Technical communiqueNew cut-balance conditions in networks of clusters☆
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 and , respectively. A non trivial subset of a set , denoted as , is a non-empty set with .
Section snippets
Problem formulation
Let be a set of agents. By abuse of notation we denote both the agent and its index by the same symbol . Each agent is characterized by a scalar state that evolves according to the following model where are measurable functions of time representing the communication weights/interaction strength. Let be the overall state of the network collecting the states of all the agents. It is noteworthy that
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 .
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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Lyapunov-based synchronization of networked systems: From continuous-time to hybrid dynamics
2020, Annual Reviews in ControlCitation 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).
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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.