A Symmetry-Preserving Reduced-Order Observer

TL;DR

This paper introduces a symmetry-preserving reduced-order observer for nonlinear systems with Lie group symmetries, ensuring invariant and stable estimation of unmeasured states, demonstrated through rigid-body velocity estimation.

Contribution

It develops a novel observer design leveraging system symmetries with a moving frame, providing conditions for asymptotic stability and simplifying error stabilization.

Findings

Observer maintains invariance under Lie group actions.

Stability conditions ensure asymptotic convergence.

Application to rigid-body velocity estimation validates approach.

Abstract

A symmetry-preserving, reduced-order state observer is presented for the unmeasured part of a system's state, where the nonlinear system dynamics exhibit symmetry under the action of a Lie group. Leveraging this symmetry with a moving frame, the observer dynamics are constructed such that they are invariant under the Lie group's action. Sufficient conditions for the observer to be asymptotically stable are developed by studying the stability of an invariant error system. As an illustrative example, the observer is applied to the problem of rigid-body velocity estimation, which demonstrates how exploiting the symmetry of the system can simplify the stabilization of the estimation error dynamics.