An ensemble Kushner-Stratonovich-Poisson filter for recursive estimation in nonlinear dynamical systems

TL;DR

This paper introduces a novel ensemble Kushner-Stratonovich-Poisson filter for recursive estimation in nonlinear dynamical systems with Poisson measurements, addressing particle collapse issues and demonstrating versatility through numerical examples.

Contribution

It develops a new Monte Carlo filter that effectively handles Poisson-type observations and avoids particle collapse, extending filtering techniques to nonlinear systems with counting process data.

Findings

Successfully filters Poisson and diffusive measurements

Eliminates particle collapse in ensemble filtering

Demonstrates application to control problems

Abstract

Despite the numerous applications that may be expeditiously modelled by counting processes, stochastic filtering strategies involving Poisson-type observations still remain somewhat poorly developed. In this work, we propose a Monte Carlo stochastic filter for recursive estimation in the context of linear/nonlinear dynamical systems with Poisson-type measurements. A key aspect of the present development is the filter-update scheme, derived from an ensemble approximation of the time-discretized nonlinear filtering equation, modified to account for Poisson-type measurements. Specifically, the additive update through a gain-like correction term, empirically approximated from the innovation integral in the filtering equation, eliminates the problem of particle collapse encountered in many conventional particle filters. Through a few numerical demonstrations, the versatility of the proposed filter is brought forth, first with application to filtering problems with diffusive or Poisson-type measurements and then to an automatic control problem wherein the extremization of the associated cost functional is achieved simply by an appropriate redefinition of the innovation process.