# Design of Constant Speed Controller for Hydraulic Retarder Based on Robust Control
## 2 Dynamic Model of Vehicle Driving Downhill
### Hydraulic Retarder Model
The literature [12] adopts the uniform density method and gas-liquid layering method to establish mathematical models for the braking performance of the hydraulic reducer under partially liquid-filled working conditions, and the comparative analysis of example calculations and experimental data shows that the uniform density method is simple and easy to carry out, while the calculation method of the gas-liquid layering method is relatively complex and theoretically more accurate, but its reasonableness still needs to be further verified through the partially liquid-filled braking performance test. Using the uniform density method, the braking torque of the hydraulic retarder under a partially filled condition can be approximated and simplified as the product of the filling rate and the full-filled braking torque at this speed:  
\[M_{r}=\lambda\rho n^{2}D^{5}q \tag{2}\]  
where \(q\) (%) is the filling rate of the working cavity. When the hydraulic retarder is working, its working cavity can convert into the kinetic energy of the vehicle into the internal energy of the working fluid, embodied in the hydraulic retarder outlet of the high-temperature working fluid, and the high-temperature working fluid enters the radiator of the cooling system and then dissipates the heat. If the heat dissipation capacity of the cooling system is unable to dissipate all the heat generated by the hydraulic retarder, the overall temperature of the coolant will rise. When the temperature rises above the permissible value, the controller needs to reduce the braking power output of the hydraulic retarder to ensure the safety and stability of the cooling system. That is, the maximum braking power that the hydraulic retarder can continuously output and the corresponding braking torque should depend on the maximum cooling power that can be dissipated by the engine cooling system. As previously demonstrated, the hydraulic retarder braking torque formula can be used to calculate the corresponding retarder braking power; its equation is as follows:  
\[P_{r}=\frac{M_{r}n}{9549} \tag{3}\]  
where \(M_{r}\) (N-m) is the hydraulic retarder braking torque when the retarder working cavity is partially filled with oil. \(n\) (rpm) is the rotor speed of the hydraulic retarder. When the working cavity of the hydraulic retarder is entirely filled with oil, the maximum braking power that the hydraulic retarder can generate at the present vehicle speed is expressed as follows:  
\[P_{\text{max}}=\frac{\lambda\rho n^{3}D^{5}}{9549} \tag{4}\]  
When the vehicle needs a hydraulic retarder for deceleration braking in downhill driving, the driver starts the retarder, and the oil begins to flow into the working cavity, generating braking torque, which enters the deceleration braking stage of the vehicle downhill braking process, and the controller calculates the maximum braking power of the hydraulic retarder under the present speed and the filling rate to compare it with the maximum radiator cooling power, so as to fulfill the braking requirement of the entire vehicle downhill deceleration braking process under the premise of ensuring the vehicle's radiator system safety. The controller will calculate the maximum braking power of the retarder at the present speed with the filling rate and compare it to the maximum cooling power of the radiator to achieve the braking requirement of the downhill deceleration braking process while ensuring the vehicle's cooling system safety. When the vehicle has decelerated to the vicinity of the driver's desired target speed, the driver opens the constant speed switch, at which time the braking process enters the constant-speed braking stage, and the electronic control unit records the present vehicle speed as the target speed. The error signal between the present speed and the target speed is input to the constant speed controller. The controller changes the braking torque of the retarder by changing the retarder filling rate to achieve the goal of driving downhill at a constant speed.