Synchronization of dynamical systems on Riemannian manifolds by an extended PID-type control theory: Numerical evaluation

Abstract

<p style='text-indent:20px;'>The present document outlines a non-linear control theory, based on the PID regulation scheme, to synchronize two second-order dynamical systems insisting on a Riemannian manifold. The devised extended PID scheme, referred to as M-PID, includes an unconventional component, termed 'canceling component', whose purpose is to cancel the natural dynamics of a system and to replace it with a desired dynamics. In addition, this document presents numerical recipes to implement such systems, as well as the devised control scheme, on a computing platform and a large number of numerical simulation results focused on the synchronization of Duffing-like non-linear oscillators on the unit sphere. Detailed numerical evaluations show that the canceling contribution of the M-PID control scheme is not critical to the synchronization of two oscillators, however, it possesses the beneficial effect of speeding up their synchronization. Simulation results obtained in non-ideal conditions, namely in the presence of additive disturbances and delays, reveal that the devised synchronization scheme is robust against high-frequency additive disturbances as well as against observation delays.</p>