## 3 MPC Strategy
### MPCSSystem
#### 3.1.2 Rolling Optimization  
Before applying the MPC, the objective function should be selected first. In order to obtain feasible solutions to the optimization objective, a relaxation factor is added to the objective function [36] as follows:  
\[\mathbf{J}(\xi(t),\mathbf{u}(t-1),\Delta\mathbf{U}_{t})=\sum_{i=1}^{Np}\parallel\mathbf{\eta}( t+\mathrm{i}|t)-\mathbf{\eta}_{ref}(t+\mathrm{i}|t)\parallel^{2}\mathbf{{}_{Q}}+\sum_{i= 1}^{Nc-1}\parallel\Delta\mathbf{u}(t+\mathrm{i}|t)\parallel^{2}\mathbf{{}_{RR}}+\rho \varepsilon^{2} \tag{15}\]  
where \(\rho\) is the weight coefficient (\(\rho>0\)) and \(\varepsilon\) is the relaxation factor. The selection of \(\mathbf{Q}\) and \(\mathbf{RR}\) matrices directly affects the control effect. In this study, through simulation experiments, the \(\mathbf{RR}\) matrix is set as a zero matrix (\(\mathbf{RR}=\mathbf{zeros}\) (3,3)) which can obtain a significant control effect. The selection of the \(\mathbf{Q}\) matrix directly affects the augmented state quantity tracked by the strategy. During the braking process of the vehicle, it is expected to achieve the same vertical velocity as the stationary state--that is, \([\dot{z}_{1},\dot{z}_{2},\dot{z}_{3},\dot{z}_{4}]=[0,0,0,0]\); therefore, set the output reference \(\mathbf{\eta}_{ref}(t+i\!\upharpoonright\!t)\)(1:4,1) \(\equiv\mathbf{zeros}\)(4,1). The weight matrix \(\mathbf{Q}\) needs to make the solution of the objective function related only to the first four items of the augmented state. Combining the simulation experiment to determine the \(\mathbf{Q}\) matrix, \(\mathbf{Q}\)\(\in\mathbf{R}_{11\times 11}\), \(\mathbf{Q}\) (1,1) = 10,000, \(\mathbf{Q}\) (2,2) = 10,000, \(\mathbf{Q}\) (3,3) = 10,000, \(\mathbf{Q}\) (4,4) = 100,000, and the remaining elements are zero. Substituting Equation (14) into Equation (15), it can be obtained that:  
\[\mathbf{J}(\mathbf{\xi}(t),\mathbf{u}(t-1),\Delta\mathbf{U}_{t})=[\Delta\mathbf{U}_{t}{}^{\mathrm{T} },\varepsilon]\mathbf{H}[\Delta\mathbf{U}_{t}{}^{\mathrm{T}},\varepsilon]^{\mathrm{T} }+\mathbf{G}_{t}[\Delta\mathbf{U}_{t}{}^{\mathrm{T}},\varepsilon]^{\mathrm{T}}+\mathbf{P}_ {t} \tag{16}\]where \(\mathbf{P}_{t}\) is a constant. A description of \(\mathbf{H}\), \(\mathbf{G}_{t}\), and \(\mathbf{P}_{t}\) is provided in Appendix C. The problem can be equivalent to the quadratic programming problem, that is:  
\[ \begin{array}{l}\min[\Delta\mathbf{U}_{t}\text{T},\varepsilon]\mathbf{H}_{t}[\Delta \mathbf{U}_{t}\text{T},\varepsilon]^{\text{T}}+\mathbf{G}_{t}[\Delta\mathbf{U}_{t}\text{T},\varepsilon]^{\text{T}}\\ \mathbf{s}.\mathbf{t}\quad\quad\Delta\mathbf{U}_{\min}(kk)\leq\Delta\mathbf{U}_{t}(kk)\leq \ \Delta\mathbf{U}_{\max}(kk),kk=1,\cdots,3\text{Nc}\\ \quad\quad\mathbf{Y}_{\min}(kk_{1})\leq\mathbf{Y}_{\max}(kk_{1}),kk_{1}=1,\cdots,11 Np \\ \quad\quad\mathbf{U}_{\min}(kk_{2})\leq\mathbf{u}\left(t-1|t\right)(kk_{2})+\Delta\mathbf{U }_{t}(kk_{2})\leq\ \mathbf{U}_{\max}(kk_{2}),kk_{2}=1,2,3\end{array}  \tag{17}\]  
Because the half-vehicle model uses the first control vector of the optimized control sequence, only the first control vector is constrained here--that is, \(\mathbf{U}_{min}\)(2:3,1) = [-1500, \(-1500\)], \(\mathbf{U}_{max}\)(2:3,1) = [1500, 1500]--and there is no restriction on other elements of the first control vector. In this study, the first control quantity (i.e., the braking intensity \(z\)) is treated as interference. This goal can be achieved through constraint restrictions. The specific constraints are:  
\[\left\{ \begin{array}{l}\Delta\mathbf{U}_{\min}(1)\ =\ [z(t)-z(t-1)]\\ \Delta\mathbf{U}_{\max}(1)\ =\ [z(t)-z(t-1)]\end{array} \right. \tag{18}\]  
where \(z(t)\) and \(z(t-1)\) are the braking intensity at time \(t\) and \(t-1\), respectively. In other control time domains, the braking intensity increment constraints are all zero. Regarding the output quantity, only the first four augmented state quantities in the last predicted time domain are restricted, and the rest of the augmented state quantities are not restricted; that is:  
\[\left\{