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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_66</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_65.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_66</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_67.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 66</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">To show that all equivalent state-space representations have the same input/output responses, consider the transfer function corresponding to Eqs. (3.45) and (3.46):</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="47c2c653e34dc476fa1f2c2479549690.gif" border="0" alt="0066-01.GIF" width="364" height="116" /></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 the property (Is-PFP</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3">)</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3"> = P(Is-F)</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3">p</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3">, which can be demonstrated by direct multiplication, has been used.</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"><i>3.3.3<br />State-Transition Matrix Properties and Calculation</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"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td> <td colspan="3" height="12"></td> <td rowspan="5"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="2">The homogeneous part of Eqs. (3.41) 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"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></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="4daed39384701b7829e719bd1467c65a.gif" border="0" alt="0066-02.GIF" width="287" height="17" /></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"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td> <td colspan="3" height="12"></td> <td rowspan="5"><img src="f7703d30723feae8ee39d997c6419c20.gif" border="0" width="24" height="1" alt="f7703d30723feae8ee39d997c6419c20.gif" /></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="2">Definition 3.3.2 A matrix function </font><font face="Symbol" size="2">F</font><font face="Times New Roman, Times, Serif" size="2">(<i>t</i>) : <i>R</i><sup>1</sup></font><font face="Symbol" size="2">庐</font><font face="Times New Roman, Times, Serif" size="2"> <i>R<sup>n</sup></i><sup></sup></font><sup><font face="Symbol" size="2">麓</font><font face="Times New Roman, Times, Serif" size="2"><i>n</i></font></sup><font face="Times New Roman, Times, Serif" size="2"> is the fundamental solution of Eq. (3.50) on <i>t</i> </font><font face="Symbol" size="2">脦</font><font face="Times New Roman, Times, Serif" size="2"> [0, <i>T</i>] if and only if <img src="75008f98778337da0746e1a42badbf43.gif" border="0" alt="C0066-01.GIF" width="236" height="17" />.</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 fundamental solution </font><font face="Symbol" size="3">F</font><font face="Times New Roman, Times, Serif" size="3"> is important since it will serve as the basis for finding the solution to both Eqs. (3.41) and (3.50).</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">If x(t) = </font><font face="Symbol" size="3">F</font><font face="Times New Roman, Times, Serif" size="3">(<i>t</i>)x(0), then x(t) satisfies the initial value problem corresponding to Eq. (3.50) with initial condition x(0):</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="d45dc5e1a3e0f33dad7deb3bd9e470c6.gif" border="0" alt="0066-03.GIF" width="122" height="106" /></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">If </font><font face="Symbol" size="3">F</font><font face="Times New Roman, Times, Serif" size="3">(<i>t</i>) is nonsingular, then </font><font face="Symbol" size="3">F</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3">(<i>t</i>)x(<i>t</i>) = x(0) and</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="4886b7ed39bb864eaf9b1424772eb196.gif" border="0" alt="0066-04.GIF" width="311" height="67" /></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 </font><font face="Symbol" size="3">F</font><font face="Times New Roman, Times, Serif" size="3">(</font><font face="Symbol" size="3"><i>t</i></font><font face="Times New Roman, Times, Serif" size="3">, <i>t</i>) = </font><font face="Symbol" size="3">F</font><font face="Times New Roman, Times, Serif" size="3">(</font><font face="Symbol" size="3"><i>t</i></font><font face="Times New Roman, Times, Serif" size="3">)</font><font face="Symbol" size="3">F</font><font face="Times New Roman, Times, Serif" size="2"><sup>-1</sup></font><font face="Times New Roman, Times, Serif" size="3">(<i>t</i>) is called the <i>state-transition matrix</i>. The state-transition matrix transforms the solution at time <i>t</i> to time </font><font face="Symbol" size="3"><i>t</i></font><font face="Times New Roman, Times, Serif" size="3">. The state-transition</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_65.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_66</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_67.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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