Stable approximation schemes for optimal filters

TL;DR

This paper develops a truncation-based method to construct stable approximate filters that remain close to possibly unstable optimal filters, providing a topological characterization of the set of stable filters.

Contribution

It introduces a general truncation scheme for creating stable filters near unstable ones and characterizes the topological properties of the set of optimal filters.

Findings

Stable filters asymptotically forget initial perturbations.

The set of stable filters is dense in the set of optimal filters under a natural topology.

The proposed truncation scheme yields uniformly convergent approximations.

Abstract

A stable filter has the property that it asymptotically `forgets' initial perturbations. As a result of this property, it is possible to construct approximations of such filters whose errors remain small in time, in other words approximations that are uniformly convergent in the time variable. As uniform approximations are ideal from a practical perspective, finding criteria for filter stability has been the subject of many papers. In this paper we seek to construct approximate filters that stay close to a given (possibly) unstable filter. Such filters are obtained through a general truncation scheme and, under certain constraints, are stable. The construction enables us to give a characterisation of the topological properties of the set of optimal filters. In particular, we introduce a natural topology on this set, under which the subset of stable filters is dense.