Probabilistic Performance Analysis of Fault Diagnosis Schemes

tic metrics that rigorously quantify the performance of a reasonably general class of fault
diagnosis schemes that includes many model-based schemes. Of course, such metrics
are only useful if they are efficiently computable. Monte Carlo methods [79] provide a
general-purpose solution to this problem, but it can be difficult to quantify the error present
in the results. Moreover, component failures are inherently rare by design, so a thorough
Monte Carlo analysis would entail the subtleties and complications of rare-event simulation
[1]. In this dissertation, we take a more practical approach—we establish a class
of linear systems and fault diagnosis schemes for which the performance metrics can be
efficiently computed without resorting to approximations. We also consider the effects of
adding uncertainty to various aspects of the fault diagnosis problem. Again, emphasizing
the need for computational tractability, we describe a set of uncertainty models for which
the worst-case performance can be efficiently and accurately computed without the need
for approximation.
1.1 Thesis Overview
The terminology and notation used throughout this dissertation are established in Chapter 2.
For the sake of brevity, only the most basic concepts of probability and reliability theory
are introduced. In addition to the core definitions, we present two probabilistic models for
component failure times. In this chapter, we also give a brief survey of the field of fault
diagnosis. After defining the key terminology used in fault diagnosis, we present a survey
of some of the most popular techniques used to design fault diagnosis schemes, and we
discuss some of the strategies used to design more reliable systems. Finally, we present a
survey of the existing performance analysis techniques that can be found in the literature.
Chapter 3 examines the quantitative performance analysis of a class of fault diagnosis
problems, in which faults affect the system via a stochastic parameter. First, we cast the
problem of fault detection as a sequence of hypothesis tests regarding the value of the fault
parameter at each time. Building on the vast hypothesis testing literature, we establish a
set of joint probabilities that fully quantify the time-varying performance of a given fault
diagnosis scheme. Bayes’ rule is then used to decompose these performance metrics into
two parts: conditional probabilities that characterize the performance of the fault diagnosis
scheme and marginal probabilities that characterize the reliability of the underlying system.
The receiver operating characteristic, a popular tool in hypothesis testing, medical diagnostic
testing, and signal detection theory, is used to develop a set of informative visualizations.
Finally, the performance analysis framework is extended to the more general problems of
fault isolation and fault identification.
In Chapter 4, we examine the computational issues involved in evaluating the performance
metrics. By examining each component of the fault diagnosis problem separately,
we arrive at a set of sufficient conditions and assumptions, which guarantee that the per-
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