5 Smoothing

Abstract

Up to this point we have not considered reduced order smoothing problems, where data over an entire interval may affect an estimate at any time during the interval. Such problems have the characteristics that they have non causal solutions and so may not be implemented in real time. Since this is the case, the reader may wonder why we would be interested in a reduced order sub optimal solution, as obtained in [1] instead of a full order optimal solution as presented in [2] and [3]. The answer is simply that complexity of the solution is still a factor, even when the signal processing is done off-line. If one has a state model of very high order, one does not want to be required to have a smoother of corresponding high order due to the high complexity of such a solution. The nicest situation one can have is when both the processing equations and design equations are of limited complexity. It should be noted, however, that there is a difference between the two categories even for off line processing, because the Riccati (design) equation is solved only once, but the smoothing (processing) equations could be used repeatedly on vast amounts of data.