# Fuzzy Sliding Mode Control of Vehicle Magnetorheological Semi-Active Air Suspension
## 2 Damping Performance Test and Mechanical Modeling of MR Damper
### Improved Parameter Identification of Hyperbolic Tangent Model  
Figure 4 shows the parameter identification process using the genetic algorithm. Taking the mean square deviation of the difference between the actual damping force and the simulated damping force as the objective function, the smallest mean square deviation is the optimal solution, and the optimal model of the MR damper is obtained through genetic operation selection, crossover and mutation. Figure 3: Improved hyperbolic tangent model. Figure 2: Mechanical characteristics of MR damper. (**a**) Force displacement, and (**b**) force velocity. The fitness function of the improved hyperbolic tangent model is denoted as follows:  
\[fitnessfunc=\frac{1}{m}\sum_{i=1}^{m}\left(F_{simti}-F_{exp\,i}\right)^{2} \tag{2}\]  
where \(F_{simi}\) and \(F_{expi}\) are the simulation damping forces and experimental damping forces of the \(i\)th point, respectively, and \(m\) is the number of experimental points. The identification parameters were processed by using the real coding method. The scattered crossover method was used to determine the crossover probability by repeated experiments, and the crossover probability was set to 0.8. To improve the local search ability of the genetic algorithm, the Gaussian function was used for mutation with a scale of 0.5 as well as a shrink of 0.7, and the number of iterations was 100. The improved hyperbolic tangent model has five parameters to be identified, which is expressed as follows:  
\[\Theta=\left[a_{1},a_{2},k,a_{3},f_{0}\right] \tag{3}\]  
The test results of the mechanical characteristics of the MR damper under sinusoidal excitation with an amplitude of 10 mm and frequency of 1 Hz were selected as the identification data. Parameter identification was carried out for the data of 5 groups under different control currents so as to obtain parameter identification results under different current, as shown in Table 1. By comparing and analyzing the identification results of different current parameters, the variation rules of each parameter were found, and the fitting values of each parameter in the improved hyperbolic tangent model were also obtained. Observing Table 1, parameters \(a_{2}\), \(k\), and \(f_{0}\) are not changed significantly under the different current. Therefore, \(a_{2}\), \(k\), and \(f_{0}\) can be considered constant values, and their mean values \(a_{2}=3874.38\), \(k=0.31\), and \(f_{0}=36.12\) are selected. Parameters \(a_{1}\) and \(a_{3}\) increase linearly with the increase in current, and the relationship between the parameters and current is obtained through fitting, as shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Applied Current/(A)**} & \multicolumn{5}{c}{**Parameter Values**} \\ \cline{2-6}  & \(\mathbf{a_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{k}\) & \(\mathbf{a_{3}}\) & \(\mathbf{f_{0}}\) \\ \hline
0 & 165.94 & 3890.24 & 0.3006 & 849.617 & 35.68 \\
0.25 & 366.04 & 3859.98 & 0.3001 & 1548.13 & 39.69 \\
0.5 & 483.51 & 3895.93 & 0.3004 & 2045.21 & 35.06 \\
0.75 & 615.02 & 3859.26 & 0.3000 & 2573.25 & 35.07 \\
1.0 & 805.25 & 3866.50 & 0.3001 & 3301.38 & 35.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter identification results of MR damper model. Figure 4: Parameter identification process of MR damper. As shown in Figure 5, parameters \(a_{1}\) and \(a_{3}\) are linearly correlated with the current, which can be obtained by linear fitting as shown below:  
\[\left\{ \begin{array}{l}a_{1}=m_{1}I+n_{1}=611.04I+181.63\\ a_{3}=m_{2}I+n_{2}=2371.46I+877.79\end{array} \right.