Optimal non-linear estimation†

For the non-linear estimation problem with non-linear 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 non-linear estimation problems with linear models excited by white gaussian noise, and with non-gaussian 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.