The two-step optical estimator and example applications

The two-step filter is a new approach for nonlinear recursive estimation that substantially improves the estimate error relative to the extended Kalman Filter (EKF) or the iterated extended Kalman filter(IEKF). Historically, when faced with an optimal estimation problem involving a set of nonlinear measurements, designers have been forced to choose between optimal, but off-line, iterative batch techniques or sub-optimal, approximate techniques, typically the EKF or IEKF. These techniques linearize the measurements and dynamics to take advantage of the well known Kalman filter equations. While broadly used, these filters typically result in sub-optimal and biased estimates and often can go unstable. The two-step estimator, introduced in 1996, provides a dramatic improvement over these filters for situations with nonlinear measurements. It accomplishes this by dividing the estimation problem (a quadratic minimization) into two-steps - a linear first step and a non-linear second step. The result is a filter that comes much closer to minimizing the desired cost, virtually eliminating any biases and dramatically reducing the mean-square error relative to the EKF. This paper presents an overview of the two-step estimator, outlining the derivation of the two-step measurement update and cost function minimization. It also presents the newest time update, resulting in a robust and accurate estimation technique. This presentation is followed by several simple aerospace examples to illustrate the utility of the filter and its improvement over the EKF and IEKF. These include both open loop estimation and closed loop control applications.