Observer-Based Controller Design for Systems on Manifolds in Euclidean Space

TL;DR

This paper introduces a novel method for designing observers and controllers for nonlinear systems on manifolds by embedding them into Euclidean space, enabling the use of standard Euclidean control techniques.

Contribution

It proposes a unified approach to extend system dynamics into Euclidean space, ensuring manifold stability and simplifying controller design on nonlinear manifolds.

Findings

Successfully applied to rigid body systems

Ensures manifold invariance and stability

Facilitates global control design in Euclidean space

Abstract

A method of designing observers and observer-based tracking controllers is proposed for nonlinear systems on manifolds via embedding into Euclidean space and transversal stabilization. Given a system on a manifold, we first embed the manifold and the system into Euclidean space and extend the system dynamics to the ambient Euclidean space in such a way that the manifold becomes an invariant attractor of the extended system, thus securing the transversal stability of the manifold in the extended dynamics. After the embedding, we design state observers and observer-based controllers for the extended system in one single global coordinate system in the ambient Euclidean space, and then restrict them to the original state-space manifold to produce observers and observer-based controllers for the original system on the manifold. This procedure has the merit that any existing control method that has been developed in Euclidean space can be applied globally to systems defined on nonlinear manifolds, thus making nonlinear controller design on manifolds easier. The detail of the method is demonstrated on the fully actuated rigid body system.