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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 - F Delebecque
*
* 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="lqg2stan">
<refnamediv>
<refname>lqg2stan</refname>
<refpurpose>LQG to standard problem</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[P,r]=lqg2stan(P22,bigQ,bigR)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>P22</term>
<listitem>
<para>
<literal>syslin</literal> list (nominal plant) in state-space form
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>bigQ</term>
<listitem>
<para>
<literal>[Q,S;S',N]</literal> (symmetric) weighting matrix
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>bigR</term>
<listitem>
<para>
<literal>[R,T;T',V]</literal> (symmetric) covariance matrix
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>r</term>
<listitem>
<para>
<literal>1</literal>x<literal>2</literal> row vector = (number of measurements, number of inputs) (dimension of the 2,2 part of <literal>P</literal>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>P</term>
<listitem>
<para>
<literal>syslin</literal> list (augmented plant)
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
<literal>lqg2stan</literal> returns the augmented plant for linear LQG (H2) controller
design.
</para>
<para>
<literal>P22=syslin(dom,A,B2,C2)</literal> is the nominal plant; it can be in continuous
time (<literal>dom='c'</literal>) or discrete time (<literal>dom='d'</literal>).
</para>
<programlisting role=""><![CDATA[
.
x = Ax + w1 + B2u
y = C2x + w2
]]></programlisting>
<para>
for continuous time plant.
</para>
<programlisting role=""><![CDATA[
x[n+1]= Ax[n] + w1 + B2u
y = C2x + w2
]]></programlisting>
<para>
for discrete time plant.
</para>
<para>
The (instantaneous) cost function is <literal>[x' u'] bigQ [x;u]</literal>.
</para>
<para>
The covariance of <literal>[w1;w2]</literal> is <literal>E[w1;w2] [w1',w2'] = bigR</literal>
</para>
<para>
If <literal>[B1;D21]</literal> is a factor of <literal>bigQ</literal>, <literal>[C1,D12]</literal>
is a factor of <literal>bigR</literal> and <literal>[A,B2,C2,D22]</literal> is
a realization of P22, then <literal>P</literal> is a realization of
<literal>[A,[B1,B2],[C1,-C2],[0,D12;D21,D22]</literal>.
The (negative) feedback computed by <literal>lqg</literal> stabilizes <literal>P22</literal>,
i.e. the poles of <literal>cl=P22/.K</literal> are stable.
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
ny=2;nu=3;nx=4;
P22=ssrand(ny,nu,nx);
bigQ=rand(nx+nu,nx+nu);bigQ=bigQ*bigQ';
bigR=rand(nx+ny,nx+ny);bigR=bigR*bigR';
[P,r]=lqg2stan(P22,bigQ,bigR);K=lqg(P,r); //K=LQG-controller
spec(h_cl(P,r,K)) //Closed loop should be stable
//Same as Cl=P22/.K; spec(Cl('A'))
s=poly(0,'s')
lqg2stan(1/(s+2),eye(2,2),eye(2,2))
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="lqg">lqg</link>
</member>
<member>
<link linkend="lqr">lqr</link>
</member>
<member>
<link linkend="lqe">lqe</link>
</member>
<member>
<link linkend="obscont">obscont</link>
</member>
<member>
<link linkend="h_inf">h_inf</link>
</member>
<member>
<link linkend="augment">augment</link>
</member>
<member>
<link linkend="fstabst">fstabst</link>
</member>
<member>
<link linkend="feedback">feedback</link>
</member>
</simplelist>
</refsection>
</refentry>
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