## Copyright (C) 1993, 1994, 1995 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 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 USA.

## -*- texinfo -*-
## @deftypefn {Function File} {[@var{retval}, @var{u}] =} is_stabilizable (@var{sys}, @var{tol})
## @deftypefnx {Function File} {[@var{retval}, @var{u}] =} is_stabilizable (@var{a}, @var{b}, @var{tol})
## Logical check for system stabilizability (i.e., all unstable modes are controllable).
##
## Test for stabilizability is performed via an ordered Schur decomposition
## that reveals the unstable subspace of the system @var{a} matrix.
##
## Returns @code{retval} = 1 if the system, @var{a}, is stabilizable,
## if the pair  (@var{a}, @var{b}) is stabilizable, or 0 if not.
## @var{u} = orthogonal basis of controllable subspace.
##
## Controllable subspace is determined by applying Arnoldi iteration with
## complete re-orthogonalization to obtain an orthogonal basis of the
## Krylov subspace.
## @example
##   span ([b,a*b,...,a^   b]).
## @end example
## tol is a roundoff paramter, set to 200*eps if omitted.
## @end deftypefn

## See also: size, rows, columns, length, ismatrix, isscalar, isvector
##     is_observable, is_stabilizable, is_detectable

## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
## Created: August 1993
## Updated by A. S. Hodel (scotte@eng.auburn.edu) Aubust, 1995 to use krylovb
## Updated by John Ingram (ingraje@eng.auburn.edu) July, 1996 to accept systems

function [retval, U] = is_stabilizable (a, b, tol)

  if(nargin < 1)        usage("[retval,U] = is_stabilizable(a {, b ,tol})");
  elseif(isstruct(a))
    ## sustem passed.
    if(nargin == 2)
      tol = b;          % get tolerance
    elseif(nargin > 2)
      usage("[retval,U] = is_stabilizable(sys{,tol})");
    endif
    [a,b] = sys2ss(a);
  else
    ## a,b arguments sent directly.
    if(nargin > 3)
      usage("[retval,U] = is_stabilizable(a {, b ,tol})");
    endif
  endif

  if(exist("tol"))
    [retval,U] = is_controllable(a,b,tol);
  else
    [retval,U] = is_controllable(a,b);
    tol = 1e2*rows(b)*eps;
  endif

  if( !retval & columns(U) > 0)
    ## now use an ordered Schur decomposition to get an orthogonal
    ## basis of the unstable subspace...
    n = rows(a);
    [ua, s] = schur (-(a+eye(n)*tol), "A");
    k = sum( real(eig(a)) >= 0 );       # count unstable poles

    if( k > 0 )
      ua = ua(:,1:k);
      ## now see if span(ua) is contained in span(U)
      retval = (norm(ua - U*U'*ua) < tol);
    else
      retval = 1;                       # all poles stable
    endif
  endif

endfunction