--- /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{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @dots{})
+## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{ncr}, @dots{})
+## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{opt}, @dots{})
+## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} btaconred (@var{G}, @var{K}, @var{ncr}, @var{opt}, @dots{})
+##
+## Controller reduction by frequency-weighted Balanced Truncation Approximation (BTA).
+## Given a plant @var{G} and a stabilizing controller @var{K}, determine a reduced
+## order controller @var{Kr} such that the closed-loop system is stable and closed-loop
+## performance is retained.
+##
+## The algorithm tries to minimize the frequency-weighted error
+## @iftex
+## @tex
+## $$ || V \\ (K - K_r) \\ W ||_{\\infty} = min $$
+## @end tex
+## @end iftex
+## @ifnottex
+## @example
+## ||V (K-Kr) 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 of the plant.
+## It has m inputs, p outputs and n states.
+## @item K
+## LTI model of the controller.
+## It has p inputs, m outputs and nc states.
+## @item ncr
+## The desired order of the resulting reduced order controller @var{Kr}.
+## If not specified, @var{ncr} 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 Kr
+## State-space model of reduced order controller.
+## @item info
+## Struct containing additional information.
+## @table @var
+## @item info.ncr
+## The order of the obtained reduced order controller @var{Kr}.
+## @item info.ncs
+## The order of the alpha-stable part of original controller @var{K}.
+## @item info.hsvc
+## The Hankel singular values of the alpha-stable part of @var{K}.
+## The @var{ncs} Hankel singular values are ordered decreasingly.
+## @end table
+## @end table
+##
+## @strong{Option Keys and Values}
+## @table @var
+## @item 'order', 'ncr'
+## The desired order of the resulting reduced order controller @var{Kr}.
+## If not specified, @var{ncr} is chosen automatically such that states with
+## Hankel singular values @var{info.hsvc} > @var{tol1} are retained.
+##
+## @item 'method'
+## Order reduction approach 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 'weight'
+## Specifies the type of frequency-weighting as follows:
+## @table @var
+## @item 'none'
+## No weightings are used (V = I, W = I).
+##
+## @item 'left', 'output'
+## Use stability enforcing left (output) weighting
+## @iftex
+## @tex
+## $$ V = (I - G K)^{-1} G, \\qquad W = I $$
+## @end tex
+## @end iftex
+## @ifnottex
+## @example
+## -1
+## V = (I-G*K) *G , W = I
+## @end example
+## @end ifnottex
+##
+## @item 'right', 'input'
+## Use stability enforcing right (input) weighting
+## @iftex
+## @tex
+## $$ V = I, \\qquad W = (I - G K)^{-1} G $$
+## @end tex
+## @end iftex
+## @ifnottex
+## @example
+## -1
+## V = I , W = (I-G*K) *G
+## @end example
+## @end ifnottex
+##
+## @item 'both', 'performance'
+## Use stability and performance enforcing weightings
+## @iftex
+## @tex
+## $$ V = (I - G K)^{-1} G, \\qquad W = (I - G K)^{-1} $$
+## @end tex
+## @end iftex
+## @ifnottex
+## @example
+## -1 -1
+## V = (I-G*K) *G , W = (I-G*K)
+## @end example
+## @end ifnottex
+## Default value.
+## @end table
+##
+## @item 'feedback'
+## Specifies whether @var{K} is a positive or negative feedback controller:
+## @table @var
+## @item '+'
+## Use positive feedback controller. Default value.
+## @item '-'
+## Use negative feedback controller.
+## @end table
+##
+## @item 'alpha'
+## Specifies the ALPHA-stability boundary for the eigenvalues
+## of the state dynamics matrix @var{K.A}. For a continuous-time
+## controller, ALPHA <= 0 is the boundary value for
+## the real parts of eigenvalues, while for a discrete-time
+## controller, 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 controllers and
+## 1 for discrete-time controllers.
+##
+## @item 'tol1'
+## If @var{'order'} is not specified, @var{tol1} contains the tolerance for
+## determining the order of the reduced controller.
+## For model reduction, the recommended value of @var{tol1} is
+## c*info.hsvc(1), where c lies in the interval [0.00001, 0.001].
+## Default value is info.ncs*eps*info.hsvc(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
+## controller. TOL2 <= TOL1.
+## If not specified, ncs*eps*info.hsvc(1) is chosen.
+##
+## @item 'gram-ctrb'
+## Specifies the choice of frequency-weighted controllability
+## Grammian as follows:
+## @table @var
+## @item 'standard'
+## Choice corresponding to standard Enns' method [1]. Default method.
+## @item 'enhanced'
+## Choice corresponding to the stability enhanced
+## modified Enns' method of [2].
+## @end table
+##
+## @item 'gram-obsv'
+## Specifies the choice of frequency-weighted observability
+## Grammian as follows:
+## @table @var
+## @item 'standard'
+## Choice corresponding to standard Enns' method [1]. Default method.
+## @item 'enhanced'
+## Choice corresponding to the stability enhanced
+## modified Enns' method of [2].
+## @end table
+##
+## @item 'equil', 'scale'
+## Boolean indicating whether equilibration (scaling) should be
+## performed on @var{G} and @var{K} prior to order reduction.
+## Default value is false if both @code{G.scaled == true, K.scaled == true}
+## and true otherwise.
+## 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
+##
+## @strong{Algorithm}@*
+## Uses SLICOT SB16AD by courtesy of
+## @uref{http://www.slicot.org, NICONET e.V.}
+## @end deftypefn
+
+## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
+## Created: December 2011
+## Version: 0.1
+
+function [Kr, info] = btaconred (varargin)
+
+ [Kr, info] = __conred_sb16ad__ ("bta", varargin{:});
+
+endfunction
+
+
+%!shared Mo, Me, Info, HSVCe
+%! A = [ -1. 0. 4.
+%! 0. 2. 0.
+%! 0. 0. -3. ];
+%!
+%! B = [ 1.
+%! 1.
+%! 1. ];
+%!
+%! C = [ 1. 1. 1. ];
+%!
+%! D = [ 0. ];
+%!
+%! G = ss (A, B, C, D, "scaled", true);
+%!
+%! AC = [ -26.4000, 6.4023, 4.3868;
+%! 32.0000, 0, 0;
+%! 0, 8.0000, 0 ];
+%!
+%! BC = [ -16
+%! 0
+%! 0 ];
+%!
+%! CC = [ 9.2994 1.1624 0.1090 ];
+%!
+%! DC = [ 0 ];
+%!
+%! K = ss (AC, BC, CC, DC, "scaled", true);
+%!
+%! [Kr, Info] = btaconred (G, K, 2, "weight", "input", "feedback", "+");
+%! [Ao, Bo, Co, Do] = ssdata (Kr);
+%!
+%! Ae = [ 9.1900 0.0000
+%! 0.0000 -34.5297 ];
+%!
+%! Be = [ -11.9593
+%! 86.3137 ];
+%!
+%! Ce = [ 2.8955 -1.3566 ];
+%!
+%! De = [ 0.0000 ];
+%!
+%! HSVCe = [ 3.8253 0.2005 ].';
+%!
+%! Mo = [Ao, Bo; Co, Do];
+%! Me = [Ae, Be; Ce, De];
+%!
+%!assert (Mo, Me, 1e-4);
+%!assert (Info.hsvc, HSVCe, 1e-4);
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff