Environmental Engineering Reference
In-Depth Information
Let us therefore consider the following observer
y
¼
B
2
sgn
ð
e
Þ
W
¼
y
B
1
jj
2
sgn
ð
e
Þ
ð
2
:
40
Þ
where the constants B
1
and B
2
are defined as
(
B
2
[ /
B
1
[
4 B
2
þð Þ
2
ð
2
:
41
Þ
2 B
2
/
ð
Þ
Lets us consider the following tracking error.
8
<
e
¼
W
z
e
¼
y
B
1
jj
2
sgn
ð
e
Þ
z
e
¼
B
2
sgn
ð
e
Þ
B
1
ð
2
:
42
Þ
:
e
2
jj
2
z
Assume now, for simplicity, that the initial values are e
¼
0 and e
¼
e
0
[ 0at
t = 0. Let e
M
be the intersection of the curve e
¼
B
2
/
ð
Þ
with e
¼
0,
jj
\/.
We have then
Þ
e
0
2e
M
B
2
/
ð
ð
2
:
43
Þ
Þ
2e
1
=
2
M
B
1
e [ 0
;
e\
B
2
þ
/
ð
)
e [ 0
Thus, the majorant curve with e [ 0 may be taken as
8
<
e
0
¼
2 B
2
/
ð
Þ
e
M
e
ð
Þ
with e [ 0
Þ
2e
2
B
1
e
¼
e
M
with 0
e [
B
2
þ
/
ð
ð
2
:
44
Þ
:
Þ
2e
2
M
B
1
Þ
2e
2
B
1
e
¼
e
M
¼
B
2
þ
/
ð
with e\
B
2
þ
/
ð
Let the trajectory next intersection with e
¼
0 axis be e
1
. Then, obviously
t
2
B
1
2
Þ
2
ð
B
2
þ
/
j
e
1
=
e
0
j
q
with q
¼
e
M
=
e
0
j
j ¼
ð
2
:
45
Þ
2 B
2
/
ð
Þ
Extending the trajectory into the half plane e\0 and carrying-out a similar
reasoning show that successive crossings of the e
¼
0 axis satisfy the inequality
