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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 ?

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  • $\begingroup$ 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. $\endgroup$ – robert bristow-johnson Apr 25 '16 at 3:17
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The only quick reference I can find is this one, which states:

enter image description here

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  • $\begingroup$ The link to the quick reference is timing out for me (not a 404, but no response). Any chance you can fix it? $\endgroup$ – Reb.Cabin Apr 4 '17 at 22:45
  • $\begingroup$ @Reb.Cabin : The link still opens directly for me. I've made a copy of it on my private Dropbox. Please try this link. $\endgroup$ – Peter K. Apr 5 '17 at 7:42
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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.

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