Predicting remaining useful life based on a generalized degradation with fractional Brownian motion

For data-driven remaining useful life (RUL) prediction, an appropriate degradation model is critically important to achieve accurate prediction. The degradation processes in some practical systems are not only related to the age but also related to the current degradation state, and the degradation processes may be non-Markovian processes. However, most existing stochastic process-based degradation models only depend on the age, and simply assume that the increments are independent. In this paper, an age- and state-dependent degradation model with long-range dependence is developed, which is more general than most of the existing models based on either Brownian motions (BMs) or fractional Brownian motions (FBMs). The Radon-Nikodym derivative is utilized to obtain a likelihood ratio function of unknown parameters, and the estimates are obtained by maximizing the likelihood ratio function. A weak convergence theorem is introduced to approximate the FBM by a BM with a time-varying coefficient. A time-space transformation is further utilized to obtain an approximate explicit solution of the RUL. At last, numerical simulations and two real case studies of blast furnace walls and ball bearings are adopted to verify the effectiveness of the proposed model.