Synchronization analysis for complex networks with coupling delay based on T–S fuzzy theory

Applied Mathematical Modelling 36 (2012) 6215–6224

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Synchronization analysis for complex networks with coupling delay based on T–S fuzzy theory Dawei Gong ⇑, Huaguang Zhang, Zhanshan Wang, Jinhai Liu College of Information Science and Engineering, Northeastern University, Shenyang 110819, China

a r t i c l e

i n f o

Article history: Received 22 September 2011 Received in revised form 12 January 2012 Accepted 17 January 2012 Available online 11 February 2012 Keywords: Complex networks Synchronization Coupling delay T–S fuzzy theory

a b s t r a c t In this paper, synchronization problems for a general complex networks are investigated by Takagi–Sugeno (T–S) fuzzy theory. A novel concept named linear approximation method is firstly proposed to solve the synchronization problems for T–S fuzzy complex networks. This novel method can linearize the system into some time-delay subsystems, which can effectively simplify the complicated system and then be easy to acquire the synchronization results. Since the expression based on linear matrix inequality (LMI) is used, the synchronization criteria can be easily checked in practice. Numerical simulation examples are provided to verify the effectiveness and the applicability of the proposed method. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction In recent years, complex networks have attracted increasing attention in various fields for its large size and complex topology. Many systems in science and technology can be modeled as complex networks, such as sociology, biology, mathematics and physics, and so on (see [1–4] and references therein). The most typical collective behavior in complex networks is the synchronization phenomena. Many problems in real world have close relationship with synchronization, for example, the synchronous phenomena on the World Wide Web, social networks, genetic regulatory networks, and so on (see [5–7] and references therein). On the other hand, time delays occur commonly in complex networks because of the network traffic congestions as well as the finite speed of signal transmission over the links. Time delay may decrease the quality of the system and even lead to oscillation, divergence, and instability. Therefore, the synchronization stability of complex networks with delays has become a topic of both theoretical and practical importance (see [8–13]). However, in mathematical modeling of real world problems, the uncertainty or vagueness is unavoidable. In order to take vagueness into consideration, fuzzy theory is considered as a suitable method. Based on traditional complex networks, the fuzzy complex networks have advantages over pure complex networks since they incorporates the capability of fuzzy reasoning in handling uncertain information. For example, in [14], it was found that the small-world networks can be modeled by fuzzy logic. In [15], by employing the stochastic analysis techniques and Kronecker product, delay-dependent synchronization criteria is obtained to ensure the mean-square synchronization of the addressed T–S fuzzy delayed complex networks with stochastic disturbances. In [16], it utilized some stochastic analysis techniques and Kronecker product to deduce delaydependent synchronization criteria. In [17], by employing the information of probability distribution of time delays and the properties of the Kronecker product, the original system is transformed into a T–S fuzzy model with stochastic parameter

⇑ Corresponding author. Tel.: +86 138 42 091060. E-mail addresses: [email protected] (D. Gong), [email protected] (H. Zhang). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2012.01.041

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D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224

matrices. These references can handle more complex situations, and can describe many new synchronization problems. However, the models in these references are too complicated, which can greatly increase the difficulty of research, and the methods in the above references are all based on Kronecker product operation, which can greatly increase the computational complexity. So how to solve the synchronization problem for general fuzzy complex networks still remains largely unsolved and challenging. Motivated by the mentioned discussions, we propose a novel linearized method to solve the problem of synchronization for fuzzy complex networks with coupling delay. The main contributions of this paper are as follows: (1) A new general class of complex networks with T–S fuzzy logic is proposed and its synchronization is investigated; (2) By using a novel method, the complicated system can be divided into some time-delay subsystems, which can be easily analyzed; (3) The proposed method can also be applied in many exiting references (such as [15–17]), and can effectively reduce the computational complexity. The rest of this paper are organized as follows. Section 2 presents preliminaries and some new lemmas to facilitate the proof. Synchronization conditions are derived in Section 3. Numerical examples are given to illustrate the effectiveness of the proposed methods in Section 4. Finally, concluding is offered in Section 5. Notations: Rn is the n-dimensional Euclidean space; Rmn denotes the set of m  n real matrices. X P 0 (respectively, X > 0) means that X is positive semidefinite (respectively, positive definite). In represents the n-dimensional identity matrix.     X Y X Y . Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic stands for T Z  Z Y operations. 2. Problem formulation In this section, we will consider complex networks with coupling delay which is represented by a T–S fuzzy model composed of a set of fuzzy implications. Based on the T–S fuzzy model concept, a general class of T–S fuzzy complex networks is described as follows: Rule l: If h1 ðtÞ is F l1 ; h2 ðtÞ is F l2 ,    ; hg ðtÞ is F lg , then

x_ i ðtÞ ¼ f ðxi ðtÞÞ þ c

N X

g ij Al xj ðt  sÞ; ði ¼ 1; 2; . . . ; NÞ;

ð1Þ

j¼1

where F lj ðl ¼ 1; . . . ; r; j ¼ 1; . . . ; gÞ are the fuzzy sets; r is the number of IF-THEN rules and hj ðj ¼ 1; . . . ; gÞ are the premise variables; xi ðtÞ ¼ ðxi1 ðtÞ; xi2 ðtÞ;    ; xin ðtÞÞT 2 Rn is the state vector of the ith node; f ðxi ðtÞÞ 2 Rn is a continuously differentiable vector function; constant c > 0 represents the coupling strength; Al are inner-coupling matrices; G ¼ ðg ij ÞNN represents the outer-coupling connections; s is a constant time delay in system. By using the standard fuzzy inference method, fuzzy system (1) can be expressed by the following model:

x_ i ðtÞ ¼

r X

ll ðhðtÞÞ½f ðxi ðtÞÞ þ c

N X

g ij Al xj ðt  sÞ; i ¼ 1; 2; . . . ; N;

where hðtÞ ¼ ½h1 ðtÞ; h2 ðtÞ;    ; hg ðtÞ, and

ll ðhðtÞÞ ¼ Pxr l ðhðtÞÞ , in which F lj ðhj ðtÞÞ is the grade of membership of hj ðtÞ in F lj . It is x ðhðtÞÞ l¼1

obvious that the fuzzy weighting functions

ll ðhðtÞÞ P 0;

ð2Þ

j¼1

l¼1

r X

l

ll ðhðtÞÞ satisfy:

ll ðhðtÞÞ ¼ 1

ð3Þ

l¼1

In this following, some elementary situations are introduced, which play an important role in proving the main result. Throughout this paper, the following assumption is needed. Assumption (H1) The outer-coupling configuration matrices of the complex network satisfy

8 > < g ij ¼ g ji P 0; i – j; N P > g ij ; i; j ¼ 1; 2; . . . ; N: : g ii ¼  j¼1;j–i

Next, we give some useful definitions and lemmas. Definition 1. Dynamical network (1) is said to achieve asymptotic synchronization if

x1 ðtÞ ¼ x2 ðtÞ ¼    ¼ xN ðtÞ ¼ sðtÞ;

t ! 1;

where sðtÞ is a solution of an isolate node, satisfying s_ ðtÞ ¼ f ðsðtÞÞ.

ð4Þ

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D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224

Lemma 1 [18]. If G ¼ ðg ij ÞNN satisfies the above conditions, then there exists a unitary matrix, U ¼ ð/1 ; . . . ; /N Þ, such that

GT /k ¼ kk /k ;

k ¼ 1; 2; . . . ; N;

where kk ; k ¼ 1; 2; . . . ; N are the eigenvalues of G. Lemma 2 [19]. Let X and Y be arbitrary n-dimensional real vectors, and K be an n  n positive definite matrix. P 2 Rnn is an arbitrary real matrix. Then, the following matrix inequality holds:

2X T PY 6 X T PK 1 PT X þ Y T KY: Lemma 3 [20]. Let yðtÞ 2 Rn be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices X; M 1 ; M 2 2 Rnn and Z 2 R2n2n , and a scalar s P 0:



Z

t

_ y_ T ðsÞX yðsÞds 6 nT ðtÞ!nðtÞ þ snT ðtÞZnðtÞ

ts

where

"

! :¼

MT1 þ M1

M T1 þ M 2



M T2  M 2

#

 nðtÞ :¼

;

"



yðtÞ yðt  sÞ

;

Z¼

M T1 M T2

# X 1 ½ M 1

M 2 :

Lemma 4. Consider delayed dynamical network (1), the eigenvalues of outer coupling matrix G are denoted by

0 ¼ k1 > k2 P k3 P    P kN ; if the following N  1 of n-dimensional differential equations are asymptotically stable about their zero solution:

_ k ðtÞ ¼ w

r X

ll ðhðtÞÞ½JðtÞwk ðtÞ þ ckk Al wk ðt  sÞ; k ¼ 2; . . . ; N;

ð5Þ

l¼1

where JðtÞ is the Jacobian of f ðxðtÞÞ at sðtÞ, then synchronized states in expression (1) are asymptotically stable. Proof. To investigate the synchronized states of system (1), let

xi ðtÞ ¼ sðtÞ þ ei ðtÞ;

i ¼ 1; 2; . . . ; N:

ð6Þ

Substituting (6) into (2), we have r X

e_ i ðtÞ ¼

ll ðhðtÞÞ½f ðsðtÞ þ ei ðtÞÞ  f ðsðtÞÞ þ c

N X

g ij Al ej ðt  sÞ;

1 6 i 6 N:

ð7Þ

j¼1

l¼1

Since f is continuous differentiable, it is easy to know that the origin of the nonlinear system (5) is an asymptotically stable equilibrium point if it is an asymptotically stable equilibrium point of the following linear time-delay systems r X

e_ i ðtÞ ¼

ll ðhðtÞÞ½JðtÞei ðtÞ þ c

g ij Al ej ðt  sÞ;

j¼1

l¼1

¼

N X

r X

ð8Þ

ll ðhðtÞÞ½JðtÞei ðtÞ þ cAl ðe1 ðt  sÞ; . . . ; eN ðt  sÞÞðg i1 ; . . . ; g iN ÞT :

l¼1

Let eðtÞ ¼ ðe1 ðtÞ; . . . ; eN ðtÞÞ 2 RnN , we can obtain

_ eðtÞ ¼

r X

ll ðhðtÞÞ½JðtÞeðtÞ þ cAl eðt  sÞGT :

ð9Þ

l¼1

According to Lemma 1, there exists a nonsingular matrix, U ¼ ð/1 ; . . . ; /N Þ 2 RNN , such that GT U ¼ UK, with K ¼ diagðk1 ; . . . ; kN Þ. Using the nonsingular transform eðtÞU ¼ wðtÞ ¼ ðw1 ðtÞ; . . . ; wN ðtÞÞ 2 RnN , eðt  sÞU ¼ wðt  sÞ ¼ ðw1 ðt  sÞ; . . . ; wN ðt  sÞÞ 2 RnN , then we have the following matrix equation

_ wðtÞ ¼

r X l¼1

ll ðhðtÞÞ½JðtÞwðtÞ þ cAl wðt  sÞK;

ð10Þ

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D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224

namely,

_ i ðtÞ ¼ w

r X

ll ðhðtÞÞ½JðtÞwi ðtÞ þ cki Al wi ðt  sÞ; i ¼ 1; . . . ; N:

ð11Þ

l¼1

Note that k1 ¼ 0 corresponding to the synchronization of the system states (2), where the state sðtÞ is an orbitally stable solution of the isolate node as assumed above in (2). If the following N  1 pieces of n-dimensional linear multiple time-delays differential equations

_ i ðtÞ ¼ w

r X

ll ðhðtÞÞ½JðtÞwi ðtÞ þ cki Al wi ðt  sÞ; i ¼ 2; . . . ; N:

ð12Þ

l¼1

are asymptotically stable, then eðtÞ will tend to the origin asymptotically, which implies that the synchronized states (4) are asymptotically stable. The proof is thus completed. h Remark 1. For the highly interconnected dynamical units of complex networks, it is difficult to deal with the complex networks by using fuzzy theory. In order to solve this problem, we use Lemma 4 to linearize the complicated system, then some new subsystems will be acquired, which are easy to be analyzed or tested.

3. Main results In this section, we are in the position to present our main results for synchronization criteria for fuzzy complex networks. 3.1. Synchronization criteria for fuzzy complex networks Theorem 1. If there exist positive definite symmetric matrices Pk ; Q k and Sk (2 6 k 6 N) such that the following LMIs hold for all l ¼ 1; 2;    ; r:

"

Xk ¼

P k JðtÞ þ J T ðtÞPk þ Q k þ sSk

kk cP k Al



Q k

# <0

ð13Þ

then the asymptotic synchronization in Definition 1 is achieved. Proof. Select a Lyapunov–Krasovskii function as

V k ðtÞ ¼ wTk ðtÞPk wk ðtÞ þ

Z

t

ts

wTk ðsÞQ k wk ðsÞds þ

Z

t

ts

Z

t

wTk ðsÞSk wk ðsÞ dsdh:

ð14Þ

h

According to Lemma 4, calculating the time derivative of V k ðtÞ along the trajectories of system (5), we can obtain

_ k ðtÞ þ wTk ðtÞQ k wk ðtÞ  wTk ðt  sÞQ k wk ðt  sÞ þ swTk ðtÞSk wk ðtÞ  V_ k ðtÞ ¼ 2wTk ðtÞPk w ¼

r n X

o

ll ðhðtÞÞðwTk ðtÞ; wTk ðt  sÞÞXk ðwTk ðtÞ; wTk ðt  sÞÞT 

l¼1

<

r n X

Z

Z

t

ts

wTk ðsÞSk wk ðsÞ ds

t

ts

wTk ðsÞSk wk ðsÞ ds

o

ll ðhðtÞÞ½ðwTk ðtÞ; wTk ðt  sÞÞXk ðwTk ðtÞ; wTk ðt  sÞÞT 

ð15Þ

l¼1

where Xk is defined as (13). It’s easy to see that system (1) is globally synchronized.

h

Theorem 2. If there exist positive definite symmetric matrices P k ; Q k and Sk (2 6 k 6 N), two arbitrary matrices X 1 ; X 2 with appropriate dimensions, such that the following LMIs hold for all l ¼ 1; 2; . . . ; r:

2

N1 kk cPk Al  X 1 þ X T2

6 6 

Xk ¼ 6 6

4  



sJT ðtÞSk sckk ATl Sk sSk





Q k  X 2  X T2

where

N1 ¼ Pk JðtÞ þ JT ðtÞPk þ Q k þ X T1 þ X 1

3

sX 1 7 sX 2 7 7<0 7

0 5 sSk

ð16Þ

D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224

6219

then the asymptotic synchronization in Definition 1 is achieved. Proof. Select a Lyapunov–Krasovskii function as

V k ðtÞ ¼ wTk ðtÞPk wk ðtÞ þ

Z

t

ts

wTk ðsÞQ k wk ðsÞds þ

Z

Z

t

ts

t

_ Tk ðsÞSk w _ k ðsÞ dsdh: w

ð17Þ

h

According to Lemma 4, calculating the time derivative of V k ðtÞ along the trajectories of system (5), we can obtain

_ k þ wTk ðtÞQ k wk ðtÞ  wTk ðt  sÞQ k wk ðt  sÞ þ sw _ Tk ðtÞSk w _ k ðtÞ  V_ k ðtÞ ¼ 2wTk ðtÞPk w

Z

t

ts

_ Tk ðsÞSk w _ k ðsÞ ds: w

ð18Þ

According to Leibniz–Newton formula, the following equations hold for arbitrary matrices Z k with appropriate dimensions:

 Z 2fTk ðtÞZ k wk ðtÞ  wk ðt  sÞ 

 _ k ðsÞ ds ¼ 0; w

t

ð19Þ

ts

where fTk ðtÞ ¼ ðwTk ðtÞ; wTk ðt  sÞÞ, Z Tk ¼ ðX T1 ; X T2 Þ. It is easy to obtain the following inequality by using Lemma 2:

2fTk ðtÞZ k

Z

Z

t

_ k ðsÞ ds 6 w

ts

t

ts

T _ Tk ðsÞSk w _ k ðsÞ ds þ sfTk ðtÞZ k S1 w k Z k fk ðtÞ;

ð20Þ

where Sk is positive definite matrix. Combing expressions (18)–(20), we obtain

_ k þ wTk ðtÞQ k wk ðtÞ  wTk ðt  sÞQ k wk ðt  sÞ þ sw _ Tk ðtÞSk w _ k ðtÞ  V_ k ðtÞ < 2wTk ðtÞPk w

Z

t

ts

_ Tk ðsÞSk w _ k ðsÞ ds w

T þ 2fTk ðtÞZ k ½wk ðtÞ  wk ðt  sÞ þ sfTk ðtÞZ k S1 k Z k fk ðtÞ;

ð21Þ

(21) is equal to the following inequality

V_ k ðtÞ 6

r n X

o

~ k þ sfT ðtÞZ k S1 Z T fk ðtÞ ; ll ðhðtÞÞfTk ðtÞ½X k k k

ð22Þ

l¼1

~ k is defined as X 2

6

N1 kk cPk Al  X 1 þ X T2

~k ¼ 4  X

Q k  X 2  X T2





3

sJT ðtÞSk 7 sckk ATl Sk 5 < 0; sSk

where

N1 ¼ Pk JðtÞ þ JT ðtÞPk þ Q k þ X T1 þ X 1 : From the well-known Schur complement, we can obtain the result of globally synchronization. h Theorem 3. For given scalars s > 0, the fuzzy system (2) is synchronized if there exist P k ¼ PTk > 0; Q k ¼ Q Tk > 0 (k ¼ 2; . . . ; N), W k ¼ W Tk > 0, and any appropriately dimensioned matrices M k ¼ ½M T1 ; M T2 T such that the LMIS are feasible for l ¼ 1; 2; . . . ; r:

2

H 6 H¼4 

3

sCT1 W k sW k

sCT2



sW k

7 5 < 0;

0

ð23Þ

where

C1 ¼ ½JðtÞ; ckk Al ; " H¼

C2 ¼ ½M1 ; M2 ;

Pk JðtÞ þ JðtÞT Pk þ Q k þ MT1 þ M 1

Pk ckk Al  M T1 þ M 2



Q k  M T2  M 2

# :

Proof. Select a Lyapunov–Krasovskii function as

V k ðtÞ ¼ wTk ðtÞPk wk ðtÞ þ

Z

t

ts

wTk ðsÞQ k wk ðsÞds þ

Z

t

ts

Z h

t

_ Tk ðsÞW k w _ k ðsÞ dsdh: w

ð24Þ

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D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224

According to Lemma 4, calculating the time derivative of V k ðtÞ along the trajectories of system (5), we can obtain

_ k þ wTk ðtÞQ k wk ðtÞ  wTk ðt  sÞQ k wk ðt  sÞ þ sw _ Tk ðtÞW k w _ k ðtÞ  V_ k ðtÞ ¼ 2wTk ðtÞPk w

Z

t

ts

_ Tk ðsÞW k w _ k ðsÞ ds: w

ð25Þ

From Lemma 3, for any M 1 ; M2 with appropriate dimensions yields the following integral inequality:



Z

"

t

_T

T k ðtÞ

_ ds 6 g x ðsÞW k xðsÞ

ts

M T1 þ M1

M T1 þ M 2 M T2



 M2

#

" T k ðtÞ

gk ðtÞ þ sg

M T1 M T2

#

 T W 1 M1 k

 M 2 gk ðtÞ;

ð26Þ

where

gTk ðtÞ ¼ ½wTk ðtÞ; wTk ðt  sÞ: Substituting the proposed equalities, we can get r X ½ll ðhðtÞÞgTk ðtÞðH þ hCT1 W k C1 þ hCT2 W 1 k C2 Þgk ðtÞ:

_ VðtÞ 6

ð27Þ

l¼1

_ By Schur complement, H þ hCT1 W k C1 þ hCT2 W 1 k C2 is equivalent to (23). we can easily get that if ð23Þ < 0, then VðtÞ < q jjxi ðtÞjj2 < 0, which means that the system is asymptotically stable. Then the proof is completed. h Remark 2. From Lemma 3, three delay-dependent conditions are acquired. As is well known, delay-dependent criteria make use of information on the length of delay, and are usually less conservative than delay-independent ones especially when the size of the time delay is small.

4. Numerical simulations In this section, some numerical examples are given to demonstrate the usefulness of the obtained results. Example 1. Consider a lower-dimensional network model with 4 nodes, where each node is a simple three-dimensional stable linear system described as:

3 3 2 x1 x_ 1 ðtÞ 7 6_ 7 6 4 x2 ðtÞ 5 ¼ 4 2x2 5; 2

x_ 3 ðtÞ

ð28Þ

3x3

which is asymptotically stable at sðtÞ ¼ 0, and its Jacobian is J ¼ diagð1; 2; 3Þ. We assume that c ¼ 0:1; s ¼ 0:1, the inner-coupling matrix Al ; ðl ¼ 1; 2Þ, the outer-coupling matrix G and the fuzzy weighting functions ll ðhðtÞÞ are defined as

2

1 0 2

6 Al ¼ 4 0 0

3

7 2 1 5; 1 3

3 1 1 0 0 7 6 6 1 3 1 1 7 7; G¼6 6 0 1 2 1 7 5 4 0 1 1 2 2

l1 ðhðtÞÞ ¼

1 ; 1 þ expð2x1 ðtÞÞ

l2 ¼ 1  l1 :

ð29Þ

The eigenvalues of G are respectively kðGÞ ¼ f0; 1:000; 3:000; 4:000g. By using the MatlabLMI Toolbox, according to Theorem 1, we can obtain the feasible solutions with the following matrix variables as follows:

2

3 49:0866 0:0759 0:0205 6 7 P2 ¼ 4 0:0759 28:4886 0:0221 5; 0:0205 0:0221 2

47:3977

6 S2 ¼ 4 0:0431

0:0899 0:0353 2

38:8890

3 46:4637 0:2185 0:0194 6 7 Q 2 ¼ 4 0:2185 51:9836 0:0243 5;

19:8318

0:0431 0:0899 48:4650

2

0:0194 3

7 0:0353 5; 48:6857

2:3181 0:5545

2

44:9657

6 P3 ¼ 4 0:7176 0:0928

3

6 Q 3 ¼ 4 2:3181

51:4759

7 0:1622 5;

0:5545

0:1622

61:1427

2

45:0708

6 S3 ¼ 4 0:5544

0:0243

54:1969

0:7176 0:0928 28:3159 0:0001

3

7 0:0001 5; 20:9463

0:5544 1:0729 47:9180

1:0729 0:4504

3

7 0:4504 5; 48:0604

D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224

2

39:3422

6 P4 ¼ 4 1:8330

7 0:1629 5;

28:1291

42:8813

6 S4 ¼ 4 1:0798

2

1:0798 2:0499

32:6955

4:2612 2:1706

3

6 Q 4 ¼ 4 4:2612

50:9573

7 0:8568 5;

2:1706

0:8568

64:4220

21:4451

0:8200 0:1629 2

3

1:8330 0:8200

6221

3

7 0:9228 5:

47:3086

47:3821

2:0499 0:9228

By using the MatlabLMI Toolbox, according to Theorem 2, we can obtain the feasible solutions with the following matrix variables as follows:

2

29:6828

6 P2 ¼ 4 0:1517

7 1:1151 5;

19:8165

93:5786

6 S2 ¼ 4 2:4643

2:4643 1:7281

3

3

0:1819

6 Q 3 ¼ 4 0:2536

32:8953

7 0:8417 5;

0:8417

38:2783

25:3379

6 P4 ¼ 4 0:3438

29:6469 6 S4 ¼ 4 0:0900

7 0:9929 5;

19:0975

0:7920

3

0:0900

0:1953

30:0841

7 0:1078 5;

0:0036

31:4187

2

27:0571

6 P3 ¼ 4 0:1558 2

29:9508

30:1482 3:0185

19:4258

0:0141

0:0026

27:7767

3

7 0:0762 5; 34:1001 3

7 0:9290 5; 15:2696

0:0211 0:0141 30:2279

2

0:1719

0:1558 2:1209

6 S3 ¼ 4 0:0211

3

7 0:0026 5; 30:3456

0:6104 0:1308

3

6 Q 4 ¼ 4 0:6104

33:0633

7 1:1209 5;

0:1308

1:1209

40:2301

15:4034

0:1953 0:1078 2

3

0:3438 2:4151

2:4151 0:9929 2

6 Q 2 ¼ 4 0:3098

2:1209 0:9290

0:2536

2

0:3098

36:0800

29:0798 0:1819

29:7604

0:1719 0:0762

7 1:7323 5;

64:6693

1:7281 1:7323 2

2

14:8178

2:2586 1:1151 2

3

0:1517 2:2586

2

3 6:4315 0:2176 6:4025 6 7 X 1 ¼ 4 0:0695 2:1552 2:4779 5; 2:5168 2:7804

0:5357

3

6 7 X 2 ¼ 4 0:3776 1:8623 0:2763 5: 2:4163

0:0371

3:6381

By using the MatlabLMI Toolbox, according to Theorem 3, we can obtain the feasible solutions with the following matrix variables as follows:

2

24:2186

6 P2 ¼ 4 0:1988

0:1988 2:0683 16:4343

2:0683 0:9100

3

7 0:9100 5;

2

27:2713

6 Q 3 ¼ 4 0:3848 0:0192

0:2587

6 Q 2 ¼ 4 0:3686

28:9162

7 0:0995 5;

3

29:8821

7 0:9704 5;

0:9704

33:8632

2

3 22:1478 0:2502 2:2193 6 7 P4 ¼ 4 0:2502 16:0260 0:7707 5; 13:6397

29:3443

2

3 23:9558 0:1330 1:9208 6 7 P 3 ¼ 4 0:1330 16:5424 0:6858 5;

31:7392

0:3848 0:0192

2:2193 0:7707

0:3686

0:2587 0:0995

3 55:6221 0:4662 0:3088 6 7 W 2 ¼ 4 0:4662 43:3512 0:3414 5;

3

29:4168

13:1510

2

0:3088 0:3414

2

1:9208 0:6858 2

54:6652

6 W 3 ¼ 4 0:7697

13:6546

0:7697 1:0406 42:6143

1:0406 0:7957

3

7 0:7957 5; 30:6188

2

3 25:8247 0:7097 0:3313 6 7 Q 4 ¼ 4 0:7097 29:8096 1:2510 5; 0:3313

1:2510

35:4508

6222

D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224

2

53:1825

6 W 4 ¼ 4 1:2302

1:2302 1:9106 41:6044

3

2

7 1:3773 5;

4:7875

6 M 1 ¼ 4 0:2248

29:5727 3 0:0093 0:4557 1:9623 6 7 M 2 ¼ 4 0:0963 2:0155 0:0460 5: 1:9106 1:3773

0:0669

2:0286

3

7 1:5046 2:3151 5; 0:8729

5:7869 1:9918

2

2:5028 0:1644 2:8929 To illustrate the efficiency of our method, we plot the synchronization errors between the states of nodes in Fig. 1. In this simulation, ej ðtÞ ¼ x1j ðtÞ  xij for i ¼ 2; 3; 4; j ¼ 1; 2; 3. From Fig. 1, it is easy to see that the system is synchronized. Example 2. In order to further illustrate the effectiveness of the obtained results, a typical chaotic system will be discussed in this example. Consider the following Lorenz system

8 > < x_ 1 ¼ aðx2  x1 Þ; x_ 2 ¼ ^cx1  x1 x3  x2 ; > :_ x3 ¼ x1 x2  bx3 ;

ð30Þ

e1(t)

0.3 0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2

e2(t)

where parameters a > 0; b > 0; ^c > 0. It is chaotic when a ¼ 10; b ¼ 8=3; ^c ¼ 28, with the initial condition x ¼ ð1; 1; 1Þ. The chaotic behavior is shown in Fig. 2.

4

6

t

8

0.3 0.25 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 0

10

0

2

4

t

e3(t)

2

2

4

6

8

10

t Fig. 1. Synchronization errors in Example 1.

50 40 30

x3

0

0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.3

20 10 0 40

20

0

x

−20 2

−40

−20

−10

0

x1

Fig. 2. The chaotic trajectory of Lorenz system.

10

20

6

8

10

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D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224

Now we consider the 3-dimensional nodes, which is asymptotically synchronized at sðtÞ ¼ ð8:4853; 8:4853; 27ÞT , and its Jacobian is

2

3 10 10 0 6 7 JðtÞ ¼ 4 1 1 8:4853 5: 8:4853 8:4853 8=3 We assume that c ¼ 1:2;

2

8 0

6 A1 ¼ 4 0 0

0

3

s ¼ 0:1, the inner-coupling matrix Al ; ðl ¼ 1; 2; 3Þ and the outer-coupling matrix G are defined as 2

7 2 0 5; 0 5

ð31Þ

3 6 0 0 6 7 A2 ¼ 4 0 3 0 5 ; 0 0 7

2

3 0:2 0:1 0:1 6 7 G ¼ 4 0:1 0:2 0:1 5: 0:1 0:1 0:2

2

3 5 0 0 6 7 A3 ¼ 4 0 3 0 5; 0 0 4

ð32Þ

The membership functions are selected as follows: 2

l1 ¼

1 þ1Þ expððx0:8 Þ 2 2

x2

2

;

2

;

1 þ1Þ 1 1Þ Þ þ expð0:812 Þ þ expððx0:8 Þ expððx0:8 2 2

x2

l2 ¼

expð0:812 Þ 2

x2

1 þ1Þ 1 1Þ expððx0:8 Þ þ expð0:812 Þ þ expððx0:8 Þ 2 2

µ1

−1.5

−1

µ3

µ2

−0.5

0

0.5

1

1.5

2

0.3

0.3

0.2

0.2

0.1

0.1

0

e2(t)

e1(t)

Fig. 3. The membership functions.

−0.1

0 −0.1

−0.2

−0.2

−0.3

−0.3 5

10

15

0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0

5

t

−0.4 0

5

t

e3(t)

−0.4 0

10 t

Fig. 4. Chaotic synchronization errors in Example 2.

15

10

15

6224

D. Gong et al. / Applied Mathematical Modelling 36 (2012) 6215–6224 2

l3 ¼

1 1Þ expððx0:8 Þ 2 2

x2

2

1 þ1Þ 1 1Þ expððx0:8 Þ þ expð0:812 Þ þ expððx0:8 Þ 2 2

;

which are shown in Fig. 3. We plot the synchronization errors between the states of nodes in Fig. 4. In this simulation, ej ðtÞ ¼ x1j ðtÞ  xij for i ¼ 2; 3; j ¼ 1; 2; 3. From Fig. 4, it is easy to see that the fuzzy system is chaotic synchronized. 5. Conclusion A novel linear approximation method which can linearize the complicated system into some subsystems has been proposed to solve the synchronization problem for general fuzzy complex networks. We established some sufficiency conditions for chaotic synchronization, and the simulation results can demonstrate the effectiveness of the obtained result. Compared with the exiting methods, our method can greatly reduce the computational complexity, and can be easily applied in many references. It is believed that the idea and approach developed in this paper could be further generalized to deal with some other problems on chaos control and synchronization for more general complex dynamical networks. Acknowledgements This work was supported by the National Natural Science Foundation of China (50977008, 61034005, and 61104021), the National High Technology Research and Development Program of China (2009AA04Z127), National Basic Research Program of China (2009CB320601). References [1] I. Trestian, S. Ranjan, A. Kuzmanovic, A. Nucci, Googling the internet: profiling internet endpoints via the world wide web, IEEE Trans. Neural Networks 18 (2010) 666–679. [2] Y. Wang, J. Xia, Unified framework for robust estimation of brain networks from fMRI using temporal and spatial correlation analyses, IEEE Trans. Med. Imag. 28 (2009) 1296–1307. [3] F. Nian, X. Wang, Efficient immunization strategies on complex networks, J. Theor. Biol. 264 (2010) 77–83. [4] M. Wang, X. Wang, Z. Liu, A new complex network model with hierarchical and modular structures, Chinese J. Phys. 48 (2010) 805–813. [5] F. Nian, X. Wang, Chaotic synchronization of hybrid state on complex networks, Int. J. Mod. Phys. C 21 (2010) 457–469. [6] T. Zhou, J. L, G. Chen, Y. Tang, Synchronization stability of three chaotic systems with linear coupling, Phys. Lett. A 301 (2002) 231–240. [7] X. Yang, J. Cao, Stochastic synchronization of coupled neural networks with intermittent control, Phys. Lett. A 373 (2009) 3259–3272. [8] Z.Y. Fei, H.J. Gao, W.X. Zheng, New synchronization stability of complex networks with an interval time-varying coupling delay, IEEE Trans. Circuits Syst. II 56 (2009) 499–503. [9] J. Cao, L. Li, Cluster synchronization in an array of hybrid coupled neural networks with delay, Neural Networks 22 (2009) 335–342. [10] C. Li, G.R. Chen, Synchronization in general complex dynamical networks with coupling delays, Physica A 343 (2004) 263–278. [11] J. Cao, G. Chen, P. Li, Global synchronization in an array of delayed neural networks with hybrid coupling, IEEE Trans. Syst. Man Cybern. B, Cybern. 38 (2008) 488–498. [12] Y. Liu, Z. Wang, J. Liang, X. Liu, Synchronization and state estimation for discrete-time complex networks with distributed delays, IEEE Trans. Syst. Man Cybern. B, Cybern. 38 (2008) 1314–1325. [13] Z. Wang, Y. Wang, Y. Liu, Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays, IEEE Trans. Neural Networks 21 (2010) 11–25. [14] M. Bucolo, S. Fazzino, M. Rosa, L. Fortuna, Small-world networks of fuzzy chaotic oscillators, Chaos Solitons Fract. 17 (2003) 557–565. [15] Y. Tang, J. Fang, M. Xia, Synchronization of Takagi–Sugeno fuzzy stochastic complex networks with mixed delays, Intell. Comput. Technol. Automat. 2 (2009) 810–813. [16] Y. Tang, J. Fang, M. Xia, X. Gu, Synchronization of Takagi–Sugeno fuzzy stochastic discrete-time complex networks with mixed time-varying delays, Appl. Math. Modell. 34 (2010) 843–857. [17] H. Li, Delay-distribution-dependent synchronization of T–S fuzzy stochastic complex networks with mixed time delays, in: Control and Decision Conference (CCDC), 2011, pp. 1629–1634 (in Chinese). [18] X. Wang, G. Chen, Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Trans. Circuits Syst. I 49 (2002) 54–62. [19] Y. Liu, Z. Wang, J. Liang, X. Liu, Synchronization and state estimation for discrete-time complex networks with distributed delays, IEEE Trans. Syst. Man Cybern. B, Cybern. 38 (2008) 1314–1325. [20] X.M. Zhang, M. Wu, J.H. She, Y. He, Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica 41 (2005) 1405–1412.