Environmental Engineering Reference
In-Depth Information
where t x
ðÞ
is the time derivative of the candidate Lyapunov function
ð
t x
ðÞ¼
x
a
Px
a
;
where P [ 0
Þ
for the augmented system 27. Using Eq. (
7.27
), inequality 30
becomes:
t x
ðÞ¼
X
r
x
a
þ
x
a
P N
i
d
þ
d
T
N
i
Px
a
A
i
P
þ
PA
i
x
a
ð
7
:
31
Þ
i
¼
1
After simple manipulation, inequality Eq. (
7.30
) implies that the inequality
Eq. (
7.32
) must hold:
"
#
\0
A
ij
P
þ
PA
ij
þ
c
I
P N
ij
ð
7
:
32
Þ
N
ij
P
cI
To be consistent with [
27
] P is structured as follows:
[ 0
P
1
0
P
¼
ð
7
:
33
Þ
0
P
2
Then after simple manipulation and using the variable change
ð
H
ai
¼
P
2
L
a
ð
p
ÞÞ
the inequality [
32
] can be re-formulated as:
2
3
P
1
½
BK
j
0
P
1
E
ð
p
Þ
X
11
P
1
R
0
4
5
X
22
0
0
P
2
G
P
ij
¼
cI
0
0
\0
ð
7
:
34
Þ
cI
0
ð
P
2
G
Þ
T
0
0
0
cI
where:
X
11
¼
A
i
X
1
þð
A
i
X
1
Þ
T
þ
BY
i
þð
BY
i
Þ
T
þ
1
c
C
p
C
p
Þ
T
H
i
C
a
H
i
C
a
Þ
T
X
22
¼
P
2
A
ai
þ
P
2
A
ai
ð
ð
A single step design formulation of the matrix inequality in [
34
] is proposed to
avoid the complexity of separate design steps characterised by repeated iteration
to determine the gains required. Hence, P
ij
as shown in [
34
] becomes:
P
11
P
12
P
ij
¼
ð
7
:
35
Þ
P
22
where
