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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_113</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_112.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_113</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_114.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 113</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">and the error variance, the Kalman filter is propagating the distribution of the state conditioned on all available measurements. Since the conditional distribution is Gaussian, the mean, mode, and median are identical.</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">3. The conditional mean (hence the Kalman filter estimate) is the minimum of any positive definite quadratic cost function.</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">4. The conditional mean (hence, the Kalman filter estimate) is the minimum of almost any reasonable nondecreasing function of the estimation error [23].</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">5. The Kalman filter, although a linear algorithm, is the optimal (linear or nonlinear) state-estimation algorithm for a linear system.</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">6. The Kalman filter residual error <img src="28365a9638c9372aa7938c0b88db7bde.gif" border="0" alt="C0113-01.GIF" width="68" height="15" /> at time step <i>k</i> is orthogonal (stochastically) to all previous measurements y(<i>k</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"><img src="c6729731b8d9ba4baa8c8beac875c80f.gif" border="0" alt="0113-01.GIF" width="247" height="20" /></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">7. The Kalman filter is asymptotically stable. Therefore, for an observable system, the effects of the initial conditions [x</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">(0) and 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">(0)] decay away and do not affect the solution as <i>k</i> </font><font face="Symbol" size="3">庐</font><font face="Times New Roman, Times, Serif" size="3"></font><font face="Symbol" size="3">楼</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">Since the above properties rely on three assumptions, it is natural to consider the reasonableness of these assumptions. Although most physical systems are in fact nonlinear, the linearity assumption can be locally applied when the system nonlinearities are linearizable and the distance from the linearizing trajectory is small. These conditions are usually valid in navigation problems, especially when aiding information such as GPS is available. The white-noise assumption is also valid, since colored driving noise can be modeled by augmentation of a linear system with white driving noise to the system model. The Gaussian assumption is valid for most driving noise sources as expected based on the <i>Central Limit Theorem</i> [121]. Even when an application involves non-Gaussian noise, it is typical to proceed as if the noise source were Gaussian with appropriately defined first and second moments. In such cases, although a better nonlinear estimator may exist, the Kalman filter will provide the minimum-variance, <i>linear</i>, unbiased state estimate.</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">4.4<br />Performance Analysis</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 previous presentation has assumed that the estimator design model is an exact model of the actual system, which is optimistic. In the design process (see Sec. 4.7) a series of design tradeoffs may be considered. As a result, the estimator design model may not exactly match the real system (i.e., truth model). This section is concerned with methods to study estimator performance under these more realistic conditions.</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_112.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_113</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_114.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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