# Design of Constant Speed Controller for Hydraulic Retarder Based on Robust Control
## 2 Dynamic Model of Vehicle Driving Downhill
### Hydraulic Retarder Model  
Figure 1 shows the open working chamber and hydraulic control model. The input, output, and interaction parameters between the two models are defined in Figure 1 as well. The retarder working chamber is constituted by the rotor 1.1 and the stator 1.2 with blades; the rotor is connected to the output shaft, whereas the stator is fixed to a fixed component. When the retarder is operational, the system oil controlled by the switch valve 3 flows from the system oil circuit into the retarder working cavity between the rotor and stator. At this time, the rotor moves the hydraulic fluid in the working chamber. Meanwhile, the stator 1.2 is fixed, which will produce the reaction force. The rotor receives the reaction force to impede its rotation, which generates the braking torque to slow down the vehicle [8]. For the working chamber, oil discharged from outlet A will flow into the circulating cooling circuit, which could be cooled by the heat exchanger. Then, oil returns to the working chamber through the inlet oil circuit, thus completing the self-circulating cooling process. Additionally, oil outflowing from outlet B will enter the charging regulation circuit. The ECU transmits the control signal to the charging regulation valve 4, containing a two-way cartridge valve 4.1 and an electromagnetic proportional pressure relief valve 4.2, which adjusts the hydraulic retarder filling rate by changing the oil pressure of outlet B. The hydraulic retarders can be divided into two types according to their installation positions: one is installed behind the engine, and the other is behind the transmission [26]. Compared with the installation behind the transmission, the hydraulic retarder installed behind the engine may cause power interruption and impact in the process of shifting gears when the vehicle is braking. However, hydraulic retarders in this installation position tend to have stronger low-speed braking performance, which can better satisfy the braking requirements of the vehicle traveling slowly downhill. In this study, the hydraulic retarder is installed behind the engine. Figure 2 shows the hydraulic retarder mounting position. The amount of retarder braking torque is mainly dependent on the vehicle speed, the amount of working fluid, and other factors. Based on one-dimensional flow theory and the structural parameters of the hydraulic retarder, the braking torque of a fully oil-filled hydraulic retarder at a constant revolution could be estimated as follows [27]:  
\[M_{r}=\lambda\rho n^{2}D^{5} \tag{1}\]  
where \(\lambda\) is the braking torque coefficient of the hydraulic retarder, the value of \(\lambda\) in this paper is taken to be 0.001. \(\rho\) (kg/m\({}^{3}\)) is the hydraulic retarder working oil density, the fluid  
Figure 1: Structure of the retarder control system. Figure 2: Mounting positions of hydraulic retarder. of the retarder is commonly used mineral hydraulic oil, and the density of the fluid is set as 860 kg/m\({}^{3}\). \(n\) (rpm) is the hydraulic retarder rotor speed. \(D\) (m) is the effective diameter of the retarder circuit; the value of \(D\) in this paper is taken to be 0.293 m.  
In the actual working process, when the hydraulic reducer is involved in braking, its internal state is usually in a partially filled state. Under this condition, the flow field inside the hydraulic reducer presents complex gas-liquid two-phase flow characteristics. Compared with the full-flooded condition, the braking performance prediction and calculation process in this partially-flooded condition has higher complexity, which is mainly reflected in the two-phase flow interactions, dynamic changes of the phase interface, and energy dissipation mechanisms.