Mercurial > hg > octave-nkf
view scripts/control/base/dare.m @ 11651:74de76325d12 release-3-0-x
more xGELSD workspace fixes
author | John W. Eaton <jwe@octave.org> |
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date | Fri, 15 Feb 2008 18:55:45 -0500 |
parents | aeeb646f6538 |
children |
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## Copyright (C) 1996, 1997, 2000, 2002, 2003, 2004, 2005, 2006, 2007 ## Auburn University. All rights reserved. ## ## This file is part of Octave. ## ## Octave 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. ## ## Octave 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 Octave; see the file COPYING. If not, see ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {Function File} {@var{x} =} dare (@var{a}, @var{b}, @var{q}, @var{r}, @var{opt}) ## ## Return the solution, @var{x} of the discrete-time algebraic Riccati ## equation ## @iftex ## @tex ## $$ ## A^TXA - X + A^TXB (R + B^TXB)^{-1} B^TXA + Q = 0 ## $$ ## @end tex ## @end iftex ## @ifinfo ## @example ## a' x a - x + a' x b (r + b' x b)^(-1) b' x a + q = 0 ## @end example ## @end ifinfo ## @noindent ## ## @strong{Inputs} ## @table @var ## @item a ## @var{n} by @var{n} matrix; ## ## @item b ## @var{n} by @var{m} matrix; ## ## @item q ## @var{n} by @var{n} matrix, symmetric positive semidefinite, or a @var{p} by @var{n} matrix, ## In the latter case @math{q:=q'*q} is used; ## ## @item r ## @var{m} by @var{m}, symmetric positive definite (invertible); ## ## @item opt ## (optional argument; default = @code{"B"}): ## String option passed to @code{balance} prior to ordered @var{QZ} decomposition. ## @end table ## ## @strong{Output} ## @table @var ## @item x ## solution of @acronym{DARE}. ## @end table ## ## @strong{Method} ## Generalized eigenvalue approach (Van Dooren; @acronym{SIAM} J. ## Sci. Stat. Comput., Vol 2) applied to the appropriate symplectic pencil. ## ## See also: Ran and Rodman, @cite{Stable Hermitian Solutions of Discrete ## Algebraic Riccati Equations}, Mathematics of Control, Signals and ## Systems, Vol 5, no 2 (1992), pp 165--194. ## @seealso{balance, are} ## @end deftypefn ## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu> ## Created: August 1993 ## Adapted-By: jwe function x = dare (a, b, q, r, opt) if (nargin == 4 || nargin == 5) if (nargin == 5) if (! (ischar (opt) && (strcmp (opt, "N") || strcmp (opt, "P") || strcmp (opt, "S") || strcmp (opt, "B")))) warning ("dare: opt has an invalid value -- setting to B"); opt = "B"; endif else opt = "B"; endif if ((p = issquare (q)) == 0) q = q'*q; endif ##Checking positive definiteness if (isdefinite (r) <= 0) error ("dare: r not positive definite"); endif if (isdefinite (q) < 0) error ("dare: q not positive semidefinite"); endif ## Check r dimensions. [n, m] = size (b); if ((m1 = issquare (r)) == 0) error ("dare: r is not square"); elseif (m1 != m) error ("b,r are not conformable"); endif s1 = [a, zeros(n) ; -q, eye(n)]; s2 = [eye(n), (b/r)*b' ; zeros(n), a']; [c, d, s1, s2] = balance (s1, s2, opt); [aa, bb, u, lam] = qz (s1, s2, "S"); u = d*u; n1 = n+1; n2 = 2*n; x = u (n1:n2, 1:n)/u(1:n, 1:n); else print_usage (); endif endfunction