<?xml version="1.0" encoding="UTF-8"?>
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 *
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<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="frep2tf">
    <refnamediv>
        <refname>frep2tf</refname>
        <refpurpose>transfer function realization from frequency response</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>
            [h, err] = frep2tf(frq, repf, dg)
            [h, err] = frep2tf(frq, repf, dg, dom, tols, weight)
        </synopsis>
    </refsynopsisdiv>
    <refsection role="parameters">
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>frq</term>
                <listitem>
                    <para>vector of frequencies in Hz.</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>repf</term>
                <listitem>
                    <para>vector of frequency response</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>dg</term>
                <listitem>
                    <para>degree of linear system</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>dom</term>
                <listitem>
                    <para>
                        time domain (<literal>'c'</literal> or <literal>'d'</literal> or <literal>dt</literal>)
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>tols</term>
                <listitem>
                    <para>
                        a vector of size 3 giving the relative and absolute tolerance  and the maximum number of iterations (default values are <literal>rtol=1.e-2; atol=1.e-4, N=10</literal>).
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>weight</term>
                <listitem>
                    <para>vector of weights on frequencies</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>h</term>
                <listitem>
                    <para>SISO transfer function</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>err</term>
                <listitem>
                    <para>
                        error (for example if <literal>dom='c'</literal> <literal>sum(abs(h(2i*pi*frq) - rep)^2)/size(frq,*)</literal>)
                    </para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection role="description">
        <title>Description</title>
        <para>
            Frequency response to  transfer function conversion. The order of <literal>h</literal>
            is a priori given in <literal>dg</literal> which must be provided.
            The following linear system is solved in the least square sense.
        </para>
        <programlisting role="no-scilab-exec"><![CDATA[
 weight(k)*(n( phi_k) - d(phi_k)*rep_k)=0, k=1,..,n
 ]]></programlisting>
        <para>
            where <literal>phi_k= 2*%i*%pi*frq</literal> when <literal>dom='c'</literal> and
            <literal>phi_k=exp(2*%i*%pi*dom*frq</literal> if not. If the <literal>weight</literal>
            vector is not given a default
            penalization is used (when <literal>dom='c'</literal>).
        </para>
        <para>
            A stable and minimum phase system can be obtained by using function <literal>factors</literal>.
        </para>
    </refsection>
    <refsection role="examples">
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
s=poly(0,'s');
h=syslin('c',(s-1)/(s^3+5*s+20))
frq=0:0.05:3;
repf=repfreq(h,frq);
clean(frep2tf(frq,repf,3))
 ]]></programlisting>
        <programlisting role="example"><![CDATA[

Sys=ssrand(1,1,10);
frq=logspace(-3,2,200);
[frq,rep]=repfreq(Sys,frq);  //Frequency response of Sys
[Sys2,err]=frep2tf(frq,rep,10);
Sys2=clean(Sys2) //Sys2 obtained from freq. resp of Sys

[frq,rep2]=repfreq(Sys2,frq); //Frequency response of Sys2
clf();
bode(frq,[rep;rep2])   //Responses of Sys and Sys2
[gsort(spec(Sys('A'))), gsort(roots(Sys2('den')))] //poles
 ]]></programlisting>
        <scilab:image>
            Sys=ssrand(1,1,10);
            frq=logspace(-3,2,200);
            [frq,rep]=repfreq(Sys,frq);
            [Sys2,err]=frep2tf(frq,rep,10);
            Sys2=clean(Sys2)

            [frq,rep2]=repfreq(Sys2,frq);
            bode(frq,[rep;rep2])
            [gsort(spec(Sys('A'))), gsort(roots(Sys2('den')))]
        </scilab:image>
        <programlisting role="example"><![CDATA[
dom = 1/1000; // Sampling time
z = poly(0,'z');
h = syslin(dom,(z^2+0.5)/(z^3+0.1*z^2-0.5*z+0.08))
frq = (0:0.01:0.5)/dom;
repf = repfreq(h,frq);
[Sys2,err] = frep2tf(frq,repf,3,dom);
[frq,rep2] = repfreq(Sys2,frq); //Frequency response of Sys2
clf();
plot2d(frq, abs([repf;rep2])')
 ]]></programlisting>
        <scilab:image>
            dom=1/1000;
            z=poly(0,'z');
            h=syslin(dom,(z^2+0.5)/(z^3+0.1*z^2-0.5*z+0.08))
            frq=(0:0.01:0.5)/dom;
            repf=repfreq(h,frq);
            [Sys2,err]=frep2tf(frq,repf,3,dom);
            [frq,rep2]=repfreq(Sys2,frq);
            plot2d(frq, abs([repf;rep2])')
        </scilab:image>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="imrep2ss">imrep2ss</link>
            </member>
            <member>
                <link linkend="arl2">arl2</link>
            </member>
            <member>
                <link linkend="time_id">time_id</link>
            </member>
            <member>
                <link linkend="armax">armax</link>
            </member>
            <member>
                <link linkend="frfit">frfit</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
