## 4 Design Example and Simulation Results
### Tracking of the Current Reference  
The current reference tracking evaluates the ability of the HESS to provide the high-frequency transients using the supercapacitor. In addition to the adaptive LQG-feed-forward controller (LQG-FF) proposed in this paper, this test also includes two traditional control strategies for comparison purposes: the classical PI controller widely adopted in literature for this same application, and the classical LQG controller. The tests consider step perturbations on the DC bus current (\(i_{dc}\)) from 0 A to 1 A. In these conditions, the battery voltage (\(v_{dc}\)) is equal to 12 V, and three levels of supercapacitor voltage (\(v_{sc}\)) are considered: 9 V (buck mode), 12 V (unitary gain), and 15 V (boost mode). The results are shown in Figure 12 (boost mode), Figure 13 (unitary gain) and Figure 14 (buck mode). In these simulations, the proposed LQG-FF controller exhibits the faster tracking of the reference (\(i_{ref}\)) (yellow/dashed line), thus imposing an output current profile (\(i_{sc}\)) (magenta/dashed line) close to the desired waveform (\(i_{ref}\)). In addition, the transitory peak occurring in the battery current (magenta/solid line) is the lowest one in comparison with the other control systems, where the classical PI controller exhibits the worst performance in terms of maximum overshoot and reference tracking. This is expected since the parameters of the proposed LQG-FF controller are adapted in real time to ensure the tracking of the high-frequency reference, and hence lower oscillations in the battery current occur. The classical controllers (PI and LQG) were designed for a particular condition in the middle of the operation range, and hence these controllers were not adapted during the transient conditions. Figure 11: Pole-zero mapping considering parametric variations with \(v_{dc}=12\) V, \(v_{sc}=12\) V, and \(i_{dc}=1\) A. The Root-Mean-Square-Error (RMSE) between \(i_{ref}\) and \(i_{sz}\) is calculated during the transitory response to be used as a performance indicator. The RMSE for the three controllers are shown in Table 2, where different operating points are tested. In this test, the proposed LQG-FF controller provides the best performance in the complete range of \(v_{sc}\), and the PI controller exhibits the worst behavior. The last two columns of the table show the error comparison between the proposed LQG-FF solution and the two classical approaches,  
Figure 14: Dynamic performance of reference tracking for \(v_{sc}=15\) V and \(v_{dc}=12\) V (buck mode). Figure 12: Dynamic performance of reference tracking for \(v_{sc}=9\) V and \(v_{dc}=12\) V (boost mode). Figure 13: Dynamic performance of reference tracking for \(v_{sc}=12\) V and \(v_{dc}=12\) V (unitary gain). where the LQG-FF controller provides a fraction of the error introduced by the other two solutions. The controller's performances for longer profiles were evaluated considering a load current that forces both charge (\(i_{dc}<0\)) and discharge (\(i_{dc}>0\)) conditions. The tests results are reported in Figures 15-17, where the DC bus current (\(i_{dc}\)) is presented in green color. The bus current starts at 0 A, and after 4 s the DC bus current is increased to 1 A, and thus the battery enters discharging mode (\(i_{dc}>0\) A) and the supercapacitor provides the fast current transient to the DC bus using \(i_{sz}\). At 10 s, the DC bus current is decreased to 0 A, and in this case the supercapacitor absorbs the negative current transient. At 16 s, the DC bus current is changed to \(-1\) A, and hence the battery enters charging mode (\(i_{dc}<0\) A) and the supercapacitor absorbs the negative current transient. Finally, the test ends by driving the DC bus current to zero, which sets the battery into stand-by mode (\(i_{dc}=0\) A). For all bus current perturbations, the battery current reaches the stable condition after 1 s with null steady-stable error.