Fixed Interval Smoothing of Nonlinear/Non-Gaussian Dynamic Systems in Cell Space

State estimation problems such as optimal filtering and smoothing do not lend themselves to analytical treatment in general nonlinear/non-Gaussian dynamic systems. The fixed interval smoothing problem aims to construct the marginal conditional probability density function of the state given past and future measurements relative to the state. For linear Gaussian systems it is derived as the Rauch-Tung-Striebel smoother. Theoretical solutions are available for nonlinear systems in the form of forward-filter-backward-smoother strategy and the two-filter-smoother formula that combines the results of two independent filters. Recently, these methods have been implemented in the context of sequential Monte Carlo filters with importance sampling. In this paper the fixed interval smoothing is numerically approximated by recursive computation of the conditional density as a piecewise constant function. The key for this algorithm is a coarse-grained representation of the system dynamics as an approximate aggregate Markov chain in discretized state space or cell space. The proposed approach is demonstrated with a simulation example involving a nonlinear CSTR model.