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−<!-- $Id: sdomainfit_content.html,v 1.3 2009/08/27 11:38:58 luigi Exp $ -->
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−<!-- ===== link box: Begin ===== -->
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−<p>
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−  <table border="1"  width="80%">
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−    <tr>
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−      <td>
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−        <table border="0" cellpadding="5" class="categorylist" width="100%">
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−          <colgroup>
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−            <col width="37%"/>
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−            <col width="63%"/>
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−          </colgroup>
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−          <tbody>
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−            <tr valign="top">
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−              <td>
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−                <a href="#description">Description</a>
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−              </td>
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−              <td>S-domain system identification in LTPDA.</td>
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−            </tr>
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−            <tr valign="top">
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−              <td>
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−                <a href="#algorithm">Algorithm</a>
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−              </td>
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−              <td>Fit Algorithm.</td>
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−            </tr>
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−            <tr valign="top">
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−              <td>
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−                <a href="#examples">Examples</a>
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−              </td>
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−              <td>Usage example of s-domain system identification tool.</td>
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−            </tr>
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−            <tr valign="top">
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−              <td>
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−                <a href="#references">References</a>
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−              </td>
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−              <td>Bibliographic references.</td>
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−            </tr>
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−          </tbody>
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−        </table>
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−      </td>
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−    </tr>
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−  </table>
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−</p>
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−<!-- ===== link box: End ====== -->
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−
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−<h2><a name="description">S-domain system identification in LTPDA</a></h2>
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−<p>
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−  System identification in s-domain is performed with the function 
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−  <a href="matlab:doc('ao/sDomainFit')">sDomainFit</a>. 
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−  It is based on a modeified version of the vector fitting algorithm. 
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−  Details on the core agorithm can be found in [1 - 2].
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−</p>
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−
+
−
+
−
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−<h2><a name="algorithm">Fit Algorithm</a></h2>
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−
+
−<p>
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−  The function performs a fitting loop to automatically identify model 
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−  order and parameters in s-domain. Output is a s-domain model expanded 
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−  in partial fractions:
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−</p>
+
−<div class="fragment"><pre>
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−              r1              rN
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−    f(s) = ------- + ... + ------- + d
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−            s - p1          s - pN
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−</pre></div>
+
−<p>
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−  Since the function can fit more than one input analysis object at a time
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−  with a common set of poles, output 
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−  <a href="parfrac.html">parfrac</a> are embedded in a 
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−  <a href="class_desc_matrix.html">matrix</a> (note that this characteristic
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−  will be probably changed becausse of the introduction of the
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−  <a href="class_desc_collection.html">collection</a> class).
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−</p>
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−<p>
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−  Identification loop stops when the stop condition is reached. 
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−  Stop criterion is based on three different approachs:
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−  <ol>
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−    <li> Mean Squared Error and variation <br>
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−      Check if the normalized mean squared error is lower than the value specified in
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−      <tt>FITTOL</tt> and if the relative variation of the mean squared error is lower
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−      than the value specified in <tt>MSEVARTOL</tt>.
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−      E.g. <tt>FITTOL = 1e-3</tt>, <tt>MSEVARTOL = 1e-2</tt> search for a fit with
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−      normalized meam square error lower than <tt>1e-3</tt> and <tt>MSE</tt> relative
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−      variation lower than <tt>1e-2</tt>. 
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−    </li>
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−    <li> Log residuals difference and root mean squared error
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−      <ul>
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−        <li> Log Residuals difference </br>
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−        Check if the minimum of the logarithmic difference between data and 
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−        residuals is larger than a specified value. ie. if the conditioning 
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−        value is <tt>2</tt>, the function ensures that the difference between data and 
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−        residuals is at lest two order of magnitude lower than data itsleves. 
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−        <li> Root Mean Squared Error </br>
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−        Check that the variation of the root mean squared error is lower than 
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−        <tt>10^(-1*value)</tt>.
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−      </ul>
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−    </li>
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−    <li> Residuals spectral flatness and root mean squared error
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−      <ul>
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−        <li> Residuals Spectral Flatness </br>
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−        In case of a fit on noisy data, the residuals from a good fit are
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−        expected to be as much as possible similar to a white noise. This
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−        property can be used to test the accuracy of a fit procedure. In
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−        particular it can be tested that the spectral flatness coefficient of
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−        the residuals is larger than a certain qiantity sf such that <tt>0 < sf < 1</tt>.
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−        <li> Root Mean Squared Error </br>
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−        Check that the variation of the root mean squared error is lower than 
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−        <tt>10^(-1*value)</tt>.
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−      </ul>
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−    </li>
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−  </ol>
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−  
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−</p>
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−<p>
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−  The function can also perform a single loop without taking care of
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−  the stop conditions. This happens when <span class="string">'AutoSearch'</span> parameter is
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−  set to <span class="string">'off'</span>.
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−</p>
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−
+
−
+
−<h2><a name="examples">Usage example of s-domain system identification tool</a></h2>
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−<p>
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−  In this example we fit a given frequency response to get a partial fraction model.
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−  For the meaning of any parameter please refer to
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−  <a href="matlab:doc('ao')">ao</a> and
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−  <a href="matlab:doc('ao/sDomainFit')">sDomainFit</a>
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−  documentation pages.
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−</p>
+
−
+
−<div class="fragment"><pre>
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−    pl = plist(...
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−      <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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−      <span class="string">'f1'</span>, 1e-6,...
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−      <span class="string">'f2'</span>, 5,...
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−      <span class="string">'nf'</span>, 100);
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−    
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−    a = ao(pl);
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−    a.setName;
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−
+
−    <span class="comment">% Fit parameter list</span>
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−    pl_fit = plist(...
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−      <span class="string">'AutoSearch'</span>,<span class="string">'on'</span>,...
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−      <span class="string">'StartPolesOpt'</span>,<span class="string">'clog'</span>,...
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−      <span class="string">'maxiter'</span>,50,...
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−      <span class="string">'minorder'</span>,7,...
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−      <span class="string">'maxorder'</span>,15,...
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−      <span class="string">'weightparam'</span>,<span class="string">'abs'</span>,...
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−      <span class="string">'CONDTYPE'</span>,<span class="string">'MSE'</span>,...
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−      <span class="string">'FITTOL'</span>,1e-3,...
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−      <span class="string">'MSEVARTOL'</span>,1e-2,...
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−      <span class="string">'Plot'</span>,<span class="string">'on'</span>,...
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−      <span class="string">'ForceStability'</span>,<span class="string">'off'</span>);
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−
+
−    <span class="comment">% Do fit</span>
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−    mod = sDomainFit(a, pl_fit);
+
−</pre></div>
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−
+
−<p>
+
−  <tt>mod</tt> is a <tt>matrix</tt> object containing a <tt>parfrac</tt> object.
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−</p>
+
−
+
−<div class="fragment"><pre>
+
−    >> mod
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−    ---- matrix 1 ----
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−           name: fit(a)
+
−           size: 1x1
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−             01: parfrac | parfrac(fit(a))
+
−    description: 
+
−           UUID: 2dc1ac28-4199-42d2-9b1a-b420252b3f8c
+
−    ------------------
+
−</pre></div> 
+
−    
+
−<div class="fragment"><pre>
+
−    >> mod.objs
+
−    ---- parfrac 1 ----
+
−    model:       fit(a)
+
−    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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−    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]
+
−    dir:         0
+
−    pmul:        [1;1;1;1;1;1;1;1;1;1;1]
+
−    iunits:      []
+
−    ounits:      []
+
−    description: 
+
−    UUID:        2afc4c82-7c2a-4fe3-8910-d8590884d58c
+
−    -------------------
+
−</pre></div>
+
−
+
−
+
−<h2><a name="references">References</a></h2>
+
−<p>
+
−  <ol>
+
−    <li> B. Gustavsen and A. Semlyen, "Rational approximation of frequency
+
−    domain responses by Vector Fitting", IEEE Trans. Power Delivery
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−    vol. 14, no. 3, pp. 1052-1061, July 1999.
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−    <li> B. Gustavsen, "Improving the Pole Relocating Properties of Vector
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−    Fitting", IEEE Trans. Power Delivery vol. 21, no. 3, pp.
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−    1587-1592, July 2006.
+
−  </ol>
+
−</p>