LSR/env/lib/python3.6/site-packages/control/nichols.py
2020-06-04 17:24:47 +02:00

307 lines
11 KiB
Python

"""nichols.py
Functions for plotting Black-Nichols charts.
Routines in this module:
nichols.nichols_plot aliased as nichols.nichols
nichols.nichols_grid
"""
# nichols.py - Nichols plot
#
# Contributed by Allan McInnes <Allan.McInnes@canterbury.ac.nz>
#
# This file contains some standard control system plots: Bode plots,
# Nyquist plots, Nichols plots and pole-zero diagrams
#
# 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: freqplot.py 139 2011-03-30 16:19:59Z murrayrm $
import scipy as sp
import numpy as np
import matplotlib.pyplot as plt
from .ctrlutil import unwrap
from .freqplot import default_frequency_range
from . import config
__all__ = ['nichols_plot', 'nichols', 'nichols_grid']
# Default parameters values for the nichols module
_nichols_defaults = {
'nichols.grid':True,
}
def nichols_plot(sys_list, omega=None, grid=None):
"""Nichols plot for a system
Plots a Nichols plot for the system over a (optional) frequency range.
Parameters
----------
sys_list : list of LTI, or LTI
List of linear input/output systems (single system is OK)
omega : array_like
Range of frequencies (list or bounds) in rad/sec
grid : boolean, optional
True if the plot should include a Nichols-chart grid. Default is True.
Returns
-------
None
"""
# Get parameter values
grid = config._get_param('nichols', 'grid', grid, True)
# If argument was a singleton, turn it into a list
if not getattr(sys_list, '__iter__', False):
sys_list = (sys_list,)
# Select a default range if none is provided
if omega is None:
omega = default_frequency_range(sys_list)
for sys in sys_list:
# Get the magnitude and phase of the system
mag_tmp, phase_tmp, omega = sys.freqresp(omega)
mag = np.squeeze(mag_tmp)
phase = np.squeeze(phase_tmp)
# Convert to Nichols-plot format (phase in degrees,
# and magnitude in dB)
x = unwrap(sp.degrees(phase), 360)
y = 20*sp.log10(mag)
# Generate the plot
plt.plot(x, y)
plt.xlabel('Phase (deg)')
plt.ylabel('Magnitude (dB)')
plt.title('Nichols Plot')
# Mark the -180 point
plt.plot([-180], [0], 'r+')
# Add grid
if grid:
nichols_grid()
def nichols_grid(cl_mags=None, cl_phases=None, line_style='dotted'):
"""Nichols chart grid
Plots a Nichols chart grid on the current axis, or creates a new chart
if no plot already exists.
Parameters
----------
cl_mags : array-like (dB), optional
Array of closed-loop magnitudes defining the iso-gain lines on a
custom Nichols chart.
cl_phases : array-like (degrees), optional
Array of closed-loop phases defining the iso-phase lines on a custom
Nichols chart. Must be in the range -360 < cl_phases < 0
line_style : string, optional
.. seealso:: https://matplotlib.org/gallery/lines_bars_and_markers/linestyles.html
Returns
-------
None
"""
# Default chart size
ol_phase_min = -359.99
ol_phase_max = 0.0
ol_mag_min = -40.0
ol_mag_max = default_ol_mag_max = 50.0
# Find bounds of the current dataset, if there is one.
if plt.gcf().gca().has_data():
ol_phase_min, ol_phase_max, ol_mag_min, ol_mag_max = plt.axis()
# M-circle magnitudes.
if cl_mags is None:
# Default chart magnitudes
# The key set of magnitudes are always generated, since this
# guarantees a recognizable Nichols chart grid.
key_cl_mags = np.array([-40.0, -20.0, -12.0, -6.0, -3.0, -1.0, -0.5, 0.0,
0.25, 0.5, 1.0, 3.0, 6.0, 12.0])
# Extend the range of magnitudes if necessary. The extended arange
# will end up empty if no extension is required. Assumes that closed-loop
# magnitudes are approximately aligned with open-loop magnitudes beyond
# the value of np.min(key_cl_mags)
cl_mag_step = -20.0 # dB
extended_cl_mags = np.arange(np.min(key_cl_mags),
ol_mag_min + cl_mag_step, cl_mag_step)
cl_mags = np.concatenate((extended_cl_mags, key_cl_mags))
# N-circle phases (should be in the range -360 to 0)
if cl_phases is None:
# Choose a reasonable set of default phases (denser if the open-loop
# data is restricted to a relatively small range of phases).
key_cl_phases = np.array([-0.25, -45.0, -90.0, -180.0, -270.0, -325.0, -359.75])
if np.abs(ol_phase_max - ol_phase_min) < 90.0:
other_cl_phases = np.arange(-10.0, -360.0, -10.0)
else:
other_cl_phases = np.arange(-10.0, -360.0, -20.0)
cl_phases = np.concatenate((key_cl_phases, other_cl_phases))
else:
assert ((-360.0 < np.min(cl_phases)) and (np.max(cl_phases) < 0.0))
# Find the M-contours
m = m_circles(cl_mags, phase_min=np.min(cl_phases), phase_max=np.max(cl_phases))
m_mag = 20*sp.log10(np.abs(m))
m_phase = sp.mod(sp.degrees(sp.angle(m)), -360.0) # Unwrap
# Find the N-contours
n = n_circles(cl_phases, mag_min=np.min(cl_mags), mag_max=np.max(cl_mags))
n_mag = 20*sp.log10(np.abs(n))
n_phase = sp.mod(sp.degrees(sp.angle(n)), -360.0) # Unwrap
# Plot the contours behind other plot elements.
# The "phase offset" is used to produce copies of the chart that cover
# the entire range of the plotted data, starting from a base chart computed
# over the range -360 < phase < 0. Given the range
# the base chart is computed over, the phase offset should be 0
# for -360 < ol_phase_min < 0.
phase_offset_min = 360.0*np.ceil(ol_phase_min/360.0)
phase_offset_max = 360.0*np.ceil(ol_phase_max/360.0) + 360.0
phase_offsets = np.arange(phase_offset_min, phase_offset_max, 360.0)
for phase_offset in phase_offsets:
# Draw M and N contours
plt.plot(m_phase + phase_offset, m_mag, color='lightgray',
linestyle=line_style, zorder=0)
plt.plot(n_phase + phase_offset, n_mag, color='lightgray',
linestyle=line_style, zorder=0)
# Add magnitude labels
for x, y, m in zip(m_phase[:][-1] + phase_offset, m_mag[:][-1], cl_mags):
align = 'right' if m < 0.0 else 'left'
plt.text(x, y, str(m) + ' dB', size='small', ha=align, color='gray')
# Fit axes to generated chart
plt.axis([phase_offset_min - 360.0, phase_offset_max - 360.0,
np.min(cl_mags), np.max([ol_mag_max, default_ol_mag_max])])
#
# Utility functions
#
# This section of the code contains some utility functions for
# generating Nichols plots
#
def closed_loop_contours(Gcl_mags, Gcl_phases):
"""Contours of the function Gcl = Gol/(1+Gol), where
Gol is an open-loop transfer function, and Gcl is a corresponding
closed-loop transfer function.
Parameters
----------
Gcl_mags : array-like
Array of magnitudes of the contours
Gcl_phases : array-like
Array of phases in radians of the contours
Returns
-------
contours : complex array
Array of complex numbers corresponding to the contours.
"""
# Compute the contours in Gcl-space. Since we're given closed-loop
# magnitudes and phases, this is just a case of converting them into
# a complex number.
Gcl = Gcl_mags*sp.exp(1.j*Gcl_phases)
# Invert Gcl = Gol/(1+Gol) to map the contours into the open-loop space
return Gcl/(1.0 - Gcl)
def m_circles(mags, phase_min=-359.75, phase_max=-0.25):
"""Constant-magnitude contours of the function Gcl = Gol/(1+Gol), where
Gol is an open-loop transfer function, and Gcl is a corresponding
closed-loop transfer function.
Parameters
----------
mags : array-like
Array of magnitudes in dB of the M-circles
phase_min : degrees
Minimum phase in degrees of the N-circles
phase_max : degrees
Maximum phase in degrees of the N-circles
Returns
-------
contours : complex array
Array of complex numbers corresponding to the contours.
"""
# Convert magnitudes and phase range into a grid suitable for
# building contours
phases = sp.radians(sp.linspace(phase_min, phase_max, 2000))
Gcl_mags, Gcl_phases = sp.meshgrid(10.0**(mags/20.0), phases)
return closed_loop_contours(Gcl_mags, Gcl_phases)
def n_circles(phases, mag_min=-40.0, mag_max=12.0):
"""Constant-phase contours of the function Gcl = Gol/(1+Gol), where
Gol is an open-loop transfer function, and Gcl is a corresponding
closed-loop transfer function.
Parameters
----------
phases : array-like
Array of phases in degrees of the N-circles
mag_min : dB
Minimum magnitude in dB of the N-circles
mag_max : dB
Maximum magnitude in dB of the N-circles
Returns
-------
contours : complex array
Array of complex numbers corresponding to the contours.
"""
# Convert phases and magnitude range into a grid suitable for
# building contours
mags = sp.linspace(10**(mag_min/20.0), 10**(mag_max/20.0), 2000)
Gcl_phases, Gcl_mags = sp.meshgrid(sp.radians(phases), mags)
return closed_loop_contours(Gcl_mags, Gcl_phases)
# Function aliases
nichols = nichols_plot