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"""statesp.py
State space representation and functions.
This file contains the StateSpace class, which is used to represent linear
systems in state space. This is the primary representation for the
python-control library.
"""
"""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
$Id$
"""
import math
import numpy as np
from numpy import any, asarray, concatenate, cos, delete, \
empty, exp, eye, isinf, ones, pad, sin, zeros, squeeze
from numpy.random import rand, randn
from numpy.linalg import solve, eigvals, matrix_rank
from numpy.linalg.linalg import LinAlgError
import scipy as sp
import scipy.linalg
from scipy.signal import cont2discrete
from scipy.signal import StateSpace as signalStateSpace
from warnings import warn
from .exception import ControlSlycot
from .frdata import FrequencyResponseData
from .lti import LTI, _process_frequency_response
from .namedio import common_timebase, isdtime, _process_namedio_keywords, \
_process_dt_keyword, NamedIOSystem
from . import config
from copy import deepcopy
try:
from slycot import ab13dd
except ImportError:
ab13dd = None
__all__ = ['StateSpace', 'tf2ss', 'ssdata', 'linfnorm']
# Define module default parameter values
_statesp_defaults = {
'statesp.use_numpy_matrix': False, # False is default in 0.9.0 and above
'statesp.remove_useless_states': False,
'statesp.latex_num_format': '.3g',
'statesp.latex_repr_type': 'partitioned',
'statesp.latex_maxsize': 10,
}
def _ssmatrix(data, axis=1):
"""Convert argument to a (possibly empty) 2D state space matrix.
The axis keyword argument makes it convenient to specify that if the input
is a vector, it is a row (axis=1) or column (axis=0) vector.
Parameters
----------
data : array, list, or string
Input data defining the contents of the 2D array
axis : 0 or 1
If input data is 1D, which axis to use for return object. The default
is 1, corresponding to a row matrix.
Returns
-------
arr : 2D array, with shape (0, 0) if a is empty
"""
# Convert the data into an array or matrix, as configured
# If data is passed as a string, use (deprecated?) matrix constructor
if config.defaults['statesp.use_numpy_matrix']:
arr = np.matrix(data, dtype=float)
elif isinstance(data, str):
arr = np.array(np.matrix(data, dtype=float))
else:
arr = np.array(data, dtype=float)
ndim = arr.ndim
shape = arr.shape
# Change the shape of the array into a 2D array
if (ndim > 2):
raise ValueError("state-space matrix must be 2-dimensional")
elif (ndim == 2 and shape == (1, 0)) or \
(ndim == 1 and shape == (0, )):
# Passed an empty matrix or empty vector; change shape to (0, 0)
shape = (0, 0)
elif ndim == 1:
# Passed a row or column vector
shape = (1, shape[0]) if axis == 1 else (shape[0], 1)
elif ndim == 0:
# Passed a constant; turn into a matrix
shape = (1, 1)
# Create the actual object used to store the result
return arr.reshape(shape)
def _f2s(f):
"""Format floating point number f for StateSpace._repr_latex_.
Numbers are converted to strings with statesp.latex_num_format.
Inserts column separators, etc., as needed.
"""
fmt = "{:" + config.defaults['statesp.latex_num_format'] + "}"
sraw = fmt.format(f)
# significand-exponent
se = sraw.lower().split('e')
# whole-fraction
wf = se[0].split('.')
s = wf[0]
if wf[1:]:
s += r'.&\hspace{{-1em}}{frac}'.format(frac=wf[1])
else:
s += r'\phantom{.}&\hspace{-1em}'
if se[1:]:
s += r'&\hspace{{-1em}}\cdot10^{{{:d}}}'.format(int(se[1]))
else:
s += r'&\hspace{-1em}\phantom{\cdot}'
return s
class StateSpace(LTI):
r"""StateSpace(A, B, C, D[, dt])
A class for representing state-space models.
The StateSpace class is used to represent state-space realizations of
linear time-invariant (LTI) systems:
.. math::
dx/dt &= A x + B u \\
y &= C x + D u
where `u` is the input, `y` is the output, and `x` is the state.
Parameters
----------
A, B, C, D: array_like
System matrices of the appropriate dimensions.
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, nstates : int
Number of input, output and state variables.
A, B, C, D : 2D arrays
System matrices defining the input/output dynamics.
dt : None, True or float
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).
Notes
-----
The main data members in the ``StateSpace`` class are the A, B, C, and D
matrices. The class also keeps track of the number of states (i.e.,
the size of A). The data format used to store state space matrices is
set using the value of `config.defaults['use_numpy_matrix']`. If True
(default), the state space elements are stored as `numpy.matrix` objects;
otherwise they are `numpy.ndarray` objects. The
:func:`~control.use_numpy_matrix` function can be used to set the storage
type.
A discrete time system 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']``.
Note: timebase processing has moved to namedio.
A state space system is callable and returns the value of the transfer
function evaluated at a point in the complex plane. See
:meth:`~control.StateSpace.__call__` for a more detailed description.
StateSpace instances have support for IPython LaTeX output,
intended for pretty-printing in Jupyter notebooks. The LaTeX
output can be configured using
`control.config.defaults['statesp.latex_num_format']` and
`control.config.defaults['statesp.latex_repr_type']`. The LaTeX output is
tailored for MathJax, as used in Jupyter, and may look odd when
typeset by non-MathJax LaTeX systems.
`control.config.defaults['statesp.latex_num_format']` is a format string
fragment, specifically the part of the format string after `'{:'`
used to convert floating-point numbers to strings. By default it
is `'.3g'`.
`control.config.defaults['statesp.latex_repr_type']` must either be
`'partitioned'` or `'separate'`. If `'partitioned'`, the A, B, C, D
matrices are shown as a single, partitioned matrix; if
`'separate'`, the matrices are shown separately.
"""
# Allow ndarray * StateSpace to give StateSpace._rmul_() priority
__array_priority__ = 11 # override ndarray and matrix types
def __init__(self, *args, init_namedio=True, **kwargs):
"""StateSpace(A, B, C, D[, dt])
Construct a state space object.
The default constructor is StateSpace(A, B, C, D), where A, B, C, D
are matrices or equivalent objects. To create a discrete time system,
use StateSpace(A, B, C, D, dt) where `dt` is the sampling time (or
True for unspecified sampling time). To call the copy constructor,
call StateSpace(sys), where sys is a StateSpace object.
The `remove_useless_states` keyword can be used to scan the A, B, and
C matrices for rows or columns of zeros. If the zeros are such that a
particular state has no effect on the input-output dynamics, then that
state is removed from the A, B, and C matrices. If not specified, the
value is read from `config.defaults['statesp.remove_useless_states']`
(default = False).
The `init_namedio` keyword can be used to turn off initialization of
system and signal names. This is used internally by the
:class:`LinearIOSystem` class to avoid renaming.
"""
#
# Process positional arguments
#
if len(args) == 4:
# The user provided A, B, C, and D matrices.
(A, B, C, D) = args
elif len(args) == 5:
# Discrete time system
(A, B, C, D, 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], StateSpace):
raise TypeError(
"The one-argument constructor can only take in a "
"StateSpace object. Received %s." % type(args[0]))
A = args[0].A
B = args[0].B
C = args[0].C
D = args[0].D
else:
raise TypeError(
"Expected 1, 4, or 5 arguments; received %i." % len(args))
# Convert all matrices to standard form
A = _ssmatrix(A)
# if B is a 1D array, turn it into a column vector if it fits
if np.asarray(B).ndim == 1 and len(B) == A.shape[0]:
B = _ssmatrix(B, axis=0)
else:
B = _ssmatrix(B)
if np.asarray(C).ndim == 1 and len(C) == A.shape[0]:
C = _ssmatrix(C, axis=1)
else:
C = _ssmatrix(C, axis=0) # if this doesn't work, error below
if np.isscalar(D) and D == 0 and B.shape[1] > 0 and C.shape[0] > 0:
# If D is a scalar zero, broadcast it to the proper size
D = np.zeros((C.shape[0], B.shape[1]))
D = _ssmatrix(D)
# Matrices defining the linear system
self.A = A
self.B = B
self.C = C
self.D = D
#
# Process keyword arguments
#
remove_useless_states = kwargs.pop(
'remove_useless_states',
config.defaults['statesp.remove_useless_states'])
# Initialize the instance variables
if init_namedio:
# Process namedio keywords
defaults = args[0] if len(args) == 1 else \
{'inputs': D.shape[1], 'outputs': D.shape[0],
'states': A.shape[0]}
name, inputs, outputs, states, dt = _process_namedio_keywords(
kwargs, defaults, static=(A.size == 0), end=True)
# Initialize LTI (NamedIOSystem) object
super().__init__(
name=name, inputs=inputs, outputs=outputs,
states=states, dt=dt)
elif kwargs:
raise TypeError("unrecognized keyword(s): ", str(kwargs))
# Reset shapes (may not be needed once np.matrix support is removed)
if self._isstatic():
# static gain
# matrix's default "empty" shape is 1x0
A.shape = (0, 0)
B.shape = (0, self.ninputs)
C.shape = (self.noutputs, 0)
#
# Check to make sure everything is consistent
#
# Check that the matrix sizes are consistent
if A.shape[0] != A.shape[1] or self.nstates != A.shape[0]:
raise ValueError("A must be square.")
if self.nstates != B.shape[0]:
raise ValueError("A and B must have the same number of rows.")
if self.nstates != C.shape[1]:
raise ValueError("A and C must have the same number of columns.")
if self.ninputs != B.shape[1]:
raise ValueError("B and D must have the same number of columns.")
if self.noutputs != C.shape[0]:
raise ValueError("C and D must have the same number of rows.")
#
# Final processing
#
# Check for states that don't do anything, and remove them
if remove_useless_states:
self._remove_useless_states()
#
# 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 = 0
#: Number of system outputs.
#:
#: :meta hide-value:
noutputs = 0
#: Number of system states.
#:
#: :meta hide-value:
nstates = 0
#: Dynamics matrix.
#:
#: :meta hide-value:
A = []
#: Input matrix.
#:
#: :meta hide-value:
B = []
#: Output matrix.
#:
#: :meta hide-value:
C = []
#: Direct term.
#:
#: :meta hide-value:
D = []
#
# Getter and setter functions for legacy state attributes
#
# For this iteration, generate a deprecation warning whenever the
# getter/setter is called. For a future iteration, turn it into a
# future warning, so that users will see it.
#
def _get_states(self):
warn("The StateSpace `states` attribute will be deprecated in a "
"future release. Use `nstates` instead.",
DeprecationWarning, stacklevel=2)
return self.nstates
def _set_states(self, value):
warn("The StateSpace `states` attribute will be deprecated in a "
"future release. Use `nstates` instead.",
DeprecationWarning, stacklevel=2)
self.nstates = value
#: Deprecated attribute; use :attr:`nstates` instead.
#:
#: The ``state`` attribute was used to store the number of states for : a
#: state space system. It is no longer used. If you need to access the
#: number of states, use :attr:`nstates`.
states = property(_get_states, _set_states)
def _remove_useless_states(self):
"""Check for states that don't do anything, and remove them.
Scan the A, B, and C matrices for rows or columns of zeros. If the
zeros are such that a particular state has no effect on the input-
output dynamics, then remove that state from the A, B, and C matrices.
"""
# Search for useless states and get indices of these states.
#
# Note: shape from np.where depends on whether we are storing state
# space objects as np.matrix or np.array. Code below will work
# correctly in either case.
ax1_A = np.where(~self.A.any(axis=1))[0]
ax1_B = np.where(~self.B.any(axis=1))[0]
ax0_A = np.where(~self.A.any(axis=0))[-1]
ax0_C = np.where(~self.C.any(axis=0))[-1]
useless_1 = np.intersect1d(ax1_A, ax1_B, assume_unique=True)
useless_2 = np.intersect1d(ax0_A, ax0_C, assume_unique=True)
useless = np.union1d(useless_1, useless_2)
# Remove the useless states.
self.A = delete(self.A, useless, 0)
self.A = delete(self.A, useless, 1)
self.B = delete(self.B, useless, 0)
self.C = delete(self.C, useless, 1)
# Remove any state names that we don't need
self.set_states(
[self.state_labels[i] for i in range(self.nstates)
if i not in useless])
def __str__(self):
"""Return string representation of the state space system."""
string = "\n".join([
"{} = {}\n".format(Mvar,
"\n ".join(str(M).splitlines()))
for Mvar, M in zip(["A", "B", "C", "D"],
[self.A, self.B, self.C, self.D])])
if self.isdtime(strict=True):
string += f"\ndt = {self.dt}\n"
return string
# represent to implement a re-loadable version
# TODO: remove the conversion to array when matrix is no longer used
def __repr__(self):
"""Print state-space system in loadable form."""
return "StateSpace({A}, {B}, {C}, {D}{dt})".format(
A=asarray(self.A).__repr__(), B=asarray(self.B).__repr__(),
C=asarray(self.C).__repr__(), D=asarray(self.D).__repr__(),
dt=(isdtime(self, strict=True) and ", {}".format(self.dt)) or '')
def _latex_partitioned_stateless(self):
"""`Partitioned` matrix LaTeX representation for stateless systems
Model is presented as a matrix, D. No partition lines are shown.
Returns
-------
s : string with LaTeX representation of model
"""
lines = [
r'$$',
(r'\left('
+ r'\begin{array}'
+ r'{' + 'rll' * self.ninputs + '}')
]
for Di in asarray(self.D):
lines.append('&'.join(_f2s(Dij) for Dij in Di)
+ '\\\\')
lines.extend([
r'\end{array}'
r'\right)'
+ self._latex_dt(),
r'$$'])
return '\n'.join(lines)
def _latex_partitioned(self):
"""Partitioned matrix LaTeX representation of state-space model
Model is presented as a matrix partitioned into A, B, C, and D
parts.
Returns
-------
s : string with LaTeX representation of model
"""
if self.nstates == 0:
return self._latex_partitioned_stateless()
lines = [
r'$$',
(r'\left('
+ r'\begin{array}'
+ r'{' + 'rll' * self.nstates + '|' + 'rll' * self.ninputs + '}')
]
for Ai, Bi in zip(asarray(self.A), asarray(self.B)):
lines.append('&'.join([_f2s(Aij) for Aij in Ai]
+ [_f2s(Bij) for Bij in Bi])
+ '\\\\')
lines.append(r'\hline')
for Ci, Di in zip(asarray(self.C), asarray(self.D)):
lines.append('&'.join([_f2s(Cij) for Cij in Ci]
+ [_f2s(Dij) for Dij in Di])
+ '\\\\')
lines.extend([
r'\end{array}'
+ r'\right)'
+ self._latex_dt(),
r'$$'])
return '\n'.join(lines)
def _latex_separate(self):
"""Separate matrices LaTeX representation of state-space model
Model is presented as separate, named, A, B, C, and D matrices.
Returns
-------
s : string with LaTeX representation of model
"""
lines = [
r'$$',
r'\begin{array}{ll}',
]
def fmt_matrix(matrix, name):
matlines = [name
+ r' = \left(\begin{array}{'
+ 'rll' * matrix.shape[1]
+ '}']
for row in asarray(matrix):
matlines.append('&'.join(_f2s(entry) for entry in row)
+ '\\\\')
matlines.extend([
r'\end{array}'
r'\right)'])
return matlines
if self.nstates > 0:
lines.extend(fmt_matrix(self.A, 'A'))
lines.append('&')
lines.extend(fmt_matrix(self.B, 'B'))
lines.append('\\\\')
lines.extend(fmt_matrix(self.C, 'C'))
lines.append('&')
lines.extend(fmt_matrix(self.D, 'D'))
lines.extend([
r'\end{array}'
+ self._latex_dt(),
r'$$'])
return '\n'.join(lines)
def _latex_dt(self):
if self.isdtime(strict=True):
if self.dt is True:
return r"~,~dt=~\mathrm{True}"
else:
fmt = config.defaults['statesp.latex_num_format']
return f"~,~dt={self.dt:{fmt}}"
return ""
def _repr_latex_(self):
"""LaTeX representation of state-space model
Output is controlled by config options statesp.latex_repr_type,
statesp.latex_num_format, and statesp.latex_maxsize.
The output is primarily intended for Jupyter notebooks, which
use MathJax to render the LaTeX, and the results may look odd
when processed by a 'conventional' LaTeX system.
Returns
-------
s : string with LaTeX representation of model, or None if
either matrix dimension is greater than
statesp.latex_maxsize
"""
syssize = self.nstates + max(self.noutputs, self.ninputs)
if syssize > config.defaults['statesp.latex_maxsize']:
return None
elif config.defaults['statesp.latex_repr_type'] == 'partitioned':
return self._latex_partitioned()
elif config.defaults['statesp.latex_repr_type'] == 'separate':
return self._latex_separate()
else:
raise ValueError(
"Unknown statesp.latex_repr_type '{cfg}'".format(
cfg=config.defaults['statesp.latex_repr_type']))
# Negation of a system
def __neg__(self):
"""Negate a state space system."""
return StateSpace(self.A, self.B, -self.C, -self.D, self.dt)
# Addition of two state space systems (parallel interconnection)
def __add__(self, other):
"""Add two LTI systems (parallel connection)."""
# Check for a couple of special cases
if isinstance(other, (int, float, complex, np.number)):
# Just adding a scalar; put it in the D matrix
A, B, C = self.A, self.B, self.C
D = self.D + other
dt = self.dt
else:
# Check to see if the right operator has priority
if getattr(other, '__array_priority__', None) and \
getattr(self, '__array_priority__', None) and \
other.__array_priority__ > self.__array_priority__:
return other.__radd__(self)
# Convert the other argument to state space
other = _convert_to_statespace(other)
# Check to make sure the dimensions are OK
if ((self.ninputs != other.ninputs) or
(self.noutputs != other.noutputs)):
raise ValueError("Systems have different shapes.")
dt = common_timebase(self.dt, other.dt)
# Concatenate the various arrays
A = concatenate((
concatenate((self.A, zeros((self.A.shape[0],
other.A.shape[-1]))), axis=1),
concatenate((zeros((other.A.shape[0], self.A.shape[-1])),
other.A), axis=1)), axis=0)
B = concatenate((self.B, other.B), axis=0)
C = concatenate((self.C, other.C), axis=1)
D = self.D + other.D
return StateSpace(A, B, C, D, dt)
# Right addition - just switch the arguments
def __radd__(self, other):
"""Right add two LTI systems (parallel connection)."""
return self + other
# Subtraction of two state space systems (parallel interconnection)
def __sub__(self, other):
"""Subtract two LTI systems."""
return self + (-other)
def __rsub__(self, other):
"""Right subtract two LTI systems."""
return other + (-self)
# Multiplication of two state space systems (series interconnection)
def __mul__(self, other):
"""Multiply two LTI objects (serial connection)."""
# Check for a couple of special cases
if isinstance(other, (int, float, complex, np.number)):
# Just multiplying by a scalar; change the output
A, B = self.A, self.B
C = self.C * other
D = self.D * other
dt = self.dt
else:
# Check to see if the right operator has priority
if getattr(other, '__array_priority__', None) and \
getattr(self, '__array_priority__', None) and \
other.__array_priority__ > self.__array_priority__:
return other.__rmul__(self)
# Convert the other argument to state space
other = _convert_to_statespace(other)
# Check to make sure the dimensions are OK
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))
dt = common_timebase(self.dt, other.dt)
# Concatenate the various arrays
A = concatenate(
(concatenate((other.A,
zeros((other.A.shape[0], self.A.shape[1]))),
axis=1),
concatenate((self.B @ other.C, self.A), axis=1)),
axis=0)
B = concatenate((other.B, self.B @ other.D), axis=0)
C = concatenate((self.D @ other.C, self.C), axis=1)
D = self.D @ other.D
return StateSpace(A, B, C, D, dt)
# Right multiplication of two state space systems (series interconnection)
# Just need to convert LH argument to a state space object
# TODO: __rmul__ only works for special cases (??)
def __rmul__(self, other):
"""Right multiply two LTI objects (serial connection)."""
# Check for a couple of special cases
if isinstance(other, (int, float, complex, np.number)):
# Just multiplying by a scalar; change the input
A, C = self.A, self.C
B = self.B * other
D = self.D * other
return StateSpace(A, B, C, D, self.dt)
# is lti, and convertible?
if isinstance(other, LTI):
return _convert_to_statespace(other) * self
# try to treat this as a matrix
try:
X = _ssmatrix(other)
C = X @ self.C
D = X @ self.D
return StateSpace(self.A, self.B, C, D, self.dt)
except Exception as e:
print(e)
pass
raise TypeError("can't interconnect systems")
# TODO: general __truediv__, and __rtruediv__; requires descriptor system support
def __truediv__(self, other):
"""Division of StateSpace systems
Only division by TFs, FRDs, scalars, and arrays of scalars is
supported.
"""
if not isinstance(other, (LTI, NamedIOSystem)):
return self * (1/other)
else:
return NotImplemented
def __call__(self, x, squeeze=None, warn_infinite=True):
"""Evaluate system's frequency response at complex frequencies.
Returns the complex frequency response `sys(x)` where `x` is `s` for
continuous-time systems and `z` for discrete-time systems.
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:`StateSpace.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'].
warn_infinite : bool, optional
If set to `False`, don't warn if frequency response is infinite.
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.
"""
# Use Slycot if available
out = self.horner(x, warn_infinite=warn_infinite)
return _process_frequency_response(self, x, out, squeeze=squeeze)
def slycot_laub(self, x):
"""Evaluate system's transfer function at complex frequency
using Laub's method from Slycot.
Expects inputs and outputs to be formatted correctly. Use ``sys(x)``
for a more user-friendly interface.
Parameters
----------
x : complex array_like or complex
Complex frequency
Returns
-------
output : (number_outputs, number_inputs, len(x)) complex ndarray
Frequency response
"""
from slycot import tb05ad
# 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")
# preallocate
n = self.nstates
m = self.ninputs
p = self.noutputs
out = np.empty((p, m, len(x_arr)), dtype=complex)
# The first call both evaluates C(sI-A)^-1 B and also returns
# Hessenberg transformed matrices at, bt, ct.
result = tb05ad(n, m, p, x_arr[0], self.A, self.B, self.C, job='NG')
# When job='NG', result = (at, bt, ct, g_i, hinvb, info)
at = result[0]
bt = result[1]
ct = result[2]
# TB05AD frequency evaluation does not include direct feedthrough.
out[:, :, 0] = result[3] + self.D
# Now, iterate through the remaining frequencies using the
# transformed state matrices, at, bt, ct.
# Start at the second frequency, already have the first.
for kk, x_kk in enumerate(x_arr[1:]):
result = tb05ad(n, m, p, x_kk, at, bt, ct, job='NH')
# When job='NH', result = (g_i, hinvb, info)
# kk+1 because enumerate starts at kk = 0.
# but zero-th spot is already filled.
out[:, :, kk+1] = result[0] + self.D
return out
def horner(self, x, warn_infinite=True):
"""Evaluate system's transfer function at complex frequency
using Laub's or 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
Complex frequencies
Returns
-------
output : (self.noutputs, self.ninputs, len(x)) complex ndarray
Frequency response
Notes
-----
Attempts to use Laub's method from Slycot library, with a
fall-back to python code.
"""
# Make sure the argument is a 1D array of complex numbers
x_arr = np.atleast_1d(x).astype(complex, copy=False)
# return fast on systems with 0 or 1 state
if not config.defaults['statesp.use_numpy_matrix']:
if self.nstates == 0:
return self.D[:, :, np.newaxis] \
* np.ones_like(x_arr, dtype=complex)
if self.nstates == 1:
with np.errstate(divide='ignore', invalid='ignore'):
out = self.C[:, :, np.newaxis] \
/ (x_arr - self.A[0, 0]) \
* self.B[:, :, np.newaxis] \
+ self.D[:, :, np.newaxis]
out[np.isnan(out)] = complex(np.inf, np.nan)
return out
try:
out = self.slycot_laub(x_arr)
except (ImportError, Exception):
# Fall back because either Slycot unavailable or cannot handle
# certain cases.
# Make sure that we are operating on a simple list
if len(x_arr.shape) > 1:
raise ValueError("input list must be 1D")
# Preallocate
out = empty((self.noutputs, self.ninputs, len(x_arr)),
dtype=complex)
# TODO: can this be vectorized?
for idx, x_idx in enumerate(x_arr):
try:
xr = solve(x_idx * eye(self.nstates) - self.A, self.B)
out[:, :, idx] = self.C @ xr + self.D
except LinAlgError:
# Issue a warning messsage, for consistency with xferfcn
if warn_infinite:
warn("singular matrix in frequency response",
RuntimeWarning)
# Evaluating at a pole. Return value depends if there
# is a zero at the same point or not.
if x_idx in self.zeros():
out[:, :, idx] = complex(np.nan, np.nan)
else:
out[:, :, idx] = complex(np.inf, np.nan)
return out
def freqresp(self, omega):
"""(deprecated) Evaluate transfer function at complex frequencies.
.. deprecated::0.9.0
Method has been given the more pythonic name
:meth:`StateSpace.frequency_response`. Or use
:func:`freqresp` in the MATLAB compatibility module.
"""
warn("StateSpace.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)
# Compute poles and zeros
def poles(self):
"""Compute the poles of a state space system."""
return eigvals(self.A).astype(complex) if self.nstates \
else np.array([])