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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="lft">
<refnamediv>
<refname>lft</refname>
<refpurpose>linear fractional transformation</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[P1]=lft(P,K)
[P1]=lft(P,r,K)
[P1,r1]=lft(P,r,Ps,rs)
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>P</term>
<listitem>
<para>
linear system (<literal>syslin</literal> list), the ``augmented'' plant, implicitly partitioned into four blocks (two input ports and two output ports).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>K</term>
<listitem>
<para>
linear system (<literal>syslin</literal> list), the controller (possibly an ordinary gain).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>r</term>
<listitem>
<para>
1x2 row vector, dimension of <literal>P22</literal>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Ps </term>
<listitem>
<para>
linear system (<literal>syslin</literal> list), implicitly partitioned into four blocks (two input ports and two output ports).
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>rs </term>
<listitem>
<para>
1x2 row vector, dimension of <literal>Ps22</literal>
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Linear fractional transform between two standard plants
<literal>P</literal> and <literal>Ps</literal> in state space form or in
transfer form (<literal>syslin</literal> lists).
</para>
<para>
<literal>r= size(P22) rs=size(P22s)</literal>
</para>
<para>
<literal>lft(P,r, K)</literal> is the linear fractional transform
between <literal>P</literal> and a controller <literal>K</literal>
(<literal>K</literal> may be a gain or a controller in state space form
or in transfer form);
</para>
<para>
<literal>lft(P,K)</literal> is <literal>lft(P,r,K)</literal> with
<literal>r</literal>=size of <literal>K</literal> transpose;
</para>
<para>
<literal>P1= P11+P12*K* (I-P22*K)^-1 *P21</literal>
</para>
<para>
<literal>[P1,r1]=lft(P,r,Ps,rs)</literal> returns the generalized (2
ports) lft of <literal>P</literal> and <literal>Ps</literal>.
</para>
<para>
<literal>P1</literal> is the pair two-port interconnected plant and the
partition of <literal>P1</literal> into 4 blocks in given by
<literal>r1</literal> which is the dimension of the <literal>22</literal>
block of <literal>P1</literal>.
</para>
<para>
<literal>P</literal> and <literal>R</literal> can be PSSDs i.e. may admit a
polynomial <literal>D</literal> matrix.
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
s=poly(0,'s');
P=[1/s, 1/(s+1); 1/(s+2),2/s]; K= 1/(s-1);
lft(P,K)
lft(P,[1,1],K)
P(1,1)+P(1,2)*K*inv(1-P(2,2)*K)*P(2,1) //Numerically dangerous!
ss2tf(lft(tf2ss(P),tf2ss(K)))
lft(P,-1)
f=[0,0;0,1];w=P/.f; w(1,1)
//Improper plant (PID control)
W=[1,1;1,1/(s^2+0.1*s)];K=1+1/s+s
lft(W,[1,1],K); ss2tf(lft(tf2ss(W),[1,1],tf2ss(K)))
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="sensi">sensi</link>
</member>
<member>
<link linkend="augment">augment</link>
</member>
<member>
<link linkend="feedback">feedback</link>
</member>
<member>
<link linkend="sysdiag">sysdiag</link>
</member>
</simplelist>
</refsection>
</refentry>
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