Gaussian Sum Approximations in Nonlinear Filtering and Control

Abstract This paper presents a survey of some recent results in the fields of nonlinear filtering and stochastic control resulting from the use of Gaussian sum approximations to certain density functions. When trying to estimate and control the state of a nonlinear stochastic dynamic system from noisy measurement data, the a posteriori density of the state conditioned on the measurement data contains all of the available information about the system state. If these densities were available, it should be possible to develop “optimal” estimation and control policies. The well-known Bayesian recursion relations describe the evolution of this a posteriori density in terms of given a priori densities, the system dynamics and measurement equations and the available data. Unfortunately, it is seldom possible to solve these closed forms for the required a posteriori densities. Over the last few years it has been shown that when the a priori densities are approximated by a sum of Gaussian densities with positive weighting coefficients that linear non-Gaussian problems can be solved analytically and that new natural solutions to nonlinear problems are available. Recently this method has been extended to dual deterministic optimal control problems.

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