# Fuzzy Sliding Mode Control of Vehicle Magnetorheological Semi-Active Air Suspension
## 5 Simulation Analysis of Fuzzy Sliding Mode Control  
### Random Road Excitation Response  
In order to verify the effectiveness of the FSMC method, the random road excitation was selected as the road input for simulation analysis. The white noise signal after the forming filter was usually used as the time domain signal of the random road surface. The real road surface of vehicle driving can be regarded as follows:  
\[\dot{q}(t)=-2\pi f_{1}q(t)+2\pi n_{0}\sqrt{G_{q}(n_{0})vw(t)} \tag{45}\]  
where \(f_{1}\) is the lower cutoff frequency of the road filter, \(q(t)\) donates the road surface excitation, \(G_{q}(n_{0})\) is the road roughness coefficient, \(w(t)\) is the white noise with a covariance of 1, and \(v\) is the vehicle speed. In the simulation analysis, MATLAB software was used for numerical simulation of the system, in which the sampling period is 0.001 s. Figure 16 shows the random input waveform of the road surface, using the filtering white noise generation method, where \(f_{1}\) is 0.01 Hz, the C-level road roughness coefficient \(G_{q}(n_{0})\) is 256 \(\times\) 10\({}^{-6}\) m\({}^{3}\), and the vehicle velocity \(v\) is 20 m/s. Table 3 shows the parameters of semi-active air suspension system required for simulation. Figure 17 shows the time domain of the 2-DOF quarter vehicle semi-active air suspension controlled by the FSMC method. The random road excitations of passive suspension, SMC control and FSMC control suspension are the same, shown in Figure 16. It can be seen that the semi-active air suspension controlled by FSMC switching has significantly improved the damping performance, compared with the passive suspension and the semi active air suspension controlled by SMC. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**S** & **SC** & **NB** & **NM** & **NS** & **ZE** & **PS** & **PM** & **PB** \\ \hline NB & NB & NB & NM & NM & NS & NS & ZE \\ NM & NB & NB & NM & NS & NS & ZE \\ NS & NM & NM & NS & ZE & ZE & ZE \\ ZE & NM & NM & NS & ZE & ZE & PS & PS \\ PS & NS & NS & ZE & ZE & PS & PS & PM \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fuzzy control rules. Figure 16: Random input waveform of road surface. Table 4 shows the root mean square (RMS) values of the air suspension performance indexes under a random road surface. Compared with passive suspension, the body acceleration under SMC control and FSMC control decreases by 15.9% and 26%, respectively, the suspension dynamic travel decreases by 16.9% and 29.4%, respectively, and the wheel dynamic load decreases by 7.4% and 8.4%, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Unit** & **Values** \\ \hline Sprung mass & \(m_{b}\) & kg & 400 \\ Unsprung mass & \(m_{w}\) & kg & 40 \\ Tire damping & \(K_{t}\) & N/m & 170,000 \\ Damping coefficient & \(C_{b1}\) & N/(m/s) & 3550 \\ Damping coefficient & \(C_{w1}\) & N/(m/s) & 3550 \\ Suspension stiffness & \(K\) & N/m & 17,000 \\ Initial volume of air spring & \(V_{0}\) & m\({}^{3}\) & 0.0078 \\ Initial area of air spring & \(A_{0}\) & m\({}^{2}\) & 0.0381 \\ Initial pressure of air spring & \(P_{0}\) & MPa & 0.382 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model parameters of two-degree-of-freedom quarter vehicle passive air suspension system. Figure 17: Time domain simulation curve of air suspension system under random road excitation. (**a**) Body acceleration, (**b**) suspension dynamic deflection, (**c**) tire dynamic load. Figure 18 shows the power spectral density (PSD) under the input of random road surface. In the high frequency range, the power spectral density of FSMC control and sliding mode control is similar to that of the passive system in terms of body acceleration, suspension dynamic travel and tire dynamic load, due to frequency conversion interference.