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

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.

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. Let 𝐼 be an identity matrix with appropriate dimension. i is the imaginary unit. For a given matrix 𝐴, 𝐴^𝑇 denotes its transpose; ‖𝐴‖ denotes its spectral norm; Λ(𝐴) denotes the set of all eigenvalues of 𝐴; 𝜆_max(𝐴) and 𝜆_min(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 of 𝑛 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 (𝑖_𝑘,𝑖_𝑘+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 𝒩_𝑖={𝑗∈𝒱∣(𝑖,𝑗)∈ℰ}.

The weighted adjacency matrix of the digraph 𝒢 is denoted by 𝐴=[𝑎_𝑖𝑗]∈ℝ^𝑛×𝑛, where 𝑎_𝑖𝑗>0 if (𝑖,𝑗)∈ℰ and 𝑎_𝑖𝑗=0, otherwise. Moreover, we assume that 𝑎_𝑖𝑖=0 for all 𝑖∈𝒱. The indegree and outdegree of node 𝑖 are defined as deg_in(𝑖)=∑^𝑛_𝑗=1𝑎_𝑗𝑖 and deg_out(𝑖)=∑^𝑛_𝑗=1𝑎_𝑖𝑗, respectively. A digraph is said to be balanced if deg_in(𝑖)=deg_out(𝑖). The Laplacian matrix 𝐿=[𝑙_𝑖𝑗] associated with digraph 𝒢 is defined as𝑙_𝑖𝑗=−𝑎_𝑖𝑗,𝑖≠𝑗,_{𝑛𝑘=1,𝑘≠𝑖}𝑎_𝑖𝑘,𝑖=𝑗. (2.1)

The definition of 𝐿 clearly implies that 𝐿 must have a zero eigenvalue corresponding to a eigenvector 𝟏, where 𝟏≜[1,…,1]^𝑇∈ℝ^𝑛. 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 digraph 𝒢, 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 𝑏_𝑖, where 𝑏_𝑖>0 if (𝑖,0) is an edge of 𝒢 and 𝑏_𝑖=0, otherwise.

In this paper, we consider the following double integrator system of 𝑛 agents:𝑥̇_𝑖=𝑣_𝑖,𝑣_𝑖̇=𝑢_𝑖,𝑖=1,2,…,𝑛, (2.2) where 𝑥_𝑖, 𝑣_𝑖, 𝑢_𝑖∈ℝ denote the position, velocity, and control input of agent 𝑖, respectively. The dynamics of the leader is expressed as follows:𝑥̇₀=𝑣₀, (2.3)

where 𝑣₀ is the desired constant velocity.

Let 𝜏_𝑖𝑗(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, 𝜏_𝑖𝑗(𝑡)=𝜏_𝑗𝑖(𝑡). Our control goal is to let the 𝑛 agents follow the considered leader in the sense of both position and velocity, namely, 𝑥_𝑖→𝑥₀,𝑣_𝑖→𝑣₀(𝑖=1,…,𝑛) as 𝑡→∞. For this end, we study the following neighbor-based protocol:𝑢_𝑖(𝑡)=^𝑛_𝑖=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡)+𝑏_𝑖(𝑡)𝑥₀𝑡−𝜏_𝑖0(𝑡)−𝑥_𝑖𝑡−𝜏_𝑖0(𝑡𝑣)+𝑘₀−𝑣_𝑖(𝑡), (2.4) 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 signal 𝜎(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 {𝒢₁,…,𝒢_𝑁} is a finite set. For convenience, let𝒟=𝜏_𝑝(𝑡)∶𝑝∈ℐ,ℐ={1,…,𝑚}, (2.5)

be the collection of independent time-delays affecting the communication links, where 𝜏_𝑝(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≤𝜏_𝑝(𝑡)≤𝜏, ∀𝑝∈ℐ. The associated edges, with the time-delay 𝜏_𝑝(𝑡) and switching signal 𝜎, define a subgraph 𝒢_𝑝,𝜎. The corresponding Laplacian matrix associated with 𝒢_𝑝,𝜎 and the leader adjacency matrix associated with 𝒢_𝑝,𝜎 are denoted by 𝐿_𝑝,𝜎 and 𝐵_𝑝,𝜎, respectively. It is clear that𝐿_𝑝,𝜎𝟏=0,𝑝∈ℐ,^𝑚_𝑝=1𝐿_𝑝,𝜎=𝐿_𝜎,^𝑚_𝑝=1𝐵_𝑝,𝜎=𝐵_𝜎. (2.6) 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 topology 𝒢₁ shown in Figure 1. We assume that 𝒢₁ has 0−1 weights, and 𝜏₁₀(𝑡)=𝜏₄₀(𝑡)=𝜏₁(𝑡), 𝜏₂₁(𝑡)=𝜏₃₄(𝑡)=𝜏₂(𝑡), 𝜏₄₁(𝑡)=𝜏₃(𝑡). The subgraphs 𝒢₁₁, 𝒢₂₁, 𝒢₃₁ are shown in Figure 2, and one can obtain the following: 𝐿₁=0000−1100001−1−1001,𝐿₁₁=0,𝐿₂₁=0000−1100001−10000,𝐿₃₁=000001000000−1001, (2.7) and 𝐵₁=𝐵₁₁=diag{1,0,0,1}, 𝐵₂₁=𝐵₃₁=0. Obviously, (2.6) is true.
Write 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇, 𝑣=(𝑣₁,…,𝑣_𝑛)^𝑇. With protocol (2.4), (2.2) can be written in the following matrix form:𝑥̇=𝑣,𝑣̇=−^𝑚_𝑝=1𝐿_𝑝,𝜎+𝐵_𝑝,𝜎𝑥𝑡−𝜏_𝑝(𝑡)+^𝑚_𝑝=1𝐵_𝑝,𝜎𝟏𝑥₀𝑡−𝜏_𝑝(𝑡)−𝑘𝑣−𝑣₀𝟏. (2.8)
Let 𝑥=𝑥−𝑥₀𝟏, 𝑣=𝑣−𝑣₀𝟏. We can obtain an error dynamics of system (2.8) as follows:𝜀̇=𝒜₀𝜀(𝑡)+^𝑚_𝑝=1𝒜_𝑝,𝜎𝜀𝑡−𝜏_𝑝(𝑡), (2.9) where 𝜀=𝑥𝑣,𝒜₀=0𝐼0−𝑘𝐼,𝒜_𝑝,𝜎=00−𝐻_𝑝,𝜎0,𝐻_𝑝,𝜎=𝐿_𝑝,𝜎+𝐵_𝑝,𝜎. (2.10)
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: 𝑥̇=𝑓𝑥_𝑡𝑥(𝜃)(𝜃)[],,𝑡>0,=𝜑,𝜃∈−𝑟,0 (2.11) where 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃), ∀𝜃∈[−𝑟,0] and 𝑓(0)=0. Let 𝐶([−𝑟,0],ℝ^𝑛) be a Banach space of continuous functions defined on an interval [−𝑟,0], taking values in ℝ^𝑛 with the topology of uniform convergence, and with a norm ‖𝜑‖_𝑐=max_{𝜃∈[−𝑟,0]}‖𝜑(𝜃)‖.

Figure 1 

Interconnection topology 𝒢1.

(a) Subgraph 𝒢11

(b) Subgraph 𝒢21

(c) Subgraph 𝒢31

Figure 2 

The subgraphs of 𝒢1 associated to delays 𝜏1, 𝜏2, and 𝜏3.

Lemma 2.2 (Lyapunov-Razumikhin Theorem [15]). Let 𝜙₁, 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],ℝ^𝑛)→ℝ takes bounded sets of 𝐶([−𝑟,0],ℝ^𝑛) in bounded sets of ℝ^𝑛. If there is a continuous function 𝑉(𝑡,𝑥) such that 𝜙₁(‖𝑥‖)≤𝑉(𝑡,𝑥)≤𝜙₂(‖𝑥‖),𝑡∈ℝ,𝑥∈ℝ^𝑛. (2.12) 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 𝑉̇(𝑡,𝑥)≤−𝜙₃(‖𝑥‖)if𝑉(𝑡+𝜃,𝑥(𝑡+𝜃))<𝜙(𝑉(𝑡,𝑥(𝑡))),𝜃∈[−𝑟,0], (2.13) 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 𝜀̇=𝒜₀𝜀(𝑡)+^𝑚_𝑝=1𝒜_𝑝𝜀𝑡−𝜏_𝑝(𝑡). (3.1)

To derive a delay-dependent stability criteria, we make the following model transformation. With the observation that𝜀𝑡−𝜏_𝑝(𝑡)=𝜀(𝑡)−⁰_{−𝜏𝑝(𝑡)}𝜀̇(𝑡+𝑠)𝑑𝑠, (3.2) it follows from (3.1) that𝜀𝑡−𝜏_𝑝(𝑡)=𝜀(𝑡)−^𝑚_𝑖=0𝒜_𝑖⁰_{−𝜏𝑝(𝑡)}𝜀𝑡+𝑠−𝜏_𝑖(𝑡)𝑑𝑠, (3.3) where 𝜏₀(𝑡)≡0. Substituting (3.3) into system (3.1) leads to𝜀̇=^𝑚_𝑝=0𝒜_𝑝𝜀(𝑡)−^𝑚𝑚_𝑝=1𝑖=0𝒜_𝑝𝒜_𝑖⁰_{−𝜏𝑝(𝑡)}𝜀𝑡+𝑠−𝜏_𝑖(𝑡)𝑑𝑠. (3.4) Noting that 𝒜_𝑝𝒜_𝑖=0 for 𝑝,𝑖=1,…,𝑚, we have𝜀̇=𝐹𝜀(𝑡)−^𝑚_𝑝=1𝐶_𝑝⁰_{−𝜏𝑝(𝑡)}𝜀(𝑡+𝑠)𝑑𝑠, (3.5) where𝐹=^𝑚_𝑝=0𝒜_𝑝=0𝐼−𝐻−𝑘𝐼,𝐶_𝑝=𝒜_𝑝𝒜₀=000−𝐻_𝑝,𝐻=𝐿+𝐵. (3.6)

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, 𝑓(𝑠)=𝑠²+(𝑎+i𝑏)𝑠+𝑐+i𝑑, (3.7) 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 𝐹=0𝐼_𝑛−𝐻−𝑘𝐼_𝑛,𝑘>0. (3.8) Then 𝐹 is Hurwitz stable if and only if 𝐻 is positive stable and 𝑘>max_{𝜇𝑖(𝐻)∈Λ}Im𝜇_𝑖Re𝜇_𝑖. (3.9)

Proof. Note that the characteristic polynomial of 𝐹 is given by det𝑠𝐼_2𝑛−𝐹=det𝑠𝐼_𝑛−𝐼_𝑛𝐻(𝑠+𝑘)𝐼_𝑛=det𝑠(𝑠+𝑘)𝐼_𝑛+𝐻, (3.10)where we have used Schur formula [18] to obtain the second equality. Let 𝜇_𝑖 be the 𝑖th eigenvalue of 𝐻, and we have det𝑠𝐼_2𝑛−𝐹=^𝑛_𝑖=1𝑠²+𝑘𝑠+𝜇_𝑖. (3.11)
Denote that 𝑓(𝑠,𝜇_𝑖)=𝑠²+𝑘𝑠+𝜇_𝑖. It follows from Lemma 3.1 that 𝑓(𝑠,𝜇_𝑖) is Hurwitz stable if and only if Re(𝜇_𝑖)>0 and 𝑘>|Im(𝜇_𝑖)|/Re(𝜇_𝑖). Therefore, all eigenvalues of 𝐹 have negative real parts if and only if Re(𝜇_𝑖)>0 and 𝑘>|Im(𝜇_𝑖)|/Re(𝜇_𝑖) for any 𝜇_𝑖∈Λ(𝐻), which implies the conclusion.

Lemma 3.3 (see [11]). The matrix 𝐻=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 𝑘>𝑘^∗=max_{𝜇𝑖(𝐻)∈Λ}Im𝜇_𝑖Re𝜇_𝑖, (3.12) 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^∗, lim_𝑡→∞𝜀(𝑡)=0; (3.13) 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 in 𝒢, 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∈ℝ^2𝑛×2𝑛 such that 𝑃𝐹+𝐹^𝑇𝑃=−𝐼_2𝑛. (3.14) Take a Lyapunov-Razumikhin function 𝑉(𝜀)=𝜀^𝑇𝑃𝜀. (3.15) Along the solution of system (2.9), from (3.5), we have 𝑉̇(𝜀)=𝜀^𝑇𝑃𝐹+𝐹^𝑇𝑃𝜀−^𝑚_𝑝=12𝜀^𝑇𝑃𝐶_𝑝⁰_{−𝜏𝑝(𝑡)}𝜀(𝑡+𝑠)𝑑𝑠. (3.16) Note that 2𝑎^𝑇𝑏≤𝑎^𝑇Ψ𝑎+𝑏^𝑇Ψ⁻¹𝑏 holds for any appropriate positive definite matrix Ψ. Then, we have 𝑉̇(𝜀)≤𝜀^𝑇𝑃𝐹+𝐹^𝑇𝑃𝜀+^𝑚_𝑝=1𝜏_𝑝(𝑡)𝜀^𝑇𝑃𝐶_𝑝𝑃⁻¹𝐶^𝑇_𝑝𝑃𝜀+⁰_{−𝜏𝑝(𝑡)}𝜀^𝑇(𝑡+𝑠)𝑃𝜀(𝑡+𝑠)𝑑𝑠. (3.17) Take 𝜙(𝑠)=𝑞𝑠 for some constant 𝑞>1. In the case of 𝑉(𝜀(𝑡+𝜃))<𝑞𝑉(𝜀(𝑡)),−𝜏≤𝜃≤0, (3.18)we have 𝑉̇(𝜀)≤−𝜀^𝑇𝜀+𝜏^𝑚_𝑝=1𝜀^𝑇𝑃𝐶_𝑝𝑃⁻¹𝐶^𝑇_𝑝𝑃+𝑞𝑃𝜀, (3.19) by recalling that 𝜏_𝑝(𝑡)≤𝜏. As a result, if 𝜏<𝜏^∗=1∑^𝑚_𝑝=1𝑃𝐶_𝑝𝑃⁻¹𝐶^𝑇_𝑝,𝑃+𝑞𝑃 (3.20) then 𝑉̇(𝜀)≤−𝜂𝜀^𝑇𝜀 for some constant 𝜂>0. Therefore, the conclusion follows by Lemma 2.2.
(Necessity). System (3.1) is asymptotically stable for any time delays 𝜏_𝑝(𝑡)<𝜏^∗, 𝑝∈ℐ. In particular, let 𝜏_𝑝(𝑡)≡0, 𝑝∈ℐ. By (3.1), the system 𝜀̇=𝐹𝜀(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 Theorem 3.4, 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 that𝜀𝑡−𝜏_𝑝(𝑡)=𝜀(𝑡)−⁰_{−𝜏𝑝(𝑡)}𝒜₀𝜀(𝑡+𝑠)+^𝑚_𝑖=1𝒜_𝑖,𝜎𝜀𝑡+𝑠−𝜏_𝑖(𝑡)𝑑𝑠. (3.21) Then, from (2.9), we have𝜀̇=𝒜₀+^𝑚_𝑝=1𝒜_𝑝,𝜎𝜀(𝑡)−^𝑚_𝑝=1𝒜_𝑝,𝜎𝒜₀⁰_{−𝜏𝑝(𝑡)}𝜀(𝑡+𝑠)𝑑𝑠, (3.22) by noting that 𝒜_𝑝,𝜎𝒜_𝑖,𝜎=0 for 𝑝, 𝑖=1,…,𝑚. Denote the following:𝐹_𝜎=𝒜₀+^𝑚_𝑝=1𝒜_𝑝,𝜎=0𝐼_𝑛−𝐻_𝜎−𝑘𝐼_𝑛,𝐶_𝑝,𝜎=𝒜_𝑝,𝜎𝒜₀=000−𝐻_𝑝,𝜎, (3.23)

where 𝐻_𝑝,𝜎=𝐿_𝑝,𝜎+𝐵_𝑝,𝜎. Then (3.22) can be rewritten as𝜀̇=𝐹_𝜎𝜀(𝑡)+^𝑚_𝑝=1𝐶_𝑝,𝜎⁰_{−𝜏𝑝(𝑡)}𝜀(𝑡+𝑠)𝑑𝑠. (3.24)

To obtain the main result of this subsection, we introduce the following assumption.

Assumption 3.6. The weights of digraph 𝒢 satisfy the following conditions:

(1)∑^𝑛_𝑗=1𝑎_𝑘𝑗≥∑^𝑛_𝑗=1𝑎_𝑗𝑘, 𝑘∈𝒱,𝑘∉ℐ,(2)2𝑏_𝑘+∑^𝑛_𝑗=1𝑎_𝑘𝑗>∑^𝑛_𝑗=1𝑎_𝑗𝑘, ∀𝑘∈ℐ,

where ℐ≜{𝑖∣𝑏_𝑖>0,𝑖∈𝒱} namely, ℐ denotes the index set of neighbors of vertex 0.

Lemma 3.7 (see [20]). Assume that the weights of 𝒢 satisfy (Assumption 3.6), and node 0 is globally reachable in 𝒢. Then 𝐻+𝐻^𝑇 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+𝐻^𝑇 is positive definite if 𝒢 is balanced, and vertex 0 is globally reachable in 𝒢. Noticing that ∑^𝑛+1_𝑗=1𝑎_𝑘𝑗 and ∑^𝑛+1_𝑗=1𝑎_𝑗𝑘 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 that 𝜇=max_𝑡≥0{𝜆_max(𝐻_𝜎(𝑡)𝐻^𝑇_𝜎(𝑡))}, 𝜆=min_𝑡≥0{𝜆_min(𝐻_𝜎(𝑡)+𝐻^𝑇_𝜎(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 of 𝒢_𝜎 satisfy (Assumption 3.6) and node 0 is globally reachable in 𝒢_𝜎 for any 𝑡≥0. Consider system (2.9) and take 𝑘>𝑘^∗=𝜇2𝜆+1. (3.25) If 𝜏<𝜏^∗₁ (which will be defined in the following (3.32)), then lim_𝑡→∞𝜀(𝑡)=0. (3.26)

Proof. Take a Lyapunov-Razumikhin function 𝑉(𝜀)=𝜀^𝑇Φ𝜀 with positive definite matrix Φ=𝑘𝐼_𝑛𝐼_𝑛𝐼_𝑛𝐼_𝑛,𝑘>1. (3.27)
Similar to the analysis in the proof of Theorem 3.4, we have 𝑉̇(𝜀)≤𝜀^𝑇Φ𝐹_𝜎+𝐹^𝑇_𝜎Φ𝜀+^𝑚_𝑝=1𝜏_𝑝(𝑡)𝜀^𝑇Φ𝐶_𝑝,𝜎Φ⁻¹𝐶^𝑇_𝑝,𝜎Φ𝜀+⁰_{−𝜏𝑝(𝑡)}𝜀^𝑇(𝑡+𝑠)Φ𝜀(𝑡+𝑠)𝑑𝑠. (3.28)
Take 𝜙(𝑠)=𝑞𝑠 for some constant 𝑞>1. In the case of 𝑉(𝜀(𝑡+𝜃))<𝑞𝑉(𝜀(𝑡)),−𝜏≤𝜃≤0, (3.29)we have 𝑉̇(𝜀)≤−𝜀^𝑇𝑄_𝜎𝜀+𝜏^𝑚_𝑝=1𝜀^𝑇Φ𝐶_𝑝,𝜎Φ⁻¹𝐶^𝑇_𝑝,𝜎Φ+𝑞Φ𝜀, (3.30) where 𝑄_𝜎=−Φ𝐹_𝜎+𝐹^𝑇_𝜎Φ=𝐻_𝜎+𝐻^𝑇_𝜎𝐻^𝑇_𝜎𝐻_𝜎2(𝑘−1)𝐼. (3.31) According to Schur complement [19], 𝑄_𝜎 is positive definite for any 𝑡≥0 if 𝑘 satisfies (3.25). Hence, if 𝜏<𝜏^∗₁=min_𝑡≥0𝜆_min𝑄_𝜎max_𝑡≥0∑^𝑚_𝑝=1Φ𝐶_𝑝,𝜎Φ⁻¹𝐶^𝑇_𝑝,𝜎,Φ+𝑞Φ (3.32) which is well defined by noting that the set 𝒫 is finite, then 𝑉̇(𝜀)≤−𝜂₁𝜀^𝑇𝜀 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 topology 𝒢₁ given in Figure 1. It is clear that node 0 is globally reachable in 𝒢₁. 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.

(a)

(b)

Figure 3 

Position errors and velocity errors of Example 4.1.

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 ⟨𝒢₂,𝒢₃⟩ described as Figure 4. It is clear that the weights of 𝒢₂ and 𝒢₃ satisfy (Assumption 3.6) and node 0 is globally reachable in 𝒢₂ and 𝒢₃. 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.

(a)

(b)

Figure 4 

Interconnection topologies 𝒢2 and 𝒢3.

(a)

(b)

Figure 5 

Position errors and velocity errors of Example 4.2.

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.

Acknowledgment

The authors would like to thank the anonymous reviewers for their insightful comments and suggestions. This work was supported by NSFC (60973015, 61170311, 61104104), NSFC Tianyuan foundation (11126104), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Sci.&Tech. Research Project (12ZC1802).