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1 ## Copyright (C) 1996 Auburn University. All rights reserved. |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by the |
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7 ## Free Software Foundation; either version 2, or (at your option) any |
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8 ## later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but WITHOUT |
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11 ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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12 ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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13 ## for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301 USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {[@var{rldata}, @var{k}] =} rlocus (@var{sys}[, @var{increment}, @var{min_k}, @var{max_k}]) |
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22 ## |
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23 ## Display root locus plot of the specified @acronym{SISO} system. |
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24 ## @example |
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25 ## @group |
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26 ## ----- --- -------- |
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27 ## --->| + |---|k|---->| SISO |-----------> |
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28 ## ----- --- -------- | |
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29 ## - ^ | |
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30 ## |_____________________________| |
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31 ## @end group |
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32 ## @end example |
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33 ## |
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34 ## @strong{Inputs} |
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35 ## @table @var |
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36 ## @item sys |
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37 ## system data structure |
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38 ## @item min_k |
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39 ## Minimum value of @var{k} |
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40 ## @item max_k |
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41 ## Maximum value of @var{k} |
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42 ## @item increment |
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43 ## The increment used in computing gain values |
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44 ## @end table |
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45 ## |
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46 ## @strong{Outputs} |
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47 ## |
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48 ## Plots the root locus to the screen. |
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49 ## @table @var |
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50 ## @item rldata |
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51 ## Data points plotted: in column 1 real values, in column 2 the imaginary values. |
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52 ## @item k |
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53 ## Gains for real axis break points. |
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54 ## @end table |
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55 ## @end deftypefn |
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56 |
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57 ## Author: David Clem |
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58 ## Author: R. Bruce Tenison <btenison@eng.auburn.edu> |
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59 ## Updated by Kristi McGowan July 1996 for intelligent gain selection |
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60 ## Updated by John Ingram July 1996 for systems |
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61 |
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62 function [rldata, k_break, rlpol, gvec, real_ax_pts] = rlocus (sys, increment, min_k, max_k) |
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63 |
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64 if (nargin < 1 || nargin > 4) |
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65 print_usage (); |
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66 endif |
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67 |
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68 ## Convert the input to a transfer function if necessary |
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69 [num,den] = sys2tf(sys); # extract numerator/denom polyomials |
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70 lnum = length(num); |
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71 lden = length(den); |
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72 # equalize length of num, den polynomials |
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73 if(lden < 2) |
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74 error("system has no poles"); |
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75 elseif(lnum < lden) |
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76 num = [zeros(1,lden-lnum), num]; # so that derivative is shortened by one |
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77 endif |
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78 |
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79 olpol = roots(den); |
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80 olzer = roots(num); |
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81 nas = lden -lnum; # number of asymptotes |
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82 maxk = 0; |
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83 if(nas > 0) |
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84 cas = ( sum(olpol) - sum(olzer) )/nas; |
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85 angles = (2*[1:nas]-1)*pi/nas; |
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86 # printf("rlocus: there are %d asymptotes centered at %f\n", nas, cas); |
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87 else |
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88 cas = angles = []; |
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89 maxk = 100*den(1)/num(1); |
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90 endif |
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91 |
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92 |
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93 # compute real axis break points and corresponding gains |
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94 dnum=polyderiv(num); |
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95 dden=polyderiv(den); |
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96 brkp = conv(den, dnum) - conv(num, dden); |
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97 real_ax_pts = roots(brkp); |
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98 real_ax_pts = real_ax_pts(find(imag(real_ax_pts) == 0)); |
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99 k_break = -polyval(den,real_ax_pts) ./ polyval(num, real_ax_pts); |
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100 idx = find(k_break >= 0); |
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101 k_break = k_break(idx); |
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102 real_ax_pts = real_ax_pts(idx); |
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103 if(!isempty(k_break)) |
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104 maxk = max(max(k_break),maxk); |
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105 endif |
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106 |
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107 if(nas == 0) |
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108 maxk = max(1, 2*maxk); % get at least some root locus |
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109 else |
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110 # get distance from breakpoints, poles, and zeros to center of asymptotes |
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111 dmax = 3*max(abs( [vec(olzer); vec(olpol); vec(real_ax_pts)] - cas )); |
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112 if(dmax == 0) |
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113 dmax = 1; |
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114 endif |
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115 |
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116 # get gain for dmax along each asymptote, adjust maxk if necessary |
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117 svals = cas + dmax*exp(j*angles); |
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118 kvals = -polyval(den,svals) ./ polyval(num, svals); |
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119 maxk = max(maxk, max(real(kvals))); |
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120 end |
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121 |
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122 ## check for input arguments: |
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123 if (nargin > 2) |
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124 mink = min_k; |
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125 else |
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126 mink = 0; |
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127 endif |
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128 if (nargin > 3) |
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129 maxk = max_k; |
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130 endif |
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131 if (nargin > 1) |
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132 if(increment <= 0) |
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133 error("increment must be positive"); |
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134 else |
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135 ngain = (maxk-mink)/increment; |
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136 endif |
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137 else |
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138 ngain = 30; |
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139 endif |
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140 |
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141 ## vector of gains |
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142 ngain = max(30,ngain); |
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143 gvec = linspace(mink,maxk,ngain); |
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144 if(length(k_break)) |
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145 gvec = sort([gvec, vec(k_break)']); |
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146 endif |
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147 |
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148 ## Find the open loop zeros and the initial poles |
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149 rlzer = roots(num); |
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150 |
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151 ## update num to be the same length as den |
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152 lnum = length(num); |
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153 if(lnum < lden) |
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154 num = [zeros(1,lden - lnum),num]; |
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155 endif |
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156 |
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157 ## compute preliminary pole sets |
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158 nroots = lden-1; |
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159 for ii=1:ngain |
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160 gain = gvec(ii); |
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161 rlpol(1:nroots,ii) = vec(sortcom(roots(den + gain*num))); |
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162 endfor |
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163 |
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164 ## set smoothing tolerance |
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165 smtolx = 0.01*( max(max(real(rlpol))) - min(min(real(rlpol)))); |
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166 smtoly = 0.01*( max(max(imag(rlpol))) - min(min(imag(rlpol)))); |
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167 smtol = max(smtolx, smtoly); |
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168 rlpol = sort_roots(rlpol,smtolx, smtoly); % sort according to nearest-neighbor |
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169 |
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170 done=(nargin == 4); # perform a smoothness check |
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171 while((!done) & ngain < 1000) |
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172 done = 1 ; # assume done |
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173 dp = abs(diff(rlpol'))'; |
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174 maxdp = max(dp); |
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175 |
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176 ## search for poles whose neighbors are distant |
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177 if(lden == 2) |
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178 idx = find(dp > smtol); |
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179 else |
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180 idx = find(maxdp > smtol); |
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181 endif |
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182 |
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183 for ii=1:length(idx) |
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184 i1 = idx(ii); |
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185 g1 = gvec(i1); |
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186 p1 = rlpol(:,i1); |
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187 |
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188 i2 = idx(ii)+1; |
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189 g2 = gvec(i2); |
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190 p2 = rlpol(:,i2); |
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191 |
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192 ## isolate poles in p1, p2 |
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193 if( max(abs(p2-p1)) > smtol) |
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194 newg = linspace(g1,g2,5); |
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195 newg = newg(2:4); |
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196 gvec = [gvec,newg]; |
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197 done = 0; # need to process new gains |
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198 endif |
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199 endfor |
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200 |
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201 ## process new gain values |
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202 ngain1 = length(gvec); |
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203 for ii=(ngain+1):ngain1 |
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204 gain = gvec(ii); |
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205 rlpol(1:nroots,ii) = vec(sortcom(roots(den + gain*num))); |
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206 endfor |
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207 |
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208 [gvec,idx] = sort(gvec); |
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209 rlpol = rlpol(:,idx); |
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210 ngain = length(gvec); |
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211 rlpol = sort_roots(rlpol,smtolx, smtoly); % sort according to nearest-neighbor |
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212 endwhile |
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213 rldata = rlpol; |
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214 |
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215 ## Plot the data |
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216 if(nargout == 0) |
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217 rlpolv = vec(rlpol); |
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218 axdata = [real(rlpolv),imag(rlpolv); real(olzer), imag(olzer)]; |
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219 axlim = axis2dlim(axdata); |
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220 rldata = [real(rlpolv), imag(rlpolv) ]; |
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221 [stn,inname,outname] = sysgetsignals(sys); |
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222 |
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223 ## build plot command args pole by pole |
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224 |
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225 n_rlpol = rows(rlpol); |
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226 nelts = n_rlpol+1; |
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227 if (! isempty (rlzer)) |
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228 nelts++; |
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229 endif |
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230 # add asymptotes |
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231 n_A = length (olpol) - length (olzer); |
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232 if (n_A > 0) |
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233 nelts += n_A; |
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234 endif |
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235 args = cell (3, nelts); |
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236 kk = 0; |
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237 # asymptotes first |
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238 if (n_A > 0) |
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239 len_A = 2*max(abs(axlim)); |
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240 sigma_A = (sum(olpol) - sum(olzer))/n_A; |
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241 for i_A=0:n_A-1 |
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242 phi_A = pi*(2*i_A + 1)/n_A; |
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243 args{1,++kk} = [sigma_A sigma_A+len_A*cos(phi_A)]; |
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244 args{2,kk} = [0 len_A*sin(phi_A)]; |
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245 if (i_A == 1) |
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246 args{3,kk} = "k--;asymptotes;"; |
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247 else |
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248 args{3,kk} = "k--"; |
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249 endif |
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250 endfor |
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251 endif |
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252 # locus next |
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253 for ii=1:rows(rlpol) |
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254 args{1,++kk} = real (rlpol (ii,:)); |
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255 args{2,kk} = imag (rlpol (ii,:)); |
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256 if (ii == 1) |
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257 args{3,kk} = "b-;locus;"; |
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258 else |
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259 args{3,kk} = "b-"; |
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260 endif |
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261 endfor |
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262 # poles and zeros last |
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263 args{1,++kk} = real(olpol); |
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264 args{2,kk} = imag(olpol); |
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265 args{3,kk} = "rx;open loop poles;"; |
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266 if (! isempty(rlzer)) |
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267 args{1,++kk} = real(rlzer); |
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268 args{2,kk} = imag(rlzer); |
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269 args{3,kk} = "go;zeros;"; |
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270 endif |
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271 |
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272 set (gcf,"visible","off"); |
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273 hplt = plot (args{:}); |
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274 set (hplt(kk--), "markersize", 2); |
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275 if (! isempty(rlzer)) |
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276 set(hplt(kk--), "markersize", 2); |
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277 endif |
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278 for ii=1:rows(rlpol) |
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279 set (hplt(kk--), "linewidth", 2); |
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280 endfor |
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281 legend ("boxon", 2); |
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282 grid ("on"); |
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283 axis (axlim); |
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284 xlabel (sprintf ("Root locus from %s to %s, gain=[%f,%f]: Real axis", |
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285 inname{1}, outname{1}, gvec(1), gvec(ngain))); |
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286 ylabel ("Imag. axis"); |
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287 set (gcf,"visible","on"); |
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288 rldata = []; |
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289 endif |
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290 endfunction |
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291 |
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292 function rlpol = sort_roots (rlpol,tolx, toly) |
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293 # no point sorting of you've only got one pole! |
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294 if(rows(rlpol) == 1) |
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295 return |
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296 endif |
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297 |
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298 # reorder entries in each column of rlpol to be by their nearest-neighbors |
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299 dp = diff(rlpol')'; |
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300 drp = max(real(dp)); |
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301 dip = max(imag(dp)); |
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302 idx = find( drp > tolx | dip > toly ); |
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303 if(isempty(idx) ) |
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304 return |
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305 endif |
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306 |
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307 [np,ng] = size(rlpol); # num poles, num gains |
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308 for jj = idx |
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309 vals = rlpol(:,[jj,jj+1]); |
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310 jdx = (jj+1):ng; |
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311 for ii=1:rows(rlpol-1) |
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312 rdx = ii:np; |
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313 dval = abs(rlpol(rdx,jj+1)-rlpol(ii,jj)); |
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314 mindist = min(dval); |
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315 sidx = min( find ( dval == mindist)) + ii - 1; |
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316 if( sidx != ii) |
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317 c1 = norm(diff(vals')); |
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318 [vals(ii,2), vals(sidx,2)] = swap( vals(ii,2), vals(sidx,2)); |
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319 c2 = norm(diff(vals')); |
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320 if(c1 > c2 ) |
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321 # perform the swap |
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322 [rlpol(ii,jdx), rlpol(sidx,jdx)] = swap( rlpol(ii,jdx), rlpol(sidx,jdx)); |
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323 vals = rlpol(:,[jj,jj+1]); |
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324 endif |
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325 endif |
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326 endfor |
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327 endfor |
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328 |
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329 endfunction |
