## 4 Simulation Results
### Experimental Result  
In this subsection, the performance of the modified HOSM controller using a PID-sliding surface (14) applied to a real-time ABS laboratory setup is shown. Moreover, the real-time results obtained with the proposed controller (14) are compared with another controller available in the literature, the PID-like controller (26) proposed in [11,26,37]. The variables and the values of the ABS laboratory setup (5) are given in Table 1. Moreover, the parameters used in the controllers (14) and (26) are given in Tables 1 and 2, respectively. To test the proposed controller robustness, a reader could consider that the coefficients in Table 1 may vary in practice due to natural wear for the mechatronic device but may remain close to these values. Similar to the numerical simulations, the tests considered \(\omega_{1}(0)=\omega_{2}(0)=178\) rad/s (1700 RPM) and \(T_{b}(0)=0\) as initial conditions for (5). These settings simulated a vehicle at a velocity of 63.43 km/h, and the brake system is abruptly activated, sending a control input to the actuator to start a braking phase. It is worth remarking that for this setup, the nominal value of the friction coefficient between the two wheels used in (4) is given by the manufacturer as \(\mu=1\). The experimental results are summarized in Figures 5-7, where the proposed controller, (14), ensures a better performance with respect to the PID-like controller, (26). Figure 5 shows the wheel velocity \(v_{w}\) and the vehicle longitudinal velocity \(v_{x}\). The wheel slip \(\lambda\) and the tracking error \(\lambda-\lambda_{\text{ref}}\) are shown in Figure 6. The applied input \(T_{b}\) is shown in Figure 7.  
Analyzing Figure 7, we emphasize that controller (14) imposes higher control values than the PI-like controller (26). In this sense, the reduction in the braking time and space, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(\gamma_{1}\) & Gain in modified HOSM & 2.62 \\ \(\gamma_{2}\) & Gain in modified HOSM & 0.9 \\ \(\gamma_{3}\) & Gain in modified HOSM & 1.7 \\ \(\gamma_{4}\) & Gain in modified HOSM & 10 \\ \(k_{p}\) & PID-sliding surface & 5.5 \\ \(k_{i}\) & PID-sliding surface & 20 \\ \(k_{d}\) & PID-sliding surface & 0.015 \\ \(k_{p,c}\) & PID-like controller gain & 32 \\ \(k_{i,c}\) & PID-like controller gain & 15 \\ \(k_{d,c}\) & PID-like controller gain & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients of controllers (14) and (26) used in the numerical and real-time simulations. as indicated in Figure 5, is quite natural. The latter could induce higher stress in the actuator. However, since the ABS is an emergency device that does not take any actions during normal braking but rather during only emergency braking, this aspect and the energy minimization are marginal. After the braking process taking place at 5.5 and 7.4 s, corresponding to the peak braking efficiency, we highlight that the performance is no longer considered as the velocity is insignificant at this point and the ABS is not operating in the appropriate scope of velocities anymore. Figure 5 shows that both controllers maintain deceleration at a constant pace. The difference lies in the time that the controllers take to complete the braking phase; the proposed controller (14) manages to stop a few fractions of a second before the PID-like controller due to the proposed controller maintaining a \(\lambda\) close to the reference, as shown in Figure 6, compared with the PID-like controller. After the braking phase, the reference tracking is pointless because the longitudinal velocity is too low and the ABS is rendered ineffective. Figure 5: **(a)** Wheel velocity \(v_{w}\): a modified HOSM controller using PID-sliding surface (14) (black) or PID-like controller (26) (blue) and (**b**) vehicle longitudinal velocity \(v_{x}\): a modified HOSM controller using PID-sliding surface (14) (black) or PID-like controller (26) (blue).