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"""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 <[email protected]>

# This file contains some standard control system plots: Bode plots,

# Nyquist plots, Nichols plots and pole-zero diagrams

# 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 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.frequency_response(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(np.degrees(phase), 360)

y = 20*np.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

:doc:`Matplotlib linestyle \

<matplotlib:gallery/lines_bars_and_markers/linestyles>`

"""

# 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*np.log10(np.abs(m))

m_phase = np.mod(np.degrees(np.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*np.log10(np.abs(n))

n_phase = np.mod(np.degrees(np.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*np.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 = np.radians(np.linspace(phase_min, phase_max, 2000))

Gcl_mags, Gcl_phases = np.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 = np.linspace(10**(mag_min/20.0), 10**(mag_max/20.0), 2000)

Gcl_phases, Gcl_mags = np.meshgrid(np.radians(phases), mags)

return closed_loop_contours(Gcl_mags, Gcl_phases)

# Function aliases

nichols = nichols_plot