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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_112</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_111.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_112</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_113.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 112</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">ordinary difference equations in the presence of additive-process driving noise. In this context, the Kalman filter is presented as an optimal combination (which turns out to be linear) between the time-propagated estimate from a previous time instant and the measurement at the present time instant. The optimal combination is dependent on the error variance of both the prior estimate and the current measurement.</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">This presentation method was selected to present the Kalman filter as an optimal estimation algorithm, in a context with which many readers are already familiar. In particular, the reader should understand that if the process-noise variance Q</font><font face="Times New Roman, Times, Serif" size="1"><sub>d</sub></font><font face="Times New Roman, Times, Serif" size="1"><sub></sub></font><font face="Times New Roman, Times, Serif" size="3"> is identically zero, then the Kalman filter reduces to RLS. Alternatively, the Kalman filter can be derived and presented entirely in the more rigorous framework of stochastic processes as an unbiased, minimum-variance, linear stochastic estimator. The results of this method of analysis yield valuable insight into the Kalman filter and its performance [23, 53, 74, 109].</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 sum of the state error variances after measurement correction is the sum of the diagonal elements of P</font><font face="Times New Roman, Times, Serif" size="2"><sup>+</sup></font><font face="Times New Roman, Times, Serif" size="3"> or trace(P</font><font face="Times New Roman, Times, Serif" size="2"><sup>+</sup></font><font face="Times New Roman, Times, Serif" size="3">). Therefore the minimum-variance estimator is determined by the solution of (</font><font face="Symbol" size="3">露</font><font face="Times New Roman, Times, Serif" size="3">/</font><font face="Symbol" size="3">露</font><font face="Times New Roman, Times, Serif" size="3">L) trace (P</font><font face="Times New Roman, Times, Serif" size="2"><sup>+</sup></font><font face="Times New Roman, Times, Serif" size="3">) = 0, where P</font><font face="Times New Roman, Times, Serif" size="2"><sup>+</sup></font><font face="Times New Roman, Times, Serif" size="3"> is defined in Eq. (3.167). The result 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="6eb6c0097194864de2241ffd845e5eb4.gif" border="0" alt="0112-01.GIF" width="399" height="58" /></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="b1e96f036a61e616a6b2dec62da62575.gif" border="0" alt="0112-02.GIF" width="413" height="22" /></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">which is identical to the previously derived expression for the Kalman filter gain. The Kalman gain is a minimum since the second derivative [i.e., H(<i>k</i>)P</font><font face="Times New Roman, Times, Serif" size="2"><sup>-</sup></font><font face="Times New Roman, Times, Serif" size="2"><sup></sup></font><font face="Times New Roman, Times, Serif" size="3">(<i>k</i>)H</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">(<i>k</i>)] is positive semidefinite.</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">A detailed stochastic analysis of the Kalman filter and its properties is not possible herein, since a presentation of the required theory of stochastic processes is tangential to the main directions of this book. Several excellent texts already present such an analysis, for example Refs. 23, 53, 58, 74, and 109. The presentation herein is meant to give the reader the theoretical understanding and practical knowledge of the Kalman filter necessary for a successful implementation. Hopefully, such implementation experience will motivate readers to further their understanding by learning more about stochastic processes. With such studies it is possible to show that, when the process noise and the measurement noise are white and Gaussian, the initial state is Gaussian, and the system is linear, the Kalman filter has the following properties:</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. The Kalman filter estimate is <i>unbiased</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="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. The Kalman filter estimate is the <i>maximum likelihood estimate</i>. Under the given assumptions, the filter state is Gaussian. By propagating the mean</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_111.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_112</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_113.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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