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## Copyright (C) 1996, 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} {} d2c (@var{sys}, @var{tol})
## @deftypefnx {Function File} {} d2c (@var{sys}, @var{opt})
## Convert a discrete (sub)system into a purely continuous one.
## The sampling time used is @code{sysgettsam(@var{sys})}.
##
## @strong{Inputs}
## @table @var
## @item sys
## system data structure with discrete components
## @item tol
## Scalar value.
## Tolerance for convergence of default @code{"log"} option (see below)
## @item opt
## conversion option. Choose from:
## @table @code
## @item "log"
## (default) Conversion is performed via a matrix logarithm.
## Due to some problems with this computation, it is
## followed by a steepest descent algorithm to identify continuous time
## @var{a}, @var{b}, to get a better fit to the original data.
##
## If called as @code{d2c (@var{sys}, @var{tol})}, with @var{tol}
## positive scalar, the @code{"log"} option is used. The default value
## for @var{tol} is @code{1e-8}.
## @item "bi"
## Conversion is performed via bilinear transform
## @math{z = (1 + s T / 2)/(1 - s T / 2)} where @math{T} is the
## system sampling time (see @code{sysgettsam}).
##
## FIXME: bilinear option exits with an error if @var{sys} is not purely
## discrete
## @end table
## @end table
## @strong{Output}
## @table @var
## @item csys
## continuous time system (same dimensions and signal names as in @var{sys}).
## @end table
## @end deftypefn
## Author: R. Bruce Tenison <btenison@eng.auburn.edu>
## Created: August 23, 1994
## Updated by John Ingram for system data structure August 1996
function csys = d2c (sys, opt)
## SYS_INTERNAL accesses members of system data structure
if (nargin != 1 && nargin != 2)
print_usage ();
elseif (! isstruct (sys))
error ("sys must be in system data structure");
elseif (nargin == 1)
opt = "log";
tol = 1e-12;
elseif (ischar (opt)) # all remaining cases are for nargin == 2
tol = 1e-12;
if (! (strcmp (opt, "log") || strcmp (opt, "bi")))
error ("d2c: invalid opt passed=%s", opt);
endif
elseif (! is_sample (opt))
error ("tol must be a positive scalar")
elseif (opt > 1e-2)
warning ("d2c: ridiculous error tolerance passed=%g, intended c2d call?",
opt);
else
tol = opt;
opt = "log";
endif
T = sysgettsam (sys);
if (isa (sys.a, "single") || isa (sys.b, "single") || isa (sys.c, "single") ||
isa (sys.d, "single"))
myeps = eps ("single");
else
myeps = eps;
endif
if (strcmp (opt, "bi"))
## bilinear transform
## convert with bilinear transform
if (! is_digital (sys) )
error ("d2c requires a discrete time system for input")
endif
[a, b, c, d, tsam, n, nz, stname, inname, outname, yd] = sys2ss (sys);
poles = eig (a);
if (find (abs (poles-1) < 200*(n+nz)*myeps))
warning ("d2c: some poles very close to one. May get bad results.");
endif
I = eye (size (a));
tk = 2 / sqrt (T);
A = (2/T)*(a-I)/(a+I);
iab = (I+a)\b;
B = tk*iab;
C = tk*(c/(I+a));
D = d- (c*iab);
stnamec = strappend (stname, "_c");
csys = ss (A, B, C, D, 0, rows (A), 0, stnamec, inname, outname);
elseif (strcmp (opt, "log"))
sys = sysupdate (sys, "ss");
[n, nz, m, p] = sysdimensions (sys);
if (nz == 0)
warning ("d2c: all states continuous; setting outputs to agree");
csys = syssetsignals (sys, "yd", zeros (1, 1:p));
return;
elseif (n != 0)
warning ("d2c: n=%d > 0; performing c2d first", n);
sys = c2d (sys, T);
endif
[a, b] = sys2ss (sys);
[ma, na] = size (a);
[mb, nb] = size (b);
if (isempty (b))
warning ("d2c: empty b matrix");
Amat = a;
else
Amat = [a, b; zeros(nb,na), eye(nb)];
endif
poles = eig (a);
if (find (abs (poles) < 200*(n+nz)*myeps))
warning ("d2c: some poles very close to zero. logm not performed");
Mtop = zeros (ma, na+nb);
elseif (find (abs (poles-1) < 200*(n+nz)*myeps))
warning ("d2c: some poles very close to one. May get bad results.");
logmat = real (logm (Amat) / T);
Mtop = logmat(1:na,:);
else
logmat = real (logm (Amat) / T);
Mtop = logmat(1:na,:);
endif
## perform simplistic, stupid optimization approach.
## should re-write with a Davidson-Fletcher CG approach
mxthresh = norm (Mtop);
if (mxthresh == 0)
mxthresh = 1;
endif
eps1 = mxthresh; #gradient descent step size
cnt = max (20, (n*nz)*4); #max number of iterations
newgrad=1; #signal for new gradient
while ((eps1/mxthresh > tol) && cnt)
cnt--;
## calculate the gradient of error with respect to Amat...
geps = norm (Mtop) * 1e-8;
if (geps == 0)
geps = 1e-8;
endif
DMtop = Mtop;
if (isempty (b))
Mall = Mtop;
DMall = DMtop;
else
Mall = [Mtop; zeros(nb,na+nb)];
DMall = [DMtop; zeros(nb,na+nb) ];
endif
if (newgrad)
GrMall = zeros (size (Mall));
for ii = 1:rows(Mtop)
for jj = 1:columns(Mtop)
DMall(ii,jj) = Mall(ii,jj) + geps;
GrMall(ii,jj) = norm (Amat - expm (DMall*T), "fro") ...
- norm (Amat - expm (Mall*T), "fro");
DMall(ii,jj) = Mall(ii,jj);
endfor
endfor
GrMall = GrMall/norm(GrMall,1);
newgrad = 0;
endif
## got a gradient, now try to use it
DMall = Mall-eps1*GrMall;
FMall = expm (Mall*T);
FDMall = expm (DMall*T);
FmallErr = norm (Amat - FMall);
FdmallErr = norm (Amat - FDMall);
if (FdmallErr < FmallErr)
Mtop = DMall(1:na,:);
eps1 = min (eps1*2, 1e12);
newgrad = 1;
else
eps1 = eps1/2;
endif
if (FmallErr == 0)
eps1 = 0;
endif
endwhile
[aa, bb, cc, dd, tsam, nn, nz, stnam, innam, outnam, yd] = sys2ss (sys);
aa = Mall(1:na,1:na);
if (! isempty (b))
bb = Mall(1:na,(na+1):(na+nb));
endif
csys = ss (aa, bb, cc, dd, 0, na, 0, stnam, innam, outnam);
## update names
nn = sysdimensions (sys);
for ii = (nn+1):na
strval = sprintf ("%s_c", sysgetsignals (csys, "st", ii, 1));
csys = syssetsignals (csys, "st", strval, ii);
endfor
endif
endfunction
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