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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="feedback">
<refnamediv>
<refname>feedback</refname>
<refpurpose>feedback operation</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>Sl=Sl1/.Sl2</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>Sl1,Sl2</term>
<listitem>
<para>
linear systems (<literal>syslin</literal> list) in state-space or transfer form, or ordinary gain matrices.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Sl</term>
<listitem>
<para>
linear system (<literal>syslin</literal> list) in state-space or transfer form
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
The feedback operation is denoted by <literal> /. </literal> (slashdot).
This command returns <literal>Sl=Sl1*(I+Sl2*Sl1)^-1</literal>, i.e the (negative)
feedback of <literal>Sl1</literal> and <literal>Sl2</literal>. <literal>Sl</literal> is the transfer
<literal> v -> y</literal> for <literal> y = Sl1 u</literal>, <literal>u = v - Sl2 y</literal>.
</para>
<para>
The result is the same as <literal>Sl=LFT([0,I;I,-Sl2],Sl1)</literal>.
</para>
<para>
<warning>
Caution: do not use with decimal point (e.g. <literal>1/.1</literal> is ambiguous!)
</warning>
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
S1=ssrand(2,2,3);S2=ssrand(2,2,2);
W=S1/.S2;
ss2tf(S1/.S2)
//Same operation by LFT:
ss2tf(lft([zeros(2,2),eye(2,2);eye(2,2),-S2],S1))
//Other approach: with constant feedback
BigS=sysdiag(S1,S2); F=[zeros(2,2),eye(2,2);-eye(2,2),zeros(2,2)];
Bigclosed=BigS/.F;
W1=Bigclosed(1:2,1:2); //W1=W (in state-space).
ss2tf(W1)
//Inverting
ss2tf(S1*inv(eye()+S2*S1))
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="lft">lft</link>
</member>
<member>
<link linkend="sysdiag">sysdiag</link>
</member>
<member>
<link linkend="augment">augment</link>
</member>
<member>
<link linkend="obscont">obscont</link>
</member>
</simplelist>
</refsection>
</refentry>
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