## 3 Design of Control Strategy
### Calculation of Nominal Value  
The linear two-degree-of-freedom model is used as the reference model for calculating the nominal value [11; 12]. As shown in Figure 4. The differential equation of its motion is shown in Equation (1). \[\left\{ \begin{array}{l}\left(C_{f}+C_{r}\right)\beta+\frac{1}{u}\big{(}aC_{f} -bC_{r}\big{)}-C_{f}\delta=m\Big{(}\dot{v}+u\alpha\Big{)}\\ \big{(}aC_{f}-bC_{r}\big{)}\beta+\frac{1}{u}\big{(}a^{2}C_{f}+b^{2}C_{r}\big{)} \omega-aC_{f}\delta=I_{z}\dot{\omega}\end{array} \right. \tag{1}\]  
Including, \(C_{f}\) is the cornering stiffness for the front axis; \(C_{r}\) is the cornering stiffness for the rear axis; \(\beta\) is side slip angle; \(u\) is longitudinal speed; \(a\) is the distance from the center of mass to the front axis; \(b\) is the distance from the center of mass to the rear axis; \(\delta\) is the steering angle of front wheels; \(m\) is the mass of the vehicle; \(\omega\) is the yaw rate; \(I_{z}\) is the moment of inertia of the vehicle around the \(Z\)-axis; \(\dot{\omega}\) is the yaw acceleration. Figure 4: The linear two-degree-of-freedom model. Figure 3: Controller system structure. Assuming the vehicle is in a steady state, \(\dot{v}\) and \(\dot{\omega}\) are both equal to zero, meaning \(\omega_{d}=G_{\omega}\cdot\delta\), \(\beta_{d}=G_{\beta}\cdot\delta\). In simultaneous Equation (1), there are transfer functions characterizing the steady-state yaw rate and the steady-state side slip angle as shown in Equations (2) and (3). \[G_{\omega}=\frac{u}{L(1+ku^{2})} \tag{2}\]  
\[G_{\beta}=\frac{u^{2}}{L(1+ku^{2})}\left(\frac{b}{u^{2}}+\frac{ma}{C_{r}L}\right) \tag{3}\]  
where \(G_{\omega}\) is the transfer function of the yaw rate; \(G_{\beta}\) is the transfer function of the side slip angle; \(L\) is the wheelbase; \(k\) is the stability factor; \(\omega_{d}\) is the expected value of the yaw rate; \(\beta_{d}\) is the expected value of the side slip angle. If the value of the state maximum lateral force is limited according to the pavement friction condition [13; 14], the maximum lateral force can be defined as Equation (4) and the centripetal acceleration can be defined as Equation (5). Without regard to the effect of longitudinal forces on the tires, according to the Equations (4) and (5), the expected yaw rate \(\omega_{d}\) can be limited as Equation (6). Combining equations \(\omega_{d}=G_{\omega}\cdot\delta\) and \(\beta_{d}=G_{\beta}\cdot\delta\) can obtain Equation (7), and the limits of the desired side slip angle can be obtained and defined as Equation (8). \[m\big{|}a_{y}\big{|}\leq m\mu g \tag{4}\]  
\[a_{y}=\dot{v}+u\omega \tag{5}\]  
\[|\omega_{d}|\leq\frac{\mu g}{u} \tag{6}\]  
\[\beta_{d}=\frac{G_{\beta}}{G_{\omega}}\omega_{d} \tag{7}\]  
\[\big{|}\beta_{d}\big{|}\leq\left|\mu g\left(\frac{b}{u^{2}}+\frac{ma}{k_{2}L} \right)\right| \tag{8}\]  
where \(a_{y}\) is the lateral acceleration of the vehicle; \(\mu\) is the coefficient of adhesion between tire and road surface; \(g\) is the acceleration due to gravity; \(v\) is the lateral speed of vehicle. The controller in this paper considers the effect of longitudinal force on the lateral force of the tire, and the longitudinal and tangential forces of the tire at the limit of adhesion form a friction circle with a radius less than \(\mu\)[15], as shown in Figure 5. With the introduction of tire force feedback, the tire force vector and its longitudinal and lateral components are known, limiting the maximum value of the state according to the friction circle. Expressed in Equation (9). Figure 3:0  
Figure 3:1 Friction circle diagram. In the Equation (9), \(F_{ymax}\) is the maximum lateral force of vehicle; \(F_{zi}\) is the vertical force of tire; \(F_{x}\) is the longitudinal force of tire; in addition, \(i=1,2,3,4\), They represent left front, right front, left rear and right rear wheels respectively.