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1 ## Copyright (C) 1996, 1998 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} {} d2c (@var{sys}, @var{tol}) |
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22 ## @deftypefnx {Function File} {} d2c (@var{sys}, @var{opt}) |
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23 ## Convert a discrete (sub)system into a purely continuous one. |
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24 ## The sampling time used is @code{sysgettsam(@var{sys})}. |
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25 ## |
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26 ## @strong{Inputs} |
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27 ## @table @var |
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28 ## @item sys |
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29 ## system data structure with discrete components |
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30 ## @item tol |
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31 ## Scalar value. |
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32 ## Tolerance for convergence of default @code{"log"} option (see below) |
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33 ## @item opt |
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34 ## conversion option. Choose from: |
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35 ## @table @code |
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36 ## @item "log" |
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37 ## (default) Conversion is performed via a matrix logarithm. |
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38 ## Due to some problems with this computation, it is |
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39 ## followed by a steepest descent algorithm to identify continuous time |
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40 ## @var{a}, @var{b}, to get a better fit to the original data. |
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41 ## |
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42 ## If called as @code{d2c (@var{sys}, @var{tol})}, with @var{tol} |
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43 ## positive scalar, the @code{"log"} option is used. The default value |
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44 ## for @var{tol} is @code{1e-8}. |
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45 ## @item "bi" |
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46 ## Conversion is performed via bilinear transform |
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47 ## @math{z = (1 + s T / 2)/(1 - s T / 2)} where @math{T} is the |
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48 ## system sampling time (see @code{sysgettsam}). |
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49 ## |
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50 ## FIXME: bilinear option exits with an error if @var{sys} is not purely |
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51 ## discrete |
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52 ## @end table |
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53 ## @end table |
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54 ## @strong{Output} |
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55 ## @table @var |
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56 ## @item csys |
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57 ## continuous time system (same dimensions and signal names as in @var{sys}). |
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58 ## @end table |
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59 ## @end deftypefn |
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60 |
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61 ## Author: R. Bruce Tenison <btenison@eng.auburn.edu> |
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62 ## Created: August 23, 1994 |
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63 ## Updated by John Ingram for system data structure August 1996 |
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64 |
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65 function csys = d2c (sys, opt) |
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66 |
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67 ## SYS_INTERNAL accesses members of system data structure |
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68 |
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69 if( (nargin != 1) & (nargin != 2) ) |
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70 usage("csys = d2c(sys[,tol]), csys = d2c(sys,opt)"); |
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71 elseif (!isstruct(sys)) |
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72 error("sys must be in system data structure"); |
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73 elseif(nargin == 1) |
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74 opt = "log"; |
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75 tol = 1e-12; |
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76 elseif(isstr(opt)) # all remaining cases are for nargin == 2 |
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77 tol = 1e-12; |
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78 if( !(strcmp(opt,"log") | strcmp(opt,"bi") ) ) |
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79 error(["d2c: invalid opt passed=",opt]); |
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80 endif |
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81 elseif(!is_sample(opt)) |
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82 error("tol must be a postive scalar") |
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83 elseif(opt > 1e-2) |
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84 warning(["d2c: ridiculous error tolerance passed=",num2str(opt); ... |
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85 ", intended c2d call?"]) |
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86 else |
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87 tol = opt; |
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88 opt = "log"; |
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89 endif |
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90 T = sysgettsam(sys); |
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91 |
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92 if(strcmp(opt,"bi")) |
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93 ## bilinear transform |
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94 ## convert with bilinear transform |
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95 if (! is_digital(sys) ) |
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96 error("d2c requires a discrete time system for input") |
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97 endif |
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98 [a,b,c,d,tsam,n,nz,stname,inname,outname,yd] = sys2ss(sys); |
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99 |
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100 poles = eig(a); |
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101 if( find(abs(poles-1) < 200*(n+nz)*eps) ) |
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102 warning("d2c: some poles very close to one. May get bad results."); |
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103 endif |
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104 |
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105 I = eye(size(a)); |
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106 tk = 2/sqrt(T); |
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107 A = (2/T)*(a-I)/(a+I); |
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108 iab = (I+a)\b; |
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109 B = tk*iab; |
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110 C = tk*(c/(I+a)); |
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111 D = d- (c*iab); |
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112 stnamec = strappend(stname,"_c"); |
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113 csys = ss(A,B,C,D,0,rows(A),0,stnamec,inname,outname); |
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114 elseif(strcmp(opt,"log")) |
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115 sys = sysupdate(sys,"ss"); |
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116 [n,nz,m,p] = sysdimensions(sys); |
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117 |
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118 if(nz == 0) |
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119 warning("d2c: all states continuous; setting outputs to agree"); |
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120 csys = syssetsignals(sys,"yd",zeros(1,1:p)); |
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121 return; |
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122 elseif(n != 0) |
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123 warning(["d2c: n=",num2str(n),">0; performing c2d first"]); |
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124 sys = c2d(sys,T); |
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125 endif |
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126 [a,b] = sys2ss(sys); |
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127 |
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128 [ma,na] = size(a); |
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129 [mb,nb] = size(b); |
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130 |
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131 if(isempty(b) ) |
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132 warning("d2c: empty b matrix"); |
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133 Amat = a; |
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134 else |
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135 Amat = [a, b; zeros(nb,na), eye(nb)]; |
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136 endif |
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137 |
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138 poles = eig(a); |
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139 if( find(abs(poles) < 200*(n+nz)*eps) ) |
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140 warning("d2c: some poles very close to zero. logm not performed"); |
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141 Mtop = zeros(ma, na+nb); |
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142 elseif( find(abs(poles-1) < 200*(n+nz)*eps) ) |
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143 warning("d2c: some poles very close to one. May get bad results."); |
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144 logmat = real(logm(Amat)/T); |
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145 Mtop = logmat(1:na,:); |
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146 else |
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147 logmat = real(logm(Amat)/T); |
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148 Mtop = logmat(1:na,:); |
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149 endif |
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150 |
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151 ## perform simplistic, stupid optimization approach. |
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152 ## should re-write with a Davidson-Fletcher CG approach |
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153 mxthresh = norm(Mtop); |
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154 if(mxthresh == 0) |
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155 mxthresh = 1; |
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156 endif |
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157 eps1 = mxthresh; #gradient descent step size |
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158 cnt = max(20,(n*nz)*4); #max number of iterations |
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159 newgrad=1; #signal for new gradient |
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160 while( (eps1/mxthresh > tol) & cnt) |
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161 cnt = cnt-1; |
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162 ## calculate the gradient of error with respect to Amat... |
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163 geps = norm(Mtop)*1e-8; |
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164 if(geps == 0) |
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165 geps = 1e-8; |
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166 endif |
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167 DMtop = Mtop; |
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168 if(isempty(b)) |
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169 Mall = Mtop; |
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170 DMall = DMtop; |
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171 else |
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172 Mall = [Mtop; zeros(nb,na+nb)]; |
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173 DMall = [DMtop; zeros(nb,na+nb) ]; |
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174 endif |
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175 |
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176 if(newgrad) |
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177 GrMall = zeros(size(Mall)); |
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178 for ii=1:rows(Mtop) |
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179 for jj=1:columns(Mtop) |
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180 DMall(ii,jj) = Mall(ii,jj) + geps; |
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181 GrMall(ii,jj) = norm (Amat - expm (DMall*T), "fro") ... |
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182 - norm (Amat - expm (Mall*T), "fro"); |
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183 DMall(ii,jj) = Mall(ii,jj); |
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184 endfor |
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185 endfor |
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186 GrMall = GrMall/norm(GrMall,1); |
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187 newgrad = 0; |
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188 endif |
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189 |
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190 ## got a gradient, now try to use it |
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191 DMall = Mall-eps1*GrMall; |
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192 |
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193 FMall = expm(Mall*T); |
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194 FDMall = expm(DMall*T); |
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195 FmallErr = norm(Amat - FMall); |
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196 FdmallErr = norm(Amat - FDMall); |
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197 if( FdmallErr < FmallErr) |
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198 Mtop = DMall(1:na,:); |
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199 eps1 = min(eps1*2,1e12); |
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200 newgrad = 1; |
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201 else |
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202 eps1 = eps1/2; |
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203 endif |
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204 |
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205 if(FmallErr == 0) |
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206 eps1 = 0; |
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207 endif |
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208 |
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209 endwhile |
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210 |
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211 [aa,bb,cc,dd,tsam,nn,nz,stnam,innam,outnam,yd] = sys2ss(sys); |
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212 aa = Mall(1:na,1:na); |
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213 if(!isempty(b)) |
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214 bb = Mall(1:na,(na+1):(na+nb)); |
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215 endif |
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216 csys = ss(aa,bb,cc,dd,0,na,0,stnam,innam,outnam); |
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217 |
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218 ## update names |
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219 nn = sysdimensions(sys); |
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220 for ii = (nn+1):na |
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221 strval = sprintf("%s_c",sysgetsignals(csys,"st",ii,1)); |
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222 csys = syssetsignals(csys,"st",strval,ii); |
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223 endfor |
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224 endif |
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225 |
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226 endfunction |
