Couple-group consensus for multi-agent networks of agents with discrete-time second-order dynamics

https://doi.org/10.1016/j.jfranklin.2013.07.004Get rights and content

Abstract

This paper considers the couple-group consensus problem for multi-agent networks with fixed and directed communication topology, where all agents are described by discrete-time second-order dynamics. Consensus protocol is designed such that some agents in a network reach a consistent value, while other agents reach another consistent value. The convergence of the system matrix is discussed based on the tools from matrix theory. An algebraic condition is established to guarantee couple-group consensus. Moreover, for a given communication topology, a theorem is derived on how to select proper control parameters and sampling period for couple-group consensus to be reached. Finally, simulation examples are presented to validate the effectiveness of the theoretical results.

Introduction

Recent years have witnessed the growing interest in cooperative control of multi-agent systems due to its wide applications in synchronization of coupled oscillators, rendezvous in space, and other areas. In these applications, the states of all agents are required to reach an agreement on some consistent value, which is known as the consensus problem.

Up to now, many theoretical results have been achieved on consensus problems for multi-agent systems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. A simple but compelling discrete-time model was proposed in [1], where a local control law was designed for all agents to move towards the same direction. Simulations were given to show that the headings of all agents will reach an agreement even if the neighbor sets of all agents may change as system evolves. By using the Lyapunov theory, consensus criteria were established in [3], under which multi-agent systems with switching topologies and time delays will reach average consensus asymptotically. In [7], the authors investigated the consensus problem for multi-agent systems with external disturbances and model uncertainty, where a model transformation was introduced and consensus conditions were given for multi-agent systems to reach consensus with desired H performance. Consensus protocols were designed and some necessary and/or sufficient conditions were established to guarantee the second-order consensus in [9]. By introducing generalized algebraic connectivity, sufficient conditions were derived in [13] for multi-agent systems with nonlinear dynamics to reach consensus. The second-order consensus problem for discrete-time multi-agent systems was addressed in [15], [16], respectively.

Note that the above references all consider such consensus where the states of all agents converge to the same consensus value. Due to the changes of situations or cooperative tasks, the consensus values may be different for agents from different sub-networks [17], [18], [19]. Group consensus, which includes the aforementioned consensus as a special case, was first introduced in [17] to represent such consensus where the states of all agents in the same sub-network reach the same consistent value while there is no agreement among different sub-networks. Average group consensus problems for multi-agent systems with undirected and balanced topologies were investigated in [17], [18], respectively, where all agents have first-order dynamics. By applying double-tree-form transformation and Lyapunov direct method, the work in [17], [18] was extended to the case of networks with switching topologies and time delays in [19]. Couple-group consensus of high-order multi-agent systems was addressed in [20], where couple-group consensus was guaranteed by the Hurwitz stability of the coefficient matrix.

In this paper, we follow the work of [17], [18], [21] to study the couple-group consensus problem for multi-agent systems with fixed communication topology. Different from [17], [18], [19], all agents considered here have discrete-time second-order dynamics. As a comparison, a necessary and sufficient condition was given in [20] to ensure couple-group consensus of high-order multi-agent systems. However, how to select proper control gains still remains unknown for the consensus conditions in [20] to be satisfied. The group consensus problem for multi-agent systems with continuous-time second-order dynamics was considered in [21]. The main contribution of this paper is as follows. Distributed consensus protocol is designed and some necessary and sufficient conditions are established for multi-agent systems to reach couple-group consensus. Furthermore, for a given communication topology, we provide an efficient way on how to select appropriate control parameters and sampling period for couple-group consensus to be achieved, which shows that couple-group consensus will be reached only if the real parts of the non-zero eigenvalues of the Laplacian matrix are positive.

The rest of this paper is organized as follows. In Section 2, some preliminaries and the problem formulation are introduced. Couple-group consensus protocol is proposed and the eigenvalues as well as the corresponding eigenvectors of the system matrix are analyzed in Section 3. Main results are given in Section 4. Simulation results are presented in Section 5 to demonstrate the effectiveness of the theoretical results. Section 6 concludes this paper with a discussion.

Notation. The following notations will be used throughout the paper. Let In (On) be the n×n unit (zero) matrix; Om×n be the m×n zero matrix and 1n (0n) be the n-dimensional column vector with all elements being 1 (0). The subscript will be omitted if it is clear from the context. Re(λ) and Im(λ) denote the real and imaginary part of a complex number λ, respectively. Cn×n represents the set of all n×n complex matrices. triag{A1,A2,,Am} is used to denote the block lower triangular matrix with diagonal blocks A1 to Am. The superscript T means the transpose of a column vector or a matrix. · refers to the standard Euclidean norm of vectors. |·| denotes the absolute value of a real number or the modulus of a complex number.

Section snippets

Preliminaries

In this subsection, we will review some basic concepts in graph theory and introduce some lemmas which will be used in this paper.

Let G=(V,E,A) be a directed graph with a finite nonempty set of nodes V={v1,v2,,vn}, a set of edges EV×V and a weighted adjacency matrix A. An edge eij=(vj,vi)E means that node vi can receive information from node vj. A=[aij]n×n is defined as aij0 if eijE and aij=0 if eijE. Moreover, aii=0 is assumed for all i. A graph is called undirected if aij=aji. The

Consensus protocol and spectrum analysis for the system matrix

In this section, consensus protocol will be designed and the spectrum of the system matrix will be analyzed.

Main results

In this section, necessary and sufficient conditions will be derived to solve the couple-group consensus problem for multi-agent system (3) with fixed communication topology.

Theorem 1

By applying consensus protocol (2), multi-agent system (1) achieves couple-group consensus asymptotically if and only if Φ has exactly an eigenvalue 1 of multiplicity four and all the other eigenvalues lie inside the unit circle. Furthermore, we have the following results about the final consensus values:|ξi(k)p11Tξ1(0)p

Couple-group consensus of a multi-agent system with undirected topology

Consider a multi-agent system (3) with undirected topology, where L=[4311133011101001102211022].

Numerical computation shows that L has eigenvalues μ1=0,μ2=0,μ3=1.2984,μ4=3 and μ5=7.7016. The states of f(τ,μi)(i=3,4,5) vs. τ are shown in Fig. 1, where α=0.8 and β=1.0. It can be seen from Fig. 1 that f(τ,μi)>0(i=3,4) for any τ and f(0.2943,μ5)=0. From Corollary 2, couple-group consensus will be achieved if we take τ=0.2. Φ has an eigenvalue 1 of multiplicity four and the rest

Conclusion

In this paper, couple-group consensus problem for discrete-time second-order multi-agent systems is investigated for networks with fixed communication topology. Consensus protocol is designed and some necessary and/or sufficient conditions are established to ensure second-order couple-group consensus. It is found that couple-group consensus will be reached only if the non-zero eigenvalues of the Laplacian matrix all have positive real parts. Simulation examples are presented to demonstrate the

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants 61074043 and 61174038, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20113219110026, the 333 Project (BRA2011143), and the Qing Lan Project.

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