On the evaluation of uncertainties for state estimation with the Kalman filter

The Kalman filter is an established tool for the analysis of dynamic systems with normally distributed noise, and it has been successfully applied in numerous application areas. It provides sequentially calculated estimates of the system states along with a corresponding covariance matrix. For nonlinear systems, the extended Kalman filter is often used which is derived from the Kalman filter by linearization around the current estimate. A key issue in metrology is the evaluation of the uncertainty associated with the Kalman filter state estimates. The "Guide to the Expression of Uncertainty in Measurements" (GUM) and its supplements serve as the de facto standard for uncertainty evaluation in metrology. We explore the relationship between the covariance matrix produced by the Kalman filter and a GUM-compliant uncertainty analysis. In addition, also the results of a Bayesian analysis are considered. For the case of linear systems with known system matrices, we show that all three approaches are compatible. When the system matrices are not precisely known, however, or when the system is nonlinear, this equivalence breaks down and different results can be reached then. Though for precisely known nonlinear systems the result of the extended Kalman filter still corresponds to the linearized uncertainty propagation of GUM. The extended Kalman filter can suffer from linearization and convergence errors. These disadvantages can be avoided to some extent by applying Monte Carlo procedures, and we propose such a method which is GUM-compliant and can also be applied online during the estimation. We illustrate all procedures in terms of a two-dimensional dynamic system and compare the results with those obtained by particle filtering, which has been proposed for the approximate calculation of a Bayesian solution. Finally, we give some recommendations based on our findings.