Group consensus in multi-agent systems with switching topologies and communication delays

Abstract

Group consensus problems in networks of dynamic agents are addressed for two cases: (i) communication topologies are switching and the switching occurs among finite topologies arbitrarily; (ii) communication topologies are switching and the switching occurs among finite topologies arbitrarily, and there exist communication delays. We introduce double-tree-form transformations under which dynamic equations of agents are transformed into reduced-order systems. Based on the reduced-order systems, we obtain some analysis results for the two cases. In addition, we further investigate multi-group consensus as an extension of the group consensus, and present some analysis results by similar techniques. Simulation results are presented to demonstrate the effectiveness of the theoretical results.

Introduction

As one type of critical problems for cooperative control of multiple agents, consensus problems have been found to possess broad applications in many areas such as computer science, vehicle systems and unmanned air vehicles. This has resulted in tremendous amount of interest in consensus problems of multi-agent systems [1], [2], [3], [4], [5], [6] in recent years. Generally speaking, the main objective in the consensus problems is to design appropriate protocols and algorithms such that a group of agents converges to some consistent value under exchanged information between each other. The consistent value might represent physical quantities such as attitude, position, temperature, voltage and so on.

Vicsek et al. proposed a discrete-time model of autonomous agents [4]. It was assumed that agents move in the plane with the same speed but with different headings. The concept of neighbors of agents was introduced and some simulation results to demonstrate the nearest neighbor rule were presented. Consequently, Jadbabaie et al. provided a theoretical explanation for the observed behavior of the Vicsek model in [7]. It was shown that consensus can be achieved if the union of the interaction graphs for the team are connected frequently enough as the system evolves. Olfati-Saber and Murray established a systematical framework of a consensus problem in continuous-time multi-agent systems with fixed/switching topology and communication time-delays in [8]. Two consensus protocols have been introduced for networks with and without time delays to solve an average-consensus problem with directed graphs, and sufficient and/or necessary algebraic criterions were established based on algebraic graph theories.

In recent years, much interest has been primarily originated from the aforementioned papers. More specifically, in [9], Ren and Beard studied more comprehensive discrete-time consensus scheme which includes the results in [7] as special cases. For discrete-time models, Xiao and Wang discussed a consensus problem in the existence of time delays when agents exchange information between each other in [10]. Second-order dynamics of continuous-time multi-agent systems were addressed in [11], [12]. Moreau introduced a set-valued Lyapunov function to deal with consensus problems for multiple agents with undirected communication links in [13]. A linear matrix inequality (LMI) approach was adopted to study consensus problems in [14], where it was proved that all the agents in a network achieve average consensus asymptotically for appropriate communication delays if the network topology is connected.

Moreover, there has been an increasing attention in extended consensus problems: asynchronous consensus problems were studied for continuous-time multi-agent systems with discontinuous information transmission in [15] and discrete-time multi-agent systems in [6]; finite-time state consensus problems were investigated for continuous-time multi-agent systems by using a finite-time Lyapunov function method [5]; stochastic consensus problems were studied in [16], [17]. For details, please refer to survey papers [18], [19] and the references therein.

Note that all the aforementioned results were concerned with such consensus that all the agents in a network share a common value. In cooperative control, for cooperative control strategies to be effective, agents need to reach consensus on a shared data. A group of agents must be able to respond to unanticipated situations or any changes when a cooperative task is carried out. This might result in that the agreements are different with the changes of environments, situations, cooperative tasks or even time. Motivated by the analysis above, we study a group consensus problem. It aims to design appropriate protocols and algorithms such that agents in a network reach more than one consistent state, i.e., to find some appropriate control inputs such that some agents in a network reach a consistent state, while others reach other consistent states. In contrast to the networks studied in the aforementioned results, the group consensus problem concerns a network which is divided into multiple sub-networks, and information exchange exists not only two agents in a group but also in different groups. This is more suitable for complex practical applications, since many applicable networks tend to be complex.

The motivation of this work is to extend the consensus in the existing results to more general consensus–group consensus. The work mainly builds on [20], [21]. As a comparison, [20] studied a consensus problem for continuous-time multi-agent systems in undirected networks with switching topologies and time-varying delays. Some conditions guaranteeing all the agents reaching a consistent state were established under the assumption that each topology graph has a spanning tree. In addition, [21] solved a group average-consensus problem with undirected graphs, and it required the interaction between any two sub-graphs to be balanced. [21] only discussed a group average-consensus problem for networks with fixed topologies, and provided sufficient and/or necessary conditions for agents achieving group average consensus. In practical applications, the interaction topology between agents may change dynamically, which has resulted in a tremendous amount of interest in switching topologies such as [9], [8], [20] and so on. In this paper, we study a group consensus problem for a multi-agent network with time-varying topologies. To solve the group consensus problem, we introduce a double-tree-form transformation under which the dynamic equation of agents is transformed into a reduced-order system. Some necessary and/or sufficient conditions are presented for the agents achieving the group consensus.

The paper is organized as follows. Section 2 contains the problem formulation as well as some definitions. Section 3 is the main results. Simulation results are presented in Section 4. The conclusion is given in Section 5.

Notations. Throughout this paper, the following notations are used: the superscripts “T” and “$-1$” stand for matrix transposition and matrix inverse, respectively; $1={[1\dots 1]}^T$ with proper dimension and $1_n={[1\dots 1]}^T\in R^n$; $0$ represents any zero matrix with an appropriate dimension; $diag\{M_1,M_2,\dots ,M_n\}$ denotes a block diagonal matrix whose diagonal blocks are given by $M_1,M_2,\dots ,M_n$; in symmetric block matrices, “$\ast$” represents an ellipsis for the term introduced by symmetry.