# A Fault-Signal-Based Generalizing Remaining Useful Life Prognostics Method for Wheel Hub Bearings
## 3 Methodology
### Method Solution  
To simplify the study work, let \(m=1\), \(n=1\), and \(\Delta t=1\). Then, Equation (17) is changed to Equation (20), and Equation (18) is changed to Equation (21).  
\[\frac{dx(t)}{dt}+ax(t)=b \tag{20}\]  
\[x(t)-x(t-1)+ax(t)=b \tag{21}\]  
The parameters \(a\), \(b\) can be acquired with the least square method. The solution of RUL prognostics method in Equations (19) and (21) can be expressed as Equation (22).  
\[\hat{x}(t+1)=\hat{c}e^{-it}+\hat{d}=[x(1)-\frac{\hat{b}}{\hat{a}}]e^{-it}+ \frac{\hat{b}}{\hat{a}} \tag{22}\]  
Considering Equation (16), \(\overleftarrow{x}(t)\) can be expressed as Equation (23).  
\[\overleftarrow{x}  
In Equation (25), Equation (26) can be obtained.  
\[\begin{array}{l}\frac{1}{x}(t)-\frac{1}{x}(t-1)=\sum\limits_{l=1}^{t}\rho(t,l, \alpha_{l})\overset{0}{x}(l)-\sum\limits_{l=1}^{t-1}\rho(t-1,l,\alpha_{l-1}) \overset{0}{x}(l)\\ =\sum\limits_{l=1}^{t-1}\left(\rho(t,l,\alpha_{l})-\rho(t-1,l,\alpha_{l}) \right)\overset{0}{x}(l)+\rho(t,t,\alpha_{l})\overset{0}{x}(t)\end{array} \tag{26}\]  
Equation (27) can be obtained in Equations (21) and (26).  
\[\sum\limits_{l=1}^{t-1}\left(\rho(t,l,\alpha_{l})-\rho(t-1,l,\alpha_{l}) \right)\overset{0}{x}(l)+\rho(t,t,\alpha_{l})\overset{0}{x}(t)+\overset{1}{ax }(t)=b \tag{27}\]  
Equation (28) can be obtained by replacing \(t\) with \(t+1\) in Equation (27).  
\[\sum\limits_{l=1}^{t}\left(\rho(t+1,l,\alpha_{l})-\rho(t,l,\alpha_{l}) \right)\overset{0}{x}(l)+\rho(t+1,t+1,\alpha_{l})\overset{0}{x}(t+1)+ax(t+1) =b \tag{28}\]  
Equation (29) can be obtained by replacing \(t\) with \(t+1\) in Equation (21).  
\[\overset{1}{x}(t+1)-\overset{1}{x}(t)+ax(t+1)=b \tag{29}\]  
Equation (29) can be obtained in Equation (30).  
\[b-ax(t+1)=\overset{1}{x}(t+1)-\overset{1}{x}(t) \tag{30}\]  
Equation (31) is obtained by substituting Equation (30) into Equation (28).  
\[\overset{0}{x}(t+1)=\rho^{-1}(t+1,t+1,\alpha_{l})\cdot[\overset{1}{x}(t+1)- \overset{1}{x}(t)+\sum\limits_{l=1}^{t}\left(\rho(t,l,\alpha_{l})-\rho(t+1,l, \alpha_{l})\right)\overset{0}{x}(l)] \tag{31}\]  
Equations (22) and (31) are the solution of the simplified RUL prognostics method in Equation (20). The RUL of wheel hub bearings can be obtained in Equation (31). In Equation (31), the \((t+1)^{th}\) signal \(x(t+1)\) can be prognosticated with the signals from \(x(1)\) to \(x(t)\); \(\rho(t,l,\alpha_{l})\) reflects the impact of past historical signals on prognostics, called memory item. The attenuation factor \(\alpha_{l}\) reflects the attenuation of the influence of the past signal on the prognostics. In this study, Equation (31) will be used to verify the experiment in Section 4.  
This proposed RUL prognostics method only utilizes the information of fault signal, so directly obtaining parameters of the fracture mechanics can be avoided. The RUL can be dynamically prognosticated in real time with respect to time change. When the amplitude of the prognosticated signal \(x(t+1)\) reaches the given maximum value, the wheel hub bearings can be considered damaged, and time \(t\) is the RUL of the wheel hub bearings. The proposed RUL prognostics method can prognosticate the RUL with the information from the fault signal in the initial running stage, which is verified in Section 4.