A simple intrinsic reduced-observer for geodesic flow

TL;DR

This paper introduces a globally convergent intrinsic reduced-observer for geodesic flow on Riemannian manifolds, improving velocity estimation in nonlinear mechanical systems and fluid flow sensing, especially under negative curvature conditions.

Contribution

It develops a new globally convergent reduced-observer based on the Jacobi metric for geodesic flow, extending previous local methods to broader classes of systems.

Findings

The reduced-observer achieves faster convergence in negatively curved spaces.

It can be used as a fluid flow soft sensor for incompressible fluids.

The method complements existing observers for non-conservative systems.

Abstract

Aghannan and Rouchon proposed a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. The observer is based on the Riemannian structure of the configuration manifold endowed with the kinetic energy metric and is intrinsic. They proved local convergence. When the system is conservative, we propose a globally convergent intrinsic reduced-observer based on the Jacobi metric. For non-conservative systems the observer can be used as a complement to the one of Aghannan and Rouchon. More generally the reduced-observer provides velocity estimation for geodesic flow with position measurements. Thus it can be (formally) used as a fluid flow soft sensor in the case of a perfect incompressible fluid. When the curvature is negative in all planes the geodesic flow is sensitive to initial conditions. Surprisingly this instability yields faster convergence.