Kalman filtering for general discrete-time LTI systems

Recursive state estimation problems for explicit and implicit time-invariant linear systems, both for systems with and without unknown inputs, can be formulated as a single problem usually referred to as descriptor Kalman filtering. Solutions to this problem have been proposed in the literature, however, these solutions either neglect possible contributions of future dynamics to the current estimate or make unnecessary assumptions on the structure of the system. We propose a solution to this problem which leads to a constructive method lifting these unnecessary assumptions. This method uses a generalization of the shuffle algorithm