Linear Filtering for Discrete Time Systems Driven by Fractional Noises

TL;DR

This paper develops a novel approach to filtering for discrete-time linear systems driven by fractional noises, revealing fundamental differences from classical filters and extending applicability to other colored noises.

Contribution

It introduces a new method for filtering in systems driven by fractional noises, showing the non-existence of a consistent optimal filter unlike the Kalman filter.

Findings

No consistent optimal filter exists for fractional noise-driven systems.

The method extends to systems with other colored noises with known self-covariation.

A simple example illustrates the main theoretical results.

Abstract

In this paper, we study the discrete time filtering problems for linear systems driven by fractional noises. The main difficulty comes from the non-Markovian of the noises. We construct the difference equation of the covariance process through the properties of the noises and transform the filtering problem to an optimal control problem. We obtain the necessary condition that the coefficients of the optimal filter should satisfy and show there is no consistent optimal filter, which is a significant difference from the classical Kalman filter. Finally, a simple example is considered to illustrate the main results. Further more, our method could also deal with systems driven by any other colored noises, as long as the self-covariation function is known.