Consensus Analysis of Second-Order Multiagent Systems with General Topology and Time Delay

Abstract

This paper addresses the consensus of second-order multiagent systems with general topology and time delay based on the nearest neighbor rule. By using the Laplace transform technique, it is proved that the second-order multi-agent system in the presence of time-delay can reach consensus if the network topology contains a globally reachable node and time delay is bounded. The bound of time-delay only depends on eigenvalues of the Laplacian matrix of the system. The main contribution of this paper is that the accurate state of the consensus center and the upper bound of the communication delay to make the agents reach consensus are given. Some numerical simulations are given to illustrate the theoretical results.

1. Introduction

The wide applications of multiagent systems abounded in nature and engineering area have stimulated a great deal of interests in studying cooperative and coordinated control problems. The mechanisms operational principles of multi-agent systems can provide useful ideas for developing distributed cooperative control, formation control, unmanned air vehicles, and sensing networks, and so forth. Recently, the study of collective dynamics and coordination of multi-agent systems becomes a hot topic and has attracted many researchers from mathematics, physics, biology, sociology, control science, computer science, artificial intelligence, and so forth [1–25].

However, study of some fundamental issues concerning the coordinated control of multi-agent systems, such as consensus, stability, synchronization, and controllability analysis. As in its usual sense, consensus means multiple agents can reach an agreement on a common value by regulating their neighbors. The consensus problem was firstly studied by DeGroot [14], who proposed a weighted average method to estimate probability distribution function of unknown variables. Vicsek and Czirok [10] established a discrete model of the multi-agents system called “Vicsek Model.” Jadbabaie et al. [11] developed “Vicsek Model,” which had been linearized by graph theory and matrix theory. Usually, a multi-agent system is interconnected through neighbor rules and hence has a local and time-varying communication topology. In [4], Olfati-Saber and Murray discussed linear systems with time-delays.

Many papers on consensus problems of first-order multi-agent system were presented (see, e.g., [1–4, 10, 11, 20, 21] and the references therein). In [1], the first-order multi-agent systems were firstly studied by the Laplace transform technique and it was proved that the system could reach consensus under certain conditions. In [2], the average consensus problem was investigated in multi-agent systems with undirected network topology and multiple time-varying communication delays by the linear matrix inequality (LMI) method. Liu et al. [23] considered the consensus of multi-agent systems with an active leader. The second-order multi-agent systems were studied in [6, 7, 12, 13, 16–19, 22]. Yu et al. [6] discussed the second-order system with time-delays. They obtained the necessary and sufficient condition and gave upper bound of time-delays. In [7], Ren considered a consensus algorithm for second-order dynamics systems with Cartesian coordinate coupling in three-dimensional space. In [12], the authors discussed the consensus of continuous second-order multi-agent system in a sampled data set. Lin and Jia [22] investigated the consensus of a class of second-order multi-agent systems with time-delays and jointly connected topologies.

In the view of [1], the Laplace transform has many advantages in solving consensus problems, such as simplifying the model of first-order systems and obtaining the accurate state of the consensus center. Therefore, we will study the consensus of the second-order multi-agent system with time-delay and general topology by the Laplace transform in this paper. We will prove the consensus can be achieved when network topology contains a globally reachable node, and the time-delays must be less than a critical value which depends on the eigenvalues of the Laplacian matrix. Furthermore, the accurate state of the consensus center of the second-order multi-agent system and the upper bound of the communication delay to make the agents reach consensus are obtained.

The rest of this paper is organized as follows. In Section 2, we present the mathematical model of the second-order multi-agent system with time-delay and formulate it by matrix form. In Section 3, the detailed theoretical analysis for the model is given and the main results are derived. Then some numerical simulations are given to verify the theoretical results in Section 4. Finally, the conclusion is given in Section 5.

2. Model

Consider the second-order multi-agent system with time-delay and general topology is described by 𝑥̇_𝑖=𝑣_𝑖𝑣̇_𝑖=−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡−𝜏)−𝑥_𝑗(𝑡−𝜏)−𝑣_𝑖, (1) where 𝑥_𝑖∈ℜ^𝑛 and 𝑣_𝑖∈ℜ^𝑛 are position and velocity of agent 𝑖, respectively, and 𝑖=1,…,𝑁; 𝒩_𝑖 is the neighbor set of agent 𝑖; 𝐴=[𝑎_𝑖𝑗]∈ℜ^𝑛×𝑛 (𝑖,𝑗=1,…,𝑁) with each 𝑎_𝑖𝑗≥0 and 𝑎_𝑖𝑖=0 is the interaction (or coupling) matrix; 𝜏>0 is the time-delay.

Let 𝑋=(𝑥₁,…,𝑥_𝑁)^𝑇 and 𝑉(𝑡)=(𝑣₁,𝑣₂,…,𝑣_𝑁)^𝑇 be the stack vector of all the agent states and velocities; it follows that ̇̇()𝑋=𝑉𝑉=−𝐿𝑋𝑡−𝜏−𝑉, (2) where the matrix 𝐿 is the Laplacian and has the following properties: (i) the off-diagonal elements are all negative or zero, and (ii) the row sums are all equal to zero. And we can formulate the model with time-delay in the following form: 𝑋̈(𝑡)+𝑋̇(𝑡)=−𝐿𝑋(𝑡−𝜏). (3)

For the Laplacian matrix 𝐿, we can have the following lemmas.

Lemma 1 (see [7]). Laplacian matrix 𝐿 has at least one zero eigenvalue with 𝟏=(1,1,…,1)^𝑇 as its eigenvector, and all nonzero eigenvalues have positive real parts. Laplacian matrix 𝐿 has a simple zero eigenvalue if and only if 𝒢 has a globally reachable node.

Lemma 2 (see [1]). For Laplacian matrix 𝐿, the algebraic complements of the elements in the same row are equal.

Lemma 3 (see [1]). For a column vector denoted as 𝑧, if a column of Laplacian matrix 𝐿 is replaced by 𝑧, the new matrix is denoted as 𝐿₁, and if another column of 𝐿 is replaced by 𝑧, the new matrix is denoted as 𝐿₂, then det(𝐿₁)=det(𝐿₂).

3. Main Results

In this section, we will analyze the consensus of second-order multi-agent systems with general topology and time-delay based on the nearest neighbor rule. For this, we use the Laplace transform technique into (3), then we can have 𝑠²+𝑠𝐼+𝑒^−𝑠𝜏𝐿ℒ[𝑋(𝑡)]=𝑠𝑋(0)+𝑋(0)+𝑉(0), (4) where 𝑋(0)=(𝑥₁(0),𝑥₂(0),…,𝑥_𝑁(0))^𝑇 and 𝑉(0)=(𝑣₁(0),𝑣₂(0),…,𝑣_𝑁(0))^𝑇 are initial values.

For simplicity and without loss of generality, we will discuss the case 𝑋(0)=0 and 𝑉(0)≠0 throughout this paper.

Theorem 4. For second-order multi-agent system (1), if there exists at least one globally reachable node in the network topology, and the time-delay 𝜏 satisfies 𝜏<min_{{}𝑘=2,3,...,𝑁}arccos𝜆_𝑘−2/𝜆_𝑘−arg𝜆_𝑘𝜆_𝑘,𝜆−1𝜋/2−arg_𝑘𝜆_𝑘, (5) then system (1) can reach consensus, and as 𝑡→∞, 𝑋(𝑡)⟶𝑐(1,1,…,1)^𝑇,𝑉(𝑡)⟶(0,0,…,0)^𝑇, (6) where 𝑐=^{𝑣1(0)𝑙12⋯𝑙1𝑁𝑣2(0)𝑙22⋯𝑙2𝑁𝑣⋮⋮⋮⋮𝑁(0)𝑙𝑁2⋯𝑙𝑁𝑁}_{𝜆2⋯𝜆ℎ𝜆𝑝𝑗⋯𝜆𝑞𝑙}, (7) and 𝜆_𝑖 is the nonzero eigenvalue of the Laplacian matrix 𝐿.

Proof. Denoting 𝒳(𝑠)=ℒ[𝑋(𝑡)], then (4) can be rewritten as 𝑠²+𝑠𝐼+𝑒^−𝑠𝜏𝐿𝒳(𝑠)=𝑉(0). (8) According to Cramer’s rule, we can have 𝒳_𝑖(𝑠)=det𝑠²+𝑠𝐼+𝑒^−𝑠𝜏𝐿^𝑖det𝑠²+𝑠𝐼+𝑒^−𝑠𝜏𝐿, (9) for 𝑖=1,2,…,𝑁, where det[(𝑠²+𝑠)𝐼+𝑒^−𝑠𝜏𝐿] is the determinant of matrix [(𝑠²+𝑠)𝐼+𝑒^−𝑠𝜏𝐿], and det[(𝑠²+𝑠)𝐼+𝑒^−𝑠𝜏𝐿]^𝑖 is the determinant of matrix [(𝑠²+𝑠)𝐼+𝑒^−𝑠𝜏𝐿] in which the 𝑖th column has been replaced by 𝑉(0). For convenience, we denote 𝑀_𝑖(𝑠)=det[(𝑠²+𝑠)𝐼+𝑒^−𝑠𝜏𝐿]^𝑖 and 𝑀(𝑠)=det[(𝑠²+𝑠)𝐼+𝑒^−𝑠𝜏𝐿]. By Lemma 1, the Laplacian matrix only has a zero eigenvalue, and the real parts of nonzero eigenvalues are positive, denoted as follows: 0,𝜆₂,𝜆₃,…,𝜆_ℎ,𝜆_𝑗,…,𝜆_𝑙, (10) where 𝜆_𝑗,…,𝜆_𝑙 are the multiple eigenvalues with multiplicity as 𝑝,…,𝑞, then 𝑀(𝑠)=𝑠²+𝑠𝑠²+𝑠+𝑒^−𝑠𝜏𝜆₂⋯𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_ℎ×𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_𝑗^𝑝⋯𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_𝑙^𝑞. (11) Applying Heaviside’s method, then 𝒳_𝑖(𝑠)=𝑐^𝑖₁𝑠²+𝑐+𝑠𝑠²+𝑠+^𝑖2𝑒^−𝑠𝜏𝜆₂+𝑐+⋯_𝑖ℎ𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_ℎ+𝑐^𝑖_𝑗𝑝𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_𝑗^𝑝+𝑐^𝑖_𝑗𝑝−1[𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_𝑗]^(𝑝−1)+𝑐+⋯_𝑖𝑗1𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_𝑗¹+𝑐+⋯_𝑖𝑙𝑞𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_𝑙^𝑞+𝑐^𝑖_{𝑙(𝑞−1)}𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_𝑙^(𝑞−1)𝑐+⋯+^𝑖𝑙1𝑠²+𝑠+𝑒^−𝑠𝜏𝜆_𝑙¹, (12) where 𝑐^𝑖₁𝑀=𝑠^𝑖(𝑠)𝑀(𝑠)_𝑠=0,𝑐^𝑖_𝑒=𝑠+𝜆_𝑒𝑀_𝑖(𝑠)𝑀(𝑠)_{𝑠=𝜆𝑒}𝑐,𝑒=2,3,…,ℎ,_𝑖𝑗𝑓=𝑑^𝑝−𝑓𝑑𝑠^𝑝−𝑓𝑠+𝜆_𝑗^{(𝑝−𝑓)}𝑀_𝑖(𝑠)𝑀(𝑠)_{𝑠=𝜆𝑗}𝑐,𝑓=1,2,…,𝑝,⋮⋮⋮_𝑖𝑘𝑔=𝑑^𝑞−𝑔𝑑𝑠^𝑞−𝑔𝑠+𝜆_𝑘^{(𝑞−𝑔)}𝑀_𝑖(𝑠)𝑀(𝑠)_{𝑠=𝜆𝑘},𝑔=1,2,…,𝑞. (13) Therefore, 𝒳_𝑖(𝑠)≥𝐆_𝑖(𝑠)=𝑐^𝑖₁𝑠²+𝑐+𝑠(𝑠+1)²+^𝑖2𝑒^−𝑠𝜏𝜆₂𝑐+⋯+^𝑖ℎ(𝑠+1)²+𝑒^−𝑠𝜏𝜆_ℎ+𝑐^𝑖_𝑗𝑝(𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑗^𝑝+𝑐^𝑖_𝑗𝑝−1(𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑗^(𝑝−1)𝑐+⋯+^𝑖𝑗1(𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑗¹𝑐+⋯+^𝑖𝑙𝑞(𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑙^𝑞+𝑐^𝑖_𝑙𝑞−1(𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑙^(𝑞−1)+𝑐+⋯_𝑖𝑙1(𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑙¹,𝒳_𝑖(𝑠)≤𝐇_𝑖(𝑠)=𝑐^𝑖₁𝑠²+𝑐+𝑠𝑠+𝑒^{𝑖2−𝑠𝜏}𝜆₂𝑐+⋯+^𝑖ℎ𝑠+𝑒^−𝑠𝜏𝜆_ℎ+𝑐^𝑖_𝑗𝑝𝑠+𝑒^−𝑠𝜏𝜆_𝑗^𝑝+𝑐^𝑖_𝑗𝑝−1𝑠+𝑒^−𝑠𝜏𝜆_𝑗^(𝑝−1)𝑐+⋯+^𝑖𝑗1𝑠+𝑒^−𝑠𝜏𝜆_𝑗¹𝑐+⋯+^𝑖𝑙𝑞𝑠+𝑒^−𝑠𝜏𝜆_𝑙^𝑞+𝑐^𝑖_𝑙𝑞−1𝑠+𝑒^−𝑠𝜏𝜆_𝑙^(𝑞−1)+𝑐+⋯_𝑖𝑙1𝑠+𝑒^−𝑠𝜏𝜆_𝑙¹. (14) In order to make 𝐆_𝑖(𝑠) converge, the real parts of the following equation’s roots (𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑘=0 (15) must be negative. Denoting 𝑠=𝑥+𝑗𝑦, 𝜆_𝑘=𝛼_𝑘+𝑗𝛽_𝑘, and 𝑗²=−1, as well as using Euler’s formula, then (𝑥+1)²−𝑦²+𝛼_𝑘𝑒^−𝑥𝜏cos−𝑦𝜏−𝛽_𝑘𝑒^−𝑥𝜏2()sin−𝑦𝜏+𝑗𝑥+1𝑦+𝛼_𝑘𝑒^−𝑥𝜏sin−𝑦𝜏−𝛽_𝑘𝑒^−𝑥𝜏cos−𝑦𝜏=0; (16) that is, (𝑥+1)²−𝑦²+𝛼_𝑘𝑒^−𝑥𝜏cos−𝑦𝜏−𝛽_𝑘𝑒^−𝑥𝜏2()sin−𝑦𝜏=0𝑥+1𝑦+𝛼_𝑘𝑒^−𝑥𝜏sin−𝑦𝜏−𝛽_𝑘𝑒^−𝑥𝜏cos−𝑦𝜏=0. (17) And then (𝑥+1)²−𝑦²+𝑒^−𝑥𝜏𝛼²_𝑘+𝛽²_𝑘cos−𝑦𝜏+𝜓_𝑘2()=0𝑥+1𝑦+𝑒^−𝑥𝜏𝛼²_𝑘+𝛽²_𝑘sin−𝑦𝜏+𝜓_𝑘=0, (18) where 𝜓_𝑘=arg(𝜆_𝑘)=arctan(𝛽_𝑘/𝛼_𝑘) and 𝜓_𝑘∈(−𝜋/2,𝜋/2). Then we can have (𝑥+1)²+𝑦²²=𝛼²_𝑘+𝛽²_𝑘𝑒^−2𝑥𝜏, (19) that is, 𝑦=±𝛼²_𝑘+𝛽²_𝑘𝑒^−𝑥𝜏−(𝑥+1)². (20) Form (18) and (20), we can obtain 2(𝑥+1)²−𝛼²_𝑘+𝛽²_𝑘𝑒^−𝑥𝜏+𝛼²_𝑘+𝛽²_𝑘𝑒^−𝑥𝜏×cos±𝛼²_𝑘+𝛽²_𝑘𝑒^−𝑥𝜏−(𝑥+1)²+𝜓_𝑘2(±=0,𝑥+1)𝛼²_𝑘+𝛽²_𝑘𝑒^−𝑥𝜏−(𝑥+1)²+𝛼²_𝑘+𝛽²_𝑘𝑒^−𝑥𝜏×sin±𝜏𝛼²_𝑘+𝛽²_𝑘𝑒^−𝑥𝜏−(𝑥+1)²+𝜓_𝑘=0. (21) Considering the case 𝑥=0, we can obtain the upper bound of time-delay cos𝜏𝛼²_𝑘+𝛽²_𝑘−1±𝜓_𝑘=𝛼²_𝑘+𝛽²_𝑘−2𝛼²_𝑘+𝛽²_𝑘, (22) that is, 𝜏𝜆_𝑘−1±arg𝜆_𝑘𝜆=arccos^𝑘−2𝜆_𝑘+2𝑚𝜋,𝑚=0,±1,±2,…. (23) To obtain the minimum critical value 𝜏₁, we take 𝑚=0, then 𝜏₁=min_{{}𝑘=2,3,...,𝑁}arccos𝜆_𝑘−2/𝜆_𝑘−arg𝜆_𝑘𝜆_𝑘−1. (24) Then we can obtain the real parts of roots of (𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑘=0 are negative as 𝜏<𝜏₁.
Similarly, for 𝐇_𝑖(𝑠), we can also get the upper bound of time-delay 𝜏₂; that is, for 𝑠+𝜆_𝑘𝑒^−𝑠𝜏=0, (25) the real parts of roots are negative. So 𝜏₂=min_{{}𝑘=2,3,...,𝑁}𝜋/2−arg𝜆_𝑘𝜆_𝑘. (26) When 𝜏<min_{{}𝑘=2,3,…,𝑁}arccos𝜆_𝑘−2/𝜆_𝑘−arg𝜆_𝑘𝜆_𝑘,𝜆−1𝜋/2−arg_𝑘𝜆_𝑘, (27) then the roots of equations (𝑠+1)²+𝑒^−𝑠𝜏𝜆_𝑘=0 and 𝑠+𝜆_𝑘𝑒^−𝑠𝜏=0 are negative real parts.
By the inverse Laplace transform, we can obtain ℒ⁻¹𝐆_𝑖(𝑠)⟶𝑐^𝑖₁,ℒ⁻¹𝐇_𝑖(𝑠)⟶𝑐^𝑖₁, (28) as 𝑡→∞.
Since 𝐆_𝑖(𝑠)≤𝒳_𝑖(𝑠)≤𝐇_𝑖(𝑠), as 𝑡→∞ℒ⁻¹𝒳_𝑖(𝑠)⟶𝑐^𝑖₁. (29) According to Lemma 3, we can know that 𝑐¹₁=𝑐²₁=⋯=𝑐^𝑁₁𝑀=𝑠^𝑖(𝑠)𝑀(𝑠)_𝑠=0=𝑐=^{𝑣1(0)𝑙12⋯𝑙1𝑁𝑣2(0)𝑙22⋯𝑙2𝑁𝑣⋮⋮⋱⋮𝑁(0)𝑙𝑁2⋯𝑙𝑁𝑁}_{𝜆2⋯𝜆ℎ𝜆𝑝𝑗⋯𝜆𝑞𝑙}. (30) Thus, 𝑥₁(𝑡)=𝑥₂(𝑡)=⋯=𝑥_𝑁(𝑡)𝑣=𝑐,₁(𝑡)=𝑣₂(𝑡)=⋯=𝑣_𝑁(𝑡)=0; (31) that is, as 𝑡→∞, 𝑋(𝑡)⟶𝑐(1,1,…,1)^𝑇,𝑉(𝑡)⟶(0,0,…,0)^𝑇. (32)(33) This completes the proof.

Notice that if the network topology is undirected with a globally reachable node and the adjacency matrix is symmetric, then the eigenvalues of the Laplacian matrix 𝐿 are real, so that 𝜓_𝑘=0.

Remark 5. Compared with the eigenvalue analysis in the time and frequency domain, such as the [24], using Cramer rule and Heavisides method, we can have obtained the accurate state of the consensus center and the upper bound of the communication delay to make the agents reach consensus.

Corollary 6. For second-order multi-agent system (1), if there exists at least one globally reachable node in the undirected and symmetric network topology, and the time-delay 𝜏 satisfies 𝜏<min_{{}𝑘=2,3,...,𝑁}arccos𝜆_𝑘−2/𝜆_𝑘𝜆_𝑘𝜋−12𝜆_𝑘, (34) then system (1) can reach consensus.

3.1. Further Extension

Consider the second-order multi-agent system with time-delay and general topology is described by 𝑥̇_𝑖=𝑣_𝑖𝑣̇_𝑖=−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑥_𝑖(𝑡−𝜏)−𝑥_𝑗(𝑡−𝜏)−𝛾𝑣_𝑖, (35) where 𝛾>0 is the feedback gain.

By the same way, the similar results are easy to obtain by the Laplace transform.

Theorem 7. For second-order multi-agent system (35), if there exists at least one globally reachable node in the network topology, and the time-delay 𝜏 satisfies 𝜏<min_{{}𝑘=2,3,...,𝑁}arccos𝜆_𝑘−2𝛾²/𝜆_𝑘−arg𝜆_𝑘𝜆_𝑘−𝛾²,𝛾𝜋/2−arg𝜆_𝑘𝜆_𝑘, (36) then system (35) can reach consensus, and as 𝑡→∞, 𝑋(𝑡)⟶𝑐(1,1,…,1)^𝑇,𝑉(𝑡)⟶(0,0,…,0)^𝑇. (37)

Corollary 8. For second-order multi-agent system (35), if there exists at least one globally reachable node in the undirected and symmetric network topology, and the time-delay 𝜏 satisfies 𝜏<min_{{}𝑘=2,3,...,𝑁}arccos𝜆_𝑘−2𝛾²/𝜆_𝑘𝜆_𝑘−𝛾²,𝛾𝜋2𝜆_𝑘, (38) then system (35) can reach consensus.

4. Numerical Simulations

In this section, we present some numerical simulations to illustrate the consensus of second-order multi-agent systems with time delay. These simulations are given with ten agents, whose initial positions are zero and velocities are given randomly. The network topologies contain at least one globally reachable node in these simulations. Figure 1 describes positions and velocities of ten agents in system (1) with the general topology. From Figure 1, it can be seen that the positions and velocities of all agents can reach consensus as 𝜏=0.5.

(a)

(b)

Figure 1 

The consensus in system (1) with the general topology as 𝜏=0.5.

Figure 2 shows that positions and velocities can converge to the common value in system (1) with symmetrical matrix as 𝜏=0.1.

(a)

(b)

Figure 2 

The consensus in system (1) with the symmetric topology as 𝜏=0.1.

In Figure 3, the consensus can be achieved in system (35) with the general topology as 𝜏=5,𝛾=10. As we can see from Figures 1–3, the consensus of systems can be achieved when the time-delay is an appropriate value. The positions of all agent converge on a fixed value and velocities reach 0.

(a)

(b)

Figure 3 

The consensus in system (35) with the general topology as 𝜏=5,𝛾=10.

Figure 4 presents that the positions of all agents with the general topology can reach consensus as 𝜏=5,𝛾=10, while the positions diverge as 𝜏=10,𝛾=5. From Figure 4, it can be seen that the system can reach consensus under a certain condition. According to the numerical simulations, we can obtain that the topology of network topology, time-delay, and feedback gain all affect consensus of multi-agent system in the simulations.

(a)

(b)

Figure 4 

The position states in system (35) with the symmetric topology as 𝜏=5,𝛾=10 and 𝜏=10,𝛾=5.

5. Conclusion

In this paper, we have introduced the Laplace transform to investigate the consensus of second-order multi-agent system with time-delay in general topology. We have proved that the system can achieve consensus when the network topology contains a globally reachable node as well as when time-delay is less than a critical value which only depends on eigenvalues of the Laplacian matrix of network topology.

Acknowledgments

The authors would like to thank the associate editor and the anonymous reviewer for their valuable comments and suggestions, which significantly contributed to improving the quality of the paper. This work was supported by the National Natural Science Foundation of China under Grants nos. 60774089, 10972003, and 61203150, the Beijing Natural Science Foundation Program (1102016, 4122019), Science and Technology Development Plan Project of Beijing Education Commission (no. KM201310009011); and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR201108055). This work was also supported by the Foundation Grant of Guangxi Key Laboratory of Automobile Components and Vehicle Technology (13-A-03-01), the Opening Project of Guangxi Key Laboratory of Automobile Components and Vehicle Technology (2012KFZD03).