## 4 Simulation
### ESN Prediction Results  
The training dataset for the ESN prediction model included 40 realization samples of the engine torque waveform. After learning the dataset, 10 untrained datasets were used for the validation. To verify the application of the ESN, simulation parameters were applied to the model as follows: input unit \(K\ =\ 1\), output unit \(L\ =\ 1\), reservoir units \(N\ =\ 500\), and leaking decay rate \(a\ =\ 0.3\). These parameters were determined empirically as the optimal values for the studied benchmarks. The input data includes the time-series waveform. One input unit received the past waveform, and one output unit determined the value for the next time step. By repeating this operation, the trajectory tracking of the waveform could be predicted.  
Figure 5 shows the prediction results of the ESN for the trajectory tracking of the waveform. Each figure displays the estimation performed by the ESN for different forward steps from the current time. As expected, the ESN estimation is nearly similar to the actual waveform shown in Figure 5a (forward step in 1 ms). However, as the forward time steps increased to 10 and 20 ms, as shown in Figure 5b,c, the difference between the estimated and true state waveforms increased. To examine this aspect, the total error in one waveform was estimated for each condition. The arithmetic mean of the total error over all points of one waveform was calculated as:  
\[e=\frac{1}{N_{sample}}\sum_{i=1}^{N_{sample}}\frac{|T_{i}^{ estimated}-T_{i}^{ actual}|}{|T_{i}^{actual}|} \tag{38}\]  
where \(N_{sample}\) is the number of sampling data points in one target waveform, \(T_{i}^{estimated}\) is the estimated result from the ESN, and \(T_{i}^{actual}\) is the target waveform. \(T_{i}^{actual}\) is the output torque from the ICE and includes pumping loss. Pumping loss is energy loss due to the reciprocating motion of the piston. Since part of the energy generated by combustion is consumed in the intake and exhaust stroke, \(T_{i}^{actual}\) takes either a positive or negative value. The results are shown in Figure 6. The error bars show the maximum and minimum data for all 10 runs. The error increased monotonically, thereby indicating that the predictions for the distant future tended to be less accurate.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(I_{m1}\) & M1 inertia & 0.0265 kg m\({}^{2}\) \\ \(I_{m2}\) & M2 inertia & 0.035 kg m\({}^{2}\) \\ \(I_{e}\) & Engine inertia & 0.20 (0.08) kg m\({}^{2}\) \\ \(I_{c}\) & Carrier inertia & 0.01 kg m\({}^{2}\) \\ \(I_{s}\) & Sun gear inertia & 0.01 kg m\({}^{2}\) \\ \(I_{r}\) & Ring gear inertia & 0.005 kg m\({}^{2}\) \\ \(I_{p}\) & Pinion gear inertia & 0.001 kg m\({}^{2}\) \\ \(R_{s}\) & Sun radius & 0.0477 m \\ \(R_{p}\) & Pinion radius & 0.0382 m \\ \(n_{p}\) & Number of pinions & 4 \\ \(\lambda\) & Planetary gear ratio & 0.3846 \\ \(k_{dp}\) & Damper stiffness & 700 Nm/rad \\ \(c_{dp}\) & Damping coefficient & 10 Nm s/rad \\ \(t_{s}\) & Sampling time & 1 ms \\ \(H_{u}\) & Control horizon & 5 steps \\ \(H_{p}\) & Prediction horizon & 6 steps \\ \(T_{m1}^{min}\,\,T_{m1}^{max}\) & M1 torque limitation & \(-\)150, +150 Nm \\ \(T_{m2}^{min}\,\,T_{m2}^{max}\) & M2 torque limitation & \(-\)100, +100 (\(-\)200, +200) Nm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Specifications of the active vibration controller for the reference HEV powertrain. The control performances of the proposed ESN-MPC were compared using different design variables. ( ) represents the case where the inertia is 0.08 kg m\({}^{2}\).