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lqr.m

## Copyright (C) 1993, 1994, 1995, 2000, 2002, 2004, 2005, 2007
##               Auburn University.  All rights reserved.
##
##
## This program 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.
##
## This program 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 this program; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {[@var{k}, @var{p}, @var{e}] =} lqr (@var{a}, @var{b}, @var{q}, @var{r}, @var{z})
## construct the linear quadratic regulator for the continuous time system
## @iftex
## @tex
## $$
##  {dx\over dt} = A x + B u
## $$
## @end tex
## @end iftex
## @ifinfo
##
## @example
## dx
## -- = A x + B u
## dt
## @end example
##
## @end ifinfo
## to minimize the cost functional
## @iftex
## @tex
## $$
##  J = \int_0^\infty x^T Q x + u^T R u
## $$
## @end tex
## @end iftex
## @ifinfo
##
## @example
##       infinity
##       /
##   J = |  x' Q x + u' R u
##      /
##     t=0
## @end example
## @end ifinfo
##
## @noindent
## @var{z} omitted or
## @iftex
## @tex
## $$
##  J = \int_0^\infty x^T Q x + u^T R u + 2 x^T Z u
## $$
## @end tex
## @end iftex
## @ifinfo
##
## @example
##       infinity
##       /
##   J = |  x' Q x + u' R u + 2 x' Z u
##      /
##     t=0
## @end example
##
## @end ifinfo
## @var{z} included.
##
## The following values are returned:
##
## @table @var
## @item k
## The state feedback gain,
## @iftex
## @tex
## $(A - B K)$
## @end tex
## @end iftex
## @ifinfo
## (@var{a} - @var{b}@var{k})
## @end ifinfo
## is stable and minimizes the cost functional
##
## @item p
## The stabilizing solution of appropriate algebraic Riccati equation.
##
## @item e
## The vector of the closed loop poles of
## @iftex
## @tex
## $(A - B K)$.
## @end tex
## @end iftex
## @ifinfo
## (@var{a} - @var{b}@var{k}).
## @end ifinfo
## @end table
##
## @strong{Reference}
## Anderson and Moore, @cite{Optimal control: linear quadratic methods},
## Prentice-Hall, 1990, pp. 56--58.
## @end deftypefn

## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
## Created: August 1993.

function [k, p, e] = lqr (a, b, q, r, s)

  ## disp("lqr: entry");

  if (nargin != 4 && nargin != 5)
    error ("lqr: invalid number of arguments");
  endif

  ## Check a.
  if ((n = issquare (a)) == 0)
    error ("lqr: requires 1st parameter(a) to be square");
  endif

  ## Check b.
  [n1, m] = size (b);
  if (n1 != n)
    error ("lqr: a,b not conformal");
  endif

  ## Check q.
  if ((n1 = issquare (q)) == 0 || n1 != n)
    error ("lqr: q must be square and conformal with a");
  endif

  ## Check r.
  if ((m1 = issquare(r)) == 0 || m1 != m)
    error ("lqr: r must be square and conformal with column dimension of b");
  endif

  ## Check if n is there.
  if (nargin == 5)
    [n1, m1] = size (s);
    if (n1 != n || m1 != m)
      error ("lqr: z must be identically dimensioned with b");
    endif

    ## Incorporate cross term into a and q.
    ao = a - (b/r)*s';
    qo = q - (s/r)*s';
  else
    s = zeros (n, m);
    ao = a;
    qo = q;
  endif

  ## Check that q, (r) are symmetric, positive (semi)definite

  if (issymmetric (q) && issymmetric (r)
      && all (eig (q) >= 0) && all (eig (r) > 0))
    p = are (ao, (b/r)*b', qo);
    k = r\(b'*p + s');
    e = eig (a - b*k);
  else
    error ("lqr: q (r) must be symmetric positive (semi) definite");
  endif

endfunction

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