SVD-based square-root Kalman filtering: A survey of existing implementation methods and novel techniques

Abstract

Singular value decomposition (SVD) is well known to be successfully utilized in the Kalman filtering realm for deriving numerically stable square-root implementation methods. It is as a powerful alternative to the traditional Cholesky factorization-based square-root approach, which has been in use in the engineering literature since the early 1960s. In this paper, we explore all existing SVD factorization-based square-root methods derived for the discrete-time Kalman filtering (KF). We examine time-invariant state-space models and, as a consequence, our survey includes both the Riccati and Chandrasekhar recursion-based KF methodologies. Each approach additionally contains the covariance-type algorithms, information-type methods and the mixed-type variants when they exist. We also propose two novel Riccati-based algorithms that belong to the information-type filtering. One of them is derived by using the hyperbolic SVD to create the homogeneous information-type SVD filter unlike the previously derived method. In our overview, we discuss the properties and difference in implementation ways, we provide the summary of each algorithm and discuss the problems that are still open in this realm for a future research. The numerical tests are also given. They exhibit a numerical behavior of the implementation methods on both well- and ill-conditioned problems.