# Sliding Mode Control of Laterally Interconnected Air Suspensions
## Full Vehicle Modeling and Road Excitations
### Generation of Road Excitations  
In order to accurately describe the performance of laterally interconnected air suspensions, a reasonable road excitation model should be built. According to [49; 50; 51], the road roughness input in the time domain can be expressed by  
\[q(t)=\sum_{i=1}^{M}\sqrt{2G_{q}(f_{mid-i}).\frac{f_{b}-f_{a}}{M}\sin(2\pi f_{ mid-i}t+\Phi_{i})}, \tag{12}\]where \(q(t)\) represents the road vertical displacement in time domain. \(f_{mid\_i}\) is the \(i^{\text{th}}\) middle frequency of \(i^{\text{th}}\) part. \(G_{q}(f_{min\_i}\)) is PSD (Power spectral density) of \(f_{mid\_i}\). \(\Phi_{i}\) is the independent and identically distributed (IID) random phase within (0.2 \(\pi\)). \(f_{a}\) and \(f_{b}\) are lower and upper frequency bounds, respectively. Considering that the commonly used vehicle speed \(v\) ranges from 10-30 m/s, and the lower and upper spatial frequency are set to 0.011/m and 2.83/m according to ISO 8608, the \(f_{a}\) and \(f_{b}\) should be equal to 0.33 Hz and 28.3 Hz, respectively. Road profiles of left and right wheels are different, but not independent. [PERSON] presents a widely used method for calculating the coherence \(\gamma(\Omega\),\(T_{w})\) of left and right side of the road profiles, as [52; 53; 54]  
\[\gamma(\Omega,T_{w})=\left[1+\left(\frac{\Omega T_{w}}{\Omega_{p}}^{a}\right) ^{W}\right]^{-p}, \tag{13}\]  
where \(\Omega\) is the angular spatial frequency of the road, \(T_{w}\) is the track width, \(W\) is the frequency exponent and equals 2, \(p\) is the reference coefficient, \(\Omega_{p}\) is the reference angular spatial frequency, and \(a\) is a constant exponent of \(T_{w}\). For the isotropic process, \(a=1\). Considering two given road vertical displacements \(q_{1}(t)\) and \(q_{2}(t)\) for left and right side of wheels, the transfer function \(G(s)\) of \(q_{1}(t)\) and \(q_{2}(t)\) can be written as  
\[G(s)=\frac{Q_{2}(s)}{Q_{1}(s)}, \tag{14}\]  
where \(Q_{1}(s)\) and \(Q_{2}(s)\) are the Laplace transform of \(q_{1}(t)\) and \(q_{2}(t)\), respectively. Taking the coherence \(\gamma(\Omega\),\(T_{w})\) in Equation (13) into consideration, the norm of this transfer function should be equal to the norm of \(\gamma(\Omega\),\(T_{w})\). Thus, a Pade algorithm is applied and a GA (genetic algorithm) method is used to obtain acceptable parameters of this transfer function [46]. The result of the transfer functions is marked as \(G_{12}(s)\) in this paper. Assuming that the vehicle runs along under a straight-line condition, the road profiles for rear wheels are roughly the same as those of front wheels, except a time delay is involved. Therefore, the left and right road vertical displacements \(q_{3}(t)\) and \(q_{4}(t)\) for rear axle can be expressed as follows  
\[\left\{ \begin{array}{l}q_{3}(t)=q_{1}(t-\tau)\\ q_{4}(t)=q_{2}(t-\tau)\end{array} \right., \tag{15}\]  
where \(t\geq\tau\) and \(\tau\) are the delay time lying on the track width and vehicle speed, respectively, and \(\tau=(l_{f}+l_{r})/v\) in this paper. Recall the road vertical displacement for one wheel in time domain can be calculated with Equation (12); firstly, two road vertical displacements \(q1\)\(0(t)\) and \(q2\)\(0(t)\) in the time domain are generated separately, then the road excitations for four wheels can be generated with \(q1\)\(0(t)\), \(q2\)\(0(t)\), and the coherence transfer function above, as Figure 2a shows the generated random C-class road excitations at a 20 m/s vehicle speed. The PSD of roads for the front left and front right wheels are illustrated in log-log scale in Figure 2b, which also contains the roughness indexes information classified by ISO. Those two figures illustrate the model the road excitations are current enough to use.