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2034 lines (1666 loc) · 75.3 KB
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# xferfcn.py - transfer function class and related functions
#
# Initial author: Richard M. Murray
# Creation date: 24 May 2009
# Pre-2014 revisions: Kevin K. Chen, Dec 2010
# Use `git shortlog -n -s xferfcn.py` for full list of contributors
"""Transfer function class and related functions.
This module contains the `TransferFunction` class and also functions
that operate on transfer functions.
"""
import sys
from collections.abc import Iterable
from copy import deepcopy
from itertools import chain, product
from re import sub
from warnings import warn
import numpy as np
import scipy as sp
# float64 needed in eval() call
from numpy import float64 # noqa: F401
from numpy import array, delete, empty, exp, finfo, ndarray, nonzero, ones, \
poly, polyadd, polymul, polyval, real, roots, sqrt, where, zeros
from scipy.signal import TransferFunction as signalTransferFunction
from scipy.signal import cont2discrete, tf2zpk, zpk2tf
from . import bdalg, config
from .exception import ControlMIMONotImplemented
from .frdata import FrequencyResponseData
from .iosys import InputOutputSystem, NamedSignal, _process_iosys_keywords, \
_process_subsys_index, common_timebase
from .lti import LTI, _process_frequency_response
__all__ = ['TransferFunction', 'tf', 'zpk', 'ss2tf', 'tfdata']
# Define module default parameter values
_xferfcn_defaults = {
'xferfcn.display_format': 'poly',
'xferfcn.floating_point_format': '.4g'
}
class TransferFunction(LTI):
"""TransferFunction(num, den[, dt])
Transfer function representation for LTI input/output systems.
The TransferFunction class is used to represent systems in transfer
function form. Transfer functions are usually created with the
`tf` factory function.
Parameters
----------
num : 2D list of coefficient arrays
Polynomial coefficients of the numerator.
den : 2D list of coefficient arrays
Polynomial coefficients of the denominator.
dt : None, True or float, optional
System timebase. 0 (default) indicates continuous time, True
indicates discrete time with unspecified sampling time, positive
number is discrete time with specified sampling time, None indicates
unspecified timebase (either continuous or discrete time).
Attributes
----------
ninputs, noutputs : int
Number of input and output signals.
shape : tuple
2-tuple of I/O system dimension, (noutputs, ninputs).
input_labels, output_labels : list of str
Names for the input and output signals.
name : string, optional
System name.
num_array, den_array : 2D array of lists of float
Numerator and denominator polynomial coefficients as 2D array
of 1D array objects (of varying length).
num_list, den_list : 2D list of 1D array
Numerator and denominator polynomial coefficients as 2D lists
of 1D array objects (of varying length).
display_format : None, 'poly' or 'zpk'
Display format used in printing the TransferFunction object.
Default behavior is polynomial display and can be changed by
changing `config.defaults['xferfcn.display_format']`.
s : `TransferFunction`
Represents the continuous-time differential operator.
z : `TransferFunction`
Represents the discrete-time delay operator.
See Also
--------
tf, InputOutputSystem, FrequencyResponseData
Notes
-----
The numerator and denominator polynomials are stored as 2D arrays
with each element containing a 1D array of coefficients. These data
structures can be retrieved using `num_array` and `den_array`. For
example,
>>> sys.num_array[2, 5] # doctest: +SKIP
gives the numerator of the transfer function from the 6th input to the
3rd output. (Note: a single 3D array structure cannot be used because
the numerators and denominators can have different numbers of
coefficients in each entry.)
The attributes `num_list` and `den_list` are properties that return
2D nested lists containing MIMO numerator and denominator coefficients.
For example,
>>> sys.num_list[2][5] # doctest: +SKIP
For legacy purposes, this list-based representation can also be
obtained using `num` and `den`.
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 other
system. Similarly, a system with timebase None can be combined with a
system having any timebase; the result will have the timebase of the
other system. The default value of dt can be changed by changing the
value of `config.defaults['control.default_dt']`.
A transfer function is callable and returns the value of the transfer
function evaluated at a point in the complex plane. See
`TransferFunction.__call__` for a more detailed description.
Subsystems corresponding to selected input/output pairs can be
created by indexing the transfer function::
subsys = sys[output_spec, input_spec]
The input and output specifications can be single integers, lists of
integers, or slices. In addition, the strings representing the names
of the signals can be used and will be replaced with the equivalent
signal offsets.
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 = ct.TransferFunction.s # or ct.tf('s')
>>> G = (s + 1)/(s**2 + 2*s + 1)
"""
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 2D arrays of arrays containing polynomial coefficients.
To create a discrete-time transfer function, 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).
See `TransferFunction` and `tf` for more information.
"""
#
# Process positional arguments
#
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
if 'dt' in kwargs:
warn("received multiple dt arguments, "
"using positional arg dt = %s" % dt)
kwargs['dt'] = dt
args = args[:-1]
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 TypeError("Needs 1, 2 or 3 arguments; received %i."
% len(args))
num = _clean_part(num, "numerator")
den = _clean_part(den, "denominator")
#
# Process keyword arguments
#
# During module init, TransferFunction.s and TransferFunction.z
# get initialized when defaults are not fully initialized yet.
# Use 'poly' in these cases.
self.display_format = kwargs.pop('display_format', None)
if self.display_format not in (None, 'poly', 'zpk'):
raise ValueError("display_format must be 'poly' or 'zpk',"
" got '%s'" % self.display_format)
#
# Determine if the transfer function is static (memoryless)
#
# True if and only if all of the numerator and denominator
# polynomials of the (MIMO) transfer function are zeroth order.
#
static = True
for arr in [num, den]:
# Iterate using refs_OK since num and den are ndarrays of ndarrays
for poly_ in np.nditer(arr, flags=['refs_ok']):
if poly_.item().size > 1:
static = False
break
if not static:
break
self._static = static # retain for later usage
defaults = args[0] if len(args) == 1 else \
{'inputs': num.shape[1], 'outputs': num.shape[0]}
name, inputs, outputs, states, dt = _process_iosys_keywords(
kwargs, defaults, static=static)
if states:
raise TypeError(
"states keyword not allowed for transfer functions")
# Initialize LTI (InputOutputSystem) object
super().__init__(
name=name, inputs=inputs, outputs=outputs, dt=dt, **kwargs)
#
# Check to make sure everything is consistent
#
# Make sure numerator and denominator matrices have consistent sizes
if self.ninputs != den.shape[1]:
raise ValueError(
"The numerator has %i input(s), but the denominator has "
"%i input(s)." % (self.ninputs, den.shape[1]))
if self.noutputs != den.shape[0]:
raise ValueError(
"The numerator has %i output(s), but the denominator has "
"%i output(s)." % (self.noutputs, den.shape[0]))
# Additional checks/updates on structure of the transfer function
for i in range(self.noutputs):
# 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(self.ninputs):
# Check that we don't have any zero denominators.
zeroden = True
for k in den[i, j]:
if np.any(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 np.any(k):
zeronum = False
break
if zeronum:
den[i][j] = ones(1)
# Store the numerator and denominator
self.num_array = num
self.den_array = den
#
# Final processing
#
# Truncate leading zeros
self._truncatecoeff()
#
# Class attributes
#
# These attributes are defined as class attributes so that they are
# documented properly. They are "overwritten" in __init__.
#
#: Number of system inputs.
#:
#: :meta hide-value:
ninputs = 1
#: Number of system outputs.
#:
#: :meta hide-value:
noutputs = 1
#: Numerator polynomial coefficients as a 2D array of 1D coefficients.
#:
#: :meta hide-value:
num_array = None
#: Denominator polynomial coefficients as a 2D array of 1D coefficients.
#:
#: :meta hide-value:
den_array = None
# Numerator and denominator as lists of lists of lists
@property
def num_list(self):
"""Numerator polynomial (as 2D nested list of 1D arrays)."""
return self.num_array.tolist()
@property
def den_list(self):
"""Denominator polynomial (as 2D nested lists of 1D arrays)."""
return self.den_array.tolist()
# Legacy versions (TODO: add DeprecationWarning in a later release?)
num, den = num_list, den_list
def __call__(self, x, squeeze=None, warn_infinite=True):
"""Evaluate system transfer function at point in complex plane.
Returns the value of the system's transfer function at a point `x`
in the complex plane, where `x` is `s` for continuous-time systems
and `z` for discrete-time systems.
See `LTI.__call__` for details.
"""
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 value of transfer function using Horner's method.
Evaluates ``sys(x)`` where `x` is a complex number `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
Complex frequency at which the transfer function is evaluated.
warn_infinite : bool, optional
If True (default), generate a warning if `x` is a pole.
Returns
-------
complex
"""
# 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_array[i, j], x_arr) /
polyval(self.den_array[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_array, self.den_array]
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_array, self.den_array] = data
def __str__(self, var=None):
"""String representation of the transfer function.
Based on the display_format property, the output will be formatted as
either polynomials or in zpk form.
"""
display_format = config.defaults['xferfcn.display_format'] if \
self.display_format is None else self.display_format
mimo = not self.issiso()
if var is None:
var = 's' if self.isctime() else 'z'
outstr = f"{InputOutputSystem.__str__(self)}"
for ni in range(self.ninputs):
for no in range(self.noutputs):
outstr += "\n"
if mimo:
outstr += "\nInput %i to output %i:\n" % (ni + 1, no + 1)
# Convert the numerator and denominator polynomials to strings.
if display_format == 'poly':
numstr = _tf_polynomial_to_string(
self.num_array[no, ni], var=var)
denstr = _tf_polynomial_to_string(
self.den_array[no, ni], var=var)
elif display_format == 'zpk':
num = self.num_array[no, ni]
if num.size == 1 and num.item() == 0:
# Catch a special case that SciPy doesn't handle
z, p, k = tf2zpk([1.], self.den_array[no, ni])
k = 0
else:
z, p, k = tf2zpk(
self.num[no][ni], self.den_array[no, ni])
numstr = _tf_factorized_polynomial_to_string(
z, gain=k, var=var)
denstr = _tf_factorized_polynomial_to_string(p, 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
return outstr
def _repr_eval_(self):
# Loadable format
if self.issiso():
out = "TransferFunction(\n{num},\n{den}".format(
num=self.num_array[0, 0].__repr__(),
den=self.den_array[0, 0].__repr__())
else:
out = "TransferFunction(\n["
for entry in [self.num_array, self.den_array]:
for i in range(self.noutputs):
out += "[" if i == 0 else "\n ["
linelen = 0
for j in range(self.ninputs):
out += ", " if j != 0 else ""
numstr = np.array_repr(entry[i, j])
if linelen + len(numstr) > 72:
out += "\n "
linelen = 0
out += numstr
linelen += len(numstr)
out += "]," if i < self.noutputs - 1 else "]"
out += "],\n[" if entry is self.num_array else "]"
out += super()._dt_repr(separator=",\n", space="")
if len(labels := self._label_repr()) > 0:
out += ",\n" + labels
out += ")"
return out
def _repr_html_(self, var=None):
"""HTML/LaTeX representation of xferfcn, for Jupyter notebook."""
display_format = config.defaults['xferfcn.display_format'] if \
self.display_format is None else self.display_format
mimo = not self.issiso()
if var is None:
var = 's' if self.isctime() else 'z'
out = [super()._repr_info_(html=True), '\n$$']
if mimo:
out.append(r"\begin{bmatrix}")
for no in range(self.noutputs):
for ni in range(self.ninputs):
# Convert the numerator and denominator polynomials to strings.
if display_format == 'poly':
numstr = _tf_polynomial_to_string(
self.num_array[no, ni], var=var)
denstr = _tf_polynomial_to_string(
self.den_array[no, ni], var=var)
elif display_format == 'zpk':
z, p, k = tf2zpk(
self.num_array[no, ni], self.den_array[no, ni])
numstr = _tf_factorized_polynomial_to_string(
z, gain=k, var=var)
denstr = _tf_factorized_polynomial_to_string(p, var=var)
numstr = _tf_string_to_latex(numstr, var=var)
denstr = _tf_string_to_latex(denstr, var=var)
out += [r"\dfrac{", numstr, "}{", denstr, "}"]
if mimo and ni < self.ninputs - 1:
out.append("&")
if mimo:
out.append(r"\\")
if mimo:
out.append(r" \end{bmatrix}")
out.append("$$")
return ''.join(out)
def __neg__(self):
"""Negate a transfer function."""
num = deepcopy(self.num_array)
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 isinstance(other, (int, float, complex, np.number, np.ndarray)):
other = _convert_to_transfer_function(other, inputs=self.ninputs,
outputs=self.noutputs)
if not isinstance(other, TransferFunction):
return NotImplemented
# Promote SISO object to compatible dimension
if self.issiso() and not other.issiso():
self = np.ones((other.noutputs, other.ninputs)) * self
elif not self.issiso() and other.issiso():
other = np.ones((self.noutputs, self.ninputs)) * other
# 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 = _create_poly_array((self.noutputs, self.ninputs))
den = _create_poly_array((self.noutputs, self.ninputs))
for i in range(self.noutputs):
for j in range(self.ninputs):
num[i, j], den[i, j] = _add_siso(
self.num_array[i, j], self.den_array[i, j],
other.num_array[i, j], other.den_array[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)."""
from .statesp import StateSpace
# Convert the second argument to a transfer function.
if isinstance(other, (StateSpace, np.ndarray)):
other = _convert_to_transfer_function(other)
elif isinstance(other, (int, float, complex, np.number)):
# Multiply by a scaled identity matrix (transfer function)
other = _convert_to_transfer_function(np.eye(self.ninputs) * other)
if not isinstance(other, TransferFunction):
return NotImplemented
# Promote SISO object to compatible dimension
if self.issiso() and not other.issiso():
self = bdalg.append(*([self] * other.noutputs))
elif not self.issiso() and other.issiso():
other = bdalg.append(*([other] * self.ninputs))
# 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))
ninputs = other.ninputs
noutputs = self.noutputs
dt = common_timebase(self.dt, other.dt)
# Preallocate the numerator and denominator of the sum.
num = _create_poly_array((noutputs, ninputs), [0])
den = _create_poly_array((noutputs, ninputs), [1])
# 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(noutputs):
for col in range(ninputs):
for k in range(self.ninputs):
num_summand[k] = polymul(
self.num_array[row, k], other.num_array[k, col])
den_summand[k] = polymul(
self.den_array[row, k], other.den_array[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)):
# Multiply by a scaled identity matrix (transfer function)
other = _convert_to_transfer_function(np.eye(self.noutputs) * other)
else:
other = _convert_to_transfer_function(other)
# Promote SISO object to compatible dimension
if self.issiso() and not other.issiso():
self = bdalg.append(*([self] * other.ninputs))
elif not self.issiso() and other.issiso():
other = bdalg.append(*([other] * self.noutputs))
# 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))
ninputs = self.ninputs
noutputs = other.noutputs
dt = common_timebase(self.dt, other.dt)
# Preallocate the numerator and denominator of the sum.
num = _create_poly_array((noutputs, ninputs), [0])
den = _create_poly_array((noutputs, ninputs), [1])
# 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(noutputs): # Iterate through rows of product.
for j in range(ninputs): # Iterate through columns of product.
for k in range(other.ninputs): # Multiply & add.
num_summand[k] = polymul(
other.num_array[i, k], self.num_array[k, j])
den_summand[k] = polymul(
other.den_array[i, k], self.den_array[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)):
# Multiply by a scaled identity matrix (transfer function)
other = _convert_to_transfer_function(np.eye(self.ninputs) * other)
else:
other = _convert_to_transfer_function(other)
# Special case for SISO ``other``
if not self.issiso() and other.issiso():
other = bdalg.append(*([other**-1] * self.noutputs))
return self * other
if (self.ninputs > 1 or self.noutputs > 1 or
other.ninputs > 1 or other.noutputs > 1):
# TransferFunction.__truediv__ is currently implemented only for
# SISO systems.
return NotImplemented
dt = common_timebase(self.dt, other.dt)
num = polymul(self.num_array[0, 0], other.den_array[0, 0])
den = polymul(self.den_array[0, 0], other.num_array[0, 0])
return TransferFunction(num, den, dt)
# 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)
# Special case for SISO ``self``
if self.issiso() and not other.issiso():
self = bdalg.append(*([self**-1] * other.ninputs))
return other * self
if (self.ninputs > 1 or self.noutputs > 1 or
other.ninputs > 1 or other.noutputs > 1):
# TransferFunction.__rtruediv__ is currently implemented only for
# SISO systems
return NotImplemented
return other / self
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):
if not isinstance(key, Iterable) or len(key) != 2:
raise IOError(
"must provide indices of length 2 for transfer functions")
# Convert signal names to integer offsets (via NamedSignal object)
iomap = NamedSignal(
np.empty((self.noutputs, self.ninputs)),
self.output_labels, self.input_labels)
indices = iomap._parse_key(key, level=1) # ignore index checks
outdx, outputs = _process_subsys_index(
indices[0], self.output_labels, slice_to_list=True)
inpdx, inputs = _process_subsys_index(
indices[1], self.input_labels, slice_to_list=True)
# Construct the transfer function for the subsystem
num = _create_poly_array((len(outputs), len(inputs)))
den = _create_poly_array(num.shape)
for row, i in enumerate(outdx):
for col, j in enumerate(inpdx):
num[row, col] = self.num_array[i, j]
den[row, col] = self.den_array[i, j]
col += 1
row += 1
# Create the system name
sysname = config.defaults['iosys.indexed_system_name_prefix'] + \
self.name + config.defaults['iosys.indexed_system_name_suffix']
return TransferFunction(
num, den, self.dt, inputs=inputs, outputs=outputs, name=sysname)
def freqresp(self, omega):
"""Evaluate transfer function at complex frequencies.
.. deprecated::0.9.0
Method has been given the more Pythonic name
`TransferFunction.frequency_response`. Or use
`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", FutureWarning)
return self.frequency_response(omega)
def poles(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).astype(complex)
def zeros(self):
"""Compute the zeros of a transfer function."""
if self.ninputs > 1 or self.noutputs > 1:
raise NotImplementedError(
"TransferFunction.zeros is currently only implemented "
"for SISO systems.")
else:
# for now, just give zeros of a SISO tf
return roots(self.num_array[0, 0]).astype(complex)
def feedback(self, other=1, sign=-1):
"""Feedback interconnection between two LTI objects.
Parameters
----------
other : `InputOutputSystem`
System in the feedback path.
sign : float, optional
Gain to use in feedback path. Defaults to -1.
"""
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 ControlMIMONotImplemented(
"TransferFunction.feedback is currently not implemented for "
"MIMO systems.")
dt = common_timebase(self.dt, other.dt)
num1 = self.num_array[0, 0]
den1 = self.den_array[0, 0]
num2 = other.num_array[0, 0]
den2 = other.den_array[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 append(self, other):
"""Append a second model to the present model.
The second model is converted to a transfer function if necessary,
inputs and outputs are appended and their order is preserved.
Parameters
----------
other : `StateSpace` or `TransferFunction`
System to be appended.
Returns
-------
sys : `TransferFunction`
System model with `other` appended to `self`.
"""
other = _convert_to_transfer_function(other)
new_tf = bdalg.combine_tf([
[self, np.zeros((self.noutputs, other.ninputs))],
[np.zeros((other.noutputs, self.ninputs)), other],
])
return new_tf
def minreal(self, tol=None):
"""Remove canceling pole/zero pairs from a transfer function.
Parameters
----------
tol : float
Tolerance for determining whether poles and zeros overlap.
"""
# based on octave minreal
# default accuracy
from sys import float_info
sqrt_eps = sqrt(float_info.epsilon)
# pre-allocate arrays
num = _create_poly_array((self.noutputs, self.ninputs))
den = _create_poly_array((self.noutputs, self.ninputs))
for i in range(self.noutputs):
for j in range(self.ninputs):
# split up in zeros, poles and gain
newzeros = []
zeros = roots(self.num_array[i, j])
poles = roots(self.den_array[i, j])
gain = self.num_array[i, j][0] / self.den_array[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 2D array of `scipy.signal.lti` objects.
For instance,
>>> out = tfobject.returnScipySignalLTI() # doctest: +SKIP
>>> out[3, 5] # doctest: +SKIP
is a `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
`scipy.signal.lti` objects are returned
Returns
-------
out : list of list of `scipy.signal.TransferFunction`
Continuous time (inheriting from `scipy.signal.lti`)
or discrete time (inheriting from `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):