# Design of Constant Speed Controller for Hydraulic Retarder Based on Robust Control
## 2 Dynamic Model of Vehicle Driving Downhill
### Vehicle Dynamics Model  
The downhill braking force analysis of heavy vehicles with hydraulic retarders is presented in Figure 4. Let the heavy vehicles along the slope of the ramp be straight downhill travelling; downhill travelling ramp length is long enough; in the process of travelling, the vehicle wheels are pure rolling, the main brake and accelerator pedal are not working, and the hydraulic retarder is in the working state. According to [PERSON]'s Second Law of Motion, the establishment of the vehicle downhill driving dynamics equation for Equation (4). \[m\frac{dv}{dt}=F_{i}-F_{r}-F_{w}-F_{f}-F_{ec} \tag{5}\]  
Figure 3: Vehicle slowing downhill control process. where \(m\) (kg) is the mass of vehicle, \(v\) (km/h) is the current vehicle velocity, \(F_{i}\) (N) is the gravity fraction in direction of vehicle travel along the ramp, \(F_{r}\) (N) is the braking force produced by the hydraulic retarder on the wheels, \(F_{w}\) (N) is the air resistance, \(F_{f}\) (N) is the rolling resistance, \(F_{cc}\) (N) is the engine braking force. The gravity fraction in direction of vehicle travel along the ramp is expressed as follows:  
\[F_{i}=G\sin\alpha \tag{6}\]  
where \(G\) (N) is the gravity of the vehicle, \(\alpha\) (%) is the slope of the road. According to China's Expressway Design Specification, the mountain ridge heavy slope area of the expressway is 5%, and the maximum slope of the plain micro-hill area is 3%; the first-class automobile special highway mountain ridge heavy slope area is 6%, and the maximum slope of the plain micro-hill area is 4%; the general four-level highway mountain ridge heavy slope area is 9%, and the plain micro-hill area is 5%. Therefore, the slope of the general roads can be obtained as follows:  
\[\sin\alpha\approx\tan\alpha=i,\cos\alpha=1 \tag{7}\]  
The force of gravity can be written as follows:  
\[F_{i}=G\sin\alpha\approx Gi \tag{8}\]  
\(F_{r}\) is the braking force created by the hydraulic retarder on the wheels; from the foregoing part of the hydraulic retarder braking torque formula, the braking force expression may be determined as follows:  
\[F_{r}=\frac{M_{r}\eta_{i}i_{0}i_{g}}{r}=\frac{\lambda\rho\eta_{t}n^{2}D^{5}i_{ 0}i_{g}q}{r}=\frac{\lambda\rho\eta_{t}n^{2}D^{5}i_{0}{}^{3}i_{g}{}^{3}q}{(0.37 7)^{2}r^{3}} \tag{9}\]  
where \(i_{0}\) is the gear ratio of the main reducer, \(i_{g}\) is the gear ratio of the transmission, \(\eta_{t}\)(%) is the vehicle mechanical transmission efficiency, \(r\) (m) is the wheel radius. The air resistance of the vehicle travelling downhill can be expressed as follows:  
\[F_{w}=C_{D}A\
u^{2}/21.15 \tag{10}\]  
where \(C_{D}\) is the aerodynamic resistance coefficient, \(A\)(m\({}^{2}\)) is the windward area. Figure 4: Force analysis of downhill-traveling vehicle. The rolling resistance is expressed as follows:  
\[F_{f}=Gf\cos\alpha \tag{11}\]  
where \(f\) is the rolling resistance coefficient equal to \(0.0076+0.000056u_{a}\), \(u_{a}\) is the present vehicle velocity. Vehicles are prohibited from coasting in neutral when travelling long distances downhill. Therefore, when descending a hill, the clutch must remain engaged with the vehicle driveline, as the vehicle is not disengaged from the engine. During this process, the engine's compression stroke, influenced by compression resistance, internal friction, and intake and exhaust resistance, generates a certain amount of braking force on the wheels. The mathematical formula for the braking torque of the motor is expressed as follows:  
\[m\frac{dv}{dt}=F_{i}-F_{r}-F_{w}-F_{f}-F_{ec} \tag{5}\]0  
where \(V_{h}\) (m\({}^{2}\)) is the working volume of the engine cylinders, \(\tau\) is the number of engine strokes, \(p_{m}\) (Pa) is the average mechanical loss pressure of the engine. The average mechanical loss pressure of the engine is influenced by various factors, including the engine's crankshaft, pistons, and other mechanical components. Additionally, it is affected by the engine's load, speed, and numerous other operational parameters.