On controller design for systems on manifolds in Euclidean space

Abstract

A new method is developed to design controllers in Euclidean space for systems defined onmanifolds. The idea is to embed the state-space manifoldM of a given control system into some Euclidean space R-n, extend the system from M to the ambient space R-n, and modify it outsideMto add transversal stability toMin the final dynamics in R-n. Controllers are designed for the final system in the ambient space R-n. Then, their restriction to M produces controllers for the original system on M. This method has the merit that only one single global Cartesian coordinate system in the ambient space R-n is used for controller synthesis, and any controller design method in R-n, such as the linearization method, can be globally applied for the controller synthesis. The proposed method is successfully applied to the tracking problem for the following two benchmark systems: the fully actuated rigid body system and the quadcopter drone system.