## 3 Main Result
### Estimator Design
\begin{pmatrix}-\delta_{u}^{\text{T}}I_{1}^{\text{T}}\\ -L_{u}^{\text{T}}I_{1}^{\text{T}}\end{pmatrix} ,J_{1}= \begin{pmatrix}\tilde{J} _{1}&0\\ 0&0\end{pmatrix} ,\ J_{2}= \begin{pmatrix}\tilde{J}_{2}&0\\ 0&0\end{pmatrix} ,\]  
\[\tilde{J}_{1}=(I_{1}+J_{2})/2\,\ \tilde{J}_{2}=(J_{1}^{\text{T}}I_{2}+J_{2}^{ \text{T}}I_{1})/2,\]  
\[\tilde{\xi}_{u}= \begin{pmatrix}\xi_{1u}&0&\xi_{2u}&0\\ *&0&0&0\\ *&*&I&0\\ *&*&*\end{pmatrix} ,\xi_{1u}= \begin{pmatrix}\delta_{u-\tau_{1}}^{\text{T}} \tilde{S}_{2}o_{u-\tau_{1}}&0&\delta_{u-\tau_{1}}^{\text{T}}\tilde{S}_{2}L_{u -\tau_{1}}\\ *&0&0\\ *&*&L_{u-\tau_{1}}^{\text{T}}\tilde{S}_{2}L_{u-\tau_{1}}\end{pmatrix} ,\]  
\[\tilde{\xi}_{2u}= \begin{pmatrix}-\delta_{u-\tau_{1}}^{\text{T}}S_{1}^{\text{T }}\\ 0\\ -L_{u-\tau_{1}}^{\text{T}}S_{1}^{\text{T}}\end{pmatrix} ,S_{1}= \begin{pmatrix} \tilde{\xi}_{1}&0\\ 0&0\end{pmatrix} ,S_{2}= \begin{pmatrix}\tilde{\xi}_{2}&0\\ 0&0\end{pmatrix} ,\]  
\[\tilde{S}_{1}=(S_{1}+S_{2})/2,\ \tilde{S}_{2}=(S_{1}^{\text{T}}S_{2}+S_{2}^{\text{T }}S_{1})/2,\ \pi_{1u}= \begin{pmatrix}-\frac{1}{2}\tilde{\xi}_{1}^{\text{T}}U_{2}^{\text{T}} \\ -\frac{1}{2}L_{u}^{\text{T}}U_{2}^{\text{T}}\end{pmatrix} ,\]  
\[U_{2}=(\tilde{\Omega}_{2}\tilde{\mathbb{C}}\quad 0),\pi_{u}= \begin{pmatrix}0&0& \pi_{1u}&0&0&0&0\\ *&0&0&0&0&0\\ *&*&I&0&0&0\\ *&*&*&0&0&0\\ *&*&*&*&*&0\end{pmatrix} .\]  
Next, we can obtain the following formula from (10) that  
\[(\tilde{\tilde{q}}_{u}-\dot{q}_{u-1})^{\text{T}}\tilde{\omega}(\Xi_{j}-\Xi_{l_ {u}})(\tilde{\tilde{q}}_{u}-\dot{q}_{u-1})\leq 0, \tag{32}\]where  
\[\tilde{\bar{q}}_{u}-\dot{q}_{u-1}\] \[=\alpha_{u}\tilde{\bar{U}}_{1}\bar{\zeta}\delta_{u}+\alpha_{u}\theta( \tilde{\bar{\zeta}}\delta_{u})+\alpha_{u}\tilde{\bar{H}}_{v}\tilde{\bar{V}}_{u}- \dot{q}_{u-1}\] \[=\zeta_{3}\delta_{u}+\theta(C_{1}o_{u})+H_{3}V_{u},\] \[C_{3}=(\alpha_{u}\tilde{\bar{U}}_{1}\tilde{\bar{C}}\quad-1),\ H_{3 }=(0\quad\alpha_{u}\tilde{\bar{H}}_{v}),\ \tilde{q}_{u}-\dot{q}_{u-1}=Q_{u}y_{u},\] \[Q_{u}=(C_{3}\delta_{u}\quad C_{3}L_{u}\quad 0\quad 0\quad 0 \quad\alpha_{u}\quad H_{3}\quad 0\quad 0\quad),\] \[\tilde{\omega}=\mathrm{diag}\{\omega_{1},\ \omega_{2},\ \cdots,\ \omega_{n_{y}}\}.\]  
Inequality (32) is equivalent to  
\[y_{u}^{\mathrm{T}}Q_{u}^{\mathrm{T}}\tilde{\omega}(\[PERSON]_{j}-\[PERSON]_{l_{u}})Q_{u}y _{u}\leq 0. \tag{33}\]  
Based on Lemma 2, if there exist positive scalars \(\dot{\gamma}\ (i=1,2,\cdots,8)\) and \(\varphi_{j}\ (j=1,\ 2,\cdots,\ n_{y})\), such that  
\[ \begin{split}&\Im_{u}-\kappa_{1}-\gamma_{1}\kappa_{2}-\gamma_{2} \kappa_{3}-\gamma_{3}\kappa_{4}-\gamma_{4}\kappa_{5}-\gamma_{5}\kappa_{6}- \gamma_{6}\bar{\xi}_{u}\\ &-\gamma_{7}\pi_{u}-\gamma_{8}Y_{u}-\sum_{j=1}^{n_{y}}\varphi_{j} Q_{u}^{\mathrm{T}}\tilde{\omega}(\Xi_{j}-\Xi_{l_{u}})Q_{u}\leq 0,\end{split}  \tag{34}\]  
where  
\[ \begin{split}&\kappa_{1}=\mathrm{diag}\{1,\ 0,\...,\ 0\},\\ &\kappa_{2}=\mathrm{diag}\{-1,\ I,\ 0,\...,\ 0\},\\ &\kappa_{3}=\mathrm{diag}\{-1,\ 0,\ I,\ 0,\...,\ 0\},\\ &\kappa_{4}=\mathrm{diag}\{-\tau_{2},\ 0,\ 0,\ I,\ 0,\...,\ 0\},\\ &\kappa_{5}=\mathrm{diag}\{-2,\ 0,\...,\ 0,\ R_{u}^{-1},\ 0,\ 0\},\\ &\kappa_{6}=\mathrm{diag}\{-1,\ 0,\...,\ 0,\ \Omega_{u}^{-1},\ 0\},\end{split} \]  
then the inequality (27) holds, and the following condensed form of Inequality (34) holds as well:  
\[\Im_{u}+\Lambda_{u}\leq 0, \tag{35}\]  
where \(\Lambda_{u}\) is given in (20). The following contents are used to handle the uncertainty of parameters and gain perturbations. According to Lemma 1, the inequality (35) can be expressed as an LMI:  
\[ \begin{pmatrix}-P_{u+1}&\varepsilon_{u}\\ 0&\Lambda_{u}\end{pmatrix} \leq 0, \tag{36}\]  
which can be decomposed into the following forms:  
\[\widetilde{\gamma}+ \begin{pmatrix}0&\widetilde{\epsilon}_{u}\\ *&0\end{pmatrix} \leq 0, \tag{37}\]  
where \(\tilde{\gamma}\) and \(\widetilde{\epsilon}_{u}\) are defined in (20).