## 4 Control System and Problem Statement
### SMC-Based Controller Design  
To accomplish the lane change process, the vehicle control system will restrain the vehicle's movement and force it to follow the intended trajectory strategy. By regulating the steering wheel angle, the lane change process allows the vehicle to follow the intended lane change trajectory. The ultimate goal of a lane change controller is to manage the steering wheel angle of the vehicle so that it follows the proper lane change trajectory. The vehicle's tracking error and its derivative are defined as follows:  
\[e=\theta-\theta_{d} \tag{5}\]  
Figure 3: Diagram of a cosine lane change trajectory. \[\dot{e} =\dot{\theta}-\dot{\theta}_{d} \tag{6}\] \[\ddot{e} =\dot{\theta}-\ddot{\theta}_{d} \tag{7}\]  
If the traditional terminal sliding surface \(s=\dot{e}+\frac{1}{\beta}\dot{e}^{\frac{p}{q}}\), \(1<\frac{p}{q}<2\), \(\beta>0\) is adopted, the controller design will have a singularity problem when the system converges on the sliding surface. Singularity issues will cause system instability, and the traditional terminal sliding mode function's convergence rate is slow when the mistake is distant from the phase plane. The non-singular fast terminal sliding mode surface is proposed in this study to address the slow convergence speed and singularity difficulties of standard terminal sliding mode, as indicated in Equation (8):  
\[s =e+\frac{1}{\alpha}|e^{\gamma}|sign(e)+\frac{1}{\beta}\Big{|}e^{ \frac{p}{q}}\Big{|}sign(\dot{e}) \tag{8}\] \[=e+\frac{1}{\alpha}sig^{\gamma}(e)+\frac{1}{\beta}sig^{\frac{p}{ q}}(\dot{e})\]  
where \(|e^{\gamma}|sign(e)=sig^{\gamma}(e)\), \(\Big{|}e^{\frac{p}{q}}\Big{|}sign(\dot{e})=sig^{\frac{p}{q}}(\dot{e})\), \(\alpha\), \(\beta>0\), \(1<\frac{p}{q}<2\) and \(\gamma>\frac{p}{q}\). The improved non-singular fast terminal sliding mode control introduces a linear sliding mode and a fast terminal attractor, allowing the system state to swiftly and precisely settle to equilibrium. Assuming the time from state \(s\
eq 0\) to state \(s=0\) is \(t_{r}\), and the time from state \(e(t_{r})\
eq 0\) to state \(e(t_{r}+t_{s})=0\) is \(t_{s}\). \(t_{s}\) denotes the time to reach equilibrium state. Along the sliding mode (8), the time from any initial condition \(e\
eq 0\) to equilibrium state \(e=0\) is:  
\[t_{s}=-\Big{(}\frac{1}{\beta}\Big{)}^{\frac{q}{p}}\cdot\int_{e(t_{r})}^{0} \frac{1}{(u-\frac{1}{\alpha}u^{\gamma})^{\frac{p}{p}}}du \tag{9}\]  
The system's control rate is made up of two components: equivalent control rate \(u_{eq}\) and switching control rate \(u_{s}\):  
\[u_{eq}=\Im^{-1}\cdot\Big{[}\ddot{\theta}_{d}-f-\beta\frac{q}{p} sig^{2-\frac{p}{q}}(\dot{e})-\frac{\gamma}{\alpha}sig^{\gamma-1}(e)\cdot \beta\frac{q}{p}sig^{2-\frac{p}{q}}(\dot{e})\Big{]} \tag{10}\] \[u_{s}=\Im^{-1}(B_{du}+\hbar)\text{sgn}(s) \tag{11}\]  
where \(\hbar>0\); therefore, the system's control rate can be determined as follows:  
\[u=u_{eq}+u_{s}\] (12) Figure 3:0 The Lyapunov function is chosen as:  
Figure 3:1  
Differentiating the Lyapunov function from (13), we have  
Figure 3:2  
**Theorem 1**. : _Assume that the upper bound of lumped interference is \(F\). Let \(\Delta f\) satisfy the hypothesis of \(\Delta f\leq F\), and \(\alpha\), \(\beta>0\), \(1<\frac{p}{q}<2\) and \(\gamma>\frac{p}{q}\), where \(p\) and \(q\) are both positive numbers. Then, for nonlinear system model (7), the stability of the closed-loop system can be guaranteed by adopting the sliding mode surface function (8) and the control law (12), and the tracking error can converge to 0 asymptotically in a finite time._  
**Proof of Theorem 1. ** [PERSON]'s theorem is used to prove stability. We can deduce the following from Equation (13):  
Figure 3:3  
Figure 3:4  
Figure 3:5  
Figure 3:6  
Figure 3:7  
Figure 3:8  
The above finishes the proof. The system is stable, according to [PERSON]'s second technique. Due to the unknown value of the interference upper bound, a huge interference upper bound must be intentionally imposed, resulting in poor robustness and severe control chattering.