# Adaptive Neural Network Control of Zero-Speed Vessel Fin Stabilizer Based on Command Filter
## 4 Simulation  
This section takes a vessel as the simulation object to verify the effectiveness of the designed control strategy. The displacement of the vessel is 1458 t, the length between the two perpendiculars of the vessel is 98 m, the width of the vessel is 10.2 m, the draft depth of the vessel is 3.1 m, the fin force arm of the fin stabilizer is 3.46 m, the flooding angle of the vessel is 0.75 rad and the initial metacentric height of the vessel is 1.15 m. The experimental coefficients \(n_{1}\) and \(n_{2}\) varying with the vessel are 0.031 and 0.0152, respectively. The span of the fin stabilizer is 1.8 m, the chord length is 2.5 m, the thickness ratio is 0.2, and the density of water is set to 1020 kg/m\({}^{3}\). The working angle of the fin stabilizer at zero speed is between \(\pm 1.05\) rad and the maximum rotation speed is 0.79 rad/s. By using the above parameters for the calculation, the relevant parameters of the vessel rolling motion mathematical model can be obtained as follows: \(a_{1}=-0.7542\), \(a_{2}=1.3403\), \(a_{3}=-0.0171\), \(a_{4}=-0.0114\), \(c=-0.7542\). The design parameters in the simulation are as follows: The initial value of the roll angle is 0.6 rad, the initial value of the angular velocity is 0 rad/s, and the expected output \(y_{d}\) is 0 rad.The number of hidden layer nodes \(m\) of the RBF neural network is 30, the center point \(\zeta_{j}\) of the Gaussian function is evenly distributed on \([-2,2]\times[-2,2]\), the width \(\omega_{j}\) is 0.8, the learning rate \(\Gamma\) is \(0.00001I_{30\times 30}\), and \(I_{30\times 30}\) is an identity matrix with a dimension of 30. \(W(0)=0\), \(s_{1}(0)=0\), \(s_{2}(0)=0\), \(\lambda_{1}(0)=0\), \(\lambda_{2}(0)=0\), \(\sigma_{1}=0.5\), \(\sigma_{2}=0.000001\), \(\sigma_{3}=0.00001\), \(L=5\), \(\dot{\varphi}_{1}=1\), \(\dot{\varphi}_{2}=1.5\), \(\omega_{n}=100\), \(\zeta=0.9\), \(k_{1}=3\), \(k_{2}=60\), \(\alpha_{1}=-1.8\), \(\alpha_{1}^{0}=-1.8\), \(\alpha_{2}=0\) and the disturbance \(e\) is \(0.1 sin(0.5t)-0.11 cos(0.35t)+0.2 sin(0.4t)\). In order to further verify the superiority of the proposed control strategy in roll reduction, it is compared with the adaptive neural network control strategy without command filters. The virtual control law and control law are as follows:  
\[\alpha_{1}=-c_{1}z_{1} \tag{72}\]  
\[v=\frac{1}{b}(-c_{2}z_{2}-z_{1}-\hat{W}^{\rm T}\psi(x)+\dot{\alpha}_{1}) \tag{73}\]  
\[\hat{W}=\Gamma\big{(}z_{2}\psi(x)-\sigma\hat{W}\big{)} \tag{74}\]In simulation, \(c_{1}=0.5\), \(c_{2}=1\), \(b=-0.031\), \(\Gamma=0.00001I_{30\times 30}\), \(\sigma=0.005\). The other parameters are consistent with those in the scheme of this paper. The simulation results under the control proposed scheme in this paper and the compared control schemes are shown in Figure 1. In Figure 1, \(\varphi\), \(\alpha\), \(\hat{e}\), \(\omega\), and \(F_{z}\) are the simulation results of the control strategy designed in this paper, and \(\varphi^{*}\), \(\alpha^{*}\), \(\hat{e}^{*}\), \(\omega^{*}\), \(F_{z}^{*}\) are the simulation results of the same simulation objects in the compared control strategies, respectively. From Figure 1a, it can be seen that under both control strategies, the vessel's roll angle \(\varphi\) and \(\varphi^{*}\) can quickly approach the stabilization angle \(y_{d}\) in a short time and reach a stable state. However, after enlarging the image, it is obvious that the control scheme designed in this paper has a better roll reduction effect and smaller roll angle after the system is stable. In Figure 1b, the changes of fin angle \(\alpha\) and \(\alpha^{*}\) during operation are shown. It can be seen from the image that the fin angle of the contrast control scheme is larger than that of the control scheme in this paper at the same time, which is more biased in the positive direction of fin angle motion.