# Fault Estimation for Semi-Markov Jump Neural Networks Based on the Extended State Method
## 2 Preliminaries
The incorporating the RRP is reformulated as follows:  
\[\overline{x}(g+1)= \overline{\mathcal{A}}_{p,\sigma(g)}\overline{x}(g)+\overline{ \mathcal{B}}_{p}f(\overline{x}(g))+\overline{\mathcal{C}}_{p}f(\overline{x} (g-d(g)))+\overline{\mathcal{D}}_{p}\overline{\omega}(g)+\overline{\mathcal{H }}_{p}fa(g),\] \[\overline{y}(g)= \overline{\mathcal{E}}_{p,\sigma(g)}\overline{x}(g)+\overline{ \mathcal{G}}_{p}fs(g), \tag{4}\] \[\overline{z}(g)= \overline{\mathcal{F}}_{p}x(g),\]  
where \(\overline{y}\) is the measurement output after joining the RRP, \(\overline{x}\) is the state vector after joining the RRP, \(\overline{z}\) is the output with estimation after joining the RRP. \(\overline{x}(g)= \begin{bmatrix}x\\ \widetilde{y}(g-1)\end{bmatrix} \), \(\overline{\omega}(g)= \begin{bmatrix}\omega\\ 0\end{bmatrix} \), \(f(\overline{x}(g))= \begin{bmatrix}f(x(g))\\ 0\end{bmatrix} \), \(\overline{\mathcal{A}}_{p,\sigma(g)}= \begin{bmatrix}A_{p}&0\\ \psi_{\sigma(g)}E_{p}&I-\psi_{\sigma(g)}\end{bmatrix} \), \(\overline{\mathcal{B}}_{p}= \begin{bmatrix}B_{p}&0\\ 0&B_{p}\end{bmatrix} \), \(\overline{\mathcal{C}}_{p}= \begin{bmatrix}C_{p}&0\\ 0&C_{p}\end{bmatrix} \), \(\overline{\mathcal{D}}_{p}= \begin{bmatrix}C_{p}&0\\ 0&C_{p}\end{bmatrix} \), \(\overline{\mathcal{D}}_{p}= \begin{bmatrix}D_{p}&0\\ 0&D_{p}\end{bmatrix} \), \(\overline{\mathcal{F}}_{p}= \begin{bmatrix}P_{p}&0\\ 0&H_{p}\end{bmatrix} \), \(\overline{\mathcal{F}}_{p}= \begin{bmatrix}P_{p}&0\end{bmatrix} \). By utilizing the extended state method, an observer is constructed to effectively and accurately estimate both the system states and faults. Set \(\overrightarrow{x}(g)= \begin{bmatrix}\overline{x}(g)\\ fs(g)\end{bmatrix} \). Then, the extended equation is (5), as follows:  
\[\overset{\rightarrow}{E}_{0}\overset{\rightarrow}{x}(g+1)= \overset{\rightarrow}{\mathcal{A}}_{p,\sigma(g)}\overset{\rightarrow}{x}(g) +\overset{\rightarrow}{\mathcal{B}}_{p}f(\overset{\rightarrow}{x}(g))+ \overset{\rightarrow}{\mathcal{C}}_{p}f(\overset{\rightarrow}{x}(g-d(g)))+ \overset{\rightarrow}{\mathcal{D}}_{p}\overline{\omega}(g)+\overset{ \rightarrow}{\mathcal{H}}_{p}fa(g)+\overset{\rightarrow}{\mathcal{G}}_{p}fs(g) \tag{5}\] \[\text{where }\overset{\rightarrow}{E}_{0}= \begin{bmatrix}I&0\\ 0&0\end{bmatrix} \overset{\rightarrow}{\mathcal{A}}_{p,\sigma(g)}= \begin{bmatrix} \overline{\mathcal{A}}_{p,\sigma(g)}&0\\ 0&-\overline{\mathcal{G}}_{p}\end{bmatrix} \), \(\overset{\rightarrow}{\mathcal{B}}_{p,\sigma(g)}= \begin{bmatrix}\overline{ \mathcal{B}}_{p}&0\end{bmatrix} \), \(\overset{\rightarrow}{\mathcal{C}}_{p,\sigma(g)}= \begin{bmatrix}\overline{ \mathcal{C}}_{p}&0\end{bmatrix} \), \(\overset{\rightarrow}{\mathcal{D}}_{p,\sigma(g)}= \begin{bmatrix}\overline{ \mathcal{C}}_{p}&0\end{bmatrix} \), \(\overset{\rightarrow}{\mathcal{D}}_{p,\sigma(g)}= \begin{bmatrix}\overline{ \mathcal{D}}_{p}&0\end{bmatrix} \), \(\overset{\rightarrow}{\mathcal{H}}_{p}= \begin{bmatrix}\overline{\mathcal{H}}_{p} &0\end{bmatrix} \), \(\overset{\rightarrow}{\mathcal{G}}_{p}= \begin{bmatrix}0&\overline{\mathcal{G}}_{p} \end{bmatrix} \).\]  
The corresponding \(\overset{\rightarrow}{y}(g)\) and \(\overset{\rightarrow}{z}(g)\) in Equation (4) can be transformed into Equation (6):  
\[ \begin{array}{rl}\overline{y}(g)&=\overline{\mathcal{E}}_{p,\sigma(g)} \overline{x}(g)+\ \overline{\mathcal{G}}_{p}fs(g)\\ &=\overset{\rightarrow}{\mathcal{E}}_{p0,\sigma(g)}\overset{\rightarrow}{x}(g) \\ &=\overset{\rightarrow}{\mathcal{E}}_{p,\sigma(g)}\overset{\rightarrow}{x}(g)+ \ \overline{\mathcal{G}}_{p}fs(g),\\ \overline{z}(g)&=\overset{\rightarrow}{\mathcal{F}}_{p}\overset{\rightarrow}{x}(g),\end{array}  \tag{6}\]  
where \(\overset{\rightarrow}{\mathcal{I}}=\overset{\rightarrow}{E}_{2}\overset{ \rightarrow}{I}_{1}\). The fault estimator observer is designed as shown in Equations (15)-(19). Based on the extended state method, it enables the simultaneous estimation of system states and faults, improving fault estimator accuracy. \[