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<title>MvTools Help : Synthesis ---> EMFC</title>
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<p align=center style='text-align:center'><span lang=EN-US><img width=371
height=51 id="_x0000_i1025" src="Immagini\logosyn.jpg"></span></p>
<p align=center style='text-align:center'><span lang=EN-US><a
href="HelpDeskmvt.htm" title=" MVTOOLS HELP DESK "><b><i>MvTools Help Desk</i></b></a></span></p>
<p align=center style='text-align:center'><span lang=EN-US><img border=0
width=536 height=5 id="_x0000_i1026" src="Immagini\riga.gif"></span></p>
<p align=center style='text-align:center'><b><u><span lang=EN-US
style='font-size:13.5pt;color:red'>EMFC<o:p></o:p></span></u></b></p>
<p align=center style='text-align:center'><span lang=EN-US>(Pseudo State
Feedback)</span></p>
<p style='text-align:justify'><span lang=EN-US>The Explicit Model Following
Control belongs to the group of control laws that solves the general tracking
problem using the LQR (Linear Quadratic Regulator) approach. The purpose of
EMFC (and also of <a href="syn03.htm" title=" IMFC CONTROL help page">IMFC</a>)
is to find a closed loop control law that lets the plant (the current system)
track the dynamics of a reference model.</span></p>
<p style='text-align:justify'><span lang=EN-US>The EMFC technique computes the
optimal state feedback control law that minimizes the cost function </span></p>
<p align=center style='text-align:center'><span lang=EN-US style='color:red'>J
= integral{ (Y - Y</span><span lang=EN-US style='font-size:7.5pt;color:red'>m</span><span
lang=EN-US style='color:red'>)</span><sup><span lang=EN-US style='font-size:
10.0pt;mso-bidi-font-size:12.0pt;color:red'>T</span></sup><span lang=EN-US
style='font-size:7.5pt;color:red'> </span><span lang=EN-US style='color:red'>Q
(Y - Y</span><span lang=EN-US style='font-size:7.5pt;color:red'>m</span><span
lang=EN-US style='color:red'>) + U</span><sup><span lang=EN-US
style='font-size:10.0pt;mso-bidi-font-size:12.0pt;color:red'>T</span></sup><span
lang=EN-US style='color:red'> R U }dt<o:p></o:p></span></p>
<p><span lang=EN-US>with</span></p>
<div align=center>
<table border=1 cellpadding=0 width=451 style='width:270.6pt;mso-cellspacing:
1.2pt;mso-padding-alt:3.0pt 3.0pt 3.0pt 3.0pt'>
<tr>
<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 lang=EN-US 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 lang=EN-US style='color:green'>Model</span></p>
</td>
</tr>
<tr>
<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 lang=EN-US>X<b><sup>'</sup></b>
= A X + B U</span></p>
<p align=center style='text-align:center'><span lang=EN-US>Y = C</span><span
lang=EN-US style='font-size:7.5pt'> </span><span lang=EN-US>X</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 lang=EN-US>X<b><sup>'</sup></b></span><span
lang=EN-US style='font-size:7.5pt'>m</span><span lang=EN-US> = A</span><span
lang=EN-US style='font-size:7.5pt'>m</span><span lang=EN-US> X</span><span
lang=EN-US style='font-size:7.5pt'>m</span><span lang=EN-US> + B</span><span
lang=EN-US style='font-size:7.5pt'>m</span><span lang=EN-US> U</span><span
lang=EN-US style='font-size:7.5pt'>m<o:p></o:p></span></p>
<p align=center style='text-align:center'><span lang=EN-US>Y</span><span
lang=EN-US style='font-size:7.5pt'>m</span><span lang=EN-US> = C</span><span
lang=EN-US style='font-size:7.5pt'>m </span><span lang=EN-US>X</span><span
lang=EN-US style='font-size:7.5pt'>m</span></p>
</td>
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<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US>The vector Y</span><span lang=EN-US style='font-size:7.5pt'>m</span><span
lang=EN-US> = C</span><span lang=EN-US style='font-size:7.5pt'>m </span><span
lang=EN-US>X</span><span lang=EN-US style='font-size:7.5pt'>m </span><span
lang=EN-US>represents the reference command for the plant outputs. The cost
function weighs, by means of the matrix Q, the error between the output vectors
of the two systems. It's important to remember that </span><span lang=EN-US
style='color:red'>the model is really present</span><span lang=EN-US> inside
the control loop so the vector X</span><span lang=EN-US style='font-size:7.5pt'>m</span><span
lang=EN-US> is actually available.</span></p>
<p style='margin:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US>For this reason, the augmented system has state </span><span
lang=EN-US style='color:red'>x = [ X ; X</span><span lang=EN-US
style='font-size:7.5pt;color:red'>m</span><span lang=EN-US style='color:red'> ]</span><span
lang=EN-US> and output </span><span lang=EN-US style='color:red'>e = [ Y - Y</span><span
lang=EN-US style='font-size:7.5pt;color:red'>m</span><span lang=EN-US
style='color:red'> ]</span><span lang=EN-US style='color:windowtext'>, so</span><span
lang=EN-US> the cost function J can be transformed in</span></p>
<p align=center style='text-align:center'><span lang=EN-US style='color:red'>J
= integral{ x</span><sup><span lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:
12.0pt;color:red'>T</span></sup><span lang=EN-US style='color:red'>[ C , -C</span><span
lang=EN-US style='font-size:7.5pt;color:red'>m</span><span lang=EN-US
style='color:red'> ]</span><sup><span lang=EN-US style='font-size:10.0pt;
mso-bidi-font-size:12.0pt;color:red'>T</span></sup><b><sup><span lang=EN-US
style='font-size:7.5pt;color:red'> </span></sup></b><span lang=EN-US
style='color:red'>Q [ C , -C</span><span lang=EN-US style='font-size:7.5pt;
color:red'>m</span><span lang=EN-US style='color:red'> ] x + U</span><sup><span
lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:12.0pt;color:red'>T</span></sup><span
lang=EN-US style='color:red'> R U }dt<o:p></o:p></span></p>
<p style='text-align:justify'><span lang=EN-US>In order that </span><span
lang=EN-US style='color:red'>Y </span><span lang=EN-US style='font-family:Symbol;
color:red;mso-char-type:symbol;mso-symbol-font-family:Symbol'><span
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