# Adaptive Neural Network Control of Zero-Speed Vessel Fin Stabilizer Based on Command Filter
## 2 Problem Formulation and Preliminaries  
### Problem Formulation  
In the application of vessel engineering, the mathematical models of vessel rolling motion mostly use the vessel rolling linear model based on the Conolly theory, which can better describe the vessel rolling motion when the vessel roll angle is small. However, when the vessel roll angle is too large, it will cause a nonlinear change of the parameters of the vessel rolling mathematical model and a nonlinear change of the rolling moment, which makes the vessel rolling linear model unable to accurately reflect the vessel motion. Therefore, this paper uses the nonlinear rolling mathematical model to describe the vessel rolling motion. The model is as follows [1]:  
\[(I_{xx}+J_{xx})\ddot{\varphi}+\delta_{N}\dot{\varphi}+\delta_{W}\dot{\varphi}| \dot{\varphi}|+Dh\varphi\Big{[}1-(\varphi/\varphi_{\varphi})^{2}\Big{]}=M_{c}+ M_{W} \tag{1}\]  
\[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]  
\[\delta_{N}=2n_{1}\sqrt{Dh(I_{xx}+J_{xx})}/\pi \tag{3}\]  
\[\delta_{W}=3n_{1}(I_{xx}+J_{xx})/4 \tag{4}\]  
\[M_{c}=-2L_{f}F_{z} \tag{5}\]  
\[M_{W}=-Dha_{e} \tag{6}\]  
where \(\varphi\) is the roll angle of the vessel, \(I_{xx}\) and \(J_{xx}\) are the inertia moment of the vessel and the additional inertia moment of the vessel, \(\delta_{N}\) and \(\delta_{W}\) are the damping coefficients, \(D\) is the displacement of the vessel, \(h\) is the initial metacentric height of the vessel, \(\varphi_{\varphi}\) is the inlet angle of the vessel, \(M_{c}\) is the control moment of the vessel's fin stabilizer and \(M_{W}\) is the wave moment acting on the vessel. \(g\) is the acceleration of gravity, \(B\) is the width of the vessel, \(L\) is the length between the two perpendiculars of the vessel, \(d\) is the draft of the vessel, \(n_{1}\) and \(n_{2}\) are the experimental coefficients changing with the vessel, \(L_{f}\) is the righting arm, \(F_{z}\) is the rolling stability force generated on the fin stabilizer and \(a_{e}\) is the effective wave angle. When the vessel is at low speed down to zero, the zero-speed fin stabilizer obtains the lift force acting on the fin by rapidly and actively tapping the water, i.e., the rolling stability force. If we choose a position where the distance between the fin axis and the leading edge is \(1/5\) chord length, the lift generated on the fin stabilizer at zero speed at this time is:  
\[F_{z}=F_{1}+F_{2}\cos\alpha \tag{7}\]\[F_{1}=7.2(\gamma+{x_{0}})^{2}(v-x_{0})l\rho\omega\|\omega\|\sin\alpha\] \[+11.58(\gamma+{x_{0}})^{2}(v-x_{0})l\rho\omega\|\omega\|\cos\alpha \tag{8}\] \[F_{2}=(12\gamma^{2}v+v^{3})l\rho\omega\|\omega\|/3\] \[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]0  
where \(\gamma\) is the \(1/4\) fin chord length of the fin surface, \(v\) is the distance between the fin axis and the midpoint of the chord length, \(\omega\) is the angular velocity of the fin, \(\alpha\) is the angle of the fin, \(l\) is the span length of the fin type, \(\rho\) is the density of the fluid and \(\kappa\) is the thickness ratio of the fin type. When the vessel is at anchor, the anchor chain forces caused by the sea disturbance influence the vessel roll. However, this paper studies roll stability of a vessel at low speed down to zero, so the influence of anchor chain force is not considered. In addition, the hydrodynamic forces are omitted in Equations (7)-(9). However, these forces can be regarded as unknown dynamics which can be reconstructed by the adaptive neural network and the disturbance observer.