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GeezerVison is showing github.com — Describing functions by murrayrm · Pull Request #521 · python-control/python-control

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@murrayrm

@murrayrm murrayrm commented Jan 28, 2021 •
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This PR adds a basic describing function capability, mainly captured in the new describing_function_plot command. Almost all of the new code is in the descfcn.py file, with a few changes elsewhere to support some of the patterns. There is a Jupyter notebook demonstrating the use of the code.

Additional changes:

  • Added a mirror keyword to nyquist_plot that sets the line style used for plotting the complementary portion of the Nyquist plot. This defaults to --, which matches the FBS convention.
  • Added an _isstatic method to the NonlinearIOSystem class that allows passing a nonlinear I/O object as the argument for computing describing functions.
  • Added a __call__ method to the NonlinearIOSystem class that direct calling of a static nonlinear input/output function (returning the value of the function at the given input).

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@murrayrm
murrayrm force-pushed the describing_functions branch from e180716 to e82e2af Compare January 28, 2021 20:42
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coveralls commented Jan 28, 2021 •
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Coverage Status

Coverage increased (+0.3%) to 88.024% when pulling 93d97cc on murrayrm:describing_functions into d75c395 on python-control:master.
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sawyerbfuller added a commit to sawyerbfuller/python-control that referenced this pull request Jan 28, 2021
@sawyerbfuller
revert some freqplot.nyquist_plot changes because they turned out to …
d195e06 
…be unneccesary and to avoid a merge conbflict with python-control#521
@sawyerbfuller sawyerbfuller mentioned this pull request Jan 28, 2021
Freqplot improvements #522
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sawyerbfuller reviewed Jan 28, 2021
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Comment thread control/iosys.py Outdated
"""Check to see if a system is single input, single output"""
return self.ninputs == 1 and self.noutputs == 1

def isstatic(self):

@sawyerbfuller sawyerbfuller Jan 28, 2021 •
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statespace and xferfcn have methods is_static_gain() I added. should we choose a standard conventiuon? I would lean toward isstatic() because it is shorter and matches issiso(),

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@murrayrm murrayrm Jan 28, 2021

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I agree we should be uniform. I'm not sure that is_static_gain() would make sense for a nonlinear system, so that is another reason to opt for isstatic().

Also: these are mainly internally used functions, so I think it is OK to stick with isstatic/issiso rather than is_static/is_siso.

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@murrayrm murrayrm Jan 29, 2021

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Converted to _isstatic in latest push since this really is an internal function, not intended to be seen by the user.

@sawyerbfuller: when we update is_static_gain, we can either change to _isstatic if all usage is internal or change to isstatic (and update descfcn.py for consistency).

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renamed in #524

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roryyorke
roryyorke reviewed Jan 30, 2021
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I'll try to review from describing_function_plot on later this weekend.

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Comment thread control/descfcn.py Outdated

# Class for nonlinearities with a built-in describing function
class DescribingFunctionNonlinearity():
"""Base class for nonlinear functions with a describing function

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Suggested change
"""Base class for nonlinear functions with a describing function
"""Base class for nonlinear systems with a describing function

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Comment thread control/descfcn.py Outdated
"""Base class for nonlinear functions with a describing function

This class is intended to be used as a base class for nonlinear functions
that have a analytically defined describing function (accessed via the

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that have a analytically defined describing function (accessed via the
that have an analytically defined describing function (accessed via the

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Comment thread control/descfcn.py Outdated
#

"""The :mod:~control.descfcn` module contains function for performing
closed loop analysis of systems with static nonlinearities using

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I find the static terminology a little confusing, especially given that some of the describing functions declared here, e.g. hysteresis, return False in their _isstatic methods. Maybe "non-linearities with frequency-independent describing functions", though that is rather long-winded.

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Comment thread control/descfcn.py
implement a `call` method that evaluates the nonlinearity at a given point
and an `_isstatic` method that is `True` if the nonlinearity has no
internal state.

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How about: "Subclasses should override the describing_function, __call__, and _isstatic methods."

It looks like __call__ 's inputs and outputs must be scalar, since __call__ is allowed to change instance memory, so the sequence of calls matters.

Can describing_function be called with a numpy.ndarray of amplitude values? This seems like a reasonable thing to want to do.

If I understand correctly, if __call__ updates any instance state, then _isstatic must return False -- it would be good to mention that too.

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@murrayrm murrayrm Jan 31, 2021

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I've updated the documentation a bit to address the comments above.

In terms of calling DescribingFunctionNonlinearity.description_function with an array: the intent is that this is a low level function that just evaluates the describing function at a scalar value. The describing_function function has the code to evaluate on an array.

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Comment thread control/descfcn.py

"""
# If there is an analytical solution, trying using that first
if try_method and hasattr(F, 'describing_function'):

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the hasattr check will succeed for any base class of DescribingFunctionNonlinearity - shouldn't you also catch the NotImplementedError ?

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Good catch. Fixed.

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Comment thread control/descfcn.py Outdated
#

# Evaluate over a full range of angles
theta = np.linspace(0, 2*np.pi, num_points)

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this is not something I'm not at all sure about: should this omit the final point, so it's more like a discrete Fourier transform?

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Good catch. We are approximating the integral by a discrete sum with the assumption that the function is constant on each small interval. But this means that we don't want to include the value at 2pi (since we already captured the segment just before 2pi in the previous evaluation).

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Comment thread control/descfcn.py Outdated
# Save the scaling factor for to make the formulas simpler
scale = dtheta / np.pi / a

# Evaluate the function (twice) along a sinusoid (for internal state)

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I don't see how this is evaluating the function twice over the sinusoid (assume that means two periods of sinusoid).

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Left over from an earlier iteration (before I implemented _isstatic). Fixed.

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Comment thread control/descfcn.py Outdated
#
# N(A) = M_1(A) e^{j \phi_1(A)} / A
#
# To compute this, we compute F(\theta) for \theta between 0 and 2 \pi,

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Suggested change
# To compute this, we compute F(\theta) for \theta between 0 and 2 \pi,
# To compute this, we compute F(A\sin\theta) for \theta between 0 and 2 \pi,

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Comment thread control/descfcn.py Outdated
#
# and then integrate the product against \sin\theta and \cos\theta to obtain
#
# \int_0^{2\pi} F(a\sin\theta) \sin\theta d\theta = M_1 \pi \cos\phi

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# \int_0^{2\pi} F(a\sin\theta) \sin\theta d\theta = M_1 \pi \cos\phi
# \int_0^{2\pi} F(A\sin\theta) \sin\theta d\theta = M_1 \pi \cos\phi

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Comment thread control/descfcn.py Outdated
# and then integrate the product against \sin\theta and \cos\theta to obtain
#
# \int_0^{2\pi} F(a\sin\theta) \sin\theta d\theta = M_1 \pi \cos\phi
# \int_0^{2\pi} F(a\sin\theta) \cos\theta d\theta = M_1 \pi \sin\phi

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Suggested change
# \int_0^{2\pi} F(a\sin\theta) \cos\theta d\theta = M_1 \pi \sin\phi
# \int_0^{2\pi} F(A\sin\theta) \cos\theta d\theta = M_1 \pi \sin\phi

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roryyorke
roryyorke reviewed Jan 30, 2021
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Searching for the intersections will be O(m * n).

We could look libraries we can pull in to find the intersection of H(jw) and N(A). I googled "numpy polyline intersection" and found pointers to https://pypi.org/project/Shapely/, but that may be over the top.

We could also look at using scipy.optimize.root, though I don't see an obvious way to bound the domain of search. And we'd have to handle the multiple roots ourselves.

When N(A) is strictly real (saturation, deadzone, relay), we could probably specialize the solver.

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Comment thread control/descfcn.py Outdated
self.ub = ub

def __call__(self, x):
return np.maximum(self.lb, np.minimum(x, self.ub))

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return np.maximum(self.lb, np.minimum(x, self.ub))
return np.clip(x, self.lb, self.ub)

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Comment thread control/freqplot.py Outdated
def nyquist_plot(syslist, omega=None, plot=True, label_freq=0,
arrowhead_length=0.1, arrowhead_width=0.1,
color=None, *args, **kwargs):
mirror='--', color=None, *args, **kwargs):

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This new argument should be documented. Are we going to allow None for no mirror? Below you checked for False.

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@murrayrm murrayrm Jan 31, 2021

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Updated documentation. Should either be a linestyle or False if you don't want to plot the mirror image at all. Setting it to (explicitly) to None will generate an error (from matplotlib).

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Comment thread control/descfcn.py
if self.lb <= A and A <= self.ub:
return 1.
else:
alpha, beta = math.asin(self.ub/A), math.asin(-self.lb/A)

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FWIW, the expression in Slotine & Li, and also https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-estimation-and-control-of-aerospace-systems-spring-2004/readings/gelb_ch2_ocr.pdf and https://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/textbook/MITRES_6-010S13_chap06.pdf are different to what you have here, but they may be equivalent. All these references restrict themselves to saturation with odd symmetry, though.

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@murrayrm murrayrm Jan 31, 2021

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It's equivalent (I checked, but also the unit tests check this).

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Comment thread control/descfcn.py

# Backlash nonlinearity (#48 in Gelb and Vander Velde, 1968)
class backlash_nonlinearity(DescribingFunctionNonlinearity):
"""Backlash nonlinearity for use in describing function analysis

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Friction-controlled backlash? (https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-30-estimation-and-control-of-aerospace-systems-spring-2004/readings/gelb_ch2_ocr.pdf p. 68?)

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@murrayrm murrayrm Jan 31, 2021

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I saw that, but couldn't figure out what the "friction-controlled" had to do with it exactly. It seemed to me that you just had backlash as in a gear and so the output just followed the input (modulo the backlash action).

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Maybe it's not important, but I remember a colleague always talking about the two backlash types (friction-controlled and inertia-controlled), and I thought it was standard nomenclature. The inertia-controlled case is discussed in the reference above immediately after friction-controlled.

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Sorry, I was looking at something different (probably just the table in Appendix B, which didn't have the explanation). I'll update.

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Comment thread control/descfcn.py Outdated
# Convert input to an array
x_array = np.array(x)

y = []

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there's an inconsistency here: you've allowed a vector input for this __call__, but not the others.

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@murrayrm murrayrm Jan 31, 2021

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Good catch.

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Comment thread control/tests/descfcn_test.py Outdated


# Static function via a class
class saturation_class():

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Suggested change
class saturation_class():
class saturation_class:

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Comment thread control/tests/descfcn_test.py Outdated
np.testing.assert_array_equal(satsys([0.]), [0.])

# Test SIMO nonlinearity
def _simofcn(t, x, u, params={}):

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Suggested change
def _simofcn(t, x, u, params={}):
def _simofcn(t, x, u, params):

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Comment thread control/tests/descfcn_test.py Outdated
return np.array([np.sin(u[0]) * np.cos(u[1])])
miso_sys = ct.NonlinearIOSystem(None, outfcn=_misofcn, input=2, output=1)
np.testing.assert_array_equal(miso_sys([0, 0]), [0])
np.testing.assert_array_equal(miso_sys([0, 0]), [0])

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np.testing.assert_array_equal(miso_sys([0, 0]), [0])

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Comment thread control/tests/descfcn_test.py


# Test saturation describing function in multiple ways
def test_saturation_describing_function(satsys):

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I think you mean to test with the function saturation here, but this tests saturation_class twice.

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Comment thread doc/descfcn.rst Outdated
linear system and a static nonlinearity, it is possible to obtain a
generalization of Nyquist's stability criterion based on the idea of
describing functions. The basic concept involves approximating the
response of a static nonlinearity to an input :math:`u = A e^{\omega

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e^{j\omega t} ?

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@murrayrm
murrayrm force-pushed the describing_functions branch 3 times, most recently from aba5b43 to f5bca43 Compare January 31, 2021 18:50
@murrayrm murrayrm mentioned this pull request Feb 6, 2021
Nyquist plot improvements #534
Merged
murrayrm added 8 commits February 6, 2021 13:09
@murrayrm
initial implementation of describing function computation
cda5640
@murrayrm
add intersection computation + Jupyter notebook, documentation
c85491e
@murrayrm
change nltools to descfcn, update docs/ and docstrings
df43e69
@murrayrm
add unit tests for exceptions/warnings + cleanup
776a814
@murrayrm
make _isstatic internal; updated doc/ for descfcn
2ac5d9d
@murrayrm
address @roryyorke review comments
17ed781
@murrayrm
additional updates based on @roryyorke review feedback
b29cedf
@murrayrm
small fixes rebasing against master
93d97cc
@murrayrm
murrayrm force-pushed the describing_functions branch from f5bca43 to 93d97cc Compare February 6, 2021 21:23
@murrayrm
murrayrm merged commit c04431f into python-control:master Feb 7, 2021
@murrayrm
murrayrm deleted the describing_functions branch February 8, 2021 00:25
@murrayrm murrayrm mentioned this pull request Feb 11, 2021
fix isssues arising in merge of descfcn/iosys changes #540
Merged
@murrayrm murrayrm added this to the 0.9.0 milestone Mar 20, 2021
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