def subspace_identification(self):
"""
Perform subspace identification based on the PO-MOESP method.
The instrumental variable contains past outputs and past inputs.
The implementation uses a QR-decomposition for numerical efficiency and is based on page 329 of [1].
A key result of this function is the eigenvalue decomposition of the :math:`R_{32}` matrix
``self.R32_decomposition``, based on which the order of the system should be determined.
[1] Verhaegen, Michel, and Vincent Verdult. *Filtering and system identification: a least squares approach.*
Cambridge university press, 2007.
"""
u_hankel = Utils.block_hankel_matrix(self.u_array, self.num_block_rows)
y_hankel = Utils.block_hankel_matrix(self.y_array, self.num_block_rows)
u_past, u_future = u_hankel[:, :-self.
num_block_rows], u_hankel[:, self.
num_block_rows:]
y_past, y_future = y_hankel[:, :-self.
num_block_rows], y_hankel[:, self.
num_block_rows:]
u_instrumental_y = np.concatenate([u_future, u_past, y_past, y_future])
q, r = map(lambda matrix: matrix.T,
np.linalg.qr(u_instrumental_y.T, mode='reduced'))
y_rows, u_rows = self.y_dim * self.num_block_rows, self.u_dim * self.num_block_rows
self.R32 = r[-y_rows:, u_rows:-y_rows]
self.R22 = r[u_rows:-y_rows, u_rows:-y_rows]
self.R32_decomposition = Utils.eigenvalue_decomposition(self.R32)
def _get_observability_matrix_decomposition(self) -> Decomposition:
"""
Calculate the eigenvalue decomposition of the estimate of the observability matrix as per N4SID.
"""
u_hankel = Utils.block_hankel_matrix(self.u_array, self.num_block_rows)
y_hankel = Utils.block_hankel_matrix(self.y_array, self.num_block_rows)
u_and_y = np.concatenate([u_hankel, y_hankel])
observability = self.R32 @ np.linalg.pinv(self.R22) @ u_and_y
observability_decomposition = Utils.reduce_decomposition(
Utils.eigenvalue_decomposition(observability), self.x_dim)
return observability_decomposition
def test_eigenvalue_decomposition(self):
matrix = np.fliplr(np.diag(range(1, 3)))
decomposition = Utils.eigenvalue_decomposition(matrix)
self.assertTrue(
np.all(
np.isclose([[0, -1], [-1, 0]], decomposition.left_orthogonal)))
self.assertTrue(
np.all(np.isclose([2, 1], np.diagonal(decomposition.eigenvalues))))
self.assertTrue(
np.all(
np.isclose([[-1, 0], [0, -1]],
decomposition.right_orthogonal)))
reduced_decomposition = Utils.reduce_decomposition(decomposition, 1)
self.assertTrue(
np.all(
np.isclose([[0], [-1]],
reduced_decomposition.left_orthogonal)))
self.assertTrue(
np.all(np.isclose([[2]], reduced_decomposition.eigenvalues)))
self.assertTrue(
np.all(
np.isclose([[-1, 0]], reduced_decomposition.right_orthogonal)))