# Path Tracking Control Strategy Based on Adaptive MPC for Intelligent Vehicles
## 3 MPC Controller Design Based on Stability Boundary  
### Vehicle Stability Boundary  
In order to ensure the stability and precision of path tracking of the MPC controller based on the dynamic model, precision constraints are incorporated into the controller design. The yaw stability of vehicles under extreme conditions holds particular significance. Currently, the phase plane method is frequently employed to analyze the yaw stability boundary of vehicles. Specifically, the phase plane method based on yaw rate/lateral velocity can accurately depict the yaw motion characteristics of the vehicle and the constraints imposed on the tire's sideslip angle, yielding promising results in real vehicle testing. Consequently, based on this method, the yaw stability criterion for vehicles with front wheel steering is derived, followed by the establishment of adaptive constraint conditions. According to the vehicle yaw dynamic model (1) under the small angle assumption  
\[\left\{ \begin{array}{l}\dot{y}=-\dot{\varphi}\dot{x}+\frac{2(F_{y,t}+F_{y,x} )}{m}\\ \ddot{\varphi}=\frac{2(aF_{y,t}-bF_{y,x})}{I_{z}}\end{array} \right. \tag{9}\]  
Given the rear axle sideslip poses a greater risk during vehicle yaw instability [33], this paper examines a scenario where the rear wheel side deflection angle attains its steady-state limit value. The calculation formulae for the front and rear wheel slip angles are as follows:  
\[\left\{ \begin{array}{l}\alpha_{\text{f}}=\frac{\dot{y}+a\dot{\varphi}}{\dot{ x}}-\delta_{\text{f}}\\ \alpha_{\text{r}}=\frac{\dot{y}-b\dot{\varphi}}{\dot{x}}\end{array} \right.\]  
The simplified expressions of the lateral forces acting on the front and rear wheels of a vehicle are as follows:  
\[\left\{ \begin{array}{l}F_{c,t}=C_{c,t}(\frac{\dot{y}+a\dot{\varphi}}{\dot{x} }-\delta_{\text{f}})\\ F_{c,r}=C_{c,r}\frac{\dot{y}-b\dot{\varphi}}{\dot{x}}\end{array} \right.\]During the driving process of a vehicle, there are the following transformation relationships between the forces acting on the front and rear wheels along the x-axis and y-axis directions and the tire lateral forces:  
\[\left\{ \begin{array}{l}F_{y,t}=F_{I,t}\sin\delta_{t}+F_{c,t}\cos\delta_{t}\\ F_{y,r}=F_{I,r}\sin\delta_{t}+F_{c,r}\cos\delta_{t}\end{array} \right.\]  
Thus, by combining Equations (1) and (9), the maximum control law of the yaw rate at any k time can be obtained:  
\[\dot{\varphi}_{\text{ss,max}}(k)=\frac{C_{c,r}\alpha_{\text{r,max}}(1+\frac{b }{a})}{m\dot{x}(k)} \tag{10}\]  
Here, \(\alpha_{\text{r,max}}\) represents the steady-state limit value of the sideslip angle for the rear wheel when an impending sideslip occurs. According to the literature [34], when the tire sideslip angle is \(\leq\)5\({}^{\circ}\), the lateral force and sideslip angle exhibit an approximately linear relationship. To ensure good driving stability of the vehicle on low-adhesion roads, this paper further restricts the tire sideslip angle to \(-2.5^{\circ}\leq\alpha_{t}\leq 2.5^{\circ}\). Consequently, the constraint relationship for the yaw rate is as follows:  
\[-\frac{C_{c,r}\alpha_{\text{r,max}}(1+\frac{b}{a})}{m\dot{x}(k)}\leq\dot{ \varphi}_{\text{ss}}(k)\leq\frac{C_{c,r}\alpha_{\text{r,max}}(1+\frac{b}{a})}{ m\dot{x}(k)} \tag{11}\]  
The restraint relationship of vehicle lateral speed meets  
\[-\alpha_{\text{r,max}}\leq\frac{\dot{y}-b\dot{\varphi}}{\dot{x}(k)}\leq\alpha_ {\text{r,max}} \tag{12}\]  
Based on the aforementioned equations, the stability boundary constraint condition derived from the relationship between the yaw rate and lateral velocity can be reformulated into the subsequent matrix inequality form:  
\[-G_{\text{sh}}(k)\leq H_{\text{sh}}(k)\xi(k)\leq G_{\text{sh}}(k) \tag{13}\]  
\[\text{Here, }G_{\text{sh}}(k)=\left[ \begin{array}{ccccc}\alpha_{\text{r,max}} \\ \frac{C_{c,r}\alpha_{\text{r,max}}(1+\frac{b}{a})}{m\dot{x}(k)}\end{array}  \right],H_{\text{sh}}(k)=\left[ \begin{array}{ccccc}\frac{1}{x(k)}&-\frac{b }{x(k)}&0&0&0&0\\ 0&1&\frac{g}{x(k)}&0&0\end{array}