Optimal nonlinear estimation

TLDR: For the nonlinear estimation problem with nonlinear plant and observation models, white gaussian excitations and continuous data, the state-vector a-posteriori probabilities for prediction, and smoothing are obtained via the "partition theorem", and optimal linear smoothing algorithms are obtained in a new form.

Abstract: For the nonlinear estimation problem with nonlinear plant and observation models, white gaussian excitations and continuous data, the state-vector a-posteriori probabilities for prediction, and smoothing are obtained via the "partition theorem". Moreover, for the special class of nonlinear estimation problems with linear models excited by white gaussian noise, and with nongaussian initial state, explicit results are obtained for the a-posteriori probabilities, the optimal estimates, and the corresponding error-covariance matrices for filtering, prediction, and smoothing. In addition, for the latter problem, approximate but simpler expressions are obtained by using a gaussian sum approximation of the initial state-vector probability density. As a special case of the above results, optimal linear smoothing algorithms are obtained in a new form.