Null-Space Square-Root Information Filtering and Smoothing for Singular Problems

Abstract

A new algorithm that does square-root information filtering and fixed-interval smoothing has been developed for singular discrete-time estimation problems. Singular measurements arise if one uses a Markov measurement error model or if one uses the quaternion normalization constraint in an attitude determination problem. The dynamics model becomes singular if delay states are included or if a system time constant is short compared to the sample interval. The new algorithm uses the lower-triangular/orthonormal factorization null-space method to deal with the estimation problem's linear equality constraints. This method is applied recursively and transforms the estimated quantities into components that are exactly determined by the constraints and components in the null space of the constraints. These latter components are estimated using linear least-squares techniques, which is the standard approach of square -root information filtering and smoothing methods. The new algorithm uses more computation and more memory than the only known competitor that can solve similar problems, but it has superior numerical stability. This improvement is demonstrated using an example one-dimensional tracking problem that includes Markov models and dynamics and measurement singularities. The new algorithm's computations nearly achieve machine precision even when the competing algorithm produces meaningless results.