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## 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}] =} bstmodred (@var{G}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{nr}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{opt}, @dots{})
## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} bstmodred (@var{G}, @var{nr}, @var{opt}, @dots{})
##
## Model order reduction by Balanced Stochastic Truncation (BST) 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}.
##
## BST is a relative error method which tries to minimize
## @iftex
## @tex
## $$ || G^{-1} (G-G_r) ||_{\\infty} = min $$
## @end tex
## @end iftex
## @ifnottex
## @example
## -1
## ||G (G-Gr)|| = min
## inf
## @end example
## @end ifnottex
##
##
##
## @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 phase system corresponding
## to 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 NU and the number of
## Hankel singular values greater than @code{MAX(TOL1,NS*EPS)};
## @var{nr} can be further reduced to ensure that
## @code{HSV(NR-NU) > HSV(NR+1-NU)}.
##
## @item 'method'
## Approximation method for the H-infinity norm.
## Valid values corresponding to this key are:
## @table @var
## @item 'sr-bta', 'b'
## Use the square-root Balance & Truncate method.
## @item 'bfsr-bta', 'f'
## Use the balancing-free square-root Balance & Truncate method. Default method.
## @item 'sr-spa', 's'
## Use the square-root Singular Perturbation Approximation method.
## @item 'bfsr-spa', 'p'
## Use the balancing-free square-root Singular Perturbation Approximation 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 'beta'
## Use @code{[G, beta*I]} as new system @var{G} to combine
## absolute and relative error methods.
## BETA > 0 specifies the absolute/relative error weighting
## parameter. A large positive value of BETA favours the
## minimization of the absolute approximation error, while a
## small value of BETA is appropriate for the minimization
## of the relative error.
## BETA = 0 means a pure relative error method and can be
## used only if rank(G.D) = rows(G.D) which means that
## the feedthrough matrice must not be rank-deficient.
## Default value is 0.
##
## @item 'tol1'
## If @var{'order'} is not specified, @var{tol1} contains the tolerance for
## determining the order of reduced system.
## For model reduction, the recommended value of @var{tol1} lies
## in the interval [0.00001, 0.001]. @var{tol1} < 1.
## If @var{tol1} <= 0 on entry, the used default value is
## @var{tol1} = NS*EPS, where NS is the number of
## ALPHA-stable eigenvalues of A and EPS is the machine
## precision.
## 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 phase system (see METHOD) corresponding
## to the ALPHA-stable part of the given system.
## The recommended value is TOL2 = NS*EPS. TOL2 <= TOL1 < 1.
## This value is used by default if @var{'tol2'} is not specified
## or if TOL2 <= 0 on entry.
##
## @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
##
##
## For the H-infinity norm, the best approximation problem is
## unsolved so far. Nevertheless, balanced truncation and related
## methods can be used to obtain good approximations using this measure.
##
## Available approximation methods are the accuracy-enhancing square-root (SR)
## or the balancing-free square-root (BFSR) versions of
## the Balance & Truncate (BTA) or Singular Perturbation Approximation (SPA)
## model reduction methods for the ALPHA-stable part of the system.
##
## Unstable models are handled by separating the stable and unstable
## parts additively, applying the model reduction only to the stable
## part and by joining the reduced stable with the original unstable part.
## The order of the reduced system can be selected by the user or
## can be determined automatically on the basis of the computed
## Hankel singular values.
##
## For MIMO models, proper scaling of input-output channels is of
## utmost importance. This can @strong{not} be done by the equilibration
## option or the @command{prescale} command because these perform state
## transformations only. While enhancing numerics, state transformations
## have no influence on the input-output behaviour and the magnitude of
## the corresponding signals. Since the algorithm calculates the
## H-infinity norm of these signals, important behaviour of @var{G}
## could be neglected just because the corresponding signals have smaller
## numbers than those of other, less important effects of @var{G}.
##
## BST is often suitable to perform model reduction in order to obtain
## low order design models for controller synthesis.
##
## Approximation Properties:
## @itemize @bullet
## @item
## Guaranteed stability of reduced models
## @item
## Approximates simultaneously gain and phase
## @item
## Preserves non-minimum phase zeros
## @item
## Guaranteed a priori error bound
## @iftex
## @tex
## $$ || G^{-1} (G-G_r) ||_{\\infty} \\leq 2 \\sum_{j=r+1}^{n} \\frac{1+\\sigma_j}{1-\\sigma_j} - 1 $$
## @end tex
## @end iftex
## @end itemize
##
## @strong{Algorithm}@*
## Uses SLICOT AB09HD 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] = bstmodred (G, varargin)
if (nargin == 0)
print_usage ();
endif
if (! isa (G, "lti"))
error ("bstmodred: first argument must be an LTI system");
endif
if (nargin > 1) # bstmodred (G, ...)
if (is_real_scalar (varargin{1})) # bstmodred (G, nr)
varargin = horzcat (varargin(2:end), {"order"}, varargin(1));
endif
if (isstruct (varargin{1})) # bstmodred (G, opt, ...), bstmodred (G, nr, opt, ...)
varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end));
endif
## order placed at the end such that nr from bstmodred (G, nr, ...)
## and bstmodred (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 ("bstmodred: keys and values must come in pairs");
endif
[a, b, c, d, tsam, scaled] = ssdata (G);
dt = isdt (G);
## default arguments
alpha = __modred_default_alpha__ (dt);
beta = 0;
tol1 = 0;
tol2 = 0;
ordsel = 1;
nr = 0;
job = 1;
## handle keys and values
for k = 1 : 2 : nkv
key = lower (varargin{k});
val = varargin{k+1};
switch (key)
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 "beta"
if (! issample (val, 0))
error ("bstmodred: argument %s must be BETA >= 0", varargin{k});
endif
beta = val;
case "method" # approximation method
switch (tolower (val))
case {"sr-bta", "b"} # 'B': use the square-root Balance & Truncate method
job = 0;
case {"bfsr-bta", "f"} # 'F': use the balancing-free square-root Balance & Truncate method
job = 1;
case {"sr-spa", "s"} # 'S': use the square-root Singular Perturbation Approximation method
job = 2;
case {"bfsr-spa", "p"} # 'P': use the balancing-free square-root Singular Perturbation Approximation method
job = 3;
otherwise
error ("bstmodred: '%s' is an invalid approximation method", val);
endswitch
case {"equil", "equilibrate", "equilibration", "scale", "scaling"}
scaled = __modred_check_equil__ (val);
otherwise
warning ("bstmodred: invalid property name '%s' ignored", key);
endswitch
endfor
## perform model order reduction
[ar, br, cr, dr, nr, hsv, ns] = slab09hd (a, b, c, d, dt, scaled, job, nr, ordsel, alpha, beta, \
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 = [ -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000
%! -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000
%! 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000
%! 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000
%! 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200
%! 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000
%! 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 ];
%!
%! B = [ 0.0000 0.0000
%! 12.500 0.0000
%! 0.0000 0.0000
%! 0.0000 0.0000
%! 0.0000 0.0000
%! 0.0000 12.500
%! 0.0000 0.0000 ];
%!
%! C = [ 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
%! 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000
%! 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 ];
%!
%! D = [ 0.0000 0.0000
%! 0.0000 0.0000
%! 0.0000 0.0000 ];
%!
%! G = ss (A, B, C, D, "scaled", true);
%!
%! [Gr, Info] = bstmodred (G, "beta", 1.0, "tol1", 0.1, "tol2", 0.0);
%! [Ao, Bo, Co, Do] = ssdata (Gr);
%!
%! Ae = [ 1.2729 0.0000 6.5947 0.0000 -3.4229
%! 0.0000 0.8169 0.0000 2.4821 0.0000
%! -2.9889 0.0000 -2.9028 0.0000 -0.3692
%! 0.0000 -3.3921 0.0000 -3.1126 0.0000
%! -1.4767 0.0000 -2.0339 0.0000 -0.6107 ];
%!
%! Be = [ 0.1331 -0.1331
%! -0.0862 -0.0862
%! -2.6777 2.6777
%! -3.5767 -3.5767
%! -2.3033 2.3033 ];
%!
%! Ce = [ -0.6907 -0.6882 0.0779 0.0958 -0.0038
%! 0.0676 0.0000 0.6532 0.0000 -0.7522
%! 0.6907 -0.6882 -0.0779 0.0958 0.0038 ];
%!
%! De = [ 0.0000 0.0000
%! 0.0000 0.0000
%! 0.0000 0.0000 ];
%!
%! HSVe = [ 0.8803 0.8506 0.8038 0.4494 0.3973 0.0214 0.0209 ].';
%!
%! Mo = [Ao, Bo; Co, Do];
%! Me = [Ae, Be; Ce, De];
%!
%!assert (Mo, Me, 1e-4);
%!assert (Info.hsv, HSVe, 1e-4);