Sampled-data synchronization for complex networks based on discontinuous LKF and mixed convex combination

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Sampled-data synchronization for complex networks based on discontinuous LKF and mixed convex combination Junyi Wang, Huaguang Zhang, Zhanshan Wang

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S0016-0032(15)00282-3 http://dx.doi.org/10.1016/j.jfranklin.2015.07.007 FI2397

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26 November 2014 16 May 2015 9 July 2015

Cite this article as: Junyi Wang, Huaguang Zhang, Zhanshan Wang, Sampled-data synchronization for complex networks based on discontinuous LKF and mixed convex combination, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.07.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Sampled-data synchronization for complex networks based on discontinuous LKF and mixed convex combination Junyi Wang, Huaguang Zhang∗, Zhanshan Wang College of Information Science and Engineering, Northeastern University, Box 134, 110819, Shenyang, China

Abstract This paper investigates the sampled-data synchronization problem of delayed complex networks with aperiodic sampling interval based on enhanced input delay approach. By introducing an improved discontinuous LyapunovKrasovskii functional (LKF), new delay-dependent synchronization criteria are obtained using Wirtinger’s integral inequality and mixed convex combination, which fully utilize the upper bound on variable sampling interval and the sawtooth structure information of varying input delay. The derived criteria are less conservative than the existing ones. In addition, the desired sampled-data controllers are obtained by solving a set of linear matrix inequalities. Finally, numerical examples are provided to demonstrate the feasibility of the proposed method. Keywords: Sampled-data synchronization, Wirtinger’s integral inequality, discontinuous Lyapunov-Krasovskii functional, mixed convex combination 1. Introduction Complex networks are a large set of interconnected nodes, in which each node represents a dynamical system and the edges represent the connections. ∗

Corresponding author Email addresses: [email protected] (Junyi Wang), [email protected] (Huaguang Zhang), [email protected] (Zhanshan Wang)

Preprint submitted to Journal of the Franklin Institute

July 24, 2015

They are ubiquitous in the real world, such as food-webs, ecosystems, biological neural networks, the Internet, social networks, and global economic markets [1–4]. Synchronization is one of the most typical collective behaviors in nature, which has received a lot of interests among complex networks [2–20]. Many natural phenomena can be explained by the synchronization of coupled oscillators. Furthermore, some synchronization phenomena are very useful in our daily life, such as the synchronous transfer of digital or analog signals in communication networks. Nowadays, many synchronization problems, including pinning synchronization [8–10, 15, 20], stochastic synchronization [3, 12, 19], impulsive synchronization [13, 14], adaptive synchronization [16], distributed synchronization [17], for complex dynamical networks have been investigated, and corresponding synchronization criteria have been obtained. With the rapid development of computer hardware, the sampled-data control technology has shown more and more superiority over other control approaches [25]. In the sampled-data control, the control signals are updated only at sampling instants and kept constant during the sampling period. Moreover, choosing proper sampling interval in sampled-data control systems is more important for designing suitable controllers. Consequently, sampled-data systems have been investigated extensively [21–31]. In [22], sampled-data control problem of linear systems has been investigated by using input delay method. In [23], the less conservative sampled-data stabilization criteria have been obtained by using Wirtinger’s integral inequality. In [30], the synchronization problem of neural networks with time-varying delay under sampled-data control has been studied. After that, the sampled-data control has been investigated for T-S fuzzy systems with aperiodic sampling interval in [21]. In [25], the sampled-data synchronization control has been investigated for complex networks with time-varying coupling delay based on input delay approach. Based on [25], the sampled-data synchronization of complex dynamical networks with time-varying coupling delay has been studied in [26], and the less conservative synchronization results have been obtained. However, in [25, 26], there is no constraint on the input delay derivative induced by sampling, and only the traditional LKF and Jensen’s inequality are used to obtain the synchronization criteria. In this case, the sawtooth structure and all available information about the actual sampling pattern are overlooked, which inevitably leads to the conservatism of the results. After that, the less conservative sampled-data synchronization criteria have been obtained for complex dynamical networks with time-varying delay 2

and uncertain sampling using novel time-dependent LKF in [27]. In this paper, the discontinuous LKF Vq (e(t))(q = 5, 6, 7) that contain tp and tp+1 are adopted, which could make full use of the sawtooth structure characteristic of sampling input delay. In addition, the mixed convex combination technique is utilized, which could introduce the free matrices. Hence, the less conservative synchronization criteria are obtained in this paper. Motivated by the above discussions, we study the problem of sampleddata synchronization for complex dynamical networks based on enhanced input delay approach. In addition, the discontinuous LKF, Wirtinger’s integral inequality and mixed convex combination included linear convex combination and reciprocally convex combination are employed in order to fully utilize the information of actual sampling pattern. Hence, the less conservative sampled-data synchronization criteria are obtained. Finally, numerical examples are given to show the effectiveness of the theoretical results. The rest of this paper is organized as follows. In Section 2, some preliminaries and complex networks with time-varying are introduced. In Section 3, the novel sampled-data synchronization criteria for complex networks are obtained. In Section 4, numerical simulations are given to demonstrate the effectiveness of the proposed results. Finally, conclusions are drawn in Section 5. Notation: Throughout this paper, Rn and Rn×n denote the n-dimensional Euclidean space and the set of all n × n real matrices respectively. The  ·  stands for the Euclidean vector norm. The symbol ⊗ represents Kronecker product. X T denotes the transpose of matrix X. X ≥ 0 (X < 0), where X ∈ Rn×n , means that X is real positive semidefinite matrix (negative defiFor  a matrix nite matrix). In×n represents the n-dimensional identity matrix.  X Y A ∈ Rn×n , λmin (A) denotes the minimum eigenvalue of A. stands ∗ Z   X Y . Matrices, if their dimensions are not explicitly stated, are for YT Z assumed to be compatible for algebraic operations. 2. Problem formulation Consider the following complex dynamical networks consisting of N dynamical nodes. Each node of the networks is an n-dimensional dynamical

3

system. x˙ i (t) = f (xi (t)) + c

N 

Gij Γxj (t − τ (t)) + ui (t), i = 1, 2, · · · , N,

(1)

j=1

where xi (t) = [xi1 (t), xi2 (t), . . . , xin (t)]T ∈ Rn is the state vector of the ith node. f (xi (t)) = [f1 (xi1 (t)), f2 (xi2 (t)), . . . , fn (xin (t))]T ∈ Rn is continuous vector valued function. c > 0 is coupling strength. τ (t) is interval timevarying delay and satisfies 0 ≤ τ1 ≤ τ (t) ≤ τ2 , τ˙ (t) ≤ h, where τ1 , τ2 (τ1 < τ2 ), and h are constants. Γ = diag[γ1 , γ2, · · · , γn ] is the inner coupling matrix with γi ≥ 0, i = 1, 2, · · · , n. G is the coupling configuration matrix representing the topological structure of network and satisfies the following condition: if there exists a connection from node j to node i (j = i), then Gij > 0; otherwise, Gij = 0; and the diagonal elements of matrix G are defined by N  Gij , i = 1, 2, · · · , N. Gii = − j=1,j=i

In this paper, consider the following state trajectory of the isolate node s(t) ˙ = f (s(t)).

(2)

By defining the error signal as ei (t) = xi (t) − s(t), the error dynamic of complex networks can be obtained as follows e˙ i (t) = g(ei (t)) + c

N 

Gij Γej (t − τ (t)) + ui (t), i = 1, 2, · · · , N,

(3)

j=1

where g(ei (t)) = f (xi (t)) − f (s(t)). The control signal is generated by using a zero-order-hold (ZOH) function with a sequence of hold times 0 = t0 < t1 < · · · < tp < · · · . The sampleddata control input may be represented as discrete control signal ui (t) = udi (tp ) = udi (t − (t − tp )) = udi (t − d(t)), t ∈ [tp , tp+1 ),

(4)

where i = 1, 2, · · · , N, d(t) = t − tp . The delay d(t) is a piecewise linear ˙ = 1. Sampling interval dp satisfies the following condition function and d(t) 0 ≤ d(t) = t − tp ≤ dp = tp+1 − tp ≤ d, 4

(5)

where d > 0 is the largest sampling interval and bounded. The sampled-data state feedback controllers are designed ui (t) = Li ei (tp ), t ∈ [tp , tp+1 ), i = 1, 2, · · · , N,

(6)

where Li (i = 1, 2, · · · , N) are the feedback controller gain matrices to be determined. In addition, we don’t require the sampling to be periodic. Substituting (6) into (3), we have the following closed-loop system e˙ i (t) = g(ei (t)) + c

N 

Gij Γej (t − τ (t)) + Li ei (t − d(t)), t ∈ [tp , tp+1 ),

(7)

j=1

where i = 1, 2, · · · , N. Assumption 1. (see [32]). The nonlinear function f : Rn → Rn satisfies [f (x) − f (y) − U(x − y)]T [f (x) − f (y) − V (x − y)] ≤ 0, ∀x, y ∈ Rn where U and V are constant matrices of appropriate dimensions. Lemma 1. (see [21]). Let Y > 0 and ω(s) be appropriate dimensional vect tor, respectively. Then, we have the following inequality: − t12 ω T (s)Y ω(s)ds ≤ t (t2 −t1 )η T (t)F T Y −1 F η(t) + 2η T (t)F T t12 ω(s)ds, where the matrix F and the vector η(t) independent on the integral variable are arbitrary appropriate dimensional ones. Lemma 2. (see [33, 34]). For matrices Zi (i = 1, 2, 3) with proper dimensions, Z1 + αZ2 + (1 − α)Z3 < 0 holds for ∀α ∈ [0, 1] if and only if the following set of inequalities hold Z1 + Z2 < 0, Z1 + Z3 < 0. Lemma 3. (see [35]) For matrices T , R = RT > 0, scalars d1 ≤ d(t) ≤ d2 , a vector function x˙ : [−d2 , −d1 ] → Rn such that the integration in the following inequality is well defined, then it holds that ⎤ ⎡  t−d1 −R R+T −T (d1 − d2 ) x˙ T (s)Rx(s)ds ˙ ≤ υ T (t) ⎣ ∗ −2R − T − T T R + T ⎦ υ(t), t−d2 ∗ ∗ −R

where υ T (t)  = xT (t − d1 ) xT (t − d(t)) xT (t − d2 ) ,  −R T ≤ 0. ∗ −R 5

Lemma 4. (see [23]). Let z(t) ∈ W [a, b) that denotes the space of functions φ : [a, b] → Rn , which are absolutely continuous on [a, b), have a finite limθ→b− φ(θ) and have square integrable first-order derivatives with the norm b 2 ˙ ds]1/2 . Furthermore, if z(a) = 0, then φW = maxθ∈[a,b] |φ(θ)| + [ a |φ(s)| forany n × n matrix R > 0 thefollowing inequality holds: b b π 2 a z T (s)Rz(s)ds ≤ 4(b − a)2 a z˙ T (s)Rz(s)ds. ˙ In this paper, the purpose is to design a set of sampled-data controls Li (i = 1, 2, · · · , N) such that the error system (7) is asymptotically stable, that is, system (1) is sampled-data synchronization. 3. Main results In this section, the sampled-data synchronization criteria of complex dynamical networks (1) are derived based on a novel discontinuous LKF and mixed convex combination technique. Let e(t) = [eT1 (t), eT2 (t), . . . , eTN (t)]T , g(e(t)) = [g T (e1 (t)), g T (e2 (t)), . . . , ¯ = G ⊗ Γ, L = diag[L1 , L2 , · · · , LN ], U¯ = (IN ⊗U )T (IN ⊗V ) g T (eN (t))]T , Γ 2 T T T N ⊗V ) + (IN ⊗V ) 2 (IN ⊗U ) , V¯ = − (IN ⊗U ) +(I . Then, error dynamical (7) can be 2 rewritten as follows ¯ − τ (t)) + Le(tp ). (8) e(t) ˙ = g(e(t)) + cΓe(t Theorem 1 Under Assumption 1, error dynamical system (8) is asymptotically stable if there exist matrices P > 0, Q1 > 0, Q2 > 0, Q3 > 0, W1 > 0, W2 > 0, Z1 > 0, Z2 > 0, N1 > 0, N2 > 0, N3 > 0, and matrices Y , X1 , X2 , F1 , F2 , and a scalar σ > 0, such that the following matrix inequalities are satisfied

T

N1 0nN ×nN InN ×nN 0nN ×6nN Π1 =Π + d 0nN ×nN InN ×nN 0nN ×6nN

T

+ 2d 0nN ×2nN InN ×nN 0nN ×5nN N2 0nN ×2nN InN ×nN 0nN ×5nN <0, (9)   Π Y < 0, (10) Π2 = ∗ − 1d N1   W2 X1 ≥ 0, (11) ∗ W2   Z 2 X2 ≥ 0, (12) ∗ Z2 6

where Π = Ψ + Y8nN ×nN 08nN ×nN −Y8nN ×nN 08nN ×5nN

T + Y8nN ×nN 08nN ×nN −Y8nN ×nN 08nN ×5nN , ⎛ ⎞ Ψ11 Ψ12 Ψ13 Ψ14 Ψ15 Ψ16 0 Ψ18 ⎜ ∗ Ψ22 Ψ23 0 0 Ψ26 0 Ψ28 ⎟ ⎜ ⎟ ⎜ ∗ ⎟ ∗ Ψ Ψ 0 0 0 0 33 34 ⎜ ⎟ ⎜ ∗ ⎟ ∗ ∗ Ψ 0 0 0 0 44 ⎟, Ψ=⎜ ⎜ ∗ ∗ ∗ ∗ Ψ55 Ψ56 Ψ57 0 ⎟ ⎜ ⎟ ⎜ ∗ ⎟ Ψ 0 ∗ ∗ ∗ ∗ Ψ 66 67 ⎜ ⎟ ⎝ ∗ ∗ ∗ ∗ ∗ ∗ Ψ77 0 ⎠ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ψ88 1 π2 ¯ Ψ12 = −F1 + P, Ψ11 = Q1 + Q2 − W1 + Z1 − d Z2 − 4 N3 − σ U, 2 ¯ Ψ13 = 1d Z2 − d1 X2 + π4 N3 + F1 L, Ψ14 = 1d X2 , Ψ15 = W1 , Ψ16 = cF1 Γ, 2 2 2 T ¯ Ψ18 = −σ V +F1 , Ψ22 = τ1 W1 +(τ2 −τ1 ) W2 +dZ2 +d N3 −F2 −F2 , Ψ23 = F2 L, ¯ Ψ28 = F2 , Ψ33 = − 1 Z2 + 1 (X2 + X T ) − 1 Z2 − dN2 − π2 N3 , Ψ26 = cF2 Γ, 2 d d d 4 Ψ34 = − 1d X2 + 1d Z2 , Ψ44 = −Z1 − d1 Z2 , Ψ55 = −Q2 + Q3 − W1 − W2 , Ψ56 = W2 − X1 , Ψ57 = X1 , Ψ66 = −(1 − h)Q1 − W2 + X1 + X1T − W2 , Ψ67 = −X1 + W2 , Ψ77 = −Q3 − W2 , Ψ88 = −σI, Proof Consider the following discontinuous LKF for error dynamical system (8) as V (e(t)) =

7 

[Vq (e(t))], t ∈ [tp , tp+1 ).

(13)

q=1

V1 (e(t)) = eT (t)P e(t),  t  T V2 (e(t)) = e (s)Q1 e(s)ds + t−τ (t)

 V3 (e(t)) = τ1 

0

−τ1



t

t−τ1



t

t+θ

eT (s)Q2 e(s)ds + 

e˙ T (s)W1 e(s)dsdθ ˙ + (τ2 − τ1 ) 

t

0



−τ1 

−τ2

(14) t−τ1

t−τ2 t t+θ

eT (s)Q3 e(s)ds, (15)

e˙ T (s)W2 e(s)dsdθ, ˙ (16)

t

eT (s)Z1 e(s)ds + e˙ T (s)Z2 e(s)dsdθ, ˙ t−d −d t+θ  t V5 (e(t)) = (d − t + tp ) e˙ T (s)N1 e(s)ds, ˙

V4 (e(t)) =

tp

7

(17) (18)

V6 (e(t)) = (tp+1 − t)(t − tp )eT (tp )N2 e(tp ), (19)  t  π2 t V7 (e(t)) = d2 e˙ T (s)N3 e(s)ds ˙ − [e(s) − e(tp )]T N3 [e(s) − e(tp )]ds. 4 tp tp (20) Calculating the time derivative of V (e(t)), one has V˙ 1 (e(t)) =2eT (t)P e(t). ˙ (21) V˙ 2 (e(t)) ≤eT (t)Q1 e(t) − (1 − h)eT (t − τ (t))Q1 e(t − τ (t)) + eT (t)Q2 e(t) − eT (t − τ1 )Q2 e(t − τ1 ) + eT (t − τ1 )Q3 e(t − τ1 ) − eT (t − τ2 )Q3 e(t − τ2 ).

(22)

˙ − eT (t)W1 e(t) + 2eT (t)W1 e(t − τ1 ) V˙ 3 (e(t)) ≤τ12 e˙ T (t)W1 e(t) − eT (t − τ1 )W1 e(t − τ1 ) + (τ2 − τ1 )2 e˙ T (t)W2 e(t) ˙  t−τ1 e˙ T (s)W2 e(s)ds. ˙ − (τ2 − τ1 )

(23)

t−τ2

According to Lemma 3, one obtains  t−τ1 e˙ T (s)W2 e(s)ds ˙ − (τ2 − τ1 ) t−τ2 ⎡ ⎤    I 0 W X I −I 0 2 1 T

1 (t), ≤ − 1 (t) ⎣ −I I ⎦ ∗ W2 0 I −I 0 −I

where 1T (t) = eT (t − τ1 ) eT (t − τ (t)) eT (t − τ2 ) . V˙ 4 (e(t)) =eT (t)Z1 e(t) − eT (t − d)Z1e(t − d) + de˙ T (t)Z2 e(t) ˙  t − e˙ T (s)Z2 e(s)ds. ˙ t−d  t T ˙ ˙ − e˙ T (s)N1 e(s)ds. ˙ V5 (e(t)) =(d − d(t))e˙ (t)N1 e(t)

(24)

(25) (26)

t−d(t)

V˙ 6 (e(t)) =(dp − 2d(t))eT (t − d(t))N2 e(t − d(t)) ≤(d − 2d(t))eT (t − d(t))N2 e(t − d(t))

T = − de(tp )N2 e(tp ) + 2(d − d(t))ξ T (t) 0nN ×2nN InN ×nN 0nN ×5nN

(27) N2 0nN ×2nN InN ×nN 0nN ×5nN ξ(t). 8

π2 ˙ − (e(t) − e(tp ))T N3 (e(t) − e(tp )). V˙ 7 (e(t)) = d2 e˙ T (t)N3 e(t) 4

(28)

From Lemma 1, we obtain  t e˙ T (s)N1 e(s)ds ˙ − t−d(t) T

≤ d(t)ξ (t)Y N1−1 Y T ξ(t) + 2ξ T (t)Y ( InN ×nN 0nN ×7nN

(29) − 0nN ×2nN InN ×nN 0nN ×5nN )ξ(t).  t − e˙ T (s)Z2 e(s)ds ˙ t−d ⎡ ⎤    I 0 1 T ⎣ Z I −I 0 X 2 2 (30)

2 (t), ≤ − 2 (t) −I I ⎦ ∗ Z2 0 I −I d 0 −I

where 2T (t) = eT (t) eT (tp ) eT (t − d) . From Assumption 1, for any scalar σ > 0, the nonlinear function g(e(t)) satisfies the following inequality 

e(t) σ g(e(t))

T 

U¯ V¯ ∗ I



e(t) g(e(t))

 ≤ 0.

(31)

For any appropriate dimensional matrices F1 and F2 , the following equation holds ¯ − τ (t)) + Le(tp )] = 0. (eT (t)F1 + e˙ T (t)F2 )[−e(t) ˙ + g(e(t)) + cΓe(t

(32)

Then, for t ∈ [tp , tp+1), combining(21)-(32), we get

V˙ (e(t)) ≤ ξ T (t) Ξ1 + (d − d(t))Ξ2 + d(t)Y N1−1 Y T ξ(t), d(t) ∈ [0, d], (33)

where Ξ1 = Ψ + Y8nN ×nN 08nN ×nN −Y8nN ×nN 08nN ×5nN

T + Y8nN ×nN 08nN ×nN −Y8nN ×nN 08nN ×5nN ,

T

Ξ2 = 0nN ×nN InN ×nN 0nN ×6nN N1 0nN ×nN InN ×nN 0nN ×6nN

T

+ 2 0 N2 0nN ×2nN InN ×nN 0nN ×5nN ,  nN ×2nN InN ×nN 0nN ×5nN ξ(t) = eT (t), e˙ T (t), eT (tp ), eT (t−d), eT (t−τ1 ), eT (t−τ (t)), eT (t−τ2 ), g T (e(s)) . 9

Let Φ(d(t)) = Ξ1 + (d − d(t))Ξ2 + d(t)Y N1−1 Y T . According to Lemma 2 and d(t) ∈ [0, d], we have Φ(d(t)) < 0 if and only if Φ(0) = Ξ1 + dΞ2 < 0,

(34)

Φ(d) = Ξ1 + dY N1−1 Y T < 0.

(35)

According to the Schur complement, (34) is equivalent to (9), and (35) is equivalent to (10). V˙ (e(t)) ≤ −γ{e(t)2 }, t ∈ [tp , tp+1 ) where γ = min{λmin (−Φ(0)), λmin (−Φ(d))} > 0. According to [23] and V˙ (e(t)) < 0, error dynamical system (8) is asymptotically stable. The proof is completed. Remark 1. It can be found that V5 (e(t)) is discontinuous at tp , and lim− V5 (e(t)) t→tp

≥ 0, V5 (e(tp )) = 0. Hence, the condition lim− V (e(t)) ≥ V (e(tp )) holds. From t→tp

Wirtinger’s integral inequality in Lemma 4, we obtain lim− V7 (e(t)) ≥ 0 and t→tp

V7 (e(tp )) = 0. The domain of definition of V (e(t)) is t ∈ [tp , tp+1). V (e(t)) is absolutely continuous for t = tp and satisfies lim− V (e(t)) ≥ V (e(tp )). t→tp

Remark 2. In LKF, Vq (e(t))(q = 5, 6, 7) have been used in order to make full use of the sawtooth structure characteristic of sampling input delay. Thus, the conservatism of the synchronization criteria is further reduced. Remark 3. The delayed coupling term in [19] involves transmission delay and self-feedback delay, which is feedback with non-identical delay. In system (1) of this paper, the transmission delay and self-feedback delay in the coupling term are feedback with identical delay τ (t). Inspired by [19], if the coupling term in system (1) is feedback with non-identical delays, we have N  x˙ i (t) = f (xi (t)) + c Gij Γ(xj (t − τ (t)) − xi (t − σ(t))) + ui (t), j=1,j=i

˙ ≤ h1 . where σ(t) is self-feedback delay, and 0 ≤ σ1 ≤ σ(t) ≤ σ2 , σ(t) N  Gij , we have According to Gii = − x˙ i (t) = f (xi (t))+c

N 

j=1,j=i

j=1

Gij Γxj (t−τ (t))−cGii Γxi (t−τ (t))+cGii Γxi (t−σ(t))+ 10

ui (t), ¯ = I ⊗ Γ. We can have error system Let Gii = −α and Γ N ¯ − σ(t)) + Le(t ). ¯ − τ (t)) − cαΓe(t ¯ e(t) ˙ = g(e(t)) + cΓe(t − τ (t)) + cαΓe(t p In addition, we can obtain the corresponding synchronization criteria when we choose the following LKF V˜ (e(t)) = V (e(t))+ V˜1 (e(t)) + V˜2 (e(t)),  t−σ t t where V˜1 (e(t)) = t−σ(t) eT (s)Q1 e(s)ds+ t−σ1 eT (s)Q2 e(s)ds+ t−σ21 eT (s)Q3 e(s)ds, 0 t T  −σ1  t T e˙ (s)W1 e(s)dsdθ+(σ ˙ e˙ (s)W2 e(s)dsdθ. ˙ V˜2 (e(t)) = σ1 2 −σ1 ) −σ1

−σ2

t+θ

t+θ

The matrix inequalities in Theorem 1 are nonlinear when Li (i = 1, 2, · · · , N) are not given. In this case, they cannot be directly solved by Matlab LMI toolbox. Hence, Theorem 2 is given to convert nonlinear matrix inequalities into linear matrix inequalities. At the same time, desired sampled-data controllers are obtained in Theorem 2. Theorem 2 Under Assumption 1 and for given scalar m, error dynamical system (8) is asymptotically stable if there exist matrices P > 0, Q1 > 0, Q2 > 0, Q3 > 0, W1 > 0, W2 > 0, Z1 > 0, Z2 > 0, N1 > 0, N2 > 0, N3 > 0, and matrices Y , X1 , X2 , F = diag{F11 , F22 , · · · , FN N }, M = diag{M11, M22 , · · · , MN N }, and a scalar σ > 0, such that the following matrix inequalities are satisfied



˜ 1 =Π ˜ + d 0nN ×nN InN ×nN 0nN ×6nN T N1 0nN ×nN InN ×nN 0nN ×6nN Π

T

+ 2d 0nN ×2nN InN ×nN 0nN ×5nN N2 0nN ×2nN InN ×nN 0nN ×5nN <0, (36)   ˜ Y ˜2 = Π < 0, (37) Π ∗ − 1d N1   W2 X1 ≥ 0, (38) ∗ W2   Z 2 X2 ≥ 0, (39) ∗ Z2

˜ =Ψ ˜ + Y8nN ×nN 08nN ×nN −Y8nN ×nN 08nN ×5nN where Π

T + Y8nN ×nN 08nN ×nN −Y8nN ×nN 08nN ×5nN ,

11

⎛

⎞ ˜ 18 ˜ 12 Ψ ˜ 13 Ψ14 Ψ15 Ψ ˜ 16 0 Ψ Ψ11 Ψ ⎜ ∗ Ψ ˜ 23 0 ˜ 22 Ψ ˜ 26 0 Ψ ˜ 28 ⎟ 0 Ψ ⎜ ⎟ ⎜ ∗ ⎟ ∗ Ψ Ψ 0 0 0 0 33 34 ⎜ ⎟ ⎜ ∗ ∗ ∗ Ψ44 0 0 0 0 ⎟ ⎜ ⎟, ˜ Ψ=⎜ ⎟ Ψ Ψ 0 ∗ ∗ ∗ ∗ Ψ 55 56 57 ⎜ ⎟ ⎜ ∗ ∗ ∗ ∗ ∗ Ψ66 Ψ67 0 ⎟ ⎜ ⎟ ⎝ ∗ ∗ ∗ ∗ ∗ ∗ Ψ77 0 ⎠ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ψ88 1 1 π2 ˜ ˜ ˜ 16 = cF Γ, ¯ Ψ ˜ 18 = −σ V¯ + F , Ψ12 = −F + P, Ψ13 = d Z2 − d X2 + 4 N3 + M, Ψ ˜ 23 = mM, Ψ ˜ 26 = ˜ 22 = τ 2 W1 + (τ2 − τ1 )2 W2 + dZ2 + d2 N3 − mF − mF T , Ψ Ψ 1 ¯ Ψ ˜ 28 = mF . c × mF Γ, Moreover, the desired controller gain matrices in (6) can be given as Li = Fii−1 Mii , i = 1, 2, · · · , N.

(40)

Proof Let F1 = F , F2 = mF , and F L = M. From Theorem 1, we know that (36)-(39) hold. The proof is completed. Remark 4. In this paper, the sampled-data synchronization problem of complex networks with fixed coupling has been investigated, and the less conservative results have been obtained. The method in this paper is not suitable for complex networks with time-varying coupling because the sampled-data controller gain matrices need to be obtained by using system parameters. How to obtain the sampled-data synchronization criteria of complex networks with time-varying coupling is our future research. If the coupling matrix is changed with Markovian parameter, the method in this paper is also available. The stochastic sampled-data synchronization criteria can be obtained when we choose mode-dependent LKF. Remark 5. Compared with [25–27], mixed convex combination and Wirtinger’s integral inequality have been adopted in this paper, which effectively utilize the sawtooth structure and all available information about the actual aperiodic sampling pattern. Hence, the less conservative sampled-data synchronization criteria can be obtained. Remark 6. In this paper, in order to reduce conservatism, some free matrices are introduced using discontinuous LKF and mixed convex combination technique. As a result, the computation complexity increases because of many decision variables in the obtained results. 12

4. Simulation In order to demonstrate the validity of the proposed method, numerical examples are given to demonstrate the effectiveness. Example 1: Consider the following complex networks with time-varying delay borrowed form [25] x˙ i (t) = f (xi (t)) + c

N 

Gij Γxj (t − τ (t)) + ui (t), i = 1, 2, 3.

(41)

j=1

where

⎛

⎞   −1 0 1 1 0 τ (t) = 0.2 + 0.05sin(10t), G = ⎝ 0 −1 1 ⎠, Γ = . 0 1 1 1 −2   −0.5xi1 + tanh(0.2xi1 ) + 0.2xi2 The activation function is given as: f (xi (t)) = , i2 − tanh(0.75x2 )   0.95x  −0.5 0.2 −0.3 0.2 which satisfies Assumption 1, and U = ,V = . 0 0.95 0 0.2 We obtain τ1 = 0.15, τ2 = 0.25, h = 0.5. When c = 0.5, m = 1 and using Theorem 2 in our paper, we can get that the maximum value of sampling interval is d = 0.9225. The following state feedback controllers are obtained under the maximum sampling   interval d = 0.9225  −0.5193 −0.1604 −0.5193 −0.1604 L1 = , L2 = , 0.0009 −1.2166  0.0009 −1.2166  −0.3675 −0.1590 L3 = . In addition, Table 1 and Table 2 provide the 0.0046 −1.0777 maximum sampling interval and the comparisons of complexity, respectively. It can be seen that the synchronization criteria in this paper are less conservative than the existing ones. Table 1. Maximum sampling interval d for c = 0.5. Methods

Theorem 2

[25]

[26]

[27]

d

0.9225

0.5409

0.5573

0.9016

Improvement rates

70.55%

65.53%

2.32%

13

3 2 1 0

e(t)

−1 −2

e11(t) e12(t)

−3

e21(t) −4

e22(t) e31(t)

−5

e32(t) −6

0

5

10 t

15

20

Figure 1: State trajectories of the error system (41)

2 1.5 1 0.5

u(t)

0 −0.5

u11(t) u12(t)

−1

u21(t) −1.5

u22(t) u31(t)

−2

u32(t) −2.5

0

5

10 t

15

Figure 2: Responses of the control inputs

14

20

Table 2. The complexity of different methods. Methods

Theorem 2

[25]

[26]

[27]

Decision variables

568

148

148

418

The maximum order of LMI

54

42

42

42

The number of LMIs

4

2

5

5

From Table 1 and Table 2, the conservatism of this method is reduced on the basis of increasing complexity. The state trajectories of the error system and control inputs are presented  T  T in Figs. 1 and 2 with initial values x1 (0) = 3 −1 , x2 (0) = 0 1 ,  T  T x3 (0) = −6 2 , s(0) = 2 3 . t ˙ In discontinuous LKF, tp and tp+1 are contained in (d−t+tp ) tp e˙ T (s)N1 e(s)ds,   2 t t π (tp+1 −t)(t−tp )eT (tp )N2 e(tp ), and d2 tp e˙ T (s)N3 e(s)ds− ˙ [e(s)−e(tp )]T N3 [e(s)− 4 tp e(tp )]ds, which could make full use of the sawtooth structure characteristic of sampling input delay. In addition, unlike [25,  t 26], we use recip˙ and rocally convex combination technique to deal with t−d e˙ T (s)Z2 e(s)ds  t−τ1 T e˙ (s)W2 e(s)ds ˙ in the LKF, which provides tighter lower bound than t−τ2 Jensen’s inequality. Hence, the conservatism of the sampled-data synchronization criteria is further reduced. Table 1 provides the comparisons of the maximum sampling interval d for different methods in this paper and [25–27]. It is clear that, for this example, the synchronization criteria in this paper are more effective than existing results. Example 2: Consider Chua’s circuit in [26], which is described as ⎧ ⎨ s˙1 = σ1 (−s1 + s2 − ψ(s1 )) s˙2 = s1 − s2 + s3 (42) ⎩ s˙3 = −σ2 s2 where σ1 = 10, σ2 = 14.87, ψ(s1 ) = −0.68s1⎛+ 0.5(−1.27 + 0.68)(|s ⎞ 1+ 2.7 10 0 −1 1 ⎠, V = 1| − |s1 − 1|), τ (t) = 0.03 + 0.01sin(t), U = ⎝ 1 0 −14.87 0 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −3.2 10 0 −2 1 1 0.9 0 0 ⎝ 1 −1 1 ⎠ . G = ⎝ 1 −1 0 ⎠, Γ = ⎝ 0 0.9 0 ⎠. 0 −14.87 0 1 0 −1 0 0 0.9 15

3

2

1

0 e(t)

e11(t) e12(t)

−1

e13(t) e21(t)

−2

e22(t) e23(t)

−3

e31(t) e32(t)

−4

e33(t) −5

0

1

2

3

4 t

5

6

7

8

Figure 3: State trajectories of the error system

80 u11(t) u12(t) 60

u13(t) u21(t) u22(t)

40

u23(t) u31(t)

20 u(t)

u32(t) u33(t)

0

−20

−40

−60

0

1

2

3

4 t

5

6

7

Figure 4: Responses of the control inputs

16

8

We can get that the maximum sampling interval is d = 0.1120. In [26], the maximum sampling interval is d = 0.0711. It can be calculated that the maximum sampling interval d = 0.1120, which is 57.52% larger than that in [26]. Hence, the method in this paper is less conservative than existing one. In addition, the state feedback controllers can be obtained under the maximum ⎛ value of sampling interval d⎞= 0.1120 −8.8783 −7.7467 0.4037 L1 = ⎝ 0.2107 −2.9613 −1.3997 ⎠ , ⎛ 4.1609 11.6779 −6.1945 ⎞ −9.0658 −7.7945 0.4076 ⎝ L2 = 0.2124 −3.0817 −1.3947 ⎠ , 4.4108 11.6875 −6.2911 ⎛ ⎞ −9.0658 −7.7945 0.4076 L3 = ⎝ 0.2124 −3.0817 −1.3947 ⎠ . 4.4108 11.6875 −6.2911 According to the state feedback controllers, the corresponding state trajectories of the error system and control inputs are presented in Figs. 3 and  T  T 4 with initial values x1 (0) = −5 −4 −3 , x2 (0) = −2 −1 0 ,  T  T x3 (0) = 1 2 3 , s(0) = 1 −2 5 . Example 3: Consider the following delayed complex networks with fifteen nodes x˙ i (t) = f (xi (t)) + c

N 

Gij Γxj (t − τ (t)) + ui (t), i = 1, 2, · · · , 15.

j=1

where

17

(43)

G= ⎛ −2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎜ 1 −3 0 0 1 1 0 0 0 0 0 0 0 0 0 ⎜ ⎜ 1 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 ⎜ ⎜ 1 0 0 −3 1 1 0 0 0 0 0 0 0 0 0 ⎜ ⎜ 0 0 0 0 −3 1 1 1 0 0 0 0 0 0 0 ⎜ ⎜ 1 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 ⎜ ⎜ 1 0 0 0 0 0 −2 0 0 0 0 0 0 0 1 ⎜ ⎜ 1 0 0 0 0 0 0 −3 0 0 0 0 0 1 1 ⎜ ⎜ 1 1 1 0 0 0 0 0 −5 0 0 0 0 1 1 ⎜ ⎜ 1 0 0 0 0 0 1 0 0 −3 0 0 0 0 1 ⎜ ⎜ 1 0 0 0 0 0 0 0 0 0 −2 0 0 0 1 ⎜ ⎜ 1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 ⎜ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 −2 1 1 ⎜ ⎝ 1 1 1 0 0 0 0 0 0 0 0 0 0 −3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 f (xi (t)), τ (t), Γ, U, and V are the same as Example 1. When c = 0.5, m = 1 and using Theorem 2 in our paper, we can get the following state feedback controllers = 0.6325  under  sampling interval d  −0.6110 −0.1819 −0.4918 −0.1219 , L2 = , L1 = 0.0015 −1.2133 −0.0078 −0.9889     −0.8455 −0.1832 −0.5698 −0.1350 L3 = , L4 = , 0.0045 −1.6162    0.0112 −1.3117  −0.5667 −0.1098 −0.8839 −0.1800 L5 = , L6 = ,  0.0070 −1.0665   0.0088 −1.7040  −0.7256 −0.1468 −0.5582 −0.1001 L7 = , L8 = , 0.0097 −1.3704 −0.0060 −1.0356     −0.3085 −0.0768 −0.5821 −0.1309 L9 = , L10 = ,  −0.0204 −0.8184   0.0052 −1.2936  −0.7307 −0.1542 −0.8709 −0.1717 , L12 = , L11 = 0.0232 −1.5421    0.0247 −1.7693  −0.7355 −0.1411 −0.5556 −0.1347 L13 = , L14 = , −0.0205 −1.2689  0.0205 −1.4609  −0.8895 −0.1899 L15 = . 0.0037 −1.6716 18

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

6

4

e(t)

2

0

−2

−4

−6

0

2

4

6

8

10

t

Figure 5: State trajectories of the error system (43)

10 8 6 4

u(t)

2 0 −2 −4 −6 −8

0

2

4

6

8

t

Figure 6: Responses of the control inputs

19

10

The state trajectories of the error system and control inputs are presented  T  T in Figs. 5 and 6 with initial values x1 (0) = 3 2 , x2 (0) = −1 0 ,  T  T  T x3 (0) = 1 −2 , x4 (0) = 2 1 , x5 (0) = −3 3 , x6 (0) =  T  T  T 1.5 −1.5 , x7 (0) = −2.5 2.5 , x8 (0) = 1.2 −1.2 , x9 (0) =  T  T  T 4.8 −4.5 , x10 (0) = −4.8 3.2 , x11 (0) = −3.4 −3.2 , x12 (0) =  T  T  T −5 −5.2 , x13 (0) = 5.2 4.8 , x14 (0) = 4.6 4.7 , x15 (0) =  T −5.5 −5.4 . 5. Conclusion In this paper, the problem of sampled-data synchronization has been studied for complex networks under aperiodic sampling intervals via inputdelay approach. The novel delay-dependant sampled-data synchronization criteria have been obtained by using the discontinuous LKF, Wirtinger’s integral inequality and mixed convex combination technique. At the same time, the desired controllers have been obtained by solving the LMIs. Finally, numerical examples have been given to demonstrate the effectiveness of the theoretical results. Further research includes the study of finite-time sampled-data of complex networks with Markovian parameters and stochastic perturbations, and the study of sampled-data synchronization of complex networks with uncertain parameters and time-varying coupling. Acknowledgement This work was supported by the National Natural Science Foundation of China (61433004, 61473070), and the National High Technology Research and Development Program of China (2012AA040104) and IAPI Fundamental Research Funds 2013ZCX14. This work was supported also by the development project of key laboratory of Liaoning province. References [1] S. H. Strogatz, Exploring complex networks, Nature 410 (2001) 268-276. [2] C. Hu, J. Yu, H. Jiang, Z. Teng, Exponential synchronization of complex networks with finite distributed delays coupling, IEEE Transactions on Neural Networks 22 (2011) 1999-2010. 20

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