# Multi-Objective Collaborative Control Method for Multi-Axle Distributed Vehicle Assisted Driving
## 2 Fuzzy Cooperative Control Strategy Based on LTV-MPC
### LTV-MPC Control Strategy Design
\(\mathbf{Q}_{r}\), \(\mathbf{Q}_{\beta}\) and \(\mathbf{Q}_{\omega_{z}}\) are the error weight matrices of the corresponding output variable. \(\mathbf{R}_{r}\) and \(\mathbf{R}_{dy}\) are the weight matrices of the corresponding control variable, and \(\mathbf{R}_{r2}\) and \(R_{dy2}\) are the weight matrices of the corresponding control increment. \(\lambda_{r}\) is the assisted driving weight adjustment coefficient, \(\lambda_{\beta}\) is the stability weight adjustment coefficient, and their range is 0-1. \(a_{r}\sim d_{r}\) are constants, where the purpose of \(d_{r}\) is to ensure that the cost function is meaningful when \(\lambda_{r}=0\). \(\xi\) is the relaxation weight coefficient, and \(\varepsilon\) is the relaxation variable of the state constraint. The weight matrices \(\mathbf{Q}_{i}\) and \(\mathbf{R}_{i}\) involved in cost function \(J\) remain unchanged, and in multi-objective optimization, the coefficients \(\lambda_{r}\) and \(\lambda_{\beta}\) play a regulatory role in the cost function. The cost of path tracking \(J_{r}\) is mainly used to adjust the effect of human-machine collaborative steering. When there is a significant human-machine conflict, the weight of path tracking is reduced, while the relevant penalties for control are increased to suppress the machine's effort in steering for path tracking. \(J_{dy}\) is used to balance and improve the dynamic performance of the vehicle. Generally, priority is given to ensuring the maneuverability of the vehicle. When the vehicle is in an unstable state, the weight of the vehicle sideslip angle error can be increased to maintain vehicle stability. The path tracking error reference \(\mathbf{y}_{k,t}^{r,ref}= \begin{bmatrix}y_{e,ref}&\psi_{e,ref}\end{bmatrix} ^{T}\) satisfies \(y_{e,ref}=0\) and \(\psi_{e,ref}=0\). Stability control takes zero sideslip angle of the vehicle as the control objective, meeting \(\mathbf{y}_{k,t}^{\beta,ref}=\beta_{ref}=0\). In the process of assisting obstacle avoidance, good vehicle handling can enhance the driving experience. A steady-state reference model is used as the reference for yaw rate control, meeting \(\mathbf{y}_{k,t}^{\omega_{z,ref}}=\omega_{z,ref}\). The reference yaw rate is represented as:  
\[\omega_{z,ref}=max\left(min\left(\frac{\frac{\psi_{e}}{K\omega_{z}^{2}+1}} \delta_{1},\frac{\psi_{e}}{v_{x}}\right),-\frac{\psi_{e}}{v_{x}}\right) \tag{17}\]  
where \(K\) and \(L_{e}\) are the stability factor and equivalent wheelbase, respectively. In the setting of constraints, it is necessary to constrain the control variable \(\mathbf{u}\) and the control increment \(\Delta\mathbf{u}\). In the constraint of state variables, the constraints of the vehicle sideslip angle and yaw rate are satisfied \(\beta_{max}=-\beta_{min}=\arctan(0.02\ \mathrm{\SIUnitSymbolMicro g})\), \(\omega_{z,max}=-\omega_{z,min}=\frac{\psi_{e}}{v_{x}}\). The optimization problem of MPC collaborative assisted driving control strategy can be summarized as follows:  
\[ \begin{array}{ll}\text{Min}\;J=J_{r}+J_{dy}+\tilde{\zeta}\cdot\varepsilon\\ st.&\mathbf{x}_{k+1,t}=\mathbf{A}_{k,t}\mathbf{x}_{k,t}+\mathbf{B}_{k,t}\mathbf{u}_{k,t}+\mathbf{C}_{k,t }\mathbf{w}_{k,t}+\mathbf{d}_{k,t}\\ &\mathbf{y}_{k,t}=\mathbf{D}_{k,t}\mathbf{x}_{k,t}\\ &\mathbf{u}_{min}\leq\mathbf{u}\leq\mathbf{u}_{max}\\ &\mathbf{u}_{min}\leq\mathbf{u}\leq\mathbf{u}_{max}\\ &\mathbf{x}_{min}-\mathbf{\varepsilon}\leq\mathbf{x}\leq\mathbf{x}_{max}+\varepsilon\end{array}  \tag{18}\]  
where \(\mathbf{u}_{min},\mathbf{u}_{max}\), \(\Delta t_{min}\), \(\Delta t_{max}\), \(\mathbf{x}_{min}\), and \(\mathbf{x}_{max}\) is the specific constraint matrix.