Sparse-grid Quadrature Kalman Filter based on the Kronrod-Patterson Rule

For the state estimation of nonlinear systems, especially high-dimensional systems, using existing nonlinear Gaussian filters it is hard to seek a balance between accuracy and efficiency. This paper proposes a novel Sparse-grid Quadrature Kalman Filter (SGQKF), an algorithm that utilizes the Kronrod-Patterson rule to determine the univariate quadrature point sets with a range of accuracy levels, which then are extended for the multi-dimensional point sets using the Sparse-grid technique. The Sparse-grid point sets generated by the Kronrod-Patterson rule, compared to those generated by the Gauss-Hermite rule and the Moment Matching method, not only has high precision, but also nested properties, which remarkably improve the estimation accuracy and efficiency of the new algorithm. Compared with conventional point-based algorithms like Quadrature Kalman Filter (QKF), the SGQKF can achieve close, even higher, accuracy with significantly less number of quadrature points, which effectively alleviate the "curse of dimensionality" for high-dimensional problems. Finally, the performance of this filter is demonstrated by a satellite orbit estimation problem. The simulation results verify the merits of the new algorithm.