# Bipartite Consensus for Multiagent Systems via Event-Based Control

Authors: Cui-Qin Ma; Wei-Guo Sun
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper studies bipartite consensus for first-order multiagent systems. To improve resource utilization, event-based protocols are considered for bipartite consensus. A new type of control gain is designed in the proposed protocols. By appropriate selection of control gains, the convergence rate of the closed-loop system can be adjusted. Firstly, for structural balance case, necessary and sufficient conditions are given on communication relations and consensus gains to achieve bipartite consensus. Secondly, for structural unbalance case, necessary and sufficient conditions are proposed to ensure the stabilizing of the system. It can be found that the system will not show Zeno behavior. Numerical simulations are used to demonstrate the theoretical results.


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## Body

## 1. Introduction

Consensus problem is one of the hot topics in coordination of multiagent systems (MASs). In the last few years, research on consensus has received considerable attention [1–9]. Consensus means that agents can reach a common value through cooperative relations among agents. However, in real applications, not only cooperative but also competitive relations among agents exist. In these circumstances, bipartite consensus is studied [10–20]. Based on the cooperation and competition among agents, bipartite consensus can be achieved if agents agree upon a certain value with the same quantity and different signs. In [10], necessary and sufficient conditions for bipartite consensus of the single-integrator MASs are given. In [12], the communication condition is first reduced to be containing a spanning tree. In [14], the communication topology is extended to the time-varying case. In [15–20], bipartite consensus with measurement noise is considered.It is worth pointing out that the above literature adopts a time-driven control pattern, where the state of the agents is monitored continuously and the control law updates are done at any moment. In practical implementation, the embedded processors are often resource-limited and thus an event-based control fashion is more beneficial in MASs. For conventional consensus of MASs, an event-based control fashion was thoroughly studied [21–31]. In the pioneering work [21], an event-based feedback protocol was proposed. In [23], the event-based protocols for both fixed and switching topologies have been considered. In [25], the self-triggered protocol of MASs was taken into account. Then, in [28], a new event-based protocol for average consensus of MASs was proposed and continuous monitoring of agents’ states was not required. The event-based consensus for general linear MASs can be found in [29–31]. Despite these productive results, works on bipartite consensus with event-based control strategy are still rare.In this paper, we consider event-based bipartite consensus for first-order MASs. In contrast to [32, 33], a new function is introduced into the event-based protocol, such that the Laplacian-like event-based bipartite consensus protocols in [32, 33] are special cases of this paper. Due to the new function gain, the closed-loop system is time-varying. By use of state transition matrix, the closed-loop system is analyzed. For structural balance case, necessary and sufficient conditions are given on communication relations and consensus gains to achieve bipartite consensus. For structural unbalance case, necessary and sufficient conditions are proposed to ensure the MAS stabilizing.Organization.In Section 2, we give some basic concepts on signed graph and formulate the problem. In Section 3, we prove the main results. In Section 4, we show the validity of theoretical analysis through the simulation results. In Section 5, we conclude this paper and put forward further research directions.Notation. R n × m represents all real matrices of n×m order. 0 denotes vector or matrix whose elements are 0. 1n represents column vector whose elements are 1. sgn(·) represents sign function.


## 2. Problem Formulation

The communication relations amongN agents are expressed by the signed digraph G=(V,E,A), where V={1,…,N} is the node set and E is the edge set. A=(aij)∈RN×N is the weighted adjacency matrix of G, where aij>0 and aij<0 represent competition relationship and cooperation relationship between i and j, respectively. Throughout this paper, we always assume that aij≠0⇔(j,i)∈E, aii=0, and aijaji≥0, ∀i,j∈V. L=Cr-A is the Laplacian of G, where Cr=diag(∑p=1Na1p,…,∑p=1NaNp). G=(V,E,A) is structurally balanced if V can be divided into two subsets V1,V2,V1∪V2=V,V1∩V2=⌀, satisfying aij≥0 for ∀i,j∈Vt(t∈{1,2}), and aij≤0 for ∀i∈Vt,j∈Ve(t≠e∈{1,2}). G is structurally unbalanced otherwise.Consider an MAS withN agents, whose dynamics obey the following equation:(1)x˙it=uit,i=1,…,N,where xi(t)∈R and ui(t)∈R are the state and control input of the ith agent.We useG=(V,E,A) to express the communication relations among the N agents.Considering the limit resources, people would like to reduce the frequency of control law updates. In this case, an event-based control law is more favorable. Our aim here is to provide an event-based control in order that all agents’ states converge to values with the same modulus and different signs regardless of initial states.To achieve this goal, we assume that each agent only updates its control law at discrete times indexed byt0,t1,…. We define the event-based control as follows:(2)uit=ct∑j=1Naijsgnaijxjtl-xitl,∀t∈tl,tl+1,where i=1,…,N,l=0,1,…,c(t)>0 is a piecewise continuous function.Remark 1.
The control law will be actuated at discrete event times. A proper functionc(t) will be designed to improve the convergence performance of the closed-loop system. In particular, when c(t)=1, protocol (2) is reduced to event-based bipartite consensus protocols in [32, 33].The state measurement error of theith agent is denoted by ϵi(t)=xi(tl)-xi(t),i=1,…,N,t∈[tl,tl+1),l=0,1,…. Let Ω(t)=(ϵ1(t),…,ϵN(t))T and, hence,(3)X˙t=-ctLXt+Ωt,where X(t)=x1(t),…,xN(t)T.We introduce the following definition to characterize the behavior of (3).Definition 2.
System (1) is said to achieve bipartite consensus via event-based protocol U={ui,i=1,…,N}, if for system (1) with any given X(0)∈RN, there exists f=(f1,…,fN)T∈RN(fi∈{±1},i=1,…,N) and μ∗∈R such that limt→∞⁡Xt-μ∗f=0, where μ∗∈R depends on X(0) and the communication relations among agents.We provide the following event triggering conditions:(4)ϵit≤Me-γt,i=1,…,N,where M>0,0<γ<minλ(L)≠0⁡{Reλ(L)}.When the measurement errorϵi(t) is over the threshold, the controller is triggered and updates itself. To analyze (3), we introduce assumptions as follows:(T1)
G=(V,E,A) is structurally balanced.(T2)
G=(V,E,A) contains a spanning tree.(T3)
∫0∞c(s)ds=∞.The following lemma is highly related to the subsequent results.Lemma 3.
If system (1) can achieve bipartite consensus via event-based protocol (2), then there exist f=(f1,…,fN)T∈RN,fi∈{±1},i=1,…,N, and ϕ=(ϕ1,…,ϕN)T∈RN, such that limt→∞⁡Θ(t,0)=fϕT, where Θ(t,0) is the state transition matrix of (3).Proof.
The proof is omitted due to space limit.


## 3. Main Results

The following result is the main result of this section.Theorem 4.
System (1) can achieve bipartite consensus via event-based protocol (2) if and only if (T1),(T2), and (T3) hold.Proof.
⁡
Sufficiency. Assume Θ(t,t0) is the state transition matrix of (3). Then, Θ(t,t0)=e-∫t0tc(s)dsL. Since (T1) and (T2) hold, Lemma 1 of [17] implies that L has exactly one zero eigenvalue and all nonzero eigenvalues are in the open right half plane. Thus, there exists Q satisfying(5)Q-1LQ=B=diag0,B2,…,Bw,where Bi(i=2,…,w) is Ri×Ri dimensional Jordan block with ςi on its diagonal, R2+⋯+Rw=N-1. Obviously, ς2,…,ςw are the eigenvalues of L, and Re(ςi)>0,i=2,…,w. Combining this, we can get the state transition matrix:(6)Θt,t0=Qdiag1,ΘB2ς2t,t0,…,ΘBwςwt,t0Q-1,where ΘBkςk,k=2,…,w, is defined as in Lemma 3 of [17]. Thus, from (T3), we know that limt→∞⁡Θ(t,t0)=Qdiag(1,0,…,0)Q-1. By (3), X(t)=Θ(t,0)X(0)-XΩ, where XΩ=∫0tc(s)Θ(t,s)LΩ(s)ds. By (6), one has(7)XΩ=Q∫0tcsdiag1,ΘB2ς2t,s,…,ΘBwςwt,sQ-1LΩsds.Furthermore, by (5), one gets(8)Q-1LΩs=diag0,B2,…,BwQ-1Ωs.Assume Q-1=(Qij). Then, Q-1Ω(s)=∑j=1NQ1jϵj(s),…,∑j=1NQNjϵj(s)T. This together with (8) gives(9)Q-1LΩs=0,Ω2∗s,…,ΩN∗sT,where Ωi∗(s)(i=2,…,N) is the linear combination of ς2,…,ςw and ∑j=1NQkjϵj(s),k=1,…,N. Then, by (4), there must exist q∗>0, such that(10)Ωi∗s≤q∗e-γs,i=2,…,N.
Considering the specific form ofΘBiςi(t,s)(i=2,…,w) in Lemma 3 of [17], we get that XΩ=Q0,Λ2Ω(s),…,ΛNΩ(s)T, where ΛpΩ(s)(p=2,…,N) is the linear combination of ∫0tc(s)e-ςi∫stc(τ)dτ∫stc(τ)dτmΩq∗(s)ds,m=0,1,…,Ri-1;i=2,…,w;q=2,…,N.
By direct calculation, one obtains that(11)limt→∞∫0tcse-ςi∫stcτdτ∫stcτdτme-γsds=limt→∞m!ςim+1eγt=0,m=0,1,…,Ri-1.This together with (10) leads to(12)limt→∞∫0tcse-ςi∫stcτdτ∫stcτdτmΩq∗sds=0,m=0,1,…,Ri-1.Therefore, limt→∞⁡XΩ=0, and thus limt→∞⁡X(t)=Qdiag(1,0,…,0)Q-1X(0)=QcQrTX(0), where Qc is the first column of Q, QrT is the first row of Q-1, and QrTQc=1. By (5), one gets that LQc=QrTL=0. From (T1), we know that there exists f such that Lf=0. Since the eigenspace corresponding to eigenvalue 0 is 1 dimensional, Qc=m0f, where m0≠0 is a constant. Denote μ∗=m0QrTX(0). Then, limt→∞⁡X(t)=μ∗f and μ∗ depends on X(0) and communication relations among agents. Therefore, Sufficiency holds.
Necessity. Assume by contradiction that (T3) does not hold. Without loss of generality, let ∫0∞c(s)ds=c->0. Then, limt→∞⁡Θ(t,0)=e-c-L. It is an invertible matrix and thus rank(limt→∞⁡Θ(t,0))=N. This contradicts Lemma 3. Therefore, (T3) holds. Next, we prove that L has exactly one zero eigenvalue. If 0 is not eigenvalue L, then limt→∞⁡Θ(t,0)=0. This implies that limt→∞⁡X(t) is independent of X(0). It contradicts Definition 2. Thus, 0 is eigenvalue L. If the geometric multiplicity of eigenvalue 0 is less than the algebraic multiplicity of 0, then by (T3) and direct calculation, one has the notion that limt→∞⁡Θ(t,0) does not exist. It is a contradiction with Lemma 3. So, the geometric multiplicity equals the algebraic multiplicity of eigenvalue 0. Assume 0 is p dimensional and p>1. Then, with abuse of notation, Q-1LQ=diag(0,…,0︸p,Bp+1,…,Bw). Hence, limt→∞⁡Θ(t,0)=Qdiag(1,…,1︸p,ΘBp+1ςp+1(t,0),…,ΘBwςw(t,0))Q-1 and rank(limt→∞⁡Θ(t,0))=p>1. This contradicts Lemma 3. So, p=1; i.e., L has exactly one zero eigenvalue. With abuse of notation, (5) and (6) hold. Noticing Lemma 3, one gets Qdiag(1,0,0,…,0)Q-1=fϕT. Therefore, Qc=fm1, where m1=ϕTQc≠0. Since LQc=0, Lf=0. By the definition of Laplacian L, for any q, one gets fq∑k≠qaqk=fkaqk,q=1,…,N. Noticing fq=±1, one has fqfkaqk=aqk≥0. Let V1={q∣fq=1} and V2={q∣fq=-1}; then V1∩V2=⌀,V1∪V2=V. If q∈Vj,j∈{1,2}, aqk≥0,k∈Vj or aqk≤0,k∈Vr,r≠j,r∈{1,2}. By definition, G is structurally balanced; i.e., (T1) holds. From Lemma 1 of [17], G has a spanning tree; i.e., (T2) holds.Remark 5.
From (11), one obtains that the convergence rate of the closed-loop system is closely related to eigenvalues of L and the rate of ∫0tc(s)ds converging to infinity. It follows that, by appropriate selection of c(t), the convergence rate of (3) can be adjusted.From Theorem4, we can see that structural balance is a necessary and sufficient condition to ensure bipartite consensus. When G is structurally unbalanced, we investigate the evolution of the MAS:T1′
G=(V,E,A) is structurally unbalanced.T2′
G=(V,E,A) does not contain structurally balanced input solitary subgraphs.An input solitary subgraph ofG indicates that agents of the subgraph cannot obtain information from other agents of G.Theorem 6.
System (2) can be stabilizing via event-based protocol (2), i.e., limt→∞⁡X(t)=0 if and only if (T1′), (T2′), and (T3) hold.Proof.
⁡
Necessity. The necessity is similar to the proof of Theorem 4.
Sufficiency. From (T1′), (T2′), and Lemma 2 of [17], we can see that all eigenvalues of L have positive real parts. Since (T3) holds, one has limt→∞⁡X(t)=0 by repeating the sufficiency proof of Theorem 4.


## 4. Numerical Simulation

Example 1.
Six agents’ communication relations are expressed by Figure1, where G1=(V,E1,A1), V={1,…,6},a12=a21=a36=2, a45=a61=-1, and a24=a53=1. Obviously, G1 satisfies assumptions (T1) and (T2). Laplacian L1 has eigenvalues λ1(L1)=0, λ2(L1)=4.5698, λ3(L1)=0.6852+0.8449ι, λ4(L1)=0.6852-0.8449ι, λ5(L1)=2.0299+0.5637ι, and λ6(L1)=2.0299-0.5637ι(ι2=-1). It can be seen that 0 is a simple eigenvalue of L1, and minλ(L1)≠0{Reλ(L1)}=0.6852. Agents’ dynamics satisfy (1). Assume X(0)=(1,2,-1.5,2.5,-3,-0.5). If we choose M=1.5, γ=0.65, and c(t)=1 in (2), then c(t) satisfies (T3) and state trajectories are given in Figure 2. One can find that agents 1, 2, and 4 converge to 1.7 while agents 3, 5, and 6 converge to -1.7; i.e., system (1) achieves bipartite consensus via event-based protocol (2). From Figure 3, we know that the inputs are constants between the event triggering time intervals. Furthermore, as shown in Figure 4, error norm of each agent converges to zero. This means that the MAS avoids the Zeno behavior.Figure 1
The communication relationsG1.Figure 2
Evolution of the states for six agents.Figure 3
Evolution of control inputs for six agents.Figure 4
Six agents’ error norms.Example 2.
When communication relationship among the six agents is given byG2=(V,E2,A2) as shown in Figure 5, where a12=a21=a32=a43=a54=1, a13=a31=-2 and a65=2. Eigenvalues of Laplacian L2 are λ1(L2)=0.7639, λ2(L2)=λ3(L2)=λ4(L2)=1, λ5(L2)=2, and λ6(L2)=5.2316, respectively. G2 satisfies (T1′) and (T2′). It is easy to know that minλ(L2)≠0{Reλ(L2)}=0.7639. Let M=1.2, γ=0.6, and c(t)=1; the state trajectories of the MAS can be obtained as shown in Figure 6. All states eventually converge to 0. This is consistent with Theorem 6. The evolution of control input and error norm are shown in Figures 7 and 8, respectively.Figure 5
The communication relationsG2.Figure 6
Six agents’ states underG2.Figure 7
Six agents’ control inputs underG2.Figure 8
Six agents’ error norms underG2.


## 5. Conclusion

In this paper, event-driven protocols are considered for bipartite consensus of MASs. Based on them, the number of controller updates is reduced. Under necessary and sufficient conditions on protocol gain and communication topology, the MAS is shown to reach event-based bipartite consensus. When the graph is structurally unbalanced, the MAS is proved to be stabilizing. The further research is related to MASs with time-varying topology and time delays.



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