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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
parents f0afece42f48
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1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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2 "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd">
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3
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4 <html lang="en">
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5 <head>
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6 <meta name="generator" content=
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7 "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org">
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8 <meta http-equiv="Content-Type" content=
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9 "text/html; charset=us-ascii">
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10
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11 <title>S-Domain Fit (LTPDA Toolbox)</title>
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12 <link rel="stylesheet" href="docstyle.css" type="text/css">
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13 <meta name="generator" content="DocBook XSL Stylesheets V1.52.2">
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14 <meta name="description" content=
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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16 </head>
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17
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18 <body>
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19 <a name="top_of_page" id="top_of_page"></a>
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20
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21 <p style="font-size:1px;">&nbsp;</p>
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22
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23 <table class="nav" summary="Navigation aid" border="0" width=
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24 "100%" cellpadding="0" cellspacing="0">
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25 <tr>
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26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
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27
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28 <td valign="baseline" align="right"><a href=
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29 "zdomainfit.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Z-Domain Fit"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "gui.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Graphical User Interfaces in LTPDA"></a></td>
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33 </tr>
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34 </table>
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35
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36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>S-Domain Fit</h1>
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37 <hr>
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38
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39 <p>
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40
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41 <!-- ================================================== -->
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42 <!-- BEGIN CONTENT FILE -->
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43 <!-- ================================================== -->
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44 <!-- ===== link box: Begin ===== -->
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45 <p>
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46 <table border="1" width="80%">
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47 <tr>
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48 <td>
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49 <table border="0" cellpadding="5" class="categorylist" width="100%">
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50 <colgroup>
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51 <col width="37%"/>
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52 <col width="63%"/>
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53 </colgroup>
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54 <tbody>
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55 <tr valign="top">
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56 <td>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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57 <a href="#description">Description</a>
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58 </td>
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59 <td>S-domain system identification in LTPDA.</td>
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60 </tr>
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61 <tr valign="top">
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62 <td>
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63 <a href="#algorithm">Algorithm</a>
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64 </td>
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65 <td>Fit Algorithm.</td>
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66 </tr>
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67 <tr valign="top">
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68 <td>
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69 <a href="#examples">Examples</a>
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70 </td>
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71 <td>Usage example of s-domain system identification tool.</td>
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72 </tr>
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73 <tr valign="top">
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74 <td>
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75 <a href="#references">References</a>
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76 </td>
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77 <td>Bibliographic references.</td>
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78 </tr>
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79 </tbody>
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80 </table>
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81 </td>
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82 </tr>
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83 </table>
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84 </p>
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85 <!-- ===== link box: End ====== -->
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86
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87 <h2><a name="description">S-domain system identification in LTPDA</a></h2>
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88 <p>
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89 System identification in s-domain is performed with the function
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Daniele Nicolodi <nicolodi@science.unitn.it>
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90 <a href="matlab:doc('ao/sDomainFit')">sDomainFit</a>.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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91 It is based on a modeified version of the vector fitting algorithm.
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92 Details on the core agorithm can be found in [1 - 2].
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93 </p>
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94
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95
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96
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97 <h2><a name="algorithm">Fit Algorithm</a></h2>
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98
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99 <p>
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100 The function performs a fitting loop to automatically identify model
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Daniele Nicolodi <nicolodi@science.unitn.it>
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101 order and parameters in s-domain. Output is a s-domain model expanded
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Daniele Nicolodi <nicolodi@science.unitn.it>
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102 in partial fractions:
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103 </p>
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104 <div class="fragment"><pre>
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105 r1 rN
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106 f(s) = ------- + ... + ------- + d
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107 s - p1 s - pN
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108 </pre></div>
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109 <p>
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110 Since the function can fit more than one input analysis object at a time
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111 with a common set of poles, output
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112 <a href="parfrac.html">parfrac</a> are embedded in a
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113 <a href="class_desc_matrix.html">matrix</a> (note that this characteristic
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114 will be probably changed becausse of the introduction of the
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115 <a href="class_desc_collection.html">collection</a> class).
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116 </p>
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117 <p>
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118 Identification loop stops when the stop condition is reached.
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119 Stop criterion is based on three different approachs:
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120 <ol>
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121 <li> Mean Squared Error and variation <br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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122 Check if the normalized mean squared error is lower than the value specified in
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Daniele Nicolodi <nicolodi@science.unitn.it>
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123 <tt>FITTOL</tt> and if the relative variation of the mean squared error is lower
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124 than the value specified in <tt>MSEVARTOL</tt>.
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125 E.g. <tt>FITTOL = 1e-3</tt>, <tt>MSEVARTOL = 1e-2</tt> search for a fit with
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126 normalized meam square error lower than <tt>1e-3</tt> and <tt>MSE</tt> relative
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Daniele Nicolodi <nicolodi@science.unitn.it>
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127 variation lower than <tt>1e-2</tt>.
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128 </li>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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129 <li> Log residuals difference and root mean squared error
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130 <ul>
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131 <li> Log Residuals difference </br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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132 Check if the minimum of the logarithmic difference between data and
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Daniele Nicolodi <nicolodi@science.unitn.it>
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133 residuals is larger than a specified value. ie. if the conditioning
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Daniele Nicolodi <nicolodi@science.unitn.it>
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134 value is <tt>2</tt>, the function ensures that the difference between data and
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Daniele Nicolodi <nicolodi@science.unitn.it>
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135 residuals is at lest two order of magnitude lower than data itsleves.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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136 <li> Root Mean Squared Error </br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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137 Check that the variation of the root mean squared error is lower than
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Daniele Nicolodi <nicolodi@science.unitn.it>
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138 <tt>10^(-1*value)</tt>.
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139 </ul>
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140 </li>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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141 <li> Residuals spectral flatness and root mean squared error
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Daniele Nicolodi <nicolodi@science.unitn.it>
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142 <ul>
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143 <li> Residuals Spectral Flatness </br>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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144 In case of a fit on noisy data, the residuals from a good fit are
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Daniele Nicolodi <nicolodi@science.unitn.it>
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145 expected to be as much as possible similar to a white noise. This
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Daniele Nicolodi <nicolodi@science.unitn.it>
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146 property can be used to test the accuracy of a fit procedure. In
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Daniele Nicolodi <nicolodi@science.unitn.it>
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147 particular it can be tested that the spectral flatness coefficient of
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Daniele Nicolodi <nicolodi@science.unitn.it>
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148 the residuals is larger than a certain qiantity sf such that <tt>0 < sf < 1</tt>.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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149 <li> Root Mean Squared Error </br>
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150 Check that the variation of the root mean squared error is lower than
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Daniele Nicolodi <nicolodi@science.unitn.it>
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151 <tt>10^(-1*value)</tt>.
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152 </ul>
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153 </li>
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154 </ol>
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155
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156 </p>
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157 <p>
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158 The function can also perform a single loop without taking care of
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Daniele Nicolodi <nicolodi@science.unitn.it>
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159 the stop conditions. This happens when <span class="string">'AutoSearch'</span> parameter is
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160 set to <span class="string">'off'</span>.
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161 </p>
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162
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163
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164 <h2><a name="examples">Usage example of s-domain system identification tool</a></h2>
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165 <p>
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166 In this example we fit a given frequency response to get a partial fraction model.
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167 For the meaning of any parameter please refer to
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168 <a href="matlab:doc('ao')">ao</a> and
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169 <a href="matlab:doc('ao/sDomainFit')">sDomainFit</a>
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170 documentation pages.
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171 </p>
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172
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173 <div class="fragment"><pre>
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174 pl = plist(...
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175 <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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176 <span class="string">'f1'</span>, 1e-6,...
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177 <span class="string">'f2'</span>, 5,...
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178 <span class="string">'nf'</span>, 100);
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179
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180 a = ao(pl);
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181 a.setName;
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182
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183 <span class="comment">% Fit parameter list</span>
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184 pl_fit = plist(...
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185 <span class="string">'AutoSearch'</span>,<span class="string">'on'</span>,...
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186 <span class="string">'StartPolesOpt'</span>,<span class="string">'clog'</span>,...
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187 <span class="string">'maxiter'</span>,50,...
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188 <span class="string">'minorder'</span>,7,...
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189 <span class="string">'maxorder'</span>,15,...
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190 <span class="string">'weightparam'</span>,<span class="string">'abs'</span>,...
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191 <span class="string">'CONDTYPE'</span>,<span class="string">'MSE'</span>,...
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192 <span class="string">'FITTOL'</span>,1e-3,...
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193 <span class="string">'MSEVARTOL'</span>,1e-2,...
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194 <span class="string">'Plot'</span>,<span class="string">'on'</span>,...
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195 <span class="string">'ForceStability'</span>,<span class="string">'off'</span>);
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196
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197 <span class="comment">% Do fit</span>
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198 mod = sDomainFit(a, pl_fit);
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199 </pre></div>
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200
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201 <p>
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202 <tt>mod</tt> is a <tt>matrix</tt> object containing a <tt>parfrac</tt> object.
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203 </p>
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204
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205 <div class="fragment"><pre>
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206 >> mod
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207 ---- matrix 1 ----
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208 name: fit(a)
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209 size: 1x1
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210 01: parfrac | parfrac(fit(a))
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211 description:
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212 UUID: 2dc1ac28-4199-42d2-9b1a-b420252b3f8c
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213 ------------------
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214 </pre></div>
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215
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216 <div class="fragment"><pre>
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217 >> mod.objs
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218 ---- parfrac 1 ----
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219 model: fit(a)
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220 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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221 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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222 dir: 0
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223 pmul: [1;1;1;1;1;1;1;1;1;1;1]
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224 iunits: []
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225 ounits: []
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226 description:
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227 UUID: 2afc4c82-7c2a-4fe3-8910-d8590884d58c
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228 -------------------
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229 </pre></div>
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230
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231
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232 <h2><a name="references">References</a></h2>
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233 <p>
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234 <ol>
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235 <li> B. Gustavsen and A. Semlyen, "Rational approximation of frequency
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236 domain responses by Vector Fitting", IEEE Trans. Power Delivery
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237 vol. 14, no. 3, pp. 1052-1061, July 1999.
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238 <li> B. Gustavsen, "Improving the Pole Relocating Properties of Vector
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239 Fitting", IEEE Trans. Power Delivery vol. 21, no. 3, pp.
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240 1587-1592, July 2006.
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241 </ol>
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242 </p>
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243 </p>
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244
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245 <br>
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246 <br>
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247 <table class="nav" summary="Navigation aid" border="0" width=
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248 "100%" cellpadding="0" cellspacing="0">
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249 <tr valign="top">
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250 <td align="left" width="20"><a href="zdomainfit.html"><img src=
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251 "b_prev.gif" border="0" align="bottom" alt=
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252 "Z-Domain Fit"></a>&nbsp;</td>
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253
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254 <td align="left">Z-Domain Fit</td>
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255
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256 <td>&nbsp;</td>
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257
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258 <td align="right">Graphical User Interfaces in LTPDA</td>
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259
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260 <td align="right" width="20"><a href=
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261 "gui.html"><img src="b_next.gif" border="0" align=
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262 "bottom" alt="Graphical User Interfaces in LTPDA"></a></td>
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263 </tr>
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264 </table><br>
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265
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Daniele Nicolodi <nicolodi@science.unitn.it>
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266 <p class="copy">&copy;LTP Team</p>
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267 </body>
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268 </html>