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Global Synchronization of Clocks in Directed Rooted Acyclic Graphs: A Hybrid Systems Approach
In this paper, we study the problem of robust global synchronization of
resetting clocks in multi-agent networked systems, where by robust global
synchronization we mean synchronization that is insensitive to arbitrarily
small disturbances, and which is achieved from all initial conditions. In
particular, we aim to address the following question: Given a set of
homogeneous agents with periodic clocks sharing the same parameters, what kind
of information flow topologies will guarantee that the resulting networked
systems can achieve robust global synchronization? To address this question, we
rely on the framework of robust hybrid dynamical systems and a class of
distributed hybrid resetting algorithms. Using the hybrid-system approach, we
provide a partial solution to the question: Specifically, we show that one can
achieve robust global synchronization with no purely discrete-time solutions in
any networked system whose underlying information flow topology is a rooted
acyclic digraph. Such a result is complementary to the existing result [1] in
which strongly connected digraphs are considered as the underlying information
flow topologies of the networked systems. We have further computed in the paper
the convergence time for a networked system to reach global synchronization. In
particular, the computation reveals the relationship between convergence time
and the structure of the underlying digraph. We illustrate our theoretical
findings via numerical simulations towards the end of the paper.Comment: Updated version is available at: arXiv:2006.1185