## 3 Main Results
### Stability Analysis
 \tag{35}\]  
Consider the following inequality of the hyperbolic tangent function holds for any \(\chi>0\) and for any \(\varpi\in\mathbb{R}\)[29]  
\[0\leqslant|\varpi|-\varpi\tan\left(\frac{\varpi}{\chi}\right)\leqslant\kappa_ {\chi}\chi \tag{36}\]  
where \(\kappa_{\chi}\) is a constant that satisfies \(\kappa_{\chi}=e^{-(\kappa_{\chi}+1)}\), i.e., \(\kappa_{\chi}=0.2785\). It is worth noting that the following equalities hold  
\[-\sum_{i=1}^{2}\iota_{i}\tilde{\mathbf{W}}_{i}^{T}\tilde{\mathbf{W}}_{i} \leqslant-\sum_{i=1}^{2}\frac{\iota_{i}}{2}\tilde{\mathbf{W}}_{i}^{T}\tilde{ \mathbf{W}}_{i}+\sum_{i=1}^{2}\frac{\iota_{i}}{2}\|\mathbf{W}_{i}^{*}\|^{2} \tag{37}\]  
\[-\sum_{i=1}^{2}\lambda_{i}\tilde{\delta}_{i}\left(\delta_{i}-\delta_{i}^{0} \right)\leqslant-\sum_{i=1}^{2}\frac{\lambda_{i}}{2}\tilde{\delta}_{i}^{2}+ \sum_{i=1}^{2}\frac{\lambda_{i}}{2}\left(\delta_{i}-\delta_{i}^{0}\right)^{2} \tag{38}\]  
\[\tilde{r}\left(\delta_{1}-\delta_{1}\tan\left(\frac{\tilde{r}}{\chi_{r}} \right)\right)+\tilde{\delta}_{1}\tan\left(\frac{\tilde{r}}{\chi_{r}}\right) \tilde{r}\leqslant\delta_{1}\left(|\tilde{r}|-\tilde{r}\tan\left(\frac{\tilde{r }}{\chi_{r}}\right)\right)\leqslant 0.2785\chi_{r}\delta_{1} \tag{39}\]  
\[\tilde{u}_{r}\left(\delta_{2}-\delta_{2}\tan\left(\frac{\tilde{u}_{r}}{\chi_{u}} \right)\right)+\tilde{\delta}_{2}\tan\left(\frac{\tilde{u}_{r}}{\chi_{u}} \right)\tilde{u}_{r}\leqslant\delta_{2}\left(|\tilde{u}_{r}|-\tilde{u}_{r} \tan\left(\frac{\tilde{u}_{r}}{\chi_{u}}\right)\right)\leqslant 0.2785\chi_{u}\delta_{2} \tag{40}\]Substituting Equations (36)-(39) into Equation (35), we have  
\[ \begin{split}\dot{V}&\leqslant-\left(k_{1}-\frac{1}{2} \right)x_{e}^{2}-\left(\frac{U_{r}}{\sqrt{y_{e}^{2}+\Delta^{2}}}-\frac{1}{2k_{ 1}}\right)y_{e}^{2}-k_{2}\dot{V}^{2}-\left(k_{3}-\frac{1}{2}\right)\dot{r}^{2 }-\left(k_{4}-\frac{1}{2}\right)\dot{u}_{r}^{2}\\ &\quad-\sum\limits_{i=1}^{2}\left(k_{ei}-\frac{1}{2}-\frac{1}{2}k_ {ei}^{2}\right)\sigma_{i}^{2}-\sum\limits_{i=1}^{2}\frac{\iota_{i}}{2}\dot{ \mathbf{W}}_{i}^{T}\dot{\mathbf{W}}_{1}-\sum\limits_{i=1}^{2}\frac{\lambda_{i} }{2}\delta_{i}^{2}+\sum\limits_{i=1}^{2}\frac{1}{2}\Big{(}\delta_{i}-\delta_{ i}^{0}\Big{)}^{2}\\ &\quad+\sum\limits_{i=1}^{2}\frac{\iota_{i}}{2}W_{iM}{}^{2}+0.278 5\left(\chi_{r}\delta_{1}+\chi_{u}\delta_{2}\right)+\frac{1+k_{1}}{2}\theta^{2 }\\ &\leqslant-\kappa_{1}V+\vartheta_{1}\end{split}  \tag{41}\]  
where \(\kappa_{1}=\min\left\{2k_{1}-1,\frac{2U_{r}}{\sqrt{y_{e}^{2}+\Delta^{2}}}- \frac{1}{k_{1}},2k_{2},\frac{2k_{3}-1}{m_{3}},\frac{2k_{4}-1}{m_{1}},i=1,2 \left(2k_{ei}-1-k_{ei}^{2}\right),\min\limits_{i=1,2}\left(\iota_{i}\lambda_{ \min}(\Gamma_{i})\right),\min\limits_{i=1,2}\left(\lambda_{i}\vec{\xi}_{i} \right)\right\}\), \(k_{1}=\max\left\{\frac{1}{2},\frac{\sqrt{y_{e}^{2}+\Delta^{2}}}{2U_{r}}\right\}\), \(\lambda_{\min}(\bullet)\) denotes the minimum eigenvalue of a matrix, and \(\vartheta_{1}=\sum\limits_{i=1}^{2}\frac{1}{2}\big{(}\delta_{i}-\delta_{i}^{0} \big{)}^{2}+\sum\limits_{i=1}^{2}\frac{\iota}{2}W_{iM}{}^{2}+0.2785\left(\chi _{r}\delta_{1}+\chi_{u}\delta_{2}\right)+\frac{1+k_{1}}{2}\theta^{2}\). (2) When \(|\sigma_{i}|<\mu_{i}(i=1,2)\), we have \(\sum\limits_{i=1}^{2}\sigma_{i}\sigma_{i}=0\). According to Equations (30) and (31) and the inequalities \(\sum\limits_{i=1}^{2}\frac{1}{2}k_{ei}^{2}\sigma_{i}^{2}\leqslant-\sum\limits _{i=1}^{2}\frac{1}{2}k_{ei}^{2}\sigma_{i}^{2}+\sum\limits_{i=1}^{2}k_{ei}^{2} \mu_{i}^{2}\), \(\bar{r}\Delta\tau_{r}\leqslant\frac{1}{2}\bar{r}^{2}+\frac{1}{2}\Delta\tau_{r}^ {2}\), and \(\bar{u}_{r}\Delta\tau_{u}\leqslant\frac{1}{2}\bar{u}_{r}^{2}+\frac{1}{2}\Delta \tau_{u}^{2}\), Equation (34) becomes  
\[