# 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. Apr 25, 2016 at 3:17

The only quick reference I can find is this one, which states: • The link to the quick reference is timing out for me (not a 404, but no response). Any chance you can fix it? Apr 4, 2017 at 22:45
• @Reb.Cabin : The link still opens directly for me. I've made a copy of it on my private Dropbox. Please try this link.
– Peter K.
Apr 5, 2017 at 7:42

My answer refers to your first question whether it matters to use the usual observability matrix to test for observability in a system with noise. According to this paper it matters. Their justification is the following:

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.

With respect to the above, they propose a stochastic observability test for which you can find more details about it, in the paper above.