## 3 Design of Control Strategy
### Upper Control Strategy  
The longitudinal force of the tire is directly changed by braking to control the stability of the vehicle. According to the left and right sides control, different differential equations are listed: the resultant lateral force is shown in Equation (16), and the total moment of inertia in the vertical direction is shown in Equation (17). \[m\Big{(}\dot{v}+um\Big{)}=\sum F_{y} \tag{16}\]  
\[I_{z}\dot{\omega}=\sum M_{z} \tag{17}\]  
where, \(\sum\ F_{y}\) is the resultant lateral force of vehicle; \(\sum\ M_{z}\) is the total moment about the vertical direction. From the equation of equilibrium for yaw motion, as shown in Equation (18), left wheel control can be derived as shown in Equations (19) and (20). \[ \begin{array}{c}I_{z}\dot{\omega}=\Big{[}\Big{(}F_{xfl}+F_{xfl}\Big{)}sin \delta+\Big{(}F_{yfl}+F_{yfl}\Big{)}cos\delta\Big{]}a\\ +\Big{[}\Big{(}F_{xfl}-F_{xfl}\Big{)}cos\delta+\Big{(}F_{yfl}-F_{yfl}\Big{)} sin\delta\Big{]}d+\left(F_{xfl}-F_{xfl}\right)d\\ -\Big{(}F_{yfl}-F_{yfl}\Big{)}d\end{array}  \tag{18}\]  
\[\dot{\beta}=-\omega+\frac{\sum\ F_{y}}{mu} \tag{19}\]  
\[\dot{\omega}=\frac{z_{1}}{I_{z}}+\frac{F_{xfl}d+F_{xfl}(cos\delta\cdot d-sin \delta\cdot a)}{I_{z}} \tag{20}\]  
Inside, \(F_{xfl}\) is the longitudinal force on the left front wheel; \(F_{xfl}\) is the longitudinal force on the left rear wheel; \(z_{1}\) is the measurable distractors moment; \(z_{1}=-\Big{[}F_{xfl}sin\delta+\Big{(}F_{yfl}+F_{yfl}\Big{)}cos\delta\Big{]}a -\Big{[}F_{xfl}cos\delta+\Big{(}F_{yfl}-F_{yfl}\Big{)}sin\delta\Big{]}d-F_{xfl }+d\Big{(}F_{yfl}-F_{yfl}\Big{)}b\). Inside, \(F_{xfl}\) is the longitudinal force on the right front wheel; \(F_{xtr}\) is the longitudinal force on the right rear wheel; \(F_{yfl}\) is the lateral force on the left front wheel; \(F_{yrf}\) is the lateral force on the left rear wheel; \(F_{yrr}\) is the lateral force on the right rear wheel; \(d\) is the half of wheelbase. Figure 6: The switching logic of the left and right sides wheel braking control. As shown in Figure 5, the ordinate of the coordinate system is \(\omega\) and the abscissa is \(\omega_{NO}\). The red line is determined by the formula \(\Big{|}\frac{\Delta\omega}{|\omega_{NO}|}\Big{|}=P\), the blue line is determined by the formula \(|\Delta\omega|=D\), and the blue dotted line distinguishes the area where the left and right wheels brake when there is no brake switching logic. Therefore, unilateral braking needs to be performed in areas where the red line and blue line are unexpected, and the controller does not work in the inner area. Obviously, whether it is switching from the left braking control to the right braking control or from the right braking control to the left braking control, it will pass through an area where the controller does not work. Right-hand wheel control is as shown in Equations (21) and (22):  
\[\dot{\beta}=-\omega+\frac{\sum\ F_{y}}{mu} \tag{21}\]  
\[\dot{\omega}=\frac{z_{2}}{I_{z}}+\frac{-F_{xrr}d-F_{xrl}(cos\delta\cdot d+sin \delta\cdot a)}{I_{z}} \tag{22}\]  
Inside, is the measurable distractors moment, and \(z_{2}=-\big{[}F_{xfl}sin\delta+\big{(}F_{yfl}+F_{yfr}\big{)}cos\delta\big{]}a+ \big{[}F_{xfl}-\big{(}F_{yfl}-F_{yfr}\big{)}sin\delta\big{]}d+F_{xrl}d+\big{(}F _{yrl}-F_{yrr}\big{)}b\). To facilitate the design of the controller, the left- and right-side controls are written as a unified expression. The equation of state is shown in Equations (23) and (24) as follows. \[\dot{\beta}=-\omega+s \tag{23}\]  
\[\dot{\omega}=z+U \tag{24}\]  
where \(s\) is the interference term of side slip angle, and \(s=\frac{\sum\ F_{y}}{mu}\); \(z\) is the measurable interference of the yaw rate, and the expression is shown in Equation (25); \(U\) is the virtual control unit, shown in Equation (26). \[z=\left\{ \begin{array}{c}\frac{z_{1}}{I_{z}}\ ;\ \text{{control}}\ \text{{of}}\ \text{{left}\ \text{{side}}}\\ \frac{z_{2}}{I_{z}}\ ;\ \text{{control}}\ \text{{of}}\ \text{{right}}\ \text{{side}} \end{array} \right. \tag{25}\]  
\[U=\left\{