Relationship between steady state Kalman filter gain and noise variances

Abstract

The Kalman filter requires an exact knowledge of the noise covariance matrices Theoretically, they may take arbitrary values under some restrictions ; positive semidefinite or positive definite. Values of the noise covariance matrices have direct effects upon the Kalman filter gain, and therefore affect the final result, of estimation. Practically, the noise covariance matrices are either unknown or are known only approximately, so they are often determined in a rule of trial and error. In this paper, we discover interesting relations between an index λ and the noise covariance matrices for multivariable identity transition systems and for general linear dynamic systems, through an algorithm of the exponential weighted least squares method. These relations are useful for determining the noise covariance matrices. New results are summarized into three points. The first main result is the relations between the steady state Kalman filter gain and the noise covariance matrices for multivariable identity trans...