Hybrid coupling rules for leaderless heterogeneous oscillators: uniform global asymptotic and finite-time synchronization

TL;DR

This paper introduces hybrid coupling rules for leaderless heterogeneous oscillators that achieve uniform global synchronization, including finite-time stability, using local information and novel mathematical tools.

Contribution

It proposes a new class of hybrid coupling rules that enable synchronization in heterogeneous oscillators, surpassing limitations of traditional models like Kuramoto.

Findings

Achieves uniform global practical and asymptotic synchronization.

Enables finite-time stability with discontinuous coupling functions.

Demonstrates effectiveness through power grid simulations.

Abstract

We investigate the engineering scenario where the objective is to synchronize heterogeneous oscillators in a distributed fashion. The internal dynamics of each oscillator are general enough to capture their time-varying natural frequency as well as physical couplings and unknown bounded terms. A communication layer is set in place to allow the oscillators to exchange synchronizing coupling actions through a tree-like leaderless network. In particular, we present a class of hybrid coupling rules depending only on local information to ensure uniform global practical or asymptotic synchronization, which is impossible to obtain by using the Kuramoto model customarily used in the literature. We further show that the synchronization set can be made uniformly globally prescribed finite-time stable by selecting the coupling function to be discontinuous at the origin. Novel mathematical tools on non-pathological functions and set-valued Lie derivatives are developed to carry out the stability analysis. The effectiveness of the approach is illustrated in simulations where we apply our synchronizing hybrid coupling rules to models of power grids previously used in the literature.