Synchronization for complex dynamical networks with time delay and discrete-time information

Applied Mathematics and Computation 258 (2015) 1–11

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Synchronization for complex dynamical networks with time delay and discrete-time information Mei Fang Business and Trade School, Ningbo City College of Vocational Technology, Ningbo 315000, China

a r t i c l e

i n f o

Keywords: Complex dynamical networks Discrete-time communication Synchronization Time delay

a b s t r a c t The synchronization problem is studied for complex dynamical networks (CDNs) with time delay based on discrete-time communication pattern. A new Lyapunov functional is proposed, which is positive definite at communication instants but not necessarily positive definite inside the communication intervals, and can make full use of the available information about the discrete-time communication pattern. Based on the Lyapunov functional, an exponential synchronization criterion is proposed to ensure the synchronization of the considered CDNs. The effectiveness of the developed results are shown by the numerical simulation. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction In the past decades, complex dynamical networks (CDNs) which consist of interacting dynamical entities with an interplay between dynamical states and interaction patterns has been intensively studied by many researchers and numerous results have been reported due to the fact that many systems in nature can be modelled by CNDs, for example, power grids, the Internet, electrical power grids, food webs and the World Wide Web. As a result, a great number of important research results have been published on this topic [2,1,3–14]. For example, in [8], the synchronization problem has been studied for discrete-time delayed complex networks with stochastic nonlinearities, multiple stochastic disturbances, and mixed time delays, and by utilizing the properties of Kronecker product, the free-weighting matrix method, and the stochastic techniques, the synchronization stability criteria have been proposed, which can be readily checked by using standard numerical software. In [9], the synchronization problem has been considered for discrete-time stochastic complex networks over a finite horizon, and based on a time-varying real-valued function and the Kronecker product, the criteria have been established that ensure the bounded H1 synchronization. The distributed adaptive control of synchronization in complex networks has been considered in [10], where an effective distributed adaptive strategy to tune the coupling weights of a network has been designed based on local information of node dynamics. On the other hand, much attention has been drawn to the study of sampled-data control systems because modern control systems usually employ digital technology for controller implementation [15–22]. For example, in [15] the synchronization problem of a complex dynamical network with coupling time-varying delays via delayed sampled-data controller has been investigated and a stability condition has been proposed to find the controller which achieves the synchronization of a complex dynamical network with coupling time-varying delay. In [17], the sampled-data control of linear systems under uncertain sampling with the known upper bound on the sampling intervals has been considered and for the first time, the time-dependent Lyapunov functionals has been proposed to guarantee the stability of systems under the time-varying E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2015.01.106 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.

2

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

sampling. In [21], a novel approach to assess the stability of continuous linear systems with sampled-data inputs has been given, which had provided easy tractable stability conditions for the continuous-time model, and sufficient conditions for asymptotic and exponential stability have been provided dealing with synchronous and asynchronous samplings and uncertain systems. Recently, the discrete-time communication pattern has been proposed for continuous-time CDNs based on the concept of sampled-data control systems in [23], where a piecewise Lyapunov–Krasovskii functional has been employed to govern the characteristics of the discrete communication instants, and a synchronization criterion has been derived and an upper bound of the communication intervals has been obtained. However, the time delay has not been considered in [23]. It is well-known that time-delay is frequently encountered in a variety of engineering systems [24–36], and a relatively small time-delay may lead to non-synchronization or significantly deteriorated performances for the CDNs. Therefore, it is necessary to study the synchronization for CDNs with time delay based on discrete-time communication pattern, which is the motivation for this paper. The synchronization problem of CDNs with time delay is investigated based on discrete-time communication pattern. A new time-dependent Lyapunov functional is proposed, which make full use of the available information about the discretetime communication pattern. Based on the Lyapunov functional, an exponential synchronization criterion is proposed to ensure the synchronization of the considered CDNs. A simulation example is given to show the efficiency of the proposed method. 2. Preliminaries Consider a CDN consisting of N identical nodes coupled in the discrete-time way:

x_ i ðtÞ ¼ Axi ðtÞ þ Bf ðxi ðtÞÞ þ Cf ðxi ðt  dÞÞ þ c

N X

g ij Cxj ðt k Þ; t 2 ½tk ; tkþ1 Þ;

i ¼ 1; 2; . . . ; N;

ð1Þ

j¼1

where xi ðtÞ ¼ ½ xi1 ðtÞ xi2 ðtÞ . . . xin ðtÞ T 2 Rn is the state vector of node i; f : Rn ! Rn is a nonlinear vector-valued function, and c is a constant denoting the coupling strength. A; B; C; C 2 Rnn are constant matrices and C denotes the inner-coupling matrix. G ¼ ðg ij ÞNN is the coupling configuration matrix. If there is a connection from node j and node i (i – j), then g ij > 0, otherwise, g ij ¼ 0 (i – j). The diagonal elements of matrix G are defined by

g ii ¼ 

N X

g ij :

ð2Þ

j¼1;j–i

On the other hand, tk ; k ¼ 0; 1; 2; . . ., are the communication instants satisfying 0 ¼ t 0 < t1 <    < t k <    < limk!þ1 tk ¼ þ1. In this paper, the distance between any two consecutive communication instants is assumed to be bounded. Specifically, it is assumed that t kþ1  t k ¼ hk 6 h for all k P 0, where h > 0 represents the largest communication interval. Let 3 2

3 f ðx1 ðtÞÞ x1 ðtÞ 7 6 6 x2 ðtÞ 7 6 f ðx2 ðtÞÞ 7 7 6 7 7 2 RNn ; gðxðtÞÞ ¼ 6 xðtÞ ¼ 6 7 2 RNn ; 6 6 .. 7 .. 7 6 4 . 5 . 5 4  xN ðtÞ f ðxN ðtÞÞ 2

then CDN (1) can be written as

_ xðtÞ ¼ ðIN  AÞxðtÞ þ ðIN  BÞgðxðtÞÞ þ ðIN  CÞ  gðxðt  dÞÞ þ cðG  CÞxðt k Þ;

t 2 ½t k ; tkþ1 Þ:

ð3Þ

Throughout this paper, we make the following assumption on f ðÞ. Assumption 1. For 8u; v 2 Rn , the nonlinear function f ðÞ is continuous and assumed to satisfy the following sector-bounded nonlinearity condition: T ðf ðuÞ  f ðv Þ  F 1 ðu  v ÞÞ ðf ðuÞ  f ðv Þ  F 2 ðu  v ÞÞ 6 0;

ð4Þ

where F 1 and F 2 are known constant real matrices. Remark 1. It is noted that such type of description of nonlinear function is known as the sector-like condition, which is originated from [37] and more general than the commonly used Lipschitz conditions. It is not difficult to prove that for any nonlinear function f ðÞ satisfying (4), there exists a scalar q > 0 such that

kf ðuÞ  f ðv Þk2 6 qku  v k2 :

ð5Þ

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

3

The following definition and lemmas will be needed in the derivations of our main results. Definition 1 [38]. CDN (1) is said to be globally asymptotically synchronized if the following holds:

lim jjxi ðtÞ  xj ðtÞjj ¼ 0;

i; j ¼ 1; 2; . . . ; N:

t!þ1

ð6Þ

Furthermore, CDN (1) is said to be globally exponentially synchronized if there exist constants a > 0 and b > 0, such that for any initial values and a sufficient large T > 0, the following inequality holds,

jjxi ðtÞ  xj ðtÞjj 6 beat ;

i; j ¼ 1; 2; . . . ; N;

ð7Þ

for all t > T. Lemma 1 [23]. Let G be an N  N matrix, in which each row has the same sum. Then,

e ¼ MG e U M; e MG

ð8Þ

where

2

1 1

0



1

0 .. .



0

1

61 6 6 e ¼ 6 ... M 6 6 41

0



1

0



2

0

0

0

0

0 

6 1 0 0  6 6 6 0 1 0    6 U¼6 .. .. 6 6 0 . 0 . 6 .. 6 .. 4 . . 1 0



0

0

0

0

3

7 7 7 7 2 RðN1ÞN ; 7 7 0 5 0 .. .

1 3

7 7 7 7 7 7 2 RNðN1Þ : 7 0 7 7 7 0 5 0 .. .

1

Denote

3 x1 ðtÞ  x2 ðtÞ 6 x1 ðtÞ  x3 ðtÞ 7 7 6 7; yðtÞ ¼ 6 . 7 6 . 5 4 . 2

x1 ðtÞ  xN ðtÞ

3 f ðx1 ðtÞÞ  f ðx2 ðtÞÞ 7 6 6 f ðx1 ðtÞÞ  f ðx3 ðtÞÞ 7 7 6 f ðyðtÞÞ ¼ 6 7; .. 7 6 . 5 4 2

f ðx1 ðtÞÞ  f ðxN ðtÞÞ

e  In . It can be seen from (5) that which imply yðtÞ ¼ MxðtÞ and f ðyðtÞÞ ¼ MgðxðtÞÞ, where M ¼ M

kf ðyðtÞÞk2 6 qkyðtÞk2 :

ð9Þ

Based on Lemma 1, it can be found that

^ ^ ðyðt  dÞÞ þ Dyðt ^ ðyðtÞÞ þ Cf ^ k Þ; _ þ Bf yðtÞ ¼ AyðtÞ

t 2 ½t k ; t kþ1 Þ;

ð10Þ

where

^ ¼ IN1  A; A ^ ¼ IN1  C; C

^ ¼ IN1  B; B ^ ¼ cMðGU  CÞ: D

Lemma 2 ((Jensen Inequality) [39]). For any matrix W > 0, scalars c1 and c2 satisfying c2 > c1 , and a vector function x : ½ c1 ; c2  ! Rn , if the following integrations concerned are well defined, then

ðc2  c1 Þ

Z

c2

c1

xðaÞT W xðaÞ da P

"Z

c2

c1

#T

xðaÞ da W

"Z

c2

c1

#

xðaÞ da :

ð11Þ

4

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

Lemma 3. Consider system (10). Then there exists two scalars h1 and h2 such that

jjyðtÞjj2 6 h1 jjyðt k Þjj2 þ h2

Z

tk

jjyðaÞjj2 da;

t k 6 t < tkþ1 :

ð12Þ

t k d

Proof. The proof of Lemma 3 is similar to the one proposed in Lemma 2 of [22]. This completes the proof. This completes the proof. h In this paper, we aim to establish a sufficient condition for the globally exponential synchronization between the nodes of CDN (1). Based on the time-dependent Lyapunov functional method, we construct a new Lyapunov functional to derive the criterion to ensure the addressed CDN to be globally exponentially synchronized. 3. Main results In this section, we will propose a novel globally exponential synchronization criterion for CDN (1).

2

X 1 þX T1 6 2

X 1 þ X 2

6 H¼6  4 

X 2  X T2 þ

3

X3

7 7 ; X4 7 5

X 1 þX T1 2

X 5 þX T5 2



 T h1 ðtÞ ¼ yðtÞT f ðyðtÞÞT yðt  dÞT ; h iT Rt T yðsÞT ds yðtk ÞT ; h2 ðtÞ ¼ f ðyðt  dÞÞ tk 

T

xðtÞ ¼ h1 ðtÞT h2 ðtÞT ; m ¼ nðN  1Þ; 

el ¼ 0mðl1Þm X1 ¼ I 

F T1 F 2

þ 2

Im

 0mð6lÞm ;

F T2 F 1

;

l ¼ 1; 2; 3; 4; 5; 6;

X2 ¼ I 

F T1 þ F T2 ; 2

Then system (10) can be rewritten as

_ yðtÞ ¼ NxðtÞ; t 2 ½tk ; tkþ1 Þ;

ð13Þ

where

^ 1 þ Be ^ 4 þ De ^ 2 þ Ce ^ 6: N ¼ Ae

Theorem 1. Given scalars a and h, CDN (1) is globally exponentially synchronized with convergence rate a, if there exist matrices     Q1 Q2 R1 R2 P > 0; Q ¼ > 0; R ¼ > 0; Z > 0; X 1 ; X 2 ; X 3 ; X 4 ; X 5 ; W, diagonal matrices K1 > 0; K2 > 0; K3 > 0 and K4 > 0  Q3  R3 such that

N þ Pl < 0;

l ¼ 1; 2;

ð14Þ

2

h N þ P1 þ H1 ðhÞ þ NT SN < 0; 4 pffiffiffi 2 hW N þ P2 þ H2 ðhÞ þ H3 6 4  e2ah R1 

ð15Þ hG 0

3 7 5 < 0;

2e2ah S



where

T    T       e1  e6 e1 e3 e3 e1  e6 T  e2ad þW  e2ah Q Q R2 e6 e2 e2 e4 e4 e5 e5 2 3T 2 3 e1 e1     e  e e1  e6 T T 1 6 6 7 6 7 2 T  4 e6 5 H4 e6 5 þ  e2ah eT6 R2 W  Ge5  eT5 GT  e2ad ðe1  e3 ÞT Zðe1  e3 Þ þ d NT Z N; e5 e5 e5 e5

N ¼ eT1 PN þ NT Pe1 þ 2aeT1 Pe1 þ



e1

ð16Þ

5

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

P1 ¼ eT1 ðK1  IÞX1 e1  eT2 ðK1  IÞe2 þ eT1 ðK1  IÞX2 e2 þ eT2 XT2 ðK1  IÞe1  eT3 ðK2  IÞX1 e3  eT4 ðK2  IÞe4 þ eT3 ðK2  IÞX2 e4 þ eT4 XT2 ðK2  IÞe3 ;

P2 ¼ eT1 ðK3  IÞX1 e1  eT2 ðK3  IÞe2 þ eT1 ðK3  IÞX2 e2 þ eT2 XT2 ðK3  IÞe1  eT3 ðK4  IÞX1 e3  eT4 ðK4  IÞe4 þ eT3 ðK4  IÞX2 e4 þ eT4 XT2 ðK4  IÞe3 ; 2 3T 2 3 2 3T 2 3 2 3T 2 3 2 3T 2 3 N  e1 N N e1 e1 e1  N R R 1 2 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 H1 ðhÞ ¼ h4 e1 5 4 e1 5 þ 2ah4 e6 5 H4 e6 5 þ h4 e6 5 H4 0 5 þ h4 0 5 H4 e6 5;  R3 e1 e1 e6 e6 e5 e5 e5 e5 and

H2 ðhÞ ¼ hGe1 þ heT1 GT ; 2

2ah T e6 R3 e6

H3 ¼ he

þ

h T N SN: 4

Proof. Consider the following Lyapunov functional for system (13):

VðtÞ ¼

5 X V l ðtÞ; t 2 ½t k ; t kþ1 Þ;

ð17Þ

l¼1

where

V 1 ðtÞ ¼ e2at yðtÞT PyðtÞ; " #T " # Z t yðsÞ yðsÞ ds; V 2 ðtÞ ¼ e2as Q f ðyðsÞÞ f ðyðsÞÞ td Z 0Z t _ T Z yðsÞ _ ds dh; V 3 ðtÞ ¼ d e2as yðsÞ d

tþh

V 4 ðtÞ ¼ ðt kþ1  tÞ

Z

t

tk

2 _ 3T 2 _ 3 yðsÞ yðsÞ 6 7 6 7 e2as 4 yðsÞ 5 R4 yðsÞ 5 ds; yðtk Þ

yðt k Þ 3T 2 3 yðtÞ yðtÞ 6 7 6 7 yðt k Þ 7 H6 yðt k Þ 7; V 5 ðtÞ ¼ ðt kþ1  tÞe2at 6 4 5 4 5 Rt Rt yðsÞ ds yðsÞ ds tk tk Z 0 Z t _ T SyðsÞ _ ds dh: e2as yðsÞ V 6 ðtÞ ¼ ðt kþ1  tÞ 2

tþt k

tþh

Calculating the time derivative of VðtÞ along the trajectories of (13), we obtain

_ þ 2ae2at yðtÞT PyðtÞ ¼ 2e2at xðtÞT eT1 PNxðtÞ þ 2ae2at xðtÞT eT1 Pe1 xðtÞ; V_ 1 ðtÞ ¼ 2e2at yðtÞT P yðtÞ T    T   yðtÞ yðt  dÞ yðt  dÞ  e2aðtdÞ Q Q f ðyðtÞÞ f ðyðtÞÞ f ðyðt  dÞÞ f ðyðt  dÞÞ  T    T   e1 e e1 e Q xðtÞ  e2at e2ad xðtÞT 3 Q 3 xðtÞ; ¼ e2at xðtÞT e2 e2 e4 e4 Z t Z 2 _ T Z yðtÞ _ d _ T Z yðsÞ _ ds 6 d2 e2at xðtÞT NT Z NxðtÞ  de2at e2as yðsÞ V_ 3 ðtÞ ¼ d e2at yðtÞ V_ 2 ðtÞ ¼e2at



ð18Þ

yðtÞ

td

t

_ T Z yðsÞ _ ds e2ad yðsÞ

td 2

6 e2at e2ad xðtÞT ðe1  e3 ÞT Zðe1  e3 ÞxðtÞ þ d e2at xðtÞT NT Z NxðtÞ; where Lemma 1 is adopted, and

2

ð19Þ

3T 2 3 2 3T 2 3 _ _ _ _ yðtÞ yðtÞ yðsÞ yðsÞ Z t 6 7 6 7 6 7 6 7 V_ 4 ðtÞ ¼ ðt kþ1  tÞe2at 4 yðtÞ 5 R4 yðtÞ 5  e2as 4 yðsÞ 5 R4 yðsÞ 5 ds tk yðt k Þ yðt k Þ yðtk Þ yðtk Þ

ð20Þ

6

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

2

N

3T

6 7 6 ðtkþ1  tÞe2at xðtÞT 4 e1 5 2

e6

N

 2e2at e2ah xðtÞT 2 6 V_ 5 ðtÞ ¼  e2at 4

Rt

e1  e6

R1  T

e5

3T 2

yðt k Þ

7 6 5 H4

yðsÞds 2



e6

yðtÞ

tk

R1 

3T

6 7 ¼ ðt kþ1  tÞe2at xðtÞT 4 e1 5 



Rt

2 3 2 3T 2 3 _ _ yðsÞ  N   yðsÞ Z t R2 6 7 6 7 R1 R2 6 7 e2ah 4 yðsÞ 5 4 e1 5xðtÞ  e2at 4 yðsÞ 5 ds R3  R3 tk e6 yðtk Þ yðt k Þ 2 3  N  T   Z t _ _ R2 6 7 yðsÞ yðsÞ ds e2ah R1 4 e1 5xðtÞ  e2at R3 yðsÞ yðsÞ tk e6

R2 e6 xðtÞ  ðt  t k Þe2at e2ah xðtÞT eT6 R3 e6 xðtÞ;

yðtÞ

3

2

yðt k Þ

7 6 5 þ 2aðt kþ1  tÞe2at 4

yðsÞds 3T 2

tk

yðtÞ yðtk Þ

Rt

yðtÞ

3T 2

yðt k Þ

7 6 5 H4

tk

3

yðsÞds

Rt

ð21Þ

yðtÞ

3

yðtk Þ

7 5

tk

yðsÞds

_ yðtÞ 7 6 7 þ 2ðtkþ1  tÞe 4 5 H4 0 5 Rt yðsÞds yðtÞ tk 2 3T 2 3 2 3T 2 3 2 3T 2 3 e1 e1 e1 e1 e1 N 6 7 6 7 6 7 6 7 6 7 6 7 ¼  e2at xðtÞT 4 e6 5 H4 e6 5xðtÞ þ 2aðt kþ1  tÞe2at xðtÞT 4 e6 5 H4 e6 5xðtÞ þ 2ðtkþ1  tÞe2at xðtÞT 4 e6 5 H4 0 5xðtÞ; 2at 6

e5

e5

e5

e5

e5

e1 ð22Þ

_ T SyðtÞ _  V_ 6 ðtÞ ¼ ðt kþ1  tÞðt  tk Þe2at yðtÞ

Z

Z

0

tþt k

6 6 ¼

ððt kþ1  tÞ þ ðt  t k ÞÞ2 2at _ T SyðtÞ _  e yðtÞ 4

Z

t

_ T SyðsÞ _ ds dh e2as yðsÞ

tþh

Z

0

Z

_ T SyðsÞ _ ds dh e2as yðsÞ

tþh

tþt k

hððt kþ1  tÞ þ ðt  t k ÞÞ _ T SyðtÞ _  e2at yðtÞ 4

t

Z

0

t

_ T SyðsÞ _ ds dh e2ah yðsÞ

tþh

tþt k

h h ðt kþ1  tÞe2at xðtÞT NT SNxðtÞ þ ðt  tk Þe2at xðtÞT NT SNxðtÞ  e2at 4 4

Z

0

tþt k

Z

t

_ T SyðsÞ _ ds d:h e2ah yðsÞ

ð23Þ

tþh

On the other hand, based on the free-weighting matrix method [40,41], for any matrices W and G with appropriate dimensions, the following equations hold:

"" 0 ¼ 2xðtÞT W

# Z  "  #   # Z t t yðtÞ  yðtk Þ _ _ e1  e6 yðsÞ yðsÞ Rt  ds ¼ 2xðtÞT W ds ; xðtÞ  yðsÞ ds e5 yðsÞ yðsÞ tk tk tk

ð24Þ

and

" T

0 ¼ 2xðtÞ G  ðt  t k ÞyðtÞ 

Z

t

yðsÞ ds 

Z

tk

Z

0

#

t

_ ds dh yðsÞ

tþh

tþtk

¼ 2ðt  tk ÞxðtÞT Ge1 xðtÞ  2xðtÞT Ge5 xðtÞ  2xðtÞT G

Z

Z

0

tþt k

t

_ ds dh: yðsÞ

ð25Þ

tþh

Moreover, based on the fact that for any matrix G > 0; 2yT x 6 yT Gy þ xT G1 x, it is easy to obtain that

Z t

2xðtÞT W

_ yðsÞ

tk

yðsÞ

0

Z



ds 6 e2ah

Z t tk

_ yðsÞ

T

yðsÞ

 R1

_ yðsÞ



yðsÞ

T ds þ ðt  tk Þe2ah xðtÞT WR1 1 W xðtÞ;

ð26Þ

and

2xðtÞT G

Z

tþt k

t

_ ds dh 6 yðsÞ

tþh

ðt  t k Þ2 2ah e xðtÞT GS1 GT xðtÞ þ 2

Z

0

tþtk

Z

t

_ T SyðsÞ _ ds dh: e2ah yðsÞ

ð27Þ

tþh

Combining (24) and (26) leads to

0 6 d1 ðtÞ , 2xðtÞT W



e1  e6 e5



xðtÞ þ e2ah

Z t tk

_ yðsÞ yðsÞ

T

R1



_ yðsÞ yðsÞ



T ds þ ðt  t k Þe2ah xðtÞT WR1 1 W xðtÞ;

ð28Þ

7

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

and combining (25) and (27) leads to

0 6 d2 ðtÞ , 2ðt  tk ÞxðtÞT Ge1 xðtÞ  2xðtÞT Ge5 xðtÞ þ

hðt  tk Þ 2ah e xðtÞT GS1 GT xðtÞ þ 2

Z

0

Z

tþt k

t

_ T SyðsÞ _ ds dh: e2ah yðsÞ

tþh

ð29Þ Furthermore, it is noted that based on (4), the following inequality holds: T

ðf ðx1 ðtÞÞ  f ðx2 ðtÞÞ  F 1 ðx1 ðtÞ  x2 ðtÞÞÞ  ðf ðx1 ðtÞÞ  f ðx2 ðtÞÞ  F 2 ðx1 ðtÞ  x2 ðtÞÞÞ 6 0; which implies for any scalar



v1 > 0, the following inequality holds:

T 

x ðtÞ  x ðtÞ

ð30Þ

x ðtÞ  x ðtÞ



2 1 2 P 0; v1  1 C  f ðx1 ðtÞÞ  f ðx2 ðtÞÞ f ðx1 ðtÞÞ  f ðx2 ðtÞÞ

ð31Þ

where

"

C¼



F T1 F 2 þF T2 F 1 2

F T1 þF T2 2



# :

I

Similarly, it can be seen that for l ¼ 2; 3; . . . ; N  1,



x ðtÞ  x

ðtÞ

T 

x ðtÞ  x



ðtÞ

1 lþ1 lþ1 P 0: vl  1 C  f ðx1 ðtÞÞ  f ðxlþ1 ðtÞÞ f ðx1 ðtÞÞ  f ðxlþ1 ðtÞÞ

ð32Þ

Thus, we can conclude from (31) and (32) that for any diagonal matrix K1 ¼ diagfv1 ; v2 ; . . . ; vN1 g > 0, the following inequality holds:



T 

yðtÞ f ðyðtÞÞ

ðK1  IÞX1

ðK1  IÞX2



K1  I



yðtÞ f ðyðtÞÞ



P 0:

ð33Þ

Rewriting (33), we have

0 6 d3 ðtÞ ,  xðtÞT eT1 ðK1  IÞX1 e1 xðtÞ  xðtÞT eT2 ðK1  IÞe2 xðtÞ þ 2xðtÞT eT1 ðK1  IÞX2 e2 xðtÞ:

ð34Þ

Similarly, we also have that any diagonal matrices K2 ; K3 and K4 with appropriate dimensions, the following inequalities hold:

0 6 d4 ðtÞ ,  xðtÞT eT3 ðK2  IÞX1 e3 xðtÞ  xðtÞT eT4 ðK2  IÞe4 xðtÞ þ 2xðtÞT eT3 ðK2  IÞX2 e4 xðtÞ;

ð35Þ

0 6 d5 ðtÞ ,  xðtÞT eT1 ðK3  IÞX1 e1 xðtÞ  xðtÞT eT2 ðK3  IÞe2 xðtÞ þ 2xðtÞT eT1 ðK3  IÞX2 e2 xðtÞ;

ð36Þ

0 6 d6 ðtÞ ,  xðtÞT eT3 ðK4  IÞX1 e3 xðtÞ  xðtÞT eT4 ðK4  IÞe4 xðtÞ þ 2xðtÞT eT3 ðK4  IÞX2 e4 xðtÞ:

ð37Þ

Then, we obtain from (18)–(22) (34)–(37) that for t 2 ½tk ; t kþ1 Þ,

t kþ1  t t  tk _ _ VðtÞ 6 VðtÞ þ e2at ðd1 ðtÞ þ d2 ðtÞÞ þ e2at ðd3 ðtÞ þ d4 ðtÞÞÞ þ e2at ðd5 ðtÞ þ d6 ðtÞÞÞ hk hk tkþ1  t t  tk _ þ e2at d1 ðtÞ þ e2at xðtÞT P1 xðtÞ þ e2at xðtÞT P2 xðtÞ ¼ VðtÞ hk hk   tkþ1  t t  tk X1 ðhk Þ þ X2 ðhk Þ xðtÞ; 6 e2at xðtÞT hk hk

ð38Þ

where

X1 ðhk Þ ¼ N þ P1 þ H1 ðhk Þ þ

hhk T N SNX2 ðhk Þ 4

¼ N þ P2 þ H2 ðhk Þ  hk e2ah eT6 R3 e6 þ

hhk T hhk 2ah 1 T T N SN þ e2ah hk WR1 e GS G : 1 W þ 4 2

On the other hand, we can deduce from (14) and (15) that

X1 ðhk Þ ¼

h  hk hk ðN þ P1 Þ þ ðN þ P1 þ HðhÞÞ < 0: h h

ð39Þ

Applying Schur complement to (16) leads to T N þ P2  he2ah eT6 R3 e6 þ e2ah hWR1 1 W < 0:

ð40Þ

8

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

Based on this together with (14), we obtain

h  hk hk 2ah T 1 ðN þ P2 Þ þ ðN þ P2  he e6 R3 e6 þ e2ah hWR1 W T Þ: < 0 h h

X2 ðhk Þ ¼

ð41Þ

Now, from (38), (39) and (41), we have

_ VðtÞ < 0; t 2 ½t k ; t kþ1 Þ:

ð42Þ

Noting that

V 4 ðt k Þ ¼ V 4 ðtk Þ ¼ 0; V 5 ðt k Þ ¼ V 5 ðtk Þ ¼ 0:

ð43Þ

Therefore,

Vðtk Þ ¼ Vðtk Þ:

ð44Þ

It is easy to see from (42) and (44) that for t 2 ½tk ; tkþ1 Þ,

VðtÞ 6 Vðtk Þ ¼ Vðtk Þ 6 Vðt k1 Þ ¼ Vðtk1 Þ 6    6 Vð0Þ:

ð45Þ

On the other hand, applying Schur complement to

Q¼



Q1

Q2



Q3



> 0;

ð46Þ

we have

^ , Q  Q Q 1 Q T > 0; Q 1 2 3 2

ð47Þ

and

" Q

^ Q

0



0

#

" ¼

T Q 2 Q 1 3 Q2

Q2



Q3

# P 0:

ð48Þ

By Lemma 3, it follows readily that for tk 6 t < t kþ1 ,

jjyðtÞjj2 6 h1 jjyðt k Þjj2 þ h2

Z

tk

jjyðaÞjj2 da

t k d

Z tk h1 h2 2 2at k 2at k ^Þ e k ðPÞjjyðt Þjj þ e k ð jjyðaÞjj2 da Q min min k ^ 2atk kmin ðPÞe2atk kmin ðQÞe t k d Z tk h1 h2 e2ad T 2at k ^ yðsÞ ds: 6 e yðt Þ Pyðt Þ þ e2as yðsÞT Q k k ^ Þe2atk tk d kmin ðPÞe2atk kmin ðQ ¼

ð49Þ

Condition (48) indicates that



T   yðsÞ ^ P yðsÞT QyðsÞ: Q f ðyðsÞÞ f ðyðsÞÞ yðsÞ

ð50Þ

Then by (49) and (50), we have that for tk 6 t < tkþ1 ,

jjyðtÞjj2 6 qe2atk ðV 1 ðtk Þ þ V 2 ðt k ÞÞ 6 qe2atk ðV 1 ðt k Þ þ V 2 ðtk Þ þ V 3 ðtk ÞÞ ¼ qe2atk Vðt k Þ 6 qe2atk Vð0Þ ¼ qe2at e2aðttk Þ Vð0Þ 6 e2ah qe2at Vð0Þ;

ð51Þ

where

(

h1 q ¼ max ; kmin ðPÞ

) h2 e2ad : ^Þ kmin ðQ

It follows from (51) that

jjx1 ðtÞ  xl ðtÞjj2 6 e2ah qe2at Vð0Þ; l ¼ 2; 3; . . . ; N:

ð52Þ

jjxi ðtÞ  xj ðtÞjj2 6 ðjjx1 ðtÞ  xi ðtÞjj þ jjx1 ðtÞ  xj ðtÞjjÞ2 6 2jjx1 ðtÞ  xi ðtÞjj2 þ 2jjx1 ðtÞ  xj ðtÞjj2 6 4e2ah qe2at Vð0Þ;

ð53Þ

Thus,

which implies

jjxi ðtÞ  xj ðtÞjj 6 2eah

pffiffiffiffiffiffiffiffiffiffiffiffiffi at qVð0Þe :

ð54Þ

9

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

Hence, eat jjxi ðtÞ  xj ðtÞjj is bounded, and we conclude that CDN (1) is globally exponentially stable with convergence rate a, namely

jjxi ðtÞ  xj ðtÞjj ¼ Oðeat Þ:

ð55Þ

This completes the proof. h Remark 2. It is noted that an exponential synchronization criterion is given for CDN (1) via a new time-dependent Lyapunov functional, which is positive definite at communication instants but not necessarily positive definite inside the communication intervals, and thus can make full use of the available information about the discrete-time communication pattern and lead to less conservative result.

4. Numerical example In this section, we will use a numerical example to illustrate the advantages of the proposed result in this paper. Consider the isolated node of CDN (1) with the following parameters:



A¼

B¼

 1 0 ; 0 1



2

0:11

5

3:2

 C¼

1:6

 ;

0:1

 ;

0:18 2:5

and the inner-coupling matrix



C¼

1 0



0 1

:

In this example, we choose

F1 ¼



0 0 0 0



; F2 ¼



1 0 0 1

 :

First we take the coupling configuration matrix

2

2 1 6 G ¼ 4 1 2 1

1

3 1 7 1 5: 2

The allowable bound of communication intervals h for different coupling strength c ¼ by Theorem 1 can be found in Table 1, from which we can find that the allowable bound of communication intervals h depends the coupling strength c. Specifically, a larger c corresponds to a smaller h. Next we choose the coupling strength c ¼ 3 and the coupling configuration matrix

2

6 1 2 6 4 15 5 6 G¼6 4 5 0 5 6

0

0

3 3 6 7 7 7: 0 5 6

By applying MATLAB LMI toolbox to solve the LMIs in Theorem 1, we can find that The allowable bound of communication interval, which ensures CND (1) is synchronized, is 0.1509.

Table 1 Allowable bound of communication intervals h. c

1

3

5

7

Theorem 1

0.2931

0.2922

0.2563

0.2271

10

M. Fang / Applied Mathematics and Computation 258 (2015) 1–11

5. Conclusion The synchronization problem has been considered for CDNs with time delay based on discrete-time communication pattern. A new Lyapunov functional has been proposed, which is positive definite at communication instants but not necessarily positive definite inside the communication intervals, in order to make full use of the available information about the discrete-time communication pattern. Based on the Lyapunov functional, a criterion has been proposed to ensure the synchronization of the considered CDNs. A simulation result has demonstrated the successful application of the proposed design methods. It should be mentioned that the given results can be extended to more complex systems, such as CDNs with time-varying delay [29], CDNs with Markov jumping parameters [42], CDNs with singular model [43,44], and stochastic CDNs [45]. References [1] Y. Tang, H. Gao, W. Zou, J. 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