# Hybrid Model Predictive Control of Semiactive Suspension in Electric Vehicle with Hub-Motor
## 2 Electromechanical Coupling Model
### Quarter Vehicle Model
The air spring is not inflated or deflated in this paper, so the air spring model can be represented as:  
\[\dot{p}_{as}A_{as}z_{as0}=-\kappa p_{as0}A_{as}\Big{(}\dot{z}_{s}-\dot{z}_{us }\Big{)} \tag{12}\]  
Figure 2: Hub-motor electric vehicle (HM-EV) system: (**a**) physical structure; (**b**) equivalent model. From the above equation, it follows that  
\[p_{\text{as}}A_{\text{as}}{=-\kappa p_{\text{as0}}{A_{\text{as}}}^{2}(z_{s}{-}z_{ \text{us}})/V_{\text{as0}}} \tag{13}\]  
The gas spring stiffness can therefore be expressed as:  
\[k_{s}{=\kappa p_{\text{as0}}{A_{\text{as}}}^{2}/V_{\text{as0}}} \tag{14}\]  
As the main components of the suspension system, dampers have a significant effect on the smoothness and handling stability of the vehicle. The damping coefficient of traditional dampers is generally fixed, making it difficult to meet the performance requirements of vehicles under different working conditions. In this paper, a controllable dampener was used, which can achieve a constantly adjustable damping coefficient by adjusting the opening of the damping valve. The controllable damper force \(F_{\text{cs}}\) can be expressed as a function of the control current \(i\) and the relative velocity of the suspension. The formulas are described as follows [28]:  
\[F_{\text{cs}}{=f(i,\dot{x}_{s}-\dot{x}_{\text{us}})} \tag{15}\]  
Many methods can be used to describe such a function and this paper adopts a nonparametric model [29], in which the controllable damper force is expressed as:  
\[\left\{ \begin{array}{c}F_{\text{cs}}{=A(i)S_{b}\Big{(}\dot{x}_{s}-\dot{x}_{ \text{us}}\Big{)}}\\ A(i)=\sum\limits_{n=0}^{k}{a_{n}}^{in}\\ S_{b}\Big{(}\dot{x}_{s}-\dot{x}_{\text{us}}\Big{)}{=sgn\Big{(}\dot{x}_{s}-\dot {x}_{\text{us}}\Big{)}[1-e^{\frac{(-b_{0}\dot{x}_{s}-\dot{x}_{\text{us}})}{V_{ 0}}}]}\end{array} \right. \tag{16}\]  
where \(A(i)\) is the maximum damping force at a given control current, \(a_{n}\) is a polynomial parameter, \(k\) is the largest order of the polynomial, \(S_{b}\) is a shape function, \(b_{0}\) and \(V_{0}\) are the shape function parameter. The parameters are shown in Table 1 [28] with the force-velocity characteristics depicted in Figure 3.  
Think of the damping coefficient of a system as a combination of uncontrollable \(c_{min}\) and controllable parts [0, \(c_{max}\)-\(c_{min}\)]. According to JASOC602:2001 [30], the damping force when the piston speed is 0.31 m/s represents the equivalent damping force, and the corresponding equivalent damping coefficient can be obtained accordingly. For a controllable damper, the nonadjustable part of the damper is always at work, so it is assumed that \(c_{s}{=}c_{min}\). Parameters of the quarter HM-EV model are shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Tension (\(\dot{x}_{s}-\dot{x}_{\text{us}}{\geq}0\))**} & \multicolumn{2}{c}{**Compressed (\(\dot{x}_{s}-\dot{x}_{\text{us}}{<}0\))**} \\ \hline \(a_{0}^{+}\) & 4002.72 & \(a_{0}^{-}\) & \(-2002.45\) \\ \(a_{1}^{+}\) & \(-1567.91\) & \(a_{1}^{-}\) & 801.58 \\ \(b_{0}^{+}\) & 3.41 & \(b_{0}\) & 9.48 \\ \(V_{0}^{+}\) & 1.31 & \(V_{0}^{-}\) & 3.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for damping force equation.