# Adaptive Neural Network Control of Zero-Speed Vessel Fin Stabilizer Based on Command Filter
## 3 Controller Design and Stability Analysis
### Stability Analysis  
Based on the above analysis, the main conclusions are given in the form of a theorem. **Theorem 1**. : _Considering the Assumptions 1-3,the control law (40) based on the virtual control law (32) and the parameter adaptation law (41)-(43) can ensure that the vessel fin stabilizer control system described by (10) has the following properties:_  
1. _The closed-loop control system is stable and all signals of the closed-loop control system are ultimately uniformly bounded;_
2. _When appropriate design parameters_ \(k_{1}\)_,_ \(k_{2}\)_,_ \(\sigma_{1}\)_,_ \(\sigma_{2}\)_,_ \(\sigma_{3}\) _and_ \(L\) _are selected, the error_ \(\varphi-y_{d}\) _between the actual roll angle_ \(\varphi\) _of the vessel roll control system and the expected roll angle_ \(y_{d}\) _can converge to a small residual set;_
3. _Under the influence of input saturation, the error_ \(\varphi-y_{d}\) _satisfies:_ \[\lim_{t\to\infty}|\varphi-y_{d}|\leq\sqrt{\frac{2M}{c}}+\frac{\bar{\Theta}}{ \sqrt{2k_{1}-1}}+\sqrt{\frac{2}{\varphi_{1}}}|\Delta u|\] (59) _where_ \(\varphi_{1}>0\) _and_ \(\bar{\Theta}>\bigl{|}\alpha_{1}-\alpha_{1}^{0}\bigr{|}\)_. The boundedness of_ \(\alpha_{1}\) _and_ \(\alpha_{1}^{0}\) _will be obtained through the following proof process._  
Proof. : Solving (58), one has  
\[V_{2}(t)\leq\frac{M}{c}+\Bigl{[}V_{2}(0)-\frac{M}{c}\Bigr{]}e^{-ct}\leq\frac{ M}{c}+V_{2}(0)e^{-ct} \tag{60}\]  
where \(V_{2}(0)\) is the initial value of \(V_{2}\). According to (60), we know that \(V_{2}\) is bounded. Then, from (47), \(\tilde{z}_{1}\), \(\tilde{z}_{2}\), \(\tilde{W}\), \(\tilde{e}\), \(\bar{\beta}\) and \(\tilde{b}\) are also bounded. Because of \(\lim_{t\to\infty}e^{-ct}=0\), and according to (47), the following equations hold:  
\[\lim_{t\to\infty}|\tilde{z}_{1}|\leq\sqrt{\frac{2M}{c}} \tag{61}\] \[\lim_{t\to\infty}|\tilde{z}_{2}|\leq\sqrt{\frac{2M}{c}} \tag{62}\]  
Therefore, it can be concluded that \(|\tilde{z}_{1}|\) and \(|\tilde{z}_{2}|\) converge to the compact set \(\Omega_{\tilde{z}_{1}}=\{\tilde{z}_{1}||\tilde{z}_{1}|\leq\sqrt{2M/c}\}\) and \(\Omega_{\tilde{z}_{2}}=\{\tilde{z}_{2}||\tilde{z}_{2}|\leq\sqrt{2M/c}\}\) as \(t\to\infty\), respectively. Since\(\tilde{W}\) is bounded, it follows that \(\tilde{W}\) is also bounded. Under Assumptions 1-3 and \(e=d_{e}+\varepsilon\), both \(e\) and \(\tilde{e}\) are bounded. According to Theorem 2 in [41], the error variable \(z_{b,1}\), the virtual control law \(\check{\kappa}_{1}\) and its time derivative \(\dot{\kappa}_{1}\) are bounded (its boundedness can be proved by the backstepping design method, and the detailed proof process can be referred to the appendix in literature [35]), i.e., \(z_{1}-z_{b,1}\), \(\alpha_{1}-\check{\alpha}_{1}\), and \(\alpha_{2}-\check{\alpha}_{1}\) are also bounded. Thus, \(z_{1}\), \(\alpha_{1}\), \(\alpha_{2}\) are bounded. Meanwhile, it follows from the boundedness of (28), (37) and \(\check{z}_{2}\) that \(z_{2}\) is bounded. Based on the boundedness of \(z_{1}\), \(\check{z}_{1}\) and (28), it can be proved that the compensation signal \(s_{1}\) in (30) is bounded. According to (26) and Assumption 3, the state variable \(\lambda_{1}\) in (25) is bounded, and the state variable \(\alpha_{1}^{0}\) in (32) is bounded. Therefore, it can be proved that the state variable \(\lambda_{2}\) in (25) is bounded. In addition, the boundedness of control law \(v\) in (40) can also be determined. According to the input saturation, it can be determined that the actual control input \(u\) is bounded. From \(\Delta u=u-v\), we can know that \(\Delta u\) is also bounded. Through the above analysis, all signals of the closed-loop system are bounded. Let \(\Theta=\alpha_{1}-\alpha_{1}^{0}\); then, by the boundedness of \(\alpha_{1}\) and \(\alpha_{1}^{0}\), \(\Theta\) is also bounded, that is, when \(\forall t_{f}\in[0,t)\), \(|\Theta|\leq\tilde{\Theta}\) and \(\tilde{\Theta}>0\) are satisfied. According to (28) and (30), we get \(\dot{s}_{1}=-k_{1}s_{1}+\tilde{\Theta}\). Let \(V_{s,1}=0.5s_{1}^{2}\).