A duality principle for state estimation with partially noise corrupted measurements

In derivations of the Kalman-Bucy filter, one generally assumes that the measurement noise process possesses a nonsingular covariance matrix. Some authors, [1], [2], have derived formulas for a reduced order optimal estimator with a singular covariance matrix. Their solutions require coordinate transformations in both state and output noise variables. Here we employ the Moore-Penrose generalized inverse of the noise covariance matrix to obtain a full order optimal filter without the use of coordinate transformations. Two different optimal estimators corresponding to two different optimization criteria are found. One of these estimators provides a duality principle relating the optimal estimation problem with a partially singular optimal control problem.