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1656 lines (1452 loc) · 65.3 KB
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"""Tests for the StateSpace class.
RMM, 30 Mar 2011 based on TestStateSp from v0.4a)
RMM, 14 Jun 2019 statesp_array_test.py coverted from statesp_test.py to test
with use_numpy_matrix(False)
BG, 26 Jul 2020 merge statesp_array_test.py differences into statesp_test.py
convert to pytest
"""
import operator
import numpy as np
import pytest
from numpy.linalg import solve
from numpy.testing import assert_array_almost_equal
from scipy.linalg import block_diag, eigvals
import control as ct
from control.config import defaults
from control.dtime import sample_system
from control.lti import LTI, evalfr
from control.statesp import StateSpace, _convert_to_statespace, \
_rss_generate, _statesp_defaults, drss, linfnorm, rss, ss, tf2ss
from control.xferfcn import TransferFunction, ss2tf
from .conftest import assert_tf_close_coeff, slycotonly
class TestStateSpace:
"""Tests for the StateSpace class."""
@pytest.fixture
def sys322ABCD(self):
"""Matrices for sys322"""
A322 = [[-3., 4., 2.],
[-1., -3., 0.],
[2., 5., 3.]]
B322 = [[1., 4.],
[-3., -3.],
[-2., 1.]]
C322 = [[4., 2., -3.],
[1., 4., 3.]]
D322 = [[-2., 4.],
[0., 1.]]
return (A322, B322, C322, D322)
@pytest.fixture
def sys322(self, sys322ABCD):
"""3-states square system (2 inputs x 2 outputs)"""
return StateSpace(*sys322ABCD, name='sys322')
@pytest.fixture
def sys121(self):
"""2 state, 1 input, 1 output (siso) system"""
A121 = [[4., 1.],
[2., -3]]
B121 = [[5.],
[-3.]]
C121 = [[2., -4]]
D121 = [[3.]]
return StateSpace(A121, B121, C121, D121)
@pytest.fixture
def sys222(self):
"""2-states square system (2 inputs x 2 outputs)"""
A222 = [[4., 1.],
[2., -3]]
B222 = [[5., 2.],
[-3., -3.]]
C222 = [[2., -4],
[0., 1.]]
D222 = [[3., 2.],
[1., -1.]]
return StateSpace(A222, B222, C222, D222)
@pytest.fixture
def sys623(self):
"""sys3: 6 states non square system (2 inputs x 3 outputs)"""
A623 = np.array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 3, 0, 0, 0],
[0, 0, 0, -4, 0, 0],
[0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 0, 3]])
B623 = np.array([[0, -1],
[-1, 0],
[1, -1],
[0, 0],
[0, 1],
[-1, -1]])
C623 = np.array([[1, 0, 0, 1, 0, 0],
[0, 1, 0, 1, 0, 1],
[0, 0, 1, 0, 0, 1]])
D623 = np.zeros((3, 2))
return StateSpace(A623, B623, C623, D623)
@pytest.mark.parametrize(
"dt",
[(), (None, ), (0, ), (1, ), (0.1, ), (True, )],
ids=lambda i: "dt " + ("unspec" if len(i) == 0 else str(i[0])))
@pytest.mark.parametrize(
"argfun",
[pytest.param(
lambda ABCDdt: (ABCDdt, {}),
id="A, B, C, D[, dt]"),
pytest.param(
lambda ABCDdt: (ABCDdt[:4], {'dt': dt_ for dt_ in ABCDdt[4:]}),
id="A, B, C, D[, dt=dt]"),
pytest.param(
lambda ABCDdt: ((StateSpace(*ABCDdt), ), {}),
id="sys")
])
def test_constructor(self, sys322ABCD, dt, argfun):
"""Test different ways to call the StateSpace() constructor"""
args, kwargs = argfun(sys322ABCD + dt)
sys = StateSpace(*args, **kwargs)
dtref = defaults['control.default_dt'] if len(dt) == 0 else dt[0]
np.testing.assert_almost_equal(sys.A, sys322ABCD[0])
np.testing.assert_almost_equal(sys.B, sys322ABCD[1])
np.testing.assert_almost_equal(sys.C, sys322ABCD[2])
np.testing.assert_almost_equal(sys.D, sys322ABCD[3])
assert sys.dt == dtref
@pytest.mark.parametrize(
"args, exc, errmsg",
[((True, ), TypeError, "(can only take in|sys must be) a StateSpace"),
((1, 2), TypeError, "1, 4, or 5 arguments"),
((np.ones((3, 2)), np.ones((3, 2)),
np.ones((2, 2)), np.ones((2, 2))), ValueError,
r"A must be a square matrix"),
((np.ones((3, 3)), np.ones((2, 2)),
np.ones((2, 3)), np.ones((2, 2))), ValueError,
r"Incompatible dimensions of B matrix; expected \(3, 2\)"),
((np.ones((3, 3)), np.ones((3, 2)),
np.ones((2, 2)), np.ones((2, 2))), ValueError,
r"Incompatible dimensions of C matrix; expected \(2, 3\)"),
((np.ones((3, 3)), np.ones((3, 2)),
np.ones((2, 3)), np.ones((2, 3))), ValueError,
r"Incompatible dimensions of D matrix; expected \(2, 2\)"),
])
def test_constructor_invalid(self, args, exc, errmsg):
"""Test invalid input to StateSpace() constructor"""
with pytest.raises(exc, match=errmsg):
StateSpace(*args)
with pytest.raises(exc, match=errmsg):
ss(*args)
def test_constructor_warns(self, sys322ABCD):
"""Test ambiguos input to StateSpace() constructor"""
with pytest.warns(UserWarning, match="received multiple dt"):
sys = StateSpace(*(sys322ABCD + (0.1, )), dt=0.2)
np.testing.assert_almost_equal(sys.A, sys322ABCD[0])
np.testing.assert_almost_equal(sys.B, sys322ABCD[1])
np.testing.assert_almost_equal(sys.C, sys322ABCD[2])
np.testing.assert_almost_equal(sys.D, sys322ABCD[3])
assert sys.dt == 0.1
def test_copy_constructor(self):
"""Test the copy constructor"""
# Create a set of matrices for a simple linear system
A = np.array([[-1]])
B = np.array([[1]])
C = np.array([[1]])
D = np.array([[0]])
# Create the first linear system and a copy
linsys = StateSpace(A, B, C, D)
cpysys = StateSpace(linsys)
# Change the original A matrix
A[0, 0] = -2
np.testing.assert_allclose(linsys.A, [[-1]]) # original value
np.testing.assert_allclose(cpysys.A, [[-1]]) # original value
# Change the A matrix for the original system
linsys.A[0, 0] = -3
np.testing.assert_allclose(cpysys.A, [[-1]]) # original value
@pytest.mark.skip("obsolete test")
def test_copy_constructor_nodt(self, sys322):
"""Test the copy constructor when an object without dt is passed"""
sysin = sample_system(sys322, 1.)
del sysin.dt # this is a nonsensical thing to do
sys = StateSpace(sysin)
assert sys.dt == defaults['control.default_dt']
# test for static gain
sysin = StateSpace([], [], [], [[1, 2], [3, 4]], 1.)
del sysin.dt # this is a nonsensical thing to do
sys = StateSpace(sysin)
assert sys.dt is None
def test_D_broadcast(self, sys623):
"""Test broadcast of D=0 to the right shape"""
# Giving D as a scalar 0 should broadcast to the right shape
sys = StateSpace(sys623.A, sys623.B, sys623.C, 0)
np.testing.assert_allclose(sys623.D, sys.D)
# Giving D as a matrix of the wrong size should generate an error
with pytest.raises(ValueError):
sys = StateSpace(sys.A, sys.B, sys.C, np.array([[0]]))
# Make sure that empty systems still work
sys = StateSpace([], [], [], 1)
np.testing.assert_allclose(sys.D, [[1]])
sys = StateSpace([], [], [], [[0]])
np.testing.assert_allclose(sys.D, [[0]])
sys = StateSpace([], [], [], [0])
np.testing.assert_allclose(sys.D, [[0]])
sys = StateSpace([], [], [], 0)
np.testing.assert_allclose(sys.D, [[0]])
def test_pole(self, sys322):
"""Evaluate the poles of a MIMO system."""
p = np.sort(sys322.poles())
true_p = np.sort([3.34747678408874,
-3.17373839204437 + 1.47492908003839j,
-3.17373839204437 - 1.47492908003839j])
np.testing.assert_array_almost_equal(p, true_p)
def test_zero_empty(self):
"""Test to make sure zero() works with no zeros in system."""
sys = _convert_to_statespace(TransferFunction([1], [1, 2, 1]))
np.testing.assert_array_equal(sys.zeros(), np.array([]))
@slycotonly
def test_zero_siso(self, sys222):
"""Evaluate the zeros of a SISO system."""
# extract only first input / first output system of sys222. This system is denoted sys111
# or tf111
tf111 = ss2tf(sys222)
sys111 = tf2ss(tf111[0, 0])
# compute zeros as root of the characteristic polynomial at the numerator of tf111
# this method is simple and assumed as valid in this test
true_z = np.sort(tf111[0, 0].zeros())
# Compute the zeros through ab08nd, which is tested here
z = np.sort(sys111.zeros())
np.testing.assert_almost_equal(true_z, z)
def test_zero_mimo_sys322_square(self, sys322):
"""Evaluate the zeros of a square MIMO system."""
z = np.sort(sys322.zeros())
true_z = np.sort([44.41465, -0.490252, -5.924398])
np.testing.assert_array_almost_equal(z, true_z)
def test_zero_mimo_sys222_square(self, sys222):
"""Evaluate the zeros of a square MIMO system."""
z = np.sort(sys222.zeros())
true_z = np.sort([-10.568501, 3.368501])
np.testing.assert_array_almost_equal(z, true_z)
@slycotonly
def test_zero_mimo_sys623_non_square(self, sys623):
"""Evaluate the zeros of a non square MIMO system."""
z = np.sort(sys623.zeros())
true_z = np.sort([2., -1.])
np.testing.assert_array_almost_equal(z, true_z)
def test_add_ss(self, sys222, sys322):
"""Add two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., 2., -4.], [1., 4., 3., 0., 1.]]
D = [[1., 6.], [1., 0.]]
sys = sys322 + sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_subtract_ss(self, sys222, sys322):
"""Subtract two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., -2., 4.], [1., 4., 3., 0., -1.]]
D = [[-5., 2.], [-1., 2.]]
sys = sys322 - sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_multiply_ss(self, sys222, sys322):
"""Multiply two MIMO systems."""
A = [[4., 1., 0., 0., 0.], [2., -3., 0., 0., 0.], [2., 0., -3., 4., 2.],
[-6., 9., -1., -3., 0.], [-4., 9., 2., 5., 3.]]
B = [[5., 2.], [-3., -3.], [7., -2.], [-12., -3.], [-5., -5.]]
C = [[-4., 12., 4., 2., -3.], [0., 1., 1., 4., 3.]]
D = [[-2., -8.], [1., -1.]]
sys = sys322 * sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_add_sub_mimo_siso(self):
# Test SS with SS
ss_siso = StateSpace(
np.array([
[1, 2],
[3, 4],
]),
np.array([
[1],
[4],
]),
np.array([
[1, 1],
]),
np.array([
[0],
]),
)
ss_siso_1 = StateSpace(
np.array([
[1, 1],
[3, 1],
]),
np.array([
[3],
[-4],
]),
np.array([
[-1, 1],
]),
np.array([
[0.1],
]),
)
ss_siso_2 = StateSpace(
np.array([
[1, 0],
[0, 1],
]),
np.array([
[0],
[2],
]),
np.array([
[0, 1],
]),
np.array([
[0],
]),
)
ss_mimo = ss_siso_1.append(ss_siso_2)
expected_add = ct.combine_tf([
[ss2tf(ss_siso_1 + ss_siso), ss2tf(ss_siso)],
[ss2tf(ss_siso), ss2tf(ss_siso_2 + ss_siso)],
])
expected_sub = ct.combine_tf([
[ss2tf(ss_siso_1 - ss_siso), -ss2tf(ss_siso)],
[-ss2tf(ss_siso), ss2tf(ss_siso_2 - ss_siso)],
])
for op, expected in [
(StateSpace.__add__, expected_add),
(StateSpace.__radd__, expected_add),
(StateSpace.__sub__, expected_sub),
(StateSpace.__rsub__, -expected_sub),
]:
result = op(ss_mimo, ss_siso)
assert_tf_close_coeff(
expected.minreal(),
ss2tf(result).minreal(),
)
# Test SS with array
expected_add = ct.combine_tf([
[ss2tf(1 + ss_siso), ss2tf(ss_siso)],
[ss2tf(ss_siso), ss2tf(1 + ss_siso)],
])
expected_sub = ct.combine_tf([
[ss2tf(-1 + ss_siso), ss2tf(ss_siso)],
[ss2tf(ss_siso), ss2tf(-1 + ss_siso)],
])
for op, expected in [
(StateSpace.__add__, expected_add),
(StateSpace.__radd__, expected_add),
(StateSpace.__sub__, expected_sub),
(StateSpace.__rsub__, -expected_sub),
]:
result = op(ss_siso, np.eye(2))
assert_tf_close_coeff(
expected.minreal(),
ss2tf(result).minreal(),
)
@slycotonly
@pytest.mark.parametrize(
"left, right, expected",
[
(
TransferFunction([2], [1, 0]),
TransferFunction(
[
[[2], [1]],
[[-1], [4]],
],
[
[[10, 1], [20, 1]],
[[20, 1], [30, 1]],
],
),
TransferFunction(
[
[[4], [2]],
[[-2], [8]],
],
[
[[10, 1, 0], [20, 1, 0]],
[[20, 1, 0], [30, 1, 0]],
],
),
),
(
TransferFunction(
[
[[2], [1]],
[[-1], [4]],
],
[
[[10, 1], [20, 1]],
[[20, 1], [30, 1]],
],
),
TransferFunction([2], [1, 0]),
TransferFunction(
[
[[4], [2]],
[[-2], [8]],
],
[
[[10, 1, 0], [20, 1, 0]],
[[20, 1, 0], [30, 1, 0]],
],
),
),
(
TransferFunction([2], [1, 0]),
np.eye(3),
TransferFunction(
[
[[2], [0], [0]],
[[0], [2], [0]],
[[0], [0], [2]],
],
[
[[1, 0], [1], [1]],
[[1], [1, 0], [1]],
[[1], [1], [1, 0]],
],
),
),
]
)
def test_mul_mimo_siso(self, left, right, expected):
result = tf2ss(left).__mul__(right)
assert_tf_close_coeff(
expected.minreal(),
ss2tf(result).minreal(),
)
@slycotonly
@pytest.mark.parametrize(
"left, right, expected",
[
(
TransferFunction([2], [1, 0]),
TransferFunction(
[
[[2], [1]],
[[-1], [4]],
],
[
[[10, 1], [20, 1]],
[[20, 1], [30, 1]],
],
),
TransferFunction(
[
[[4], [2]],
[[-2], [8]],
],
[
[[10, 1, 0], [20, 1, 0]],
[[20, 1, 0], [30, 1, 0]],
],
),
),
(
TransferFunction(
[
[[2], [1]],
[[-1], [4]],
],
[
[[10, 1], [20, 1]],
[[20, 1], [30, 1]],
],
),
TransferFunction([2], [1, 0]),
TransferFunction(
[
[[4], [2]],
[[-2], [8]],
],
[
[[10, 1, 0], [20, 1, 0]],
[[20, 1, 0], [30, 1, 0]],
],
),
),
(
np.eye(3),
TransferFunction([2], [1, 0]),
TransferFunction(
[
[[2], [0], [0]],
[[0], [2], [0]],
[[0], [0], [2]],
],
[
[[1, 0], [1], [1]],
[[1], [1, 0], [1]],
[[1], [1], [1, 0]],
],
),
),
]
)
def test_rmul_mimo_siso(self, left, right, expected):
result = tf2ss(right).__rmul__(left)
assert_tf_close_coeff(
expected.minreal(),
ss2tf(result).minreal(),
)
@slycotonly
@pytest.mark.parametrize("power", [0, 1, 3, -3])
@pytest.mark.parametrize("sysname", ["sys222", "sys322"])
def test_pow(self, request, sysname, power):
"""Test state space powers."""
sys = request.getfixturevalue(sysname)
result = sys**power
if power == 0:
expected = StateSpace([], [], [], np.eye(sys.ninputs), dt=0)
else:
sign = 1 if power > 0 else -1
expected = sys**sign
for i in range(1,abs(power)):
expected *= sys**sign
np.testing.assert_allclose(expected.A, result.A)
np.testing.assert_allclose(expected.B, result.B)
np.testing.assert_allclose(expected.C, result.C)
np.testing.assert_allclose(expected.D, result.D)
@slycotonly
@pytest.mark.parametrize("order", ["left", "right"])
@pytest.mark.parametrize("sysname", ["sys121", "sys222", "sys322"])
def test_pow_inv(self, request, sysname, order):
"""Check for identity when multiplying by inverse.
This holds approximately true for a few steps but is very
unstable due to numerical precision. Don't assume this in
real life. For testing purposes only!
"""
sys = request.getfixturevalue(sysname)
if order == "left":
combined = sys**-1 * sys
else:
combined = sys * sys**-1
combined = combined.minreal()
np.testing.assert_allclose(combined.dcgain(), np.eye(sys.ninputs),
atol=1e-7)
T = np.linspace(0., 0.3, 100)
U = np.random.rand(sys.ninputs, len(T))
R = combined.forced_response(T=T, U=U, squeeze=False)
# Check that the output is the same as the input
np.testing.assert_allclose(R.outputs, U)
@slycotonly
def test_truediv(self, sys222, sys322):
"""Test state space truediv"""
for sys in [sys222, sys322]:
# Divide by self
result = (sys.__truediv__(sys)).minreal()
expected = StateSpace([], [], [], np.eye(2), dt=0)
assert_tf_close_coeff(
ss2tf(expected).minreal(),
ss2tf(result).minreal(),
)
# Divide by TF
result = sys.__truediv__(TransferFunction.s)
expected = ss2tf(sys) / TransferFunction.s
assert_tf_close_coeff(
expected.minreal(),
ss2tf(result).minreal(),
)
@slycotonly
def test_rtruediv(self, sys222, sys322):
"""Test state space rtruediv"""
for sys in [sys222, sys322]:
result = (sys.__rtruediv__(sys)).minreal()
expected = StateSpace([], [], [], np.eye(2), dt=0)
assert_tf_close_coeff(
ss2tf(expected).minreal(),
ss2tf(result).minreal(),
)
# Divide TF by SS
result = sys.__rtruediv__(TransferFunction.s)
expected = TransferFunction.s / sys
assert_tf_close_coeff(
expected.minreal(),
result.minreal(),
)
# Divide array by SS
sys = tf2ss(TransferFunction([1, 2], [2, 1]))
result = sys.__rtruediv__(np.eye(2))
expected = TransferFunction([2, 1], [1, 2]) * np.eye(2)
assert_tf_close_coeff(
expected.minreal(),
ss2tf(result).minreal(),
)
@pytest.mark.parametrize("k", [2, -3.141, np.float32(2.718), np.array([[4.321], [5.678]])])
def test_truediv_ss_scalar(self, sys322, k):
"""Divide SS by scalar."""
sys = sys322 / k
syscheck = sys322 * (1/k)
np.testing.assert_array_almost_equal(sys.A, syscheck.A)
np.testing.assert_array_almost_equal(sys.B, syscheck.B)
np.testing.assert_array_almost_equal(sys.C, syscheck.C)
np.testing.assert_array_almost_equal(sys.D, syscheck.D)
@pytest.mark.parametrize("omega, resp",
[(1.,
np.array([[ 4.37636761e-05-0.01522976j,
-7.92603939e-01+0.02617068j],
[-3.31544858e-01+0.0576105j,
1.28919037e-01-0.14382495j]])),
(32,
np.array([[-1.16548243e-05-3.13444825e-04j,
-7.99936828e-01+4.54201816e-06j],
[-3.00137118e-01+3.42881660e-03j,
6.32015038e-04-1.21462255e-02j]]))])
@pytest.mark.parametrize("dt", [None, 0, 1e-3])
def test_call(self, dt, omega, resp):
"""Evaluate the frequency response at single frequencies"""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
if dt:
sys = sample_system(sys, dt)
s = np.exp(omega * 1j * dt)
else:
s = omega * 1j
# Correct versions of the call
np.testing.assert_allclose(evalfr(sys, s), resp, atol=1e-3)
np.testing.assert_allclose(sys(s), resp, atol=1e-3)
# Deprecated name of the call (should generate error)
with pytest.raises(AttributeError):
sys.evalfr(omega)
def test_freq_resp(self):
"""Evaluate the frequency response at multiple frequencies."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
true_mag = [[[0.0852992637230322, 0.00103596611395218],
[0.935374692849736, 0.799380720864549]],
[[0.55656854563842, 0.301542699860857],
[0.609178071542849, 0.0382108097985257]]]
true_phase = [[[-0.566195599644593, -1.68063565332582],
[3.0465958317514, 3.14141384339534]],
[[2.90457947657161, 3.10601268291914],
[-0.438157380501337, -1.40720969147217]]]
true_omega = [0.1, 10.]
mag, phase, omega = sys.frequency_response(true_omega)
np.testing.assert_almost_equal(mag, true_mag)
np.testing.assert_almost_equal(phase, true_phase)
np.testing.assert_almost_equal(omega, true_omega)
# Deprecated version of the call (should return warning)
with pytest.warns(FutureWarning, match="will be removed"):
mag, phase, omega = sys.freqresp(true_omega)
np.testing.assert_almost_equal(mag, true_mag)
@slycotonly
def test_minreal(self):
"""Test a minreal model reduction."""
# A = [-2, 0.5, 0; 0.5, -0.3, 0; 0, 0, -0.1]
A = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
# B = [0.3, -1.3; 0.1, 0; 1, 0]
B = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
# C = [0, 0.1, 0; -0.3, -0.2, 0]
C = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
# D = [0 -0.8; -0.3 0]
D = [[0., -0.8], [-0.3, 0.]]
# sys = ss(A, B, C, D)
sys = StateSpace(A, B, C, D)
sysr = sys.minreal()
assert sysr.nstates == 2
assert sysr.ninputs == sys.ninputs
assert sysr.noutputs == sys.noutputs
np.testing.assert_array_almost_equal(
eigvals(sysr.A), [-2.136154, -0.1638459])
def test_append_ss(self):
"""Test appending two state-space systems."""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
A2 = [[-1.]]
B2 = [[1.2]]
C2 = [[0.5]]
D2 = [[0.4]]
A3 = [[-2, 0.5, 0, 0], [0.5, -0.3, 0, 0], [0, 0, -0.1, 0],
[0, 0, 0., -1.]]
B3 = [[0.3, -1.3, 0], [0.1, 0., 0], [1.0, 0.0, 0], [0., 0, 1.2]]
C3 = [[0., 0.1, 0.0, 0.0], [-0.3, -0.2, 0.0, 0.0], [0., 0., 0., 0.5]]
D3 = [[0., -0.8, 0.], [-0.3, 0., 0.], [0., 0., 0.4]]
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = StateSpace(A2, B2, C2, D2)
sys3 = StateSpace(A3, B3, C3, D3)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys3.A, sys3c.A)
np.testing.assert_array_almost_equal(sys3.B, sys3c.B)
np.testing.assert_array_almost_equal(sys3.C, sys3c.C)
np.testing.assert_array_almost_equal(sys3.D, sys3c.D)
def test_append_tf(self):
"""Test appending a state-space system with a tf"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
s = TransferFunction([1, 0], [1])
h = 1 / (s + 1) / (s + 2)
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = _convert_to_statespace(h)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys1.A, sys3c.A[:3, :3])
np.testing.assert_array_almost_equal(sys1.B, sys3c.B[:3, :2])
np.testing.assert_array_almost_equal(sys1.C, sys3c.C[:2, :3])
np.testing.assert_array_almost_equal(sys1.D, sys3c.D[:2, :2])
np.testing.assert_array_almost_equal(sys2.A, sys3c.A[3:, 3:])
np.testing.assert_array_almost_equal(sys2.B, sys3c.B[3:, 2:])
np.testing.assert_array_almost_equal(sys2.C, sys3c.C[2:, 3:])
np.testing.assert_array_almost_equal(sys2.D, sys3c.D[2:, 2:])
np.testing.assert_array_almost_equal(sys3c.A[:3, 3:], np.zeros((3, 2)))
np.testing.assert_array_almost_equal(sys3c.A[3:, :3], np.zeros((2, 3)))
def test_array_access_ss_failure(self):
sys1 = StateSpace(
[[1., 2.], [3., 4.]],
[[5., 6.], [6., 8.]],
[[9., 10.], [11., 12.]],
[[13., 14.], [15., 16.]], 1,
inputs=['u0', 'u1'], outputs=['y0', 'y1'])
with pytest.raises(IOError):
sys1[0]
@pytest.mark.parametrize(
"outdx, inpdx",
[(0, 1),
(slice(0, 1, 1), 1),
(0, slice(1, 2, 1)),
(slice(0, 1, 1), slice(1, 2, 1)),
(slice(None, None, -1), 1),
(0, slice(None, None, -1)),
(slice(None, 2, None), 1),
(slice(None, None, 1), slice(None, None, 2)),
(0, slice(1, 2, 1)),
(slice(0, 1, 1), slice(1, 2, 1)),
# ([0, 1], [0]), # lists of indices
])
@pytest.mark.parametrize("named", [False, True])
def test_array_access_ss(self, outdx, inpdx, named):
sys1 = StateSpace(
[[1., 2.], [3., 4.]],
[[5., 6.], [7., 8.]],
[[9., 10.], [11., 12.]],
[[13., 14.], [15., 16.]], 1,
inputs=['u0', 'u1'], outputs=['y0', 'y1'])
if named:
# Use names instead of numbers (and re-convert in statesp)
outnames = sys1.output_labels[outdx]
inpnames = sys1.input_labels[inpdx]
sys1_01 = sys1[outnames, inpnames]
else:
sys1_01 = sys1[outdx, inpdx]
# Convert int to slice to ensure that numpy doesn't drop the dimension
if isinstance(outdx, int): outdx = slice(outdx, outdx+1, 1)
if isinstance(inpdx, int): inpdx = slice(inpdx, inpdx+1, 1)
np.testing.assert_array_almost_equal(sys1_01.A, sys1.A)
np.testing.assert_array_almost_equal(sys1_01.B, sys1.B[:, inpdx])
np.testing.assert_array_almost_equal(sys1_01.C, sys1.C[outdx, :])
np.testing.assert_array_almost_equal(sys1_01.D, sys1.D[outdx, inpdx])
assert sys1.dt == sys1_01.dt
assert sys1_01.input_labels == sys1.input_labels[inpdx]
assert sys1_01.output_labels == sys1.output_labels[outdx]
assert sys1_01.name == sys1.name + "$indexed"
def test_dc_gain_cont(self):
"""Test DC gain for continuous-time state-space systems."""
sys = StateSpace(-2., 6., 5., 0)
np.testing.assert_allclose(sys.dcgain(), 15.)
sys2 = StateSpace(-2, [6., 4.], [[5.], [7.], [11]], np.zeros((3, 2)))
expected = np.array([[15., 10.], [21., 14.], [33., 22.]])
np.testing.assert_allclose(sys2.dcgain(), expected)
sys3 = StateSpace(0., 1., 1., 0.)
np.testing.assert_equal(sys3.dcgain(), np.inf)
def test_dc_gain_discr(self):
"""Test DC gain for discrete-time state-space systems."""
# static gain
sys = StateSpace([], [], [], 2, True)
np.testing.assert_allclose(sys.dcgain(), 2)
# averaging filter
sys = StateSpace(0.5, 0.5, 1, 0, True)
np.testing.assert_allclose(sys.dcgain(), 1)
# differencer
sys = StateSpace(0, 1, -1, 1, True)
np.testing.assert_allclose(sys.dcgain(), 0)
# summer
sys = StateSpace(1, 1, 1, 0, True)
np.testing.assert_equal(sys.dcgain(), np.inf)
@pytest.mark.parametrize("outputs", range(1, 6))
@pytest.mark.parametrize("inputs", range(1, 6))
@pytest.mark.parametrize("dt", [None, 0, 1, True],
ids=["dtNone", "c", "dt1", "dtTrue"])
def test_dc_gain_integrator(self, outputs, inputs, dt):
"""DC gain w/ pole at origin returns appropriately sized array of inf.
the SISO case is also tested in test_dc_gain_{cont,discr}
time systems (dt=0)
"""
states = max(inputs, outputs)
# a matrix that is singular at DC, and has no "useless" states as in
# _remove_useless_states
a = np.triu(np.tile(2, (states, states)))
# eigenvalues all +2, except for ...
a[0, 0] = 0 if dt in [0, None] else 1
b = np.eye(max(inputs, states))[:states, :inputs]
c = np.eye(max(outputs, states))[:outputs, :states]
d = np.zeros((outputs, inputs))
sys = StateSpace(a, b, c, d, dt)
dc = np.full_like(d, np.inf, dtype=float)
if sys.issiso():
dc = dc.squeeze()
try:
np.testing.assert_array_equal(dc, sys.dcgain())
except NotImplementedError:
# Skip MIMO tests if there is no slycot
pytest.skip("slycot required for MIMO dcgain")
def test_scalar_static_gain(self):
"""Regression: can we create a scalar static gain?
make sure StateSpace internals, specifically ABC matrix
sizes, are OK for LTI operations
"""
g1 = StateSpace([], [], [], [2])
g2 = StateSpace([], [], [], [3])
assert g1.dt == None
assert g2.dt == None
g3 = g1 * g2
assert 6 == g3.D[0, 0]
assert g3.dt == None
g4 = g1 + g2
assert 5 == g4.D[0, 0]
assert g4.dt == None
g5 = g1.feedback(g2)
np.testing.assert_allclose(2. / 7, g5.D[0, 0])
assert g5.dt == None
g6 = g1.append(g2)
np.testing.assert_allclose(np.diag([2, 3]), g6.D)
assert g6.dt == None
def test_matrix_static_gain(self):
"""Regression: can we create matrix static gains?"""
d1 = np.array([[1, 2, 3], [4, 5, 6]])
d2 = np.array([[7, 8], [9, 10], [11, 12]])
g1 = StateSpace([], [], [], d1)
# _remove_useless_states was making A = [[0]]
assert (0, 0) == g1.A.shape
g2 = StateSpace([], [], [], d2)
g3 = StateSpace([], [], [], d2.T)
h1 = g1 * g2
np.testing.assert_allclose(d1 @ d2, h1.D)
h2 = g1 + g3
np.testing.assert_allclose(d1 + d2.T, h2.D)
h3 = g1.feedback(g2)
np.testing.assert_array_almost_equal(
solve(np.eye(2) + d1 @ d2, d1), h3.D)
h4 = g1.append(g2)
np.testing.assert_allclose(block_diag(d1, d2), h4.D)
def test_remove_useless_states(self):
"""Regression: _remove_useless_states gives correct ABC sizes."""
g1 = StateSpace(np.zeros((3, 3)), np.zeros((3, 4)),
np.zeros((5, 3)), np.zeros((5, 4)),
remove_useless_states=True)
assert (0, 0) == g1.A.shape
assert (0, 4) == g1.B.shape
assert (5, 0) == g1.C.shape
assert (5, 4) == g1.D.shape
assert 0 == g1.nstates
@pytest.mark.parametrize("A, B, C, D",
[([1], [], [], [1]),
([1], [1], [], [1]),
([1], [], [1], [1]),
([], [1], [], [1]),
([], [1], [1], [1]),
([], [], [1], [1]),
([1], [1], [1], [])])
def test_bad_empty_matrices(self, A, B, C, D):
"""Mismatched ABCD matrices when some are empty."""
with pytest.raises(ValueError):
StateSpace(A, B, C, D)
def test_minreal_static_gain(self):
"""Regression: minreal on static gain was failing."""
g1 = StateSpace([], [], [], [1])
g2 = g1.minreal()
np.testing.assert_array_equal(g1.A, g2.A)
np.testing.assert_array_equal(g1.B, g2.B)
np.testing.assert_array_equal(g1.C, g2.C)
np.testing.assert_allclose(g1.D, g2.D)
def test_empty(self):
"""Regression: can we create an empty StateSpace object?"""
g1 = StateSpace([], [], [], [])
assert 0 == g1.nstates
assert 0 == g1.ninputs
assert 0 == g1.noutputs
def test_matrix_to_state_space(self):
"""_convert_to_statespace(matrix) gives ss([],[],[],D)"""
with pytest.deprecated_call():
D = np.matrix([[1, 2, 3], [4, 5, 6]])
g = _convert_to_statespace(D)
np.testing.assert_array_equal(np.empty((0, 0)), g.A)