# Hybrid Model Predictive Control of Semiactive Suspension in Electric Vehicle with Hub-Motor
## 2 Electromechanical Coupling Model
### BLDC Motor Model  
In recent years, the PMBLDC motor has received more interest in automotive applications due to the higher reliability, lower maintenance and quieter operation the BLDC motor has compared to its brushed DC counterpart [22]. This paper uses an outer rotor PMBLDC motor as the hub drive motor.  
The current equation of the outer rotor PMBLDC is as follows:  
\[\left[\begin{array}{c}u_{a}\\ u_{b}\\ u_{c}\end{array}\right]=\left[\begin{array}{ccc}R&0&0\\ 0&R&0\\ 0&0&R\end{array}\right]\left[\begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\end{array}\right]+\left[\begin{array}{ccc}L-M&0&0\\ 0&L-M&0\\ 0&0&L-M\end{array}\right]p_{op}\left[\begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\end{array}\right]+\left[\begin{array}{c}e_{a}\\ e_{b}\\ e_{c}\end{array}\right] \tag{1}\]  
where \(u_{a}\), \(u_{b}\), and \(u_{c}\) represent the three-phase winding phase voltage, \(R\) is the stator phase resistance, \(L\) is the winding inductance, \(M\) is the mutual inductance, \(p_{op}\) is the differential operator, \(i_{a}\), \(i_{b}\), and \(i_{c}\) represent the three-phase winding phase current, and \(e_{a}\), \(e_{b}\), and \(e_{c}\) represent the three-phase winding back electromotive force.  
According to Equation (1), the electromagnetic torque equation and mechanical motion equation of motor are as follows:  
\[T_{e}=\frac{1}{\omega_{n}}(e_{a}i_{a}+e_{b}i_{b}+e_{c}i_{c}) \tag{2}\]  
\[J_{n}\frac{d\omega_{n}}{dt}+B\omega_{n}=(T_{e}-T_{L}) \tag{3}\]  
where \(T_{e}\) is the electromagnetic torque of the motor, \(T_{L}\) is the load torque of the motor, \(J_{n}\) is the moment of inertia of the motor, \(B\) is the friction coefficient, and \(\omega_{n}\) is the angular velocity of the rotor.  
Due to the complex installation conditions and mechanical environment of the hub motor on an electric vehicle, the motor is under vertical load during operation, which causes the motor stator-rotor to be eccentric and thus generates unbalanced electromagnetic force. The motor noneccentricity and eccentricity are shown in Figure 1.  
The air gap magnetic field of a PMBLDC motor includes the permanent magnet magnetic field and the armature reaction magnetic field, which can be added together by the linear superposition method to obtain the radial air gap magnetic field and the tangential air gap magnetic field of a PMBLDC motor--the equations are as follows [23]:  
\[B_{er}(r,\alpha,t)=[B_{mr}(r,\alpha,t)+B_{ar}(r,\alpha,t)]\varepsilon_{\delta} \tag{4}\]  
\[B_{et}(r,\alpha,t)=[B_{mt}(r,\alpha,t)+B_{at}(r,\alpha,t)]\varepsilon_{\delta} \tag{5}\]  
where \(r\) is the radius at the intended position, \(\alpha\) is the stator angle, \(t\) is time, \(B_{er}\) is the radial air gap magnetic field in the eccentric state, \(B_{mr}\) is the radial magnetic field of the permanent magnet, \(B_{ar}\) is the radial magnetic field of armature reaction, \(B_{et}\) is the tangential air gap magnetic field in the eccentric state, \(B_{mt}\) is the tangential magnet field of the permanent magnet, \(B_{at}\) is the tangential magnetic field of armature reaction, and \(\varepsilon_{\delta}\) is the correction coefficient of magnetic conductivity in the eccentric state.  
Finally, the vertical and longitudinal unbalanced electromagnetic forces caused by the eccentricity between the stator and rotor of the motor are calculated as follows:  
\[\left\{\begin{array}{l}F_{\mathrm{ez}}=\frac{Lr}{2t_{0}}\int_{0}^{2\pi}\left\{ \Big{[}B_{\sigma r}(r,\alpha,t)^{2}-B_{\sigma t}(r,\alpha,t)^{2}\Big{]}sin \alpha+2[B_{\sigma r}(r,\alpha,t)\cdot B_{\sigma t}(r,\alpha,t)]cos\alpha\right\} \mathrm{d}\alpha\\ F_{\mathrm{ex}}=\frac{Lr}{2t_{0}}\int_{0}^{2\pi}\left\{\Big{[}B_{\sigma r}(r, \alpha,t)^{2}-B_{\sigma t}(r,\alpha,t)^{2}\Big{]}cos\alpha-2[B_{\sigma r}(r, \alpha,t)\cdot B_{\sigma t}(r,\alpha,t)]sin\alpha\right\}\mathrm{d}\alpha\end{array}\right. \tag{6}\]  
where \(L\) is the axial length of the motor and \(\mu_{0}\) is vacuum permeability.