## 4 Control System and Problem Statement
### Enhanced SMC-Based Controller Design  
Estimating the upper bound of system lumped disturbance with RBF neural network is a crucial challenge for the enhanced SMC-Based Controller. The upper bound of system lumped interference is estimated using a neural network. Due to the uncertainty of system parameters and external disturbances, robust terms are designed to compensate for the influence caused by disturbances. However, if the robust items are too large, serious chattering will occur in the system. Therefore, this study uses a neural network to estimate the upper bound of system lumped interference. The new proposed self-learning framework is named as NNTSMC, which means a neural network modified version of TSMC (non-singular fast terminal sliding mode control). Assume that the estimation value of the interference upper bound is set, as the neural network is used to approximate the estimation value of interference upper bound, which can be expressed as: Assuming that the estimation value of the upper bound of interference is set to \(\hat{k}\), the neural network is used to approximate the estimation value of the upper bound of interference, which may be expressed as:  
\[\hat{k}=\hat{\varphi}^{T}\phi \tag{15}\]  
where \(\varphi\) is the weight vector of neural network, and \(n\) represents the number of nodes of hidden layer. \(\phi=\left[\phi_{1}\ \ \phi_{2}\ \ \ldots\ \ \phi_{n}\right]^{T}\) denotes the output vector of the hidden layer, the Gaussian basis function is used as the membership function, and the output of the \(i\)-th node in the hidden layer is:  
\[\phi_{i}(x)=\exp(-\frac{\left\|x-\text{H}_{i}\right\|^{2}}{\phi_{i}^{2}}) \tag{16}\]where H\({}_{i}\) is the center vector of the \(i\)-th node in the hidden layer, and \(\sigma^{2}_{i}\) is the base width of the \(i\)-th node in the hidden layer, \(i=1,\ldots,n\). When the neural network is employed to approach the interference upper bound \(\hat{k}\), it is assumed that an ideal weight \(\varphi^{*}\) exists, satisfying condition \(\varphi^{*T}\hat{\varphi}-\Delta f=\varepsilon\), and \(\varepsilon\) is the difference between the ideal and real output, satisfying condition \(\hat{k}-\Delta f\geq\varepsilon\geq 0\). The weight error of the upper bound of interference estimation by neural network is defined as:  
\[\hat{\varphi}=\varphi^{*}-\hat{\varphi}>0 \tag{17}\]  
Based on the above discussion, the improved driving control law can be deduced as follows:  
\[u=u_{eq}+u_{s}=\Im^{-1}\cdot\left[\hat{\theta}_{d}-f+\beta\frac{q}{p}sig^{2- \frac{p}{q}}(\varepsilon)+\frac{\gamma}{\alpha}sig^{\gamma-1}(\varepsilon) \cdot\beta\frac{q}{p}sig^{2-\frac{p}{q}}(\varepsilon)\right]-\Im^{-1}\hat{k} \text{sgn}(s) \tag{18}\]  
**Theorem 2**. : _For the vehicle model under consideration, after the adaptive neural network was added into the original sliding mode control strategy, the designed vehicle lateral control rate was (18), which could still ensure the stability of the controlled system._  
The Lyapunov function is designed as:  
\[V=\frac{1}{2}s^{2}+\frac{1}{2\zeta}\hat{\varphi}^{T}\hat{\varphi} \tag{19}\]  
where \(\zeta>0\). Differentiating the Lyapunov function from (19), we have  
\[\dot{V}=s\hat{s}+\frac{1}{\zeta}\hat{\varphi}^{T}\hat{\varphi} \tag{20}\]  
**Proof of Theorem 2. ** [PERSON]'s theorem is used to prove stability. The Lyapunov function is negative semidefinite, according to the second method of Lyapunov, and the control rate proposed in this study can ensure system stability. It can be seen from the above proof process that the disturbance upper bound is estimated, and the disturbance compensation is carried out by using an adaptive neural network, which ensures the stability of the system and solves the chattering phenomenon caused by excessive robust gain. Through the above analysis, it can be seen that, based on the vehicle dynamics model and road condition information, the lane changing control strategy oriented to the desired lane-changing trajectory can be designed.