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 can be achieved from all initial conditions and is insensitive to small perturbations. In particular, we address the following question: Given a set of homogeneous agents with periodic clocks, what kind of information flow topologies will guarantee that the resulting networked systems can achieve robust global synchronization? To address the question, we rely on the use of robust hybrid dynamical systems. Using the hybrid-system approach, we provide a partial solution to the question: Specifically, we show that one can achieve robust global synchronization if the underlying information flow topology is a rooted acyclic digraph. Such a result is complementary to the existing results in [1] and [2] by Poveda & Teel, in which strongly connected digraphs are considered as the underlying information flow topologies of the networked systems. We have further computed an upper bound on 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 toward the end of the paper.