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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="inistate">
    <refnamediv>
        <refname>inistate</refname>
        <refpurpose>Estimates the initial state of a discrete-time system</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>
            X0 = inistate(A, C, Y)
            X0 = inistate(A, B, C, Y, U)
            X0 = inistate(SYS, Y, U, TOL, PRINTW)

            [x0, V, rcnd] = inistate(SYS, Y, U, TOL, PRINTW)
        </synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>SYS</term>
                <listitem>
                    <para>given system, syslin(dt,A,B,C,D)</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>Y</term>
                <listitem>
                    <para>the output of the system</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>U</term>
                <listitem>
                    <para>the input of the system</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>TOL</term>
                <listitem>
                    <para>TOL is 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.</para>
                    <para>
                        Default:    prod(size(matrix))*epsilon_machine where epsilon_machine is the relative machine precision.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>PRINTW</term>
                <listitem>
                    <para>PRINTW is a switch for printing the warning messages.</para>
                    <variablelist>
                        <varlistentry>
                            <term>=  </term>
                            <listitem>
                                <para>1: print warning messages;</para>
                            </listitem>
                        </varlistentry>
                        <varlistentry>
                            <term>=  </term>
                            <listitem>
                                <para>0: do not print warning messages.</para>
                            </listitem>
                        </varlistentry>
                    </variablelist>
                    <para>
                        Default:    PRINTW = 0.
                    </para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>X0</term>
                <listitem>
                    <para>the estimated initial state vector</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>V</term>
                <listitem>
                    <para>orthogonal matrix which reduces the system state matrix A to  a real Schur form</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>rcnd</term>
                <listitem>
                    <para>estimate of the reciprocal condition number of the coefficient matrix of the least squares problem solved.</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            inistate  Estimates the initial state of a discrete-time system, given the
            (estimated) system matrices, and a set of input/output data.
        </para>
        <para>
            X0 = inistate(SYS,Y,U,TOL,PRINTW)  estimates the initial state X0 of
            the discrete-time system SYS = (A,B,C,D), using the output data Y
            and the input data U. The model structure is :
        </para>
        <programlisting role=""><![CDATA[
x(k+1) = Ax(k) + Bu(k),   k >= 1,
y(k)   = Cx(k) + Du(k),
 ]]></programlisting>
        <para>
            The vectors y(k) and u(k) are transposes of the k-th rows of Y and U,
            respectively.
        </para>
        <para>
            Instead of the first input parameter SYS (an syslin object), equivalent
            information may be specified using matrix parameters, for instance,
            X0 = inistate(A,B,C,Y,U);   or   X0 = inistate(A,C,Y);
        </para>
        <para>
            [x0,V,rcnd] = inistate(SYS,Y,U,TOL,PRINTW) returns, besides x0,
            the orthogonal matrix V which reduces the system state matrix A to
            a real Schur form, as well as an estimate of the reciprocal condition
            number of the coefficient matrix of the least squares problem solved.
        </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;
[R,N1,SVAL] = findR(S,Y',U');
N=3;
SYS1 = findABCD(S,N,1,R) ;
SYS1.dt=0.1;

inistate(SYS1,Y',U')

]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="findBD">findBD</link>
            </member>
            <member>
                <link linkend="findx0BD">findx0BD</link>
            </member>
        </simplelist>
    </refsection>
</refentry>
