Elsevier

Automatica

Volume 46, Issue 12, December 2010, Pages 1994-1999
Automatica

Brief paper
Leader-following consensus of second-order agents with multiple time-varying delays

https://doi.org/10.1016/j.automatica.2010.08.003Get rights and content

Abstract

In this paper, a leader-following consensus problem of second-order multi-agent systems with fixed and switching topologies as well as non-uniform time-varying delays is considered. For the case of fixed topology, a necessary and sufficient condition is obtained. For the case of switching topology, a sufficient condition is obtained under the assumption that the total period over which the leader is globally reachable is sufficiently large. We not only prove that a consensus is reachable asymptotically but also give an estimation of the convergence rate. An example with simulation is presented to illustrate the theoretical results.

Introduction

In recent years, the problem of coordinating the motion of multiple agents has attracted great attention. This is due to the fact that multi-agent systems can be found in many application areas, such as formation control (Fax & Murray, 2004) and flocking (Jadbabaie et al., 2003, Vicsek et al., 1995). A critical problem for coordinated control is to design appropriate protocols and algorithms such that the group of agents can reach a consensus on the shared information in the presence of limited and unreliable information exchange as well as communication delays (Sun, Wang, & Xie, 2008). The effect of communication time delays on agents’ consensus behavior was first analyzed for a continuous-time model in Saber and Murray (2004). Based on a reduced-order Lyapunov–Krasovskii functional and linear matrix inequalities (LMIs), Lin and Jia (2008) studied the average consensus problem with switching topology and uniform time delays. Seuret, Dimarogonas, and Johansson (2008) investigated the consensus under non-uniform communication constant delays by using Lyapunov–Krasovskii techniques given in terms of LMIs. Moreau (2005) discussed the stability of multi-agent systems with time-dependent communication links.

In this paper, we consider the leader-following consensus problem of a group of second-order dynamic agents with non-uniform multiple time-varying delays as well as fixed and switching topologies. In Saber and Murray (2004), the authors also discussed the consensus problem of multi-agent systems with communication time delays by a frequency domain approach, but the systems are of first order and without a leader, and the time delay is uniform. Moreover, as said in Lin and Jia (2009), the frequency domain approach is limited to the fixed topology case and is invalid when the topologies dynamically change, whereas the Lyapunov-based approach and the passivity-based approach are hard to apply to the case of general directed graphs with time delay and switching topologies. This paper proposes an inequality technique, which can not only prove that the consensus of multi-agent is reachable asymptotically, but also give an estimate of the convergence rate.

Section snippets

Model description

We consider a system consisting of N agents and a leader which is depicted by a graph G¯. It contains N agents (related to an undirected or directed graph G) and a leader (labeled by 0) with directed edges from some agents to the leader. The set of neighbors of vertex i is denoted by Ni={jV:(i,j)E,ji}, where V={1,2,,N} and EV×V, are the sets of vertices and edges of graph G, respectively. (i,j)E means that agent i can directly receive information from agent j; i is the head and j is the

Consensus with fixed topology

This section focuses on the convergence analysis of (6) with fixed interconnection topology. In this case, the subscript σ(t) can be dropped. Rewrite (6) as ε̇=Dε+r=1mArε(tτr(t)), where D=(0N×NIN(D+B)kIN),Ar=(0N×N0N×NAr0N×N).

Lemma 1

LetQ=(0N×NINFkIN).Assume that the control parameter k satisfies k2>(maxμρ(F){|Imμ|})2minμρ(F)Reμ,(minμρ(F)Reμ0), where ρ(F) denotes the set of all eigenvalues of F ; F is an N×N matrix. Then maxθρ(Q)Reθ<0 if and only if F is positive stable (i.e., all eigenvalues

Consensus with switching topology

Consider system (6) with switching graphs {G¯p,pΛ}. The corresponding Laplacian, degree matrix, adjacency matrix and leader adjacency matrix are denoted by Lp, Dp, Ap and Bp, respectively, and Hp=Lp+Bp, pΛ. Next, we consider the case that the node 0 is not always globally reachable in every graph G¯p, pΛ. Let Gs be the set over which the node 0 is globally reachable for any G¯pGs, Gu be the set over which the node 0 is not globally reachable for any G¯pGu, and denote the cardinality of Λ

Example and simulation result

Example 1

Consider a leader-following consensus with the following switching topologies; see Fig. 1.

It is clear that node 0 is not globally reachable in G¯2 and G¯3.

For simplicity, let the activated time be T=10 for each subsystem.

Let τ101=0.1, τ121=0.2, τ211=0.3, τ341=0.4, τ411=0.5; τ102=0.1, τ122=0.2, τ142=0.3, τ342=0.4; τ123=0.1, τ143=0.2, τ213=0.3, τ233=0.4, τ343=0.5, τ413=0.6, τ414=0.1, τ214=0.2, τ314=0.3, τ404=0.4, τ105=0.1, τ235=0.2, τ315=0.4, τ435=0.4, where τijp denotes the time delay from i to j

Conclusion

Based on the inequality technique and algebraic graph theory, the consensus of multi-agent systems with fixed and variable interaction topologies was studied. The non-uniform communication delays between different agents were considered and a new neighbor-based feedback control rule for each agent was designed. Under the assumption that node 0 is globally reachable (for fixed topology) or the total period over which the node 0 is globally reachable is sufficiently large (for switching

Acknowledgements

The authors gratefully acknowledge suggestions and comments by the associate editor and anonymous reviewers.

Wei Zhu received his Ph.D. degree in applied mathematics from Sichuan University in 2007. He is currently a post-doctoral fellow at the Key Laboratory of Systems and Control and an associate professor with Key Laboratory of Network Control & Intelligent Instrument of Ministry of Education and the College of Mathematics and Physics, Chongqing University of Posts and Telecommunications. His research interests include switched systems, multi-agent systems, and stability theory of functional

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Wei Zhu received his Ph.D. degree in applied mathematics from Sichuan University in 2007. He is currently a post-doctoral fellow at the Key Laboratory of Systems and Control and an associate professor with Key Laboratory of Network Control & Intelligent Instrument of Ministry of Education and the College of Mathematics and Physics, Chongqing University of Posts and Telecommunications. His research interests include switched systems, multi-agent systems, and stability theory of functional differential equations.

Daizhan Cheng received his Ph.D. degree from Washington University, St. Louis, in 1985. He is currently a professor with the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chairman of Technical Committee on Control Theory (2003), Chinese Association of Automation, Fellow of IEEE and Fellow of IFAC. His research interests include nonlinear systems, numerical methods, switched systems, and systems biology.

This work is supported partly by the National Natural Science Foundation of China under Grant 60736022, 60821091, 10971240, and the Natural Science Foundation Project of CQ CSTC 2009BB2417, 2008BB2364. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hideaki Ishii under the direction of Editor Ian R. Petersen.

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