## 3 RUL Prediction Methodology
### Bayesian RUL Prediction Based on Extended Kalman Filter
#### 3.4.2 Extended Kalman Filter  
Kalman filtering is one of the Bayesian filtering approaches commonly used for state estimation, the problem of which consists of estimating the state vector \(x\) that describes the true behaviours of a dynamical system based on limited and noisy observations \(z\) which are often imperfect or even unknown due to the inaccuracy of sensor measurements and the restrictions of data acquisition [22]. The Kalman filter (KF) has been applied to many engineering applications such as target tracking [28], navigation [29] and system identification [30]. Recently, the KF has also been introduced to the field of equipment condition monitoring and predictive maintenance. For example, [PERSON] et al. proposed the RUL prediction model for aircraft power generators based on the Kalman filter by assuming the time-dependent state-space model to relate the estimated state with the RUL [31]. However, as the degradation of machine equipment often involves nonlinear dynamics, the performance of the KF may be restricted by the assumption of a linear system and Gaussian noise. The extended Kalman filter (EKF) is one of the generalizations of the Kalman filter to nonlinear state estimation and has been applied by [PERSON] et al. to the health management of the Li-ion battery pack for the state of charge estimation [32]. In bearing health management, [PERSON] et al. used the EKF for estimating the RUL of rolling bearings based on the exponential state model assumed to govern the trends of the statistical indicators of the vibration signals [16]. In EKF, the state being estimated is assumed to follow a nonlinear discrete-time stochastic process [19],  
\[\mathbf{x}_{t}=f(\mathbf{x}_{t-1},\mathbf{u}_{t-1},\mathbf{v}_{t-1}), \tag{9}\]  
where \(\mathbf{x}_{t}\) and \(\mathbf{u}_{t}\) are the state vector and the input at the time step \(t\),\(f\) is the nonlinear state transition function and \(\mathbf{v}_{t}\) is the random process noise represented by the Gaussian process with zero mean and covariance matrix \(\mathbf{Q}_{t}\). The sensor measurements can be modelled by  
\[\mathbf{z}_{t}=h(\mathbf{x}_{t},\mathbf{w}_{t}), \tag{10}\]  
where \(\mathbf{z}_{t}\) is the measurement vector at time step \(t\), \(h\) is the nonlinear transition function of measurements relating \(\mathbf{z}_{t}\) with the system state \(\mathbf{x}_{t}\), and \(\mathbf{w}_{t}\) denotes the measurement noise with zero mean and covariance matrix \(\mathbf{R}_{t}\). Specifically, \(\mathbf{v}_{t}\) and \(\mathbf{w}_{t}\) are the uncorrelated processes of white noises defined as follows [33],  
\[\mathbf{v_{t}}\sim N(\mathbf{0},\mathbf{Q}_{t}),with\ E[\mathbf{v}_{t}]=\mathbf{0},and\ E\Big{[} \mathbf{v}_{t}\mathbf{v}_{t}^{T}\Big{]}=\mathbf{Q}_{t}\delta_{t}, \tag{11}\]  
\[\mathbf{w_{t}}\sim N(\mathbf{0},\mathbf{R}_{t}),with\ E[\mathbf{w}_{t}]=\mathbf{0},and\ E\Big{[} \mathbf{w}_{t}\mathbf{w}_{t}^{T}\Big{]}=\mathbf{R}_{t}\delta_{t}, \tag{12}\]  
\[E\Big{[}\mathbf{w}_{t}\mathbf{v}_{t}^{T}\Big{]}=\mathbf{0}, \tag{13}\]  
where \(\delta_{t}\) denotes the Kronecker delta function which is equal to 1 only if \(t=t^{\prime}\), and 0 otherwise. Assuming a sufficient description of the system's nonlinearity via local linearization, the EKF approximates the nonlinear state model \(f\) and the measurement model \(h\) around the current estimate based on the first-order Taylor series expansion.