Observability for Kalman Filtering?

I wanted to know how observability of a stochastic state space system affects the performance of a Kalman Filter. Do we check for the usual observability matrix involving $\mathbf{C}$ (observation matrix) and $\mathbf{A}$ (state transition matrix) or there is a newer notion of stochastic observability? In case of non-observability, does the error between true and estimated states go out of bound ?

• it's one of the few things i remember about the Kalman filter from grad school. the notion of observability for the KF is precisely the same notion of observability from state-variable control system theory. (it's the $\mathbf{A}$ and $\mathbf{C}$ matrix thing.) and, the result of the KF are estimates of the states of the state-variable system. not directly an estimate of the signal, which you can get with the estimate of the states and the $\mathbf{C}$ matrix. – robert bristow-johnson Apr 25 '16 at 3:17

1) The "usual" tests for observability do not incorporate the noise covariance matrices, $Q_k$, $R_k$ or $P_0$.
2) Regardless of whether the $A$ and $C$ matrices satisfy the observability condition, $Q_k$, $R_k$, or $P_0$ could cause the state covariance matrix to become unbounded or exceed a predefined threshold value.