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# Author: M.M. (Rene) van Paassen (using xferfcn.py as basis)
# Date: 02 Oct 12
from __future__ import division
"""
Frequency response data representation and functions.
This module contains the FRD class and also functions that operate on
FRD data.
"""
# External function declarations
from warnings import warn
import numpy as np
from numpy import angle, array, empty, ones, \
real, imag, absolute, eye, linalg, where, dot
from scipy.interpolate import splprep, splev
from .lti import LTI
__all__ = ['FrequencyResponseData', 'FRD', 'frd']
class FrequencyResponseData(LTI):
"""FrequencyResponseData(d, w)
A class for models defined by frequency response data (FRD)
The FrequencyResponseData (FRD) class is used to represent systems in
frequency response data form.
The main data members are 'omega' and 'fresp', where `omega` is a 1D array
with the frequency points of the response, and `fresp` is a 3D array, with
the first dimension corresponding to the output index of the FRD, the
second dimension corresponding to the input index, and the 3rd dimension
corresponding to the frequency points in omega. For example,
>>> frdata[2,5,:] = numpy.array([1., 0.8-0.2j, 0.2-0.8j])
means that the frequency response from the 6th input to the 3rd
output at the frequencies defined in omega is set to the array
above, i.e. the rows represent the outputs and the columns
represent the inputs.
"""
# Allow NDarray * StateSpace to give StateSpace._rmul_() priority
# https://docs.scipy.org/doc/numpy/reference/arrays.classes.html
__array_priority__ = 11 # override ndarray and matrix types
epsw = 1e-8
def __init__(self, *args, **kwargs):
"""Construct an FRD object.
The default constructor is FRD(d, w), where w is an iterable of
frequency points, and d is the matching frequency data.
If d is a single list, 1d array, or tuple, a SISO system description
is assumed. d can also be
To call the copy constructor, call FRD(sys), where sys is a
FRD object.
To construct frequency response data for an existing LTI
object, other than an FRD, call FRD(sys, omega)
"""
smooth = kwargs.get('smooth', False)
if len(args) == 2:
if not isinstance(args[0], FRD) and isinstance(args[0], LTI):
# not an FRD, but still a system, second argument should be
# the frequency range
otherlti = args[0]
self.omega = array(args[1], dtype=float)
self.omega.sort()
numfreq = len(self.omega)
# calculate frequency response at my points
self.fresp = empty(
(otherlti.outputs, otherlti.inputs, numfreq),
dtype=complex)
for k, w in enumerate(self.omega):
self.fresp[:, :, k] = otherlti._evalfr(w)
else:
# The user provided a response and a freq vector
self.fresp = array(args[0], dtype=complex)
if len(self.fresp.shape) == 1:
self.fresp = self.fresp.reshape(1, 1, len(args[0]))
self.omega = array(args[1], dtype=float)
if len(self.fresp.shape) != 3 or \
self.fresp.shape[-1] != self.omega.shape[-1] or \
len(self.omega.shape) != 1:
raise TypeError(
"The frequency data constructor needs a 1-d or 3-d"
" response data array and a matching frequency vector"
" size")
elif len(args) == 1:
# Use the copy constructor.
if not isinstance(args[0], FRD):
raise TypeError(
"The one-argument constructor can only take in"
" an FRD object. Received %s." % type(args[0]))
self.omega = args[0].omega
self.fresp = args[0].fresp
else:
raise ValueError("Needs 1 or 2 arguments; receivd %i." % len(args))
# create interpolation functions
if smooth:
self.ifunc = empty((self.fresp.shape[0], self.fresp.shape[1]),
dtype=tuple)
for i in range(self.fresp.shape[0]):
for j in range(self.fresp.shape[1]):
self.ifunc[i, j], u = splprep(
u=self.omega, x=[real(self.fresp[i, j, :]),
imag(self.fresp[i, j, :])],
w=1.0/(absolute(self.fresp[i, j, :]) + 0.001), s=0.0)
else:
self.ifunc = None
LTI.__init__(self, self.fresp.shape[1], self.fresp.shape[0])
def __str__(self):
"""String representation of the transfer function."""
mimo = self.inputs > 1 or self.outputs > 1
outstr = ['frequency response data ']
mt, pt, wt = self.freqresp(self.omega)
for i in range(self.inputs):
for j in range(self.outputs):
if mimo:
outstr.append("Input %i to output %i:" % (i + 1, j + 1))
outstr.append('Freq [rad/s] Response ')
outstr.append('------------ ---------------------')
outstr.extend(
['%12.3f %10.4g%+10.4gj' % (w, m, p)
for m, p, w in zip(real(self.fresp[j, i, :]),
imag(self.fresp[j, i, :]), wt)])
return '\n'.join(outstr)
def __neg__(self):
"""Negate a transfer function."""
return FRD(-self.fresp, self.omega)
def __add__(self, other):
"""Add two LTI objects (parallel connection)."""
if isinstance(other, FRD):
# verify that the frequencies match
if len(other.omega) != len(self.omega) or \
(other.omega != self.omega).any():
warn("Frequency points do not match; expect "
"truncation and interpolation.")
# Convert the second argument to a frequency response function.
# or re-base the frd to the current omega (if needed)
other = _convertToFRD(other, omega=self.omega)
# 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))
return FRD(self.fresp + other.fresp, other.omega)
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)):
return FRD(self.fresp * other, self.omega,
smooth=(self.ifunc is not None))
else:
other = _convertToFRD(other, omega=self.omega)
# Check that the input-output sizes are consistent.
if self.inputs != other.outputs:
raise ValueError(
"H = G1*G2: input-output size mismatch: "
"G1 has %i input(s), G2 has %i output(s)." %
(self.inputs, other.outputs))
inputs = other.inputs
outputs = self.outputs
fresp = empty((outputs, inputs, len(self.omega)),
dtype=self.fresp.dtype)
for i in range(len(self.omega)):
fresp[:, :, i] = dot(self.fresp[:, :, i], other.fresp[:, :, i])
return FRD(fresp, self.omega,
smooth=(self.ifunc is not None) and
(other.ifunc is not None))
def __rmul__(self, other):
"""Right Multiply two LTI objects (serial connection)."""
# Convert the second argument to an frd function.
if isinstance(other, (int, float, complex, np.number)):
return FRD(self.fresp * other, self.omega,
smooth=(self.ifunc is not None))
else:
other = _convertToFRD(other, omega=self.omega)
# Check that the input-output sizes are consistent.
if self.outputs != other.inputs:
raise ValueError(
"H = G1*G2: input-output size mismatch: "
"G1 has %i input(s), G2 has %i output(s)." %
(other.inputs, self.outputs))
inputs = self.inputs
outputs = other.outputs
fresp = empty((outputs, inputs, len(self.omega)),
dtype=self.fresp.dtype)
for i in range(len(self.omega)):
fresp[:, :, i] = dot(other.fresp[:, :, i], self.fresp[:, :, i])
return FRD(fresp, self.omega,
smooth=(self.ifunc is not None) and
(other.ifunc is not None))
# 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)):
return FRD(self.fresp * (1/other), self.omega,
smooth=(self.ifunc is not None))
else:
other = _convertToFRD(other, omega=self.omega)
if (self.inputs > 1 or self.outputs > 1 or
other.inputs > 1 or other.outputs > 1):
raise NotImplementedError(
"FRD.__truediv__ is currently only implemented for SISO "
"systems.")
return FRD(self.fresp/other.fresp, self.omega,
smooth=(self.ifunc is not None) and
(other.ifunc is not None))
# TODO: Remove when transition to python3 complete
def __div__(self, other):
return self.__truediv__(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)):
return FRD(other / self.fresp, self.omega,
smooth=(self.ifunc is not None))
else:
other = _convertToFRD(other, omega=self.omega)
if (self.inputs > 1 or self.outputs > 1 or
other.inputs > 1 or other.outputs > 1):
raise NotImplementedError(
"FRD.__rtruediv__ is currently only implemented for "
"SISO systems.")
return other / self
# TODO: Remove when transition to python3 complete
def __rdiv__(self, other):
return self.__rtruediv__(other)
def __pow__(self, other):
if not type(other) == int:
raise ValueError("Exponent must be an integer")
if other == 0:
return FRD(ones(self.fresp.shape), self.omega,
smooth=(self.ifunc is not None)) # unity
if other > 0:
return self * (self**(other-1))
if other < 0:
return (FRD(ones(self.fresp.shape), self.omega) / 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 frequency response
at frequency omega.
Note that a "normal" FRD only returns values for which there is an
entry in the omega vector. An interpolating FRD can return
intermediate values.
"""
warn("FRD.evalfr(omega) will be deprecated in a future release "
"of python-control; use sys.eval(omega) instead",
PendingDeprecationWarning) # pragma: no coverage
return self._evalfr(omega)
# Define the `eval` function to evaluate an FRD at a given (real)
# frequency. Note that we choose to use `eval` instead of `evalfr` to
# avoid confusion with :func:`evalfr`, which takes a complex number as its
# argument. Similarly, we don't use `__call__` to avoid confusion between
# G(s) for a transfer function and G(omega) for an FRD object.
def eval(self, omega):
"""Evaluate a transfer function at a single angular frequency.
self.evalfr(omega) returns the value of the frequency response
at frequency omega.
Note that a "normal" FRD only returns values for which there is an
entry in the omega vector. An interpolating FRD can return
intermediate values.
"""
return self._evalfr(omega)
# Internal function to evaluate the frequency responses
def _evalfr(self, omega):
"""Evaluate a transfer function at a single angular frequency."""
# Preallocate the output.
if getattr(omega, '__iter__', False):
out = empty((self.outputs, self.inputs, len(omega)), dtype=complex)
else:
out = empty((self.outputs, self.inputs), dtype=complex)
if self.ifunc is None:
try:
out = self.fresp[:, :, where(self.omega == omega)[0][0]]
except Exception:
raise ValueError(
"Frequency %f not in frequency list, try an interpolating"
" FRD if you want additional points" % omega)
else:
if getattr(omega, '__iter__', False):
for i in range(self.outputs):
for j in range(self.inputs):
for k, w in enumerate(omega):
frraw = splev(w, self.ifunc[i, j], der=0)
out[i, j, k] = frraw[0] + 1.0j * frraw[1]
else:
for i in range(self.outputs):
for j in range(self.inputs):
frraw = splev(omega, self.ifunc[i, j], der=0)
out[i, j] = frraw[0] + 1.0j * frraw[1]
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))
omega.sort()
for k, w in enumerate(omega):
fresp = self._evalfr(w)
mag[:, :, k] = abs(fresp)
phase[:, :, k] = angle(fresp)
return mag, phase, omega
def feedback(self, other=1, sign=-1):
"""Feedback interconnection between two FRD objects."""
other = _convertToFRD(other, omega=self.omega)
if (self.outputs != other.inputs or self.inputs != other.outputs):
raise ValueError(
"FRD.feedback, inputs/outputs mismatch")
fresp = empty((self.outputs, self.inputs, len(other.omega)),
dtype=complex)
# TODO: vectorize this
# TODO: handle omega re-mapping
# TODO: is there a reason to use linalg.solve instead of linalg.inv?
# https://github.com/python-control/python-control/pull/314#discussion_r294075154
for k, w in enumerate(other.omega):
fresp[:, :, k] = np.dot(
self.fresp[:, :, k],
linalg.solve(
eye(self.inputs)
+ np.dot(other.fresp[:, :, k], self.fresp[:, :, k]),
eye(self.inputs))
)
return FRD(fresp, other.omega, smooth=(self.ifunc is not None))
#
# Allow FRD as an alias for the FrequencyResponseData class
#
# Note: This class was initially given the name "FRD", but this caused
# problems with documentation on MacOS platforms, since files were generated
# for control.frd and control.FRD, which are not differentiated on most MacOS
# filesystems, which are case insensitive. Renaming the FRD class to be
# FrequenceResponseData and then assigning FRD to point to the same object
# fixes this problem.
#
FRD = FrequencyResponseData
def _convertToFRD(sys, omega, inputs=1, outputs=1):
"""Convert a system to frequency response data form (if needed).
If sys is already an frd, and its frequency range matches or
overlaps the range given in omega then it is returned. If sys is
another LTI object or a transfer function, then it is converted to
a frequency response data at the specified omega. If sys is a
scalar, then the number of inputs and outputs can be specified
manually, as in:
>>> frd = _convertToFRD(3., omega) # Assumes inputs = outputs = 1
>>> frd = _convertToFRD(1., omegs, inputs=3, outputs=2)
In the latter example, sys's matrix transfer function is [[1., 1., 1.]
[1., 1., 1.]].
"""
if isinstance(sys, FRD):
omega.sort()
if len(omega) == len(sys.omega) and \
(abs(omega - sys.omega) < FRD.epsw).all():
# frequencies match, and system was already frd; simply use
return sys
raise NotImplementedError(
"Frequency ranges of FRD do not match, conversion not implemented")
elif isinstance(sys, LTI):
omega.sort()
fresp = empty((sys.outputs, sys.inputs, len(omega)), dtype=complex)
for k, w in enumerate(omega):
fresp[:, :, k] = sys._evalfr(w)
return FRD(fresp, omega, smooth=True)
elif isinstance(sys, (int, float, complex, np.number)):
fresp = ones((outputs, inputs, len(omega)), dtype=float)*sys
return FRD(fresp, omega, smooth=True)
# try converting constant matrices
try:
sys = array(sys)
outputs, inputs = sys.shape
fresp = empty((outputs, inputs, len(omega)), dtype=float)
for i in range(outputs):
for j in range(inputs):
fresp[i, j, :] = sys[i, j]
return FRD(fresp, omega, smooth=True)
except Exception:
pass
raise TypeError('''Can't convert given type "%s" to FRD system.''' %
sys.__class__)
def frd(*args):
"""frd(d, w)
Construct a frequency response data model
frd models store the (measured) frequency response of a system.
This function can be called in different ways:
``frd(response, freqs)``
Create an frd model with the given response data, in the form of
complex response vector, at matching frequency freqs [in rad/s]
``frd(sys, freqs)``
Convert an LTI system into an frd model with data at frequencies
freqs.
Parameters
----------
response: array_like, or list
complex vector with the system response
freq: array_lik or lis
vector with frequencies
sys: LTI (StateSpace or TransferFunction)
A linear system
Returns
-------
sys: FRD
New frequency response system
See Also
--------
FRD, ss, tf
"""
return FRD(*args)
