Information Technology Reference
In-Depth Information
⎡
⎤
z
2
+
l
cos(
α
)
⎣
⎦
.
z
=
z
2
z
2
−
l
cos(
α
)
It is important to remark that by definition one gets
l
<<
z
2
. Hence, the following
approximation can be done:
l
cos(
α
)
l
cos(
α
)
1
1
z
2
(1
)=
z
2
−
z
1
−
−
=
p
1
p
2
;
z
2
z
2
1
z
2
=
p
1
;
1
z
3
(15.25)
z
2
(1 +
l
cos(α)
)=
z
2
+
l
cos(α)
1
=
p
1
+
p
2
.
z
2
z
2
From the definition of the admissible intervals relative to
z
2
and
α
given by (15.2)
and (15.5), it follows that the scalars
p
1
and
p
2
in (15.25) satisfy
p
jmin
≤
p
j
≤
p
jmax
,
j
= 1
,
2
,
with the bounds
1
d
2
max
;
1
d
2
min
cos
(
p
1
min
=
p
1
max
=
η
max
)
;
(15.26)
lcos
(
−
π
+
α
min
)
lcos
(
−
α
min
)
p
2
min
=
η
max
))
2
;
p
2
max
=
η
max
))
2
.
(
d
2
min
cos
(
(
d
2
min
cos
(
Thus, the matrices
L
(
z
x
),
depend on two uncertain parameters
p
1
and
p
2
. Thus, from (15.25),
B
1
(
z
),
B
2
(
z
),
B
4
(
z
) and
B
5
(
z
) depend on two uncertain parameters
p
1
and
p
2
. By using the clas-
sical framework of uncertain systems [2], it follows that these matrices belong to a
polytope with 4 vertices given by the combinations of value of
p
1
and
p
2
in their
definition interval:
,
e
) and
B
(
z
,
e
), and therefore the matrices
A
(
z
,
x
) and
B
2
(
z
,
B
k
(
z
)
∈
Co
{
B
kj
,
j
= 1
,...,
4
},
for
k
= 1
,
2
,
4
,
5
.
(15.27)
Consequently, the closed-loop system (15.21) or (15.24) can be written as
4
j
=1
λ
j
{
[
A
1
j
+
R
T
(2)
B
2
j
R
+
R
T
(3)
(
B
3
+
D
(
e
))
R
+
B
1
K
]
x
+
x
=
(15.28)
R
(
B
4
j
+
D
(
e
)
B
5
j
)
ω
}
+
B
1
φ
(
K
x
)
,
4
j
=1
λ
j
= 1, λ
j
≥
0and
A
1
j
=
R
B
1
j
C
.
with
15.4
Control Design Results
This section is dedicated to the presentation of the main results of the chapter. Ex-
istence conditions are provided through Theorem 15.1 and Proposition 15.1. Then,
Search WWH ::
Custom Search