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#! TODO: add module docstring
# modelsimp.py - tools for model simplification
#
# Author: Steve Brunton, Kevin Chen, Lauren Padilla
# Date: 30 Nov 2010
#
# This file contains routines for obtaining reduced order models
#
# 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.
#
# $Id$
# External packages and modules
import numpy as np
import warnings
from .exception import ControlSlycot, ControlMIMONotImplemented, \
ControlDimension
from .namedio import isdtime, isctime
from .statesp import StateSpace
from .statefbk import gram
__all__ = ['hsvd', 'balred', 'modred', 'era', 'markov', 'minreal']
# Hankel Singular Value Decomposition
#
# The following returns the Hankel singular values, which are singular values
# of the matrix formed by multiplying the controllability and observability
# Gramians
def hsvd(sys):
"""Calculate the Hankel singular values.
Parameters
----------
sys : StateSpace
A state space system
Returns
-------
H : array
A list of Hankel singular values
See Also
--------
gram
Notes
-----
The Hankel singular values are the singular values of the Hankel operator.
In practice, we compute the square root of the eigenvalues of the matrix
formed by taking the product of the observability and controllability
gramians. There are other (more efficient) methods based on solving the
Lyapunov equation in a particular way (more details soon).
Examples
--------
>>> H = hsvd(sys)
"""
# TODO: implement for discrete time systems
if (isdtime(sys, strict=True)):
raise NotImplementedError("Function not implemented in discrete time")
Wc = gram(sys, 'c')
Wo = gram(sys, 'o')
WoWc = Wo @ Wc
w, v = np.linalg.eig(WoWc)
hsv = np.sqrt(w)
hsv = np.array(hsv)
hsv = np.sort(hsv)
# Return the Hankel singular values, high to low
return hsv[::-1]
def modred(sys, ELIM, method='matchdc'):
"""
Model reduction of `sys` by eliminating the states in `ELIM` using a given
method.
Parameters
----------
sys: StateSpace
Original system to reduce
ELIM: array
Vector of states to eliminate
method: string
Method of removing states in `ELIM`: either ``'truncate'`` or
``'matchdc'``.
Returns
-------
rsys: StateSpace
A reduced order model
Raises
------
ValueError
Raised under the following conditions:
* if `method` is not either ``'matchdc'`` or ``'truncate'``
* if eigenvalues of `sys.A` are not all in left half plane
(`sys` must be stable)
Examples
--------
>>> rsys = modred(sys, ELIM, method='truncate')
"""
# Check for ss system object, need a utility for this?
# TODO: Check for continous or discrete, only continuous supported for now
# if isCont():
# dico = 'C'
# elif isDisc():
# dico = 'D'
# else:
if (isctime(sys)):
dico = 'C'
else:
raise NotImplementedError("Function not implemented in discrete time")
# Check system is stable
if np.any(np.linalg.eigvals(sys.A).real >= 0.0):
raise ValueError("Oops, the system is unstable!")
ELIM = np.sort(ELIM)
# Create list of elements not to eliminate (NELIM)
NELIM = [i for i in range(len(sys.A)) if i not in ELIM]
# A1 is a matrix of all columns of sys.A not to eliminate
A1 = sys.A[:, NELIM[0]].reshape(-1, 1)
for i in NELIM[1:]:
A1 = np.hstack((A1, sys.A[:, i].reshape(-1, 1)))
A11 = A1[NELIM, :]
A21 = A1[ELIM, :]
# A2 is a matrix of all columns of sys.A to eliminate
A2 = sys.A[:, ELIM[0]].reshape(-1, 1)
for i in ELIM[1:]:
A2 = np.hstack((A2, sys.A[:, i].reshape(-1, 1)))
A12 = A2[NELIM, :]
A22 = A2[ELIM, :]
C1 = sys.C[:, NELIM]
C2 = sys.C[:, ELIM]
B1 = sys.B[NELIM, :]
B2 = sys.B[ELIM, :]
if method == 'matchdc':
# if matchdc, residualize
# Check if the matrix A22 is invertible
if np.linalg.matrix_rank(A22) != len(ELIM):
raise ValueError("Matrix A22 is singular to working precision.")
# Now precompute A22\A21 and A22\B2 (A22I = inv(A22))
# We can solve two linear systems in one pass, since the
# coefficients matrix A22 is the same. Thus, we perform the LU
# decomposition (cubic runtime complexity) of A22 only once!
# The remaining back substitutions are only quadratic in runtime.
A22I_A21_B2 = np.linalg.solve(A22, np.concatenate((A21, B2), axis=1))
A22I_A21 = A22I_A21_B2[:, :A21.shape[1]]
A22I_B2 = A22I_A21_B2[:, A21.shape[1]:]
Ar = A11 - A12 @ A22I_A21
Br = B1 - A12 @ A22I_B2
Cr = C1 - C2 @ A22I_A21
Dr = sys.D - C2 @ A22I_B2
elif method == 'truncate':
# if truncate, simply discard state x2
Ar = A11
Br = B1
Cr = C1
Dr = sys.D
else:
raise ValueError("Oops, method is not supported!")
rsys = StateSpace(Ar, Br, Cr, Dr)
return rsys
def balred(sys, orders, method='truncate', alpha=None):
"""Balanced reduced order model of sys of a given order.
States are eliminated based on Hankel singular value.
If sys has unstable modes, they are removed, the
balanced realization is done on the stable part, then
reinserted in accordance with the reference below.
Reference: Hsu,C.S., and Hou,D., 1991,
Reducing unstable linear control systems via real Schur transformation.
Electronics Letters, 27, 984-986.
Parameters
----------
sys: StateSpace
Original system to reduce
orders: integer or array of integer
Desired order of reduced order model (if a vector, returns a vector
of systems)
method: string
Method of removing states, either ``'truncate'`` or ``'matchdc'``.
alpha: float
Redefines the stability boundary for eigenvalues of the system
matrix A. By default for continuous-time systems, alpha <= 0
defines the stability boundary for the real part of A's eigenvalues
and for discrete-time systems, 0 <= alpha <= 1 defines the stability
boundary for the modulus of A's eigenvalues. See SLICOT routines
AB09MD and AB09ND for more information.
Returns
-------
rsys: StateSpace
A reduced order model or a list of reduced order models if orders is
a list.
Raises
------
ValueError
If `method` is not ``'truncate'`` or ``'matchdc'``
ImportError
if slycot routine ab09ad, ab09md, or ab09nd is not found
ValueError
if there are more unstable modes than any value in orders
Examples
--------
>>> rsys = balred(sys, orders, method='truncate')
"""
if method != 'truncate' and method != 'matchdc':
raise ValueError("supported methods are 'truncate' or 'matchdc'")
elif method == 'truncate':
try:
from slycot import ab09md, ab09ad
except ImportError:
raise ControlSlycot(
"can't find slycot subroutine ab09md or ab09ad")
elif method == 'matchdc':
try:
from slycot import ab09nd
except ImportError:
raise ControlSlycot("can't find slycot subroutine ab09nd")
# Check for ss system object, need a utility for this?
# TODO: Check for continous or discrete, only continuous supported for now
# if isCont():
# dico = 'C'
# elif isDisc():
# dico = 'D'
# else:
dico = 'C'
job = 'B' # balanced (B) or not (N)
equil = 'N' # scale (S) or not (N)
if alpha is None:
if dico == 'C':
alpha = 0.
elif dico == 'D':
alpha = 1.
rsys = [] # empty list for reduced systems
# check if orders is a list or a scalar
try:
order = iter(orders)
except TypeError: # if orders is a scalar
orders = [orders]
for i in orders:
n = np.size(sys.A, 0)
m = np.size(sys.B, 1)
p = np.size(sys.C, 0)
if method == 'truncate':
# check system stability
if np.any(np.linalg.eigvals(sys.A).real >= 0.0):
# unstable branch
Nr, Ar, Br, Cr, Ns, hsv = ab09md(
dico, job, equil, n, m, p, sys.A, sys.B, sys.C,
alpha=alpha, nr=i, tol=0.0)
else:
# stable branch
Nr, Ar, Br, Cr, hsv = ab09ad(
dico, job, equil, n, m, p, sys.A, sys.B, sys.C,
nr=i, tol=0.0)
rsys.append(StateSpace(Ar, Br, Cr, sys.D))
elif method == 'matchdc':
Nr, Ar, Br, Cr, Dr, Ns, hsv = ab09nd(
dico, job, equil, n, m, p, sys.A, sys.B, sys.C, sys.D,
alpha=alpha, nr=i, tol1=0.0, tol2=0.0)
rsys.append(StateSpace(Ar, Br, Cr, Dr))
# if orders was a scalar, just return the single reduced model, not a list
if len(orders) == 1:
return rsys[0]
# if orders was a list/vector, return a list/vector of systems
else:
return rsys
def minreal(sys, tol=None, verbose=True):
'''
Eliminates uncontrollable or unobservable states in state-space
models or cancelling pole-zero pairs in transfer functions. The
output sysr has minimal order and the same response
characteristics as the original model sys.
Parameters
----------
sys: StateSpace or TransferFunction
Original system
tol: real
Tolerance
verbose: bool
Print results if True
Returns
-------
rsys: StateSpace or TransferFunction
Cleaned model
'''
sysr = sys.minreal(tol)
if verbose:
print("{nstates} states have been removed from the model".format(
nstates=len(sys.poles()) - len(sysr.poles())))
return sysr
def era(YY, m, n, nin, nout, r):
"""Calculate an ERA model of order `r` based on the impulse-response data
`YY`.
.. note:: This function is not implemented yet.
Parameters
----------
YY: array
`nout` x `nin` dimensional impulse-response data
m: integer
Number of rows in Hankel matrix
n: integer
Number of columns in Hankel matrix
nin: integer
Number of input variables
nout: integer
Number of output variables
r: integer
Order of model
Returns
-------
sys: StateSpace
A reduced order model sys=ss(Ar,Br,Cr,Dr)
Examples
--------
>>> rsys = era(YY, m, n, nin, nout, r)
"""
raise NotImplementedError('This function is not implemented yet.')
def markov(Y, U, m=None, transpose=False):
"""Calculate the first `m` Markov parameters [D CB CAB ...]
from input `U`, output `Y`.
This function computes the Markov parameters for a discrete time system
.. math::
x[k+1] &= A x[k] + B u[k] \\\\
y[k] &= C x[k] + D u[k]
given data for u and y. The algorithm assumes that that C A^k B = 0 for
k > m-2 (see [1]_). Note that the problem is ill-posed if the length of
the input data is less than the desired number of Markov parameters (a
warning message is generated in this case).
Parameters
----------
Y : array_like
Output data. If the array is 1D, the system is assumed to be single
input. If the array is 2D and transpose=False, the columns of `Y`
are taken as time points, otherwise the rows of `Y` are taken as
time points.
U : array_like
Input data, arranged in the same way as `Y`.
m : int, optional
Number of Markov parameters to output. Defaults to len(U).
transpose : bool, optional
Assume that input data is transposed relative to the standard
:ref:`time-series-convention`. Default value is False.
Returns
-------
H : ndarray
First m Markov parameters, [D CB CAB ...]
References
----------
.. [1] J.-N. Juang, M. Phan, L. G. Horta, and R. W. Longman,
Identification of observer/Kalman filter Markov parameters - Theory
and experiments. Journal of Guidance Control and Dynamics, 16(2),
320-329, 2012. http://doi.org/10.2514/3.21006
Notes
-----
Currently only works for SISO systems.
This function does not currently comply with the Python Control Library
:ref:`time-series-convention` for representation of time series data.
Use `transpose=False` to make use of the standard convention (this
will be updated in a future release).
Examples
--------
>>> T = numpy.linspace(0, 10, 100)
>>> U = numpy.ones((1, 100))
>>> T, Y, _ = forced_response(tf([1], [1, 0.5], True), T, U)
>>> H = markov(Y, U, 3, transpose=False)
"""
# Convert input parameters to 2D arrays (if they aren't already)
Umat = np.array(U, ndmin=2)
Ymat = np.array(Y, ndmin=2)
# If data is in transposed format, switch it around
if transpose:
Umat, Ymat = np.transpose(Umat), np.transpose(Ymat)
# Make sure the system is a SISO system
if Umat.shape[0] != 1 or Ymat.shape[0] != 1:
raise ControlMIMONotImplemented
# Make sure the number of time points match
if Umat.shape[1] != Ymat.shape[1]:
raise ControlDimension(
"Input and output data are of differnent lengths")
n = Umat.shape[1]
# If number of desired parameters was not given, set to size of input data
if m is None:
m = Umat.shape[1]
# Make sure there is enough data to compute parameters
if m > n:
warnings.warn("Not enough data for requested number of parameters")
#
# Original algorithm (with mapping to standard order)
#
# RMM note, 24 Dec 2020: This algorithm sets the problem up correctly
# until the final column of the UU matrix is created, at which point it
# makes some modifications that I don't understand. This version of the
# algorithm does not seem to return the actual Markov parameters for a
# system.
#
# # Create the matrix of (shifted) inputs
# UU = np.transpose(Umat)
# for i in range(1, m-1):
# # Shift previous column down and add a zero at the top
# newCol = np.vstack((0, np.reshape(UU[0:n-1, i-1], (-1, 1))))
# UU = np.hstack((UU, newCol))
#
# # Shift previous column down and add a zero at the top
# Ulast = np.vstack((0, np.reshape(UU[0:n-1, m-2], (-1, 1))))
#
# # Replace the elements of the last column new values (?)
# # Each row gets the sum of the rows above it (?)
# for i in range(n-1, 0, -1):
# Ulast[i] = np.sum(Ulast[0:i-1])
# UU = np.hstack((UU, Ulast))
#
# # Solve for the Markov parameters from Y = H @ UU
# # H = [[D], [CB], [CAB], ..., [C A^{m-3} B], [???]]
# H = np.linalg.lstsq(UU, np.transpose(Ymat))[0]
#
# # Markov parameters are in rows => transpose if needed
# return H if transpose else np.transpose(H)
#
# New algorithm - Construct a matrix of control inputs to invert
#
# This algorithm sets up the following problem and solves it for
# the Markov parameters
#
# [ y(0) ] [ u(0) 0 0 ] [ D ]
# [ y(1) ] [ u(1) u(0) 0 ] [ C B ]
# [ y(2) ] = [ u(2) u(1) u(0) ] [ C A B ]
# [ : ] [ : : : : ] [ : ]
# [ y(n-1) ] [ u(n-1) u(n-2) u(n-3) ... u(n-m) ] [ C A^{m-2} B ]
#
# Note: if the number of Markov parameters (m) is less than the size of
# the input/output data (n), then this algorithm assumes C A^{j} B = 0
# for j > m-2. See equation (3) in
#
# J.-N. Juang, M. Phan, L. G. Horta, and R. W. Longman, Identification
# of observer/Kalman filter Markov parameters - Theory and
# experiments. Journal of Guidance Control and Dynamics, 16(2),
# 320-329, 2012. http://doi.org/10.2514/3.21006
#
# Create matrix of (shifted) inputs
UU = Umat
for i in range(1, m):
# Shift previous column down and add a zero at the top
new_row = np.hstack((0, UU[i-1, 0:-1]))
UU = np.vstack((UU, new_row))
UU = np.transpose(UU)
# Invert and solve for Markov parameters
YY = np.transpose(Ymat)
H, _, _, _ = np.linalg.lstsq(UU, YY, rcond=None)
# Return the first m Markov parameters
return H if transpose else np.transpose(H)