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<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>Syntax</title>
        <synopsis>
            sl = lin(sim,x0,u0)
            [A,B,C,D] = 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>
