# Event-Sampled Adaptive Neural Course Keeping Control for USVs Using Intermittent Course Data
## 3 Observer Design
### Stability Analysis of State Observer Design  
**Theorem 1**.: _For the CKC system (1), under Assumptions 1-3, the NN-based observer (10) with the adaptive law (11) can estimate the states of the system (1) if the design parameters \(\iota\), \(k\), \(\eta\), \(k_{1}\), \(k_{2}\), and \(\Lambda\) can guarantee the following conditions:_  
\[\begin{cases}kk_{1}>\iota-k\\ k_{2}>2(l+k)\\ \eta>2+0.5a^{-1}\end{cases} \tag{14}\]  
Proof.: Consider the following Lyapunov function for the NN-based state observer  
\[L_{V_{o}}=\frac{k}{2}\hat{\chi}_{1}^{2}+\frac{1}{2}\hat{\chi}_{2}^{2}+\frac{1 }{2}\tilde{\vartheta}^{T}\Lambda^{-1}\tilde{\vartheta} \tag{15}\]  
Differentiating \(L_{V_{0}}\) and using (13), yield  
\[\dot{L}_{V_{0}}= -kk_{1}\tilde{\chi}_{1}^{2}-kk_{1}\hat{\chi}_{1}(\tilde{\chi}_{1 }-\chi_{1})-k_{2}\tilde{\chi}_{2}^{2}+(\tilde{\chi}_{2}-\tilde{\chi}_{1}) \tilde{\vartheta}^{T}\xi(\hat{s})+\eta\tilde{\vartheta}^{T}\tilde{\vartheta}+ \rho_{d}\tilde{\chi}_{2}\] \[-\tilde{\vartheta}^{T}\xi(\hat{s})(\tilde{\chi}_{1}-\chi_{1})+k_{ 2}\chi_{2}\tilde{\chi}_{2}+k_{1}k_{2}\tilde{\chi}_{1}\tilde{\chi}_{2}-(k-k_{1 }k_{2})(\tilde{\chi}_{1}-\chi_{1})\tilde{\chi}_{2} \tag{16}\]  
According to the ETM (3), one has \(|\tilde{\chi}_{1}-\chi_{1}|\geq\beta\) for \(\forall t\in[t_{f},t_{f+1})\). In addition, from the feature of the basis function \(\xi(\hat{s})\), one has \(\|\xi(\hat{s})\|\leq\sqrt{\iota}\). Furthermore, according to Assumption 2, one can obtain \(|\rho_{d}|\leq\varsigma\) with \(\varsigma\) being a constant. In practice, the steering \(\delta\) is a constraint, which implies that \(\chi_{2}\) satisfies \(|\chi_{2}|\leq\theta\) with \(\theta\) being a constant. Therefore, the following inequalities hold using [PERSON]'s inequality [25]:  
\[-kk_{1}\tilde{\chi}_{1}(\tilde{\chi}_{1}-\chi_{1})\leq\frac{kk_{1}}{2}\hat{ \chi}_{1}^{2}+\frac{kk_{1}}{2}\beta^{2} \tag{17}\]  
\[(\tilde{\chi}_{1}-\tilde{\chi}_{2})\tilde{\vartheta}^{T}\xi(\hat{s})\leq\frac {l}{2}\Big{(}\hat{\chi}_{2}^{2}+\hat{\chi}_{1}^{2}\Big{)}+\|\tilde{\vartheta}\| ^{2} \tag{18}\]  
\[\tilde{\vartheta}^{T}\xi(\hat{s})(\tilde{\chi}_{1}-\chi_{1})\leq\frac{1}{4a}\| \tilde{\vartheta}\|^{2}+a\mu\beta^{2} \tag{19}\]  
\[\rho_{d}\tilde{\chi}_{2}\leq\frac{1}{4}\hat{\chi}_{2}^{2}+4\varsigma^{2} \tag{20}\]  
\[\tilde{\vartheta}^{T}\tilde{\vartheta}\leq-\frac{1}{2}\|\tilde{\vartheta}\|^{2 }+\frac{1}{2}\|\vartheta\|^{2} \tag{21}\]\[k\tilde{\chi}_{1}\tilde{\chi}_{2}\leq\frac{k}{2}\tilde{\chi}_{1}^{2}+\frac{k}{2} \tilde{\chi}_{2}^{2} \tag{22}\]  
\[k_{1}k_{2}(\tilde{\chi}_{1}-\chi_{1})\tilde{\chi}_{2}\leq\frac{k_{2}}{4}\tilde{ \chi}_{2}^{2}+k_{2}k_{1}^{2}\beta^{2} \tag{23}\]  
where \(a\) is a design parameter.  
Using (17)-(23), (16) can be rewritten as  
\[\begin{split}\dot{L}_{V_{0}}\leq&-\frac{kk_{1}- \iota-k}{2}\tilde{\chi}_{1}^{2}-\frac{k_{2}-2\iota-2k}{4}\tilde{\chi}_{2}^{2}- \left(\frac{\eta-2}{2}-\frac{1}{4a}\right)\|\tilde{\vartheta}\|^{2}+\frac{ \eta}{2}\|\vartheta\|^{2}\\ &+4\varsigma^{2}+a\mu\beta^{2}+k_{2}k_{1}^{2}\beta^{2}+\frac{kk_{1 }}{2}\beta^{2}\\ \leq&-\varphi L_{V_{0}}+\varpi\end{split} \tag{24}\]  
where \(\varphi=\min\Bigl{\{}kk_{1}-\iota-k,\frac{k_{2}-2\iota-2k}{2},\Bigl{(}\eta-2 -\frac{1}{2\iota}\Bigr{)}\lambda_{\min}(\Lambda)\Bigr{\}}\) and \(\varpi=\frac{\eta}{2}\|\vartheta\|^{2}+4\varsigma^{2}+a\mu\beta^{2}+k_{2}k_{1} ^{2}\beta^{2}+\frac{kk_{1}}{2}\beta^{2}\). Recalling (24), one can obtain  
\[L_{V_{o}}\leq-\biggl{(}\frac{\varpi}{\varphi}-L_{V_{o}}(0)\biggr{)}\exp(-\varphi t )+\frac{\varpi}{\varphi} \tag{25}\]  
where \(L_{V_{o}}(0)\) is the initial value of \(L_{V_{0}}\). This implies that \(L_{V_{0}}\) is bounded and its upper bound can be adjusted to be as small as desired. Furthermore, from (15), one can understand that \(\tilde{\chi}_{1}\), \(\tilde{\chi}_{2}\), and \(\tilde{\vartheta}\) are bounded, which indicates that the NN-based state observer (10) can estimate the states of the CKC system (1) in real time as long as Equation (24) can be held. Thus, Theorem 1 is proved.