# Stabilization of Underactuated Surface Vessels: A Continuous Fractional Power Control Method
## 3 Main Results
### A Continuous Fractional Power Control Framework  
The traditional time-varying control laws in previous works can guarantee all the states of system (9) to converge to zero. However, with these control laws, high-order nonlinear terms may appear in the closed loop system dynamics, partly due to the fact that \(\dot{\theta}_{2}=\theta_{5}\cdot\vartheta_{6}\). This means when the magnitude of states \(\theta_{5}\) and \(\vartheta_{6}\) is less than one near the origin, the convergence rate of \(\vartheta_{2}\) will become negligible. To this end, we present the following control method, which allows us to improve the system performance by using arbitrary order state feedback control inputs. \[ \begin{cases}\sigma_{1}=-\kappa_{1}sign(\vartheta_{2})|\vartheta_{2}|^{\xi_{ 1}}\vartheta_{6}-\kappa_{2}\vartheta_{5},\\ \sigma_{2}=-\kappa_{3}\vartheta_{3}-\kappa_{4}\vartheta_{6}+\lambda sign( \vartheta_{2})|\vartheta_{2}|^{\xi_{2}}\sin t,\end{cases}  \tag{10}\]  
where \(\kappa_{1}>0\), \(\kappa_{2}>0\), \(\kappa_{3}>0\), \(\kappa_{4}>0\), \(\xi_{1}>0\), \(\xi_{2}>0\) and \(\lambda\
eq 0\) are the constant parameters satisfying \(\kappa_{4}>\kappa_{2}+1\) and \(\kappa_{3}>\max\{2\), \(\frac{(\kappa_{2}+\kappa_{4})^{2}}{2(\kappa_{4}-\kappa_{2}-1)}\}\). Here, \(\xi_{1}=\frac{p_{1}}{q_{1}}\) and \(\xi_{2}=\frac{p_{2}}{q_{2}}\), where \(p_{1}\), \(q_{1}\), \(q_{2}\) are positive odds and \(q_{2}\) is positive even. With this fractional power controller, we have the following result. **Theorem 1**. : _System (9) can be globally asymptotically stabilized with the fractional power controller in (10). Namely, the states \(\vartheta_{2},\vartheta_{3},\vartheta_{5}\) and \(\vartheta_{6}\) converge to zero globally and asymptotically._  
Proof. : Substituting (10) to (9), one can have  
\[ \begin{cases}\dot{\vartheta}_{2}=\vartheta_{5}\vartheta_{6},\ \ \dot{\vartheta}_{3}=\vartheta_{6},\\ \dot{\vartheta}_{5}=-\kappa_{1}sign(\vartheta_{2})|\vartheta_{2}|^{\xi_{1}} \vartheta_{6}-\kappa_{2}\vartheta_{5},\\ \dot{\vartheta}_{6}=-\kappa_{3}\vartheta_{3}-\kappa_{4}\vartheta_{6}+\lambda| \vartheta_{2}|^{\xi_{2}}\sin t.\end{cases}  \tag{11}\]  
Consider the following Lyapunov function  
\[V_{1}=\frac{\kappa_{1}}{1+\xi_{1}}\mid\vartheta_{2}\mid^{1+\xi_{1}}+\frac{1}{ 2}\vartheta_{5}^{2}\geq 0, \tag{12}\]  
whose derivative satisfies  
\[\dot{V}_{1}=-\kappa_{2}\vartheta_{5}^{2}\leq 0. \tag{13}\]  
The Lyapunov function \(V_{1}\) is monotonically decreasing for \(\vartheta_{5}\
eq 0\) and bounded, there must be a minimum value of \(V_{1}\). Since \(\vartheta_{2}\) and \(\vartheta_{5}\) are bounded, and the dynamics of \(\vartheta_{3}\) and \(\vartheta_{6}\),  
\[ \begin{cases}\dot{\vartheta}_{3}=\vartheta_{6},\\ \dot{\vartheta}_{6}=-\kappa_{3}\vartheta_{3}-\kappa_{4}\vartheta_{6}+\lambda |\vartheta_{2}|^{\xi_{2}}\sin t.\end{cases}  \tag{14}\]  
It is easy to know that states \(\vartheta_{3}\) and \(\vartheta_{6}\) are bounded. Take the derivative of \(\dot{V}_{1}\),  
\[\ddot{V}_{1}=-2\kappa_{2}\vartheta_{5}(-\kappa_{1}sign(\vartheta_{2})|\vartheta _{2}|^{\xi_{1}}\vartheta_{6}-\kappa_{2}\vartheta_{5}). \tag{15}\]Since states \(\theta_{2}\) and \(\theta_{5}\) are bounded, then \(\dot{V}_{1}\) and \(\dot{V}_{1}\) are continuous and bounded. This implies that \(\dot{V}_{1}\) is uniformly continuous, according to the Lemma 3, we can have that \(\lim\limits_{t\rightarrow+\infty}\theta_{5}=0\). Define positive odd numbers \(p_{3}\) and \(q_{3}\) as \(\dfrac{p_{3}}{q_{3}}-\xi_{1}>2\). Since the state \(\dot{\theta}_{2}\) is bounded and \(\lim\limits_{t\rightarrow+\infty}\theta_{5}=0\), there must be a positive real number \(K_{1}=\xi_{1}+\dfrac{p_{3}}{q_{3}}\) satisfying that \(\lim\limits_{t\rightarrow+\infty}\theta_{2}^{K_{1}-\xi_{1}}\dot{\theta}_{5}=0\). Define \(\varGamma_{1}=\theta_{2}^{K_{1}-\xi_{1}}\dot{\theta}_{5}\), according to Appendix A, \(\dot{\varGamma_{1}}\) is uniformly continuous.