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# rlocus.py - code for computing a root locus plot
# Code contributed by Ryan Krauss, 2010
#
# Copyright (c) 2010 by Ryan Krauss
# 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.
#
# RMM, 17 June 2010: modified to be a standalone piece of code
# * Added BSD copyright info to file (per Ryan)
# * Added code to convert (num, den) to poly1d's if they aren't already.
# This allows Ryan's code to run on a standard signal.ltisys object
# or a control.TransferFunction object.
# * Added some comments to make sure I understand the code
#
# RMM, 2 April 2011: modified to work with new LTI structure (see ChangeLog)
# * Not tested: should still work on signal.ltisys objects
#
# $Id$
# Packages used by this module
import numpy as np
from scipy import array, poly1d, row_stack, zeros_like, real, imag
import scipy.signal # signal processing toolbox
import pylab # plotting routines
from . import xferfcn
from .exception import ControlMIMONotImplemented
from functools import partial
__all__ = ['root_locus', 'rlocus']
# Main function: compute a root locus diagram
def root_locus(sys, kvect=None, xlim=None, ylim=None, plotstr='-', Plot=True,
PrintGain=True):
"""Root locus plot
Calculate the root locus by finding the roots of 1+k*TF(s)
where TF is self.num(s)/self.den(s) and each k is an element
of kvect.
Parameters
----------
sys : LTI object
Linear input/output systems (SISO only, for now)
kvect : list or ndarray, optional
List of gains to use in computing diagram
xlim : tuple or list, optional
control of x-axis range, normally with tuple (see matplotlib.axes)
ylim : tuple or list, optional
control of y-axis range
Plot : boolean, optional (default = True)
If True, plot magnitude and phase
PrintGain: boolean (default = True)
If True, report mouse clicks when close to the root-locus branches,
calculate gain, damping and print
Returns
-------
rlist : ndarray
Computed root locations, given as a 2d array
klist : ndarray or list
Gains used. Same as klist keyword argument if provided.
"""
# Convert numerator and denominator to polynomials if they aren't
(nump, denp) = _systopoly1d(sys)
if kvect is None:
kvect = _default_gains(sys)
# Compute out the loci
mymat = _RLFindRoots(sys, kvect)
mymat = _RLSortRoots(sys, mymat)
# Create the plot
if (Plot):
f = pylab.figure()
if PrintGain:
f.canvas.mpl_connect(
'button_release_event', partial(_RLFeedbackClicks, sys=sys))
ax = pylab.axes()
# plot open loop poles
poles = array(denp.r)
ax.plot(real(poles), imag(poles), 'x')
# plot open loop zeros
zeros = array(nump.r)
if zeros.any():
ax.plot(real(zeros), imag(zeros), 'o')
# Now plot the loci
for col in mymat.T:
ax.plot(real(col), imag(col), plotstr)
# Set up plot axes and labels
if xlim:
ax.set_xlim(xlim)
if ylim:
ax.set_ylim(ylim)
ax.set_xlabel('Real')
ax.set_ylabel('Imaginary')
return mymat, kvect
def _default_gains(sys):
# TODO: update with a smart calculation of the gains using sys poles/zeros
return np.logspace(-3, 3)
# Utility function to extract numerator and denominator polynomials
def _systopoly1d(sys):
"""Extract numerator and denominator polynomails for a system"""
# Allow inputs from the signal processing toolbox
if (isinstance(sys, scipy.signal.lti)):
nump = sys.num
denp = sys.den
else:
# Convert to a transfer function, if needed
sys = xferfcn._convertToTransferFunction(sys)
# Make sure we have a SISO system
if (sys.inputs > 1 or sys.outputs > 1):
raise ControlMIMONotImplemented()
# Start by extracting the numerator and denominator from system object
nump = sys.num[0][0]
denp = sys.den[0][0]
# Check to see if num, den are already polynomials; otherwise convert
if (not isinstance(nump, poly1d)):
nump = poly1d(nump)
if (not isinstance(denp, poly1d)):
denp = poly1d(denp)
return (nump, denp)
def _RLFindRoots(sys, kvect):
"""Find the roots for the root locus."""
# Convert numerator and denominator to polynomials if they aren't
(nump, denp) = _systopoly1d(sys)
roots = []
for k in kvect:
curpoly = denp + k * nump
curroots = curpoly.r
curroots.sort()
roots.append(curroots)
mymat = row_stack(roots)
return mymat
def _RLSortRoots(sys, mymat):
"""Sort the roots from sys._RLFindRoots, so that the root
locus doesn't show weird pseudo-branches as roots jump from
one branch to another."""
sorted = zeros_like(mymat)
for n, row in enumerate(mymat):
if n == 0:
sorted[n, :] = row
else:
# sort the current row by finding the element with the
# smallest absolute distance to each root in the
# previous row
available = list(range(len(prevrow)))
for elem in row:
evect = elem-prevrow[available]
ind1 = abs(evect).argmin()
ind = available.pop(ind1)
sorted[n, ind] = elem
prevrow = sorted[n, :]
return sorted
def _RLFeedbackClicks(event, sys):
"""Print root-locus gain feedback for clicks on the root-locus plot
"""
s = complex(event.xdata, event.ydata)
K = -1./sys.horner(s)
if abs(K.real) > 1e-8 and abs(K.imag/K.real) < 0.04:
print("Clicked at %10.4g%+10.4gj gain %10.4g damp %10.4g" %
(s.real, s.imag, K.real, -1*s.real/abs(s)))
rlocus = root_locus