--- /dev/null
+## Copyright (C) 1996, 2000, 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} rlocus (@var{sys})
+## @deftypefnx {Function File} {[@var{rldata}, @var{k}] =} rlocus (@var{sys}, @var{increment}, @var{min_k}, @var{max_k})
+## Display root locus plot of the specified @acronym{SISO} system.
+##
+## @strong{Inputs}
+## @table @var
+## @item sys
+## LTI model. Must be a single-input and single-output (SISO) system.
+## @item min_k
+## Minimum value of @var{k}.
+## @item max_k
+## Maximum value of @var{k}.
+## @item increment
+## The increment used in computing gain values.
+## @end table
+##
+## @strong{Outputs}
+## @table @var
+## @item rldata
+## Data points plotted: in column 1 real values, in column 2 the imaginary values.
+## @item k
+## Gains for real axis break points.
+## @end table
+##
+## @strong{Block Diagram}
+## @example
+## @group
+## u + +---+ +------+ y
+## ------>(+)----->| k |----->| SISO |-------+------->
+## ^ - +---+ +------+ |
+## | |
+## +---------------------------------+
+## @end group
+## @end example
+## @end deftypefn
+
+## Author: David Clem
+## Author: R. Bruce Tenison <btenison@eng.auburn.edu>
+## Updated by Kristi McGowan July 1996 for intelligent gain selection
+## Updated by John Ingram July 1996 for systems
+
+## Adapted-By: Lukas Reichlin <lukas.reichlin@gmail.com>
+## Date: December 2009
+## Version: 0.4
+
+## TODO: Improve compatibility
+
+function [rldata_r, k_break, rlpol, gvec, real_ax_pts] = rlocus (sys, increment, min_k, max_k)
+
+ ## TODO: multiplot feature: rlocus (sys1, "b", sys2, "r", ...)
+
+ if (nargin < 1 || nargin > 4)
+ print_usage ();
+ endif
+
+ if (! isa (sys, "lti") || ! issiso (sys))
+ error ("rlocus: first argument must be a SISO LTI model");
+ endif
+
+ ## Convert the input to a transfer function if necessary
+ [num, den] = tfdata (sys, "vector"); # extract numerator/denominator polynomials
+ lnum = length (num);
+ lden = length (den);
+ ## equalize length of num, den polynomials
+ ## TODO: handle case lnum > lden (non-proper models)
+ if (lden < 2)
+ error ("rlocus: system has no poles");
+ elseif (lnum < lden)
+ num = [zeros(1,lden-lnum), num]; # so that derivative is shortened by one
+ endif
+
+ olpol = roots (den);
+ olzer = roots (num);
+ nas = lden - lnum; # number of asymptotes
+ maxk = 0;
+ if (nas > 0)
+ cas = (sum (olpol) - sum (olzer)) / nas;
+ angles = (2*[1:nas]-1)*pi/nas;
+ ## printf("rlocus: there are %d asymptotes centered at %f\n", nas, cas);
+ else
+ cas = angles = [];
+ maxk = 100*den(1)/num(1);
+ endif
+
+
+ ## compute real axis break points and corresponding gains
+ dnum = polyder (num);
+ dden = polyder (den);
+ brkp = conv (den, dnum) - conv (num, dden);
+ real_ax_pts = roots (brkp);
+ real_ax_pts = real_ax_pts(find (imag (real_ax_pts) == 0));
+ k_break = -polyval (den, real_ax_pts) ./ polyval (num, real_ax_pts);
+ idx = find (k_break >= 0);
+ k_break = k_break(idx);
+ real_ax_pts = real_ax_pts(idx);
+ if (! isempty (k_break))
+ maxk = max (max (k_break), maxk);
+ endif
+
+ if (nas == 0)
+ maxk = max (1, 2*maxk); # get at least some root locus
+ else
+ ## get distance from breakpoints, poles, and zeros to center of asymptotes
+ dmax = 3*max (abs ([vec(olzer); vec(olpol); vec(real_ax_pts)] - cas));
+ if (dmax == 0)
+ dmax = 1;
+ endif
+
+ ## get gain for dmax along each asymptote, adjust maxk if necessary
+ svals = cas + dmax * exp (j*angles);
+ kvals = -polyval (den, svals) ./ polyval (num, svals);
+ maxk = max (maxk, max (real (kvals)));
+ endif
+
+ ## check for input arguments:
+ if (nargin > 2)
+ mink = min_k;
+ else
+ mink = 0;
+ endif
+ if (nargin > 3)
+ maxk = max_k;
+ endif
+ if (nargin > 1)
+ if (increment <= 0)
+ error ("rlocus: increment must be positive");
+ else
+ ngain = (maxk-mink)/increment;
+ endif
+ else
+ ngain = 30;
+ endif
+
+ ## vector of gains
+ ngain = max (30, ngain);
+ gvec = linspace (mink, maxk, ngain);
+ if (length (k_break))
+ gvec = sort ([gvec, reshape(k_break, 1, [])]);
+ endif
+
+ ## Find the open loop zeros and the initial poles
+ rlzer = roots (num);
+
+ ## update num to be the same length as den
+ lnum = length (num);
+ if (lnum < lden)
+ num = [zeros(1,lden - lnum),num];
+ endif
+
+ ## compute preliminary pole sets
+ nroots = lden - 1;
+ for ii = 1:ngain
+ gain = gvec(ii);
+ rlpol(1:nroots,ii) = vec(sort (roots (den + gain*num)));
+ endfor
+
+ ## set smoothing tolerance
+ smtolx = 0.01*(max (max (real (rlpol))) - min (min (real (rlpol))));
+ smtoly = 0.01*(max (max (imag (rlpol))) - min (min (imag (rlpol))));
+ smtol = max (smtolx, smtoly);
+ ## sort according to nearest-neighbor
+ rlpol = sort_roots (rlpol, smtolx, smtoly);
+
+ done = (nargin == 4); # perform a smoothness check
+ while (! done && ngain < 1000)
+ done = 1 ; # assume done
+ dp = abs (diff (rlpol.')).';
+ maxdp = max (dp);
+
+ ## search for poles whose neighbors are distant
+ if (lden == 2)
+ idx = find (dp > smtol);
+ else
+ idx = find (maxdp > smtol);
+ endif
+
+ for ii = 1:length(idx)
+ i1 = idx(ii);
+ g1 = gvec(i1);
+ p1 = rlpol(:,i1);
+
+ i2 = idx(ii)+1;
+ g2 = gvec(i2);
+ p2 = rlpol(:,i2);
+
+ ## isolate poles in p1, p2
+ if (max (abs (p2-p1)) > smtol)
+ newg = linspace (g1, g2, 5);
+ newg = newg(2:4);
+ gvec = [gvec,newg];
+ done = 0; # need to process new gains
+ endif
+ endfor
+
+ ## process new gain values
+ ngain1 = length (gvec);
+ for ii = (ngain+1):ngain1
+ gain = gvec(ii);
+ rlpol(1:nroots,ii) = vec(sort (roots (den + gain*num)));
+ endfor
+
+ [gvec, idx] = sort (gvec);
+ rlpol = rlpol(:,idx);
+ ngain = length (gvec);
+ ## sort according to nearest-neighbor
+ rlpol = sort_roots (rlpol, smtolx, smtoly);
+ endwhile
+ rldata = rlpol;
+
+ ## Plot the data
+ if (nargout == 0)
+ rlpolv = vec(rlpol);
+ axdata = [real(rlpolv), imag(rlpolv); real(olzer), imag(olzer)];
+ axlim = __axis_limits__ (axdata);
+ rldata = [real(rlpolv), imag(rlpolv) ];
+
+ %inname = get (sys, "inname");
+ %outname = get (sys, "outname");
+
+ ## build plot command args pole by pole
+
+ n_rlpol = rows (rlpol);
+ nelts = n_rlpol+1;
+ if (! isempty (rlzer))
+ nelts++;
+ endif
+ ## add asymptotes
+ n_A = length (olpol) - length (olzer);
+ if (n_A > 0)
+ nelts += n_A;
+ endif
+ args = cell (3, nelts);
+ kk = 0;
+ ## asymptotes first
+ if (n_A > 0)
+ len_A = 2*max (abs (axlim));
+ sigma_A = (sum(olpol) - sum(olzer))/n_A;
+ for i_A=0:n_A-1
+ phi_A = pi*(2*i_A + 1)/n_A;
+ args{1,++kk} = [sigma_A sigma_A+len_A*cos(phi_A)];
+ args{2,kk} = [0 len_A*sin(phi_A)];
+ if (i_A == 1)
+ args{3,kk} = "k--;asymptotes;";
+ else
+ args{3,kk} = "k--";
+ endif
+ endfor
+ endif
+ ## locus next
+ for ii = 1:rows(rlpol)
+ args{1,++kk} = real (rlpol (ii,:));
+ args{2,kk} = imag (rlpol (ii,:));
+ if (ii == 1)
+ args{3,kk} = "b-;locus;";
+ else
+ args{3,kk} = "b-";
+ endif
+ endfor
+ ## poles and zeros last
+ args{1,++kk} = real (olpol);
+ args{2,kk} = imag (olpol);
+ args{3,kk} = "rx;open loop poles;";
+ if (! isempty (rlzer))
+ args{1,++kk} = real (rlzer);
+ args{2,kk} = imag (rlzer);
+ args{3,kk} = "go;zeros;";
+ endif
+
+ set (gcf,"visible","off");
+ hplt = plot (args{:});
+ set (hplt(kk--), "markersize", 2);
+ if (! isempty (rlzer))
+ set (hplt(kk--), "markersize", 2);
+ endif
+ for ii = 1:rows(rlpol)
+ set (hplt(kk--), "linewidth", 2);
+ endfor
+ legend ("boxon", 2);
+ grid ("on");
+ axis (axlim);
+ title (["Root Locus of ", inputname(1)]);
+ xlabel (sprintf ("Real Axis gain = [%g, %g]", gvec(1), gvec(ngain)));
+ ylabel ("Imaginary Axis");
+ set (gcf (), "visible", "on");
+ else
+ rldata_r = rldata;
+ endif
+endfunction
+
+
+function rlpol = sort_roots (rlpol, tolx, toly)
+ ## no point sorting of you've only got one pole!
+ if (rows (rlpol) == 1)
+ return;
+ endif
+
+ ## reorder entries in each column of rlpol to be by their nearest-neighbors rlpol
+ dp = diff (rlpol.').';
+ drp = max (real (dp));
+ dip = max (imag (dp));
+ idx = find (drp > tolx | dip > toly);
+ if (isempty (idx))
+ return;
+ endif
+
+ [np, ng] = size (rlpol); # num poles, num gains
+ for jj = idx
+ vals = rlpol(:,[jj,jj+1]);
+ jdx = (jj+1):ng;
+ for ii = 1:rows(rlpol-1)
+ rdx = ii:np;
+ dval = abs (rlpol(rdx,jj+1)-rlpol(ii,jj));
+ mindist = min (dval);
+ sidx = min (find (dval == mindist)) + ii - 1;
+ if (sidx != ii)
+ c1 = norm (diff(vals.'));
+ [vals(ii,2), vals(sidx,2)] = swap (vals(ii,2), vals(sidx,2));
+ c2 = norm (diff (vals.'));
+ if (c1 > c2)
+ ## perform the swap
+ [rlpol(ii,jdx), rlpol(sidx,jdx)] = swap (rlpol(ii,jdx), rlpol(sidx,jdx));
+ vals = rlpol(:,[jj,jj+1]);
+ endif
+ endif
+ endfor
+ endfor
+
+endfunction
+
+
+function [b, a] = swap (a, b)
+
+endfunction
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff