# Leader-Following Consensus in Networks of Agents with Nonuniform Time-Varying Delays

Authors: Zhao-Jun Tang; Ting-Zhu Huang; Jiang-Ping Hu; Jin-Liang Shao
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper is concerned with a leader-following consensus problem for networks of agents with fixed and switching topologies as well as nonuniform time-varying communication delays. By employing Lyapunov-Razumikhin function, a necessary and sufficient condition is derived in the case of fixed topology, and a sufficient condition is obtained in the case when the interconnection topology is switched and satisfies certain condition. Simulation results are provided to illustrate the theoretical results.


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## Body

## 1. Introduction

In recent years, consensus problems of multiagent systems have received compelling attention from various research communities. This is mainly due to their wide applications in many areas such as synchronization of coupled oscillators, flocking, rendezvous, distributed sensor fusion in sensor networks, and distributed formation control (see [1]). The study of consensus problems is of great importance to understand many complex phenomena related to animal behaviors, such as flocking of birds, schooling of fish, and swarming of bees.Consensus problems have a long history in the field of computer science, particularly in automata theory and distributed computation [2]. Vicsek et al. [3] proposed a simple model of a system of several autonomous agents, and demonstrated by simulation that all agents eventually reach an agreement. Jadbabaie et al. [4] provided a theoretical explanation for the observed behavior of the Vicsek model. Up to now, a variety of topics related to consensus problems have been addressed such as consensus with switching topologies, consensus with time-delays, finite-time consensus, consensus over random networks, consensus with measurement noises.In multiagent systems, time-varying delays may arise naturally, for example, because of the moving of the agents, the congestion of the communication channels, the asymmetry of interactions, and the finite transmission speed due to the physical characteristic of the medium transmitting the information [5]. It has been observed from numerical experiments that consensus protocols without considering time delays may lead to unexpected instability. Therefore, it is important and meaningful to consider the consensus problems when communication is affected by time-delays. Consensus problems with communication delays have been addressed by many researchers. Olfati-Saber and Murray [2] studied the average consensus of first-order multiagent systems with constant and uniform communication time delays under fixed topology. The upper bound on the admissible delays was derived by means of frequency domain approach, and it was shown that it is inversely proportional to the largest eigenvalue of the Laplacian matrix of the network topology. Bliman and Ferrari-Trecate [6] generalized the results of Reference [2] in considering uniform and nonuniform time-varying time-delays. Consensus in networks of agents with single-integrator dynamics and double-integrator dynamics as well as multiple time-varying delays was addressed in [5, 7–10]. Note that the maximum allowable upper bound of time-delays in these literatures is presented in terms of linear matrix inequalities (LMIs). By employing Lyapunov-Razuminkhin function, Hu and Hong [11] and Hu and Lin [12] investigated the consensus problems of second-order multiagent systems with fixed and switching topologies in the presence of uniform communication delays.In this paper, we are interested in the leader-following consensus of multiagent systems with fixed and switching topologies as well as multiple time-varying delays. With the help of Lyapunov-Razumikhin function, we derive the maximal allowable upper bound of communication delays such that all the agents can follow the considered leader. In [13], the authors also studied the leader-following consensus problem with multiple time-varying delays, but the proposed protocol requires that the time-delays can be detectable. We assume that the time-delays are all unknown in the present paper. The objective of this paper is to generalize the results of [11] by considering nonuniform time-varying communication delays. Obviously, it is more practical to consider nonuniform time-delays than uniform time-delays. It is worthy to note that we derive an explicit formula for the bound of the allowable time-delays by means of Lyapunov-Razumikhin function whereas the bound is presented in terms of LMIs in [5, 7–10].The following notations will be used throughout this paper. LetI be an identity matrix with appropriate dimension. i is the imaginary unit. For a given matrix A, AT denotes its transpose; ∥A∥ denotes its spectral norm; Λ(A) denotes the set of all eigenvalues of A; λmax(A) and λmin(A) denote its maximum and minimum eigenvalues, respectively. ∥·∥ denotes the Euclidean norm for a given vector. Given a complex number μ∈ℂ, Re(μ) and Im(μ) are its real part and imaginary part, respectively. A matrix A is said to be positive stable if all its eigenvalues have positive real parts.


## 2. Problem Formulation

Consider a multiagent system consisting ofn agents, and a leader. We first describe the interconnection topology among the n agents by a simple digraph 𝒢=(𝒱,ℰ), where 𝒱={1,…,n} is the set of nodes representing the n agents and ℰ⊆𝒱×𝒱 is the set of edges of the graph. An edge of 𝒢 is denoted by (i,j), representing that agent i can directly receive information from agent j. A path in a digraph is a sequence of ordered edges of the form (ik,ik+1), k=1,…,m-1. We say that node j is reachable from node i if there is a path from node i to node j. The set of neighbors of node i is denoted by 𝒩i={j∈𝒱∣(i,j)∈ℰ}.The weighted adjacency matrix of the digraph𝒢 is denoted by A=[aij]∈ℝn×n, where aij>0 if (i,j)∈ℰ and aij=0, otherwise. Moreover, we assume that aii=0 for all i∈𝒱. The indegree and outdegree of node i are defined as degin(i)=∑j=1naji and degout(i)=∑j=1naij, respectively. A digraph is said to be balanced if degin(i)=degout(i). The Laplacian matrix L=[lij] associated with digraph 𝒢 is defined as(2.1)lij={-aij,i≠j,∑k=1,k≠inaik,i=j.The definition ofL clearly implies that L must have a zero eigenvalue corresponding to a eigenvector 1, where 1≜[1,…,1]T∈ℝn. Moreover, 0 is a simple eigenvalue of L if and only if 𝒢 has a spanning tree [14].In order to study a leader-following problem, we also concern another digraph𝒢¯, which consists of digraph 𝒢, node 0, and edges from some nodes to node 0. We say that node 0 is globally reachable in 𝒢¯ if node 0 is reachable from any node in 𝒢. The leader adjacency matrix associated with 𝒢¯ is defined as a diagonal matrix B with diagonal elements bi, where bi>0 if (i,0) is an edge of 𝒢¯ and bi=0, otherwise.In this paper, we consider the following double integrator system ofn agents:(2.2)ẋi=vi,v̇i=ui,i=1,2,…,n,
where xi, vi, ui∈ℝ denote the position, velocity, and control input of agent i, respectively. The dynamics of the leader is expressed as follows:(2.3)ẋ0=v0,wherev0 is the desired constant velocity.Letτij(t) denote the communication time-delay from agent j to agent i. Similarly to [2], we assume that communication delays between agents are symmetrical, that is, τij(t)=τji(t). Our control goal is to let the n agents follow the considered leader in the sense of both position and velocity, namely, xi→x0,vi→v0(i=1,…,n) as t→∞. For this end, we study the following neighbor-based protocol:(2.4)ui(t)=∑i=1naij(t)[xj(t-τij(t))-xi(t-τij(t))]+bi(t)[x0(t-τi0(t))-xi(t-τi0(t))]+k(v0-vi(t)),
where k is a control parameter.The communication topology among the group of agents may change dynamically due to link failure or creation, for instance, because of the limited detection range of agents, existence of the obstacles. In order to describe the switching topologies, we define a piecewise constant switching signalσ(t) (σ in short):[0,∞)→𝒫={1,2,…,N}, where N denotes the total number of all possible interaction topologies. The collection of all possible interaction topologies {𝒢¯1,…,𝒢¯N} is a finite set. For convenience, let(2.5)D={τp(t):p∈I},I={1,…,m},be the collection of independent time-delays affecting the communication links, whereτp(t) are piecewise continuous functions. It is clear that m≤((n+1)n)/2 because the delays are symmetrical. We assume that the nonuniform time-varying delays are uniformly bounded; namely, there exists a constant τ≥0 such that 0≤τp(t)≤τ, ∀p∈ℐ. The associated edges, with the time-delay τp(t) and switching signal σ, define a subgraph 𝒢¯p,σ. The corresponding Laplacian matrix associated with 𝒢p,σ and the leader adjacency matrix associated with 𝒢¯p,σ are denoted by Lp,σ and Bp,σ, respectively. It is clear that(2.6)Lp,σ1=0,p∈I,∑p=1mLp,σ=Lσ,∑p=1mBp,σ=Bσ.
To illustrate these relationships, an example is given as follows.Example 2.1.
Consider a multiagent system consisting of four agents and a leader with the interconnection topology𝒢¯1 shown in Figure 1. We assume that 𝒢¯1 has 0-1 weights, and τ10(t)=τ40(t)=τ1(t), τ21(t)=τ34(t)=τ2(t), τ41(t)=τ3(t). The subgraphs 𝒢¯11, 𝒢¯21, 𝒢¯31 are shown in Figure 2, and one can obtain the following:
(2.7)L1=(0000-1100001-1-1001),L11=0,L21=(0000-1100001-10000),L31=(000001000000-1001),
and B1=B11=diag{1,0,0,1}, B21=B31=0. Obviously, (2.6) is true.
Writex=(x1,…,xn)T, v=(v1,…,vn)T. With protocol (2.4), (2.2) can be written in the following matrix form:(2.8)ẋ=v,v̇=-∑p=1m(Lp,σ+Bp,σ)x(t-τp(t))+∑p=1mBp,σ1x0(t-τp(t))-k(v-v01).
Letx¯=x-x01, v¯=v-v01. We can obtain an error dynamics of system (2.8) as follows:(2.9)ɛ̇=A0ɛ(t)+∑p=1mAp,σɛ(t-τp(t)),
where
(2.10)ɛ=(x¯v¯),A0=(0I0-kI),Ap,σ=(00-Hp,σ0),Hp,σ=Lp,σ+Bp,σ.
Before ending this section, we introduce Lyapunov-Razumikhin Theorem, which plays a key role in the convergence analysis of system (2.9).
Consider the following system:(2.11)ẋ=f(xt),t>0,x(θ)=φ(θ),θ∈[-r,0],
where xt(θ)=x(t+θ), ∀θ∈[-r,0] and f(0)=0. Let C([-r,0],ℝn) be a Banach space of continuous functions defined on an interval [-r,0], taking values in ℝn with the topology of uniform convergence, and with a norm ∥φ∥c=maxθ∈[-r,0]∥φ(θ)∥.Figure 1
Interconnection topology𝒢¯1.The subgraphs of𝒢¯1 associated to delays τ1, τ2, and τ3.
(a)
Subgraph𝒢¯11(b)
Subgraph𝒢¯21(c)
Subgraph𝒢¯31Lemma 2.2 (Lyapunov-Razumikhin Theorem [15]).
Letϕ1, ϕ2, and ϕ3 be continuous, nonnegative, nondecreasing functions with ϕ1(s)>0, ϕ2(s)>0, ϕ3(s)>0 for s>0 and ϕ1(0)=ϕ2(0)=0. For system (2.11), suppose that the function f:C([-r,0],ℝn)→ℝ takes bounded sets of C([-r,0],ℝn) in bounded sets of ℝn. If there is a continuous function V(t,x) such that
(2.12)ϕ1(‖x‖)≤V(t,x)≤ϕ2(‖x‖),t∈R,x∈Rn.
In addition, there exists a continuous nondecreasing function ϕ(s) with ϕ(s)>s, s>0 such that the derivative of V along the solution x(t) of (2.11) satisfies
(2.13)V̇(t,x)≤-ϕ3(‖x‖)ifV(t+θ,x(t+θ))<ϕ(V(t,x(t))),θ∈[-r,0],
then the solution x=0 is uniformly asymptotically stable.Usually,V(t,x) is called a Lyapunov-Razumikhin function if it satisfies (2.12) and (2.13) in Lemma 2.2.


## 3. Main Results

### 3.1. Fixed Interconnection Topology

Consider system (2.9) with fixed interconnection topology. In this case, the subscript σ can be dropped. Rewritte (2.9) as
(3.1)ɛ̇=A0ɛ(t)+∑p=1mApɛ(t-τp(t)).To derive a delay-dependent stability criteria, we make the following model transformation. With the observation that(3.2)ɛ(t-τp(t))=ɛ(t)-∫-τp(t)0ɛ̇(t+s)ds,
it follows from (3.1) that(3.3)ɛ(t-τp(t))=ɛ(t)-∑i=0mAi∫-τp(t)0ɛ(t+s-τi(t))ds,
where τ0(t)≡0. Substituting (3.3) into system (3.1) leads to(3.4)ɛ̇=∑p=0mApɛ(t)-∑p=1m∑i=0mApAi∫-τp(t)0ɛ(t+s-τi(t))ds.
Noting that 𝒜p𝒜i=0 for p,i=1,…,m, we have(3.5)ɛ̇=Fɛ(t)-∑p=1mCp∫-τp(t)0ɛ(t+s)ds,
where(3.6)F=∑p=0mAp=(0I-H-kI),Cp=ApA0=(000-Hp),H=L+B.The process of transforming a system represented by (3.1) to one represented by (3.5) is known as a model transformation. The stability of the system represented by (3.5) implies the stability of the original system [16].To get the main result of this subsection, we need the following lemmas.Lemma 3.1 (see [17]).
Given a complex-coefficient polynomial,(3.7)f(s)=s2+(a+ib)s+c+id,
where a, b, c, d∈ℝ, f(s) is Hurwitz stable if and only if a>0 and abd+a2c-d2>0.Lemma 3.2.
Let(3.8)F=(0In-H-kIn),k>0.
Then F is Hurwitz stable if and only if H is positive stable and
(3.9)k>max⁡μi∈Λ(H){|Im⁡(μi)|Re(μi)}.Proof.
Note that the characteristic polynomial ofF is given by
(3.10)det⁡(sI2n-F)=det⁡([sIn-InH(s+k)In])=det⁡(s(s+k)In+H),where we have used Schur formula [18] to obtain the second equality. Let μi be the ith eigenvalue of H, and we have
(3.11)det⁡(sI2n-F)=∏i=1n(s2+ks+μi).
Denote thatf(s,μi)=s2+ks+μi. It follows from Lemma 3.1 that f(s,μi) is Hurwitz stable if and only if Re(μi)>0 and k>|Im(μi)|/Re(μi). Therefore, all eigenvalues of F have negative real parts if and only if Re(μi)>0 and k>|Im(μi)|/Re(μi) for any μi∈Λ(H), which implies the conclusion.Lemma 3.3 (see [11]).
The matrixH=L+B is positive stable if and only if node 0 is globally reachable in 𝒢¯.Now we state one of our main results.Theorem 3.4.
Consider system (3.1) and take
(3.12)k>k*=max⁡μi∈Λ(H){|Im⁡(μi)|Re(μi)},
where H=L+B, k is the control parameter in protocol (2.4). Then, there exists a constant τ*>0(which will be defined in the following (3.20)) such that when τ<τ*,
(3.13)lim⁡t→∞ɛ(t)=0;
namely, the n agents can follow the leader (in the sense of both position and velocity), if and only if node 0 is globally reachable in 𝒢¯.Proof.
(Sufficiency). Since node 0 is globally reachable in𝒢¯, it follows from Lemma 3.3 that H is positive stable. Thus, it follows from (3.12) and Lemma 3.2 that F is Hurwitz stable. Hence, by Lyapunov theorem [19], there exists a positive definite matrix P∈ℝ2n×2n such that
(3.14)PF+FTP=-I2n.
Take a Lyapunov-Razumikhin function
(3.15)V(ɛ)=ɛTPɛ.
Along the solution of system (2.9), from (3.5), we have
(3.16)V̇(ɛ)=ɛT(PF+FTP)ɛ-∑p=1m2ɛTPCp∫-τp(t)0ɛ(t+s)ds.
Note that 2aTb≤aTΨa+bTΨ-1b holds for any appropriate positive definite matrix Ψ. Then, we have
(3.17)V̇(ɛ)≤ɛT(PF+FTP)ɛ+∑p=1m[τp(t)ɛTPCpP-1CpTPɛ+∫-τp(t)0ɛT(t+s)Pɛ(t+s)ds].
Take ϕ(s)=qs for some constant q>1. In the case of
(3.18)V(ɛ(t+θ))<qV(ɛ(t)),-τ≤θ≤0,we have
(3.19)V̇(ɛ)≤-ɛTɛ+τ∑p=1mɛT(PCpP-1CpTP+qP)ɛ,
by recalling that τp(t)≤τ. As a result, if
(3.20)τ<τ*=1‖∑p=1m(PCpP-1CpTP+qP)‖,
then V̇(ɛ)≤-ηɛTɛ for some constant η>0. Therefore, the conclusion follows by Lemma 2.2.
(Necessity). System (3.1) is asymptotically stable for any time delays τp(t)<τ*, p∈ℐ. In particular, let τp(t)≡0, p∈ℐ. By (3.1), the system ɛ̇=Fɛ(t) is asymptotically stable, and hence all eigenvalues of F have negative real parts. Therefore, it follows from Lemma 3.2 that H is positive stable, and the conclusion follows by Lemma 3.3.Remark 3.5.
From the proof of Theorem3.4, we can see that many zoom techniques have to be applied during the derivation of τ*, and hence our estimate τ* may be very conservative.


### 3.2. Time-Varying Topology

In this subsection, we consider the case of switching topologies. Similar to the case of fixed topology, we can obtain that(3.21)ɛ(t-τp(t))=ɛ(t)-∫-τp(t)0(A0ɛ(t+s)+∑i=1mAi,σɛ(t+s-τi(t)))ds.
Then, from (2.9), we have(3.22)ɛ̇=(A0+∑p=1mAp,σ)ɛ(t)-∑p=1mAp,σA0∫-τp(t)0ɛ(t+s)ds,
by noting that 𝒜p,σ𝒜i,σ=0 for p, i=1,…,m. Denote the following:(3.23)Fσ=A0+∑p=1mAp,σ=(0In-Hσ-kIn),Cp,σ=Ap,σA0=(000-Hp,σ),whereHp,σ=Lp,σ+Bp,σ. Then (3.22) can be rewritten as(3.24)ɛ̇=Fσɛ(t)+∑p=1mCp,σ∫-τp(t)0ɛ(t+s)ds.To obtain the main result of this subsection, we introduce the following assumption.Assumption 3.6.
The weights of digraph𝒢¯ satisfy the following conditions:(1)
∑j=1nakj≥∑j=1najk, k∈𝒱,k∉ℐ,(2)
2bk+∑j=1nakj>∑j=1najk, ∀k∈ℐ,whereℐ≜{i∣bi>0,i∈𝒱} namely, ℐ denotes the index set of neighbors of vertex 0.Lemma 3.7 (see [20]).
Assume that the weights of𝒢¯ satisfy (Assumption 3.6), and node 0 is globally reachable in 𝒢¯. Then H+HT is positive definite, where H=L+B.Remark 3.8.
In the study of leader-following consensus for second-order multiagent systems with switching topologies [11, 21–24], it was assumed that 𝒢 is balanced and vertex 0 is globally reachable in 𝒢¯ so that a common Lyapunov function can be established. The theoretical base is the conclusion that H+HT is positive definite if 𝒢 is balanced, and vertex 0 is globally reachable in 𝒢¯. Noticing that ∑j=1n+1akj and ∑j=1n+1ajk denote the out-degree and in-degree of node k in 𝒢¯, respectively, (Assumption 3.6) contains that the out-degree of node k is greater or equal to its in-degree for each k∈𝒱 as a special case. Then (Assumption 3.6) is much weaker than the balanced constraint on 𝒢, and the corresponding results in the above literatures can be improved accordingly.
For convenience, denote thatμ=maxt≥0{λmax(Hσ(t)Hσ(t)T)}, λ=mint≥0{λmin(Hσ(t)+Hσ(t)T)}, which are well defined by noting that the set 𝒫 is finite. The main result of this subsection is as follows.Theorem 3.9.
Suppose that the weights of𝒢¯σ satisfy (Assumption 3.6) and node 0 is globally reachable in 𝒢¯σ for any t≥0. Consider system (2.9) and take
(3.25)k>k*=μ2λ+1.
If τ<τ1* (which will be defined in the following (3.32)), then
(3.26)lim⁡t→∞ɛ(t)=0.Proof.
Take a Lyapunov-Razumikhin functionV(ɛ)=ɛTΦɛ with positive definite matrix
(3.27)Φ=(kInInInIn),k>1.
Similar to the analysis in the proof of Theorem3.4, we have
(3.28)V̇(ɛ)≤ɛT(ΦFσ+FσTΦ)ɛ+∑p=1m[τp(t)ɛTΦCp,σΦ-1Cp,σTΦɛ+∫-τp(t)0ɛT(t+s)Φɛ(t+s)ds].
Takeϕ(s)=qs for some constant q>1. In the case of
(3.29)V(ɛ(t+θ))<qV(ɛ(t)),-τ≤θ≤0,we have
(3.30)V̇(ɛ)≤-ɛTQσɛ+τ∑p=1mɛT(ΦCp,σΦ-1Cp,σTΦ+qΦ)ɛ, where
(3.31)Qσ=-(ΦFσ+FσTΦ)=(Hσ+HσTHσTHσ2(k-1)I).
According to Schur complement [19], Qσ is positive definite for any t≥0 if k satisfies (3.25). Hence, if
(3.32)τ<τ1*=min⁡t≥0{λmin⁡(Qσ)}max⁡t≥0‖∑p=1m(ΦCp,σΦ-1Cp,σTΦ+qΦ)‖,
which is well defined by noting that the set 𝒫 is finite, then V̇(ɛ)≤-η1ɛTɛ for some η1. Therefore, the conclusion follows by Lemma 2.2.Remark 3.10.
For the first-order multiagent systems, it was shown that the consensus can be achieved provided that the network topology jointly contains a spanning tree [14, 25]. However, if the group of agents is governed by second-order dynamics, the consensus depends not only on the topology condition but also on the coupling strength between neighboring agents, and it was shown that consensus may fail to be achieved even if the network topology contains a spanning tree [26]. It should be pointed out that (Assumption 3.6) is not necessary to ensure the consensus, and it is of great interest to consider the more general condition on the network topology.


## 3.1. Fixed Interconnection Topology

Consider system (2.9) with fixed interconnection topology. In this case, the subscript σ can be dropped. Rewritte (2.9) as
(3.1)ɛ̇=A0ɛ(t)+∑p=1mApɛ(t-τp(t)).To derive a delay-dependent stability criteria, we make the following model transformation. With the observation that(3.2)ɛ(t-τp(t))=ɛ(t)-∫-τp(t)0ɛ̇(t+s)ds,
it follows from (3.1) that(3.3)ɛ(t-τp(t))=ɛ(t)-∑i=0mAi∫-τp(t)0ɛ(t+s-τi(t))ds,
where τ0(t)≡0. Substituting (3.3) into system (3.1) leads to(3.4)ɛ̇=∑p=0mApɛ(t)-∑p=1m∑i=0mApAi∫-τp(t)0ɛ(t+s-τi(t))ds.
Noting that 𝒜p𝒜i=0 for p,i=1,…,m, we have(3.5)ɛ̇=Fɛ(t)-∑p=1mCp∫-τp(t)0ɛ(t+s)ds,
where(3.6)F=∑p=0mAp=(0I-H-kI),Cp=ApA0=(000-Hp),H=L+B.The process of transforming a system represented by (3.1) to one represented by (3.5) is known as a model transformation. The stability of the system represented by (3.5) implies the stability of the original system [16].To get the main result of this subsection, we need the following lemmas.Lemma 3.1 (see [17]).
Given a complex-coefficient polynomial,(3.7)f(s)=s2+(a+ib)s+c+id,
where a, b, c, d∈ℝ, f(s) is Hurwitz stable if and only if a>0 and abd+a2c-d2>0.Lemma 3.2.
Let(3.8)F=(0In-H-kIn),k>0.
Then F is Hurwitz stable if and only if H is positive stable and
(3.9)k>max⁡μi∈Λ(H){|Im⁡(μi)|Re(μi)}.Proof.
Note that the characteristic polynomial ofF is given by
(3.10)det⁡(sI2n-F)=det⁡([sIn-InH(s+k)In])=det⁡(s(s+k)In+H),where we have used Schur formula [18] to obtain the second equality. Let μi be the ith eigenvalue of H, and we have
(3.11)det⁡(sI2n-F)=∏i=1n(s2+ks+μi).
Denote thatf(s,μi)=s2+ks+μi. It follows from Lemma 3.1 that f(s,μi) is Hurwitz stable if and only if Re(μi)>0 and k>|Im(μi)|/Re(μi). Therefore, all eigenvalues of F have negative real parts if and only if Re(μi)>0 and k>|Im(μi)|/Re(μi) for any μi∈Λ(H), which implies the conclusion.Lemma 3.3 (see [11]).
The matrixH=L+B is positive stable if and only if node 0 is globally reachable in 𝒢¯.Now we state one of our main results.Theorem 3.4.
Consider system (3.1) and take
(3.12)k>k*=max⁡μi∈Λ(H){|Im⁡(μi)|Re(μi)},
where H=L+B, k is the control parameter in protocol (2.4). Then, there exists a constant τ*>0(which will be defined in the following (3.20)) such that when τ<τ*,
(3.13)lim⁡t→∞ɛ(t)=0;
namely, the n agents can follow the leader (in the sense of both position and velocity), if and only if node 0 is globally reachable in 𝒢¯.Proof.
(Sufficiency). Since node 0 is globally reachable in𝒢¯, it follows from Lemma 3.3 that H is positive stable. Thus, it follows from (3.12) and Lemma 3.2 that F is Hurwitz stable. Hence, by Lyapunov theorem [19], there exists a positive definite matrix P∈ℝ2n×2n such that
(3.14)PF+FTP=-I2n.
Take a Lyapunov-Razumikhin function
(3.15)V(ɛ)=ɛTPɛ.
Along the solution of system (2.9), from (3.5), we have
(3.16)V̇(ɛ)=ɛT(PF+FTP)ɛ-∑p=1m2ɛTPCp∫-τp(t)0ɛ(t+s)ds.
Note that 2aTb≤aTΨa+bTΨ-1b holds for any appropriate positive definite matrix Ψ. Then, we have
(3.17)V̇(ɛ)≤ɛT(PF+FTP)ɛ+∑p=1m[τp(t)ɛTPCpP-1CpTPɛ+∫-τp(t)0ɛT(t+s)Pɛ(t+s)ds].
Take ϕ(s)=qs for some constant q>1. In the case of
(3.18)V(ɛ(t+θ))<qV(ɛ(t)),-τ≤θ≤0,we have
(3.19)V̇(ɛ)≤-ɛTɛ+τ∑p=1mɛT(PCpP-1CpTP+qP)ɛ,
by recalling that τp(t)≤τ. As a result, if
(3.20)τ<τ*=1‖∑p=1m(PCpP-1CpTP+qP)‖,
then V̇(ɛ)≤-ηɛTɛ for some constant η>0. Therefore, the conclusion follows by Lemma 2.2.
(Necessity). System (3.1) is asymptotically stable for any time delays τp(t)<τ*, p∈ℐ. In particular, let τp(t)≡0, p∈ℐ. By (3.1), the system ɛ̇=Fɛ(t) is asymptotically stable, and hence all eigenvalues of F have negative real parts. Therefore, it follows from Lemma 3.2 that H is positive stable, and the conclusion follows by Lemma 3.3.Remark 3.5.
From the proof of Theorem3.4, we can see that many zoom techniques have to be applied during the derivation of τ*, and hence our estimate τ* may be very conservative.


## 3.2. Time-Varying Topology

In this subsection, we consider the case of switching topologies. Similar to the case of fixed topology, we can obtain that(3.21)ɛ(t-τp(t))=ɛ(t)-∫-τp(t)0(A0ɛ(t+s)+∑i=1mAi,σɛ(t+s-τi(t)))ds.
Then, from (2.9), we have(3.22)ɛ̇=(A0+∑p=1mAp,σ)ɛ(t)-∑p=1mAp,σA0∫-τp(t)0ɛ(t+s)ds,
by noting that 𝒜p,σ𝒜i,σ=0 for p, i=1,…,m. Denote the following:(3.23)Fσ=A0+∑p=1mAp,σ=(0In-Hσ-kIn),Cp,σ=Ap,σA0=(000-Hp,σ),whereHp,σ=Lp,σ+Bp,σ. Then (3.22) can be rewritten as(3.24)ɛ̇=Fσɛ(t)+∑p=1mCp,σ∫-τp(t)0ɛ(t+s)ds.To obtain the main result of this subsection, we introduce the following assumption.Assumption 3.6.
The weights of digraph𝒢¯ satisfy the following conditions:(1)
∑j=1nakj≥∑j=1najk, k∈𝒱,k∉ℐ,(2)
2bk+∑j=1nakj>∑j=1najk, ∀k∈ℐ,whereℐ≜{i∣bi>0,i∈𝒱} namely, ℐ denotes the index set of neighbors of vertex 0.Lemma 3.7 (see [20]).
Assume that the weights of𝒢¯ satisfy (Assumption 3.6), and node 0 is globally reachable in 𝒢¯. Then H+HT is positive definite, where H=L+B.Remark 3.8.
In the study of leader-following consensus for second-order multiagent systems with switching topologies [11, 21–24], it was assumed that 𝒢 is balanced and vertex 0 is globally reachable in 𝒢¯ so that a common Lyapunov function can be established. The theoretical base is the conclusion that H+HT is positive definite if 𝒢 is balanced, and vertex 0 is globally reachable in 𝒢¯. Noticing that ∑j=1n+1akj and ∑j=1n+1ajk denote the out-degree and in-degree of node k in 𝒢¯, respectively, (Assumption 3.6) contains that the out-degree of node k is greater or equal to its in-degree for each k∈𝒱 as a special case. Then (Assumption 3.6) is much weaker than the balanced constraint on 𝒢, and the corresponding results in the above literatures can be improved accordingly.
For convenience, denote thatμ=maxt≥0{λmax(Hσ(t)Hσ(t)T)}, λ=mint≥0{λmin(Hσ(t)+Hσ(t)T)}, which are well defined by noting that the set 𝒫 is finite. The main result of this subsection is as follows.Theorem 3.9.
Suppose that the weights of𝒢¯σ satisfy (Assumption 3.6) and node 0 is globally reachable in 𝒢¯σ for any t≥0. Consider system (2.9) and take
(3.25)k>k*=μ2λ+1.
If τ<τ1* (which will be defined in the following (3.32)), then
(3.26)lim⁡t→∞ɛ(t)=0.Proof.
Take a Lyapunov-Razumikhin functionV(ɛ)=ɛTΦɛ with positive definite matrix
(3.27)Φ=(kInInInIn),k>1.
Similar to the analysis in the proof of Theorem3.4, we have
(3.28)V̇(ɛ)≤ɛT(ΦFσ+FσTΦ)ɛ+∑p=1m[τp(t)ɛTΦCp,σΦ-1Cp,σTΦɛ+∫-τp(t)0ɛT(t+s)Φɛ(t+s)ds].
Takeϕ(s)=qs for some constant q>1. In the case of
(3.29)V(ɛ(t+θ))<qV(ɛ(t)),-τ≤θ≤0,we have
(3.30)V̇(ɛ)≤-ɛTQσɛ+τ∑p=1mɛT(ΦCp,σΦ-1Cp,σTΦ+qΦ)ɛ, where
(3.31)Qσ=-(ΦFσ+FσTΦ)=(Hσ+HσTHσTHσ2(k-1)I).
According to Schur complement [19], Qσ is positive definite for any t≥0 if k satisfies (3.25). Hence, if
(3.32)τ<τ1*=min⁡t≥0{λmin⁡(Qσ)}max⁡t≥0‖∑p=1m(ΦCp,σΦ-1Cp,σTΦ+qΦ)‖,
which is well defined by noting that the set 𝒫 is finite, then V̇(ɛ)≤-η1ɛTɛ for some η1. Therefore, the conclusion follows by Lemma 2.2.Remark 3.10.
For the first-order multiagent systems, it was shown that the consensus can be achieved provided that the network topology jointly contains a spanning tree [14, 25]. However, if the group of agents is governed by second-order dynamics, the consensus depends not only on the topology condition but also on the coupling strength between neighboring agents, and it was shown that consensus may fail to be achieved even if the network topology contains a spanning tree [26]. It should be pointed out that (Assumption 3.6) is not necessary to ensure the consensus, and it is of great interest to consider the more general condition on the network topology.


## 4. Simulations

In this section, two examples are provided to illustrate the theoretical results. For simplicity, we assume that each interconnection topology has 0-1 weights in the following two examples.Example 4.1.
Consider a multiagent system consisting of a leader and four agents with fixed topology𝒢¯1 given in Figure 1. It is clear that node 0 is globally reachable in 𝒢¯1. By simple calculation, we have τ*=0.0426 and k>0. Let τ1(t)=0.02|sint|, τ2(t)=0.03|cost|, τ3(t)=0.03+0.01sint. The simulation results are obtained with k=2. Figure 3 shows that the four agents can follow the considered leader.Position errors and velocity errors of Example4.1.
(a)(b)Example 4.2.
Consider a multiagent system consisting of a leader and four agents. The interconnection topology of the multiagent system switches every 1s in the sequence 〈𝒢¯2,𝒢¯3〉 described as Figure 4. It is clear that the weights of 𝒢¯2 and 𝒢¯3 satisfy (Assumption 3.6) and node 0 is globally reachable in 𝒢¯2 and 𝒢¯3. Let τ10(t)=τ40(t)=0.1|sint|, τ21(t)=τ34(t)=0.2|cost|, τ23(t)=0.2+0.1sint. The simulation results are obtained with k=10. It can be seen from Figure 5 that the four agents can follow the considered leader.Interconnection topologies𝒢2¯ and 𝒢3¯.
(a)(b)Position errors and velocity errors of Example4.2.
(a)(b)


## 5. Conclusion

In this paper, we study a leader-following consensus problem of second-order multiagent systems with fixed and switching topologies as well as nonuniform time-varying communication delays. With the help of Lyapunov-Razumikhin function, an explicit formula for the upper bound of admissible delays is obtained for both fixed and switching topologies. Future research issues will include the cases when the communication delays are asymmetric, and the velocity of the considered leader is time-varying.



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*Source: 848942-2012-05-29.xml*