Environmental Engineering Reference
In-Depth Information
The objective here is to compute the gains L
a
(p) and K(p) such that the effect of
the input d in Eq. (
7.28
) is attenuated below the desired level c, to ensure robust
stabilisation performance.
Theorem 8.1: For t [ 0 and h
i
ðÞ
h
j
ðÞ6¼
0, The closed-loop fuzzy system in [
28
]
is stable and the H
?
performance is guaranteed with an attenuation level, provided
that the signal
ð
d
Þ
is bounded, if there exist SPD matrices P
1
,P
2
, matrices H
ai
;
Y
i
;
and scalar satisfying the following LMI constraints (
7.29
) and (
7.30
):
Minimise, such that:
P
1
[ 0
;
P
2
[ 0
ð
7
:
29
Þ
2
4
3
5
E
ðÞ
0
X
1
C
p
W
11
W
12
R
0
0
0
0
0
0
2lX
1
0
0
0
0
lI
0
0
0
0
0
2lI
0
0
0
0
lI
0
0
0
0
2lI
0
0
0
0
lI
0
0
0
2lI
0
0
0
0
lI
0
0
2lI
0
0
0
0
lI
0
\0
W
55
0
E
a
ðÞ
0
P
2
G
0
cI
0
0
0
0
cI
0
0
0
cI
0
0
cI
0
cI
ð
7
:
30
Þ
1
;
L
a
¼
P
2
H
ai
;
X
1
¼
P
1
;
X
1
¼
diagonal
ð
X
1
;
I
q
q
Þ
W
11
¼
A
i
X
1
þð
A
i
X
1
Þ
T
þ
BY
i
þð
BY
i
Þ
T
; W
12
¼
BY
i
where: K
i
¼
Y
i
X
1
½
0
;
Þ
T
H
ai
C
a
H
ai
C
a
Þ
T
.
W
55
¼
P
2
A
ai
þ
P
2
A
ai
ð
ð
Proof From Theorem 1, to achieve the performance and required closed-loop
stability of 27, the following inequality must hold [
13
]:
t x
ðÞþ
1
c
x
a
C
p
C
p
x
a
cd
T
d\0
ð
30
Þ
