Maria V. Kulikova

Maria V. Kulikova, Julia V. Tsyganova

The paper presents a new Kalman filter (KF) implementation useful in applications where the accuracy of numerical solution of the associated Riccati equation might be crucially reduced by influence of roundoff errors. Since the appearance of the KF in 1960s, it has been recognized that the factored-form of the KF is preferable for practical implementation. The most popular and beneficial techniques are found in the class of square-root algorithms based on the Cholesky decomposition of error covariance matrix. Another important matrix factorization method is the singular value decomposition (SVD) and, hence, further encouraging implementations might be found under this approach. 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 design a new SVD-based KF implementation for enhancing the robustness against roundoff errors, provide its detailed derivation, and discuss the numerical stability issues. A set of numerical experiments are performed for comparative study. The obtained results illustrate that the new SVD-based method is algebraically equivalent to the conventional KF and to the previously proposed SVD-based method, but it outperforms the mentioned techniques for estimation accuracy in ill-conditioned situations.

Maria V. Kulikova

Using the array form of numerically stable square-root implementation methods for Kalman filtering formulas, we construct a new square-root algorithm for the log-likelihood gradient (score) evaluation. This avoids the use of the conventional Kalman filter with its inherent numerical instabilities and improves the robustness of computations against roundoff errors. The new algorithm is developed in terms of covariance quantities and based on the "condensed form" of the array square-root filter.

Julia V. Tsyganova, Maria V. Kulikova

Recursive adaptive filtering methods are often used for solving the problem of simultaneous state and parameters estimation arising in many areas of research. The gradient-based schemes for adaptive Kalman filtering (KF) require the corresponding filter sensitivity computations. The standard approach is based on the direct differentiation of the KF equations. The shortcoming of this strategy is a numerical instability of the conventional KF (and its derivatives) with respect to roundoff errors. For decades, special attention has been paid in the KF community for designing efficient filter implementations that improve robustness of the estimator against roundoff. The most popular and beneficial techniques are found in the class of square-root (SR) or UD factorization-based methods. They 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 algorithms might be found under this approach. Meanwhile, the filter sensitivity computation heavily relies on the use of matrix differential calculus. Previous works on the robust KF derivative computation have produced the SR- and UD-based methodologies. Alternatively, in this paper we design the SVD-based approach. The solution is expressed in terms of the SVD-based KF covariance quantities and their derivatives (with respect to unknown system parameters). The results of numerical experiments illustrate that although the newly-developed SDV-based method is algebraically equivalent to the conventional approach and the previously derived SR- and UD-based strategies, it outperforms the mentioned techniques for estimation accuracy in ill-conditioned situations.

Julia V. Tsyganova, Maria V. Kulikova

This technical note addresses the UD factorization based Kalman filtering (KF) algorithms. Using this important class of numerically stable KF schemes, we extend its functionality and develop an elegant and simple method for computation of sensitivities of the system state to unknown parameters required in a variety of applications. For instance, it can be used for efficient calculations in sensitivity analysis and in gradient-search optimization algorithms for the maximum likelihood estimation. The new theory presented in this technical note is a solution to the problem formulated by Bierman in , which has been open since 1990s. As in the cited paper, our method avoids the standard approach based on the conventional KF (and its derivatives with respect to unknown system parameters) with its inherent numerical instabilities and, hence, improves the robustness of computations against roundoff errors.

Maria V. Kulikova

Recent developments in the realm of state estimation of stochastic dynamic systems in the presence of non-Gaussian noise have induced a new methodology called the maximum correntropy filtering. The filters designed under the maximum correntropy criterion (MCC) utilize a similarity measure (or correntropy) between two random variables as a cost function. They are shown to improve the estimators' robustness against outliers or impulsive noises. In this paper we explore the numerical stability of linear filtering technique proposed recently under the MCC approach. The resulted estimator is called the maximum correntropy criterion Kalman filter (MCC-KF). The purpose of this study is two-fold. First, the previously derived MCC-KF equations are revise and the related Kalman-like equality conditions are proved. Based on this theoretical finding, we improve the MCC-KF technique in the sense that the new method possesses a better estimation quality with the reduced computational cost compared with the previously proposed MCC-KF variant. Second, we devise some square-root implementations for the newly-designed improved estimator. The square-root algorithms are well known to be inherently more stable than the conventional Kalman-like implementations, which process the full error covariance matrix in each iteration step of the filter. Additionally, following the latest achievements in the KF community, all square-root algorithms are formulated here in the so-called array form. All the MCC-KF variants developed in this paper are demonstrated to outperform the previously proposed MCC-KF version in two numerical examples.