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speech2derive.old/lib/python3.8/site-packages/scipy/signal/tests/test_cont2discrete.py

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import numpy as np
from numpy.testing import \
assert_array_almost_equal, assert_almost_equal, \
assert_allclose, assert_equal
import pytest
from scipy.signal import cont2discrete as c2d
from scipy.signal import dlsim, ss2tf, ss2zpk, lsim2, lti
from scipy.signal import tf2ss, impulse2, dimpulse, step2, dstep
# Author: Jeffrey Armstrong <jeff@approximatrix.com>
# March 29, 2011
class TestC2D(object):
def test_zoh(self):
ac = np.eye(2)
bc = np.full((2, 1), 0.5)
cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
dc = np.array([[0.0], [0.0], [-0.33]])
ad_truth = 1.648721270700128 * np.eye(2)
bd_truth = np.full((2, 1), 0.324360635350064)
# c and d in discrete should be equal to their continuous counterparts
dt_requested = 0.5
ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, method='zoh')
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)
assert_almost_equal(dt_requested, dt)
def test_foh(self):
ac = np.eye(2)
bc = np.full((2, 1), 0.5)
cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
dc = np.array([[0.0], [0.0], [-0.33]])
# True values are verified with Matlab
ad_truth = 1.648721270700128 * np.eye(2)
bd_truth = np.full((2, 1), 0.420839287058789)
cd_truth = cc
dd_truth = np.array([[0.260262223725224],
[0.297442541400256],
[-0.144098411624840]])
dt_requested = 0.5
ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, method='foh')
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_almost_equal(dt_requested, dt)
def test_impulse(self):
ac = np.eye(2)
bc = np.full((2, 1), 0.5)
cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
dc = np.array([[0.0], [0.0], [0.0]])
# True values are verified with Matlab
ad_truth = 1.648721270700128 * np.eye(2)
bd_truth = np.full((2, 1), 0.412180317675032)
cd_truth = cc
dd_truth = np.array([[0.4375], [0.5], [0.3125]])
dt_requested = 0.5
ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
method='impulse')
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_almost_equal(dt_requested, dt)
def test_gbt(self):
ac = np.eye(2)
bc = np.full((2, 1), 0.5)
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_requested = 0.5
alpha = 1.0 / 3.0
ad_truth = 1.6 * np.eye(2)
bd_truth = np.full((2, 1), 0.3)
cd_truth = np.array([[0.9, 1.2],
[1.2, 1.2],
[1.2, 0.3]])
dd_truth = np.array([[0.175],
[0.2],
[-0.205]])
ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
method='gbt', alpha=alpha)
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)
def test_euler(self):
ac = np.eye(2)
bc = np.full((2, 1), 0.5)
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_requested = 0.5
ad_truth = 1.5 * np.eye(2)
bd_truth = np.full((2, 1), 0.25)
cd_truth = np.array([[0.75, 1.0],
[1.0, 1.0],
[1.0, 0.25]])
dd_truth = dc
ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
method='euler')
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_almost_equal(dt_requested, dt)
def test_backward_diff(self):
ac = np.eye(2)
bc = np.full((2, 1), 0.5)
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_requested = 0.5
ad_truth = 2.0 * np.eye(2)
bd_truth = np.full((2, 1), 0.5)
cd_truth = np.array([[1.5, 2.0],
[2.0, 2.0],
[2.0, 0.5]])
dd_truth = np.array([[0.875],
[1.0],
[0.295]])
ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
method='backward_diff')
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)
def test_bilinear(self):
ac = np.eye(2)
bc = np.full((2, 1), 0.5)
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_requested = 0.5
ad_truth = (5.0 / 3.0) * np.eye(2)
bd_truth = np.full((2, 1), 1.0 / 3.0)
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]])
dd_truth = np.array([[0.291666666666667],
[1.0 / 3.0],
[-0.121666666666667]])
ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
method='bilinear')
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_almost_equal(dt_requested, dt)
# Same continuous system again, but change sampling rate
ad_truth = 1.4 * np.eye(2)
bd_truth = np.full((2, 1), 0.2)
cd_truth = np.array([[0.9, 1.2], [1.2, 1.2], [1.2, 0.3]])
dd_truth = np.array([[0.175], [0.2], [-0.205]])
dt_requested = 1.0 / 3.0
ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
method='bilinear')
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_almost_equal(dt_requested, dt)
def test_transferfunction(self):
numc = np.array([0.25, 0.25, 0.5])
denc = np.array([0.75, 0.75, 1.0])
numd = np.array([[1.0 / 3.0, -0.427419169438754, 0.221654141101125]])
dend = np.array([1.0, -1.351394049721225, 0.606530659712634])
dt_requested = 0.5
num, den, dt = c2d((numc, denc), dt_requested, method='zoh')
assert_array_almost_equal(numd, num)
assert_array_almost_equal(dend, den)
assert_almost_equal(dt_requested, dt)
def test_zerospolesgain(self):
zeros_c = np.array([0.5, -0.5])
poles_c = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)])
k_c = 1.0
zeros_d = [1.23371727305860, 0.735356894461267]
polls_d = [0.938148335039729 + 0.346233593780536j,
0.938148335039729 - 0.346233593780536j]
k_d = 1.0
dt_requested = 0.5
zeros, poles, k, dt = c2d((zeros_c, poles_c, k_c), dt_requested,
method='zoh')
assert_array_almost_equal(zeros_d, zeros)
assert_array_almost_equal(polls_d, poles)
assert_almost_equal(k_d, k)
assert_almost_equal(dt_requested, dt)
def test_gbt_with_sio_tf_and_zpk(self):
"""Test method='gbt' with alpha=0.25 for tf and zpk cases."""
# State space coefficients for the continuous SIO system.
A = -1.0
B = 1.0
C = 1.0
D = 0.5
# The continuous transfer function coefficients.
cnum, cden = ss2tf(A, B, C, D)
# Continuous zpk representation
cz, cp, ck = ss2zpk(A, B, C, D)
h = 1.0
alpha = 0.25
# Explicit formulas, in the scalar case.
Ad = (1 + (1 - alpha) * h * A) / (1 - alpha * h * A)
Bd = h * B / (1 - alpha * h * A)
Cd = C / (1 - alpha * h * A)
Dd = D + alpha * C * Bd
# Convert the explicit solution to tf
dnum, dden = ss2tf(Ad, Bd, Cd, Dd)
# Compute the discrete tf using cont2discrete.
c2dnum, c2dden, dt = c2d((cnum, cden), h, method='gbt', alpha=alpha)
assert_allclose(dnum, c2dnum)
assert_allclose(dden, c2dden)
# Convert explicit solution to zpk.
dz, dp, dk = ss2zpk(Ad, Bd, Cd, Dd)
# Compute the discrete zpk using cont2discrete.
c2dz, c2dp, c2dk, dt = c2d((cz, cp, ck), h, method='gbt', alpha=alpha)
assert_allclose(dz, c2dz)
assert_allclose(dp, c2dp)
assert_allclose(dk, c2dk)
def test_discrete_approx(self):
"""
Test that the solution to the discrete approximation of a continuous
system actually approximates the solution to the continuous system.
This is an indirect test of the correctness of the implementation
of cont2discrete.
"""
def u(t):
return np.sin(2.5 * t)
a = np.array([[-0.01]])
b = np.array([[1.0]])
c = np.array([[1.0]])
d = np.array([[0.2]])
x0 = 1.0
t = np.linspace(0, 10.0, 101)
dt = t[1] - t[0]
u1 = u(t)
# Use lsim2 to compute the solution to the continuous system.
t, yout, xout = lsim2((a, b, c, d), T=t, U=u1, X0=x0,
rtol=1e-9, atol=1e-11)
# Convert the continuous system to a discrete approximation.
dsys = c2d((a, b, c, d), dt, method='bilinear')
# Use dlsim with the pairwise averaged input to compute the output
# of the discrete system.
u2 = 0.5 * (u1[:-1] + u1[1:])
t2 = t[:-1]
td2, yd2, xd2 = dlsim(dsys, u=u2.reshape(-1, 1), t=t2, x0=x0)
# ymid is the average of consecutive terms of the "exact" output
# computed by lsim2. This is what the discrete approximation
# actually approximates.
ymid = 0.5 * (yout[:-1] + yout[1:])
assert_allclose(yd2.ravel(), ymid, rtol=1e-4)
def test_simo_tf(self):
# See gh-5753
tf = ([[1, 0], [1, 1]], [1, 1])
num, den, dt = c2d(tf, 0.01)
assert_equal(dt, 0.01) # sanity check
assert_allclose(den, [1, -0.990404983], rtol=1e-3)
assert_allclose(num, [[1, -1], [1, -0.99004983]], rtol=1e-3)
def test_multioutput(self):
ts = 0.01 # time step
tf = ([[1, -3], [1, 5]], [1, 1])
num, den, dt = c2d(tf, ts)
tf1 = (tf[0][0], tf[1])
num1, den1, dt1 = c2d(tf1, ts)
tf2 = (tf[0][1], tf[1])
num2, den2, dt2 = c2d(tf2, ts)
# Sanity checks
assert_equal(dt, dt1)
assert_equal(dt, dt2)
# Check that we get the same results
assert_allclose(num, np.vstack((num1, num2)), rtol=1e-13)
# Single input, so the denominator should
# not be multidimensional like the numerator
assert_allclose(den, den1, rtol=1e-13)
assert_allclose(den, den2, rtol=1e-13)
class TestC2dLti(object):
def test_c2d_ss(self):
# StateSpace
A = np.array([[-0.3, 0.1], [0.2, -0.7]])
B = np.array([[0], [1]])
C = np.array([[1, 0]])
D = 0
A_res = np.array([[0.985136404135682, 0.004876671474795],
[0.009753342949590, 0.965629718236502]])
B_res = np.array([[0.000122937599964], [0.049135527547844]])
sys_ssc = lti(A, B, C, D)
sys_ssd = sys_ssc.to_discrete(0.05)
assert_allclose(sys_ssd.A, A_res)
assert_allclose(sys_ssd.B, B_res)
assert_allclose(sys_ssd.C, C)
assert_allclose(sys_ssd.D, D)
def test_c2d_tf(self):
sys = lti([0.5, 0.3], [1.0, 0.4])
sys = sys.to_discrete(0.005)
# Matlab results
num_res = np.array([0.5, -0.485149004980066])
den_res = np.array([1.0, -0.980198673306755])
# Somehow a lot of numerical errors
assert_allclose(sys.den, den_res, atol=0.02)
assert_allclose(sys.num, num_res, atol=0.02)
class TestC2dInvariants:
# Some test cases for checking the invariances.
# Array of triplets: (system, sample time, number of samples)
cases = [
(tf2ss([1, 1], [1, 1.5, 1]), 0.25, 10),
(tf2ss([1, 2], [1, 1.5, 3, 1]), 0.5, 10),
(tf2ss(0.1, [1, 1, 2, 1]), 0.5, 10),
]
# Some options for lsim2 and derived routines
tolerances = {'rtol': 1e-9, 'atol': 1e-11}
# Check that systems discretized with the impulse-invariant
# method really hold the invariant
@pytest.mark.parametrize("sys,sample_time,samples_number", cases)
def test_impulse_invariant(self, sys, sample_time, samples_number):
time = np.arange(samples_number) * sample_time
_, yout_cont = impulse2(sys, T=time, **self.tolerances)
_, yout_disc = dimpulse(c2d(sys, sample_time, method='impulse'),
n=len(time))
assert_allclose(sample_time * yout_cont.ravel(), yout_disc[0].ravel())
# Step invariant should hold for ZOH discretized systems
@pytest.mark.parametrize("sys,sample_time,samples_number", cases)
def test_step_invariant(self, sys, sample_time, samples_number):
time = np.arange(samples_number) * sample_time
_, yout_cont = step2(sys, T=time, **self.tolerances)
_, yout_disc = dstep(c2d(sys, sample_time, method='zoh'), n=len(time))
assert_allclose(yout_cont.ravel(), yout_disc[0].ravel())
# Linear invariant should hold for FOH discretized systems
@pytest.mark.parametrize("sys,sample_time,samples_number", cases)
def test_linear_invariant(self, sys, sample_time, samples_number):
time = np.arange(samples_number) * sample_time
_, yout_cont, _ = lsim2(sys, T=time, U=time, **self.tolerances)
_, yout_disc, _ = dlsim(c2d(sys, sample_time, method='foh'), u=time)
assert_allclose(yout_cont.ravel(), yout_disc.ravel())
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