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# linflat.py - FlatSystem subclass for linear systems
# RMM, 10 November 2012
#
# This file defines a FlatSystem class for a linear system.
#
# Copyright (c) 2012 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.
import numpy as np
import control
from .flatsys import FlatSystem
from ..iosys import LinearIOSystem
class LinearFlatSystem(FlatSystem, LinearIOSystem):
"""Base class for a linear, differentially flat system.
This class is used to create a differentially flat system representation
from a linear system.
Parameters
----------
linsys : StateSpace
LTI StateSpace system to be converted
inputs : int, list of str or None, optional
Description of the system inputs. This can be given as an integer
count or as a list of strings that name the individual signals.
If an integer count is specified, the names of the signal will be
of the form `s[i]` (where `s` is one of `u`, `y`, or `x`). If
this parameter is not given or given as `None`, the relevant
quantity will be determined when possible based on other
information provided to functions using the system.
outputs : int, list of str or None, optional
Description of the system outputs. Same format as `inputs`.
states : int, list of str, or None, optional
Description of the system states. Same format as `inputs`.
dt : None, True or float, optional
System timebase. None (default) indicates continuous
time, True indicates discrete time with undefined sampling
time, positive number is discrete time with specified
sampling time.
params : dict, optional
Parameter values for the systems. Passed to the evaluation
functions for the system as default values, overriding internal
defaults.
name : string, optional
System name (used for specifying signals)
"""
def __init__(self, linsys, inputs=None, outputs=None, states=None,
name=None):
"""Define a flat system from a SISO LTI system.
Given a reachable, single-input/single-output, linear time-invariant
system, create a differentially flat system representation.
"""
# Make sure we can handle the system
if (not control.isctime(linsys)):
raise control.ControlNotImplemented(
"requires continuous time, linear control system")
elif (not control.issiso(linsys)):
raise control.ControlNotImplemented(
"only single input, single output systems are supported")
# Initialize the object as a LinearIO system
LinearIOSystem.__init__(
self, linsys, inputs=inputs, outputs=outputs, states=states,
name=name)
# Find the transformation to chain of integrators form
# Note: store all array as ndarray, not matrix
zsys, Tr = control.reachable_form(linsys)
Tr = np.array(Tr[::-1, ::]) # flip rows
# Extract the information that we need
self.F = np.array(zsys.A[0, ::-1]) # input function coeffs
self.T = Tr # state space transformation
self.Tinv = np.linalg.inv(Tr) # compute inverse once
# Compute the flat output variable z = C x
Cfz = np.zeros(np.shape(linsys.C)); Cfz[0, 0] = 1
self.Cf = Cfz @ Tr
# Compute the flat flag from the state (and input)
def forward(self, x, u, params):
"""Compute the flat flag given the states and input.
See :func:`control.flatsys.FlatSystem.forward` for more info.
"""
x = np.reshape(x, (-1, 1))
u = np.reshape(u, (1, -1))
zflag = [np.zeros(self.nstates + 1)]
zflag[0][0] = self.Cf @ x
H = self.Cf # initial state transformation
for i in range(1, self.nstates + 1):
zflag[0][i] = H @ (self.A @ x + self.B @ u)
H = H @ self.A # derivative for next iteration
return zflag
# Compute state and input from flat flag
def reverse(self, zflag, params):
"""Compute the states and input given the flat flag.
See :func:`control.flatsys.FlatSystem.reverse` for more info.
"""
z = zflag[0][0:-1]
x = self.Tinv @ z
u = zflag[0][-1] - self.F @ z
return np.reshape(x, self.nstates), np.reshape(u, self.ninputs)
# Update function
def _rhs(self, t, x, u, params={}):
# Use LinearIOSystem._rhs instead of default (MRO) NonlinearIOSystem
return LinearIOSystem._rhs(self, t, x, u)
# output function
def _out(self, t, x, u, params={}):
# Use LinearIOSystem._out instead of default (MRO) NonlinearIOSystem
return LinearIOSystem._out(self, t, x, u)