On Controller Design for Systems on Manifolds in Euclidean Space
This addresses controller synthesis for nonlinear systems on manifolds, offering a global coordinate approach, but it appears incremental as it adapts existing Euclidean methods to manifold contexts.
The paper tackles controller design for systems on manifolds by embedding them into Euclidean space, adding transversal stability, and designing controllers in the ambient space before restricting them back to the manifold. It successfully applies this method to tracking problems for a rigid body system and a quadcopter drone system, demonstrating feasibility but without providing concrete performance numbers.
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 $\mathbb R^n$, extend the system from $M$ to the ambient space $\mathbb R^n$, and modify it outside $M$ to add transversal stability to $M$ in the final dynamics in $\mathbb R^n$. Controllers are designed for the final system in the ambient space $\mathbb R^n$. 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 $\mathbb R^n$ is used for controller synthesis, and any controller design method in $\mathbb R^n$, 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.