On filter consistency of discrete-time nonlinear systems with partial-state measurements

Linearized filters, such as the extended Kalman filter (EKF), often become inconsistent, when applied to observable nonlinear systems with partial-state measurements (where the full state cannot be reconstructed from the measurements at each time step). Relying on a novel decomposition of the observability matrix with respect to different measurement sources (sensors), we show that the standard EKF acquires spurious information from the measurements of each source, which erroneously reduces the uncertainty of the state estimates and hence causes inconsistency. With this key insight, we propose two EKF algorithms which compute the Jacobians in such as way so as to ensure that all decompositions of the observability matrix have nullspace of correct dimension. In the first, the linearization points are selected so as to minimize linearization errors under the constraints that the decompositions of the observability matrix have the appropriate nullspace. In the second, we project the most accurate measurement Jacobian (i.e., computed using the latest, and thus the best, state estimates as in the standard EKF) onto the actual information-available directions. We test the proposed algorithms in a two-radar target tracking example and show that significant performance improvement over the standard EKF is attained.