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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="lin">
<refnamediv>
<refname>lin</refname>
<refpurpose>linearization</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[A,B,C,D]=lin(sim,x0,u0)
[sl]=lin(sim,x0,u0)
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>sim</term>
<listitem>
<para>function</para>
</listitem>
</varlistentry>
<varlistentry>
<term>x0, u0</term>
<listitem>
<para>vectors of compatible dimensions</para>
</listitem>
</varlistentry>
<varlistentry>
<term>A,B,C,D</term>
<listitem>
<para>real matrices</para>
</listitem>
</varlistentry>
<varlistentry>
<term>sl</term>
<listitem>
<para>
<literal>syslin</literal> list
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
linearization of the non-linear system <literal>[y,xdot]=sim(x,u)</literal>
around <literal>x0,u0</literal>.
</para>
<para>
<literal>sim</literal> is a function which computes <literal>y</literal> and <literal>xdot</literal>.
</para>
<para>
The output is a linear system (<literal>syslin</literal> list) <literal>sl</literal> or the
four matrices <literal>(A,B,C,D)</literal>
</para>
<para>
For example, if <literal>ftz</literal> is the function passed to ode e.g.
</para>
<programlisting role=""><![CDATA[
[zd]=ftz(t,z,u)
]]></programlisting>
<para>
and if we assume that <literal>y=x</literal>
</para>
<para>
<literal>[z]=ode(x0,t0,tf,list(ftz,u)</literal> compute <literal>x(tf)</literal>.
</para>
<para>
If <literal>simula</literal> is the following function:
</para>
<programlisting role=""><![CDATA[
deff('[y,xd]=simula(x,u)','xd=ftz(tf,x,u); y=x;');
]]></programlisting>
<para>
the tangent linear system <literal>sl</literal> can be obtained by:
</para>
<programlisting role=""><![CDATA[
[A,B,C,D]=lin(simula,z,u)
sl = syslin('c',A,B,C,D,x0)
]]></programlisting>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
deff('[y,xdot]=sim(x,u)','xdot=[u*sin(x);-u*x^2];y=xdot(1)+xdot(2)')
sl=lin(sim,1,2);
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="external">external</link>
</member>
<member>
<link linkend="derivat">derivat</link>
</member>
</simplelist>
</refsection>
</refentry>
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