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
|
.TH ctr_gram 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
ctr_gram - controllability gramian
.SH CALLING SEQUENCE
.nf
[Gc]=ctr_gram(A,B [,dom])
[Gc]=ctr_gram(sl)
.fi
.SH PARAMETERS
.TP 10
A,B
: two real matrices of appropriate dimensions
.TP
dom
: character string (\fV'c'\fR (default value) or \fV'd'\fR)
.TP
sl
: linear system, \fVsyslin\fR list
.SH DESCRIPTION
Controllability gramian of \fV(A,B)\fR or \fVsl\fR (a \fVsyslin\fR
linear system).
.LP
\fVdom\fR character string giving the time domain : \fV"d"\fR for a
discrete time system and \fV"c"\fR for continuous time (default case).
.LP
.IG
.nf
/+inf ---+inf
[ At ' A't \\ k k
Gc = | e BB e dt Gc = > A BB' A'
] /
/0 ---0
.fi
.FI
.LA $$
.LA Gc = \int_0^{\infty} e^{At}BB'e^{A't}dt
.LA \qquad Gc = \sum_{0}^{\infty} A^k B B'{A'}^k
.LA $$
.SH EXAMPLE
.nf
A=diag([-1,-2,-3]);B=rand(3,2);
Wc=ctr_gram(A,B)
U=rand(3,3);A1=U*A/U;B1=U*B;
Wc1=ctr_gram(A1,B1) //Not invariant!
.fi
.SH SEE ALSO
equil1, obs_gram, contr, cont_mat, cont_frm, contrss
.SH AUTHOR
S. Steer INRIA 1988
|
