46. Optimal control

• Deterministic/stochastic signals

• Full-state feedback, observer and full-state on observed state, output feedback

• Regulator (targeting zero state \(\mathbf{x} = \mathbf{0}\)), or reference tracking (targeting zero error of the output w.r.t. a reference input of the closed-loop system \(\mathbf{y}_{\text{ref}}\))

Deterministic signals.

• Full-state feedback, \(\mathbf{u} = - \mathbf{K} \mathbf{x}\)

• Observer, for state estimation

• Full-state feed back on the observed state, \(\mathbf{u} = - \mathbf{K} \mathbf{o}\)

• Output feedback, \(\mathbf{u} = - \mathbf{K} \mathbf{y}\)

Stochastic signals.

• Kalman filter, i.e. optimal observer w.r.t. white noise system exogenous input and measurement noise