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<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>st_ility</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="st_ility"> stabilizability test</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>[ns, [nc, [,U [,Slo] ]]]=st_ility(Sl [,tol])  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <PARAM>
    <PARAM_INDENT>
      <PARAM_ITEM>
        <PARAM_NAME>Sl</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: <VERB>syslin</VERB> list (linear system)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>ns</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>:  integer (dimension of stabilizable subspace)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>nc</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>:  integer (dimension of controllable subspace <VERB>nc &lt;= ns</VERB>)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>U</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: basis such that its <VERB>ns</VERB> (resp. <VERB>nc</VERB>) first components span the stabilizable (resp. controllable) subspace</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>Slo</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: a linear system (<VERB>syslin</VERB> list)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>tol</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: threshold for controllability detection (see contr)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
    </PARAM_INDENT>
  </PARAM>
  <DESCRIPTION>
    <P><VERB> Slo=( U'*A*U, U'*B, C*U, D, U'*x0 )</VERB> (<VERB>syslin</VERB> list)
    displays the stabilizable form of <VERB>Sl</VERB>. Stabilizability means
    <VERB>ns=nx</VERB> (dim. of <VERB>A</VERB> matrix).</P>
    <VERBATIM>
<![CDATA[
         [*,*,*]            [*]
U'*A*U = [0,*,*]     U'*B = [0]
         [0,0,*]            [0]
   ]]>
    </VERBATIM>
    <P>
    where <VERB> (A11,B1) </VERB> (dim(A11)= <VERB>nc</VERB>) is controllable and <VERB>A22</VERB> 
    (dim(A22)=<VERB>ns-nc</VERB>) is stable.
    &quot;Stable&quot; means real part of eigenvalues negative for a continuous
    linear system, and magnitude of eigenvalues lower than one for a
    discrete-time system (as defined by <VERB>syslin</VERB>).</P>
  </DESCRIPTION>
  <EXAMPLE>
<![CDATA[
A=diag([0.9,-2,3]);B=[0;0;1];Sl=syslin('c',A,B,[]);
[ns,nc,U]=st_ility(Sl);
U'*A*U
U'*B
[ns,nc,U]=st_ility(syslin('d',A,B,[]));
U'*A*U
U'*B
 ]]>
  </EXAMPLE>
  <SEE_ALSO>
    <SEE_ALSO_ITEM>
      <LINK>dt_ility</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>contr</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>stabil</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>ssrand</LINK>
    </SEE_ALSO_ITEM>
  </SEE_ALSO>
  <AUTHOR>S. Steer INRIA 1988</AUTHOR>
</MAN>
