control.design_kalman_filter

control.design_kalman_filter(
    A,
    C,
    Q,
    R,
    system_type='discrete',
    backend='numpy',
)

Design Kalman filter for optimal state estimation.

For linear system with Gaussian noise: x[k+1] = Ax[k] + Bu[k] + w[k], w ~ N(0, Q) (process noise) y[k] = Cx[k] + v[k], v ~ N(0, R) (measurement noise)

Kalman filter provides optimal state estimate: Discrete: x̂[k+1] = Ax̂[k] + Bu[k] + L(y[k] - Cx̂[k]) Continuous: ˙x̂ = Ax̂ + Bu + L(y - Cx̂)

Minimizes steady-state estimation error covariance.

Args: A: State matrix (nx, nx) C: Output matrix (ny, nx) Q: Process noise covariance (nx, nx), Q ≥ 0 R: Measurement noise covariance (ny, ny), R > 0 system_type: ‘continuous’ or ‘discrete’ backend: Computational backend

Returns: KalmanFilterResult containing: - gain: Kalman gain L (nx, ny) - estimation_error_covariance: Steady-state error covariance P (nx, nx) - innovation_covariance: Innovation covariance S = CPC’ + R (ny, ny) - estimator_eigenvalues: Eigenvalues of (A - LC)

Raises: ValueError: If matrices have incompatible shapes or invalid system_type LinAlgError: If Riccati equation has no solution

Examples

>>> # Discrete Kalman filter
>>> A = np.array([[1, 0.1], [0, 0.95]])  # Slightly unstable
>>> C = np.array([[1, 0]])  # Measure position only
>>> Q = 0.01 * np.eye(2)    # Small process noise
>>> R = np.array([[0.1]])   # Measurement noise
>>>
>>> result = design_kalman_filter(A, C, Q, R, system_type='discrete')
>>> L = result['gain']
>>> print(f"Kalman gain: {L}")
>>>
>>> # Use in estimation loop
>>> x_hat = np.zeros(2)
>>> for k in range(N):
...     # Prediction
...     x_hat_pred = A @ x_hat + B @ u[k]
...
...     # Correction
...     innovation = y[k] - C @ x_hat_pred
...     x_hat = x_hat_pred + L @ innovation
>>>
>>> # Check estimator stability
>>> print(f"Estimator stable: {np.all(np.abs(result['estimator_eigenvalues']) < 1)}")
>>>
>>> # Continuous Kalman filter
>>> result_c = design_kalman_filter(A, C, Q, R, system_type='continuous')
>>> L_c = result_c['gain']

Notes

For observability, (A, C) must be observable
For detectability, unstable modes must be observable
Q must be positive semi-definite (process noise)
R must be positive definite (measurement noise)
Kalman filter is optimal for linear Gaussian systems
For nonlinear systems, use Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF)

# control.design_kalman_filter { #cdesym.control.design_kalman_filter } ```python control.design_kalman_filter( A, C, Q, R, system_type='discrete', backend='numpy', ) ``` Design Kalman filter for optimal state estimation. For linear system with Gaussian noise: x[k+1] = Ax[k] + Bu[k] + w[k], w ~ N(0, Q) (process noise) y[k] = Cx[k] + v[k], v ~ N(0, R) (measurement noise) Kalman filter provides optimal state estimate: Discrete: x̂[k+1] = Ax̂[k] + Bu[k] + L(y[k] - Cx̂[k]) Continuous: ˙x̂ = Ax̂ + Bu + L(y - Cx̂) Minimizes steady-state estimation error covariance. Args: A: State matrix (nx, nx) C: Output matrix (ny, nx) Q: Process noise covariance (nx, nx), Q ≥ 0 R: Measurement noise covariance (ny, ny), R > 0 system_type: 'continuous' or 'discrete' backend: Computational backend Returns: KalmanFilterResult containing: - gain: Kalman gain L (nx, ny) - estimation_error_covariance: Steady-state error covariance P (nx, nx) - innovation_covariance: Innovation covariance S = CPC' + R (ny, ny) - estimator_eigenvalues: Eigenvalues of (A - LC) Raises: ValueError: If matrices have incompatible shapes or invalid system_type LinAlgError: If Riccati equation has no solution ## Examples {.doc-section .doc-section-examples} ```python >>> # Discrete Kalman filter >>> A = np.array([[1, 0.1], [0, 0.95]]) # Slightly unstable >>> C = np.array([[1, 0]]) # Measure position only >>> Q = 0.01 * np.eye(2) # Small process noise >>> R = np.array([[0.1]]) # Measurement noise >>> >>> result = design_kalman_filter(A, C, Q, R, system_type='discrete') >>> L = result['gain'] >>> print(f"Kalman gain: {L}") >>> >>> # Use in estimation loop >>> x_hat = np.zeros(2) >>> for k in range(N): ... # Prediction ... x_hat_pred = A @ x_hat + B @ u[k] ... ... # Correction ... innovation = y[k] - C @ x_hat_pred ... x_hat = x_hat_pred + L @ innovation >>> >>> # Check estimator stability >>> print(f"Estimator stable: {np.all(np.abs(result['estimator_eigenvalues']) < 1)}") >>> >>> # Continuous Kalman filter >>> result_c = design_kalman_filter(A, C, Q, R, system_type='continuous') >>> L_c = result_c['gain'] ``` ## Notes {.doc-section .doc-section-notes} - For observability, (A, C) must be observable - For detectability, unstable modes must be observable - Q must be positive semi-definite (process noise) - R must be positive definite (measurement noise) - Kalman filter is optimal for linear Gaussian systems - For nonlinear systems, use Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF)