Global Synchronization of Clocks in Directed Rooted Acyclic Graphs: A Hybrid Systems Approach

Abstract

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 toward the end of the paper.

Scalable Resetting Algorithms for Synchronization of Pulse-Coupled Oscillators over Rooted Directed Graphs

“…which, by the first inequality in (17), triggers the jumps τ + 1 = 0, τ + 2 = τ 2 , and τ + 3 = 0. The system will now flow for τ 3,0 + (1 − τ 1,0 ) seconds, until the following conditions hold:…”

“…which, by the third condition in (17), trigger jumps τ + 1 = 0, τ + 2 = 0 and τ + 3 = τ 3 . Using (18) we can observer that this is the exact same state obtained after (19).…”

“…Furthermore, we provide theoretical bounds for the stochastic synchronization time of our algorithms, as functions of the structures of the underlying digraphs. Our preliminary results, reported in the conference paper [17], considered only deterministic algorithms and presented results only for rooted acyclic digraphs, a subclass of the digraphs considered in this paper. Analysis and proofs of the results were also omitted in the conference version.…”