Global Observer Design for a Class of Linear Observed Systems on Groups

TL;DR

This paper introduces a novel observer framework for linear systems on Lie groups, enabling global exponential stability under certain conditions, with applications to navigation problems.

Contribution

It proposes a new observer design leveraging system embedding and optimization for systems on Lie groups, addressing topological challenges.

Findings

Achieves global exponential stability under an observability condition.

Provides a systematic approach for observer design on Lie groups.

Demonstrates effectiveness through simulations on navigation models.

Abstract

Linear observed systems on groups encode the geometry of a variety of practical state estimation problems. In this paper, we propose an observer framework for a class of linear observed systems by restricting a bi-invariant system on a Lie group to its normal subgroup. This structural property enables a system embedding of the original system into a linear time-varying system. An observer is constructed by first designing a Kalman-like observer for the embedded system and then reconstructing the group-valued state via optimization. Under an extrinsic observability rank condition, global exponential stability (GES) is achieved provided that one global optimum of the reconstruction optimization is found, reflecting the topological difficulties inherent to the non-Euclidean state space. Semi-global stability is guaranteed when input biases are jointly estimated. The theory is applied to the GES observer design for two-frame systems, capable of modeling a family of navigation problems. Simulations are provided to illustrate the implementation details.