Environmental Engineering Reference
In-Depth Information
We can assume that (triangular structure and the Lipschitz assumption on u)
D
h
u
ð
x
Þ
u
ð
x
Þ
ð
Þ
k
n
kk
with n
¼
K
J
k
f
ð
t
kk
d
It comes that
V
hV
þ
2
kk
2
k
max
ð
S
Þ
n
þ
2
kk
k
max
ð
S
Þ
d
h
then,
p
V
V
hV
þ
c
1
V
þ
c
2
d
h
with
<
c
1
¼
2
k
max
ð
S
Þ
k
min
ð
S
Þ
n
k
max
ð
S
Þ
p
k
min
ð
S
Þ
:
c
2
¼
2
Now taking
8
<
h
0
¼
max 1
;
c
fg
k
¼
q
k
max
ð
S
Þ
k
min
ð
S
Þ
l
h
¼
h
c
1
:
2
M
h
¼
2
k
max
ð
S
Þ
k
min
ð
S
Þ
h
c
1
ð
Þ
and h [ h
0
, we obtain
e
ð
t
kk
hk exp
ð
l
h
t
Þ
e
ð
0
k þ
M
h
d
ð
2
:
25
Þ
With T
a
¼
Jx
2
, it comes that
T
a
¼
T
a
T
a
J hk exp
ð
l
h
t
Þ
e
ð
0
k þ
M
h
d
½
ð
2
:
26
Þ
A practical estimate of the aerodynamic torque is then obtained as M
h
decreases
when h increases. The asymptotic estimation error can be made as small as desired
by choosing high enough values of h. However, very large values of h are to be
avoided in practice since the estimator may become noise sensitive.
Now, the control objective can be formulated by the following tracking errors
e
T
¼
T
opt
T
a
ð
2
:
27
Þ
