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1431 lines (1226 loc) · 55.5 KB
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# freqplot.py - frequency domain plots for control systems
#
# Author: Richard M. Murray
# Date: 24 May 09
#
# This file contains some standard control system plots: Bode plots,
# Nyquist plots and pole-zero diagrams. The code for Nichols charts
# is in nichols.py.
#
# Copyright (c) 2010 by California Institute of Technology
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions
# are met:
#
# 1. Redistributions of source code must retain the above copyright
# notice, this list of conditions and the following disclaimer.
#
# 2. Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in the
# documentation and/or other materials provided with the distribution.
#
# 3. Neither the name of the California Institute of Technology nor
# the names of its contributors may be used to endorse or promote
# products derived from this software without specific prior
# written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
# FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL CALTECH
# OR THE CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
# USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
# ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
# OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
# SUCH DAMAGE.
#
# $Id$
import math
import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
import warnings
from .ctrlutil import unwrap
from .bdalg import feedback
from .margins import stability_margins
from .exception import ControlMIMONotImplemented
from .statesp import StateSpace
from .xferfcn import TransferFunction
from . import config
__all__ = ['bode_plot', 'nyquist_plot', 'gangof4_plot', 'singular_values_plot',
'bode', 'nyquist', 'gangof4']
# Default values for module parameter variables
_freqplot_defaults = {
'freqplot.feature_periphery_decades': 1,
'freqplot.number_of_samples': 1000,
'freqplot.dB': False, # Plot gain in dB
'freqplot.deg': True, # Plot phase in degrees
'freqplot.Hz': False, # Plot frequency in Hertz
'freqplot.grid': True, # Turn on grid for gain and phase
'freqplot.wrap_phase': False, # Wrap the phase plot at a given value
# deprecations
'deprecated.bode.dB': 'freqplot.dB',
'deprecated.bode.deg': 'freqplot.deg',
'deprecated.bode.Hz': 'freqplot.Hz',
'deprecated.bode.grid': 'freqplot.grid',
'deprecated.bode.wrap_phase': 'freqplot.wrap_phase',
}
#
# Main plotting functions
#
# This section of the code contains the functions for generating
# frequency domain plots
#
#
# Bode plot
#
def bode_plot(syslist, omega=None,
plot=True, omega_limits=None, omega_num=None,
margins=None, method='best', *args, **kwargs):
"""Bode plot for a system
Plots a Bode plot for the system over a (optional) frequency range.
Parameters
----------
syslist : linsys
List of linear input/output systems (single system is OK)
omega : array_like
List of frequencies in rad/sec to be used for frequency response
dB : bool
If True, plot result in dB. Default is false.
Hz : bool
If True, plot frequency in Hz (omega must be provided in rad/sec).
Default value (False) set by config.defaults['freqplot.Hz']
deg : bool
If True, plot phase in degrees (else radians). Default value (True)
config.defaults['freqplot.deg']
plot : bool
If True (default), plot magnitude and phase
omega_limits : array_like of two values
Limits of the to generate frequency vector.
If Hz=True the limits are in Hz otherwise in rad/s.
omega_num : int
Number of samples to plot. Defaults to
config.defaults['freqplot.number_of_samples'].
margins : bool
If True, plot gain and phase margin.
method : method to use in computing margins (see :func:`stability_margins`)
*args : :func:`matplotlib.pyplot.plot` positional properties, optional
Additional arguments for `matplotlib` plots (color, linestyle, etc)
**kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional
Additional keywords (passed to `matplotlib`)
Returns
-------
mag : ndarray (or list of ndarray if len(syslist) > 1))
magnitude
phase : ndarray (or list of ndarray if len(syslist) > 1))
phase in radians
omega : ndarray (or list of ndarray if len(syslist) > 1))
frequency in rad/sec
Other Parameters
----------------
grid : bool
If True, plot grid lines on gain and phase plots. Default is set by
`config.defaults['freqplot.grid']`.
initial_phase : float
Set the reference phase to use for the lowest frequency. If set, the
initial phase of the Bode plot will be set to the value closest to the
value specified. Units are in either degrees or radians, depending on
the `deg` parameter. Default is -180 if wrap_phase is False, 0 if
wrap_phase is True.
wrap_phase : bool or float
If wrap_phase is `False`, then the phase will be unwrapped so that it
is continuously increasing or decreasing. If wrap_phase is `True` the
phase will be restricted to the range [-180, 180) (or [:math:`-\\pi`,
:math:`\\pi`) radians). If `wrap_phase` is specified as a float, the
phase will be offset by 360 degrees if it falls below the specified
value. Default to `False`, set by config.defaults['freqplot.wrap_phase'].
The default values for Bode plot configuration parameters can be reset
using the `config.defaults` dictionary, with module name 'bode'.
Notes
-----
1. Alternatively, you may use the lower-level methods
:meth:`LTI.frequency_response` or ``sys(s)`` or ``sys(z)`` or to
generate the frequency response for a single system.
2. If a discrete time model is given, the frequency response is plotted
along the upper branch of the unit circle, using the mapping ``z =
exp(1j * omega * dt)`` where `omega` ranges from 0 to `pi/dt` and `dt`
is the discrete timebase. If timebase not specified (``dt=True``),
`dt` is set to 1.
Examples
--------
>>> sys = ss("1. -2; 3. -4", "5.; 7", "6. 8", "9.")
>>> mag, phase, omega = bode(sys)
"""
# Make a copy of the kwargs dictionary since we will modify it
kwargs = dict(kwargs)
# Check to see if legacy 'Plot' keyword was used
if 'Plot' in kwargs:
import warnings
warnings.warn("'Plot' keyword is deprecated in bode_plot; use 'plot'",
FutureWarning)
# Map 'Plot' keyword to 'plot' keyword
plot = kwargs.pop('Plot')
# Get values for params (and pop from list to allow keyword use in plot)
dB = config._get_param(
'freqplot', 'dB', kwargs, _freqplot_defaults, pop=True)
deg = config._get_param(
'freqplot', 'deg', kwargs, _freqplot_defaults, pop=True)
Hz = config._get_param(
'freqplot', 'Hz', kwargs, _freqplot_defaults, pop=True)
grid = config._get_param(
'freqplot', 'grid', kwargs, _freqplot_defaults, pop=True)
plot = config._get_param('freqplot', 'plot', plot, True)
margins = config._get_param(
'freqplot', 'margins', margins, False)
wrap_phase = config._get_param(
'freqplot', 'wrap_phase', kwargs, _freqplot_defaults, pop=True)
initial_phase = config._get_param(
'freqplot', 'initial_phase', kwargs, None, pop=True)
omega_num = config._get_param('freqplot', 'number_of_samples', omega_num)
# If argument was a singleton, turn it into a tuple
if not hasattr(syslist, '__iter__'):
syslist = (syslist,)
omega, omega_range_given = _determine_omega_vector(
syslist, omega, omega_limits, omega_num)
if plot:
# Set up the axes with labels so that multiple calls to
# bode_plot will superimpose the data. This was implicit
# before matplotlib 2.1, but changed after that (See
# https://github.com/matplotlib/matplotlib/issues/9024).
# The code below should work on all cases.
# Get the current figure
if 'sisotool' in kwargs:
fig = kwargs['fig']
ax_mag = fig.axes[0]
ax_phase = fig.axes[2]
sisotool = kwargs['sisotool']
del kwargs['fig']
del kwargs['sisotool']
else:
fig = plt.gcf()
ax_mag = None
ax_phase = None
sisotool = False
# Get the current axes if they already exist
for ax in fig.axes:
if ax.get_label() == 'control-bode-magnitude':
ax_mag = ax
elif ax.get_label() == 'control-bode-phase':
ax_phase = ax
# If no axes present, create them from scratch
if ax_mag is None or ax_phase is None:
plt.clf()
ax_mag = plt.subplot(211, label='control-bode-magnitude')
ax_phase = plt.subplot(
212, label='control-bode-phase', sharex=ax_mag)
mags, phases, omegas, nyquistfrqs = [], [], [], []
for sys in syslist:
if not sys.issiso():
# TODO: Add MIMO bode plots.
raise ControlMIMONotImplemented(
"Bode is currently only implemented for SISO systems.")
else:
omega_sys = np.asarray(omega)
if sys.isdtime(strict=True):
nyquistfrq = math.pi / sys.dt
if not omega_range_given:
# limit up to and including nyquist frequency
omega_sys = np.hstack((
omega_sys[omega_sys < nyquistfrq], nyquistfrq))
else:
nyquistfrq = None
mag, phase, omega_sys = sys.frequency_response(omega_sys)
mag = np.atleast_1d(mag)
phase = np.atleast_1d(phase)
#
# Post-process the phase to handle initial value and wrapping
#
if initial_phase is None:
# Start phase in the range 0 to -360 w/ initial phase = -180
# If wrap_phase is true, use 0 instead (phase \in (-pi, pi])
initial_phase = -math.pi if wrap_phase is not True else 0
elif isinstance(initial_phase, (int, float)):
# Allow the user to override the default calculation
if deg:
initial_phase = initial_phase/180. * math.pi
else:
raise ValueError("initial_phase must be a number.")
# Shift the phase if needed
if abs(phase[0] - initial_phase) > math.pi:
phase -= 2*math.pi * \
round((phase[0] - initial_phase) / (2*math.pi))
# Phase wrapping
if wrap_phase is False:
phase = unwrap(phase) # unwrap the phase
elif wrap_phase is True:
pass # default calculation OK
elif isinstance(wrap_phase, (int, float)):
phase = unwrap(phase) # unwrap the phase first
if deg:
wrap_phase *= math.pi/180.
# Shift the phase if it is below the wrap_phase
phase += 2*math.pi * np.maximum(
0, np.ceil((wrap_phase - phase)/(2*math.pi)))
else:
raise ValueError("wrap_phase must be bool or float.")
mags.append(mag)
phases.append(phase)
omegas.append(omega_sys)
nyquistfrqs.append(nyquistfrq)
# Get the dimensions of the current axis, which we will divide up
# TODO: Not current implemented; just use subplot for now
if plot:
nyquistfrq_plot = None
if Hz:
omega_plot = omega_sys / (2. * math.pi)
if nyquistfrq:
nyquistfrq_plot = nyquistfrq / (2. * math.pi)
else:
omega_plot = omega_sys
if nyquistfrq:
nyquistfrq_plot = nyquistfrq
phase_plot = phase * 180. / math.pi if deg else phase
mag_plot = mag
if nyquistfrq_plot:
# append data for vertical nyquist freq indicator line.
# if this extra nyquist lime is is plotted in a single plot
# command then line order is preserved when
# creating a legend eg. legend(('sys1', 'sys2'))
omega_nyq_line = np.array((np.nan, nyquistfrq, nyquistfrq))
omega_plot = np.hstack((omega_plot, omega_nyq_line))
mag_nyq_line = np.array((
np.nan, 0.7*min(mag_plot), 1.3*max(mag_plot)))
mag_plot = np.hstack((mag_plot, mag_nyq_line))
phase_range = max(phase_plot) - min(phase_plot)
phase_nyq_line = np.array(
(np.nan,
min(phase_plot) - 0.2 * phase_range,
max(phase_plot) + 0.2 * phase_range))
phase_plot = np.hstack((phase_plot, phase_nyq_line))
#
# Magnitude plot
#
if dB:
ax_mag.semilogx(omega_plot, 20 * np.log10(mag_plot),
*args, **kwargs)
else:
ax_mag.loglog(omega_plot, mag_plot, *args, **kwargs)
# Add a grid to the plot + labeling
ax_mag.grid(grid and not margins, which='both')
ax_mag.set_ylabel("Magnitude (dB)" if dB else "Magnitude")
#
# Phase plot
#
# Plot the data
ax_phase.semilogx(omega_plot, phase_plot, *args, **kwargs)
# Show the phase and gain margins in the plot
if margins:
# Compute stability margins for the system
margin = stability_margins(sys, method=method)
gm, pm, Wcg, Wcp = (margin[i] for i in (0, 1, 3, 4))
# Figure out sign of the phase at the first gain crossing
# (needed if phase_wrap is True)
phase_at_cp = phases[0][(np.abs(omegas[0] - Wcp)).argmin()]
if phase_at_cp >= 0.:
phase_limit = 180.
else:
phase_limit = -180.
if Hz:
Wcg, Wcp = Wcg/(2*math.pi), Wcp/(2*math.pi)
# Draw lines at gain and phase limits
ax_mag.axhline(y=0 if dB else 1, color='k', linestyle=':',
zorder=-20)
ax_phase.axhline(y=phase_limit if deg else
math.radians(phase_limit),
color='k', linestyle=':', zorder=-20)
mag_ylim = ax_mag.get_ylim()
phase_ylim = ax_phase.get_ylim()
# Annotate the phase margin (if it exists)
if pm != float('inf') and Wcp != float('nan'):
if dB:
ax_mag.semilogx(
[Wcp, Wcp], [0., -1e5],
color='k', linestyle=':', zorder=-20)
else:
ax_mag.loglog(
[Wcp, Wcp], [1., 1e-8],
color='k', linestyle=':', zorder=-20)
if deg:
ax_phase.semilogx(
[Wcp, Wcp], [1e5, phase_limit + pm],
color='k', linestyle=':', zorder=-20)
ax_phase.semilogx(
[Wcp, Wcp], [phase_limit + pm, phase_limit],
color='k', zorder=-20)
else:
ax_phase.semilogx(
[Wcp, Wcp], [1e5, math.radians(phase_limit) +
math.radians(pm)],
color='k', linestyle=':', zorder=-20)
ax_phase.semilogx(
[Wcp, Wcp], [math.radians(phase_limit) +
math.radians(pm),
math.radians(phase_limit)],
color='k', zorder=-20)
# Annotate the gain margin (if it exists)
if gm != float('inf') and Wcg != float('nan'):
if dB:
ax_mag.semilogx(
[Wcg, Wcg], [-20.*np.log10(gm), -1e5],
color='k', linestyle=':', zorder=-20)
ax_mag.semilogx(
[Wcg, Wcg], [0, -20*np.log10(gm)],
color='k', zorder=-20)
else:
ax_mag.loglog(
[Wcg, Wcg], [1./gm, 1e-8], color='k',
linestyle=':', zorder=-20)
ax_mag.loglog(
[Wcg, Wcg], [1., 1./gm], color='k', zorder=-20)
if deg:
ax_phase.semilogx(
[Wcg, Wcg], [0, phase_limit],
color='k', linestyle=':', zorder=-20)
else:
ax_phase.semilogx(
[Wcg, Wcg], [0, math.radians(phase_limit)],
color='k', linestyle=':', zorder=-20)
ax_mag.set_ylim(mag_ylim)
ax_phase.set_ylim(phase_ylim)
if sisotool:
ax_mag.text(
0.04, 0.06,
'G.M.: %.2f %s\nFreq: %.2f %s' %
(20*np.log10(gm) if dB else gm,
'dB ' if dB else '',
Wcg, 'Hz' if Hz else 'rad/s'),
horizontalalignment='left',
verticalalignment='bottom',
transform=ax_mag.transAxes,
fontsize=8 if int(mpl.__version__[0]) == 1 else 6)
ax_phase.text(
0.04, 0.06,
'P.M.: %.2f %s\nFreq: %.2f %s' %
(pm if deg else math.radians(pm),
'deg' if deg else 'rad',
Wcp, 'Hz' if Hz else 'rad/s'),
horizontalalignment='left',
verticalalignment='bottom',
transform=ax_phase.transAxes,
fontsize=8 if int(mpl.__version__[0]) == 1 else 6)
else:
plt.suptitle(
"Gm = %.2f %s(at %.2f %s), "
"Pm = %.2f %s (at %.2f %s)" %
(20*np.log10(gm) if dB else gm,
'dB ' if dB else '',
Wcg, 'Hz' if Hz else 'rad/s',
pm if deg else math.radians(pm),
'deg' if deg else 'rad',
Wcp, 'Hz' if Hz else 'rad/s'))
# Add a grid to the plot + labeling
ax_phase.set_ylabel("Phase (deg)" if deg else "Phase (rad)")
def gen_zero_centered_series(val_min, val_max, period):
v1 = np.ceil(val_min / period - 0.2)
v2 = np.floor(val_max / period + 0.2)
return np.arange(v1, v2 + 1) * period
if deg:
ylim = ax_phase.get_ylim()
ax_phase.set_yticks(gen_zero_centered_series(
ylim[0], ylim[1], 45.))
ax_phase.set_yticks(gen_zero_centered_series(
ylim[0], ylim[1], 15.), minor=True)
else:
ylim = ax_phase.get_ylim()
ax_phase.set_yticks(gen_zero_centered_series(
ylim[0], ylim[1], math.pi / 4.))
ax_phase.set_yticks(gen_zero_centered_series(
ylim[0], ylim[1], math.pi / 12.), minor=True)
ax_phase.grid(grid and not margins, which='both')
# ax_mag.grid(which='minor', alpha=0.3)
# ax_mag.grid(which='major', alpha=0.9)
# ax_phase.grid(which='minor', alpha=0.3)
# ax_phase.grid(which='major', alpha=0.9)
# Label the frequency axis
ax_phase.set_xlabel("Frequency (Hz)" if Hz
else "Frequency (rad/sec)")
if len(syslist) == 1:
return mags[0], phases[0], omegas[0]
else:
return mags, phases, omegas
#
# Nyquist plot
#
# Default values for module parameter variables
_nyquist_defaults = {
'nyquist.mirror_style': '--',
'nyquist.arrows': 2,
'nyquist.arrow_size': 8,
'nyquist.indent_radius': 1e-1,
'nyquist.indent_direction': 'right',
}
def nyquist_plot(syslist, omega=None, plot=True, omega_limits=None,
omega_num=None, label_freq=0, color=None,
return_contour=False, warn_nyquist=True, *args, **kwargs):
"""Nyquist plot for a system
Plots a Nyquist plot for the system over a (optional) frequency range.
The curve is computed by evaluating the Nyqist segment along the positive
imaginary axis, with a mirror image generated to reflect the negative
imaginary axis. Poles on or near the imaginary axis are avoided using a
small indentation. The portion of the Nyquist contour at infinity is not
explicitly computed (since it maps to a constant value for any system with
a proper transfer function).
Parameters
----------
syslist : list of LTI
List of linear input/output systems (single system is OK). Nyquist
curves for each system are plotted on the same graph.
plot : boolean
If True, plot magnitude
omega : array_like
Set of frequencies to be evaluated, in rad/sec.
omega_limits : array_like of two values
Limits to the range of frequencies. Ignored if omega is provided, and
auto-generated if omitted.
omega_num : int
Number of frequency samples to plot. Defaults to
config.defaults['freqplot.number_of_samples'].
color : string
Used to specify the color of the line and arrowhead.
mirror_style : string or False
Linestyle for mirror image of the Nyquist curve. If `False` then
omit completely. Default linestyle ('--') is determined by
config.defaults['nyquist.mirror_style'].
return_contour : bool
If 'True', return the contour used to evaluate the Nyquist plot.
label_freq : int
Label every nth frequency on the plot. If not specified, no labels
are generated.
arrows : int or 1D/2D array of floats
Specify the number of arrows to plot on the Nyquist curve. If an
integer is passed. that number of equally spaced arrows will be
plotted on each of the primary segment and the mirror image. If a 1D
array is passed, it should consist of a sorted list of floats between
0 and 1, indicating the location along the curve to plot an arrow. If
a 2D array is passed, the first row will be used to specify arrow
locations for the primary curve and the second row will be used for
the mirror image.
arrow_size : float
Arrowhead width and length (in display coordinates). Default value is
8 and can be set using config.defaults['nyquist.arrow_size'].
arrow_style : matplotlib.patches.ArrowStyle
Define style used for Nyquist curve arrows (overrides `arrow_size`).
indent_radius : float
Amount to indent the Nyquist contour around poles that are at or near
the imaginary axis.
indent_direction : str
For poles on the imaginary axis, set the direction of indentation to
be 'right' (default), 'left', or 'none'.
warn_nyquist : bool, optional
If set to 'False', turn off warnings about frequencies above Nyquist.
*args : :func:`matplotlib.pyplot.plot` positional properties, optional
Additional arguments for `matplotlib` plots (color, linestyle, etc)
**kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional
Additional keywords (passed to `matplotlib`)
Returns
-------
count : int (or list of int if len(syslist) > 1)
Number of encirclements of the point -1 by the Nyquist curve. If
multiple systems are given, an array of counts is returned.
contour : ndarray (or list of ndarray if len(syslist) > 1)), optional
The contour used to create the primary Nyquist curve segment. To
obtain the Nyquist curve values, evaluate system(s) along contour.
Notes
-----
1. If a discrete time model is given, the frequency response is computed
along the upper branch of the unit circle, using the mapping ``z =
exp(1j * omega * dt)`` where `omega` ranges from 0 to `pi/dt` and `dt`
is the discrete timebase. If timebase not specified (``dt=True``),
`dt` is set to 1.
2. If a continuous-time system contains poles on or near the imaginary
axis, a small indentation will be used to avoid the pole. The radius
of the indentation is given by `indent_radius` and it is taken to the
right of stable poles and the left of unstable poles. If a pole is
exactly on the imaginary axis, the `indent_direction` parameter can be
used to set the direction of indentation. Setting `indent_direction`
to `none` will turn off indentation. If `return_contour` is True, the
exact contour used for evaluation is returned.
Examples
--------
>>> sys = ss([[1, -2], [3, -4]], [[5], [7]], [[6, 8]], [[9]])
>>> count = nyquist_plot(sys)
"""
# Check to see if legacy 'Plot' keyword was used
if 'Plot' in kwargs:
warnings.warn("'Plot' keyword is deprecated in nyquist_plot; "
"use 'plot'", FutureWarning)
# Map 'Plot' keyword to 'plot' keyword
plot = kwargs.pop('Plot')
# Check to see if legacy 'labelFreq' keyword was used
if 'labelFreq' in kwargs:
warnings.warn("'labelFreq' keyword is deprecated in nyquist_plot; "
"use 'label_freq'", FutureWarning)
# Map 'labelFreq' keyword to 'label_freq' keyword
label_freq = kwargs.pop('labelFreq')
# Check to see if legacy 'arrow_width' or 'arrow_length' were used
if 'arrow_width' in kwargs or 'arrow_length' in kwargs:
warnings.warn(
"'arrow_width' and 'arrow_length' keywords are deprecated in "
"nyquist_plot; use `arrow_size` instead", FutureWarning)
kwargs['arrow_size'] = \
(kwargs.get('arrow_width', 0) + kwargs.get('arrow_length', 0)) / 2
kwargs.pop('arrow_width', False)
kwargs.pop('arrow_length', False)
# Get values for params (and pop from list to allow keyword use in plot)
omega_num = config._get_param('freqplot', 'number_of_samples', omega_num)
mirror_style = config._get_param(
'nyquist', 'mirror_style', kwargs, _nyquist_defaults, pop=True)
arrows = config._get_param(
'nyquist', 'arrows', kwargs, _nyquist_defaults, pop=True)
arrow_size = config._get_param(
'nyquist', 'arrow_size', kwargs, _nyquist_defaults, pop=True)
arrow_style = config._get_param('nyquist', 'arrow_style', kwargs, None)
indent_radius = config._get_param(
'nyquist', 'indent_radius', kwargs, _nyquist_defaults, pop=True)
indent_direction = config._get_param(
'nyquist', 'indent_direction', kwargs, _nyquist_defaults, pop=True)
# If argument was a singleton, turn it into a list
if not hasattr(syslist, '__iter__'):
syslist = (syslist,)
omega, omega_range_given = _determine_omega_vector(
syslist, omega, omega_limits, omega_num)
if not omega_range_given:
# Start contour at zero frequency
omega[0] = 0.
# Go through each system and keep track of the results
counts, contours = [], []
for sys in syslist:
if not sys.issiso():
# TODO: Add MIMO nyquist plots.
raise ControlMIMONotImplemented(
"Nyquist plot currently only supports SISO systems.")
# Figure out the frequency range
omega_sys = np.asarray(omega)
# Determine the contour used to evaluate the Nyquist curve
if sys.isdtime(strict=True):
# Transform frequencies in for discrete-time systems
nyquistfrq = math.pi / sys.dt
if not omega_range_given:
# limit up to and including nyquist frequency
omega_sys = np.hstack((
omega_sys[omega_sys < nyquistfrq], nyquistfrq))
# Issue a warning if we are sampling above Nyquist
if np.any(omega_sys * sys.dt > np.pi) and warn_nyquist:
warnings.warn("evaluation above Nyquist frequency")
# Transform frequencies to continuous domain
contour = np.exp(1j * omega * sys.dt)
else:
contour = 1j * omega_sys
# Bend the contour around any poles on/near the imaginary axis
if isinstance(sys, (StateSpace, TransferFunction)) \
and sys.isctime() and indent_direction != 'none':
poles = sys.pole()
if contour[1].imag > indent_radius \
and 0. in poles and not omega_range_given:
# add some points for quarter circle around poles at origin
contour = np.concatenate(
(1j * np.linspace(0., indent_radius, 50),
contour[1:]))
for i, s in enumerate(contour):
# Find the nearest pole
p = poles[(np.abs(poles - s)).argmin()]
# See if we need to indent around it
if abs(s - p) < indent_radius:
if p.real < 0 or \
(p.real == 0 and indent_direction == 'right'):
# Indent to the right
contour[i] += \
np.sqrt(indent_radius ** 2 - (s-p).imag ** 2)
elif p.real > 0 or \
(p.real == 0 and indent_direction == 'left'):
# Indent to the left
contour[i] -= \
np.sqrt(indent_radius ** 2 - (s-p).imag ** 2)
else:
ValueError("unknown value for indent_direction")
# TODO: add code to indent around discrete poles on unit circle
# Compute the primary curve
resp = sys(contour)
# Compute CW encirclements of -1 by integrating the (unwrapped) angle
phase = -unwrap(np.angle(resp + 1))
count = int(np.round(np.sum(np.diff(phase)) / np.pi, 0))
counts.append(count)
contours.append(contour)
if plot:
# Parse the arrows keyword
if isinstance(arrows, int):
N = arrows
# Space arrows out, starting midway along each "region"
arrow_pos = np.linspace(0.5/N, 1 + 0.5/N, N, endpoint=False)
elif isinstance(arrows, (list, np.ndarray)):
arrow_pos = np.sort(np.atleast_1d(arrows))
elif not arrows:
arrow_pos = []
else:
raise ValueError("unknown or unsupported arrow location")
# Set the arrow style
if arrow_style is None:
arrow_style = mpl.patches.ArrowStyle(
'simple', head_width=arrow_size, head_length=arrow_size)
# Save the components of the response
x, y = resp.real, resp.imag
# Plot the primary curve
p = plt.plot(x, y, '-', color=color, *args, **kwargs)
c = p[0].get_color()
ax = plt.gca()
_add_arrows_to_line2D(
ax, p[0], arrow_pos, arrowstyle=arrow_style, dir=1)
# Plot the mirror image
if mirror_style is not False:
p = plt.plot(x, -y, mirror_style, color=c, *args, **kwargs)
_add_arrows_to_line2D(
ax, p[0], arrow_pos, arrowstyle=arrow_style, dir=-1)
# Mark the -1 point
plt.plot([-1], [0], 'r+')
# Label the frequencies of the points
if label_freq:
ind = slice(None, None, label_freq)
for xpt, ypt, omegapt in zip(x[ind], y[ind], omega_sys[ind]):
# Convert to Hz
f = omegapt / (2 * np.pi)
# Factor out multiples of 1000 and limit the
# result to the range [-8, 8].
pow1000 = max(min(get_pow1000(f), 8), -8)
# Get the SI prefix.
prefix = gen_prefix(pow1000)
# Apply the text. (Use a space before the text to
# prevent overlap with the data.)
#
# np.round() is used because 0.99... appears
# instead of 1.0, and this would otherwise be
# truncated to 0.
plt.text(xpt, ypt, ' ' +
str(int(np.round(f / 1000 ** pow1000, 0))) + ' ' +
prefix + 'Hz')
if plot:
ax = plt.gca()
ax.set_xlabel("Real axis")
ax.set_ylabel("Imaginary axis")
ax.grid(color="lightgray")
# "Squeeze" the results
if len(syslist) == 1:
counts, contours = counts[0], contours[0]
# Return counts and (optionally) the contour we used
return (counts, contours) if return_contour else counts
# Internal function to add arrows to a curve
def _add_arrows_to_line2D(
axes, line, arrow_locs=[0.2, 0.4, 0.6, 0.8],
arrowstyle='-|>', arrowsize=1, dir=1, transform=None):
"""
Add arrows to a matplotlib.lines.Line2D at selected locations.
Parameters:
-----------
axes: Axes object as returned by axes command (or gca)
line: Line2D object as returned by plot command
arrow_locs: list of locations where to insert arrows, % of total length
arrowstyle: style of the arrow
arrowsize: size of the arrow
transform: a matplotlib transform instance, default to data coordinates
Returns:
--------
arrows: list of arrows
Based on https://stackoverflow.com/questions/26911898/
"""
if not isinstance(line, mpl.lines.Line2D):
raise ValueError("expected a matplotlib.lines.Line2D object")
x, y = line.get_xdata(), line.get_ydata()
arrow_kw = {
"arrowstyle": arrowstyle,
}
color = line.get_color()
use_multicolor_lines = isinstance(color, np.ndarray)
if use_multicolor_lines:
raise NotImplementedError("multicolor lines not supported")
else:
arrow_kw['color'] = color
linewidth = line.get_linewidth()
if isinstance(linewidth, np.ndarray):
raise NotImplementedError("multiwidth lines not supported")
else:
arrow_kw['linewidth'] = linewidth
if transform is None:
transform = axes.transData
# Compute the arc length along the curve
s = np.cumsum(np.sqrt(np.diff(x) ** 2 + np.diff(y) ** 2))
arrows = []
for loc in arrow_locs:
n = np.searchsorted(s, s[-1] * loc)
# Figure out what direction to paint the arrow
if dir == 1:
arrow_tail = (x[n], y[n])
arrow_head = (np.mean(x[n:n + 2]), np.mean(y[n:n + 2]))
elif dir == -1:
# Orient the arrow in the other direction on the segment
arrow_tail = (x[n + 1], y[n + 1])
arrow_head = (np.mean(x[n:n + 2]), np.mean(y[n:n + 2]))
else:
raise ValueError("unknown value for keyword 'dir'")
p = mpl.patches.FancyArrowPatch(
arrow_tail, arrow_head, transform=transform, lw=0,
**arrow_kw)
axes.add_patch(p)
arrows.append(p)
return arrows
#
# Gang of Four plot
#
# TODO: think about how (and whether) to handle lists of systems
def gangof4_plot(P, C, omega=None, **kwargs):
"""Plot the "Gang of 4" transfer functions for a system
Generates a 2x2 plot showing the "Gang of 4" sensitivity functions
[T, PS; CS, S]
Parameters
----------
P, C : LTI
Linear input/output systems (process and control)
omega : array
Range of frequencies (list or bounds) in rad/sec
**kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional
Additional keywords (passed to `matplotlib`)
Returns
-------
None
"""
if not P.issiso() or not C.issiso():
# TODO: Add MIMO go4 plots.
raise ControlMIMONotImplemented(
"Gang of four is currently only implemented for SISO systems.")
# Get the default parameter values
dB = config._get_param(
'freqplot', 'dB', kwargs, _freqplot_defaults, pop=True)
Hz = config._get_param(
'freqplot', 'Hz', kwargs, _freqplot_defaults, pop=True)
grid = config._get_param(
'freqplot', 'grid', kwargs, _freqplot_defaults, pop=True)
# Compute the senstivity functions
L = P * C
S = feedback(1, L)
T = L * S
# Select a default range if none is provided
# TODO: This needs to be made more intelligent
if omega is None:
omega = _default_frequency_range((P, C, S))
# Set up the axes with labels so that multiple calls to
# gangof4_plot will superimpose the data. See details in bode_plot.
plot_axes = {'t': None, 's': None, 'ps': None, 'cs': None}
for ax in plt.gcf().axes:
label = ax.get_label()
if label.startswith('control-gangof4-'):
key = label[len('control-gangof4-'):]
if key not in plot_axes:
raise RuntimeError(
"unknown gangof4 axis type '{}'".format(label))
plot_axes[key] = ax
# if any of the axes are missing, start from scratch
if any((ax is None for ax in plot_axes.values())):
plt.clf()
plot_axes = {'s': plt.subplot(221, label='control-gangof4-s'),
'ps': plt.subplot(222, label='control-gangof4-ps'),
'cs': plt.subplot(223, label='control-gangof4-cs'),
't': plt.subplot(224, label='control-gangof4-t')}
#
# Plot the four sensitivity functions
#
omega_plot = omega / (2. * math.pi) if Hz else omega
# TODO: Need to add in the mag = 1 lines
mag_tmp, phase_tmp, omega = S.frequency_response(omega)
mag = np.squeeze(mag_tmp)
if dB:
plot_axes['s'].semilogx(omega_plot, 20 * np.log10(mag), **kwargs)
else:
plot_axes['s'].loglog(omega_plot, mag, **kwargs)
plot_axes['s'].set_ylabel("$|S|$" + " (dB)" if dB else "")
plot_axes['s'].tick_params(labelbottom=False)
plot_axes['s'].grid(grid, which='both')
mag_tmp, phase_tmp, omega = (P * S).frequency_response(omega)
mag = np.squeeze(mag_tmp)
if dB:
plot_axes['ps'].semilogx(omega_plot, 20 * np.log10(mag), **kwargs)
else:
plot_axes['ps'].loglog(omega_plot, mag, **kwargs)
plot_axes['ps'].tick_params(labelbottom=False)
plot_axes['ps'].set_ylabel("$|PS|$" + " (dB)" if dB else "")