Probabilistic Performance Analysis of Fault Diagnosis Schemes

The dissertation explores the problem of rigorously quantifying the performance of a fault diagnosis scheme in terms of probabilistic performance metrics. Typically, when the performance of a fault diagnosis scheme is of utmost importance, physical redundancy is used to create a highly reliable system that is easy to analyze. However, in this dissertation, we provide a general framework that applies to more complex analytically redundant or model-based fault diagnosis schemes. For each fault diagnosis problem in this framework, our performance metrics can be computed accurately in polynomial-time.First, we cast the fault diagnosis problem as a sequence of hypothesis tests. At each time, the performance of a fault diagnosis scheme is quantified by the probability that the scheme has chosen the correct hypothesis. The resulting performance metrics are joint probabilities. Using Bayes rule, we decompose these performance metrics into two parts: marginal probabilities that quantify the reliability of the system and conditional probabilities that quantify the performance of the fault diagnosis scheme. These conditional probabilities are used to draw connections between the fault diagnosis and the fields of medical diagnostic testing, signal detection, and general statistical decision theory.Second, we examine the problem of computing the performance metrics efficiently and accurately. To solve this problem, we examine each portion of the fault diagnosis problem and specify a set of sufficient assumptions that guarantee efficient computation. In particular, we provide a detailed characterization of the class of finite-state Markov chains that lead to tractable fault parameter models. To demonstrate that these assumptions enable efficient computation, we provide pseudocode algorithms and prove that their running time is indeed polynomial.Third, we consider fault diagnosis problems involving uncertain systems. The inclusion of uncertainty enlarges the class of systems that may be analyzed with our framework. It also addresses the issue of model mismatch between the actual system and the system used to design the fault diagnosis scheme. For various types of uncertainty, we present convex optimization problems that yield the worst-case performance over the uncertainty set.Finally, we discuss applications of the performance metrics and compute the performance for two fault diagnosis problems. The first problem is based on a simplified air-data sensor model, and the second problem is based on a linearized vertical take-off and landing aircraft model.