syn12.htm

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

components</span></p>

<p align=center style='text-align:center'><span lang=EN-US style='color:red'>X
= [X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;
color:red'>p1</span><span lang=EN-US style='color:red'> , X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>p2</span><span
lang=EN-US style='color:red'> , X</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:red'>r</span><span lang=EN-US style='color:
red'> ]'<o:p></o:p></span></p>

<p style='text-align:justify'><span lang=EN-US>where both the X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>p1</span><span
lang=EN-US> and X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt'>p2</span><span lang=EN-US> are the states that have to track the
reference command vector, and</span></p>

<p align=center style='margin-bottom:0cm;margin-bottom:.0001pt;text-align:center'><span
lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:12.0pt;color:red'>d (</span><span
lang=EN-US style='color:red'> Z</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:red'>p</span><span lang=EN-US style='color:
red'> ) </span><span lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:
12.0pt;color:red'>/dt</span><span lang=EN-US style='color:red'> = X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>p1</span><span
lang=EN-US style='color:red'><o:p></o:p></span></p>

<p align=center style='margin:0cm;margin-bottom:.0001pt;text-align:center'><span
lang=EN-US><img border=0 width=436 height=166 id="_x0000_i1236"
src="Immagini\synlqs.00.gif"></span></p>

<p style='margin-top:0cm;text-align:justify'><span lang=EN-US>Then, considering
the new state vector</span></p>

<p align=center style='text-align:center'><span lang=EN-US>X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>aug</span><span
lang=EN-US> = [ Z</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt'>p</span><span lang=EN-US> , X</span><span lang=EN-US style='font-size:
8.0pt;mso-bidi-font-size:12.0pt'>p1</span><span lang=EN-US> , X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>p2</span><span
lang=EN-US> , X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt'>r</span><span lang=EN-US> ]'</span></p>

<p style='text-align:justify'><span lang=EN-US>and using the Linear Quadratic
approach, we can compute 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:8.0pt;mso-bidi-font-size:
12.0pt'>aug</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><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:red'>aug</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:8.0pt;
mso-bidi-font-size:12.0pt;color:red'>aug</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='margin-bottom:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US>where Q and R, which are the square matrices of the weighs for the
(augmented) state and input, must be symmetric and respectively positive semi
definite and positive definite. Then </span></p>

<p align=center style='margin-bottom:0cm;margin-bottom:.0001pt;text-align:center'><span
lang=EN-US style='color:blue'>U = - K</span><span lang=EN-US style='font-size:
8.0pt;mso-bidi-font-size:12.0pt;color:blue'>sf</span><span lang=EN-US
style='color:blue'> X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:blue'>aug</span><span lang=EN-US style='color:blue'> = - K</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:blue'>z</span><span
lang=EN-US style='color:blue'> Z</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:blue'>p</span><span lang=EN-US
style='color:blue'> - K</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:blue'>p1</span><span lang=EN-US
style='color:blue'> X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:blue'>p1</span><span lang=EN-US style='color:blue'> - K</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:blue'>p2</span><span
lang=EN-US style='color:blue'> X</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:blue'>p2</span><span lang=EN-US
style='color:blue'> - K</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:blue'>r</span><span lang=EN-US
style='color:blue'> X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:blue'>r</span><span lang=EN-US style='color:blue'><o:p></o:p></span></p>

<p style='margin-bottom:0cm;margin-bottom:.0001pt;text-align:justify'><span
lang=EN-US>and, in the presence of the reference commands for X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>p1</span><span
lang=EN-US> and X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt'>p2</span><span lang=EN-US>, the closed loop system results: </span><span
lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:12.0pt'>(note that the
structure showed below has been simplified because the error vector should be
subdivided in its components relative to X</span><span lang=EN-US
style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>p1</span><span lang=EN-US
style='font-size:10.0pt;mso-bidi-font-size:12.0pt'> and X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>p2</span><span
lang=EN-US style='font-size:10.0pt;mso-bidi-font-size:12.0pt'>)<o:p></o:p></span></p>

<p align=center style='margin-top:0cm;text-align:center'><span lang=EN-US><img
border=0 width=500 height=260 id="_x0000_i1235" src="Immagini\synlqs01.gif"></span></p>

<p style='margin-top:0cm;text-align:justify'><span lang=EN-US>The user can set
the vectors Z</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt'>p</span><span lang=EN-US>, X</span><span lang=EN-US style='font-size:
8.0pt;mso-bidi-font-size:12.0pt'>p1</span><span lang=EN-US>, X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>p2</span><span
lang=EN-US> and X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt'>r</span><span lang=EN-US> (some of them can also be empty) by means of
the check buttons of the first window:</span></p>

<p align=center style='margin-top:0cm;text-align:center'><span lang=EN-US><img
border=0 width=371 height=249 id="_x0000_i1253" src="Immagini\synlqs02.gif"></span></p>

<p style='margin-top:0cm;text-align:justify'><span lang=EN-US>if we consider
the selection made in the example before, the new augmented state vector will
be equal to</span></p>

<p align=center style='margin-top:0cm;text-align:center'><span lang=EN-US
style='color:red'>X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:red'>aug</span><span lang=EN-US style='color:red'> = [ Z</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>3</span><span
lang=EN-US style='color:red'> , Z</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:red'>6</span><span lang=EN-US style='color:
red'> , X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:red'>3</span><span lang=EN-US style='color:red'> , X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>6</span><span
lang=EN-US style='color:red'> , X</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:red'>2</span><span lang=EN-US style='color:
red'> , X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:red'>4</span><span lang=EN-US style='color:red'> , X</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt;color:red'>1</span><span
lang=EN-US style='color:red'> , X</span><span lang=EN-US style='font-size:8.0pt;
mso-bidi-font-size:12.0pt;color:red'>5</span><span lang=EN-US style='color:
red'> , X</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt;color:red'>7</span><span lang=EN-US style='color:red'> ]'<o:p></o:p></span></p>

<p align=center style='margin-top:0cm;text-align:center'><span lang=EN-US>Z</span><span
lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:12.0pt'>p</span><span
lang=EN-US> = [ Z</span><span lang=EN-US style='font-size:8.0pt;mso-bidi-font-size:
12.0pt'>3</span><span lang=EN-US> , Z</span><span lang=EN-US style='font-size:
8.0pt;mso-bidi-font-size:12.0pt'>6</span><span lang=EN-US> ]'<span
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