Second-Order Leader-Following Consensus of Multiagent Systems with Time Delays

Abstract

This paper is concerned with a leader-following consensus problem of second-order multiagent systems with a constant acceleration leader and time-varying delays. At first, a distributed control protocol for every agent to track the leader is proposed; then by utilizing the Lyapunov-Razumikhin function, the convergence analysis under both fixed and switching interconnection topologies is investigated. For the case of fixed topology, a sufficient and necessary condition is obtained, and for the case of switching topology, a sufficient condition is derived under some assumptions. Finally, simulation examples are provided to demonstrate the effectiveness of the theoretical results.

1. Introduction

Recent years have witnessed dramatic advances concerned with distributed coordination of multiple agents due to rapid developments of computer science and communication technologies. As an important issue of distributed coordination, the consensus problem has attracted more and more attention from researchers in various fields [1–5]. A special case of the issue is known as the leader-following consensus problems and has been investigated from different perspectives. Hong et al. in [6], for instance, designed distributed observers to track an active leader for second-order continuous multi-agents systems. Also, Tang et al. studied the leader-following consensus problem via sampled-data control in [7]. Besides, the authors in [8] explored first-order leader-following consensus algorithms under a directed fixed topology with a time-varying leader. Moreover, the results in [8] were furthered extended to the case of actuator saturation and switching topology in [9].

Due to the limited communication capacity, time delays are sometimes unavoidable in multiagent systems. Olfati-saber and Murray first analyzed the consensus problem with fixed undirected topology and time delays by a frequency domain approach in [1]. Based on the reduced-order Lyapunov-Krasovskii function and linear matrix inequalities (LMIs), Lin and Jia considered the averaged consensus problem with a switching topology and time-varying delays in [10]. In addition, Hu and Lin analyzed the second-order consensus problem for multiple agents with time-varying delays in terms of the Lyapunov-Razumikhin method in [11]. Note that [1, 10, 11] studied leaderless consensus problem with time delays. For leader-follower networks, Hu and Hong [12] investigated the second-order consensus problems with fixed and switching topologies with time delays, and Tang et al. generalized the results of [12] to the case of nonuniform time delays in [13], while the scenario in the presence of detectable time delays was addressed in [14]. It is worth pointing out that the velocity of the leader in [12–14] is a constant, and the time delays exist only in the transmission of position between neighbors.

In this paper, we consider the second-order leader-following consensus problem for multi-agent systems, with time delays existing in the transmission of both velocity and position. Additionally, the velocity of the leader is not restricted to be a constant but a linear function of time; that is, the acceleration of the leader can be a nonzero constant. Furthermore, we investigate the consensus problem under both fixed and switching directed interconnection topologies by means of the Lyapunov-Razumikhin function and provide the bounds of the allowable time delays.

The rest of this paper is organized as follows. Section 2 provides some preliminaries and formulates the leader-following consensus problem. With the proposed consensus protocol, Section 3 presents the convergence analysis of the case of fixed topology, and Section 4 deals with the case of switching topology. Then simulation results are presented in Section 5. Finally, the conclusion is drawn in Section 6.

The following notations will be used throughout this paper. Let 𝐼_𝑛 be an 𝑛×𝑛 identity matrix, 𝟏_𝑛=[1⋯1]^𝑇∈ℝ^𝑛, 0_𝑛×𝑛 represents an 𝑛×𝑛 zero matrix. Given a matrix, the superscripts “𝑇", and “−1" stand for matrix transposition and matrix inverse respectively; Λ(⋅) denotes the set of all eigenvalues of the matrix; ‖⋅‖ denotes the spectral norm; 𝜆_max(⋅) and 𝜆_min(⋅) denote its maximum and minimum eigenvalues, respectively. diag{𝑏₁,…,𝑏_𝑛} denotes a diagonal matrix with diagonal elements being 𝑏₁,…,𝑏_𝑛. Given a complex number 𝜇∈ℂ,Re(𝜇),Im(𝜇) and |𝜇| are its real part, imaginary part, and modulus, respectively. ⊗ denotes the Kronecker product.

2. Preliminaries and Formulation

Let 𝒢=(𝒱,ℰ,𝐴) be a directed graph with a node set 𝒱={1,2,…,𝑛}, an edge set ℰ⊆𝒱×𝒱, and a weighted adjacency matrix 𝐴=[𝑎_𝑖𝑗] with nonnegative elements 𝑎_𝑖𝑗. Node 𝑖 represents agent 𝑖, and an edge in ℰ is denoted by an ordered pair (𝑗,𝑖), where (𝑗,𝑖)∈ℰ if and only if that agent 𝑖 can access the state information of agent 𝑗. The index set of neighbors of node 𝑖 is denoted by 𝒩_𝑖={𝑗∈𝒱∶(𝑗,𝑖)∈ℰ}. For 𝑖,𝑗∈𝒱, 𝑗∈𝒩_𝑖⇔𝑎_𝑖𝑗>0. Moreover, we assume that 𝑎_𝑖𝑖=0 for all 𝑖∈𝒱. The indegree and outdegree of node 𝑖 are, respectively, defined as 𝑑_in(𝑖)=∑^𝑖_𝑗=1𝑎_𝑖𝑗, 𝑑_out(𝑖)=∑^𝑖_𝑗=1𝑎_𝑗𝑖. A digraph is called balanced if ∑^𝑖_𝑗=1𝑎_𝑖𝑗=∑^𝑖_𝑗=1𝑎_𝑖𝑗 for all of its nodes. The Laplacian matrix 𝐿=[𝑙_𝑖𝑗]∈ℝ^𝑛×𝑛 associated with 𝐴 is defined as 𝑙_𝑖𝑖=∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗 and 𝑙_𝑖𝑗=−𝑎_𝑖𝑗. Clearly, 𝐿 has at least one zero eigenvalue with a corresponding eigenvector 𝟏_𝑛.

A directed path is a sequence of edges in a directed graph of the form (𝑖₁,𝑖₂), (𝑖₂,𝑖₃),…, where 𝑖_𝑗∈𝒱. If there exists a path from node 𝑗 to node 𝑖, we say that 𝑖 is reachable from 𝑗. Moreover, 𝑖 is to said be globally reachable if there is a path from any other node to it. A directed tree is a directed graph, where there exists a node, called the root, such that any other node of the digraph can be reached by one and only one path starting at the root. A spanning tree of a digraph is a directed tree formed by graph edges that connect all the nodes of the graph.

Consider a multiagent system depicted by a graph 𝒢, which consists of 𝑛 followers (related to graph 𝒢) and one leader (labeled as node 0) with directed edges from node 0 to some nodes. The leader adjacency matrix associated with graph 𝒢 is defined as a matrix 𝐵=diag{𝑏₁,…,𝑏_𝑛}, where 𝑏_𝑖>0 if node 0 is a neighbor of node 𝑖 and 𝑏_𝑖=0 otherwise.

The matrix 𝐻=𝐿+𝐵 plays an important role in the analysis of leader-following consensus problem, the following lemma shows a relationship between the positive stability of 𝐻 and the connectedness of graph 𝒢.

Lemma 1 (see [12]). The matrix 𝐻=𝐿+𝐵 is positive stable if and only if node 0 is globally reachable in 𝒢.

Remark 2. It is straightforward to verify that 𝐻=𝐿+𝐵 is positive stable if and only if the opposite graph, which is formed by changing the orientation of each arc of 𝒢, has a spanning tree rooted at node 0.

All the considered 𝑛 followers in this paper move in a 𝑝-dimensional space, with the kinematic of each follower being described by a double integrator of the form: 𝑥̇_𝑖=𝑣_𝑖,𝑣̇_𝑖=𝑢_𝑖, (1) where 𝑥_𝑖, 𝑣_𝑖, 𝑢_𝑖∈ℝ^𝑝 denote the position, velocity, and control input of follower 𝑖, respectively. The dynamics of the leader is expressed as follows: 𝑥̇₀=𝑣₀,𝑣̇₀=𝑎₀, (2) where 𝑎₀∈ℝ^𝑝 is the constant acceleration. The leader-following consensus of system (1)-(2) is said to be achieved if the states of followers satisfy lim_𝑡→∞(𝑥_𝑖(𝑡)−𝑥₀(𝑡))=0 and lim_𝑡→∞(𝑣_𝑖(𝑡)−𝑣₀(𝑡))=0, 𝑖=1,…,𝑛.

In this paper, we are interested in discussing the leader-following consensus problem in multi-agent systems for both fixed and switching topologies, with the transmitted information time delay 𝜏(𝑡). To solve such a problem, the following neighbored-based protocol is used: 𝑢_𝑖(𝑡)=𝑎₀+_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗(𝑡)𝑘₁𝑥_𝑗(𝑡−𝜏)−𝑥_𝑖(𝑡−𝜏)+𝑘₂𝑣_𝑗(𝑡−𝜏)−𝑣_𝑖(𝑡−𝜏)+𝑏_𝑖(𝑡)𝑘₁𝑥₀(𝑡−𝜏)−𝑥_𝑖(𝑡−𝜏)+𝑘₂𝑣₀(𝑡−𝜏)−𝑣_𝑖(𝑡−𝜏), (3) where 𝑘₁,𝑘₂>0 are control parameters, 𝑎₀ is the constant acceleration of the leader, and the time-varying delay 𝜏(𝑡)>0 is a continuously differentiable and bounded function. Note that 𝑢_𝑖 in (3) depends on the position information and velocity information, all with time delay 𝜏(𝑡), of its neighbors and itself. When each follower can get the velocity feedback without time delay, only from the leader and itself, and when the leader moves at a constant velocity (i.e., 𝑎₀≡0), protocol (3) becomes 𝑢_𝑖(𝑡)=_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗(𝑡)𝑘₁𝑥_𝑗(𝑡−𝜏)−𝑥_𝑖(𝑡−𝜏)+𝑏_𝑖(𝑡)𝑘₁𝑥₀(𝑡−𝜏)−𝑥_𝑖(𝑣𝑡−𝜏)+𝑘₀(𝑡)−𝑣_𝑖(𝑡), (4) which is equivalent to protocol (6) in [12].

To describe the variable topologies, we define a piecewise constant switching function 𝜎(𝑡)∶[0,∞]→ℐ_Γ, where ℐ_Γ⊂ℕ is the index set associated with the elements of Γ. The set Γ is finite because at most a digraph with 𝑛 nodes is complete and has 𝑛(𝑛−1) edges.

Let 𝑥_𝑖=𝑥_𝑖−𝑥₀, 𝑣_𝑖=𝑥_𝑖−𝑣₀; then (3) can be written as 𝑢_𝑖(𝑡)=𝑎₀+_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗(𝑡)𝑘₁𝑥_𝑗(𝑡−𝜏)−𝑥_𝑖(𝑡−𝜏)+𝑘₂𝑣_𝑗(𝑡−𝜏)−𝑣_𝑖(𝑡−𝜏)−𝑏_𝑖(𝑡)𝑘₁𝑥_𝑖(𝑡−𝜏)+𝑘₂𝑣_𝑖(𝑡−𝜏). (5) By taking 𝑥=(𝑥^𝑇₁,…,𝑥^𝑇_𝑛)^𝑇, 𝑣=(𝑣^𝑇₁,…,𝑣^𝑇_𝑛)^𝑇, the closed-loop system (1) with protocol (3) can be expressed in a matrix form: 𝑥̇‌=̇𝑣,𝑣=−𝑘₁𝐿_𝜎+𝐵_𝜎⊗𝐼_𝑝𝑥(𝑡−𝜏)−𝑘₂𝐿_𝜎+𝐵_𝜎⊗𝐼_𝑝𝑣(𝑡−𝜏), (6) where ⊗ denotes the Kronecker product and 𝐿_𝜎,𝐵_𝜎 are the Laplacian matrix and the leader adjacency matrix associated with 𝒢_𝜎, respectively.

Before discussing the convergence of system (6), we first introduce the following result on the solutions for general functional differential equations [15]. Let 𝐶([−𝜏,0],ℝ^𝑛) be a Banach space of continuous functions 𝜑∶[−𝜏,0]→ℝ^𝑛 with a norm ‖𝜑‖_𝑐=max_{𝜃∈[−𝜏,0]}‖𝜑(𝜃)‖. Consider the following time-delay system: 𝑥̇=𝑓𝑡,𝑥_𝑡𝑥(𝜃)(𝜃)[],,𝑡>0,=𝜑,𝜃∈−𝜏,0 (7) where 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃), ∀𝜃∈[−𝜏,0], 𝑓∶ℝ×𝐶([−𝜏,0],ℝ^𝑛)→ℝ^𝑛 is a continuous function and 𝑓(𝑡,0) = 0, ∀𝑡∈ℝ.

Lemma 3 (the 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. If there is a continuous function 𝑉(𝑡,𝑥) such that 𝜙₁(‖𝑥‖)≤𝑉(𝑡,𝑥)≤𝜙₂(‖x‖),𝑡∈ℝ,𝑥∈ℝ^𝑛. (8) 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 (7) satisfies 𝑉̇(𝑡,𝑥)≤−𝜙₃𝑥(((𝑉((𝑡[];(‖‖)if𝑉𝑡+𝜃,𝑥𝑡+𝜃))<𝜙𝑡,x))),𝜃∈−𝜏,0 (9) then the solution 𝑥=0 is uniformly asymptotically stable.

The function 𝑉(𝑡,𝑥) which satisfies (8) and (9) in Lemma 3 is usually called the Lyapunov-Razumikhin function.

The following lemmas are needed in the subsequent sections.

Lemma 4 (see [16]). For any two real vectors 𝑎 and 𝑏 with the same dimension, one has 2𝑎^𝑇𝑏≤𝑎^𝑇Ψ⁻¹𝑎+𝑏^𝑇Ψ𝑏, (10) where Ψ is any positive definite matrix with an appropriate dimension.

Lemma 5 (the Schur Complement Theorem [17]). Let 𝑆 be a symmetric matrix partitioned into blocks: 𝐵𝑆=𝐴𝐵_𝑇𝐶, (11) where both 𝐴 and 𝐶 are square; then the following properties are equivalent:(1)𝑆is positive definite;(2)𝐶 and 𝐴−𝐵𝐶⁻¹𝐵^𝑇 are positive definite;(3)𝐴 and 𝐶−𝐵^𝑇𝐴⁻¹𝐵 are positive definite.

Give a complex polynomial 𝑅(𝑧)=𝜌₀+𝜌₁𝑧+⋯+𝜌_𝑛𝑧^𝑛, where 𝜌_𝑖∈ℂ, 𝑖=0,1,…,𝑛. Substituting 𝑧=i𝜔 into 𝑅(𝑧), we have 𝑅(i𝜔)=𝛼(𝜔)+i𝛽(𝜔), where 𝛼(𝜔),𝛽(𝜔)∈ℝ, and i is the imaginary unit. The following lemma shows the relationship between the Hurwitz stability of 𝑅(𝑧) and the related pair 𝛼(𝜔), 𝛽(𝜔).

Lemma 6 (the Hermite-Biehler Theorem [18]). The complex polynomial 𝑅(𝑧) is Hurwitz stable if and only if the related pair 𝛼(𝜔), 𝛽(𝜔) is interlaced and 𝛼(0)𝛽^′(0)−𝛼^′(0)𝛽(0)>0.

3. Consensus with Fixed Topology

This section focuses on the convergence analysis of (6) with fixed topology. In this case, the subscript 𝜎 can be dropped for simplicity. Denote 𝜀=(𝑥^𝑇,𝑣^𝑇)^𝑇; we can obtain an error dynamics of (3) as follows: 𝜀̇=𝑌𝜀+𝑍𝜀(𝑡−𝜏), (12) where 0𝑌=^𝑛×𝑛𝐼_𝑛0_𝑛×𝑛0_𝑛×𝑛⊗𝐼_𝑝0,𝑍=^𝑛×𝑛0_𝑛×𝑛−𝑘₁𝐻−𝑘₂𝐻⊗𝐼_𝑝,𝐻=𝐿+𝐵. (13)

By the Newton-Leibnitz formula, we have 𝜀(𝑡−𝜏)=𝜀(𝑡)−⁰_−𝜏𝜀̇()(𝑡)𝑡+𝑠𝑑𝑠=𝜀−𝑌⁰_−𝜏𝜀(𝑡+𝑠)𝑑𝑠−𝑍⁰_−𝜏𝜀(𝑡−𝜏+𝑠)𝑑𝑠. (14) Thus, system (12) can be rewritten as 𝜀̇=𝐹𝜀−𝑍𝑌⁰_−𝜏𝜀(𝑡+𝑠)𝑑𝑠−𝑍²⁰_−𝜏𝜀(𝑡−𝜏+𝑠)𝑑𝑠, (15) where 𝐹=𝑌+𝑍.

The matrix 𝐹 plays a key role in the convergence analysis of (15); the following lemma shows a sufficient and necessary condition that can guarantee the Hurwitz stability of 𝐹.

Lemma 7. The matrix 𝐹 is Hurwitz stable if and only if 𝐻 is positive stable and the control parameters satisfy 𝑘²₂/𝑘₁>max_{𝜆∈Λ(𝐻)}(Im²(𝜆)/Re(𝜆)|𝜆|²).

Proof. Let 𝑧 be an eigenvalue of 𝐹. Then, one has det𝑧𝐼_2𝑛𝑝−𝐹=det𝑧𝐼_𝑛−𝐼_𝑛𝑘₁𝐻𝑧𝐼_𝑛+𝑘₂𝐻⊗𝐼_𝑝=det𝑧𝑧𝐼_𝑛+𝑘₂𝐻+𝑘₁𝐻⊗𝐼_𝑝=^𝑛_𝑖=1𝑧²+𝑘₂𝜆_𝑖𝑧+𝑘₁𝜆_𝑖^𝑝, (16) where 𝜆_𝑖 is the 𝑖th eigenvalue of 𝐻. Therefore, the Hurwitz stability of matrix 𝐹 is equivalent to that of the polynomial: 𝑅(𝑧)=𝑧²+𝑘₂𝜆_𝑖𝑧+𝑘₁𝜆_𝑖. Substituting 𝑧=i𝜔 into 𝑅(𝑧), one gets 𝑅(i𝜔) = −𝜔² + 𝑘₂(Re(𝜆_𝑖)+iIm(𝜆_𝑖))i𝜔+𝑘₁(Re(𝜆_𝑖)+iIm(𝜆_𝑖)). Namely, 𝛼(𝜔) = −𝜔² − 𝑘₂Im(𝜆_𝑖)𝜔 + 𝑘₁Re(𝜆_𝑖) and 𝛽(𝜔) = 𝑘₂Re(𝜆_𝑖)𝜔+𝑘₁Im(𝜆_𝑖). According to Lemma 6, 𝑅(𝑧) is Hurwitz stable if and only if the following conditions hold:(i)𝛼(𝜔)=0 has two distinct real roots 𝛼₁<𝛼₂;(ii)the interlaced condition holds; that is, 𝛼₁<𝛽₁<𝛼₂, where 𝛽₁ is the unique root of 𝛽(𝜔)=0;(iii)𝛼(0)𝛽^′(0)−𝛼^′(0)𝛽(0)>0.
Considering condition (i), 𝛼(𝜔)=0 has two distinct real roots if and only if Δ_𝛼 = 𝑘²₂Im²(𝜆_𝑖) + 4𝑘₁Re(𝜆_𝑖)>0. Noting that Re(𝜆_𝑖)>0, since 𝐻 is positive stable, condition (i) holds obviously. In addition, the roots in condition (ii) can be expressed as 𝛼₁=(−𝑘₂Im(𝜆_𝑖)−Δ_𝛼)/2, 𝛼₂=(−𝑘₂Im(𝜆_𝑖)+Δ_𝛼)/2 and 𝛽₁=−𝑘₁Im(𝜆_𝑖)/𝑘₂Re(𝜆_𝑖). It is easy to check 𝛼₁<𝛽₁<𝛼₂⇔𝑘²₂/𝑘₁>Im²(𝜆)/R𝑒(𝜆)|𝜆|². Furthermore, using the fact that |𝜆_𝑖|>0, the condition (iii) 𝛼(0)𝛽^′(0) − 𝛼^′(0)𝛽(0)=Re²(𝜆_𝑖)𝑘₁𝑘₂+𝑘₁𝑘₂Im²(𝜆_𝑖)=𝑘₁𝑘₂|𝜆_𝑖|²>0 holds.

Theorem 8. For system (12), suppose that the control parameters satisfy 𝑘²₂𝑘₁>𝑘₀=max_{(𝐻)𝜆∈Λ}Im²(𝜆)Re(𝜆)|𝜆|². (17) Then, there exists a constant 𝜏₀>0 (which will be estimated in the following (22)) such that when 𝜏<𝜏₀, lim_𝑡→∞𝜀(𝑡)=0; (18) namely, the leader-following consensus is reached asymptotically if and only if the graph 𝒢 has a spanning tree.

Proof. Sufficiency. Since the graph 𝒢 has a spanning tree, by Remark 2, 𝐻 is positive stable. Then, it follows from Lemma 7 that 𝐹 is Hurwitz stable. Therefore, there exists a positive definite matrix 𝑃∈ℝ^{2𝑛𝑝×2𝑛𝑝} such that 𝑃𝐹+𝐹^𝑇𝑃=−𝐼_2𝑛𝑝. (19) Take the Lyapunov-Razumikhin function 𝑉(𝜀)=𝜀^𝑇𝑃𝜀. Obviously, 𝜆_min(𝑃)‖𝜀‖²≤𝑉(𝜀(𝑡))≤𝜆_max(𝑃)‖𝜀‖². In addition, based on (15), we have 𝑉̇(𝜀)=𝜀^𝑇𝑃𝐹+𝐹^𝑇𝑃𝜀−2⁰_−𝜏𝜀^𝑇𝑃𝑍𝑌𝜀(𝑡+𝑠)𝑑𝑠−2⁰_−𝜏𝜀^𝑇𝑃𝑍²𝜀(𝑡−𝜏+𝑠)𝑑𝑠≤𝜀^𝑇𝑃𝐹+𝐹^𝑇𝑃𝜀+𝜏𝜀^𝑇𝑃𝑍𝑌𝑃⁻¹𝑌^𝑇𝑍^𝑇+𝑃𝜀_0−𝜏𝜀^𝑇(𝑡−𝑟+𝑠)𝑃𝜀(𝑡−𝜏+𝑠)𝑑𝑠+𝜏𝜀^𝑇𝑃𝑍²𝑃⁻¹𝑍^2𝑇+𝑃𝜀_0−𝜏𝜀^𝑇(𝑡−𝜏+𝑠)𝑃𝜀(𝑡−𝜏+𝑠)𝑑𝑠, (20) where the inequality is obtained from Lemma 4 by setting 𝑎₁=−𝑌^𝑇𝑍^𝑇𝑃𝜀, Ψ=𝑃, 𝑏₁=𝜀(𝑡+𝑠), 𝑎₂=−(𝑍²)^𝑇𝑃𝜀, 𝑏₂=𝜀(𝑡−𝑟+𝑠). Then, take 𝜙(𝑠)=𝑞𝑠 for some constant 𝑞>1. In the case of 𝑉(𝜀(𝑡+𝜃))<𝑞𝑉(𝜀(𝑡)), −2𝜏≤𝜃≤0, the time derivative of 𝑉(𝜀(𝑡)) gives 𝑉̇(𝜀)≤−𝜀^𝑇𝜀+𝜏𝜀^𝑇𝑃𝑍𝑌𝑃⁻¹𝑌^𝑇𝑍^𝑇𝑃+𝑃𝑍²𝑃⁻¹𝑍^2𝑇𝑃+2𝑞𝑃𝜀. (21) Consequently, 𝑉̇(𝜀)≤−𝜂𝜀^𝑇𝜀 for some constant 𝜂>0 if 𝜏<𝜏₀=1𝑃𝑍𝑌𝑃⁻¹𝑌^𝑇𝑍^𝑇𝑃+𝑃𝑍²𝑃⁻¹𝑍^2𝑇.𝑃+2𝑞𝑃 (22) Hence, system (12) is asymptotically stable, invoking Lemma 3.
Necessity. As a special case of (15), the system 𝜀̇=𝐹𝜀 is asymptotical stable, which implies that 𝐹 is Hurwitz stable. Hence, by Lemma 7, 𝐻 is positive stable, and then it follows from Remark 2 that the graph 𝒢 has a spanning tree.

Remark 9. In the special case where 𝜏≡0 and 𝑏_𝑖(𝑡)>0 for 𝑖=1,…,𝑛, that is, the system is free of time delay and all the followers have full access to the leader, system (3) becomes 𝑢_𝑖(𝑡)=𝑎₀+_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗(𝑡)𝑘₁𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+𝑘₂𝑣_𝑗(𝑡)−𝑣_𝑖(𝑡)+𝑏_𝑖𝑘₁𝑥₀(𝑡)−𝑥_𝑖(𝑡)+𝑘₂𝑣₀(𝑡)−𝑣_𝑖(𝑡), (23) which is equivalent to the protocol (9) in [19].

4. Consensus with Switching Topology

Consider system (6) with switching topology. Denote 𝜀=(𝑥^𝑇,𝑣^𝑇)^𝑇; (6) can be equivalently expressed as follows: 𝜀̇=𝑌𝜀+𝑍_𝜎𝜀(𝑡−𝜏), (24) where 0𝑌=^𝑛×𝑛𝐼_𝑛0_𝑛×𝑛0_𝑛×𝑛⊗𝐼_𝑝,𝑍_𝜎=0_𝑛×𝑛0_𝑛×𝑛−𝑘₁𝐻_𝜎−𝑘₂𝐻_𝜎⊗𝐼_𝑝,𝐻_𝜎=𝐿_𝜎+𝐵_𝜎. (25) Following the lines of the analysis of the case with fixed topology, we have 𝜀(𝑡−𝜏)=𝜀(𝑡)−⁰_−𝜏𝜀̇()(𝑡)𝑡+𝑠𝑑𝑠=𝜀−𝑌⁰_−𝜏𝜀(𝑡+𝑠)𝑑𝑠−𝑍_𝜎⁰_−𝜏𝜀(𝑡−𝜏+𝑠)𝑑𝑠. (26) Then, (24) can be written as 𝜀̇=𝐹_𝜎𝜀−𝑍_𝜎𝑌⁰_−𝜏𝜀(𝑡+𝑠)𝑑𝑠−𝑍²_𝜎⁰_−𝜏𝜀(𝑡−𝜏+𝑠)𝑑𝑠, (27) where 𝐹_𝜎=𝑌+𝑍_𝜎.

Before moving on, we first recall some conditions under which the matrix 𝐻_𝜎+𝐻^𝑇_𝜎 is positive definite. In [12], Hu and Hong proved that 𝐻_𝜎+𝐻^𝑇_𝜎 is positive definite when the graph 𝒢 is balanced and node 0 is globally reachable, while the authors presented the corresponding improved result in [7].

Lemma 10 (see [7]). Suppose that the weights for edges of 𝒢_𝜎 satisfy the following assumptions.(A1)∑_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗≥∑_{𝑗∈𝒩𝑖}𝑎_𝑗𝑖, 𝑖∉ℐ,𝑖∈𝒱,(A2)2𝑏_𝑖+∑_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗>∑_{𝑗∈𝒩𝑖}𝑎_𝑗𝑖, 𝑖∈ℐ,(A3) node 0 is globally reachable in 𝒢_𝜎,where ℐ denotes the index set of neighbors of node 0, that is, ℐ={𝑏_𝑖>0,𝑖∈𝒱}; then 𝐻_𝜎+𝐻^𝑇_𝜎 is positive definite.

Note that ∑_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗 and ∑_{𝑗∈𝒩𝑖}𝑎_𝑗𝑖 are the indegree and outdegree of node 𝑖 in 𝒢, respectively. Then, the assumption (A1) in Lemma 10 implies that the in-degree of node 𝑖 is no less than its out-degree for 𝑖∉ℐ. Clearly, the condition that 𝒢 is balanced is a special case of assumptions (A1) and (A2). Also, changing the orientation of each arc of the graph, the assumption (A3) is equivalent to a spanning tree (rooted at node 0) contained in 𝒢.

Based on Lemma 10 and the fact that the set ℐ_Γ is finite, when the weights for edges of 𝒢_𝜎 satisfy the assumptions of Lemma 10, 𝜇=min_𝜎{𝜆_min(𝐻_𝜎+𝐻^𝑇_𝜎)}>0 is well defined. Now, we give the main result of leader-following consensus problem with switching topology.

Theorem 11. For system (24), suppose that the control parameters satisfy 𝑘₂>𝑘₁>𝑘^∗=1,2𝜇 (28) and the graph 𝒢 satisfies the assumptions in Lemma 10. Take 𝜏<𝜏₁ (which will be estimated in the following (34)); then one has lim_𝑡→∞𝜀(𝑡)=0; (29) that is, the leader-following consensus is reached asymptotically.

Proof. Take the Lyapunov-Razumikhin function 𝑉(𝜀)=𝜀^𝑇Φ𝜀, where 𝑘Φ=²𝐼_𝑛𝑘₁𝐼_𝑛𝑘₁𝐼_𝑛𝑘₂𝐼_𝑛⊗𝐼_𝑝 (30) is positive definite for 𝑘₂>𝑘₁>0.
Following a similar line to the proof of Theorem 8, we can obtain 𝑉̇(𝜀)≤𝜀^𝑇Φ𝐹_𝜎+𝐹^𝑇_𝜎Φ𝜀+𝜏𝜀^𝑇Φ𝑍_𝜎𝑌Φ⁻¹𝑌^𝑇𝑍𝜎^𝑇+Ψ𝜀_0−𝜏𝜀^𝑇(𝑡−𝑟+𝑠)Ψ𝜀(𝑡−𝜏+𝑠)𝑑𝑠+𝜏𝜀^𝑇Φ𝑍²_𝜎Ψ⁻¹𝑍²_𝜎^𝑇+Ψ𝜀_0−𝜏𝜀^𝑇(𝑡−𝜏+𝑠)Ψ𝜀(𝑡−𝜏+𝑠)𝑑𝑠. (31) Take 𝜙(𝑠)=𝑞𝑠 for some constant 𝑞>1. In the case of 𝑉(𝜀(𝑡+𝜃))<𝑞𝑉(𝜀(𝑡)), −2𝜏≤𝜃≤0, we have 𝑉̇(𝜀)≤−𝜀^𝑇𝑀_𝜎𝜀+𝜏𝜀^𝑇Ψ𝑍_𝜎𝑌Φ⁻¹𝑌^𝑇𝑍^𝑇_𝜎Φ+Φ𝑍²_𝜎Φ⁻¹𝑍²_𝜎^𝑇Φ+2𝑞Φ𝜀, (32) where 𝑀_𝜎=−Φ𝐹_𝜎+𝐹^𝑇_𝜎Φ=𝑘²₁𝐻_𝜎+𝐻^𝑇_𝜎𝑘₁𝑘₂𝐻_𝜎+𝐻^𝑇_𝜎−𝑘₂𝐼_𝑛𝑘₁𝑘₂𝐻_𝜎+𝐻^𝑇_𝜎−𝑘₂𝐼_𝑛𝑘²₂𝐻_𝜎+𝐻^𝑇_𝜎−2𝑘₁𝐼_𝑛⊗𝐼_𝑝. (33) From Lemma 5, 𝑀_𝜎 is positive definite if the control parameters satisfy (28). Furthermore, 𝑉̇(𝜀) is negative definite if 𝜏<𝜏₁=min_𝜎𝜆_min𝑀_𝜎max_𝜎Φ𝑍_𝜎𝑌Φ⁻¹𝑌^𝑇𝑍^𝑇_𝜎Φ+Φ𝑍²_𝜎Φ⁻¹𝑍²_𝜎^𝑇.Φ+2𝑞Φ (34) Therefore, the conclusion follows from Lemma 3.

5. Simulations

In this section, to illustrate our theoretical results derived in the above section, we provide two numerical simulations. For simplicity, let 𝑎_𝑖𝑗=1 if agent 𝑗 is a neighbor of agent 𝑖 and 𝑎_𝑖𝑗=0 otherwise.

Example 12. For the fixed topology case, we consider a multi-agent system consisting of one leader and five followers with the interaction graph 𝒢₁ given in Figure 1. It can be noted that 𝒢₁ has a spanning tree. With simple calculations, we can obtain that 𝑘₀=0.1295 and 𝜏₀=0.0717. Take 𝑘₂=0.5 and 𝑘₁=1; then 𝑘²₂/𝑘₁=0.25>0.1295. Figures 2(a) and 2(b) show, respectively, the position errors and velocity errors when 𝜏(𝑡)=0.07|sin𝑡|. We can see that the five followers can track the leader; Figures 3(a) and 3(b) show, respectively, the position errors and velocity errors when 𝑘₂=0.3592 and 𝑘₁=1, and in this case 𝑘²₂/𝑘₁=0.129<0.1295. From Figure 3, it can be seen that the tracking errors become unbounded.

Figure 1 

Interaction topology 𝒢1.

(a)

(b)

Figure 2 

Position errors and velocity errors.

(a)

(b)

Figure 3 

Position errors and velocity errors.

Example 13. For the switching topology case with 𝑝=2 (i.e., the agents move in a plane), we consider a multiagent system consisting of one leader and four followers. The interaction topology switches from a set {𝒢₂,𝒢₃} as shown in Figure 4. Clearly, the weights of both 𝒢₂ and 𝒢₃ satisfy the assumptions in Lemma 10. With simple calculations, we can obtain that 𝑘^∗=0.6063. Take 𝑘₂=0.65,𝑘₁=0.61; Figures 5(a) and 5(b) illustrate, respectively, the position evolutions and velocity evolutions with 𝜏(𝑡)=0.015|cos𝑡|. we can see that the four followers can track the leader.

(a)

(b)

Figure 4 

Interconnection topologies 𝒢2 and 𝒢3.

(a)

(b)

Figure 5 

Position and velocity evolutions.

6. Conclusion

In this paper, we consider a second-order consensus problem for a multi-agent system with a constant acceleration leader and time-varying delays. Based on the Lyapunov-Razumikhin function and algebraic theory, we give a sufficient and necessary condition when the interaction topology is fixed and present a sufficient condition when the interaction topology is switching and the weights of graph satisfy some assumptions. Furthermore, two numerical simulations are provided to illustrate the results.

Acknowledgments

The authors would like to thank the anonymous reviewers and Editor Professor Lenci for their insightful comments and suggestions. This research is supported by NSFC (61104141, 61170309), NSFC Tianyuan foundation (11126104), and Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020).