## 3 Controller Design
The stabilizing function \(\beta_{i,3}\) is chosen as:  
\[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]  
where \(k_{i,2}\) is a positive constant. **Remark 5**. : _The virtual controllers (46) and (49) are composed of the error surfaces (20) and (21), the distributed vectors \(P_{i}\), and matrices \(\Xi_{i,1}\) and \(\Xi_{i,2}\). The link weights \(c_{ij}\) of the distributed error surfaces (20) and (21) depict the directed graph among USVs that satisfies Assumption 1. Thus, the proposed formation approach is in a fully distributed formation manner._  
**Step 3:** Differentiating (22)-(24)  
\[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\] \[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\] (51) \[\dot{r}_{i,e} =\zeta_{i,3}+(\tau_{i,rc}-\varpi_{i,r})/m_{i,33}-\dot{\beta}_{i,3 }-\gamma_{1}\frac{\dot{\alpha}_{i,3}}{\cosh^{2}(\alpha_{i,3})}. \tag{52}\]  
The update law of additional controls \(\alpha_{i,1}\), \(\alpha_{i,2}\) and \(\alpha_{i,3}\) is given by  
\[\dot{\alpha}_{i,1} =\cosh^{2}(\alpha_{i,1})\{-T_{u}\alpha_{i,1}-\varpi_{i,u}/m_{i,1 1}\}/\gamma_{1}, \tag{53}\] \[\dot{\alpha}_{i,2} =\cosh^{2}(\alpha_{i,2})(\dot{\xi}_{i,2}+k_{i,4}v_{i,e}-f_{i,2}- \dot{\beta}_{i,2}),\] (54) \[\dot{\alpha}_{i,3} =\cosh^{2}(\alpha_{i,3})\{-T_{r}\alpha_{i,3}-\varpi_{i,r}/m_{i,3 3}\}/\gamma_{1}, \tag{55}\]  
where \(k_{i,A}\), \(T_{u}\) and \(T_{r}>0\) are positive constants. **Step 4:** To stabilize the system, kinetic control laws are designed as  
\[\tau_{i,\mu c} =-k_{i,3}u_{i,e}-m_{i,11}(\dot{\xi}_{i,1}+T_{u}\alpha_{i,1}-\dot{ \beta}_{i,1})+f_{i,1}, \tag{56}\] \[\tau_{i,rc} =-k_{i,5}r_{i,e}-m_{i,33}(\dot{\xi}_{i,3}+T_{r}\alpha_{i,3}-\dot{ \beta}_{i,3})-\psi_{i,e}\vartheta_{i,1}, \tag{57}\]  
where \(k_{i,3}\) and \(k_{i,5}\) are positive constants. **Theorem 2**. : _Consider that the system consist of USV (1), the control law (56) and (57), the observer (6) and the update law of additional controls (53)-(55), with environmental disturbances, input saturation and limited communication ranges under Assumptions 1-5. The proposed controller guarantees all signals in the closed-loop system are bounded and two objectives are achieved._  
Proof. : Consider the Lyapunov function as follows:  
\[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]0along with (40), (48) and (50)-(52), the differentiation of \(V\) is given as  
\[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]1 \[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]2 \[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]3 \[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]4 \[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]5  
where \(h=1,2,3\). Furthermore, substituting virtual controls (46) and (49), additional controls (53)-(55), control laws (56) and (57) into (59) leads to  
\[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]6 \[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]7 \[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]8 \[\beta_{i,3}=\tfrac{1}{\vartheta_{i,1}}(-k_{i,2}\psi_{i,e}+\vartheta_{i,2}), \tag{49}\]9 \[\dot{u}_{i,e} =\zeta_{i,1}+(\tau_{i,\mu c}-\varpi_{i,u})/m_{i,11}-\dot{\beta}_{ i,1}-\gamma_{1}\frac{\dot{\alpha}_{i,1}}{\cosh^{2}(\alpha_{i,1})}, \tag{50}\]0  
where \(h=1,2,3\), \(\bar{\alpha}_{i}=[0,\ \tanh(\alpha_{i,2})]^{T}\), \(z_{i,e}=z_{i}-\bar{\alpha}_{i}-\omega_{i}-\beta_{i}-\gamma_{1}\bar{\alpha}_{i,1}\), \(\omega_{i}=[\omega_{i,1},\omega_{i,2}]^{T}\) and \(\bar{\alpha}_{i,1}=[\tanh(\alpha_{i,1}),0]^{T}\). From the definition of \(f_{i}\) and \(z_{i,e}\), the following can be attained.