1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
|
<?xml version="1.0" encoding="UTF-8"?>
<!--
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) INRIA -
*
* This file must be used under the terms of the CeCILL.
* This source file is licensed as described in the file COPYING, which
* you should have received as part of this distribution. The terms
* are also available at
* http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
*
-->
<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="augment">
<refnamediv>
<refname>augment</refname>
<refpurpose>augmented plant</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[P,r]=augment(G)
[P,r]=augment(G,flag1)
[P,r]=augment(G,flag1,flag2)
</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>G</term>
<listitem>
<para>
linear system (<literal>syslin</literal> list), the nominal plant
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>flag1</term>
<listitem>
<para>
one of the following (upper case) character string: <literal> 'S' </literal>, <literal> 'R' </literal>, <literal> 'T' </literal> <literal> 'SR' </literal>, <literal> 'ST' </literal>, <literal> 'RT' </literal> <literal> 'SRT' </literal>
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>flag2</term>
<listitem>
<para>
one of the following character string: <literal> 'o' </literal> (stands for 'output', this is the default value) or <literal>'i'</literal> (stands for 'input').
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>P</term>
<listitem>
<para>
linear system (<literal>syslin</literal> list), the ``augmented'' plant
</para>
</listitem>
</varlistentry>
<varlistentry>
<term>r</term>
<listitem>
<para>
1x2 row vector, dimension of <literal>P22 = G</literal>
</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
If <literal>flag1='SRT'</literal> (default value), returns the "full" augmented plant
</para>
<programlisting role=""><![CDATA[
[ I | -G] -->'S'
[ 0 | I] -->'R'
P = [ 0 | G] -->'T'
[-------]
[ I | -G]
]]></programlisting>
<para>
<literal> 'S' </literal>, <literal> 'R' </literal>, <literal> 'T' </literal> refer to the first three (block) rows
of <literal>P</literal> respectively.
</para>
<para>
If one of these letters is absent in <literal>flag1</literal>, the corresponding
row in <literal>P</literal> is missing.
</para>
<para>
If <literal>G</literal> is given in state-space form, the returned <literal>P</literal> is minimal.
<literal>P</literal> is calculated by: <literal>[I,0,0;0,I,0;-I,0,I;I,0,0]*[I,-G;0,I;I,0]</literal>.
</para>
<para>
The augmented plant associated with input sensitivity functions, namely
</para>
<programlisting role=""><![CDATA[
[ I | -I] -->'S' (input sensitivity)
[ G | -G] -->'R' (G*input sensitivity)
P = [ 0 | I] -->'T' (K*G*input sensitivity)
[-------]
[ G | -G]
]]></programlisting>
<para>
is obtained by the command <literal>[P,r]=augment(G,flag,'i')</literal>. For
state-space <literal>G</literal>, this <literal>P</literal>
is calculated by: <literal>[I,-I;0,0;0,I;0,0]+[0;I;0;I]*G*[I,-I]</literal>
and is thus generically minimal.
</para>
<para>
Note that weighting functions can be introduced by left-multiplying
<literal>P</literal> by a diagonal system of appropriate dimension, e.g.,
<literal> P = sysdiag(W1,W2,W3,eye(G))*P</literal>.
</para>
<para>
Sensitivity functions can be calculated by <literal>lft</literal>. One has:
</para>
<para>
For output sensitivity functions [P,r]=augment(P,'SRT'):
lft(P,r,K)=[inv(eye()+G*K);K*inv(eye()+G*K);G*K*inv(eye()+G*K)];
</para>
<para>
For input sensitivity functions [P,r]=augment(P,'SRT','i'):
lft(P,r,K)=[inv(eye()+K*G);G*inv(eye()+K*G);K*G*inv(eye()+G*K)];
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
G=ssrand(2,3,2); //Plant
K=ssrand(3,2,2); //Compensator
[P,r]=augment(G,'T');
T=lft(P,r,K); //Complementary sensitivity function
Ktf=ss2tf(K);Gtf=ss2tf(G);
Ttf=ss2tf(T);T11=Ttf(1,1);
Oloop=Gtf*Ktf;
Tn=Oloop*inv(eye(Oloop)+Oloop);
clean(T11-Tn(1,1));
//
[Pi,r]=augment(G,'T','i');
T1=lft(Pi,r,K);T1tf=ss2tf(T1); //Input Complementary sensitivity function
Oloop=Ktf*Gtf;
T1n=Oloop*inv(eye(Oloop)+Oloop);
clean(T1tf(1,1)-T1n(1,1))
]]></programlisting>
</refsection>
<refsection role="see also">
<title>See Also</title>
<simplelist type="inline">
<member>
<link linkend="lft">lft</link>
</member>
<member>
<link linkend="sensi">sensi</link>
</member>
</simplelist>
</refsection>
</refentry>
|
