## 3 Design of the Adaptive Sliding Mode Observer
**_The adaptive adjustment law of the adjustable gain parameter in (14) should observe the following condition: the upper limit of \(\rho_{i}(t)\) is \(\rho_{i}^{*}\) and \(\rho_{i}^{*}>\gamma_{i}\)[23]._  
The state estimation error is defined as follows:  
\[\widetilde{\varepsilon}_{i}=\overline{x}_{i}-\widehat{\overline{x}}_{i} \tag{17}\]  
By subtracting Equation (12) from Equation (11) when the \(i^{th}\) sensor fails, the deviation equation of the \(i^{th}\) observer is:  
\[\dot{\widetilde{\varepsilon}}_{i}=(\overline{A}_{i}-L_{i}\overline{C}_{i}) \widetilde{\varepsilon}_{i}+\overline{H}_{i}d(t)+\overline{F}_{i}(f_{i}-v_{i}) \tag{18}\]  
**Theorem 1. **_Based on the above definitions, when the \(i^{th}\) sensor fails and the following conditions are met:_  
(1) \(\|\widetilde{\varepsilon}_{i}\|>2\|P_{i}\|\|\overline{H}_{i}\|\omega/\lambda_ {\min}\left(Q_{i}\right)\)__  
(2) \(LMI\):\(\left[ \begin{array}{cc}(\overline{A}_{i}-L_{i}\overline{C}_{i})^{T}P_{i}+P_{i}( \overline{A}_{i}-L_{i}\overline{C}_{i})&0\\ 0&-P_{i}\end{array} \right]\leq 0\)_,_  
_the dynamic systems of the state estimation error are asymptotically stable._  
**Proof. ** The Lyapunov function is defined as follows:  
\[V(\widetilde{\varepsilon}_{i})=\widetilde{\varepsilon}_{i}^{T}P\widetilde{ \varepsilon}_{i}+\frac{1}{2}(\rho_{i}-\rho_{i}^{*})^{2} \tag{19}\]and its derivative is calculated as:  
\[ \begin{array}{l}\dot{V}(\overline{\epsilon}_{i})=\overline{\dot{e}}_{i}^{T} \Big{[}(\overline{A}_{i}-L_{i}\overline{\dot{C}}_{i})^{T}P_{i}+P_{i}(\overline{ A}_{i}-L_{i}\overline{\dot{C}}_{i})\Big{]}\overline{\dot{e}}_{i}\\ +2\overline{\dot{e}}_{i}^{T}P_{i}\overline{\dot{F}}_{i}(\dot{f}_{i}-v_{i})+ \dot{\rho}_{i}(\rho_{i}-\rho_{i}^{*})+2\overline{\dot{e}}_{i}^{T}P_{i}\overline {\dot{H}}_{i}d(t)\\ =-\|\overline{\dot{e}}_{i}\|\Big{[}\lambda_{\min}(Q_{i})\|\overline{\dot{e}}_{ i}\|-2\|P_{i}\|\|\overline{\dot{H}}_{i}\|\omega\Big{]}\\ +2\overline{\dot{e}}_{i}^{T}P_{i}\overline{\dot{F}}_{i}(\dot{f}_{i}-v_{i})+ \eta_{i}(\rho_{i}-\rho_{i}^{*})\|D_{i}\overline{\dot{e}}_{i}\|\text{sgn}( \|D_{i}\overline{\dot{e}}_{i}\|-\lambda_{i})\\ \leq\ -\|\overline{\dot{e}}_{i}\|\Big{[}\lambda_{\min}(Q_{i})\|\overline{\dot{e}} _{i}\|-2\|P_{i}\|\|\overline{\dot{H}}_{i}\|\omega\Big{]}\\ +\big{[}2\gamma_{i}-2\rho_{i}+\eta_{i}(\rho_{i}-\rho_{i}^{*})\big{]}\|D_{i} \overline{\dot{e}}_{i}\|\\ =-\|\overline{\dot{e}}_{i}\|\Big{[}\lambda_{\min}(Q_{i})\|\overline{\dot{e}}_{ i}\|-2\|P_{i}\|\|\overline{\dot{H}}_{i}\|\omega\Big{]}\\ +\big{(}2\gamma_{i}-2\rho_{i}+2\rho_{i}^{*}-2\rho_{i}^{*}+\eta_{i}(\rho_{i}- \rho_{i}^{*})\big{)}\|D_{i}\overline{\dot{e}}_{i}\|\\ =-\|\overline{\dot{e}}_{i}\|\Big{[}\lambda_{\min}(Q_{i})\|\overline{\dot{e}}_{ i}\|-2\|P_{i}\|\|\overline{\dot{H}}_{i}\|\omega\Big{]}\\ +\big{[}2(\gamma_{i}-\rho_{i}^{*})-\big{|}\rho_{i}-\rho_{i}^{*}\big{]}\big{(} \eta_{i}-2\big{)}\|D_{i}\overline{\dot{e}}_{i}\|\\ \end{array}  \tag{20}\]  
Combined with the condition in Theorem 1, \(\dot{V}(\overline{\dot{e}}_{i})<0\) and \(\|\overline{\dot{e}}_{i}\|\) converges in the following domain:  
\[U(\overline{\dot{e}}_{i})=\left\{\|\overline{\dot{e}}_{i}\|\leq 2\|P_{i}\|\| \overline{\dot{H}}_{i}\|\omega/\lambda_{\min}(Q_{i})\right\} \tag{21}\]