Elsevier

Automatica

Volume 77, March 2017, Pages 180-183
Automatica

Technical communique
New cut-balance conditions in networks of clusters

https://doi.org/10.1016/j.automatica.2016.11.043Get rights and content

Abstract

Existing results in the literature guarantee that the state of multi-agent systems interacting over networks that satisfy the cut-balance assumption asymptotically converges to a constant vector. Furthermore, when the network is persistently connected the agents reach a common value called consensus. Many real large-scale networks are obtained by sparsely connecting subnetworks of densely connected agents. In this context, our objective is to provide new cut-balance assumptions that are adapted to networks of clusters. They are useful for consensus and agreement in clusters in situations when network topology is such that clusters are given or can be easily identified. In this case our new cut-balance assumptions can be checked by realizing a smaller number of operations.

Introduction

The multi-agent framework is widely used to model the dynamics of large numbers of interconnected systems. The most studied problem in this context is the consensus or synchronization of all agents in the network. The convergence to consensus is typically characterized by conditions that depend on the communication graph between agents. Basic results concern fixed undirected topologies but notable advances towards directed and time varying topologies have been provided in Hendrickx and Tsitsiklis (2013), Jadbabaie, Lin, and Morse (2003), Moreau (2005) and Ren and Beard (2005) for discrete time dynamics and (Hendrickx and Tsitsiklis, 2013, Martin and Girard, 2013, Martin and Hendrickx, 2016, Olfati-Saber and Murray, 2004, Ren and Beard, 2005) for continuous time algorithms.

In Hendrickx and Tsitsiklis (2013), the authors introduced the assumption of cut-balance communication which is a general form of communication reciprocity among the agents. Under the cut-balance assumption, convergence is ensured, and consensus may occur in groups or globally. The cut-balance assumption was extended in Martin and Girard (2013) where the authors also provided a convergence rate when global consensus takes place. One drawback of the cut-balance assumption is that it is a global assumption which may be hard to verify when not ensured by design.

A direction to search for a local assumption is to split the agents into clusters. It is reported in the literature that large scale networks often consist of sparsely interconnected clusters of densely coupled agents (Bıyık and Arcak, 2007, Chow and Kokotović, 1985, Morărescu et al., 2016). Different algorithms have been developed to detect the clusters in such networks (Blondel et al., 2008, Morărescu and Girard, 2011, Newman and Girvan, 2004). In the sequel we take advantage of the partition of network in clusters to state new conditions for consensus.

Consequently, the contribution of the present study is that, under stronger assumptions on the interaction graph, we provide a new assumption on reciprocity of communication which can be verified in a local manner. Therefore, we provide conditions for consensus that can be checked by performing a reduced number of operations.

Notation

The following notation will be used throughout the paper. The set of nonnegative integers, real and nonnegative real numbers is denoted by N,R and R+, respectively. A non trivial subset S of a set C, denoted as SC, is a non-empty set with SC.

Section snippets

Problem formulation

Let N{1,,n} be a set of n agents. By abuse of notation we denote both the agent and its index by the same symbol iN. Each agent is characterized by a scalar state xiR,iN that evolves according to the following model ẋi(t)=j=1naij(t)(xj(t)xi(t)),iN where aij(t)0 are measurable functions of time representing the communication weights/interaction strength. Let x(t)=(x1(t),,xn(t))Rn be the overall state of the network collecting the states of all the agents. It is noteworthy that x(t)

Asymptotic behavior: consensus and clustering

In this section, we suppose that we deal with a network partitioned in clusters satisfying the communication pattern introduced by Assumption 1, Assumption 2, Assumption 3. Our main result can be stated as follows.

Proposition 1

Under   Assumption 1, Assumption 2, Assumption 3, the communication weights satisfy   Hypothesis  1   with reciprocity constant K=(KI+ρ+KEmax(ρ,1)).

Notice that the global cut-balance Hypothesis 1 does not guarantee the existence of a non-trivial partition into clusters for which

Conclusion

In this note we investigated consensus in network structured in clusters. Our main assumptions guaranteeing consensus in this framework are adaptations of the global cut-balance assumption. We believe that the conditions we propose are better adapted to the clustered communications case because they are local and consequently their verification requires a smaller number of operations.

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Cited by (1)

  • Lyapunov-based synchronization of networked systems: From continuous-time to hybrid dynamics

    2020, Annual Reviews in Control
    Citation Excerpt :

    In this case, sub-networks are formed due to their strong interconnections. The nodes within the same sub-network tend to synchronize quickly (fast dynamics) before interacting with the other sub-networks (slow dynamics) (Lu, Liu, & Chen, 2010; Martin, Morărescu, & Nes̆ić, 2017; Su et al., 2013; Wu, Zhou, & Chen, 2009). The global hybrid network exhibits a multi-time-scale dynamics (Chow & Kokotovic, 1985; Kokotović, Khalil, & O’reilly, 1999).

The work of S. Martin and I-C. Morărescu was supported by the ANR project COMPACS “Computation Aware Control Systems”, ANR-13-BS03-004, the CNRS PEPS “CONAS” and the CNRS PICS No. 6614 “AICONS”. The work of D. Nešić was funded by the Australian Research Council under the Discovery Project DP1094326. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Carlo Fischione under the direction of Editor André L. Tits.

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