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## -*- 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, OPTIMAL CONTROL: LINEAR QUADRATIC METHODS,
## Prentice-Hall, 1990, pp. 56-58
## @end deftypefn

function [k, p, e] = lqr (a, b, q, r, s)
# Written by A. S. Hodel (scotte@eng.auburn.edu) August 1993.

  # disp("lqr: entry");

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

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

# Check r.
  if ( ((m1 = is_square(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 (is_symmetric (q) && is_symmetric (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

  # disp("lqr: exit");
endfunction