## 5 Closed-Loop System's Stability Analysis
\begin{cases}\mathbf{e}_{1}^{T}\mathbf{e}_{2}=\mathbf{e}_{1}^{T}(\mathbf{e}_{2}-\mathbf{\xi}_{1})+ \mathbf{e}_{1}^{T}\mathbf{\xi}_{1}\leq\parallel\mathbf{e}_{1}\parallel\parallel\mathbf{e}_{2 }-\mathbf{\xi}_{1}\parallel+\parallel\mathbf{e}_{1}\parallel\parallel\mathbf{\xi}_{1} \parallel\\ \mathbf{e}_{4}^{T}\mathbf{e}_{5}=\mathbf{e}_{4}^{T}(\mathbf{e}_{5}-\mathbf{\xi}_{2})+\mathbf{e}_{4}^{T }\mathbf{\xi}_{2}\leq\parallel\mathbf{e}_{4}\parallel\parallel\mathbf{e}_{5}-\mathbf{\xi}_{2 }\parallel+\parallel\mathbf{e}_{4}\parallel\parallel\mathbf{\xi}_{2}\parallel\\ \mathbf{\xi}_{1}^{T}\Delta\mathbf{F}\leq\frac{\mathbf{\zeta}_{1}}{2}\parallel\mathbf{\xi}_{1 }\parallel^{2}+\frac{1}{2c_{1}}\mathbf{v}_{1}^{2}\\ \mathbf{\xi}_{2}^{T}\Delta\mathbf{U}_{2}\leq\frac{c_{2}}{2}\parallel\mathbf{\xi}_{2} \parallel^{2}+\frac{1}{2c_{2}}\mathbf{v}_{2}^{2}\\ \beta\parallel\mathbf{\vartheta}\parallel\leq\frac{c_{3}}{4}\parallel\mathbf{ \vartheta}\parallel^{2}+\frac{1}{c_{3}}\beta^{2}\\ -(\mathbf{e}_{2}-\mathbf{\xi}_{1})^{T}\mathbf{e}_{2}\leq\frac{c_{4}}{2}\parallel\mathbf{e}_{ 2}-\mathbf{\xi}_{1}\parallel^{2}+\frac{1}{2c_{4}}\mathbf{E}_{2}^{2}\\ -(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}\mathbf{e}_{5}\leq\frac{c_{5}}{2}\parallel\mathbf{e}_{5}- \mathbf{\xi}_{2}\parallel^{2}+\frac{1}{2c_{5}}\mathbf{E}_{5}^{2}\end{cases}  \tag{55}\]  
By making use of (54), (55) can be rewritten as  
\[\dot{V} \leq-k_{1}\parallel\mathbf{e}_{1}\parallel^{2}-(k_{2}-\frac{c_{4}}{2 })\parallel\mathbf{e}_{2}-\mathbf{\xi}_{1}\parallel^{2}-(k_{3}-\frac{c_{1}}{2}) \parallel\mathbf{\xi}_{1}\parallel^{2} \tag{56}\] \[-k_{4}\parallel\mathbf{e}_{4}\parallel^{2}-(k_{5}-\frac{c_{5}}{2}) \parallel\mathbf{e}_{5}-\mathbf{\xi}_{2}\parallel^{2}-(k_{6}-\frac{c_{2}}{2}) \parallel\mathbf{\xi}_{2}\parallel^{2}\] \[-(\frac{\alpha}{2}-\frac{c_{3}}{4})\parallel\mathbf{\vartheta} \parallel^{2}+(\gamma_{2}+\frac{k_{2}}{\omega_{1}})\parallel\mathbf{e}_{2}-\mathbf{ \xi}_{1}\parallel\parallel\mathbf{\vartheta}_{2}\parallel\] \[+\parallel\mathbf{e}_{2}-\mathbf{\xi}_{1}\parallel\parallel\mathbf{\vartheta} _{3}\parallel+\frac{k_{4}}{\omega_{2}^{2}}\parallel\mathbf{e}_{4}\parallel\parallel \mathbf{\vartheta}_{4}\parallel\] \[+(\gamma_{5}+\frac{k_{5}}{\omega_{2}})\parallel\mathbf{e}_{5}-\mathbf{ \xi}_{2}\parallel\parallel\mathbf{\vartheta}_{5}\parallel+\parallel\mathbf{e}_{5}-\bm {\xi}_{2}\parallel\parallel\mathbf{\vartheta}_{6}\parallel\] \[+\parallel\mathbf{e}_{1}\parallel\parallel\mathbf{\xi}_{1}\parallel+ \parallel\mathbf{e}_{4}\parallel\parallel\mathbf{\xi}_{2}\parallel+(\frac{v_{1}^{2}}{2c_{ 1}}+\frac{v_{2}^{2}}{2c_{2}}+\frac{\beta^{2}}{c_{3}}+\frac{1}{2c_{4}}\mathbf{E}_{2 }^{2}+\frac{1}{2c_{5}}\mathbf{E}_{5}^{2})\] \[\leq-\mathbf{\eta}^{T}\Pi\mathbf{\eta}+\sigma\]  
where \(\sigma=\frac{v_{1}^{2}}{2c_{1}}+\frac{v_{2}^{2}}{2c_{2}}+\frac{\beta^{2}}{c_{3}}+ \frac{1}{2c_{4}}\mathbf{E}_{2}^{2}+\frac{1}{2c_{5}}\mathbf{E}_{5}^{2}\) and  
\[\mathbf{\eta}=[\parallel\mathbf{With \(\mathbf{e}=[\mathbf{e}_{1}^{T},(\mathbf{e}_{2}-\mathbf{\xi}_{1})^{T},\mathbf{\xi}_{1}^{T},\mathbf{e}_{4}^{T},(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T},\mathbf{\xi}_{2}^{T}]^{T}\) and matrix \(\mathbf{\Pi}\) is positive definite, we have  
\[\dot{V} \leq-\lambda_{min}(\mathbf{\Pi})(\parallel\mathbf{e}\parallel^{2}+ \parallel\mathbf{\vartheta}\parallel^{2})+\sigma \tag{58}\] \[\leq-\lambda_{min}(\mathbf{\Pi})[\mathbf{e}_{1}^{T}\mathbf{e}_{1}+(\mathbf{e}_{2 }-\mathbf{\xi}_{1})^{T}(\mathbf{e}_{2}-\mathbf{\xi}_{1})\mathbf{\xi}_{1}^{T}\mathbf{\xi}_{1}+\mathbf{ e}_{4}^{T}\mathbf{e}_{4}\] \[\quad+(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}(\mathbf{e}_{5}-\mathbf{\xi}_{2})+\mathbf{ \xi}_{2}^{T}\mathbf{\xi}_{2}+\frac{1}{\lambda_{max}(\mathbf{P})}\mathbf{\vartheta}^{T} \mathbf{P}\mathbf{\vartheta}]+\sigma\] \[-k_{4}\parallel\mathbf{e}_{4}\parallel^{2}-(k_{5}-\frac{c_{5}}{2}) \parallel\mathbf{e}_{5}-\mathbf{\xi}_{2}\parallel^{2}-(k_{6}-\frac{c_{2}}{2}) \parallel\mathbf{\xi}_{2}\parallel^{2}\]0  
where \(a=-2\lambda_{min}(\mathbf{\Pi})\text{min}\{1,\frac{1}{\lambda_{max}(\mathbf{P})}\}\).