1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
|
import warnings
import numpy as np
from numpy.testing import assert_almost_equal, assert_equal, run_module_suite
from scipy.signal.ltisys import ss2tf, lsim2, impulse2, step2, lti
from scipy.signal.filter_design import BadCoefficients
class TestSS2TF:
def tst_matrix_shapes(self, p, q, r):
ss2tf(np.zeros((p, p)),
np.zeros((p, q)),
np.zeros((r, p)),
np.zeros((r, q)), 0)
def test_basic(self):
for p, q, r in [
(3, 3, 3),
(1, 3, 3),
(1, 1, 1)]:
yield self.tst_matrix_shapes, p, q, r
class Test_lsim2(object):
def test_01(self):
t = np.linspace(0,10,1001)
u = np.zeros_like(t)
# First order system: x'(t) + x(t) = u(t), x(0) = 1.
# Exact solution is x(t) = exp(-t).
system = ([1.0],[1.0,1.0])
tout, y, x = lsim2(system, u, t, X0=[1.0])
expected_x = np.exp(-tout)
assert_almost_equal(x[:,0], expected_x)
def test_02(self):
t = np.array([0.0, 1.0, 1.0, 3.0])
u = np.array([0.0, 0.0, 1.0, 1.0])
# Simple integrator: x'(t) = u(t)
system = ([1.0],[1.0,0.0])
tout, y, x = lsim2(system, u, t, X0=[1.0])
expected_x = np.maximum(1.0, tout)
assert_almost_equal(x[:,0], expected_x)
def test_03(self):
t = np.array([0.0, 1.0, 1.0, 1.1, 1.1, 2.0])
u = np.array([0.0, 0.0, 1.0, 1.0, 0.0, 0.0])
# Simple integrator: x'(t) = u(t)
system = ([1.0],[1.0, 0.0])
tout, y, x = lsim2(system, u, t, hmax=0.01)
expected_x = np.array([0.0, 0.0, 0.0, 0.1, 0.1, 0.1])
assert_almost_equal(x[:,0], expected_x)
def test_04(self):
t = np.linspace(0, 10, 1001)
u = np.zeros_like(t)
# Second order system with a repeated root: x''(t) + 2*x(t) + x(t) = 0.
# With initial conditions x(0)=1.0 and x'(t)=0.0, the exact solution
# is (1-t)*exp(-t).
system = ([1.0], [1.0, 2.0, 1.0])
tout, y, x = lsim2(system, u, t, X0=[1.0, 0.0])
expected_x = (1.0 - tout) * np.exp(-tout)
assert_almost_equal(x[:,0], expected_x)
def test_05(self):
# The call to lsim2 triggers a "BadCoefficients" warning from
# scipy.signal.filter_design, but the test passes. I think the warning
# is related to the incomplete handling of multi-input systems in
# scipy.signal.
# A system with two state variables, two inputs, and one output.
A = np.array([[-1.0, 0.0], [0.0, -2.0]])
B = np.array([[1.0, 0.0], [0.0, 1.0]])
C = np.array([1.0, 0.0])
D = np.zeros((1,2))
t = np.linspace(0, 10.0, 101)
warnings.simplefilter("ignore", BadCoefficients)
try:
tout, y, x = lsim2((A,B,C,D), T=t, X0=[1.0, 1.0])
finally:
del warnings.filters[0]
expected_y = np.exp(-tout)
expected_x0 = np.exp(-tout)
expected_x1 = np.exp(-2.0*tout)
assert_almost_equal(y, expected_y)
assert_almost_equal(x[:,0], expected_x0)
assert_almost_equal(x[:,1], expected_x1)
def test_06(self):
"""Test use of the default values of the arguments `T` and `U`."""
# Second order system with a repeated root: x''(t) + 2*x(t) + x(t) = 0.
# With initial conditions x(0)=1.0 and x'(t)=0.0, the exact solution
# is (1-t)*exp(-t).
system = ([1.0], [1.0, 2.0, 1.0])
tout, y, x = lsim2(system, X0=[1.0, 0.0])
expected_x = (1.0 - tout) * np.exp(-tout)
assert_almost_equal(x[:,0], expected_x)
class Test_impulse2(object):
def test_01(self):
# First order system: x'(t) + x(t) = u(t)
# Exact impulse response is x(t) = exp(-t).
system = ([1.0],[1.0,1.0])
tout, y = impulse2(system)
expected_y = np.exp(-tout)
assert_almost_equal(y, expected_y)
def test_02(self):
"""Specify the desired time values for the output."""
# First order system: x'(t) + x(t) = u(t)
# Exact impulse response is x(t) = exp(-t).
system = ([1.0],[1.0,1.0])
n = 21
t = np.linspace(0, 2.0, n)
tout, y = impulse2(system, T=t)
assert_equal(tout.shape, (n,))
assert_almost_equal(tout, t)
expected_y = np.exp(-t)
assert_almost_equal(y, expected_y)
def test_03(self):
"""Specify an initial condition as a scalar."""
# First order system: x'(t) + x(t) = u(t), x(0)=3.0
# Exact impulse response is x(t) = 4*exp(-t).
system = ([1.0],[1.0,1.0])
tout, y = impulse2(system, X0=3.0)
expected_y = 4.0*np.exp(-tout)
assert_almost_equal(y, expected_y)
def test_04(self):
"""Specify an initial condition as a list."""
# First order system: x'(t) + x(t) = u(t), x(0)=3.0
# Exact impulse response is x(t) = 4*exp(-t).
system = ([1.0],[1.0,1.0])
tout, y = impulse2(system, X0=[3.0])
expected_y = 4.0*np.exp(-tout)
assert_almost_equal(y, expected_y)
def test_05(self):
# Simple integrator: x'(t) = u(t)
system = ([1.0],[1.0,0.0])
tout, y = impulse2(system)
expected_y = np.ones_like(tout)
assert_almost_equal(y, expected_y)
def test_06(self):
# Second order system with a repeated root: x''(t) + 2*x(t) + x(t) = u(t)
# The exact impulse response is t*exp(-t).
system = ([1.0], [1.0, 2.0, 1.0])
tout, y = impulse2(system)
expected_y = tout * np.exp(-tout)
assert_almost_equal(y, expected_y)
class Test_step2(object):
def test_01(self):
# First order system: x'(t) + x(t) = u(t)
# Exact step response is x(t) = 1 - exp(-t).
system = ([1.0],[1.0,1.0])
tout, y = step2(system)
expected_y = 1.0 - np.exp(-tout)
assert_almost_equal(y, expected_y)
def test_02(self):
"""Specify the desired time values for the output."""
# First order system: x'(t) + x(t) = u(t)
# Exact step response is x(t) = 1 - exp(-t).
system = ([1.0],[1.0,1.0])
n = 21
t = np.linspace(0, 2.0, n)
tout, y = step2(system, T=t)
assert_equal(tout.shape, (n,))
assert_almost_equal(tout, t)
expected_y = 1 - np.exp(-t)
assert_almost_equal(y, expected_y)
def test_03(self):
"""Specify an initial condition as a scalar."""
# First order system: x'(t) + x(t) = u(t), x(0)=3.0
# Exact step response is x(t) = 1 + 2*exp(-t).
system = ([1.0],[1.0,1.0])
tout, y = step2(system, X0=3.0)
expected_y = 1 + 2.0*np.exp(-tout)
assert_almost_equal(y, expected_y)
def test_04(self):
"""Specify an initial condition as a list."""
# First order system: x'(t) + x(t) = u(t), x(0)=3.0
# Exact step response is x(t) = 1 + 2*exp(-t).
system = ([1.0],[1.0,1.0])
tout, y = step2(system, X0=[3.0])
expected_y = 1 + 2.0*np.exp(-tout)
assert_almost_equal(y, expected_y)
def test_05(self):
# Simple integrator: x'(t) = u(t)
# Exact step response is x(t) = t.
system = ([1.0],[1.0,0.0])
tout, y = step2(system, atol=1e-10, rtol=1e-8)
expected_y = tout
assert_almost_equal(y, expected_y)
def test_06(self):
# Second order system with a repeated root: x''(t) + 2*x(t) + x(t) = u(t)
# The exact step response is 1 - (1 + t)*exp(-t).
system = ([1.0], [1.0, 2.0, 1.0])
tout, y = step2(system, atol=1e-10, rtol=1e-8)
expected_y = 1 - (1 + tout) * np.exp(-tout)
assert_almost_equal(y, expected_y)
if __name__ == "__main__":
run_module_suite()
|
