Modeling deterioration and predicting remaining useful life using stochastic differential equations

The deterioration of engineering systems might reduce the system reliability and prompt maintenance operations that may disrupt the ability of the systems to provide regular service. Estimating the Remaining Useful Life (RUL) of the system requires an understanding of the deterioration processes acting on it. Recent formulations have shifted the focus of deterioration modeling from the system as a whole to the individual, time-varying state-variables that define the characteristics of the system. These state-dependent formulations depend on the selected models for the evolution of the state-variables over time. However, most available models rely on simplifying assumptions that disregard the true nature of the processes, either by discretizing the time domain or by assuming independence among the several processes acting on the system. This paper proposes using a system of Stochastic Differential Equations (SDEs) to model the state variables' evolution. The proposed formulation captures the continuous nature of the processes and accounts for the possible interactions among them. In addition, results from stochastic calculus can be used to facilitate the simulation of the processes and to obtain closed-form solutions for the distribution of the state variables over time and the RUL of the system. Moreover, the proposed SDEs can be calibrated based on observations of the state variables, should those be obtained via testing/monitoring or simulations. A procedure for calibration is introduced, and several numerical examples are investigated to highlight the different scenarios encountered in practice. Finally, the proposed SDE formulation is used to predict the RUL of lithium-ion batteries using actual Structural Health Monitoring data.