## 4 Friction Coefficient Compensation Control Strategy  
### Optimal Control  
In the synchronization process, the synchronization ring and the target gear ring are compressed by the shifting force, and the speed difference between them is eliminated by friction torque. The synchronization process takes a certain amount of time, the sliding-friction work and synchronization impact are also produced. The length of the synchronization time affects the shift interruption time. The magnitude of sliding-friction work affects the service life of the synchronizer, and the magnitude of shock degree affects the smoothness of synchronization process. The friction coefficient compensation control strategy of the synchronization process studied in this paper was bound to cause an increase in the shifting force. Increasing the shifting force would shorten the synchronization time, but due to the limitation of friction pair material and other two evaluation indexes, a greater shifting force would not be better. When the initial speed difference was fixed, shortening the synchronization time meant that the speed difference decreased faster and the shock degree was higher. However, in order to reduce the shock degree and reduce the shifting force, the synchronization time was prolonged and the synchronizer wear was increased. Therefore, in order to balance the evaluation indexes of each synchronization performance, it was necessary to limit the magnitude of the shifting force acquired by the friction coefficient compensation control strategy. Other evaluation indicators of synchronization performance were also optimized, while reducing synchronization time. According to the principle of minimum value, the optimal control method was adopted, and the evaluation indexes of the synchronization process of the synchronizer were optimized [29; 30]. The mathematical model of the shift synchronization process is expressed as:  
\[\left\{ \begin{array}{c}J_{n}\dot{\omega}_{r}=-\frac{T_{f}}{i_{g}}\\ J_{s}\dot{\omega}_{c}=T_{f}-\frac{T_{n}}{i_{n}}\end{array} \right., \tag{7}\]  
where \(J_{n}=[J_{c}+J_{m}(i_{g},i_{n})]\), \(J_{m}\) is the rotational inertia of the rotating part of the transmission equivalent to the input shaft, \(J_{c}\) is the rotational inertia of clutch driven disk, \(i_{g}\) is transmission ratio, \(i_{n}\) is the final drive ratio, \(T_{m}\) is the external resistance moment of the vehicle. The state vector is taken as:  
\[X=\left( \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right)=\left( \begin{array}{c}\omega_{r}\\ \omega_{c}\\ \dot{\omega}_{c}\end{array} \right). \tag{8}\]  
Figure 7: The corresponding diagram of temperature and friction coefficient. The control vector is:  
\[Y=\left( \begin{array}{c}y_{1}\\ y_{2}\end{array} \right)=\left( \begin{array}{c}T_{f}\\ \dot{T}_{f}\end{array} \right). \tag{9}\]  
The state equation is established as:  
\[\left( \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\end{array} \right)=\left( \begin{array}{ccc}0&0&0\\ 0&0&1\\ 0&0&0\end{array} \right)\left( \begin{array}{c}x_{1}\\ x_{2}\\ 0&0\\ 0&\frac{1}{s_{s}}\end{array} \right)\left( \begin{array}{c}y_{1}\\ y_{2}\end{array} \right)+\left( \begin{array}{c}0\\ 0\\ -\frac{\dot{T}_{m}}{\dot{n}\cdot\dot{n}}\end{array} \right) \tag{10}\]  
The maximum driving force produced by the electromagnetic linear actuator determined the range of synchronous moment. The maximum driving force of the actuator used in this paper was 1000 N, so the maximum synchronous torque was 13.2 N-m, that \(0\leq y_{1}\leq[y_{1}]=13.2\) N-m. \(y_{2}\) stands for the change rate of synchronous torque, while the magnitude of the shock degree in the synchronous phase is determined by the change rate of synchronous torque. Therefore, the shock degree can be well limited by limiting the change rate of synchronous torque in synchronization phase. The constraint condition of calculating \(y_{2}\) is \(0\leq y_{2}\leq[y_{2}]=1596\).