Information Technology Reference
In-Depth Information
T
where
z
1
¼½
x
1
;
x
2
;
x
4
;
x
5
;
v
2
; C
l
and
e
3
is the tracking error of
x
3
.Itis
given by
e
3
¼
x
3
x
3d
ð
32
Þ
The desired signal
x
3d
is given by the
expression (
15
),
i.e.
.
The uncertain continuous function f
1
ð
a
2
a
1
u
s
þ u
s
1
x
3d
¼
z
1
Þ
can be approximated by the fuzzy
system (
12
) as follows
f
1
ð
T
z
1
; h
1
Þ
¼
h
1
w
1
ð
z
1
Þ
ð
33
Þ
where
fixed a priori by the designer, and
h
1
is the adjustable parameter vector of the fuzzy system. Furthermore, the
functions
f
1
ð
w
1
ð
z
1
Þ
is the FBF vector, which is
z
1
Þ
can be approximated optimally (Wang
1993
,
1994
)as
follows
f
1
ð
z
1
Þ
¼f
1
ð
z
1
; h
1
Þþe
1
ð
z
1
Þ
¼
h
T
1
ð
34
Þ
w
1
ð
z
1
Þþe
1
ð
z
1
Þ
h
1
is the optimal parameter vector and
where
is the unavoidable
fuzzy approximation error which is generally assumed to be bounded
(Boulkroune et al.
2008
,
2009
,
2010a
,
b
; Wang
1993
,
1994
) as follows
e
1
ð
z
1
Þ
j
e
1
ð
z
1
Þ
j e
1
;
8
z
1
2 X
z
1
e
1
is an unknown constant.
Since the input vector z
1
¼½
where
T
x
1
;
x
2
;
x
4
;
x
5
;
v
2
; C
l
is not available, it must
v
2
; C
l
T
be replaced by its estimate
^
z
1
¼½
x
1
;
x
2
;
x
4
;
x
5
;
in (
33
). Thus, the
fuzzy system (
33
) used to approximate f
1
ð
z
1
Þ
is replaced by the following
fuzzy system:
f
1
ð^
T
1
z
1
; h
1
Þ
¼
h
w
1
ð^
z
1
Þ
ð
35
Þ
From (
33
-
35
), we have
z
1
Þ
f
1
ð
z
1
; h
1
Þþ
f
1
ð
z
1
; h
1
Þ
f
1
ð^
z
1
; h
1
Þþ
f
1
ð^
z
1
; h
1
Þ
f
1
ð
z
1
Þ
¼
f
1
ð
¼ f
1
ð^
z
1
Þ
f
1
ð
z
1
; h
1
Þþ
f
1
ð
z
1
; h
1
Þ
f
1
ð^
z
1
; h
1
Þþ
z
1
; h
1
Þ
f
1
ð
¼
h
1
wð^
z
1
Þþ
½
h
1
w
1
ð
z
1
Þh
1
w
1
ð^
z
1
Þþe
1
ð
z
1
Þ
¼
h
1
wð
z
1
Þþ#
1
ð
z
1
;
z
1
Þ
ð
36
Þ
Search WWH ::
Custom Search