Probabilistic Performance Analysis of Fault Diagnosis Schemes

v
w
u
G θ
y
F
r
δ
V
d
Figure 3.1. General parametric fault diagnosis problem. Faults affect the physical parameters of
the system, which in turn affect the system model parameter θ. The plant G is subject to a known
deterministic input u, a random input v, and a deterministic disturbance w. The residual generator
uses the plant input u and output y to produce a residual r , and the decision function δ uses the
residual r to produce a decision d about the current value of θ. Together, F and δ form a fault
diagnosis scheme, denoted V = (F,δ).
corresponding set of possible decisions is defined as
D := {0,1,..., q}.
The purpose of the fault diagnosis scheme V = (F,δ) is to produce a decision d k ∈ D, at each
time k, indicating which subset Θ dk ⊂ Θ most likely contains the parameter θ k . Of course,
the scheme V does not have direct access to the parameter. Instead, V must make a decision
based on the known input {u k } and the measured output {y k }, which is corrupted by the
noise signal {v k } and the disturbance {w k }. Therefore, the performance of the scheme V is
quantified by the probability that the correct decision is made.
The number of partitions q determines what type fault diagnosis problem the scheme V
is designed to address. If q = 1, the set Θ 1 contains all faulty parameter values, and V is
interpreted as a fault detection scheme. If q > 1, each subset Θ i ⊂ Θ represents a different
class of faulty behavior, and V is interpreted as a fault isolation scheme. If the parameter
space Θ is finite and each partition Θ i a singleton set, then V achieves fault identification,
as well. In Section 3.3, we define probabilistic performance metrics for the fault detection
problem (q = 1). Then, in Section 3.6, these results are extended to the more general fault
isolation problem (q > 1).
In this chapter and in Chapter 4, we assume that the deterministic input {u k } is known
and fixed, that there is no deterministic disturbance {w k }, and that G θ is a known function of
the parameter {θ k }. Chapter 5 extends these results by considering how uncertainty impacts
the performance metrics. In particular, Chapter 5 presents some techniques for computing
25