Lyapunov Stability Analysis of an Orbit Determination Problem

Statistical orbit determination is an important part of tracking satellites and predicting their future location. This is accomplished by processing the raw observation data from ground stations tracking satellites in orbit and estimating the location of these satellites. Kalman filter theory is a popular statistical process but it is not commonly used for satellite orbit determination because it exhibits poor stability characteristics in this application. In particular, an Extended Kalman Filter (EKF) is needed for this application because the orbit determination problem is a nonlinear one. The EKF does not lend itself well to the traditional eigenvalue stability analysis, which is applicable for linear, time-invariant systems. If the stability characteristics of the EKF used for the orbit determination problem were better understood, it could become useful for these applications. A stability analysis concept which has received much attention is Lyapunov’s stability theory. The focus of this paper is to apply the second method of Lyapunov (or the direct method) to evaluate the stability characteristics of an EKF estimating a satellite’s orbit using simulated radar data from ground tracking stations at Mahe Island and Thule. Specifically, a Lyapunov function is used to analyze the stability characteristics of that EKF while selecting the weighting matrices of the filter for the satellite orbit determination problem. © 1997, American Astronautical Society.