## 3 Analysis of the Fault State and Establishment of the Fault Data Sample
### Creation of the Fault Diagnosis Data Sample  
According to the analysis results of the characteristic signals in each fault state in Section 3.1, it is found that the characteristic signal can establish a single mapping relationship with each fault sample. The correspondence between fault characteristics and characteristic signals is shown in Table 1.  
Data samples for fault diagnosis is established by finding analog signals such as vibration signals in the frequency domain of the corresponding speed. The 8-dimensional vectors \((\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},\mathbf{x}_{4},\mathbf{x}_{5}, \mathbf{x}_{6},\mathbf{x}_{7},\mathbf{x}_{8})\). are used as inputs to probabilistic neural networks and support vector machines. Among them, \(\mathbf{x}_{1}\) represents the current rotating speed, \(\mathbf{x}_{2}\) represents the throttle opening, \(\mathbf{x}_{3}\) represents the frequency signal, \(\mathbf{x}_{4}\) represents the vibration peak, \(\mathbf{x}_{5}\) and \(\mathbf{x}_{6}\) represent temperatures of the two cylinders, \(\mathbf{x}_{7}\) and \(\mathbf{x}_{8}\) represent the two-cylinder air-fuel ratios, respectively. During the experiment, the rotating speed  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Fault Type** & **The Signature Signals Required for Establishing the** \\  & **Mapping** \\ \hline The fuel injection failure of the injector & Throttle opening, air-fuel ratio signal and vibration peak \\ \hline Low fuel supply pressure & Rotating speed signal, vibration peak, air-fuel ratio signal, \\  & throttle opening, torque signal and cylinder temperature signal \\ \hline Leaks of the exhaust pipe & Air-fuel ratio signal \\ \hline The intermittent flameout of engine & Air-fuel ratio signal, rotating speed signal and vibration frequency domain signal \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mapping relationship between fault types and characteristic signals.  
Figure 9: Time-domain and frequency-domain analysis chart of vibration signals. (**a**) Vibration waveform in normal state, (**b**) Vibration waveform of flameout fault, (**c**) Frequency domain diagram of normal state, (**d**) Frequency domain diagram of flameout fault.  
was within 2500-5500 r/min, and the rotating speed was increased at 500 r/min intervals. In each fault state, 80 sets of data were recorded and screened out. The data of each state is collected as data samples, and some typical data of each fault state with an engine speed of about 3000 r/min are shown in Table 2.  
The status bar in the table indicates the state of the engine control system, state 1 represents the engine control system is normal, state 2 represents the fuel injection failure of the injector, state 3 represents the low fuel supply pressure, state 4 represents the leaks of the exhaust pipe, and state 5 represents the intermittent flameout of engine. With the data shown in Table 2, it can be seen that when an injector failure occurs, the throttle increases due to an increase in the amount of fuel injection, the air-fuel ratio of the two cylinders increases, and the vibration peak increases, which is consistent with the actual situation. When the fuel pressure is too low, due to the decrease of the fuel supply amount, the throttle opening increases, the vibration peak decreases, and the air-fuel ratio of the two cylinders increase at the same rotation speed, which is also consistent with the actual situation. Similarly, when the number 2-cylinder exhaust pipe leaks, the air-fuel ratio of the number 2 cylinder increases due to air leakage from the exhaust pipe, which is also consistent with the actual situation. In the state where the engine is intermittently flameout, since the injector of the 3 W engine does not inject fuel, the air-fuel ratio of the two cylinders increases compared with the normal condition, which is consistent with the actual situation. Therefore, a conclusion can be gotten that all the fault states' data filtered by the fault sample are truly valid.