## 4 Simulation
### ESN-Based MPC Results  
After training the ESN, we obtained the trained weight matrix \(W^{out}\) and applied it to the proposed ESN-MPC scheme. Figure 7 shows the results of the proposed active vibration control for the ICE start condition compared to that for the HEV drivetrain without a controller. The untrained samples were applied to the proposed ESN-MPC scheme. For comparison, the time margin between the current and MPC optimization times (as illustrated in Figure 2) was set as one control period for the proposed controller Figure 7a,c,e show the results for the proposed controller, whereas Figure 7b,d,f show the results for the case without a controller. Figure 7c,d show the results for the output torque. The proposed controller reduces the torque oscillation compared to that in the case without the controller, despite the occurrence of sudden torque fluctuations with a large amplitude. The rotation speeds \(\omega_{e}-\omega_{c}\) and \(\omega_{m2}\) were nearly stable at 0 rpm during the entire engine starting period, as shown in Figure 7a. Generally, \(\omega_{m2}\) is related to the output speed, and 0 rpm indicates that the vehicle is not moving. In contrast, in the case without a controller, as shown in Figure 7b, \(\omega_{m2}\) is stable at 0 rpm, whereas \(\omega_{e}-\omega_{c}\) exhibits the typical characteristics of an oscillating waveform. Generally, \(\omega_{e}-\omega_{c}\) indicates the difference in the engine rotation and planetary carrier rotation, which corresponds to the angular speed of the torsional damper. Considering the function of the torsional damper, when the input speed is equal to the output speed (\(\omega_{e}-\omega_{c}~{}=~{}0\)), the torsional damper does not have a twist angle. In this case, the HEV powertrain does not transmit any torque to drive the output shaft. Therefore, the proposed ESN-MPC controller can reduce the torque oscillation. For the control input, the proposed controller inputs the torque of the two motors, i.e., \(T_{m1}\) and \(T_{m2}\), as shown in Figure 7e. By comparing Figure 7d,e, it can be noted that the proposed controller tends to generate an inverse torque for the oscillation. Figure 5: ESN simulations and predictions of the forward time steps for (**a**) 1 ms, (**b**) 10 ms, and (**c**) 20 ms. The gray line and blue line represent the true state and the ESN prediction, respectively. Figure 6: ESN estimation error for 10 variations of the test data. We evaluated EDT-MPC using the conventional approach and compared it with the proposed ESN-MPC. Both the controllers considered the effect of the dead-time and time lag behavior of the actuator in different ways. The time margin applied to the proposed ESN-MPC and the artificial dead-time applied to the conventional EDT-MPC indicates the time difference between the current and actual control times. Generally, this time difference affects the prediction accuracy of the disturbance and compensation performance. Figure 8 shows the results for the proposed ESN-MPC and conventional EDT-MPC during the ICE start condition. Three different time margins (1, 10, and 20 ms) are compared in the results. Figure 8a-c present the output torque of the HEV powertrain. In the 1 and 10 ms cases as shown in Figure 8a,b, the output torque is stable at 0 Nm when the proposed ESN-MPC is used; however, in the case of the conventional EDT-MPC, the torque tends to converge with minor oscillations. In the 20 ms case Figure 8c, the result for the proposed controller is stable, whereas that of EDT-MPC exhibits increased fluctuation, thereby indicating the higher performance of the ESN\(-\)MPC controller compared to that of EDT-MPC. Figure 8d-i presents two control inputs. These control inputs remain within the corresponding limits in all the cases owing to the MPC constraints. The control inputs for the proposed method do not exhibit a large difference as the time margin increases, as shown in Figure 8d-f.