# Fault Estimation for Semi-Markov Jump Neural Networks Based on the Extended State Method
## 2 Preliminaries
\begin{array}{ll}\mathcal{V}(g)&=\widetilde{\mathcal{A}}_{p,\sigma(g)} \mathcal{V}(g)+\widetilde{\mathcal{B}}_{p}f(\mathcal{V}(g))+\widetilde{ \mathcal{C}}_{p}f(\mathcal{V}(g-d(g)))+\widetilde{\mathcal{D}}_{p}\widetilde{ \omega}(g)\\ \mathcal{J}(g)&=\widetilde{\mathcal{F}}_{p}\mathcal{V}(g)\\ &=\overset{\rightarrow}{\mathcal{F}}_{p}\overset{\rightarrow}{x}(g)- \overset{\rightarrow}{\mathcal{F}}_{p}\overset{\rightarrow}{x}(g)\\ \mathcal{V}(g)&=\widetilde{\mathcal{V}}(g),g=-d_{2},-d_{2}+1,\ldots,0,\end{array}  \tag{22}\]\[\text{where }\widetilde{\mathcal{A}}_{p,\sigma(g)}=\left[ \begin{matrix} \overset{\rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{A}}_{p,\sigma(g)}+ \widetilde{\mathcal{K}}_{p,\sigma(g)}&\overset{\rightarrow}{E}_{2}\overset{ \rightarrow}{\mathcal{H}}_{p}\\ 0&1\end{matrix} \right]\widetilde{B}_{p}=\left[ \begin{matrix}\overset{ \rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{B}}_{p}&0\\ 0&0\end{matrix} \right]\widetilde{C}_{p}=\left[ \begin{matrix}\overset{ \rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{C}}_{p}&0\\ 0&0\end{matrix} \right]\] \[\widetilde{\mathcal{D}}_{p}=\left[ \begin{matrix}\overset{ \rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{D}}_{p}&0\\ 0&0\end{matrix} \right]\!,\,f(\mathcal{V}(g-d(g)))=f\!\left(\left[ \begin{matrix} \widetilde{\mathcal{Z}}(g-d(g))\\ \widetilde{fa}(g-d(g))\end{matrix} \right]\right)\!,\,f(\mathcal{V}(g))=f\! \left(\left[ \begin{matrix}\widetilde{\mathcal{Z}}(g)\\ \widetilde{fa}(g)\end{matrix} \right]\right)\!,\] \[\widetilde{\omega}(g)=\left[ \begin{matrix}\overline{\omega}(g)\\ 0\end{matrix} \right]\!.\]  
This paper presents a novel observer designed for discrete-time hybrid systems, based on the design methodology of discrete-time Markov jump systems [23; 24] and the extended state method. The proposed observer extends the range of applicability, offering broader potential for practical applications. **Assumption 1** [31]. : _The transition probability \(\pi_{pq}(\zeta)\) is bounded and satisfies the following assumption:_  
\[\pi_{pq}(\zeta)=\sum\limits_{k=1}^{K}\xi_{k}\pi_{pq,k},\sum\limits_{k=1}^{K} \xi_{k}=1\zeta_{k}\geq 0 \tag{23}\]  
\[\pi_{pq,k}= \begin{cases}\pi_{pq}^{-}+(k-1)\frac{\pi_{pq}^{-}-\pi_{pq}^{-}}{K-1},p\
eq q,q\in Q\\ \pi_{pq}^{+}-(k-1)\frac{\pi_{pq}^{-}-\pi_{pq}^{-}}{K-1},p=q,q\in Q\end{cases}  \tag{24}\]  
where \(\pi_{pq}^{+}\) and \(\pi_{pq}^{-}\) are the upper and lower bounds of \(\pi_{pq}(\zeta)\). **Assumption 2**. : _The nonlinear neuron activation function \(f_{l}(\cdot)\) in Equation (1) is continuous, and \(f_{l}(0)=0\) satisfies the following assumptions:_  
\[a_{l}^{-}\leq\frac{f_{l}(\alpha)-f_{l}(\beta)}{\alpha-\beta}\leq a_{l}^{+},l=1,2,3,\ldots,n\;\alpha,\beta\in R \tag{25}\]  
where \(\alpha\
eq\beta\), \(a_{l}^{-}\) and \(a_{l}^{+}\) are known scalars. **Lemma 1** [32]. : _Let \(R\) be a matrix, and \(d_{1}\), \(d_{2}\) be non-negative integers that satisfy the condition \(d_{1}\leq d_{2}\leq g\),_  
\[\Gamma(g,d_{1},d_{2})= \begin{cases}\frac{1}{d_{2}-d_{1}}\left[\begin{matrix}2 \overset{g-d_{1}-1}{\sum\limits_{s=g-d_{2}}^{-}}\mathcal{V}(s)+\mathcal{V}(g- d_{1})-\mathcal{V}(g-d_{2})\\ 2\mathcal{V}(g-d_{1}),d_{1}=d_{2}\end{matrix}\right],d_{1}<d_{2}\\ \\ \end{cases}  \tag{26}\]  
Then,  
\[-(d_{2}-d_{1})\sum\limits_{s=g-d_{2}}^{g-d_{1}-1}\mathfrak{M}^{T}(s) \mathbb{R}\mathfrak{M}(s)\leq- \begin{bmatrix}\circ_{1}\\ \circ_{2}\end{bmatrix} ^{T} \begin{bmatrix}\mathbb{R}&0\\ 0&3\mathbb{R}\end{bmatrix}\begin{bmatrix}\circ_{1}\\ \circ_{2}\end{bmatrix}  \tag{27}\]  
where \(\mathfrak{M}(s)=\)\(\mathcal{V}(s+1)-\mathcal{V}(s)\), \(\circ_{1}=\mathcal{V}(g-d_{1})-\mathcal{V}(g-d_{2})\), \(\circ_{2}=\mathcal{V}(g-d_{1})+\mathcal{V}(g-d_{2})-\Gamma(g,d_{1},d_{2})\). **Definition 1** [33].