## 4 Simulation
### ESN-Based MPC Results
In contrast, the control inputs when using the conventional EDT-MPC, as shown in Figure 8g-i, exhibit a considerable change as the setting dead-time increases. In particular, as the setting dead-time increases, the amplitude of the control input increases, and the shape of the control input changes. In the 20 ms case, the waveforms of the two control inputs are inverted compared to those at 10 ms, demonstrating the occurrence of a phase shift. The fluctuations of the output torque are larger in the conventional EDT-MPC than those in the proposed method. Owing to the phase shift of the control input and larger amplitude, a larger compensation error occurs while using the conventional controller. The proposed ESN-MPC outperformed the conventional EDT-MPC even when the time margins were large. The execution time for the optimization computation for the two controllers is compared in Table 2. The histograms of the computation time are shown in Figure 9. The measured computation time includes all the 10 patterns of test data. The proposed ESN-MPC and conventional EDT-MPC were tested for three values of the time margin and setting dead-time, respectively, 1, 10, and 20 ms. In Table 2, the average computation time  
Figure 7: The results of the simulation during the ICE start. (**a**,**c**,**e**) correspond to the proposed ESN-MPC active vibration control, with a time margin of 1 ms. (**b**,**d**,**f**) correspond to the case without a controller. when using the proposed ESN-MPC is the same, regardless of the time margin. Similarly, the histograms of the computation time shown in Figure 9 exhibit nearly the same shape when using the proposed method. In contrast, the average computation time when using the conventional EDT-MPC increases with the setting dead-time. The distribution of the calculation time when using the conventional method tends to shift towards a larger computation time as the setting dead-time increases. Owing to the dead-time required for the recursive calculation in the conventional EDT-MPC, a larger dead-time corresponds to a larger computation time. Moreover, the maximum computation time is different under different simulation conditions. Although the frequency is extremely low, it is assumed that rare conditions exist in the optimization calculation in which a significant amount of time is required to identify the optimal solution in the test case for both models. Figure 8: A comparison of the proposed ESN-MPC and conventional EDT-MPC models under different time margins during ICE start conditions: (**a**,**d**,**g**) correspond to 1 ms; (**b**,**e**,**h**) correspond to 10 ms; and (**c**,**f**,**i**) correspond to 20 ms. In (**a**–**c**), ESN-MPC and EDT-MPC results are represented by the black and red lines, respectively. The control inputs of ESN-MPC are shown in (**d**–**f**), whereas those for EDT-MPC are shown in (**g**–**i**). The blue and magenta lines indicate the torque of electric motors M1 and M2, respectively, and the dotted lines indicate the limitations of the motor torque. Figure 9: A comparison of the computation time for the optimization calculation in terms of relative frequency distribution. (**a**–**c**) and (**d**–**f**) correspond to the proposed ESN-MPC and conventional EDT-MPC, respectively. The dotted black and magenta lines indicate the minimum and maximum computation time, respectively.