A Practical Approach to Nonlinear Estimation by Using the Maximum Entropy Principle

Abstract

The problem of estimating the state of a continuous markovian process in the presence of nonlinear observation (nonlinear filtering) may be considered as being completely solved on a theoretical standpoint. All the difficulties arise in the practical applications which require new ways of investigation: search for special approaches related to special problems, and search for improvement of the numerical techniques which are now available. In fact, nonlinear filtering is basically an infinite dimensional problem, and any approximation should work in a finite dimensional space. The paper proposes an approach without using stochastic differential equations. The continuous markovian process is defined by its transition moments only and therefore one can derive the equation of state moments. When the transition moments are polynomials, the state moments are then given by an infinite set of linear differential equations. Likewise when the observation is polynomial, an infinite set of linear equations provides estimates of the state moments in terms of the observation moments. Given the estimates of the state moments, and using the maximum entropy principle we will obtain the corresponding probability density, and therefore the estimate of the state. When the nonlinear functions are not polynomials, it will be possible to apply the method above, using a polynomial approximation.