# Fault Estimation for Semi-Markov Jump Neural Networks Based on the Extended State Method
## 3 Main Results
\begin{split}&\mathrm{E}\{\Delta\mathbb{V}_{3}(\mathcal{V}(g), \sigma(g),p)\}\\ &=\mathrm{E}\{\sum\limits_{h=-d_{2}+1}^{-d_{1}}\rho^{d_{2}-1} \mathcal{V}^{\mathrm{T}}(g+1-d_{2})X_{2}\mathcal{V}(g+1-d_{2})+d_{12}\sum \limits_{h=-d_{2}}^{-d_{1}-1}\rho^{d_{2}-1}\mathfrak{M}^{\mathrm{T}}(g+1-d_{2 })X_{4}\mathfrak{M}(g+1-d_{2})\\ &-\left(\sum\limits_{h=-d_{2}+1}^{-d_{1}}\sum\limits_{s=g+h}^{g- 1}\rho^{g-s-1}\mathcal{V}^{\mathrm{T}}(s)X_{2}\mathcal{V}(s)+(d_{2}-d_{1}) \sum\limits_{h=-d_{2}}^{-d_{1}-1}\sum\limits_{s=g+h}^{g-1}\rho^{g-s-1} \mathcal{V}^{\mathrm{T}}(s)X_{2}\mathcal{V}(s)\right)\\ &+(\rho-1)\mathbb{V}_{3}(\mathcal{V}(g),\sigma(g),p)\}\\ &\leq\mathrm{E}\{d_{12}\theta^{\mathrm{T}}(g)X_{2}\mathcal{V}(g)- \sum\limits_{h=g+1-d_{2}}^{g-d_{1}}\rho^{d_{2}-1}\theta^{\mathrm{T}}(g+1-d_{2 })X_{2}\mathcal{V}(g+1-d_{2})+d_{12}{}^{2}\mathfrak{M}^{\mathrm{T}}(g)X_{4} \mathfrak{M}(g)\\ &-d_{12}\rho^{d_{1}+1}\sum\limits_{h=g-d_{2}}^{g-d_{1}-1} \mathfrak{M}^{\mathrm{T}}(g-d_{2})X_{4}\mathfrak{M}(g-d_{2})+(\rho-1)\mathbb{ V}_{3}(\mathcal{V}(g),\sigma(g),p)\}\end{split}  \tag{35}\]  
Based on Lemma 1[32], it is easy to obtain that  
\[-d_{12}\rho^{d_{1}+1}\sum\limits_{h=g-d_{2}}^{g-d_{1}-1}\mathfrak{M}^{\mathrm{ T}}(g-d_{2})X_{4}\mathfrak{M}(g-d_{2})\leq- \begin{bmatrix}\circ_{1}\\ \circ_{2}\end{bmatrix} ^{T} \begin{bmatrix}X_{4}&0\\ 0&3X_{4}\end{bmatrix}\begin{bmatrix}\circ_{1}\\ \circ_{2}\end{bmatrix}  \tag{36}\]  
In the Equation (35): \(\circ_{1}=\mathcal{V}(g-d_{1})-\mathcal{V}(g-d_{2})\), \(\circ_{2}=\mathcal{V}(g-d_{1})+\mathcal{V}(g-d_{2})\)\(-\)\(\Gamma(g,d_{1},d_{2})\). Because condition \(d_{1}<d_{2}\) is satisfied,  
\[\Gamma(g,d_{1},d_{2})=\frac{1}{d_{2}-d_{1}}\left[2\sum\limits_{s=g-d_{1}-1}^{g- d_{1}-1}V(s)+V(g-d_{1})-V(g-d_{2})\right].\]  
According to Equation (35), it has  
\[ \begin{split}&\left[\begin{array}{c}\circ_{1}\\ \circ_{2}\end{array}\right]^{T}\!\!\left[\begin{array}{cc}X_{4}&0\\ 0&3X_{4}\end{array}\right]\left[\begin{array}{c}\circ_{1}\\ \circ_{2}\end{array}\right]\\ &=\left[\begin{array}{cc}\mathcal{V}(g-d_{1})-\mathcal{V}(g-d_{2})\\ \mathcal{V}(g-d_{1})+\mathcal{V}(g-d_{2})-\Gamma(g,d_{1},d_{2})\end{array} \right]^{T}\!\!\left[\begin{array}{cc}X_{4}&0\\ 0&3X_{4}\end{array}\right]\left[\begin{array}{c}\mathcal{V}(g-d_{1})-\mathcal{ V}(g-d_{2})\\ \mathcal{V}(g-d_{1})+\mathcal{V}(g-d_{2})-\Gamma(g,d_{1},d_{2})\end{array}\right]\end{split}  \tag{37}\]  
To achieve a unified matrix, the inequalities are extended to the entire system's state equations. Based on Assumption 2 and the given matrices \(\mathcal{L}_{1}>0\), \(\mathcal{L}_{2}>0\), we can derive the following:  
\[a_{l}^{-}\leq\frac{f_{l}(\mathcal{V}(g))-f_{l}(\mathcal{V}(g-1))}{\mathcal{V}(g )-\mathcal{V}(g-1)}\leq a_{l}^{+} \tag{38}\]  
where \(a_{l}^{-}=\mathrm{diag}\{a_{1}{}^{-},a_{2}{}^{-},a_{3}{}^{-},a_{4}{}^{-},\ldots a _{n}{}^{-}\},a_{l}{}^{+}=\mathrm{diag}\{a_{1}{}^{+},a_{2}{}^{+},a_{3}{}^{+},a_ {4}{}^{+},\ldots a_{n}{}^{+}\}\). According to Equations (36) and (37), the following can be obtained:  
\[ \begin{bmatrix}\mathcal{V}(g)\\ f(\mathcal{V}(g))\end{bmatrix} ^{T}\!\!\left[ \begin{array}{cc}-a_{1}\mathcal{L}_{ 1}&a_{2}\mathcal{L}_{1}\\ a_{2}\mathcal{L}&-\mathcal{L}_{1}\end{array} \right]\!\!\left[ \begin{array}{c} \mathcal{V}(g)\\ f(\mathcal{V}(g))\end{array} \right]\geq 0\end{split} \tag{39}\]where \(a_{1}=\text{diag}\{a_{1}{}^{-}a_{1}{}^{+},a_{2}{}^{-}a_{2}{}^{+},a_{3}{}^{-}a_{3}{}^ {+},\ldots a_{n}{}^{-}a_{n}{}^{+}\},a_{2}=\text{diag}\Big{\{}\frac{a_{1}{}^{-}a_{1 }{}^{+}},\frac{a_{2}{}^{-}a_{2}{}^{+}}{2},\frac{a_{3}{}^{-}a_{3}{}^{+}}{2}, \ldots\frac{a_{n}{}^{-}a_{n}{}^{+}}{2}\Big{\}}\). By applying the inequalities derived in Equation (38) and the matrix conditions established in Equation (39), we obtain the following result, which further constrains the system's state evolution and ensures the feasibility of the proposed estimation approach. \[ \begin{bmatrix}\mathcal{V}(g-d(g))\\ f(\mathcal{V}(g-d(g)))\end{bmatrix} ^{T}