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Input-Output Models

Every identified linear input-output model is returned according to the following structure:

\[ y_k = G(z)u_k + H(z)e_k \]

where \( G(z) \) and \( H(z) \) are transfer function matrices of polynomials in \( z \), which is the forward shift operator (see Figure~ ef{fig:gen_model}).

Modules:

Name	Description
armax	Created on Fri Jul 28 2017
arx	Created on Wed Jul 26 2017
base	Helper functions for nonlinear optimization problem used by some of the identification functions.
opt	Created on 2021
rls	Created on 2021

Classes:

Name	Description
Armax

Functions:

Name	Description
ARX_MISO_id	Auto-Regressive with eXogenous Inputs model (ARX) identification.
ARX_id	Auto-Regressive with eXogenous Inputs model (ARX) identification.

Armax

Armax(
    G: TransferFunction,
    H: TransferFunction,
    *order_bounds: tuple[int, int],
    Vn: float | floating,
    y_id: ndarray,
    method: ICMethods = "AIC"
)

The AutoRegressive-Moving-Average with eXogenous inputs model is computed based on a recursive lest-square regression between the input data (U) and the measured output data (Y). As Y is noisy in practice, a white noise (E) is identified within the model. This model is designed to deal with potential time-delays between U and Y.

SIPPY implements an iterative procedure, with extit{iterative least-square regression} = ILLS.

The following equations summarize the equations involved in the model:

$$ Y = G.U + H.E

G = B / A H = C / A

A = 1 + a_1z^(-1) + ... + a_naz^(-na) B = b_1z^(-1-theta) + ... + b_nbz^(-nb-theta) C = c_1z^(-1) + ... + c_ncz^(-nc) $$

Parameters:

Name	Type	Description	Default
G	TransferFunction	output response	required
H	TransferFunction	noise response	required
order_bounds	tuple[int, int]	extended range of the order of: - na_bounds: the common denominator - nb_bounds: the G numerator - nc_bounds: the H numerator - theta_bounds: the discrete theta in B	()
Vn	float | floating	The estimated error norm.	required
y_id	ndarray	The model output including non-identified outputs.	required
method	ICMethods	Method used of to attribute a performance to the model	'AIC'

References

https://ieeexplore.ieee.org/abstract/document/8516791

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.