# Synchronization of Complex Dynamical Networks with Hybrid Time Delay under Event-Triggered Control: The Threshold Function Method

Authors: Fei Wang; Zhaowen Zheng; Yongqing Yang
Journal: Complexity
 (2019)

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## Abstract

This paper investigates the synchronization of general complex dynamical networks (CDNs) with both internal delay and transmission delay. Event-triggered mechanism is applied for the feedback controllers, in which the triggered function is formed as a nonincreasing function. Both continuous feedback and sampled-data feedback methods are studied. According to Lyapunov stability theorem and generalized Halanay’s inequality, quasi-synchronization criteria are derived at first. The synchronization error is bounded with some parameters of the triggered function. Then, the completed synchronization can be guaranteed as a special case. Finally, coupled neural networks as numerical simulation examples are given to verify the theoretical results.


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## Body

## 1. Introduction

In the last few decades, models formed as coupled networks have been applied to many fields, such as biology neural networks, ecological networks, Internet, WWW (World Wide Web), and so on [1–3]. Consequently, the CDNs (complex dynamical networks) have attracted much attention from a large number of researchers. There were also some topics in the investigations of CDNs. Among which, synchronization behaviors is one of the most important topics due to its applications in communication system, consensus, distributed computing, formation control and image processing, and so on [4–6]. Many results about synchronization of CDNs have been published in recent years, which could be seen in [7–14] and referenced there in.However, there are some networks that cannot be synchronized by their own internal structure; thus, some controllers have been designed to force them to synchronize. Some works have investigated the synchronization of the complex dynamical networks via control strategies in recent years [15–18]. For example, in [19], the problem of synchronization control of complex dynamical networks subject to nonlinear couplings and uncertainties has been investigated. In [20], the pinning impulsive synchronization problem for a class of complex dynamical networks with time-varying delay has been summarized. The exponential synchronization of Lur’e type complex dynamical networks with uncertain coupling strength has been studied in [21]. Synchronization of CDNs under actuator saturation has been investigated in [9, 22]. In the above results, signals were transferred real-time among networks. However, it may not fit for the real world systems. With the fast development of digital signal technologies, the sampled-data control has catered to people by its lower cost than continuous control approaches. Some results about synchronization of CDNs with sampled-data controllers were existed. The first result about the sampled-data synchronization control of CDNs was investigated in [23]. In [24], the stochastic sampled-data control method has been applied to the synchronization of CDNs. The synchronization problem for a class of complex delayed dynamical networks by using the sampled-data feedback control has been studied in [25], in which, reciprocally convex technique and a novel class of integral inequalities have also been used. The sampled-data synchronization problem of chaotic Lur’e systems by using sampled output of the systems with variable sampling rates has been studied in [26], and so on.Results in the aforementioned sampled-data feedback control, data of controllers often update continuous or period. Note that the signals of controllers were updated at a fixed rate regardless, whether they are really necessary or not. Naturally, there must be some unnecessary updates in the sampled-data control strategy. Recently, a new effective strategy has been presented, which was named as event-triggered control or event-driven control. Under the event-triggered mechanism, controllers will be updated as well as the occurrence of a designed event. It is obviously that the energy can be saved compared with the continuous updating or period updating. The analysis of the feedback control system with event-triggered scheme has drawn much attention in the last few years. For example, event-triggered sliding mode control for uncertain stochastic systems has been studied in [27]. In [28], the leader-following consensus for multiagent systems with general linear dynamics by means of event-triggered scheme has been investigated. The networked synchronization communication for nonlinear uncertain fractional-order chaotic systems via the event-triggered control method was investigated in [29]. Two survey papers survey recent advances in event-triggered communication and control has been published recently [30, 31]. There were also a number of works concerning the event-triggered control for the synchronization control problem, which can be found in [32–39] and references there in.According to the above existing works on the event-triggered consensus of multiagent systems, one can conclude that the event-triggered conditions are designed to estimate the change of system’s errors, which means that when errors converge to zeros, the event would not be triggered. It is a natural idea to design an event with a threshold about system’s errors. In [40], a constant event threshold has been designed, when measure errors are larger than the threshold, the event would be triggered. After that, a new exponentially decreasing event threshold condition has been proposed in [41]. Compared to some other designed events in above results, this kind of event condition is easier to analyse stability due to some fine properties of the exponential function.This paper investigates the synchronization problem for complex dynamical networks; both continuous feedback control and sampled-data feedback control will be studied. Inspired by the results of [41, 42], a nonincreasing event threshold function is used as the condition to define an event, quasi-synchronization can be reached. The trajectory of the error system exponentially converges to the bounded region which is related to the parameters of the threshold function. Furthermore, complete synchronization can be guaranteed with appropriate parameters in the threshold function.Indeed, our recent work has been published in [43], which has considered a simple CDN model without time delay. However, time delays cannot be unavoidable in real-world systems, which also could lead to some complex dynamical behaviors such as chaos. In the CDNs, the time delays also exist when information is transferring among nodes. Consequently, this paper will consider the internal delays of each node and transmission delays among nodes. The main contribution of this paper could be concluded as follows: first, a general CDN model with hybrid time delay has been considered in this paper, the model in [43] was a special case of this paper. Second, the synchronization can be reached under the event-triggered control method, in which, a kind of monotone nonincreasing threshold function is applied for the triggered function. Third, both continuous feedback and sampled-data feedback method are studied, and the Zeno behavior has been excluded in the continuous case.The rest of the paper is organized as follows: in Section2, we introduce some definitions and some lemmas which are necessary for presenting our results in the following. The main results about the synchronization control problem are presented in Section 3. Then, some examples are given to demonstrate the effectiveness of our results in Section 4. Conclusions are finally drawn in Section 5.Notations. Let ℝ be the set of real numbers. ℝn and ℝn1×n2 refer to the n-dimensional real vector and n1×n2 real matrices. The superscript “T” denotes matrix transposition. In denotes the n-dimensional identify matrix. For a vector x∈ℝn, x is defined as x=xTx. For P∈ℝn×n, λmaxP represents the maximum eigenvalue of P. ⊗ denotes Kronecker product. C−τ,0,ℝn denotes the set of all n-dimensional continuous functions defined on the interval −τ,0.


## 2. Preliminaries

Consider a complex dynamical network withN nodes, and the dynamics of the ith node is described by(1)x˙it=ft,xit,xit−τ1t+c1∑j=1NaijΓ1xjt−xit+c2∑j=1NbijΓ2xjt−τ2t−xit−τ2t+uit,where xit=xi1t,xi2t,…,xintT∈ℝni=1,2,…,N is the state variables of the ith node. f:ℝ×ℝn×ℝn⟶ℝn is a continuously vector value function. τ1t∈0,τ1 represents the internal delay occurring inside the node; τ2t∈0,τ2 denotes the transmission delay for the signal sent from jth node to ith node, where τ1,τ2 are known constants. uit is the control input which will be given later. Γ1,Γ2∈ℝn×n are diagonal matrices with positive diagonal elements. A=aij∈ℝN×N and B=bij∈ℝN×N are the weight configuration matrices. If there is a connection from node i to node j, then aij=aji>0, bij=bji>0; otherwise, aij=aji=bij=bji=0. The diagonal elements of matrix A and B are defined by(2)aii=−∑j=1,j≠iNaij,bii=−∑j=1,j≠iNbij.The initial values associated with system (1) are xit=ϕit∈ℂ−τ,0;ℝni=1,2,…,N, where τ=maxτ1,τ2. Let st be a solution of the isolated dynamic system:(3)s˙t=ft,st,st−τ1t.The initial values of the isolated dynamic system is defined asst=ϕst∈ℂ−τ,0;ℝn. Here, st may be an equilibrium point, a periodic orbit, or even a chaotic orbit. In this paper, the objective trajectory that the nonlinear dynamical network (1) will be forced to st.Leteit=xit−st, for i=1,2,…,N. According to (2), one has(4)∑j=1NaijΓ1st=∑j=1NbijΓ2st−τ2t=0,∑j=1NbijΓ2xit−τ2t=0.Then,(5)e˙it=x˙it−s˙t=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1xjt−st−xit−st+c2∑j=1NbijΓ2xjt−τ2t−xit−τ3t+c2∑j=1NbijΓ1st−τ2t−st−τ2t+uit=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt−eit+c2∑j=1NbijΓ2ejt−τ2t−eit−τ2t+uit,where f˜t,eit,eit−τ1t=ft,xit,xit−τ1t−ft,st,st−τ1t. Then, the error dynamics could be described by(6)e˙it=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t+uit.The initial conditions associated with the error systems areeit=ψit=ϕit−ϕst∈ℂ−τ,0;ℝn. It is easy to see that the globally exponential synchronization of the controlled delayed dynamical network (1) is achieved if the zero solution of the error dynamical system is globally exponentially stable. To proceed with the following definition, assumptions and lemmas are given.Definition 1 (see [13]).
The dynamic network (1) is said to achieve quasi-synchronization with an error bound ∈>0, if there exists a compact set Ω such that, for any ϕit,ϕst∈ℂ−τ,0;ℝn, the error signal eit=xit−st converges into the set Ω=et∣et≤∈ as t⟶+∞.Assumption 1 (see [20]).
For the vector-valued functionft,xt,xt−τ1t, suppose for any xt∈ℝn, yt∈ℝn, there exists positive constants L1>0 and L2>0 such that(7)xt−ytTft,xt,xt−τ1t−ft,yt,yt−τ1t≤L1xt−yt2+L2xt−τ1t−yt−τ1t2.Assumption 2 (see [44]).
For the vector-valued functionft,xt,xt−τ1t, suppose for any xt∈ℝn, yt∈ℝn, there exists positive constants Lf1>0 and Lf20 such that(8)ft,xt,xt−τ1t−ft,yt,yt−τ1t≤Lf1xt−yt+Lf2xt−τ1t−yt−τ1t.Lemma 1 (see [45]).
For matricesA,B,C, and D with appropriate dimensions, the Kronecker product ⊗ satisfies(9)A+B⊗C=A⊗C+B⊗C;A⊗BC⊗D=AC⊗BD;AT⊗BT=A⊗BT;λmaxA⊗B=λmaxAλmaxB.Lemma 2 (generalized Halanay’s inequality) (see [46]).
Ifut≥0, t∈−∞+∞, and(10)u˙t≤γt+αtut+βtsupt−τt≤ξ≤tuξt≥t0,ut=Ψtt≤t0,where Ψt is bounded and continuous for t≤t0, continuous functions γt≥0, βt≥0, and αt≤0 for t∈t0,+∞, τt≥0, and if there exists σ>0 such that(11)αt+βt≤−σ<0fort≥t0,then we have(12)ut≤γ∗σ+Ge−μ∗t−t0,t≥t0,where G, γ∗, and μ∗ are defined as the following: G=sup−∞<ξ≤t0Ψt, γ∗=supt0≤t<+∞γt, and μ∗=inft≥t0μt:μt+αt+βteμtτt=0.


## 3. Main Results

In this paper, both continuous feedback controllers and sampled-data feedback controllers are adopted; meanwhile, the controller considered in this paper is event-triggered.


### 3.1. Event-Triggered Control Method Based on Continuous Feedback

A continuous feedback controller is defined as(13)uit=−dixit−st.When we consider the event-triggered strategy, controllers will be updated when an event has been triggered instead of in a continuous time. More specifically, let us denote the time instants forith node when an event happens by t0i,t1i,… with tki<tk+1i. The controller for ith can be described as(14)uit=−dixitki−stki,t∈tki,tk+1i,where di is the feedback gain to be determined. Without loss generality, the first event is generated at time t0 for all nodes. For ith node, the event-triggering time instants are defined as follows:(15)tk+1i=inft>tki∣δit≥θt,where δit=eitki−eit and θt=θ0+αε−βt is the exponentially decreasing event threshold with given parameters ε>1, β≥0, α>0, and θ0≥0. Under controller (14), the synchronization error system can be rewritten as(16)e˙it=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieitki,t∈tki,tk+1i.Theorem 1.
Suppose that Assumption1 holds. If there exist positive constants p, q, ω1, ω2, and di such that(1)
a+L1+pIN−2−ω2D<0,(2)
p−q>0,where a=2c1λmaxA⊗Γ1+c2ω1λmaxBBT⊗Γ2Γ2T, D=diagd1,d2,…,dN, and q=L2+c2/ω1. Then, the controlled delayed dynamical network (1) can achieve quasi-synchronization, and the trajectory of the error system exponentially converges to the bounded region Ωθ0=et∣et≤d∗Nθ0/ω2p−q, where d∗=maxd1,d2,…,dN.Proof.
Considering the following Lyapunov function(17)Vt=et2=∑i=1Neit2=eTtet=∑i=1NeiTteit,calculating the derivatives of Vt along the solutions of system (16), one has(18)V˙t=2∑i=1NeiTte˙it=2∑i=1NeiTf˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieitki=2∑i=1NeiTtf˜t,eit,eit−τ1t+2c1∑i=1NeiTt∑j=1NaijΓ1ejt+2c2∑i=1NeiTt∑j=1NbijΓ2ejt−τ2t−2∑i=1NdieiTteitki.
Based on Assumption1, we have(19)2∑i=1NeiTtf˜t,eit,eit−τ1t≤L1∑i=1Neit2+L2∑i=1Neit−τ1t2=L1eTtet+L2eTt−τ1tet−τ1t.
By using the Kronecker product of the matrices, the second term and the third term can be written in the following compact matrix form:(20)2c1∑i=1NeiTt∑j=1NaijΓ1ejt=2c1etTA⊗Γ1et,2c2∑i=1NeiTt∑j=1NbijΓ2ejt−τ2t=2c2etTB⊗Γ2et−τ2t.
According toxTy=yTx=1/2xTy+yTx≤ε/2xTx+1/2εyTy, the following inequalities can be derived:(21)2c2etTB⊗Γ2et−τ2t≤c2ω1eTtB⊗Γ2B⊗Γ2Tet+c2ω1eTt−τ2tet−τ2t=c2ω1eTtBBT⊗Γ2Γ2Tet+c2ω1eTt−τ2tIN⊗Inet−τ2t.
The last term will be handled as follows:(22)−2∑i=1NdieiTteitki=−2∑i=1NdieiTteit+δit=−2∑i=1NdieiTteit−2∑i=1NdieiTtδit≤−2eTtD⊗Inet+∑i=1Ndiω2eiTteit+1ω2δiTtδit=−2−ω2eTtD⊗Inet+1ω2∑i=1Ndiδit2≤−2−ω2eTtD⊗Inet+d∗Nω2αε−βt+d∗Nθ0ω2.
Then, one can derive(23)V˙t≤L1eTtIN⊗Inet+L2eTt−τ1tIN⊗Inet−τ1t+2c1etTA⊗Γ1et+c2ω1eTtBBT⊗Γ2Γ2Tet+c2ω1eTt−τ2tIN⊗Inet−τ2t−2−ω2eTtD⊗Inet+d∗Nω2αε−βt+d∗Nθ0ω2=eTtL1IN⊗In+2c1A⊗Γ1+c2ω1BBT⊗Γ2Γ2T−2−ω2D⊗Inet+eTt−τ2tL2IN⊗In+c2ω1IN⊗Inet−τ2t+d∗Nω2αε−βt+d∗Nθ0ω2≤−pVt+qsupt−τ≤s≤tVs+γt,where γt=d∗N/ω2αε−βt+d∗Nθ0/ω2; it is easy to get γ∗=supt0≤t<+∞γt=d∗Nθ0/ω2. By Lemma 2 and condition (2) of Theorem 1, the following result can be obtained:(24)Vt≤γ∗p−q+Ψe−μ∗t−t0,t≥t0,where Ψ=sup−τ≤s≤t0Vs and μ∗ is a root of the equation μ−p+qeτμ=0. It is obvious that μ∗>0; then, we can conclude that the error system converges exponentially to the set Ωθ0=et∣et≤d∗Nθ0/ω2p−q can be guaranteed, and according to Definition 1, the controlled delayed dynamical network (1) can achieve quasi-synchronization. The proof is thus completed.
In order to exclude Zeno behavior, the following theorem shows that the interevent times are lower-bounded by a positive constant in the following theorem.Theorem 2.
Suppose that Assumption2 and all conditions of Theorem 1 hold. With the triggered instants determined by (15), the Zeno behavior can be excluded for all plants in the network, which means that the minimum interevent interval for all plants is lower-bounded by a positive scalar.Proof.
To prove this theorem,e˙it is bounded which will be derived at first, and then the Zeno behavior will be excluded.
By using the triangular inequality, it is easy to get(25)e˙it=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieitki≤ft,xit,xit−τ1t−ft,st,st−τ1t+c1∑j=1Naijγ1ejt+c2∑j=1Nbijγ2ejt−τ2t+dieit+δit≤Lf1+dieit+Lf2eit−τ1t+c1∑j=1Naijγ1ejt+c2∑j=1Nbijγ2ejt−τ2t+diδit,where γ1=Γ1 and γ2=Γ2. According to the results in Theorem 1, we have(26)eit≤et≤Vt≤γ∗p−q+Ψ,t∈−τ,+∞,where p, q, γ∗, and Ψ have been defined in Theorem 1. Then,(27)e˙it≤Lf1+Lf2+di−2c1γ1aii−2c2γ2biiγ∗p−q+Ψ+diδit≤M+diθt,where M=Lf1+Lf2+di−2c1γ1aii−2c2γ2biiγ∗/p−q+Ψ is a bounded constant. For t∈tki,tk+1i, noting that δit=∫tkitδ˙isds=−∫tkite˙isds, one has(28)δit=∫tkite˙isds≤∫tkite˙isds≤Mt−tki+di∫tkitθsds≤M+θ0+αdit−tki.
According to the triggered instants determined by (15), the next event will not be generated before δit∗=θt∗; denote T=t∗−tki, then,(29)θ0+αε−βT+tki≤M+θ0+αdiT.
LetqT=θ0+αε−βT+tki−M+θ0+αdiT, one has q0>0, and when T⟶+∞, qT<0; furthermore, qT is monotonically decreasing. Thus, to obtain the above inequality, there must be T≥T∗>0. It can be concluded that there exists a positive lower bound of the minimum interevent interval. This completes the proof.Remark 1.
Note thatε, β, and α in the threshold function θt cannot affect the region of synchronization error. However, from (29), one can find that those parameters may affect the lower bound of the minimum interevent interval. The parameter M has included aii and bii, which are determined by the topology of networks.Remark 2.
θ
0 is not a parameter of controllers, and it can be adjusted free. According to the result in Theorem 1, the synchronization error will be bounded by a parameter related with θ0. In addition, the complete synchronization can be guaranteed when θ0=0, which will be checked in the simulation part. Furthermore, when θ0=0, the Zeno behavior still can be excluded.


### 3.2. Event-Triggered Control Method Based on Sampled-Data Feedback

A sampled-data feedback controller is defined as(30)uit=−diximT−smT,where T is the sampling period, m=0,1,…. Under the event-triggered strategy, the data will be updated when the designed event was triggered, instead of in every sampled time. The controller for ith can be described as(31)uit=−dixitki−stki,t∈tki,tk+1i,where tki=hkiT means that kth trigger for ith node at hkith sample, hki=0,1,… and h0i=0. For ith node, the event-triggering time instants are defined as follows:(32)hk+1i=minj∈ℤj>hkiδijT≥θj+1T,where δishki+jT=eitki−eihki+jT, j=1,2,…. Let us define τ^it=t−hki+jT for ith node during t∈hki+jT,hki+j+1T. Under controller (14), the synchronization error system can be rewritten as(33)e˙it=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieit−τ^it+δist−τ^it.The following vector symbols will be used later:(34)et−τ^t=e1t−τ^1t,e2t−τ^2t,…,eNt−τ^NtT,et−τ2t=e1t−τ2t,e2t−τ2t,…,eNt−τ2t,F˜t,et,et−τ1t=f˜t,e1t,e1t−τ1t,f˜t,eit,e2t−τ1t,…,f˜t,eNt,eNt−τ1tT,e˙t=e˙1t,e˙2t,…,e˙NtT.Theorem 3.
The controlled delayed dynamical network (1) can achieve quasi-synchronization under controller (31), and the trajectory of the error system exponentially converges to the bounded region Ωθ0=etet≤Nθ0/2ξ if there exist matrices P>0, Q>0, R>0, and S>0 such that(35)Ξ0=Ξ11Ξ12Ξ13Ξ14Ξ15∗000Ξ25∗∗00Ξ35∗∗∗0Ξ45∗∗∗∗Ξ550<0,(36)ΞT=Ξ11Ξ12Ξ13Ξ14Ξ150∗000Ξ250∗∗00Ξ350∗∗∗0Ξ450∗∗∗∗Ξ55T0∗∗∗∗∗Ξ66T<0,where ξ>0 is a given decay rate,(37)Ξ11=2ξIN⊗P+2c1A⊗RΓ1+DDT⊗RRT,Ξ12=c2B⊗RΓ2,Ξ13=−D⊗R,Ξ14=IN⊗R,Ξ15=−IN⊗R+c1A⊗SΓ1T+IN⊗P2,Ξ25=c2B⊗SΓ2T,Ξ35=−D⊗S,Ξ45=IN⊗S,Ξ550=TIN⊗Q−2IN⊗S+DDT⊗SST,Ξ55T=−2IN⊗S+DDT⊗SST,Ξ66T=Te−2ξTIN⊗Q.Proof.
Considering the following Lyapunov function(38)Vt=∑i=1NeiTPeit+∑i=1NT−τ^it∫t−τ^itte2ξs−te˙iTsQe˙isds,calculating the derivatives of Vt along the solutions of system (33), one has(39)V˙t=∑i=1NeiTPe˙it+∑i=1NT−τ^ite2ξt′∫t−τ^itte2ξse˙iTsQe˙isds+∑i=1NT−τ^it∫t−τ^itte2ξs−te˙iTsQe˙isds′.
Noting thatτ^i⋅t=1 and deit−τ^it/dt=e˙it−τ^itt−τ^it′=0, then we have(40)V˙t=∑i=1NeiTPe˙it−2ξ∑i=1NT−τ^it∫t−τ^itte2ξs−te˙iTsQe˙isds−∑i=1N∫t−τ^itte2ξs−te˙iTsQe˙isds+∑i=1NT−τ^ite˙iTsQe˙is≤∑i=1NeiTPe˙it−2ξ∑i=1NT−τ^it∫t−τ^itte2ξs−te˙iTsQe˙isds−e−2ξT∑i=1N∫t−τ^itte˙iTsQe˙isds+∑i=1NT−τ^ite˙iTsQe˙is.
Thus, by using Kronecker product, we have(41)V˙t+2ξVt≤eTtIN⊗Pe˙t+e˙TtTIN−τ^t⊗Qe˙t−e−2ξT∑i=1N∫t−τ^itte˙iTsQe˙isds+2ξeTtIN⊗Pet.
Letωit=1/τ^it∫t−τ^itte˙isds; according to Jensen’s inequality, one has(42)∫t−τ^itte˙iTsQe˙isds≥τ^itωiTtQωit.
Thus,(43)−e−2ξT∑i=1N∫t−τ^itte˙iTsQe˙isds≤−e−2ξT∑i=1Nτ^itωiTtQωit=−e−2ξTωTtτ^t⊗Qωt,where ωTt=ωiTt,ω2Tt,…,ωNTt. From (33), the following equality can be derived:(44)0=∑i=1N2eiTtR+e˙iTtSf˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieit−τ^it+δist−τ^it−e˙it.
By using Kronecker product again, we have(45)0=2eTtIN⊗RF˜+2c1eTtA⊗RΓ1et+2c2eTtB⊗RΓ2et−τ2t−2eTtD⊗Rδst−τ^t−2eTtD⊗Ret−τ^t−2eTtIN⊗Re˙t+2e˙TtIN⊗SF˜+2c1e˙TtA⊗SΓ1et+2c2e˙TtB⊗SΓ2et−τ2t−2e˙TtD⊗Sδst−τ^t−2e˙TtD⊗Set−τ^t−2e˙TtIN⊗Se˙t.
Among the aforementioned equation, note that(46)−2eTtD⊗Rδst−τ^t≤eTtD⊗RD⊗RTet+δst−τ^t2,(47)−2e˙TtD⊗Sδst−τ^t≤e˙TtD⊗RD⊗STe˙t+δst−τ^t2.
We can find from (41), (43), and (45)–(47) that(48)V˙t+2ξVt≤eTtIN⊗Pe˙t+e˙TtTIN−τ^t⊗Qe˙t+2ξeTIN⊗Ptet−e−2ξTωTtτ^t⊗Qωt+2eTtIN⊗RF˜+2c1eTtA⊗RΓ1et+2c2eTtB⊗RΓ2et−τ2t−2eTtD⊗Ret−τ^t−2eTtIN⊗Re˙t−2e˙TtD⊗Set−τ^t−2e˙TtIN⊗Se˙t+2e˙TtIN⊗SF˜+2c1e˙TtA⊗SΓ1et+2c2e˙TtB⊗SΓ2et−τ2t+eTtD⊗RD⊗RTet+e˙TtD⊗RD⊗STe˙t+2δst−τ^t2.
Then, one has(49)V˙t+2ξVt≤ηTtΞηt+2δst−τ^t2,where ηt=eTt,eTt−τ2t,eTt−τ^t,F˜,e˙Tt,ωTt.(50)Ξ=Ξ11Ξ12Ξ13Ξ14Ξ150∗000Ξ250∗∗00Ξ350∗∗∗0Ξ450∗∗∗∗Ξ550∗∗∗∗∗Ξ66,where Ξ55=TIN−τ^t⊗Q−2IN⊗S+DDT⊗SST, Ξ66=e−2ξTτ^t⊗Q, and Ξij has been given in Theorem 3. Let η0t=eTt,eTt−τ2t,eTt−τ^t,F˜,e˙Tt, and based on the definition of Ξ0 and ΞT, we have(51)TIN−τ^tTη0TtΞ0η0t+τ^tTηTtΞTηt=ηTtΞηt.
Thus, (35) and (36) imply that Ξ<0; based on the definition of event-triggered instants, we have(52)V˙t≤−2ξVt+Nαε−βt+Nθ0.
According to Lemma2, we have(53)Vt≤Nθ02ξ+Ψe−2ξt−t0,where Ψ=sup−τ≤s≤t0Vs. This completes our proof.Remark 3.
Note that the event-triggered instantstki will be updated in some sampled instants; thus, it is easy to see the Zeno behavior will be excluded. Meanwhile, there were some published results which had investigated the synchronization problem about complex dynamical networks under the sampled control [23–25]. Compared with those results, the event-triggered strategy can reduce transmission frequency.Remark 4.
Some conditions formed as LMI have been obtained to guarantee the quasi-synchronization of CDNs under event-triggered with sampled data in Theorem3. In which, the conditions can be checked by LMI toolbox of Matlab software directly and easily. However, the sampled period T needs to be given in the conditions; otherwise, the conditions cannot be regarded as LMI. It is obviously that the sampled-data period T cannot be chosen arbitrarily. Thus, it would be a significant problem that how to determine the maximum and minimum of sampled period T. This will be our future work.Remark 5.
This paper has investigated a CDN model with time-varying delays. However, it is not evident that how the time delays effect the synchronization behavior. Indeed, in the case of continuous data, we have used the generalized Halanay’s inequality; this powerful tool just requires time delays to be bounded and the time delays can effect the rate of convergenceμ∗. In the case of sampled data, to deal with the time delay, we have applied the well-known free-weighting matrix method, and one can see (41) and (43) in the proof of Theorem 3, then, the time delay has not appeared on the conditions. However, the conditions formed as LMIs in Theorem 3 are conservative. It is an interesting problem to analyse how can the time delays affect the synchronization in the sampled-data case and how to reduce the conservatism of the conditions in Theorem 3. They would be our future works.


## 3.1. Event-Triggered Control Method Based on Continuous Feedback

A continuous feedback controller is defined as(13)uit=−dixit−st.When we consider the event-triggered strategy, controllers will be updated when an event has been triggered instead of in a continuous time. More specifically, let us denote the time instants forith node when an event happens by t0i,t1i,… with tki<tk+1i. The controller for ith can be described as(14)uit=−dixitki−stki,t∈tki,tk+1i,where di is the feedback gain to be determined. Without loss generality, the first event is generated at time t0 for all nodes. For ith node, the event-triggering time instants are defined as follows:(15)tk+1i=inft>tki∣δit≥θt,where δit=eitki−eit and θt=θ0+αε−βt is the exponentially decreasing event threshold with given parameters ε>1, β≥0, α>0, and θ0≥0. Under controller (14), the synchronization error system can be rewritten as(16)e˙it=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieitki,t∈tki,tk+1i.Theorem 1.
Suppose that Assumption1 holds. If there exist positive constants p, q, ω1, ω2, and di such that(1)
a+L1+pIN−2−ω2D<0,(2)
p−q>0,where a=2c1λmaxA⊗Γ1+c2ω1λmaxBBT⊗Γ2Γ2T, D=diagd1,d2,…,dN, and q=L2+c2/ω1. Then, the controlled delayed dynamical network (1) can achieve quasi-synchronization, and the trajectory of the error system exponentially converges to the bounded region Ωθ0=et∣et≤d∗Nθ0/ω2p−q, where d∗=maxd1,d2,…,dN.Proof.
Considering the following Lyapunov function(17)Vt=et2=∑i=1Neit2=eTtet=∑i=1NeiTteit,calculating the derivatives of Vt along the solutions of system (16), one has(18)V˙t=2∑i=1NeiTte˙it=2∑i=1NeiTf˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieitki=2∑i=1NeiTtf˜t,eit,eit−τ1t+2c1∑i=1NeiTt∑j=1NaijΓ1ejt+2c2∑i=1NeiTt∑j=1NbijΓ2ejt−τ2t−2∑i=1NdieiTteitki.
Based on Assumption1, we have(19)2∑i=1NeiTtf˜t,eit,eit−τ1t≤L1∑i=1Neit2+L2∑i=1Neit−τ1t2=L1eTtet+L2eTt−τ1tet−τ1t.
By using the Kronecker product of the matrices, the second term and the third term can be written in the following compact matrix form:(20)2c1∑i=1NeiTt∑j=1NaijΓ1ejt=2c1etTA⊗Γ1et,2c2∑i=1NeiTt∑j=1NbijΓ2ejt−τ2t=2c2etTB⊗Γ2et−τ2t.
According toxTy=yTx=1/2xTy+yTx≤ε/2xTx+1/2εyTy, the following inequalities can be derived:(21)2c2etTB⊗Γ2et−τ2t≤c2ω1eTtB⊗Γ2B⊗Γ2Tet+c2ω1eTt−τ2tet−τ2t=c2ω1eTtBBT⊗Γ2Γ2Tet+c2ω1eTt−τ2tIN⊗Inet−τ2t.
The last term will be handled as follows:(22)−2∑i=1NdieiTteitki=−2∑i=1NdieiTteit+δit=−2∑i=1NdieiTteit−2∑i=1NdieiTtδit≤−2eTtD⊗Inet+∑i=1Ndiω2eiTteit+1ω2δiTtδit=−2−ω2eTtD⊗Inet+1ω2∑i=1Ndiδit2≤−2−ω2eTtD⊗Inet+d∗Nω2αε−βt+d∗Nθ0ω2.
Then, one can derive(23)V˙t≤L1eTtIN⊗Inet+L2eTt−τ1tIN⊗Inet−τ1t+2c1etTA⊗Γ1et+c2ω1eTtBBT⊗Γ2Γ2Tet+c2ω1eTt−τ2tIN⊗Inet−τ2t−2−ω2eTtD⊗Inet+d∗Nω2αε−βt+d∗Nθ0ω2=eTtL1IN⊗In+2c1A⊗Γ1+c2ω1BBT⊗Γ2Γ2T−2−ω2D⊗Inet+eTt−τ2tL2IN⊗In+c2ω1IN⊗Inet−τ2t+d∗Nω2αε−βt+d∗Nθ0ω2≤−pVt+qsupt−τ≤s≤tVs+γt,where γt=d∗N/ω2αε−βt+d∗Nθ0/ω2; it is easy to get γ∗=supt0≤t<+∞γt=d∗Nθ0/ω2. By Lemma 2 and condition (2) of Theorem 1, the following result can be obtained:(24)Vt≤γ∗p−q+Ψe−μ∗t−t0,t≥t0,where Ψ=sup−τ≤s≤t0Vs and μ∗ is a root of the equation μ−p+qeτμ=0. It is obvious that μ∗>0; then, we can conclude that the error system converges exponentially to the set Ωθ0=et∣et≤d∗Nθ0/ω2p−q can be guaranteed, and according to Definition 1, the controlled delayed dynamical network (1) can achieve quasi-synchronization. The proof is thus completed.
In order to exclude Zeno behavior, the following theorem shows that the interevent times are lower-bounded by a positive constant in the following theorem.Theorem 2.
Suppose that Assumption2 and all conditions of Theorem 1 hold. With the triggered instants determined by (15), the Zeno behavior can be excluded for all plants in the network, which means that the minimum interevent interval for all plants is lower-bounded by a positive scalar.Proof.
To prove this theorem,e˙it is bounded which will be derived at first, and then the Zeno behavior will be excluded.
By using the triangular inequality, it is easy to get(25)e˙it=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieitki≤ft,xit,xit−τ1t−ft,st,st−τ1t+c1∑j=1Naijγ1ejt+c2∑j=1Nbijγ2ejt−τ2t+dieit+δit≤Lf1+dieit+Lf2eit−τ1t+c1∑j=1Naijγ1ejt+c2∑j=1Nbijγ2ejt−τ2t+diδit,where γ1=Γ1 and γ2=Γ2. According to the results in Theorem 1, we have(26)eit≤et≤Vt≤γ∗p−q+Ψ,t∈−τ,+∞,where p, q, γ∗, and Ψ have been defined in Theorem 1. Then,(27)e˙it≤Lf1+Lf2+di−2c1γ1aii−2c2γ2biiγ∗p−q+Ψ+diδit≤M+diθt,where M=Lf1+Lf2+di−2c1γ1aii−2c2γ2biiγ∗/p−q+Ψ is a bounded constant. For t∈tki,tk+1i, noting that δit=∫tkitδ˙isds=−∫tkite˙isds, one has(28)δit=∫tkite˙isds≤∫tkite˙isds≤Mt−tki+di∫tkitθsds≤M+θ0+αdit−tki.
According to the triggered instants determined by (15), the next event will not be generated before δit∗=θt∗; denote T=t∗−tki, then,(29)θ0+αε−βT+tki≤M+θ0+αdiT.
LetqT=θ0+αε−βT+tki−M+θ0+αdiT, one has q0>0, and when T⟶+∞, qT<0; furthermore, qT is monotonically decreasing. Thus, to obtain the above inequality, there must be T≥T∗>0. It can be concluded that there exists a positive lower bound of the minimum interevent interval. This completes the proof.Remark 1.
Note thatε, β, and α in the threshold function θt cannot affect the region of synchronization error. However, from (29), one can find that those parameters may affect the lower bound of the minimum interevent interval. The parameter M has included aii and bii, which are determined by the topology of networks.Remark 2.
θ
0 is not a parameter of controllers, and it can be adjusted free. According to the result in Theorem 1, the synchronization error will be bounded by a parameter related with θ0. In addition, the complete synchronization can be guaranteed when θ0=0, which will be checked in the simulation part. Furthermore, when θ0=0, the Zeno behavior still can be excluded.


## 3.2. Event-Triggered Control Method Based on Sampled-Data Feedback

A sampled-data feedback controller is defined as(30)uit=−diximT−smT,where T is the sampling period, m=0,1,…. Under the event-triggered strategy, the data will be updated when the designed event was triggered, instead of in every sampled time. The controller for ith can be described as(31)uit=−dixitki−stki,t∈tki,tk+1i,where tki=hkiT means that kth trigger for ith node at hkith sample, hki=0,1,… and h0i=0. For ith node, the event-triggering time instants are defined as follows:(32)hk+1i=minj∈ℤj>hkiδijT≥θj+1T,where δishki+jT=eitki−eihki+jT, j=1,2,…. Let us define τ^it=t−hki+jT for ith node during t∈hki+jT,hki+j+1T. Under controller (14), the synchronization error system can be rewritten as(33)e˙it=f˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieit−τ^it+δist−τ^it.The following vector symbols will be used later:(34)et−τ^t=e1t−τ^1t,e2t−τ^2t,…,eNt−τ^NtT,et−τ2t=e1t−τ2t,e2t−τ2t,…,eNt−τ2t,F˜t,et,et−τ1t=f˜t,e1t,e1t−τ1t,f˜t,eit,e2t−τ1t,…,f˜t,eNt,eNt−τ1tT,e˙t=e˙1t,e˙2t,…,e˙NtT.Theorem 3.
The controlled delayed dynamical network (1) can achieve quasi-synchronization under controller (31), and the trajectory of the error system exponentially converges to the bounded region Ωθ0=etet≤Nθ0/2ξ if there exist matrices P>0, Q>0, R>0, and S>0 such that(35)Ξ0=Ξ11Ξ12Ξ13Ξ14Ξ15∗000Ξ25∗∗00Ξ35∗∗∗0Ξ45∗∗∗∗Ξ550<0,(36)ΞT=Ξ11Ξ12Ξ13Ξ14Ξ150∗000Ξ250∗∗00Ξ350∗∗∗0Ξ450∗∗∗∗Ξ55T0∗∗∗∗∗Ξ66T<0,where ξ>0 is a given decay rate,(37)Ξ11=2ξIN⊗P+2c1A⊗RΓ1+DDT⊗RRT,Ξ12=c2B⊗RΓ2,Ξ13=−D⊗R,Ξ14=IN⊗R,Ξ15=−IN⊗R+c1A⊗SΓ1T+IN⊗P2,Ξ25=c2B⊗SΓ2T,Ξ35=−D⊗S,Ξ45=IN⊗S,Ξ550=TIN⊗Q−2IN⊗S+DDT⊗SST,Ξ55T=−2IN⊗S+DDT⊗SST,Ξ66T=Te−2ξTIN⊗Q.Proof.
Considering the following Lyapunov function(38)Vt=∑i=1NeiTPeit+∑i=1NT−τ^it∫t−τ^itte2ξs−te˙iTsQe˙isds,calculating the derivatives of Vt along the solutions of system (33), one has(39)V˙t=∑i=1NeiTPe˙it+∑i=1NT−τ^ite2ξt′∫t−τ^itte2ξse˙iTsQe˙isds+∑i=1NT−τ^it∫t−τ^itte2ξs−te˙iTsQe˙isds′.
Noting thatτ^i⋅t=1 and deit−τ^it/dt=e˙it−τ^itt−τ^it′=0, then we have(40)V˙t=∑i=1NeiTPe˙it−2ξ∑i=1NT−τ^it∫t−τ^itte2ξs−te˙iTsQe˙isds−∑i=1N∫t−τ^itte2ξs−te˙iTsQe˙isds+∑i=1NT−τ^ite˙iTsQe˙is≤∑i=1NeiTPe˙it−2ξ∑i=1NT−τ^it∫t−τ^itte2ξs−te˙iTsQe˙isds−e−2ξT∑i=1N∫t−τ^itte˙iTsQe˙isds+∑i=1NT−τ^ite˙iTsQe˙is.
Thus, by using Kronecker product, we have(41)V˙t+2ξVt≤eTtIN⊗Pe˙t+e˙TtTIN−τ^t⊗Qe˙t−e−2ξT∑i=1N∫t−τ^itte˙iTsQe˙isds+2ξeTtIN⊗Pet.
Letωit=1/τ^it∫t−τ^itte˙isds; according to Jensen’s inequality, one has(42)∫t−τ^itte˙iTsQe˙isds≥τ^itωiTtQωit.
Thus,(43)−e−2ξT∑i=1N∫t−τ^itte˙iTsQe˙isds≤−e−2ξT∑i=1Nτ^itωiTtQωit=−e−2ξTωTtτ^t⊗Qωt,where ωTt=ωiTt,ω2Tt,…,ωNTt. From (33), the following equality can be derived:(44)0=∑i=1N2eiTtR+e˙iTtSf˜t,eit,eit−τ1t+c1∑j=1NaijΓ1ejt+c2∑j=1NbijΓ2ejt−τ2t−dieit−τ^it+δist−τ^it−e˙it.
By using Kronecker product again, we have(45)0=2eTtIN⊗RF˜+2c1eTtA⊗RΓ1et+2c2eTtB⊗RΓ2et−τ2t−2eTtD⊗Rδst−τ^t−2eTtD⊗Ret−τ^t−2eTtIN⊗Re˙t+2e˙TtIN⊗SF˜+2c1e˙TtA⊗SΓ1et+2c2e˙TtB⊗SΓ2et−τ2t−2e˙TtD⊗Sδst−τ^t−2e˙TtD⊗Set−τ^t−2e˙TtIN⊗Se˙t.
Among the aforementioned equation, note that(46)−2eTtD⊗Rδst−τ^t≤eTtD⊗RD⊗RTet+δst−τ^t2,(47)−2e˙TtD⊗Sδst−τ^t≤e˙TtD⊗RD⊗STe˙t+δst−τ^t2.
We can find from (41), (43), and (45)–(47) that(48)V˙t+2ξVt≤eTtIN⊗Pe˙t+e˙TtTIN−τ^t⊗Qe˙t+2ξeTIN⊗Ptet−e−2ξTωTtτ^t⊗Qωt+2eTtIN⊗RF˜+2c1eTtA⊗RΓ1et+2c2eTtB⊗RΓ2et−τ2t−2eTtD⊗Ret−τ^t−2eTtIN⊗Re˙t−2e˙TtD⊗Set−τ^t−2e˙TtIN⊗Se˙t+2e˙TtIN⊗SF˜+2c1e˙TtA⊗SΓ1et+2c2e˙TtB⊗SΓ2et−τ2t+eTtD⊗RD⊗RTet+e˙TtD⊗RD⊗STe˙t+2δst−τ^t2.
Then, one has(49)V˙t+2ξVt≤ηTtΞηt+2δst−τ^t2,where ηt=eTt,eTt−τ2t,eTt−τ^t,F˜,e˙Tt,ωTt.(50)Ξ=Ξ11Ξ12Ξ13Ξ14Ξ150∗000Ξ250∗∗00Ξ350∗∗∗0Ξ450∗∗∗∗Ξ550∗∗∗∗∗Ξ66,where Ξ55=TIN−τ^t⊗Q−2IN⊗S+DDT⊗SST, Ξ66=e−2ξTτ^t⊗Q, and Ξij has been given in Theorem 3. Let η0t=eTt,eTt−τ2t,eTt−τ^t,F˜,e˙Tt, and based on the definition of Ξ0 and ΞT, we have(51)TIN−τ^tTη0TtΞ0η0t+τ^tTηTtΞTηt=ηTtΞηt.
Thus, (35) and (36) imply that Ξ<0; based on the definition of event-triggered instants, we have(52)V˙t≤−2ξVt+Nαε−βt+Nθ0.
According to Lemma2, we have(53)Vt≤Nθ02ξ+Ψe−2ξt−t0,where Ψ=sup−τ≤s≤t0Vs. This completes our proof.Remark 3.
Note that the event-triggered instantstki will be updated in some sampled instants; thus, it is easy to see the Zeno behavior will be excluded. Meanwhile, there were some published results which had investigated the synchronization problem about complex dynamical networks under the sampled control [23–25]. Compared with those results, the event-triggered strategy can reduce transmission frequency.Remark 4.
Some conditions formed as LMI have been obtained to guarantee the quasi-synchronization of CDNs under event-triggered with sampled data in Theorem3. In which, the conditions can be checked by LMI toolbox of Matlab software directly and easily. However, the sampled period T needs to be given in the conditions; otherwise, the conditions cannot be regarded as LMI. It is obviously that the sampled-data period T cannot be chosen arbitrarily. Thus, it would be a significant problem that how to determine the maximum and minimum of sampled period T. This will be our future work.Remark 5.
This paper has investigated a CDN model with time-varying delays. However, it is not evident that how the time delays effect the synchronization behavior. Indeed, in the case of continuous data, we have used the generalized Halanay’s inequality; this powerful tool just requires time delays to be bounded and the time delays can effect the rate of convergenceμ∗. In the case of sampled data, to deal with the time delay, we have applied the well-known free-weighting matrix method, and one can see (41) and (43) in the proof of Theorem 3, then, the time delay has not appeared on the conditions. However, the conditions formed as LMIs in Theorem 3 are conservative. It is an interesting problem to analyse how can the time delays affect the synchronization in the sampled-data case and how to reduce the conservatism of the conditions in Theorem 3. They would be our future works.


## 4. Numerical Simulations

In this section, some examples will be given to check our theoretical result. An isolated dynamic behavior is described by a chaotic-delayed neural network at first. Then, quasi-synchronization for a coupled network under event-triggered controllers will be displayed. Finally, complete synchronization of the network under event-triggered controllers will be shown. Both continuous and sampled-data methods have been considered in the following examples. All the simulation results are based on the well-known fourth order Runge–Kutta method. The step in the simulation is set as 0.01. As for the sampled data, we used the input-delay method in the simulation, which means thatτt=t−tk, then tk=t−τt; consequently, the sampled-data system becomes to a time-delayed system, which also could be simulated by the Runge–Kutta method.


### 4.1. The Isolated Dynamic Behaviors

Suppose that the isolated dynamic behaviors can be described by the following delayed neural network:(54)s˙t=ft,st,st−τ1t=Cst+B1g1st+B2g1st−τ1t,where st=s1t,s2tT∈R2, g1st=g2st=tanhs1t,tanhs2tT, τ1=1, and(55)C=−100−1,B1=2−0.1−54.5,B2=−1.5−0.1−0.2−4.The delayed neural network’s state has a chaotic attractor as shown in Figure1 with initial condition ϕst=0.1,0.1T for t∈−1,0.Figure 1
Chaotic attractor ofst.Let us proveft,xt,xt−τ1t conforms with Assumption 1. In fact, for any xt∈ℝn, yt∈ℝn,(56)xt−ytTft,xt,xt−τ1t−ft,yt,yt−τ1t=xt−ytTCxt−yt+B1g1xt−g1yt+B2g2xt−τ1−g2yt−τ1=xt−ytTC+CT2xt−yt+xt−ytTB1g1xt−g1yt+xt−ytTB2g2xt−τ1−g2yt−τ1≤λmaxC+CT2+B1+∈B22I2xt−ytTxt−yt+B12g1xt−g1ytTg1xt−g1yt+B22∈g2xt−τ1−g2yt−τ1Tg2xt−g2yt≤λmaxC+CT2+B1+∈B22I2xt−ytTxt−yt+B22∈xt−τ1−yt−τ1Txt−τ1−yt−τ1.Let∈=5, L1=λmaxC+CT/2+B1+∈B2/2I2=15.9335, and L2=B2/2∈=0.4009; then, Assumption 1 can be guaranteed. Time delay can affect the dynamics of the node, and Figure 2 shows the different behaviors with different time delays.Figure 2
Different behaviors ofst with different time delays.
(a)
(b)
(c)
(d)


### 4.2. Quasi-Synchronization for a Coupled Network under Event-Triggered Controllers

Consider the information interactive network with a communication graph given in Figure3, and the corresponding adjacent matrix G is given by(57)G=0100000001101000000001000000000000110000000100001100010010000000010110000000100000001010001000100000,where the weight configuration matrices A and B are A=15G and B=0.01G, respectively. We also defined the inner coupling matrix Γ1=0.02Γ and Γ2=0.05Γ, where(58)Γ=0.2000.2.Figure 3
Interaction graph of 10 nodes.A constant coupled-delayτ2=0.1 has been considered in this example, and c1=1.8, c2=1. Without the control, the network cannot be synchronized to st by itself, and the state of its nodes and st can be seen in Figure 3.Now, let us consider the event-triggered control. Letθ0=0.5, α=0.1, β=0.5, and ε=e for the threshold function θt. Also, given D=20I20. The following two examples are shown quasi-synchronized for a coupled network under event-triggered controllers with continuous feedback and sampled-data feedback, respectively (Figure 4).Figure 4
Time evolution of nodes’ statesxi1t, xi2t, and st without the control, i=1,2,…,10.From above parameters, one can calculateλmax2c1λmaxA⊗Γ1+c2ω1λmaxBBT⊗Γ2Γ2T+L1+p−2−ω2D=−0.8687<0 with ω1=ω2=1, p=2. Let q=1.4, then, all conditions in Theorem 1 can be satisfied. Noting that, under these parameters, the quasi-synchronization can be derived, the errors are bounded. The synchronized states and synchronization errors are shown in Figures 5 and 6, respectively. Figure 7 shows the events of each node in time interval 0,1. The evolution of measurement errors under the event-triggered methods has been displayed in Figure 8.Figure 5
Time evolution of nodes’ statesxi1t, xi2t, and st with θ0=0.5 under continuous feedback, i=1,2,…,10.Figure 6
Time evolution of synchronization errorsei1t, ei2t with θ0=0.5 under continuous feedback, i=1,2,…,10.Figure 7
The events of each node withθ0=0.5 under continuous feedback, i=1,2,…,10.Figure 8
Evolution of measurement error and threshold withθ0=0.5 under continuous feedback, i=1,2,…,10.Let sampled data periodT=0.09; based on LMI toolbox of Matlab, there exists P, Q, R, and S can satisfy conditions in Theorem 2. Then, quasi-synchronization can be derived, and the errors are bounded. The synchronized states and synchronization errors are shown in Figures 9 and 10, respectively. Figure 11 shows the events of each node in time interval 5,10. The evolution of measurement errors under the event-triggered methods has been displayed in Figure 12.Figure 9
Time evolution of nodes’ statesxi1t, xi2t, and st with θ0=0.5 under sampled-data feedback, i=1,2,…,10.Figure 10
Time evolution of synchronization errorsei1t, ei2t with θ0=0.5 under sampled-data feedback, i=1,2,…,10.Figure 11
The events of each node withθ0=0.5 under sampled-data feedback, i=1,2,…,10.Figure 12
Evolution of measurement error and threshold withθ0=0.5 under sampled-data feedback, i=1,2,…,10.In the following, we will compare the event-triggered method that has been used in this paper with some existed results. We will consider three cases: the sampled-data control method, event-triggered method with constant event threshold and based on sampled data, and event-triggered method with exponentially decreasing event threshold function and based on sampled data. The number of triggers with corresponding parameters has been listed in Table1, in which we set the maximum of time as 10, and other parameters which are not mentioned in the table are the same as above. The synchronization errors are shown in Figure 13, respectively.Table 1
T
θ
0
α
β
The number of trigger (sum of all nodes)
0.05
0
0
0
2000
0.05
0
1
0
1134
0.05
0
0.01
0
1277
0.05
0
0.5
0.5
1076Figure 13
Time evolution of synchronization errorsei1t, ei2t with different parameters in the above table, i=1,2,…,10.
(a)
(b)
(c)
(d)Remark 6.
According to above table, one has the number of trigger under the event-triggered method that has been used in this paper which is less than some other methods. It is obvious that less trigger times can provide a less update of information, which could reduce channel blockage. In addition, from Figure13, one can conclude that there is a better synchronization performance which can be obtained under our method.


### 4.3. Complete Synchronization of a Small-World Network under Event-Triggered Controllers

A small-world network withN=200 nodes will be considered in this simulation. The small-world network is generated by taking initial neighboring nodes k=8 and the edge adding probability p=0.1. Suppose that the adjacent matrix of the small-world network is G, which is defined as follows: if there is a connection between nodes i and j, then gij=gji=1; otherwise gij=gji=0. In this example, the weight configuration matrices A and B are A=15G and B=0.01G, respectively. We also defined the inner coupling matrix Γ1=0.02Γ and Γ2=0.05Γ, where(59)Γ=0.01000.02.A constant coupled-delayτ2=0.1 has been considered in this example, and c1=1.8, c2=1. Without the control, the network cannot be synchronized to st by itself, and the state of its nodes and st can be seen in Figure 14.Figure 14
Time evolution of nodes’ statesxi1t, xi2t, and st without the control, i=1,2,…,200.Now, let us consider the event-triggered control. Letθ0=0, α=0.1, β=0.5, and ε=e for the threshold function θt. Also, given D=25I200. Then, one can calculate λmax2c1λmaxA⊗Γ1+c2ω1λmaxBBT⊗Γ2Γ2T+L1+p−2−ω2D=−0.4<0 with ω1=ω2=0, p=2. Let q=1.4, then, all conditions in Theorem 1 can be satisfied. Noting that, under these parameters, the completed synchronization can be derived due to θ0=0. The synchronized states and synchronization errors under continuous feedback and sampled-data feedback with T=0.09 are shown in Figures 15–18, respectively.Figure 15
Time evolution of nodes’ statesxi1t, xi2t, and st under continuous feedback, i=1,2,…,200.Figure 16
Time evolution of synchronization errorsei1t, ei2t under continuous feedback, i=1,2,…,200.Figure 17
Time evolution of nodes’ statesxi1t, xi2t, and st under sampled-data feedback, i=1,2,…,200.Figure 18
Time evolution of synchronization errorsei1t, ei2t under sampled-data feedback, i=1,2,…,200.


## 4.1. The Isolated Dynamic Behaviors

Suppose that the isolated dynamic behaviors can be described by the following delayed neural network:(54)s˙t=ft,st,st−τ1t=Cst+B1g1st+B2g1st−τ1t,where st=s1t,s2tT∈R2, g1st=g2st=tanhs1t,tanhs2tT, τ1=1, and(55)C=−100−1,B1=2−0.1−54.5,B2=−1.5−0.1−0.2−4.The delayed neural network’s state has a chaotic attractor as shown in Figure1 with initial condition ϕst=0.1,0.1T for t∈−1,0.Figure 1
Chaotic attractor ofst.Let us proveft,xt,xt−τ1t conforms with Assumption 1. In fact, for any xt∈ℝn, yt∈ℝn,(56)xt−ytTft,xt,xt−τ1t−ft,yt,yt−τ1t=xt−ytTCxt−yt+B1g1xt−g1yt+B2g2xt−τ1−g2yt−τ1=xt−ytTC+CT2xt−yt+xt−ytTB1g1xt−g1yt+xt−ytTB2g2xt−τ1−g2yt−τ1≤λmaxC+CT2+B1+∈B22I2xt−ytTxt−yt+B12g1xt−g1ytTg1xt−g1yt+B22∈g2xt−τ1−g2yt−τ1Tg2xt−g2yt≤λmaxC+CT2+B1+∈B22I2xt−ytTxt−yt+B22∈xt−τ1−yt−τ1Txt−τ1−yt−τ1.Let∈=5, L1=λmaxC+CT/2+B1+∈B2/2I2=15.9335, and L2=B2/2∈=0.4009; then, Assumption 1 can be guaranteed. Time delay can affect the dynamics of the node, and Figure 2 shows the different behaviors with different time delays.Figure 2
Different behaviors ofst with different time delays.
(a)
(b)
(c)
(d)


## 4.2. Quasi-Synchronization for a Coupled Network under Event-Triggered Controllers

Consider the information interactive network with a communication graph given in Figure3, and the corresponding adjacent matrix G is given by(57)G=0100000001101000000001000000000000110000000100001100010010000000010110000000100000001010001000100000,where the weight configuration matrices A and B are A=15G and B=0.01G, respectively. We also defined the inner coupling matrix Γ1=0.02Γ and Γ2=0.05Γ, where(58)Γ=0.2000.2.Figure 3
Interaction graph of 10 nodes.A constant coupled-delayτ2=0.1 has been considered in this example, and c1=1.8, c2=1. Without the control, the network cannot be synchronized to st by itself, and the state of its nodes and st can be seen in Figure 3.Now, let us consider the event-triggered control. Letθ0=0.5, α=0.1, β=0.5, and ε=e for the threshold function θt. Also, given D=20I20. The following two examples are shown quasi-synchronized for a coupled network under event-triggered controllers with continuous feedback and sampled-data feedback, respectively (Figure 4).Figure 4
Time evolution of nodes’ statesxi1t, xi2t, and st without the control, i=1,2,…,10.From above parameters, one can calculateλmax2c1λmaxA⊗Γ1+c2ω1λmaxBBT⊗Γ2Γ2T+L1+p−2−ω2D=−0.8687<0 with ω1=ω2=1, p=2. Let q=1.4, then, all conditions in Theorem 1 can be satisfied. Noting that, under these parameters, the quasi-synchronization can be derived, the errors are bounded. The synchronized states and synchronization errors are shown in Figures 5 and 6, respectively. Figure 7 shows the events of each node in time interval 0,1. The evolution of measurement errors under the event-triggered methods has been displayed in Figure 8.Figure 5
Time evolution of nodes’ statesxi1t, xi2t, and st with θ0=0.5 under continuous feedback, i=1,2,…,10.Figure 6
Time evolution of synchronization errorsei1t, ei2t with θ0=0.5 under continuous feedback, i=1,2,…,10.Figure 7
The events of each node withθ0=0.5 under continuous feedback, i=1,2,…,10.Figure 8
Evolution of measurement error and threshold withθ0=0.5 under continuous feedback, i=1,2,…,10.Let sampled data periodT=0.09; based on LMI toolbox of Matlab, there exists P, Q, R, and S can satisfy conditions in Theorem 2. Then, quasi-synchronization can be derived, and the errors are bounded. The synchronized states and synchronization errors are shown in Figures 9 and 10, respectively. Figure 11 shows the events of each node in time interval 5,10. The evolution of measurement errors under the event-triggered methods has been displayed in Figure 12.Figure 9
Time evolution of nodes’ statesxi1t, xi2t, and st with θ0=0.5 under sampled-data feedback, i=1,2,…,10.Figure 10
Time evolution of synchronization errorsei1t, ei2t with θ0=0.5 under sampled-data feedback, i=1,2,…,10.Figure 11
The events of each node withθ0=0.5 under sampled-data feedback, i=1,2,…,10.Figure 12
Evolution of measurement error and threshold withθ0=0.5 under sampled-data feedback, i=1,2,…,10.In the following, we will compare the event-triggered method that has been used in this paper with some existed results. We will consider three cases: the sampled-data control method, event-triggered method with constant event threshold and based on sampled data, and event-triggered method with exponentially decreasing event threshold function and based on sampled data. The number of triggers with corresponding parameters has been listed in Table1, in which we set the maximum of time as 10, and other parameters which are not mentioned in the table are the same as above. The synchronization errors are shown in Figure 13, respectively.Table 1
T
θ
0
α
β
The number of trigger (sum of all nodes)
0.05
0
0
0
2000
0.05
0
1
0
1134
0.05
0
0.01
0
1277
0.05
0
0.5
0.5
1076Figure 13
Time evolution of synchronization errorsei1t, ei2t with different parameters in the above table, i=1,2,…,10.
(a)
(b)
(c)
(d)Remark 6.
According to above table, one has the number of trigger under the event-triggered method that has been used in this paper which is less than some other methods. It is obvious that less trigger times can provide a less update of information, which could reduce channel blockage. In addition, from Figure13, one can conclude that there is a better synchronization performance which can be obtained under our method.


## 4.3. Complete Synchronization of a Small-World Network under Event-Triggered Controllers

A small-world network withN=200 nodes will be considered in this simulation. The small-world network is generated by taking initial neighboring nodes k=8 and the edge adding probability p=0.1. Suppose that the adjacent matrix of the small-world network is G, which is defined as follows: if there is a connection between nodes i and j, then gij=gji=1; otherwise gij=gji=0. In this example, the weight configuration matrices A and B are A=15G and B=0.01G, respectively. We also defined the inner coupling matrix Γ1=0.02Γ and Γ2=0.05Γ, where(59)Γ=0.01000.02.A constant coupled-delayτ2=0.1 has been considered in this example, and c1=1.8, c2=1. Without the control, the network cannot be synchronized to st by itself, and the state of its nodes and st can be seen in Figure 14.Figure 14
Time evolution of nodes’ statesxi1t, xi2t, and st without the control, i=1,2,…,200.Now, let us consider the event-triggered control. Letθ0=0, α=0.1, β=0.5, and ε=e for the threshold function θt. Also, given D=25I200. Then, one can calculate λmax2c1λmaxA⊗Γ1+c2ω1λmaxBBT⊗Γ2Γ2T+L1+p−2−ω2D=−0.4<0 with ω1=ω2=0, p=2. Let q=1.4, then, all conditions in Theorem 1 can be satisfied. Noting that, under these parameters, the completed synchronization can be derived due to θ0=0. The synchronized states and synchronization errors under continuous feedback and sampled-data feedback with T=0.09 are shown in Figures 15–18, respectively.Figure 15
Time evolution of nodes’ statesxi1t, xi2t, and st under continuous feedback, i=1,2,…,200.Figure 16
Time evolution of synchronization errorsei1t, ei2t under continuous feedback, i=1,2,…,200.Figure 17
Time evolution of nodes’ statesxi1t, xi2t, and st under sampled-data feedback, i=1,2,…,200.Figure 18
Time evolution of synchronization errorsei1t, ei2t under sampled-data feedback, i=1,2,…,200.


## 5. Conclusion

In this paper, the event-triggered synchronization problem of the complex dynamical network has been studied. Both continuous feedback and sampled-data feedback control methods have been studied. In the case of continuous feedback control approach, the Zeno behavior has been excluded. The main theoretical results are derived based on Lyapunov stability and generalized Halanay’s inequality. In addition, both internal delay and coupling delay have been considered in the complex dynamical model. The given numerical examples illustrate the corresponding theoretical analysis. Extending the proposed approach to the consensus of multiagent systems will be our future works. In addition, in the sampled-data case, this paper just has considered the periodic situation. However, it would be more challenging and interesting to study the event-triggered method based on aperiodic sampled data. Some external disturbances also could be taken into account, such as impulsive and stochastic noise. All of them will be our future works.



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