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# 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 names of the California Institute of Technology nor

# the Delft University 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: M.M. (Rene) van Paassen (using xferfcn.py as basis)

# Date: 02 Oct 12

"""

Frequency response data representation and functions.

This module contains the FRD class and also functions that operate on

FRD data.

"""

# External function declarations

from copy import copy

from warnings import warn

import numpy as np

from numpy import angle, array, empty, ones, \

real, imag, absolute, eye, linalg, where, sort

from scipy.interpolate import splprep, splev

from .lti import LTI, _process_frequency_response

from .exception import pandas_check

from .namedio import NamedIOSystem, _process_namedio_keywords

from . import config

__all__ = ['FrequencyResponseData', 'FRD', 'frd']

class FrequencyResponseData(LTI):

"""FrequencyResponseData(d, w[, smooth])

A class for models defined by frequency response data (FRD).

The FrequencyResponseData (FRD) class is used to represent systems in

frequency response data form.

Parameters

----------

d : 1D or 3D complex array_like

The frequency response at each frequency point. If 1D, the system is

assumed to be SISO. If 3D, the system is MIMO, 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

w : iterable of real frequencies

List of frequency points for which data are available.

smooth : bool, optional

If ``True``, create an interpolation function that allows the

frequency response to be computed at any frequency within the range of

frequencies give in ``w``. If ``False`` (default), frequency response

can only be obtained at the frequencies specified in ``w``.

Attributes

----------

ninputs, noutputs : int

Number of input and output variables.

omega : 1D array

Frequency points of the response.

fresp : 3D array

Frequency response, indexed by output index, input index, and

frequency point.

Notes

-----

The main data members are 'omega' and 'fresp', where 'omega' is a 1D array

of frequency points and and 'fresp' is a 3D array of frequency responses,

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.

A frequency response data object is callable and returns the value of the

transfer function evaluated at a point in the complex plane (must be on

the imaginary access). See :meth:`~control.FrequencyResponseData.__call__`

for a more detailed description.

"""

# Allow NDarray * StateSpace to give StateSpace._rmul_() priority

# https://docs.scipy.org/doc/numpy/reference/arrays.classes.html

__array_priority__ = 13 # override ndarray, StateSpace, I/O sys

# 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

_epsw = 1e-8 #: Bound for exact frequency match

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).

"""

# TODO: discrete-time FRD systems?

smooth = kwargs.pop('smooth', False)

# Process positional arguments

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 = sort(np.asarray(args[1], dtype=float))

# calculate frequency response at my points

if otherlti.isctime():

s = 1j * self.omega

self.fresp = otherlti(s, squeeze=False)

else:

z = np.exp(1j * self.omega * otherlti.dt)

self.fresp = otherlti(z, squeeze=False)

else:

# The user provided a response and a freq vector

self.fresp = array(args[0], dtype=complex, ndmin=1)

if self.fresp.ndim == 1:

self.fresp = self.fresp.reshape(1, 1, -1)

self.omega = array(args[1], dtype=float, ndmin=1)

if self.fresp.ndim != 3 or self.omega.ndim != 1 or \

self.fresp.shape[-1] != 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; received %i." % len(args))

# Process key word arguments

# Keep track of return type

self.return_magphase=kwargs.pop('return_magphase', False)

if self.return_magphase not in (True, False):

raise ValueError("unknown return_magphase value")

# Determine whether to squeeze the output

self.squeeze=kwargs.pop('squeeze', None)

if self.squeeze not in (None, True, False):

raise ValueError("unknown squeeze value")

# Process namedio keywords

defaults = {

'inputs': self.fresp.shape[1], 'outputs': self.fresp.shape[0]}

name, inputs, outputs, states, dt = _process_namedio_keywords(

kwargs, defaults, end=True)

# Process signal names

NamedIOSystem.__init__(

self, name=name, inputs=inputs, outputs=outputs, dt=dt)

# 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

# Frequency response properties

# Different properties of the frequency response that can be used for

# analysis and characterization.

@property

def magnitude(self):

return np.abs(self.fresp)

@property

def phase(self):

return np.angle(self.fresp)

@property

def frequency(self):

return self.omega

@property

def response(self):

return self.fresp

def __str__(self):

"""String representation of the transfer function."""

mimo = self.ninputs > 1 or self.noutputs > 1

outstr = ['Frequency response data']

for i in range(self.ninputs):

for j in range(self.noutputs):

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, re, im)

for w, re, im in zip(self.omega,

real(self.fresp[j, i, :]),

imag(self.fresp[j, i, :]))])

return '\n'.join(outstr)

def __repr__(self):

"""Loadable string representation,

limited for number of data points.

"""

return "FrequencyResponseData({d}, {w}{smooth})".format(

d=repr(self.fresp), w=repr(self.omega),

smooth=(self.ifunc and ", smooth=True") or "")

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 = _convert_to_FRD(other, omega=self.omega)

# 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))

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 = _convert_to_FRD(other, omega=self.omega)

# Check that the input-output sizes are consistent.

if self.ninputs != other.noutputs:

raise ValueError(

"H = G1*G2: input-output size mismatch: "

"G1 has %i input(s), G2 has %i output(s)." %

(self.ninputs, other.noutputs))

inputs = other.ninputs

outputs = self.noutputs

fresp = empty((outputs, inputs, len(self.omega)),

dtype=self.fresp.dtype)

for i in range(len(self.omega)):

fresp[:, :, i] = 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 = _convert_to_FRD(other, omega=self.omega)

# Check that the input-output sizes are consistent.

if self.noutputs != other.ninputs:

raise ValueError(

"H = G1*G2: input-output size mismatch: "

"G1 has %i input(s), G2 has %i output(s)." %

(other.ninputs, self.noutputs))

inputs = self.ninputs

outputs = other.noutputs

fresp = empty((outputs, inputs, len(self.omega)),

dtype=self.fresp.dtype)

for i in range(len(self.omega)):

fresp[:, :, i] = 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 = _convert_to_FRD(other, omega=self.omega)

if (self.ninputs > 1 or self.noutputs > 1 or

other.ninputs > 1 or other.noutputs > 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: 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 = _convert_to_FRD(other, omega=self.omega)

if (self.ninputs > 1 or self.noutputs > 1 or

other.ninputs > 1 or other.noutputs > 1):

raise NotImplementedError(

"FRD.__rtruediv__ is currently only implemented for "

"SISO systems.")

return other / self

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))

# 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.

# update Sawyer B. Fuller 2020.08.14: __call__ added to provide a uniform

# interface to systems in general and the lti.frequency_response method

def eval(self, omega, squeeze=None):

"""Evaluate a transfer function at angular 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.

Parameters

----------

omega : float or 1D array_like

Frequencies in radians per second

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'].

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.

"""

omega_array = np.array(omega, ndmin=1) # array-like version of omega

# Make sure that we are operating on a simple list

if len(omega_array.shape) > 1:

raise ValueError("input list must be 1D")

# Make sure that frequencies are all real-valued

if any(omega_array.imag > 0):

raise ValueError("FRD.eval can only accept real-valued omega")

if self.ifunc is None:

elements = np.isin(self.omega, omega) # binary array

if sum(elements) < len(omega_array):

raise ValueError(

"not all frequencies omega are in frequency list of FRD "

"system. Try an interpolating FRD for additional points.")

else:

out = self.fresp[:, :, elements]

else:

out = empty((self.noutputs, self.ninputs, len(omega_array)),

dtype=complex)

for i in range(self.noutputs):

for j in range(self.ninputs):

for k, w in enumerate(omega_array):

frraw = splev(w, self.ifunc[i, j], der=0)

out[i, j, k] = frraw[0] + 1.0j * frraw[1]

return _process_frequency_response(self, omega, out, squeeze=squeeze)

def __call__(self, s=None, squeeze=None, return_magphase=None):

"""Evaluate system's transfer function at complex frequencies.

Returns the complex frequency response `sys(s)` of system `sys` with

`m = sys.ninputs` number of inputs and `p = sys.noutputs` number of

outputs.

To evaluate at a frequency omega in radians per second, enter

``s = omega * 1j`` or use ``sys.eval(omega)``

For a frequency response data object, the argument must be an

imaginary number (since only the frequency response is defined).

If ``s`` is not given, this function creates a copy of a frequency

response data object with a different set of output settings.

Parameters

----------

s : complex scalar or 1D array_like

Complex frequencies. If not specified, return a copy of the

frequency response data object with updated settings for output

processing (``squeeze``, ``return_magphase``).

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'].

return_magphase : bool, optional

If True, then a frequency response data object will enumerate as a

tuple of the form (mag, phase, omega) where where ``mag`` is the

magnitude (absolute value, not dB or log10) of the system

frequency response, ``phase`` is the wrapped phase in radians of

the system frequency response, and ``omega`` is the (sorted)

frequencies at which the response was evaluated.

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.

Raises

------

ValueError

If `s` is not purely imaginary, because

:class:`FrequencyDomainData` systems are only defined at imaginary

frequency values.

"""

if s is None:

# Create a copy of the response with new keywords

response = copy(self)

# Update any keywords that we were passed

response.squeeze = self.squeeze if squeeze is None else squeeze

response.return_magphase = self.return_magphase \

if return_magphase is None else return_magphase

return response

# Make sure that we are operating on a simple list

if len(np.atleast_1d(s).shape) > 1:

raise ValueError("input list must be 1D")

if any(abs(np.atleast_1d(s).real) > 0):

raise ValueError("__call__: FRD systems can only accept "

"purely imaginary frequencies")

# need to preserve array or scalar status

if hasattr(s, '__len__'):

return self.eval(np.asarray(s).imag, squeeze=squeeze)

else:

return self.eval(complex(s).imag, squeeze=squeeze)

# Implement iter to allow assigning to a tuple

def __iter__(self):

fresp = _process_frequency_response(

self, self.omega, self.fresp, squeeze=self.squeeze)

if not self.return_magphase:

return iter((self.omega, fresp))

return iter((np.abs(fresp), np.angle(fresp), self.omega))

# Implement (thin) getitem to allow access via legacy indexing

def __getitem__(self, index):

return list(self.__iter__())[index]

# Implement (thin) len to emulate legacy testing interface

def __len__(self):

return 3 if self.return_magphase else 2

def freqresp(self, omega):

"""(deprecated) Evaluate transfer function at complex frequencies.

.. deprecated::0.9.0

Method has been given the more pythonic name

:meth:`FrequencyResponseData.frequency_response`. Or use

:func:`freqresp` in the MATLAB compatibility module.

"""

warn("FrequencyResponseData.freqresp(omega) will be removed in a "

"future release of python-control; use "

"FrequencyResponseData.frequency_response(omega), or "

"freqresp(sys, omega) in the MATLAB compatibility module "

"instead", DeprecationWarning)

return self.frequency_response(omega)

def feedback(self, other=1, sign=-1):

"""Feedback interconnection between two FRD objects."""

other = _convert_to_FRD(other, omega=self.omega)

if (self.noutputs != other.ninputs or self.ninputs != other.noutputs):

raise ValueError(

"FRD.feedback, inputs/outputs mismatch")

# TODO: handle omega re-mapping

# reorder array axes in order to leverage numpy broadcasting

myfresp = np.moveaxis(self.fresp, 2, 0)

otherfresp = np.moveaxis(other.fresp, 2, 0)

I_AB = eye(self.ninputs)[np.newaxis, :, :] + otherfresp @ myfresp

resfresp = (myfresp @ linalg.inv(I_AB))

fresp = np.moveaxis(resfresp, 0, 2)

return FRD(fresp, other.omega, smooth=(self.ifunc is not None))

# Convert to pandas

def to_pandas(self):

if not pandas_check():

ImportError('pandas not installed')

import pandas

# Create a dict for setting up the data frame

data = {'omega': self.omega}

data.update(

{'H_{%s, %s}' % (out, inp): self.fresp[i, j] \

for i, out in enumerate(self.output_labels) \

for j, inp in enumerate(self.input_labels)})

return pandas.DataFrame(data)

# 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 _convert_to_FRD(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 = _convert_to_FRD(3., omega) # Assumes inputs = outputs = 1

>>> frd = _convert_to_FRD(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 = np.sort(omega)

if sys.isctime():

fresp = sys(1j * omega)

else:

fresp = sys(np.exp(1j * omega * sys.dt))

if len(fresp.shape) == 1:

fresp = fresp[np.newaxis, np.newaxis, :]

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

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