## 5 Closed-Loop System's Stability Analysis
\begin{bmatrix}k_{1}&-\frac{1}{2}&-\frac{1}{2}&\mathbf{0}&\mathbf{0}& \mathbf{0}\\ -\frac{1}{2}&k_{2}-\frac{c_{4}}{2}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ -\frac{1}{2}&\mathbf{0}&k_{3}-\frac{c_{1}}{2}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&k_{4}&-\frac{1}{2}&-\frac{1}{2}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-\frac{1}{2}&k_{5}-\frac{c_{5}}{2}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-\frac{1}{2}&\mathbf{0}&k_{6}-\frac{c_{2}}{2}\end{bmatrix} \] \[\mathbf{\Pi}_{2} = \begin{bmatrix}\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&-\frac{1}{2}(\gamma_{2}+\frac{k_{2}}{\omega_{1}})&-\frac{1}{2}&\mathbf{0}& \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-\frac{k_{4}}{2\omega_{2}^{2}}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&-\frac{1}{2}(\gamma_{5}+\frac{k_{5}}{\omega_{2}})&-\frac{1 }{2}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\end{bmatrix} \] \[\mathbf{\Pi}_{3} =(\frac{\alpha}{2}-\frac{c_{3}}{4})\text{diag}(1,1,1,1,1,1)\]  
_with \(c_{j}(j=1,...,5)\) being some positive constants to be chosen._  
Proof: Consider the following Lyapunov candidate function  
\[V=\frac{1}{2}\mathbf{e}_{1}^{T}\mathbf{e}_{1}+\frac{1}{2}(\mathbf{e}_{2}-\mathbf{\xi}_{1})^{T}( \mathbf{e}_{2}-\mathbf{\xi}_{1})+\frac{1}{2}\mathbf{\xi}_{1}^{T}\mathbf{\xi}_{1}+\frac{1}{2} \mathbf{\varepsilon}_{1}^{T}\mathbf{\epsilon}_{1}+\frac{1}{2}(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}( \mathbf{e}_{5}-\mathbf{\bar{\zeta}}_{2})+\frac{1}{2}\mathbf{\xi}_{2}^{T}\mathbf{\xi}_{2}+\frac{1 }{2}\mathbf{\vartheta}^{T}\mathbf{P}\mathbf{\vartheta} \tag{53}\]Substituting (10) and (51) into the time derivative of \(V\) gives  
\[\dot{V} \leq\mathbf{e}_{1}^{T}(-k_{1}\mathbf{e}_{1}+\mathbf{e}_{2})+(\mathbf{e}_{2}-\mathbf{ \zeta}_{1})^{T}[-k_{2}(\mathbf{e}_{2}-\mathbf{\zeta}_{1})+\tilde{f}_{2}+\mathbf{\vartheta}_{ 3}-\epsilon_{2} \tag{54}\] \[+k_{2}\frac{\mathbf{\vartheta}_{2}}{\omega_{1}}]+\mathbf{\xi}_{1}^{T}(-k_ {3}\mathbf{\xi}_{1}+\Delta F)+\mathbf{e}_{4}^{T}(-k_{4}\mathbf{e}_{4}+\mathbf{e}_{5}+k_{4}\frac {\mathbf{\vartheta}_{4}}{\omega_{2}^{2}})\] \[+(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}[-k_{5}(\mathbf{e}_{5}-\mathbf{\xi}_{2})+k_ {5}\frac{\mathbf{\vartheta}_{5}}{\omega_{2}}+\tilde{f}_{5}+\mathbf{\vartheta}_{6}- \varepsilon_{5}]+\mathbf{\xi}_{2}^{T}(-k_{6}\mathbf{\xi}_{2}+\Delta\mathbf{U}_{2})-\frac{1 }{2}\omega_{m}\parallel\mathbf{\vartheta}\parallel^{2}\] \[+(\omega_{1}\gamma_{1}\mu_{2}+\omega_{1}\gamma_{2}\mu_{2}+\omega_ {2}\gamma_{5}\mu_{5})\parallel\mathbf{\vartheta}\parallel^{2}+\mu_{3}\parallel \mathbf{\vartheta}\parallel\parallel\mathbf{h}_{p}\parallel+\mu_{5}\parallel\mathbf{ \vartheta}\parallel\parallel\mathbf{h}_{\theta}\parallel\] \[\leq-k_{1}\mathbf{e}_{1}^{T}\mathbf{e}_{1}-k_{2}(\mathbf{e}_{2}-\mathbf{\xi}_{1} )^{T}(\mathbf{e}_{2}-\mathbf{\xi}_{1})-k_{3}\mathbf{\xi}_{1}^{T}\mathbf{\xi}_{1}-k_{4}\mathbf{e}_ {4}^{T}\mathbf{e}_{4}-k_{5}(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}(\mathbf{e}_{5}-\mathbf{\xi}_{2})- k_{6}\mathbf{\xi}_{2}^{T}\mathbf{\xi}_{2}\] \[+(\mathbf{e}_{2}-\mathbf{\xi}_{1})^{T}(\tilde{f}_{2}+\frac{k_{2}}{\omega_ {1}}\mathbf{\vartheta}_{2})+(\mathbf{e}_{2}-\mathbf{\xi}_{1})^{T}\mathbf{\vartheta}_{3}+\frac {k_{4}}{\omega_{2}^{2}}\mathbf{e}_{4}^{T}\mathbf{\vartheta}_{4}\] \[+(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}(\tilde{f}_{5}+\frac{k_{5}}{\omega_ {2}}\mathbf{\vartheta}_{5})+(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}\mathbf{\vartheta}_{6}+\mathbf{ e}_{1}^{T}\mathbf{e}_{2}-(\mathbf{e}_{2}-\mathbf{\zeta}_{1})^{T}\mathbf{e}_{2}-(\mathbf{e}_{5}-\mathbf{ \zeta}_{2})^{T}\mathbf{e}_{5}\] \[+\mathbf{e}_{4}^{T}\mathbf{e}_{5}+\mathbf{\xi}_{1}^{T}\Delta F+\mathbf{\xi}_{2}^{T }\Delta\mathbf{U}_{2}-\frac{\alpha}{2}\parallel\mathbf{\vartheta}\parallel^{2}+ \beta\parallel\mathbf{\vartheta}\parallel\]  
Obviously, we have the following relationship. \[