Chapter 5: Full State Feedback Control

State Feedback Controller

$u = -Kx + r$

Where K is the feedback gain matrix.

Pole Placement

Design K to place closed-loop poles at desired locations.

from scipy import signal

def pole_placement(A, B, desired_poles):
    K = signal.place_poles(A, B, desired_poles).gain_matrix
    return K

LQR (Linear Quadratic Regulator)

Minimize cost function: $J = \int_0^\infty (x^TQx + u^TRu) dt$

from scipy import linalg

def lqr(A, B, Q, R):
    P = linalg.solve_continuous_are(A, B, Q, R)
    K = np.linalg.inv(R) @ B.T @ P
    return K

# Example
Q = np.diag([10, 1])  # State cost
R = np.array([[1]])    # Control cost
K = lqr(A, B, Q, R)

Next: Chapter 6 - Model Predictive Control!