Non-linear filtering by approximation of the a posteriori density

The problem of estimating from noisy measurement data the state of a dynamical system described by non-linear difference equations is considered. The measurement data have a non-linear relation with the state and are assumed to be available at discrete instants of time. A Bayesian approach to the problem is suggested in which the density function for the state conditioned upon the available measurement data is computed recursively. The evolution of the a posteriori density function cannot be described in a closed form for most systems; the class of linear systems with additive, white gaussian noise provides the major exception. Thus, the problem of non-linear filtering can be viewed as essentially a problem of approximating this density function. For linear systems with additive, white gaussian noise, the a posteriori density is gaussian. The results for linear systems are frequently applied to non-linear systems by introducing linear perturbation theory. Then, the linear equations and gaussian a posteriori density represent approximations to the true description of the system. A generalization of this procedure is described here. The density is approximated by an Edgeworth expansion and the plant and measurerment systems are described, using perturbation techniques, as quadratic equations with additive, white gaussian noise. The Edgeworth expansions are characterized by the central moments of the distribution. Recursion relations are derived for a finite number of these moments and these relations are assumed to describe the set of sufficient statistics for the system. The mean value gives immediately the estimate that minimizes the mean-square error. The well-known linear perturbation theory and this non-linear theory are applied to a simple system and the results are compared.