<?xml version="1.0" encoding="UTF-8"?>
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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="findAC">
    <refnamediv>
        <refname>findAC</refname>
        <refpurpose>discrete-time system subspace identification</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>
            [A, C] = findAC(S, N, L, R, METH, TOL, PRINTW)
            [A, C, RCND] = findAC(S, N, L, R, METH, TOL, PRINTW)
        </synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>S</term>
                <listitem>
                    <para>integer, the number of block rows in the block-Hankel matrices</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>N</term>
                <listitem>
                    <para>integer</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>L</term>
                <listitem>
                    <para>integer</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>R</term>
                <listitem>
                    <para>matrix, relevant part of the  R factor of the concatenated block-Hankel matrices computed by a call to findr.</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>METH</term>
                <listitem>
                    <para>integer, an option for the method to use</para>
                    <variablelist>
                        <varlistentry>
                            <term>= 1</term>
                            <listitem>
                                <para> MOESP method with past inputs and outputs;</para>
                            </listitem>
                        </varlistentry>
                        <varlistentry>
                            <term>= 2</term>
                            <listitem>
                                <para> N4SID method;</para>
                            </listitem>
                        </varlistentry>
                    </variablelist>
                    <para>
                        Default:    METH = 3.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>TOL</term>
                <listitem>
                    <para>the tolerance used for estimating the rank of matrices.  If  TOL &gt; 0,  then the given value of  TOL  is used as a lower bound for the reciprocal condition number. Default: prod(size(matrix))*epsilon_machine where epsilon_machine is the relative machine precision.</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>PRINTW</term>
                <listitem>
                    <para>integer, switch for printing the warning messages.</para>
                    <variablelist>
                        <varlistentry>
                            <term>PRINTW</term>
                            <listitem>
                                <para>= 1: print warning messages;</para>
                            </listitem>
                        </varlistentry>
                        <varlistentry>
                            <term>= 0</term>
                            <listitem>
                                <para>do not print warning messages.</para>
                            </listitem>
                        </varlistentry>
                    </variablelist>
                    <para>
                        Default:    PRINTW = 0.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>A</term>
                <listitem>
                    <para>matrix, state system matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>C</term>
                <listitem>
                    <para>matrix, output system matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>RCND</term>
                <listitem>
                    <para>vector of length 4,  condition numbers of the matrices involved in rank decision</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            finds the system matrices A and C of a discrete-time system, given the
            system order and the relevant part of the R factor of the concatenated
            block-Hankel matrices, using subspace identification techniques (MOESP
            or N4SID).
        </para>
        <itemizedlist>
            <listitem>
                <para>[A,C] = findAC(S,N,L,R,METH,TOL,PRINTW)  computes the system matrices A and C. The model structure is:       x(k+1) = Ax(k) + Bu(k) + Ke(k),   k &gt;= 1,      y(k)   = Cx(k) + Du(k) + e(k),  where x(k) and y(k) are vectors of length N and L, respectively.</para>
            </listitem>
            <listitem>
                <para>[A,C,RCND] = findAC(S,N,L,R,METH,TOL,PRINTW)  also returns the vector RCND of length 4 containing the condition numbers of the matrices involved in rank decisions.</para>
            </listitem>
        </itemizedlist>
        <para>
            Matrix R, computed by findR, should be determined with suitable arguments
            METH and JOBD.
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
//generate data from a given linear system
A = [ 0.5, 0.1,-0.1, 0.2;
      0.1, 0,  -0.1,-0.1;
     -0.4,-0.6,-0.7,-0.1;
      0.8, 0,  -0.6,-0.6];
B = [0.8;0.1;1;-1];
C = [1 2 -1 0];
SYS=syslin(0.1,A,B,C);
nsmp=100;
U=prbs_a(nsmp,nsmp/5);
Y=(flts(U,SYS)+0.3*rand(1,nsmp,'normal'));

// Compute R
S=15;L=1;
[R,N,SVAL] = findR(S,Y',U');

N=3;
METH=3;TOL=-1;
[A,C] = findAC(S,N,L,R,METH,TOL);
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="findABCD">findABCD</link>
            </member>
            <member>
                <link linkend="findBD">findBD</link>
            </member>
            <member>
                <link linkend="findBDK">findBDK</link>
            </member>
            <member>
                <link linkend="findR">findR</link>
            </member>
            <member>
                <link linkend="sorder">sorder</link>
            </member>
            <member>
                <link linkend="sident">sident</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
