Linear fixed-point smoothing by using functional analysis

A new approach to the fixed-point smoothing problem for linear stochastic distributed parameter systems is proposed by using functional analysis. The number of sensor locations is assumed to be finite and the error criterion is based on the unbiased and least-squares estimations. The algorithm for an optimal fixed-point smoothing estimate is derived by using Itô´s stochastic calculus in Hilbert spaces. By applying the kernel theorem to these results, a family of partial differential equations for the optimal fixed-point smoothing estimate is derived. The existence and uniqueness theorems concerning the solutions for both the smoothing gain and the smoothing estimator equations are proved. Finally, usefulness of the algorithm is illustrated with a numerical example.