<?xml version="1.0" encoding="UTF-8"?>
<!--
 * Scilab ( https://www.scilab.org/ ) - This file is part of Scilab
 * Copyright (C) INRIA -
 * Copyright (C) 2012 - 2016 - Scilab Enterprises
 *
 * This file is hereby licensed under the terms of the GNU GPL v2.0,
 * pursuant to article 5.3.4 of the CeCILL v.2.1.
 * This file was originally licensed under the terms of the CeCILL v2.1,
 * and continues to be available under such terms.
 * For more information, see the COPYING file which you should have received
 * along with this program.
 *
 -->
<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="ricc">
    <refnamediv>
        <refname>ricc</refname>
        <refpurpose>Riccati equation</refpurpose>
    </refnamediv>
    <refsynopsisdiv>
        <title>Syntax</title>
        <synopsis>
            [X, RCOND, FERR] = ricc(A, B, C, "cont", method)
            [X, RCOND, FERR] = ricc(F, G, H, "disc", method)
        </synopsis>
    </refsynopsisdiv>
    <refsection>
        <title>Arguments</title>
        <variablelist>
            <varlistentry>
                <term>A,B,C</term>
                <listitem>
                    <para>real matrices of appropriate dimensions</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>F,G,H</term>
                <listitem>
                    <para>real matrices of appropriate dimensions</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>X</term>
                <listitem>
                    <para>real matrix</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>"cont","disc"'  </term>
                <listitem>
                    <para>imposed string (flag for continuous or discrete)</para>
                </listitem>
            </varlistentry>
            <varlistentry>
                <term>method</term>
                <listitem>
                    <para>'schr' or 'sign' for continuous-time systems and 'schr' or 'invf' for discrete-tyme systems</para>
                </listitem>
            </varlistentry>
        </variablelist>
    </refsection>
    <refsection>
        <title>Description</title>
        <para>
            Riccati solver.
        </para>
        <para>
            Continuous time:
        </para>
        <programlisting role=""><![CDATA[
X=ricc(A,B,C,'cont')
 ]]></programlisting>
        <para>
            gives a solution to the continuous time ARE
        </para>
        <programlisting role=""><![CDATA[
A'*X+X*A-X*B*X+C=0 .
 ]]></programlisting>
        <para>
            <literal>B</literal> and <literal>C</literal> are assumed to be nonnegative definite.
            <literal>(A,G)</literal> is assumed to be stabilizable with <literal>G*G'</literal> a full rank
            factorization of <literal>B</literal>.
        </para>
        <para>
            <literal>(A,H)</literal> is assumed to be detectable with <literal>H*H'</literal> a full rank
            factorization of <literal>C</literal>.
        </para>
        <para>
            Discrete time:
        </para>
        <programlisting role=""><![CDATA[
X=ricc(F,G,H,'disc')
 ]]></programlisting>
        <para>
            gives a solution  to the discrete time ARE
        </para>
        <programlisting role=""><![CDATA[
X=F'*X*F-F'*X*G1*((G2+G1'*X*G1)^-1)*G1'*X*F+H
 ]]></programlisting>
        <para>
            <literal>F</literal> is assumed invertible and   <literal>G = G1*inv(G2)*G1'</literal>.
        </para>
        <para>
            One  assumes  <literal>(F,G1)</literal> stabilizable and <literal>(C,F)</literal> detectable
            with <literal>C'*C</literal> full rank factorization of <literal>H</literal>. Use preferably
            <literal>riccati()</literal>.
        </para>
        <para>
            C, D are symmetric .It is assumed that the matrices A, C and D are such that the corresponding matrix pencil has N eigenvalues with moduli less than one.
        </para>
        <para>
            Error bound on the solution and a condition estimate are also provided.
            It is assumed that the matrices A, C and D are such that the
            corresponding Hamiltonian matrix has N eigenvalues with negative real parts.
        </para>
    </refsection>
    <refsection>
        <title>Examples</title>
        <programlisting role="example"><![CDATA[
//Standard formulas to compute Riccati solutions
A=rand(3,3);
B=rand(3,2);
C=rand(3,3);
C=C*C';
R=rand(2,2);
R=R*R'+eye();
B=B*inv(R)*B';
X=ricc(A,B,C,'cont');
norm(A'*X+X*A-X*B*X+C,1)
H=[A -B;-C -A'];
[T,d]=schur(eye(H),H,'cont');
T=T(:,1:d);
X1=T(4:6,:)/T(1:3,:);
norm(X1-X,1)
[T,d]=schur(H,'cont');
T=T(:,1:d);
X2=T(4:6,:)/T(1:3,:);
norm(X2-X,1)
//       Discrete time case
F=A;
B=rand(3,2);
G1=B;
G2=R;
G=G1/G2*G1';
H=C;
X=ricc(F,G,H,'disc');
norm(F'*X*F-(F'*X*G1/(G2+G1'*X*G1))*(G1'*X*F)+H-X)
H1=[eye(3,3) G;zeros(3,3) F'];
H2=[F zeros(3,3);-H eye(3,3)];
[T,d]=schur(H2,H1,'disc');
T=T(:,1:d);
X1=T(4:6,:)/T(1:3,:);
norm(X1-X,1)
Fi=inv(F);
Hami=[Fi Fi*G;H*Fi F'+H*Fi*G];
[T,d]=schur(Hami,'d');
T=T(:,1:d);
Fit=inv(F');
Ham=[F+G*Fit*H -G*Fit;-Fit*H Fit];
[T,d]=schur(Ham,'d');
T=T(:,1:d);
X2=T(4:6,:)/T(1:3,:);
norm(X2-X,1)
 ]]></programlisting>
    </refsection>
    <refsection role="see also">
        <title>See also</title>
        <simplelist type="inline">
            <member>
                <link linkend="riccati">riccati</link>
            </member>
            <member>
                <link linkend="schur">schur</link>
            </member>
        </simplelist>
    </refsection>
    <refsection>
        <title>Used Functions</title>
        <para>See SCI/modules/cacsd/src/slicot/riccpack.f</para>
    </refsection>
</refentry>
