In this dissertation we study the problem of fault diagnosis in both discrete event systems and cyber physical systems. Discrete event systems (DESs) are event-driven systems with discrete states that evolve in response to abrupt occurrences of discrete changes (called events). The stochastic DESs are used to characterize the quantitative behavior of the system, by modeling the uncertainty on the occurrence of events as random variables with certain distribution. A stochastic DES is similar to the Markov chain models, with the difference being that, in stochastic DESs, the transition is labeled with the event while the event information is omitted in a Markov chain. Many physical systems, such as manufacturing systems, communication protocols, reactive software, telephone networks, traffic systems, robotics and digital hardware, can be modeled as DESs at a certain level of abstraction.

Fault diagnosis is to detect the occurrence of a fault so as to enable any fault tolerant actions. It is a crucial and challenging problem that has attracted considerable attentions in the literature of software engineering, automotive systems, power systems and nuclear engineering. In this dissertation, we propose the online detection schemes for stochastic DESs and also introduce the notions of missed detections (MDs) and false alarms (FAs), or equivalently, false-negatives and false-positives, for the schemes. The idea is that given any observation (of partially observed events), the detector recursively computes the conditional probability of the nonoccurrence of a fault and issues a "fault" decision if the probability of the nonoccurrence of a fault falls below an appropriately chosen threshold, and issues "no-decision" otherwise. We establish that S-Diagnosability is a necessary and sufficient condition for achieving any desired levels of MD and FA rates, where the notion of S-Diagnosability was proposed by Thorsley, et al. in 2005, requiring that given any tolerable ambiguity level &rho and error bound &tau , there

must exist a delay bound n such that for any fault trace, its extensions, longer than n and probability of ambiguity higher than &rho, occur with probability smaller than &tau . Algorithms for determining the detection scheme parameters of detection threshold and detection delay bound for the specified MD and FA rates requirement are also presented, based on the construction of an extended observer, which computes, for each observation sequence, the set of states reached in the system model, along with their probabilities and the number of post-fault transitions executed.

This dissertation also studies the fault diagnosis in cyber physical systems, where the dynamics of the physical systems over discrete sample instances are described by stochastic difference equations, and the nonfault behaviors are specified by linear-time temporal logic (LTL) formulas over sequences of requirement variables that are functions of inputs and states (just as the outputs). We first introduce the notion of an input-output stochastic hybrid automaton (I/O-SHA), and then show that it can be used to model the refinement of a given discrete-time stochastic system against its LTL specification so as to identify the system behaviors that satisfy the nonfault specification versus the ones that violate it in form of reachability of a fault location. For this we propose a refinement algorithm that refines the system model in form of discrete-time stochastic equations with respect to its specification model in form of a B uchi acceptor, and the resulting refinement can be modeled as an I/O-SHA. We further show that the fault detection problem then reduces to a state estimation problem for the I/O-SHA. The performance of the detection protocol is evaluated in terms of its FA and MD rates. We additionally propose the notion of S-Diagnosability for I/O-SHA, which can guarantee the existence of detectors that can achieve any desired FA and MD rates.

We further consider the fault prognosis problem, where the goal is to predict a fault prior to its occurrence, for stochastic DESs. We introduce m-steps Stochastic-Prognosability, or simply S_m-Prognosability, requiring for any tolerance level &rho and error bound &tau , there exists a reaction bound k &ge m, such that the set of fault traces for which a fault cannot be predicted k steps in advance with tolerance level &rho, occurs with probability smaller than &tau . Similar to the fault diagnosis problem, we formalize the notion of a prognoser that maps observations to decisions by comparing a suitable statistic with a threshold, and show that S_m-Prognosability is a necessary and sufficient condition for the existence of a prognoser with reaction bound at least _m (i.e., prediction at least _m-steps prior to the occurrence of a fault) that can achieve any specified FA and MD rate requirement. Moreover, we provide a polynomial algorithm for verifying S_m-Prognosability.