A Stochastic Observability Test for Discrete-Time Kalman Filters

Stochastic observability refers to the existence of a filter for which the errors of the estimated state mean vector have bounded variance. In this paper, we derive a test to assess the stochastic observability of a Kalman filter implemented for discrete linear time-varying stochastic systems. This test is derived with the assumptions that the system matrices consist of known, deterministic parameters and that there is complete uncertainty in the statistics of the initial state vector. This test can also be used to assess the stochastic observability of extended Kalman filters implemented for nonlinear stochastic systems linearized about the true state vector trajectory. We illustrate the utility of the stochastic observability test using an aided INS. We also provide a counterexample to illustrate that observability is a necessary, but not sufficient, condition for the stochastic observability of a Kalman filter implemented for a system.