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given by the observer ( 5 ) and fuzzy bilinear model
( 2 ), the state estimation error ( 6 ) becomes
Using the expression of
x
^
ð
t
Þ
e
ð
t
Þ ¼ ð
I n þ
EC
Þ
x
ð
t
Þ
z
ð
t
Þ
ð
7
Þ
According to ( 2 ), ( 5 ) and ( 7 ), the dynamics of the state estimation error is
given by
0
1
H i e
ð
t
Þþ
ð
TA i
H i T
L i C
Þ
x
ð
t
Þ
X
r
@
A
þ
ð
TN i
M i C
Þ
x
ð
t
Þ
u
ð
t
Þþ
TF i d
ð
t
Þ
e
_
ð
t
Þ ¼
h i ðnð
t
ÞÞ
ð
8
Þ
þ
ð
TB i
J i
Þ
u
ð
t
Þ
1
with
¼
I n þ
ð
Þ
T
EC
9
If the following conditions hold true
8
i
¼
f
1
; ...;
r
g
,
TA i H i T L i C ¼
0
ð
10
Þ
TN i
M i C
¼
0
ð
11
Þ
TB i
J i ¼
0
ð
12
Þ
TF i ¼
0
ð
13
Þ
Then the equation of the observing error becomes
X
r
_
e
ð
t
Þ ¼
h i ðnð
t
ÞÞ
H i e
ð
t
Þ
ð
14
Þ
1
The problem of designing the fuzzy bilinear observer for the fuzzy bilinear
model with unknown input is reduced to find the observer gains such that the
equation of the dynamic estimation error ( 7 ) is stable. This aspect will be subject the
next subsection.
3.2 Design and Stability Analysis
This subsection proposes suf
cient linear design conditions to guarantee the global
asymptotic convergence of state estimation error. Therefore, the stability problem is
studied for two cases:
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