A Low-Complexity Sliding-Window Kalman FIR Smoother for Discrete-Time Models

The information filter is a form of the Kalman filter that, in many of its realizations, allows optimal, unbiased, recursive state estimation without an initial state estimate. We review a number of forms of the information filter. We then derive the coefficients for the sliding-window Kalman finite impulse response (FIR) smoother (also known as a receding or moving horizon Kalman FIR smoother) starting from the equations for the information filter. The resulting FIR smoother has a simple, recursive form for calculating the coefficients, allowing them to be calculated with $O(N)$ complexity versus the $O(N^{2})$ to $O(N^{3})$ complexity of previous approaches, where $N$ is the length of the batch. It also allows for a control input, something not present in previous algorithms. This method is only limited in the assumption that the state transition matrix is invertible, which, however, is satisfied in most practical problems.