# NESO-Based Path Following Control for
## 3 Control Strategy Design
### Safety Control Strategy for the Underactuated Hovercraft
#### 3.2.1 Yaw Safe Control Law for the Underactuated Hovercraft
Finally, the safe control is realized by tracking the virtual control law._  
Consider the candidate Lyapunov function \(V_{1}=\big{(}\psi_{e}^{2}+\zeta^{2}\big{)}/2\), together with (27) and (31), the derivative of \(V_{1}\) can be obtained as  
\[\dot{V}_{1}=\psi_{e}\dot{\psi}_{e}+\dot{\zeta}\dot{\zeta}=\psi_{e}\big{(}r_{e}+ \alpha_{r}-\dot{\psi}_{d}\big{)}+\zeta\bigg{(}-k_{a}\zeta-\frac{\big{|}\Delta \alpha_{r}\psi_{e}\big{|}+\Delta_{\alpha_{r}}^{2}/2}{\zeta^{2}}\zeta+\Delta \alpha_{r}\bigg{)} \tag{33}\]  
It follows (29) that  
\[\dot{V}_{1} =\psi_{e}\big{(}r_{e}-k_{\psi}\psi_{e}+\dot{\psi}_{d}+k_{\zeta} \zeta+\Delta\alpha_{r}-\dot{\psi}_{d}\big{)}-k_{a}\zeta^{2}-\big{|}\Delta \alpha_{r}\psi_{e}\big{|}-\frac{1}{2}\Delta_{\alpha_{r}}^{2}+\Delta\alpha_{r}\zeta \tag{34}\] \[=-k_{\psi}\psi_{e}^{2}+k_{\zeta}\psi_{e}\zeta+\psi_{e}r_{e}+ \Delta\alpha_{r}\psi_{e}-k_{a}\zeta^{2}-\big{|}\Delta\alpha_{r}\psi_{e}\big{|} -\frac{1}{2}\Delta_{\alpha_{r}}^{2}+\Delta\alpha_{r}\zeta\]  
Using [PERSON]'s inequalities \(k_{\zeta}\psi_{e}\zeta\leq k_{\zeta}\psi_{e}^{2}/2\varepsilon_{1}+\varepsilon _{1}k_{\zeta}\zeta^{2}/2\), \(\psi_{e}r_{e}\leq\psi_{e}^{2}/2+r_{e}^{2}/2\), and \(\Delta\alpha_{r}\zeta\leq\Delta_{\alpha_{r}}^{2}/2+\zeta^{2}/2\), where \(\varepsilon_{1}\) is a positive constant, then we have that  
\[\dot{V}_{1}=-k_{\psi}\psi_{e}^{2}+\frac{k_{\zeta}}{2\varepsilon_{1}}\psi_{e}^ {2}+\frac{\varepsilon_{1}k_{\zeta}}{2\zeta}\zeta^{2}+\frac{1}{2}\psi_{e}^{2}+ \frac{1}{2}r_{e}^{2}+\Delta\alpha_{r}\psi_{e}-k_{a}\zeta^{2}-\big{|}\Delta \alpha_{r}\psi_{e}\big{|}+\frac{1}{2}\zeta^{2}=-\Big{(}k_{\psi}-\frac{k_{\zeta }}{2\varepsilon_{1}}-\frac{1}{2}\Big{)}\psi_{e}^{2} \tag{35}\]  
To further improve the maneuverability of designed control system, the proposed NESO and backstepping sliding mode control technologies are applied to design a disturbances rejection control law with SLC for an underactuated hovercraft as follows. **Step 2:** Define the sliding mode surface:  
\[s=k_{s}\psi_{e}+r_{e} \tag{36}\]  
where \(k_{s}\) is a designed positive constant, then considering the following Lyapunov function \(V_{r}\ =\ V_{1}\ +\ s^{2}/2\), and differentiating \(V_{r}\) respect to time, yields  
\[\dot{V}_{r}=\dot{V}_{1}+s\dot{s}=\dot{V}_{1}+s\big{(}k_{s}\dot{ \psi}_{e}+\dot{r}_{e}\big{)}=\dot{V}_{1}+s\big{(}k_{s}\dot{\psi}_{e}+\dot{r}- \dot{\alpha}_{r}\big{)} \tag{37}\] \[\leq-\Big{(}k_{\psi}-\frac{k_{\zeta}}{2\varepsilon_{1}}-\frac{1}{ 2}\Big{)}\psi_{e}^{2}-\Big{(}k_{a}-\frac{\varepsilon_{1}k_{\zeta}}{2}-\frac{1} {2}\Big{)}\zeta^{2}+\frac{1}{2}r_{e}^{2}+s\Big{(}k_{s}\big{(}r_{e}-k_{\psi} \psi_{e}\big{)}+d_{r}(t)+\frac{M_{zR}}{l_{z}}-\dot{\alpha}_{r}\Big{)}\]  
The NESO-based backstepping sliding mode control law with time-varying turning SLC is designed as  
\[M_{zR}=l_{z}\Big{[}-k_{s}\big{(}r_{e}-k_{\psi}\psi_{e}\big{)}+\dot{\alpha}_{r} -\big{|}\dot{d}_{r}\big{|}\text{sgn}(s)-hs\Big{]} \tag{38}\]  
where \(k_{u}\) is a designed control gain. Thus, the surge tracking error dynamic can be described as  
\[\dot{u}_{e}=-k_{u}u_{e}-\widetilde{d_{u}} \tag{44}\]  
Considering the candidate [PERSON] function \(V_{u}=u_{e}^{2}/2\), it follows (44) that the derivative of \(V_{u}\) can be obtained as  
\[\dot{V}_{u}=-k_{u}u_{e}^{2}-u_{e}\widetilde{d_{u}} \tag{45}\]  
and using [PERSON]'s inequality \(-u_{e}\widetilde{d_{u}}\leq u_{e}^{2}/2\varepsilon_{4}+\varepsilon_{4} \widetilde{d_{u}^{2}}/2\), where \(\varepsilon_{4}\) is a positive constant, yields  
\[\dot{V}_{1} =\psi_{e}\big{(}r_{e}-k_{\psi}\psi_{e}+\dot{\psi}_{d}+k_{\zeta} \zeta+\Delta\alpha_{r}-\dot{\psi}_{d}\big{)}-k_{a}\zeta^{2}-\big{|}\Delta \alpha_{r}\psi_{e}\big{|}-\frac{1}{2}\Delta_{\alpha_{r}}^{2}+\Delta\alpha_{r}\zeta \tag{34}\]0  
where \(\kappa_{u}=k_{u}-1/2\varepsilon_{4}>0\) and \(C_{u}=\varepsilon_{4}b_{du}/2\), indicating that the surge tracking control objective is achieved.