# Event-Triggered Average Consensus of Multiagent Systems with Switching Topologies

Authors: Gaosen Dong; Chunde Yang; Wei Zhu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Event-triggered average consensus of multiagent systems with switching topologies is studied in this paper. A distributed protocol based on event-triggered time sequences and switching time sequences is designed. Based on the inequality technique and stability theory of differential equations, a sufficient condition for achieving average consensus is obtained under the assumption that switching signal is ergodic and the total period over which connected topologies is sufficiently large. A numerical simulation is presented to show the effectiveness of the theoretical results.


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## Body

## 1. Introduction

Recently, the consensus problem of multiagent systems has attracted a great deal of attention in many fields such as multirobot systems [1], sensor networks [2], and unmanned air vehicles (UAVs). This hot topic has been widely discussed in the literature by different methods, such as Lyapunov function methods [3, 4], linear matrix inequality methods [5–7], matrix decomposition approaches [8], and impulsive control methods [9, 10], just to name a few.In practical engineering applications, connections between agents often change due to obstacles in the environment or limitations of sensor communication distance. Hence, consensus of multiagent systems with switching topologies has attracted considerable attention [11–23]. In particular, the average consensus problem with switching topologies was studied in [17]. The leader-following consensus of second-order agents with switching topologies was studied in [18], which proved that the consensus of multiagent systems was asymptotically reachable and gave an estimate of the convergence rate. In [19], for controllability of multiagent systems with periodically switching topologies was studied and a criterion for m-periodic controllability was proposed. In [20], the guaranteed-performance consensualization for high-order linear and nonlinear multiagent systems with switching topologies was studied. In [21], the consensus of multiagent systems with switching jointly reachable interconnection was studied. In [22], the consensus problem of multiagent systems with jointly connected switching topologies was studied by adding adaptive control. However, the control protocol used in the above literature requires continuous communication among agents, which is hard to realize due to the limited communication bandwidth. It may also result in the waste of computing resources as well as the consumption of a large amount of energy.To improve the usage of limited bandwidth resource, event-triggered consensus of multiagent systems with switching topologies has been extensively studied [24–29]. In [30], the event-triggered leaderless and leader-following consensus problems of multiagent systems with jointly connected topology were investigated. In [31], an event-triggered protocol for networks with switching topologies was proposed where the triggering functions were designed based on continuous information. In [32], event-triggered control for pinning cluster synchronization in an array of coupled neural networks was studied. In [33], event-triggered control leader-following consensus problems of multiagent systems with semi-Markov switching topologies were discussed. The event-triggered consensus problems of multiagent systems with jointly connected switching topologies were also presented in [34–37].By the above discussion, how to define the control protocol, which updates at both the triggering time instants and switching time instants, is an interesting topic for multiagent systems with switching topologies. In this paper, event-triggered consensus of first-order multiagent systems with switching topologies is studied. The contribution of this paper is as follows: (1) The event-triggered consensus of multiagent systems with switching communication topologies is considered. The update time instants of the controller are determined by both triggering time instants and switching time instants. (2) The Zeno-behavior of the sampling time sequences of the controller can be excluded directly and the infimum of the sampling interval can be bigger than a given positive constant. (3) A sufficient condition on consensus is presented under the assumption that the switching communication topologies are ergodic.The rest of this paper is organized as follows. Section2 introduces some preliminaries and the first-order multiagent model. The main results are presented in Section 3. A numerical example is presented in Section 4 to illustrate the effectiveness of theoretical results. Section 5 concludes the paper and offers suggestions for future work.Notations.
letx be the Euclidean norm of vector x and A be the corresponding induced matrix norm for a matrix A. λminA and λmaxA are the minimum and maximum eigenvalues of a symmetric real matrix A. 1n=1,1,…,1T.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

ConsiderΛ=1,2,…,M as index set, where M is the total number of all possible interconnection graphs. To consider the consensus problem of multiagent systems with switching topologies, we define Gp=V,εp,Ap, p∈Λ as graphs of order n with the set of nodes V=1,2,…,n (denoting the n agents) for any p∈Λ, the set of edges εp⊂V×V, and adjacency matrices Ap=aijpn×n, where aijp is the weight of the edge i,j∈εp in the graph Gp. aijp>0 if and only if there is an edge i,j in Gp. Moreover, we assume aiip=0 for all i=1,…,n, p∈Λ. The degree matrices Dp=diagd1p,…,dnp are diagonal matrices, whose diagonal elements are given by dip=∑j=1naijp. The Laplacian of the weighted graph is defined as Lp=Dp−Ap for every p. The set of neighbors of node i is denoted by Ni=j∈V:i,j∈εp.LetGs be the set of connected graphs in Gp and Gu be the set of unconnected graphs in Gp. The cardinality of Λ is denoted by Λ. Without loss of generality, we assume that Gs=Gp,p=1,2,…,ν and Gu=Gp,p=ν+1,ν+2,…,Λ for some ν∈Λ. For any t>t0, let Tpt0,t denote the total activation time of Gp, p∈Λ, and Tst=∑p=1νTpt0,t, and Tut=∑p=ν+1ΛTpt0,t.In the following, we assume that switching is “ergodic switching” [13], i.e., each graph will be activated infinite times.


### 2.2. Problem Formulation

Consider the first-order multiagent systems described by(1)x˙it=uit,i=1,2,…,n,where xit and uit∈R denote the position and control input of agent i, respectively.In order to discuss the consensus problem of multiagent systems (1) and avoid the Zeno-behavior directly, i.e., there is no infinite sampling in a finite interval, the following distributed event-triggered control protocol will be used:(2)uit=0,t∈tki,tki+ς,−∑j∈Niaijpxitki−xjtk′j,t∈tki+ς,tk+1i,where ς>0,tk′j=argminl∈Ni:t≥tljt−tlj. tki is the kth triggering time instant for agent i and is defined iteratively as(3)tk+1i=inft>tki+ςt:fit≥0∪tp,in which tp is the switching time and fit is the triggering function defined as follows:(4)fit=eit−β∑j∈Niaijpxitki−xjtk′j,for some β>0 and eit=xitki−xit.Remark 1.
From formulas (2) and (3), one can see that if tk+1i−tki<ς, then uit=0, which means that the agent will not sample the information of its neighboring agents. The controller update only occurs when tk+1i−tki≥ς. Thus, the Zeno-behavior of the sampling time sequences of the controller can be excluded directly.
Noticing thatxitki=xit+eit, event-triggered control protocol (2) can be rewritten as(5)uit=−∑j∈Niaijpxit−xjt+eit−ejt.
Denotingxt=x1t,…,xntT and et=e1t,…,entT, by (1) and (5), we have the following equation:(6)x˙t=−Lpxt−Lpet.
Definingx¯t=1/n∑i=1nxit, by simple computation, one can easily obtain that x¯˙t=0, which shows that x¯t is constant.
Denotingδt=xt−x¯t1n, by (6), we have the following equation:(7)δ˙t=−Lpδt−Lpet.Definition 1.
The MASs (1) is said to achieve average consensus under designed control protocol, if limt⟶∞xit−x¯t=0 holds for any initial conditions.


## 2.1. Preliminaries

ConsiderΛ=1,2,…,M as index set, where M is the total number of all possible interconnection graphs. To consider the consensus problem of multiagent systems with switching topologies, we define Gp=V,εp,Ap, p∈Λ as graphs of order n with the set of nodes V=1,2,…,n (denoting the n agents) for any p∈Λ, the set of edges εp⊂V×V, and adjacency matrices Ap=aijpn×n, where aijp is the weight of the edge i,j∈εp in the graph Gp. aijp>0 if and only if there is an edge i,j in Gp. Moreover, we assume aiip=0 for all i=1,…,n, p∈Λ. The degree matrices Dp=diagd1p,…,dnp are diagonal matrices, whose diagonal elements are given by dip=∑j=1naijp. The Laplacian of the weighted graph is defined as Lp=Dp−Ap for every p. The set of neighbors of node i is denoted by Ni=j∈V:i,j∈εp.LetGs be the set of connected graphs in Gp and Gu be the set of unconnected graphs in Gp. The cardinality of Λ is denoted by Λ. Without loss of generality, we assume that Gs=Gp,p=1,2,…,ν and Gu=Gp,p=ν+1,ν+2,…,Λ for some ν∈Λ. For any t>t0, let Tpt0,t denote the total activation time of Gp, p∈Λ, and Tst=∑p=1νTpt0,t, and Tut=∑p=ν+1ΛTpt0,t.In the following, we assume that switching is “ergodic switching” [13], i.e., each graph will be activated infinite times.


## 2.2. Problem Formulation

Consider the first-order multiagent systems described by(1)x˙it=uit,i=1,2,…,n,where xit and uit∈R denote the position and control input of agent i, respectively.In order to discuss the consensus problem of multiagent systems (1) and avoid the Zeno-behavior directly, i.e., there is no infinite sampling in a finite interval, the following distributed event-triggered control protocol will be used:(2)uit=0,t∈tki,tki+ς,−∑j∈Niaijpxitki−xjtk′j,t∈tki+ς,tk+1i,where ς>0,tk′j=argminl∈Ni:t≥tljt−tlj. tki is the kth triggering time instant for agent i and is defined iteratively as(3)tk+1i=inft>tki+ςt:fit≥0∪tp,in which tp is the switching time and fit is the triggering function defined as follows:(4)fit=eit−β∑j∈Niaijpxitki−xjtk′j,for some β>0 and eit=xitki−xit.Remark 1.
From formulas (2) and (3), one can see that if tk+1i−tki<ς, then uit=0, which means that the agent will not sample the information of its neighboring agents. The controller update only occurs when tk+1i−tki≥ς. Thus, the Zeno-behavior of the sampling time sequences of the controller can be excluded directly.
Noticing thatxitki=xit+eit, event-triggered control protocol (2) can be rewritten as(5)uit=−∑j∈Niaijpxit−xjt+eit−ejt.
Denotingxt=x1t,…,xntT and et=e1t,…,entT, by (1) and (5), we have the following equation:(6)x˙t=−Lpxt−Lpet.
Definingx¯t=1/n∑i=1nxit, by simple computation, one can easily obtain that x¯˙t=0, which shows that x¯t is constant.
Denotingδt=xt−x¯t1n, by (6), we have the following equation:(7)δ˙t=−Lpδt−Lpet.Definition 1.
The MASs (1) is said to achieve average consensus under designed control protocol, if limt⟶∞xit−x¯t=0 holds for any initial conditions.


## 3. Main Results

Lemma 1 (see [23]).
SupposeL is the Laplacian of an undirected and connected graph G, then(8)e−Lt−t0v≤e−λ2Lt−t0v,t≥t0,holds for v∈Rn and 1nTv=0.Theorem 1.
Consider multiagent systems (1) with control strategy (2) and triggering time sequence (3). The average consensus can be achieved asymptotically, if the switching guarantees that(9)inft>t0TstTut>γλ,where 0<λ<min1≤p≤vλ2p and λ2p is the minimum nonzero eigenvalue of undirected connected graph Gp:(10)maxv+1≤p≤ΛLp<γ<∞,0<β<min1≤p≤Λλ2pnLp2+nLpλ2p.Proof.
Since the switching is “ergodic switching,” one can see that the estimating process is independent of the order of the activated topology. Without loss of generality, assume that activate topology ontq−1,tq,q=1,2,…, is Gp, where p∈Λ and p=q mod Λ, and set p=Λ if p=0.
From equation (7), by the variation of parameter formula, for t∈tp−1,tp, p=1,2,…,v, we have(11)δt=e−Lpt−tp−1δtp−1+∫tp−1te−Lpt−s−Lpesds.
Then, since1nTδtp−1=0, by Lemma 1, we can derive that(12)δt≤e−λ2pt−tp−1δtp−1+∫tp−1te−λ2pt−s⋅Lp⋅esds.
On the other hand, fort∈tp−1,tp, p=v+1,v+2,…,Λ, we have(13)δt≤eLpt−tp−1δtp−1+∫tp−1teLpt−s⋅Lp⋅esds.
It follows from (4) that fit≤0, that is,(14)eit≤β∑j∈Niaijpxitki−xjtk′j≤β∑j∈Niaijpδit−δjt+∑j∈Niaijpeit−ejt≤βLpet+βLpδt.
Then,(15)et≤nβLp1−nβLpδt.
Due to the fact that1−nβLpλ2p−nβLp2>0, there is a positive constant λ such that(16)1−nβLpλ2p−λ−nβLp2>0.
In the following, forp=1,2,…,v, it will be proved that(17)δt≤δtp−1e−λt−tp−1,t∈tp−1,tp.
In order to prove (17), we first claim that(18)δt<μδtp−1e−λt−tp−1≜ψt,t∈tp−1,tp,holds for any μ>1.
Otherwise, by the continuity ofδt, there must exist a t∗>tp−1 such that(19)δt<ψt,fort∈tp−1,t∗,δt∗=ψt∗.
Then, we have(20)ψt∗=δt∗<μe−λ2pt∗−tp−1δtp−1+μ∫tp−1t∗e−λ2pt∗−s⋅Lp⋅esds<μe−λ2pt∗−tp−1δtp−1+μnβLp21−nβLp∫tp−1t∗e−λ2pt∗−sδsds=μe−λ2pt∗−tp−1δtp−1+μnβδtp−1Lp2λ2p−λ1−nβLpe−λt∗−tp−1−e−λ2pt∗−tp−1=μδtp−1e−λ2pt∗−tp−1+nβLp2λ2p−λ1−nβLpe−λt∗−tp−1−e−λ2pt∗−tp−1<μδtp−1e−λ2pt∗−tp−1+e−λt∗−tp−1−e−λ2pt∗−tp−1=μδtp−1e−λt∗−tp−1=ψt∗,which is a contradiction. Hence, (18) holds for any number μ>1. Let μ⟶1, we have(21)δt≤δtp−1e−λt−tp−1,t∈tp−1,tp,p=1,2,…,v.
Therefore, fort∈t0,t1, one can derive that(22)δt≤δt0e−λt−t0.
Then, fort∈t1,t2, we have(23)δt≤δt1e−λt−t1≤δt0e−λt1−t0e−λt−t1.
Repeating the above procedure, fort∈tv−1,tv, one has(24)δt≤δt0e−∑p=1v−1λtp−tp−1−λt−tv−1.
Forp=v+1,v+2,…,Λ, in terms of (13), we have(25)δt≤eLpt−tp−1δtp−1+∫tp−1teLpt−s⋅Lp⋅esds.
Sincemaxv+1≤p≤ΛLp<γ<∞ and nβLp2<λ2p1−nβLp, then there is γ such that nβLp2<γ−Lp1−nβLp,p=v+1,v+2,…,Λ. By a similar argument to that in the proof of (17), we have(26)δt≤φeγt−tv,t∈tv,tv+1,where φ=δt0e∑p=1v−λtp−tp−1.
Repeating the above procedure, we can derive that(27)δt≤δt≤φe∑p=v+1Λ−1γtp−tp−1+γt−tΛ−1,t∈tΛ−1,tΛ.
In general, sinceGp is finite and switching is “ergodic switching,” by (24), (27), and mathematical induction, we have(28)δt≤δt0e−∑p=1vλTpt0,t+∑p=v+1ΛγTpt0,t≤δt0e−λTst+γTut,t≥t0.
Noticing thatinfTst/Tut>γ/λ, there is a constant ρ>0 such that γ+ρTut≤λ−ρTst. Thus, γTut−λTst≤−ρTut+Tst=−ρt−t0. Then, we have(29)δt≤δt0e−ρt−t0,t≥t0,which implies that the consensus is reached asymptotically and the exponential convergence rate is ρ.Remark 2.
(1)
Compared with the work in [18], a novel distributed event-triggered consensus protocol with switching topologies is proposed in this paper. Our control protocol does not require continuous communication among agents and a sufficient condition on consensus is presented under the assumption that the switching topologies are ergodic.(2)
Compared with the works in [26, 27, 34], the update time instants of the controller are determined by both triggering time instants and switching time instants. Furthermore, the Zeno-behavior of the sampling time sequences of the controller can be excluded directly and the infimum of the sampling interval can be bigger than a given positive constant.


## 4. Simulations

In this section, a numerical example is given to illustrate the feasibility and effectiveness of the theoretical results.Consider the multiagent systems with five agents, where the communication topologies are described by Figure1. Obviously, G1,G2,G3 are connected and G4,G5 are disconnected. For each communication topology, the corresponding Laplacian is given as follows, respectively:(30)L1=2−100−1−12−1000−12−1000−110−10001,L2=2−100−1−11000001−1000−12−1−100−12,L3=1−100−101−1000−12−1000−12−1−100−12,L4=1−1000−12−1000−12−1000−11000000,L5=1000−1000000−12−1000−110−10001.Figure 1
Switching topologies:Gp,p=1,2,3,4,5. (a) G1, (b) G2, (c) G3, (d) G4, and (e) G5.
(a)
(b)
(c)
(d)
(e)Then, we can getmin1≤p≤vλ2p=0.382, maxv+1≤p≤ΛLp=3.4142, and min1≤p≤vλ2p/nLp2+nLpλ2p≈0.0118. We assume β=0.01<0.0118, λ=0.3<0.382, and γ=3.5>3.4142, which satisfies the parameter requirements of Theorem 1. Let ς=0.5 and the dwell time is a random number which is greater than ς. In order to guarantee that(31)inft>t0TstTut>γλ,we assume that the total activated time for connected graphs is about 93 percent of the total time. Hence, the consensus can be achieved by Theorem 1.The initial states are denoted byx0=1,2,3,4,5T. The state responses xit and the controllers uit are depicted in Figures 2 and 3, respectively. The switching sequence is depicted in Figure 4. The triggering time sequences for each agent are shown in Figure 5.Figure 2
State for each agent.Figure 3
Controller for each agent.Figure 4
Switching sequence.Figure 5
Triggering time sequences.


## 5. Conclusions

Based on the inequality technique and stability theory of differential equations, the event-triggered average consensus of multiagent systems with switching topologies is studied. A sufficient condition for achieving average consensus is obtained under the assumption that switching signal is ergodic and the total period over which connected topologies is sufficiently large. Moreover, the designed control protocol can exclude Zeno-behavior directly. It should be noted that the main results are derived only for a first-order multiagent system and an undirected graph. More general linear system models and directed graph scenarios will be addressed in future study.



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