def test_simple_lft():
H = Transfer([1, 0.5, 9], [1, 5, 9])
assert_raises(ValueError, discretize, H, dt=0.25, method='lft', q=4)
assert_raises(ValueError, discretize, H, dt=0.25, method='lft', q=eye(3))
Hd = discretize(H, dt=0.25, method="lft", q=np.array([[1, .5], [.5, 0]]))
Hdf = discretize(H, dt=0.25, method=">>")
assert_array_almost_equal(Hd.num, Hdf.num)
assert_array_almost_equal(Hd.den, Hdf.den)
def test_simple_foh():
G = Transfer(1, [1, 0, 0])
dt_ = 0.1
H = discretize(G, dt=dt_, method='foh')
n, d = H.polynomials
assert_equal(H.SamplingPeriod, dt_)
assert_array_almost_equal(n, np.array([[5, 20, 5]]) / 3000)
assert_array_almost_equal(d, np.array([[1., -2., 1.]]))
G = Transfer(1, [1, 1])
dt_ = 0.1
H = discretize(G, dt=dt_, method='foh')
n, d = H.polynomials
assert_equal(H.SamplingPeriod, dt_)
assert_array_almost_equal(n, np.array([[0.04837418, 0.0467884]]))
assert_array_almost_equal(d, np.array([[1., -0.90483742]]))
def test_simple_tustin_prewarp():
# Example from B. de Moor's slides
H = Transfer([1, 0.5, 9], [1, 5, 9])
# prewarp at 3 rad/s
Hd = discretize(H, dt=0.5, prewarp_at=3 / 2 / np.pi, method='bilinear')
assert_array_almost_equal(
Hd.num, np.array([[0.591468698033, -0.0772558231247, 0.500683964262]]))
assert_array_almost_equal(
Hd.den, np.array([[1.0, -0.0772558231247, 0.0921526622952]]))
# while we are at it test the upper Nyquist limit on prewarping
assert_raises(ValueError, discretize, H, dt=0.5, prewarp_at=2)
def test_simple_tustin():
ac = np.eye(2)
bc = 0.5 * np.ones((2, 1))
cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
dc = np.array([[0.0], [0.0], [-0.33]])
dt_ = 0.5
G = State(ac, bc, cc, dc)
ad_truth = (5.0 / 3.0) * np.eye(2)
bd_truth = (1.0 / 3.0) * np.ones((2, 1)) / np.sin(np.pi/4)
cd_truth = np.array([[1.0, 4.0 / 3.0],
[4.0 / 3.0, 4.0 / 3.0],
[4.0 / 3.0, 1.0 / 3.0]]) * np.sin(np.pi/4)
dd_truth = np.array([[0.291666666666667],
[1.0 / 3.0],
[-0.121666666666667]])
H = discretize(G, dt_, method='bilinear')
ad, bd, cd, dd = H.matrices
assert_array_almost_equal(ad_truth, ad)
assert_array_almost_equal(bd_truth, bd)
assert_array_almost_equal(cd_truth, cd)
assert_array_almost_equal(dd_truth, dd)
assert_equal(H.SamplingPeriod, dt_)
# Same continuous system again, but change sampling rate
ad_truth = 1.4 * np.eye(2)
bd_truth = 0.2 * np.ones((2, 1)) / np.sqrt(1/3)
cd_truth = np.array([[0.9, 1.2], [1.2, 1.2], [1.2, 0.3]]) * np.sqrt(1/3)
dd_truth = np.array([[0.175], [0.2], [-0.205]])
dt_ = 1.0 / 3.0
H = discretize(G, dt_, method='bilinear')
ad, bd, cd, dd = H.matrices
assert_array_almost_equal(ad_truth, ad)
assert_array_almost_equal(bd_truth, bd)
assert_array_almost_equal(cd_truth, cd)
assert_array_almost_equal(dd_truth, dd)
assert_equal(H.SamplingPeriod, dt_)
def test_simple_zoh():
ac = eye(2)
bc = 0.5 * ones((2, 1))
cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
dc = np.array([[0.0], [0.0], [-0.33]])
G = State(ac, bc, cc, dc)
ad_truth = 1.648721270700128 * eye(2)
bd_truth = 0.324360635350064 * np.ones((2, 1))
# c and d in discrete should be equal to their continuous counterparts
dt_ = 0.5
H = discretize(G, dt=dt_, method='zoh')
ad, bd, cd, dd = H.matrices
assert_equal(H.SamplingPeriod, dt_)
assert_array_almost_equal(ad_truth, ad)
assert_array_almost_equal(bd_truth, bd)
assert_array_almost_equal(cc, cd)
assert_array_almost_equal(dc, dd)
def test_undiscretize():
ac = eye(2)
bc = 0.5 * ones((2, 1))
cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
dc = np.array([[0.0], [0.0], [-0.33]])
G = State(ac, bc, cc, dc)
for method in ('zoh', 'foh', 'tustin', '>>', '<<'):
print(method)
Gd = discretize(G, dt=0.5, method=method)
Gu = undiscretize(Gd)
assert_array_almost_equal(Gu.a, ac)
assert_array_almost_equal(Gu.b, bc)
assert_array_almost_equal(Gu.c, cc)
assert_array_almost_equal(Gu.d, dc)
Gd = State(zeros([2, 2]), ones([2, 1]), ones([1, 2]), dt=1)
Gd.DiscretizedWith = '<<'
assert_raises(ValueError, undiscretize, Gd)
def test_static_gain_discretization():
Gd = discretize(State(np.eye(5)), dt=1)
assert_equal(Gd._dt, 1)
def test_simple_lft():
H = Transfer([1, 0.5, 9], [1, 5, 9])
Hd = discretize(H, dt=0.25, method="lft", q=np.array([[1, .5], [.5, 0]]))
Hdf = discretize(H, dt=0.25, method=">>")
assert_array_almost_equal(Hd.num, Hdf.num)
assert_array_almost_equal(Hd.den, Hdf.den)
