Prognostic Algorithms Design Based on Predictive Bayesian Cramér-Rao Lower Bounds * *This work has been partially supported by FONDECYT Chile Grant Nr. 1170044, CONICYT PIA Project ACT1405, and the Advanced Center for Electrical and Electronic Engineering, Basal Project FB0008.

Abstract

Abstract In the area of failure prognosis, system states are typically related to critical variables whose future evolution in time might significantly affect the health condition of the process, thus yielding into a critical failure at a particular time instant -typically referred to as the Time-of-Failure (ToF). Prognostic frameworks based on Bayesian processors, such as particle filtering, have already demonstrated their efficiency when trying to estimate the probability of failure in nonlinear, non-Gaussian, systems with stochastic operating profiles. However, even in those cases, it is still unclear how to measure the efficacy of the obtained results. For this purpose, it is first necessary to establish adequate performance metrics, and the Prognostics and Health Management (PHM) community has not found a convincing theory that could help to provide adequate performance indicators yet. This article represents a first step towards the solution of this problem by focusing on a rigorous mathematical definition of the prognostic problem, and defining novel performance metrics based on Bayesian Cramer-Rao Lower Bounds for the predicted state mean square error (MSE) conditional to measurement data and model dynamics. Furthermore, we also use these performance metrics to design a step-by-step methodology aimed at tuning the parameters of prognostic algorithms; guaranteeing that the precision of obtained results does not violate these fundamental bounds. This new metric is applied on the design of prognostic algorithms for the problem of State-of-Health monitoring on lithium-ion batteries.