# Hybrid Model Predictive Control of Semiactive Suspension in Electric Vehicle with Hub-Motor
## 3 Hybrid Model Predictive Controller Design  
Suspension performance directly affects passenger comfort and vehicle driving safety. Conventional suspensions have difficulty adapting to changes in the road surface, speed and load due to their stiffness and damping, making it difficult to meet the high demands of vehicle performance in changing conditions. This paper proposes a hybrid model predictive controller for the nonlinear constraint of damping forces to improve the overall system vertical performance deterioration due to the HM. For traditional control methods, the optimization problem with state constraints has been a research challenge [32]. Model predictive control (MPC) can handle optimization problems with constraints in a finite time domain. At each sampling time interval, MPC transforms the system control problem into an open-loop optimal control problem solving a finite time domain with a step length of \(N\) according to the amount of state of the system \(x(k)\) at the current moment, obtains the optimal control variables \(\{u(k),u(k+1),...,u(k+N-1)\}\) within N steps by an online calculation method, applies the first step control amount u(k) to the system and corrects the future output at the next moment according to the error between the system output and the predicted output to realize the closed-loop of the overall rolling optimization. The control procedure is shown in the Figure 6.  
For linear MPC, the optimization problem can be described by the following equations:  
\[ \begin{split}\min\ J=\sum\limits_{k=1}^{N}y^{\prime}(k)Qy(k)+ \sum\limits_{k=1}^{N}u^{\prime}(k)Ru(k)\\ \text{Subject to}\left\{\begin{array}{l}x(k+1)\!=Ax(k)\!+\!B_{u }u(k)\\ y(k)\!=Cx(k)\!+\!D_{u}u(k)\\ u_{min}\leq u(k)\leq u_{max}\\ x_{min}\leq x(k)\leq x_{max}\end{array}\right.\end{split}  \tag{21}\]  
where \(Q\) and \(R\) are the corresponding weighting matrixes. The optimization problem described above can be translated into a standard form of linear quadratic programming. \[ \begin{split}\min\ \frac{1}{2}x^{T}Gx+f^{T}x\\ \text{Subject to}\left\{\begin{array}{l}A_{e}x=b_{e}\\ A_{j}x\leq b_{j}\end{array}\right.\end{split}  \tag{22}\]  
Equation (7) can be expressed in the form of a state-space:  
\[\left\{ \begin{array}{l}\dot{x}(t)=Ax(t)+B_{u}u(t)+B_{\omega}\omega(t)\\ y(t)=Cx(t)+D_{u}u(t)\end{array} \right. \tag{23}\]  
Figure 6: Model predictive control (MPC) control procedure. with the state variable \(x=\left[\dot{z_{s}}\;\dot{z_{us}}\;\dot{z_{ur}}\;z_{s}{-}z_{us}\;z_{us}{-}z_{ur}\;z _{ur}-q\;\right]^{T}\), \(u=\left[u_{\;dam}\right]\), unmeasured disturbance \(\omega=\left[\dot{qF}_{\;ump}\right]\) and output \(y=\left[\dot{z_{s}}\;z_{us}{-}z_{ur}\;k_{t}(z_{ur}-q)\;\right]^{T}\). Since MPC solves the problem in the discrete time domain, it is necessary to discretize the existing continuous time state equations. The equation of state after discretization can be written as follows:  
\[\left\{ \begin{array}{rl}x(k+1)=&A_{d}x(k)+B_{du}u(k)+B_{du}\omega(k)\\ y(k)=&C_{d}x(k)+D_{du}u(k)\end{array} \right. \tag{24}\]  
where Ts is the sampling time, \(A_{d}\)= \(e^{A\texttt{Ts}}\), \(B_{d}=\left(\int_{\tau}^{T\texttt{Ts}}e^{A\tau}d\tau\right)\left[B_{u}\;B_{w} \right]^{T}\), \(C_{d}\)= \(C\), \(D_{d}\)= \(D\). For the quarter vehicle model for HM-EVs, the evaluation indicators are usually the acceleration of vertical vibration of the sprung mass \(\widetilde{z}_{s}\), the eccentricity of the motor \(z_{us}-z_{ur}\) and tire dynamic load \(k_{t}(z_{us}-q)\). The acceleration of the vertical vibration of the sprung mass represents the comfort of the vehicle; the eccentricity of the motor represents the safety of the motor--too large an eccentricity will damage the motor, but also increase the electromagnetic torque fluctuation and deteriorate the comfort and handling stability of the vehicle; the dynamic load of the tire represents the grounding of the tire, and its value is smaller, indicating that the vehicle handling stability is better.