types.control_classical.LQGResult

types.control_classical.LQGResult()

Linear Quadratic Gaussian (LQG) controller design result.

LQG combines optimal control (LQR) with optimal estimation (Kalman): - LQR: Optimal state feedback u = -Kx (if state known) - Kalman: Optimal state estimate x̂ from measurements - LQG: Certainty equivalence u = -Kx̂

Separation Principle: - Design LQR and Kalman independently - Combine for optimal performance under Gaussian noise

Fields

control_gain : GainMatrix LQR feedback gain K (nu, nx) estimator_gain : GainMatrix Kalman gain L (nx, ny) control_cost_to_go : CovarianceMatrix Controller Riccati solution P_control estimation_error_covariance : CovarianceMatrix Estimator Riccati solution P_estimate separation_verified : bool Confirms separation principle holds closed_loop_stable : bool Combined controller-estimator system is stable controller_eigenvalues : np.ndarray Eigenvalues of (A - BK) - control loop estimator_eigenvalues : np.ndarray Eigenvalues of (A - LC) - estimation loop

Examples

>>> # System matrices
>>> A = np.array([[1, 0.1], [0, 0.9]])
>>> B = np.array([[0], [0.1]])
>>> C = np.array([[1, 0]])  # Measure position only
>>>
>>> # Design weights
>>> Q_control = np.diag([10, 1])   # State cost
>>> R_control = np.array([[0.1]])  # Control cost
>>> Q_process = 0.01 * np.eye(2)   # Process noise
>>> R_meas = 0.1 * np.eye(1)       # Measurement noise
>>>
>>> result: LQGResult = design_lqg(A, B, C, Q_control, R_control, Q_process, R_meas)
>>>
>>> K = result['control_gain']
>>> L = result['estimator_gain']
>>> print(result['closed_loop_stable'])  # True
>>> print(result['separation_verified'])  # True
>>>
>>> # Implement LQG controller
>>> x_hat = np.zeros(2)  # Initial estimate
>>> for k in range(N):
...     # Control (certainty equivalence)
...     u[k] = -K @ x_hat
...
...     # Prediction
...     x_hat = A @ x_hat + B @ u[k]
...
...     # Measurement update
...     innovation = y[k] - C @ x_hat
...     x_hat = x_hat + L @ innovation
>>>
>>> # Check eigenvalues
>>> print("Controller poles:", result['controller_eigenvalues'])
>>> print("Estimator poles:", result['estimator_eigenvalues'])