--- /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}] =} btamodred (@var{G}, @dots{})
+## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{nr}, @dots{})
+## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{opt}, @dots{})
+## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} btamodred (@var{G}, @var{nr}, @var{opt}, @dots{})
+##
+## Model order reduction by frequency weighted Balanced Truncation Approximation (BTA) 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}.
+##
+## BTA 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', 'b'
+## Use the square-root Balance & Truncate method.
+## @item 'bfsr', 'f'
+## Use the balancing-free square-root Balance & Truncate 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.
+## This is done by state transformations.
+## 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{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] = btamodred (varargin)
+
+ [Gr, info] = __modred_ab09id__ ("bta", varargin{:});
+
+endfunction
+
+
+%!shared Mo, Me, Info, HSVe
+%! A = [ -26.4000, 6.4023, 4.3868;
+%! 32.0000, 0, 0;
+%! 0, 8.0000, 0 ];
+%!
+%! B = [ 16
+%! 0
+%! 0 ];
+%!
+%! C = [ 9.2994 1.1624 0.1090 ];
+%!
+%! D = [ 0 ];
+%!
+%! G = ss (A, B, C, D); % "scaled", false
+%!
+%! AV = [ -1.0000, 0, 4.0000, -9.2994, -1.1624, -0.1090;
+%! 0, 2.0000, 0, -9.2994, -1.1624, -0.1090;
+%! 0, 0, -3.0000, -9.2994, -1.1624, -0.1090;
+%! 16.0000, 16.0000, 16.0000, -26.4000, 6.4023, 4.3868;
+%! 0, 0, 0, 32.0000, 0, 0;
+%! 0, 0, 0, 0, 8.0000, 0 ];
+%!
+%! BV = [ 1
+%! 1
+%! 1
+%! 0
+%! 0
+%! 0 ];
+%!
+%! CV = [ 1 1 1 0 0 0 ];
+%!
+%! DV = [ 0 ];
+%!
+%! V = ss (AV, BV, CV, DV);
+%!
+%! [Gr, Info] = btamodred (G, 2, "left", V);
+%! [Ao, Bo, Co, Do] = ssdata (Gr);
+%!
+%! Ae = [ 9.1900 0.0000
+%! 0.0000 -34.5297 ];
+%!
+%! Be = [ 11.9593
+%! 16.9329 ];
+%!
+%! Ce = [ 2.8955 6.9152 ];
+%!
+%! De = [ 0.0000 ];
+%!
+%! HSVe = [ 3.8253 0.2005 ].';
+%!
+%! 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