Using the controller \(u=-K\hat{x}\) in the system \(\dot{x}=Ax+Bu\) Gives us
$$\dot{\hat{x}}=A\hat{x}+Bu+L(y-C\hat{x})$$
which has two sets of eigenvalues
$$\begin{matrix}
eig(A-BK) \
eig(A-LC)
\end{matrix}$$
This shows that the controller does not affect the estimator eigenvalues and the estimator eigenvalues don’t change the controller eigenvalues.
- bevlyMECH4420Lectureb