# Distributed Control for Multiagent Consensus Motions with Nonuniform Time Delays

Authors: Mengji Shi; Kaiyu Qin
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper solves control problems of agents achieving consensus motions in presence of nonuniform time delays by obtaining the maximal tolerable delay value. Two types of consensus motions are considered: the rectilinear motion and the rotational motion. Unlike former results, this paper has remarkably reduced conservativeness of the consensus conditions provided in such form: for each system, if all the nonuniform time delays are bounded by the maximal tolerable delay value which is referred to as “delay margin,” the system will achieve consensus motion; otherwise, if all the delays exceed the delay margin, the system will be unstable. When discussing the system which is intended to achieve rotational consensus motion, an expanded system whose state variables are real numbers (those of the original system are complex numbers) is introduced, and corresponding consensus condition is given also in the form of delay margin. Numerical examples are provided to illustrate the results.


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## Body

## 1. Introduction

As one of the fundamental problems in the study of the cooperative control for multiagent systems, consensus problem has received a fair amount of recent attention, because of its applications in formation control [1], target tracking [2], and sensor networks [3]. When researching consensus problems, time delay is hardly a matter of rarity, and numerous examples [4–16] have attested to its prevalence. It then raises a question: are the consensus conditions for the nondelayed systems still applicable in presence of time delays?The treatments of systems with time delays have been widely studied, and researchers have developed diverse applicable approaches to analyzing the system stability under the occurrence of time delays, such as Lyapunov approach, the approach based on the properties of nonnegative matrices, and frequency domain approach. Lyapunov approach is a powerful tool of time domain analysis, with which one can easily handle nonlinearities and time-varying uncertainties. The consensus problems of delayed systems in [4, 5, 7, 8, 10, 12, 13, 16] have been worked out by establishing Lyapunov functions (functionals). Approaches based on the properties of nonnegative matrices can be available to the cases of dynamically changing topologies [15], but they are commonly limited to discrete-time systems. Linear delay systems are discussed in greater length with frequency domain approaches with which [6, 9, 11] have studied the stability of multiagent systems with time delays, and necessary and sufficient conditions for these systems to reach consensus have been derived. Usually, consensus conditions derived by Lyapunov-based approaches are more likely to be conservative than the ones obtained by frequency domain approaches.Taking nonuniform time delays into account, the paper investigates the consensus problems of multiagent systems in terms of motion. We consider two types of consensus motions: one is rectilinear motion, and the other is rotational motion. In the two systems, the agents aim to collectively move at a uniform velocity and angular speed, respectively, depending on their initial states. These multiagent models have found various applications, such as formation control for unmanned aerial vehicles (UAVs), spacecraft docking, and circular mobile sensor networks. The second order control algorithms are extended from the previous works [11, 17], but nonuniform time delays are considered.An efficient frequency domain approach is employed to derived sufficient conditions for these consensus problems. The results are presented in the form of “delay margins.” Each delay margin is a derived value that if all the time delays in the corresponding system are bounded by this value the system will reach consensus, but if all the delays exceed this value the agents will not achieve collective motion. Consequently, one cannot find out a greater value that if all the delays are bounded by that value the system still reaches consensus always. In [18], the author discussed the stabilities of second-order multiagent systems with nonuniform delays using Lyapunov methods, and the results are presented in the form of Linear Matrix Inequalities (LMIs), which are different from our work. The previous publications [11, 19] have studied multiagent systems with unstable modes, but nonuniform delays have not been considered. Reference [17] has proposed the rotating consensus algorithm, without consideration of delay effects. The rotating consensus algorithm we adopt enables the agents to achieve rotational consensus motion in the complex plane. However, in many cases, complex numbers cannot be processed by the processor. We expand the original system by separating the real part and imaginary part of each state variable. The new system has real state variables and coefficient matrices. We finally provide consensus conditions for the expanded system in the form of delay margin.The remainder of this paper is organized as follows. Section2 presents some mathematical preliminaries for the later proof, and with their help, we define the consensus problems and then protocols are given for the second-order multiagent systems with multiple time delays. Section 3 shows the stability analysis of the systems and discusses what will happen to the system to achieve rectilinear consensus motion if the input delays are absent; this section also provides an expansion to the system to reach rotating consensus, and this additional result allows us to simulate the algorithms in an easier way. Section 4 gives numerical simulations. Finally in Section 5, some conclusions are drawn.


## 2. Model

Before presenting the model, we first provide a brief narrative on some definition and results in graph theory, which are used extensively in the rest of this paper.Consider a multiagent system consisting ofn agents. The communication network topology among them is represented by an undirected graph G(V,E,A), which consists of a set of nodes V={si},i∈I={1,2,…,n}, a set of edges E⊆V×V, and a weighted adjacency matrix A=[aik], where aii=0, and aik=aki≥0 (for G is undirected). aik>0 if and only if there exists an edge eik∈E between the ith and kth nodes, which implies that they can get information from each other. The set of neighbours of node si is denoted by Ni≜{sk∈V:eki∈E}. The Laplacian associated with the graph G is defined as Lσ≜[lik], where lii=∑k=1naik and lik=-aik,i≠k. A path is a sequence of ordered edges er1r2,er2r3,…, where eriri+1∈E. If there is a path between every pair of nodes in graph G, the graph is said to be connected. The lemma below is given by [20].Lemma 1.
If the undirected graphG is connected, then its Laplacian Lσ has one singleton zero eigenvalue (with eigenvector 1), and the rest of n-1 eigenvalues of Lσ are all positive.Considern-agent systems with second-order dynamics:(1)x˙it=vit,v˙it=uit,where xi, vi denote the position and velocity of the ith agent, respectively, and ui(t) is the control input. Denoting the state vector ζi(t)≜[xi(t),vi(t)]T, we assume that the initial condition is ζi(p)=ζi(0), for p∈-∞,0.Two goals have been set for the system to achieve: rectilinear consensus motion and rotational consensus motion. Clearly, the control inputui(t) decides the behavior of the agents. We will propose two algorithms, respectively, for the two goals; before that, we use mathematical symbols to represent what will happen to the agents when the goals are achieved.For the first goal, rectilinear consensus motion, iflimt→+∞⁡ζit-ζkt=0 is satisfied for all i,k∈I, the control input ui(t) is said to solve the consensus problem asymptotically [11]. System (1) under this control input is referred to as “system Y1” hereinafter. In fact, in some cases (very extreme cases), the agents will not move collectively but all stay at a point as a final state; it depends on the initial conditions.For the second goal, rotational consensus motion, the consensus is reached if(2)limt→+∞⁡vit-vkt=0,limt→+∞⁡xit+w-1jvit-xkt+w-1jvkt=0,limt→+∞⁡v˙it-jwvit=0for all i,k∈I, where w∈R+ is the constant angular velocity in which all agents move around a common point together [17]. We assume w=1 to simplify the future discussion. System (1) under a control input to satisfy (2) is called “system Y2” hereunder.The state variables of agents in systemY1 are considered in the real domain for convenience of calculation; that is, xi,vi∈R in Y1. However, the agents in system Y2 are intended to achieve rotational consensus motion in the complex plane; thus the state variables xi,vi∈C are complex values for system Y2.Taking nonuniform time delays into account, we modify the corresponding protocols proposed in [11, 17] and give the control inputs of our systems Y1 and Y2 as(3)ForY1:uit=∑sk∈Niaikk1xit-τik-xkt-τik+k2vit-τik-vkt-τik,ForY2:uit=jvit-∑sk∈Niaikvit-τik-vkt-τik,-∑sk∈Niaikxit-τik+jvit-τik-xkt-τik+jvkt-τik,for any i,k∈I, where aik>0 denotes the edge weight, and τik=τki is the time delay for the ith agent to get the state information of the kth agent. We also consider the system to have M different time delays, denoted by τm∈{τik,i,k∈I}(m=1,2,…,M).We use matrix notation to write the models of both systems in compact form. To unify the form, we define(4)ψ1t≜x1t,v1t,x2t,v2t,…,xnt,vntT,ψ2t≜x1t,c1t,x2t,c2t,…,xnt,cnt,where ci(t)=xi(t)+jvi(t) for all i∈I. ψ1(t) and ψ2(t) are the state vectors of system Y1 and Y2, respectively. By defining coefficient matrices (5)A1≜0100,B1≜00k1k2,A2≜j-j00,B2≜00-11+j, system (1) under the control inputs (3) can be rewritten as(6)Yq:ψ˙qt=In⊗Aqψqt-∑m=1MLσm⊗Bqψqt-τm,for q=1,2, with the initial condition ψq(p)=ψq(0),p∈-∞,0, where Lσm denotes the Laplacian of a subgraph associated with the delay τm. Clearly, Lσ=∑m=1MLσm.In this paper, the graphG is assumed to be connected and undirected. Then according to Lemma 1, we suppose that the eigenvalues of Lσ are 0=λ1<λ2≤λ3≤⋯≤λn.


## 3. Main Results

In this section, we derive consensus conditions for systemsYq,q=1,2, with a frequency domain approach. The main idea of this method is to firstly prove the consensus achievement of each system when all the time delays are equal to zero and then to calculate the delay margin to ensure that if all the delays are under the bound of it, the stability of the system will not be affected.


### 3.1. Consensus Conditions for the Original Systems

Firstly, we investigate the stability ofY1 in absence of time delays. Let (7)Φ≜In⊗A1-Lσ⊗B1, and then Y1 given by (6) (q=1) with zero time delays can be written as(8)ψ˙1t=Φψ1t.Lemma 2.
The matrixΦ has a double eigenvalue at zero with a single Jordan block, and all other eigenvalues have negative real parts.Proof.
According to Lemma1, there exists an orthogonal matrix W, such that (9)WTLσW=diag⁡0,λ2,…,λn,and then it follows that (10)W⊗IlTΦW⊗Il=diag⁡A1,A1-λ2B1,…,A1-λnB1.Through simple calculations, we have(11)det⁡diag⁡A1,A1-λ2B1,…,A1-λnB1=s2∏i=2ns2+λik2s+λik1=0.The roots of each s2+λik2s+λik1=0(i≥2) are(12)-λik2±λik22-4λik12.Since λi>0 when i≥2, if (λik2)2<4λik1, the real parts of (12) are -λik2/2<0, and when (λik2)2≥4λik1, (12) apparently are negative real numbers. Thus the polynomial (11) has a double root at zero, and the real part of all other roots is negative; then the lemma is proven.Remark 3.
According to Lemma2, system (8) is unstable in the sense of traditional control due to the existence of the zero Jordan block in the system matrix. However, one can easily obtain that the vectors 1n⊗[1,0]T and 1n⊗[0,1]T are the eigenvectors associated with the zero eigenvalue. Therefore the nondelayed system (8) finally converges to the space spanned by the two vectors; that is, limt→+∞⁡ζit-ζkt=0, for all i,k∈I. This has illustrated that the consensus is reached.Reference [17] has proven that system Y2 will achieve rotational consensus motion when all the time delays are absent. Then we summarize the results in the following lemma.Lemma 4.
Consider a network of second-order agents with fixed topology that is connected. Both of the multiagent systemsYq  (q=1,2) given by (6) will reach consensus if all the delays τm=0,m=1,2,…,M.The next step is to find out the conditions ensuring stable consensus motions of both systems.Theorem 5.
Consider the second-order systemsYq,q=1,2, that, given by (6), the consensus problems are solved, if in each system all the delays τm are smaller than a corresponding value τ¯q,q=1,2, where (13)forY1:τ¯1=arctan⁡k2/k1ω¯1ω¯1,ω¯1=λn2k22+λn2k222+4λn2k122;forY2:τ¯2=π-arctan⁡ω¯2/ω¯2-1ω¯2,ω¯2=1+1+4λn2+4λn4+λn22.Proof.
Consider the multiagent systems (6) with nonuniform time delays. Take Ψq(s) as the Laplace transform of ψq(t); we have Ψq(s)=Gτq-1Ψq(0), where (14)Gτqs=sI2n-In⊗Aq+∑m=1MLσm⊗Bqe-τms,for q=1,2. According to the foregoing discussions, to study the stability of the delayed system, we only need to investigate the values of τm that guarantee the existence of nonzero roots of Gτq(s) on imaginary axis, which represents the crossing of the characteristic roots from the stable region to the unstable one. The roots of characteristic polynomials such as Gτq(s) are hereinafter referred to as “the eigenvalues of the system.”
Suppose that, for eachq∈{1,2}, sq=jωq≠0 is an imaginary root of Gτq(s), and uq=uq1⊗1,0T+uq2⊗0,1T is a corresponding eigenvector, where uq=1,uq1,uq2∈Cn. Then we have(15)jωqI2n-In⊗Aq+∑m=1MLσm⊗Bqe-jωqτmuq=0.Note that all the complex roots of each Gτq(s) appeared in conjugated pairs; we only need to study the situation that ωq>0. Since the elements of the vector obtained by calculating the left part of (15) equal zero, we have(16)jω1u11=u12,1-ω2u21=u22.Multiplying by uqH (the conjugate transpose of uq) on the left side of the left part of (15), and with (16) substituted, it yields(17)∑m=1Mαme-jω1τm=ω12k1+jω1k2,∑m=1Mαme-jω2τm=-jω2ω2-1ω2+jω2-1,where (18)αm=uqHLσm⊗I2uquqHuq.Equation (17) can be rewritten as(19)∑m=1Mαmejω1τm=ω12k1-jω1k2≜F1ω1,∑m=1Mαmejω2τm=jω2ω2-1ω2-jω2-1≜F2ω2.
Take modulus of both sides of (19) to obtain(20)Mqωq≜Fqωq=∑m=1Mαmejωqτm≤∑m=1Mαm=uqHLσ⊗IluquqHuq≤λn,for q=1,2. After some calculation, we find that the inequality Mq(ωq)≤λn from (20) holds true if and only if ωq≤ω¯q. This is a necessary condition for our hypothesis to hold.
The idea now is to analyze the argument of eachFq(ωq), to find out how τm affects the validity of our hypothesis that the imaginary eigenvalues jωq exist. Calculate the arguments of F1(ωq); we get (21)θ1ω1≜arg⁡F1ω1=arctan⁡k2k1ω2,θ2ω2≜arg⁡F2ω2=arctan⁡ω21-ω2,ω2<1,π2,ω2=1,π-arctan⁡ω2ω2-1,ω2>1,where θqω1∈[0,2π). Define τq(ωq)≜θq(ωq)/ωq. Then the following proves that, for each q=1,2, τ¯q=τq(ω¯q) is the minimal value of τq(ωq) when ωq≤ω¯q. For system Y1, Let a≜k2/k1. Calculate derivative of τ1(ω1): (22)D1ω1≜dτ1ω1dω1=aω1a2ω12+1-arctan⁡aω1ω12=1ω12aω1a2ω12+1-arctan⁡aω1︸D2ω1,and the derivative of D2(ω1) is (23)dD2ω1dω1=-2a3ω12a2ω12+12<0.Thus D2(ω1)<D2(0)=0 when ω1>0. Consequently, D1(ω1)=D2(ω1)/ω12<0 when ω1>0, which means τ1(ω1) decreases as ω1 grows. Therefore we have τ¯1=τ1(ω¯1)≤τ1(ω1), when ω1≤ω¯1. For system Y2, when ω2<1, one can easily calculate that τ2(ω2)>1 always holds; and when ω2≥1,  τ2(ω2) is monotonically decreasing from π/2; moreover, when ω>2, τ2(ω2)<1 (we omit the proof which is similar to the one given above). Because the network of the system is connected, λn>1 holds true; it follows that ω¯2>2. Therefore, one cannot find a smaller value of τ2(ω2) than τ¯2=τ2(ω¯2) when ω2≤ω¯2.
Note thatθq(ωq) are also the arguments of the left parts of (19); if for each system we set all τm<τ¯q, we have (24)τqωq=θqωqωq=arg⁡∑m=1Mαmejωqτmωq≤max⁡ωqτmωq<ωqτ¯qωq=τ¯q,for q=1,2, which conflicts with the idea that τ¯q=τq(ω¯q) is the minimal value of τq(ωq) when ωq≤ω¯q; and this rejects our hypothesis. Consequently, each of the two systems has no nonzero imaginary eigenvalue when all the delays τm<τ¯q. Following the continuity of function, the zero crossing of eigenvalues from the left half plane to the right half plane has been prevented, and the bounded delays will not bring about instabilities to the systems. According to Lemma 4, the system can reach consensus with bounded delays, and the theorem is proven.In fact, for each system, if allτm=τ¯q, then there must exist an imaginary eigenvalue jω¯q and corresponding eigenvector uq that satisfy ∑m=1Mαm=λn. That is, according to (19),(25)∑m=1Mαmejω¯qτ¯q=Fqω¯qholds true.Remark 6.
τ ¯ q are termed the delay margins. If we draw trajectories of the delays τm that each value grows from 0 to +∞, then the touch of τm=τ¯q leads to the first contact of the eigenvalues from the stable region to the unstable one. When all the delays exceed τ¯q, the system will be unstable, because there must exist at least one eigenvalue in the unstable region (the RHP). The instabilities get worse with increasing τm.Remark 7.
Reference [9] has provided a general method of obtaining delay margins for delayed systems (including nonuniform delays) to solve consensus problems, and the method proposed in this paper is inspired by that idea. However, only first-order systems have been considered in [9], where the arguments of ∑αmejωqτm are bounded by the simple π/2. In our paper, the arguments θq(ωq) are bounded by variables changing with ωq; thus we have to find possible maximal delay values related to ωq to cause the contradictions. Reference [21] has discussed the consensus problem of multiagent systems with delays using frequency approaches, but the agent dynamics are only considered in the form of single integrator, and the methods can not be directly used in second-order systems. In the literatures [11, 22], the authors studied the consensus problems of second-order systems with uniform delays; however, the situations of multiple time delays were not discussed. In [18], the stability analyses of second-order multiagent systems with nonuniform delays were performed using Lyapunov methods, and the results were presented in the form of Linear Matrix Inequalities (LMIs).


### 3.2. Consensus Phenomenon of SystemY1 without Input Delays

In many cases, the consensus is reached when all agents converge on a point and stay. References [11, 19] have proposed consensus algorithms enabling the vehicles to drive together at a uniform velocity after their convergence. These classical algorithms are then extensively applied in the subsequent works, such as the generalized use in [23] to solve the tracking problem. Therefore exploring such basic algorithms is very useful and the results can be generalized for many applications. In this paper, the idea of Y1 comes from the algorithm proposed in [11]; but we consider nonuniform time delays. In our systems, each agent gets delayed information from its neighbors, and besides these communication delays, input delays also occur. Consequently, one may come up with a question that what if the input delays are absent while communication delays still occur. In this situation, the control input of (1) for Y1 becomes(26)uit=∑sk∈Niaikk1xit-xkt-τik+k2vit-vjt-τik.Suppose that there is a time when all the agents are moving forward together at the same point; then the value ofxi(t)-xk(t-τik)>0 depends on the value of τik and how fast the agents move. If this happens, (26), which determines the acceleration of each agent, will not equal zero, and it is difficult for ui(t) and uk(t) (i≠k) to reach the same value due to the nonuniform delays. Then, in the next period of time, the agents have to separate from each other because of their different accelerations.Simulations (see in Section4) have shown a reasonable consensus phenomenon when applying (26) to the system: when appropriate delays are set to the system, the agents move to get together at first, and then slow down until they stop at the same point, which means that the agents will not finally achieve a collective motion when the input delays are absent while communication delays still occur.


### 3.3. Conditions for an Expanded System to Achieve Rotational Consensus Motion

SystemY2 given by (6) (q=2) has provided a solution that the agents are intended to perform a rotational consensus motion in the complex plane. However, in many practical cases, the processors of the systems do not accept complex numbers directly; as an example, it is difficult to draw a simulation model in the MATLAB for such a system involving complex state variables. Commonly, we “expand” the system by replacing each complex state variable with two real variables: the real part and the imaginary part of the original variable, respectively. The new system is equipped with real coefficient matrices.Denote the new state vector and coefficient matrices:(27)ψ2¯t≜Re⁡x1,Im⁡x1,Re⁡c1,Im⁡c1,Re⁡x2,Im⁡x2,Re⁡c2,Im⁡c2,…,Re⁡xn,Im⁡xn,Re⁡cn,Im⁡cn,A2¯≜Re⁡A2⊗I2+Im⁡A2⊗E2,B2¯≜Re⁡B2⊗I2+Im⁡B2⊗E2,where Re⁡(·) and Im⁡(·) mean the real and imaginary part of the term enclosed, respectively, and (28)E2≜0-110.We get the new system expanded from the system given by (6) (q=2):(29)ψ2¯˙t=In⊗A2¯ψ2¯t-∑m=1MLσm⊗B2¯ψ2¯t-τm.Equation (29) is another form of Y2, and its state variables are real values. Agents in system (29) behave like those in Y2 given by (6) (q=2) when all the time delays are absent. Therefore, in the light of the real state variables, we usually simulate the expanded system instead of the original one. However, there is a notable difference between the two systems. The amount of eigenvalues of the new system (29) doubles that of (6) (q=2). We introduce another related system for reference:(30)ψ˙2t=In⊗A2∗ψ2t-∑m=1MLσm⊗B2∗ψ2t-τm,where A2∗ and B2∗ are the the conjugate matrices of A2 and B2, respectively. The subsequent lemma illustrates the relationships among the three systems (6) (q=2), (30), and (29).Lemma 8.
There are4n eigenvalues of system (29), which consist of all the 2n eigenvalues of (6) (q=2) and all the 2n eigenvalues of (30).Proof.
Suppose thatγ is one eigenvalue of (6) (q=2), and the corresponding eigenvector is α. Then we have(31)γI2n-In⊗A2+∑m=1MLσm⊗B2e-τmγα=0.Substituting γ into the characteristic polynomial of system (29), and multiplying by α⊗[j,1], it yields (32)γI4n-In⊗A2¯+∑m=1MLσm⊗B2¯e-τmγα⊗j,1=γI2n-In⊗A2+∑m=1MLσm⊗B2e-τmγα⊗j,1=0,which implies that γ is an eigenvalue of system (29), and α⊗[j,1] a corresponding eigenvector. It follows that the eigenvalues of (6) (q=2) also are the eigenvalues of (29). By the same way, one can prove that each eigenvalue of system (30), whose corresponding eigenvector is here denoted as αp, is also an eigenvalue of (29), and αp⊗[-j,1] can be a corresponding eigenvector. As the 2n+2n=4n eigenvalues fit the system (29) in amount, and (29) has no other eigenvalues, the lemma is proven.Lemma 9.
The expanded multiagent system (29) with real state variables will reach consensus if all the time delays are absent; that is, τm=0,m=1,2,…,M.Proof.
When all the time delays equal zero, the eigenvalues of system (30) are the conjugate numbers of the eigenvalues of (6) (q=2). As each conjugated pair has the same real parts, the expanded system (29) has the same stability properties as (6) (q=2) when all time delays are absent, and the agents will achieve rotational consensus motion.Theorem 10.
Considering the multiagent system (29) expanded from (6) (q=2), it will reach consensus if all the delays τm<τ¯, where (33)τ¯=min⁡τd1,τd2,τd1=π-arctan⁡ωd1/ωd1-1ωd1,ωd1=1+1+4λn2+4λn4+λn22,τd2=arctan⁡ωd2/ωd2+1ωd2,ωd2=-1+1+4λn2+4λn4+λn22.Proof.
Theorem5 has proven that τd1 is the delay margin ensuring that the eigenvalues of (6) (q=2) are in the stable region. We can also prove that the eigenvalues of system (30) are in the stable region if all the delays are bounded by τd2, and the system will be unstable if all the delays exceed τd2, by the same way as in the proof of Theorem 5. Thus the proof is omitted.
The foregoing discussions have helped us to understand that when all the delaysτm<τ¯, the eigenvalues of both systems (6) (q=2) and (30) are in the stable region. According to Lemmas 8 and 9, system (30) can reach consensus with nonuniform time delays bounded by τ¯.Remark 11.
The theorem above has provided us with a delay bound ensuring the consensus achievement. Note that bothτd1 and τd2 are delay margins of the corresponding systems. We set all the time delays of the expanded system (29) greater than τ¯; when these delays are put into the systems (6) (q=2) and (30), at least one of the systems will have eigenvalues in the unstable region; this will bring about the failure of the expanded system (29) to achieve consensus motion. Hence τ¯ is also the delay margin for the expanded system.


## 3.1. Consensus Conditions for the Original Systems

Firstly, we investigate the stability ofY1 in absence of time delays. Let (7)Φ≜In⊗A1-Lσ⊗B1, and then Y1 given by (6) (q=1) with zero time delays can be written as(8)ψ˙1t=Φψ1t.Lemma 2.
The matrixΦ has a double eigenvalue at zero with a single Jordan block, and all other eigenvalues have negative real parts.Proof.
According to Lemma1, there exists an orthogonal matrix W, such that (9)WTLσW=diag⁡0,λ2,…,λn,and then it follows that (10)W⊗IlTΦW⊗Il=diag⁡A1,A1-λ2B1,…,A1-λnB1.Through simple calculations, we have(11)det⁡diag⁡A1,A1-λ2B1,…,A1-λnB1=s2∏i=2ns2+λik2s+λik1=0.The roots of each s2+λik2s+λik1=0(i≥2) are(12)-λik2±λik22-4λik12.Since λi>0 when i≥2, if (λik2)2<4λik1, the real parts of (12) are -λik2/2<0, and when (λik2)2≥4λik1, (12) apparently are negative real numbers. Thus the polynomial (11) has a double root at zero, and the real part of all other roots is negative; then the lemma is proven.Remark 3.
According to Lemma2, system (8) is unstable in the sense of traditional control due to the existence of the zero Jordan block in the system matrix. However, one can easily obtain that the vectors 1n⊗[1,0]T and 1n⊗[0,1]T are the eigenvectors associated with the zero eigenvalue. Therefore the nondelayed system (8) finally converges to the space spanned by the two vectors; that is, limt→+∞⁡ζit-ζkt=0, for all i,k∈I. This has illustrated that the consensus is reached.Reference [17] has proven that system Y2 will achieve rotational consensus motion when all the time delays are absent. Then we summarize the results in the following lemma.Lemma 4.
Consider a network of second-order agents with fixed topology that is connected. Both of the multiagent systemsYq  (q=1,2) given by (6) will reach consensus if all the delays τm=0,m=1,2,…,M.The next step is to find out the conditions ensuring stable consensus motions of both systems.Theorem 5.
Consider the second-order systemsYq,q=1,2, that, given by (6), the consensus problems are solved, if in each system all the delays τm are smaller than a corresponding value τ¯q,q=1,2, where (13)forY1:τ¯1=arctan⁡k2/k1ω¯1ω¯1,ω¯1=λn2k22+λn2k222+4λn2k122;forY2:τ¯2=π-arctan⁡ω¯2/ω¯2-1ω¯2,ω¯2=1+1+4λn2+4λn4+λn22.Proof.
Consider the multiagent systems (6) with nonuniform time delays. Take Ψq(s) as the Laplace transform of ψq(t); we have Ψq(s)=Gτq-1Ψq(0), where (14)Gτqs=sI2n-In⊗Aq+∑m=1MLσm⊗Bqe-τms,for q=1,2. According to the foregoing discussions, to study the stability of the delayed system, we only need to investigate the values of τm that guarantee the existence of nonzero roots of Gτq(s) on imaginary axis, which represents the crossing of the characteristic roots from the stable region to the unstable one. The roots of characteristic polynomials such as Gτq(s) are hereinafter referred to as “the eigenvalues of the system.”
Suppose that, for eachq∈{1,2}, sq=jωq≠0 is an imaginary root of Gτq(s), and uq=uq1⊗1,0T+uq2⊗0,1T is a corresponding eigenvector, where uq=1,uq1,uq2∈Cn. Then we have(15)jωqI2n-In⊗Aq+∑m=1MLσm⊗Bqe-jωqτmuq=0.Note that all the complex roots of each Gτq(s) appeared in conjugated pairs; we only need to study the situation that ωq>0. Since the elements of the vector obtained by calculating the left part of (15) equal zero, we have(16)jω1u11=u12,1-ω2u21=u22.Multiplying by uqH (the conjugate transpose of uq) on the left side of the left part of (15), and with (16) substituted, it yields(17)∑m=1Mαme-jω1τm=ω12k1+jω1k2,∑m=1Mαme-jω2τm=-jω2ω2-1ω2+jω2-1,where (18)αm=uqHLσm⊗I2uquqHuq.Equation (17) can be rewritten as(19)∑m=1Mαmejω1τm=ω12k1-jω1k2≜F1ω1,∑m=1Mαmejω2τm=jω2ω2-1ω2-jω2-1≜F2ω2.
Take modulus of both sides of (19) to obtain(20)Mqωq≜Fqωq=∑m=1Mαmejωqτm≤∑m=1Mαm=uqHLσ⊗IluquqHuq≤λn,for q=1,2. After some calculation, we find that the inequality Mq(ωq)≤λn from (20) holds true if and only if ωq≤ω¯q. This is a necessary condition for our hypothesis to hold.
The idea now is to analyze the argument of eachFq(ωq), to find out how τm affects the validity of our hypothesis that the imaginary eigenvalues jωq exist. Calculate the arguments of F1(ωq); we get (21)θ1ω1≜arg⁡F1ω1=arctan⁡k2k1ω2,θ2ω2≜arg⁡F2ω2=arctan⁡ω21-ω2,ω2<1,π2,ω2=1,π-arctan⁡ω2ω2-1,ω2>1,where θqω1∈[0,2π). Define τq(ωq)≜θq(ωq)/ωq. Then the following proves that, for each q=1,2, τ¯q=τq(ω¯q) is the minimal value of τq(ωq) when ωq≤ω¯q. For system Y1, Let a≜k2/k1. Calculate derivative of τ1(ω1): (22)D1ω1≜dτ1ω1dω1=aω1a2ω12+1-arctan⁡aω1ω12=1ω12aω1a2ω12+1-arctan⁡aω1︸D2ω1,and the derivative of D2(ω1) is (23)dD2ω1dω1=-2a3ω12a2ω12+12<0.Thus D2(ω1)<D2(0)=0 when ω1>0. Consequently, D1(ω1)=D2(ω1)/ω12<0 when ω1>0, which means τ1(ω1) decreases as ω1 grows. Therefore we have τ¯1=τ1(ω¯1)≤τ1(ω1), when ω1≤ω¯1. For system Y2, when ω2<1, one can easily calculate that τ2(ω2)>1 always holds; and when ω2≥1,  τ2(ω2) is monotonically decreasing from π/2; moreover, when ω>2, τ2(ω2)<1 (we omit the proof which is similar to the one given above). Because the network of the system is connected, λn>1 holds true; it follows that ω¯2>2. Therefore, one cannot find a smaller value of τ2(ω2) than τ¯2=τ2(ω¯2) when ω2≤ω¯2.
Note thatθq(ωq) are also the arguments of the left parts of (19); if for each system we set all τm<τ¯q, we have (24)τqωq=θqωqωq=arg⁡∑m=1Mαmejωqτmωq≤max⁡ωqτmωq<ωqτ¯qωq=τ¯q,for q=1,2, which conflicts with the idea that τ¯q=τq(ω¯q) is the minimal value of τq(ωq) when ωq≤ω¯q; and this rejects our hypothesis. Consequently, each of the two systems has no nonzero imaginary eigenvalue when all the delays τm<τ¯q. Following the continuity of function, the zero crossing of eigenvalues from the left half plane to the right half plane has been prevented, and the bounded delays will not bring about instabilities to the systems. According to Lemma 4, the system can reach consensus with bounded delays, and the theorem is proven.In fact, for each system, if allτm=τ¯q, then there must exist an imaginary eigenvalue jω¯q and corresponding eigenvector uq that satisfy ∑m=1Mαm=λn. That is, according to (19),(25)∑m=1Mαmejω¯qτ¯q=Fqω¯qholds true.Remark 6.
τ ¯ q are termed the delay margins. If we draw trajectories of the delays τm that each value grows from 0 to +∞, then the touch of τm=τ¯q leads to the first contact of the eigenvalues from the stable region to the unstable one. When all the delays exceed τ¯q, the system will be unstable, because there must exist at least one eigenvalue in the unstable region (the RHP). The instabilities get worse with increasing τm.Remark 7.
Reference [9] has provided a general method of obtaining delay margins for delayed systems (including nonuniform delays) to solve consensus problems, and the method proposed in this paper is inspired by that idea. However, only first-order systems have been considered in [9], where the arguments of ∑αmejωqτm are bounded by the simple π/2. In our paper, the arguments θq(ωq) are bounded by variables changing with ωq; thus we have to find possible maximal delay values related to ωq to cause the contradictions. Reference [21] has discussed the consensus problem of multiagent systems with delays using frequency approaches, but the agent dynamics are only considered in the form of single integrator, and the methods can not be directly used in second-order systems. In the literatures [11, 22], the authors studied the consensus problems of second-order systems with uniform delays; however, the situations of multiple time delays were not discussed. In [18], the stability analyses of second-order multiagent systems with nonuniform delays were performed using Lyapunov methods, and the results were presented in the form of Linear Matrix Inequalities (LMIs).


## 3.2. Consensus Phenomenon of SystemY1 without Input Delays

In many cases, the consensus is reached when all agents converge on a point and stay. References [11, 19] have proposed consensus algorithms enabling the vehicles to drive together at a uniform velocity after their convergence. These classical algorithms are then extensively applied in the subsequent works, such as the generalized use in [23] to solve the tracking problem. Therefore exploring such basic algorithms is very useful and the results can be generalized for many applications. In this paper, the idea of Y1 comes from the algorithm proposed in [11]; but we consider nonuniform time delays. In our systems, each agent gets delayed information from its neighbors, and besides these communication delays, input delays also occur. Consequently, one may come up with a question that what if the input delays are absent while communication delays still occur. In this situation, the control input of (1) for Y1 becomes(26)uit=∑sk∈Niaikk1xit-xkt-τik+k2vit-vjt-τik.Suppose that there is a time when all the agents are moving forward together at the same point; then the value ofxi(t)-xk(t-τik)>0 depends on the value of τik and how fast the agents move. If this happens, (26), which determines the acceleration of each agent, will not equal zero, and it is difficult for ui(t) and uk(t) (i≠k) to reach the same value due to the nonuniform delays. Then, in the next period of time, the agents have to separate from each other because of their different accelerations.Simulations (see in Section4) have shown a reasonable consensus phenomenon when applying (26) to the system: when appropriate delays are set to the system, the agents move to get together at first, and then slow down until they stop at the same point, which means that the agents will not finally achieve a collective motion when the input delays are absent while communication delays still occur.


## 3.3. Conditions for an Expanded System to Achieve Rotational Consensus Motion

SystemY2 given by (6) (q=2) has provided a solution that the agents are intended to perform a rotational consensus motion in the complex plane. However, in many practical cases, the processors of the systems do not accept complex numbers directly; as an example, it is difficult to draw a simulation model in the MATLAB for such a system involving complex state variables. Commonly, we “expand” the system by replacing each complex state variable with two real variables: the real part and the imaginary part of the original variable, respectively. The new system is equipped with real coefficient matrices.Denote the new state vector and coefficient matrices:(27)ψ2¯t≜Re⁡x1,Im⁡x1,Re⁡c1,Im⁡c1,Re⁡x2,Im⁡x2,Re⁡c2,Im⁡c2,…,Re⁡xn,Im⁡xn,Re⁡cn,Im⁡cn,A2¯≜Re⁡A2⊗I2+Im⁡A2⊗E2,B2¯≜Re⁡B2⊗I2+Im⁡B2⊗E2,where Re⁡(·) and Im⁡(·) mean the real and imaginary part of the term enclosed, respectively, and (28)E2≜0-110.We get the new system expanded from the system given by (6) (q=2):(29)ψ2¯˙t=In⊗A2¯ψ2¯t-∑m=1MLσm⊗B2¯ψ2¯t-τm.Equation (29) is another form of Y2, and its state variables are real values. Agents in system (29) behave like those in Y2 given by (6) (q=2) when all the time delays are absent. Therefore, in the light of the real state variables, we usually simulate the expanded system instead of the original one. However, there is a notable difference between the two systems. The amount of eigenvalues of the new system (29) doubles that of (6) (q=2). We introduce another related system for reference:(30)ψ˙2t=In⊗A2∗ψ2t-∑m=1MLσm⊗B2∗ψ2t-τm,where A2∗ and B2∗ are the the conjugate matrices of A2 and B2, respectively. The subsequent lemma illustrates the relationships among the three systems (6) (q=2), (30), and (29).Lemma 8.
There are4n eigenvalues of system (29), which consist of all the 2n eigenvalues of (6) (q=2) and all the 2n eigenvalues of (30).Proof.
Suppose thatγ is one eigenvalue of (6) (q=2), and the corresponding eigenvector is α. Then we have(31)γI2n-In⊗A2+∑m=1MLσm⊗B2e-τmγα=0.Substituting γ into the characteristic polynomial of system (29), and multiplying by α⊗[j,1], it yields (32)γI4n-In⊗A2¯+∑m=1MLσm⊗B2¯e-τmγα⊗j,1=γI2n-In⊗A2+∑m=1MLσm⊗B2e-τmγα⊗j,1=0,which implies that γ is an eigenvalue of system (29), and α⊗[j,1] a corresponding eigenvector. It follows that the eigenvalues of (6) (q=2) also are the eigenvalues of (29). By the same way, one can prove that each eigenvalue of system (30), whose corresponding eigenvector is here denoted as αp, is also an eigenvalue of (29), and αp⊗[-j,1] can be a corresponding eigenvector. As the 2n+2n=4n eigenvalues fit the system (29) in amount, and (29) has no other eigenvalues, the lemma is proven.Lemma 9.
The expanded multiagent system (29) with real state variables will reach consensus if all the time delays are absent; that is, τm=0,m=1,2,…,M.Proof.
When all the time delays equal zero, the eigenvalues of system (30) are the conjugate numbers of the eigenvalues of (6) (q=2). As each conjugated pair has the same real parts, the expanded system (29) has the same stability properties as (6) (q=2) when all time delays are absent, and the agents will achieve rotational consensus motion.Theorem 10.
Considering the multiagent system (29) expanded from (6) (q=2), it will reach consensus if all the delays τm<τ¯, where (33)τ¯=min⁡τd1,τd2,τd1=π-arctan⁡ωd1/ωd1-1ωd1,ωd1=1+1+4λn2+4λn4+λn22,τd2=arctan⁡ωd2/ωd2+1ωd2,ωd2=-1+1+4λn2+4λn4+λn22.Proof.
Theorem5 has proven that τd1 is the delay margin ensuring that the eigenvalues of (6) (q=2) are in the stable region. We can also prove that the eigenvalues of system (30) are in the stable region if all the delays are bounded by τd2, and the system will be unstable if all the delays exceed τd2, by the same way as in the proof of Theorem 5. Thus the proof is omitted.
The foregoing discussions have helped us to understand that when all the delaysτm<τ¯, the eigenvalues of both systems (6) (q=2) and (30) are in the stable region. According to Lemmas 8 and 9, system (30) can reach consensus with nonuniform time delays bounded by τ¯.Remark 11.
The theorem above has provided us with a delay bound ensuring the consensus achievement. Note that bothτd1 and τd2 are delay margins of the corresponding systems. We set all the time delays of the expanded system (29) greater than τ¯; when these delays are put into the systems (6) (q=2) and (30), at least one of the systems will have eigenvalues in the unstable region; this will bring about the failure of the expanded system (29) to achieve consensus motion. Hence τ¯ is also the delay margin for the expanded system.


## 4. Numerical Examples

In this section, some simulations are carried out to evidence the theoretical results obtained by the previous analysis.Consider that a multiagent system consists of 4 agents, whose communication topology is described in graphG shown in Figure 1. The 1st-2nd and 3rd-4th agent pairs are connected and the communication delay between each of them is τ1; the 2nd-3rd and 4th-1st agent pairs are connected in presence of communication delay τ2. Suppose the weight of each edge on G is 1; then λn=4.Figure 1
The communication topology of the 4-agent system.We start withY1 given by (6) (q=1). Let k1=1.2 and k2=1; then according to Theorem 5, the delay bound is τ¯=0.31. Firstly, we set the nonuniform time delays labelled on graph G as τ1(t)=0.30, and τ2t=0.29, which are under the bound. Apparently, from the simulation results shown in Figure 2, the consensus is reached. As a contrast example, we change the value of delays to see whether the agents could reach consensus again when the delays exceed the bound. Here τ1(t)=0.31 and τ2(t)=0.32. Figure 3 has shown that the agents fail to reach consensus.Figure 2
The trajectories of the position and velocity inY1 given by (6) (q=1) when delays are under the bound.Figure 3
The trajectories of the position and velocity inY1 given by (6) (q=1) when delays exceed the bound.To understand what actually happens to the multiagent system when (26) is the control input of system (1), we set k1=1.2,k2=1, and the time delays τ1(t)=0.01,τ2(t)=0.04. The simulation results shown in Figure 4 indicate the situation that is described in Section 3.2.Figure 4
The trajectories of the position and velocity in the system (1) applying control input (26) where input delays are absent.For the system to achieve rotational consensus motion, we use the expanded system (29) as the simulation model. According to Theorem 10, τ¯=0.134. We also execute the simulation twice: for the first time, we set τ1(t)=0.12,τ2(t)=0.13 as the bounded delays; and for the second time, the delays are set as τ1(t)=0.14,τ2(t)=0.15 which exceed the delay margin. The results are shown in the first and second pictures in Figure 5 separately, and they have validated Theorem 10.Figure 5
The trajectories of the agents in the system to achieve rotational consensus motion given by (29).


## 5. Conclusions

In this work, we studied two types of consensus problems of multiagent systems with nonuniform time delays: one makes the agents achieve a collective rectilinear motion, and the other demands the agents to finally move rotationally. We provided sufficient conditions in the form of delay margins for the systems to reach consensus and carried out simulations to show the instance of the results.Frequency domain approaches are commonly limited to linear systems. The proposed method is not compatible with nonlinear systems. Further research will focus on consensus problems of delayed multiagent systems with high-order dynamics and directed topologies. High-order dynamics will cause the occurrence of high-order polynomials in (20); directed topologies will bring complex eigenvalues to Laplacian Lσ and break monotonicity of θq(ωq). Nevertheless, it is realistic to overcome these difficulties and adapt the approach to such systems in the future as we explore proper mathematical tools.



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