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<?xml version="1.0" encoding="UTF-8"?>
<!--
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) ENPC - JPC
*
* 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="armax">
<refnamediv>
<refname>armax</refname>
<refpurpose>armax identification</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[arc,la,lb,sig,resid]=armax(r,s,y,u,[b0f,prf])</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>y</term>
<listitem>
<para>output process y(ny,n); ( ny: dimension of y , n : sample size)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>u</term>
<listitem>
<para>input process u(nu,n); ( nu: dimension of u , n : sample size)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>r and s</term>
<listitem>
<para>auto-regression orders r >=0 et s >=-1</para>
</listitem>
</varlistentry>
<varlistentry>
<term>b0f</term>
<listitem>
<para>optional parameter. Its default value is 0 and it means that the coefficient b0 must be identified. if bof=1 the b0 is supposed to be zero and is not identified</para>
</listitem>
</varlistentry>
<varlistentry>
<term>prf</term>
<listitem>
<para>optional parameter for display control. If prf =1, the default value, a display of the identified Arma is given.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>arc</term>
<listitem>
<para>a Scilab arma object (see armac)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>la</term>
<listitem>
<para>is the list(a,a+eta,a-eta) ( la = a in dimension 1) ; where eta is the estimated standard deviation. , a=[Id,a1,a2,...,ar] where each ai is a matrix of size (ny,ny)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>lb</term>
<listitem>
<para>is the list(b,b+etb,b-etb) (lb =b in dimension 1) ; where etb is the estimated standard deviation. b=[b0,.....,b_s] where each bi is a matrix of size (nu,nu)</para>
</listitem>
</varlistentry>
<varlistentry>
<term>sig</term>
<listitem>
<para>is the estimated standard deviation of the noise and resid=[ sig*e(t0),....] (</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
armax is used to identify the coefficients of a n-dimensional
ARX process
</para>
<programlisting role=""><![CDATA[
A(z^-1)y= B(z^-1)u + sig*e(t)
]]></programlisting>
<para>
where e(t) is a n-dimensional white noise with variance I.
sig an nxn matrix and A(z) and B(z):
</para>
<programlisting role=""><![CDATA[
A(z) = 1+a1*z+...+a_r*z^r; ( r=0 => A(z)=1)
B(z) = b0+b1*z+...+b_s z^s ( s=-1 => B(z)=0)
]]></programlisting>
<para>
for the method see Eykhoff in trends and progress in system identification, page 96.
with
<literal>z(t)=[y(t-1),..,y(t-r),u(t),...,u(t-s)]</literal>
and
<literal>coef= [-a1,..,-ar,b0,...,b_s] </literal>
we can write
<literal>y(t)= coef* z(t) + sig*e(t) </literal> and the algorithm minimises
<literal>sum_{t=1}^N ( [y(t)- coef'z(t)]^2)</literal>
where t0=max(max(r,s)+1,1))).
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
//-Ex1- Arma model : y(t) = 0.2*u(t-1)+0.01*e(t-1)
ny=1,nu=1,sig=0.01;
Arma=armac(1,[0,0.2],[0,1],ny,nu,sig) //defining the above arma model
u=rand(1,1000,'normal'); //a random input sequence u
y=arsimul(Arma,u); //simulation of a y output sequence associated with u.
Armaest=armax(0,1,y,u); //Identified model given u and y.
Acoeff=Armaest('a'); //Coefficients of the polynomial A(x)
Bcoeff=Armaest('b') //Coefficients of the polynomial B(x)
Dcoeff=Armaest('d'); //Coefficients of the polynomial D(x)
[Ax,Bx,Dx]=arma2p(Armaest) //Results in polynomial form.
]]></programlisting>
<programlisting role="example"><![CDATA[
//-Ex2- Arma1: y_t -0.8*y_{t-1} + 0.2*y_{t-2} = sig*e(t)
ny=1,nu=1;sig=0.001;
// First step: simulation the Arma1 model, for that we define
// Arma2: y_t -0.8*y_{t-1} + 0.2*y_{t-2} = sig*u(t)
// with normal deviates for u(t).
Arma2=armac([1,-0.8,0.2],sig,0,ny,nu,0);
//Definition of the Arma2 arma model (a model with B=sig and without noise!)
u=rand(1,10000,'normal'); // An input sequence for Arma2
y=arsimul(Arma2,u); // y = output of Arma2 with input u
// can be seen as output of Arma1.
// Second step: identification. We look for an Arma model
// y(t) + a1*y(t-1) + a2 *y(t-2) = sig*e(t)
Arma1est=armax(2,-1,y,[]);
[A,B,D]=arma2p(Arma1est)
]]></programlisting>
<programlisting role="example"><![CDATA[
a = [1, -2.851, 2.717, -0.865];
b = [0, 1, 1, 1];
d = [1, 0.7, 0.2];
ar = armac(a, b, d, 1, 1, 1);
disp(_("Simulation of an ARMAX process:"));
disp(ar);
// The input
n = 300;
u = -prbs_a(n, 1, int([2.5,5,10,17.5,20,22,27,35]*100/12));
// simulation
zd = narsimul(ar, u);
// visualization
plot2d(1:n,[zd',1000*u'],style=[1,3]);curves = gce();
legend(["Simulated output";"Input [scaled]"])
]]></programlisting>
<scilab:image>
a = [1, -2.851, 2.717, -0.865];
b = [0, 1, 1, 1];
d = [1, 0.7, 0.2];
ar = armac(a, b, d, 1, 1, 1);
n = 300;
u = -prbs_a(n, 1, int([2.5,5,10,17.5,20,22,27,35]*100/12));
zd = narsimul(ar, u);
plot2d(1:n,[zd',1000*u'],style=[1,3]);curves = gce();
legend(["Simulated output";"Input [scaled]"]);
</scilab:image>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="imrep2ss">imrep2ss</link>
</member>
<member>
<link linkend="time_id">time_id</link>
</member>
<member>
<link linkend="arl2">arl2</link>
</member>
<member>
<link linkend="armax">armax</link>
</member>
<member>
<link linkend="frep2tf">frep2tf</link>
</member>
</simplelist>
</refsection>
</refentry>
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