Probabilistic Performance Analysis of Fault Diagnosis Schemes

metrics at time k are defined as
Ĵ k (i , j ) := P( ˆD j,k ∩ H i ,k ), i , j ∈ D.
For each k, the value Ĵ k (i , j ) is the probability that the system is in configuration s i when it
should be in configuration s j . Note that the event ˆD j,k ∩H i ,k may or may not represent a safe
state of affairs, depending on the values of i and j . For example, when the j th fault occurs
(i.e., {θ k } enters the set Θ j ), the system is designed to reconfigure to a back-up mode s j .
Hence, it would be unsafe to continue operation in the nominal configuration s 0 when the
j th fault occurs. In any case, the probability that the system is in a safe configuration at
time k can be computed by summing the appropriate entries of Ĵ k .
4.5 Algorithms for Computing Performance
In this section, we present high-level algorithms for computing the performance metrics.
First, we consider systems that satisfy the restrictions discussed in Sections 4.2–4.4. Then,
we consider a special case, based on Sections 4.2.2 and 4.3.3, that consists of an ltv system
with L independent additive faults. Finally, this special case is further simplified by assuming
that the dynamics are lti. For each system class, the time-complexity of computing the
performance metrics is analyzed.
4.5.1 Sufficiently Structured Systems
Suppose that the fault parameter sequence θ is a tractable Markov chain satisfying the
conditions of Theorem 4.11 or 4.12. Also, assume that the combined clg dynamics of G θ
and F can be written in the form of equation (4.12), and assume that the decision function
δ is such that the probability
P(D 0,k | θ 0:k = ϑ 0:k )
can be computed in O(1) time. The most common class of decision functions meeting this
last criterion is the class of threshold functions.
If all these assumptions hold, then the joint probability performance metrics {P tn,k },
{P fp,k }, {P fn,k }, and {P tp,k } are computed using Algorithm 4.1. This algorithm consists of two
nested for-loops. The outer loop (Lines 1–21) considers all possible mode sequences, while
the inner loop (Lines 2–20) updates the performance metrics at each time step. The inner
loop can be divided into three parts, as follows:
• Lines 3–7 compute the probability of the fault parameter sequence ϑ 0:N .
• Lines 8–11 update the recurrences for the mean ˆr k and variance Σ k of the residual,
conditional on the event {θ 0:k = ϑ 0:k }.
68