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## Copyright (C) 2010 Benjamin Fernandez
## 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{sysbar}, @var{T}, @var{K}] =} ctrbf (@var{sys})
## @deftypefnx{Function File} {[@var{sysbar}, @var{T}, @var{K}] =} ctrbf (@var{sys}, @var{tol})
## @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} ctrbf (@var{A}, @var{B}, @var{C})
## @deftypefnx{Function File} {[@var{Abar}, @var{Bbar}, @var{Cbar}, @var{T}, @var{K}] =} ctrbf (@var{A}, @var{B}, @var{C}, @var{TOL})
## If Co=ctrb(A,B) has rank r <= n = SIZE(A,1), then there is a
## similarity transformation Tc such that Tc = [t1 t2] where t1
## is the controllable subspace and t2 is orthogonal to t1
##
## @example
## @group
## Abar = Tc \ A * Tc , Bbar = Tc \ B , Cbar = C * Tc
## @end group
## @end example
##
## and the transformed system has the form
##
## @example
## @group
## | Ac A12| | Bc |
## Abar = |----------|, Bbar = | ---|, Cbar = [Cc | Cnc].
## | 0 Anc| | 0 |
## @end group
## @end example
##
## where (Ac,Bc) is controllable, and Cc(sI-Ac)^(-1)Bc = C(sI-A)^(-1)B.
## and the system is stabilizable if Anc has no eigenvalues in
## the right half plane. The last output K is a vector of length n
## containing the number of controllable states.
## @end deftypefn
## Author: Benjamin Fernandez <mail@benjaminfernandez.info>
## Created: 2010-04-30
## Version: 0.1
function [ac, bc, cc, z, ncont] = ctrbf (a, b = [], c, tol = [])
if (nargin < 1 || nargin > 4)
print_usage ();
endif
islti = isa (a, "lti");
if (islti)
if (nargin > 2)
print_usage ();
endif
sys = a;
tol = b;
[a, b, c] = ssdata (sys);
else
if (nargin < 3)
print_usage ();
endif
sys = ss (a, b, c);
[a, b, c] = ssdata (sys);
endif
if (isempty (tol))
tol = 0; # default tolerance
elseif (! is_real_scalar (tol))
error ("ctrbf: tol must be a real scalar");
endif
[ac, bc, cc, z, ncont] = sltb01ud (a, b, c, tol);
if (islti)
ac = set (sys, "a", ac, "b", bc, "c", cc, "scaled", false);
bc = z;
cc = ncont;
endif
endfunction
%!shared Ao, Bo, Co, Zo, Ae, Be, Ce, Ze, NCONT
%! A = [ -1.0 0.0 0.0
%! -2.0 -2.0 -2.0
%! -1.0 0.0 -3.0 ];
%!
%! B = [ 1.0 0.0 0.0
%! 0.0 2.0 1.0 ].';
%!
%! C = [ 0.0 2.0 1.0
%! 1.0 0.0 0.0 ];
%!
%! [Ao, Bo, Co, Zo, NCONT] = ctrbf (A, B, C);
%!
%! Ae = [ -3.0000 2.2361
%! 0.0000 -1.0000 ];
%!
%! Be = [ 0.0000 -2.2361
%! 1.0000 0.0000 ];
%!
%! Ce = [ -2.2361 0.0000
%! 0.0000 1.0000 ];
%!
%! Ze = [ 0.0000 1.0000 0.0000
%! -0.8944 0.0000 -0.4472
%! -0.4472 0.0000 0.8944 ];
%!
%!assert (Ao(1:NCONT, 1:NCONT), Ae, 1e-4);
%!assert (Bo(1:NCONT, :), Be, 1e-4);
%!assert (Co(:, 1:NCONT), Ce, 1e-4);
%!assert (Zo, Ze, 1e-4);
|
