598 lines
21 KiB
Python
598 lines
21 KiB
Python
# Author: Jeffrey Armstrong <jeff@approximatrix.com>
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# April 4, 2011
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import numpy as np
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from numpy.testing import (assert_equal,
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assert_array_almost_equal, assert_array_equal,
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assert_allclose, assert_, assert_almost_equal,
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suppress_warnings)
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from pytest import raises as assert_raises
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from scipy.signal import (dlsim, dstep, dimpulse, tf2zpk, lti, dlti,
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StateSpace, TransferFunction, ZerosPolesGain,
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dfreqresp, dbode, BadCoefficients)
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class TestDLTI(object):
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def test_dlsim(self):
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a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
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b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
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c = np.asarray([[0.1, 0.3]])
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d = np.asarray([[0.0, -0.1, 0.0]])
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dt = 0.5
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# Create an input matrix with inputs down the columns (3 cols) and its
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# respective time input vector
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u = np.hstack((np.linspace(0, 4.0, num=5)[:, np.newaxis],
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np.full((5, 1), 0.01),
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np.full((5, 1), -0.002)))
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t_in = np.linspace(0, 2.0, num=5)
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# Define the known result
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yout_truth = np.array([[-0.001,
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-0.00073,
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0.039446,
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0.0915387,
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0.13195948]]).T
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xout_truth = np.asarray([[0, 0],
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[0.0012, 0.0005],
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[0.40233, 0.00071],
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[1.163368, -0.079327],
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[2.2402985, -0.3035679]])
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tout, yout, xout = dlsim((a, b, c, d, dt), u, t_in)
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assert_array_almost_equal(yout_truth, yout)
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assert_array_almost_equal(xout_truth, xout)
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assert_array_almost_equal(t_in, tout)
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# Make sure input with single-dimension doesn't raise error
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dlsim((1, 2, 3), 4)
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# Interpolated control - inputs should have different time steps
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# than the discrete model uses internally
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u_sparse = u[[0, 4], :]
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t_sparse = np.asarray([0.0, 2.0])
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tout, yout, xout = dlsim((a, b, c, d, dt), u_sparse, t_sparse)
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assert_array_almost_equal(yout_truth, yout)
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assert_array_almost_equal(xout_truth, xout)
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assert_equal(len(tout), yout.shape[0])
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# Transfer functions (assume dt = 0.5)
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num = np.asarray([1.0, -0.1])
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den = np.asarray([0.3, 1.0, 0.2])
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yout_truth = np.array([[0.0,
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0.0,
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3.33333333333333,
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-4.77777777777778,
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23.0370370370370]]).T
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# Assume use of the first column of the control input built earlier
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tout, yout = dlsim((num, den, 0.5), u[:, 0], t_in)
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assert_array_almost_equal(yout, yout_truth)
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assert_array_almost_equal(t_in, tout)
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# Retest the same with a 1-D input vector
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uflat = np.asarray(u[:, 0])
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uflat = uflat.reshape((5,))
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tout, yout = dlsim((num, den, 0.5), uflat, t_in)
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assert_array_almost_equal(yout, yout_truth)
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assert_array_almost_equal(t_in, tout)
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# zeros-poles-gain representation
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zd = np.array([0.5, -0.5])
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pd = np.array([1.j / np.sqrt(2), -1.j / np.sqrt(2)])
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k = 1.0
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yout_truth = np.array([[0.0, 1.0, 2.0, 2.25, 2.5]]).T
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tout, yout = dlsim((zd, pd, k, 0.5), u[:, 0], t_in)
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assert_array_almost_equal(yout, yout_truth)
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assert_array_almost_equal(t_in, tout)
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# Raise an error for continuous-time systems
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system = lti([1], [1, 1])
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assert_raises(AttributeError, dlsim, system, u)
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def test_dstep(self):
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a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
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b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
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c = np.asarray([[0.1, 0.3]])
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d = np.asarray([[0.0, -0.1, 0.0]])
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dt = 0.5
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# Because b.shape[1] == 3, dstep should result in a tuple of three
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# result vectors
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yout_step_truth = (np.asarray([0.0, 0.04, 0.052, 0.0404, 0.00956,
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-0.036324, -0.093318, -0.15782348,
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-0.226628324, -0.2969374948]),
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np.asarray([-0.1, -0.075, -0.058, -0.04815,
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-0.04453, -0.0461895, -0.0521812,
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-0.061588875, -0.073549579,
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-0.08727047595]),
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np.asarray([0.0, -0.01, -0.013, -0.0101, -0.00239,
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0.009081, 0.0233295, 0.03945587,
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0.056657081, 0.0742343737]))
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tout, yout = dstep((a, b, c, d, dt), n=10)
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assert_equal(len(yout), 3)
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for i in range(0, len(yout)):
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assert_equal(yout[i].shape[0], 10)
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assert_array_almost_equal(yout[i].flatten(), yout_step_truth[i])
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# Check that the other two inputs (tf, zpk) will work as well
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tfin = ([1.0], [1.0, 1.0], 0.5)
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yout_tfstep = np.asarray([0.0, 1.0, 0.0])
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tout, yout = dstep(tfin, n=3)
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assert_equal(len(yout), 1)
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assert_array_almost_equal(yout[0].flatten(), yout_tfstep)
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zpkin = tf2zpk(tfin[0], tfin[1]) + (0.5,)
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tout, yout = dstep(zpkin, n=3)
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assert_equal(len(yout), 1)
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assert_array_almost_equal(yout[0].flatten(), yout_tfstep)
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# Raise an error for continuous-time systems
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system = lti([1], [1, 1])
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assert_raises(AttributeError, dstep, system)
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def test_dimpulse(self):
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a = np.asarray([[0.9, 0.1], [-0.2, 0.9]])
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b = np.asarray([[0.4, 0.1, -0.1], [0.0, 0.05, 0.0]])
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c = np.asarray([[0.1, 0.3]])
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d = np.asarray([[0.0, -0.1, 0.0]])
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dt = 0.5
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# Because b.shape[1] == 3, dimpulse should result in a tuple of three
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# result vectors
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yout_imp_truth = (np.asarray([0.0, 0.04, 0.012, -0.0116, -0.03084,
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-0.045884, -0.056994, -0.06450548,
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-0.068804844, -0.0703091708]),
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np.asarray([-0.1, 0.025, 0.017, 0.00985, 0.00362,
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-0.0016595, -0.0059917, -0.009407675,
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-0.011960704, -0.01372089695]),
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np.asarray([0.0, -0.01, -0.003, 0.0029, 0.00771,
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0.011471, 0.0142485, 0.01612637,
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0.017201211, 0.0175772927]))
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tout, yout = dimpulse((a, b, c, d, dt), n=10)
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assert_equal(len(yout), 3)
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for i in range(0, len(yout)):
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assert_equal(yout[i].shape[0], 10)
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assert_array_almost_equal(yout[i].flatten(), yout_imp_truth[i])
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# Check that the other two inputs (tf, zpk) will work as well
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tfin = ([1.0], [1.0, 1.0], 0.5)
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yout_tfimpulse = np.asarray([0.0, 1.0, -1.0])
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tout, yout = dimpulse(tfin, n=3)
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assert_equal(len(yout), 1)
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assert_array_almost_equal(yout[0].flatten(), yout_tfimpulse)
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zpkin = tf2zpk(tfin[0], tfin[1]) + (0.5,)
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tout, yout = dimpulse(zpkin, n=3)
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assert_equal(len(yout), 1)
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assert_array_almost_equal(yout[0].flatten(), yout_tfimpulse)
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# Raise an error for continuous-time systems
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system = lti([1], [1, 1])
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assert_raises(AttributeError, dimpulse, system)
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def test_dlsim_trivial(self):
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a = np.array([[0.0]])
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b = np.array([[0.0]])
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c = np.array([[0.0]])
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d = np.array([[0.0]])
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n = 5
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u = np.zeros(n).reshape(-1, 1)
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tout, yout, xout = dlsim((a, b, c, d, 1), u)
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assert_array_equal(tout, np.arange(float(n)))
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assert_array_equal(yout, np.zeros((n, 1)))
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assert_array_equal(xout, np.zeros((n, 1)))
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def test_dlsim_simple1d(self):
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a = np.array([[0.5]])
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b = np.array([[0.0]])
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c = np.array([[1.0]])
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d = np.array([[0.0]])
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n = 5
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u = np.zeros(n).reshape(-1, 1)
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tout, yout, xout = dlsim((a, b, c, d, 1), u, x0=1)
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assert_array_equal(tout, np.arange(float(n)))
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expected = (0.5 ** np.arange(float(n))).reshape(-1, 1)
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assert_array_equal(yout, expected)
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assert_array_equal(xout, expected)
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def test_dlsim_simple2d(self):
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lambda1 = 0.5
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lambda2 = 0.25
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a = np.array([[lambda1, 0.0],
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[0.0, lambda2]])
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b = np.array([[0.0],
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[0.0]])
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c = np.array([[1.0, 0.0],
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[0.0, 1.0]])
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d = np.array([[0.0],
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[0.0]])
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n = 5
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u = np.zeros(n).reshape(-1, 1)
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tout, yout, xout = dlsim((a, b, c, d, 1), u, x0=1)
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assert_array_equal(tout, np.arange(float(n)))
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# The analytical solution:
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expected = (np.array([lambda1, lambda2]) **
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np.arange(float(n)).reshape(-1, 1))
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assert_array_equal(yout, expected)
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assert_array_equal(xout, expected)
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def test_more_step_and_impulse(self):
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lambda1 = 0.5
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lambda2 = 0.75
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a = np.array([[lambda1, 0.0],
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[0.0, lambda2]])
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b = np.array([[1.0, 0.0],
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[0.0, 1.0]])
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c = np.array([[1.0, 1.0]])
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d = np.array([[0.0, 0.0]])
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n = 10
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# Check a step response.
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ts, ys = dstep((a, b, c, d, 1), n=n)
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# Create the exact step response.
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stp0 = (1.0 / (1 - lambda1)) * (1.0 - lambda1 ** np.arange(n))
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stp1 = (1.0 / (1 - lambda2)) * (1.0 - lambda2 ** np.arange(n))
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assert_allclose(ys[0][:, 0], stp0)
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assert_allclose(ys[1][:, 0], stp1)
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# Check an impulse response with an initial condition.
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x0 = np.array([1.0, 1.0])
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ti, yi = dimpulse((a, b, c, d, 1), n=n, x0=x0)
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# Create the exact impulse response.
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imp = (np.array([lambda1, lambda2]) **
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np.arange(-1, n + 1).reshape(-1, 1))
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imp[0, :] = 0.0
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# Analytical solution to impulse response
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y0 = imp[:n, 0] + np.dot(imp[1:n + 1, :], x0)
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y1 = imp[:n, 1] + np.dot(imp[1:n + 1, :], x0)
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assert_allclose(yi[0][:, 0], y0)
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assert_allclose(yi[1][:, 0], y1)
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# Check that dt=0.1, n=3 gives 3 time values.
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system = ([1.0], [1.0, -0.5], 0.1)
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t, (y,) = dstep(system, n=3)
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assert_allclose(t, [0, 0.1, 0.2])
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assert_array_equal(y.T, [[0, 1.0, 1.5]])
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t, (y,) = dimpulse(system, n=3)
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assert_allclose(t, [0, 0.1, 0.2])
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assert_array_equal(y.T, [[0, 1, 0.5]])
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class TestDlti(object):
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def test_dlti_instantiation(self):
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# Test that lti can be instantiated.
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dt = 0.05
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# TransferFunction
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s = dlti([1], [-1], dt=dt)
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assert_(isinstance(s, TransferFunction))
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assert_(isinstance(s, dlti))
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assert_(not isinstance(s, lti))
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assert_equal(s.dt, dt)
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# ZerosPolesGain
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s = dlti(np.array([]), np.array([-1]), 1, dt=dt)
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assert_(isinstance(s, ZerosPolesGain))
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assert_(isinstance(s, dlti))
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assert_(not isinstance(s, lti))
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assert_equal(s.dt, dt)
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# StateSpace
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s = dlti([1], [-1], 1, 3, dt=dt)
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assert_(isinstance(s, StateSpace))
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assert_(isinstance(s, dlti))
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assert_(not isinstance(s, lti))
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assert_equal(s.dt, dt)
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# Number of inputs
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assert_raises(ValueError, dlti, 1)
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assert_raises(ValueError, dlti, 1, 1, 1, 1, 1)
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class TestStateSpaceDisc(object):
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def test_initialization(self):
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# Check that all initializations work
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dt = 0.05
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StateSpace(1, 1, 1, 1, dt=dt)
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StateSpace([1], [2], [3], [4], dt=dt)
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StateSpace(np.array([[1, 2], [3, 4]]), np.array([[1], [2]]),
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np.array([[1, 0]]), np.array([[0]]), dt=dt)
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StateSpace(1, 1, 1, 1, dt=True)
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def test_conversion(self):
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# Check the conversion functions
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s = StateSpace(1, 2, 3, 4, dt=0.05)
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assert_(isinstance(s.to_ss(), StateSpace))
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assert_(isinstance(s.to_tf(), TransferFunction))
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assert_(isinstance(s.to_zpk(), ZerosPolesGain))
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# Make sure copies work
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assert_(StateSpace(s) is not s)
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assert_(s.to_ss() is not s)
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def test_properties(self):
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# Test setters/getters for cross class properties.
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# This implicitly tests to_tf() and to_zpk()
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# Getters
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s = StateSpace(1, 1, 1, 1, dt=0.05)
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assert_equal(s.poles, [1])
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assert_equal(s.zeros, [0])
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class TestTransferFunction(object):
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def test_initialization(self):
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# Check that all initializations work
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dt = 0.05
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TransferFunction(1, 1, dt=dt)
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TransferFunction([1], [2], dt=dt)
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TransferFunction(np.array([1]), np.array([2]), dt=dt)
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TransferFunction(1, 1, dt=True)
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def test_conversion(self):
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# Check the conversion functions
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s = TransferFunction([1, 0], [1, -1], dt=0.05)
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assert_(isinstance(s.to_ss(), StateSpace))
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assert_(isinstance(s.to_tf(), TransferFunction))
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assert_(isinstance(s.to_zpk(), ZerosPolesGain))
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# Make sure copies work
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assert_(TransferFunction(s) is not s)
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assert_(s.to_tf() is not s)
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def test_properties(self):
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# Test setters/getters for cross class properties.
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# This implicitly tests to_ss() and to_zpk()
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# Getters
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s = TransferFunction([1, 0], [1, -1], dt=0.05)
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assert_equal(s.poles, [1])
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assert_equal(s.zeros, [0])
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class TestZerosPolesGain(object):
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def test_initialization(self):
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# Check that all initializations work
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dt = 0.05
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ZerosPolesGain(1, 1, 1, dt=dt)
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ZerosPolesGain([1], [2], 1, dt=dt)
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ZerosPolesGain(np.array([1]), np.array([2]), 1, dt=dt)
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ZerosPolesGain(1, 1, 1, dt=True)
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def test_conversion(self):
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# Check the conversion functions
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s = ZerosPolesGain(1, 2, 3, dt=0.05)
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assert_(isinstance(s.to_ss(), StateSpace))
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assert_(isinstance(s.to_tf(), TransferFunction))
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assert_(isinstance(s.to_zpk(), ZerosPolesGain))
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# Make sure copies work
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assert_(ZerosPolesGain(s) is not s)
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assert_(s.to_zpk() is not s)
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class Test_dfreqresp(object):
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def test_manual(self):
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# Test dfreqresp() real part calculation (manual sanity check).
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# 1st order low-pass filter: H(z) = 1 / (z - 0.2),
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system = TransferFunction(1, [1, -0.2], dt=0.1)
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w = [0.1, 1, 10]
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w, H = dfreqresp(system, w=w)
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# test real
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expected_re = [1.2383, 0.4130, -0.7553]
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assert_almost_equal(H.real, expected_re, decimal=4)
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# test imag
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expected_im = [-0.1555, -1.0214, 0.3955]
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assert_almost_equal(H.imag, expected_im, decimal=4)
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def test_auto(self):
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# Test dfreqresp() real part calculation.
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# 1st order low-pass filter: H(z) = 1 / (z - 0.2),
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system = TransferFunction(1, [1, -0.2], dt=0.1)
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w = [0.1, 1, 10, 100]
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w, H = dfreqresp(system, w=w)
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jw = np.exp(w * 1j)
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y = np.polyval(system.num, jw) / np.polyval(system.den, jw)
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# test real
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expected_re = y.real
|
|
assert_almost_equal(H.real, expected_re)
|
|
|
|
# test imag
|
|
expected_im = y.imag
|
|
assert_almost_equal(H.imag, expected_im)
|
|
|
|
def test_freq_range(self):
|
|
# Test that freqresp() finds a reasonable frequency range.
|
|
# 1st order low-pass filter: H(z) = 1 / (z - 0.2),
|
|
# Expected range is from 0.01 to 10.
|
|
system = TransferFunction(1, [1, -0.2], dt=0.1)
|
|
n = 10
|
|
expected_w = np.linspace(0, np.pi, 10, endpoint=False)
|
|
w, H = dfreqresp(system, n=n)
|
|
assert_almost_equal(w, expected_w)
|
|
|
|
def test_pole_one(self):
|
|
# Test that freqresp() doesn't fail on a system with a pole at 0.
|
|
# integrator, pole at zero: H(s) = 1 / s
|
|
system = TransferFunction([1], [1, -1], dt=0.1)
|
|
|
|
with suppress_warnings() as sup:
|
|
sup.filter(RuntimeWarning, message="divide by zero")
|
|
sup.filter(RuntimeWarning, message="invalid value encountered")
|
|
w, H = dfreqresp(system, n=2)
|
|
assert_equal(w[0], 0.) # a fail would give not-a-number
|
|
|
|
def test_error(self):
|
|
# Raise an error for continuous-time systems
|
|
system = lti([1], [1, 1])
|
|
assert_raises(AttributeError, dfreqresp, system)
|
|
|
|
def test_from_state_space(self):
|
|
# H(z) = 2 / z^3 - 0.5 * z^2
|
|
|
|
system_TF = dlti([2], [1, -0.5, 0, 0])
|
|
|
|
A = np.array([[0.5, 0, 0],
|
|
[1, 0, 0],
|
|
[0, 1, 0]])
|
|
B = np.array([[1, 0, 0]]).T
|
|
C = np.array([[0, 0, 2]])
|
|
D = 0
|
|
|
|
system_SS = dlti(A, B, C, D)
|
|
w = 10.0**np.arange(-3,0,.5)
|
|
with suppress_warnings() as sup:
|
|
sup.filter(BadCoefficients)
|
|
w1, H1 = dfreqresp(system_TF, w=w)
|
|
w2, H2 = dfreqresp(system_SS, w=w)
|
|
|
|
assert_almost_equal(H1, H2)
|
|
|
|
def test_from_zpk(self):
|
|
# 1st order low-pass filter: H(s) = 0.3 / (z - 0.2),
|
|
system_ZPK = dlti([],[0.2],0.3)
|
|
system_TF = dlti(0.3, [1, -0.2])
|
|
w = [0.1, 1, 10, 100]
|
|
w1, H1 = dfreqresp(system_ZPK, w=w)
|
|
w2, H2 = dfreqresp(system_TF, w=w)
|
|
assert_almost_equal(H1, H2)
|
|
|
|
|
|
class Test_bode(object):
|
|
|
|
def test_manual(self):
|
|
# Test bode() magnitude calculation (manual sanity check).
|
|
# 1st order low-pass filter: H(s) = 0.3 / (z - 0.2),
|
|
dt = 0.1
|
|
system = TransferFunction(0.3, [1, -0.2], dt=dt)
|
|
w = [0.1, 0.5, 1, np.pi]
|
|
w2, mag, phase = dbode(system, w=w)
|
|
|
|
# Test mag
|
|
expected_mag = [-8.5329, -8.8396, -9.6162, -12.0412]
|
|
assert_almost_equal(mag, expected_mag, decimal=4)
|
|
|
|
# Test phase
|
|
expected_phase = [-7.1575, -35.2814, -67.9809, -180.0000]
|
|
assert_almost_equal(phase, expected_phase, decimal=4)
|
|
|
|
# Test frequency
|
|
assert_equal(np.array(w) / dt, w2)
|
|
|
|
def test_auto(self):
|
|
# Test bode() magnitude calculation.
|
|
# 1st order low-pass filter: H(s) = 0.3 / (z - 0.2),
|
|
system = TransferFunction(0.3, [1, -0.2], dt=0.1)
|
|
w = np.array([0.1, 0.5, 1, np.pi])
|
|
w2, mag, phase = dbode(system, w=w)
|
|
jw = np.exp(w * 1j)
|
|
y = np.polyval(system.num, jw) / np.polyval(system.den, jw)
|
|
|
|
# Test mag
|
|
expected_mag = 20.0 * np.log10(abs(y))
|
|
assert_almost_equal(mag, expected_mag)
|
|
|
|
# Test phase
|
|
expected_phase = np.rad2deg(np.angle(y))
|
|
assert_almost_equal(phase, expected_phase)
|
|
|
|
def test_range(self):
|
|
# Test that bode() finds a reasonable frequency range.
|
|
# 1st order low-pass filter: H(s) = 0.3 / (z - 0.2),
|
|
dt = 0.1
|
|
system = TransferFunction(0.3, [1, -0.2], dt=0.1)
|
|
n = 10
|
|
# Expected range is from 0.01 to 10.
|
|
expected_w = np.linspace(0, np.pi, n, endpoint=False) / dt
|
|
w, mag, phase = dbode(system, n=n)
|
|
assert_almost_equal(w, expected_w)
|
|
|
|
def test_pole_one(self):
|
|
# Test that freqresp() doesn't fail on a system with a pole at 0.
|
|
# integrator, pole at zero: H(s) = 1 / s
|
|
system = TransferFunction([1], [1, -1], dt=0.1)
|
|
|
|
with suppress_warnings() as sup:
|
|
sup.filter(RuntimeWarning, message="divide by zero")
|
|
sup.filter(RuntimeWarning, message="invalid value encountered")
|
|
w, mag, phase = dbode(system, n=2)
|
|
assert_equal(w[0], 0.) # a fail would give not-a-number
|
|
|
|
def test_imaginary(self):
|
|
# bode() should not fail on a system with pure imaginary poles.
|
|
# The test passes if bode doesn't raise an exception.
|
|
system = TransferFunction([1], [1, 0, 100], dt=0.1)
|
|
dbode(system, n=2)
|
|
|
|
def test_error(self):
|
|
# Raise an error for continuous-time systems
|
|
system = lti([1], [1, 1])
|
|
assert_raises(AttributeError, dbode, system)
|
|
|
|
|
|
class TestTransferFunctionZConversion(object):
|
|
"""Test private conversions between 'z' and 'z**-1' polynomials."""
|
|
|
|
def test_full(self):
|
|
# Numerator and denominator same order
|
|
num = [2, 3, 4]
|
|
den = [5, 6, 7]
|
|
num2, den2 = TransferFunction._z_to_zinv(num, den)
|
|
assert_equal(num, num2)
|
|
assert_equal(den, den2)
|
|
|
|
num2, den2 = TransferFunction._zinv_to_z(num, den)
|
|
assert_equal(num, num2)
|
|
assert_equal(den, den2)
|
|
|
|
def test_numerator(self):
|
|
# Numerator lower order than denominator
|
|
num = [2, 3]
|
|
den = [5, 6, 7]
|
|
num2, den2 = TransferFunction._z_to_zinv(num, den)
|
|
assert_equal([0, 2, 3], num2)
|
|
assert_equal(den, den2)
|
|
|
|
num2, den2 = TransferFunction._zinv_to_z(num, den)
|
|
assert_equal([2, 3, 0], num2)
|
|
assert_equal(den, den2)
|
|
|
|
def test_denominator(self):
|
|
# Numerator higher order than denominator
|
|
num = [2, 3, 4]
|
|
den = [5, 6]
|
|
num2, den2 = TransferFunction._z_to_zinv(num, den)
|
|
assert_equal(num, num2)
|
|
assert_equal([0, 5, 6], den2)
|
|
|
|
num2, den2 = TransferFunction._zinv_to_z(num, den)
|
|
assert_equal(num, num2)
|
|
assert_equal([5, 6, 0], den2)
|
|
|