### Copyright (C) 1996 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 = are (a, b, c {,opt})
##
## Solves algebraic riccati equation
##
##   a' x + x a - x b x + c = 0
##
## for identically dimensioned square matrices a, b, c.  If b (c) is not
## square, then the function attempts to use b * b' (c' * c) instead.
##
## Solution method: apply Laub's Schur method (IEEE Trans. Auto. Contr,
## 1979) to the appropriate Hamiltonian matrix.
##
## opt is an option passed to the eigenvalue balancing routine default is "B".
##
## See also: balance

## Author: A. S. Hodel <scotte@eng.auburn.edu>
## Created: August 1993
## Adapted-By: jwe

function x = are (a, b, c, opt)

  if (nargin == 3 || nargin == 4)
    if (nargin == 4)
      if (! (strcmp (opt, "N") || strcmp (opt, "P") ...
	     || strcmp (opt, "S") || strcmp (opt, "B") ...
	     || strcmp (opt, "n") || strcmp (opt, "p") ...
	     || strcmp (opt, "s") || strcmp (opt, "b")))
	warning ("are: opt has an invalid value; setting to B");
	opt = "B";
      endif
    else
      opt = "B";
    endif
    if ((n = is_square(a)) == 0)
      error ("are: a is not square");
    endif

    if (is_controllable(a,b) == 0)
      warning ("are: a, b are not controllable");
    endif
    if ((m = is_square (b)) == 0)
      b = b * b';
      m = rows (b);
    endif
    if (is_observable (a, c) == 0)
      warning ("are: a,c are not observable");
    endif
    if ((p = is_square (c)) == 0)
      c = c' * c;
      p = rows (c);
    endif
    if (n != m || n != p)
      error ("are: a, b, c not conformably dimensioned.");
    endif

    ## Should check for controllability/observability here
    ## use Boley-Golub (Syst. Contr. Letters, 1984) method, not the
    ##
    ##                     n-1
    ## rank ([ B A*B ... A^   *B]) method 

    [d, h] = balance ([a, -b; -c, -a'], opt);
    [u, s] = schur (h, "A");
    u = d * u;
    n1 = n + 1;
    n2 = 2 * n;
    x = u (n1:n2, 1:n) / u (1:n, 1:n);
  else
    usage ("x = are (a, b, c)");
  endif

endfunction
