# Design of Constant Speed Controller for Hydraulic Retarder Based on Robust Control
## 4 Simulation Results and Analysis  
### Heavy-Duty Vehicle Simulation Conditions  
According to the vehicle downhill dynamic equations established by Equation (14), the constant speed control strategy during continuous braking is simulated and analyzed in the MATLAB/Simulink environment, and the overall simulation model of heavy-duty vehicle downhill braking is established as shown in Figure 5. Figure 6 shows the constant speed control logic for the vehicle travelling downhill. The controller control structure schematic is shown in Figure 7. In order to test the viability of the controller created in this work, simulation experiments were carried out utilizing the sliding mode control algorithm and fuzzy control algorithm models [Appendix A] under varied operating conditions. Assuming the vehicle is traveling downhill in third gear, with a speed of 60 km/h, the vehicle descends without involving mechanical brakes, travels in a straight line on a flat road with adequate grip, and the wheels do not lock or skid. The hydraulic retarder filling delay time of 0.1 s is considered in the simulation. The simulation time was set to 80 s, and the simulation step was set to 0.1 s. The main simulation parameters for the system are detailed in Table 1. To test the performance of the control strategy, a fixed vehicle mass M1 and a varying vehicle mass M2 were set to complete the simulation, as shown in Figure 8. In addition, three road gradient conditions, S1, S2, and S3, were designed to verify the performance of the controller, as shown in Figure 9.  
Figure 5: Overall simulation architecture. Figure 6: Feedback control system with the controller. Figure 7: Schematic diagram of control structure. This paper compares the performance of different hydraulic retarder filling rate controllers by varying the vehicle payload and the gradient conditions of the road during the long downhill drive. As shown in Table 2, three different simulation conditions, C1, C2, and C3, are designed in this paper. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Initial vehicle velocity & 60 km/h \\ \hline Target vehicle velocity & 30 km/h \\ \hline Aerodynamic resistance coefficient & 0.65 \\ \hline Windward area & 5 m\({}^{2}\) \\ \hline Density of fluid medium & 860 kg/m\({}^{3}\) \\ \hline Gear ratio of transmission & 1.646 \\ \hline Gear ratio of main reducer & 4.111 \\ \hline Wheel radius & 0.554 m \\ \hline Simulation step & 0.1 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of simulation model. Figure 8: Vehicle mass conditions in the simulation: (**a**) Fixed vehicle mass, (**b**) Varying vehicle mass. Figure 9: Road slope conditions in the simulation: (**a**) Fixed slope, (**b**) Sinusoidal change slope. In Section 3 high-frequency controller design, the values of the two hyperparameters, \(k\) and \(\varepsilon\), will have a great impact on the high-frequency controller speed tracking performance and control outputs. In this paper, three different combinations of \(k\) and \(\varepsilon\) are selected, respectively, and simulation experiments are carried out under C2 conditions, and the control performance of high-frequency controllers with different parameter combinations is compared and analyzed, and the simulation results are shown in Figure 9. The RMSE (Root Mean Square Error) and Max (Maximum Absolute Speed Error) of the speeds are shown in Table 3.  
Figure 10a illustrates that an increase in the value of \(k\) within the controller enhances the system's response speed, hence accelerating the regulation of the vehicle's speed. Table 3 shows that when the value of \(k\) is fixed, the value of \(\varepsilon\) is directly proportional to the RMSE and Max of the vehicle speed. This indicates that the smaller the value of \(\varepsilon\), the better the speed tracking performance of the controller.