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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="lqg_ltr">
    <refnamediv>
        <refname>lqg_ltr</refname>
        <refpurpose>LQG with loop transform recovery</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>[kf, kc] = lqg_ltr(sl, mu, ro)</synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>sl</term>
                <listitem>
                    <para>
                        linear system in state-space form (<literal>syslin</literal> list)
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>mu,ro</term>
                <listitem>
                    <para> real positive numbers chosen ``small enough''</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>kf,kc</term>
                <listitem>
                    <para>controller and observer Kalman gains.</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            returns the Kalman gains for:
        </para>
        <programlisting role=""><![CDATA[
         x = a*x + b*u + l*w1
(sl)
         y = c*x + mu*I*w2

         z = h*x
 ]]></programlisting>
        <para>
            Cost function:
        </para>
        <programlisting role=""><![CDATA[
         /+oo
         |
J    = E(| [z(t)'*z(t) + ro^2*u(t)'*u(t)]dt)
 lqg     |
         / 0
  ]]></programlisting>
        <para>
            The lqg/ltr approach looks for <literal>L,mu,H,ro</literal> such that:
            J(lqg) = J(freq) where
        </para>
        <programlisting role=""><![CDATA[
        /+oo        *  *           *
J    =  | tr[S  W  W  S ] + tr[T  T]dw
 freq   |
        /0
 ]]></programlisting>
        <para>
            and
        </para>
        <programlisting role=""><![CDATA[
 S = (I + G*K)^(-1)
 T = G*K*(I+G*K)^(-1)
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="syslin">syslin</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
