## 5 Simulation  
### Parameter of the Extension Controller  
In order to determine the parameters of the extension controller, the extension controller is simulated, compared with passive suspension and the sky-hook control. Because the extension controller is a nonlinear time-varying system, the time-domain analysis is applied to the extension controller [42]. The amplitude responses of each control method are obtained by MATLAB/Simulink under sinusoidal excitation. The parameters of the vehicle model depend on the data in Table 1. The parameters of the extension controller are determined in Table 2 based on the low-frequency signal and high-frequency signal simulation. When the parameters of the extension controller are determined as Table 2, the suspension system could obtain good control performances. The Simulink model of hybrid damping extension control for the suspension system is illustrated in Figure 13.  
Figure 12: Flowchart of hybrid control based on extension theory. Figure 13: Simulink model of suspension extension control system. The control targets are \(\ddot{z}_{2}\) and \(F_{d}\), which are sensitive to the low-frequency excitation and the high-frequency excitation respectively. Therefore, low-frequency excitation (frequency 1 Hz, amplitude 0.02 m) and high-frequency excitation (frequency 11.1 Hz, amplitude 0.02 m) are used as inputs of the Simulink model respectively. The time responses of body acceleration, suspension dynamic deflection, and tire dynamic deformation are compared under passive suspension, sky-hook control and extension control. The simulation time is set to 20 s, as shown in Figure 14.  
As shown in Figure 14, all indexes of the sky-hook control and the extension control are better than that of the passive suspension under low-frequency excitation. Compared with the sky-hook control, the amplitude of body acceleration, suspension dynamic displacement, and tire dynamic deformation of the extension control is similar, that the difference is not significant. Table 3 describes the RMS of each index under different control methods quantitatively. The RMS of the body acceleration under extension control increases 6% from that under the sky-hook control. The responses of suspension dynamic displacement and the tire dynamic deformation decrease 0.5% and 9%, respectively. It means that compared with the sky-hook control, the ride comfort performance of extension control reduces slightly, while the drive safety performance is improved. However, compared with the passive suspension, body acceleration under extension control decreases by 55%. The value of suspension dynamic displacement increases 12.3% and the tire dynamic deformation decreases 57%, respectively. It reflects compared with the passive suspension under low-frequency excitation, the performance of the ride comfort can be significantly improved by extension control. Figure 14: Comparison of system responses under low-frequency excitation: (**a**) Body acceleration; (**b**) Suspension dynamic displacement; (**c**) Tire dynamic deformation. Figure 15 illustrates the weight \(\alpha\) change of the extension controller under low frequency. The working range of \(\alpha\) is above 0.8. It shows that the weight of the upper controller under low frequency makes the lower controller similar to sky-hook control, which improves ride comfort and driving safety performance. With sky-hook control and extension control, the RMS value of body acceleration, tire dynamic deformation and suspension dynamic displacement under high frequency are significantly higher than the passive suspension. However, compared with sky-hook control, extension control has a significant decrease in ride comfort and driving safety performance, as shown in Figure 16.  
Table 4 reflects the RMS of ride comfort and driving safety indexes under high frequency.