**3. Results**

Simulation of the models was performed using MATLAB/Simulink. According to the differential equations of motion and the 1/4 suspension system's stochastic road input model, the simulation models of the 1/4 active and passive suspension systems were created in Simulink. Table 2 illustrates the fundamental parameters for the 1/4 suspension model.

**Table 2.** Fundamental data of suspension system.


For the sake of highlighting the optimization effect of the FNN-PID control strategy and to verify its effectiveness, the passive suspension, PID, and FNN-PID control active suspension were emulated and analyzed, respectively. The PID controller's parameters were KP = 5, KI = 430, and KD = 0.1. The structure of the FNN was designed as 2-14-49-49-3, and the number of network parameters to be adjusted was 14 × 2 + 49 × 3 = 175. Therefore, the dimension of the particles is set to 175, and then the following other pertinent settings are made to the particle swarm algorithm: the overall population size is 150, the learning factor is c1 = c2 = 2, and the inertia weight is 0.8. The particle velocity interval for the width of the affiliation function, bij, and the center value, cij, is set to [−3, 3], and the particle velocity interval for the connection weight, ωjs, of the FNN is [−1, 1]. The learning rate of the FNN is η = 0.5 and the momentum factor is α = 0.2.

When vehicles are driven on actual highways and dirt roads, they are often subjected to impact-type road surfaces, such as gravel and speed bumps, which affect all vehicle driving performance. In order to study the control effect of the FNN-PID-controlled active suspension under such operating conditions, a stepped road model was established to examine the vibration response properties of the suspension under such conditions. The step excitation with a step amplitude of 0.01 m was selected, as well as the suspension system's vibration response curve, which is displayed in Figures 5–7.

**Figure 5.** Step response diagram of Sprung Mass Acceleration.

**Figure 6.** Step response diagram of Dynamic Deflection of Suspension.

**Figure 7.** Step response diagram of Dynamic Tire Deformation.

Figure 5 demonstrates that the vehicle suspension system with the FNN-PID controller has better Sprung Mass Acceleration (SMA) than the conventional suspension system, which can make the body more stable with good control effect and can make the vehicle amplitude stable in a short time and quickly converge to 0. Additionally, Figures 6 and 7 show that the Dynamic Deflection of Suspension (DDS) and Dynamic Tire Deformation (DTD) can also reduce the amplitude under FNN-PID control and cause it to quickly converge to 0. Therefore, the active suspension controlled by FNN-PID can effectively reduce vibration and recover quickly, which greatly improves the passenger's ride experience.

Meanwhile, Table 3 shows the root mean square of each suspension index, demonstrating that the SMA, DDS, and DTD of the active suspension system with the FNN-PID controller are improved to some extent. The SMA, DDS, and DTD are decreased by 30.7%, 23.4%, and 16.3%, respectively, when compared to passive suspension. Compared to the PID-controlled active suspension system, the three performance indicators are reduced by 14.6%, 11.3%, and 8.2% respectively. The FNN-PID controller clearly has the potential to significantly lower the suspension's performance indices and improve the vehicle's passenger comfort.


**Table 3.** Comparison of root-mean-square suspension performance under step excitation.

In addition, the issue of the time between the change in road conditions and the response achieved by the suspension system is taken into account. A set of control tests was set up with the objective of achieving a steady state of vehicle vertical displacement under step response. As shown in Figure 8, the suspension is given a step signal of 0.01 m, 0.05 m, and 0.08 m at 1 s, and a reasonable steady-state error, Δ, is set. When the step signal is 0.01 m, Δ = 0.0002, and when the step signal is 0.05 m and 0.08 m, Δ = 0.001. When the step signal is 0.01 m and 0.05 m, the time for the FNN-PID controller to reach steady state is approximately 2.1 s, and the overall response time is 1.1 s. The time for the PID controller to reach steady state is approximately 2.4 s, with an overall response time of 1.4 s, and the passive system reaches steady state in approximately 3.1 s, with an overall response time of 2.1 s. When the step signal is 0.08, the time for the FNN-PID controller to reach steady-state is approximately 2.6 s, and the overall response time is 1.6 s. The time for the PID controller to reach steady-state is approximately 2.7 s, the overall response time is 1.7 s, and the passive system reaches steady state in approximately 3.4 s, with an overall response time of 2.4 s. The results show that the response time of the FNN-PID controller is reduced by 21.4% compared to the PID controller and 47.6% compared to the passive suspension when the road conditions are less variable. When the road conditions vary widely, the response time of the FNN-PID controller is reduced by only 5.9% compared to the PID controller and 33.3% compared to the passive suspension.

In order to obtain each suspension performance index under normal vehicle driving, the proposed random road excitation is used for simulation analysis. It is assumed that the vehicle is driven in a straight line at 30 km/h on the Class B road, with a simulation time of 20 s. The simulation is performed under the control of the PID controller and FNN-PID controller, respectively. The SMA, DDS, and DTD were still chosen as the main indexes to evaluate the performance of suspension, and the simulation result curves are shown in Figures 9–11.

**Figure 8.** Plot of vehicle vertical displacement change under step signal. (**a**) The step signal is 0.01 m. (**b**) The step signal is 0.05 m. (**c**) The step signal is 0.08 m.

**Figure 9.** Comparison of Sprung Mass Acceleration simulation results.

From Figures 9–11, it is clear that, in comparison with the passive suspension system,the SMA, DDS, and DTD of the active suspension system with the PID controller and the FNN-PID controller are reduced to a certain extent, indicating that both designed active suspension control systems are able to curb the overall vehicle vibration.

To make the analysis of the control effect of different controllers on the suspension system more intuitive, the above graphs were data processed to obtain the root mean square values of each curve, as demonstrated in Table 4.

**Figure 10.** Comparison of Dynamic Deflection of Suspension simulation results.

**Figure 11.** Comparison of Dynamic Tire Deformation simulation results.


**Table 4.** Root mean square comparison of suspension performance indexes under random.

Table 4 illustrates that, in comparison with the passive suspension, the active suspension with FNN-PID control has 30.4%, 17.8%, and 15.5% reduction in SMA, DDS, and DTD,

respectively. On the other hand, the SMA, DDS, and DTD of the active suspension with FNN-PID control were reduced by 14.6%, 12.1%, and 11.2%, respectively, compared to the active suspension with PID control. From these data, it is possible to draw the conclusion that, when compared with the other two suspension systems, the FNN-PID controller is able to suppress the variation of SMA, so that the wheels can closely follow the road, ensuring good maneuverability while giving passengers a more comfortable ride.
