Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations

TLDR

The resulting nonlinear filter is referred to as the continuous‑discrete cubature Kalman filter (CD‑CKF) to align with existing literature. The paper extends the cubature Kalman filter to handle nonlinear continuous‑discrete state‑space models. The authors transform the continuous‑discrete stochastic differential equations via a 1.5‑order Itô‑Taylor expansion, assume Gaussian densities, compute Gaussian‑weighted integrals with a third‑degree cubature rule, and implement a square‑root version for finite‑word‑length machines, validated on a high‑dimensional radar example. The results indicate that the CD‑CKF markedly outperforms existing continuous‑discrete filters.

Abstract

In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Itô-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters.