A new sampling interval fragmentation approach to synchronization of chaotic Lur’e systems

Applied Mathematics and Computation 348 (2019) 12–24

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

A new sampling interval fragmentation approach to synchronization of chaotic Lur’e systems Huilan Yang a,∗, Xin Wang b, Lan Shu a, Guozhu Zhao c, Shouming Zhong a a

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China c Civil Aviation Flight University of China Simulator Training Center, Guanghan, Sichuan 618300, PR China b

a r t i c l e

i n f o

Keywords: Synchronization Chaotic Lur’e systems Sampling interval fragmentation approach Sampled-data control

a b s t r a c t This paper investigates the issue of sampled-data synchronization for a class of chaotic Lur’e systems (CLSs), where a novel sampling interval fragmentation approach (SIFA) is proposed. To this end, first, by partitioning sampling interval into several nonuniform segments based on a geometric series and taking advantage of the convex combination technique, a newly discontinuous Lyapunov–Krasovskii functional (LKF) is developed for the first time to analyze the synchronization problem of such systems, which significantly uses more information on actual sampling behavior of the system. Meanwhile, an uniform sampling interval fragmentation approach (USIFA) is also taken into account. Then, some relaxed sampled-data synchronization criteria of concerned systems are formulated in framework of matrix inequalities with a larger sampling period. Two numerical simulations are provided to demonstrate the superiority and effectiveness of the derived results. © 2018 Published by Elsevier Inc.

1. Introduction In recent years, much attention has been attracted to the CLSs. The main reason is that many nonlinear practical systems such as neural networks, Chua’s circuit systems and hyper chaotic attractors can be described as Lur’e form. Especially, chaotic synchronization is an important research topic and has many applications in various disciplines including information processing, secure communication, pattern recognition and so on [1–10]. Naturally, the issue of synchronization for CLSs has been widely studied in [11,12] and references therein. On the other hand, with the development of both computer and digital technology, digital control plays a key role in speed, accuracy, small size and low cost. As known to us, when a discrete-time control signal is produced by a digital computer, and then it will be further converted as a continuous-time control input signal by a zero-order-holder, in which the overall system turns into a sampled-data control system. Furthermore, compared with continuous control, the sampled-data control can effectively save the communication bandwidth, which takes advantage of the merits of efficiency high reliability, easy installation and maintenance with low cost. Hence, the investigation of synchronization for CLSs by sampled-data control has gained significance and many outstanding achievements have been reported in [13–19] and references therein. For example, Chen et al. [13] analyzed the problem of master-slave synchronization for CLSs with sampled-data control, in which

∗

Corresponding author. E-mail address: [email protected] (H. Yang).

https://doi.org/10.1016/j.amc.2018.11.009 0 096-30 03/© 2018 Published by Elsevier Inc.

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

13

the constructed LKF was continuous at sampling instants but not necessarily positive definite inside the sampling intervals. By modeling the synchronization error system as an input-delay system and constructing a new LKF, the master-slave synchronization problem of CLSs was considered in [15]. Meanwhile, since a larger sampling interval will make it possible for operating conditions such as limitation of load, computational burden, and communication channel occupation to be relaxed [20,21]. Therefore, it is very meaningful and important to suggest synchronization criteria of CLSs for a sampling interval as large as possible. With this regard, by making full use of the available information about the actual sampling pattern, the stabilization criteria were developed in terms of linear matrix inequalities in [22], which not only depended on the upper bounds but also on lower bounds in a sampling interval. Lately, Park et al. investigated the sampled-data synchronization problem for two identical CLSs under a fragmentation approach in [23], where the fragmentation scheme divided a interval between last sampling instants tk and present time t, and hence more information on inner sampling behavior of the systems could be used for sampled-data control systems. However, these works suffer a common drawback, e.g., the considered LKF is only decreasing during the sampling intervals but not always decreasing in time. Recently, in order to overcome this difficulty and better handle the discontinuous control signals in sampled-data systems, there are two discontinuous types of LKF and their effectiveness have been demonstrated by many outstanding achievements. One is a free-matrix-based time-dependent discontinuous LKF [24–26], and the other is a special discontinuous term based on Wirtinger’s inequality [27] in LKF [19,28,30]. Thus, how to create a new discontinuous form and utilize more fragmented state information to derive less conservative synchronization criteria for CLSs is the motivation of this paper. Motivated by the above discussions, we investigate, in this paper, the problem of sampled-data stabilization for a class of CLSs. The main contributions of this paper are highlighted as follows. • Different from the existing discontinuous LKF methods in [24–26,28–30], by means of partitioning sampling interval into several nonuniform segments based on a geometric series and taking advantage of the convex combination technique, a novel discontinuous LKF is constructed for the first time in our work, which significantly contains more information on actual sampling behavior of the system. • An uniform sampling interval fragmentation approach (USIFA) is also taken into account. • In comparison to existing works in [3,13–19,23,26], the proposed SIFA can enhance the feasible region of the synchronization criteria. Two numerical simulations are provided to demonstrate the superiority and effectiveness of the derived results. Notations: throughout this paper, Rn stands for n-dimensional Euclidean space and Rn×m is the set of all n × m real matrices. The notation X > Y (X ≥ Y ) implies that the matrix X − Y is positive definite (semidefinite). Sym{X } means X + X T . The superscript T and −1 stand for the transpose and inverse, respectively, the symmetric term in a matrix is denoted by ∗ . C = diag(c1 , c2 , . . . , cn ) means C is a diagonal matrix. 2. Preliminaries Consider the following drive and slave Lur’e systems:



U :

 V :

x˙ (t ) = A x(t ) + B ω (Dx(t )), p(t ) = H x(t ),

y˙ (t ) = A y(t ) + B ω (Dy(t )) + u(t ), q(t ) = H y(t ),

W : u(t ) = G ( p(tk ) − q(tk )), tk ≤ t < tk+1 ,

(1)

where U is the drive system, V is the slave system, and W is the sampled-data controller. U and V under u(t ) = 0 are identical chaotic Lur’e systems (CLSs) with state vectors x(t), y(t ) ∈ Rn and subsystem outputs p(t), q(t ) ∈ Rn h¯ . A ∈ Rn×n , H ∈ Rn h¯ ×n , D ∈ Rm h¯ ×n , B ∈ Rn×m h¯ are known constant matrices. u(t ) ∈ Rn is the control input on the slave system, tk is the updated instant time by using a zero-order-hold (ZOH) function and G ∈ Rn×n h¯ is the sampling control gain matrix to be designed. We assume that, in this paper, ω (· ) : Rm h¯ → Rm h¯ is a nonlinear vector valued function and ωρ ( · ) satisfying

ρ − ≤

ωρ ( a ) − ωρ ( b ) a−b

≤ ρ + ,

f or any a = b,

ρ = 1, 2, . . . , m h¯ .

(2)

Moreover, it is assumed that the sampling interval hk satisfies the following condition:

hmin ≤ tk+1 − tk = hk ≤ hmax ,

(3)

where hmax ≥ hmin > 0. Let e(t ) = x(t ) − y(t ), according to (1), the synchronization error system (SES) under sampled-data controller is described as:

e˙ (t ) = A e(t ) + B f (De(t )) − G H e(tk ), tk ≤ t < tk+1 , where f (De(t )) = ω (D (e(t ) + y(t ))) − ω (Dy(t )).

(4)

14

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

Review (2), since ωρ ( · ) belongs to sector [ρ − , ρ + ], and hence one has

ρ − ≤

ωρ (Dρ (e(t ) + y(t ))) − ωρ (Dρ y(t )) ≤ ρ + , Dρ e(t ) f or any a = b, ρ = 1, 2, . . . , m h¯ .

fρ (Dρ e(t )) = Dρ e(t )

(5)

where Dρ stands for the ρ th row of D. In this paper, for the sake of simplicity, E j T = [0n×( j−1 )n In 0n×(4− j )n 0n×m h¯ ]( j = 1, . . . , 4 ), E 5 T = [0m h ×4n Im h¯ ], E i T = ¯

[0n×(i−1 )n In 0n×(5−i )n 0n×nm 0n×m h¯ ], (i = 1, . . . , 5 ) and m ≥ 2, E 6 T = [0nm×5n Inm 0nm×m h¯ ], E 7 T = [0m h ×5n 0m h¯ ×nm Im h¯ ], ¯ are defined as block entry matrices. 3. Main results 3.1. A nonuniform sampling interval fragmentation approach We will first introduce, in this section, a nonuniform sampling interval fragmentation approach (NSIFA) to address the CLSs: Firstly, for a given positive integer H, the interval [tk , tk+1 ) is partitioned into H + 1 parts. Meanwhile, let tk, = tk + hmin (1 − γ  )/(1 − γ H ), where  = 0, 1, . . . , H − 1, γ ∈ (0, 1). Thus, one has tk,0 = tk and tk,H = tk + hmin . That is to say, this approach partitions the interval [tk , tk + hmin ) based on a geometric series, and γ is the common ratio of the geometric series. Remark 1. Actually, when γ > 1, one has

1−γ = 1−γH

γ −1 1 − (1/γ ) −H 1 − ρ  H− = γ = ρ , where 1/γ = ρ ∈ (0, 1 ). γ H − 1 1 − (1/γ )H 1 − ρH

Thus, we can take into consideration of the scenario of common ratio γ > 1 with the same approach as above. Denote Uk  = [tk, , tk,+1 ),

P (t ) = (1 − ϑ )P + ϑ P+1 = P (ϑ ), where P is the positive definite matrix, ϑ = (t − tk, )/H , H = H γ  , H = ((1 − γ )hmin )/(1 − γ H ),  = 0, 1, . . . , H − 1. Moreover, when t ∈ [tk,H , tk+1 ), we denote P (t ) = PH , where PH is a fixed positive definite matrix. Now, consider the following discontinuous LKF,

V (t ) =



where

4

 V1 (t ) =

p=1 V p

(t ), t ∈ [t k , t k+1 ),

(6)

eT (t )P (ϑ )e(t ), t ∈ Uk  , eT (t )PH e(t ), t ∈ [tk,H , tk+1 ),

V 2 (t ) = (tk+1 − t )(t − tk )α T (t )Qα (t ),   t T V 3 (t ) = (tk+1 − t ) β (s )Rβ (s )ds + ηT (t )S η (t ) , V 4 (t ) = 2



m h¯

λρ

ρ =1



tk

Dρ e(t )

0

( fρ (s ) − ρ − s )ds + 2

 ρ =1

m h¯

σρ

 0

Dρ e(t )

(ρ + s − fρ (s ))ds,

with α (t ) = [eT (t ), eT (tk )]T , β (t ) = [eT (tk ), e˙ T (t ), eT (t )]T , η (t ) = [eT (t ) − eT (tk ),

 tk

t eT (s )ds]T .

Theorem 1. Given scalars γ ∈ (0, 1), H ≥ 2, εi (i = 1, 2, 3 ), the error system (4) is asymptotically stable under the NSIFA,  if there exist positive definite matrices P ( = 0, 1, . . . , H ), R, X , positive diagonal matrices , , , symmetric matrices Q = S=





S1 S2 , R= ∗ S3



R

1

∗ ∗

1, + hk 3 ∗

1, + hk 2, ∗ ∗

R2 R4 ∗ hk ˜ Q 2

−X





Q1 Q2 , ∗ Q3

R3 R5 , any appropriate dimensional matrices Y, N , G satisfying the following LMIs: R6

< 0,

hk Y −R ∗

(7)

hk ˜ Q 2

0 −X

 < 0,

(8)

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24



{ h k , 0 }

hk ˜ Q 2

∗

−X

{hk ,hk }



 < 0,

hk Y −R ∗

∗ ∗

15

(9)

hk ˜ Q 2



0 −X

< 0,

(10)



PH ∗

P0 ≥ 0, P0

(11)

where, the designed control gain matrix is given by G = N −1 G , and

1, = E 1 (P+1 − P )/(H γ  )E 1 T + Sym{E 1 [−(1 − γ  )/((1 − γ )γ  )(P+1 − P ) + P ]E T2 } +E 2 (hk 2 /4 )X E 2 T − Sym[E 3 R2 [E 1 − E 3 ]T + E 3 R3 E 4 T } − [E 1 − E 3 , E 4 ]S[E 1 − E 3 , E 4 ]T +Sym{Y[E 1 − E 3 , E 4 ]T } − Sym{[E 5 − E 1 D T L− ][E 5 T − L+ DE 1 T ]} +Sym{[E 5 − E 1 D T L− ] DE 2 T + [E 1 D T L+ − E 5 ]DE 2 T } + Sym{(ε1 E 1 + ε2 E 2 + ε3 E 3 ) ×(−N E 2 T + N A E 1 T + N B E 5 T − G H E 3 T )}, 2, = Sym{E 1 (P+1 − P )/(H γ  )E T2 } − [E 1 , E 3 ]Q[E 1 , E 3 ]T − E 3 R1 E 3 T , 3 = [E 1 , E 3 ]Q[E 1 , E 3 ]T + [E 3 , E 2 , E 1 ]R[E 3 , E 2 , E 1 ]T + Sym{[E 1 − E 3 , E 4 ]S[E 2 , E 1 ]T }, {hk ,h(t )} = Sym{E 1 PM E T2 } + E 2 (hk 2 /4 )X E 2 T − Sym[E 3 R2 [E 1 − E 3 ]T + E 3 R3 E 4 T } −[E 1 − E 3 , E 4 ]S[E 1 − E 3 , E 4 ]T + Sym{Y[E 1 − E 3 , E 4 ]T } −Sym{[E 5 − E 1 D T L− ][E 5 T − L+ DE 1 T ]} + Sym{[E 5 − E 1 D T L− ] DE 2 T + [E 1 D T L+ − E 5 ]DE 2 T } hk 2 Q˜ X −1 Q˜ T 4 + Y R−1 Y T } + (hk − h(t )){[E 1 , E 3 ]Q[E 1 , E 3 ]T

+Sym{(ε1 E 1 + ε2 E 2 + ε3 E 3 )(−N E 2 T + N A E 1 T + N B E 5 T − G H E 3 T )} + −h(t ){[E 1 , E 3 ]Q[E 1 , E 3 ]T − E 3 R1 E 3 T

+[E 3 , E 2 , E 1 ]R[E 3 , E 2 , E 1 ]T + [E 1 − E 3 , E 4 ]S[E 2 , E 1 ]T }, Proof. Case 1, when t ∈ Uk  , differentiating the time derivative of V (t ) along the trajectory of the system (4) yields

V˙ 1 (t ) = 2eT (t )P (ϑ )e˙ (t ) + eT (t )P˙  (ϑ )e(t ), = 2eT (t )[(1 − ϑ )P + ϑ P+1 ]e˙ (t ) + eT (t )[(P+1 − P )/(H γ  )]e(t ) = 2eT (t )[(t − tk )/(H γ  )(P+1 − P ) − (1 − γ  )/((1 − γ )γ  )(P+1 − P ) + P ]e˙ (t ) +eT (t )[(P+1 − P )/(H γ  )]e(t ),

(12)

V˙ 2 (t ) = −(t − tk )α T (t )Qα (t ) + (tk+1 − t )α T (t )Qα (t ) +2(tk+1 − t )(t − tk )ξ T (t )Q˜ e˙ (t ), where ξ (t ) =

[eT (t ),

e˙ T (t ),

eT (t

k ),



tk

t eT (s )ds,

f T (De(t ))]T ,

(13) Q˜ = [Q1 , 0, Q2 , 0, 0]T .



For any positive matrix X , we have

hk 2 T (ξ (t )Q˜ X −1 Q˜ T ξ (t ) + e˙ T (t )X e˙ (t )), 4

2(tk+1 − t )(t − tk )ξ T (t )Q˜ e˙ (t ) ≤

V˙ 3 (t ) = −



tk

 =−

t

t

tk

β T (s )Rβ (s )ds + (tk+1 − t )β T (t )Rβ (t ) + 2(tk+1 − t )ηT (t )S η1 (t ) − ηT (t )S η (t ), η1 T (s )Rη1 (s )ds − 2eT (tk )[R2 (e(t ) − e(tk )) + R3



t

tk

e(s )ds] − (t − tk )eT (tk )R1 e(tk )

+(tk+1 − t )β T (t )Rβ (t ) + 2(tk+1 − t )ηT (t )S η1 (t ) − ηT (t )S η (t ),





R R where η1 (t ) = [e˙ T (t ), eT (t )]T , R = ∗4 R5 . 6

On the other hand, for any appropriate dimensional matrix Y, it is calculated that



0 = 2ξ

T

(14)



e(t ) − e(tk ) (t )Y[  t − tk e (s )ds



t

tk

η1 (s )ds]

(15)

16

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24



≤ (t − tk )ξ (t )Y R

Y

T

ξ (t ) + 2ξ



e(t ) − e(tk ) (t )Y  t + tk e (s )ds



t

η1 T (s )ds(R/(t − tk )) tk

  t e(t ) − e(tk ) ≤ (t − tk )ξ T (t )Y R−1 Y T ξ (t ) + η1 T (s )Rη1 (s )ds + 2ξ T (t )Y  t , tk tk e (s )ds −1

T

T

so, one has

 −

t

tk



η1 (s )Rη1 (s )ds ≤ (t − tk )ξ (t )Y R Y ξ (t ) + 2ξ T

−1

T

T

T



Moreover, for any positive diagonal matrix  ∈ R

tk

η1 (s )ds (16)



e(t ) − e(t ) (t )Y  t e(s )dsk . tk

V˙ 4 (t ) = 2( f (De(t )) − L− D e(t ))T D e˙ (t ) + 2(L+ De(t ) − f (D e(t )))T D e˙ (t ). m h¯ ×m h¯

t

(17)

(18)

, we have

2( f (De(t )) − L− De(t ))T (L+ De(t ) − f (De(t ))) ≥ 0.

(19)

Based on the system (4), for any appropriately dimensioned matrix N , one has

2[ε1 eT (t )N + ε2 e˙ T (t )N + ε3 eT (tk )N ][−e˙ (t ) + A e(t ) + B f (De(t )) − G H e(tk )] = 0.

(20)

Define G = N G, and hence from (12)–(20) we get

V˙ (t ) ≤ ξ T (t ) ξ (t ),

(21)

2 (t − tk )(2, + Y R−1 Y T ) + h4k Q˜ X −1 Q˜ T + (hk − (t − tk ))3 . from inequalities (7) and (8) that  < 0. Therefore, one can

where  = 1, + It can be found for t ∈ Uk  . Case 2, when t ∈ [tk,H , tk+1 ), we have

V˙ 1 (t ) = 2eT (t )PH e˙ (t ).

derive from (21) that for ξ (t) = 0, V˙ (t ) < 0,

(22)

Thus, combine (22) and (13)–(20), we have

ξ T (t ){hk ,h(t )} ξ (t )   hk − h(t ) h(t ) T = ξ (t ) { h k , 0 } + {hk ,hk } ξ (t ),

V˙ (t ) ≤

hk

(23)

hk

where h(t ) = t − tk . It is not difficult to see that {hk ,h(t )} < 0 if and only if {hk ,0} < 0 and {hk ,hk } < 0. Hence, one can derive from inequalities (9) and (10) that for ξ (t) = 0, V˙ (t ) < 0, t ∈ [tk,H , tk+1 ). On the other hand, when t = tk , based on (6), we have

lim V j (t ) = lim+ V j (t ) = V j (tk ) = 0, j = 2, 3,

t→tk −

t→tk

lim− V j (t ) = lim+ V j (t ) = V j (tk ) > 0, j = 4, tk ≤ t < tk+1 , k = 1, 2, . . . .

t→tk

t→tk

(24)

Furthermore, pre and post multiplication in (11) by diag{I, P −1 }, one has 0



I ∗

0 P −1 0



PH ∗

P0 P0



I ∗



0 ≥ 0, ⇐⇒ P −1 0



PH ∗

I



P0 −1

≥ 0,

⇐⇒ PH − P0 ≥ 0. Hence, from (6), we have

V1 (tk ) = eT (tk )P0 e(tk ) ≤ eT (tk − )PH e(tk − ) = V1 (tk − ), which means that

V (tk ) ≤ V (tk − ), tk ≤ t < tk+1 , k = 1, 2, . . . .

(25)

This completes the proof. 3.2. An uniform sampling interval fragmentation approach Next, we will give an uniform sampling interval fragmentation approach (USIFA) to analyze the chaotic Lur’e systems. For a given positive integer M, this approach divides a interval [tk , tk+1 ) into M + 1 parts, and the first M parts are the same segments. Let tk, = tk + hmin /M, where  = 0, 1, . . . , M − 1. Then, we have tk,0 = tk , tk,M = tk + hmin . Denote Wk  = [tk, , tk,+1 ).

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

17

Similarly, when t ∈ Wk  , we denote

P (t ) = (1 − ϑ )P + ϑ P+1 = P (ϑ ), where P ( = 0, 1, . . . , M − 1 ) is the positive definite matrix, ϑ = (t − tk, )/M, and M = hmin /M. On the other side, when t ∈ [tk,M , tk+1 ), we denote P (t ) = PM , where PM is a fixed positive definite matrix. Consider the following LKF,

V (t ) = V 1 (t ) + where

 V 1 (t ) =



4 p=2 V p

(t ), t ∈ [tk , tk+1 ),

(26)

eT (t )P (ϑ )e(t ), t ∈ Wk  ,  = 0, 1, 2, . . . , M − 1, eT (t )PM e(t ), t ∈ [tk,M , tk+1 ).

Theorem 2. Given scalars M ≥ 2, ε i (i = 1, 2, 3 ), the error system (4) is asymptotically stable under the USIFA, if there exist Q Q positive definite matrices P ( = 0, 1, . . . , M ), R, X , positive diagonal matrices , , , symmetric matrices Q = ∗1 Q2 , S =









3

R1 R2 R3 S1 S2 , R = ∗ R4 R5 , any appropriate dimensional matrices Y, N , G satisfying the following LMIs hold: ∗ S3 ∗ ∗ R6



1, + hk 3

hk ˜ Q 2

∗

hk ˜ Q 2

∗

−X

{hk ,hk }



PM ∗



0 −X

< 0,

(28)

 < 0,

hk Y −R ∗

∗ ∗

(27) hk ˜ Q 2

hk Y −R ∗

∗ ∗

{hk ,0}

< 0,

−X

1, + hk 2,



hk ˜ Q 2

0 −X

(29)

 < 0,

(30)



P0 ≥ 0, P0

(31)

where

1, = E 1 ((P +1 − P  )/M )E 1 T + Sym{E 1 [−(P +1 − P ) + P ]E T2 } + E 2 (hk 2 /4 )X E 2 T −Sym[E 3 R2 [E 1 − E 3 ]T + E 3 R3 E 4 T } − [E 1 − E 3 , E 4 ]S[E 1 − E 3 , E 4 ]T +Sym{Y[E 1 − E 3 , E 4 ]T } − Sym{[E 5 − E 1 D T L− ][E 5 T − L+ DE 1 T ]} +Sym{[E 5 − E 1 D T L− ] DE 2 T + [E 1 D T L+ − E 5 ]DE 2 T } + Sym{(ε1 E 1 + ε2 E 2 + ε3 E 3 ) ×(−N E 2 T + N A E 1 T + N B E 5 T − G H E 3 T )},

2, = Sym{E 1 ((P+1 − P )/M )E T2 } − [E 1 , E 3 ]Q[E 1 , E 3 ]T − E 3 R1 E 3 T , {hk ,h(t )} = Sym{E 1 PH E T2 } + E 2 (hk 2 /4 )X E 2 T − Sym[E 3 R2 [E 1 − E 3 ]T + E 3 R3 E 4 T } −[E 1 − E 3 , E 4 ]S[E 1 − E 3 , E 4 ]T + Sym{Y[E 1 − E 3 , E 4 ]T } −Sym{[E 5 − E 1 D T L− ][E 5 T − L+ DE 1 T ]} + Sym{[E 5 − E 1 D T L− ] DE 2 T + [E 1 D T L+ − E 5 ]DE 2 T } hk 2 Q˜ X −1 Q˜ T 4 + Y R−1 Y T } + (hk − h(t )){[E 1 , E 3 ]Q[E 1 , E 3 ]T

+Sym{(ε1 E 1 + ε2 E 2 + ε3 E 3 )(−N E 2 T + N A E 1 T + N B E 5 T − G H E 3 T )} + −h(t ){[E 1 , E 3 ]Q[E 1 , E 3 ]T − E 3 R1 E 3 T

+[E 3 , E 2 , E 1 ]R[E 3 , E 2 , E 1 ]T + [E 1 − E 3 , E 4 ]S[E 2 , E 1 ]T }, and 3 has the same definition as in Theorem 1. Furthermore, the designed control gain matrix is given by G = N −1 G . Proof. When t ∈ Wk  , differentiating the time derivative of V (t ) along the trajectory of the system (4) yields

V˙ 1 (t ) = 2eT (t )P (ϑ )e˙ (t ) + eT (t )P˙  (ϑ )e(t )

18

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

= 2eT (t )[(1 − ϑ )P + ϑ P+1 ]e˙ (t ) + eT (t )[(P+1 − P )/M]e(t ) = 2eT (t )[((t − tk )/M )(P+1 − P ) − (P+1 − P ) + P ]e˙ (t ) + eT (t )[(P+1 − P )/M]e(t ),

(32)

 Then, with the same method as in proof of Theorem 1, we have Theorem 2. This completes the proof. Remark 2. It is noted that if the common ratio of the geometric series γ = 1, then Theorem 1 can be transformed into Theorem 2. In other words, the USIFA is a special case of NSIFA. Remark 3. It is worth noting that the involved LMIs are (hmin , hmax )-dependent, that is to say, the derived criteria depend not only on the upper bound, but also on the lower bound of the sampling interval. Therefore, the proposed approach is not only less conservative but has wider application scopes. More details, please refer to the Remark 4 in [25]. Finally, in order to show the superiority of the NSIFA, the case of constant sampling periods is taken into account. Consider the following LKF:



Vˆ (t ) = Vˆ 1 (t ) +

4 p=2 V p

(t ) + Vˆ 5 (t ) + Vˆ 6 (t ), t ∈ [tk , tk+1 ),

(33)

where

Vˆ 1 (t ) = eT (t )P (ϑ )e(t ), H−1  t Vˆ 5 (t ) = eT (s − tk, )Z1, e(s − tk, )ds, t−H

=0

Vˆ 6 (t ) =

H−1 



−tk,



H

t+θ

−tk,+1

=0

t

e˙ T (s )Z2, e˙ (s )dsdθ .

Corollary 1. Given scalars γ ∈ (0, 1), H ≥ 2, ε i (i = 1, 2, 3 ), the error system (4) is asymptotically stable under the NSIFA, if there exist positive definite matrices P , Z1, , Z2, , ( = 0, 1, . . . , H − 1 ), R, X , positive diagonal matrices , , , symmetric matrices









Q Q S S Q = ∗1 Q2 , S = ∗1 S2 , R = 3 3



ˆ 1, + hk  ˆ3  ∗

ˆ 1, + hk  ˆ 2, 

hk ˆ Q 2

−X







R2 R3 ∗ R4 R5 , any appropriate dimensional matrices Y, N , G satisfying the following LMIs hold: ∗ ∗ R6 1

< 0,

hk Y −R ∗

∗ ∗

R

(34)

hk ˆ Q 2

0 −X

 < 0,

(35)



PH−1 ∗

P0 ≥ 0, P0

(36)

where, the designed control gain matrix is given by G = N −1 G , and

ˆ 1, = E 1 (P+1 − P )/H E 1 T + Sym{E 1 [−(1 − γ  )/((1 − γ )γ  )(P+1 − P ) + P ]E 2 T }  +E 2 (hk 2 /4 )X E 2 T − Sym[E 3 R2 [E 1 − E 3 ]T + E 3 R3 E 4 T } − [E 1 − E 3 , E 4 ]S[E 1 − E 3 , E 4 ]T +Sym{Y[E 1 − E 3 , E 4 ]T } − Sym{[E 7 − E 1 D T L− ][E 7 T − L+ D E 1 T ]} +Sym{[E 7 − E 1 D T L− ] D E 2 T + [E 1 D T L+ − E 7 ]D E 2 T } + E 5 (Z1,0 − Z2,0 )E 5 T + E 5 [Z2,0 , 0]E 6 T +E 2 (



H−1

H 2 Z2, )E 2 T + E 6 Zˆ E 6 T + Sym{(ε1 E 1 + ε2 E 2 + ε3 E 3 )(−N E 2 T + N A E 1 T + N B E 7 T − G H E 3 T )},

=0

ˆ 2, = Sym{E 1 (P+1 − P )/H E 2 T } − [E 1 , E 3 ]Q[E 1 , E 3 ]T − E 3 R1 E 3 T ,  ˆ 3 = [E 1 , E 3 ]Q[E 1 , E 3 ]T + [E 3 , E 2 , E 1 ]R[E 3 , E 2 , E 1 ]T + Sym{[E 1 − E 3 , E 4 ]S[E 2 , E 1 ]T },  ⎡ ⎤

⎢ ⎢ Zˆ = ⎢ ⎢ ⎣

Zˆ1,1 ∗ ... ∗ ∗

Z 2,1 Zˆ2,2 .. . ∗ ∗

∗ Z 2,2 .. . ∗ ∗

∗ ∗ .. .

∗ ∗ .. .

ZˆH −1,H −1 ∗

Z2,H−1 −Z1,H−1 − Z2,H−1

⎥ ⎥ ⎥, Zˆ1,1 = Z1,1 − Z1,0 − Z2,1 − Z2,0 , ⎥ ⎦

Zˆ2,2 = Z1,2 − Z1,1 − Z2,2 − Z2,1 , ZˆH −1,H −1 = Z1,H−1 − Z1,H−2 − Z2,H−1 − Z2,H−2 .

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

19

Proof. Taking the time derivative of V (ˆt ) along the trajectory of the system (4) yields

Vˆ˙ 1 (t ) = 2eT (t )P (ϑ )e˙ (t ) + eT (t )P˙  (ϑ )e(t ), = 2eT (t )[(1 − ϑ )P + ϑ P+1 ]e˙ (t ) + eT (t )[(P+1 − P )/H ]e(t ) = 2eT (t )[(t − tk )/H (P+1 − P ) − (1 − γ  )/((1 − γ )γ  )(P+1 − P ) + P ]e˙ (t ) +eT (t )[(P+1 − P )/H ]e(t ),

Vˆ˙ 5 (t ) =

H−1 

(37)

eT (t − tk, )Z1, e(t − tk, ) −

=0

Vˆ˙ 6 (t ) = ≤ =

H−1 

H−1 

eT (t − tk,+1 )Z1, e(t − tk,+1 ),

(38)

=0

H 2 e˙ T (t )Z2, e˙ (t ) −

H−1 

=0

=0 H−1 

=0

=0 H−1 

H e˙ (t )Z2, e˙ (t ) +

e˙ T (s )Z2, e˙ (s )ds

[e(t − tk, ) − e(t − tk,+1 )]T Z2, [e(t − tk, ) − e(t − tk,+1 )]

H−1 

2 T

t−tk,

t−tk,+1

H−1 

H 2 e˙ T (t )Z2, e˙ (t ) −

 H

=0



=0



e(t − tk, ) e(t − tk,+1 )

T

−Z2, ∗

Z2, −Z2,





e(t − tk, ) . e(t − tk,+1 )

(39)

Then, by using the same method as in proof of Theorem 1, we have

ˆ  ξ (t ), Vˆ˙ (t ) ≤ ξ T (t )

(40)

where ξ (t ) = [eT (t ), e˙ T (t ), eT (tk ),

 tk

t eT (s )ds,

ˆ = ˆ 1, + (t − tk )( ˆ 2, + Y R−1 Y T ) + tk,H )]T ,  pletes the proof.

eT (t − tk ), ζ T (t ), f T (De(t ))]T , ζ (t ) = [eT (t − tk,1 ), eT (t − tk,2 ), . . . , eT (t −

hk 2 ˆ −1 Qˆ T 4 QX

ˆ 3 , Qˆ = [Q1 , 0, Q2 , 0, 0, 0, 0]T . This com+ (hk − (t − tk ))

Remark 4. Different from the existing discontinuous LKF methods in [24–26,28–30], this paper introduces a new discontinuous form in (6) based on NSIFA and convex combination technique due to its superiority in reducing conservatism. 4. Numerical examples In this section, two numerical examples are provided to demonstrate the superiority and less conservatism of the obtained synchronization criteria. Example 1. We apply the derived results to the Chua’s circuit system, which can be described as:



x˙ 1 (t ) = ζ1 (x2 (t ) − η1 x1 (t ) − h(x1 (t ))), x˙ 2 (t ) = x1 (t ) − x2 (t ) + x3 (t ), x˙ 3 (t ) = −ζ2 x2 (t ),

(41)

where the nonlinear function is given by

h(x1 (t )) = 0.5(η1 − η0 )(|x1 (t ) + 1| − |x1 (t ) − 1| ), with η1 = 2/7, η0 = −1/7, ζ1 = 9, ζ2 = 14.28. Then, the system (41) can be represented as a Lur’e system (1) with following parameters:



A =

−ζ1 η1 1 0

ζ1

−1 −ζ2



0 1 , B= 0





−ζ1 (η0 − η1 ) 0 , H = D = [1, 0, 0]. 0

In this example, we first consider the Theorem 1 with ε1 = 1, ε2 = 2, ε3 = 1.5, H = 4. It can be calculated that hmax = 0.46 with hmin = 0.12. In order to show our proposed approach can achieve less conservative results than those in [13–18,23,26]. Next, we only consider the constant sampling case, i.e., hmin = hmax , and the comparative works of ours and existing results are given in Table 1. From the Table 1, we can see that the proposed NSIFA approach can enhance the feasible region of the synchronization criteria. Moreover, when hmin = hmax = 0.6153, the corresponding control gain matrices are obtained as G = [1.9829, 0.1793, −1.9525]T . Let the initial conditions x(0 ) = [1.5, −2, 0.5]T , y(0 ) = [−1, 0.1, −2]T , the state responses of error system (4) without control input and with u(t) are showed in Figs. 1 and 2, respectively. Fig. 3 displays the control input curve. From Fig. 2, it is observed that the synchronization between chaotic Lur’e drive and slave systems (1) is achieved.

20

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

600

400

e (t), i=1,2,3.

200

i

0

−200

−400

−600

0

5

10 Time t

15

20

Fig. 1. The state response of error system without u(t). Table 1 The maximum allowable upper bound of h. Synchronization criteria

MASI

Degree of improvement (%)

Corollary 1 (H = 4) [13] [14] [15] [16] [17] [18] [26] [23]

0.6153 0.3914 0.3981 0.45 0.48 0.5144 0.5147 0.5368 0.5965

57.20 54.55 36.73 28.18 19.61 19.54 14.62 3.15

Table 2 The maximum allowable upper bound of h. Synchronization criteria

MASI

Degree of improvement (%)

Corollary 1 (H = 4) [3] [14] [17] [26] [19] [23]

1.0574 0.3212 0.3981 0.9121 0.9488 1.0347 1.0380

229.20 165.61 15.93 11.44 2.19 1.86

Example 2. Consider the chaotic Lur’e systems (1) with the following parameters in [3,14,17,19,23,26]:

A =

−1 0 0

0 −1 0



0 0 , B= −1



1.2 1.24 0

−1.6 1 2.2



0 0.9 , H = D = 1.5



1 0 0

0 1 0



0 0 . 1

As discussed in [17,23,26], it can be viewed as a neural network, where neural activation function is ωρ = 0.5(|x1 (t ) + 1| − |x1 (t ) − 1| )(ρ = 1, 2, 3 ) and the neural activation function belongs to sector [0,1]. In this example, we only consider the constant sampling case. By using various approaches, the comparative results are listed in Table 2.

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

21

10

ei(t), i=1,2,3.

5

0

−5

−10

−15

0

5

10

15

20

25

30

35

30

35

Time t Fig. 2. The state response of error system with u(t).

15

10

ui(t), i=1,2,3.

5

0

−5

−10

−15

0

5

10

15

20

25

Time t Fig. 3. The control input ui (t) curve.

From the Table 2, we can see that the proposed NSIFA approach can enhance the feasible region of the synchronization criteria. Moreover, when the sampling interval is 1.0574,

consider the Corollary 1 with ε1 = 1, ε2 = 1.5, ε3 = 0.5, H = 4, the corresponding control gain matrix is obtained as G =

1.0621 −0.6490 −0.2396 0.5478 0.8605 0.5875 . −0.4909 1.1425 1.1227

22

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

5 4 3

ei(t), i=1,2,3.

2 1 0 −1 −2 −3 −4 −5

0

50

100 Time t

150

200

Fig. 4. The state response of error system without u(t).

2 1.5 1

ei(t), i=1,2,3.

0.5 0 −0.5 −1 −1.5 −2 −2.5

0

5

10

15 Time t

20

25

30

Fig. 5. The state response of error system with u(t).

Under the initial conditions x(0 ) = [1, −2, 0.5]T , y(0 ) = [−1, 0.5, 1]T , the state responses of error system (4) without control input and with u(t) are showed in Figs. 4 and 5, respectively. Fig. 6 displays the control input curve. From Fig. 5, it is observed that the synchronization between chaotic drive and slave Lur’e systems (1) is achieved.

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

23

4 3 2

ui(t), i=1,2,3.

1 0 −1 −2 −3 −4 −5

0

5

10

15 Time t

20

25

30

Fig. 6. The control input ui (t) curve.

5. Conclusion In this paper, we have studied the issue of sampled-data synchronization for a class of CLSs, where a novel (SIFA) is proposed. First, by partitioning sampling interval into several nonuniform segments based on a geometric series and taking advantage of the convex combination technique, a newly discontinuous LKF is designed for the first time to analyze the synchronization problem of such systems, which significantly uses more information on actual sampling behavior of the system. Moveover, an USIFA is also taken into account. Then, some sufficient sampled-data synchronization criteria of concerned systems are formulated in framework of matrix inequalities with a larger sampling period. Finally, two numerical simulations are provided to demonstrate the superiority and effectiveness of the derived results. In future work, we will further research the topic on sampled-data synchronization of multi-agent systems and complex dynamical networks by pinning control. Acknowledgments The authors would like to thank the referee and the editor for their useful comments. This work was supported by the National Natural Science Foundation of China (61771004 and 61533006). References [1] X. Yang, Z. Yang, X. Nie, Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication, Commun. Nonlinear Sci. Numer. Simul. 19 (5) (2014) 1529–1543. [2] W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Annu. Rev. Physiol. 55 (1) (1993) 349–374. [3] Z. Wu, P. Shi, H. Su, J. Chu, Exponential synchronization of neural networks with discrete and distributed delays under time-varying sampling, IEEE Trans. Neural Netw. Learning Syst. 23 (9) (2012) 1368–1376. [4] J. Xiao, S. Zhong, Y. Li, F. Xu, Finite-time Mittag–Leffler synchronization of fractional-order memristive BAM neural networks with time delays, Neurocomputing 219 (2017) 431–439. [5] X. Wang, X. Liu, K. She, S. Zhong, Pinning impulsive synchronization of complex dynamical networks with various time-varying delay sizes, Nonlinear Anal. Hybrid Syst. 26 (2017) 307–318. [6] R. Zhang, D. Zeng, S. Zhong, Y. Yu, Event-triggered sampling control for stability and stabilization of memristive neural networks with communication delays, Appl. Math. Comput. 310 (2017) 57–74. [7] H. Yang, X. Wang, S. Zhong, L. Shu, Synchronization of nonlinear complex dynamical systems via delayed impulsive distributed control, Appl. Math. Comput. 320 (2018) 75–85. [8] J.H. Park, C.H. Park, O.M. Kwon, S.M. Lee, A new stability criterion for bidirectional associative memory neural networks of neutral-type, Appl. Math. Comput. 199 (2008) 716–722. [9] J.H. Park, O.M. Kwon, Further results on state estimation for neural networks of neutral-type with time-varying delay, Appl. Math. Comput. 208 (2009) 69–75. [10] X. Wang, X. Liu, K. She, S. Zhong, L. Shi, Delay-dependent impulsive distributed synchronization of stochastic complex dynamical networks with timevarying delays, IEEE Trans. Syst. Man Cybern. Syst. (2018), doi:10.1109/TSMC.2018.2812895.

24

H. Yang, X. Wang and L. Shu et al. / Applied Mathematics and Computation 348 (2019) 12–24

[11] R. Rakkiyappan, G. Velmurugan, J.N. George, R. Selvamani, Exponential synchronization of Lur’e complex dynamical networks with uncertain inner coupling and pinning impulsive control, Appl. Math. Comput. 307 (2017) 217–231. [12] Z. Tang, J.H. Park, J. Feng, Novel approaches to pin cluster synchronization on complex dynamical networks in Lur’e forms, Commun. Nonlinear Sci. Numer. Simul. 57 (2018) 422–438. [13] W. Chen, Z. Wang, X. Lu, On sampled-data control for master-slave synchronization of chaotic Lur’e systems, IEEE Trans. Circuits Syst. II. 59 (8) (2012) 515–519. [14] Z. Wu, P. Shi, H. Su, J. Chu, Sampled-data synchronization of chaotic Lur’e systems with time delays, IEEE Trans. Neural Netw. Learn. Syst. 24 (3) (2013) 410–421. [15] X. Xiao, L. Zhou, Z. Zhang, Synchronization of chaotic Lur’e systems with quantized sampled-data controller, Commun. Nonlinear Sci. Numer. Simul. 19 (6) (2014) 2039–2047. [16] C. Zhang, L. Jiang, Y. He, Q. Wu, M. Wu, Asymptotical synchronization for chaotic Lur’e systems using sampled-data control, Commun. Nonlinear Sci. Numer. Simul. 18 (10) (2013) 2743–2751. [17] C. Hua, C. Ge, X. Guan, Synchronization of chaotic Lur’e systems with time delays using sampled-data control, IEEE Trans. Neural Netw. Learn. Syst 26 (6) (2015) 1214–1221. [18] C. Ge, W. Zhang, W. Li, X. Sun, Improved stability criteria for synchronization of chaotic Lur’e systems using sampled-data control, Neurocomputing 151 (2015) 215–222. [19] D. Zeng, R. Zhang, Y. Liu, S. Zhong, Sampled-data synchronization of chaotic Lur’e systems via input-delay-dependent-free-matrix zero equality approach, Appl. Math. Comput. 315 (2017) 34–46. [20] S.H. Lee, M.J. Park, O.M. Kwon, R. Sakthivel, Advanced sampled-data synchronization control for complex dynamical networks with coupling time– varying delays, Inf. Sci. 420 (2017) 454–465. [21] H. Yang, L. Shu, S. Zhong, X. Wang, Extended dissipative exponential synchronization of complex dynamical systems with coupling delay and sampled-data control, J. Frankl. Instit. 353 (8) (2016) 1829–1847. [22] Z. Wu, P. Shi, H. Su, J. Chu, Sampled-data fuzzy control of chaotic systems based on a t-s fuzzy model, IEEE Trans. Fuzzy Syst. 22 (1) (2014) 153–163. [23] J. Park, S.Y. Lee, P. Park, An improved fragmentation approach to sampled-data synchronization of chaotic Lur’e systems, Nonlinear Anal. Hybrid Syst. 29 (2018) 333–347. [24] T.H. Lee, J.H. Park, Stability analysis of sampled-data systems via free-matrix-based time-dependent discontinuous Lyapunov approach, IEEE Trans. Automa. Control. 62 (2017) 3653–3657. [25] T.H. Lee, J.H. Park, New methods of fuzzy sampled-data control for stabilization of chaotic systems, IEEE Trans. Syst. Man Cybern. Syst. (2017), doi:10. 1109/TSMC.2017.2690803. [26] T.H. Lee, J.H. Park, Improved criteria for sampled-data synchronization of chaotic Lur’e systems using two new approaches, Nonlinear Anal. Hybrid Syst. 24 (2017) 132–145. [27] K. Liu, E. Fridman, Wirtinger’s inequality and Lyapunov-based sampled-data stabilization, Automatica 48 (1) (2012) 102–108. [28] P. Naghshtabrizi, J.P. Hespanha, A.R. Teel, Exponential stability of impulsive systems with application to uncertain sampled-data systems, Syst. Control Lett. 57 (5) (2008) 378–385. [29] T.H. Lee, Z. Wu, J.H. Park, Synchronization of a complex dynamical network with coupling time-varying delays via sampled-data control, Appl. Math. Comput. 219 (3) (2012) 1354–1366. [30] Y. Liu, B. Guo, J.H. Park, S.M. Lee, Nonfragile exponential synchronization of delayed complex dynamical networks with memory sampled-data control, IEEE Trans. Neural Netw. Learning Syst. 29 (1) (2018) 118–128.