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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="rtitr">
<refnamediv>
<refname>rtitr</refname>
<refpurpose>discrete time response (transfer matrix) </refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[y]=rtitr(Num,Den,u [,up,yp])</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>Num,Den</term>
<listitem>
<para>
polynomial matrices (resp. dimensions : <literal>n</literal>x<literal>m</literal> and <literal>n</literal>x<literal>n</literal>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>u</term>
<listitem>
<para>
real matrix (dimension <literal>m</literal>x<literal>(t+1)</literal>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>up,yp</term>
<listitem>
<para>
real matrices (<literal>up</literal> dimension <literal>m</literal>x<literal>(max(degree(Den)))</literal> (default values=<literal>0</literal>) ,
<literal>yp</literal> dimension <literal>n</literal>x
<literal>(max(degree(Den)))</literal>)
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>y</term>
<listitem>
<para>real matrix</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
<literal>y=rtitr(Num,Den,u [,up,yp])</literal> returns the time response of
the discrete time linear system with transfer matrix <literal>Den^-1 Num</literal>
for the input <literal>u</literal>, i.e <literal>y</literal> and <literal>u</literal> are such that <literal>Den y = Num u</literal> at t=0,1,...
</para>
<para>
If <literal>d1=max(degree(Den))</literal>, and <literal>d2=max(degree(Num))</literal> the polynomial
matrices <literal>Den(z)</literal> and <literal>Num(z)</literal> may be written respectively as:
</para>
<programlisting role=""><![CDATA[
D(z) = D_0 + D_1 z + ... + D_d1 z^d1
N(z) = N_0 + N_1 z + ... + N_d2 z^d2
]]></programlisting>
<para>
and <literal>Den y = Num u</literal> is interpreted as the recursion:
</para>
<programlisting role=""><![CDATA[
D(0)y(t)+D(1)y(t+1)+...+ D(d1)y(t+d1)= N(0) u(t) +....+ N(d2) u(t+d2)
]]></programlisting>
<para>
It is assumed that <literal>D(d1)</literal> is non singular.
</para>
<para>
The columns of u are the inputs of the system at t=0,1,...,T:
</para>
<programlisting role=""><![CDATA[
u=[u(0) , u(1),...,u(T)]
]]></programlisting>
<para>
The outputs at <literal>t=0,1,...,T+d1-d2</literal> are the columns of the matrix <literal>y</literal>:
</para>
<programlisting role=""><![CDATA[
y = [y(0), y(1), .... y(T+d1-d2)]
]]></programlisting>
<para>
<literal>up</literal> and <literal>yp</literal> define the initial conditions for t < 0 i.e
</para>
<programlisting role=""><![CDATA[
up = [u(-d1), ..., u(-1) ]
yp = [y(-d1), ... y(-1) ]
]]></programlisting>
<para>
Depending on the relative values of <literal>d1</literal> and <literal>d2</literal>, some of the
leftmost components of <literal>up</literal>, <literal>yp</literal> are ignored.
The default values of <literal>up</literal> and <literal>yp</literal> are zero:
<literal>up = 0*ones(m,d1), yp=0*ones(n,d1)</literal>
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
z=poly(0,'z');
Num=1+z;
Den=1+z;
u=[1,2,3,4,5];
rtitr(Num,Den,u)-u
]]></programlisting>
<programlisting role="example"><![CDATA[
//Other examples
//siso
//causal
n1=1;
d1=poly([1 1],'z','coeff'); // y(j)=-y(j-1)+u(j-1)
r1=[0 1 0 1 0 1 0 1 0 1 0];
r=rtitr(n1,d1,ones(1,10));
norm(r1-r,1)
//hot restart
r=rtitr(n1,d1,ones(1,9),1,0);
norm(r1(2:11)-r)
//non causal
n2=poly([1 1 1],'z','coeff');
d2=d1; // y(j)=-y(j-1)+u(j-1)+u(j)+u(j+1)
r2=[2 1 2 1 2 1 2 1 2];
r=rtitr(n2,d2,ones(1,10));
norm(r-r2,1)
//hot restart
r=rtitr(n2,d2,ones(1,9),1,2);
norm(r2(2:9)-r,1)
//
//MIMO example
//causal
d1=d1*diag([1 0.5]);
n1=[1 3 1;2 4 1];
r1=[5;14]*r1;
r=rtitr(n1,d1,ones(3,10));
norm(r1-r,1)
//
r=rtitr(n1,d1,ones(3,9),[1;1;1],[0;0]);
norm(r1(:,2:11)-r,1)
//polynomial n1 (same ex.)
n1(1,1)=poly(1,'z','c');
r=rtitr(n1,d1,ones(3,10));
norm(r1-r,1)
//
r=rtitr(n1,d1,ones(3,9),[1;1;1],[0;0]);
norm(r1(:,2:11)-r,1)
//non causal
d2=d1;n2=n2*n1;
r2=[5;14]*r2;
r=rtitr(n2,d2,ones(3,10));
norm(r2-r)
//
r=rtitr(n2,d2,ones(3,9),[1;1;1],[10;28]);
norm(r2(:,2:9)-r,1)
]]></programlisting>
<programlisting role="example"><![CDATA[
//
// State-space or transfer
a = [0.21 , 0.63 , 0.56 , 0.23 , 0.31
0.76 , 0.85 , 0.66 , 0.23 , 0.93
0 , 0.69 , 0.73 , 0.22 , 0.21
0.33 , 0.88 , 0.2 , 0.88 , 0.31
0.67 , 0.07 , 0.54 , 0.65 , 0.36];
b = [0.29 , 0.5 , 0.92
0.57 , 0.44 , 0.04
0.48 , 0.27 , 0.48
0.33 , 0.63 , 0.26
0.59 , 0.41 , 0.41];
c = [0.28 , 0.78 , 0.11 , 0.15 , 0.84
0.13 , 0.21 , 0.69 , 0.7 , 0.41];
d = [0.41 , 0.11 , 0.56
0.88 , 0.2 , 0.59];
s=syslin('d',a,b,c,d);
h=ss2tf(s);num=h('num');
den=h('den');
den=den(1,1)*eye(2,2);
u=1;u(3,10)=0;
r3=flts(u,s);
r=rtitr(num,den,u);
norm(r3-r,1)
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="ltitr">ltitr</link>
</member>
<member>
<link linkend="exp">exp</link>
</member>
<member>
<link linkend="flts">flts</link>
</member>
</simplelist>
</refsection>
</refentry>
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