# Research on Active Anti-Slip Control of High-Speed Trains Based on High-Order Sliding Mode
## 3 Controller Design and Convergence Analysis
### Proactive Anti-Diversion Strategies  
For a specific train route, the wheel-rail adhesion state can be categorized into five typical conditions, ranging from good to poor. The adhesion characteristics between the wheel and rail under these varying conditions at a fixed speed are illustrated in Figure 3.  
Based on the basic theory of adhesion discussed in Section 2.1, it is understood that when the braking force applied to the wheels of a train does not exceed the maximum traction force provided by the adhesion between the wheels and rails, the wheel-rail system operates on the left side of the peak adhesion point, preventing wheelset slip. Therefore, when the train is operating under specific track conditions, the maximum traction force that the rail surface can provide under slightly poorer adhesion conditions can be selected as the control force output to avoid wheel slip. For example, in the case of road condition 3 shown in Figure 3, the controller output is limited by the maximum traction force corresponding to the optimal adhesion point of road condition 4, thereby preventing wheelset slip under the poorer adhesion conditions of road condition 3.  
Considering factors such as axle load and environmental elements affecting torque, the actual wheel torque characteristics will inevitably fall within a specific torque characteristic range. For example, the shaded area in Figure 3. In this section, the torque observer is used to obtain the actual wheel torque coefficient \(\mu_{real}\). By selecting \(|\mu_{real}-\mu_{i}|\) as the typical road condition that is most similar to the current road condition, the train is actively prevented from slipping by selecting the typical road condition that is a little bit poorer as the output of the train's maximum traction force limiting controller, where \(\mu_{i}\) represents the typical condition torque coefficient, and \(i\) represents the typical condition number from 1 to 5. The researched typical conditions are represented by a 10-segment axis, so comparing the actual torque characteristics to this typical condition will yield a maximum vehicle adhesion force of \(10 Wg\mu_{max}\). \(\mu_{max}\) is the maximum observed torque coefficient for this condition.  
Through the torque characteristic model, the maximum adhesion force \(\mu_{max}\) can be obtained. The adopted O-Polach torque characteristic model cannot directly calculate the eigenvalue to obtain the maximum torque coefficient \(\mu_{max}\), so the calculation of the maximum vehicle adhesion force needs to be continuously approximated to approach  
Figure 3: Adhesion characteristics under different typical road conditions.  
the eigenvalue information expressed by the analytical formula. Therefore, this section adopts the approximation method to obtain the maximum torque coefficient \(\mu_{max}\) for the typical condition closest to the actual observed value and calculates the maximum vehicle adhesion force.  
During the implementation process, sensor data (such as wheel speed, traction force sensors, etc.) is used to monitor the adhesion between the wheels and the track, providing real-time feedback to the controller. When insufficient adhesion is detected, the controller immediately adjusts the traction output and limits it to the maximum value determined by the current track conditions. In this way, the controller can quickly respond to external disturbances, ensuring stable operation of the train under low adhesion conditions.  
The specific active control strategy can be designed as follows (see Figure 4):