Convergence of Nonlinear Observers on R^n with a Riemannian Metric (Part I)

TL;DR

This paper explores how Riemannian metrics can be used to analyze the convergence of nonlinear observers on b^n, linking metric properties to system observability and observer design.

Contribution

It establishes a connection between Riemannian metrics, system observability, and observer convergence, providing conditions for the existence of observers with infinite gain margins.

Findings

Existence of a nonincreasing Riemannian distance implies a conditionally negative Lie derivative.

Complete Riemannian metrics with negative Lie derivatives relate to observer existence.

Level sets of output functions are geodesically convex under certain observer gain conditions.

Abstract

We study how convergence of an observer whose state lives in a copy of the given system's space can be established using a Riemannian metric. We show that the existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing implies that the Lie derivative of the Riemannian metric along the system vector field is conditionally negative. Moreover, we establish that the existence of this metric is related to the observability of the system's linearization along its solutions. Moreover, if the observer has an infinite gain margin then the level sets of the output function are geodesically convex. Conversely, we establish that, if a complete Riemannian metric has a Lie derivative along the system vector field that is conditionally negative and is such that the output function has a monotonicity property, then there exists an observer with an infinite gain margin.