# An Improved FFIP Method Based on Mathematical Logic and SysML
## 6 Case Study
### Further Analysis  
According to the results of the single-failure and double-failure simulations, the fault propagation paths of the brake system can be deduced. Through analyzing the fault propagation paths, the fault tree of the brake system can be obtained, as shown in Figure 9. The propagation of single failure is represented by an OR gate in the fault tree, while the propagation of double failure is represented by an AND gate. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & **Failure 1** & **Failure 2** & **Failure 3** & **Failure 4** & **Failure 5** & **Failure 6** & **Failure 7** & **Failure 8** & **Failure 9** \\ \hline \multirow{4}{*}{BSCU} & Command & \(\times\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \cline{2-11}  & Monitor & \(Z_{1}\) & \(\times\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \cline{2-11}  & SystemModel\_SetCmd & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \cline{2-11}  & AutoBraking\_PressureOut & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \hline \multirow{4}{*}{Green Pump} & \(Z_{0}\) & \(Z_{0}\) & \(\times\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \cline{2-11}  & Green Pump Isolation Valve & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \hline \multirow{4}{*}{Selector Valve} & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(\times\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \cline{2-11}  & CMD/AS Meter Valve & \(Z_{1}\) & \(Z_{1}\) & \(Z_{0}\) & \(Z_{2}\) & \(\times\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \hline \multirow{4}{*}{Blue Pump} & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{1}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \cline{2-11}  & Blue Pump Isolation Valve & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{1}\) & \(Z_{0}\) & \(\times\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) \\ \hline \multirow{4}{*}{Accumulator Valve} & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{1}\) & \(\times\) & \(Z_{0}\) & \(Z_{1}\) \\ \cline{2-11}  & AS Meter Valve & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{2}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{1}\) & \(\times\) & \(Z_{0}\) \\ \hline \multirow{4}{*}{Manual Meter Valve} & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{2}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{1}\) & \(Z_{0}\) \\ \cline{2-11}  & Accumulator Pump & \(Z_{0}\) & \(Z_{0}\) & \(Z_{0}\) & \(Z_{1}\) & \(Z_{0}\) & \(Z_{1}\) & \(Z_{0}\) & \(Z_{1}\) & \(\times\) \\ \hline \multirow{4}{*}{System} & \(Z_{1}\) & \(Z_{1}\) & \(Z_{0}\) & \(Z_{2}\) & \(Z_{1}\) & \(Z_{1}\) & \(Z_{1}\) & \(Z_{1}\) & \(Z_{1}\) & \(Z_{1}\) \\ \hline \end{tabular} Where: \(\times\) indicates the failure location, \(Z_{0}\) indicates normal function, \(Z_{1}\) indicates degradation function, and \(Z_{2}\) indicates failure functional.  
\end{table} 
Table 5: Component States under single fault simulation. Next, we use letters and numbers to present the failure events, as shown in Figure 10. The purpose is to facilitate the calculation of the minimum cut set. Through calculation, the minimum cut set is obtained as follows: {A1}, {A2}, {A3}, {A4}, {A6, a7}, {A5, A8, A9}, {A5, A8, A12}, {A5, A8, A13}, {A5, A9, A10}, {A5, A10, A12}, {A5, A10, A13}, {A5, A11, a11}, {A5, a11, A13}. Based on the minimum cut set, it can be concluded that the key events are A1, A2, A3, A4 and A5. They represent the failure of the selection valve, power, System Model_SelCmd module, Auto Braking_Pressure Out module and accumulator pump. SystemModel_SelCmd and AutoBraking_PressureOut are two modules in BSCU modeling, so it is not considered as the key failure position. In addition, power-off belongs to external factors of equipment, so it is not considered.