# Stabilization of Underactuated Surface Vessels: A Continuous Fractional Power Control Method
## 3 Main Results
### Convergence Analysis  
Compared with the methods in [10,12], the presented control law can improve the system performance by designing the order of \(\phi_{2}\). In this subsection, the roles what parameters \(\xi_{1}\) and \(\xi_{2}\) play on improving the convergence rate. As we know, in the system (11), if we define  
\[V_{2}=\frac{\kappa_{1}}{1+\xi_{1}}\mid\phi_{2}\mid^{1+\xi_{1}}+\frac{1}{2}\phi _{5}^{2}, \tag{23}\]  
we can then have  
\[\dot{V}_{2}=-\kappa_{2}\theta_{5}^{2}. \tag{24}\]  
Define \(\beta\) as \(\sin\beta=\frac{\phi_{5}}{\sqrt{V_{2}}}\), \(\sin\beta=\frac{\phi_{2}^{\frac{1+\xi_{1}}{2}}}{\sqrt{V_{2}}}\), Equation (24) can be written as  
\[V_{2}=-\kappa_{2}\sin^{2}\beta V_{2}. \tag{25}\]  
Obviously, when \(V_{2}>0\), a larger \(\sin^{2}\beta(t)\) implies a faster convergence rate of \(V_{2}\). Therefore, we can improve the convergence rate by increasing \(\sin^{2}\beta=\frac{\phi_{5}^{2}}{V_{2}}\). For this matter, define \(\Theta_{3}=\frac{\phi_{3}}{V_{2}^{\frac{1}{2}-\frac{\xi_{1}}{1+\xi_{1}}}}\), \(\Theta_{6}=\frac{\theta_{6}}{V_{2}^{\frac{1}{2}-\frac{\xi_{1}}{1+\xi_{1}}}}\). Since the effects of the parameters are different in each cases, the analysis is divided into two cases: \(V_{2}\) near and far from the origin. When \(V_{2}\) converge to zero, the dynamics of \(\beta\), \(\Theta_{3}\) and \(\Theta_{6}\) can be  
\[ \begin{cases}\frac{d\sin\beta}{dt}=-\kappa_{2}\cos^{2}\beta\sin \beta+\frac{\kappa_{1}}{2}\cdot\Theta_{6}\Big{(}\cos\beta\Big{)}^{\frac{2\xi_ {1}}{1+\xi_{1}}},\\ \dot{\Theta}_{3}=\Theta_{6}-\kappa_{2}\epsilon_{1}\Theta_{3}\sin^{2}\beta,\\ \dot{\Theta}_{6}=-\kappa_{3}\Theta_{3}-\epsilon_{2}\Theta_{6}+\epsilon_{3}\sin t.\end{cases}  \tag{26}\]  
where  
\[\epsilon_{1}=\frac{1}{2}-\frac{\xi_{1}}{1+\xi_{1}},\ \ \ \ \ \ \epsilon_{2}= \kappa_{4}-\kappa_{2}\epsilon_{1}\sin^{2}\beta, \tag{27}\] \[\epsilon_{3}=\lambda V_{2}(0)^{(\xi_{1}+\xi_{2}-1)}\epsilon^{(1- \xi_{1}-\xi_{2})\epsilon_{s}},\ \ \ \epsilon_{s}=\int_{0}^{t}[\sin\beta(s)]^{2}ds. \tag{28}\]  
Since \(V_{2}\) converges to zero and \(V_{2}=V_{2}(0)\epsilon^{\epsilon_{s}}\), \(\epsilon_{s}\) must be infinite. Define the Lyapunov function \(V_{3}\) as  
\[V_{3}=\frac{\kappa_{3}}{2}\Theta_{3}^{2}+\frac{1}{2}\Theta_{6}^{2}+\Theta_{6} \Theta_{3}. \tag{29}\]  
We can then have  
\[\dot{V}_{3}= -(\kappa_{3}+\kappa_{2}\kappa_{3}\epsilon_{1}\Theta_{5}^{2}) \Theta_{3}^{2}-(\epsilon_{2}-1)\Theta_{6}^{2}-(\kappa_{4}+\kappa_{2}\epsilon_ {1}\Theta_{5}^{2})\Theta_{3}\Theta_{6}+(\Theta_{6}+\Theta_{3})\epsilon_{3}\sin t \tag{30}\] \[\leq -\epsilon_{6}V_{3}+\epsilon_{3}\epsilon_{5}\sqrt{V_{3}}\]where  
\[\epsilon_{4}=\kappa_{4}-\kappa_{2}-1-\frac{(\kappa_{2}+\kappa_{4})^{2}}{2\kappa_{3} },\quad\epsilon_{5}=\sqrt{2}\Bigg{(}\min\left\{\frac{1}{4},\frac{\kappa_{3}-2}{ 2}\right\}\Bigg{)}^{-\frac{1}{2}},\quad\epsilon_{6}=\min\Bigg{\{}\frac{\kappa_ {3}}{\kappa_{3}-2},4\epsilon_{4}\Bigg{\}}. \tag{31}\]  
Since \(\kappa_{4}>\kappa_{2}+1\) and \(\kappa_{3}>\max\{2,\frac{(\kappa_{2}+\kappa_{4})^{2}}{2(\kappa_{4}-\kappa_{2} -1)}\}\), \(\epsilon_{4}\), \(\epsilon_{5}\) and \(\epsilon_{6}\) are positive. If \(\xi_{1}+\xi_{2}>1\), we can have \(\lim\limits_{t\to\infty}\epsilon_{3}\sin t=0\), which implies that  
\[\dot{V}_{3}\leq-\epsilon_{6}V_{3}+o_{1}\cdot\sqrt{V_{3}} \tag{32}\]  
where \(\lim\limits_{t\to\infty}o_{1}(t)=0\). This means that \(\lim\limits_{t\to\infty}\Theta_{6}=0\), hence the dynamics of \(\beta\) can be  
\[\frac{d\sin\beta}{dt}=-\kappa_{2}\cos^{2}\beta\sin\beta+\frac{\kappa_{1}}{2} \Big{(}\cos\beta\Big{)}^{\frac{2\xi_{1}}{1+\xi_{1}}}\cdot o_{2}. \tag{33}\]  
where \(\lim\limits_{t\to\infty}o_{2}(t)=0\). Therefore, \(\beta\) converges to zero implying a slow convergence rate of \(V_{2}\). If \(\xi_{1}+\xi_{2}<1\), we can have \(\lim\limits_{t\to\infty}\epsilon_{3}(t)=+\infty\). This means that  
\[\frac{d\sin\beta}{dt}=-\kappa_{2}\cos^{2}\beta\sin\beta+\frac{\kappa_{1}}{2} \Big{(}\cos\beta\Big{)}^{\frac{2\xi_{1}}{1+\xi_{1}}}\cdot f(t), \tag{34}\]  
which \(f(t)\) is periodically oscillating between \((-\infty,+\infty)\).