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##               Auburn University.  All rights reserved.
##
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## your option) any later version.
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## General Public License for more details.
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## along with Octave; see the file COPYING.  If not, see
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## -*- texinfo -*-
## @deftypefn {Function File} {[@var{tvals}, @var{plist}] =} dre (@var{sys}, @var{q}, @var{r}, @var{qf}, @var{t0}, @var{tf}, @var{ptol}, @var{maxits})
## Solve the differential Riccati equation
## @ifinfo
## @example
##   -d P/dt = A'P + P A - P B inv(R) B' P + Q
##   P(tf) = Qf
## @end example
## @end ifinfo
## @iftex
## @tex
## $$ -{dP \over dt} = A^T P+PA-PBR^{-1}B^T P+Q $$
## $$ P(t_f) = Q_f $$
## @end tex
## @end iftex
## for the @acronym{LTI} system sys.  Solution of 
## standard @acronym{LTI} state feedback optimization
## @ifinfo
## @example
##   min int(t0, tf) ( x' Q x + u' R u ) dt + x(tf)' Qf x(tf)
## @end example
## @end ifinfo
## @iftex
## @tex
## $$ \min \int_{t_0}^{t_f} x^T Q x + u^T R u dt + x(t_f)^T Q_f x(t_f) $$
## @end tex
## @end iftex
## optimal input is
## @ifinfo
## @example
##   u = - inv(R) B' P(t) x
## @end example
## @end ifinfo
## @iftex
## @tex
## $$ u = - R^{-1} B^T P(t) x $$
## @end tex
## @end iftex
## @strong{Inputs}
## @table @var
## @item sys
## continuous time system data structure
## @item q
## state integral penalty
## @item r
## input integral penalty
## @item qf
## state terminal penalty
## @item t0
## @itemx tf
## limits on the integral
## @item ptol
## tolerance (used to select time samples; see below); default = 0.1
## @item maxits
## number of refinement iterations (default=10)
## @end table
## @strong{Outputs}
## @table @var
## @item tvals
## time values at which @var{p}(@var{t}) is computed
## @item plist
## list values of @var{p}(@var{t}); @var{plist} @{ @var{i} @}
## is @var{p}(@var{tvals}(@var{i}))
## @end table
## @var{tvals} is selected so that:
## @iftex
## @tex
## $$ \Vert plist_{i} - plist_{i-1} \Vert < ptol $$
## @end tex
## @end iftex
## @ifinfo
## @example
## || Plist@{i@} - Plist@{i-1@} || < Ptol
## @end example
## @end ifinfo
## for every @var{i} between 2 and length(@var{tvals}).
## @end deftypefn

function [tvals, Plist] = dre (sys, Q, R, Qf, t0, tf, Ptol, maxits)

  if (nargin < 6 || nargin > 8 || nargout != 2)
    print_usage ();
  elseif (! isstruct (sys))
    error ("sys must be a system data structure")
  elseif (is_digital (sys))
    error ("sys must be a continuous time system")
  elseif (! ismatrix (Q) || ! ismatrix (R) || ! ismatrix (Qf))
    error ("Q, R, and Qf must be matrices");
  elseif (! isscalar (t0) || ! isscalar (tf))
    error ("t0 and tf must be scalars")
  elseif (t0 >= tf)
    error ("t0=%e >= tf=%e", t0, tf);
  elseif (nargin < 7)
    Ptol = 0.1;
  elseif (! isscalar (Ptol))
    error ("Ptol must be a scalar");
  elseif (Ptol <= 0)
    error ("Ptol must be positive");
  endif

  if (nargin < 8)
    maxits = 10;
  elseif (! isscalar (maxits))
    error ("maxits must be a scalar");
  elseif (maxits <= 0)
    error ("maxits must be positive");
  endif
  maxits = ceil (maxits);

  [aa, bb] = sys2ss (sys);
  nn = sysdimensions (sys, "cst");
  mm = sysdimensions (sys, "in");
  pp = sysdimensions (sys, "out");

  if (size (Q) != [nn, nn])
    error ("Q(%dx%d); sys has %d states", rows (Q), columns (Q), nn);
  elseif (size (Qf) != [nn, nn])
    error ("Qf(%dx%d); sys has %d states", rows (Qf), columns (Qf), nn);
  elseif (size (R) != [mm, mm])
    error ("R(%dx%d); sys has %d inputs", rows (R), columns (R), mm);
  endif

  ## construct Hamiltonian matrix
  H = [aa , -(bb/R)*bb' ; -Q, -aa'];

  ## select time step to avoid numerical overflow
  fast_eig = max (abs (eig (H)));
  tc = log (10) / fast_eig;
  nst = ceil ((tf-t0)/tc);
  tvals = -linspace (-tf, -t0, nst);
  Plist = list (Qf);
  In = eye (nn);
  n1 = nn+1;
  n2 = nn+nn;
  done = 0;
  while (! done)
    done = 1;      # assume this pass will do the job
    ## sort time values in reverse order
    tvals = -sort (-tvals);
    tvlen = length (tvals);
    maxerr = 0;
    ## compute new values of P(t); recompute old values just in case
    for ii = 2:tvlen
      uv_i_minus_1 = [In; Plist{ii-1}];
      delta_t = tvals(ii-1) - tvals(ii);
      uv = expm (-H*delta_t)*uv_i_minus_1;
      Qi = uv(n1:n2,1:nn)/uv(1:nn,1:nn);
      Plist(ii) = (Qi+Qi')/2;
      ## check error
      Perr = norm (Plist{ii} - Plist{ii-1})/norm(Plist{ii});
      maxerr = max (maxerr,Perr);
      if (Perr > Ptol)
        new_t = mean (tvals([ii,ii-1]));
        tvals = [tvals, new_t];
        done = 0;
      endif
    endfor

    ## check number of iterations
    maxits = maxits - 1;
    done = done + (maxits == 0);
  endwhile
  if (maxerr > Ptol)
    warning ("dre: exiting with %d points, max rel chg. = %e, Ptol = %e",
             tvlen, maxerr, Ptol);
    tvals = tvals(1:length(Plist));
  endif

endfunction