## Appendix B  
This is a study on the time derivative of \(V_{3}\):  
\[\dot{V}_{3}=-\sum_{i=1}^{n}\widetilde{\[PERSON]}_{i}^{T}J_{i}(H)e_{2i}-\sum_{i=1}^{ n}\widetilde{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}\hat{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}-\sum_{i=1}^{ n}\varepsilon_{i}\widetilde{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i} \tag{11}\]  
Given \(-\varepsilon_{i}\widetilde{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}\) in Equation (11), the following inequality holds:  
\[-\varepsilon_{i}\widetilde{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}=-\varepsilon_{i} \widetilde{\[PERSON]}_{i}^{T}\widetilde{\[PERSON]}_{i}-\varepsilon_{i}\widetilde{\[PERSON]}_ {i}^{T}\[PERSON]_{i}\leq-\frac{\varepsilon_{i}}{2}\widetilde{\[PERSON]}_{i}^{T} \widetilde{\[PERSON]}_{i}+\frac{\varepsilon_{i}}{2}\[PERSON]_{i}^{T}\[PERSON]_{i}\leq- \varepsilon_{i}\bigg{(}\widetilde{\[PERSON]}_{i}^{T}\widetilde{\[PERSON]}_{i}\bigg{)}^ {\frac{1}{2}}+\frac{\varepsilon_{i}}{2}\bigg{(}\[PERSON]_{i}^{T}\[PERSON]_{i}+1\bigg{)} \tag{12}\]  
The expression for \(\widetilde{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}\hat{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}\) in Equation (11) can be presented as follows:  
\[-\widetilde{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}\hat{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}=- \left\|\widetilde{\[PERSON]}_{i}\right\|^{4}-\|\[PERSON]_{i}\|^{2}\bigg{(}\left\| \widetilde{\[PERSON]}_{i}\right\|^{2}\widetilde{\[PERSON]}_{i}^{T}\hat{\[PERSON]}_{i}\bigg{)} -3\left\|\widetilde{\[PERSON]}_{i}\right\|^{2}\widetilde{\[PERSON]}_{i}^{T}\[PERSON]_{i}-2 \left\|\widetilde{\[PERSON]}_{i}^{T}\[PERSON]_{i}\right\|^{2} \tag{13}\]  
Using Lemma 3 results in the following:  
\[-3\left\|\widetilde{\[PERSON]}_{i}\right\|^{2}\widetilde{\[PERSON]}_{i}^{T}\[PERSON]_{i}\leq 3 \left\|\widetilde{\[PERSON]}_{i}\right\|^{3}\|\[PERSON]_{i}\|\leq 3\Bigg{(}\frac{3}{4}K_{1i} \frac{4}{3}\Bigg{(}\left\|\widetilde{\[PERSON]}_{i}\right\|^{3}\Bigg{)}^{\frac{4} {3}}+\frac{1}{4K_{1i}^{4}}\|\[PERSON]_{i}\|^{4}\Bigg{)} \tag{14}\]  
\[-\|\[PERSON]_{i}\|^{2}\widetilde{\[PERSON]}_{i}^{T}\[PERSON]_{i}\leq\|\[PERSON]_{i}\|^{3}\bigg{\|} \widetilde{\[PERSON]}_{i}\bigg{\|}\leq\frac{3}{4}K_{2i}\frac{4}{3}\Big{(}\|\[PERSON]_{i }\|^{3}\Big{)}^{\frac{4}{3}}+\frac{1}{4K_{2i}^{4}}\left\|\widetilde{\[PERSON]}_{i} \right\|^{4} \tag{15}\]  
\[-\|\[PERSON]_{i}\|^{2}\bigg{\|}\widetilde{\[PERSON]}_{i}\bigg{\|}^{2}\leq-\|\[PERSON]_{i}\|^ {2}\bigg{(}2\bigg{\|}\widetilde{\[PERSON]}_{i}\bigg{\|}-1\bigg{)} \tag{16}\]  
Moreover, \(-2\left\|\widetilde{\[PERSON]}_{i}^{T}\[PERSON]_{i}\right\|^{2}<0\) is clear. Considering the inequalities (14)-(16), we can rewrite Equation (13) as follows:  
\[-\widetilde{\[PERSON]}_{i}^{T}\[PERSON]_{i}\[PERSON]_{i}^{T}\[PERSON]_{i}\leq-\bigg{(}1-\frac{9}{ 4}K_{1i}^{\frac{4}{3}}-\frac{1}{4K_{2i}^{4}}\bigg{)}\left\|\widetilde{\[PERSON]}_{i }\right\|^{4}-2\|\[PERSON]_{i}\|^{2}\bigg{\|}\widetilde{\[PERSON]}_{i}\bigg{\|} \tag{17}\]  
Upon substituting Equations (12) and (17) into Equation (11), the ensuing inequality emerges:  
\[\dot{V}_{3}\leq-\sum_{i=1}^{n}\widetilde{\[PERSON]}_{i}^{T}J_{i}(H)e_{2i}-\sum_{i= 1}^{n}\bigg{(}1-\frac{9}{4}K_{1i}^{\frac{4}{3}}-\frac{1}{4K_{2i}^{2}}\bigg{)} \left\|\widetilde{\[PERSON]}_{i}\right\|^{4}-\sum_{i=1}^{n}\Big{(}2\|\[PERSON]_{i}\|^{2}+ \varepsilon_{i}\Big{)}\bigg{\|}\widetilde{\[PERSON]}_{i}\bigg{\|}+B_{1} \tag{18}\]  
Per Lemma 4, Equation (18) is reformulated as follows:\[\dot{V}_{3}\leq-\sum\limits_{i=1}^{n}\frac{1}{\sqrt{\lambda_{\text{arr}} \left(\Lambda_{i}^{-1}\right)}}\Big{(}2\|\[PERSON]_{i}\|^{2}+\varepsilon_{i}\Big{)} \left(\widetilde{\[PERSON]}_{i}^{T}\Lambda_{i}^{-1}\widetilde{\[PERSON]}_{i}\right)^{ \frac{1}{2}}-\sum\limits_{i=1}^{n}\frac{1}{\left(\Lambda_{\text{arr}}\left( \Lambda_{i}^{-1}\right)\right)^{2}}\Big{(}1-\frac{9}{2}\mathbb{K}_{1i}\frac{4} {4}-\frac{1}{4\mathbb{K}_{2}^{2}}\Big{)}\left(\widetilde{\[PERSON]}_{i}^{T}\Lambda_ {i}^{-1}\widetilde{\[PERSON]}_{i}\right)^{2}-\sum\limits_{i=1}^{n}\widetilde{\Phi}_ {i}^{T}J_{i}(H)e_{2i}+B_{1}} \tag{101}\]  
According to Lemmas 1 and 2, (101) is written as follows:  
\[\dot{V}_{3}\leq-K_{31}\sqrt{V_{3}}-\frac{K_{32}}{n}V_{3}^{2}-\sum\limits_{i=1 }^{n}\widetilde{\Phi}_{i}^{T}J_{i}(H)e_{2i}+B_{1} \tag{102}\]