The equations for a state-observer can be represented in state-space form. The equation for the plant is
$$\begin{matrix}
\dot{x}=Ax+Bu \\
y=Cx
\end{matrix}$$
and the equations for the plant model are
$$\begin{matrix}
\dot{\hat{x}}=A\hat{x}+Bu \\
\hat{y}=C\hat{x}
\end{matrix}$$
Where $x$ and $\hat{x}$ are the unobservable internal states, and $y$ and $\hat{y}$ are the observable output states. When we add in the closed-loop gain $K$ we get an updated equation for $\dot{\hat{x}}$
$$\dot{\hat{x}}=A\hat{x}+Bu+K(y-\hat{y})$$
When we subtract the equations for $\dot{x}$ and $\dot{\hat{x}}$ from each other we get the equations for the error of the system
$$
\begin{matrix}
\dot{x}-\dot{\hat{x}}=Ax-A\hat{x}+Bu-Bu-K(y-\hat{y}) \\
y-\hat{y}=C(x-\hat{x})
\end{matrix}$$
These can be simplified down to
$$
\begin{matrix}
\dot{x}-\dot{\hat{x}}=A(x-\hat{x})-KC(x-\hat{x}) \\
y-\hat{y}=C(x-\hat{x})
\end{matrix}$$
Because $x-\hat{x}$ is the observation error we can simplify these further to
$$
\begin{matrix}
\dot{e}=(A-KC)e \\
y-\hat{y}=Ce
\end{matrix}$$
The solution of which is an exponential function of time
$$e(t)=e^{(A-KC)t}e(0)$$
If $A-KC$ is less than zero then we know that the error will converge to zero. The faster that this function converges to zero the faster that the estimated state converges to the actual state.
- UnderstandingKalmanFiltersa