Fault Detection with Uncertain Priors

Abstract

In this paper we discuss the design of a failure detection algorithm based on highly uncertain probability distributions for the ‘fail’ (F) and ‘no-fail’ (NF) cases. The structure of the algorithm is that NF is declared if and only if the measurement vector x falls in a pre-specifled domain D. The designer’s task is to choose the domain D so that the probability of missed detection is less than P0c and the probability of false alarm is less than P1c. Severe uncertainty in the F and NF prior probabilities is represented with info-gap models of uncertainty. The design procedure is developed and the trade-ofis between performance (in terms of P0c and P1c) and robustness to uncertainty are explored. An heuristic example is presented. I. Theoretical Background Measurements are made in order to decide whether the measured system is in the no-fail (NF) or fail (F) state. The measurement vector x is a random variable conditioned on the state of the system: NF or F. No-fail is declared if and only if x falls in a pre-specifled domain D. The best-estimated (but highly uncertain) probability density functions (pdfs) of x under NF and F are e 0 (x) and e 1 (x) respectively. These estimated pdfs are highly uncertain because of limited data, surprises in operating conditions, variability of the system, unanticipated efiects of failure, etc. We use info-gap models to represent this uncertainty 1;2 . An info-gap model is an unbounded family of nested sets of uncertain events, in our case, pdfs. The info-gap models for uncertainty in NF and F are denoted U0(fi; e