# Consensus Analysis for High-Order Multi-Agent Systems without or with Delays

Authors: Zhengxin Wang; Yang Cao
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

This paper studies the consensus problem for a high-order multi-agent systems without or with
delays. Consensus protocols, which only depend on the own partial information of agents and partial
relative information with its neighbors, are proposed for consensus and quasi-consensus, respectively.
Firstly, some lemmas are presented, and then a necessary and sufficient condition for guaranteeing the
consensus is established under the consensus protocol without delays. Furthermore, communication delays
are considered. Some necessary and sufficient conditions for solving quasi-consensus problem with delays
are obtained. Finally, some simulations are given to verify the theoretical results.


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## Body

## 1. Introduction

In recent years, the consensus problem of multi-agent systems has received a great deal of attention due to its broad applications in formation control of unmanned air vehicles, the design of sensor networks, the cooperative control of mobile robots, and automated highway system. In a word, we say all agents reach the consensus upon certain quantities of interest if all agents agree on a common state eventually; the precise definition will be introduced in the following sections. Many researches have been devoted to the consensus problem of multi-agent systems. In 1995, Vicsek et al. [1] proposed a discrete-time model and concluded that all the headings of the agents converged to a common value by simulations. In 2003, Jadbabaie et al. [2] provided a theoretical explanation for Vicsek’s simulation results by graph theory. Gradually, the consensus problem of multi-agent systems received a growing attention. For example, Olfati-Saber and Murray [3] discussed consensus problems for networks of dynamic agents with fixed and switching topologies under three cases. Ren and Beard [4] considered the problem of information consensus under dynamically changing interaction topologies and weighting factors. Recently, the second-order consensus problem of multi-agent systems has attracted more and more researchers’ attention [5–14]. Different from [5–7, 10], Yu et al. [9], Song et al. [11], and Wen et al. [14] all studied the second-order consensus problem of multi-agent systems with nonlinear dynamics. Compared with synchronization in complex networks which are of nonlinear dynamics, a generalized consensus was investigated. From a unified viewpoint, [15] provided a novel framework for the consensus of multi-agent systems and the synchronization of complex networks. Viewed from the leader-following consensus, both Song et al. [11] and Zhu and Cheng [12] considered the consensus problem with a leader. In general, the second-order consensus problem pays attention to whether the relative position and velocity of each agent will converge to common state.At the same time, the high-order consensus problem of multi-agent systems is also worth being studied. Generally speaking, the first-order consensus problem of multi-agent systems can be considered as a special case of the synchronization of complex dynamic networks. Moreover, second-order consensus problem of multi-agent system means that the relative position and velocity of each agent will be converged to common. However, the birds achieving the consensus should be on the acceleration besides the relative position and velocity for the flocks of birds [16], which means that all the position, velocity, and acceleration will be converged to common. Recently, there are more and more studies on the high-order consensus or swarming problem (see [16–20] and the references therein).Motivated by the previously mentioned results, consensus of high-order multi-agent systems is investigated in this paper. A new consensus protocol which only depends on the own partial information of agents and partial relative information with its neighbors is proposed. In other words, the consensus protocol in this paper does not need the global information of agents and the global relative information. It is convenient to be designed when the global information states cannot be available. Furthermore, we consider the quasi-consensus problem of high-order multi-agent systems with a time-delay protocol. The delay is nonnegligible in the information exchanges due to the limited transmission speed or memory effect. Therefore, there are many papers focusing on the consensus problem with delay [21–25]. It is worth noting that Yu et al. [25] considered the quasi-consensus problem of second-order multi-agent systems with delay under the protocol
(1)ui(t)=α∑j=1,j≠iN‍Gij(xj(t)-xi(t))-α∑j=1,j≠iNGij(xj(t-τ)-xi(t-τ)).
This delay is caused by the fact that the agent needs some memory to store the outdated information from its neighboring agents. It found that this delay can induce quasi-consensus; that is, all agents have the same velocity but keep a distance from each other eventually. Therefore, the delay in this sense looks very interesting. This paper considers quasi-consensus of high-order multi-agent systems with the delay in the sense of memory effect. Based on the Nyquist stability criterion, some results on the quasi-consensus of high-order multi-agent systems are obtained. It is worth noting that the results on the consensus of high-order multi-agent systems with a protocol of communication delay hold similarly. Furthermore, some simulations are given to verify the theoretical results.Notations. Let ℝn denote the n-dimensional Euclidian space. Let 0N and IN be the N×N zero matrix and identity matrix, respectively. If M is a matrix, then rank M and σ(M) denote the rank and the spectrum of M, respectively. Let 1 and 0 denote the N×1 column vectors of all ones and all zeros, respectively. The symbol ⊗ denotes the Kronecker product.


## 2. Preliminaries and Model

Consider a multi-agent systems ofN agents. The dynamics of each agent is governed by the following general ℓth-order integrator:
(2)ξ˙i(0)=ξi(1),⋮ξ˙i(ℓ-2)=ξi(ℓ-1),ξ˙i(ℓ-1)=ui,
where i=1,2,…,N,  ℓ≥1 is a positive integer and denotes the order of the differential equations, ξi(k)∈ℝm,k=0,1,…,ℓ-1, are the states of agent i, ui∈ℝm is the consensus protocol, and ξi(k) denotes the kth-order derivative of ξi. The initial state denotes ξi0=[ξi(0),ξi(1)(0),…,ξi(ℓ-1)(0)].We use the weighted undirected graph𝒢 to model the interaction topology of the multi-agent systems; that is, each vertex represents an agent of systems; each arc (edge) eji represents that there is a communication link from agent j to agent i. aij>0 is the weight of the communication link eji. One can see [26] for more details. Now, we propose the following consensus protocol:
(3)ui=-∑k∈M1‍ckξi(k)1-α∑j=1N‍aij(∑k∈M2‍γk(ξi(k)-ξj(k))),
where M={0,1,2,…,ℓ-1}. Let M1,M2⊂M be two nonempty subsets such that M1∪M2=M,M1∩M2=∅, α is the coupling strength and γk>0 is the feedback gain of relative information. ck≠0⇔k∈M1, otherwise, ck=0 and let c0=0; γk≠0⇔k∈M2, otherwise, γk=0 and let γ0≠0. Let n0=min{k:k∈M1}.Letξ(0)=[ξ1(0)T,ξ2(0)T,…,ξN(0)T]T and ξ(n+1)=ξ˙(n), n=0,1,…,ℓ-2. By employing the consensus protocol (3), the dynamics equations of all agents can be written as follows:
(4)[ξ˙(0)ξ˙(1)⋮ξ˙(ℓ-1)]=(Φ⊗Im)[ξ(0)ξ(1)⋮ξ(ℓ-1)],
where
(5)Φ=A⊗IN-αB⊗L,(6)A=[010⋯0001⋯0⋮⋮⋮⋮⋮000⋯10-c1-c2⋯-cℓ-1]ℓ×ℓ,B=[00⋯000⋯0⋮⋮⋮⋮00⋯0γ0γ1⋯γℓ-1]ℓ×ℓ,L=[ℓij] is the associated Laplacian of 𝒢; that is ℓii=∑j≠i‍aij, ℓij=-aij. It is easy to note that
(7)Φ=[0NIN0N⋯0N0N0NIN⋯0N⋮⋮⋮⋮⋮0N0N0N⋯IN-αγ0L-Δ1-Δ2⋯-Δℓ-1],
where Δk=ckIN+αγkL.Definition 1.
The multi-agent systems governed by (4) are said to achieve consensus if limt→∞∥ξi(k)(t)-ξj(k)(t)∥=0,  k=0,1,2,…,ℓ-1, i,j=1,2,…,N,i≠j, for any initial values.Remark 2.
For convenience, we assumem=1 in the following discussion; however, all the results hereafter remain valid for m>1 by the Kronecker product.


## 3. Consensus Analysis

In this section, we consider the consensus problem defined in the Section2 for the multi-agent systems (4) with fixed topology. To this end, we establish the following lemmas which are needed for the main results.DenoteΓ=IN⊗A-αL⊗B. One has the following lemmas.Lemma 3.
Φ is similar to Γ.Proof.
See the appendix.Lemma 4.
Assume thatΦ is defined in (5) and λ1,λ2,…,λN∈σ(L); then the characteristic polynomial of Φ is
(8)det(μINℓ-Φ)=∏i=1N‍(μℓ+δℓ-1iμℓ-11111+⋯+δ1iμ+αλiγ0),
where δki=ck+αλiγk,k=1,2,…,ℓ-1,i=1,2,…,N.Proof.
See the appendix.Lemma 5.
Φ has at least n0 zero eigenvalues. It has exactly n0 zero eigenvalues if and only if L has a simple zero eigenvalue. If L has a simple zero eigenvalue, then the geometric multiplicity of the zero eigenvalue of Φ equals one. Moreover, the left and right eigenvector of Φ associated with eigenvalue zero are as follows:
(9)vl=1cn0[c1,…,cℓ-1,1]⊗1T,wr=1Ne1⊗1.Proof.
See the appendix.Based on the previous lemmas, here we give the main result.Theorem 6.
Under the consensus protocol (3), the multi-agent systems (4) achieve the consensus if and only if Φ has exactly n0 zero eigenvalues and all of other eigenvalues are negative.Proof (Sufficiency).
IfΦ has exactly n0 zero eigenvalues, then Φ has the Jordan canonical form [J1J2], where
(10)J1=[010⋯0001⋯0⋮⋮⋮⋮⋮000⋯1000⋯0]n0×n0;J2 is the upper diagonal block matrix with diagonal entries being the nonzero eigenvalues of Φ. Moreover, one has
(11)Φ=[w1,w2,…,wℓN]×[J10n0×(Nℓ-n0)0(Nℓ-n0)×n0J2]×[v1Tv2T⋮vℓNT],
where wi∈ℝℓN,i=1,2,…,ℓN, can be chosen to be the right eigenvectors or generalized eigenvectors of Φ, vi∈ℝℓN, i=1,2,…,ℓN, can be chosen to be the left eigenvectors or generalized eigenvectors of Φ. Without of loss generality, assume that M2={0,1,…,n0-1} and M1={n0,n0+1,…,ℓ-1}. Let w1=[1T,0T,…,0T]T, w2=[0T,1T,…,0T]T,…,wn0=[0T,0T,…,1T︸n0,0T,…,0T︸ℓ-n0]T; one can verify that w1,w2,…,wn0 are a right eigenvector and n0-1 generalized right eigenvectors of Φ associated with zero eigenvalue, respectively. Since Φ has exactly n0 zero eigenvalues, without loss of generality, one can denote them by μ1=μ2=⋯=μn0=0. By Lemma 5, we know that L has a simple zero eigenvalue, which implies that there is a nonnegative vector p=(1/N)1 such that pTL=0, pT1=1 by [16]. Then we choose, v1=(1/cn0)[cn0pT,…,cℓ-1pT,pT,0T,…,0T︸n0-1]T,…, v2=(1/cn0)[0T,cn0pT,…,cℓ-1pT,pT,0T,…,0T︸n0-2]T,…,  vn0=(1/cn0)[0T,…,0T︸n0-1,cn0pT,…,cℓ-1pT,pT]T. It follows from Lemma 5 that they are n0-1 generalized left eigenvectors and a left eigenvector of Φ associated with eigenvalue zero, respectively. Moreover, viTwi=1, i=1,2,…,n0. It is noted that
(12)eΦt=[w1,w2,…,wℓN]×[eJ1t0n0×(Nℓ-n0)0(Nℓ-n0)×n0eJ2t][v1Tv2T⋮vℓNT],
where limt→∞eJ2t=0Nℓ-n0 since the nonzero eigenvalues have negative real parts and
(13)eJ1t=[1tt22!⋯tn0-1(n0-1)!01t⋯tn0-2(n0-2)!⋮⋮⋮⋮⋮000⋯t000⋯1]n0×n0.
Therefore, for enough large t, the dominant terms of eΦt denoted by Φd is(14)[1pT(cn0+1cn0+t)1pT⋯∑k=0n0-1‍cn0+ktn0-k-1cn0(n0-k-1)!1pT∑k=1n0‍cn0+ktn0-kcn0(n0-k)!1pT⋯tn0-11pTcn0(n0-1)!0N1pT⋯∑k=0n0-2‍cn0+ktn0-k-2cn0(n0-k-2)!1pT∑k=1n0-1‍cn0+ktn0-k-1cn0(n0-k-1)!1pT⋯tn0-21pTcn0(n0-2)!⋮⋮⋮⋮⋮⋮⋮0N0N⋯1pTcn0+1cn01pT⋯1pTcn00N0N⋯0N0N⋯0N⋮⋮⋮⋮⋮⋮⋮0N0N⋯0N0N⋯0N]ℓ×ℓ.Thus, we write Φd by a compact form
(15)[1pTr12(t)1pT⋯r1n0(t)1pT⋯r1ℓ(t)1pT0N1pT⋯r2n0(t)1pT⋯r2ℓ(t)1pT⋮⋮⋮⋮0N0N⋯1pT⋯rn0ℓ(t)1pT0N0N⋯0N⋯0N⋮⋮⋮⋮0N0N⋯0N⋯0N],
where rij(t),j>i is a polynomial of t with degree j-i. It follows from (4) that
(16)[ξ(0)(t)ξ(1)(t)⋮ξ(ℓ-1)(t)]=eΦt[ξ(0)(0)ξ(1)(0)⋮ξ(ℓ-1)(0)],
that is, for enough large t,
(17)ξ(k)(t)=[ξ1(k)(t),ξ2(k)(t),…,ξN(k)(t)]T=1pTξ(k)(0)+rk+1,k+2(t)1pTξ(k+1)(0)+⋯+rk+1,ℓ(t)1pTξ(ℓ-1)(0),k≤n0-1,
which implies that ξi(k)(t)→ξj(k)(t) as t→∞ for k≤n0-1.
Ifk≥n0, then ξ(k)(t)→0 as t→∞. Therefore, ξi(k)(t)→ξj(k)(t) as t→∞, for all i≠j, k=0,1,…,ℓ-1. This implies that the multi-agent systems achieve the consensus.
(Necessity). By reduction to absurdity, suppose that the sufficient condition dose not hold; then there are two cases may be hold.
Case  1.
Φ has more than n0 zero eigenvalues.
Case  2.
Φ has exactly n0 zero eigenvalues, but it has at least one eigenvalue with nonnegative real part.
If Case  1 holds, thenrank(limt→∞eΦt)≥rank(limt→∞eJ1t)>n0.
If Case  2 holds, thenlimt→∞eJ2t≠0Nℓ-n0, and rank(limt→∞eΦt)≥rank(limt→∞eJ1t)+rank(limt→∞eJ2t)>n0. However, it follows from Φd (the dominant terms of eΦt) that if the consensus is achieved, then rank(limt→∞eΦt)≤n0. This is a contradiction.Remark 7.
From Theorem6, if (4) achieves the consensus, then Φ has exactly n0 zero eigenvalues. Therefore, L has a simple zero eigenvalue, which implies that the information exchange topology 𝒢 is connected.Remark 8.
Compared to the consensus protocols proposed in [16–19], protocol (3) only depends on the own partial information of agents and the partial relative states with its neighbors, which does not need the global information of agents or the global relative information. It is convenient to design when global information states cannot be available.To illustrate the previous result, we consider the following example.Example 9.
Considering the following third-order system of8 agents, where the dynamics of each agent are governed by
(18)x˙i=vi,v˙i=ai,a˙i=ui,
where xi, vi, ai, i=1,2,…,8, denote the position, velocity, and acceleration of agent i, respectively, let Laplacian matrix L be
(19)[10-10000002-100-100-1-14-100-1000-12-1000000-110000-100010000-100100000000-11].
It follows from the L that the graph is connected. Moreover, the eigenvalues of L are 0, 0.382, 0.6086, 1, 1, 2.2271, 2.618, and 5.1642. If we choose α=1, γ0=1, γ1=2, and c2=2, then n0=2. Furthermore, Φ has 2 zero eigenvalues and the real parts of all other eigenvalues are negative. Let the initial values be the x(0)=[10,20,30,5,-10,-20,-30,-5], v(0)=[2,-8,3,-3,-2,5,8,-5], a(0)=[-2,2,3,-3,5,-5,1,-1]. By Theorem 6, consensus in dynamical multi-agent systems (18) with network topology (19) can be achieved. Figures 1, 2, and 3 show the plots of xi, vi, ai, i=1,2,…,8, respectively.Figure 1
The trajectories ofxi, i=1,2,…,8.Figure 2
The trajectories ofvi, i=1,2,…,8.Figure 3
The trajectories ofai, i=1,2,…,8.


## 4. Quasi-Consensus Problem with Delays

Because of the limited transmission speed or memory effect, there exist nonnegligible delays in the communication process. Therefore, it is necessary to consider the influence of delays on the consensus. In [25], the authors considered the delays in the sense of memory store by the fact that the agent uses the memory to store the outdated information of its neighbors. Such delays are so interesting that can deduce quasi-consensus. In this section, we consider the consensus problem with the protocol of delays in the sense of memory store as follows:
(20)ui=-∑k∈M1‍ckξi(k)+α∑j=1N‍aij∑k∈M2‍γk(ξj(k)(t)-ξi(k)(t))+∑k∈M1‍ckξi(k)(t-τ)-α∑j=1N‍aij∑k∈M2‍γk(ξj(k)(t-τ)-ξi(k)(t-τ)),
where τ>0. Let ηi=[ξi(0),ξi(1),…,ξi(ℓ-1)], η=[η1T,η2T,…,ηNT]T.With the consensus protocol (20), the closed loop of multi-agent systems (2) can be written as
(21)η˙i=Aηi-A~ηi(t-τ)-α∑j=1N‍lijBηj+α∑j=1N‍lijBηj(t-τ),
where A, B are defined as (5) and
(22)A~=[00⋯000⋯0⋮⋮⋮⋮00⋯00-c1⋯-cℓ-1]ℓ×ℓ.Therefore,(23)η˙=(IN⊗A)η-(IN⊗A~)η(t-τ)-α(L⊗B)η+α(L⊗B)η(t-τ).
Let J be the Jordan matrix associate with the L. Therefore, there exists orthogonal matrix Q such that QTLQ=J=diag{λ1,λ2,…,λN}. Then,
(24)(QT⊗Iℓ)η˙(t)11=(QT⊗Iℓ)[(IN⊗A)-α(L⊗B)]η(t)11-(QT⊗Iℓ)[(IN⊗A~)-α(L⊗B)]η(t-τ)11=[(IN⊗A)-α(J⊗B)](QT⊗Iℓ)η(t)11-[(IN⊗A~)-α(J⊗B)](QT⊗Iℓ)η(t-τ).
Let θ(t)=(QT⊗Iℓ)η(t)=[θ1T(t),θ2T(t),…,θNT(t)]T. Then, it follows from (24) that
(25)θ˙(t)=[(IN⊗A)-α(J⊗B)]θ(t)-[(IN⊗A~)-α(J⊗B)]θ(t-τ);
that is
(26)θ˙i(t)=(A-αλiB)θi(t)-(A~-αλiB)θi(t-τ).
The characteristic equation of the multi-agent system (26) is
(27)λℓ+[cℓ-1+αλiγℓ-1](1-e-λτ)λℓ-1+⋯+[c1+αλiγ1](1-e-λτ)λ+αλiγ0(1-e-λτ)=0,i=1,2,…,N.Definition 10.
The multi-agent systems (2) are said to achieve quasi-consensus if
(28)limt→∞∥ξi(0)(t)-ξj(0)(t)∥=cij,limt→∞∥ξi(k)(t)-ξj(k)(t)∥=0,
for any initial conditions, where cij are constants, for all k=1,2,…,ℓ-1 and i,j=1,2,…,N.Theorem 11.
Assume that𝒢 is connected. Quasi-consensus of multi-agent systems (2) with protocol (20) can be achieved if and only if each equation of (27) (i=2,3,…,N) has only simple zero root and the real parts of all other roots are negative.Proof (Sufficiency).
If𝒢 is connected, then L has only a simple zero eigenvalue with associated right eigenvector q1=(1/N)1, where QTLQ=J, Q=[q1,q2,…,qN], qi=[qi1,qi2,…,qiN]T, i=1,2,…,N. If each equation of (27) (i=2,3,…,N) has only simple zero root and the real parts of all other roots are negative, then the states in (26) converge to some constants. It follows from the θ(t)=(QT⊗Iℓ)η(t) that θi(t)=(qiT⊗Iℓ)η(t)=∑j=1Nqijηj(t). Denote θi(t)=[θi0(t),θi1(t),…,θiℓ-1(t)]T; then one has
(29)θik(t)=∑j=1N‍qijξj(k)(t),k=0,1,…,ℓ-1.
Noting that θ˙ik(t)=θik+1(t), if one of the θi1(t),θi2(t),…,θiℓ-1(t) does not converge to zero, then limt→∞∥θi0(t)∥=∞. Therefore, limt→∞∥θik(t)∥=0, i=2,3,…,N, k=1,…,ℓ-1. Moreover, one has ηi(t)=∑j=1Nqjiθj(t); therefore,
(30)ξi(k)(t)=∑j=1N‍qjiθjk(t),k=0,1,…,ℓ-1.
Thus,
(31)limt→∞∥ξ(k)(t)-1N[θ1k(t),θ1k(t),…,θ1k(t)]T∥=0,
where k=1,2,…,ℓ-1,  θ˙1(t)=Aθ1(t). Therefore, the quasi-consensus is reached.
(Necessity). If the quasi-consensus can be reached, then there existsσ(t)=[σ0(t),σ1(t),…,σℓ-1(t)]T such that limt→∞∥ξ(k)(t)-1N⊗σk(t)∥=0. Since G is connected, L has only one zero eigenvalue with associated eigenvector q1=(1/N)1. Thus, 0=QTL1=JQT1=[λ1q1T1,λ2q2T1,…,λNqNT1]T; that is, qiT1=0,i=2,3,…,N. Therefore, it follows from θi(t)=qi1η1(t)+qi2η2(t)+⋯+qiNηN(t)=(qiT⊗Iℓ)η(t) that
(32)limt→∞∥θik(t)∥=limt→∞∥(qiT1)⊗σk(t)∥=0,i=2,3,…,N, k=1,2,…,ℓ-1, θ˙1(t)=Aθ1(t). By way of contradiction, assume that the sufficient condition does not hold; then there are two cases.
Case  1.Each of N-1 equations (27) (i=2,3,…,N) has more than one zero roots.
Case  2. Each of N-1 equations (27) (i=2,3,…,N) has simple zero root; however, there exists nonnegative real part.
If one of the previous two cases holds, then there exists a1≤k≤ℓ-1 such that limt→∞∥θik(t)∥≠0 for i=2,3,…,N, which contradicts formula (32).e
-
λ
τ can be expressed by Taylor expansion; therefore, (27) can be rewritten as follows:
(33)λℓ+λ[τ-∑k=2∞‍(-τ)kλk-1k!]∑k=0ℓ-1‍δkiλk=0.Sinceτ>0 and δ0i=αλiγ0>0, i=2,3,…,N, (33) has only simple zero root for i=2,3,…,N. When s≠0, let Fi(λ)=1+Gi(λ), where
(34)Gi(λ)=∑k=0ℓ-1δkiλkλℓ(1-e-λτ),i=2,3,…,N.
Based on the Nyquist stability criterion, therefore, the real parts of nonzero roots of (27) are negative if and only if Gi(jω) does not enclose the point (-1,j0) for ω∈ℝ, where j  is imaginary unit.Without loss of generality, assume thatℓ is an odd number. Denote S1={0,2,…,ℓ-1} and S2={1,3,…,ℓ-2}. By some calculations, one can obtain
(35)Gi(jω)=∑k=0ℓ-1δki(jω)k(jω)ℓ(1-e-jωτ)=∑k=0ℓ-1δkiωkjk+ℓ(-ω)ℓ2sinωτ2ej((π-ωτ)/2)=((j∑k∈S1‍δkiωk(-1)(k+ℓ-1)/2111+∑k∈S2‍δkiωk(-1)(k+ℓ)/2)11×(-ωℓ)-1∑k∈S2‍)2sinωτ2ej((π-ωτ)/2).Letai=∑k∈S2δkiωk(-1)(k+ℓ)/2,  bi=∑k∈S1δkiωk(-1)(k+ℓ-1)/2.Therefore,(36)Gi(jω)=ai+bij-ωℓ2sinωτ2ej((π-ωτ)/2)=2sin(ωτ/2)ai2+bi2ωℓej(((-π-ωτ)/2)+arctan(bi/ai)).Therefore, Theorem11 is equivalent to the following theorem.Theorem 12.
Assume that𝒢 is connected. Quasi-consensus of multi-agent systems (2) with protocol (20) can be achieved if and only if
(37)Gi(jω)=2sin(ωτ/2)ai2+bi2ωℓej(((-π-ωτ)/2)+arctan(bi/ai))
does not enclose the point (-1,j0) for i∈{2,3,…,N} and ω∈ℝ, where  ai=∑k∈S2δkiωk(-1)(k+ℓ)/2, bi=∑k∈S1δkiωk(-1)(k+ℓ-1)/2.As the symmetric property of the Nyquist plot, we only need to consider the case ofω>0. When ω∈(0,+∞), |Gi(jω)|≤(2∑k=0ℓ-1δkiωk)/ωℓ is monotonously decreasing; furthermore,
(38)arg(Gi(jω))={arctanbiai-π2-ωτ2,2nπ≤ωτ2<2nπ+π;arctanbiai+π2-ωτ2,2nπ+π≤ωτ2<2nπ+2π,
where n≥0 is the nonnegative integer.Whenω∈(0,+∞), Gi(jω) crosses the negative real axis for the first time at ω1i, one has
(39)ω1iτ2=arctanbi1ai1+π2,|Gi(jω1i)|≤2∑k∈S2δki(ω1i)k(ω1i)ℓ,
where ai1=∑k∈S2δki(ω1i)k(-1)(k+ℓ)/2, bi1=∑k∈S1δki(ω1i)k(-1)(k+ℓ-1)/2.Therefore, one has the following result equivalently.Theorem 13.
Assume that𝒢 is connected. Quasi-consensus of multi-agent systems (2) with protocol (20) can be achieved if and only if
(40)2∑k∈S2δki(ω1i)k<(ω1i)ℓ,fori=2,3,…,N,
where ω1i is defined by ω1iτ/2=arctan(bi1/ai1)+π/2.Denoteδmax=maxi,kδki. If all ω1i satisfy
(41)2∑k∈S2δmax(ω1i)k≤(ω1i)ℓ,fori=2,3,…,N,
then Theorem 13 holds.Remark 14.
If one considers consensus protocol with communication delays as follows:(42)ui=-∑k∈M1ckξi(k)+α∑j=1Naij∑k∈M2γk(ξj(k)(t)-ξi(k)(t))-∑k∈M1ckξi(k)(t-τ)+α∑j=1Naij∑k∈M2γk(ξj(k)(t-τ)-ξi(k)(t-τ)),
then the similar results can be obtained.Example 15.
Considering the multi-agent systems which is similar to example 1, here we take (20) as the quasi-consensus protocol. We take the same parameters; that is, the Laplacian matrix L is also
(43)[10-10000002-100-100-1-14-100-1000-12-1000000-110000-100010000-100100000000-11],α=1, γ0=1, γ1=2 and c2=2 and n0=2. Then, eigenvalues of L are λ1=0, λ2=0.382, λ3=0.6086, λ4=λ5=1, λ6=2.2271, λ7=2.618, and λ8=5.1642. By simulation, τmax=1.3224. Moreover, we choose τ=0.5. By Theorem 13, one knows that the quasi-consensus can be reached. Figures 4, 5 and 6 show that the quasi-consensus is reached.Figure 4
The trajectories ofxi, i=1,2,…,8.Figure 5
The trajectories ofvi, i=1,2,…,8.Figure 6
The trajectories ofai, i=1,2,…,8.


## 5. Conclusions

When the global states of agents and global relative information are unavailable, this paper designs a consensus protocol which only depends on the own partial information of agents and partial relative information with its neighbors. Under the consensus protocol, a necessary and sufficient condition for guaranteeing the consensus for a high-order integrator multi-agent systems is established. Moreover, the consensus protocol shows that the consensus can be reached when the own states of agents are complementary to the relative information with its neighbors; the consensus may not be reached when there are only a few information exchanges. Moreover, the paper also considers the quasi-consensus problem under the protocol with the delays. It is interesting that multi-agent systems can reach consensus without delays and quasi-consensus with delays.



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*Source: 269235-2013-10-24.xml*