## 4 Gearshift Control Problem
### Simulation Results  
A model is developed in MATLAB/SIMULINK (version R2018a) to validate the proposed feedforward-feedback shift controller with disturbance compensation. It mainly consists of four parts: submodels for the motor, the transmission, the vehicle body, and the shift controller. Stiffness and damping on the output shafts are taken into consideration. Firstly, the constant-output-torque control  
Figure 9: Schematic diagram of the feedforward–feedback slip speed controller.  
(COTC) is compared to the conventional constant-input-torque control (CITC), which keeps the input shaft torque constant along the whole shifting process. Simulation results are shown in Figure 10. In COTC, during the torque phase, the motor torque is increased to compensate the reduction of output torque due to the change of gear ratio. In contrast, CITC simply maintains the motor torque. As a result, the output shaft torque decreases during the upshift. The pressure level of the oncoming brake in COTC is higher than that of CITC to keep the output torque constant. In the inertia phase, COTC relies on the motor torque for slip speed regulation, whereas CITC changes the pressure of the oncoming brake to accomplish this.  
Advantages of the feedforward-feedback scheme are validated through comparison with pure feedforward and feedback ones, as are shown in Figure 11. Both pure feedback and feedforward control fail to follow the slip speed reference. Under the pure feedback one, the slip speed falls to zero quickly and its derivative fails to meet the GV no-lurch condition at the synchronization point. As a result, significant torque fluctuations are observed on the output shaft. For the pure feedforward control, since model simplification is applied in derivation of the control law, the tracking error is quite large at the end of the interval, and there are also output torque fluctuations. On the other hand, if the feedforward and the feedback control are combined, the slip speed reference is then well followed and the output torque is successfully kept constant and smooth.  
Figure 10: Comparison with constant-input-shaft torque control.  
Simulations above are based on the nominal model, i.e., there are no disturbances. Necessity of the disturbance observer is evaluated by adding a friction torque (1.5 Nm) on the input shaft. Performances of the three controllers are compared, namely, the conventional proportional-integral control (PI), the feedforward-feedback control, and the feedforward-feedback control with disturbance compensation. Tuning of the PI controller is a time-consuming process, and in this paper, we set the proportional gain to be consistent with the feedback control. We perform simulations multiple times with different integrating gains and take the parameter values corresponding to the best results. The proportional and integrating gains are set to be 0.5 and 1, respectively, in this test scenario. Results are shown in Figure 12. As can be seen, all the three controllers follow the slip speed reference generally well. However, at the synchronization point (when the rotational speed of the oncoming brake decreases to zero), slip speed from the feedforward-feedback controller with DO comes with a more \"gentle landing\" (derivative of the slip speed has a smaller magnitude) than those of the other two. Consequently, the driving torque on the output shaft is smoother and without a sudden change (though it is small for PI and feedforward-feedback without DO). Therefore, for practical engineering use where there is bound to be unknown disturbances and uncertainties, DO should always be integrated in the controller.  
Figure 11: Comparison with pure feedforward and feedback controller.