A Hybrid, Coupled Approach to the Continuous-Discrete Kalman Filter

Abstract

In this letter we present a novel approach to continuous-discrete (CD) Kalman filtering. Unlike the EKF or other hybrid UKF-EKF filters, the novel approach does not require direct calculation of system Jacobians and instead uses unscented transforms (UTs) to extract a pair of matrices, each made up of a linear combination of derivatives (with respect to the state), that are used in its place. More specifically, they are used (1) in parts of the filtering process, and (2) in the linearly implicit numerical integration scheme of the filter's state propagation stage. Extracting these matrices from the filter's UTs and using them in both the filtering process and model simulation, or what we refer to as coupling the filter to model simulation, avoids having to calculate further function evaluations for standard implicit methods, making the process of state estimation for stiff systems more efficient. Another benefit of the proposed approach is that it offers UKF accuracy levels but improved numerical stability over the UKF. This is because it uses symmetric and positive-definite (PD) representations of the state covariance propagation and measurement update equations.