## 3 Lateral Control Strategy
### Lateral Control Strategy Prediction Modeling  
The vehicle's dynamic control heavily relies on the vehicle dynamic model, which incorporates tire models such as that of [PERSON] and [PERSON]. Reference [34] proposes a method to estimate the sideslip angle of a smart car by integrating vehicle kinematics and dynamic Kalman filtering based on consensus, which improves the precision of the estimation. Figure 8: Lateral control strategy diagram. The three-degrees-of-freedom monorail model was selected as the prediction model of the controller, as described in Figure 9.  
By simplifying the tire model as well as using minor angles' assumptions, the forces on the tire are obtained as Equation (5):  
\[\left\{ \begin{array}{l}F_{lf}=C_{lf}S_{f}\\ F_{lr}=C_{lr}S_{r}\\ F_{cf}=C_{cf}(\delta_{f}-\frac{\dot{y}+a\dot{\varphi}}{\dot{x}})\\ F_{cr}=C_{cr}(\frac{\dot{y}-\dot{y}}{\dot{x}})\end{array} \right. \tag{5}\]  
In Equation (5), \(F_{lf}\) and \(F_{lr}\) are the fore and att fires vertical force; \(C_{lf}\) and \(C_{lr}\) are the fore and att fires vertical stiffness; \(S_{f}\), \(S_{r}\) are the fore and att fires slip rate; \(F_{cf}\) and \(F_{cr}\) are the fore and att fires lateral force; \(C_{cf}\) and \(C_{cr}\) are the fore and att fires lateral stiffness; \(\delta_{f}\) is the vehicle fore and tire turning angle; \(\dot{x}\) and \(\dot{y}\) are the car velocity in the \(x\) and \(y\) axis orientation. \[\left\{ \begin{array}{l}\ddot{x}=\frac{2}{n}[C_{lf}S_{f}+C_{lr}S_{r}-C_{cf}( \delta_{f}-\frac{\dot{y}+a\dot{\varphi}}{\dot{x}})\delta_{f}-C_{cr}(\delta_{r} -\frac{\dot{y}-b\dot{\varphi}}{\dot{x}})\delta_{f}]+\dot{y}\dot{\varphi}\\ \ddot{y}=\frac{2}{n}[C_{cf}(\delta_{f}-\frac{\dot{y}+a\dot{\varphi}}{\dot{x}} )+C_{cr}(\delta_{r}-\frac{\dot{y}-b\dot{\varphi}}{\dot{x}})]-\dot{x}\dot{ \varphi}\\ \ddot{\varphi}=\frac{2}{n}[aC_{cf}(\delta_{f}-\frac{\dot{y}+a\dot{\varphi}}{ \dot{x}})-bC_{cr}(\delta_{r}-\frac{\dot{y}-b\dot{\varphi}}{\dot{x}})]\\ \dot{Y}=\dot{x}\sin\varphi+\dot{y}\cos\varphi\\ \dot{X}=\dot{x}\cos\varphi-\dot{y}\sin\varphi\end{array} \right. \tag{6}\]  
In Equation (6), \(\ddot{x}\) is the increase in speed of the vehicle in the orientation of \(x\); \(\ddot{y}\) is the increase in speed of the vehicle in the orientation of \(y\); \(\dot{X}\) and \(\dot{Y}\) are the velocity of the vehicle in the orientation of the \(X\) and \(Y\) axes of the geodetic coordinate system; \(\dot{\varphi}\) is the yaw rate; \(\delta_{f}\) is the angle of the fore wheel; \(\delta_{f}\) is the angle of rotation of the att wheel; \(\alpha_{f}\) and \(\alpha_{r}\) are the sideslip angles of the fore and att wheels. Rewrite Equation (6) as a state space expression, firstly, linearize the expression using Taylor series expansion; secondly, use forward Eulerian discretization for the constituted linear state space equation; finally, obtain the state space equation as Equation (7):  
\[\dot{\xi}(t)=f(\xi(t),u(t)) \tag{7}\]  
In Equation (7), \(\xi(t)\) denotes the system state quantity, \(u(t)\) denotes the system control quantity, \(\xi(t)=[\dot{y},\,\dot{x},\,\varphi,\,\dot{\varphi},\,Y,\,X]^{T}\), \(u(t)=[\delta_{f},\delta_{r}]\). Linearizing Equation (7) at time t using Taylor expansion:  
\[\dot{\xi}(t)=\dot{\xi}(t)+A_{t}\xi(t)+B_{t}u(t) \tag{8}\]  
In Equation (8), the Jacobi matrix \(A_{t}=\frac{\partial f(\xi(t),u(t))}{\partial\xi}\), \(B_{t}=\frac{\partial f(\xi(t),u(t))}{\partial u}\). Figure 9: Three-degrees-of-freedom monorail model of vehicle. Equation (9) reveals the discrete linear time-varying system obtained by discretizing Equation (8) via the forward Euler formula:  
\[\xi(k+1)=A_{k}\xi(k)+B_{k}u(k) \tag{9}\]  
In Equation (9), \(A_{k}=I+TA_{t}\), \(B_{k}=TB_{t}\). The Jacobi matrix, \(A_{t}\) and \(B_{t}\), the calculation equations are as follows:  
\[A(t)=\frac{\partial f(\xi(t),u(t))}{\partial\xi}=