Parameter identification and adaptive–impulsive synchronization of uncertain complex networks with nonidentical topological structures

Accepted Manuscript Title: Parameter identification and adaptive-impulsive synchronization of uncertain complex networks with nonidentical topological structures Author: Hong-Li Li Yao-Lin Jiang Zuolei Wang Long Zhang Zhidong Teng PII: DOI: Reference:

S0030-4026(15)00958-4 http://dx.doi.org/doi:10.1016/j.ijleo.2015.08.191 IJLEO 56115

To appear in: Received date: Accepted date:

29-8-2014 25-8-2015

Please cite this article as: Hong-Li Li, Yao-Lin Jiang, Zuolei Wang, Long Zhang, Zhidong Teng, Parameter identification and adaptive-impulsive synchronization of uncertain complex networks with nonidentical topological structures, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.08.191 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Hong-Li Li∗, Yao-Lin Jiang, Zuo-Lei Wang

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Parameter identification and adaptive-impulsive synchronization of uncertain complex networks with nonidentical topological structures

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Urumqi 830046, P.R. China

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College of Mathematics and System Sciences, Xinjiang University

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d

M

an

Abstract: In this paper, we integrate impulsive control and adaptive control methods, based on the stability theory of impulsive differential equations, synchronization of uncertain complex networks with nonidentical topological structures is investigated. By designing effective adaptive-impulsive controllers, we achieve synchronization criteria of uncertain complex networks with nonidentical topological structures, parameter identification is realized simultaneously as the synchronization occurs. Moreover, the case of identical topological structures is considered. Finally, numerical simulations are given to illustrate the theoretical results.

1

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Keywords: Parameter identification; Synchronization; Uncertain complex network; Nonidentical topological structure; Adaptive-impulsive control.

Introduction

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Complex networks widely exist in our life, such as the Internet, the social networks, the transportation networks, the phone call networks, the food webs and so on. They are receiving more and more attention from various fields of science and engineering, including mathematics, physics, ecology science and social science [1-4]. Synchronization of complex networks has been one of the focal points in many research and application fields. Synchronization has been studied from various angles and a variety Foundation item: This work is supported by National Natural Science Foundation of China (Grant No. 11102180, 11371287, 61273106, 61379064) and the International Science and Technology Cooperation Program of China (Grant No. 2010DFA14700). ∗ Corresponding author. E-mail: [email protected]

1 Page 1 of 13

of different synchronization phenomena have been discovered, such as complete synchronization [5,6], projective synchronization [7], phase synchronization [8], lag synchronization [9], cluster synchronization [10], and so on. Correspondingly, there are several control schemes have been introduced to realize the network synchronization, for example, the adaptive synchronization [11,12], pining control [13,14] and impulsive control [15-18].

ip t

In real systems, not all model parameters are known due to various difficulties in practical applications, i.e., there exist unknown parameters or parameter perturbations. Therefore, unknown parameter estimation in complex dynamical networks, as an inter-

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esting research subject, has drawn increasing attention from different fields [19-22]. In [21,22], Zhang et al. investigated the synchronization of uncertain networks with identical

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topological structures via adaptive-impulsive control. In fact, in engineering, as we all known, sometimes it is unrealistic to assume that the topological structures of complex networks are identical. Motivated by the above reasons, in this paper, we will study the

an

synchronization of uncertain complex networks with nonidentical topological structures via adaptive-impulsive control.

The rest of this paper is organized as follows. In section 2, model description and

M

preliminaries are given. In Section 3, a set of novel adaptive-impulsive laws and synchronization criteria are given. In Section 4, numerical simulations are given to show the effectiveness of the proposed synchronization scheme. Finally, a conclusion is given in

te

2

d

Section 5.

Models and preliminaries

ce p

Some notations used throughout this paper are introduced firstly as follows: kxk denotes the 2-norm of the vector x; AT (or xT ) means the transpose of the matrix A (or vector x); In ∈ Rn×n represents the identity matrix with dimension n; ⊗ denotes the Kronecker product of two matrices, and λmax (A) represents the maximum eigenvalue of a square matrix A.

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Consider a class of n-dimensional dynamical system, which is described by the following form of the differential equation: x˙ i (t) = Fi (t, xi (t), Θi ),

(1)

Rewrite system (1) in the following form x˙ i (t) = fi (t, xi (t)) + gi (t, xi (t)) · Θi ,

(2)

where xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T ∈ Rn is state vector, Θi ∈ Rmi is the unknown parameter vector, fi (t, xi (t)) : R+ × Rn → Rn is the continuous nonlinear function vector 2 Page 2 of 13

without unknown parameters and gi (t, xi (t)) : R+ × Rn → Rn×mi is a continuous function matrix. Assumption 1. For any xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T and yi (t) = (yi1 (t), yi2 (t), · · · , yin (t))T , there exists a positive constant Li > 0 such that

ip t

kFi (t, yi (t), Θi ) − Fi (t, xi (t), Θi )k ≤ Li kyi (t) − xi (t)k. Remark 1. In fact, there are many classical chaotic systems, such as Lorenz system, Chen system, L¨ u system and Chua’s circuit system, their corresponding dynamical functions all

cr

satisfy the above assumption.

Now, we consider a general complex network consisting of N dynamical nodes. Each is described by aij Cxj (t),

j=1

i = 1, 2, · · · , N,

an

x˙ i (t) = Fi (t, xi (t), Θi ) +

N ∑

us

isolate node of the network is an n-dimensional dynamical system. Then the drive network

(3)

where xi (t) = (xi1 (t), xi2 (t), · · · , xin (t))T ∈ Rn is the state variables of node i; C =

M

(cij )n×n is an inner coupling matrix determining the interaction of variables; A = (aij )N ×N is the coupling configuration matrix representing the coupling strength and topological structure of the network, in which aij is defined as follows: If there is a connection between

d

node i and node j (i 6= j), then aij 6= 0; otherwise, aij = 0.

te

Remark 2. The network model (3) is more popular and generous because it is composed with N different node dynamics. If there are some kinds of the same nodes, it degenerates

ce p

into the cluster network. If all the nodes have the same dynamics, it simplified to the general network.

We take the network given by (3) as the drive network, and another impulsively controlled response network is given by  N ∑    ˆ y ˙ (t) = F (t, y (t), Θ ) + bij Cyj (t) + Ui ,  i i i i 

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

t 6= tk ,

j=1

  ∆yi (t+ ) = Dik (yi (t) − xi (t)),    yi (t+ = yi0 , 0)

t = tk ,

k = 1, 2, · · · ,

(4)

where yi (t) = (yi1 (t), yi2 (t), · · · , yin (t))T ∈ Rn is the response state variables of node i, ˆ i is the estimation of the unknown Θi . The matrices Dik ∈ Rn×n (k = 1, 2, · · · ) and Θ are the impulsive feedback matrices received by the ith node at the moment tk , Ui is + − the adaptive controller received by the ith node. Moreover, ∆yi (t+ k ) = yi (tk ) − yi (tk ), − yi (t+ k ) = limt→t+ yi (t) and any solution of (4) is left continuous at each tk , i.e. yi (tk ) = k

yi (tk ). The moments of impulse satisfy 0 ≤ t1 < t2 < · · · < tk < · · · , limk→∞ tk = +∞.

3 Page 3 of 13

3

Synchronization criteria In this section, our goal is to design appropriate adaptive controllers Ui and the cor-

responding updating laws which make the impulsively controlled network (4) and (3) be asymptotically synchronous. (4) as ei (t) = yi (t) − xi (t), i = 1, 2, · · · , N . Then, from (1)-(4), the error systems are described by N ∑

bij Cyj (t) − Fi (t, xi (t), Θi ) −

j=1

N ∑

aij Cxj (t) + Ui

cr

ˆ i) + e˙ i (t) = Fi (t, yi (t), Θ

ip t

We define the synchronization errors between drive network (3) and response network

j=1

bij Cyj (t) −

j=1

N ∑

aij Cxj (t) + Ui

j=1

an

+

N ∑

us

ˆ i − gi (t, yi (t))Θi + gi (t, yi (t))Θi − Fi (t, xi (t), Θi ) = fi (t, yi (t)) + gi (t, yi (t))Θ

ˆ i − Θi ) + = Fi (t, yi (t), Θi ) − Fi (t, xi (t), Θi ) + gi (t, yi (t))(Θ aij Cej (t) +

j=1

N ∑ j=1

and

bij Cxj (t) −

N ∑

N ∑

bij Cej (t)

j=1

aij Cyj (t) + Ui ,

M

−

N ∑

j=1

(5)

d

+ + e˙ i (t+ k ) = yi (tk ) − xi (tk )

(6)

te

= yi (tk ) + Dik (tk )ei (tk ) − xi (tk ) = (In + Dik )ei (tk ).

ce p

Theorem 1. Suppose that Assumption 1 holds. Let ∑ ∑ ˜ 2 ei − Ui = −d∗ ei + µkΘk b Cx (t) + aij Cyj (t), ij j kEk2 {

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

N

N

j=1

j=1

ˆ˙ i = −giT (t, yi (t))ei (t), t 6= tk , Θ ˆ i = Fik Θ ˆ i , t = tk , k = 1, 2, · · · , ∆Θ

(7)

(8)

ˆ ˜ =Θ ˆ − Θ, Θ ˆ = (Θ ˆT,Θ ˆT,··· ,Θ ˆ T )T , Θ = (ΘT , ΘT , · · · , ΘT )T , Θ where d∗ > 0, µ > 0, Θ 1 2 1 2 N N is an estimation vector to Θ, E(t) = (eT1 (t), eT2 (t), · · · , eTN )T . Let µk = max{αk , βk }, αk = max{α1k , α2k , · · · , αnk } < 1, αik = λmax {(In +Fik )T (In +Fik )}, βk = max{β1k , β2k , · · · , βnk } < 1, βik = λmax {(In + Dik )T (In + Dik )}, ι = λmax { 21 ((A ⊗ C) + (A ⊗ C)T )}, κ = λmax { 21 ((B ⊗ C) + (B ⊗ C)T )}. If there exists a constant ξ > 1 such that 2%(tk − tk−1 ) + ln(ξµk ) < 0,

(9)

where % = max{ν, µ} and ν = L − d∗ + κ + ι. Then, the drive network (3) and the response ˆ → Θ. network (4) can be synchronized. Moreover, Θ 4 Page 4 of 13

Proof. Construct the Lyapunov function as follows: V (t) = =

1 1 ˜ 2 kE(t)k2 + kΘk 2 2 (10)

1∑ T 1∑ ˆ ˆ i − Θi ). ei (t)ei (t) + (Θi − Θi )T (Θ 2 2 N

N

i=1

i=1

ip t

When t ∈ (tk−1 , tk ](k = 1, 2, · · · ), from Assumption 1, (7) and (8), the derivative of V (t) with respect to (5) is

i=1

=

N ∑

i=1

[

ˆ i − Θi ) eTi Fi (t, yi (t), Θi ) − Fi (t, xi (t), Θi ) + gi (t, yi (t))(Θ

i=1 N ∑

+

bij Cej (t) +

j=1

−

N ∑ ˆ˙ Ti (Θ ˆ i − Θi ) + Θ

cr

eTi e˙ i (t)

N ∑

N ∑

aij Cej (t) +

j=1

ˆ i − Θi ) ei (t)T gi (t, yi (t))(Θ

j=1 N ∑ N ∑

Li eTi (t)ei (t) +

i=1

−d∗

N ∑ i=1

i=1 j=1 N ∑

M

=

]

aij Cyj (t) + Ui

N N ] [ ∑ ∑ ˆ − Θk2 ei eTi Li ei (t) + bij Cej (t) + aij Cej (t) − d∗ ei + µkΘ kEk2 j=1

eTi (t)bij Cej (t) +

eTi (t)ei (t) + µ

N ∑ N ∑

eTi (t)aij Cej (t)

i=1 j=1

d

i=1 N ∑

N ∑ j=1

ˆ − Θk2 eTi kΘ

te

≤

bij Cxj (t) −

j=1

i=1

N ∑

N ∑

us

N ∑

an

V˙ (t) =

i=1

ei kEk2

ce p

≤ (L − d∗ )E T (t)E(t) + E T (t)(B ⊗ C)E(t) + E T (t)(A ⊗ C)E(t) +µ

N ∑

ˆ − Θk2 eTi kΘ

i=1

ei , kEk2

where L = max{Li }, and notice that

(11)

∑N

T ei i=1 ei kEk2

= 1, we further have

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[ (B ⊗ C) + (B ⊗ C)T (A ⊗ C) + (A ⊗ C)T ] V˙ (t) ≤ (L − d∗ )E T (t)E(t) + + 2 2 ˆ − Θk2 ×E T (t)E(t) + µkΘ ˆ − Θk2 ≤ (L − d∗ + κ + ι)kE(t)k2 + µkΘ

(12)

≤ 2%V (t). This implies that V (t) ≤ V (t+ k−1 ) exp(2%(t − tk−1 )),

t ∈ (tk−1 , tk ], k = 1, 2, · · · .

(13)

5 Page 5 of 13

On the other hand, when t = tk , k = 1, 2, · · · , we have = =

1∑ T + 1 ∑ ˆT + ˆ + ei (tk )ei (t+ Θi (tk )Θi (tk ) ) + k 2 2 N

N

i=1

i=1

N 1∑

2

eTi (tk )(In + Bik )T (In + Bik )ei (tk )

i=1 N

1 ∑ ˆT ˆ i (tk ) Θi (tk )(In + Fik )T (In + Fik )Θ + 2 ≤

1 λmax [(In + Bik )T (In + Bik )] 2

N ∑

i=1

an

Thus, let k = 1 in the inequality (13), we obtain

ˆ Ti (tk )Θ ˆ i (tk ) Θ

us

≤ µk V (tk ).

eTi (tk )ei (tk )

i=1 N ∑

1 + λmax [(In + Fik )T (In + Fik )] 2

(14)

cr

i=1

ip t

V

(t+ k)

V (t) ≤ V (t+ 0 ) exp[(2%(t − t0 )],

(15)

V (t1 ) ≤ V (t+ 0 ) exp[(2%(t1 − t0 )].

(16)

+ V (t+ 1 ) ≤ µ1 V (t1 ) ≤ µ1 V (t0 ) exp[(2%(t1 − t0 )].

(17)

M

which leads to

te

Therefore, for t ∈ (t1 , t2 ],

d

Then from (14), we have

V (t) ≤ V (t+ 1 ) exp[(2%(t − t1 )]

ce p

≤ µ1 V (t+ 0 ) exp[(2%(t − t0 )].

(18)

Repeating the same process, for t ∈ (tk , tk+1 ], V (t) ≤ µ1 µ2 · · · µk V (t+ 0 ) exp[(2%(t − t0 )].

(19)

In virtue of the inequality (9), we know that

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

1 µk exp[2%(tk+1 − tk )] ≤ . ξ

(20)

Notice that t − t0 = (t − tk ) + (tk − tk−1 ) + · · · + (t2 − t1 ) + (t1 − t0 ). Thus, the inequality (19) can be further rewritten as V (t) ≤ µ1 µ2 · · · µk V (t+ 0 ) exp[(2%(t − t0 )] = V (t+ 0 ){µ1 exp[(2%(t1 − t0 )]} × {µ2 exp[(2%(t2 − t1 )]} × · · · × {µk exp[(2%(tk − tk−1 )]} exp[2%(t − tk )] ≤ V (t+ 0)

(21)

1 exp[2%(t − tk )]. ξk 6 Page 6 of 13

Therefore V (t) → 0 as k → ∞ because ξ > 1, which implies that all the errors ei (t) → 0 ˆ → Θ. So the adaptive-impulsive synchronization between complex network (3) and and Θ (4) is realized and the unknown system parameters are identified simultaneously. This completes the proof. Remark 3. From proof of Theorem 1, we know that the synchronization error will con-

ip t

verge to zero, that is, ei (t) → 0 as t → ∞. From error system (5) and the condition (7), ˆ i − Θi ) = 0. Therefore, if all the column vectors of gi (t, yi (t)) are we can get gi (t, yi (t))(Θ ˆ i as t → ∞. linear independent, then the unknown parameters Θi can be identified by Θ

cr

Remark 4. In our work, the coupling matrices A and B are not necessarily to be symmetric and irreducible, which means that the drive and response networks can be undirected

us

or directed networks. In addition, there is not any constraint imposed on the inner coupling matrix C. by x˙ i (t) = f (t, xi (t)) + g(t, xi (t)) · Θ +

an

Corollary 1. If a complex network consists of N identical nodes, which can be described N ∑

aij Cxj (t),

i = 1, 2, · · · , N,

(22)

j=1

M

ˆ with the unknown system parameters Θ can be identified by using the estimated values Θ

d

the following impulsively controlled response network  N  ∑   ˆ  y˙ i (t) = f (t, yi (t)) + g(t, yi (t)) · Θ + bij Cyj (t) + Ui ,   

ce p

te

∆yi (t+ ) = Dik (yi (t) − xi (t)),        y (t+ ) = y , i 0 i0

t 6= tk ,

j=1

t = tk ,

k = 1, 2, · · · ,

(23)

under the following restriction conditions ˜ 2 Ui = −d∗ ei + µkΘk

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

∑ ∑ ei − b Cx (t) + aij Cyj (t), ij j kEk2 N

j=1

N

(24)

j=1

 N ∑   Θ ˆ˙ = − g T (t, yi (t))ei (t), t 6= tk , i=1   ˆ ˆ ∆Θ = Fk Θ, t = tk , k = 1, 2, · · · ,

(25)

where d∗ > 0, µ > 0 are defined as in Theorem 1. If there exist a constant ξ > 1 such that 2%(tk − tk−1 ) + ln(ξµk ) < 0,

(26)

where % and µk are defined as in Theorem 1. Then the drive network (22) and response network (23) can achieve synchronization. Corollary 2. Suppose that Assumption 1 holds. If the two complex networks have the

7 Page 7 of 13

same configuration matrices, i.e., A = B. Under the following restriction conditions ∑ ˜ 2 ei + Ui = −d∗ ei + µkΘk aij Cej (t), 2 kEk

(27)

 N ∑   Θ ˆ˙ = − g T (t, yi (t))ei (t), t 6= tk , i=1   ˆ ˆ ∆Θ = Fk Θ, t = tk , k = 1, 2, · · · ,

(28)

N

ip t

j=1

where d∗ > 0, µ > 0 are defined as in Theorem 1. If there exist a constant ξ > 1 such that 2%(tk − tk−1 ) + ln(ξµk ) < 0,

cr

(29)

us

where % and µk are defined as in Theorem 1. The drive network (22) and response network ˆ → Θ. (23) can achieve synchronization. Moreover, Θ Remark 5. Note that the difference in the topological structures between the two networks makes synchronization difficult, and the controllers needed for the response network

an

are more complex than those for the case of identical topological structures. However, it

4

Numerical simulation

M

is obvious that Corollary 1 is more practically meaningful than Corollary 2.

In this section, we present several numerical examples to show the effectiveness of

d

above synchronization criteria. Consider a simple network with 3 identical nodes. The

te

Lorenz system is taken as the node’s dynamical function in this example, which is given by





a(xi2 − xi1 )

ce p

  F (t, xi (t), Θ) =  oxi1 − xi1 xi3 − xi2 

   

xi1 xi2 − uxi3



Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0

  =  −xi1 xi3 − xi2  xi1 xi2





    +  

xi2 − xi1

0

0

0

0

−xi3

0

 

a



      xi1  ·  u     0 o

, f (t, xi (t)) + g(t, xi (t)) · Θ, where i = 1, 2, 3. In numerical simulations, the unknown parameters are set as (a, u, o)T = (10, 8/3, 28). For any two state vectors xi and xj of Lorenz system, the assumption 1 is easily satisfied since the chaotic attractor is bounded in a certain region, see Fig.1. By simple calculation, we obtain L = 70.4277. In the following, we discuss the synchronization of two networks consisting of the above Lorenz system under the two cases of A = B and A 6= B. 8 Page 8 of 13

4.1

Identical topological structures: A = B

In this subsection, we consider the case in which both networks (22) and (23) have the same configuration matrices, i.e. A = B. For simplicity, we assume that C = I3 , and the control of the response system (23) is given by (24) and (25). Suppose that the

−2

2

  P1 =  2  0

−3 1

0



  1 .  −1

cr



where

ip t

configuration matrices of systems (22) and (23) are 3 × 3 dimensional and A = B = P1 ,

Since the eigenvalues of the matrix 12 (A + AT ) are -4.7321, -1.2679 and 0, λ = λmax ( 12 (A +

us

AT )) = 0. Let d∗ = 70.43, µ = 1.5, Bik = Fk = diag{−0.6, −0.6, −0.6}, τ = tk+1 − tk , with the condition in Corollary 1, the impulsive interval can be estimated as follows: ln 1.1 + ln 0.16 = 0.5791. 3

an

0≤τ ≤−

To test Corollary 1, we set τ = 0.1. The initial state of the ith node of the drive network

M

is xi (0) = (−3(i − 1) − 1, −3(i − 1) − 2, −3(i − 1) − 3)T , while the state state of the ith node of the response network is yi (0) = (3(i − 1) + 1, 3(i − 1) + 2, 3(i − 1) + 3)T , i = 1, 2, 3. The initial estimated parameters are (ˆ a(0), u ˆ(0), oˆ(0)) = (1.0, 30.0, 10.0). Fig.2 gives the

d

synchronization errors between the derive-response networks (22) and (23). Fig.3 presents

Nonidentical topological structures: A 6= B

ce p

4.2

te

the identification of unknown parameters Θ.

Here, we consider networks (22) and (23) with 3 nodes, but the topological structures A and B are nonidentical. Suppose that the configuration matrix of network (22) A = P1 as in the above subsection, and the configuration  −3 2   P2 =  2 −3  1 1

Ac

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

matrix of network (23) B = P2 is  1   1 .  −2

Obviously, P1 6= P2 , and we also assume that C = I3 . Since the eigenvalues of the matrix 1 2 (B

+ B T ) are -5, -3 and 0, ι = λmax ( 21 (B + B T )) = 0. Let d∗ = 70.43, µ = 1.5,

Bik = Fk = diag{−0.6, −0.6, −0.6}, τ = tk+1 − tk , with the condition in Corollary 1, the impulsive interval can be estimated as follows: 0≤τ ≤−

ln 1.1 + ln 0.16 = 0.5791. 3

To test Corollary 2, we set τ = 0.1. The initial values of state variables and the initial estimated parameters are identical to all above. Fig.4 gives the synchronization errors 9 Page 9 of 13

between the derive-response networks (22) and (23). Fig.5 presents the identification of unknown parameters Θ.

5

Conclusion In this paper, we have proposed an adaptive-impulsive control approach to simultane-

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ously identify the uncertain complex networks. By designing effective adaptive-impulsive controllers, we realize the synchronization between two networks with identical and non-

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identical topological structures. Moreover, parameter identification is realized simultaneously as the synchronization occurs. Particularly, the outer-coupling matrixes do not to

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be symmetric and there is not any constraint on the inner-coupling matrix. The proposed adaptive-impulsive controllers are simple and can be readily applied in practical applications. Finally, numerical simulations have been given to verify the effectiveness of the

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proposed synchronization criteria.

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References

[1] S.H. Strogatz, Exploring complex network, Nature 410 (2001) 268-276.

74 (2002) 47-97.

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[2] R. Albert, A.L. Barab´asi, Statistical mechanics of complex networks, Rev. Mod. Phys.

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[3] R. Pastor-Satorras, E. Smith, R.V. Sol´e, Evolving protein interaction networks

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through gene duplication, J. Theor. Biol. 222 (2003) 199-210. [4] A.L. Barab´asi, H. Jeong, Z. N´eda, E. Ravasz, A. Schubert, T. Vicsek, Evolution of the social network of scientific collaborations, Physica A 311 (2002) 590-614. [5] C. Li, G. Chen, Synchronization in general complex dynamical networks with coupling delays, Physica A 343 (2004) 263-278.

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[6] Y.Q. Gu, C. Shao, X.C. Fu, Complete synchronization and stability of star-shaped complex networks, Chaos Soliton Fract. 28 (2006) 480-488.

[7] M.F. Hu, Y.Q. Yang, Z.Y. Xu, R. Zhang, L.X. Guo, Projective synchronization in drive-response dynamical networks, Physica A 381 (2007) 457-466. [8] X. Li, Phase synchronization in complex networks with decayed long-range interactions, Physica D 223 (2006) 242-247. [9] Q.J. Zhang, J.C. Zhao, Projective and lag synchronization between general complex networks via impulsive control, Nonlinear Dyn. 67 (2012) 2519-2525. 10 Page 10 of 13

[10] E. Guirey, M. Martin, M. Srokosz, Persistence of cluster synchronization under the influence of advection, Phys. Rev. E 81 (2010) 051902. [11] Y.H. Xu, C.R. Xie, D.B. Tong, Adaptive synchronization for dynamical networks of neutral type with time-delay, Optik 125 (2014) 380-385.

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[12] H.W. Tang, L. Chen, J.A. Lu, C.K. Tse, Adaptive synchronization between two complex networks with nonidentical topological structures, Physica A 387 (2008) 5623-5630.

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[13] X.F. Wang, G. Chen, Pinning control of scale-free dynamical networks, Physica A

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[14] L. Xiang, J. Zhu, On pinning synchronization of general coupled networks, Nonlinear

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Dyn. 64 (2011) 339-348.

[15] J. Zhou, Q.J. Wu, L. Xiang, S.M. Cai, Z.R. Liu, Impulsive synchronization seeking in general complex delayed dynamical networks, Nonlinear Anal. Hybrid Syst. 5 (2011)

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513-524.

[16] S. Zheng, G.G. Dong, Q.S. Bi, Impulsive synchronization of complex networks with

d

non-delayed and delayed coupling, Phys. Lett. A 373 (2009) 4255-4259.

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[17] S.M. Cai, J. Zhou, L. Xiang, Z.R. Liu, Robust impulsive synchronization of complex delayed dynamical networks, Phys. Lett. A 372 (2008) 4990-4995.

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[18] S. Zheng, Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling, Nonlinear Dyn. 67 (2012) 2621-2630.

[19] J.Q. Lu, J.D. Cao, Synchronization-based approach for parameters identification in delayed chaotic neural networks, Physica A 382 (2007) 672-682.

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[20] L. L¨ u, L. Meng, Parameter identification and synchronization of spatiotemporal chaos in uncertain complex network, Nonlinear Dyn. 66 (2011) 489-495.

[21] Q.J. Zhang, J. Luo, L. Wan, Parameter identification and synchronization of uncertain general complex networks via adaptive-impulsive control, Nonlinear Dyn. 71 (2013) 353-359. [22] Q.J. Zhang, J. Luo, L. Wan, Erratum to: Parameter identification and synchronization of uncertain general complex networks via adaptive-impulsive control, Nonlinear Dyn. 75 (2014) 403-405.

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(a)

(b)

30

50

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3

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20

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Fig.1 The chaotic attractor of Lorenz system: (a)(x1 , x2 ); (b)(x1 , x3 ).

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18 16 14 12

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Fig.2. Synchronization errors of drive network (22) and response network (23) with A = B = P1 (1 ≤ i, n ≤ 3).

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

0

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4

6

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10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 t

Fig.3. Identification of unknown parameters (a, u, o) for drive network (22) and response network (23) with A = B = P1 .

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18 16 14 12

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(1 ≤ i, n ≤ 3).

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Fig.4. Synchronization errors of drive network (22) and response network (23) with A 6= B(A = P1 , B = P2 )

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Fig.5. Identification of unknown parameters (a, u, o) for drive network (22) and response network (23) with A 6= B(A = P1 , B = P2 ).

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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