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c2d.m

## Copyright (C) 1993, 1994, 1995, 2000, 2001, 2002, 2004, 2005, 2006,
##               2007 John W. Eaton
##
##
## 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} {} c2d (@var{sys}, @var{opt}, @var{t})
## @deftypefnx {Function File} {} c2d (@var{sys}, @var{t})
##
## Converts the system data structure describing:
## @iftex
## @tex
## $$ \dot x = A_cx + B_cu $$
## @end tex
## @end iftex
## @ifinfo
## @example
## .
## x = Ac x + Bc u
## @end example
## @end ifinfo
## into a discrete time equivalent model:
## @iftex
## @tex
## $$ x_{n+1} = A_dx_n + B_du_n $$
## @end tex
## @end iftex
## @ifinfo
## @example
## x[n+1] = Ad x[n] + Bd u[n]
## @end example
## @end ifinfo
## via the matrix exponential or bilinear transform.
##
## @strong{Inputs}
## @table @var
## @item sys
## system data structure (may have both continuous time and discrete
## time subsystems)
## @item opt
## string argument; conversion option (optional argument;
## may be omitted as shown above)
## @table @code
## @item "ex"
## use the matrix exponential (default)
## @item "bi"
## use the bilinear transformation
## @iftex
## @tex
## $$ s = { 2(z-1) \over T(z+1) } $$
## @end tex
## @end iftex
## @ifinfo
## @example
##     2(z-1)
## s = -----
##     T(z+1)
## @end example
## @end ifinfo
## FIXME: This option exits with an error if @var{sys} is not purely
## continuous. (The @code{ex} option can handle mixed systems.)
## @item "matched"
## Use the matched pole/zero equivalent transformation (currently only
## works for purely continuous @acronym{SISO} systems).
## @end table
## @item t
## sampling time; required if @var{sys} is purely continuous.
## 
## @strong{Note} that if the second argument is not a string, @code{c2d()}
## assumes that the second argument is @var{t} and performs 
## appropriate argument checks.
## @end table
##
## @strong{Output}
## @table @var
## @item dsys 
## Discrete time equivalent via zero-order hold, sample each @var{t} sec.
## @end table
##
## This function adds the suffix  @code{_d}
## to the names of the new discrete states.
## @end deftypefn

## Author: R. Bruce Tenison <btenison@eng.auburn.edu>
## Created: October 1993
## Updated by John Ingram for system data structure August 1996

function dsys = c2d (sys, opt, T)

  ## parse input arguments
  if (nargin < 1 || nargin > 3)
    print_usage ();
  elseif (! isstruct (sys))
    error ("sys must be a system data structure");
  elseif (nargin == 1)
    opt = "ex";
  elseif (nargin == 2 && ! ischar (opt))
    T = opt;
    opt = "ex";
  endif

  if (! ischar (opt))
    error ("expecting option as a string");
  endif

  ## check if sampling period T was passed.
  Ts = sysgettsam (sys);
  if (! exist ("T"))
    T = Ts;
    if (T == 0)
      error ("sys is purely continuous; no sampling period T provided");
    endif
  elseif (T != Ts && Ts > 0)
    warning ("c2d: T=%g, system tsam=%g: using T=", T, Ts, min (T, Ts));
    T = min (T, Ts);
  endif

  if (! is_sample (T))
    error ("sampling period T must be a positive, real scalar");
  elseif (! (strcmp (opt, "ex")
           || strcmp (opt, "bi")
           || strcmp (opt, "matched")))
    error ("invalid option passed: %s", opt);
  endif

  sys = sysupdate (sys, "ss");
  [n, nz, m, p] = sysdimensions (sys);
  if (n == 0)
    dsys = syssetsignals (sys, "yd", ones(1:p));
  elseif (strcmp (opt, "ex"));
    [aa, bb, cc, dd] = sys2ss (sys);
    crng= 1:n;
    drng = n+(1:nz);

    ## partition state equations into continuous, imaginary subsystems
    Ac = aa(crng,crng);
    Bc = bb(crng,:);
    if (nz == 0)
      Acd = Adc = Add = Bd = 0;
    else
      Acd = aa(crng,drng);
      Adc = aa(drng,crng);
      Add = aa(drng,drng);
      Bd  = bb(drng,:);
      Bc  = [Bc, Acd];   ## append discrete states as inputs to cont system
    endif

    ## convert state equations
    mat = ;
    matexp = expm (mat * T);

    ## replace Ac
    aa(crng,crng) = matexp(crng,crng);    ## discretized homegenous diff eqn

    ## replace Bc
    bb(crng,:) = matexp(crng,n+(1:m));

    ## replace Acd
    if (nz)
      aa(crng,drng) = matexp(crng,n+m+(1:nz));
    endif

    stnames = sysgetsignals (sys, "st");   ## continuous states renamed below
    innames = sysgetsignals (sys, "in");
    outnames = sysgetsignals (sys, "out");
    outlist = 1:p;
    dsys = ss (aa, bb, cc, dd, T, 0, n+nz, stnames, innames,
             outnames, outlist);
    ## rename states
    for ii = 1:n
      strval = sprintf ("%s_d", sysgetsignals (dsys, "st", ii, 1));
      dsys = syssetsignals (dsys, "st", strval, ii);
    endfor

  elseif (strcmp (opt, "bi"))
    if (is_digital (sys))
      error ("c2d: system is already digital")
    else
      ## convert with bilinear transform
      [a, b, c, d, tsam, n, nz, stname, inname, outname, yd] = sys2ss (sys);
      IT = (2/T) * eye (size (a));
      A = (IT+a)/(IT-a);
      iab = (IT-a)\b;
      tk = 2 / sqrt (T);
      B = tk*iab;
      C = tk*(c/(IT-a));
      D = d + (c*iab);
      stnamed = strappend (stname, "_d");
      dsys = ss (A, B, C, D, T, 0, rows (A), stnamed, inname, outname);
     endif
   elseif (strcmp (opt, "matched"))
     if (is_digital (sys))
       error ("c2d: system is already digital");
     elseif (length (sys.inname) != 1 || length (sys.outname) != 1)
       error ("c2d: system in not single input, single output");
     else
       sys = sysupdate (sys, "zp");
       p = exp (sys.pol*T);
       z = exp (sys.zer*T);
       infinite_zeros = max (size (sys.pol)) - max (size (sys.zer)) - 1;
       for i = 1:infinite_zeros
       z = [z ; -1];
       endfor
       ## Should the freaquency we adjust around always be 1?   
       [cmag, cphase, cw] = bode (sys, 1);
       [dmag, dpahse, dw] = bode (zp (z, p, 1, T), 1);
       dsys = zp (z, p, cmag/dmag, T);
     endif
  else
    error ("invalid option = %s", opt);
  endif

endfunction

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