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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="rowinout">
<refnamediv>
<refname>rowinout</refname>
<refpurpose>inner-outer factorization</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[Inn,X,Gbar]=rowinout(G)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>G</term>
<listitem>
<para>
linear system (<literal>syslin</literal> list) <literal>[A,B,C,D]</literal>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Inn</term>
<listitem>
<para>
inner factor (<literal>syslin</literal> list)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Gbar</term>
<listitem>
<para>
outer factor (<literal>syslin</literal> list)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>X</term>
<listitem>
<para>
row-compressor of <literal>G</literal> (<literal>syslin</literal> list)
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Inner-outer factorization (and row compression) of (<literal>l</literal>x<literal>p</literal>) <literal>G =[A,B,C,D]</literal> with <literal>l>=p</literal>.
</para>
<para>
<literal>G</literal> is assumed to be tall (<literal>l>=p</literal>) without zero on the imaginary axis
and with a <literal>D</literal> matrix which is full column rank.
</para>
<para>
<literal>G</literal> must also be stable for having <literal>Gbar</literal> stable.
</para>
<para>
<literal>G</literal> admits the following inner-outer factorization:
</para>
<programlisting role=""><![CDATA[
G = [ Inn ] | Gbar |
| 0 |
]]></programlisting>
<para>
where <literal>Inn</literal> is square and inner (all pass and stable) and <literal>Gbar</literal>
square and outer i.e:
Gbar is square bi-proper and bi-stable (Gbar inverse is also proper
and stable);
</para>
<para>
Note that:
</para>
<programlisting role=""><![CDATA[
[ Gbar ]
X*G = [ - ]
[ 0 ]
]]></programlisting>
<para>
is a row compression of <literal>G</literal> where <literal>X</literal> = <literal>Inn</literal> inverse is all-pass i.e:
</para>
<programlisting role=""><![CDATA[
T
X (-s) X(s) = Identity
]]></programlisting>
<para>
(for the continuous time case).
</para>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="syslin">syslin</link>
</member>
<member>
<link linkend="colinout">colinout</link>
</member>
</simplelist>
</refsection>
</refentry>
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