<!-- $Id: sdomainfit_content.html,v 1.3 2009/08/27 11:38:58 luigi Exp $ -->

<!-- ================================================== -->
<!--                 BEGIN CONTENT FILE                 -->
<!-- ================================================== -->
<!-- ===== link box: Begin ===== -->
<p>
  <table border="1"  width="80%">
    <tr>
      <td>
        <table border="0" cellpadding="5" class="categorylist" width="100%">
          <colgroup>
            <col width="37%"/>
            <col width="63%"/>
          </colgroup>
          <tbody>
            <tr valign="top">
              <td>
                <a href="#description">Description</a>
              </td>
              <td>S-domain system identification in LTPDA.</td>
            </tr>
            <tr valign="top">
              <td>
                <a href="#algorithm">Algorithm</a>
              </td>
              <td>Fit Algorithm.</td>
            </tr>
            <tr valign="top">
              <td>
                <a href="#examples">Examples</a>
              </td>
              <td>Usage example of s-domain system identification tool.</td>
            </tr>
            <tr valign="top">
              <td>
                <a href="#references">References</a>
              </td>
              <td>Bibliographic references.</td>
            </tr>
          </tbody>
        </table>
      </td>
    </tr>
  </table>
</p>
<!-- ===== link box: End ====== -->

<h2><a name="description">S-domain system identification in LTPDA</a></h2>
<p>
  System identification in s-domain is performed with the function 
  <a href="matlab:doc('ao/sDomainFit')">sDomainFit</a>. 
  It is based on a modeified version of the vector fitting algorithm. 
  Details on the core agorithm can be found in [1 - 2].
</p>



<h2><a name="algorithm">Fit Algorithm</a></h2>

<p>
  The function performs a fitting loop to automatically identify model 
  order and parameters in s-domain. Output is a s-domain model expanded 
  in partial fractions:
</p>
<div class="fragment"><pre>
              r1              rN
    f(s) = ------- + ... + ------- + d
            s - p1          s - pN
</pre></div>
<p>
  Since the function can fit more than one input analysis object at a time
  with a common set of poles, output 
  <a href="parfrac.html">parfrac</a> are embedded in a 
  <a href="class_desc_matrix.html">matrix</a> (note that this characteristic
  will be probably changed becausse of the introduction of the
  <a href="class_desc_collection.html">collection</a> class).
</p>
<p>
  Identification loop stops when the stop condition is reached. 
  Stop criterion is based on three different approachs:
  <ol>
    <li> Mean Squared Error and variation <br>
      Check if the normalized mean squared error is lower than the value specified in
      <tt>FITTOL</tt> and if the relative variation of the mean squared error is lower
      than the value specified in <tt>MSEVARTOL</tt>.
      E.g. <tt>FITTOL = 1e-3</tt>, <tt>MSEVARTOL = 1e-2</tt> search for a fit with
      normalized meam square error lower than <tt>1e-3</tt> and <tt>MSE</tt> relative
      variation lower than <tt>1e-2</tt>. 
    </li>
    <li> Log residuals difference and root mean squared error
      <ul>
        <li> Log Residuals difference </br>
        Check if the minimum of the logarithmic difference between data and 
        residuals is larger than a specified value. ie. if the conditioning 
        value is <tt>2</tt>, the function ensures that the difference between data and 
        residuals is at lest two order of magnitude lower than data itsleves. 
        <li> Root Mean Squared Error </br>
        Check that the variation of the root mean squared error is lower than 
        <tt>10^(-1*value)</tt>.
      </ul>
    </li>
    <li> Residuals spectral flatness and root mean squared error
      <ul>
        <li> Residuals Spectral Flatness </br>
        In case of a fit on noisy data, the residuals from a good fit are
        expected to be as much as possible similar to a white noise. This
        property can be used to test the accuracy of a fit procedure. In
        particular it can be tested that the spectral flatness coefficient of
        the residuals is larger than a certain qiantity sf such that <tt>0 < sf < 1</tt>.
        <li> Root Mean Squared Error </br>
        Check that the variation of the root mean squared error is lower than 
        <tt>10^(-1*value)</tt>.
      </ul>
    </li>
  </ol>
  
</p>
<p>
  The function can also perform a single loop without taking care of
  the stop conditions. This happens when <span class="string">'AutoSearch'</span> parameter is
  set to <span class="string">'off'</span>.
</p>


<h2><a name="examples">Usage example of s-domain system identification tool</a></h2>
<p>
  In this example we fit a given frequency response to get a partial fraction model.
  For the meaning of any parameter please refer to
  <a href="matlab:doc('ao')">ao</a> and
  <a href="matlab:doc('ao/sDomainFit')">sDomainFit</a>
  documentation pages.
</p>

<div class="fragment"><pre>
    pl = plist(...
      <span class="string">'fsfcn'</span>, <span class="string">'(1e-3./(f).^2 + 1e3./(0.001+f) + 1e5.*f.^2).*1e-10'</span>,...
      <span class="string">'f1'</span>, 1e-6,...
      <span class="string">'f2'</span>, 5,...
      <span class="string">'nf'</span>, 100);
    
    a = ao(pl);
    a.setName;

    <span class="comment">% Fit parameter list</span>
    pl_fit = plist(...
      <span class="string">'AutoSearch'</span>,<span class="string">'on'</span>,...
      <span class="string">'StartPolesOpt'</span>,<span class="string">'clog'</span>,...
      <span class="string">'maxiter'</span>,50,...
      <span class="string">'minorder'</span>,7,...
      <span class="string">'maxorder'</span>,15,...
      <span class="string">'weightparam'</span>,<span class="string">'abs'</span>,...
      <span class="string">'CONDTYPE'</span>,<span class="string">'MSE'</span>,...
      <span class="string">'FITTOL'</span>,1e-3,...
      <span class="string">'MSEVARTOL'</span>,1e-2,...
      <span class="string">'Plot'</span>,<span class="string">'on'</span>,...
      <span class="string">'ForceStability'</span>,<span class="string">'off'</span>);

    <span class="comment">% Do fit</span>
    mod = sDomainFit(a, pl_fit);
</pre></div>

<p>
  <tt>mod</tt> is a <tt>matrix</tt> object containing a <tt>parfrac</tt> object.
</p>

<div class="fragment"><pre>
    >> mod
    ---- matrix 1 ----
           name: fit(a)
           size: 1x1
             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]
    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
    vol. 14, no. 3, pp. 1052-1061, July 1999.
    <li> B. Gustavsen, "Improving the Pole Relocating Properties of Vector
    Fitting", IEEE Trans. Power Delivery vol. 21, no. 3, pp.
    1587-1592, July 2006.
  </ol>
</p>