# Research on Active Anti-Slip Control of High-Speed Trains Based on High-Order Sliding Mode
## 3 Controller Design and Convergence Analysis
### Higher-Order Sliding Mode-Based Active Anti-Slip Controller  
Traditional sliding mode control often results in large oscillations, which can undermine the stability of the entire vehicle system. To address this issue, higher-order sliding mode theory is introduced. The higher-order sliding mode controller can be designed as follows:  
\[\left\{ \begin{array}{l}F_{\text{HO}}=M(-k_{1}e_{2}+\frac{F_{d}}{M}+\frac{dr_{ d}}{dt}+u_{HO})\\ u_{HO}=-k_{1\_fst}\phi_{1\_HO}+v_{HO}\\ \frac{dr_{HO}}{dt}=-k_{2\_HO}\phi_{2\_HO}\\ \phi_{1\_HO}=|s_{HO}|^{0.5}sign(s_{HO})+s_{HO}\\ \phi_{2\_HO}=0.5 sign(s_{HO})+1.5|s_{HO}|^{0.5}sign(s_{HO})+s_{HO}\\ \end{array} \right. \tag{29}\]  
In the above equations, \(k_{1\_HO}\) and \(k_{2\_HO}\) are determined by the adaptive laws in Equations (30) and (31):  
\[\left\{ \begin{array}{ll}\frac{dk_{1\_HO}}{dt}=\phi_{1\_HO}\phi_{2\_HO}\frac {2\pi}{1+4\beta^{2}}sign(|s_{HO}|-\sigma_{HO})&k_{1\_HO}>k_{4\_HO}\\ k_{1\_HO}=k_{4\_HO}&k_{1\_HO}<k_{4\_HO}\\ \end{array} \right. \tag{30}\]  
\[k_{2\_HO}=2\beta k_{1\_HO} \tag{31}\]  
To maintain consistent controller speed, \(k_{4\_HO}\) generally assumes a larger value. Based on the analysis in Section 3.2, limiting the controller output to be lower than the maximum adhesion force can prevent wheel slipping and enhance vehicle operational safety. Consequently, the maximum adhesion force derived from the active anti-slip control strategy in Section 3.2 is used to cap the controller output, thereby achieving active anti-slip control. Figure 4: Flowchart of active anti-short turn strategy. Compared to traditional vehicle controllers, this approach minimizes the impact of wheel torque variations on the controller's output. The actual controller output is  
\[F_{contr}=sat(F_{HO})=\left\{ \begin{array}{cc}F_{max}&F_{HO}>F_{max}\\ F_{HO}&F_{HO}<F_{max}\end{array} \right. \tag{32}\]  
\(F_{max}\) is defined as \(\min(10\mu_{max}Wg,F(v))\), where \(F(v)\) is the vehicle adhesion characteristic given by Equation (33) \(10\mu_{max}Wg\) represents the control limit derived from the active anti-slip strategy in Section 3.2. \[F(v)=\left\{ \begin{array}{cc}302000&v<18\;\text{m/s}\\ 5500000/v&v>18\;\text{m/s}\end{array} \right. \tag{33}\]  
Additionally, the higher-order sliding mode control law incorporates an integral term and includes a parametric adaptive law in integral form. When the anti-slip control strategy and traction characteristics impose limits on the controller output, the convergence of the sliding mode variables may be compromised. As a result, the adaptive and integral components of the control law may increase, potentially affecting the stability of the controller. To address this issue, this section combines the saturation compensation control law with the higher-order sliding mode train controller, ensuring system stability when the controller output is constrained. The train saturation compensation law is designed as follows:  
\[\left\{ \begin{array}{l}\frac{d\lambda_{HO}}{dt}=-k_{3\_HO}\lambda_{HO}- \Delta F_{HO}/M\\ \Delta F_{HO}=F_{HO}-F_{contr}\end{array} \right. \tag{34}\]  
Specifically, the main function of the saturation compensation control law is to prevent the controller output from exceeding the maximum available traction force, thereby avoiding wheel slip. When the train operates under low adhesion conditions, traditional control strategies may lead to outputs exceeding the system's physical limits, causing instability or slip. The saturation compensation control law corrects the control output in real time to ensure it does not exceed these physical limits. With the inclusion of the saturation compensation control law, the train sliding mode variables are redefined as follows:  
\[s_{HO}=k_{1}e_{1}+e_{2}-\lambda_{HO} \tag{35}\]  
By integrating Equations (29)-(35), the active anti-slip control law can be formulated as  
\[\left\{