## 3 Main Result
### Estimator Design
 \tag{23}\]Then, the estimation error can be rewritten as the below compact form:  
\[o_{u+1}-\partial_{u+1}=\varepsilon_{u}y_{u}, \tag{24}\]  
where  
\[\varepsilon_{u}=\left( \begin{matrix}\varepsilon_{u}&\bar{\varepsilon}_{u} \end{matrix} \right),\] \[\varepsilon_{u}=\left( \begin{matrix}\varepsilon_{1,u}&\varepsilon_{2,u}&0&E(o_{u})\sum\limits_{i=1}^{\tau_{2}}\mu_{i}L_{u-i}&D(o_{u})\end{matrix}  \right),\] \[\bar{\varepsilon}_{u}=\left( \begin{matrix}\varepsilon_{3,u}& \varepsilon_{4,u}&\varepsilon_{5,u}&B(o_{u})\end{matrix} \right),\] \[\varepsilon_{1,u}=\Delta A_{u}\partial_{u}-BF(\delta_{u})-DG( \delta_{u-\tau_{1}})+\Delta E_{u}\sum\limits_{i=1}^{\tau_{2}}\mu_{i}\partial_{ u-i},\] \[\varepsilon_{2,u}=(\aleph_{u}A(o_{u})-\alpha_{u}(\Delta K_{u}+K_{u })C_{2})L_{u},\] \[\varepsilon_{3,u}=\alpha_{u}l_{u}-\alpha_{u}(\Delta K_{u}+K_{u}) \Xi_{l_{u}},\] \[\varepsilon_{4,u}=\aleph_{u}H_{1}-\alpha_{u}(\Delta K_{u}+K_{u}) H_{2},\] \[\varepsilon_{5,u}=-\beta_{u}(\Delta K_{u}+K_{u})\Xi_{l_{u}}.\]  
Next, the stochastic matrix \(\varepsilon_{u}\) can be decomposed through a specific process:  
\[\varepsilon_{u}=\varepsilon_{u}^{1}+\varepsilon_{u}^{2}, \tag{25}\]  
where  
\[\varepsilon_{u}^{1}=(\varepsilon_{u}^{1}\quad\bar{\varepsilon}_{u}^{1}),\] \[\varepsilon_{u}^{1}=\left( \begin{matrix}\varepsilon_{1,u}&\varepsilon _{2,u}&0&E(o_{u})\sum\limits_{i=1}^{\tau_{2}}\mu_{i}L_{u-i}&D(o_{u})\end{matrix}  \right),\] \[\tilde{\varepsilon}_{u}^{1}=\left( \begin{matrix}\varepsilon_{3,u}& \varepsilon_{4,u}&\varepsilon_{5,u}^{1}&B(o_{u})\end{matrix} \right),\] \[\varepsilon_{5,u}^{1}=-\tilde{\beta}(\Delta K_{u}+K_{u})\Xi_{l_{ u}},\] \[\varepsilon_{u}^{2}=\left( \begin{matrix}0&0&0&0&0&0&\varepsilon_{5,u }^{2}&0\end{matrix} \right),\] \[\varepsilon_{5,u}^{2}=-\tilde{\beta}_{u}(\Delta K_{u}+K_{u})\Xi_{ l_{u}},\] \[\beta_{u}=\tilde{\beta}+\tilde{\beta}_{u},\]  
Subsequently, \(y_{u}^{\mathrm{T}}\varepsilon_{u}^{\mathrm{T}}y_{u+1}^{-1}\varepsilon_{u}y_{u}\) can be rewritten as a new form:  
\[y_{u}^{\mathrm{T}}\varepsilon_{u}^{\mathrm{T}}y_{u+1}^{-1} \varepsilon_{u}y_{u}\] \[=y_{u}^{\mathrm{T}}(\varepsilon_{u}^{\mathrm{T}}y_{u+1}^{-1} \varepsilon_{u}^{1}+\varepsilon_{u}^{2\mathrm{T}}y_{u+1}^{-1}\varepsilon_{u} ^{2})y_{u} \tag{26}\] \[=y_{u}^{\mathrm{T}}\zeta_{u}y_{u}.\]  
Therefore, \(y_{u}^{\mathrm{T}}\varepsilon_{u}^{\mathrm{T}}y_{u+1}^{-1}\varepsilon_{u}y_{u}\leq 1\) can be rewritten as  
\[y_{u}^{\mathrm{T}}(\mathbb{S}_{u}-\mathrm{diag}\{1,0,\,...\,,\,0\})y_{u}\leq 0. \tag{27}\]  
According to inequalities (2) and (10), the following constraint conditions can be obtained:  
\[ \begin{cases}\|M_{u}\|^{2}\leq 1\\ \|M_{u-\tau_{1}}\|^{2}\leq 1\\ \|M_{\tau u}\|^{2}\leq\tau_{2}\\ W_{u}^{\mathrm{T}}R_{u}^{-1}W_{u}\leq 2\\ \rho_{u}^{\mathrm{T}}\Omega_{u}^{-1}\rho_{u}\leq 1\end{cases}  \tag{28}\]which can be rewritten as  
\[ \begin{cases}y_{u}^{\text{T}}\text{diag}\{-1,\ I,\ 0,\...\,\ 0\}y_{u}\leq 0\\ y_{u}^{\text{T}}\text{diag}\{-1,\ 0,\ I,\ 0,\...\,\ 0\}y_{u}\leq 0\\ y_{u}^{\text{T}}\text{diag}\{-\tau_{2},\ 0,\ 0,\ I,\ 0,\...\,\ 0\}y_{u}\leq 0\\ y_{u}^{\text{T}}\text{diag}\{-2,\ 0,\...\,\ 0,\ R_{u}^{-1},\ 0,\ 0\}y_{u}\leq 0\\ y_{u}^{\text{T}}\text{diag}\{-1,\ 0,\...\,\ 0,\ \Omega_{u}^{-1},\ 0\}y_{u}\leq 0 \end{cases}  \tag{29}\]  
According to Inequalities (5) and (7), one obtains  
\[ \begin{cases}(F(o_{u})-J_{1}o_{u})^{\text{T}}(F(o_{u})-J_{2}o_{u})\leq 0\\ (G(o_{u-\tau_{1}})-S_{1}o_{u-\tau_{1}})^{\text{T}}(G(o_{u-\tau_{1}})-S_{2}o_{u -\tau_{1}})\leq 0\\ \theta^{\text{T}}(C_{1}o_{u})(\theta(C_{1}o_{u})-U_{2}o_{u})\leq 0\end{cases}  \tag{30}\]  
which can be further described as  
\[ \begin{cases}y_{u}^{\text{T}}Y_{u}y_{u}\leq 0\\ y_{u}^{\text{T}}\xi_{u}y_{u}\leq 0\\ y_{u}^{\text{T}}\pi_{u}y_{u}\leq 0\end{cases}  \tag{31}\]  
where  
\[Y_{u}= \begin{pmatrix}Y_{1u}&0&Y_{2u}\\ *&0&0\\ *&*&I\end{pmatrix} ,Y_{1u}= \begin{pmatrix}\delta_{u}^{\text{T}}I_{2}\delta_{u}& \delta_{u}^{\text{T}}I_{2}L_{u}\\ *&L_{u}^{\text{T}}I_{2}L_{u}\end{pmatrix} ,\]  
\[Y_{2u}=