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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd"> <html> <head> <title>page_96</title> <link rel="stylesheet" href="reset.css" type="text/css" media="all"> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> </head> <body> <table summary="top nav" border="0" width="100%"> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_95.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_96</strong></td> <td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_97.html">next page ></a></td> </tr> <tr> <td id="ebook_page" align="left" colspan="3" style="background: #ffffff; padding: 20px;"> <table border="0" width="100%" cellpadding="0"><tr><td align="center"> <table border="0" cellpadding="2" cellspacing="0" width="100%"><tr><td align="left"></td> <td align="right"></td> </tr></table></td></tr><tr><td align="left"><p></p><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="right"><font face="Times New Roman, Times, Serif" size="2" color="#FF0000">Page 96</font></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">for an in-depth discussion of <i>stochastic processes</i> and <i>optimal estimation theory</i>.</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="17"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">4.1<br />Weighted Least Squares (WLS)</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">In linear least-squares estimation, the problem setting is an available set of noisy measurements <img src="27c3843742197fec763bf5648a5ad39f.gif" border="0" alt="C0096-01.GIF" width="72" height="18" /> where <img src="c250496dbd388a5c33bc5a7ca1372bbe.gif" border="0" alt="C0096-02.GIF" width="177" height="19" />. The variable x </font><font face="Symbol" size="3">脦</font><font face="Times New Roman, Times, Serif" size="3"> <i>R</i></font><i><font face="Times New Roman, Times, Serif" size="2"><sup>n</sup></font></i><font face="Times New Roman, Times, Serif" size="2"><sup></sup></font><font face="Times New Roman, Times, Serif" size="3"> is assumed to be unknown. For each <img src="7c8c1d6465bafea961874ed8fbe9e084.gif" border="0" alt="C0096-03.GIF" width="71" height="21" /> is assumed to be known, and <i>n</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>i</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3"> is a random variable representing measurement noise. Define n = [<i>n</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">,聽.聽.聽.聽, <i>n</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font><font face="Times New Roman, Times, Serif" size="3">]. The vector n is assumed to have <i>E</i><n> = 0 and <i>E</i><nn</font><font face="Times New Roman, Times, Serif" size="2"><sup><i>T</i></sup></font><font face="Times New Roman, Times, Serif" size="3">> = R </font><font face="Symbol" size="3">脦</font><font face="Times New Roman, Times, Serif" size="3"> <i>R</i></font><i><font face="Times New Roman, Times, Serif" size="2"><sup>m</sup></font></i><font face="Times New Roman, Times, Serif" size="2"><sup></sup></font><sup><font face="Symbol" size="2">麓</font><font face="Times New Roman, Times, Serif" size="2"><i>m</i></font></sup><font face="Times New Roman, Times, Serif" size="2"></font><font face="Times New Roman, Times, Serif" size="3">, where R is positive definite (see App. B).</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">The objective is to find an estimate <img src="f090754f50daa0e52ebe28b2762c1c4a.gif" border="0" alt="XCIRC.GIF" width="15" height="16" /> to minimize</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="760e8cbc22030e4d62dc2bd9f784813b.gif" border="0" alt="0096-01.GIF" width="337" height="34" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">where W </font><font face="Symbol" size="3">脦</font><font face="Times New Roman, Times, Serif" size="3"> <i>R</i></font><i><font face="Times New Roman, Times, Serif" size="2"><sup>m</sup></font></i><font face="Times New Roman, Times, Serif" size="2"><sup></sup></font><sup><font face="Symbol" size="2">麓</font><font face="Times New Roman, Times, Serif" size="2"><i>m</i></font></sup><font face="Times New Roman, Times, Serif" size="2"></font><font face="Times New Roman, Times, Serif" size="3"> is a positive definite matrix, <img src="e90bfe5d5bed20f97a2514abceb910ac.gif" border="0" alt="ytildequation.GIF" width="111" height="22" /> and <img src="ba8c85f9455fad55cd67e5e626163978.gif" border="0" alt="C0096-04.GIF" width="129" height="21" />. This cost function can be motivated in two contexts. First, in a deterministic sense, it is desirable to minimize some norm of the error between the measurements <img src="b483774d122316adcadf7f8da4665204.gif" border="0" alt="YTILDE.GIF" width="16" height="18" /> and the estimated measurements <img src="3ce65568a258817ddfb3cb984c32fd3f.gif" border="0" alt="C0096-05.GIF" width="53" height="19" />. The cost function is a weighted two norm. In a general approach, some measurements may be known to be more accurate than others. The measurement accuracy is characterized by R</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3">. Therefore it is natural to select</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="f9f9e87ac11d6cc8f92adbf8a9d7c341.gif" border="0" alt="0096-02.GIF" width="271" height="21" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">to give the least weighting to the most uncertain measurements. Second, in a probabilistic sense, given the assumption that the measurement noise is normally distributed, the probability density of x, given the measurements <img src="b483774d122316adcadf7f8da4665204.gif" border="0" alt="YTILDE.GIF" width="16" height="18" />, is</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="86569ffb00dabddb6f90aa8cf71ed3a8.gif" border="0" alt="0096-03.GIF" width="373" height="39" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">The value of x that maximizes <i>p</i>(x:<img src="b483774d122316adcadf7f8da4665204.gif" border="0" alt="YTILDE.GIF" width="16" height="18" />) is the maximum-likelihood estimate of x and coincides with the minimum of <i>J</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>WLS</sub></font><font face="Times New Roman, Times, Serif" size="3"> when W = R</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3">.</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">For <i>m</i> < <i>n</i>, the problem will be underdetermined and there will be an infinite number of solutions. If <i>m </i>> <i>n</i> then the problem may be overdetermined, in which case no exact solution will exist. In the latter case, the objective function in Eq. (4.1) will result in the estimate <img src="f090754f50daa0e52ebe28b2762c1c4a.gif" border="0" alt="XCIRC.GIF" width="15" height="16" /> that minimizes the weighted two norm of </font><font face="Symbol" size="3">d</font><font face="Times New Roman, Times, Serif" size="3"><i>y</i> = [</font><font face="Symbol" size="3">d</font><font face="Times New Roman, Times, Serif" size="3"><i>y</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">,聽.聽.聽.聽, </font><font face="Symbol" size="3">d</font><font face="Times New Roman, Times, Serif" size="3"><i>y</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3">]</font><font face="Times New Roman, Times, Serif" size="2"><sup><i>T</i></sup></font><font face="Times New Roman, Times, Serif" size="3">, where <img src="69132b9bce4b905af3ae594bbb2fc0f7.gif" border="0" alt="c0096-06.GIF" width="88" height="16" />.</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">In the following text we build up to the Kalman filter by discussing sequentially the following set of questions:</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">1. What is the formula for the WLS estimate of x [i.e., the minimum of Eq. (4.1)]?</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">2. Given that the WLS estimate <img src="59d309f86810e74b55acdfae11c7d758.gif" border="0" alt="XCIRCM.GIF" width="20" height="16" /> has been calculated for a set of <i>m</i> measurements, if an additional measurement <i>y</i></font><i><font face="Times New Roman, Times, Serif" size="1"><sub>m</sub></font></i><font face="Times New Roman, Times, Serif" size="1"><sub>+1</sub></font><font face="Times New Roman, Times, Serif" size="3"> is taken, can <img src="59d309f86810e74b55acdfae11c7d758.gif" border="0" alt="XCIRCM.GIF" width="20" height="16" /> be adjusted to efficiently produce the new WLS estimate <img src="ce2972e199967fc7e99ac49c68c7aa6f.gif" border="0" alt="XCIRCMPLUS.GIF" width="34" height="17" />?</font><font face="Times New Roman, Times, Serif" size="3" color="#FFFF00"></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr></table><p><font size="0"></font></p>聽 </td> </tr> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_95.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_96</strong></td> <td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_97.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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