Comparison of nonlinear estimation for ballistic missile tracking

Approaches towards nonlinear state estimation have been recently advanced to include more accurate and stable alternatives. The Extended Kalman Filter (EKF), the first and most widely used approach (applied as early as the late 1960's and developed into the early 1980's), uses potentially unstable derivative-based linearization of nonlinear process and/or measurement dynamics. The Unscented Kalman Filter (UKF), developed after around 1994, approximates a distribution about the mean using a set of calculated sigma points. The Central Difference Filter (CDF), or Divided Difference Filter (DDF), developed after around 1997, uses divided difference approximations of derivatives based on Stirling's interpolation formula and results in a similar mean, but a different covariance from the EKF and using techniques based on similar principles to those of the UKF. This paper compares the performance of the three approaches above to the problem of Ballistic Missile tracking under various sensor configurations, target dynamics, measurement update / sensor communication rate and measurement noise. The importance of filter stability in some cases is emphasized as the EKF shows possible divergence due to linearization errors and overconfident state covariance while the UKF shows possibly slow convergence due to overly large state covariance approximations. The CDF demonstrates relatively consistent stability, despite its similarities to the UKF. The requirement that the UKF expected state covariance is positive definite is demonstrated to be unrealistic in a case involving multi-sensor fusion, indicating the necessity for its reportedly more robust and efficient square-root implementation. Strategies for taking advantage of the strengths (and avoiding the weaknesses) of each filter are proposed.