This file is indexed.

/usr/share/octave/packages/control-3.0.0/hnamodred.m is in octave-control 3.0.0-5.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  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
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
## Copyright (C) 2009-2015   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}] =} hnamodred (@var{G}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{nr}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{opt}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} hnamodred (@var{G}, @var{nr}, @var{opt}, @dots{})
##
## Model order reduction by frequency weighted optimal Hankel-norm (HNA) method.
## The aim of model reduction is to find an @acronym{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}.
##
## HNA is an absolute error method which tries to minimize
## @iftex
## @tex
## $$ || G - G_r ||_H = \\min $$
## $$ || V \\ (G - G_r) \\ W ||_H = \\min $$
## @end tex
## @end iftex
## @ifnottex
## @example
## ||G-Gr||  = min
##         H
##
## ||V (G-Gr) W||  = min
##               H
## @end example
## @end ifnottex
## where @var{V} and @var{W} denote output and input weightings.
##
##
## @strong{Inputs}
## @table @var
## @item G
## @acronym{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 function @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 corresponding to the projection @code{op(V)*G1*op(W)},
## where G1 denotes the @var{alpha}-stable part of the original system @var{G}. 
## The @var{ns} Hankel singular values are 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 the sum of @var{info.nu} and the number of
## Hankel singular values greater than @code{max(tol1, ns*eps*info.hsv(1)};
##
## @item 'method'
## Specifies the computational approach to be used.
## Valid values corresponding to this key are:
## @table @var
## @item 'descriptor'
## Use the inverse free descriptor system approach.
## @item 'standard'
## Use the inversion based standard approach.
## @item 'auto'
## Switch automatically to the inverse free
## descriptor approach in case of badly conditioned
## feedthrough matrices in V or W.  Default method.
## @end table
##
##
## @item 'left', 'v'
## @acronym{LTI} model of the left/output frequency weighting.
## The weighting must be antistable.
## @iftex
## @math{|| V \\ (G-G_r) \\dots ||_H = \\min}
## @end iftex
## @ifnottex
## @example
## || V (G-Gr) . ||  = min
##                 H
## @end example
## @end ifnottex
##
## @item 'right', 'w'
## @acronym{LTI} model of the right/input frequency weighting.
## The weighting must be antistable.
## @iftex
## @math{|| \\dots (G-G_r) \\ W ||_H = \\min}
## @end iftex
## @ifnottex
## @example
## || . (G-Gr) W ||  = min
##                 H
## @end example
## @end ifnottex
##
##
## @item 'left-inv', 'inv-v'
## @acronym{LTI} model of the left/output frequency weighting.
## The weighting must have only antistable zeros.
## @iftex
## @math{|| inv(V) \\ (G-G_r) \\dots ||_H = \\min}
## @end iftex
## @ifnottex
## @example
## || inv(V) (G-Gr) . ||  = min
##                      H
## @end example
## @end ifnottex
##
## @item 'right-inv', 'inv-w'
## @acronym{LTI} model of the right/input frequency weighting.
## The weighting must have only antistable zeros.
## @iftex
## @math{|| \\dots (G-G_r) \\ inv(W) ||_H = \\min}
## @end iftex
## @ifnottex
## @example
## || . (G-Gr) inv(W) ||  = min
##                      H
## @end example
## @end ifnottex
##
##
## @item 'left-conj', 'conj-v'
## @acronym{LTI} model of the left/output frequency weighting.
## The weighting must be stable.
## @iftex
## @math{|| conj(V) \\ (G-G_r) \\dots ||_H = \\min}
## @end iftex
## @ifnottex
## @example
## || V (G-Gr) . ||  = min
##                 H
## @end example
## @end ifnottex
##
## @item 'right-conj', 'conj-w'
## @acronym{LTI} model of the right/input frequency weighting.
## The weighting must be stable.
## @iftex
## @math{|| \\dots (G-G_r) \\ conj(W) ||_H = \\min}
## @end iftex
## @ifnottex
## @example
## || . (G-Gr) W ||  = min
##                 H
## @end example
## @end ifnottex
##
##
## @item 'left-conj-inv', 'conj-inv-v'
## @acronym{LTI} model of the left/output frequency weighting.
## The weighting must be minimum-phase.
## @iftex
## @math{|| conj(inv(V)) \\ (G-G_r) \\dots ||_H = \\min}
## @end iftex
## @ifnottex
## @example
## || V (G-Gr) . ||  = min
##                 H
## @end example
## @end ifnottex
##
## @item 'right-conj-inv', 'conj-inv-w'
## @acronym{LTI} model of the right/input frequency weighting.
## The weighting must be minimum-phase.
## @iftex
## @math{|| \\dots (G-G_r) \\ conj(inv(W)) ||_H = \\min}
## @end iftex
## @ifnottex
## @example
## || . (G-Gr) W ||  = min
##                 H
## @end example
## @end ifnottex
##
##
## @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].
## @var{tol1} < 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.  @var{tol2} <= @var{tol1} < 1.
## If not specified, ns*eps*info.hsv(1) is chosen.
##
## @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}.
## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs
## is of utmost importance.  The input and output scaling can @strong{not}
## be done by the equilibration option or the @command{prescale} function
## because these functions perform state transformations only.
## Furthermore, signals should not be scaled simply to a certain range.
## For all inputs (or outputs), a certain change should be of the same
## importance for the model.
## @end table
##
##
## Approximation Properties:
## @itemize @bullet
## @item
## Guaranteed stability of reduced models
## @item
## Lower guaranteed error bound
## @item
## Guaranteed a priori error bound
## @iftex
## @tex
## $$ \\sigma_{r+1} \\leq || (G-G_r) ||_{\\infty} \\leq 2 \\sum_{j=r+1}^{n} \\sigma_j $$
## @end tex
## @end iftex
## @end itemize
##
## @strong{Algorithm}@*
## Uses SLICOT AB09JD by courtesy of
## @uref{http://www.slicot.org, NICONET e.V.}
## @end deftypefn

## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
## Created: October 2011
## Version: 0.1

function [Gr, info] = hnamodred (G, varargin)

  if (nargin == 0)
    print_usage ();
  endif
  
  if (! isa (G, "lti"))
    error ("hnamodred: first argument must be an LTI system");
  endif

  if (nargin > 1)                                  # hnamodred (G, ...)
    if (is_real_scalar (varargin{1}))              # hnamodred (G, nr)
      varargin = horzcat (varargin(2:end), {"order"}, varargin(1));
    endif
    if (isstruct (varargin{1}))                    # hnamodred (G, opt, ...), hnamodred (G, nr, opt, ...)
      varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end));
    endif
    ## order placed at the end such that nr from hnamodred (G, nr, ...)
    ## and hnamodred (G, nr, opt, ...) overrides possible nr's from
    ## key/value-pairs and inside opt struct (later keys override former keys,
    ## nr > key/value > opt)
  endif

  nkv = numel (varargin);                          # number of keys and values

  if (rem (nkv, 2))
    error ("hnamodred: keys and values must come in pairs");
  endif

  [a, b, c, d, tsam, scaled] = ssdata (G);
  [p, m] = size (G);
  dt = isdt (G);
  
  ## default arguments
  alpha = __modred_default_alpha__ (dt);
  av = bv = cv = dv = [];
  jobv = 0;
  aw = bw = cw = dw = [];
  jobw = 0;
  jobinv = 2;
  tol1 = 0; 
  tol2 = 0;
  ordsel = 1;
  nr = 0;

  ## handle keys and values
  for k = 1 : 2 : nkv
    key = lower (varargin{k});
    val = varargin{k+1};
    switch (key)
      case {"left", "v", "wo"}
        [av, bv, cv, dv, jobv] = __modred_check_weight__ (val, dt, p, p);
        ## TODO: correct error messages for non-square weights

      case {"right", "w", "wi"}
        [aw, bw, cw, dw, jobw] = __modred_check_weight__ (val, dt, m, m);

      case {"left-inv", "inv-v"}
        [av, bv, cv, dv] = __modred_check_weight__ (val, dt, p, p);
        jobv = 2;

      case {"right-inv", "inv-w"}
        [aw, bw, cw, dw] = __modred_check_weight__ (val, dt, m, m);
        jobv = 2

      case {"left-conj", "conj-v"}
        [av, bv, cv, dv] = __modred_check_weight__ (val, dt, p, p);
        jobv = 3;

      case {"right-conj", "conj-w"}
        [aw, bw, cw, dw] = __modred_check_weight__ (val, dt, m, m);
        jobv = 3

      case {"left-conj-inv", "conj-inv-v"}
        [av, bv, cv, dv] = __modred_check_weight__ (val, dt, p, p);
        jobv = 4;

      case {"right-conj-inv", "conj-inv-w"}
        [aw, bw, cw, dw] = __modred_check_weight__ (val, dt, m, m);
        jobv = 4

      case {"order", "nr"}
        [nr, ordsel] = __modred_check_order__ (val, rows (a));

      case "tol1"
        tol1 = __modred_check_tol__ (val, "tol1");

      case "tol2"
        tol2 = __modred_check_tol__ (val, "tol2");

      case "alpha"
        alpha = __modred_check_alpha__ (val, dt);

      case "method"
        switch (tolower (val(1)))
          case {"d", "n"}      # "descriptor"
            jobinv = 0;
          case {"s", "i"}      # "standard"
            jobinv = 1;
          case "a"             # {"auto", "automatic"}
            jobinv = 2;
          otherwise
            error ("hnamodred: invalid computational approach");
        endswitch

      case {"equil", "equilibrate", "equilibration", "scale", "scaling"}
        scaled = __modred_check_equil__ (val);

      otherwise
        warning ("hnamodred: invalid property name '%s' ignored", key);
    endswitch
  endfor

  
  ## perform model order reduction
  [ar, br, cr, dr, nr, hsv, ns] = __sl_ab09jd__ (a, b, c, d, dt, scaled, nr, ordsel, alpha, ...
                                            jobv, av, bv, cv, dv, ...
                                            jobw, aw, bw, cw, dw, ...
                                            jobinv, tol1, tol2);

  ## assemble reduced order model
  Gr = ss (ar, br, cr, dr, tsam);

  ## assemble info struct  
  n = rows (a);
  nu = n - ns;
  info = struct ("n", n, "ns", ns, "hsv", hsv, "nu", nu, "nr", nr);

endfunction


%!shared Mo, Me, Info, HSVe
%! A =  [ -3.8637   -7.4641   -9.1416   -7.4641   -3.8637   -1.0000
%!         1.0000,         0         0         0         0         0
%!              0    1.0000         0         0         0         0
%!              0         0    1.0000         0         0         0
%!              0         0         0    1.0000         0         0
%!              0         0         0         0    1.0000         0 ];
%!
%! B =  [       1
%!              0
%!              0
%!              0
%!              0
%!              0 ];
%!
%! C =  [       0         0         0         0         0         1 ];
%!
%! D =  [       0 ];
%!
%! G = ss (A, B, C, D);  # "scaled", false
%!
%! AV = [  0.2000   -1.0000
%!         1.0000         0 ];
%!
%! BV = [       1
%!              0 ];
%!
%! CV = [ -1.8000         0 ];
%!
%! DV = [       1 ];
%!
%! V = ss (AV, BV, CV, DV);
%!
%! [Gr, Info] = hnamodred (G, "left", V, "tol1", 1e-1, "tol2", 1e-14);
%! [Ao, Bo, Co, Do] = ssdata (Gr);
%!
%! Ae = [ -0.2391   0.3072   1.1630   1.1967
%!        -2.9709  -0.2391   2.6270   3.1027
%!         0.0000   0.0000  -0.5137  -1.2842
%!         0.0000   0.0000   0.1519  -0.5137 ];
%!
%! Be = [ -1.0497
%!        -3.7052
%!         0.8223
%!         0.7435 ];
%!
%! Ce = [ -0.4466   0.0143  -0.4780  -0.2013 ];
%!
%! De = [  0.0219 ];
%!
%! HSVe = [  2.6790   2.1589   0.8424   0.1929   0.0219   0.0011 ].';
%!
%! Mo = [Ao, Bo; Co, Do];
%! Me = [Ae, Be; Ce, De];
%!
%!assert (Mo, Me, 1e-4);
#%!assert (Info.hsv, HSVe, 1e-4);