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from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import TestCase, run_module_suite, \
assert_array_almost_equal, assert_almost_equal, \
assert_allclose
from scipy.signal import cont2discrete as c2d
from scipy.signal import dlsim, ss2tf, ss2zpk, lsim2
# Author: Jeffrey Armstrong <jeff@approximatrix.com>
# March 29, 2011
class TestC2D(TestCase):
def test_zoh(self):
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]])
ad_truth = 1.648721270700128 * np.eye(2)
bd_truth = 0.324360635350064 * np.ones((2, 1))
# 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_gbt(self):
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_requested = 0.5
alpha = 1.0 / 3.0
ad_truth = 1.6 * np.eye(2)
bd_truth = 0.3 * np.ones((2, 1))
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 = 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_requested = 0.5
ad_truth = 1.5 * np.eye(2)
bd_truth = 0.25 * np.ones((2, 1))
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 = 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_requested = 0.5
ad_truth = 2.0 * np.eye(2)
bd_truth = 0.5 * np.ones((2, 1))
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 = 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_requested = 0.5
ad_truth = (5.0 / 3.0) * np.eye(2)
bd_truth = (1.0 / 3.0) * np.ones((2, 1))
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 = 0.2 * np.ones((2, 1))
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)
if __name__ == "__main__":
run_module_suite()
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