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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="cainv">
<refnamediv>
<refname>cainv</refname>
<refpurpose>Dual of abinv</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[X,dims,J,Y,k,Z]=cainv(Sl,alfa,beta,flag)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>Sl</term>
<listitem>
<para>
<literal>syslin</literal> list containing the matrices <literal>[A,B,C,D]</literal>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>alfa</term>
<listitem>
<para>real number or vector (possibly complex, location of closed loop poles)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>beta</term>
<listitem>
<para>real number or vector (possibly complex, location of closed loop poles)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>flag</term>
<listitem>
<para>
(optional) character string <literal>'ge'</literal> (default) or <literal>'st'</literal> or <literal>'pp'</literal>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>X</term>
<listitem>
<para>orthogonal matrix of size nx (dim of state space).</para>
</listitem>
</varlistentry>
<varlistentry>
<term>dims</term>
<listitem>
<para>
integer row vector <literal>dims=[nd1,nu1,dimS,dimSg,dimN]</literal> (5 entries, nondecreasing order).If <literal>flag='st'</literal>, (resp. <literal>'pp'</literal>), <literal>dims</literal> has 4 (resp. 3) components.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>J</term>
<listitem>
<para>real matrix (output injection)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Y</term>
<listitem>
<para>orthogonal matrix of size ny (dim of output space).</para>
</listitem>
</varlistentry>
<varlistentry>
<term>k</term>
<listitem>
<para>
integer (normal rank of <literal>Sl</literal>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Z</term>
<listitem>
<para>
non-singular linear system (<literal>syslin</literal> list)
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
<literal>cainv</literal> finds a bases <literal>(X,Y)</literal> (of state space and output space resp.)
and output injection matrix <literal>J</literal> such that the matrices of Sl in
bases (X,Y) are displayed as:
</para>
<programlisting role=""><![CDATA[
[A11,*,*,*,*,*] [*]
[0,A22,*,*,*,*] [*]
X'*(A+J*C)*X = [0,0,A33,*,*,*] X'*(B+J*D) = [*]
[0,0,0,A44,*,*] [0]
[0,0,0,0,A55,*] [0]
[0,0,0,0,0,A66] [0]
Y*C*X = [0,0,C13,*,*,*] Y*D = [*]
[0,0,0,0,0,C26] [0]
]]></programlisting>
<para>
The partition of <literal>X</literal> is defined by the vector
<literal>dims=[nd1,nu1,dimS,dimSg,dimN]</literal> and the partition of <literal>Y</literal>
is determined by <literal>k</literal>.
</para>
<para>
Eigenvalues of <literal>A11</literal> <literal>(nd1 x nd1)</literal> are unstable.
Eigenvalues of <literal>A22</literal> <literal>(nu1-nd1 x nu1-nd1)</literal> are stable.
</para>
<para>
The pair <literal>(A33, C13)</literal> <literal>(dimS-nu1 x dimS-nu1, k x dimS-nu1)</literal> is observable,
and eigenvalues of <literal>A33</literal> are set to <literal>alfa</literal>.
</para>
<para>
Matrix <literal>A44</literal> <literal>(dimSg-dimS x dimSg-dimS)</literal> is unstable.
Matrix <literal>A55</literal> <literal>(dimN-dimSg,dimN-dimSg)</literal> is stable
</para>
<para>
The pair <literal>(A66,C26)</literal> <literal>(nx-dimN x nx-dimN)</literal> is observable,
and eigenvalues of <literal>A66</literal> set to <literal>beta</literal>.
</para>
<para>
The <literal>dimS</literal> first columns of <literal>X</literal> span S= smallest (C,A) invariant
subspace which contains Im(B), <literal>dimSg</literal> first columns of <literal>X</literal>
span Sg the maximal "complementary detectability subspace" of <literal>Sl</literal>
</para>
<para>
The <literal>dimN</literal> first columns of <literal>X</literal> span the maximal
"complementary observability subspace" of <literal>Sl</literal>.
(<literal>dimS=0</literal> if B(ker(D))=0).
</para>
<para>
If <literal>flag='st'</literal> is given, a five blocks partition of the matrices is
returned and <literal>dims</literal> has four components. If <literal>flag='pp'</literal> is
given a four blocks partition is returned (see abinv).
</para>
<para>
This function can be used to calculate an unknown input observer:
</para>
<programlisting role=""><![CDATA[
// DDEP: dot(x)=A x + Bu + Gd
// y= Cx (observation)
// z= Hx (z=variable to be estimated, d=disturbance)
// Find: dot(w) = Fw + Ey + Ru such that
// zhat = Mw + Ny
// z-Hx goes to zero at infinity
// Solution exists iff Ker H contains Sg(A,C,G) inter KerC (assuming detectability)
//i.e. H is such that:
// For any W which makes a column compression of [Xp(1:dimSg,:);C]
// with Xp=X' and [X,dims,J,Y,k,Z]=cainv(syslin('c',A,G,C));
// [Xp(1:dimSg,:);C]*W = [0 | *] one has
// H*W = [0 | *] (with at least as many aero columns as above).
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="abinv">abinv</link>
</member>
<member>
<link linkend="dt_ility">dt_ility</link>
</member>
<member>
<link linkend="ui_observer">ui_observer</link>
</member>
</simplelist>
</refsection>
</refentry>
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