--- /dev/null
+## 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}] =} 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 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
+## 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 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'
+## 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'
+## 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'
+## 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'
+## 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'
+## 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'
+## 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'
+## 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'
+## 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} command
+## 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] = slab09jd (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);
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff