Synchronization criterion for Lur'e type complex dynamical networks with time-varying delay

Physics Letters A 374 (2010) 1218–1227

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Physics Letters A www.elsevier.com/locate/pla

Synchronization criterion for Lur’e type complex dynamical networks with time-varying delay D.H. Ji a , Ju H. Park b,∗ , W.J. Yoo c , S.C. Won c , S.M. Lee d a

Mobile Communication Division, Digital Media and Communications, Samsung Electronics, Co. Ltd., 461-2 Maetan-Dong, Suwon 443-803, Republic of Korea Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea c Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, San 31 Hyoja-Dong, Pohang 790-784, Republic of Korea d School of Electronics Engineering, Daegu University, Kyongsan, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 19 September 2009 Received in revised form 22 November 2009 Accepted 6 January 2010 Available online 12 January 2010 Communicated by A.R. Bishop Keywords: Complex dynamical networks Lur’e systems Synchronization Delay dependent criterion Absolute stability LMIs

a b s t r a c t In this Letter, the synchronization problem for a class of complex dynamical networks in which every identical node is a Lur’e system with time-varying delay is considered. A delay-dependent synchronization criterion is derived for the synchronization of complex dynamical network that represented by Lur’e system with sector restricted nonlinearities. The derived criterion is a sufficient condition for absolute stability of error dynamics between the each nodes and the isolated node. Using a convex representation of the nonlinearity for error dynamics, the stability condition based on the discretized Lyapunov–Krasovskii functional is obtained via LMI formulation. The proposed delaydependent synchronization criterion is less conservative than the existing ones. The effectiveness of our work is verified through numerical examples. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Complex dynamical networks have received a great deal of attention since they are shown to widely exist in various fields of real world [1–22]. A complex network is a set of interconnected nodes, in which a node is a basic unit with specific contents or dynamics. Examples of complex networks include the Internet, the World Wide Web (WWW), food chain, electricity distribution networks, relationship networks and disease transmission networks, etc. Many of these networks exhibit complexity in the overall topological properties and dynamical properties of the network nodes and the coupled units. The complex nature of the networks has results in a series of important research problems. In particular, one significant and interesting phenomenon is the synchronization of all its dynamics. Therefore, many researchers have focused on this topic and have developed several efficient synchronization techniques for complex dynamical networks [3–9]. Wang and Chen introduced a uniform dynamical network model and investigated its synchronization and control [6]. Li and Chen [7] further extended the network model to include coupling delays among the network nodes and studied its synchronization. Gao et al. [8] considered the synchronization stability of both continuous- and discrete-time networks with coupling delays. Li et al. [9] proposed the synchronization criterion of both continuous- and discrete-time networks with time varying delays. However, the obtained results by using the linearization are not guaranteed the globally asymptotic stability. Recently, many researches for the synchronization of Lur’e systems were presented, because various chaotic systems such as Chua’s circuit [10] can be modeled as Lur’e systems. The Lur’e system is continuum of a linear system and a feedback nonlinearity satisfying sector bound constraints. The stability of the Lur’e system is called absolute stability that means global asymptotic stability [11]. For these reason, many researcher have studied the synchronization for Lur’e systems and applied to various applications. Suykens et al. [12–14] investigated extensively synchronization problem including H∞ synchronization for Lur’e systems. More recently, practical issues of the synchronization such as propagation delay have been considered since Chen and Liu introduced the delay on the chaotic synchronization and showed that the delay may break synchronization [15]. There are several studies that considered the delay effect on the synchronization of Lur’s systems [16–21]. In those researches, synchronization criteria were derived for given gain matrices of the synchronization controller and

*

Corresponding author. Tel.: +82 53 810 2491; fax: +82 53 810 4767. E-mail addresses: [email protected] (D.H. Ji), [email protected] (J.H. Park).

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D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227

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time delay, which are sufficient conditions for absolute stability of the synchronization. However, the model transformation technique used in [16,17] can lead to conservative conditions by inducing additional dynamics as addressed in [18]. In order to derive less conservative conditions for synchronization, synchronization criteria that does not use the model transformation were presented independently in [18–20]. Guo et al. [18] applied a free weighting matrix approach employed on the Libnitz–Newton formula and an equality constraint for the synchronization criterion. Xiang et al. [19] used the integral inequality and a free weighting matrix approach. In [20], a more general Lur’e–Postnikov Lyapunov functional was presented to derive a less conservative criterion. Furthermore, a synchronization method for the chaotic Lur’e system was extended for time-varying delay in [21]. Very recently, the synchronization problem is addressed for complex dynamical networks in which every identical node is a time-delayed Lur’e system [22]. However, it is worth noting that the time-delays of system dynamical states considered in these works are assumed to be constant. Time-varying delay case, which is more general than the constant one should be considered. Moreover, in much of the literature, time delays in the couplings are considered. However, the time delays in the dynamical nodes, which are more complex, are still relatively unexplored. Therefore, the synchronization problem of nonlinear Lur’e type complex dynamical networks with time-varying delay still remains challenging. In this Letter, we will propose a delay dependent synchronization criterion for Lur’e type complex dynamical networks with a time varying delay. Convex representation of the nonlinearity of the Lur’e networks is introduced, and then, a sector bounded constraint of the nonlinearity is converted to an equality constraint. The projection lemma [23] is utilized for handling the equality constraint so that a less conservative delay-dependent criterion is obtained. Furthermore, the discretized Lyapunov–Krasovskii functional that employs redundant state of differential equations shifted in time by a fraction of the time delay is also applied to reduce conservatism in searching the maximum allowable delay such that the error dynamics of synchronization are absolutely stable. The derived criterion is formulated by LMIs that are easily solvable using various numerical methods [24]. Notations. The following notations are used in the Letter. Rn denotes the n-dimensional Euclidean space. Rn×m is the set of all n × m real matrices. For a real matrix X , X > 0 and X < 0 mean that X is a positive/negative definite symmetric matrix, respectively. I is an identity matrix with appropriate dimension and 0 is a null matrix with appropriate dimension. For given matrix A ∈ Rn×m such that rank( A ) = r, we define A ⊥ ∈ Rn×(n−r ) as the right orthogonal complement of A by A A ⊥ = 0. diag(· · ·) represents a block diagonal matrix. A ⊗ B indicates the Kronecker product of a n × m matrix A and a p × q matrix B, i.e.,

⎡a B ··· a B ⎤ 11 1m . .. ⎦ .. A ⊗ B = ⎣ .. . . . an1 B · · · anm B

2. Problem formulation Consider a dynamical network consisting of N identical linearly and diffusively coupled nodes and each node consisting of a ndimensional nonlinear Lur’e dynamical networks with time-delay described by the following state-space model M:



M:

x˙ i = Axi (t ) + A d xi (t − τ (t )) + B f (μi (t )),

(1)

μi (t ) = C xi (t ),

where τ (t ) is the time-delay satisfying 0  τ (t )  τ and τ˙ (t )  τd , xi (t ) = [xi1 , xi2 , . . . , xin ] T ∈ Rn are state vectors and μi (t ) ∈ R p are the output vectors of the Lur’e systems respectively, and A ∈ Rn×n , B ∈ Rn×m and C ∈ Rm×n are constant matrices. The nonlinearity of the Lur’e system f (·) : Rm → Rm is a memoryless vector valued function of which l-th element f l (·) is in a certain sector such that

bl 

f l (μi (t ))

μi (t )

 al ,

l = 1, . . . , m ,

(2)

where μi (t ) is i-th element of the μ(·), bl and al are lower/upper bound of the sector nonlinearity, respectively. We assume that the nonlinearity f (·) also satisfies a slope constraint such that

βl 

df l (μi (t )) dμi (t )

 αl ,

l = 1, . . . , m ,

(3)

βl and αl are lower/upper bound of the slope nonlinearity, respectively. Suppose the nodes are coupled with states xi (t ), i = 1, 2, . . . , N, then the nonlinear time-delay Lur’e network model N can be written as

 N:

x˙ i = Axi (t ) + A d xi (t − τ (t )) + B f (μi (t )) + c μi (t ) = C xi (t ),

N

j =1 h i j Γ

x j (t ),

(4)

where the constant c > 0 represents the coupling strength of the network, and Γ = (γi j )n×n is the inner-coupling matrix. H = (h i j ) N × N is the coupling matrix, standing for the coupling configuration of the network. If there Nis a connection between node i and node j (i = j), then h i j = h ji = 1; otherwise, h i j = h ji = 0 (i = j). The row sum of H are zero, i.e., j ,i = j h i j = −h ii , i = 1, 2, . . . , N. Lemma 2.1. (See [4].) The eigenvalues of an irreducible matrix H = (h i j ) ∈ R N × N with properties:



j =1, j =i

h i j = −h ii , i = 1, 2, . . . , N, satisfy the following

• 0 is an eigenvalue of H associated the eigenvector [1, 1, . . . , 1] T . • If h i j  0 for all 1  i, j  N, i = j, then the real parts of all eigenvalues of H are less than or equal to 0 and all possible eigenvalues with zero part are 0. In fact, 0 is its eigenvalue of multiplicity 1.

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D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227

Assume the network (4) has no isolate clusters, i.e., the network is connected. Under this circumstance, the coupling matrix H is symmetric and irreducible, and it will satisfy all the properties in Lemma 2.1. Also, suppose that the coupling matrix has q distinct different eigenvalues, i.e.,

Λ = diag{λ1 , λ2 , . . . , λ2 , λ3 , . . . , λ3 , . . . , λq , . . . , λq }

(5)

where λ1 = 0 is the maximum eigenvalue of multiply 1 and λi is the eigenvalue of multiply mi , i = 2, 3, . . . , q, satisfying m2 + · · · + mq and λ2 > λ3 > · · · > λq . Dynamical network (4) is said to achieve global (asymptotically) synchronization if





lim xi (t ) − s(t )2 = 0,

i = 1, 2, . . . , N ,

t →∞

(6)

where · 2 means the Euclidean norm, and s(t ) ∈ Rn is a solution of an isolate node with s˙ (t ) = As + A d s(t − τ (t )) + B f (σ (t )), which can be an equilibrium point, a periodic orbit, or even a non-periodic orbit where σ (t ) = C s(t ) ∈ R p are the output vectors of the isolated node. From the properties of the coupling matrix H , the following condition holds





s˙ (t ) = As(t ) + A d s t − τ (t ) + B f





σ (t ) + c

N 

h i j Γ s(t ).

(7)

j =1

By subtracting (7) from (4), one arrives at the error dynamical networks E



E:

e˙ i = Ae i (t ) + A d e i (t − τ (t )) + B φ(νi (t ); s(t )) + c νi (t ) = C e i (t ),

N

j =1 h i j Γ e j (t ),

(8)

where e i (t ) = xi (t ) − s(t ) and φ(νi (t ); s(t )) = f (μi (t ) + σ (t )) − f (σ ). Using the slope constraint of the nonlinear function in Eq. (3), we can derive new sector bounds for the error of the nonlinear functions f (μi (t )) and f (σ (t )). By the mean value theorem, there exists a constant δ ∈ (μi (t ), κi (t )) such that

fl









μi (t ) − f l σ (t ) =

df l (δ) d νi



μi (t ) − σ (t ) .

(9)

From the slope bounds given in Eq. (3), we have

βi  Since

df l (δ) d νi

 αi .

μ(t ) − κ (t ) = C (x(t ) − y (t )) = C e (t ), we also have



βl cl e i (t )  f l μi (t ) − f l σ (t )  αl cl e i (t )

(10)

where cl is l-th row vector of the matrix C . Let us denote νi (t ) = cl e i (t ), then we obtain a new nonlinear function φl (νi (t )) bounded by a sector that belongs to [βl , αl ] such that

βl 

φl (νi (t ))  αl , νi (t )

(11)

where φl (νi (t ))  f l (μi (t )) − f l (σ (t )). Reformulating system (8) in virtue of the Kronecker product as





¯ (t ) + A¯ d e t − τ (t ) + B¯ Φ C¯ e (t ); S (t ) E : e˙ (t ) = Ae

(12)

where

¯ = I N ⊗ A + c H ⊗ Γ, A e (t ) =



T e 1T (t ), . . . , e TN (t ) ,

¯ d = I N ⊗ Ad , A

B¯ = ( I N ⊗ B ),

T S (t ) = s (t ), . . . , s (t ) . T

C¯ = I N ⊗ C ,

T

(13)

Take Θ1 = I N ⊗ β , Θ2 = I N ⊗ α where β = diag(β1 , . . . , βm ), α = diag(α1 , . . . , αm ), then nonlinearity Φ(C¯ e (t ); S (t )) = [φ T (C e 1 (t ); s(t )), . . . , φ T (C e N (t ); s(t ))] belongs to the sector [Θ1 , Θ2 ]. The nonlinear function Φ(·) can be represented by a convex combination of the sector bounds such as Θ1 and Θ2 . We can rewrite the Φ(·) as below







Φ C¯ e (t ); S (t ) = λl νi (t ) Θ1 + λu νi (t ) Θ2 νi (t )

(14)

where



Φ(C¯ e (t ); S (t )) − Θ1 νi (t ) λl νi (t ) = , (Θ2 − Θ1 )νi (t )

Θ1 νi (t ) − Φ(C¯ e (t ); S (t )) λu νi (t ) = . (Θ2 − Θ1 )νi (t ) Since λl (νi ) + λu (νi ) = 1, λl (νi )  0 and λu (νi )  0, the Φ(·) can be represented using a convex hull:

(15)

D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227





Φ C¯ e (t ); S (t ) = i νi (t ) νi (t )

1221

(16)

where i (νi (t )) is an element of a convex hull Co{Θ1 , Θ2 }. Then, the nonlinear function φ(·) can be represented by





Φ C¯ e (t ); S (t ) =  ν (t ) ν (t )

(17)

where (ν (t )) belongs to the following set

   Φ := (ν )  (ν ) ∈ Co{Θ1 , Θ2 } .

(18)

Consequently, the error dynamical networks (12) can be treated as a (N × n)-dimensional nonlinear time-delay Lur’e system, and the synchronization problem of the nonlinear time-delay Lur’e dynamical networks can be transformed into the stabilization problem of the corresponding error dynamical networks. Lemma 2.2. (See [20].) For any constant matrix R ∈ Rn×n , R > 0, scalar τ > 0 and a vector function e (t ) : R → Rn such that the following integration is well defined, then

0



e˙ T (t + ξ ) R e˙ (t + ξ ) dξ  e (t ) T

−τ

e (t − τ ) T



−R R

−τ

R −R





e (t ) . e (t − τ )

(19)

Lemma 2.3. (See Projection Lemma [23].) Let x ∈ Rn , Ψ = Ψ T ∈ Rn×n , Γ ∈ Rm×n . The following statements are equivalent (i) x T Ψ x < 0 s.t. Γ x = 0, ∀x = 0, T

(ii) Γ ⊥ Ψ Γ ⊥ < 0. 3. Main results

In this section, we derive LMI conditions for the absolute stability for complex dynamical networks in form of (12) with time-varying delay in this section. A delay-dependent criterion will be proposed in the next theorem, which can be further simplified to other equivalent conditions. The convex representation (17) of the nonlinearity can be used to establish equality constraints. From (17), we have the following equality constraint as







Φ C¯ e (t ); S (t ) −  ν (t ) ν (t ) = Φ C¯ e (t ); S (t ) − C e (t ) = 0,

∀ ∈ Φ.

(20)

Furthermore, we can establish an additional equality constraint from the error dynamics (12) and the definition of e (t ) as follows:









¯ (t ) + A¯ d e t − τ (t ) + B¯ Φ C¯ e (t ); S (t ) = 0. e˙ (t ) − Ae

(21)

Next, define the augmented vectors for simplicity,

ζ (t ) = e˙ T (t ) e T (t ) e T (t − τ2 ) e T (t − τ (t )) e T (t − τ ) Φ(C¯ e (t ); S (t )) .

(22)

In the following theorem, a synchronization criterion for the Lur’e complex dynamical network (12) under equality constraints described above is presented using the projection lemma. Theorem 3.1. The error system described as Eq. (12) is absolutely stable for any delay τ (t ) such that 0  τ (t )  τ , τ˙ (t )  τd if there exist positive G G definite matrices P , Q 1 , Q 2 , G = 11 G 12 , R 1 , R 2 ∈ R Nn× Nn and a positive definite diagonal matrix S satisfying the following LMIs: 22

⊥T

⊥

Y 1 Γk T Γk⊥ Y 2 Γk⊥

Γk

< 0, <0

(23)

where Γk⊥ is a right orthogonal complement of



Γk =

¯ I −A 0 Θk C¯

and Y 1 and Y 2 are

¯d 0 −A 0 0

0 0

 − B¯ , −I

k = 1, 2

⎤ 0 0 0 Ξ11 Ξ12 0 ⎢  Ξ22 Ξ23 Ξ24 0 Ξ26 ⎥ ⎥ ⎢  Ξ33 Ξ34 Ξ35 0 ⎥ ⎢  Y1 = ⎢ ⎥,   Ξ44 0 0 ⎥ ⎢  ⎣     Ξ55 0 ⎦      Ξ66 ⎡

in which

(24)

⎤ 0 0 0 0 Π11 Π12 0 0 Π26 ⎥ ⎢  Π22 Π23 ⎥ ⎢  Π33 Π34 Π35 0 ⎥ ⎢  Y2 = ⎢ ⎥   Π44 Π45 0 ⎥ ⎢  ⎦ ⎣     Π55 0      Π66 ⎡

(25)

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D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227

Ξ11 =

τ 2

( R 1 + R 2 ),

Ξ12 = P ,

Ξ26 = C¯ T S (Θ1 + Θ2 ),

2

4

τ

Π33 = G 22 − G 11 − Π45 =

2

τ

τ

2

τ

R1 −

2

τ

2

τ

Π34 =

R2,

τ 2

τ

2

τ

R 1 + G 11 − 2C¯ T Θ1 S Θ2 C¯ ,

2

τ

2

Ξ34 =

R 2,

Ξ66 = −2S ,

τ

τ

Ξ23 = G 12 ,

Ξ35 = −G 12 +

R1,

Π11 =

Π23 = G 12 +

Π35 = −G 12 ,

R2,

2

R2,

R 1 + G 11 − 2C¯ T Θ1 S Θ2 C¯ ,

Π55 = − Q 2 − G 22 −

R2,

τ

2

R1 −

Ξ55 = − Q 2 − G 22 −

R1,

Π22 = Q 1 + Q 2 −

R1,

2

Ξ33 = G 22 − G 11 −

Ξ44 = −(1 − τd ) Q 1 − Ξ14 =

Ξ22 = Q 1 + Q 2 −

2

τ

τ 2

2

τ

Ξ24 =

τ

R1,

R2,

( R 1 + R 2 ),

Π12 = P ,

Π26 = C¯ T S (Θ1 + Θ2 ),

R1,

Π44 = −(1 − τd ) Q 1 −

4

τ

R2,

Π66 = −2S .

R2,

2

(26)

Proof. Consider the following Lyapunov–Krasovskii functional:





















V e (t ) = V 1 e (t ) + V 2 e (t ) + V 3 e (t ) + V 4 e (t )

(27)

where





V 1 e (t ) = e (t ) T P e (t ),

t



V 2 e (t ) =

t T

t −τ

t −τ (t )



e T (ξ ) Q 2 e (ξ ) dξ,

e (ξ ) Q 1 e (ξ ) dξ +



t



V 3 e (t ) =

T 

e (ξ )

e (ξ − τ2 )

t −τ /2

G 11



G 12 G 22

V 4 e (t ) =



e (ξ )

e (ξ − τ2 )

dξ,

t− τ /2 t

t t





e˙ (κ ) R 1 e˙ (κ ) dξ dκ +

e˙ T (κ ) R 2 e˙ (κ ) dξ dκ .

T

t −τ

t −τ /2 ξ

ξ

First, the time derivative of V 1 (e (t )) with respect to time along the trajectory of (12) is





V˙ 1 e (t ) = e˙ T (t ) P e (t ) + e T (t ) P e˙ (t ) = ζ T (t )Ω1 ζ (t ) where

⎡

0

⎢ ⎢ ⎢ Ω1 = ⎢ ⎢ ⎣ 

P 0

0 0 0 0 0 0  0

   

0 0 0 0 0

    

(28)

⎤

0 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎦ 0 0

(29)

Secondly, time derivative of V 2 (e (t )) is as follows:

V˙ 2 (e (t )) =



T 

e (t ) e (t − τ (t ))

Q1 0



0

−(1 − τd ) Q 1

e (t )



e (t − τ (t ))

 +

e (t ) e (t − τ )

T 

Q2 0

0 −Q 2



e (t ) e (t − τ )



T

= ζ (t )Ω2 ζ (t ) where

⎡

0

⎢ ⎢ ⎢ Ω2 = ⎢ ⎢ ⎣ 

0 Q1 + Q2

   

(30)

0 0 0

0 0 0

 −(1 − τd ) Q 1    

0 0 0 0 −Q 2



⎤

0 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎦ 0 0

(31)

Similarly, time derivative of V 3 (e (t )) is found as



V˙ 3 e (t ) = where



e (t ) e (t − τ2 )

T 

G 11



G 12 G 22







e (t ) e (t − τ2 ) − τ e (t − 2 ) e (t − τ )

T 

G 11



G 12 G 22





e (t − τ2 ) = ζ T (t )Ω3 ζ (t ), e (t − τ )

(32)

D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227

⎡

0

⎢ ⎢ ⎢ Ω3 = ⎢ ⎢ ⎣ 

0 G 11

0 G 12 G 22 − G 11

   

  

0 0 0 0 0 −G 12 0 0  −G 22





1223

⎤

0 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎦ 0 0

(33)

Next, the time-varying delays τ (t ) can be considered in two intervals [0, τ /2) and (τ /2, τ ], thus the time derivative of V 4 (e (t )) is obtained as two cases at each interval by using Jensen’s inequality (Lemma 2.2). (i) Case 1: 0  τ (t )  τ /2. For this case, we have





V˙ 4 e (t ) 

τ 2

e˙ T (t )( R 1 + R 2 )˙e (t ) + 2

+

τ 2

+

 

τ

e (t − τ (t )) e (t − τ2 ) e (t − τ2 ) e (t − τ )

T 

T 

2



T 

e (t )

τ e(t − τ (t ))

−R1 R1

−R2

R1

−R1 

R2 −R2

R2



−R1 R1

e (t − τ (t )) e (t − τ2 )

e (t − τ2 ) e (t − τ )



R1 −R1



e (t )



e (t − τ (t )) (34)



T

= ζ (t )Ω4 ζ (t ) where

(35)

⎡ τ (R + R ) 0 0 0 0 1 2 2 2 ⎢  − τ2 R 1 0 R 0 τ 1 ⎢ 2 ⎢   − τ2 R 1 − τ2 R 2 τ2 R 1 ⎢ τ R2 Ω4 = ⎢ 4    − τ R1 0 ⎢ ⎣     − τ2 R 2     

(ii) Case 2:

0⎤ 0⎥

⎥

0⎥ ⎥ 0⎥ ⎥

0 0

(36)

.

⎦

τ /2  τ (t )  τ .





V˙ 4 e (t ) 

τ 2

+ +

e˙ T (t )( R 1 + R 2 )˙e (t ) + 2



2

τ



τ

T  e (t − τ2 ) −R2

τ e(t − τ (t )) 

2

e (t − τ (t )) e (t − τ )

T 

R2

−R2 R2

e (t ) e (t − τ2 ) R2 −R2 R2

−R2

T 

 

−R1 R1

e (t − τ2 ) e (t − τ (t )) e (t − τ (t )) e (t − τ )

R1 −R1





e (t ) e (t − τ2 )



(37)



= ζ T (t )Ω5 ζ (t ) where

(38)

⎡ τ (R + R ) 0 0 0 0 1 2 2 2 ⎢  − τ2 R 1 0 R 0 1 τ ⎢ 2 ⎢   − τ2 R 1 − τ2 R 2 τ2 R 1 τ R2 Ω5 = ⎢ ⎢ 4    − τ R1 0 ⎢ ⎣ 2     − τ R2     

0⎤ 0⎥

⎥

0⎥ ⎥ 0⎥ ⎥

0 0

(39)

.

⎦

From the sector constraint of the nonlinearity φ(·), the following inequality is obtained



T



Φ C¯ e (t ); S (t ) − Θ2 ν (t ) Φ C¯ e (t ); S (t ) − Θ1 ν (t )  0.

(40)

By applying the well-known S-procedure [24] to Eq. (40) and utilizing Eqs. (37)–(41), we have the following inequalities at each intervals: (i) Case 1: 0  τ (t )  τ /2

V˙ e (t )  ζ T (t )(Ω1 + Ω2 + Ω3 + Ω4 )ζ (t ) − 2 Φ C¯ e (t ); S (t ) − Θ2 ν (t )



T



S Φ C¯ e (t ); S (t ) − Θ1 ν (t ) .

(41)

τ /2  τ (t )  τ



T



V˙ e (t )  ζ T (t )(Ω1 + Ω2 + Ω3 + Ω5 )ζ (t ) − 2 Φ C¯ e (t ); S (t ) − Θ2 ν (t ) S Φ C¯ e (t ); S (t ) − Θ1 ν (t ) .

(42)







(ii) Case 2:

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D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227

By rewriting (41) and (42) for ζ (t ) at each intervals, we have





V˙ e (t )  ζ T (t )Y 1 ζ (t ),

for 0  τ (t )  τ /2,



V˙ e (t )  ζ T (t )Y 2 ζ (t ),

for τ /2  τ (t )  τ .

(43)

Next, the equality constraints (20), (21) for the augmented state (22) and the nonlinear function Φ(·) are rewritten with the extended vector ζ (t ) as below

Γ ζ (t ) = 0 where

(44)



¯ I −A Γ = 0 C¯

 − A¯ d 0 − B¯ . 0 0 −I

0 0

(45)

Therefore, a sufficient condition for stability of the error dynamics (12) with a given time delay is that









V˙ e (t )  ζ T (t )Y 1 ζ (t ) < 0, V˙ e (t )  ζ T (t )Y 2 ζ (t ) < 0,

∀ζ (t ) = 0

(46)

subject to

Γ ζ (t ) = 0.

(47)

Applying the projection lemma to Eqs. (46) and (47), we obtain the LMIs:

ζ T (t )Γ ⊥T Y 1 Γ ⊥ ζ (t ) < 0, ζ T (t )Γ ⊥T Y 2 Γ ⊥ ζ (t ) < 0,

∀ζ (t ) = 0.

(48)

Since  is element of the convex set Φ , Eq. (48) is satisfied if the inequality (23) holds for each point pairs k = 1, 2.

2

Corollary 3.1. For fixed delay τ > 0, the error system described as Eq. (12) is absolutely stable for any delay τ if there exist positive definite matrices G G P , Q 1 , G = 11 G 12 , R 1 , R 2 ∈ R Nn× Nn and a positive definite diagonal matrix S satisfying the following LMI: 22

Γk⊥ Y Γk⊥ < 0 T

(49)

where Γk⊥ is a right orthogonal complement of

⎡

− A¯ −I

I ⎢0 Γk = ⎣ 0 0 and Y is

0

Θk C¯

0 I −I 0

⎤ − A¯ d 0 0 B¯ 0 I 0 0⎥ ⎦, I 0

0 I 0 0

0 I

k = 1, 2,

(50)

⎤ 0 0 0 0 Ξ11 Ξ12 0 0 0 Ξ27 ⎥ ⎢  Ξ22 Ξ23 0 ⎥ ⎢  Ξ33 Ξ34 0 0 0 ⎥ ⎢  ⎥ ⎢ Y =⎢    Ξ44 0 0 0 ⎥ ⎥ ⎢    Ξ55 0 0 ⎥ ⎢  ⎣      Ξ66 0 ⎦       Ξ77 ⎡

(51)

where

Ξ11 =

τ

R1 +

τ

R2, 2 2 Ξ27 = C¯ T S (Θ1 + Θ2 ), 2 2 Ξ55 = − R 1 − R 2 ,

τ

Ξ22 = Q 1 + G 11 + C¯ T Θ1 S Θ2 C¯ ,

Ξ12 = P ,

Ξ33 = G 22 − G 11 , 2

Ξ66 = − R 2 ,

τ

τ

Proof. First, let us define an extended vector

Ξ34 = −G 12 ,

Ξ23 = G 12 ,

Ξ44 = − Q 1 − G 22 ,

Ξ77 = −2S .

(52)

χ (t ) as



χ (t ) = e˙ T (t ) e T (t ) e T (t − τ2 ) e T (t − τ ) e T (t ) − e T (t − τ2 ) e T (t − τ2 ) − e T (t − τ ) Φ(C¯ e(t ); S (t )) .

(53)

Next, consider the following Lyapunov–Krasovskii functional:

















V e (t ) = V 1 e (t ) + V 2 e (t ) + V 3 e (t ) where

(54)

D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227



1225



V 1 e (t ) = e (t ) T P e (t ),

t 

t



V 2 e (t ) =

T

e (ξ ) Q 1 e (ξ ) dξ +

t −τ

t −τ /2

t



V 3 e (t ) =

T 

e (ξ )

e (ξ − τ2 )

G 11



G 12 G 22



e (ξ )

 dξ,

e (ξ − τ2 )

t− τ /2 t

t e˙ (κ ) R 1 e˙ (κ ) dξ dκ +

e˙ T (κ ) R 2 e˙ (κ ) dξ dκ .

T

t −τ

t −τ /2 ξ

ξ

Time derivative of V 1 (e (t )) with respect to time along the trajectory of (12) is





V˙ 1 e (t ) = e˙ T (t ) P e (t ) + e T (t ) P e˙ (t ).

(55)

Next, time derivative of V 3 (e (t )) is found as







V˙ 2 e (t ) =

e (t ) e (t − τ )

T 

Q2 0

0 −Q 2







e (t ) e (t ) + e (t − τ ) e (t − τ2 )

T 



 T   e (t − τ2 ) G 11 G 12 e (t − τ2 ) . − e (t − τ )  G 22 e (t − τ ) 

G 11

G 12 G 22



e (t ) e (t − τ2 )



(56)

The following inequality for time derivative of V 3 (e (t )) can be obtained by using Jensen’s inequality:





τ

V˙ 3 e (t ) 

2

e˙ T (t ) R 1 e˙ (t ) +



τ 2

e˙ T (t ) R 2 e˙ (t ) +

2

τ



e (t ) e (t − τ2 )

T 

 T   2 e (t − τ ) e (t − τ2 ) −R2 R2 2 . + R2 −R2 e (t − τ ) τ e(t − τ )

−R1 R1

R1 −R1



e (t ) e (t − τ2 )



(57)

From the sector constraint of the nonlinearity φ(·), the following inequality is obtained



T



Φ C¯ e (t ); S (t ) − Θ2 ν (t ) Φ C¯ e (t ); S (t ) − Θ1 ν (t )  0.

(58)

By applying the well-known S-procedure [24] to (58) and utilizing Eqs. (55)–(57), we have





V˙ e (t )  χ T (t )Y χ (t ).

(59)

Next, the equality constraints (20), (21) for the augmented state (53) and the nonlinear function Φ(·) are rewritten with the extended vector χ (t ) as below



I 0

 − A¯ 0 − A¯ d 0 0 − B¯ χ (t ) = 0. C¯ 0 0 0 0 −I

(60)

Moreover, two additional equality constraints can be derived from e (t ) − e (t − τ2 ) and e (t − τ2 ) − e (t − τ ) as below



0 −I 0 0

I

−I

0 I

I 0 0 0 I 0



χ (t ) = 0.

(61)

Combining (60) with (61) yields the following equality constraint

Γ χ (t ) = 0 where

⎡

¯ I −A ⎢ 0 −I Γ =⎣ 0 0 0 C¯

(62)

0 I −I 0

⎤ − A¯ d 0 0 B¯ 0 I 0 0⎥ ⎦. I 0

0 I 0 0

(63)

0 I

Therefore, a sufficient condition for stability of the error dynamics (12) with a given time delay is that





V˙ e (t )  χ T (t )Y χ (t ) < 0,

∀χ (t ) = 0

(64)

subject to

Γ χ (t ) = 0.

(65)

Applying the projection lemma to (64) and (65), we obtain the LMI.

χ T (t )Γ ⊥T Y Γ ⊥ χ (t ) < 0, ∀χ (t ) = 0. Since  is element of the convex set Φ , Eq. (66) is satisfied if the inequality (49) holds for each point pairs k = 1, 2.

(66)

2

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D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227

(a) For

τ = 2.8 at τd = 0

(b) For

τ = 3.1 at τd = 0

Fig. 1. Synchronization errors for the network with c = 1,

τ = 2.8 and τ = 3.1.

4. Numerical example In this section, a numerical example is used to illustrate the effectiveness of the proposed synchronization criterion given in Theorem 3.1. For the sake of simplicity, consider the network (12) consisted of SISO Lur’e system with time-delay ( N = 5) and the parameters of these identical nodes can be chosen as:



A=

 −1.2 0.1 , −0.1 −1



Ad =

 −0.6 0.7 , −1 −0.8

B = [ 0.2

0.3 ] ,

C = [1

1]

(67)

and f (μ) = 12 (|μ + 1| − |μ − 1|) belonging to sector b = 0, a = 1. Suppose that each pair of two connected time-delay Lur’e system are linked together through their identical sub-state variables, i.e., Γ = I . The coupling strength c = 1, and assume that the coupling matrix of the entire dynamical network is described in the following form, which is a so-called star-like network:

⎡

−4

⎢ 1 ⎢ H =⎢ 1 ⎣ 1 1

1 −1 0 0 0

⎤

1 1 1 0 0 0 ⎥ ⎥ −1 0 0 ⎥. ⎦ 0 −1 0 0 0 −1

(68)

By using Theorem 1 in [22], it is found that the maximum delay bound is τ = 1.221, for which the synchronized states of the network are asymptotically stable. By Corollary 3.1 in this Letter, however, it is found that the maximum delay bound for the synchronized states to be asymptotically stable is τ = 1.670, much larger than that obtained in [22]. In Table 1, we found the maximum allowable time delay τ¯ using Corollary 3.1 and method in [22] are compared with our result. From Table 1, one can see that the proposed method presented in this Letter provides less conservative result than the previous result. Fig. 1(a) shows the synchronization errors between the

D.H. Ji et al. / Physics Letters A 374 (2010) 1218–1227

1227

Table 1 The maximum allowable delay for fixed delay. For fixed delay

The maximum allowable delay τ¯

τd

τd = 0

Xu et al. [22] Corollary 3.1

Table 2 The maximum allowable delay for different The upper bound for

τd = 0.1 τd = 0.5 τd = 0.9

τd

1.221 1.670

τd .

The maximum allowable delay τ¯ 1.462 1.131 1.108

states of node i and node i + 1 for given chosen initial conditions for the case with τ = 2.8 and c = 1. This figure shows that the errors between the synchronized states converge to zero under the above conditions. It is important to note that the obtained maximum delay bound τ = 2.8 by Corollary 3.1 is very close to the true value of the maximum delay bound beyond which the synchronized states is not asymptotically stable. To show this, we assume the time-delay in the network to be τ = 3.1. Fig. 1(b) shows the synchronization errors between the states of node i and node i + 1 for given initial conditions with τ = 3.1 and c = 1. This figure shows that the errors between the synchronized states do not converge to zero under the above conditions. Moreover, we obtained the maximum allowable delay for stability with different value τd from synchronization criteria in Table 2 by Theorem 3.1. 5. Conclusion This Letter has presented new synchronization stability criteria for Lur’e type complex dynamical networks with coupling delays. The delay-dependent conditions in terms of LMIs have been derived, which guarantee the synchronized states to be asymptotically stable. It has been shown less conservative than the existing result. Acknowledgements The authors would like to thank the editor and a reviewer for their valuable comments and suggestions. This research was supported by the Yeungnam University research Grant (209-A-380-120) in 2009. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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