Probabilistic Graphical Models for Fault Diagnosis in Complex Systems

In this chapter, we discuss the problem of fault diagnosis for complex systems in two different contexts: static and dynamic probabilistic graphical models of systems. The fault diagnosis problem is represented using a tripartite probabilistic graphical model. The first layer of this tripartite graph is composed of components of the system, which are the potential sources of failures. The condition of each component is represented by a binary state variable which is zero if the component is healthy and one otherwise. The second layer is composed of tests with binary outcomes (pass or fail) and the third layer is the noisy observations associated with the test outcomes. The cause–effect relations between the states of components and the observed test outcomes can be compactly modeled in terms of detection and false alarm probabilities. For a failure source and an observed test outcome, the probability of fault detection is defined as the probability that the observed test outcome is a fail given that the component is faulty, and the probability of false alarm is defined as the probability that the observed test outcome is a fail given that the component is healthy. When the probability of fault detection is one and the probability of false alarm is zero, the test is termed perfect; otherwise, it is deemed imperfect. In static models, the diagnosis problem is formulated as one of maximizing the posterior probability of component states given the observed fail or pass outcomes of tests. Since the solution to this problem is known to be NP-hard, to find near-optimal diagnostic solutions, we use a Lagrangian (dual) relaxation technique, which has the desirable property of providing a measure of suboptimality in terms of the approximate duality gap. Indeed, the solution would be optimal if the approximate duality gap is zero. The static problem is discussed in detail and some interesting properties, such as the reduction of the problem to a set covering problem in the case of perfect tests, are discussed. We also visualize the dual function graphically and introduce some insights into the static fault diagnosis problem. In the context of dynamic probabilistic graphical models, it is assumed that the states of components evolve as independent Markov chains and that, at each time epoch, we have access to some of the observed test outcomes. Given the observed test outcomes at different time epochs, the goal is to determine the most likely evolution of the states of components over time. The application of dual relaxation techniques results in significant reduction in the computational burden as it transforms the original coupled problem into separable subproblems, one for each component, which are solved using a Viterbi decoding algorithm. The problems, as stated above, can be regarded as passive monitoring, which relies on synchronous or asynchronous availability of sensor results to infer the most likely state evolution of component states. When information is sequentially acquired to isolate the faults in minimum time, cost, or other economic factors, the problem of fault diagnosis can be viewed as active probing (also termed sequential testing or troubleshooting). We discuss the solution of active probing problems using the information heuristic and rollout strategies of dynamic programming. The practical applications of passive monitoring and active probing to fault diagnosis problems in automotive, aerospace, power, and medical systems are briefly mentioned.