Kalman Filter Sensitivity Evaluation with Orthogonal and J-Orthogonal Transformations

IEEE Transactions on Automatic Control, 58(7) (2013), 1798-1804
http://dx.doi.org/10.1109/TAC.2012.2231572

This technical note addresses the array square-root Kalman filtering/smoothing algorithms with the conventional orthogonal and J-orthogonal transformations. In the adaptive filtering context, J-orthogonal matrices arise in computation of the square-root of the covariance (or smoothed covariance) by solving an equation of the form CCT=DDT-BBT. The latter implies an application of the QR decomposition with J-orthogonal transformations in each iteration step. In this paper, we extend functionality of array square-root Kalman filtering schemes and develop an elegant and simple method for computation of the derivatives of the filter variables to unknown system parameters required in a variety of applications. For instance, our result can be implemented for an efficient sensitivity analysis, and in gradient-search optimization algorithms for the maximum likelihood estimation of unknown system parameters. It also replaces the standard approach based on direct differentiation of the conventional Kalman filtering equations (with their inherent numerical instability) and, hence, improves the robustness of computations against roundoff errors.