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Dynamic multiple fault diagnosis (DMFD) is a challenging and difficult problem due to coupling effects of the states of components and imperfect test outcomes that manifest themselves as missed detections and false alarms. The objective of the DMFD problem is to determine the most likely temporal evolution of fault states, the one that best explains the observed test outcomes over time.Here, we discuss four formulations of the DMFD problem. These include the deterministic situation corresponding to a perfectlyobserved coupled Markov decision processes, to several partiallyobserved factorial hidden Markov models ranging from the case where the imperfect test outcomes are functions of tests only to the case where the test outcomes are functions of faults and tests, as well as the case where the false alarms are associated with the nominal (fault-free) case only. All these formulations are intractable NP-hard combinatorial optimization problems. We solve each of the DMFD problems by decomposing them into separable subproblems, one for each component state sequence.Our solution scheme can be viewed as a two-level coordinated solution framework for the DMFD problem. At the top (coordination) level, we update the Lagrange multipliers (coordination variables, dual variables) using the subgradient method. The top level facilitates coordination among each of the subproblems, and can thus reside in a vehicle-level diagnostic control unit. At the bottom level, we use a dynamic programming technique (specifically, the Viterbi decoding or Max-sum algorithm) to solve each of the subproblems. The key advantage of our approach is that it provides an approximate duality gap, which is a measure of suboptimality of the DMFD solution. Interestingly, the perfectly-observed DMFD problem leads to a dynamic set covering problem, which can be approximately solved via Lagrangian relaxation and Viterbi decoding. Computational results on real-world problems are presented.

Abstract-Imperfect test outcomes, due to factors such as unreliable sensors, electromagnetic interference, and environmental conditions, manifest themselves as missed detections and false alarms. The main objective of our research on on-board diagnostic inference is to develop near-optimal algorithms for dynamic multiple fault diagnosis (DMFD) problems in the presence of imperfect test outcomes. Our problem is to determine the most likely evolution of fault states, the one that best explains the observed test outcomes. Here, we develop a primal-dual algorithm for solving the DMFD problem by combining Lagrangian relaxation and the Viterbi decoding algorithm in an iterative way. A novel feature of our approach is that the approximate duality gap provides a measure of suboptimality of the DMFD solution.

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The Set-covering problem has been widely used to model many real world applications. In this paper, we formulate a time-dependent set covering problem, viz., dynamic set-covering (DSC), which involves a series of coupled set-covering problems over time. We motivate the DSC problem from the viewpoint of a fault diagnosis problem, wherein multiple faults may evolve over time and the fault-test dependencies are deterministic. The objective of the DSC problem is to evaluate the most likely evolution of the minimal set of columns (component fault states) covering the rows (failed tests) of the DSC constraint matrix at a minimum cost or maximum reward. The DSC problem is an NP-hard and intractable due to the coupling among the rows and columns via the constraint matrix, and the temporal dependence of columns over time. By relaxing the constraints using Lagrange multipliers, the DSC problem can be decoupled into several subproblems; each corresponding to a column of the constraint matrix. Each subproblem is solved using the Viterbi decoding algorithm, and a primal feasible solution is constructed by modifying the Viterbi solutions via a heuristic. The Lagrange multipliers are updated using the subgradient method. The proposed primal-dual optimization framework provides a measure of suboptimality via approximate duality gap.