Environmental Engineering Reference
In-Depth Information
where the exponent
'
q refers to the log-likelihood ratio between hypotheses
'
and q.
As above, the maximization over j aims at determining the most likely fault
occurrence time. An alarm for fault
' 2f
1
;
...
;
n
f
g
will be triggered at time t
a
such
that
t
a
¼
inf
f
k
1 : g
'
ð
k
Þ
[ h
'
g
where h
'
is a user-defined threshold that depends on the specifications regarding
the probability of false alarm and missed isolation. It turns out that the decision
function can be implemented in a recursive way while guaranteeing attractive
optimality properties of the algorithm [
9
,
10
]. Let us introduce the following
notation for the decision function of the CUSUM algorithm between hypothesis
H
'
and H
0
:
g
'
0
ð
k
Þ¼
max
ð
0
;
g
'
0
ð
k
1
Þþ
s
'
0
ð
k
ÞÞ
ð
10
:
11
Þ
where s
'
0
ð
k
Þ¼
ln
p
h
'
ð
r
ð
k
ÞÞ
p
h
0
ð
r
ð
k
ÞÞ
. Then recursive computation of the decision functions
can be written:
g
'
ð
k
Þ¼
min
0
q
6¼'
n
f
ð
g
'
0
ð
k
Þ
g
q0
ð
k
ÞÞ
' ¼
1
;
...
;
n
f
ð
10
:
12
Þ
where g
0
;
0
ð
k
Þ¼
0, and an alarm is generated when
g
'
ð
k
Þ
[ h
'
for some
' ¼
1
;
...
;
n
f
ð
10
:
13
Þ
The detection/isolation algorithm can be summarized as follows:
• Initialization:
Set g
'
0
ð
0
Þ¼
0
;'¼
1
;
...
;
n
f
.
• Upon receipt of the kthresidual sample,
perform the following operations:
-
Compute the n
f
CUSUM test functions according to Eq. (
10.11
).
-
Decision
Compute g
'
ð
k
Þ
from Eq. (
10.12
), for
' ¼
1
;
...
;
n
f
.
If g
'
ð
k
Þ
[ h
'
, then an alarm for fault
'
is issued at the kth time instant
and the algorithm stops.
We now turn to some practical issues.
