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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="ddp">
<refnamediv>
<refname>ddp</refname>
<refpurpose>disturbance decoupling</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[Closed,F,G]=ddp(Sys,zeroed,B1,D1)
[Closed,F,G]=ddp(Sys,zeroed,B1,D1,flag,alfa,beta)
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>Sys</term>
<listitem>
<para>
<literal>syslin</literal> list containing the matrices <literal>(A,B2,C,D2)</literal>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>zeroed</term>
<listitem>
<para>
integer vector, indices of outputs of <literal>Sys</literal> which are zeroed.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>B1</term>
<listitem>
<para>real matrix</para>
</listitem>
</varlistentry>
<varlistentry>
<term>D1</term>
<listitem>
<para>
real matrix. <literal>B1</literal> and <literal>D1</literal> have the same number of columns.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>flag</term>
<listitem>
<para>
string <literal>'ge'</literal> or <literal>'st'</literal> (default) or <literal>'pp'</literal>.
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>alpha</term>
<listitem>
<para>real or complex vector (loc. of closed loop poles)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>beta</term>
<listitem>
<para>real or complex vector (loc. of closed loop poles)</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
Exact disturbance decoupling (output nulling algorithm).
Given a linear system, and a subset of outputs, z, which are to
be zeroed, characterize the inputs w of Sys such that the
transfer function from w to z is zero.
<literal>Sys</literal> is a linear system {A,B2,C,D2} with one input and two outputs
( i.e. Sys: u-->(z,y) ), part the following system defined from <literal>Sys</literal>
and <literal>B1,D1</literal>:
</para>
<programlisting role=""><![CDATA[
xdot = A x + B1 w + B2 u
z = C1 x + D11 w + D12 u
y = C2 x + D21 w + D22 u
]]></programlisting>
<para>
outputs of Sys are partitioned into (z,y) where z is to be zeroed,
i.e. the matrices C and D2 are:
</para>
<programlisting role=""><![CDATA[
C=[C1;C2] D2=[D12;D22]
C1=C(zeroed,:) D12=D2(zeroed,:)
]]></programlisting>
<para>
The matrix <literal>D1</literal> is partitioned similarly as <literal>D1=[D11;D21]</literal>
with <literal>D11=D1(zeroed,:)</literal>.
The control is u=Fx+Gw and one looks for matriced <literal>F,G</literal> such that the
closed loop system: w-->z given by
</para>
<programlisting role=""><![CDATA[
xdot= (A+B2*F) x + (B1 + B2*G) w
z = (C1+D12F) x + (D11+D12*G) w
]]></programlisting>
<para>
has zero transfer transfer function.
</para>
<para>
<literal>flag='ge'</literal>no stability constraints.
<literal>flag='st'</literal> : look for stable closed loop system (A+B2*F stable).
<literal>flag='pp'</literal> : eigenvalues of A+B2*F are assigned to <literal>alfa</literal> and
<literal>beta</literal>.
</para>
<para>
Closed is a realization of the <literal>w-->y</literal> closed loop system
</para>
<programlisting role=""><![CDATA[
xdot= (A+B2*F) x + (B1 + B2*G) w
y = (C2+D22*F) x + (D21+D22*G) w
]]></programlisting>
<para>
Stability (resp. pole placement) requires stabilizability
(resp. controllability) of (A,B2).
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
rand('seed',0);nx=6;nz=3;nu=2;ny=1;
A=diag(1:6);A(2,2)=-7;A(5,5)=-9;B2=[1,2;0,3;0,4;0,5;0,0;0,0];
C1=[zeros(nz,nz),eye(nz,nz)];D12=[0,1;0,2;0,3];
Sys12=syslin('c',A,B2,C1,D12);
C=[C1;rand(ny,nx)];D2=[D12;rand(ny,size(D12,2))];
Sys=syslin('c',A,B2,C,D2);
[A,B2,C1,D12]=abcd(Sys12); //The matrices of Sys12.
my_alpha=-1;my_beta=-2;flag='ge';
[X,dims,F,U,k,Z]=abinv(Sys12,my_alpha,my_beta,flag);
clean(X'*(A+B2*F)*X)
clean(X'*B2*U)
clean((C1+D12*F)*X)
clean(D12*U);
//Calculating an ad-hoc B1,D1
G1=rand(size(B2,2),3);
B1=-B2*G1;
D11=-D12*G1;
D1=[D11;rand(ny,size(B1,2))];
[Closed,F,G]=ddp(Sys,1:nz,B1,D1,'st',my_alpha,my_beta);
closed=syslin('c',A+B2*F,B1+B2*G,C1+D12*F,D11+D12*G);
ss2tf(closed)
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="abinv">abinv</link>
</member>
<member>
<link linkend="ui_observer">ui_observer</link>
</member>
</simplelist>
</refsection>
</refentry>
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