Decentralized guaranteed cost dynamic control for synchronization of a complex dynamical network with randomly switching topology

Applied Mathematics and Computation 219 (2012) 996–1010

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

Decentralized guaranteed cost dynamic control for synchronization of a complex dynamical network with randomly switching topology Tae H. Lee a, D.H. Ji c, Ju H. Park a,⇑, H.Y. Jung b a

Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea Nonlinear Dynamics Group, Department of Information and Communication Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea c Mobile communication Division, Digital Media and Communications, Samsung Electronics, Co. Ltd., Maetan-dong, Suwon 416-2, Republic of Korea b

a r t i c l e

i n f o

Keywords: Complex network Synchronization Guaranteed cost problem Decentralized dynamic controller Randomly switching topology

a b s t r a c t This paper considers synchronization problem of a complex dynamical network with randomly switching topology which means that the topology of a complex network probabilistically switches. For this problem, a decentralized guaranteed cost dynamic feedback controller is designed to achieve the synchronization of the network. Based on Lyapunov stability theory and linear matrix inequality framework, the existence condition for feasible controllers is derived in terms of linear matrix inequalities. Finally, the proposed method is applied to two numerical examples in order to show the effectiveness of our result. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction During last half a century, complex dynamical networks have been received much attention from researchers working in different fields and their important research results have been noted [1–4]. Complex dynamical networks are a set of interconnected nodes, in which a node is a basic unit with specific contents or dynamics. As well known, the properties of such a complex dynamical network depend on the structure of coupling matrix (or network topology), so network topology is most important information in the field of complex dynamical networks. There are some useful network topologies such as smallworld network, scale-free network and so on which have been defined from random-graph model built by Erdös and Rényi [5,6]. After above works, these network topologies have been extensively investigated and become basic assumption in complex dynamical networks [7,8]. In particular, the synchronization of complex dynamical networks with these network topologies is one of the key issues that has been extensively addressed in several books and reviews [9–11]. Therefore, many researchers have focused on this topic and have developed several efficient synchronization techniques for complex dynamical networks. The classical constant connection topology is of course very restrictive and only reflects a few ideal situations. Timevarying connection topology is more realistic and covers various situations in practice. Further, the study about the synchronization of complex dynamical networks with switching topology have been attracting increasing research attention recently. In [12], a sensor network has been shown to have jumping behavior due to the network’s working environment and the mobility of sensor node. In [13], the synchronization of complex dynamical networks with arbitrary switching topology has been studied by using multiple Lyapunov functions. In [14], the global and local synchronization criteria in the form

⇑ Corresponding author. E-mail address: [email protected] (J.H. Park). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.07.004

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

997

of the second smallest eigenvalue of a switching coupling matrix have been proposed for complex dynamical networks with switching topology. More recently, the concept of Markovian jumping topology is adopted to synchronization problems of complex dynamical networks. The Markovian jump systems have the advantage of modeling the complex dynamical networks subject to abrupt variation in their communication topologies, such as component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, and operating in different points of a nonlinear plant. In [15], the exponential synchronization of complex dynamical networks with Markovian jump and mixed delay has been studied. In [16], the asymptotic synchronization problem has been investigated for a class of discrete-time stochastic Markovian complex dynamical networks with discrete and distributed time delays. In [17], the synchronization of Markovian jumping stochastic complex dynamical networks with distributed and probabilistic interval time delays was treated. Furthermore, in many real situations, networks can not synchronize by themselves. Therefore, some control schemes are adopted to design controllers, such as linear sate feedback control [18], adaptive control [19], state observer-based control [20], impulsive control [21], H1 control [22] and pinning control [23]. However, most of works in the literature on the controller design problem has focused on static controllers. Unlikely the static controllers, the dynamic control method means that the controller has their own dynamics. In some real control situations, there are some strong needs to construct dynamic feedback controllers in order to obtain a better performance and dynamical behavior of the state response. The dynamic controller will provide more flexibility compared to the static controller and the apparent advantage of this type of controller is that it provides more free parameters for selection [24]. In addition, designing a stabilizing controller for a complex network with N nodes is often difficult if not impossible, because of their complexity and strong interconnection. Therefore, it is more efficient to construct decentralized controllers to achieve control objectives for the network. So it is very worth to consider the design problem of dynamic and decentralized controller for synchronization in a complex dynamical network. On the other hand, when controlling a real plant, it is also desirable to design a control system which is not only stable but also guarantees an adequate level of performance. There is a solution so called guaranteed cost control approach [25–27]. This approach has the advantage of providing an upper bound on a given linear quadratic cost function. Up to date, some researchers in diverse fields have applied the approach to achieve both stability and performance of dynamic systems. Unfortunately, there are a few paper about the topic of guaranteed cost control for complex dynamical network [28]. In this paper, we consider the synchronization of a complex network with randomly switching topology. From above mentioned, most of researches about synchronization of complex dynamical networks with switching topology have assumed that switching trajectory is known or satisfies Markovian process. The randomly switching topology, however, means that the topology is switched at the random time. Unlikely other switching concept, randomly switching concept have not any switching rules, so it is said that this concept has more generality in switching system. To the best of authors’ knowledge, the concept of randomly switching topology for complex dynamical network is still an open area and worth considering. Furthermore, in order to consider both stability and performance of the network, the guaranteed cost control scheme for the synchronization problem is also applied. Additionally, we apply the problem of decentralized control via dynamic feedback controllers. The existence condition of such controller is derived in terms of linear matrix inequalities which can be easily solved by standard convex optimization algorithms [29]. This paper is organized as follows. A problem statement is described in Section 2. Section 3 provides the design method of a stabilizing controller for synchronization of a complex network with randomly switching topology. Two numerical examples are given in Section 4 to show the effectiveness of the derived results. Conclusions are drawn in Section 5. Notation: Rn is the n-dimensional Euclidean space, Rmn denotes the set of m  n real matrix. X > 0 (respectively, X P0) means that the matrix X is a real symmetric positive definite matrix (respectively, positive semi-definite). In denotes the ndimensional identity matrix. k  k refers to the Euclidean vector norm and induced matrix norm. H in a matrix represents the elements below the main diagonal of a symmetric matrix. Efxg and Efxjyg, respectively, mean the expectation of the stochastic variable x and the expectation of the stochastic variable x conditional on the stochastic variable y. diagf  g denotes the block diagonal matrix. Prfag means the occurrence probability of the event a. 2. Problem formulation Consider following a complex dynamical network with randomly switching topology:

x_ i ðtÞ ¼ Axi ðtÞ þ f ðxi ðtÞÞ þ

N X rðtÞ cij xj ðtÞ þ ui ðtÞ;

i ¼ 1; . . . ; N;

ð1Þ

j¼1

where xi ¼ ðxi1 ; xi2 ; . . . ; xin ÞT 2 Rn is the state vector of the ith node, A is a known constant matrix, f : Rn ! Rn is a smooth nonlinear vector field, ui ðtÞ is the control input of ith node, rðtÞ : ½0; 1Þ ! M ¼ f1; 2; . . . ; mg is a switching signal, and rðtÞ rðtÞ C rðtÞ ¼ ðcij ÞNN is the coupling matrix function of the network, where the coupling configuration parameter, cij , is defined rðtÞ rðtÞ as follows: If there is a connection from node i to node j ði – jÞ then cij is 1; otherwise cij ¼ 0 ði – jÞ, and the diagonal eleP P rðtÞ rðtÞ rðtÞ ments of matrix function C rðtÞ are assumed cii ¼  Nj¼1;j–i cij ¼  Nj¼1;j–i cji ; ði ¼ 1; . . . ; NÞ. It is noted that the switching function rðtÞ is allowed to change only at such instant which is randomly occurred and is kept some constant values k 2 M for the rest which is called mode. For example, if the values of rðtÞ is randomly changed among three constants k1 ; k2 ; k3 , then it can be said that the system (1) has three mode and belongs to the first mode when rðtÞ ¼ k1 . For simplicity of notation, C rðtÞ associated with the kth mode is defined by

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T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

C k ¼ C rðtÞ ¼ the system belongs to kth mode: In order to deal with randomly switching topology in a complex network, we introduce stochastic variables



ak ðtÞ ¼

1;

rðtÞ ¼ k;

0;

otherwise;

k ¼ 1; . . . ; m;

with the following probability:

Prfak ðtÞ ¼ 1g ¼ ak ;

k ¼ 1; . . . ; m;

ð2Þ

where the probability of stochastic variables ak ðtÞ satisfies Then Eq. (1) can be rewritten by

x_ i ðtÞ ¼ Axi ðtÞ þ f ðxi ðtÞÞ þ

Pm

k¼1

ak ¼ 1.

N X m X

ak ðtÞckij xj ðtÞ þ ui ðtÞ; i ¼ 1; . . . ; N;

ð3Þ

j¼1 k¼1

where ckij is the ith row, jth column element of matrix C k . Remark 1. The random variable ak ðtÞ which satisfies Efak ðtÞg ¼ ak are used to model the probability distribution of the randomly switching topology in a complex network. This kind of stochastic variables is applied to deal with randomly occurring uncertainties, nonlinearities and time-delays. There are many existing works about switching systems [12–17,30– 32]. However, most of previous works about switching systems only considers that switching trajectory of the system is known or satisfies Markovian process. To the best of our knowledge, no related results have been established for the concept of randomly switching topology in complex networks. Our objective of this paper is to design stabilizing controllers ui ðtÞ for each nodes which render asymptotic synchronization between all nodes of the complex network and a target node. For this synchronization scheme, let us define error vectors as follows:

ei ðtÞ ¼ sðtÞ  xi ðtÞ;

ð4Þ

n

where sðtÞ 2 R is a solution of a target node, satisfying

s_ ðtÞ ¼ f ðsðtÞÞ: Definition 1. A complex network is said to achieve asymptotically synchronization, if limt!1 kei ðtÞk ¼ 0 for all 0 < i 6 N. From Eq. (4), the error dynamics is given to

e_ i ðtÞ ¼ Aei ðtÞ þ f ðsðtÞÞ  f ðxi ðtÞÞ 

N X m X

N X m X

j¼1 k¼1

j¼i k¼1

ak ðtÞckij ej ðtÞ  ui ðtÞ ¼ Aei ðtÞ þ f i ðei ðtÞÞ 

ak ðtÞckij ej ðtÞ  ui ðtÞ;

i ¼ 1; . . . ; N;

ð5Þ

where f i ðtÞ ¼ f ðsðtÞÞ  f ðxi ðtÞÞ. In order to stabilize the error system given in Eq. (5), the following dynamic feedback controllers are proposed:

f_ i ðtÞ ¼ Aci fi ðtÞ þ Bci ei ðtÞ; ui ðtÞ ¼ C ci fi ðtÞ;

ð6Þ

fi ð0Þ ¼ 0;

where fi ðtÞ 2 Rn is the state vector of controller, and Aci ; Bci and C ci are constant gain matrices of ith node with n  n dimensions to be determined later. The performance index associated with ith node is the following quadratic function:

Z Ji ¼ E

1

0



  eTi ðtÞQ i ei ðtÞ þ uTi ðtÞRi ui ðtÞ dt ;

ð7Þ

where Q i and Ri 2 Rnn are given positive-definite matrices. Applying this controller (6) to error system (5) results in the following closed-loop system

z_ i ðtÞ ¼ Hi zi ðtÞ 

"

N X m X

ak ðtÞC kij zj ðtÞ þ

f ðtÞ i 0

j¼1 k¼1

# ð8Þ

;

where

zi ðtÞ ¼



ei ðtÞ fi ðtÞ



" 2 R2n ;

C kij ¼

ckij In

0

0

0

# 2 R2n2n

Hi ¼



A

C ci

Bci

Aci



2 R2n2n :

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

999

The corresponding closed-loop cost function can be written

Ji ¼ E

Z

1

0

zTi ðtÞ diagfQ i ; C Tci Ri C ci g zi ðtÞdt

 E

Z 0

1

 zTi ðtÞQ i zi ðtÞdt :

ð9Þ

Here, the objective of this paper is to develop a procedure to design a dynamic feedback controller (6) for system (5) and performance index (7), such that the resulting closed-loop system is asymptotically stable and the closed-loop value of H the cost function (7) satisfies J i 6 J H i , where J i is some specified constant. Definition 2. For the error system (5) and cost function (7), if there exist a control law ui ðtÞ and a positive constant J H i such that the closed-loop system (8) is asymptotically stable and the closed-loop value of the cost function (7) satisfies J i 6 J H i , then J H i is said to be a guaranteed cost and ui ðtÞ is said to be decentralized guaranteed cost synchronization controller. Before proceeding further, a assumption and two well-known facts are given, below. Assumption 1. The smooth nonlinear function f ðÞ is satisfied the following Lipschitz condition:

kf ðaÞ  f ðbÞk 6 lka  bk; where l is a positive constant. Fact 1. For any real vectors a, b and positive constant , it follows that: T

2aT b 6 aaT þ 1 bb ;

 > 0:

Fact 2 (Schur complements). Given constant symmetric matrices R1 ; R2 ; R3 where R1 ¼ RT1 and 0 < R2 ¼ RT2 , then R1 þ RT3 R1 2 R3 < 0 if and only if

"

#

R1 RT3 < 0; R3 R2

 or

R2

RT3



R3 < 0: R1

3. Main results In this section, the existence criterion for a decentralized guaranteed cost dynamic controller (6) for error system (5) will be derived by use of Lyapunov theory and linear matrix inequality framework with convex optimization. The following is a main result of this paper. Theorem 1. For given a positive constant k; Q i > 0; Ri > 0 and a known Lipschitz constant l, the dynamic controller (6) is the guaranteed cost synchronization controller for the complex network (3) if there exist positive-definite matrices Si ; Y i ; X i4 2 Rnn and matrices X i1 ; X i2 ; X i3 2 Rnn satisfying the following linear matrix inequalities:

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2 4

Ci2 ci In

Ci1

bY i

YiQ i

X i1 Ri

0

In

0

H

In

0

0

0

0

0

0

0

H

H

Q i

0

0

0

0

0

0

H

H

H

Ri

0

0

0

0

0

H

H

H

H

Ci3

Si ci

0

Si

0

H

H

H

H

H

In

0

0

0

H

H

H

H

H

H

In

0

0

H

H

H

H

H

H

H

k2 In

0

H

H

H

H

H

H

H

H

k2 In

Yi

In

In

Si

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 < 0; 7 7 7 7 7 7 7 7 7 7 7 7 5

ð10Þ

3 5 > 0;

i ¼ 1; . . . ; N;

ð11Þ

1000

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

where

pffiffiffiffi

l ¼ N;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 k 2 l þ l2 ; 0 !2 112 N m X X k ci ¼ @ ak cij A ; b¼

j¼1

k¼1

bi  A b T þ l2 D b i; Ci1 ¼ Y i A þ AY i  A i T

Ci2 ¼ A þ Cb i þ b2 Y i þ Y i Q i ; bi þ B b T þ b2 In þ Q i : Ci3 ¼ AT Si þ Si A þ B i Then, the upper bound of cost function is given by

J i 6 eTi ð0ÞSi ei ð0Þ , J i :

ð12Þ

Proof. Consider a Lyapunov function for system (8)

Vðzt Þ ¼

N N X X  T  V i ðzt Þ ¼ zi ðtÞPi zi ðtÞ ; i¼1

ð13Þ

i¼1

where P i 2 R2n2n > 0; i ¼ 1; . . . ; N. The infinitesimal operator L of Vðzt Þ is defined as follows:

1 L ¼ limþ ½EfVðztþh Þjzt g  Vðzt Þ: h!0 h Using the infinitesimal operator L of Vðzt Þ and Assumption 1 to derive the condition for stochastic stability, we obtain

"

( N X

zTi ðtÞðP i Hi þ HTi Pi Þzi ðtÞ þ 2zTi ðtÞPi I

EfLVðzt Þg ¼ E

i¼1

f ðtÞ i 0

#  2zTi ðtÞPi

N X m X

ak C kij zj ðtÞ

!) ;

ð14Þ

j¼1 k¼1

where I ¼ diagfIn ; 0g. By Fact 1, we have following two equations:

( N X

E

" 2zTi ðtÞPi I

f ðtÞ i

#!)

0

i¼1

( ) ( ) N N X X  2 T   2 T  k zi ðtÞPi IIPi zi ðtÞ þ k2 f Ti ðtÞf i ðtÞ 6E k zi ðtÞPi IIPi zi ðtÞ þ k2 zTi ðtÞLzi ðtÞ ;

6E

i¼1

i¼1

ð15Þ ( !) ( ) N N X m N N X N X X X X T T T T 2zi ðtÞPi ak C ij zj ðtÞ 6E zi ðtÞPi C di C di Pi zi ðtÞ þ zj ðtÞzj ðtÞ ; E i¼1

j¼1 k¼1

i¼1

i¼1

ð16Þ

j¼1

2

where L ¼ diagfl In ; 0g and C di ¼ diagfci In ; 0g. Here, by the following relationship N X N N X X zTj ðtÞzj ðtÞ ¼ NzTi ðtÞzi ðtÞ: i¼1 j¼1

i¼1

Eq. (16) can be rewritten as

( !) ( ) N N X m N

X X X T T T 2 T E 2zi ðtÞPi ak C ij zj ðtÞ 6E zi ðtÞPi C di C di P i zi ðtÞ þ l zi ðtÞzi ðtÞ : i¼1

j¼1 k¼1

ð17Þ

i¼1

Thus, from Eqs. (15) and (17), Eq. (14) satisfies the inequality:

( ) N

X zTi ðtÞðPi Hi þ HTi Pi þ Pi ðk2 II þ C di C Tdi ÞPi þ k2 L þ l2 I2n Þzi ðtÞ

EfLVðzt Þg 6 E

i¼1

( ) N N X X ¼E zTi ðtÞRi zi ðtÞ  eTi ðtÞQ i ei ðtÞ ; i¼1

i¼1

where Ri ¼ P i Hi þ HTi Pi þ Pi ðk2 II þ C di C Tdi ÞPi þ k2 L þ l2 I2n þ Q i and Q i ¼ diagfQ i ; 0g.

ð18Þ

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

1001

Therefore, if Ri < 0; 8i, there exist a positive scalar ci such that

( N X

EfLVðzt Þg 6 E

( N X

)

 zTi ðtÞQ i zi ðtÞ

6E

i¼1

)

 ci kei ðtÞk2 ;

i¼1

which guarantees the stochastic stability of the system by Lyapunov stability theory. It should be pointed out that the matrix Pi > 0 and the controller parameters Aci ; Bci and C ci in the matrix Ri are unknown and occur in nonlinear fashion. Hence, the inequality Ri < 0 cannot be considered as a linear matrix inquality problem. So, we will use a method of changing variables such that the inequality can be solved as convex optimization algorithm [33]. First, partition the matrix Pi and its inverse as

 Pi ¼

Si

Ti

T Ti

Oi

 ;



P1 ¼ i

Yi

Gi

GTi

Ui

 ;

where Si ; Y i are positive-definite matrices, and Gi ; T i 2 Rnn are invertible matrices. It should be noted that the equality P 1 i P i ¼ I2n gives that

Gi T Ti ¼ In  Y i Si :

ð19Þ

Define two matrices as

X 1i ¼



Yi GTi

 In ; 0

X 2i ¼



In 0

 Si : T Ti

Then, it follows that

X T1i Pi X 1i ¼ X T1i X 2i ¼

Pi X 1i ¼ X 2i ;



Yi

In

In

Si



> 0:

Now, postmultiplying and premultiplying the matrix inequality, Ri < 0, by the matrix X T1i and by its transpose, respectively, gives





X T2i Hi X 1i þ X T1i HTi X 2i þ X T1i k2 L þ l2 I2n þ Q i X 1i þ X T2i k2 II þ C di C Tdi X 2i < 0:

ð20Þ

By Fact 2, the inequality (20) is derived to

2 6 6 6 6 6 6 6 6 6 6 6 4

ð1; 1Þ ð1; 2Þ

ci In Si ci

0

In

0

0

Si

0

H

ð2; 2Þ

H

H

0

0

0

0

H H

H H

In H

0 In

0 0

0 0

H

H

H

H

k2 In

0

H

H

H

H

H

k2 In

3 7 7 7 7 7 7 7 < 0; 7 7 7 7 5

ð21Þ

where

ð1; 1Þ ¼ Y i AT þ AY i  Gi C Tci  C ci GTi þ b2 Y i Y i þ Y i Q i Y i þ l2 Gi GTi þ Gi C Tci Ri C ci GTi ; ð1; 2Þ ¼ A þ Y i AT Si þ Y i BTci T Ti  Gi C Tci S þ Gi ATci T Ti þ b2 Y i þ Y i Q i ; ð2; 2Þ ¼ AT Si þ Si A þ T i Bci þ BTci T Ti þ b2 In þ Q i : By defining a new set of variables as follows:

b i ¼ Gi C T ; A ci b i ¼ T i Bci ; B b i Si þ G i A T T T ; b i ¼ Y i AT Si þ Y i B bT  A C i ci i T b D i ¼ Gi G : i

By Fact 2, inequality (21) is equivalent to the linear matrix inequality (10). On the other hand, from Eq. (18), we get

n o EfLV i g 6 E zTi ðtÞQ i zi ðtÞ : Integrating both sides of the above inequality from 0 to T f leads to

ð22Þ

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T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

Z E

Tf

0

zTi ðtÞQ i zi ðtÞ



< EfVð0Þ  VðT f Þg:

Since the asymptotic stability of the system has already been established, we conclude that EfVðT f Þg ! 0 as t ! 1. Hence, we have

J i 6 zTi ð0ÞQ i zi ð0Þ ¼ eTi ð0ÞSi ei ð0Þ ¼ J i ; which completes the proof.

h

Remark 2. Given any solution of the linear matrix inequalities (10) and (11) in Theorem 1, a corresponding controller of the form Eq. (6) be constructed as follows: Compute the invertible matrices Gi and T i satisfying Eq. (19) using matrix algebra. bi; B b i; D bi; C b i for Bci ; C ci and Aci (in this order). Utilizing the matrices Gi and T i obtained above, solve the equations A Theorem 1 presents a method of designing a dynamic feedback controller for synchronization of a complex network (3). In the following, we will present a method of selecting the optimal controller minimizing the upper bound of the guaranteed cost (12).

Theorem 2. Consider error system (5) and cost function (7). For all i, if the following linear matrix inequality optimization problem,

min

di ;Si ;Y i ;b A i ;b B i ;b C i ;b Di

ð23Þ

di ;

subject to ðiÞ linear matrix inequalities ð10Þ and ð11Þ;   bi eTi ð0ÞSi <0 ðiiÞ H Si

ð24Þ ð25Þ

bi; B b i; D bi; C b i ), the controller (6) is the optimal guaranteed cost dynamic controller which ensures the has the solution set (di ; Si ; Y i ; A P P minimization of the guaranteed cost (12) of the system. The optimal cost is Ni¼1 J i ¼ Ni¼1 di  d. Proof. By Theorem 1, Eq. (24) is clear, and from Fact 1, Eq. (25) is equivalent to eTi ð0ÞSi ei ð0Þ < di . So, it follows from (12). Thus, the minimization of di implies the minimization of the guaranteed cost (7). It is well-known that the convexity of the linear matrix inequality optimization problem ensures that a global optimum, when it exists, is reachable. This completes the proof. h

4. Numerical examples In this section, two simulation results are presented to show the effectiveness of the proposed controller for synchronizing all nodes of a complex network to a target node. In two examples, the parameters associated with cost function are chosen as Q i ¼ In and Ri ¼ In . Example 1 (Chua’s circuit system). The first example is about synchronization of a complex dynamical network with five linearly coupled identical nodes which are Chua’s circuit [34] which is typical benchmark three dimensional chaotic system. Fig. 1 depicts its chaotic behavior. Thus, the dynamic equation of a complex network of this example is described by:

x_ i ðtÞ ¼ Axi ðtÞ þ f ðxi ðtÞÞ þ

N X

qðtÞ

cij xj ðtÞ þ ui ðtÞ;

i ¼ 1; . . . ; 5;

ð26Þ

j¼1

where

2 6 A¼4

am1

a

0

3

1

7 1 1 5;

0

b

0

2a 6 f ðxi ðtÞÞ ¼ 4

2

ðm0  m1 Þðjxi1 ðtÞ þ cj  jxi1 ðtÞ  cjÞ 0 0

with the parameters a ¼ 9; b ¼ 14:28; c ¼ 1; m0 ¼ 1=7; m1 ¼ 2=7.

3 7 5;

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T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

4 3 2 1 3

s (t)

0 −1 −2 −3 −4 3 2

0.6

1

0.4

0

0.2

−1

0

−2 −3

s (t) 1

−0.2 −0.4

s (t) 2

Fig. 1. The chaotic behavior of Chua’s circuit.

5

4

e2(t)

e1(t)

2 0

0 −2

−5

0

2

4

time

6

8

−4

10

4

0

2

4

0

2

4

time

6

8

10

6

8

10

5

e (t)

0

4

e3(t)

2 0

−2 −4

0

2

4

time

6

8

10

i1

e

i2

2 5

time

e

4

e (t)

−5

e

i3

0 −2 −4

0

2

4

tme

6

8

10

Fig. 2. The uncontrolled error signals of Example 1.

It is noted that the Lipschitz constant of Chua’s circuit is l ¼ 3 and initial conditions of each nodes are chosen: x1 ð0Þ ¼ ½ 1; 0:5; 1:7 , x2 ð0Þ ¼ ½ 0:5; 0:4; 0:3 , x3 ð0Þ ¼ ½ 0:6; 0:5; 0 , x4 ð0Þ ¼ ½ 1:1; 1:5; 1:3 , x5 ð0Þ ¼ ½ 0:1; 0:5; 1:7 , sð0Þ ¼ ½ 0:1; 0:1; 0:1 . The switching nodes are assumed three with probabilities a1 ¼ 0:1; a2 ¼ 0:4; a3 ¼ 0:5, then coupling matrices, C k , are given by

1004

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

2

e(t)

1 0 −1 −2

0

0.5

1

0

0.5

1

1.5

2

2.5

3

1.5

2

2.5

3

100 80

ζ(t)

60 40 20 0 −20

time Fig. 3. The controlled error signals and control states of Example 1.

3 3 1 1 0 1 7 6 1 1 7 6 1 4 1 7 6 C 1 ¼ 0:4  6 1 3 1 0 7 7; 6 1 7 6 1 1 3 1 5 4 0 1 1 0 1 3 3 2 4 1 1 1 1 7 6 1 0 7 6 1 2 0 7 6 2 C ¼ 0:2  6 0 3 1 1 7 7; 6 1 7 6 1 1 4 1 5 4 1 1 0 1 1 3 3 2 4 1 1 1 1 7 6 1 1 7 6 1 4 1 7 6 3 C ¼ 0:2  6 1 4 1 1 7 7: 6 1 7 6 1 1 4 1 5 4 1 1 1 1 1 4 2

In order to show original behavior of the complex network (26), the trajectories of the uncontrolled error signals of each nodes is depicted in Fig. 2. Now, by consideration of linear matrix inequality problem given in (10) and (11) in Theorem 1, we can calculate feasible solution set from Eq. (22) by use of MATLAB LMI Toolbox. Then, we found gain matrices of the controller (6) (see Appendix A). The simulation result with control inputs which are calculated by Theorem 1 is presented in Fig. 3. As seen in Fig. 3, the trajectories of error systems are indeed well stabilized and also the state orbits of controller approach to zero. It can be concluded that our proposed dynamic controller guarantees asymptotic synchronization of the complex network (26) under some value of performance index. In addition, Fig. 4 indicates the stochastic switching variable rðtÞ which associates with system mode in this example. Example 2 (Hyperchaotic Chua’s circuit). Next, let us consider a complex dynamical network with three identical nodes which consist of six dimensional hyperchaotic Chua’s circuits [35]:

x_ i ðtÞ ¼ Axi ðtÞ þ f ðxi ðtÞÞ þ

N X j¼1

qðtÞ

cij xj ðtÞ þ ui ðtÞ;

i ¼ 1; 2; 3;

ð27Þ

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T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

SYSTEM MODE

3

2

1

0

1

2

3

4

5 time

6

7

8

9

10

Fig. 4. The curve of the system mode in Example 1.

where

2

am1 a 0

6 6 1 6 6 6 0 6 A¼6 6  6 6 6 0 4

0

0 0

3

2a

2

ðm0 þ m1 Þðjxi1 ðtÞ þ cj  jxi1 ðtÞ  cjÞ

3

7 7 6 7 6 0 07 0 7 7 6 7 7 6 7 6 b 0 0 0 07 0 7 7 6 ; f ðx ðtÞÞ ¼ 7 7; 6 i 6 a ðm0 þ m1 Þðjxi4 ðtÞ þ cj  jxi4 ðtÞ  cjÞ 7 0 0   am1 a 0 7 7 7 62 7 7 6 7 6 0 0 1 1 1 7 0 5 5 4

1 1

0

0 0

0

0

b 0

0

with the parameters a ¼ 9; b ¼ 14:28; c ¼ 1; m0 ¼ 1=7; m1 ¼ 2=7;  ¼ 0:01. The system f ðÞ in (27) consisted of two unidirectionally coupled Chua’s circuits and exhibits hyperchaotic behavior with a double-double scroll attractor which is shown in Fig. 5. This system satisfies Assumption 1 with Lipschitz constant l ¼ 5. The initial conditions of each noes are chosen:

x1 ð0Þ ¼ ½ 0:9; 0:5; 0:7; 0; 1; 1 ; x3 ð0Þ ¼ ½ 0:6; 0:5; 0; 0; 0; 3 ;

x2 ð0Þ ¼ ½ 0:1; 0:4; 0:3; 0:1; 0:5; 0:7 ;

sð0Þ ¼ ½ 0:1; 0:5; 0:7; 0:1; 0:1; 0:1 

and we consider the number of switching nodes is two with the following coupling matrices C k and probabilities:

2

3 2 1 1 6 7 C ¼ 0:5  4 1 1 0 5; 1 0 2 1

2

3 2 1 1 6 7 C ¼ 0:2  4 1 2 1 5; 1 1 2 2

a1 ¼ 0:2; a2 ¼ 0:8:

In order to find the feasible solution set and control parameters, we follow same procedure of Example 1. Then, the suitable control parameters are obtained (see Appendix A). The trajectories of the uncontrolled error signals are displayed in Fig. 6. Fig. 7 shows trajectories of the controlled error signals and state of controller. As compared with Figs. 6 and 7, it is clear that our proposed method ensures to achieve the guaranteed cost synchronization of a complex dynamical network. qðtÞ is depicted in Fig. 8. Finally, by solving the optimization problem (23) given in Theorem 2, the optimal guaranteed cost of closed-loop system are obtained and listed in Table 1.

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

4

4

3

3

2

2

1

1

0

0

6

s (t)

3

s (t)

1006

−1

−1

−2

−2

−3

−3

−4

−4

0.4

0.5 0.2

4

4

2

0

0

−0.2 s2(t)

−2 −0.4

s5(t)

s1(t)

−4

2

0

0 −2 −0.5

s4(t)

−4

Fig. 5. The chaotic behavior of hyperchaotic Chua’s circuit.

5

e1(t)

0 −5 −10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 time

6

7

8

9

10

2

e (t)

5

0

−5

e3(t)

5 0 −5 −10

Fig. 6. The uncontrolled error signals of Example 2.

5. Conclusions In this paper, the decentralized guaranteed cost dynamic controller for synchronization of a complex dynamical network with randomly switching topology has been designed based on the Lyapunov method and linear matrix inequality framework. Unlike other works, randomly switching topology was considered for the synchronization problem instead of known switching topology trajectory or Markovian jumping topology, so it is said that this concept has more generality in switching system. Then, two criteria expressed by linear matrix inequalities for achieving both stability and performance of error

1007

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

4 3

e(t)

2 1 0 −1 −2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

5

30 20

ζ(t)

10 0 −10 −20 −30

Fig. 7. The controlled error signals and control states of Example 2.

SYSTEM MODE

2

1

0

1

2

3

4

5 time

6

7

8

9

10

Fig. 8. The curve of the system mode in Example 2.

Table 1 The cost value of each method. d Example 1

Theorem 1

Example 2

Theorem 2 Theorem 1 Theorem 2

9:4248  103 129.8939 1:6394  103 58.7485

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T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

dynamics were derived. Finally, two numerical examples were illustrated to show the effectiveness of the designed controller and the minimum guaranteed cost values of examples were calculated. Acknowledgement This work was supported by Yeungnam University Research Grant. Appendix A The parameters of controller of Example 1.

2

Ac1

C c1

3 49:2340 6:9000 9:9257 6 7 ¼ 4 158:3105 318:5246 127:9737 5; 19:0575 43:5158 80:4290 2 3 4:3102 1:9898 1:7164 6 7 ¼ 4 1:9877 12:6273 1:1094 5 1:7238 1:1292 5:6013 2

49:6761

7:2159

6 Ac2 ¼ 4 42:8792 92:4075 2

8:9567 3:3671

6 C c2 ¼ 4 1:0751 1:1013 2

52:0509 9:8499 3:5761

6 C c3 ¼ 4 1:2476 1:2272 2

7:0988

49:7999 19:5832

4:2989

6 C c4 ¼ 4 1:9766

5:8743

44:3975 1:9740

12:6920

1:7259 1:1612 2

52:0373 9:8385 3:5757

6 C c5 ¼ 4 1:2474 1:2281

46:5910

6 Bc2 ¼ 4 86:9512

433:2343

2

6 Bc3 ¼ 4 102:7672

2

42:6535 532:8556

0:0591

6 Bc4 ¼ 103  4 0:2905

80:9709 3

3

7 156:6515 5 128:6597

1:6229

0:0204 0:2564

0:0357

3

7 0:3917 5 0:1563

7 1:1404 5 5:6178

7:0945

0:4978

3

7 45:9048 5; 20:3327 64:9111 3 1:2440 1:2265 7 8:3250 0:7613 5 0:7693 4:0801

2

69:4257

6 Bc5 ¼ 4 102:7615

42:5931 532:0702

10:2625 151:2770

2

39:5840 5:0898 13:5592 0:2580 0:0062 6 90:3189 56:2379 95:7998 0:3931 0:4596 6 6 6 40:7824 13:4287 73:1394 0:5351 0:0771 ¼6 6 0:0122 0:1307 0:0111 39:9719 5:2202 6 6 4 0:0118 0:8633 0:0614 90:8427 56:3104 0:6851

3:6915

0:0437

The parameters of controller of Example 2.

Ac1

3

122:8671

10:1956 151:4661

3

7 132:9846 5;

69:4269

0:4569

7 136:0418 5

10:3377 136:9562

3

9:4030

67:7845

1:7200

6 Ac5 ¼ 4 51:5171 111:3121 2

0:4514

Bc1

3 0:0580 0:0511 0:0378 6 7 ¼ 103  4 0:2829 1:5940 0:3836 5 0:0189 0:2541 0:1563

2

7 46:2601 5; 20:2708 64:9394 3 1:2443 1:2254 7 8:3242 0:7604 5 0:7681 4:0766

6 Ac4 ¼ 4 161:3150 324:6500 2

3

7 39:5712 5; 17:4016 61:0600 3 1:0720 1:1000 7 7:3464 0:6896 5 0:6937 3:7156

6 Ac3 ¼ 4 51:6281 111:2262 2

0:3715

2

0:1412

0:8048

40:9170

13:4255

3 0:1387 0:5249 7 7 7 0:1790 7 7 14:1835 7 7 7 97:3516 5 73:7084

3:7732

3

7 155:7319 5 128:3552

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

Bc1

3 57:7592 90:8749 39:4603 0:4366 2:4916 1:3749 6 170:5164 552:1323 291:1761 0:5075 6:0069 3:1139 7 7 6 6 70:7948 269:4487 188:1032 0:8462 4:9677 2:8282 7 7 6 ¼6 7 6 0:2221 0:8373 0:3167 58:8944 94:2024 41:1028 7 7 6 4 1:4817 5:7556 2:1177 173:4389 557:9791 293:4236 5 2:1955 7:6995 3:6881 72:0440 270:8043 188:2732

C c1

3 2:1387 0:5117 0:6494 0:0001 0:0072 0:0044 6 0:5257 3:8401 0:2309 0:0024 0:0171 0:0131 7 7 6 6 0:6470 0:2292 2:1257 0:0001 0:0004 0:0062 7 7 6 ¼6 7 6 0:0004 0:0015 0:0001 2:1378 0:5026 0:6515 7 7 6 4 0:0083 0:0168 0:0001 0:5144 3:8318 0:2367 5 0:0045 0:0120 0:0061 0:6498 0:2340 2:1307

Ac2

3 31:7664 2:5386 10:4556 0:1802 0:0090 0:1738 6 57:3484 33:9594 64:0287 0:5148 0:0818 0:4634 7 7 6 6 25:7329 4:7829 53:8973 0:5344 0:0567 0:5606 7 7 6 ¼6 7 6 0:1940 0:0038 0:1805 31:9798 2:5373 10:6514 7 7 6 4 0:6352 0:0998 0:4591 57:9536 33:7834 64:5257 5 0:0460 0:5554 26:2626 4:7846 54:4289 0:5820

Bc2

3 44:0401 59:5378 26:9172 0:4049 1:4834 0:8394 6 105:4855 297:2272 173:1005 1:2901 5:8740 3:2214 7 7 6 6 42:7254 150:1987 122:0063 1:1507 4:6135 2:6536 7 7 6 ¼6 7 6 0:4310 1:4292 0:8486 44:5373 61:0299 27:7866 7 7 6 4 1:4486 5:6956 3:1759 106:9722 302:4871 176:1727 5 43:8832 154:2961 124:4481 1:1985 4:4910 2:6212

C c2

3 2:0174 0:4444 0:5943 0:0012 0:0009 0:0014 6 0:4531 3:5062 0:2298 0:0006 0:0083 0:0061 7 7 6 6 0:5921 0:2261 1:9983 0:0005 0:0059 0:0064 7 7 6 ¼6 7 6 0:0011 0:0008 0:0007 2:0199 0:4433 0:5961 7 7 6 4 0:0008 0:0082 0:0061 0:4520 3:4986 0:2244 5 0:0012 0:0060 0:0064 0:5938 0:2209 1:9930

Ac3

3 40:1978 5:1778 14:4064 0:0153 0:0810 0:2080 6 92:6909 56:1220 99:2212 0:6603 0:6677 3:0318 7 7 6 6 42:2133 13:4841 75:0642 0:4090 0:3296 1:6742 7 7 6 ¼6 7 6 0:7100 0:0664 0:3246 39:5869 5:6610 12:8941 7 7 6 4 4:4684 0:3474 1:9971 89:4665 59:5684 90:5033 5 1:3792 41:1776 15:2269 72:2381 2:3435 0:1817

Bc3

3 59:2098 95:4148 42:0802 0:1449 0:6395 0:2164 6 176:3935 567:1867 300:7083 2:3604 9:6799 5:7243 7 7 6 6 74:2326 277:8507 193:3172 1:3984 5:2067 3:0852 7 7 6 ¼6 7 6 1:1050 1:0741 0:6892 58:5551 94:3399 38:6390 7 7 6 4 6:8062 4:9088 3:4942 173:6973 566:5259 281:5816 5 73:9508 281:5814 187:3118 3:5396 2:3353 2:3238

2

2

2

2

2

2

2

2

C c3

2:1345

0:5023

0:6505

0:0032

6 0:5142 3:8256 0:2324 0:0048 6 6 6 0:6485 0:2302 2:1247 0:0040 ¼6 6 0:0036 0:0046 0:0037 2:1426 6 6 4 0:0014 0:0008 0:0031 0:5185 0:0025

0:0082

0:0058

0:6543

0:0014 0:0010 0:0032 0:5059 3:8232 0:2242

0:0017

3

0:0090 7 7 7 0:0062 7 7 0:6566 7 7 7 0:2270 5 2:1304

1009

1010

T.H. Lee et al. / Applied Mathematics and Computation 219 (2012) 996–1010

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