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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="trzeros">
<refnamediv>
<refname>trzeros</refname>
<refpurpose>transmission zeros and normal rank</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[tr]=trzeros(Sl)
[nt,dt,rk]=trzeros(Sl)
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>Sl</term>
<listitem>
<para>
linear system (<literal>syslin</literal> list)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>nt</term>
<listitem>
<para>complex vectors</para>
</listitem>
</varlistentry>
<varlistentry>
<term>dt</term>
<listitem>
<para>real vector</para>
</listitem>
</varlistentry>
<varlistentry>
<term>rk</term>
<listitem>
<para>integer (normal rank of Sl)</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Called with one output argument, <literal>trzeros(Sl)</literal> returns the
transmission zeros of the linear system <literal>Sl</literal>.
</para>
<para>
<literal>Sl</literal> may have a polynomial (but square) <literal>D</literal> matrix.
</para>
<para>
Called with 2 output arguments, <literal>trzeros</literal> returns the
transmission zeros of the linear system <literal>Sl</literal> as <literal>tr=nt./dt</literal>;
</para>
<para>
(Note that some components of <literal>dt</literal> may be zeros)
</para>
<para>
Called with 3 output arguments, <literal>rk</literal> is the normal rank of <literal>Sl</literal>
</para>
<para>
Transfer matrices are converted to state-space.
</para>
<para>
If <literal>Sl</literal> is a (square) polynomial matrix <literal>trzeros</literal> returns the
roots of its determinant.
</para>
<para>
For usual state-space system <literal>trzeros</literal> uses the state-space
algorithm of Emami-Naeni and Van Dooren.
</para>
<para>
If <literal>D</literal> is invertible the transmission zeros are the eigenvalues
of the "<literal>A</literal> matrix" of the inverse system : <literal>A - B*inv(D)*C</literal>;
</para>
<para>
If <literal>C*B</literal> is invertible the transmission zeros are the eigenvalues
of <literal>N*A*M</literal> where <literal>M*N</literal> is a full rank factorization of
<literal>eye(A)-B*inv(C*B)*C</literal>;
</para>
<para>
For systems with a polynomial <literal>D</literal> matrix zeros are
calculated as the roots of the determinant of the system matrix.
</para>
<para>
<warning>
Caution: the computed zeros are not always reliable, in particular
in case of repeated zeros.
</warning>
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
W1=ssrand(2,2,5);trzeros(W1) //call trzeros
roots(det(systmat(W1))) //roots of det(system matrix)
s=poly(0,'s');W=[1/(s+1);1/(s-2)];W2=(s-3)*W*W';[nt,dt,rk]=trzeros(W2);
St=systmat(tf2ss(W2));[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(St);
St1=Q*St*Z;rowf=(Qd(1)+Qd(2)+1):(Qd(1)+Qd(2)+Qd(3));
colf=(Zd(1)+Zd(2)+1):(Zd(1)+Zd(2)+Zd(3));
roots(St1(rowf,colf)), nt./dt //By Kronecker form
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="gspec">gspec</link>
</member>
<member>
<link linkend="kroneck">kroneck</link>
</member>
</simplelist>
</refsection>
</refentry>
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