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1354 lines (1096 loc) · 49 KB
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"""xferfcn.py
Transfer function representation and functions.
This file contains the TransferFunction class and also functions
that operate on transfer functions. This is the primary representation
for the python-control library.
"""
# Python 3 compatibility (needs to go here)
from __future__ import print_function
from __future__ import division
"""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
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: 24 May 09
Revised: Kevin K. Chen, Dec 10
$Id$
"""
# External function declarations
import numpy as np
from numpy import angle, any, array, empty, finfo, insert, ndarray, ones, \
polyadd, polymul, polyval, roots, sort, sqrt, zeros, squeeze, exp, pi, \
where, delete, real, poly, poly1d, nonzero
import scipy as sp
from numpy.polynomial.polynomial import polyfromroots
from scipy.signal import lti, tf2zpk, zpk2tf, cont2discrete
from copy import deepcopy
import warnings
from warnings import warn
from itertools import chain
from .lti import LTI, timebaseEqual, timebase, isdtime
__all__ = ['TransferFunction', 'tf', 'ss2tf', 'tfdata']
class TransferFunction(LTI):
"""TransferFunction(num, den[, dt])
A class for representing transfer functions
The TransferFunction class is used to represent systems in transfer function
form.
The main data members are 'num' and 'den', which are 2-D lists of arrays
containing MIMO numerator and denominator coefficients. For example,
>>> num[2][5] = numpy.array([1., 4., 8.])
means that the numerator of the transfer function from the 6th input to the
3rd output is set to s^2 + 4s + 8.
Discrete-time transfer functions are implemented by using the 'dt' instance
variable and setting it to something other than 'None'. If 'dt' has a
non-zero value, then it must match whenever two transfer functions are
combined. If 'dt' is set to True, the system will be treated as a
discrete time system with unspecified sampling time.
"""
def __init__(self, *args):
"""TransferFunction(num, den[, dt])
Construct a transfer function.
The default constructor is TransferFunction(num, den), where num and
den are lists of lists of arrays containing polynomial coefficients.
To create a discrete time transfer funtion, use TransferFunction(num,
den, dt) where 'dt' is the sampling time (or True for unspecified
sampling time). To call the copy constructor, call
TransferFunction(sys), where sys is a TransferFunction object
(continuous or discrete).
"""
args = deepcopy(args)
if len(args) == 2:
# The user provided a numerator and a denominator.
(num, den) = args
dt = None
elif len(args) == 3:
# Discrete time transfer function
(num, den, dt) = args
elif len(args) == 1:
# Use the copy constructor.
if not isinstance(args[0], TransferFunction):
raise TypeError("The one-argument constructor can only take \
in a TransferFunction object. Received %s."
% type(args[0]))
num = args[0].num
den = args[0].den
try:
dt = args[0].dt
except NameError:
dt = None
else:
raise ValueError("Needs 1, 2 or 3 arguments; received %i."
% len(args))
num = _cleanPart(num)
den = _cleanPart(den)
inputs = len(num[0])
outputs = len(num)
# Make sure numerator and denominator matrices have consistent sizes
if inputs != len(den[0]):
raise ValueError("The numerator has %i input(s), but the \
denominator has %i\ninput(s)." % (inputs, len(den[0])))
if outputs != len(den):
raise ValueError("The numerator has %i output(s), but the \
denominator has %i\noutput(s)." % (outputs, len(den)))
# Additional checks/updates on structure of the transfer function
for i in range(outputs):
# Make sure that each row has the same number of columns
if len(num[i]) != inputs:
raise ValueError("Row 0 of the numerator matrix has %i \
elements, but row %i\nhas %i." % (inputs, i, len(num[i])))
if len(den[i]) != inputs:
raise ValueError("Row 0 of the denominator matrix has %i \
elements, but row %i\nhas %i." % (inputs, i, len(den[i])))
# Check for zeros in numerator or denominator
# TODO: Right now these checks are only done during construction.
# It might be worthwhile to think of a way to perform checks if the
# user modifies the transfer function after construction.
for j in range(inputs):
# Check that we don't have any zero denominators.
zeroden = True
for k in den[i][j]:
if k:
zeroden = False
break
if zeroden:
raise ValueError("Input %i, output %i has a zero \
denominator." % (j + 1, i + 1))
# If we have zero numerators, set the denominator to 1.
zeronum = True
for k in num[i][j]:
if k:
zeronum = False
break
if zeronum:
den[i][j] = ones(1)
LTI.__init__(self, inputs, outputs, dt)
self.num = num
self.den = den
self._truncatecoeff()
def __call__(self, s):
"""Evaluate the system's transfer function for a complex variable
For a SISO transfer function, returns the value of the
transfer function. For a MIMO transfer fuction, returns a
matrix of values evaluated at complex variable s."""
if self.issiso():
# return a scalar
return self.horner(s)[0][0]
else:
# return a matrix
return self.horner(s)
def _truncatecoeff(self):
"""Remove extraneous zero coefficients from num and den.
Check every element of the numerator and denominator matrices, and
truncate leading zeros. For instance, running self._truncatecoeff()
will reduce self.num = [[[0, 0, 1, 2]]] to [[[1, 2]]].
"""
# Beware: this is a shallow copy. This should be okay.
data = [self.num, self.den]
for p in range(len(data)):
for i in range(self.outputs):
for j in range(self.inputs):
# Find the first nontrivial coefficient.
nonzero = None
for k in range(data[p][i][j].size):
if (data[p][i][j][k]):
nonzero = k
break
if nonzero is None:
# The array is all zeros.
data[p][i][j] = zeros(1)
else:
# Truncate the trivial coefficients.
data[p][i][j] = data[p][i][j][nonzero:]
[self.num, self.den] = data
def __str__(self, var=None):
"""String representation of the transfer function."""
mimo = self.inputs > 1 or self.outputs > 1
if (var is None):
#! TODO: replace with standard calls to lti functions
var = 's' if self.dt is None or self.dt == 0 else 'z'
outstr = ""
for i in range(self.inputs):
for j in range(self.outputs):
if mimo:
outstr += "\nInput %i to output %i:" % (i + 1, j + 1)
# Convert the numerator and denominator polynomials to strings.
numstr = _tfpolyToString(self.num[j][i], var=var)
denstr = _tfpolyToString(self.den[j][i], var=var)
# Figure out the length of the separating line
dashcount = max(len(numstr), len(denstr))
dashes = '-' * dashcount
# Center the numerator or denominator
if len(numstr) < dashcount:
numstr = (' ' * int(round((dashcount - len(numstr))/2)) +
numstr)
if len(denstr) < dashcount:
denstr = (' ' * int(round((dashcount - len(denstr))/2)) +
denstr)
outstr += "\n" + numstr + "\n" + dashes + "\n" + denstr + "\n"
# See if this is a discrete time system with specific sampling time
if (not (self.dt is None) and type(self.dt) != bool and self.dt > 0):
#! TODO: replace with standard calls to lti functions
outstr += "\ndt = " + self.dt.__str__() + "\n"
return outstr
# represent as string, makes display work for IPython
__repr__ = __str__
def __neg__(self):
"""Negate a transfer function."""
num = deepcopy(self.num)
for i in range(self.outputs):
for j in range(self.inputs):
num[i][j] *= -1
return TransferFunction(num, self.den, self.dt)
def __add__(self, other):
"""Add two LTI objects (parallel connection)."""
from .statesp import StateSpace
# Convert the second argument to a transfer function.
if (isinstance(other, StateSpace)):
other = _convertToTransferFunction(other)
elif not isinstance(other, TransferFunction):
other = _convertToTransferFunction(other, inputs=self.inputs,
outputs=self.outputs)
# Check that the input-output sizes are consistent.
if self.inputs != other.inputs:
raise ValueError("The first summand has %i input(s), but the \
second has %i." % (self.inputs, other.inputs))
if self.outputs != other.outputs:
raise ValueError("The first summand has %i output(s), but the \
second has %i." % (self.outputs, other.outputs))
# Figure out the sampling time to use
if (self.dt is None and other.dt is not None):
dt = other.dt # use dt from second argument
elif (other.dt is None and self.dt is not None) or \
(timebaseEqual(self, other)):
dt = self.dt # use dt from first argument
else:
raise ValueError("Systems have different sampling times")
# Preallocate the numerator and denominator of the sum.
num = [[[] for j in range(self.inputs)] for i in range(self.outputs)]
den = [[[] for j in range(self.inputs)] for i in range(self.outputs)]
for i in range(self.outputs):
for j in range(self.inputs):
num[i][j], den[i][j] = _addSISO(self.num[i][j], self.den[i][j],
other.num[i][j],
other.den[i][j])
return TransferFunction(num, den, dt)
def __radd__(self, other):
"""Right add two LTI objects (parallel connection)."""
return self + other
def __sub__(self, other):
"""Subtract two LTI objects."""
return self + (-other)
def __rsub__(self, other):
"""Right subtract two LTI objects."""
return other + (-self)
def __mul__(self, other):
"""Multiply two LTI objects (serial connection)."""
# Convert the second argument to a transfer function.
if isinstance(other, (int, float, complex, np.number)):
other = _convertToTransferFunction(other, inputs=self.inputs,
outputs=self.inputs)
else:
other = _convertToTransferFunction(other)
# Check that the input-output sizes are consistent.
if self.inputs != other.outputs:
raise ValueError("C = A * B: A has %i column(s) (input(s)), but B \
has %i row(s)\n(output(s))." % (self.inputs, other.outputs))
inputs = other.inputs
outputs = self.outputs
# Figure out the sampling time to use
if (self.dt is None and other.dt is not None):
dt = other.dt # use dt from second argument
elif (other.dt is None and self.dt is not None) or \
(self.dt == other.dt):
dt = self.dt # use dt from first argument
else:
raise ValueError("Systems have different sampling times")
# Preallocate the numerator and denominator of the sum.
num = [[[0] for j in range(inputs)] for i in range(outputs)]
den = [[[1] for j in range(inputs)] for i in range(outputs)]
# Temporary storage for the summands needed to
# find the (i, j)th element of the product.
num_summand = [[] for k in range(self.inputs)]
den_summand = [[] for k in range(self.inputs)]
for i in range(outputs): # Iterate through rows of product.
for j in range(inputs): # Iterate through columns of product.
for k in range(self.inputs): # Multiply & add.
num_summand[k] = polymul(self.num[i][k], other.num[k][j])
den_summand[k] = polymul(self.den[i][k], other.den[k][j])
num[i][j], den[i][j] = _addSISO(
num[i][j], den[i][j],
num_summand[k], den_summand[k])
return TransferFunction(num, den, dt)
def __rmul__(self, other):
"""Right multiply two LTI objects (serial connection)."""
# Convert the second argument to a transfer function.
if isinstance(other, (int, float, complex, np.number)):
other = _convertToTransferFunction(other, inputs=self.inputs,
outputs=self.inputs)
else:
other = _convertToTransferFunction(other)
# Check that the input-output sizes are consistent.
if other.inputs != self.outputs:
raise ValueError("C = A * B: A has %i column(s) (input(s)), but B \
has %i row(s)\n(output(s))." % (other.inputs, self.outputs))
inputs = self.inputs
outputs = other.outputs
# Figure out the sampling time to use
if (self.dt is None and other.dt is not None):
dt = other.dt # use dt from second argument
elif (other.dt is None and self.dt is not None) \
or (self.dt == other.dt):
dt = self.dt # use dt from first argument
else:
raise ValueError("Systems have different sampling times")
# Preallocate the numerator and denominator of the sum.
num = [[[0] for j in range(inputs)] for i in range(outputs)]
den = [[[1] for j in range(inputs)] for i in range(outputs)]
# Temporary storage for the summands needed to find the
# (i, j)th element
# of the product.
num_summand = [[] for k in range(other.inputs)]
den_summand = [[] for k in range(other.inputs)]
for i in range(outputs): # Iterate through rows of product.
for j in range(inputs): # Iterate through columns of product.
for k in range(other.inputs): # Multiply & add.
num_summand[k] = polymul(other.num[i][k], self.num[k][j])
den_summand[k] = polymul(other.den[i][k], self.den[k][j])
num[i][j], den[i][j] = _addSISO(
num[i][j], den[i][j],
num_summand[k], den_summand[k])
return TransferFunction(num, den, dt)
# TODO: Division of MIMO transfer function objects is not written yet.
def __truediv__(self, other):
"""Divide two LTI objects."""
if isinstance(other, (int, float, complex, np.number)):
other = _convertToTransferFunction(
other, inputs=self.inputs,
outputs=self.inputs)
else:
other = _convertToTransferFunction(other)
if (self.inputs > 1 or self.outputs > 1 or
other.inputs > 1 or other.outputs > 1):
raise NotImplementedError(
"TransferFunction.__truediv__ is currently \
implemented only for SISO systems.")
# Figure out the sampling time to use
if (self.dt is None and other.dt is not None):
dt = other.dt # use dt from second argument
elif (other.dt is None and self.dt is not None)\
or (self.dt == other.dt):
dt = self.dt # use dt from first argument
else:
raise ValueError("Systems have different sampling times")
num = polymul(self.num[0][0], other.den[0][0])
den = polymul(self.den[0][0], other.num[0][0])
return TransferFunction(num, den, dt)
# TODO: Remove when transition to python3 complete
def __div__(self, other):
return TransferFunction.__truediv__(self, other)
# TODO: Division of MIMO transfer function objects is not written yet.
def __rtruediv__(self, other):
"""Right divide two LTI objects."""
if isinstance(other, (int, float, complex, np.number)):
other = _convertToTransferFunction(
other, inputs=self.inputs,
outputs=self.inputs)
else:
other = _convertToTransferFunction(other)
if (self.inputs > 1 or self.outputs > 1 or
other.inputs > 1 or other.outputs > 1):
raise NotImplementedError(
"TransferFunction.__rtruediv__ is currently \
implemented only for SISO systems.")
return other / self
# TODO: Remove when transition to python3 complete
def __rdiv__(self, other):
return TransferFunction.__rtruediv__(self, other)
def __pow__(self, other):
if not type(other) == int:
raise ValueError("Exponent must be an integer")
if other == 0:
return TransferFunction([1], [1]) # unity
if other > 0:
return self * (self**(other-1))
if other < 0:
return (TransferFunction([1], [1]) / self) * (self**(other+1))
def evalfr(self, omega):
"""Evaluate a transfer function at a single angular frequency.
self._evalfr(omega) returns the value of the transfer function
matrix with input value s = i * omega.
"""
warn("TransferFunction.evalfr(omega) will be deprecated in a "
"future release of python-control; use evalfr(sys, omega*1j) "
"instead", PendingDeprecationWarning)
return self._evalfr(omega)
def _evalfr(self, omega):
"""Evaluate a transfer function at a single angular frequency."""
# TODO: implement for discrete time systems
if isdtime(self, strict=True):
# Convert the frequency to discrete time
dt = timebase(self)
s = exp(1.j * omega * dt)
if (omega * dt > pi):
warn("_evalfr: frequency evaluation above Nyquist frequency")
else:
s = 1.j * omega
return self.horner(s)
def horner(self, s):
"""Evaluate the systems's transfer function for a complex variable
Returns a matrix of values evaluated at complex variable s.
"""
# Preallocate the output.
if getattr(s, '__iter__', False):
out = empty((self.outputs, self.inputs, len(s)), dtype=complex)
else:
out = empty((self.outputs, self.inputs), dtype=complex)
for i in range(self.outputs):
for j in range(self.inputs):
out[i][j] = (polyval(self.num[i][j], s) /
polyval(self.den[i][j], s))
return out
# Method for generating the frequency response of the system
def freqresp(self, omega):
"""Evaluate a transfer function at a list of angular frequencies.
mag, phase, omega = self.freqresp(omega)
reports the value of the magnitude, phase, and angular frequency of the
transfer function matrix evaluated at s = i * omega, where omega is a
list of angular frequencies, and is a sorted
version of the input omega.
"""
# Preallocate outputs.
numfreq = len(omega)
mag = empty((self.outputs, self.inputs, numfreq))
phase = empty((self.outputs, self.inputs, numfreq))
# Figure out the frequencies
omega.sort()
if isdtime(self, strict=True):
dt = timebase(self)
slist = np.array([exp(1.j * w * dt) for w in omega])
if (max(omega) * dt > pi):
warn("freqresp: frequency evaluation above Nyquist frequency")
else:
slist = np.array([1j * w for w in omega])
# Compute frequency response for each input/output pair
for i in range(self.outputs):
for j in range(self.inputs):
fresp = (polyval(self.num[i][j], slist) /
polyval(self.den[i][j], slist))
mag[i, j, :] = abs(fresp)
phase[i, j, :] = angle(fresp)
return mag, phase, omega
def pole(self):
"""Compute the poles of a transfer function."""
num, den, denorder = self._common_den()
rts = []
for d, o in zip(den,denorder):
rts.extend(roots(d[:o+1]))
return np.array(rts)
def zero(self):
"""Compute the zeros of a transfer function."""
if self.inputs > 1 or self.outputs > 1:
raise NotImplementedError("TransferFunction.zero is currently \
only implemented for SISO systems.")
else:
#for now, just give zeros of a SISO tf
return roots(self.num[0][0])
def feedback(self, other=1, sign=-1):
"""Feedback interconnection between two LTI objects."""
other = _convertToTransferFunction(other)
if (self.inputs > 1 or self.outputs > 1 or
other.inputs > 1 or other.outputs > 1):
# TODO: MIMO feedback
raise NotImplementedError("TransferFunction.feedback is currently \
only implemented for SISO functions.")
# Figure out the sampling time to use
if (self.dt is None and other.dt is not None):
dt = other.dt # use dt from second argument
elif (other.dt is None and self.dt is not None) \
or (self.dt == other.dt):
dt = self.dt # use dt from first argument
else:
raise ValueError("Systems have different sampling times")
num1 = self.num[0][0]
den1 = self.den[0][0]
num2 = other.num[0][0]
den2 = other.den[0][0]
num = polymul(num1, den2)
den = polyadd(polymul(den2, den1), -sign * polymul(num2, num1))
return TransferFunction(num, den, dt)
# For MIMO or SISO systems, the analytic expression is
# self / (1 - sign * other * self)
# But this does not work correctly because the state size will be too
# large.
def minreal(self, tol=None):
"""Remove cancelling pole/zero pairs from a transfer function"""
# based on octave minreal
# default accuracy
from sys import float_info
sqrt_eps = sqrt(float_info.epsilon)
# pre-allocate arrays
num = [[[] for j in range(self.inputs)] for i in range(self.outputs)]
den = [[[] for j in range(self.inputs)] for i in range(self.outputs)]
for i in range(self.outputs):
for j in range(self.inputs):
# split up in zeros, poles and gain
newzeros = []
zeros = roots(self.num[i][j])
poles = roots(self.den[i][j])
gain = self.num[i][j][0] / self.den[i][j][0]
# check all zeros
for z in zeros:
t = tol or \
1000 * max(float_info.epsilon, abs(z) * sqrt_eps)
idx = where(abs(z - poles) < t)[0]
if len(idx):
# cancel this zero against one of the poles
poles = delete(poles, idx[0])
else:
# keep this zero
newzeros.append(z)
# poly([]) returns a scalar, but we always want a 1d array
num[i][j] = np.atleast_1d(gain * real(poly(newzeros)))
den[i][j] = np.atleast_1d(real(poly(poles)))
# end result
return TransferFunction(num, den, self.dt)
def returnScipySignalLTI(self):
"""Return a list of a list of scipy.signal.lti objects.
For instance,
>>> out = tfobject.returnScipySignalLTI()
>>> out[3][5]
is a signal.scipy.lti object corresponding to the
transfer function from the 6th input to the 4th output.
"""
# TODO: implement for discrete time systems
if (self.dt != 0 and self.dt is not None):
raise NotImplementedError("Function not \
implemented in discrete time")
# Preallocate the output.
out = [[[] for j in range(self.inputs)] for i in range(self.outputs)]
for i in range(self.outputs):
for j in range(self.inputs):
out[i][j] = lti(self.num[i][j], self.den[i][j])
return out
def _common_den(self, imag_tol=None):
"""
Compute MIMO common denominators; return them and adjusted numerators.
This function computes the denominators per input containing all
the poles of sys.den, and reports it as the array den. The
output numerator array num is modified to use the common
denominator for this input/column; the coefficient arrays are also
padded with zeros to be the same size for all num/den.
num is an sys.outputs by sys.inputs
by len(d) array.
Parameters
----------
imag_tol: float
Threshold for the imaginary part of a root to use in detecting
complex poles
Returns
-------
num: array
Multi-dimensional array of numerator coefficients. num[i][j]
gives the numerator coefficient array for the ith input and jth
output, also prepared for use in td04ad; matches the denorder
order; highest coefficient starts on the left.
den: array
Multi-dimensional array of coefficients for common denominator
polynomial, one row per input. The array is prepared for use in
slycot td04ad, the first element is the highest-order polynomial
coefficiend of s, matching the order in denorder, if denorder <
number of columns in den, the den is padded with zeros
denorder: array of int, orders of den, one per input
Examples
--------
>>> num, den, denorder = sys._common_den()
"""
# Machine precision for floats.
eps = finfo(float).eps
# Decide on the tolerance to use in deciding of a pole is complex
if (imag_tol is None):
imag_tol = 1e-8 # TODO: figure out the right number to use
# A list to keep track of cumulative poles found as we scan
# self.den[..][..]
poles = [ [] for j in range(self.inputs) ]
# RvP, new implementation 180526, issue #194
# pre-calculate the poles for all num, den
# has zeros, poles, gain, list for pole indices not in den,
# number of poles known at the time analyzed
# do not calculate minreal. Rory's hint .minreal()
poleset = []
for i in range(self.outputs):
poleset.append([])
for j in range(self.inputs):
if abs(self.num[i][j]).max() <= eps:
poleset[-1].append( [array([], dtype=float),
roots(self.den[i][j]), 0.0, [], 0 ])
else:
z, p, k = tf2zpk(self.num[i][j], self.den[i][j])
poleset[-1].append([ z, p, k, [], 0])
# collect all individual poles
epsnm = eps * self.inputs * self.outputs
for j in range(self.inputs):
for i in range(self.outputs):
currentpoles = poleset[i][j][1]
nothave = ones(currentpoles.shape, dtype=bool)
for ip, p in enumerate(poles[j]):
idx, = nonzero(
(abs(currentpoles - p) < epsnm) * nothave)
if len(idx):
nothave[idx[0]] = False
else:
# remember id of pole not in tf
poleset[i][j][3].append(ip)
for h, c in zip(nothave, currentpoles):
if h:
poles[j].append(c)
# remember how many poles now known
poleset[i][j][4] = len(poles[j])
# figure out maximum number of poles, for sizing the den
npmax = max([len(p) for p in poles])
den = zeros((self.inputs, npmax+1), dtype=float)
num = zeros((max(1,self.outputs,self.inputs),
max(1,self.outputs,self.inputs), npmax+1), dtype=float)
denorder = zeros((self.inputs,), dtype=int)
for j in range(self.inputs):
if not len(poles[j]):
# no poles matching this input; only one or more gains
den[j,0] = 1.0
for i in range(self.outputs):
num[i,j,0] = poleset[i][j][2]
else:
# create the denominator matching this input
# polyfromroots gives coeffs in opposite order from what we use
# coefficients should be padded on right, ending at np
np = len(poles[j])
den[j,np::-1] = polyfromroots(poles[j]).real
denorder[j] = np
# now create the numerator, also padded on the right
for i in range(self.outputs):
# start with the current set of zeros for this output
nwzeros = list(poleset[i][j][0])
# add all poles not found in the original denominator,
# and the ones later added from other denominators
for ip in chain(poleset[i][j][3],
range(poleset[i][j][4],np)):
nwzeros.append(poles[j][ip])
numpoly = poleset[i][j][2] * polyfromroots(nwzeros).real
# print(numpoly, den[j])
# polyfromroots gives coeffs in opposite order => invert
# numerator polynomial should be padded on left and right
# ending at np to line up with what td04ad expects...
num[i, j, np+1-len(numpoly):np+1] = numpoly[::-1]
# print(num[i, j])
if (abs(den.imag) > epsnm).any():
print("Warning: The denominator has a nontrivial imaginary part: %f"
% abs(den.imag).max())
den = den.real
return num, den, denorder
def sample(self, Ts, method='zoh', alpha=None):
"""Convert a continuous-time system to discrete time
Creates a discrete-time system from a continuous-time system by
sampling. Multiple methods of conversion are supported.
Parameters
----------
Ts : float
Sampling period
method : {"gbt", "bilinear", "euler", "backward_diff", "zoh", "matched"}
Which method to use:
* gbt: generalized bilinear transformation
* bilinear: Tustin's approximation ("gbt" with alpha=0.5)
* euler: Euler (or forward differencing) method ("gbt" with alpha=0)
* backward_diff: Backwards differencing ("gbt" with alpha=1.0)
* zoh: zero-order hold (default)
alpha : float within [0, 1]
The generalized bilinear transformation weighting parameter, which
should only be specified with method="gbt", and is ignored otherwise
Returns
-------
sysd : StateSpace system
Discrete time system, with sampling rate Ts
Notes
-----
1. Available only for SISO systems
2. Uses the command `cont2discrete` from `scipy.signal`
Examples
--------
>>> sys = TransferFunction(1, [1,1])
>>> sysd = sys.sample(0.5, method='bilinear')
"""
if not self.isctime():
raise ValueError("System must be continuous time system")
if not self.issiso():
raise NotImplementedError("MIMO implementation not available")
if method == "matched":
return _c2dmatched(self, Ts)
sys = (self.num[0][0], self.den[0][0])
numd, dend, dt = cont2discrete(sys, Ts, method, alpha)
return TransferFunction(numd[0,:], dend, dt)
def dcgain(self):
"""Return the zero-frequency (or DC) gain
For a continous-time transfer function G(s), the DC gain is G(0)
For a discrete-time transfer function G(z), the DC gain is G(1)
Returns
-------
gain : ndarray
The zero-frequency gain
"""
if self.isctime():
return self._dcgain_cont()
else:
return self(1)
def _dcgain_cont(self):
"""_dcgain_cont() -> DC gain as matrix or scalar
Special cased evaluation at 0 for continuous-time systems"""
gain = np.empty((self.outputs, self.inputs), dtype=float)
for i in range(self.outputs):
for j in range(self.inputs):
num = self.num[i][j][-1]
den = self.den[i][j][-1]
if den:
gain[i][j] = num / den
else:
if num:
# numerator nonzero: infinite gain
gain[i][j] = np.inf
else:
# numerator is zero too: give up
gain[i][j] = np.nan
return np.squeeze(gain)
# c2d function contributed by Benjamin White, Oct 2012
def _c2dmatched(sysC, Ts):
# Pole-zero match method of continuous to discrete time conversion
szeros, spoles, sgain = tf2zpk(sysC.num[0][0], sysC.den[0][0])
zzeros = [0] * len(szeros)
zpoles = [0] * len(spoles)
pregainnum = [0] * len(szeros)
pregainden = [0] * len(spoles)
for idx, s in enumerate(szeros):
sTs = s*Ts
z = exp(sTs)
zzeros[idx] = z
pregainnum[idx] = 1-z
for idx, s in enumerate(spoles):
sTs = s*Ts
z = exp(sTs)
zpoles[idx] = z
pregainden[idx] = 1-z
zgain = np.multiply.reduce(pregainnum)/np.multiply.reduce(pregainden)
gain = sgain/zgain
sysDnum, sysDden = zpk2tf(zzeros, zpoles, gain)
return TransferFunction(sysDnum, sysDden, Ts)
# Utility function to convert a transfer function polynomial to a string
# Borrowed from poly1d library
def _tfpolyToString(coeffs, var='s'):
"""Convert a transfer function polynomial to a string"""
thestr = "0"
# Compute the number of coefficients
N = len(coeffs)-1
for k in range(len(coeffs)):
coefstr = '%.4g' % abs(coeffs[k])
if coefstr[-4:] == '0000':
coefstr = coefstr[:-5]
power = (N-k)
if power == 0:
if coefstr != '0':
newstr = '%s' % (coefstr,)
else:
if k == 0:
newstr = '0'
else:
newstr = ''
elif power == 1:
if coefstr == '0':
newstr = ''
elif coefstr == '1':
newstr = var
else:
newstr = '%s %s' % (coefstr, var)
else:
if coefstr == '0':
newstr = ''
elif coefstr == '1':
newstr = '%s^%d' % (var, power,)
else:
newstr = '%s %s^%d' % (coefstr, var, power)
if k > 0:
if newstr != '':
if coeffs[k] < 0:
thestr = "%s - %s" % (thestr, newstr)
else:
thestr = "%s + %s" % (thestr, newstr)
elif (k == 0) and (newstr != '') and (coeffs[k] < 0):
thestr = "-%s" % (newstr,)
else:
thestr = newstr
return thestr
def _addSISO(num1, den1, num2, den2):
"""Return num/den = num1/den1 + num2/den2.
Each numerator and denominator is a list of polynomial coefficients.
"""