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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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