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"""iosys_test.py - test input/output system operations
RMM, 17 Apr 2019
This test suite checks to make sure that basic input/output class
operations are working. It doesn't do exhaustive testing of
operations on input/output systems. Separate unit tests should be
created for that purpose.
"""
import re
import warnings
import numpy as np
import pytest
import control as ct
from control import iosys as ios
from control.tests.conftest import matrixfilter
class TestIOSys:
@pytest.fixture
def tsys(self):
class TSys:
pass
T = TSys()
"""Return some test systems"""
# Create a single input/single output linear system
T.siso_linsys = ct.StateSpace(
[[-1, 1], [0, -2]], [[0], [1]], [[1, 0]], [[0]])
# Create a multi input/multi output linear system
T.mimo_linsys1 = ct.StateSpace(
[[-1, 1], [0, -2]], [[1, 0], [0, 1]],
[[1, 0], [0, 1]], np.zeros((2, 2)))
# Create a multi input/multi output linear system
T.mimo_linsys2 = ct.StateSpace(
[[-1, 1], [0, -2]], [[0, 1], [1, 0]],
[[1, 0], [0, 1]], np.zeros((2, 2)))
# Create simulation parameters
T.T = np.linspace(0, 10, 100)
T.U = np.sin(T.T)
T.X0 = [0, 0]
return T
def test_linear_iosys(self, tsys):
# Create an input/output system from the linear system
linsys = tsys.siso_linsys
iosys = ios.LinearIOSystem(linsys)
# Make sure that the right hand side matches linear system
for x, u in (([0, 0], 0), ([1, 0], 0), ([0, 1], 0), ([0, 0], 1)):
np.testing.assert_array_almost_equal(
np.reshape(iosys._rhs(0, x, u), (-1, 1)),
linsys.A @ np.reshape(x, (-1, 1)) + linsys.B * u)
# Make sure that simulations also line up
T, U, X0 = tsys.T, tsys.U, tsys.X0
lti_t, lti_y = ct.forced_response(linsys, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys, T, U, X0)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y, atol=0.002, rtol=0.)
def test_tf2io(self, tsys):
# Create a transfer function from the state space system
linsys = tsys.siso_linsys
tfsys = ct.ss2tf(linsys)
iosys = ct.tf2io(tfsys)
# Verify correctness via simulation
T, U, X0 = tsys.T, tsys.U, tsys.X0
lti_t, lti_y = ct.forced_response(linsys, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys, T, U, X0)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y, atol=0.002, rtol=0.)
# Make sure that non-proper transfer functions generate an error
tfsys = ct.tf('s')
with pytest.raises(ValueError):
iosys=ct.tf2io(tfsys)
def test_ss2io(self, tsys):
# Create an input/output system from the linear system
linsys = tsys.siso_linsys
iosys = ct.ss2io(linsys)
np.testing.assert_allclose(linsys.A, iosys.A)
np.testing.assert_allclose(linsys.B, iosys.B)
np.testing.assert_allclose(linsys.C, iosys.C)
np.testing.assert_allclose(linsys.D, iosys.D)
# Try adding names to things
iosys_named = ct.ss2io(linsys, inputs='u', outputs='y',
states=['x1', 'x2'], name='iosys_named')
assert iosys_named.find_input('u') == 0
assert iosys_named.find_input('x') is None
assert iosys_named.find_output('y') == 0
assert iosys_named.find_output('u') is None
assert iosys_named.find_state('x0') is None
assert iosys_named.find_state('x1') == 0
assert iosys_named.find_state('x2') == 1
np.testing.assert_allclose(linsys.A, iosys_named.A)
np.testing.assert_allclose(linsys.B, iosys_named.B)
np.testing.assert_allclose(linsys.C, iosys_named.C)
np.testing.assert_allclose(linsys.D, iosys_named.D)
def test_iosys_unspecified(self, tsys):
"""System with unspecified inputs and outputs"""
sys = ios.NonlinearIOSystem(secord_update, secord_output)
np.testing.assert_raises(TypeError, sys.__mul__, sys)
def test_iosys_print(self, tsys, capsys):
"""Make sure we can print various types of I/O systems"""
# Send the output to /dev/null
# Simple I/O system
iosys = ct.ss2io(tsys.siso_linsys)
print(iosys)
# I/O system without ninputs, noutputs
ios_unspecified = ios.NonlinearIOSystem(secord_update, secord_output)
print(ios_unspecified)
# I/O system with derived inputs and outputs
ios_linearized = ios.linearize(ios_unspecified, [0, 0], [0])
print(ios_linearized)
@pytest.mark.parametrize("ss", [ios.NonlinearIOSystem, ct.ss])
def test_nonlinear_iosys(self, tsys, ss):
# Create a simple nonlinear I/O system
nlsys = ios.NonlinearIOSystem(predprey)
T = tsys.T
# Start by simulating from an equilibrium point
X0 = [0, 0]
ios_t, ios_y = ios.input_output_response(nlsys, T, 0, X0)
np.testing.assert_array_almost_equal(ios_y, np.zeros(np.shape(ios_y)))
# Now simulate from a nonzero point
X0 = [0.5, 0.5]
ios_t, ios_y = ios.input_output_response(nlsys, T, 0, X0)
#
# Simulate a linear function as a nonlinear function and compare
#
# Create a single input/single output linear system
linsys = tsys.siso_linsys
# Create a nonlinear system with the same dynamics
nlupd = lambda t, x, u, params: \
np.reshape(linsys.A @ np.reshape(x, (-1, 1))
+ linsys.B @ np.reshape(u, (-1, 1)),
(-1,))
nlout = lambda t, x, u, params: \
np.reshape(linsys.C @ np.reshape(x, (-1, 1))
+ linsys.D @ np.reshape(u, (-1, 1)),
(-1,))
nlsys = ios.NonlinearIOSystem(nlupd, nlout, inputs=1, outputs=1)
# Make sure that simulations also line up
T, U, X0 = tsys.T, tsys.U, tsys.X0
lti_t, lti_y = ct.forced_response(linsys, T, U, X0)
ios_t, ios_y = ios.input_output_response(nlsys, T, U, X0)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y,atol=0.002,rtol=0.)
@pytest.fixture
def kincar(self):
# Create a simple nonlinear system to check (kinematic car)
def kincar_update(t, x, u, params):
return np.array([np.cos(x[2]) * u[0], np.sin(x[2]) * u[0], u[1]])
def kincar_output(t, x, u, params):
return np.array([x[0], x[1]])
return ios.NonlinearIOSystem(
kincar_update, kincar_output,
inputs = ['v', 'phi'],
outputs = ['x', 'y'],
states = ['x', 'y', 'theta'])
def test_linearize(self, tsys, kincar):
# Create a single input/single output linear system
linsys = tsys.siso_linsys
iosys = ios.LinearIOSystem(linsys)
# Linearize it and make sure we get back what we started with
linearized = iosys.linearize([0, 0], 0)
np.testing.assert_array_almost_equal(linsys.A, linearized.A)
np.testing.assert_array_almost_equal(linsys.B, linearized.B)
np.testing.assert_array_almost_equal(linsys.C, linearized.C)
np.testing.assert_array_almost_equal(linsys.D, linearized.D)
# Create a simple nonlinear system to check (kinematic car)
iosys = kincar
linearized = iosys.linearize([0, 0, 0], [0, 0])
np.testing.assert_array_almost_equal(linearized.A, np.zeros((3,3)))
np.testing.assert_array_almost_equal(
linearized.B, [[1, 0], [0, 0], [0, 1]])
np.testing.assert_array_almost_equal(
linearized.C, [[1, 0, 0], [0, 1, 0]])
np.testing.assert_array_almost_equal(linearized.D, np.zeros((2,2)))
@pytest.mark.usefixtures("editsdefaults")
def test_linearize_named_signals(self, kincar):
# Full form of the call
linearized = kincar.linearize([0, 0, 0], [0, 0], copy=True,
name='linearized')
assert linearized.name == 'linearized'
assert linearized.find_input('v') == 0
assert linearized.find_input('phi') == 1
assert linearized.find_output('x') == 0
assert linearized.find_output('y') == 1
assert linearized.find_state('x') == 0
assert linearized.find_state('y') == 1
assert linearized.find_state('theta') == 2
# If we copy signal names w/out a system name, append '$linearized'
linearized = kincar.linearize([0, 0, 0], [0, 0], copy=True)
assert linearized.name == kincar.name + '$linearized'
# Test legacy version as well
ct.use_legacy_defaults('0.8.4')
ct.config.use_numpy_matrix(False) # np.matrix deprecated
linearized = kincar.linearize([0, 0, 0], [0, 0], copy=True)
assert linearized.name == kincar.name + '_linearized'
# If copy is False, signal names should not be copied
lin_nocopy = kincar.linearize(0, 0, copy=False)
assert lin_nocopy.find_input('v') is None
assert lin_nocopy.find_output('x') is None
assert lin_nocopy.find_state('x') is None
def test_connect(self, tsys):
# Define a couple of (linear) systems to interconnection
linsys1 = tsys.siso_linsys
iosys1 = ios.LinearIOSystem(linsys1, name='iosys1')
linsys2 = tsys.siso_linsys
iosys2 = ios.LinearIOSystem(linsys2, name='iosys2')
# Connect systems in different ways and compare to StateSpace
linsys_series = linsys2 * linsys1
iosys_series = ios.InterconnectedSystem(
[iosys1, iosys2], # systems
[[1, 0]], # interconnection (series)
0, # input = first system
1 # output = second system
)
# Run a simulation and compare to linear response
T, U = tsys.T, tsys.U
X0 = np.concatenate((tsys.X0, tsys.X0))
ios_t, ios_y, ios_x = ios.input_output_response(
iosys_series, T, U, X0, return_x=True)
lti_t, lti_y = ct.forced_response(linsys_series, T, U, X0)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y,atol=0.002,rtol=0.)
# Connect systems with different timebases
linsys2c = tsys.siso_linsys
linsys2c.dt = 0 # Reset the timebase
iosys2c = ios.LinearIOSystem(linsys2c)
iosys_series = ios.InterconnectedSystem(
[iosys1, iosys2c], # systems
[[1, 0]], # interconnection (series)
0, # input = first system
1 # output = second system
)
assert ct.isctime(iosys_series, strict=True)
ios_t, ios_y, ios_x = ios.input_output_response(
iosys_series, T, U, X0, return_x=True)
lti_t, lti_y = ct.forced_response(linsys_series, T, U, X0)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y,atol=0.002,rtol=0.)
# Feedback interconnection
linsys_feedback = ct.feedback(linsys1, linsys2)
iosys_feedback = ios.InterconnectedSystem(
[iosys1, iosys2], # systems
[[1, 0], # input of sys2 = output of sys1
[0, (1, 0, -1)]], # input of sys1 = -output of sys2
0, # input = first system
0 # output = first system
)
ios_t, ios_y, ios_x = ios.input_output_response(
iosys_feedback, T, U, X0, return_x=True)
lti_t, lti_y = ct.forced_response(linsys_feedback, T, U, X0)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y,atol=0.002,rtol=0.)
@pytest.mark.parametrize(
"connections, inplist, outlist",
[pytest.param([[(1, 0), (0, 0, 1)]], [[(0, 0, 1)]], [[(1, 0, 1)]],
id="full, raw tuple"),
pytest.param([[(1, 0), (0, 0, -1)]], [[(0, 0)]], [[(1, 0, -1)]],
id="full, raw tuple, canceling gains"),
pytest.param([[(1, 0), (0, 0)]], [[(0, 0)]], [[(1, 0)]],
id="full, raw tuple, no gain"),
pytest.param([[(1, 0), (0, 0)]], [(0, 0)], [(1, 0)],
id="full, raw tuple, no gain, no outer list"),
pytest.param([['sys2.u[0]', 'sys1.y[0]']], ['sys1.u[0]'],
['sys2.y[0]'], id="named, full"),
pytest.param([['sys2.u[0]', '-sys1.y[0]']], ['sys1.u[0]'],
['-sys2.y[0]'], id="named, full, caneling gains"),
pytest.param([['sys2.u[0]', 'sys1.y[0]']], 'sys1.u[0]', 'sys2.y[0]',
id="named, full, no list"),
pytest.param([['sys2.u[0]', ('sys1', 'y[0]')]], [(0, 0)], [(1,)],
id="mixed"),
pytest.param([[1, 0]], 0, 1, id="minimal")])
def test_connect_spec_variants(self, tsys, connections, inplist, outlist):
# Define a couple of (linear) systems to interconnection
linsys1 = tsys.siso_linsys
iosys1 = ios.LinearIOSystem(linsys1, name="sys1")
linsys2 = tsys.siso_linsys
iosys2 = ios.LinearIOSystem(linsys2, name="sys2")
# Simple series connection
linsys_series = linsys2 * linsys1
# Create a simulation run to compare against
T, U = tsys.T, tsys.U
X0 = np.concatenate((tsys.X0, tsys.X0))
lti_t, lti_y, lti_x = ct.forced_response(
linsys_series, T, U, X0, return_x=True)
# Create the input/output system with different parameter variations
iosys_series = ios.InterconnectedSystem(
[iosys1, iosys2], connections, inplist, outlist)
ios_t, ios_y, ios_x = ios.input_output_response(
iosys_series, T, U, X0, return_x=True)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y, atol=0.002, rtol=0.)
@pytest.mark.parametrize(
"connections, inplist, outlist",
[pytest.param([['sys2.u[0]', 'sys1.y[0]']],
[[('sys1', 'u[0]'), ('sys1', 'u[0]')]],
[('sys2', 'y[0]', 0.5)], id="duplicated input"),
pytest.param([['sys2.u[0]', ('sys1', 'y[0]', 0.5)],
['sys2.u[0]', ('sys1', 'y[0]', 0.5)]],
'sys1.u[0]', 'sys2.y[0]', id="duplicated connection"),
pytest.param([['sys2.u[0]', 'sys1.y[0]']], 'sys1.u[0]',
[[('sys2', 'y[0]', 0.5), ('sys2', 'y[0]', 0.5)]],
id="duplicated output")])
def test_connect_spec_warnings(self, tsys, connections, inplist, outlist):
# Define a couple of (linear) systems to interconnection
linsys1 = tsys.siso_linsys
iosys1 = ios.LinearIOSystem(linsys1, name="sys1")
linsys2 = tsys.siso_linsys
iosys2 = ios.LinearIOSystem(linsys2, name="sys2")
# Simple series connection
linsys_series = linsys2 * linsys1
# Create a simulation run to compare against
T, U = tsys.T, tsys.U
X0 = np.concatenate((tsys.X0, tsys.X0))
lti_t, lti_y, lti_x = ct.forced_response(
linsys_series, T, U, X0, return_x=True)
# Set up multiple gainst and make sure a warning is generated
with pytest.warns(UserWarning, match="multiple.*Combining"):
iosys_series = ios.InterconnectedSystem(
[iosys1, iosys2], connections, inplist, outlist)
ios_t, ios_y, ios_x = ios.input_output_response(
iosys_series, T, U, X0, return_x=True)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y, atol=0.002, rtol=0.)
def test_static_nonlinearity(self, tsys):
# Linear dynamical system
linsys = tsys.siso_linsys
ioslin = ios.LinearIOSystem(linsys)
# Nonlinear saturation
sat = lambda u: u if abs(u) < 1 else np.sign(u)
sat_output = lambda t, x, u, params: sat(u)
nlsat = ios.NonlinearIOSystem(None, sat_output, inputs=1, outputs=1)
# Set up parameters for simulation
T, U, X0 = tsys.T, 2 * tsys.U, tsys.X0
Usat = np.vectorize(sat)(U)
# Make sure saturation works properly by comparing linear system with
# saturated input to nonlinear system with saturation composition
lti_t, lti_y, lti_x = ct.forced_response(
linsys, T, Usat, X0, return_x=True)
ios_t, ios_y, ios_x = ios.input_output_response(
ioslin * nlsat, T, U, X0, return_x=True)
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_array_almost_equal(lti_y, ios_y, decimal=2)
@pytest.mark.filterwarnings("ignore:Duplicate name::control.iosys")
def test_algebraic_loop(self, tsys):
# Create some linear and nonlinear systems to play with
linsys = tsys.siso_linsys
lnios = ios.LinearIOSystem(linsys)
nlios = ios.NonlinearIOSystem(None, \
lambda t, x, u, params: u*u, inputs=1, outputs=1)
nlios1 = nlios.copy(name='nlios1')
nlios2 = nlios.copy(name='nlios2')
# Set up parameters for simulation
T, U, X0 = tsys.T, tsys.U, tsys.X0
# Single nonlinear system - no states
ios_t, ios_y = ios.input_output_response(nlios, T, U)
np.testing.assert_array_almost_equal(ios_y, U*U, decimal=3)
# Composed nonlinear system (series)
ios_t, ios_y = ios.input_output_response(nlios1 * nlios2, T, U)
np.testing.assert_array_almost_equal(ios_y, U**4, decimal=3)
# Composed nonlinear system (parallel)
ios_t, ios_y = ios.input_output_response(nlios1 + nlios2, T, U)
np.testing.assert_array_almost_equal(ios_y, 2*U**2, decimal=3)
# Nonlinear system composed with LTI system (series) -- with states
ios_t, ios_y = ios.input_output_response(
nlios * lnios * nlios, T, U, X0)
lti_t, lti_y = ct.forced_response(linsys, T, U*U, X0)
np.testing.assert_array_almost_equal(ios_y, lti_y*lti_y, decimal=3)
# Nonlinear system in feeback loop with LTI system
iosys = ios.InterconnectedSystem(
[lnios, nlios], # linear system w/ nonlinear feedback
[[1], # feedback interconnection (sig to 0)
[0, (1, 0, -1)]],
0, # input to linear system
0 # output from linear system
)
ios_t, ios_y = ios.input_output_response(iosys, T, U, X0)
# No easy way to test the result
# Algebraic loop from static nonlinear system in feedback
# (error will be due to no states)
iosys = ios.InterconnectedSystem(
[nlios1, nlios2], # two copies of a static nonlinear system
[[0, 1], # feedback interconnection
[1, (0, 0, -1)]],
0, 0
)
args = (iosys, T, U)
with pytest.raises(RuntimeError):
ios.input_output_response(*args)
# Algebraic loop due to feedthrough term
linsys = ct.StateSpace(
[[-1, 1], [0, -2]], [[0], [1]], [[1, 0]], [[1]])
lnios = ios.LinearIOSystem(linsys)
iosys = ios.InterconnectedSystem(
[nlios, lnios], # linear system w/ nonlinear feedback
[[0, 1], # feedback interconnection
[1, (0, 0, -1)]],
0, 0
)
args = (iosys, T, U, X0)
# ios_t, ios_y = ios.input_output_response(iosys, T, U, X0)
with pytest.raises(RuntimeError):
ios.input_output_response(*args)
def test_summer(self, tsys):
# Construct a MIMO system for testing
linsys = tsys.mimo_linsys1
linio1 = ios.LinearIOSystem(linsys, name='linio1')
linio2 = ios.LinearIOSystem(linsys, name='linio2')
linsys_parallel = linsys + linsys
iosys_parallel = linio1 + linio2
# Set up parameters for simulation
T = tsys.T
U = [np.sin(T), np.cos(T)]
X0 = 0
lin_t, lin_y = ct.forced_response(linsys_parallel, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_parallel, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
def test_rmul(self, tsys):
# Test right multiplication
# TODO: replace with better tests when conversions are implemented
# Set up parameters for simulation
T, U, X0 = tsys.T, tsys.U, tsys.X0
# Linear system with input and output nonlinearities
# Also creates a nested interconnected system
ioslin = ios.LinearIOSystem(tsys.siso_linsys)
nlios = ios.NonlinearIOSystem(None, \
lambda t, x, u, params: u*u, inputs=1, outputs=1)
sys1 = nlios * ioslin
sys2 = ios.InputOutputSystem.__rmul__(nlios, sys1)
# Make sure we got the right thing (via simulation comparison)
ios_t, ios_y = ios.input_output_response(sys2, T, U, X0)
lti_t, lti_y = ct.forced_response(ioslin, T, U*U, X0)
np.testing.assert_array_almost_equal(ios_y, lti_y*lti_y, decimal=3)
def test_neg(self, tsys):
"""Test negation of a system"""
# Set up parameters for simulation
T, U, X0 = tsys.T, tsys.U, tsys.X0
# Static nonlinear system
nlios = ios.NonlinearIOSystem(None, \
lambda t, x, u, params: u*u, inputs=1, outputs=1)
ios_t, ios_y = ios.input_output_response(-nlios, T, U)
np.testing.assert_array_almost_equal(ios_y, -U*U, decimal=3)
# Linear system with input nonlinearity
# Also creates a nested interconnected system
ioslin = ios.LinearIOSystem(tsys.siso_linsys)
sys = (ioslin) * (-nlios)
# Make sure we got the right thing (via simulation comparison)
ios_t, ios_y = ios.input_output_response(sys, T, U, X0)
lti_t, lti_y = ct.forced_response(ioslin, T, U*U, X0)
np.testing.assert_array_almost_equal(ios_y, -lti_y, decimal=3)
def test_feedback(self, tsys):
# Set up parameters for simulation
T, U, X0 = tsys.T, tsys.U, tsys.X0
# Linear system with constant feedback (via "nonlinear" mapping)
ioslin = ios.LinearIOSystem(tsys.siso_linsys)
nlios = ios.NonlinearIOSystem(None, \
lambda t, x, u, params: u, inputs=1, outputs=1)
iosys = ct.feedback(ioslin, nlios)
linsys = ct.feedback(tsys.siso_linsys, 1)
ios_t, ios_y = ios.input_output_response(iosys, T, U, X0)
lti_t, lti_y = ct.forced_response(linsys, T, U, X0)
np.testing.assert_allclose(ios_y, lti_y,atol=0.002,rtol=0.)
def test_bdalg_functions(self, tsys):
"""Test block diagram functions algebra on I/O systems"""
# Set up parameters for simulation
T = tsys.T
U = [np.sin(T), np.cos(T)]
X0 = 0
# Set up systems to be composed
linsys1 = tsys.mimo_linsys1
linio1 = ios.LinearIOSystem(linsys1)
linsys2 = tsys.mimo_linsys2
linio2 = ios.LinearIOSystem(linsys2)
# Series interconnection
linsys_series = ct.series(linsys1, linsys2)
iosys_series = ct.series(linio1, linio2)
lin_t, lin_y = ct.forced_response(linsys_series, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_series, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Make sure that systems don't commute
linsys_series = ct.series(linsys2, linsys1)
lin_t, lin_y = ct.forced_response(linsys_series, T, U, X0)
assert not (np.abs(lin_y - ios_y) < 1e-3).all()
# Parallel interconnection
linsys_parallel = ct.parallel(linsys1, linsys2)
iosys_parallel = ct.parallel(linio1, linio2)
lin_t, lin_y = ct.forced_response(linsys_parallel, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_parallel, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Negation
linsys_negate = ct.negate(linsys1)
iosys_negate = ct.negate(linio1)
lin_t, lin_y = ct.forced_response(linsys_negate, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_negate, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Feedback interconnection
linsys_feedback = ct.feedback(linsys1, linsys2)
iosys_feedback = ct.feedback(linio1, linio2)
lin_t, lin_y = ct.forced_response(linsys_feedback, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_feedback, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
def test_algebraic_functions(self, tsys):
"""Test algebraic operations on I/O systems"""
# Set up parameters for simulation
T = tsys.T
U = [np.sin(T), np.cos(T)]
X0 = 0
# Set up systems to be composed
linsys1 = tsys.mimo_linsys1
linio1 = ios.LinearIOSystem(linsys1)
linsys2 = tsys.mimo_linsys2
linio2 = ios.LinearIOSystem(linsys2)
# Multiplication
linsys_mul = linsys2 * linsys1
iosys_mul = linio2 * linio1
lin_t, lin_y = ct.forced_response(linsys_mul, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_mul, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Make sure that systems don't commute
linsys_mul = linsys1 * linsys2
lin_t, lin_y = ct.forced_response(linsys_mul, T, U, X0)
assert not (np.abs(lin_y - ios_y) < 1e-3).all()
# Addition
linsys_add = linsys1 + linsys2
iosys_add = linio1 + linio2
lin_t, lin_y = ct.forced_response(linsys_add, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_add, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Subtraction
linsys_sub = linsys1 - linsys2
iosys_sub = linio1 - linio2
lin_t, lin_y = ct.forced_response(linsys_sub, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_sub, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Make sure that systems don't commute
linsys_sub = linsys2 - linsys1
lin_t, lin_y = ct.forced_response(linsys_sub, T, U, X0)
assert not (np.abs(lin_y - ios_y) < 1e-3).all()
# Negation
linsys_negate = -linsys1
iosys_negate = -linio1
lin_t, lin_y = ct.forced_response(linsys_negate, T, U, X0)
ios_t, ios_y = ios.input_output_response(iosys_negate, T, U, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
def test_nonsquare_bdalg(self, tsys):
# Set up parameters for simulation
T = tsys.T
U2 = [np.sin(T), np.cos(T)]
U3 = [np.sin(T), np.cos(T), T]
X0 = 0
# Set up systems to be composed
linsys_2i3o = ct.StateSpace(
[[-1, 1, 0], [0, -2, 0], [0, 0, -3]], [[1, 0], [0, 1], [1, 1]],
[[1, 0, 0], [0, 1, 0], [0, 0, 1]], np.zeros((3, 2)))
iosys_2i3o = ios.LinearIOSystem(linsys_2i3o)
linsys_3i2o = ct.StateSpace(
[[-1, 1, 0], [0, -2, 0], [0, 0, -3]],
[[1, 0, 0], [0, 1, 0], [0, 0, 1]],
[[1, 0, 1], [0, 1, -1]], np.zeros((2, 3)))
iosys_3i2o = ios.LinearIOSystem(linsys_3i2o)
# Multiplication
linsys_multiply = linsys_3i2o * linsys_2i3o
iosys_multiply = iosys_3i2o * iosys_2i3o
lin_t, lin_y = ct.forced_response(linsys_multiply, T, U2, X0)
ios_t, ios_y = ios.input_output_response(iosys_multiply, T, U2, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
linsys_multiply = linsys_2i3o * linsys_3i2o
iosys_multiply = iosys_2i3o * iosys_3i2o
lin_t, lin_y = ct.forced_response(linsys_multiply, T, U3, X0)
ios_t, ios_y = ios.input_output_response(iosys_multiply, T, U3, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Right multiplication
# TODO: add real tests once conversion from other types is supported
iosys_multiply = ios.InputOutputSystem.__rmul__(iosys_3i2o, iosys_2i3o)
ios_t, ios_y = ios.input_output_response(iosys_multiply, T, U3, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Feedback
linsys_multiply = ct.feedback(linsys_3i2o, linsys_2i3o)
iosys_multiply = iosys_3i2o.feedback(iosys_2i3o)
lin_t, lin_y = ct.forced_response(linsys_multiply, T, U3, X0)
ios_t, ios_y = ios.input_output_response(iosys_multiply, T, U3, X0)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Mismatch should generate exception
args = (iosys_3i2o, iosys_3i2o)
with pytest.raises(ValueError):
ct.series(*args)
def test_discrete(self, tsys):
"""Test discrete time functionality"""
# Create some linear and nonlinear systems to play with
linsys = ct.StateSpace(
[[-1, 1], [0, -2]], [[0], [1]], [[1, 0]], [[0]], True)
lnios = ios.LinearIOSystem(linsys)
# Set up parameters for simulation
T, U, X0 = tsys.T, tsys.U, tsys.X0
# Simulate and compare to LTI output
ios_t, ios_y = ios.input_output_response(lnios, T, U, X0)
lin_t, lin_y = ct.forced_response(linsys, T, U, X0)
np.testing.assert_allclose(ios_t, lin_t,atol=0.002,rtol=0.)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
# Test MIMO system, converted to discrete time
linsys = ct.StateSpace(tsys.mimo_linsys1)
linsys.dt = tsys.T[1] - tsys.T[0]
lnios = ios.LinearIOSystem(linsys)
# Set up parameters for simulation
T = tsys.T
U = [np.sin(T), np.cos(T)]
X0 = 0
# Simulate and compare to LTI output
ios_t, ios_y = ios.input_output_response(lnios, T, U, X0)
lin_t, lin_y = ct.forced_response(linsys, T, U, X0)
np.testing.assert_allclose(ios_t, lin_t,atol=0.002,rtol=0.)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
def test_discrete_iosys(self, tsys):
"""Create a discrete time system from scratch"""
linsys = ct.StateSpace(
[[-1, 1], [0, -2]], [[0], [1]], [[1, 0]], [[0]], True)
# Create nonlinear version of the same system
def nlsys_update(t, x, u, params):
A, B = params['A'], params['B']
return A @ x + B @ u
def nlsys_output(t, x, u, params):
C = params['C']
return C @ x
nlsys = ct.NonlinearIOSystem(
nlsys_update, nlsys_output, inputs=1, outputs=1, states=2, dt=True)
# Set up parameters for simulation
T, U, X0 = tsys.T, tsys.U, tsys.X0
# Simulate and compare to LTI output
ios_t, ios_y = ios.input_output_response(
nlsys, T, U, X0,
params={'A': linsys.A, 'B': linsys.B, 'C': linsys.C})
lin_t, lin_y = ct.forced_response(linsys, T, U, X0)
np.testing.assert_allclose(ios_t, lin_t,atol=0.002,rtol=0.)
np.testing.assert_allclose(ios_y, lin_y,atol=0.002,rtol=0.)
def test_find_eqpts(self, tsys):
"""Test find_eqpt function"""
# Simple equilibrium point with no inputs
nlsys = ios.NonlinearIOSystem(predprey)
xeq, ueq, result = ios.find_eqpt(
nlsys, [1.6, 1.2], None, return_result=True)
assert result.success
np.testing.assert_array_almost_equal(xeq, [1.64705879, 1.17923874])
np.testing.assert_array_almost_equal(
nlsys._rhs(0, xeq, ueq), np.zeros((2,)))
# Ducted fan dynamics with output = velocity
nlsys = ios.NonlinearIOSystem(pvtol, lambda t, x, u, params: x[0:2])
# Make sure the origin is a fixed point
xeq, ueq, result = ios.find_eqpt(
nlsys, [0, 0, 0, 0], [0, 4*9.8], return_result=True)
assert result.success
np.testing.assert_array_almost_equal(
nlsys._rhs(0, xeq, ueq), np.zeros((4,)))
np.testing.assert_array_almost_equal(xeq, [0, 0, 0, 0])
# Use a small lateral force to cause motion
xeq, ueq, result = ios.find_eqpt(
nlsys, [0, 0, 0, 0], [0.01, 4*9.8], return_result=True)
assert result.success
np.testing.assert_array_almost_equal(
nlsys._rhs(0, xeq, ueq), np.zeros((4,)), decimal=5)
# Equilibrium point with fixed output
xeq, ueq, result = ios.find_eqpt(
nlsys, [0, 0, 0, 0], [0.01, 4*9.8],
y0=[0.1, 0.1], return_result=True)
assert result.success
np.testing.assert_array_almost_equal(
nlsys._out(0, xeq, ueq), [0.1, 0.1], decimal=5)
np.testing.assert_array_almost_equal(
nlsys._rhs(0, xeq, ueq), np.zeros((4,)), decimal=5)
# Specify outputs to constrain (replicate previous)
xeq, ueq, result = ios.find_eqpt(
nlsys, [0, 0, 0, 0], [0.01, 4*9.8], y0=[0.1, 0.1],
iy = [0, 1], return_result=True)
assert result.success
np.testing.assert_array_almost_equal(
nlsys._out(0, xeq, ueq), [0.1, 0.1], decimal=5)
np.testing.assert_array_almost_equal(
nlsys._rhs(0, xeq, ueq), np.zeros((4,)), decimal=5)
# Specify inputs to constrain (replicate previous), w/ no result
xeq, ueq = ios.find_eqpt(
nlsys, [0, 0, 0, 0], [0.01, 4*9.8], y0=[0.1, 0.1], iu = [])
np.testing.assert_array_almost_equal(
nlsys._out(0, xeq, ueq), [0.1, 0.1], decimal=5)
np.testing.assert_array_almost_equal(
nlsys._rhs(0, xeq, ueq), np.zeros((4,)), decimal=5)
# Now solve the problem with the original PVTOL variables
# Constrain the output angle and x velocity
nlsys_full = ios.NonlinearIOSystem(pvtol_full, None)
xeq, ueq, result = ios.find_eqpt(
nlsys_full, [0, 0, 0, 0, 0, 0], [0.01, 4*9.8],
y0=[0, 0, 0.1, 0.1, 0, 0], iy = [2, 3],
idx=[2, 3, 4, 5], ix=[0, 1], return_result=True)
assert result.success
np.testing.assert_array_almost_equal(
nlsys_full._out(0, xeq, ueq)[[2, 3]], [0.1, 0.1], decimal=5)
np.testing.assert_array_almost_equal(
nlsys_full._rhs(0, xeq, ueq)[-4:], np.zeros((4,)), decimal=5)
# Fix one input and vary the other
nlsys_full = ios.NonlinearIOSystem(pvtol_full, None)
xeq, ueq, result = ios.find_eqpt(
nlsys_full, [0, 0, 0, 0, 0, 0], [0.01, 4*9.8],
y0=[0, 0, 0.1, 0.1, 0, 0], iy=[3], iu=[1],
idx=[2, 3, 4, 5], ix=[0, 1], return_result=True)
assert result.success
np.testing.assert_almost_equal(ueq[1], 4*9.8, decimal=5)
np.testing.assert_array_almost_equal(
nlsys_full._out(0, xeq, ueq)[[3]], [0.1], decimal=5)
np.testing.assert_array_almost_equal(
nlsys_full._rhs(0, xeq, ueq)[-4:], np.zeros((4,)), decimal=5)
# PVTOL with output = y velocity
xeq, ueq, result = ios.find_eqpt(
nlsys_full, [0, 0, 0, 0.1, 0, 0], [0.01, 4*9.8],
y0=[0, 0, 0, 0.1, 0, 0], iy=[3],
dx0=[0.1, 0, 0, 0, 0, 0], idx=[1, 2, 3, 4, 5],
ix=[0, 1], return_result=True)
assert result.success
np.testing.assert_array_almost_equal(
nlsys_full._out(0, xeq, ueq)[-3:], [0.1, 0, 0], decimal=5)
np.testing.assert_array_almost_equal(
nlsys_full._rhs(0, xeq, ueq)[-5:], np.zeros((5,)), decimal=5)
# Unobservable system
linsys = ct.StateSpace(
[[-1, 1], [0, -2]], [[0], [1]], [[0, 0]], [[0]])
lnios = ios.LinearIOSystem(linsys)
# If result is returned, user has to check
xeq, ueq, result = ios.find_eqpt(
lnios, [0, 0], [0], y0=[1], return_result=True)
assert not result.success
# If result is not returned, find_eqpt should return None
xeq, ueq = ios.find_eqpt(lnios, [0, 0], [0], y0=[1])
assert xeq is None
assert ueq is None
def test_params(self, tsys):
# Start with the default set of parameters
ios_secord_default = ios.NonlinearIOSystem(
secord_update, secord_output, inputs=1, outputs=1, states=2)
lin_secord_default = ios.linearize(ios_secord_default, [0, 0], [0])
w_default, v_default = np.linalg.eig(lin_secord_default.A)
# New copy, with modified parameters
ios_secord_update = ios.NonlinearIOSystem(
secord_update, secord_output, inputs=1, outputs=1, states=2,
params={'omega0':2, 'zeta':0})
# Make sure the default parameters haven't changed
lin_secord_check = ios.linearize(ios_secord_default, [0, 0], [0])
w, v = np.linalg.eig(lin_secord_check.A)
np.testing.assert_array_almost_equal(np.sort(w), np.sort(w_default))
# Make sure updated system parameters got set correctly
lin_secord_update = ios.linearize(ios_secord_update, [0, 0], [0])
w, v = np.linalg.eig(lin_secord_update.A)
np.testing.assert_array_almost_equal(np.sort(w), np.sort([2j, -2j]))
# Change the parameters of the default sys just for the linearization
lin_secord_local = ios.linearize(ios_secord_default, [0, 0], [0],
params={'zeta':0})
w, v = np.linalg.eig(lin_secord_local.A)
np.testing.assert_array_almost_equal(np.sort(w), np.sort([1j, -1j]))
# Change the parameters of the updated sys just for the linearization
lin_secord_local = ios.linearize(ios_secord_update, [0, 0], [0],
params={'zeta':0, 'omega0':3})
w, v = np.linalg.eig(lin_secord_local.A)
np.testing.assert_array_almost_equal(np.sort(w), np.sort([3j, -3j]))
# Make sure that changes propagate through interconnections
ios_series_default_local = ios_secord_default * ios_secord_update
lin_series_default_local = ios.linearize(
ios_series_default_local, [0, 0, 0, 0], [0])
w, v = np.linalg.eig(lin_series_default_local.A)
np.testing.assert_array_almost_equal(
np.sort(w), np.sort(np.concatenate((w_default, [2j, -2j]))))
# Show that we can change the parameters at linearization
lin_series_override = ios.linearize(
ios_series_default_local, [0, 0, 0, 0], [0],
params={'zeta':0, 'omega0':4})
w, v = np.linalg.eig(lin_series_override.A)
np.testing.assert_array_almost_equal(w, [4j, -4j, 4j, -4j])
# Check for warning if we try to set params for LinearIOSystem
linsys = tsys.siso_linsys
iosys = ios.LinearIOSystem(linsys)
T, U, X0 = tsys.T, tsys.U, tsys.X0
lti_t, lti_y = ct.forced_response(linsys, T, U, X0)
with pytest.warns(UserWarning, match="LinearIOSystem.*ignored"):
ios_t, ios_y = ios.input_output_response(
iosys, T, U, X0, params={'something':0})
# Check to make sure results are OK
np.testing.assert_array_almost_equal(lti_t, ios_t)
np.testing.assert_allclose(lti_y, ios_y,atol=0.002,rtol=0.)
def test_named_signals(self, tsys):
sys1 = ios.NonlinearIOSystem(
updfcn = lambda t, x, u, params: np.array(
tsys.mimo_linsys1.A @ np.reshape(x, (-1, 1)) \
+ tsys.mimo_linsys1.B @ np.reshape(u, (-1, 1))
).reshape(-1,),
outfcn = lambda t, x, u, params: np.array(
tsys.mimo_linsys1.C @ np.reshape(x, (-1, 1)) \
+ tsys.mimo_linsys1.D @ np.reshape(u, (-1, 1))
).reshape(-1,),
inputs = ['u[0]', 'u[1]'],
outputs = ['y[0]', 'y[1]'],
states = tsys.mimo_linsys1.nstates,
name = 'sys1')
sys2 = ios.LinearIOSystem(tsys.mimo_linsys2,
inputs = ['u[0]', 'u[1]'],
outputs = ['y[0]', 'y[1]'],
name = 'sys2')
# Series interconnection (sys1 * sys2) using __mul__
ios_mul = sys1 * sys2
ss_series = tsys.mimo_linsys1 * tsys.mimo_linsys2
lin_series = ct.linearize(ios_mul, 0, 0)
np.testing.assert_array_almost_equal(ss_series.A, lin_series.A)
np.testing.assert_array_almost_equal(ss_series.B, lin_series.B)
np.testing.assert_array_almost_equal(ss_series.C, lin_series.C)
np.testing.assert_array_almost_equal(ss_series.D, lin_series.D)
# Series interconnection (sys1 * sys2) using series
ios_series = ct.series(sys2, sys1)
ss_series = ct.series(tsys.mimo_linsys2, tsys.mimo_linsys1)
lin_series = ct.linearize(ios_series, 0, 0)
np.testing.assert_array_almost_equal(ss_series.A, lin_series.A)
np.testing.assert_array_almost_equal(ss_series.B, lin_series.B)
np.testing.assert_array_almost_equal(ss_series.C, lin_series.C)
np.testing.assert_array_almost_equal(ss_series.D, lin_series.D)
# Series interconnection (sys1 * sys2) using named + mixed signals
ios_connect = ios.InterconnectedSystem(
[sys2, sys1],
connections=[
[('sys1', 'u[0]'), 'sys2.y[0]'],
['sys1.u[1]', 'sys2.y[1]']
],
inplist=['sys2.u[0]', ('sys2', 1)],
outlist=[(1, 'y[0]'), 'sys1.y[1]']
)
lin_series = ct.linearize(ios_connect, 0, 0)
np.testing.assert_array_almost_equal(ss_series.A, lin_series.A)
np.testing.assert_array_almost_equal(ss_series.B, lin_series.B)
np.testing.assert_array_almost_equal(ss_series.C, lin_series.C)
np.testing.assert_array_almost_equal(ss_series.D, lin_series.D)
# Try the same thing using the interconnect function
# Since sys1 is nonlinear, we should get back the same result
ios_connect = ios.interconnect(
(sys2, sys1),
connections=(
[('sys1', 'u[0]'), 'sys2.y[0]'],
['sys1.u[1]', 'sys2.y[1]']
),
inplist=['sys2.u[0]', ('sys2', 1)],
outlist=[(1, 'y[0]'), 'sys1.y[1]']
)
lin_series = ct.linearize(ios_connect, 0, 0)
np.testing.assert_array_almost_equal(ss_series.A, lin_series.A)
np.testing.assert_array_almost_equal(ss_series.B, lin_series.B)
np.testing.assert_array_almost_equal(ss_series.C, lin_series.C)
np.testing.assert_array_almost_equal(ss_series.D, lin_series.D)
# Try the same thing using the interconnect function
# Since sys1 is nonlinear, we should get back the same result
# Note: use a tuple for connections to make sure it works
ios_connect = ios.interconnect(
(sys2, sys1),
connections=(
[('sys1', 'u[0]'), 'sys2.y[0]'],
['sys1.u[1]', 'sys2.y[1]']
),
inplist=['sys2.u[0]', ('sys2', 1)],
outlist=[(1, 'y[0]'), 'sys1.y[1]']
)