Consensus of the Multiagent System with a Dynamic Leader Based on Directed Topology Using Laplace Transform

Abstract

The consensus of the multiagent system with directed topology and a leader is investigated, in which the leader is dynamic. Based on Laplace transform method, the accurate upper error bound between the leader and the followers can be obtained. It is also proved that all agents of the system will aggregate and eventually form a cohesive cluster following the leader if the leader is globally reachable. Finally, some simulation examples are given to illustrate the theoretical results.

1. Introduction

In recent years, the consensus problem of multiagent systems has become a hot topic due to their broad applications, such as cooperative unmanned air vehicles, automated highway systems, air traffic control, and autonomous underwater vehicles [1–22].

Various algorithms and models about multiagent systems have been discussed based on different tasks or interests. The leader-following system is one of the most interesting topics in the motion control of the multiagent systems. Vicsek et al. [1] proposed a simple model about autonomous agents moving with a constant identical speed and tending to the average direction of its neighbors. It is demonstrated numerically that all the agents will move in the same direction at the same speed eventually. Jadbabaie et al. [2] gave the theoretical explanation for the numerical results of the Vicsek model by algebraic graph theory. Based on the Vicsek model, some researchers designed the leader-following model, where the leader agent can be regarded as the control input used to control the other agents in such system. Shi et al. [5] regarded the reference signal as a virtual leader for guiding the agent group to move at the desired constant velocity. Hong et al. [7] discussed the leader-following system with variable coupling topology. Olfati-Saber [3] used virtual leaders to accomplish obstacle avoidance.

The coupling topology plays an important role in the studies of multiagent systems. Because of the complexity in the consensus analysis with directed topology, most researchers focus on the undirected topology or balance topology to simplify the problem [7–22]. In this paper, we will discuss the consensus problem with directed topology and give a weak condition for reaching network consensus. According to Laplace transform, the accurate upper error bound between the leader and followers can be calculated and some assumptions needed in Lyapunov approach can be simplified. Finally, some simulation examples are provided to illustrate the theoretical results.

This paper is organized as follows. Section 2 proposes the model formulation and some preliminaries of graph theory. Section 3 gives the analysis of the convergence of the model. Section 4 gives simulations of the theoretical results. Finally, the main contribution of this paper is summarized in Section 5.

2. Model and Preliminaries

2.1. Model

Consider a multiagent system of 𝑛+1 agents, where the leader is labeled with 0 and the followers are labeled with 1,2,…,𝑛. The motion of the system is described by 𝑥̇_𝑖=−_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗𝑥_𝑖−𝑥_𝑗+𝑏_𝑖𝑥₀−𝑥_𝑖𝑥̇,𝑖=1,2,…,𝑛,₀=𝑣(𝑡), (1) where 𝑥_𝑖∈ℝ^𝑚 represents the state of agent 𝑖; 𝒩_𝑖 is the neighbor set of agent 𝑖. 𝐴=[𝑎_𝑖𝑗]∈ℝ^𝑛×𝑛 is the coupling matrix, where 𝑎_𝑖𝑗 is the weight parameter with 𝑎_𝑖𝑗≥0 and 𝑎_𝑖𝑖=0. 𝑏_𝑖(𝑥₀−𝑥_𝑖) is the term performing attractive effect from leader agent 0 to agent 𝑖. Define 𝐵=diag{𝑏₁,𝑏₂,…,𝑏_𝑛}∈ℝ^𝑛×𝑛. 𝑏_𝑖>0 means agent 𝑖 can measure the information of the leader with strength 𝑏_𝑖, while 𝑏_𝑖=0 means there is no information from the leader to agent 𝑖. 𝑣(𝑡) is the velocity of the leader, which is a bounded function.

2.2. Preliminaries

To discuss the coordinated control among the agents, graph theory is a very effective tool. Regard the agent as a node and the connection link between any two agents as an edge; the coupling topology is conveniently described by a directed graph. Let 𝒢=(𝒱,ℰ,𝐴) be a weighted digraph of order 𝑛 with the set of nodes 𝒱={1,2,3,…,𝑛} and set of arcs ℰ⊆𝒱×𝒱. 𝐴=[𝑎_𝑖𝑗] is adjacency matrix of graph 𝒢. An arc of 𝒢 is denoted by (𝑖,𝑗), which starts from 𝑖 and ends at 𝑗. The set of neighbors of node 𝑖 is denoted by 𝓝_𝑖={𝑗∈𝒱∶(𝑖,𝑗)∈ℰ}. A digraph 𝒢 is strongly connected if there exists a path between any two distinct nodes. For a node 𝑗, if there exists at least a path from every other node 𝑖 in 𝒢 to node 𝑗, we say that node 𝑗 is globally reachable.

A diagonal matrix 𝐷=diag{𝑑₁,𝑑₂,…,𝑑_𝑛}∈ℝ^𝑛×𝑛 is a degree matrix of 𝒢 with its diagonal elements 𝑑_𝑖=∑_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗, 𝑖=1,2,…,𝑛. Then the Laplacian of the weighted digraph 𝒢(or matrix 𝐴) is defined as 𝐿=𝐷−𝐴∈ℝ^𝑛×𝑛. (2) In order to study the leader-following problem, a multiagent system with directed topology is considered, which is consisting of 𝑛 agents and one leader. In 𝒢, if there exists at least one path from every node in 𝒢 to the leader, we say that the leader is globally reachable in 𝒢.

It is easy to see that 𝐿 has a zero eigenvalue corresponding to the right eigenvector 1 =(1,1,…,1)^𝑇.

Lemma 1 (see [9]). A digraph 𝒢 has a globally reachable node if and only if 0 is a simple eigenvalue of 𝐿 (i.e., rank(𝐿)=𝑛−1).

Lemma 2. The nonzero eigenvalues of 𝐿 have positive real parts.

Proof. The proof is similar to that of [3]; it is omitted here.

Lemma 3. The eigenvalues of 𝐿+𝐵 have positive real parts if and only if the leader is a globally reachable node in digraph 𝒢.

Proof. If node 0 is globally reachable, the Laplacian matrix of 𝒢 can be written as 𝑙𝐿=000…0₁₀𝑙₁₁𝑙₁₂…𝑙_1𝑛𝑙₂₀𝑙₂₁𝑙₂₂…𝑙_2𝑛𝑙⋮⋮⋮⋱⋮_𝑛0𝑙_𝑛1𝑙_𝑛2…𝑙_𝑛𝑛∈ℝ^{(𝑛+1)×(𝑛+1)}. (3) According to Lemma 1, then rank(𝐿)=𝑛.
Notice that the block matrix of 𝐿 is 𝑙₁₁𝑙₁₂…𝑙_1𝑛𝑙₂₁𝑙₂₂…𝑙_2𝑛𝑙⋮⋮⋱⋮_𝑛1𝑙_𝑛2…𝑙_𝑛𝑛=𝐿+𝐵∈ℝ^𝑛×𝑛. (4) Then, we can have rank(𝐿+𝐵)=𝑛, where 𝐿 is the Laplacian matrix of 𝒢.

3. Coordinated Control Analysis

In this section, we focus on the coordinated control problem of model (1). Denote 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇, 𝑋₀=(𝑥₀,𝑥₀,…,𝑥₀)^𝑇, and 𝑉=(𝑣,𝑣,…,𝑣)^𝑇= 1𝑣; then the model can be rewritten into matrix form 𝑋̇=−𝐿𝑋−𝐵𝑋−𝑋₀. (5) Based on system (5), we can derive the following theorem.

Theorem 4. If leader 0 is globally reachable in digraph 𝒢, all the agents described by (1) will converge and form a cohesive cluster following the leader asymptotically. Moreover, the errors between the leader and followers will be included in a fixed bound.

Proof. To solve this problem, we introduce the Laplace transform, succinctly denoted by ℒ(⋅) in this paper. System (5) can be transformed as 𝑋̇−𝑋̇₀=−(𝐿+𝐵)𝑋−𝑋₀−𝑉. (6) Define 𝑋−𝑋₀=𝑌 as the error vector; then (6) can be written as 𝑌̇=−(𝐿+𝐵)𝑌−𝑉. (7) Applying ℒ to (7), then we have ℒ𝑌̇=ℒ(−(𝐿+𝐵)𝑌−𝑉). (8) That is, 𝑠𝐼_𝑛+(𝐿+𝐵)ℒ(𝑌)=𝑌(0)−ℒ(𝑉). (9) For simplicity, denote ℒ(𝑌)=𝒴(𝑠) and ℒ(𝑉)=𝒱(𝑠), where 𝑠 is the Laplace variable; then 𝑠𝐼_𝑛+(𝐿+𝐵)𝒴(𝑠)=𝑌(0)−𝒱(𝑠). (10) According to Cramer rule, we can get solutions of (10): 𝒴_𝑖(𝑠)=Δ𝑠𝐼_𝑛+(𝐿+𝐵)^𝑖Δ𝑠𝐼_𝑛+(𝐿+𝐵),𝑖=1,2,…,𝑛, (11) where 𝐼_𝑛 is the identity matrix of order 𝑛; Δ[𝑠𝐼_𝑛+(𝐿+𝐵)]^𝑖 is the determinant of matrix 𝑠𝐼_𝑛+(𝐿+𝐵) in which the 𝑖th column has been replaced by 𝑌(0)−𝒱(𝑠); and Δ[𝑠𝐼_𝑛+(𝐿+𝐵)] is the determinant of matrix 𝑠𝐼_𝑛+(𝐿+𝐵). We can separate (11) into two parts as follows: 𝒴_𝑖(𝑠)=Δ𝑠𝐼_𝑛+(𝐿+𝐵)^𝑖₁Δ𝑠𝐼_𝑛+(−Δ𝐿+𝐵)𝑠𝐼_𝑛+(𝐿+𝐵)^𝑖₂Δ𝑠𝐼_𝑛+(,𝐿+𝐵) (12) where Δ[𝑠𝐼_𝑛+(𝐿+𝐵)]^𝑖₁ is the determinant of matrix 𝑠𝐼_𝑛+(𝐿+𝐵) in which the 𝑖th column has been replaced by 𝑌(0); Δ[𝑠𝐼_𝑛+(𝐿+𝐵)]^𝑖₂ is the determinant of matrix 𝑠𝐼_𝑛+(𝐿+𝐵) in which the 𝑖th column has been replaced by 𝒱(𝑠). Because 𝒱(𝑠)=ℒ[𝑣(𝑡)]1, determinant Δ[𝑠𝐼_𝑛+(𝐿+𝐵)]^𝑖₂ can be transformed as Δ𝑠𝐼_𝑛+(𝐿+𝐵)^𝑖₂=ℒ[𝑣(𝑡)]⋅Δ𝑠𝐼_𝑛+(𝐿+𝐵)^𝑖₃, (13) where Δ[𝑠𝐼_𝑛+(𝐿+𝐵)]^𝑖₃ is the determinant of matrix 𝑠𝐼_𝑛+(𝐿+𝐵) in which the 𝑖th column has been replaced by 1. Since Δ[𝑠𝐼_𝑛+(𝐿+𝐵)]^𝑖₁, Δ[𝑠𝐼_𝑛+(𝐿+𝐵)]^𝑖₃, and Δ[𝑠𝐼_𝑛+(𝐿+𝐵)] are polynomials about 𝑠, for simplicity, we denote 𝑃_𝑖1(𝑠)=Δ(𝑠𝐼+𝐿)^𝑖1, 𝑃_𝑖3(𝑠)=Δ[𝑠𝐼_𝑛+(𝐿+𝐵)]^𝑖₃, and 𝑄(𝑠)=Δ(𝑠𝐼+𝐿), respectively. Then (12) can be rewritten as 𝒴_𝑖(𝑠)=Δ𝑠𝐼_𝑛+(𝐿+𝐵)^𝑖₁Δ𝑠𝐼_𝑛+(−ℒ[𝑣(𝑡Δ𝐿+𝐵))]𝑠𝐼_𝑛+(𝐿+𝐵)^𝑖₃Δ𝑠𝐼_𝑛+(=𝑃𝐿+𝐵)_𝑖1(𝑠)𝑄(𝑠)−ℒ[𝑣(𝑡)]𝑃_𝑖3(𝑠)𝑄(𝑠). (14) According to Lemma 3, the eigenvalues of 𝐿+𝐵 have positive real parts, denoted as 𝜆₁,𝜆₂,…,𝜆_𝑗, (15) with multiplicity as 𝑝,𝑞,…,𝑟. So we have Δ(𝑠𝐼+𝐿+𝐵)=𝑠+𝜆₁^𝑝𝑠+𝜆₂^𝑞⋯𝑠+𝜆_𝑗^𝑟. (16)
If the fractional expressions in (14) are reducible, we should make reduction of the fraction expressions. In the following discussion, we assume that the fractional expressions in (14) are irreducible. Applying Heaviside’s Method, (14) can be expanded into 𝒴_𝑖(𝑠)=𝑐^𝑖_1𝑝𝑠+𝜆₁^𝑝+𝑐^𝑖_1(𝑝−1)𝑠+𝜆₁^(𝑝−1)𝑐+⋯+^𝑖11𝑠+𝜆₁+𝑐^𝑖_2𝑞𝑠+𝜆₂^𝑞+𝑐^𝑖_2(𝑞−1)𝑠+𝜆₂^(𝑞−1)𝑐+⋯+^𝑖21𝑠+𝜆₂𝑐+⋯+^𝑖𝑗𝑟𝑠+𝜆_𝑗^𝑟+𝑐^𝑖_{𝑗(𝑟−1)}𝑠+𝜆_𝑗^(𝑟−1)𝑐+⋯+^𝑖𝑗1𝑠+𝜆_𝑗[𝑣(𝑡×𝑑−ℒ)]_𝑖1𝑝𝑠+𝜆₁^𝑝+𝑑^𝑖_1(𝑝−1)𝑠+𝜆₁^(𝑝−1)𝑑+⋯+^𝑖11𝑠+𝜆₁+𝑑^𝑖_2𝑞𝑠+𝜆₂^𝑞+𝑑^𝑖_2(𝑞−1)𝑠+𝜆₂^(𝑞−1)+𝑑+⋯_𝑖21𝑠+𝜆₂𝑑+⋯+^𝑖𝑗𝑟𝑠+𝜆_𝑗^𝑟+𝑑^𝑖_{𝑗(𝑟−1)}𝑠+𝜆_𝑗^(𝑟−1)𝑑+⋯+^𝑖𝑗1𝑠+𝜆_𝑗, (17) where 𝑐^𝑖_1𝑘=𝑑^𝑝−𝑘𝑑𝑠^𝑝−𝑘𝑠+𝜆₁^{(𝑝−𝑘)}𝑃_𝑖1(𝑠)𝑄(𝑠)_𝑠=𝜆1𝑐,𝑘=1,2,…,𝑝,_𝑖2𝑘=𝑑^𝑞−𝑘𝑑𝑠^𝑞−𝑘𝑠+𝜆₂^{(𝑞−𝑘)}𝑃_𝑖1(𝑠)𝑄(𝑠)_𝑠=𝜆2⋮𝑐,𝑘=1,2,…,𝑞,_𝑖𝑗𝑘=𝑑^𝑟−𝑘𝑑𝑠^𝑟−𝑘𝑠+𝜆_𝑗^{(𝑟−𝑘)}𝑃_𝑖1(𝑠)𝑄(𝑠)_{𝑠=𝜆𝑗}𝑑,𝑘=1,2,…,𝑟,_𝑖1𝑘=𝑑^𝑝−𝑘𝑑𝑠^𝑝−𝑘𝑠+𝜆₁^{(𝑝−𝑘)}𝑃_𝑖3(𝑠)𝑄(𝑠)_𝑠=𝜆1𝑑,𝑘=1,2,…,𝑝,_𝑖2𝑘=𝑑^𝑞−𝑘𝑑𝑠^𝑞−𝑘𝑠+𝜆₂^{(𝑞−𝑘)}𝑃_𝑖3(𝑠)𝑄(𝑠)_𝑠=𝜆2⋮𝑑,𝑘=1,2,…,𝑞,_𝑖𝑗𝑘=𝑑^𝑟−𝑘𝑑𝑠^𝑟−𝑘𝑠+𝜆_𝑗^{(𝑟−𝑘)}𝑃_𝑖3(𝑠)𝑄(𝑠)_{𝑠=𝜆𝑗},𝑘=1,2,…,𝑟. (18) Applying Inverse Laplace transform to (17), we obtain 𝑦_𝑖(𝑡)=𝑐^𝑖₁₁+𝑐^𝑖₁₂𝑐1!𝑡+⋯+^𝑖1𝑝!𝑡𝑝−1^(𝑝−1)𝑒^{(−𝜆1𝑡)}+𝑐^𝑖₂₁+𝑐^𝑖₂₂𝑐1!𝑡+⋯+^𝑖2𝑞!𝑡𝑞−1^(𝑞−1)𝑒^{(−𝜆2𝑡)}+𝑐+⋯_𝑖𝑗1+𝑐^𝑖_𝑗2𝑐1!𝑡+⋯+^𝑖𝑗𝑟()!𝑡𝑟−1^(𝑟−1)𝑒^{(−𝜆𝑗𝑡)}+^𝑡₀𝑑^𝑖₁₁+𝑑^𝑖₁₂𝑑1!𝜉+⋯+^𝑖1𝑝!𝜉𝑝−1^(𝑝−1)𝑒^{(−𝜆1𝜉)}+𝑑^𝑖₂₁+𝑑^𝑖₂₂𝑑1!𝜉+⋯+^𝑖2𝑞!𝜉𝑞−1^(𝑞−1)×𝑒^{(−𝜆2𝜉)}+𝑑+⋯_𝑖𝑗1+𝑑^𝑖_𝑗2𝑑1!𝜉+⋯+^𝑖𝑗𝑟()!𝜉𝑟−1^(𝑟−1)×𝑒^{(−𝜆𝑗𝜉)}𝑣(𝑡−𝜉)𝑑𝜉. (19) According to integral property, it is easy to have 𝑦_𝑖(𝑡)≤𝑐^𝑖₁₁+𝑐^𝑖₁₂𝑐1!𝑡+⋯+^𝑖1𝑝!𝑡𝑝−1^(𝑝−1)𝑒^{(−𝜆1𝑡)}+𝑐^𝑖₂₁+𝑐^𝑖₂₂𝑐1!𝑡+⋯+^𝑖2𝑞!𝑡𝑞−1^(𝑞−1)𝑒^{(−𝜆2𝑡)}+𝑐+⋯_𝑖𝑗1+𝑐^𝑖_𝑗2𝑐1!𝑡+⋯+^𝑖𝑗𝑟()!𝑡𝑟−1^(𝑟−1)𝑒^{(−𝜆𝑗𝑡)}+|𝑣(𝑡×)|_𝑡0𝑑^𝑖₁₁+𝑑^𝑖₁₂𝑑1!𝜉+⋯+^𝑖1𝑝!𝜉𝑝−1^(𝑝−1)𝑒^{(−𝜆1𝜉)}+𝑑^𝑖₂₁+𝑑^𝑖₂₂𝑑1!𝜉+⋯+^𝑖2𝑞!𝜉𝑞−1^(𝑞−1)×𝑒^{(−𝜆2𝜉)}+𝑑+⋯_𝑖𝑗1+𝑑^𝑖_𝑗2𝑑1!𝜉+⋯+^𝑖𝑗𝑟()!𝜉𝑟−1^(𝑟−1)×𝑒^{(−𝜆𝑗𝜉)}𝑑𝜉. (20) Since as 𝑡→+∞, we have 𝑒^{(−𝜆𝑡)}→0, then we can get the following inequality: 𝑦_𝑖(𝑡𝑑)≤𝜀+𝐻^𝑖11𝜆₁+𝑑^𝑖₁₂𝜆²₁𝑑+⋯+^𝑖1𝑝𝜆^𝑝₁+𝑑^𝑖₂₁𝜆₂+𝑑^𝑖₂₂𝜆²₂𝑑+⋯+^𝑖2𝑞𝜆^𝑞₂𝑑+⋯+^𝑖𝑗1𝜆_𝑗+𝑑^𝑖_𝑗2𝜆²_𝑗𝑑+⋯+^𝑖𝑗𝑟𝜆^𝑟_𝑗=𝜀+𝐻𝑀_𝑖, (21) as 𝑡→+∞ where 𝜀 is a function close to 0, 𝐻 is the upper bound of 𝑣(𝑡), and 𝑀_𝑖=|(𝑑^𝑖₁₁/𝜆₁)+(𝑑^𝑖₁₂/𝜆²₁)+⋯+(𝑑^𝑖_1𝑝/𝜆^𝑝₁)+(𝑑^𝑖₂₁/𝜆₂)+(𝑑^𝑖₂₂/𝜆²₂)+⋯+(𝑑^𝑖_2𝑞/𝜆^𝑞₂)+⋯+(𝑑^𝑖_𝑗1/𝜆_𝑗)+(𝑑^𝑖_𝑗2/𝜆²_𝑗)+⋯+(𝑑^𝑖_𝑗𝑟/𝜆^𝑟_𝑗)|. It shows that the upper bound of 𝑦_𝑖(𝑡) is influenced by 𝑣(𝑡) and the coupling matrix 𝐴. Define 𝑀=max_{𝑖=1,2,…,𝑛}{𝑀_𝑖} and choose 𝜀=0; we have 𝑥_𝑖(𝑡)−𝑥₀(𝑡)=𝑦_𝑖(𝑡)≤𝐻𝑀, (22) as 𝑡→+∞. Then, any agent of the system will enter into a region with the leader as the center bounded by 𝐻𝑀 within sufficient time.
This completes the proof.

Remark 1. In Theorem 4, the error between the leader 0 and any agent is bounded. If the velocity of leader is zero, we can get that the error will tend to zero, which implies that all the agents will asymptotically approach the same states of the leader. If the velocity of the leader is nonzero and bounded, it is easy to see that the states and velocities of all the followers will keep in a bounded region following the leader.

4. Numerical Simulations

In order to verify the above theoretical analysis, we present some numerical simulations to illustrate the systems. These simulations are performed with ten followers and one leader, and the initial positions of the agents are chosen randomly. The coupling matrix 𝐴 is given based on certain conditions.

Figures 1, 2, and 3 show the errors between leader and followers about system (1), under different coupling topology and control. It is easy to see that the states of the error are contained in a bounded region. Figures 4 and 5 depict the trajectories of the agents in 2-dimensional Euclidian space. If leader agent in the coupling topology is globally reachable, then the followers in the system will reach synchronization with the leader within sufficient time. Figures 6 and 7 show the trajectories of the agents in 3-dimensional Euclidian space, in which the states of the followers are influenced by the state of leader, and all agents in the system can achieve consensus eventually.

Figure 1 

Errors between leader and followers about system (1), in which 0 is globally reachable in 𝒢 and 𝑣(𝑡)=1.3.

Figure 2 

Errors between leader and followers about system (1), in which 0 is globally reachable in 𝒢 and 𝑣(𝑡)=1.5+2sin(4𝑡).

Figure 3 

Errors between leader and followers about system (1), in which 0 is globally reachable in 𝒢 and 𝑣(𝑡)=0.7.

Figure 4 

Trajectories of the agents about system (1), in which 0 is globally reachable in 𝒢 and 𝑣(𝑡)=0.7+sin(𝑡).

Figure 5 

Trajectories of the agents about system (1), in which 0 is globally reachable in 𝒢 and 𝑣(𝑡)=0.7.

Figure 6 

Trajectories of the agents about system (1), in which 0 is globally reachable in 𝒢.

Figure 7 

Trajectories of the agents about system (1), in which 0 is globally reachable in 𝒢.

5. Conclusion

In this paper, we have investigated the coordinated control of leader-following multiagent systems. It is proved that the agents of the systems will aggregate and form a cluster following the leader asymptotically. Meanwhile, we have studied the coupling topology among the agents in the general case. The systems considered in this paper can better reflect the collective behavior in practice. It is clear that the ideas and approaches about graph theory and linear dynamical system theory will play an important role in the analysis of the multiagent system. Finally, simulations give an effective demonstration of the leader-following systems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 61304049, the Science and Technology Development Plan Project of Beijing Education Commission (no. KM201310009011), and the Plan training project of excellent young teacher of North China University of Technology.