## 3 Theoretical Analysis
### Formation Analysis
From Equation (36), we obtain the following:  
\[=\overline{u}_{1}\otimes\xi(t)\]2  
Note that \(\widetilde{U}1_{N}=0\), and then:  
\[=\overline{u}_{1}\otimes\xi(t)\]3  
Define the following:  
\[\left\{ \begin{array}{l}\overline{m}_{x}(t)=\Big{[}m_{1x}(t),m_{2x}(t),\dots,m_{(N-1)x}(t)\Big{]}^{T}\\ \overline{m}_{v}(t)=\Big{[}m_{1v}(t),m_{2v}(t),\dots,m_{(N-1)v}(t)\Big{]}^{T} \end{array} \right., \tag{42}\]  
Based on Equations (40) and (41), we obtain the following:  
\[\lim_{t\to\infty}\big{(}\hat{U}\otimes I_{2}\big{)}\Big{(}(I_{N-1}\otimes B_{ 1})\Big{(}\overline{m}_{v}(t)-\dot{\overline{m}}_{x}(t)\Big{)}-((1_{N-1} \otimes B_{1})\times)\Big{)}\big{(}m_{Nb}(t)-\dot{m}_{Nx}(t)\big{)}=0 \tag{43}\]  
Repeat the same procedure by \(\hat{U}^{-1}\otimes I_{2}\), and then for \(\forall i\in\{1,\dots,\ N-1\}\), as follows:  
\[\lim_{t\to\infty}\big{(}(m_{ip}(t)-m_{Nb}(t))-\big{(}\dot{m}_{ix}(t)-\dot{m}_{ Nx}(t)\big{)}\big{)}=0, \tag{44}\]  
Equation (12) is held by Equation (44). Equations (12) and (36) are equivalent under Theorem 1. So, based on the structure of \(\overline{J}\), the stability of Equation (35) is equivalent to the \(N-1\) subsystem, as follows:  
\[\dot{\overline{c}}_{i}(t)=\Big{(}B_{2}(K_{1}-\lambda_{i}K_{2})+B_{1}B_{2}^{T} \Big{)}\overline{c}(t)(i=2,3,\dots,N), \tag{45}\]  
and  
\[ \begin{array}{l}B_{2}(K_{1}-\lambda_{i}K_{2})+B_{1}B_{2}^{T}\\ =\left[\begin{array}{c}0\\ 1\end{array}\right]\big{[}k_{11}-\lambda_{i}k_{21}\;k_{12}-\lambda_{i}k_{22}]+ \left[\begin{array}{c}1\\ 0\end{array}\right]\big{[}\begin{array}{cc}0&1\end{array}\right]\\ =\left[\begin{array}{cc}0&1\\ k_{11}-\lambda_{i}k_{21}&k_{12}-\lambda_{i}k_{22}\end{array}\right]\end{array} , \tag{46}\]  
Now, the characteristic polynomial of the state matrix of the \(N-1\) subsystem in Equation (45) can be obtained by \(f_{i}(s)=s^{2}-(k_{12}-\lambda_{i}k_{22})s-(k_{11}-\lambda_{i}k_{21})(i=2,3, \dots,N)\), where \(s\) is a complex variable. Thus, Equation (34) is asymptotically stable when Equation (13) holds. Now, according to the above, there is an achievable formation constraint from Equation (28), and the internal stability of the subsystem in Equation (24) can be ensured by Equations (29) and (30), and the formation error converges to 0. If the multi-AUV in Equation (17) obtains \(m(t)\), it means that \(\widetilde{\theta}_{i}(t)-\xi(t)\to 0\) as \(t\to\infty\) under Theorem 1. Thus, we can establish the formation center function through Equation (25). The explicit expression of the formation center function is shown as follows:  
**Lemma 5**. : _Multi-AUV in Equation (17) obtains \(m(t)\) with the function \(c(t)\) of the formation center:_  
\[\lim_{t\to\infty}(c(t)-c_{0}(t)-c_{h}(t))=0, \tag{47}\]  
_where_  
\[c_{0}(t)=e^{(B_{2}K_{1}+B_{1}B_{2}^{T})t}\Big{(}\widetilde{u}_{1}^{H}\otimes I _{2}\Big{)}\theta(0), \tag{48}\]  
\[c_{0}(t)=\int_{0}^{t}e^{(B_{2}K_{1}+B_{1}B_{2}^{T})(t-\tau)}\Big{(}\widetilde{ u}_{1}^{H}\otimes I_{2}\Big{)}\times(m_{v}(\tau)-k_{12}m_{v}(\tau)-k_{11}m_{x}( \tau))d\tau-\Big{(}\widetilde{u}_{1}^{H}\otimes I_{2}\Big{)}m(t) \tag{49}\]  
\(c_{0}(t)\) _is consensus function and time-varying formation center function for the multi-AUV system in Equation (_17_) when \(m(t)\equiv 0\). \(c_{h}(t)\) shows the effect of \(m(t)\). If \(m(t)\equiv 0\), the multi-AUV system's explicit formula of consensus function is \(c(t)\). According to Theorem 2, the motion modes of the formation center informing the design of \(K_{1}\) can be selected by the eigenvalues of \(B_{2}K_{1}+B_{1}B_{2}^{T}\)._