# Remaining Useful Life Prediction of Broken Rotor Bar Based on Data-Driven and Degradation Model
## 4 Proposed Prognostics Methodology  
The hybrid algorithm proposed in this paper is shown in Figure 3. It is composed of three modules, i.e., data acquisition, HI construction, and RUL estimation. Synthetic failure datasets are generated from a Matlab/Simulink model through simulations in the data acquisition module. In the HI construction module, original features are extracted from the rotating machine's torque, stator current, and speed signals. Then, the degradation trends of these features are evaluated, and the most meaningful ones are selected. Furthermore, the selected features are compressed using the PCA technique to reflect the degradation evolving. In the third module, each constructed HI is given as an input to an exponential degradation model. The RUL estimation is based on the predicted values of each HI using the parameters of the model. More details about these modules are presented in the following sections.  
Figure 2: Simulation model of an IM with broken rotor fault.  
RUL Estimation Based on the Exponential Degradation Model  
To use the calculated HIs, which are supposed to incorporate useful information about degradation, it is necessary to fit them into a mathematical model for the RUL estimation step. This fitting is performed by using an exponential degradation model. This kind of model is one of the most widely used stochastic process models for rotating machines prognostic. The first version of the model was established by [40] using a Bayesian approach to update the model parameters. Many improvements have been developed and implemented in the RUL prediction of rotating machines compared with the first version. [PERSON] et al. [41] used the Bayesian approach with the expectation-maximization (EM) technique to estimate the parameters and to offer a closed-form RUL distribution. In addition, the authors improved the model using historical degradation signals obtained from condition monitoring to update the model parameters. In this work, the improved degradation model allows for fitting each HI time evolution with the ability to indicate when the constructed HI crosses the threshold, indicating the failure threshold (FT).  
The degradation process using the exponential degradation model is described using a stochastic process \(\{h_{k}=h(t_{k}),t_{k}\geqslant 0\}\), where \(h_{k}\) is the HI system state at time \(t_{k}\) and  
\[h_{k}=\phi+\theta(t_{k})e^{(\beta(t_{k})+\zeta(t_{k})-\frac{\sigma^{2}}{2})} \tag{4}\]  
where \(h_{k}\) represents the HI expressed in a function of time; \(\phi\) is a known constant; and \(\theta(t_{k})\) is a lognormal distribution while \(\beta(t_{k})\) is a Gaussian distribution, both being random variables characterizing the stochastic part of the model. At each time step \(t_{k}\), these two distributions capture the degradation process variation and update the posterior based on the latest observation of \(h_{k}\). The \(\zeta(t_{k})\) represents a Gaussian additive noise and is modeled as a normal distribution with zero mean and variance. The RUL of the IM is defined as \"the length from the current time to the end of the useful life\", which is expressed as \(l_{k}=t_{EoL}-t_{k}\), where \(t_{EoL}\) is the end of life (EOL), \(t_{k}\) is the current time, and \(l_{k}\) is the RUL at \(t_{k}\)[25].  
In this study, the RUL refers to the time left before rotating machines lose their operation ability, giving the machine's state, the current age, and the past operating profile. This means the time left before each HI of the machine reaches a predefined FT.  
The RUL \(l_{k}\) at current time \(t_{k}\) is expressed as follow:  
\[l_{k}=inf\{l_{k}:h(l_{k}+t_{k})\geqslant\lambda\ |\ h_{0k}\} \tag{5}\]  
Figure 3: Flowchart of the proposed prognostics algorithm.  
where \(h(l_{k}+t_{k})\) represents the HI state at the future time \(l_{k}+t_{k}\), \(inf(.)\) represents the inferior limit of a variable, and \(\lambda\) is a prespecified FT.