Environmental Engineering Reference
In-Depth Information
)
x
¼
A
ð
p
Þþ
Bu
þ
E
ðÞ
v
y
¼
Cx
þ
D
f
f
s
ð
7
:
22
Þ
A
ðÞ2R
n
n
ð¼
P
i
¼
1
h
i
ð
p
Þ
A
i
Þ
;
B
2R
n
m
;
E
ðÞ2R
n
m
v
ð¼
P
i
¼
1
h
i
ð
p
Þ
E
i
Þ
;
D
f
2
R
l
g
and C
2R
l
n
are known system matrices. r is the number of fuzzy rules and
) satisfying
P
i
¼
1
h
i
ðÞ¼
1, and 1
h
i
ðÞ
0
;
for all i.
An augmented system consisting of the Eq. (
7.22
) and the tracking error
integral
ð
e
t
¼
R
ð
y
r
Sy
ÞÞ
is defined as:
x
¼
A
ð
p
Þ
x
þ
Bu
þ
E
ð
p
Þ
v
þ
Ry
r
y
¼
Cx
þ
D
f
f
s
_
ð
7
:
23
Þ
;
x
¼
;
B
¼
;
E
ðÞ¼
0
E
ðÞ
;
R
¼
A
ðÞ¼
0
SC
0
e
t
x
0
B
I
0
A
ðÞ
;
D
f
¼
0
D
f
I
q
0
C
¼
0
C
where S
2R
w
l
is used to define which output variable is considered to track the
reference signal. Hence, the tracking problem is transferred to a fuzzy state
feedback control, for which the proposed control signal is:
u
¼
K
ð
p
Þ
^
x
ð
7
:
24
Þ
where K
ðÞ2R
m
ð
n
þ
w
Þ
ð¼
P
i
¼
1
h
i
ð
p
Þ
K
i
Þ
is the controller gain and
^
x
2R
n
þ
w
ð
Þ
is
the estimated augmented state vector.
As described in [
16
], if it can be assumed that the q
th
derivative of the sensor
fault signal is bounded, then an augmented state system comprising the states of
the original local linear system and the q
th
derivative of the f
s
, is given as follows:
u
i
¼
f
q
i
Þ
;
u
1
¼
f
s
ð
i
¼
1
;
2
;
...
;
q
; u
2
¼
u
1
; u
3
¼
u
2
;...; u
q
¼
u
q
1
s
Then the system of Eq. (
7.22
) with the augmented fault derivative states will
become:
x
a
¼
A
a
ðÞ
x
a
þ
B
a
u
þ
E
a
ðÞ
v
þ
R
a
y
r
þ
Gf
s
y
a
¼
C
a
x
a
ð
7
:
25
Þ
where
