Bounding Linearization Errors with Sets of Densities in Approximate Kalman Filtering

Abstract – Applying the Kalman filtering scheme to linearized system dynamics and observation models does in general not yield optimal state estimates. More pre-cisely, inconsistent state estimates and covariance ma-trices are caused by neglected linearization errors. This paper introduces a concept for systematically predicting and updating bounds for the linearization errors within the Kalman filtering framework. To achieve this, an uncertain quantity is not characterized by a single prob-ability density anymore, but rather by a set of densities and accordingly, the linear estimation framework is gen-eralized in order to process sets of probability densities. By means of this generalization, the Kalman filter may then not only be applied to stochastic quantities, but also to unknown but bounded quantities. In order to improve the reliability of Kalman filtering results, the last-mentioned quantities are utilized to bound the typi-cally neglected nonlinear parts of a linearized mapping