An Approach to Duality in Nonlinear Filtering

This paper revisits the question of duality between minimum variance estimation and optimal control first described for the linear Gaussian case in the celebrated paper of Kalman and Bucy. A duality result is established for nonlinear filtering, mirroring closely the original Kalman-Bucy duality of control and estimation for linear systems. The result for the finite state-space continuous time Markov chain is presented. It's solution is used to derive the classical Wonham filter.

I. Introduction

In Kalman's celebrated paper with Bucy, it is shown that the problem of optimal estimation is dual to an optimal control problem [1]. A striking example of the dual relationship is that, with the time arrow reversed, the dynamic Riccati equation (DRE) of the optimal control is the same as the covariance update equation of the Kalman filter. The relationship is useful, e.g., to derive results on asymptotic stability of the linear filter based on asymptotic properties of the solution of the DRE [2].