## 3 Controller Design and Stability Analysis
If the sliding mode surface is selected as Equation (13) and the control law is designed as Equation (18), then all the signals of the closed-loop train system are bounded and asymptotically stable position and velocity tracking to the desired trajectories can be guaranteed._  
\[F=F_{0}+F_{1}+F_{2}+F_{3}, \tag{18a}\] \[F_{0}=\hat{\partial}\varsigma^{T}(v)-\hat{m}(\delta_{1}\dot{e}+\delta_{2}e-\dot{ v}_{d})-\iota\kappa\text{sign}(\overline{s})-\alpha\varphi,\] (18b) \[F_{1}=-\hat{T}\eta^{T}(v)\text{sign}(\overline{s}),\] (18c) \[F_{2}=-\hat{T}_{4}|F_{0}|\text{sign}(\overline{s}),\] (18d) \[F_{3}=-k\overline{s}. \tag{18e}\]  
_where \(\hat{\theta}\) and \(\hat{m}\) are the estimation of \(\theta\) and \(m\); \(\iota\) and \(\kappa\) are positive constants satisfied \(\left|\frac{\hat{m}}{\hat{m}}\right|\leq\iota\) and \(\left|\alpha\varphi-\sigma(t)\right|\leq\kappa\), respectively; \(k>0\) is a design parameter chosen by the designers; \(\hat{T}\) and \(\hat{T}_{4}\) are the estimation of \(T\) and \(T_{4}\) with the definition of \(T=T^{\prime}/(1-\delta)\) and \(T_{4}=\delta/(1-\delta)\), respectively. The parameter adaptive laws are selected as follows:_  
\[\dot{s}=\ddot{e}+\hat{\delta}_{1}e+\hat{\delta}_{1}\dot{e}+\hat{\delta}_{2} \int_{0}^{t}e(\tau)d\tau+\hat{\delta}_{2}e. \tag{14}\]0 \[\dot{s}=\ddot{e}+\hat{\delta}_{1}e+\hat{\delta}_{1}\dot{e}+\hat{\delta}_{2} \int_{0}^{t}e(\tau)d\tau+\hat{\delta}_{2}e. \tag{14}\]1 (19b) \[\dot{s}=\ddot{e}+\hat{\delta}_{1}e+\hat{\delta}_{1}\dot{e}+\hat{\delta}_{2} \int_{0}^{t}e(\tau)d\tau+\hat{\delta}_{2}e. \tag{14}\]2 (19c) \[\dot{s}=\ddot{e}+\hat{\delta}_{1}e+\hat{\delta}_{1}\dot{e}+\hat{\delta}_{2} \int_{0}^{t}e(\tau)d\tau+\hat{\delta}_{2}e. \tag{14}\]3  
_where \(\Gamma_{1}=\text{diag}[k_{0},k_{1},k_{2}]\) and \(\Gamma_{2}=\text{diag}[l_{0},l_{1},l_{2},l_{3}]\) are two positive definite matrices; \(k_{3}>0\); \(l_{4}>0\)._  
The gains of the selected sliding mode surface are estimated by  
\[\dot{s}=\ddot{e}+\hat{\delta}_{1}e+\hat{\delta}_{1}\dot{e}+\hat{\delta}_{2} \int_{0}^{t}e(\tau)d\tau+\hat{\delta}_{2}e. \tag{14}\]4 \[\dot{s}=\ddot{e}+\hat{\delta}_{1}e+\hat{\delta}_{1}\dot{e}+\hat{\delta}_{2} \int_{0}^{t}e(\tau)d\tau+\hat{\delta}_{2}e. \tag{14}\]5  
where \(\xi_{1}\) and \(\xi_{2}\) are two positive design parameters. The block diagram of the proposed controller is depicted in \[\dot{s}=\ddot{e}+\hat{\delta}_{1}e+\hat{\delta}_{1}\dot{e}+\hat{\delta}_{2} \int_{0}^{t}e(\tau)d\tau+\hat{\delta}_{2}e. \tag{14}\]6  
\[\dot{s}=\ddot{e}+\hat{\delta}_{1}e+\hat{\delta}_{1}\dot{e}+\hat{\delta}_{2} \int_{0}^{t}e(\tau)d\tau+\hat{\delta}_{2}e. \tag{14}\]7 Block diagram of the proposed controller. the curve length \(l_{s}=200\) m, the centric angle of the curve \(\alpha_{s}=2\pi/3\), and the ramp angle \(\theta=\pi/180\). \(o(\cdot)\) stands for other resistance with the given value of \(o(\cdot)=1500\sin(0.02 vt)\). It is noteworthy that force of train gravity can help or hinder train movement depending on whether the ramp angle is positive or negative. However, the projection of gravity on the direction parallel to the track, which is unknown to the proposed controller, has been included in the lumped uncertainty, so no matter whether the force of gravity hinders or helps train movement, it would be compensated by the compensation term \(F_{1}\) in our algorithm. Therefore, no matter whether gravity hinders or helps train movement, the train model and proposed algorithm are applicable. It should be noted that the train parameters, parametric uncertainties, and additional disturbance given above, the actuator's effective coefficients used in the following simulation, are unknown and not required in the design of the controller.