Nonexistence of finite-dimensional estimation algebras on closed smooth manifolds

TL;DR

This paper extends the theory of estimation algebras to filtering on closed Riemannian manifolds and proves that, unlike in Euclidean space, finite-dimensional estimation algebras do not exist for systems with non-constant observation functions.

Contribution

It establishes that finite-dimensional estimation algebras cannot exist on closed Riemannian manifolds with non-constant observation functions, extending filtering theory beyond Euclidean spaces.

Findings

Estimation algebras are infinite-dimensional on closed Riemannian manifolds.

Finite-dimensional estimation algebras do not exist with non-constant observation functions.

The result generalizes the understanding of filtering in curved spaces.

Abstract

Estimation algebras have been extensively studied in Euclidean space, where finite-dimensional estimation algebras form the foundation of the Kalman and Benes filters, and have contributed to the discovery of many other finite-dimensional filters. This work extends the theory of estimation algebras to filtering problems on Riemannian manifolds in continuous time. Our main result demonstrates that, with non-constant observation functions, the estimation algebra associated with the system on closed Riemannian manifolds is infinite-dimensional.