456 lines
16 KiB
Python
456 lines
16 KiB
Python
from __future__ import division, print_function, absolute_import
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import numpy as np
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from numpy.testing import \
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assert_array_almost_equal, assert_almost_equal, \
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assert_allclose, assert_equal
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import pytest
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from scipy.signal import cont2discrete as c2d
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from scipy.signal import dlsim, ss2tf, ss2zpk, lsim2, lti
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from scipy.signal import tf2ss, impulse2, dimpulse, step2, dstep
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# Author: Jeffrey Armstrong <jeff@approximatrix.com>
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# March 29, 2011
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class TestC2D(object):
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def test_zoh(self):
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ac = np.eye(2)
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bc = np.full((2, 1), 0.5)
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cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
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dc = np.array([[0.0], [0.0], [-0.33]])
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ad_truth = 1.648721270700128 * np.eye(2)
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bd_truth = np.full((2, 1), 0.324360635350064)
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# c and d in discrete should be equal to their continuous counterparts
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dt_requested = 0.5
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ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, method='zoh')
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assert_array_almost_equal(ad_truth, ad)
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assert_array_almost_equal(bd_truth, bd)
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assert_array_almost_equal(cc, cd)
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assert_array_almost_equal(dc, dd)
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assert_almost_equal(dt_requested, dt)
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def test_foh(self):
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ac = np.eye(2)
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bc = np.full((2, 1), 0.5)
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cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
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dc = np.array([[0.0], [0.0], [-0.33]])
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# True values are verified with Matlab
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ad_truth = 1.648721270700128 * np.eye(2)
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bd_truth = np.full((2, 1), 0.420839287058789)
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cd_truth = cc
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dd_truth = np.array([[0.260262223725224],
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[0.297442541400256],
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[-0.144098411624840]])
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dt_requested = 0.5
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ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested, method='foh')
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assert_array_almost_equal(ad_truth, ad)
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assert_array_almost_equal(bd_truth, bd)
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assert_array_almost_equal(cd_truth, cd)
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assert_array_almost_equal(dd_truth, dd)
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assert_almost_equal(dt_requested, dt)
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def test_impulse(self):
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ac = np.eye(2)
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bc = np.full((2, 1), 0.5)
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cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
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dc = np.array([[0.0], [0.0], [0.0]])
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# True values are verified with Matlab
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ad_truth = 1.648721270700128 * np.eye(2)
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bd_truth = np.full((2, 1), 0.412180317675032)
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cd_truth = cc
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dd_truth = np.array([[0.4375], [0.5], [0.3125]])
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dt_requested = 0.5
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ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
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method='impulse')
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assert_array_almost_equal(ad_truth, ad)
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assert_array_almost_equal(bd_truth, bd)
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assert_array_almost_equal(cd_truth, cd)
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assert_array_almost_equal(dd_truth, dd)
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assert_almost_equal(dt_requested, dt)
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def test_gbt(self):
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ac = np.eye(2)
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bc = np.full((2, 1), 0.5)
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cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
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dc = np.array([[0.0], [0.0], [-0.33]])
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dt_requested = 0.5
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alpha = 1.0 / 3.0
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ad_truth = 1.6 * np.eye(2)
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bd_truth = np.full((2, 1), 0.3)
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cd_truth = np.array([[0.9, 1.2],
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[1.2, 1.2],
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[1.2, 0.3]])
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dd_truth = np.array([[0.175],
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[0.2],
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[-0.205]])
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ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
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method='gbt', alpha=alpha)
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assert_array_almost_equal(ad_truth, ad)
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assert_array_almost_equal(bd_truth, bd)
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assert_array_almost_equal(cd_truth, cd)
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assert_array_almost_equal(dd_truth, dd)
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def test_euler(self):
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ac = np.eye(2)
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bc = np.full((2, 1), 0.5)
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cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
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dc = np.array([[0.0], [0.0], [-0.33]])
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dt_requested = 0.5
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ad_truth = 1.5 * np.eye(2)
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bd_truth = np.full((2, 1), 0.25)
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cd_truth = np.array([[0.75, 1.0],
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[1.0, 1.0],
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[1.0, 0.25]])
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dd_truth = dc
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ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
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method='euler')
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assert_array_almost_equal(ad_truth, ad)
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assert_array_almost_equal(bd_truth, bd)
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assert_array_almost_equal(cd_truth, cd)
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assert_array_almost_equal(dd_truth, dd)
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assert_almost_equal(dt_requested, dt)
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def test_backward_diff(self):
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ac = np.eye(2)
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bc = np.full((2, 1), 0.5)
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cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
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dc = np.array([[0.0], [0.0], [-0.33]])
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dt_requested = 0.5
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ad_truth = 2.0 * np.eye(2)
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bd_truth = np.full((2, 1), 0.5)
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cd_truth = np.array([[1.5, 2.0],
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[2.0, 2.0],
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[2.0, 0.5]])
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dd_truth = np.array([[0.875],
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[1.0],
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[0.295]])
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ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
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method='backward_diff')
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assert_array_almost_equal(ad_truth, ad)
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assert_array_almost_equal(bd_truth, bd)
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assert_array_almost_equal(cd_truth, cd)
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assert_array_almost_equal(dd_truth, dd)
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def test_bilinear(self):
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ac = np.eye(2)
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bc = np.full((2, 1), 0.5)
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cc = np.array([[0.75, 1.0], [1.0, 1.0], [1.0, 0.25]])
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dc = np.array([[0.0], [0.0], [-0.33]])
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dt_requested = 0.5
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ad_truth = (5.0 / 3.0) * np.eye(2)
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bd_truth = np.full((2, 1), 1.0 / 3.0)
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cd_truth = np.array([[1.0, 4.0 / 3.0],
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[4.0 / 3.0, 4.0 / 3.0],
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[4.0 / 3.0, 1.0 / 3.0]])
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dd_truth = np.array([[0.291666666666667],
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[1.0 / 3.0],
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[-0.121666666666667]])
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ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
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method='bilinear')
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assert_array_almost_equal(ad_truth, ad)
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assert_array_almost_equal(bd_truth, bd)
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assert_array_almost_equal(cd_truth, cd)
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assert_array_almost_equal(dd_truth, dd)
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assert_almost_equal(dt_requested, dt)
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# Same continuous system again, but change sampling rate
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ad_truth = 1.4 * np.eye(2)
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bd_truth = np.full((2, 1), 0.2)
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cd_truth = np.array([[0.9, 1.2], [1.2, 1.2], [1.2, 0.3]])
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dd_truth = np.array([[0.175], [0.2], [-0.205]])
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dt_requested = 1.0 / 3.0
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ad, bd, cd, dd, dt = c2d((ac, bc, cc, dc), dt_requested,
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method='bilinear')
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assert_array_almost_equal(ad_truth, ad)
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assert_array_almost_equal(bd_truth, bd)
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assert_array_almost_equal(cd_truth, cd)
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assert_array_almost_equal(dd_truth, dd)
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assert_almost_equal(dt_requested, dt)
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def test_transferfunction(self):
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numc = np.array([0.25, 0.25, 0.5])
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denc = np.array([0.75, 0.75, 1.0])
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numd = np.array([[1.0 / 3.0, -0.427419169438754, 0.221654141101125]])
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dend = np.array([1.0, -1.351394049721225, 0.606530659712634])
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dt_requested = 0.5
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num, den, dt = c2d((numc, denc), dt_requested, method='zoh')
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assert_array_almost_equal(numd, num)
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assert_array_almost_equal(dend, den)
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assert_almost_equal(dt_requested, dt)
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def test_zerospolesgain(self):
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zeros_c = np.array([0.5, -0.5])
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poles_c = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)])
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k_c = 1.0
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zeros_d = [1.23371727305860, 0.735356894461267]
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polls_d = [0.938148335039729 + 0.346233593780536j,
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0.938148335039729 - 0.346233593780536j]
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k_d = 1.0
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dt_requested = 0.5
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zeros, poles, k, dt = c2d((zeros_c, poles_c, k_c), dt_requested,
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method='zoh')
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assert_array_almost_equal(zeros_d, zeros)
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assert_array_almost_equal(polls_d, poles)
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assert_almost_equal(k_d, k)
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assert_almost_equal(dt_requested, dt)
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def test_gbt_with_sio_tf_and_zpk(self):
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"""Test method='gbt' with alpha=0.25 for tf and zpk cases."""
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# State space coefficients for the continuous SIO system.
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A = -1.0
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B = 1.0
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C = 1.0
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D = 0.5
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# The continuous transfer function coefficients.
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cnum, cden = ss2tf(A, B, C, D)
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# Continuous zpk representation
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cz, cp, ck = ss2zpk(A, B, C, D)
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h = 1.0
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alpha = 0.25
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# Explicit formulas, in the scalar case.
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Ad = (1 + (1 - alpha) * h * A) / (1 - alpha * h * A)
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Bd = h * B / (1 - alpha * h * A)
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Cd = C / (1 - alpha * h * A)
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Dd = D + alpha * C * Bd
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# Convert the explicit solution to tf
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dnum, dden = ss2tf(Ad, Bd, Cd, Dd)
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# Compute the discrete tf using cont2discrete.
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c2dnum, c2dden, dt = c2d((cnum, cden), h, method='gbt', alpha=alpha)
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assert_allclose(dnum, c2dnum)
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assert_allclose(dden, c2dden)
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# Convert explicit solution to zpk.
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dz, dp, dk = ss2zpk(Ad, Bd, Cd, Dd)
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# Compute the discrete zpk using cont2discrete.
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c2dz, c2dp, c2dk, dt = c2d((cz, cp, ck), h, method='gbt', alpha=alpha)
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assert_allclose(dz, c2dz)
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assert_allclose(dp, c2dp)
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assert_allclose(dk, c2dk)
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def test_discrete_approx(self):
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"""
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Test that the solution to the discrete approximation of a continuous
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system actually approximates the solution to the continuous system.
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This is an indirect test of the correctness of the implementation
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of cont2discrete.
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"""
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def u(t):
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return np.sin(2.5 * t)
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a = np.array([[-0.01]])
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b = np.array([[1.0]])
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c = np.array([[1.0]])
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d = np.array([[0.2]])
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x0 = 1.0
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t = np.linspace(0, 10.0, 101)
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dt = t[1] - t[0]
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u1 = u(t)
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# Use lsim2 to compute the solution to the continuous system.
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t, yout, xout = lsim2((a, b, c, d), T=t, U=u1, X0=x0,
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rtol=1e-9, atol=1e-11)
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# Convert the continuous system to a discrete approximation.
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dsys = c2d((a, b, c, d), dt, method='bilinear')
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# Use dlsim with the pairwise averaged input to compute the output
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# of the discrete system.
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u2 = 0.5 * (u1[:-1] + u1[1:])
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t2 = t[:-1]
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td2, yd2, xd2 = dlsim(dsys, u=u2.reshape(-1, 1), t=t2, x0=x0)
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# ymid is the average of consecutive terms of the "exact" output
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# computed by lsim2. This is what the discrete approximation
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# actually approximates.
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ymid = 0.5 * (yout[:-1] + yout[1:])
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assert_allclose(yd2.ravel(), ymid, rtol=1e-4)
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def test_simo_tf(self):
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# See gh-5753
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tf = ([[1, 0], [1, 1]], [1, 1])
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num, den, dt = c2d(tf, 0.01)
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assert_equal(dt, 0.01) # sanity check
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assert_allclose(den, [1, -0.990404983], rtol=1e-3)
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assert_allclose(num, [[1, -1], [1, -0.99004983]], rtol=1e-3)
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def test_multioutput(self):
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ts = 0.01 # time step
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tf = ([[1, -3], [1, 5]], [1, 1])
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num, den, dt = c2d(tf, ts)
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tf1 = (tf[0][0], tf[1])
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num1, den1, dt1 = c2d(tf1, ts)
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tf2 = (tf[0][1], tf[1])
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num2, den2, dt2 = c2d(tf2, ts)
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# Sanity checks
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assert_equal(dt, dt1)
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assert_equal(dt, dt2)
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# Check that we get the same results
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assert_allclose(num, np.vstack((num1, num2)), rtol=1e-13)
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# Single input, so the denominator should
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# not be multidimensional like the numerator
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assert_allclose(den, den1, rtol=1e-13)
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assert_allclose(den, den2, rtol=1e-13)
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class TestC2dLti(object):
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def test_c2d_ss(self):
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# StateSpace
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A = np.array([[-0.3, 0.1], [0.2, -0.7]])
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B = np.array([[0], [1]])
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C = np.array([[1, 0]])
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D = 0
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A_res = np.array([[0.985136404135682, 0.004876671474795],
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[0.009753342949590, 0.965629718236502]])
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B_res = np.array([[0.000122937599964], [0.049135527547844]])
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sys_ssc = lti(A, B, C, D)
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sys_ssd = sys_ssc.to_discrete(0.05)
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assert_allclose(sys_ssd.A, A_res)
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assert_allclose(sys_ssd.B, B_res)
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assert_allclose(sys_ssd.C, C)
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assert_allclose(sys_ssd.D, D)
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def test_c2d_tf(self):
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sys = lti([0.5, 0.3], [1.0, 0.4])
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sys = sys.to_discrete(0.005)
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# Matlab results
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num_res = np.array([0.5, -0.485149004980066])
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den_res = np.array([1.0, -0.980198673306755])
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# Somehow a lot of numerical errors
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assert_allclose(sys.den, den_res, atol=0.02)
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assert_allclose(sys.num, num_res, atol=0.02)
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class TestC2dLti(object):
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def test_c2d_ss(self):
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# StateSpace
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A = np.array([[-0.3, 0.1], [0.2, -0.7]])
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B = np.array([[0], [1]])
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C = np.array([[1, 0]])
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D = 0
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A_res = np.array([[0.985136404135682, 0.004876671474795],
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[0.009753342949590, 0.965629718236502]])
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B_res = np.array([[0.000122937599964], [0.049135527547844]])
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sys_ssc = lti(A, B, C, D)
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sys_ssd = sys_ssc.to_discrete(0.05)
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assert_allclose(sys_ssd.A, A_res)
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assert_allclose(sys_ssd.B, B_res)
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assert_allclose(sys_ssd.C, C)
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assert_allclose(sys_ssd.D, D)
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def test_c2d_tf(self):
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sys = lti([0.5, 0.3], [1.0, 0.4])
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sys = sys.to_discrete(0.005)
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# Matlab results
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num_res = np.array([0.5, -0.485149004980066])
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den_res = np.array([1.0, -0.980198673306755])
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# Somehow a lot of numerical errors
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assert_allclose(sys.den, den_res, atol=0.02)
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assert_allclose(sys.num, num_res, atol=0.02)
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class TestC2dInvariants:
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# Some test cases for checking the invariances.
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# Array of triplets: (system, sample time, number of samples)
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cases = [
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(tf2ss([1, 1], [1, 1.5, 1]), 0.25, 10),
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(tf2ss([1, 2], [1, 1.5, 3, 1]), 0.5, 10),
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(tf2ss(0.1, [1, 1, 2, 1]), 0.5, 10),
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]
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# Some options for lsim2 and derived routines
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tolerances = {'rtol': 1e-9, 'atol': 1e-11}
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# Check that systems discretized with the impulse-invariant
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# method really hold the invariant
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@pytest.mark.parametrize("sys,sample_time,samples_number", cases)
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def test_impulse_invariant(self, sys, sample_time, samples_number):
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time = np.arange(samples_number) * sample_time
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_, yout_cont = impulse2(sys, T=time, **self.tolerances)
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_, yout_disc = dimpulse(c2d(sys, sample_time, method='impulse'),
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n=len(time))
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assert_allclose(sample_time * yout_cont.ravel(), yout_disc[0].ravel())
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# Step invariant should hold for ZOH discretized systems
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@pytest.mark.parametrize("sys,sample_time,samples_number", cases)
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def test_step_invariant(self, sys, sample_time, samples_number):
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time = np.arange(samples_number) * sample_time
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_, yout_cont = step2(sys, T=time, **self.tolerances)
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_, yout_disc = dstep(c2d(sys, sample_time, method='zoh'), n=len(time))
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assert_allclose(yout_cont.ravel(), yout_disc[0].ravel())
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# Linear invariant should hold for FOH discretized systems
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@pytest.mark.parametrize("sys,sample_time,samples_number", cases)
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def test_linear_invariant(self, sys, sample_time, samples_number):
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time = np.arange(samples_number) * sample_time
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_, yout_cont, _ = lsim2(sys, T=time, U=time, **self.tolerances)
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_, yout_disc, _ = dlsim(c2d(sys, sample_time, method='foh'), u=time)
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assert_allclose(yout_cont.ravel(), yout_disc.ravel())
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