# Event-Sampled Adaptive Neural Course Keeping Control for USVs Using Intermittent Course Data
## 4 Control Design
### Simulations  
In the simulation, the model parameters of the Norbbin model are \(h=-0.00463\), \(a_{1}=1\), \(a_{3}=30\), \(b=0.00221\)[12]. The reference course angle \(y_{d}\) is generated using the following dynamic equation \(y_{d}=8 sin(0.015t)\), and the disturbance is chosen as \(d=2\Big{(}sin(0.2t)+cos(0.3t)\big{)}\). The node number of RBF-NN for \(H(\chi_{2},\tau)\) and \(L\) is set as \(20\), the base function centers are evenly spaced in the range \([-2\times 2]\times[-2\times 2]\) and all RBF-NN base function widths are set as \(1\). In addition, the initial states and design parameters are shown in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline
**Index** & **Items** & **Value** \\ \hline \multirow{4}{*}{Initial states} & \(\psi(0)\) & \(-1\) \\  & \(\hat{\chi}_{1}\) & \(0\) \\  & \(\hat{\chi}_{2}\) & \(0\) \\  & \(\hat{\theta}\) & \([0,0]\) \\ \hline \end{tabular}
\end{table}
Table 1: The values of initial states and design parameters.  
Figure 6: The actual control input.  
Figure 8: Sampled event.  
Figure 7: The varying of adaptive parameter \(\hat{\theta}\).  
Figure 9: Triggered event.  
**Remark 11**.: _In this work, the whole CLCS involves several units, such as ESM, NN-SO, and ETM, which leads to a number of design parameters. As a result, the simulation presents some challenges. To address this, we initially determined the design parameters \(K_{1}\) and \(K_{2}\) through trial and error. It should be pointed out that \(kk_{1}-\iota-k>0\), \(k_{2}-2\iota-2k>0\) and \(\eta>2+(2a)^{-1}\) should be met. Subsequently, one adjusts \(K_{1}\) and \(K_{2}\) to achieve system stability. Following this, one adjusts the other design parameters to meet specific control performance objectives, such as satisfying tracking performance and NN-SO estimation performance. Extensive numerical simulations indicate that, initially, the selection of \(K_{1}\) and \(K_{2}\) is a significant challenge, which relies on the experience of the designer. The relatively large values of \(K_{1}\) and \(K_{2}\) can improve the control performance, but they should not be too large. Otherwise, this causes system oscillation. In the initial phase, it is advisable to endow appropriate values for \(K_{1}\) and \(K_{2}\). And then, according to the control performance, one should increase or decrease the value size appropriately until satisfactory results are achieved. The final tracking performance is influenced by each of the design parameters. When determining these design parameters, it is important to not only adjust one or two parameters separately, but also to tune all the parameters together._  
The simulation results under our proposed method are shown in Figures 2-9. Specifically, Figure 2 shows that the proposed method is able to force USVs to track the desired course \(y_{d}\) despite equivalent disturbance. Figure 3 presents the varying of the course tracking error. Figures 4 and 5 show the observation performance of the proposed observer for state variables, which is able to achieve the refactoring of the absence of course state data. Figure 6 plots the changing curve of the control input \(\tau\), which indicates that the actual steering angle is bounded and reasonable, and that the control input cannot be triggered infinitely within a finite time interval, i.e., the Zeno behavior has been avoided. Figure 7 plots the changing curve of adaptive parameter \(\hat{\theta}\), which indicates that the \(\hat{\theta}\) is bounded. Figure 8 plots the event-sampled instants and times, from which one can understand that the maximal sampling instant is about 5.5 s. Figure 9 shows the event-triggered instant and times in control input channel, from which one can understand that the maximal sampling instant is about 27 s. According to the statistics, the event numbers of sampled events and triggered events are 1251 and 270, respectively. These above results demonstrate that all signals in the CLCS are bounded and that the Zeno behavior has been avoided successfully. Thus, Theorems 1 and 2 are proven.