# 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  
In the design of the upper layer controller, it is necessary to control the steering torque of the assisted motor and the additional yaw torque of the vehicle based on the vehicle status, the planned auxiliary obstacle avoidance path, and the driver's input, in order to ensure that the intelligent assisted system can track the auxiliary obstacle avoidance path while avoiding human-machine conflicts and taking into account vehicle handling stability. When constructing MPC performance indicators, weight adjustment coefficients are considered to balance path tracking and vehicle handling stability. Here, the weight adjustment coefficients are obtained by mapping fuzzy logic rules with human-machine collaboration indicators, steering angle, and steering angle change rate as inputs. Firstly, the model (9) is discretization by zero order preserving method, which is expressed as follows:  
\[ \begin{cases}\mathbf{x}_{k+1,t}=\mathbf{A}_{k,t}\mathbf{x}_{k,t}+\mathbf{B}_{k,t}\mathbf{u}_{k,t}+ \mathbf{C}_{k,t}\mathbf{w}_{k,t}+\mathbf{d}_{k,t}\\ \mathbf{y}_{k,t}=\mathbf{D}_{k,t}\mathbf{x}_{k,t}\end{cases}  \tag{13}\]  
where \(\mathbf{A}_{k,t}=e^{\mathbf{A}_{k}T}\), \(\mathbf{B}_{k,t}=\int_{0}^{T}e^{\mathbf{A}_{k}\mathbf{\tau}}\cdot\mathbf{B}_{t}d\tau\), \(\mathbf{C}_{k,t}=\int_{0}^{T}e^{\mathbf{A}_{k}\mathbf{\tau}}\cdot\mathbf{C}_{t}d\tau\), \(\mathbf{d}_{k,t}=\int_{0}^{T}e^{\mathbf{A}_{k}\mathbf{\tau}}\cdot\mathbf{d}_{t}d\tau\), \(\mathbf{D}_{k,t}=\mathbf{D}_{t}\). \(T\) is the sampling interval, and matrices \(\mathbf{A}_{k,t}\), \(\mathbf{C}_{k,t}\), and \(\mathbf{d}_{k,t}\) are calculated in real time at each control time based on the longitudinal vehicle speed \(\overline{v}_{x}\), \(\overline{\mathbf{C}}_{i}\), and \(\overline{\mathbf{K}}_{i}\). The values of \(\overline{v}_{x}\), \(\overline{\mathbf{C}}_{i}\), and \(\overline{\mathbf{K}}_{i}\) are considered to remain constant over the prediction horizon, and the driver torque \(T_{h}\) in the measurable disturbance matrix \(\mathbf{w}_{k,t}\) is also considered constant in the prediction horizon. Figure 6: Schematic diagram of phase plane analysis results. (**a**) \(\delta_{1}=0.203\) rad. (**b**) \(\delta_{1}=0.138\) rad. The cost function \(J\) of the assisted driving controller mainly consists of two parts, namely the path tracking cost \(J_{r}\) and the handling stability cost \(J_{dy}\). The cost function is specifically represented as follows:  
\[J=J_{r}+J_{dy}+\xi\cdot\varepsilon \tag{14}\]  
\[J_{r}=\lambda_{r}{}^{\sigma_{r}}\cdot\sum\limits_{k=1}^{N_{p}}\|\mathbf{H}_{r} \cdot\mathbf{y}_{k,t}-\mathbf{y}_{k,t}^{r,ref}\|_{\mathbf{Q}_{r}}^{2}+\frac{1}{\lambda_{r} {}^{\eta}+d_{r}}\cdot\sum\limits_{k=1}^{N_{c}}\|\mathbf{G}_{r}\cdot u_{k,t}\|_{\mathbf{ R}_{r}}^{2}+\frac{1}{\lambda_{r}{}^{\eta}+d_{r}}\cdot\sum\limits_{k=1}^{N_{c}}\| \mathbf{G}_{r}\cdot\Delta u_{k,t}\|_{\mathbf{R}_{r}2}^{2} \tag{15}\]  
\[J_{dy}=\sum\limits_{k=1}^{N_{p}}\left(\lambda_{\beta}\cdot\|\mathbf{H}_{\beta} \cdot\mathbf{y}_{k,t}-\mathbf{y}_{k,t}^{g,ref}\|_{\mathbf{Q}_{\beta}}^{2}+\left(1-\lambda _{\beta}\right)\|\mathbf{H}_{\omega_{z}}\cdot\mathbf{y}_{k,t}-\mathbf{y}_{k,t}^{\omega_{z,ref}}\|_{\mathbf{Q}_{\omega_{z}}}^{2}\right)+ \tag{16}\]  
where \(N_{p}\) and \(N_{c}\) represent the prediction horizon and the control horizon, respectively. \(\mathbf{H}_{r}\), \(\mathbf{H}_{\beta}\), \(\mathbf{H}_{\omega_{z}}\), \(\mathbf{G}_{r}\) and \(\mathbf{G}_{dy}\) are coefficient matrices that satisfy \(\mathbf{H}_{r}\cdot\mathbf{y}= \begin{bmatrix}y_{e}&\psi_{e}\end{bmatrix} ^{T}\mathbf{H}_{ \beta}\cdot\mathbf{y}=\beta\), \(\mathbf{H}_{\omega_{z}}\cdot\mathbf{y}=\omega_{z}\), \(\mathbf{G}_{r}\cdot u=T_{a2}\), \(\mathbf{G}_{dy}\cdot u=M_{d}\). \(\Delta u\) is the control increment that satisfies \(\Delta u_{k}=u_{k+1}-u_{k}\). \(\mathbf{y}_{k,t}^{r,ref}\), \(\mathbf{y}_{k,t}^{g,ref}\) and \(\mathbf{y}_{k,t}^{\omega_{z,ref}}\) are the reference values of the output variable.