Factorized Smoothing with Weighted Hyperbolic Householder Reflectors

The goal of filtering is to estimate the current state using past and present information. In contrast, smoothing refers to estimating past states using current and future information. Under certain assumptions, the Kalman filter (KF) provides a closed-form solution to the filtering problem, from which fixed-interval smoothers such as the Fraser–Potter (FP) and Rauch–Tung–Striebel (RTS) algorithms can be derived. To address numerical instabilities in the KF and associated smoothers, covariance factorization techniques, such as UDU factorization, are used. In this work, different versions of the FP and RTS smoothers are derived and analyzed. Specifically, an FP UDU smoother and an RTS UDU smoother are proposed. For the derivation of the RTS smoother, the weighted hyperbolic Householder reflector is introduced as a generalization of both the Householder reflector and the hyperbolic Householder reflector. The UDU smoothers are compared with traditional and stable formulations in a numerical example, demonstrating their numerical equivalence and validating their implementation.