# Hybrid Model Predictive Control of Semiactive Suspension in Electric Vehicle with Hub-Motor
## 2 Electromechanical Coupling Model
### Road Excitation  
The road excitation is the main excitation of the suspension system, so a suitable road excitation model is needed for control effect verification. The change in the height of the pavement relative to the reference surface along the length of the roadway is commonly referred to as the pavement unevenness function. In this paper, two types of road excitation were used: random road excitation and bump road excitation.  
Random road surface unevenness can be represented by the power spectral density (PSD) function [31]. The PSD of velocity in the spatial frequency domain of the road surface is shown in the following equation  
\[G_{v}(n) =4\pi^{2}n_{0}G_{q}(n_{0})v \tag{17}\]  
where \(n\) is the spatial frequency, \(n_{0}\) is the reference spatial frequency, \(G_{q}(n_{0})\) is the power spectral density at the reference spatial frequency and \(v\) is the vehicle velocity.  
The road speed signal is white noise with a finite bandwidth, so the white noise can be passed through a filter to generate a time-domain signal of the road elevation. The filter is shown in the following equation  
\[G(s)=\frac{2\pi\sqrt{G_{q}(n_{0})v}}{s+2\pi f_{0}} \tag{18}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline Sprung mass, \(m_{s}\) (kg) & 335 & Stator and residual unsprung mass, \(m_{us}\) (kg) & 57.5 \\ Rotor, rim and tire mass, \(m_{tar}\) (kg) & 66.5 & Tire rolling radius, \(R_{t}\) (m) & 0.256 \\ Bearing stiffness, \(k_{b}\) (N/m) & \(5\times 10^{6}\) & Tire stiffness, \(k_{t}\) (N/m) & 2.5 \(\times 10^{5}\) \\ Initial air spring sectional area, \(A_{0}\) (m\({}^{2}\)) & 0.009 & Maximum damping coefficient, \(c_{max}\) (Ns/m) & 9000 \\ Initial relative air spring pressure, \(P_{0}\) (pa) & \(4.6478\times 10^{5}\) & Minimum Damping coefficient, \(c_{min}\) (Ns/m) & 1000 \\ Atmospheric pressure, \(Pa\) (pa) & \(1\times 10^{5}\) & Air polytropic exponent, \(k\) (-) & 1.4 \\ Initial air spring volume, \(V_{0}\) (m\({}^{3}\)) & 0.0024 & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of quarter HM-EV model.  
Figure 3: The force-velocity characteristics of controllable damper.  
where \(f_{0}\) is the cutting frequency and \(f_{0}=0.01\) HZ. Road excitation at 20 m/s on a Class C road and on a Class B road is shown in Figure 4. Then, the road time-domain signal can be expressed as follows:  
\[\dot{q}(t)=-2\pi f_{0}q(t)+2\pi\sqrt{G_{q}(n_{0})vw(t)} \tag{19}\]  
In order to study the dynamic response characteristics of the HM-EV in this paper introduces bump road excitation as shown in the following equation.  
\[z(t)= \left\{\begin{array}{l}\frac{\pi}{2}\big{(}1-\cos\frac{2\pi t }{Lz}t\big{)},0\leq t\leq\frac{l}{v}\\ 0\hskip 28.452756 pt,t>\frac{l}{v}\end{array}\right. \tag{20}\]  
where a represents the height of the bump, \(l\) represents the length of the bump and \(v\) represents the vehicle velocity. In this paper, two different heights of bump road were selected as excitations: \(a\) = 0.1 m and \(a\) = 0.02 m. For other parameters, \(v\) = 1.5 m/s, \(l\) = 0.85 m was chosen. Bump road excitation is shown in Figure 5.  
Figure 4: Road excitation: (**a**) C-class 20 m/s road excitation; (**b**) B-class 20 m/s road excitation.  
Figure 5: Bump excitation: (**a**) \(a\) = 0.1 m; (**b**) \(a\) = 0.02 m.