## 2 RDFTA (Repairable Dynamic Fault Tree Analysis)
### Feedback Logic Gate of RDFTA  
The failure of a component within a diesel engine system occurs and triggers event \(x_{3}(t)\), i.e., when event \(x_{3}(t)\) occurs, the failure of components \(x_{1}(t)\) and \(x_{2}(t)\) also occurs. This is generally used to describe the relationship between the feedback link and the failure of components in the pathway. For example, in a diesel engine system, the feedback control relationship between the electronic control system and each subsystem of the diesel engine uses a feedback logic gate. The electronic control system is represented by event \(x_{3}(t)\), while the subsystem is represented by events such as \(x_{1}(t)\) and \(x_{2}(t)\). \(S_{x(i)}(t)\) represents the normal working probability of the FB gate input event \(x_{i}(t)\) at time \(t\), \(F_{x(i)}(t)\) is the failure state probability of the FB gate input event \(x_{i}(t)\) at time \(t\), \(\mu_{x(i)}\) is the maintenance rate of the input event \(x_{i}(t)\), and \(\lambda_{x(i)}\) is the failure rate of the input event \(x_{i}(t)(i=3)\). \(S_{y}(t)\) represents the normal working probability of the FB gate output event \(y(t)\) at time \(t\), \(F_{y}(t)\) is the failure state probability of the FB gate output event \(y(t)\) at time \(t\), \(\mu_{y}\) is the maintenance rate of the output event \(y(t)\), and \(\lambda_{y}\) is the failure rate of the output event \(y(t)\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number** & \(x_{1}(t)\) & \(x_{2}(t)\) & \(y(t)\) \\ \hline
2 & 1 & 0 & 0 \\
3 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The quantitative operation rules of the CSP gate. Figure 3: The transformation of the CSP gate into a Markov model. A conversion rate matrix can be derived from Figure 4 as follows:  
\[\mathbf{A}= \begin{bmatrix}-\lambda_{1}-\lambda_{2}-\lambda_{3}&\lambda_{2}&0& \lambda_{1}&\lambda_{3}\\ \mu_{2}&-\lambda_{1}-\mu_{2}-\lambda_{3}&\lambda_{1}&0&\lambda_{3}\\ 0&\mu_{1}&-\mu_{1}-\lambda_{3}-\lambda_{2}&\mu_{2}&\lambda_{3}\\ \mu_{1}&0&\lambda_{2}&-\mu_{1}-\lambda_{3}-\lambda_{2}&\lambda_{3}\\ \mu_{3}&\mu_{3}&\mu_{3}&\mu_{3}&-4\mu_{3}\end{bmatrix} . \tag{7}\]  
According to the Formula State Matrix (7), the differential Equation (8) can be solved:  
\[\left\{ \begin{array}{l}\left(P_{0}^{\prime}(t),P_{1}^{\prime}(t),P_{2}^{ \prime}(t),P_{3}^{\prime}(t),P_{4}^{\prime}(t)\right)=\left(P_{0}(t),P_{1}(t),P _{2}(t),P_{3}(t),P_{4}(t)\right)A\\ \left(P_{0}(t),P_{1}(t),P_{2}(t),P_{3}(t),P_{4}(t)\right)=\left(1,0,0,0,0\right) \end{array} \right.. \tag{8}\]  
According to the definition of availability, the system fault status is \(P_{4}(t)\), i.e., only \(P_{4}(t)\) can be solved:  
\[P_{4}(t)=\lambda_{3}\frac{e^{-t(\lambda_{2}+\mu_{3})}\left(e^{t(\lambda_{2}+ \mu_{2})}-1\right)}{4(\lambda_{3}+\mu_{3})}. \tag{9}\]  
Quantitative operation rule  
In the FB logic gate, the input event \(x_{i}(t)(i=1,2,\ldots,n)\) and output event \(y(t)\) have two states: state \(0\) is the normal working state, while state \(1\) is the fault state. According to the relationship between the input and output events \(x_{i}(t)\) and \(y(t)\), a list of the quantitative operation rules of the FB gate can be obtained, as shown in Table 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & \(x_{1}(t)\) & \(x_{2}(t)\) & \(x_{3}(t)\) & \(y(t)\) \\ \hline
2 & 0 & 1 & 0 & 0 \\
3 & 1 & 1 & 0 & 1 \\
4 & 1 & 0 & 0 & 1 \\
5 & 0 & 0 & 1 & 1 \\
6 & 0 & 1 & 1 & 1 \\
7 & 1 & 0 & 1 & 1 \\
8 & 1 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The quantitative operation rules of the FB gate. Figure 4: The transformation of the feedback gate into a Markov model. 2. The quantitative calculation formula of the FB gate is as follows: \[\left\{