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<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>dt_ility</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="dt_ility"> detectability test</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>[k, [n [,U [,Sld ] ] ]]=dt_ility(Sl [,tol])  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <PARAM>
    <PARAM_INDENT>
      <PARAM_ITEM>
        <PARAM_NAME>Sl</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: linear system (<VERB>syslin</VERB> list)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>n</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: dimension of unobservable subspace</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>k</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: dimension of unstable, unobservable subspace (<VERB> k&lt;=n</VERB>).</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>U</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: orthogonal matrix</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>Sld</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: linear system (<VERB>syslin</VERB> list)</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
      <PARAM_ITEM>
        <PARAM_NAME>tol</PARAM_NAME>
        <PARAM_DESCRIPTION>
          <SP>: threshold for controllability test.</SP>
        </PARAM_DESCRIPTION>
      </PARAM_ITEM>
    </PARAM_INDENT>
  </PARAM>
  <DESCRIPTION>
    <P>
    Detectability test for <VERB>sl</VERB>, a linear system in state-space
    representation. 
    <VERB>U</VERB> is a basis whose <VERB>k</VERB> first columns span the
    unstable, unobservable subspace of <VERB>Sl</VERB> (intersection
    of unobservable subspace of <VERB>(A,C)</VERB> and unstable subspace
    of <VERB>A</VERB>). Detectability means <VERB>k=0</VERB>.</P>
    <P><VERB>Sld = (U'*A*U,U'*B,C*U,D)</VERB> displays the &quot;detectable part&quot;
    of <VERB>Sl=(A,B,C,D)</VERB>, i.e.</P>
    <VERBATIM>
<![CDATA[
         [*,*,*]
U'*A*U = [0,*,*]
         [0,0,*]
 
C*U = [0,0,*]
   ]]>
    </VERBATIM>
    <P>
    with <VERB>(A33,C3)</VERB> observable (dimension <VERB>nx-n</VERB>), <VERB>A22</VERB> stable 
    (dimension <VERB>n-k</VERB>) and <VERB>A11</VERB> unstable (dimension <VERB>k</VERB>).</P>
  </DESCRIPTION>
  <EXAMPLE>
<![CDATA[
A=[2,1,1;0,-2,1;0,0,3];
C=[0,0,1];
X=rand(3,3);A=inv(X)*A*X;C=C*X;
W=syslin('c',A,[],C);
[k,n,U,W1]=dt_ility(W);
W1("A")
W1("C")
 ]]>
  </EXAMPLE>
  <SEE_ALSO>
    <SEE_ALSO_ITEM>
      <LINK>contr</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>st_ility</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>unobs</LINK>
    </SEE_ALSO_ITEM>
    <SEE_ALSO_ITEM>
      <LINK>stabil</LINK>
    </SEE_ALSO_ITEM>
  </SEE_ALSO>
</MAN>
