[doc] misc. patchs & small improvements
[scilab.git] / scilab / modules / cacsd / help / en_US / linear_system_factorization / rowinout.xml
1 <?xml version="1.0" encoding="UTF-8"?>
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16 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink"
17           xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML"
18           xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org"
19           xml:lang="en" xml:id="rowinout">
20     <refnamediv>
21         <refname>rowinout</refname>
22         <refpurpose>inner-outer factorization</refpurpose>
23     </refnamediv>
24     <refsynopsisdiv>
25         <title>Syntax</title>
26         <synopsis>[Inn, X, Gbar] = rowinout(G)</synopsis>
27     </refsynopsisdiv>
28     <refsection>
29         <title>Arguments</title>
30         <variablelist>
31             <varlistentry>
32                 <term>G</term>
33                 <listitem>
34                     <para>
35                         linear system (<literal>syslin</literal> list) <literal>[A,B,C,D]</literal>
36                     </para>
37                 </listitem>
38             </varlistentry>
39             <varlistentry>
40                 <term>Inn</term>
41                 <listitem>
42                     <para>
43                         inner factor (<literal>syslin</literal> list)
44                     </para>
45                 </listitem>
46             </varlistentry>
47             <varlistentry>
48                 <term>Gbar</term>
49                 <listitem>
50                     <para>
51                         outer factor (<literal>syslin</literal> list)
52                     </para>
53                 </listitem>
54             </varlistentry>
55             <varlistentry>
56                 <term>X</term>
57                 <listitem>
58                     <para>
59                         row-compressor of <literal>G</literal> (<literal>syslin</literal> list)
60                     </para>
61                 </listitem>
62             </varlistentry>
63         </variablelist>
64     </refsection>
65     <refsection>
66         <title>Description</title>
67         <para>
68             Inner-outer factorization (and row compression) of (<literal>l</literal>x<literal>p</literal>) <literal>G =[A,B,C,D]</literal> with <literal>l&gt;=p</literal>.
69         </para>
70         <para>
71             <literal>G</literal> is assumed to be tall (<literal>l&gt;=p</literal>) without zero on the imaginary axis
72             and with a <literal>D</literal> matrix which is full column rank.
73         </para>
74         <para>
75             <literal>G</literal> must also be stable for having <literal>Gbar</literal> stable.
76         </para>
77         <para>
78             <literal>G</literal> admits the following inner-outer factorization:
79         </para>
80         <screen><![CDATA[
81  G = [ Inn ] | Gbar |
82              |  0   |
83 ]]></screen>
84         <para>
85             where <literal>Inn</literal> is square and inner (all pass and stable) and <literal>Gbar</literal>
86             square and outer i.e:
87             Gbar is square bi-proper and bi-stable (Gbar inverse is also proper
88             and stable);
89         </para>
90         <para>
91             Note that:
92         </para>
93         <screen><![CDATA[
94       [ Gbar ]
95 X*G = [  -   ]
96       [  0   ]
97 ]]></screen>
98         <para>
99             is a row compression of <literal>G</literal> where <literal>X</literal> = <literal>Inn</literal>
100           inverse is all-pass i.e: <literal>X<superscript>t</superscript>(-s).X(s) = Identity</literal>
101           (for the continuous time case).
102         </para>
103     </refsection>
104     <refsection role="see also">
105         <title>See also</title>
106         <simplelist type="inline">
107             <member>
108                 <link linkend="syslin">syslin</link>
109             </member>
110             <member>
111                 <link linkend="colinout">colinout</link>
112             </member>
113         </simplelist>
114     </refsection>
115 </refentry>