On stability of a class of filters for non-linear stochastic systems

TL;DR

This paper provides a unified stability analysis framework for a broad class of non-linear stochastic filters, including Kalman variants, under less restrictive assumptions, with theoretical guarantees and numerical validation.

Contribution

It introduces a comprehensive stability analysis applicable to many Gaussian assumed density filters without requiring exponential stability or full observability.

Findings

Time-uniform mean square bounds for filtering error

Exponential concentration inequalities for error analysis

Validation through numerical experiments with synthetic data

Abstract

This article develops a comprehensive framework for stability analysis of a broad class of commonly used continuous and discrete time-filters for stochastic dynamic systems with non-linear state dynamics and linear measurements under certain strong assumptions. The class of filters encompasses the extended and unscented Kalman filters and most other Gaussian assumed density filters and their numerical integration approximations. The stability results are in the form of time-uniform mean square bounds and exponential concentration inequalities for the filtering error. In contrast to existing results, it is not always necessary for the model to be exponentially stable or fully observed. We review three classes of models that can be rigorously shown to satisfy the stringent assumptions of the stability theorems. Numerical experiments using synthetic data validate the derived error bounds.