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<html><head><title>Synthesis Methods :: Introduction to PID Autotuning (AutotunerPID Toolkit)</title>
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<!-- CHAPNAME: Introduction to PID Autotuning -->
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<td valign=baseline bgcolor="#ffe4b0"><b>AutotunerPID Toolkit</b></td>
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<a name="process_identification"></a><!-- H1 -->
<p><font size=+2 color="#990000"><b>Synthesis methods</b></font><br>
<p>Once that a model (or some characteristic) of the plant have been identified,
the second step is to tune the parameters of the PID regulator in order to
achieve the best performances. In the following a brief overview of the
synthesis methods implemented in the Toolkit is given.</p>
<p><a href="autopid_in4.html#t1">Ziegler & Nichols with Step Identification [ZN(OL)]</a></p>
<p><a href="autopid_in4.html#t2">Kappa-Tau [KT]</a></p>
<p><a href="autopid_in4.html#t3">Internal Model Control [IMC]</a></p>
<p><a href="autopid_in4.html#t4">Ziegler & Nichols with Relay Identification [ZN(CL)]</a></p>
<a name="znol_method"></a><!-- H2 --><a name="t1"></a>
<p><font size=+1 color="#990000"><b>Ziegler & Nichols with Step Identification [ZN(OL)]</b></font><br>
<p> The first Ziegler and Nichols method tunes the parameter of the PID according to the following table,
on the basis of the parameters identified for a FOPDT model</p>
<table width="400" border="1" align="center">
<tr bgcolor="#CCCCCC">
<th width="100" scope="col"> </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">T/mL</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">0.9T/mL</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">3L</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">1.2T/mL</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">2L</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">0.5L</font></td>
</tr>
</table>
<p>In the original version of method, the tuning formulas are given with respect
to some characteristic of the process identified in terms of the
points where the tangent to the step response in the point of
maximum slope intersect the step response. However here the modified version
has been used since it is more robust with respect to noise.</p>
<br>
<a name="kt_method"></a><!-- H2 --><a name="t2"></a>
<p><font size=+1 color="#990000"><b>Kappa-Tau [KT]</b></font><br>
<p>The Kappa-Tau method computes the parameters of the 2-d.o.f. ISA PID
control law apart from <font face="Courier New, Courier, mono">N</font> and
in the output derivation case (i.e. <font face="Courier New, Courier, mono">c=0</font>).
It requires to identify a FOPDT model if the process is not integrating, or
a FOPDT one plus a factor <font face="Courier New, Courier, mono">1/s</font> if
it is (not yet considered in the Toolkit).</p>
<p>The information used is then given by the model parameters <font face="Courier New, Courier, mono">m</font>,
<font face="Courier New, Courier, mono">T</font> and <font face="Courier New, Courier, mono">L</font>,
and by the request of a PI or PID regulator. A further specification is the required magnitude margin <font face="Courier New, Courier, mono">M<sub>s</sub></font>,
defined as</p>
<img src="images/ktFormula.gif">
<p>for which the two values of 1.4 (conservative tuning) or 2.0 (more aggressive tuning) are advised.
Defining the <em>process normalized gain</em> <font face="Courier New, Courier, mono">alpha</font> and <em>normalized
delay</em>
<font face="Courier New, Courier, mono">tau</font> as</p>
<img src="images/ktFormula1.gif">
<p>the PI(D) regulator parameters are computed as</p>
<img src="images/ktFormula2.gif">
<p> The coefficients <font face="Courier New, Courier, mono">A<sub>i</sub></font>,
<font face="Courier New, Courier, mono">B<sub>i</sub></font>, <font face="Courier New, Courier, mono">C<sub>i</sub></font>,
<font face="Courier New, Courier, mono">D<sub>i</sub></font> come from the following
table. These coefficients were derived by applying dominant pole design to many
different processes and then interpolating the results to obtain compact tuning
relationships. Thus, this is a model following method with the peculiarity
of using interpolation. One important remark is that the normalized delay, sometimes
called the "controllability index", can be taken as a measure of
how difficult to control a process is.</p>
<table width="534" border="1" align="center">
<tr bgcolor="#CCCCCC">
<th width="100" bgcolor="#CCCCCC" scope="row">Structure</th>
<th width="100" bgcolor="#FFFFFF" scope="col">PI</th>
<th width="100" bgcolor="#FFFFFF" scope="col">PI</th>
<th width="100" bgcolor="#FFFFFF" scope="col">PID</th>
<th width="100" bgcolor="#FFFFFF" scope="col">PID</th>
</tr>
<tr>
<th width="100" bgcolor="#CCCCCC" scope="row">M<sub>s</sub></th>
<td width="100" align="center"><font face="Courier New, Courier, mono">1.4</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">2.0</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">1.4</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">2.0</font></td>
</tr>
<tr>
<th width="100" bgcolor="#CCCCCC" scope="row">A<sub>0</sub></th>
<td width="100" align="center"><font face="Courier New, Courier, mono">0.29</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">0.78</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">3.8</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">8.4</font></td>
</tr>
<tr>
<th bgcolor="#CCCCCC" scope="row">A<sub>1</sub></th>
<td width="100" align="center"><font face="Courier New, Courier, mono">-2.7</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">-4.1</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">-8.4</font></td>
<td width="100" align="center"><font face="Courier New, Courier, mono">-9.6</font></td>
</tr>
<tr>
<th bgcolor="#CCCCCC" scope="row">A<sub>2</sub></th>
<td align="center"><font face="Courier New, Courier, mono">3.7</font></td>
<td align="center"><font face="Courier New, Courier, mono">5.7</font></td>
<td align="center"><font face="Courier New, Courier, mono">7.3</font></td>
<td align="center"><font face="Courier New, Courier, mono">9.8</font></td>
</tr>
<tr>
<th bgcolor="#CCCCCC" scope="row">B<sub>0</sub></th>
<td align="center"><font face="Courier New, Courier, mono">8.9</font></td>
<td align="center"><font face="Courier New, Courier, mono">8.9</font></td>
<td align="center"><font face="Courier New, Courier, mono">5.2</font></td>
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