Sensitivity of discrete-time Kalman filter to statistical modeling errors

Abstract

The optimum filtering results of Kalman filtering for linear dynamic systems require an exact knowledge of the process noise covariance matrix Qk, the measurement noise covariance matrix Rk and the initial error covariance matrix P0. In a number of practical solutions, Qk, Rk and P0, are either unknown or are known only approximately. In this paper the sensitivity due to a class of errors in statistical modelling employing a Kalman Filter is discussed. In particular, we present a special case where it is shown that Kalman filter gains can be insensitive to scaling of covariance matrices. Some basic results are derived to describe the mutual relations among the three covariance matrices (actual and perturbed covariance matrices), their respective Kalman gain Kk and the error covariance matrices Pk. It is also shown that system modelling errors, particularly scaling errors of the input matrix, do not perturb the Kalman gain. A numerical example is presented to illustrate the theoretical results, and also to show the Kalman gain insensitivity to less restrictive statistical uncertainties in an approximate sense.