## 4 Illustrative Experiment
### Experiment Process  
Step 1: Build the dependency model of the system to obtain the dependency matrix. Build the fault-test dependency matrix of the control system based on the reachability algorithm [26]. The aforementioned control system is not that complex; its fault-test dependency matrix is formulated as Table 1. In Figure 2 and Table 1, \(t_{1}\), \(t_{2}\), \(t_{3}\), and \(t_{4}\) denote 4 independent tests, while \(f_{1}\), \(f_{2}\), \(f_{3}\), and \(f_{4}\) denote 4 independent fault types. In detail, the 4 independent tests and the 4 fault types are shown as follows:  
* \(t_{1}\) denotes a signal acquisition test,
* \(t_{2}\) denotes a controller unit efficiency test,  
Figure 2: The dependency model of a control system. \(t_{3}\) denotes an actuator action range test,
* \(t_{4}\) denotes a feature signal output test,
* \(f_{1}\) denotes a signal collector fault,
* \(f_{2}\) denotes a controller unit fault,
* \(f_{3}\) denotes an actuator fault, and
* \(f_{4}\) denotes a feature signal output unit fault. Step 2: Simplify the dependency matrix according to the test results of the system. The system only 'Pass' the test \(t_{1}\) according to the observation results of the test, where \(t_{1}\)=0, \(t_{2}\)= 1, \(t_{3}\)= 1, and \(t_{4}\)= 1. According to the dependency model in Table 1, among all the column vectors correlated to the test \(t_{1}\), only \(ft_{11}=1\), which means only \(f_{1}\) is correlated to \(t_{1}\) (\(ft_{21}=ft_{31}=ft_{41}=0\)); thus, the fault type \(f_{1}\) cannot be wrong. Delete the column vector corresponding to \(t_{1}\) and the row vector corresponding to \(f_{1}\) to get the simplified dependency matrix is shown in Table 2.  
Step 3: Formulate the multiple fault dependency matrix. Perform the logical OR operation on the eigenvector (the row vector) of each fault type based on the simplified dependency matrix presented in Table 2. The eigenvector of multiple faults can be formulated as follows:  
* Fault in Controller and Actuator unit \(f_{(2,3)}\). \(f_{(2,3)}=f_{2}\left\|f_{3}\right.=\left[1\ \ 1\ \ 0\right]\left\|\left[0\ 1\ \ 0\right] =\left[1\ \ 1\ \ 0\right]\right.\)
* Fault in Controller and Feature signal output unit \(f_{(2,4)}\). \(f_{(2,4)}=f_{2}\left\|f_{4}\right.=\left[1\ \ 1\ \ 0\right]\left\|\left[0\ \ 0\ \ 1\right] =\left[1\ \ 1\ \ 1\right]\right.\). * Fault in Actuator and Feature signal output unit \(f_{(3,4)}\). \(f_{(3,4)}=f_{3}\left\|f_{4}\right.=\left[0\ \ 1\ \ 0\right]\left\|\left[0\ \ 0\ \ 1\right] =\left[0\ \ 1\ \ 1\right]\right.\)
* Fault in Controller unit, Actuator and Feature signal output unit \(f_{(2,3,4)}\). \(f_{(2,3,4)}=f_{2}\left\|f_{3}\right.\left\|f_{4}\right.=\left[1\ \ 1\ \ 0\right]\left\|\left[0\ \ 1\ \ 0\right] \right\|\left[0\ \ 0\ \ 1\right] =\left[1\ \ 1\ \ 1\right]\). The multiple fault dependency matrix is formulated as Table 3. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Fault Type** & \(\mathbf{t_{2}}\) & \(\mathbf{t_{3}}\) & \(\mathbf{t_{4}}\) \\ \hline Controller unit fault \(f_{2}\) & 1 & 1 & 0 \\ Actuator fault \(f_{3}\) & 0 & 1 & 0 \\ Feature signal output unit fault \(f_{4}\) & 0 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simplified dependency matrix of the control system. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Fault Type** & \(\mathbf{t_{1}}\) & \(\mathbf{t_{2}}\) & \(\mathbf{t_{3}}\) & \(\mathbf{t_{4}}\) \\ \hline Signal collector fault \(f_{1}\) & 1 & 1 & 1 \\ Controller unit fault \(f_{2}\) & 0 & 1 & 1 & 0 \\ Actuator fault \(f_{3}\) & 0 & 0 & 1 & 0 \\ Feature signal output unit fault \(f_{4}\) & 0 & 0 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fault-test dependency matrix of the control system. Step 4: Fault diagnosis of the system based on the multiple fault dependency matrix. For the multiple fault dependency matrix shown in Table 3, delete the row vector of the fault type with '0' including \(f_{2}\), \(f_{3}\), \(f_{4}\), \(f_{(2,3)}\), and \(f_{(2,3,4)}\).