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Hidden or partially observable degradation state of the equipment is frequently encountered in many engineering practices. This may encourage the state space modeling technique as a feasible way to estimate equipment's remaining useful life (RUL). However, most of the existing state space models falling into this category are based on the assumptions that the degradation process is linear or can be linearized. Therefore, modeling the hidden degradation process under a general nonlinear function and deriving the corresponding analytical form of the RUL distribution are still challenging and have not been well solved in literature. In this paper, we present a state-space-based prognostic model to address the above issues, in which the nonlinearity is characterized by an age-dependent general nonlinear function. Specifically, we model the degradation process as the unobservable nonlinear drift-based Brownian motion (BM) and apply extended Kalman filter (EKF) and expectation-maximization (EM) algorithm to estimate and update the degradation state and the unknown parameters of the established model jointly. Furthermore, we derive the analytical form of the RUL distribution approximately which incorporates the uncertainty of the estimation for hidden state and can be real-time updated based on the available observations. For verifying our approach, a numerical example and a case study for a NASA battery are provided, and the results show that both the parameters and the RUL are estimated accurately. We also consider several different nonlinear functions and compare them with the linear model. The comparative results demonstrate our approach is better than the results in the linear case.