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Implement databases listing in database connection dialog
author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
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  <title>Modelling a system (LTPDA Toolbox)</title>
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  <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Modelling a system</h1>
  <hr>
  
  <p>
	<p>
  To show some of the possibilities of the toolbox to model digital system we
  introduce the usual notation for a closed loop model
</p>
<div align="center">
  <img src="images/ltpda_training_1/topic4/ClosedLoop.png" alt="Closed Loop" border="3">
</div>
<p>
  In our example we will assume that we know the <tt>pzmodel</tt> of the filter,
  <tt>H</tt>, and the open loop gain (OLG). These are related with the closed
  loop gain (CLG) by the following equation
</p>
<br>
<div align="center">
  <img src="images/ltpda_training_1/topic4/ClosedLoop_eq.png" border="3">
</div>
<br>
<p>
  We want to determine H and CLG. We would also like to find a digital filter
  for H, but we will deal with this in the following section.
</p>
<h2>Loading the starting models</h2>
<p>
  Imagine that we have somehow managed to find the following model for OLG
</p>
<table cellspacing="0" class="body" cellpadding="2" border="0" width="50%">
  <colgroup>
    <col width="15%"/>
    <col width="35%"/>
  </colgroup>
  <thead>
    <tr valign="top">
      <th class="categorylist">Key</th>
      <th class="categorylist">Value</th>
    </tr>
  </thead>
  <tbody>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt>GAIN</tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        4e6
      </td>
    </tr>
    <!-- Key 'filename' -->
    <tr valign="top">
      <td bgcolor="#f3f4f5">
        <p><tt>POLES</tt></p>
      </td>
      <td bgcolor="#f3f4f5">
        1e-6
      </td>
    </tr>
  </tbody>
</table>
<p>
  then we can create a <tt>pzmodel</tt> with these parameters as follows
</p>
<div class="fragment"><pre>
    OLG = pzmodel(4e6,1e-6,[],<span class="string">'OLG'</span>)
    ---- pzmodel 1 ----
           name: OLG
           gain: 4000000
          delay: 0
         iunits: []
         ounits: []
    description:
           UUID: 3d8bce32-a9a9-4e72-ab4d-183da69a9b5d
    pole 001: (f=1e-06 Hz,Q=NaN)
    -------------------
</pre></div>
<p>
  To introduce the second model, the one describing <tt>H</tt>, we will show another feature
  of the <tt>pzmodel</tt> constructor. We will read it from a LISO file,
  since this contructor accepts this files as inputs. We can then type
</p>
<div class="fragment"><pre>
    H = pzmodel(<span class="string">'topic4/LISOfile.fil'</span>)
    ---- pzmodel 1 ----
           name: none
           gain: 1000000000
          delay: 0
         iunits: []
         ounits: []
    description:
           UUID: e683b32f-4653-474e-b457-939d33aeb63c
    pole 001: (f=1e-06 Hz,Q=NaN)
    pole 002: (f=1e-06 Hz,Q=NaN)
    zero 001: (f=0.001 Hz,Q=NaN)
-------------------
</pre></div>
<pp>
  and we see how the constructor recognizes and translates the poles and zeros in the file.
  The model gets the name from the file but we can easily change it to have
  the name of our model
</pp>
<div class="fragment"><pre>
    H.setName;
</pre></div>
<pp>
  According to our previous definition we can get the <tt>plant</tt> by dividing the
  OLG by H. We can do so directly when dealing with <tt>pzmodel</tt> objects
  since multiplication and division are allowed for these objects, then
</pp>
<div class="fragment"><pre>
    G = OLG/H
    ---- pzmodel 1 ----
           name: (OLG./H)
           gain: 0.004
          delay: 0
         iunits: []
         ounits: []
    description:
           UUID: fcd166f7-d726-4d39-ad2e-0f3b7141415b
    pole 001: (f=1e-06 Hz,Q=NaN)
    pole 002: (f=0.001 Hz,Q=NaN)
    zero 001: (f=1e-06 Hz,Q=NaN)
    zero 002: (f=1e-06 Hz,Q=NaN)
    -------------------
</pre></div>
<pp>
  which we need to simplify to get rid of cancelling poles and zeros. We also
  set the model name here.
</pp>
<div class="fragment"><pre>
    G.setName;
    G.simplify
    ---- pzmodel 1 ----
           name: simplify(G)
           gain: 0.004
          delay: 0
         iunits: []
         ounits: []
    description:
           UUID: c1883713-b860-4942-a127-e42ea565460f
    pole 001: (f=0.001 Hz,Q=NaN)
    zero 001: (f=1e-06 Hz,Q=NaN)
    -------------------
</pre></div>
<pp>
  The CLG requires more than a simple multiplication or division between models
  and we will not be able to derive a <tt>pzmodel</tt> for it. However, we can
  evaluate the response of this object as follows
</pp>
<div class="fragment"><pre>
    pl = plist(<span class="string">'f1'</span>,1e-3,<span class="string">'f2'</span>,5,<span class="string">'nf'</span>,100);
    CLG = 1/(1-resp(OLG,pl));
    CLG.setName();
    CLG.iplot();
</pre></div>
<pp>
  which gives us an AO that we can plot
</pp>
<div align="center">
  <img src="images/ltpda_training_1/topic4/ClosedLoop_resp.png" alt="Closed Loop response" width="800px" border="1">
</div>
<h2>Fine, but my (real) system has a delay...</h2>
<p>
  You can now repeat the same procedure but loading a <tt>H</tt> model with a delay
  from the LISO file 'LISOFileDelay.fil'
</p>
<div class="fragment"><pre>
    >> HDel = pzmodel(<span class="string">'topic4/LISOfileDelay.fil'</span>)
    ---- pzmodel 1 ----
           name: none
           gain: 1000000000
          delay: 0.125
         iunits: []
         ounits: []
    description:
           UUID: 6dfbddc5-b186-4405-8f6e-02a2822a22c5
    pole 001: (f=1e-06 Hz,Q=NaN)
    pole 002: (f=1e-06 Hz,Q=NaN)
    zero 001: (f=0.001 Hz,Q=NaN)
    -------------------
    HDel.setName();
    pl = plist(<span class="string">'f1'</span>,1e-3,<span class="string">'f2'</span>,5,<span class="string">'nf'</span>,100);
    resp([H, HDel],pl)
</pre></div>
<p>
  you will see how the delay is correctly handled, meaning that it is added when
  we multiply two models and substracted if the models are divided.
</p>
<div align="center">
  <img src="images/ltpda_training_1/topic4/ClosedLoop_H_delay.png" alt="Response with delay" width="800px" border="1">
  <img src="images/ltpda_training_1/topic4/ClosedLoop_G_delay.png" alt="Response with delay" width="800px" border="1">
</div>







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