An Integral Characterization Of Optimal Error Covariance By Kalman Filtering

In this paper, we discover that the determinant of the optimal output estimation error covariance attained by the Kalman filter can be expressed explicitly in terms of the plant dynamics and noise statistics in an integral characterization. Towards this end, we examine the algebraic Riccati equation associated with Kalman filtering using analytic function theory and relate it to the Bode integral. This result may be interpreted as a generalization of the Kolmogorov-Szego formula to the nonstationary case. In addition, the integral characterization is applicable to Kalman filtering with correlated noises and that with intermittent observations as well.