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1 ## Copyright (C) 1993, 1994, 1995, 2000, 2001, 2002, 2004, 2005, 2006, |
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2 ## 2007 John W. Eaton |
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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} {} c2d (@var{sys}, @var{opt}, @var{t}) |
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22 ## @deftypefnx {Function File} {} c2d (@var{sys}, @var{t}) |
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23 ## |
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24 ## Converts the system data structure describing: |
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25 ## @iftex |
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26 ## @tex |
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27 ## $$ \dot x = A_cx + B_cu $$ |
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28 ## @end tex |
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29 ## @end iftex |
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30 ## @ifinfo |
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31 ## @example |
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32 ## . |
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33 ## x = Ac x + Bc u |
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34 ## @end example |
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35 ## @end ifinfo |
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36 ## into a discrete time equivalent model: |
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37 ## @iftex |
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38 ## @tex |
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39 ## $$ x_{n+1} = A_dx_n + B_du_n $$ |
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40 ## @end tex |
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41 ## @end iftex |
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42 ## @ifinfo |
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43 ## @example |
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44 ## x[n+1] = Ad x[n] + Bd u[n] |
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45 ## @end example |
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46 ## @end ifinfo |
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47 ## via the matrix exponential or bilinear transform. |
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48 ## |
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49 ## @strong{Inputs} |
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50 ## @table @var |
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51 ## @item sys |
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52 ## system data structure (may have both continuous time and discrete |
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53 ## time subsystems) |
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54 ## @item opt |
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55 ## string argument; conversion option (optional argument; |
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56 ## may be omitted as shown above) |
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57 ## @table @code |
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58 ## @item "ex" |
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59 ## use the matrix exponential (default) |
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60 ## @item "bi" |
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61 ## use the bilinear transformation |
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62 ## @iftex |
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63 ## @tex |
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64 ## $$ s = { 2(z-1) \over T(z+1) } $$ |
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65 ## @end tex |
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66 ## @end iftex |
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67 ## @ifinfo |
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68 ## @example |
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69 ## 2(z-1) |
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70 ## s = ----- |
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71 ## T(z+1) |
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72 ## @end example |
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73 ## @end ifinfo |
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74 ## FIXME: This option exits with an error if @var{sys} is not purely |
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75 ## continuous. (The @code{ex} option can handle mixed systems.) |
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76 ## @item "matched" |
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77 ## Use the matched pole/zero equivalent transformation (currently only |
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78 ## works for purely continuous @acronym{SISO} systems). |
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79 ## @end table |
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80 ## @item t |
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81 ## sampling time; required if @var{sys} is purely continuous. |
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82 ## |
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83 ## @strong{Note} that if the second argument is not a string, @code{c2d()} |
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84 ## assumes that the second argument is @var{t} and performs |
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85 ## appropriate argument checks. |
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86 ## @end table |
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87 ## |
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88 ## @strong{Output} |
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89 ## @table @var |
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90 ## @item dsys |
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91 ## Discrete time equivalent via zero-order hold, sample each @var{t} sec. |
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92 ## @end table |
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93 ## |
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94 ## This function adds the suffix @code{_d} |
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95 ## to the names of the new discrete states. |
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96 ## @end deftypefn |
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97 |
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98 ## Author: R. Bruce Tenison <btenison@eng.auburn.edu> |
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99 ## Created: October 1993 |
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100 ## Updated by John Ingram for system data structure August 1996 |
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101 |
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102 function dsys = c2d (sys, opt, T) |
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103 |
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104 ## parse input arguments |
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105 if (nargin < 1 || nargin > 3) |
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106 print_usage (); |
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107 elseif (! isstruct (sys)) |
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108 error ("sys must be a system data structure"); |
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109 elseif (nargin == 1) |
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110 opt = "ex"; |
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111 elseif (nargin == 2 && ! ischar (opt)) |
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112 T = opt; |
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113 opt = "ex"; |
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114 endif |
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115 |
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116 if (! ischar (opt)) |
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117 error ("expecting option as a string"); |
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118 endif |
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119 |
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120 ## check if sampling period T was passed. |
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121 Ts = sysgettsam (sys); |
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122 if (! exist ("T")) |
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123 T = Ts; |
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124 if (T == 0) |
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125 error ("sys is purely continuous; no sampling period T provided"); |
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126 endif |
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127 elseif (T != Ts && Ts > 0) |
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128 warning ("c2d: T=%g, system tsam=%g: using T=", T, Ts, min (T, Ts)); |
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129 T = min (T, Ts); |
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130 endif |
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131 |
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132 if (! is_sample (T)) |
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133 error ("sampling period T must be a positive, real scalar"); |
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134 elseif (! (strcmp (opt, "ex") |
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135 || strcmp (opt, "bi") |
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136 || strcmp (opt, "matched"))) |
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137 error ("invalid option passed: %s", opt); |
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138 endif |
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139 |
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140 sys = sysupdate (sys, "ss"); |
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141 [n, nz, m, p] = sysdimensions (sys); |
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142 if (n == 0) |
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143 dsys = syssetsignals (sys, "yd", ones(1:p)); |
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144 elseif (strcmp (opt, "ex")); |
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145 [aa, bb, cc, dd] = sys2ss (sys); |
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146 crng= 1:n; |
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147 drng = n+(1:nz); |
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148 |
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149 ## partition state equations into continuous, imaginary subsystems |
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150 Ac = aa(crng,crng); |
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151 Bc = bb(crng,:); |
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152 if (nz == 0) |
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153 Acd = Adc = Add = Bd = 0; |
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154 else |
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155 Acd = aa(crng,drng); |
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156 Adc = aa(drng,crng); |
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157 Add = aa(drng,drng); |
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158 Bd = bb(drng,:); |
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159 Bc = [Bc, Acd]; ## append discrete states as inputs to cont system |
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160 endif |
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161 |
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162 ## convert state equations |
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163 mat = [Ac, Bc; zeros(m+nz,n+nz+m)]; |
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164 matexp = expm (mat * T); |
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165 |
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166 ## replace Ac |
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167 aa(crng,crng) = matexp(crng,crng); ## discretized homegenous diff eqn |
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168 |
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169 ## replace Bc |
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170 bb(crng,:) = matexp(crng,n+(1:m)); |
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171 |
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172 ## replace Acd |
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173 if (nz) |
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174 aa(crng,drng) = matexp(crng,n+m+(1:nz)); |
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175 endif |
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176 |
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177 stnames = sysgetsignals (sys, "st"); ## continuous states renamed below |
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178 innames = sysgetsignals (sys, "in"); |
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179 outnames = sysgetsignals (sys, "out"); |
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180 outlist = 1:p; |
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181 dsys = ss (aa, bb, cc, dd, T, 0, n+nz, stnames, innames, |
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182 outnames, outlist); |
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183 ## rename states |
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184 for ii = 1:n |
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185 strval = sprintf ("%s_d", sysgetsignals (dsys, "st", ii, 1)); |
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186 dsys = syssetsignals (dsys, "st", strval, ii); |
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187 endfor |
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188 |
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189 elseif (strcmp (opt, "bi")) |
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190 if (is_digital (sys)) |
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191 error ("c2d: system is already digital") |
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192 else |
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193 ## convert with bilinear transform |
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194 [a, b, c, d, tsam, n, nz, stname, inname, outname, yd] = sys2ss (sys); |
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195 IT = (2/T) * eye (size (a)); |
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196 A = (IT+a)/(IT-a); |
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197 iab = (IT-a)\b; |
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198 tk = 2 / sqrt (T); |
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199 B = tk*iab; |
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200 C = tk*(c/(IT-a)); |
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201 D = d + (c*iab); |
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202 stnamed = strappend (stname, "_d"); |
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203 dsys = ss (A, B, C, D, T, 0, rows (A), stnamed, inname, outname); |
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204 endif |
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205 elseif (strcmp (opt, "matched")) |
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206 if (is_digital (sys)) |
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207 error ("c2d: system is already digital"); |
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208 elseif (length (sys.inname) != 1 || length (sys.outname) != 1) |
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209 error ("c2d: system in not single input, single output"); |
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210 else |
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211 sys = sysupdate (sys, "zp"); |
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212 p = exp (sys.pol*T); |
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213 z = exp (sys.zer*T); |
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214 infinite_zeros = max (size (sys.pol)) - max (size (sys.zer)) - 1; |
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215 for i = 1:infinite_zeros |
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216 z = [z ; -1]; |
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217 endfor |
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218 ## Should the freaquency we adjust around always be 1? |
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219 [cmag, cphase, cw] = bode (sys, 1); |
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220 [dmag, dpahse, dw] = bode (zp (z, p, 1, T), 1); |
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221 dsys = zp (z, p, cmag/dmag, T); |
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222 endif |
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223 else |
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224 error ("invalid option = %s", opt); |
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225 endif |
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226 |
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227 endfunction |
