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)))