Abstract :
Using the "partition theorem" - an explicit Bayes theorem -- fundamentally new linear filtering and smoothing algorithms both for continuous as well as discrete data have been obtained. The new algorithms are given in explicit, integral expressions of a "partitioned" form, and in terms of decoupled forward filters. The "partitioned" algorithms were shown to be especially advantageous from a computational as well as from an analysis standpoint. The "partitioned" algorithms are the natural framework in which to study such important concepts as observability, controllability, unbiasedness, and the solution of Riccati equations. Specifically, they yield further insight as well as significant new results on: a) unbiased estimation and filter initialization procedures; b) stochastic observability and stochastic controllability; c) the interconnection between stochastic observability, Fisher information matrix, and the Cramer-Rao bound; d) estimation errorbounds; and most importantly, e) computationally effective "partitioned" solutions of time-varying matrix Riccati equations. In fact, all of the above results have been obtained for general, time-varying lumped, linear systems.
