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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:ns4="http://www.w3.org/1999/xhtml"
          xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook"
          xmlns:scilab="http://www.scilab.org" xml:id="routh_t" xml:lang="en">
    <refnamediv>
        <refname>routh_t</refname>
        <refpurpose>Routh's table</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>
            [r [,num] ] = routh_t(p)
            [r [,num] ] = routh_t(h ,kp)
            r = routh_t(h, k)
            r = routh_t(h, k, normalized)
        </synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>p</term>
                <listitem>
                    <para>a real polynomial</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>h</term>
                <listitem>
                    <para>a real SISO transfer system</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>k</term>
                <listitem>
                    <para>a real polynomial or a scalar</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>kp</term>
                <listitem>
                    <para>a scalar: proportional controller constant</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>normalized</term>
                <listitem>
                    <para>a boolean (%t (default value) or %f)</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>r</term>
                <listitem>
                    <para>a matrix or a list: Routh array elements</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>num</term>
                <listitem>
                    <para>a scalar: the number of sign changes</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            <literal>r=routh_t(p)</literal> computes Routh's table of the
            polynomial <literal>p</literal>.
        </para>
        <para>
            <literal>r=routh_t(h,k)</literal> computes Routh's table of
            denominator of the system described by transfer matrix SISO
            <literal>h</literal> with the feedback by the gain
            <literal>k</literal>.
        </para>
        <para>
            If <literal>k=poly(0,'k')</literal> we will have a polynomial or
            a rational matrix with dummy variable <literal>k</literal>,
            formal expression of the Routh table.
        </para>
        <para>
            If <literal>normalized=%f</literal> we will have a polynomial matrix
            with non normalized elements. In the other case, we will have a rational
            and normalized matrix.
        </para>
        <para>
            The second output argument <literal>num</literal> returns the number of sign changes
            in the first column of Routh's table. Note that, this argument value will only have sense
            when the table is normalized.
        </para>
        <para>
            <note>
                Hint: If <literal>h=1/p</literal>, then <literal>routh_t(h, kp)</literal> is equivalent to
                <literal>routh_t(p+kp)</literal> .
            </note>
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
s=%s;
P=5*s^3-10*s^2+7*s+20;
routh_t(P)

// Transfer function with formal feedback, normalized case
routh_t((1+s)/P,poly(0,'k'))

// Transfer function with formal feedback, non normalized case
routh_t((1+s)/P,poly(0,'k'),%f)

// One of the coefficients in the polynomial equals zero
P1=2*s^3-24*s+32;
routh_t(P1)

// A row full of zeros
P2=s^4-6*s^3+10*s^2-6*s+9;
routh_t(P2)

// The number of roots in the rhp could be retrieved as the second output argument
P3=5*s^3-10*s^2+7*s;
[r,num]=routh_t(1/P3,20)
if num==0
   disp("System is stable")
else
   mprintf("There is %g sign changes in entries of first column.\nTherefore, system is unstable.", num)
end

//
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="roots">roots</link>
            </member>
            <member>
                <link linkend="spec">spec</link>
            </member>
            <member>
                <link linkend="evans">evans</link>
            </member>
            <member>
                <link linkend="kpure">kpure</link>
            </member>
            <member>
                <link linkend="krac2">krac2</link>
            </member>
        </simplelist>
    </refsection>
    <refsection>
        <title>Bibliography</title>
        <para>
            <ulink url="http://controls.engin.umich.edu/wiki/index.php/RouthStability">http://controls.engin.umich.edu/wiki/index.php/RouthStability</ulink>
        </para>
        <para>
            <ulink url="http://www.jdotec.net/s3i/TD_Info/Routh/Routh.pdf">http://www.jdotec.net/s3i/TD_Info/Routh/Routh.pdf</ulink>
        </para>
        <para>Comments on the Routh-Hurwitz criterion, Shamash, Y.,Automatic
            Control, IEEE T.A.C Volume 25, Issue 1, Feb 1980 Page(s): 132 - 133
        </para>
    </refsection>
    <refsection>
        <title>History</title>
        <revhistory>
            <revision>
                <revnumber>5.5.0</revnumber>
                <revremark>New output argument added: num (SEP #104).</revremark>
            </revision>
            <revision>
                <revnumber>5.4.0</revnumber>
                <revremark>New input argument added: normalized (SEP #89).</revremark>
            </revision>
        </revhistory>
    </refsection>
</refentry>
