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1569 lines (1275 loc) · 57.6 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, array, empty, finfo, ndarray, ones, \
polyadd, polymul, polyval, roots, sqrt, zeros, squeeze, exp, pi, \
where, delete, real, poly, nonzero
import scipy as sp
from scipy.signal import tf2zpk, zpk2tf, cont2discrete
from scipy.signal import TransferFunction as signalTransferFunction
from copy import deepcopy
from warnings import warn
from itertools import chain
from re import sub
from .lti import LTI, common_timebase, isdtime, _process_frequency_response
from . import config
__all__ = ['TransferFunction', 'tf', 'ss2tf', 'tfdata']
# Define module default parameter values
_xferfcn_defaults = {}
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.
A discrete time transfer function is created by specifying a nonzero
'timebase' dt when the system is constructed:
* dt = 0: continuous time system (default)
* dt > 0: discrete time system with sampling period 'dt'
* dt = True: discrete time with unspecified sampling period
* dt = None: no timebase specified
Systems must have compatible timebases in order to be combined. A discrete
time system with unspecified sampling time (`dt = True`) can be combined
with a system having a specified sampling time; the result will be a
discrete time system with the sample time of the latter system. Similarly,
a system with timebase `None` can be combined with a system having any
timebase; the result will have the timebase of the latter system.
The default value of dt can be changed by changing the value of
``control.config.defaults['control.default_dt']``.
The TransferFunction class defines two constants ``s`` and ``z`` that
represent the differentiation and delay operators in continuous and
discrete time. These can be used to create variables that allow algebraic
creation of transfer functions. For example,
>>> s = TransferFunction.s
>>> G = (s + 1)/(s**2 + 2*s + 1)
"""
# Give TransferFunction._rmul_() priority for ndarray * TransferFunction
__array_priority__ = 11 # override ndarray and matrix types
def __init__(self, *args, **kwargs):
"""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
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
else:
raise ValueError("Needs 1, 2 or 3 arguments; received %i."
% len(args))
num = _clean_part(num)
den = _clean_part(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 input(s)." % (inputs, len(den[0])))
if outputs != len(den):
raise ValueError(
"The numerator has %i output(s), but the denominator has "
"%i output(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 has %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 has %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)
self.num = num
self.den = den
self._truncatecoeff()
# get dt
if len(args) == 2:
# no dt given in positional arguments
if 'dt' in kwargs:
dt = kwargs['dt']
elif self._isstatic():
dt = None
else:
dt = config.defaults['control.default_dt']
elif len(args) == 3:
# Discrete time transfer function
if 'dt' in kwargs:
warn('received multiple dt arguments, '
'using positional arg dt=%s' % dt)
elif len(args) == 1:
# TODO: not sure this can ever happen since dt is always present
try:
dt = args[0].dt
except AttributeError:
if self._isstatic():
dt = None
else:
dt = config.defaults['control.default_dt']
self.dt = dt
def __call__(self, x, squeeze=None, warn_infinite=True):
"""Evaluate system's transfer function at complex frequencies.
Returns the complex frequency response `sys(x)` where `x` is `s` for
continuous-time systems and `z` for discrete-time systems.
In general the system may be multiple input, multiple output
(MIMO), where `m = self.ninputs` number of inputs and `p =
self.noutputs` number of outputs.
To evaluate at a frequency omega in radians per second, enter
``x = omega * 1j``, for continuous-time systems, or
``x = exp(1j * omega * dt)`` for discrete-time systems. Or use
:meth:`TransferFunction.frequency_response`.
Parameters
----------
x : complex or complex 1D array_like
Complex frequencies
squeeze : bool, optional
If squeeze=True, remove single-dimensional entries from the shape
of the output even if the system is not SISO. If squeeze=False,
keep all indices (output, input and, if omega is array_like,
frequency) even if the system is SISO. The default value can be
set using config.defaults['control.squeeze_frequency_response'].
If True and the system is single-input single-output (SISO),
return a 1D array rather than a 3D array. Default value (True)
set by config.defaults['control.squeeze_frequency_response'].
warn_infinite : bool, optional
If set to `False`, turn off divide by zero warning.
Returns
-------
fresp : complex ndarray
The frequency response of the system. If the system is SISO and
squeeze is not True, the shape of the array matches the shape of
omega. If the system is not SISO or squeeze is False, the first
two dimensions of the array are indices for the output and input
and the remaining dimensions match omega. If ``squeeze`` is True
then single-dimensional axes are removed.
"""
out = self.horner(x, warn_infinite=warn_infinite)
return _process_frequency_response(self, x, out, squeeze=squeeze)
def horner(self, x, warn_infinite=True):
"""Evaluate system's transfer function at complex frequency
using Horner's method.
Evaluates `sys(x)` where `x` is `s` for continuous-time systems and `z`
for discrete-time systems.
Expects inputs and outputs to be formatted correctly. Use ``sys(x)``
for a more user-friendly interface.
Parameters
----------
x : complex array_like or complex scalar
Complex frequencies
Returns
-------
output : (self.noutputs, self.ninputs, len(x)) complex ndarray
Frequency response
"""
# Make sure the argument is a 1D array of complex numbers
x_arr = np.atleast_1d(x).astype(complex, copy=False)
# Make sure that we are operating on a simple list
if len(x_arr.shape) > 1:
raise ValueError("input list must be 1D")
# Initialize the output matrix in the proper shape
out = empty((self.noutputs, self.ninputs, len(x_arr)), dtype=complex)
# Set up error processing based on warn_infinite flag
with np.errstate(all='warn' if warn_infinite else 'ignore'):
for i in range(self.noutputs):
for j in range(self.ninputs):
out[i][j] = (polyval(self.num[i][j], x_arr) /
polyval(self.den[i][j], x_arr))
return out
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.noutputs):
for j in range(self.ninputs):
# 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.ninputs > 1 or self.noutputs > 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.ninputs):
for j in range(self.noutputs):
if mimo:
outstr += "\nInput %i to output %i:" % (i + 1, j + 1)
# Convert the numerator and denominator polynomials to strings.
numstr = _tf_polynomial_to_string(self.num[j][i], var=var)
denstr = _tf_polynomial_to_string(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 = ' ' * ((dashcount - len(numstr)) // 2) + numstr
if len(denstr) < dashcount:
denstr = ' ' * ((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 to implement a re-loadable version
def __repr__(self):
"""Print transfer function in loadable form"""
if self.issiso():
return "TransferFunction({num}, {den}{dt})".format(
num=self.num[0][0].__repr__(), den=self.den[0][0].__repr__(),
dt=(isdtime(self, strict=True) and ', {}'.format(self.dt)) or '')
else:
return "TransferFunction({num}, {den}{dt})".format(
num=self.num.__repr__(), den=self.den.__repr__(),
dt=(isdtime(self, strict=True) and ', {}'.format(self.dt)) or '')
def _repr_latex_(self, var=None):
"""LaTeX representation of transfer function, for Jupyter notebook"""
mimo = self.ninputs > 1 or self.noutputs > 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'
out = ['$$']
if mimo:
out.append(r"\begin{bmatrix}")
for i in range(self.noutputs):
for j in range(self.ninputs):
# Convert the numerator and denominator polynomials to strings.
numstr = _tf_polynomial_to_string(self.num[i][j], var=var)
denstr = _tf_polynomial_to_string(self.den[i][j], var=var)
numstr = _tf_string_to_latex(numstr, var=var)
denstr = _tf_string_to_latex(denstr, var=var)
out += [r"\frac{", numstr, "}{", denstr, "}"]
if mimo and j < self.noutputs - 1:
out.append("&")
if mimo:
out.append(r"\\")
if mimo:
out.append(r" \end{bmatrix}")
# See if this is a discrete time system with specific sampling time
if not (self.dt is None) and type(self.dt) != bool and self.dt > 0:
out += [r"\quad dt = ", str(self.dt)]
out.append("$$")
return ''.join(out)
def __neg__(self):
"""Negate a transfer function."""
num = deepcopy(self.num)
for i in range(self.noutputs):
for j in range(self.ninputs):
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 = _convert_to_transfer_function(other)
elif not isinstance(other, TransferFunction):
other = _convert_to_transfer_function(other, inputs=self.ninputs,
outputs=self.noutputs)
# Check that the input-output sizes are consistent.
if self.ninputs != other.ninputs:
raise ValueError(
"The first summand has %i input(s), but the second has %i."
% (self.ninputs, other.ninputs))
if self.noutputs != other.noutputs:
raise ValueError(
"The first summand has %i output(s), but the second has %i."
% (self.noutputs, other.noutputs))
dt = common_timebase(self.dt, other.dt)
# Preallocate the numerator and denominator of the sum.
num = [[[] for j in range(self.ninputs)] for i in range(self.noutputs)]
den = [[[] for j in range(self.ninputs)] for i in range(self.noutputs)]
for i in range(self.noutputs):
for j in range(self.ninputs):
num[i][j], den[i][j] = _add_siso(
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 = _convert_to_transfer_function(other, inputs=self.ninputs,
outputs=self.ninputs)
else:
other = _convert_to_transfer_function(other)
# Check that the input-output sizes are consistent.
if self.ninputs != other.noutputs:
raise ValueError(
"C = A * B: A has %i column(s) (input(s)), but B has %i "
"row(s)\n(output(s))." % (self.ninputs, other.noutputs))
inputs = other.ninputs
outputs = self.noutputs
dt = common_timebase(self.dt, other.dt)
# 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.ninputs)]
den_summand = [[] for k in range(self.ninputs)]
# Multiply & add.
for row in range(outputs):
for col in range(inputs):
for k in range(self.ninputs):
num_summand[k] = polymul(
self.num[row][k], other.num[k][col])
den_summand[k] = polymul(
self.den[row][k], other.den[k][col])
num[row][col], den[row][col] = _add_siso(
num[row][col], den[row][col],
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 = _convert_to_transfer_function(other, inputs=self.ninputs,
outputs=self.ninputs)
else:
other = _convert_to_transfer_function(other)
# Check that the input-output sizes are consistent.
if other.ninputs != self.noutputs:
raise ValueError(
"C = A * B: A has %i column(s) (input(s)), but B has %i "
"row(s)\n(output(s))." % (other.ninputs, self.noutputs))
inputs = self.ninputs
outputs = other.noutputs
dt = common_timebase(self.dt, other.dt)
# 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.ninputs)]
den_summand = [[] for k in range(other.ninputs)]
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.ninputs): # 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] = _add_siso(
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 = _convert_to_transfer_function(
other, inputs=self.ninputs,
outputs=self.ninputs)
else:
other = _convert_to_transfer_function(other)
if (self.ninputs > 1 or self.noutputs > 1 or
other.ninputs > 1 or other.noutputs > 1):
raise NotImplementedError(
"TransferFunction.__truediv__ is currently \
implemented only for SISO systems.")
dt = common_timebase(self.dt, other.dt)
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 = _convert_to_transfer_function(
other, inputs=self.ninputs,
outputs=self.ninputs)
else:
other = _convert_to_transfer_function(other)
if (self.ninputs > 1 or self.noutputs > 1 or
other.ninputs > 1 or other.noutputs > 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 __getitem__(self, key):
key1, key2 = key
# pre-process
if isinstance(key1, int):
key1 = slice(key1, key1 + 1, 1)
if isinstance(key2, int):
key2 = slice(key2, key2 + 1, 1)
# dim1
start1, stop1, step1 = key1.start, key1.stop, key1.step
if step1 is None:
step1 = 1
if start1 is None:
start1 = 0
if stop1 is None:
stop1 = len(self.num)
# dim1
start2, stop2, step2 = key2.start, key2.stop, key2.step
if step2 is None:
step2 = 1
if start2 is None:
start2 = 0
if stop2 is None:
stop2 = len(self.num[0])
num = []
den = []
for i in range(start1, stop1, step1):
num_i = []
den_i = []
for j in range(start2, stop2, step2):
num_i.append(self.num[i][j])
den_i.append(self.den[i][j])
num.append(num_i)
den.append(den_i)
if self.isctime():
return TransferFunction(num, den)
else:
return TransferFunction(num, den, self.dt)
def freqresp(self, omega):
"""(deprecated) Evaluate transfer function at complex frequencies.
.. deprecated::0.9.0
Method has been given the more pythonic name
:meth:`TransferFunction.frequency_response`. Or use
:func:`freqresp` in the MATLAB compatibility module.
"""
warn("TransferFunction.freqresp(omega) will be removed in a "
"future release of python-control; use "
"sys.frequency_response(omega), or freqresp(sys, omega) in the "
"MATLAB compatibility module instead", DeprecationWarning)
return self.frequency_response(omega)
def pole(self):
"""Compute the poles of a transfer function."""
_, den, denorder = self._common_den(allow_nonproper=True)
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.ninputs > 1 or self.noutputs > 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 = _convert_to_transfer_function(other)
if (self.ninputs > 1 or self.noutputs > 1 or
other.ninputs > 1 or other.noutputs > 1):
# TODO: MIMO feedback
raise NotImplementedError(
"TransferFunction.feedback is currently only implemented "
"for SISO functions.")
dt = common_timebase(self.dt, other.dt)
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.ninputs)] for i in range(self.noutputs)]
den = [[[] for j in range(self.ninputs)] for i in range(self.noutputs)]
for i in range(self.noutputs):
for j in range(self.ninputs):
# 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, strict=True):
"""Return a list of a list of :class:`scipy.signal.lti` objects.
For instance,
>>> out = tfobject.returnScipySignalLTI()
>>> out[3][5]
is a :class:`scipy.signal.lti` object corresponding to the
transfer function from the 6th input to the 4th output.
Parameters
----------
strict : bool, optional
True (default):
The timebase `tfobject.dt` cannot be None; it must be
continuous (0) or discrete (True or > 0).
False:
if `tfobject.dt` is None, continuous time
:class:`scipy.signal.lti` objects are returned
Returns
-------
out : list of list of :class:`scipy.signal.TransferFunction`
continuous time (inheriting from :class:`scipy.signal.lti`)
or discrete time (inheriting from :class:`scipy.signal.dlti`)
SISO objects
"""
if strict and self.dt is None:
raise ValueError("with strict=True, dt cannot be None")
if self.dt:
kwdt = {'dt': self.dt}
else:
# scipy convention for continuous time lti systems: call without
# dt keyword argument
kwdt = {}
# Preallocate the output.
out = [[[] for j in range(self.ninputs)] for i in range(self.noutputs)]
for i in range(self.noutputs):
for j in range(self.ninputs):
out[i][j] = signalTransferFunction(self.num[i][j],
self.den[i][j],
**kwdt)
return out
def _common_den(self, imag_tol=None, allow_nonproper=False):
"""
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.
Parameters
----------
imag_tol: float
Threshold for the imaginary part of a root to use in detecting
complex poles
allow_nonproper : boolean
Do not enforce proper transfer functions
Returns
-------
num: array
n by n by kd where n = max(sys.noutputs,sys.ninputs)
kd = max(denorder)+1
Multi-dimensional array of numerator coefficients. num[i,j]
gives the numerator coefficient array for the ith output and jth
input; padded for use in td04ad ('C' option); matches the
denorder order; highest coefficient starts on the left.
If allow_nonproper=True and the order of a numerator exceeds the
order of the common denominator, num will be returned as None
den: array
sys.ninputs by kd
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
coefficient 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
real_tol = sqrt(eps * self.ninputs * self.noutputs)
# The tolerance to use in deciding if a pole is complex
if (imag_tol is None):
imag_tol = 2 * real_tol
# A list to keep track of cumulative poles found as we scan
# self.den[..][..]
poles = [[] for j in range(self.ninputs)]
# RvP, new implementation 180526, issue #194
# BG, modification, issue #343, PR #354
# 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.noutputs):
poleset.append([])
for j in range(self.ninputs):
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
for j in range(self.ninputs):
for i in range(self.noutputs):
currentpoles = poleset[i][j][1]
nothave = ones(currentpoles.shape, dtype=bool)
for ip, p in enumerate(poles[j]):
collect = (np.isclose(currentpoles.real, p.real,
atol=real_tol) &
np.isclose(currentpoles.imag, p.imag,
atol=imag_tol) &
nothave)
if np.any(collect):
# mark first found pole as already collected
nothave[nonzero(collect)[0][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:
if abs(c.imag) < imag_tol:
c = c.real
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
maxindex = max([len(p) for p in poles])
den = zeros((self.ninputs, maxindex + 1), dtype=float)
num = zeros((max(1, self.noutputs, self.ninputs),
max(1, self.noutputs, self.ninputs),
maxindex + 1),
dtype=float)
denorder = zeros((self.ninputs,), dtype=int)
havenonproper = False
for j in range(self.ninputs):
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.noutputs):
num[i, j, 0] = poleset[i][j][2]
else:
# create the denominator matching this input
# coefficients should be padded on right, ending at maxindex
maxindex = len(poles[j])
den[j, :maxindex+1] = poly(poles[j])
denorder[j] = maxindex
# now create the numerator, also padded on the right
for i in range(self.noutputs):
# 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], maxindex)):
nwzeros.append(poles[j][ip])
numpoly = poleset[i][j][2] * np.atleast_1d(poly(nwzeros))
# td04ad expects a proper transfer function. If the
# numerater has a higher order than the denominator, the
# padding will fail
if len(numpoly) > maxindex + 1:
if allow_nonproper:
havenonproper = True
break
raise ValueError(
self.__str__() +
"is not a proper transfer function. "
"The degree of the numerators must not exceed "
"the degree of the denominators.")
# numerator polynomial should be padded on left and right
# ending at maxindex to line up with what td04ad expects.
num[i, j, maxindex+1-len(numpoly):maxindex+1] = numpoly
# print(num[i, j])
if havenonproper:
num = None
return num, den, denorder
def sample(self, Ts, method='zoh', alpha=None, prewarp_frequency=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.