## Copyright (C) 1996, 1997 John W. Eaton
##
## 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 2, 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, write to the Free
## Software Foundation, 59 Temple Place - Suite 330, Boston, MA
## 02111-1307, USA.

## Usage: x = dare (a, b, c, r {,opt})
##
## Solves discrete-time algebraic riccati equation
##
##   a' x a - x + a' x b (r + b' x b)^{-1} b' x a + c = 0
##
## for
##
##   a: nxn
##   b: nxm
##   c: nxn, symmetric positive semidefinite
##   r: mxm, invertible
##
## If c is not square, then the function attempts to use c'*c instead.
##
## Solution method: Generalized eigenvalue approach (Van Dooren; SIAM J.
## Sci. Stat. Comput., Vol 2) applied  to the appropriate symplectic pencil.
##
## See also: Ran and Rodman, "Stable Hermitian Solutions of Discrete
## Algebraic Riccati Equations," Mathematics of Control, Signals and
## Systems, Vol 5, no 2 (1992)  pp 165-194.
##
## opt is an option passed to the eigenvalue balancing routine default
## is "B".
##
## See also: balance, are

## Author: A. S. Hodel <scotte@eng.auburn.edu>
## Created: August 1993
## Adapted-By: jwe
## $Revision: 1.15 $
## $Log: dare.m,v $
## Revision 1.15  1998-11-06 16:15:36  jwe
## *** empty log message ***
##

function x = dare (a, b, c, r, opt)

  if (nargin == 4 | nargin == 5)
    if (nargin == 5)
      if (opt != "N" || opt != "P" || opt != "S" || opt != "B")
	warning ("dare: opt has an invalid value -- setting to B");
	opt = "B";
      endif
    else
      opt = "B";
    endif

    # dimension checks are done in is_controllable, is_observable
    if (is_controllable (a, b) == 0)
      warning ("dare: a,b are not controllable");
    elseif (is_observable (a, c) == 0)
      warning ("dare: a,c are not observable");
    endif

    if ((p = is_square (c)) == 0)
      c = c'*c;
      p = rows (c);
    endif

    ## Check r dimensions.
    n = rows(a);
    m = columns(b);
    if ((m1 = is_square (r)) == 0)
      warning ("dare: r is not square");
    elseif (m1 != m)
      warning ("b,r are not conformable");
    endif

    s1 = [a, zeros(n) ; -c, 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
    usage ("x = dare (a, b, c, r {,opt})");
  endif

endfunction