# Research on Active Anti-Slip Control of High-Speed Trains Based on High-Order Sliding Mode
## 3 Controller Design and Convergence Analysis  
### Robust Adhesion Observer  
The adhesion between the wheels and rails can be estimated using an observer, with the extended state observer being one of the most commonly used methods. Dividing both sides of (3) by \(r\) and \(J\), and then rewriting the wheel motion equation gives  
\[\dfrac{dw_{wi}}{dt}=\dfrac{1}{J}T-\dfrac{r}{J}F_{\mu i}-\dfrac{{{B_{t}}}}{J}{w _{wi}} \tag{16}\]  
Therefore, the extended state observer can be designed as  
\[\dot{z}_{1}=\dfrac{{{R_{g}}}}{J}T_{m}-\dfrac{{{B_{t}}}}{J}{w_{w}}+{z_{2}}-{K_{ r01}}{\phi_{1}} \tag{17}\]  
\[\dot{z}_{2}=-{K_{r02}}{\phi_{2}} \tag{18}\]  
where \(z_{1}\) is the observed value of the wheel speed \(w_{w}\), and \(z_{2}\) is the observed value of \(-rF_{\mu i}/J\).The functions \(\phi_{1}\) and \(\phi_{2}\) are the design functions of the observer, where \({K_{r01}}\) and \({K_{r02}}\) are the design parameters of the observer. By defining \(f=-rF_{\mu i}/J\), \(e_{1}=z_{1}-w_{w}\), and \(e_{2}=z_{2}-f\), then substituting Equations (16) and (17) into Equation (18) while considering \(\dot{f}\), we obtain the error differential equations:  
\[\dot{e}_{1}=e_{2}-{K_{r01}}{\phi_{1}} \tag{19}\]  
\[\dot{e}_{2}=-{K_{r02}}{\phi_{2}}-\dot{f} \tag{20}\]  
Traditional extended state observers assume \(\dot{f}=0\) and design \(\phi_{1}\) and \(\phi_{2}\) to make the observer error gradually converge to zero, thus observing the system's state. However, from Equation (20), it can be seen that even if the extended state observer makes \(e_{2}=0\), there still exists an observation error in the observer. This indicates that the dynamic observation accuracy of the traditional extended state observer requires improvement. Inaccurate torque estimation can impact the performance of subsequent control strategies. Therefore, it is crucial to design a torque observer that accounts for changes in torque. Referring to the design flow of the fixed-time robust differentiator, a robust adhesion observer is designed in this section. Under the assumption \(\dot{f}\
eq 0\), the design of \(\phi_{1}\) and \(\phi_{2}\) in (17) and (18) is given by Equations (21) and (22). Thus, a robust adhesion observer is constructed:  
\[{\phi_{1}}(e_{1})={|{e_{1}}|^{\frac{1}{2}}}{sign(e_{1})}+{K_{r03}}{|{e_{1}}|^ {\frac{3}{2}}}{sign(e_{1})} \tag{21}\]  
\[{\phi_{2}}(e_{1})=\dfrac{1}{2}{sign(e_{1})}+2{K_{r03}}{e_{1}}+\dfrac{3}{2}{K_ {r03}}^{2}{|{e_{1}}|^{2}}{sign(e_{1})} \tag{22}\]where \(K_{r03}\) is a constant and \(K_{r03}>0\). In (21) and (22), the terms \(K_{r03}{}^{2}|e_{1}|^{2}sign(e_{1})\) and \(K_{r03}{}|e_{1}|^{3/2}sign(e_{1})\) act to accelerate the convergence speed when the observer has a relatively large observation error. **Theorem 1**. : _If \(\dot{f}\
eq 0\) and \(\left|\dot{f}\right|<L\), where \(L\) is a positive constant, and if the conditions \(K_{r01}>2\sqrt{K_{r02}}K_{r02}>2L\), and \(K_{r03}>0\), the observer can observe the torque within a finite time._  
Proof. : Let \(\zeta=\left[\phi_{1}(e_{1})\;\;e_{2}\right]^{T}\), and define the Lyapunov function \(V_{1}=\zeta^{T}P\zeta\), where \(P\) is a positive definite symmetric matrix. \(\square\)  
From \(\phi_{1}^{\prime}(e_{1})=\frac{1}{2}\left|\;e_{1}\left|\;{}^{-\frac{1}{2}}+ \frac{3}{2}K_{r03}\right|e_{1}\left|\;{}^{\frac{1}{2}}\), the following expression can be derived:  
\[\phi_{1}(e_{1})\phi_{1}^{\prime}(e_{1})=\phi_{2}(e_{1})\]  
Combining Equations (19) and (20), the expression becomes  
\[\dot{\zeta}= \begin{bmatrix}\phi_{1}^{\prime}(e_{1})(e_{2}-K_{r01}\phi_{1})\\ -K_{r02}\phi_{2}-\dot{f}\end{bmatrix} =\phi_{1}^{\prime}(e_{1})(A\zeta+B\rho) \tag{23}\]  
where \(A= \begin{bmatrix}-K_{r01}&1\\ -K_{r02}&0\end{bmatrix} \), \(B= \begin{bmatrix}0\\ 1\end{bmatrix} \), and \(\rho=\frac{-\dot{f}}{\phi_{1}^{\prime}}=\frac{-2\dot{f}sign(e_{1})}{(1+3K_{r0 3}|e_{1}|)(1+K_{r03}|e_{1}|)}\phi_{1}(e_{1})\). According to \(\left|\dot{f}\right|<L\), where \(L\) is a positive constant, it follows that \(\left|\rho\right|\leq 2L|\phi_{1}(e_{1})|\). Letting \(C= \begin{bmatrix}1&0\end{bmatrix}