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<?xml version="1.0" encoding="UTF-8"?>
<!--
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) 2008 - 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-en.txt
*
-->
<refentry version="5.0-subset Scilab" xml:id="lmisolver" xml:lang="en"
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">
<info>
<pubdate>$LastChangedDate$</pubdate>
</info>
<refnamediv>
<refname>lmisolver</refname>
<refpurpose>linear matrix inequation solver</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[XLISTF[,OPT]] = lmisolver(XLIST0,evalfunc [,options])</synopsis>
</refsynopsisdiv>
<refsection>
<title>Parameters</title>
<variablelist>
<varlistentry>
<term>XLIST0</term>
<listitem>
<para>a list of containing initial guess (e.g.
<literal>XLIST0=list(X1,X2,..,Xn)</literal>)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>evalfunc</term>
<listitem>
<para>a Scilab function ("external" function with specific
syntax)</para>
<para>The syntax the function <literal>evalfunc</literal> must be as
follows:</para>
<para><literal>[LME,LMI,OBJ]=evalfunct(X)</literal> where
<literal>X</literal> is a list of matrices, <literal>LME,
LMI</literal> are lists and <literal>OBJ</literal> a real
scalar.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>XLISTF</term>
<listitem>
<para>a list of matrices (e.g.
<literal>XLIST0=list(X1,X2,..,Xn)</literal>)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>options</term>
<listitem>
<para>optional parameter. If given, <literal>options</literal> is a
real row vector with 5 components
<literal>[Mbound,abstol,nu,maxiters,reltol]</literal></para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para><literal>lmisolver</literal> solves the following problem:</para>
<para>minimize <literal>f(X1,X2,...,Xn)</literal> a linear function of
Xi's</para>
<para>under the linear constraints: <literal>Gi(X1,X2,...,Xn)=0</literal>
for i=1,...,p and LMI (linear matrix inequalities) constraints:</para>
<para><literal>Hj(X1,X2,...,Xn) > 0</literal> for j=1,...,q</para>
<para>The functions f, G, H are coded in the Scilab function
<literal>evalfunc</literal> and the set of matrices Xi's in the list X
(i.e. <literal>X=list(X1,...,Xn)</literal>).</para>
<para>The function <literal>evalfun</literal> must return in the list
<literal>LME</literal> the matrices <literal>G1(X),...,Gp(X)</literal>
(i.e. <literal>LME(i)=Gi(X1,...,Xn),</literal> i=1,...,p).
<literal>evalfun</literal> must return in the list <literal>LMI</literal>
the matrices <literal>H1(X0),...,Hq(X)</literal> (i.e.
<literal>LMI(j)=Hj(X1,...,Xn)</literal>, j=1,...,q).
<literal>evalfun</literal> must return in <literal>OBJ</literal> the value
of <literal>f(X)</literal> (i.e.
<literal>OBJ=f(X1,...,Xn)</literal>).</para>
<para><literal>lmisolver</literal> returns in <literal>XLISTF</literal>, a
list of real matrices, i. e. <literal>XLIST=list(X1,X2,..,Xn)</literal>
where the Xi's solve the LMI problem:</para>
<para>Defining <literal>Y,Z</literal> and <literal>cost</literal>
by:</para>
<para><literal>[Y,Z,cost]=evalfunc(XLIST)</literal>, <literal>Y</literal>
is a list of zero matrices, <literal>Y=list(Y1,...,Yp)</literal>,
<literal>Y1=0, Y2=0, ..., Yp=0</literal>.</para>
<para><literal> Z</literal> is a list of square symmetric matrices,
<literal> Z=list(Z1,...,Zq) </literal>, which are semi positive definite
<literal> Z1>0, Z2>0, ..., Zq>0</literal> (i.e.
<literal>spec(Z(j))</literal> > 0),</para>
<para><literal>cost</literal> is minimized.</para>
<para><literal>lmisolver</literal> can also solve LMI problems in which
the <literal>Xi's</literal> are not matrices but lists of matrices. More
details are given in the documentation of LMITOOL.</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
//Find diagonal matrix X (i.e. X=diag(diag(X), p=1) such that
//A1'*X+X*A1+Q1 < 0, A2'*X+X*A2+Q2 < 0 (q=2) and trace(X) is maximized
n = 2;
A1 = rand(n,n);
A2 = rand(n,n);
Xs = diag(1:n);
Q1 = -(A1'*Xs+Xs*A1+0.1*eye());
Q2 = -(A2'*Xs+Xs*A2+0.2*eye());
deff('[LME,LMI,OBJ]=evalf(Xlist)','X = Xlist(1); ...
LME = X-diag(diag(X));...
LMI = list(-(A1''*X+X*A1+Q1),-(A2''*X+X*A2+Q2)); ...
OBJ = -sum(diag(X)) ');
X=lmisolver(list(zeros(A1)),evalf);
X=X(1)
[Y,Z,c]=evalf(X)
]]></programlisting>
</refsection>
<refsection>
<title>See Also</title>
<simplelist type="inline">
<member><link linkend="lmitool">lmitool</link></member>
</simplelist>
</refsection>
</refentry>
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