## 3 Controller Design
Choosing \(k_{i,n}=\frac{1}{2}+k_{i,n}^{*}\) with \(n=2,3,4,5\), and constants \(k_{i,n}^{*}\geq 0\) and \(\frac{1}{l_{i,h}}=s_{i,h}^{2}/(2\varrho_{i})+l_{i,h}^{*}\) with \(l_{i,h}^{*}>0\) is a constants. The following can be attained:  
\[\dot{v}_{i,e} =\zeta_{i,2}-\dot{\beta}_{i,2}-\frac{\dot{\alpha}_{i,2}}{\cosh^{2 }(\alpha_{i,2})},\]9 \[\dot{r}_{i,e} =\zeta_{i,3}+(\tau_{i,rc}-\varpi_{i,r})/m_{i,33}-\dot{\beta}_{i,3 }-\gamma_{1}\frac{\dot{\alpha}_{i,3}}{\cosh^{2}(\alpha_{i,3})}. \tag{52}\]0 \[\dot{r}_{i,e} =\zeta_{i,3}+(\tau_{i,rc}-\varpi_{i,r})/m_{i,33}-\dot{\beta}_{i,3 }-\gamma_{1}\frac{\dot{\alpha}_{i,3}}{\cosh^{2}(\alpha_{i,3})}. \tag{52}\]1  
Owing to \(|\Omega_{i,h}|\leq s_{i,h}\) on \(V=\rho_{\rho}\), the inequality (72) becomes  
\[\dot{r}_{i,e} =\zeta_{i,3}+(\tau_{i,rc}-\varpi_{i,r})/m_{i,33}-\dot{\beta}_{i,3 }-\gamma_{1}\frac{\dot{\alpha}_{i,3}}{\cosh^{2}(\alpha_{i,3})}. \tag{52}\]2where \(\mu_{0}=\min\{2k_{i,1},2k_{i,2},2k_{i,3}^{*}/m_{i,11},2k_{i,4}^{*},2k_{i,5}^{*}/m_{ i,33},2l_{i,h}^{*},\lambda_{\min}(A_{i})-\frac{1}{2}-\tilde{m}\}\). This implies that:  
\[\dot{r}_{i,e} =\zeta_{i,3}+(\tau_{i,rc}-\varpi_{i,r})/m_{i,33}-\dot{\beta}_{i,3 }-\gamma_{1}\frac{\dot{\alpha}_{i,3}}{\cosh^{2}(\alpha_{i,3})}. \tag{52}\]3  
From the definition of \(V_{t}\), it can be concluded that \(q_{i,e}\) is bounded. From Assumption 1, there exists \(i\) and \(j\) such that \(\underline{R}<\rho_{ij}(0)<\underline{L}\) for \(j\in N_{i}(0)\) where \(j=0,\ldots,N\), and \(i\
eq j\). From the definition of \(q_{i,e}\), the boundedness of \(q_{i,e}(t)\) leads to \(\underline{R}<\rho_{ij}<\underline{L}\) and \(\underline{\theta}<\lambda_{ij}<\overline{\theta}\). Thus, if \(\underline{R}<\rho_{ij}(0)<\underline{L}\) for \(j\in N_{i}(0)\), then \(\underline{R}<\rho_{ij}(t)<\underline{L}\) and \(\underline{\theta}<\lambda_{ij}(t)<\overline{\theta}\). This completes the proof of (1). From (74), \(q_{e}=[q_{1,e},\ldots,q_{N,e}]^{T}\) exponentially converges to the compact set \(\prod=\{q_{e}|\|q_{e}\|\leq 2\mu_{2}/\mu_{0}\}\) that can be made arbitrarily small by adjusting \(u_{0}\). From (20), \(e_{ij,1}\) and \(e_{ij,2}\) can be reduced to be arbitrarily small; this leads to the conclusion that \(\rho_{ij,e}\) and \(\lambda_{ij,e}\) can be also reduced to be arbitrarily small. Then, it holds that \(\lim_{x\rightarrow+\infty}\lvert\rho_{ij}(t)-\rho_{ij,d}\rvert\leq c_{1}\) and \(\lim_{x\rightarrow+\infty}\lvert\lambda_{ij}(t)-\lambda_{ij,d}\rvert\leq c_{2}\), and the proof of (2) is completed. The distributed controller is composed of the local virtual control laws (46) and (49) and the auxiliary dynamics and the actual control laws (56) and (57). In (1), there is no control input on the sway dynamics, and this causes difficulty in the control of sway. To make sure of the stability of the sway dynamics, auxiliary variable \(\alpha_{i,2}\) is introduced to solve the problem, and auxiliary variables \(\alpha_{i,1}\) and \(\alpha_{i,3}\) are introduced to solve the problem of input saturation. The \(\tanh(. )\) function guarantees that the introduced variables \(\tanh(\alpha_{i,h})\), \(h=1,2,3\) are smooth and differentiable. By designing \(\dot{\alpha}_{i,2}\) in (54) and substituting it into (59), the boundedness and convergence of \(v_{i,e}\) can be achieved. According to (72), the following conditions determine the stability of the entire system: \(k_{i,1}>0\), \(k_{i,2}>\frac{1}{2}\), \(k_{i,3}>\frac{1}{2}\), \(k_{i,4}>\frac{1}{2}\), \(k_{i,5}>\frac{1}{2}\), and \(\lambda_{\min}(A_{i})>\frac{1}{2}+\tilde{m}\). According to (13), the size of the eigenvalues of parameters \(K_{i,\rho 1}\), \(K_{i,\rho 2}\) and \(K_{i,\rho 3}\) determines the convergence speed of estimation error. Note that \(\lambda_{\min}(A_{i})\) affects both the stability of the observer and of the whole system.