📄 syn02.htm
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<title>MvTools Help : Synthesis ---> LQR</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'>LQR<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 Linear Quadratic Regulator
design is represented by the computation of the optimal gain matrix K</span><span
lang=EN-US style='font-size:10.0pt'>sf</span><span lang=EN-US> such that the
state feedback law u</span><span lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:
12.0pt'>(t)</span><span lang=EN-US> = -K</span><span lang=EN-US
style='font-size:10.0pt'>sf</span><span lang=EN-US style='font-size:7.5pt'> </span><span
lang=EN-US>x</span><span lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:
12.0pt'>(t)</span><span lang=EN-US> minimizes the cost function J</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:9.0pt;mso-bidi-font-size:
12.0pt;color:red'>T</span></sup><span lang=EN-US style='font-size:10.0pt;
mso-bidi-font-size:12.0pt;color:red'>(t)</span><span lang=EN-US
style='color:red'> Q x</span><span lang=EN-US style='font-size:10.0pt;
mso-bidi-font-size:12.0pt;color:red'>(t)</span><span lang=EN-US
style='color:red'> + u</span><sup><span lang=EN-US style='font-size:9.0pt;
mso-bidi-font-size:12.0pt;color:red'>T</span></sup><span lang=EN-US
style='font-size:10.0pt;mso-bidi-font-size:12.0pt;color:red'>(t)</span><span
lang=EN-US style='color:red'> R u</span><span lang=EN-US style='font-size:10.0pt;
mso-bidi-font-size:12.0pt;color:red'>(t)</span><span lang=EN-US
style='color:red'> }dt<o:p></o:p></span></p>
<p style='text-align:justify'><span lang=EN-US>where x</span><span lang=EN-US
style='font-size:10.0pt;mso-bidi-font-size:12.0pt'>(t)</span><span lang=EN-US>
and u</span><span lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:12.0pt'>(t)</span><span
lang=EN-US> are respectively the state and input vector. Q and R, which are the
square matrices of the weighs for the plant state and input, must be symmetric
and respectively positive semi definite and positive definite.</span></p>
<p style='text-align:justify'><span lang=EN-US>The LQR subsection is made up of
the only following window</span></p>
<p align=center style='text-align:center'><span lang=EN-US><img border=0
width=371 height=248 id="_x0000_i1027" src="Immagini\synflqr00.gif"></span></p>
<p style='text-align:justify'><span lang=EN-US>If a matrix has size less or
equal to 10x10, the relative button allows to visualize all its coefficients,
and the user is able to modify any of them; instead if a matrix can't be
displayed on the window, it's possible to fix only the value T such that </span><i><span
lang=EN-US style='color:green'>mat_name </span></i><span lang=EN-US
style='color:green'>= T*eye(size(<i>mat_name</i>))</span><span lang=EN-US>. The
changes introduced become effective only through the function associated with
the <b>SAVE MATRIX</b> button that, if the matrix is correct, save it modifying
the string of the relative button in <b>[<i>mat_name</i>]</b>.</span></p>
<p style='text-align:justify'><span lang=EN-US>When both the matrices have been
saved, the button <b>COMPUTE LQR</b> appears so the user can start the
controller computation pressing it. The activation of the two buttons <a
href="eva01.htm" title=" EVALUATION SECTION help page ">EVALUATION</a> and <a
href="sim01.htm" title=" SIMULATION SECTION help page">SIMULATION</a> notifies
that the state feedback gain matrix K</span><span lang=EN-US style='font-size:
10.0pt'>sf</span><span lang=EN-US> has been correctly computed and <a
href="syn01.htm#pseudo" title=" PSEUDO STATE FEEDBACK help note">remapped</a>
in the output feedback controller K</span><span lang=EN-US style='font-size:
10.0pt'>of</span><span lang=EN-US>.</span></p>
<p align=center style='text-align:center'><span lang=EN-US><img border=0
width=536 height=4 id="_x0000_i1028" src="Immagini\riga.gif"></span></p>
<p align=center style='text-align:center'><span lang=EN-US><img border=0
width=536 height=4 id="_x0000_i1029" src="Immagini\riga.gif"></span></p>
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