Kalman Filter in Continuous Time

If we find the system error, $e$ and $\dot{e}$, in a similar fashion as we did in the Closed-loop Estimator - Luenberger Observer, we get.

$$\begin{array}{r}\dot{e}=(A-LC)e+Gw-Lv\end{array}$$

Where $A-LC$ determines the eigenvalues.

If we take the expected value of the system, $E[\cdot ]$, we get

$${\dot{m}}_e=(A-LC)m_e$$

Where $m_e$ is the mean of the error.

IMPORTANT: This means that if $m_e(0)=0$, we get that $m_e(t)=0$. This isn’t usually the case, and normally it converges towards $m_e(0)$ then stays there.