# Average Consensus in Networks of Neutral Dynamical Agents with Fixed and Switching Topologies

Authors: Yingying Wu; Yuangong Sun
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper addresses the average consensus problem of neutral multiagent systems in undirected
networks with fixed and switching topologies. For the case of fixed topology, necessary and sufficient conditions to average consensus are established by decoupling the neutral multiagent system in terms of the eigenvalues of the Laplacian matrix. For the case of switching topology, sufficient conditions to average consensus are given in terms of linear matrix inequalities to determine the allowable upper bound of the time-varying communication delay. Finally, two examples are worked out to explain the effectiveness of the theoretical results.


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## Body

## 1. Introduction

In the last few years, more and more researchers have focused their attention on the study of multiagent systems due to many applications in unmanned aerial vehicles, satellite cluster, automated highways, and mobile robots. In all cases, the aim is to control a group of agents connected through a wireless network. Many profound theoretical results have been established for cooperative control of multiagent systems. In cooperative control of multiagent systems, an important issue is to design appropriate control strategies or protocols based on local information such that the group of dynamic agents can reach an agreement on certain quantities of interest. Such a problem is usually called a consensus problem, which has attracted many people in recent years.Consensus problem has a long history in the field of computer science. In the field of control theory, a discrete-time model ofn agents all moving in the plane with the same speed was proposed by Vicsek et al. [1]. A theoretical explanation of the consensus behavior in the Vecsek model was put forward by Jadbabaie et al. [2], where the alignment of a network of agent with switching topologies that are periodically connected was analyzed. Olfati-Saber and Murray [3] discussed the consensus problem for networks of dynamic agents with switching topologies and time delays in a continuous-time model by defining a disagreement function. Some more relaxable consensus conditions were presented by Ren and Beard [4] under dynamically changing interaction topologies.In recent years, more and more consensus results with single integrator models were established. Xiao and Wang studied the consensus problem of discrete-time multiagent systems with time delay [5, 6]. Sun et al. [7, 8] studied the average consensus problem in undirected networks of multiagent systems with fixed and switching topologies as well as multiple time-varying communication delays. Yu et al. [9] investigated the average consensus of multiagent systems with the problem of packet losses when both the first-order neighbors’ information and the second-order neighbors’ information are used. The consensus problem in directed networks via nonlinear protocols was premeditated by Liu et al. [10], and consensus of nonlinear multiagent systems with observer-based protocols was studied by Hu et al. [11].Unlike the single integrator case, a spanning tree is a necessary rather than a sufficient condition for consensus seeking with double integrator dynamics. However, the extension of consensus algorithms from the first-order case to the second-order case is nontrivial [12]. The general consensus protocol of multiagent systems with second-order dynamics was investigated by Zhu et al. [13], where necessary and sufficient conditions to consensus were given. Tian and Liu [14] considered the robust consensus of second-order multiagent systems with different input delays and asymmetric interconnection perturbations based on the frequency-domain analysis method. Yu et al. [15] investigated the second-order consensus for nonlinear multiagent systems with directed topologies. Song et al. [16] studied the second-order leader-following consensus of nonlinear multiagent systems by pinning control. Consensus algorithms designed for double integrator dynamics were provided by Abdessameud and Tayebi [17]. Zheng et al. [18] investigated the consensus problem of heterogeneous multiagent systems. Some sufficient conditions to consensus were established for both cases of undirected graphs and leader-following networks.It is well known that neutral systems play an important role in engineering area. If the agent’s dynamical equation takes the neutral form, what about the consensus of multiple agents in networks? To the best of our knowledge, consensus problem of neutral multiagent systems is not resolved by now. Therefore, it is necessary to study the consensus of neutral multiagent systems, which is the main purpose of this paper. We make the following contributions. Firstly, average consensus for neutral multiagent systems in undirected networks is proposed. Secondly, the convergence of the given proposed protocols is analyzed, and sufficient and necessary or sufficient conditions to average consensus of neutral multiagent systems have been established.The remainder of this paper is organized as follows. In Section2, graph theory and some important definitions and lemmas are given. The neutral multiagent system model is also established. The average consensus analysis of the neutral multiagent system with fixed and switching topologies is given in Section 3. Two examples are provided in Section 4. Finally, conclusions are drawn in Section 5.


## 2. Preliminaries and Problem Statement

As usual, the communication network is modeled through an undirected weighted graphG=(V,E,A) defined by a set of nodes V={v1,v2,…,vn}, a set of edges E⊂N×N, and a weighted adjacency matrix A=[aij] with nonnegative adjacency elements aij. The node indexes belong to a finite index set N={1,2,…,n}, and vi represents the ith agent. An edge of G is denoted by eij=(vi,vj). The adjacency elements associated with the edges of the graph are positive, that is, eij∈E if and only if aij>0. Moreover, we assume aii=0 and aij=aji, for i,j∈N. The set of neighbors of node vi is denoted by Ni={vj∈V:(vi,vj)∈E}. An undirected graph is called connected if any two distinct nodes of the graph can be connected via a path that follows the edges of the graph. The Laplacian matrix L(G)=[lij] of the graph is defined by (1)lij=-∑j=1,j≠inaij,j=i,aij,j≠i.It is easy to see thatL(G) is symmetric and 1TL(G)=L(G)1=0, where 1 and 0 are appropriate dimensional vectors with entries 1 and 0. We now define an operator Δ as (2)Δx=Lx,x∈Rn.Let H1 represent the space collecting all vectors in Rn with zero average. The eigenstructure of the operator Δ is completely characterized by the next lemma proved in Bensoussan and Menaldi [19].Lemma 1.
(a) The operatorΔ:H1→H1 is symmetric. It has n-1 strictly negative eigenvalues and the corresponding eigenvectors form an orthogonal basis for H1; (b) for x∈Rn, Δx=0 if and only if x∈H1⊥.Lemma1 shows that the operator Δ or the Laplacian matrix L(G) is invertible on the subspace H1. Now, consider the following neutral multiagent system model:(3)x˙it+cx˙it-σ=uit,i∈N,where xi(t)∈R and ui(t)∈R, i∈N, t∈R+, denote the state and the control input of the ith agent at time t, respectively, and σ>0 and c are constants. Consider the delayed Laplacian protocol as follows:(4)uit=∑vj∈Niaijxjt-τt-xit-τt,i∈N,where the time-varying delay τ:R+→R+ is a piecewise continuous function satisfying 0≤τ(t)≤h, for h>0. Then, the neutral multiagent system takes the form(5)x˙it+cx˙it-σ=∑vj∈Niaijxjt-τt-xit-τt,i∈N.Rewrite (5) in the following operator form:(6)X˙t+cX˙t-σ=ΔXt-τt,where X(t)=[x1(t),x2(t),…,xn(t)]T.As a result of the supposed piecewise continuity of the delay, the Cauchy problem associated with (6) is defined because it admits unique solution, depending on the initial condition. Set μ=max⁡{σ,h}. The initial condition may be given by a continuously differentiable function X(t)=X~(t) defined on [-μ,0]. In this paper, assume without loss of generality that xi(t)≡xi(0), for t∈[-μ,0].The average consensus for the neutral multiagent system (6) is defined as follows.Definition 2.
The neutral multiagent system (6) is said to achieve consensus if(7)limt→∞⁡Xt=α1,where α is a constant. If α=1/n∑i=1nxi(0), the neutral multiagent system (6) is said to achieve average consensus.Remark 3.
For the neutral multiagent system (6), once it achieves consensus, it must achieve average consensus since the involved graph is undirected. In fact, if (7) holds, noting that 1TΔ=0, we have (8)1TX˙t+cX˙t-σ=1TΔXt-τt=0,t≥0,which implies that 1T[X(t)+cX(t-σ)] is an invariant quantity. Therefore, (9)1TXt+cXt-σ=1TX0+cX-σ,t≥0.Since X(t)≡X(0), for t∈[-σ,0], we have that α=1/n∑i=1nxi(0).By Lemma1 and the following decomposition X=X1+X¯, where X1∈H1, X¯∈H1⊥, we can easily get the following lemma.Lemma 4.
The functionX is a solution of (6) if and only if X1 and X¯ are solutions of the following: (10)X˙1t+cX˙1t-σ=ΔX1t-τt,(11)X¯˙t+cX¯˙t-σ=0.Remark 5.
By Lemma4, we know that the spaces H1 and H1⊥ are positively invariant for (6). Furthermore, by Remark 3, the average state X¯ is a constant in time and equals 1/n∑i=1nxi(0). Thus, the problem of average consensus for (6) is simplified to prove that X1(t)→0 as t→∞; that is, the zero solution of (10) is asymptotically stable.


## 3. Main Results

In this section, we will consider the average consensus of neutral multiagent systems with fixed and switching topologies, respectively.


### 3.1. Networks with Fixed Topology

Theorem 6.
LetG=(V,E) be an undirected connected weighted graph. The neutral multiagent system (6) achieves average consensus if and only if all the following scalar differential equations (12)α˙kt+cα˙kt-σ=λkαkt-τt,k=1,2,…,n-1are asymptotically stable, where λ1≤λ2≤⋯≤λn-1<0 are eigenvalues of the operator Δ.Proof.
Let{φk}k=1n-1 be a set of eigenfunctions of Δ corresponding to the eigenvalues {λk}k=1n-1 that forms a basis for H1. Then, there exist suitable functions αk:R→R such that (13)X1t=∑k=1N-1αktφk.By multiplying φm (m=1,2,…,n-1) on both sides of the first equation in (10), we have that the left-hand side and the right-hand side of (10), respectively, take the following forms:(14)φm,X˙1t+cX˙1t-σ=φm,∑k=1n-1α˙kt+cα˙kt-σφk=α˙mt+cα˙mt-σ,(15)φm,ΔX1t-τt=φm,∑k=1n-1αkt-τtΔφk=φm,∑k=1n-1αkt-τtλkφk=λmαmt-τt.By (14) and (15), one can obtain(16)α˙mt+cα˙mt-σ=λmαmt-τt,m=1,2,…,n-1.Therefore, the neutral multiagent system (6) can achieve average consensus; that is, (10) is asymptotically stable, if and only if all equations (16) are asymptotically stable for m=1,…,n-1. The proof of Theorem 6 is completed.For the particular caseτ(t)≡σ, the following lemma is required.Lemma 7 (see [20]).
The following scalar neutral differential equation(17)x˙t+cx˙t-σ=bxt-σis asymptotically stable if and only if (18)0<c<1,b<0,σ<1-c2b2arccos⁡-c.Based on Lemma7, we have the following explicit result.Corollary 8.
LetG=(V,E) be an undirected connected weighted graph. The neutral multiagent system(19)X˙t+cX˙t-σ=ΔXt-σcan achieve average consensus if and only if 0<|c|<1 and σ<τ¯, where(20)τ¯=1-c2λ12arccos⁡-c.Proof.
By Theorem6, system (19) can achieve average consensus if and only if all the following scalar differential equations(21)α˙kt+cα˙kt-σ=λkαt-σ,k=1,…,n-1are asymptotically stable. By Lemma 7, all equations (21) are asymptotically stable if and only if (22)0<c<1,λk<0,σ<1-c2λk2arccos⁡-c,k=1,…,n-1.Noting that λk<0, for k=1,…,N-1, all equations (21) are asymptotically stable if and only if (23)0<c<1,σ<1-c2λ12arccos⁡-c.The proof of Corollary 8 is completed.


### 3.2.  Networks with Switching Topologies

Note that system (5) can be rewritten in the following matrix form:(24)X˙t+cX˙t-σ=LGXt-τt,where L(G) and σ are defined as above and the time-varying delay τ(t) satisfies the assumptions (H) 0≤τ(t)≤h and τ˙(t)≤d with h>0 and 0≤d<1.We are here interested in discussing such a problem of whether a network with switching topology can still solve average consensus. In this case, the following switched system is considered:(25)X˙t+cX˙t-σ=LGsXt-τt,s=θt∈ϑ0,where L(Gs) is the Laplacian matrix of the undirected graph Gs=(Vs,Es) that belongs to set Γ and Γ is a finite collection of graphs of order n with an index set ϑ0⊂Z. The map θ(t):R→ϑ0 is a switching signal that determines the network topology.Definition 9.
The neutral multiagent system (25) is said to achieve average consensus under arbitrary switching if each solution of system (25) satisfies limt→∞X(t)=α1 with α=1/n∑i=1nxi(0).Let us first take the following reduced-order transformation:(26)y12=x1-x2,y13=x1-x3,⋮y1n=x1-xn.The system (25) can be transformed into the following reduced-order disagreement system:(27)Y˙t+cY˙t-σ=AsYt-τt,s=θt∈ϑ0,where Y=[y12,y13,…,y1n]T and As=EL(Gs)F with (28)E=1-In-1,F=0T-In-1.Define the operator B:R→R as (29)BYt=Yt+cYt-σ.The operator B is stable when 0<|c|<1 and (30)B˙Yt=AsYt-τt.Before giving the main result, we need the following lemma.Lemma 10 (see [7]).
For any real differentiable vector functionx(t)∈Rn and any n×n constant matrix W=WT>0, the following inequality holds: (31)h-1xt-xt-τtTWxt-xt-τt≤∫t-τttx˙TsWx˙sds,t≥0,where τ(t) satisfies 0≤τ(t)≤h.Theorem 11.
Assume that (H) holds and 0<|c|<1. System (25) can achieve average consensus under arbitrary switching, if there exist symmetric positive definite matrices P, Q, M, S, and R of appropriate dimensions such that the following linear matrix inequality holds:(32)Ωs=-Q+S+MPAs+Q00AsTP+QΥcAsTP-ch2AsTQ-cAsTR0cPAs-S00-ch2QAs-cRAs0-R+c2h2Q+c2R<0,s∈ϑ0,where Υ=-Q-(1-d)M+h2AsTQAs+AsTRAs.Proof.
In spite of the existence of switching topology, similar to the discussion given in Remark3, system (25) achieves average consensus once it solves consensus. Therefore, based on the above analysis, it is sufficient to prove that system (27) is asymptotically stable. Let V(t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t) be the common Lyapunov function for system (27), where (33)V1t=BYtTPBYt,V2t=h∫t-hth-t+ξY˙TξQY˙ξdξ,V3t=∫t-τttYTξMYξdξ,V4t=∫t-σtYTξSYξdξ,V5t=∫t-σtY˙TξRY˙ξdξ.Assume that the sth subsystem is activated at time t; that is, θ(t)=s. In the sequel, by considering the right-hand side derivative of V(t), that is, D+V(t), along the trajectory of the solution of system (27), we have (34)D+V1t=2BYtTPB˙Yt=2YTtPAsYt-τt+2cYTt-τtAsTPYt-σ,D+V2t≤h2Y˙TtQY˙t-h∫t-τttY˙TξQY˙ξdξ.Noting that Y˙(t)=AsY(t-τ(t))-cY˙(t-σ) and following Lemma 10, we have (35)D+V2t≤h2YTt-τtAsTQAsYt-τt-2ch2YTt-τtAsTQY˙Tt-σ+c2h2Y˙Tt-σQY˙t-σ-YTtQYt+2YTtQYt-τt-YTt-τtQYt-τt.In addition, (36)D+V3t≤YTtMYt-1-dYTt-τtMt-τt,D+V4t=YTtSYt-YTt-σSYt-σ,D+V5t=Y˙TtRY˙t-Y˙Tt-σRY˙t-σ=YTt-τtAsTRAsYt-τt-2cYTt-τtAsTRY˙Tt-σ+c2Y˙Tt-σRY˙t-σ-Y˙Tt-σRY˙t-σ.Therefore, (37)D+Vt≤YTt-Q+S+MYt+2YTtPAs+QYt-τt+YTt-τt-Q-1-dM+h2AsTQAs+AsTRAsYt-τt-YTt-σSYt-σ+2cYTt-τtAsTPYt-σY˙Tt-σ-R+c2h2+c2RY˙t-σ-2cYTt-τth2AsTQ+AsTRY˙t-σ.Then, a new bound of V˙ is given as follows: (38)D+Vt≤UΩU,where U=[YT(t),YT(t-τ(t)),YT(t-σ),Y˙T(t-σ)]T, and Ω is defined by (32). Noting that the matrix Ωs is negative definite, there exists ηs>0 such that D+V(t)≤-ηsY(t)2. Let η=min{ηs:s∈ϑ0}. Then, for any switching signal, the zero solution of (27) is asymptotically stable since the operator B is stable. Thus, for any time-varying delay τ(t) satisfying (H), system (25) achieves average consensus under any switching. The proof of Theorem 11 is completed.Whend≥1 or nothing has been known about the derivative of τ(t), construct the common Lyapunov function V(t)=V1(t)+V2(t)+V4(t)+V5(t), where V1(t), V2(t), V4(t), and V5(t) are defined as above. Similar to the proof of Theorem 11, we can easily obtain the following corollary.Corollary 12.
Assume that0≤τ(t)≤h for some h>0 and 0<|c|<1. System (25) achieves average consensus under arbitrary switching, if there exist symmetric positive definite matrices P, Q, S, and R of appropriate dimensions such that the following linear matrix inequality holds:(39)Ω=-Q+SPAs+Q00AsTP+Q-Q+h2AsTQAs+AsTRAscAsTP-ch2AsTQ-cAsTR0cPAs-S00-ch2QAs-cRAs0-R+c2h2Q+c2R<0,s∈ϑ0.Remark 13.
In Theorem11 and Corollary 12, average consensus of the neutral multiagent system is determined by a linear matrix inequality rather than the connectivity of each switching topology. Similar to the analysis given in [7], it is not surprising that the connectivity of each switching topology is necessary to guarantee the feasibility of linear matrix inequalities (32) and (39).Remark 14.
Theorem11 can also be extended to the case of directed graphs. Following the proof of Theorem 11, if each switching topology is a directed graph, the given linear matrix inequalities (32) guarantee consensus of the neutral multiagent system (25).


## 3.1. Networks with Fixed Topology

Theorem 6.
LetG=(V,E) be an undirected connected weighted graph. The neutral multiagent system (6) achieves average consensus if and only if all the following scalar differential equations (12)α˙kt+cα˙kt-σ=λkαkt-τt,k=1,2,…,n-1are asymptotically stable, where λ1≤λ2≤⋯≤λn-1<0 are eigenvalues of the operator Δ.Proof.
Let{φk}k=1n-1 be a set of eigenfunctions of Δ corresponding to the eigenvalues {λk}k=1n-1 that forms a basis for H1. Then, there exist suitable functions αk:R→R such that (13)X1t=∑k=1N-1αktφk.By multiplying φm (m=1,2,…,n-1) on both sides of the first equation in (10), we have that the left-hand side and the right-hand side of (10), respectively, take the following forms:(14)φm,X˙1t+cX˙1t-σ=φm,∑k=1n-1α˙kt+cα˙kt-σφk=α˙mt+cα˙mt-σ,(15)φm,ΔX1t-τt=φm,∑k=1n-1αkt-τtΔφk=φm,∑k=1n-1αkt-τtλkφk=λmαmt-τt.By (14) and (15), one can obtain(16)α˙mt+cα˙mt-σ=λmαmt-τt,m=1,2,…,n-1.Therefore, the neutral multiagent system (6) can achieve average consensus; that is, (10) is asymptotically stable, if and only if all equations (16) are asymptotically stable for m=1,…,n-1. The proof of Theorem 6 is completed.For the particular caseτ(t)≡σ, the following lemma is required.Lemma 7 (see [20]).
The following scalar neutral differential equation(17)x˙t+cx˙t-σ=bxt-σis asymptotically stable if and only if (18)0<c<1,b<0,σ<1-c2b2arccos⁡-c.Based on Lemma7, we have the following explicit result.Corollary 8.
LetG=(V,E) be an undirected connected weighted graph. The neutral multiagent system(19)X˙t+cX˙t-σ=ΔXt-σcan achieve average consensus if and only if 0<|c|<1 and σ<τ¯, where(20)τ¯=1-c2λ12arccos⁡-c.Proof.
By Theorem6, system (19) can achieve average consensus if and only if all the following scalar differential equations(21)α˙kt+cα˙kt-σ=λkαt-σ,k=1,…,n-1are asymptotically stable. By Lemma 7, all equations (21) are asymptotically stable if and only if (22)0<c<1,λk<0,σ<1-c2λk2arccos⁡-c,k=1,…,n-1.Noting that λk<0, for k=1,…,N-1, all equations (21) are asymptotically stable if and only if (23)0<c<1,σ<1-c2λ12arccos⁡-c.The proof of Corollary 8 is completed.


## 3.2.  Networks with Switching Topologies

Note that system (5) can be rewritten in the following matrix form:(24)X˙t+cX˙t-σ=LGXt-τt,where L(G) and σ are defined as above and the time-varying delay τ(t) satisfies the assumptions (H) 0≤τ(t)≤h and τ˙(t)≤d with h>0 and 0≤d<1.We are here interested in discussing such a problem of whether a network with switching topology can still solve average consensus. In this case, the following switched system is considered:(25)X˙t+cX˙t-σ=LGsXt-τt,s=θt∈ϑ0,where L(Gs) is the Laplacian matrix of the undirected graph Gs=(Vs,Es) that belongs to set Γ and Γ is a finite collection of graphs of order n with an index set ϑ0⊂Z. The map θ(t):R→ϑ0 is a switching signal that determines the network topology.Definition 9.
The neutral multiagent system (25) is said to achieve average consensus under arbitrary switching if each solution of system (25) satisfies limt→∞X(t)=α1 with α=1/n∑i=1nxi(0).Let us first take the following reduced-order transformation:(26)y12=x1-x2,y13=x1-x3,⋮y1n=x1-xn.The system (25) can be transformed into the following reduced-order disagreement system:(27)Y˙t+cY˙t-σ=AsYt-τt,s=θt∈ϑ0,where Y=[y12,y13,…,y1n]T and As=EL(Gs)F with (28)E=1-In-1,F=0T-In-1.Define the operator B:R→R as (29)BYt=Yt+cYt-σ.The operator B is stable when 0<|c|<1 and (30)B˙Yt=AsYt-τt.Before giving the main result, we need the following lemma.Lemma 10 (see [7]).
For any real differentiable vector functionx(t)∈Rn and any n×n constant matrix W=WT>0, the following inequality holds: (31)h-1xt-xt-τtTWxt-xt-τt≤∫t-τttx˙TsWx˙sds,t≥0,where τ(t) satisfies 0≤τ(t)≤h.Theorem 11.
Assume that (H) holds and 0<|c|<1. System (25) can achieve average consensus under arbitrary switching, if there exist symmetric positive definite matrices P, Q, M, S, and R of appropriate dimensions such that the following linear matrix inequality holds:(32)Ωs=-Q+S+MPAs+Q00AsTP+QΥcAsTP-ch2AsTQ-cAsTR0cPAs-S00-ch2QAs-cRAs0-R+c2h2Q+c2R<0,s∈ϑ0,where Υ=-Q-(1-d)M+h2AsTQAs+AsTRAs.Proof.
In spite of the existence of switching topology, similar to the discussion given in Remark3, system (25) achieves average consensus once it solves consensus. Therefore, based on the above analysis, it is sufficient to prove that system (27) is asymptotically stable. Let V(t)=V1(t)+V2(t)+V3(t)+V4(t)+V5(t) be the common Lyapunov function for system (27), where (33)V1t=BYtTPBYt,V2t=h∫t-hth-t+ξY˙TξQY˙ξdξ,V3t=∫t-τttYTξMYξdξ,V4t=∫t-σtYTξSYξdξ,V5t=∫t-σtY˙TξRY˙ξdξ.Assume that the sth subsystem is activated at time t; that is, θ(t)=s. In the sequel, by considering the right-hand side derivative of V(t), that is, D+V(t), along the trajectory of the solution of system (27), we have (34)D+V1t=2BYtTPB˙Yt=2YTtPAsYt-τt+2cYTt-τtAsTPYt-σ,D+V2t≤h2Y˙TtQY˙t-h∫t-τttY˙TξQY˙ξdξ.Noting that Y˙(t)=AsY(t-τ(t))-cY˙(t-σ) and following Lemma 10, we have (35)D+V2t≤h2YTt-τtAsTQAsYt-τt-2ch2YTt-τtAsTQY˙Tt-σ+c2h2Y˙Tt-σQY˙t-σ-YTtQYt+2YTtQYt-τt-YTt-τtQYt-τt.In addition, (36)D+V3t≤YTtMYt-1-dYTt-τtMt-τt,D+V4t=YTtSYt-YTt-σSYt-σ,D+V5t=Y˙TtRY˙t-Y˙Tt-σRY˙t-σ=YTt-τtAsTRAsYt-τt-2cYTt-τtAsTRY˙Tt-σ+c2Y˙Tt-σRY˙t-σ-Y˙Tt-σRY˙t-σ.Therefore, (37)D+Vt≤YTt-Q+S+MYt+2YTtPAs+QYt-τt+YTt-τt-Q-1-dM+h2AsTQAs+AsTRAsYt-τt-YTt-σSYt-σ+2cYTt-τtAsTPYt-σY˙Tt-σ-R+c2h2+c2RY˙t-σ-2cYTt-τth2AsTQ+AsTRY˙t-σ.Then, a new bound of V˙ is given as follows: (38)D+Vt≤UΩU,where U=[YT(t),YT(t-τ(t)),YT(t-σ),Y˙T(t-σ)]T, and Ω is defined by (32). Noting that the matrix Ωs is negative definite, there exists ηs>0 such that D+V(t)≤-ηsY(t)2. Let η=min{ηs:s∈ϑ0}. Then, for any switching signal, the zero solution of (27) is asymptotically stable since the operator B is stable. Thus, for any time-varying delay τ(t) satisfying (H), system (25) achieves average consensus under any switching. The proof of Theorem 11 is completed.Whend≥1 or nothing has been known about the derivative of τ(t), construct the common Lyapunov function V(t)=V1(t)+V2(t)+V4(t)+V5(t), where V1(t), V2(t), V4(t), and V5(t) are defined as above. Similar to the proof of Theorem 11, we can easily obtain the following corollary.Corollary 12.
Assume that0≤τ(t)≤h for some h>0 and 0<|c|<1. System (25) achieves average consensus under arbitrary switching, if there exist symmetric positive definite matrices P, Q, S, and R of appropriate dimensions such that the following linear matrix inequality holds:(39)Ω=-Q+SPAs+Q00AsTP+Q-Q+h2AsTQAs+AsTRAscAsTP-ch2AsTQ-cAsTR0cPAs-S00-ch2QAs-cRAs0-R+c2h2Q+c2R<0,s∈ϑ0.Remark 13.
In Theorem11 and Corollary 12, average consensus of the neutral multiagent system is determined by a linear matrix inequality rather than the connectivity of each switching topology. Similar to the analysis given in [7], it is not surprising that the connectivity of each switching topology is necessary to guarantee the feasibility of linear matrix inequalities (32) and (39).Remark 14.
Theorem11 can also be extended to the case of directed graphs. Following the proof of Theorem 11, if each switching topology is a directed graph, the given linear matrix inequalities (32) guarantee consensus of the neutral multiagent system (25).


## 4. Examples

We now work out two examples to illustrate the theoretical results in this paper.Example 1.
Consider the neutral multiagent system (19) with the topology given by Figure 1.
The Laplacian matrixL(G) takes the form (40)LG=-21011-21001-21101-2.We can get λ1=-4, λ2=λ3=-2, and λ4=0. When c=0.45, we have σ<0.4549 by Corollary 8.Figure 1
An undirected connected weighted graph.Example 2.
Consider an undirected network with the following switching topologies shown in Figure2.
Solving (32) gives the following estimate on h for different d and c, which guarantees system (25) associated with the switching topology given in Figure 2 achieves average consensus under arbitrary switching. When c is fixed to 0.1, the estimate on h for different d is shown in Table 1.
Whend is fixed to 0.1, the estimate on h for different c is shown in Table 2.Table 1
d
0
0.1
0.2
0.3
0.4
0.5
Nothing
h
0.3431
0.3337
0.3249
0.3162
0.3081
0.2975
0.2612Table 2
c
0.1
0.2
0.3
0.4
0.5
0.6
h
0.3337
0.2798
0.2294
0.1816
0.1348
0.0874Figure 2
Two examples of undirected connected graphs.


## 5. Conclusion

The average consensus problem of neutral multiagent systems in undirected networks with fixed and switching topologies is studied in this paper. Firstly, for the case of fixed topology, we establish sufficient and necessary conditions to average consensus by decoupling the neutral multiagent system in terms of the eigenvalues of the Laplacian matrix. Secondly, by using a linear matrix inequality method, average consensus criteria in terms of linear matrix inequality are given to determine the allowable upper bound of the time-varying communication delay for the case of switching topology. Finally, two examples are worked out to illustrate the theoretical results.



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*Source: 252050-2015-10-05.xml*