"""margin.py
Functions for computing stability margins and related functions.
Routines in this module:
margin.stability_margins
margin.phase_crossover_frequencies
margin.margin
"""
# Python 3 compatibility (needs to go here)
from __future__ import print_function
"""Copyright (c) 2011 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.
Author: Richard M. Murray
Date: 14 July 2011
$Id$
"""
import math
import numpy as np
import scipy as sp
from . import xferfcn
from .lti import issiso
from . import frdata
__all__ = ['stability_margins', 'phase_crossover_frequencies', 'margin']
# helper functions for stability_margins
def _polyimsplit(pol):
"""split a polynomial with (iw) applied into a real and an
imaginary part with w applied"""
rpencil = np.zeros_like(pol)
ipencil = np.zeros_like(pol)
rpencil[-1::-4] = 1.
rpencil[-3::-4] = -1.
ipencil[-2::-4] = 1.
ipencil[-4::-4] = -1.
return pol * rpencil, pol*ipencil
def _polysqr(pol):
"""return a polynomial squared"""
return np.polymul(pol, pol)
# Took the framework for the old function by
# Sawyer B. Fuller <minster@caltech.edu>, removed a lot of the innards
# and replaced with analytical polynomial functions for LTI systems.
#
# idea for the frequency data solution copied/adapted from
# https://github.com/alchemyst/Skogestad-Python/blob/master/BODE.py
# Rene van Paassen <rene.vanpaassen@gmail.com>
#
# RvP, July 8, 2014, corrected to exclude phase=0 crossing for the gain
# margin polynomial
# RvP, July 8, 2015, augmented to calculate all phase/gain crossings with
# frd data. Correct to return smallest phase
# margin, smallest gain margin and their frequencies
# RvP, Jun 10, 2017, modified the inclusion of roots found for phase
# crossing to include all >= 0, made subsequent calc
# insensitive to div by 0
# also changed the selection of which crossings to
# return on basis of "A note on the Gain and Phase
# Margin Concepts" Journal of Control and Systems
# Engineering, Yazdan Bavafi-Toosi, Dec 2015, vol 3
# issue 1, pp 51-59, closer to Matlab behavior, but
# not completely identical in edge cases, which don't
# cross but touch gain=1
def stability_margins(sysdata, returnall=False, epsw=0.0):
"""Calculate stability margins and associated crossover frequencies.
Parameters
----------
sysdata: LTI system or (mag, phase, omega) sequence
sys : LTI system
Linear SISO system
mag, phase, omega : sequence of array_like
Arrays of magnitudes (absolute values, not dB), phases (degrees),
and corresponding frequencies. Crossover frequencies returned are
in the same units as those in `omega` (e.g., rad/sec or Hz).
returnall: bool, optional
If true, return all margins found. If False (default), return only the
minimum stability margins. For frequency data or FRD systems, only
margins in the given frequency region can be found and returned.
epsw: float, optional
Frequencies below this value (default 0.0) are considered static gain,
and not returned as margin.
Returns
-------
gm: float or array_like
Gain margin
pm: float or array_loke
Phase margin
sm: float or array_like
Stability margin, the minimum distance from the Nyquist plot to -1
wg: float or array_like
Frequency for gain margin (at phase crossover, phase = -180 degrees)
wp: float or array_like
Frequency for phase margin (at gain crossover, gain = 1)
ws: float or array_like
Frequency for stability margin (complex gain closest to -1)
"""
try:
if isinstance(sysdata, frdata.FRD):
sys = frdata.FRD(sysdata, smooth=True)
elif isinstance(sysdata, xferfcn.TransferFunction):
sys = sysdata
elif getattr(sysdata, '__iter__', False) and len(sysdata) == 3:
mag, phase, omega = sysdata
sys = frdata.FRD(mag * np.exp(1j * phase * math.pi/180),
omega, smooth=True)
else:
sys = xferfcn._convert_to_transfer_function(sysdata)
except Exception as e:
print (e)
raise ValueError("Margin sysdata must be either a linear system or "
"a 3-sequence of mag, phase, omega.")
# calculate gain of system
if isinstance(sys, xferfcn.TransferFunction):
# check for siso
if not issiso(sys):
raise ValueError("Can only do margins for SISO system")
# real and imaginary part polynomials in omega:
rnum, inum = _polyimsplit(sys.num[0][0])
rden, iden = _polyimsplit(sys.den[0][0])
# test (imaginary part of tf) == 0, for phase crossover/gain margins
test_w_180 = np.polyadd(np.polymul(inum, rden), np.polymul(rnum, -iden))
w_180 = np.roots(test_w_180)
# first remove imaginary and negative frequencies, epsw removes the
# "0" frequency for type-2 systems
w_180 = np.real(w_180[(np.imag(w_180) == 0) * (w_180 >= epsw)])
# evaluate response at remaining frequencies, to test for phase 180 vs 0
with np.errstate(all='ignore'):
resp_w_180 = np.real(
np.polyval(sys.num[0][0], 1.j*w_180) /
np.polyval(sys.den[0][0], 1.j*w_180))
# only keep frequencies where the negative real axis is crossed
w_180 = w_180[np.real(resp_w_180) <= 0.0]
# and sort
w_180.sort()
# test magnitude is 1 for gain crossover/phase margins
test_wc = np.polysub(np.polyadd(_polysqr(rnum), _polysqr(inum)),
np.polyadd(_polysqr(rden), _polysqr(iden)))
wc = np.roots(test_wc)
wc = np.real(wc[(np.imag(wc) == 0) * (wc > epsw)])
wc.sort()
# stability margin was a bitch to elaborate, relies on magnitude to
# point -1, then take the derivative. Second derivative needs to be >0
# to have a minimum
test_wstabd = np.polyadd(_polysqr(rden), _polysqr(iden))
test_wstabn = np.polyadd(_polysqr(np.polyadd(rnum,rden)),
_polysqr(np.polyadd(inum,iden)))
test_wstab = np.polysub(
np.polymul(np.polyder(test_wstabn),test_wstabd),
np.polymul(np.polyder(test_wstabd),test_wstabn))
# find the solutions, for positive omega, and only real ones
wstab = np.roots(test_wstab)
wstab = np.real(wstab[(np.imag(wstab) == 0) *
(np.real(wstab) >= 0)])
# and find the value of the 2nd derivative there, needs to be positive
wstabplus = np.polyval(np.polyder(test_wstab), wstab)
wstab = np.real(wstab[(np.imag(wstab) == 0) * (wstab > epsw) *
(wstabplus > 0.)])
wstab.sort()
else:
# a bit coarse, have the interpolated frd evaluated again
def mod(w):
"""to give the function to calculate |G(jw)| = 1"""
return np.abs(sys._evalfr(w)[0][0]) - 1
def arg(w):
"""function to calculate the phase angle at -180 deg"""
return np.angle(-sys._evalfr(w)[0][0])
def dstab(w):
"""function to calculate the distance from -1 point"""
return np.abs(sys._evalfr(w)[0][0] + 1.)
# Find all crossings, note that this depends on omega having
# a correct range
widx = np.where(np.diff(np.sign(mod(sys.omega))))[0]
wc = np.array(
[ sp.optimize.brentq(mod, sys.omega[i], sys.omega[i+1])
for i in widx if i+1 < len(sys.omega)])
# find the phase crossings ang(H(jw) == -180
widx = np.where(np.diff(np.sign(arg(sys.omega))))[0]
widx = widx[np.real(sys._evalfr(sys.omega[widx])[0][0]) <= 0]
w_180 = np.array(
[ sp.optimize.brentq(arg, sys.omega[i], sys.omega[i+1])
for i in widx if i+1 < len(sys.omega) ])
# find all stab margins?
widx = np.where(np.diff(np.sign(np.diff(dstab(sys.omega)))))[0]
wstab = np.array([ sp.optimize.minimize_scalar(
dstab, bracket=(sys.omega[i], sys.omega[i+1])).x
for i in widx if i+1 < len(sys.omega) and
np.diff(np.diff(dstab(sys.omega[i-1:i+2])))[0] > 0 ])
wstab = wstab[(wstab >= sys.omega[0]) *
(wstab <= sys.omega[-1])]
# margins, as iterables, converted frdata and xferfcn calculations to
# vector for this
with np.errstate(all='ignore'):
gain_w_180 = np.abs(sys._evalfr(w_180)[0][0])
GM = 1.0/gain_w_180
SM = np.abs(sys._evalfr(wstab)[0][0]+1)
PM = np.remainder(np.angle(sys._evalfr(wc)[0][0], deg=True), 360.0) - 180.0
if returnall:
return GM, PM, SM, w_180, wc, wstab
else:
if GM.shape[0] and not np.isinf(GM).all():
with np.errstate(all='ignore'):
gmidx = np.where(np.abs(np.log(GM)) ==
np.min(np.abs(np.log(GM))))
else:
gmidx = -1
if PM.shape[0]:
pmidx = np.where(np.abs(PM) == np.amin(np.abs(PM)))[0]
return (
(not gmidx != -1 and float('inf')) or GM[gmidx][0],
(not PM.shape[0] and float('inf')) or PM[pmidx][0],
(not SM.shape[0] and float('inf')) or np.amin(SM),
(not gmidx != -1 and float('nan')) or w_180[gmidx][0],
(not wc.shape[0] and float('nan')) or wc[pmidx][0],
(not wstab.shape[0] and float('nan')) or wstab[SM==np.amin(SM)][0])
# Contributed by Steffen Waldherr <waldherr@ist.uni-stuttgart.de>
#! TODO - need to add test functions
def phase_crossover_frequencies(sys):
"""Compute frequencies and gains at intersections with real axis
in Nyquist plot.
Call as:
omega, gain = phase_crossover_frequencies()
Returns
-------
omega: 1d array of (non-negative) frequencies where Nyquist plot
intersects the real axis
gain: 1d array of corresponding gains
Examples
--------
>>> tf = TransferFunction([1], [1, 2, 3, 4])
>>> PhaseCrossoverFrequenies(tf)
(array([ 1.73205081, 0. ]), array([-0.5 , 0.25]))
"""
# Convert to a transfer function
tf = xferfcn._convert_to_transfer_function(sys)
# if not siso, fall back to (0,0) element
#! TODO: should add a check and warning here
num = tf.num[0][0]
den = tf.den[0][0]
# Compute frequencies that we cross over the real axis
numj = (1.j)**np.arange(len(num)-1,-1,-1)*num
denj = (-1.j)**np.arange(len(den)-1,-1,-1)*den
allfreq = np.roots(np.imag(np.polymul(numj,denj)))
realfreq = np.real(allfreq[np.isreal(allfreq)])
realposfreq = realfreq[realfreq >= 0.]
# using real() to avoid rounding errors and results like 1+0j
# it would be nice to have a vectorized version of self.evalfr here
gain = np.real(np.asarray([tf._evalfr(f)[0][0] for f in realposfreq]))
return realposfreq, gain
def margin(*args):
"""margin(sysdata)
Calculate gain and phase margins and associated crossover frequencies
Parameters
----------
sysdata : LTI system or (mag, phase, omega) sequence
sys : StateSpace or TransferFunction
Linear SISO system
mag, phase, omega : sequence of array_like
Input magnitude, phase (in deg.), and frequencies (rad/sec) from
bode frequency response data
Returns
-------
gm : float
Gain margin
pm : float
Phase margin (in degrees)
wg: float
Frequency for gain margin (at phase crossover, phase = -180 degrees)
wp: float
Frequency for phase margin (at gain crossover, gain = 1)
Margins are calculated for a SISO open-loop system.
If there is more than one gain crossover, the one at the smallest
margin (deviation from gain = 1), in absolute sense, is
returned. Likewise the smallest phase margin (in absolute sense)
is returned.
Examples
--------
>>> sys = tf(1, [1, 2, 1, 0])
>>> gm, pm, wg, wp = margin(sys)
"""
if len(args) == 1:
sys = args[0]
margin = stability_margins(sys)
elif len(args) == 3:
margin = stability_margins(args)
else:
raise ValueError("Margin needs 1 or 3 arguments; received %i."
% len(args))
return margin[0], margin[1], margin[3], margin[4]
