Non-linear Symmetry-preserving Observer on Lie Groups

TL;DR

This paper introduces a geometric framework for designing symmetry-preserving observers on Lie groups, ensuring autonomous error dynamics and convergence properties independent of specific trajectories.

Contribution

It provides an explicit, intrinsic method for observer design on Lie groups with symmetry properties, including a special case with autonomous error equations.

Findings

Observers have locally convergent error dynamics

Error equations are autonomous and trajectory-independent

Framework applies to systems with specific symmetry structures

Abstract

In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which reminds of the linear stationary case.