# Consensus Conditions for High-Order Multiagent Systems with Nonuniform Delays

Authors: Mengji Shi; Kaiyu Qin; Ping Li; Jun Liu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Consensus of first-order and second-order multiagent systems has been wildly studied. However, the convergence of high-order (especially the third-order to the sixth-order) state variables is also ubiquitous in various fields. The paper handles consensus problems of high-order multiagent systems in the presence of multiple time delays. Obtained by a novel frequency domain approach which properly resolves the challenges associated with nonuniform time delays, the consensus conditions for the first-order and second-order systems are proven to be nonconservative, and those for the third-order to the sixth-order systems are provided in the form of simple inequalities. The method revealed in this article is applicable to arbitrary-order systems, and the results are less conservative than those based on Lyapunov approaches, because it roots in sufficient and necessary criteria of stabilities. Simulations are carried out to validate the theoretical results.


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## Body

## 1. Introduction

Consensus problems of multiagent systems have found many applications in the fields that hold great promise, including (but not limited to) biosciences, robotics, and computer sciences. Consensus is the agreement regarding a certain quality of interest on specific states of all the agents, which is widely demanded in the engineering applications. The research on consensus problems has lasted for decades. Various techniques are developed to solve consensus problems of numerous multiagent systems [1–23].This paper addresses the consensus control problems of high-order multiagents systems with nonuniform time delays. One motivation for studying high-order systems is to achieve accurate control of complex motion: for example, when performing consensus motion that requires abrupt change of heading, a team of vehicles should maintain consistency of acceleration (as well as position and velocity) among them by controlling the third-order state (acceleration), while lower-order (first-order and second-order) consensus protocols are usually designed for more regular motion (e.g., rectilinear [16] and rotational [23] motion). Besides high-order dynamics [3–9], nonlinearities [1, 2, 24], time delays [8–21, 24, 25] and fuzziness [26, 27] also bring complexities to the control systems, which often lead to difficulties in stability analysis.A novel frequency-domain-based method is developed to challenge the system complexities and derive the consensus conditions. Comparing to the universal stability analysis tool Lyapunov approaches, frequency domain methods are more possibly conducing to less conservative results as it roots in sufficient and necessary stability criteria. On the other hand, Lyapunov approaches applied in many literatures yield consensus conditions in the form of Linear Matrix Inequalities (LMIs) [7–9], while with frequency domain methods authors of [12–16, 22] as well as this note obtain consensus conditions in the form of inequalities which are more perspicuous and simple to calculate. However, frequency domain methods are limited to linear and time-invariant systems, and consequently most of the aforementioned articles [1–11, 24, 25] especially those coping with nonlinear and high-order systems have adopted Lyapunov approaches instead. Moreover, both high-order dynamics and nonuniform time delays give rise to dramatic increment of the systems’ dimensionality, which makes it a knotty problem. In existing literatures, methods based on the properties of nonnegative matrices [17, 18] and Nyquist stability theories [19–21] are introduced as alternative stability analysis tools.The main idea of the proposed approach is to transform the high-order systems’ dynamics into high-degree polynomials with respect to hypothetically existing imaginary eigenvalues of the systems. By studying the monotonicity of the polynomials and their derivatives, consensus conditions can be figured out in the form of inequalities. The present work first brings out sufficient consensus conditions for the first-order to sixth-order nonuniformly delayed systems which are most likely to apply to practical engineering applications [28]: if all the delays are bounded by a given value and all the parameters agree to corresponding inequalities, the systems can achieve consensus and in addition, for the sake of nonconservativeness, the paper provides stronger conditions for the first-order and second-order systems by thorough derivation: the states converge when all the delays are bounded by a given value but diverge when all the delays exceed that value.Literature [29] has proposed a high-order nonlinear consensus tracking algorithm with unmeasurable system states which applies to wide-range multiagent systems and proven the achievement of consensus by constructing Lyapunov functions, while time delays are not considered. In [9], Zhang et al. have solved average consensus problem of high-order multiagent system with time-varying delays and provided stability conditions in the form of LMIs via a Lyapunov-Krasovskii approach. The authors of [19] have derived necessary and sufficient consensus conditions for large-scale high-order linear multiagent systems with heterogeneous communication delays by using the generalized Nyquist criterion; however, the derived consensus conditions are set-valued graphical conditions, and inequality conditions are only derived for the first-order system. The previous work in [30] has studied consensus motion of delayed second-order multiagent systems by a nonconservative frequency-domain-based method, while this article will present stationary consensus conditions for more complex high-order multiagent systems.The remainder of this note is organized as follows: Section2 states the consensus problem with the help of graph theory; Section 3 presents the main results by demonstrating the stability analysis; Section 4 depicts the selected simulation experiments; Section 5 draws conclusions with future research directions.


## 2. Problem Statement

This section starts with some definitions and results in graph theory.Consider ann-agent system. 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=[aij], where aii=0 and aij=aji≥0 (for G is undirected). aij>0 if and only if there exists an edge eij∈E between the ith and jth nodes, which implies that they can get information from each other. The set of neighbors of node si is denoted by Ni≜{sj∈V:eji∈E}. The Laplacian corresponding to the graph G is defined as Lσ≜[lij], where lii=∑j=1naij and lij=-aij, i≠j. A path is a sequence of indexed edges ek1k2,ek2k3,…, where ekiki+1∈E. If there is a path between every pair of nodes in graph G, the graph is said to be connected. The following lemma is given by [31].Lemma 1.
If the undirected graphG is connected, then its Laplacian Lσ has one singleton zero eigenvalue (with eigenvector 1), and the rest n-1 eigenvalues of Lσ are all positive.Consider anlth-order multiagent system consisting of n agents. The dynamics of the ith agent (i∈I) is(1)ψ˙i0t=ψi1t,⋮ψ˙il-2t=ψil-1t,ψ˙il-1t=uit,where ψi(k)∈R is the kth state variable of the ith agent, k=0,1,…,l-1, and ui(t)∈R is the control input. Let ψi≜[ψi(0),ψi(1),…,ψi(l-1)]T be the state vector of the ith agent; we assume that the initial conditions are ψi(0)(s)=ψi(0)(0) and ψi(k)(s)=ψi(k)(0)=0, k=1,2,…,l-1, for s∈(-∞,0]. The control input ui(t) is said to solve the consensus problem asymptotically, if and only if limt→+∞⁡[ψi(t)-ψj(t)]=0 for all i,j∈I.In [8], a discrete-time control input was introduced as (2)uik=-∑j=1l-1pjψijk-∑sj∈Nikaijkψi0k-ψj0k-τij.In this paper, we introduce an continuous-time consensus algorithm for system (1) with multiple time delays, and the input delays are supposed to occur. The protocol is(3)uit=-∑j=1l-1pjψijt-∑sj∈Niaijψi0t-τij-ψj0t-τij,for any i∈I, where pj>0 for j=1,2,…,l-1; aij>0 denotes the edge weight, and τij=τji is the time delay for the ith agent to get the state information of the jth agent. We assume that the system has M different time delays, denoted by τm∈(τij,i,j∈I(m=1,2,…,M).Letψt≜ψ1t,ψ2t,…,ψnt, and (4)A≜0100000100⋮⋱⋱⋱00⋮0010-p1…-pl-2-pl-1∈Rl×l,B≜00…0⋮⋮⋮⋮00…010…0∈Rl×l.Under the control input given by (3), the network dynamics of the multiagent system becomes(5)ψ˙t=In⊗Aψt-∑m=1MLσm⊗Bψt-τmwith the initial condition ψ(s)=ψ(0), s∈(-∞,0], where Lσm denotes the Laplacian of a subgraph associated with the delay τm. Clearly, Lσ=∑m=1MLσm. If all the time delays are equal to zero, system (5) could be rewritten as(6)ψ˙t=In⊗A-Lσ⊗Bψt.This paper assumes that the graphG is always connected and undirected.


## 3. Main Results

The following lemma presents a sufficient condition for the stability of high-degree polynomials given by [32], which is helpful in the present work.Lemma 2.
Consider a polynomialf(x)=a0+a1x+⋯+anxn, where ak>0, k=0,1,…,n, n≥3, with coefficients of determination defined as (7)μi≜ai-1ai+2aiai+1, if all the coefficients of determination satisfy that μi<β/2≈0.4655, where i=1,2,…,n-2, n≥3, and β is the only real root of equation(8)β34+β2+β-2=0;then all the roots of f(x)=0 have negative real parts.Let(9)Φ≜In⊗A-Lσ⊗Band suppose that the eigenvalues of Lσ are 0=λ1<λ2≤λ3≤⋯≤λn according to Lemma 1.Assumption 3.
Assume forl≥3 that all pj>0 (j=1,2,…,l-1) satisfy the following:(10)pl-1pl-2>3plpl-3pl-2pl-3>3pl-1pl-4⋮p3p2>3p4p1p2p1>2.15p3λn,where pl=1 and λn is the largest eigenvalue of Lσ.Under Assumption3, the following lemma can be proven.Lemma 4.
MatrixΦ has a singleton zero eigenvalue and all other eigenvalues have negative real parts if Assumption 3 is satisfied.Proof.
According to Lemma1, there exists an orthogonal matrix W, such that (11)WTLσW=diag⁡0,λ2,…,λn,and then it follows that(12)W⊗IlTΦW⊗Il=diag⁡A,A-λ2B,…,A-λnB.Through simple calculations, the eigenpolynomial of (A-λjB) could be obtained; then we get(13)det⁡sI2-A+λjB=sl+∑i=1l-1pisi+λj=0.To simplify the following statements, let pl=1, and (13) could be written as(14)∑i=1lpisi+λj=0.For the first-order system, since l=1, it is evident that -λj are the eigenvalues of the system; thus the lemma is proven. For the second-order system, (14) becomes s2+p1s+λj=0 and its roots are(15)-p1±p12-4λj2.If p12<4λj, the real part of (15) is -p1/2<0, and when p12≥4λj, (15) apparently is a pair of nonpositive real numbers, and the bigger one equals zero if and only if λj=0; then the lemma is proven.
Whenl≥3, note that pi>0 for i=1,2,…,l, and λn is the largest eigenvalue of Lσ. It is apparent that 3>2.15>2.1482≈1/0.4655; according to Lemma 2, (10) is a group of more conservative condition, which ensures that all the roots of (14) have negative real parts except that there exists one singleton zero root for λj=0. Thus, matrix Φ has a singleton zero eigenvalue and all its other n∗l-1 eigenvalues have negative real parts.Remark 5.
Lemma4 has shown that each of the nondelayed multiagent systems given by (6) has all its eigenvalues on the open LHP except one equals zero. That implies that system (6) is stable, and all the states of each agent will reach a common value. The existence of the only zero root indicates that only the first-order state variable of each agent reaches a value that is decided by the initial state and all the other high-order state variables return to zero at last; that is, if all τm=0, multiagent system (5) will reach consensus.By analyzing the effect of nonuniform time delays on the stability of the systems, we will give a proof to the ensuing theorem.Theorem 6.
Considerlth-order (l=1,2,…,6) system given by (5) that satisfies Assumption 3, and the following inequalities (16), (17), and (18) are satisfied for l=4,5,6, respectively:(16)p2>λnp3p1+2p1p3.(17)p1>14p2p42(18)p1>max⁡p54p2p42,λn3p4.Define functions θl(ω) and Tl(ω) (l=1,2,…,6) as follows:(19)θlω≜arg⁡Flω=arg⁡-∑i=1l-jωipi∈0,2π,(20)Tlω≜1ωθlω,where Fl(ω) is defined in (27). If all τm satisfy τm<τ¯ for lth-order multiagent system, where(21)τ¯=Tlλnl=1Tlλn/p2l=2,3Tl2λn/p2l=4minTl2λn/p2,Tlpl-2/l-3l=5,6,then control input (3) can solve the consensus problem of system (5).Proof.
Consider the network of high-order multiagents with nonuniform time delays. LetΨ(s)=Gτ-1Ψ(0), where Ψ(s) is the Laplace transform of ψ(t), and (22)Gτs=sIn×l-In⊗A+∑m=1MLσm⊗Be-τms.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 Gτ(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 Gτ(s) are hereinafter referred to as “the eigenvalues of the system.”
Supposes=jω≠0 is an imaginary root of Gτ(s), and u=u1⊗1,0,…,0T+u2⊗0,1,…,0T+⋯+ul⊗0,0,…,1T is a corresponding eigenvector, where u=1, ui∈Cn, i=1,2,…,n. Then we have(23)jωIn×l-In⊗A+∑m=1MLσm⊗Be-jωτmu=0.Note that all the complex roots of each Gτ(s) appeared in conjugated pairs; we only need to study the situation that ωq>0. Since all of the first l-1 elements of the vector obtained by calculating the left part of (23) are equal to zero, we get(24)jωui=ui+1,for all i=1,2,…,l-1. Multiplied by u∗ (the conjugate transpose of u) on the left side of the left part of (23), and with (24) substituted, we obtain(25)∑m=1Mαme-jωτm+∑i=1ljωipi=0,where pl=1, and (26)αm=u∗Lσm⊗Iluu∗u.
Rewrite (25) as(27)∑m=1Mαmejωτm=-∑i=1l-jωipi≜Flω.
Take modulus of both sides of (27); then, we have(28)Mlω≜Flω=∑m=1Mαmejωτm≤∑m=1Mαm=u∗Lσ⊗Iluu∗u≤λn.Another necessary condition of (27) is(29)arg⁡∑m=1Mαmejωτm=arg⁡Flω=θlω.
Asωτm>0, θl(ω) should be discussed in the positive interval 0,2π]. From (29), it is obvious that(30)θlω=arg⁡∑m=1Mαmejωτm≤max⁡ωτm.Define(31)Rlω≜Im⁡FlωRe⁡Flω=tan⁡θlω,where Im(c) and Re(c) denote the imaginary part and real part of the complex number c, respectively. Rl(ω) is an access to θl(ω).
Consider the first-order system; then, we haveF1(ω)=0+jω and R1(ω)=ω/0. It is apparent that θ1(ω)=π/2, and M1(ω)=ω; according to (28), we should only consider ω≤λn; if we set all τm<τ¯=T1(λn)=π/(2λn), then (32)ωτm<λnτ¯=π/2=θ1ω,which contradicts (30). Then when all τm<τ¯, the first-order system is impossible to have an imaginary eigenvalue which presents the first contact of the eigenvalues from the stable region to the unstable one. Hence the system is still stable then, and it can reach consensus, and the theorem is proven for l=1.
Unlikeθ1(ω), θlωl≥2 are not a fixed value. But we have found that the derivatives of Rlω2≤l≤6 listed below are negative values when Assumption 3 is applied:(33)R2′=-p1p2p2ω2<0,R3′=-p2p3ω2-p1p2p2ω2<0,R4′=-p3p4ω4-p2p3-3p1p4ω2-p1p2p2ω-p4ω32<0,R5′=-p4p5ω6-p3p4-3p2p5ω4-p2p3-3p1p4ω2-p1p2p2ω-p4ω32<0,R6′=-p5p6ω8-p4p5-3p3p6ω6-p3p4-3p2p5+5p1p6ω4-p2p3-3p1p4ω2-p1p2p2ω-p4ω32<0.Rl′<0 means that Rl=tan[θl(ω)] are monotonically decreasing with the growth of ω. Then it can be deduced that the arguments θl(ω) also decrease monotonically and continuously because the values of Fl(ω) vary smoothly. Evidently, we have(34)limω→0+⁡Rl=p1ω+oωp2ω2+oω=+∞,for l≥2, which implies that the arguments θl(ω) start at π/2 when ω→0+.
By investigating the locus ofF6(ω) on a complex plane, we have found that the value of the argument θ6(ω) first falls from π/2, tending to 0; as the trajectory of F6(ω) passes from the first quadrant to the fourth, θ6(ω) does not really turn negative but jumps to 2π instead and then it falls again, without second “jump” to perform. Similar phenomena occur in the other high-order systems when l<6; in each system, the argument θl(ω) performs a jump at most once and no jumps for some of these systems, such as the second-order system, because the real and imaginary parts of F2(ω) are always positive when ω>0. The trajectories of F2(ω) and F6(ω) on the complex plane are shown in Figure 1 to give evidence of the variation on their arguments. For those whose arguments perform jumps, we consider the value of each θl(ω) in two continuous intervals: (0,π/2) and (0,2π].
Forl=2,3, if ω>λn/p2 then Re[Fl(ω)]=p2ω2>λn and Ml(ω)>λn, which we do not want. Thus, θl(λn/p2) is the minimum to consider. Then p1ω-p3ω3>0 holds when l=3, which implies θ3(ω)∈(0,π/2). Setting all τm<τ¯=Tl(λn/p2), we have (35)ωτm<λnp2Tlλnp2=θlλnp2<θlω,which contradicts (30); then by the same idea of the proof for the first-order system, the theorem for l=2,3 is proven.
Investigate the following parabola whenω∈[2λn/p2,p2/p4-2λn/p2]: (36)ξω=-p4ω4+p2ω2-λn.Apparently, its maximal value comes at the point when ω=p2/2p4. According to the symmetry of parabolas, ξ(2λn/p2)=ξ(p2/p4-2λn/p2) are the minima of ξ(ω). According to Assumption 3 that p2/p4>3p1/p3>6λn/p2, which implies p22>4λnp4, we have (37)ξ2λnp2=2λn-4p4λn2p22-λn=λnp22p22-4λnp4>0.Thus, ξ(ω)≥ξ(2λn/p2)>0. Then we have Ml(ω)≥Re⁡[Fl(ω)]≥p2ω2-p4ω4>λn for l≥4, when ω∈[2λn/p2,p2/p4-2λn/p2].
LetFl(ω0) be the first contact of its trajectory from the first quadrant to the fourth, as is shown in Figure 1. We analyze ω in two intervals: (0,ω0) and [ω0,+∞). Obviously, ω0 is the smallest positive real root of Im[Fl(ω0)]=0.
Consider the intervalω<ω0. If ω0∈(2λn/p2,p2/p4-2λn/p2), then θl(2λn/p2) would be smaller than the minimal possible value of θl(ω) to satisfy (27), because M(ω)>λn if 2λn/p2<ω<ω0.
Taking the fourth-order system into account, we have(38)ω0=p1p3∈2λnp2,p2p4-2λnp2.According to (16), one can obtain(39)ImF4p2p4-2λnp2=p3p2p4-2λnp2p2p4-2λnp2-p1p3>p3p1p3p2p4-2p1p3=p3p1p2p4-2p1p3>λn,where p4=1. Because |Im[F4(ω)]|=p3ω3-p1ω is monotonically increasing when ω≥ω0, we have |Im[F4(ω)]|>|Im[F4(p2/p4-2λn/p2)]|>λn for ω>p2/p4-2λn/p2. Then we know that M4(ω)>λn holds if ω>2λn/p2. The only situation that needs to be considered is ω≤2λn/p2, where θl(2λn/p2) is the smallest value of θl(ω); if we set all τm<τ¯=T4(2λn/p2), it can be obtained that(40)ωτm<2λnp2Tl2λnp2=θl2λnp2<θlω,which brings about the impossibility of (30) and provides the fourth-order system with consensus achievement.
Forl=5,6, Im[Fl(ω)]=p1ω-p3ω3+p5ω5, then (41)ω0=p3-p32-4p1p52p5.
Comparingω02 and p2/p4-2λn/p2, we have(42)p2p4-2λnp2-p3-p32-4p1p52p5=p32p52-p1p5-p32p5-p2p4+2λnp2.According to (17) and (18), 4-p5p22/p1p42>0, so that (43)p32p52-p1p5-p32p5-p2p4+2λnp22=p1p52p2p3p1p4-6+4p5λnp1p2p2p4-λnp2+4-p5p22p1p42+1-2p3λnp1p2>0under Assumption 3. Thus (42) is also positive, which means ω0<p2/p4-2λn/p2. When ω<ω0, if we set all τm<Tl(2λn/p2), for l=5,6, (30) is impossible.
Consider the intervalω≥ω0. Evidently, when ω≥p3/2>p2/p4, |Re[F5(ω)]|=p4ω4-p2ω2 is monotonically increasing, According to Assumption 3, we have p2p3>18λn. Then (44)ReF5p32=p4p334-p2p32>3p2p34-p2p32=p2p34>λn.Therefore, all τm<Tl(p3/2) would make (30) impossible when ω≥ω0. As is described in Theorem 6, if all τm<min{Tl(2λn/p2),Tl(p3/2)}, the consensus problem of the fifth-order system is solved.
Likewise, forl=6, according to (18) we have(45)ReF6p43-λn=p43p1-p3p43+p5p432-λn=p43313p5p4-3p3+p1p43-λn>0;when ω>p4/3>p3/p5, Re[F6(ω)] is a monotonically increasing function. Since Re[F6(p4/3)]>λn is positive already, M6(ω) would be further greater than λn as the growth of ω has passed through p4/3 where the situation should be ignored. Consequently, if all τm<min{Tl(2λn/p2),Tl(p4/3)}, the sixth-order system is stable, and the theorem is proven out.Figure 1
The trajectories ofF2(ω) and F6(ω) on the complex plane.Remark 7.
Despite the fact that it is difficult to present a general solution in the form of inequality to consensus problems of all high-order multiagent systems because the monotonicity of each high-degree polynomial that the solution relies on requires specialized derivation to figure out, the process of deriving the consensus conditions for the first-to-sixth-order systems has demonstrated a general approach, with which one can work out consensus conditions for an arbitrary-order system. Moreover, by further calculation, we provide stronger consensus conditions for the first-order and second-order multiagent systems in the following theorem.Theorem 8.
Consider the first-order and second-order multiagent system (5) with the control input (3) (l=1,2); then, the systems reach consensus if all the delays τm<τ¯l, where(46)τ¯1=π2λn,τ¯2=2p14+4λn2-p12arctan⁡2p12p14+4λn2-p12.Moreover, if all the delays τm≥τ¯l, the systems will be unstable, and the agents will fail to reach consensus.Proof.
For the first-order multiagent system, we have already proven in Theorem6 that τm<τ¯1 is a sufficient condition for the system to reach consensus. If all τm=τ¯1, jλn is the imaginary eigenvalue and its corresponding eigenvector u is the one ensuring ∑m=1Mαm=λn≜ω¯1. With these values substituted into (27) (l=1), the equation (47)∑m=1Mαmejω¯1τ¯1=F1ω¯1holds true; that is, τ¯1 is the “delay margin” equal to which the delays of the first-order system present the first contact of the eigenvalues from the stable region to the unstable one. When all the delays exceed the delay margin, the systems will be unstable, because there then exist eigenvalues in the unstable region (the RHP). The instabilities become stronger and the delays get greater distances exceeding the delay margin.
Forl=2, M2(ω)=ω4+p12ω2. Let ω∗=p14+4λn2-p12/2 be the only positive root of M2(ω)=λn; thus M2(ω)≤λn as long as ω≤ω∗. Owing to the decline of θ(ω), we can set all τm<θ(ω∗)/ω∗=τ¯2 to avoid the possibility of (27) (l=2). Therefore the sufficiency of the condition has been proven. By the same principle as that for the first-order system, τ¯2 is also the delay margin bringing about the failure of the system to reach consensus.


## 4. Numerical Simulations

In this section, some simulations are provided to illustrate the theoretical results obtained by the previous analysis.Consider a multiagent system consisting of 4 agents. Figure2 shows two different communication topologies described with undirected graphs G1 and G2, respectively. With three different time delays on each connection, G1 presents the simplest connected topology, while G2 having six different time delays displays full connectivity. All the delays are marked with τij, where i and j are the indices of the connected agents. Suppose the weight of each edge on both graphs is 1; then, for G1, λn≈3.4142, and, for G2, λn=4. Selected experiments will be carried out to validate the obtained results on 4-agent systems with both topologies illustrated by G1 and G2.Figure 2
Two communication topologies of the 4-agent system.For a third-order multiagent system with simply connected graphG1, set p1=9 and p2=6; thus τ¯=1.43 according to Theorem 6. Let τ12=1.42, τ23=1.40, and τ34=1.38. Seen from the simulation results shown in Figure 3, it is apparent that agents have reached consensus. And for the system with topology G2 showing full connectivity, let p1=9 and p2=6, and then one gets τ¯=1.2729. With time delays τ12=1.27, τ23=1.26, τ34=1.25, τ14=1.24, τ13=1.23, and τ24=1.22, simulation results are depicted in Figure 4, where the consensus achievement recurs.Figure 3
The trajectories of agents in the third-order system (the topology isG1).Figure 4
The trajectories of agents in the third-order system (the topology isG2).For a sixth-order system connected asG1, the parameters p1=12, p2=27, p3=40, p4=28, and p5=12 yield τ¯=0.51. Figure 5 has shown that the consensus achievement is guaranteed, where τ12=0.50, τ23=0.48, and τ34=0.46. With connections in G2, the system reaches consensus as shown in Figure 6, where p1=12, p2=27, p3=35, p4=28, and p5=12 and time delays τ12=0.47, τ23=0.46, τ34=0.45, τ14=0.44, τ13=0.43, and τ24=0.42 which are bounded by τ¯=0.4713 calculated according to Theorem 6.Figure 5
The trajectories of agents in the sixth-order system (the topology isG1).Figure 6
The trajectories of agents in the sixth-order system (the topology isG2).To examine Theorem8, we set p1=0.4 for the second-order system. According to Theorem 8, for G1, the delay margin τ¯=0.1181. Two groups of delays are set: τ12=0.11, τ23=0.10, τ34=0.09, which are bounded by the delay margin, and τ12=0.12, τ23=0.13, τ34=0.14, which exceed the delay margin. The results are shown in Figure 7: the first two figures are the trajectories of agents when time delays are bounded, which indicates that and consensus is reached; the last two figures are the trajectories of agents when all time delays exceed the delay margin, where the system is unstable. These phenomena have attested the theorem. And for G2, similar experiment is carried out: the delay margin is obtained as τ¯=0.1007, and then, respectively, gives the bounded delays τ12=0.10, τ23=0.09, τ34=0.08, τ14=0.07, τ13=0.06, and τ24=0.05 and the exceeded delays τ12=0.101, τ23=0.104, τ34=0.107, τ14=0.11, τ13=0.113, and τ24=0.116. As shown in Figure 8, the simulation results validate the theorem again.Figure 7
The trajectories of agents in the second-order system (the topology isG1).Figure 8
The trajectories of agents in the second-order system (the topology isG2).


## 5. Conclusions

This paper has studied consensus problem of high-order multiagent systems with nearest-neighbor control rules in the presence of nonuniform time delays. For each delayedlth-order (l=1,2,…,6) system, a sufficient condition has been provided in the form of inequalities and, for the first-order and second-order system, consensus conditions have been presented in the form of delay margins, which are less conservative and simpler in calculation than the existing results by Lyapunov methods in the form of LMIs. Numerical simulations on systems with two sets of different topologies have been carried out to testify the theorems. The simulation results show that the selected experiments have reached expected effect: the systems achieve consensus under given conditions.Future research will seek solutions to consensus problems of nonuniformly delayed high-order systems with directed topologies by applying this method. The main challenge is the calculation ofFl(ω)’s arguments (see (27)), because the eigenvalues of systems with directed topologies are complex values even in the absence of time delays and it is hard to tell the relationship among the arguments of several complex values and their summation.



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