If state feedback requires $x$ but only an estimate $\hat x$ is available, the controller can use

$$u=-K\hat x.$$

The plant and observer then operate together: the observer reconstructs the state from measurements, while the controller feeds that estimate back.

For a linear time-invariant system, the separation principle states that state-feedback poles and observer-error poles can be designed independently when the required controllability and observability conditions hold.

The combined closed-loop eigenvalues are the union of the eigenvalues of

$$A-BK$$

and

$$A-LC.$$

This does not mean estimation and control are physically unrelated. Measurement noise, actuator limits and modelling errors can still couple their practical performance.

The principle means something more precise: the nominal pole-placement calculations for controller gain $K$ and observer gain $L$ can be carried out separately and then combined.