# Event-Sampled Adaptive Neural Course Keeping Control for USVs Using Intermittent Course Data
## 3 Observer Design  
### NN-Based State Observer Design  
In this subsection, to solve the issue of the constraint of bandwidth, a neural-network-based state observer using only intermittent course data is designed, and the stability analysis is also presented. The detailed processes are presented below. From (2), the nonlinear model of Norbbin can be rewritten as  
\[ \begin{cases}\dot{\chi}_{1}=\chi_{2}\\ \dot{\chi}_{2}=H(\chi_{2},\tau)+d\end{cases}  \tag{8}\]  
Here, \(H(\chi_{2},\tau)=f(\chi_{2})+b\tau\). According to Assumption 1, one can establish that \(H(\chi_{2},\tau)\) is unknown. In addition, recalling Lemma 1, one can see that \(H(\chi_{2},\tau)\) can be approximated by RBF-NN \(\theta^{*}T_{\xi}^{*}(x)\), i.e.,  
\[H(s)=\theta^{*T}\xi(s)+\varepsilon_{h} \tag{9}\]  
where \(s=[x^{T},\tau]^{T}\) and \(\varepsilon_{h}\) is the approximate error, which satisfies \(|\varepsilon_{h}|\leq\tilde{\varepsilon}_{h}\). Furthermore, one can obtain  
\[ \begin{cases}\dot{\chi}_{1}=\chi_{2}\\ \dot{\chi}_{2}=\theta^{*T}\tilde{\xi}(s)+d_{h}\end{cases}  \tag{10}\]  
where \(d_{h}=d+\varepsilon_{h}\). According to (10), design the following the NN-based state observer:  
\[ \begin{cases}\dot{\tilde{\chi}}_{1}=\hat{\chi}_{2}+k_{1}(\tilde{\chi}_{1}- \hat{\chi}_{1})\\ \dot{\mu}=\hat{\vartheta}^{T}\xi(s)+k(\tilde{\chi}_{1}-\hat{\chi}_{1})\\ \hat{\chi}_{2}=\mu+k_{2}(\hat{\chi}_{1}-\hat{\chi}_{1})\end{cases}  \tag{11}\]  
with the adaptive law  
\[\dot{\hat{\vartheta}}=\Lambda\Big{(}\tilde{\xi}^{T}(\hat{s})(\hat{\chi}_{1}- \hat{\chi}_{1})-\eta\hat{\vartheta}\Big{)} \tag{12}\]  
where \(\tilde{\chi}_{1}=\tilde{\psi}\) and \(\delta=[x^{T},\tau]^{T};\hat{\chi}_{1},\hat{\chi}_{2}\) and \(\hat{\vartheta}\) are, respectively, the estimates of \(\chi_{1},\chi_{2}\) and \(\vartheta\); \(k\), \(\eta\), \(k_{1}\), and \(k_{2}\) are the design parameters and \(\Lambda\) is a design matrix. **Remark 5**. : _The CKC of USVs under a network environment is susceptible to influences from various sources. Under the influence of a network environment and an external environment, the course and yaw data collected by the sensors of USVs are difficult to continuously transmit to the control center at the shore end. Then, the continuous transmission of course and yaw data is the key to ensuring the safety of the course of USVs. To do this, an observer is undoubtedly an effective tool. The observer can be used to improve the accuracy of the transmitted data, thus improving the control performance. At the same time, the observer can also indirectly obtain the signal of \(\dot{\psi}\) using the intermittent course data of the USV, which not only saves the cost of installing yaw sensors, but also greatly improves the efficiency of yaw data transmission. Nowadays, there are many kinds of observers, such as state observers, disturbance observers [27], sliding mode observers [28], extended state observers [29], and so on. All of these observers have the function of reconstructing the uncertain state signals inside the system, but they all have to be based on the assumption that the course and yaw are continuous, which undoubtedly makes the work lose effectiveness. Therefore, an NN-SO (11) is designed to resolve the lose effectiveness problem, which does not depend on continuous course data._  
**Remark 6**. : _In references [30; 31; 32; 33], several various observers are also designed to achieve the refactoring of the system state, in which the signal \(\hat{\chi}_{1}\) must be continuous. In addition, the control gain \(b\) must be known if the observer [30; 31; 32; 33] is employed. From (11) and (12), the difference between the proposed observer and the existing observer is that \(\hat{\chi}_{1}\) can be discontinuous, which implies that the proposed observer is more practical._Let \(\hat{\chi}_{1}=\chi_{1}-\hat{\chi}_{1}\), \(\hat{\chi}_{2}=\chi_{2}-\hat{\chi}_{2}\) and \(\tilde{\vartheta}=\vartheta-\hat{\vartheta}\).