control.design_lqg

control.design_lqg(
    A,
    B,
    C,
    Q_state,
    R_control,
    Q_process,
    R_measurement,
    N=None,
    system_type='discrete',
    backend='numpy',
)

Design Linear Quadratic Gaussian (LQG) controller.

Combines LQR controller with Kalman filter estimator. Separation principle allows independent design of controller and estimator.

System: x[k+1] = Ax[k] + Bu[k] + w[k], w ~ N(0, Q_process) y[k] = Cx[k] + v[k], v ~ N(0, R_measurement)

Controller: u[k] = -Kx̂[k] (feedback on estimate) Estimator: x̂[k+1] = Ax̂[k] + Bu[k] + L(y[k] - Cx̂[k])

Args: A: State matrix (nx, nx) B: Input matrix (nx, nu) C: Output matrix (ny, nx) Q_state: LQR state cost matrix (nx, nx) R_control: LQR control cost matrix (nu, nu) Q_process: Process noise covariance (nx, nx) R_measurement: Measurement noise covariance (ny, ny) system_type: ‘continuous’ or ‘discrete’ backend: Computational backend

Returns: LQGResult containing: - controller_gain: LQR gain K (nu, nx) - estimator_gain: Kalman gain L (nx, ny) - controller_riccati: LQR Riccati solution P_c (nx, nx) - estimator_covariance: Kalman covariance P_e (nx, nx) - controller_eigenvalues: Controller eigenvalues of (A - BK) - estimator_eigenvalues: Estimator eigenvalues of (A - LC)

Examples

>>> # Design LQG controller
>>> A = np.array([[1, 0.1], [0, 0.9]])
>>> B = np.array([[0], [0.1]])
>>> C = np.array([[1, 0]])  # Measure position
>>>
>>> # LQR weights
>>> Q_state = np.diag([10, 1])
>>> R_control = np.array([[0.1]])
>>>
>>> # Noise covariances
>>> Q_process = 0.01 * np.eye(2)
>>> R_measurement = np.array([[0.1]])
>>>
>>> result = design_lqg(
...     A, B, C,
...     Q_state, R_control,
...     Q_process, R_measurement,
...     system_type='discrete'
... )
>>>
>>> K = result['controller_gain']
>>> L = result['estimator_gain']
>>>
>>> # Implementation
>>> x_hat = np.zeros(2)
>>> for k in range(N):
...     # Control
...     u = -K @ x_hat
...
...     # Estimation
...     x_hat_pred = A @ x_hat + B @ u
...     innovation = y[k] - C @ x_hat_pred
...     x_hat = x_hat_pred + L @ innovation

Notes

• Separation principle: LQR and Kalman can be designed independently
• LQG is optimal for linear systems with Gaussian noise
• Closed-loop has eigenvalues of both controller and estimator
• Controller must stabilize system
• Estimator must converge faster than controller for good performance
• Trade-off: Lower Q_process/R_measurement → more aggressive estimator