## 4 Experiments and Discussions
### Sliding Form Controller for Ship Dynamic Positioning System
#### 4.1.2 Terminal Sliding Mode Control  
Terminal sliding mode control (TSMC) is a control theory method commonly used for stability and robustness control of nonlinear systems. The goal of terminal sliding mode control is to achieve stable control of the system by guiding the system state trajectory to a specific surface called the terminal sliding mode. It is well known that certain non-smooth characteristics in dynamics can be utilized to achieve superior performance. For instance, research by [PERSON] in 1988 demonstrated that introducing a terminal attractor (i.e., first-order dynamics with fractional power) in neural learning rules results in finite-time convergence. Inspired by this, the concept of terminal sliding mode control (TSMC) was proposed in 1992 by [PERSON] and [PERSON] for second-order systems and later extended to handle higher-order single-input single-output (sIsO) systems and a class of multi-input multi-output (MIMO) systems in the studies of [PERSON] and [PERSON], among others. Leveraging the finite-time convergence property, TSMC, as highlighted by [PERSON], can amplify control forces to expedite convergence [13]. In the early 21 st century, a series of theoretical breakthroughs were achieved. [PERSON], [PERSON], and [PERSON] [14] introduced the nonsingular TSMC to address singularity issues in TSMC systems. Additionally, a fast TSMC was developed to accelerate the convergence of basic TSMC for higher-order SISO systems. Continuous TSMC was also introduced for robot control. Since then, the theory and applications of TSMC have gained significant momentum, experiencing substantial growth in publications, with an annual increase of over 20% in the past five years according to Google Scholar's data [14]. 1.3 Terminal Sliding Mode Control Design for Dynamic Positioning of Dual-Purpose Ship for Intelligent Research and Training  
The mathematical model of the ship with three degrees of freedom derived from the previous is  
\[\left\{ \begin{array}{c}\dot{\eta}=R(\psi)v\\ M\dot{v}+Dv=\tau+d\end{array} \right. \tag{30}\]  
Let the error of the system be \(e_{2}\) and the speed error be \(\dot{e}_{2}\), to obtain  
\[\left\{ \begin{array}{c}e_{2}=\eta-\eta_{d}\\ \dot{e}_{2}=\dot{\eta}-\dot{\eta}_{d}\end{array} \right. \tag{31}\]  
Combined with the three-degree-of-freedom mathematical model, the following can be obtained:  
\[\ddot{\eta}=\left(MR^{-1}\right)^{-1}\left(\tau-\dot{\eta}\left(M\dot{R}^{-1} +DR^{-1}\right)\right) \tag{32}\]  
Define the end die face:  
\[s_{2}=\dot{e}_{2}+\lambda sig(e_{2})^{a}  
1.4 Simulation of Terminal Slip Film Controller for Power Positioning of Intelligent Research and Training Dual Purpose Vessels  
The three-degree-of-freedom model of the dual-use ship for intelligent research and training at Dalian Maritime University is combined with the terminal sliding film controller to build the ship power positioning terminal sliding mode control system in Figure 3. The * in the figure is the multiplication sign [15]  
The dual-purpose ship for intelligent research and training of Dalian Maritime University was used as the simulation object to carry out the simulation. The initial position of the ship was set to [0 m, 0 m, 0\({}^{\circ}\)], and the desired position was set to [80 m, 20 m, 10\({}^{\circ}\)]. The parameter settings of the controller were \(\mathcal{C}_{1}=1,k_{1}=5,\xi_{1}=3,\text{and}\ \delta=30\). The simulation time was 600 s.  
From Figure 4, it can be seen that the speed of the ship gradually increases from 0 and then gradually decreases, indicating that the speed control of controlled positioning  
Figure 3: Dynamic positioning terminal sliding mode control system simulation model. Figure 2: Terminal sliding mode controller simulation model. is ideal. From Figure 5, it can be seen that the ship's position reaches the desired position from the initial position after a period of time under the control of the terminal sliding film controller, and the process of controlled positioning is precise, smooth, and fast.