## 3 Theoretical Analysis
### Formation Analysis  
Define \(\lambda_{i}(i=1,2\ldots,N)\) as the eigenvalues of \(L\) corresponding to \(G\), where \(\lambda_{1}=0\), with the associated eigenvector \(\overline{u}_{1}=1_{N}\) and \(0<\text{Re}(\lambda_{2})\leq\ldots\leq\text{Re}(\lambda_{N})\). Define the following:  
\[U^{-1}LU=J, \tag{18}\]  
where \(U=[\overline{u}_{1},\overline{u}_{2}\ldots\overline{u}_{N}]\), \(U^{-1}=[\widetilde{u}_{1},\widetilde{u}_{2}\ldots\widetilde{u}_{N}]^{H}\), and the \(J\) is the form of Jordan canonical of \(L\), \(\overline{u}_{i}\in\mathbb{C}^{N}\), and \(\widetilde{u}_{i}\in\mathbb{C}^{N}(i=1,2\ldots,N)\). Figure 4: AUV formation on the X–Y plane with \(N=5\). Let  
\[\widetilde{\theta}_{i}(t)=\theta_{i}(t)-m_{i}(t)(i=1,2\ldots,N), \tag{19}\]  
\[\widetilde{\theta}(t)=\Big{[}\widetilde{\theta}_{1}^{T}(t),\widetilde{\theta}_{ 2}^{T}(t),\ldots\widetilde{\theta}_{N}^{T}(t)\Big{]}^{T}, \tag{20}\]  
So, the multi-AUV system in Equation (17) can be transformed into the following:  
\[ \begin{array}{l}\dot{\widetilde{\theta}}(t)=\big{(}I_{N}\otimes\big{(}B_{2} K_{1}+B_{1}B_{2}^{T}\big{)}-L\otimes(B_{2}K_{2})\big{)}\widetilde{\theta}(t)+ \big{(}I_{N}\otimes\big{(}B_{1}B_{2}^{T}\big{)}\big{)}m(t)\\ +(I_{N}\otimes B_{2})\dot{m}_{v}(t)-\dot{m}(t)\end{array} , \tag{21}\]  
As \(B_{1}=[1,0]^{T}\), \(B_{2}=[0,1]^{T}\), and \(\dot{m}(t)=(I_{N}\otimes I_{2})\dot{m}(t)\), the multi-AUV system in Equation (21) can be rewritten as follows:  
\[\dot{\widetilde{\theta}}(t)=\Big{(}I_{N}\otimes\Big{(}B_{2}K_{1}+B_{1}B_{2}^{ T}\Big{)}-L\otimes(B_{2}K_{2})\Big{)}\widetilde{\theta}(t)+(I_{N}\otimes B_{1}) \big{(}m_{v}(t)-\dot{m}_{x}(t)\big{)}, \tag{22}\]  
The following lemma holds directly. Lemma 2: _The multi-AUV system in Equation (17) establishes time-varying formation \(m(t)\) if and only if Equation (22) establishes consensus._  
Let \(c_{1}\in\mathbb{R}^{2}\) and \(c_{2}\in\mathbb{R}^{2}\) be linearly independent vectors. \[p_{j}=\overline{u}_{i}\otimes c_{q}(j=2(i-1)+q;i=1,2\ldots,N;q=1,2), \tag{23}\]  
A consensus subspace (CS) can be defined as the subspace \(\mathbb{C}(\mathrm{U})\) spanned by \(p_{1}=\overline{u}_{1}\otimes c_{1}=1_{N}\otimes c_{1}\) and \(p_{2}=\overline{u}_{1}\otimes c_{2}=1_{N}\otimes c_{2}\). Moreover, a complement of the CS can be represented as the subspace \(\overline{\mathbb{C}}(\mathrm{U})\) spanned by \(p_{3},p_{4},\ldots,p_{2N}\). Since \(p_{j}(j=1,2,\ldots,2N)\) is linearly independent, the following lemma can be obtained. Lemma 3: \(\mathbb{C}(\mathrm{U})\oplus\overline{\mathbb{C}}(\mathrm{U})=\mathbb{C}^{2 \mathbb{N}}\)_._  
By Lemma 1, one obtains \(J=\mathrm{diag}\big{\{}0,\overline{J}\big{\}}\), where \(\overline{J}\) consists of Jordan blocks corresponding to \(\lambda_{i}(i=2,3,\ldots,N)\). Let  
\[ \begin{cases}\dot{\overline{U}}=\big{[}\widetilde{u}_{2},\widetilde{u}_{3}, \ldots\widetilde{u}_{N}\big{]}^{H}\\ \dot{\xi}(t)=\big{(}\widetilde{u}_{1}^{H}\otimes I_{2}\big{)}\widetilde{ \theta}(t)\\ \xi(t)=\big{(}\widetilde{U}\otimes I_{2}\big{)}\theta(t)\end{cases} , \tag{24}\]  
The multi-AUV system in Equation (22) thus becomes the following:  
\[\dot{\xi}(t)=\Big{(}B_{2}K_{1}+B_{1}B_{2}^{T}\Big{)}\dot{\xi}(t)+\Big{(} \widetilde{u}_{1}^{H}\otimes B_{1}\Big{)}\big{(}m_{v}(t)-\dot{m}_{x}(t)\big{)}, \tag{25}\]  
\[\dot{\xi}(t)=\Big{(}I_{N-1}\otimes\Big{(}B_{2}K_{1}+B_{1}B_{2}^{T}\Big{)} \Big{)}-\overline{J}\otimes(B_{2}K_{2})_{\xi}(t)+\Big{(}\widetilde{U}\otimes B _{1}\Big{)}\big{(}m_{v}(t)-\dot{m}_{x}(t)\big{)} \tag{26}\]  
Lemma 4: _Based on [43]\(\dot{\varphi}(t)=Y\varphi(t)\), we know that \(Y\) is a \(2\times 2\) complex matrix with the characteristic polynomial \(f(q)=q^{2}+a_{1}q+a_{2}\), so the system is asymptotically stable when the only condition is the following_:__  
\[\left\{ \begin{array}{l}\mathrm{Re}(a_{1})>0\\ \mathrm{Re}(a_{1})\mathrm{Re}(a_{1}\overline{u}_{2})-\mathrm{Im}(a_{2})^{2}>0 \end{array}