# Hybrid Model Predictive Control of Semiactive Suspension in Electric Vehicle with Hub-Motor
## 2 Electromechanical Coupling Model
### Quarter Vehicle Model  
Despite its simplicity, the quarter vehicle model is widely used in vertical dynamics analysis and controller design [24]. Figure 2 depicts the physical structure and equivalent model of the quarter vehicle model for HM-EV. The dynamics equations can be described according to [PERSON]'s law of motion as follows:  
\[\left\{ \begin{array}{l}\ddot{z}_{s}=\frac{k_{s}(z_{s}-z_{us})}{m_{s}}-\frac{ c_{s}\Big{(}\dot{z}_{s}-\dot{z}_{us}\Big{)}}{m_{s}}-\frac{u_{dam}}{m_{s}}\\ \ddot{z}_{us}=\frac{k_{s}(z_{s}-z_{us})}{m_{us}}+\frac{c_{s}\Big{(}\dot{z}_{s} -\dot{z}_{us}\Big{)}}{m_{us}}-\frac{k_{b}(z_{us}-z_{ur})}{m_{us}}+\frac{F_{ump }}{m_{us}}+\frac{u_{dam}}{m_{s}}\\ \ddot{z}_{ur}=\frac{k_{b}(z_{us}-z_{ur})}{m_{ur}}-\frac{k_{t}(z_{ur}-q)}{m_{ur }}-\frac{F_{ump}}{m_{ur}}\end{array} \right. \tag{7}\]  
where \(m_{s}\) is sprung mass, \(\dot{z}_{s}\) is vertical acceleration of sprung mass, \(\dot{z}_{s}\) is vertical velocity of sprung mass, \(z_{s}\) is vertical displacement of sprung mass, \(m_{us}\) is stator and residual unsprung mass, \(\ddot{z}_{us}\) is vertical acceleration of stator and residual unsprung mass, \(z_{us}\) is vertical displacement of stator and residual unsprung mass, \(m_{ur}\) is rotor, rim and tire mass, \(\ddot{z}_{ur}\) is vertical acceleration of rotor, rim and tire mass, \(\dot{z}_{ur}\) is vertical velocity of rotor, rim and tire mass, \(z_{ur}\) is vertical displacement rotor, rim and tire mass, \(c_{s}\) is damping coefficient, \(k_{s}\) is spring stiffness, \(k_{b}\) is spring stiffness, \(k_{t}\) is tire stiffness, \(q\) is road excitation and \(u_{dam}\) is damping force. Figure 1: Eccentricity diagram of stator and rotor. (**a**) Noneccentricity. (**b**) Eccentricity. The air suspension has a low frequency of vibration, can change the height of the vehicle and can improve control stability and ride comfort [25]. Therefore, in this paper, air springs have been chosen as the elastic element of the suspension system. [PERSON] states that the differential equation for the change in air pressure in an air spring with time can be expressed as [26]:  
\[\dot{p}_{as}=\frac{\kappa p_{as}q_{m}}{m_{as}}-\frac{\kappa p_{as}\dot{V}_{as}}{ V_{as}} \tag{8}\]  
Assuming that the air pressure, volume and air mass changes inside the air spring are much less than its initial state, the above equation can be linearized as:  
\[\dot{p}_{as}=\frac{\kappa p_{as0}q_{m}}{m_{as0}}-\frac{\kappa p_{as0}\dot{V}_{as}} {V_{as0}} \tag{9}\]  
Xinbo indicates that the air spring model can be ultimately represented as [27]:  
\[\dot{p}_{as}A_{as}(z_{as0}+z_{s}-z_{us})=\kappa RTq_{m}-\kappa p_{as}A_{as} \Big{(}\dot{z}_{s}-\dot{z}_{us}\Big{)} \tag{10}\]  
Considering that the rate of change in air pressure and suspension dynamic deflection has a large influence on the dynamic characteristics of the system and therefore is retained, and the current air pressure and current suspension dynamic deflection are replaced by the initial air pressure and initial suspension dynamic deflection, the linearized air spring model can be represented as:  
\[\dot{p}_{as}A_{as}z_{as0}=\kappa RTq_{m}-\kappa p_{as0}A_{as}\Big{(}\dot{z}_{s }-\dot{z}_{us}\Big{)} \tag{11}\]  
where \(p_{as}\) is the air pressure of the air spring, \(V_{as}\) is the volume of the air spring, \(V_{as0}\) is the initial volume of the air spring, \(m_{as}\) is the air mass in the air spring, \(m_{as0}\) is the initial air mass of the air spring, \(\kappa\) is the air polytropic exponent, \(R\) is the perfect gas constant, \(T\) is the air temperature, \(z_{as0}\) is the initial air spring displacement, \(A_{as}\) is the air spring cross-section area (assuming it remains constant) and \(q_{m}\) is mass flow of air flowing into the air spring.