autopid_in4.html

这是一个PID自动调节的程序

    <td align="center"><font face="Courier New, Courier, mono">3.2</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">-6.6</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">-2.5</font></td>
    <td align="center"><font face="Courier New, Courier, mono">-1.5</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">3.0</font></td>
    <td align="center"><font face="Courier New, Courier, mono">3.0</font></td>
    <td align="center"><font face="Courier New, Courier, mono">-1.4</font></td>
    <td align="center"><font face="Courier New, Courier, mono">-0.93</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">0.89</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">-0.37</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">-4.1</font></td>
    <td align="center"><font face="Courier New, Courier, mono">-0.44</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">0.81</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">0.73</font></td>
    <td align="center"><font face="Courier New, Courier, mono">0.78</font></td>
    <td align="center"><font face="Courier New, Courier, mono">0.18</font></td>
    <td align="center"><font face="Courier New, Courier, mono">0.65</font></td>
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    <td align="center"><font face="Courier New, Courier, mono">1.9</font></td>
    <td align="center"><font face="Courier New, Courier, mono">-0.45</font></td>
    <td align="center"><font face="Courier New, Courier, mono">2.8</font></td>
    <td align="center"><font face="Courier New, Courier, mono">0.051</font></td>
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<a name="imc_method"></a><!-- H2 --><a name="t3"></a>
<p><font size=+1 color="#990000"><b>Internal Model Control [IMC]</b></font><br>
<p>To explain the main idea behind the Internal Model Control consider the block diagram below</p>
<table width="100%" border="0">
  <tr><td align="center"><img src="images/imcSchema.png"></td></tr>
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<p>where P(s) is the transfer function of the process (which we assume to be
  asymptotically stable, thus excluding integrating processes), <font face="Courier New, Courier, mono">M(s)</font>
	is the process model, <font face="Courier New, Courier, mono">Q(s)</font> and
	<font face="Courier New, Courier, mono">F(s)</font> are asymptotically stable transfer functions, at this
  stage arbitrary; <font face="Courier New, Courier, mono">Y<sub>m</sub>(s)</font> and
	<font face="Courier New, Courier, mono">Y<sub>n</sub>(s)</font> are the true (measured) and nominal controlled variables.</p>
<p>Coming to its practical use, the IMC synthesis method is a two-step procedure.
  First <font face="Courier New, Courier, mono">Q(s)</font> is determined as an approximated
	inverse of <font face="Courier New, Courier, mono">M(s)</font>, namely that of its minimum-phase
	part. Then, to ensure robustness, the low-pass &quot;IMC filter&quot; <font face="Courier New, Courier, mono">F(s)</font>
	is introduced. The structure and the parameters of F(s) are chosen to achieve a reasonable balance between
	robust stability and performance.<br></p>
<p>Being in the ideal case <font face="Courier New, Courier, mono">T(s) = F(s)Q(s)M(s)</font>, the IMC is a model-following
 	method trying (roughly speaking) to cancel <font face="Courier New, Courier, mono">M(s)</font> with
	<font face="Courier New, Courier, mono">Q(s)</font> so as to impose the closed-loop dynamics
	<font face="Courier New, Courier, mono">F(s)</font>.</p>
<p>In synthesis, the method consists of identifying a FOPDT model and then applying the IMC technique by choosing</p>
<img src="images/imcFormula.gif">
<p>and by replacing the process delay by its <font face="Courier New, Courier, mono">(1,1)</font> Pade approximation
 	<font face="Courier New, Courier, mono">(1-sL/2)/(1+sL/2)</font>. The regulator turns out to be a real PID given by</p>
<p><img src="images/imcFormula1.gif"> </p>
<p>The main concern in using the IMC-PID method is the choice of <font face="Courier New, Courier, mono">lambda</font>,
 	which is a knob for trading stability and robustness against performance.</p>

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<a name="zncl_method"></a><!-- H2 --><a name="t4"></a>
<p><font size=+1 color="#990000"><b>Ziegler &amp; Nichols with Relay Identification [ZN(CL)]</b></font><br>
<p>The second Ziegler and Nichols method tunes the parameter of the PID according
 	to the following table, on the basis of a point of the frequency response identified by a relay experiment.
	The period of oscillation is denoted as <font face="Courier New, Courier, mono">T<sub>u</sub></font>, while the
	gain margin is <font face="Courier New, Courier, mono">Ku = 4As/pi*A</font>.</p>
<table width="400" border="1" align="center">
  <tr bgcolor="#CCCCCC">
    <th width="100" scope="col">&nbsp;</th>
    <th width="100" scope="col">K</th>
    <th width="100" scope="col">Ti</th>
    <th width="100" scope="col">Td</th>
  </tr>
  <tr>
    <th width="100" bgcolor="#CCCCCC" scope="row">P</th>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0.5K<sub>u</sub></font></td>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0</font></td>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0</font></td>
  </tr>
  <tr>
    <th width="100" bgcolor="#CCCCCC" scope="row">PI</th>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0.4K<sub>u</sub></font></td>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0.8T<sub>u</sub></font></td>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0</font></td>
  </tr>
  <tr>
    <th width="100" bgcolor="#CCCCCC" scope="row">PID</th>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0.6K<sub>u</sub></font></td>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0.5T<sub>u</sub></font></td>
    <td width="100" align="center"><font face="Courier New, Courier, mono">0.125T<sub>u</sub></font></td>
  </tr>
</table>
<p>In the original version of method, the identification of the point requires to apply proportional control
 	and increase the controller gain until the process output reaches a sustained oscillation. However it is a dangerous
	practice since it leads the model near the stability limit.</p>
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