Robust Kalman Smoothers With Linear Equality Constraints

It is well known that the Kalman filter (KF) performance is clearly degraded in real-life applications where the system model is misspecified to a certain extent, that is, when the assumed knowledge does not perfectly match the true system dynamics. A possible way to mitigate the impact of a system model mismatch is to resort to linear constraints. In that perspective, a general linearly constrained KF (LCKF) formulation has been recently derived and shown to provide an effective robust filtering solution. In this contribution, we extend the LCKF framework to linear smoothers. Adapting the Fixed-Interval (FI) smoother is easy with sequential state augmentation, but the obtained linearly constrained FI (LCFI) has quadratic complexity in the interval length. Hence the need for an efficient alternative: the linearly constrained Rauch-Tung-Striebel (LCRTS) smoother. Introduced in an intuitive manner, its consistency is proved by showing that the LCFI and LCRTS coincide. The mismatch mitigation capabilities and performance of the LCRTS, with respect to unconstrained smoothers, are shown through illustrative examples with different types of mismatch in both measurement and process equations.