base

Helper functions used by the Subspace Identification Methods and other useful functions for State Space models.

@author: Giuseppe Armenise

Functions:

Name	Description
Vn_mat	Compute the variance of the model residuals
Z_dot_PIort	Compute the scalar product between a vector z and \(I - x^T \cdot pinv(X^T)\), avoiding the direct computation of the matrix
lsim_innovation_form	Simulate system in a innovation form.
lsim_predictor_form	Simulate system in a predictor form.
lsim_process_form	Simulate system in a process form.

Vn_mat

Vn_mat(y: ndarray, yest: ndarray) -> ndarray

Compute the variance of the model residuals

Parameters:

Name	Type	Description	Default
y		(L*l,1) vectorized matrix of output of the process	required
yest		(L*l,1) vectorized matrix of output of the estimated model	required

Z_dot_PIort

Z_dot_PIort(z: ndarray, X: ndarray) -> ndarray

Compute the scalar product between a vector z and \(I - x^T \cdot pinv(X^T)\), avoiding the direct computation of the matrix

PI = np.dot(X.T, np.linalg.pinv(X.T)), causing high memory usage

Parameters:

Name	Type	Description	Default
z		(...) vector array_like	required
X		(...) matrix array_like	required

lsim_innovation_form

lsim_innovation_form(
    A: ndarray,
    B: ndarray,
    C: ndarray,
    D: ndarray,
    K: ndarray,
    y: ndarray,
    u: ndarray,
    x0: ndarray | None = None,
)

Simulate system in a innovation form.

This function performs a simulation in the innovation form. This function is analogous to the previous one, using the system matrices $ A $ and $ B $ instead of $ A_K $ and $ B_K $

lsim_predictor_form

lsim_predictor_form(
    A_K: ndarray,
    B_K: ndarray,
    C: ndarray,
    D: ndarray,
    K: ndarray,
    y: ndarray,
    u: ndarray,
    x0: ndarray | None = None,
)

Simulate system in a predictor form.

This function performs a simulation in the predictor form, given the identified system matrices, the Kalman filter gain, the real output sequence (array with \(n_y\) rows and L columns, the input sequence (an array with \(n_u\) rows and L columns) and the initial state estimate (array with \(n\) rows and one column). The state sequence and the estimated output sequence are returned.

lsim_process_form

lsim_process_form(
    A: ndarray,
    B: ndarray,
    C: ndarray,
    D: ndarray,
    u: ndarray,
    x0: ndarray | None = None,
)

Simulate system in a process form.

This function performs a simulation in the process form, given the identified system matrices, the input sequence (an array with \(n_u\) rows and L columns) and the initial state estimate (array with \(n\) rows and one column).