On the relationship between the Lagrange multiplier method and the two-filter smoother

Some relationships between the Lagrange multiplier method and a two-filter smoother are explored using orthogonal projection lemmas for linear discrele-time systems. The obtained expectation results are applied to algebraically derive a forward-pass fixed-interval smoothing algorithm that utilizes backward-pass information filter outputs. Furthermore, a convenient form for numerically efficient and stable algorithms is developed and the derived equation can also be used to quantitatively evaluate the effect of initial statistics on the smoothed estimate. Numerical results, pertaining to a linearized version for the in-track motion of a satellite travelling in a circular orbit, are included to illustrate the use of the given algorithms.