Fixed Interval Smoothing: Revisited

Gaps in the derivation of early fixed-interval smoothers are filled in. Particular attention is paid to the use of the yariational approach introduced by Bryson and Frazier. The notion of complementary models introduced by Weinert and Desai is used to provide an immediate derivation of the continuous Rauch-Tung-Striebel (R-T-S) smoother from the continuous sweep solution of the Bryson-Frazier two-point boundary-value problem. The discrete version of the sweep is also derived, which leads to a fundamental simplification of Bryson's discrete algorithm discovered by Bierman. Relatively new transformations of the R-T-S smoother that may offer some computational advantages are also discussed. Computational comparisons of different algorithms are given. A simple derivation of Bierman's smoother for the mixed continuous discrete problem is given. Nomenclature A = 2n x2n matrix defined by Eq. (13), used in the continuous two-point boundary-value problem (TPBVP) [Eq. (14)] B = /I x n symmetric matrix defined by Eq. (98), appears in the propagation of A/ [Eq. (97)]; alternative expressions are given by Eqs. (99) and (107)

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