## 3 Extended State Observer
\begin{split}\dot{V}_{1}(\mathbf{\vartheta})&=\mathbf{ \vartheta}^{T}\mathbf{P}\mathbf{\vartheta}+\mathbf{\vartheta}^{T}\mathbf{P}\mathbf{\vartheta}\\ &=\mathbf{\vartheta}^{T}\mathbf{P}[\mathbf{\omega}\mathbf{A}\mathbf{\vartheta}+ \omega_{1}\delta_{2}(\tilde{f}_{2}+\tilde{\mathbf{F}})+\delta_{3}\mathbf{h}_{p}+\omega _{2}\delta_{5}\tilde{f}_{5}+\delta_{6}\mathbf{h}_{\theta}]\\ &\quad+[\omega\mathbf{A}\mathbf{\vartheta}+\omega_{1}\delta_{2}(\tilde{f }_{2}+\tilde{\mathbf{F}})+\delta_{3}\mathbf{h}_{p}+\omega_{2}\delta_{5}\tilde{f}_{5}+ \delta_{6}\mathbf{h}_{\theta}]^{T}\mathbf{P}\mathbf{\vartheta}\\ &=\mathbf{\vartheta}^{T}(\mathbf{P}\mathbf{\omega}A+\mathbf{A}^{T}\mathbf{\omega}^{ T}\mathbf{P})\mathbf{\vartheta}+2\omega_{1}\mathbf{\vartheta}^{T}\mathbf{P}\mathbf{\delta}_{2}( \tilde{f}_{2}+\tilde{\mathbf{F}})+2\omega_{2}\mathbf{\vartheta}^{T}\mathbf{P}\mathbf{\delta}_{ 5}\tilde{f}_{5}+2\mathbf{\vartheta}^{T}\mathbf{P}\mathbf{\delta}_{3}\mathbf{h}_{p}+2\mathbf{ \vartheta}^{T}\mathbf{P}\mathbf{\delta}_{5}\mathbf{h}_{\theta}\\ &\leq-\omega_{m}\parallel\mathbf{\vartheta}\parallel^{2}+2\omega_{1 }(\gamma_{1}+\gamma_{2})\parallel\mathbf{\vartheta}^{T}\parallel\mathbf{P}\mathbf{\delta} _{2}\parallel\parallel\mathbf{\vartheta}\parallel\\ &\quad+2\omega_{2}\gamma_{5}\parallel\mathbf{\vartheta}^{T}\parallel \parallel\mathbf{P}\mathbf{\delta}_{5}\parallel\parallel\mathbf{\vartheta}\parallel+2H_{ 1}\parallel\mathbf{\vartheta}^{T}\parallel\mathbf{P}\mathbf{\delta}_{3}\parallel+2H_{2} \parallel\mathbf{\vartheta}\parallel\mathbf{P}\mathbf{\delta}_{6}\parallel\\ &\leq-(\omega_{m}-2\omega_{1}\gamma_{1}\mu_{2}-2\omega_{1} \gamma_{2}\mu_{2}-2\omega_{2}\gamma_{5}\mu_{5})\parallel\mathbf{\vartheta} \parallel^{2}+2(H_{1}\mu_{3}\ +H_{2}\mu_{6})\parallel\mathbf{\vartheta}\parallel\\ &\leq-\frac{\omega_{m}-2\omega_{1}\gamma_{1}\mu_{2}-2\omega_{1} \gamma_{2}\mu_{2}-2\omega_{2}\gamma_{5}\mu_{5}}{\lambda_{max}(P)}V_{1}+\frac{ 2(H_{1}\mu_{3}\ +H_{2}\mu_{6})}{\sqrt{\lambda_{min}(P)}}\sqrt{V}_{1}\\ &\leq-\frac{\alpha}{\lambda_{max}(P)}V_{1}+\frac{2\beta}{\sqrt{ \lambda_{min}(P)}}\sqrt{V}_{1}\end{split}  \tag{14}\]  
where \(\beta=H_{1}\mu_{3}\ +H_{2}\mu_{6}\). To obtain a linear differential inequality from Equation (14), let us introduce the following function. \[W_{1}(t)=\sqrt{V_{1}} \tag{15}\]  
Then, when \(V_{1}\
eq 0\), we have  
\[\dot{W}_{1}(t)=\frac{\dot{V}_{1}}{2\sqrt{V_{1}}}\leq-\frac{\alpha}{2\lambda_{ max}(P)}W_{1}+\frac{\beta}{\sqrt{\lambda_{min}(P)}} \tag{16}\]  
Following the comparison principle [52], the following inequality can be derived from Equation (16). \[W_{1}\leq[W_{1}(0)-C_{0}]\text{exp}[-\frac{\alpha}{2\lambda_{max}(P)}t]+C_{0} \tag{17}\]  
where  
\[C_{0}=\frac{2\lambda_{max}(P)\beta}{\alpha\sqrt{\lambda_{min}(P)}} \tag{18}\]  
Invoking (13), one obtains the upper bound of \(\parallel\mathbf{\vartheta}\parallel\) as  
\[\parallel\mathbf{\vartheta}\parallel\leq\frac{\sqrt{V_{1}(0)}-C_{0}}{\sqrt{\lambda _{min}(P)}}\text{exp}[-\frac{\alpha}{2\lambda_{max}(P)}t]+\frac{C_{0}}{\sqrt{ \lambda_{min}(P)}} \tag{19}\]Then, we have:  
\[\limsup_{t\rightarrow\infty}\parallel\boldsymbol{\vartheta}\parallel\leq C=\frac{2 \lambda_{max}(P)\beta}{\alpha\lambda_{min}(P)} \tag{20}\]  
From the above analysis, it is clear that \(\boldsymbol{\vartheta}\) is uniformly ultimately bounded. Consequently, from Equation (9), the ultimate upper bound of the scaled estimation error is  
\[ \begin{split}\limsup_{t\rightarrow\infty}\parallel\tilde{\mathbf{X} }_{1}\parallel&\leq C/\omega_{1}^{2},\ \limsup_{t\rightarrow\infty}\parallel\tilde{\mathbf{X}}_{2}\parallel\leq C/ \omega_{1},\\ \limsup_{t\rightarrow\infty}\parallel\tilde{\mathbf{X}}_{3} \parallel&\leq C,\ \limsup_{t\rightarrow\infty}\parallel\tilde{\mathbf{X}}_{4}\parallel\leq C/ \omega_{2}^{2},\\ \limsup_{t\rightarrow\infty}\parallel\tilde{\mathbf{X}}_{5} \parallel&\leq C/\omega_{2},\ \limsup_{t\rightarrow\infty}\parallel\tilde{\mathbf{X}}_{6}\parallel\leq C\end{split}