--- /dev/null
+## Copyright (C) 2009, 2010 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{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r})
+## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{s})
+## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{[]}, @var{e})
+## @deftypefnx {Function File} {[@var{x}, @var{l}, @var{g}] =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{s}, @var{e})
+## Solve discrete-time algebraic Riccati equation (ARE).
+##
+## @strong{Inputs}
+## @table @var
+## @item a
+## Real matrix (n-by-n).
+## @item b
+## Real matrix (n-by-m).
+## @item q
+## Real matrix (n-by-n).
+## @item r
+## Real matrix (m-by-m).
+## @item s
+## Optional real matrix (n-by-m). If @var{s} is not specified, a zero matrix is assumed.
+## @item e
+## Optional descriptor matrix (n-by-n). If @var{e} is not specified, an identity matrix is assumed.
+## @end table
+##
+## @strong{Outputs}
+## @table @var
+## @item x
+## Unique stabilizing solution of the discrete-time Riccati equation (n-by-n).
+## @item l
+## Closed-loop poles (n-by-1).
+## @item g
+## Corresponding gain matrix (m-by-n).
+## @end table
+##
+## @strong{Equations}
+## @example
+## @group
+## -1
+## A'XA - X - A'XB (B'XB + R) B'XA + Q = 0
+##
+## -1
+## A'XA - X - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0
+##
+## -1
+## G = (B'XB + R) B'XA
+##
+## -1
+## G = (B'XB + R) (B'XA + S')
+##
+## L = eig (A - B*G)
+## @end group
+## @end example
+## @example
+## @group
+## -1
+## A'XA - E'XE - A'XB (B'XB + R) B'XA + Q = 0
+##
+## -1
+## A'XA - E'XE - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0
+##
+## -1
+## G = (B'XB + R) B'XA
+##
+## -1
+## G = (B'XB + R) (B'XA + S')
+##
+## L = eig (A - B*G, E)
+## @end group
+## @end example
+##
+## @strong{Algorithm}@*
+## Uses SLICOT SB02OD and SG02AD by courtesy of
+## @uref{http://www.slicot.org, NICONET e.V.}
+##
+## @seealso{care, lqr, dlqr, kalman}
+## @end deftypefn
+
+## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
+## Created: October 2009
+## Version: 0.5.1
+
+function [x, l, g] = dare (a, b, q, r, s = [], e = [])
+
+ ## TODO: extract feedback matrix g from SB02OD (and SG02AD)
+
+ if (nargin < 4 || nargin > 6)
+ print_usage ();
+ endif
+
+ if (! is_real_square_matrix (a, q, r))
+ ## error ("dare: a, q, r must be real and square");
+ error ("dare: %s, %s, %s must be real and square", \
+ inputname (1), inputname (3), inputname (4));
+ endif
+
+ if (! is_real_matrix (b) || rows (a) != rows (b))
+ ## error ("dare: a and b must have the same number of rows");
+ error ("dare: %s and %s must have the same number of rows", \
+ inputname (1), inputname (2));
+ endif
+
+ if (columns (r) != columns (b))
+ ## error ("dare: b and r must have the same number of columns");
+ error ("dare: %s and %s must have the same number of columns", \
+ inputname (2), inputname (4));
+ endif
+
+ if (! is_real_matrix (s) && ! size_equal (s, b))
+ ## error ("dare: s(%dx%d) must be real and identically dimensioned with b(%dx%d)",
+ ## rows (s), columns (s), rows (b), columns (b));
+ error ("dare: %s(%dx%d) must be real and identically dimensioned with %s(%dx%d)", \
+ inputname (5), rows (s), columns (s), inputname (2), rows (b), columns (b));
+ endif
+
+ if (! isempty (e) && (! is_real_square_matrix (e) || ! size_equal (e, a)))
+ ## error ("dare: a and e must have the same number of rows");
+ error ("dare: %s and %s must have the same number of rows", \
+ inputname (1), inputname (6));
+ endif
+
+ ## check stabilizability
+ if (! isstabilizable (a, b, e, [], 1))
+ ## error ("dare: (a, b) not stabilizable");
+ error ("dare: (%s, %s) not stabilizable", \
+ inputname (1), inputname (2));
+ endif
+
+ ## check positive semi-definiteness
+ if (isempty (s))
+ t = zeros (size (b));
+ else
+ t = s;
+ endif
+
+ m = [q, t; t.', r];
+
+ if (isdefinite (m) < 0)
+ ## error ("dare: require [q, s; s.', r] >= 0");
+ error ("dare: require [%s, %s; %s.', %s] >= 0", \
+ inputname (3), inputname (5), inputname (5), inputname (4));
+ endif
+
+ ## solve the riccati equation
+ if (isempty (e))
+ if (isempty (s))
+ [x, l] = slsb02od (a, b, q, r, b, true, false);
+ g = (r + b.'*x*b) \ (b.'*x*a); # gain matrix
+ else
+ [x, l] = slsb02od (a, b, q, r, s, true, true);
+ g = (r + b.'*x*b) \ (b.'*x*a + s.'); # gain matrix
+ endif
+ else
+ if (isempty (s))
+ [x, l] = slsg02ad (a, e, b, q, r, b, true, false);
+ g = (r + b.'*x*b) \ (b.'*x*a); # gain matrix
+ else
+ [x, l] = slsg02ad (a, e, b, q, r, s, true, true);
+ g = (r + b.'*x*b) \ (b.'*x*a + s.'); # gain matrix
+ endif
+ endif
+
+endfunction
+
+
+%!shared x, l, g, xe, le, ge
+%! a = [ 0.4 1.7
+%! 0.9 3.8];
+%!
+%! b = [ 0.8
+%! 2.1];
+%!
+%! c = [ 1 -1];
+%!
+%! r = 3;
+%!
+%! [x, l, g] = dare (a, b, c.'*c, r);
+%!
+%! xe = [ 1.5354 1.2623
+%! 1.2623 10.5596];
+%!
+%! le = [-0.0022
+%! 0.2454];
+%!
+%! ge = [ 0.4092 1.7283];
+%!
+%!assert (x, xe, 1e-4);
+%!assert (sort (l), sort (le), 1e-4);
+%!assert (g, ge, 1e-4);
+
+## TODO: add more tests (nonempty s and/or e)
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff