Modified Bryson-Frazier Smoother Cross-Covariance

The Expectation Maximization algorithm can be used to estimate Kalman filter parameter matrices. This requires smoother results and typically the Rauch-Tung-Striebel smoother is used for this purpose. If either of the state transition or plant noise matrices require estimation, a sequence of single time-step smoother cross-covariance matrices, based on the Rauch-Tung-Striebel smoother formulation, are also required. The modified Bryson-Frazier smoother could be used as the smoother for the Expectation Maximization algorithm, but to estimate the state transition or plant noise matrix, this smoother would first need a method to generate the sequence of single time-step smoother cross-covariance matrices based on the modified Bryson-Frazier formulation. This paper develops an expression for these modified Bryson-Frazier cross-covariance matrices, thus allowing the modified Bryson-Frazier smoother to be substituted for the Rauch-Tung-Striebel smoother in the Expectation Maximization algorithm. The modified Bryson-Frazier smoother formulation requires inversion only of filter innovation matrices, which are generally much smaller than state error covariance matrices, which allows both greater computational speed and better numerical accuracy due to smaller matrix size. This formulation can be used even when covariance and transition matrices are singular.