## 4 Real-Time Test from Slipping to Being Locked Phase
After synchronization, the normal force is actuated to maximum, but the clutch transmitted torque \(T_{c}\) is determined by the input torque and load torque of the powertrain so that it is less than that in the slipping phase. It can be seen that the driving and driven parts of the clutch can slip and synchronize smoothly, and the vehicle jerk is almost imperceptible. The clutch torque \(T_{c}\) gradually increases and decreases before synchronization. Since the transmitted torque \(T_{c}\) has a direct effect on the powertrain dynamics, the profile of \(T_{c}\) is given. These results are ideal for clutch engagement control. The results for the empirical case (\(T\) = 0.01 s) are illustrated in Figure 8b. It can be observed that the angular velocity of the driving part grows faster than that for the continuous case at the beginning of the engagement. Thus, the slipping velocity is higher, and the transient clutch torque is larger according to (13). Thereafter, the transient vehicle jerk is more intense, and the frictional loss is greater than those of the continuous case. Nevertheless, the driving and driven parts of the clutch can synchronize after slipping, and the vehicle jerk is <10 m/s\({}^{3}\). The engagement performance is acceptable. Thus, the test results were validated by the empirical results. When the task scheduling period increases to \(T\) = 0.1 s, which still falls into the stable region of \(T\) < 0.3 s, the control input \(T_{c}\) is obviously in a step shape, owing to the discretization and zero-order hold, as shown in Figure 8c. In addition, the control input \(T_{c}\) oscillates between positive and negative values, because \(e(t)\) in (13) is positive and negative. The torque \(T_{c}\) affects the dynamic response of the clutch driving part and driven part simultaneously according to (1)-(4). Because the inertia moment of the driving part (\(J_{e}\) and \(J_{c1}\)) is significantly lower than that of the driven part (\(J_{c2}\) and \(J_{c}\)), the oscillation of the driving part (as indicated by \(\dot{\theta}_{c1}\)) induced by the discretized \(T_{c}\) is evident, and is more intense than that of the driven part (as indicated by \(\dot{\theta}_{c2}\)). Nevertheless, the clutch can complete the synchronization at the end of the engagement. The effect of the discretized \(T_{c}\) on the clutch-driven part, which is also called the vehicle-body part, is reflected by vehicle jerk. The vehicle jerk is greater than 850 m/s\({}^{3}\) in the slipping stage. Such intense jerking is harmful to the powertrain components and reduces the comfort of the vehicle passengers. When the task scheduling period increases to \(T\) = 0.3 s, which is at the critical boundary, the oscillation of the clutch driving part is significantly more intense than that in the previous two cases, as shown in Figure 8d. The amplitude of the control input \(T_{c}\) increases from approximately 200 to 1000 Nm owing to the large error in the slipping velocity. Stickslip motions can be observed, which are not expected in a smooth engagement process. Because of the oscillation induced by discretization with the scheduling period at the critical boundary, the vehicle jerk is greater than \(1\times 10^{5}\) m/s\({}^{3}\), and the frictional loss is greater than 60 kJ. When the task scheduling period increases to \(T\) = 0.35 s, which falls into the unstable region of \(T\) > 0.3 s, the dynamic response is divergent, as shown in Figure 8e, and the clutch synchronization fails. From the above, the critical scheduling period (\(T\) = 0.3 s) cannot satisfy the control performance evaluated in terms of the vehicle jerk. A sub-boundary may exist. The control performance can be satisfied when the scheduling period is within the sub-boundary, for example, the empirical scheduling period of \(T\) = 0.1 s.  
Figure 8: Dynamic responses under different task scheduling periods.