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In-Depth Information
2
4
3
5
2
4
3
5
2
4
3
5
01 0
0
0
0
x
1
x
2
x
3
x
4
x
5
x
6
x
1
x
2
x
3
x
4
x
5
x
6
f
r
I
1
I
2
P
2
a
1
a
3
I
2
P
2
a
1
C
1
I
2
P
2
a
2
a
4
I
1
P
2
a
2
C
2
I
1
P
2
0
00 0
1
0
0
¼
a
3
ð
a
2
mI
2
Þ
P
1
c
1
ð
a
2
mI
2
Þ
P
1
a
1
f
r
I
2
P
1
a
1
a
2
a
4
P
1
a
1
a
2
a
4
P
1
0
00 0
0
0
1
a
1
a
2
a
3
I
2
P
3
a
1
a
4
I
2
a
1
a
4
mI
1
I
2
P
3
a
1
C
2
I
2
þ
a
1
C
2
mI
1
I
2
P
3
a
2
f
r
I
1
P
2
a
1
a
2
C
1
P
2
0
2
4
3
5
0
I
1
I
2
P
2
0
a
1
I
2
P
1
0
a
2
I
1
P
2
þ
u
ð
Þ
25
where
a
1
I
2
a
2
þ
P
1
¼
I
1
ð
mI
2
Þ
ð
26
Þ
P
2
¼ a
2
I
1
þ
a
1
I
2
mI
1
I
2
ð
27
Þ
a
2
I
1
þ
a
1
I
2
P
3
¼
a
1
I
2
ð
mI
1
I
2
Þ
ð
28
Þ
The block diagram of the linear state feedback controller to control the parallel-
double-inverted pendulum is illustrated in Fig.
19
. The control effort of the state
feedback controller is obtained as follows
þ
þ
u
¼
K
1
x
1
x
1
;
d
K
2
x
2
x
2
;
d
K
3
x
3
x
3
;
d
þ
ð
29
Þ
þ
K
4
x
4
x
4
;
d
K
5
ð
x
5
x
5
;
d
Þþ
K
6
ð
x
6
x
6
;
d
Þ
T
where x
d
¼½
x
1
;
d
;
x
2
;
d
;
x
3
;
d
;
x
4
;
d
;
x
5
;
d
;
x
6
;
d
is the vector of the desired states and
K
1
;
K
2
;
K
3
;
K
4
;
K
5
;
K
6
K
is the vector of design variables obtained via the
optimization algorithm. The boundaries of the system are:
The boundary of the control effort is u
¼½
The boundary of the length of x
1
;
x
3
and x
5
are x
jj
jj
20
½
N
0
:
5 ½m
, x
jj
0
:
174 ½rad
,
x
jj
0
:
174
½rad
.
final state vector, and the boundaries of design variables
are as follows. Furthermore, the values of the parameters of the system of a parallel-
double-inverted pendulum are presented in Table
19
.
The initial state vector,
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