# Design of Constant Speed Controller for Hydraulic Retarder Based on Robust Control
## 4 Simulation Results and Analysis
### Simulation Results  
The RMSE and Max of the speeds of the three controllers under different vehicle loads and road gradients are shown in Table 4. Figures 10-12 show the simulation results of different controllers for long downhill slopes of heavy vehicles under three simulation conditions. Figure 10: Simulation results of controller with different control parameters: (**a**) Vehicle speed, (**b**) Hydraulic retarder filling rate. \begin{table}
\begin{tabular}{l l l} \hline  & **RMSE** & **Max** \\ \hline \(k=0.1\), \(\varepsilon=0.01\) & 0.145 & 0.165 \\ \hline \(k=1\), \(\varepsilon=0.01\) & 0.0652 & 0.0749 \\ \hline \(k=1\), \(\varepsilon=0.1\) & 0.182 & 0.194 \\ \hline \end{tabular}
\end{table} 
Table 3: Comparison of speed error results for different control parameters. Analyzing the data in Table 4, it can be observed that the fuzzy controller has the best speed tracking performance with the RMSE of speed of 0.01759 and the maximum absolute error of speed of 0.0195 in the C1 simulation scenario. However, under the C3 simulation situation, the fuzzy controller has a huge gap in speed tracking performance compared to the other two controllers. The difference in speed tracking performance between the high-frequency controller and the sliding mode controller is not substantial. In the C2 situation, the RMSE of the high-frequency controller is 7.2% less than that of the sliding-mode controller, and the highest absolute error of the speed is 14.1% smaller than that of the sliding-mode controller. The sliding mode controller, on the other hand, shows the best speed tracking performance in the C3 condition, where the RMSE of the speed of the sliding mode controller is 1.4% less than that of the high-frequency controller, and the maximum absolute error of the speed is 25.8% less than that of the high-frequency controller. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{2}{c}{**C1 (km/h)**} & \multicolumn{2}{c}{**C2 (km/h)**} & \multicolumn{2}{c}{**C3 (km/h)**} \\ \hline \multirow{2}{*}{Fuzzy control} & RMSE & Max & RMSE & Max & RMSE & Max \\ \cline{2-7}  & 0.0176 & 0.0195 & 0.0729 & 0.113 & 0.184 & 0.370 \\ \hline \multirow{2}{*}{High-frequency} & RMSE & Max & RMSE & Max & RMSE & Max \\ \cline{2-7}  & 0.0841 & 0.0842 & 0.0652 & 0.0679 & 0.0683 & 0.122 \\ \hline \multirow{2}{*}{Sliding mode} & RMSE & Max & RMSE & Max & RMSE & Max \\ \cline{2-7}  & 0.0704 & 0.0716 & 0.0703 & 0.0790 & 0.0674 & 0.0905 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of controller speed error results. Figure 11: Simulation results under C1 conditions: (**a**) Vehicle speed, (**b**) Hydraulic retarder filling rate, (**c**) Vehicle speed error and (**d**) Braking power of hydraulic retarder. Figure 11 illustrates the control effects of the three controllers under constant vehicle mass and fixed road gradient conditions. Figure 11a-d illustrate the vehicle speed, retarder filling rate, vehicle speed error, and retarder braking power, respectively. Upon activation of the retarder, the vehicle's speed diminishes to approximately the target velocity within 15 s. The saturation function of the sliding mode controller, high-frequency controller, and fuzzy controller can swiftly and precisely determine the necessary retarder filling rate under current operating conditions, thereby satisfying the requirements for the vehicle's slow downhill traveling. The fuzzy controller exhibits the highest control accuracy with the minimal speed error; nevertheless, it requires the longest duration, approximately 20.1 s, to approach the stable value. As the vehicle speed approaches the target speed, the control output (hydraulic retarder filling rate) of the sliding-mode controller with a saturation function exhibits pronounced chatter, accompanied by minor fluctuations in the high-frequency controller, whereas the fuzzy controller demonstrates superior output smoothing. The simulation results considering the change in vehicle mass are shown in Figure 12a,b.