## 4 Real-Time Test from Slipping to Being Locked Phase  
The stability analysis in the first step is useful but does not provide enough information regarding the three phases of the complete engagement process. Due to the transition of the friction state from slipping to being locked, as modeled in (5), a stable slipping phase does not necessarily indicate a smooth transition. Therefore, the second step of the control-scheduling co-design was to perform real-time hardware-in-the-loop tests aiming to examine the influence of the task scheduling period on the clutch engagement performance, especially in regard to the friction state transition. The real-time environment consists of one MicroAutobox and one hardware-in-the-loop device (dSPACE PX10), as shown in Figure 7. The MicroAutobox is used to run the control algorithm and implement the task scheduling period via timer configuration, and the PX10 is used to run the powertrain model and simulate the time delay. The two communicate through a timer-triggered input/output (I/O) connection. Because the time-related settings can be implemented by hardware, the test accuracy can be higher than that of software simulation. The dynamic responses during the clutch engagement under different task scheduling periods are illustrated in Figure 7, with the engine torque set as a constant (\(T_{e}\) = 50 Nm) and with the car parameters listed in Tables 1 and 2[25]. Regarding the control-scheduling co-design, there are three kinds of evaluation methods. The first describes the evaluation index as a function of the task scheduling period [30; 31]. The second defines the execution deadline for the control task, which represents the worst case [32; 33]. Obviously, neither of these reflect the dynamic response affected by the task scheduling period. The third type of method employs dynamic characteristics, such as the stability, vibration, or response time, according to the state equations or transfer functions of the control plant [19; 34]. In this study, the third method is used to evaluate the clutch engagement performance with regard to the vehicle jerk and frictional loss. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbol** & **Value** & **Symbol** & **Value** & **Symbol** & **Value** \\ \hline \(m\) & 1600 kg & \(R_{c}\) & 0.15 m & \(\mu_{s}\) & 0.44 \\ \(\mu\) & 0.4 & \(c_{D}\) & 0.32 & \(A_{V}\) & 1.8 m\({}^{2}\) \\ \(\rho_{air}\) & 1.205 kg/m\({}^{3}\) & \(f\) & 0.0015 & \(R\) & 0.32 m \\ \(N\) & 2 & \(i_{g}\) & 3.5 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used in real-time modeling [25]. Figure 7: Real-time test platform. The vehicle jerk describes the smoothness of the running vehicle, which is a derivative of the vehicle acceleration. The frictional loss represents the energy dissipation during the engagement process, which is calculated as follows. \[E_{D}=\int_{I_{1}}^{t_{2}}\bigl{|}(\dot{\theta}_{c1}-\dot{\theta}_{c2})\cdot T_{ c}\bigr{|}\mathrm{d}t \tag{28}\]  
The test results for the six cases are illustrated in Figure 8. The first one is the continuous case without considering the task scheduling period, represented by \(T\) = 0 s. Additionally, three typical scheduling periods were selected around the critical period, as shown in the figure: \(T\) = 0.1 s, 0.3 s, and 0.35 s. To better understand the significance of the proposed method, the results for the empirical scheduling period (\(T\) = 0.01 s) typically used in engineering are provided for comparison. The results for the continuous case (\(T\) = 0 s) are shown as the benchmark in Figure 8a. The control input, i.e., the clutch transmitted torque \(T_{c}\), gradually increases and decreases before synchronization. The clutch is slipping in this phase, and the magnitude of the clutch transmitted torque \(T_{c}\) is proportional to the normal force on the friction plates.