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## 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
## @acronym{LTI} model. Must be a single-input and single-output (SISO) system.
## @item increment
## The increment used in computing gain values.
## @item min_k
## Minimum value of @var{k}.
## @item max_k
## Maximum value of @var{k}.
## @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: 0.5
function [rldata_r, k_break, rlpol, gvec, real_ax_pts] = rlocus (sys, increment, min_k, max_k)
## TODO: improve compatibility to the "dark side"
## TODO: untangle spaghetti code
## 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));
real_ax_pts(polyval (num, real_ax_pts) == 0) = []; # avoid division by zero and therefore infinite k_break
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 = fix ((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
hplt = plot (args{:}); # yes, line 288 is a duplicate of line 290 - needed for subplots
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