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where:
"
#
A 0
00
A
0
¼
"#
B
0
ð
44
Þ
B
0
¼
Fa
1
=
EF
1
=
E
M
¼
F
0
In order to obtain the linear matrix inequality to solve the gains of the PID anti
windup compensator the following Lyapunov function must be considered:
z
T
Pz
V
ð
z
Þ
¼
ð
45
Þ
where P is a positive de
nite matrix, used in order to ensure the stability of the
system. Deriving the Lyapunov function the following result is obtained:
V
A
0
þ
B
0
M
T
Pz
A
0
þ
B
0
M
z
T
z
T
P
ð
z
Þ
¼
ð
Þ
þ
ð
Þ
z
ð
46
Þ
where in linear matrix inequality form (
45
) is represented as:
A
0
þ
B
0
M
T
P
A
0
þ
B
0
M
ð
Þ
þ
P
ð
Þ
\
0
ð
47
Þ
So by solving the following LMI the controller parameters of the equivalent
system are found (Fujimori
2004
):
T
P
ð
A
0
þ
B
0
M
Þ
þ
P
ð
A
0
þ
B
0
M
Þ
0
ð
Þ
\
0
48
0
0
For the H
∞
synthesis, a similar approach is implemented to
find the controller
gains, considering the following criteria:
k
T
2
x
ð
s
Þ
k
1
\
c
ð
49
Þ
where T
2
ˉ
T2ω(s) is the closed loop transfer function of the model (Fujimori
2004
;He
and Wang
2006
; Rehan et al.
2013
) and
> 0 is a positive constant that indicates the
desired performance. Then the respective LMI is needed to
ʳ
find the gain F and the
solutions of the anti windup PID controller.
2
3
PA
cl
þ
P
T
A
cl
0
C
cl
4
5
\
c
0
ð
50
Þ
0
I
0
C
cl
0
c
I
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