Probabilistic Performance Analysis of Fault Diagnosis Schemes

that deserve equal attention.
• Likelihood ratio tests: As stated in Chapter 3, likelihood ratio tests provide the highest
probability of detection for a given probability of false alarm (see Lemma 3.10).
A decision function based on a likelihood ratio test between two hypotheses H 0,k
and H 1,k can be written as
⎧
⎨0 if Λ(r 0:k ) > ε k
δ(k,r 0:k ) =
⎩1 otherwise,
where the likelihood ratio test statistic is defined as
Λ(r 0:k ) := p r (r 0:k | H 0,k )
p r (r 0:k | H 1,k ) .
Note that, at each time k, the decision function δ depends on the entire sequence of
residuals r 0:k . Therefore, δ must be written in terms of a dynamic decision function
with a state that “remembers” the past values of r k , or the decision function must
become increasingly complex with each time step.
• Decision functions based on norms: There are a number of decision functions in
the fault detection literature that are based on taking some norm of the residual
signal. For example, when the residual is vector-valued, the decision function may
be of the form
δ(k,r k ) := 1 ( ‖r k ‖ 2 > ε k
)
,
where 1 is the indicator function. Similarly, one may define a norm over some time
window T , as follows:
‖r 0:k ‖ 2,T :=
(
1
The corresponding decision function is
T
k∑
l=max{0,k−T +1}
‖r l ‖ 2 2
δ(k,r 0:k ) := 1 ( ‖r 0:k ‖ 2,T > ε k
)
.
Both of these norm-based decision functions can be found in the literature (see [24]
and references therein); however, neither of them fit the computational framework
presented here.
• Dynamic decision functions applied to correlated residuals: Recall that in Section
4.4.2, the state of the dynamic decision function is a Markov chain if and
only if the residuals are Gaussian and uncorrelated in time. This strong assumption
usually only occurs when the noise signal is added directly to the system output as
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