Couple-group consensus for multi-agent networks of agents with discrete-time second-order dynamics
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 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; be the zero matrix and () be the n-dimensional column vector with all elements being 1 (0). The subscript will be omitted if it is clear from the context. and denote the real and imaginary part of a complex number , respectively. represents the set of all n×n complex matrices. 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 be a directed graph with a finite nonempty set of nodes , a set of edges and a weighted adjacency matrix A. An edge means that node vi can receive information from node vj. is defined as if and if . Moreover, is assumed for all i. A graph is called undirected if . 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 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:
Couple-group consensus of a multi-agent system with undirected topology
Consider a multi-agent system (3) with undirected topology, where
Numerical computation shows that L has eigenvalues and . The states of vs. are shown in Fig. 1, where and . It can be seen from Fig. 1 that for any and . From Corollary 2, couple-group consensus will be achieved if we take . 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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2022, NeurocomputingCitation Excerpt :In [17], Feng et al. discovered that group consensus would be reached only if the nonzero eigenvalues of the Laplacian matrix locate in the right half plane for discrete-time second-order MASs. And then, the results in [17] were extended to discrete-time heterogenous MASs (DHMASs) governed by first- and second-order dynamics, two kinds of distributed group consensus algorithms in [18]. The group consensus problem was studied for second-order DHMASs with a fixed topology and stochastic switching topologies in [19].
Distributed coordination on state-dependent fuzzy graphs
2021, Journal of the Franklin InstituteAdvanced Distributed Consensus for Multiagent Systems
2020, Advanced Distributed Consensus for Multiagent SystemsCouple-group consensus conditions for general first-order multiagent systems with communication delays
2018, Systems and Control LettersConsensus of second-order multi-agent systems with nonlinear dynamics via edge-based distributed adaptive protocols
2016, Journal of the Franklin InstituteCitation Excerpt :Distributed leader-following consensus for second-order multi-agent systems with nonlinear inherent dynamics was investigated in [30]. Couple-group consensus problem for multi-agent networks with fixed and directed communication topology was investigated in [31]. Moreover, in order to achieve consensus of second-order multi-agent systems with a time-delayed and directed communication topology, a necessary and sufficient condition was obtained [32].