## 1 Introduction
KF assumes a linear system dynamic model with Gaussian noise in measurement, which is not always realistic in practical application. The extended Kalman filter (EKF) [23] is an extension of the Kalman filter to nonlinear system dynamics. EKF linearizes the nonlinear equation locally through the Taylor series expansion formula, so as to adapt to the nonlinear system. However, EKF must solve the Jacobi matrix of nonlinear function, which is complex and error prone. For systems with strong nonlinearity, the stability is relatively poor, which is easy to lead to the decline of filtering results, and can only meet the requirements of some weak nonlinear environments. The unscented Kalman filter (UKF) [22] is a combination of the unscented transform and the standard Kalman filter system. It uses the linear regression of sigma points to linearize the nonlinear function of variables, avoid the operation of derivatives, and make it more suitable for real strong nonlinear systems. However, in numerical calculations, UKF encounters some problems, such as rounding error, and negative definite covariance matrix, which leads to filter divergence. The derivative-free nonlinear Kalman filter [24] is a QR decomposition of the state variance matrix in the unscented Kalman filtering process, which can ensure the positive definiteness of the covariance matrix and overcome the divergence problem caused by the accumulation of calculation errors in the filtering processing No matter how the KF is improved, it still estimates the system under the linear and Gaussian assumptions. When the linearity of the system is very poor, the effect of the Kalman filter will be greatly reduced. Different from the Kalman filter method, particle filter (PF) [18] uses a group of particles with specific weights to approximate the state estimation. This method is not limited to Gaussian hypothesis and can approximate any system state distribution. It is especially suitable for nonlinear and non-Gaussian application scenarios, and has been widely used in residual life prediction. The standard PF algorithm uses the prior distribution of the system as the importance sampling distribution. After a certain number of iterative steps, the weight of many particles becomes minimal, resulting in particle degradation and further affecting the accuracy of residual life prediction. The unscented particle filter (UPF) [19]combines the advantages of the particle filter and the unscented Kalman filter, uses the UKF algorithm to obtain an appropriate importance sampling probability distribution, and makes full use of the latest observations, which can effectively solve the problem of particle degradation. In practical engineering applications, the environmental conditions of the bearing may change in real time, and its own operating conditions may also change, i.e., variable working conditions. Affected by environmental conditions, operating conditions and other uncertain factors, a variety of failure modes may occur during bearing operation, resulting in non-stationary vibration signals. When using the time-frequency analysis method to analyze non-stationary signals, there are some problems, such as low time-frequency resolution and false component interference, which make it difficult to extract and identify fault features. Under the condition of time-varying speed, the vibration impact interval and impact amplitude of the bearing are time-varying, resulting in the distortion of the vibration signal envelope spectrum obtained by the conventional envelope demodulation method, which makes it difficult to identify the fault characteristics. Order analysis [25], generalized demodulation (GD) [26], time frequency representation and phase-space dissimilarity measure [27] are common methods for vibration signal analysis under variable working conditions, which can accurately extract the degradation characteristics of the bearing life cycle, and are suitable for predicting the remaining useful life of rolling bearings under variable working conditions.