# Hybrid Model Predictive Control of Semiactive Suspension in Electric Vehicle with Hub-Motor
## 5 Simulation Results  
In order to verify the effectiveness of the proposed control strategy, a simulation analysis of the control strategy is required. The overall control framework is shown in Figure 17 as follows: First, the HM-EV model inputs its measurable sprung mass acceleration and suspension dynamic deflection into the Kalman filter, which estimates the system's state variables in real-time. Then, the MLD model generates the objective function from the values of state variables obtained from the Kalman filter in a predictive iterative process combined with a reference trajectory, which is then transformed into a standard form of a mixed integer quadratic program (MIQP) problem to be solved. Then, the damping reverse model generates a control current to the dampers based on the optimal damping force output from the controller and the relative velocity of the suspension. Finally, dampers generate the damping force provided to the model, thus closing the loop on the entire control strategy. For comparison purposes, the clipped optimal control (COC) was selected as the comparator. In active suspension systems, optimal control can be achieved by offline  
Figure 16: Observed results of tire deflection. (**a**) Comparison of observed and actual values. (**b**) Relative error between observed and actual values. Figure 17: Hybrid model predictive controller (HMPC) control scheme. calculations or by updating the optimal feedback matrix online according to driving conditions. For the semiactive suspension system, the equivalent mathematics is described as a complex nonlinear problem due to its controllable force energy dissipation constraint and state-dependent control force time-varying constraint [35]. Typical COC can be obtained as follows:  
\[J\ =\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}x(t)^{T}Qx(t)+u(t)^{T}Ru(t )+2x(t)^{T}Nx(t) \tag{40}\]  
where \(Q\) is the semipositive matrix and \(R\) and \(N\) are the positive definite matrixes. The system dynamics model can be represented as  
\[\dot{x}(t)=Ax(t)+Bu(t) \tag{41}\]  
\(u(t)\) can be expressed as linear state feedback. \[u(t)=-Kx(t) \tag{42}\]  
where the state feedback gain matrix can be calculated by the following equation:  
\[ \begin{gathered} K=-R^{-1}B^{T}S\\ -\dot{S}=A^{T}S+SA-SBR^{-1}B^{T}S+Q\end{gathered}  \tag{43}\]  
Thus, the optimal control when unconstrained is as follows:  
\[F=-R^{-1}B^{T}Sx(t) \tag{44}\]  
There is a constraint on the damping force of the semiactive suspension, so the final output control force after passing the constraint is expressed as  
\[F=sat(-R^{-1}B^{T}Sx(t)) \tag{45}\]  
where \(sat\) denotes a damping force constraint. Assuming the vehicle is driving at 20 m/s on a C-class road surface and at 20 m/s on a B-class road surface, the damping coefficient of the passive suspension is 3000 N\(\cdot\)s\(\cdot\)m\({}^{-1}\). In order to make the results comparable, for the selection of the weight coefficients of the controller, the trial-and-error method was used to make an appropriate selection to ensure that the performance of the system could be optimized under the control of the controller. For the weighting coefficients of the hybrid model predictive controller (HMPC) controller, this paper chose \(Q\) = diag ([1, 10 \(\times\) 10\({}^{7}\), 10 \(\times\) 10\({}^{4}\)]) and \(R\) = 0.001. For the weighting coefficients of the COC controller, this paper chose \(Q\) = diag ([2.5 \(\times\) 10\({}^{4}\), 2 \(\times\) 10\({}^{11}\), 5 \(\times\) 10\({}^{9}\)]) and \(R\) = 0.005. Figures 18-20 shows vertical acceleration of sprung mass, motor eccentricity and tire dynamic load with a time under 20 m/s of C-class road excitation. Table 3 gives the root mean square (RMS) for the suspension performance index under B-class and C-class road. As shown in Figures 18-20 and Table 3, both COC and HMPC effectively improve the system performance compared to passive suspension under, but the HMPC can improve the vertical performance of the vehicle to a greater extent in comparison.