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1078 lines (900 loc) · 41.5 KB
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"""statefbk_test.py - test state feedback functions
RMM, 30 Mar 2011 (based on TestStatefbk from v0.4a)
"""
import numpy as np
import pytest
import itertools
import warnings
from math import pi, atan
import control as ct
from control import lqe, dlqe, poles, rss, ss, tf
from control.exception import ControlDimension, ControlSlycot, \
ControlArgument, slycot_check
from control.mateqn import care, dare
from control.statefbk import (ctrb, obsv, place, place_varga, lqr, dlqr,
gram, acker)
from control.tests.conftest import (slycotonly, check_deprecated_matrix,
ismatarrayout, asmatarrayout)
@pytest.fixture
def fixedseed():
"""Get consistent test results"""
np.random.seed(0)
class TestStatefbk:
"""Test state feedback functions"""
# Maximum number of states to test + 1
maxStates = 5
# Maximum number of inputs and outputs to test + 1
maxTries = 4
# Set to True to print systems to the output.
debug = False
def testCtrbSISO(self, matarrayin, matarrayout):
A = matarrayin([[1., 2.], [3., 4.]])
B = matarrayin([[5.], [7.]])
Wctrue = np.array([[5., 19.], [7., 43.]])
with check_deprecated_matrix():
Wc = ctrb(A, B)
assert ismatarrayout(Wc)
np.testing.assert_array_almost_equal(Wc, Wctrue)
def testCtrbMIMO(self, matarrayin):
A = matarrayin([[1., 2.], [3., 4.]])
B = matarrayin([[5., 6.], [7., 8.]])
Wctrue = np.array([[5., 6., 19., 22.], [7., 8., 43., 50.]])
Wc = ctrb(A, B)
np.testing.assert_array_almost_equal(Wc, Wctrue)
# Make sure default type values are correct
assert ismatarrayout(Wc)
def testObsvSISO(self, matarrayin):
A = matarrayin([[1., 2.], [3., 4.]])
C = matarrayin([[5., 7.]])
Wotrue = np.array([[5., 7.], [26., 38.]])
Wo = obsv(A, C)
np.testing.assert_array_almost_equal(Wo, Wotrue)
# Make sure default type values are correct
assert ismatarrayout(Wo)
def testObsvMIMO(self, matarrayin):
A = matarrayin([[1., 2.], [3., 4.]])
C = matarrayin([[5., 6.], [7., 8.]])
Wotrue = np.array([[5., 6.], [7., 8.], [23., 34.], [31., 46.]])
Wo = obsv(A, C)
np.testing.assert_array_almost_equal(Wo, Wotrue)
def testCtrbObsvDuality(self, matarrayin):
A = matarrayin([[1.2, -2.3], [3.4, -4.5]])
B = matarrayin([[5.8, 6.9], [8., 9.1]])
Wc = ctrb(A, B)
A = np.transpose(A)
C = np.transpose(B)
Wo = np.transpose(obsv(A, C))
np.testing.assert_array_almost_equal(Wc,Wo)
@slycotonly
def testGramWc(self, matarrayin, matarrayout):
A = matarrayin([[1., -2.], [3., -4.]])
B = matarrayin([[5., 6.], [7., 8.]])
C = matarrayin([[4., 5.], [6., 7.]])
D = matarrayin([[13., 14.], [15., 16.]])
sys = ss(A, B, C, D)
Wctrue = np.array([[18.5, 24.5], [24.5, 32.5]])
with check_deprecated_matrix():
Wc = gram(sys, 'c')
assert ismatarrayout(Wc)
np.testing.assert_array_almost_equal(Wc, Wctrue)
@slycotonly
def testGramRc(self, matarrayin):
A = matarrayin([[1., -2.], [3., -4.]])
B = matarrayin([[5., 6.], [7., 8.]])
C = matarrayin([[4., 5.], [6., 7.]])
D = matarrayin([[13., 14.], [15., 16.]])
sys = ss(A, B, C, D)
Rctrue = np.array([[4.30116263, 5.6961343], [0., 0.23249528]])
Rc = gram(sys, 'cf')
np.testing.assert_array_almost_equal(Rc, Rctrue)
@slycotonly
def testGramWo(self, matarrayin):
A = matarrayin([[1., -2.], [3., -4.]])
B = matarrayin([[5., 6.], [7., 8.]])
C = matarrayin([[4., 5.], [6., 7.]])
D = matarrayin([[13., 14.], [15., 16.]])
sys = ss(A, B, C, D)
Wotrue = np.array([[257.5, -94.5], [-94.5, 56.5]])
Wo = gram(sys, 'o')
np.testing.assert_array_almost_equal(Wo, Wotrue)
@slycotonly
def testGramWo2(self, matarrayin):
A = matarrayin([[1., -2.], [3., -4.]])
B = matarrayin([[5.], [7.]])
C = matarrayin([[6., 8.]])
D = matarrayin([[9.]])
sys = ss(A,B,C,D)
Wotrue = np.array([[198., -72.], [-72., 44.]])
Wo = gram(sys, 'o')
np.testing.assert_array_almost_equal(Wo, Wotrue)
@slycotonly
def testGramRo(self, matarrayin):
A = matarrayin([[1., -2.], [3., -4.]])
B = matarrayin([[5., 6.], [7., 8.]])
C = matarrayin([[4., 5.], [6., 7.]])
D = matarrayin([[13., 14.], [15., 16.]])
sys = ss(A, B, C, D)
Rotrue = np.array([[16.04680654, -5.8890222], [0., 4.67112593]])
Ro = gram(sys, 'of')
np.testing.assert_array_almost_equal(Ro, Rotrue)
def testGramsys(self):
num =[1.]
den = [1., 1., 1.]
sys = tf(num,den)
with pytest.raises(ValueError):
gram(sys, 'o')
with pytest.raises(ValueError):
gram(sys, 'c')
def testAcker(self, fixedseed):
for states in range(1, self.maxStates):
for i in range(self.maxTries):
# start with a random SS system and transform to TF then
# back to SS, check that the matrices are the same.
sys = rss(states, 1, 1)
if (self.debug):
print(sys)
# Make sure the system is not degenerate
Cmat = ctrb(sys.A, sys.B)
if np.linalg.matrix_rank(Cmat) != states:
if (self.debug):
print(" skipping (not reachable or ill conditioned)")
continue
# Place the poles at random locations
des = rss(states, 1, 1)
desired = poles(des)
# Now place the poles using acker
K = acker(sys.A, sys.B, desired)
new = ss(sys.A - sys.B * K, sys.B, sys.C, sys.D)
placed = poles(new)
# Debugging code
# diff = np.sort(poles) - np.sort(placed)
# if not all(diff < 0.001):
# print("Found a problem:")
# print(sys)
# print("desired = ", poles)
np.testing.assert_array_almost_equal(
np.sort(desired), np.sort(placed), decimal=4)
def checkPlaced(self, P_expected, P_placed):
"""Check that placed poles are correct"""
# No guarantee of the ordering, so sort them
P_expected = np.squeeze(np.asarray(P_expected))
P_expected.sort()
P_placed.sort()
np.testing.assert_array_almost_equal(P_expected, P_placed)
def testPlace(self, matarrayin):
# Matrices shamelessly stolen from scipy example code.
A = matarrayin([[1.380, -0.2077, 6.715, -5.676],
[-0.5814, -4.290, 0, 0.6750],
[1.067, 4.273, -6.654, 5.893],
[0.0480, 4.273, 1.343, -2.104]])
B = matarrayin([[0, 5.679],
[1.136, 1.136],
[0, 0],
[-3.146, 0]])
P = matarrayin([-0.5 + 1j, -0.5 - 1j, -5.0566, -8.6659])
K = place(A, B, P)
assert ismatarrayout(K)
P_placed = np.linalg.eigvals(A - B @ K)
self.checkPlaced(P, P_placed)
# Test that the dimension checks work.
with pytest.raises(ControlDimension):
place(A[1:, :], B, P)
with pytest.raises(ControlDimension):
place(A, B[1:, :], P)
# Check that we get an error if we ask for too many poles in the same
# location. Here, rank(B) = 2, so lets place three at the same spot.
P_repeated = matarrayin([-0.5, -0.5, -0.5, -8.6659])
with pytest.raises(ValueError):
place(A, B, P_repeated)
@slycotonly
def testPlace_varga_continuous(self, matarrayin):
"""
Check that we can place eigenvalues for dtime=False
"""
A = matarrayin([[1., -2.], [3., -4.]])
B = matarrayin([[5.], [7.]])
P = [-2., -2.]
K = place_varga(A, B, P)
P_placed = np.linalg.eigvals(A - B @ K)
self.checkPlaced(P, P_placed)
# Test that the dimension checks work.
np.testing.assert_raises(ControlDimension, place, A[1:, :], B, P)
np.testing.assert_raises(ControlDimension, place, A, B[1:, :], P)
# Regression test against bug #177
# https://github.com/python-control/python-control/issues/177
A = matarrayin([[0, 1], [100, 0]])
B = matarrayin([[0], [1]])
P = matarrayin([-20 + 10*1j, -20 - 10*1j])
K = place_varga(A, B, P)
P_placed = np.linalg.eigvals(A - B @ K)
self.checkPlaced(P, P_placed)
@slycotonly
def testPlace_varga_continuous_partial_eigs(self, matarrayin):
"""
Check that we are able to use the alpha parameter to only place
a subset of the eigenvalues, for the continous time case.
"""
# A matrix has eigenvalues at s=-1, and s=-2. Choose alpha = -1.5
# and check that eigenvalue at s=-2 stays put.
A = matarrayin([[1., -2.], [3., -4.]])
B = matarrayin([[5.], [7.]])
P = matarrayin([-3.])
P_expected = np.array([-2.0, -3.0])
alpha = -1.5
K = place_varga(A, B, P, alpha=alpha)
P_placed = np.linalg.eigvals(A - B @ K)
# No guarantee of the ordering, so sort them
self.checkPlaced(P_expected, P_placed)
@slycotonly
def testPlace_varga_discrete(self, matarrayin):
"""
Check that we can place poles using dtime=True (discrete time)
"""
A = matarrayin([[1., 0], [0, 0.5]])
B = matarrayin([[5.], [7.]])
P = matarrayin([0.5, 0.5])
K = place_varga(A, B, P, dtime=True)
P_placed = np.linalg.eigvals(A - B @ K)
# No guarantee of the ordering, so sort them
self.checkPlaced(P, P_placed)
@slycotonly
def testPlace_varga_discrete_partial_eigs(self, matarrayin):
""""
Check that we can only assign a single eigenvalue in the discrete
time case.
"""
# A matrix has eigenvalues at 1.0 and 0.5. Set alpha = 0.51, and
# check that the eigenvalue at 0.5 is not moved.
A = matarrayin([[1., 0], [0, 0.5]])
B = matarrayin([[5.], [7.]])
P = matarrayin([0.2, 0.6])
P_expected = np.array([0.5, 0.6])
alpha = 0.51
K = place_varga(A, B, P, dtime=True, alpha=alpha)
P_placed = np.linalg.eigvals(A - B @ K)
self.checkPlaced(P_expected, P_placed)
def check_LQR(self, K, S, poles, Q, R):
S_expected = asmatarrayout(np.sqrt(Q @ R))
K_expected = asmatarrayout(S_expected / R)
poles_expected = -np.squeeze(np.asarray(K_expected))
np.testing.assert_array_almost_equal(S, S_expected)
np.testing.assert_array_almost_equal(K, K_expected)
np.testing.assert_array_almost_equal(poles, poles_expected)
def check_DLQR(self, K, S, poles, Q, R):
S_expected = asmatarrayout(Q)
K_expected = asmatarrayout(0)
poles_expected = -np.squeeze(np.asarray(K_expected))
np.testing.assert_array_almost_equal(S, S_expected)
np.testing.assert_array_almost_equal(K, K_expected)
np.testing.assert_array_almost_equal(poles, poles_expected)
@pytest.mark.parametrize("method", [None, 'slycot', 'scipy'])
def test_LQR_integrator(self, matarrayin, matarrayout, method):
if method == 'slycot' and not slycot_check():
return
A, B, Q, R = (matarrayin([[X]]) for X in [0., 1., 10., 2.])
K, S, poles = lqr(A, B, Q, R, method=method)
self.check_LQR(K, S, poles, Q, R)
@pytest.mark.parametrize("method", [None, 'slycot', 'scipy'])
def test_LQR_3args(self, matarrayin, matarrayout, method):
if method == 'slycot' and not slycot_check():
return
sys = ss(0., 1., 1., 0.)
Q, R = (matarrayin([[X]]) for X in [10., 2.])
K, S, poles = lqr(sys, Q, R, method=method)
self.check_LQR(K, S, poles, Q, R)
@pytest.mark.parametrize("method", [None, 'slycot', 'scipy'])
def test_DLQR_3args(self, matarrayin, matarrayout, method):
if method == 'slycot' and not slycot_check():
return
dsys = ss(0., 1., 1., 0., .1)
Q, R = (matarrayin([[X]]) for X in [10., 2.])
K, S, poles = dlqr(dsys, Q, R, method=method)
self.check_DLQR(K, S, poles, Q, R)
def test_DLQR_4args(self, matarrayin, matarrayout):
A, B, Q, R = (matarrayin([[X]]) for X in [0., 1., 10., 2.])
K, S, poles = dlqr(A, B, Q, R)
self.check_DLQR(K, S, poles, Q, R)
@pytest.mark.parametrize("cdlqr", [lqr, dlqr])
def test_lqr_badmethod(self, cdlqr):
A, B, Q, R = 0, 1, 10, 2
with pytest.raises(ControlArgument, match="Unknown method"):
K, S, poles = cdlqr(A, B, Q, R, method='nosuchmethod')
@pytest.mark.parametrize("cdlqr", [lqr, dlqr])
def test_lqr_slycot_not_installed(self, cdlqr):
A, B, Q, R = 0, 1, 10, 2
if not slycot_check():
with pytest.raises(ControlSlycot, match="Can't find slycot"):
K, S, poles = cdlqr(A, B, Q, R, method='slycot')
@pytest.mark.xfail(reason="warning not implemented")
def testLQR_warning(self):
"""Test lqr()
Make sure we get a warning if [Q N;N' R] is not positive semi-definite
"""
# from matlab_test siso.ss2 (testLQR); probably not referenced before
# not yet implemented check
A = np.array([[-2, 3, 1],
[-1, 0, 0],
[0, 1, 0]])
B = np.array([[-1, 0, 0]]).T
Q = np.eye(3)
R = np.eye(1)
N = np.array([[1, 1, 2]]).T
# assert any(np.linalg.eigvals(np.block([[Q, N], [N.T, R]])) < 0)
with pytest.warns(UserWarning):
(K, S, E) = lqr(A, B, Q, R, N)
@pytest.mark.parametrize("cdlqr", [lqr, dlqr])
def test_lqr_call_format(self, cdlqr):
# Create a random state space system for testing
sys = rss(2, 3, 2)
sys.dt = None # treat as either continuous or discrete time
# Weighting matrices
Q = np.eye(sys.nstates)
R = np.eye(sys.ninputs)
N = np.zeros((sys.nstates, sys.ninputs))
# Standard calling format
Kref, Sref, Eref = cdlqr(sys.A, sys.B, Q, R)
# Call with system instead of matricees
K, S, E = cdlqr(sys, Q, R)
np.testing.assert_array_almost_equal(Kref, K)
np.testing.assert_array_almost_equal(Sref, S)
np.testing.assert_array_almost_equal(Eref, E)
# Pass a cross-weighting matrix
K, S, E = cdlqr(sys, Q, R, N)
np.testing.assert_array_almost_equal(Kref, K)
np.testing.assert_array_almost_equal(Sref, S)
np.testing.assert_array_almost_equal(Eref, E)
# Inconsistent system dimensions
with pytest.raises(ct.ControlDimension, match="Incompatible dimen"):
K, S, E = cdlqr(sys.A, sys.C, Q, R)
# Incorrect covariance matrix dimensions
with pytest.raises(ct.ControlDimension, match="Q must be a square"):
K, S, E = cdlqr(sys.A, sys.B, sys.C, R, Q)
# Too few input arguments
with pytest.raises(ct.ControlArgument, match="not enough input"):
K, S, E = cdlqr(sys.A, sys.B)
# First argument is the wrong type (use SISO for non-slycot tests)
sys_tf = tf(rss(3, 1, 1))
sys_tf.dt = None # treat as either continuous or discrete time
with pytest.raises(ct.ControlArgument, match="LTI system must be"):
K, S, E = cdlqr(sys_tf, Q, R)
@pytest.mark.xfail(reason="warning not implemented")
def testDLQR_warning(self):
"""Test dlqr()
Make sure we get a warning if [Q N;N' R] is not positive semi-definite
"""
# from matlab_test siso.ss2 (testLQR); probably not referenced before
# not yet implemented check
A = np.array([[-2, 3, 1],
[-1, 0, 0],
[0, 1, 0]])
B = np.array([[-1, 0, 0]]).T
Q = np.eye(3)
R = np.eye(1)
N = np.array([[1, 1, 2]]).T
# assert any(np.linalg.eigvals(np.block([[Q, N], [N.T, R]])) < 0)
with pytest.warns(UserWarning):
(K, S, E) = dlqr(A, B, Q, R, N)
def test_care(self, matarrayin):
"""Test stabilizing and anti-stabilizing feedback, continuous"""
A = matarrayin(np.diag([1, -1]))
B = matarrayin(np.identity(2))
Q = matarrayin(np.identity(2))
R = matarrayin(np.identity(2))
S = matarrayin(np.zeros((2, 2)))
E = matarrayin(np.identity(2))
X, L, G = care(A, B, Q, R, S, E, stabilizing=True)
assert np.all(np.real(L) < 0)
if slycot_check():
X, L, G = care(A, B, Q, R, S, E, stabilizing=False)
assert np.all(np.real(L) > 0)
else:
with pytest.raises(ControlArgument, match="'scipy' not valid"):
X, L, G = care(A, B, Q, R, S, E, stabilizing=False)
@pytest.mark.parametrize(
"stabilizing",
[True, pytest.param(False, marks=slycotonly)])
def test_dare(self, matarrayin, stabilizing):
"""Test stabilizing and anti-stabilizing feedback, discrete"""
A = matarrayin(np.diag([0.5, 2]))
B = matarrayin(np.identity(2))
Q = matarrayin(np.identity(2))
R = matarrayin(np.identity(2))
S = matarrayin(np.zeros((2, 2)))
E = matarrayin(np.identity(2))
X, L, G = dare(A, B, Q, R, S, E, stabilizing=stabilizing)
sgn = {True: -1, False: 1}[stabilizing]
assert np.all(sgn * (np.abs(L) - 1) > 0)
def test_lqr_discrete(self):
"""Test overloading of lqr operator for discrete time systems"""
csys = ct.rss(2, 1, 1)
dsys = ct.drss(2, 1, 1)
Q = np.eye(2)
R = np.eye(1)
# Calling with a system versus explicit A, B should be the sam
K_csys, S_csys, E_csys = ct.lqr(csys, Q, R)
K_expl, S_expl, E_expl = ct.lqr(csys.A, csys.B, Q, R)
np.testing.assert_almost_equal(K_csys, K_expl)
np.testing.assert_almost_equal(S_csys, S_expl)
np.testing.assert_almost_equal(E_csys, E_expl)
# Calling lqr() with a discrete time system should call dlqr()
K_lqr, S_lqr, E_lqr = ct.lqr(dsys, Q, R)
K_dlqr, S_dlqr, E_dlqr = ct.dlqr(dsys, Q, R)
np.testing.assert_almost_equal(K_lqr, K_dlqr)
np.testing.assert_almost_equal(S_lqr, S_dlqr)
np.testing.assert_almost_equal(E_lqr, E_dlqr)
# Calling lqr() with no timebase should call lqr()
asys = ct.ss(csys.A, csys.B, csys.C, csys.D, dt=None)
K_asys, S_asys, E_asys = ct.lqr(asys, Q, R)
K_expl, S_expl, E_expl = ct.lqr(csys.A, csys.B, Q, R)
np.testing.assert_almost_equal(K_asys, K_expl)
np.testing.assert_almost_equal(S_asys, S_expl)
np.testing.assert_almost_equal(E_asys, E_expl)
# Calling dlqr() with a continuous time system should raise an error
with pytest.raises(ControlArgument, match="dsys must be discrete"):
K, S, E = ct.dlqr(csys, Q, R)
@pytest.mark.parametrize(
'nstates, noutputs, ninputs, nintegrators, type_',
[(2, 0, 1, 0, None),
(2, 1, 1, 0, None),
(4, 0, 2, 0, None),
(4, 3, 2, 0, None),
(2, 0, 1, 1, None),
(4, 0, 2, 2, None),
(4, 3, 2, 2, None),
(2, 0, 1, 0, 'nonlinear'),
(4, 0, 2, 2, 'nonlinear'),
(4, 3, 2, 2, 'nonlinear'),
])
def test_statefbk_iosys(
self, nstates, ninputs, noutputs, nintegrators, type_):
# Create the system to be controlled (and estimator)
# TODO: make sure it is controllable?
if noutputs == 0:
# Create a system with full state output
sys = ct.rss(nstates, nstates, ninputs, strictly_proper=True)
sys.C = np.eye(nstates)
est = None
else:
# Create a system with of the desired size
sys = ct.rss(nstates, noutputs, ninputs, strictly_proper=True)
# Create an estimator with different signal names
L, _, _ = ct.lqe(
sys.A, sys.B, sys.C, np.eye(ninputs), np.eye(noutputs))
est = ss(
sys.A - L @ sys.C, np.hstack([L, sys.B]), np.eye(nstates), 0,
inputs=sys.output_labels + sys.input_labels,
outputs=[f'xhat[{i}]' for i in range(nstates)])
# Decide whether to include integral action
if nintegrators:
# Choose the first 'n' outputs as integral terms
C_int = np.eye(nintegrators, nstates)
# Set up an augmented system for LQR computation
# TODO: move this computation into LQR
A_aug = np.block([
[sys.A, np.zeros((sys.nstates, nintegrators))],
[C_int, np.zeros((nintegrators, nintegrators))]
])
B_aug = np.vstack([sys.B, np.zeros((nintegrators, ninputs))])
C_aug = np.hstack([sys.C, np.zeros((sys.C.shape[0], nintegrators))])
aug = ss(A_aug, B_aug, C_aug, 0)
else:
C_int = np.zeros((0, nstates))
aug = sys
# Design an LQR controller
K, _, _ = ct.lqr(aug, np.eye(nstates + nintegrators), np.eye(ninputs))
Kp, Ki = K[:, :nstates], K[:, nstates:]
# Create an I/O system for the controller
ctrl, clsys = ct.create_statefbk_iosystem(
sys, K, integral_action=C_int, estimator=est,
controller_type=type_, name=type_)
# Make sure the name got set correctly
if type_ is not None:
assert ctrl.name == type_
# If we used a nonlinear controller, linearize it for testing
if type_ == 'nonlinear':
clsys = clsys.linearize(0, 0)
# Make sure the linear system elements are correct
if noutputs == 0:
# No estimator
Ac = np.block([
[sys.A - sys.B @ Kp, -sys.B @ Ki],
[C_int, np.zeros((nintegrators, nintegrators))]
])
Bc = np.block([
[sys.B @ Kp, sys.B],
[-C_int, np.zeros((nintegrators, ninputs))]
])
Cc = np.block([
[np.eye(nstates), np.zeros((nstates, nintegrators))],
[-Kp, -Ki]
])
Dc = np.block([
[np.zeros((nstates, nstates + ninputs))],
[Kp, np.eye(ninputs)]
])
else:
# Estimator
Be1, Be2 = est.B[:, :noutputs], est.B[:, noutputs:]
Ac = np.block([
[sys.A, -sys.B @ Ki, -sys.B @ Kp],
[np.zeros((nintegrators, nstates + nintegrators)), C_int],
[Be1 @ sys.C, -Be2 @ Ki, est.A - Be2 @ Kp]
])
Bc = np.block([
[sys.B @ Kp, sys.B],
[-C_int, np.zeros((nintegrators, ninputs))],
[Be2 @ Kp, Be2]
])
Cc = np.block([
[sys.C, np.zeros((noutputs, nintegrators + nstates))],
[np.zeros_like(Kp), -Ki, -Kp]
])
Dc = np.block([
[np.zeros((noutputs, nstates + ninputs))],
[Kp, np.eye(ninputs)]
])
# Check to make sure everything matches
np.testing.assert_array_almost_equal(clsys.A, Ac)
np.testing.assert_array_almost_equal(clsys.B, Bc)
np.testing.assert_array_almost_equal(clsys.C, Cc)
np.testing.assert_array_almost_equal(clsys.D, Dc)
def test_statefbk_iosys_unused(self):
# Create a base system to work with
sys = ct.rss(2, 1, 1, strictly_proper=True)
# Create a system with extra input
aug = ct.rss(2, inputs=[sys.input_labels[0], 'd'],
outputs=sys.output_labels, strictly_proper=True,)
aug.A = sys.A
aug.B[:, 0:1] = sys.B
# Create an estimator
est = ct.create_estimator_iosystem(
sys, np.eye(sys.ninputs), np.eye(sys.noutputs))
# Design an LQR controller
K, _, _ = ct.lqr(sys, np.eye(sys.nstates), np.eye(sys.ninputs))
# Create a baseline I/O control system
ctrl0, clsys0 = ct.create_statefbk_iosystem(sys, K, estimator=est)
clsys0_lin = clsys0.linearize(0, 0)
# Create an I/O system with additional inputs
ctrl1, clsys1 = ct.create_statefbk_iosystem(
aug, K, estimator=est, control_indices=[0])
clsys1_lin = clsys1.linearize(0, 0)
# Make sure the extra inputs are there
assert aug.input_labels[1] not in clsys0.input_labels
assert aug.input_labels[1] in clsys1.input_labels
np.testing.assert_allclose(clsys0_lin.A, clsys1_lin.A)
# Switch around which input we use
aug = ct.rss(2, inputs=['d', sys.input_labels[0]],
outputs=sys.output_labels, strictly_proper=True,)
aug.A = sys.A
aug.B[:, 1:2] = sys.B
# Create an I/O system with additional inputs
ctrl2, clsys2 = ct.create_statefbk_iosystem(
aug, K, estimator=est, control_indices=[1])
clsys2_lin = clsys2.linearize(0, 0)
# Make sure the extra inputs are there
assert aug.input_labels[0] not in clsys0.input_labels
assert aug.input_labels[0] in clsys1.input_labels
np.testing.assert_allclose(clsys0_lin.A, clsys2_lin.A)
def test_lqr_integral_continuous(self):
# Generate a continuous time system for testing
sys = ct.rss(4, 4, 2, strictly_proper=True)
sys.C = np.eye(4) # reset output to be full state
C_int = np.eye(2, 4) # integrate outputs for first two states
nintegrators = C_int.shape[0]
# Generate a controller with integral action
K, _, _ = ct.lqr(
sys, np.eye(sys.nstates + nintegrators), np.eye(sys.ninputs),
integral_action=C_int)
Kp, Ki = K[:, :sys.nstates], K[:, sys.nstates:]
# Create an I/O system for the controller
ctrl, clsys = ct.create_statefbk_iosystem(
sys, K, integral_action=C_int)
# Construct the state space matrices for the controller
# Controller inputs = xd, ud, x
# Controller state = z (integral of x-xd)
# Controller output = ud - Kp(x - xd) - Ki z
A_ctrl = np.zeros((nintegrators, nintegrators))
B_ctrl = np.block([
[-C_int, np.zeros((nintegrators, sys.ninputs)), C_int]
])
C_ctrl = -K[:, sys.nstates:]
D_ctrl = np.block([[Kp, np.eye(nintegrators), -Kp]])
# Check to make sure everything matches
np.testing.assert_array_almost_equal(ctrl.A, A_ctrl)
np.testing.assert_array_almost_equal(ctrl.B, B_ctrl)
np.testing.assert_array_almost_equal(ctrl.C, C_ctrl)
np.testing.assert_array_almost_equal(ctrl.D, D_ctrl)
# Construct the state space matrices for the closed loop system
A_clsys = np.block([
[sys.A - sys.B @ Kp, -sys.B @ Ki],
[C_int, np.zeros((nintegrators, nintegrators))]
])
B_clsys = np.block([
[sys.B @ Kp, sys.B],
[-C_int, np.zeros((nintegrators, sys.ninputs))]
])
C_clsys = np.block([
[np.eye(sys.nstates), np.zeros((sys.nstates, nintegrators))],
[-Kp, -Ki]
])
D_clsys = np.block([
[np.zeros((sys.nstates, sys.nstates + sys.ninputs))],
[Kp, np.eye(sys.ninputs)]
])
# Check to make sure closed loop matches
np.testing.assert_array_almost_equal(clsys.A, A_clsys)
np.testing.assert_array_almost_equal(clsys.B, B_clsys)
np.testing.assert_array_almost_equal(clsys.C, C_clsys)
np.testing.assert_array_almost_equal(clsys.D, D_clsys)
# Check the poles of the closed loop system
assert all(np.real(clsys.poles()) < 0)
# Make sure controller infinite zero frequency gain
if slycot_check():
ctrl_tf = tf(ctrl)
assert abs(ctrl_tf(1e-9)[0][0]) > 1e6
assert abs(ctrl_tf(1e-9)[1][1]) > 1e6
def test_lqr_integral_discrete(self):
# Generate a discrete time system for testing
sys = ct.drss(4, 4, 2, strictly_proper=True)
sys.C = np.eye(4) # reset output to be full state
C_int = np.eye(2, 4) # integrate outputs for first two states
nintegrators = C_int.shape[0]
# Generate a controller with integral action
K, _, _ = ct.lqr(
sys, np.eye(sys.nstates + nintegrators), np.eye(sys.ninputs),
integral_action=C_int)
Kp, Ki = K[:, :sys.nstates], K[:, sys.nstates:]
# Create an I/O system for the controller
ctrl, clsys = ct.create_statefbk_iosystem(
sys, K, integral_action=C_int)
# Construct the state space matrices by hand
A_ctrl = np.eye(nintegrators)
B_ctrl = np.block([
[-C_int, np.zeros((nintegrators, sys.ninputs)), C_int]
])
C_ctrl = -K[:, sys.nstates:]
D_ctrl = np.block([[Kp, np.eye(nintegrators), -Kp]])
# Check to make sure everything matches
assert ct.isdtime(clsys)
np.testing.assert_array_almost_equal(ctrl.A, A_ctrl)
np.testing.assert_array_almost_equal(ctrl.B, B_ctrl)
np.testing.assert_array_almost_equal(ctrl.C, C_ctrl)
np.testing.assert_array_almost_equal(ctrl.D, D_ctrl)
@pytest.mark.parametrize(
"rss_fun, lqr_fun",
[(ct.rss, lqr), (ct.drss, dlqr)])
def test_lqr_errors(self, rss_fun, lqr_fun):
# Generate a discrete time system for testing
sys = rss_fun(4, 4, 2, strictly_proper=True)
with pytest.raises(ControlArgument, match="must pass an array"):
K, _, _ = lqr_fun(
sys, np.eye(sys.nstates), np.eye(sys.ninputs),
integral_action="invalid argument")
with pytest.raises(ControlArgument, match="gain size must match"):
C_int = np.eye(2, 3)
K, _, _ = lqr_fun(
sys, np.eye(sys.nstates), np.eye(sys.ninputs),
integral_action=C_int)
with pytest.raises(TypeError, match="unrecognized keywords"):
K, _, _ = lqr_fun(
sys, np.eye(sys.nstates), np.eye(sys.ninputs),
integrator=None)
def test_statefbk_errors(self):
sys = ct.rss(4, 4, 2, strictly_proper=True)
K, _, _ = ct.lqr(sys, np.eye(sys.nstates), np.eye(sys.ninputs))
with pytest.warns(UserWarning, match="cannot verify system output"):
ctrl, clsys = ct.create_statefbk_iosystem(sys, K)
# reset the system output
sys.C = np.eye(sys.nstates)
with pytest.raises(ControlArgument, match="must be I/O system"):
sys_tf = ct.tf([1], [1, 1])
ctrl, clsys = ct.create_statefbk_iosystem(sys_tf, K)
with pytest.raises(ControlArgument,
match="estimator output must include the full"):
est = ct.rss(3, 3, 2)
ctrl, clsys = ct.create_statefbk_iosystem(sys, K, estimator=est)
with pytest.raises(ControlArgument,
match="system output must include the full state"):
sys_nf = ct.rss(4, 3, 2, strictly_proper=True)
ctrl, clsys = ct.create_statefbk_iosystem(sys_nf, K)
with pytest.raises(ControlArgument, match="gain must be an array"):
ctrl, clsys = ct.create_statefbk_iosystem(sys, "bad argument")
with pytest.warns(DeprecationWarning, match="'type' is deprecated"):
ctrl, clsys = ct.create_statefbk_iosystem(sys, K, type='nonlinear')
with pytest.raises(ControlArgument, match="duplicate keywords"):
with warnings.catch_warnings():
warnings.simplefilter("ignore")
ctrl, clsys = ct.create_statefbk_iosystem(
sys, K, type='nonlinear', controller_type='nonlinear')
with pytest.raises(TypeError, match="unrecognized keyword"):
ctrl, clsys = ct.create_statefbk_iosystem(sys, K, typo='nonlinear')
with pytest.raises(ControlArgument, match="unknown controller_type"):
ctrl, clsys = ct.create_statefbk_iosystem(sys, K, controller_type=1)
# Errors involving integral action
C_int = np.eye(2, 4)
K_int, _, _ = ct.lqr(
sys, np.eye(sys.nstates + C_int.shape[0]), np.eye(sys.ninputs),
integral_action=C_int)
with pytest.raises(ControlArgument, match="must pass an array"):
ctrl, clsys = ct.create_statefbk_iosystem(
sys, K_int, integral_action="bad argument")
with pytest.raises(ControlArgument, match="must be an array of size"):
ctrl, clsys = ct.create_statefbk_iosystem(
sys, K, integral_action=C_int)
# Kinematic car example for testing gain scheduling
@pytest.fixture
def unicycle():
# Create a simple nonlinear system to check (kinematic car)
def unicycle_update(t, x, u, params):
return np.array([np.cos(x[2]) * u[0], np.sin(x[2]) * u[0], u[1]])
return ct.NonlinearIOSystem(
unicycle_update, None,
inputs = ['v', 'phi'],
outputs = ['x', 'y', 'theta'],
states = ['x_', 'y_', 'theta_'])
from math import pi
@pytest.mark.parametrize("method", ['nearest', 'linear', 'cubic'])
def test_gainsched_unicycle(unicycle, method):
# Speeds and angles at which to compute the gains
speeds = [1, 5, 10]
angles = np.linspace(0, pi/2, 4)
points = list(itertools.product(speeds, angles))
# Gains for each speed (using LQR controller)
Q = np.identity(unicycle.nstates)
R = np.identity(unicycle.ninputs)
gains = [np.array(ct.lqr(unicycle.linearize(
[0, 0, angle], [speed, 0]), Q, R)[0]) for speed, angle in points]
#
# Schedule on desired speed and angle
#
# Create gain scheduled controller
ctrl, clsys = ct.create_statefbk_iosystem(
unicycle, (gains, points),
gainsched_indices=[3, 2], gainsched_method=method)
# Check the gain at the selected points
for speed, angle in points:
# Figure out the desired state and input
xe, ue = np.array([0, 0, angle]), np.array([speed, 0])
xd, ud = np.array([0, 0, angle]), np.array([speed, 0])
# Check the control system at the scheduling points
ctrl_lin = ctrl.linearize([], [xd, ud, xe*0])
K, S, E = ct.lqr(unicycle.linearize(xd, ud), Q, R)
np.testing.assert_allclose(
ctrl_lin.D[-xe.size:, -xe.size:], -K, rtol=1e-2)
# Check the closed loop system at the scheduling points
clsys_lin = clsys.linearize(xe, [xd, ud])
np.testing.assert_allclose(
np.sort(clsys_lin.poles()), np.sort(E), rtol=1e-2)
# Check the gain at an intermediate point and confirm stability
speed, angle = 2, pi/3
xe, ue = np.array([0, 0, angle]), np.array([speed, 0])
xd, ud = np.array([0, 0, angle]), np.array([speed, 0])
clsys_lin = clsys.linearize(xe, [xd, ud])
assert np.all(np.real(clsys_lin.poles()) < 0)
# Make sure that gains are different from 'nearest'
if method is not None and method != 'nearest':
ctrl_nearest, clsys_nearest = ct.create_statefbk_iosystem(
unicycle, (gains, points),
gainsched_indices=['ud[0]', 2], gainsched_method='nearest')
nearest_lin = clsys_nearest.linearize(xe, [xd, ud])
assert not np.allclose(
np.sort(clsys_lin.poles()), np.sort(nearest_lin.poles()), rtol=1e-2)
# Run a simulation following a curved path
T = 10 # length of the trajectory [sec]
r = 10 # radius of the circle [m]
timepts = np.linspace(0, T, 50)
Xd = np.vstack([
r * np.cos(timepts/T * pi/2 + 3*pi/2),
r * np.sin(timepts/T * pi/2 + 3*pi/2) + r,
timepts/T * pi/2
])
Ud = np.vstack([
np.ones_like(timepts) * (r * pi/2) / T,
np.ones_like(timepts) * (pi / 2) / T
])
X0 = Xd[:, 0] + np.array([-0.1, -0.1, -0.1])
resp = ct.input_output_response(clsys, timepts, [Xd, Ud], X0)
# plt.plot(resp.states[0], resp.states[1])
np.testing.assert_allclose(
resp.states[:, -1], Xd[:, -1], atol=1e-2, rtol=1e-2)
#
# Schedule on actual speed
#
# Create gain scheduled controller
ctrl, clsys = ct.create_statefbk_iosystem(
unicycle, (gains, points),
ud_labels=['vd', 'phid'], gainsched_indices=['vd', 'theta'])
# Check the gain at the selected points
for speed, angle in points:
# Figure out the desired state and input
xe, ue = np.array([0, 0, angle]), np.array([speed, 0])
xd, ud = np.array([0, 0, angle]), np.array([speed, 0])
# Check the control system at the scheduling points
ctrl_lin = ctrl.linearize([], [xd*0, ud, xe])
K, S, E = ct.lqr(unicycle.linearize(xe, ue), Q, R)
np.testing.assert_allclose(
ctrl_lin.D[-xe.size:, -xe.size:], -K, rtol=1e-2)
# Check the closed loop system at the scheduling points
clsys_lin = clsys.linearize(xe, [xd, ud])
np.testing.assert_allclose(np.sort(
clsys_lin.poles()), np.sort(E), rtol=1e-2)
# Run a simulation following a curved path
resp = ct.input_output_response(clsys, timepts, [Xd, Ud], X0)
np.testing.assert_allclose(
resp.states[:, -1], Xd[:, -1], atol=1e-2, rtol=1e-2)
@pytest.mark.parametrize("method", ['nearest', 'linear', 'cubic'])
def test_gainsched_1d(method):
# Define a linear system to test
sys = ct.ss([[-1, 0.1], [0, -2]], [[0], [1]], np.eye(2), 0)
# Define gains for the first state only
points = [-1, 0, 1]
# Define gain to be constant
K, _, _ = ct.lqr(sys, np.eye(sys.nstates), np.eye(sys.ninputs))
gains = [K for p in points]
# Define the paramters for the simulations
timepts = np.linspace(0, 10, 100)
X0 = np.ones(sys.nstates) * 1.1 # Start outside defined range
# Create a controller and simulate the initial response
gs_ctrl, gs_clsys = ct.create_statefbk_iosystem(
sys, (gains, points), gainsched_indices=[0])
gs_resp = ct.input_output_response(gs_clsys, timepts, 0, X0)