On Prognostic Algorithm Design and Fundamental Precision Limits in Long-Term Prediction

Abstract

System states are related, directly or indirectly, to health condition indicators. This fact has encouraged the development of a series of failure prognostic frameworks based on Bayesian processors (e.g., particle or unscented Kalman filters) to efficiently estimate the Time-of-Failure (ToF) Probability Mass Function (PMF) in nonlinear, non-Gaussian, systems with uncertain future operating profiles. However, the assessment of the effectiveness of these methods has always been a concern for the Prognostics and Health Management (PHM) community. This chapter tackles this issue, providing a formal mathematical definition of the prognostic problem and a rigorous analysis for performance metrics based on the concept of Bayesian Cramer–Rao Lower Bounds (BCRLBs) for the predicted state mean square error (MSE) in prognostic algorithms. Furthermore, a step-by-step design methodology to tune prognostic algorithm hyper-parameters is explored, allowing to guarantee that obtained results do not violate fundamental precision bounds for ToF estimates. The design methodology distinguishes between hyper-parameters that affect the efficiency of the implementation and those that have impact on the efficacy of obtained results, providing a structured procedure to explore different combinations that could improve the characterization of the ToF PMF. It is shown how this design procedure allows detecting situations in which the prognostic algorithm implementation generates results that violate these fundamental precision bounds. In addition, the impact of a relaxation in efficiency constraints on the outcome of the prognostic algorithm is measured, helping the designer to take an informed decision on the hardware that is required to implement the algorithm in real-time applications. These concepts are applied to the problem of End-of-Discharge (EoD) time prognostics in lithium-ion batteries as an illustrative example.