When the full state cannot be measured directly, a state observer runs a model of the system and corrects it using the difference between predicted and measured outputs.

For

$$\dot x=Ax+Bu,$$

$$y=Cx,$$

a Luenberger observer is

$$\dot{\hat x}=A\hat x+Bu+L(y-C\hat x),$$

where $\hat x$ is the state estimate and $L$ is the observer gain.

Define the estimation error

$$e_x=x-\hat x.$$

Subtracting the observer dynamics from the plant dynamics gives

$$\dot e_x=(A-LC)e_x.$$

The observer poles are therefore the eigenvalues of $A-LC$. If the system is observable, $L$ can be chosen to place these poles in stable locations so the estimation error decays.

An observer is deterministic model-based estimation. Unlike a Kalman filter, its basic form does not require probabilistic process and measurement noise models.