# Adaptive Neural Network Control of Zero-Speed Vessel Fin Stabilizer Based on Command Filter
## 3 Controller Design and Stability Analysis
### Stability Analysis
Then, the time derivative of \(V_{s,1}\) is  
\[ \begin{split}&\dot{V}_{s,1}\leq-(k_{1}-0.5)s_{1}^{2}+0.5\tilde{ \[PERSON]}^{2}\\ &\leq-(2k_{1}-1)V_{s,1}+0.5\tilde{\[PERSON]}^{2}\end{split}  \tag{63}\]  
where \(k_{1}>0.5\). Solving (63), we have  
\[ \begin{split}& V_{s,1}(t)=\bigg{[}V_{s,1}(0)-\frac{0.5\tilde{ \[PERSON]}^{2}}{2k_{1}-1}\bigg{]}e^{-(2k_{1}-1)t}+\frac{0.5\tilde{\[PERSON]}^{2}}{2 k_{1}-1}\\ &\leq\frac{0.5\tilde{\[PERSON]}^{2}}{2k_{1}-1}+V_{s,1}(0)e^{-(2k_{1 }-1)t}\end{split}  \tag{64}\]  
where \(V_{s,1}(0)\) is the initial value of \(V_{s,1}\). Because \(\lim\limits_{t\to\infty}e^{-(2k_{1}-1)t}=0\), \(s_{1}\) converges to the compact set \(\Omega_{s}=\big{\{}s_{1}\big{|}\big{|}s_{1}\big{|}\leq\tilde{\Theta}/\sqrt{2k_{ 1}-1}\big{\}}\) when \(t\to\infty\). It should be noted that the positive number \(\tilde{\Theta}\) is only used for stability analysis and not for controller design, so only \(\tilde{\Theta}\) is required to exist. Next, in order to obtain the bounds of variables \(\lambda_{1}\), \(\lambda_{2}\), a Lyapunov function \(V_{\lambda}=0.5\lambda_{1}^{2}+0.5\lambda_{2}^{2}\) is constructed, and the time derivative is as follows:  
\[ \begin{split}&\dot{V}_{\lambda}=\lambda_{1}\dot{\lambda}_{1}+ \lambda_{2}\dot{\lambda}_{2}\\ &=\lambda_{1}\lambda_{2}-\phi_{1}\lambda_{1}^{2}+\hat{b}\Delta u \lambda_{2}-\phi_{2}\lambda_{2}^{2}\\ &\leq-(\phi_{1}-0.5)\lambda_{1}^{2}-(\phi_{2}-0.5-0.25\hat{b}^{2 })\lambda_{2}^{2}+|\Delta u|^{2}\\ &\leq-\varphi_{1}V_{\lambda}+|\Delta u|^{2}\end{split}  \tag{65}\]  
where \(\varphi_{1}=\min\Big{\{}2\phi_{1}-1,2\phi_{2}-1-0.5\hat{b}^{2}\Big{\}}\), \(\phi_{1}>0.5\), \(\phi_{2}>0.5+0.25\hat{b}^{2}\). Solving (65), one can get  
\[V_{\lambda}(t)\leq\frac{|\Delta u|^{2}}{\varphi_{1}}+\bigg{[}V_{\lambda}(0)- \frac{|\Delta u|^{2}}{\varphi_{1}}\bigg{]}e^{-\varphi_{1}t} \tag{66}\]  
Since \(\lambda(0)=0\), the initial value of \(V_{\lambda}\) is \(V_{\lambda}(0)=0\). According to \(V_{\lambda}=0.5\lambda_{1}^{2}+0.5\lambda_{2}^{2}\) and (66), we have  
\[|\lambda_{1}|\leq\sqrt{\frac{2}{\varphi_{1}}}|\Delta u| \tag{67}\]\[|\lambda_{2}|\leq\sqrt{\frac{2}{\varphi_{1}}}|\Delta u| \tag{68}\]  
From (26) and (28), one can get  
\[|x_{1}-\alpha_{0}|=|z_{1}+s_{1}+\lambda_{1}|\leq|z_{1}|+|s_{1}|+|\lambda_{1}| \tag{69}\]  
where \(\alpha_{0}=y_{d}\). Combining (61), (64), (67) and (69), there is  
\[\lim_{t\rightarrow\infty}|x_{1}-\alpha_{0}|\leq\sqrt{\frac{2M}{c}}+\frac{ \tilde{\Theta}}{\sqrt{2k_{1}-1}}+\sqrt{\frac{2}{\varphi_{1}}}|\Delta u| \tag{70}\]  
According to (28), (62), (68) and \(\dot{s}_{2}\)= \(-k_{2}s_{2}\), we have  
\[\lim_{t\rightarrow\infty}|x_{2}-\alpha_{1}|\leq\sqrt{\frac{2M}{c}}+\sqrt{ \frac{2}{\varphi_{1}}}|\Delta u| \tag{71}\]  
Therefore, from (70) and (71), we know that \(x_{1}-\alpha_{0}\) and \(x_{2}-\alpha_{1}\) are bounded. According to \(x_{1}=\varphi\), \(\alpha_{0}=y_{d}\) and (70), it can be obtained that when appropriate design parameters \(k_{1}\), \(k_{2}\), \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\) and \(L\) are selected, \(|\varphi-y_{d}|\) can converge to a very small value. At the same time, the error \(\varphi-y_{d}\) satisfies (59). Theorem 1 has been proved.