Probabilistic Performance Analysis of Fault Diagnosis Schemes

s 2
. ..
s 0
s 1
s q
Figure 4.4. State-transition diagram for a system that reconfigures when a fault occurs. The state s 0
represents the nominal configuration, while state s i , i ≠ 0, represents the configuration that is used
when d k = i . Since the fault diagnosis problem essentially restarts when a reconfiguration occurs, only
one level of reconfiguration is shown.
Systems that Reconfigure when a Fault is Detected
Thus far, we have considered fault diagnosis problems in which the decision sequence {d k }
may be nonzero at one instant and then return to zero at the next. Sometimes, however,
it is useful to consider the case where some action is taken once {d k } is no longer zero. In
particular, we consider the case where the system is reconfigured when d k ≠ 0. For example,
if d k = i indicates that component i has failed at or before time k, then the system G θ
should be reconfigured to no longer use that component. Similarly, the fault diagnosis
scheme V = (F,δ) must also be reconfigured. Once the system G θ and scheme V have been
reconfigured, a new fault diagnosis problem begins. In this section, we demonstrate that
such reconfigurations can be modeled by a dynamic decision function, so that the property
of being in a given configuration can be computed efficiently using our performance analysis
framework.
Suppose that V = (F,δ) is a fault diagnosis scheme designed for the plant G θ in its
nominal configuration, such that d k = δ(k,r k ) takes values in the set D = {0,1,..., q}. Let s 0
denote the original configuration of G θ and V . Similarly, for i = 1,..., q, let s i denote the
reconfiguration of the system and scheme that takes place when d k = i . Assume that, after
reconfiguration, there is no returning to the original configuration s 0 . Hence, the set of
possible reconfigurations is governed by the state-transition diagram shown in Figure 4.4.
Let the sequence {z k } represent the configuration at each time step, and let { d ˆ k } be a
new sequence of decisions that is given by the recurrence
⎧
⎨δ(k,r k ) if z k−1 = 0,
z k =
⎩ otherwise,
ˆ d k = z k ,
z k−1
where z −1 = 0. This recurrence defines a dynamic decision function that decides which configuration
is in use at each point in time. Note that the state space of {z k } is Z = {0,1,..., q}.
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