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

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Document pages: 18 pages

Abstract: We study the problem of robust global synchronization of pulse-coupledoscillators (PCOs) over directed graphs. It is known that when the digraphs arestrongly connected, global synchronization can be achieved by using a class ofdeterministic set-valued reset controllers. However, for large-scale networks,these algorithms are not scalable because some of their tuning parameters haveupper bounds of the order of O(1 N), where N is the number of agents. Thispaper resolves this scalability issue by presenting several new results in thecontext of global synchronization of PCOs with more general network topologiesusing deterministic and stochastic hybrid dynamical systems. First, weestablish that similar deterministic resetting algorithms can achieve robust,global, and fixed-time synchronization in any rooted acyclic digraph. Moreover,in this case we show that the synchronization dynamics are now scalable as thetuning parameters of the algorithm are network independent, i.e., of orderO(1). However, the algorithms cannot be further extended to all rooteddigraphs. We establish this new impossibility result by introducing acounterexample with a particular rooted digraph for which globalsynchronization cannot be achieved, irrespective of the tuning of the resetrule. Nevertheless, we show that if the resetting algorithms are modified byaccommodating an Erdos-Renyi type random graph model, then the resultingstochastic resetting dynamics will guarantee global synchronization almostsurely for all rooted digraphs and, moreover, the tunable parameters of thedynamics are network independent. Stability and robustness properties of theresetting algorithms are studied using the tools from set-valued hybriddynamical systems. Numerical simulations are provided at the end of the paperfor demonstration of the main results.

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