def test_modal_form_empty():
# empty system should be returned as-is
# t empty matrix
insys = ss([], [], [], 123)
outsys, t = modal_form(insys)
np.testing.assert_array_equal(outsys.A, insys.A)
np.testing.assert_array_equal(outsys.B, insys.B)
np.testing.assert_array_equal(outsys.C, insys.C)
np.testing.assert_array_equal(outsys.D, insys.D)
assert t.shape == (0, 0)
def test_modal_form_condmax(condmax, len_blksizes):
# condmax passed through as expected
a = companion_from_eig([-1, -2, -3, -4, -5])
amodal, tmodal, blksizes = bdschur(a, condmax=condmax)
assert len(blksizes) == len_blksizes
xsys = ss(a, [[1],[0],[0],[0],[0]], [0,0,0,0,1], 0)
zsys, t = modal_form(xsys, condmax=condmax)
np.testing.assert_array_almost_equal(zsys.A, amodal)
np.testing.assert_array_almost_equal(t, tmodal)
np.testing.assert_array_almost_equal(zsys.B, np.linalg.solve(tmodal, xsys.B))
np.testing.assert_array_almost_equal(zsys.C, xsys.C.dot(tmodal))
np.testing.assert_array_almost_equal(zsys.D, xsys.D)
def test_modal_form_sort(sys_type):
a = companion_from_eig([0.1+0.9j,0.1-0.9j, 0.2+0.8j, 0.2-0.8j])
amodal, tmodal, blksizes = bdschur(a, sort=sys_type)
dt = 0 if sys_type == 'continuous' else True
xsys = ss(a, [[1],[0],[0],[0],], [0,0,0,1], 0, dt)
zsys, t = modal_form(xsys, sort=True)
my_amodal = np.linalg.solve(tmodal, a).dot(tmodal)
np.testing.assert_array_almost_equal(amodal, my_amodal)
np.testing.assert_array_almost_equal(t, tmodal)
np.testing.assert_array_almost_equal(zsys.A, amodal)
np.testing.assert_array_almost_equal(zsys.B, np.linalg.solve(tmodal, xsys.B))
np.testing.assert_array_almost_equal(zsys.C, xsys.C.dot(tmodal))
np.testing.assert_array_almost_equal(zsys.D, xsys.D)
def test_modal_form(A_true, B_true, C_true, D_true):
# Check modal_canonical corresponds to bdschur
# Perform a coordinate transform with a random invertible matrix
T_true = np.array([[-0.27144004, -0.39933167, 0.75634684, 0.44135471],
[-0.74855725, -0.39136285, -0.18142339, -0.50356997],
[-0.40688007, 0.81416369, 0.38002113, -0.16483334],
[-0.44769516, 0.15654653, -0.50060858, 0.72419146]])
A = np.linalg.solve(T_true, A_true) @ T_true
B = np.linalg.solve(T_true, B_true)
C = C_true @ T_true
D = D_true
# Create a state space system and convert it to modal canonical form
sys_check, T_check = modal_form(ss(A, B, C, D))
a_bds, t_bds, _ = bdschur(A)
np.testing.assert_array_almost_equal(sys_check.A, a_bds)
np.testing.assert_array_almost_equal(T_check, t_bds)
np.testing.assert_array_almost_equal(sys_check.B,
np.linalg.solve(t_bds, B))
np.testing.assert_array_almost_equal(sys_check.C, C @ t_bds)
np.testing.assert_array_almost_equal(sys_check.D, D)
# canonical_form(...,'modal') is the same as modal_form with default parameters
cf_sys, T_cf = canonical_form(ss(A, B, C, D), 'modal')
np.testing.assert_array_almost_equal(cf_sys.A, sys_check.A)
np.testing.assert_array_almost_equal(cf_sys.B, sys_check.B)
np.testing.assert_array_almost_equal(cf_sys.C, sys_check.C)
np.testing.assert_array_almost_equal(cf_sys.D, sys_check.D)
np.testing.assert_array_almost_equal(T_check, T_cf)
# Make sure Hankel coefficients are OK
for i in range(A.shape[0]):
np.testing.assert_almost_equal(
C_true @ np.linalg.matrix_power(A_true, i) @ B_true,
C @ np.linalg.matrix_power(A, i) @ B)
def test_modal_form_MIMO(self):
"""Test error because modal form only supports SISO"""
sys = tf([[[1], [1]]], [[[1, 2, 1], [1, 2, 1]]])
with pytest.raises(ControlNotImplemented):
modal_form(sys)