  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.
