Sensor Fault Diagnosis in Wind Turbines - Wind Turbine Control and Monitoring - page 264

where N ð l ; R Þ denotes the multinormal distribution with mean l and variance R.
The associated probability density function is the following:

1
ð 2p Þ n r detR

exp 1

2 ð r ð i Þ l Þ T R 1 ð r ð i Þ l Þ

p

p ð r ð i ÞÞ ¼ ð 10 : 9 Þ

Straightforward computations yield the following expression for s(i) in this
case:

r ð i Þ 1 s ð i Þ¼ð l 1 l 0 Þ T R 1

2 ð l 1 þ l 0 Þ

ð 10 : 10 Þ

Notice that the factor ð l 1 l 0 Þ T R 1 can be seen as a vector form of signal-to-
noise ratio that weights the contributions of the different components of r ð i Þ .
Let us now turn to the situation where several faults can possibly occur.

10.2.2 Detection/Isolation Algorithm

The problem amounts to generating an alarm when one out a set of n f possible
faults can occur, given a sequence of independent samples of a residual vector
f r ð 1 Þ; r ð 2 Þ; ... ; r ð k Þg . The corresponding hypothesis testing problem can be stated
as follows.
Data: A set of independent random vectors R 1 ¼f r ð 1 Þ; r ð 2 Þ; ... ; r ð k Þg where k
denotes the present time instant characterized by the probability density function
p h ð r ð i ÞÞ . The latter depends on a parameter vector h that takes value h 0 in fault free
mode and h ' ;'¼ 1 ; ... ; n f in fault mode ' (with h ' 6¼ h q ; 0 q 6¼ ' n f )
Problem: Choose between the following n f þ 1 hypotheses:

H 0 L ð r ð i ÞÞ ¼ p h 0 ð r ð i ÞÞ for i ¼ 1 ; ... ; k

H ' ;'¼ 1 ; ... ; n f L ð r ð i ÞÞ ¼ p h 0 ð r ð i ÞÞ for i ¼ 1 ; ... ; k 0 1

¼ p h ' ð r ð i ÞÞ

for i ¼ k 0 ; ... ; k

In order to decide that a fault of type ' has occurred, the log-likelihood ratio
between fault ' and all other possible fault modes, as well as the fault free mode
should be larger than a threshold h ' . The test function for fault ' can thus be
written:

ð S ' q
j

g ' ð k Þ¼ max
1 j k

ð R 1 ÞÞ min

0 q 6¼' n f