Towards consistent filtering for discrete-time partially-observable nonlinear systems

In this paper, we study the filter consistency of discrete-time nonlinear systems with partially-observable measurements, where the full state is not reconstructable from the available measurements at each time step. Linearized filters such as the extended Kalman filter (EKF) which are realized based on the corresponding linearized systems, may become inconsistent. Relying on a novel decomposition of the observability matrix based on different measurement sources, we show that the filter acquires spurious information from the measurements of each source, which erroneously reduces the uncertainty of the state estimates and hence causes inconsistency. Based on this key insight, we propose an information-aware methodology and develop two novel EKF algorithms of computing filter Jacobians which ensure that all decompositions of the observability matrix have nullspace of correct dimensions. In the first, the linearization points are selected so as to minimize their expected linearization errors under the constraints that the decompositions of the observability matrix have correct nullspace. In the second, we project the canonical measurement Jacobian onto the actual information-available directions. The proposed approaches are shown to significantly outperform the canonical EKF in the particular application of radar-based target tracking.