## 3 Control Design and Stability Analysis
### Sway Control  
The hovercraft is a kind of typical underactuated surface vessel due to the lateral axis not being directly actuated. Thus, the virtual sway control law will be designed to stabilize the sway velocity tracking error within a user-appointed time while guaranteeing that the sway speed tracking error remains within the predefined appointed-time performance constraint boundary. Consider the following candidate Lyapunov function:  
\[V_{v}=\int_{0}^{v_{e}}\frac{k_{va}^{2}\left(t\right)\xi}{k_{va}^{2}\left(t \right)-\xi^{2}}d\xi+\frac{1}{2}\tilde{W}_{v}^{T}\Gamma_{v}^{-1}\tilde{W}_{v} \tag{33}\]  
where \(\tilde{W}_{v}=W_{v}^{*}-\hat{W}_{v}\) represents the estimation error with \(\hat{W}_{v}\) representing the estimated value of \(W_{v}^{*}\) and \(\Gamma_{v}\) is a positive definite diagonal matrix. \(k_{va}\left(t\right)\) is given by:  
\[k_{va}\left(t\right)=\left\{ \begin{array}{l}\frac{k_{vb}-k_{vb}}{T_{va}^{2}} t^{2}-2\frac{k_{vb}-k_{vb}}{T_{va}}t+\bar{k}_{vb},t\in\left[0,T_{va}\right)\\ k_{vb},t\in\left[T_{va},+\infty\right)\end{array} \right. \tag{34}\]  
We know that the candidate Lyapunov function \(V_{v}\) is positive definite, continuously differentiable, and radially unbounded in the set \(\Omega_{v}=\left\{v_{e}:\left|v_{e}\right|<k_{va}\left(t\right)\right\}\); therefore, \(V_{v}\) is a valid candidate Lyapunov function. Calculating the time derivative of \(V_{v}\) yields:  
\[ \begin{split}\dot{V}_{v}=&\frac{\partial V_{v}}{\partial v _{e}}\dot{v}_{e}+\frac{\partial V_{v}}{\partial k_{va}}k_{va}-\tilde{W}_{v}^{ T}\Gamma_{v}^{-1}\dot{\hat{W}}_{v}\\ =&\frac{k_{va}^{2}\left(t\right)v_{e}}{k_{va}^{2} \left(t\right)-v_{e}^{2}}\dot{v}_{e}+v_{e}\left(\frac{k_{va}}{v_{e}}\log\frac{k _{va}^{2}\left(t\right)}{k_{va}^{2}\left(t\right)-v_{e}^{2}}-\frac{k_{va}v_{e }}{k_{va}^{2}\left(t\right)-v_{e}^{2}}\right)k_{va}-\tilde{W}_{v}^{T}\Gamma_{v }^{-1}\dot{\hat{W}}_{v}\end{split}  \tag{35}\]  
Substituting \(\dot{v}_{e}\) into (35), we have:  
\[ \begin{split}\dot{V}_{v}=&\frac{k_{va}^{2}\left(t \right)v_{e}}{k_{va}^{2}\left(t\right)-v_{e}^{2}}\left(-ur+\frac{F_{yD0}}{m_{0}} +f_{v}-\dot{\alpha}_{v}\right)\\ &+v_{e}\left(\frac{k_{va}}{v_{e}}\log\frac{k_{va}^{2}\left(t \right)u_{e}}{k_{va}^{2}\left(t\right)-v_{e}^{2}}-\frac{k_{va}v_{e}}{k_{va}^{2 }\left(t\right)-v_{e}^{2}}\right)\dot{k}_{va}-\tilde{W}_{v}^{T}\Gamma_{v}^{-1 }\dot{\hat{W}}_{v}\end{split}  \tag{36}\]According to [PERSON]'s direct method, the virtual sway control law, and the adaptive updating law are designed as:  
\[r_{d}= \frac{1}{u}\left(k_{v}v_{e}+\frac{F_{yD0}}{m_{0}}-\dot{x}_{v}+\hat{ W}_{v}^{T}H_{v}\left(Z_{v}\right)-\delta_{v}sgn\left(v_{e}\right)\right.\] \[+\frac{k_{va}^{2}\left(t\right)-v_{e}^{2}}{k_{va}^{2}\left(t \right)}\left(\frac{k_{va}}{v_{e}}\log\frac{k_{va}^{2}\left(t\right)}{k_{va}^{ 2}\left(t\right)-v_{e}^{2}}-\frac{k_{va}v_{e}}{k_{va}^{2}\left(t\right)-v_{e}^{ 2}}\right)\dot{k}_{va} \tag{37}\] \[\left.+\frac{k_{va}^{2}\left(t\right)-v_{e}^{2}}{k_{va}^{2}\left(t \right)}\left(-\sin\psi\cos\phi\frac{k_{xa}^{2}\left(t\right)x_{e}}{k_{xa}^{2} \left(t\right)-x_{e}^{2}}+\cos\psi\cos\phi\frac{k_{ya}^{2}\left(t\right)y_{e} }{k_{ya}^{2}\left(t\right)-y_{e}^{2}}\right)\right)\]  
\[\hat{W}_{v}=\Gamma_{v}\left(\frac{k_{va}^{2}\left(t\right)v_{e}}{k_{va}^{2} \left(t\right)-v_{e}^{2}}H_{v}\left(Z_{v}\right)-\sigma_{v}\hat{W}_{v}\right) \tag{38}\]  
where \(k_{v}\), \(\delta_{v}\), and \(\sigma_{v}\) are positive constants. **Remark 7**. : _Using [PERSON] rule, we have:_  
\[\lim_{v_{e}\to 0}\frac{k_{va}\left(t\right)v_{e}}{v_{e}}\log\frac{k_{va}^{2} \left(t\right)}{k_{va}^{2}\left(t\right)-v_{e}^{2}}=\frac{2k_{va}\left(t\right) v_{e}}{k_{va}^{2}\left(t\right)-v_{e}^{2}} \tag{39}\]  
_In addition, the surge speed \(u\) is always set to a positive value in the control process.