What is a state observer and how is it designed?

A state observer estimates unmeasured states from output measurements. Full-order Luenberger observer: dx_hat/dt = Ax_hat + Bu + L(y - Cx_hat). Observer gain L is chosen to make (A - LC) have desired eigenvalues, typically 2-5 times faster than controller poles. System must be observable. Reduced-order observers estimate only unmeasured states. Separation principle allows independent design of controller and observer gains. Kalman filter is optimal observer for systems with noise, minimizing estimation error covariance.