## 2 System Modeling
### Powertrain Model  
The lumped dynamic model of the powertrain is depicted in Figure 3. The model consists of the engine, the clutch (with separate driving and driven parts), the vehicle body, the elastic connection between the engine and the clutch, and the elastic connection between the clutch and the vehicle body. Figure 1: Two-step framework of the proposed control–scheduling co-design. Figure 2: Diagram of closed-loop control system during clutch engagement. The model has four degrees of freedom. The governing equations are presented below. Equations (1) and (2) represent the dynamics of the engine-side components separated by the clutch. Similarly, Equations (3) and (4) represent the dynamics of the body-side components. The time delay \(\tau\) is used to describe the inevitable delay of the actuators due to a variety of factors, such as the electromotive force of the motor-driven system and the filling process of the hydraulic system. \[J_{c}\bar{\theta}_{e}(t)=T_{e}(t)-c_{e}\Big{(}\dot{\theta}_{e}(t)-\dot{\theta }_{c1}(t)\Big{)}e-k_{e}(\theta_{e}(t)-\theta_{c1}(t)) \tag{1}\]  
\[J_{c1}\bar{\theta}_{c1}(t)=c_{e}\Big{(}\dot{\theta}_{e}(t)-\dot{\theta}_{c1}( t)\Big{)}+k_{e}(\theta_{e}(t)-\theta_{c1}(t))-T_{c}(t-\tau) \tag{2}\]  
\[J_{c2}\bar{\theta}_{c2}(t)=T_{c}(t-\tau)-c_{v}\Big{(}\dot{\theta}_{c2}(t)- \dot{\theta}_{v}(t)\Big{)}-k_{v}(\theta_{c2}(t)-\theta_{v}(t)) \tag{3}\]  
\[J_{v}\bar{\theta}_{v}(t)=c_{v}\Big{(}\dot{\theta}_{c2}(t)-\dot{\theta}_{v}(t) \Big{)}+k_{v}(\theta_{c2}(t)-\theta_{v}(t))-T_{v}(t) \tag{4}\]  
The clutch transmitted torque \(T_{c}\) is generated by the friction force between the two friction plates of the clutch. According to the Coulomb law of friction [24], \(T_{c}\) can be described with a piecewise function in terms of the status of the clutch, as follows. \[T_{c}(t)=\left\{ \begin{array}{ll}0&\mbox{open}\\ \overline{T_{c}}(t)\cdot\mbox{sign}\Big{(}\Delta\dot{\theta}_{c}(t)\Big{)}& \mbox{slipping}\\ x\mbox{ in }[\ -\overline{T_{cs}}(t)\quad\overline{T_{cs}}(t)\ ]&\mbox{locked} \end{array} \right. \tag{5}\]  
where \(\overline{T}_{c}\) is the magnitude of the clutch transmitted torque in the slipping phase, and \(\overline{T_{cs}}\) is the magnitude limitation of the clutch static friction torque in the locked phase. So, the magnitude of the clutch transmitted torque \(T_{c}\) can be controlled by a clutch actuator in the slipping phase, but it can be any value within \(\big{[}\ -\overline{T}_{cs}\quad\overline{T}_{cs}\ \big{]}\) in the locked stage. In the slipping stage, the magnitude of the clutch torque \(T_{c}\) is proportional to the magnitude of the normal force \(F_{n}\), and the direction of \(T_{c}\) is determined by the slipping speed \(\Delta\dot{\theta}_{c}\). Therefore, \(\overline{T}_{c}\) can be calculated as follows. \[\overline{T_{c}}(t)=\mu\cdot F_{n}(t)\cdot R_{c}\cdot N \tag{6}\]  
\(R_{c}\) is the effective radius of clutch plate, \(N\) is the number of friction surfaces. The definition of \(\Delta\dot{\theta}_{c}\) is given as follows. \[\Delta\dot{\theta}_{c}(t)=\dot{\theta}_{c1}(t)-\dot{\theta}_{c2}(t) \tag{7}\]  
As indicated by (5), the clutch transmitted torque \(T_{c}\) can be actively controlled only in the slipping stage by applying a proper normal force \(F_{n}\), while the other two stages cannot be controlled by the normal force \(F_{n}\). That is, the clutch torque \(T_{c}\) is zero in the open stage  
Figure 3: Powertrain model. and can be any value within the range defined by the normal force \(F_{n}\) due to the static friction in the locked stage. Therefore, the clutch transmitted torque \(T_{c}\) has a direct effect on the powertrain dynamics, but it has piecewise characteristics.