## 1 Introduction
As compared with the particle filter algorithm, their method significantly improved the prediction accuracy of the lithium battery RUL. [PERSON] et al. [25] adopted particle filtering method to estimate the system state and identify prediction model parameters. Data-driven predictors were used to learn system degradation patterns from the historical data and for predicting system evolution. Their method significantly improved the accuracy and transparency of prediction. Their investigation results demonstrated that the hybrid approach can integrate the advantages of different methods to improve the accuracy of the lithium-ion battery prognostics. However, due to various uncertainties in the battery degradation process, the prognostic performance of the lithium-ion battery will deteriorate when the prediction time is extended [7]. In addition, [PERSON] et al. [26; 27; 28; 29; 30] showed that capacity regeneration is a vital and uncertain factor affecting the RUL prediction accuracy. To describe capacity regeneration, [PERSON] et al. [30] proposed a prognostic framework based on the rest time for the SOH estimation of a lithium-ion battery. They predicted the final SOH of the lithium-ion battery by integrated the SOH value and the cycle number of the regeneration region and the global degradation trend extracted from the original SOH time sequence.Their framework described the capacity regeneration phenomenon and effectively improved the short-term RUL prediction accuracy, but the long-term prediction accuracy was still unsatisfactory. This was because the model had no measurements in the prediction stage, so the parameters of the model could not be updated. Therefore, [PERSON] et al. [15] introduced an iterative update method to improve the long-term prognostics performance of a lithium-ion battery. In the prediction stage, the measurements were obtained from the relevance vector machine (RVM), and the physical degradation model was optimized by a Kalman filter. Then, the optimized results were added into the training set to retrain the RVM, and the related vectors were dynamically adjusted to enter the next iteration cycle. The iterative updating strategy keep updating the RVM keep and improved the prediction accuracy. However, their iterative strategy could not eliminate the gradually accumulated prediction errors with cycling and aging, which would still affect the accuracy of a long-term prediction. To obtain a more accurate long-term prediction of RUL, we propose a new hybrid forecasting model using the RVM and the gray model (GM). Our work has improved on the prognostic framework proposed by [PERSON] et al. [30]. In addition, we decoupled capacity regeneration based on rest time. However, this study uses the exponential model to model the regeneration region instead of the linear model, which improves the modeling accuracy. Moreover, the RVM and the GM are used to describe the normal degradation trend. In the prediction phase, we use the RVM and the GM alternately to carry out a multi-step prediction. The prediction results of the RVM and GM model (RVM-GM) update the training dataset and the updated training dataset is used to retain the RVM-GM model. This iterative updating strategy can make the prediction results of the battery's SOH and RUL more accuracy and robust. Finally, we use the NASA battery datasets to verify the prediction performance and effectiveness of the proposed prognostic framework. The rest of this paper is organized as follows. In Section 2, the algorithm of the RVM and the GM algorithm are introduced. In Section 3, the hybrid model for prognostics of a lithium-ion battery is development. Experimental results and discussion are shown in Section 4 based on the National Aeronautics and Space Administration (NASA) battery dataset. The conclusion is discussed in Section 5.