# Fault Estimation for Semi-Markov Jump Neural Networks Based on the Extended State Method
## 2 Preliminaries
\begin{array}{ll}\hat{\mathcal{Z}}(g+1)&=\overset{\rightarrow}{E}_{2} \overset{\rightarrow}{\mathcal{A}}_{p,\sigma(g)}\hat{\mathcal{Z}}(g)+\overset{ \rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{B}}_{p}f(\hat{\mathcal{Z}} (g)+\overset{\rightarrow}{E}_{2}\overset{\rightarrow}{M}\overline{y}(g))+ \overset{\rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{C}}_{p}f(\hat{ \mathcal{Z}}(g-d(g))\\ &+\overset{\rightarrow}{E}_{2}\overset{\rightarrow}{M}\overline{y}(g-d(g)))+ \overset{\rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{H}}_{p}\overset{ \rightarrow}{f}a(g)+\overset{\rightarrow}{\mathcal{I}}\overline{y}(g)+ \overset{\rightarrow}{\mathcal{K}}_{p,\sigma(g)}(\overline{y}(g)-\widehat{y}(g) )\end{array}  \tag{15}\]  
\[\widehat{y}(g)=\overset{\rightarrow}{\mathcal{E}}_{p0,\sigma(g)}\overset{ \rightarrow}{x}(g) \tag{16}\]  
\[\hat{z}(g)=\overset{\rightarrow}{\mathcal{F}}_{p}\overset{\rightarrow}{x}(g) \tag{17}\]  
\[\overset{\rightarrow}{x}(g)=\hat{\mathcal{Z}}(g)+\overset{\rightarrow}{E}_{2 }\overset{\rightarrow}{M}\overline{y}(g) \tag{18}\]  
\[\hat{f}a(g)=\mathcal{L}-\mathcal{K}_{3}(\overline{y}(g)-\widehat{y}(g)) \tag{19}\]  
\[\mathcal{L}(g+1)=\overset{\rightarrow}{\mathcal{K}}_{pp,\sigma(g)}(\overline {y}(g)-\widehat{y}(g))\]  
where \(\overset{\rightarrow}{x}(g)\) and \(\widehat{y}(g)\) represent the estimated vectors of \(\overset{\rightarrow}{x}(g)\) and \(\overline{y}(g)\), respectively. \(\hat{z}(g)\) is the estimated of \(z(g)\). \(\hat{\mathcal{Z}}(g)\) is the estimated vector of \(\mathcal{Z}(g)\). \(\hat{f}a(g)\) is the estimated vector of \(fa(g)\). Matrices \(\overset{\rightarrow}{\mathcal{K}}_{p,\sigma(g)}=\left[\overset{\rightarrow}{ \mathcal{K}}_{p1,\sigma(g)}\right.\overset{\rightarrow}{\mathcal{K}}_{p2, \sigma(g)}\right]^{\mathrm{T}}\) and \(\overset{\rightarrow}{\mathcal{K}}_{pp,\sigma(g)}\) are gain parameters to be determined for the fault estimation process, ensuring accurate estimation convergence. Matrix \(\mathcal{K}_{3}\) is an adjustable parameter used to fine-tune the fault estimation performance. Fault estimates play a key role in accurately identifying actuator and sensor faults, ensuring the reliability of the proposed estimation framework. Therefore, the estimator must be designed to minimize errors and ensure convergence to actual fault values. According to Equations (18) and (19), the fault estimates are given by  
\[ \begin{array}{ll}fa(g+1)&=\mathcal{L}(g+1)+\mathcal{K}_{3}(\overline{y}(g)- \widehat{y}(g))\\ &=\overset{\rightarrow}{\mathcal{K}}_{pp,\sigma(g)}(\overline{y}(g)- \widehat{y}(g))+\mathcal{K}_{3}(\overline{y}(g+1)-\widehat{y}(g+1))\end{array}  \tag{20}\]  
Set \(\widetilde{\mathcal{Z}}(g)=\mathcal{Z}(g)-\hat{\mathcal{Z}}(g)\) and \(\widetilde{f}a(g)=fa(g)-\hat{f}a(g)\). Then, we have an estimation error of Equation (21):  
\[ \begin{array}{ll}\widetilde{\mathcal{Z}}(g+1)&=\mathcal{Z}(g+1)-\hat{ \mathcal{Z}}(g+1)\\ &=\overset{\rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{A}}_{p,\sigma (g)}\widetilde{\mathcal{Z}}(g)+\overset{\rightarrow}{E}_{2}\overset{ \rightarrow}{\mathcal{B}}_{p}f(\widetilde{\mathcal{Z}}(g))\ +\overset{\rightarrow}{E}_{2}\overset{ \rightarrow}{\mathcal{C}}_{p}f(\widetilde{\mathcal{Z}}(g-d(g)))\\ &+\overset{\rightarrow}{E}_{2}\overset{\rightarrow}{\mathcal{H}}_{p} \widetilde{f}a(g)-\overset{\rightarrow}{K}_{p,\sigma(g)}(\overline{y}(g)- \widehat{y}(g))+\overset{\rightarrow}{E}_{2}\overset{\rightarrow}{ \mathcal{D}}_{p}\overline{\omega}(g)\end{array}  \tag{21}\]  
Set \(\mathcal{V}(g)= \begin{bmatrix}\widetilde{\mathcal{Z}}(g)\\ \widetilde{f}a(g)\end{bmatrix} \), \(\mathcal{J}(g)=z(g)-\hat{z}(g)\). Derived from Equations (6) and (18), augmented S-MJNNs can be obtained as follows (22):  
\[