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<title>MvTools Help : Synthesis ---> LQG</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'>LQG<o:p></o:p></span></u></b></p>
<p align=center style='text-align:center'><span lang=EN-US>(Output Feedback)</span></p>
<p style='text-align:justify'><span lang=EN-US>This type of synthesis
implements the classical LQG (Linear Quadratic Gaussian) control on the working
model in presence of process and sensor noise</span></p>
<p align=center style='text-align:center'><span lang=EN-US>x'</span><span
lang=EN-US style='font-size:10.0pt'>(t)</span><span lang=EN-US> = A x</span><span
lang=EN-US style='font-size:10.0pt'>(t)</span><span lang=EN-US> + B u</span><span
lang=EN-US style='font-size:10.0pt'>(t)</span><span lang=EN-US> + w</span><span
lang=EN-US style='font-size:10.0pt'>(t)<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:10.0pt'>(t)</span><span lang=EN-US> = C x</span><span
lang=EN-US style='font-size:10.0pt'>(t)</span><span lang=EN-US> + D u</span><span
lang=EN-US style='font-size:10.0pt'>(t)</span><span lang=EN-US> + v</span><span
lang=EN-US style='font-size:10.0pt'>(t)<o:p></o:p></span></p>
<p style='text-align:justify'><span lang=EN-US>where w and v (white gaussians)
are, respectively, a process noise with mean </span><span lang=EN-US
style='font-size:10.0pt;font-family:Symbol'>h</span><span lang=EN-US
style='font-size:7.5pt'>w</span><span lang=EN-US> = 0 and power spectral
density (of the covariance matrix) C</span><span lang=EN-US style='font-size:
7.5pt'>w</span><span lang=EN-US> = W, and a sensor or measurement noise with
mean </span><span lang=EN-US style='font-size:10.0pt;font-family:Symbol'>h</span><span
lang=EN-US style='font-size:7.5pt'>v</span><span lang=EN-US> = 0 and power
spectral density (of the covariance matrix) C</span><span lang=EN-US
style='font-size:7.5pt'>v</span><span lang=EN-US> = V; considering the following
stochastic function</span></p>
<p align=center style='text-align:center'><span lang=EN-US style='color:red'>J
= E{ integral( x</span><b><sup><span lang=EN-US style='font-size:7.5pt;
color:red'>T</span></sup></b><span lang=EN-US style='color:red'> Q x + u</span><b><sup><span
lang=EN-US style='font-size:7.5pt;color:red'>T</span></sup></b><span
lang=EN-US style='color:red'> R u )dt }</span><span lang=EN-US>,</span></p>
<p style='text-align:justify'><span lang=EN-US>where Q and R are the cost
matrices for the state and input vectors, the objective of LQG is to find an
optimal full-state feedback of the </span><span lang=EN-US style='color:green'>estimated
state</span><span lang=EN-US> u</span><span lang=EN-US style='font-size:10.0pt'>(t)</span><span
lang=EN-US> = -K</span><span lang=EN-US style='font-size:7.5pt'>c </span><span
lang=EN-US>x</span><span lang=EN-US style='font-size:7.5pt'>es</span><span
lang=EN-US style='font-size:10.0pt'>(t)</span><span lang=EN-US> which minimizes
J.</span></p>
<p style='text-align:justify'><span lang=EN-US>The optimal state estimate x</span><span
lang=EN-US style='font-size:7.5pt'>es</span><span lang=EN-US style='font-size:
10.0pt'>(t)</span><span lang=EN-US> of x</span><span lang=EN-US
style='font-size:10.0pt'>(t)</span><span lang=EN-US> is produced, using the
sensor measurement y</span><span lang=EN-US style='font-size:10.0pt'>(t)</span><span
lang=EN-US>, by a Kalman estimator with gain matrix K</span><span lang=EN-US
style='font-size:7.5pt'>f</span><span lang=EN-US> and dynamic</span></p>
<p align=center style='text-align:center'><span lang=EN-US style='color:red'>x'</span><span
lang=EN-US style='font-size:7.5pt;color:red'>es</span><span lang=EN-US
style='font-size:10.0pt;color:red'>(t)</span><span lang=EN-US style='color:
red'> = A x</span><span lang=EN-US style='font-size:7.5pt;color:red'>es</span><span
lang=EN-US style='font-size:10.0pt;color:red'>(t) </span><span lang=EN-US
style='color:red'>+ B u</span><span lang=EN-US style='font-size:10.0pt;
color:red'>(t)</span><span lang=EN-US style='color:red'> + K</span><span
lang=EN-US style='font-size:7.5pt;color:red'>f </span><span lang=EN-US
style='color:red'>( y</span><span lang=EN-US style='font-size:10.0pt;
color:red'>(t)</span><span lang=EN-US style='color:red'> - C x</span><span
lang=EN-US style='font-size:7.5pt;color:red'>es</span><span lang=EN-US
style='font-size:10.0pt;color:red'>(t)</span><span lang=EN-US style='color:
red'> - D u</span><span lang=EN-US style='font-size:10.0pt;color:red'>(t)</span><span
lang=EN-US style='color:red'> )</span><span lang=EN-US>,</span></p>
<p><a name=compensator></a><span lang=EN-US>and then the closed loop system
becomes </span></p>
<p align=center style='text-align:center'><span lang=EN-US><img border=0
width=375 height=208 id="_x0000_i1027" src="Immagini\synlqg00.gif"></span></p>
<p style='text-align:justify'><span lang=EN-US>in which the output feedback
controller K</span><span lang=EN-US style='font-size:10.0pt'>of</span><span
lang=EN-US>(s) is composed by the observer and the optimal gain matrix K</span><span
lang=EN-US style='font-size:7.5pt'>c</span><span lang=EN-US>.</span></p>
<p><span lang=EN-US>The LQG 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_i1028" src="Immagini\synlqg01.gif"></span></p>
<p style='text-align:justify'><span lang=EN-US>If a matrix has size less or
equal to 10x10, the relative push 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 all the matrices have been
saved, the user can start the controller computation pressing the button <b>COMPUTE
LQG</b>, which appears on the window, and 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 output-feedback controller K</span><span lang=EN-US style='font-size:
10.0pt'>of </span><span lang=EN-US>has been correctly computed.</span></p>
<p style='text-align:justify'><span lang=EN-US>Note: </span><i><span
lang=EN-US style='font-size:10.0pt'>Q and W must be symmetric and positive semi
definite, while R and V must be symmetric and positive definite.</span></i><span
lang=EN-US style='font-size:10.0pt'><o:p></o:p></span></p>
<p align=center style='text-align:center'><span lang=EN-US><img border=0
width=536 height=5 id="_x0000_i1029" src="Immagini\riga.gif"></span></p>
<p align=center style='text-align:center'><span lang=EN-US><img border=0
width=536 height=5 id="_x0000_i1030" src="Immagini\riga.gif"></span></p>
<p align=center style='text-align:center'><span lang=EN-US><![if !supportEmptyParas]> <![endif]><o:p></o:p></span></p>
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