Adaptive Fading Kalman Filter Design Using the Geometric Mean of Normal Probability Densities

The paper extends the Kalman filter to operate with the potential process model uncertainty by relying on the use of a variable fading factor. A loss functional evaluating the prediction step of the Kalman filter is constructed based on Bayesian decision-making. This evaluation results in coupling two normal probability density functions (pdfs), defining a lower and upper bound for a state uncertainty increase. The coupling policy is identical with the geometric mean of pdfs weighted by adaptively adjusted probabilities. In this respect, the fading factor is optimally determined by being treated as a probability assigned to the more conservative pdf. The proposed schema corrects state filtering in the presence of model uncertainty through controlling the Kalman gain matrix in response to observed performance.