Observer Design for Nonlinear Systems with Equivariance

TL;DR

This paper introduces the Equivariant Filter (EqF), a novel observer design leveraging system symmetry and geometry to enhance robustness and performance in nonlinear control systems.

Contribution

It proposes a new observer framework that uses equivariance and symmetry group structures, extending EKF to better handle nonlinear systems with geometric properties.

Findings

The EqF respects system geometry and symmetry.

The error dynamics have a simplified form using equivariant lifts.

The method improves robustness and performance in practical systems.

Abstract

Equivariance is a common and natural property of many nonlinear control systems, especially those associated with models of mechatronic and navigation systems. Such systems admit a symmetry, associated with the equivariance, that provides structure enabling the design of robust and high performance observers. A key insight is to pose the observer state to lie in the symmetry group rather than on the system state space. This allows one to define a globally defined intrinsic equivariant error but poses a challenge in defining internal dynamics for the observer. By choosing an equivariant lift of the system dynamics for the observer internal model we show that the error dynamics have a particularly nice form. Applying the methodology of Extended Kalman Filtering (EKF) to the equivariant error state yields the Equivariant Filter (EqF). The geometry of the state-space manifold appears naturally as a curvature modification to the classical EKF Riccati equation. The equivariant filter exploits the symmetry and respects the geometry of an equivariant system model and yields high performance robust filters for a wide range of mechatronic and navigation systems.