Constrained Posterior Cramér-Rao Bound for Discrete-Time Systems

Abstract

This article presents a Cramér-Rao lower bound for the discrete-time filtering problem under linear state constraints. A simple recursive algorithm is presented that extends the computation of the Cramér-Rao lower bound found in previous literature by one additional step in which the full-rank Fisher Information matrix is projected onto the tangent hyperplane of the constraint set. This makes it possible to compute the constrained Cramér-Rao lower bound for the discrete-time filtering problem without reparametrization of the original problem to remove redundancies in the state vector, which improves insights into the problem. It is shown that in case of a positive-definite Fisher Information Matrix the presented constrained Cramér-Rao bound is lower than the unconstrained Cramér-Rao bound. The bound is evaluated on an example.