Probabilistic Performance Analysis of Fault Diagnosis Schemes

formance metrics can be computed in polynomial time. In particular, we state and prove a
number of theoretical results regarding Markov chains with finite state spaces. In this chapter,
we also explore a simplified class of systems, based on independent faults with additive
effects, for which the performance metrics can be computed even more efficiently. Finally,
we present pseudocode algorithms for computing the performance metrics and we prove
that their running time is indeed polynomial, given that the aforementioned conditions are
met.
Chapter 5 extends the results of Chapters 3 and 4 by considering fault diagnosis problems
with some uncertain aspect. In particular, we examine systems with uncertain inputs,
unknown disturbances, uncertain fault signals, and unmodeled or uncertain system dynamics.
For each type of uncertainty, we consider the problem of computing the worst-case
values of the performance metrics over the given uncertainty set. Hence, these performance
analyses take the form of optimization problems. We show that, under some reasonable
assumptions, these optimization problems can be written as convex programs, which are
readily solved using off-the-shelf numerical optimization packages.
Chapter 6 describes some practical applications of the performance metrics and demonstrates
these applications on numerical examples. More specifically, we discuss how the
performance metrics can be used in engineering applications such as trade studies, selecting
a fault diagnosis scheme, and safety certification. We demonstrate some of these
applications using two examples. The first is an air-data sensor system, which measures an
aircraft’s airspeed and altitude. The second example is a linearized model of the longitudinal
dynamics of a fixed-wing vertical take-off and landing (vtol) aircraft.
Finally, Chapter 7 summarizes the conclusions drawn from this research work and
discusses some avenues for future research.
1.2 Thesis Contributions
1. Performance of fault detection schemes: In Chapter 3, we present a rigorous probabilistic
framework that can be used to assess the performance of any fault diagnosis
scheme applied to a system with a parametric fault model. Unlike existing performance
analyses, the performance metrics produced by this framework capture the
time-varying nature of the fault-diagnosis problem. Moreover, this framework can be
applied to the problems of fault detection, fault isolation, and fault identification.
2. Time-complexity analysis: By closely examining the time-complexity of each step
in computing the performance metrics, we arrive at a broad class of fault diagnosis
problems for which our performance analysis is computationally tractable.
• Efficient Algorithms: We present algorithms for efficiently and accurately computing
the performance metrics without resorting to Monte Carlo methods or approxima-
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