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## Copyright (C) 2009, 2010, 2011, 2012 Lukas F. Reichlin
##
## This file is part of LTI Syncope.
##
## LTI Syncope 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.
##
## LTI Syncope 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 LTI Syncope. If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys})
## @deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}, @var{w})
## @deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}, @var{[]}, @var{ptype})
## @deftypefnx{Function File} {[@var{sv}, @var{w}] =} sigma (@var{sys}, @var{w}, @var{ptype})
## Singular values of frequency response. If no output arguments are given,
## the singular value plot is printed on the screen;
##
## @strong{Inputs}
## @table @var
## @item sys
## LTI system. Multiple inputs and/or outputs (MIMO systems) make practical sense.
## @item w
## Optional vector of frequency values. If @var{w} is not specified,
## it is calculated by the zeros and poles of the system.
## Alternatively, the cell @code{@{wmin, wmax@}} specifies a frequency range,
## where @var{wmin} and @var{wmax} denote minimum and maximum frequencies
## in rad/s.
## @item ptype = 0
## Singular values of the frequency response @var{H} of system @var{sys}. Default Value.
## @item ptype = 1
## Singular values of the frequency response @code{inv(H)}; i.e. inversed system.
## @item ptype = 2
## Singular values of the frequency response @code{I + H}; i.e. inversed sensitivity
## (or return difference) if @code{H = P * C}.
## @item ptype = 3
## Singular values of the frequency response @code{I + inv(H)}; i.e. inversed complementary
## sensitivity if @code{H = P * C}.
## @end table
##
## @strong{Outputs}
## @table @var
## @item sv
## Array of singular values. For a system with m inputs and p outputs, the array sv
## has @code{min (m, p)} rows and as many columns as frequency points @code{length (w)}.
## The singular values at the frequency @code{w(k)} are given by @code{sv(:,k)}.
## @item w
## Vector of frequency values used.
## @end table
##
## @seealso{bodemag, svd}
## @end deftypefn
## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
## Created: May 2009
## Version: 0.6
function [sv_r, w_r] = sigma (sys, w = [], resptype = 0)
## TODO: multiplot feature: sigma (sys1, "b", sys2, "r", ...)
if (nargin == 0 || nargin > 3)
print_usage ();
endif
[H, w] = __frequency_response__ (sys, w, true, resptype, "std", true);
sv = cellfun (@svd, H, "uniformoutput", false);
sv = horzcat (sv{:});
if (! nargout) # plot the information
## convert to dB for plotting
sv_db = 20 * log10 (sv);
## determine xlabel
if (isct (sys))
xl_str = "Frequency [rad/s]";
else
xl_str = sprintf ("Frequency [rad/s] w_N = %g", pi / get (sys, "tsam"));
endif
## plot results
semilogx (w, sv_db, "b")
axis ("tight")
ylim (__axis_margin__ (ylim))
grid ("on")
title (["Singular Values of ", inputname(1)])
xlabel (xl_str)
ylabel ("Singular Values [dB]")
else # return values
sv_r = sv;
w_r = reshape (w, [], 1);
endif
endfunction
%!shared sv_exp, w_exp, sv_obs, w_obs
%! A = [1, 2; 3, 4];
%! B = [5, 6; 7, 8];
%! C = [4, 3; 2, 1];
%! D = [8, 7; 6, 5];
%! w = [2, 3, 4];
%! sv_exp = [7.9176, 8.6275, 9.4393;
%! 0.6985, 0.6086, 0.5195];
%! w_exp = [2; 3; 4];
%! [sv_obs, w_obs] = sigma (ss (A, B, C, D), w);
%!assert (sv_obs, sv_exp, 1e-4);
%!assert (w_obs, w_exp, 1e-4);
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