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<div class=Section1>
<p align=center style='text-align:center'><img width=371 height=55
id="_x0000_i1025" src="Immagini\logoana.jpg"></p>
<p align=center style='text-align:center'><a href="HelpDeskmvt.htm"
title="MVTOOLS HELP DESK"><b><i>MvTools Help Desk</i></b></a><span lang=EN-US
style='mso-ansi-language:EN-US'><o:p></o:p></span></p>
<p align=center style='text-align:center'><img border=0 width=530 height=5
id="_x0000_i1026" src="Immagini\riga.gif"></p>
<p align=center style='text-align:center'>Subsection<span style='font-size:
13.5pt'> </span><b><u><span style='font-size:13.5pt;color:red'>POLES and ZEROS<o:p></o:p></span></u></b></p>
<p><img border=0 width=13 height=12 id="_x0000_i1027" src="Immagini\Bullet1.gif"><span
style='color:blue'> Poles and Zeros<o:p></o:p></span></p>
<p align=center style='text-align:center'><img border=0 width=371 height=249
id="_x0000_i1028" src="Immagini\anaf4.gif"></p>
<p style='text-align:justify'>The main window of this subsection resumes the frequency
domain structure of the LTI current model; the graphic (pzmap) shows the
positions of its poles and transmission zeros in the complex plane (the poles
are plotted as x's and the zeros are plotted as o's), while the buttons <b>POLES</b>
and <b>TZEROS</b> allow the visualization of their values in the text-fields on
the right of the window (a yellow background points out the instability of a
pole or zero).</p>
<p style='text-align:justify'>Note:<span style='font-size:10.0pt'> the button <b>POLES</b>
is disabled if the total number of poles is more than 16, while the button <b>TZEROS</b>
is disabled if the total number of transmission zeros is more than 8 or equal
to 0 (in the last case, also the button <b>Zeros analysis</b> is inactive).<o:p></o:p></span></p>
<p><span style='font-size:10.0pt'> <o:p></o:p></span></p>
<p><img border=0 width=13 height=13 id="_x0000_i1029" src="Immagini\Bullet1.gif"><span
style='font-size:10.0pt'> </span><span style='color:blue'>Eigenstructure
(or poles analysis)<a name=eigenst></a><o:p></o:p></span></p>
<p style='text-align:justify'>Pressing the button <b>Eigenstructure</b>, it's
possible to open the following window in which is performed the analysis of the
poles and of the corresponding eigenvectors.</p>
<p align=center style='text-align:center'><img border=0 width=371 height=248
id="_x0000_i1030" src="Immagini\anaf5.gif"></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'>For each
eigenvalue, which can be selected with the buttons <b><<</b> and <b>>></b>,
the corresponding eigenvector is computed, then the absolute values of its
components along the state space are displayed in a bar graphic; at the top of
this diagram are also pointed out the value of the pole in the complex plane
and its values of controllability and observability (<a
href="ana05.htm#gramians" title=" CTRB-OBSV GRAMIANS ">gramians</a>) that are
computed as the magnitude of the intersection point between the eigenvector
corresponding to the pole and the controllability/observability ellipsoid.</p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'>The button <b>CTRB-OBSV</b>
allows to visualize the values of ctrb/obsv gramian of every poles in a
3-dimensional graphic, treating them as heights above the complex plane: the
values of the considered pole are pointed out with circles.</p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'> </p>
<p><img border=0 width=13 height=13 id="_x0000_i1041" src="Immagini\Bullet1.gif">
<span style='color:blue'>Zeros analysis<a name=zeros></a><o:p></o:p></span></p>
<p style='text-align:justify'>The function associated with the <b>Zeros
Analysis</b> button starts the examination of the model transmission zeros (if
they exist).</p>
<p align=center style='text-align:center'><img border=0 width=371 height=249
id="_x0000_i1032" src="Immagini\anaf6.gif"></p>
<p style='text-align:justify'>As usual, the user can select a transmission zero
with the pushbuttons <b><<</b> and <b>>></b>: the index and the
value of the considered zero are indicated in the first two text-fields above
these buttons. Then, for every zero (z<span style='font-size:8.0pt;mso-bidi-font-size:
12.0pt'>o</span>), the transfer matrix G(z<span style='font-size:8.0pt;
mso-bidi-font-size:12.0pt'>o</span>) of the current model and the parameter x<span
style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>o</span> are evaluated in the
following way:</p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center;
page-break-before:always'><span style='color:red'>G(z</span><span
style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>o</span><span
style='color:red'>) = C pinv( z</span><span style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:red'>o</span><span style='color:red'> I - A ) B +D<o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center'><span
style='color:red'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center'>[U S
V] = svd( G(z<span style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>o</span>)
)</p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center'><![if !supportEmptyParas]> <![endif]><o:p></o:p></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center'><span
style='color:red'>Un = U( : , #outputs )<span style='mso-tab-count:2'>牋牋牋牋牋牋牋 </span>Vn
= V( : , #inputs )<o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center'><span
style='color:red'><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center'><span
style='color:red'>x</span><span style='font-size:8.0pt;mso-bidi-font-size:12.0pt;
color:red'>o</span><span style='color:red'> = pinv( z</span><span
style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>o</span><span
style='color:red'> I - A ) B Vn<o:p></o:p></span></p>
<p style='text-align:justify'>where the parameter x<span style='font-size:8.0pt;
mso-bidi-font-size:12.0pt'>o</span> represents the initial condition that,
together with the input signal</p>
<p align=center style='text-align:center'>u(t) = u<span style='font-size:8.0pt;
mso-bidi-font-size:12.0pt'>o</span> exp(z<span style='font-size:8.0pt;
mso-bidi-font-size:12.0pt'>o</span> t) <span style='font-family:Symbol;
mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman";
mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type:
symbol;mso-symbol-font-family:Symbol'>⌨️ 快捷键说明
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