<BODY bgcolor="#aaeebb"> <CENTER> <H3>An Example: Simple LQG Problem</H3> </CENTER> <B><u>Problem</u></B>: (a simple Linear-Quadratic Problem)<BR> <P> <CENTER> Find <i>u</i><sub>t</sub> = <i>g</i>(<i>x<sub>t</sub></i>) to minimize<BR> <BR> subject to <i>x</i><sub><i>t</>+1</sub> = -<i>x<sub>t</sub></i> + <i>u<sub>t</sub></i> + <i>w<sub>t</sub></i> <BR> </CENTER> where <i>w<sub>t</sub></i> is GWN(0,25) (gaussian white noise).<BR> <P> <B><u>Strategy Space G</u></B>: (enormous search space)<BR> <P> <CENTER> Quantize <i>x</i> to <i>n</i> values and <i>u</i> to <i>m</i> values ==&gt; |<b>G</b>| = <i>m<sup>n</sup></i><BR> e.g., <i>n</i> = <i>m</i> = 100 , |<b>G</b><sub>100</sub>| = 100<sup>100</sup>, <i>n</i> = 5, |<b>G</b><sub>5</sub>| = 100<sup>5</sup> </CENTER> <P> <B><u>Known Optimal Solution <i>g</i>*</u></B>: (optimal solution is linear)<BR> <P> <CENTER> <i>u<sub>t</sub></i> = 0.618<i>x<sub>t</sub></>; <i>J</i>*= 40.45 </CENTER> <P> <B><u>Plan</u></b>:<BR> <CENTER> Sample 5,000 g's UNIFORMLY in <b>G</b><BR> </CENTER> <BR> <tt> <A HREF="Slide02.html" target="_parent">To Next Slide</A> / <A HREF="../OOTOC.html" target="_parent">Table of Content</A> </tt> <BR> <hr> <hr> <br> <P> This discussion also brings forth another important concept in OO, namely, the Performance Density/Distribution Function (PDF). This is a plot of the sample cumulative distribution of performances observed. We display in the next slide the PDF of the LQG problem above sampled in the spaces <b>G</b><sub>5</sub>, <b>G</b><sub>10</sub>, <b>G</b><sub>25</sub>, <b>G</b><sub>50</sub>, <b>G</b><sub>100</sub>. The control law with the best observed performance in <b>G</b><sub>10</sub> and <b>G</b><sub>100</sub> are also display along side. The results are very instructive.<BR> <BR> </BODY> </HTML>