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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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1 <!-- $Id: sdomainfit_content.html,v 1.3 2009/08/27 11:38:58 luigi Exp $ -->
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2
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3 <!-- ================================================== -->
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4 <!-- BEGIN CONTENT FILE -->
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5 <!-- ================================================== -->
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6 <!-- ===== link box: Begin ===== -->
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7 <p>
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8 <table border="1" width="80%">
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9 <tr>
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10 <td>
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11 <table border="0" cellpadding="5" class="categorylist" width="100%">
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12 <colgroup>
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13 <col width="37%"/>
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14 <col width="63%"/>
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15 </colgroup>
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16 <tbody>
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17 <tr valign="top">
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18 <td>
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19 <a href="#description">Description</a>
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20 </td>
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21 <td>S-domain system identification in LTPDA.</td>
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22 </tr>
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23 <tr valign="top">
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24 <td>
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25 <a href="#algorithm">Algorithm</a>
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26 </td>
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27 <td>Fit Algorithm.</td>
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28 </tr>
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29 <tr valign="top">
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30 <td>
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31 <a href="#examples">Examples</a>
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32 </td>
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33 <td>Usage example of s-domain system identification tool.</td>
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34 </tr>
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35 <tr valign="top">
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36 <td>
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37 <a href="#references">References</a>
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38 </td>
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39 <td>Bibliographic references.</td>
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40 </tr>
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41 </tbody>
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42 </table>
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43 </td>
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44 </tr>
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45 </table>
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46 </p>
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47 <!-- ===== link box: End ====== -->
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48
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49 <h2><a name="description">S-domain system identification in LTPDA</a></h2>
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50 <p>
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51 System identification in s-domain is performed with the function
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52 <a href="matlab:doc('ao/sDomainFit')">sDomainFit</a>.
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53 It is based on a modeified version of the vector fitting algorithm.
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54 Details on the core agorithm can be found in [1 - 2].
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55 </p>
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56
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57
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58
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59 <h2><a name="algorithm">Fit Algorithm</a></h2>
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60
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61 <p>
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62 The function performs a fitting loop to automatically identify model
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63 order and parameters in s-domain. Output is a s-domain model expanded
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64 in partial fractions:
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65 </p>
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66 <div class="fragment"><pre>
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67 r1 rN
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68 f(s) = ------- + ... + ------- + d
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69 s - p1 s - pN
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70 </pre></div>
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71 <p>
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72 Since the function can fit more than one input analysis object at a time
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73 with a common set of poles, output
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74 <a href="parfrac.html">parfrac</a> are embedded in a
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75 <a href="class_desc_matrix.html">matrix</a> (note that this characteristic
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76 will be probably changed becausse of the introduction of the
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77 <a href="class_desc_collection.html">collection</a> class).
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78 </p>
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79 <p>
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80 Identification loop stops when the stop condition is reached.
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81 Stop criterion is based on three different approachs:
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82 <ol>
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83 <li> Mean Squared Error and variation <br>
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84 Check if the normalized mean squared error is lower than the value specified in
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85 <tt>FITTOL</tt> and if the relative variation of the mean squared error is lower
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86 than the value specified in <tt>MSEVARTOL</tt>.
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87 E.g. <tt>FITTOL = 1e-3</tt>, <tt>MSEVARTOL = 1e-2</tt> search for a fit with
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88 normalized meam square error lower than <tt>1e-3</tt> and <tt>MSE</tt> relative
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89 variation lower than <tt>1e-2</tt>.
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90 </li>
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91 <li> Log residuals difference and root mean squared error
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92 <ul>
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93 <li> Log Residuals difference </br>
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94 Check if the minimum of the logarithmic difference between data and
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95 residuals is larger than a specified value. ie. if the conditioning
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96 value is <tt>2</tt>, the function ensures that the difference between data and
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97 residuals is at lest two order of magnitude lower than data itsleves.
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98 <li> Root Mean Squared Error </br>
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99 Check that the variation of the root mean squared error is lower than
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100 <tt>10^(-1*value)</tt>.
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101 </ul>
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102 </li>
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103 <li> Residuals spectral flatness and root mean squared error
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104 <ul>
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105 <li> Residuals Spectral Flatness </br>
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106 In case of a fit on noisy data, the residuals from a good fit are
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107 expected to be as much as possible similar to a white noise. This
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108 property can be used to test the accuracy of a fit procedure. In
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109 particular it can be tested that the spectral flatness coefficient of
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110 the residuals is larger than a certain qiantity sf such that <tt>0 < sf < 1</tt>.
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111 <li> Root Mean Squared Error </br>
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112 Check that the variation of the root mean squared error is lower than
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113 <tt>10^(-1*value)</tt>.
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114 </ul>
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115 </li>
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116 </ol>
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117
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118 </p>
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119 <p>
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120 The function can also perform a single loop without taking care of
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121 the stop conditions. This happens when <span class="string">'AutoSearch'</span> parameter is
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122 set to <span class="string">'off'</span>.
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123 </p>
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124
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125
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126 <h2><a name="examples">Usage example of s-domain system identification tool</a></h2>
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127 <p>
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128 In this example we fit a given frequency response to get a partial fraction model.
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129 For the meaning of any parameter please refer to
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130 <a href="matlab:doc('ao')">ao</a> and
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131 <a href="matlab:doc('ao/sDomainFit')">sDomainFit</a>
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132 documentation pages.
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133 </p>
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134
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135 <div class="fragment"><pre>
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136 pl = plist(...
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137 <span class="string">'fsfcn'</span>, <span class="string">'(1e-3./(f).^2 + 1e3./(0.001+f) + 1e5.*f.^2).*1e-10'</span>,...
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138 <span class="string">'f1'</span>, 1e-6,...
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139 <span class="string">'f2'</span>, 5,...
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140 <span class="string">'nf'</span>, 100);
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141
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142 a = ao(pl);
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143 a.setName;
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144
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145 <span class="comment">% Fit parameter list</span>
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146 pl_fit = plist(...
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147 <span class="string">'AutoSearch'</span>,<span class="string">'on'</span>,...
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148 <span class="string">'StartPolesOpt'</span>,<span class="string">'clog'</span>,...
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149 <span class="string">'maxiter'</span>,50,...
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150 <span class="string">'minorder'</span>,7,...
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151 <span class="string">'maxorder'</span>,15,...
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152 <span class="string">'weightparam'</span>,<span class="string">'abs'</span>,...
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153 <span class="string">'CONDTYPE'</span>,<span class="string">'MSE'</span>,...
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154 <span class="string">'FITTOL'</span>,1e-3,...
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155 <span class="string">'MSEVARTOL'</span>,1e-2,...
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156 <span class="string">'Plot'</span>,<span class="string">'on'</span>,...
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157 <span class="string">'ForceStability'</span>,<span class="string">'off'</span>);
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158
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159 <span class="comment">% Do fit</span>
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160 mod = sDomainFit(a, pl_fit);
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161 </pre></div>
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162
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163 <p>
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164 <tt>mod</tt> is a <tt>matrix</tt> object containing a <tt>parfrac</tt> object.
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165 </p>
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166
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167 <div class="fragment"><pre>
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168 >> mod
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169 ---- matrix 1 ----
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170 name: fit(a)
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171 size: 1x1
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172 01: parfrac | parfrac(fit(a))
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173 description:
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174 UUID: 2dc1ac28-4199-42d2-9b1a-b420252b3f8c
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175 ------------------
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176 </pre></div>
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177
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178 <div class="fragment"><pre>
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179 >> mod.objs
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180 ---- parfrac 1 ----
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181 model: fit(a)
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182 res: [1.69531090137847e-006;-1.69531095674486e-006;1.39082537801437e-007;-1.39094453401266e-007;3.9451875151135e-007;-3.94524993613367e-007;4.53671387948961e-007;-4.53664974359603e-007;1124.81020427899;0.000140057852149302-i*0.201412268649905;0.000140057852149302+i*0.201412268649905]
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183 poles: [-1.18514026248382e-006;1.18514354570495e-006;-0.00457311582050939;0.0045734088943545;-0.0316764149343339;0.0316791653277322;-0.276256442292693;0.27627799022013;330754.550617933;-0.0199840558095427+i*118.439896186467;-0.0199840558095427-i*118.439896186467]
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184 dir: 0
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185 pmul: [1;1;1;1;1;1;1;1;1;1;1]
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186 iunits: []
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187 ounits: []
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188 description:
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189 UUID: 2afc4c82-7c2a-4fe3-8910-d8590884d58c
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190 -------------------
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191 </pre></div>
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192
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193
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194 <h2><a name="references">References</a></h2>
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195 <p>
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196 <ol>
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197 <li> B. Gustavsen and A. Semlyen, "Rational approximation of frequency
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198 domain responses by Vector Fitting", IEEE Trans. Power Delivery
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199 vol. 14, no. 3, pp. 1052-1061, July 1999.
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200 <li> B. Gustavsen, "Improving the Pole Relocating Properties of Vector
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201 Fitting", IEEE Trans. Power Delivery vol. 21, no. 3, pp.
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202 1587-1592, July 2006.
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203 </ol>
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204 </p>