Analysis of Riccati equations for the linear system with unobservable biases and its application on the INS

This paper studies the existence of the ARE solution and the convergence of the DRE solution for the linear system with an unobservable bias, while it is assumed that bias free system is observable and controllable. Until now, it has been believed that the sufficient condition for an optimal estimator to be stable is that the system is at least detectable. The result of this study shows that for LSUB, the ARE has infinite number of strong solutions, and the limiting solution of the DRE depends on the initial condition unlike a detectable system. It is also shown that the estimation errors are biased. The validity of these results are illustrated through a 2nd order LSUB example and an INS alignment for more real application.