# Bipartite Consensus of Heterogeneous Multiagent Systems with Diverse Input Delays

Authors: Hongtao Ye; Zhongqiu Chen; Wenguang Luo; Jiayan Wen; Kene Li
Journal: Complexity
 (2020)

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## Abstract

This paper investigates the bipartite consensus problem of heterogeneous multiagent systems with diverse input delays. Based on the systems composed of first-order and second-order agents, the novel control protocols are designed. Using frequency-domain analysis and matrix theory, the corresponding upper bounds of the allowable delays are obtained under the undirected topology and directed topology, respectively. Finally, simulation examples are given to verify the theoretical analysis.


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## Body

## 1. Introduction

Multiagent systems and its cooperative control [1] are widely used in many practical systems such as unmanned aerial vehicles (UAV) [2], monitoring and security [3], sensor network [4], and delay system [5–11].It is noteworthy that most of the problems about the consensus of multiagent systems are focused on cooperative network. In fact, cooperative and competitive relationships exist extensively in both natural and engineered network systems, such as opinion dynamics in social networks [12] and biological systems [13]. The bipartite consensus was firstly proposed in [14], which defined a signed graph the edges with positive and negative weights to describe the cooperative and competitive relations between agents. Bipartite consensus can be used for formation control [15], obstacle avoidance of wheeled robots [16], and nanoquadcopters formation [17].There are many factors that affect the stability of agents. The time delay problem is one of the important problems that affect the consensus of a multiagent system. The bipartite consensus problem of second-order multiagent systems with fixed time delays was studied in [18]. Based on the second-order multiagent systems, Tian et al. [19] discussed the bipartite consensus problem of the system under different disturbances. The bipartite consensus with arbitrary finite communication delay was discussed in [20]. Most of the above work use undirected graphs as communication networks. Compared with undirected graphs [21], directed graphs [22–25] are more versatile and cost effective when the edge weights can be arbitrary between two agents.This paper pays attention to bipartite consensus of heterogeneous MAS [26–28] with diverse input delays. As far as we know, there are few studies on this aspect, which is the motivation of this work. The theoretical analysis and simulation are presented under undirected topology and directed topology. The upper bounds of the allowable delays are given.The rest of this article is structured as follows. Section2 introduces some concepts and basic lemmas of graph theory. In Section 3, the bipartite consensus analysis of heterogeneous multiagent systems with multiple input delays under undirected and directed topologies is presented. In Section 4, two numerical examples are given to illustrate the validity of theoretical analysis. Finally, Section 5 draws some conclusions.


## 2. Preliminaries and Problem Formulation

### 2.1. Preliminaries

In this section, the concepts and lemmas of some preliminary diagrams are introduced. We consider a heterogeneous multiagent system withn agents. The connection between agents is represented by an undirected graph or directed graph G=V,ε,A, where V=ζ1,ζ2,…,ζn represents the set of nodes, ε⊆V×V represents the set of edges, and A=aij∈Rn×n represents the adjacency matrix of G. As the multiagent system of cooperation and competition is studied in this paper, the value can be either positive or negative. Here, we choose aii≠0 for all i∈1,2,…,n. If aij≠0, nodes ζj and ζi have information exchange; then, node ζj is said to be the neighbor of node ζi. The set of neighbors of node ζi is denoted by Ni=ζjζj,ζi∈ε. Graph G is strongly connected if there is a path between any two nodes in graph G. The Laplacian matrix of G is defined as Ls=C−A, where C=diag∑j=1na1j,∑j=1na2j,…,∑j=1nanj. Therefore, the elements of Laplacian matrix Ls are(1)Ls=∑ζi∈Niaij,j=i,−aij,j≠i.Assumption 1 (see [29]).
∑j∈Nlaij=0,∀i∈Ni, and ∑j∈Niaij=0,∀i∈Nl.Lemma 1 (see [30]).
For Assumption1, Laplace’s matrix Ls has at least two roots of zero. Give the time-delay system(2)y˙t=∑i=1NAiyt−τi,where yt∈Rn, Ai∈Rn×n, τi∈R, and N is a positive integer.Taking the Laplace transform, we can get the characteristic equation:(3)detsI−∑i=1NAiyt−τi=0.Lemma 2 (see [30]).
According to the characteristic equation of Lemma1, if it has only two zero roots and the rest of its roots are on the left half-plane of the complex plane, then it has limt⟶∞yt=α+βt, where α∈Rn and β∈Rn are constant vectors.Lemma 3 (see [31]).
Lethλ=arctanδμλ/μλ, where δ>0. When λ>0, μλ>0 increases monotonically; then, hλ decreases monotonically.Lemma 4 (see [31]).
y/1+y2<arctany holds for y>0.Lemma 5 (see [30]).
Ifm>0, then the inequality y/1+y2<arctany+m holds for y>0. If −π/2<m<0, then there exists y0>0, such that the inequality y/1+y2<arctany+m holds for y>y0.Lemma 6 (see [32]).
If and only if the connected graphG has a globally accessible node, Laplace’s matrix Ls has a simple root 0. The corresponding right eigenvector is 1n=1,…,1T, that is, L1n=0.


### 2.2. Problem Formulation

In this section, we will consider a heterogeneous multiagent system composed of agents, wherem agents are second-order agents and n-m agents are first-order agents. The information transmission of each agent in the heterogeneous system is represented by G, and each agent represents a node.Suppose the input delay of each agent in the system is not consistent. The dynamics of each second-order agent is given as follows:(4)x˙it=vit,v˙it=uit−Tii=1,2,…,m,where xi,vi,ui∈R represent the position, velocity, and control input of the second-order agent i, respectively. Ti>0 represents the input delay. The dynamics of each first-order agent is given as follows:(5)x˙l=ult−Tll=m+1,m+2,…,n,where xl,ul∈R represent the position and control input of the first-order agent l, respectively. Tl>0 represents the input delay.The bipartite consensus means all agents converge to a value which is the same for all in modulus but not in sign through distributed protocols. Similar to [33], the bipartite consensus protocols for the second-order agents are given by(6)ui=k1∑j∈Ni∪Nlaijsgnaijxj−xi+k2∑j∈Ni∪Nlaijsgnaijvj−vii=1,2,…,m,where k1,k2>0.Based on the dynamic neighbor estimation rule in [31, 34–36], an estimated speed is added to the first-order agent. The bipartite consensus protocols are given by(7)u˙l=vl+k2∑j∈Ni∪Nlaljsgnaljxj−xlv˙l=k1∑j∈Ni∪Nlaljsgnaljxj−xll=m+1,m+2,…,n,where k1,k2>0.The main purpose of this paper is to study the bipartite consensus protocol for heterogeneous multiagents with diverse input delays under the undirected topology and directed topology, respectively.


## 2.1. Preliminaries

In this section, the concepts and lemmas of some preliminary diagrams are introduced. We consider a heterogeneous multiagent system withn agents. The connection between agents is represented by an undirected graph or directed graph G=V,ε,A, where V=ζ1,ζ2,…,ζn represents the set of nodes, ε⊆V×V represents the set of edges, and A=aij∈Rn×n represents the adjacency matrix of G. As the multiagent system of cooperation and competition is studied in this paper, the value can be either positive or negative. Here, we choose aii≠0 for all i∈1,2,…,n. If aij≠0, nodes ζj and ζi have information exchange; then, node ζj is said to be the neighbor of node ζi. The set of neighbors of node ζi is denoted by Ni=ζjζj,ζi∈ε. Graph G is strongly connected if there is a path between any two nodes in graph G. The Laplacian matrix of G is defined as Ls=C−A, where C=diag∑j=1na1j,∑j=1na2j,…,∑j=1nanj. Therefore, the elements of Laplacian matrix Ls are(1)Ls=∑ζi∈Niaij,j=i,−aij,j≠i.Assumption 1 (see [29]).
∑j∈Nlaij=0,∀i∈Ni, and ∑j∈Niaij=0,∀i∈Nl.Lemma 1 (see [30]).
For Assumption1, Laplace’s matrix Ls has at least two roots of zero. Give the time-delay system(2)y˙t=∑i=1NAiyt−τi,where yt∈Rn, Ai∈Rn×n, τi∈R, and N is a positive integer.Taking the Laplace transform, we can get the characteristic equation:(3)detsI−∑i=1NAiyt−τi=0.Lemma 2 (see [30]).
According to the characteristic equation of Lemma1, if it has only two zero roots and the rest of its roots are on the left half-plane of the complex plane, then it has limt⟶∞yt=α+βt, where α∈Rn and β∈Rn are constant vectors.Lemma 3 (see [31]).
Lethλ=arctanδμλ/μλ, where δ>0. When λ>0, μλ>0 increases monotonically; then, hλ decreases monotonically.Lemma 4 (see [31]).
y/1+y2<arctany holds for y>0.Lemma 5 (see [30]).
Ifm>0, then the inequality y/1+y2<arctany+m holds for y>0. If −π/2<m<0, then there exists y0>0, such that the inequality y/1+y2<arctany+m holds for y>y0.Lemma 6 (see [32]).
If and only if the connected graphG has a globally accessible node, Laplace’s matrix Ls has a simple root 0. The corresponding right eigenvector is 1n=1,…,1T, that is, L1n=0.


## 2.2. Problem Formulation

In this section, we will consider a heterogeneous multiagent system composed of agents, wherem agents are second-order agents and n-m agents are first-order agents. The information transmission of each agent in the heterogeneous system is represented by G, and each agent represents a node.Suppose the input delay of each agent in the system is not consistent. The dynamics of each second-order agent is given as follows:(4)x˙it=vit,v˙it=uit−Tii=1,2,…,m,where xi,vi,ui∈R represent the position, velocity, and control input of the second-order agent i, respectively. Ti>0 represents the input delay. The dynamics of each first-order agent is given as follows:(5)x˙l=ult−Tll=m+1,m+2,…,n,where xl,ul∈R represent the position and control input of the first-order agent l, respectively. Tl>0 represents the input delay.The bipartite consensus means all agents converge to a value which is the same for all in modulus but not in sign through distributed protocols. Similar to [33], the bipartite consensus protocols for the second-order agents are given by(6)ui=k1∑j∈Ni∪Nlaijsgnaijxj−xi+k2∑j∈Ni∪Nlaijsgnaijvj−vii=1,2,…,m,where k1,k2>0.Based on the dynamic neighbor estimation rule in [31, 34–36], an estimated speed is added to the first-order agent. The bipartite consensus protocols are given by(7)u˙l=vl+k2∑j∈Ni∪Nlaljsgnaljxj−xlv˙l=k1∑j∈Ni∪Nlaljsgnaljxj−xll=m+1,m+2,…,n,where k1,k2>0.The main purpose of this paper is to study the bipartite consensus protocol for heterogeneous multiagents with diverse input delays under the undirected topology and directed topology, respectively.


## 3. Main Results

### 3.1. Bipartite Consensus under Undirected Topology

We will consider the bipartite consensus of heterogeneous multiagent systems with diverse input delays under undirected topology in this section. This paper uses neighborhood estimation rules to estimate the speed of first-order agents. Using the dynamic change of the position of the agent around the first-order agents, the estimated velocity is substituted for its actual velocity. Based on the heterogeneous system with both cooperative and competitive relations, the second-order agent control protocol is designed as follows:(8)x˙it=vit,v˙it=k1∑j∈Ni∪Nlaijsgnaijxjt−τi−xit−τi+k2∑j∈Ni∪Nlaijsgnaijvjt−τi−vit−τi,i=1,2,…,m,and the first-order agent control protocol is designed as follows:(9)x˙lt=vlt−τl+k2∑j∈Ni∪Nlaljsgnaljxjt−τl−xlt−τl,v˙lt=k1∑j∈Ni∪Nlaljsgnaljxjt−xlt,l=m+1,m+2,…,n,where sgn⋅ is the sign function, k1>0andk2>0 are the control gains, Ni and Nl represent the neighborhood of the agent i and l, and τi and τl represent the input delay of the agent i and l.Lemma 7.
For the heterogeneous multiagent systems (8) and (9), if and only if the Laplacian matrix Ls has at least one zero eigenvalue, the real parts of the rest of eigenvalues are positive, and satisfy the condition of inequality (10), and the systems can achieve bipartite consensus.(10)λik12+ωci2k22ωci2<1,where λi is the eigenvalue of Ls and ωci satisfies ωci=arctanωcik2/k1.Proof.
Take the Laplace transform on (8) and (9); then, we can obtain(11)sXis=Vis,sVis=k1∑j∈Ni∪NlaijsgnaijXjs−Xise−τis+k2∑j∈Ni∪NlaijsgnaijVjs−Vise−τis,i=1,2,⋯,m,(12)sXls=Vlse−τls+k2∑j∈Ni∪NlaljsgnaljXjs−Xlse−τls,sVls=k1∑j∈Ni∪NlaljsgnaljXjs−Xls,l=m+1,m+2,…,n.
From (11) and (12), we can obtain(13)s2Xs=−k1+k2sXsLse−τs,where Xs=X1s,X2s,…,XnsT is the Laplace transform of xt=x1t,x2t,…,xntT. Thus, the characteristic equation of (8) and (9) is given by(14)dets2I+k1+k2sLse−τs=0.
Because the topology is undirected connected, 0 is a simple root of matrixLs, and the rest of the roots are positive. Hence, rankLs=n−1. Let λi,i=1,2,…,n be the eigenvalue of Ls. Assume λ1=0,λi>0,i=2,3,…,n, then (14) is equal to(15)s2∏i=2,…,ns2+λik1+k2se−τs=0.
So, the equation has two roots of zero. Analyze the rest of the roots of the equation. For(16)s2+λik1+k2se−τs=0,i=2,3,…,n,let fs=1+gijω=0. Equation (16) can be written as 1+λik1+k2se−τs/s2=0, where gijω=λik1+k2se−τs/s2. Based on the Nyquist criterion, if and only if the curve gijω does not include the point (−1, j0), the characteristic root of equation (16) is located on the left half-plane of the complex plane. Then,(17)gijω=λik12+ω2k22ω2e−jωτ+π−arctanωk2/k1.
Then,(18)gijω=λik12+ωci2k22ωci2,arggijω=−ωτ+arctanωk2/k1,where arg⋅ represents the phase. We know that when ω∈0,+∞, gijω is monotonically decreasing. When curve gijω crosses the real axis for the first time, ωci satisfies −ωτ+arctanωk2/k1=0. Therefore,(19)gijω=λik12+ωci2k22ωci2<1.
In other words, (10) is true. Therefore, except for the two zero roots, all the characteristic roots of (14) are located in the left half-plane. Based on Lemma 2, it is true that limt⟶∞xt=α+βt, where α=α1,α2,…,αnT∈Rn and β=β1,β2,⋯,βnT∈Rn are constant vectors. From the control protocols (8) and (9), it is true that limt⟶∞vt=β, where vt=v1t,v2t,…,vntT. For (8),(20)k1∑j∈Ni∪Nlaijαj−αi+k1t−τ+k2∑j∈Ni∪Nlaijβj−βi=0,i=1,2,…,m.
And (20) is also true for any t, and we have(21)∑j∈Ni∪Nlaijαj−αi=0,i=1,2,…,m.(22)∑j∈Ni∪Nlaijβj−βi=0,i=1,2,…,m.
For (9),(23)∑j∈Ni∪Nlaljαj−αl+t∑j∈Ni∪Nlaljβj−βl=0,l=m+1,m+2,…,n.
And (23) is also true for any t, and we have(24)∑j∈Ni∪Nlaljαj−αl=0,l=m+1,m+2,…,n,(25)∑j∈Ni∪Nlaljβj−βl=0,l=m+1,m+2,…,n.
From (21) and (24), it is true that Lα=0. From (22) and (25), it is true that Lβ=0. Because rankL=n−1 and L1,…,1T=0. According to Lemma 6, we have α=a1,…,1T, β=b1,…,1T.Theorem 1.
For the heterogeneous multiagent systems (8) and (9) under undirected communication topology, the systems can achieve bipartite consensus if the largest input delay satisfies τmax<τ∗, where τ∗=arctank2/k1λmax2k22+λmax4k24+4λmax2k12/2/λmax2k22+λmax4k24+4λmax2k12/2, and λmax is the maximum eigenvalue of Ls.Proof.
For Lemma7, we can get ωciτ=arctanωcik2/k1. Namely,(26)τ=arctanωcik2/k1ωci.
Taking the derivative ofτ with respect to ωci for equation (24), we can obtain(27)dτdωci=1ωci2ηωci1+ηωci2−arctanηωci,where η=k2/k1. According to Lemma 4, τ decreases as ωci increases.
Then,(28)ωci>λi2k22+λi4k24+λi2k122.
From (26), (28), and Lemma 3, we can obtain(29)τ∗<arctank2/k1λmax2k22+λmax4k24+4λmax2k12/2λmax2k22+λmax4k24+4λmax2k12/2,where λmax is the maximum eigenvalue of Ls.


### 3.2. Bipartite Consensus under Directed Topology

We will study the bipartite consensus of heterogeneous multiagent systems with diverse input delays under directed topology in this section. The control protocols are designed as follows:(30)x˙it=vit,v˙it=k1∑j∈Niaijsgnaijxjt−τi−xit−τi+k1∑j∈Nlaijsgnaijxjt−τi+k2∑j∈Niaijsgnaijvjt−τi−vit−τi+k2∑j∈Nlaijsgnaijvjt−τi,i=1,2,…,m,(31)x˙lt=vlt−τl+k2∑j∈Nlaljsgnaljxjt−τl−xlt−τl+k2∑j∈Nialjsgnaljxjt−τl,v˙lt=k1∑j∈Nlaljsgnaljxjt−xlt+k1∑j∈Nialjsgnaljxjt,l=m+1,m+2,…,n,where k1>0,k2>0 are the control gains, Ni and Nl represent the neighborhood of the agent i and l, and τi and τl represent the input delay of the agent i and l.Lemma 8.
For the heterogeneous multiagent systems (30) and (31), if and only if the Laplacian matrix Ls has at least two zero eigenvalues, the real parts of the rest of eigenvalues are positive, and satisfy the condition of inequality (32), and the systems can achieve bipartite consensus:(32)λik12+ωci2k22ωci2<1,where λi is the eigenvalue of Ls, and ωci satisfies(33)ωciτ=arctanωcik2k1+arctanImλiReλi.Proof.
The Laplace transform of (30) and (31):(34)sXis=Vis,sVis=k1∑j∈NiaijsgnaijXjs−Xise−τis+k1∑j∈NlaijsgnaijXjse−τis+k2∑j∈NiaijsgnaijVjs−Vise−τis+k2∑j∈NlaijsgnaijXjse−τis,i=1,2,…,m,(35)sXls=Vse−τls+k2∑j∈NlaljsgnaljXjs−Xlse−τls+k2∑j∈NialjsgnaljXjse−τls,sVls=k1∑j∈NlaljsgnaljXjs−Xls+k1∑j∈NialjsgnaljVjs,l=m+1,m+2,…,n.
From Assumption1, (34), and (35),(36)s2Xs=−k1+k2sXsLse−τs.
Then, the characteristic root of systems (31) and (32) is(37)dets2I+k1+k2sLse−τs=0.
Letλi,i=1,2,…,n be the eigenvalue of Ls. Assume λ1=λ2=0,λi>0,i=3,4,…,n; then, (37) is equal to(38)s4∏i=3,…,ns2+λik1+k2se−τs=0.
Obviously, we know that the equation has four zero roots, and then we can analyze the rest of the characteristic roots of the equation. For(39)s2+λik1+k2se−τs=0,i=3,4,…,n,let hs=1+gijω=0. Therefore, equation (39) can be written as 1+λik1+k2se−τs/s2=0, where gijω=λik1+k2se−τs/s2. Based on Nyquist’s criterion, if and only if curve gijω does not include point −1,j0, the characteristic root of equation (39) is located on the left half-plane of the complex plane. Then,(40)gijω=λik12+ωci2k22ωci2e−jωτ+π−arctanωk2/k1−arctanImλi/Reλi.
Therefore,(41)gijω=λik12+ωci2k22ωci2,arggijω=−ωτ+arctanωk2k1+arctanImλiReλi,where arg⋅ represents the phase. And it can be seen that gijω monotonically decreases for ω∈0,+∞. When curve gijω crosses the real axis for the first time, ωci satisfies −ωτ+arctanωk2/k1+arctanImλi/Reλi=0.
In addition, if the characteristic root of (37) falls on the left half-plane, there is(42)gijω=λik12+ωci2k22ωci2<1.
In other words,(43)λik12+ωci2k22ωci2<1.
Therefore, except for the four zero roots, all the characteristic roots of (37) are located in the left half-plane. According to Lemma 6 and Theorem 1, we can get α=a1,…,1T and β=b1,…,1T.Theorem 2.
For the heterogeneous multiagent systems (30) and (31) under directed communication topology, the systems can achieve bipartite consensus if the largest input delay satisfies τmax<τ∗=minτa,τb, where τa=arctank2/k1λmax2k22+λmax4k24+4λmax2k12/2+θmax/λmax2k22+λmax4k24+4λmax2k12/2.
θmax=arctanImλi/Reλi, λmax is the maximum modulus of λi,τb=arctanω˜maxk2/k1−θmax/ω˜max,ω˜max=max3≤i≤nω˜ci, ω˜ci is the root of equation, 1/ωciηωci/1+ηωci2−arctanηωci−arctanImλi/Reλi=0, and η=k2/k1.Proof.
From (33), we can obtain(44)τ=arctanωcik2/k1+arctanImλi/Reλiωci.
Taking the derivative ofτ with respect to ωci for equation (44), we can obtain
dτ/dωci=1/ωci2ηωci/1+ηωci2−arctanηωci−arctanImλi/Reλi, where η=k2/k1.
According to Lemma5, if m=arctanImλi/Reλi>0, then ωci>0 and dτ/dωci<0. According to (32), we can obtain ωci>λi2k22+λi4k24+λi2k12/2.
Then, using Lemma8, we have. τ=arctank2/k1λi2k22+λi4k24+4λi2k12/2+arctanImλi/Reλi/λi2k22+λi4k24+4λi2k12/2.
According to Lemma5, if −π/2<m=arctanImλi/Reλi<0, then there exists ω˜ci>0, when ωci>ω˜ci and dτ/dωci<0. ω˜ci is the root of(45)1ωciηωci/1+ηωci2−arctanηωci−arctanImλi/Reλi=0,where η=k2/k1.
Similarly, using Lemma8, we have τ=arctanωcik2/k1+arctanImλi/Reλi/ωci.
In conclusion, according to Lemma3, the maximum allowable upper bound of the system is τ∗=minτa,τb, where(46)τa=arctank2/k1λmax2k22+λmax4k24+4λmax2k12/2+θmaxλmax2k22+λmax4k24+4λmax2k12/2,τb=arctanω˜maxk2/k1−θmaxω˜max,
θmax=arctanImλi/Reλi,ω˜max=max3≤i≤nω˜ci,ω˜ci is the root of equation 1/ωciηωci/1+ηωci2−arctanηωci−arctanImλi/Reλi=0, and η=k2/k1.


## 3.1. Bipartite Consensus under Undirected Topology

We will consider the bipartite consensus of heterogeneous multiagent systems with diverse input delays under undirected topology in this section. This paper uses neighborhood estimation rules to estimate the speed of first-order agents. Using the dynamic change of the position of the agent around the first-order agents, the estimated velocity is substituted for its actual velocity. Based on the heterogeneous system with both cooperative and competitive relations, the second-order agent control protocol is designed as follows:(8)x˙it=vit,v˙it=k1∑j∈Ni∪Nlaijsgnaijxjt−τi−xit−τi+k2∑j∈Ni∪Nlaijsgnaijvjt−τi−vit−τi,i=1,2,…,m,and the first-order agent control protocol is designed as follows:(9)x˙lt=vlt−τl+k2∑j∈Ni∪Nlaljsgnaljxjt−τl−xlt−τl,v˙lt=k1∑j∈Ni∪Nlaljsgnaljxjt−xlt,l=m+1,m+2,…,n,where sgn⋅ is the sign function, k1>0andk2>0 are the control gains, Ni and Nl represent the neighborhood of the agent i and l, and τi and τl represent the input delay of the agent i and l.Lemma 7.
For the heterogeneous multiagent systems (8) and (9), if and only if the Laplacian matrix Ls has at least one zero eigenvalue, the real parts of the rest of eigenvalues are positive, and satisfy the condition of inequality (10), and the systems can achieve bipartite consensus.(10)λik12+ωci2k22ωci2<1,where λi is the eigenvalue of Ls and ωci satisfies ωci=arctanωcik2/k1.Proof.
Take the Laplace transform on (8) and (9); then, we can obtain(11)sXis=Vis,sVis=k1∑j∈Ni∪NlaijsgnaijXjs−Xise−τis+k2∑j∈Ni∪NlaijsgnaijVjs−Vise−τis,i=1,2,⋯,m,(12)sXls=Vlse−τls+k2∑j∈Ni∪NlaljsgnaljXjs−Xlse−τls,sVls=k1∑j∈Ni∪NlaljsgnaljXjs−Xls,l=m+1,m+2,…,n.
From (11) and (12), we can obtain(13)s2Xs=−k1+k2sXsLse−τs,where Xs=X1s,X2s,…,XnsT is the Laplace transform of xt=x1t,x2t,…,xntT. Thus, the characteristic equation of (8) and (9) is given by(14)dets2I+k1+k2sLse−τs=0.
Because the topology is undirected connected, 0 is a simple root of matrixLs, and the rest of the roots are positive. Hence, rankLs=n−1. Let λi,i=1,2,…,n be the eigenvalue of Ls. Assume λ1=0,λi>0,i=2,3,…,n, then (14) is equal to(15)s2∏i=2,…,ns2+λik1+k2se−τs=0.
So, the equation has two roots of zero. Analyze the rest of the roots of the equation. For(16)s2+λik1+k2se−τs=0,i=2,3,…,n,let fs=1+gijω=0. Equation (16) can be written as 1+λik1+k2se−τs/s2=0, where gijω=λik1+k2se−τs/s2. Based on the Nyquist criterion, if and only if the curve gijω does not include the point (−1, j0), the characteristic root of equation (16) is located on the left half-plane of the complex plane. Then,(17)gijω=λik12+ω2k22ω2e−jωτ+π−arctanωk2/k1.
Then,(18)gijω=λik12+ωci2k22ωci2,arggijω=−ωτ+arctanωk2/k1,where arg⋅ represents the phase. We know that when ω∈0,+∞, gijω is monotonically decreasing. When curve gijω crosses the real axis for the first time, ωci satisfies −ωτ+arctanωk2/k1=0. Therefore,(19)gijω=λik12+ωci2k22ωci2<1.
In other words, (10) is true. Therefore, except for the two zero roots, all the characteristic roots of (14) are located in the left half-plane. Based on Lemma 2, it is true that limt⟶∞xt=α+βt, where α=α1,α2,…,αnT∈Rn and β=β1,β2,⋯,βnT∈Rn are constant vectors. From the control protocols (8) and (9), it is true that limt⟶∞vt=β, where vt=v1t,v2t,…,vntT. For (8),(20)k1∑j∈Ni∪Nlaijαj−αi+k1t−τ+k2∑j∈Ni∪Nlaijβj−βi=0,i=1,2,…,m.
And (20) is also true for any t, and we have(21)∑j∈Ni∪Nlaijαj−αi=0,i=1,2,…,m.(22)∑j∈Ni∪Nlaijβj−βi=0,i=1,2,…,m.
For (9),(23)∑j∈Ni∪Nlaljαj−αl+t∑j∈Ni∪Nlaljβj−βl=0,l=m+1,m+2,…,n.
And (23) is also true for any t, and we have(24)∑j∈Ni∪Nlaljαj−αl=0,l=m+1,m+2,…,n,(25)∑j∈Ni∪Nlaljβj−βl=0,l=m+1,m+2,…,n.
From (21) and (24), it is true that Lα=0. From (22) and (25), it is true that Lβ=0. Because rankL=n−1 and L1,…,1T=0. According to Lemma 6, we have α=a1,…,1T, β=b1,…,1T.Theorem 1.
For the heterogeneous multiagent systems (8) and (9) under undirected communication topology, the systems can achieve bipartite consensus if the largest input delay satisfies τmax<τ∗, where τ∗=arctank2/k1λmax2k22+λmax4k24+4λmax2k12/2/λmax2k22+λmax4k24+4λmax2k12/2, and λmax is the maximum eigenvalue of Ls.Proof.
For Lemma7, we can get ωciτ=arctanωcik2/k1. Namely,(26)τ=arctanωcik2/k1ωci.
Taking the derivative ofτ with respect to ωci for equation (24), we can obtain(27)dτdωci=1ωci2ηωci1+ηωci2−arctanηωci,where η=k2/k1. According to Lemma 4, τ decreases as ωci increases.
Then,(28)ωci>λi2k22+λi4k24+λi2k122.
From (26), (28), and Lemma 3, we can obtain(29)τ∗<arctank2/k1λmax2k22+λmax4k24+4λmax2k12/2λmax2k22+λmax4k24+4λmax2k12/2,where λmax is the maximum eigenvalue of Ls.


## 3.2. Bipartite Consensus under Directed Topology

We will study the bipartite consensus of heterogeneous multiagent systems with diverse input delays under directed topology in this section. The control protocols are designed as follows:(30)x˙it=vit,v˙it=k1∑j∈Niaijsgnaijxjt−τi−xit−τi+k1∑j∈Nlaijsgnaijxjt−τi+k2∑j∈Niaijsgnaijvjt−τi−vit−τi+k2∑j∈Nlaijsgnaijvjt−τi,i=1,2,…,m,(31)x˙lt=vlt−τl+k2∑j∈Nlaljsgnaljxjt−τl−xlt−τl+k2∑j∈Nialjsgnaljxjt−τl,v˙lt=k1∑j∈Nlaljsgnaljxjt−xlt+k1∑j∈Nialjsgnaljxjt,l=m+1,m+2,…,n,where k1>0,k2>0 are the control gains, Ni and Nl represent the neighborhood of the agent i and l, and τi and τl represent the input delay of the agent i and l.Lemma 8.
For the heterogeneous multiagent systems (30) and (31), if and only if the Laplacian matrix Ls has at least two zero eigenvalues, the real parts of the rest of eigenvalues are positive, and satisfy the condition of inequality (32), and the systems can achieve bipartite consensus:(32)λik12+ωci2k22ωci2<1,where λi is the eigenvalue of Ls, and ωci satisfies(33)ωciτ=arctanωcik2k1+arctanImλiReλi.Proof.
The Laplace transform of (30) and (31):(34)sXis=Vis,sVis=k1∑j∈NiaijsgnaijXjs−Xise−τis+k1∑j∈NlaijsgnaijXjse−τis+k2∑j∈NiaijsgnaijVjs−Vise−τis+k2∑j∈NlaijsgnaijXjse−τis,i=1,2,…,m,(35)sXls=Vse−τls+k2∑j∈NlaljsgnaljXjs−Xlse−τls+k2∑j∈NialjsgnaljXjse−τls,sVls=k1∑j∈NlaljsgnaljXjs−Xls+k1∑j∈NialjsgnaljVjs,l=m+1,m+2,…,n.
From Assumption1, (34), and (35),(36)s2Xs=−k1+k2sXsLse−τs.
Then, the characteristic root of systems (31) and (32) is(37)dets2I+k1+k2sLse−τs=0.
Letλi,i=1,2,…,n be the eigenvalue of Ls. Assume λ1=λ2=0,λi>0,i=3,4,…,n; then, (37) is equal to(38)s4∏i=3,…,ns2+λik1+k2se−τs=0.
Obviously, we know that the equation has four zero roots, and then we can analyze the rest of the characteristic roots of the equation. For(39)s2+λik1+k2se−τs=0,i=3,4,…,n,let hs=1+gijω=0. Therefore, equation (39) can be written as 1+λik1+k2se−τs/s2=0, where gijω=λik1+k2se−τs/s2. Based on Nyquist’s criterion, if and only if curve gijω does not include point −1,j0, the characteristic root of equation (39) is located on the left half-plane of the complex plane. Then,(40)gijω=λik12+ωci2k22ωci2e−jωτ+π−arctanωk2/k1−arctanImλi/Reλi.
Therefore,(41)gijω=λik12+ωci2k22ωci2,arggijω=−ωτ+arctanωk2k1+arctanImλiReλi,where arg⋅ represents the phase. And it can be seen that gijω monotonically decreases for ω∈0,+∞. When curve gijω crosses the real axis for the first time, ωci satisfies −ωτ+arctanωk2/k1+arctanImλi/Reλi=0.
In addition, if the characteristic root of (37) falls on the left half-plane, there is(42)gijω=λik12+ωci2k22ωci2<1.
In other words,(43)λik12+ωci2k22ωci2<1.
Therefore, except for the four zero roots, all the characteristic roots of (37) are located in the left half-plane. According to Lemma 6 and Theorem 1, we can get α=a1,…,1T and β=b1,…,1T.Theorem 2.
For the heterogeneous multiagent systems (30) and (31) under directed communication topology, the systems can achieve bipartite consensus if the largest input delay satisfies τmax<τ∗=minτa,τb, where τa=arctank2/k1λmax2k22+λmax4k24+4λmax2k12/2+θmax/λmax2k22+λmax4k24+4λmax2k12/2.
θmax=arctanImλi/Reλi, λmax is the maximum modulus of λi,τb=arctanω˜maxk2/k1−θmax/ω˜max,ω˜max=max3≤i≤nω˜ci, ω˜ci is the root of equation, 1/ωciηωci/1+ηωci2−arctanηωci−arctanImλi/Reλi=0, and η=k2/k1.Proof.
From (33), we can obtain(44)τ=arctanωcik2/k1+arctanImλi/Reλiωci.
Taking the derivative ofτ with respect to ωci for equation (44), we can obtain
dτ/dωci=1/ωci2ηωci/1+ηωci2−arctanηωci−arctanImλi/Reλi, where η=k2/k1.
According to Lemma5, if m=arctanImλi/Reλi>0, then ωci>0 and dτ/dωci<0. According to (32), we can obtain ωci>λi2k22+λi4k24+λi2k12/2.
Then, using Lemma8, we have. τ=arctank2/k1λi2k22+λi4k24+4λi2k12/2+arctanImλi/Reλi/λi2k22+λi4k24+4λi2k12/2.
According to Lemma5, if −π/2<m=arctanImλi/Reλi<0, then there exists ω˜ci>0, when ωci>ω˜ci and dτ/dωci<0. ω˜ci is the root of(45)1ωciηωci/1+ηωci2−arctanηωci−arctanImλi/Reλi=0,where η=k2/k1.
Similarly, using Lemma8, we have τ=arctanωcik2/k1+arctanImλi/Reλi/ωci.
In conclusion, according to Lemma3, the maximum allowable upper bound of the system is τ∗=minτa,τb, where(46)τa=arctank2/k1λmax2k22+λmax4k24+4λmax2k12/2+θmaxλmax2k22+λmax4k24+4λmax2k12/2,τb=arctanω˜maxk2/k1−θmaxω˜max,
θmax=arctanImλi/Reλi,ω˜max=max3≤i≤nω˜ci,ω˜ci is the root of equation 1/ωciηωci/1+ηωci2−arctanηωci−arctanImλi/Reλi=0, and η=k2/k1.


## 4. Numerical Examples and Simulations

Example 1.
We will test and verify the results obtained by a multiagent system with five agents in this section. The initial conditions are randomly set and the topology is shown in Figure1. The maximum eigenvalue of Ls is λmax=4.0445. k1=0.45 and k2=0.3. According to Theorem 1, we obtain τ∗<0.5057. Let τ1=0.5,τ2=0.4,τ3=0.3,τ4=0.2,andτ5=0.1. It is clear that bipartite consensus can be achieved when the input delays are below the upper bound of the allowable delay (see Figure 2). Let τ1=0.4,τ2=0.3,τ3=0.4,τ4=0.3,andτ5=0.6. The bipartite consensus cannot be achieved when one of input delays exceeds the upper bound of the allowable delay (see Figure 3).Figure 1
Undirected topology of a multiagent system.Figure 2
τ1=0.5,τ2=0.4,τ3=0.3,τ4=0.2,andτ5=0.1. (a) Position trajectories of all agents. (b) Velocity trajectories of all agents.
(a)(b)Figure 3
τ1=0.4,τ2=0.3,τ3=0.4,τ4=0.3,andτ5=0.6. (a) Position trajectories of all agents. (b) Velocity trajectories of all agents.
(a)(b)Example 2.
We will test and verify the results obtained by a multiagent system with six agents in this section. The initial conditions are randomly set and the topology is shown in Figure4. The maximum eigenvalue of Ls is λmax=7.9101+1.3498i. Let k1=1 and k2=1.According to Theorem 2, we obtain τ∗<0.2. When τ1=0.17,τ2=0.16,τ3=0.15,τ4=0.14,τ5=0.13,andτ6=0.12, bipartite consensus can be achieved (see Figure 5). When τ1=0.22,τ2=0.14,τ3=0.15,τ4=0.16,τ5=0.15,andτ6=0.14, bipartite consensus cannot be achieved (see Figure 6).Figure 4
Directional topology of a multiagent system.Figure 5
τ1=0.17,τ2=0.16,τ3=0.15,τ4=0.14,τ5=0.13,andτ6=0.12. (a) Position trajectories of all agents. (b) Velocity trajectories of all agents.
(a)(b)Figure 6
τ1=0.22,τ2=0.14,τ3=0.15,τ4=0.16,τ5=0.15,andτ6=0.14. (a) Position trajectories of all agents. (b) Velocity trajectories of all agents.
(a)(b)


## 5. Conclusions

Different from the previous work, we consider bipartite consensus of heterogeneous multiagent systems with diverse input delays. Based on the matrix theory and the frequency domain theory, the maximum input delay for the systems to achieve bipartite consensus is obtained. The future work will extend the existing work to time-varying input delays.



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