arx

Created on Wed Jul 26 2017

@author: Giuseppe Armenise

Functions:

Name	Description
ARX_MISO_id	Auto-Regressive with eXogenous Inputs model (ARX) identification.
ARX_id	Auto-Regressive with eXogenous Inputs model (ARX) identification.
compute_num_den	Compute the numerator and denominator coefficients.
compute_phi	Compute the regressor matrix PHI.
compute_theta	Computes the parameter vector THETA, the model output y_id, and the estimated error norm Vn.

ARX_MISO_id

ARX_MISO_id(
    y: ndarray,
    u: ndarray,
    na: int,
    nb: ndarray,
    theta: ndarray,
) -> tuple[
    ndarray, ndarray, ndarray, ndarray, floating, ndarray
]

Auto-Regressive with eXogenous Inputs model (ARX) identification.

Identified through the computation of the pseudo-inverse of the regressor matrix ($ \phi $).

Parameters:

Name	Type	Description	Default
y	ndarray	Measured data	required
u	ndarray	Input data	required
na	int	Order of the autoregressive part.	required
nb	ndarray	Order of the exogenous part.	required
theta	ndarray	Delay of the exogenous part.	required

Returns:

Name	Type	Description
numerator	ndarray
denominator	ndarray
numerator_h	ndarray
denominator_h	ndarray
Vn	floating	The estimated error norm.
y_id	ndarray	The model output including non-identified outputs.

ARX_id

ARX_id(
    y: ndarray,
    u: ndarray,
    na: int,
    nb: ndarray,
    theta: ndarray,
    y_std: float = 1.0,
    U_std: ndarray = array(1.0),
) -> tuple[
    ndarray, ndarray, ndarray, ndarray, floating, ndarray
]

Auto-Regressive with eXogenous Inputs model (ARX) identification.

Identified through the computation of the pseudo-inverse of the regressor matrix ($ \phi $).

Parameters:

Name	Type	Description	Default
y	ndarray	Measured data	required
u	ndarray	Input data	required
na	int	Order of the autoregressive part.	required
nb	ndarray	Order of the exogenous part.	required
theta	ndarray	Delay of the exogenous part.	required
y_std	float	Standard deviation of the output data.	1.0
U_std	ndarray	Standard deviation of the input data.	array(1.0)

Returns:

Name	Type	Description
numerator	ndarray
denominator	ndarray
numerator_h	ndarray
denominator_h	ndarray
Vn	floating	The estimated error norm.
y_id	ndarray	The model output including non-identified outputs.

compute_num_den

compute_num_den(
    THETA: ndarray,
    na: int,
    nb: ndarray,
    theta: ndarray,
    val: int,
    udim: int = 1,
    y_std: ndarray | float = 1.0,
    U_std: ndarray | float = array(1.0),
) -> tuple[ndarray, ndarray]

Compute the numerator and denominator coefficients.

Parameters:

Name	Type	Description	Default
THETA	ndarray	Coefficient vector.	required
na	int	Order of the autoregressive part.	required
nb	ndarray	Order of the exogenous part.	required
theta	ndarray	Delay of the exogenous part.	required
val	int	Maximum predictable order.	required
udim	int	Dimension of the input data.	1
y_std	ndarray | float	Standard deviation of the output data.	1.0
U_std	ndarray | float	Standard deviation of the input data.	array(1.0)

Returns:

Name	Type	Description
tuple	tuple[ndarray, ndarray]	Denominator coefficients, numerator coefficients, and numerator_h.

Examples:

>>> THETA = np.array([0.5, -0.2, 0.3, 0.1])
>>> na = 2
>>> nb = np.array([2])
>>> theta = np.array([1])
>>> val = 3
>>> udim = 1
>>> compute_num_den(THETA, na, nb, theta, val, udim)
(array([0. , 0.3, 0.1]), array([ 1. ,  0.5, -0.2,  0. ]))

>>> THETA = np.array([0.5, -0.2, 0.3, 0.1, 0.4, 0.2])
>>> na = 2
>>> nb = np.array([2, 2])
>>> theta = np.array([1, 1])
>>> val = 3
>>> udim = 2
>>> compute_num_den(THETA, na, nb, theta, val, udim, y_std=1, U_std=[1.0, 1.0])
(array([[0. , 0.3, 0.1],
    [0. , 0.4, 0.2]]), array([[ 1. ,  0.5, -0.2,  0. ],
    [ 1. ,  0.5, -0.2,  0. ]]))

compute_phi

compute_phi(
    y: ndarray,
    u: ndarray,
    na: int,
    nb: ndarray,
    theta: ndarray,
    val: int,
    N: int,
    udim: int = 1,
) -> ndarray

Compute the regressor matrix PHI.

Parameters:

Name	Type	Description	Default
y	ndarray	Output data.	required
u	ndarray	Input data.	required
na	int	Order of the autoregressive part.	required
nb	ndarray	Order of the exogenous part.	required
theta	ndarray	Delay of the exogenous part.	required
val	int	Maximum predictable order.	required
N	int	Number of data points.	required
udim	int	Dimension of the input data.	1

Returns:

Type	Description
ndarray	Regressor matrix PHI.

Examples:

>>> y = np.array([1, 2, 3, 4, 5])
>>> u = np.array([1, 2, 3, 4, 5])
>>> na = 2
>>> nb = np.array([2])
>>> theta = np.array([1])
>>> val = 2
>>> N = 3
>>> compute_phi(y, u, na, nb, theta, val, N, udim=1)
array([[-2., -1.,  2.,  1.],
    [-3., -2.,  3.,  2.],
    [-4., -3.,  4.,  3.]])

>>> y = np.array([1, 2, 3, 4, 5])
>>> u = np.array([[1, 2, 3, 4, 5], [5, 4, 3, 2, 1]])
>>> na = 2
>>> nb = np.array([2, 2])
>>> theta = np.array([1, 1])
>>> val = 2
>>> N = 3
>>> compute_phi(y, u, na, nb, theta, val, N, udim=2)
array([[-2., -1.,  1.,  1.,  5.,  5.],
    [-3., -2.,  2.,  1.,  4.,  5.],
    [-4., -3.,  3.,  2.,  3.,  4.]])

compute_theta

compute_theta(
    PHI: ndarray, y: ndarray, val: int, y_std=1.0
) -> tuple[ndarray, ndarray, floating]

Computes the parameter vector THETA, the model output y_id, and the estimated error norm Vn.

Parameters:

Name	Type	Description	Default
PHI	ndarray	The regression matrix.	required
y	ndarray	The output vector.	required
val	int	The index from which to start the validation.	required

Returns:

Name	Type	Description
THETA	ndarray	The parameter vector.
y_id	ndarray	The model output including non-identified outputs.
Vn	floating	The estimated error norm.

Examples:

>>> import numpy as np
>>> PHI = np.array([[1, 2], [3, 4], [5, 6]])
>>> y = np.array([1, 2, 3, 4])
>>> val = 1
>>> THETA, y_id, Vn = compute_theta(PHI, y, val)
>>> THETA
array([-1. ,  1.5])
>>> y_id
array([1., 2., 3., 4.])
>>> round(Vn, 6)
np.float64(0.0)