## 3 Main Result  
### Estimator Design  
The subsequent theorem gives the sufficient conditions for constraining estimation error within desired ellipsoidal regions by employing the LMI approach. **Theorem 1**. : _Considering the augmented MNNs in (13) and the estimator in (15), the estimation error can still be constrained within the required ellipsoidal region at the sampling instant \(u+1\)  
\[ \begin{split}\hat{T}&=(T^{T}\quad 0\quad 0\quad 0\quad 0\quad 0 \quad 0\quad 0\quad 0\quad 0\quad 0\quad 0\quad)^{T},\\ T&=\begin{pmatrix}T\quad T\quad T\quad T\quad T\quad \hat{T}\quad 0\\ 0\quad 0\quad 0\quad 0\quad 0\quad\hat{T}\end{pmatrix},\ \hat{X}=(0\quad X),\\ \\ X&=\begin{pmatrix}X_{11}\quad X_{12}\\ X_{21}\quad X_{22}\end{pmatrix},\ X_{11}=\begin{pmatrix}X_{1}\theta_{u}\quad X_ {1}L_{u}\quad 0\quad 0\\ 0\quad 0\quad 0\quad 0\\ 0\quad 0\quad 0\quad 0\end{pmatrix},\\ X_{21}=\begin{pmatrix}X_{4}\quad\sum\limits_{i=1}^{T_{0}}\mu_{i}\partial_{u -i}\quad 0\quad 0\quad X_{4}\sum\limits_{i=1}^{T_{0}}\mu_{i}L_{u-i}\\ 0\quad\quad-\alpha_{u}\hat{X}C_{2}L_{u}\quad 0\quad 0\\ 0\quad\quad 0\quad 0\quad 0\end{pmatrix},\\ X_{12}=\begin{pmatrix}0\quad 0\quad 0\quad 0\quad 0\quad 0\\ 0\quad 0\quad 0\quad 0\quad 0\quad X_{22}=\begin{pmatrix}0\quad 0\quad 0 \quad 0\quad 0\quad 0\quad 0\\ 0\quad-\alpha_{u}\hat{X}\Xi_{l_{u}}\quad-\alpha_{u}\hat{X}H_{2}\quad-\beta \hat{X}\Xi_{l_{u}}\quad 0\\ 0\quad 0\quad 0\quad 0\quad-\beta_{u}\hat{X}\Xi_{l_{u}}\quad 0\end{pmatrix}, \\ \\ X_{3}\quad 0\quad 0\quad 0\quad 0\end{pmatrix}\end{split}  \tag{20}\]  
_in which \(L_{u}\) is a Cholesky factorization of \(P_{u}=L_{u}L_{u}^{T}\)._  
Proof. : The mathematical induction is introduced to prove Theorem 1. First, we can know that \((o_{0}-\theta_{0})^{T}p_{0}^{-1}(o_{0}-\theta_{0})\leq 1\) from Assumption 2 when the sampling instant \(u=0\). Next, we assume that \((o_{u}-\tilde{o}_{u})^{T}p_{u}^{-1}\)\((o_{u}-\tilde{o}_{u})\leq 1\) holds when the sampling instant \(u\geq 0\). Finally, we need to prove that \((o_{u+1}-\tilde{o}_{u+1})^{T}p_{u+1}^{-1}\)\((o_{u+1}-\tilde{o}_{u+1})\leq 1\) also holds when the sampling instant \(u+1\geq 0\). The proof process is shown as follows. As mentioned in [62], there exists a vector \(M_{u}\) with \(\|M_{u}\|\leq 1\) satisfying  
\[e_{u}=o_{u}-\theta_{u}=L_{u}M_{u}, \tag{21}\]  
where \(L_{u}\) is given in (20). Therefore, the estimation error can be obtained by the process of calculation as follows:  
\[ \begin{split}& o_{u+1}-\tilde{o}_{u+1}\\ &=\aleph_{u}A(o_{u})o_{u}+B(o_{u})F(o_{u})+D(o_{u})G(o_{u-\tau_{1}}) \\ &\quad+E(o_{u})\sum\limits_{i=1}^{T_{0}}\mu_{i}o_{u-i}+\aleph_{u} H_{1}W_{u}+\alpha_{u}I_{1u}\theta(C_{1}o_{u})\\ &\quad-\aleph_{u}A\partial_{u}-BF(\partial_{u})-DG(\partial_{u- \tau_{1}})-E\sum\limits_{i=1}^{T_{0}}\mu_{i}\partial_{u-i}\\ &\quad+(K_{u}+\Delta K_{u})(q_{u}-\alpha_{u}C_{2}\partial_{u})\\ &=\aleph_{u}A(o_{u})e_{u}+\Delta A_{u}\partial_{u}+B(o_{u})F( \partial_{u})\\ &\quad-BF(\partial_{u})+D(o_{u})G(o_{u-\tau_{1}})-DG(\partial_{u- \tau_{1}})\\ &\quad+E(o_{u})\sum\limits_{i=1}^{T_{0}}\mu_{i}e_{u-i}+\Delta E_{u} \sum\limits_{i=1}^{T_{0}}\mu_{i}\tilde{o}_{u-i}\\ &\quad+(\aleph_{u}H_{1}-\alpha_{u}(\Delta K_{u}+K_{u})H_{2})W_{u} \\ &\quad+(\alpha_{u}I_{u}-\alpha_{u}(\Delta K_{u}+K_{u})\Xi_{l_{u}}) \theta(C_{1}o_{u})\\ &\quad-\beta_{u}(\Delta K_{u}+K_{u})\Xi_{l_{u}}\rho_{u}-\alpha_{u}( \Delta K_{u}+K_{u})C_{2}e_{u}.\end{split}  \tag{22}\]  
For convenience, the following notations are defined:  
\[ \begin{split}& y_{u}=\left(\begin{matrix}\dot{y}_{u}\quad\dot{y}_{ u}\end{matrix}\right)^{\mathsf{T}},\\ &\dot{y}_{u}=\left(\begin{matrix}1&M_{u}^{\mathsf{T}}\quad M_{u- \tau_{1}}^{\mathsf{T}}\quad M_{Tu}^{\mathsf{T}}\quad G^{\mathsf{T}}(o_{u- \tau_{1}})\end{matrix}\right),\\ &\ddot{y}_{u}=\left(\theta^{\mathsf{T}}(C_{1}o_{u})\quad W_{u}^{ \mathsf{T}}\quad\rho_{u}^{\mathsf{T}}\quad F^{\mathsf{T}}(o_{u})\right),\\ & M_{Tu}=\left(M_{u-1}^{\mathsf{T}}\quad M_{u-2}^{\mathsf{T}} \quad\cdots\quad M_{u-\tau_{2}}^{\mathsf{T}}\right)^{\mathsf{T}}.\end{split}