direct_data_driven_mpc.utilities.hankel_matrix

direct_data_driven_mpc.utilities.hankel_matrix#

Functions for constructing Hankel matrices and evaluating persistent excitation.

This module provides functions for constructing Hankel matrices from multidimensional data sequences and for evaluating whether a given data sequence is persistently exciting of a given order based on the rank of its Hankel matrix.

Functions

evaluate_persistent_excitation(X, order)

Evaluate whether a data sequence X is persistently exciting of a given order based on the rank of its Hankel matrix.

hankel_matrix(X, L)

Construct a Hankel matrix from the input data matrix X with a window length L.
hankel_matrix(X: ndarray, L: int) → ndarray[source]#

Construct a Hankel matrix from the input data matrix X with a window length L. The matrix X consists of a sequence of N elements, each of length n.

Parameters:

  • X (numpy.ndarray) – Input data matrix of shape (N, n), where N is the number of elements, and n is the length of each element.

  • L (int) – Window length for the Hankel matrix.

Returns:

np.ndarray – A Hankel matrix of shape (L * n, N - L + 1), where each column represents a flattened window of length L sliding over the N data elements.

Raises:

ValueError – If the number of elements N is less than the window length L, indicating that the window length exceeds the available data length.

Examples

>>> import numpy as np
>>> N = 4  # Data length
>>> L = 2  # Hankel matrix window length
>>> n = 2  # Data vector length
>>> rng = np.random.default_rng(0)  # RNG for reproducibility
>>> u_d = rng.uniform(-1, 1, (N, n))  # Generate data matrix
>>> print(hankel_matrix(u_d, L))
[[ 0.27392337 -0.91805295  0.62654048]
 [-0.46042657 -0.96694473  0.82551115]
 [-0.91805295  0.62654048  0.21327155]
 [-0.96694473  0.82551115  0.45899312]]
evaluate_persistent_excitation(X: ndarray, order: int) → tuple[int, bool][source]#

Evaluate whether a data sequence X is persistently exciting of a given order based on the rank of its Hankel matrix. The matrix X consists of a sequence of N elements, each of length n.

This is determined by checking if the rank of the Hankel matrix constructed from X is greater than or equal to the expected rank n * order.

Parameters:

  • X (numpy.ndarray) – Input data matrix of shape (N, n), where N is the number of elements, and n is the length of each element.

  • order (int) – The order of persistent excitation to evaluate.

Returns:

tuple[int, bool] – A tuple containing the rank of the Hankel matrix and a boolean indicating whether X is persistently exciting of the given order.

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