Progressive Bayesian estimation for nonlinear discrete-time systems: the measurement step

This paper is concerned with estimating the internal state of a dynamic system by processing measurements taken from the system output. An exact analytic representation of the probability density functions characterizing the estimate may not be possible to obtain. Even when available, it may be too complex or not practical because, for example, recursive application is required. Hence, approximations are generally inevitable. Gaussian mixture approximations are convenient for a number of reasons. However, calculating appropriate mixture parameters that minimize a global measure of deviation from the true density is a tough optimization task. Here, we propose a approximation method that minimizes the squared integral deviation between the true density and its mixture approximation. Rather than trying to solve the original problem, it is converted into a corresponding system of explicit ordinary first-order differential equations. This system of differential equations is then solved over a finite "time" interval, which is an efficient way of calculating the desired optimal parameter values. For polynomial measurement nonlinearities, closed-form analytic expressions for the coefficients of the system of differential equations are derived.