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<h2><font color="#0000FF">PolyX Browser Watch</font></h2> <p>This site uses frames and JavaScript features, but your browser doesn't support them.<br> To take full advantage of the PolyX Web site, it is highly recommended that you download<br> an up-to-date frames and JavaScript-friendly browser (Microsoft Internet Explorer 4.0<br> or Netscape Navigator 3.0+ / Communicator):</p> <p align="center"><MAP NAME="FrontPageMap"><AREA SHAPE="RECT" COORDS="1, 0, 87, 30" HREF="http://www.microsoft.com/ie/download/" TARGET="_blank"></MAP><a href="_vti_bin/shtml.dll/frm-main-product.htm/map"><img ismap usemap="#FrontPageMap" border="0" height="31" alt="ie.gif (723 bytes)" src="images/ie.gif" width="88"></a>                  <MAP NAME="FrontPageMap1"><AREA SHAPE="RECT" COORDS="3, 0, 87, 30" HREF="http://www.netscape.com/computing/download/index.html" TARGET="_blank"></MAP><a href="_vti_bin/shtml.dll/frm-main-product.htm/map1"><img ismap usemap="#FrontPageMap1" border="0" height="31" alt="ns.gif (786 bytes)" src="images/ns.gif" width="88"></a> <p align="center"><font face="Verdana, Arial, Helvetica, SANS-SERIF" size="4">Polynomial Methods for Systems, Signals and Control</font></p> <div align="center"><center> <table border="0" cellPadding="0" cellSpacing="0" width="100%" height="39"> <tbody> <tr> <td bgColor="#52A9AB" width="100%" height="39" valign="top"><p align="center"><font face="Verdana, Arial, Helvetica, SANS-SERIF" color="#FFFFFF"><strong><big><big>Polynomial Toolbox 2.0 for MATLAB 5</big></big></strong></font></td> </tr> </tbody> </table> </center></div> <p><font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </p> <p>The Polynomial Toolbox 2.0 is a package for systems, signals and control analysis and design based on advanced polynomial methods. It consists of as many as 222 M-files in MATLAB code and is easy to use.</p> <h2><font color="#52A9AB"><b>Background</b></font></h2> <p>Polynomials and polynomial matrices play an important role in linear system theory. They do not only arise because polynomials cannot be avoided, but also because from first principles multivariable linear systems are modeled by sets of differential equations in the input <font face="Times, Times New Roman, SERIF" size="2"><em>u</em></font> and the output <font face="Times, Times New Roman, SERIF" size="2"><em>y</em></font> of the form <dir> <p><img src="images/InfoSheet/Image12.gif" alt="polynomial equations" WIDTH="150" HEIGHT="22"></p> </dir> <p><img height="6" alt="polynomial matrices" src="images/InfoSheet/Image14.gif" width="1">where <font face="Times, Times New Roman, SERIF" size="2"><em>A</em></font> and <font face="Times, Times New Roman, SERIF" size="2"><em>B</em></font> are polynomial matrices in the differential operator <font face="Times, Times New Roman, SERIF" size="2"><em>d/dt </em></font>(or the delay operator for discrete-time systems). Polynomial matrix models do not replace state space and frequency domain descriptions but provide a powerful additional tool. </p> <h2><font color="#52A9AB"><b>Polynomial Matrix Operations</b></font></h2> <p>To define polynomial matrices is as simple as typing </p> <table border="0" width="49%"> <tr> <td width="100%"></font><font face="Courier New" size="2" color="navy">   » A=[1 1+s; 1-s 2*s]; </font></td> </tr> <tr> <td width="100%"><font face="Courier New" size="2" color="navy">   » B=[s 0; 0 s];</font></td> </tr> </table> <p>It is even easier to compute with them</p> <font face="Courier New" size="2"> <table border="0" width="50%"> <tr> </font><td width="39%"><font face="Courier New" size="2" color="#000080">   » A+B</font></td> <td width="61%"></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"><font face="Courier New" size="2" color="#000080">   ans = </font></td> <td width="61%"></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"></td> <td width="61%"><font face="Courier New" size="2" color="#000080">1 + s   1 + s</font></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"></td> <td width="61%"><font face="Courier New" size="2" color="#000080">1 - s      3s</font></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"></td> <td width="61%"></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"><font face="Courier New" size="2" color="#000080">   » A*B</font></td> <td width="61%"></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"><font face="Courier New" size="2" color="#000080">   ans = </font></td> <td width="61%"></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"></td> <td width="61%"><font face="Courier New" size="2" color="#000080">s       s + s^2</font></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"></td> <td width="61%"><font face="Courier New" size="2" color="#000080">s - s^2    2s^2 </font></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"></td> <td width="61%"></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"><font face="Courier New" size="2" color="#000080">   » det(A)</font></td> <td width="61%"></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"><font face="Courier New" size="2" color="#000080">   ans = </font></td> <td width="61%"></td> <font face="Courier New" size="2"> </tr> <tr> </font><td width="39%"></td> <td width="61%"><font face="Courier New" size="2" color="#000080">-1 + 2s + s^2</font></td> <font face="Courier New" size="2"> </tr> </table> <p>For larger matrices the Polynomial Matrix Editor is available.</p> </font> <p align="center"><img src="images/InfoSheet/Image13.gif" alt="polynomial toolbox" WIDTH="488" HEIGHT="237"> </p> <p align="center">Polynomial Matrix Editor</p> <h2><font color="#52A9AB"><b>Algorithms</b></font></h2> <p>The Polynomial Toolbox implements new original algorithms that are fast and reliable. This includes linear matrix polynomial equation solvers based on Sylvester matrices, the application of FFT for rank, determinant and other functions, a variety of new algorithms for spectral factorization and much more.</p> <h2><font color="#52A9AB">Polynomial Matrix Fractions</font></h2> <p>The modeling of single-input single-output LTI systems often leads to differential equations <dir> <p><img src="images/InfoSheet/Image17.gif" alt="polynomial methods" WIDTH="148" HEIGHT="22"></p> </dir> <p>with <em>A</em> and <em>B</em> polynomials or, equivalently, to the transfer function <dir> <p><img src="images/InfoSheet/Image20.gif" alt="polynomials" WIDTH="98" HEIGHT="22"></p> </dir> <p>of the system. For multi-input multi-output systems <em>A</em> and <em>B</em> become polynomial matrices and the transfer matrix is expressed in polynomial matrix fraction (PMF) form <dir> <p><img src="images/InfoSheet/Image22.gif" alt="control design" WIDTH="106" HEIGHT="22"></p> </dir> <p>The Polynomial Toolbox provides many macros for PMFs such as conversion between left and right fractions (lmf2rmf, rmf2lmf), properties testing (isprime, ispropper, isstable), and zero-pole plots (zpplot).</p> <h2><font color="#52A9AB"><b>Polynomial Equations</b></font></h2> <p>Feedback control system design by polynomial methods naturally introduces linear polynomial matrix equations such as <dir> <p><img src="images/InfoSheet/Image23.gif" alt="control systems analysis" WIDTH="156" HEIGHT="17"></p> </dir> <p>where</p> <blockquote> <p><img align="bottom" alt="signal processing" src="images/InfoSheet/Image24.gif" WIDTH="108" HEIGHT="22"></p> </blockquote> <p>is the controller transfer matrix. The Polynomial Toolbox offers numerous solvers for the equations suitably named axbyc, xaybc, axybc, and alike.</p> <p>Next to the linear equations, optimum design problems naturally call for quadratic equations with polynomial matrices such as spectral, <i>J</i>-spectral or even nonsymmetrical factorizations (spf, spcof, fact).</p> <h2><font color="#52A9AB"><b>Analysis</b></font></h2> <p>The Polynomial Toolbox offers simple programs for classical analysis. Its built-in convertors make all the tools of Control System Toolbox available for systems described by PMFs. In addition a wide range of macros is provided to test robustness of various kinds for systems with parametric uncertainties, including single parameter stability margins (stabint), interval polynomials (kharit, khplot), and polytopic uncertainties (ptopplot, ptopex, etc.) </p> <p align="center"><img height="404" src="images/InfoSheet/Image18.gif" alt="control engineers" width="492"></p> <p align="center">Robust stability analysis for interval polynomials</p> <h2><font color="#52A9AB"><b>Design</b></font></h2> <p>Frequency domain solutions of many famous and proven design methods are directly provided, including</p> <div align="center"><center> <table border="0" cellPadding="0" cellSpacing="0" width="73%"> <tbody> <tr> <td width="31%"><font face="Courier New" size="2" color="#000080">plqg</font></td> <td width="69%"><font face="Verdana" size="2">LQG design</font></td> </tr> <tr> <td width="31%"><font face="Courier New" size="2" color="#000080">dsshinf</font></td> <td width="69%"><font face="Verdana" size="2">sub- and optimal H-infinity design</font></td> </tr> <tr> <td width="31%"><font face="Courier New" size="2" color="#000080">mixeds</font></td> <td width="69%"><font face="Verdana" size="2">mixed sensitivity problem</font></td> </tr> <tr> <td width="31%"><font face="Courier New" size="2" color="#000080">debe</font></td> <td width="69%"><font face="Verdana" size="2">deadbeat control</font></td> </tr> <tr> <td width="31%"><font face="Courier New" size="2" color="#000080">pplace</font></td> <td width="69%"><font face="Verdana" size="2">pole placement</font></td> </tr> <tr> <td width="31%"><font face="Courier New" size="2" color="#000080">stab</font></td> <td width="69%"><font face="Verdana" size="2">all stabilizing controllers</font></td> </tr> </tbody> </table> </center></div> <p>Many other design routines can easily be developed based upon the basic polynomial matrix macros.</p> <h2><font color="#52A9AB"><b>Links to Other Packages</b></font></h2> <p>Numerous convertors enable direct cooperation with the <i><font color="#000080">Control System Toolbox</font><font color="#0000ff" face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </font></i><font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"></p> <div align="center"><center> <table border="0" cellPadding="0" cellSpacing="0" width="73%"> <tbody> <tr> </font><td width="31%"><font face="Courier New" size="2" color="#000080">lti2lmf, lti2rmf</font></td> <td width="69%"><font face="Verdana" size="2">LTI objects to PMF </font></td> <font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </tr> <tr> </font><td width="31%"><font face="Courier New" size="2" color="#000080">ss, tf, zpk</font></td> <td width="69%"><font face="Verdana" size="2">PMF to LTI objects</font></td> <font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </tr> <tr> </font><td width="31%"><font face="Courier New" size="2" color="#000080">lmf2dss, rmf2dss</font></td> <td width="69%"><font face="Verdana" size="2">PMF to descriptor systems</font></td> <font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </tr> <tr> </font><td width="31%"><font face="Courier New" size="2" color="#000080">dss2lmf, dss2rmf</font></td> <td width="69%"><font face="Verdana" size="2">descriptor systems to PMF</font></td> <font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </tr> </tbody> </table> </center></div> <p align="justify">and the</font><i><font color="#0000ff" face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </font><font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"><font color="#000080">Symbolic Math Toolbox</font></i></p> <div align="center"><center> <table border="0" cellPadding="0" cellSpacing="0" width="73%"> <tbody> <tr> </font><td width="31%"><font face="Courier New" size="2" color="#000080">sym</font></td> <td width="69%"><font face="Verdana" size="2">polynomial matrix to symbolic format</font></td> <font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </tr> <tr> </font><td width="31%"><font face="Courier New" size="2" color="#000080">sym2pol </font></td> <td width="69%"><font face="Verdana" size="2">symbolic format to polynomial matrix</font></td> <font face="Verdana, Arial, Helvetica, SANS-SERIF" size="2"> </tr> </tbody> </table> </center></div> <p>A <font color="#000080"><em>SIMULINK 2</em> </font>block set for LTI systems described by polynomial matrix fractions is also provided.</p> <h2><font color="#52A9AB"><b>Users</b></font></h2> <p>Users of the Polynomial Toolbox include control engineers involved in control systems analysis and design, communication engineers with an interest in filter design, and university teachers engaged in a variety of courses in linear systems, signals, and control.</p> <hr> <p align="left"><font face="Verdana, Arial, Helvetica, SANS-SERIF" size="1">© Copyright 1998 by PolyX, Ltd. All rights reserved. Polynomial Toolbox is a trademark of PolyX, Ltd. MATLAB and SIMULINK are registered trademarks of The MathWorks, Inc. Other product or brand names are trademarks or registered trademarks of their respective holders.</font></font></p> <table BORDER="0" CELLSPACING="0" CELLPADDING="2" COLS="2" WIDTH="315" BGCOLOR="#96CBC4"> <tr> <td width="163" align="center" colspan="2"><strong><font color="#FFFFFF" face="Verdana" size="3">Product Info - </font><a href="mailto:info@polyx.cz"><font face="Verdana" size="3" color="white">info@polyx.cz</font></a></strong></td> </tr> <tr> <td width="145" align="left"><a href="ProdInfo2column.htm" target="_blank"><font face="Verdana" size="2" color="#0000A0"><strong>Two-column Format</strong></font></a></td> <td width="18" align="right"><a href="documentation.htm"><font face="Verdana" size="2" color="#0000A0"><strong>Documentation</strong></font></a></td> </tr> <tr ALIGN="CENTER" VALIGN="CENTER"> <td width="145" align="left"><a href="download/handout.pdf" target="_blank"><font face="Verdana" size="2" color="#0000A0"><strong><b>Download Info Sheet</b></strong></font></a></td> <td width="18" align="right"><b><font face="Verdana" size="2" color="#0000A0"><strong><a href="ListOfCommands.htm">Commands</a> (list)</strong></font></b></td> </tr> </table>