<?xml version="1.0" encoding="ISO-8859-1" standalone="no"?>
<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>fstabst</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="fstabst"> Youla's parametrization</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>[J]=fstabst(P,r)  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <PARAM>
    <PARAM_INDENT>
      <PARAM_ITEM>
        <PARAM_NAME>P</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: <VERB>syslin</VERB> list (linear system)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>r</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: 1x2 row vector, dimension of <VERB>P22</VERB></SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>J</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: <VERB>syslin</VERB> list (linear system in state-space representation)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
    </PARAM_INDENT>
  </PARAM>
  <DESCRIPTION>
    <P>
    Parameterization of all stabilizing feedbacks.</P>
    <P><VERB>P</VERB> is partitioned as follows:</P>
    <VERBATIM>
<![CDATA[
P=[ P11 P12;
    P21 P22]  
   ]]>
    </VERBATIM>
    <P>
    (in state-space or transfer form: automatic conversion in state-space is
    done for the computations)</P>
    <P><VERB>r</VERB> = size of  <VERB>P22</VERB> subsystem, (2,2) block of <VERB>P</VERB></P>
    <VERBATIM>
<![CDATA[
          J =[ J11 J12;
               J21 J22]
   ]]>
    </VERBATIM>
    <P><VERB>K</VERB> is a stabilizing controller for <VERB>P</VERB> (i.e. <VERB>P22</VERB>) iff 
    <VERB>K=lft(J,r,Q)</VERB> with <VERB>Q</VERB> stable.</P>
    <P>
    The central part of <VERB>J</VERB> , <VERB>J11</VERB> is the lqg regulator for <VERB>P</VERB></P>
    <P>
    This <VERB>J</VERB> is such that defining <VERB>T</VERB> as the 2-port <VERB>lft</VERB> of <VERB>P</VERB>
    and <VERB>J</VERB> : <VERB>[T,rt]=lft(P,r,J,r)</VERB> one has that <VERB>T12</VERB> is inner
    and <VERB>T21</VERB> is co-inner.</P>
  </DESCRIPTION>
  <EXAMPLE>
<![CDATA[
ny=2;nu=3;nx=4;
P22=ssrand(ny,nu,nx);
bigQ=rand(nx+nu,nx+nu);bigQ=bigQ*bigQ';
bigR=rand(nx+ny,nx+ny);bigR=bigR*bigR';
[P,r]=lqg2stan(P22,bigQ,bigR);
J=fstabst(P,r);
Q=ssrand(nu,ny,1);Q('A')=-1;  //Stable Q
K=lft(J,r,Q);
A=h_cl(P,r,K); spec(A)
 ]]>
  </EXAMPLE>
  <SEE_ALSO>
    <SEE_ALSO_ITEM>
      <LINK>obscont</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>lft</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>lqg</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>lqg2stan</LINK>
    </SEE_ALSO_ITEM>
  </SEE_ALSO>
</MAN>
