Observability analysis for tracking of coordinated turn maneuvers

Various models for tracking coordinated turn maneuvers where suggested in the literature for two-dimensional (2D) or three-dimensional (3D) tracking with known or unknown target turn rate. In some papers a comparison of the estimation performance between several different models is made. Yet, it seems, that an observability check of coordinated turn (CT) models is lacking in the literature. In this paper to fill this gap. To begin with, we extend the survey paper of tracking maneuvering targets with recent new CT models. For each model, the observability matrix is constructed and a binary check of yes/no observability is made. In the linear models this check is a sufficient condition for global observability, yet in the nonlinear case it is only a necessary condition for observability at the current time step. Therefore, for the nonlinear models we calculate the observability Gramian. Whereas a full rank Gramian matrix indicates an observable system, a rank deficiency indicates that only a sub-space of the state-vector is observable. For proper estimation analysis, the question of which states (or some combination of them) are observable and which are not is of great importance. This question may be answered numerically or analytically by the null-space of the Gramian matrix. Herein, we employ the latter approach and derive the unobservable subspace for the models found to be incompletely observable.