Extension of a PID control theory to Lie groups applied to synchronising satellites and drones. Issue 17 (8th October 2020) - Fiori, Simone; Cervigni, Italo; Ippoliti, Mattia; Menotta, Claudio
Abstract : The authors tackle time synchronisation of two (possibly non‐identical) second‐order dynamical systems whose state space possess the structure of a Lie group. Synchronisation is treated as a non‐linear control problem on Lie groups. They present a control theory inspired by the proportional–integral–derivative (PID) regulation principle. The resulting L‐PID control scheme is applied to sync the rotational component of motions in physical systems. A model‐dependent version of the L‐PID, equipped with a dynamics‐cancelling component, is theoretically proven to converge to zero control error by a Lyapunov stability analysis. Furthermore, they present Lie‐group‐tailored numerical techniques to implement the devised PID regulation theory and test such numerical schemes on two dynamical systems, namely, a gyrostat satellite and a quadrotor drone. As a further extension, they present an empirical translational synchronisation algorithm for two drones based on a pair of concurring L‐PID and projected‐PID controllers.