A note on two-filter smoothing formulas

In this note, the two-filter smoothing formulas are obtained from general optimal smoothers in consideration of the control inputs and the correlation between system and measurement noises. The inverse of the system matrix and any backwards time model for a backwards filter are not required. Besides, we do not take heuristic approaches such as infinite covariances. We first propose a new batch form of the general optimal smoother, and then represent it with two filters without any assumptions and heuristic approaches. Even in the case that the lag size is small, the proposed formulas in this note still work without any singularity problem, and thus, are guaranteed to be optimal even for a singular system matrix. A recursive form is introduced for efficient calculation and is shown to be reduced to the existing two-filter formulas if some condition is satisfied.