# Research on Ride Comfort Control of Air Suspension Based on Genetic Algorithm Optimized Fuzzy PID
## 2 Dynamic Modeling of Air Suspension Systems  
### Vehicle Model Construction  
The air suspension system is a rather complex nonlinear system, making it difficult to establish an accurate model. In a study of vehicle performance, the suspension system should be simplified based on the specific research problem. Key aspects that require detailed investigation should be thoroughly analyzed, while factors with a minimal impact on certain performance characteristics should be simplified or omitted. This approach ensures that the simplified system model retains the main features pertinent to the research problem while remaining both straightforward and practical.  
Within the scope of vehicle ride comfort research, focusing primarily on the vertical dynamics of the vehicle, the following assumptions are made based on the operational characteristics of the suspension system.  
1. During vehicle operation, the road surface unevenness excitation experienced by the left and right tires is identical;
2. The effects of the engine, steering, and drivetrain systems on vibrations are neglected in the modeling process;
3. The damping of the tires is significantly less than that of the suspension dampers; hence, only the stiffness of the tires is considered, and tire damping is ignored in the modeling;
4. The time delay effect of the suspension is neglected in the modeling.  
Based on these assumptions, when the vehicle is symmetrical about its longitudinal axis and the left and right wheels experience identical road surface unevenness, the vehicle exhibits only vertical and pitch vibrations. These two degrees of freedom have the most significant impact on ride comfort. In this scenario, the vehicle can be approximated as a planar model. When the mass distribution coefficient of the vehicle's sprung mass approaches 1, the vertical motion of the concentrated masses above the front and rear axles can be considered independent of each other [22]. Thus, the study of vertical dynamics can be simplified to the vertical motion of a single-axle body. For the analysis of ride comfort, the vehicle model considers one degree of freedom in the vertical direction for the sprung mass and one vertical degree of freedom for the wheel mass, totaling two degrees of freedom. Therefore, the vehicle model is simplified to a quarter-vehicle model. Despite representing only a quarter of the entire vehicle, the quarter-vehicle model retains all the characteristics relevant to suspension studies in practical research contexts.  
The 1/4 vehicle model can accurately describe the basic motion state of the vehicle. Moreover, when applying control, it exhibits higher credibility in control effectiveness, and the control effort required for controller design can be simplified. Figure 1 depicts the two-degree-of-freedom 1/4 vehicle model, wherein linear springs replace elastic tries, and tire damping is neglected.  
Based on Newtonian mechanics laws, combined with the air spring model, a \(1/4\) vehicle dynamic model was established.  
\[\begin{cases}m_{1}\dot{z}_{1}-c\left(\dot{z}_{2}-\dot{z}_{1}\right)+k_{t}(z_{1}-q )+F-m_{2}g+U=0\\ m_{2}\ddot{z}_{2}+c\left(\dot{z}_{2}-\dot{z}_{1}\right)-\left(F-m_{2}g\right)-U= 0\end{cases} \tag{1}\]  
In Equation (1), \"\(m_{1}\)\" represents the unsprung mass, \"\(m_{2}\)\" represents the sprung mass, \"\(k_{t}\)\" is the tire stiffness coefficient, \"\(c\)\" is the suspension damping coefficient, \"\(F\)\" denotes the air spring force, \"\(U\)\" represents the active control force, \"\(Z_{1}\)\", \"\(Z_{2}\)\", and \"\(q\)\" respectively denote the displacement of the unsprung mass, the displacement of the sprung mass, and the road input excitation.