# Fault Estimation for Semi-Markov Jump Neural Networks Based on the Extended State Method
## 3 Main Results
\begin{split}&\overline{\Phi}_{p,\sigma(g)}=\Big{[}\overline{ \Theta}_{p,\sigma(g)}\Big{]}_{1111}<0,\\ &\Big{[}\overline{\Theta}_{p,\sigma(g)}\Big{]}_{19}=\Bigg{[} \overset{\rightarrow}{\widetilde{\mathcal{A}}}_{p,\sigma(g)}0,\;0,\;0,\;0, \overset{\rightarrow}{\widetilde{\mathcal{Z}}}_{p},\overset{\rightarrow}{ \widetilde{\mathcal{C}}}_{p},\overset{\rightarrow}{\widetilde{\mathcal{D}}}_{p} \Bigg{]}^{\mathsf{T}},\\ &\Big{[}\overline{\Theta}_{p,\sigma(g)}\Big{]}_{110}=\Bigg{[}d_ {12}\overset{\rightarrow}{\overline{\widetilde{\mathcal{A}}}}_{p,\sigma(g)}-1,\;0,\;0,\;0,\;0,\;d_{12}\overset{\rightarrow}{\widetilde{\widetilde{ \mathcal{Z}}}}_{p},\;d_{12}\overset{\rightarrow}{\widetilde{\mathcal{C}}}_{p},d _{12}\overset{\rightarrow}{\widetilde{\mathcal{D}}}_{p}\Bigg{]}^{\mathsf{T}},\\ &\Big{[}\overline{\Theta}_{p,\sigma(g)}\Big{]}_{99}=\overset{ \rightarrow}{\widetilde{\mathcal{D}}}_{p,\sigma(g+1)}-\mathfrak{J}_{p,\sigma(g) }^{\mathsf{T}}-\mathfrak{J}_{p,\sigma(g)},\\ &\Big{[}\overline{\Theta}_{p,\sigma(g)}\Big{]}_{1010}=\mathsf{X}_{ 4}-\mathfrak{J}_{p,\sigma(g)}^{\mathsf{T}}-\mathfrak{J}_{p,\sigma(g)},\end{split}  \tag{50}\]Proof. : Define \(\hbar=diag\Big{\{}I,I,I,I,I,I,I,I,\Im_{p,\sigma(g)},\Im_{p,\sigma(g)},I\Big{\}}, \overset{\rightarrow}{K}_{p1,\sigma(g)}=\Im^{-1}{}_{p1,\sigma(g)}\wp_{p1,\sigma( g)}\), and \(\overset{\rightarrow}{K}_{p2,\sigma(g)}=\Im^{-1}{}_{p2,\sigma(g)}\wp_{p2, \sigma(g)}\). Here, the two matrices \(\Im_{p,\sigma(g)}\) act on two different parts of the observer gain, which may correspond to the state feedback and delay feedback components, respectively. To incorporate \(\Im_{pb,\sigma(g)}\), we use a diagonal matrix \(\hbar\) to pre-multiply \(\Phi_{p,\sigma(g)}\) by \(\hbar^{T}\) and post-multiply it by \(\hbar\), leading to the following:  
\[ \begin{array}{ll}\overline{\Phi}_{p,\sigma(g)}&=\hbar^{T}\Phi_{p,\sigma(g) }\hbar\\ &=\begin{bmatrix}\left[\overline{\Theta}_{p,\sigma(g)}\right]^{88\delta}& \boldsymbol{\Lambda}^{T}_{p,\sigma(g)}\widetilde{\mathrm{P}}_{p,\sigma(g+1)} &\circ^{T}_{p,\sigma(g)}\mathrm{X}_{4}&\Gamma^{T}_{p}\\ *&-\Im^{T}{}_{p,\sigma(g)}\widetilde{\mathrm{P}}_{p,\sigma(g+1)}\Im_{p, \sigma(g)}&0&0\\ *&0&-\Im^{T}{}_{p,\sigma(g)}\mathrm{X}_{4}\Im_{p,\sigma(g)}&0\\ *&0&0&1\end{bmatrix}\end{array}  \tag{51}\]  
Utilizing the Schur complement lemma, and considering that \(\mathrm{P}_{p,\sigma(g)}>0(p\in Q)\) is a symmetric positive definite matrix and \(\Im_{pb,\sigma(g)}\) is an arbitrary nonzero matrix, the following is obtained:  
\[-\Im^{T}{}_{pb,\sigma(g)}\mathrm{P}^{-1}{}_{p,\sigma(g)}\Im_{pb,\sigma(g)}<0 \tag{52}\]  
Similarly, since \(\mathrm{X}_{4}\) is also a symmetric positive definite matrix, we have:  
\[-\Im^{T}{}_{pb,\sigma(g)}\mathrm{X}_{4}{}^{-1}\Im_{pb,\sigma(g)}<0 \tag{53}\]  
From Equations (52) and (53), the following can be written:  
\[-\Im^{T}{}_{pb,\sigma(g)}\mathrm{P}^{-1}{}_{p,\sigma(g)}\Im_{pb,\sigma(g)}< \mathrm{P}_{p,\sigma(g)}-\Im^{T}{}_{pb,\sigma(g)}-\Im_{pb,\sigma(g)} \tag{54}\]  
\[-\Im^{T}{}_{pb,\sigma(g)}\mathrm{X}_{4}{}^{-1}\Im_{pb,\sigma(g)}<\mathrm{X}_ {4}-\Im^{T}{}_{pb,\sigma(g)}-\Im_{pb,\sigma(g)} \tag{55}\]  
Hence, if \(\overline{\Phi}_{p,\sigma(g)}=\left[\overline{\Theta}_{p,\sigma(g)}\right]^{ 11\lambda 11}<0\) hold, (22) is mean square exponentially stable (MSES), thereby completing the proof. \(\square\)  
Building upon the stability analysis in Theorem 1, Theorem 2 further extends the robustness analysis by considering the case where the control gain parameters are unknown. It demonstrates that the augmented s-MJNN (22) maintains strict \((\chi_{1},\chi_{2},\chi_{3})-\alpha\) dissipative performance, ensuring the system's robustness against parameter variations. This result implies that the dissipative performance condition inherently guarantees MSES, reinforcing that the proposed estimation framework is resilient to uncertainties and capable of maintaining stable and reliable performance under different conditions.