Universal MATLAB‐based square‐root solutions in the family of continuous‐discrete Gaussian filters for state estimation in nonlinear stochastic dynamic systems

Abstract

This article concerns with the issue of square-rooting in continuous-discrete Gaussian filters intended for state estimation in nonlinear stochastic dynamic systems of continuous-discrete sort. These cover all methods devised within the quadrature, cubature, and unscented Kalman filtering approaches as well as those that can be constructed in the future. Based on the universal moment differential equations developed by Särkkä and Sarmavuori in 2013, we advance further that study and design two square-root solutions grounded on MAT-LAB ODE solvers in the mentioned Gaussian filtering framework. The main problem addressed is a potential negativity of some weights utilized in calculations of the predicted and filtering means and covariances, which precludes from orthogonal square-rooting schemes to be applied. In practice, such square-root implementations are often requested because of their exceptional numerical robustness to round-off and other disturbances. These also preserve the symmetry and positivity of the covariances computed, automatically. Here, we employ the recently-devised J-orthogonal square-rooting technique for designing our universal MATLAB-based square-root solutions in the realm of quadrature, cubature, and unscented Kalman filters, which are easily adjusted to any particular method by using its weights and deterministically selected samples exploited in calculations of the sampled means and covariances. Such a J-orthogonal square-rooting approach is grounded on hyperbolic QR factorizations. It leads to two novel algorithms covering any continuous-discrete Gaussian filter of the quadrature, cubature, or unscented Kalman-like kind. Practical performances of our square-root solutions are validated, assessed, and compared within two simulated ill-conditioned scenarios in aeronautical and chemical engineering.