[43ce9a]: inst / spamodred.m Maximize Restore History

Download this file

spamodred.m    225 lines (219 with data), 7.9 kB

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
## Copyright (C) 2011 Lukas F. Reichlin
##
## This file is part of LTI Syncope.
##
## LTI Syncope is free software: you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## LTI Syncope is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with LTI Syncope. If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{opt}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @var{opt}, @dots{})
##
## Model order reduction by frequency weighted Singular Perturbation Approximation (SPA).
## The aim of model reduction is to find an LTI system @var{Gr} of order
## @var{nr} (nr < n) such that the input-output behaviour of @var{Gr}
## approximates the one from original system @var{G}.
##
## SPA is an absolute error method which tries to minimize
## @iftex
## @tex
## $$ || G - G_r ||_{\\infty} = min $$
## $$ || V \\ (G - G_r) \\ W ||_{\\infty} = min $$
## @end tex
## @end iftex
## @ifnottex
## @example
## ||G-Gr|| = min
## inf
##
## ||V (G-Gr) W|| = min
## inf
## @end example
## @end ifnottex
## where @var{V} and @var{W} denote output and input weightings.
##
##
## @strong{Inputs}
## @table @var
## @item G
## LTI model to be reduced.
## @item nr
## The desired order of the resulting reduced order system @var{Gr}.
## If not specified, @var{nr} is chosen automatically according
## to the description of key @var{'order'}.
## @item @dots{}
## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}.
## @item opt
## Optional struct with keys as field names.
## Struct @var{opt} can be created directly or
## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
## @end table
##
## @strong{Outputs}
## @table @var
## @item Gr
## Reduced order state-space model.
## @item info
## Struct containing additional information.
## @table @var
## @item info.n
## The order of the original system @var{G}.
## @item info.ns
## The order of the @var{alpha}-stable subsystem of the original system @var{G}.
## @item info.hsv
## The Hankel singular values of the @var{alpha}-stable part of
## the original system @var{G}, ordered decreasingly.
## @item info.nu
## The order of the @var{alpha}-unstable subsystem of both the original
## system @var{G} and the reduced-order system @var{Gr}.
## @item info.nr
## The order of the obtained reduced order system @var{Gr}.
## @end table
## @end table
##
##
## @strong{Option Keys and Values}
## @table @var
## @item 'order', 'nr'
## The desired order of the resulting reduced order system @var{Gr}.
## If not specified, @var{nr} is chosen automatically such that states with
## Hankel singular values @var{info.hsv} > @var{tol1} are retained.
##
## @item 'left', 'output'
## LTI model of the left/output frequency weighting @var{V}.
## Default value is an identity matrix.
##
## @item 'right', 'input'
## LTI model of the right/input frequency weighting @var{W}.
## Default value is an identity matrix.
##
## @item 'method'
## Approximation method for the L-infinity norm to be used as follows:
## @table @var
## @item 'sr', 's'
## Use the square-root Singular Perturbation Approximation method.
## @item 'bfsr', 'p'
## Use the balancing-free square-root Singular Perturbation Approximation method. Default method.
## @end table
##
## @item 'alpha'
## Specifies the ALPHA-stability boundary for the eigenvalues
## of the state dynamics matrix @var{G.A}. For a continuous-time
## system, ALPHA <= 0 is the boundary value for
## the real parts of eigenvalues, while for a discrete-time
## system, 0 <= ALPHA <= 1 represents the
## boundary value for the moduli of eigenvalues.
## The ALPHA-stability domain does not include the boundary.
## Default value is 0 for continuous-time systems and
## 1 for discrete-time systems.
##
## @item 'tol1'
## If @var{'order'} is not specified, @var{tol1} contains the tolerance for
## determining the order of the reduced model.
## For model reduction, the recommended value of @var{tol1} is
## c*info.hsv(1), where c lies in the interval [0.00001, 0.001].
## Default value is info.ns*eps*info.hsv(1).
## If @var{'order'} is specified, the value of @var{tol1} is ignored.
##
## @item 'tol2'
## The tolerance for determining the order of a minimal
## realization of the ALPHA-stable part of the given
## model. TOL2 <= TOL1.
## If not specified, ns*eps*info.hsv(1) is chosen.
##
## @item 'gram-ctrb'
## Specifies the choice of frequency-weighted controllability
## Grammian as follows:
## @table @var
## @item 'standard'
## Choice corresponding to a combination method [4]
## of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method.
## @item 'enhanced'
## Choice corresponding to the stability enhanced
## modified combination method of [4].
## @end table
##
## @item 'gram-obsv'
## Specifies the choice of frequency-weighted observability
## Grammian as follows:
## @table @var
## @item 'standard'
## Choice corresponding to a combination method [4]
## of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method.
## @item 'enhanced'
## Choice corresponding to the stability enhanced
## modified combination method of [4].
## @end table
##
## @item 'alpha-ctrb'
## Combination method parameter for defining the
## frequency-weighted controllability Grammian.
## abs(alphac) <= 1.
## If alphac = 0, the choice of
## Grammian corresponds to the method of Enns [1], while if
## alphac = 1, the choice of Grammian corresponds
## to the method of Lin and Chiu [2,3].
## Default value is 0.
##
## @item 'alpha-obsv'
## Combination method parameter for defining the
## frequency-weighted observability Grammian.
## abs(alphao) <= 1.
## If alphao = 0, the choice of
## Grammian corresponds to the method of Enns [1], while if
## alphao = 1, the choice of Grammian corresponds
## to the method of Lin and Chiu [2,3].
## Default value is 0.
##
## @item 'equil', 'scale'
## Boolean indicating whether equilibration (scaling) should be
## performed on system @var{G} prior to order reduction.
## Default value is true if @code{G.scaled == false} and
## false if @code{G.scaled == true}.
## @end table
##
##
## @strong{References}@*
## [1] Enns, D.
## Model reduction with balanced realizations: An error bound
## and a frequency weighted generalization.
## Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984.
##
## [2] Lin, C.-A. and Chiu, T.-Y.
## Model reduction via frequency-weighted balanced realization.
## Control Theory and Advanced Technology, vol. 8,
## pp. 341-351, 1992.
##
## [3] Sreeram, V., Anderson, B.D.O and Madievski, A.G.
## New results on frequency weighted balanced reduction
## technique.
## Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995.
##
## [4] Varga, A. and Anderson, B.D.O.
## Square-root balancing-free methods for the frequency-weighted
## balancing related model reduction.
## (report in preparation)
##
##
## @strong{Algorithm}@*
## Uses SLICOT AB09ID by courtesy of
## @uref{http://www.slicot.org, NICONET e.V.}
## @end deftypefn
## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
## Created: November 2011
## Version: 0.1
function [Gr, info] = spamodred (varargin)
[Gr, info] = __modred_ab09id__ ("spa", varargin{:});
endfunction
## TODO: add a test