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<?xml version="1.0" encoding="UTF-8"?>
<!--
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) INRIA -
*
* This file must be used under the terms of the CeCILL.
* This source file is licensed as described in the file COPYING, which
* you should have received as part of this distribution. The terms
* are also available at
* http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
*
-->
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:lang="en" xml:id="fstabst">
<refnamediv>
<refname>fstabst</refname>
<refpurpose>Youla's parametrization of continuous time linear dynmaical systems</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>J = fstabst(P,r)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>P</term>
<listitem>
<para>a continuous time linear dynamical system.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>r</term>
<listitem>
<para>
1x2 row vector, dimension of <literal>P22</literal>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>J</term>
<listitem>
<para>
a continuous time linear dynamical system (with same representation as <literal>P</literal>.
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Parameterization of all stabilizing feedbacks.
</para>
<para>
<literal>P</literal> is partitioned as follows:
</para>
<programlisting role=""><![CDATA[
P=[ P11 P12;
P21 P22]
]]></programlisting>
<para>
(in state-space or transfer form: automatic conversion in state-space is
done for the computations)
</para>
<para>
<literal>r</literal> = size of <literal>P22</literal> subsystem, (2,2) block of <literal>P</literal>
</para>
<programlisting role=""><![CDATA[
J =[J11 J12;
J21 J22]
]]></programlisting>
<para>
<literal>K</literal> is a stabilizing controller for <literal>P</literal> (i.e. <literal>P22</literal>) iff
<literal>K=lft(J,r,Q)</literal> with <literal>Q</literal> stable.
</para>
<para>
The central part of <literal>J</literal> , <literal>J11</literal> is the lqg regulator for <literal>P</literal>
</para>
<para>
This <literal>J</literal> is such that defining <literal>T</literal> as the 2-port <literal>lft</literal> of <literal>P</literal>
and <literal>J</literal> : <literal>[T,rt]=lft(P,r,J,r)</literal> one has that <literal>T12</literal> is inner
and <literal>T21</literal> is co-inner.
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="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)
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="obscont">obscont</link>
</member>
<member>
<link linkend="lft">lft</link>
</member>
<member>
<link linkend="lqg">lqg</link>
</member>
<member>
<link linkend="lqg2stan">lqg2stan</link>
</member>
</simplelist>
</refsection>
<refsection>
<title>History</title>
<revhistory>
<revision>
<revnumber>5.4.0</revnumber>
<revremark>
<literal>Sl</literal> is now checked for
continuous time linear dynamical system. This modification
has been introduced by this <ulink url="http://gitweb.scilab.org/?p=scilab.git;a=commit;h=3d7083daae3339813ba747c8adcda1f9599bb80d">commit</ulink>
</revremark>
</revision>
</revhistory>
</refsection>
</refentry>
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