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# 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 ctrlutil
from control.exception import *
# 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 grammians
def hsvd(sys):
"""Calculate the Hankel singular values
Usage
=====
H = hsvd(sys)
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).
Inputs
------
sys : a state space system
Outputs
-------
H : a list of Hankel singular values
"""
Wc = gram(sys,'c')
Wo = gram(sys,'o')
WoWc = np.dot(Wo, Wc)
w, v = LA.eig(WoWc)
hsv = np.sqrt(w)
hsv = np.matrix(hsv)
# Return the Hankel singular values
return hsv
def era(YY,m,n,nin,nout,r):
"""Calculate an ERA model of order r based on the impulse-response data YY
Usage
=====
sys = era(YY,m,n,nin,nout,r)
Inputs
------
YY : nout x nin dimensional impulse-response data
m : number of rows in Hankel matrix
n : number of columns in Hankel matrix
nin : number of input variables
nout : number of output variables
r : order of model
Outputs
-------
sys : a reduced order model sys=ss(Ar,Br,Cr,Dr)
"""
def markov(Y,U,M):
"""Calculate the first M Markov parameters [D CB CAB ...] from input U, output Y
Usage
=====
H = markov(Y,U,M)
Currently only works for SISO
Inputs
------
Y : output data
U : input data
M : number of Markov parameters to output
Outputs
-------
H : first M Markov parameters
"""
# Convert input parameters to matrices (if they aren't already)
Ymat = np.mat(Y)
Umat = np.mat(U)
n = np.size(U)
# Construct a matrix of control inputs to invert
UU = Umat
for i in range(1, M-1):
newCol = np.vstack((0, UU[0:n-1,i-2]))
UU = np.hstack((UU, newCol))
Ulast = np.vstack((0, UU[0:n-1,M-2]))
for i in range(n-1,0,-1):
Ulast[i] = np.sum(Ulast[0:i-1])
UU = np.hstack((UU, Ulast))
# Invert and solve for Markov parameters
H = UU.I
H = np.dot(H, Y)
return H