syn00.htm

vTools is a toolbox for Matlab 5.3 developed within the Department of Electrical Systems and A

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  <p align=center style='text-align:center'><span lang=EN-US>Sensitivity</span></p>
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  <td width="33%" style='width:33.16%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
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  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>i</span><span lang=EN-US
  style='color:red'> = (I + K G)<sup>-1</sup> = </span><span lang=EN-US
  style='mso-bidi-font-size:7.5pt;color:blue'>e/r</span></p>
  </td>
  <td width="33%" style='width:33.08%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
  height:26.4pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>o</span><span lang=EN-US
  style='color:red'> = (I + G</span><span lang=EN-US style='font-size:10.0pt;
  color:red'> </span><span lang=EN-US style='color:red'>K)<sup>-1</sup> = </span><span
  lang=EN-US style='mso-bidi-font-size:7.5pt;color:blue'>e/r</span></p>
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  <p align=center style='text-align:center'><span lang=EN-US>Complementary
  Sensitivity</span></p>
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  <td width="33%" style='width:33.16%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
  height:26.4pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>T</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>i</span><span lang=EN-US
  style='color:red'> = S</span><span lang=EN-US style='font-size:7.5pt;
  color:red'>i </span><span lang=EN-US style='color:red'>K G</span></p>
  </td>
  <td width="33%" style='width:33.08%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
  height:26.4pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>T</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>o</span><span lang=EN-US
  style='color:red'> =</span><span lang=EN-US style='font-size:10.0pt;
  color:red'> </span><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>o </span><span lang=EN-US
  style='color:red'>G</span><span lang=EN-US style='font-size:10.0pt;
  color:red'> </span><span lang=EN-US style='color:red'>K = </span><span
  lang=EN-US style='mso-bidi-font-size:7.5pt;color:blue'>y/r</span></p>
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  <p align=center style='text-align:center'><span lang=EN-US>Control
  Sensitivity</span></p>
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  <td width="33%" style='width:33.16%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
  height:26.4pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>M</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>i</span><span lang=EN-US
  style='color:red'> = G</span><span lang=EN-US style='font-size:10.0pt;
  color:red'> </span><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>i </span><span lang=EN-US
  style='mso-bidi-font-size:7.5pt;color:red'>= </span><span lang=EN-US
  style='mso-bidi-font-size:7.5pt;color:blue'>y/r</span></p>
  </td>
  <td width="33%" style='width:33.08%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
  height:26.4pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>M</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>o</span><span lang=EN-US
  style='color:red'> = K</span><span lang=EN-US style='font-size:10.0pt;
  color:red'> </span><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:7.5pt;color:red'>o</span></p>
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</div>

<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>

<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US>As we can see from the previous relations, the Sensitivity and the
Complementary Sensitivity transfer matrices are such that:</span></p>

<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>

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  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>i</span><span
  lang=EN-US style='color:red'> + T</span><span lang=EN-US style='font-size:
  8.0pt;mso-bidi-font-size:12.0pt;color:red'>i</span><span lang=EN-US
  style='color:red'> = I<o:p></o:p></span></p>
  </td>
  <td width=212 style='width:127.3pt;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
  height:19.05pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>o</span><span
  lang=EN-US style='color:red'> + T</span><span lang=EN-US style='font-size:
  8.0pt;mso-bidi-font-size:12.0pt;color:red'>o</span><span lang=EN-US
  style='color:red'> = I<o:p></o:p></span></p>
  </td>
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  <td width=212 style='width:127.25pt;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
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  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>i</span><sup><span
  lang=EN-US style='color:red'>-1</span></sup><span lang=EN-US
  style='color:red'> = I + F</span><span lang=EN-US style='font-size:8.0pt;
  mso-bidi-font-size:12.0pt;color:red'>i</span></p>
  </td>
  <td width=212 style='width:127.3pt;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
  height:19.8pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>S</span><span
  lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>o</span><sup><span
  lang=EN-US style='color:red'>-1</span></sup><span lang=EN-US
  style='color:red'> = I + F</span><span lang=EN-US style='font-size:8.0pt;
  mso-bidi-font-size:12.0pt;color:red'>o</span></p>
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  height:20.2pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>T</span><span
  lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>i</span><sup><span
  lang=EN-US style='color:red'>-1</span></sup><span lang=EN-US
  style='color:red'> = I + F</span><span lang=EN-US style='font-size:8.0pt;
  mso-bidi-font-size:12.0pt;color:red'>i</span><sup><span lang=EN-US
  style='color:red'>-1</span></sup></p>
  </td>
  <td width=212 style='width:127.3pt;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt;
  height:20.2pt'>
  <p align=center style='text-align:center'><span lang=EN-US style='color:red'>T</span><span
  lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>o</span><sup><span
  lang=EN-US style='color:red'>-1</span></sup><span lang=EN-US
  style='color:red'> = I + F</span><span lang=EN-US style='font-size:8.0pt;
  mso-bidi-font-size:12.0pt;color:red'>o</span><sup><span lang=EN-US
  style='color:red'>-1</span></sup></p>
  </td>
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</div>

<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>

<p style='text-align:justify;page-break-before:always'><span lang=EN-US>In
MvTools, the conventional closed loop configuration shown in <i>Fig.2</i> is
considered and the relative design specifications are: <a href="biblo.htm#b3"
title=" REFERENCE ">[3]</a> <a href="biblo.htm#b4" title=" REFERENCE ">[4]</a></span></p>

<p style='text-align:justify'><a name=matrici></a><span lang=EN-US
style='color:red'><img border=0 width=13 height=13 id="_x0000_i1185"
src="Immagini\Bullet1.gif"><span style="mso-spacerun: yes">牋
</span>PERFORMANCE requirement</span><b><span lang=EN-US style='font-size:10.0pt;
color:red'><o:p></o:p></span></b></p>

<p style='margin-top:0cm;margin-right:0cm;margin-bottom:0cm;margin-left:17.6pt;
margin-bottom:.0001pt;text-align:justify'><span lang=EN-US>At </span><span
lang=EN-US style='color:green'>low frequencies</span><span lang=EN-US>, where a
satisfying knowledge of the plant is assumed, the most important requirements
are the tracking of the reference signal and a good disturbance rejection,
which can be both meet by <b>keeping the Sensitivity low, and a Complementary
Sensitivity close to the identity matrix,</b> which implies the output signal
to be close to the reference one:</span></p>

<p style='margin-top:0cm;margin-right:0cm;margin-bottom:0cm;margin-left:17.6pt;
margin-bottom:.0001pt;text-align:justify'><span lang=EN-US><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>

<p align=center style='margin-top:0cm;margin-right:0cm;margin-bottom:0cm;
margin-left:17.6pt;margin-bottom:.0001pt;text-align:center'><span lang=EN-US
style='mso-bidi-font-size:10.0pt;color:red'>S</span><span lang=EN-US
style='font-size:8.0pt;mso-bidi-font-size:10.0pt;color:red'>o</span><span
lang=EN-US style='mso-bidi-font-size:10.0pt;color:red'> </span><span
lang=EN-US style='mso-bidi-font-size:10.0pt;font-family:Symbol;mso-ascii-font-family:
"Times New Roman";mso-hansi-font-family:"Times New Roman";color:red;mso-char-type:
symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type:symbol;
mso-symbol-font-family:Symbol'>