## 2 Basic Handling Stability Control Strategy of ESP
### Construction of Direct Yaw Moment Controller  
#### 2.1.1 Linear Monorail Standard Equation with DYC  
The stability of a vehicle when turning can be significantly improved by DYC, which is designed to boost vehicle stability. The basic idea of DYC is to change the ratio of the driving or braking force between the inner and outer wheels of a vehicle so as to make the vehicle produce an additional restoring yaw moment, which prevents the instability of the vehicle as much as possible. The research and development of handling and stability systems is mainly completed by tracking the ideal vehicle model on the basis of the actual motion state of vehicles. However, the definition of an ideal vehicle model remains unclear. In the whole process of vehicle motion, numerous uncertain parameters affect the modeling accuracy of complex vehicle systems. Although the design of vehicle controllers is considerably difficult, the variations of bounded uncertain parameters make the whole vehicle system controllable. The linear monorail model of vehicle motion with two degrees of freedom includes several important parameters, such as body mass, yaw rate, sideslip angle, wheelbase, and lateral stiffness of the front and rear axles. Collectively, these parameters can reflect the lateral motion of vehicles. Therefore, such linear monorail model with two degrees of freedom is usually used as an ideal model for the design of stability control strategies, and its steering characteristics are considered as vehicle stability control objectives. When the vehicle is regarded as a monorail model with two degrees of freedom (Figure 1), the following equilibrium relationships of the system dynamic force of the vehicle can be established as follows:  
\[m_{t}\big{(}\dot{v}+u\big{)}=F_{y1}+F_{y2} \tag{1}\]  
\[I_{z}\dot{r}=L_{1}F_{y1}-L_{2}F_{y2} \tag{2}\]  
\[F_{yj}=-K_{aj}\alpha_{j} \tag{3}\]  
\[v_{1}=v+L_{1}r=u\tan(\delta_{1}-\alpha_{1}) \tag{4}\]  
\[v_{2}=v-L_{2}r=u\tan(-\alpha_{2}) \tag{5}\]  
\[\frac{v}{u}=\tan\beta \tag{6}\]  
where \(m_{t}\) is the total mass of the vehicle; \(F_{yj}\) is the lateral force of each vehicle axle; \(u\), \(v\) are the longitudinal and lateral speeds of the vehicle, respectively; \(r\) is the yaw angle rate of the vehicle; \(K_{aj}\) is the lateral slip stiffness of each axle; \(\alpha_{j}\) is the slip angle of each axle; \(v_{j}\) is the longitudinal speed of each axle; \(L_{j}\) is the longitudinal distance from each axle to the mass center; \(\delta_{1}\) is the front wheel angle; \(\beta\) is the sideslip angle of mass center; and subscript \(j\) indicates front axle (subscript 1) and rear axle (subscript 2). When an angle such as \(\alpha_{j}\), \(\delta_{1}\), and \(\beta\) is considerably small, the tangent value is approximately equal to the angle radian value. Therefore, Equations (4) and (5) can be respectively simplified into the following:  
\[\alpha_{1}=\delta_{1}-\frac{v+L_{1}r}{u} \tag{7}\]  
\[\alpha_{2}=-\frac{v-L_{2}r}{u} \tag{8}\]By substituting Equations (3), (6)-(8) into Equations (1) and (2) and with the application of the DYC moment \(M_{\geq d-ESP}\), the following are obtained:  
\[um_{t}\Big{(}\hat{\beta}+r\Big{)}=(K_{\alpha 1}+K_{\alpha 2})\beta+\frac{1}{u}(L_{ 1}K_{\alpha 1}-L_{2}K_{\alpha 2})r-K_{\alpha 1}\delta_{1} \tag{9}\]  
\[I_{z}\dot{r}=(L_{1}K_{\alpha 1}-L_{2}K_{\alpha 2})\beta+\frac{1}{u}\Big{(}L_{ 1}^{2}K_{\alpha 1}+L_{2}^{2}K_{\alpha 2}\Big{)}r-L_{1}K_{\alpha 1}\delta_{1}+M_{zd- ESP} \tag{10}\]  
Equations (9) and (10) can be transformed into the following standard state space equation:  
\[I_{z}\dot{r}=L_{1}F_{y1}-L_{2}F_{y2} \tag{2}\]0  
where state vector, input vector, and relevant matrix are:  
\[\mathbf{X}=\Big{(} \begin{array}{cc}\beta&r\end{array} \Big{)}^{T}\] \[\mathbf{U}=\Big{(} \begin{array}{cc}\delta_{1}&M_{zd-ESP}\end{array}  \Big{)}^{T}\] \[\mathbf{A}=\left[