📄 syn03.htm
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<title>MvTools Help : Synthesis ---> IMFC</title>
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<p align=center style='text-align:center'><img width=371 height=51
id="_x0000_i1025" src="Immagini\logosyn.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></p>
<p align=center style='text-align:center'><img border=0 width=536 height=5
id="_x0000_i1026" src="Immagini\riga.gif"></p>
<p align=center style='text-align:center'><b><u><span style='font-size:13.5pt;
color:red'>IMFC<o:p></o:p></span></u></b></p>
<p align=center style='text-align:center'>(Pseudo State Feedback)</p>
<p style='text-align:justify'>The Implicit Model Following Control belongs to
the group of control laws that solve the general tracking problem using the LQR
(Linear Quadratic Regulator) approach. The purpose of IMFC (and also of <a
href="syn04.htm" title=" EMFC CONTROL help page ">EMFC</a>) is to find a closed
loop control law that lets the plant (the current system) track the dynamics of
a reference model.</p>
<p style='text-align:justify'>The IMFC technique computes the optimal state
feedback control law that minimizes the cost function </p>
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<table border=0 cellspacing=0 cellpadding=0 style='border-collapse:collapse;
mso-padding-alt:0cm 5.4pt 0cm 5.4pt'>
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<td valign=top style='padding:0cm 5.4pt 0cm 5.4pt'>
<p align=center style='text-align:center'><span style='color:red'>J =
integral{ ( Y<b><sup>'</sup></b></span><span style='font-size:10.0pt;
mso-bidi-font-size:12.0pt;color:red'> </span><span style='color:red'>- Y<b><sup>'</sup></b></span><span
style='font-size:7.5pt;color:red'>m</span><span style='font-size:10.0pt;
mso-bidi-font-size:12.0pt;color:red'> </span><span style='color:red'>)</span><sup><span
style='font-size:10.0pt;mso-bidi-font-size:12.0pt;color:red'>T</span></sup><span
style='font-size:7.5pt;color:red'> </span><span style='color:red'>Q ( Y<b><sup>'</sup></b></span><span
style='font-size:10.0pt;mso-bidi-font-size:12.0pt;color:red'> </span><span
style='color:red'>- Y<b><sup>'</sup></b></span><span style='font-size:7.5pt;
color:red'>m</span><span style='font-size:10.0pt;mso-bidi-font-size:12.0pt;
color:red'> </span><span style='color:red'>) + U</span><sup><span
style='font-size:10.0pt;mso-bidi-font-size:12.0pt;color:red'>T</span></sup><span
style='color:red'> R U }dt</span></p>
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<p style='text-align:justify'>with</p>
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1.2pt;mso-padding-alt:3.0pt 3.0pt 3.0pt 3.0pt'>
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<td width="48%" style='width:48.0%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt'>
<p align=center style='text-align:center'><span style='color:green'>Plant</span></p>
</td>
<td width="52%" style='width:52.0%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt'>
<p align=center style='text-align:center'><span style='color:green'>Model</span></p>
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<p align=center style='text-align:center'>X<b><sup>'</sup></b> = A X + B U</p>
<p align=center style='text-align:center'>Y = C<span style='font-size:7.5pt'>
</span>X</p>
</td>
<td width="52%" style='width:52.0%;background:white;padding:3.0pt 3.0pt 3.0pt 3.0pt'>
<p align=center style='text-align:center'>X<b><sup>'</sup></b><span
style='font-size:7.5pt'>m</span> = A<span style='font-size:7.5pt'>m</span> X<span
style='font-size:7.5pt'>m</span> + B<span style='font-size:7.5pt'>m</span> U<span
style='font-size:7.5pt'>m<o:p></o:p></span></p>
<p align=center style='text-align:center'><span style='color:red'>Y</span><span
style='font-size:7.5pt;color:red'>m</span> = C<span style='font-size:7.5pt'>m
</span>X<span style='font-size:7.5pt'>m</span> = I <span style='color:red'>X</span><span
style='font-size:7.5pt;color:red'>m</span></p>
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<p style='text-align:justify'>The vector Y<span style='font-size:7.5pt'>m</span>
= X<span style='font-size:7.5pt'>m </span>represents the reference command for
the plant outputs. The cost function weighs, by means of the matrix Q, the
tracking error between the output vectors of the two systems. It's important to
remember that <span style='color:red'>the model is not really present</span> in
the control loop but it appears, with its matrices, only inside the cost
function J; therefore the vector X<span style='font-size:7.5pt'>m</span> is not
actually available and the controller has to be computed just in accordance
with the plant states X.</p>
<p style='text-align:justify'>In order that <span style='color:red'>Y<b><sup>'</sup></b>
</span><span style='font-family:Symbol;color:red;mso-char-type:symbol;
mso-symbol-font-family:Symbol'><span style='mso-char-type:symbol;mso-symbol-font-family:
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