Oscillatory Kalman filtering for Duffing, Coulomb, and Van der Pol oscillators

Abstract

The popularly known Gaussian filtering witnesses intractable integrals numerically approximated during the filtering. However, the numerical approximation methods used in the existing Gaussian filters are generally inaccurate for oscillatory and chaotic (OC) systems, resulting in poor accuracy. In this paper, we propose a new Gaussian filter named oscillatory spherical-radial Kalman filter (OSRKF) to improve the accuracy of OC systems. The proposed OSRKF decomposes the intractable integral into spherical and radial integrals. The spherical integral is approximated using the higher-degree spherical cubature rule, while the radial integral is approximated using exponentially-fitted Gauss–Laguerre quadrature rule. We also formulate the state estimation problems for three OC dynamical systems: the Duffing, Coulomb, and Van-der Pol oscillators. Subsequently, we validate the improved accuracy of the proposed OSRKF for all three OC dynamical systems.