## 3 Improved PIM under Dynamic Conditions
### Improved PIM Which Considers Past Effects  
The PIM assumes that the intensity function in different WCs has a proportional relationship. This study seeks to apply the influence of WCs over time through transformations. It is noticed that when the baseline has the form of Weibull process, the transformation can be implemented in PIM. Thus, Equation (13) can be re-expressed as:  
\[w(t|e_{i})=a_{0}b\cdot\left(\left(\exp\left(\sum_{l=1}^{m}k_{l}\cdot\left(Z_{ l}(e_{i})-Z_{l}(e_{0})\right)\right)\right)^{\frac{1}{l}}\cdot t\right)^{b-1} \cdot\left(\exp\left(\sum_{l=1}^{m}k_{l}\cdot\left(Z_{l}(e_{i})-Z_{l}(e_{0}) \right)\right)\right)^{\frac{1}{b}}, \tag{15}\]  
In Equation (15), the effects of WCs on the FI are divided into two parts. One of which acts directly on the FI, the other indirectly affects the FI by changing the time scale. When these two parts have the same form, they have the basic characteristics of the AM. Compared with Equation (10), we can regard this segmented effect as the unit time damage in \(e_{k}\), which takes \(e_{0}\) as the standard WC. When the segmented effect is multiplied by time \(t\), it can be regarded as the cumulative damage up to \(t\), as shown in Equation (16). \[w(t|e_{i})=a_{0}b\cdot\left(M(0,t|e_{i})\right)^{b-1}\cdot m(0,t|e_{i}), \tag{16}\]  
Here, the standard WC is simply a reference standard, and there are no specific requirements of the WC itself. Any WC can be set as the standard WC, all other WCs thereafter are based on it as the reference standard. Given that Equation (10) shows that the FI is a function of cumulative damage, assuming that the unit damage under any WC is 1, it will only lead to a change in the function between the cumulative damage and the FI and will not affect the reference of other functions. Therefore, the dynamic form of WCs needs discussion. Supposing that the system runs in a dynamic WC \(E_{2}\), where the system works in \(e_{1}\) from \(t_{0}(t_{0}=0)\) to \(t_{1}\), in \(e_{2}\) from \(t_{1}\) to \(t_{2}\). As the unit time damage of the systems is not equal in each WC, the FI function also has unequal characteristics in each WC. When \(t_{0}\leq t<t_{1}\), the cumulative damage of the system can be expressed as \(M(0,t|e_{1})\). Since it is assumed that the damage per unit time under standard WC is 1, then the cumulative damage can be expressed as:  
\[M(0,t)=M(0,t|e_{1})=\left(\exp\left(\sum_{l=1}^{m}k_{l}\cdot\left(Z_{l}(e_{1} )-Z_{l}(e_{0})\right)\right)\right)^{\frac{1}{b}}\cdot t, \tag{17}\]  
The unit time damage is:  
\[m(0,t|e_{1})=\left(\exp\left(\sum_{l=1}^{m}k_{l}\cdot\left(Z_{l}(e_{1})-Z_{l}( e_{0})\right)\right)\right)^{\frac{1}{b}}, \tag{18}\]Applying Equations (17) and (18) into Equation (16) results in:  
\[w(t|e_{1})=a_{0}b\cdot(t)^{b-1}\cdot\left(\exp\left(\sum\limits_{l=1}^{m}k_{l} \cdot(Z_{l}(e_{1})-Z_{l}(e_{0}))\right)\right), \tag{19}\]  
When \(t_{1}\leq t<t_{2}\), the cumulative damage can be divided into two parts: the first part is the cumulative damage in \(e_{1}\) from \(t_{0}(t_{0}=0)\) to \(t_{1}\), the second part is the cumulative damage in \(e_{2}\) from \(t_{1}\) to \(t_{2}\). FI function can be derived as:  
\[ \begin{array}{l}w(t|E_{2})=\lim\limits_{\Delta t\to 0}\frac{W(M(0,t_{1}|e_{1})+M(t_{1},t|e_{ 2})+M((t,t+\Delta t|e_{2}))-W(M(0,t_{1}|e_{1})+M(t_{1},t|e_{2}))}{\Delta t}\\ =\lim\limits_{\Delta t\to 0}\frac{W(M(0,t_{1}|e_{1})+M(t_{1},t|e_{2})+M(t,t+ \Delta t|e_{2}))-W(M(0,t_{1}|e_{1})+M(t_{1},t|e_{2}))}{\Delta t\cdot M(t,t+ \Delta t|e_{2})}\cdot\frac{M(t+\Delta t|e_{2})}{\Delta t}\\ =m(t|e_{2})\cdot w_{0}(M(0,t_{1}|e_{1})+M(t_{1},t|e_{2}))\end{array} , \tag{20}\]  
So, Equation (16) can be rewritten as follows:  
\[w(t|e_{i})=a_{0}b\cdot(M(0,t_{1}|e_{1})+M(t_{1},t|e_{2}))^{b-1}\cdot m(0,t|e_{ 2}), \tag{21}\]  
The total cumulative damage is the sum of the two parts of damage, which is shown in Equation (22):  
\[