45258b650de4aec87af53962940af6fa14ff7045
[scilab.git] / scilab / modules / cacsd / help / en_US / linear_analysis / freq_domain / trzeros.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" 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="trzeros">
17     <refnamediv>
18         <refname>trzeros</refname>
19         <refpurpose>transmission zeros and normal rank</refpurpose>
20     </refnamediv>
21     <refsynopsisdiv>
22         <title>Syntax</title>
23         <synopsis>[tr]=trzeros(Sl)
24             [nt,dt,rk]=trzeros(Sl)
25         </synopsis>
26     </refsynopsisdiv>
27     <refsection>
28         <title>Arguments</title>
29         <variablelist>
30             <varlistentry>
31                 <term>Sl</term>
32                 <listitem>
33                     <para>
34                         linear system (<literal>syslin</literal> list)
35                     </para>
36                 </listitem>
37             </varlistentry>
38             <varlistentry>
39                 <term>nt</term>
40                 <listitem>
41                     <para>complex vectors</para>
42                 </listitem>
43             </varlistentry>
44             <varlistentry>
45                 <term>dt</term>
46                 <listitem>
47                     <para>real vector</para>
48                 </listitem>
49             </varlistentry>
50             <varlistentry>
51                 <term>rk</term>
52                 <listitem>
53                     <para>integer (normal rank of Sl)</para>
54                 </listitem>
55             </varlistentry>
56         </variablelist>
57     </refsection>
58     <refsection>
59         <title>Description</title>
60         <para>
61             Called with one output argument, <literal>trzeros(Sl)</literal> returns the 
62             transmission zeros of the linear system <literal>Sl</literal>.
63         </para>
64         <para>
65             <literal>Sl</literal> may have a polynomial (but square) <literal>D</literal> matrix.
66         </para>
67         <para>
68             Called with 2 output arguments, <literal>trzeros</literal> returns the 
69             transmission zeros of the linear system <literal>Sl</literal> as <literal>tr=nt./dt</literal>;
70         </para>
71         <para>
72             (Note that some components of <literal>dt</literal> may be zeros)
73         </para>
74         <para>
75             Called with 3 output arguments, <literal>rk</literal>  is the normal rank of <literal>Sl</literal>
76         </para>
77         <para>
78             Transfer matrices are converted to state-space.
79         </para>
80         <para>
81             If <literal>Sl</literal> is a (square) polynomial matrix <literal>trzeros</literal> returns the 
82             roots of its determinant.
83         </para>
84         <para>
85             For usual state-space system <literal>trzeros</literal> uses the state-space 
86             algorithm of Emami-Naeni and Van Dooren.
87         </para>
88         <para>
89             If <literal>D</literal> is invertible the transmission zeros are the eigenvalues
90             of the "<literal>A</literal> matrix" of the inverse system : <literal>A - B*inv(D)*C</literal>;
91         </para>
92         <para>
93             If <literal>C*B</literal> is invertible the transmission zeros are the eigenvalues
94             of <literal>N*A*M</literal> where <literal>M*N</literal> is a full rank factorization of 
95             <literal>eye(A)-B*inv(C*B)*C</literal>;
96         </para>
97         <para>
98             For systems with a polynomial <literal>D</literal> matrix zeros are 
99             calculated as the roots of the determinant of the system matrix.
100         </para>
101         <para>
102             <warning>
103                 Caution: the computed zeros are not always reliable, in particular
104                 in case of repeated zeros.
105             </warning>
106         </para>
107     </refsection>
108     <refsection>
109         <title>Examples</title>
110         <programlisting role="example"><![CDATA[ 
111 W1=ssrand(2,2,5);trzeros(W1)    //call trzeros
112 roots(det(systmat(W1)))         //roots of det(system matrix)
113 s=poly(0,'s');W=[1/(s+1);1/(s-2)];W2=(s-3)*W*W';[nt,dt,rk]=trzeros(W2);
114 St=systmat(tf2ss(W2));[Q,Z,Qd,Zd,numbeps,numbeta]=kroneck(St);
115 St1=Q*St*Z;rowf=(Qd(1)+Qd(2)+1):(Qd(1)+Qd(2)+Qd(3));
116 colf=(Zd(1)+Zd(2)+1):(Zd(1)+Zd(2)+Zd(3));
117 roots(St1(rowf,colf)), nt./dt     //By Kronecker form
118  ]]></programlisting>
119     </refsection>
120     <refsection role="see also">
121         <title>See Also</title>
122         <simplelist type="inline">
123             <member>
124                 <link linkend="gspec">gspec</link>
125             </member>
126             <member>
127                 <link linkend="kroneck">kroneck</link>
128             </member>
129         </simplelist>
130     </refsection>
131 </refentry>