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ltisys.py
2448 lines (2020 loc) · 78 KB
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ltisys.py
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"""
ltisys -- a collection of classes and functions for modeling linear
time invariant systems.
"""
from __future__ import division, print_function, absolute_import
#
# Author: Travis Oliphant 2001
#
# Feb 2010: Warren Weckesser
# Rewrote lsim2 and added impulse2.
# Aug 2013: Juan Luis Cano
# Rewrote abcd_normalize.
# Jan 2015: Irvin Probst irvin DOT probst AT ensta-bretagne DOT fr
# Added pole placement
# Mar 2015: Clancy Rowley
# Rewrote lsim
# May 2015: Felix Berkenkamp
# Split lti class into subclasses
#
import warnings
import numpy as np
#np.linalg.qr fails on some tests with LinAlgError: zgeqrf returns -7
#use scipy's qr until this is solved
from scipy.linalg import qr as s_qr
import numpy
from numpy import (r_, eye, real, atleast_1d, atleast_2d, poly,
squeeze, asarray, product, zeros, array,
dot, transpose, ones, zeros_like, linspace, nan_to_num)
import copy
from scipy import integrate, interpolate, linalg
from scipy._lib.six import xrange
from .filter_design import tf2zpk, zpk2tf, normalize, freqs
__all__ = ['tf2ss', 'ss2tf', 'abcd_normalize', 'zpk2ss', 'ss2zpk', 'lti',
'TransferFunction', 'ZerosPolesGain', 'StateSpace', 'lsim',
'lsim2', 'impulse', 'impulse2', 'step', 'step2', 'bode',
'freqresp', 'place_poles']
def tf2ss(num, den):
r"""Transfer function to state-space representation.
Parameters
----------
num, den : array_like
Sequences representing the numerator and denominator polynomials.
The denominator needs to be at least as long as the numerator.
Returns
-------
A, B, C, D : ndarray
State space representation of the system, in controller canonical
form.
Examples
--------
Convert the transfer function:
.. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
>>> num = [1, 3, 3]
>>> den = [1, 2, 1]
to the state-space representation:
.. math::
\dot{\textbf{x}}(t) =
\begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
\begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\
\textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
\begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)
>>> from scipy.signal import tf2ss
>>> A, B, C, D = tf2ss(num, den)
>>> A
array([[-2., -1.],
[ 1., 0.]])
>>> B
array([[ 1.],
[ 0.]])
>>> C
array([[ 1., 2.]])
>>> D
array([ 1.])
"""
# Controller canonical state-space representation.
# if M+1 = len(num) and K+1 = len(den) then we must have M <= K
# states are found by asserting that X(s) = U(s) / D(s)
# then Y(s) = N(s) * X(s)
#
# A, B, C, and D follow quite naturally.
#
num, den = normalize(num, den) # Strips zeros, checks arrays
nn = len(num.shape)
if nn == 1:
num = asarray([num], num.dtype)
M = num.shape[1]
K = len(den)
if M > K:
msg = "Improper transfer function. `num` is longer than `den`."
raise ValueError(msg)
if M == 0 or K == 0: # Null system
return (array([], float), array([], float), array([], float),
array([], float))
# pad numerator to have same number of columns has denominator
num = r_['-1', zeros((num.shape[0], K - M), num.dtype), num]
if num.shape[-1] > 0:
D = num[:, 0]
else:
D = array([], float)
if K == 1:
return array([], float), array([], float), array([], float), D
frow = -array([den[1:]])
A = r_[frow, eye(K - 2, K - 1)]
B = eye(K - 1, 1)
C = num[:, 1:] - num[:, 0] * den[1:]
return A, B, C, D
def _none_to_empty_2d(arg):
if arg is None:
return zeros((0, 0))
else:
return arg
def _atleast_2d_or_none(arg):
if arg is not None:
return atleast_2d(arg)
def _shape_or_none(M):
if M is not None:
return M.shape
else:
return (None,) * 2
def _choice_not_none(*args):
for arg in args:
if arg is not None:
return arg
def _restore(M, shape):
if M.shape == (0, 0):
return zeros(shape)
else:
if M.shape != shape:
raise ValueError("The input arrays have incompatible shapes.")
return M
def abcd_normalize(A=None, B=None, C=None, D=None):
"""Check state-space matrices and ensure they are two-dimensional.
If enough information on the system is provided, that is, enough
properly-shaped arrays are passed to the function, the missing ones
are built from this information, ensuring the correct number of
rows and columns. Otherwise a ValueError is raised.
Parameters
----------
A, B, C, D : array_like, optional
State-space matrices. All of them are None (missing) by default.
See `ss2tf` for format.
Returns
-------
A, B, C, D : array
Properly shaped state-space matrices.
Raises
------
ValueError
If not enough information on the system was provided.
"""
A, B, C, D = map(_atleast_2d_or_none, (A, B, C, D))
MA, NA = _shape_or_none(A)
MB, NB = _shape_or_none(B)
MC, NC = _shape_or_none(C)
MD, ND = _shape_or_none(D)
p = _choice_not_none(MA, MB, NC)
q = _choice_not_none(NB, ND)
r = _choice_not_none(MC, MD)
if p is None or q is None or r is None:
raise ValueError("Not enough information on the system.")
A, B, C, D = map(_none_to_empty_2d, (A, B, C, D))
A = _restore(A, (p, p))
B = _restore(B, (p, q))
C = _restore(C, (r, p))
D = _restore(D, (r, q))
return A, B, C, D
def ss2tf(A, B, C, D, input=0):
r"""State-space to transfer function.
A, B, C, D defines a linear state-space system with `p` inputs,
`q` outputs, and `n` state variables.
Parameters
----------
A : array_like
State (or system) matrix of shape ``(n, n)``
B : array_like
Input matrix of shape ``(n, p)``
C : array_like
Output matrix of shape ``(q, n)``
D : array_like
Feedthrough (or feedforward) matrix of shape ``(q, p)``
input : int, optional
For multiple-input systems, the index of the input to use.
Returns
-------
num : 2-D ndarray
Numerator(s) of the resulting transfer function(s). `num` has one row
for each of the system's outputs. Each row is a sequence representation
of the numerator polynomial.
den : 1-D ndarray
Denominator of the resulting transfer function(s). `den` is a sequence
representation of the denominator polynomial.
Examples
--------
Convert the state-space representation:
.. math::
\dot{\textbf{x}}(t) =
\begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
\begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\
\textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
\begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)
>>> A = [[-2, -1], [1, 0]]
>>> B = [[1], [0]] # 2-dimensional column vector
>>> C = [[1, 2]] # 2-dimensional row vector
>>> D = 1
to the transfer function:
.. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
>>> from scipy.signal import ss2tf
>>> ss2tf(A, B, C, D)
(array([[1, 3, 3]]), array([ 1., 2., 1.]))
"""
# transfer function is C (sI - A)**(-1) B + D
# Check consistency and make them all rank-2 arrays
A, B, C, D = abcd_normalize(A, B, C, D)
nout, nin = D.shape
if input >= nin:
raise ValueError("System does not have the input specified.")
# make SIMO from possibly MIMO system.
B = B[:, input:input + 1]
D = D[:, input:input + 1]
try:
den = poly(A)
except ValueError:
den = 1
if (product(B.shape, axis=0) == 0) and (product(C.shape, axis=0) == 0):
num = numpy.ravel(D)
if (product(D.shape, axis=0) == 0) and (product(A.shape, axis=0) == 0):
den = []
return num, den
num_states = A.shape[0]
type_test = A[:, 0] + B[:, 0] + C[0, :] + D
num = numpy.zeros((nout, num_states + 1), type_test.dtype)
for k in range(nout):
Ck = atleast_2d(C[k, :])
num[k] = poly(A - dot(B, Ck)) + (D[k] - 1) * den
return num, den
def zpk2ss(z, p, k):
"""Zero-pole-gain representation to state-space representation
Parameters
----------
z, p : sequence
Zeros and poles.
k : float
System gain.
Returns
-------
A, B, C, D : ndarray
State space representation of the system, in controller canonical
form.
"""
return tf2ss(*zpk2tf(z, p, k))
def ss2zpk(A, B, C, D, input=0):
"""State-space representation to zero-pole-gain representation.
A, B, C, D defines a linear state-space system with `p` inputs,
`q` outputs, and `n` state variables.
Parameters
----------
A : array_like
State (or system) matrix of shape ``(n, n)``
B : array_like
Input matrix of shape ``(n, p)``
C : array_like
Output matrix of shape ``(q, n)``
D : array_like
Feedthrough (or feedforward) matrix of shape ``(q, p)``
input : int, optional
For multiple-input systems, the index of the input to use.
Returns
-------
z, p : sequence
Zeros and poles.
k : float
System gain.
"""
return tf2zpk(*ss2tf(A, B, C, D, input=input))
class lti(object):
"""
Linear Time Invariant system base class.
Parameters
----------
*system : arguments
The `lti` class can be instantiated with either 2, 3 or 4 arguments.
The following gives the number of arguments and the corresponding
subclass that is created:
* 2: `TransferFunction`: (numerator, denominator)
* 3: `ZerosPolesGain`: (zeros, poles, gain)
* 4: `StateSpace`: (A, B, C, D)
Each argument can be an array or a sequence.
Notes
-----
`lti` instances do not exist directly. Instead, `lti` creates an instance
of one of its subclasses: `StateSpace`, `TransferFunction` or
`ZerosPolesGain`.
Changing the value of properties that are not directly part of the current
system representation (such as the `zeros` of a `StateSpace` system) is
very inefficient and may lead to numerical inaccuracies.
"""
def __new__(cls, *system):
"""Create an instance of the appropriate subclass."""
if cls is lti:
N = len(system)
if N == 2:
return super(lti, cls).__new__(TransferFunction)
elif N == 3:
return super(lti, cls).__new__(ZerosPolesGain)
elif N == 4:
return super(lti, cls).__new__(StateSpace)
else:
raise ValueError('Needs 2, 3 or 4 arguments.')
# __new__ was called from a subclass, let it call its own functions
return super(lti, cls).__new__(cls)
def __init__(self, *system):
"""
Initialize the `lti` baseclass.
The heavy lifting is done by the subclasses.
"""
self.inputs = None
self.outputs = None
@property
def num(self):
"""Numerator of the `TransferFunction` system."""
return self.to_tf().num
@num.setter
def num(self, num):
obj = self.to_tf()
obj.num = num
source_class = type(self)
self._copy(source_class(obj))
@property
def den(self):
"""Denominator of the `TransferFunction` system."""
return self.to_tf().den
@den.setter
def den(self, den):
obj = self.to_tf()
obj.den = den
source_class = type(self)
self._copy(source_class(obj))
@property
def zeros(self):
"""Zeros of the `ZerosPolesGain` system."""
return self.to_zpk().zeros
@zeros.setter
def zeros(self, zeros):
obj = self.to_zpk()
obj.zeros = zeros
source_class = type(self)
self._copy(source_class(obj))
@property
def poles(self):
"""Poles of the `ZerosPolesGain` system."""
return self.to_zpk().poles
@poles.setter
def poles(self, poles):
obj = self.to_zpk()
obj.poles = poles
source_class = type(self)
self._copy(source_class(obj))
@property
def gain(self):
"""Gain of the `ZerosPolesGain` system."""
return self.to_zpk().gain
@gain.setter
def gain(self, gain):
obj = self.to_zpk()
obj.gain = gain
source_class = type(self)
self._copy(source_class(obj))
@property
def A(self):
"""State matrix of the `StateSpace` system."""
return self.to_ss().A
@A.setter
def A(self, A):
obj = self.to_ss()
obj.A = A
source_class = type(self)
self._copy(source_class(obj))
@property
def B(self):
"""Input matrix of the `StateSpace` system."""
return self.to_ss().B
@B.setter
def B(self, B):
obj = self.to_ss()
obj.B = B
source_class = type(self)
self._copy(source_class(obj))
@property
def C(self):
"""Output matrix of the `StateSpace` system."""
return self.to_ss().C
@C.setter
def C(self, C):
obj = self.to_ss()
obj.C = C
source_class = type(self)
self._copy(source_class(obj))
@property
def D(self):
"""Feedthrough matrix of the `StateSpace` system."""
return self.to_ss().D
@D.setter
def D(self, D):
obj = self.to_ss()
obj.D = D
source_class = type(self)
self._copy(source_class(obj))
def impulse(self, X0=None, T=None, N=None):
"""
Return the impulse response of a continuous-time system.
See `scipy.signal.impulse` for details.
"""
return impulse(self, X0=X0, T=T, N=N)
def step(self, X0=None, T=None, N=None):
"""
Return the step response of a continuous-time system.
See `scipy.signal.step` for details.
"""
return step(self, X0=X0, T=T, N=N)
def output(self, U, T, X0=None):
"""
Return the response of a continuous-time system to input `U`.
See `scipy.signal.lsim` for details.
"""
return lsim(self, U, T, X0=X0)
def bode(self, w=None, n=100):
"""
Calculate Bode magnitude and phase data of a continuous-time system.
Returns a 3-tuple containing arrays of frequencies [rad/s], magnitude
[dB] and phase [deg]. See `scipy.signal.bode` for details.
Notes
-----
.. versionadded:: 0.11.0
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> s1 = signal.lti([1], [1, 1])
>>> w, mag, phase = s1.bode()
>>> plt.figure()
>>> plt.semilogx(w, mag) # Bode magnitude plot
>>> plt.figure()
>>> plt.semilogx(w, phase) # Bode phase plot
>>> plt.show()
"""
return bode(self, w=w, n=n)
def freqresp(self, w=None, n=10000):
"""
Calculate the frequency response of a continuous-time system.
Returns a 2-tuple containing arrays of frequencies [rad/s] and
complex magnitude.
See `scipy.signal.freqresp` for details.
"""
return freqresp(self, w=w, n=n)
class TransferFunction(lti):
r"""Linear Time Invariant system class in transfer function form.
Represents the system as the transfer function
:math:`H(s)=\sum_{i=0}^N b[N-i] s^i / \sum_{j=0}^M a[M-j] s^j`, where :math:`b` are
elements of the numerator `num`, :math:`a` are elements of the denominator
`den`, and ``N == len(b) - 1``, ``M == len(a) - 1``.
Parameters
----------
*system : arguments
The `TransferFunction` class can be instantiated with 1 or 2 arguments.
The following gives the number of input arguments and their
interpretation:
* 1: `lti` system: (`StateSpace`, `TransferFunction` or
`ZerosPolesGain`)
* 2: array_like: (numerator, denominator)
See Also
--------
ZerosPolesGain, StateSpace, lti
tf2ss, tf2zpk, tf2sos
Notes
-----
Changing the value of properties that are not part of the
`TransferFunction` system representation (such as the `A`, `B`, `C`, `D`
state-space matrices) is very inefficient and may lead to numerical
inaccuracies.
Examples
--------
Construct the transfer function:
.. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
>>> from scipy import signal
>>> num = [1, 3, 3]
>>> den = [1, 2, 1]
>>> signal.TransferFunction(num, den)
TransferFunction(
array([ 1., 3., 3.]),
array([ 1., 2., 1.])
)
"""
def __new__(cls, *system):
"""Handle object conversion if input is an instance of lti."""
if len(system) == 1 and isinstance(system[0], lti):
return system[0].to_tf()
# No special conversion needed
return super(TransferFunction, cls).__new__(cls)
def __init__(self, *system):
"""Initialize the state space LTI system."""
# Conversion of lti instances is handled in __new__
if isinstance(system[0], lti):
return
super(TransferFunction, self).__init__(self, *system)
self._num = None
self._den = None
self.num, self.den = normalize(*system)
def __repr__(self):
"""Return representation of the system's transfer function"""
return '{0}(\n{1},\n{2}\n)'.format(
self.__class__.__name__,
repr(self.num),
repr(self.den),
)
@property
def num(self):
"""Numerator of the `TransferFunction` system."""
return self._num
@num.setter
def num(self, num):
self._num = atleast_1d(num)
# Update dimensions
if len(self.num.shape) > 1:
self.outputs, self.inputs = self.num.shape
else:
self.outputs = 1
self.inputs = 1
@property
def den(self):
"""Denominator of the `TransferFunction` system."""
return self._den
@den.setter
def den(self, den):
self._den = atleast_1d(den)
def _copy(self, system):
"""
Copy the parameters of another `TransferFunction` object
Parameters
----------
system : `TransferFunction`
The `StateSpace` system that is to be copied
"""
self.num = system.num
self.den = system.den
def to_tf(self):
"""
Return a copy of the current `TransferFunction` system.
Returns
-------
sys : instance of `TransferFunction`
The current system (copy)
"""
return copy.deepcopy(self)
def to_zpk(self):
"""
Convert system representation to `ZerosPolesGain`.
Returns
-------
sys : instance of `ZerosPolesGain`
Zeros, poles, gain representation of the current system
"""
return ZerosPolesGain(*tf2zpk(self.num, self.den))
def to_ss(self):
"""
Convert system representation to `StateSpace`.
Returns
-------
sys : instance of `StateSpace`
State space model of the current system
"""
return StateSpace(*tf2ss(self.num, self.den))
class ZerosPolesGain(lti):
"""
Linear Time Invariant system class in zeros, poles, gain form.
Represents the system as the transfer function
:math:`H(s)=k \prod_i (s - z[i]) / \prod_j (s - p[j])`, where :math:`k` is
the `gain`, :math:`z` are the `zeros` and :math:`p` are the `poles`.
Parameters
----------
*system : arguments
The `ZerosPolesGain` class can be instantiated with 1 or 3 arguments.
The following gives the number of input arguments and their
interpretation:
* 1: `lti` system: (`StateSpace`, `TransferFunction` or
`ZerosPolesGain`)
* 3: array_like: (zeros, poles, gain)
See Also
--------
TransferFunction, StateSpace, lti
zpk2ss, zpk2tf, zpk2sos
Notes
-----
Changing the value of properties that are not part of the
`ZerosPolesGain` system representation (such as the `A`, `B`, `C`, `D`
state-space matrices) is very inefficient and may lead to numerical
inaccuracies.
"""
def __new__(cls, *system):
"""Handle object conversion if input is an instance of `lti`"""
if len(system) == 1 and isinstance(system[0], lti):
return system[0].to_zpk()
# No special conversion needed
return super(ZerosPolesGain, cls).__new__(cls)
def __init__(self, *system):
"""Initialize the zeros, poles, gain LTI system."""
# Conversion of lti instances is handled in __new__
if isinstance(system[0], lti):
return
super(ZerosPolesGain, self).__init__(self, *system)
self._zeros = None
self._poles = None
self._gain = None
self.zeros, self.poles, self.gain = system
def __repr__(self):
"""Return representation of the `ZerosPolesGain` system"""
return '{0}(\n{1},\n{2},\n{3}\n)'.format(
self.__class__.__name__,
repr(self.zeros),
repr(self.poles),
repr(self.gain),
)
@property
def zeros(self):
"""Zeros of the `ZerosPolesGain` system."""
return self._zeros
@zeros.setter
def zeros(self, zeros):
self._zeros = atleast_1d(zeros)
# Update dimensions
if len(self.zeros.shape) > 1:
self.outputs, self.inputs = self.zeros.shape
else:
self.outputs = 1
self.inputs = 1
@property
def poles(self):
"""Poles of the `ZerosPolesGain` system."""
return self._poles
@poles.setter
def poles(self, poles):
self._poles = atleast_1d(poles)
@property
def gain(self):
"""Gain of the `ZerosPolesGain` system."""
return self._gain
@gain.setter
def gain(self, gain):
self._gain = gain
def _copy(self, system):
"""
Copy the parameters of another `ZerosPolesGain` system.
Parameters
----------
system : instance of `ZerosPolesGain`
The zeros, poles gain system that is to be copied
"""
self.poles = system.poles
self.zeros = system.zeros
self.gain = system.gain
def to_tf(self):
"""
Convert system representation to `TransferFunction`.
Returns
-------
sys : instance of `TransferFunction`
Transfer function of the current system
"""
return TransferFunction(*zpk2tf(self.zeros, self.poles, self.gain))
def to_zpk(self):
"""
Return a copy of the current 'ZerosPolesGain' system.
Returns
-------
sys : instance of `ZerosPolesGain`
The current system (copy)
"""
return copy.deepcopy(self)
def to_ss(self):
"""
Convert system representation to `StateSpace`.
Returns
-------
sys : instance of `StateSpace`
State space model of the current system
"""
return StateSpace(*zpk2ss(self.zeros, self.poles, self.gain))
class StateSpace(lti):
"""
Linear Time Invariant system class in state-space form.
Represents the system as the first order differential equation
:math:`\dot{x} = A x + B u`.
Parameters
----------
*system : arguments
The `StateSpace` class can be instantiated with 1 or 4 arguments.
The following gives the number of input arguments and their
interpretation:
* 1: `lti` system: (`StateSpace`, `TransferFunction` or
`ZerosPolesGain`)
* 4: array_like: (A, B, C, D)
See Also
--------
TransferFunction, ZerosPolesGain, lti
ss2zpk, ss2tf, zpk2sos
Notes
-----
Changing the value of properties that are not part of the
`StateSpace` system representation (such as `zeros` or `poles`) is very
inefficient and may lead to numerical inaccuracies.
"""
def __new__(cls, *system):
"""Handle object conversion if input is an instance of `lti`"""
if len(system) == 1 and isinstance(system[0], lti):
return system[0].to_ss()
# No special conversion needed
return super(StateSpace, cls).__new__(cls)
def __init__(self, *system):
"""Initialize the state space LTI system."""
# Conversion of lti instances is handled in __new__
if isinstance(system[0], lti):
return
super(StateSpace, self).__init__(self, *system)
self._A = None
self._B = None
self._C = None
self._D = None
self.A, self.B, self.C, self.D = abcd_normalize(*system)
def __repr__(self):
"""Return representation of the `StateSpace` system."""
return '{0}(\n{1},\n{2},\n{3},\n{4}\n)'.format(
self.__class__.__name__,
repr(self.A),
repr(self.B),
repr(self.C),
repr(self.D),
)
@property
def A(self):
"""State matrix of the `StateSpace` system."""
return self._A
@A.setter
def A(self, A):
self._A = _atleast_2d_or_none(A)
@property
def B(self):
"""Input matrix of the `StateSpace` system."""
return self._B
@B.setter
def B(self, B):
self._B = _atleast_2d_or_none(B)
self.inputs = self.B.shape[-1]
@property
def C(self):
"""Output matrix of the `StateSpace` system."""
return self._C
@C.setter
def C(self, C):
self._C = _atleast_2d_or_none(C)
self.outputs = self.C.shape[0]
@property
def D(self):
"""Feedthrough matrix of the `StateSpace` system."""
return self._D
@D.setter
def D(self, D):
self._D = _atleast_2d_or_none(D)
def _copy(self, system):
"""
Copy the parameters of another `StateSpace` system.
Parameters
----------
system : instance of `StateSpace`
The state-space system that is to be copied
"""
self.A = system.A
self.B = system.B
self.C = system.C
self.D = system.D
def to_tf(self, **kwargs):
"""
Convert system representation to `TransferFunction`.
Parameters
----------
kwargs : dict, optional
Additional keywords passed to `ss2zpk`