Explicit formulas for two state Kalman, H2 and H∞ target tracking filters

The continuous time, two state, target tracking problem is considered from the Kalman, H₂, and H_∞ filter viewpoint. While previous treatments were numerical in nature, analytic transient responses and infinite horizon solutions with analytic performance expressions are presented here. Tracking indices, involving the maneuver and measurement uncertainties, are shown to have a role for both the steady state and transient responses. In addition, the H_∞ tracker has a sensor index involving the performance bound and measurement uncertainty, which, along with the tracking index, plays a significant role in the H_∞ tracker expressions. Analytical expressions for the probability of target escape, the probability that the target position will be outside the radar beamwidth (BW), are developed not only to compare the performance of various trackers, but also as a design tool to meet tracking specifications. Examples illustrate the performance of the target trackers as a function of the error gain upper bound.