# Bipartite Consensus of Heterogeneous Multiagent Systems Based on Distributed Event-Triggered Control

Authors: Xiaoyu Wang; Kaien Liu; Zhijian Ji; Shitao Han
Journal: Complexity
 (2020)

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## Abstract

In this paper, the bipartite consensus problem of heterogeneous multiagent systems composed of first-order and second-order agents is considered by utilizing the event-triggered control scheme. Under structurally balanced directed topology, event-triggered bipartite consensus protocol is put forward, and event-triggering functions consisting of measurement error and threshold are designed. To exclude Zeno behavior, an exponential function is introduced in the threshold. The bipartite consensus problem is transformed into the corresponding stability problem by means of gauge transformation and model transformation. By virtue of Lyapunov method, sufficient conditions for systems without input delay are obtained to guarantee bipartite consensus. Furthermore, for the case with input delay, sufficient conditions which include an admissible upper bound of the delay are obtained to guarantee bipartite consensus. Finally, numerical simulations are provided to illustrate the effectiveness of the obtained theoretical results.


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## Body

## 1. Introduction

Due to the wide applications on multirobot collaboration, cooperative control of unmanned aerial vehicles, and attitude alignment of satellites, many researchers have devoted to the study of coordination control of multiagent systems [1–4]. As one of the fundamental problems for distributed coordination in multiagent systems, the consensus problem has attracted much attention from multidisciplinary researchers due to its theoretical and practical significance. Consensus means states of all agents can reach a common value under an appropriate control protocol. During the past decade, a vast amount of theoretical achievements have been made on consensus problems [5–15]. Among them, when the single final consensus state cannot satisfy the control requirement with the increase of system size and complexity, group consensus, which means states of different groups of agents may approach to different values, has been put forward [13–15]. In [13], Yu and Wang studied the group consensus problem with both switching topologies and transmission delays. In [14], Altafini first proposed a sufficient and necessary condition for first-order multiagent systems to achieve bipartite consensus. In [15], Jiang et al. extended the results in [14] to general linear multiagent systems with signs.Due to the complexity of real systems, it is possible for agents to have different dynamics. Hence, it is meaningful to study heterogeneous multiagent systems composed of first-order and second-order agents [16–20]. In [16], based on the properties of nonnegative matrices, sufficient consensus criterion was obtained for the agents with bounded communication delays under fixed topology and switching topologies, respectively. In [17], the consensus problem for continuous-time heterogeneous multiagent systems under directed graphs was investigated. In [18], by designing novel consensus protocols for continuous and discrete heterogeneous multiagent systems under directed topology, Liu et al. proved that the corresponding system asymptotically achieves consensus if and only if the fixed directed topology contains a directed spanning tree. In [19], sufficient and necessary condition for consensus of heterogeneous multiagent systems was given under certain assumptions on control parameters. In [20], some sufficient group consensus conditions which depend on input delays and control parameters were obtained for heterogeneous multiagent systems by using the frequency-domain analysis method and matrix theory.However, all the control schemes in the aforementioned works depend on continuous or periodic communication among agents. In practical systems, as the communication bandwidth and system energy are usually limited, it is a waste of energy to communicate with others when it is unnecessary. In order to reduce the frequency of communication controller updates, the event-triggered scheme has been developed [21–33]. The basic idea of the event-triggered control is to replace the paradigm of periodic sampling (or continuous control) by aperiodic sampling. In [21], Dimarogonas et al. proposed distributed event-triggered consensus strategies for first-order multiagent systems, whose updates depended on the ratio of a certain measurement error with respect to the norm of a function of the states. Different from [21], event-triggering conditions based on sampled data were designed in [22–25], where energy can be saved by checking the event condition only at periodic sampling instants. In [23], Fan et al. proposed a self-triggered consensus protocol to further reduce the amount of state sampling. In [25], Liu et al. investigated the periodic event-triggered consensus problem of multiagent systems under directed topology which contains a spanning tree. In [26], Xie et al. put forward a novel event-triggered control strategy for second-order multiagent systems. In [27, 28], event-triggered consensus of multiagent systems with general linear dynamics was considered. Event-triggered consensus of multiagent systems with nonlinear dynamics and directed network topology was investigated in [29]. In [30], by using a distributed event-triggered control strategy, Tan et al. addressed the mean square consensus problem of leader-following stochastic multiagent systems with input delay. In [31], the bipartite consensus problem for first-order multiagent systems via the event-triggered control was investigated. In [32], Yin et al. considered the event-triggered consensus of discrete heterogeneous multiagent systems and designed event-triggering functions by using the positions and velocities of agents separately under a directed topology. After that, Yin et al. investigated event-triggered discrete-time heterogeneous multiagent systems with random communication delays represented by a Markov chain in [33].Motivated by the aforementioned works, the distributed event-triggered scheme is applied to solve the bipartite consensus problem of heterogeneous multiagent systems in this paper. Different from [32], instead of designing the event-triggering function for the second-order agents by using positions and velocities of agents separately, we introduce a variable yi=xi+1/αvi in the event-triggering function, which has special use in analysis. Moreover, variable yi is also used in designing control protocol, which can weaken the requirement of control gain. Compared with the threshold designed in [32, 33], where each agent still needs to continuously monitor its neighbors’ states, the threshold in this paper only depends on the latest event-triggered states of agents. To exclude Zeno behavior, we introduce an exponential function in the threshold. As the input delay between the controller and the actuator is ubiquitous in practical systems, the consensus problem of multiagent systems with input delay has been investigated in [10, 20, 30]. To the best of our knowledge, there are no results concerning bipartite consensus of heterogeneous multiagent systems with input delay under the event-triggered scheme. Therefore, we also consider the case with input delay in this study.The paper is organized as follows. The graph theory and the problem formulation are given in Section2. Some important definitions and lemmas are also presented in Section 2. The main results of this paper are given in Section 3. Both cases without input delay and with input delay are investigated under topology which contains a directed spanning tree. Numerical examples are given in Section 4 to validate the effectiveness of the obtained results. Finally, conclusions are given in Section 5.Notations:ℝ and ℕ represent the set of real and natural numbers, respectively. ℝn and ℝn×m denote n-dimensional real vector space and the n × m real matrix space, respectively. 1n (0n) denotes a column vector with all 1(0) elements. 0 indicates a zero matrix with a proper order. ℐ=1,…,n indicates an index set.Notation diag{b1, …, bN} denotes a diagonal matrix. For a symmetric matrix P, P>0 means P is positive definite. λ (B) represents the eigenvalue of matrix B. Re (λ(B)) represents the real part of λ (B). λmax (B) and λmin (B) represent the maximum eigenvalue and the minimum eigenvalue of matrix B, respectively. |⋅| and ⋅ denote 1-norm and Euclidean norm, respectively, both for vectors and matrices.


## 2. Preliminaries

In this section, we first give a brief review of graph theory and introduce some lemmas which will be used. Then, the model is formulated.


### 2.1. Graph Theory

A weighted directed graph (digraph)G=V,ℰ,A consists of a node set V=v1,…,vn, an edge set ℰ⊆V×V, and a weighted adjacency matrix A=aij∈ℝn×n satisfying aij ≠ 0 iff vi,vj∈ℰ. An edge eij=vi,vj∈ℰ means that node vi can receive information from node vj. We assume that vi,vi∉ℰ; hence, aii = 0 for all i∈ℐ. Define node vj as a neighbor of node vi if aij ≠ 0. The set of neighbors of node vi is denoted by Ni=vj∈V:eij∈ℰ. The in-degree of node vi is defined as Λii=∑j=1naij, i∈ℐ. The Laplacian matrix L of a weighted digraph G is defined as L = Λ − A, where Λ = diag{Λ11, Λ22, …, Λnn}. A directed path from vj to vi is a finite ordered sequence of distinct edges of G with the form vi,vk1,vk1,vk2,…,vkl,vj. A digraph contains a directed spanning tree if there exists a node called the root node such that there exists a directed path from it to every other node.Definition 1.
(structural balance, see [14]). A digraph G is said to be structurally balanced if all the nodes of G can be partitioned into two nonempty subsets V1 and V2 such that V1∪V2=V,V1∩V2=∅, and the following two conditions hold:(1)
aij≥0,∀vi,vj∈Vq, where q ∈{1, 2}(2)
aij≤0,∀vi∈Vq,vj∈Vr, where q ≠ r and q, r ∈{1, 2}Definition 2.
(gauge transformation, see [14]). Gauge transformation is a change of orthant order in ℝn performed by an orthogonal matrix D which is defined as D∈D≜diagσ1,…,σn,σi∈1,−1.
Obviously,D = D−1. When the digraph is structurally balanced, the entries of DAD can be guaranteed to be nonnegative by selecting appropriate D. Consequently, the nondiagonal entries of DLD are nonpositive, and the sum of each row is zero.
The following are some lemmas which will be used in Section 3.Lemma 1 (see [5]).
Given a matrixB=bij∈ℝn×n, where bii ≥ 0, bij ≤ 0 for ∀i ≠ j, and ∑j=1nbij=0 for ∀i, B has at least one zero eigenvalue, and all of the nonzero eigenvalues are in the open right half plane. Furthermore, B has exactly one simple zero eigenvalue if and only if the digraph associated with B contains a directed spanning tree.Lemma 2 (Young’s inequality, see [34]).
Givenx,y∈ℝ, for ∀ϵ∈ℝ>0, xy≤ϵx2/2+y2/2ϵ.Lemma 3 (comparison principle, see [35]).
Consider a differential equationdu/dt=ft,u,t≥t0, where f (t, u) is continuous and satisfies local Lipschitz condition in u. Let [t0, T) be the maximum existence interval of the solution u (t), where T can be infinite. If, for ∀t ∈ [t0, T), v = vt satisfies dv/dt≤ft,v,t≥t0, and vt0 ≤ u (t0), then vt ≤ u (t), t ∈ [t0, T).Lemma 4 (see [36]).
Assuming that all the eigenvalues of matrixP1 are in the open left half plane, then for all t ≥ 0, it holds that(1)eP1t≤βe−rt,where β ≥ 1 and r is any positive constant which is smaller than Re (λmin (−P1)).


### 2.2. Problem Formulation

Suppose that the considered heterogeneous multiagent system has a communication topology represented by a digraphG, which means each agent is regarded as a node of the interaction digraph. Assume that the system is composed of m (1 ≤ m < n) agents with second-order dynamics and n − m agents with first-order dynamics. Without loss of generality, we assume that agents v1 to vm are second-order agents, whose index set is denoted as ℐ1=1,2,…,m, and agents vm+1 to vn are first-order agents, whose index set is denoted as ℐ2=m+1,m+2,…,n.Assumption 1.
The communication topology is structural balanced with the following groups: the second-order agents constituteV1 and the first-order agents constitute V2.
The dynamics of the system are described as(2)x˙it=vit,v˙it=uit,i∈ℐ1,x˙it=uit,i∈ℐ2,t≥0,where xi, vi, and ui∈ℝ are the position, velocity, and control input of the ith agent, respectively. Based on the event-triggered control scheme, each agent broadcasts its state information at the event-triggering time instants. The sequence of event-triggering time instants of agent vi is denoted as tmi,m∈ℕ, with t0i=0.
The definition of bipartite consensus of (2) is given as follows.Definition 3.
Bipartite consensus of system (2) can be achieved for any initial conditions if it holds that(3)limt⟶∞xit−xjt=0,limt⟶∞vit=vjt=0,∀i,j∈ℐ1,limt⟶∞xit−xjt=0,∀i,j∈ℐ2,limt⟶∞xit+xjt=0,∀i∈ℐ1,j∈ℐ2.


## 2.1. Graph Theory

A weighted directed graph (digraph)G=V,ℰ,A consists of a node set V=v1,…,vn, an edge set ℰ⊆V×V, and a weighted adjacency matrix A=aij∈ℝn×n satisfying aij ≠ 0 iff vi,vj∈ℰ. An edge eij=vi,vj∈ℰ means that node vi can receive information from node vj. We assume that vi,vi∉ℰ; hence, aii = 0 for all i∈ℐ. Define node vj as a neighbor of node vi if aij ≠ 0. The set of neighbors of node vi is denoted by Ni=vj∈V:eij∈ℰ. The in-degree of node vi is defined as Λii=∑j=1naij, i∈ℐ. The Laplacian matrix L of a weighted digraph G is defined as L = Λ − A, where Λ = diag{Λ11, Λ22, …, Λnn}. A directed path from vj to vi is a finite ordered sequence of distinct edges of G with the form vi,vk1,vk1,vk2,…,vkl,vj. A digraph contains a directed spanning tree if there exists a node called the root node such that there exists a directed path from it to every other node.Definition 1.
(structural balance, see [14]). A digraph G is said to be structurally balanced if all the nodes of G can be partitioned into two nonempty subsets V1 and V2 such that V1∪V2=V,V1∩V2=∅, and the following two conditions hold:(1)
aij≥0,∀vi,vj∈Vq, where q ∈{1, 2}(2)
aij≤0,∀vi∈Vq,vj∈Vr, where q ≠ r and q, r ∈{1, 2}Definition 2.
(gauge transformation, see [14]). Gauge transformation is a change of orthant order in ℝn performed by an orthogonal matrix D which is defined as D∈D≜diagσ1,…,σn,σi∈1,−1.
Obviously,D = D−1. When the digraph is structurally balanced, the entries of DAD can be guaranteed to be nonnegative by selecting appropriate D. Consequently, the nondiagonal entries of DLD are nonpositive, and the sum of each row is zero.
The following are some lemmas which will be used in Section 3.Lemma 1 (see [5]).
Given a matrixB=bij∈ℝn×n, where bii ≥ 0, bij ≤ 0 for ∀i ≠ j, and ∑j=1nbij=0 for ∀i, B has at least one zero eigenvalue, and all of the nonzero eigenvalues are in the open right half plane. Furthermore, B has exactly one simple zero eigenvalue if and only if the digraph associated with B contains a directed spanning tree.Lemma 2 (Young’s inequality, see [34]).
Givenx,y∈ℝ, for ∀ϵ∈ℝ>0, xy≤ϵx2/2+y2/2ϵ.Lemma 3 (comparison principle, see [35]).
Consider a differential equationdu/dt=ft,u,t≥t0, where f (t, u) is continuous and satisfies local Lipschitz condition in u. Let [t0, T) be the maximum existence interval of the solution u (t), where T can be infinite. If, for ∀t ∈ [t0, T), v = vt satisfies dv/dt≤ft,v,t≥t0, and vt0 ≤ u (t0), then vt ≤ u (t), t ∈ [t0, T).Lemma 4 (see [36]).
Assuming that all the eigenvalues of matrixP1 are in the open left half plane, then for all t ≥ 0, it holds that(1)eP1t≤βe−rt,where β ≥ 1 and r is any positive constant which is smaller than Re (λmin (−P1)).


## 2.2. Problem Formulation

Suppose that the considered heterogeneous multiagent system has a communication topology represented by a digraphG, which means each agent is regarded as a node of the interaction digraph. Assume that the system is composed of m (1 ≤ m < n) agents with second-order dynamics and n − m agents with first-order dynamics. Without loss of generality, we assume that agents v1 to vm are second-order agents, whose index set is denoted as ℐ1=1,2,…,m, and agents vm+1 to vn are first-order agents, whose index set is denoted as ℐ2=m+1,m+2,…,n.Assumption 1.
The communication topology is structural balanced with the following groups: the second-order agents constituteV1 and the first-order agents constitute V2.
The dynamics of the system are described as(2)x˙it=vit,v˙it=uit,i∈ℐ1,x˙it=uit,i∈ℐ2,t≥0,where xi, vi, and ui∈ℝ are the position, velocity, and control input of the ith agent, respectively. Based on the event-triggered control scheme, each agent broadcasts its state information at the event-triggering time instants. The sequence of event-triggering time instants of agent vi is denoted as tmi,m∈ℕ, with t0i=0.
The definition of bipartite consensus of (2) is given as follows.Definition 3.
Bipartite consensus of system (2) can be achieved for any initial conditions if it holds that(3)limt⟶∞xit−xjt=0,limt⟶∞vit=vjt=0,∀i,j∈ℐ1,limt⟶∞xit−xjt=0,∀i,j∈ℐ2,limt⟶∞xit+xjt=0,∀i∈ℐ1,j∈ℐ2.


## 3. Consensus Analysis

For system (2), we propose the following bipartite consensus protocol:(4)uit=−αvitmi−τ+α2∑j∈Ni1aijxjtm∗j−τ−yitmi−τ−∑j∈Ni2aijyitmi−τ+xjtm∗j−τ,i∈ℐ1,uit=α∑j∈Ni2aijxjtm∗j−τ−xitmi−τ−∑j∈Ni1aijxitmi−τ+xjtm∗j−τ,i∈ℐ2,t≥τ,where yit−τ=xit−τ+1/αvit−τ,t∈tmi,tm+1i,i∈ℐ1, τ ≥ 0 represents the input delay, tm∗j≜argminm∈ℕt−tmjtmj≤t, α > 0 is the control gain, and Ni1 and Ni2 represent the set of neighbors of the ith agent in V1 and V2, respectively. Note that when τ > 0, the control input ui (t) is chosen as zero for t ∈ [0, τ).Remark 1.
Compared with [17] where the condition k1>2maxi∈ℐ1lii was required to guarantee consensus of heterogeneous multiagent systems, the control gain α is only required to be positive, even though the event-triggered scheme is applied here and it was absent in [17]. More specifically, from the following analysis, we can see that heterogenous system (2) under (4) can be transformed into an equivalent homogeneous system containing n + m agents by virtue of variable yi, which makes the analysis easier.
Fort∈tmi,tm+1i, denote measurement errors as eixt=xitmi−τ−xit−τ,i∈ℐ, eivt=vitmi−τ−vit−τ,i∈ℐ1, and eiyt=yitmi−τ−yit−τ,i∈ℐ1. Event-triggering function for each agent is designed as(5)fi1t=eixteiyt2−σpit2−ζt,i∈ℐ1,fi2t=eixt2−σqit2−ζt,i∈ℐ2,where pi(t) = −vitmi−τ+α∑j∈Ni1aijxjtm∗j−τ−yitmi−τ−∑j∈Ni2aijyitmi−τ+xjtm∗j−τ, i∈ℐ1; qit=α∑j∈Ni2aijxjtm∗j−τ−xitmi−τ−∑j∈Ni1aijxitmi−τ+xjtm∗j−τ,i∈ℐ2;ζt=c1e−c2t is an exponential function with c1 > 0, c2 > 0; σ > 0 is the event-triggering parameter which will be further defined later. When fi1(t) ≥ 0 or fi2(t) ≥ 0, the event of corresponding agent vi is triggered immediately.


### 3.1. Without Input Delay

First, we investigate bipartite consensus of (2) without input delay, i.e., τ = 0. According to the definitions of measurement errors, bipartite consensus protocol (4) can be rewritten as(6)uit=−αvit+eivt+α2∑j∈Ni1aijxjt−yit+ejxt−eiyt−∑j∈Ni2aijyit+xjt+eiyt+ejxt,i∈ℐ1,uit=α∑j∈Ni2aijxjt−xit+ejxt−eixt−∑j∈Ni1aijxit+xjt+eixt+ejxt,i∈ℐ2.Substituting (6) into system (2), we can obtain(7)x˙it=αyit−xit,i∈ℐ1,y˙it=−αeiyt−eixt+α∑j∈Ni1aijxjt−yit+ejxt−eiyt−∑j∈Ni2aijyit+xjt+eiyt+ejxt,i∈ℐ1,x˙it=α∑j∈Ni2aijxjt−xit+ejxt−eixt−∑j∈Ni1aijxit+xjt+eixt+ejxt,i∈ℐ2.Based on the grouping of multiagent systems, the adjacency matrixA of the digraph G can be described as(8)A=A1A2A3A4,where A1∈ℝm×m, A2∈ℝm×n−m, A3∈ℝn−m×m, and A4∈ℝn−m×n−m. Set xmt=x1t,…,xmtT, ymt=y1t,…,ymtT, xn−mt=xm+1t,…,xntT, emxt=e1xt,…,emxtT,emyt=e1yt,…,emytT,en−mxt=em+1xt,…,enxtT, and ξt=xmTt,ymTt,xn−mTtT,et=emxTt,emyTt,en−mxTtT. Thus, (7) can be written in a compact form as(9)ξ˙t=−Ξξ˙t−Ψet,where(10)Ξ=αIm−αIm0−αA˜1αΛ1+Λ2αA˜2αA˜30−αA˜4+αΛ3+Λ4,Ψ=000−αIm−αA˜1αIm+αΛ1+Λ2αA˜2αA˜30−αA˜4+αΛ3+Λ4,in which Ai˜=Ai for i = 1, 4, Ai˜=−Ai for i = 2, 3,(11)Λ1=diag∑j=1ma1j,…,∑j=1mamj,Λ2=−diag∑j=m+1na1j,…,∑j=m+1namj,Λ3=−diag∑j=1mam+1j,…,∑j=1manj,Λ4=diag∑j=m+1nam+1j,…,∑j=m+1nanj.Obviously, all entries ofAi˜ (i = 1, 2, 3, 4) are nonnegative.Based on Assumption1, (9) can be written as(12)ξ^˙t=−Φ^ξ^t−Ψ^e^t,where ξ^t=Dξt,e^t=Det,Ξ^=DΞD,Ψ^=DΨD with D = diag{Im, Im, −In−m}. Obviously, the sum of entries in each row of Ξ^ is zero. Meanwhile, the diagonal entries are all nonnegative. Therefore, Ξ^ can be regarded as the Laplacian matrix corresponding to a digraph G′ with n + m nodes.Lemma 5 (see [18]).
The digraphG′ corresponding to the matrix Φ^ contains a directed spanning tree if and only if digraph G contains a directed spanning tree.It is easy to see that the bipartite consensus problem of system (2) is equivalent to the consensus problem of system (12). Denote ξ¯t=Eξ^t and e¯t=Ee^t, where E = [−1n+m−1In+m−1]. Correspondingly, ξ^t=Fξ¯t1n+mξ^1t and e^t=Fe¯t+1n+me^1t, where F=0n+m−1In+m−1T. Then, (12) can be rewritten as(13)ξ¯˙t=−EΦ^Fξ¯t−EΨ^Fe¯t.Then, from [7], we have the consensus problem of system (12) which can be solved asymptotically if and only if system (13) achieves stability asymptotically. Moreover, when the digraph G′ contains a directed spanning tree, all the eigenvalues of the matrix −EΞ^F have negative real parts. So, there exists a positive definite matrix P satisfying(14)P−EΦ^F+−EΦ^FTP=−In+m−1.For further analysis, we select a positive numberϵ > 0 such that(15)γ=1−ϵP−EΨ^F2>0.Theorem 1.
Suppose that Assumption1 holds and the digraph G contains a directed spanning tree. The bipartite consensus problem of heterogeneous system (2) can be solved under event-triggered control protocol (4) and event-triggering functions (5) when event-triggering parameter σ satisfies(16)σ<γϵ2γϵ−Ψ^2+2λmaxETE−Ψ^F2.Proof.
Construct the following Lyapunov function:(17)Vt=ξ¯TtPξ¯t,where P is the positive definite matrix satisfying (14). The time derivative of V (t) is as follows:(18)V˙t=ξ¯TtPξ¯˙t+ξ¯˙TtPξ¯t=ξ¯TtP−EΞ^F+−EΞ^FTPξ¯t+2ξ¯TtP−EΨ^Fe¯t≤−ξ¯Ttξ¯t+εξ¯TtP−EΨ^F−EΨ^FTPξ¯t+1εe¯Tte¯t≤−ξ¯t2+εP−EΨ^F2ξ¯t2+1ελmaxETEe^t2.
Based on event-triggering functions (5), we can get that(19)et2≤σ−Ψξt+et2+nc1e−c2t≤2σ−Ψξt2+−Ψet2+nc1e−c2t.
According to the definitions ofξ^t,e^t,andΨ^ in (12), we have(20)e^t2≤2σ−Ψ^ξ^t2+−Ψ^e^t2+nc1e−c2t.
Combining with (16), it is easy to get that σ<1/2−Ψ^2. Thus, e^t2≤2σ−Ψ^ξ^t2+nc1e−c2t/1−2σ−Ψ^2. Therefore, (18) can be further extended to the following inequality:(21)V˙t≤−γ−λmaxETE2σ−Ψ^F2ϵ1−2σ−Ψ^F2ξ¯2+λmaxETEϵ1−2σ−Ψ^2nc1e−c2t.
From (16), we have γ−λmaxETE2σ−Ψ^F2/ϵ1−2σ−Ψ^2>0. Then, combining (21) with (17), we can get that(22)V˙t≤−ϱV+ϱξ¯TtPξ¯t−γ−λmaxETE2σ−Ψ^F2ϵ1−2σ−Ψ^2ξ¯2+λmaxETEϵ1−2σ−Ψ^2nc1e−c2t≤−ϱV+ϱλmaxPξ¯2−γ−λmaxETE2σ−Ψ^F2ϵ1−2σ−Ψ^2ξ¯2+λmaxETEϵ1−2σ−Ψ^2nc1e−c2t≤−ϱV+λmaxETEϵ1−2σ−Ψ^2nc1e−c2t,where ϱ=γ/λmaxP−λmaxETE2σ∥−Ψ^F∥2/λmaxPϵ1−2σ∥−Ψ^∥2. According to Lemma 3, we have 0 ≤ V(t) ≤ φ(t), where φ(t) satisfies φ.t = − ϱφ(t) + λmaxETEn/ϵ1−2σ∥−Ψ^∥2c1e−c2t with φ(0) = V(0), that is,(23)φt​​=​​e−ϱtφ0+λmaxETEnc1tϵ1−2σ∥−Ψ^∥2e−ϱt,ifϱ=c2,e−ϱtφ0​​+​​λmaxETEnc1ϵ1−2σ∥−Ψ^∥2ϱ−c2e−c2t​−​e−ϱt,ifϱ≠c2.
Obviously,limt⟶∞V (t) = 0. Hence, limt⟶∞ξ¯t=0, i.e., system (2) achieves bipartite consensus asymptotically.
The following theorem illustrates that Zeno behavior does not exist, i.e., there is no trajectory with infinite event-triggered time instants in a finite time interval.Theorem 2.
System (2) does not exhibit Zeno behavior under conditions of Theorem 1.Proof.
First, we consider agents inℐ1. Assume that tmi and tm+1i are two adjacent event-triggering time instants of agent vi, i.e., eixtmi=eixtm+1i=0 and eivtmi=eivtm+1i=0. Thus, eiytmi=eiytm+1i=0.
Considering the derivative ofeixt,eiytT over the interval tmi,tm+1i, we have(24)ddteixteiyt=eixte˙ixt+eiyte˙iyteix2t+eiy2t≤e˙ixt+e˙iyt=−αyit−xit+−αyit−xit+pit=−αyit−x1t−xit−x1t+−αyit−x1t−xit−x1t+pit.
From the proof of Theorem1, we know that limt⟶∞ξ¯t=0. Thus, ξ¯t is bounded. According to the definition of ξ¯t and gauge transformation, for ∀t∈tmi,tm+1i, xit−x1t,yit−x1t,i∈ℐ1, and −xit−x1t,i∈ℐ2 are all bounded. Thus, pit = −αyit−x1t−xit−x1t+α∑j∈Ni1aijxjt−x1t−yit−x1t−∑j∈Ni2aijyit−x1t−−xjt−x1t is bounded for t∈tmi,tm+1i. Therefore, d/dteixt,eiytT is bounded, i.e., there exists a constant M > 0 such that d/dteixt,eiytT≤M. Then,(25)eixteiyt≤Mt−tmi,t∈tmi,tm+1i,i∈ℐ1.
Fori∈ℐ1, since the event will not be triggered before eixt,eiytT2=c1e−c2t, by (25), a lower bound on tmi,tm+1i, which is denoted as τmi, satisfies c1e−c2tmi=Mτmi2e−c2τmi. That is, τmi=c1e−c2tmi+τmi/M. Assume that Zeno behavior occurs, which indicates that there exists a positive constant t∗ such that limm⟶∞tmi=t∗. Let ε0=c1e−c2tmi+τmi/2M. According to the definition of sequence limit, for ɛ0 > 0, there exists a positive integer N0 such that t∗−ε0<tmi≤t∗ for m ≥ N0. Therefore, when m≥N0,tm+1i≥tmi+τmi=tmi+2ε0≥t∗+ε0. This contradicts with the fact that tm+1i≤t∗ for m ≥ N0. Hence, Zeno behavior is strictly excluded.
Next, we consider agents inℐ2. Similarly, assume that tmi and tm+1i are two adjacent event-triggering time instants of agent vi, i.e., eixtmi=eixtm+1i=0. Since e˙ixt=−x˙it=−uit,i∈ℐ2, it holds that, for ∀t∈tmi,tm+1i, eixt≤∫tmituisds. Similar to the above analysis, it is easy to get that ui(t) is bounded for i∈ℐ2, i.e., there exists a constant M′ > 0 such that uit≤M′. Thus,(26)eixt≤M′t−tmi,i∈ℐ2.
The event will not be triggered beforeeixt2=c1e−c2t for i∈ℐ2. Combining with (26), the Zeno behavior can be excluded. Since the proof is just as same as that for agents in ℐ1, we omit it here.
In conclusion, all the agents of system (2) do not exhibit Zeno behavior under conditions of Theorem 1.Remark 2.
From the proof of Theorem2, we can see that, for second-order agents, to prove that d/dteixt,eiytT is bounded by virtue of variable yi which plays a key role to exclude Zeno behavior. This further shows the importance of introducing yi.


### 3.2. With Input Delay

In this section, we investigate bipartite consensus of (2) with input delay, i.e., τ > 0. Substituting (4) into system (2) and according to the definitions of measurement errors, we can get that(27)x˙it=αyit−xit,i∈ℐ1,y˙it=−αeiyt−eixt+α∑j∈Ni1xjt−τ−yit−τ+ejxt−eiyt−∑j∈Ni2aijyit−τ+xjt−τ+eiyt+ejxt,i∈ℐ1,x˙it=α∑j∈Ni2aijxjt−τ−xit−τ+ejxt−eixt−∑j∈Ni1aijxit−τ+xjt−τ+eixt+ejxt,i∈ℐ2.Then, (27) can be rewritten in a compact form as(28)ξ˙t=−Hξt−Ξξt−τ−Ψet,where(29)H=αIm−αIm0000000,Ξ=000−αA˜1αΛ1+Λ2αA˜2αA˜30−αA˜4+αΛ3+Λ4,and Ai˜, Λi, i = {1, 2, 3, 4} and Ψ have the same definitions as in (9). Analogously, according to Assumption 1, system (28) can be written as(30)ξ^˙t=−H^ξ^t−Ξ^ξ^t−τ−Ψ^e^t,where H^=DHD,Ξ^=DΞD, and ξ^t,e^t,Ψ^,D have the same definitions as in (12). By Newton–Leibniz formula, (30) can be rewritten as(31)ξ^˙t=−H^+Ξ^ξ^t−Ψ^e^t+Ξ^∫t−τtξ^˙sds.Remark 3.
Based on Lemma5, we can get the following result similarly. The digraph G′ corresponding to the matrix H^+Ξ^ has a directed spanning tree if and only if graph G has a directed spanning tree. And the bipartite consensus problem of system (2) under control protocol (4) can be solved asymptotically if and only if system (31) achieves consensus asymptotically.
By using vectorsξ¯t and e¯t which have been denoted in (13), system (31) can be written as (32)ξ¯˙t=−EH^+Ξ^Fξ¯t−EΨ^Fe¯t+EΞ^F∫t−τtξ¯˙sds.
Then, system (30) achieves consensus asymptotically if and only if system (32) achieves stability asymptotically.
The following notations will be used in the next analysis. Seta1 = ∥−EĤF ∥, a2=−EΞ^F, a3=−EΨ^F, b1=λmaxETE2σ−Ψ^F2/1−2σ−Ψ^2, and b2=λmaxETEnc1/1−2σ−Ψ^2.Theorem 3.
Suppose that Assumption1 holds and the digraph G contains a directed spanning tree. The bipartite consensus problem of system (2) can be solved under event-triggered control protocol (4) and event-triggering functions (5) when σ, τ, and c2 satisfy 0 < σ < σ0, 0 ≤ τ < τ0, and 0 < c2 < 2r, where(33)σ0=r22β2a32λmaxETE−Ψ^F2+2r2−Ψ^2,τ0=r−βa3b1βa1a2+a2a2+a3b1,β and r are the numbers obtained from Lemma 4 with P1=−EH^+Ξ^F. Moreover, system (2) does not exhibit Zeno behavior.Proof.
By integrating (32), we have (34)ξ¯t=e−EH^+Ξ^Ftξ0+∫0te−EH^+Ξ^Ft−θ×−EΨ^Fe¯θ+EΞ^F∫θ−τθ−EH^Fξ¯sEΞ^Fξ¯s−τ−EΨ^Fe¯sdsdθ.
According to [7], we know that all the eigenvalues of the matrix −EH^+Ξ^F have negative real parts. Therefore, based on Lemma 4, it follows that(35)ξ¯t≤βe−rtξ¯0+β∫0te−rt−θ×−EΨ^Fe¯θ+EΞ^F×∫θ−τθ−EH^Fξ¯s+−EΞ^Fξ¯s−τ+−EΨ^Fe¯sdsdθ.
Based on event-triggering functions (5) and the requirement of σ in Theorem 3, we can get that (36)e¯t≤b1ξ¯t−τ+b2e−c2/2t.
By replacing (36) into (35), we can derive that (37)ξ¯t≤βe−rtξ¯0+β∫0te−rt−θa3b1ξ¯θ−τ+a3b2e−c2/2θ+a2∫θ−τθa1ξ¯s+a2ξ¯s−τ+a3b1ξ¯s−τ+a3b2e−c2/2sdsdθ.
Next, we prove that(38)ξ¯t<wt≜βξ¯0e−λt+χe−c2/2t,t≥0,where λ ∈ (0, r) satisfies (39)βλa3b1eλτ+βa1a2+a2a2+a3b1eλτeλτ−1λr−λ<1χ≜βc2/2a3b2+a3b2ec2/2τ−1c2/2r−c2/2−βa1a2+a2a2+a3b1ec2/2τec2/2τ−1+c2/2a3b1ec2/2τ.
First, we prove the existence ofλ. Define a function h (λ) = βλa3b1eλτ + β [a1a2 + a2 (a2 + a3b1) eλτ] (eλτ − 1) − λ (r − λ). It is easy to get that h (0) = 0 and ḣ (0) = β [a1a2 + a2 (a2 + a3b1)]τ + βa3b1 − r. Since σ < σ0, we can get that βa3b1 < r. Meanwhile, ḣ (0) < 0 since τ < τ0. Therefore, there must exists a positive constant λ ∈ (0, r) such that h (λ) < 0. Consequently, (39) holds. Similarly, we can prove that the denominator of χ is positive when 0 < c2 < 2r, which implies that χ > 0. Next, we prove that (38) holds. Note that ξ¯t≤βξ¯0e−rt≤βξ¯0e−λt<wt for t = 0. If (38) does not hold, there must exist t∗ > 0 such that (40)ξ¯t∗=wt∗=βξ¯0e−λt∗+χe−c2/2t∗ξ¯t<wt,0≤t<t∗.
Replacing (38) into (37), we have (41)ξ¯t∗<βe−rt∗ξ¯0+β∫0t∗e−rt∗−θa3b1βξ¯0e−λθ−τ+χe−c2/2θ−τ+a3b2e−c2/2θ+a2∫θ−τθa1βξ¯0e−λs+χe−c2/2s+a2βξ¯0e−λs−τ+χe−c2/2s−τ+a3b1βξ¯0e−λs−τ+χe−c2/2s−τa3b2e−c2/2sdsdθ.
By some direct calculations, it follows that(42)ξ¯t∗<βξ¯0e−rt∗+e−λt∗−e−rt∗×βλa3b1eλτ+βa1a2+a2a2+a3b1eλτeλτ−1λr−λ+e−c2/2t∗−e−rt∗×βc2/2a3b2+a3b2ec2/2τ−1c2/2r−c2/2−βa1a2+a2a2+a3b1ec2/2τec2/2τ−1+c2/2a3b1ec2/2τ<βξ¯0e−λt∗+χe−c2/2t∗=wt∗.
This contradicts with (40). Hence, (38) holds, and bipartite consensus of system (2) under control protocol (4) and event-triggering function (5) can be solved asymptotically.
The proof of exclusion of Zeno behavior is just as same as that for Theorem2. The details are omitted here for brevity.


## 3.1. Without Input Delay

First, we investigate bipartite consensus of (2) without input delay, i.e., τ = 0. According to the definitions of measurement errors, bipartite consensus protocol (4) can be rewritten as(6)uit=−αvit+eivt+α2∑j∈Ni1aijxjt−yit+ejxt−eiyt−∑j∈Ni2aijyit+xjt+eiyt+ejxt,i∈ℐ1,uit=α∑j∈Ni2aijxjt−xit+ejxt−eixt−∑j∈Ni1aijxit+xjt+eixt+ejxt,i∈ℐ2.Substituting (6) into system (2), we can obtain(7)x˙it=αyit−xit,i∈ℐ1,y˙it=−αeiyt−eixt+α∑j∈Ni1aijxjt−yit+ejxt−eiyt−∑j∈Ni2aijyit+xjt+eiyt+ejxt,i∈ℐ1,x˙it=α∑j∈Ni2aijxjt−xit+ejxt−eixt−∑j∈Ni1aijxit+xjt+eixt+ejxt,i∈ℐ2.Based on the grouping of multiagent systems, the adjacency matrixA of the digraph G can be described as(8)A=A1A2A3A4,where A1∈ℝm×m, A2∈ℝm×n−m, A3∈ℝn−m×m, and A4∈ℝn−m×n−m. Set xmt=x1t,…,xmtT, ymt=y1t,…,ymtT, xn−mt=xm+1t,…,xntT, emxt=e1xt,…,emxtT,emyt=e1yt,…,emytT,en−mxt=em+1xt,…,enxtT, and ξt=xmTt,ymTt,xn−mTtT,et=emxTt,emyTt,en−mxTtT. Thus, (7) can be written in a compact form as(9)ξ˙t=−Ξξ˙t−Ψet,where(10)Ξ=αIm−αIm0−αA˜1αΛ1+Λ2αA˜2αA˜30−αA˜4+αΛ3+Λ4,Ψ=000−αIm−αA˜1αIm+αΛ1+Λ2αA˜2αA˜30−αA˜4+αΛ3+Λ4,in which Ai˜=Ai for i = 1, 4, Ai˜=−Ai for i = 2, 3,(11)Λ1=diag∑j=1ma1j,…,∑j=1mamj,Λ2=−diag∑j=m+1na1j,…,∑j=m+1namj,Λ3=−diag∑j=1mam+1j,…,∑j=1manj,Λ4=diag∑j=m+1nam+1j,…,∑j=m+1nanj.Obviously, all entries ofAi˜ (i = 1, 2, 3, 4) are nonnegative.Based on Assumption1, (9) can be written as(12)ξ^˙t=−Φ^ξ^t−Ψ^e^t,where ξ^t=Dξt,e^t=Det,Ξ^=DΞD,Ψ^=DΨD with D = diag{Im, Im, −In−m}. Obviously, the sum of entries in each row of Ξ^ is zero. Meanwhile, the diagonal entries are all nonnegative. Therefore, Ξ^ can be regarded as the Laplacian matrix corresponding to a digraph G′ with n + m nodes.Lemma 5 (see [18]).
The digraphG′ corresponding to the matrix Φ^ contains a directed spanning tree if and only if digraph G contains a directed spanning tree.It is easy to see that the bipartite consensus problem of system (2) is equivalent to the consensus problem of system (12). Denote ξ¯t=Eξ^t and e¯t=Ee^t, where E = [−1n+m−1In+m−1]. Correspondingly, ξ^t=Fξ¯t1n+mξ^1t and e^t=Fe¯t+1n+me^1t, where F=0n+m−1In+m−1T. Then, (12) can be rewritten as(13)ξ¯˙t=−EΦ^Fξ¯t−EΨ^Fe¯t.Then, from [7], we have the consensus problem of system (12) which can be solved asymptotically if and only if system (13) achieves stability asymptotically. Moreover, when the digraph G′ contains a directed spanning tree, all the eigenvalues of the matrix −EΞ^F have negative real parts. So, there exists a positive definite matrix P satisfying(14)P−EΦ^F+−EΦ^FTP=−In+m−1.For further analysis, we select a positive numberϵ > 0 such that(15)γ=1−ϵP−EΨ^F2>0.Theorem 1.
Suppose that Assumption1 holds and the digraph G contains a directed spanning tree. The bipartite consensus problem of heterogeneous system (2) can be solved under event-triggered control protocol (4) and event-triggering functions (5) when event-triggering parameter σ satisfies(16)σ<γϵ2γϵ−Ψ^2+2λmaxETE−Ψ^F2.Proof.
Construct the following Lyapunov function:(17)Vt=ξ¯TtPξ¯t,where P is the positive definite matrix satisfying (14). The time derivative of V (t) is as follows:(18)V˙t=ξ¯TtPξ¯˙t+ξ¯˙TtPξ¯t=ξ¯TtP−EΞ^F+−EΞ^FTPξ¯t+2ξ¯TtP−EΨ^Fe¯t≤−ξ¯Ttξ¯t+εξ¯TtP−EΨ^F−EΨ^FTPξ¯t+1εe¯Tte¯t≤−ξ¯t2+εP−EΨ^F2ξ¯t2+1ελmaxETEe^t2.
Based on event-triggering functions (5), we can get that(19)et2≤σ−Ψξt+et2+nc1e−c2t≤2σ−Ψξt2+−Ψet2+nc1e−c2t.
According to the definitions ofξ^t,e^t,andΨ^ in (12), we have(20)e^t2≤2σ−Ψ^ξ^t2+−Ψ^e^t2+nc1e−c2t.
Combining with (16), it is easy to get that σ<1/2−Ψ^2. Thus, e^t2≤2σ−Ψ^ξ^t2+nc1e−c2t/1−2σ−Ψ^2. Therefore, (18) can be further extended to the following inequality:(21)V˙t≤−γ−λmaxETE2σ−Ψ^F2ϵ1−2σ−Ψ^F2ξ¯2+λmaxETEϵ1−2σ−Ψ^2nc1e−c2t.
From (16), we have γ−λmaxETE2σ−Ψ^F2/ϵ1−2σ−Ψ^2>0. Then, combining (21) with (17), we can get that(22)V˙t≤−ϱV+ϱξ¯TtPξ¯t−γ−λmaxETE2σ−Ψ^F2ϵ1−2σ−Ψ^2ξ¯2+λmaxETEϵ1−2σ−Ψ^2nc1e−c2t≤−ϱV+ϱλmaxPξ¯2−γ−λmaxETE2σ−Ψ^F2ϵ1−2σ−Ψ^2ξ¯2+λmaxETEϵ1−2σ−Ψ^2nc1e−c2t≤−ϱV+λmaxETEϵ1−2σ−Ψ^2nc1e−c2t,where ϱ=γ/λmaxP−λmaxETE2σ∥−Ψ^F∥2/λmaxPϵ1−2σ∥−Ψ^∥2. According to Lemma 3, we have 0 ≤ V(t) ≤ φ(t), where φ(t) satisfies φ.t = − ϱφ(t) + λmaxETEn/ϵ1−2σ∥−Ψ^∥2c1e−c2t with φ(0) = V(0), that is,(23)φt​​=​​e−ϱtφ0+λmaxETEnc1tϵ1−2σ∥−Ψ^∥2e−ϱt,ifϱ=c2,e−ϱtφ0​​+​​λmaxETEnc1ϵ1−2σ∥−Ψ^∥2ϱ−c2e−c2t​−​e−ϱt,ifϱ≠c2.
Obviously,limt⟶∞V (t) = 0. Hence, limt⟶∞ξ¯t=0, i.e., system (2) achieves bipartite consensus asymptotically.
The following theorem illustrates that Zeno behavior does not exist, i.e., there is no trajectory with infinite event-triggered time instants in a finite time interval.Theorem 2.
System (2) does not exhibit Zeno behavior under conditions of Theorem 1.Proof.
First, we consider agents inℐ1. Assume that tmi and tm+1i are two adjacent event-triggering time instants of agent vi, i.e., eixtmi=eixtm+1i=0 and eivtmi=eivtm+1i=0. Thus, eiytmi=eiytm+1i=0.
Considering the derivative ofeixt,eiytT over the interval tmi,tm+1i, we have(24)ddteixteiyt=eixte˙ixt+eiyte˙iyteix2t+eiy2t≤e˙ixt+e˙iyt=−αyit−xit+−αyit−xit+pit=−αyit−x1t−xit−x1t+−αyit−x1t−xit−x1t+pit.
From the proof of Theorem1, we know that limt⟶∞ξ¯t=0. Thus, ξ¯t is bounded. According to the definition of ξ¯t and gauge transformation, for ∀t∈tmi,tm+1i, xit−x1t,yit−x1t,i∈ℐ1, and −xit−x1t,i∈ℐ2 are all bounded. Thus, pit = −αyit−x1t−xit−x1t+α∑j∈Ni1aijxjt−x1t−yit−x1t−∑j∈Ni2aijyit−x1t−−xjt−x1t is bounded for t∈tmi,tm+1i. Therefore, d/dteixt,eiytT is bounded, i.e., there exists a constant M > 0 such that d/dteixt,eiytT≤M. Then,(25)eixteiyt≤Mt−tmi,t∈tmi,tm+1i,i∈ℐ1.
Fori∈ℐ1, since the event will not be triggered before eixt,eiytT2=c1e−c2t, by (25), a lower bound on tmi,tm+1i, which is denoted as τmi, satisfies c1e−c2tmi=Mτmi2e−c2τmi. That is, τmi=c1e−c2tmi+τmi/M. Assume that Zeno behavior occurs, which indicates that there exists a positive constant t∗ such that limm⟶∞tmi=t∗. Let ε0=c1e−c2tmi+τmi/2M. According to the definition of sequence limit, for ɛ0 > 0, there exists a positive integer N0 such that t∗−ε0<tmi≤t∗ for m ≥ N0. Therefore, when m≥N0,tm+1i≥tmi+τmi=tmi+2ε0≥t∗+ε0. This contradicts with the fact that tm+1i≤t∗ for m ≥ N0. Hence, Zeno behavior is strictly excluded.
Next, we consider agents inℐ2. Similarly, assume that tmi and tm+1i are two adjacent event-triggering time instants of agent vi, i.e., eixtmi=eixtm+1i=0. Since e˙ixt=−x˙it=−uit,i∈ℐ2, it holds that, for ∀t∈tmi,tm+1i, eixt≤∫tmituisds. Similar to the above analysis, it is easy to get that ui(t) is bounded for i∈ℐ2, i.e., there exists a constant M′ > 0 such that uit≤M′. Thus,(26)eixt≤M′t−tmi,i∈ℐ2.
The event will not be triggered beforeeixt2=c1e−c2t for i∈ℐ2. Combining with (26), the Zeno behavior can be excluded. Since the proof is just as same as that for agents in ℐ1, we omit it here.
In conclusion, all the agents of system (2) do not exhibit Zeno behavior under conditions of Theorem 1.Remark 2.
From the proof of Theorem2, we can see that, for second-order agents, to prove that d/dteixt,eiytT is bounded by virtue of variable yi which plays a key role to exclude Zeno behavior. This further shows the importance of introducing yi.


## 3.2. With Input Delay

In this section, we investigate bipartite consensus of (2) with input delay, i.e., τ > 0. Substituting (4) into system (2) and according to the definitions of measurement errors, we can get that(27)x˙it=αyit−xit,i∈ℐ1,y˙it=−αeiyt−eixt+α∑j∈Ni1xjt−τ−yit−τ+ejxt−eiyt−∑j∈Ni2aijyit−τ+xjt−τ+eiyt+ejxt,i∈ℐ1,x˙it=α∑j∈Ni2aijxjt−τ−xit−τ+ejxt−eixt−∑j∈Ni1aijxit−τ+xjt−τ+eixt+ejxt,i∈ℐ2.Then, (27) can be rewritten in a compact form as(28)ξ˙t=−Hξt−Ξξt−τ−Ψet,where(29)H=αIm−αIm0000000,Ξ=000−αA˜1αΛ1+Λ2αA˜2αA˜30−αA˜4+αΛ3+Λ4,and Ai˜, Λi, i = {1, 2, 3, 4} and Ψ have the same definitions as in (9). Analogously, according to Assumption 1, system (28) can be written as(30)ξ^˙t=−H^ξ^t−Ξ^ξ^t−τ−Ψ^e^t,where H^=DHD,Ξ^=DΞD, and ξ^t,e^t,Ψ^,D have the same definitions as in (12). By Newton–Leibniz formula, (30) can be rewritten as(31)ξ^˙t=−H^+Ξ^ξ^t−Ψ^e^t+Ξ^∫t−τtξ^˙sds.Remark 3.
Based on Lemma5, we can get the following result similarly. The digraph G′ corresponding to the matrix H^+Ξ^ has a directed spanning tree if and only if graph G has a directed spanning tree. And the bipartite consensus problem of system (2) under control protocol (4) can be solved asymptotically if and only if system (31) achieves consensus asymptotically.
By using vectorsξ¯t and e¯t which have been denoted in (13), system (31) can be written as (32)ξ¯˙t=−EH^+Ξ^Fξ¯t−EΨ^Fe¯t+EΞ^F∫t−τtξ¯˙sds.
Then, system (30) achieves consensus asymptotically if and only if system (32) achieves stability asymptotically.
The following notations will be used in the next analysis. Seta1 = ∥−EĤF ∥, a2=−EΞ^F, a3=−EΨ^F, b1=λmaxETE2σ−Ψ^F2/1−2σ−Ψ^2, and b2=λmaxETEnc1/1−2σ−Ψ^2.Theorem 3.
Suppose that Assumption1 holds and the digraph G contains a directed spanning tree. The bipartite consensus problem of system (2) can be solved under event-triggered control protocol (4) and event-triggering functions (5) when σ, τ, and c2 satisfy 0 < σ < σ0, 0 ≤ τ < τ0, and 0 < c2 < 2r, where(33)σ0=r22β2a32λmaxETE−Ψ^F2+2r2−Ψ^2,τ0=r−βa3b1βa1a2+a2a2+a3b1,β and r are the numbers obtained from Lemma 4 with P1=−EH^+Ξ^F. Moreover, system (2) does not exhibit Zeno behavior.Proof.
By integrating (32), we have (34)ξ¯t=e−EH^+Ξ^Ftξ0+∫0te−EH^+Ξ^Ft−θ×−EΨ^Fe¯θ+EΞ^F∫θ−τθ−EH^Fξ¯sEΞ^Fξ¯s−τ−EΨ^Fe¯sdsdθ.
According to [7], we know that all the eigenvalues of the matrix −EH^+Ξ^F have negative real parts. Therefore, based on Lemma 4, it follows that(35)ξ¯t≤βe−rtξ¯0+β∫0te−rt−θ×−EΨ^Fe¯θ+EΞ^F×∫θ−τθ−EH^Fξ¯s+−EΞ^Fξ¯s−τ+−EΨ^Fe¯sdsdθ.
Based on event-triggering functions (5) and the requirement of σ in Theorem 3, we can get that (36)e¯t≤b1ξ¯t−τ+b2e−c2/2t.
By replacing (36) into (35), we can derive that (37)ξ¯t≤βe−rtξ¯0+β∫0te−rt−θa3b1ξ¯θ−τ+a3b2e−c2/2θ+a2∫θ−τθa1ξ¯s+a2ξ¯s−τ+a3b1ξ¯s−τ+a3b2e−c2/2sdsdθ.
Next, we prove that(38)ξ¯t<wt≜βξ¯0e−λt+χe−c2/2t,t≥0,where λ ∈ (0, r) satisfies (39)βλa3b1eλτ+βa1a2+a2a2+a3b1eλτeλτ−1λr−λ<1χ≜βc2/2a3b2+a3b2ec2/2τ−1c2/2r−c2/2−βa1a2+a2a2+a3b1ec2/2τec2/2τ−1+c2/2a3b1ec2/2τ.
First, we prove the existence ofλ. Define a function h (λ) = βλa3b1eλτ + β [a1a2 + a2 (a2 + a3b1) eλτ] (eλτ − 1) − λ (r − λ). It is easy to get that h (0) = 0 and ḣ (0) = β [a1a2 + a2 (a2 + a3b1)]τ + βa3b1 − r. Since σ < σ0, we can get that βa3b1 < r. Meanwhile, ḣ (0) < 0 since τ < τ0. Therefore, there must exists a positive constant λ ∈ (0, r) such that h (λ) < 0. Consequently, (39) holds. Similarly, we can prove that the denominator of χ is positive when 0 < c2 < 2r, which implies that χ > 0. Next, we prove that (38) holds. Note that ξ¯t≤βξ¯0e−rt≤βξ¯0e−λt<wt for t = 0. If (38) does not hold, there must exist t∗ > 0 such that (40)ξ¯t∗=wt∗=βξ¯0e−λt∗+χe−c2/2t∗ξ¯t<wt,0≤t<t∗.
Replacing (38) into (37), we have (41)ξ¯t∗<βe−rt∗ξ¯0+β∫0t∗e−rt∗−θa3b1βξ¯0e−λθ−τ+χe−c2/2θ−τ+a3b2e−c2/2θ+a2∫θ−τθa1βξ¯0e−λs+χe−c2/2s+a2βξ¯0e−λs−τ+χe−c2/2s−τ+a3b1βξ¯0e−λs−τ+χe−c2/2s−τa3b2e−c2/2sdsdθ.
By some direct calculations, it follows that(42)ξ¯t∗<βξ¯0e−rt∗+e−λt∗−e−rt∗×βλa3b1eλτ+βa1a2+a2a2+a3b1eλτeλτ−1λr−λ+e−c2/2t∗−e−rt∗×βc2/2a3b2+a3b2ec2/2τ−1c2/2r−c2/2−βa1a2+a2a2+a3b1ec2/2τec2/2τ−1+c2/2a3b1ec2/2τ<βξ¯0e−λt∗+χe−c2/2t∗=wt∗.
This contradicts with (40). Hence, (38) holds, and bipartite consensus of system (2) under control protocol (4) and event-triggering function (5) can be solved asymptotically.
The proof of exclusion of Zeno behavior is just as same as that for Theorem2. The details are omitted here for brevity.


## 4. Simulation

In this section, we give some numerical simulations to demonstrate the effectiveness of the obtained theoretical results. Consider a heterogeneous multiagent system composed of four agents and under a direct topology shown in Figure1. Obviously, G is structurally balanced with V1=v1,v2,V2=v3,v4, where V1 is composed of second-order agents and V2 is composed of first-order agents. For simplicity, let aij∣ = 1 if aij ≠ 0. Let the initial states of system (2) be x (0) = [2,−2,−4,−1]T, v0 = [−2, 3]T and choose α = 0.18. According to the topology in Figure 1, we have(43)Φ^=0.180−0.1800000.180−0.1800000.360−0.18−0.18−0.18000.18000−0.18000.36−0.180−0.180000.18,Ψ^=000000000000−0.1800.540−0.18−0.18−0.18−0.1800.36000−0.18000.36−0.180−0.180000.18.Figure 1
Directed topology which contains a directed spanning tree.Example. 1.
This example is for the case without input delay, i.e.,τ = 0. By solving the Lyapunov equation in (14), we can get that(44)P=3.742−1.5761.3570.0910.874−1.5762.034−1.5100.113−0.3511.357−1.5104.135−0.473−0.5550.0910.113−0.4731.4460.4030.874−0.351−0.5550.4032.829.
By selectingϵ = 0.1, we can get γ = 0.39 from (15). By substituting parameters ϵ and γ into (16), we can get that 0 < σ < 0.0088. Therefore, set σ = 0.007 and ζ (t) = e−0.1t. Figures 2 and 3 show that the heterogeneous multiagent system can achieve bipartite consensus under control protocol (4) and event-triggering functions (5). Figure 4 shows that limt⟶∞yit=limt⟶∞xit=−limt⟶∞xjt,∀i∈ℐ1,j∈ℐ2, which means the introduction of yi does not change bipartite consensus of the heterogeneous multiagent system. Figure 5 exhibits that all states can achieve consensus after being applied with gauge transformation. Figure 6 shows the event-triggering time instants of all agents. Figure 7 shows the evolution of ξ¯t.Figure 2
The position trajectory of each agent whenτ = 0.Figure 3
The velocity trajectory of each agent whenτ = 0.Figure 4
The state trajectories whenτ = 0.Figure 5
The state trajectories applied with gauge transformation whenτ = 0.Figure 6
Event-triggering time instants of each agent whenτ = 0.Figure 7
Evolution ofξ¯t when τ = 0.Example 2.
This example is for the case with input delay. SinceReλminEH^+Ξ^F=0.161, we can choose r = 0.08, β = 1 and c2 = 0.1 according to Lemma 4 and conditions of Theorem 3. Therefore, set ζ (t) = e−0.1t. By substituting the above parameters into (33), we can get that σ0 = 0.0077, τ0 = 0.0314. By choosing σ = 0.007 and τ = 0.01, the following numerical simulations are carried out. Figures 8 and 9 show that the heterogeneous multiagent system can achieve bipartite consensus under control protocol (4) and event-triggering functions (5). By comparing Figures 2 and 8 or Figures 3 and 9, we can see that the case with input delay takes longer time to achieve bipartite consensus than the case without input delay. Figure 10 shows the event-triggering time instants of all agents under event-triggered control protocol (4). Figure 11 shows the evolution of ξ¯t.Figure 8
The position trajectory of each agent whenτ = 0.01.Figure 9
The velocity trajectory of each agent whenτ = 0.01.Figure 10
Event-triggering time instants of each agent whenτ = 0.01.Figure 11
Evolution ofξ¯t when τ = 0.01.


## 5. Conclusions and Future Work

In this paper, we have investigated event-triggered bipartite consensus for heterogeneous multiagent systems composed of first-order agents and second-order agents under a structurally balanced topology which contains a directed spanning tree. A special variableyi=xi+1/αvi is used in designing event-triggered control protocol. By using gauge transformation, we transform a bipartite consensus problem under a structurally balanced signed network into a standard consensus problem under a nonnegative network. Sufficient conditions are obtained to guarantee bipartite consensus for both cases without input delay and with input delay. For the case with input delay, an upper bound of the delay is given to ensure bipartite consensus. It has been shown that no Zeno behavior occurs for heterogeneous multiagent systems under the presented event-triggered control protocol. Investigating bipartite consensus for heterogeneous multiagent systems under adaptive event-triggered control will be our future work.



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