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## Copyright (C) 1998, 2000, 2002, 2004, 2005, 2006, 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{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)
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
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