On Controller Design for Systems on Manifolds in Euclidean Space
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A new method is developed to design controllers in Euclidean space for systems defined on manifolds. The idea is to embed the state-space manifold M 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 outside M to add transversal stability to M in 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.
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