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# Author: Jeffrey Armstrong <jeff@approximatrix.com>
# April 4, 2011
import numpy as np
from numpy.testing import TestCase, run_module_suite, assert_equal, \
assert_array_almost_equal, assert_array_equal, \
assert_allclose
from scipy.signal import dlsim, dstep, dimpulse, tf2zpk
class TestDLTI(TestCase):
def test_dlsim(self):
a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
c = np.asarray([[0.1, 0.3]])
d = np.asarray([[0.0, -0.1, 0.0]])
dt = 0.5
# Create an input matrix with inputs down the columns (3 cols) and its
# respective time input vector
u = np.hstack((np.asmatrix(np.linspace(0, 4.0, num=5)).transpose(),
0.01 * np.ones((5, 1)),
-0.002 * np.ones((5, 1))))
t_in = np.linspace(0, 2.0, num=5)
# Define the known result
yout_truth = np.asmatrix([-0.001,
-0.00073,
0.039446,
0.0915387,
0.13195948]).transpose()
xout_truth = np.asarray([[0, 0],
[0.0012, 0.0005],
[0.40233, 0.00071],
[1.163368, -0.079327],
[2.2402985, -0.3035679]])
tout, yout, xout = dlsim((a, b, c, d, dt), u, t_in)
assert_array_almost_equal(yout_truth, yout)
assert_array_almost_equal(xout_truth, xout)
assert_array_almost_equal(t_in, tout)
# Interpolated control - inputs should have different time steps
# than the discrete model uses internally
u_sparse = u[[0, 4], :]
t_sparse = np.asarray([0.0, 2.0])
tout, yout, xout = dlsim((a, b, c, d, dt), u_sparse, t_sparse)
assert_array_almost_equal(yout_truth, yout)
assert_array_almost_equal(xout_truth, xout)
assert_equal(len(tout), yout.shape[0])
# Transfer functions (assume dt = 0.5)
num = np.asarray([1.0, -0.1])
den = np.asarray([0.3, 1.0, 0.2])
yout_truth = np.asmatrix([0.0,
0.0,
3.33333333333333,
-4.77777777777778,
23.0370370370370]).transpose()
# Assume use of the first column of the control input built earlier
tout, yout = dlsim((num, den, 0.5), u[:, 0], t_in)
assert_array_almost_equal(yout, yout_truth)
assert_array_almost_equal(t_in, tout)
# Retest the same with a 1-D input vector
uflat = np.asarray(u[:, 0])
uflat = uflat.reshape((5,))
tout, yout = dlsim((num, den, 0.5), uflat, t_in)
assert_array_almost_equal(yout, yout_truth)
assert_array_almost_equal(t_in, tout)
# zeros-poles-gain representation
zd = np.array([0.5, -0.5])
pd = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)])
k = 1.0
yout_truth = np.asmatrix([0.0, 1.0, 2.0, 2.25, 2.5]).transpose()
tout, yout = dlsim((zd, pd, k, 0.5), u[:, 0], t_in)
assert_array_almost_equal(yout, yout_truth)
assert_array_almost_equal(t_in, tout)
def test_dstep(self):
a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
c = np.asarray([[0.1, 0.3]])
d = np.asarray([[0.0, -0.1, 0.0]])
dt = 0.5
# Because b.shape[1] == 3, dstep should result in a tuple of three
# result vectors
yout_step_truth = (np.asarray([0.0, 0.04, 0.052, 0.0404, 0.00956,
-0.036324, -0.093318, -0.15782348,
-0.226628324, -0.2969374948]),
np.asarray([-0.1, -0.075, -0.058, -0.04815,
-0.04453, -0.0461895, -0.0521812,
-0.061588875, -0.073549579,
-0.08727047595]),
np.asarray([0.0, -0.01, -0.013, -0.0101, -0.00239,
0.009081, 0.0233295, 0.03945587,
0.056657081, 0.0742343737]))
tout, yout = dstep((a, b, c, d, dt), n=10)
assert_equal(len(yout), 3)
for i in range(0, len(yout)):
assert_equal(yout[i].shape[0], 10)
assert_array_almost_equal(yout[i].flatten(), yout_step_truth[i])
# Check that the other two inputs (tf, zpk) will work as well
tfin = ([1.0], [1.0, 1.0], 0.5)
yout_tfstep = np.asarray([0.0, 1.0, 0.0])
tout, yout = dstep(tfin, n=3)
assert_equal(len(yout), 1)
assert_array_almost_equal(yout[0].flatten(), yout_tfstep)
zpkin = tf2zpk(tfin[0], tfin[1]) + (0.5,)
tout, yout = dstep(zpkin, n=3)
assert_equal(len(yout), 1)
assert_array_almost_equal(yout[0].flatten(), yout_tfstep)
def test_dimpulse(self):
a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
c = np.asarray([[0.1, 0.3]])
d = np.asarray([[0.0, -0.1, 0.0]])
dt = 0.5
# Because b.shape[1] == 3, dimpulse should result in a tuple of three
# result vectors
yout_imp_truth = (np.asarray([0.0, 0.04, 0.012, -0.0116, -0.03084,
-0.045884, -0.056994, -0.06450548,
-0.068804844, -0.0703091708]),
np.asarray([-0.1, 0.025, 0.017, 0.00985, 0.00362,
-0.0016595, -0.0059917, -0.009407675,
-0.011960704, -0.01372089695]),
np.asarray([0.0, -0.01, -0.003, 0.0029, 0.00771,
0.011471, 0.0142485, 0.01612637,
0.017201211, 0.0175772927]))
tout, yout = dimpulse((a, b, c, d, dt), n=10)
assert_equal(len(yout), 3)
for i in range(0, len(yout)):
assert_equal(yout[i].shape[0], 10)
assert_array_almost_equal(yout[i].flatten(), yout_imp_truth[i])
# Check that the other two inputs (tf, zpk) will work as well
tfin = ([1.0], [1.0, 1.0], 0.5)
yout_tfimpulse = np.asarray([0.0, 1.0, -1.0])
tout, yout = dimpulse(tfin, n=3)
assert_equal(len(yout), 1)
assert_array_almost_equal(yout[0].flatten(), yout_tfimpulse)
zpkin = tf2zpk(tfin[0], tfin[1]) + (0.5,)
tout, yout = dimpulse(zpkin, n=3)
assert_equal(len(yout), 1)
assert_array_almost_equal(yout[0].flatten(), yout_tfimpulse)
def test_dlsim_trivial(self):
a = np.array([[0.0]])
b = np.array([[0.0]])
c = np.array([[0.0]])
d = np.array([[0.0]])
n = 5
u = np.zeros(n).reshape(-1, 1)
tout, yout, xout = dlsim((a, b, c, d, 1), u)
assert_array_equal(tout, np.arange(float(n)))
assert_array_equal(yout, np.zeros((n, 1)))
assert_array_equal(xout, np.zeros((n, 1)))
def test_dlsim_simple1d(self):
a = np.array([[0.5]])
b = np.array([[0.0]])
c = np.array([[1.0]])
d = np.array([[0.0]])
n = 5
u = np.zeros(n).reshape(-1, 1)
tout, yout, xout = dlsim((a, b, c, d, 1), u, x0=1)
assert_array_equal(tout, np.arange(float(n)))
expected = (0.5 ** np.arange(float(n))).reshape(-1, 1)
assert_array_equal(yout, expected)
assert_array_equal(xout, expected)
def test_dlsim_simple2d(self):
lambda1 = 0.5
lambda2 = 0.25
a = np.array([[lambda1, 0.0],
[0.0, lambda2]])
b = np.array([[0.0],
[0.0]])
c = np.array([[1.0, 0.0],
[0.0, 1.0]])
d = np.array([[0.0],
[0.0]])
n = 5
u = np.zeros(n).reshape(-1, 1)
tout, yout, xout = dlsim((a, b, c, d, 1), u, x0=1)
assert_array_equal(tout, np.arange(float(n)))
# The analytical solution:
expected = (np.array([lambda1, lambda2]) **
np.arange(float(n)).reshape(-1, 1))
assert_array_equal(yout, expected)
assert_array_equal(xout, expected)
def test_more_step_and_impulse(self):
lambda1 = 0.5
lambda2 = 0.75
a = np.array([[lambda1, 0.0],
[0.0, lambda2]])
b = np.array([[1.0, 0.0],
[0.0, 1.0]])
c = np.array([[1.0, 1.0]])
d = np.array([[0.0, 0.0]])
n = 10
# Check a step response.
ts, ys = dstep((a, b, c, d, 1), n=n)
# Create the exact step response.
stp0 = (1.0 / (1 - lambda1)) * (1.0 - lambda1 ** np.arange(n))
stp1 = (1.0 / (1 - lambda2)) * (1.0 - lambda2 ** np.arange(n))
assert_allclose(ys[0][:, 0], stp0)
assert_allclose(ys[1][:, 0], stp1)
# Check an impulse response with an initial condition.
x0 = np.array([1.0, 1.0])
ti, yi = dimpulse((a, b, c, d, 1), n=n, x0=x0)
# Create the exact impulse response.
imp = (np.array([lambda1, lambda2]) **
np.arange(-1, n + 1).reshape(-1, 1))
imp[0, :] = 0.0
# Analytical solution to impulse response
y0 = imp[:n, 0] + np.dot(imp[1:n + 1, :], x0)
y1 = imp[:n, 1] + np.dot(imp[1:n + 1, :], x0)
assert_allclose(yi[0][:, 0], y0)
assert_allclose(yi[1][:, 0], y1)
if __name__ == "__main__":
run_module_suite()
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