# Adaptive Neural Network Control of Zero-Speed Vessel Fin Stabilizer Based on Command Filter
## 2 Problem Formulation and Preliminaries
### Problem Formulation
According to (1)-(6), one can obtain  
\[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]1  
\[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]2  
\[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]3  
\[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]4  
\[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]5  
\[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]6  
\[I_{xx}+J_{xx}=\frac{DB^{2}}{g}(0.3085+\frac{0.0227B}{d}-\frac{0.0043L}{100})^{2} \tag{2}\]7  
Let \(\varphi=x_{1}\), \(\dot{\varphi}=x_{2}\), which yields \(x=\left[x_{1}\quad x_{2}\right]^{\mathsf{T}}=\left[\varphi\quad\dot{\varphi} \right]^{\mathsf{T}}\). Let \(u=F_{z}\), \(d_{e}=ca_{e}\). In practical engineering applications, according to factors such as the structure of the vessel's fin stabilizer and the physical limitations of the power system and other factors, it can be known that the rolling stability force generated on the fin stabilizer will be constrained by input saturation. The nonlinearity of input saturation is described as follows:  
\[u=u(v)=\left\{ \begin{array}{ll}u_{\max},&v>u_{\max}\\ v,&u_{\min}\leq v\leq u_{\max}\\ u_{\min},&v<u_{\min}\end{array} \right. \tag{17}\]  
where \(v\) is the command control input, \(u\) is the actual control input, and \(u_{\max}>0\) and \(u_{\min}<0\) are the known maximum control input and minimum control input, respectively. Then, we have  
\[\left\{ \begin{array}{ll}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f(x_{1},x_{2})+bu+d_{e}\\ y=x_{1}\end{array} \right. \tag{18}\]  
where \(f(x_{1},x_{2})=a_{1}x_{1}+a_{2}{x_{1}}^{3}+a_{3}x_{2}+a_{4}x_{2}|x_{2}|\). **Assumption 1**. : _The unknown external disturbance \(d_{e}\) and its derivative \(\dot{d}_{e}\) are bounded and satisfy \(|d_{e}|\leq d_{e}\), \(\left|d_{e}\right|\leq\bar{d}_{e}\). \(\bar{d}_{e}\) and \(\bar{d}_{e}\) are unknown positive constants._  
**Assumption 2**. : \(f(x_{1},x_{2})\) _and \(b\) are unknown, but the sign of \(b\) is known._  
**Assumption 3**. : _The (10) is input-to-state stable (ISS)._  
**Remark 1**. : _Assumption 1 is formulated to ensure the feasibility of the designed control strategy, and a similar assumption is made in [35]. The inertia matrix in the vessel mathematical model can be obtained by semi-physical experiments, but the coefficients of parameters in the model cannot be obtained in this way. The vessel's mathematical model is vulnerable to disturbance, and model oscillation occurs. Therefore, \(f(x_{1},x_{2})\) and \(b\) are unknown. Since the control gain has an upper bound of 0, the sign of \(b\) is known. In the case of input saturation, the stability of the device must be ensured in order to satisfy the global stability, so Assumption 3 is reasonable [37; 38]._  
The control objective of this paper is to find an adaptive neural network control law for the zero-speed vessel fin stabilizer control system (10), so that the output \(\varphi\) approaches the expected value \(y_{d}\) and ensures that all signals in the closed-loop control system are uniformly bounded.