Relationship between steady state Kalman filter gain and noise variances
Masatoshi Nakamura · International Journal of Systems Science · 1982
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 transition systems and for general linear dynamic systems. The second is the algorithm of the exponential weighted least squares method for general noisy linear dynamic systems. A remarkable feature of this derivation is the introduction of the index λ into the criterion for this problem. The third is concerned with the correspondence between the Kalman filter and the least squares method. The correspondence has been considered by many investigators only for process noise free systems. The results, derived here, are an extension of the correspondence for general systems.