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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 .
+
+## -*- 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
+## 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
+## 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] = slsb16bd (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);