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Event-Triggered Master-Slave Synchronization With Sampled-Data Communication Guanghui Wen, Member, IEEE, Michael Z. Q. Chen, Member, IEEE, and Xinghuo Yu, Fellow, IEEE

Abstract—In this brief, event-triggered synchronization is investigated for a class of heterogeneous master-slave coupling systems consisting of a high-dimensional master system but a low-dimensional slave system. An event-triggered sampleddata transmission strategy is designed, which allows the eventtriggered condition to be examined only at sampling instants. With the assumption that only some sampled-data outputs of the master system are available to the slave system, a kind of observer-based synchronization controller is constructed and utilized. On the basis of a delayed-input approach, the masterslave synchronization problem is equivalently converted into the asymptotical stability problem of a time-delay system. By constructing an appropriate Lyapunov-Krasovskii functional and employing the free-weighting-matrix approach, some sufficient synchronization criteria in terms of linear matrix inequalities (LMIs) are derived. Numerical simulations are finally provided to verify the theoretical results. Index Terms—Event-triggered communication, master-slave synchronization, sampled-data, reduced-order observer.

I. I NTRODUCTION Synchronization is a ubiquitous phenomenon in both Nature and human society. The last decades have witnessed significant progress in our understanding of master-slave synchronization in coupled chaotic systems [1]. Note that the importance of master-slave synchronization does not only lie in its important theoretical value within chaotic theory, but also in its practical applications in such fields as symmetry and pattern formation, image processing, and secure communication [1]–[4]. A great number of control strategies have been proposed for achieving master-slave synchronization, including adaptive control [5], linear output control [6], [7], intermittent control [8] and so forth. However, it is assumed in most of the aforementioned works that full states or outputs of the master system can be continuously measured by the slave system over at least a sequence of uniformly bounded time intervals. Thus, the existing controllers acting on the slave system in the above-mentioned literature are indeed executed by analog signals. However, with the rapid developments This research was supported by the National Nature Science Foundation of China under Grant 61304168 and the Natural Science Foundation of Jiangsu Province of China under Grant BK20130595. G. Wen is with the Department of Mathematics, Southeast University, Nanjing 210096, China (e-mail: [email protected]). M. Z. Q. Chen is with the Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong SAR, China (e-mail: [email protected]). X. Yu is with the School of Electrical and Computer Engineering, RMIT University, Melbourne VIC 3001, Australia. He is also with the School of Automation, Southeast University, Nanjing 210096, China (e-mail: [email protected]). Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to [email protected].

of high-performance computing technology and modern digital communication technique, the designed continuous-time feedback controllers are usually implemented in digital form for superior reliability, lower cost and easier implementation. Partly motivated by this observation, sampled-data-based control strategies for synchronization of master-slave Lur’etype nonlinear systems were proposed in [9]–[12]. In [13], [14], master-slave synchronization for neural networks with sampled-data coupling was respectively studied. These works improve our understanding on how to design the controllers to make the states of the slave system track those of the master system by using only sampled-data coupling. However, it should be noted that the controllers for the salve system in the aforementioned references are time-triggered. Furthermore, a minimum sampling interval should be carefully selected to ensure master-slave synchronization in these works. Thus, these time-triggered design methods could lead to unnecessary utilization of the communication bandwidth in practice, especially in the case where there is very little fluctuation between successive sampling packets. Note that these drawbacks could be overcome by designing some event-triggered controllers for the slave system. Compared with the time-triggered sampling scheme, event-triggered sampling scheme typically requires less information transmission since the sampler is triggered only when the value of some carefully designed function of the system states exceeds a fixed or time-varying threshold value. It is also worth noting that event-triggered stabilization and its related problems for dynamic systems have received much attention in the past few years [15]-[17]. Another common assumption in most existing literature on masterslave synchronization is that the dimensions of master and slave systems are the same. From a practical viewpoint, it is favorable to allow the dimensions of master and slave systems to be different. Motivated by the above-mentioned observations, this brief studies the event-triggered sampled-data synchronization for a class of master-salve systems with a high-dimensional master system. Specifically, the dimensions of the considered master system are higher than those of the slave system. Thus, the existing synchronization criteria for master-slave systems with homogeneous dimensions are inapplicable for solving the considered synchronization problem. Furthermore, to make the model more practical, it is assumed that only some sampleddata outputs of the master system are available for controller design. To reduce communication costs, the sampled-data outputs of the master system will be sent out only when the value of some predesigned function on synchronization error is larger than a threshold value. By constructing a LyapunovKrasovskii functional, some sufficient synchronization condi-

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tions in terms of LMIs are derived. The effectiveness of the analytical analysis is verified by numerical simulations. The rest of this paper is organized as follows. The considered synchronization problem is formulated in Section II. The main theoretical results are provided in Section III. In Section IV, some numerical simulations are given for illustration. Conclusions are finally drawn in Section V. Notations. Let N and Rn be the sets of positive natural numbers and n-dimensional real vectors. Let Rn×m be the set of n × m real matrices. Notations In and On×m indicate the n × n identity and n × m zero matrices, respectively. Symbols ⊗ and k · k denote respectively the Kronecker product and the Euclidian norm. Symbol ∗ represents the symmetric term in a symmetric matrix.

constant vector until the next sample-data from the controller arrives at the actuator. The goal of this paper is to design a new kind of event-triggered controller u(t) such that the states of the slave system (2) can asymptotically track the output of the master system (1), i.e., limt→∞ kz(t) − y(t)k = 0 for any given initial conditions x(0) ∈ Rs , z(0) ∈ Rmn . To this end, an observer with order s−mn is first designed for the slave system to estimate the unavailable states of the master system. Then, an observer-based controller is constructed for the slave system. Let ζ(t) = (ζ1 (t)T , . . . , ζn−1 (t)T )T ∈ Rs−mn be the state vector of the reduced-order observer at time t, with ζi (t) ∈ Rmi for each i = 1, . . . , n−1. For convenience, one e into the following form may partition A  ˜ A11 ˜21  A e= A  .  .. A˜n1

II. P ROBLEM F ORMULATION Consider a master-slave system with heterogeneous dimensions where the dynamics of the master system are given as ( e x(t) ˙ = Ax(t) + f (x(t)), (1) e y(t) = Cx(t),

where x(t) ∈ Rs and y(t) ∈ Rmn represent respectively the state vector and output vector of the master system at time t, constant scalar h > 0 represents the sampling period for sampler embedded at the master system. The state vector is assumed to be partitioned as x(t) = (x1 (t)T , x2 (t)T , . . . , xn (t)T )T , where xi (t) ∈ Rmi , mi ∈ N e ∈ Rs×s with for i P = 1, 2, . . . , n, n ∈ N, matrix A n T T T ∈ s = i=1 mi , f (x(t)) = (f1 (x(t)) , . . . , fn (x(t)) ) s R is a continuously differential vector-valued function with fi (x(t)) ∈ Rmi for i = 1, . . . , n. Furthermore, it is assumed e = [Om ×(s−m ) , Imn ] ∈ Rmn ×s . that y(t) = xn (t), i.e., C n n e A) e is detectable. Note that the Assume that the matrix pair (C, system (1) covers many well-known chaotic systems such as Chua’s circuit system, the Duffing system, etc. The dynamics of the slave system are given as z(t) ˙ = Az(t) + f˜(z(t)) + u(t),

(2)

where z(t) ∈ Rmn is the state vector, system matrix A ∈ Rmn ×mn , f˜(z(t)) : Rmn 7→ Rmn is a continuously differential vector-valued function, and u(t) is the control input that will be designed later. To facilitate the analysis, it is assumed that the state matrix e and nonlinear function fe(·) of (1) are available to controller A design. However, the initial condition x(0) and the exact value of fe(x(t)) are both unavailable. Furthermore, suppose that there is a sampler embedded at the master system which samples the master system’s output with a constant periodic h > 0. Then, the available output information yb(t) of the master system is defined as yb(t) = y(kh), for t ∈ [kh, (k+1)h), k ∈ N. However, whether the sampled-data output y(kh) should be sent to the controller of the slave system or not depends on whether or not the designed event actually occurs. On the other hand, the output of the controller acting on the slave system is sent to the actuator equipped with a zero-order hold (ZOH) circuit. Roughly speaking, the ZOH circuit smoothes the control inputs of the slave systems by keeping them as a

˜12 A ˜ A22 .. . ˜n2 A

··· ··· .. . ···

˜1n A ˜ A2n .. . ˜nn A



  , 

(3)

e − x(t), where of which A˜ij ∈ Rmi ×mj . Let e(t) = ζ(t) T T T e ζ(t) = (ζ(t) , z(t) ) . Then, it is sufficient to show that synchronization in master-salve systems (1) and (2) will be achieved if limt→∞ ke(t)k = 0. The sampled-data output of the slave system is transmitted to the controller for the slave system based on the violation of some well designed conditions rather than the elapse of certain time. Let tk h be the kth (k ≥ 1, k ∈ N) transmission time instant of the sampleddata output of the slave system and t0 = 0. Then, the (k +1)th transmission instant of the sampled-data output of the slave system is defined as tk+1 h = tk h + τk h,

(4)

where

n e (e(tk h + rh) − e(tk h)) k τk = argminr∈N kC o e k h + rh)k ≥ δkCe(t

(5)

for some δ ≥ 0. To construct the controller for the salve system, the following state estimator is first presented e ζ˙i (t) = A˜in z(t) + fi (ζ(t)) +

n−1 X j=1

A˜ij ζj (t)

(6)

− Fi (z(tk h) − y(tk h)),

where t ∈ [tk h, tk+1 h), k ∈ N, gain matrix Fi ∈ Rmi ×mn will be designed later, i = 1, . . . , n − 1. Then, for t ∈ [tk h, tk+1 h) and k ∈ N, the following event-triggered controller is designed for the slave system ˇ u(t)=Az(t)+ fˇn (t)+

n−1 X

A˜nj ζj (t)−Fn (z(tk h)−y(tk h)), (7)

j=1

where Aˇ = A˜nn − A, A˜ij , i, j = 1, 2, . . . , n, are defined in e (3), fˇn (t) = fn (ζ(t)) − f˜(z(t)), gain matrix Fn ∈ Rmn ×mn will be designed later. To derive the main results, the following assumption is made.

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Assumption 1: There exists a positive semi-definite matrix M ∈ Rs×s such that (f (ρ) − f (̺))T (f (ρ) − f (̺)) ≤ (ρ − ̺)T M (ρ − ̺), for any given ρ, ̺ ∈ Rs . Remark 1: It can be seen from (6) and (7) that only the sampled-data outputs of the master system are employed in designing the observer and the controller. Thus, the present design method is more favorable than those depending on the sampled-data states of the master system. The design procedure is partly motivated by [18] where the distributed consensus tracking problem for networked agents with continuous information transmission was addressed. Note that the e and nonlinear function fe(·) of structures of state matrix A (1) are assumed to be available for designing the transmission strategy in this brief. Whether it is possible to solve the considered master-slave synchronization problem without employing the aforementioned assumption remains an open issue. III. M AIN R ESULTS e According to the definition of ζ(t) = (ζ(t)T , z(t)T )T , it can be obtained from (1), (2), (6) and (7) that e˙ e + f (ζ(t)) e eζ(t) ζ(t) =A − F (z(tk h)−y(tk h)), (8)

where F = (F1T , . . . , FnT )T ∈ Rs×mn . It thus follows from (8) that, for t ∈ [tk h, tk+1 h) and an arbitrarily given k ∈ N, e e e k h). e(t) ˙ = Ae(t) + f (ζ(t)) − f (x(t)) − F Ce(t

(9)

Motivated by the methods provided in [15], define the auxiliary error eb(lh) = e(lh) − e(tk h), where tk ≤ l < tk+1 , l ∈ N, tk is defined in (4). For an arbitrarily given k ∈ N, tk+1 −1 one has [tk h, tk+1 h) = ∪l=t [lh, (l+1)h). Then, for t ∈ k [lh, (l+1)h), system (9) can be rewritten as e e e − τ (t)) e(t) ˙ = Ae(t) + f (ζ(t)) − f (x(t)) − F Ce(t e e(t − τ (t)), + F Cb

(10)

(11)

for all t ∈ [lh, (l+1)h). The initial condition of e(t) in system (10) is supplemented by setting e(θ) = ϕ(θ), θ ∈ [−h, 0] with constraints that ϕ(0) = e(0) and ϕ(·) : [−h, 0] 7→ Rnm is an absolutely continuous function with square-integrable derivatives. It can be concluded that the considered master-slave synchronization is achieved if system (10) is asymptotically stable. We are now in a position to present the main results of this paper. Theorem 1: Suppose that Assumption 1 holds. If for some given scalars c1 , c2 , ǫ > 0 and h > 0, there exist matrices P ∈ Rs×s > 0, Q ∈ Rs×s > 0, Ξ1 ∈ Rs×s , Ξ2 ∈ Rs×s , Ei ∈ Rs×s , i = 1, 2, 3, Fb ∈ Rs×mn , and L ∈ Rs×s , such that 

    

Ω11 ∗ ∗ ∗ ∗

P +hΞ1 +hΞT1 ∗

e 12 Ω12 + Ω e 22 Ω22 + Ω ∗ ∗ ∗

−hΞ1 −hΞ2 hΞ2 +hΞT2

Ω13 e 23 Ω23 + Ω Ω33 ∗ ∗

Ω14 Ω24 Ω34 Ω44 ∗



>0

Ω15 Ω25 Ω35 Ω45 Ω55



   < 0, 

     

Ω11 ∗ ∗ ∗ ∗ ∗

Ω12 Ω22 ∗ ∗ ∗ ∗

Ω13 Ω23 Ω33 ∗ ∗ ∗

Ω14 Ω24 Ω34 Ω44 ∗ ∗

Ω15 Ω25 Ω35 Ω45 Ω55 ∗

hE1T hE2T hE3T O O −hQ



    < 0,  

(14)

where e 1A eT LT +E1 +E1T −Ξ1 −ΞT1 +ǫM, Ω11 = c1 LA+c eT LT , Ω e 12 = hΞ1 +hΞT1 , Ω12 = P +E2 −c1 L+A eTLT−c1 FbC, e Ω14 =c1 L, Ω15 =c1 Fb , Ω13 = Ξ1 +Ξ2 −E1T +E3 +c2 A T e 22 = hQ, Ω23 = −E2T −FbC−c e 2 LT , Ω22 = −L−L , Ω

e 23 = −hΞT1 −hΞT2 , Ω24 = L, Ω25 = Fb , Ω T e T C−E e be eT b T Ω33 = −Ξ2 −ΞT2 +δ 2 C 3 −E3 − c2 F C − c2 C F , Ω34 = c2 L, Ω35 =c2 Fb , Ω44 =−ǫIs , Ω45 =Os×mn , Ω55 =−Imn ,

then the event-triggered master-slave synchronization in systems (1) and (2) can be achieved. Moreover, the desired feedback gain matrix F is given by F = L−1 Fb . Proof: Construct the following Lyapunov-Krasovskii functional V (t) for (10): 3 X Vi (t), t ∈ [lh, (l+1)h), (15) V (t) = i=1

where

V1 (t) = e(t)T P e(t), V2 (t) = ((l+1)h − t)

Z

t

e(s) ˙ T Qe(s)ds, ˙

lh

 T   e(t) e(t) V3 (t) = ((l+1)h − t) Ξ , e(lh) e(lh) # " Ξ1 +ΞT1 − Ξ1 −Ξ2 . Ξ= ∗ Ξ2 +ΞT2 It can be obtained from (15) that

where τ (t) = t − lh. Obviously, τ (t) is piecewise linear over time t and 0 ≤ τ (t) < h. Furthermore, by the event-triggered condition defined in (4), one obtains that e e(t − τ (t))k < δkCe(t e − τ (t))k, kCb



(12)

V (t) ≥ V1 (t) + V3 (t)     T  e(t) e(t) P O + [(l+1)h − t]Ξ = ∗ O e(lh) e(lh)    T  e(t) t − lh e(t) (l + 1)h − t P O = + ∗ O h h e(lh) e(lh)     T  e(t) e(t) P O + hΞ > 0, (16) · ∗ O e(lh) e(lh) for t ∈ [lh, (l+1)h), where the last inequality of (16) is derived by using (12) and the fact that P > 0. The above analysis indicates that V (t) given in (15) is a valid LyapunovKrasovskii functional for analyzing the stability of (10). For t ∈ [lh, (l+1)h) and an arbitrarily given k ∈ N, taking the timeP derivative of V (t) along the trajectory of (10) gives 3 V˙ (t) = i=1 V˙ i (t), where Vi (t), i = 1, 2, 3, are respectively given as: V˙ 1 (t) = 2e(t)T P e(t), ˙ Z t V˙ 2 (t) = − e(s) ˙ T Qe(s)ds ˙ + ((l+1)h − t)e(t) ˙ T Qe(t), ˙ lh

(13)

T  T      e(t) e(t) e(t) e(t) ˙ ˙ V3 (t) = − +2((l+1)h−t) . Ξ Ξ e(lh) e(lh) e(lh) 0

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According to the fact that



E T Q−1 E ∗

4

ET Q



≥ 0 for any

given matrix E = [E1 , E2 , E3 ] ∈ Rs×3s , one obtains Z t ˙ V2 (t)≤ − e(s) ˙ T Qe(s)ds ˙ + ((l+1)h − t)e(t) ˙ T Qe(t) ˙ lh

 Z t π(t) T E T Q−1 E + ∗ ˙ lh e(s)

ET Q





π(t) ds, e(s) ˙

(17)

of which π(t) = [e(t)T , e(t) ˙ T , e(lh)T ]T . Furthermore, for any s×s given matrix L ∈ R and scalars c1 , c2 , the following equality holds
 e 0 = 2 −c1 e(t)TL−e(t) ˙ TL−c2 e(lh)T L e(t)− ˙ Ae(t)
 e e e e(lh) . + f (ζ(t))−f (x(t))−F Ce(lh)+F Cb (18)

e For notational brevity, let fe(e(t)) = f (ζ(t))−f (x(t)). It can thus be obtained from Assumption 1 that 0 ≤ e(t)T M e(t)−fe(e(t))T fe(e(t)).

(19)

Furthermore, according to (11), one knows

e T Ce(lh) e eT Cb e e(lh) > 0. δ 2 e(lh)T C − eb(lh)T C

(20)

V˙ (t) ≤ 2e(t)T P e(t) ˙ + (t − lh)π(t)T E T Q−1 Eπ(t) T T + 2π(t) E (e(t)−e(lh))+((l+1)h−t)e(t) ˙ T Qe(t) ˙ T T T − e(t) (Ξ1 + Ξ1 )e(t) − 2e(t) (−Ξ1 − Ξ2 )e(lh) − e(lh)T (Ξ2 + ΞT2 )e(lh) + 2((l+1)h−t)e(t)T (Ξ1 + ΞT1 )e(t) ˙ + 2((l+1)h−t)e(lh)T (−Ξ1 − Ξ2 )e(t) ˙ 2 T eT e T Te e + e(t) M e(t)−f (e(t)) f (e(t))+δ e(lh) C Ce(lh)

(21)

of which

e θ1 (t)=−2e(t) c1 Le(t)+2e(t) ˙ c1 LAe(t)+2e(t) c1 Lfe(e(t)) e e e(lh), −2e(t)Tc1 LF Ce(lh) + 2e(t)Tc1 LF Cb T

T

V˙ (t) ≤ µ(t)T {Ω + [(l+1)h−t]Φ} µ(t)   T −1 E Q E O µ(t), +(t − lh)µ(t)T ∗ O where



Ω11  ∗ Ω=  ∗ ∗ ∗

Ω12 Ω22 ∗ ∗ ∗

Ω13 Ω23 Ω33 ∗ ∗

Ω14 Ω24 Ω34 Ω44 ∗



Ω15 Ω25  Ω35  , Ω45 Ω55

Ωij , i, j = 1, . . . , 5, are defined in (13), and  T O  ∗ Φ=  ∗ ∗ ∗

Ξ1 + Ξ1 Q ∗ ∗ ∗

O −ΞT1 − ΞT2 O ∗ ∗

O O O O ∗

The above analysis indicates that V˙ (t) ≤

According to (17)–(20), one obtains that

eT Cb e e(lh) + θ1 (t) + θ2 (t) + θ3 (t), −b e(lh)T C

i.e.,

(l+1)h−t µ(t)T (Ω + hΦ)µ(t) h   T −1

· Ω+h

E Q E ∗

(23)

(24)



O O  O  . O O

+ t−lh µ(t)T h O µ(t). O

(25)

(26)

According to (13), one obtains Ω + hΦ < 0.

(27)

In addition, by using the Schur complement lemma, it can be yielded from (14) that   T −1 E Q E O < 0. (28) Ω+h ∗ O Combining (26)–(28) leads to V˙ (t) < 0,

∀ t ∈ [lh, (l+1)h).

(29)

T

e θ2 (t)=−2e(t) ˙ TLe(t)+2 ˙ e(t) ˙ TLAe(t)+2 e(t) ˙ TLfe(e(t)) e e e(lh), −2e(t) ˙ TLF Ce(lh) + 2e(t) ˙ TLF Cb

T e θ3 (t)= − 2e(lh)Tc2 Le(t)+2e(lh) ˙ c2 LAe(t) T e + 2e(lh) c2 Lfe(e(t)) − 2e(lh)Tc2 LF Ce(lh)

ee(lh), t ∈ [lh, (l+1)h). + 2e(lh)Tc2 LF Cb

e e(lh)]T and Let µ(t) = [e(t)T , e(t) ˙ T , e(lh)T , fe(e(t))T , Cb b F = LF . It can be obtained from (21) that   Ω11 Ω12 Ω13 Ω14 Ω15    ∗ Ω22 Ω23 Ω24 Ω25  ∗ ∗ Ω33 Ω34 Ω35  V˙ (t) ≤ µ(t)T  +[(l+1)h   ∗ ∗ ∗ Ω44 Ω45   ∗ ∗ ∗ ∗ Ω55   O Ξ1 + ΞT1 O O O   Q −ΞT1 − ΞT2 O O   ∗   − t] ∗ ∗ O O O  µ(t)  ∗ ∗ ∗ O O   ∗ ∗ ∗ ∗ O   T −1 E Q E O µ(t), (22) + (t − lh)µ(t)T ∗ O

It thus can be concluded from (16) and (29) that the considered master-slave synchronization problem is solved. ■ Remark 2: Obviously, the Zeno-behavior of system (10) is excluded since the minimum inter-event time is lower bounded by h, i.e., the number of event-triggered data transmissions between the master and slave systems is finite over finite time intervals. Furthermore, it can be seen that the proposed eventtriggered communication strategy will be reduced to the timetriggered communication strategy when the parameter δ in (5) is set as 0. A larger δ implies lower communication frequency in strategy (4) and generally leads to a slower convergence rate for synchronization. It is also worth noting that matrix L is invertible under condition (14). At last, it should be noted that the specific construction of Lyapunov-Krasovskii functional (15) is motivated by [19]. IV. N UMERICAL E XAMPLE In simulation, the master system is assumed to be the followingChua’s circuit system  x˙ 1 (t) = ̺1 (−x1 (t) + x2 (t) − φ(x1 (t))) ,  x˙ 2 (t) = x1 (t) − x2 (t) + x3 (t), (30)   x˙ 3 (t) = −̺2 x2 (t),

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with ̺1 = 9, ̺2 = −14.28, φ(x1 (t)) = 57 x1 (t) + 3 14 (|x1 (t) + 1| − |x1 (t) − 1|). According to (1) and Assumption 1, one obtains f (x(t)) = (φ(x1 (t)), 0, 0)T , # " " # −9 9 0 10.28572 0 0 e 1 −1 1 , M = 0 0 0 . A= 0

−14.28

0

0

0

0

The slave system is chosen as a second-order integrator:  z˙1 (t) = z2 (t) + u1 (t), (31) z˙2 (t) = u2 (t), where u(t) = (u1 (t), u2 (t))T ∈ R2 is the control input. e = [O2×1 , I2 ], h = 0.02, c1 = 20, c2 = 18, Set C ǫ = 0.05 and δ = 1. It can be obtained from Theorem 1 that the event-triggered master-slave synchronization in systems (30) and (31) can be feedback # " solved. The desirable matrix is given as F =

0.0945 1.9296 0.5508

0.1443 −3.3795 . Profiles of 3.0957

e = (ζ(t), z1 (t), z2 (t))T and x(t) = (x1 (t), x2 (t), x3 (t))T ζ(t) are given in Fig. 1. The transmission instants and release intervals for slave system and the profiles of synchronization error are shown in Fig. 2, which verifies the analytic results very well. 1

0

0

0

−0.5

x1(t)

−1

x2(t)

−1

z1(t)

ζ(t) 0

5

10

0

5

10

−2

x3(t)

−4 −6

z (t) 2

0

5

10

e and x(t). Profiles of ζ(t)

Fig. 1.

release intervals

2

1

0.5

10 5 0

0

2

4

6

8

10

6

8

10

t

||e(t)||

3 2 1 0

2

4

t

Fig. 2. Transmission instants and release intervals for the slave system and the profiles of ke(t)k.

V. C ONCLUSIONS Event-triggered synchronization has been studied in this brief for coupling systems with a high-dimensional master system. To reduce the communication cost, a new type of event-triggered sampled-data transmission strategy has been proposed. One favorable property of the present transmission strategy is that the event-triggering condition may be only checked periodically with a certain time interval. Furthermore, it is only assumed that some partial states of the master system can be sensed at sampling time points. By transforming the synchronization problem into stability problem and using tools

from Lyapunov-Krasovskii stability theory, some synchronization criteria have been yielded. At last, numerical simulations have been given for illustration. R EFERENCES [1] G. Chen and X. Yu, (Eds.) Chaos Control: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 292, Berlin, Germany: Springer-Verlag, 2003. [2] K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, “Synchronization of Lorenz-based chaotic circuits with applications to communications,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 40, no. 10, pp. 626-633, Oct. 1993. [3] J. Lu, X. Wu, and J. L¨u, “Synchronization of a unified chaotic system and the application in secure communication,” Phys. Lett. A, vol. 305, no. 6, pp. 365-370, Dec. 2002. [4] H. Dimassi and A. Lor´ıa, “Adaptive unknown-input observers-based synchronization of chaotic systems for telecommunication,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 4, pp. 800-812, Apr. 2011. [5] H. R. Karimi, M. Zapateiro, and N. Luo, “Adaptive synchronization of master-slave systems with mixed neutral and discrete time-delays and nonlinear perturbations,” Asian J. Control, vol. 14, no. 1, pp. 251-257, Jan. 2012. [6] A. Lor´ıa, “Master-slave synchronization of fourth-order L¨u chaotic oscillators via linear output feedback,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 57, no. 3, pp. 213-217, Mar. 2010. [7] T.-L. Liao and N.-S. Huang, “An observer-based approach for chaotic synchronization with applications to secure communications,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 9, pp. 1144-1150, Sept. 1999. [8] Y. Wang, J. Hao, and Z. Zuo, “A new method for exponential synchronization of chaotic delayed systems via intermittent control,” Phys. Lett. A, vol. 374, nos. 19-20, pp. 2024-2029, Apr. 2010. [9] J.-G. Lu and D. J. Hill, “Global asymptotical synchronization of chaotic Lur’e systems using sampled-data: A linear matrix inequality approach,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 6, pp. 586-590, Jun. 2008. [10] C.-K. Zhang, Y. He, and M. Wu, “Improved global asymptotical synchronization of chaotic Lur’e systems with sampled-data control,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 4, pp. 320-324, Mar. 2009. [11] W.-H. Chen, Z. Wang, and X. Lu, “On sampled-data control for masterslave synchronization of chaotic Lur’e systems,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 59, no. 8, pp. 515-519, Aug. 2012. [12] Y.-Q. Wu, H. Su, and Z.-G. Wu, “Asymptotical synchronization of chaotic Lur’e systems under time-varying sampling,” Circ. Syst. Signal Pr., vol. 33, no. 3, pp. 699-712, Mar. 2014. [13] C.-K. Zhang, Y. He, and M. Wu, “Exponential synchronization of neural networks with time-varying mixed delays and sampled-data,” Neurocomputing, vol. 74, nos. 1-3, pp. 265-273, Dec. 2010. [14] Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Exponential synchronization of neural networks with discrete and distributed delays under time-varying sampling,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 9, pp. 1368-1376, Sept. 2012. [15] D. Yue, E. Tian, and Q.-L. Han, “A delay system method for designing event-triggered controllers of networked control systems,” IEEE Trans. Autom. Control, vol. 58, no. 2, pp. 475-481, Feb. 2013. [16] C. Peng and Q.-L. Han, “A novel event-triggered transmission scheme and control co-design for sampled-data control systems,” IEEE Trans. Autom. Control, vol. 58, no. 10, pp. 2620-2626, Oct. 2013. [17] G. Guo, L. Ding and Q.-L. Han, “A distributed event-triggered transmission strategy for sampled-data consensus of multi-agent systems,” Automatica, vol. 50, no. 5, pp. 1489-1496, May 2014. [18] Y. Hong and X. Wang, “Multi-agent tracking of a high-dimensional active leader with switching topology,” J. Syst. Sci. Complex., vol. 22, no. 4, pp. 722-731, Nov. 2009. [19] E. Fridman, “A refined input delay approach to sampled-data control,” Automatica, vol. 46, no. 2, pp. 421-427, Feb. 2010.

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