Gradient-like observers for invariant dynamics on a Lie group

TL;DR

This paper introduces a new methodology for designing nonlinear state observers on Lie groups, leveraging invariant system structures and gradient-like innovation terms to achieve strong convergence.

Contribution

It presents a novel design framework for invariant observers on Lie groups, including a factorization theorem and gradient-based innovation design, with proven convergence properties.

Findings

Observers exhibit strong (almost) global convergence.

The methodology applies to invariant kinematic systems on Lie groups.

Examples demonstrate the effectiveness of the proposed approach.

Abstract

This paper proposes a design methodology for non-linear state observers for invariant kinematic systems posed on finite dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialised to systems on Lie groups. For invariant systems this leads to a general factorisation theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach.