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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="stabil">
<refnamediv>
<refname>stabil</refname>
<refpurpose>stabilization</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>F=stabil(A,B,alfa)
K=stabil(Sys,alfa,beta)
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>A</term>
<listitem>
<para>
square real matrix (<literal>nx x nx</literal>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>B</term>
<listitem>
<para>
real matrix (<literal>nx x nu</literal>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>alfa, beta</term>
<listitem>
<para> real or complex vector (in conjugate pairs) or real number.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>F</term>
<listitem>
<para>
real matrix (<literal>nx x nu</literal>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Sys</term>
<listitem>
<para>
linear system (<literal>syslin</literal> list) (<literal>m</literal> inputs, <literal>p</literal> outputs).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>K</term>
<listitem>
<para>
linear system (<literal>p</literal> inputs, <literal>m</literal> outputs)
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
<literal>F=stabil(A,B,alfa)</literal> returns a gain matrix <literal>F</literal> such that
<literal>A+B*F</literal> is stable if pair <literal>(A,B)</literal> is stabilizable.
Assignable poles are set to <literal>alfa(1),alfa(2),...</literal>.
If <literal>(A,B)</literal> is not stabilizable a warning is given
and assignable poles are set to <literal>alfa(1),alfa(2),...</literal>.
If <literal>alfa</literal> is a number all eigenvalues are set to this
<literal>alfa</literal> (default value is <literal>alfa=-1</literal>).
</para>
<para>
<literal>K=stabil(Sys,alfa,beta)</literal> returns <literal>K</literal>, a compensator for <literal>Sys</literal>
such that <literal>(A,B)</literal>-controllable eigenvalues are set to
<literal>alfa</literal> and <literal>(C,A)</literal>-observable eigenvalues are set to <literal>beta</literal>.
</para>
<para>
All assignable closed loop poles (which are given by the
eigenvalues of <literal>Aclosed=h_cl(Sys,K)</literal> are set to <literal>alfa(i)</literal>'s
and <literal>beta(j)</literal>'s.
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
// Gain:
Sys=ssrand(0,2,5,list('st',2,3,3));
A=Sys('A');B=Sys('B');F=stabil(A,B);
spec(A) //2 controllable modes 2 unstable uncontrollable modes
//and one stable uncontrollable mode
spec(A+B*F) //the two controllable modes are set to -1.
// Compensator:
Sys=ssrand(3,2,5,list('st',2,3,3)); //3 outputs, 2 inputs, 5 states
//2 controllables modes, 3 controllable or stabilizable modes.
K=stabil(Sys,-2,-3); //Compensator for Sys.
spec(Sys('A'))
spec(h_cl(Sys,K)) //K Stabilizes what can be stabilized.
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="st_ility">st_ility</link>
</member>
<member>
<link linkend="contr">contr</link>
</member>
<member>
<link linkend="ppol">ppol</link>
</member>
</simplelist>
</refsection>
</refentry>
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