Non-uniform Observability for Moving Horizon Estimation and stability with respect to additive perturbation

TL;DR

This paper introduces new concepts of weak persistence in input trajectories, linking them to observability and stability in Moving Horizon Estimation, with proofs of stability under small measurement and dynamic perturbations.

Contribution

It formalizes weak persistence notions, connects them to the Observability Grammian, and proves stability of MHE solutions using a novel time-uniform Implicit Function Theorem.

Findings

Weakly persistent trajectories ensure MHE stability.

Counter-examples highlight limitations of persistence conditions.

Stability results apply to 2D bearing-only navigation scenarios.

Abstract

This paper formalises the concepts of weakly and weakly regularly persistent input trajectory as well as their link to the Observability Grammian and the existence and uniqueness of solutions of Moving Horizon Estimation (MHE) problems. Additionally, thanks to a new time-uniform Implicit Function Theorem, these notions are proved to imply the stability of MHE solutions with respect to small additive perturbation in the measurements and in the dynamics, both uniformly and non-uniformly in time. Finally, examples and counter-examples of weakly persistent and weakly regularly persistent input trajectories are given in the case of 2D bearing-only navigation.