Measurement updating using the U-D factorization

Abstract

In this paper we describe a fresh approach to the discrete linear filtering problem. Our method involves an upper triangular factorization of the filter error covariance matrix, i.e. P = UDUT. Efficient and stable measurement updating recursions are developed for the unit upper triangular factor, U, and the diagonal factor, D. This paper treats only the parameter estimation problem; effects of mapping, inclusion of process noise and other aspects of filtering are treated in separate publications. The algorithm is surprisingly simple and, except for the fact that square roots are not involved, can be likened to square root filtering. Indeed, like the square root filter our algorithm guarantees nonnegativity of the computed covariance matrix. As in the case of the Kalman filter, our algorithm is well suited for use in real time. Attributes of our factorization update include: efficient one point at a time processing that requires little more computation than does the optimal but numerically unstable conventional Kalman measurement update algorithm; stability that compares with the square root filter and the variable dimension flexibility that is enjoyed by the square root information filter. These properties are the subject of this paper.