An Age- and State-Dependent Nonlinear Prognostic Model for Degrading Systems

Nonlinearity and stochasticity are two important factors contributing to the degradation processes of complicated systems, and thus have to be taken into account in stochastic degradation modeling based prognostics. However, current studies almost always focus on age-dependent stochastic degradation models, most of which are linear, or can be transformed into linear models. In this paper, we propose a general age- and state-dependent nonlinear degradation model for prognostics. In the presented model, a diffusion process with age- and state-dependent nonlinear drift and volatility coefficients is utilized to characterize the dynamics and nonlinearity of the degradation progression. To derive the estimated remaining useful life distribution, the considered diffusion process is first converted into a diffusion process with age- or state-dependent nonlinear drift but constant volatility through Lamperti transformation. Then, based on a well-known time-space transformation, we obtain an analytical approximated remaining useful life distribution in the concept of the first passage time. Furthermore, a maximum likelihood estimation method for unknown parameters in the concerned model is presented on the basis of closed-form approximated degradation state transition density functions by the Hermite-expansion method. An illustrative example is provided to show how the obtained results can be applied to a specific age- and state-dependent nonlinear degradation model. Finally, the presented model is fitted to bearing degradation data. Comparative results suggest the necessity of age- and state-dependent nonlinear degradation modeling in prognostics.