A unified square-root approach for the score and Fisher information matrix computation in linear dynamic systems

One of the most frequently encountered problems in practice is to combine a priori knowledge about a physical system with experimental data to provide on-line estimation of an unknown dynamic state and system parameters. The classical way for solving this problem is to use adaptive filtering techniques. The adaptive schemes for the maximum likelihood estimation based on gradient-based optimization methods are, in general, preferable. They require the likelihood function and its gradient evaluation (score), and might demand the Fisher information matrix (FIM) computation. All techniques for the score and the FIM calculation in linear dynamic systems yield the implementation of the Kalman filter (KF) and its derivatives (with respect to unknown system parameters), which is known to be numerically unstable. An alternative solution can be found among algorithms developed in the KF community for solving ill conditioned problems: the square-root algorithms, the UD-based factorization methods and the fast SR Chandrasekhar–Kailath–Morf–Sidhu techniques. Recently, these advanced KF implementations have been extended on the filter derivatives computation. However there is no systematic way of designing the robust “differentiated” methods. In this paper, we develop a unified square-root methodology of generating the computational techniques for the filter/smoother derivatives evaluation required in gradient-based adaptive schemes for the score and the FIM computation.