# Design of Constant Speed Controller for Hydraulic Retarder Based on Robust Control
## 1 Introduction
[22] discussed in depth how to achieve the constant-speed downhill purpose through reasonable hydraulic retarder filling rate adjustment under different road conditions and adopted the graded variable domain fuzzy control strategy based on the fuzzy control theory to achieve the constant-speed downhill control of a vehicle equipped with a hydraulic retarder. The hydraulic retarder filling rate control method that sets the step value in the above control strategies is difficult to ensure the constant-speed control when the vehicle speed or road gradient changes during the downhill process. The fuzzy control is adaptive to the model uncertainty, but its rule-based design is complicated, and the dynamic response is insufficient. In contrast, sliding mode control is an effective means to solve such problems due to its strong robustness in sliding mode. However, although the sliding mode control is excellent in dealing with parameter uncertainty, external perturbations, and nonlinear systems, the classic sliding mode control often suffers from high-frequency jitter problems. In order to suppress jitter while retaining the robustness of SMC, the mainstream solutions can be divided into three categories:  
1. Boundary layer saturation function method [23], which achieves smooth transition in the neighborhood of the sliding mode surface by replacing the sign function with a continuous saturation function, but needs to trade off between the steady-state error and the smoothness. 2. Higher-order sliding mode control [24], which fundamentally eliminates jitter by transferring the discontinuous terms to higher-order derivatives through differentiation of the sliding mode surface, but the computational complexity increases significantly. 3. Perturbation observer compensation method [25], which reduces the switching gain requirement by estimating the perturbation through feed-forward compensation. However, existing methods still face challenges when applied to the long downhill filling rate control of heavy vehicles: the saturation function method is difficult to adapt to the optimal boundary layer thickness under time-varying perturbations; the higher-order sliding mode computation is complicated and not easy to implement; and the observer method relies on an accurate perturbation model. This study presents a new solution targeting these issues. Firstly, an enhanced robust control strategy based on high-frequency switching is devised to address the difficulties of nonlinearity and time lag in the retarder filling rate management, the core of which resides in the optimal design of the high-frequency switching and dynamic adjustment mechanism. Secondly, in order to verify the control performance of the high-frequency robust controller, this study carried out a detailed analysis of the long downhill braking process of the hydraulic retarder for heavy-duty vehicles and designed and compared various hydraulic retarder filling rate controllers by using sliding-mode control with a saturation function, high-frequency robust control, and fuzzy control methods. The results show that the high-frequency robust controller designed in this study achieves a constant speed control accuracy of 0.0683 km/h for the root-mean-square error of vehicle speed and 0.122 km/h for the maximum absolute error of vehicle speed under the conditions of variation of vehicle mass and road gradient and is capable of suppressing jitter during external disturbances and system parameter variations while still retaining strong robustness and speed tracking performance. This paper is arranged as follows. Section 2 displays the simulation model of the hydraulic retarder and the vehicle in Simulink. Section 3 focuses on the controller design and stability proof. Section 4 displays the simulation results and their analysis. Section 5 addresses the conclusion and future work.