Real-Time Robust Multivariate Estimator for Dynamic Systems with Heavy-Tailed Additive Uncertainties

A recursive, analytic, real-time state estimation algorithm for linear and nonlinear systems, referred to as the Multivariate Cauchy Estimator (MCE), is presented. The algorithm enables robust state estimation performance for applications where the system noises are more volatile than the Gaussian distribution suggests. This is achieved by over-bounding realistic process and measurement noises with additive, heavy-tailed Cauchy random variables. The characteristic function of the un-normalized conditional probability density function is propagated as a growing sum of terms in the MCE due to the closed form of a convolution integral. Each term of the characteristic function is propagated and updated through an enumeration table generated from a central arrangement of hyperplanes and consistent with the convolution integrals solution. Besides being a great numeric simplification, many terms can be combined, thereby eliminating over 99% of terms that previously comprised this characteristic function. To completely truncate the growing sum, a sliding, fixed measurement window is developed such that each window is initialized from the conditional mean and variance from the last completed window through a simple rotation matrix of a positive definite symmetric matrix. Through the use of graphical processing units, the MCE can exploit its parallel mathematical structure and achieve real-time performance. Nonlinearities are included by linearization about the current conditional mean. Three illustrations are presented. For a lightly damped pendulum, robustness to parameter variations is demonstrated. For a homing missile engagement in different levels of radar clutter, Monte Carlo simulations reveal that the estimation performance is notably robust. To demonstrate estimation performance in large dimensional problems, a seven state low Earth orbital dynamic system with three GPS pseudo-range measurements at each measurement time is forced by impulsive atmospheric density uncertainty.