Synchronization of Kuramoto oscillators via HEOL, and a discussion on AI
Emmanuel Delaleau, Cédric Join, Michel Fliess
TL;DR
The paper addresses robust synchronization of Kuramoto oscillators under multiplicative control by combining differential flatness with HEOL (model-free control). It establishes $\theta_i$ as flat outputs, constructs open-loop reference trajectories via a second-order filter to achieve finite-time synchronization, and closes the loop with a data-driven estimator $F_i^{est}$ and an intelligent proportional controller, enabling stable tracking even with model mismatches. Simulations on a three-oscillator network confirm finite-time synchronization, angular-frequency following, and robustness to disturbances, and the additive-control variant is shown to yield negligible tracking error while preserving the same synchronization objective. The work highlights a bridge between AI-inspired control concepts and classical control, suggesting that AI-like paradigms can inform robust, computation-efficient control of networked dynamical systems.
Abstract
Artificial neural networks and their applications in deep learning have recently made an incursion into the field of control. Deep learning techniques in control are often related to optimal control, which relies on Pontryagin maximum principle or the Hamilton-Jacobi-Bellman equation. They imply control schemes that are tedious to implement. We show here that the new HEOL setting, resulting from the fusion of the two established approaches, namely differential flatness and model-free control, provides a solution to control problems that is more sober in terms of computational resources. This communication is devoted to the synchronization of the popular Kuramoto's coupled oscillators, which was already considered via artificial neural networks (Böttcher et al., Nature Communications 2022), where, contrarily to this communication, only the single control variable is examined. One establishes the flatness of Kuramoto's coupled oscillator model with multiplicative control and develops the resulting HEOL control. Unlike many exemples, this system reveals singularities that are avoided by a clever generation of phase angle trajectories. The results obtained, verified in simulation, show that it is not only possible to synchronize these oscillators in finite time, and even to follow angular frequency profiles, but also to exhibit robustness concerning model mismatches. To the best of our knowledge this has never been done before. Concluding remarks advocate a viewpoint, which might be traced back to Wiener's cybernetics: control theory belongs to AI.