Initialization of the Kalman filter with partially diffuse initial conditions

The problem of computing estimates of the state vector when the Kalman filter is seeded with an arbitrarily large variance is considered. To date the response in the literature has been the development of a number of relatively complex hybrid filters, usually involving additional quantities and equations over and above the conventional filter. We show, however, that a certain square root covariance filter is capable of handling the complete range of variances (zero, positive and infinite) without modification to the filtering equations themselves and without additional computation loads. Instead of the more conventional Cholesky factorization, our filter employs an alternative matrix factorization procedure based on a unit lower triangular matrix and a diagonal matrix. This permits the use of a modified form of fast Givens transformations, central to the development of an efficient algorithm.