# 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

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#

# $Id$

import matplotlib as mpl

import matplotlib.pyplot as plt

import scipy as sp

import numpy as np

import math

from .ctrlutil import unwrap

from .bdalg import feedback

from .margins import stability_margins

from . import config

__all__ = ['bode_plot', 'nyquist_plot', 'gangof4_plot',

'bode', 'nyquist', 'gangof4']

# Default values for module parameter variables

_freqplot_defaults = {

'freqplot.feature_periphery_decades': 1,

'freqplot.number_of_samples': None,

}

#

# Main plotting functions

#

# This section of the code contains the functions for generating

# frequency domain plots

#

#

# Bode plot

#

# Default values for Bode plot configuration variables

_bode_defaults = {

'bode.dB': False, # Plot gain in dB

'bode.deg': True, # Plot phase in degrees

'bode.Hz': False, # Plot frequency in Hertz

'bode.grid': True, # Turn on grid for gain and phase

}

def bode_plot(syslist, omega=None,

Plot=True, omega_limits=None, omega_num=None,

margins=None, *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 : list

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['bode.Hz']

deg : bool

If True, plot phase in degrees (else radians). Default value (True)

config.defaults['bode.deg']

Plot : bool

If True, plot magnitude and phase

omega_limits: tuple, list, ... 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.

\*args, \**kwargs:

Additional options to matplotlib (color, linestyle, etc)

Returns

-------

mag : array (list if len(syslist) > 1)

magnitude

phase : array (list if len(syslist) > 1)

phase in radians

omega : array (list 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['bode.grid'].

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 method (mag, phase, freq)

= sys.freqresp(freq) to generate the frequency response for a system,

but it returns a MIMO response.

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(j

\omega dt) where omega ranges from 0 to pi/dt and dt is the discrete

timebase. If not timebase is 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 dictonary since we will modify it

kwargs = dict(kwargs)

# Get values for params (and pop from list to allow keyword use in plot)

dB = config._get_param('bode', 'dB', kwargs, _bode_defaults, pop=True)

deg = config._get_param('bode', 'deg', kwargs, _bode_defaults, pop=True)

Hz = config._get_param('bode', 'Hz', kwargs, _bode_defaults, pop=True)

grid = config._get_param('bode', 'grid', kwargs, _bode_defaults, pop=True)

Plot = config._get_param('bode', 'grid', Plot, True)

margins = config._get_param('bode', 'margins', margins, False)

# If argument was a singleton, turn it into a list

if not getattr(syslist, '__iter__', False):

syslist = (syslist,)

if omega is None:

if omega_limits is None:

# Select a default range if none is provided

omega = default_frequency_range(syslist, Hz=Hz,

number_of_samples=omega_num)

else:

omega_limits = np.array(omega_limits)

if Hz:

omega_limits *= 2. * math.pi

if omega_num:

omega = sp.logspace(np.log10(omega_limits[0]),

np.log10(omega_limits[1]),

num=omega_num,

endpoint=True)

else:

omega = sp.logspace(np.log10(omega_limits[0]),

np.log10(omega_limits[1]),

endpoint=True)

mags, phases, omegas, nyquistfrqs = [], [], [], []

for sys in syslist:

if sys.inputs > 1 or sys.outputs > 1:

# TODO: Add MIMO bode plots.

raise NotImplementedError(

"Bode is currently only implemented for SISO systems.")

else:

omega_sys = np.array(omega)

if sys.isdtime(True):

nyquistfrq = 2. * math.pi * 1. / sys.dt / 2.

omega_sys = omega_sys[omega_sys < nyquistfrq]

# TODO: What distance to the Nyquist frequency is appropriate?

else:

nyquistfrq = None

# Get the magnitude and phase of the system

mag_tmp, phase_tmp, omega_sys = sys.freqresp(omega_sys)

mag = np.atleast_1d(np.squeeze(mag_tmp))

phase = np.atleast_1d(np.squeeze(phase_tmp))

phase = unwrap(phase)

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

# 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)

# Magnitude plot

if dB:

pltline = ax_mag.semilogx(omega_plot, 20 * np.log10(mag),

*args, **kwargs)

else:

pltline = ax_mag.loglog(omega_plot, mag, *args, **kwargs)

if nyquistfrq_plot:

ax_mag.axvline(nyquistfrq_plot,

color=pltline[0].get_color())

# 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

if deg:

phase_plot = phase * 180. / math.pi

else:

phase_plot = phase

ax_phase.semilogx(omega_plot, phase_plot, *args, **kwargs)

# Show the phase and gain margins in the plot

if margins:

margin = stability_margins(sys)

gm, pm, Wcg, Wcp = \

margin[0], margin[1], margin[3], margin[4]

# TODO: add some documentation describing why this is here

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)

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()

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)

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], [1e-8, phase_limit],

color='k', linestyle=':', zorder=-20)

else:

ax_phase.semilogx(

[Wcg, Wcg], [1e-8, 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 '\b',

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'))

if nyquistfrq_plot:

ax_phase.axvline(

nyquistfrq_plot, color=pltline[0].get_color())

# 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

#

def nyquist_plot(syslist, omega=None, Plot=True, color=None,

labelFreq=0, *args, **kwargs):

"""

Nyquist plot for a system

Plots a Nyquist plot for the system over a (optional) frequency range.

Parameters

----------

syslist : list of LTI

List of linear input/output systems (single system is OK)

omega : freq_range

Range of frequencies (list or bounds) in rad/sec

Plot : boolean

If True, plot magnitude

color : string

Used to specify the color of the plot

labelFreq : int

Label every nth frequency on the plot

\*args, \**kwargs:

Additional options to matplotlib (color, linestyle, etc)

Returns

-------

real : array

real part of the frequency response array

imag : array

imaginary part of the frequency response array

freq : array

frequencies

Examples

--------

>>> sys = ss("1. -2; 3. -4", "5.; 7", "6. 8", "9.")

>>> real, imag, freq = nyquist_plot(sys)

"""

# If argument was a singleton, turn it into a list

if not getattr(syslist, '__iter__', False):

syslist = (syslist,)

# Select a default range if none is provided

if omega is None:

omega = default_frequency_range(syslist)

# Interpolate between wmin and wmax if a tuple or list are provided

elif isinstance(omega, list) or isinstance(omega, tuple):

# Only accept tuple or list of length 2

if len(omega) != 2:

raise ValueError("Supported frequency arguments are (wmin,wmax)"

"tuple or list, or frequency vector. ")

omega = np.logspace(np.log10(omega[0]), np.log10(omega[1]),

num=50, endpoint=True, base=10.0)

for sys in syslist:

if sys.inputs > 1 or sys.outputs > 1:

# TODO: Add MIMO nyquist plots.

raise NotImplementedError(

"Nyquist is currently only implemented for SISO systems.")

else:

# 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)

# Compute the primary curve

x = sp.multiply(mag, sp.cos(phase))

y = sp.multiply(mag, sp.sin(phase))

if Plot:

# Plot the primary curve and mirror image

p = plt.plot(x, y, '-', color=color, *args, **kwargs)

c = p[0].get_color()

ax = plt.gca()

# Plot arrow to indicate Nyquist encirclement orientation

ax.arrow(x[0], y[0], (x[1]-x[0])/2, (y[1]-y[0])/2, fc=c, ec=c,

head_width=0.2, head_length=0.2)

plt.plot(x, -y, '-', color=c, *args, **kwargs)

ax.arrow(

x[-1], -y[-1], (x[-1]-x[-2])/2, (y[-1]-y[-2])/2,

fc=c, ec=c, head_width=0.2, head_length=0.2)

# Mark the -1 point

plt.plot([-1], [0], 'r+')

# Label the frequencies of the points

if labelFreq:

ind = slice(None, None, labelFreq)

for xpt, ypt, omegapt in zip(x[ind], y[ind], omega[ind]):

# Convert to Hz

f = omegapt / (2 * sp.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")

return x, y, omega

#

# Gang of Four plot

#

# TODO: think about how (and whether) to handle lists of systems

def gangof4_plot(P, C, omega=None):

"""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

Returns

-------

None

"""

if P.inputs > 1 or P.outputs > 1 or C.inputs > 1 or C.outputs > 1:

# TODO: Add MIMO go4 plots.

raise NotImplementedError(

"Gang of four is currently only implemented for SISO systems.")

else:

# 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))

# Compute the senstivity functions

L = P * C

S = feedback(1, L)

T = L * 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 = {'t': plt.subplot(221, label='control-gangof4-t'),

'ps': plt.subplot(222, label='control-gangof4-ps'),

'cs': plt.subplot(223, label='control-gangof4-cs'),

's': plt.subplot(224, label='control-gangof4-s')}

#

# Plot the four sensitivity functions

#

# TODO: Need to add in the mag = 1 lines

mag_tmp, phase_tmp, omega = T.freqresp(omega)

mag = np.squeeze(mag_tmp)

plot_axes['t'].loglog(omega, mag)

mag_tmp, phase_tmp, omega = (P * S).freqresp(omega)

mag = np.squeeze(mag_tmp)

plot_axes['ps'].loglog(omega, mag)

mag_tmp, phase_tmp, omega = (C * S).freqresp(omega)

mag = np.squeeze(mag_tmp)

plot_axes['cs'].loglog(omega, mag)

mag_tmp, phase_tmp, omega = S.freqresp(omega)

mag = np.squeeze(mag_tmp)

plot_axes['s'].loglog(omega, mag)

#

# Utility functions

#

# This section of the code contains some utility functions for

# generating frequency domain plots

#

# Compute reasonable defaults for axes

def default_frequency_range(syslist, Hz=None, number_of_samples=None,

feature_periphery_decades=None):

"""Compute a reasonable default frequency range for frequency

domain plots.

Finds a reasonable default frequency range by examining the features

(poles and zeros) of the systems in syslist.

Parameters

----------

syslist : list of LTI

List of linear input/output systems (single system is OK)

Hz : bool

If True, the limits (first and last value) of the frequencies

are set to full decades in Hz so it fits plotting with logarithmic

scale in Hz otherwise in rad/s. Omega is always returned in rad/sec.

number_of_samples : int, optional

Number of samples to generate. The default value is read from

``config.defaults['freqplot.number_of_samples']. If None, then the

default from `numpy.logspace` is used.

feature_periphery_decades : float, optional

Defines how many decades shall be included in the frequency range on

both sides of features (poles, zeros). The default value is read from

``config.defaults['freqplot.feature_periphery_decades']``.

Returns

-------

omega : array

Range of frequencies in rad/sec

Examples

--------

>>> from matlab import ss

>>> sys = ss("1. -2; 3. -4", "5.; 7", "6. 8", "9.")

>>> omega = default_frequency_range(sys)

"""

# This code looks at the poles and zeros of all of the systems that

# we are plotting and sets the frequency range to be one decade above

# and below the min and max feature frequencies, rounded to the nearest

# integer. It excludes poles and zeros at the origin. If no features

# are found, it turns logspace(-1, 1)

# Set default values for options

number_of_samples = config._get_param(

'freqplot', 'number_of_samples', number_of_samples)

feature_periphery_decades = config._get_param(

'freqplot', 'feature_periphery_decades', feature_periphery_decades, 1)

# Find the list of all poles and zeros in the systems

features = np.array(())

freq_interesting = []

# detect if single sys passed by checking if it is sequence-like

if not getattr(syslist, '__iter__', False):

syslist = (syslist,)

for sys in syslist:

try:

# Add new features to the list

if sys.isctime():

features_ = np.concatenate((np.abs(sys.pole()),

np.abs(sys.zero())))

# Get rid of poles and zeros at the origin

features_ = features_[features_ != 0.0]

features = np.concatenate((features, features_))

elif sys.isdtime(strict=True):

fn = math.pi * 1. / sys.dt

# TODO: What distance to the Nyquist frequency is appropriate?

freq_interesting.append(fn * 0.9)

features_ = np.concatenate((sys.pole(),

sys.zero()))

# Get rid of poles and zeros

# * at the origin and real <= 0 & imag==0: log!

# * at 1.: would result in omega=0. (logaritmic plot!)

features_ = features_[

(features_.imag != 0.0) | (features_.real > 0.)]

features_ = features_[

np.bitwise_not((features_.imag == 0.0) &

(np.abs(features_.real - 1.0) < 1.e-10))]

# TODO: improve

features__ = np.abs(np.log(features_) / (1.j * sys.dt))

features = np.concatenate((features, features__))

else:

# TODO

raise NotImplementedError(

"type of system in not implemented now")

except:

pass

# Make sure there is at least one point in the range

if features.shape[0] == 0:

features = np.array([1.])

if Hz:

features /= 2. * math.pi

features = np.log10(features)

lsp_min = np.floor(np.min(features) - feature_periphery_decades)

lsp_max = np.ceil(np.max(features) + feature_periphery_decades)

lsp_min += np.log10(2. * math.pi)

lsp_max += np.log10(2. * math.pi)

else:

features = np.log10(features)

lsp_min = np.floor(np.min(features) - feature_periphery_decades)

lsp_max = np.ceil(np.max(features) + feature_periphery_decades)

if freq_interesting:

lsp_min = min(lsp_min, np.log10(min(freq_interesting)))

lsp_max = max(lsp_max, np.log10(max(freq_interesting)))

# TODO: Add a check in discrete case to make sure we don't get aliasing

# (Attention: there is a list of system but only one omega vector)

# Set the range to be an order of magnitude beyond any features

if number_of_samples:

omega = sp.logspace(

lsp_min, lsp_max, num=number_of_samples, endpoint=True)

else:

omega = sp.logspace(lsp_min, lsp_max, endpoint=True)

return omega

#

# KLD 5/23/11: Two functions to create nice looking labels

#

def get_pow1000(num):

"""Determine exponent for which significand of a number is within the

range [1, 1000).

"""

# Based on algorithm from http://www.mail-archive.com/matplotlib-users@lists.sourceforge.net/msg14433.html, accessed 2010/11/7

# by Jason Heeris 2009/11/18

from decimal import Decimal

from math import floor

dnum = Decimal(str(num))

if dnum == 0:

return 0

elif dnum < 0:

dnum = -dnum

return int(floor(dnum.log10() / 3))

def gen_prefix(pow1000):

"""Return the SI prefix for a power of 1000.

"""

# Prefixes according to Table 5 of [BIPM 2006] (excluding hecto,

# deca, deci, and centi).

if pow1000 < -8 or pow1000 > 8:

raise ValueError(

"Value is out of the range covered by the SI prefixes.")

return ['Y', # yotta (10^24)

'Z', # zetta (10^21)

'E', # exa (10^18)

'P', # peta (10^15)

'T', # tera (10^12)

'G', # giga (10^9)

'M', # mega (10^6)

'k', # kilo (10^3)

'', # (10^0)

'm', # milli (10^-3)

r'$\mu$', # micro (10^-6)

'n', # nano (10^-9)

'p', # pico (10^-12)

'f', # femto (10^-15)

'a', # atto (10^-18)

'z', # zepto (10^-21)

'y'][8 - pow1000] # yocto (10^-24)

def find_nearest_omega(omega_list, omega):

omega_list = np.asarray(omega_list)

return omega_list[(np.abs(omega_list - omega)).argmin()]

# Function aliases

bode = bode_plot

nyquist = nyquist_plot

gangof4 = gangof4_plot