Acuña Ureta, David E.
(2020)
Theoretical foundations of uncertain event prognosis and predictive Bayesian Cramer-Rao bounds.
PhD thesis, University of Nottingham.
Abstract
The reliability and operational continuity of systems have become increasingly important as technological progress translates into products that are gradually being incorporated and standardised in society. Some typical examples that can be named are the use of satellites for communications and location, biomedical engineering devices such as magnetic resonance, computers, machinery for manufacturing processes such as lathes or milling machines, vehicles, defense, among others.
Hereby, the probabilistic fundamentals of the failure prognosis problem are revisited, presenting constructive grievance to inconsistencies found in strategies that have been followed by many researchers. Moreover, a theoretically rigorous formalisation is developed, thus presenting new semi-closed probability distributions for the first occurrence time of any sort of event (not necessarily a failure) in both discrete- and continuous-time dynamic systems. A new concept of ''uncertain event'' is introduced as well, generalising the typical threshold-crossing approach to declare the occurrence of events. This generalisation allows uncertainty in the definition of events, making them uncertain. These new concepts are illustrated through a case study of fatigue crack growth prognosis.
Due to a lack of mathematical formalism with respect to the failure prognosis problem, heuristic methods were developed to assess quality of results. In this regard, another two contributions presented hereby aim at tackling this issue in a more formal fashion, namely, by using Bayesian Cramer-Rao Lower Bounds (BCRLBs) for the Error Covariance Matrix (ECM) of predictive estimates. Both contributions are illustrated in the light of the problem of End-of-Discharge (EoD) time prognosis of Lithium-Ion batteries.
Within the prognosis problem, uncertainty of system states is propagated over time, yielding probability distributions that end up characterising the first occurrence time of future events. Therefore, there are novel BCRLBs associated with the ECM of future system states (at each future time instant) as well as to the Mean Squared Error (MSE) of the first occurrence time of future events (the concept of ECM reduces to MSE in one dimension). The former (related to future system states) are used to propose a step-by-step design methodology to adjust prognostic algorithms hyper-parameters, permitting to guarantee that results do not violate fundamental precision bounds. The latter (related to the first occurrence time of future events), however, are presented and analysed in terms of potential use in the analysis and design of prognostic algorithms.
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