 共3兲

where xi苸Rmis the state variable of node i and fi: R⫻Rm

→Rm_{is a continuous function,}_⑀_{⬎0 is the known coupling}

strength, and Hij is the nonlinear inner-coupling function.

The outer-coupling matrix C =共cij兲N_⫻N is defined as

C = A − D, where A =共aij兲N⫻N is the adjacency matrix of the

graph of the network, and D =共dij兲 is the diagonal matrix of

vertex with dii=兺j=1 N

aijfor all i = 1 , 2 , . . . , N. In this paper, we

assume that the elements of the matrix C are unknown. Our purpose is to estimate these unknown parameters in an asymptotic manner.

Assumption 1: For all i , j = 1 , . . . , N, the functions fi共t,x兲

and Hij共t,x兲 are continuous in t and satisfy that there exist

constants␣₁,␣₂⬎0 such that

储fi共t,x兲 − fi共t,y兲储 ⱕ␣1储x − y储共4兲

and

储Hij共t,x兲 − Hij共t,y兲储 ⱕ␣2储x − y储共5兲

for all t and all x , y苸Rm

Design the response network as

y˙i= fi共t,yi兲 +

兺

j=1 N

dij共t兲Hij共t,yj兲 − ki共t兲ei, i = 1,2, . . . ,N,

共6兲 with the adaptive feedback law

再

d˙ij= −␦ijei T Hij共t,yj兲 k˙i= siei T ei,

冎

共7兲 where ei= yi− xi is called the synchronization error, ki is

adaptive feedback gain, dijis the adaptive parameters of the

response system, and ␦ijand siare two positive constants.

In the following, we show how the unknown parameters cijcan be asymptotically estimated from dij. The

synchroni-zation errors between network 共3兲 and network 共6兲 can be written as e˙i共t兲 = fi共t,yi兲 − fi共t,xi兲 +

兺

j=1 N dij共t兲Hij共t,yj兲 −⑀

兺

j=1 N cijHij共t,xj兲 − ki共t兲ei, i = 1,2, . . . ,N. 共8兲

Theorem 1: Suppose that the functions fiand Hijsatisfy

Assumption 1, and Hijis bounded. Then, by the updated law

共7兲, limt→⬁储xi− yi储=0 for all i. Furthermore, if for any i,

g_iT共t兲=关H_i1共t,y₁共t兲兲, ... ,HiN共t,yN共t兲兲兴T is PE, and g˙i共t兲 is

bounded, then limt→⬁共dij−⑀cij兲=0.

Proof: Construct the function

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j=1 N ␣2⑀兩cij兩 · 储ei T_{储 · 储e} j储 = ETQE,

where E =共储e1储, ... ,储eN储兲T苸RN, and

Q =

再

qij=

1

2␣2⑀共兩cij兩 + 兩cji兩兲, i ⫽ j

qii=␣2⑀·兩cii兩 +␣1− kⴱ.

冎

It is obvious that there exists a sufficiently large positive constant kⴱ such that Q is negative definite. Namely,

V˙ ⱕ ␭_max共Q兲ET_E_{ⱕ 0,} _共9兲

which means that 储E储, 兩dij共t兲−⑀cij兩 are bounded. From Eq.

共9兲, we have for any t,

冕

0 t ET共␶兲E共␶兲d␶ⱕ 1 ␭max共Q兲

冕

0 t V˙ 共␶兲d␶ = − 1 ␭max共Q兲 共V共0兲 − V共t兲兲 ⱕ − 1 ␭max共Q兲 V共0兲, then E共t兲苸L2_{, so e} i共t兲苸L2.

From Eq.共8兲and conditions共4兲and共5兲, where we have that e˙i 共i=1, ... ,N兲 is bounded, it follows from Barbalat’s

lemma22 that limt_→⬁储ei共t兲储=0 for i=1, ... ,N.

Note that e˙i= fi共t,yi兲 − fi共t,xi兲 +

兺

j=1 N ⑀cij共Hij共t,yj兲 − Hij共t,xj兲兲 − kiei +关Hi1共t,y1兲, ... ,HiN共t,yN兲兴␰i, ␰˙_i_{= −}_关_␦ i1ei T Hi1共t,y1兲, ... ,␦iNei T HiN共t,yN兲兴T,

where ␰i=关di1共t兲−⑀ci1, . . . , diN共t兲−⑀ciN兴T. Let ␩i共t兲= fi共t,yi兲

− fi共t,xi兲+兺j=1 N _⑀

cij共Hij共t,yj兲−Hij共t,xj兲兲−kiei. It is easy to

know that and limt_→⬁␩i共t兲=0 as limt_→⬁关yi共t兲−xi共t兲兴

= limt_→⬁ei共t兲=0. Since gi T_共t兲=关H

i1共t,y1共t兲兲, ... ,

HiN共t,yN共t兲兲兴T is PE, and gi共t兲 and g˙i共t兲 are bounded, by

Lemma 1, we can conclude that limt→⬁共dij共t兲−⑀cij兲=0 for all

i , j = 1 , 2 . . . , N. 䊏

Remark 2: Since the system described by Eqs. 共7兲 and 共8兲 is time varying共even if the functions fi and Hij do not

explicitly depend on time t兲, LaSalle’s invariance principle cannot be applied directly. Meanwhile, the feedback gains ki 共i=1, . . ,N兲 are convergent 共see the proof in the

Appen-dix兲.

Remark 3: In our method, the outer-coupling matrix C needs not to be symmetric or irreducible. So this theorem can be applied to a wide class of complex dynamical networks.

Remark 4: Consider that the functions fiand Hijdo not

explicitly depend on time t and xi, yiare bounded for all i.

Combine the ith node system of the drive network 共3兲 and response network 共6兲into the system as follows:

z˙i= Fi+ ki共t兲

兺

j=1 2 a ˜ijzj, i = 1,2, where z1= xi, z2= yi, F1= fi共xi兲+⑀兺j=1 N cijHij共xj兲, F2= fi共yi兲

+兺_j=1N dij共t兲Hij共yj兲, ki共t兲 is as in Eq. 共6兲, and the coupling

matrix A˜ =关a˜ij兴=关0,0;1,−1兴. Note that all the elements of z1

and z₂ are coupled in z₂ system. From Theorem 1 and Re-mark 1 in Ref. 13, we can obtain that a sufficiently large coupling strength kiwill lead to synchronization. However,

with only a partial coupling of z₁ and z₂, for example, Rössler system with the first component coupling,9the large coupling strength may lead to desynchronization. In this pa-per, all the states of the ith node system of the drive and response networks are coupled, thus avoid the case that the large coupling strength may lead to desynchronization. Therefore, our approach is consistent with the master stabil-ity function method.3

IV. TOPOLOGY IDENTIFICATION OF DYNAMICAL NETWORKS WITH COUPLING DELAY

In this section, we consider a dynamical network with coupling delay. Each node in the network represents an m-dimensional dynamical system described by

x˙i= fi共t,xi兲 +⑀

兺

j=1 N cijHij共t,xj共t −␶j i_{共t兲兲兲, i = 1,2, ... ,N,} 共10兲 where xi苸Rm is the state variables of node i, fi: R⫻Rm

→Rm

is a continuous function,⑀⬎0 is the known coupling strength, Hij is the nonlinear inner-coupling function,

␶j

i_{共t兲ⱖ0 represents the transmission delay of the data sent}

from agent j to agent i, and the outer-coupling matrix C =共cij兲N⫻Nis defined the same as in Sec. III.

Assumption 2: For any i , j, the delay function ␶j i_{共t兲 is}

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differentiable with bounded derivative and satisfies ␶˙i_j共t兲⬍r ⬍1.

Remark 5: Assumption 2 requires that the function

␾ij共t兲=t−␶j

i_{共t兲 is increasing with respect to t. This coincides}

with the physical constraint that the data sent earlier from system j to system i will be received earlier by system i.

Design the response network as

y˙i= fi共t,yi兲 +

兺

j=1 N dij共t兲Hij共t,yj共t −␶j i_{共t兲兲兲 − k} iei, i = 1,2, . . . ,N, 共11兲

with the adaptive feedback law

再

d˙ij= −␦ijei T Hij共t,yj共t −␶j i_{共t兲兲兲} k˙i= siei T ei,

冎

共12兲 where ei= yi− xiis the synchronization error, kiand dijare the

adaptive parameters of the response system, and␦ijand siare

two positive constants.

In the following, we show that cijcan be identified using

the response networks 共11兲 and 共12兲. The synchronization errors between network共10兲and network共11兲can be written as follows: for all i = 1 , 2 , . . . , N,

e˙i= fi共t,yi兲 − fi共t,xi兲 +

兺

j=1 N dij共t兲Hij共t,yj共t −␶j i_{共t兲兲兲} −⑀

兺

j=1 N cijHij共t,xj共t −␶j i_{共t兲兲兲 − k} i共t兲ei. 共13兲

Theorem 2: Suppose Assumptions 1 and 2 hold, and Hij

is bounded. Then, by updated law共12兲, limt→⬁储xi− yi储=0 for

all i. Furthermore, if for any i, ˜g_iT共t兲=关H_i1共t,y₁共t −␶₁i共t兲兲兲, ... ,HiN共t,yN共t−␶N

i_{共t兲兲兲兴}T

is PE, and g˜˙i共t兲 is

bounded, then limt_→⬁共dij−⑀cij兲=0.

储ei共t兲储 2 −

冉

␤共1 − r兲 −␥ 2

冊

˜i共t兲,g˜˙i共t兲 are bounded, by Lemma 1, we can conclude that

limt→⬁共dij共t兲−⑀cij兲=0 for i, j=1,2... ,N.

Remark 6: By Lemma 2 in Ref.34, we can conclude in the proof of Theorem 2 that limt→⬁˜gi共t兲␰i共t兲=0. It can also

be proved that limt→⬁␰i共t兲=␰iⴱ. Therefore, the result of

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rem 2 can be justified if, instead of the PE assumption of g

˜_iT共t兲, we have the following condition H1.

共H1兲 For any N⫻1 vector␩⫽0, g˜i共t兲␩y0 as t→⬁.

In addition, the PE condition implies the condition H1. See the Appendix for the detailed proof of the claims above. According to Lemma 8 in Ref.23, it can be shown that H1 also implies the PE condition.

V. NUMERICAL SIMULATIONS

In this section, we show three illustrative examples that validate our results in Secs. III and IV. In all the three ex-amples, each node in the network represents a three-dimensional neural system, which is described by

dxi dt = − I3xi+ TM共xi兲, 共20兲 where xi=关xi1, xi2, xi3兴T苸R3, T =

冤

1.25 − 3.2 − 3.2 − 3.2 1.1 − 4.4 − 3.2 4.4 1

冥

, 共21兲

I₃ is the 3⫻3 unity matrix, and M共xi兲

=关m共xi1兲,m共xi2兲,m共xi3兲兴T with m共u兲=共兩u+1兩−兩u−1兩兲/2. As

indicated in Ref.35, system共20兲has a double-scrolling cha-otic attractor.

Example 1: First, let us consider a weighted complex dynamical network being composed of N = 6 nodes and with-out coupling delay. The with-outer-coupling matrix is chosen as

C =

冤

− 4 0 1 1 1 1 1 − 4 1 0 2 0 0 1 − 4 1 2 0 1 0 4 − 6 0 1 0 0 0 1 − 2 1 1 0 0 4 3 − 8

冥

.

For simplicity, let Hij共t,xj兲=xjand⑀= 0.01 in Eq.共3兲. In the

simulations, the parameters in the adaptive feedback law共7兲 are set as␦ij= 1 and si= 1. The initial conditions of the drive

network xij共0兲, i=1,2, ... ,N, j=1,2,3 are randomly

se-lected in 关⫺10,10兴; and the initial condition of the response network yij共0兲, i=1,2, ... ,N, j=1,2,3 are randomly

se-lected in关0,1兴. In addition, we set ki共0兲=0,dij共0兲=0. Figures

1共a兲and1共b兲show that the subsystems of response network do not achieve synchronization and the response network synchronizes the drive network asymptotically, respectively. Meanwhile, the feedback gains ki, i = 1 , . . . , N, are

conver-gent, as shown in Fig.1共c兲. Define Wi共t兲=兰t t+T0_g

i共␶兲gi T_共_␶_兲d_␶

. Figure 2共a兲 shows that ␭min共Wi共t兲兲⬎0 for all i, t, and T0

= 15, which indicates that gi=关y1, . . . , yN兴T satisfies the PE

condition for all i. The parameter identification errors vanish as time increases, as can be seen in Fig. 2共b兲, which means that dij共t兲 approaches⑀cijasymptotically.

Example 2: Now we choose⑀= 0.5 in Eq. 共3兲and other conditions the same as in Example 1. In this case, the coupling strength of the drive network is sufficiently large to result in complete inner synchronization, i.e., limt→⬁储xi共t兲−xj共t兲储=0 for all i, j. By the first part of Theorem

1, the response network still synchronizes the drive network and, as a consequence, also asymptotically achieves com-plete inner synchronization. However, it can be easily veri-fied that the PE condition is not satisveri-fied, and it is seen from Figs.3and4 that the identification procedure fails.

Example 3: We consider another network with coupling delays, whose outer-coupling matrix is chosen as

C =

冤

− 4 0 1 1 1 1 1 − 3 1 0 0 1 0 1 − 5 1 2 1 1 0 4 − 6 0 1 0 0 0 1 − 2 1 1 0 0 4 2 − 7

冥

. We assume that Hij共t,xj共t−␶j i_{共t兲兲兲=x} j共t−1兲 for all i, j

= 1 , 2 , . . . , N and ⑀= 0.01 in Eq. 共10兲. The initial conditions are set as follows: ki共0兲=0, dij共0兲=0, xij共0兲, j=1,2,3 are

randomly selected in 关–5,5兴, and yij共0兲, j=1,2,3 are

ran-domly selected in 关0,1兴. Additionally,␦ijand si are selected

0 100 200 300 400 500 600 100 (a) t Ψ 0 100 200 300 400 500 600 10−20 100 1020 (b) t max{ |e1 |,..., |eN |} 0 100 200 300 400 500 600 0 2 4 6 8 (c) t ki

FIG. 1. 共Color online兲共a兲 The maximum inner synchronization error ⌿ of the response network, where the coupling strength⑀= 0.01 in Eq.共3兲.共b兲 The synchronization error between the drive network and the response net-work.共c兲 Time evolution of adaptive feedback gains.

0 200 400 600 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 (a) t λmin (W i (t)) 0 200 400 600 10−10 10−8 10−6 10−4 10−2 100 102 104 (b) t Max{|d ij (t)− ε cij |}

FIG. 2.共Color online兲共a兲 The minimum eigenvalue of the matrix Wi共t兲. 共b兲

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the same as in Example 1. Figures 5共a兲and 5共b兲 show that the subsystems of response network do not achieve synchro-nization and the response network synchronizes the drive network, respectively. Meanwhile, the feedback gains are convergent, as shown in Fig.5共c兲. Similarly as in Example 1, the PE condition is verified with T0= 10.5, as shown in

Fig. 6共a兲. The parameter identification error is depicted in Fig.6共b兲.

VI. CONCLUSIONS

In this paper, we have proposed adaptive feedback laws to identify the exact topology of the weighted complex dy-namical networks. PE conditions prove to guarantee the ef-ficiency of our method for networks with and without cou-pling delays. The results are applicable to a large class of networks since the coupling matrix need not to be symmetri-cal or irreducible, and the inner coupling can be nonlinear and time varying.

ACKNOWLEDGMENTS

This work was supported by the National Natural Sci-ence Foundation of China 共Grant Nos. 70771084 and 60974081兲, the National Basic Research Program of China 共973兲共Grant No. 2007CB310805兲, the 2010 Natural Science Foundation of Hubei Provincial Department of Education 共Grant No. D20101605兲, and the 2010 Key Project of Chi-nese Ministry of Education.

APPENDIX: PROOFS OF THE CLAIMS IN REMARK 6

共1兲 limt_→⬁˜gi共t兲␰i共t兲=0. From Eq.共18兲, we know that

e˙i共t兲 = f˜i共t兲 + g˜i共t兲␰i共t兲, where f˜i共t兲= fi共t,yi兲− fi共t,xi兲+兺j=1 N _⑀ cij共Hij共t,yj共t −␶j i_{共t兲兲兲−H} ij共t,xj共t−␶j i_{共t兲兲兲兲−k} iei. By the

bounded-ness of g˜i共t兲, g˜˙i共t兲, ␰i共t兲, and ␰˙i共t兲, we have that

g

˜i共t兲␰i共t兲 is uniformly continuous. In addition, it

fol-lows from limt→⬁ei共t兲=0 that 兰⬁t₀共f˜i共t兲+g˜i共t兲␰i共t兲兲

exists and limt→⬁f˜i共t兲=0. Hence, by Lemma 2 in

Ref.34, we conclude that limt→⬁˜gi共t兲␰i共t兲=0.

0 10 20 30 40 50 60 70 80 90 100 10−20 100 1020 (a) t Ψ 0 10 20 30 40 50 60 70 80 90 100 10−20 100 1020 (b) t max{|e 1 |,...,|e N |} 0 10 20 30 40 50 60 70 80 90 100 0 5 (c) t ki

FIG. 3. 共Color online兲共a兲 The maximum inner synchronization error ⌿ of the response network, where the coupling strength⑀= 0.5 in Eq.共3兲.共b兲 The synchronization error between the drive network and the response network. 共c兲 Time evolution of adaptive feedback gains.

0 20 40 60 80 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 (a) t λmin (W i (t)) 0 50 100 100.85 100.87 100.89 100.91 100.93 100.95 (b) t Max{|d ij (t)− ε cij |}

Parameter identification error.

0 50 100 150 200 250 300 350 400 100 (a) t Ψ 0 50 100 150 200 250 300 350 400 10−20 100 1020 (b) t max{|e 1 |,...,|e N |} 0 50 100 150 200 250 300 350 400 0 2 4 6 (c) t ki

FIG. 5. 共Color online兲共a兲 The maximum inner synchronization error ⌿ of the response network, where the coupling strength⑀= 0.01 in Eq.共10兲.共b兲 The synchronization error between the drive network and the response net-work.共c兲 Time evolution of adaptive feedback gains.

0 100 200 300 400 10−10 10−8 10−6 10−4 10−2 100 102 104 (b) t max{|d ij (t)− ε cij |} 0 100 200 300 400 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a) t λmin (W i (t))

Parameter identification error.

(8)

共2兲 limt→⬁␰i共t兲 exists. From Eq.共12兲, we see that kiis

in-creasing as t increases. However, since V is bounded, we know that limt→⬁ki共t兲 exists. On the other hand, from

Eq. 共17兲, we have兰t⬁₀储ei共t兲储2dt⬍⬁. This, together with

the fact that limt→⬁共t−␶j

i_{共t兲兲=⬁, gives that the third term}

on the right-hand side of Eq.共14兲converges to zero as t goes to infinity. Then, by Eq.共14兲, the convergence of␰i

follows from that of the Lyapunov function V. 共3兲 g˜i

T_{共t兲 is PE⇒H1. Since g˜}

i共t兲 is bounded, let 储g˜i共t兲储⬍b,

where b is some positive number. According to Defini-tion 1, we have that if g˜_iT共t兲 is PE, then there exist T₀,␦₁⬎0 such that for any 储␩储=1,

冕

t t+T0 ␩T g ˜i T_共_␶_兲g˜ i共␶兲␩d␶ⱖ␦1 ∀ t ⱖ 0, which leads to

冕

t t+T0 储g˜i共␶兲␩储d␶ⱖ ␦1 b ∀ t ⱖ 0. Therefore, g˜i共t兲␩y0.

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