## 4 Results and Discussion
For case II, the SMC method was used to regulate the motor torque, and the engine still followed control commands that came from the vehicle energy management strategy. It can be seen from Figure 11 that the speed of driven plate decreased by about 43 r/min. This still adversely affected the vehicle performance during the clutch slipping process. Although the control strategy for the motor had a positive effect on depressing the disturbance torque, the fluctuation in engine speed cannot be ignored in this case. Hence, the dynamic responses of the engine should be taken into consideration. The proposed coordinated control strategy was verified and taken as case III, whose verification results are shown in Figure 12. Similarly, when the velocity was greater than 20 km/h, the engine was dragged to the idle speed by the starting motor. At the engine side, an adaptive PI control strategy based on fuzzy algorithm was adopted to adjust engine speed to synchronize with the driven plate speed and reduce the speed difference of the clutch. The motor was controlled to compensate for the disturbance torque based on the SMC-GP method. From Figure 12, it is seen that the proposed coordinated control strategy exhibited a better control performance in terms of the speed and torque output from both power sources among the parallel HEV. \begin{table}
\begin{tabular}{c c} \hline \hline
**Cases** & **Control Strategy** \\ \hline I & Without coordinated control \\ II & Motor control with SMC \\ III & Motor control with GP-SMC and engine control with Fuzzy-PI \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Description of three test cases. Figure 9: HIL test bench. Figure 11: HIL results for case II. Figure 10: HIL results for case I. As is well-known, vehicle jerk and clutch frictional loss are two important criteria that are widely used to quantitatively evaluate the control strategy performance for mode transitions [44; 45; 46]. Jerk is defined as the derivative of vehicle acceleration, as shown in Equation (25). Vehicle jerk with a large value denotes that the driveline torque occurs as an obvious fluctuation phenomenon, and it will result in the deterioration of the vehicle's drivability and smoothness. The recommended value is less than 17.64 m/s\({}^{3}\) in China and 10 m/s\({}^{3}\) in Germany, respectively [47; 48; 49]. \[|J|=\frac{da}{dt}=\frac{d^{2}v}{dt^{2}} \tag{25}\]  
where \(J\) is the vehicle jerk, \(a\) is the vehicle acceleration, and \(v\) is the vehicle velocity determined by the driving cycle. Clutch frictional loss indicates the work done by friction during the clutch slipping process, which is closely related to the clutch lifespan. The clutch frictional loss \(W_{f}\) can be written as:  
\[W_{f}=\int_{t_{1}}^{t_{2}}T_{c}\cdot\Delta\omega dt \tag{26}\]  
where \(t_{1}\) and \(t_{2}\) stand for initial time and end time of the clutch slipping process, respectively. Figure 13 shows the vehicle jerks for the three cases, and their performance comparisons are summarized in Table 5. Due to the influence of disturbance torque and diversity of dynamic responses between the two power sources, the maximum vehicle jerk during the clutch slipping process reached up to 18.81 m/s\({}^{3}\) for case I. In addition, vehicle jerk fluctuations further deteriorated the drivability and smoothness performance. In case II, the maximum vehicle jerk could be reduced to 7.65 m/s\({}^{3}\) when the SMC method was used to control motor to compensate for the disturbance torque. However, the clutch frictional loss reached up to 4.983 kJ. The black dashed line represents the jerk for case III in Figure 13. It can be observed that by using the proposed strategy, the vehicle jerk was significantly reduced. The clutch frictional loss in case III was only 1.897 kJ, which avoids the unnecessary energy loss and prolongs the clutch lifespan. Additionally, the maximum  
Figure 12: HIL results for case III. vehicle velocity error was also restricted to an acceptable range.