## 4 The Yaw Tracking Control System
\begin{array}{l}\dot{V}_{4}=s_{a}\big{(}J\lambda_{1}\widetilde{\psi}+J \lambda_{2}r+\alpha_{v}v+\alpha_{r}r+\alpha_{y}y\big{)}\\ +\big{(}-|s_{a}|\hat{\Gamma}_{0}-(\lambda_{1}\widetilde{\psi}+\lambda_{2}r) \big{)}s_{a}-\hat{\alpha}_{v}vs_{a}-\hat{\alpha}_{r}rs_{a}-\hat{\alpha}_{y}ys_{ a}-\tau_{0}{s_{a}}^{2}\big{)}\\ +s_{a}\Gamma_{0}(\cdot)-p_{1}\widetilde{J}-p_{2}\widetilde{\alpha}_{v}\dot{ \widetilde{\alpha}}_{v}-p_{3}\widetilde{\alpha}_{r}\dot{\widetilde{\alpha}}_{ r}-p_{4}\widetilde{\alpha}_{y}\dot{\widetilde{\alpha}}_{y}-p_{5}\widetilde{\Gamma}_{0} \hat{\Gamma}_{0}\\ =s_{a}\big{(}\lambda_{1}\widetilde{\psi}+\lambda_{2}r\big{)}J-s_{a}(\lambda_ {1}\widetilde{\psi}+\lambda_{2}r)\hat{J}+s_{a}\Gamma_{0}(\cdot)-|s_{a}|\hat{ \Gamma}_{0}\\ +s_{a}\big{(}\alpha_{v}v+\alpha_{r}r+\alpha_{y}y\big{)}-s_{a}\big{(}\hat{ \alpha}_{v}v+\hat{\alpha}_{r}r+\hat{\alpha}_{y}y\big{)}-\tau_{0}{s_{a}}^{2}\\ -\widetilde{J}s_{a}\big{(}\lambda_{1}\widetilde{\psi}+\lambda_{2}r\big{)}- \widetilde{\alpha}_{v}s_{a}v-\widetilde{\alpha}_{r}s_{a}r-\widetilde{\alpha}_ {y}s_{a}y-\widetilde{\Gamma}_{0}|s_{a}\\ =s_{a}\big{(}\lambda_{1}\widetilde{\psi}+\lambda_{2}r\big{)}\widetilde{J}+s_{ a}\Gamma_{0}(\cdot)-|s_{a}|\hat{\Gamma}_{0}+s_{a}\big{(}\widetilde{\alpha}_{v}v+ \widetilde{\alpha}_{r}r+\widetilde{\alpha}_{y}y\big{)}-\tau_{0}{s_{a}}^{2}\\ -s_{a}\big{(}\lambda_{1}\widetilde{\psi}+\lambda_{2}r\big{)}\widetilde{J}- \widetilde{\alpha}_{v}s_{a}v-\widetilde{\alpha}_{r}s_{a}r-\widetilde{\alpha}_ {y}s_{a}y-\widetilde{\Gamma}_{0}|s_{a}\\ =s_{a}(-\tau_{0}s_{a}+\Gamma_{0}(\cdot)-\Gamma_{0}(\cdot)sign(s_{a}))\\ \leq 0.\end{array}  \tag{53}\]  
Thus, the yaw tracking control system is GUAS. According to [PERSON]'s Lemma [29], the system states move to and stay on the sliding surface \(\underset{t\rightarrow\infty}{\lim}s_{a}(t)=0\). In addition, the objective \(\underset{t\rightarrow\infty}{\lim}\widetilde{\psi}=0\) can be achieved. **Remark 2**. : _The control law (48) is independent of the model coefficients and the control coefficient. Thus, the proposed control law can be used to design the yaw tracking control system with large model uncertainties. Moreover, the controller does not depend on the disturbance bound and the derivative terms (\(\dot{\psi}_{d}\) and \(\ddot{\psi}_{d}\)). Therefore, all problems existing in the standard SMC can be solved by the modified control._  
**Remark 3**. : _The parameters \(\lambda_{1}>0\), \(\lambda_{2}>0\) in the control law (48) are very important, for they determine the convergence rate of the sliding surface. It is clear that the farther the eigenvalues of \(\
abla^{2}+\lambda_{2}\
abla+\lambda_{1}=0\) away from the imaginary axis on the left half complex plane, the faster the convergence to the origin. While fast convergence speeds need very high control input, which is always bounded in real life, thus \(\lambda_{1}>0\), \(\lambda_{2}>0\) must be selected reasonably._  
**Remark 4**. : _The control law (48) is discontinuous when crossing the sliding surface \(s_{a}=0\), which may lead to undesirable chattering. A solution to eliminate chattering is to introduce a bounded layer around the sliding surface, replace the signum function by a high-slope saturation [30]:_  
\[\delta_{p}=-\hat{\Gamma}_{0}\text{sat}\Big{(}\frac{s_{a}}{\varepsilon}\Big{)}, \tag{54}\]_where \(\varepsilon>0\) is the bounded layer; \(\text{sat}(\cdot)\) is the saturation function defined by_  
\[\text{sat}(ui)=\left\{ \begin{array}{l}ui,\,\text{if}|ui|\leq 1\\ \text{sgn}(ui),\,\text{if}|ui|>1\end{array} \right. \tag{55}\]  
_The benefit of the smooth technique is that it offers a continuous approximation to the discontinuous SMC law inside the bounded layer, and guarantees the output tracking error within any neighborhood of zero._