## 4 Controller Design
According to USV dynamics, the time derivative of \(\psi_{i,e}\) is  
\[\dot{\psi}_{i,e}=r_{i}-\dot{\psi}_{ij,a}. \tag{60}\]  
The virtual control \(\beta_{i,3}\) is chosen as:  
\[\beta_{i,3}=-k_{i,2}\psi_{i,e}+\dot{\psi}_{ij,a}, \tag{61}\]  
where \(k_{i,2}\) is a positive constant. Step 3: Error functions are defined as follows:  
\[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]0 \[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]1 (63) \[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]2 (64) \[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]3  
where \(\hat{z}_{i}=[\hat{a}_{i},\hat{\rho}_{i}]^{T}\); \(\tilde{\beta}_{i}=[\hat{\beta}_{i,1},\hat{\beta}_{i,2}]^{T}\); \(\gamma_{1}\) is a positive constant; \(\tilde{\beta}_{i,b}\), \(h=1,2,3\), is derived from the first-order filter \(l_{i,h}\hat{\beta}_{i,h}+\hat{\beta}_{i,h}=\beta_{i,h};l_{i,h}>0\) is a constant; \(\bar{\kappa}_{i}=[\alpha_{i,1},\,\tanh(\alpha_{i,2})/\gamma_{1}]^{T}\)\(\alpha_{i,h}\), \(h=1,2,3\), is the auxiliary variable derived to deal with the underactuated problem and input saturation. \(\omega_{i}=[\omega_{i,1},\omega_{i,2}]^{T}\), \(\omega_{i,h}\), \(h=1,2,3\), is the error caused by the first order filter. The updated laws of extra variables are given by  
\[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]4 \[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]5 (67) \[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]6  
where \(T_{u}\), \(k_{i,4}\) and \(T_{r}\) are positive constants. Step 4: A control law at the kinetic level is designed as  
\[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]7 \[\hat{\Theta}_{i}=-K_{\Theta}\hat{\Theta}_{i}-\tilde{P}_{ij}, \tag{54}\]8where \(k_{i,3}\) and \(k_{i,5}\) are positive constants. **Remark 5**. : _In the common continuous control schemes, the controller design has already been completed in Step 4. Here, an event-triggered controller is designed, and the event-triggered condition is be established in the following step. During the flow period between two successive triggering instants, a zero-order holder is used to keep input signals unchanged. The key to a successful ETC design is to choose an appropriate event-triggered strategy. In [23], the triggered condition of a time-varying threshold event-triggered controller (TTETC) is constructed based on flow time and the Lyapunov function. Therefore, the action times of the actuators are limited, and Zeno behavior can be avoided. The signals calculated by controller should be closely related to the triggered condition, its size change determines whether or not to update the control inputs and the change is used to design the proposed triggered condition. It is necessary to provide proof that Zeno behavior is avoided. The appropriate event-triggered strategy reduces the action times of the actuators._  
An event-triggered controller is chosen as  
\[\left\{ \begin{array}{ll}\tau_{i,uc}(t)=\omega_{\tau n}(t_{k}^{u}),&\forall t \in[t_{k}^{u},t_{k+1}^{u})\\ \tau_{i,rc}(t)=\omega_{\tau r}(t_{k}^{r}),&\forall t\in[t_{k}^{r},t_{k+1}^{r}) \end{array} \right. \tag{71}\]  
where  
\[\left\{ \begin{array}{ll}t_{k}^{u}=\inf\{t\in\mathbb{R}||e_{u}(t)|\geq a_{1} \},&t_{1}^{u}=0\\ t_{k}^{r}=\inf\{t\in\mathbb{R}||e_{r}(t)|\geq a_{2}\},&t_{1}^{r}=0\\ e_{u}=\omega_{\tau u}-\tau_{i,uc},&e_{r}=\omega_{\tau r}-\tau_{i,rc},\end{array} \right. \tag{72}\]  
and \(a_{1}\) and \(a_{2}\) are positive constants. The following theorem is provided to demonstrate the stability of the overall closed-loop system. **Theorem 3**. : _Consider a system consisting of USV dynamics ((7) and (8)), an observer ((15)-(17)), auxiliary variables ((66)-(68)), a control law (71) and environmental disturbances under Assumptions 1-3. Then, the proposed control scheme guarantees that (1) all signals in the closed-loop system are bounded and (2) all USVs can track the leader with a bounded tracking error._  
Proof.