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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="gfrancis">
<refnamediv>
<refname>gfrancis</refname>
<refpurpose>Francis equations for tracking</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[L,M,T]=gfrancis(Plant,Model)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>Plant</term>
<listitem>
<para> a continuous time dynamical system in state-space representation.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Model</term>
<listitem>
<para> a continuous time dynamical system in state-space representation.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>L,M,T</term>
<listitem>
<para>real matrices</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Given the linear plant:
</para>
<programlisting role=""><![CDATA[
x'= F*x + G*u
y = H*x + J*u
]]></programlisting>
<para>
and the linear model
</para>
<programlisting role=""><![CDATA[
xm'= A*xm + B*um
ym = C*xm + D*um
]]></programlisting>
<para>
the goal is for the plant to track the model i.e. <literal>e = y - ym ---> 0</literal>
while keeping stable the state x(t) of the plant.
<literal>u</literal> is given by feedforward and feedback
</para>
<programlisting role=""><![CDATA[
u = L*xm + M*um + K*(x-T*xm) = [K , L-K*T] *(x,xm) + M*um
]]></programlisting>
<para>
The matrices T,L,M satisfy generalized Francis equations
</para>
<programlisting role=""><![CDATA[
F*T + G*L = T*A
H*T + J*L = C
G*M = T*B
J*M = D
]]></programlisting>
<para>
The matrix <literal>K</literal> must be chosen as stabilizing the pair <literal>(F,G)</literal>
See example of use in directory <literal>demos/tracking</literal>.
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
Plant=ssrand(1,3,5);
[F,G,H,J]=abcd(Plant);
nw=4;nuu=2;A=rand(nw,nw);
st=max(real(spec(A)));A=A-st*eye(A);
B=rand(nw,nuu);C=2*rand(1,nw);D=0*rand(C*B);
Model=syslin('c',A,B,C,D);
[L,M,T]=gfrancis(Plant,Model);
norm(F*T+G*L-T*A,1)
norm(H*T+J*L-C,1)
norm(G*M-T*B,1)
norm(J*M-D,1)
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="lqg">lqg</link>
</member>
<member>
<link linkend="ppol">ppol</link>
</member>
</simplelist>
</refsection>
<refsection>
<title>History</title>
<revhistory>
<revision>
<revnumber>5.4.0</revnumber>
<revremark>
<literal>Sl</literal> is now checked for
continuous time linear dynamical system. This modification
has been introduced by this <ulink url="http://gitweb.scilab.org/?p=scilab.git;a=commit;h=3d7083daae3339813ba747c8adcda1f9599bb80d">commit</ulink>
</revremark>
</revision>
</revhistory>
</refsection>
</refentry>
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