# Event-Triggered Bipartite Consensus of Single-Integrator Multi-Agent Systems with Measurement Noise

Authors: Cui-Qin Ma; Yun-Bo Zhao; Wei-Guo Sun
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

Event-triggered bipartite consensus of single-integrator multi-agent systems is investigated in the presence of measurement noise. A time-varying gain function is proposed in the event-triggered bipartite consensus protocol to reduce the negative effects of the noise corrupted information processed by the agents. Using the state transition matrix, Ito^ formula, and the algebraic graph theory, necessary and sufficient conditions are given for the proposed protocol to yield mean square bipartite consensus. We find that the weakest communication requirement to ensure the mean square bipartite consensus under event-triggered protocol is that the signed digraph is structurally balanced and contains a spanning tree. Numerical examples validated the theoretical findings where the system shows no Zeno behavior.


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## Body

## 1. Introduction

Recent years have witnessed the great achievements in studying the consensus problem of multi-agent systems (MASs) which has broad applications in various fields [1–8]. We notice that in these mentioned works interactions among agents are all assumed to be cooperative to achieve consensus. However, it is very natural to see, in many real examples, that in MASs some agents cooperate while others compete, and MASs with competitive interactions can introduce more complex behaviors. To quantitatively model such a scenario, the concept of bipartite consensus, i.e., agents agree on a certain quantity with the equal modulus but different signs, has been proposed [9], and many achievements have been made [9–18]. In [9], for single-integrator MASs, a linear feedback protocol is designed and under the assumption that the communication topology G is strongly connected, the MAS is proved to achieve bipartite consensus if and only if G is structurally balanced. Then, in [10], the communication condition in [9] is relaxed to containing a spanning tree. In [11], the communication topology in [9] is extended to the time-varying case.It is worth noting that the above literatures mainly focus on continuous feedback protocols, where the agent state is monitored continuously and its controller is updated all the time. However, updating the controller in real-time easily increases the computational burden. Therefore, reducing the update frequency for a trade-off between the system performance and the resource usage is usually desired. This requirement then naturally brings event-triggered schemes into consideration, which updates only at some predetermined discrete time instants. Event-triggered techniques have already been widely used in traditional consensus problems of MASs [19–27]. For example, a self-triggered protocol is proposed in [19] and a decentralized event-triggered protocol ensuring average consensus is proposed in [20] for single-integrator MASs, time-dependent triggering functions are investigated in [24] for second-order MASs, and event-triggered consensus problems are considered in [25, 26] for general linear systems, just name a few. Despite these achievements, event-triggered protocols have not been well studied for bipartite consensus [28, 29], which thus motivates the present study.In another parallel line, measurement noise is unavoidable in practice, making the investigation on the event-triggered bipartite consensus of MASs with noise even interesting. In fact, studies on bipartite consensus with measurement noise can be found in [13, 16–18], which are however all with time-triggered controllers. Event-triggered bipartite consensus for MASs with measurement noise still remains to tackle.In this paper, we investigate event-triggered bipartite consensus for single-integrator MASs with measurement noise. A time-varying control gain is introduced into the event-triggered protocols, leading to a time-varying closed-loop system. With the help of the state transition matrix and stochastic analysis theory, the closed-loop system is analyzed. Necessary and sufficient conditions for the system to achieve mean square bipartite consensus based on event-triggered protocols are given. We find that the communication topology being structurally balanced and containing a spanning tree are necessary and sufficient for ensuring a mean square bipartite consensus based on event-triggered protocols.Organization. Section 2 gives the algebraic graph preliminaries and the problem in question. Section 3 contains the main results of the paper. Section 4 applies the results to examples of MASs with six agents. Section 5 closes this paper.Notations. R n × m represents the real matrix of n×m order. 0 denotes vector or matrix whose elements are 0. 1n represents column vector whose elements are 1. sgn⁡(·) represents the sign function. ⊗ represents Kronecker product. For a given matrix or vector X, XT, and X represent the transpose and European norm of X, respectively. XF, X1, and X∞ represent the Frobenius norm, 1-norm, and ∞-norm, respectively. Re⁡(λ) is the real part of λ.


## 2. Problem Statement

The communication relations amongN agents are described by the signed digraph G=(V,E,A), where V=1,…,N and E⊆V×V represent the node set and the edge set, respectively. A=(aij)∈RN×N, where aij>0 and aij<0 represent cooperation and competition between agents i and j, respectively. aij≠0⇔(j,i)∈E. We assume that aii=0 and aijaji≥0, ∀i,j∈V. L=Cr-A is the Laplacian matrix of G, where Cr=diag⁡∑j=1Na1j,…,∑j=1NaNj. A signed digraph G=(V,E,A) is said structurally balanced if V can be divided into two subsets V1,V2,V1∪V2=V,V1∩V2=⌀, such that aij≥0, ∀i,j∈Vp(p∈{1,2}), and aij≤0, ∀i∈Vp,j∈Vq(p≠q,p,q∈{1,2}). It is said structurally unbalanced otherwise.Lemma 1 (see [12]).
IfG is structurally balanced, Laplacian L of G has at least one zero eigenvalue and all of the nonzero eigenvalues have positive real parts. Furthermore, L has only one zero eigenvalue if and only if G has a spanning tree.Consider a MAS described by(1)x˙it=uit,i=1,…,N,where xi(t)∈Rn is the state of the ith agent and ui(t)∈Rn is the control input. A signed digraph G=(V,E,A) is used to describe interactions among the N agents.Since communication is often disturbed by measurement noise, we assume theith agent receives information from its neighbors with measurement noise xj(t)+φji(t),j∈Ni,i=1,…,N. In order to reduce the frequency of controller updates, we design the following event-triggered protocol for the ith agent:(2)uit=bt∑j=1Naijsgn⁡aijxjtk-xitk+∑j=1Naijφjit,∀t∈tk,tk+1,where i=1,…,N,k=0,1,…,b(t)>0 is a piecewise continuous function. {φji(t)} is n dimensional independent standard white noise.Remark 2.
As far as we know the existing results [28, 29] for event-triggered bipartite consensus did not consider measurement noise. Here, we take noise into consideration. If we take b(t)≡1, then (2) is reduced to the protocols in [28, 29] without measurement noise.LetX(t)=(x1T(t),…,xNT(t))T and J=diag⁡(ς1Tt,…,ςNTt)T be N×N2 dimensional block diagonal matrix, where ςiT(t)=(ai1,…,aiN) is the ith row element of matrix A. Then the closed-loop system is(3)dXt=-btL⊗InXtkdt+btJ⊗IndΛt,t∈tk,tk+1,k=0,1,⋯where Λ(t)=Λ1T(t),…,ΛNT(t)T and Λi(t)=Λ1iT(t),…,ΛNiT(t)T, i=1,…,N. For i,j=1,…,N, ∫0tφji(s)ds=Λji(t) is n dimensional standard Brownian motion. Let e(t)=(e1(t),…,eN(t))T be the measurement error, where ei(t)=xi(tk)-xi(t),t∈[tk,tk+1), k=0,1,⋯. Then (3) is changed to(4)dXt=-btL⊗InXt+etdt+btJ⊗IndΛt,t∈tk,tk+1,k=0,1,⋯We present the following definition of event-triggered bipartite consensus for the stochastic system.Definition 3.
LetU={ui,i=1,…,N} be an event-triggered protocol. If for any given X(0)∈RnN, there exist g=(g1,…,gN)T∈RN, gi∈{±1}, i=1,…,N and n dimensional random vector ν∗, (5)limt→∞⁡EXt-g⊗ν∗2=0,where Eν∗2<∞, Eν∗ is dependent on communication relations among agents and X(0), which is deterministic. Then, event-triggered protocol U is called a mean square bipartite consensus protocol.We introduce the event-triggered condition(6)et≤c1e-αt,where c1>0,0<α<minλ(L)≠0⁡{Re⁡(λ(L))}. When the measurement error e(t) is over the threshold, the controller is triggered and updates itself.To analyze the closed-loop system in (4), we make the following assumptions:(Q1)G=(V,E,A) is structurally balanced.(Q2)G=(V,E,A) contains a spanning tree.(Q3)∫0∞b(s)ds=∞.(Q4)∫0∞b2(s)ds<∞.The following lemma plays an important role in the following section.Lemma 4 (see [16]).
Given linear time-varying system(7)dYldt=-btFlλYl,l∈N,λ∈C,t≥t0≥0,where Yl=(yl1,…,ylRl)T∈RRl and Flλ is the Rl×Rl dimensional Jordan block, which λ is the diagonal element. Then the state transition matrix of (7) is ΨFlλ(t,t0)=e-∫t0tb(s)dsFlλ. In addition, we can obtain limt→∞⁡ΨFlλ(t,t0)=0 if ∫0∞b(s)ds=∞ and Re⁡(λ)>0.Lemma 5.
If the event-triggered protocol (2) is a mean square bipartite consensus protocol, then ∃g=(g1,…,gN)T∈RN, gi∈{±1}, i=1,…,N, and θ=(θ1,…,θN)T∈RN, such that limt→∞⁡Ψ(t,0)=gθT⊗In, where Ψ(t,0) is the state transition matrix of (4).Proof.
From the above condition, Definition3 implies that for any given initial state X(0), there exist a vector g and a random vector ν∗ so that limt→∞⁡EXt-g⊗ν∗2=0. Obviously, (8)Xt=Ψt,0X0-∫0tbsΨt,sL⊗Inesds+∫0tbsΨt,sJ⊗IndΛs.Without loss of generality, we assume ∫0tb(s)Ψ(t,s)(L⊗In)e(s)ds and ∫0tb(s)Ψ(t,s)(J⊗In)dΛ(s) converge to Y∗ and Z∗ in mean square sense, respectively. Then, (9)g⊗Eν∗=limt→∞⁡EXt=Ψ∞⊗InX0-EY∗+EZ∗,where limt→∞⁡Ψ(t,0)=Ψ∞⊗In. According to Definition 3 and the arbitrariness of X(0), one obtains (Ψ∞⊗In)X(0)=g⊗Ed∗, where Ed∗∈RN.
LetΨ∞=(ϱ1,…,ϱN). Then, all elements of ϱi have the same absolute value. The same applies for ∑j=1N(ϱj⊗τj), where τj∈Rn,j=1,…,N. If Ψ∞=0, then by making θ=0, Lemma 5 holds. If Ψ∞ has at least one nonzero column, without loss of generality, we assume ϱ1≠0. Then ϱ1=θ1g. Without loss of generality, we assume θ1>0. For any α,β∈Rn,a,b∈R,a,b≠0, aα+bβ≠±(aα-bβ). If ϱj≠0 for some j≠1, then all n dimensional components of ϱj⊗α-ϱ1⊗β have the same modulus if and only if ϱj=θjg,θj>0. If ϱj=0, we have ϱj=θjg by taking θj=0. Then Ψ∞=(ϱ1,…,ϱN)=(θ1g,…,θNg)=g(θ1,…,θn). In addition, limt→∞⁡Ψ(t,0)=Ψ∞⊗IN, so limt→∞⁡Ψ(t,0)=g(θ1,θ2,…,θN)⊗In=gθT⊗In.Lemma 6.
If(Q1)-(Q4) hold, then for any given initial state X(0), there is a random vector X∗ such that X(t) converges to X∗ in mean square sense, i.e., limt→∞⁡EX(t)-X∗2=0.Proof.
If(Q1) and (Q2) hold, then Laplacian L has exactly one zero eigenvalue and all nonzero eigenvalues have positive real parts by Lemma 1. Thus, there exists an invertible matrix D, such that(10)D-1LD=F=diag⁡0,F2,…,Fγ,where Fi(i=2,…,γ) is the Ri×Ri dimensional Jordan block, which λi is the diagonal element, and R2+⋯+Rγ=N-1. Obviously, λ2,…,λγ are eigenvalues of L and Re⁡(λi)>0,i=2,…,γ.
SinceΨ(t,t0)(t0≥0) is the state transition matrix of (4), Ψ(t,t0)=e-∫t0tb(s)dsL⊗In. From Lemma 4,(11)Ψt,t0=D⊗Indiag⁡In,ΨF2λ2t,t0⊗In,…,ΨFγλγt,t0⊗InD-1⊗In.Combining this with (Q3), one has(12)limt→∞⁡Ψt,t0=Ddiag⁡1,0,0,…,0D-1⊗In.Thus, there exists T>0 so that for any t≥t0>0,(13)max⁡Ψt,t01,Ψt,t0∞≤T<∞.
By Ito^ formula, the solution of (4) is given by (14)Xt=Ψt,0X0-∫0tbsΨt,sL⊗Inesds+∫0tbsΨt,sJ⊗IndΛs.
By(Q4), one obtains that, ∀ε>0, ∃Γ0>0, ∫Γ0∞b2(s)ds<ε. By (12), ∃Γ1>Γ0, such that max⁡Ψt2,t3-Ψt1,t31,Ψt2,t3-Ψt1,t3∞<ε, t2≥t1>Γ1, ∀t3∈[0,Γ0].
LetX2(t)=∫0tb(s)Ψ(t,s)(L⊗In)e(s)ds, then by (10) and (11), one has(15)X2t=∫0tbsD⊗Indiag⁡In,ΨF2λ2t,s⊗In,…,ΨFγλγt,s⊗InD-1L⊗Inesds.By (6), (10), and direct calculation, one has D-1L⊗Ine(s)=(0,D2T(s),…,DNT(s))T, where Di(s)(i=2,…,N) is the linear combination of e1(s),…,eN(s). By L’Hospital and direct calculation, one obtains (16)limt→∞⁡∫0tbse-λi∫stbτdτ∫stbτdτme-αsds=0,m=0,1,…,Ri-1;i=2,…,γ.Noticing that ΨFiλi(t,s)=∑d=0∞-∫stbτdτdFiλid/d!, one has limt→∞⁡X2(t)=0.
LetX3(t)=∫0tb(s)Ψ(t,s)(J⊗In)dΛ(s), then (17)X3t2-X3t1=∫0t2bsΨt2,s-Ψt1,sJ⊗IndΛs+∫t1t2bsΨt1,sJ⊗IndΛs≜X31+X32.Therefore EX3(t2)-X3(t1)2≤2EX312+2EX322. It is easy to obtain (18)EX312=∫0t2b2sΨt2,s-Ψt1,sJ⊗InF2ds.Noting(19)∫0T0b2sΨt2,s-Ψt1,sJ⊗InF2ds≤nNJ⊗In12∫0∞b2sds≜M4ε2and (20)∫Γ0t2b2sΨt2,s-Ψt1,sJ⊗InF2ds≤4nNL2J⊗In12ε≜M5ε,one has EX312≤M4ε2+M5ε. Similarly, one obtains (21)EX322=∫t1t2b2sΨt1,sJ⊗InF2ds≤nNL2J⊗In12ε≜M54ε.So EX3(t2)-X3(t1)2≤2M4ε2+5/2M5ε. By Cauchy criterion and the arbitrariness of ε, there exists X3∗ such that X3(t) converges to X3∗ in mean square sense. So there exists X∗ such that X(t) converges to X∗ in mean square sense. By (12), X∗=Ddiag⁡(1,0,…,0)D-1⊗InX(0)+X3∗.


## 3. Main Results

In this section, we give necessary and sufficient conditions for the proposed event-triggered protocols to guarantee a mean square bipartite consensus.Theorem 7.
The event-triggered protocol in (2) is a mean square bipartite consensus protocol for the system in (1) if and only if (Q1)-(Q4) hold.Proof (sufficiency).
(S.1) Construct a Bipartition for the MAS. By (Q3), V can be decomposed into two disjoint subsets Vv1,Vv2, Vv1∪Vv2=V, Vv1∩Vv2=⌀, and ast≤0 for s∈Vf,t∈Vg,f≠g,f,g∈{v1,v2}, and ast≥0 for s,t∈Vp,p∈{v1,v2}. Without loss of generality, we assume Vv1={1,…,m}, Vv2={m+1,…,N}. Let gi=1 for i∈Vv1 and gj=-1 for j∈Vv2. By definition, one has Lg=0, where g=(g1,…,gN)T.
(S.2) Prove lim t → ∞ ⁡ E X ( t ) - g ⊗ ν ∗ 2 = 0 . From Lemma 6, ∃X∗, limt→∞⁡EX(t)-X∗2=0. Without loss of generality, we assume limt→∞⁡Ex1(t)-ν∗2=0. Next, we will prove limt→∞⁡Exi(t)-giν∗2=0,i=2,…,N.
Letϕ(t)=ϕ2T(t),…,ϕNT(t)T, where ϕi(t)=xi(t)-gix1(t),i=2,…,N. Now we prove that limt→∞⁡Eϕ(t)2=0. For this purpose, We assume ϕ~(t)≜(Q⊗In)X(t), where(22)Q≜1N1N⋯1N-1N⋯-1N-1m-1Im-10-1N-m0-IN-m=1NgTQ2.
Thenϕ~t=χTt,ϕTtT, where χt≜1/NgT⊗InXt. Since(23)QLQ-1=0ϖT0L2,
by (4), one has(24)dϕt=-btL2⊗Inϕtdt-btQ2L⊗Inetdt+btQ2J⊗IndΛt.By (23), S-1L2S=diag⁡(F2,…,Fγ), where S is invertible and F2,…,Fγ are given in (10). The state transition matrix of the system in (24) is (25)Ψ2t,t0=S⊗Indiag⁡ΨF2λ2t,t0⊗In,…,ΨFγλγt,t0⊗InS-1⊗In,where ΨFqλqt,t0,q=2,…,γ are defined as in Lemma 4. Hence, limt→∞⁡Ψ2(t,t0)=0, i.e., ∀ε>0, ∃Γ2>Γ1, such that Ψ2(t,t0)<ε, ∀t>Γ2. Furthermore, ∃TL2>0, such that, ∀t>t0≥0, max⁡(Ψ2t,t01,Ψ2t,t0∞)≤TL2<∞.
By Ito^ formula, it can be seen that the state of the system in (24) can be described as (26)ϕt=Ψ2t,0ϕ0-∫0tbsΨ2t,sQ2L⊗Inesds+∫0tbsΨ2t,sQ2J⊗IndΛs.Therefore, (27)Eϕt2=Ψ2t,0ϕ02-2ϕT0Ψ2Tt,0∫0tbsΨ2t,sQ2L⊗Inesds+∫0tbsΨ2t,sQ2L⊗Inesds2+∫0tb2sΨ2t,sQ2J⊗InF2ds,and hence, (28)Eϕt2≤2Ψ2t,0ϕ02+2∫0tbsΨ2t,sQ2L⊗Inesds2+∫0tb2sΨ2t,sQ2J⊗InF2ds.Since (29)∫0tbsΨ2t,sQ2L⊗Inesds2≤∫0tb2sΨ2t,s2ds∫0tQ2L⊗In2es2dsand (30)∫0tQ2L⊗In2es2ds≤Q2L⊗In2∫0tc12e-2αsds≤c122αQ2L⊗In2,there exists β3>0 such that ∫0t(Q2L⊗In)2e(s)2ds≤β3. Then (31)∫0tbsΨ2t,sQ2L⊗Inesds2⩽β3∫0Γ0b2sΨ2t,s2ds+β3∫Γ0tb2sΨ2t,s2ds.Since ∫0Γ0b2(s)Ψ2(t,s)2ds=Ψ2(t,η)2∫0Γ0b2(s)ds≤Ψ2(t,η)2∫0∞b2(s)ds, where η∈(0,Γ0) and ∫Γ0∞b2(s)ds<ε, one has ∫Γ0tb2(s)Ψ2(t,s)2ds≤TL22ε≜M6ε. Therefore, ∀t>T2(32)∫0tbsΨ2t,sQ2L⊗Inesds2≤β3Ψ2t,η2∫0∞b2sds+M6β3ε≤ε2β3∫0∞b2sds+M6β3ε.From ∫Γ0∞b2(s)ds<ε, one gets (33)∫Γ0tb2sΨ2t,sQ2J⊗InF2ds<nNTL22Q2J⊗In12ε≜M7ε.
Combining this with(34)∫0Γ0b2sΨ2t,sQ2J⊗InF2ds≤nNΨ2t,ξ12Q2J⊗In12∫0∞b2sds,ξ∈0,Γ0,one has (35)Eϕt2≤2ϕ02ε2+2β3ε2∫0∞b2sds+nNε2Q2J⊗In12∫0∞b2sds+2M6β3ε+M7ε. By the arbitrariness of ε, one gets limt→∞⁡Eϕ(t)2=0. Hence, limt→∞⁡EX(t)-g⊗ν∗2=0.
(S.3) Analyze the Statistical Characteristics of ν ∗ . By Lemma 6, g⊗ν∗=X∗=Ddiag⁡(1,0,0,…,0)D-1⊗InX(0)+X3∗. So g⊗Eν∗=EX∗=Ddiag⁡(1,0,0,…,0)D-1⊗InX(0).
We assumemr, mlT represent the first column of D and the first row of D-1, respectively. Then, g⊗Eν∗=(mrmlT⊗In)X(0). Since D-1LD=F, LD=DF, and D-1L=FD-1, Lmr=0 and mlTL=0. By (S.1), Lg=0. Therefore, mr=κg(κ∈R and κ≠0) and g⊗Eν∗=g⊗κ(mlT⊗In)X(0). Then Eν∗=κ(mlT⊗In)X(0). Clearly, ml is concerned with communication topology. Thus, Eν∗ is determined by X(0) and communication topology of MASs.
It is easy to obtain thatΨ(,) is uniformly bounded. Therefore, ∀ε>0, ∃Γ3>Γ2,(36)∫Γ3∞b2sΨt,sJ⊗InJ⊗InTΨTt,sds<ε,∫Γ3∞b2smrmlT⊗InJ⊗InJ⊗InTmrmlT⊗InTds<ε.
LetX4t=∫0Γ3b2(s)Ψ(t,s)(J⊗In)(J⊗In)TΨT(t,s)ds. Then for any t>Γ4, X4(t)-∫0Γ3b2(s)(mrmlT⊗In)(J⊗In)(J⊗In)T(mrmlT⊗In)Tds<ε. This together with (36) leads to limt→∞⁡∫0tb2(s)Ψ(t,s)(J⊗In)(J⊗In)TΨT(t,s)ds=∫0∞b2(s)(mrmlTJJTmlmrT⊗In)ds=Θ(ggT⊗In), where Θ=κ2(mlTJJTml)∫0∞b2(s)ds. Combining this with D(X∗)=limt→∞⁡∫0tb2(s)×Ψ(t,s)(J⊗In)(J⊗In)TΨT(t,s)ds, one gets D(ν∗)=ΘIn. Therefore, Eν∗2<∞. By Definition 3, the sufficiency is established.
Necessity.
(B.1) Prove ( Q 3 ) , Namely, ∫ 0 ∞ b ( s ) d s = ∞ . By contradiction, we assume that (Q3) does not hold. Then, ∃ϵ¯>0, ∫0∞b(s)ds=ϵ¯, and limt→∞⁡e-∫0tb(s)dsL=e-ϵ¯L. Therefore, rank⁡limt→∞⁡Ψt,0=rank⁡(e-ϵ¯L⊗In)=nN. However, by Lemma 5, limt→∞⁡Ψ(t,0)=gθT⊗In and rank⁡(limt→∞⁡Ψ(t,0))≤n. This is a contradiction. So (Q3) holds.
(B.2) Prove That Laplacian Matrix L Has Exactly One Zero Eigenvalue. By contradiction, we assume that 0 is not an eigenvalue of L. Then all the eigenvalues of L have positive real part and -L is a Hurwitz matrix. By (Q3) and Lemma 4, limt→∞⁡Ψ(t,0)=0. Combining this with Lemma 5, one has g⊗Eν∗=EZ∗-EY∗. Since EZ∗ and EY∗ are independent of X(0), Eν∗ is independent of X(0). This contradicts Definition 3. So 0 is an eigenvalue of L.
LetF10 be a Jordan block with eigenvalue 0. Then it is 1 dimensional. Otherwise, we assume F10 is R1 dimensional and R1>1. Then, by (Q3) and the definition of matrix exponent function, one gets that limt→∞⁡e-∫0tb(s)dsF10 does not exist, and hence, limt→∞⁡Ψ(t,0) does not exist. This contradicts Lemma 5. So F10 is 1 dimensional.
Let algebra multiplicity of eigenvalue 0 bew. Then w=1. Otherwise, w>1. Take w=2 as an example. Since each Jordan block corresponding to eigenvalue 0 is 1 dimensional, (37)Ψ∞⊗In≜limt→∞⁡Ψt,0=limt→∞⁡e-∫0tbsdsL⊗In=Ddiag⁡1,1,0,…,0D-1⊗In.Thus, rank⁡(Ψ∞)=2. This contradicts rank⁡(Ψ∞)≤1 from Lemma 6. So Laplacian L has exactly one zero eigenvalue.
(B.3) Prove ( Q 1 ) and ( Q 2 ) . By (B.2) and (Q3), one has (12). By Lemma 5, one gets(38)Ddiag⁡1,0,0,…,0D-1=gθT.
Noticing thatmr is the first column of D, one has Lmr=0. By (38), one obtains mr=gκ∗, where κ∗=θTmr∈R. Then, Lg=0. By the definition of L, for any j, we obtain gj∑k≠jajk=∑k≠jgkajk,j=1,…,N. Since gj=±1 and gj2=1,j=1,2,…,N, ∑k≠jajk=∑k≠jgjgkajk. So gjgkajk=ajk≥0. Let V1=j∣gj=1 and V2=j∣gj=-1, then V1∩V2=⌀, V1∪V2=V. If j∈Vp,p∈{1,2}, then ajk≥0,k∈Vp or ajk≤0,k∈Vr,r≠p,r∈{1,2}. By definition, G is structurally balanced, that is, (Q1) holds.
By(B.2) and (Q1), Lemma 1 implies that (Q2) holds.
(B.4) Prove ( Q 4 ) . Assume ∫0∞b2(s)ds=∞. Due to the first row of D-1 which is mlT, mlTL=0. By (4), we obtain d((mlT⊗In)X(t))=b(t)(mlT⊗In)(J⊗In)dΛ(t), i.e., (39)mlT⊗InXt=mlT⊗InX0+mlT⊗InJ⊗In∫0tbsdΛs.From Definition 3, it is known that X(t) converges to g⊗ν∗ in mean square sense, where Eν∗2<∞. Thus, when t→∞, (mlT⊗In)(J⊗In)∫0tb(s)dΛ(s) converges to a random variable Xm in mean square sense with EXm2<∞. Then limt→∞⁡E(mlT⊗In)(J⊗In)∫0tb(s)dΛ(s)2=limt→∞⁡tr⁡(mlTJJTml⊗In)∫0tb2(s)ds=∞. This leads to a contradiction. So (Q4) holds.Remark 8.
From Theorem7 it can be seen that under (Q1)-(Q4) the event-triggered protocol in (2) ensures agents converging to ν∗ or -ν∗ under measurement noise.Remark 9.
From Theorem7 one sees that to guarantee the mean square bipartite consensus, (Q1)-(Q2) are requirements for time-varying gain b(t) while (Q3)-(Q4) are the weakest connectivity assumptions.


## 4. Numerical Simulation

To demonstrate the developed result in the preceding, we consider an MAS of six agents, whose dynamics satisfy the system in (1). The communication graph that connects the six agents is illustrated in Figure 1. Clearly, V={1,…,6}, A=(aij), a16=a61=a54=1, a21=a35=-1, and a42=a63=2 in G=(V,E,A). From Figure 1, G satisfies (Q1) and (Q2). Furthermore, all eigenvalues of Laplacian L are λ1=0, λ2=0.6733+0.9192ȷ, λ3=0.6733-0.9192ȷ, λ4=2.0887+0.7157ȷ, λ5=2.0887-0.7157ȷ, and λ6=3.4760(ȷ2=-1). Obviously, minλ(L)≠0⁡{Re⁡λ(L)}=0.6733. The initial state of the MAS is given by X(0)=(10,-15,20,-25,30,-5). Choose b(t)=ln⁡(t+1)/t+1. By direct calculation we know that b(t) satisfies (Q3)-(Q4). Assume event-triggered condition (6) is satisfied by taking c1=1.2 and α=0.6. Applying protocol (2) to the system in (1), we get the six agents’ state trajectories. As shown in Figure 2 one can see that the states of agents 1, 3, and 6 converge to 5 in mean square sense while the states of agents 2, 4, and 5 converge to -5 in mean square sense. Thus, mean square bipartite consensus is achieved with event-triggered protocol (2). On the other hand, from Figure 3 we know that the inputs are constants between the event triggering time interval. Moreover, from Figure 4, it can be seen that the absolute value of the measurement error of each agent converges to zero. This means that the MAS does not exhibit Zeno behavior.Figure 1
Communication graphG among the 6 agents.Figure 2
State trajectories of six agents.Figure 3
Control inputs of six agents.Figure 4
The evolution of error norm.


## 5. Conclusion

Mean square bipartite consensus problem of single-integrator MASs is investigated in the context of event-triggered control and measurement noise. By using time-varying gain, an event-triggered bipartite consensus protocol is proposed under measurement noise, with which the controller update frequency is reduced. With given necessary and sufficient conditions on protocol gain and communication topology, the MAS is proved to achieve event-triggered bipartite consensus. The simulation shows that the system will not show Zeno behavior.



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*Source: 7186737-2018-08-06.xml*