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We investigate the engineering scenario where the objective is to synchronizeheterogeneous oscillators in a distributed fashion. The internal dynamics ofeach oscillator are general enough to capture their time-varying naturalfrequency as well as physical couplings and unknown bounded terms. Acommunication layer is set in place to allow the oscillators to exchangesynchronizing coupling actions through a tree-like leaderless network. Inparticular, we present a class of hybrid coupling rules depending only on localinformation to ensure uniform global practical or asymptotic synchronization,which is impossible to obtain by using the Kuramoto model customarily used inthe literature. We further show that the synchronization set can be madeuniformly globally prescribed finite-time stable by selecting the couplingfunction to be discontinuous at the origin. Novel mathematical tools onnon-pathological functions and set-valued Lie derivatives are developed tocarry out the stability analysis. The effectiveness of the approach isillustrated in simulations where we apply our synchronizing hybrid couplingrules to models of power grids previously used in the literature.

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