Improved Discrete-Time Kalman Filtering within Singular Value Decomposition

The paper is concerned with the hidden dynamic state estimation in linear discrete-time stochastic systems in presence of Gaussian noises. The associated with the state-space model estimator is known as the Kalman filter (KF). One of the shortcomings of this recursive algorithm is its numerical instability with respect to roundoff errors. Since the appearance of the KF in 1960s, much effort has been made to design numerically stable filter implementations. The most popular and beneficial techniques are found in the class of square-root (SR) methods and imply the Cholesky decomposition of the corresponding error covariance matrix. Another important matrix factorization method is the singular value decomposition (SVD) and, hence, further encouraging KF implementations might be found under this approach. Motivated by previous studies in the nonlinear SVD-based filtering realm, we aim at exploring the SVD-based strategy for linear filtering problem examined in this paper. The analysis presented here exposes that the previously proposed SVD-based KF variant is still sensitive to roundoff errors and poorly treats ill-conditioned situations, although the SVD-based strategy is inherently more stable than the conventional KF approach. In this paper we explain how it can be further improved for enhancing the numerical robustness against roundoff errors. We design some new SVD-based KF implementations, provide their detailed derivations, and discuss the numerical stability and computational complexity issues. All new SVD-based KF variants are derived here in the covariance form. A set of numerical experiments are performed for comparative study.