"""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 , 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 # # 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 #! 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]