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1 ## Copyright (C) 1996, 1998, 2000, 2004, 2005, 2006, 2007 |
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2 ## Auburn University. All rights reserved. |
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3 ## |
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4 ## This file is part of Octave. |
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5 ## |
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6 ## Octave is free software; you can redistribute it and/or modify it |
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7 ## under the terms of the GNU General Public License as published by |
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8 ## the Free Software Foundation; either version 3 of the License, or (at |
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9 ## your option) any later version. |
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10 ## |
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11 ## Octave is distributed in the hope that it will be useful, but |
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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14 ## General Public License for more details. |
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15 ## |
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16 ## You should have received a copy of the GNU General Public License |
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17 ## along with Octave; see the file COPYING. If not, see |
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18 ## <http://www.gnu.org/licenses/>. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {[@var{K}, @var{gain}, @var{kc}, @var{kf}, @var{pc}, @var{pf}] = } h2syn (@var{asys}, @var{nu}, @var{ny}, @var{tol}) |
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22 ## Design |
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23 ## @iftex |
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24 ## @tex |
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25 ## $ { \cal H }_2 $ |
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26 ## @end tex |
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27 ## @end iftex |
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28 ## @ifinfo |
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29 ## H-2 |
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30 ## @end ifinfo |
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31 ## optimal controller per procedure in |
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32 ## Doyle, Glover, Khargonekar, Francis, @cite{State-Space Solutions to Standard} |
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33 ## @iftex |
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34 ## @tex |
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35 ## $ { \cal H }_2 $ @cite{and} $ { \cal H }_\infty $ |
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36 ## @end tex |
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37 ## @end iftex |
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38 ## @ifinfo |
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39 ## @cite{H-2 and H-infinity} |
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40 ## @end ifinfo |
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41 ## @cite{Control Problems}, @acronym{IEEE} @acronym{TAC} August 1989. |
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42 ## |
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43 ## Discrete-time control per Zhou, Doyle, and Glover, @cite{Robust and optimal control}, Prentice-Hall, 1996. |
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44 ## |
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45 ## @strong{Inputs} |
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46 ## @table @var |
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47 ## @item asys |
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48 ## system data structure (see ss, sys2ss) |
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49 ## @itemize @bullet |
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50 ## @item controller is implemented for continuous time systems |
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51 ## @item controller is @strong{not} implemented for discrete time systems |
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52 ## @end itemize |
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53 ## @item nu |
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54 ## number of controlled inputs |
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55 ## @item ny |
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56 ## number of measured outputs |
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57 ## @item tol |
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58 ## threshold for 0. Default: 200*@code{eps} |
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59 ## @end table |
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60 ## |
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61 ## @strong{Outputs} |
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62 ## @table @var |
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63 ## @item k |
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64 ## system controller |
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65 ## @item gain |
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66 ## optimal closed loop gain |
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67 ## @item kc |
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68 ## full information control (packed) |
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69 ## @item kf |
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70 ## state estimator (packed) |
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71 ## @item pc |
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72 ## @acronym{ARE} solution matrix for regulator subproblem |
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73 ## @item pf |
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74 ## @acronym{ARE} solution matrix for filter subproblem |
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75 ## @end table |
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76 ## @end deftypefn |
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77 |
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78 ## Updated for System structure December 1996 by John Ingram |
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79 |
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80 function [K, gain, Kc, Kf, Pc, Pf] = h2syn (Asys, nu, ny, tol) |
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81 |
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82 if (nargin < 3 || nargin > 4) |
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83 print_usage (); |
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84 elseif (nargin == 3) |
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85 [chkdgkf, dgs] = is_dgkf (Asys, nu, ny); |
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86 elseif (nargin == 4) |
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87 [chkdgkf, dgs] = is_dgkf (Asys, nu, ny, tol); |
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88 endif |
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89 |
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90 if (! chkdgkf) |
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91 error ("h2syn: system does not meet required assumptions") |
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92 endif |
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93 |
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94 ## extract dgs information |
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95 nw = dgs.nw; |
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96 nu = dgs.nu; |
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97 nz = dgs.nz; |
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98 ny = dgs.ny; |
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99 |
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100 A = dgs.A; |
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101 |
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102 Bw = dgs.Bw; |
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103 Bu = dgs.Bu; |
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104 |
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105 Cz = dgs.Cz; |
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106 Cy = dgs.Cy; |
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107 |
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108 Dzw = dgs.Dzw; |
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109 Dzu = dgs.Dzu; |
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110 |
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111 Dyw = dgs.Dyw; |
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112 Dyu = dgs.Dyu; |
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113 |
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114 d22nz = dgs.Dyu_nz; |
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115 |
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116 dflg = dgs.dflg; |
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117 |
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118 if (norm (Dzw, Inf) > norm ([Dzw, Dzu; Dyw, Dyu], Inf)*1e-12) |
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119 warning ("h2syn: Dzw nonzero; feedforward not implemented") |
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120 Dzw |
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121 D = [Dzw, Dzu ; Dyw, Dyu] |
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122 endif |
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123 |
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124 ## recover i/o transformations |
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125 Ru = dgs.Ru; |
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126 Ry = dgs.Ry; |
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127 |
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128 [ncstates, ndstates, nout, nin] = sysdimensions (Asys); |
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129 Atsam = sysgettsam (Asys); |
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130 [Ast, Ain, Aout] = sysgetsignals (Asys); |
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131 |
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132 if (dgs.dflg == 0) |
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133 Pc = are (A, Bu*Bu', Cz'*Cz); # solve control, filtering ARE's |
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134 Pf = are(A', Cy'*Cy, Bw*Bw'); |
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135 F2 = -Bu'*Pc; # calculate feedback gains |
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136 L2 = -Pf*Cy'; |
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137 |
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138 AF2 = A + Bu*F2; |
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139 AL2 = A + L2*Cy; |
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140 CzF2 = Cz + (Dzu/Ru)*F2; |
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141 BwL2 = Bw+L2*(Ry\Dyw); |
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142 |
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143 else |
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144 ## discrete time solution |
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145 error ("h2syn: discrete-time case not yet implemented") |
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146 Pc = dare (A, Bu*Bu', Cz'*Cz); |
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147 Pf = dare (A', Cy'*Cy, Bw*Bw'); |
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148 endif |
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149 |
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150 nn = ncstates + ndstates; |
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151 In = eye (nn); |
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152 KA = A + Bu*F2 + L2*Cy; |
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153 Kc1 = ss (AF2, Bw, CzF2, zeros (nz, nw)); |
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154 Kf1 = ss (AL2, BwL2, F2, zeros (nu, nw)); |
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155 |
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156 g1 = h2norm (Kc1); |
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157 g2 = h2norm (Kf1); |
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158 |
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159 ## compute optimal closed loop gain |
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160 gain = sqrt (g1*g1 + g2*g2); |
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161 |
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162 if (nargout) |
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163 Kst = strappend (Ast, "_K"); |
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164 Kin = strappend (Aout((nout-ny+1):(nout)), "_K"); |
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165 Kout = strappend (Ain((nin-nu+1):(nin)), "_K"); |
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166 |
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167 ## compute systems for return |
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168 K = ss (KA, -L2/Ru, Ry\F2, zeros(nu,ny), Atsam, ncstates, |
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169 ndstates, Kst, Kin, Kout); |
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170 endif |
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171 |
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172 if (nargout > 2) |
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173 ## system full information control state names |
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174 stname2 = strappend (Ast, "_FI"); |
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175 |
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176 ## system full information control input names |
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177 inname2 = strappend (Ast, "_FI_in"); |
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178 |
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179 ## system full information control output names |
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180 outname2 = strappend (Aout(1:(nout-ny)), "_FI_out"); |
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181 |
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182 nz = rows (Cz); |
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183 nw = columns (Bw); |
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184 |
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185 Kc = ss (AF2, In, CzF2, zeros(nz,nn), Atsam, |
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186 ncstates, ndstates, stname2, inname2, outname2); |
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187 endif |
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188 |
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189 if (nargout >3) |
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190 ## fix system state estimator state names |
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191 stname3 = strappend (Ast, "_Kf"); |
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192 |
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193 ## fix system state estimator input names |
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194 inname3 = strappend (Ast, "_Kf_noise"); |
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195 |
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196 ## fix system state estimator output names |
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197 outname3 = strappend (Ast, "_est"); |
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198 |
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199 Kf = ss (AL2, BwL2, In, zeros(nn,nw),Atsam, |
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200 ncstates, ndstates, stname3, inname3, outname3); |
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201 endif |
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202 |
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203 endfunction |
