# Couple-Group Consensus for Multiagent Systems via Time-Dependent Event-Triggered Control

Authors: Huiyang Liu; Xiaoshuang Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper investigates couple-group consensus problems for multiagent first-order and second-order systems. Several consensus protocols are proposed based on the time-dependent distributed event-triggered control. For the case of no communication delays, the time-dependent event-triggered strategies are applied to couple-group consensus problems. Based on the matrix theory, algebraic conditions for couple-group consensus are established. For the system with communication delays, based on event-triggered strategies, a Lyapunov-Krasovskii functional is constructed to prove the input-to-state stability of the systems. Moreover, Zeno behavior is excluded. Finally, numeral examples are given to illustrate the effectiveness of these results.


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## Body

## 1. Introduction

In recent years, many researchers have paid attention to cooperative control of multiagent systems. This is because of the wide applications, such as formation control of satellite clusters, unmanned air vehicles, flocking, and swarming [1–5]. The objective in coordination control is to design appropriate control protocols such that the agents in a multiagent team can achieve a coordination objective asymptotically by using exchanged information between each other. The consensus problem is one type of the critical coordination control problems in the field of multiagent systems. It is simple in mathematics but represents a lot of real engineering systems, e.g., the attitude consensus and velocity consensus. Therefore, many researchers have focused on this topic and there emerge in large numbers of literatures [6–13].In practical engineering applications, every autonomous agent is equipped with a microprocesser, which has limited resources and energy, to collect and process information. To reduce data transmission and save energy, event-triggered control strategies have been employed. A proper designed event-triggered strategy can also serve the desired properties of the ideal system, such as stability and convergence. Up to now, many theoretical results have been derived on event-triggered consensus problems by combining algebraic graph theory with matrix theory. The authors of [14] proposed event-triggered control strategies for first-order dynamics. A centralized formulation was considered, and then the results were extended to the decentralized counterparts. Some further results on event-based control for first-order dynamics were obtained in [15], including the analysis of discrete systems and continuous systems with high dimensions. The event-triggered tracking control protocol for first-order dynamics was proposed in [16]. The input-to-state stability (ISS) of the closed-loop multiagent system was analyzed by employing an ISS Lyapunov function. The consensus problem for second-order dynamics via event-triggered control was addressed in [17], where a centralized event-triggered strategy was designed, and the bound of interevent times was ensured. To overcome the limitation of the state-dependent event-triggered control to its practical implementation, a modified version of the event-based consensus scheme was proposed in [18]. A combinational measurement approach to designing the event-triggered scheme was developed in [19]. Among these literatures mentioned above, the data transmission only occurs when the measurement error exceeds the state-dependent threshold. However, the authors of [20, 21] designed event-triggered protocols where the event occurs when measurement error exceeds the time-dependent threshold. Moreover, in [22], flocking of multiagent systems with multiple groups was studied in fixed and switching heterogeneous networks. In [23], an event-triggered based protocol was designed for first-order discrete-time multiagent systems with time-varying topology. In [24], a distributed sampled-data based event-triggered consensus protocol was proposed. In [25], based on weighted average consensus, distributed event-triggered cubature information filtering was studied. There are a lot of results on the event-triggered consensus scheme reported in [26–32]. For more literatures, please refer to the survey paper [33].To our best knowledge, all the aforementioned literatures have focused on consensus problems where all the agents can reach an agreement, even a dynamic agreement. In practical engineering applications, there is a phenomenon that the agreements are changed with the varying of environment and cooperative tasks. A novel consensus control strategy is introduced to multiagent systems, where agents in the same subnetwork reach a consistent value while no agreement between any two different subnetworks. It is called a couple-group consensus if there are only two subnetworks. In [34], a double-tree-form transformation was introduced and the first-order couple-group consensus problem was studied with finite switching topologies and bounded communication delays. In multiagent systems, communication delays are a kind of nonnegligible factors when they exist in the communication channel between any two agents. The authors of [35, 36] investigated this problem with undirected and directed topology. Moreover, the convergence analysis was discussed, and some criteria were established. The extension of this work to the second-order dynamics was addressed in [37], where two different kinds of consensus protocols were proposed for networks with fixed topology. In [38], the group consensus problem of second-order multiagent systems with time-delays was studied. In [39], the authors studied time-varying group formation analysis and design for general linear multiagent systems with directed topologies. In [40], group tracking control of second-order multiagent systems with fixed and Markovian switching topologies was studied.In this paper, we want to concern the following problems. (1) The state-dependent threshold and the time-dependent threshold are usually used in the event-triggered control systems [33]. Since the time-dependent threshold has an advantage to avoid the continuous communication and also has a simpler form, we will consider the time-dependent threshold in this paper. (2) Although the authors of [20] adopted the time-dependent event-triggered strategy, single consensus was considered. It makes great sense to investigate the couple-group consensus in engineering applications. So far, however, few works have focused on the couple-group control problem via event-triggered control, especially the time-dependent event-triggered control. Motivated by the above discussion, the couple-group consensus problem via time-dependent event-triggered is considered in this paper. The contributions of this work are three-fold: (1) both first-order and second-order dynamics in the framework of event-triggered based control are considered in this paper. In addition, an important factor that may exist in the communication channel is considered; i.e., communication delays are considered when the event-triggered based protocols are proposed; (2) the time-dependent event-triggered protocol is introduced to deal with energy consumption and communication constraints; (3) we discuss the distributed event-based couple-group consensus in presence of both positive and negative adjacent weights. Zeno behavior is excluded for all cases.The rest of this paper is organized as follows. Section2 gives some preliminaries on graph theory, assumptions, and lemmas. Event-based couple-group consensus problems for first-order dynamics and second-order dynamics are presented in Sections 3 and 4, respectively. Numerical examples are provided in Section 5. Section 6 draws the conclusions and develops the vision for future work.Notations. In this paper, standard notations are used to illustrate the problem. Rn represents the set of n-dimensional vectors. R(·) and I(·) are the real part and the imaginary part of a complex number, respectively. The vector 1n=[1,1,…,1]T∈Rn has all elements equal to 1. In×n and 0n×n are n-dimensional identity matrix and n-dimensional zero matrix, respectively. · denotes Euclidean norm, and ∗ in this paper stands for a term of block that is induced by symmetry.


## 2. Preliminaries

LetG=(V,E,A) be a weighted graph to model the information communication among n+m agents with the node set V={ν1,ν2,…,νn+m}, the edge set E⊆V×V, and the weighted adjacency matrix A. The node indexes belong to a finite index set P={1,2,…,n+m}. An edge eij∈E means that node νi can receive information from node νj. A=[aij]∈R(n+m)×(n+m) is defined as aij≠0 if eij∈E and aij=0 if eij∉E. Moreover, in this paper, there is no self-loop, i.e., aii=0. Let Ni be the neighbor set of agent νi, denoted by Ni={νj∣eij=(νj,νi)∈ε}. An undirected graph G is said to be connected if there is a path between any two nodes of G.The following definitions, notations, and lemmas are borrowed from [36]. Suppose that the network can be separated into two subnetworks G1 and G2. Denote P1={1,2,…,n}, P2={n+1,n+2,…,n+m}, V1={v1,v2,…,vn}, and V2={vn+1,vn+2,…,vn+m}. Then V=V1∪V2. Furthermore, letting N1i={vj∈V1∣(vi,vj)∈E} and N2i={vj∈V2∣(vi,vj)∈E}, we have Ni={vj∈V∣(vi,vj)∈E}=N1i∪N2i. The Laplacian matrix L=[lij] is defined as(1)lij=-aij,j≠i,∑k=1,k≠in+maik,j=i.For the two subgroups, we propose two assumptions as follows:A1∑j=n+1n+maij=0;∀i∈P1;A2∑j=1naij=0;∀i∈P2,which means G1 and G2 are in-degree balanced to each other.In this case, the Laplacian matrix has the following partition:(2)L=L11L12L12TL22,where L11∈Rn×n and L22∈Rm×m. Since G is undirected, L is symmetric. Therefore, all the eigenvalues of L have real values.Lemma 1 (see [36]).
L has a zero eigenvalue whose geometric multiplicity is at least two. If L only has two-simple zero eigenvalues and the rest of the eigenvalues of L are positive, then(3)R=limt→+∞e-Lt=ω1rω1lT+ω2rω2lT,where ωilTL=0, Lωir=0, and ωilTωir=1 for any i=1,2.


## 3. Event-Triggered Group Consensus for First-Order Multiagent Systems

In this section, event-triggered control strategy is employed for first-order multiagent system without and with communication delays.Considering a network withn+m agents, each agent has the following dynamics:(4)x˙it=uit,i=1,2,…,n+m,where xi,ui∈R represent the state and control input of the ith agent, respectively.


### 3.1. Without Communication Delays

In this subsection, we consider the distributed event-triggered consensus scheme for first-order multiagent systems.The distributed event-triggered control law is obtained:(5)
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,where x^j(t)=xj(tk′(t)j), k′(t)=argminl∈N:t>tlj{t-tlj}. Define the measurement error ei(t)=x^i(t)-xi(t)=xi(tki)-xi(t), t∈[tki,tk+1i), where xi(tki) represents the kth triggered instant of agent i.Definition 2 (see [35]).
Couple-group consensus is asymptotically solved under protocol (5) if the states of agents in G satisfy limt→∞xi(t)-xj(t)=0; ∀i,j∈P1; limt→∞xi(t)-xj(t)=0; ∀i,j∈P2.Remark 3.
It is worth noting that the weightaij between two groups is allowed to be negative in the couple-group consensus problems. The term aijx^j for i∈P1 and j∈P2 can be regarded as disturbances from the other subnetwork. In addition, the existence of negative factors complicates the dynamics of the multiagent systems. Particularly, the nonzero eigenvalues of Laplacian matrix L does not satisfy λi(L)>0 any more when the undirected topology is connected.Letx1(t)=[x1(t),x2(t),…,xn(t)]T and x2(t)=[xn+1(t),xn+2(t),…,xn+m(t)]T. Correspondingly, e(t)=((e1)T(t),(e2)T(t))T=(e1(t),e2(t),…,en+m(t))T. With the protocol (5), the matrix form of the closed-loop system (4) is given as follows: (6)x˙1t=-L11x1t-L11e1t-L12x2t-L12e2t,x˙2t=-L21x1t-L21e1t-L22x2t-L22e2t.Defining the average value ofx1(t) as x¯1(t), then (7)x¯˙1t=1n1nTx˙1t=1n1nT-L11x1t+e1t-L12x2t+e2t.Under assumptions A1 and A2, G1 and G2 are a balanced couple. Note that the Laplacian matrix L is symmetric. It follows that 1n+mTL=0, 1nTL11=0, 1nTL12=0, and x¯˙1(t)=0. Normally, x¯1(t)=x¯1(0)=a1 is calculated. The state of agents can be described as x1(t)=a11n+θ1(t), where θ1(t) is the disagreement vector, satisfying 1nTθ1(t)=0. Similarly, x2(t)=a21m+θ2(t), where 1mTθ2(t)=0. Therefore(8)θ˙t=x˙t=-Lθt+et,where 1n+mTθ(t)=0 and θ(t)=[θ1T(t),θ2T(t)]T. Obviously, the couple-group consensus problem of multiagent system is equivalent to the asymptotic stability of system (8).Adopt the following event-triggered function [20]:(9)feit=eit-c1e-ϱ1t,where c1>0.Theorem 4.
Under assumptionsA1 and A2, multiagent system (4) with protocol (5) and distributed event-triggered function (9) asymptotically reaches couple-group consensus if L has two-simple zero eigenvalues and all the other nonzero eigenvalues are positive, where the parameter ϱ1 in the event-triggered function satisfies 0<ϱ1<λ3(L), 0=λ1,2L<λ3L≤⋯≤λn+m(L). Furthermore, the system does not exhibit Zeno behavior for all initial conditions x(0)∈Rn+m.Proof.
From (8), it follows that(10)θt=e-Ltθ0-∫0te-Lt-sLesds.Therefore, θ(t) is bounded by (11)θt≤e-Ltθ0+∫0te-Lt-sLesds.From the exponential estimation of solutions, we have the following inequality: (12)θt≤cθe-λ3Ltθ0+cθ∫0te-λ3Lt-sLesds,where cθ is a constant.
According to event-triggered function (9), it follows that (13)et≤n+mc1e-ϱ1t.Therefore,(14)θt≤cθe-λ3Ltθ0+n+mc1cθL∫0te-λ3Lt-se-ϱ1sds,≤cθe-λ3Ltθ0+n+mc1cθLe-ϱ1t-e-λ3Ltλ3L-ϱ1.We obtain limt→∞θ(t)=0 if 0<ϱ1<λ3(L).
Thus, the group consensus problem is solved. The verification of no Zeno behavior is similar to that of Sections3.2 and 4 and is thus omitted here. When the absolute value of the measurement error of agent i exceeds a certain value, the event occurs and the corresponding controller updates, which keeps constant value until the next event occurs.Remark 5.
The results can be extended to the more general directed graphs. Since the eigenvalues of the Laplacian matrix associated a directed graph may be complex values. For the directed graph case,λ3(L) is replaced with R(λ3(L)), where 0=λ1,2L<Rλ3L≤⋯≤R(λn+m(L)).


### 3.2. With Communication Delays

Assume that the communication delays in all channels are uniform, bounded but time-varying, which satisfy0<τ(t)<r. The couple-group consensus protocol is written as(15)
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.System (4) with protocol (15) and event error ei(t-τ(t))=x^i(t-τ(t))-xi(t-τ(t)) can be written in matrix form as(16)x˙t=-Lxt-τt-Let-τt.According to the analysis of system without communication delays, it is obvious thatθ˙(t)=x˙(t). Then the dynamics of θ(t) is given as follows:(17)θ˙t=-Lθt-τt-Let-τt.Theorem 6.
Under assumptionsA1 and A2, multiagent system (4) with protocol (15) and event-triggered function (9) reaches couple-group consensus asymptotically for bounded communication delay 0<τ(t)<r if L has two-simple zero eigenvalues while all the other nonzero eigenvalues are positive, and there exist matrices P¯>0, R¯>0, Q, and Y and constants ϵ, μ>0, η>0, and μ+η>2ϱ1, satisfying the following matrix inequality:(18)κ11κ12rLR¯-L0n+m×n+m∗-ϵQT-ϵQrϵLR¯-ϵLrQT∗∗-rR¯1-rμ+η0n+m×n+m0n+m×n+m∗∗∗-ηIn+m×n+m0n+m×n+m∗∗∗∗-rR¯<0,where κ11=(μ+η)P¯-Y-YT and κ12=P¯-Q-ϵYT. Furthermore, the system does not exhibit Zeno behavior for all initial conditions x(0)∈Rn+m.Proof.
Construct the following Lyapunov-Krasovskii functional:(19)Vθt,θ˙t=θTtPθt+∫t-rt∫stθ˙TξRθ˙ξdξds.We verify that (20)∫t-rt∫stθ˙TξRθ˙ξdξds=∫t-rtξ-t+rθ˙TξRθ˙ξdξ≤r∫t-rtθ˙TξRθ˙ξdξ.For some constant μ>0, let V˙(θt,θ˙t)+μV(θt,θ˙t)<0, for V(θt,θ˙t)≥e(t-τ(t))2. Denote U=V˙(θt,θ˙t)+μV(θt,θ˙t)+η(V(θt,θ˙t)-e(t-τ(t))2), where η>0. Hence, taking the derivative of V(θt,θ˙t), it can be obtained that (21)U≤θ˙TPθ+θTPθ˙+rθ˙TRθ˙-∫t-rtθ˙TξRθ˙ξdξ+μ+ηθTPθ+r∫t-rtθ˙TξRθ˙ξdξ-ηet-τtTet-τt.
Denotingδ=-θ˙-Lθ(t-τ(t))-Le(t-τ(t)), then 0=[θTP2T+θ˙TP3T]δ+δT[P2θ+P3θ˙]. We can get(22)U≤θ˙TPθ+θTPθ˙+rθ˙TRθ˙-∫t-rtθ˙TξRθ˙ξdξ+μ+ηθTPθ+r∫t-rtθ˙TξRθ˙ξdξ-ηet-τtTet-τt+θTP2T+θ˙TP3Tδ+δTP2θ+P3θ˙.
We apply the relation(23)θt-τt=θt-∫t-τttθ˙sds,∫t-rtθ˙TsRθ˙sds≥1r∫t-τttθ˙sdsTrR1r∫t-τttθ˙sdsand let ψT(t)=[θT(t),θ˙T(t),1/r∫t-τ(t)t‍θ˙T(s)ds,eT(t)]; then U≤ψT(t)ϕψ(t)+rθ˙T(t)Rθ˙(t) holds, where (24)Φ=χ11χ12rP2TL-P2TL∗-P3T-P3rP3TL-P3TL∗∗-rR1-rμ+η0n+m×n+m∗∗∗-ηIn+m×n+m,with χ11=(μ+η)P-P2TL-LTP2 and χ12=P-P2T-LTP3. Applying Schur Complements Lemma [41], U<0 if the following linear matrix inequality holds:(25)χ11χ12rP2TL-P2TL0n+m×n+m∗-P3T-P3rP3TL-P3TLrR∗∗-rR1-rμ+η0n+m×n+m0n+m×n+m∗∗∗-ηIn+m×n+m0n+m×n+m∗∗∗∗-rR<0.
We multiply (25) by diag{P2-1,P2-1,R-1,Ip,R-1} and its transpose on the left and the right, respectively, and denote P3=ϵP2, Q=P2-1, P¯=QTPQ, R¯=R-1, and Y=LQ; then inequality (25) can be rewritten as (18). Therefore, limt→∞θ(t)=0, i.e., limt→∞x(t)=[a11nT,a21mT]T, which means that the couple-group consensus problem is solved. Furthermore, we will prove that the event-triggered function does not have Zeno behavior. Because (26)V˙θt,θ˙t+μ+ηVθt,θ˙t-ηet-τt2<0,we have(27)θTtPθt≤Vθt,θ˙t≤e-μ+ηt-t0Vθt0,θ˙t0+∫t0te-μ+ηt-sηes-τs2ds.Furthermore, λmin(P)θ(t)2≤θT(t)Pθ(t)≤λmax(P)θ(t)2, and thus (28)θt2≤λmaxPe-μ+ηtλminPθ02+1λminP∫0te-μ+ηt-sηes-τs2ds,that is,(29)θt≤λmaxPe-μ+ηt/2λminPθ0+1λminP∫0te-μ+ηt-sηdset-τt∞,≤λmaxPe-μ+ηt/2λminPθ0+1λminPημ+ηet-τt∞.
From the input-to-state stability (ISS) theory [42], the system (17) is ISS with the disturbance e(t-τ(t)). It can reach couple-group consensus via event-triggered function (9) without Zeno behavior when μ+η>2ϱ1>0.


## 3.1. Without Communication Delays

In this subsection, we consider the distributed event-triggered consensus scheme for first-order multiagent systems.The distributed event-triggered control law is obtained:(5)
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t
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,where x^j(t)=xj(tk′(t)j), k′(t)=argminl∈N:t>tlj{t-tlj}. Define the measurement error ei(t)=x^i(t)-xi(t)=xi(tki)-xi(t), t∈[tki,tk+1i), where xi(tki) represents the kth triggered instant of agent i.Definition 2 (see [35]).
Couple-group consensus is asymptotically solved under protocol (5) if the states of agents in G satisfy limt→∞xi(t)-xj(t)=0; ∀i,j∈P1; limt→∞xi(t)-xj(t)=0; ∀i,j∈P2.Remark 3.
It is worth noting that the weightaij between two groups is allowed to be negative in the couple-group consensus problems. The term aijx^j for i∈P1 and j∈P2 can be regarded as disturbances from the other subnetwork. In addition, the existence of negative factors complicates the dynamics of the multiagent systems. Particularly, the nonzero eigenvalues of Laplacian matrix L does not satisfy λi(L)>0 any more when the undirected topology is connected.Letx1(t)=[x1(t),x2(t),…,xn(t)]T and x2(t)=[xn+1(t),xn+2(t),…,xn+m(t)]T. Correspondingly, e(t)=((e1)T(t),(e2)T(t))T=(e1(t),e2(t),…,en+m(t))T. With the protocol (5), the matrix form of the closed-loop system (4) is given as follows: (6)x˙1t=-L11x1t-L11e1t-L12x2t-L12e2t,x˙2t=-L21x1t-L21e1t-L22x2t-L22e2t.Defining the average value ofx1(t) as x¯1(t), then (7)x¯˙1t=1n1nTx˙1t=1n1nT-L11x1t+e1t-L12x2t+e2t.Under assumptions A1 and A2, G1 and G2 are a balanced couple. Note that the Laplacian matrix L is symmetric. It follows that 1n+mTL=0, 1nTL11=0, 1nTL12=0, and x¯˙1(t)=0. Normally, x¯1(t)=x¯1(0)=a1 is calculated. The state of agents can be described as x1(t)=a11n+θ1(t), where θ1(t) is the disagreement vector, satisfying 1nTθ1(t)=0. Similarly, x2(t)=a21m+θ2(t), where 1mTθ2(t)=0. Therefore(8)θ˙t=x˙t=-Lθt+et,where 1n+mTθ(t)=0 and θ(t)=[θ1T(t),θ2T(t)]T. Obviously, the couple-group consensus problem of multiagent system is equivalent to the asymptotic stability of system (8).Adopt the following event-triggered function [20]:(9)feit=eit-c1e-ϱ1t,where c1>0.Theorem 4.
Under assumptionsA1 and A2, multiagent system (4) with protocol (5) and distributed event-triggered function (9) asymptotically reaches couple-group consensus if L has two-simple zero eigenvalues and all the other nonzero eigenvalues are positive, where the parameter ϱ1 in the event-triggered function satisfies 0<ϱ1<λ3(L), 0=λ1,2L<λ3L≤⋯≤λn+m(L). Furthermore, the system does not exhibit Zeno behavior for all initial conditions x(0)∈Rn+m.Proof.
From (8), it follows that(10)θt=e-Ltθ0-∫0te-Lt-sLesds.Therefore, θ(t) is bounded by (11)θt≤e-Ltθ0+∫0te-Lt-sLesds.From the exponential estimation of solutions, we have the following inequality: (12)θt≤cθe-λ3Ltθ0+cθ∫0te-λ3Lt-sLesds,where cθ is a constant.
According to event-triggered function (9), it follows that (13)et≤n+mc1e-ϱ1t.Therefore,(14)θt≤cθe-λ3Ltθ0+n+mc1cθL∫0te-λ3Lt-se-ϱ1sds,≤cθe-λ3Ltθ0+n+mc1cθLe-ϱ1t-e-λ3Ltλ3L-ϱ1.We obtain limt→∞θ(t)=0 if 0<ϱ1<λ3(L).
Thus, the group consensus problem is solved. The verification of no Zeno behavior is similar to that of Sections3.2 and 4 and is thus omitted here. When the absolute value of the measurement error of agent i exceeds a certain value, the event occurs and the corresponding controller updates, which keeps constant value until the next event occurs.Remark 5.
The results can be extended to the more general directed graphs. Since the eigenvalues of the Laplacian matrix associated a directed graph may be complex values. For the directed graph case,λ3(L) is replaced with R(λ3(L)), where 0=λ1,2L<Rλ3L≤⋯≤R(λn+m(L)).


## 3.2. With Communication Delays

Assume that the communication delays in all channels are uniform, bounded but time-varying, which satisfy0<τ(t)<r. The couple-group consensus protocol is written as(15)
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^
j
t
-
τ
t
,
i
∈
P
2
.System (4) with protocol (15) and event error ei(t-τ(t))=x^i(t-τ(t))-xi(t-τ(t)) can be written in matrix form as(16)x˙t=-Lxt-τt-Let-τt.According to the analysis of system without communication delays, it is obvious thatθ˙(t)=x˙(t). Then the dynamics of θ(t) is given as follows:(17)θ˙t=-Lθt-τt-Let-τt.Theorem 6.
Under assumptionsA1 and A2, multiagent system (4) with protocol (15) and event-triggered function (9) reaches couple-group consensus asymptotically for bounded communication delay 0<τ(t)<r if L has two-simple zero eigenvalues while all the other nonzero eigenvalues are positive, and there exist matrices P¯>0, R¯>0, Q, and Y and constants ϵ, μ>0, η>0, and μ+η>2ϱ1, satisfying the following matrix inequality:(18)κ11κ12rLR¯-L0n+m×n+m∗-ϵQT-ϵQrϵLR¯-ϵLrQT∗∗-rR¯1-rμ+η0n+m×n+m0n+m×n+m∗∗∗-ηIn+m×n+m0n+m×n+m∗∗∗∗-rR¯<0,where κ11=(μ+η)P¯-Y-YT and κ12=P¯-Q-ϵYT. Furthermore, the system does not exhibit Zeno behavior for all initial conditions x(0)∈Rn+m.Proof.
Construct the following Lyapunov-Krasovskii functional:(19)Vθt,θ˙t=θTtPθt+∫t-rt∫stθ˙TξRθ˙ξdξds.We verify that (20)∫t-rt∫stθ˙TξRθ˙ξdξds=∫t-rtξ-t+rθ˙TξRθ˙ξdξ≤r∫t-rtθ˙TξRθ˙ξdξ.For some constant μ>0, let V˙(θt,θ˙t)+μV(θt,θ˙t)<0, for V(θt,θ˙t)≥e(t-τ(t))2. Denote U=V˙(θt,θ˙t)+μV(θt,θ˙t)+η(V(θt,θ˙t)-e(t-τ(t))2), where η>0. Hence, taking the derivative of V(θt,θ˙t), it can be obtained that (21)U≤θ˙TPθ+θTPθ˙+rθ˙TRθ˙-∫t-rtθ˙TξRθ˙ξdξ+μ+ηθTPθ+r∫t-rtθ˙TξRθ˙ξdξ-ηet-τtTet-τt.
Denotingδ=-θ˙-Lθ(t-τ(t))-Le(t-τ(t)), then 0=[θTP2T+θ˙TP3T]δ+δT[P2θ+P3θ˙]. We can get(22)U≤θ˙TPθ+θTPθ˙+rθ˙TRθ˙-∫t-rtθ˙TξRθ˙ξdξ+μ+ηθTPθ+r∫t-rtθ˙TξRθ˙ξdξ-ηet-τtTet-τt+θTP2T+θ˙TP3Tδ+δTP2θ+P3θ˙.
We apply the relation(23)θt-τt=θt-∫t-τttθ˙sds,∫t-rtθ˙TsRθ˙sds≥1r∫t-τttθ˙sdsTrR1r∫t-τttθ˙sdsand let ψT(t)=[θT(t),θ˙T(t),1/r∫t-τ(t)t‍θ˙T(s)ds,eT(t)]; then U≤ψT(t)ϕψ(t)+rθ˙T(t)Rθ˙(t) holds, where (24)Φ=χ11χ12rP2TL-P2TL∗-P3T-P3rP3TL-P3TL∗∗-rR1-rμ+η0n+m×n+m∗∗∗-ηIn+m×n+m,with χ11=(μ+η)P-P2TL-LTP2 and χ12=P-P2T-LTP3. Applying Schur Complements Lemma [41], U<0 if the following linear matrix inequality holds:(25)χ11χ12rP2TL-P2TL0n+m×n+m∗-P3T-P3rP3TL-P3TLrR∗∗-rR1-rμ+η0n+m×n+m0n+m×n+m∗∗∗-ηIn+m×n+m0n+m×n+m∗∗∗∗-rR<0.
We multiply (25) by diag{P2-1,P2-1,R-1,Ip,R-1} and its transpose on the left and the right, respectively, and denote P3=ϵP2, Q=P2-1, P¯=QTPQ, R¯=R-1, and Y=LQ; then inequality (25) can be rewritten as (18). Therefore, limt→∞θ(t)=0, i.e., limt→∞x(t)=[a11nT,a21mT]T, which means that the couple-group consensus problem is solved. Furthermore, we will prove that the event-triggered function does not have Zeno behavior. Because (26)V˙θt,θ˙t+μ+ηVθt,θ˙t-ηet-τt2<0,we have(27)θTtPθt≤Vθt,θ˙t≤e-μ+ηt-t0Vθt0,θ˙t0+∫t0te-μ+ηt-sηes-τs2ds.Furthermore, λmin(P)θ(t)2≤θT(t)Pθ(t)≤λmax(P)θ(t)2, and thus (28)θt2≤λmaxPe-μ+ηtλminPθ02+1λminP∫0te-μ+ηt-sηes-τs2ds,that is,(29)θt≤λmaxPe-μ+ηt/2λminPθ0+1λminP∫0te-μ+ηt-sηdset-τt∞,≤λmaxPe-μ+ηt/2λminPθ0+1λminPημ+ηet-τt∞.
From the input-to-state stability (ISS) theory [42], the system (17) is ISS with the disturbance e(t-τ(t)). It can reach couple-group consensus via event-triggered function (9) without Zeno behavior when μ+η>2ϱ1>0.


## 4. Event-Triggered Group Consensus for Second-Order Multiagent Systems

In this section, distributed event-triggered couple-group consensus problems for second-order multiagent systems are addressed.Consider a multiagent network consisting ofn+m agents. The dynamics of a second-order agent is modeled by(30)ξ˙it=ζit,ζ˙it=uit,i=1,2,…,n+m,where ξi,ζi,ui∈R.Definition 7 (see [43]).
Couple-group consensus for second-order multiagent system (30) is asymptotically solved if the states of agents in G satisfy limt→∞ξi(t)-ξj(t)=0; limt→∞ζi(t)-ζj(t)=0, ∀i,j∈P1, and limt→∞ξi(t)-ξj(t)=0; limt→∞ζi(t)-ζj(t)=0, ∀i,j∈P2.


### 4.1. Without Communication Delays

Similar to the first-order multiagent systems, the broadcast states are described byξ^i(t)=ξi(tki) and ζ^i(t)=ζi(tki), t∈[tki,tk+1i). Then an event-based control strategy is proposed as follows:(31)
u
i
t
=
∑
j
∈
N
1
i
a
i
j
α
ξ
^
j
t
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ξ
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i
t
+
β
ζ
^
j
t
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ζ
^
i
t
+
∑
j
∈
N
2
i
a
i
j
α
ξ
^
j
t
+
β
ζ
^
j
t
,
i
∈
P
1
;
∑
j
∈
N
2
i
a
i
j
α
ξ
^
j
t
-
ξ
^
i
t
+
β
ζ
^
j
t
-
ζ
^
i
t
+
∑
j
∈
N
1
i
a
i
j
α
ξ
^
j
t
+
β
ζ
^
j
t
,
i
∈
P
2
,where α>0, β>0, aij≥0, ∀i,j∈P1, and ∀i,j∈P2; otherwise, aij∈R.We define the measurement errorseξ(t)=ξ^(t)-ξ(t) and eζ(t)=ζ^(t)-ζ(t). This yields u(t)=-αLξ(t)-αLeξ(t)-βLζ(t)-βLeζ(t). The matrix form of (30) with protocol (31) is given as(32)ξ˙tζ˙t=Γξtζt+Ωeξteζt,where (33)Γ=0n+m×n+mIn+m×n+m-αL-βL,Ω=0n+m×n+m0n+m×n+m-αL-βL.Letξ(t)=ξ1(t)ξ2(t)=b11nb21m+c11nc21mt+ϕξ1(t)ϕξ2(t) and ζ(t)=c11nc21m+ϕζ1(t)ϕζ2(t). Denote ϕξ(t)=[ϕξ1T(t),ϕξ2T(t)]T, ϕζ(t)=[ϕζ1T(t),ϕζ2T(t)]T, e(t)=[eξT(t),eζT(t)]T, and ϕ(t)=[ϕξT(t),ϕζT(t)]T. The disagreement dynamics is given by(34)ϕ˙t=Γϕt+Ωet.In this section, we adopt the following event-triggered function [20]:(35)feit=eit-c2e-ϱ2t,where ei(t)=eξi(t)eζi(t), c2>0.Theorem 8.
Under assumptionsA1 and A2, multiagent system (30) reaches couple-group consensus asymptotically with protocol (31) and distributed event-triggered function (35) if L has two-simple zero eigenvalues and the rest of nonzero eigenvalues are positive together with ∀α>0, β>0, where the parameter ϱ2 satisfies 0<ϱ2<R(λ5(Γ)), 0=λ1,2,3,4Γ>Rλ5Γ≥⋯R(λ2(n+m)(Γ)). Furthermore, the system does not exhibit Zeno behavior for any initial conditions ξ(0)∈Rn+m and ζ(0)∈Rn+m.Proof.
From (34), the following analytical solution is obtained:(36)ϕt=eΓtϕ0+∫0teΓt-sΩesds.According to the event-triggered function (35) and the exponential estimation of solutions, it follows that(37)ϕt≤eRλ5Γtcϕϕ0+∫0teRλ5Γt-scϕΩesds,≤eRλ5Γtcϕϕ0+n+mcϕc22L∫0teRλ5Γt-se-ϱ2sds,≤eRλ5Γtcϕϕ0+n+mcϕc22LeRλ5Γt-e-ϱ2tRλ5Γ+ϱ2,where cϕ is a constant. Because 0<ϱ2<R(λ5(Γ)), limt→∞ϕ(t)=0. Then couple-group consensus of system (30) is asymptotically reached.
Next, we will show that Zeno behavior is excluded. Assume that the agents trigger at timet∗≥0 and e(t∗)=0, for t≥t∗. Before the next trigger time, the following inequality holds: (38)et=eξteζt≤∫t∗te˙sds=∫t∗teζs-usds≤∫t∗tesusds≤∫t∗tes+usds.
From (31), (35), and (37), we obtain (39)ut≤-αLξt-βLζt-αLeξt-βLeζt=-αLϕξt-βLϕζt-αLeξt-βLeζt≤α2+β2Lϕt+et,et≤n+mc2e-ϱ2t,ϕt≤k1eRλ5Γt+k2e-ϱ2t,where k1=cϕϕ(0)+n+mcϕc22L/R(λ5(Γ))+ϱ2and k2=n+mcϕc22L/R(λ5(Γ))+ϱ2. Then it can be deduced that(40)et≤∫t∗tk1∗eRλ5Γs+k2∗e-ϱ2sds,where k1∗=α2+β2Lk1 and k2∗=α2+β2L(k2+n+mc2)+1. The next event will not be triggered until (t-t∗)(k1∗eR(λ5(Γ))t∗+k2∗e-ϱ2t∗)=n+mc2e-ϱ2t. Then we can calculate the trigger time interval ς=t-t∗ from (k1∗eR(λ5(Γ))+ϱ2t∗)+k2∗)ς=n+mc2e-ϱ2ς, which implies that Zeno behavior is excluded for the lower bound of the interevent times ς.


### 4.2. With Communication Delays

In this subsection, we analyze couple-group consensus for second-order multiagent systems with communication delays via event-triggered control.Assume that the communication delay between any two neighbors isd(t), satisfying 0<d(t)<h, h>0, and t>0. The event-based couple-group consensus protocol is given as(41)
u
i
t
=
∑
j
∈
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1
i
a
i
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ξ
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j
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ξ
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β
ζ
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t
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a
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ξ
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ζ
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j
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t
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i
∈
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1
;
∑
j
∈
N
2
i
a
i
j
α
ξ
^
j
t
-
d
t
-
ξ
^
i
t
-
d
t
+
β
ζ
^
j
t
-
d
t
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ζ
^
i
t
-
d
t
+
∑
j
∈
N
1
i
a
i
j
α
ξ
^
j
t
-
d
t
+
β
ζ
^
j
t
-
d
t
,
i
∈
P
2
,where α>0, β>0, aij≥0, ∀i,j∈P1, and ∀i,j∈P2; otherwise, aij∈R.Definingeξ(t) and eζ(t) as the measurement errors, system (30) with protocol (41) can be written as(42)ξ˙tζ˙t=Ψξtζt+Υ1ξt-dtζt-dt+Υ1eξt-dteζt-dt.With the definition of ϕ(t), the system (42) is further given as(43)ϕ˙t=Ψϕt+Υ1ϕt-dt+Υ1et-dt,where (44)Ψ=0n+m×n+mIn+m×n+m0n+m×n+m0n+m×n+m,Υ1=0n+m×n+m0n+m×n+m-αL-βL.Theorem 9.
Under assumptionsA1 and A2, multiagent system (30) with protocol (41) and event-triggered function (35) asymptotically reaches couple-group consensus under bounded communication delay 0<d(t)<h for ∀α>0, β>0 if L has two-simple zero eigenvalues while all the other nonzero eigenvalues are positive and there exist matrices P¯1>0, R¯1>0, Q1, and Y1 and constants ϵ1, μ1>0, η1>0, and μ1+η1>2ϱ2, satisfying the following matrix inequality:(45)Ξ11Ξ12hΥ1R¯Υ10n+m×n+m∗-ϵ1QT-ϵ1QhϵΥ1R¯ϵΥ1hQ1T∗∗-hR¯11-hμ1+η10n+m×n+m0n+m×n+m∗∗∗-η1In+m×n+m0n+m×n+m∗∗∗∗-hR¯1<0,where Ξ11=(μ1+η1)P¯1+Y1+Y1T and Ξ12=P¯1-Q1+ϵ1Y1T. Furthermore, the system does not exhibit Zeno behavior for all initial conditions ξ(0)∈Rn+m and ζ(0)∈Rn+m.The proof is similar to that of Theorem6 and thus is omitted here.


## 4.1. Without Communication Delays

Similar to the first-order multiagent systems, the broadcast states are described byξ^i(t)=ξi(tki) and ζ^i(t)=ζi(tki), t∈[tki,tk+1i). Then an event-based control strategy is proposed as follows:(31)
u
i
t
=
∑
j
∈
N
1
i
a
i
j
α
ξ
^
j
t
-
ξ
^
i
t
+
β
ζ
^
j
t
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ζ
^
i
t
+
∑
j
∈
N
2
i
a
i
j
α
ξ
^
j
t
+
β
ζ
^
j
t
,
i
∈
P
1
;
∑
j
∈
N
2
i
a
i
j
α
ξ
^
j
t
-
ξ
^
i
t
+
β
ζ
^
j
t
-
ζ
^
i
t
+
∑
j
∈
N
1
i
a
i
j
α
ξ
^
j
t
+
β
ζ
^
j
t
,
i
∈
P
2
,where α>0, β>0, aij≥0, ∀i,j∈P1, and ∀i,j∈P2; otherwise, aij∈R.We define the measurement errorseξ(t)=ξ^(t)-ξ(t) and eζ(t)=ζ^(t)-ζ(t). This yields u(t)=-αLξ(t)-αLeξ(t)-βLζ(t)-βLeζ(t). The matrix form of (30) with protocol (31) is given as(32)ξ˙tζ˙t=Γξtζt+Ωeξteζt,where (33)Γ=0n+m×n+mIn+m×n+m-αL-βL,Ω=0n+m×n+m0n+m×n+m-αL-βL.Letξ(t)=ξ1(t)ξ2(t)=b11nb21m+c11nc21mt+ϕξ1(t)ϕξ2(t) and ζ(t)=c11nc21m+ϕζ1(t)ϕζ2(t). Denote ϕξ(t)=[ϕξ1T(t),ϕξ2T(t)]T, ϕζ(t)=[ϕζ1T(t),ϕζ2T(t)]T, e(t)=[eξT(t),eζT(t)]T, and ϕ(t)=[ϕξT(t),ϕζT(t)]T. The disagreement dynamics is given by(34)ϕ˙t=Γϕt+Ωet.In this section, we adopt the following event-triggered function [20]:(35)feit=eit-c2e-ϱ2t,where ei(t)=eξi(t)eζi(t), c2>0.Theorem 8.
Under assumptionsA1 and A2, multiagent system (30) reaches couple-group consensus asymptotically with protocol (31) and distributed event-triggered function (35) if L has two-simple zero eigenvalues and the rest of nonzero eigenvalues are positive together with ∀α>0, β>0, where the parameter ϱ2 satisfies 0<ϱ2<R(λ5(Γ)), 0=λ1,2,3,4Γ>Rλ5Γ≥⋯R(λ2(n+m)(Γ)). Furthermore, the system does not exhibit Zeno behavior for any initial conditions ξ(0)∈Rn+m and ζ(0)∈Rn+m.Proof.
From (34), the following analytical solution is obtained:(36)ϕt=eΓtϕ0+∫0teΓt-sΩesds.According to the event-triggered function (35) and the exponential estimation of solutions, it follows that(37)ϕt≤eRλ5Γtcϕϕ0+∫0teRλ5Γt-scϕΩesds,≤eRλ5Γtcϕϕ0+n+mcϕc22L∫0teRλ5Γt-se-ϱ2sds,≤eRλ5Γtcϕϕ0+n+mcϕc22LeRλ5Γt-e-ϱ2tRλ5Γ+ϱ2,where cϕ is a constant. Because 0<ϱ2<R(λ5(Γ)), limt→∞ϕ(t)=0. Then couple-group consensus of system (30) is asymptotically reached.
Next, we will show that Zeno behavior is excluded. Assume that the agents trigger at timet∗≥0 and e(t∗)=0, for t≥t∗. Before the next trigger time, the following inequality holds: (38)et=eξteζt≤∫t∗te˙sds=∫t∗teζs-usds≤∫t∗tesusds≤∫t∗tes+usds.
From (31), (35), and (37), we obtain (39)ut≤-αLξt-βLζt-αLeξt-βLeζt=-αLϕξt-βLϕζt-αLeξt-βLeζt≤α2+β2Lϕt+et,et≤n+mc2e-ϱ2t,ϕt≤k1eRλ5Γt+k2e-ϱ2t,where k1=cϕϕ(0)+n+mcϕc22L/R(λ5(Γ))+ϱ2and k2=n+mcϕc22L/R(λ5(Γ))+ϱ2. Then it can be deduced that(40)et≤∫t∗tk1∗eRλ5Γs+k2∗e-ϱ2sds,where k1∗=α2+β2Lk1 and k2∗=α2+β2L(k2+n+mc2)+1. The next event will not be triggered until (t-t∗)(k1∗eR(λ5(Γ))t∗+k2∗e-ϱ2t∗)=n+mc2e-ϱ2t. Then we can calculate the trigger time interval ς=t-t∗ from (k1∗eR(λ5(Γ))+ϱ2t∗)+k2∗)ς=n+mc2e-ϱ2ς, which implies that Zeno behavior is excluded for the lower bound of the interevent times ς.


## 4.2. With Communication Delays

In this subsection, we analyze couple-group consensus for second-order multiagent systems with communication delays via event-triggered control.Assume that the communication delay between any two neighbors isd(t), satisfying 0<d(t)<h, h>0, and t>0. The event-based couple-group consensus protocol is given as(41)
u
i
t
=
∑
j
∈
N
1
i
a
i
j
α
ξ
^
j
t
-
d
t
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ξ
^
i
t
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d
t
+
β
ζ
^
j
t
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t
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ζ
^
i
t
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t
+
∑
j
∈
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2
i
a
i
j
α
ξ
^
j
t
-
d
t
+
β
ζ
^
j
t
-
d
t
,
i
∈
P
1
;
∑
j
∈
N
2
i
a
i
j
α
ξ
^
j
t
-
d
t
-
ξ
^
i
t
-
d
t
+
β
ζ
^
j
t
-
d
t
-
ζ
^
i
t
-
d
t
+
∑
j
∈
N
1
i
a
i
j
α
ξ
^
j
t
-
d
t
+
β
ζ
^
j
t
-
d
t
,
i
∈
P
2
,where α>0, β>0, aij≥0, ∀i,j∈P1, and ∀i,j∈P2; otherwise, aij∈R.Definingeξ(t) and eζ(t) as the measurement errors, system (30) with protocol (41) can be written as(42)ξ˙tζ˙t=Ψξtζt+Υ1ξt-dtζt-dt+Υ1eξt-dteζt-dt.With the definition of ϕ(t), the system (42) is further given as(43)ϕ˙t=Ψϕt+Υ1ϕt-dt+Υ1et-dt,where (44)Ψ=0n+m×n+mIn+m×n+m0n+m×n+m0n+m×n+m,Υ1=0n+m×n+m0n+m×n+m-αL-βL.Theorem 9.
Under assumptionsA1 and A2, multiagent system (30) with protocol (41) and event-triggered function (35) asymptotically reaches couple-group consensus under bounded communication delay 0<d(t)<h for ∀α>0, β>0 if L has two-simple zero eigenvalues while all the other nonzero eigenvalues are positive and there exist matrices P¯1>0, R¯1>0, Q1, and Y1 and constants ϵ1, μ1>0, η1>0, and μ1+η1>2ϱ2, satisfying the following matrix inequality:(45)Ξ11Ξ12hΥ1R¯Υ10n+m×n+m∗-ϵ1QT-ϵ1QhϵΥ1R¯ϵΥ1hQ1T∗∗-hR¯11-hμ1+η10n+m×n+m0n+m×n+m∗∗∗-η1In+m×n+m0n+m×n+m∗∗∗∗-hR¯1<0,where Ξ11=(μ1+η1)P¯1+Y1+Y1T and Ξ12=P¯1-Q1+ϵ1Y1T. Furthermore, the system does not exhibit Zeno behavior for all initial conditions ξ(0)∈Rn+m and ζ(0)∈Rn+m.The proof is similar to that of Theorem6 and thus is omitted here.


## 5. Simulations and Analysis

In this section, several examples are given to verify the effectiveness of the main results.Example 1.
Consider a multiagent team with five agents, and each agent is modeled by system (4). The communication topology is given by Figure 1 with initial state x(0)=[-5,2,3,-8,1]T. Agent 1, agent 2, and agent 3 are in one group while agent 4 and agent 5 are in another group. From the definition of Laplacian matrix, we can easily obtain the Laplacian matrix L. The eigenvalues of L are λ1,2=0, λ3=2, λ4=3.55, and λ5=8.45, which implies that L has two-simple zero eigenvalues and all the other nonzero eigenvalues are positive. By tuning c1=[0.12,0.19,0.2,0.1,0.2] and ϱ1=[1.2,1.35,1.4,1.3,1.4], which are the parameters in event-triggered function (9), the trajectory and event times of every agent under event-triggered function (9) are shown by Figures 2 and 3, respectively. The evolution of ei(t) is plotted in Figure 4. It is clear that system (8) achieves couple-group consensus and the event-triggered scheme reduces the number of actuator updating to a great extent.Figure 1
Topology 1.Figure 2
The state trajectories of five agents without communication delay.Figure 3
The event times of five agents without communication delay.Figure 4
Evolution of the measurement error absoluteei(t) with event-triggered scheme (9).Example 2.
Consider a multiagent system that has the same topology, initial state, and group division as that of Example1. To solve the inequality in Theorem 6, we choose ϵ=0.2, η=3.2, and μ=0.0001, satisfying μ+η>2ϱ1>0, and then the feasible solution is given as (46)P¯=108×0.67730.42380.2510-0.20270.20130.42380.86620.01420.0881-0.10380.25100.01421.13450.0883-0.0800-0.20270.08810.08831.12740.20990.2013-0.1038-0.08000.20991.1454,R¯=108×1.85671.80221.5618-0.17650.16341.80222.56410.91500.3275-0.33221.56180.91502.5323-0.19480.1877-0.17650.3275-0.19482.73201.83140.1634-0.33220.18771.83142.7118,and rmax=0.18s is obtained. When r=0.1s<rmax, the trajectories of all the agents are shown by Figure 5. It is clear that the couple-group consensus is achieved when r<rmax.Figure 5
The state trajectories of five agents with communication delayr=0.1s.Example 3.
Consider multiagent system (30) with five agents. The communication topology is given by Figure 6 with initial state ξ(0)=[-5,1,2,-1,6]T and ζ(0)=[0,0.3,0.2,0.8,0.4]T. Agent 1, agent 2, and agent 3 are in one group while agent 4 and agent 5 are in another group. The eigenvalues of the Laplacian matrix are μ1,2=0, μ3=1.2984, μ4=3, and μ5=7.7016, and the eigenvalues of Γ are λ1,2,3,4=0, λ5=-6.5204, λ6=-1.1811, λ7,8=-1.5±0.866i, and λ9,10=-0.6492±0.9365i. It is obvious that Γ has a zero eigenvalue whose algebraic multiplicity is four and the real parts of all nonzero eigenvalues are negative. Choose α=1 and β=1 as the parameter values of second-order couple-group consensus protocol (31). By tuning c2=[0.28,0.2,0.19,0.28,0.3] and ϱ2=[0.246,0.241,0.227,0.246,0.192], which are the parameter values of event-triggered function (35), the trajectories of position, velocity, and event times of all agents under event-triggered control protocol (31) are shown by Figures 7 and 8, respectively. The evolution of ei(t) is plotted by Figure 9. It is clear that system (34) achieves couple-group consensus and the event-triggered scheme reduces the number of actuator updating to a great extent. This is consistent with the theoretical results.Figure 6
Topology 3.Figure 7
The position and velocity trajectories of five agents without communication delay.Figure 8
The event times of five agents without communication delays.Figure 9
Evolution of the measurement error normei(t) with event-triggered scheme (35).Example 4.
Consider a multiagent system that has the same topology, initial state, and group division as that of Example3. To solve the inequality in Theorem 9, we choose ϵ1=0.1, η1=3.6, and μ1=0.0002, satisfying μ1+η1>2ϱ2>0. The feasible solution is given as (47)P¯=103×1.72890.34630.1825-0.03990.0399-0.93280.34700.1832-0.04080.04080.34631.83180.07960.0609-0.06090.3470-0.83050.08090.0613-0.06130.18250.07961.9956-0.02100.02100.18320.0809-0.6667-0.02050.0205-0.03990.0609-0.02101.90100.3567-0.04080.0613-0.0205-0.75950.35690.0399-0.06090.02100.35671.90100.0408-0.06130.02050.3569-0.7595-0.93280.34700.1832-0.04080.04081.73280.34150.1834-0.03690.03690.3470-0.83050.08090.0613-0.06130.34151.83290.08330.0580-0.05800.18320.0809-0.6667-0.02050.02050.18340.08331.9909-0.02110.0211-0.04080.0613-0.0205-0.75950.3569-0.03690.0580-0.02111.90090.35680.0408-0.06130.02050.3569-0.75950.0369-0.05800.02110.35681.9009,R¯=103×4.02630.47360.2916-0.03180.0318-3.04030.46880.2831-0.03260.03260.47364.14690.17090.0614-0.06140.4688-2.91730.16010.0627-0.06270.29160.17094.3290-0.02960.02960.28310.1601-2.7316-0.03010.0301-0.03180.0614-0.02964.24940.5421-0.03260.0627-0.0301-2.81980.53130.0318-0.06140.02960.54214.24940.0326-0.06270.03010.5313-2.8198-3.04030.46880.2831-0.03260.03264.05620.43380.2751-0.02510.02510.4688-2.91730.16010.0627-0.06270.43384.16270.16860.0522-0.05220.28310.1601-2.7316-0.03010.03010.27510.16864.3214-0.02710.0271-0.03260.0627-0.0301-2.81980.5313-0.02510.0522-0.02714.25850.50660.0326-0.06270.03010.5313-2.81980.0251-0.05220.02710.50664.2585,and hmax=0.21s is obtained. When h=0.135s<hmax, the trajectories of all the agents are shown by Figure 10. It is clear that the couple-group consensus is achieved when h<hmax.Figure 10
The state trajectories of five agents with communication delayh=0.135s.


## 6. Conclusion

In this paper, the couple-group consensus problems of first-order and second-order multiagent systems with undirected topology via time-dependent event-triggered control protocols have been considered. Based on algebraic graph theory and matrix theory, the conditions for the cases without delays have been established to ensure couple-group consensus. For the cases with communication delays, the Lyapunov method has been used to guarantee the stability of the disagreement dynamics. The presence of time-dependent event-triggered function has saved energy consumption to a great extent. In addition, Zeno behavior has been excluded in all of those cases. In future work, we will focus on couple-group consensus of multiagent systems with directed topologies.



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*Source: 2947643-2018-09-09.xml*