# 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
(1) The stability of the controller when \(k_{1\_HO}=k_{4\_HO}\) is analyzed as follows:  
\[\frac{d\upsilon_{2}}{dt}=\frac{d\phi_{1\_HO}}{ds_{HO}}(X_{HO}{}^{T}(A_{HO}{}^{ T}P_{HO}+P_{HO}A_{HO})X_{HO}+2\overline{\delta}B_{HO}{}^{T}P_{HO}X_{HO}) \tag{44}\]\[\text{Since}\ \overline{\delta}^{2}=4\overline{\delta}^{2}(\frac{ds_{HO}}{d\phi_{1,HO}})^{2 }=4\overline{\delta}^{2}\frac{\phi_{1,HO}}{\phi_{2,HO}}^{2}\text{, and when }s_{HO}\
eq 0\]  
\[\phi_{2,HO}{}^{2}=0.25+3.25|s_{HO}|+s_{HO}{}^{2}+1.5|s_{HO}|^{0.5}+3|s_{HO}|^{1. 5}>0.25 \tag{45}\]  
Therefore, \(\overline{\delta}^{2}<16\overline{\delta}^{2}\phi_{1,HO}{}^{2}<16\overline{ \eta}^{2}\phi_{1,HO}{}^{2}\). Furthermore, according to  
\[2\overline{\delta}B_{HO}{}^{T}P_{HO}X_{HO}-\overline{\delta}^{2}=-(\overline {\delta}-B_{HO}{}^{T}P_{HO}X_{HO})^{2}+X_{HO}{}^{T}P_{HO}B_{HO}B_{HO}{}^{T}P_{ HO}X_{HO} \tag{46}\]  
Equation (44) can be simplified as  
\[\frac{dV_{2}}{dt}\leq\frac{d\phi_{1,HO}}{ds_{HO}}X_{HO}{}^{T}(A_{HO}{}^{T}P_{ HO}+P_{HO}A_{HO}+16\overline{\eta}^{2}C_{HO}{}^{T}C_{HO}+P_{HO}B_{HO}B_{HO}{}^{T}P_{ HO})X_{HO} \tag{47}\]  
where \(C_{HO}= \begin{bmatrix}1&0\end{bmatrix} \). By substituting the expressions for \(Q_{HO}=A_{HO}{}^{T}P_{HO}+P_{HO}A_{HO}+16\overline{\eta}^{2}C_{HO}{}^{T}C_{HO }+P_{HO}B_{HO}B_{HO}{}^{T}P_{HO}\), \(k_{2,HO}=2\beta k_{1,HO}\), \(Q_{HO}\) simplifies to  
\[Q_{HO}= \begin{bmatrix}-2k_{1,HO}\alpha+16\overline{\eta}^{2}+\beta^{2}&\alpha +4\beta^{2}-0.5\beta\\ \alpha+4\beta^{2}-0.5\beta&-4\beta+0.25\end{bmatrix}  \tag{48}\]  
By [PERSON]'s complementary lemma, \(Q_{HO}\) is a negative definite matrix when the following conditions are satisfied:  
\[\left\{ \begin{array}{l}\beta>\frac{1}{16}\\ k_{1,HO}>\frac{1}{2\alpha}(16\overline{\eta}^{2}+\beta^{2}-\frac{1}{0.25-4 \beta}(\alpha+4\beta^{2}-0.5\beta)^{2})\end{array} \right. \tag{49}\]  
Therefore, if \(k_{4,HO}\) satisfies the condition of Equation (49), then \(\frac{dV_{2}}{dt}<0\), ensuring the system's stability, and the sliding mode variable \(s_{HO}\) will converge to zero. When \(k_{4,HO}\) is not satisfied, the adaptive law for \(k_{1,HO}\) will increase, ensuring \(k_{1,HO}>k_{4,HO}\). (2) The following analysis demonstrates that when \(k_{1,HO}>k_{4,HO}\) and \(|s_{HO}|>\sigma_{HO}\), the controllers remain stable. Define the Lyapunov function as  
\[V_{3}=X_{HO}{}^{T}P_{HO}X_{HO}+0.5(k_{1,HO}-k_{1,HO}{}^{*})^{2}+0.5(k_{2,HO}-k_ {2,HO}{}^{*})^{2} \tag{50}\]  
where \(k_{1,HO}{}^{*}\) and \(k_{2,HO}{}^{*}\) are the target values of \(k_{1,HO}\) and \(k_{2,HO}\), respectively. Satisfying the convergence condition of (49), let \(\widetilde{k}_{1,HO}=k_{1,HO}-k_{1,HO}{}^{*}\) and \(\widetilde{k}_{2,HO}=k_{2,HO}-k_{2,HO}{}^{*}\). The derivation of Equation (50) gives:  
\[\frac{dV_{3}}{dt}\leq\frac{d\phi_{1,HO}}{ds_{HO}}X_{HO}{}^{T}Q_{HO}X_{HO}+ \frac{dk_{1,HO}}{dt}\widetilde{k}_{1,HO}+\frac{dk_{2,HO}}{dt}\widetilde{k}_ {2,HO} \tag{51}\]  
where \(A_{HO}{}^{*}=\left[ \begin{array}{cc}-k_{1,HO}{}^{*}&1\\ -k_{2,HO}{}^{*}&0\end{array} \right]\). Let \(\widetilde{A}_{HO}=A_{HO}-A_{HO}{}^{*}\) and \(Q_{HO}{}^{*}\) be \(Q_{HO}\) when \(k_{1,HO}\) and \(k_{2,HO}\) take \(k_{1,HO}{}^{*}\) and \(k_{2,HO}{}^{*}\). Then, Equation (51) reduces to  
\[ \begin{array}{l}\frac{dV_{3}}{dt}\leq\frac{d\phi_{1,HO}}{ds_{HO}}X_{HO}{}^{ T}Q_{HO}{}^{*}X_{HO}+\frac{d\phi_{1,HO}}{ds_{HO}}X_{HO}{}^{T}(\widetilde{A}_{HO}^{T}P_{HO}+P_{HO} \widetilde{A}_{HO})X_{HO}\\ \quad+\frac{dk_{1,HO}}{dt}\widetilde{k}_{1,HO}+\frac{dk_{2,HO}}{dt}\widetilde{ k}_{2,HO}\end{array}  \tag{52}\]  
Since \(\frac{dk_{2,HO}}{dt}=2\beta\frac{dk_{1,HO}}{dt},\widetilde{k}_{2,HO}=2\beta \widetilde{k}_{1,HO}\), it is possible to calculate:  
\[\widetilde{A}_{HO}^{T}P_{HO}+P_{HO}\widetilde{A}_{HO}= \begin{bmatrix}-2\alpha \widetilde{k}_{1,HO}&0\\ 0&0\end{bmatrix}