## 2 Preliminaries and Problem Formulation
### The Finite-Time Velocity Constraint Controller Design  
Define the surge velocity tracking error as follows:  
\[u_{e}=u-u_{d} \tag{58}\]  
where \(u_{d}\) is the desired surge velocity. Consider a Lyapunov function as follows:  
\[V_{5}=\frac{1}{2}m_{11}u_{e}^{2} \tag{59}\]  
Taking the time derivative of \(V_{5}\) along (5) and (58), we have  
\[\dot{V}_{5}=u_{e}(d_{u}+\tau_{u}-m_{11}\dot{u}_{d}) \tag{60}\]  
Next, a finite-time disturbance observer is designed to estimate the compound disturbance \(d_{u}\):  
\[ \begin{cases}\dot{u}=\frac{1}{m_{11}}\left[\tau_{u}-b_{3}\dot{u}-b_{4}|\dot{u }|^{\gamma_{2}}\operatorname{sgn}(\dot{u})-\bar{d}_{u}\operatorname{sgn}( \dot{u})\right]\\ \hat{d}_{u}=m_{11}\dot{u}-\tau_{u}=-b_{3}\dot{u}-b_{4}|\dot{u}|^{\gamma_{2}} \operatorname{sgn}(\dot{u})-\bar{d}_{u}\operatorname{sgn}(\dot{u})\end{cases}  \tag{61}\]  
where \(\hat{d}_{u}\) is the observation value of the disturbance, \(0<\gamma_{2}<1\), \(b_{3}\), \(b_{4}\) are positive design constants and define the estimation error as \(\tilde{u}=\hat{u}-u\). Then, taking the time derivative of \(\tilde{u}\) results in  
\[\dot{u}=\frac{1}{m_{11}}\left[-b_{3}\tilde{u}-b_{4}|\tilde{u}|^{\gamma_{2}} \operatorname{sgn}(\tilde{u})-\bar{d}_{u}\operatorname{sgn}(\tilde{u})-d_{u}\right] \tag{62}\]  
**Theorem 2**. : _For the mathematical model of an underactuated MSV (4) and (5), considering the compound disturbance \(d_{u}\), a finite-time disturbance observer is designed as (61), then the disturbance observation error \(\bar{d}_{u}\) can converge to 0 in finite time._  
Proof. : Define the Lyapunov function as follows:  
\[V_{d_{u}}=\frac{1}{2}\tilde{u}^{2} \tag{63}\]Taking the time derivative of \(V_{d_{u}}\) results in  
\[ \begin{split}\dot{V}_{d_{u}}&=\ddot{u}\frac{1}{m_{11}} \left[-b_{3}\ddot{u}-b_{4}|\ddot{u}|^{\gamma_{2}}\operatorname{sgn}(\ddot{u})- \tilde{d}_{u}\operatorname{sgn}(\ddot{u})-d_{u}\right]\\ &\leq-b_{3}\frac{1}{m_{11}}\ddot{u}^{2}-b_{4}\frac{1}{m_{11}}| \dot{u}|^{\gamma_{2}+1}\\ &=-2\frac{b_{3}}{m_{11}}V_{d_{u}}-2^{(\gamma_{2}+1)/2}\frac{b_{4} }{m_{11}}V_{d_{u}}^{(\gamma_{2}+1)/2}\\ &=-\chi_{3}V_{d_{u}}-\chi_{4}V_{d_{u}}^{(\gamma_{2}+1)/2}\end{split}  \tag{64}\]  
where \(\chi_{3}=2\frac{b_{3}}{m_{11}}\) and \(\chi_{4}=2^{(\gamma_{2}+1)/2}\frac{b_{4}}{m_{11}}\). According to the Lemma 1, the estimation error \(\ddot{u}\) converges to \(0\) in finite time, and the convergence time is: \(T_{2}\leq\frac{\ln\left(1+\frac{\chi_{3}}{\chi_{4}}V_{d_{u}}^{1-(\gamma_{2}+1) /2}\right)}{\chi_{3}(1-(\gamma_{2}+1)/2)}\). The observation error of the compound disturbance \(d_{u}\) is  
\[ \begin{split}\tilde{d}_{u}&=\hat{d}_{u}-d_{u}=m_{11 }\dot{u}-\tau_{u}-(m_{11}\dot{u}-\tau_{u})\\ &=m_{11}\dot{u}\end{split}  \tag{65}\]  
Because it is proven that the estimation error \(\ddot{u}\) converges to zero in finite time, the disturbance observation error \(\tilde{d}_{u}\) converges to \(0\) in finite time, that is \(\tilde{d}_{u}=0\), \(\forall t\geq T_{2}\). Theorem 2 has been proven. To handle the actuator saturation (6), a finite-time velocity input saturation compensator is designed as follows:  
\[\dot{\chi}_{u}=-k_{\chi_{xu}}\chi_{u}-k_{\chi_{xu}}S_{\chi_{u}}+\Delta\tau_{u} \tag{66}\]  
where \(\chi_{u}\) is the output of the finite-time velocity input saturation compensator, \(k_{\chi_{xu}}>0\), \(k_{\chi_{xu}}>0\), \(\Delta\tau_{u}=\tau_{u}-\tau_{uc}\). \(S_{\chi_{u}}\) is designed as follows:  
\[S_{\chi_{u}}= \begin{cases}|\chi_{u}|^{\frac{1}{2}}\operatorname{sgn}(\chi_{u }),&\text{if }|\chi_{u}|\geq\varepsilon_{\chi_{u}}\\ \frac{3}{2}\varepsilon_{\chi_{u}}^{-\frac{1}{2}}\chi_{u}-\frac{1}{2}\varepsilon _{\chi_{u}}^{-\frac{3}{2}}\chi_{u}^{2},&\text{else}\end{cases}  \tag{67}\]  
Then, the finite-time surge velocity constraint controller is designed as follows:  
\[\tau_{uc}=m_{11}\dot{u}_{d}-k_{u}u_{e}-k_{u1}|u_{e}|^{\frac{1}{2}}-\tilde{d}_{u }+k_{\chi_{u}}\chi_{u} \tag{68}\]  
where \(k_{u}\), \(k_{u1}\) and \(k_{\chi_{u}}\) are control parameters.