## 3 Stability Analysis for the Slipping Phase
### Sensitivity Analysis  
It is of interest that the critical period \(T_{\text{max}}\) may be sensitive to some parameters. The key parameters of the three subsystems other than the zero-order holder of the overall system, as shown in Figure 2, were selected as influencing factors. Regarding the power-errain subsystem, the natural frequency \(\omega_{n}\) (defined as \(w_{n}=\sqrt{k_{v}/J_{v}}\)) and the damping coefficient \(\xi_{n}\) (defined as \(\xi_{n}=c_{v}/\sqrt{k_{v}J_{v}}\)) were selected since the two kinetic parameters are related to one performance evaluation index of clutch engagement, i.e., vehicle jerk. Regarding the control algorithm subsystem, two control gains, \(k_{p}\) and \(k_{i}\), were selected because they are used in the feedback loop. In addition, the delay \(\tau\) was selected from the time delay subsystem. In total, five influencing factors were examined.  
Using the data in Table 1, the base values of \(\omega_{n}\) and \(\xi_{n}\) were calculated to be \(\omega_{n}=32\) rad/s and \(\xi_{n}=0.32\), respectively. The base values of \(k_{p}\), \(k_{i}\), and the time delay are \(k_{p}=5\), \(k_{i}=1\), and \(\tau=0.01\) s, respectively, as shown in Table 1. With the other four parameters unchanged, the critical period \(T_{\text{max}}\) for task scheduling changes with respect to \(\omega_{n}\), \(\xi_{n}\), \(k_{p}\), \(k_{i}\), and \(\tau\), as shown in Figure 6.  
As shown in Figure 6a, the critical period \(T_{\text{max}}\) decreases with the increase in \(\omega_{n}\). As we know that the increase in \(\omega_{n}\) means a decrease in the natural oscillation period of the powertrain. Therefore, the critical period \(T_{\text{max}}\) decreases as the natural oscillation period decreases, and vice versa. In other words, a shorter oscillation period tolerates a shorter critical period \(T_{\text{max}}\).  
On the other hand, as shown in Figure 6b, the critical period \(T_{\text{max}}\) increases with the increase in the damping \(\xi_{n}\). This trend agrees with the common sense that more damping helps to prevent system instability because damping can absorb a lot of vibration energy. Nevertheless, as indicated by the range of \(T_{\text{max}}\), the critical period \(T_{\text{max}}\) is more sensitive to the natural frequency \(\omega_{n}\) than to the damping coefficient \(\xi_{n}\).  
Figure 4: Maximum module \(\left|z\right|_{\text{max}}\) of the poles vs. the task scheduling period \(T\).  
Figure 5: Poles location with different task scheduling period \(T\).  
As shown in Figure 6c,d, the critical period \(T_{\text{max}}\) generally decreases with the increase in the gains \(k_{p}\) and \(k_{i}\). In real applications, the two gains are always tuned to large value to obtain a small steady-state error and short settling time. However, the analysis in this study indicates that the two gains should be small when the task scheduling period is large.  
As shown in Figure 6e, the critical period \(T_{\text{max}}\) decreases with the increase in the delay \(\tau\). For the ideal actuation system without a time delay, the critical period is 0.31 s. However, in a real actuation system, time delay is inevitable. Thus, the critical period should be less than 0.31 s in real cases.  
A comparison of the sensitivity of the five factors shows that the critical period is the most sensitive to the natural frequency \(\omega_{n}\); on the contrary, it is the least sensitive to the damping coefficient \(\xi_{n}\). In addition, the critical period is more sensitive to the proportional gain \(k_{p}\) than the integral gain \(k_{i}\). The sensitivity of the time delay \(\tau\) is similar to that of the proportional gain \(k_{p}\).  
Figure 6: Factors influencing the critical period \(T_{\text{max}}\).