Adaptive Hybrid Control for Robust Global Phase Synchronization of Kuramoto Oscillators

A distributed controller is designed for the robust adaptive global phase synchronization of a network of uncertain second-order Kuramoto oscillators with a leader system, modeled as an autonomous nonlinear exosystem that communicates the reference signals only to a subset of the oscillators. We propose an adaptive strategy, only assuming knowledge of upper bounds on the unknown oscillators parameters, that exploits a hybrid hysteresis mechanism to obtain global synchronization despite the well-known topological obstructions with the phases (which evolve on the unit circle). A distributed observer of the leader exosystem is key to overcoming these topological obstructions combined with the generic graph topology we consider. Leveraging the results of hybrid systems theory, including reduction theorems, Lyapunov techniques, and properties of $\omega$ -limit sets, we prove global convergence of the phases to the leader reference and robust global asymptotic stability of the closed-loop dynamics, despite the presence of an adaptive control law.