Synchronization for Complex Dynamic Networks with

No. E-PSS-1994-15

Synchronization for Complex Dynamic Networks with State and Coupling Time-Delays Ali Kazemy Department of Electrical Engineering Tafresh University Tafresh, IRAN [email protected] Abstract— This paper is concerned with the problem of synchronization for complex dynamic networks with state and coupling time-delays. Therefore, larger class and more complicated complex dynamic networks can be considered for the synchronization problem. Based on the Lyapunov-Krasovskii functional, a delay-independent criterion is obtained and formulated in the form of linear matrix inequalities (LMIs) to ascertain the synchronization between each nodes of the complex dynamic network. The effectiveness of the proposed method is illustrated using a numerical simulation. Keywords— Complex Dynamic Network; Synchronization; Lyapunov–Krasovskii; Time-Delayed Systems

I.

INTRODUCTION

Many systems in the real-world can be modeled by networks, such as the electrical power grids, neural networks, social network, communication networks, the Internet, and the World Wide Web [1]. Complex networks are made up of interconnected nodes interacting with the others via a topology defined on the network edges [2-5]. These nodes are representing the individuals in the network with different meaning in different situations [6]. Each node of the network can be a nonlinear dynamical system and create a complex dynamic network (CDN), which have been widely applied to model many complex systems. In the past few decades, the study of CDNs has received increasing attention from researchers in various disciplines, such as physics, mathematics, engineering, biology, and sociology [7-11]. Synchronization among all network’s dynamical nodes is one of the most typical collective behavior and basic motions in nature and is one of the most interesting and significant phenomena in CDNs [12-18]. In general, time delays occur commonly in networks because of the network traffic congestion as well as the finite speed of signal transmission over the links. Hence, the synchronization study of CDNs with coupling time delays is quite important [19-22]. Exponential synchronization in CDNs with time-varying delay and hybrid coupling is investigated in [21]. Guaranteed cost synchronization of CDNs is introduced in [23-25]. To the best of the author’s knowledge, complex dynamic networks with time-delay in the states of dynamical nodes have been rarely

studied. In [26], synchronization criterion for Lur’e type complex dynamical networks are considered with time-delay in the states of nodes, but the coupling delay between nodes is not considered. In this paper, synchronization criteria for complex dynamic networks with state and coupling time-delays are presented. Therefore, larger class and more complicated CDNs can be considered for the synchronization problem. Based on the Lyapunov-Krasovskii functional approach, a delayindependent criterion is obtained and formulated in the form of LMIs. The effectiveness of the proposed method is illustrated using some numerical simulations. The organization of this paper is as follows. In Section 2, the problem formulation for the complex dynamic network structure with state and coupling time-delays are presented. In Section 3, based on the Lyapunov–Krasovskii functional and LMI, a criterion is given to ascertain the synchronization between the nodes of complex dynamic network. Section 4 provides simulation results. Finally, section 5 concludes the paper. Notations. Throughout this paper, ¡ n denotes the ndimensional Euclidean space and ¡ n ´m is the set of real n ´ m matrices. P > 0 means that P is a real positive definite and symmetric matrix. I is the identity matrix with appropriate dimensions and diag{W1 , K , Wm } refers to a real matrix with diagonal elements W1 , K, Wm . AT denotes the transpose of the real matrix A . Symmetric terms in a symmetric matrix are denoted by * and the sign Ä is stand for the Kronecker product. II.

PROBLEM STATEMENT AND PRELIMINARIES

Consider a complex dynamic network with N delayed identical nodes with coupling delay: x& i (t ) = Axi (t ) + Ad xi (t - t ) + Bf ( Mxi (t ) ) + Cf ( Dxi (t - t ) ) N

N

+ åG ij( ) Γ1xi (t ) + åG ij( ) Γ 2 xi (t - t c ) , i = 1, 2,K, N j =1

where

1

2

(1)

j =1

xi (t ) = éë x i 1 (t ) x i 2 (t ) L x in (t ) ùû Î ¡ n T

denote the

Synchronization for Complex Dynamic Networks with State and Coupling Time-Delays 30th Power System Conference - 2015 Tehran, Iran

(3) ( A Ä B )( C Ä D ) = ( AC ) Ä ( BD ) (4) A Ä B Ä C = ( A Ä B ) Ä C = A Ä ( B Ä C ) Lemma 2. ((Jensen Inequality), [19]). Assume that the vector function ω : [ 0, r ] ® ¡ n is well defined for the following integrations. For any symmetric matrix R Î ¡ n ´ n and scalar r > 0 , one has

state vector of node i , f : ¡ n ® ¡ n is a nonlinear vectorvalued function, A, Ad , B, M, C, D Î ¡ n ´ n are constant matrices, t > 0 denotes the state delay and t c > 0 is the coupling delay. G ( ) = (G ij( ) ) N ´ N , (q = 1, 2 ) denotes the coupling connections and q

q

Γ1 , Γ 2 Î ¡ n ´ n represent the inner coupling matrices.

Remark 1. To the best of the author knowledge, this model is not considered yet. In this model, the state delay ( t ) could be different to coupling delay ( t c ). Hence, more general complex dynamic networks could be modeled in this way.

r

r ò ωT ( s ) Rω ( s ) ds ³ 0

( ò ω (s )ds ) R ( ò ω (s )ds ) . T

r

r

0

0

Lemma 3. According to [27] and Assumption 2, for any diagonal matrices J > 0, L > 0 , and constant matrices M and D with appropriate dimensions, it follows that

Assumption 1. The coupling connection matrices should satisfy

é -MT JΔ1M MT JΔ 2 ù θT (t ) ê ú θ (t ) -J û * ë é -MT LΔ1M MT LΔ 2 ù + θT (t - t ) ê ú θ (t - t ) ³ 0, * -L û ë

ìG ij(q ) = G ji(q ) ³ 0, i ¹ j , q = 1, 2, ï N í (q ) (q ) ïG ii = - å G ij ³ 0, i , j = 1,K, N , q = 1, 2. j =1, j ¹ i î

(3)

Throughout this paper, we make the following assumption on f ( . ) .

where

Assumption 2. For any x 1 , x 2 Î ¡ there are some constants, s r- , s r+ , which the nonlinear function satisfies

é xi (t ) - x j (t ) ù θ (t ) = ê ú , Δ1 = diag éës 1+s 1- ,K, s n+s n- ùû , f Mx t f Mx t ( ) ( ) ( ) ( ) i j ëê ûú

s r- £

f r (x1 ) - f r (x 2 ) £ s r+ , x1 - x 2

é s + + s 1s + + s n- ù ,K, n Δ 2 = diag ê 1 ú. 2 û ë 2

r = 1, 2,K, n .

For notation simplicity, let

Lemma 4. ([28]). Let

x (t ) = éë x (t ) x (t ) L x T 1

T 2

T N

T

(t )ùû

T

y = éë yT1 ,K, yTN ùû

T

With the help of the matrix Kronecker product, the network (1) can be written as the following form: F ( ( I N Ä M ) x (t ) ) + ( I N Ä C ) F ( ( I N Ä D ) x (t - t ) )

(

)

(

2

and

- x j ) P ( yi - y j ) . T

i

Lemma 5. ([28]). Let H , S be n ´ n any real matrix, T

T

)

T

y = éë yT1 ,K, yTN ùû

xk , y k Î ¡n ,

with

(k

= 1, 2,K , N ) ,

and

f ( . ) and F ( . ) are functions and defined in (2). Then, for any vectors x and y with appropriate dimensions, the following matrix inequality holds:

Definition1. The system (1) is said to be globally synchronized for any initial conditions P i 0 ( s ) , ( i = 1, 2,K , N ) , if the following holds: lim xi (t ) - x j (t ) = 0,

T

x = éë xT1 ,K, xTN ùû ,

e = [1,1,K,1] , EN = eeT , U = NI N - E N , x = éë xT1 ,K, xTN ùû , and

(2)

The following definition and lemmas will be needed in the derivations of our main results.

t ®¥

N

å (x

1£ i < j £ N

+ G ( ) Ä Γ1 x (t ) + G ( ) Ä Γ 2 x (t - t c ) , 1

EN = eeT , and

with x k , y k Î ¡ n , ( k = 1, 2,K , N ) , then

xT ( U Ä P ) y =

x& (t ) = ( I N Ä A ) x (t ) + ( I N Ä Ad ) x (t - t ) + ( I N Ä B ) ´

T

P Î ¡ n ´n ,

U = NI N - E N ,

F ( x (t ) ) = éëf T ( x1 (t ) ) f T ( x 2 (t ) ) L f T ( x N (t ) ) ùû .

e = [1,1,K ,1] ,

xT ( U Ä P ) F ( ( I N Ä S ) y ) =

"i , j = 1, 2,K, N ,

N

å (x

1£ i < j £ N

where . denotes Euclidean norm.

III.

(

- x j ) H f ( Sy i ) - f ( Sy j ) T

i

)

MAIN RESULT

In this chapter, a sufficient condition based on the Lyapunov-Krasovskii functional method will be presented for the synchronization between the nodes of complex dynamic network (2).

Lemma 1. ([8]). Let a Î ¡ and A, B, C, D be matrices with appropriate dimensions. The following properties can be proved (1) (a A ) Ä B = A Ä (a B )

Theorem 1. The system (2) is globally synchronized if there exist positive definite matrices P > 0 , Q > 0 , R > 0 ,

(2) ( A Ä B ) = AT Ä BT T

2


and positive diagonal matrices J1 > 0 , J 2 > 0 , L1 > 0 , L 2 > 0 , such that the following LMIs hold for all 1 £ i < j £ N : éP11 ê* ê ê* ê Ξij = ê* ê* ê ê* ê* ë

P13 P14 0 P16 P 25 0 0 0 -R 0 0 0 * P 44 0 Q 23 * * P 55 0 * * * P 66 * * * *

PAd P 22 * * * * *

éQ11 Q12 Q = êê * Q 22 êë * *

PC ù P 27 úú ú 0 ú 0 ú 0 , Q33 úû

1

N

å éêë2 ( x (t ) - x (t ))

T

i

(5)

j

i

13

j

T

i

where

j

i

22

j

T

i

V1 (t ) = xT (t )( U Ä P ) x (t ) ,

j

i

23

j

T

i

j

i

33

j

- ( xi (t - t ) - x j (t - t ) ) Q11 ( xi (t - t ) - x j (t - t ) )

V 2 (t ) =

T

T

é x (s ) ù ú t ê òt -t êêF ( Mx ( s ) )úú ëê F ( Dx ( s ) ) ûú

V 3 (t ) = ò

t

t -t c

T

x

é U Ä Q11 U Ä Q12 ê * U Ä Q 22 ê * ëê *

( ) - 2 ( x (t - t ) - x (t - t ) ) Q ( f ( Dx (t - t ) ) - f ( Dx (t - t ) ) ) - ( f ( Mx (t - t ) ) - f ( Mx (t - t ) ) ) Q ( f ( Mx (t - t ) ) - f ( Mx (t - t ) ) ) - 2 ( f ( Mx (t - t ) ) - f ( Mx (t - t ) ) ) Q ( f ( Dx (t - t ) ) - f ( Dx (t - t ) ) ) - ( f ( Dx (t - t ) ) - f ( Dx (t - t ) ) ) Q ( f ( Dx (t - t ) ) - f ( Dx (t - t ) ) ) ù úû - 2 ( xi (t - t ) - x j (t - t ) ) Q12 f ( Mxi (t - t ) ) - f ( Mx j (t - t ) ) T

U Ä Q13 ù é x ( s ) ù ê ú , U Ä Q 23 úú êF ( Mx ( s ) ) ú ds U Ä Q33 ûú êê F ( Dx ( s ) ) úú ë û

T

i

j

i

13

j

T

i

j

i

22

j

T

( s )( U Ä R )x ( s )ds ,

i

j

i

23

j

T

where U is defined in Lemma 4.

i

(

N -1

=å

)

+ ( I N Ä C ) F ( ( I N Ä D ) x (t - t ) ) + G Ä Γ1 x (t )

(

)

j

V&3 (t ) = xT (t )( U Ä R ) x (t ) - xT (t - t c )( U Ä R ) x (t - t c )

+ ( I N Ä A d ) x (t - t ) + ( I N Ä B ) F ( ( I N Ä M ) x (t ) )

+ G

i

The third term of (5) becomes

V&1 (t ) = 2xT (t )( U Ä P ) x& (t ) = 2xT (t )( U Ä P ) ëé( I N Ä A ) x (t )

Ä Γ 2 x (t - t c ) ù û

33

(9)

Taking the derivative of V1 (t ) with respect to t yields:

( 2)

j

(1)

N

å éêë( x (t ) - x (t ) ) R ( x (t ) - x (t ) )

i =1 j = i +1

T

i

j

i

j

(10)

- ( xi (t - t c ) - x j (t - t c ) ) R ( xi (t - t c ) - x j (t - t c ) )ù úû T

(6)

According to Lemma 3 and Assumption 2, for any positive diagonal matrices J1 , J 2 , L1 , L 2 , one has

According to Lemma 4, (6) can be written as the following:

3


é -MT J1Δ1M MT J1Δ 2 ù θT (t ) ê ú θ (t ) - J1 û * ë é -MT L1Δ1M MT L1Δ 2 ù + θ (t - t ) ê ú θ (t - t ) ³ 0 * -L1 û ë

,

(11)

G( ) = G( 1

T

é -DT J 2Δ1D DT J 2Δ 2 ù βT (t ) ê ú β (t ) -J 2 û * ë , é -DT L 2Δ1D DT L 2Δ 2 ù β +βT (t - t ) ê t t ³ 0 ( ) ú -L 2 û * ë

2)

é -5 1 1 1 1 1 ù ê 1 -5 1 1 1 1 ú ê ú ê 1 1 -5 1 1 1 ú =ê ú. ê 1 1 1 -5 1 1 ú ê 1 1 1 1 -5 1 ú ê ú êë 1 1 1 1 1 -5úû

By applying Theorem 1 into this example, it is shown that this system can achieve global synchronization with any admissible time delay. For f ( x ) = 0.25 ( x + 1 - x - 1 ) , t = 0.6 , and t c = 0.7 , the states trajectories and the synchronization errors are shown in figures 1 and 2, where e j (t ) = ( xij (t ) - x1 j (t ) ) , i = 2,K,6; j = 1, 2 .

(12)

where é xi (t ) - x j (t ) ù θ (t ) = ê ú, f Mx t f Mx t ( j ( ))ûú i ( )) ëê (

1

é xi (t ) - x j (t ) ù β (t ) = ê ú. êëf ( Dxi (t ) ) - f ( Dx j (t ) )úû

0.5

N -1

N

V& (t ) £ å

å éëξ (t ) Ξ ξ (t )ùû

i =1 j = i +1

T ij

ij

Amplitude

Considering (6)--(12), it is straightforward to show that (13)

ij

0

-0.5

where Ξij is defined in (4) and ξ ij (t ) = éê( xi (t ) - x j (t ) ) , ( xi (t - t ) - x j (t - t ) ) ë T

T

-1

( x (t - t ) - x (t - t )) , (f ( Mx (t ) ) - f ( Mx (t ) ) ) ( f (Mx (t - t ) ) - f ( Mx (t - t ) ) ) , ( f ( Dx (t ) ) - f ( Dx (t ) ) ) i

c

j

c

i

j

T

i

T

j

( f ( Dx (t - t ) ) - f ( Dx

i

(t - t ) ) )

T

j

2

3

4

5

Fig. 1. State trajectories: xi (t ) ; i = 1, 2 .

If

j

T

0.8 0.6 Amplitude

this implies that the system (2) has a global synchronization. This completes the proof. □ Remark 2. Theorem 1 provides delay-independent criterion. If Theorem 1 satisfied for a system, then the system has global synchronization for any state delay ( t ) and coupling delay ( t c ). Obviously this criterion is a conservative condition. IV.

1

Time (sec)

ù . úû Ξij < 0 for "1 £ i < j £ N , then V& (t ) < 0 . From Definition 1, i

0

T

T

0.4 0.2 0 -0.2 0

1

2

3

4

5

3

4

5

Time(sec)

0.6

ILLUSTRATIVE EXAMPLE

0.4

Example. Consider the system (2) with the following parameters [28]:

0.2 0

0ù , 1 úû

Amplitude

é -1 0 ù é -1.2 0.8 ù é1 A=ê Ad = ê B=C=ê ú, ú, ë 0 -1û ë -0.2 -0.2û ë0 1 ù é3.8 2 ù é -3.5 é3 0ù é1 M=ê ú , D = ê 0.1 -1.5ú , Γ1 = ê0 3ú , Γ 2 = ê0 0.1 1.8 ë û ë û ë û ë é0 0 ù é0.5 0 ù Δ1 = ê ú , Δ 2 = ê 0 0.5ú , ë0 0 û ë û

0ù , 1 úû

-0.2 -0.4 -0.6 -0.8 -1 -1.2

0

1

2 Time(sec)

and

Fig. 2. Synchronization errors for the network: e1 (t ) ,e 2 (t )

4


V.

CONCLUSION

This paper is considered the problem of synchronization for complex dynamic networks with state and coupling timedelays. Based on the Lyapunov-Krasovskii functional, a delayindependent criterion was obtained and formulated in the form of linear matrix inequalities (LMIs) to ascertain the synchronization between each nodes of the complex dynamic network. The effectiveness of the proposed method was illustrated using a numerical simulation. REFERENCES

[14]

T. Liu, J. Zhao, and D. J. Hill, "Exponential synchronization of complex delayed dynamical networks with switching topology," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 57, 2010, pp. 2967-2980.

[15]

B. Wang and Z.-H. Guan, "Chaos synchronization in general complex dynamical networks with coupling delays," Nonlinear Analysis: Real World Applications, vol. 11, 2010, pp. 1925-1932.

[16]

D. Yue and H. Li, "Synchronization stability of continuous/discrete complex dynamical networks with interval time-varying delays," Neurocomputing, vol. 73, 2010, pp. 809-819.

[17]

Y. Zhang, D.-W. Gu, and S. Xu, "Global exponential adaptive synchronization of complex dynamical networks with neutral-type neural network nodes and stochastic disturbances," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, 2013, pp. 2709-2718.

[1]

R. Lu, W. Yu, J. Lu, and A. Xue, "Synchronization on complex networks of networks," IEEE Transactions on Neural Networks and Learning Systems, vol. 25, 2014, pp. 2110-2118.

[2]

L. Cui, S. Kumara, and R. Albert, "Complex networks: An engineering view," IEEE Circuits and Systems Magazine, vol. 10, 2010, pp. 10-25.

[18]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, "Synchronization in complex networks," Physics Reports, vol. 469, 2008, pp. 93-153.

[3]

S. Manaffam and A. Seyedi, "Synchronization Probability in Large Complex Networks," IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 60, 2013, pp. 697-701.

[19]

M. Fang, "Synchronization for complex dynamical networks with time delay and discrete-time information," Applied Mathematics and Computation, vol. 258, 2015, pp. 1-11.

[4]

M. E. Newman, "The structure and function of complex networks," SIAM review, vol. 45, 2003, pp. 167-256.

[20]

[5]

W. Yu, P. DeLellis, G. Chen, M. D. Bernardo, and J. Kurths, "Distributed adaptive control of synchronization in complex networks," IEEE Transactions on Automatic Control, vol. 57, 2012, pp. 2153-2158.

Y. Sun, W. Li, and J. Ruan, "Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation," Communications in Nonlinear Science and Numerical Simulation, vol. 18, 2013, pp. 989-998.

[21]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, "Complex networks: Structure and dynamics," Physics reports, vol. 424, 2006, pp. 175-308.

J. Wang, H. Zhang, Z. Wang, and B. Wang, "Local exponential synchronization in complex dynamical networks with time-varying delay and hybrid coupling," Applied Mathematics and Computation, vol. 225, 2013, pp. 16-32.

[22]

C. Cai, Z. Wang, J. Xu, X. Liu, and F. E. Alsaadi, "An Integrated Approach to Global Synchronization and State Estimation for Nonlinear Singularly Perturbed Complex Networks," IEEE Transactions on Cybernetics, vol. 45, 2014, pp. 1597-1609.

L. Zhang, Y. Wang, Y. Huang, and X. Chen, "Delay-dependent synchronization for non-diffusively coupled time-varying complex dynamical networks," Applied Mathematics and Computation, vol. 259, 2015, pp. 510-522.

[23]

T. H. Lee, D. Ji, J. H. Park, and H. Y. Jung, "Decentralized guaranteed cost dynamic control for synchronization of a complex dynamical network with randomly switching topology," Applied Mathematics and Computation, vol. 219, 2012, pp. 996-1010.

[24]

T. H. Lee, J. H. Park, D. Ji, O. Kwon, and S.-M. Lee, "Guaranteed cost synchronization of a complex dynamical network via dynamic feedback control," Applied Mathematics and Computation, vol. 218, 2012, pp. 6469-6481.

[25]

L. Yi‐ping and Z. Bi‐feng, "Guaranteed Cost Synchronization of Complex Network Systems with Delay," Asian Journal of Control, vol. 17, 2015, pp. 1274–1284.

[26]

D. Ji, J. H. Park, W. Yoo, S. Won, and S. Lee, "Synchronization criterion for Lur'e type complex dynamical networks with timevarying delay," Physics Letters A, vol. 374, 2010, pp. 1218-1227.

[27]

Y. Liu, Z. Wang, J. Liang, and X. Liu, "Synchronization and state estimation for discrete-time complex networks with distributed delays," IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 38, 2008, pp. 1314-1325.

[28]

B. Huang, H. Zhang, D. Gong, and J. Wang, "Synchronization analysis for static neural networks with hybrid couplings and time delays," Neurocomputing, vol. 148, 2015, pp. 288-293.

[6]

[7]

[8]

H. Li, Z. Ning, Y. Yin, and Y. Tang, "Synchronization and state estimation for singular complex dynamical networks with timevarying delays," Communications in Nonlinear Science and Numerical Simulation, vol. 18, 2013, pp. 194-208.

[9]

J. Lü and G. Chen, "A time-varying complex dynamical network model and its controlled synchronization criteria," IEEE Transactions on Automatic Control, vol. 50, 2005, pp. 841-846.

[10]

N. Mahdavi, M. B. Menhaj, J. Kurths, and J. Lu, "Fuzzy complex dynamical networks and its synchronization," IEEE Transactions on Cybernetics, vol. 43, 2013, pp. 648-659.

[11]

B. Shen, Z. Wang, and X. Liu, "Bounded H∞ synchronization and state estimation for discrete time-varying stochastic complex for discrete time-varying stochastic complex networks over a finite horizon," IEEE Transactions on Neural Networks, vol. 22, 2011, pp. 145-157.

[12]

[13]

C. Li, W. Sun, and J. Kurths, "Synchronization of complex dynamical networks with time delays," Physica A: Statistical Mechanics and its Applications, vol. 361, 2006, pp. 24-34. H. Li and D. Yue, "Synchronization stability of general complex dynamical networks with time-varying delays: A piecewise analysis method," Journal of computational and applied mathematics, vol. 232, 2009, pp. 149-158.

5