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1215 lines (995 loc) · 41.1 KB
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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.
"""
# Python 3 compatibility (needs to go here)
from __future__ import print_function
from __future__ import division # for _convertToStateSpace
"""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 all, angle, any, array, asarray, concatenate, cos, delete, \
dot, empty, exp, eye, matrix, ones, poly, poly1d, roots, shape, 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
from scipy.signal import lti, cont2discrete
# from exceptions import Exception
import warnings
from .lti import LTI, timebase, timebaseEqual, isdtime
from .xferfcn import _convertToTransferFunction
from copy import deepcopy
__all__ = ['StateSpace', 'ss', 'rss', 'drss', 'tf2ss', 'ssdata']
class StateSpace(LTI):
"""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:
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.
The main data members are the A, B, C, and D matrices. The class also
keeps track of the number of states (i.e., the size of A).
Discrete-time state space system are implemented by using the 'dt' instance
variable and setting it to the sampling period. If 'dt' is not None,
then it must match whenever two state space systems are combined.
Setting dt = 0 specifies a continuous system, while leaving dt = None
means the system timebase is not specified. If 'dt' is set to True, the
system will be treated as a discrete time system with unspecified
sampling time.
"""
def __init__(self, *args):
"""
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.
"""
if len(args) == 4:
# The user provided A, B, C, and D matrices.
(A, B, C, D) = args
dt = None;
elif len(args) == 5:
# Discrete time system
(A, B, C, D, dt) = args
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
try:
dt = args[0].dt
except NameError:
dt = None;
else:
raise ValueError("Needs 1 or 4 arguments; received %i." % len(args))
A, B, C, D = map(matrix, [A, B, C, D])
# TODO: use super here?
LTI.__init__(self, inputs=D.shape[1], outputs=D.shape[0], dt=dt)
self.A = A
self.B = B
self.C = C
self.D = D
self.states = A.shape[1]
if 0 == self.states:
# static gain
# matrix's default "empty" shape is 1x0
A.shape = (0,0)
B.shape = (0,self.inputs)
C.shape = (self.outputs,0)
# Check that the matrix sizes are consistent.
if self.states != A.shape[0]:
raise ValueError("A must be square.")
if self.states != B.shape[0]:
raise ValueError("A and B must have the same number of rows.")
if self.states != C.shape[1]:
raise ValueError("A and C must have the same number of columns.")
if self.inputs != B.shape[1]:
raise ValueError("B and D must have the same number of columns.")
if self.outputs != C.shape[0]:
raise ValueError("C and D must have the same number of rows.")
# Check for states that don't do anything, and remove them.
self._remove_useless_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.
"""
# Indices of useless states.
useless = []
# Search for useless states.
for i in range(self.states):
if (all(self.A[i, :] == zeros((1, self.states))) and
all(self.B[i, :] == zeros((1, self.inputs)))):
useless.append(i)
# To avoid duplucate indices in useless, jump to the next
# iteration.
continue
if (all(self.A[:, i] == zeros((self.states, 1))) and
all(self.C[:, i] == zeros((self.outputs, 1)))):
useless.append(i)
# 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)
self.states = self.A.shape[0]
self.inputs = self.B.shape[1]
self.outputs = self.C.shape[0]
def __str__(self):
"""String representation of the state space."""
str = "A = " + self.A.__str__() + "\n\n"
str += "B = " + self.B.__str__() + "\n\n"
str += "C = " + self.C.__str__() + "\n\n"
str += "D = " + self.D.__str__() + "\n"
#! TODO: replace with standard calls to lti functions
if (type(self.dt) == bool and self.dt == True):
str += "\ndt unspecified\n"
elif (not (self.dt is None) and type(self.dt) != bool and self.dt > 0):
str += "\ndt = " + self.dt.__str__() + "\n"
return str
# represent as string, makes display work for IPython
__repr__ = __str__
# 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:
other = _convertToStateSpace(other)
# Check to make sure the dimensions are OK
if ((self.inputs != other.inputs) or
(self.outputs != other.outputs)):
raise ValueError("Systems have different shapes.")
# Figure out the sampling time to use
if (self.dt == None and other.dt != None):
dt = other.dt # use dt from second argument
elif (other.dt == None and self.dt != None) or \
(timebaseEqual(self, other)):
dt = self.dt # use dt from first argument
else:
raise ValueError("Systems have different sampling times")
# 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:
other = _convertToStateSpace(other)
# Check to make sure the dimensions are OK
if self.inputs != other.outputs:
raise ValueError("C = A * B: A has %i column(s) (input(s)), \
but B has %i row(s)\n(output(s))." % (self.inputs, other.outputs))
# Figure out the sampling time to use
if (self.dt == None and other.dt != None):
dt = other.dt # use dt from second argument
elif (other.dt == None and self.dt != None) or \
(timebaseEqual(self, other)):
dt = self.dt # use dt from first argument
else:
raise ValueError("Systems have different sampling times")
# 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 _convertToStateSpace(other) * self
# try to treat this as a matrix
try:
X = matrix(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: __div__ and __rdiv__ are not written yet.
def __div__(self, other):
"""Divide two LTI systems."""
raise NotImplementedError("StateSpace.__div__ is not implemented yet.")
def __rdiv__(self, other):
"""Right divide two LTI systems."""
raise NotImplementedError("StateSpace.__rdiv__ is not implemented yet.")
# TODO: add discrete time check
def evalfr(self, omega):
"""Evaluate a SS system's transfer function at a single frequency.
self.evalfr(omega) returns the value of the transfer function matrix with
input value s = i * omega.
"""
# Figure out the point to evaluate the transfer function
if isdtime(self, strict=True):
dt = timebase(self)
s = exp(1.j * omega * dt)
if (omega * dt > math.pi):
warnings.warn("evalfr: frequency evaluation above Nyquist frequency")
else:
s = omega * 1.j
return self.horner(s)
def horner(self, s):
'''Evaluate the systems's transfer function for a complex variable
Returns a matrix of values evaluated at complex variable s.
'''
resp = self.C * solve(s * eye(self.states) - self.A,
self.B) + self.D
return array(resp)
# Method for generating the frequency response of the system
def freqresp(self, omega):
"""Evaluate the system's transfer func. at a list of ang. frequencies.
mag, phase, omega = self.freqresp(omega)
reports the value of the magnitude, phase, and angular frequency of the
system's 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.
"""
# when evaluating at many frequencies, much faster to convert to
# transfer function first and then evaluate, than to solve an
# n-dimensional linear system at each frequency
tf = _convertToTransferFunction(self)
return tf.freqresp(omega)
# Compute poles and zeros
def pole(self):
"""Compute the poles of a state space system."""
return eigvals(self.A) if self.states else np.array([])
def zero(self):
"""Compute the zeros of a state space system."""
if self.inputs > 1 or self.outputs > 1:
raise NotImplementedError("StateSpace.zeros is currently \
implemented only for SISO systems.")
den = poly1d(poly(self.A))
# Compute the numerator based on zeros
#! TODO: This is currently limited to SISO systems
num = poly1d(poly(self.A - dot(self.B, self.C)) + ((self.D[0, 0] - 1) *
den))
return roots(num)
# Feedback around a state space system
def feedback(self, other=1, sign=-1):
"""Feedback interconnection between two LTI systems."""
other = _convertToStateSpace(other)
# Check to make sure the dimensions are OK
if ((self.inputs != other.outputs) or (self.outputs != other.inputs)):
raise ValueError("State space systems don't have compatible \
inputs/outputs for feedback.")
# Figure out the sampling time to use
if (self.dt == None and other.dt != None):
dt = other.dt # use dt from second argument
elif (other.dt == None and self.dt != None) or \
timebaseEqual(self, other):
dt = self.dt # use dt from first argument
else:
raise ValueError("Systems have different sampling times")
A1 = self.A
B1 = self.B
C1 = self.C
D1 = self.D
A2 = other.A
B2 = other.B
C2 = other.C
D2 = other.D
F = eye(self.inputs) - sign * D2 * D1
if matrix_rank(F) != self.inputs:
raise ValueError("I - sign * D2 * D1 is singular to working precision.")
# Precompute F\D2 and F\C2 (E = inv(F))
# We can solve two linear systems in one pass, since the
# coefficients matrix F is the same. Thus, we perform the LU
# decomposition (cubic runtime complexity) of F only once!
# The remaining back substitutions are only quadratic in runtime.
E_D2_C2 = solve(F, concatenate((D2, C2), axis=1))
E_D2 = E_D2_C2[:, :other.inputs]
E_C2 = E_D2_C2[:, other.inputs:]
T1 = eye(self.outputs) + sign * D1 * E_D2
T2 = eye(self.inputs) + sign * E_D2 * D1
A = concatenate(
(concatenate(
(A1 + sign * B1 * E_D2 * C1, sign * B1 * E_C2), axis=1),
concatenate(
(B2 * T1 * C1, A2 + sign * B2 * D1 * E_C2), axis=1)),
axis=0)
B = concatenate((B1 * T2, B2 * D1 * T2), axis=0)
C = concatenate((T1 * C1, sign * D1 * E_C2), axis=1)
D = D1 * T2
return StateSpace(A, B, C, D, dt)
def minreal(self, tol=0.0):
"""Calculate a minimal realization, removes unobservable and
uncontrollable states"""
if self.states:
try:
from slycot import tb01pd
B = empty((self.states, max(self.inputs, self.outputs)))
B[:,:self.inputs] = self.B
C = empty((max(self.outputs, self.inputs), self.states))
C[:self.outputs,:] = self.C
A, B, C, nr = tb01pd(self.states, self.inputs, self.outputs,
self.A, B, C, tol=tol)
return StateSpace(A[:nr,:nr], B[:nr,:self.inputs],
C[:self.outputs,:nr], self.D)
except ImportError:
raise TypeError("minreal requires slycot tb01pd")
else:
return StateSpace(self)
# TODO: add discrete time check
def returnScipySignalLTI(self):
"""Return a list of a list of scipy.signal.lti objects.
For instance,
>>> out = ssobject.returnScipySignalLTI()
>>> out[3][5]
is a signal.scipy.lti object corresponding to the transfer function from
the 6th input to the 4th output."""
# Preallocate the output.
out = [[[] for j in range(self.inputs)] for i in range(self.outputs)]
for i in range(self.outputs):
for j in range(self.inputs):
out[i][j] = lti(asarray(self.A), asarray(self.B[:, j]),
asarray(self.C[i, :]), asarray(self.D[i, j]))
return out
def append(self, other):
"""Append a second model to the present model. The second
model is converted to state-space if necessary, inputs and
outputs are appended and their order is preserved"""
if not isinstance(other, StateSpace):
other = _convertToStateSpace(other)
if self.dt != other.dt:
raise ValueError("Systems must have the same time step")
n = self.states + other.states
m = self.inputs + other.inputs
p = self.outputs + other.outputs
A = zeros( (n, n) )
B = zeros( (n, m) )
C = zeros( (p, n) )
D = zeros( (p, m) )
A[:self.states,:self.states] = self.A
A[self.states:,self.states:] = other.A
B[:self.states,:self.inputs] = self.B
B[self.states:,self.inputs:] = other.B
C[:self.outputs,:self.states] = self.C
C[self.outputs:,self.states:] = other.C
D[:self.outputs,:self.inputs] = self.D
D[self.outputs:,self.inputs:] = other.D
return StateSpace(A, B, C, D, self.dt)
def __getitem__(self, indices):
"""Array style acces"""
if len(indices) != 2:
raise IOError('must provide indices of length 2 for state space')
i = indices[0]
j = indices[1]
return StateSpace(self.A,
self.B[:,j],
self.C[i,:],
self.D[i,j], self.dt)
def sample(self, Ts, method='zoh', alpha=None):
"""Convert a continuous time system to discrete time
Creates a discrete-time system from a continuous-time system by
sampling. Multiple methods of conversion are supported.
Parameters
----------
Ts : float
Sampling period
method : {"gbt", "bilinear", "euler", "backward_diff", "zoh"}
Which method to use:
* gbt: generalized bilinear transformation
* bilinear: Tustin's approximation ("gbt" with alpha=0.5)
* euler: Euler (or forward differencing) method ("gbt" with alpha=0)
* backward_diff: Backwards differencing ("gbt" with alpha=1.0)
* zoh: zero-order hold (default)
alpha : float within [0, 1]
The generalized bilinear transformation weighting parameter, which
should only be specified with method="gbt", and is ignored otherwise
Returns
-------
sysd : StateSpace system
Discrete time system, with sampling rate Ts
Notes
-----
Uses the command 'cont2discrete' from scipy.signal
Examples
--------
>>> sys = StateSpace(0, 1, 1, 0)
>>> sysd = sys.sample(0.5, method='bilinear')
"""
if not self.isctime():
raise ValueError("System must be continuous time system")
sys = (self.A, self.B, self.C, self.D)
Ad, Bd, C, D, dt = cont2discrete(sys, Ts, method, alpha)
return StateSpace(Ad, Bd, C, D, dt)
def dcgain(self):
"""Return the zero-frequency gain
The zero-frequency gain of a continuous-time state-space
system is given by:
.. math: G(0) = - C A^{-1} B + D
and of a discrete-time state-space system by:
.. math: G(1) = C (I - A)^{-1} B + D
Returns
-------
gain : ndarray
An array of shape (outputs,inputs); the array will either
be the zero-frequency (or DC) gain, or, if the frequency
response is singular, the array will be filled with np.nan.
"""
try:
if self.isctime():
gain = np.asarray(self.D -
self.C.dot(np.linalg.solve(self.A, self.B)))
else:
gain = self.horner(1)
except LinAlgError:
# eigenvalue at DC
gain = np.tile(np.nan,(self.outputs,self.inputs))
return np.squeeze(gain)
# TODO: add discrete time check
def _convertToStateSpace(sys, **kw):
"""Convert a system to state space form (if needed).
If sys is already a state space, then it is returned. If sys is a transfer
function object, then it is converted to a state space and returned. If sys
is a scalar, then the number of inputs and outputs can be specified
manually, as in:
>>> sys = _convertToStateSpace(3.) # Assumes inputs = outputs = 1
>>> sys = _convertToStateSpace(1., inputs=3, outputs=2)
In the latter example, A = B = C = 0 and D = [[1., 1., 1.]
[1., 1., 1.]].
"""
from .xferfcn import TransferFunction
import itertools
if isinstance(sys, StateSpace):
if len(kw):
raise TypeError("If sys is a StateSpace, _convertToStateSpace \
cannot take keywords.")
# Already a state space system; just return it
return sys
elif isinstance(sys, TransferFunction):
try:
from slycot import td04ad
if len(kw):
raise TypeError("If sys is a TransferFunction, _convertToStateSpace \
cannot take keywords.")
# Change the numerator and denominator arrays so that the transfer
# function matrix has a common denominator.
num, den = sys._common_den()
# Make a list of the orders of the denominator polynomials.
index = [len(den) - 1 for i in range(sys.outputs)]
# Repeat the common denominator along the rows.
den = array([den for i in range(sys.outputs)])
#! TODO: transfer function to state space conversion is still buggy!
#print num
#print shape(num)
ssout = td04ad('R',sys.inputs, sys.outputs, index, den, num,tol=0.0)
states = ssout[0]
return StateSpace(ssout[1][:states, :states],
ssout[2][:states, :sys.inputs],
ssout[3][:sys.outputs, :states],
ssout[4], sys.dt)
except ImportError:
# No Slycot. Scipy tf->ss can't handle MIMO, but static
# MIMO is an easy special case we can check for here
maxn = max(max(len(n) for n in nrow)
for nrow in sys.num)
maxd = max(max(len(d) for d in drow)
for drow in sys.den)
if 1==maxn and 1==maxd:
D = empty((sys.outputs,sys.inputs),dtype=float)
for i,j in itertools.product(range(sys.outputs),range(sys.inputs)):
D[i,j] = sys.num[i][j][0] / sys.den[i][j][0]
return StateSpace([], [], [], D, sys.dt)
else:
if (sys.inputs != 1 or sys.outputs != 1):
raise TypeError("No support for MIMO without slycot")
# TODO: do we want to squeeze first and check dimenations?
# I think this will fail if num and den aren't 1-D after
# the squeeze
A, B, C, D = sp.signal.tf2ss(squeeze(sys.num), squeeze(sys.den))
return StateSpace(A, B, C, D, sys.dt)
elif isinstance(sys, (int, float, complex, np.number)):
if "inputs" in kw:
inputs = kw["inputs"]
else:
inputs = 1
if "outputs" in kw:
outputs = kw["outputs"]
else:
outputs = 1
# Generate a simple state space system of the desired dimension
# The following Doesn't work due to inconsistencies in ltisys:
# return StateSpace([[]], [[]], [[]], eye(outputs, inputs))
return StateSpace(0., zeros((1, inputs)), zeros((outputs, 1)),
sys * ones((outputs, inputs)))
# If this is a matrix, try to create a constant feedthrough
try:
D = matrix(sys)
outputs, inputs = D.shape
return StateSpace(0., zeros((1, inputs)), zeros((outputs, 1)), D)
except Exception(e):
print("Failure to assume argument is matrix-like in" \
" _convertToStateSpace, result %s" % e)
raise TypeError("Can't convert given type to StateSpace system.")
# TODO: add discrete time option
def _rss_generate(states, inputs, outputs, type):
"""Generate a random state space.
This does the actual random state space generation expected from rss and
drss. type is 'c' for continuous systems and 'd' for discrete systems.
"""
# Probability of repeating a previous root.
pRepeat = 0.05
# Probability of choosing a real root. Note that when choosing a complex
# root, the conjugate gets chosen as well. So the expected proportion of
# real roots is pReal / (pReal + 2 * (1 - pReal)).
pReal = 0.6
# Probability that an element in B or C will not be masked out.
pBCmask = 0.8
# Probability that an element in D will not be masked out.
pDmask = 0.3
# Probability that D = 0.
pDzero = 0.5
# Check for valid input arguments.
if states < 1 or states % 1:
raise ValueError("states must be a positive integer. states = %g." %
states)
if inputs < 1 or inputs % 1:
raise ValueError("inputs must be a positive integer. inputs = %g." %
inputs)
if outputs < 1 or outputs % 1:
raise ValueError("outputs must be a positive integer. outputs = %g." %
outputs)
# Make some poles for A. Preallocate a complex array.
poles = zeros(states) + zeros(states) * 0.j
i = 0
while i < states:
if rand() < pRepeat and i != 0 and i != states - 1:
# Small chance of copying poles, if we're not at the first or last
# element.
if poles[i-1].imag == 0:
# Copy previous real pole.
poles[i] = poles[i-1]
i += 1
else:
# Copy previous complex conjugate pair of poles.
poles[i:i+2] = poles[i-2:i]
i += 2
elif rand() < pReal or i == states - 1:
# No-oscillation pole.
if type == 'c':
poles[i] = -exp(randn()) + 0.j
elif type == 'd':
poles[i] = 2. * rand() - 1.
i += 1
else:
# Complex conjugate pair of oscillating poles.
if type == 'c':
poles[i] = complex(-exp(randn()), 3. * exp(randn()))
elif type == 'd':
mag = rand()
phase = 2. * math.pi * rand()
poles[i] = complex(mag * cos(phase),
mag * sin(phase))
poles[i+1] = complex(poles[i].real, -poles[i].imag)
i += 2
# Now put the poles in A as real blocks on the diagonal.
A = zeros((states, states))
i = 0
while i < states:
if poles[i].imag == 0:
A[i, i] = poles[i].real
i += 1
else:
A[i, i] = A[i+1, i+1] = poles[i].real
A[i, i+1] = poles[i].imag
A[i+1, i] = -poles[i].imag
i += 2
# Finally, apply a transformation so that A is not block-diagonal.
while True:
T = randn(states, states)
try:
A = dot(solve(T, A), T) # A = T \ A * T
break
except LinAlgError:
# In the unlikely event that T is rank-deficient, iterate again.
pass
# Make the remaining matrices.
B = randn(states, inputs)
C = randn(outputs, states)
D = randn(outputs, inputs)
# Make masks to zero out some of the elements.
while True:
Bmask = rand(states, inputs) < pBCmask
if any(Bmask): # Retry if we get all zeros.
break
while True:
Cmask = rand(outputs, states) < pBCmask
if any(Cmask): # Retry if we get all zeros.
break
if rand() < pDzero:
Dmask = zeros((outputs, inputs))
else:
Dmask = rand(outputs, inputs) < pDmask
# Apply masks.
B = B * Bmask
C = C * Cmask
D = D * Dmask
return StateSpace(A, B, C, D)
# Convert a MIMO system to a SISO system
# TODO: add discrete time check
def _mimo2siso(sys, input, output, warn_conversion=False):
#pylint: disable=W0622
"""
Convert a MIMO system to a SISO system. (Convert a system with multiple
inputs and/or outputs, to a system with a single input and output.)
The input and output that are used in the SISO system can be selected
with the parameters ``input`` and ``output``. All other inputs are set
to 0, all other outputs are ignored.
If ``sys`` is already a SISO system, it will be returned unaltered.
Parameters
----------
sys: StateSpace
Linear (MIMO) system that should be converted.
input: int
Index of the input that will become the SISO system's only input.
output: int
Index of the output that will become the SISO system's only output.
warn_conversion: bool
If True: print a warning message when sys is a MIMO system.
Warn that a conversion will take place.
Returns:
sys: StateSpace
The converted (SISO) system.
"""
if not (isinstance(input, int) and isinstance(output, int)):
raise TypeError("Parameters ``input`` and ``output`` must both "
"be integer numbers.")
if not (0 <= input < sys.inputs):
raise ValueError("Selected input does not exist. "
"Selected input: {sel}, "
"number of system inputs: {ext}."
.format(sel=input, ext=sys.inputs))
if not (0 <= output < sys.outputs):
raise ValueError("Selected output does not exist. "
"Selected output: {sel}, "
"number of system outputs: {ext}."
.format(sel=output, ext=sys.outputs))
#Convert sys to SISO if necessary
if sys.inputs > 1 or sys.outputs > 1:
if warn_conversion:
warnings.warn("Converting MIMO system to SISO system. "
"Only input {i} and output {o} are used."
.format(i=input, o=output))
# $X = A*X + B*U
# Y = C*X + D*U
new_B = sys.B[:, input]
new_C = sys.C[output, :]
new_D = sys.D[output, input]
sys = StateSpace(sys.A, new_B, new_C, new_D, sys.dt)
return sys
def _mimo2simo(sys, input, warn_conversion=False):
#pylint: disable=W0622
"""
Convert a MIMO system to a SIMO system. (Convert a system with multiple
inputs and/or outputs, to a system with a single input but possibly
multiple outputs.)
The input that is used in the SIMO system can be selected with the
parameter ``input``. All other inputs are set to 0, all other
outputs are ignored.
If ``sys`` is already a SIMO system, it will be returned unaltered.
Parameters
----------
sys: StateSpace
Linear (MIMO) system that should be converted.
input: int
Index of the input that will become the SIMO system's only input.
warn_conversion: bool
If True: print a warning message when sys is a MIMO system.
Warn that a conversion will take place.
Returns:
--------
sys: StateSpace
The converted (SIMO) system.
"""
if not (isinstance(input, int)):
raise TypeError("Parameter ``input`` be an integer number.")
if not (0 <= input < sys.inputs):
raise ValueError("Selected input does not exist. "
"Selected input: {sel}, "
"number of system inputs: {ext}."
.format(sel=input, ext=sys.inputs))
#Convert sys to SISO if necessary
if sys.inputs > 1:
if warn_conversion:
warnings.warn("Converting MIMO system to SIMO system. "
"Only input {i} is used."
.format(i=input))
# $X = A*X + B*U
# Y = C*X + D*U
new_B = sys.B[:, input]
new_D = sys.D[:, input]
sys = StateSpace(sys.A, new_B, sys.C, new_D, sys.dt)
return sys
def ss(*args):
"""ss(A, B, C, D[, dt])
Create a state space system.
The function accepts either 1, 4 or 5 parameters:
``ss(sys)``
Convert a linear system into space system form. Always creates a
new system, even if sys is already a StateSpace object.
``ss(A, B, C, D)``
Create a state space system from the matrices of its state and
output equations:
.. math::
\dot x = A \cdot x + B \cdot u
y = C \cdot x + D \cdot u
``ss(A, B, C, D, dt)``
Create a discrete-time state space system from the matrices of
its state and output equations:
.. math::
x[k+1] = A \cdot x[k] + B \cdot u[k]
y[k] = C \cdot x[k] + D \cdot u[ki]
The matrices can be given as *array like* data types or strings.
Everything that the constructor of :class:`numpy.matrix` accepts is
permissible here too.
Parameters
----------
sys: StateSpace or TransferFunction
A linear system