## 3 Improved PIM under Dynamic Conditions  
### Traditional PIM and Its Limitations  
As a widely used statistical model, the PIM does not need to understand the physical failure mechanism of the equipment. Thus, it is suitable for complex electromechanical equipment that often operates in various WCs. The PIM can be written as the product of a baseline function and a covariant function, such as:  
\[w(t)=h(t)\cdot g(x), \tag{12}\]  
where \(h(t)\) is baseline FI function, and \(g(x)\) is the function of WC covariates. For complex electromechanical equipment, the baseline intensity can be considered as a Weibull process. So, the baseline FI function can be written in the form of the Weibull function. If the WC covariates are in the form of an exponential function, Equation (12) is equivalent to Equation (13).  
\[w(t|e_{i})=abt^{b-1}\cdot\exp\left(\sum_{l=1}^{m}k_{l}\cdot Z_{l}(e_{i})\right) \tag{13}\]  
where \(a\) is the scale parameter, \(a>0\); \(b\) is the shape parameter, \(b>0\). \(Z_{l}(e_{i})\) is the \(l\)-\(th\) WC factor in WC \(e_{i}\), \(k_{l}\) is the corresponding coefficient of the \(l\)-\(th\) factor, and \(m\) indicates the total number of WC factors. When \(k_{l}>0\), the corresponding WC factor has promoted the FI. When \(k_{l}=0\), the corresponding WC factor does not affect FI. When \(k_{l}<0\), the corresponding WC factor has restrained the FI. For the standard WC \(e_{0}\), the covariant function can be seen as 1. Thus:  
\[w(t|e_{0})=abt^{b-1}\cdot\exp\left(\sum_{l=1}^{m}k_{l}\cdot Z_{l}(e_{0}) \right)=a_{0}bt^{b-1},a_{0}=a\cdot\exp\left(\sum_{l=1}^{m}k_{l}\cdot Z_{l}(e_{ 0})\right) \tag{14}\]  
By using the PIM, the WCs act on the FI function with the form of covariant functions which makes modeling and computing easier. However, an obvious flaw remains and limits its further application. It is known that when electromechanical systems are in operation the WCs often show the characteristics of dynamic changes, which rarely retain the same state for a long time. The PIM can only describe the difference between WCs, but it cannot describe the FI in dynamic WCs because it cannot reflect the past cumulative effect of influencing factors [28]. Consider the following situation as shown in Figure 1.  
It is assumed that an electromechanical system operates in three different WCs, namely, \(e_{1}\), \(e_{2}\), \(e_{3}\). According to the PIM, the FI curves under these three WCs are \(S_{1}\), \(S_{2}\), and  
Figure 1: FIs in three different WCs.  
respectively. If the system operates in \(e_{1}\) from 0 to \(t_{1}\), in \(e_{2}\) from \(t_{1}\) to \(t_{2}\), and in \(e_{3}\) from \(t_{2}\) to \(t_{3}\), the FI can only be represented in the form of a piecewise function, which curves \(OA_{1}\) from 0 to \(t_{1}\), \(A_{2}B_{2}\) from \(t_{1}\) to \(t_{2}\), and \(B_{3}C_{3}\) from \(t_{2}\) to \(t_{3}\) in Figure 1. However, this result is inconsistent with the actual situation. The premise of the curve \(A_{2}B_{2}\) representing the FI from \(t_{1}\) to \(t_{2}\) is that the system operates in \(e_{2}\) from 0 to \(t_{1}\), but the actual situation is that the system operates in \(e_{1}\) from 0 to \(t_{1}\). The premise of the curve \(B_{3}C_{3}\) representing the FI from \(t_{2}\) to \(t_{3}\) is that the system operates in \(e_{3}\) from 0 to \(t_{2}\), but the system actually operates in \(e_{1}\) from 0 to \(t_{1}\) and in \(e_{2}\) from \(t_{1}\) to \(t_{2}\). Therefore, the curve \(B_{3}C_{3}\) cannot accurately reflect the change of the FI from \(t_{2}\) to \(t_{3}\) because the FI is related to the degree of cumulative damage, which differs in varying WCs. The PIM does not have the ability to calculate the cumulative damage, so it cannot describe FI in dynamic WCs. Faced with such problems, it is necessary to consider the damage degree in different WCs to improve the shortcomings of traditional methods.