Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable approximation schemes for optimal filters

Dan Crisan, Alberto Lopez-Yela, Joaquin Miguez

TL;DR

The methods and results obtained in this paper can be applied to determine whether a prescribed system $\mS$ yields a sequence of stable filters and to investigate topological properties of classes of optimal filters.

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.

Stable approximation schemes for optimal filters

TL;DR

The methods and results obtained in this paper can be applied to determine whether a prescribed system yields a sequence of stable filters and to investigate topological properties of classes of optimal filters.

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.

Paper Structure

This paper contains 27 sections, 14 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. State space models and optimal filters
  3. Stability of the optimal filter
  4. Contributions
  5. Organisation of the paper
  6. Notation
  7. State space models and optimal filters
  8. Markov state-space models in discrete time
  9. The optimal filter
  10. The prediction-update operator
  11. A topological structure on the set of state space Markov models
  12. Stability of the optimal filter
  13. Truncated filters
  14. Truncation of state space models
  15. Stability of truncated PU operators
  16. ...and 12 more sections

Key Result

Lemma 2.4

\newlabellmMarginals0 Let ${\mathcal{S}}^n=\{\pi_0^n,\kappa^n,g^n\}$, $n\ge 0$, and ${\mathcal{S}}=\{\pi_0,\kappa,g\}$ be elements of $\mathfrak{M}$ with corresponding PU operators $\Phi_t^n$ and $\Phi_t$, respectively. If $\lim_{n \rightarrow \infty} {\mathcal{S}}^n=\mathcal{S}$, then $\lim_{n \r

Theorems & Definitions (38)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proof 1
  • Remark 2.5
  • Definition 2.6
  • Lemma 3.1
  • Proof 2
  • Theorem 3.2
  • ...and 28 more