# Event-Based Consensus for General Linear Multiagent Systems under Switching Topologies

Authors: Yinshuang Sun; Zhijian Ji; Kaien Liu
Journal: Complexity
 (2020)

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## Abstract

In this paper, event-triggered leader-following consensus of general linear multiagent systems under both fixed topology and switching topologies is studied. First, centralised and decentralised event-triggered control strategies based on neighbors’ state estimation are proposed under fixed topology, in which the controller is only updated at the time of triggering. Obviously, compared with the continuous time control algorithms, the event-triggered control strategies can reduce the communication frequency among agents effectively. Meanwhile, event-triggering conditions are derived for systems to achieve consensus by using the Lyapunov stability theory and model transformation method. Then, the theoretical results obtained under the fixed topology are extended to the switching topologies, and the sufficient conditions for the system to achieve leader-following consensus under the switching topologies are given. However, different from fixed topology, the control input of each agent is updated both at event-triggering and topology switching time. Finally, Zeno behaviors can be excluded by proving that the minimum triggering interval of each agent is strictly positive, and the effectiveness of the event-triggered protocol is verified by simulation experiments.


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## Body

## 1. Introduction

In recent years, multi-agent systems have great potential of application in the fields of biology, engineering, physics, society and so on. Also its distributed cooperative control [1, 2] has attracted more and more researchers' attention. For example, the controllability [3–11] and the consensus problem [12–19] are widely studied in multi-agent systems. Among them, as the basic problem of multi-agent cooperative control, consensus is widely used in formation control [20], cluster control, sensor network and other aspects, which is a research hot issue in the control discipline at present.In practical applications, the information needed for cooperative control among agents is transmitted through the network. It is necessary to design a reasonable controller to ensure the control performance of the system due to the finite energy of the agent and the limited network bandwidth. It is well known that periodic sampling control [21–23] can save resources, but when the system runs in an ideal environment or the system state tends to be consensus gradually, it will cause unnecessary resource waste if the control task is executed periodically. In order to reduce this unnecessary waste of resources, a new simple event-triggered control strategy based on feedback mechanism was proposed in [24]. In short, the event-triggered control strategy means that the control task is executed as required. On the premise of ensuring the closed-loop system has certain performance, only can the task be executed once when a specific event occurs (such as the state error exceeds the preset threshold value). The advantage of event-triggered control strategies is that it can not only guarantee the performance of the system but also save the network and computing resources. At present, event-triggered mechanism has been applied to the research of consensus for multiagent systems effectively. For example, Dimarogonas and Frazoli and Dimarogonas et al. [25, 26] studied the consensus of a first-order multiagent system in undirected topology, and Yan et al. [27] investigated the consensus of a second-order multiagent system based on event-triggered mechanism in directed topology. Event-triggering conditions based on composite measurements were designed in [28] to address the consensus of a multiagent system. Hu et al. and Li et al. [29, 30] discussed the leader-following consensus of the second-order multiagent system. The centralized and decentralized event-triggered strategies were proposed in [31], which made the state of all agents converge to the same value gradually. In [32], a consensus control algorithm based on event-triggered mechanism was proposed for the nonlinear multiagent system. In [33], a new average consensus problem under event-triggered control strategy was set up, which limits the measurement error of each agent to a threshold that changes with time. Meng and Chen [34] studied the average consensus of the event-triggered multi-integrator systems under fixed and switching topologies and designed an event-triggered scheme based on quadratic Lyapunov function, which made each agent’s state converge to it’s initial average eventually. Xiao et al. [35] discussed the average consensus of the network of the integrator system with unidirectional information link. In order to reduce the communication cost, a distributed state consensus sampling data control scheme based on edge-event was proposed. In recent years, more and more attention has been paid to general linear systems. In [36], the consensus of general linear multiagent systems under integral event-triggered strategy has been considered. In [37], the consensus of general linear systems under fixed topology and switching topologies has been investigated. Hu et al. [38] studied the leader-following consensus of general linear multiagent system under fixed topology, and Zhu et al. [39] studied the consensus of general linear multiagent system under mixed event-triggering conditions.With this background, we consider the event-triggered leader-following consensus of the general linear multiagent system under fixed topology and switching topologies. Event-triggered control mechanism is designed for each agent, respectively. Under the mechanism, multiagent systems can achieve leader-following consensus gradually. Moreover, a continuous event-triggering condition is proposed, which uses the state error between the follower and leader to design the triggering conditions of each agent under the fixed topology and switching topologies, respectively. The consensus problem is transformed into the stability problem by the method of model transformation, and the sufficient conditions for the system to achieve leader-following consensus are obtained by using Lyapunov stability theory. In addition, all the proposed event-triggered mechanism can exclude Zeno behavior. Finally, the accuracy of the conclusion is verified by simulation experiment.The structure of this paper is as follows. Section2 introduces some concepts of the graph theory and system model. The leader-following consensus of the systems under the fixed/switching topologies is considered in Sections 3 and 4 In addition, the effectiveness of the results is shown through simulation experiment in Section 5. Section 6 summarizes this paper.The following notations are used in this paper.⊗ denotes the Kronecker product. IN denotes the N-dimensional identity matrix. ℝn and ℝm×n indicate the set of n dimensional real vectors and m×n dimensional real matrices, respectively.


## 2. Preliminaries

### 2.1. Theory of Graph

For a multiagent system composed of a leader and follower agents, its communication topology can be represented by an undirected graphG¯=V¯,E¯, where V¯=0,1,…,N. 0 denotes the leader, and 1,…,N denote the followers. E¯⊆V¯×V¯ denotes the edges set. The connection matrix between the follower agent ii=1,…,N and leader 0 is D¯=diaga10,…,aN0, where ai0 is the connection weight between the leader 0 and follower i. If ai0>0, the follower agent i can receive state information of leader 0; otherwise, ai0=0.The communication network among the followers is denoted byG=V,E,A, where V=1,2,…,N and E⊆V×V are obtained from E¯ by removing all edges among the leader 0 and followers in V¯, and A=aij∈ℝN×N is the weighted adjacency matrix of graph G, where aij>0 for i,j∈E if agent i obtains information from agent j. We assume that i,i∉E, and hence aii=0. For a given graph G with the adjacency matrix A, the Laplacian matrix used in this paper is L=D−A, where D is a diagonal matrix, its diagonal elements are dii=∑j∈Niaij, and define Ni as the neighbor set of agent i in V. Therefore, the elements of L are(1)Lik=∑j∈Niaij,k=i,−aik,k≠i.A path from the vertexi to vertex k is a sequence of adjacent edges in the form i,i+1,i+1,i+2,…,k−1,k. The undirected graph is said to be connected if there exists a path between any two distinct vertices.


### 2.2. System Model

Consider a multiagent system composed of the leader 0 andN followers. The dynamics of leader 0 is(2)x˙0t=Ax0t,where x0t∈ℝn is the state and A∈ℝn×n is constant matrix.Accordingly, each follower has the following linear dynamic equation:(3)x˙it=Axit+Buit,i=1,…,N,where xit∈ℝn and uit∈ℝp are the state and input of the ith follower agent, respectively. A∈ℝn×nandB∈ℝn×p are constant matrices. Denote the initial state of the ith follower as xi0.Definition 1.
If there is a control inputuit, the leader 0 and follower i for any initial state satisfy the following conditions:(4)limt⟶∞xit−x0t=0,i=1,2,…,N.
Then, the leader (2) is said to be successfully tracked by follower (3).Assumption 1.
The communication network topologyG among followers is connected.Assumption 2.
The pairA,B is stabilizable.
Based on Assumption2, there is a symmetric positive definite matrix P that satisfies the following algebraic Riccati and Lyapunov inequality with β>0:(5)ATP+PA−2βPBBTP+βI<0,(6)ATP+PA<0.Lemma 1.
(see [37]). For an undirected and connected graph G, the eigenvalues of L are real and can be labelled as(7)0=λ1L<λ2L≤⋯≤λNL.Lemma 2.
(see [40]). For any x,y∈ℝ and β>0, one has the following property:(8)xy≤β2x2+12βy2.Lemma 3.
(see [41]). (Comparison Principle). Consider a differential equation du/dt=ft,u,ut0=u0, where t>0, ft,u is continuous and satisfies the local Lipschitz condition in t. Let t0,T be the maximum existence interval of the solution u, where T can be infinite. If, for any t∈t0,T satisfies(9)dvdt≤ft,v,vt0≤u0,then vt≤ut,t∈t0,T.


## 2.1. Theory of Graph

For a multiagent system composed of a leader and follower agents, its communication topology can be represented by an undirected graphG¯=V¯,E¯, where V¯=0,1,…,N. 0 denotes the leader, and 1,…,N denote the followers. E¯⊆V¯×V¯ denotes the edges set. The connection matrix between the follower agent ii=1,…,N and leader 0 is D¯=diaga10,…,aN0, where ai0 is the connection weight between the leader 0 and follower i. If ai0>0, the follower agent i can receive state information of leader 0; otherwise, ai0=0.The communication network among the followers is denoted byG=V,E,A, where V=1,2,…,N and E⊆V×V are obtained from E¯ by removing all edges among the leader 0 and followers in V¯, and A=aij∈ℝN×N is the weighted adjacency matrix of graph G, where aij>0 for i,j∈E if agent i obtains information from agent j. We assume that i,i∉E, and hence aii=0. For a given graph G with the adjacency matrix A, the Laplacian matrix used in this paper is L=D−A, where D is a diagonal matrix, its diagonal elements are dii=∑j∈Niaij, and define Ni as the neighbor set of agent i in V. Therefore, the elements of L are(1)Lik=∑j∈Niaij,k=i,−aik,k≠i.A path from the vertexi to vertex k is a sequence of adjacent edges in the form i,i+1,i+1,i+2,…,k−1,k. The undirected graph is said to be connected if there exists a path between any two distinct vertices.


## 2.2. System Model

Consider a multiagent system composed of the leader 0 andN followers. The dynamics of leader 0 is(2)x˙0t=Ax0t,where x0t∈ℝn is the state and A∈ℝn×n is constant matrix.Accordingly, each follower has the following linear dynamic equation:(3)x˙it=Axit+Buit,i=1,…,N,where xit∈ℝn and uit∈ℝp are the state and input of the ith follower agent, respectively. A∈ℝn×nandB∈ℝn×p are constant matrices. Denote the initial state of the ith follower as xi0.Definition 1.
If there is a control inputuit, the leader 0 and follower i for any initial state satisfy the following conditions:(4)limt⟶∞xit−x0t=0,i=1,2,…,N.
Then, the leader (2) is said to be successfully tracked by follower (3).Assumption 1.
The communication network topologyG among followers is connected.Assumption 2.
The pairA,B is stabilizable.
Based on Assumption2, there is a symmetric positive definite matrix P that satisfies the following algebraic Riccati and Lyapunov inequality with β>0:(5)ATP+PA−2βPBBTP+βI<0,(6)ATP+PA<0.Lemma 1.
(see [37]). For an undirected and connected graph G, the eigenvalues of L are real and can be labelled as(7)0=λ1L<λ2L≤⋯≤λNL.Lemma 2.
(see [40]). For any x,y∈ℝ and β>0, one has the following property:(8)xy≤β2x2+12βy2.Lemma 3.
(see [41]). (Comparison Principle). Consider a differential equation du/dt=ft,u,ut0=u0, where t>0, ft,u is continuous and satisfies the local Lipschitz condition in t. Let t0,T be the maximum existence interval of the solution u, where T can be infinite. If, for any t∈t0,T satisfies(9)dvdt≤ft,v,vt0≤u0,then vt≤ut,t∈t0,T.


## 3. Leader-Following Control of Multiagent Systems under Fixed Topology

In this part, we consider the leader-following control of multiagent systems (2) and (3) under the event-triggered strategy. Based on the general event-triggered control law, we put forward two kinds of piecewise continuous control mechanisms, which are centralized event-triggered mechanism and decentralized event-triggered mechanism with state estimation in order to minimize the frequency of controller updating. The analysis shows that under the two control mechanisms, multiagent system (3) can track the system (2) successfully with appropriate event-triggering function. The minimum interval between any two consecutive event-triggering instants under the two control mechanisms is greater than 0, and Zeno behavior can be excluded.


### 3.1. Centralized Event-Triggered Control Strategy

Under the centralized event-triggered strategy, all agentsi in system (3) are triggered synchronously at the time tkk=0,1,…,. At the triggering instants, all agents send their states information to neighbours and update the control law with the received state information. Compared with the control protocol in continuous time, each agent i only updates the control input at the event instants under the event-triggered mechanism. So, ui is a piecewise continuous function, and the updating frequency can be reduced.We consider the following control input for theith follower:(10)uit=−K∑j∈Nitaijtxitk−xjtk−Kai0txitk−x0tk,where t∈tk,tk+1, K∈ℝp×n is the control gain matrix to be designed, and xitk is the sampling state of agent i at the kth triggering instant. Since there does not exist control input for leader 0, we take x0tk=x0t,t∈tk,tk+1. For convenience, we make t0=0.The event-triggering time sequencetk is determined by the following triggering functions:(11)ft=−κai0minλminW−αλmaxPai0maxλmaxW+2λmaxDλmaxW∑i=1Nx^iTx^i+∑i=1NeiTei≥0.That is,tk+1=inft>tkft≥0, where 0<κ<1,0<α<ai0minλminW/λmaxP,W=PBBTP.The state error between the follower and leader is defined asx^it=xit−x0t,x^t=x^1T,…,x^NTT. For agent i, the measurement error is defined as eit=xitk−xit,t∈tk,tk+1. Then, formula (10) is converted to(12)uit=−K∑j=0Naijx^it−x^jt+eit−ejt=−K∑j=1Naijx^it−x^jt+eit−ejt−Kai0x^it+eit.Combining (2), (3), and (12), we get(13)x^˙t=IN⊗Ax^t−IN⊗BKL+D¯⊗Inx^t+et=IN⊗Ax^t−L+D¯⊗BKx^t+et.Remark 1.
Through the model transformation, the leader-following control problem between systems (2) and (3) can be interpreted by the stability problem of system (13).
Next, we will give the following consensus conditions under the centralized event-triggering protocol (10).Theorem 1.
Under Assumptions1 and 2, centralized event-triggered control strategy (10) can make multiagent system (3) track system (2) successfully under event-triggering condition (4), where feedback gain matrix K satisfies K=BTP and W=PBK.Proof.
We consider a candidate Lyapunov function, as follows:(14)V1=eαt∑i=1Nx^iTPx^i.
Along with the trajectories of the state as described in (8), the time derivative of Lyapunov function is(15)V˙1=αeαt∑i=1Nx^iTPx^i+2eαt∑i=1Nx^iTPx^˙i=eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i+Bui=eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i+2eαt∑i=1Nx^iTPBui,where(16)eαt∑i=1Nx^iTPBui=−eαt∑i=1Nx^iTPBKai0x^i+ei+∑j=1Naijx^i−x^j+ei−ej=−eαt∑i=1Nx^iTWai0x^i+ei−eαt∑i=1Nx^iTW∑j=1Naijx^i−x^j−eαt∑i=1Nx^iTW∑j=1Naijei−ej,where W=PBK.
According to the property ofL=LT in undirected graph G, we can deduce(17)eαt∑i=1Nx^iTW∑j=1Naijx^i−x^j=eαt∑i=1N∑j=1Naijx^iTWx^i−x^j=eαt∑i=1N∑j=1Najix^jTWx^j−x^i=−eαt∑i=1N∑j=1Naijx^jTWx^i−x^j=12eαt∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j.
Similarly,(18)eαt∑i=1Nx^iTW∑j=1Naijei−ej=12eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej.
Hence,(19)eαt∑i=1Nx^iTPBui=−eαt∑i=1Nx^iTWai0x^i+ei−12eαt∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j−12eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej=−eαt∑i=1Nx^iTWai0x^i+ei−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j−12eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej.
Combining equality (15) yields(20)V˙1=eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−2eαt∑i=1N∑j=1Naijx^iTWx^i−x^j−eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej−2eαt∑i=1Nx^iTWai0x^i+ei.
In the light of Lemma2, we have(21)−eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej≤12eαt∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j+12eαt∑i=1N∑j=1Naijei−ejTWei−ej.
By substituting the abovementioned formula into equation (20), we obtain(22)V˙1≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−2eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt2∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j+eαt2∑i=1N∑j=1Naijei−ejTWei−ej−2eαt∑i=1Nx^iTWai0x^i+ei≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−2eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt2∑i=1N∑j=1Naijei−ejTWei−ej−2eαt∑i=1Nx^iTWai0x^i+ei≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt2∑i=1N∑j=1Naijei−ejTWei−ej−2eαt∑i=1Nx^iTWai0x^i+ei.
From Lemma2, we have(23)eαt2∑i=1N∑j=1Naijei−ejTWei−ej≤2eαt∑i=1N∑j=1NaijeiTWei.
Together with (22), we can get that(24)V˙1≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+2eαt∑i=1N∑j=1NaijeiTWei−2eαt∑i=1Nx^iTWai0x^i+ei,(25)−eαt∑i=1Nx^iTWai0x^i+ei=−eαt∑i=1Nx^iTWai0x^i−eαt∑i=1Nx^iTWai0ei≤−eαt∑i=1Nx^iTWai0x^i+12eαt∑i=1Nx^iTWai0x^i+12eαt∑i=1NeiTWai0ei≤−12eαt∑i=1Nx^iTWai0x^i+12eαt∑i=1NeiTWai0ei.
Combining (24) and (25), we arrive at(26)V˙1≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j−eαt∑i=1Nx^iTWai0x^i+eαt∑i=1NeiTWai0ei+2eαt∑i=1N∑j=1NaijeiTWei≤eαtx^TIN⊗αPx^+eαtx^TIN⊗PA+ATPx^−eαtx^TL⊗Wx^−eαtx^TD¯⊗Wx^+eαteTD¯⊗We+2eαteTD⊗We.
Under Assumption1, by using Lemma 1, x^TL⊗Wx^≥λ2Lx^TIN⊗Wx^ holds. Consequently,(27)V˙1≤eαtx^TIN⊗PA+ATP−λ2LWx^+eαtαλmaxP−ai0minλminWx^2+eαtai0maxλmaxW+2λmaxDλmaxWe2.
Using inequality (5) and event-triggering condition (11), we claim that the following inequality holds:(28)V˙1≤κ−1eαtai0minλminW−αλmaxPx^2−eαtλ2L2x^Tx^≤−eαtλ2L2x^Tx^.
It can be seen from (28) that V1 is not increasing; therefore,(29)V10≥V1t=eαt∑i=1Nx^itTPx^ii≥eαtλminPx^t2.
That is to say,x^t≤V10/λminPe−α/2t, i.e., limt⟶∞x^t=0 is equivalent to limt⟶∞x^it=0, which means limt⟶∞xit−x0t=0,i=1,2,…,N holds.Theorem 2.
Under the conditions of Theorem1, system (13) does not exhibit Zeno behavior. The interval between any two consecutive event-triggering instants of the system is not less than(30)IN⊗A+L+D¯⊗BK3×1+κai0minλminW−αλmaxPai0maxλmaxW+2λmaxDλmaxW3−1.Proof.
From the mechanism of event-triggering strategy, the event interval betweentk and tk+1 is the time that et/x^t grows from 0 to κai0minλminW−αλmaxP/ai0maxλmaxW+2λmaxDλmaxW. The time derivative of et/x^t has(31)ddtetx^t=ddtetTet1/2x^tTx^t1/2=etTet1/2′x^t−etTet1/2x^tTx^t1/2′x^t2=etTe˙tx^tet−etx^tTx^˙tx^t2x^t=−etTx^˙tx^tet−etx^tTx^˙tx^t2x^t≤x^˙tx^t+x^˙tetx^t2=x^˙tx^t1+etx^t≤IN⊗A+L+D¯⊗BK1+etx^t+L+D¯⊗BKex^t1+etx^t≤IN⊗A+L+D¯⊗BK1+etx^t+IN⊗A+L+D¯⊗BKex^t1+etx^t=IN⊗A+L+D¯⊗BK1+etx^t2.
Denotez=et/x^t, then(32)z˙≤IN⊗A+L+D¯⊗BK1+z2.
Consider that a nonnegative functionψt,ψ0 satisfies ψ˙=IN⊗A+L+D¯⊗BK1+ψ2, and ψ0=0. Then, from Lemma 3, z≤ψt,0. It can be seen from (11) that(33)ψτ,0=κai0minλminW−αλmaxPai0maxλmaxW+2λmaxDλmaxW.
Therefore,(34)τ=IN⊗A+L+D¯⊗BK31+ψτ3−1=IN⊗A+L+D¯⊗BK3×1+κai0minλminW−αλmaxPai0maxλmaxW+2λmaxDλmaxW3−1.
Obviously,τ>0.
It is assumed that the Zeno behavior occurs, which means that there exists a positive constantt∗ such that limk⟶∞tk=t∗. Let ε0=1/2τ. There exists a positive integer N0 such that t∗−ε0≤tk≤t∗ for the abovementioned ε0>0 according to the definition of sequence limit, where k≥N0. Therefore, t∗+ε0≤tk+2ε0≤tk+1 holds when k≥N0. This contradicts with t∗≥tk+1 for k≥N0. Thus, Zeno behavior is strictly excluded.


### 3.2. Decentralized Event-Triggered Control Strategy

The centralized event-triggered mechanism given in the previous section sets a global state error threshold for all agents. Once the system error reaches the threshold, all agents in the system perform control tasks at the same time. In this section, an error threshold based on the state of its neighbor node is set for each agent. When the state error of the agent reaches the set threshold, the agent triggers the event independently and executes the control task.The triggering time of thekth event of the i agent is defined as tkik=0,1,…. In the design of this section, it should be noted that the agent triggers asynchronously, that is, each agent has its own event-triggering sequence. The measurement error of agent i is defined as eit=xitki−xit,t∈tki,tk+1i. It is clear that eitki=0 when t=tki.For a multiagent system composed of (2) and (3), we consider the following decentralized event-triggered control protocol:(35)uit=−K∑j∈Nitaijtxitki−xjtk′i−Kai0txitki−x0t,where t∈tki,tk+1i, tk′j=argminl∈N,t≥tljt−tlj represents the latest event-triggering time before t for agent j. According to (35), agent i will update control input ui at both its triggering instants t0i,t1i,… and neighbor agent j event instants t0j,t1j,…. The event-triggering instant sequence tki for agent i is determined by the following decentralized event-triggering function:(36)fit=−1−κ∑j=1Naijx^i−x^jTWx^i−x^j+λmaxWei2∑j=1N4aij+2ai0≥0,where 0<κ<1,W=PBBTP. According to the definition of measurement error and x^it=xit−x0t, (35) can be rewritten as(37)uit=−K∑j=0Naijxit−xjt+eit−ejt=−K∑j=0Naijxit−x0t−xjt−x0t−K∑j=0Naijeit−ejt=−K∑j=1Naijx^it−x^jt+eit−ejt−Kai0x^it+eit.Combining (2), (3), with (37) yields(38)x^˙t=IN⊗Ax^t−L+D¯⊗BKx^t+et.Theorem 3.
Under Assumption1, the multiagent systems (3) with protocol (35) can track system (2) successfully under the event-triggering condition (36), where K=BTP and W=PBBTP.Proof.
Define the Lyapunov function(39)V2=12∑i=1Nx^iTPx^i.
Following the same proof as that of Theorem1, the time derivation of V2 along the trajectory of system (38) is obtained:(40)V˙2≤∑i=1Nx^iTPAx^i−12∑i=1N∑j=1Naijx^iTWx^i−x^j−12∑i=1Nx^iTWai0x^i+12∑i=1NeiTWai0ei+∑i=1N∑j=1NaijeiTWei≤x^TIN⊗PAx^−14∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j−12∑i=1Nx^iTWai0x^i+∑i=1N∑j=1NaijeiTWei+12eiTWai0ei≤x^TIN⊗PAx^+14∑i=1N∑j=1NeiT4aij+2ai0Wei−14∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j≤x^TIN⊗PA+ATPx^+14∑i=1NλmaxWei2∑j=1N4aij+2ai0−14∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j.
According to (6) and event-triggering condition (36), we can find that(41)V˙2≤12x^TIN⊗PA+ATPx^−κ4∑j=1N∑i=1Naijx^i−x^jTWx^i−x^j≤−κ2x^TL⊗Wx^≤−κ2λmaxLλmaxWx^2≤0.
It can be seen from the abovementioned formula thatV2 is not increasing; therefore,(42)V20≥V2t=12∑i=1Nx^itTPx^it≥12λminPx^t2.That is to say, x^t≤2V20/λminP=0. According to LaSalle’s invariance principle, we can obtain that system (38) can achieve consensus, that is, limt⟶∞x^i=0, which is equivalent to limt⟶∞xit−x0t=0,i=1,2,…,N. The proof is completed.Theorem 4.
Under the conditions of Theorem3, system (38) does not exhibit Zeno behavior. The interval between any two consecutive event-triggering instants of the system is not less than(43)A+BKLi+ai0⊗In3×1+1−κai0λminW2dii+ai0λmaxW1/23−1.Proof.
It is similar to the proof of Theorem2. The event interval between tki and tk+1i is eit/x^it which grows from 0 to 1−κai0λminW/2dii+ai0λmaxW1/2. The time derivative of eit/x^it is(44)ddteix^i≤x^˙itx^it+x^˙iteitx^it2=x^˙itx^it1+eitx^it=A+BKLi+ai0⊗In1+eitx^it+BKLi+ai0⊗Ineix^it1+eitx^it≤A+BKLi+ai0⊗In1+eitx^it+A+BKLi+ai0⊗Ineix^it1+eitx^it=A+BKLi+ai0⊗In1+eitx^it2,where Li is the row i of the Laplace matrix L.
Letzi=eit/x^it, then(45)z˙i≤A+BKLi+ai0⊗In1+zi2.
Consider that a nonnegative functionψt,ψ0 satisfies ψ˙=A+BKLi+ai0⊗In1+ψ2 and ψ0=0, according to Lemma 3, zi≤ψt,0. It can be seen from (36),(46)ψτki,0=1−κai0λminW2dii+ai0λmaxW1/2.
Hence,(47)τki=A+BKLi+ai0⊗In31+ψτ3−1,=A+BKLi+ai0⊗In3×1+1−κai0λminW2dii+ai0λmaxW1/23−1.
Similar to Theorem2, that Zeno behavior that does not occur can be proved by contradiction, which is omitted here.


## 3.1. Centralized Event-Triggered Control Strategy

Under the centralized event-triggered strategy, all agentsi in system (3) are triggered synchronously at the time tkk=0,1,…,. At the triggering instants, all agents send their states information to neighbours and update the control law with the received state information. Compared with the control protocol in continuous time, each agent i only updates the control input at the event instants under the event-triggered mechanism. So, ui is a piecewise continuous function, and the updating frequency can be reduced.We consider the following control input for theith follower:(10)uit=−K∑j∈Nitaijtxitk−xjtk−Kai0txitk−x0tk,where t∈tk,tk+1, K∈ℝp×n is the control gain matrix to be designed, and xitk is the sampling state of agent i at the kth triggering instant. Since there does not exist control input for leader 0, we take x0tk=x0t,t∈tk,tk+1. For convenience, we make t0=0.The event-triggering time sequencetk is determined by the following triggering functions:(11)ft=−κai0minλminW−αλmaxPai0maxλmaxW+2λmaxDλmaxW∑i=1Nx^iTx^i+∑i=1NeiTei≥0.That is,tk+1=inft>tkft≥0, where 0<κ<1,0<α<ai0minλminW/λmaxP,W=PBBTP.The state error between the follower and leader is defined asx^it=xit−x0t,x^t=x^1T,…,x^NTT. For agent i, the measurement error is defined as eit=xitk−xit,t∈tk,tk+1. Then, formula (10) is converted to(12)uit=−K∑j=0Naijx^it−x^jt+eit−ejt=−K∑j=1Naijx^it−x^jt+eit−ejt−Kai0x^it+eit.Combining (2), (3), and (12), we get(13)x^˙t=IN⊗Ax^t−IN⊗BKL+D¯⊗Inx^t+et=IN⊗Ax^t−L+D¯⊗BKx^t+et.Remark 1.
Through the model transformation, the leader-following control problem between systems (2) and (3) can be interpreted by the stability problem of system (13).
Next, we will give the following consensus conditions under the centralized event-triggering protocol (10).Theorem 1.
Under Assumptions1 and 2, centralized event-triggered control strategy (10) can make multiagent system (3) track system (2) successfully under event-triggering condition (4), where feedback gain matrix K satisfies K=BTP and W=PBK.Proof.
We consider a candidate Lyapunov function, as follows:(14)V1=eαt∑i=1Nx^iTPx^i.
Along with the trajectories of the state as described in (8), the time derivative of Lyapunov function is(15)V˙1=αeαt∑i=1Nx^iTPx^i+2eαt∑i=1Nx^iTPx^˙i=eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i+Bui=eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i+2eαt∑i=1Nx^iTPBui,where(16)eαt∑i=1Nx^iTPBui=−eαt∑i=1Nx^iTPBKai0x^i+ei+∑j=1Naijx^i−x^j+ei−ej=−eαt∑i=1Nx^iTWai0x^i+ei−eαt∑i=1Nx^iTW∑j=1Naijx^i−x^j−eαt∑i=1Nx^iTW∑j=1Naijei−ej,where W=PBK.
According to the property ofL=LT in undirected graph G, we can deduce(17)eαt∑i=1Nx^iTW∑j=1Naijx^i−x^j=eαt∑i=1N∑j=1Naijx^iTWx^i−x^j=eαt∑i=1N∑j=1Najix^jTWx^j−x^i=−eαt∑i=1N∑j=1Naijx^jTWx^i−x^j=12eαt∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j.
Similarly,(18)eαt∑i=1Nx^iTW∑j=1Naijei−ej=12eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej.
Hence,(19)eαt∑i=1Nx^iTPBui=−eαt∑i=1Nx^iTWai0x^i+ei−12eαt∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j−12eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej=−eαt∑i=1Nx^iTWai0x^i+ei−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j−12eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej.
Combining equality (15) yields(20)V˙1=eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−2eαt∑i=1N∑j=1Naijx^iTWx^i−x^j−eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej−2eαt∑i=1Nx^iTWai0x^i+ei.
In the light of Lemma2, we have(21)−eαt∑i=1N∑j=1Naijx^i−x^jTWei−ej≤12eαt∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j+12eαt∑i=1N∑j=1Naijei−ejTWei−ej.
By substituting the abovementioned formula into equation (20), we obtain(22)V˙1≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−2eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt2∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j+eαt2∑i=1N∑j=1Naijei−ejTWei−ej−2eαt∑i=1Nx^iTWai0x^i+ei≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−2eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt2∑i=1N∑j=1Naijei−ejTWei−ej−2eαt∑i=1Nx^iTWai0x^i+ei≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt2∑i=1N∑j=1Naijei−ejTWei−ej−2eαt∑i=1Nx^iTWai0x^i+ei.
From Lemma2, we have(23)eαt2∑i=1N∑j=1Naijei−ejTWei−ej≤2eαt∑i=1N∑j=1NaijeiTWei.
Together with (22), we can get that(24)V˙1≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+2eαt∑i=1N∑j=1NaijeiTWei−2eαt∑i=1Nx^iTWai0x^i+ei,(25)−eαt∑i=1Nx^iTWai0x^i+ei=−eαt∑i=1Nx^iTWai0x^i−eαt∑i=1Nx^iTWai0ei≤−eαt∑i=1Nx^iTWai0x^i+12eαt∑i=1Nx^iTWai0x^i+12eαt∑i=1NeiTWai0ei≤−12eαt∑i=1Nx^iTWai0x^i+12eαt∑i=1NeiTWai0ei.
Combining (24) and (25), we arrive at(26)V˙1≤eαt∑i=1Nx^iTαPx^i+2eαt∑i=1Nx^iTPAx^i−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j−eαt∑i=1Nx^iTWai0x^i+eαt∑i=1NeiTWai0ei+2eαt∑i=1N∑j=1NaijeiTWei≤eαtx^TIN⊗αPx^+eαtx^TIN⊗PA+ATPx^−eαtx^TL⊗Wx^−eαtx^TD¯⊗Wx^+eαteTD¯⊗We+2eαteTD⊗We.
Under Assumption1, by using Lemma 1, x^TL⊗Wx^≥λ2Lx^TIN⊗Wx^ holds. Consequently,(27)V˙1≤eαtx^TIN⊗PA+ATP−λ2LWx^+eαtαλmaxP−ai0minλminWx^2+eαtai0maxλmaxW+2λmaxDλmaxWe2.
Using inequality (5) and event-triggering condition (11), we claim that the following inequality holds:(28)V˙1≤κ−1eαtai0minλminW−αλmaxPx^2−eαtλ2L2x^Tx^≤−eαtλ2L2x^Tx^.
It can be seen from (28) that V1 is not increasing; therefore,(29)V10≥V1t=eαt∑i=1Nx^itTPx^ii≥eαtλminPx^t2.
That is to say,x^t≤V10/λminPe−α/2t, i.e., limt⟶∞x^t=0 is equivalent to limt⟶∞x^it=0, which means limt⟶∞xit−x0t=0,i=1,2,…,N holds.Theorem 2.
Under the conditions of Theorem1, system (13) does not exhibit Zeno behavior. The interval between any two consecutive event-triggering instants of the system is not less than(30)IN⊗A+L+D¯⊗BK3×1+κai0minλminW−αλmaxPai0maxλmaxW+2λmaxDλmaxW3−1.Proof.
From the mechanism of event-triggering strategy, the event interval betweentk and tk+1 is the time that et/x^t grows from 0 to κai0minλminW−αλmaxP/ai0maxλmaxW+2λmaxDλmaxW. The time derivative of et/x^t has(31)ddtetx^t=ddtetTet1/2x^tTx^t1/2=etTet1/2′x^t−etTet1/2x^tTx^t1/2′x^t2=etTe˙tx^tet−etx^tTx^˙tx^t2x^t=−etTx^˙tx^tet−etx^tTx^˙tx^t2x^t≤x^˙tx^t+x^˙tetx^t2=x^˙tx^t1+etx^t≤IN⊗A+L+D¯⊗BK1+etx^t+L+D¯⊗BKex^t1+etx^t≤IN⊗A+L+D¯⊗BK1+etx^t+IN⊗A+L+D¯⊗BKex^t1+etx^t=IN⊗A+L+D¯⊗BK1+etx^t2.
Denotez=et/x^t, then(32)z˙≤IN⊗A+L+D¯⊗BK1+z2.
Consider that a nonnegative functionψt,ψ0 satisfies ψ˙=IN⊗A+L+D¯⊗BK1+ψ2, and ψ0=0. Then, from Lemma 3, z≤ψt,0. It can be seen from (11) that(33)ψτ,0=κai0minλminW−αλmaxPai0maxλmaxW+2λmaxDλmaxW.
Therefore,(34)τ=IN⊗A+L+D¯⊗BK31+ψτ3−1=IN⊗A+L+D¯⊗BK3×1+κai0minλminW−αλmaxPai0maxλmaxW+2λmaxDλmaxW3−1.
Obviously,τ>0.
It is assumed that the Zeno behavior occurs, which means that there exists a positive constantt∗ such that limk⟶∞tk=t∗. Let ε0=1/2τ. There exists a positive integer N0 such that t∗−ε0≤tk≤t∗ for the abovementioned ε0>0 according to the definition of sequence limit, where k≥N0. Therefore, t∗+ε0≤tk+2ε0≤tk+1 holds when k≥N0. This contradicts with t∗≥tk+1 for k≥N0. Thus, Zeno behavior is strictly excluded.


## 3.2. Decentralized Event-Triggered Control Strategy

The centralized event-triggered mechanism given in the previous section sets a global state error threshold for all agents. Once the system error reaches the threshold, all agents in the system perform control tasks at the same time. In this section, an error threshold based on the state of its neighbor node is set for each agent. When the state error of the agent reaches the set threshold, the agent triggers the event independently and executes the control task.The triggering time of thekth event of the i agent is defined as tkik=0,1,…. In the design of this section, it should be noted that the agent triggers asynchronously, that is, each agent has its own event-triggering sequence. The measurement error of agent i is defined as eit=xitki−xit,t∈tki,tk+1i. It is clear that eitki=0 when t=tki.For a multiagent system composed of (2) and (3), we consider the following decentralized event-triggered control protocol:(35)uit=−K∑j∈Nitaijtxitki−xjtk′i−Kai0txitki−x0t,where t∈tki,tk+1i, tk′j=argminl∈N,t≥tljt−tlj represents the latest event-triggering time before t for agent j. According to (35), agent i will update control input ui at both its triggering instants t0i,t1i,… and neighbor agent j event instants t0j,t1j,…. The event-triggering instant sequence tki for agent i is determined by the following decentralized event-triggering function:(36)fit=−1−κ∑j=1Naijx^i−x^jTWx^i−x^j+λmaxWei2∑j=1N4aij+2ai0≥0,where 0<κ<1,W=PBBTP. According to the definition of measurement error and x^it=xit−x0t, (35) can be rewritten as(37)uit=−K∑j=0Naijxit−xjt+eit−ejt=−K∑j=0Naijxit−x0t−xjt−x0t−K∑j=0Naijeit−ejt=−K∑j=1Naijx^it−x^jt+eit−ejt−Kai0x^it+eit.Combining (2), (3), with (37) yields(38)x^˙t=IN⊗Ax^t−L+D¯⊗BKx^t+et.Theorem 3.
Under Assumption1, the multiagent systems (3) with protocol (35) can track system (2) successfully under the event-triggering condition (36), where K=BTP and W=PBBTP.Proof.
Define the Lyapunov function(39)V2=12∑i=1Nx^iTPx^i.
Following the same proof as that of Theorem1, the time derivation of V2 along the trajectory of system (38) is obtained:(40)V˙2≤∑i=1Nx^iTPAx^i−12∑i=1N∑j=1Naijx^iTWx^i−x^j−12∑i=1Nx^iTWai0x^i+12∑i=1NeiTWai0ei+∑i=1N∑j=1NaijeiTWei≤x^TIN⊗PAx^−14∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j−12∑i=1Nx^iTWai0x^i+∑i=1N∑j=1NaijeiTWei+12eiTWai0ei≤x^TIN⊗PAx^+14∑i=1N∑j=1NeiT4aij+2ai0Wei−14∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j≤x^TIN⊗PA+ATPx^+14∑i=1NλmaxWei2∑j=1N4aij+2ai0−14∑i=1N∑j=1Naijx^i−x^jTWx^i−x^j.
According to (6) and event-triggering condition (36), we can find that(41)V˙2≤12x^TIN⊗PA+ATPx^−κ4∑j=1N∑i=1Naijx^i−x^jTWx^i−x^j≤−κ2x^TL⊗Wx^≤−κ2λmaxLλmaxWx^2≤0.
It can be seen from the abovementioned formula thatV2 is not increasing; therefore,(42)V20≥V2t=12∑i=1Nx^itTPx^it≥12λminPx^t2.That is to say, x^t≤2V20/λminP=0. According to LaSalle’s invariance principle, we can obtain that system (38) can achieve consensus, that is, limt⟶∞x^i=0, which is equivalent to limt⟶∞xit−x0t=0,i=1,2,…,N. The proof is completed.Theorem 4.
Under the conditions of Theorem3, system (38) does not exhibit Zeno behavior. The interval between any two consecutive event-triggering instants of the system is not less than(43)A+BKLi+ai0⊗In3×1+1−κai0λminW2dii+ai0λmaxW1/23−1.Proof.
It is similar to the proof of Theorem2. The event interval between tki and tk+1i is eit/x^it which grows from 0 to 1−κai0λminW/2dii+ai0λmaxW1/2. The time derivative of eit/x^it is(44)ddteix^i≤x^˙itx^it+x^˙iteitx^it2=x^˙itx^it1+eitx^it=A+BKLi+ai0⊗In1+eitx^it+BKLi+ai0⊗Ineix^it1+eitx^it≤A+BKLi+ai0⊗In1+eitx^it+A+BKLi+ai0⊗Ineix^it1+eitx^it=A+BKLi+ai0⊗In1+eitx^it2,where Li is the row i of the Laplace matrix L.
Letzi=eit/x^it, then(45)z˙i≤A+BKLi+ai0⊗In1+zi2.
Consider that a nonnegative functionψt,ψ0 satisfies ψ˙=A+BKLi+ai0⊗In1+ψ2 and ψ0=0, according to Lemma 3, zi≤ψt,0. It can be seen from (36),(46)ψτki,0=1−κai0λminW2dii+ai0λmaxW1/2.
Hence,(47)τki=A+BKLi+ai0⊗In31+ψτ3−1,=A+BKLi+ai0⊗In3×1+1−κai0λminW2dii+ai0λmaxW1/23−1.
Similar to Theorem2, that Zeno behavior that does not occur can be proved by contradiction, which is omitted here.


## 4. Leader-Following Control of Multiagent Systems under Switching Topologies

In this part, we consider the extended case that the interconnection network switches according to signalσt and is not connected all the time. It is worth noting that, unlike the fixed topology, the controller updates only when the event is triggered. In the switching topologies, the controller updates in the following two cases: (1) event-triggering instant. (2) Communication topology switching instant.The control input of theith agent is defined as follows:(48)uit=−K∑j∈Nitaijtxitki−xjtk′i−Kai0txitki−x0t,where t∈tki,tk+1i. Different from control protocols (10) and (35), Nit and aijt in (48) are changed under the switching topologies. Matrices L¯σt and D¯σt in G¯σt represent Laplacian matrix and connection matrix between leader and agent, respectively. Switching signal σt:0,∞⟶P is a piecewise continuous constant function, which is used to describe the switching law of communication topology. Also, G¯σt:p∈P is a set of graphs that are switched within a finite set P=1,2,… in any finite time interval. Consider a nonempty and continuous infinite sequence ts,ts+1, where k=0,1,… and t0=0. Suppose that G¯σt is switched only at and remains unchanged in t∈ts,ts+1.Remark 2.
It should be noted that graphG¯σt may be connected or unconnected in interval ts,ts+1.
By replacing the similar variables in Section3.2, we can derive that(49)x^˙t=IN⊗Ax^t−Lσt+D¯σt⊗BKx^t+et.Theorem 5.
Under Assumptions1 and 2, if feedback gain matrix K satisfies K=BTP and W=PBK, then the protocol (48) still makes the multiagent system with (3) track the system (2) successfully if the event-triggering condition satisfies(50)fit=−κai0minλminW−αλmaxPai0maxλmaxW+2diimaxλmaxW∑i=1Nx^iTx^i+∑i=1NeiTei≥0,where 0<κ<1,0<α≤ai0σtλminW/λmaxP.Proof.
Construct the Lyapunov function for system (49) as follows:(51)V3=eαt∑i=1Nx^iTPx^i.
Similar to Section3.2, taking the derivative of V3 along the trajectory of system (49) yields(52)V˙3≤2eαt∑i=1Nx^iTPAx^i−eαt∑i=1N∑j=1Naijx^iTWx^i−x^j+eαt∑i=1NeiTWai0ei+2eαt∑i=1N∑j=1NaijeiTWei+eαt∑i=1Nx^iTαPx^i−eαt∑i=1Nx^iTWai0x^i≤eαtx^TIN⊗PA+ATPx^−eαtx^TLσt⊗Wx^+eαt∑i=1NeiTai0σtW+2diiσtWei+eαt∑i=1Nx^iTαP−ai0σtWx^i.(i)
If the graphG¯p is not connected during t∈ts,ts+1, according to the event-triggering condition (50) and equation (6), one has(53)V˙3≤eαtx^TIN⊗PA+ATPx^+eαt∑i=1NαλmaxP−ai0σtλminWx^i2+eαt∑i=1Nai0σtλmaxW+2diiσtλmaxWei2≤0.
It can be seen from the abovementioned formula thatV3 is not increasing; hence,(54)V3t≥V3ts+1=eαts+1∑i=1Nx^its+1TPx^its+1≥eαts+1λminPx^ts+12,i.e., x^ts+1≤V3t/λminPe−α/2ts+1≤V30/λminPe−α/2ts+1.(ii)
If the graphG¯p is connected during t∈ts,ts+1, then(55)V˙3≤eαtx^TIN⊗PA+ATP−λ2LσtWx^+eαt∑i=1NαλmaxP−ai0σtλminWx^i2+eαt∑i=1Nai0σtλmaxW+2diiσtλmaxWei2.
According to event-triggering condition (50) and equation (5),(56)V˙3≤eαtx^TIN⊗PA+ATP−λ2LσtWx^≤−eαtλ2Lσt2x^Tx^.
It can be seen from (56) that V3 is not increasing; hence,(57)V3t≥V3ts+1=eαts+1∑i=1Nx^its+1TPx^its+1≥eαts+1λminPx^ts+12,i.e., x^ts+1≤V3t/λminPe−α/2ts+1≤V30/λminPe−α/2ts+1.
In summary,x^ts+n≤V3ts+n−1/λminPe−α/2ts+n≤⋯≤V30/λminPe−α/2ts+n, i.e., x^t≤V3t/λminPe−α/2t≤⋯≤V30/λminPe−α/2t, so limt⟶∞x^t=0 is equivalent to limt⟶∞x^it=0, and accordingly limt⟶∞xit−x0t=0,i=1,2,…,N is established.Remark 3.
Indexα/2 can be approximated as the convergence rate of multiagent system (49), and the convergence rate can be changed by adjusting α.Theorem 6.
Under the conditions of Theorem5, system (49) does not have Zeno behavior. The interval between any two consecutive event-triggering instants of the system is not less than(58)IN⊗A+L+D¯⊗BK3×1+κai0σtλminW−αλmaxPai0σtλmaxW+2diiσtλmaxW1/23−1.Proof.
The proof is similar to that of Theorem2.


## 5. Simulation

In this part, we consider the trajectories of the state errors between the follower and leader under the fixed topology and the switching topology, respectively, where the dynamic equations of the leader and the follower are given by (2) and (3), respectively, and the communication network topology among agents is shown in Figures 1 and 2. Assume that xi=xi1,xi2T, and A and B are chosen as follows: A=00.5−4.80,B=0−0.5, it is easy to prove that the Assumption 2 is satisfied. By solving Riccati equation by MATLAB, we know that feedback gain matrix K=BTP=−0.4995,−1.1343T. Let the leader’s initial state be x00=2,3T and the follower’s initial state be x10=−1,1T,x20=−2,−3T,x30=5,−6T,x40=4,2T.Figure 1
Communication topologyG¯.Figure 2
Communication topologyG¯1 and G¯2.Example 1.
Under the centralized event-triggering protocol (10), the leader-following consensus of the multiagent system composed of (2) and (3) is considered. The communication network among agents is shown in Figure 1, and the corresponding weights are all 1. It can be seen from Figures 3 and 4 that followers can successfully follow the leader. Figure 5 shows the event instants of each follower with the centralized event-triggering protocol (10). It can be seen that protocol (10) can effectively reduce the number of communications among agents, thus reducing the waste of resources. Also, there is no Zeno behavior.Figure 3
The 1st state error trajectory of each agent under protocol (10).Figure 4
The 2st state error trajectory of each agent under protocol (10).Figure 5
Event times instants for four agents in Theorem1.Example 2.
In this example, we illustrate the leader-following consensus of the multiagent system under the distributed event-triggering protocol (35). The communication network among agents is shown in Figure 1. It can be seen from Figures 6 and 7 that followers can successfully follow the leader. Figure 8 shows the event triggering time of each follower under the decentralized event triggering protocol (35), and Zeno behavior is excluded. The simulation results verify Theorems 3 and 4.Figure 6
The 1st state error trajectory of each agent under protocol (35).Figure 7
The 2nd state error trajectory of each agent under protocol (35).Figure 8
Event times instants for four agents in Theorem3.Example 3.
Finally, the leader-following consensus of the multiagent system under the control protocol (48) is considered. The communication network among agents will randomly switch between G¯1 and G¯2, as shown in Figure 2, where G¯1 is a connected graph and G¯2 is an unconnected graph. The state errors between the follower agent i and leader 0 are shown in Figures 9 and 10, respectively. It indicates that all followers can successfully follow the leader. Figure 11 shows the event-triggering instants of each follower under (48), and there is no Zeno behavior.Figure 9
The 1st state error trajectory of each agent under protocol (48).Figure 10
The 2nd state error trajectory of each agent under protocol (48).Figure 11
Event times instants for four agents in Theorem5.


## 6. Conclusions and Future Work

In this paper, the leader-following control of general linear multiagent systems based on event-triggering mechanism under both fixed topology and switching topologies have been studied. Under the fixed topology, two different control protocols are designed in order to reduce waste of resources. Based on these two control protocols, we propose two different triggering functions, i.e., centralized event-triggering function and decentralized event-triggering function with state error between the follower and leader. When the triggering function exceeds 0, the agent will update the control input at the triggering instants. Through theoretical analysis, the sufficient conditions are derived for the system to achieve leader-following consensus under two control protocols and event-triggering conditions. The conditions obtained under fixed topology are extended to switching topologies (different from the fixed topology, the controller update at the triggering time, and also the switching time). The results show that the conditions to achieve leader-following are also valid under switching topologies. Finally, it is proved that the system can realize leader-following control without Zeno behavior. The simulation results verify the effectiveness of the theoretical analysis. In the future, we will further study the leader-following control of the linear multiagent system with interference, delay, and other factors.



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*Source: 5972749-2020-03-28.xml*