Overall hyperbolic-singular-value-decomposition-based square-root solutions in Kalman filters with deterministically sampled mean and covariance for state estimation in continuous-discrete nonlinear stochastic systems

Abstract

The paper aims at square-rooting continuous-discrete Kalman filters with deterministically sampled mean and covariance. These cover all methods devised within the quadrature, cubature and unscented Kalman filtering frameworks. The main problem addressed is a potential negativity of some weights used in calculations of the sampled expectations and covariances, which precludes from an orthogonal square-rooting procedure to be designed and applied. However, square-root solutions are often requested because of their exceptional robustness to round-off and other errors committed. These also preserve the symmetry and positivity of the covariances computed in automatic mode. Here, we apply the J-orthogonal square-rooting approach devised earlier to designing our overall square-root solutions in the realm of quadrature, cubature and unscented Kalman filtering techniques, which are adjusted easily to any state estimator in use by setting its weights and deterministically selected nodes. Our J-orthogonal square-rooting procedure is grounded on the hyperbolic singular value decomposition. It leads to two algorithms covering all Itô-Taylor-approximation-based Kalman filters with deterministically sampled mean and covariance. Performances of such square-root solutions are assessed and compared to their non-square-root counterparts in a simulated flight control scenario with ill-conditioned measurements.