# 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
\begin{array}{l}F_{HO}=M(-k_{1}e_{2}+\frac{F_{H}}{M}+\frac{dt\sigma_{ H}}{dt}-k_{3\_HO}\lambda_{HO}+u_{HO})\\ \frac{d\lambda_{HO}}{dt}=-k_{3\_HO}\lambda_{HO}-\Delta F_{HO}/M\\ \Delta F_{HO}=F_{HO}-F_{contr}\\ u_{HO}=-k_{1\_HO}\phi_{1\_HO}+v_{HO}\\ \frac{d\varphi_{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{36}\]  
\[\left\{ \begin{array}{cc}\frac{dk_{1\_HO}}{dt}=\phi_{1\_HO}\phi_{2\_HO}\frac {2\_{\text{a}}}{1+4\beta^{2}}sign(|s_{HO}|-\sigma_{HO})&k_{1\_HO}>k_{4\_HO}\\ k_{1\_HO}=k_{4\_HO}&k_{1\_HO}<k_{4\_HO}\\ k_{2\_HO}=2\beta k_{1\_HO}\\ F_{contr}=sat(F_{HO})\end{array} \right. \tag{37}\]  
\[\left\{ \begin{array}{cc}\frac{dk_{1\_HO}}{dt}=\phi_{1\_HO}\phi_{2\_HO}\frac {2\_{\text{a}}}{1+4\beta^{2}}sign(|s_{HO}|-\sigma_{HO})&k_{1\_HO}>k_{4\_HO}\\ k_{1\_HO}=k_{4\_HO}&k_{1\_HO}<k_{4\_HO}\\ k_{2\_HO}=2\beta k_{1\_HO}\\ F_{contr}=sat(F_{HO})\end{array} \right. \tag{38}\]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. Compared to traditional vehicle controllers, this approach minimizes the impact of wheel torque variations on the controller's output. The system is stabilized under the control law in Equations (36)-(38), and the sliding mode variable \(s_{HO}\) will converge within a small range defined by \(|s_{HO}|<\sigma_{HO}\). If \(\Delta F_{HO}=0\), the displacement and vehicle speed tracking errors will converge. If \(\Delta F_{HO}>0\), the following condition is satisfied:  
\[sat(F_{HO})>M(-k_{1}e_{2}+\frac{F_{d}}{M}+\frac{d\upsilon_{d}}{dt}+\frac{1}{M}\delta) \tag{40}\]  
**Theorem 2**. : _The position and speed tracking errors \(e_{1}\) and \(e_{2}\) will still converge. The train will then track up the target velocity-displacement curve. Furthermore, the designed controller is capable of providing the maximum vehicle adhesion force, thereby preventing wheel slipping._  
Proof. : Taking the derivative of the sliding mode variable \(s_{HO}\) and combining the control laws (36)-(39) and the vehicle model (15), the following is obtained:  
\[\frac{ds_{HO}}{dt}==-k_{1\_HO}\phi_{1\_HO}+\upsilon_{HO}-\frac{1}{M}\delta \tag{41}\]  
From Equation (41), the following can be obtained:  
\[\left\{ \begin{array}{l}\frac{ds_{HO}}{dt}=-k_{1\_HO}\phi_{1\_HO}+\varsigma_{ HO}\\ \frac{ds_{HO}}{dt}=-k_{2\_HO}\phi_{2\_HO}+\widetilde{\delta}\end{array} \right. \tag{42}\]  
where \(\widetilde{\delta}=-\frac{1}{M}\cdot\frac{ds}{dt}\). \(\square\)  
The Lyapunov function is chosen as \(V_{2}={X_{HO}}^{T}P_{HO}X_{HO}\), where \(X_{HO}=\left[ \begin{array}{cc}\phi_{1\_HO}&\varsigma_{HO}\\ \end{array} \right]^{T}\). To facilitate proving the adaptive law, the controller stability is chosen as \(P_{HO}=\left[ \begin{array}{cc}\alpha+4\beta^{2}&-2\beta\\ -2\beta&1\end{array} \right]\). When \(s_{HO}\
eq 0\), \(d\phi_{1\_HO}/ds_{HO}=0.5|s_{HO}|^{-0.5}+1\) and \(\phi_{2\_HO}=\phi_{1\_HO}d\phi_{1\_HO}/ds_{HO}\). Combining this with (42), the result is  
\[\frac{dX_{HO}}{dt}=\frac{d\phi_{1\_HO}}{ds_{HO}}(A_{HO}X_{HO}+B_{HO}\overline{ \delta}) \tag{43}\]  
where \(A_{HO}=\left[ \begin{array}{cc}-k_{1\_HO}&1\\ -k_{2\_HO}&0\end{array} \right]\) and \(B_{HO}=\left[ \begin{array}{cc}0&0.5\end{array} \right]^{T}\), with \(\overline{\delta}=2\widetilde{\delta}ds_{HO}/d\phi_{1\_HO}\). The purpose of reducing \(B_{HO}\) in the above equation is to lower \(k_{2\_HO}\), preventing the integral term from becoming excessively high and adversely affecting control performance.