# Observer-Based Synchronization and Quasi-Synchronization for Multiple Neural Networks with Time-Varying Delays

Authors: Biwen Li; Donglun Wang; Jingjing Huang
Journal: Complexity
 (2022)

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## Abstract

In this paper, we study the synchronization of a class of multiple neural networks (MNNs) with delay and directed disconnected switching topology based on state observer via impulsive coupling control. The coupling topology is connected sequentially, and the controller adjusts the state value through event-triggering strategies. Different from the related works on MNNs, its state in this paper is assumed to be unmeasurable, and the time delay is also unmeasurable. Therefore, the observer does not contain the time-delay term. The impulsive switching controller and observer controller adjust the system through the observed value. By constructing the corresponding augmented matrix, the system can finally achieve quasi-synchronization (synchronization). Through derivation, we give the sufficient conditions ensuring quasi-synchronization (synchronization) via the event-triggered impulse control mechanism. In addition, numerical simulation examples are given to test our results of the theorem.


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## Body

## 1. Introduction

There has been rapid development of multi-agent systems (MASs). In practical application, the application of MAS mainly includes power engineering [1], bioengineering [2], robot formation control [3], vehicle formation control [4], and some other fields. Theoretically, the dynamic behavior of MAS, such as stability [5], robustness [6], synchronization [7], and so on, has become the basis of various theories and greatly promoted the development of MAS. So far, a number of achievements have been made in the study of MAS.In addition, as a complex network, the topology of MAS plays an important role in the dynamic behavior. For example, in [8], second-order leaderless and leader-following consensus algorithms with communication and input delays in directed network topology are studied. In addition, this paper involves three different situations: leaderless consumption, consumption regulation, and consumption tracking. On the other hand, the network topology plays an important role in the asymptotical stability scheme. In the leader-following problem of multi-agent network, it is assumed that the network topology switches arbitrarily between limited topology sets and there is a time-varying delay in the coupling of agents [9]. Different from the general topology, the switching topology in this paper is of great significance to the sudden change or failure of the environment, so switching topology widely exists in MAS.In recent years, MNNs have been widely used, especially in automatic control [10], signal processing [11], optimization [12], and so on. Such complex systems are extremely dependent on the synchronization and stability of MNNs. Therefore, synchronization problem is receiving more and more attention and has always been a very important research direction [13–16]. Especially, Chen et al. [16] considered synchronization for nonlinear neural complex networks by a switching topology. On the other hand, different from synchronous, quasi-synchronous is a special form of dynamical behavior, where all of the control systems in networks are almost synchronized with a given synchronization error, which could not tend to zero with time. Chen et al. [17] discussed the quasi-synchronization problem through a coupled memristor neural network with time-varying delay. The quasi-synchronization problem in fractional-order multi-layer networks with fractional mismatch is studied in [18].With the development of industrial demand and information technology, some traditional control strategies have been replaced by other control schemes. The feasibility and advantages of event-triggered control (ETC) have been proposed for the first time since 1999. Different from traditional control scheme, ETC can ensure system performance while effectively reducing the execution of control tasks. In recent years, ETC has become a popular research subject [19–22]. Because we only need to adjust the state of the controller at the event-trigger instants via setting an appropriate event-trigger mechanism. Different from continuous control and ETC, controller status is updated only at the moment of event trigger, it is adjusted to meet the needs of the system. At present, existing work of ETC in the multiagent field (see [23, 24]). Especially, [25] the recurrent neural network triggered by finite-time event-triggered strategy is studied, and the stability of finite-time systems is proved by novel inequality methods such as, LyapunovCKrasovskii functional and Wirtinger single and double integral inequality. Compared with static trigger conditions, dynamic trigger conditions have more advantages. For example, a new fuzzy filter error system model under dynamic event-triggered control strategy is considered. In addition, there are different triggered thresholds for different fuzzy rules, which can save communication resources more effectively in [26].In order to realize the synchronization of MNNs, we often add appropriate controllers to the system. According to Tang et al. [27], the leader following consistency problem for a class of nonlinear multiagent systems with mixed impulses and time-varying bounded delays is studied. The time-varying impulses in this paper is not only composed of synchronization impulses and desynchronization impulses but also placed in some nodes of the system. Based on Riemann Liouville derivative, Lyapunov functional method and comparison theorem, we can get the global synchronization problem of time-varying delay neural networks with impulsive fractional complex memristor [10]. The impulse controller is one of the most widely used controllers in recent years. Different from the traditional continuous control strategy, impulse control mechanism has the advantage of short action time, which makes it possible to use the impulse controller to occupy less communication resources for a system with a very large amount of information transmission (see [28, 29]). In order to reduce communication bandwidth and save communication costs, a new control strategy based on event-trigger impulse is given. For example, Yi et al. [30] proposed an impulsive control mechanism based on ETC. Except for above control strategy, the impulse coupling protocol is also studied. The coupling between neural networks only occurs at some discrete-time instants, that is, impulse instants. Consequently, the impulse coupling scheme is naturally proposed.Observer-based output feedback control is one of the traditional hot topics. It can be divided into two categories according to whether the variables are measurable or not. For the former, a relaxed stability condition based on state observer is proposed [31], and for the latter, a scheme based on fuzzy controller for a class of nonlinear systems is presented [32]. More recently, it is usually presumed that MAS state is measurable. Due to the limitations of measurement methods, many states cannot be measured. On the other hand, system states are unavailable for the state feedback control or too expensive to measure. Thus, it is imperative to research the observer for the system state is not measurable.In general, we design an observer to estimate the value of different MNNs and then use the information to establish an observer based on feedback controller. However, the measured value is usually collected in discrete time. Therefore, an impulse observer is promoted. It was first proposed by Raf and Allgower in [33]. The observer is updated in the form of impulsive; hence, the measured output is discrete. So, use the impulse observer to estimate the error. Apart from this, designing a suitable control scheme based on impulsive observer is still a challenging problem.Motivated by the previous research, this paper studies synchronization problem of MNNs with observer via an ETC mechanism. The main contributions of this paper are as follows:(1)
An impulsive switching controller is designed via the event-triggered strategy of MNNs with disconnection switching topology. Considering the practical needs, the actual state may be unpredictable in reality. Thus, the system state in this paper is assumed to be unmeasurable.(2)
A particular observer is constructed. Considering the unknown time delay in practical application, the observer does not exhibit time delay. Through the observation value of synchronization error and the tracking error of synchronization error, an augmented system is formed. The synchronization (quasi-synchronization) of the augmented system is also of the MNN system.(3)
The MNNs with a switching topology is studied and the topology is disconnect. At the same time, impulse control, event-triggered strategies, and observers are used to study synchronization (quasisynchronization) issues. In the real system, the sufficient conditions of the synchronization (quasi-synchronization) are proved. We discuss this kind of question and give the relevant theorems. The Zeno behavior can be ruled out.The remainder of this article is organized as follows. Section2 describes problem formulation and some necessary preliminaries. In Section 3, a number of results are presented. In Section 4, a numerical simulation is presented to test the obtained theoretical analysis. Some conclusions are drawn in Section 5.


## 2. Preparation and Modeling

Notations. Throughout this study, sign( ) is the standard sign function. ℤ and ℤ+ represent a set of integer and positive integer. ℝn represents n-dimensional Euclid space. ‖⋅‖ represents 1-norm. signx=signx1,signx2,…,signxnT, and thus x=signxTx where x∈ℝn, and I is an identity matrix. Let G=ℱ,S denote a graph. ℱ is the set of nodes. S⊆ℱ×ℱ is the set of edges. An arbitrary matrix A=aijN×N∈RN×N is given, corresponding to A, and thus from GA=ℱ,S, the graph is indicated, where i,j∈S if and only if aij>0.Also, a subsetK⊆ℱ and a graph G=ℱ,S are given. Define the neighborhood of ∂K,Gk∈ℱ/K|∃i∈K, such that i,k∈S}. If K is a singleton set, ∂K,G represents the neighborhood of one single point.Whenj,i=i,j, the graph G is undirected. G=ℱ,S has a directed path from node i to j if there is a sequence of edges in the form i,i1, i,i2, ⋯, i,ik and the ip∈N, wherep=1,2,…,k, G is called connected if there exists a directed path between each pair of nodes. The node r^ is called a root of G if has a directed path from r^ to every other node G contains a directed spanning tree if there exists at least one root.For the graphsG1=ℱ,S1 and G2=ℱ,S2, G1∪G2=ℱ,S1∪S2 is the union of G1, G2. A sequence of graphs with common nodesGii=1m is jointly connected if ∪i=1mGi contains a spanning tree. A sequence of graphs with common nodesGii=1m is sequentially connected if there exist m+1node sets ℱ0,ℱ1,…,ℱmsuch that ℱk+1⊆∂Gk+1,Ωk and Ωk=∪l=0kℱl , Ω0=ℱ0is a set of Singleton, Ωm=ℱ.Then, by the following dynamics:(1)dxidt=Axit+Bxit−τt+uit+It,where t⩾t0, i∈ℱ=1,2,…,N, xit=xi1t,xi2t,…,xintT, A=aijn×n and B=bijn×n are weight matrices; τt is transmission delay and satisfies τ⩾τt⩾0; uit is a controller; and It∈ℝn indicates external input.In the actual situation, considering that the system state value cannot be measured, we give the observer of the correspondingith node as follows:(2)dx^itdt=Ax^it+vit+It.x^it=x^i1t,x^i2t,…,x^intT is the estimated value of the corresponding ith node. vit∈ℝn is the controller of the observer. Under (2), we can see that the observer does not contain time-delay term. Considering the fact that the time delay is unknown in practical application, observer (2) does not contain time delay.LetS⊂ℤ+ be a limited set of index and Gs:s∈S be a directed graph set. σt:t0,+∞⟶S represent function of switching in Gs:s∈S, and tp=qh,q∈ℤ+ (where h>0 ) represent instants of switching impulsive time. Let Gσt indicate the graph of directed at t, where t⩾t0. Thus, obviously t=tq is the switch time unchanged for t∈tq,tq+1. We use ϖijσtN×N to express Gσt of adjacent matrix, where ϖijσt=1 when the system sends information from node j to node i and ϖijσt=0 otherwise. In addition, we give ϖiiσt=0, namely, there is no self-loop in Gσt where t⩾t0. In order to convenient calculation, we assume that ϖiσt=∑j=1Nϖijσt and ϖ=supt⩾t0maxi∈ℱϖiσtConsider the following assumptions:(A1)
The set of discrete graphsGσt:tmT⩽t<tm+1T is sequentially connected, if there exists a positive integer T∈ℤ+, where m∈ℤ+.(A2)
The set of discrete graphsGσt:tmT0⩽t<tm+1T0 is jointly connected, if there exists a positive integer T0∈ℤ+, where m∈ℤ+.tqiq=1+∞ denotes a sequence of triggering time. Hence, we will give the event-trigger protocol (ETP). In order to make the system achieve synchronization (quasi-synchronization), we design the impulsive switching controller with ETP and the controller of the corresponding observer. In order to make the system achieve synchronization (quasi-synchronization), we design the impulsive switching controller and the corresponding observer controller of the ith node as follows:(3)uit=γ∑q=1+∞δt−tk∑j=1Nϖijσtx^jtqi−x^itqi,(4)vit=η∑q=1+∞δt−tk∑j=1Nϖijσtx^jtqi−x^itqi,where γ>0,η>0.The state is not measurable.Therefore, (3) and (4) are only related to the observed values. Fori∈V, the measurement error is defined as follows:(5)Λit=∑j=1Nϖijσtx^jtqi−x^itqi−∑j=1Nϖijσtx^jt−x^it.Meanwhile, by Figure1, we can get the block diagram for ETC and the ETP:(6)tq+1i=inft>tqi,Λit>βe−ςt−t0+α,where ς>0 , α2+β2≠0; moreover, α⩾0 and β⩾0. They are both threshold parameters.Figure 1
The block diagram for ETC.From (1)–(4), we have(7)dxidt=Axit+Beijt−τt+I,t≠tk,x^itk+=xitk+γ∑j=1Nϖijσtx^jtqi−x^itqi,t=tk,(8)dx^itdt=Ax^it+I,t≠tk,xitk+=xitk+η∑j=1Nϖijσtx^jtqi−x^itqi,t=tk,where k⩾1, i∈ℱ, tqi⩽tk<tq+1i, xitk+=limt⟶tk+0xit, and xitk−=xitk.LetCτ be a Banach space and Cτ=C−τ,0,ℝn. Let φ:−τ,0⟶ℝn represent all continuity functions, and thus the initial value of (7) and (8) can be given:(9)xit0+θ=φiθ,θ∈−τ,0,x^it0=φ^it0,i=1,…,N,for φi∈Cτ and φ^it0∈ℝn. Let eijt=xit−xjt denote the synchronization error for i,j∈ℱ, where t⩾t0 and e^ijt=x^it−x^jt denotes the observed value of eijt.Then,(10)deijtdt=Aeijt+Beijt−τt,t≠tk,eijtk+=eijtk+γ∑v=1Nϖivσtx^vtqi−x^itqi−∑v=1Nϖjvσtx^vtqj−x^jtqj=eijtk+γ∑v=1Nϖivσte^vjtk−e^ijtk+∑v=1Nϖjvσte^ivtk−e^ijtk+γΛitk−Λjtk,t=tk,(11)de^ijtdt=Ae^ijt,t≠tk,e^ijtk+=e^ijtk+η∑v=1Nϖivσtx^vtqi−x^itqi−∑v=1Nϖjvσtx^vtqj−x^jtqj=e^ijtk+η∑v=1Nϖivσte^vjtk−e^ijtk+∑v=1Nϖjvσte^ivtk−e^ijtk+ηΛitk−Λjtk,t=tk.Letξijt=eijt−e^ijt denote tracking error of synchronization error of ith node and jth node.Then,(12)dξijtdt=Aξijt+Beijt−τt,t≠tk,ξijtk+=ξijtk+γ−η∑v=1Nϖivσte^vjtk−e^ijtk+∑v=1Nϖjvσte^ivtk−e^ijtk+γ−ηΛitk−Λjtk,t=tk.LetWijt=e^ijtξijt; from (10) and (11), there are the following augmentation systems:(13)dWijtdt=A00Ae^ijtξijt+00BBe^ijt−τtξijt−τt,t≠tk,Wijtk+=I00Ie^ijtkξijtk+Iη0Iγ−η0∑v=1Nϖivσte^vjtkξvjtk−e^ijtkξijtk+Iη0Iγ−η0∑v=1Nϖjvσte^ivtkξivtk−e^ijtkξijtk+Iη0Iγ−η0Λitk−ΛjtkΛitk−Λjtk,t=tk.In addition, assume thatC1=A00A, C2=00BB,C3=Iη0Iγ−η0 and ∇it=ΛitΛit where i∈ℱ andt⩾t0.Thus, from (13), it follows that(14)dWijtdt=C1Wijt+C2Wijt−τt,t≠tk,Wijtk+=Wijtk+C3∑v=1NϖivσtWvjtk−Wijtk+∑v=1NϖjvσtWivtk−Wijtk+C3∇itk−∇jtk,t=tk.Definition 1.
(14) is called to achieve final synchronization, if limt⟶∞Wijt=0.Definition 2.
(14) is called to achieve final quasi-synchronization, if limt⟶∞Wijt⩽a, where a>0.Remark 1.
From augmented matrix (14), we can find that there are limt⟶∞eijt=0limt⟶∞‖eijt‖⩽a if limt⟶∞Wijt=0limt⟶∞‖Wijt‖⩽a for any i,j∈ℱ. The synchronization (quasi-synchronization) of augmented system (13) is also of (1). Moreover, limt⟶∞eijt=0 by observer of error systems e^ijt and ξijt of tracking error of error system. We do not directly quote the state of the error system, and thus it has certain significance for the system. Its state is not measurable in practical application. For notational convenience, we denote ϖijσtk=dijtk at any time instant tk for i,j∈ℱ, and Gk denotes Gσtk.


## 3. Main Results

whereW˜t=sup−τ⩽θ⩽0Wt+θ.where 0<ζ⩽min2C3,1−4C3ϖ.Lemma 1.
A sequence of graphsGjj=1T0 is jointly connected if a sequence of graphs Gjj=1T is sequentially connected.Lemma 2.
There existsl>0such that(15)l+C2elτ+d1⩽0,for anyi,j∈ℱandt∈tk,tk+1whered1denotes the largest eigenvalue ofC1, and we obtain that(16)Wijt⩽W˜ijt⩽W˜ijtk+e−lt−tk.Proof.
LetVt=Wijtelt−tk for t∈tk,tk+1. From (14), we can deduce(17)dVtdt=signWijtTdWijtdtelt−tk+lVt=signWijtTC1Wijtelt−tk+signWijtTC2Wijt−τtelt−tk+lVt⩽d1Vt+C2Vt−τtelτ+lVt.
Here letV˜t=sup−τ⩽θ⩽0Vt+θ.
We have(18)dVtdt⩽d1+C2elτ+lV˜t⩽0.
Then, one hasdV˜t/dt⩽0, and so Wijtelt−tk=Vt⩽Vtk+⩽V˜tk+=W˜ijtk+, i.e., Wijt⩽W˜ijtk+e−lt−tk. The proof is completed.
From Assumption (A2), there exists a sequence of graphsGjj=mT+1m+1T which is sequentially connected where m∈ℤ+. Hereafter, Grm represents GmT+r. A sequence of graphs Grmr=1T is sequentially connected for any m∈ℤ+; meanwhile, ΩTm=ℱ and Ω0m is a set of singleton.
Let(19)Ht=maxi,j∈ℱWijt,ℋ˜t=supθ∈−τ,0ℋt+θ,ℋrm=ℋtmT+r,ℋrm+=ℋtmT+r+,Hrmt=maxi,j∈ΩrmWijt,H˜rmt=supθ∈−τ,0Hrmt+θ,Hrm=HtmT+r,Hrm+=Hr+1tmT+r+,where 0⩽r⩽T−1 and m∈ℤ+.Lemma 3.
By Assumption (A2), if0<C3<1/2ϖ, then(20)Wijtk+⩽ζHrmtk+1−ζℋtk+4C3βe−ςmT+rh+α,wherem∈ℤ+,0⩽r⩽T−1,tqi⩽tmT+r<tq+1i,q∈ℤ+, and0<ζ⩽minC3,1−2ϖC3.Proof.
First, we review the state equation of (14) at the impulse instant.(21)Wijtk+=Wijtk+C3∑v=1NϖivσtWvjtk−Wijtk+∑v=1NϖjvσtWivtk−Wijtk+C3∇itk−∇jtk,for tqi⩽tk<tq+1i and q∈ℤ+.
Consideringi,j∈Ωr+1m and tk=mT+r, where Ωr+1m=Ωrm∪ℱr+1m, we give the following three cases.Case 1.
Fori,j∈Ωrm, under (21), we get(22)Wijtk+⩽signWijtk+TWijtk+2C3ϖℋtk−signWijtk+TWijtk+C3∇itk+∇jtk.
From (6), ETP, and ∇i, we deduce(23)∇i⩽2βe−ςt−t0+α,for t∈tqi,tq+1i. Hence, due to (22), we have(24)Wijtk+⩽1−2C3ϖWijtk+2C3ϖℋtk+4C3βe−ςtk−t0+α.
From0<ζ⩽minC3,1−2ϖC3 and Hrmtk⩽ℋtk, we can deduce(25)Wijtk+⩽ζHrmtk+1−ζℋtk+4C3βe−ςtk−t0+α.
Therefore, (20) is established.Case 2.
Leti∈ℱr+1m,j∈Ωrm, and distk=1 if a, s ∈Ωrm.
Under (21), one has(26)Wijtk+=Wijtk+C3Wsjtk−Wijtk+C3∑v=1,v≠sNϖivσtWvjtk−Wijtk+∑v=1NϖjvσtWivtk−Wijtk+C3∇itk−∇jtk.
From (26), we can deduce(27)Wijtk+=1−C3signWijtk+TWijtk+C3Hrmtk+C32ϖ−1ℋtk−signWijtk+TWijtk+C3∇itk+∇jtk⩽ζHrmtk+1−ζℋtk+4C3βe−ςtk−t0+α.
Hence, (20) is established.Case 3.
i,j∈ℱr+1m, distk=1, and djqtk=1 if s,q∈Ωrm.
Because(28)Wsjtk+Wiqtk=Wsqtk+Wijtk,from (21), we have(29)Wijtk+=Wijtk+C3Wsjtk−Wijtk+C3Wiqtk−Wijtk+C3∑v=1,v≠sNϖivσtWvjtk−Wijtk+∑v=1,v≠qNϖjvσtWivtk−Wijtk+C3∇itk−∇jtk=Wijtk+C3Wsqtk−Wijtk+C3∑v=1,v≠sNϖivσtWvjtk−Wijtk+∑v=1,v≠qNϖjvσtWivtk−Wijtk+C3∇itk−∇jtk.
By (29),(30)Wijtk+≤1−C3signWijtk+TWijtk+C3Hrmtk+C32ϖ−2ℋtk−signWijtk+TWijtk+C3∇itk+∇jtk≤C3Hrmtk+1−2C3ϖ+C3Wijtk+C32ϖ−2ℋtk+4C3βe−ςtk−t0+α≤ζHrmtk+1−ζℋtk+4C3βe−ςtk−t0+α.
Obviously, (20) holds. Now the proof is completed.
It is always important to assure that Zeno behavior can not be occurred under ETC (3) and (4) in order to prove the synchronization or quasi-synchronization that can be reached by (14).
For error observation (11), it follows that(31)Gt=maxi,j∈ℱe^ijt,G˜t=supθ∈−τ,0Gt+θ,Grm=GtmT+r,Grm+=GtmT+r+,Qrmt=maxi,j∈Ωrme^ijt,Q˜rmt=supθ∈−τ,0Qrmt+θ,Qrm=QtmT+r,Qrm+=QtmT+r+.Theorem 1.
Assume that all the conditions ofLemma 2, hold, and satisfy following conditions:C1: l2>0 and l2+c1⩽0C2: 0<η<1/2ϖwherec1is the maximum eigenvalue of matrixA, and thus (14) does not exhibit Zeno behavior. Event-triggered time sequencetqiis generated under event-triggered strategy (6), which satisfiesq⟶+∞whentqi⟶+∞.Proof.
Lettqi be a bounded set and Gtqi also be a bounded set. It is assumed that Gtqi<D2.
For anyi∈ℱ, t∈tqi,tq+1i, there are(32)Λit=∑v=1Nϖivσte^vitqi−∑v=1Nϖivσte^vit=∑v=1Nϖivσte^vitqi−ϖivσte^vit⩽∑v=1Nϖivσt∫tqite^˙vitdt.
Letm∈ℤ+ such that tqi,tq+1i⊂tmT,tm+1T. By (11), e^˙vis⩽D1G˜s where D1=A.
Similar to Lemma2, there is l2+c1⩽0, where l2>0 and c1 is maximum eigenvalue of A. Thus, we can deduce G˜s⩽G˜tqi+el2s−tqi, where s∈tqi,t and G˜tqi+=lims⟶tqi+0G˜s.
Then, we have(33)e^˙vis⩽D1G˜tqi+el2s−tqi,where s∈tqi,t. Obviously, there are G˜tqi+=G˜tqi⩽D2 for tqi∈tmT,tm+1T.
Similar to Lemma 3, at impulse instants, (11) has(34)Qrmtk+=ψQrmtk+1−ψGrmtk+2ηβe−ςmT+rh+α,if exists 0<η<21ϖ, where0<ψ⩽minη,1−2ϖη, iftqi=tmT, Q˜0mtmT=G˜0mtmTthus(35)G˜tqi+=G˜0m+⩽G˜0m+2ηβe−ςmTh+α⩽D2+2ηβe−ςmTh+α.
LetD2+2ηβe−ςmTh+α=Π. Combine (32)–(35), and we have(36)Λit⩽D1Πϖ∫tqitel2s−tqids=D1Πϖl2el2t−tqi−1.
From (6), event-triggered strategy is triggered to update the controller when Λit=βe−ςt−t0+α; in other words, tqi⟶tq+1i. Hence, from (36), we obtain that(37)Λitq+1i=βe−ςtq+1i−t0+α⩽D1Πϖl2el2tq+1i−tqi−1,which means(38)βe−ςtq+1i−t0+αl2D1Πϖ+1⩽el2tq+1i−tqi,and then(39)tq+1i−tqi⩾lnβe−ςtq+1i−t0+αl2/D1Πϖ+1l2,such that the event-triggered sequence tqi has time interval. This is contrary to the assumption that the sequence tqi is bounded. This ends the proof.Theorem 2.
ByAssumption (A1), ifTheorem 1,Lemma 2, andLemma 3hold, then the quasi-synchronization of (7) can be obtained based on observer ifl≠0satisfies (15) besides(40)e−lTh1−ζT<1,where0<ζ⩽min2C3,1−4C3ϖ.Proof.
According to Lemma2, we can see that(41)ℋr+1m⩽ℋ˜r+1m⩽ℋ˜rm+e−lh,(42)Hr+1m⩽H˜r+1m⩽H˜rm+e−lh,where m∈ℤ+,0≤r≤T−1. By Theorem 1, we can know that Zeno behavior cannot occur.
Meanwhile, according to Lemma3, it follows that(43)H˜rm+⩽ζH˜rm+r−ζℋ˜rm+4C3βe−ςmT+rh+α,ℋ˜rm+⩽ℋ˜rm+4C3βe−ςmT+rh+α.
Combining (41) and (43), we can obtain that(44)ℋ˜r+1m⩽e−lhℋ˜rm+4C3βe−ςmT+rh+α.
Then, from (41)–(44), through the iterative method, we can obtain that(45)H˜rm+⩽e−lhζH˜r−1m++1−ζℋ˜r−1m+β1e−ςmT+r−1h+α1⩽e−2lhζ2H˜r−2m++1−ζ2ℋ˜r−2m+β2e−ςmT+r−2h+α2⋯⩽e−lvhζvH˜r−vm++1−ζvℋ˜r−vm+βve−ςmT+r−vh+αv,and(46)α1=4C3α1−ζe−lh+1,α2=e−2lhζ2+1α1+4C31−ζe−2lhα,⋯αv=e−vlhζv+1αv+4C31−ζv−1e−vlhα,β1=4C31−ζe−lh+e−ςhβ,β2=e−2lhζ2+e−2ςhβ1+4C31−ζe−2lhβ,⋯βv=e−vlhζv+e−vςhβv−1+4C31−ζv−1e−vlhβ,and thus(47)H˜Tm⩽e−lhH˜T−1m+⩽e−lThζT−1H˜0m++1−ζT−1ℋ˜0m+e−lhβT−1e−ςmTh+e−lhαT−1.
From the introduction of the previous preparation part, we haveΩT−1=ℱ and H˜Tm=ℋ˜Tm=ℋ˜0m+1. Meanwhile, from Ω0 which is a single point, we can obtain H˜0m=0. By using (47), one has(48)H˜0m+⩽ζH˜0m+1−ζℋ˜0m+4C3βe−ςmTh+α⩽1−ζℋ˜0m+4C3βe−ςmTh+α.
By applying (47) and (48), we can prove the following result:(49)ℋ˜0m+1⩽εℋ˜0m+α¯+β¯e−mςTh,where 0<ε=e−lTh1−ζT<1, α¯=4C3αe−lTh+e−lhαT−1+, and β¯=e−lhβT−1+4C3βe−lTh. From (43), it follows that(50)ℋ˜0m+1⩽εℋ˜0m+β¯e−mςTh+α¯⩽ε2ℋ˜0m−1+β¯εe−m−1ςTh+β¯e−mςTh+εα¯+α¯⋯⩽εm−1ℋ˜02+β¯εm−2e−2ςTh+⋯+β¯εe−m−1ςTh+β¯e−mςTh+εm−1+⋯+ε+1α¯⩽εmℋ˜01+e−mςTh−εm1−εeςThβ¯+1−εm+11−εα¯.
Therefore, we can obtain(51)limm⟶∞supℋ˜0m⩽α¯.
(14) can achieve quasi-synchronization. In other words, (7) can also quasi-synchronization-based observer.
In addition,limm⟶∞supℋ˜0m=0 if α¯=0. From Theorem 2, (7) can reach synchronization-based observer if the sufficient conditions are satisfied.Remark 2.
Under all the conditions ofTheorem 2, (14) can be quasi-synchronized. According to the definition of (14), we can find that the system is composed of an observation error system e^ijt and a tracking error system ξijt. Therefore, when (14) can be quasi-synchronized by means of e^ijtandξijt, we can have the quasi-synchronization of (1). In this process, we do not directly use the state value of the original system (1). It is consistent with the situation that the state value of the system is unknown in practical application.Theorem 3.
UnderAssumption (A3), usingLemma 2,Lemma 3, andTheorem 1, system (7) can obtain quasi-synchronization-based observer if(52)e−lN−12T0h1−ζN−12T0<1.Proof.
From Lemma1, when the sequence of graphs Gjj=1T0 satisfies e−lN−12T0h1−ζN−12T0<1, the sequence of graphs Gjj=1T satisfies e−lTh1−ζT<1. Therefore, under Lemma 2, system (7) can have quasi-synchronization-based observer and can have synchronization if α=0.


## 4. Numerical Simulations

In this section, a numerical example is given to verify the validity of theory analyses.Review original system (7) with controller and observer system (8):(53)dxitdt=Axit+Bxit−τt+I,t≠tk,xitk+=xitk+γ∑j=1Nϖijσtx^jtqi−x^itqi,t=tk,dx^itdt=Ax^it+I,t≠tk,x^itk+=x^itk+η∑j=1Nϖijσtx^jtqi−x^itqi,t=tk.Leti∈ℱ=1,2,3,4,5,6,7,8, and the parameters in the system are designed as follows: γ=0.12 and η=0.05, and the weight matrix is designed as (54)A=−412−3,B=1−0.5−0.50.8.Meanwhile, the initial function is given as follows:φ1=0.8,−0.3T, φ2=−0.7,0.4T, φ3=1.2,−0.5T, φ4=1.3,−0.2T, φ5=0.3,−0.3T, φ6=0.78,−0.5T, φ7=0.42,−0.32T, φ8=−0.4,−0.6T, φ^1=1.1,−0.8T, φ^2=0.7,−0.78T, φ^3=0.45,−1T, φ^4=−0.63,0.9T, φ^5=−0.43,−0.56T, φ^6=0.7,−1.3T, φ^7=0.3,−0.4T, φ^8=0.86,−0.3T, and τt=sint, and we set tk=0.5k, k∈+. Figures 2 and 3 show the switching topology.Figure 2
G2m.Figure 3
G2m+1.From Figure2 (G2m) and Figure 3 (G2m+1), we can find that G2m,G2m+1 is sequential connection. In the switching period T=2, it is also a joint connection and T0=2. The change of node set is as follows: Ω0=1, Ω1=1,4,5,8, and Ω2=1,2,3,4,5,6,7,8=V. Satisfy Assumptions (A1) and (A2), and the coupling matrix is as follows:(55)G2m=0001100100000000000000000000000000000000000000000000000000000000,G2m+1=0000000000000000000000000110000010000100000000000000000000000010.Thus, we knowω=3, T=2.From the above data, augmented matrix (14), we have(56)C1=A00A=−41002−30000−41002−3,C2=00BB=000000001−0.51−0.5−0.50.8−0.50.8,C3=Iη0Iγ−η0=0.0500000.05000.0700000.0700.For Lemma2, l+C2elτ+d1=l+1.5er−2≤0 by solution of the integral equation reach l<=0.188. Let l=0.15, and the condition of Lemma 2 can be satisfied.For Lemma3, we can obtain C3=0.12≤1/2ω=1/6, and Lemma 3 is also satisfied. According to the conditions in Theorem 1, we have l2+c1≤0 and l2≤2. Let l2=0.15 and satisfy the condition of Theorem 1. For C2, we have η=0.05<1/2ϖ=1/6. Thus, according to Theorem 1, there is no Zeno behavior.For Theorem2, 0<ζ≤min2C3,1−4C3ϖ obtain ζ=0.3. According to e−lTh1−ζT=e−0.15×2×0.00251−0.32<1, the condition of Theorem 2 is also established where h=0.0025. Then, (14) can reach the quasi-synchronization, namely, (1) achieves quasi-synchronization based on observers.Takeα>0, β=2, and ς=0.8.Under quasi-synchronization, the state diagram of 8 nodes of (1) and (2) when α=0.2 is shown in Figure 4. Under synchronization, the state diagram of 8 nodes of (1) and (2) when α=0 is shown in Figure 5.Figure 4
The quasi-synchronization. State diagram of 8 nodes of observation system (1) and system (2), when α=0.2.Figure 5
The quasi-synchronization. State diagram of 8 nodes of observation system (1) and system (2), when α=0.Figure6 shows the state diagram of 8 nodes of system (14) when α=0.2.Figure 6
The state diagram of 8 nodes of system (14) when α=0.2.Figure7 shows the state diagram of 8 nodes of system (14) when α=0.Figure 7
The state diagram of 8 nodes of system (14) when α=0.In Figure8, the state of sampling value is given when α=0.2. In Figure 9, the state of sampling value is given when α=0. When the system meets the conditions given in this paper, all nodes reach synchronization. Thus, we can find that the theorem given in this article is valid.Figure 8
The state of sampling value is given whenα=0.2.Figure 9
The state of sampling value is given whenα=0.


## 5. Conclusion

The question of quasi-synchronization (synchronization) in MNNs with observers in impulsive coupling controller via event-trigger strategy is discussed. An event-triggering mechanism is designed by using the combination measurement method. The real system state in this paper is assumed to be unmeasurable, and the system time delay is also unmeasurable. The state of (1) is measured by observer. The time delay is unknown, so the observer does not have time delay too. The augmented system is composed of the observer and the tracking error system of the error system. In the real system, the sufficient conditions of the quasi-synchronization and synchronization are proved. Compared with existing works, this paper considers the real state and unmeasurable time delay, and the controller used in this paper is a impulsive controller with event-triggered mechanism, so it plays a significant role in saving communication resources. In addition, we consider trying to spread it to the more general system and more complex systems.



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