/usr/share/octave/packages/control-2.6.2/cfconred.m is in octave-control 2.6.2-1build1.
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##
## 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}] =} cfconred (@var{G}, @var{F}, @var{L}, @dots{})
## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @dots{})
## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{opt}, @dots{})
## @deftypefnx{Function File} {[@var{Kr}, @var{info}] =} cfconred (@var{G}, @var{F}, @var{L}, @var{ncr}, @var{opt}, @dots{})
##
## Reduction of state-feedback-observer based controller by coprime factorization (CF).
## Given a plant @var{G}, state feedback gain @var{F} and full observer gain @var{L},
## determine a reduced order controller @var{Kr}.
##
## @strong{Inputs}
## @table @var
## @item G
## @acronym{LTI} model of the open-loop plant (A,B,C,D).
## It has m inputs, p outputs and n states.
## @item F
## Stabilizing state feedback matrix (m-by-n).
## @item L
## Stabilizing observer gain matrix (n-by-p).
## @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.hsv
## The Hankel singular values of the extended system?!?.
## The @var{n} Hankel singular values are ordered decreasingly.
## @item info.ncr
## The order of the obtained reduced order controller @var{Kr}.
## @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.hsv} > @var{tol1} are retained.
##
## @item 'method'
## Order reduction approach to be used as follows:
## @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 'cf'
## Specifies whether left or right coprime factorization is
## to be used as follows:
## @table @var
## @item 'left', 'l'
## Use left coprime factorization. Default method.
## @item 'right', 'r'
## Use right coprime factorization.
## @end table
##
## @item 'feedback'
## Specifies whether @var{F} and @var{L} are fed back positively or negatively:
## @table @var
## @item '+'
## A+BK and A+LC are both Hurwitz matrices.
## @item '-'
## A-BK and A-LC are both Hurwitz matrices. Default value.
## @end table
##
## @item 'tol1'
## If @var{'order'} is not specified, @var{tol1} contains the tolerance for
## determining the order of the reduced system.
## 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 n*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 coprime factorization controller.
## TOL2 <= TOL1.
## If not specified, n*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
##
## @strong{Algorithm}@*
## Uses SLICOT SB16BD 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] = cfconred (G, F, L, varargin)
if (nargin < 3)
print_usage ();
endif
if (! isa (G, "lti"))
error ("cfconred: first argument must be an LTI system");
endif
if (! is_real_matrix (F))
error ("cfconred: second argument must be a real matrix");
endif
if (! is_real_matrix (L))
error ("cfconred: third argument must be a real matrix");
endif
if (nargin > 3) # cfconred (G, F, L, ...)
if (is_real_scalar (varargin{1})) # cfconred (G, F, L, nr)
varargin = horzcat (varargin(2:end), {"order"}, varargin(1));
endif
if (isstruct (varargin{1})) # cfconred (G, F, L, opt, ...), cfconred (G, F, L, nr, opt, ...)
varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end));
endif
## order placed at the end such that nr from cfconred (G, F, L, nr, ...)
## and cfconred (G, F, L, 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 ("cfconred: keys and values must come in pairs");
endif
[a, b, c, d, tsam, scaled] = ssdata (G);
[p, m] = size (G);
n = rows (a);
[mf, nf] = size (F);
[nl, pl] = size (L);
dt = isdt (G);
jobd = any (d(:));
if (mf != m || nf != n)
error ("cfconred: dimensions of state-feedback matrix (%dx%d) and plant (%dx%d, %d states) don't match", ...
mf, nf, p, m, n);
endif
if (nl != n || pl != p)
error ("cfconred: dimensions of observer matrix (%dx%d) and plant (%dx%d, %d states) don't match", ...
nl, pl, p, m, n);
endif
## default arguments
tol1 = 0.0;
tol2 = 0.0;
jobcf = 0;
jobmr = 2; # balancing-free BTA
equil = scaled; # equil: 0 means "S", 1 means "N"
ordsel = 1;
ncr = 0;
negfb = true; # A-BK, A-LC Hurwitz
## handle keys and values
for k = 1 : 2 : nkv
key = lower (varargin{k});
val = varargin{k+1};
switch (key)
case {"order", "ncr", "nr"}
[ncr, ordsel] = __modred_check_order__ (val, n);
case "tol1"
tol1 = __modred_check_tol__ (val, "tol1");
case "tol2"
tol2 = __modred_check_tol__ (val, "tol2");
case "cf"
switch (lower (val(1)))
case "l"
jobcf = 0;
case "r"
jobcf = 1;
otherwise
error ("cfconred: '%s' is an invalid coprime factorization", val);
endswitch
case "method" # approximation method
switch (tolower (val))
case {"sr-bta", "b"} # 'B': use the square-root Balance & Truncate method
jobmr = 0;
case {"bfsr-bta", "f"} # 'F': use the balancing-free square-root Balance & Truncate method
jobmr = 1;
case {"sr-spa", "s"} # 'S': use the square-root Singular Perturbation Approximation method
jobmr = 2;
case {"bfsr-spa", "p"} # 'P': use the balancing-free square-root Singular Perturbation Approximation method
jobmr = 3;
otherwise
error ("cfconred: '%s' is an invalid approach", val);
endswitch
case {"equil", "equilibrate", "equilibration", "scale", "scaling"}
equil = __modred_check_equil__ (val);
case "feedback"
negfb = __conred_check_feedback_sign__ (val);
otherwise
warning ("cfconred: invalid property name '%s' ignored", key);
endswitch
endfor
## A - B*F --> A + B*F ; A - L*C --> A + L*C
if (negfb)
F = -F;
L = -L;
endif
## perform model order reduction
[acr, bcr, ccr, dcr, ncr, hsv] = __sl_sb16bd__ (a, b, c, d, dt, equil, ncr, ordsel, jobd, jobmr, ...
F, L, jobcf, tol1, tol2);
## assemble reduced order controller
Kr = ss (acr, bcr, ccr, dcr, tsam);
## assemble info struct
info = struct ("ncr", ncr, "hsv", hsv);
endfunction
%!shared Mo, Me, Info, HSVe
%! A = [ 0 1.0000 0 0 0 0 0 0
%! 0 0 0 0 0 0 0 0
%! 0 0 -0.0150 0.7650 0 0 0 0
%! 0 0 -0.7650 -0.0150 0 0 0 0
%! 0 0 0 0 -0.0280 1.4100 0 0
%! 0 0 0 0 -1.4100 -0.0280 0 0
%! 0 0 0 0 0 0 -0.0400 1.850
%! 0 0 0 0 0 0 -1.8500 -0.040 ];
%!
%! B = [ 0.0260
%! -0.2510
%! 0.0330
%! -0.8860
%! -4.0170
%! 0.1450
%! 3.6040
%! 0.2800 ];
%!
%! C = [ -.996 -.105 0.261 .009 -.001 -.043 0.002 -0.026 ];
%!
%! D = [ 0.0 ];
%!
%! G = ss (A, B, C, D); % "scaled", false
%!
%! F = [ 4.4721e-002 6.6105e-001 4.6986e-003 3.6014e-001 1.0325e-001 -3.7541e-002 -4.2685e-002 3.2873e-002 ];
%!
%! L = [ 4.1089e-001
%! 8.6846e-002
%! 3.8523e-004
%! -3.6194e-003
%! -8.8037e-003
%! 8.4205e-003
%! 1.2349e-003
%! 4.2632e-003 ];
%!
%! [Kr, Info] = cfconred (G, F, L, 4, "method", "bfsr-bta", "cf", "left", "feedback", "+");
%! [Ao, Bo, Co, Do] = ssdata (Kr);
%!
%! Ae = [ 0.5946 -0.7336 0.1914 -0.3368
%! 0.5960 -0.0184 -0.1088 0.0207
%! 1.2253 0.2043 0.1009 -1.4948
%! -0.0330 -0.0243 1.3440 0.0035 ];
%!
%! Be = [ 0.0015
%! -0.0202
%! 0.0159
%! -0.0544 ];
%!
%! Ce = [ 0.3534 0.0274 0.0337 -0.0320 ];
%!
%! De = [ 0.0000 ];
%!
%! HSVe = [ 4.9078 4.8745 3.8455 3.7811 1.2289 1.1785 0.5176 0.1148 ].';
%!
%! Mo = [Ao, Bo; Co, Do];
%! Me = [Ae, Be; Ce, De];
%!
%!assert (Mo, Me, 1e-4);
%!assert (Info.hsv, HSVe, 1e-4);
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